[cvxopt] 05/64: Imported Upstream version 0.9
Andreas Tille
tille at debian.org
Wed Jul 20 11:23:47 UTC 2016
This is an automated email from the git hooks/post-receive script.
tille pushed a commit to branch master
in repository cvxopt.
commit bf979b38d1255fa32de28cb28881986ddddab5ab
Author: Andreas Tille <tille at debian.org>
Date: Wed Jul 20 08:26:49 2016 +0200
Imported Upstream version 0.9
---
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src/C/SuiteSparse/UMFPACK/Include/umfpack_solve.h | 2 +-
.../SuiteSparse/UMFPACK/Include/umfpack_symbolic.h | 2 +-
src/C/SuiteSparse/UMFPACK/Include/umfpack_tictoc.h | 2 +-
src/C/SuiteSparse/UMFPACK/Include/umfpack_timer.h | 2 +-
.../UMFPACK/Include/umfpack_transpose.h | 2 +-
.../UMFPACK/Include/umfpack_triplet_to_col.h | 2 +-
src/C/SuiteSparse/UMFPACK/Include/umfpack_wsolve.h | 2 +-
.../UMFPACK/{Source => Lib}/GNUmakefile | 97 +-
src/C/SuiteSparse/UMFPACK/Lib/Makefile | 483 +++
src/C/SuiteSparse/UMFPACK/Makefile | 14 +-
src/C/SuiteSparse/UMFPACK/README.txt | 8 +-
src/C/SuiteSparse/UMFPACK/Source/Makefile | 482 ---
src/C/SuiteSparse/UMFPACK/Source/cholmod_blas.h | 4 +-
src/C/SuiteSparse/UMFPACK/Source/umf_2by2.c | 2 +-
src/C/SuiteSparse/UMFPACK/Source/umf_2by2.h | 2 +-
src/C/SuiteSparse/UMFPACK/Source/umf_analyze.c | 2 +-
src/C/SuiteSparse/UMFPACK/Source/umf_analyze.h | 2 +-
src/C/SuiteSparse/UMFPACK/Source/umf_apply_order.c | 2 +-
src/C/SuiteSparse/UMFPACK/Source/umf_apply_order.h | 2 +-
src/C/SuiteSparse/UMFPACK/Source/umf_assemble.c | 2 +-
src/C/SuiteSparse/UMFPACK/Source/umf_assemble.h | 2 +-
.../SuiteSparse/UMFPACK/Source/umf_blas3_update.c | 2 +-
.../SuiteSparse/UMFPACK/Source/umf_blas3_update.h | 2 +-
.../SuiteSparse/UMFPACK/Source/umf_build_tuples.c | 2 +-
.../SuiteSparse/UMFPACK/Source/umf_build_tuples.h | 2 +-
src/C/SuiteSparse/UMFPACK/Source/umf_colamd.c | 2 +-
src/C/SuiteSparse/UMFPACK/Source/umf_colamd.h | 2 +-
src/C/SuiteSparse/UMFPACK/Source/umf_config.h | 2 +-
.../UMFPACK/Source/umf_create_element.c | 2 +-
.../UMFPACK/Source/umf_create_element.h | 2 +-
src/C/SuiteSparse/UMFPACK/Source/umf_dump.c | 2 +-
src/C/SuiteSparse/UMFPACK/Source/umf_dump.h | 2 +-
.../SuiteSparse/UMFPACK/Source/umf_extend_front.c | 2 +-
.../SuiteSparse/UMFPACK/Source/umf_extend_front.h | 2 +-
src/C/SuiteSparse/UMFPACK/Source/umf_free.c | 2 +-
src/C/SuiteSparse/UMFPACK/Source/umf_free.h | 2 +-
src/C/SuiteSparse/UMFPACK/Source/umf_fsize.c | 2 +-
src/C/SuiteSparse/UMFPACK/Source/umf_fsize.h | 2 +-
.../UMFPACK/Source/umf_garbage_collection.c | 2 +-
.../UMFPACK/Source/umf_garbage_collection.h | 2 +-
src/C/SuiteSparse/UMFPACK/Source/umf_get_memory.c | 2 +-
src/C/SuiteSparse/UMFPACK/Source/umf_get_memory.h | 2 +-
src/C/SuiteSparse/UMFPACK/Source/umf_grow_front.c | 2 +-
src/C/SuiteSparse/UMFPACK/Source/umf_grow_front.h | 2 +-
src/C/SuiteSparse/UMFPACK/Source/umf_init_front.c | 2 +-
src/C/SuiteSparse/UMFPACK/Source/umf_init_front.h | 2 +-
src/C/SuiteSparse/UMFPACK/Source/umf_internal.h | 2 +-
.../UMFPACK/Source/umf_is_permutation.c | 2 +-
.../UMFPACK/Source/umf_is_permutation.h | 2 +-
src/C/SuiteSparse/UMFPACK/Source/umf_kernel.c | 2 +-
src/C/SuiteSparse/UMFPACK/Source/umf_kernel.h | 2 +-
src/C/SuiteSparse/UMFPACK/Source/umf_kernel_init.c | 2 +-
src/C/SuiteSparse/UMFPACK/Source/umf_kernel_init.h | 2 +-
.../SuiteSparse/UMFPACK/Source/umf_kernel_wrapup.c | 2 +-
.../SuiteSparse/UMFPACK/Source/umf_kernel_wrapup.h | 2 +-
.../SuiteSparse/UMFPACK/Source/umf_local_search.c | 2 +-
.../SuiteSparse/UMFPACK/Source/umf_local_search.h | 2 +-
src/C/SuiteSparse/UMFPACK/Source/umf_lsolve.c | 2 +-
src/C/SuiteSparse/UMFPACK/Source/umf_lsolve.h | 2 +-
src/C/SuiteSparse/UMFPACK/Source/umf_ltsolve.c | 2 +-
src/C/SuiteSparse/UMFPACK/Source/umf_ltsolve.h | 2 +-
src/C/SuiteSparse/UMFPACK/Source/umf_malloc.c | 2 +-
src/C/SuiteSparse/UMFPACK/Source/umf_malloc.h | 2 +-
.../UMFPACK/Source/umf_mem_alloc_element.c | 2 +-
.../UMFPACK/Source/umf_mem_alloc_element.h | 2 +-
.../UMFPACK/Source/umf_mem_alloc_head_block.c | 2 +-
.../UMFPACK/Source/umf_mem_alloc_head_block.h | 2 +-
.../UMFPACK/Source/umf_mem_alloc_tail_block.c | 2 +-
.../UMFPACK/Source/umf_mem_alloc_tail_block.h | 2 +-
.../UMFPACK/Source/umf_mem_free_tail_block.c | 2 +-
.../UMFPACK/Source/umf_mem_free_tail_block.h | 2 +-
.../UMFPACK/Source/umf_mem_init_memoryspace.c | 2 +-
.../UMFPACK/Source/umf_mem_init_memoryspace.h | 2 +-
.../SuiteSparse/UMFPACK/Source/umf_multicompile.c | 2 +-
src/C/SuiteSparse/UMFPACK/Source/umf_realloc.c | 2 +-
src/C/SuiteSparse/UMFPACK/Source/umf_realloc.h | 2 +-
src/C/SuiteSparse/UMFPACK/Source/umf_report_perm.c | 2 +-
src/C/SuiteSparse/UMFPACK/Source/umf_report_perm.h | 2 +-
.../SuiteSparse/UMFPACK/Source/umf_report_vector.c | 2 +-
.../SuiteSparse/UMFPACK/Source/umf_report_vector.h | 2 +-
src/C/SuiteSparse/UMFPACK/Source/umf_row_search.c | 2 +-
src/C/SuiteSparse/UMFPACK/Source/umf_row_search.h | 2 +-
src/C/SuiteSparse/UMFPACK/Source/umf_scale.c | 2 +-
src/C/SuiteSparse/UMFPACK/Source/umf_scale.h | 2 +-
.../SuiteSparse/UMFPACK/Source/umf_scale_column.c | 2 +-
.../SuiteSparse/UMFPACK/Source/umf_scale_column.h | 2 +-
src/C/SuiteSparse/UMFPACK/Source/umf_set_stats.c | 2 +-
src/C/SuiteSparse/UMFPACK/Source/umf_set_stats.h | 2 +-
src/C/SuiteSparse/UMFPACK/Source/umf_singletons.c | 23 +-
src/C/SuiteSparse/UMFPACK/Source/umf_singletons.h | 2 +-
src/C/SuiteSparse/UMFPACK/Source/umf_solve.c | 2 +-
src/C/SuiteSparse/UMFPACK/Source/umf_solve.h | 2 +-
src/C/SuiteSparse/UMFPACK/Source/umf_start_front.c | 2 +-
src/C/SuiteSparse/UMFPACK/Source/umf_start_front.h | 2 +-
src/C/SuiteSparse/UMFPACK/Source/umf_store_lu.c | 2 +-
src/C/SuiteSparse/UMFPACK/Source/umf_store_lu.h | 2 +-
.../UMFPACK/Source/umf_symbolic_usage.c | 2 +-
.../UMFPACK/Source/umf_symbolic_usage.h | 2 +-
src/C/SuiteSparse/UMFPACK/Source/umf_transpose.c | 2 +-
src/C/SuiteSparse/UMFPACK/Source/umf_transpose.h | 2 +-
src/C/SuiteSparse/UMFPACK/Source/umf_triplet.c | 2 +-
src/C/SuiteSparse/UMFPACK/Source/umf_triplet.h | 2 +-
.../SuiteSparse/UMFPACK/Source/umf_tuple_lengths.c | 2 +-
.../SuiteSparse/UMFPACK/Source/umf_tuple_lengths.h | 2 +-
src/C/SuiteSparse/UMFPACK/Source/umf_usolve.c | 2 +-
src/C/SuiteSparse/UMFPACK/Source/umf_usolve.h | 2 +-
src/C/SuiteSparse/UMFPACK/Source/umf_utsolve.c | 2 +-
src/C/SuiteSparse/UMFPACK/Source/umf_utsolve.h | 2 +-
.../SuiteSparse/UMFPACK/Source/umf_valid_numeric.c | 2 +-
.../SuiteSparse/UMFPACK/Source/umf_valid_numeric.h | 2 +-
.../UMFPACK/Source/umf_valid_symbolic.c | 2 +-
.../UMFPACK/Source/umf_valid_symbolic.h | 2 +-
src/C/SuiteSparse/UMFPACK/Source/umf_version.h | 2 +-
.../UMFPACK/Source/umfpack_col_to_triplet.c | 2 +-
.../SuiteSparse/UMFPACK/Source/umfpack_defaults.c | 2 +-
.../UMFPACK/Source/umfpack_free_numeric.c | 2 +-
.../UMFPACK/Source/umfpack_free_symbolic.c | 2 +-
.../UMFPACK/Source/umfpack_get_determinant.c | 2 +-
.../SuiteSparse/UMFPACK/Source/umfpack_get_lunz.c | 2 +-
.../UMFPACK/Source/umfpack_get_numeric.c | 2 +-
.../UMFPACK/Source/umfpack_get_symbolic.c | 2 +-
src/C/SuiteSparse/UMFPACK/Source/umfpack_global.c | 2 +-
.../UMFPACK/Source/umfpack_load_numeric.c | 2 +-
.../UMFPACK/Source/umfpack_load_symbolic.c | 2 +-
src/C/SuiteSparse/UMFPACK/Source/umfpack_numeric.c | 2 +-
.../SuiteSparse/UMFPACK/Source/umfpack_qsymbolic.c | 2 +-
.../UMFPACK/Source/umfpack_report_control.c | 2 +-
.../UMFPACK/Source/umfpack_report_info.c | 2 +-
.../UMFPACK/Source/umfpack_report_matrix.c | 2 +-
.../UMFPACK/Source/umfpack_report_numeric.c | 2 +-
.../UMFPACK/Source/umfpack_report_perm.c | 2 +-
.../UMFPACK/Source/umfpack_report_status.c | 2 +-
.../UMFPACK/Source/umfpack_report_symbolic.c | 2 +-
.../UMFPACK/Source/umfpack_report_triplet.c | 2 +-
.../UMFPACK/Source/umfpack_report_vector.c | 2 +-
.../UMFPACK/Source/umfpack_save_numeric.c | 2 +-
.../UMFPACK/Source/umfpack_save_symbolic.c | 2 +-
src/C/SuiteSparse/UMFPACK/Source/umfpack_scale.c | 2 +-
src/C/SuiteSparse/UMFPACK/Source/umfpack_solve.c | 2 +-
.../SuiteSparse/UMFPACK/Source/umfpack_symbolic.c | 2 +-
src/C/SuiteSparse/UMFPACK/Source/umfpack_tictoc.c | 2 +-
src/C/SuiteSparse/UMFPACK/Source/umfpack_timer.c | 2 +-
.../SuiteSparse/UMFPACK/Source/umfpack_transpose.c | 2 +-
.../UMFPACK/Source/umfpack_triplet_to_col.c | 2 +-
src/C/amd.c | 16 +-
src/C/base.c | 16 +-
src/C/blas.c | 24 +-
src/C/cholmod.c | 16 +-
src/C/cvxopt.h | 16 +-
src/C/dense.c | 21 +-
src/C/dsdp.c | 16 +-
src/C/fftw.c | 16 +-
src/C/glpk.c | 16 +-
src/C/lapack.c | 16 +-
src/C/misc.h | 16 +-
src/C/mosek.c | 16 +-
src/C/random.c | 16 +-
src/C/sparse.c | 16 +-
src/C/umfpack.c | 16 +-
src/python/coneprog.py | 4108 ++++++++++++--------
src/python/cvxprog.py | 16 +-
src/python/info.py | 56 +-
src/python/misc.py | 16 +-
src/python/modeling.py | 16 +-
src/python/solvers.py | 23 +-
src/setup.py | 20 +-
588 files changed, 13967 insertions(+), 24462 deletions(-)
diff --git a/INSTALL b/INSTALL
index a9ffc1a..73352f8 100644
--- a/INSTALL
+++ b/INSTALL
@@ -1,4 +1,4 @@
-Installation instructions for CVXOPT Version 0.8.2.
+Installation instructions for CVXOPT Version 0.9.
The package requires version 2.3 or newer of Python, and is built
from source, so the header files and libraries for Python must be
diff --git a/LICENSE b/LICENSE
index 62afd96..ebaf7d6 100644
--- a/LICENSE
+++ b/LICENSE
@@ -1,9 +1,8 @@
-CVXOPT version 0.8.2.
-Copyright (C) 2004-2007 J. Dahl and L. Vandenberghe.
+CVXOPT version 0.9. Copyright (c) 2004-2007 J. Dahl and L. Vandenberghe.
This program is free software; you can redistribute it and/or modify
it under the terms of the GNU General Public License as published by
-the Free Software Foundation; either version 2 of the License, or
+the Free Software Foundation; either version 3 of the License, or
(at your option) any later version.
This program is distributed in the hope that it will be useful,
@@ -11,309 +10,707 @@ but WITHOUT ANY WARRANTY; without even the implied warranty of
MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the
GNU General Public License for more details.
-A copy of the GNU General Public License is enclosed below.
-See also www.gnu.org/copyleft/gpl.html.
-------------------------------------------------------------------------
+A copy of the GNU General Public License is included below.
+For further information, see <http://www.gnu.org/licenses/>.
+---------------------------------------------------------------------------
-The CVXOPT distribution includes source code for the following software
-libraries.
+The CVXOPT distribution includes source code for the following software
+libraries.
-AMD Version 2.0. Copyright (c) 2006 by Timothy A. Davis, Patrick R.
-Amestoy, and Iain S. Duff.
-See www.cise.ufl.edu/research/sparse/amd.
+1. Part of the SuiteSparse suite of sparse matrix algorithms, including:
+ - AMD Version 2.1. Copyright (c) 2007 by Timothy A. Davis, Patrick R.
+ Amestoy, and Iain S. Duff.
+ - CHOLMOD Version 1.5. Copyright (c) 2005-2007 by University of Florida,
+ Timothy A. Davis and W. Hager.
+ - COLAMD version 2.7. Copyright (c) 1998-2007 by Timothy A. Davis.
+ - UMFPACK Version 5.0.2. Copyright (c) 1995-2006 by Timothy A. Davis.
-CHOLMOD Version 1.4. Copyright (c) 2005-2006 by University of Florida,
-Timothy A. Davis and William W. Hager.
-See www.cise.ufl.edu/research/sparse/cholmod.
+ These packages are licensed under the terms of the GNU Lesser General
+ Public License, version 2.1 or higher (UMFPACK, parts of CHOLMOD, AMD,
+ COLAMD) and the GNU General Public License, version 2 or higher
+ (the Supernodal module of CHOLMOD). For copyright and license details,
+ consult the README files in the source directories or the website
+ listed below.
-COLAMD version 2.6. Copyright (c) 1998-2006 by Timothy A. Davis.
-See www.cise.ufl.edu/research/sparse/colamd.
+ Availability: www.cise.ufl.edu/research/sparse.
-UMFPACK Version 5.0.2. Copyright (c) 1995-2006 by Timothy A. Davis.
-See www.cise.ufl.edu/research/sparse/umfpack.
+2. RNGS Random Number Generation -- Multiple Streams (Sep. 22, 1998) by
+ Steve Park & Dave Geyer.
-RNGS Random Number Generation - Multiple Streams (Sep. 22, 1998) by
-Steve Park & Dave Geyer.
-See www.cs.wm.edu/~va/software/park/park.html.
-------------------------------------------------------------------------
+ Availability: www.cs.wm.edu/~va/software/park/park.html.
+----------------------------------------------------------------------------
- GNU GENERAL PUBLIC LICENSE
- Version 2, June 1991
+ GNU GENERAL PUBLIC LICENSE
+ Version 3, 29 June 2007
- Copyright (C) 1989, 1991 Free Software Foundation, Inc.
- 59 Temple Place, Suite 330, Boston, MA 02111-1307 USA
+ Copyright (C) 2007 Free Software Foundation, Inc. <http://fsf.org/>
Everyone is permitted to copy and distribute verbatim copies
of this license document, but changing it is not allowed.
- Preamble
+ Preamble
- The licenses for most software are designed to take away your
-freedom to share and change it. By contrast, the GNU General Public
-License is intended to guarantee your freedom to share and change free
-software--to make sure the software is free for all its users. This
-General Public License applies to most of the Free Software
-Foundation's software and to any other program whose authors commit to
-using it. (Some other Free Software Foundation software is covered by
-the GNU Library General Public License instead.) You can apply it to
+ The GNU General Public License is a free, copyleft license for
+software and other kinds of works.
+
+ The licenses for most software and other practical works are designed
+to take away your freedom to share and change the works. By contrast,
+the GNU General Public License is intended to guarantee your freedom to
+share and change all versions of a program--to make sure it remains free
+software for all its users. We, the Free Software Foundation, use the
+GNU General Public License for most of our software; it applies also to
+any other work released this way by its authors. You can apply it to
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+ To protect your rights, we need to prevent others from denying you
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+ If the Program specifies that a proxy can decide which future
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+public statement of acceptance of a version permanently authorizes you
+to choose that version for the Program.
+
+ Later license versions may give you additional or different
+permissions. However, no additional obligations are imposed on any
+author or copyright holder as a result of your choosing to follow a
+later version.
+
+ 15. Disclaimer of Warranty.
+
+ THERE IS NO WARRANTY FOR THE PROGRAM, TO THE EXTENT PERMITTED BY
+APPLICABLE LAW. EXCEPT WHEN OTHERWISE STATED IN WRITING THE COPYRIGHT
+HOLDERS AND/OR OTHER PARTIES PROVIDE THE PROGRAM "AS IS" WITHOUT WARRANTY
+OF ANY KIND, EITHER EXPRESSED OR IMPLIED, INCLUDING, BUT NOT LIMITED TO,
+THE IMPLIED WARRANTIES OF MERCHANTABILITY AND FITNESS FOR A PARTICULAR
+PURPOSE. THE ENTIRE RISK AS TO THE QUALITY AND PERFORMANCE OF THE PROGRAM
+IS WITH YOU. SHOULD THE PROGRAM PROVE DEFECTIVE, YOU ASSUME THE COST OF
+ALL NECESSARY SERVICING, REPAIR OR CORRECTION.
+
+ 16. Limitation of Liability.
+
+ IN NO EVENT UNLESS REQUIRED BY APPLICABLE LAW OR AGREED TO IN WRITING
+WILL ANY COPYRIGHT HOLDER, OR ANY OTHER PARTY WHO MODIFIES AND/OR CONVEYS
+THE PROGRAM AS PERMITTED ABOVE, BE LIABLE TO YOU FOR DAMAGES, INCLUDING ANY
+GENERAL, SPECIAL, INCIDENTAL OR CONSEQUENTIAL DAMAGES ARISING OUT OF THE
+USE OR INABILITY TO USE THE PROGRAM (INCLUDING BUT NOT LIMITED TO LOSS OF
+DATA OR DATA BEING RENDERED INACCURATE OR LOSSES SUSTAINED BY YOU OR THIRD
+PARTIES OR A FAILURE OF THE PROGRAM TO OPERATE WITH ANY OTHER PROGRAMS),
+EVEN IF SUCH HOLDER OR OTHER PARTY HAS BEEN ADVISED OF THE POSSIBILITY OF
+SUCH DAMAGES.
+
+ 17. Interpretation of Sections 15 and 16.
+
+ If the disclaimer of warranty and limitation of liability provided
+above cannot be given local legal effect according to their terms,
+reviewing courts shall apply local law that most closely approximates
+an absolute waiver of all civil liability in connection with the
+Program, unless a warranty or assumption of liability accompanies a
+copy of the Program in return for a fee.
+
+ END OF TERMS AND CONDITIONS
+
+ How to Apply These Terms to Your New Programs
+
+ If you develop a new program, and you want it to be of the greatest
+possible use to the public, the best way to achieve this is to make it
+free software which everyone can redistribute and change under these terms.
+
+ To do so, attach the following notices to the program. It is safest
+to attach them to the start of each source file to most effectively
+state the exclusion of warranty; and each file should have at least
+the "copyright" line and a pointer to where the full notice is found.
+
+ <one line to give the program's name and a brief idea of what it does.>
+ Copyright (C) <year> <name of author>
+
+ This program is free software: you can redistribute it and/or modify
+ it under the terms of the GNU General Public License as published by
+ the Free Software Foundation, either version 3 of the License, or
+ (at your option) any later version.
+
+ This program is distributed in the hope that it will be useful,
+ but WITHOUT ANY WARRANTY; without even the implied warranty of
+ MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the
+ GNU General Public License for more details.
+
+ You should have received a copy of the GNU General Public License
+ along with this program. If not, see <http://www.gnu.org/licenses/>.
+
+Also add information on how to contact you by electronic and paper mail.
+
+ If the program does terminal interaction, make it output a short
+notice like this when it starts in an interactive mode:
+
+ <program> Copyright (C) <year> <name of author>
+ This program comes with ABSOLUTELY NO WARRANTY; for details type `show w'.
+ This is free software, and you are welcome to redistribute it
+ under certain conditions; type `show c' for details.
+
+The hypothetical commands `show w' and `show c' should show the appropriate
+parts of the General Public License. Of course, your program's commands
+might be different; for a GUI interface, you would use an "about box".
+
+ You should also get your employer (if you work as a programmer) or school,
+if any, to sign a "copyright disclaimer" for the program, if necessary.
+For more information on this, and how to apply and follow the GNU GPL, see
+<http://www.gnu.org/licenses/>.
+
+ The GNU General Public License does not permit incorporating your program
+into proprietary programs. If your program is a subroutine library, you
+may consider it more useful to permit linking proprietary applications with
+the library. If this is what you want to do, use the GNU Lesser General
+Public License instead of this License. But first, please read
+<http://www.gnu.org/philosophy/why-not-lgpl.html>.
diff --git a/doc/.latex2html-init b/doc/.latex2html-init
new file mode 100644
index 0000000..8345f62
--- /dev/null
+++ b/doc/.latex2html-init
@@ -0,0 +1,198 @@
+#LaTeX2HTML Version 96.1 : dot.latex2html-init
+#
+### Command Line Argument Defaults #######################################
+
+$IMAGE_TYPE = $IMAGE_TYPES[1]; # use gif (see /etc/latex2html.conf)
+
+$MAX_SPLIT_DEPTH = 8; # Stop making separate files at this depth
+
+$MAX_LINK_DEPTH = 4; # Stop showing child nodes at this depth
+
+$NOLATEX = 0; # 1 = do not pass unknown environments to Latex
+
+$EXTERNAL_IMAGES = 0; # 1 = leave the images outside the document
+
+$ASCII_MODE = 0; # 1 = do not use any icons or internal images
+
+# 1 = use links to external postscript images rather than inlined bitmap
+# images.
+$PS_IMAGES = 0;
+
+$TITLE = $default_title; # The default is "No Title"
+
+$DESTDIR = ''; # Put the result in this directory
+
+# When this is set, the generated HTML files will be placed in the
+# current directory. If set to 0 the default behaviour is to create (or reuse)
+# another file directory.
+$NO_SUBDIR = 0;
+
+
+# Supply your own string if you don't like the default <Name> <Date>
+## $ADDRESS = "<I>$address_data[0] <BR>\n$address_data[1]</I>";
+$ADDRESS = "";
+
+$NO_NAVIGATION = 0; # 1 = do not put a navigation panel at the top of each page
+
+# Put navigation links at the top of each page. If the page exceeds
+# $WORDS_IN_PAGE number of words then put one at the bottom of the page.
+$AUTO_NAVIGATION = 1;
+
+# Put a link to the index page in the navigation panel
+$INDEX_IN_NAVIGATION = 1;
+
+# Put a link to the table of contents in the navigation panel
+$CONTENTS_IN_NAVIGATION = 1;
+
+# Put a link to the next logical page in the navigation panel
+$NEXT_PAGE_IN_NAVIGATION = 1;
+
+# Put a link to the previous logical page in the navigation panel
+$PREVIOUS_PAGE_IN_NAVIGATION = 1;
+
+$INFO = 0; # 0 = do not make a "About this document..." section
+
+# Reuse images generated during previous runs
+$REUSE = 2;
+
+# When this is 1, the section numbers are shown. The section numbers should
+# then match those that would have bee produced by LaTeX.
+# The correct section numbers are obtained from the $FILE.aux file generated
+# by LaTeX.
+# Hiding the seciton numbers encourages use of particular sections
+# as standalone documents. In this case the cross reference to a section
+# is shown using the default symbol rather than the section number.
+$SHOW_SECTION_NUMBERS = 1;
+
+### Other global variables ###############################################
+$CHILDLINE = "<BR> <HR>\n";
+
+# This is the line width measured in pixels and it is used to right justify
+# equations and equation arrays;
+$LINE_WIDTH = 500;
+
+# Used in conjunction with AUTO_NAVIGATION
+$WORDS_IN_PAGE = 300;
+
+# Affects ONLY the way accents are processed
+$default_language = 'english';
+
+# The value of this variable determines how many words to use in each
+# title that is added to the navigation panel (see below)
+#
+$WORDS_IN_NAVIGATION_PANEL_TITLES = 4;
+
+# This number will determine the size of the equations, special characters,
+# and anything which will be converted into an inlined image
+# *except* "image generating environments" such as "figure", "table"
+# or "minipage".
+# Effective values are those greater than 0.
+# Sensible values are between 0.1 - 4.
+$MATH_SCALE_FACTOR = 1.6;
+
+# This number will determine the size of
+# image generating environments such as "figure", "table" or "minipage".
+# Effective values are those greater than 0.
+# Sensible values are between 0.1 - 4.
+$FIGURE_SCALE_FACTOR = 1.6;
+
+
+# If this is set then intermediate files are left for later inspection.
+# This includes $$_images.tex and $$_images.log created during image
+# conversion.
+# Caution: Intermediate files can be *enormous*.
+$DEBUG = 0;
+
+# If both of the following two variables are set then the "Up" button
+# of the navigation panel in the first node/page of a converted document
+# will point to $EXTERNAL_UP_LINK. $EXTERNAL_UP_TITLE should be set
+# to some text which describes this external link.
+$EXTERNAL_UP_LINK = "";
+$EXTERNAL_UP_TITLE = "";
+
+# If this is set then the resulting HTML will look marginally better if viewed
+# with Netscape.
+$NETSCAPE_HTML = 0;
+
+# Valid paper sizes are "letter", "legal", "a4","a3","a2" and "a0"
+# Paper sizes has no effect other than in the time it takes to create inlined
+# images and in whether large images can be created at all ie
+# - larger paper sizes *MAY* help with large image problems
+# - smaller paper sizes are quicker to handle
+$PAPERSIZE = "a4";
+
+# Replace "english" with another language in order to tell LaTeX2HTML that you
+# want some generated section titles (eg "Table of Contents" or "References")
+# to appear in a different language. Currently only "english" and "french"
+# is supported but it is very easy to add your own. See the example in the
+# file "latex2html.config"
+$TITLES_LANGUAGE = "english";
+
+### Navigation Panel ##########################################################
+#
+# The navigation panel is constructed out of buttons and section titles.
+# These can be configured in any combination with arbitrary text and
+# HTML tags interspersed between them.
+# The buttons available are:
+# $PREVIOUS - points to the previous section
+# $UP - points up to the "parent" section
+# $NEXT - points to the next section
+# $NEXT_GROUP - points to the next "group" section
+# $PREVIOUS_GROUP - points to the previous "group" section
+# $CONTENTS - points to the contents page if there is one
+# $INDEX - points to the index page if there is one
+#
+# If the corresponding section exists the button will contain an
+# active link to that section. If the corresponding section does
+# not exist the button will be inactive.
+#
+# Also for each of the $PREVIOUS $UP $NEXT $NEXT_GROUP and $PREVIOUS_GROUP
+# buttons there are equivalent $PREVIOUS_TITLE, $UP_TITLE, etc variables
+# which contain the titles of their corresponding sections.
+# Each title is empty if there is no corresponding section.
+#
+# The subroutine below constructs the navigation panels in each page.
+# Feel free to mix and match buttons, titles, your own text, your logos,
+# and arbitrary HTML (the "." is the Perl concatenation operator).
+sub top_navigation_panel {
+
+ # Now add a few buttons with a space between them
+# "$NEXT $UP $PREVIOUS $CONTENTS $INDEX $CUSTOM_BUTTONS" .
+
+# "<BR>\n" . # Line break
+
+ # If ``next'' section exists, add its title to the navigation panel
+ ($NEXT_TITLE ? "<B> Next:</B> $NEXT_TITLE\n" : undef) .
+
+ # Similarly with the ``up'' title ...
+ ($UP_TITLE ? "<B>Up:</B> $UP_TITLE\n" : undef) .
+
+ # ... and the ``previous'' title
+ ($PREVIOUS_TITLE ? "<B> Previous:</B> $PREVIOUS_TITLE\n" : undef) .
+
+ # Line Break, horizontal rule (3-d dividing line) and new paragraph
+ "<BR> <P>\n"
+}
+
+sub bot_navigation_panel {
+
+ # Start with a horizontal rule (3-d dividing line)
+ "<HR>".
+
+ # Now add a few buttons with a space between them
+# "$NEXT $UP $PREVIOUS $CONTENTS $INDEX $CUSTOM_BUTTONS" .
+
+ "<BR>\n" . # Line break
+
+ # If ``next'' section exists, add its title to the navigation panel
+ ($NEXT_TITLE ? "<B> Next:</B> $NEXT_TITLE\n" : undef) .
+
+ # Similarly with the ``up'' title ...
+ ($UP_TITLE ? "<B>Up:</B> $UP_TITLE\n" : undef) .
+
+ # ... and the ``previous'' title
+ ($PREVIOUS_TITLE ? "<B> Previous:</B> $PREVIOUS_TITLE\n" : undef)
+
+}
+
+1; # This must be the last line
diff --git a/doc/Makefile b/doc/Makefile
index 00879d3..26e9576 100644
--- a/doc/Makefile
+++ b/doc/Makefile
@@ -1,14 +1,15 @@
-MKHOWTO = /usr/lib/python2.4/doc/tools/mkhowto
SOURCES = cvxopt.tex intro.tex base.tex blas.tex lapack.tex \
- spsolvers.tex modeling.tex solvers.tex c-api.tex
+ spsolvers.tex modeling.tex coneprog.tex solvers.tex c-api.tex
all: html
-ps: Makefile $(SOURCES)
- $(MKHOWTO) --ps cvxopt.tex
+latex: Makefile $(SOURCES)
+ latex cvxopt.tex
html: Makefile $(SOURCES)
- $(MKHOWTO) --html --image-type gif cvxopt.tex
+ latex cvxopt.tex
+ latex2html -ps_images cvxopt
+ rm -rf *.dvi *.idx *.ind *.l2h *.log *.toc *.aux *~
clean:
rm -rf *.dvi *.idx *.ind *.l2h *.log *.toc *.aux *~ *.ps cvxopt
diff --git a/doc/README b/doc/README
deleted file mode 100644
index f47740c..0000000
--- a/doc/README
+++ /dev/null
@@ -1,2 +0,0 @@
-python.sty in this directory is the style file from the standard Python
-distribution with lines 555--563 commented out.
diff --git a/doc/base.tex b/doc/base.tex
index c723608..450a299 100644
--- a/doc/base.tex
+++ b/doc/base.tex
@@ -18,7 +18,7 @@ The number of rows and/or the number of columns can be zero.
\ztc, for integer, real (double) and complex matrices, respectively.
\var{x} can be a number, a sequence of numbers, a dense or sparse
-matrix, a two-dimensional \module{numarray} array, or a list of
+matrix, a one- or two-dimensional \program{NumPy} array, or a list of
lists of matrices and numbers.
\BIT
\item If \var{x} is a number (Python \intgr, \flt\ or \cmplx), a matrix
@@ -44,10 +44,10 @@ double to complex when used to create a matrix of type \ztc).
\end{verbatim}
\item If \var{x} is a sequence of numbers (list, tuple, \module{array}
-array, xrange object, one-dimensional \module{numarray} array, \ldots),
+array, xrange object, one-dimensional \program{NumPy} array, \ldots),
then the numbers are interpreted as the coefficients of a matrix in
column-major order. The length of \var{x} must be equal to the product
-of \var{size}[0] and \var{size}[1].
+of \code{size[0]} and \code{size[1]}.
If \var{size} is not specified, a matrix with one column is created.
If \var{tc} is not specified, it is determined from the elements of
\var{x} (and if that is impossible, for example because \var{x} is
@@ -68,11 +68,11 @@ matrix.
\end{verbatim}
\item If \var{x} is a dense or sparse matrix (a \mtrx\ or a \spmtrx\
-object), or a two-dimensional \module{numarray} array of type \itc,
+object), or a two-dimensional \program{NumPy} array of type \itc,
\dtc\ or \ztc, then the coefficients of \var{x} are copied, in
column-major order, to a new matrix of the given size.
The total number of elements in the new matrix (the product of
-\var{size}[0] and \var{size}[1]) must be the same as the product of
+\code{size[0]} and \code{size[1]}) must be the same as the product of
the dimensions of \var{x}. If \var{size} is not specified, the
dimensions of \var{x} are used.
The default value of \var{tc} is the type of \var{x}.
@@ -94,8 +94,8 @@ Type conversion takes place when the type of \var{x} differs from
1.0000e+00-j0.0000e+00 4.0000e+00-j0.0000e+00
2.0000e+00-j0.0000e+00 5.0000e+00-j0.0000e+00
3.0000e+00-j0.0000e+00 6.0000e+00-j0.0000e+00
->>> from numarray import array
->>> x = array([1., 2., 3., 4., 5., 6.], shape=(2,3))
+>>> from numpy import array
+>>> x = array([[1., 2., 3.], [4., 5., 6.]])
>>> print x
[[ 1. 2. 3.]
[ 4. 5. 6.]]
@@ -153,50 +153,50 @@ list (\ie, when the length of \var{x} is one, it can be replaced with
\section{Attributes and Methods}
A \mtrx\ has the following attributes.
-\begin{memberdesc}[tuple]{size}
+\begin{memberdesc}{size}
A tuple with the dimensions of the matrix. This is a read-only
attribute; operations that change the size of a matrix are not
permitted.
\end{memberdesc}
-\begin{memberdesc}[char]{typecode}
+\begin{memberdesc}{typecode}
A \ctype{char}, either \itc, \dtc, or \ztc, for integer, real
and complex matrices, respectively. A read-only attribute.
\end{memberdesc}
-\begin{methoddesc}{trans}{}
+\begin{methoddesc}{matrix}{trans}
Returns the transpose of the matrix as a new matrix.
One can also use \code{A.T} instead of \code{A.trans()}.
\end{methoddesc}
-\begin{methoddesc}{ctrans}{}
+\begin{methoddesc}{matrix}{ctrans}
Returns the conjugate transpose of the matrix as a new matrix.
One can also use \code{A.H} instead of \code{A.ctrans()}.
\end{methoddesc}
-\begin{methoddesc}{real}{}
+\begin{methoddesc}{matrix}{real}
For complex matrices, returns the real part as a real matrix.
For integer and real matrices, returns a copy of the matrix.
\end{methoddesc}
-\begin{methoddesc}{imag}{}
+\begin{methoddesc}{matrix}{imag}
For complex matrices, returns the imaginary part as a real matrix.
For integer and real matrices, returns an integer or real zero matrix.
\end{methoddesc}
-\begin{memberdesc}[PyCObject]{\_\_array\_struct\_\_}
-A PyCObject implementing the NumPy array interface
+\begin{memberdesc}{\_\_array\_struct\_\_}
+A PyCObject implementing the \program{NumPy} array interface
(see section~\ref{s-array-interface} for details).
\end{memberdesc}
-\begin{methoddesc}{tofile}{f}
+\begin{funcdesc}{tofile}{f}
Writes the elements of the matrix in column-major order to a binary
file \var{f}.
-\end{methoddesc}
+\end{funcdesc}
-\begin{methoddesc}{fromfile}{f}
+\begin{funcdesc}{fromfile}{f}
Reads the contents of a binary file \var{f} into the matrix object.
-\end{methoddesc}
+\end{funcdesc}
The last two methods are illustrated in the following example.
\begin{verbatim}
@@ -236,7 +236,7 @@ Addition & \code{\var{A}+\var{B}}, \code{\var{A}+\var{c}},
\code{\var{c}+\var{A}}\\
Subtraction & \code{\var{A}-\var{B}}, \code{\var{A}-\var{c}},
\code{\var{c}-\var{A}}\\
-Matrix multiplication & {\var{A}*\var{B}} \\
+Matrix multiplication & \code{{\var{A}*\var{B}}} \\
Scalar multiplication and division & \code{\var{c}*\var{A}},
\code{\var{A}*\var{c}}, \code{\var{A}/\var{c}} \\
Remainder after division & \code{\var{A}\%\var{c}} \\
@@ -657,10 +657,9 @@ The pseudo-random number generators used to
generate the random matrices are from the package described in the
references below.
\begin{seealso}
-\seelink{http://www.cs.wm.edu/\~{}va/software/park/park.html}
-{S. Park, Random Number Generators.}{}
-
-\seealso{S. Park, D. Geyer, Random Number Generators: Good Ones Are
+\item \seelink{http://www.cs.wm.edu/\~{}va/software/park/park.html}
+{S. Park, Random Number Generators.}{}
+\item \seetext{S. Park, D. Geyer, Random Number Generators: Good Ones Are
Hard To Find,
Communications of the ACM, October 1988.}
\end{seealso}
@@ -696,43 +695,55 @@ from the system clock.
\section{The NumPy Array Interface} \label{s-array-interface}
-The CVXOPT \mtrx\ object is compatible with the NumPy Array Interface,
+The CVXOPT \mtrx\ object is compatible with the \program{NumPy} Array
+Interface,
which allows Python objects that represent multidimensional
arrays to exchange data using information stored in the
attribute \code{\_\_array\_struct\_\_}.
\begin{seealso}
-\seelink{http://numpy.scipy.org/array_interface.shtml}
+\item \seelink{http://numpy.scipy.org/array_interface.shtml}
{NumPy Array Interface Specification}{}
-\seelink{http://numpy.scipy.org}{NumPy home page}{}
+\item \seelink{http://numpy.scipy.org}{NumPy home page}{}
\end{seealso}
As already mentioned in section~\ref{s-creating-matrices},
-a two-dimensional array object (for example, a 2D \module{numarray}
-array) can be converted to a \mtrx\ object by using the
-\function{matrix()} constructor.
+a two-dimensional array object (for example, a \program{NumPy} matrix or
+two-dimensional array) can be converted to a CVXOPT \mtrx\ object by using
+the \function{matrix()} constructor.
Conversely, CVXOPT matrices can be used as array-like objects
-in \module{numarray}. The following example illustrates the
-compatibility of CVXOPT matrices and \module{numarray} arrays.
+in \program{NumPy}. The following example illustrates the
+compatibility of CVXOPT matrices and \program{NumPy} arrays.
\begin{verbatim}
>>> from cvxopt import matrix
>>> a = matrix(range(6), (2,3), 'd')
>>> print a
0.0000e+00 2.0000e+00 4.0000e+00
1.0000e+00 3.0000e+00 5.0000e+00
->>> from numarray import array
+>>> from numpy import array
>>> b = array(a)
->>> print b
-[[ 0. 2. 4.]
- [ 1. 3. 5.]]
+>>> b
+array([[ 0. 2. 4.]
+ [ 1. 3. 5.]])
>>> print a*b
-[[ 0. 4. 16.]
- [ 1. 9. 25.]]
+array([[ 0. 4. 16.]
+ [ 1. 9. 25.]])
+>>> from numpy import mat
+>>> c = mat(a)
+>>> c
+matrix([[ 0. 2. 4.]
+ [ 1. 3. 5.]])
+>>> a.T * c
+matrix([[ 1., 3., 5.],
+ [ 3., 13., 23.],
+ [ 5., 23., 41.]])
\end{verbatim}
-In the last expression \code{a*b} is interpreted as a \module{numarray}
-multiplication (\ie, componentwise) even though one operand is a
-\mtrx\ object.
+In the first product, \code{a*b} is interpreted as \program{NumPy} array
+multiplication, \ie, componentwise multiplication.
+The second product \code{a.T*c} is interpreted as \program{NumPy} matrix
+multiplication, \ie, standard matrix multiplication.
+
\section{Printing Options}
The format used for printing dense matrices (and the sparse matrices
diff --git a/doc/base_sparse.tex b/doc/base_sparse.tex
index d0d7c6b..a14961f 100644
--- a/doc/base_sparse.tex
+++ b/doc/base_sparse.tex
@@ -187,7 +187,7 @@ SIZE: (6,3)
\section{Attributes and Methods}
The following attributes and methods are defined for \spmtrx\ objects.
-\begin{memberdesc}[matrix]{V}
+\begin{memberdesc}{V}
A single-column dense matrix containing the numerical values of the
nonzero entries in column-major order. Making an assignment to
the attribute is an efficient way of changing the values of the sparse
@@ -202,26 +202,26 @@ will be read and returned as a new matrix; then the elements of this
new matrix are modified.)
\end{memberdesc}
-\begin{memberdesc}[spmatrix]{I}
+\begin{memberdesc}{I}
A single-column integer matrix with the row indices of the entries in
\code{V}. A read-only attribute.
\end{memberdesc}
-\begin{memberdesc}[matrix]{J}
+\begin{memberdesc}{J}
A single-column integer matrix with the column indices of the entries
in \code{V}. A read-only attribute.
\end{memberdesc}
-\begin{memberdesc}[tuple]{size}
+\begin{memberdesc}{size}
A tuple with the dimensions of the matrix. A read-only attribute.
\end{memberdesc}
-\begin{methoddesc}{trans}{}
+\begin{methoddesc}{spmatrix}{trans}
Returns the transpose of a sparse matrix as a new sparse matrix.
One can also use \code{A.T} instead of \code{A.trans()}.
\end{methoddesc}
-\begin{methoddesc}{ctrans}{}
+\begin{methoddesc}{spmatrix}{ctrans}
Returns the complex conjugate transpose of a sparse matrix as a
new sparse matrix.
One can also use \code{A.H} instead of \code{A.ctrans()}.
@@ -347,7 +347,7 @@ Python number or a 1 by 1 dense matrix.
If \var{B} is a dense matrix, then the result of
\code{\var{A}+\var{B}}, \code{\var{B}+\var{A}}, \code{\var{A}-\var{B}},
\code{\var{B}-\var{A}} is a dense matrix.
-The typecode of the result is \dtc\ if \var{A} has typcode \dtc\ and
+The typecode of the result is \dtc\ if \var{A} has typecode \dtc\ and
\var{B} has typecode \itc\ or \dtc,
and it is \ztc\ if \var{A} and/or \var{B} have typecode \ztc.
@@ -688,7 +688,7 @@ SIZE: (3,3)
(0, 2) 2.0000e+00
(2, 2) 2.0000e+00
\end{verbatim}
-Now suppose we want to replace {\it C}
+Now suppose we want to replace {\it C} with
\[
C = A^TD, \qquad
D = \left[ \begin{array}{ccc}
diff --git a/doc/blas.tex b/doc/blas.tex
index 2d4531a..0dc383f 100644
--- a/doc/blas.tex
+++ b/doc/blas.tex
@@ -27,13 +27,13 @@ matrix addition and scalar multiplication of
section~\ref{s-arithmetic} to more complicated operations.
\begin{seealso}
-\seetext{C. L. Lawson, R. J. Hanson, D. R. Kincaid, F. T. Krogh,
+\item \seetext{C. L. Lawson, R. J. Hanson, D. R. Kincaid, F. T. Krogh,
Basic Linear Algebra Subprograms for Fortran Use,
ACM Transactions on Mathematical Software, 5(3), 309-323, 1975.}
-\seetext{J. J. Dongarra, J. Du Croz, S. Hammarling, R. J. Hanson,
+\item \seetext{J. J. Dongarra, J. Du Croz, S. Hammarling, R. J. Hanson,
An Extended Set of Fortran Basic Linear Algebra Subprograms,
ACM Transactions on Mathematical Software, 14(1), 1-17, 1988.}
-\seetext{J. J. Dongarra, J. Du Croz, S. Hammarling, I. Duff,
+\item \seetext{J. J. Dongarra, J. Du Croz, S. Hammarling, I. Duff,
A Set of Level 3 Basic Linear Algebra Subprograms,
ACM Transactions on Mathematical Software, 16(1), 1-17, 1990.}
\end{seealso}
@@ -85,7 +85,7 @@ X[0,n-1] & X[1,n-1] & X[2,n-1] & \cdots & X[n-1,n-1]
\item[Complex Hermitian matrix]
A complex Hermitian matrix of order {\it n} is represented
by a \mtrx\ of type \ztc\ and size ({\it n}, {\it n}), and
-a character argument \var{uplo} with the ame meaning as for symmetric
+a character argument \var{uplo} with the same meaning as for symmetric
matrices.
A complex \mtrx\ {\var X} of size ({\it n}, {\it n}) can
represent the Hermitian matrices
@@ -301,7 +301,7 @@ Euclidean norm of a vector: returns
\end{funcdesc}
\begin{funcdesc}{asum}{x}
-L-1 norm of a vector: returns
+1-Norm of a vector: returns
\[
\|x\|_1 \quad \mbox{($x$ real)}, \qquad
\|\Re x\|_1 + \|\Im x\|_1 \quad \mbox{($x$ complex)}.
diff --git a/doc/c-api.tex b/doc/c-api.tex
index a836b51..b25b5b7 100644
--- a/doc/c-api.tex
+++ b/doc/c-api.tex
@@ -6,8 +6,8 @@ objects defined in \module{cvxopt.base} must include the
\file{cvxopt.h} header file in the \file{src} directory of the
distribution.
-Before the C-API can be used in an extension module it must be
-initialized by calling the macro \cfunction{import\_cvxopt}.
+Before the C API can be used in an extension module it must be
+initialized by calling the macro \function{import\_cvxopt}.
As an example
we show the module initialization from the \module{cvxopt.blas} module,
which itself uses the API:
@@ -35,18 +35,18 @@ The \cdata{buffer} field is an array that contains the matrix elements
stored contiguously in column-major order.
The following C functions can be used to create matrices.
-\begin{cfuncdesc}{matrix*}{Matrix\_New}{int nrows, int ncols, int id}
+\begin{cfuncdesc}{matrix *}{Matrix\_New}{int nrows, int ncols, int id}
Returns a \mtrx\ object of type \var{id} with \var{nrows} rows
and \var{ncols} columns. The elements of the matrix are uninitialized.
\end{cfuncdesc}
-\begin{cfuncdesc}{matrix*}{Matrix\_NewFromMatrix}{matrix *src, int id}
+\begin{cfuncdesc}{matrix *}{Matrix\_NewFromMatrix}{matrix *src, int id}
Returns a copy of the matrix \var{src} converted to type \var{id}.
The following type conversions are allowed: \itc\ to \dtc,
\itc\ to \ztc\ and \dtc\ to \ztc.
\end{cfuncdesc}
-\begin{cfuncdesc}{matrix*}{Matrix\_NewFromSequence}{PyListObject *x, int id}
+\begin{cfuncdesc}{matrix *}{Matrix\_NewFromSequence}{PyListObject *x, int id}
Creates a matrix of type \var{id} from the Python sequence type \var{x}. The
returned matrix has size \code{(len(\var{x}),1)}.
The size can be changed by modifying the \member{nrows} and
@@ -144,7 +144,7 @@ The \cdata{id} field controls the type of the matrix and can have
values \constant{DOUBLE} and \constant{COMPLEX}.
Sparse matrices are created using the following functions from the API.
-\begin{cfuncdesc}{spmatrix*}{SpMatrix\_New}{int nrows, int ncols, int
+\begin{cfuncdesc}{spmatrix *}{SpMatrix\_New}{int nrows, int ncols, int
nzmax, int id}
Returns a sparse zero matrix with \var{nrows} rows and
\var{ncols} columns. \var{nzmax} is the number of elements that will
@@ -152,12 +152,12 @@ Sparse matrices are created using the following functions from the API.
fields).
\end{cfuncdesc}
-\begin{cfuncdesc}{spmatrix*}{SpMatrix\_NewFromMatrix}{spmatrix *src, int
+\begin{cfuncdesc}{spmatrix *}{SpMatrix\_NewFromMatrix}{spmatrix *src, int
id}
Returns a copy the sparse matrix \var{src}.
\end{cfuncdesc}
-\begin{cfuncdesc}{spmatrix*}{SpMatrix\_NewFromIJV}{matrix *I, matrix *J,
+\begin{cfuncdesc}{spmatrix *}{SpMatrix\_NewFromIJV}{matrix *I, matrix *J,
matrix *V, int nrows, int ncols, int nzmax, int id}
Creates a sparse matrix with \var{nrows} rows and \var{ncols}
columns from a triplet description. \var{I} and \var{J}
@@ -170,7 +170,7 @@ Sparse matrices are created using the following functions from the API.
\end{cfuncdesc}
We illustrate use of the sparse matrix class by listing the source
-code for the \method{real()} method, which returns the real part of
+code for the \function{real()} method, which returns the real part of
a sparse matrix:
\begin{verbatim}
diff --git a/doc/coneprog.tex b/doc/coneprog.tex
new file mode 100644
index 0000000..64d23c7
--- /dev/null
+++ b/doc/coneprog.tex
@@ -0,0 +1,1255 @@
+\chapter{Cone Programming (\module{cvxopt.solvers})}
+\label{chap:coneprog}
+
+A \emph{cone (linear) program} is an optimization problem of the form
+\[
+ \begin{array}{ll}
+ \mbox{minimize} & c^T x \\
+ \mbox{subject to} & G x \preceq h \\ & Ax = b.
+ \end{array}
+\]
+The inequality is a generalized inequality with respect to a proper convex
+cone. The \module{cvxopt.solvers} module provides functions for solving
+cone programs with constraints that include (scalar) linear inequalities,
+second-order cone inequalities, and linear matrix inequalities.
+The main solver, described in section~\ref{s-conelp}, is
+\function{conelp()}.
+For convenience (and backward compatibility), simpler interfaces to this
+function are also provided that handle pure linear programs, second-order
+cone programs, and semidefinite programs. These are described in
+sections~\ref{s-lpsolver}--\ref{s-sdpsolver}.
+In section~\ref{s-conelp-struct} we explain how customized solvers
+can be implemented that exploit structure in specific classes of problems.
+The last two sections describe optional interfaces to external solvers,
+and the algorithm parameters that control the cone programming solvers.
+
+\section{General Solver} \label{s-conelp}
+\begin{funcdesc}{conelp}{c, G, h, dims\optional{, A, b\optional{,
+primalstart\optional{, dualstart\optional{, kktsolver}}}}}
+Solves a pair of primal and dual cone programs
+\BEQ \label{e-conelp}
+ \begin{array}[t]{ll}
+ \mbox{minimize} & c^T x \\
+ \mbox{subject to} & G x + s = h \\ & Ax = b \\ & s \succeq 0
+ \end{array}
+ \qquad\qquad\qquad\qquad
+ \begin{array}[t]{ll}
+ \mbox{maximize} & -h^T z - b^T y \\
+ \mbox{subject to} & G^T z + A^T y + c = 0 \\
+ & z \succeq 0.
+ \end{array}
+\EEQ
+The primal variables are {\it x} and the slack variable {\it s}.
+The dual variables are {\it y} and {\it z}. The inequalities are
+interpreted as $s \in C$, $z\in C$, where $C$ is a cone defined as a
+Cartesian product of a nonnegative orthant, a number of second-order
+cones, and a number of positive semidefinite cones:
+\[
+C = C_0 \times C_1 \times \cdots \times C_M \times C_{M+1} \times
+ \cdots \times C_{M+N}
+\]
+with
+\[
+C_0 =
+ \{ u \in \reals^l \;| \; u_k \geq 0, \; k=1, \ldots,l\}, \qquad
+C_{k+1} = \{ (u_0, u_1) \in \reals \times \reals^{q_{k}-1} \; | \;
+ u_0 \geq \|u_1\|_2 \}, \quad k=0,\ldots, M-1, \qquad
+C_{k+M+1} = \left\{ \svec(u) \; | \;
+ u \in \symm^{p_k}_+ \right\}, \quad k=0,\ldots,N-1.
+\]
+Here $\svec(u)$ denotes a symmetric matrix {\it u} stored as a vector
+in column major order.
+
+The arguments \var{c}, \var{h} and \var{b} are real single-column dense
+matrices. \var{G} and \var{A} are real dense or sparse matrices.
+The default values for \var{A} and \var{b} are sparse matrices with
+zero rows, meaning that there are no equality constraints.
+The number of rows of \var{G} and \var{h} is equal to
+\[
+ K = l + \sum_{k=0}^{M-1} q_k + \sum_{k=0}^{N-1} p_k^2.
+\]
+The columns of \var{G} and \var{h} are vectors in
+\[
+\reals^l \times \reals^{q_0} \times \cdots \times
+\reals^{q_{M-1}} \times \reals^{p_0^2} \times \cdots \times
+\reals^{p_{N-1}^2},
+\]
+where the last {\it N} components represent symmetric matrices stored in
+column major order. The strictly upper triangular entries of these
+matrices are not accessed ({\it i.e.}, the symmetric matrices are stored
+in the 'L'-type column major order used in the \module{blas} and
+\module{lapack} modules).
+
+The argument \var{dims} is a dictionary with the dimensions of the
+cones. It has three fields.
+\begin{description}
+\item[\var{dims['l']}:] $l$, the dimension of the nonnegative orthant
+ (a nonnegative integer).
+\item[\var{dims['q']}:] \code{[q\_0, \ldots, q\_\{M-1\}]},
+a list with the dimensions of the second-order cones (positive integers).
+\item[\var{dims['s']}:]
+\code{[p\_0, \ldots, p\_\{N-1\}]},
+a list with the dimensions of the positive semidefinite cones
+(nonnegative integers).
+\end{description}
+
+\var{primalstart} is a dictionary with keys \code{'x'} and \code{'s'},
+used as an optional primal starting point.
+\code{primalstart['x']} and
+\code{primalstart['s']} are real dense matrices of size
+\code{(n,1)} and \code{(K,1)}, respectively, where {\it n} is the
+length of \var{c}.
+The vector \code{primalstart['s']} must be strictly positive with respect
+to the cone $C$.
+
+\var{dualstart} is a dictionary with keys \code{'y'} and \code{'z'},
+used as an optional dual starting point.
+\code{dualstart['y']} and
+\code{dualstart['z']} are real dense matrices of size
+\code{(p,1)} and \code{(K,1)}, respectively, where {\it p} is the
+number of rows in \var{A}.
+The vector \code{dualstart['s']} must be strictly positive with respect
+to the cone $C$.
+
+The role of the optional argument \var{kktsolver} is explained in
+section~\ref{s-conelp-struct}.
+
+\function{conelp()} returns a dictionary with keys \code{'status'},
+\code{'x'}, \code{'s'}, \code{'y'}, \code{'z'}.
+The \code{'status'} field is a string with possible values
+\code{'optimal'}, \code{'primal infeasible'}, \code{'dual infeasible'}
+and \code{'unknown'}. The meaning of the other fields depends on the
+value of \code{'status'}.
+\begin{description}
+\item[\code{'optimal'.}] In this case the \code{'x'}, \code{'s'},
+\code{'y'} and \code{'z'} entries contain the primal and dual solutions,
+which approximately satisfy
+\[
+ Gx + s = h, \qquad Ax=b, \qquad G^T z + A^T y + c = 0, \qquad
+ s \succeq 0, \qquad z \succeq 0, \qquad s^T z =0.
+\]
+
+\item[\code{'primal infeasible'.}]
+The \code{'x'} and \code{'s'} entries are \None, and the \code{'y'},
+\code{'z'} entries provide an approximate certificate of
+infeasibility, \ie, vectors that approximately satisfy
+\[
+ G^T z + A^T y = 0, \qquad h^T z + b^T y = -1, \qquad z \succeq 0.
+\]
+
+\item[\code{'dual infeasible'.}]
+The \code{'y'} and \code{'z'} entries are \None, and the \code{'x'}
+and \code{'s'} entries contain an approximate certificate of dual
+infeasibility
+\[
+ Gx + s = 0, \qquad Ax=0, \qquad c^T x = -1, \qquad s \succeq 0.
+\]
+
+\item[\code{'unknown'}.] The \code{'x'}, \code{'s'}, \code{'y'},
+\code{'z'} entries are \None.
+\end{description}
+
+It is required that
+\[
+\Rank(A) = p, \qquad
+\Rank(\left[\begin{array}{c} G \\ A \end{array}\right]) = n,
+\]
+where {\it p} is the number or rows of \var{A} and {\it n}
+is the number of columns of \var{G} and \var{A}.
+\end{funcdesc}
+
+As an example we solve the problem
+\[
+\begin{array}{ll}
+\mbox{minimize} & -6x_1 - 4x_2 - 5x_3 \\*[1ex]
+\mbox{subject to} & 16x_1 - 14x_2 + 5x_3 \leq -3 \\*[1ex]
+ & 7x_1 + 2x_2 \leq 5 \\*[1ex]
+ & \left( (8x_1 + 13x_2 - 12x_3 - 2)^2 + (-8x_1 + 18x_2 + 6x_3 - 14)^2
+ + (x_1 - 3x_2 - 17x_3 - 13)^2\right)^{1/2} \leq -24x_1 - 7x_2 + 15x_3 + 12 \\*[1ex]
+ & (x_1^2 + x_2^2 + x_3^2)^{1/2} \leq 10 \\*[1ex]
+ & \left[\begin{array}{ccc}
+ 7x_1 + 3x_2 + 9x_3 & -5x_1 + 13x_2 + 6x_3 & x_1 - 6x_2 -6x_3 \\
+-5x_1 + 13x_2 + 6x_3 & x_1 + 12x_2 -7x_3 & -7x_1 -10x_2 - 7x_3 \\
+ x_1 - 6x_2 -6x_3 & -7x_1 -10x_2 -7 x_3 &
+-4x_1 -28 x_2 -11x_3 \end{array}\right]
+\preceq
+\left[\begin{array}{ccc}
+68 & -30 & -19 \\
+-30 & 99 & 23 \\
+-19 & 23 & 10 \end{array}\right].
+\end{array}
+\]
+
+\begin{verbatim}
+>>> from cvxopt.base import matrix
+>>> from cvxopt import solvers
+>>> c = matrix([-6., -4., -5.])
+>>> G = matrix([[ 16., 7., 24., -8., 8., -1., 0., -1., 0., 0., 7., -5., 1., -5., 1., -7., 1., -7., -4.],
+ [-14., 2., 7., -13., -18., 3., 0., 0., -1., 0., 3., 13., -6., 13., 12., -10., -6., -10., -28.],
+ [ 5., 0., -15., 12., -6., 17., 0., 0., 0., -1., 9., 6., -6., 6., -7., -7., -6., -7., -11.]])
+>>> h = matrix( [ -3., 5., 12., -2., -14., -13., 10., 0., 0., 0., 68., -30., -19., -30., 99., 23., -19., 23., 10.] )
+>>> dims = {'l': 2, 'q': [4, 4], 's': [3]}
+>>> sol = solvers.conelp(c, G, h, dims)
+>>> print sol['status']
+optimal
+>>> print sol['x']
+ -1.2209e+00
+ 9.6633e-02
+ 3.5775e+00
+>>> print sol['z']
+ 9.2985e-02
+ 2.0401e-08
+ 2.3534e-01
+ 1.3339e-01
+ -4.7354e-02
+ 1.8801e-01
+ 2.7871e-08
+ 1.8544e-09
+ -6.3156e-10
+ -7.5921e-09
+ 1.2558e-01
+ 8.7775e-02
+ -8.6652e-02
+ 8.7775e-02
+ 6.1349e-02
+ -6.0564e-02
+ -8.6652e-02
+ -6.0564e-02
+ 5.9790e-02
+\end{verbatim}
+
+Only the entries of \var{G} and \var{h} defining the lower triangular
+portions of the coefficients in the linear matrix inequalities are
+accessed. This means we
+obtain the same result if we define \var{G} and \var{h} as below.
+\begin{verbatim}
+>>> G = matrix([[ 16., 7., 24., -8., 8., -1., 0., -1., 0., 0., 7., -5., 1., 0., 1., -7., 0., 0., -4.],
+ [-14., 2., 7., -13., -18., 3., 0., 0., -1., 0., 3., 13., -6., 0., 12., -10., 0., 0., -28.],
+ [ 5., 0., -15., 12., -6., 17., 0., 0., 0., -1., 9., 6., -6., 0., -7., -7., 0., 0., -11.]])
+>>> h = matrix( [ -3., 5., 12., -2., -14., -13., 10., 0., 0., 0., 68., -30., -19., 0., 99., 23., 0., 0., 10.] )
+\end{verbatim}
+
+
+\section{Linear Programming} \label{s-lpsolver}
+The function \function{lp()} is an interface to \function{conelp()} for
+linear programs. It also provides the option of using the linear programming
+solvers from GLPK or MOSEK.
+
+\begin{funcdesc}{lp}{c, G, h\optional{, A, b\optional{,
+solver\optional{, primalstart\optional{, dualstart}}}}}
+Solves the pair of primal and dual linear programs
+\[
+ \begin{array}[t]{ll}
+ \mbox{minimize} & c^T x \\
+ \mbox{subject to} & G x + s = h \\ & Ax = b \\ & s \succeq 0
+ \end{array}
+ \qquad\qquad
+ \begin{array}[t]{ll}
+ \mbox{maximize} & -h^T z - b^T y \\
+ \mbox{subject to} & G^T z + A^T y + c = 0 \\
+ & z \succeq 0.
+ \end{array}
+\]
+All inequalities are componentwise vector inequalities.
+
+The \var{solver} argument is used to choose among three solvers.
+When it is omitted or \None, the CVXOPT function
+\function{solvers.conelp()} is used.
+The external solvers GLPK and MOSEK (if installed) can be
+selected by setting \code{\var{solver} = 'glpk'} or
+\code{\var{solver} = 'mosek'}; see section~\ref{s-external}.
+
+The meaning of the other arguments and the return value are the same as
+for \function{conelp()} called with
+\code{dims = \{'l': G.size[0], 'q': [], 's': []\}}.
+No certificates of primal or dual infeasibility are returned with the
+\code{solver = 'glpk'} option.
+\end{funcdesc}
+
+As a simple example we solve the LP
+\[
+ \begin{array}[t]{ll}
+ \mbox{minimize} & -4x_1 - 5x_2 \\
+ \mbox{subject to} & 2x_1 + x_2 \leq 3 \\
+ & x_1 + 2x_2 \leq 3 \\
+ & x_1 \geq 0, \quad x_2 \geq 0.
+ \end{array}
+\]
+\begin{verbatim}
+>>> from cvxopt.base import matrix
+>>> from cvxopt import solvers
+>>> c = matrix([-4., -5.])
+>>> G = matrix([[2., 1., -1., 0.], [1., 2., 0., -1.]])
+>>> h = matrix([3., 3., 0., 0.])
+>>> sol = solvers.lp(c, G, h)
+>>> print sol['x']
+ 1.0000e-00
+ 1.0000e-00
+\end{verbatim}
+
+\section{Second-Order Cone Programming} \label{s-socpsolver}
+The function \function{socp()} is a simpler interface to
+\function{conelp()} for cone programs with no linear matrix inequality
+constraints.
+
+\begin{funcdesc}{socp}{c\optional{, Gl, hl\optional{, Gq, hq\optional{,
+ A, b\optional{, primalstart\optional{, dualstart}}}}}}
+Solves the pair of primal and dual second-order cone programs
+\[
+ \begin{array}[t]{ll}
+ \mbox{minimize} & c^T x \\
+ \mbox{subject to}
+ & G_k x + s_k = h_k, \quad k = 0, \ldots, M \\
+ & Ax = b \\
+ & s_0 \succeq 0 \\
+ & s_{k0} \geq \|s_{k1}\|_2, \quad k = 1, \ldots, M
+ \end{array}
+ \qquad\qquad
+ \begin{array}[t]{ll}
+ \mbox{maximize} & - \sum_{k=0}^M h_k^Tz_k - b^T y \\
+ \mbox{subject to} & \sum_{k=0}^M G_k^T z_k + A^T y + c = 0 \\
+ & z_0 \succeq 0 \\
+ & z_{k0} \geq \|z_{k1}\|_2, \quad k=1,\ldots,M.
+ \end{array}
+\]
+The inequalities
+\[
+ s_0 \succeq 0, \qquad z_0 \succeq 0
+\]
+are componentwise vector inequalities.
+In the other inequalities, it is assumed that the variables are partitioned
+as
+\[
+ s_k = (s_{k0}, s_{k1}) \in\reals\times\reals^{q_{k}-1}, \qquad
+ z_k = (z_{k0}, z_{k1}) \in\reals\times\reals^{q_{k}-1}.
+\]
+
+The input argument \var{c} is a real single-column dense matrix.
+The arguments \var{Gl} and \var{hl} are the coefficient matrix $G_0$
+and the righthand side $h_0$ of the componentwise inequalities.
+\var{Gl} is a real dense or sparse matrix; \var{hl} is a real single-column
+dense matrix. The default values for \var{Gl} and \var{hl} are matrices
+with zero rows.
+
+The argument \var{Gq} is a list of $M$ dense or sparse matrices
+\var{G\_1}, \ldots, \var{G\_M}.
+The argument \var{hq} is a list of $M$ dense single-column matrices
+\var{h\_1}, \ldots, \var{h\_M}.
+The elements of \var{Gq} and \var{hq} must have at least one row.
+The default values of \var{Gq} and \var{hq} are empty lists.
+
+\var{A} is dense or sparse matrix and \var{b} is a single-column dense
+matrix. The default values for \var{A} and \var{b} are matrices with
+zero rows.
+
+\var{primalstart} and \var{dualstart} are dictionaries with optional
+primal, respectively, dual starting points.
+\var{primalstart} has elements \code{'x'}, \code{'sl'}, \code{'sq'}.
+\code{primalstart['x']} and \code{primalstart['sl']} are single-column
+dense matrices with the initial values of {\it x} and {\it s\_0};
+\code{primalstart['sq']} is a list of single-column matrices with the
+initial values of {\it s\_1}, \ldots, {\it s\_M}.
+The initial values must satisfy the inequalities in the primal problem
+strictly, but not necessarily the equality constraints.
+
+\var{dualstart} has elements \code{'y'}, \code{'zl'}, \code{'zq'}.
+\code{dualstart['y']} and \code{dualstart['zl']} are single-column dense
+matrices with the initial values of {\it y} and {\it z\_0}.
+\code{dualstart['zq']} is a list of single-column matrices with the
+initial values of {\it z\_1}, \ldots, {\it z\_M}. These values must
+satisfy the dual inequalities strictly, but not necessarily the equality
+constraint.
+
+\function{socp()} returns a dictionary with keys \var{'status'},
+\var{'x'}, \var{'sl'}, \var{'sq'}, \var{'y'}, \var{'zl'}, \var{'zq'}.
+The meaning is similar to the output of \function{conelp()}.
+The \var{'sl'} and \var{'zl'} fields are matrices with the primal
+slacks and dual variables associated with the componentwise linear
+inequalities.
+The \var{'sq'} and \var{'zq'} fields are lists with the primal slacks and
+dual variables associated with the second-order cone inequalities.
+\end{funcdesc}
+
+
+As an example, we solve the second-order cone program
+\[
+\begin{array}{ll}
+\mbox{minimize} & -2x_1 + x_2 + 5x_3 \\*[2ex]
+\mbox{subject to}
+ & \left\| \left[\begin{array}{c}
+ -13 x_1 + 3x_2 + 5x_3 - 3 \\ -12 x_1 + 12x_2 - 6x_3 - 2
+ \end{array}\right] \right\|_2 \leq -12 x_1 - 6 x_2 + 5x_3 - 12 \\*[2ex]
+ & \left\| \left[\begin{array}{c}
+ -3x_1 + 6x_2 + 2x_3 \\ x_1 + 9x_2 + 2x_3 + 3 \\ -x_1 - 19 x_2 + 3 x_3 - 42
+ \end{array}\right] \right\|_2 \leq
+ -3x_1 + 6x_2 - 10x_3 + 27.
+\end{array}
+\]
+\begin{verbatim}
+>>> from cvxopt.base import matrix
+>>> from cvxopt import solvers
+>>> c = matrix([-2., 1., 5.])
+>>> G = [ matrix( [[12., 13., 12.], [6., -3., -12.], [-5., -5., 6.]] ) ]
+>>> G += [ matrix( [[3., 3., -1., 1.], [-6., -6., -9., 19.], [10., -2., -2., -3.]] ) ]
+>>> h = [ matrix( [-12., -3., -2.] ), matrix( [27., 0., 3., -42.] ) ]
+>>> sol = solvers.socp(c, Gq = G, hq = h)
+>>> sol['status']
+optimal
+>>> print sol['x']
+ -5.0150e+00
+ -5.7670e+00
+ -8.5219e+00
+>>> print sol['zq'][0]
+ 1.3423e+00
+ -7.6286e-02
+ -1.3401e+00
+>>> print sol['zq'][1]
+ 1.0185e+00
+ 4.0234e-01
+ 7.7996e-01
+ -5.1681e-01
+\end{verbatim}
+
+
+\section{Semidefinite Programming} \label{s-sdpsolver}
+The function \function{sdp()} is a simple interface to
+\function{conelp()} for cone programs with no second-order cone
+constraints.
+It also provides the option of using the DSDP semidefinite programming
+solver.
+
+\begin{funcdesc}{sdp}{c\optional{, Gl, hl\optional{,
+ Gs, hs\optional{, A, b\optional{, solver\optional{,
+ primalstart\optional{, dualstart}}}}}}}
+
+Solves the pair of primal and dual semidefinite programs
+\[
+ \begin{array}[t]{ll}
+ \mbox{minimize} & c^T x \\
+ \mbox{subject to}
+ & G_0 x + s_0 = h_0 \\
+ & G_k x + \svec{(s_k)} = \svec{(h_k)}, \quad k = 1, \ldots, N \\
+ & Ax = b \\
+ & s_0 \succeq 0 \\
+ & s_k \succeq 0, \quad k=1,\ldots,N
+ \end{array}
+ \qquad\qquad
+ \begin{array}[t]{ll}
+ \mbox{maximize} & - h_0^Tz_0 - \sum_{k=1}^N \Tr(h_kz_k) - b^T y \\
+ \mbox{subject to} &
+ G_0^Tz_0 + \sum_{k=1}^N G_k^T \svec(z_k) + A^T y + c = 0 \\
+ & z_0 \succeq 0 \\
+ & z_k \succeq 0, \quad k=1,\ldots,N.
+ \end{array}
+\]
+The inequalities
+\[
+ s_0 \succeq 0, \qquad z_0 \succeq 0
+\]
+are componentwise vector inequalities. The other inequalities
+are matrix inequalities (\ie, the require the lefthand sides
+to be positive semidefinite).
+We use the notation $\svec(z)$ to denote a symmetric matrix $z$
+stored in column major order as a column vector.
+
+The input argument \var{c} is a dense real matrix with one column of
+length {\it n}.
+The arguments \var{Gl} and \var{hl} are the coefficient matrix $G_0$
+and the righthand side $h_0$ of the componentwise inequalities.
+\var{Gl} is a real dense or sparse matrix; \var{hl} is a real single-column
+dense matrix. The default values for \var{Gl} and \var{hl} are matrices
+with zero rows.
+
+\var{Gs} and \var{hs} are lists of length $N$ that specify the
+linear matrix inequality constraints.
+\var{Gs} is a list of $N$ dense or sparse real matrices
+\code{G\_1}, \ldots, \code{G\_M}.
+The columns of these matrices can be interpreted as
+symmetric matrices stored in column major order, using the BLAS 'L'-type
+storage (\ie, only the entries corresponding to lower triangular positions
+are accessed).
+\var{hs} is a list of $N$ dense symmetric matrices \code{h\_1},
+\ldots, \code{h\_N}.
+Only the lower triangular elements of these matrices are accessed.
+The default values for \code{Gs} and \code{hs} are empty lists.
+
+\var{A} is a dense or sparse matrix and \var{b} is a single-column dense
+matrix. The default values for \var{A} and \var{b} are matrices with
+zero rows.
+
+The \var{solver} argument is used to choose between two
+solvers: the CVXOPT \function{conelp()} solver
+(used when \var{solver} is absent or
+equal to \None) and the external solver DSDP5
+(\code{\var{solver}='dsdp'}); see section~\ref{s-external}.
+With the \code{'dsdp'} option the code does not accept problems with
+equality constraints.
+
+The optional argument \var{primalstart} is a dictionary with keys
+\code{'x'}, \code{'sl'}, and \code{'ss'}, used as an optional primal
+starting point.
+\code{primalstart['x']} and \code{primalstart['sl']} are single-column
+dense matrices with the initial values of {\it x} and {\it s\_0};
+\code{primalstart['ss']} is a list of square matrices with the
+initial values of {\it s\_1}, \ldots, {\it s\_N}.
+The initial values must satisfy the inequalities in the primal problem
+strictly, but not necessarily the equality constraints.
+
+\var{dualstart} is a dictionary with keys \code{'y'}, \code{'zl'},
+\code{'zs'}, used as an optional dual starting point.
+\code{dualstart['y']} and \code{dualstart['zl']} are single-column dense
+matrices with the initial values of {\it y} and {\it z\_0}.
+\code{dualstart['zs']} is a list of square matrices with the
+initial values of {\it z\_1}, \ldots, {\it z\_N}. These values must
+satisfy the dual inequalities strictly, but not necessarily the equality
+constraint.
+
+The arguments \var{primalstart} and \var{dualstart} are ignored when
+the DSDP solver is used.
+
+\function{sdp()} returns a dictionary with keys \code{'status'},
+\code{'x'}, \code{'sl'}, \code{'ss'}, \code{'y'}, \code{'zl'},
+\code{'ss'}.
+The \var{'sl'} and \var{'zl'} fields are matrices with the primal
+slacks and dual variables associated with the componentwise linear
+inequalities.
+The \var{'ss'} and \var{'zs'} fields are lists with the primal slacks and
+dual variables associated with the second-order cone inequalities.
+\end{funcdesc}
+
+We illustrate the calling sequence with a small example.
+\[
+\begin{array}{ll}
+\mbox{minimize} & x_1 - x_2 + x_3 \\
+\mbox{subject to} &
+ x_1 \left[ \begin{array}{cc} -7 & -11 \\ -11 & 3
+ \end{array}\right] +
+ x_2 \left[ \begin{array}{cc}
+ 7 & -18 \\ -18 & 8 \end{array}\right] +
+ x_3 \left[ \begin{array}{cc}
+ -2 & -8 \\ -8 & 1
+ \end{array}\right] \preceq
+ \left[ \begin{array}{cc}
+ 33 & -9 \\ -9 & 26 \end{array}\right] \\*[1ex]
+& x_1 \left[ \begin{array}{ccc}
+ -21 & -11 & 0 \\ -11 & 10 & 8 \\ 0 & 8 & 5
+ \end{array}\right] +
+ x_2 \left[ \begin{array}{ccc}
+ 0 & 10 & 16 \\
+10 & -10 & -10 \\
+16 & -10 & 3
+ \end{array}\right] +
+ x_3 \left[ \begin{array}{ccc}
+ -5 & 2 & -17 \\
+ 2 & -6 & -7 \\
+ -17 & 8 & 6
+ \end{array}\right]
+\preceq \left[ \begin{array}{ccc}
+ 14 & 9 & 40 \\
+ 9 & 91 & 10 \\
+ 40 & 10 & 15
+\end{array} \right]
+\end{array}
+\]
+\begin{verbatim}
+>>> from cvxopt.base import matrix
+>>> from cvxopt import solvers
+>>> c = matrix([1.,-1.,1.])
+>>> G = [ matrix([[-7., -11., -11., 3.],
+ [ 7., -18., -18., 8.],
+ [-2., -8., -8., 1.]]) ]
+>>> G += [ matrix([[-21., -11., 0., -11., 10., 8., 0., 8., 5.],
+ [ 0., 10., 16., 10., -10., -10., 16., -10., 3.],
+ [ -5., 2., -17., 2., -6., 8., -17., -7., 6.]]) ]
+>>> h = [ matrix([[33., -9.], [-9., 26.]]) ]
+>>> h += [ matrix([[14., 9., 40.], [9., 91., 10.], [40., 10., 15.]]) ]
+>>> sol = solvers.sdp(c, Gs=G, hs=h)
+>>> print sol['x']
+ -3.6767e-01
+ 1.8983e+00
+ -8.8755e-01
+>>> print sol['zs'][0]
+ 3.9611e-03 -4.3384e-03
+ -4.3384e-03 4.7516e-03
+>>> print sol['zs'][1]
+ 5.5801e-02 -2.4091e-03 2.4215e-02
+ -2.4091e-03 1.0402e-04 -1.0454e-03
+ 2.4215e-02 -1.0454e-03 1.0508e-02
+\end{verbatim}
+Only the entries in \var{Gs} and \var{hs} that correspond to lower
+triangular entries need to be provided, so in the example \var{h} and
+\var{G} may also be defined as follows.
+\begin{verbatim}
+>>> G = [ matrix([[-7., -11., 0., 3.],
+ [ 7., -18., 0., 8.],
+ [-2., -8., 0., 1.]]) ]
+>>> G += [ matrix([[-21., -11., 0., 0., 10., 8., 0., 0., 5.],
+ [ 0., 10., 16., 0., -10., -10., 0., 0., 3.],
+ [ -5., 2., -17., 0., -6., 8., 0., 0., 6.]]) ]
+>>> h = [ matrix([[33., -9.], [0., 26.]]) ]
+>>> h += [ matrix([[14., 9., 40.], [0., 91., 10.], [0., 0., 15.]]) ]
+\end{verbatim}
+
+
+\section{Exploiting Structure} \label{s-conelp-struct}
+By default, the \function{conelp()} exploits no problem structure except
+(to some limited extent) sparsity.
+Two mechanisms are provided for implementing customized solvers that
+take advantage of problem structure.
+
+\begin{description}
+\item[\emph{Providing a function for solving KKT equations.}]
+The most expensive step of each iteration of \function{conelp()} is the
+solution of a set of linear equations (`KKT equations') of the form
+\BEQ \label{e-conelp-kkt}
+ \left[\begin{array}{ccc}
+ 0 & A^T & G^T \\
+ A & 0 & 0 \\
+ G & 0 & -W^T W \end{array}\right]
+ \left[\begin{array}{c} u_x \\ u_y \\ u_z \end{array}\right]
+ = \left[\begin{array}{c} b_x \\ b_y \\ b_z \end{array}\right].
+\EEQ
+The matrix $W$ depends on the current iterates and is defined as follows.
+We use the notation of~section~\ref{s-conelp}. Suppose
+\[
+ u = \left(u_\mathrm{l}, \; u_{\mathrm{q},0}, \; \ldots, \;
+ u_{\mathrm{q},M-1}, \; \svec{(u_{\mathrm{s},0})}, \; \ldots, \;
+ \svec{(u_{\mathrm{s},N-1})}\right), \qquad
+ u_\mathrm{l} \in\reals^l, \qquad
+ u_{\mathrm{q},k} \in\reals^{q_k}, \quad k = 0,\ldots,M-1, \qquad
+ u_{\mathrm{s},k} \in\symm^{p_k}, \quad k = 0,\ldots,N-1.
+\]
+Then {\it W} is a block-diagonal matrix,
+\[
+ Wu = \left( W_\mathrm{l} u_\mathrm{l}, \;
+ W_{\mathrm{q},0} u_{\mathrm{q},0}, \; \ldots, \;
+ W_{\mathrm{q},M-1} u_{\mathrm{q},M-1},\;
+ W_{\mathrm{s},0} \svec{(u_{\mathrm{s},0})}, \; \ldots, \;
+ W_{\mathrm{s},N-1} \svec{(u_{\mathrm{s},N-1})} \right)
+\]
+with the following diagonal blocks.
+\BIT
+\item The first block is a \emph{positive diagonal scaling} with a
+ vector {\it d}:
+\[
+ W_\mathrm{l} = \diag(d), \qquad W_\mathrm{l}^{-1} = \diag(d)^{-1}.
+\]
+This transformation is symmetric:
+\[
+ W_\mathrm{l}^T = W_\mathrm{l}.
+\]
+
+\item The next $M$ blocks are positive multiples of \emph{hyperbolic
+ Householder transformations}:
+\[
+ W_{\mathrm{q},k} = \beta_k ( 2 v_k v_k^T - J),
+ \qquad
+ W_{\mathrm{q},k}^{-1} = \frac{1}{\beta_k} ( 2 Jv_k v_k^T J - J),
+ \qquad k = 0,\ldots,M-1,
+\]
+where
+\[
+ \beta_k > 0, \qquad v_{k0} > 0, \qquad
+ v_k^T Jv_k = 1, \qquad J = \left[\begin{array}{cc}
+ 1 & 0 \\ 0 & -I \end{array}\right].
+\]
+These transformations are also symmetric:
+\[
+ W_\mathrm{q,k}^T = W_\mathrm{q,k}.
+\]
+
+\item The last $N$ blocks are \emph{congruence transformations} with
+ nonsingular matrices:
+\[
+ W_{\mathrm{s},k} \svec{(u_{\mathrm{s},k})} =
+ \svec{(r_k^T u_{\mathrm{s},k} r_k)}, \qquad
+ W_{\mathrm{s},k}^{-1} \svec{(u_{\mathrm{s},k})} =
+ \svec{(r_k^{-T} u_{\mathrm{s},k} r_k^{-1})}, \qquad
+ k = 0,\ldots,N-1.
+\]
+In general, this operation is not symmetric, and
+\[
+ W_{\mathrm{s},k}^T \svec{(u_{\mathrm{s},k})} =
+ \svec{(r_k u_{\mathrm{s},k} r_k^T)}, \qquad
+ \qquad
+ W_{\mathrm{s},k}^{-T} \svec{(u_{\mathrm{s},k})} =
+ \svec{(r_k^{-1} u_{\mathrm{s},k} r_k^{-T})}, \qquad
+ \qquad
+ k = 0,\ldots,N-1.
+\]
+\EIT
+It is often possible to exploit structure in the coefficient matrices
+$G$ and $A$ to solve~(\ref{e-conelp-kkt}) faster than by
+standard methods. The last argument \var{kktsolver} of
+\function{conelp()} allows the user to supply a Python function for
+solving the KKT equations.
+This function will be called as \samp{f = kktsolver(W)}, where
+\var{W} is a dictionary that contains the parameters of the scaling:
+
+\BIT
+\item \code{W['d']} is the positive vector that defines the diagonal
+ scaling. \code{W['di']} is its componentwise inverse.
+\item \code{W['beta']} and \code{W['v']} are lists of length $M$ with
+ the coefficients and vectors that define the hyperbolic Householder
+ transformations.
+\item \code{W['r']} is a list of length $N$ with the matrices that
+ define the the congruence transformations.
+ \code{W['rti']} is a list of length $N$ with the transposes of the
+ inverses of the matrices in \code{W['r']}.
+\EIT
+
+The function call \samp{f = kktsolver(W)} should return a routine
+for solving the KKT system~(\ref{e-conelp-kkt}) defined by \var{W}.
+It will be called as \samp{f(bx, by, bz)}.
+On entry, \var{bx}, \var{by}, \var{bz} contain the righthand side.
+On exit, they should contain the solution of the KKT system, with
+the last component scaled, \ie, on exit,
+\[
+ b_x := u_x, \qquad b_y := u_y, \qquad b_z := W u_z.
+\]
+
+\item[\emph{Specifying constraints via Python functions}.]
+In the default use of \function{conelp()}, the arguments \var{G} and
+\var{A} are the coefficient matrices in the constraints
+of~(\ref{e-conelp}).
+It is also possible to specify these matrices by providing Python functions
+that evaluate the corresponding matrix-vector products and their adjoints.
+
+If the argument \var{G} of \function{conelp()} is a Python
+function, it should be defined as follows:
+\begin{funcdesc}{\var{G}}{\var{x}, \var{y} \optional{,
+\var{alpha}\optional{, \var{beta}\optional{, \var{trans}}}}}
+This evaluates the matrix-vector products
+\[
+y := \alpha Gx + \beta y \quad
+ (\mathrm{trans} = \mathrm{'N'}), \qquad
+y := \alpha G^T x + \beta y \quad
+ (\mathrm{trans} = \mathrm{'T'}).
+\]
+The default values of the optional arguments must be
+\code{alpha = 1.0}, \code{beta = 0.0}, \code{trans = 'N'}.
+\end{funcdesc}
+
+Similarly, if the argument \var{A} is a Python function, then it must
+be defined as follows.
+\begin{funcdesc}{\var{A}}{\var{x}, \var{y} \optional{,
+\var{alpha}\optional{, \var{beta}\optional{, \var{trans}}}}}
+This evaluates the matrix-vector products
+\[
+y := \alpha Ax + \beta y \quad (\mathrm{trans} = \mathrm{'N'}), \qquad
+y := \alpha A^T x + \beta y \quad (\mathrm{trans} = \mathrm{'T'}).
+\]
+The default values of the optional arguments must be
+\code{alpha = 1.0}, \code{beta = 0.0}, \code{trans = 'N'}.
+\end{funcdesc}
+
+If \var{G} or \var{A} are Python functions, then the argument
+\var{kktsolver} must also be provided.
+\end{description}
+
+We illustrate these features with three applications.
+
+\begin{description}
+\item[Example: 1-norm approximation]
+
+The optimization problem
+\[
+ \begin{array}{ll}
+ \mbox{minimize} & \|Pu-q\|_1
+ \end{array}
+\]
+can be formulated as a linear program
+\[
+ \begin{array}{ll}
+ \mbox{minimize} & \ones^T v \\
+ \mbox{subject to} & -v \preceq Pu - q \preceq v.
+ \end{array}
+\]
+By exploiting the structure in the inequalities, the cost of
+an iteration of an interior-point method can be reduced
+to the cost of least-squares problem of the same dimensions.
+(See section 11.8.2 in the book
+\citetitle{http://www.ee.ucla.edu/\~{}vandenbe/cvxbook}{Convex Optimization}.)
+The code belows takes advantage of this fact.
+
+\begin{verbatim}
+from cvxopt import base, blas, lapack, solvers
+from cvxopt.base import matrix, spmatrix, mul, div
+
+def l1(P, q):
+ """
+
+ Returns the solution u, w of the l1 approximation problem
+
+ (primal) minimize ||P*u - q||_1
+
+ (dual) maximize q'*w
+ subject to P'*w = 0
+ ||w||_infty <= 1.
+ """
+
+ m, n = P.size
+
+ # Solve the equivalent LP
+ #
+ # minimize [0; 1]' * [u; v]
+ # subject to [P, -I; -P, -I] * [u; v] <= [q; -q]
+ #
+ # maximize -[q; -q]' * z
+ # subject to [P', -P']*z = 0
+ # [-I, -I]*z + 1 = 0
+ # z >= 0.
+
+ c = matrix(n*[0.0] + m*[1.0])
+
+ def G(x, y, alpha = 1.0, beta = 0.0, trans = 'N'):
+
+ if trans=='N':
+ # y := alpha * [P, -I; -P, -I] * x + beta*y
+ u = P*x[:n]
+ y[:m] = alpha * ( u - x[n:]) + beta * y[:m]
+ y[m:] = alpha * (-u - x[n:]) + beta * y[m:]
+
+ else:
+ # y := alpha * [P', -P'; -I, -I] * x + beta*y
+ y[:n] = alpha * P.T * (x[:m] - x[m:]) + beta * y[:n]
+ y[n:] = -alpha * (x[:m] + x[m:]) + beta * y[n:]
+
+ h = matrix([q, -q])
+ dims = {'l': 2*m, 'q': [], 's': []}
+
+ def F(W):
+
+ """
+ Returns a function f(x, y, z) that solves
+
+ [ 0 0 P' -P' ] [ x[:n] ] [ bx[:n] ]
+ [ 0 0 -I -I ] [ x[n:] ] [ bx[n:] ]
+ [ P -I -D1^{-1} 0 ] [ z[:m] ] = [ bz[:m] ]
+ [-P -I 0 -D2^{-1} ] [ z[m:] ] [ bz[m:] ]
+
+ where D1 = diag(di[:m])^2, D2 = diag(di[m:])^2 and di = W['di'].
+ """
+
+ # Factor A = 4*P'*D*P where D = d1.*d2 ./(d1+d2) and
+ # d1 = di[:m].^2, d2 = di[m:].^2.
+
+ di = W['di']
+ d1, d2 = di[:m]**2, di[m:]**2
+ D = div( mul(d1,d2), d1+d2 )
+ A = P.T * spmatrix(4*D, range(m), range(m)) * P
+ lapack.potrf(A)
+
+ def f(x, y, z):
+
+ """
+ On entry bx, bz are stored in x, z.
+ On exit x, z contain the solution, with z scaled: z./di is
+ returned instead of z.
+ """"
+
+ # Solve for x[:n]:
+ #
+ # A*x[:n] = bx[:n] + P' * ( ((D1-D2)*(D1+D2)^{-1})*bx[n:]
+ # + (2*D1*D2*(D1+D2)^{-1}) * (bz[:m] - bz[m:]) ).
+
+ x[:n] += P.T * ( mul(div(d1-d2, d1+d2), x[n:]) + mul(2*D, z[:m]-z[m:]) )
+ lapack.potrs(A, x)
+
+ # x[n:] := (D1+D2)^{-1} * (bx[n:] - D1*bz[:m] - D2*bz[m:] + (D1-D2)*P*x[:n])
+
+ u = P*x[:n]
+ x[n:] = div(x[n:] - mul(d1, z[:m]) - mul(d2, z[m:]) + mul(d1-d2, u), d1+d2)
+
+ # z[:m] := d1[:m] .* ( P*x[:n] - x[n:] - bz[:m])
+ # z[m:] := d2[m:] .* (-P*x[:n] - x[n:] - bz[m:])
+
+ z[:m] = mul(d[:m], u - x[n:] - z[:m])
+ z[m:] = mul(d[m:], -u - x[n:] - z[m:])
+
+ return f
+
+ sol = solvers.conelp(c, G, h, dims, kktsolver = F)
+ return sol['x'][:n], sol['z'][m:] - sol['z'][:m]
+\end{verbatim}
+
+\item[Example: SDP with diagonal linear term]
+
+The SDP
+\[
+ \begin{array}{ll}
+ \mbox{minimize} & \ones^T x \\
+ \mbox{subject to} & W + \diag(x) \succeq 0
+ \end{array}
+\]
+can be solved efficiently by exploiting properties of the diag operator.
+
+\begin{verbatim}
+from cvxopt import base, blas, lapack, solvers
+from cvxopt.base import matrix
+
+def mcsdp(w):
+ """
+ Returns solution x, z to
+
+ (primal) minimize sum(x)
+ subject to w + diag(x) >= 0
+
+ (dual) maximize -tr(w*z)
+ subject to diag(z) = 1
+ z >= 0.
+ """
+
+ n = w.size[0]
+ c = matrix(1.0, (n,1))
+
+ def G(x, y, alpha = 1.0, beta = 0.0, trans = 'N'):
+ """
+ y := alpha*(-diag(x)) + beta*y.
+ """
+
+ if trans=='N':
+ # x is a vector; y is a symmetric matrix in column major order.
+ y *= beta
+ y[::n+1] -= alpha * x
+
+ else:
+ # x is a symmetric matrix in column major order; y is a vector.
+ y *= beta
+ y -= alpha * x[::n+1]
+
+
+ def cngrnc(r, x, alpha = 1.0):
+ """
+ Congruence transformation
+
+ x := alpha * r'*x*r.
+
+ r is a matrix of size (n, n).
+ x is a matrix of size (n**2, 1), representing a symmetric matrix stored in column major order.
+ """
+
+ # Scale diagonal of x by 1/2.
+ x[::n+1] *= 0.5
+
+ # a := tril(x)*r
+ a = +r
+ blas.trmm(x, a, side = 'L')
+
+ # x := alpha*(a*r' + r*a')
+ blas.syr2k(r, a, x, trans = 'T', alpha = alpha)
+
+ dims = {'l': 0, 'q': [], 's': [n]}
+
+ def F(W):
+ """
+ Returns a function f(x, y, z) that solves
+
+ -diag(z) = bx
+ -diag(x) - r*r'*z*r*r' = bz
+
+ where r = W['r'][0] = W['rti'][0]^{-T}.
+ """
+
+ rti = W['rti'][0]
+
+ # t = rti*rti' as a nonsymmetric matrix.
+ t = matrix(0.0, (n,n))
+ blas.gemm(rti, rti, t, transB = 'T')
+
+ # Cholesky factorization of tsq = t.*t.
+ tsq = t**2
+ lapack.potrf(tsq)
+
+ def f(x, y, z):
+ """
+ On entry, x contains bx, y is empty, and z contains bz stored
+ in column major order.
+ On exit, they contain the solution, with z scaled
+ (vec(r'*z*r) is returned instead of z).
+
+ We first solve
+
+ ((rti*rti') .* (rti*rti')) * x = bx - diag(t*bz*t)
+
+ and take z = - rti' * (diag(x) + bz) * rti.
+ """
+
+ # tbst := t * bz * t
+ tbst = +z
+ cngrnc(t, tbst)
+
+ # x := x - diag(tbst) = bx - diag(rti*rti' * bz * rti*rti')
+ x -= tbst[::n+1]
+
+ # x := (t.*t)^{-1} * x = (t.*t)^{-1} * (bx - diag(t*bz*t))
+ lapack.potrs(tsq, x)
+
+ # z := z + diag(x) = bz + diag(x)
+ z[::n+1] += x
+
+ # z := -vec(rti' * z * rti)
+ # = -vec(rti' * (diag(x) + bz) * rti
+ cngrnc(rti, z, alpha = -1.0)
+
+ return f
+
+ sol = solvers.conelp(c, G, w[:], dims, kktsolver = F)
+ return sol['x'], sol['z']
+
+\end{verbatim}
+
+\item[Example: Minimizing 1-norm subject to a 2-norm constraint]
+In the second example, we use a similar trick to solve the problem
+\[
+ \begin{array}{ll}
+ \mbox{minimize} & \|u\|_1 \\
+ \mbox{subject to} & \|Au - b\|_2 \leq 1.
+ \end{array}
+\]
+The code below is efficient, if we assume that the number of rows in $A$
+is greater than or equal to the number of columns.
+
+\begin{verbatim}
+def qcl1(A, b):
+ """
+ Returns the solution u, z of
+
+ (primal) minimize || u ||_1
+ subject to || A * u - b ||_2 <= 1
+
+ (dual) maximize b^T z - ||z||_2
+ subject to || A'*z ||_inf <= 1.
+
+ Exploits structure, assuming A is m by n with m >= n.
+ """
+
+ m, n = A.size
+
+ # Solve equivalent cone LP with variables x = [u; v].
+ #
+ # minimize [0; 1]' * x
+ # subject to [ I -I ] * x <= [ 0 ] (componentwise)
+ # [-I -I ] * x <= [ 0 ] (componentwise)
+ # [ 0 0 ] * x <= [ 1 ] (SOC)
+ # [-A 0 ] [ -b ]
+ #
+ # maximize -t + b' * w
+ # subject to z1 - z2 = A'*w
+ # z1 + z2 = 1
+ # z1 >= 0, z2 >=0, ||w||_2 <= t.
+
+ c = matrix(n*[0.0] + n*[1.0])
+ h = matrix( 0.0, (2*n + m + 1, 1))
+ h[2*n] = 1.0
+ h[2*n+1:] = -b
+
+ def G(x, y, alpha = 1.0, beta = 0.0, trans = 'N'):
+ y *= beta
+ if trans=='N':
+ # y += alpha * G * x
+ y[:n] += alpha * (x[:n] - x[n:2*n])
+ y[n:2*n] += alpha * (-x[:n] - x[n:2*n])
+ y[2*n+1:] -= alpha * A*x[:n]
+
+ else:
+ # y += alpha * G'*x
+ y[:n] += alpha * (x[:n] - x[n:2*n] - A.T * x[-m:])
+ y[n:] -= alpha * (x[:n] + x[n:2*n])
+
+
+ def Fkkt(W):
+ """
+ Returns a function f(x, y, z) that solves
+
+ [ 0 G' ] [ x ] = [ bx ]
+ [ G -W'*W ] [ z ] [ bz ].
+ """
+
+ # First factor
+ #
+ # S = G' * W**-1 * W**-T * G
+ # = [0; -A]' * W3^-2 * [0; -A] + 4 * (W1**2 + W2**2)**-1
+ #
+ # where
+ #
+ # W1 = diag(d1) with d1 = W['d'][:n] = 1 ./ W['di'][:n]
+ # W2 = diag(d2) with d2 = W['d'][n:] = 1 ./ W['di'][n:]
+ # W3 = beta * (2*v*v' - J), W3^-1 = 1/beta * (2*J*v*v'*J - J)
+ # with beta = W['beta'][0], v = W['v'][0], J = [1, 0; 0, -I].
+
+ # As = W3^-1 * [ 0 ; -A ] = 1/beta * ( 2*J*v * v' - I ) * [0; A]
+ beta, v = W['beta'][0], W['v'][0]
+ As = 2 * v * (v[1:].T * A)
+ As[1:,:] *= -1.0
+ As[1:,:] -= A
+ As /= beta
+
+ # S = As'*As + 4 * (W1**2 + W2**2)**-1
+ S = As.T * As
+ d1, d2 = W['d'][:n], W['d'][n:]
+ d = 4.0 * (d1**2 + d2**2)**-1
+ S[::n+1] += d
+ lapack.potrf(S)
+
+ def f(x, y, z):
+
+ # z := - W**-T * z
+ z[:n] = -div( z[:n], d1 )
+ z[n:2*n] = -div( z[n:2*n], d2 )
+ z[2*n:] -= 2.0*v*( v[0]*z[2*n] - blas.dot(v[1:], z[2*n+1:]) )
+ z[2*n+1:] *= -1.0
+ z[2*n:] /= beta
+
+ # x := x - G' * W**-1 * z
+ x[:n] -= div(z[:n], d1) - div(z[n:2*n], d2) + As.T * z[-(m+1):]
+ x[n:] += div(z[:n], d1) + div(z[n:2*n], d2)
+
+ # Solve for x[:n]:
+ #
+ # S*x[:n] = x[:n] - (W1**2 - W2**2)(W1**2 + W2**2)^-1 * x[n:]
+
+ x[:n] -= mul( div(d1**2 - d2**2, d1**2 + d2**2), x[n:])
+ lapack.potrs(S, x)
+
+ # Solve for x[n:]:
+ #
+ # (d1**-2 + d2**-2) * x[n:] = x[n:] + (d1**-2 - d2**-2)*x[:n]
+
+ x[n:] += mul( d1**-2 - d2**-2, x[:n])
+ x[n:] = div( x[n:], d1**-2 + d2**-2)
+
+ # z := z + W^-T * G*x
+ z[:n] += div( x[:n] - x[n:2*n], d1)
+ z[n:2*n] += div( -x[:n] - x[n:2*n], d2)
+ z[2*n:] += As*x[:n]
+
+ return f
+
+ dims = {'l': 2*n, 'q': [m+1], 's': []}
+ sol = solvers.conelp(c, G, h, dims, kktsolver = Fkkt)
+ if sol['status'] == 'optimal':
+ return sol['x'][:n], sol['z'][-m:]
+ else:
+ return None, None
+\end{verbatim}
+
+
+
+\end{description}
+
+
+\section{Optional Solvers} \label{s-external}
+CVXOPT includes optional interfaces to several other optimization
+libraries.
+
+\begin{description}
+\item[GLPK] \function{lp()} with the \code{solver='glpk'} option uses
+the simplex algorithm in
+\ulink{GLPK (GNU Linear Programming Kit)}{http://www.gnu.org/software/glpk/glpk.html}.
+
+\item[MOSEK] \function{lp()} with the \code{solver='mosek'} option uses
+\ulink{MOSEK}{http://www.mosek.com} version 4.
+
+\item[DSDP] \function{sdp()} with the \code{solver='dsdp'} option uses
+the \ulink{DSDP5.8}{http://www-unix.mcs.anl.gov/DSDP} solver.
+\end{description}
+GLPK, MOSEK and DSDP are not included in the CVXOPT distribution and
+need to be installed separately.
+
+
+\section{Algorithm Parameters} \label{s-parameters}
+In this section we list some algorithm control parameters that can
+be modified without editing the source code.
+These control parameters are accessible via the dictionary
+\member{solvers.options}. By default the dictionary
+is empty and the default values of the parameters are used.
+
+One can change the parameters in the default solvers by
+adding entries with the following key values.
+\begin{description}
+\item[\code{'show\_progress'}]
+\True\ or \False; turns the output to the screen on or off
+(default: \True).
+\item[\code{'maxiters'}] maximum number of iterations (default: 100).
+\item[\code{'abstol'}] absolute accuracy (default: \code{1e-7}).
+\item[\code{'reltol'}] relative accuracy (default: \code{1e-6}).
+\item[\code{'feastol'}] tolerance for feasibility conditions (default:
+\code{1e-7}).
+\end{description}
+For example the command
+\begin{verbatim}
+>>> from cvxopt import solvers
+>>> solvers.options['show_progress'] = False
+\end{verbatim}
+turns off the screen output during calls to the solvers.
+The tolerances \var{abstol}, \var{reltol} and \var{feastol} have the
+following meaning. \function{conelp()} terminates with
+status \code{'optimal'} if
+\[
+s \succeq 0, \qquad z \succeq 0, \qquad
+\qquad
+ \frac{\|Gx + s - h\|_2} {\max\{1,\|h\|_2\}} \leq \epsilon_\mathrm{feas},
+\qquad
+\frac{\|Ax-b\|_2}{\max\{1,\|b\|_2\}} \leq \epsilon_\mathrm{feas},
+\qquad
+\frac{\|G^Tz + A^Ty + c\|_2}{\max\{1,\|c\|_2\}} \leq
+ \epsilon_\mathrm{feas},
+\]
+and
+\[
+ s^T z \leq \epsilon_\mathrm{abs} \qquad \mbox{or} \qquad
+\left( \min\left\{c^Tx, h^T z + b^Ty \right\} < 0, \quad
+ \frac{s^Tz} {-\min\{c^Tx, h^Tz + b^T y\}} \leq \epsilon_\mathrm{rel}
+\right).
+\]
+It returns with status \code{'primal infeasible'} if
+\[
+z \succeq 0, \qquad
+\qquad \frac{\|G^Tz +A^Ty\|_2}{\max\{1, \|c\|_2\}} \leq
+ \epsilon_\mathrm{feas},
+ \qquad h^Tz +b^Ty = -1.
+\]
+It returns with status \code{'dual infeasible'} if
+\[
+s \succeq 0, \qquad
+\qquad
+\frac{\|Gx+s\|_2}{\max\{1, \|h\|_2\}} \leq \epsilon_\mathrm{feas}, \qquad
+\frac{\|Ax\|_2}{\max\{1, \|b\|_2\}} \leq \epsilon_\mathrm{feas}, \qquad
+c^Tx = -1.
+\]
+The functions \function{lp()}, \function{socp()} and \function{sdp()} call
+\function{conelp()} and hence use the same stopping criteria.
+
+The control parameters listed in the GLPK documentation are
+set to their default values and can also be customized by making
+an entry in \member{solvers.options}.
+The keys in the dictionary are strings with the name of the GLPK
+parameter. The command
+\begin{verbatim}
+>>> from cvxopt import solvers
+>>> solvers.options['LPX_K_MSGLEV'] = 0
+\end{verbatim}
+turns off the screen output subsequent calls \function{lp()} with
+the \code{'glpk'} option.
+
+The MOSEK \ulink{control parameters}{http://www.mosek.com/fileadmin/products/3/tools/doc/html/tools/node22.html}
+are set to their default values.
+The corresponding keys in \code{solvers.options} are strings with the
+name of the MOSEK parameter. For example the command
+\begin{verbatim}
+>>> from cvxopt import solvers
+>>> solvers.options['MSK_IPAR_LOG'] = 0
+\end{verbatim}
+turns off the screen output during calls of \function{lp()}
+with the \code{'mosek'} option.
+
+The following control parameters affect the DSDP algorithm:
+\begin{description}
+\item[\code{'DSDP\_Monitor'}] the interval (in number of iterations)
+ at which output is printed to the screen
+(default: 0).
+\item[\code{'DSDP\_MaxIts'}] maximum number of iterations.
+\item[\code{'DSDP\_GapTolerance'}] relative accuracy (default:
+\code{1e-5}).
+\end{description}
diff --git a/doc/cvxopt.tex b/doc/cvxopt.tex
index f0d4888..b6e6bd4 100644
--- a/doc/cvxopt.tex
+++ b/doc/cvxopt.tex
@@ -1,5 +1,5 @@
-\documentclass{manual}
-\usepackage{graphicx}
+\documentclass{book}
+\usepackage{html,graphicx}
\def\BIT{\begin{itemize}}
\def\EIT{\end{itemize}}
@@ -21,6 +21,39 @@
\newcommand{\argmax}{\mathop{\rm argmax}}
\newcommand{\symm}{{\mbox{\bf S}}}
\newcommand{\op}{\mathop{\mathrm{op}}}
+\newcommand{\svec}{\mathop{\mathbf{vec}}}
+
+% redefine Python markup
+\newcommand{\code}[1]{{\tt #1}}
+\newcommand{\pytype}[1]{{\tt #1}}
+\newcommand{\cdata}[1]{{\tt #1}}
+\newcommand{\var}[1]{{\tt #1}}
+\newcommand{\samp}[1]{"{\tt #1}"}
+\newcommand{\module}[1]{{\tt #1}}
+\newcommand{\class}[1]{{\tt #1}}
+\newcommand{\function}[1]{{\tt #1}}
+\newcommand{\member}[1]{{\tt #1}}
+\newcommand{\program}[1]{{\rm #1}}
+\newcommand{\optional}[1]{{\rm [}#1{\rm ]}}
+\newcommand{\file}[1]{{\tt #1}}
+\newcommand{\ctype}[1]{{\tt #1}}
+\newcommand{\constant}[1]{{\tt #1}}
+\newenvironment{cfuncdesc}[3]{\par{\tt #1}%
+ {\bf #2}({\tt #3})\begin{list}{}{}\item[]}{\end{list}}
+\newenvironment{classdesc}[2]{\par{\bf #1}%
+ ({\tt #2})\begin{list}{}{}\item[]}{\end{list}}
+\newenvironment{funcdesc}[2]{\par{\bf #1}%
+ ({\tt #2})\begin{list}{}{}\item[]}{\end{list}}
+\newenvironment{memberdesc}[3]{\par{\bf #2}%
+ \begin{list}{}{}\item[]}{\end{list}}
+\newenvironment{methoddesc}[3]{\par{\bf #2}()%
+ \begin{list}{}{}\item[]}{\end{list}}
+\newenvironment{seealso}{\par{\bf See also:}\begin{list}{}{}\item[]}{\end{list}}
+%\renewcommand{\seealso}[1]{#1}
+\newcommand{\seetext}[1]{#1}
+\newcommand{\ulink}[2]{\htmladdnormallink{#1}{#2}}
+\newcommand{\seelink}[3]{\htmladdnormallink{#2}{#1}}
+\newcommand{\citetitle}[2]{\htmladdnormallink{#2}{#1}}
% constants
\newcommand{\dtc}{\code{'d'}}
@@ -38,13 +71,13 @@
\newcommand{\mtrx}{\class{matrix}}
\newcommand{\spmtrx}{\class{spmatrix}}
-%\makeindex
-\title{CVXOPT: A Python Package for Convex Optimization}
-\author{Joachim Dahl \& Lieven Vandenberghe \\
-{\tt joachim at kom.aau.dk}, {\tt vandenbe at ee.ucla.edu}}
-\date{February 6, 2007}
-\release{0.8.2}
+%\title{CVXOPT: A Python Package for Convex Optimization}
+\title{CVXOPT User's Guide}
+\author{Joachim Dahl \& Lieven Vandenberghe}
+%{\tt joachim at kom.aau.dk}, {\tt vandenbe at ee.ucla.edu}}
+\date{Release 0.9 -- August 10, 2007}
+%\release{0.8.2}
\begin{document}
\maketitle
@@ -52,43 +85,45 @@
\chapter*{Copyright and License}
Copyright \copyright{2004-2007} J. Dahl \& L. Vandenberghe.
-This program is free software; you can redistribute it and/or modify
+CVXOPT is free software; you can redistribute it and/or modify
it under the terms of the
-\ulink{GNU General Public License}{http://www.gnu.org/copyleft/gpl.html}
-as published by
-the Free Software Foundation; either version 2 of the License, or
-(at your option) any later version.
+\ulink{GNU General Public License}{http://www.gnu.org/licenses/gpl-3.0.html}
+as published by the Free Software Foundation; either version 3 of the
+License, or (at your option) any later version.
-This program is distributed in the hope that it will be useful,
+CVXOPT is distributed in the hope that it will be useful,
but WITHOUT ANY WARRANTY; without even the implied warranty of
-MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the
-\ulink{GNU General Public License}{http://www.gnu.org/copyleft/gpl.html}
+MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE.
+See the
+\ulink{GNU General Public License}{http://www.gnu.org/licenses/gpl-3.0.html}
for more details.
\hrule
The CVXOPT distribution includes source code for the following
-software libraries.
+software libraries.
\BIT
\item Part of the SuiteSparse suite of sparse matrix algorithms,
including:
\BIT
-\item AMD Version 2.0. Copyright (c) 2006 by Timothy A.\ Davis,
+\item AMD Version 2.1. Copyright (c) 2007 by Timothy A.\ Davis,
Patrick R.\ Amestoy, and Iain S.\ Duff.
-\item CHOLMOD Version 1.4.
- Copyright (c) 2005-2006 by University of Florida, Timothy A. Davis
+\item CHOLMOD Version 1.5.
+ Copyright (c) 2005-2007 by University of Florida, Timothy A. Davis
and W. Hager.
-\item COLAMD version 2.6. Copyright (c) 1998-2006 by Timothy A.\ Davis.
+\item COLAMD version 2.7. Copyright (c) 1998-2007 by Timothy A.\ Davis.
\item UMFPACK Version 5.0.2.
Copyright (c) 1995-2006 by Timothy A.\ Davis.
\EIT
These packages are licensed under the terms of the
-\ulink{GNU Lesser General Public License}
-{http://www.gnu.org/copyleft/lesser.html} (UMFPACK, parts of CHOLMOD,
-AMD, COLAMD) and the \ulink{GNU General Public License}
-{http://www.gnu.org/copyleft/gpl.html} (parts of CHOLMOD).
-For details, consult the README files in the source directories or
-the website listed below.
+\ulink{GNU Lesser General Public License, version 2.1 or higher}
+{http://www.gnu.org/licenses/old-licenses/lgpl-2.1.html}
+(UMFPACK, parts of CHOLMOD, AMD, COLAMD) and the
+\ulink{GNU General Public License, version 2 or higher}
+{http://www.gnu.org/licenses/old-licenses/gpl-2.0.html}
+(the Supernodal module of CHOLMOD).
+For copyright and license details, consult the README files in the source
+directories or the website listed below.
\begin{quote}
Availability: \ulink{www.cise.ufl.edu/research/sparse}
@@ -105,7 +140,7 @@ Availability: \ulink{www.cs.wm.edu/\~{}va/software/park/park.html}
\EIT
-\tableofcontents
+%\tableofcontents
\input{intro}
\input{base}
@@ -114,8 +149,9 @@ Availability: \ulink{www.cs.wm.edu/\~{}va/software/park/park.html}
\input{fftw}
\input{base_sparse}
\input{spsolvers}
+\input{coneprog}
\input{solvers}
\input{modeling}
\input{c-api}
-\input{cvxopt.ind}
+%\input{cvxopt.ind}
\end{document}
diff --git a/doc/cvxopt/WARNINGS b/doc/cvxopt/WARNINGS
new file mode 100644
index 0000000..486bcce
--- /dev/null
+++ b/doc/cvxopt/WARNINGS
@@ -0,0 +1 @@
+No implementation found for style `graphicx'
diff --git a/doc/cvxopt/about.html b/doc/cvxopt/about.html
deleted file mode 100644
index 5295a32..0000000
--- a/doc/cvxopt/about.html
+++ /dev/null
@@ -1,114 +0,0 @@
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-<H1><A NAME="SECTION007000000000000000000"></A> <A NAME="c-fftw"></A>
-<BR>
-5. Discrete Transforms (<tt class="module">cvxopt.fftw</tt>)
-</H1>
-
-<P>
-The <tt class="module">cvxopt.fftw</tt> module is an interface to the FFTW library
-and contains routines for discrete Fourier, cosine, and sine
-transforms. This module is optional, and only installed when
-the FFTW library is made available during the CVXOPT installation.
-
-<P>
-<div class="seealso">
- <p class="heading">See Also:</p>
-
-<dl compact="compact" class="seeurl">
- <dt><a href='http://www.fftw.org'
- >FFTW3 code, documentation, copyright and
-license.</a></dt>
- <dd></dd>
- </dl>
-</div>
-
-<P>
-
-<p><br /></p><hr class='online-navigation' />
-<div class='online-navigation'>
-<!--Table of Child-Links-->
-<A NAME="CHILD_LINKS"><STRONG>Subsections</STRONG></a>
-
-<UL CLASS="ChildLinks">
-<LI><A href="node30.html">5.1 Discrete Fourier Transform</a>
-<LI><A href="node31.html">5.2 Discrete Cosine Transform</a>
-<LI><A href="node32.html">5.3 Discrete Sine Transform</a>
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-<html>
-<head>
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-<link rel="first" href="cvxopt.html" title='CVXOPT: A Python Package for Convex Optimization' />
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-<title>7. Sparse Linear Equation Solvers</title>
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-<body>
-<DIV CLASS="navigation">
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-<table align="center" width="100%" cellpadding="0" cellspacing="2">
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-
-<H1><A NAME="SECTION009000000000000000000"></A> <A NAME="c-spsolvers"></A>
-<BR>
-7. Sparse Linear Equation Solvers
-</H1>
-In this section we describe routines for solving sparse sets of linear
-equations.
-
-<P>
-A real symmetric or complex Hermitian sparse matrix is stored as
-an <tt class="class">spmatrix</tt> object <var>X</var> of size (<I>n</I>, <I>n</I>) and an
-additional character argument <code>uplo</code> with possible values
-<code>'L'</code> and <code>'U'</code>.
-If <code>uplo</code> is <code>'L'</code>, the lower triangular part
-of <var>X</var> contains the lower triangular part of the
-symmetric or Hermitian matrix, and the upper triangular matrix
-of <var>X</var> is ignored.
-If <code>uplo</code> is <code>'U'</code>, the upper triangular part
-of <var>X</var> contains the upper triangular part of the
-matrix, and the lower triangular matrix of <var>X</var> is ignored.
-
-<P>
-A general sparse square matrix of order <I>n</I> is represented by an
-<tt class="class">spmatrix</tt> object of size (<I>n</I>, <I>n</I>).
-
-<P>
-Dense matrices, which appear as righthand sides of equations, are
-stored using the same conventions as in the BLAS and LAPACK modules.
-
-<P>
-
-<p><br /></p><hr class='online-navigation' />
-<div class='online-navigation'>
-<!--Table of Child-Links-->
-<A NAME="CHILD_LINKS"><STRONG>Subsections</STRONG></a>
-
-<UL CLASS="ChildLinks">
-<LI><A href="s-orderings.html">7.1 Matrix Orderings (<tt class="module">cvxopt.amd</tt>)</a>
-<LI><A href="s-umfpack.html">7.2 General Linear Equations (<tt class="module">cvxopt.umfpack</tt>)</a>
-<LI><A href="s-cholmod.html">7.3 Positive Definite Linear Equations (<tt class="module">cvxopt.cholmod</tt>)</a>
-<LI><A href="e-covsel.html">7.4 Example: Covariance Selection</a>
-</ul>
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-<p></p><hr />
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-<!DOCTYPE html PUBLIC "-//W3C//DTD HTML 4.0 Transitional//EN">
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-</div>
-<hr /></div>
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-<BR><h2><A NAME="SECTION002000000000000000000">
-Contents</A>
-</h2>
-<!--Table of Contents-->
-
-<UL CLASS="TofC">
-<LI><A href="node3.html">1. Introduction</a>
-<LI><A href="node4.html">2. Dense Matrices (cvxopt.base)</a>
-<UL>
-<LI><A href="s-creating-matrices.html">2.1 Creating Matrices</a>
-<LI><A href="node6.html">2.2 Attributes and Methods</a>
-<LI><A href="s-arithmetic.html">2.3 Arithmetic Operations</a>
-<LI><A href="s-indexing.html">2.4 Indexing and Slicing</a>
-<LI><A href="s-builtinfuncs.html">2.5 Built-in Functions</a>
-<LI><A href="s-otherfuncs.html">2.6 Other Matrix Functions</a>
-<LI><A href="s-random.html">2.7 Randomly Generated Matrices</a>
-<LI><A href="s-array-interface.html">2.8 The NumPy Array Interface</a>
-<LI><A href="node13.html">2.9 Printing Options</a>
-</ul>
-<LI><A href="node14.html">3. The BLAS Interface (cvxopt.blas)</a>
-<UL>
-<LI><A href="s-conventions.html">3.1 Matrix Classes</a>
-<LI><A href="s-blas1.html">3.2 Level 1 BLAS</a>
-<LI><A href="s-blas2.html">3.3 Level 2 BLAS</a>
-<LI><A href="s-blas3.html">3.4 Level 3 BLAS</a>
-</ul>
-<LI><A href="node19.html">4. The LAPACK Interface (cvxopt.lapack)</a>
-<UL>
-<LI><A href="node20.html">4.1 General Linear Equations</a>
-<LI><A href="e-kkt-example.html">4.2 Positive Definite Linear Equations</a>
-<LI><A href="node22.html">4.3 Symmetric and Hermitian Linear Equations</a>
-<LI><A href="node23.html">4.4 Triangular Linear Equations</a>
-<LI><A href="node24.html">4.5 Least-Squares and Least-Norm Problems</a>
-<LI><A href="node25.html">4.6 Symmetric and Hermitian Eigenvalue Decomposition</a>
-<LI><A href="e-gevd.html">4.7 Generalized Symmetric Definite Eigenproblems</a>
-<LI><A href="node27.html">4.8 Singular Value Decomposition</a>
-<LI><A href="node28.html">4.9 Example: Analytic Centering</a>
-</ul>
-<LI><A href="c-fftw.html">5. Discrete Transforms (cvxopt.fftw)</a>
-<UL>
-<LI><A href="node30.html">5.1 Discrete Fourier Transform</a>
-<LI><A href="node31.html">5.2 Discrete Cosine Transform</a>
-<LI><A href="node32.html">5.3 Discrete Sine Transform</a>
-</ul>
-<LI><A href="node33.html">6. Sparse Matrices (cvxopt.base)</a>
-<UL>
-<LI><A href="s-creating-spmatrix.html">6.1 Creating Sparse Matrices</a>
-<LI><A href="e-spA-example.html">6.2 Attributes and Methods</a>
-<LI><A href="s-spmatrix-arith.html">6.3 Arithmetic Operations</a>
-<LI><A href="node37.html">6.4 Indexing and Slicing</a>
-<LI><A href="node38.html">6.5 Built-In Functions</a>
-<LI><A href="node39.html">6.6 Sparse BLAS Functions</a>
-</ul>
-<LI><A href="c-spsolvers.html">7. Sparse Linear Equation Solvers</a>
-<UL>
-<LI><A href="s-orderings.html">7.1 Matrix Orderings (cvxopt.amd)</a>
-<LI><A href="s-umfpack.html">7.2 General Linear Equations (cvxopt.umfpack)</a>
-<LI><A href="s-cholmod.html">7.3 Positive Definite Linear Equations (cvxopt.cholmod)</a>
-<LI><A href="e-covsel.html">7.4 Example: Covariance Selection</a>
-</ul>
-<LI><A href="node45.html">8. Optimization Routines (cvxopt.solvers)</a>
-<UL>
-<LI><A href="s-lpsolver.html">8.1 Linear Programming</a>
-<LI><A href="node47.html">8.2 Quadratic Programming</a>
-<LI><A href="node48.html">8.3 Geometric Programming</a>
-<LI><A href="s-sdpsolver.html">8.4 Semidefinite Programming</a>
-<LI><A href="e-nlcp.html">8.5 Nonlinear Convex Programming</a>
-<LI><A href="node51.html">8.6 Exploiting Structure in LPs and SDPs</a>
-<LI><A href="node52.html">8.7 Exploiting Structure in Nonlinear Convex Programs</a>
-<LI><A href="s-external.html">8.8 Optional Solvers</a>
-<LI><A href="s-parameters.html">8.9 Algorithm Parameters</a>
-</ul>
-<LI><A href="node55.html">9. Modeling (cvxopt.modeling)</a>
-<UL>
-<LI><A href="s-variables.html">9.1 Variables</a>
-<LI><A href="s-functions.html">9.2 Functions</a>
-<LI><A href="node58.html">9.3 Constraints</a>
-<LI><A href="s-lp.html">9.4 Optimization Problems</a>
-<LI><A href="node60.html">9.5 Examples</a>
-</ul>
-<LI><A href="node61.html">10. C API</a>
-<UL>
-<LI><A href="node62.html">10.1 Dense Matrices</a>
-<LI><A href="node63.html">10.2 Sparse Matrices</a>
-</ul>
-<LI><A href="genindex.html">Index</a>
-</ul>
-<!--End of Table of Contents-->
-<P>
-
-<DIV CLASS="navigation">
-<div class='online-navigation'>
-<p></p><hr />
-<table align="center" width="100%" cellpadding="0" cellspacing="2">
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diff --git a/doc/cvxopt/cvxopt.html b/doc/cvxopt/cvxopt.html
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-<div class='center'>
-<h1>CVXOPT: A Python Package for Convex Optimization</h1>
-<p><b><font size="+2">Joachim Dahl & Lieven Vandenberghe</font></b></p>
-<p><i><TT>joachim at kom.aau.dk</TT>, <TT>vandenbe at ee.ucla.edu</TT></i></p>
-<p><strong>Release 0.8.2</strong><br />
-<strong>February 6, 2007</strong></p>
-<p></p>
-</div>
-</div>
-
-<P>
-
-<p><br /></p><hr class='online-navigation' />
-<div class='online-navigation'>
-<!--Table of Child-Links-->
-<A NAME="CHILD_LINKS"></a>
-
-<UL CLASS="ChildLinks">
-<LI><A href="node1.html">Copyright and License</a>
-<LI><A href="contents.html">Contents</a>
-<LI><A href="node3.html">1. Introduction</a>
-<LI><A href="node4.html">2. Dense Matrices (<tt class="module">cvxopt.base</tt>)</a>
-<UL>
-<LI><A href="s-creating-matrices.html">2.1 Creating Matrices</a>
-<LI><A href="node6.html">2.2 Attributes and Methods</a>
-<LI><A href="s-arithmetic.html">2.3 Arithmetic Operations</a>
-<LI><A href="s-indexing.html">2.4 Indexing and Slicing</a>
-<LI><A href="s-builtinfuncs.html">2.5 Built-in Functions</a>
-<LI><A href="s-otherfuncs.html">2.6 Other Matrix Functions</a>
-<LI><A href="s-random.html">2.7 Randomly Generated Matrices</a>
-<LI><A href="s-array-interface.html">2.8 The NumPy Array Interface</a>
-<LI><A href="node13.html">2.9 Printing Options</a>
-</ul>
-<LI><A href="node14.html">3. The BLAS Interface (<tt class="module">cvxopt.blas</tt>)</a>
-<UL>
-<LI><A href="s-conventions.html">3.1 Matrix Classes</a>
-<LI><A href="s-blas1.html">3.2 Level 1 BLAS</a>
-<LI><A href="s-blas2.html">3.3 Level 2 BLAS</a>
-<LI><A href="s-blas3.html">3.4 Level 3 BLAS</a>
-</ul>
-<LI><A href="node19.html">4. The LAPACK Interface (<tt class="module">cvxopt.lapack</tt>)</a>
-<UL>
-<LI><A href="node20.html">4.1 General Linear Equations</a>
-<LI><A href="e-kkt-example.html">4.2 Positive Definite Linear Equations</a>
-<LI><A href="node22.html">4.3 Symmetric and Hermitian Linear Equations</a>
-<LI><A href="node23.html">4.4 Triangular Linear Equations</a>
-<LI><A href="node24.html">4.5 Least-Squares and Least-Norm Problems</a>
-<LI><A href="node25.html">4.6 Symmetric and Hermitian Eigenvalue Decomposition</a>
-<LI><A href="e-gevd.html">4.7 Generalized Symmetric Definite Eigenproblems</a>
-<LI><A href="node27.html">4.8 Singular Value Decomposition</a>
-<LI><A href="node28.html">4.9 Example: Analytic Centering</a>
-</ul>
-<LI><A href="c-fftw.html">5. Discrete Transforms (<tt class="module">cvxopt.fftw</tt>)</a>
-<UL>
-<LI><A href="node30.html">5.1 Discrete Fourier Transform</a>
-<LI><A href="node31.html">5.2 Discrete Cosine Transform</a>
-<LI><A href="node32.html">5.3 Discrete Sine Transform</a>
-</ul>
-<LI><A href="node33.html">6. Sparse Matrices (<tt class="module">cvxopt.base</tt>)</a>
-<UL>
-<LI><A href="s-creating-spmatrix.html">6.1 Creating Sparse Matrices</a>
-<LI><A href="e-spA-example.html">6.2 Attributes and Methods</a>
-<LI><A href="s-spmatrix-arith.html">6.3 Arithmetic Operations</a>
-<LI><A href="node37.html">6.4 Indexing and Slicing</a>
-<LI><A href="node38.html">6.5 Built-In Functions</a>
-<LI><A href="node39.html">6.6 Sparse BLAS Functions</a>
-</ul>
-<LI><A href="c-spsolvers.html">7. Sparse Linear Equation Solvers</a>
-<UL>
-<LI><A href="s-orderings.html">7.1 Matrix Orderings (<tt class="module">cvxopt.amd</tt>)</a>
-<LI><A href="s-umfpack.html">7.2 General Linear Equations (<tt class="module">cvxopt.umfpack</tt>)</a>
-<LI><A href="s-cholmod.html">7.3 Positive Definite Linear Equations (<tt class="module">cvxopt.cholmod</tt>)</a>
-<LI><A href="e-covsel.html">7.4 Example: Covariance Selection</a>
-</ul>
-<LI><A href="node45.html">8. Optimization Routines (<tt class="module">cvxopt.solvers</tt>)</a>
-<UL>
-<LI><A href="s-lpsolver.html">8.1 Linear Programming</a>
-<LI><A href="node47.html">8.2 Quadratic Programming</a>
-<LI><A href="node48.html">8.3 Geometric Programming</a>
-<LI><A href="s-sdpsolver.html">8.4 Semidefinite Programming</a>
-<LI><A href="e-nlcp.html">8.5 Nonlinear Convex Programming</a>
-<LI><A href="node51.html">8.6 Exploiting Structure in LPs and SDPs</a>
-<LI><A href="node52.html">8.7 Exploiting Structure in Nonlinear Convex Programs</a>
-<LI><A href="s-external.html">8.8 Optional Solvers</a>
-<LI><A href="s-parameters.html">8.9 Algorithm Parameters</a>
-</ul>
-<LI><A href="node55.html">9. Modeling (<tt class="module">cvxopt.modeling</tt>)</a>
-<UL>
-<LI><A href="s-variables.html">9.1 Variables</a>
-<LI><A href="s-functions.html">9.2 Functions</a>
-<LI><A href="node58.html">9.3 Constraints</a>
-<LI><A href="s-lp.html">9.4 Optimization Problems</a>
-<LI><A href="node60.html">9.5 Examples</a>
-</ul>
-<LI><A href="node61.html">10. C API</a>
-<UL>
-<LI><A href="node62.html">10.1 Dense Matrices</a>
-<LI><A href="node63.html">10.2 Sparse Matrices</a>
-</ul>
-<LI><A href="genindex.html">Index</a>
-<LI><A href="about.html">About this document ...</a>
-</ul>
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- href="contents.html"><img src='contents.gif'
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diff --git a/doc/cvxopt/e-covsel.html b/doc/cvxopt/e-covsel.html
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-<!DOCTYPE html PUBLIC "-//W3C//DTD HTML 4.0 Transitional//EN">
-<html>
-<head>
-<link rel="STYLESHEET" href="cvxopt.css" type='text/css' />
-<link rel="first" href="cvxopt.html" title='CVXOPT: A Python Package for Convex Optimization' />
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-<link rel='index' href='genindex.html' title='Index' />
-<link rel='last' href='about.html' title='About this document...' />
-<link rel='help' href='about.html' title='About this document...' />
-<link rel="prev" href="s-cholmod.html" />
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-<title>7.4 Example: Covariance Selection</title>
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-<td align="center" width="100%">CVXOPT: A Python Package for Convex Optimization</td>
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-<H1><A NAME="SECTION009400000000000000000">
-7.4 Example: Covariance Selection</A>
-</H1>
-This example illustrates the use of the routines for sparse Cholesky
-factorization. We consider the problem
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-(7.5)</TD></TR>
-</TABLE>
-<BR CLEAR="ALL"></DIV><P></P>
-The optimization variable is a symmetric matrix <I>K</I> of order <I>n</I>
-and the domain of the problem is the set of positive definite matrices.
-The matrix <SPAN CLASS="MATH"><IMG
- WIDTH="17" HEIGHT="14" ALIGN="BOTTOM" BORDER="0"
- SRC="img105.gif"
- ALT="$Y$"></SPAN> and the index set <SPAN CLASS="MATH"><IMG
- WIDTH="15" HEIGHT="14" ALIGN="BOTTOM" BORDER="0"
- SRC="img106.gif"
- ALT="$S$"></SPAN> are given. We assume that all
-the diagonal positions are included in <SPAN CLASS="MATH"><IMG
- WIDTH="15" HEIGHT="14" ALIGN="BOTTOM" BORDER="0"
- SRC="img106.gif"
- ALT="$S$"></SPAN>.
-This problem arises in maximum likelihood estimation of the covariance
-matrix of a zero-mean normal distribution, with constraints
-that specify that pairs of variables are conditionally independent.
-
-<P>
-We can express <I>K</I> as
-<BR><P></P>
-<DIV ALIGN="CENTER" CLASS="mathdisplay">
-<!-- MATH
- \begin{displaymath}
-K(x) = E_1\mbox{\bf diag}\,(x)E_2^T+E_2\mbox{\bf diag}\,(x)E_1^T
-\end{displaymath}
- -->
-
-<IMG
- WIDTH="265" HEIGHT="28" BORDER="0"
- SRC="img107.gif"
- ALT="\begin{displaymath}
-K(x) = E_1\mbox{\bf diag}\,(x)E_2^T+E_2\mbox{\bf diag}\,(x)E_1^T
-\end{displaymath}">
-</DIV>
-<BR CLEAR="ALL">
-<P></P>
-where <I>x</I> are the nonzero elements in the lower triangular part
-of <I>K</I>, with the diagonal elements scaled by 1/2,
-and
-<BR><P></P>
-<DIV ALIGN="CENTER" CLASS="mathdisplay">
-<!-- MATH
- \begin{displaymath}
-E_1 = \left[ \begin{array}{cccc}
- e_{i_1} & e_{i_2} & \cdots & e_{i_q} \end{array}\right], \qquad
- E_2 = \left[ \begin{array}{cccc}
- e_{j_1} & e_{j_2} & \cdots & e_{j_q} \end{array}\right],
-\end{displaymath}
- -->
-
-<IMG
- WIDTH="436" HEIGHT="30" BORDER="0"
- SRC="img108.gif"
- ALT="\begin{displaymath}
-E_1 = \left[ \begin{array}{cccc}
-e_{i_1} & e_{i_2} & \cdot...
-...cc}
-e_{j_1} & e_{j_2} & \cdots & e_{j_q} \end{array}\right],
-\end{displaymath}">
-</DIV>
-<BR CLEAR="ALL">
-<P></P>
-where (<I>i_k</I>, <I>j_k</I>) are the positions of the nonzero
-entries in the lower-triangular part of <I>K</I>.
-With this notation, we can solve problem (<A HREF="#e-covsel">7.5</A>) by solving
-the unconstrained problem
-<BR><P></P>
-<DIV ALIGN="CENTER" CLASS="mathdisplay">
-<!-- MATH
- \begin{displaymath}
-\begin{array}{ll}
- \mbox{minimize} & f(x) = -\log\det K(x) + \mathop{\bf tr}(K(x)Y).
- \end{array}
-\end{displaymath}
- -->
-
-<IMG
- WIDTH="325" HEIGHT="30" BORDER="0"
- SRC="img109.gif"
- ALT="\begin{displaymath}
-\begin{array}{ll}
-\mbox{minimize} & f(x) = -\log\det K(x) + \mathop{\bf tr}(K(x)Y).
-\end{array}\end{displaymath}">
-</DIV>
-<BR CLEAR="ALL">
-<P></P>
-The code below implements Newton's method with a backtracking line
-search. The gradient and Hessian of the objective function are given
-by
-<P></P>
-<DIV ALIGN="CENTER" CLASS="mathdisplay">
-<!-- MATH
- \begin{eqnarray*}
-\nabla f(x)
-&=& 2 \mbox{\bf diag}\,( E_1^T (Y - K(x)^{-1}) E_2)) \\
-&=& 2\mbox{\bf diag}\,(Y_{IJ} - \left(K(x)^{-1}\right)_{IJ})\\
-\nabla^2 f(x) & = &
- 2 (E_1^T K(x)^{-1} E_1) \circ (E_2^T K(x)^{-1} E_2)
- + 2 (E_1^T K(x)^{-1} E_2) \circ (E_2^T K(x)^{-1} E_1) \\
-&=& 2 \left(K(x)^{-1}\right)_{II} \circ
- \left(K(x)^{-1}\right)_{JJ}
- +2 \left(K(x)^{-1}\right)_{IJ} \circ
- \left(K(x)^{-1}\right)_{JI},
-\end{eqnarray*}
- -->
-<IMG
- WIDTH="588" HEIGHT="105" BORDER="0"
- SRC="img110.gif"
- ALT="\begin{eqnarray*}
-\nabla f(x)
-&=& 2 \mbox{\bf diag}\,( E_1^T (Y - K(x)^{-1}) E_...
-...\left(K(x)^{-1}\right)_{IJ} \circ
-\left(K(x)^{-1}\right)_{JI},
-\end{eqnarray*}"></DIV>
-<BR CLEAR="ALL"><P></P>
-<BR CLEAR="ALL"><P></P>
-where <code>o</code> denotes Hadamard product.
-
-<P>
-<div class="verbatim"><pre>
-from cvxopt.base import matrix, spmatrix, log, mul
-from cvxopt import blas, lapack, amd, cholmod
-
-def covsel(Y):
- """
- Returns the solution of
-
- minimize -logdet K + Tr(KY)
- subject to K_{ij}=0, (i,j) not in indices listed in I,J.
-
- Y is a symmetric sparse matrix with nonzero diagonal elements.
- I = Y.I, J = Y.J.
- """
-
- I, J = Y.I, Y.J
- n, m = Y.size[0], len(I)
- N = I + J*n # non-zero positions for one-argument indexing
- D = [k for k in xrange(m) if I[k]==J[k]] # position of diagonal elements
-
- # starting point: symmetric identity with nonzero pattern I,J
- K = spmatrix(0, I, J)
- K[::n+1] = 1
-
- # Kn is used in the line search
- Kn = spmatrix(0, I, J)
-
- # symbolic factorization of K
- F = cholmod.symbolic(K)
-
- # Kinv will be the inverse of K
- Kinv = matrix(0.0, (n,n))
-
- for iters in xrange(100):
-
- # numeric factorization of K
- cholmod.numeric(K, F)
- d = cholmod.diag(F)
-
- # compute Kinv by solving K*X = I
- Kinv[:] = 0
- Kinv[::n+1] = 1
- cholmod.solve(F, Kinv)
-
- # solve Newton system
- grad = 2*(Y.V - Kinv[N])
- hess = 2*(mul(Kinv[I,J],Kinv[J,I]) + mul(Kinv[I,I],Kinv[J,J]))
- v = -grad
- lapack.posv(hess,v)
-
- # stopping criterion
- sqntdecr = -blas.dot(grad,v)
- print "Newton decrement squared:%- 7.5e" %sqntdecr
- if (sqntdecr < 1e-12):
- print "number of iterations: ", iters+1
- break
-
- # line search
- dx = +v
- dx[D] *= 2 # scale the diagonal elems
- f = -2.0 * sum(log(d)) # f = -log det K
- s = 1
- for lsiter in xrange(50):
- Kn.V = K.V + s*dx
- try:
- cholmod.numeric(Kn, F)
- except ArithmeticError:
- s *= 0.5
- else:
- d = cholmod.diag(F)
- fn = -2.0 * sum(log(d)) + 2*s*blas.dot(v,Y.V)
- if (fn < f - 0.01*s*sqntdecr):
- break
- s *= 0.5
-
- K.V = Kn.V
-
- return K
-</pre></div>
-
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-
-<H1><A NAME="SECTION006700000000000000000">
-4.7 Generalized Symmetric Definite Eigenproblems</A>
-</H1>
-Three types of generalized eigenvalue problems can be solved:
-<BR>
-<DIV ALIGN="RIGHT" CLASS="mathdisplay">
-
-<!-- MATH
- \begin{equation}
-AZ = BZ\mbox{\bf diag}\,(\lambda)\quad \mbox{(type 1)}, \qquad
- ABZ = Z\mbox{\bf diag}\,(\lambda) \quad \mbox{(type 2)}, \qquad
- BAZ = Z\mbox{\bf diag}\,(\lambda) \quad \mbox{(type 3)},
-\end{equation}
- -->
-<A NAME="e-gevd"></A>
-<TABLE WIDTH="100%" ALIGN="CENTER">
-<TR VALIGN="MIDDLE"><TD></TD><TD ALIGN="CENTER" NOWRAP><A NAME="e-gevd"></A><IMG
- WIDTH="649" HEIGHT="28" BORDER="0"
- SRC="img66.gif"
- ALT="\begin{displaymath}
-AZ = BZ\mbox{\bf diag}\,(\lambda)\quad \mbox{(type 1)}, \qq...
-...ad
-BAZ = Z\mbox{\bf diag}\,(\lambda) \quad \mbox{(type 3)},
-\end{displaymath}"></TD>
-<TD CLASS="eqno" WIDTH=10 ALIGN="RIGHT">
-(4.2)</TD></TR>
-</TABLE>
-<BR CLEAR="ALL"></DIV><P></P>
-with <SPAN CLASS="MATH"><IMG
- WIDTH="16" HEIGHT="14" ALIGN="BOTTOM" BORDER="0"
- SRC="img53.gif"
- ALT="$A$"></SPAN> and <SPAN CLASS="MATH"><IMG
- WIDTH="17" HEIGHT="14" ALIGN="BOTTOM" BORDER="0"
- SRC="img67.gif"
- ALT="$B$"></SPAN> real symmetric or complex Hermitian, and <SPAN CLASS="MATH"><IMG
- WIDTH="17" HEIGHT="14" ALIGN="BOTTOM" BORDER="0"
- SRC="img67.gif"
- ALT="$B$"></SPAN> positive
-definite.
-The matrix of eigenvectors is normalized as follows:
-<BR><P></P>
-<DIV ALIGN="CENTER" CLASS="mathdisplay">
-<!-- MATH
- \begin{displaymath}
-Z^H BZ = I \quad \mbox{(types 1 and 2)}, \qquad
- Z^H B^{-1}Z = I \quad \mbox{(type 3)}.
-\end{displaymath}
- -->
-
-<IMG
- WIDTH="392" HEIGHT="27" BORDER="0"
- SRC="img68.gif"
- ALT="\begin{displaymath}
-Z^H BZ = I \quad \mbox{(types 1 and 2)}, \qquad
-Z^H B^{-1}Z = I \quad \mbox{(type 3)}.
-\end{displaymath}">
-</DIV>
-<BR CLEAR="ALL">
-<P></P>
-
-<P>
-<dl><dt><table cellpadding="0" cellspacing="0"><tr valign="baseline">
- <td><nobr><b><tt id='l2h-105' xml:id='l2h-105' class="function">sygv</tt></b>(</nobr></td>
- <td><var>A, B, W</var><big>[</big><var>, itype=1</var><big>[</big><var>,
-jobz='N'</var><big>[</big><var>, uplo='L'</var><big>]</big><var></var><big>]</big><var></var><big>]</big><var></var>)</td></tr></table></dt>
-<dd>
-Solves the generalized eigenproblem (<A HREF="#e-gevd">4.2</A>) for real symmetric
-matrices of order <I>n</I>, stored in real matrices <var>A</var> and <var>B</var>.
-<var>itype</var> is an integer with possible values 1, 2, 3, and specifies
-the type of eigenproblem.
-<var>W</var> is a real matrix of length at least <I>n</I>.
-On exit, it contains the eigenvalues in ascending order.
-On exit, <var>B</var> contains the Cholesky factor of <I>B</I>.
-If <var>jobz</var> is <code>'V'</code>, the eigenvectors are computed
-and returned in <var>A</var>.
-If <var>jobz</var> is <code>'N'</code>, the eigenvectors are not returned and the
-contents of <var>A</var> are destroyed.
-</dl>
-
-<P>
-<dl><dt><table cellpadding="0" cellspacing="0"><tr valign="baseline">
- <td><nobr><b><tt id='l2h-106' xml:id='l2h-106' class="function">hegv</tt></b>(</nobr></td>
- <td><var>A, B, W</var><big>[</big><var>, itype=1</var><big>[</big><var>,
-jobz='N'</var><big>[</big><var>, uplo='L'</var><big>]</big><var></var><big>]</big><var></var><big>]</big><var></var>)</td></tr></table></dt>
-<dd>
-Generalized eigenvalue problem (<A HREF="#e-gevd">4.2</A>) of real symmetric or
-complex Hermitian matrix of order <var>n</var>.
-The calling sequence is identical to <tt class="function">sygv()</tt>,
-except that <var>A</var> and <var>B</var> can be real or complex.
-</dl>
-
-<P>
-
-<DIV CLASS="navigation">
-<div class='online-navigation'>
-<p></p><hr />
-<table align="center" width="100%" cellpadding="0" cellspacing="2">
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-<H1><A NAME="SECTION006200000000000000000">
-4.2 Positive Definite Linear Equations</A>
-</H1>
-<dl><dt><table cellpadding="0" cellspacing="0"><tr valign="baseline">
- <td><nobr><b><tt id='l2h-72' xml:id='l2h-72' class="function">posv</tt></b>(</nobr></td>
- <td><var>A, B</var><big>[</big><var>, uplo='L'</var><big>]</big><var></var>)</td></tr></table></dt>
-<dd>
-Solves
-<BR><P></P>
-<DIV ALIGN="CENTER" CLASS="mathdisplay">
-<!-- MATH
- \begin{displaymath}
-A X = B,
-\end{displaymath}
- -->
-
-<IMG
- WIDTH="64" HEIGHT="27" BORDER="0"
- SRC="img41.gif"
- ALT="\begin{displaymath}
-A X = B,
-\end{displaymath}">
-</DIV>
-<BR CLEAR="ALL">
-<P></P>
-where <I>A</I> is a real symmetric or complex Hermitian positive
-definite matrix.
-On exit, <var>B</var> is replaced by the solution, and <var>A</var> is
-overwritten with the Cholesky factor.
-The matrices <var>A</var> and <var>B</var> must have the same type (<code>'d'</code> or
-<code>'z'</code>).
-Raises an <code>ArithmeticError</code> if the matrix is not positive definite.
-</dl>
-
-<P>
-<dl><dt><table cellpadding="0" cellspacing="0"><tr valign="baseline">
- <td><nobr><b><tt id='l2h-73' xml:id='l2h-73' class="function">potrf</tt></b>(</nobr></td>
- <td><var>A</var><big>[</big><var>, uplo='L'</var><big>]</big><var></var>)</td></tr></table></dt>
-<dd>
-Cholesky factorization
-<BR><P></P>
-<DIV ALIGN="CENTER" CLASS="mathdisplay">
-<!-- MATH
- \begin{displaymath}
-A = LL^T \qquad \mbox{or} \qquad A = LL^H
-\end{displaymath}
- -->
-
-<IMG
- WIDTH="207" HEIGHT="24" BORDER="0"
- SRC="img46.gif"
- ALT="\begin{displaymath}
-A = LL^T \qquad \mbox{or} \qquad A = LL^H
-\end{displaymath}">
-</DIV>
-<BR CLEAR="ALL">
-<P></P>
-of a positive definite real symmetric or complex Hermitian matrix
-<I>A</I>. On exit, the lower triangular part of <var>A</var>
-(if <var>uplo</var> is <code>'L'</code>) or the upper triangular part
-(if <var>uplo</var> is <code>'U'</code>) is overwritten with the Cholesky factor
-or its (conjugate) transpose.
-Raises an <code>ArithmeticError</code> if the matrix is not positive definite.
-</dl>
-
-<P>
-<dl><dt><table cellpadding="0" cellspacing="0"><tr valign="baseline">
- <td><nobr><b><tt id='l2h-74' xml:id='l2h-74' class="function">potrs</tt></b>(</nobr></td>
- <td><var>A, B</var><big>[</big><var>, uplo='L'</var><big>]</big><var></var>)</td></tr></table></dt>
-<dd>
-Solves a set of linear equations
-<BR><P></P>
-<DIV ALIGN="CENTER" CLASS="mathdisplay">
-<!-- MATH
- \begin{displaymath}
-AX=B
-\end{displaymath}
- -->
-
-<IMG
- WIDTH="60" HEIGHT="24" BORDER="0"
- SRC="img47.gif"
- ALT="\begin{displaymath}
-AX=B
-\end{displaymath}">
-</DIV>
-<BR CLEAR="ALL">
-<P></P>
-with a positive
-definite real symmetric or complex Hermitian matrix,
-given the Cholesky factorization computed by
-<tt class="function">posv()</tt> or <tt class="function">potrf()</tt>.
-On entry, <var>A</var> contains the triangular factor, as computed by
-<tt class="function">posv()</tt> or <tt class="function">potrf()</tt>. On exit, <var>B</var> is replaced
-by the solution.
-<var>B</var> must have the same type as <var>A</var>.
-</dl>
-
-<P>
-<dl><dt><table cellpadding="0" cellspacing="0"><tr valign="baseline">
- <td><nobr><b><tt id='l2h-75' xml:id='l2h-75' class="function">potri</tt></b>(</nobr></td>
- <td><var>A</var><big>[</big><var>, uplo='L'</var><big>]</big><var></var>)</td></tr></table></dt>
-<dd>
-Computes the inverse of a positive definite matrix.
-On entry, <var>A</var> contains the Cholesky factorization computed
-by <tt class="function">potrf()</tt> or <tt class="function">posv()</tt>. On exit, it contains
-the inverse.
-</dl>
-
-<P>
-As an example, we use <tt class="function">posv()</tt> to solve the linear system
-<BR>
-<DIV ALIGN="RIGHT" CLASS="mathdisplay">
-
-<!-- MATH
- \begin{equation}
-\left[ \begin{array}{cc}
- -\mbox{\bf diag}\,(d)^2 & A \\
- A^T & 0 \end{array} \right]
- \left[ \begin{array}{c} x_1 \\x_2 \end{array} \right]
- = \left[ \begin{array}{c} b_1 \\b_2 \end{array} \right]
-\end{equation}
- -->
-<A NAME="e-kkt-example"></A>
-<TABLE WIDTH="100%" ALIGN="CENTER">
-<TR VALIGN="MIDDLE"><TD></TD><TD ALIGN="CENTER" NOWRAP><A NAME="e-kkt-example"></A><IMG
- WIDTH="244" HEIGHT="45" BORDER="0"
- SRC="img48.gif"
- ALT="\begin{displaymath}
-\left[ \begin{array}{cc}
--\mbox{\bf diag}\,(d)^2 & A \\
-...
-...ght]
-= \left[ \begin{array}{c} b_1 \\ b_2 \end{array} \right]
-\end{displaymath}"></TD>
-<TD CLASS="eqno" WIDTH=10 ALIGN="RIGHT">
-(4.1)</TD></TR>
-</TABLE>
-<BR CLEAR="ALL"></DIV><P></P>
-by block-elimination.
-We first pick a random problem.
-<div class="verbatim"><pre>
->>> from cvxopt.base import matrix, div
->>> from cvxopt.random import normal, uniform
->>> from cvxopt.blas import syrk, gemv
->>> from cvxopt.lapack import posv
->>> m, n = 100, 50
->>> A = normal(m,n)
->>> b1, b2 = normal(m), normal(n)
->>> d = uniform(m)
-</pre></div>
-We then solve the equations
-<BR><P></P>
-<DIV ALIGN="CENTER" CLASS="mathdisplay">
-<!-- MATH
- \begin{displaymath}
-A^T \mbox{\bf diag}\,(d)^{-2}A x_2 = b_2 + A^T \mbox{\bf diag}\,(d)^{-2} b_1, \qquad
- \mbox{\bf diag}\,(d)^2 x_1 = Ax_2 - b_1.
-\end{displaymath}
- -->
-
-<IMG
- WIDTH="481" HEIGHT="28" BORDER="0"
- SRC="img49.gif"
- ALT="\begin{displaymath}
-A^T \mbox{\bf diag}\,(d)^{-2}A x_2 = b_2 + A^T \mbox{\bf di...
-...(d)^{-2} b_1, \qquad
-\mbox{\bf diag}\,(d)^2 x_1 = Ax_2 - b_1.
-\end{displaymath}">
-</DIV>
-<BR CLEAR="ALL">
-<P></P>
-<div class="verbatim"><pre>
->>> Asc = div(A, d[:, n*[0]]) # Asc := diag(d)^{-1}*A
->>> B = matrix(0.0, (n,n))
->>> syrk(Asc, B, trans='T') # B := Asc^T * Asc = A^T * diag(d)^{-2} * A
->>> x1 = div(b1, d) # x1 := diag(d)^{-1}*b1
->>> x2 = +b2
->>> gemv(Asc, x1, x2, trans='T', beta=1.0) # x2 := x2 + Asc^T*x1 = b2 + A^T*diag(d)^{-2}*b1
->>> posv(B, x2) # x2 := B^{-1}*x2 = B^{-1}*(b2 + A^T*diag(d)^{-2}*b1)
->>> gemv(Asc, x2, x1, beta=-1.0) # x1 := Asc*x2 - x1 = diag(d)^{-1} * (A*x2 - b1)
->>> x1 = div(x1, d) # x1 := diag(d)^{-1}*x1 = diag(d)^{-2} * (A*x2 - b1)
-</pre></div>
-
-<P>
-There are separate routines for equations with positive definite band
-matrices.
-<dl><dt><table cellpadding="0" cellspacing="0"><tr valign="baseline">
- <td><nobr><b><tt id='l2h-76' xml:id='l2h-76' class="function">pbsv</tt></b>(</nobr></td>
- <td><var>A, B</var><big>[</big><var>, uplo='L'</var><big>]</big><var></var>)</td></tr></table></dt>
-<dd>
-Solves
-<BR><P></P>
-<DIV ALIGN="CENTER" CLASS="mathdisplay">
-<!-- MATH
- \begin{displaymath}
-AX=B
-\end{displaymath}
- -->
-
-<IMG
- WIDTH="60" HEIGHT="24" BORDER="0"
- SRC="img47.gif"
- ALT="\begin{displaymath}
-AX=B
-\end{displaymath}">
-</DIV>
-<BR CLEAR="ALL">
-<P></P>
-where <I>A</I> is a real symmetric or complex Hermitian positive definite
-band matrix. On entry, the diagonals of <I>A</I> are stored in <var>A</var>,
-using the BLAS format for symmetric or Hermitian band matrices
-(see section <A href="s-conventions.html#s-conventions">3.1</A>). On exit, <var>B</var> is replaced by the
-solution, and <var>A</var> is overwritten with the Cholesky factor (in the
-BLAS format for triangular band matrices). The matrices <var>A</var> and
-<var>B</var> must have the same type (<code>'d'</code> or <code>'z'</code>).
-Raises an <code>ArithmeticError</code> if the matrix is not positive definite.
-</dl>
-
-<P>
-<dl><dt><table cellpadding="0" cellspacing="0"><tr valign="baseline">
- <td><nobr><b><tt id='l2h-77' xml:id='l2h-77' class="function">pbtrf</tt></b>(</nobr></td>
- <td><var>A</var><big>[</big><var>, uplo='L'</var><big>]</big><var></var>)</td></tr></table></dt>
-<dd>
-Cholesky factorization
-<BR><P></P>
-<DIV ALIGN="CENTER" CLASS="mathdisplay">
-<!-- MATH
- \begin{displaymath}
-A = LL^T \qquad \mbox{or} \qquad A = LL^H
-\end{displaymath}
- -->
-
-<IMG
- WIDTH="207" HEIGHT="24" BORDER="0"
- SRC="img46.gif"
- ALT="\begin{displaymath}
-A = LL^T \qquad \mbox{or} \qquad A = LL^H
-\end{displaymath}">
-</DIV>
-<BR CLEAR="ALL">
-<P></P>
-of a positive definite real symmetric or complex Hermitian band matrix
-<I>A</I>. On entry, the diagonals of <I>A</I> are stored in <var>A</var>,
-using the BLAS format for symmetric or Hermitian band matrices.
-On exit, <var>A</var> contains the Cholesky factor, in the BLAS format
-for triangular band matrices.
-Raises an <code>ArithmeticError</code> if the matrix is not positive definite.
-</dl>
-
-<P>
-<dl><dt><table cellpadding="0" cellspacing="0"><tr valign="baseline">
- <td><nobr><b><tt id='l2h-78' xml:id='l2h-78' class="function">pbtrs</tt></b>(</nobr></td>
- <td><var>A, B</var><big>[</big><var>, uplo='L'</var><big>]</big><var></var>)</td></tr></table></dt>
-<dd>
-Solves a set of linear equations
-<BR><P></P>
-<DIV ALIGN="CENTER" CLASS="mathdisplay">
-<!-- MATH
- \begin{displaymath}
-AX=B
-\end{displaymath}
- -->
-
-<IMG
- WIDTH="60" HEIGHT="24" BORDER="0"
- SRC="img47.gif"
- ALT="\begin{displaymath}
-AX=B
-\end{displaymath}">
-</DIV>
-<BR CLEAR="ALL">
-<P></P>
-with a positive definite real symmetric or complex Hermitian band matrix,
-given the Cholesky factorization computed by <tt class="function">pbsv()</tt> or
-<tt class="function">pbtrf()</tt>.
-On entry, <var>A</var> contains the triangular factor, as computed by
-<tt class="function">pbsv()</tt> or <tt class="function">pbtrf()</tt>. On exit, <var>B</var> is replaced
-by the solution. <var>B</var> must have the same type as <var>A</var>.
-</dl>
-
-<P>
-The following functions are useful for tridiagonal systems.
-
-<P>
-<dl><dt><table cellpadding="0" cellspacing="0"><tr valign="baseline">
- <td><nobr><b><tt id='l2h-79' xml:id='l2h-79' class="function">ptsv</tt></b>(</nobr></td>
- <td><var>d, e, B</var>)</td></tr></table></dt>
-<dd>
-Solves
-<BR><P></P>
-<DIV ALIGN="CENTER" CLASS="mathdisplay">
-<!-- MATH
- \begin{displaymath}
-A X = B,
-\end{displaymath}
- -->
-
-<IMG
- WIDTH="64" HEIGHT="27" BORDER="0"
- SRC="img41.gif"
- ALT="\begin{displaymath}
-A X = B,
-\end{displaymath}">
-</DIV>
-<BR CLEAR="ALL">
-<P></P>
-where <I>A</I> is an <I>n</I> by <I>n</I> real symmetric or complex
-Hermitian tridiagonal matrix, with diagonal <var>d</var> (a <code>'d'</code> matrix of
-length <I>n</I>) and subdiagonal <var>e</var> (a <code>'d'</code> or <code>'z'</code> matrix of length
-<I>n</I>-1).
-The arguments <var>e</var> and <var>B</var> must have the same type.
-On exit <var>d</var> contains the diagonal elements of <I>D</I> in
-the LDL<SPAN CLASS="MATH"><IMG
- WIDTH="14" HEIGHT="18" ALIGN="BOTTOM" BORDER="0"
- SRC="img50.gif"
- ALT="${}\mathrm{^T}$"></SPAN> or LDL<SPAN CLASS="MATH"><IMG
- WIDTH="14" HEIGHT="18" ALIGN="BOTTOM" BORDER="0"
- SRC="img51.gif"
- ALT="${}\mathrm{^H}$"></SPAN> factorization
-of <I>A</I>, and <var>e</var> contains the subdiagonal elements of the unit
-lower bidiagonal matrix <I>L</I>.
-<var>B</var> is overwritten with the solution <I>X</I>.
-Raises an <code>ArithmeticError</code> if the matrix is singular.
-</dl>
-
-<P>
-<dl><dt><table cellpadding="0" cellspacing="0"><tr valign="baseline">
- <td><nobr><b><tt id='l2h-80' xml:id='l2h-80' class="function">pttrf</tt></b>(</nobr></td>
- <td><var>d, e</var>)</td></tr></table></dt>
-<dd>
-LDL<SPAN CLASS="MATH"><IMG
- WIDTH="14" HEIGHT="18" ALIGN="BOTTOM" BORDER="0"
- SRC="img50.gif"
- ALT="${}\mathrm{^T}$"></SPAN> or LDL<SPAN CLASS="MATH"><IMG
- WIDTH="14" HEIGHT="18" ALIGN="BOTTOM" BORDER="0"
- SRC="img51.gif"
- ALT="${}\mathrm{^H}$"></SPAN> factorization of an <I>n</I> by
-<I>n</I> real symmetric or complex Hermitian tridiagonal matrix <I>A</I>.
-On entry, the argument <var>d</var> is a <code>'d'</code> matrix with the diagonal elements
-of <I>A</I>. The argument <var>e</var> is <code>'d'</code> or <code>'z'</code> matrix with
-the subdiagonal elements of <I>A</I>.
-On exit <var>d</var> contains the diagonal elements of <I>D</I>, and <var>e</var>
-contains the subdiagonal elements of the unit lower bidiagonal matrix
-<I>L</I>. Raises an <code>ArithmeticError</code> if the matrix is singular.
-</dl>
-
-<P>
-<dl><dt><table cellpadding="0" cellspacing="0"><tr valign="baseline">
- <td><nobr><b><tt id='l2h-81' xml:id='l2h-81' class="function">gttrs</tt></b>(</nobr></td>
- <td><var>d, e, B</var><big>[</big><var>, uplo='L'</var><big>]</big><var></var>)</td></tr></table></dt>
-<dd>
-Solves a set of linear equations
-<BR><P></P>
-<DIV ALIGN="CENTER" CLASS="mathdisplay">
-<!-- MATH
- \begin{displaymath}
-AX=B
-\end{displaymath}
- -->
-
-<IMG
- WIDTH="60" HEIGHT="24" BORDER="0"
- SRC="img47.gif"
- ALT="\begin{displaymath}
-AX=B
-\end{displaymath}">
-</DIV>
-<BR CLEAR="ALL">
-<P></P>
-where <I>A</I> is an <I>n</I> by <I>n</I> real symmetric or complex
-Hermitian tridiagonal matrix, given its LDL<SPAN CLASS="MATH"><IMG
- WIDTH="14" HEIGHT="18" ALIGN="BOTTOM" BORDER="0"
- SRC="img50.gif"
- ALT="${}\mathrm{^T}$"></SPAN> or
-LDL<SPAN CLASS="MATH"><IMG
- WIDTH="14" HEIGHT="18" ALIGN="BOTTOM" BORDER="0"
- SRC="img51.gif"
- ALT="${}\mathrm{^H}$"></SPAN> factorization.
-The argument <var>d</var> is the diagonal of the diagonal matrix <I>D</I>.
-The argument <var>uplo</var> only matters for complex matrices.
-If <var>uplo</var> is <code>'L'</code>, then <var>e</var> contains the subdiagonal
-elements of the unit bidiagonal matrix <I>L</I>.
-If <var>uplo</var> is <code>'U'</code>, then <var>e</var> contains the complex
-conjugates of the elements of the unit bidiagonal matrix <I>L</I>.
-On exit, <var>B</var> is overwritten with the solution <I>X</I>.
-<var>B</var> must have the same type as <var>e</var>.
-</dl>
-
-<P>
-
-<DIV CLASS="navigation">
-<div class='online-navigation'>
-<p></p><hr />
-<table align="center" width="100%" cellpadding="0" cellspacing="2">
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- href="node20.html"><img src='previous.gif'
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+++ /dev/null
@@ -1,594 +0,0 @@
-<!DOCTYPE html PUBLIC "-//W3C//DTD HTML 4.0 Transitional//EN">
-<html>
-<head>
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-<H1><A NAME="SECTION0010500000000000000000">
-8.5 Nonlinear Convex Programming</A>
-</H1>
-<dl><dt><table cellpadding="0" cellspacing="0"><tr valign="baseline">
- <td><nobr><b><tt id='l2h-149' xml:id='l2h-149' class="function">cp</tt></b>(</nobr></td>
- <td><var>F</var><big>[</big><var>, G, h</var><big>[</big><var>, A, b</var><big>]</big><var></var><big>]</big><var></var>)</td></tr></table></dt>
-<dd>
-Solves an optimization problem
-<BR>
-<DIV ALIGN="RIGHT" CLASS="mathdisplay">
-
-<!-- MATH
- \begin{equation}
-\begin{array}{ll}
- \mbox{minimize} & f_0(x) \\
- \mbox{subject to} & f_k(x) \leq 0, \quad k=1,\ldots,m \\
- & G x \preceq h \\
- & A x = b,
- \end{array}
-\end{equation}
- -->
-<A NAME="e-nlcp"></A>
-<TABLE WIDTH="100%" ALIGN="CENTER">
-<TR VALIGN="MIDDLE"><TD></TD><TD ALIGN="CENTER" NOWRAP><A NAME="e-nlcp"></A><IMG
- WIDTH="254" HEIGHT="83" BORDER="0"
- SRC="img138.gif"
- ALT="\begin{displaymath}
-\begin{array}{ll}
-\mbox{minimize} & f_0(x) \\
-\mbox{subj...
-... k=1,\ldots,m \\
-& G x \preceq h \\
-& A x = b,
-\end{array}\end{displaymath}"></TD>
-<TD CLASS="eqno" WIDTH=10 ALIGN="RIGHT">
-(8.3)</TD></TR>
-</TABLE>
-<BR CLEAR="ALL"></DIV><P></P>
-with <!-- MATH
- $f=(f_0,\ldots,f_m)$
- -->
-<SPAN CLASS="MATH"><IMG
- WIDTH="117" HEIGHT="32" ALIGN="MIDDLE" BORDER="0"
- SRC="img139.gif"
- ALT="$f=(f_0,\ldots,f_m)$"></SPAN> convex and twice differentiable.
-
-<P>
-<var>F</var> is a function that evaluates the objective and nonlinear
-constraint functions. It must handle the following calling sequences.
-
-<P>
-
-<UL>
-<LI><code>F()</code> returns a tuple (<var>m</var>, <var>x0</var>), where <var>m</var> is
- the number of nonlinear constraints and <var>x0</var> is a point in the
- domain of <I>f</I>. <var>x0</var> is a dense real matrix of size
- (<var>n</var>,1).
-
-<P>
-</LI>
-<LI><code>F(x)</code>, with <var>x</var> a dense real matrix of size
- (<var>n</var>,1), returns a tuple (<var>f</var>, <var>Df</var>).
- <var>f</var> is a dense real matrix of size (<var>m</var>+1,1), with
- <code><var>f</var>[<var>k</var>]</code> equal to <I>f_k(x)</I>.
- (If <I>m</I> is zero, <var>f</var> can also be returned as a number.)
- <var>Df</var> is a dense or sparse real matrix of size (<var>m</var>+1,<var>n</var>)
- with <code><var>Df</var>[<var>k</var>,:]</code> equal to the transpose of the gradient
- of <I>f_k</I> at <I>x</I>.
- If <var>x</var> is not in the domain of <I>f</I>, <code>F(x)</code> returns
- <code>None</code> or a tuple (<code>None</code>,<code>None</code>).
-
-<P>
-</LI>
-<LI><code>F(x,z)</code>, with <var>x</var> a dense real matrix of size
- (<var>n</var>,1) and <var>z</var> a positive dense real matrix of size
- (<var>m</var>+1,1) returns a tuple (<var>f</var>, <var>Df</var>, <var>H</var>).
- <var>f</var> and <var>Df</var> are defined as above.
- <var>H</var> is a square dense or sparse real matrix of size
- (<I>n</I>, <I>n</I>), whose lower triangular part contains the lower
- triangular part of
- <BR><P></P>
-<DIV ALIGN="CENTER" CLASS="mathdisplay">
-<!-- MATH
- \begin{displaymath}
-z_0 \nabla^2f_0(x) + z_1 \nabla^2f_1(x) + \cdots + z_m \nabla^2f_m(x).
-\end{displaymath}
- -->
-
-<IMG
- WIDTH="303" HEIGHT="28" BORDER="0"
- SRC="img140.gif"
- ALT="\begin{displaymath}
-z_0 \nabla^2f_0(x) + z_1 \nabla^2f_1(x) + \cdots + z_m \nabla^2f_m(x).
-\end{displaymath}">
-</DIV>
-<BR CLEAR="ALL">
-<P></P>
-If <var>F</var> is called with two arguments, it can be assumed that
- <var>x</var> is in the domain of <I>f</I>.
-</LI>
-</UL>
-
-<P>
-<var>G</var> and <var>A</var> are dense or sparse real matrices with <var>n</var>
-columns. Their default values are matrices of size (0,<var>n</var>).
-<var>h</var> and <var>b</var> are dense real matrices with one column, and the
-same number of rows as <var>G</var> and <var>A</var>, respectively.
-Their default values are matrices of size (0,1).
-
-<P>
-<tt class="function">cp()</tt> returns a dictionary with keys <code>'status'</code>,
-<code>'x'</code>, <code>'snl'</code>, <code>'sl'</code>, <code>'y'</code>, <code>'znl'</code>,
-<code>'zl'</code>.
-The possible values of the <code>'status'</code> key are:
-<DL>
-<DT><STRONG><code>'optimal'</code></STRONG></DT>
-<DD>In this case the
-<code>'x'</code> entry of the dictionary is the primal optimal solution,
- the <code>'snl'</code> and <code>'sl'</code> entries are the corresponding
- slacks in the nonlinear and linear inequality constraints, and the
-<code>'znl'</code>, <code>'zl'</code> and <code>'y'</code> entries are the optimal
-values of the dual variables associated with the nonlinear
-inequalities, the linear inequalities, and the linear equality
-constraints. These vectors approximately satisfy the Karush-
-Kuhn-Tucker (KKT) conditions
-<BR><P></P>
-<DIV ALIGN="CENTER" CLASS="mathdisplay">
-<!-- MATH
- \begin{displaymath}
-\nabla f_0(x) + D\tilde f(x)^T z_\mathrm{nl} +
- G^T z_\mathrm{l} + A^T y = 0, \qquad
-\tilde f(x) + s_\mathrm{nl} = 0, \quad k=1,\ldots,m, \qquad
- Gx + s_\mathrm{l} = h, \qquad
- Ax = b,
-\end{displaymath}
- -->
-
-<IMG
- WIDTH="701" HEIGHT="28" BORDER="0"
- SRC="img141.gif"
- ALT="\begin{displaymath}
-\nabla f_0(x) + D\tilde f(x)^T z_\mathrm{nl} +
-G^T z_\mat...
-... k=1,\ldots,m, \qquad
-Gx + s_\mathrm{l} = h, \qquad
-Ax = b,
-\end{displaymath}">
-</DIV>
-<BR CLEAR="ALL">
-<P></P>
-where <!-- MATH
- $\tilde f = (f_1,\ldots, f_m)$
- -->
-<SPAN CLASS="MATH"><IMG
- WIDTH="117" HEIGHT="38" ALIGN="MIDDLE" BORDER="0"
- SRC="img142.gif"
- ALT="$\tilde f = (f_1,\ldots, f_m)$"></SPAN>,
-<BR><P></P>
-<DIV ALIGN="CENTER" CLASS="mathdisplay">
-<!-- MATH
- \begin{displaymath}
-s_\mathrm{nl}\succeq 0, \qquad s_\mathrm{l}\succeq 0, \qquad
-z_\mathrm{nl} \succeq 0, \qquad z_\mathrm{l} \succeq 0, \qquad
-s_\mathrm{nl}^T z_\mathrm{nl} + s_\mathrm{l}^T z_\mathrm{l} = 0.
-\end{displaymath}
- -->
-
-<IMG
- WIDTH="453" HEIGHT="28" BORDER="0"
- SRC="img125.gif"
- ALT="\begin{displaymath}
-s_\mathrm{nl}\succeq 0, \qquad s_\mathrm{l}\succeq 0, \qquad...
-...\mathrm{nl}^T z_\mathrm{nl} + s_\mathrm{l}^T z_\mathrm{l} = 0.
-\end{displaymath}">
-</DIV>
-<BR CLEAR="ALL">
-<P></P>
-
-<P>
-</DD>
-<DT><STRONG><code>'unknown'</code></STRONG></DT>
-<DD>This indicates that the algorithm reached
-the maximum number of iterations before a solution was found.
-The <code>'x'</code>, <code>'snl'</code>, <code>'sl'</code>,
-<code>'y'</code>, <code>'znl'</code> and <code>'zl'</code> entries are <code>None</code>.
-</DD>
-</DL>
-
-<P>
-</dl>
-
-<P>
-<DL>
-<DT><STRONG>Example: equality constrained analytic centering</STRONG></DT>
-<DD><P>
-The equality constrained analytic centering problem is defined as
-<BR><P></P>
-<DIV ALIGN="CENTER" CLASS="mathdisplay">
-<!-- MATH
- \begin{displaymath}
-\begin{array}{ll}
- \mbox{minimize} & -\sum_{i=1}^m \log x_i \\
- \mbox{subject to} & Ax = b.
- \end{array}
-\end{displaymath}
- -->
-
-<IMG
- WIDTH="169" HEIGHT="45" BORDER="0"
- SRC="img143.gif"
- ALT="\begin{displaymath}
-\begin{array}{ll}
-\mbox{minimize} & -\sum_{i=1}^m \log x_i \\
-\mbox{subject to} & Ax = b.
-\end{array}\end{displaymath}">
-</DIV>
-<BR CLEAR="ALL">
-<P></P>
-The function <tt class="function">acent()</tt> defined below solves the problem,
-assumping it is solvable.
-
-<P>
-<div class="verbatim"><pre>
-from cvxopt import solvers
-from cvxopt.base import matrix, spmatrix, log
-
-def acent(A, b):
- m, n = A.size
- def F(x=None, z=None):
- if x is None: return 0, matrix(1.0, (n,1))
- if min(x) <= 0.0: return None
- f = -sum(log(x))
- Df = -(x**-1).T
- if z is None: return f, Df
- H = z[0] * spmatrix(x**-2, range(n), range(n))
- return f, Df, H
- return solvers.cp(F, A=A, b=b)['x']
-</pre></div>
-
-<P>
-</DD>
-<DT><STRONG>Example: robust least-squares</STRONG></DT>
-<DD><P>
-The function <tt class="function">robls()</tt> defined below solves the unconstrained
-problem
-<BR><P></P>
-<DIV ALIGN="CENTER" CLASS="mathdisplay">
-<!-- MATH
- \begin{displaymath}
-\begin{array}{ll}
-\mbox{minimize} & \sum_{k=1}^m \phi((Ax-b)_k),
-\end{array} \qquad \mbox{where} \quad A \in{\mbox{\bf R}}^{m\times n}, \quad
-\phi(u) = \sqrt{\rho + u^2}.
-\end{displaymath}
- -->
-
-<IMG
- WIDTH="519" HEIGHT="30" BORDER="0"
- SRC="img144.gif"
- ALT="\begin{displaymath}
-\begin{array}{ll}
-\mbox{minimize} & \sum_{k=1}^m \phi((Ax-b)...
-...{\mbox{\bf R}}^{m\times n}, \quad
-\phi(u) = \sqrt{\rho + u^2}.
-\end{displaymath}">
-</DIV>
-<BR CLEAR="ALL">
-<P></P>
-
-<P>
-<div class="verbatim"><pre>
-from cvxopt import solvers
-from cvxopt.base import matrix, spmatrix, sqrt, div
-
-def robls(A, b, rho):
- m, n = A.size
- def F(x=None, z=None):
- if x is None: return 0, matrix(0.0, (n,1))
- y = A*x-b
- w = sqrt(rho + y**2)
- f = sum(w)
- Df = div(y, w).T * A
- if z is None: return f, Df
- H = A.T * spmatrix(z[0]*rho*(w**-3), range(m), range(m)) * A
- return f, Df, H
- return solvers.cp(F)['x']
-</pre></div>
-
-<P>
-</DD>
-<DT><STRONG>Example: floor planning</STRONG></DT>
-<DD><P>
-This example is the floor planning problem of section 8.8.2 in the book
-<em class="citetitle"><a
- href="http://www.stanford.edu/~boyd/cvxbook"
- title="Convex Optimization"
- >Convex Optimization</a></em>:
-<BR><P></P>
-<DIV ALIGN="CENTER" CLASS="mathdisplay">
-<!-- MATH
- \begin{displaymath}
-\begin{array}{ll}
- \mbox{minimize} & W + H \\
- \mbox{subject to} & A_{\mathrm{min}, k}/h_k - w_k \leq 0,
- \quad k=1,\ldots, 5 \\
- & x_1 \geq 0, \quad x_2 \geq 0, \quad x_4 \geq 0 \\
- & x_1 + w_1 + \rho \leq x_3, \quad x_2 + w_2 + \rho \leq x_3, \quad
- x_3 + w_3 + \rho \leq x_5, \quad x_4 + w_4 + \rho \leq x_5, \quad
- x_5 + w_5 \leq W \\
- & y_2 \geq 0, \quad y_3 \geq 0, \quad y_5 \geq 0 \\
- & y_2 + h_2 + \rho \leq y_1, \quad y_1 + h_1 + \rho \leq y_4, \quad
- y_3 + h_3 + \rho \leq y_4, \quad y_4 + h_4 \leq H, \quad
- y_5 + h_5 \leq H \\
- & h_k/\gamma \leq w_k \leq \gamma h_k, \quad k=1,\ldots,5.
-\end{array}
-\end{displaymath}
- -->
-
-<IMG
- WIDTH="740" HEIGHT="140" BORDER="0"
- SRC="img145.gif"
- ALT="\begin{displaymath}
-\begin{array}{ll}
-\mbox{minimize} & W + H \\
-\mbox{subjec...
-...gamma \leq w_k \leq \gamma h_k, \quad k=1,\ldots,5.
-\end{array}\end{displaymath}">
-</DIV>
-<BR CLEAR="ALL">
-<P></P>
-This problem has 22 variables
-<BR><P></P>
-<DIV ALIGN="CENTER" CLASS="mathdisplay">
-<!-- MATH
- \begin{displaymath}
-W, \qquad H, \qquad x\in{\mbox{\bf R}}^5, \qquad y\in{\mbox{\bf R}}^5, \qquad
-w\in{\mbox{\bf R}}^5, \qquad h\in{\mbox{\bf R}}^5,
-\end{displaymath}
- -->
-
-<IMG
- WIDTH="417" HEIGHT="27" BORDER="0"
- SRC="img146.gif"
- ALT="\begin{displaymath}
-W, \qquad H, \qquad x\in{\mbox{\bf R}}^5, \qquad y\in{\mbox{...
-...}^5, \qquad
-w\in{\mbox{\bf R}}^5, \qquad h\in{\mbox{\bf R}}^5,
-\end{displaymath}">
-</DIV>
-<BR CLEAR="ALL">
-<P></P>
-5 nonlinear inequality constraints, and 26 linear inequality
-constraints. The code belows defines a function
-<tt class="function">floorplan()</tt> that solves the problem by calling
-<tt class="function">cp()</tt>, then applies it to 4 instances, and creates
-a figure.
-
-<P>
-<div class="verbatim"><pre>
-import pylab
-from cvxopt import solvers
-from cvxopt.base import matrix, spmatrix, mul, div
-
-def floorplan(Amin):
-
- # minimize W+H
- # subject to Amink / hk <= wk, k = 1,..., 5
- # x1 >= 0, x2 >= 0, x4 >= 0
- # x1 + w1 + rho <= x3
- # x2 + w2 + rho <= x3
- # x3 + w3 + rho <= x5
- # x4 + w4 + rho <= x5
- # x5 + w5 <= W
- # y2 >= 0, y3 >= 0, y5 >= 0
- # y2 + h2 + rho <= y1
- # y1 + h1 + rho <= y4
- # y3 + h3 + rho <= y4
- # y4 + h4 <= H
- # y5 + h5 <= H
- # hk/gamma <= wk <= gamma*hk, k = 1, ..., 5
- #
- # 22 Variables W, H, x (5), y (5), w (5), h (5).
- #
- # W, H: scalars; bounding box width and height
- # x, y: 5-vectors; coordinates of bottom left corners of blocks
- # w, h: 5-vectors; widths and heigths of the 5 blocks
-
- rho, gamma = 1.0, 5.0 # min spacing, min aspect ratio
-
- # The objective is to minimize W + H. There are five nonlinear
- # constraints
- #
- # -wk + Amink / hk <= 0, k = 1, ..., 5
-
- def F(x=None, z=None):
- if x is None: return 5, matrix(17*[0.0] + 5*[1.0])
- if min(x[17:]) <= 0.0: return None
- f = matrix(0.0, (6,1))
- f[0] = x[0] + x[1]
- f[1:] = -x[12:17] + div(Amin, x[17:])
- Df = matrix(0.0, (6,22))
- Df[0, [0,1]] = 1.0
- Df[1:,12:17] = spmatrix(-1.0, range(5), range(5))
- Df[1:,17:] = spmatrix(-div(Amin, x[17:]**2), range(5), range(5))
- if z is None: return f, Df
- H = spmatrix( 2.0* mul(z[1:], div(Amin, x[17::]**3)), range(17,22), range(17,22) )
- return f, Df, H
-
- G = matrix(0.0, (26,22))
- h = matrix(0.0, (26,1))
- G[0,2] = -1.0 # -x1 <= 0
- G[1,3] = -1.0 # -x2 <= 0
- G[2,5] = -1.0 # -x4 <= 0
- G[3, [2, 4, 12]], h[3] = [1.0, -1.0, 1.0], -rho # x1 - x3 + w1 <= -rho
- G[4, [3, 4, 13]], h[4] = [1.0, -1.0, 1.0], -rho # x2 - x3 + w2 <= -rho
- G[5, [4, 6, 14]], h[5] = [1.0, -1.0, 1.0], -rho # x3 - x5 + w3 <= -rho
- G[6, [5, 6, 15]], h[6] = [1.0, -1.0, 1.0], -rho # x4 - x5 + w4 <= -rho
- G[7, [0, 6, 16]] = -1.0, 1.0, 1.0 # -W + x5 + w5 <= 0
- G[8,8] = -1.0 # -y2 <= 0
- G[9,9] = -1.0 # -y3 <= 0
- G[10,11] = -1.0 # -y5 <= 0
- G[11, [7, 8, 18]], h[11] = [-1.0, 1.0, 1.0], -rho # -y1 + y2 + h2 <= -rho
- G[12, [7, 10, 17]], h[12] = [1.0, -1.0, 1.0], -rho # y1 - y4 + h1 <= -rho
- G[13, [9, 10, 19]], h[13] = [1.0, -1.0, 1.0], -rho # y3 - y4 + h3 <= -rho
- G[14, [1, 10, 20]] = -1.0, 1.0, 1.0 # -H + y4 + h4 <= 0
- G[15, [1, 11, 21]] = -1.0, 1.0, 1.0 # -H + y5 + h5 <= 0
- G[16, [12, 17]] = -1.0, 1.0/gamma # -w1 + h1/gamma <= 0
- G[17, [12, 17]] = 1.0, -gamma # w1 - gamma * h1 <= 0
- G[18, [13, 18]] = -1.0, 1.0/gamma # -w2 + h2/gamma <= 0
- G[19, [13, 18]] = 1.0, -gamma # w2 - gamma * h2 <= 0
- G[20, [14, 18]] = -1.0, 1.0/gamma # -w3 + h3/gamma <= 0
- G[21, [14, 19]] = 1.0, -gamma # w3 - gamma * h3 <= 0
- G[22, [15, 19]] = -1.0, 1.0/gamma # -w4 + h4/gamma <= 0
- G[23, [15, 20]] = 1.0, -gamma # w4 - gamma * h4 <= 0
- G[24, [16, 21]] = -1.0, 1.0/gamma # -w5 + h5/gamma <= 0
- G[25, [16, 21]] = 1.0, -gamma # w5 - gamma * h5 <= 0.0
-
- # solve and return W, H, x, y, w, h
- sol = solvers.cp(F, G, h)
- return sol['x'][0], sol['x'][1], sol['x'][2:7], sol['x'][7:12], sol['x'][12:17], sol['x'][17:]
-
-pylab.figure(facecolor='w')
-pylab.subplot(221)
-Amin = matrix([100., 100., 100., 100., 100.])
-W, H, x, y, w, h = floorplan(Amin)
-for k in xrange(5):
- pylab.fill([x[k], x[k], x[k]+w[k], x[k]+w[k]],
- [y[k], y[k]+h[k], y[k]+h[k], y[k]], '#D0D0D0')
- pylab.text(x[k]+.5*w[k], y[k]+.5*h[k], "%d" %(k+1))
-pylab.axis([-1.0, 26, -1.0, 26])
-pylab.xticks([])
-pylab.yticks([])
-
-pylab.subplot(222)
-Amin = matrix([20., 50., 80., 150., 200.])
-W, H, x, y, w, h = floorplan(Amin)
-for k in xrange(5):
- pylab.fill([x[k], x[k], x[k]+w[k], x[k]+w[k]],
- [y[k], y[k]+h[k], y[k]+h[k], y[k]], '#D0D0D0')
- pylab.text(x[k]+.5*w[k], y[k]+.5*h[k], "%d" %(k+1))
-pylab.axis([-1.0, 26, -1.0, 26])
-pylab.xticks([])
-pylab.yticks([])
-
-pylab.subplot(223)
-Amin = matrix([180., 80., 80., 80., 80.])
-W, H, x, y, w, h = floorplan(Amin)
-for k in xrange(5):
- pylab.fill([x[k], x[k], x[k]+w[k], x[k]+w[k]],
- [y[k], y[k]+h[k], y[k]+h[k], y[k]],'#D0D0D0')
- pylab.text(x[k]+.5*w[k], y[k]+.5*h[k], "%d" %(k+1))
-pylab.axis([-1.0, 26, -1.0, 26])
-pylab.xticks([])
-pylab.yticks([])
-
-pylab.subplot(224)
-Amin = matrix([20., 150., 20., 200., 110.])
-W, H, x, y, w, h = floorplan(Amin)
-for k in xrange(5):
- pylab.fill([x[k], x[k], x[k]+w[k], x[k]+w[k]],
- [y[k], y[k]+h[k], y[k]+h[k], y[k]],'#D0D0D0')
- pylab.text(x[k]+.5*w[k], y[k]+.5*h[k], "%d" %(k+1))
-pylab.axis([-1.0, 26, -1.0, 26])
-pylab.xticks([])
-pylab.yticks([])
-
-pylab.show()
-</pre></div>
-
-<P>
-<DIV ALIGN="CENTER">
-<IMG
- WIDTH="680" HEIGHT="511" ALIGN="BOTTOM" BORDER="0"
- SRC="img147.gif"
- ALT="\includegraphics[width=15cm]{figures/floorplan.eps}">
-
-</DIV>
-</DD>
-</DL>
-
-<P>
-
-<DIV CLASS="navigation">
-<div class='online-navigation'>
-<p></p><hr />
-<table align="center" width="100%" cellpadding="0" cellspacing="2">
-<tr>
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-<span class="release-info">Release 0.8.2, documentation updated on February 6, 2007.</span>
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+++ /dev/null
@@ -1,278 +0,0 @@
-<!DOCTYPE html PUBLIC "-//W3C//DTD HTML 4.0 Transitional//EN">
-<html>
-<head>
-<link rel="STYLESHEET" href="cvxopt.css" type='text/css' />
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-<H1><A NAME="SECTION008200000000000000000">
-6.2 Attributes and Methods</A>
-</H1>
-The following attributes and methods are defined for <tt class="class">spmatrix</tt> objects.
-
-<P>
-<dl><dt><b><tt id='l2h-117' xml:id='l2h-117' class="member">V</tt></b></dt>
-<dd>
-A single-column dense matrix containing the numerical values of the
-nonzero entries in column-major order. Making an assignment to
-the attribute is an efficient way of changing the values of the sparse
-matrix, without changing the sparsity pattern.
-
-<P>
-When the attribute <tt class="member">V</tt> is read, a <em>copy</em> of <tt class="member">V</tt> is
-returned, as a new dense matrix.
-(This implies, for example, that an indexed assignment
-"<tt class="samp">A.V[I] = B</tt>" does not work, or at least
-cannot be used to modify <var>A</var>. Instead the attribute <code>V</code>
-will be read and returned as a new matrix; then the elements of this
-new matrix are modified.)
-</dl>
-
-<P>
-<dl><dt><b><tt id='l2h-118' xml:id='l2h-118' class="member">I</tt></b></dt>
-<dd>
-A single-column integer matrix with the row indices of the entries in
-<code>V</code>. A read-only attribute.
-</dl>
-
-<P>
-<dl><dt><b><tt id='l2h-119' xml:id='l2h-119' class="member">J</tt></b></dt>
-<dd>
-A single-column integer matrix with the column indices of the entries
-in <code>V</code>. A read-only attribute.
-</dl>
-
-<P>
-<dl><dt><b><tt id='l2h-120' xml:id='l2h-120' class="member">size</tt></b></dt>
-<dd>
-A tuple with the dimensions of the matrix. A read-only attribute.
-</dl>
-
-<P>
-<dl><dt><table cellpadding="0" cellspacing="0"><tr valign="baseline">
- <td><nobr><b><tt id='l2h-121' xml:id='l2h-121' class="method">trans</tt></b>(</nobr></td>
- <td><var></var>)</td></tr></table></dt>
-<dd>
-Returns the transpose of a sparse matrix as a new sparse matrix.
-One can also use <code>A.T</code> instead of <code>A.trans()</code>.
-</dl>
-
-<P>
-<dl><dt><table cellpadding="0" cellspacing="0"><tr valign="baseline">
- <td><nobr><b><tt id='l2h-122' xml:id='l2h-122' class="method">ctrans</tt></b>(</nobr></td>
- <td><var></var>)</td></tr></table></dt>
-<dd>
-Returns the complex conjugate transpose of a sparse matrix as a
-new sparse matrix.
-One can also use <code>A.H</code> instead of <code>A.ctrans()</code>.
-</dl>
-
-<P>
-In the following example we take the elementwise square root of
-the matrix
-<BR>
-<DIV ALIGN="RIGHT" CLASS="mathdisplay">
-
-<!-- MATH
- \begin{equation}
-A = \left[ \begin{array}{rrrrr}
- 0 & 2 & 0 & 0 & 3 \\
- 2 & 0 & 0 & 0 & 0 \\
- 1 & 2 & 0 & 4 & 0 \\
- 0 & 0 & 1 & 0 & 0 \end{array} \right]
-\end{equation}
- -->
-<A NAME="e-spA-example"></A>
-<TABLE WIDTH="100%" ALIGN="CENTER">
-<TR VALIGN="MIDDLE"><TD></TD><TD ALIGN="CENTER" NOWRAP><A NAME="e-spA-example"></A><IMG
- WIDTH="168" HEIGHT="83" BORDER="0"
- SRC="img81.gif"
- ALT="\begin{displaymath}
-A = \left[ \begin{array}{rrrrr}
-0 & 2 & 0 & 0 & 3 \\
-2 &...
-...
-1 & 2 & 0 & 4 & 0 \\
-0 & 0 & 1 & 0 & 0 \end{array} \right]
-\end{displaymath}"></TD>
-<TD CLASS="eqno" WIDTH=10 ALIGN="RIGHT">
-(6.2)</TD></TR>
-</TABLE>
-<BR CLEAR="ALL"></DIV><P></P>
-<div class="verbatim"><pre>
->>> from cvxopt.base import sqrt
->>> A = spmatrix([2,1,2,2,1,3,4], [1,2,0,2,3,0,2], [0,0,1,1,2,3,3])
->>> B = spmatrix(sqrt(A.V), A.I, A.J)
->>> print B
-SIZE: (4,4)
-(1, 0) 1.4142e+00
-(2, 0) 1.0000e+00
-(0, 1) 1.4142e+00
-(2, 1) 1.4142e+00
-(3, 2) 1.0000e+00
-(0, 3) 1.7321e+00
-(2, 3) 2.0000e+00
-</pre></div>
-
-<P>
-The next example below illustrates assignments to <tt class="member">V</tt>.
-<div class="verbatim"><pre>
->>> from cvxopt.base import spmatrix, matrix
->>> A = spmatrix(range(5), [0,1,1,2,2], [0,0,1,1,2])
->>> print A
-SIZE: (3,3)
-(0, 0) 0.0000e+00
-(1, 0) 1.0000e+00
-(1, 1) 2.0000e+00
-(2, 1) 3.0000e+00
-(2, 2) 4.0000e+00
->>> B = spmatrix(A.V, A.J, A.I, (4,4)) # transpose and add a zero row and column
->>> print B
-SIZE: (4,4)
-(0, 0) 0.0000e+00
-(0, 1) 1.0000e+00
-(1, 1) 2.0000e+00
-(1, 2) 3.0000e+00
-(2, 2) 4.0000e+00
->>> print matrix(B)
- 0.0000e+00 1.0000e+00 0.0000e+00 0.0000e+00
- 0.0000e+00 2.0000e+00 3.0000e+00 0.0000e+00
- 0.0000e+00 0.0000e+00 4.0000e+00 0.0000e+00
- 0.0000e+00 0.0000e+00 0.0000e+00 0.0000e+00
->>> B.V[:] = 1., 7., 8., 6., 4. # assign new values to nonzero entries
->>> print B
-SIZE: (4,4)
-(0, 0) 1.0000e+00
-(0, 1) 7.0000e+00
-(1, 1) 8.0000e+00
-(1, 2) 6.0000e+00
-(2, 2) 4.0000e+00
->>> B.V += 1.0 # add 1 to the nonzero entries
->>> print B
-SIZE: (4,4)
-(0, 0) 2.0000e+00
-(0, 1) 8.0000e+00
-(1, 1) 9.0000e+00
-(1, 2) 7.0000e+00
-(2, 2) 5.0000e+00
-</pre></div>
-
-<P>
-The <tt class="member">V</tt>, <tt class="member">I</tt> and <tt class="member">J</tt> attributes can be used for
-reading sparse matrices from or writing them to binary files.
-Suppose we want to write the matrix <var>A</var> defined above to a binary
-file.
-<div class="verbatim"><pre>
->>> f = open('test.bin','w')
->>> A.V.tofile(f)
->>> A.I.tofile(f)
->>> A.J.tofile(f)
->>> f.close()
-</pre></div>
-A sparse matrix can be created from this file as follows.
-<div class="verbatim"><pre>
->>> f = open('test.bin','r')
->>> V = matrix(0.0, (5,1)); V.fromfile(f)
->>> I = matrix(0, (5,1)); I.fromfile(f)
->>> J = matrix(0, (5,1)); J.fromfile(f)
->>> B = spmatrix(V, I, J)
->>> print B
-SIZE: (3,3)
-(0, 0) 0.0000e+00
-(1, 0) 1.0000e+00
-(1, 1) 2.0000e+00
-(2, 1) 3.0000e+00
-(2, 2) 4.0000e+00
-</pre></div>
-
-<P>
-Note that the <tt class="module">pickle</tt> module provides a convenient alternative
-to this method.
-
-<P>
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-<hr />
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-
-<table width="100%"><tr valign="top"><td width="50%">
-<dl compact='compact'>
-<dt><a href="s-builtinfuncs.html#l2h-15">abs()</a>,
- <a href="node38.html#l2h-127">[Link]</a>
-<dt><a href="s-lp.html#l2h-170">addconstraint() (lp method)</a>
-</dl>
-</td><td width="50%">
-<dl compact='compact'>
-<dt><a href="s-blas1.html#l2h-30">asum()</a>
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-
-<table width="100%"><tr valign="top"><td width="50%">
-<dl compact='compact'>
-<dt><a href="s-builtinfuncs.html#l2h-12">bool()</a>,
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-<dl compact='compact'>
-</dl>
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-<dl compact='compact'>
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-<table width="100%"><tr valign="top"><td width="50%">
-<dl compact='compact'>
-<dt><a href="e-spA-example.html#l2h-118">I (spmatrix attribute)</a>
-<dt><a href="s-blas1.html#l2h-31">iamax()</a>
-<dt><a href="node31.html#l2h-112">idct()</a>
-<dt><a href="node30.html#l2h-110">idft()</a>
-</dl>
-</td><td width="50%">
-<dl compact='compact'>
-<dt><a href="node32.html#l2h-114">idst()</a>
-<dt><a href="node6.html#l2h-7">imag()</a>
-<dt><a href="s-lp.html#l2h-167">inequalities() (lp method)</a>
-</dl>
-</td>
-</tr></table>
-<hr />
-<h2 id="letter-j">J</h2>
-
-<table width="100%"><tr valign="top"><td width="50%">
-<dl compact='compact'>
-<dt><a href="e-spA-example.html#l2h-119">J (matrix attribute)</a>
-</dl>
-</td><td width="50%">
-<dl compact='compact'>
-</dl>
-</td>
-</tr></table>
-<hr />
-<h2 id="letter-l">L</h2>
-
-<table width="100%"><tr valign="top"><td width="50%">
-<dl compact='compact'>
-<dt><a href="s-builtinfuncs.html#l2h-11">len()</a>,
- <a href="node38.html#l2h-123">[Link]</a>
-<dt><a href="s-umfpack.html#l2h-134">linsolve()</a>,
- <a href="s-cholmod.html#l2h-138">[Link]</a>
-</dl>
-</td><td width="50%">
-<dl compact='compact'>
-<dt><a href="s-otherfuncs.html#l2h-21">log()</a>
-<dt><a href="s-lpsolver.html#l2h-145">lp()</a>
-</dl>
-</td>
-</tr></table>
-<hr />
-<h2 id="letter-m">M</h2>
-
-<table width="100%"><tr valign="top"><td width="50%">
-<dl compact='compact'>
-<dt><a href="s-creating-matrices.html#l2h-1">matrix()</a>
-<dt><a href="node62.html#l2h-174">Matrix\_New()</a>
-<dt><a href="node62.html#l2h-175">Matrix\_NewFromMatrix()</a>
-<dt><a href="node62.html#l2h-176">Matrix\_NewFromSequence()</a>
-</dl>
-</td><td width="50%">
-<dl compact='compact'>
-<dt><a href="s-builtinfuncs.html#l2h-13">max()</a>,
- <a href="node38.html#l2h-125">[Link]</a>
-<dt><a href="s-builtinfuncs.html#l2h-14">min()</a>,
- <a href="node38.html#l2h-126">[Link]</a>
-<dt><a href="s-otherfuncs.html#l2h-22">mul()</a>
-<dt><a href="node58.html#l2h-161">multiplier (constraint attribute)</a>
-</dl>
-</td>
-</tr></table>
-<hr />
-<h2 id="letter-n">N</h2>
-
-<table width="100%"><tr valign="top"><td width="50%">
-<dl compact='compact'>
-<dt><a href="node58.html#l2h-162">name (constraint attribute)</a>
-<dt><a href="s-variables.html#l2h-153">name (variable attribute)</a>
-<dt><a href="node52.html#l2h-151">nlcp()</a>
-</dl>
-</td><td width="50%">
-<dl compact='compact'>
-<dt><a href="s-random.html#l2h-24">normal()</a>
-<dt><a href="s-blas1.html#l2h-29">nrm2()</a>
-<dt><a href="s-umfpack.html#l2h-136">numeric()</a>,
- <a href="s-cholmod.html#l2h-141">[Link]</a>
-</dl>
-</td>
-</tr></table>
-<hr />
-<h2 id="letter-o">O</h2>
-
-<table width="100%"><tr valign="top"><td width="50%">
-<dl compact='compact'>
-<dt><a href="s-lp.html#l2h-164">objective (op attribute)</a>
-<dt><a href="s-lp.html#l2h-163">op (class in )</a>
-</dl>
-</td><td width="50%">
-<dl compact='compact'>
-<dt><a href="s-orderings.html#l2h-133">order()</a>
-<dt><a href="node24.html#l2h-95">ormqr()</a>
-</dl>
-</td>
-</tr></table>
-<hr />
-<h2 id="letter-p">P</h2>
-
-<table width="100%"><tr valign="top"><td width="50%">
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-<dt><a href="e-kkt-example.html#l2h-76">pbsv()</a>
-<dt><a href="e-kkt-example.html#l2h-77">pbtrf()</a>
-<dt><a href="e-kkt-example.html#l2h-78">pbtrs()</a>
-<dt><a href="e-kkt-example.html#l2h-72">posv()</a>
-<dt><a href="e-kkt-example.html#l2h-73">potrf()</a>
-</dl>
-</td><td width="50%">
-<dl compact='compact'>
-<dt><a href="e-kkt-example.html#l2h-75">potri()</a>
-<dt><a href="e-kkt-example.html#l2h-74">potrs()</a>
-<dt><a href="e-kkt-example.html#l2h-79">ptsv()</a>
-<dt><a href="e-kkt-example.html#l2h-80">pttrf()</a>
-<dt><a href="node20.html#l2h-71">pttrs()</a>
-</dl>
-</td>
-</tr></table>
-<hr />
-<h2 id="letter-q">Q</h2>
-
-<table width="100%"><tr valign="top"><td width="50%">
-<dl compact='compact'>
-<dt><a href="node47.html#l2h-146">qp()</a>
-</dl>
-</td><td width="50%">
-<dl compact='compact'>
-</dl>
-</td>
-</tr></table>
-<hr />
-<h2 id="letter-r">R</h2>
-
-<table width="100%"><tr valign="top"><td width="50%">
-<dl compact='compact'>
-<dt><a href="node6.html#l2h-6">real()</a>
-</dl>
-</td><td width="50%">
-<dl compact='compact'>
-</dl>
-</td>
-</tr></table>
-<hr />
-<h2 id="letter-s">S</h2>
-
-<table width="100%"><tr valign="top"><td width="50%">
-<dl compact='compact'>
-<dt><a href="s-blas2.html#l2h-43">sbmv()</a>
-<dt><a href="s-blas1.html#l2h-28">scal()</a>
-<dt><a href="s-sdpsolver.html#l2h-148">sdp()</a>
-<dt><a href="s-random.html#l2h-27">setseed()</a>
-<dt><a href="s-otherfuncs.html#l2h-18">sin()</a>
-<dt><a href="node6.html#l2h-2">size (tuple attribute)</a>,
- <a href="e-spA-example.html#l2h-120">[Link]</a>
-<dt><a href="s-umfpack.html#l2h-137">solve()</a>,
- <a href="s-cholmod.html#l2h-142">[Link]</a>
-<dt><a href="s-lp.html#l2h-171">solve() (op method)</a>
-<dt><a href="s-creating-spmatrix.html#l2h-116">sparse()</a>
-<dt><a href="s-cholmod.html#l2h-139">splinsolve()</a>
-<dt><a href="s-creating-spmatrix.html#l2h-115">spmatrix()</a>
-<dt><a href="node63.html#l2h-177">SpMatrix\_New()</a>
-<dt><a href="node63.html#l2h-179">SpMatrix\_NewFromIJV()</a>
-<dt><a href="node63.html#l2h-178">SpMatrix\_NewFromMatrix()</a>
-<dt><a href="s-cholmod.html#l2h-143">spsolve()</a>
-<dt><a href="s-otherfuncs.html#l2h-17">sqrt()</a>
-<dt><a href="s-builtinfuncs.html#l2h-16">sum()</a>,
- <a href="node38.html#l2h-128">[Link]</a>,
- <a href="s-functions.html#l2h-157">[Link]</a>
-</dl>
-</td><td width="50%">
-<dl compact='compact'>
-<dt><a href="s-blas1.html#l2h-32">swap()</a>
-<dt><a href="node25.html#l2h-97">syev()</a>
-<dt><a href="node25.html#l2h-98">syevd()</a>
-<dt><a href="node25.html#l2h-100">syevr()</a>
-<dt><a href="node25.html#l2h-99">syevx()</a>
-<dt><a href="e-gevd.html#l2h-105">sygv()</a>
-<dt><a href="s-umfpack.html#l2h-135">symbolic()</a>,
- <a href="s-cholmod.html#l2h-140">[Link]</a>
-<dt><a href="s-blas3.html#l2h-54">symm()</a>
-<dt><a href="s-blas2.html#l2h-38">symv()</a>,
- <a href="node39.html#l2h-130">[Link]</a>
-<dt><a href="s-blas2.html#l2h-49">syr()</a>
-<dt><a href="s-blas2.html#l2h-51">syr2()</a>
-<dt><a href="s-blas3.html#l2h-60">syr2k()</a>
-<dt><a href="s-blas3.html#l2h-58">syrk()</a>,
- <a href="node39.html#l2h-132">[Link]</a>
-<dt><a href="node22.html#l2h-82">sysv()</a>
-<dt><a href="node22.html#l2h-83">sytrf()</a>
-<dt><a href="node22.html#l2h-85">sytri()</a>
-<dt><a href="node22.html#l2h-84">sytrs()</a>
-</dl>
-</td>
-</tr></table>
-<hr />
-<h2 id="letter-t">T</h2>
-
-<table width="100%"><tr valign="top"><td width="50%">
-<dl compact='compact'>
-<dt><a href="s-blas2.html#l2h-45">tbmv()</a>
-<dt><a href="s-blas2.html#l2h-46">tbsv()</a>
-<dt><a href="node23.html#l2h-92">tbtrs()</a>
-<dt><a href="node6.html#l2h-9">tofile()</a>
-<dt><a href="s-lp.html#l2h-172">tofile() (op method)</a>
-<dt><a href="node6.html#l2h-4">trans()</a>,
- <a href="e-spA-example.html#l2h-121">[Link]</a>
-<dt><a href="s-blas3.html#l2h-56">trmm()</a>
-</dl>
-</td><td width="50%">
-<dl compact='compact'>
-<dt><a href="s-blas2.html#l2h-40">trmv()</a>
-<dt><a href="s-blas3.html#l2h-57">trsm()</a>
-<dt><a href="s-blas2.html#l2h-41">trsv()</a>
-<dt><a href="node23.html#l2h-91">trtri()</a>
-<dt><a href="node23.html#l2h-90">trtrs()</a>
-<dt><a href="node58.html#l2h-159">type() (constraint method)</a>
-<dt><a href="node6.html#l2h-3">typecode (char attribute)</a>
-</dl>
-</td>
-</tr></table>
-<hr />
-<h2 id="letter-u">U</h2>
-
-<table width="100%"><tr valign="top"><td width="50%">
-<dl compact='compact'>
-<dt><a href="s-random.html#l2h-25">uniform()</a>
-</dl>
-</td><td width="50%">
-<dl compact='compact'>
-<dt><a href="node24.html#l2h-96">unmqr()</a>
-</dl>
-</td>
-</tr></table>
-<hr />
-<h2 id="letter-v">V</h2>
-
-<table width="100%"><tr valign="top"><td width="50%">
-<dl compact='compact'>
-<dt><a href="e-spA-example.html#l2h-117">V (matrix attribute)</a>
-<dt><a href="node58.html#l2h-160">value() (constraint method)</a>
-<dt><a href="s-variables.html#l2h-154">value (variable attribute)</a>
-<dt><a href="s-functions.html#l2h-156">value() (variable method)</a>
-</dl>
-</td><td width="50%">
-<dl compact='compact'>
-<dt><a href="s-variables.html#l2h-152">variable (class in )</a>
-<dt><a href="s-lp.html#l2h-165">variables() (lp method)</a>
-<dt><a href="s-functions.html#l2h-155">variables() (variable method)</a>
-</dl>
-</td>
-</tr></table>
-
-<DIV CLASS="navigation">
-<div class='online-navigation'>
-<p></p><hr />
-<table align="center" width="100%" cellpadding="0" cellspacing="2">
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+mr/m/n/10 ) + [] \OML/cmm/m/it/10 z[]\OMS/cmsy/m/n/10 r\OML/cmm/m/it/10 f[]\OT1
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+/cmr/m/n/10 + \OML/cmm/m/it/10 A[]y \OT1/cmr/m/n/10 = 0\OML/cmm/m/it/10 ; f[]\
+OT1/cmr/m/n/10 (\OML/cmm/m/it/10 x\OT1/cmr/m/n/10 ) + \OML/cmm/m/it/10 s[] \OT1
+/cmr/m/n/10 = 0\OML/cmm/m/it/10 ; k \OT1/cmr/m/n/10 = 1\OML/cmm/m/it/10 ; [] ;
+ m; Gx \OT1/cmr/m/n/10 + \OML/cmm/m/it/10 s[] \OT1/cmr/m/n/10 = \OML/cmm/m/it/
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+[210
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+] (./images.aux)
+
+LaTeX Warning: There were multiply-defined labels.
+
+ )
+Here is how much of TeX's memory you used:
+ 1225 strings out of 94501
+ 15203 string characters out of 1175812
+ 72667 words of memory out of 1000000
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+
+Output written on images.dvi (210 pages, 100432 bytes).
diff --git a/doc/cvxopt/images.pl b/doc/cvxopt/images.pl
new file mode 100644
index 0000000..d4d4cbd
--- /dev/null
+++ b/doc/cvxopt/images.pl
@@ -0,0 +1,1932 @@
+# LaTeX2HTML 2002-2-1 (1.71)
+# Associate images original text with physical files.
+
+
+$key = q/{displaymath}W_{mathrm{q},k}=beta_k(2v_kv_k^T-J),qquadW_{mathrm{q},k}^{-1}=frac{1}{beta_k}(2Jv_kv_k^TJ-J),qquadk=0,ldots,M-1,{displaymath};MSF=1.6;AAT/;
+$cached_env_img{$key} = q|<IMG
+ WIDTH="535" HEIGHT="41" BORDER="0"
+ SRC="|."$dir".q|img146.gif"
+ ALT="\begin{displaymath}
+W_{\mathrm{q},k} = \beta_k ( 2 v_k v_k^T - J),
+\qquad
+W_{...
+...c{1}{\beta_k} ( 2 Jv_k v_k^T J - J),
+\qquad k = 0,\ldots,M-1,
+\end{displaymath}">|;
+
+$key = q/DL^TX=B;MSF=1.6;AAT/;
+$cached_env_img{$key} = q|<IMG
+ WIDTH="87" HEIGHT="16" ALIGN="BOTTOM" BORDER="0"
+ SRC="|."$dir".q|img96.gif"
+ ALT="$DL^TX=B$">|;
+
+$key = q/mathrm{{}^H};MSF=1.6;AAT/;
+$cached_env_img{$key} = q|<IMG
+ WIDTH="15" HEIGHT="18" ALIGN="BOTTOM" BORDER="0"
+ SRC="|."$dir".q|img89.gif"
+ ALT="$\mathrm{{}^H}$">|;
+
+$key = q/{displaymath}PAP^T=LDL^T,qquadPAP^T=LDL^H,{displaymath};MSF=1.6;AAT/;
+$cached_env_img{$key} = q|<IMG
+ WIDTH="272" HEIGHT="27" BORDER="0"
+ SRC="|."$dir".q|img92.gif"
+ ALT="\begin{displaymath}
+PAP^T = LDL^T, \qquad PAP^T = LDL^H,
+\end{displaymath}">|;
+
+$key = q/{displaymath}f(x_1,ldots,x_n)=0,qquadf(x_1,ldots,x_n)preceq0,{displaymath};MSF=1.6;AAT/;
+$cached_env_img{$key} = q|<IMG
+ WIDTH="283" HEIGHT="28" BORDER="0"
+ SRC="|."$dir".q|img199.gif"
+ ALT="\begin{displaymath}
+f(x_1,\ldots,x_n) = 0, \qquad f(x_1,\ldots,x_n) \preceq 0,
+\end{displaymath}">|;
+
+$key = q/{eqnarraystar}&left[array{{ccccc}ReX[0,0]&barX[1,0]&barX[2,0]&cdots&barX[n-1,0]Xdots&ReX[n-1,n-1]array{right]&mbox{ifuplo='U'}.{eqnarraystar};MSF=1.6;TAGS=R;AAT/;
+$cached_env_img{$key} = q|<IMG
+ WIDTH="563" HEIGHT="225" BORDER="0"
+ SRC="|."$dir".q|img2.gif"
+ ALT="\begin{eqnarray*}
+&
+\left[\begin{array}{ccccc}
+\Re X[0,0] & \bar X[1,0] & \bar X...
+...dots & \Re X[n-1,n-1]
+\end{array}\right] & \mbox{if uplo = 'U'}.
+\end{eqnarray*}">|;
+
+$key = q/{eqnarraystar}mbox{DCT-I:}qquadX[k,:]&:=&X[0,:]+(-1)^kX[n-1,:]+2sum_{j=1}^{n-2}X+1slash2)(k+1slash2)slashn),qquadk=0,ldots,n-1.{eqnarraystar};MSF=1.6;TAGS=R;AAT/;
+$cached_env_img{$key} = q|<IMG
+ WIDTH="731" HEIGHT="230" BORDER="0"
+ SRC="|."$dir".q|img76.gif"
+ ALT="\begin{eqnarray*}
+\mbox{DCT-I:} \qquad
+X[k,:] & := & X[0,:] + (-1)^k X[n-1,:] +...
+...n-1} X[j,:] \cos(\pi (j+1/2)(k+1/2)/n),
+\qquad k=0,\ldots,n-1.
+\end{eqnarray*}">|;
+
+$key = q/L^TX=B;MSF=1.6;AAT/;
+$cached_env_img{$key} = q|<IMG
+ WIDTH="74" HEIGHT="16" ALIGN="BOTTOM" BORDER="0"
+ SRC="|."$dir".q|img98.gif"
+ ALT="$L^TX=B$">|;
+
+$key = q/{displaymath}mathrm{gap}=left[array{{c}s_mathrm{nl}s_mathrm{l}array{right]^Tleft+z_mathrm{nl}^Ttildef(x)+z_mathrm{l}^T(Gx-h)+y^T(Ax-b).{displaymath};MSF=1.6;AAT/;
+$cached_env_img{$key} = q|<IMG
+ WIDTH="594" HEIGHT="48" BORDER="0"
+ SRC="|."$dir".q|img193.gif"
+ ALT="\begin{displaymath}
+\mathrm{gap} =
+\left[\begin{array}{c} s_\mathrm{nl} \ s_\...
+...athrm{nl}^T \tilde f(x) +
+z_\mathrm{l}^T (Gx-h) + y^T(Ax-b).
+\end{displaymath}">|;
+
+$key = q/{displaymath}array{{ll}mbox{minimize}&-sum_{i=1}^mlogx_imbox{subjectto}&Ax=b.array{{displaymath};MSF=1.6;AAT/;
+$cached_env_img{$key} = q|<IMG
+ WIDTH="177" HEIGHT="45" BORDER="0"
+ SRC="|."$dir".q|img172.gif"
+ ALT="\begin{displaymath}
+\begin{array}{ll}
+\mbox{minimize} & -\sum_{i=1}^m \log x_i \\\\
+\mbox{subject to} & Ax = b.
+\end{array}\end{displaymath}">|;
+
+$key = q/zinC;MSF=1.6;AAT/;
+$cached_env_img{$key} = q|<IMG
+ WIDTH="45" HEIGHT="30" ALIGN="MIDDLE" BORDER="0"
+ SRC="|."$dir".q|img114.gif"
+ ALT="$z\in C$">|;
+
+$key = q/{displaymath}s^Tzleqepsilon_mathrm{abs}qquadmbox{or}qquadleft(minleft{c^Tx,h^Tz+^Tz}{-min{c^Tx,h^Tz+b^Ty}}leqepsilon_mathrm{rel}right).{displaymath};MSF=1.6;AAT/;
+$cached_env_img{$key} = q|<IMG
+ WIDTH="596" HEIGHT="46" BORDER="0"
+ SRC="|."$dir".q|img160.gif"
+ ALT="\begin{displaymath}
+s^T z \leq \epsilon_\mathrm{abs} \qquad \mbox{or} \qquad
+\l...
+...in\{c^Tx, h^Tz + b^T y\}} \leq \epsilon_\mathrm{rel}
+\right).
+\end{displaymath}">|;
+
+$key = q/{displaymath}array{{ll}mbox{minimize}&|u|_1mbox{subjectto}&|Au-b|_2leq1.array{{displaymath};MSF=1.6;AAT/;
+$cached_env_img{$key} = q|<IMG
+ WIDTH="188" HEIGHT="45" BORDER="0"
+ SRC="|."$dir".q|img158.gif"
+ ALT="\begin{displaymath}
+\begin{array}{ll}
+\mbox{minimize} & \Vert u\Vert _1 \\\\
+\mbox{subject to} & \Vert Au - b\Vert _2 \leq 1.
+\end{array}\end{displaymath}">|;
+
+$key = q/{displaymath}A=left[array{{cccc}1&2&0&03&4&5&06&7&8&90&10&11&12array{right],qquadx=left[array{{c}1111array{right].{displaymath};MSF=1.6;AAT/;
+$cached_env_img{$key} = q|<IMG
+ WIDTH="296" HEIGHT="83" BORDER="0"
+ SRC="|."$dir".q|img45.gif"
+ ALT="\begin{displaymath}
+A = \left[ \begin{array}{cccc}
+1 & 2 & 0 & 0 \\\\
+3 & 4 & ...
+...= \left[\begin{array}{c} 1 \ 1 \ 1 \ 1
+\end{array}\right].
+\end{displaymath}">|;
+
+$key = q/{displaymath}frac{|nablaf_0(x)+Dtildef(x)^Tz_mathrm{nl}+G^Tz_mathrm{l}+A^Ty|_2}{f{1},Gx_0+{{bf{1}-h,Ax_0-b)|_2}}leqepsilon_mathrm{feas}{displaymath};MSF=1.6;AAT/;
+$cached_env_img{$key} = q|<IMG
+ WIDTH="742" HEIGHT="49" BORDER="0"
+ SRC="|."$dir".q|img191.gif"
+ ALT="\begin{displaymath}
+\frac{\Vert \nabla f_0(x) + D\tilde f(x)^Tz_\mathrm{nl} +
+...
+...+ {\bf 1}-h, Ax_0-b) \Vert _2 \}} \leq \epsilon_\mathrm{feas}
+\end{displaymath}">|;
+
+$key = q/{displaymath}array{{ll}mbox{minimize}&f_0(x)mbox{subjectto}&f_k(x)leq0,quadk=1,ldots,m&Gxpreceqh&Ax=b,array{{displaymath};MSF=1.6;AAT/;
+$cached_env_img{$key} = q|<IMG
+ WIDTH="262" HEIGHT="83" BORDER="0"
+ SRC="|."$dir".q|img163.gif"
+ ALT="\begin{displaymath}
+\begin{array}{ll}
+\mbox{minimize} & f_0(x) \\\\
+\mbox{subj...
+... k=1,\ldots,m \\\\
+& G x \preceq h \\\\
+& A x = b,
+\end{array}\end{displaymath}">|;
+
+$key = q/{displaymath}A^Tmbox{{bf{diag},(d)^{-2}Ax_2=b_2+A^Tmbox{{bf{diag},(d)^{-2}b_1,qquadmbox{{bf{diag},(d)^2x_1=Ax_2-b_1.{displaymath};MSF=1.6;AAT/;
+$cached_env_img{$key} = q|<IMG
+ WIDTH="493" HEIGHT="28" BORDER="0"
+ SRC="|."$dir".q|img49.gif"
+ ALT="\begin{displaymath}
+A^T \mbox{\bf diag} (d)^{-2}A x_2 = b_2 + A^T \mbox{\bf di...
+...(d)^{-2} b_1, \qquad
+\mbox{\bf diag} (d)^2 x_1 = Ax_2 - b_1.
+\end{displaymath}">|;
+
+$key = q/{displaymath}array{{ll}mbox{minimize}&sum_kphi((Ax-b)_k),array{qquadphi(u)=left{leq|u|leq3slash22|u|-9slash4&|u|geq3slash2.array{right.{displaymath};MSF=1.6;AAT/;
+$cached_env_img{$key} = q|<IMG
+ WIDTH="507" HEIGHT="64" BORDER="0"
+ SRC="|."$dir".q|img204.gif"
+ ALT="\begin{displaymath}
+\begin{array}{ll}
+\mbox{minimize} & \sum_k \phi((Ax-b)_k)...
+...
+2\vert u\vert-9/4 & \vert u\vert \geq 3/2.\end{array}\right.
+\end{displaymath}">|;
+
+$key = q/{displaymath}nablaf_0(x)+Dtildef(x)^Tz_mathrm{nl}+G^Tz_mathrm{l}+A^Ty=0,qquadtil{nl}=0,quadk=1,ldots,m,qquadGx+s_mathrm{l}=h,qquadAx=b,{displaymath};MSF=1.6;AAT/;
+$cached_env_img{$key} = q|<IMG
+ WIDTH="717" HEIGHT="28" BORDER="0"
+ SRC="|."$dir".q|img166.gif"
+ ALT="\begin{displaymath}
+\nabla f_0(x) + D\tilde f(x)^T z_\mathrm{nl} +
+G^T z_\mat...
+... k=1,\ldots,m, \qquad
+Gx + s_\mathrm{l} = h, \qquad
+Ax = b,
+\end{displaymath}">|;
+
+$key = q/{displaymath}v:=sum_{k=0}^mu_knablaf_k(x)+v.{displaymath};MSF=1.6;AAT/;
+$cached_env_img{$key} = q|<IMG
+ WIDTH="158" HEIGHT="53" BORDER="0"
+ SRC="|."$dir".q|img188.gif"
+ ALT="\begin{displaymath}
+v := \sum_{k=0}^m u_k \nabla f_k(x) + v.
+\end{displaymath}">|;
+
+$key = q/{displaymath}left[array{{ccc}0&A^T&G^TA&0&0G&0&-W^TWarray{right]left[array{{c}u_u_yu_zarray{right]=left[array{{c}b_xb_yb_zarray{right].{displaymath};MSF=1.6;AAT/;
+$cached_env_img{$key} = q|<IMG
+ WIDTH="291" HEIGHT="64" BORDER="0"
+ SRC="|."$dir".q|img140.gif"
+ ALT="\begin{displaymath}
+\left[\begin{array}{ccc}
+0 & A^T & G^T \\\\
+A & 0 & 0 \\\\
+...
+...= \left[\begin{array}{c} b_x \ b_y \ b_z \end{array}\right].
+\end{displaymath}">|;
+
+$key = q/{displaymath}array{{ll}mbox{minimize}&-2x_1+x_2+5x_3*[2ex]mbox{subjectto}&left|lx_3-42array{right]right|_2leq-3x_1+6x_2-10x_3+27.array{{displaymath};MSF=1.6;AAT/;
+$cached_env_img{$key} = q|<IMG
+ WIDTH="498" HEIGHT="140" BORDER="0"
+ SRC="|."$dir".q|img134.gif"
+ ALT="\begin{displaymath}
+\begin{array}{ll}
+\mbox{minimize} & -2x_1 + x_2 + 5x_3 \ *[...
+...t] \right\Vert _2 \leq
+-3x_1 + 6x_2 - 10x_3 + 27.
+\end{array}\end{displaymath}">|;
+
+$key = q/{displaymath}array{{ll}mbox{minimize}&f_0(x)=mathop{{bf{lse}(F_0x+g_0)mbox{subjelse}(F_ix+g_i)leq0,quadi=1,ldots,m&Gxpreceqh&Ax=barray{{displaymath};MSF=1.6;AAT/;
+$cached_env_img{$key} = q|<IMG
+ WIDTH="368" HEIGHT="83" BORDER="0"
+ SRC="|."$dir".q|img183.gif"
+ ALT="\begin{displaymath}
+\begin{array}{ll}
+\mbox{minimize} & f_0(x) = \mathop{\bf lse...
+...,\quad i=1,\ldots,m \\\\
+& Gx \preceq h \\\\
+& Ax=b
+\end{array}\end{displaymath}">|;
+
+$key = q/{displaymath}y:=x.{displaymath};MSF=1.6;AAT/;
+$cached_env_img{$key} = q|<IMG
+ WIDTH="46" HEIGHT="27" BORDER="0"
+ SRC="|."$dir".q|img13.gif"
+ ALT="\begin{displaymath}
+y := x.
+\end{displaymath}">|;
+
+$key = q/sinC;MSF=1.6;AAT/;
+$cached_env_img{$key} = q|<IMG
+ WIDTH="44" HEIGHT="30" ALIGN="MIDDLE" BORDER="0"
+ SRC="|."$dir".q|img113.gif"
+ ALT="$s \in C$">|;
+
+$key = q/{displaymath}mathrm{gap}leqepsilon_mathrm{abs}qquadmbox{or}qquadleft(f_0(x)<0,qufrac{mathrm{gap}}{L(x,y,z)}leqepsilon_mathrm{rel}right){displaymath};MSF=1.6;AAT/;
+$cached_env_img{$key} = q|<IMG
+ WIDTH="686" HEIGHT="45" BORDER="0"
+ SRC="|."$dir".q|img192.gif"
+ ALT="\begin{displaymath}
+\mathrm{gap} \leq \epsilon_\mathrm{abs}
+\qquad \mbox{or} \qq...
+...ac{\mathrm{gap}}
+{L(x,y,z)} \leq \epsilon_\mathrm{rel} \right)
+\end{displaymath}">|;
+
+$key = q/{displaymath}A=left[array{{rrrrr}0&2&0&0&32&0&0&0&01&2&0&4&00&0&1&0&0array{right]{displaymath};MSF=1.6;AAT/;
+$cached_env_img{$key} = q|<IMG
+ WIDTH="168" HEIGHT="83" BORDER="0"
+ SRC="|."$dir".q|img81.gif"
+ ALT="\begin{displaymath}
+A = \left[ \begin{array}{rrrrr}
+0 & 2 & 0 & 0 & 3 \\\\
+2 &...
+...
+1 & 2 & 0 & 4 & 0 \\\\
+0 & 0 & 1 & 0 & 0 \end{array} \right]
+\end{displaymath}">|;
+
+$key = q/{displaymath}C:=alphaAA^T+betaCquad(mathrm{trans}=mathrm{'N'}),qquadC:=alphaA^TA+betaCquad(mathrm{trans}=mathrm{'T'}),{displaymath};MSF=1.6;AAT/;
+$cached_env_img{$key} = q|<IMG
+ WIDTH="501" HEIGHT="28" BORDER="0"
+ SRC="|."$dir".q|img36.gif"
+ ALT="\begin{displaymath}
+C := \alpha AA^T + \beta C \quad (\mathrm{trans} = \mathrm{...
+... \alpha A^TA + \beta C \quad (\mathrm{trans} = \mathrm{'T'}),
+\end{displaymath}">|;
+
+$key = q/{displaymath}(2,1,0),qquad(-1,2,0),qquad(2,0,1),qquad(-2,2,1),qquad(1,3,2),qquad(4,2,3),qquad(3,0,4).{displaymath};MSF=1.6;AAT/;
+$cached_env_img{$key} = q|<IMG
+ WIDTH="612" HEIGHT="28" BORDER="0"
+ SRC="|."$dir".q|img79.gif"
+ ALT="\begin{displaymath}
+(2,1,0), \qquad (-1,2,0), \qquad (2,0,1), \qquad (-2,2,1), \qquad
+(1,3,2), \qquad (4,2,3), \qquad (3,0,4).
+\end{displaymath}">|;
+
+$key = q/{displaymath}array{{ll}mbox{minimize}&|Ax-y|_2^2+|x|_1array{{displaymath};MSF=1.6;AAT/;
+$cached_env_img{$key} = q|<IMG
+ WIDTH="201" HEIGHT="30" BORDER="0"
+ SRC="|."$dir".q|img189.gif"
+ ALT="\begin{displaymath}
+\begin{array}{ll}
+\mbox{minimize} & \Vert Ax - y\Vert _2^2 + \Vert x\Vert _1
+\end{array}\end{displaymath}">|;
+
+$key = q/{displaymath}mathop{{bf{rank}(A)=p,qquadmathop{{bf{rank}(left[array{{c}GAarray{right])=n,{displaymath};MSF=1.6;AAT/;
+$cached_env_img{$key} = q|<IMG
+ WIDTH="257" HEIGHT="45" BORDER="0"
+ SRC="|."$dir".q|img125.gif"
+ ALT="\begin{displaymath}
+\mathop{\bf rank}(A) = p, \qquad
+\mathop{\bf rank}(\left[\begin{array}{c} G \ A \end{array}\right]) = n,
+\end{displaymath}">|;
+
+$key = q/{displaymath}C:=alphaAB^H+baralphaBA^H+betaCquad(mathrm{trans}=mathrm{'N'}),qquaA^HB+baralphaB^HA+betaCquad(mathrm{trans}=mathrm{'C'}),{displaymath};MSF=1.6;AAT/;
+$cached_env_img{$key} = q|<IMG
+ WIDTH="641" HEIGHT="28" BORDER="0"
+ SRC="|."$dir".q|img39.gif"
+ ALT="\begin{displaymath}
+C := \alpha AB^H + \bar \alpha BA^H + \beta C \quad
+(\mat...
+...alpha B^HA + \beta C \quad
+(\mathrm{trans} = \mathrm{'C'}),
+\end{displaymath}">|;
+
+$key = q/{displaymath}C_0={uin{mbox{{bf{R}}^l;|;u_kgeq0,;k=1,ldots,l},qquadC_{k+1}={(u_0,(u);|;uin{mbox{{bf{S}}^{p_k}_+right},quadk=0,ldots,N-1.{displaymath};MSF=1.6;AAT/;
+$cached_env_img{$key} = q|<IMG
+ WIDTH="1115" HEIGHT="30" BORDER="0"
+ SRC="|."$dir".q|img117.gif"
+ ALT="\begin{displaymath}
+C_0 =
+\{ u \in {\mbox{\bf R}}^l \;\vert \; u_k \geq 0, \; ...
+...
+u \in {\mbox{\bf S}}^{p_k}_+ \right\}, \quad k=0,\ldots,N-1.
+\end{displaymath}">|;
+
+$key = q/{displaymath}x=A^{-T}B^{-1}A^{-1}{{bf{1}.{displaymath};MSF=1.6;AAT/;
+$cached_env_img{$key} = q|<IMG
+ WIDTH="132" HEIGHT="24" BORDER="0"
+ SRC="|."$dir".q|img88.gif"
+ ALT="\begin{displaymath}
+x = A^{-T}B^{-1}A^{-1}{\bf 1}.
+\end{displaymath}">|;
+
+$key = q/{displaymath}x^Hy.{displaymath};MSF=1.6;AAT/;
+$cached_env_img{$key} = q|<IMG
+ WIDTH="32" HEIGHT="27" BORDER="0"
+ SRC="|."$dir".q|img15.gif"
+ ALT="\begin{displaymath}
+x^Hy.
+\end{displaymath}">|;
+
+$key = q/PX=B;MSF=1.6;AAT/;
+$cached_env_img{$key} = q|<IMG
+ WIDTH="65" HEIGHT="14" ALIGN="BOTTOM" BORDER="0"
+ SRC="|."$dir".q|img101.gif"
+ ALT="$PX=B$">|;
+
+$key = q/{displaymath}K=l+sum_{k=0}^{M-1}q_k+sum_{k=0}^{N-1}p_k^2.{displaymath};MSF=1.6;AAT/;
+$cached_env_img{$key} = q|<IMG
+ WIDTH="176" HEIGHT="56" BORDER="0"
+ SRC="|."$dir".q|img119.gif"
+ ALT="\begin{displaymath}
+K = l + \sum_{k=0}^{M-1} q_k + \sum_{k=0}^{N-1} p_k^2.
+\end{displaymath}">|;
+
+$key = q/{displaymath}A:=A+alphaxx^T,{displaymath};MSF=1.6;AAT/;
+$cached_env_img{$key} = q|<IMG
+ WIDTH="111" HEIGHT="27" BORDER="0"
+ SRC="|."$dir".q|img26.gif"
+ ALT="\begin{displaymath}
+A := A + \alpha xx^T,
+\end{displaymath}">|;
+
+$key = q/{displaymath}A:=A+alphaxy^H+baralphayx^H,{displaymath};MSF=1.6;AAT/;
+$cached_env_img{$key} = q|<IMG
+ WIDTH="172" HEIGHT="27" BORDER="0"
+ SRC="|."$dir".q|img29.gif"
+ ALT="\begin{displaymath}
+A := A + \alpha xy^H + \bar \alpha yx^H,
+\end{displaymath}">|;
+
+$key = q/mathop{mathbf{vec}}(u);MSF=1.6;AAT/;
+$cached_env_img{$key} = q|<IMG
+ WIDTH="52" HEIGHT="32" ALIGN="MIDDLE" BORDER="0"
+ SRC="|."$dir".q|img118.gif"
+ ALT="$\mathop{\mathbf{vec}}(u)$">|;
+
+$key = q/{displaymath}Gx+s=0,qquadAx=0,qquadc^Tx=-1,qquadssucceq0.{displaymath};MSF=1.6;AAT/;
+$cached_env_img{$key} = q|<IMG
+ WIDTH="351" HEIGHT="27" BORDER="0"
+ SRC="|."$dir".q|img124.gif"
+ ALT="\begin{displaymath}
+Gx + s = 0, \qquad Ax=0, \qquad c^T x = -1, \qquad s \succeq 0.
+\end{displaymath}">|;
+
+$key = q/mathop{mathbf{vec}}(z);MSF=1.6;AAT/;
+$cached_env_img{$key} = q|<IMG
+ WIDTH="51" HEIGHT="32" ALIGN="MIDDLE" BORDER="0"
+ SRC="|."$dir".q|img136.gif"
+ ALT="$\mathop{\mathbf{vec}}(z)$">|;
+
+$key = q/d_1;MSF=1.6;AAT/;
+$cached_env_img{$key} = q|<IMG
+ WIDTH="20" HEIGHT="30" ALIGN="MIDDLE" BORDER="0"
+ SRC="|."$dir".q|img170.gif"
+ ALT="$d_1$">|;
+
+$key = q/{displaymath}AX=B,{displaymath};MSF=1.6;AAT/;
+$cached_env_img{$key} = q|<IMG
+ WIDTH="64" HEIGHT="27" BORDER="0"
+ SRC="|."$dir".q|img41.gif"
+ ALT="\begin{displaymath}
+A X = B,
+\end{displaymath}">|;
+
+$key = q/{displaymath}W_{mathrm{s},k}mathop{mathbf{vec}}{(u_{mathrm{s},k})}=mathop{mathbf{(r_k^{-T}u_{mathrm{s},k}r_k^{-1})},qquadk=0,ldots,N-1.{displaymath};MSF=1.6;AAT/;
+$cached_env_img{$key} = q|<IMG
+ WIDTH="660" HEIGHT="31" BORDER="0"
+ SRC="|."$dir".q|img149.gif"
+ ALT="\begin{displaymath}
+W_{\mathrm{s},k} \mathop{\mathbf{vec}}{(u_{\mathrm{s},k})} ...
+..._k^{-T} u_{\mathrm{s},k} r_k^{-1})}, \qquad
+k = 0,\ldots,N-1.
+\end{displaymath}">|;
+
+$key = q/f;MSF=1.6;AAT/;
+$cached_env_img{$key} = q|<IMG
+ WIDTH="14" HEIGHT="30" ALIGN="MIDDLE" BORDER="0"
+ SRC="|."$dir".q|img200.gif"
+ ALT="$f$">|;
+
+$key = q/{displaymath}K(x)=E_1mbox{{bf{diag},(x)E_2^T+E_2mbox{{bf{diag},(x)E_1^T{displaymath};MSF=1.6;AAT/;
+$cached_env_img{$key} = q|<IMG
+ WIDTH="273" HEIGHT="28" BORDER="0"
+ SRC="|."$dir".q|img107.gif"
+ ALT="\begin{displaymath}
+K(x) = E_1\mbox{\bf diag} (x)E_2^T+E_2\mbox{\bf diag} (x)E_1^T
+\end{displaymath}">|;
+
+$key = q/{displaymath}A:=A+alphaxy^H,{displaymath};MSF=1.6;AAT/;
+$cached_env_img{$key} = q|<IMG
+ WIDTH="113" HEIGHT="27" BORDER="0"
+ SRC="|."$dir".q|img24.gif"
+ ALT="\begin{displaymath}
+A := A + \alpha x y^H,
+\end{displaymath}">|;
+
+$key = q/{displaymath}C:=alphaAB+betaCquad(mathrm{side}=mathrm{'L'}),qquadC:=alphaBA+betaCquad(mathrm{side}=mathrm{'R'}).{displaymath};MSF=1.6;AAT/;
+$cached_env_img{$key} = q|<IMG
+ WIDTH="463" HEIGHT="28" BORDER="0"
+ SRC="|."$dir".q|img33.gif"
+ ALT="\begin{displaymath}
+C := \alpha AB + \beta C \quad (\mathrm{side} = \mathrm{'L'...
+... := \alpha BA + \beta C \quad (\mathrm{side} = \mathrm{'R'}).
+\end{displaymath}">|;
+
+$key = q/{displaymath}W_mathrm{q,k}^T=W_mathrm{q,k}.{displaymath};MSF=1.6;AAT/;
+$cached_env_img{$key} = q|<IMG
+ WIDTH="89" HEIGHT="30" BORDER="0"
+ SRC="|."$dir".q|img148.gif"
+ ALT="\begin{displaymath}
+W_\mathrm{q,k}^T = W_\mathrm{q,k}.
+\end{displaymath}">|;
+
+$key = q/{displaymath}|x|_1quadmbox{({x{real)},qquad|Rex|_1+|Imx|_1quadmbox{({x{complex)}.{displaymath};MSF=1.6;AAT/;
+$cached_env_img{$key} = q|<IMG
+ WIDTH="347" HEIGHT="28" BORDER="0"
+ SRC="|."$dir".q|img10.gif"
+ ALT="\begin{displaymath}
+\Vert x\Vert _1 \quad \mbox{($x$ real)}, \qquad
+\Vert\Re x\Vert _1 + \Vert\Im x\Vert _1 \quad \mbox{($x$ complex)}.
+\end{displaymath}">|;
+
+$key = q/{displaymath}C:=mathop{mathrm{op}}(Q)Cquad(mathrm{side}=mathrm{'L'}),qquadC:=Cma}=mathrm{'T'}Q^H&mathrm{trans}=mathrm{'C'},array{right.{displaymath};MSF=1.6;AAT/;
+$cached_env_img{$key} = q|<IMG
+ WIDTH="663" HEIGHT="64" BORDER="0"
+ SRC="|."$dir".q|img63.gif"
+ ALT="\begin{displaymath}
+C := \mathop{\mathrm{op}}(Q)C \quad (\mathrm{side} = \mathr...
+...} \\\\
+Q^H & \mathrm{trans} = \mathrm{'C'},
+\end{array}\right.
+\end{displaymath}">|;
+
+$key = q/{displaymath}Px=0,qquadq^Tx=-1,qquadGx+s=0,qquadAx=0,qquadssucceq0.{displaymath};MSF=1.6;AAT/;
+$cached_env_img{$key} = q|<IMG
+ WIDTH="442" HEIGHT="27" BORDER="0"
+ SRC="|."$dir".q|img179.gif"
+ ALT="\begin{displaymath}
+Px = 0, \qquad q^Tx = -1, \qquad Gx + s = 0, \qquad Ax=0, \qquad
+s \succeq 0.
+\end{displaymath}">|;
+
+$key = q/{displaymath}array{{ll}mbox{minimize}&(1slash2)x^TPx+q^Txmbox{subjectto}&Gxpreceqh&Ax=b.array{{displaymath};MSF=1.6;AAT/;
+$cached_env_img{$key} = q|<IMG
+ WIDTH="207" HEIGHT="64" BORDER="0"
+ SRC="|."$dir".q|img177.gif"
+ ALT="\begin{displaymath}
+\begin{array}{ll}
+\mbox{minimize} & (1/2) x^TPx + q^T x \\\\
+\mbox{subject to} & Gx \preceq h \ & Ax = b.
+\end{array}\end{displaymath}">|;
+
+$key = q/{displaymath}(2,1,0),qquad(-1,2,0),qquad(0,3,0),qquad(2,0,1),qquad(-2,2,1),qquad(1,3,2),qquad(4,2,3),qquad(3,0,4){displaymath};MSF=1.6;AAT/;
+$cached_env_img{$key} = q|<IMG
+ WIDTH="697" HEIGHT="28" BORDER="0"
+ SRC="|."$dir".q|img80.gif"
+ ALT="\begin{displaymath}
+(2,1,0), \qquad (-1,2,0), \qquad (0,3,0), \qquad (2,0,1), \qquad
+(-2,2,1), \qquad (1,3,2), \qquad (4,2,3), \qquad (3,0,4)
+\end{displaymath}">|;
+
+$key = q/{displaymath}A=LL^Tqquadmbox{or}qquadA=LL^H{displaymath};MSF=1.6;AAT/;
+$cached_env_img{$key} = q|<IMG
+ WIDTH="209" HEIGHT="24" BORDER="0"
+ SRC="|."$dir".q|img46.gif"
+ ALT="\begin{displaymath}
+A = LL^T \qquad \mbox{or} \qquad A = LL^H
+\end{displaymath}">|;
+
+$key = q/{displaymath}A=left[array{{cccc}1&0&0&52&0&4&00&0&0&63&0&0&0array{right]{displaymath};MSF=1.6;AAT/;
+$cached_env_img{$key} = q|<IMG
+ WIDTH="144" HEIGHT="83" BORDER="0"
+ SRC="|."$dir".q|img210.gif"
+ ALT="\begin{displaymath}
+A=\left [\begin{array}{cccc}
+1 & 0 & 0 & 5\\\\
+2 & 0 & 4 & 0\\\\
+0 & 0 & 0 & 6\\\\
+3 & 0 & 0 & 0
+\end{array}\right]
+\end{displaymath}">|;
+
+$key = q/{displaymath}A=USigmaV^T,qquadA=USigmaV^H{displaymath};MSF=1.6;AAT/;
+$cached_env_img{$key} = q|<IMG
+ WIDTH="201" HEIGHT="27" BORDER="0"
+ SRC="|."$dir".q|img69.gif"
+ ALT="\begin{displaymath}
+A = U \Sigma V^T, \qquad A = U \Sigma V^H
+\end{displaymath}">|;
+
+$key = q/{displaymath}A:=A+alphaxx^H,{displaymath};MSF=1.6;AAT/;
+$cached_env_img{$key} = q|<IMG
+ WIDTH="114" HEIGHT="27" BORDER="0"
+ SRC="|."$dir".q|img27.gif"
+ ALT="\begin{displaymath}
+A := A + \alpha xx^H,
+\end{displaymath}">|;
+
+$key = q/{displaymath}C:=alpha(AB^T+BA^T)+betaCquad(mathrm{trans}=mathrm{'N'}),qquadC:=alpha(A^TB+B^TA)+betaCquad(mathrm{trans}=mathrm{'T'}).{displaymath};MSF=1.6;AAT/;
+$cached_env_img{$key} = q|<IMG
+ WIDTH="635" HEIGHT="28" BORDER="0"
+ SRC="|."$dir".q|img38.gif"
+ ALT="\begin{displaymath}
+C := \alpha (AB^T + BA^T) + \beta C \quad
+(\mathrm{trans}...
+...TB + B^TA) + \beta C \quad
+(\mathrm{trans} = \mathrm{'T'}).
+\end{displaymath}">|;
+
+$key = q/{displaymath}A=PLU{displaymath};MSF=1.6;AAT/;
+$cached_env_img{$key} = q|<IMG
+ WIDTH="69" HEIGHT="24" BORDER="0"
+ SRC="|."$dir".q|img42.gif"
+ ALT="\begin{displaymath}
+A = PLU
+\end{displaymath}">|;
+
+$key = q/{displaymath}f(x_1,ldots,x_n)=b+A_1x_1+cdots+A_nx_n+sum_{k=1}^Kmax(y_1,y_2,ldots,y_{m_k}).{displaymath};MSF=1.6;AAT/;
+$cached_env_img{$key} = q|<IMG
+ WIDTH="458" HEIGHT="56" BORDER="0"
+ SRC="|."$dir".q|img196.gif"
+ ALT="\begin{displaymath}
+f(x_1,\ldots,x_n) = b + A_1 x_1 + \cdots + A_n x_n +
+\sum_{k=1}^K \max (y_1, y_2, \ldots, y_{m_k}).
+\end{displaymath}">|;
+
+$key = q/{displaymath}array{{ll}mbox{minimize}&x_1-x_2+x_3mbox{subjectto}&x_1left[array{{eft[array{{ccc}14&9&409&91&1040&10&15array{right]array{{displaymath};MSF=1.6;AAT/;
+$cached_env_img{$key} = q|<IMG
+ WIDTH="734" HEIGHT="121" BORDER="0"
+ SRC="|."$dir".q|img139.gif"
+ ALT="\begin{displaymath}
+\begin{array}{ll}
+\mbox{minimize} & x_1 - x_2 + x_3 \\\\
+\mbo...
+...
+9 & 91 & 10 \\\\
+40 & 10 & 15
+\end{array} \right]
+\end{array}\end{displaymath}">|;
+
+$key = q/{displaymath}W,qquadH,qquadxin{mbox{{bf{R}}^5,qquadyin{mbox{{bf{R}}^5,qquadwin{mbox{{bf{R}}^5,qquadhin{mbox{{bf{R}}^5,{displaymath};MSF=1.6;AAT/;
+$cached_env_img{$key} = q|<IMG
+ WIDTH="425" HEIGHT="27" BORDER="0"
+ SRC="|."$dir".q|img175.gif"
+ ALT="\begin{displaymath}
+W, \qquad H, \qquad x\in{\mbox{\bf R}}^5, \qquad y\in{\mbox{...
+...}^5, \qquad
+w\in{\mbox{\bf R}}^5, \qquad h\in{\mbox{\bf R}}^5,
+\end{displaymath}">|;
+
+$key = q/{displaymath}y:=alphaGx+betayquad(mathrm{trans}=mathrm{'N'}),qquady:=alphaG^Tx+betayquad(mathrm{trans}=mathrm{'T'}).{displaymath};MSF=1.6;AAT/;
+$cached_env_img{$key} = q|<IMG
+ WIDTH="470" HEIGHT="28" BORDER="0"
+ SRC="|."$dir".q|img153.gif"
+ ALT="\begin{displaymath}
+y := \alpha Gx + \beta y \quad
+(\mathrm{trans} = \mathrm{'...
+...alpha G^T x + \beta y \quad
+(\mathrm{trans} = \mathrm{'T'}).
+\end{displaymath}">|;
+
+$key = q/{eqnarraystar}f(x,y)&=&left[array{{c}22array{right]x+y+left[array{{c}33array{rig&5array{right]y+left[array{{c}1317array{right].{eqnarraystar};MSF=1.6;TAGS=R;AAT/;
+$cached_env_img{$key} = q|<IMG
+ WIDTH="366" HEIGHT="134" BORDER="0"
+ SRC="|."$dir".q|img195.gif"
+ ALT="\begin{eqnarray*}
+f(x,y) & = & \left[ \begin{array}{c} 2 \ 2 \end{array}\right...
+...\right] y
++ \left[ \begin{array}{c} 13 \ 17\end{array}\right].
+\end{eqnarray*}">|;
+
+$key = q/M;MSF=1.6;AAT/;
+$cached_env_img{$key} = q|<IMG
+ WIDTH="22" HEIGHT="14" ALIGN="BOTTOM" BORDER="0"
+ SRC="|."$dir".q|img133.gif"
+ ALT="$M$">|;
+
+$key = q/{displaymath}C:=mathop{mathrm{op}}(Q)Cquad(mathrm{side}=mathrm{'L'}),qquadC:=Cma}=mathrm{'N'}Q^T&mathrm{trans}=mathrm{'T'},array{right.{displaymath};MSF=1.6;AAT/;
+$cached_env_img{$key} = q|<IMG
+ WIDTH="659" HEIGHT="45" BORDER="0"
+ SRC="|."$dir".q|img62.gif"
+ ALT="\begin{displaymath}
+C := \mathop{\mathrm{op}}(Q)C \quad (\mathrm{side} = \mathr...
+...} \\\\
+Q^T & \mathrm{trans} = \mathrm{'T'},
+\end{array}\right.
+\end{displaymath}">|;
+
+$key = q/{displaymath}array{{ll}mbox{minimize}&c^Txmbox{subjectto}&sup_{|v|_inftyleq1}(a_i+v)^Txleqb_i,qquadi=1,ldots,marray{{displaymath};MSF=1.6;AAT/;
+$cached_env_img{$key} = q|<IMG
+ WIDTH="387" HEIGHT="46" BORDER="0"
+ SRC="|."$dir".q|img206.gif"
+ ALT="\begin{displaymath}
+\begin{array}{ll}
+\mbox{minimize} & c^T x \\\\
+\mbox{subje...
+...leq 1}
+(a_i+v)^T x \leq b_i, \qquad i=1,\ldots,m
+\end{array}\end{displaymath}">|;
+
+$key = q/{displaymath}Px+q+G^Tz+A^Ty=0,qquadGx+s=h,qquadAx=b,qquadssucceq0,qquadzsucceq0,qquads^Tz=0.{displaymath};MSF=1.6;AAT/;
+$cached_env_img{$key} = q|<IMG
+ WIDTH="632" HEIGHT="27" BORDER="0"
+ SRC="|."$dir".q|img178.gif"
+ ALT="\begin{displaymath}
+Px + q + G^T z + A^T y = 0, \qquad Gx + s = h, \qquad
+Ax = b, \qquad s \succeq 0, \qquad z \succeq 0, \qquad s^T z = 0.
+\end{displaymath}">|;
+
+$key = q/{displaymath}array{[t]{ll}mbox{minimize}&c^Txmbox{subjectto}&G_0x+s_0=h_0&G_kx+m)+A^Ty+c=0&z_0succeq0&z_ksucceq0,quadk=1,ldots,N.array{{displaymath};MSF=1.6;AAT/;
+$cached_env_img{$key} = q|<IMG
+ WIDTH="828" HEIGHT="125" BORDER="0"
+ SRC="|."$dir".q|img135.gif"
+ ALT="\begin{displaymath}
+\begin{array}[t]{ll}
+\mbox{minimize} & c^T x \\\\
+\mbox{su...
+...succeq 0 \\\\
+& z_k \succeq 0, \quad k=1,\ldots,N.
+\end{array}\end{displaymath}">|;
+
+$key = q/{eqnarraystar}&left[array{{ccccc}X[0,0]&X[1,0]&X[2,0]&cdots&X[n-1,0]X[1,0]&X[1,1]&cdots&X[n-1,n-1]array{right]&mbox{ifuplo=U'}.{eqnarraystar};MSF=1.6;TAGS=R;AAT/;
+$cached_env_img{$key} = q|<IMG
+ WIDTH="550" HEIGHT="225" BORDER="0"
+ SRC="|."$dir".q|img1.gif"
+ ALT="\begin{eqnarray*}
+& \left[\begin{array}{ccccc}
+X[0,0] & X[1,0] & X[2,0] & \cdots...
+...& \cdots & X[n-1,n-1]
+\end{array}\right] & \mbox{if uplo = U'}.
+\end{eqnarray*}">|;
+
+$key = q/f=(f_0,ldots,f_m);MSF=1.6;AAT/;
+$cached_env_img{$key} = q|<IMG
+ WIDTH="118" HEIGHT="32" ALIGN="MIDDLE" BORDER="0"
+ SRC="|."$dir".q|img164.gif"
+ ALT="$f=(f_0,\ldots,f_m)$">|;
+
+$key = q/{displaymath}s_mathrm{nl}succeq0,qquads_mathrm{l}succeq0,qquadz_mathrm{nl}succeqds_mathrm{nl}^Tz_mathrm{nl}+s_mathrm{l}^Tz_mathrm{l}=0.{displaymath};MSF=1.6;AAT/;
+$cached_env_img{$key} = q|<IMG
+ WIDTH="452" HEIGHT="28" BORDER="0"
+ SRC="|."$dir".q|img168.gif"
+ ALT="\begin{displaymath}
+s_\mathrm{nl}\succeq 0, \qquad s_\mathrm{l}\succeq 0, \qquad...
+...\mathrm{nl}^T z_\mathrm{nl} + s_\mathrm{l}^T z_\mathrm{l} = 0.
+\end{displaymath}">|;
+
+$key = q/{displaymath}y:=alphaAx+betayquad(mathrm{trans}=mathrm{'N'}),qquady:=alphaA^Tx+bqquady:=alphaA^Hx+betayquad(mathrm{trans}=mathrm{'C'}).{displaymath};MSF=1.6;AAT/;
+$cached_env_img{$key} = q|<IMG
+ WIDTH="716" HEIGHT="28" BORDER="0"
+ SRC="|."$dir".q|img17.gif"
+ ALT="\begin{displaymath}
+y := \alpha Ax + \beta y \quad (\mathrm{trans} = \mathrm{'N'...
+...\alpha A^H x + \beta y \quad (\mathrm{trans} = \mathrm{'C'}).
+\end{displaymath}">|;
+
+$key = q/{displaymath}X[k,:]:=sum_{j=0}^{n-1}e^{-2pijksqrt{-1}slashn}X[j,:],qquadk=0,ldots,n-1.{displaymath};MSF=1.6;AAT/;
+$cached_env_img{$key} = q|<IMG
+ WIDTH="376" HEIGHT="58" BORDER="0"
+ SRC="|."$dir".q|img74.gif"
+ ALT="\begin{displaymath}
+X[k,:] := \sum_{j=0}^{n-1} e^{-2\pi j k \sqrt{-1}/n} X[j,:],
+\qquad k=0,\ldots,n-1.
+\end{displaymath}">|;
+
+$key = q/{displaymath}array{{ll}mbox{minimize}&W+Hmbox{subjectto}&A_{mathrm{min},k}slashhH&h_kslashgammaleqw_kleqgammah_k,quadk=1,ldots,5.array{{displaymath};MSF=1.6;AAT/;
+$cached_env_img{$key} = q|<IMG
+ WIDTH="747" HEIGHT="140" BORDER="0"
+ SRC="|."$dir".q|img174.gif"
+ ALT="\begin{displaymath}
+\begin{array}{ll}
+\mbox{minimize} & W + H \\\\
+\mbox{subjec...
+...gamma \leq w_k \leq \gamma h_k, \quad k=1,\ldots,5.
+\end{array}\end{displaymath}">|;
+
+$key = q/{displaymath}B=left[array{{rrrrr}4&3&0&0&03&0&4&0&60&-1&-3&2&00&0&1&0&00&4&2&0&2array{right],{displaymath};MSF=1.6;AAT/;
+$cached_env_img{$key} = q|<IMG
+ WIDTH="206" HEIGHT="102" BORDER="0"
+ SRC="|."$dir".q|img87.gif"
+ ALT="\begin{displaymath}
+B = \left[\begin{array}{rrrrr}
+4 & 3 & 0 & 0 & 0 \\\\
+3 & 0...
+...0 & 0 & 1 & 0 & 0 \\\\
+0 & 4 & 2 & 0 & 2
+\end{array}\right],
+\end{displaymath}">|;
+
+$key = q/{displaymath}array{{ll}mbox{minimize}&-6x_1-4x_2-5x_3*[1ex]mbox{subjectto}&16x_1ay{{ccc}68&-30&-19-30&99&23-19&23&10array{right].array{{displaymath};MSF=1.6;AAT/;
+$cached_env_img{$key} = q|<IMG
+ WIDTH="882" HEIGHT="199" BORDER="0"
+ SRC="|."$dir".q|img126.gif"
+ ALT="\begin{displaymath}
+\begin{array}{ll}
+\mbox{minimize} & -6x_1 - 4x_2 - 5x_3 \ *...
+...0 & 99 & 23 \\\\
+-19 & 23 & 10 \end{array}\right].
+\end{array}
+\end{displaymath}">|;
+
+$key = q/{displaymath}array{{ll}mbox{minimize}&c^Txmbox{subjectto}&a_i^Tx+|x|_1leqb_i,qquadi=1,ldots,m.array{{displaymath};MSF=1.6;AAT/;
+$cached_env_img{$key} = q|<IMG
+ WIDTH="324" HEIGHT="45" BORDER="0"
+ SRC="|."$dir".q|img207.gif"
+ ALT="\begin{displaymath}
+\begin{array}{ll}
+\mbox{minimize} & c^Tx \\\\
+\mbox{subjec...
+...x + \Vert x\Vert _1 \leq b_i, \qquad i=1,\ldots,m.
+\end{array}\end{displaymath}">|;
+
+$key = q/LDL^TX=B;MSF=1.6;AAT/;
+$cached_env_img{$key} = q|<IMG
+ WIDTH="98" HEIGHT="16" ALIGN="BOTTOM" BORDER="0"
+ SRC="|."$dir".q|img94.gif"
+ ALT="$LDL^TX=B$">|;
+
+$key = q/{displaymath}C=A^TD,qquadD=left[array{{ccc}0&1&03&0&-20&1&04&0&0array{right].{displaymath};MSF=1.6;AAT/;
+$cached_env_img{$key} = q|<IMG
+ WIDTH="253" HEIGHT="83" BORDER="0"
+ SRC="|."$dir".q|img84.gif"
+ ALT="\begin{displaymath}
+C = A^TD, \qquad
+D = \left[ \begin{array}{ccc}
+0 & 1 & 0 \ 3 & 0 & -2 \ 0 & 1 & 0 \ 4 & 0 & 0
+\end{array}\right].
+\end{displaymath}">|;
+
+$key = q/{displaymath}array{{ll}mbox{minimize}&c^Txmbox{subjectto}&a_i^Tx+{{bf{1}^Tyleqb_i,qquadi=1,ldots,m&-ypreceqxpreceqyarray{{displaymath};MSF=1.6;AAT/;
+$cached_env_img{$key} = q|<IMG
+ WIDTH="316" HEIGHT="64" BORDER="0"
+ SRC="|."$dir".q|img208.gif"
+ ALT="\begin{displaymath}
+\begin{array}{ll}
+\mbox{minimize} & c^Tx \\\\
+\mbox{subjec...
+...i, \qquad i=1,\ldots,m \\\\
+& -y \preceq x \preceq y
+\end{array}\end{displaymath}">|;
+
+$key = q/{displaymath}array{{ll}mbox{minimize}&|A^TX-B|_F.array{{displaymath};MSF=1.6;AAT/;
+$cached_env_img{$key} = q|<IMG
+ WIDTH="177" HEIGHT="30" BORDER="0"
+ SRC="|."$dir".q|img58.gif"
+ ALT="\begin{displaymath}
+\begin{array}{ll}
+\mbox{minimize} & \Vert A^TX-B\Vert _F.
+\end{array}\end{displaymath}">|;
+
+$key = q/{displaymath}A=Vmbox{{bf{diag},(lambda)V^H,qquadV^HV=I.{displaymath};MSF=1.6;AAT/;
+$cached_env_img{$key} = q|<IMG
+ WIDTH="237" HEIGHT="28" BORDER="0"
+ SRC="|."$dir".q|img65.gif"
+ ALT="\begin{displaymath}
+A = V\mbox{\bf diag} (\lambda)V^H,\qquad V^HV = I.
+\end{displaymath}">|;
+
+$key = q/{displaymath}left[array{{cc}-mbox{{bf{diag},(d)^2&AA^T&0array{right]left[array{{c}x_1x_2array{right]=left[array{{c}b_1b_2array{right]{displaymath};MSF=1.6;AAT/;
+$cached_env_img{$key} = q|<IMG
+ WIDTH="248" HEIGHT="45" BORDER="0"
+ SRC="|."$dir".q|img48.gif"
+ ALT="\begin{displaymath}
+\left[ \begin{array}{cc}
+-\mbox{\bf diag} (d)^2 & A \\\\
+...
+...ght]
+= \left[ \begin{array}{c} b_1 \ b_2 \end{array} \right]
+\end{displaymath}">|;
+
+$key = q/{eqnarraystar}&left[array{{ccccccc}ReX[0,0]&barX[1,0]&barX[2,0]&cdots&barX[k,0]&ts&vdots&ddots&&&array{right]&mbox{ifuplo='U'}.{eqnarraystar};MSF=1.6;TAGS=R;AAT/;
+$cached_env_img{$key} = q|<IMG
+ WIDTH="627" HEIGHT="323" BORDER="0"
+ SRC="|."$dir".q|img6.gif"
+ ALT="\begin{eqnarray*}
+& \left[ \begin{array}{ccccccc}
+\Re X[0,0] & \bar X[1,0] & \ba...
+... \vdots & \ddots & & &
+\end{array}\right] & \mbox{if uplo='U'}.
+\end{eqnarray*}">|;
+
+$key = q/{displaymath}s_k=(s_{k0},s_{k1})in{mbox{{bf{R}}times{mbox{{bf{R}}^{q_{k}-1},qquak0},z_{k1})in{mbox{{bf{R}}times{mbox{{bf{R}}^{q_{k}-1}.{displaymath};MSF=1.6;AAT/;
+$cached_env_img{$key} = q|<IMG
+ WIDTH="433" HEIGHT="28" BORDER="0"
+ SRC="|."$dir".q|img130.gif"
+ ALT="\begin{displaymath}
+s_k = (s_{k0}, s_{k1}) \in{\mbox{\bf R}}\times{\mbox{\bf R}...
+...{k0}, z_{k1}) \in{\mbox{\bf R}}\times{\mbox{\bf R}}^{q_{k}-1}.
+\end{displaymath}">|;
+
+$key = q/{displaymath}Wu=left(W_mathrm{l}u_mathrm{l},;W_{mathrm{q},0}u_{mathrm{q},0},;ldom{s},N-1}mathop{mathbf{vec}}{(u_{mathrm{s},N-1})}right){displaymath};MSF=1.6;AAT/;
+$cached_env_img{$key} = q|<IMG
+ WIDTH="608" HEIGHT="29" BORDER="0"
+ SRC="|."$dir".q|img143.gif"
+ ALT="\begin{displaymath}
+Wu = \left( W_\mathrm{l} u_\mathrm{l}, \;
+W_{\mathrm{q},0}...
+...rm{s},N-1} \mathop{\mathbf{vec}}{(u_{\mathrm{s},N-1})} \right)
+\end{displaymath}">|;
+
+$key = q/{displaymath}array{{ll}mbox{minimize}&|Ax-y|_2^2+{{bf{1}^Tumbox{subjectto}&-upreceqxprecequarray{{displaymath};MSF=1.6;AAT/;
+$cached_env_img{$key} = q|<IMG
+ WIDTH="205" HEIGHT="45" BORDER="0"
+ SRC="|."$dir".q|img190.gif"
+ ALT="\begin{displaymath}
+\begin{array}{ll}
+\mbox{minimize} & \Vert Ax - y\Vert _2^2...
+... u \\\\
+\mbox{subject to} & -u \preceq x \preceq u
+\end{array}\end{displaymath}">|;
+
+$key = q/{displaymath}AZ=BZmbox{{bf{diag},(lambda)quadmbox{(type1)},qquadABZ=Zmbox{{bf{di2)},qquadBAZ=Zmbox{{bf{diag},(lambda)quadmbox{(type3)},{displaymath};MSF=1.6;AAT/;
+$cached_env_img{$key} = q|<IMG
+ WIDTH="681" HEIGHT="28" BORDER="0"
+ SRC="|."$dir".q|img66.gif"
+ ALT="\begin{displaymath}
+AZ = BZ\mbox{\bf diag} (\lambda)\quad \mbox{(type 1)}, \qq...
+...ad
+BAZ = Z\mbox{\bf diag} (\lambda) \quad \mbox{(type 3)},
+\end{displaymath}">|;
+
+$key = q/{displaymath}PAP^T=LDL^H{displaymath};MSF=1.6;AAT/;
+$cached_env_img{$key} = q|<IMG
+ WIDTH="115" HEIGHT="24" BORDER="0"
+ SRC="|."$dir".q|img54.gif"
+ ALT="\begin{displaymath}
+PAP^T = LDL^H
+\end{displaymath}">|;
+
+$key = q/{displaymath}A=left[array{{rrrr}1&6&0&02&-4&3&00&-3&-1&1array{right]{displaymath};MSF=1.6;AAT/;
+$cached_env_img{$key} = q|<IMG
+ WIDTH="169" HEIGHT="64" BORDER="0"
+ SRC="|."$dir".q|img30.gif"
+ ALT="\begin{displaymath}
+A = \left[\begin{array}{rrrr}
+1 & 6 & 0 & 0 \\\\
+2 & -4 & 3 & 0 \\\\
+0 & -3 & -1 & 1
+\end{array}\right]
+\end{displaymath}">|;
+
+$key = q/{displaymath}array{{ll}mbox{minimize}&w^{-1}h^{-1}d^{-1}mbox{subjectto}&(2slashA-1}leq1&gammawd^{-1}leq1&(1slashdelta)dw^{-1}leq1array{{displaymath};MSF=1.6;AAT/;
+$cached_env_img{$key} = q|<IMG
+ WIDTH="297" HEIGHT="140" BORDER="0"
+ SRC="|."$dir".q|img186.gif"
+ ALT="\begin{displaymath}
+\begin{array}{ll}
+\mbox{minimize} & w^{-1} h^{-1} d^{-1} \...
+...mma wd^{-1} \leq 1 \\\\
+& (1/\delta)dw^{-1} \leq 1
+\end{array}\end{displaymath}">|;
+
+$key = q/{displaymath}array{{ll}mbox{minimize}&|x|_1+{{bf{1}^Tumbox{subjectto}&Axsucceq{{bf{1}-u&usucceq0.array{{displaymath};MSF=1.6;AAT/;
+$cached_env_img{$key} = q|<IMG
+ WIDTH="166" HEIGHT="64" BORDER="0"
+ SRC="|."$dir".q|img209.gif"
+ ALT="\begin{displaymath}
+\begin{array}{ll}
+\mbox{minimize} & \Vert x\Vert _1 + {\bf 1...
+...bject to} & Ax \succeq {\bf 1}-u \\\\
+& u \succeq 0.
+\end{array}\end{displaymath}">|;
+
+$key = q/{displaymath}W_{mathrm{s},k}^Tmathop{mathbf{vec}}{(u_{mathrm{s},k})}=mathop{math^{-1}u_{mathrm{s},k}r_k^{-T})},qquadqquadk=0,ldots,N-1.{displaymath};MSF=1.6;AAT/;
+$cached_env_img{$key} = q|<IMG
+ WIDTH="719" HEIGHT="31" BORDER="0"
+ SRC="|."$dir".q|img150.gif"
+ ALT="\begin{displaymath}
+W_{\mathrm{s},k}^T \mathop{\mathbf{vec}}{(u_{\mathrm{s},k})...
+...u_{\mathrm{s},k} r_k^{-T})}, \qquad
+\qquad
+k = 0,\ldots,N-1.
+\end{displaymath}">|;
+
+$key = q/{displaymath}C=A^TB,qquadA=left[array{{ccc}0&1&01&0&10&1&01&0&0array{right],qquadB=left[array{{ccc}0&-1&02&0&20&3&02&0&0array{right].{displaymath};MSF=1.6;AAT/;
+$cached_env_img{$key} = q|<IMG
+ WIDTH="419" HEIGHT="83" BORDER="0"
+ SRC="|."$dir".q|img83.gif"
+ ALT="\begin{displaymath}
+C = A^TB, \qquad
+A = \left[ \begin{array}{ccc}
+0 & 1 & 0 \ ...
+...0 \ 2 & 0 & 2 \ 0 & 3 & 0 \ 2 & 0 & 0
+\end{array}\right].
+\end{displaymath}">|;
+
+$key = q/{displaymath}A^Tmbox{{bf{diag},(b-Ax)^{-2}Av=-mbox{{bf{diag},(b-Ax)^{-1}{{bf{1}{displaymath};MSF=1.6;AAT/;
+$cached_env_img{$key} = q|<IMG
+ WIDTH="309" HEIGHT="28" BORDER="0"
+ SRC="|."$dir".q|img72.gif"
+ ALT="\begin{displaymath}
+A^T \mbox{\bf diag} (b-Ax)^{-2} A v = -\mbox{\bf diag} (b-Ax)^{-1}{\bf 1}
+\end{displaymath}">|;
+
+$key = q/DX=B;MSF=1.6;AAT/;
+$cached_env_img{$key} = q|<IMG
+ WIDTH="67" HEIGHT="14" ALIGN="BOTTOM" BORDER="0"
+ SRC="|."$dir".q|img99.gif"
+ ALT="$DX=B$">|;
+
+$key = q/{}mathrm{^T};MSF=1.6;AAT/;
+$cached_env_img{$key} = q|<IMG
+ WIDTH="14" HEIGHT="18" ALIGN="BOTTOM" BORDER="0"
+ SRC="|."$dir".q|img50.gif"
+ ALT="${}\mathrm{^T}$">|;
+
+$key = q/includegraphics[width=15cm]{figuresslashfloorplan.eps};FSF=1.6;AAT/;
+$cached_env_img{$key} = q|<IMG
+ WIDTH="681" HEIGHT="511" ALIGN="BOTTOM" BORDER="0"
+ SRC="|."$dir".q|img176.gif"
+ ALT="\includegraphics[width=15cm]{figures/floorplan.eps}">|;
+
+$key = q/mathrm{{}^T};MSF=1.6;AAT/;
+$cached_env_img{$key} = q|<IMG
+ WIDTH="14" HEIGHT="18" ALIGN="BOTTOM" BORDER="0"
+ SRC="|."$dir".q|img40.gif"
+ ALT="$\mathrm{{}^T}$">|;
+
+$key = q/{displaymath}PAP^T=LDL^T{displaymath};MSF=1.6;AAT/;
+$cached_env_img{$key} = q|<IMG
+ WIDTH="113" HEIGHT="24" BORDER="0"
+ SRC="|."$dir".q|img52.gif"
+ ALT="\begin{displaymath}
+PAP^T = LDL^T
+\end{displaymath}">|;
+
+$key = q/{displaymath}x:=Axquad(mathrm{trans}=mathrm{'N'}),qquadx:=A^Txquad(mathrm{trans}thrm{'T'}),qquadx:=A^Hxquad(mathrm{trans}=mathrm{'C'}),{displaymath};MSF=1.6;AAT/;
+$cached_env_img{$key} = q|<IMG
+ WIDTH="587" HEIGHT="28" BORDER="0"
+ SRC="|."$dir".q|img19.gif"
+ ALT="\begin{displaymath}
+x := Ax \quad (\mathrm{trans} = \mathrm{'N'}), \qquad
+x := A...
+...'}), \qquad
+x := A^H x \quad (\mathrm{trans} = \mathrm{'C'}),
+\end{displaymath}">|;
+
+$key = q/z;MSF=1.6;AAT/;
+$cached_env_img{$key} = q|<IMG
+ WIDTH="13" HEIGHT="13" ALIGN="BOTTOM" BORDER="0"
+ SRC="|."$dir".q|img137.gif"
+ ALT="$z$">|;
+
+$key = q/{displaymath}array{{ll}mbox{minimize}&-sum_{i=1}^mlog(b_i-a_i^Tx).array{{displaymath};MSF=1.6;AAT/;
+$cached_env_img{$key} = q|<IMG
+ WIDTH="229" HEIGHT="30" BORDER="0"
+ SRC="|."$dir".q|img71.gif"
+ ALT="\begin{displaymath}
+\begin{array}{ll}
+\mbox{minimize} & -\sum_{i=1}^m \log(b_i-a_i^Tx).
+\end{array}\end{displaymath}">|;
+
+$key = q/{displaymath}array{{ll}mbox{minimize}&|X|_Fmbox{subjectto}&AX=B.array{{displaymath};MSF=1.6;AAT/;
+$cached_env_img{$key} = q|<IMG
+ WIDTH="149" HEIGHT="45" BORDER="0"
+ SRC="|."$dir".q|img56.gif"
+ ALT="\begin{displaymath}
+\begin{array}{ll}
+\mbox{minimize} & \Vert X\Vert _F \\\\
+\mbox{subject to} & AX = B.
+\end{array}\end{displaymath}">|;
+
+$key = q/d_2;MSF=1.6;AAT/;
+$cached_env_img{$key} = q|<IMG
+ WIDTH="20" HEIGHT="30" ALIGN="MIDDLE" BORDER="0"
+ SRC="|."$dir".q|img171.gif"
+ ALT="$d_2$">|;
+
+$key = q/{displaymath}array{{ll}mbox{minimize}&c^Txmbox{subjectto}&Gxpreceqh&Ax=b.array{{displaymath};MSF=1.6;AAT/;
+$cached_env_img{$key} = q|<IMG
+ WIDTH="137" HEIGHT="64" BORDER="0"
+ SRC="|."$dir".q|img111.gif"
+ ALT="\begin{displaymath}
+\begin{array}{ll}
+\mbox{minimize} & c^T x \\\\
+\mbox{subject to} & G x \preceq h \ & Ax = b.
+\end{array}\end{displaymath}">|;
+
+$key = q/{displaymath}{mbox{{bf{R}}^ltimes{mbox{{bf{R}}^{q_0}timescdotstimes{mbox{{bf{R}}bf{R}}^{p_0^2}timescdotstimes{mbox{{bf{R}}^{p_{N-1}^2},{displaymath};MSF=1.6;AAT/;
+$cached_env_img{$key} = q|<IMG
+ WIDTH="322" HEIGHT="27" BORDER="0"
+ SRC="|."$dir".q|img120.gif"
+ ALT="\begin{displaymath}
+{\mbox{\bf R}}^l \times {\mbox{\bf R}}^{q_0} \times \cdots \...
+... R}}^{p_0^2} \times \cdots \times
+{\mbox{\bf R}}^{p_{N-1}^2},
+\end{displaymath}">|;
+
+$key = q/tildef=(f_1,ldots,f_m);MSF=1.6;AAT/;
+$cached_env_img{$key} = q|<IMG
+ WIDTH="118" HEIGHT="38" ALIGN="MIDDLE" BORDER="0"
+ SRC="|."$dir".q|img167.gif"
+ ALT="$\tilde f = (f_1,\ldots, f_m)$">|;
+
+$key = q/L^H;MSF=1.6;AAT/;
+$cached_env_img{$key} = q|<IMG
+ WIDTH="27" HEIGHT="16" ALIGN="BOTTOM" BORDER="0"
+ SRC="|."$dir".q|img103.gif"
+ ALT="$L^H$">|;
+
+$key = q/{displaymath}x:=A^{-1}xquad(mathrm{trans}=mathrm{'N'}),qquadx:=A^{-T}xquad(mathrm{'T'}),qquadx:=A^{-H}xquad(mathrm{trans}=mathrm{'C'}),{displaymath};MSF=1.6;AAT/;
+$cached_env_img{$key} = q|<IMG
+ WIDTH="624" HEIGHT="28" BORDER="0"
+ SRC="|."$dir".q|img20.gif"
+ ALT="\begin{displaymath}
+x := A^{-1}x \quad (\mathrm{trans} = \mathrm{'N'}), \qquad
+x...
+..., \qquad
+x := A^{-H}x \quad (\mathrm{trans} = \mathrm{'C'}),
+\end{displaymath}">|;
+
+$key = q/{eqnarraystar}mbox{DST-I:}qquadX[k,:]&:=&2sum_{j=0}^{n-1}X[j,:]sin(pi(j+1)(k+1)s+1slash2)(k+1slash2)slashn),qquadk=0,ldots,n-1.{eqnarraystar};MSF=1.6;TAGS=R;AAT/;
+$cached_env_img{$key} = q|<IMG
+ WIDTH="721" HEIGHT="230" BORDER="0"
+ SRC="|."$dir".q|img77.gif"
+ ALT="\begin{eqnarray*}
+\mbox{DST-I:} \qquad
+X[k,:] & := &
+2 \sum_{j=0}^{n-1} X[j,:...
+...n-1} X[j,:] \sin(\pi (j+1/2)(k+1/2)/n),
+\qquad k=0,\ldots,n-1.
+\end{eqnarray*}">|;
+
+$key = q/{displaymath}A=left[array{{rrrrr}2&3&0&0&03&0&4&0&60&-1&-3&2&00&0&1&0&00&4&2&0&1array{right].{displaymath};MSF=1.6;AAT/;
+$cached_env_img{$key} = q|<IMG
+ WIDTH="204" HEIGHT="102" BORDER="0"
+ SRC="|."$dir".q|img86.gif"
+ ALT="\begin{displaymath}
+A = \left[\begin{array}{rrrrr}
+2 & 3 & 0 & 0 & 0 \\\\
+3 & 0...
+...0 & 0 & 1 & 0 & 0 \\\\
+0 & 4 & 2 & 0 & 1
+\end{array}\right].
+\end{displaymath}">|;
+
+$key = q/{displaymath}mathop{mathrm{op}}(A)=left{array{{ll}A&mathrm{transA}=mathrm{'N'}A^=mathrm{'T'}B^H&mathrm{transB}=mathrm{'C'}.array{right.{displaymath};MSF=1.6;AAT/;
+$cached_env_img{$key} = q|<IMG
+ WIDTH="475" HEIGHT="64" BORDER="0"
+ SRC="|."$dir".q|img32.gif"
+ ALT="\begin{displaymath}
+\mathop{\mathrm{op}}(A) = \left\{ \begin{array}{ll}
+A & \ma...
+...\\\\
+B^H & \mathrm{transB} = \mathrm{'C'}. \end{array} \right.
+\end{displaymath}">|;
+
+$key = q/{displaymath}left[array{{rrrr}10&0&3&00&5&0&-23&0&5&00&-2&0&2array{right].{displaymath};MSF=1.6;AAT/;
+$cached_env_img{$key} = q|<IMG
+ WIDTH="150" HEIGHT="83" BORDER="0"
+ SRC="|."$dir".q|img85.gif"
+ ALT="\begin{displaymath}
+\left[ \begin{array}{rrrr}
+10 & 0 & 3 & 0 \\\\
+0 & 5 & 0 & -2 \\\\
+3 & 0 & 5 & 0 \\\\
+0 & -2 & 0 & 2
+\end{array}\right].
+\end{displaymath}">|;
+
+$key = q/{}mathrm{^H};MSF=1.6;AAT/;
+$cached_env_img{$key} = q|<IMG
+ WIDTH="15" HEIGHT="18" ALIGN="BOTTOM" BORDER="0"
+ SRC="|."$dir".q|img51.gif"
+ ALT="${}\mathrm{^H}$">|;
+
+$key = q/S;MSF=1.6;AAT/;
+$cached_env_img{$key} = q|<IMG
+ WIDTH="15" HEIGHT="14" ALIGN="BOTTOM" BORDER="0"
+ SRC="|."$dir".q|img106.gif"
+ ALT="$S$">|;
+
+$key = q/G_0;MSF=1.6;AAT/;
+$cached_env_img{$key} = q|<IMG
+ WIDTH="24" HEIGHT="30" ALIGN="MIDDLE" BORDER="0"
+ SRC="|."$dir".q|img131.gif"
+ ALT="$G_0$">|;
+
+$key = q/{displaymath}left[array{{rrrr}10&0&3&00&5&0&-23&0&5&00&-2&0&2array{right]X=left[array{{cc}0&41&52&63&7array{right].{displaymath};MSF=1.6;AAT/;
+$cached_env_img{$key} = q|<IMG
+ WIDTH="257" HEIGHT="83" BORDER="0"
+ SRC="|."$dir".q|img90.gif"
+ ALT="\begin{displaymath}
+\left[ \begin{array}{rrrr}
+10 & 0 & 3 & 0 \\\\
+0 & 5 & 0 & ...
+...ray}{cc}
+0 & 4 \ 1 & 5 \ 2 & 6 \ 3 & 7\end{array} \right].
+\end{displaymath}">|;
+
+$key = q/{displaymath}E_1=left[array{{cccc}e_{i_1}&e_{i_2}&cdots&e_{i_q}array{right],qqua[array{{cccc}e_{j_1}&e_{j_2}&cdots&e_{j_q}array{right],{displaymath};MSF=1.6;AAT/;
+$cached_env_img{$key} = q|<IMG
+ WIDTH="435" HEIGHT="30" BORDER="0"
+ SRC="|."$dir".q|img108.gif"
+ ALT="\begin{displaymath}
+E_1 = \left[ \begin{array}{cccc}
+e_{i_1} & e_{i_2} & \cdot...
+...cc}
+e_{j_1} & e_{j_2} & \cdots & e_{j_q} \end{array}\right],
+\end{displaymath}">|;
+
+$key = q/l;MSF=1.6;AAT/;
+$cached_env_img{$key} = q|<IMG
+ WIDTH="10" HEIGHT="14" ALIGN="BOTTOM" BORDER="0"
+ SRC="|."$dir".q|img121.gif"
+ ALT="$l$">|;
+
+$key = q/{displaymath}Gx+s=h,qquadAx=b,qquadG^Tz+A^Ty+c=0,qquadssucceq0,qquadzsucceq0,qquads^Tz=0.{displaymath};MSF=1.6;AAT/;
+$cached_env_img{$key} = q|<IMG
+ WIDTH="589" HEIGHT="27" BORDER="0"
+ SRC="|."$dir".q|img122.gif"
+ ALT="\begin{displaymath}
+Gx + s = h, \qquad Ax=b, \qquad G^T z + A^T y + c = 0, \qquad
+s \succeq 0, \qquad z \succeq 0, \qquad s^T z =0.
+\end{displaymath}">|;
+
+$key = q/V^H;MSF=1.6;AAT/;
+$cached_env_img{$key} = q|<IMG
+ WIDTH="29" HEIGHT="16" ALIGN="BOTTOM" BORDER="0"
+ SRC="|."$dir".q|img70.gif"
+ ALT="$V^H$">|;
+
+$key = q/{displaymath}y:=alphaAx+betay.{displaymath};MSF=1.6;AAT/;
+$cached_env_img{$key} = q|<IMG
+ WIDTH="106" HEIGHT="27" BORDER="0"
+ SRC="|."$dir".q|img82.gif"
+ ALT="\begin{displaymath}
+y := \alpha A x + \beta y.
+\end{displaymath}">|;
+
+$key = q/{displaymath}PAP^T=LL^T,qquadPAP^T=LL^H,{displaymath};MSF=1.6;AAT/;
+$cached_env_img{$key} = q|<IMG
+ WIDTH="244" HEIGHT="27" BORDER="0"
+ SRC="|."$dir".q|img91.gif"
+ ALT="\begin{displaymath}
+PAP^T = LL^T, \qquad PAP^T = LL^H,
+\end{displaymath}">|;
+
+$key = q/{displaymath}W_mathrm{l}^T=W_mathrm{l}.{displaymath};MSF=1.6;AAT/;
+$cached_env_img{$key} = q|<IMG
+ WIDTH="70" HEIGHT="28" BORDER="0"
+ SRC="|."$dir".q|img145.gif"
+ ALT="\begin{displaymath}
+W_\mathrm{l}^T = W_\mathrm{l}.
+\end{displaymath}">|;
+
+$key = q/{displaymath}array{{ll}mbox{minimize}&{{bf{1}^Txmbox{subjectto}&W+mbox{{bf{diag},(x)succeq0array{{displaymath};MSF=1.6;AAT/;
+$cached_env_img{$key} = q|<IMG
+ WIDTH="207" HEIGHT="45" BORDER="0"
+ SRC="|."$dir".q|img157.gif"
+ ALT="\begin{displaymath}
+\begin{array}{ll}
+\mbox{minimize} & {\bf 1}^T x \\\\
+\mbox{subject to} & W + \mbox{\bf diag} (x) \succeq 0
+\end{array}
+\end{displaymath}">|;
+
+$key = q/LX=B;MSF=1.6;AAT/;
+$cached_env_img{$key} = q|<IMG
+ WIDTH="64" HEIGHT="14" ALIGN="BOTTOM" BORDER="0"
+ SRC="|."$dir".q|img97.gif"
+ ALT="$LX=B$">|;
+
+$key = q/includegraphics[width=10cm]{figuresslashportfolio1.eps};FSF=1.6;AAT/;
+$cached_env_img{$key} = q|<IMG
+ WIDTH="453" HEIGHT="340" ALIGN="BOTTOM" BORDER="0"
+ SRC="|."$dir".q|img181.gif"
+ ALT="\includegraphics[width=10cm]{figures/portfolio1.eps}">|;
+
+$key = q/{displaymath}s_0succeq0,qquadz_0succeq0{displaymath};MSF=1.6;AAT/;
+$cached_env_img{$key} = q|<IMG
+ WIDTH="125" HEIGHT="27" BORDER="0"
+ SRC="|."$dir".q|img129.gif"
+ ALT="\begin{displaymath}
+s_0 \succeq 0, \qquad z_0 \succeq 0
+\end{displaymath}">|;
+
+$key = q/{displaymath}A:=A+alpha(xy^T+yx^T),{displaymath};MSF=1.6;AAT/;
+$cached_env_img{$key} = q|<IMG
+ WIDTH="170" HEIGHT="28" BORDER="0"
+ SRC="|."$dir".q|img28.gif"
+ ALT="\begin{displaymath}
+A := A + \alpha (xy^T + yx^T),
+\end{displaymath}">|;
+
+$key = q/{displaymath}A=QR.{displaymath};MSF=1.6;AAT/;
+$cached_env_img{$key} = q|<IMG
+ WIDTH="61" HEIGHT="27" BORDER="0"
+ SRC="|."$dir".q|img61.gif"
+ ALT="\begin{displaymath}
+A = Q R.
+\end{displaymath}">|;
+
+$key = q/{displaymath}y:=alphaAx+betayquad(mathrm{trans}=mathrm{'N'}),qquady:=alphaA^Tx+bqquady:=alphaA^Hx+betayquad(mathrm{trans}=mathrm{'C'}),{displaymath};MSF=1.6;AAT/;
+$cached_env_img{$key} = q|<IMG
+ WIDTH="717" HEIGHT="28" BORDER="0"
+ SRC="|."$dir".q|img21.gif"
+ ALT="\begin{displaymath}
+y := \alpha Ax + \beta y \quad (\mathrm{trans} = \mathrm{'N'...
+... \alpha A^H x + \beta y \quad (\mathrm{trans} = \mathrm{'C'}),
+\end{displaymath}">|;
+
+$key = q/{displaymath}b_x:=u_x,qquadb_y:=u_y,qquadb_z:=Wu_z.{displaymath};MSF=1.6;AAT/;
+$cached_env_img{$key} = q|<IMG
+ WIDTH="268" HEIGHT="29" BORDER="0"
+ SRC="|."$dir".q|img152.gif"
+ ALT="\begin{displaymath}
+b_x := u_x, \qquad b_y := u_y, \qquad b_z := W u_z.
+\end{displaymath}">|;
+
+$key = q/B;MSF=1.6;AAT/;
+$cached_env_img{$key} = q|<IMG
+ WIDTH="17" HEIGHT="14" ALIGN="BOTTOM" BORDER="0"
+ SRC="|."$dir".q|img67.gif"
+ ALT="$B$">|;
+
+$key = q/{displaymath}x:=Axquad(mathrm{trans}=mathrm{'N'}),qquadx:=A^Txquad(mathrm{trans}thrm{'T'}),qquadx:=A^Hxquad(mathrm{trans}=mathrm{'C'}).{displaymath};MSF=1.6;AAT/;
+$cached_env_img{$key} = q|<IMG
+ WIDTH="586" HEIGHT="28" BORDER="0"
+ SRC="|."$dir".q|img22.gif"
+ ALT="\begin{displaymath}
+x := Ax \quad (\mathrm{trans} = \mathrm{'N'}), \qquad
+x := A...
+...}), \qquad
+x := A^H x \quad (\mathrm{trans} = \mathrm{'C'}).
+\end{displaymath}">|;
+
+$key = q/includegraphics[width=linewidth]{figuresslashnormappr.eps};FSF=1.6;AAT/;
+$cached_env_img{$key} = q|<IMG
+ WIDTH="719" HEIGHT="539" ALIGN="BOTTOM" BORDER="0"
+ SRC="|."$dir".q|img205.gif"
+ ALT="\includegraphics[width=\linewidth]{figures/normappr.eps}">|;
+
+$key = q/{displaymath}y:=alphaAx+betay,{displaymath};MSF=1.6;AAT/;
+$cached_env_img{$key} = q|<IMG
+ WIDTH="107" HEIGHT="27" BORDER="0"
+ SRC="|."$dir".q|img18.gif"
+ ALT="\begin{displaymath}
+y := \alpha A x + \beta y,
+\end{displaymath}">|;
+
+$key = q/{displaymath}C:=alphamathop{mathrm{op}}(A)mathop{mathrm{op}}(B)+betaC{displaymath};MSF=1.6;AAT/;
+$cached_env_img{$key} = q|<IMG
+ WIDTH="177" HEIGHT="28" BORDER="0"
+ SRC="|."$dir".q|img31.gif"
+ ALT="\begin{displaymath}
+C := \alpha \mathop{\mathrm{op}}(A) \mathop{\mathrm{op}}(B) + \beta C
+\end{displaymath}">|;
+
+$key = q/{displaymath}f(x_1,ldots,x_n)=b+A_1x_1+cdots+A_nx_n+sum_{k=1}^Kmin(y_1,y_2,ldots,y_{m_k}).{displaymath};MSF=1.6;AAT/;
+$cached_env_img{$key} = q|<IMG
+ WIDTH="455" HEIGHT="56" BORDER="0"
+ SRC="|."$dir".q|img197.gif"
+ ALT="\begin{displaymath}
+f(x_1,\ldots,x_n) = b + A_1 x_1 + \cdots + A_n x_n +
+\sum_{k=1}^K \min (y_1, y_2, \ldots, y_{m_k}).
+\end{displaymath}">|;
+
+$key = q/{displaymath}array{[t]{ll}mbox{minimize}&c^Txmbox{subjectto}&G_kx+s_k=h_k,quadk=0&z_0succeq0&z_{k0}geq|z_{k1}|_2,quadk=1,ldots,M.array{{displaymath};MSF=1.6;AAT/;
+$cached_env_img{$key} = q|<IMG
+ WIDTH="669" HEIGHT="106" BORDER="0"
+ SRC="|."$dir".q|img128.gif"
+ ALT="\begin{displaymath}
+\begin{array}[t]{ll}
+\mbox{minimize} & c^T x \\\\
+\mbox{su...
+...k0} \geq \Vert z_{k1}\Vert _2, \quad k=1,\ldots,M.
+\end{array}\end{displaymath}">|;
+
+$key = q/{displaymath}x:=A^{-1}xquad(mathrm{trans}=mathrm{'N'}),qquadx:=A^{-T}xquad(mathrm{'T'}),qquadx:=A^{-H}xquad(mathrm{trans}=mathrm{'T'}),{displaymath};MSF=1.6;AAT/;
+$cached_env_img{$key} = q|<IMG
+ WIDTH="624" HEIGHT="28" BORDER="0"
+ SRC="|."$dir".q|img23.gif"
+ ALT="\begin{displaymath}
+x := A^{-1}x \quad (\mathrm{trans} = \mathrm{'N'}), \qquad
+x...
+..., \qquad
+x := A^{-H} x \quad (\mathrm{trans} = \mathrm{'T'}),
+\end{displaymath}">|;
+
+$key = q/{displaymath}z_0nabla^2f_0(x)+z_1nabla^2f_1(x)+cdots+z_mnabla^2f_m(x).{displaymath};MSF=1.6;AAT/;
+$cached_env_img{$key} = q|<IMG
+ WIDTH="304" HEIGHT="28" BORDER="0"
+ SRC="|."$dir".q|img165.gif"
+ ALT="\begin{displaymath}
+z_0 \nabla^2f_0(x) + z_1 \nabla^2f_1(x) + \cdots + z_m \nabla^2f_m(x).
+\end{displaymath}">|;
+
+$key = q/{displaymath}mathop{{bf{lse}(u)=logsum_kexp(u_k),qquadF=left[array{{cccc}F_0^T&Fleft[array{{cccc}g_0^T&g_1^T&cdots&g_m^Tarray{right]^T.{displaymath};MSF=1.6;AAT/;
+$cached_env_img{$key} = q|<IMG
+ WIDTH="649" HEIGHT="45" BORDER="0"
+ SRC="|."$dir".q|img184.gif"
+ ALT="\begin{displaymath}
+\mathop{\bf lse}(u) = \log \sum_k \exp(u_k), \qquad
+F = \l...
+...}{cccc}
+g_0^T & g_1^T & \cdots & g_m^T \end{array}\right]^T.
+\end{displaymath}">|;
+
+$key = q/A;MSF=1.6;AAT/;
+$cached_env_img{$key} = q|<IMG
+ WIDTH="16" HEIGHT="14" ALIGN="BOTTOM" BORDER="0"
+ SRC="|."$dir".q|img53.gif"
+ ALT="$A$">|;
+
+$key = q/{displaymath}u=left(u_mathrm{l},;u_{mathrm{q},0},;ldots,;u_{mathrm{q},M-1},;mathu_{mathrm{s},k}in{mbox{{bf{S}}^{p_k},quadk=0,ldots,N-1.{displaymath};MSF=1.6;AAT/;
+$cached_env_img{$key} = q|<IMG
+ WIDTH="980" HEIGHT="29" BORDER="0"
+ SRC="|."$dir".q|img142.gif"
+ ALT="\begin{displaymath}
+u = \left(u_\mathrm{l}, \; u_{\mathrm{q},0}, \; \ldots, \; ...
+...\mathrm{s},k} \in{\mbox{\bf S}}^{p_k}, \quad k = 0,\ldots,N-1.
+\end{displaymath}">|;
+
+$key = q/{displaymath}array{[t]{ll}mbox{minimize}&c^Txmbox{subjectto}&Gx+s=h&Ax=b&ssucceq-h^Tz-b^Tymbox{subjectto}&G^Tz+A^Ty+c=0&zsucceq0.array{{displaymath};MSF=1.6;AAT/;
+$cached_env_img{$key} = q|<IMG
+ WIDTH="527" HEIGHT="87" BORDER="0"
+ SRC="|."$dir".q|img112.gif"
+ ALT="\begin{displaymath}
+\begin{array}[t]{ll}
+\mbox{minimize} & c^T x \\\\
+\mbox{su...
+...ct to} & G^T z + A^T y + c = 0 \\\\
+& z \succeq 0.
+\end{array}\end{displaymath}">|;
+
+$key = q/{displaymath}array{{ll}mbox{minimize}&sum_{k=1}^mphi((Ax-b)_k),array{qquadmbox{wuadAin{mbox{{bf{R}}^{mtimesn},quadphi(u)=sqrt{rho+u^2}.{displaymath};MSF=1.6;AAT/;
+$cached_env_img{$key} = q|<IMG
+ WIDTH="525" HEIGHT="30" BORDER="0"
+ SRC="|."$dir".q|img173.gif"
+ ALT="\begin{displaymath}
+\begin{array}{ll}
+\mbox{minimize} & \sum_{k=1}^m \phi((Ax-b)...
+...{\mbox{\bf R}}^{m\times n}, \quad
+\phi(u) = \sqrt{\rho + u^2}.
+\end{displaymath}">|;
+
+$key = q/{displaymath}Z^HBZ=Iquadmbox{(types1and2)},qquadZ^HB^{-1}Z=Iquadmbox{(type3)}.{displaymath};MSF=1.6;AAT/;
+$cached_env_img{$key} = q|<IMG
+ WIDTH="409" HEIGHT="28" BORDER="0"
+ SRC="|."$dir".q|img68.gif"
+ ALT="\begin{displaymath}
+Z^H BZ = I \quad \mbox{(types 1 and 2)}, \qquad
+Z^H B^{-1}Z = I \quad \mbox{(type 3)}.
+\end{displaymath}">|;
+
+$key = q/{displaymath}x^Ty.{displaymath};MSF=1.6;AAT/;
+$cached_env_img{$key} = q|<IMG
+ WIDTH="29" HEIGHT="27" BORDER="0"
+ SRC="|."$dir".q|img16.gif"
+ ALT="\begin{displaymath}
+x^Ty.
+\end{displaymath}">|;
+
+$key = q/{displaymath}left[array{{ccccccc}X[k_u,0]&X[k_u-1,1]&X[k_u-2,2]&cdots&X[0,k_u]&0_l-1,2]&cdots&&&vdots&vdots&vdots&ddots&&&array{right].{displaymath};MSF=1.6;AAT/;
+$cached_env_img{$key} = q|<IMG
+ WIDTH="625" HEIGHT="161" BORDER="0"
+ SRC="|."$dir".q|img4.gif"
+ ALT="\begin{displaymath}
+\left[ \begin{array}{ccccccc}
+X[k_u,0] & X[k_u-1,1] & X[k_u-...
+...
+\vdots & \vdots & \vdots & \ddots & & &
+\end{array}\right].
+\end{displaymath}">|;
+
+$key = q/{displaymath}array{{ll}mbox{minimize}&|Ax-b|_infty,array{qquadarray{{ll}mbox{minimize}&|Ax-b|_1array{,{displaymath};MSF=1.6;AAT/;
+$cached_env_img{$key} = q|<IMG
+ WIDTH="368" HEIGHT="30" BORDER="0"
+ SRC="|."$dir".q|img203.gif"
+ ALT="\begin{displaymath}
+\begin{array}{ll}
+\mbox{minimize} & \Vert Ax - b\Vert _\i...
+...ay}{ll}
+\mbox{minimize} & \Vert Ax - b\Vert _1
+\end{array},
+\end{displaymath}">|;
+
+$key = q/{displaymath}array{{ll}mbox{minimize}&|AX-B|_F.array{{displaymath};MSF=1.6;AAT/;
+$cached_env_img{$key} = q|<IMG
+ WIDTH="167" HEIGHT="30" BORDER="0"
+ SRC="|."$dir".q|img55.gif"
+ ALT="\begin{displaymath}
+\begin{array}{ll}
+\mbox{minimize} & \Vert AX-B\Vert _F.
+\end{array}
+\end{displaymath}">|;
+
+$key = q/N;MSF=1.6;AAT/;
+$cached_env_img{$key} = q|<IMG
+ WIDTH="19" HEIGHT="14" ALIGN="BOTTOM" BORDER="0"
+ SRC="|."$dir".q|img138.gif"
+ ALT="$N$">|;
+
+$key = q/{eqnarraystar}nablaf(x)&=&2mbox{{bf{diag},(E_1^T(Y-K(x)^{-1})E_2))&=&2mbox{{bf{d)^{-1}right)_{IJ}circleft(K(x)^{-1}right)_{JI},{eqnarraystar};MSF=1.6;TAGS=R;AAT/;
+$cached_env_img{$key} = q|<IMG
+ WIDTH="589" HEIGHT="105" BORDER="0"
+ SRC="|."$dir".q|img110.gif"
+ ALT="\begin{eqnarray*}
+\nabla f(x)
+&=& 2 \mbox{\bf diag} ( E_1^T (Y - K(x)^{-1}) E_...
+...\left(K(x)^{-1}\right)_{IJ} \circ
+\left(K(x)^{-1}\right)_{JI},
+\end{eqnarray*}">|;
+
+$key = q/{displaymath}ssucceq0,qquadzsucceq0,qquadqquadfrac{|Gx+s-h|_2}{max{1,|h|_2}}leqe{|G^Tz+A^Ty+c|_2}{max{1,|c|_2}}leqepsilon_mathrm{feas},{displaymath};MSF=1.6;AAT/;
+$cached_env_img{$key} = q|<IMG
+ WIDTH="746" HEIGHT="45" BORDER="0"
+ SRC="|."$dir".q|img159.gif"
+ ALT="\begin{displaymath}
+s \succeq 0, \qquad z \succeq 0, \qquad
+\qquad
+\frac{\Ver...
+..._2}{\max\{1,\Vert c\Vert _2\}} \leq
+\epsilon_\mathrm{feas},
+\end{displaymath}">|;
+
+$key = q/L^T;MSF=1.6;AAT/;
+$cached_env_img{$key} = q|<IMG
+ WIDTH="25" HEIGHT="16" ALIGN="BOTTOM" BORDER="0"
+ SRC="|."$dir".q|img102.gif"
+ ALT="$L^T$">|;
+
+$key = q/{displaymath}|x|_2.{displaymath};MSF=1.6;AAT/;
+$cached_env_img{$key} = q|<IMG
+ WIDTH="33" HEIGHT="28" BORDER="0"
+ SRC="|."$dir".q|img9.gif"
+ ALT="\begin{displaymath}
+\Vert x\Vert _2.
+\end{displaymath}">|;
+
+$key = q/W;MSF=1.6;AAT/;
+$cached_env_img{$key} = q|<IMG
+ WIDTH="22" HEIGHT="14" ALIGN="BOTTOM" BORDER="0"
+ SRC="|."$dir".q|img141.gif"
+ ALT="$W$">|;
+
+$key = q/{eqnarraystar}&left[array{{ccccccc}X[0,0]&X[1,0]&X[2,0]&cdots&X[k,0]&0&cdotsX[1,ts&vdots&ddots&&&array{right]&mbox{ifuplo='U'}.{eqnarraystar};MSF=1.6;TAGS=R;AAT/;
+$cached_env_img{$key} = q|<IMG
+ WIDTH="616" HEIGHT="323" BORDER="0"
+ SRC="|."$dir".q|img5.gif"
+ ALT="\begin{eqnarray*}
+& \left[ \begin{array}{ccccccc}
+X[0,0] & X[1,0] & X[2,0] & \cd...
+... \vdots & \ddots & & &
+\end{array}\right] & \mbox{if uplo='U'}.
+\end{eqnarray*}">|;
+
+$key = q/h_0;MSF=1.6;AAT/;
+$cached_env_img{$key} = q|<IMG
+ WIDTH="21" HEIGHT="30" ALIGN="MIDDLE" BORDER="0"
+ SRC="|."$dir".q|img132.gif"
+ ALT="$h_0$">|;
+
+$key = q/{displaymath}beta_k>0,qquadv_{k0}>0,qquadv_k^TJv_k=1,qquadJ=left[array{{cc}1&00&-Iarray{right].{displaymath};MSF=1.6;AAT/;
+$cached_env_img{$key} = q|<IMG
+ WIDTH="401" HEIGHT="45" BORDER="0"
+ SRC="|."$dir".q|img147.gif"
+ ALT="\begin{displaymath}
+\beta_k > 0, \qquad v_{k0} > 0, \qquad
+v_k^T Jv_k = 1, \q...
+...= \left[\begin{array}{cc}
+1 & 0 \ 0 & -I \end{array}\right].
+\end{displaymath}">|;
+
+$key = q/{displaymath}C:=alphaAA^H+betaCquad(mathrm{trans}=mathrm{'N'}),qquadC:=alphaA^HA+betaCquad(mathrm{trans}=mathrm{'C'}),{displaymath};MSF=1.6;AAT/;
+$cached_env_img{$key} = q|<IMG
+ WIDTH="505" HEIGHT="28" BORDER="0"
+ SRC="|."$dir".q|img37.gif"
+ ALT="\begin{displaymath}
+C := \alpha AA^H + \beta C \quad (\mathrm{trans} = \mathrm{...
+...= \alpha A^HA + \beta C \quad (\mathrm{trans} = \mathrm{'C'}),
+\end{displaymath}">|;
+
+$key = q/{displaymath}A=left[array{{rrrrr}0&2&0&0&32&0&0&0&0-1&-2&0&4&00&0&1&0&0array{right]{displaymath};MSF=1.6;AAT/;
+$cached_env_img{$key} = q|<IMG
+ WIDTH="193" HEIGHT="83" BORDER="0"
+ SRC="|."$dir".q|img78.gif"
+ ALT="\begin{displaymath}
+A = \left[ \begin{array}{rrrrr}
+0 & 2 & 0 & 0 & 3 \\\\
+2 &...
+...-1 & -2 & 0 & 4 & 0 \\\\
+0 & 0 & 1 & 0 & 0 \end{array} \right]
+\end{displaymath}">|;
+
+$key = q/{displaymath}h(x)=sum_kphi(x[k]),qquadphi(u)=left{array{{ll}0&|u|leq1|u|-1&1leq|u|leq22|u|-3&|u|geq2.array{right.{displaymath};MSF=1.6;AAT/;
+$cached_env_img{$key} = q|<IMG
+ WIDTH="382" HEIGHT="64" BORDER="0"
+ SRC="|."$dir".q|img198.gif"
+ ALT="\begin{displaymath}
+h(x) = \sum_k \phi(x[k]),\qquad
+\phi(u) = \left\{\begin{ar...
+...\\\\
+2\vert u\vert-3 & \vert u\vert \geq 2. \end{array}\right.
+\end{displaymath}">|;
+
+$key = q/{displaymath}array{{ll}mbox{minimize}&-logdetK+mathop{{bf{tr}(KY)mbox{subjectto}&K_{ij}=0,quad(i,j)notinS.array{{displaymath};MSF=1.6;AAT/;
+$cached_env_img{$key} = q|<IMG
+ WIDTH="236" HEIGHT="45" BORDER="0"
+ SRC="|."$dir".q|img104.gif"
+ ALT="\begin{displaymath}
+\begin{array}{ll}
+\mbox{minimize} & -\log\det K + \mathop{...
+...box{subject to} & K_{ij}=0,\quad (i,j) \not \in S.
+\end{array}\end{displaymath}">|;
+
+$key = q/{displaymath}W_mathrm{l}=mbox{{bf{diag},(d),qquadW_mathrm{l}^{-1}=mbox{{bf{diag},(d)^{-1}.{displaymath};MSF=1.6;AAT/;
+$cached_env_img{$key} = q|<IMG
+ WIDTH="267" HEIGHT="29" BORDER="0"
+ SRC="|."$dir".q|img144.gif"
+ ALT="\begin{displaymath}
+W_\mathrm{l} = \mbox{\bf diag} (d), \qquad W_\mathrm{l}^{-1} = \mbox{\bf diag} (d)^{-1}.
+\end{displaymath}">|;
+
+$key = q/{displaymath}ssucceq0,qquadqquadfrac{|Gx+s|_2}{max{1,|h|_2}}leqepsilon_mathrm{fe|_2}{max{1,|b|_2}}leqepsilon_mathrm{feas},qquadc^Tx=-1.{displaymath};MSF=1.6;AAT/;
+$cached_env_img{$key} = q|<IMG
+ WIDTH="545" HEIGHT="44" BORDER="0"
+ SRC="|."$dir".q|img162.gif"
+ ALT="\begin{displaymath}
+s \succeq 0, \qquad
+\qquad
+\frac{\Vert Gx+s\Vert _2}{\max\{1...
+...rt b\Vert _2\}} \leq \epsilon_\mathrm{feas}, \qquad
+c^Tx = -1.
+\end{displaymath}">|;
+
+$key = q/{eqnarraystar}sum_{k=0}^mz_knabla^2f_k(x)u_x+A^Tu_y+Dtildef(x)^Tu_{z_mathrm{nl}}(d_mathrm{l})^{-2}z_mathrm{l}&=&b_{z_mathrm{l}}{eqnarraystar};MSF=1.6;TAGS=R;AAT/;
+$cached_env_img{$key} = q|<IMG
+ WIDTH="402" HEIGHT="125" BORDER="0"
+ SRC="|."$dir".q|img187.gif"
+ ALT="\begin{eqnarray*}
+\sum_{k=0}^m z_k \nabla^2 f_k(x)u_x + A^T u_y +
+D \tilde f(...
+...diag} (d_\mathrm{l})^{-2} z_\mathrm{l} & = &
+b_{z_\mathrm{l}}
+\end{eqnarray*}">|;
+
+$key = q/{displaymath}B:=alphamathop{mathrm{op}}(A)^{-1}Bquad(mathrm{side}=mathrm{'L'}),q=mathrm{'T'}A^H&mathrm{transA}=mathrm{'C'},array{right.{displaymath};MSF=1.6;AAT/;
+$cached_env_img{$key} = q|<IMG
+ WIDTH="732" HEIGHT="64" BORDER="0"
+ SRC="|."$dir".q|img35.gif"
+ ALT="\begin{displaymath}
+B := \alpha \mathop{\mathrm{op}}(A)^{-1}B \quad (\mathrm{si...
+...\\\\
+A^H & \mathrm{transA} = \mathrm{'C'}, \end{array} \right.
+\end{displaymath}">|;
+
+$key = q/{displaymath}array{{ll}mbox{minimize}&|Pu-q|_1array{{displaymath};MSF=1.6;AAT/;
+$cached_env_img{$key} = q|<IMG
+ WIDTH="149" HEIGHT="30" BORDER="0"
+ SRC="|."$dir".q|img155.gif"
+ ALT="\begin{displaymath}
+\begin{array}{ll}
+\mbox{minimize} & \Vert Pu-q\Vert _1
+\end{array}\end{displaymath}">|;
+
+$key = q/{displaymath}A:=A+alphaxy^T,{displaymath};MSF=1.6;AAT/;
+$cached_env_img{$key} = q|<IMG
+ WIDTH="111" HEIGHT="27" BORDER="0"
+ SRC="|."$dir".q|img25.gif"
+ ALT="\begin{displaymath}
+A := A + \alpha x y^T,
+\end{displaymath}">|;
+
+$key = q/{displaymath}array{{ll}mbox{minimize}&|A^HX-B|_F.array{{displaymath};MSF=1.6;AAT/;
+$cached_env_img{$key} = q|<IMG
+ WIDTH="179" HEIGHT="30" BORDER="0"
+ SRC="|."$dir".q|img60.gif"
+ ALT="\begin{displaymath}
+\begin{array}{ll}
+\mbox{minimize} & \Vert A^HX-B\Vert _F.
+\end{array}\end{displaymath}">|;
+
+$key = q/AX=B;MSF=1.6;AAT/;
+$cached_env_img{$key} = q|<IMG
+ WIDTH="65" HEIGHT="14" ALIGN="BOTTOM" BORDER="0"
+ SRC="|."$dir".q|img93.gif"
+ ALT="$AX=B$">|;
+
+$key = q/{displaymath}mathop{{rm{argmax}_{k=0,ldots,n-1}|x_k|quadmbox{({x{real)},qquadmat}_{k=0,ldots,n-1}|Rex_k|+|Imx_k|quadmbox{({x{complex)}.{displaymath};MSF=1.6;AAT/;
+$cached_env_img{$key} = q|<IMG
+ WIDTH="467" HEIGHT="41" BORDER="0"
+ SRC="|."$dir".q|img11.gif"
+ ALT="\begin{displaymath}
+\mathop{\rm argmax}_{k=0,\ldots,n-1} \vert x_k\vert \quad \...
+...e x_k\vert + \vert\Im x_k\vert \quad
+\mbox{($x$ complex)}.
+\end{displaymath}">|;
+
+$key = q/includegraphics[width=10cm]{figuresslashportfolio2.eps};FSF=1.6;AAT/;
+$cached_env_img{$key} = q|<IMG
+ WIDTH="453" HEIGHT="340" ALIGN="BOTTOM" BORDER="0"
+ SRC="|."$dir".q|img182.gif"
+ ALT="\includegraphics[width=10cm]{figures/portfolio2.eps}">|;
+
+$key = q/{displaymath}AX=Bquad(mathrm{trans}=mathrm{'N'}),qquadA^TX=Bquad(mathrm{trans}=mathrm{'T'}),qquadA^HX=Bquad(mathrm{trans}=mathrm{'C'}),{displaymath};MSF=1.6;AAT/;
+$cached_env_img{$key} = q|<IMG
+ WIDTH="602" HEIGHT="28" BORDER="0"
+ SRC="|."$dir".q|img43.gif"
+ ALT="\begin{displaymath}
+AX=B \quad (\mathrm{trans} = \mathrm{'N'}), \qquad
+A^TX=B ...
+...'T'}), \qquad
+A^HX=B \quad (\mathrm{trans} = \mathrm{'C'}),
+\end{displaymath}">|;
+
+$key = q/{displaymath}array{{ll}mbox{minimize}&-barp^Tx+mux^TSxmbox{subjectto}&{{bf{1}^Tx=1,quadxsucceq0array{{displaymath};MSF=1.6;AAT/;
+$cached_env_img{$key} = q|<IMG
+ WIDTH="203" HEIGHT="45" BORDER="0"
+ SRC="|."$dir".q|img180.gif"
+ ALT="\begin{displaymath}
+\begin{array}{ll}
+\mbox{minimize} & -\bar p^T x + \mu x^T S ...
+...ox{subject to} & {\bf 1}^T x = 1, \quad x \succeq 0
+\end{array}\end{displaymath}">|;
+
+$key = q/{displaymath}xleftrightarrowy.{displaymath};MSF=1.6;AAT/;
+$cached_env_img{$key} = q|<IMG
+ WIDTH="44" HEIGHT="27" BORDER="0"
+ SRC="|."$dir".q|img12.gif"
+ ALT="\begin{displaymath}
+x \leftrightarrow y.
+\end{displaymath}">|;
+
+$key = q/{displaymath}C=C_0timesC_1timescdotstimesC_MtimesC_{M+1}timescdotstimesC_{M+N}{displaymath};MSF=1.6;AAT/;
+$cached_env_img{$key} = q|<IMG
+ WIDTH="336" HEIGHT="28" BORDER="0"
+ SRC="|."$dir".q|img116.gif"
+ ALT="\begin{displaymath}
+C = C_0 \times C_1 \times \cdots \times C_M \times C_{M+1} \times
+\cdots \times C_{M+N}
+\end{displaymath}">|;
+
+$key = q/{displaymath}x=(A^{-1}+A^{-T})b{displaymath};MSF=1.6;AAT/;
+$cached_env_img{$key} = q|<IMG
+ WIDTH="129" HEIGHT="28" BORDER="0"
+ SRC="|."$dir".q|img44.gif"
+ ALT="\begin{displaymath}
+x = (A^{-1} + A^{-T})b
+\end{displaymath}">|;
+
+$key = q/G;MSF=1.6;AAT/;
+$cached_env_img{$key} = q|<IMG
+ WIDTH="17" HEIGHT="14" ALIGN="BOTTOM" BORDER="0"
+ SRC="|."$dir".q|img151.gif"
+ ALT="$G$">|;
+
+$key = q/{displaymath}y:=alphaAx+betayquad(mathrm{trans}=mathrm{'N'}),qquady:=alphaA^Tx+betayquad(mathrm{trans}=mathrm{'T'}).{displaymath};MSF=1.6;AAT/;
+$cached_env_img{$key} = q|<IMG
+ WIDTH="469" HEIGHT="28" BORDER="0"
+ SRC="|."$dir".q|img154.gif"
+ ALT="\begin{displaymath}
+y := \alpha Ax + \beta y \quad (\mathrm{trans} = \mathrm{'N'...
+... \alpha A^T x + \beta y \quad (\mathrm{trans} = \mathrm{'T'}).
+\end{displaymath}">|;
+
+$key = q/{displaymath}nablaf_0(x)+sum_{k=1}^mz_{mathrm{nl},k}nablaf_k(x)+G^Tz_mathrm{l}+Al},k}=0,quadk=1,ldots,m,qquadGx+s_mathrm{l}=h,qquadAx=b{displaymath};MSF=1.6;AAT/;
+$cached_env_img{$key} = q|<IMG
+ WIDTH="747" HEIGHT="53" BORDER="0"
+ SRC="|."$dir".q|img185.gif"
+ ALT="\begin{displaymath}
+\nabla f_0(x) + \sum_{k=1}^m z_{\mathrm{nl},k}
+\nabla f_k...
+...quad k=1,\ldots,m, \qquad
+Gx + s_\mathrm{l} = h, \qquad Ax=b
+\end{displaymath}">|;
+
+$key = q/Y;MSF=1.6;AAT/;
+$cached_env_img{$key} = q|<IMG
+ WIDTH="17" HEIGHT="14" ALIGN="BOTTOM" BORDER="0"
+ SRC="|."$dir".q|img105.gif"
+ ALT="$Y$">|;
+
+$key = q/{displaymath}array{{ll}mbox{minimize}&|X|_Fmbox{subjectto}&A^HX=B.array{{displaymath};MSF=1.6;AAT/;
+$cached_env_img{$key} = q|<IMG
+ WIDTH="161" HEIGHT="45" BORDER="0"
+ SRC="|."$dir".q|img59.gif"
+ ALT="\begin{displaymath}
+\begin{array}{ll}
+\mbox{minimize} & \Vert X\Vert _F \\\\
+\mbox{subject to} & A^HX=B.
+\end{array}\end{displaymath}">|;
+
+$key = q/{displaymath}array{{ll}mbox{minimize}&f(x)=-logdetK(x)+mathop{{bf{tr}(K(x)Y).array{{displaymath};MSF=1.6;AAT/;
+$cached_env_img{$key} = q|<IMG
+ WIDTH="328" HEIGHT="30" BORDER="0"
+ SRC="|."$dir".q|img109.gif"
+ ALT="\begin{displaymath}
+\begin{array}{ll}
+\mbox{minimize} & f(x) = -\log\det K(x) + \mathop{\bf tr}(K(x)Y).
+\end{array}\end{displaymath}">|;
+
+$key = q/{displaymath}B:=alphamathop{mathrm{op}}(A)Bquad(mathrm{side}=mathrm{'L'}),qquadB=mathrm{'T'}A^H&mathrm{transA}=mathrm{'C'}.array{right.{displaymath};MSF=1.6;AAT/;
+$cached_env_img{$key} = q|<IMG
+ WIDTH="697" HEIGHT="64" BORDER="0"
+ SRC="|."$dir".q|img34.gif"
+ ALT="\begin{displaymath}
+B := \alpha\mathop{\mathrm{op}}(A)B \quad (\mathrm{side} = ...
+...\\\\
+A^H & \mathrm{transA} = \mathrm{'C'}. \end{array} \right.
+\end{displaymath}">|;
+
+$key = q/{displaymath}array{[t]{ll}mbox{minimize}&-4x_1-5x_2mbox{subjectto}&2x_1+x_2leq3&x_1+2x_2leq3&x_1geq0,quadx_2geq0.array{{displaymath};MSF=1.6;AAT/;
+$cached_env_img{$key} = q|<IMG
+ WIDTH="202" HEIGHT="87" BORDER="0"
+ SRC="|."$dir".q|img127.gif"
+ ALT="\begin{displaymath}
+\begin{array}[t]{ll}
+\mbox{minimize} & -4x_1 - 5x_2 \\\\
+\...
+...2x_2 \leq 3 \\\\
+& x_1 \geq 0, \quad x_2 \geq 0.
+\end{array}
+\end{displaymath}">|;
+
+$key = q/{eqnarraystar}&left[array{{ccccc}X[0,0]&0&0&cdots&0X[1,0]&X[1,1]&0&cdots&0X[2,0]dots&1array{right]&mbox{ifuplo='U'anddiag='U'}.{eqnarraystar};MSF=1.6;TAGS=R;AAT/;
+$cached_env_img{$key} = q|<IMG
+ WIDTH="662" HEIGHT="450" BORDER="0"
+ SRC="|."$dir".q|img3.gif"
+ ALT="\begin{eqnarray*}
+& \left[\begin{array}{ccccc}
+X[0,0] & 0 & 0 & \cdots & 0 \\\\
+X...
+...ts & 1
+\end{array}\right] & \mbox{if uplo = 'U' and diag = 'U'}.
+\end{eqnarray*}">|;
+
+$key = q/{displaymath}array{{ll}mbox{minimize}&-4x-5ymbox{subjectto}&2x+yleq3&x+2yleq3&xgeq0,quadygeq0.array{{displaymath};MSF=1.6;AAT/;
+$cached_env_img{$key} = q|<IMG
+ WIDTH="187" HEIGHT="83" BORDER="0"
+ SRC="|."$dir".q|img202.gif"
+ ALT="\begin{displaymath}
+\begin{array}{ll}
+\mbox{minimize} & -4x - 5y \\\\
+\mbox{su...
+...
+& x +2y \leq 3 \\\\
+& x \geq 0, \quad y \geq 0.
+\end{array}\end{displaymath}">|;
+
+$key = q/{displaymath}G^Tz+A^Ty=0,qquadh^Tz+b^Ty=-1,qquadzsucceq0.{displaymath};MSF=1.6;AAT/;
+$cached_env_img{$key} = q|<IMG
+ WIDTH="340" HEIGHT="27" BORDER="0"
+ SRC="|."$dir".q|img123.gif"
+ ALT="\begin{displaymath}
+G^T z + A^T y = 0, \qquad h^T z + b^T y = -1, \qquad z \succeq 0.
+\end{displaymath}">|;
+
+$key = q/C;MSF=1.6;AAT/;
+$cached_env_img{$key} = q|<IMG
+ WIDTH="17" HEIGHT="14" ALIGN="BOTTOM" BORDER="0"
+ SRC="|."$dir".q|img115.gif"
+ ALT="$C$">|;
+
+$key = q/P^TX=B;MSF=1.6;AAT/;
+$cached_env_img{$key} = q|<IMG
+ WIDTH="75" HEIGHT="16" ALIGN="BOTTOM" BORDER="0"
+ SRC="|."$dir".q|img100.gif"
+ ALT="$P^TX=B$">|;
+
+$key = q/{eqnarraystar}&left[array{{cccc}X[0,0]&0&0&cdotsX[1,0]&X[0,1]&0&cdotsX[2,0]&X[1,&ddotsarray{right]&mbox{ifuplo='U'anddiag='U'}.{eqnarraystar};MSF=1.6;TAGS=R;AAT/;
+$cached_env_img{$key} = q|<IMG
+ WIDTH="700" HEIGHT="509" BORDER="0"
+ SRC="|."$dir".q|img7.gif"
+ ALT="\begin{eqnarray*}
+& \left[ \begin{array}{cccc}
+X[0,0] & 0 & 0 & \cdots \\\\
+X[1,0...
+...ddots
+\end{array}\right] & \mbox{if uplo = 'U' and diag = 'U'}.
+\end{eqnarray*}">|;
+
+$key = q/{displaymath}X[k,:]:=frac{1}{n}sum_{j=0}^{n-1}e^{2pijksqrt{-1}slashn}X[j,:],qquadk=0,ldots,n-1.{displaymath};MSF=1.6;AAT/;
+$cached_env_img{$key} = q|<IMG
+ WIDTH="382" HEIGHT="58" BORDER="0"
+ SRC="|."$dir".q|img75.gif"
+ ALT="\begin{displaymath}
+X[k,:] :=
+\frac{1}{n} \sum_{j=0}^{n-1} e^{2\pi j k \sqrt{-1}/n} X[j,:],
+\qquad k=0,\ldots,n-1.
+\end{displaymath}">|;
+
+$key = q/a_i^T;MSF=1.6;AAT/;
+$cached_env_img{$key} = q|<IMG
+ WIDTH="23" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
+ SRC="|."$dir".q|img73.gif"
+ ALT="$a_i^T$">|;
+
+$key = q/{displaymath}array{{ll}mbox{minimize}&|X|_Fmbox{subjectto}&A^TX=B.array{{displaymath};MSF=1.6;AAT/;
+$cached_env_img{$key} = q|<IMG
+ WIDTH="159" HEIGHT="45" BORDER="0"
+ SRC="|."$dir".q|img57.gif"
+ ALT="\begin{displaymath}
+\begin{array}{ll}
+\mbox{minimize} & \Vert X\Vert _F \\\\
+\mbox{subject to} & A^TX=B.
+\end{array}\end{displaymath}">|;
+
+$key = q/{displaymath}zsucceq0,qquadqquadfrac{|G^Tz+A^Ty|_2}{max{1,|c|_2}}leqepsilon_mathrm{feas},qquadh^Tz+b^Ty=-1.{displaymath};MSF=1.6;AAT/;
+$cached_env_img{$key} = q|<IMG
+ WIDTH="420" HEIGHT="45" BORDER="0"
+ SRC="|."$dir".q|img161.gif"
+ ALT="\begin{displaymath}
+z \succeq 0, \qquad
+\qquad \frac{\Vert G^Tz +A^Ty\Vert _2}{\...
+..._2\}} \leq
+\epsilon_\mathrm{feas},
+\qquad h^Tz +b^Ty = -1.
+\end{displaymath}">|;
+
+$key = q/{displaymath}array{{ll}mbox{minimize}&{{bf{1}^Tvmbox{subjectto}&-vpreceqPu-qpreceqv.array{{displaymath};MSF=1.6;AAT/;
+$cached_env_img{$key} = q|<IMG
+ WIDTH="208" HEIGHT="45" BORDER="0"
+ SRC="|."$dir".q|img156.gif"
+ ALT="\begin{displaymath}
+\begin{array}{ll}
+\mbox{minimize} & {\bf 1}^T v \\\\
+\mbox{subject to} & -v \preceq Pu - q \preceq v.
+\end{array}\end{displaymath}">|;
+
+$key = q/{displaymath}left[array{{cccc}sum_{k=0}^mz_knabla^2f_k(x)&Dtildef(x)^T&G^T&A^TDt(d_1)&0&0G&0&-mbox{{bf{diag},(d_2)&0A&0&0&0array{right]{displaymath};MSF=1.6;AAT/;
+$cached_env_img{$key} = q|<IMG
+ WIDTH="365" HEIGHT="86" BORDER="0"
+ SRC="|."$dir".q|img169.gif"
+ ALT="\begin{displaymath}
+\left[\begin{array}{cccc}
+\sum_{k=0}^m z_k \nabla^2 f_k(x) &...
+...box{\bf diag} (d_2) & 0 \\\\
+A & 0 & 0 & 0 \end{array}\right]
+\end{displaymath}">|;
+
+$key = q/{displaymath}AX=B{displaymath};MSF=1.6;AAT/;
+$cached_env_img{$key} = q|<IMG
+ WIDTH="59" HEIGHT="24" BORDER="0"
+ SRC="|."$dir".q|img47.gif"
+ ALT="\begin{displaymath}
+AX=B
+\end{displaymath}">|;
+
+$key = q/{displaymath}f(x_1,ldots,x_n)=A_1x_1+cdots+A_nx_n+b.{displaymath};MSF=1.6;AAT/;
+$cached_env_img{$key} = q|<IMG
+ WIDTH="272" HEIGHT="28" BORDER="0"
+ SRC="|."$dir".q|img194.gif"
+ ALT="\begin{displaymath}
+f(x_1,\ldots,x_n) = A_1 x_1 + \cdots + A_n x_n + b.
+\end{displaymath}">|;
+
+$key = q/{displaymath}y:=alphax+y.{displaymath};MSF=1.6;AAT/;
+$cached_env_img{$key} = q|<IMG
+ WIDTH="84" HEIGHT="27" BORDER="0"
+ SRC="|."$dir".q|img14.gif"
+ ALT="\begin{displaymath}
+y := \alpha x + y.
+\end{displaymath}">|;
+
+$key = q/{displaymath}x:=alphax.{displaymath};MSF=1.6;AAT/;
+$cached_env_img{$key} = q|<IMG
+ WIDTH="56" HEIGHT="24" BORDER="0"
+ SRC="|."$dir".q|img8.gif"
+ ALT="\begin{displaymath}
+x := \alpha x.
+\end{displaymath}">|;
+
+$key = q/{displaymath}A=Vmbox{{bf{diag},(lambda)V^T,qquadV^TV=I.{displaymath};MSF=1.6;AAT/;
+$cached_env_img{$key} = q|<IMG
+ WIDTH="233" HEIGHT="28" BORDER="0"
+ SRC="|."$dir".q|img64.gif"
+ ALT="\begin{displaymath}
+A = V\mbox{\bf diag} (\lambda)V^T,\qquad V^TV = I.
+\end{displaymath}">|;
+
+$key = q/LDLX=B;MSF=1.6;AAT/;
+$cached_env_img{$key} = q|<IMG
+ WIDTH="88" HEIGHT="14" ALIGN="BOTTOM" BORDER="0"
+ SRC="|."$dir".q|img95.gif"
+ ALT="$LDLX=B$">|;
+
+$key = q/{displaymath}0preceqxpreceq{{bf{1},qquad{{bf{1}^Tx=2,{displaymath};MSF=1.6;AAT/;
+$cached_env_img{$key} = q|<IMG
+ WIDTH="168" HEIGHT="27" BORDER="0"
+ SRC="|."$dir".q|img201.gif"
+ ALT="\begin{displaymath}
+0 \preceq x \preceq {\bf 1}, \qquad {\bf 1}^T x = 2,
+\end{displaymath}">|;
+
+1;
+
diff --git a/doc/cvxopt/images.tex b/doc/cvxopt/images.tex
new file mode 100644
index 0000000..d92f92b
--- /dev/null
+++ b/doc/cvxopt/images.tex
@@ -0,0 +1,2523 @@
+\batchmode
+\documentclass{book}
+\RequirePackage{ifthen}
+
+
+\usepackage{html,graphicx}
+
+%
+\providecommand{\eg}{{\em e.g.}}%
+\providecommand{\ie}{{\em i.e.}}%
+\providecommand{\BEA}{\begin{eqnarray}}%
+\providecommand{\EEA}{\end{eqnarray}}%
+\providecommand{\BEAS}{\begin{eqnarray*}}%
+\providecommand{\EEAS}{\end{eqnarray*}}%
+\providecommand{\Rank}{\mathop{\bf rank}}%
+\providecommand{\Tr}{\mathop{\bf tr}}%
+\providecommand{\lse}{\mathop{\bf lse}}%
+\providecommand{\diag}{\mbox{\bf diag}\,}%
+\providecommand{\ones}{{\bf 1}}%
+\providecommand{\argmax}{\mathop{\rm argmax}}%
+\providecommand{\symm}{{\mbox{\bf S}}}%
+\providecommand{\op}{\mathop{\mathrm{op}}}%
+\providecommand{\svec}{\mathop{\mathbf{vec}}}
+
+%
+\providecommand{\code}[1]{{\tt #1}}%
+\providecommand{\pytype}[1]{{\tt #1}}%
+\providecommand{\cdata}[1]{{\tt #1}}%
+\providecommand{\var}[1]{{\tt #1}}%
+\providecommand{\samp}[1]{"{\tt #1}"}%
+\providecommand{\module}[1]{{\tt #1}}%
+\providecommand{\class}[1]{{\tt #1}}%
+\providecommand{\function}[1]{{\tt #1}}%
+\providecommand{\member}[1]{{\tt #1}}%
+\providecommand{\program}[1]{{\rm #1}}%
+\providecommand{\optional}[1]{{\rm [}#1{\rm ]}}%
+\providecommand{\file}[1]{{\tt #1}}%
+\providecommand{\ctype}[1]{{\tt #1}}%
+\providecommand{\constant}[1]{{\tt #1}}
+
+%
+\newenvironment{cfuncdesc}[3]{
+
+{\tt #1}%
+ {\bf #2}({\tt #3})\begin{list}{}{}\item[]}{\end{list}}
+
+%
+\newenvironment{classdesc}[2]{
+
+{\bf #1}%
+ ({\tt #2})\begin{list}{}{}\item[]}{\end{list}}
+
+%
+\newenvironment{funcdesc}[2]{
+
+{\bf #1}%
+ ({\tt #2})\begin{list}{}{}\item[]}{\end{list}}
+
+%
+\newenvironment{memberdesc}[3]{
+
+{\bf #2}%
+ \begin{list}{}{}\item[]}{\end{list}}
+
+%
+\newenvironment{methoddesc}[3]{
+
+{\bf #2}()%
+ \begin{list}{}{}\item[]}{\end{list}}
+
+%
+\newenvironment{seealso}{
+
+{\bf See also:}\begin{list}{}{}\item[]}{\end{list}}%
+\providecommand{\seetext}[1]{#1}%
+\providecommand{\ulink}[2]{\htmladdnormallink{#1}{#2}}%
+\providecommand{\seelink}[3]{\htmladdnormallink{#2}{#1}}%
+\providecommand{\citetitle}[2]{\htmladdnormallink{#2}{#1}}
+
+%
+\providecommand{\dtc}{{\tt 'd'}}%
+\providecommand{\itc}{{\tt 'i'}}%
+\providecommand{\ztc}{{\tt 'z'}}%
+\providecommand{\None}{{\tt None}}%
+\providecommand{\True}{{\tt True}}%
+\providecommand{\False}{{\tt False}}
+
+%
+\providecommand{\flt}{{\tt float}}%
+\providecommand{\chr}{{\tt char}}%
+\providecommand{\intgr}{{\tt integer}}%
+\providecommand{\cmplx}{{\tt complex}}%
+\providecommand{\mtrx}{{\tt matrix}}%
+\providecommand{\spmtrx}{{\tt spmatrix}}
+
+
+\title{CVXOPT User's Guide}
+\author{Joachim Dahl \& Lieven Vandenberghe}
+\date{Release 0.9 -- August 10, 2007}
+
+
+
+
+\usepackage[dvips]{color}
+
+
+\pagecolor[gray]{.7}
+
+\usepackage[latin1]{inputenc}
+
+
+
+\makeatletter
+
+\makeatletter
+\count@=\the\catcode`\_ \catcode`\_=8
+\newenvironment{tex2html_wrap}{}{}%
+\catcode`\<=12\catcode`\_=\count@
+\newcommand{\providedcommand}[1]{\expandafter\providecommand\csname #1\endcsname}%
+\newcommand{\renewedcommand}[1]{\expandafter\providecommand\csname #1\endcsname{}%
+ \expandafter\renewcommand\csname #1\endcsname}%
+\newcommand{\newedenvironment}[1]{\newenvironment{#1}{}{}\renewenvironment{#1}}%
+\let\newedcommand\renewedcommand
+\let\renewedenvironment\newedenvironment
+\makeatother
+\let\mathon=$
+\let\mathoff=$
+\ifx\AtBeginDocument\undefined \newcommand{\AtBeginDocument}[1]{}\fi
+\newbox\sizebox
+\setlength{\hoffset}{0pt}\setlength{\voffset}{0pt}
+\addtolength{\textheight}{\footskip}\setlength{\footskip}{0pt}
+\addtolength{\textheight}{\topmargin}\setlength{\topmargin}{0pt}
+\addtolength{\textheight}{\headheight}\setlength{\headheight}{0pt}
+\addtolength{\textheight}{\headsep}\setlength{\headsep}{0pt}
+\setlength{\textwidth}{451pt}
+\setlength{\textheight}{554pt}
+\newwrite\lthtmlwrite
+\makeatletter
+\let\realnormalsize=\normalsize
+\global\topskip=2sp
+\def\preveqno{}\let\real at float=\@float \let\realend at float=\end at float
+\def\@float{\let\@savefreelist\@freelist\real at float}
+\def\liih at math{\ifmmode$\else\bad at math\fi}
+\def\end at float{\realend at float\global\let\@freelist\@savefreelist}
+\let\real at dbflt=\@dbflt \let\end at dblfloat=\end at float
+\let\@largefloatcheck=\relax
+\let\if at boxedmulticols=\iftrue
+\def\@dbflt{\let\@savefreelist\@freelist\real at dbflt}
+\def\adjustnormalsize{\def\normalsize{\mathsurround=0pt \realnormalsize
+ \parindent=0pt\abovedisplayskip=0pt\belowdisplayskip=0pt}%
+ \def\phantompar{\csname par\endcsname}\normalsize}%
+\def\lthtmltypeout#1{{\let\protect\string \immediate\write\lthtmlwrite{#1}}}%
+\newcommand\lthtmlhboxmathA{\adjustnormalsize\setbox\sizebox=\hbox\bgroup\kern.05em }%
+\newcommand\lthtmlhboxmathB{\adjustnormalsize\setbox\sizebox=\hbox to\hsize\bgroup\hfill }%
+\newcommand\lthtmlvboxmathA{\adjustnormalsize\setbox\sizebox=\vbox\bgroup %
+ \let\ifinner=\iffalse \let\)\liih at math }%
+\newcommand\lthtmlboxmathZ{\@next\next\@currlist{}{\def\next{\voidb at x}}%
+ \expandafter\box\next\egroup}%
+\newcommand\lthtmlmathtype[1]{\gdef\lthtmlmathenv{#1}}%
+\newcommand\lthtmllogmath{\dimen0\ht\sizebox \advance\dimen0\dp\sizebox
+ \ifdim\dimen0>.95\vsize
+ \lthtmltypeout{%
+*** image for \lthtmlmathenv\space is too tall at \the\dimen0, reducing to .95 vsize ***}%
+ \ht\sizebox.95\vsize \dp\sizebox\z@ \fi
+ \lthtmltypeout{l2hSize %
+:\lthtmlmathenv:\the\ht\sizebox::\the\dp\sizebox::\the\wd\sizebox.\preveqno}}%
+\newcommand\lthtmlfigureA[1]{\let\@savefreelist\@freelist
+ \lthtmlmathtype{#1}\lthtmlvboxmathA}%
+\newcommand\lthtmlpictureA{\bgroup\catcode`\_=8 \lthtmlpictureB}%
+\newcommand\lthtmlpictureB[1]{\lthtmlmathtype{#1}\egroup
+ \let\@savefreelist\@freelist \lthtmlhboxmathB}%
+\newcommand\lthtmlpictureZ[1]{\hfill\lthtmlfigureZ}%
+\newcommand\lthtmlfigureZ{\lthtmlboxmathZ\lthtmllogmath\copy\sizebox
+ \global\let\@freelist\@savefreelist}%
+\newcommand\lthtmldisplayA{\bgroup\catcode`\_=8 \lthtmldisplayAi}%
+\newcommand\lthtmldisplayAi[1]{\lthtmlmathtype{#1}\egroup\lthtmlvboxmathA}%
+\newcommand\lthtmldisplayB[1]{\edef\preveqno{(\theequation)}%
+ \lthtmldisplayA{#1}\let\@eqnnum\relax}%
+\newcommand\lthtmldisplayZ{\lthtmlboxmathZ\lthtmllogmath\lthtmlsetmath}%
+\newcommand\lthtmlinlinemathA{\bgroup\catcode`\_=8 \lthtmlinlinemathB}
+\newcommand\lthtmlinlinemathB[1]{\lthtmlmathtype{#1}\egroup\lthtmlhboxmathA
+ \vrule height1.5ex width0pt }%
+\newcommand\lthtmlinlineA{\bgroup\catcode`\_=8 \lthtmlinlineB}%
+\newcommand\lthtmlinlineB[1]{\lthtmlmathtype{#1}\egroup\lthtmlhboxmathA}%
+\newcommand\lthtmlinlineZ{\egroup\expandafter\ifdim\dp\sizebox>0pt %
+ \expandafter\centerinlinemath\fi\lthtmllogmath\lthtmlsetinline}
+\newcommand\lthtmlinlinemathZ{\egroup\expandafter\ifdim\dp\sizebox>0pt %
+ \expandafter\centerinlinemath\fi\lthtmllogmath\lthtmlsetmath}
+\newcommand\lthtmlindisplaymathZ{\egroup %
+ \centerinlinemath\lthtmllogmath\lthtmlsetmath}
+\def\lthtmlsetinline{\hbox{\vrule width.1em \vtop{\vbox{%
+ \kern.1em\copy\sizebox}\ifdim\dp\sizebox>0pt\kern.1em\else\kern.3pt\fi
+ \ifdim\hsize>\wd\sizebox \hrule depth1pt\fi}}}
+\def\lthtmlsetmath{\hbox{\vrule width.1em\kern-.05em\vtop{\vbox{%
+ \kern.1em\kern0.8 pt\hbox{\hglue.17em\copy\sizebox\hglue0.8 pt}}\kern.3pt%
+ \ifdim\dp\sizebox>0pt\kern.1em\fi \kern0.8 pt%
+ \ifdim\hsize>\wd\sizebox \hrule depth1pt\fi}}}
+\def\centerinlinemath{%
+ \dimen1=\ifdim\ht\sizebox<\dp\sizebox \dp\sizebox\else\ht\sizebox\fi
+ \advance\dimen1by.5pt \vrule width0pt height\dimen1 depth\dimen1
+ \dp\sizebox=\dimen1\ht\sizebox=\dimen1\relax}
+
+\def\lthtmlcheckvsize{\ifdim\ht\sizebox<\vsize
+ \ifdim\wd\sizebox<\hsize\expandafter\hfill\fi \expandafter\vfill
+ \else\expandafter\vss\fi}%
+\providecommand{\selectlanguage}[1]{}%
+\makeatletter \tracingstats = 1
+
+
+\begin{document}
+\pagestyle{empty}\thispagestyle{empty}\lthtmltypeout{}%
+\lthtmltypeout{latex2htmlLength hsize=\the\hsize}\lthtmltypeout{}%
+\lthtmltypeout{latex2htmlLength vsize=\the\vsize}\lthtmltypeout{}%
+\lthtmltypeout{latex2htmlLength hoffset=\the\hoffset}\lthtmltypeout{}%
+\lthtmltypeout{latex2htmlLength voffset=\the\voffset}\lthtmltypeout{}%
+\lthtmltypeout{latex2htmlLength topmargin=\the\topmargin}\lthtmltypeout{}%
+\lthtmltypeout{latex2htmlLength topskip=\the\topskip}\lthtmltypeout{}%
+\lthtmltypeout{latex2htmlLength headheight=\the\headheight}\lthtmltypeout{}%
+\lthtmltypeout{latex2htmlLength headsep=\the\headsep}\lthtmltypeout{}%
+\lthtmltypeout{latex2htmlLength parskip=\the\parskip}\lthtmltypeout{}%
+\lthtmltypeout{latex2htmlLength oddsidemargin=\the\oddsidemargin}\lthtmltypeout{}%
+\makeatletter
+\if at twoside\lthtmltypeout{latex2htmlLength evensidemargin=\the\evensidemargin}%
+\else\lthtmltypeout{latex2htmlLength evensidemargin=\the\oddsidemargin}\fi%
+\lthtmltypeout{}%
+\makeatother
+\setcounter{page}{1}
+\onecolumn
+
+% !!! IMAGES START HERE !!!
+
+\stepcounter{chapter}
+\stepcounter{chapter}
+\stepcounter{section}
+\stepcounter{section}
+\stepcounter{section}
+\stepcounter{section}
+\stepcounter{section}
+\stepcounter{section}
+\stepcounter{section}
+\stepcounter{section}
+\stepcounter{section}
+\stepcounter{chapter}
+\stepcounter{section}
+{\newpage\clearpage
+\lthtmldisplayA{eqnarraystar4063}%
+\begin{eqnarray*}
+& \left[\begin{array}{ccccc}
+X[0,0] & X[1,0] & X[2,0] & \cdots & X[n-1,0] \\
+X[1,0] & X[1,1] & X[2,1] & \cdots & X[n-1,1] \\
+X[2,0] & X[2,1] & X[2,2] & \cdots & X[n-1,2] \\
+\vdots & \vdots & \vdots & \ddots & \vdots \\
+X[n-1,0] & X[n-1,1] & X[n-1,2] & \cdots & X[n-1,n-1]
+\end{array}\right] & \mbox{if uplo = 'L'}, \\
+& \left[\begin{array}{ccccc}
+X[0,0] & X[0,1] & X[0,2] & \cdots & X[0,n-1] \\
+X[0,1] & X[1,1] & X[1,2] & \cdots & X[1,n-1] \\
+X[0,2] & X[1,2] & X[2,2] & \cdots & X[2,n-1] \\
+\vdots & \vdots & \vdots & \ddots & \vdots \\
+X[0,n-1] & X[1,n-1] & X[2,n-1] & \cdots & X[n-1,n-1]
+\end{array}\right] & \mbox{if uplo = U'}.
+\end{eqnarray*}%
+\lthtmldisplayZ
+\lthtmlcheckvsize\clearpage}
+
+{\newpage\clearpage
+\lthtmldisplayA{eqnarraystar4070}%
+\begin{eqnarray*}
+&
+\left[\begin{array}{ccccc}
+\Re X[0,0] & \bar X[1,0] & \bar X[2,0] & \cdots & \bar X[n-1,0] \\
+X[1,0] & \Re X[1,1] & \bar X[2,1] & \cdots & \bar X[n-1,1] \\
+X[2,0] & X[2,1] & \Re X[2,2] & \cdots & \bar X[n-1,2] \\
+\vdots & \vdots & \vdots & \ddots & \vdots \\
+X[n-1,0] & X[n-1,1] & X[n-1,2] & \cdots & \Re X[n-1,n-1]
+\end{array}\right] & \mbox{if uplo = 'L'}, \\
+& \left[\begin{array}{ccccc}
+\Re X[0,0] & X[0,1] & X[0,2] & \cdots & X[0,n-1] \\
+\bar X[0,1] & \Re X[1,1] & X[1,2] & \cdots & X[1,n-1] \\
+\bar X[0,2] & \bar X[1,2] & \Re X[2,2] & \cdots & X[2,n-1] \\
+\vdots & \vdots & \vdots & \ddots & \vdots \\
+\bar X[0,n-1] & \bar X[1,n-1] & \bar X[2,n-1] & \cdots & \Re X[n-1,n-1]
+\end{array}\right] & \mbox{if uplo = 'U'}.
+\end{eqnarray*}%
+\lthtmldisplayZ
+\lthtmlcheckvsize\clearpage}
+
+{\newpage\clearpage
+\lthtmldisplayA{eqnarraystar4081}%
+\begin{eqnarray*}
+& \left[\begin{array}{ccccc}
+X[0,0] & 0 & 0 & \cdots & 0 \\
+X[1,0] & X[1,1] & 0 & \cdots & 0 \\
+X[2,0] & X[2,1] & X[2,2] & \cdots & 0 \\
+\vdots & \vdots & \vdots & \ddots & \vdots \\
+X[n-1,0] & X[n-1,1] & X[n-1,2] & \cdots & X[n-1,n-1]
+\end{array}\right] & \mbox{if uplo = 'L' and diag = 'N'}, \\
+& \left[\begin{array}{ccccc}
+1 & 0 & 0 & \cdots & 0 \\
+X[1,0] & 1 & 0 & \cdots & 0 \\
+X[2,0] & X[2,1] & 1 & \cdots & 0 \\
+\vdots & \vdots & \vdots & \ddots & \vdots \\
+X[n-1,0] & X[n-1,1] & X[n-1,2] & \cdots & 1
+\end{array}\right] & \mbox{if uplo = 'L' and diag = 'U'}, \\
+& \left[\begin{array}{ccccc}
+X[0,0] & X[0,1] & X[0,2] & \cdots & X[0,n-1] \\
+0 & X[1,1] & X[1,2] & \cdots & X[1,n-1] \\
+0 & 0 & X[2,2] & \cdots & X[2,n-1] \\
+\vdots & \vdots & \vdots & \ddots & \vdots \\
+0 & 0 & 0 & \cdots & X[n-1,n-1]
+\end{array}\right] & \mbox{if uplo = 'U' and diag = 'N'}, \\
+& \left[\begin{array}{ccccc}
+1 & X[0,1] & X[0,2] & \cdots & X[0,n-1] \\
+0 & 1 & X[1,2] & \cdots & X[1,n-1] \\
+0 & 0 & 1 & \cdots & X[2,n-1] \\
+\vdots & \vdots & \vdots & \ddots & \vdots \\
+0 & 0 & 0 & \cdots & 1
+\end{array}\right] & \mbox{if uplo = 'U' and diag = 'U'}.
+\end{eqnarray*}%
+\lthtmldisplayZ
+\lthtmlcheckvsize\clearpage}
+
+{\newpage\clearpage
+\lthtmldisplayA{displaymath4586}%
+\begin{displaymath}
+\left[ \begin{array}{ccccccc}
+X[k_u,0] & X[k_u-1,1] & X[k_u-2,2] & \cdots & X[0,k_u] & 0 & \cdots \\
+X[k_u+1,0] & X[k_u,1] & X[k_u-1,2] & \cdots & X[1,k_u] & X[0,k_u+1] & \cdots \\
+X[k_u+2,0] & X[k_u+1,1] & X[k_u,2] & \cdots & X[2,k_u] & X[1,k_u+1] & \cdots \\
+ \vdots & \vdots & \vdots & \ddots & \vdots & \vdots & \ddots \\
+X[k_u+k_l,0] & X[k_u+k_l-1,1] & X[k_u+k_l-2,2] & \cdots & & & \\
+0 & X[k_u+k_l,1] & X[k_u+k_l-1,2] & \cdots & & & \\
+\vdots & \vdots & \vdots & \ddots & & &
+\end{array}\right].
+\end{displaymath}%
+\lthtmldisplayZ
+\lthtmlcheckvsize\clearpage}
+
+{\newpage\clearpage
+\lthtmldisplayA{eqnarraystar4098}%
+\begin{eqnarray*}
+& \left[ \begin{array}{ccccccc}
+X[0,0] & X[1,0] & X[2,0] & \cdots & X[k,0] & 0 & \cdots \\
+X[1,0] & X[0,1] & X[1,1] & \cdots & X[k-1,1] & X[k,1] & \cdots \\
+X[2,0] & X[1,1] & X[0,2] & \cdots & X[k-2,2] & X[k-1,2] & \cdots \\
+\vdots & \vdots & \vdots & \ddots & \vdots & \vdots & \ddots \\
+X[k,0] & X[k-1,1] & X[k-2,2] & \cdots & & & \\
+0 & X[k,1] & X[k-1,2] & \cdots & & & \\
+\vdots & \vdots & \vdots & \ddots & & &
+\end{array}\right] & \mbox{if uplo = 'L'}, \\
+&
+\left[ \begin{array}{ccccccc}
+X[k,0] & X[k-1,1] & X[k-2,2] & \cdots & X[0,k] & 0 & \cdots \\
+X[k-1,1] & X[k,1] & X[k-1,2] & \cdots & X[1,k] & X[0,k+1] & \cdots \\
+X[k-2,2] & X[k-1,2] & X[k,2] & \cdots & X[2,k] & X[1,k+1] & \cdots \\
+\vdots & \vdots & \vdots & \ddots & \vdots & \vdots & \ddots \\
+X[0,k] & X[1,k] & X[2,k] & \cdots & & & \\
+0 & X[0,k+1] & X[1,k+1] & \cdots & & & \\
+\vdots & \vdots & \vdots & \ddots & & &
+\end{array}\right] & \mbox{if uplo='U'}.
+\end{eqnarray*}%
+\lthtmldisplayZ
+\lthtmlcheckvsize\clearpage}
+
+{\newpage\clearpage
+\lthtmldisplayA{eqnarraystar4102}%
+\begin{eqnarray*}
+& \left[ \begin{array}{ccccccc}
+\Re X[0,0] & \bar X[1,0] & \bar X[2,0] & \cdots & \bar X[k,0] & 0 & \cdots \\
+X[1,0] & \Re X[0,1] & \bar X[1,1] & \cdots & \bar X[k-1,1] & \bar X[k,1] & \cdots \\
+X[2,0] & X[1,1] & \Re X[0,2] & \cdots & \bar X[k-2,2] & \bar X[k-1,2] & \cdots \\
+\vdots & \vdots & \vdots & \ddots & \vdots & \vdots & \ddots \\
+X[k,0] & X[k-1,1] & X[k-2,2] & \cdots & & & \\
+0 & X[k,1] & X[k-1,2] & \cdots & & & \\
+\vdots & \vdots & \vdots & \ddots & & &
+\end{array}\right] & \mbox{if uplo = 'L'}, \\
+&
+\left[ \begin{array}{ccccccc}
+\Re X[k,0] & X[k-1,1] & X[k-2,2] & \cdots & X[0,k] & 0 & \cdots \\
+\bar X[k-1,1] & \Re X[k,1] & X[k-1,2] & \cdots & X[1,k] & X[0,k+1] & \cdots \\
+\bar X[k-2,2] & \bar X[k-1,2] & \Re X[k,2] & \cdots & X[2,k] & X[1,k+1] & \cdots \\
+\vdots & \vdots & \vdots & \ddots & \vdots & \vdots & \ddots \\
+\bar X[0,k] & \bar X[1,k] & \bar X[2,k] & \cdots & & & \\
+0 & \bar X[0,k+1] & \bar X[1,k+1] & \cdots & & & \\
+\vdots & \vdots & \vdots & \ddots & & &
+\end{array}\right] & \mbox{if uplo='U'}.
+\end{eqnarray*}%
+\lthtmldisplayZ
+\lthtmlcheckvsize\clearpage}
+
+{\newpage\clearpage
+\lthtmldisplayA{eqnarraystar4107}%
+\begin{eqnarray*}
+& \left[ \begin{array}{cccc}
+X[0,0] & 0 & 0 & \cdots \\
+X[1,0] & X[0,1] & 0 & \cdots \\
+X[2,0] & X[1,1] & X[0,2] & \cdots \\
+\vdots & \vdots & \vdots & \ddots \\
+X[k,0] & X[k-1,1] & X[k-2,2] & \cdots \\
+0 & X[k,1] & X[k-1,1] & \cdots \\
+\vdots & \vdots & \vdots & \ddots
+\end{array}\right] & \mbox{if uplo = 'L' and diag = 'N'}, \\
+& \left[ \begin{array}{cccc}
+1 & 0 & 0 & \cdots \\
+X[1,0] & 1 & 0 & \cdots \\
+X[2,0] & X[1,1] & 1 & \cdots \\
+\vdots & \vdots & \vdots & \ddots \\
+X[k,0] & X[k-1,1] & X[k-2,2] & \cdots \\
+0 & X[k,1] & X[k-1,2] & \cdots \\
+\vdots & \vdots & \vdots & \ddots
+\end{array}\right] & \mbox{if uplo = 'L' and diag = 'U'}, \\
+& \left[ \begin{array}{ccccccc}
+X[k,0] & X[k-1,1] & X[k-2,3] & \cdots & X[0,k] & 0 & \cdots\\
+0 & X[k,1] & X[k-1,2] & \cdots & X[1,k] & X[0,k+1] & \cdots \\
+0 & 0 & X[k,2] & \cdots & X[2,k] & X[1,k+1] & \cdots \\
+\vdots & \vdots & \vdots & \ddots & \vdots & \vdots & \ddots
+\end{array}\right] & \mbox{if uplo = 'U' and diag = 'N'}, \\
+& \left[ \begin{array}{ccccccc}
+1 & X[k-1,1] & X[k-2,3] & \cdots & X[0,k] & 0 & \cdots\\
+0 & 1 & X[k-1,2] & \cdots & X[1,k] & X[0,k+1] & \cdots \\
+0 & 0 & 1 & \cdots & X[2,k] & X[1,k+1] & \cdots \\
+\vdots & \vdots & \vdots & \ddots & \vdots & \vdots & \ddots
+\end{array}\right] & \mbox{if uplo = 'U' and diag = 'U'}.
+\end{eqnarray*}%
+\lthtmldisplayZ
+\lthtmlcheckvsize\clearpage}
+
+\stepcounter{section}
+{\newpage\clearpage
+\lthtmldisplayA{displaymath4587}%
+\begin{displaymath}
+x := \alpha x.
+\end{displaymath}%
+\lthtmldisplayZ
+\lthtmlcheckvsize\clearpage}
+
+{\newpage\clearpage
+\lthtmldisplayA{displaymath4588}%
+\begin{displaymath}
+ \|x\|_2.
+\end{displaymath}%
+\lthtmldisplayZ
+\lthtmlcheckvsize\clearpage}
+
+{\newpage\clearpage
+\lthtmldisplayA{displaymath4589}%
+\begin{displaymath}
+\|x\|_1 \quad \mbox{($x$\ real)}, \qquad
+\|\Re x\|_1 + \|\Im x\|_1 \quad \mbox{($x$\ complex)}.
+\end{displaymath}%
+\lthtmldisplayZ
+\lthtmlcheckvsize\clearpage}
+
+{\newpage\clearpage
+\lthtmldisplayA{displaymath4590}%
+\begin{displaymath}
+ \mathop{\rm argmax}_{k=0,\ldots,n-1} |x_k| \quad \mbox{($x$\ real)}, \qquad
+ \mathop{\rm argmax}_{k=0,\ldots,n-1} |\Re x_k| + |\Im x_k| \quad
+ \mbox{($x$\ complex)}.
+\end{displaymath}%
+\lthtmldisplayZ
+\lthtmlcheckvsize\clearpage}
+
+{\newpage\clearpage
+\lthtmldisplayA{displaymath4591}%
+\begin{displaymath}
+ x \leftrightarrow y.
+\end{displaymath}%
+\lthtmldisplayZ
+\lthtmlcheckvsize\clearpage}
+
+{\newpage\clearpage
+\lthtmldisplayA{displaymath4592}%
+\begin{displaymath}
+ y := x.
+\end{displaymath}%
+\lthtmldisplayZ
+\lthtmlcheckvsize\clearpage}
+
+{\newpage\clearpage
+\lthtmldisplayA{displaymath4593}%
+\begin{displaymath}
+y := \alpha x + y.
+\end{displaymath}%
+\lthtmldisplayZ
+\lthtmlcheckvsize\clearpage}
+
+{\newpage\clearpage
+\lthtmldisplayA{displaymath4594}%
+\begin{displaymath}
+x^Hy.
+\end{displaymath}%
+\lthtmldisplayZ
+\lthtmlcheckvsize\clearpage}
+
+{\newpage\clearpage
+\lthtmldisplayA{displaymath4595}%
+\begin{displaymath}
+x^Ty.
+\end{displaymath}%
+\lthtmldisplayZ
+\lthtmlcheckvsize\clearpage}
+
+\stepcounter{section}
+{\newpage\clearpage
+\lthtmldisplayA{displaymath4596}%
+\begin{displaymath}
+y := \alpha Ax + \beta y \quad (\mathrm{trans} = \mathrm{'N'}),
+ \qquad
+y := \alpha A^T x + \beta y \quad (\mathrm{trans} = \mathrm{'T'}),
+ \qquad
+y := \alpha A^H x + \beta y \quad (\mathrm{trans} = \mathrm{'C'}).
+\end{displaymath}%
+\lthtmldisplayZ
+\lthtmlcheckvsize\clearpage}
+
+{\newpage\clearpage
+\lthtmldisplayA{displaymath4597}%
+\begin{displaymath}
+ y := \alpha A x + \beta y,
+\end{displaymath}%
+\lthtmldisplayZ
+\lthtmlcheckvsize\clearpage}
+
+{\newpage\clearpage
+\lthtmldisplayA{displaymath4599}%
+\begin{displaymath}
+x := Ax \quad (\mathrm{trans} = \mathrm{'N'}), \qquad
+x := A^T x \quad (\mathrm{trans} = \mathrm{'T'}), \qquad
+x := A^H x \quad (\mathrm{trans} = \mathrm{'C'}),
+\end{displaymath}%
+\lthtmldisplayZ
+\lthtmlcheckvsize\clearpage}
+
+{\newpage\clearpage
+\lthtmldisplayA{displaymath4600}%
+\begin{displaymath}
+x := A^{-1}x \quad (\mathrm{trans} = \mathrm{'N'}), \qquad
+x := A^{-T}x \quad (\mathrm{trans} = \mathrm{'T'}), \qquad
+x := A^{-H}x \quad (\mathrm{trans} = \mathrm{'C'}),
+\end{displaymath}%
+\lthtmldisplayZ
+\lthtmlcheckvsize\clearpage}
+
+{\newpage\clearpage
+\lthtmldisplayA{displaymath4601}%
+\begin{displaymath}
+y := \alpha Ax + \beta y \quad (\mathrm{trans} = \mathrm{'N'}), \qquad
+y := \alpha A^T x + \beta y \quad (\mathrm{trans} = \mathrm{'T'}),
+\qquad
+y := \alpha A^H x + \beta y \quad (\mathrm{trans} = \mathrm{'C'}),
+\end{displaymath}%
+\lthtmldisplayZ
+\lthtmlcheckvsize\clearpage}
+
+{\newpage\clearpage
+\lthtmldisplayA{displaymath4604}%
+\begin{displaymath}
+x := Ax \quad (\mathrm{trans} = \mathrm{'N'}), \qquad
+x := A^T x \quad (\mathrm{trans} = \mathrm{'T'}), \qquad
+x := A^H x \quad (\mathrm{trans} = \mathrm{'C'}).
+\end{displaymath}%
+\lthtmldisplayZ
+\lthtmlcheckvsize\clearpage}
+
+{\newpage\clearpage
+\lthtmldisplayA{displaymath4605}%
+\begin{displaymath}
+x := A^{-1}x \quad (\mathrm{trans} = \mathrm{'N'}), \qquad
+x := A^{-T} x \quad (\mathrm{trans} = \mathrm{'T'}), \qquad
+x := A^{-H} x \quad (\mathrm{trans} = \mathrm{'T'}),
+\end{displaymath}%
+\lthtmldisplayZ
+\lthtmlcheckvsize\clearpage}
+
+{\newpage\clearpage
+\lthtmldisplayA{displaymath4606}%
+\begin{displaymath}
+A := A + \alpha x y^H,
+\end{displaymath}%
+\lthtmldisplayZ
+\lthtmlcheckvsize\clearpage}
+
+{\newpage\clearpage
+\lthtmldisplayA{displaymath4607}%
+\begin{displaymath}
+A := A + \alpha x y^T,
+\end{displaymath}%
+\lthtmldisplayZ
+\lthtmlcheckvsize\clearpage}
+
+{\newpage\clearpage
+\lthtmldisplayA{displaymath4608}%
+\begin{displaymath}
+ A := A + \alpha xx^T,
+\end{displaymath}%
+\lthtmldisplayZ
+\lthtmlcheckvsize\clearpage}
+
+{\newpage\clearpage
+\lthtmldisplayA{displaymath4609}%
+\begin{displaymath}
+ A := A + \alpha xx^H,
+\end{displaymath}%
+\lthtmldisplayZ
+\lthtmlcheckvsize\clearpage}
+
+{\newpage\clearpage
+\lthtmldisplayA{displaymath4610}%
+\begin{displaymath}
+ A := A + \alpha (xy^T + yx^T),
+\end{displaymath}%
+\lthtmldisplayZ
+\lthtmlcheckvsize\clearpage}
+
+{\newpage\clearpage
+\lthtmldisplayA{displaymath4611}%
+\begin{displaymath}
+ A := A + \alpha xy^H + \bar \alpha yx^H,
+\end{displaymath}%
+\lthtmldisplayZ
+\lthtmlcheckvsize\clearpage}
+
+{\newpage\clearpage
+\lthtmldisplayA{displaymath4612}%
+\begin{displaymath}
+A = \left[\begin{array}{rrrr}
+ 1 & 6 & 0 & 0 \\
+ 2 & -4 & 3 & 0 \\
+ 0 & -3 & -1 & 1
+ \end{array}\right]
+\end{displaymath}%
+\lthtmldisplayZ
+\lthtmlcheckvsize\clearpage}
+
+\stepcounter{section}
+{\newpage\clearpage
+\lthtmldisplayA{displaymath4613}%
+\begin{displaymath}
+ C := \alpha \mathop{\mathrm{op}}(A) \mathop{\mathrm{op}}(B) + \beta C
+\end{displaymath}%
+\lthtmldisplayZ
+\lthtmlcheckvsize\clearpage}
+
+{\newpage\clearpage
+\lthtmldisplayA{displaymath4614}%
+\begin{displaymath}
+\mathop{\mathrm{op}}(A) = \left\{ \begin{array}{ll}
+ A & \mathrm{transA} = \mathrm{'N'} \\
+ A^T & \mathrm{transA} = \mathrm{'T'} \\
+ A^H & \mathrm{transA} = \mathrm{'C'} \end{array} \right.
+\qquad
+\mathop{\mathrm{op}}(B) = \left\{ \begin{array}{ll}
+ B & \mathrm{transB} = \mathrm{'N'} \\
+ B^T & \mathrm{transB} = \mathrm{'T'} \\
+ B^H & \mathrm{transB} = \mathrm{'C'}. \end{array} \right.
+\end{displaymath}%
+\lthtmldisplayZ
+\lthtmlcheckvsize\clearpage}
+
+{\newpage\clearpage
+\lthtmldisplayA{displaymath4615}%
+\begin{displaymath}
+ C := \alpha AB + \beta C \quad (\mathrm{side} = \mathrm{'L'}), \qquad
+ C := \alpha BA + \beta C \quad (\mathrm{side} = \mathrm{'R'}).
+\end{displaymath}%
+\lthtmldisplayZ
+\lthtmlcheckvsize\clearpage}
+
+{\newpage\clearpage
+\lthtmldisplayA{displaymath4617}%
+\begin{displaymath}
+ B := \alpha\mathop{\mathrm{op}}(A)B \quad (\mathrm{side} = \mathrm{'L'}), \qquad
+ B := \alpha B\mathop{\mathrm{op}}(A) \quad (\mathrm{side} = \mathrm{'R'}), \qquad
+ \mathop{\mathrm{op}}(A) = \left\{ \begin{array}{ll}
+ A & \mathrm{transA} = \mathrm{'N'} \\
+ A^T & \mathrm{transA} = \mathrm{'T'} \\
+ A^H & \mathrm{transA} = \mathrm{'C'}. \end{array} \right.
+\end{displaymath}%
+\lthtmldisplayZ
+\lthtmlcheckvsize\clearpage}
+
+{\newpage\clearpage
+\lthtmldisplayA{displaymath4618}%
+\begin{displaymath}
+ B := \alpha \mathop{\mathrm{op}}(A)^{-1}B \quad (\mathrm{side} = \mathrm{'L'}), \qquad
+ B := \alpha B\mathop{\mathrm{op}}(A)^{-1} \quad (\mathrm{side} = \mathrm{'R'}), \qquad
+ \mathop{\mathrm{op}}(A) = \left\{ \begin{array}{ll}
+ A & \mathrm{transA} = \mathrm{'N'} \\
+ A^T & \mathrm{transA} = \mathrm{'T'} \\
+ A^H & \mathrm{transA} = \mathrm{'C'}, \end{array} \right.
+\end{displaymath}%
+\lthtmldisplayZ
+\lthtmlcheckvsize\clearpage}
+
+{\newpage\clearpage
+\lthtmldisplayA{displaymath4619}%
+\begin{displaymath}
+ C := \alpha AA^T + \beta C \quad (\mathrm{trans} = \mathrm{'N'}),
+ \qquad
+ C := \alpha A^TA + \beta C \quad (\mathrm{trans} = \mathrm{'T'}),
+\end{displaymath}%
+\lthtmldisplayZ
+\lthtmlcheckvsize\clearpage}
+
+{\newpage\clearpage
+\lthtmldisplayA{displaymath4620}%
+\begin{displaymath}
+ C := \alpha AA^H + \beta C \quad (\mathrm{trans} = \mathrm{'N'}),
+ \qquad
+ C := \alpha A^HA + \beta C \quad (\mathrm{trans} = \mathrm{'C'}),
+\end{displaymath}%
+\lthtmldisplayZ
+\lthtmlcheckvsize\clearpage}
+
+{\newpage\clearpage
+\lthtmldisplayA{displaymath4621}%
+\begin{displaymath}
+ C := \alpha (AB^T + BA^T) + \beta C \quad
+ (\mathrm{trans} = \mathrm{'N'}), \qquad
+ C := \alpha (A^TB + B^TA) + \beta C \quad
+ (\mathrm{trans} = \mathrm{'T'}).
+\end{displaymath}%
+\lthtmldisplayZ
+\lthtmlcheckvsize\clearpage}
+
+{\newpage\clearpage
+\lthtmldisplayA{displaymath4622}%
+\begin{displaymath}
+ C := \alpha AB^H + \bar \alpha BA^H + \beta C \quad
+ (\mathrm{trans} = \mathrm{'N'}), \qquad
+ C := \alpha A^HB + \bar\alpha B^HA + \beta C \quad
+ (\mathrm{trans} = \mathrm{'C'}),
+\end{displaymath}%
+\lthtmldisplayZ
+\lthtmlcheckvsize\clearpage}
+
+\stepcounter{chapter}
+{\newpage\clearpage
+\lthtmlinlinemathA{tex2html_wrap_inline8374}%
+$\mathrm{{}^T}$%
+\lthtmlinlinemathZ
+\lthtmlcheckvsize\clearpage}
+
+\stepcounter{section}
+{\newpage\clearpage
+\lthtmldisplayA{displaymath8292}%
+\begin{displaymath}
+ A X = B,
+\end{displaymath}%
+\lthtmldisplayZ
+\lthtmlcheckvsize\clearpage}
+
+{\newpage\clearpage
+\lthtmldisplayA{displaymath8293}%
+\begin{displaymath}
+ A = PLU
+\end{displaymath}%
+\lthtmldisplayZ
+\lthtmlcheckvsize\clearpage}
+
+{\newpage\clearpage
+\lthtmldisplayA{displaymath8294}%
+\begin{displaymath}
+ AX=B \quad (\mathrm{trans} = \mathrm{'N'}), \qquad
+ A^TX=B \quad (\mathrm{trans} = \mathrm{'T'}), \qquad
+ A^HX=B \quad (\mathrm{trans} = \mathrm{'C'}),
+\end{displaymath}%
+\lthtmldisplayZ
+\lthtmlcheckvsize\clearpage}
+
+{\newpage\clearpage
+\lthtmldisplayA{displaymath8295}%
+\begin{displaymath}
+ x = (A^{-1} + A^{-T})b
+\end{displaymath}%
+\lthtmldisplayZ
+\lthtmlcheckvsize\clearpage}
+
+{\newpage\clearpage
+\lthtmldisplayA{displaymath8298}%
+\begin{displaymath}
+ A = \left[ \begin{array}{cccc}
+ 1 & 2 & 0 & 0 \\
+ 3 & 4 & 5 & 0 \\
+ 6 & 7 & 8 & 9 \\
+ 0 & 10 & 11 & 12
+ \end{array}\right], \qquad x = \left[\begin{array}{c} 1 \\1 \\1 \\1
+ \end{array}\right].
+\end{displaymath}%
+\lthtmldisplayZ
+\lthtmlcheckvsize\clearpage}
+
+\stepcounter{section}
+{\newpage\clearpage
+\lthtmldisplayA{displaymath8302}%
+\begin{displaymath}
+ A = LL^T \qquad \mbox{or} \qquad A = LL^H
+\end{displaymath}%
+\lthtmldisplayZ
+\lthtmlcheckvsize\clearpage}
+
+{\newpage\clearpage
+\lthtmldisplayA{displaymath8303}%
+\begin{displaymath}
+ AX=B
+\end{displaymath}%
+\lthtmldisplayZ
+\lthtmlcheckvsize\clearpage}
+
+{\newpage\clearpage
+\lthtmldisplayA{displaymath7646}%
+\begin{displaymath}
+ \left[ \begin{array}{cc}
+ -\mbox{\bf diag}\,(d)^2 & A \\
+ A^T & 0 \end{array} \right]
+ \left[ \begin{array}{c} x_1 \\x_2 \end{array} \right]
+ = \left[ \begin{array}{c} b_1 \\b_2 \end{array} \right]
+\end{displaymath}%
+\lthtmldisplayZ
+\lthtmlcheckvsize\clearpage}
+
+{\newpage\clearpage
+\lthtmldisplayA{displaymath8304}%
+\begin{displaymath}
+ A^T \mbox{\bf diag}\,(d)^{-2}A x_2 = b_2 + A^T \mbox{\bf diag}\,(d)^{-2} b_1, \qquad
+ \mbox{\bf diag}\,(d)^2 x_1 = Ax_2 - b_1.
+\end{displaymath}%
+\lthtmldisplayZ
+\lthtmlcheckvsize\clearpage}
+
+{\newpage\clearpage
+\lthtmlinlinemathA{tex2html_wrap_inline8376}%
+${}\mathrm{^T}$%
+\lthtmlinlinemathZ
+\lthtmlcheckvsize\clearpage}
+
+{\newpage\clearpage
+\lthtmlinlinemathA{tex2html_wrap_inline8378}%
+${}\mathrm{^H}$%
+\lthtmlinlinemathZ
+\lthtmlcheckvsize\clearpage}
+
+\stepcounter{section}
+{\newpage\clearpage
+\lthtmldisplayA{displaymath8311}%
+\begin{displaymath}
+ PAP^T = LDL^T
+\end{displaymath}%
+\lthtmldisplayZ
+\lthtmlcheckvsize\clearpage}
+
+{\newpage\clearpage
+\lthtmlinlinemathA{tex2html_wrap_inline8390}%
+$A$%
+\lthtmlinlinemathZ
+\lthtmlcheckvsize\clearpage}
+
+{\newpage\clearpage
+\lthtmldisplayA{displaymath8314}%
+\begin{displaymath}
+ PAP^T = LDL^H
+\end{displaymath}%
+\lthtmldisplayZ
+\lthtmlcheckvsize\clearpage}
+
+\stepcounter{section}
+\stepcounter{section}
+{\newpage\clearpage
+\lthtmldisplayA{displaymath8318}%
+\begin{displaymath}
+ \begin{array}{ll}
+ \mbox{minimize} & \|AX-B\|_F.
+ \end{array}
+\end{displaymath}%
+\lthtmldisplayZ
+\lthtmlcheckvsize\clearpage}
+
+{\newpage\clearpage
+\lthtmldisplayA{displaymath8319}%
+\begin{displaymath}
+ \begin{array}{ll}
+ \mbox{minimize} & \|X\|_F \\
+ \mbox{subject to} & AX = B.
+ \end{array}
+\end{displaymath}%
+\lthtmldisplayZ
+\lthtmlcheckvsize\clearpage}
+
+{\newpage\clearpage
+\lthtmldisplayA{displaymath8320}%
+\begin{displaymath}
+ \begin{array}{ll}
+ \mbox{minimize} & \|X\|_F \\
+ \mbox{subject to} & A^TX=B.
+ \end{array}
+\end{displaymath}%
+\lthtmldisplayZ
+\lthtmlcheckvsize\clearpage}
+
+{\newpage\clearpage
+\lthtmldisplayA{displaymath8321}%
+\begin{displaymath}
+ \begin{array}{ll}
+ \mbox{minimize} & \|A^TX-B\|_F.
+ \end{array}
+\end{displaymath}%
+\lthtmldisplayZ
+\lthtmlcheckvsize\clearpage}
+
+{\newpage\clearpage
+\lthtmldisplayA{displaymath8322}%
+\begin{displaymath}
+ \begin{array}{ll}
+ \mbox{minimize} & \|X\|_F \\
+ \mbox{subject to} & A^HX=B.
+ \end{array}
+\end{displaymath}%
+\lthtmldisplayZ
+\lthtmlcheckvsize\clearpage}
+
+{\newpage\clearpage
+\lthtmldisplayA{displaymath8323}%
+\begin{displaymath}
+ \begin{array}{ll}
+ \mbox{minimize} & \|A^HX-B\|_F.
+ \end{array}
+\end{displaymath}%
+\lthtmldisplayZ
+\lthtmlcheckvsize\clearpage}
+
+{\newpage\clearpage
+\lthtmldisplayA{displaymath8324}%
+\begin{displaymath}
+ A = Q R.
+\end{displaymath}%
+\lthtmldisplayZ
+\lthtmlcheckvsize\clearpage}
+
+{\newpage\clearpage
+\lthtmldisplayA{displaymath8325}%
+\begin{displaymath}
+ C := \mathop{\mathrm{op}}(Q)C \quad (\mathrm{side} = \mathrm{'L'}), \qquad
+ C := C\mathop{\mathrm{op}}(Q) \quad (\mathrm{side} = \mathrm{'R'}), \qquad
+ \mathop{\mathrm{op}}(Q) = \left\{ \begin{array}{ll}
+ Q & \mathrm{trans} = \mathrm{'N'} \\
+ Q^T & \mathrm{trans} = \mathrm{'T'},
+\end{array}\right.
+\end{displaymath}%
+\lthtmldisplayZ
+\lthtmlcheckvsize\clearpage}
+
+{\newpage\clearpage
+\lthtmldisplayA{displaymath8326}%
+\begin{displaymath}
+ C := \mathop{\mathrm{op}}(Q)C \quad (\mathrm{side} = \mathrm{'L'}), \qquad
+ C := C\mathop{\mathrm{op}}(Q) \quad (\mathrm{side} = \mathrm{'R'}), \qquad
+ \mathop{\mathrm{op}}(Q) = \left\{ \begin{array}{ll}
+ Q & \mathrm{trans} = \mathrm{'N'} \\
+ Q^T & \mathrm{trans} = \mathrm{'T'} \\
+ Q^H & \mathrm{trans} = \mathrm{'C'},
+\end{array}\right.
+\end{displaymath}%
+\lthtmldisplayZ
+\lthtmlcheckvsize\clearpage}
+
+\stepcounter{section}
+{\newpage\clearpage
+\lthtmldisplayA{displaymath8327}%
+\begin{displaymath}
+ A = V\mbox{\bf diag}\,(\lambda)V^T,\qquad V^TV = I.
+\end{displaymath}%
+\lthtmldisplayZ
+\lthtmlcheckvsize\clearpage}
+
+{\newpage\clearpage
+\lthtmldisplayA{displaymath8328}%
+\begin{displaymath}
+ A = V\mbox{\bf diag}\,(\lambda)V^H,\qquad V^HV = I.
+\end{displaymath}%
+\lthtmldisplayZ
+\lthtmlcheckvsize\clearpage}
+
+\stepcounter{section}
+{\newpage\clearpage
+\lthtmldisplayA{displaymath8126}%
+\begin{displaymath}
+ AZ = BZ\mbox{\bf diag}\,(\lambda)\quad \mbox{(type 1)}, \qquad
+ ABZ = Z\mbox{\bf diag}\,(\lambda) \quad \mbox{(type 2)}, \qquad
+ BAZ = Z\mbox{\bf diag}\,(\lambda) \quad \mbox{(type 3)},
+\end{displaymath}%
+\lthtmldisplayZ
+\lthtmlcheckvsize\clearpage}
+
+{\newpage\clearpage
+\lthtmlinlinemathA{tex2html_wrap_inline8408}%
+$B$%
+\lthtmlinlinemathZ
+\lthtmlcheckvsize\clearpage}
+
+{\newpage\clearpage
+\lthtmldisplayA{displaymath8329}%
+\begin{displaymath}
+ Z^H BZ = I \quad \mbox{(types 1 and 2)}, \qquad
+ Z^H B^{-1}Z = I \quad \mbox{(type 3)}.
+\end{displaymath}%
+\lthtmldisplayZ
+\lthtmlcheckvsize\clearpage}
+
+\stepcounter{section}
+{\newpage\clearpage
+\lthtmldisplayA{displaymath8330}%
+\begin{displaymath}
+ A = U \Sigma V^T, \qquad A = U \Sigma V^H
+\end{displaymath}%
+\lthtmldisplayZ
+\lthtmlcheckvsize\clearpage}
+
+{\newpage\clearpage
+\lthtmlinlinemathA{tex2html_wrap_inline8412}%
+$V^H$%
+\lthtmlinlinemathZ
+\lthtmlcheckvsize\clearpage}
+
+\stepcounter{section}
+{\newpage\clearpage
+\lthtmldisplayA{displaymath8331}%
+\begin{displaymath}
+ \begin{array}{ll}
+ \mbox{minimize} & -\sum_{i=1}^m \log(b_i-a_i^Tx).
+ \end{array}
+\end{displaymath}%
+\lthtmldisplayZ
+\lthtmlcheckvsize\clearpage}
+
+{\newpage\clearpage
+\lthtmldisplayA{displaymath8332}%
+\begin{displaymath}
+ A^T \mbox{\bf diag}\,(b-Ax)^{-2} A v = -\mbox{\bf diag}\,(b-Ax)^{-1}{\bf 1}
+\end{displaymath}%
+\lthtmldisplayZ
+\lthtmlcheckvsize\clearpage}
+
+{\newpage\clearpage
+\lthtmlinlinemathA{tex2html_wrap_inline8414}%
+$a_i^T$%
+\lthtmlinlinemathZ
+\lthtmlcheckvsize\clearpage}
+
+\stepcounter{chapter}
+\stepcounter{section}
+{\newpage\clearpage
+\lthtmldisplayA{displaymath10727}%
+\begin{displaymath}
+ X[k,:] := \sum_{j=0}^{n-1} e^{-2\pi j k \sqrt{-1}/n} X[j,:],
+ \qquad k=0,\ldots,n-1.
+\end{displaymath}%
+\lthtmldisplayZ
+\lthtmlcheckvsize\clearpage}
+
+{\newpage\clearpage
+\lthtmldisplayA{displaymath10728}%
+\begin{displaymath}
+ X[k,:] :=
+ \frac{1}{n} \sum_{j=0}^{n-1} e^{2\pi j k \sqrt{-1}/n} X[j,:],
+ \qquad k=0,\ldots,n-1.
+\end{displaymath}%
+\lthtmldisplayZ
+\lthtmlcheckvsize\clearpage}
+
+\stepcounter{section}
+{\newpage\clearpage
+\lthtmldisplayA{eqnarraystar10717}%
+\begin{eqnarray*}
+\mbox{DCT-I:} \qquad
+ X[k,:] & := & X[0,:] + (-1)^k X[n-1,:] +
+ 2 \sum_{j=1}^{n-2} X[j,:] \cos(\pi j k /(n-1)),
+ \qquad k=0,\ldots,n-1.\\
+\mbox{DCT-II:} \qquad
+ X[k,:] & := & 2 \sum_{j=0}^{n-1} X[j,:] \cos(\pi(j+1/2)k/n),
+ \qquad k=0,\ldots,n-1.\\
+\mbox{DCT-III:} \qquad
+ X[k,:] & := & X[0,:] + 2 \sum_{j=1}^{n-1} X[j,:] \cos(\pi j(k+1/2)/n),
+ \qquad k=0,\ldots,n-1.\\
+\mbox{DCT-IV:} \qquad
+ X[k,:] & := & 2 \sum_{j=0}^{n-1} X[j,:] \cos(\pi (j+1/2)(k+1/2)/n),
+ \qquad k=0,\ldots,n-1.
+\end{eqnarray*}%
+\lthtmldisplayZ
+\lthtmlcheckvsize\clearpage}
+
+\stepcounter{section}
+{\newpage\clearpage
+\lthtmldisplayA{eqnarraystar10723}%
+\begin{eqnarray*}
+\mbox{DST-I:} \qquad
+ X[k,:] & := &
+ 2 \sum_{j=0}^{n-1} X[j,:] \sin(\pi(j+1)(k+1)/(n+1)),
+ \qquad k=0,\ldots,n-1.\\
+\mbox{DST-II:} \qquad
+ X[k,:] & := & 2 \sum_{j=0}^{n-1} X[j,:] \sin(\pi(j+1/2)(k+1)/n),
+ \qquad k=0,\ldots,n-1.\\
+\mbox{DST-III:} \qquad
+ X[k,:] & := & (-1)^k X[n-1,:] + 2 \sum_{j=0}^{n-2}
+ X[j,:] \sin(\pi(j+1)(k+1/2)/n),
+ \qquad k=0,\ldots,n-1. \\
+\mbox{DST-IV:} \qquad
+ X[k,:] & := & 2 \sum_{j=0}^{n-1} X[j,:] \sin(\pi (j+1/2)(k+1/2)/n),
+ \qquad k=0,\ldots,n-1.
+\end{eqnarray*}%
+\lthtmldisplayZ
+\lthtmlcheckvsize\clearpage}
+
+\stepcounter{chapter}
+\stepcounter{section}
+{\newpage\clearpage
+\lthtmldisplayA{displaymath11487}%
+\begin{displaymath}
+ A = \left[ \begin{array}{rrrrr}
+ 0 & 2 & 0 & 0 & 3 \\
+ 2 & 0 & 0 & 0 & 0 \\
+ -1 & -2 & 0 & 4 & 0 \\
+ 0 & 0 & 1 & 0 & 0 \end{array} \right]
+\end{displaymath}%
+\lthtmldisplayZ
+\lthtmlcheckvsize\clearpage}
+
+{\newpage\clearpage
+\lthtmldisplayA{displaymath12088}%
+\begin{displaymath}
+(2,1,0), \qquad (-1,2,0), \qquad (2,0,1), \qquad (-2,2,1), \qquad
+(1,3,2), \qquad (4,2,3), \qquad (3,0,4).
+\end{displaymath}%
+\lthtmldisplayZ
+\lthtmlcheckvsize\clearpage}
+
+{\newpage\clearpage
+\lthtmldisplayA{displaymath12089}%
+\begin{displaymath}
+(2,1,0), \qquad (-1,2,0), \qquad (0,3,0), \qquad (2,0,1), \qquad
+ (-2,2,1), \qquad (1,3,2), \qquad (4,2,3), \qquad (3,0,4)
+\end{displaymath}%
+\lthtmldisplayZ
+\lthtmlcheckvsize\clearpage}
+
+\stepcounter{section}
+{\newpage\clearpage
+\lthtmldisplayA{displaymath11622}%
+\begin{displaymath}
+ A = \left[ \begin{array}{rrrrr}
+ 0 & 2 & 0 & 0 & 3 \\
+ 2 & 0 & 0 & 0 & 0 \\
+ 1 & 2 & 0 & 4 & 0 \\
+ 0 & 0 & 1 & 0 & 0 \end{array} \right]
+\end{displaymath}%
+\lthtmldisplayZ
+\lthtmlcheckvsize\clearpage}
+
+\stepcounter{section}
+\stepcounter{section}
+\stepcounter{section}
+\stepcounter{section}
+{\newpage\clearpage
+\lthtmldisplayA{displaymath12091}%
+\begin{displaymath}
+ y := \alpha A x + \beta y.
+\end{displaymath}%
+\lthtmldisplayZ
+\lthtmlcheckvsize\clearpage}
+
+{\newpage\clearpage
+\lthtmldisplayA{displaymath12095}%
+\begin{displaymath}
+C = A^TB, \qquad
+A = \left[ \begin{array}{ccc}
+0 & 1 & 0 \\1 & 0 & 1 \\0 & 1 & 0 \\1 & 0 & 0 \end{array}\right],
+\qquad
+B = \left[ \begin{array}{ccc}
+ 0 & -1 & 0 \\2 & 0 & 2 \\0 & 3 & 0 \\2 & 0 & 0
+ \end{array}\right].
+\end{displaymath}%
+\lthtmldisplayZ
+\lthtmlcheckvsize\clearpage}
+
+{\newpage\clearpage
+\lthtmldisplayA{displaymath12096}%
+\begin{displaymath}
+C = A^TD, \qquad
+D = \left[ \begin{array}{ccc}
+ 0 & 1 & 0 \\3 & 0 & -2 \\0 & 1 & 0 \\4 & 0 & 0
+ \end{array}\right].
+\end{displaymath}%
+\lthtmldisplayZ
+\lthtmlcheckvsize\clearpage}
+
+\stepcounter{chapter}
+\stepcounter{section}
+{\newpage\clearpage
+\lthtmldisplayA{displaymath13881}%
+\begin{displaymath}
+\left[ \begin{array}{rrrr}
+ 10 & 0 & 3 & 0 \\
+ 0 & 5 & 0 & -2 \\
+ 3 & 0 & 5 & 0 \\
+ 0 & -2 & 0 & 2
+\end{array}\right].
+\end{displaymath}%
+\lthtmldisplayZ
+\lthtmlcheckvsize\clearpage}
+
+\stepcounter{section}
+{\newpage\clearpage
+\lthtmldisplayA{displaymath13754}%
+\begin{displaymath}
+A = \left[\begin{array}{rrrrr}
+ 2 & 3 & 0 & 0 & 0 \\
+ 3 & 0 & 4 & 0 & 6 \\
+ 0 &-1 &-3 & 2 & 0 \\
+ 0 & 0 & 1 & 0 & 0 \\
+ 0 & 4 & 2 & 0 & 1
+ \end{array}\right].
+\end{displaymath}%
+\lthtmldisplayZ
+\lthtmlcheckvsize\clearpage}
+
+{\newpage\clearpage
+\lthtmldisplayA{displaymath13884}%
+\begin{displaymath}
+B = \left[\begin{array}{rrrrr}
+ 4 & 3 & 0 & 0 & 0 \\
+ 3 & 0 & 4 & 0 & 6 \\
+ 0 &-1 &-3 & 2 & 0 \\
+ 0 & 0 & 1 & 0 & 0 \\
+ 0 & 4 & 2 & 0 & 2
+ \end{array}\right],
+\end{displaymath}%
+\lthtmldisplayZ
+\lthtmlcheckvsize\clearpage}
+
+{\newpage\clearpage
+\lthtmldisplayA{displaymath13885}%
+\begin{displaymath}
+ x = A^{-T}B^{-1}A^{-1}{\bf 1}.
+\end{displaymath}%
+\lthtmldisplayZ
+\lthtmlcheckvsize\clearpage}
+
+\stepcounter{section}
+{\newpage\clearpage
+\lthtmlinlinemathA{tex2html_wrap_inline13901}%
+$\mathrm{{}^H}$%
+\lthtmlinlinemathZ
+\lthtmlcheckvsize\clearpage}
+
+{\newpage\clearpage
+\lthtmldisplayA{displaymath13792}%
+\begin{displaymath}
+\left[ \begin{array}{rrrr}
+ 10 & 0 & 3 & 0 \\
+ 0 & 5 & 0 & -2 \\
+ 3 & 0 & 5 & 0 \\
+ 0 & -2 & 0 & 2
+ \end{array}\right] X = \left[ \begin{array}{cc}
+ 0 & 4 \\1 & 5 \\2 & 6 \\3 & 7\end{array} \right].
+\end{displaymath}%
+\lthtmldisplayZ
+\lthtmlcheckvsize\clearpage}
+
+{\newpage\clearpage
+\lthtmldisplayA{displaymath13816}%
+\begin{displaymath}
+ PAP^T = LL^T, \qquad PAP^T = LL^H,
+\end{displaymath}%
+\lthtmldisplayZ
+\lthtmlcheckvsize\clearpage}
+
+{\newpage\clearpage
+\lthtmldisplayA{displaymath13818}%
+\begin{displaymath}
+ PAP^T = LDL^T, \qquad PAP^T = LDL^H,
+\end{displaymath}%
+\lthtmldisplayZ
+\lthtmlcheckvsize\clearpage}
+
+{\newpage\clearpage
+\lthtmlinlinemathA{tex2html_wrap_inline13903}%
+$AX=B$%
+\lthtmlinlinemathZ
+\lthtmlcheckvsize\clearpage}
+
+{\newpage\clearpage
+\lthtmlinlinemathA{tex2html_wrap_inline13905}%
+$LDL^TX=B$%
+\lthtmlinlinemathZ
+\lthtmlcheckvsize\clearpage}
+
+{\newpage\clearpage
+\lthtmlinlinemathA{tex2html_wrap_inline13907}%
+$LDLX=B$%
+\lthtmlinlinemathZ
+\lthtmlcheckvsize\clearpage}
+
+{\newpage\clearpage
+\lthtmlinlinemathA{tex2html_wrap_inline13909}%
+$DL^TX=B$%
+\lthtmlinlinemathZ
+\lthtmlcheckvsize\clearpage}
+
+{\newpage\clearpage
+\lthtmlinlinemathA{tex2html_wrap_inline13911}%
+$LX=B$%
+\lthtmlinlinemathZ
+\lthtmlcheckvsize\clearpage}
+
+{\newpage\clearpage
+\lthtmlinlinemathA{tex2html_wrap_inline13913}%
+$L^TX=B$%
+\lthtmlinlinemathZ
+\lthtmlcheckvsize\clearpage}
+
+{\newpage\clearpage
+\lthtmlinlinemathA{tex2html_wrap_inline13915}%
+$DX=B$%
+\lthtmlinlinemathZ
+\lthtmlcheckvsize\clearpage}
+
+{\newpage\clearpage
+\lthtmlinlinemathA{tex2html_wrap_inline13917}%
+$P^TX=B$%
+\lthtmlinlinemathZ
+\lthtmlcheckvsize\clearpage}
+
+{\newpage\clearpage
+\lthtmlinlinemathA{tex2html_wrap_inline13919}%
+$PX=B$%
+\lthtmlinlinemathZ
+\lthtmlcheckvsize\clearpage}
+
+{\newpage\clearpage
+\lthtmlinlinemathA{tex2html_wrap_inline13921}%
+$L^T$%
+\lthtmlinlinemathZ
+\lthtmlcheckvsize\clearpage}
+
+{\newpage\clearpage
+\lthtmlinlinemathA{tex2html_wrap_inline13923}%
+$L^H$%
+\lthtmlinlinemathZ
+\lthtmlcheckvsize\clearpage}
+
+\stepcounter{section}
+{\newpage\clearpage
+\lthtmldisplayA{displaymath13872}%
+\begin{displaymath}
+ \begin{array}{ll}
+ \mbox{minimize} & -\log\det K + \mathop{\bf tr}(KY)\\
+ \mbox{subject to} & K_{ij}=0,\quad (i,j) \not \in S.
+ \end{array}
+\end{displaymath}%
+\lthtmldisplayZ
+\lthtmlcheckvsize\clearpage}
+
+{\newpage\clearpage
+\lthtmlinlinemathA{tex2html_wrap_inline13927}%
+$Y$%
+\lthtmlinlinemathZ
+\lthtmlcheckvsize\clearpage}
+
+{\newpage\clearpage
+\lthtmlinlinemathA{tex2html_wrap_inline13929}%
+$S$%
+\lthtmlinlinemathZ
+\lthtmlcheckvsize\clearpage}
+
+{\newpage\clearpage
+\lthtmldisplayA{displaymath13887}%
+\begin{displaymath}
+ K(x) = E_1\mbox{\bf diag}\,(x)E_2^T+E_2\mbox{\bf diag}\,(x)E_1^T
+\end{displaymath}%
+\lthtmldisplayZ
+\lthtmlcheckvsize\clearpage}
+
+{\newpage\clearpage
+\lthtmldisplayA{displaymath13888}%
+\begin{displaymath}
+ E_1 = \left[ \begin{array}{cccc}
+ e_{i_1} & e_{i_2} & \cdots & e_{i_q} \end{array}\right], \qquad
+ E_2 = \left[ \begin{array}{cccc}
+ e_{j_1} & e_{j_2} & \cdots & e_{j_q} \end{array}\right],
+\end{displaymath}%
+\lthtmldisplayZ
+\lthtmlcheckvsize\clearpage}
+
+{\newpage\clearpage
+\lthtmldisplayA{displaymath13889}%
+\begin{displaymath}
+ \begin{array}{ll}
+ \mbox{minimize} & f(x) = -\log\det K(x) + \mathop{\bf tr}(K(x)Y).
+ \end{array}
+\end{displaymath}%
+\lthtmldisplayZ
+\lthtmlcheckvsize\clearpage}
+
+{\newpage\clearpage
+\lthtmldisplayA{eqnarraystar13565}%
+\begin{eqnarray*}
+\nabla f(x)
+&=& 2 \mbox{\bf diag}\,( E_1^T (Y - K(x)^{-1}) E_2)) \\
+&=& 2\mbox{\bf diag}\,(Y_{IJ} - \left(K(x)^{-1}\right)_{IJ})\\
+\nabla^2 f(x) & = &
+ 2 (E_1^T K(x)^{-1} E_1) \circ (E_2^T K(x)^{-1} E_2)
+ + 2 (E_1^T K(x)^{-1} E_2) \circ (E_2^T K(x)^{-1} E_1) \\
+&=& 2 \left(K(x)^{-1}\right)_{II} \circ
+ \left(K(x)^{-1}\right)_{JJ}
+ +2 \left(K(x)^{-1}\right)_{IJ} \circ
+ \left(K(x)^{-1}\right)_{JI},
+\end{eqnarray*}%
+\lthtmldisplayZ
+\lthtmlcheckvsize\clearpage}
+
+\stepcounter{chapter}
+{\newpage\clearpage
+\lthtmldisplayA{displaymath15763}%
+\begin{displaymath}
+ \begin{array}{ll}
+ \mbox{minimize} & c^T x \\
+ \mbox{subject to} & G x \preceq h \\& Ax = b.
+ \end{array}
+\end{displaymath}%
+\lthtmldisplayZ
+\lthtmlcheckvsize\clearpage}
+
+\stepcounter{section}
+{\newpage\clearpage
+\lthtmldisplayA{displaymath15279}%
+\begin{displaymath}
+ \begin{array}[t]{ll}
+ \mbox{minimize} & c^T x \\
+ \mbox{subject to} & G x + s = h \\& Ax = b \\& s \succeq 0
+ \end{array}
+ \qquad\qquad\qquad\qquad
+ \begin{array}[t]{ll}
+ \mbox{maximize} & -h^T z - b^T y \\
+ \mbox{subject to} & G^T z + A^T y + c = 0 \\
+ & z \succeq 0.
+ \end{array}
+\end{displaymath}%
+\lthtmldisplayZ
+\lthtmlcheckvsize\clearpage}
+
+{\newpage\clearpage
+\lthtmlinlinemathA{tex2html_wrap_inline15841}%
+$s \in C$%
+\lthtmlinlinemathZ
+\lthtmlcheckvsize\clearpage}
+
+{\newpage\clearpage
+\lthtmlinlinemathA{tex2html_wrap_inline15843}%
+$z\in C$%
+\lthtmlinlinemathZ
+\lthtmlcheckvsize\clearpage}
+
+{\newpage\clearpage
+\lthtmlinlinemathA{tex2html_wrap_inline15845}%
+$C$%
+\lthtmlinlinemathZ
+\lthtmlcheckvsize\clearpage}
+
+{\newpage\clearpage
+\lthtmldisplayA{displaymath15764}%
+\begin{displaymath}
+C = C_0 \times C_1 \times \cdots \times C_M \times C_{M+1} \times
+ \cdots \times C_{M+N}
+\end{displaymath}%
+\lthtmldisplayZ
+\lthtmlcheckvsize\clearpage}
+
+{\newpage\clearpage
+\lthtmldisplayA{displaymath15765}%
+\begin{displaymath}
+C_0 =
+ \{ u \in {\mbox{\bf R}}^l \;| \; u_k \geq 0, \; k=1, \ldots,l\}, \qquad
+C_{k+1} = \{ (u_0, u_1) \in {\mbox{\bf R}}\times {\mbox{\bf R}}^{q_{k}-1} \; | \;
+ u_0 \geq \|u_1\|_2 \}, \quad k=0,\ldots, M-1, \qquad
+C_{k+M+1} = \left\{ \mathop{\mathbf{vec}}(u) \; | \;
+ u \in {\mbox{\bf S}}^{p_k}_+ \right\}, \quad k=0,\ldots,N-1.
+\end{displaymath}%
+\lthtmldisplayZ
+\lthtmlcheckvsize\clearpage}
+
+{\newpage\clearpage
+\lthtmlinlinemathA{tex2html_wrap_inline15847}%
+$\mathop{\mathbf{vec}}(u)$%
+\lthtmlinlinemathZ
+\lthtmlcheckvsize\clearpage}
+
+{\newpage\clearpage
+\lthtmldisplayA{displaymath15766}%
+\begin{displaymath}
+ K = l + \sum_{k=0}^{M-1} q_k + \sum_{k=0}^{N-1} p_k^2.
+\end{displaymath}%
+\lthtmldisplayZ
+\lthtmlcheckvsize\clearpage}
+
+{\newpage\clearpage
+\lthtmldisplayA{displaymath15767}%
+\begin{displaymath}
+{\mbox{\bf R}}^l \times {\mbox{\bf R}}^{q_0} \times \cdots \times
+{\mbox{\bf R}}^{q_{M-1}} \times {\mbox{\bf R}}^{p_0^2} \times \cdots \times
+{\mbox{\bf R}}^{p_{N-1}^2},
+\end{displaymath}%
+\lthtmldisplayZ
+\lthtmlcheckvsize\clearpage}
+
+{\newpage\clearpage
+\lthtmlinlinemathA{tex2html_wrap_inline15849}%
+$l$%
+\lthtmlinlinemathZ
+\lthtmlcheckvsize\clearpage}
+
+{\newpage\clearpage
+\lthtmldisplayA{displaymath15768}%
+\begin{displaymath}
+ Gx + s = h, \qquad Ax=b, \qquad G^T z + A^T y + c = 0, \qquad
+ s \succeq 0, \qquad z \succeq 0, \qquad s^T z =0.
+\end{displaymath}%
+\lthtmldisplayZ
+\lthtmlcheckvsize\clearpage}
+
+{\newpage\clearpage
+\lthtmldisplayA{displaymath15769}%
+\begin{displaymath}
+ G^T z + A^T y = 0, \qquad h^T z + b^T y = -1, \qquad z \succeq 0.
+\end{displaymath}%
+\lthtmldisplayZ
+\lthtmlcheckvsize\clearpage}
+
+{\newpage\clearpage
+\lthtmldisplayA{displaymath15770}%
+\begin{displaymath}
+ Gx + s = 0, \qquad Ax=0, \qquad c^T x = -1, \qquad s \succeq 0.
+\end{displaymath}%
+\lthtmldisplayZ
+\lthtmlcheckvsize\clearpage}
+
+{\newpage\clearpage
+\lthtmldisplayA{displaymath15771}%
+\begin{displaymath}
+\mathop{\bf rank}(A) = p, \qquad
+\mathop{\bf rank}(\left[\begin{array}{c} G \\A \end{array}\right]) = n,
+\end{displaymath}%
+\lthtmldisplayZ
+\lthtmlcheckvsize\clearpage}
+
+{\newpage\clearpage
+\lthtmldisplayA{displaymath15772}%
+\begin{displaymath}
+\begin{array}{ll}
+\mbox{minimize} & -6x_1 - 4x_2 - 5x_3 \\*[1ex]
+\mbox{subject to} & 16x_1 - 14x_2 + 5x_3 \leq -3 \\*[1ex]
+ & 7x_1 + 2x_2 \leq 5 \\*[1ex]
+ & \left( (8x_1 + 13x_2 - 12x_3 - 2)^2 + (-8x_1 + 18x_2 + 6x_3 - 14)^2
+ + (x_1 - 3x_2 - 17x_3 - 13)^2\right)^{1/2} \leq -24x_1 - 7x_2 + 15x_3 + 12 \\*[1ex]
+ & (x_1^2 + x_2^2 + x_3^2)^{1/2} \leq 10 \\*[1ex]
+ & \left[\begin{array}{ccc}
+ 7x_1 + 3x_2 + 9x_3 & -5x_1 + 13x_2 + 6x_3 & x_1 - 6x_2 -6x_3 \\
+-5x_1 + 13x_2 + 6x_3 & x_1 + 12x_2 -7x_3 & -7x_1 -10x_2 - 7x_3 \\
+ x_1 - 6x_2 -6x_3 & -7x_1 -10x_2 -7 x_3 &
+-4x_1 -28 x_2 -11x_3 \end{array}\right]
+\preceq
+\left[\begin{array}{ccc}
+68 & -30 & -19 \\
+-30 & 99 & 23 \\
+-19 & 23 & 10 \end{array}\right].
+\end{array}
+\end{displaymath}%
+\lthtmldisplayZ
+\lthtmlcheckvsize\clearpage}
+
+\stepcounter{section}
+{\newpage\clearpage
+\lthtmldisplayA{displaymath15774}%
+\begin{displaymath}
+ \begin{array}[t]{ll}
+ \mbox{minimize} & -4x_1 - 5x_2 \\
+ \mbox{subject to} & 2x_1 + x_2 \leq 3 \\
+ & x_1 + 2x_2 \leq 3 \\
+ & x_1 \geq 0, \quad x_2 \geq 0.
+ \end{array}
+\end{displaymath}%
+\lthtmldisplayZ
+\lthtmlcheckvsize\clearpage}
+
+\stepcounter{section}
+{\newpage\clearpage
+\lthtmldisplayA{displaymath15775}%
+\begin{displaymath}
+ \begin{array}[t]{ll}
+ \mbox{minimize} & c^T x \\
+ \mbox{subject to}
+ & G_k x + s_k = h_k, \quad k = 0, \ldots, M \\
+ & Ax = b \\
+ & s_0 \succeq 0 \\
+ & s_{k0} \geq \|s_{k1}\|_2, \quad k = 1, \ldots, M
+ \end{array}
+ \qquad\qquad
+ \begin{array}[t]{ll}
+ \mbox{maximize} & - \sum_{k=0}^M h_k^Tz_k - b^T y \\
+ \mbox{subject to} & \sum_{k=0}^M G_k^T z_k + A^T y + c = 0 \\
+ & z_0 \succeq 0 \\
+ & z_{k0} \geq \|z_{k1}\|_2, \quad k=1,\ldots,M.
+ \end{array}
+\end{displaymath}%
+\lthtmldisplayZ
+\lthtmlcheckvsize\clearpage}
+
+{\newpage\clearpage
+\lthtmldisplayA{displaymath15776}%
+\begin{displaymath}
+ s_0 \succeq 0, \qquad z_0 \succeq 0
+\end{displaymath}%
+\lthtmldisplayZ
+\lthtmlcheckvsize\clearpage}
+
+{\newpage\clearpage
+\lthtmldisplayA{displaymath15777}%
+\begin{displaymath}
+ s_k = (s_{k0}, s_{k1}) \in{\mbox{\bf R}}\times{\mbox{\bf R}}^{q_{k}-1}, \qquad
+ z_k = (z_{k0}, z_{k1}) \in{\mbox{\bf R}}\times{\mbox{\bf R}}^{q_{k}-1}.
+\end{displaymath}%
+\lthtmldisplayZ
+\lthtmlcheckvsize\clearpage}
+
+{\newpage\clearpage
+\lthtmlinlinemathA{tex2html_wrap_inline15855}%
+$G_0$%
+\lthtmlinlinemathZ
+\lthtmlcheckvsize\clearpage}
+
+{\newpage\clearpage
+\lthtmlinlinemathA{tex2html_wrap_inline15857}%
+$h_0$%
+\lthtmlinlinemathZ
+\lthtmlcheckvsize\clearpage}
+
+{\newpage\clearpage
+\lthtmlinlinemathA{tex2html_wrap_inline15859}%
+$M$%
+\lthtmlinlinemathZ
+\lthtmlcheckvsize\clearpage}
+
+{\newpage\clearpage
+\lthtmldisplayA{displaymath15778}%
+\begin{displaymath}
+\begin{array}{ll}
+\mbox{minimize} & -2x_1 + x_2 + 5x_3 \\*[2ex]
+\mbox{subject to}
+ & \left\| \left[\begin{array}{c}
+ -13 x_1 + 3x_2 + 5x_3 - 3 \\-12 x_1 + 12x_2 - 6x_3 - 2
+ \end{array}\right] \right\|_2 \leq -12 x_1 - 6 x_2 + 5x_3 - 12 \\*[2ex]
+ & \left\| \left[\begin{array}{c}
+ -3x_1 + 6x_2 + 2x_3 \\x_1 + 9x_2 + 2x_3 + 3 \\-x_1 - 19 x_2 + 3 x_3 - 42
+ \end{array}\right] \right\|_2 \leq
+ -3x_1 + 6x_2 - 10x_3 + 27.
+\end{array}
+\end{displaymath}%
+\lthtmldisplayZ
+\lthtmlcheckvsize\clearpage}
+
+\stepcounter{section}
+{\newpage\clearpage
+\lthtmldisplayA{displaymath15779}%
+\begin{displaymath}
+ \begin{array}[t]{ll}
+ \mbox{minimize} & c^T x \\
+ \mbox{subject to}
+ & G_0 x + s_0 = h_0 \\
+ & G_k x + \mathop{\mathbf{vec}}{(s_k)} = \mathop{\mathbf{vec}}{(h_k)}, \quad k = 1, \ldots, N \\
+ & Ax = b \\
+ & s_0 \succeq 0 \\
+ & s_k \succeq 0, \quad k=1,\ldots,N
+ \end{array}
+ \qquad\qquad
+ \begin{array}[t]{ll}
+ \mbox{maximize} & - h_0^Tz_0 - \sum_{k=1}^N \mathop{\bf tr}(h_kz_k) - b^T y \\
+ \mbox{subject to} &
+ G_0^Tz_0 + \sum_{k=1}^N G_k^T \mathop{\mathbf{vec}}(z_k) + A^T y + c = 0 \\
+ & z_0 \succeq 0 \\
+ & z_k \succeq 0, \quad k=1,\ldots,N.
+ \end{array}
+\end{displaymath}%
+\lthtmldisplayZ
+\lthtmlcheckvsize\clearpage}
+
+{\newpage\clearpage
+\lthtmlinlinemathA{tex2html_wrap_inline15863}%
+$\mathop{\mathbf{vec}}(z)$%
+\lthtmlinlinemathZ
+\lthtmlcheckvsize\clearpage}
+
+{\newpage\clearpage
+\lthtmlinlinemathA{tex2html_wrap_inline15865}%
+$z$%
+\lthtmlinlinemathZ
+\lthtmlcheckvsize\clearpage}
+
+{\newpage\clearpage
+\lthtmlinlinemathA{tex2html_wrap_inline15871}%
+$N$%
+\lthtmlinlinemathZ
+\lthtmlcheckvsize\clearpage}
+
+{\newpage\clearpage
+\lthtmldisplayA{displaymath15781}%
+\begin{displaymath}
+\begin{array}{ll}
+\mbox{minimize} & x_1 - x_2 + x_3 \\
+\mbox{subject to} &
+ x_1 \left[ \begin{array}{cc} -7 & -11 \\-11 & 3
+ \end{array}\right] +
+ x_2 \left[ \begin{array}{cc}
+ 7 & -18 \\-18 & 8 \end{array}\right] +
+ x_3 \left[ \begin{array}{cc}
+ -2 & -8 \\-8 & 1
+ \end{array}\right] \preceq
+ \left[ \begin{array}{cc}
+ 33 & -9 \\-9 & 26 \end{array}\right] \\*[1ex]
+& x_1 \left[ \begin{array}{ccc}
+ -21 & -11 & 0 \\-11 & 10 & 8 \\0 & 8 & 5
+ \end{array}\right] +
+ x_2 \left[ \begin{array}{ccc}
+ 0 & 10 & 16 \\
+10 & -10 & -10 \\
+16 & -10 & 3
+ \end{array}\right] +
+ x_3 \left[ \begin{array}{ccc}
+ -5 & 2 & -17 \\
+ 2 & -6 & -7 \\
+ -17 & 8 & 6
+ \end{array}\right]
+\preceq \left[ \begin{array}{ccc}
+ 14 & 9 & 40 \\
+ 9 & 91 & 10 \\
+ 40 & 10 & 15
+\end{array} \right]
+\end{array}
+\end{displaymath}%
+\lthtmldisplayZ
+\lthtmlcheckvsize\clearpage}
+
+\stepcounter{section}
+{\newpage\clearpage
+\lthtmldisplayA{displaymath15597}%
+\begin{displaymath}
+ \left[\begin{array}{ccc}
+ 0 & A^T & G^T \\
+ A & 0 & 0 \\
+ G & 0 & -W^T W \end{array}\right]
+ \left[\begin{array}{c} u_x \\u_y \\u_z \end{array}\right]
+ = \left[\begin{array}{c} b_x \\b_y \\b_z \end{array}\right].
+\end{displaymath}%
+\lthtmldisplayZ
+\lthtmlcheckvsize\clearpage}
+
+{\newpage\clearpage
+\lthtmlinlinemathA{tex2html_wrap_inline15877}%
+$W$%
+\lthtmlinlinemathZ
+\lthtmlcheckvsize\clearpage}
+
+{\newpage\clearpage
+\lthtmldisplayA{displaymath15782}%
+\begin{displaymath}
+ u = \left(u_\mathrm{l}, \; u_{\mathrm{q},0}, \; \ldots, \;
+ u_{\mathrm{q},M-1}, \; \mathop{\mathbf{vec}}{(u_{\mathrm{s},0})}, \; \ldots, \;
+ \mathop{\mathbf{vec}}{(u_{\mathrm{s},N-1})}\right), \qquad
+ u_\mathrm{l} \in{\mbox{\bf R}}^l, \qquad
+ u_{\mathrm{q},k} \in{\mbox{\bf R}}^{q_k}, \quad k = 0,\ldots,M-1, \qquad
+ u_{\mathrm{s},k} \in{\mbox{\bf S}}^{p_k}, \quad k = 0,\ldots,N-1.
+\end{displaymath}%
+\lthtmldisplayZ
+\lthtmlcheckvsize\clearpage}
+
+{\newpage\clearpage
+\lthtmldisplayA{displaymath15783}%
+\begin{displaymath}
+ Wu = \left( W_\mathrm{l} u_\mathrm{l}, \;
+ W_{\mathrm{q},0} u_{\mathrm{q},0}, \; \ldots, \;
+ W_{\mathrm{q},M-1} u_{\mathrm{q},M-1},\;
+ W_{\mathrm{s},0} \mathop{\mathbf{vec}}{(u_{\mathrm{s},0})}, \; \ldots, \;
+ W_{\mathrm{s},N-1} \mathop{\mathbf{vec}}{(u_{\mathrm{s},N-1})} \right)
+\end{displaymath}%
+\lthtmldisplayZ
+\lthtmlcheckvsize\clearpage}
+
+{\newpage\clearpage
+\lthtmldisplayA{displaymath15784}%
+\begin{displaymath}
+ W_\mathrm{l} = \mbox{\bf diag}\,(d), \qquad W_\mathrm{l}^{-1} = \mbox{\bf diag}\,(d)^{-1}.
+\end{displaymath}%
+\lthtmldisplayZ
+\lthtmlcheckvsize\clearpage}
+
+{\newpage\clearpage
+\lthtmldisplayA{displaymath15785}%
+\begin{displaymath}
+ W_\mathrm{l}^T = W_\mathrm{l}.
+\end{displaymath}%
+\lthtmldisplayZ
+\lthtmlcheckvsize\clearpage}
+
+{\newpage\clearpage
+\lthtmldisplayA{displaymath15786}%
+\begin{displaymath}
+ W_{\mathrm{q},k} = \beta_k ( 2 v_k v_k^T - J),
+ \qquad
+ W_{\mathrm{q},k}^{-1} = \frac{1}{\beta_k} ( 2 Jv_k v_k^T J - J),
+ \qquad k = 0,\ldots,M-1,
+\end{displaymath}%
+\lthtmldisplayZ
+\lthtmlcheckvsize\clearpage}
+
+{\newpage\clearpage
+\lthtmldisplayA{displaymath15787}%
+\begin{displaymath}
+ \beta_k > 0, \qquad v_{k0} > 0, \qquad
+ v_k^T Jv_k = 1, \qquad J = \left[\begin{array}{cc}
+ 1 & 0 \\0 & -I \end{array}\right].
+\end{displaymath}%
+\lthtmldisplayZ
+\lthtmlcheckvsize\clearpage}
+
+{\newpage\clearpage
+\lthtmldisplayA{displaymath15788}%
+\begin{displaymath}
+ W_\mathrm{q,k}^T = W_\mathrm{q,k}.
+\end{displaymath}%
+\lthtmldisplayZ
+\lthtmlcheckvsize\clearpage}
+
+{\newpage\clearpage
+\lthtmldisplayA{displaymath15789}%
+\begin{displaymath}
+ W_{\mathrm{s},k} \mathop{\mathbf{vec}}{(u_{\mathrm{s},k})} =
+ \mathop{\mathbf{vec}}{(r_k^T u_{\mathrm{s},k} r_k)}, \qquad
+ W_{\mathrm{s},k}^{-1} \mathop{\mathbf{vec}}{(u_{\mathrm{s},k})} =
+ \mathop{\mathbf{vec}}{(r_k^{-T} u_{\mathrm{s},k} r_k^{-1})}, \qquad
+ k = 0,\ldots,N-1.
+\end{displaymath}%
+\lthtmldisplayZ
+\lthtmlcheckvsize\clearpage}
+
+{\newpage\clearpage
+\lthtmldisplayA{displaymath15790}%
+\begin{displaymath}
+ W_{\mathrm{s},k}^T \mathop{\mathbf{vec}}{(u_{\mathrm{s},k})} =
+ \mathop{\mathbf{vec}}{(r_k u_{\mathrm{s},k} r_k^T)}, \qquad
+ \qquad
+ W_{\mathrm{s},k}^{-T} \mathop{\mathbf{vec}}{(u_{\mathrm{s},k})} =
+ \mathop{\mathbf{vec}}{(r_k^{-1} u_{\mathrm{s},k} r_k^{-T})}, \qquad
+ \qquad
+ k = 0,\ldots,N-1.
+\end{displaymath}%
+\lthtmldisplayZ
+\lthtmlcheckvsize\clearpage}
+
+{\newpage\clearpage
+\lthtmlinlinemathA{tex2html_wrap_inline15883}%
+$G$%
+\lthtmlinlinemathZ
+\lthtmlcheckvsize\clearpage}
+
+{\newpage\clearpage
+\lthtmldisplayA{displaymath15791}%
+\begin{displaymath}
+ b_x := u_x, \qquad b_y := u_y, \qquad b_z := W u_z.
+\end{displaymath}%
+\lthtmldisplayZ
+\lthtmlcheckvsize\clearpage}
+
+{\newpage\clearpage
+\lthtmldisplayA{displaymath15792}%
+\begin{displaymath}
+y := \alpha Gx + \beta y \quad
+ (\mathrm{trans} = \mathrm{'N'}), \qquad
+y := \alpha G^T x + \beta y \quad
+ (\mathrm{trans} = \mathrm{'T'}).
+\end{displaymath}%
+\lthtmldisplayZ
+\lthtmlcheckvsize\clearpage}
+
+{\newpage\clearpage
+\lthtmldisplayA{displaymath15793}%
+\begin{displaymath}
+y := \alpha Ax + \beta y \quad (\mathrm{trans} = \mathrm{'N'}), \qquad
+y := \alpha A^T x + \beta y \quad (\mathrm{trans} = \mathrm{'T'}).
+\end{displaymath}%
+\lthtmldisplayZ
+\lthtmlcheckvsize\clearpage}
+
+{\newpage\clearpage
+\lthtmldisplayA{displaymath15794}%
+\begin{displaymath}
+ \begin{array}{ll}
+ \mbox{minimize} & \|Pu-q\|_1
+ \end{array}
+\end{displaymath}%
+\lthtmldisplayZ
+\lthtmlcheckvsize\clearpage}
+
+{\newpage\clearpage
+\lthtmldisplayA{displaymath15795}%
+\begin{displaymath}
+ \begin{array}{ll}
+ \mbox{minimize} & {\bf 1}^T v \\
+ \mbox{subject to} & -v \preceq Pu - q \preceq v.
+ \end{array}
+\end{displaymath}%
+\lthtmldisplayZ
+\lthtmlcheckvsize\clearpage}
+
+{\newpage\clearpage
+\lthtmldisplayA{displaymath15796}%
+\begin{displaymath}
+ \begin{array}{ll}
+ \mbox{minimize} & {\bf 1}^T x \\
+ \mbox{subject to} & W + \mbox{\bf diag}\,(x) \succeq 0
+ \end{array}
+\end{displaymath}%
+\lthtmldisplayZ
+\lthtmlcheckvsize\clearpage}
+
+{\newpage\clearpage
+\lthtmldisplayA{displaymath15797}%
+\begin{displaymath}
+ \begin{array}{ll}
+ \mbox{minimize} & \|u\|_1 \\
+ \mbox{subject to} & \|Au - b\|_2 \leq 1.
+ \end{array}
+\end{displaymath}%
+\lthtmldisplayZ
+\lthtmlcheckvsize\clearpage}
+
+\stepcounter{section}
+\stepcounter{section}
+{\newpage\clearpage
+\lthtmldisplayA{displaymath15798}%
+\begin{displaymath}
+s \succeq 0, \qquad z \succeq 0, \qquad
+\qquad
+ \frac{\|Gx + s - h\|_2} {\max\{1,\|h\|_2\}} \leq \epsilon_\mathrm{feas},
+\qquad
+\frac{\|Ax-b\|_2}{\max\{1,\|b\|_2\}} \leq \epsilon_\mathrm{feas},
+\qquad
+\frac{\|G^Tz + A^Ty + c\|_2}{\max\{1,\|c\|_2\}} \leq
+ \epsilon_\mathrm{feas},
+\end{displaymath}%
+\lthtmldisplayZ
+\lthtmlcheckvsize\clearpage}
+
+{\newpage\clearpage
+\lthtmldisplayA{displaymath15799}%
+\begin{displaymath}
+ s^T z \leq \epsilon_\mathrm{abs} \qquad \mbox{or} \qquad
+\left( \min\left\{c^Tx, h^T z + b^Ty \right\} < 0, \quad
+ \frac{s^Tz} {-\min\{c^Tx, h^Tz + b^T y\}} \leq \epsilon_\mathrm{rel}
+\right).
+\end{displaymath}%
+\lthtmldisplayZ
+\lthtmlcheckvsize\clearpage}
+
+{\newpage\clearpage
+\lthtmldisplayA{displaymath15800}%
+\begin{displaymath}
+z \succeq 0, \qquad
+\qquad \frac{\|G^Tz +A^Ty\|_2}{\max\{1, \|c\|_2\}} \leq
+ \epsilon_\mathrm{feas},
+ \qquad h^Tz +b^Ty = -1.
+\end{displaymath}%
+\lthtmldisplayZ
+\lthtmlcheckvsize\clearpage}
+
+{\newpage\clearpage
+\lthtmldisplayA{displaymath15801}%
+\begin{displaymath}
+s \succeq 0, \qquad
+\qquad
+\frac{\|Gx+s\|_2}{\max\{1, \|h\|_2\}} \leq \epsilon_\mathrm{feas}, \qquad
+\frac{\|Ax\|_2}{\max\{1, \|b\|_2\}} \leq \epsilon_\mathrm{feas}, \qquad
+c^Tx = -1.
+\end{displaymath}%
+\lthtmldisplayZ
+\lthtmlcheckvsize\clearpage}
+
+\stepcounter{chapter}
+{\newpage\clearpage
+\lthtmldisplayA{displaymath17449}%
+\begin{displaymath}
+ \begin{array}{ll}
+ \mbox{minimize} & f_0(x) \\
+ \mbox{subject to} & f_k(x) \leq 0, \quad k=1,\ldots,m \\
+ & G x \preceq h \\
+ & A x = b,
+ \end{array}
+\end{displaymath}%
+\lthtmldisplayZ
+\lthtmlcheckvsize\clearpage}
+
+{\newpage\clearpage
+\lthtmlinlinemathA{tex2html_wrap_inline17867}%
+$f=(f_0,\ldots,f_m)$%
+\lthtmlinlinemathZ
+\lthtmlcheckvsize\clearpage}
+
+\stepcounter{section}
+{\newpage\clearpage
+\lthtmldisplayA{displaymath17815}%
+\begin{displaymath}
+ z_0 \nabla^2f_0(x) + z_1 \nabla^2f_1(x) + \cdots + z_m \nabla^2f_m(x).
+ \end{displaymath}%
+\lthtmldisplayZ
+\lthtmlcheckvsize\clearpage}
+
+{\newpage\clearpage
+\lthtmldisplayA{displaymath17816}%
+\begin{displaymath}
+ \nabla f_0(x) + D\tilde f(x)^T z_\mathrm{nl} +
+ G^T z_\mathrm{l} + A^T y = 0, \qquad
+\tilde f(x) + s_\mathrm{nl} = 0, \quad k=1,\ldots,m, \qquad
+ Gx + s_\mathrm{l} = h, \qquad
+ Ax = b,
+\end{displaymath}%
+\lthtmldisplayZ
+\lthtmlcheckvsize\clearpage}
+
+{\newpage\clearpage
+\lthtmlinlinemathA{tex2html_wrap_inline17871}%
+$\tilde f = (f_1,\ldots, f_m)$%
+\lthtmlinlinemathZ
+\lthtmlcheckvsize\clearpage}
+
+{\newpage\clearpage
+\lthtmldisplayA{displaymath17817}%
+\begin{displaymath}
+s_\mathrm{nl}\succeq 0, \qquad s_\mathrm{l}\succeq 0, \qquad
+z_\mathrm{nl} \succeq 0, \qquad z_\mathrm{l} \succeq 0, \qquad
+s_\mathrm{nl}^T z_\mathrm{nl} + s_\mathrm{l}^T z_\mathrm{l} = 0.
+\end{displaymath}%
+\lthtmldisplayZ
+\lthtmlcheckvsize\clearpage}
+
+{\newpage\clearpage
+\lthtmldisplayA{displaymath17818}%
+\begin{displaymath}
+\left[\begin{array}{cccc}
+\sum_{k=0}^m z_k \nabla^2 f_k(x) & D\tilde f(x)^T & G^T & A^T\\
+D\tilde f(x) & -\mbox{\bf diag}\,(d_1) & 0 & 0 \\
+ G & 0 & -\mbox{\bf diag}\,(d_2) & 0 \\
+ A & 0 & 0 & 0 \end{array}\right]
+\end{displaymath}%
+\lthtmldisplayZ
+\lthtmlcheckvsize\clearpage}
+
+{\newpage\clearpage
+\lthtmlinlinemathA{tex2html_wrap_inline17873}%
+$d_1$%
+\lthtmlinlinemathZ
+\lthtmlcheckvsize\clearpage}
+
+{\newpage\clearpage
+\lthtmlinlinemathA{tex2html_wrap_inline17875}%
+$d_2$%
+\lthtmlinlinemathZ
+\lthtmlcheckvsize\clearpage}
+
+{\newpage\clearpage
+\lthtmldisplayA{displaymath17819}%
+\begin{displaymath}
+ \begin{array}{ll}
+ \mbox{minimize} & -\sum_{i=1}^m \log x_i \\
+ \mbox{subject to} & Ax = b.
+ \end{array}
+\end{displaymath}%
+\lthtmldisplayZ
+\lthtmlcheckvsize\clearpage}
+
+{\newpage\clearpage
+\lthtmldisplayA{displaymath17820}%
+\begin{displaymath}
+\begin{array}{ll}
+\mbox{minimize} & \sum_{k=1}^m \phi((Ax-b)_k),
+\end{array} \qquad \mbox{where} \quad A \in{\mbox{\bf R}}^{m\times n}, \quad
+\phi(u) = \sqrt{\rho + u^2}.
+\end{displaymath}%
+\lthtmldisplayZ
+\lthtmlcheckvsize\clearpage}
+
+{\newpage\clearpage
+\lthtmldisplayA{displaymath17821}%
+\begin{displaymath}
+\begin{array}{ll}
+ \mbox{minimize} & W + H \\
+ \mbox{subject to} & A_{\mathrm{min}, k}/h_k - w_k \leq 0,
+ \quad k=1,\ldots, 5 \\
+ & x_1 \geq 0, \quad x_2 \geq 0, \quad x_4 \geq 0 \\
+ & x_1 + w_1 + \rho \leq x_3, \quad x_2 + w_2 + \rho \leq x_3, \quad
+ x_3 + w_3 + \rho \leq x_5, \quad x_4 + w_4 + \rho \leq x_5, \quad
+ x_5 + w_5 \leq W \\
+ & y_2 \geq 0, \quad y_3 \geq 0, \quad y_5 \geq 0 \\
+ & y_2 + h_2 + \rho \leq y_1, \quad y_1 + h_1 + \rho \leq y_4, \quad
+ y_3 + h_3 + \rho \leq y_4, \quad y_4 + h_4 \leq H, \quad
+ y_5 + h_5 \leq H \\
+ & h_k/\gamma \leq w_k \leq \gamma h_k, \quad k=1,\ldots,5.
+\end{array}
+\end{displaymath}%
+\lthtmldisplayZ
+\lthtmlcheckvsize\clearpage}
+
+{\newpage\clearpage
+\lthtmldisplayA{displaymath17822}%
+\begin{displaymath}
+W, \qquad H, \qquad x\in{\mbox{\bf R}}^5, \qquad y\in{\mbox{\bf R}}^5, \qquad
+w\in{\mbox{\bf R}}^5, \qquad h\in{\mbox{\bf R}}^5,
+\end{displaymath}%
+\lthtmldisplayZ
+\lthtmlcheckvsize\clearpage}
+
+{\newpage\clearpage
+\lthtmlpictureA{tex2html_wrap22797}%
+\includegraphics[width=15cm]{figures/floorplan.eps}%
+\lthtmlpictureZ
+\lthtmlcheckvsize\clearpage}
+
+\stepcounter{section}
+{\newpage\clearpage
+\lthtmldisplayA{displaymath17823}%
+\begin{displaymath}
+\begin{array}{ll}
+\mbox{minimize} & (1/2) x^TPx + q^T x \\
+\mbox{subject to} & Gx \preceq h \\& Ax = b.
+\end{array}
+\end{displaymath}%
+\lthtmldisplayZ
+\lthtmlcheckvsize\clearpage}
+
+{\newpage\clearpage
+\lthtmldisplayA{displaymath17824}%
+\begin{displaymath}
+ Px + q + G^T z + A^T y = 0, \qquad Gx + s = h, \qquad
+ Ax = b, \qquad s \succeq 0, \qquad z \succeq 0, \qquad s^T z = 0.
+\end{displaymath}%
+\lthtmldisplayZ
+\lthtmlcheckvsize\clearpage}
+
+{\newpage\clearpage
+\lthtmldisplayA{displaymath17826}%
+\begin{displaymath}
+ Px = 0, \qquad q^Tx = -1, \qquad Gx + s = 0, \qquad Ax=0, \qquad
+ s \succeq 0.
+\end{displaymath}%
+\lthtmldisplayZ
+\lthtmlcheckvsize\clearpage}
+
+{\newpage\clearpage
+\lthtmldisplayA{displaymath17827}%
+\begin{displaymath}
+\begin{array}{ll}
+\mbox{minimize} & -\bar p^T x + \mu x^T S x \\
+\mbox{subject to} & {\bf 1}^T x = 1, \quad x \succeq 0
+\end{array}
+\end{displaymath}%
+\lthtmldisplayZ
+\lthtmlcheckvsize\clearpage}
+
+{\newpage\clearpage
+\lthtmlpictureA{tex2html_wrap22809}%
+\includegraphics[width=10cm]{figures/portfolio1.eps}%
+\lthtmlpictureZ
+\lthtmlcheckvsize\clearpage}
+
+{\newpage\clearpage
+\lthtmlpictureA{tex2html_wrap22810}%
+\includegraphics[width=10cm]{figures/portfolio2.eps}%
+\lthtmlpictureZ
+\lthtmlcheckvsize\clearpage}
+
+\stepcounter{section}
+{\newpage\clearpage
+\lthtmldisplayA{displaymath17828}%
+\begin{displaymath}
+\begin{array}{ll}
+\mbox{minimize} & f_0(x) = \mathop{\bf lse}(F_0x+g_0) \\
+\mbox{subject to} & f_i(x) = \mathop{\bf lse}(F_ix+g_i) \leq 0,\quad i=1,\ldots,m \\
+ & Gx \preceq h \\
+ & Ax=b
+\end{array}
+\end{displaymath}%
+\lthtmldisplayZ
+\lthtmlcheckvsize\clearpage}
+
+{\newpage\clearpage
+\lthtmldisplayA{displaymath17829}%
+\begin{displaymath}
+ \mathop{\bf lse}(u) = \log \sum_k \exp(u_k), \qquad
+ F = \left[ \begin{array}{cccc}
+ F_0^T & F_1^T & \cdots & F_m^T \end{array}\right]^T, \qquad
+ g = \left[ \begin{array}{cccc}
+ g_0^T & g_1^T & \cdots & g_m^T \end{array}\right]^T.
+\end{displaymath}%
+\lthtmldisplayZ
+\lthtmlcheckvsize\clearpage}
+
+{\newpage\clearpage
+\lthtmldisplayA{displaymath17830}%
+\begin{displaymath}
+ \nabla f_0(x) + \sum_{k=1}^m z_{\mathrm{nl},k}
+ \nabla f_k(x) + G^T z_\mathrm{l} + A^T y = 0, \qquad
+ f_k(x) + s_{\mathrm{nl},k} = 0, \quad k=1,\ldots,m, \qquad
+ Gx + s_\mathrm{l} = h, \qquad Ax=b
+\end{displaymath}%
+\lthtmldisplayZ
+\lthtmlcheckvsize\clearpage}
+
+{\newpage\clearpage
+\lthtmldisplayA{displaymath17832}%
+\begin{displaymath}
+ \begin{array}{ll}
+ \mbox{minimize} & w^{-1} h^{-1} d^{-1} \\
+ \mbox{subject to}
+ & (2/A_\mathrm{wall}) hw + (2/A_\mathrm{wall})hd \leq 1 \\
+ & (1/A_\mathrm{flr}) wd \leq 1 \\
+ & \alpha wh^{-1} \leq 1 \\
+ & (1/\beta) hw^{-1} \leq 1 \\
+ & \gamma wd^{-1} \leq 1 \\
+ & (1/\delta)dw^{-1} \leq 1
+ \end{array}
+\end{displaymath}%
+\lthtmldisplayZ
+\lthtmlcheckvsize\clearpage}
+
+\stepcounter{section}
+{\newpage\clearpage
+\lthtmldisplayA{eqnarraystar17683}%
+\begin{eqnarray*}
+ \sum_{k=0}^m z_k \nabla^2 f_k(x)u_x + A^T u_y +
+ D \tilde f(x)^T u_{z_\mathrm{nl}} +
+ G_\mathrm{l}^T u_{z_\mathrm{l}} & = & b_x \\
+ A x & = & b_y \\
+ D\tilde f(x) x - \mbox{\bf diag}\,(d_\mathrm{nl})^{-2} z_\mathrm{nl} & = &
+ b_{z_\mathrm{nl}} \\
+ G_\mathrm{l}x - \mbox{\bf diag}\,(d_\mathrm{l})^{-2} z_\mathrm{l} & = &
+ b_{z_\mathrm{l}}
+\end{eqnarray*}%
+\lthtmldisplayZ
+\lthtmlcheckvsize\clearpage}
+
+{\newpage\clearpage
+\lthtmldisplayA{displaymath17833}%
+\begin{displaymath}
+ v := \sum_{k=0}^m u_k \nabla f_k(x) + v.
+\end{displaymath}%
+\lthtmldisplayZ
+\lthtmlcheckvsize\clearpage}
+
+{\newpage\clearpage
+\lthtmldisplayA{displaymath17836}%
+\begin{displaymath}
+\begin{array}{ll}
+\mbox{minimize} & \|Ax - y\|_2^2 + \|x\|_1
+\end{array}
+\end{displaymath}%
+\lthtmldisplayZ
+\lthtmlcheckvsize\clearpage}
+
+{\newpage\clearpage
+\lthtmldisplayA{displaymath17837}%
+\begin{displaymath}
+ \begin{array}{ll}
+ \mbox{minimize} & \|Ax - y\|_2^2 + {\bf 1}^T u \\
+ \mbox{subject to} & -u \preceq x \preceq u
+ \end{array}
+\end{displaymath}%
+\lthtmldisplayZ
+\lthtmlcheckvsize\clearpage}
+
+\stepcounter{section}
+{\newpage\clearpage
+\lthtmldisplayA{displaymath17838}%
+\begin{displaymath}
+\frac{\| \nabla f_0(x) + D\tilde f(x)^Tz_\mathrm{nl} +
+ G^Tz_\mathrm{l} + A^T y \|_2 }
+{\max\{ 1,
+\| \nabla f_0(x_0) + D\tilde f(x_0)^T{\bf 1}+ G^T{\bf 1}\|_2 \}}
+\leq \epsilon_\mathrm{feas}, \qquad
+\frac{\| ( \tilde f(x) + s_{\mathrm{nl}}, Gx + s_\mathrm{l} - h,
+ Ax-b ) \|_2}
+{\max\{1, \| ( \tilde f(x_0) + {\bf 1},
+Gx_0 + {\bf 1}-h, Ax_0-b) \|_2 \}} \leq \epsilon_\mathrm{feas}
+\end{displaymath}%
+\lthtmldisplayZ
+\lthtmlcheckvsize\clearpage}
+
+{\newpage\clearpage
+\lthtmldisplayA{displaymath17839}%
+\begin{displaymath}
+\mathrm{gap} \leq \epsilon_\mathrm{abs}
+\qquad \mbox{or} \qquad \left( f_0(x) < 0, \quad
+\frac{\mathrm{gap}} {-f_0(x)} \leq \epsilon_\mathrm{rel} \right)
+\qquad \mbox{or} \qquad
+\left( L(x,y,z) > 0, \quad \frac{\mathrm{gap}}
+{L(x,y,z)} \leq \epsilon_\mathrm{rel} \right)
+\end{displaymath}%
+\lthtmldisplayZ
+\lthtmlcheckvsize\clearpage}
+
+{\newpage\clearpage
+\lthtmldisplayA{displaymath17840}%
+\begin{displaymath}
+ \mathrm{gap} =
+\left[\begin{array}{c} s_\mathrm{nl} \\s_\mathrm{l}
+\end{array}\right]^T
+\left[\begin{array}{c} z_\mathrm{nl} \\z_\mathrm{l}
+\end{array}\right],
+\qquad
+L(x,y,z) = f_0(x) + z_\mathrm{nl}^T \tilde f(x) +
+ z_\mathrm{l}^T (Gx-h) + y^T(Ax-b).
+\end{displaymath}%
+\lthtmldisplayZ
+\lthtmlcheckvsize\clearpage}
+
+\stepcounter{chapter}
+\stepcounter{section}
+\stepcounter{section}
+{\newpage\clearpage
+\lthtmldisplayA{displaymath19892}%
+\begin{displaymath}
+ f(x_1,\ldots,x_n) = A_1 x_1 + \cdots + A_n x_n + b.
+\end{displaymath}%
+\lthtmldisplayZ
+\lthtmlcheckvsize\clearpage}
+
+{\newpage\clearpage
+\lthtmldisplayA{eqnarraystar19548}%
+\begin{eqnarray*}
+ f(x,y) & = & \left[ \begin{array}{c} 2 \\2 \end{array}\right] x
+ + y + \left[ \begin{array}{c} 3 \\3 \end{array}\right], \\
+ g(x,y) & = &
+ \left[ \begin{array}{cc} 1 & 3 \\2 & 4 \end{array}\right] f(x,y)
+ + \left[ \begin{array}{cc} 1 & 1 \\1 & 1 \end{array} \right] y +
+ \left[ \begin{array}{c} 1 \\-1 \end{array} \right] \\
+ & = & \left[ \begin{array}{c} 8 \\12 \end{array}\right] x
+ + \left[ \begin{array}{cc} 2 & 4 \\3 & 5 \end{array}\right] y
+ + \left[ \begin{array}{c} 13 \\17\end{array}\right].
+\end{eqnarray*}%
+\lthtmldisplayZ
+\lthtmlcheckvsize\clearpage}
+
+{\newpage\clearpage
+\lthtmldisplayA{displaymath19893}%
+\begin{displaymath}
+ f(x_1,\ldots,x_n) = b + A_1 x_1 + \cdots + A_n x_n +
+ \sum_{k=1}^K \max (y_1, y_2, \ldots, y_{m_k}).
+\end{displaymath}%
+\lthtmldisplayZ
+\lthtmlcheckvsize\clearpage}
+
+{\newpage\clearpage
+\lthtmldisplayA{displaymath19894}%
+\begin{displaymath}
+ f(x_1,\ldots,x_n) = b + A_1 x_1 + \cdots + A_n x_n +
+ \sum_{k=1}^K \min (y_1, y_2, \ldots, y_{m_k}).
+\end{displaymath}%
+\lthtmldisplayZ
+\lthtmlcheckvsize\clearpage}
+
+{\newpage\clearpage
+\lthtmldisplayA{displaymath19895}%
+\begin{displaymath}
+ h(x) = \sum_k \phi(x[k]),\qquad
+ \phi(u) = \left\{\begin{array}{ll}
+ 0 & |u| \leq 1 \\
+ |u|-1 & 1 \leq |u| \leq 2 \\
+ 2|u|-3 & |u| \geq 2. \end{array}\right.
+\end{displaymath}%
+\lthtmldisplayZ
+\lthtmlcheckvsize\clearpage}
+
+\stepcounter{section}
+{\newpage\clearpage
+\lthtmldisplayA{displaymath19896}%
+\begin{displaymath}
+ f(x_1,\ldots,x_n) = 0, \qquad f(x_1,\ldots,x_n) \preceq 0,
+\end{displaymath}%
+\lthtmldisplayZ
+\lthtmlcheckvsize\clearpage}
+
+{\newpage\clearpage
+\lthtmlinlinemathA{tex2html_wrap_inline19918}%
+$f$%
+\lthtmlinlinemathZ
+\lthtmlcheckvsize\clearpage}
+
+{\newpage\clearpage
+\lthtmldisplayA{displaymath19897}%
+\begin{displaymath}
+ 0 \preceq x \preceq {\bf 1}, \qquad {\bf 1}^T x = 2,
+\end{displaymath}%
+\lthtmldisplayZ
+\lthtmlcheckvsize\clearpage}
+
+\stepcounter{section}
+{\newpage\clearpage
+\lthtmldisplayA{displaymath19898}%
+\begin{displaymath}
+ \begin{array}{ll}
+ \mbox{minimize} & -4x - 5y \\
+ \mbox{subject to} & 2x +y \leq 3 \\
+ & x +2y \leq 3 \\
+ & x \geq 0, \quad y \geq 0.
+ \end{array}
+\end{displaymath}%
+\lthtmldisplayZ
+\lthtmlcheckvsize\clearpage}
+
+\stepcounter{section}
+{\newpage\clearpage
+\lthtmldisplayA{displaymath19899}%
+\begin{displaymath}
+ \begin{array}{ll}
+ \mbox{minimize} & \|Ax - b\|_\infty,
+ \end{array} \qquad
+ \begin{array}{ll}
+ \mbox{minimize} & \|Ax - b\|_1
+ \end{array},
+\end{displaymath}%
+\lthtmldisplayZ
+\lthtmlcheckvsize\clearpage}
+
+{\newpage\clearpage
+\lthtmldisplayA{displaymath19900}%
+\begin{displaymath}
+ \begin{array}{ll}
+ \mbox{minimize} & \sum_k \phi((Ax-b)_k),
+ \end{array} \qquad
+ \phi(u) = \left\{\begin{array}{ll}
+ 0 & |u| \leq 3/4 \\
+ |u|-3/4 & 3/4 \leq |u| \leq 3/2 \\
+ 2|u|-9/4 & |u| \geq 3/2.\end{array}\right.
+\end{displaymath}%
+\lthtmldisplayZ
+\lthtmlcheckvsize\clearpage}
+
+{\newpage\clearpage
+\lthtmlpictureA{tex2html_wrap22898}%
+\includegraphics[width=\linewidth]{figures/normappr.eps}%
+\lthtmlpictureZ
+\lthtmlcheckvsize\clearpage}
+
+{\newpage\clearpage
+\lthtmldisplayA{displaymath19901}%
+\begin{displaymath}
+ \begin{array}{ll}
+ \mbox{minimize} & c^T x \\
+ \mbox{subject to} & \sup_{\|v\|_\infty \leq 1}
+ (a_i+v)^T x \leq b_i, \qquad i=1,\ldots,m
+ \end{array}
+\end{displaymath}%
+\lthtmldisplayZ
+\lthtmlcheckvsize\clearpage}
+
+{\newpage\clearpage
+\lthtmldisplayA{displaymath19902}%
+\begin{displaymath}
+ \begin{array}{ll}
+ \mbox{minimize} & c^Tx \\
+ \mbox{subject to} & a_i^Tx + \|x\|_1 \leq b_i, \qquad i=1,\ldots,m.
+ \end{array}
+\end{displaymath}%
+\lthtmldisplayZ
+\lthtmlcheckvsize\clearpage}
+
+{\newpage\clearpage
+\lthtmldisplayA{displaymath19903}%
+\begin{displaymath}
+ \begin{array}{ll}
+ \mbox{minimize} & c^Tx \\
+ \mbox{subject to} & a_i^Tx + {\bf 1}^Ty \leq b_i, \qquad i=1,\ldots,m \\
+& -y \preceq x \preceq y
+\end{array}
+\end{displaymath}%
+\lthtmldisplayZ
+\lthtmlcheckvsize\clearpage}
+
+{\newpage\clearpage
+\lthtmldisplayA{displaymath19904}%
+\begin{displaymath}
+\begin{array}{ll}
+\mbox{minimize} & \|x\|_1 + {\bf 1}^Tu \\
+\mbox{subject to} & Ax \succeq {\bf 1}-u \\
+& u \succeq 0.
+\end{array}
+\end{displaymath}%
+\lthtmldisplayZ
+\lthtmlcheckvsize\clearpage}
+
+\stepcounter{chapter}
+\stepcounter{section}
+\stepcounter{section}
+{\newpage\clearpage
+\lthtmldisplayA{displaymath21247}%
+\begin{displaymath}
+A=\left [\begin{array}{cccc}
+ 1 & 0 & 0 & 5\\
+ 2 & 0 & 4 & 0\\
+ 0 & 0 & 0 & 6\\
+ 3 & 0 & 0 & 0
+\end{array}\right]
+\end{displaymath}%
+\lthtmldisplayZ
+\lthtmlcheckvsize\clearpage}
+
+
+\end{document}
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-<h1>CVXOPT: A Python Package for Convex Optimization</h1>
-<p><b><font size="+2">Joachim Dahl & Lieven Vandenberghe</font></b></p>
-<p><i><TT>joachim at kom.aau.dk</TT>, <TT>vandenbe at ee.ucla.edu</TT></i></p>
-<p><strong>Release 0.8.2</strong><br />
-<strong>February 6, 2007</strong></p>
-<p></p>
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-<UL CLASS="ChildLinks">
-<LI><A href="node1.html">Copyright and License</a>
-<LI><A href="contents.html">Contents</a>
-<LI><A href="node3.html">1. Introduction</a>
-<LI><A href="node4.html">2. Dense Matrices (<tt class="module">cvxopt.base</tt>)</a>
-<UL>
-<LI><A href="s-creating-matrices.html">2.1 Creating Matrices</a>
-<LI><A href="node6.html">2.2 Attributes and Methods</a>
-<LI><A href="s-arithmetic.html">2.3 Arithmetic Operations</a>
-<LI><A href="s-indexing.html">2.4 Indexing and Slicing</a>
-<LI><A href="s-builtinfuncs.html">2.5 Built-in Functions</a>
-<LI><A href="s-otherfuncs.html">2.6 Other Matrix Functions</a>
-<LI><A href="s-random.html">2.7 Randomly Generated Matrices</a>
-<LI><A href="s-array-interface.html">2.8 The NumPy Array Interface</a>
-<LI><A href="node13.html">2.9 Printing Options</a>
-</ul>
-<LI><A href="node14.html">3. The BLAS Interface (<tt class="module">cvxopt.blas</tt>)</a>
-<UL>
-<LI><A href="s-conventions.html">3.1 Matrix Classes</a>
-<LI><A href="s-blas1.html">3.2 Level 1 BLAS</a>
-<LI><A href="s-blas2.html">3.3 Level 2 BLAS</a>
-<LI><A href="s-blas3.html">3.4 Level 3 BLAS</a>
-</ul>
-<LI><A href="node19.html">4. The LAPACK Interface (<tt class="module">cvxopt.lapack</tt>)</a>
-<UL>
-<LI><A href="node20.html">4.1 General Linear Equations</a>
-<LI><A href="e-kkt-example.html">4.2 Positive Definite Linear Equations</a>
-<LI><A href="node22.html">4.3 Symmetric and Hermitian Linear Equations</a>
-<LI><A href="node23.html">4.4 Triangular Linear Equations</a>
-<LI><A href="node24.html">4.5 Least-Squares and Least-Norm Problems</a>
-<LI><A href="node25.html">4.6 Symmetric and Hermitian Eigenvalue Decomposition</a>
-<LI><A href="e-gevd.html">4.7 Generalized Symmetric Definite Eigenproblems</a>
-<LI><A href="node27.html">4.8 Singular Value Decomposition</a>
-<LI><A href="node28.html">4.9 Example: Analytic Centering</a>
-</ul>
-<LI><A href="c-fftw.html">5. Discrete Transforms (<tt class="module">cvxopt.fftw</tt>)</a>
-<UL>
-<LI><A href="node30.html">5.1 Discrete Fourier Transform</a>
-<LI><A href="node31.html">5.2 Discrete Cosine Transform</a>
-<LI><A href="node32.html">5.3 Discrete Sine Transform</a>
-</ul>
-<LI><A href="node33.html">6. Sparse Matrices (<tt class="module">cvxopt.base</tt>)</a>
-<UL>
-<LI><A href="s-creating-spmatrix.html">6.1 Creating Sparse Matrices</a>
-<LI><A href="e-spA-example.html">6.2 Attributes and Methods</a>
-<LI><A href="s-spmatrix-arith.html">6.3 Arithmetic Operations</a>
-<LI><A href="node37.html">6.4 Indexing and Slicing</a>
-<LI><A href="node38.html">6.5 Built-In Functions</a>
-<LI><A href="node39.html">6.6 Sparse BLAS Functions</a>
-</ul>
-<LI><A href="c-spsolvers.html">7. Sparse Linear Equation Solvers</a>
-<UL>
-<LI><A href="s-orderings.html">7.1 Matrix Orderings (<tt class="module">cvxopt.amd</tt>)</a>
-<LI><A href="s-umfpack.html">7.2 General Linear Equations (<tt class="module">cvxopt.umfpack</tt>)</a>
-<LI><A href="s-cholmod.html">7.3 Positive Definite Linear Equations (<tt class="module">cvxopt.cholmod</tt>)</a>
-<LI><A href="e-covsel.html">7.4 Example: Covariance Selection</a>
-</ul>
-<LI><A href="node45.html">8. Optimization Routines (<tt class="module">cvxopt.solvers</tt>)</a>
-<UL>
-<LI><A href="s-lpsolver.html">8.1 Linear Programming</a>
-<LI><A href="node47.html">8.2 Quadratic Programming</a>
-<LI><A href="node48.html">8.3 Geometric Programming</a>
-<LI><A href="s-sdpsolver.html">8.4 Semidefinite Programming</a>
-<LI><A href="e-nlcp.html">8.5 Nonlinear Convex Programming</a>
-<LI><A href="node51.html">8.6 Exploiting Structure in LPs and SDPs</a>
-<LI><A href="node52.html">8.7 Exploiting Structure in Nonlinear Convex Programs</a>
-<LI><A href="s-external.html">8.8 Optional Solvers</a>
-<LI><A href="s-parameters.html">8.9 Algorithm Parameters</a>
-</ul>
-<LI><A href="node55.html">9. Modeling (<tt class="module">cvxopt.modeling</tt>)</a>
-<UL>
-<LI><A href="s-variables.html">9.1 Variables</a>
-<LI><A href="s-functions.html">9.2 Functions</a>
-<LI><A href="node58.html">9.3 Constraints</a>
-<LI><A href="s-lp.html">9.4 Optimization Problems</a>
-<LI><A href="node60.html">9.5 Examples</a>
-</ul>
-<LI><A href="node61.html">10. C API</a>
-<UL>
-<LI><A href="node62.html">10.1 Dense Matrices</a>
-<LI><A href="node63.html">10.2 Sparse Matrices</a>
-</ul>
-<LI><A href="genindex.html">Index</a>
-<LI><A href="about.html">About this document ...</a>
-</ul>
-<!--End of Table of Child-Links-->
-</div>
-
-<DIV CLASS="navigation">
-<div class='online-navigation'>
-<p></p><hr />
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-<tr>
-<td class='online-navigation'><img src='previous.gif'
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-<td class='online-navigation'><img src='up.gif'
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- href="node1.html"><img src='next.gif'
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-<td align="center" width="100%">CVXOPT: A Python Package for Convex Optimization</td>
-<td class='online-navigation'><a rel="contents" title="Table of Contents"
- href="contents.html"><img src='contents.gif'
- border='0' height='32' alt='Contents' width='32' /></A></td>
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-<span class="release-info">Release 0.8.2, documentation updated on February 6, 2007.</span>
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-<!DOCTYPE html PUBLIC "-//W3C//DTD HTML 4.0 Transitional//EN">
-<html>
-<head>
-<link rel="STYLESHEET" href="cvxopt.css" type='text/css' />
-<link rel="first" href="cvxopt.html" title='CVXOPT: A Python Package for Convex Optimization' />
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-<body>
-<DIV CLASS="navigation">
-<div id='top-navigation-panel' xml:id='top-navigation-panel'>
-<table align="center" width="100%" cellpadding="0" cellspacing="2">
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-<td align="center" width="100%">CVXOPT: A Python Package for Convex Optimization</td>
-<td class='online-navigation'><a rel="contents" title="Table of Contents"
- href="contents.html"><img src='contents.gif'
- border='0' height='32' alt='Contents' width='32' /></A></td>
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- border='0' height='32' alt='Index' width='32' /></A></td>
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-</div>
-<hr /></div>
-</DIV>
-<!--End of Navigation Panel-->
-
-<H1><A NAME="SECTION001000000000000000000">
-Copyright and License</A>
-</H1>
-Copyright ©2004-2007 J. Dahl & L. Vandenberghe.
-
-<P>
-This program is free software; you can redistribute it and/or modify
-it under the terms of the
-<a class="ulink" href="http://www.gnu.org/copyleft/gpl.html"
- >GNU General Public License</a>
-as published by
-the Free Software Foundation; either version 2 of the License, or
-(at your option) any later version.
-
-<P>
-This program is distributed in the hope that it will be useful,
-but WITHOUT ANY WARRANTY; without even the implied warranty of
-MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the
-<a class="ulink" href="http://www.gnu.org/copyleft/gpl.html"
- >GNU General Public License</a>
-for more details.
-
-<P>
-<HR>The CVXOPT distribution includes source code for the following
-software libraries.
-
-<UL>
-<LI>Part of the SuiteSparse suite of sparse matrix algorithms,
- including:
-
-<UL>
-<LI>AMD Version 2.0. Copyright (c) 2006 by Timothy A. Davis,
- Patrick R. Amestoy, and Iain S. Duff.
-</LI>
-<LI>CHOLMOD Version 1.4.
- Copyright (c) 2005-2006 by University of Florida, Timothy A. Davis
- and W. Hager.
-</LI>
-<LI>COLAMD version 2.6. Copyright (c) 1998-2006 by Timothy A. Davis.
-</LI>
-<LI>UMFPACK Version 5.0.2.
- Copyright (c) 1995-2006 by Timothy A. Davis.
-</LI>
-</UL>
-
-<P>
-These packages are licensed under the terms of the
-<a class="ulink" href="http://www.gnu.org/copyleft/lesser.html"
- >GNU Lesser General Public License</a> (UMFPACK, parts of CHOLMOD,
-AMD, COLAMD) and the <a class="ulink" href="http://www.gnu.org/copyleft/gpl.html"
- >GNU General Public License</a> (parts of CHOLMOD).
-For details, consult the README files in the source directories or
-the website listed below.
-
-<P>
-<BLOCKQUOTE>
-Availability: <a class="ulink" href="http://www.cise.ufl.edu/research/sparse"
- >www.cise.ufl.edu/research/sparse</a>.
-
-</BLOCKQUOTE>
-
-<P>
-</LI>
-<LI>RNGS Random Number Generation -- Multiple Streams
-(Sep. 22, 1998) by Steve Park & Dave Geyer.
-
-<P>
-<BLOCKQUOTE>
-Availability: <a class="ulink" href="http://www.cs.wm.edu/~va/software/park/park.html"
- >www.cs.wm.edu/~va/software/park/park.html</a>.
-
-</BLOCKQUOTE>
-
-<P>
-</LI>
-</UL>
-
-<P>
-
-<DIV CLASS="navigation">
-<div class='online-navigation'>
-<p></p><hr />
-<table align="center" width="100%" cellpadding="0" cellspacing="2">
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- href="genindex.html"><img src='index.gif'
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-</div>
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-<span class="release-info">Release 0.8.2, documentation updated on February 6, 2007.</span>
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diff --git a/doc/cvxopt/node13.html b/doc/cvxopt/node13.html
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--- a/doc/cvxopt/node13.html
+++ /dev/null
@@ -1,123 +0,0 @@
-<!DOCTYPE html PUBLIC "-//W3C//DTD HTML 4.0 Transitional//EN">
-<html>
-<head>
-<link rel="STYLESHEET" href="cvxopt.css" type='text/css' />
-<link rel="first" href="cvxopt.html" title='CVXOPT: A Python Package for Convex Optimization' />
-<link rel='contents' href='contents.html' title="Contents" />
-<link rel='index' href='genindex.html' title='Index' />
-<link rel='last' href='about.html' title='About this document...' />
-<link rel='help' href='about.html' title='About this document...' />
-<link rel="prev" href="s-array-interface.html" />
-<link rel="parent" href="node4.html" />
-<link rel="next" href="node14.html" />
-<meta name='aesop' content='information' />
-<title>2.9 Printing Options</title>
-</head>
-<body>
-<DIV CLASS="navigation">
-<div id='top-navigation-panel' xml:id='top-navigation-panel'>
-<table align="center" width="100%" cellpadding="0" cellspacing="2">
-<tr>
-<td class='online-navigation'><a rel="prev" title="2.8 The NumPy Array"
- href="s-array-interface.html"><img src='previous.gif'
- border='0' height='32' alt='Previous Page' width='32' /></A></td>
-<td class='online-navigation'><a rel="parent" title="2. Dense Matrices (cvxopt.base)"
- href="node4.html"><img src='up.gif'
- border='0' height='32' alt='Up One Level' width='32' /></A></td>
-<td class='online-navigation'><a rel="next" title="3. The BLAS Interface"
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-<td align="center" width="100%">CVXOPT: A Python Package for Convex Optimization</td>
-<td class='online-navigation'><a rel="contents" title="Table of Contents"
- href="contents.html"><img src='contents.gif'
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-<td class='online-navigation'><a rel="index" title="Index"
- href="genindex.html"><img src='index.gif'
- border='0' height='32' alt='Index' width='32' /></A></td>
-</tr></table>
-<div class='online-navigation'>
-<b class="navlabel">Previous:</b>
-<a class="sectref" rel="prev" href="s-array-interface.html">2.8 The NumPy Array</A>
-<b class="navlabel">Up:</b>
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-<b class="navlabel">Next:</b>
-<a class="sectref" rel="next" href="node14.html">3. The BLAS Interface</A>
-</div>
-<hr /></div>
-</DIV>
-<!--End of Navigation Panel-->
-
-<H1><A NAME="SECTION004900000000000000000">
-2.9 Printing Options</A>
-</H1>
-The format used for printing dense matrices (and the sparse matrices
-discussed in chapter <A HREF="node33.html#chap:spmatrix">6</A>) is controlled
-by the dictionary <tt class="member">cvxopt.base.print_options</tt>. The dictionary
-has three keys, <code>'iformat'</code>, <code>'dformat'</code>, <code>'zformat'</code>
-that control, respectively, how integer, double and complex numbers are
-printed. The fields are C printf format strings with default
-values <code>'5.4e'</code> for <code>'d'</code> and <code>'z'</code> matrices and <code>'5i'</code> for
-<code>'i'</code> matrices.
-<div class="verbatim"><pre>
->>> from cvxopt.base import matrix, print_options
->>> print_options
-{'zformat': '5.4e', 'iformat': '5i', 'dformat': '5.4e'}
->>> A = matrix([1., 2., 3.])
->>> print A
- 1.0000e+00
- 2.0000e+00
- 3.0000e+00
->>> print_options['dformat'] = 'f'
->>> print A
- 1.000000
- 2.000000
- 3.000000
->>> print_options['dformat'] = '5.2e'
->>> print A
- 1.00e+00
- 2.00e+00
- 3.00e+00
-</pre></div>
-
-<DIV CLASS="navigation">
-<div class='online-navigation'>
-<p></p><hr />
-<table align="center" width="100%" cellpadding="0" cellspacing="2">
-<tr>
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- href="s-array-interface.html"><img src='previous.gif'
- border='0' height='32' alt='Previous Page' width='32' /></A></td>
-<td class='online-navigation'><a rel="parent" title="2. Dense Matrices (cvxopt.base)"
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-</div>
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-</DIV>
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diff --git a/doc/cvxopt/node14.html b/doc/cvxopt/node14.html
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--- a/doc/cvxopt/node14.html
+++ /dev/null
@@ -1,155 +0,0 @@
-<!DOCTYPE html PUBLIC "-//W3C//DTD HTML 4.0 Transitional//EN">
-<html>
-<head>
-<link rel="STYLESHEET" href="cvxopt.css" type='text/css' />
-<link rel="first" href="cvxopt.html" title='CVXOPT: A Python Package for Convex Optimization' />
-<link rel='contents' href='contents.html' title="Contents" />
-<link rel='index' href='genindex.html' title='Index' />
-<link rel='last' href='about.html' title='About this document...' />
-<link rel='help' href='about.html' title='About this document...' />
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-<title>3. The BLAS Interface (cvxopt.blas)</title>
-</head>
-<body>
-<DIV CLASS="navigation">
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-<table align="center" width="100%" cellpadding="0" cellspacing="2">
-<tr>
-<td class='online-navigation'><a rel="prev" title="2.9 Printing Options"
- href="node13.html"><img src='previous.gif'
- border='0' height='32' alt='Previous Page' width='32' /></A></td>
-<td class='online-navigation'><a rel="parent" title="CVXOPT: A Python Package"
- href="cvxopt.html"><img src='up.gif'
- border='0' height='32' alt='Up One Level' width='32' /></A></td>
-<td class='online-navigation'><a rel="next" title="3.1 Matrix Classes"
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-<td align="center" width="100%">CVXOPT: A Python Package for Convex Optimization</td>
-<td class='online-navigation'><a rel="contents" title="Table of Contents"
- href="contents.html"><img src='contents.gif'
- border='0' height='32' alt='Contents' width='32' /></A></td>
-<td class='online-navigation'><img src='blank.gif'
- border='0' height='32' alt='' width='32' /></td>
-<td class='online-navigation'><a rel="index" title="Index"
- href="genindex.html"><img src='index.gif'
- border='0' height='32' alt='Index' width='32' /></A></td>
-</tr></table>
-<div class='online-navigation'>
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-<b class="navlabel">Next:</b>
-<a class="sectref" rel="next" href="s-conventions.html">3.1 Matrix Classes</A>
-</div>
-<hr /></div>
-</DIV>
-<!--End of Navigation Panel-->
-
-<H1><A NAME="SECTION005000000000000000000"></A><A NAME="chap:blas"></A>
-<BR>
-3. The BLAS Interface (<tt class="module">cvxopt.blas</tt>)
-</H1>
-The <tt class="module">cvxopt.blas</tt> module provides an interface to the
-double-precision real and complex Basic Linear Algebra Subprograms
-(BLAS). The names and calling sequences of the Python functions in
-the interface closely match the corresponding Fortran BLAS routines
-(described in the references below) and their functionality is exactly
-the same.
-
-<P>
-Many of the operations performed by the BLAS routines can be
-implemented in a more straightforward way by using the matrix
-arithmetic of section <A href="s-arithmetic.html#s-arithmetic">2.3</A>, combined with the slicing
-and indexing of section <A href="s-indexing.html#s-indexing">2.4</A>.
-As an example, "<tt class="samp">C = A*B</tt>" gives the same result as the BLAS
-call "<tt class="samp">gemm(A,B,C)</tt>".
-The BLAS interface offers two advantages. First, some of the
-functions it includes are not easily implemented using the basic
-matrix arithmetic. For example, BLAS includes functions that
-efficiently exploit symmetry or triangular matrix structure.
-Second, there is a performance difference that can be significant for
-large matrices. Although our implementation of the basic matrix
-arithmetic makes internal calls to BLAS, it also often requires
-creating temporary matrices to store intermediate results.
-The BLAS functions on the other hand always operate directly
-on their matrix arguments and never require any copying to temporary
-matrices. Thus they can be viewed as generalizations of the in-place
-matrix addition and scalar multiplication of
-section <A href="s-arithmetic.html#s-arithmetic">2.3</A> to more complicated operations.
-
-<P>
-<div class="seealso">
- <p class="heading">See Also:</p>
-
-<div class="seetext"><p>C. L. Lawson, R. J. Hanson, D. R. Kincaid, F. T. Krogh,
-Basic Linear Algebra Subprograms for Fortran Use,
-ACM Transactions on Mathematical Software, 5(3), 309-323, 1975.</p></div>
-<div class="seetext"><p>J. J. Dongarra, J. Du Croz, S. Hammarling, R. J. Hanson,
-An Extended Set of Fortran Basic Linear Algebra Subprograms,
-ACM Transactions on Mathematical Software, 14(1), 1-17, 1988.</p></div>
-<div class="seetext"><p>J. J. Dongarra, J. Du Croz, S. Hammarling, I. Duff,
-A Set of Level 3 Basic Linear Algebra Subprograms,
-ACM Transactions on Mathematical Software, 16(1), 1-17, 1990.</p></div>
-</div>
-
-<P>
-
-<p><br /></p><hr class='online-navigation' />
-<div class='online-navigation'>
-<!--Table of Child-Links-->
-<A NAME="CHILD_LINKS"><STRONG>Subsections</STRONG></a>
-
-<UL CLASS="ChildLinks">
-<LI><A href="s-conventions.html">3.1 Matrix Classes</a>
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-<H1><A NAME="SECTION006000000000000000000"></A>
-<A NAME="chap:lapack"></A>
-<BR>
-4. The LAPACK Interface (<tt class="module">cvxopt.lapack</tt>)
-</H1>
-
-<P>
-The module <tt class="module">cvxopt.lapack</tt> includes functions for
-solving dense sets of linear equations, for the corresponding matrix
-factorizations (LU, Cholesky, LDL<SPAN CLASS="MATH"><IMG
- WIDTH="14" HEIGHT="18" ALIGN="BOTTOM" BORDER="0"
- SRC="img40.gif"
- ALT="$\mathrm{{}^T}$"></SPAN>),
-for solving least-squares and least-norm problems, for QR
-factorization, for symmetric eigenvalue problems and for singular
-value decomposition.
-
-<P>
-In this chapter we briefly describe the Python calling sequences.
-For further details on the underlying LAPACK functions we refer to the
-LAPACK Users' Guide and manual pages.
-
-<P>
-The BLAS conventional storage scheme of section <A href="s-conventions.html#s-conventions">3.1</A> is
-used. As in the previous chapter, we omit from the function definitions
-less important arguments that are useful for selecting submatrices.
-The complete definitions are documented in the docstrings in the
-source code.
-
-<P>
-<div class="seealso">
- <p class="heading">See Also:</p>
-
-<dl compact="compact" class="seeurl">
- <dt><a href='http://www.netlib.org/lapack/lug/lapack_lug.html'
- >LAPACK
-Users' Guide, Third Edition, SIAM, 1999.</a></dt>
- <dd></dd>
- </dl>
-</div>
-
-<P>
-
-<p><br /></p><hr class='online-navigation' />
-<div class='online-navigation'>
-<!--Table of Child-Links-->
-<A NAME="CHILD_LINKS"><STRONG>Subsections</STRONG></a>
-
-<UL CLASS="ChildLinks">
-<LI><A href="node20.html">4.1 General Linear Equations</a>
-<LI><A href="e-kkt-example.html">4.2 Positive Definite Linear Equations</a>
-<LI><A href="node22.html">4.3 Symmetric and Hermitian Linear Equations</a>
-<LI><A href="node23.html">4.4 Triangular Linear Equations</a>
-<LI><A href="node24.html">4.5 Least-Squares and Least-Norm Problems</a>
-<LI><A href="node25.html">4.6 Symmetric and Hermitian Eigenvalue Decomposition</a>
-<LI><A href="e-gevd.html">4.7 Generalized Symmetric Definite Eigenproblems</a>
-<LI><A href="node27.html">4.8 Singular Value Decomposition</a>
-<LI><A href="node28.html">4.9 Example: Analytic Centering</a>
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-<H1><A NAME="SECTION006100000000000000000">
-4.1 General Linear Equations</A>
-</H1>
-<dl><dt><table cellpadding="0" cellspacing="0"><tr valign="baseline">
- <td><nobr><b><tt id='l2h-62' xml:id='l2h-62' class="function">gesv</tt></b>(</nobr></td>
- <td><var>A, B</var><big>[</big><var>, ipiv=None</var><big>]</big><var></var>)</td></tr></table></dt>
-<dd>
-Solves
-<BR><P></P>
-<DIV ALIGN="CENTER" CLASS="mathdisplay">
-<!-- MATH
- \begin{displaymath}
-A X = B,
-\end{displaymath}
- -->
-
-<IMG
- WIDTH="64" HEIGHT="27" BORDER="0"
- SRC="img41.gif"
- ALT="\begin{displaymath}
-A X = B,
-\end{displaymath}">
-</DIV>
-<BR CLEAR="ALL">
-<P></P>
-where <I>A</I> and <I>B</I> are real or complex matrices, with <I>A</I>
-square and nonsingular. On exit, <var>B</var> is replaced by the solution.
-The arguments <var>A</var> and <var>B</var> must have the same type (<code>'d'</code> or
-<code>'z'</code>).
-The optional argument <var>ipiv</var> is an integer matrix of length at
-least <var>n</var>.
-If <var>ipiv</var> is provided, then <tt class="function">gesv()</tt> solves the system,
-replaces <var>A</var> with its triangular factors, and returns the
-permutation matrix in <var>ipiv</var>.
-If <var>ipiv</var> is not specified, then <tt class="function">gesv()</tt> solves the
-system but does not return the LU factorization and does not
-modify <var>A</var>.
-For example,
-<div class="verbatim"><pre>
->>> gesv(A, B)
-</pre></div>
-solves the system without modifying <var>A</var> and returns the solution
-in <var>B</var>.
-<div class="verbatim"><pre>
->>> gesv(A, B, ipiv)
-</pre></div>
-returns the solution in <var>B</var> and also returns the details of the
-LU factorization in <var>A</var> and <var>ipiv</var>.
-
-<P>
-Raises an <code>ArithmeticError</code> if the matrix is singular.
-</dl>
-
-<P>
-<dl><dt><table cellpadding="0" cellspacing="0"><tr valign="baseline">
- <td><nobr><b><tt id='l2h-63' xml:id='l2h-63' class="function">getrf</tt></b>(</nobr></td>
- <td><var>A, ipiv</var>)</td></tr></table></dt>
-<dd>
-LU factorization of a general, possibly rectangular, real or
-complex matrix,
-<BR><P></P>
-<DIV ALIGN="CENTER" CLASS="mathdisplay">
-<!-- MATH
- \begin{displaymath}
-A = PLU
-\end{displaymath}
- -->
-
-<IMG
- WIDTH="68" HEIGHT="24" BORDER="0"
- SRC="img42.gif"
- ALT="\begin{displaymath}
-A = PLU
-\end{displaymath}">
-</DIV>
-<BR CLEAR="ALL">
-<P></P>
-where <I>A</I> is <var>m</var> by <var>n</var>.
-The argument <var>ipiv</var> is an integer matrix of length at least
-min{<var>m</var>, <var>n</var>}.
-On exit, the lower triangular part of <var>A</var> is replaced by <I>L</I>,
-the upper triangular part by <I>U</I>, and the permutation matrix is
-returned in <var>ipiv</var>.
-Raises an <code>ArithmeticError</code> if the matrix is not full rank.
-</dl>
-
-<P>
-<dl><dt><table cellpadding="0" cellspacing="0"><tr valign="baseline">
- <td><nobr><b><tt id='l2h-64' xml:id='l2h-64' class="function">getrs</tt></b>(</nobr></td>
- <td><var>A, ipiv, B</var><big>[</big><var>, trans='N'</var><big>]</big><var></var>)</td></tr></table></dt>
-<dd>
-Solves a general set of linear equations
-<BR><P></P>
-<DIV ALIGN="CENTER" CLASS="mathdisplay">
-<!-- MATH
- \begin{displaymath}
-AX=B \quad (\mathrm{trans} = \mathrm{'N'}), \qquad
- A^TX=B \quad (\mathrm{trans} = \mathrm{'T'}), \qquad
- A^HX=B \quad (\mathrm{trans} = \mathrm{'C'}),
-\end{displaymath}
- -->
-
-<IMG
- WIDTH="601" HEIGHT="28" BORDER="0"
- SRC="img43.gif"
- ALT="\begin{displaymath}
-AX=B \quad (\mathrm{trans} = \mathrm{'N'}), \qquad
-A^TX=B ...
-...'T'}), \qquad
-A^HX=B \quad (\mathrm{trans} = \mathrm{'C'}),
-\end{displaymath}">
-</DIV>
-<BR CLEAR="ALL">
-<P></P>
-given the LU factorization computed by <tt class="function">gesv()</tt> or
-<tt class="function">getrf()</tt>.
-On entry, <var>A</var> and <var>ipiv</var> must contain the factorization
-as computed by <tt class="function">gesv()</tt> or <tt class="function">getrf()</tt>.
-On exit, <var>B</var> is overwritten with the solution.
-<var>B</var> must have the same type as <var>A</var>.
-</dl>
-
-<P>
-<dl><dt><table cellpadding="0" cellspacing="0"><tr valign="baseline">
- <td><nobr><b><tt id='l2h-65' xml:id='l2h-65' class="function">getri</tt></b>(</nobr></td>
- <td><var>A, ipiv</var>)</td></tr></table></dt>
-<dd>
-Computes the inverse of a matrix.
-On entry, <var>A</var> and <var>ipiv</var> must contain the factorization
-as computed by <tt class="function">gesv()</tt> or <tt class="function">getrf()</tt>. On exit,
-<var>A</var> contains the inverse.
-</dl>
-
-<P>
-In the following example we compute
-<BR><P></P>
-<DIV ALIGN="CENTER" CLASS="mathdisplay">
-<!-- MATH
- \begin{displaymath}
-x = (A^{-1} + A^{-T})b
-\end{displaymath}
- -->
-
-<IMG
- WIDTH="129" HEIGHT="28" BORDER="0"
- SRC="img44.gif"
- ALT="\begin{displaymath}
-x = (A^{-1} + A^{-T})b
-\end{displaymath}">
-</DIV>
-<BR CLEAR="ALL">
-<P></P>
-for randomly generated problem data, factoring the coefficient matrix
-once.
-<div class="verbatim"><pre>
->>> from cvxopt.base import matrix
->>> from cvxopt.random import normal
->>> from cvxopt.lapack import gesv, getrs
->>> n = 10
->>> A = normal(n,n)
->>> b = normal(n)
->>> ipiv = matrix(0, (n,1))
->>> x = +b
->>> gesv(A, x, ipiv) # x = A^{-1}*b
->>> x2 = +b
->>> getrs(A, ipiv, x2, trans='T') # x2 = A^{-T}*b
->>> x += x2
-</pre></div>
-
-<P>
-Separate functions are provided for equations with band matrices.
-<dl><dt><table cellpadding="0" cellspacing="0"><tr valign="baseline">
- <td><nobr><b><tt id='l2h-66' xml:id='l2h-66' class="function">gbsv</tt></b>(</nobr></td>
- <td><var>A, kl, B</var><big>[</big><var>, ipiv=None</var><big>]</big><var></var>)</td></tr></table></dt>
-<dd>
-Solves
-<BR><P></P>
-<DIV ALIGN="CENTER" CLASS="mathdisplay">
-<!-- MATH
- \begin{displaymath}
-A X = B,
-\end{displaymath}
- -->
-
-<IMG
- WIDTH="64" HEIGHT="27" BORDER="0"
- SRC="img41.gif"
- ALT="\begin{displaymath}
-A X = B,
-\end{displaymath}">
-</DIV>
-<BR CLEAR="ALL">
-<P></P>
-where <I>A</I> and <I>B</I> are real or complex matrices, with <I>A</I>
-<I>n</I> by <I>n</I> and banded with <var>kl</var> subdiagonals.
-The arguments <var>A</var> and <var>B</var> must have the same type (<code>'d'</code> or
-<code>'z'</code>).
-
-<P>
-The optional argument <var>ipiv</var> is an integer matrix of length at least
-<I>n</I>.
-If <var>ipiv</var> is provided, then <var>A</var> must have
-2<var>kl</var> + <var>ku</var> + 1 rows. On entry the diagonals of <I>A</I> are
-stored in rows <var>kl</var> + 1 to 2<var>kl</var> + <var>ku</var> +1 of the <var>A</var>, using
-the BLAS format for general band matrices (see section <A href="s-conventions.html#s-conventions">3.1</A>).
-On exit, the factorization is returned in <var>A</var> and <var>ipiv</var>.
-
-<P>
-If <var>ipiv</var> is not provided, then <var>A</var> must have <var>kl</var> + <var>ku</var> +
-1 rows. On entry the diagonals of <I>A</I> are stored in the rows of
-<var>A</var>, following the standard format for general band matrices.
-In this case, <tt class="function">gbsv()</tt> does not modify <var>A</var> on exit and does
-not return the factorization.
-
-<P>
-On exit, <var>B</var> is replaced by the solution <I>X</I>.
-Raises an <code>ArithmeticError</code> if the matrix is singular.
-</dl>
-
-<P>
-<dl><dt><table cellpadding="0" cellspacing="0"><tr valign="baseline">
- <td><nobr><b><tt id='l2h-67' xml:id='l2h-67' class="function">gbtrf</tt></b>(</nobr></td>
- <td><var>A, m, kl, ipiv</var>)</td></tr></table></dt>
-<dd>
-LU factorization of a general <I>m</I> by <I>n</I> real or complex band
-matrix with <var>kl</var> subdiagonals.
-The matrix is stored using the BLAS format for general band matrices
-(see section <A href="s-conventions.html#s-conventions">3.1</A>), by providing the
-diagonals (stored as rows of a <var>ku</var> + <var>kl</var> + 1 by <I>n</I> matrix),
-the number of rows <var>m</var>, and the number of subdiagonals <var>kl</var>.
-The argument <var>ipiv</var> is an integer matrix of length at least
-min{<I>m</I>, <I>n</I>}.
-On exit, <var>A</var> and <var>ipiv</var> contain the details of the factorization.
-Raises an <code>ArithmeticError</code> if the matrix is not full rank.
-</dl>
-
-<P>
-<dl><dt><table cellpadding="0" cellspacing="0"><tr valign="baseline">
- <td><nobr><b><tt id='l2h-68' xml:id='l2h-68' class="function">gbtrs</tt></b>(</nobr></td>
- <td><var>A, kl, ipiv, B</var><big>[</big><var>, trans='N'</var><big>]</big><var></var>)</td></tr></table></dt>
-<dd>
-Solves a set of linear equations
-<BR><P></P>
-<DIV ALIGN="CENTER" CLASS="mathdisplay">
-<!-- MATH
- \begin{displaymath}
-AX=B \quad (\mathrm{trans} = \mathrm{'N'}), \qquad
- A^TX=B \quad (\mathrm{trans} = \mathrm{'T'}), \qquad
- A^HX=B \quad (\mathrm{trans} = \mathrm{'C'}),
-\end{displaymath}
- -->
-
-<IMG
- WIDTH="601" HEIGHT="28" BORDER="0"
- SRC="img43.gif"
- ALT="\begin{displaymath}
-AX=B \quad (\mathrm{trans} = \mathrm{'N'}), \qquad
-A^TX=B ...
-...'T'}), \qquad
-A^HX=B \quad (\mathrm{trans} = \mathrm{'C'}),
-\end{displaymath}">
-</DIV>
-<BR CLEAR="ALL">
-<P></P>
-with <I>A</I> a general band matrix with <var>kl</var> subdiagonals, given the
-LU factorization computed by <tt class="function">gbsv()</tt> or <tt class="function">gbtrf()</tt>.
-On entry, <var>A</var> and <var>ipiv</var> must contain the factorization
-as computed by <tt class="function">gbsv()</tt> or <tt class="function">gbtrf()</tt>.
-On exit, <var>B</var> is overwritten with the solution.
-<var>B</var> must have the same type as <var>A</var>.
-</dl>
-
-<P>
-As an example, we solve a linear equation with
-<BR><P></P>
-<DIV ALIGN="CENTER" CLASS="mathdisplay">
-<!-- MATH
- \begin{displaymath}
-A = \left[ \begin{array}{cccc}
- 1 & 2 & 0 & 0 \\
- 3 & 4 & 5 & 0 \\
- 6 & 7 & 8 & 9 \\
- 0 & 10 & 11 & 12
- \end{array}\right], \qquad x = \left[\begin{array}{c} 1 \\1 \\1 \\1
- \end{array}\right].
-\end{displaymath}
- -->
-
-<IMG
- WIDTH="297" HEIGHT="83" BORDER="0"
- SRC="img45.gif"
- ALT="\begin{displaymath}
-A = \left[ \begin{array}{cccc}
-1 & 2 & 0 & 0 \\
-3 & 4 & ...
-...= \left[\begin{array}{c} 1 \\ 1 \\ 1 \\ 1
-\end{array}\right].
-\end{displaymath}">
-</DIV>
-<BR CLEAR="ALL">
-<P></P>
-<div class="verbatim"><pre>
->>> from cvxopt.base import matrix
->>> from cvxopt.lapack import gbsv, gbtrf, gbtrs
->>> n, kl, ku = 4, 2, 1
->>> A = matrix([[0., 1., 3., 6.], [2., 4., 7., 10.], [5., 8., 11., 0.], [9., 12., 0., 0.]])
->>> x = matrix(1.0, (4,1))
->>> gbsv(A, kl, x)
->>> print x
- 7.1429e-02
- 4.6429e-01
- -2.1429e-01
- -1.0714e-01
-</pre></div>
-The code below illustrates how one can reuse the factorization returned
-by <tt class="function">gbsv()</tt>.
-<div class="verbatim"><pre>
->>> Ac = matrix(0.0, (2*kl+ku+1,n))
->>> Ac[kl:,:] = A
->>> ipiv = matrix(0, (n,1))
->>> x = matrix(1.0, (4,1))
->>> gbsv(Ac, kl, x, ipiv) # solves A*x = 1
->>> print x
- 7.1429e-02
- 4.6429e-01
- -2.1429e-01
- -1.0714e-01
->>> x = matrix(1.0, (4,1))
->>> gbtrs(Ac, kl, ipiv, x, trans='T') # solve A^T*x = 1
->>> print x
- 7.1429e-02
- 2.3810e-02
- 1.4286e-01
- -2.3810e-02
-</pre></div>
-An alternative method uses <tt class="function">getrf()</tt> for the factorization.
-<div class="verbatim"><pre>
->>> Ac[kl:,:] = A
->>> gbtrf(Ac, n, kl, ipiv)
->>> x = matrix(1.0, (4,1))
->>> gbtrs(Ac, kl, ipiv, x) # solve A^T*x = 1
->>> print x
- 7.1429e-02
- 4.6429e-01
- -2.1429e-01
- -1.0714e-01
->>> x = matrix(1.0, (4,1))
->>> gbtrs(Ac, kl, ipiv, x, trans='T') # solve A^T*x = 1
->>> print x
- 7.1429e-02
- 2.3810e-02
- 1.4286e-01
- -2.3810e-02
-</pre></div>
-
-<P>
-The following functions can be used for tridiagonal matrices They use a
-simpler matrix format, that stores the diagonals in three separate
-vectors.
-
-<P>
-<dl><dt><table cellpadding="0" cellspacing="0"><tr valign="baseline">
- <td><nobr><b><tt id='l2h-69' xml:id='l2h-69' class="function">gtsv</tt></b>(</nobr></td>
- <td><var>dl, d, du, B)</var>)</td></tr></table></dt>
-<dd>
-Solves
-<BR><P></P>
-<DIV ALIGN="CENTER" CLASS="mathdisplay">
-<!-- MATH
- \begin{displaymath}
-A X = B,
-\end{displaymath}
- -->
-
-<IMG
- WIDTH="64" HEIGHT="27" BORDER="0"
- SRC="img41.gif"
- ALT="\begin{displaymath}
-A X = B,
-\end{displaymath}">
-</DIV>
-<BR CLEAR="ALL">
-<P></P>
-where <I>A</I> is an <I>n</I> by <I>n</I> tridiagonal matrix,
-with subdiagonal <var>dl</var> (a matrix of length <I>n</I>-1),
-diagonal <var>d</var> (a matrix of length <I>n</I>), and superdiagonal
-<var>du</var> (a matrix of length <I>n</I>-1).
-The four arguments must have the same type (<code>'d'</code> or <code>'z'</code>).
-On exit <var>dl</var>, <var>d</var>, <var>du</var> are overwritten with the details of
-the LU factorization of <I>A</I>, and <var>B</var> is overwritten with the
-solution <I>X</I>.
-Raises an <code>ArithmeticError</code> if the matrix is singular.
-</dl>
-
-<P>
-<dl><dt><table cellpadding="0" cellspacing="0"><tr valign="baseline">
- <td><nobr><b><tt id='l2h-70' xml:id='l2h-70' class="function">gttrf</tt></b>(</nobr></td>
- <td><var>dl, d, du, du2, ipiv</var>)</td></tr></table></dt>
-<dd>
-LU factorization of an <I>n</I> by <I>n</I> tridiagonal matrix with
-subdiagonal <var>dl</var>, diagonal <var>d</var> and superdiagonal <var>du</var>.
-<var>dl</var>, <var>d</var> and <var>du</var> must have the same type.
-<var>du2</var> is a matrix of length <I>n</I>-2, and of the same type as
-<var>dl</var>.
-<var>ipiv</var> is an <code>'i'</code> matrix of length <I>n</I>.
-On exit, the five arguments contain the details of the factorization.
-Raises an <code>ArithmeticError</code> if the matrix is singular.
-</dl>
-
-<P>
-<dl><dt><table cellpadding="0" cellspacing="0"><tr valign="baseline">
- <td><nobr><b><tt id='l2h-71' xml:id='l2h-71' class="function">pttrs</tt></b>(</nobr></td>
- <td><var>dl, d, du, du2, ipiv, B</var><big>[</big><var>, trans='N'</var><big>]</big><var></var>)</td></tr></table></dt>
-<dd>
-Solves a set of linear equations
-<BR><P></P>
-<DIV ALIGN="CENTER" CLASS="mathdisplay">
-<!-- MATH
- \begin{displaymath}
-AX=B \quad (\mathrm{trans} = \mathrm{'N'}), \qquad
- A^TX=B \quad (\mathrm{trans} = \mathrm{'T'}), \qquad
- A^HX=B \quad (\mathrm{trans} = \mathrm{'C'}),
-\end{displaymath}
- -->
-
-<IMG
- WIDTH="601" HEIGHT="28" BORDER="0"
- SRC="img43.gif"
- ALT="\begin{displaymath}
-AX=B \quad (\mathrm{trans} = \mathrm{'N'}), \qquad
-A^TX=B ...
-...'T'}), \qquad
-A^HX=B \quad (\mathrm{trans} = \mathrm{'C'}),
-\end{displaymath}">
-</DIV>
-<BR CLEAR="ALL">
-<P></P>
-where <I>A</I> is an <I>n</I> by <I>n</I> tridiagonal matrix.
-The arguments <var>dl</var>, <var>d</var>, <var>du</var>, <var>du2</var> and <var>ipiv</var>
-contain the details of the LU factorization as returned by
-<tt class="function">gttrf()</tt>.
-On exit, <var>B</var> is overwritten with the solution <I>X</I>.
-<var>B</var> must have the same type as <var>dl</var>.
-</dl>
-
-<P>
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-<H1><A NAME="SECTION006300000000000000000">
-4.3 Symmetric and Hermitian Linear Equations</A>
-</H1>
-<dl><dt><table cellpadding="0" cellspacing="0"><tr valign="baseline">
- <td><nobr><b><tt id='l2h-82' xml:id='l2h-82' class="function">sysv</tt></b>(</nobr></td>
- <td><var>A, B</var><big>[</big><var>, ipiv=None</var><big>[</big><var>, uplo='L'</var><big>]</big><var></var><big>]</big><var></var>)</td></tr></table></dt>
-<dd>
-Solves
-<BR><P></P>
-<DIV ALIGN="CENTER" CLASS="mathdisplay">
-<!-- MATH
- \begin{displaymath}
-AX=B
-\end{displaymath}
- -->
-
-<IMG
- WIDTH="60" HEIGHT="24" BORDER="0"
- SRC="img47.gif"
- ALT="\begin{displaymath}
-AX=B
-\end{displaymath}">
-</DIV>
-<BR CLEAR="ALL">
-<P></P>
-where <I>A</I> is a real or complex symmetric matrix of order <I>n</I>.
-On exit, <var>B</var> is replaced by the solution.
-The matrices <var>A</var> and <var>B</var> must have the same type (<code>'d'</code> or
-<code>'z'</code>).
-The optional argument <var>ipiv</var> is an integer matrix of length at
-least equal to <I>n</I>.
-If <var>ipiv</var> is provided, <tt class="function">sysv()</tt> solves the system and
-returns the factorization in <var>A</var> and <var>ipiv</var>.
-If <var>ipiv</var> is not specified, <tt class="function">sysv()</tt> solves the
-system but does not return the factorization and does not modify
-<var>A</var>.
-Raises an <code>ArithmeticError</code> if the matrix is singular.
-</dl>
-
-<P>
-<dl><dt><table cellpadding="0" cellspacing="0"><tr valign="baseline">
- <td><nobr><b><tt id='l2h-83' xml:id='l2h-83' class="function">sytrf</tt></b>(</nobr></td>
- <td><var>A, ipiv</var><big>[</big><var>, uplo='L'</var><big>]</big><var></var>)</td></tr></table></dt>
-<dd>
-LDL<SPAN CLASS="MATH"><IMG
- WIDTH="14" HEIGHT="18" ALIGN="BOTTOM" BORDER="0"
- SRC="img50.gif"
- ALT="${}\mathrm{^T}$"></SPAN> factorization
-<BR><P></P>
-<DIV ALIGN="CENTER" CLASS="mathdisplay">
-<!-- MATH
- \begin{displaymath}
-PAP^T = LDL^T
-\end{displaymath}
- -->
-
-<IMG
- WIDTH="112" HEIGHT="24" BORDER="0"
- SRC="img52.gif"
- ALT="\begin{displaymath}
-PAP^T = LDL^T
-\end{displaymath}">
-</DIV>
-<BR CLEAR="ALL">
-<P></P>
-of a real or complex symmetric matrix <SPAN CLASS="MATH"><IMG
- WIDTH="16" HEIGHT="14" ALIGN="BOTTOM" BORDER="0"
- SRC="img53.gif"
- ALT="$A$"></SPAN> of order <I>n</I>.
-<var>ipiv</var> is an <code>'i'</code> matrix of length at least <I>n</I>.
-On exit, <var>A</var> and <var>ipiv</var> contain the factorization.
-Raises an <code>ArithmeticError</code> if the matrix is singular.
-</dl>
-
-<P>
-<dl><dt><table cellpadding="0" cellspacing="0"><tr valign="baseline">
- <td><nobr><b><tt id='l2h-84' xml:id='l2h-84' class="function">sytrs</tt></b>(</nobr></td>
- <td><var>A, ipiv, B</var><big>[</big><var>, uplo='L'</var><big>]</big><var></var>)</td></tr></table></dt>
-<dd>
-Solves
-<BR><P></P>
-<DIV ALIGN="CENTER" CLASS="mathdisplay">
-<!-- MATH
- \begin{displaymath}
-A X = B
-\end{displaymath}
- -->
-
-<IMG
- WIDTH="60" HEIGHT="24" BORDER="0"
- SRC="img47.gif"
- ALT="\begin{displaymath}
-AX=B
-\end{displaymath}">
-</DIV>
-<BR CLEAR="ALL">
-<P></P>
-given the LDL<SPAN CLASS="MATH"><IMG
- WIDTH="14" HEIGHT="18" ALIGN="BOTTOM" BORDER="0"
- SRC="img50.gif"
- ALT="${}\mathrm{^T}$"></SPAN> factorization computed by
-<tt class="function">sytrf()</tt> or <tt class="function">sysv()</tt>. <var>B</var> must have the same
-type as <var>A</var>.
-</dl>
-
-<P>
-<dl><dt><table cellpadding="0" cellspacing="0"><tr valign="baseline">
- <td><nobr><b><tt id='l2h-85' xml:id='l2h-85' class="function">sytri</tt></b>(</nobr></td>
- <td><var>A, ipiv</var><big>[</big><var>, uplo='L'</var><big>]</big><var></var>)</td></tr></table></dt>
-<dd>
-Computes the inverse of a real or complex symmetric matrix.
-On entry, <var>A</var> and <var>ipiv</var> contain the LDL<SPAN CLASS="MATH"><IMG
- WIDTH="14" HEIGHT="18" ALIGN="BOTTOM" BORDER="0"
- SRC="img50.gif"
- ALT="${}\mathrm{^T}$"></SPAN>
-factorization computed by <tt class="function">sytrf()</tt> or <tt class="function">sysv()</tt>.
-On exit, <var>A</var> contains the inverse.
-</dl>
-
-<P>
-<dl><dt><table cellpadding="0" cellspacing="0"><tr valign="baseline">
- <td><nobr><b><tt id='l2h-86' xml:id='l2h-86' class="function">hesv</tt></b>(</nobr></td>
- <td><var>A, B</var><big>[</big><var>, ipiv=<code>None</code></var><big>[</big><var>, uplo='L'</var><big>]</big><var></var><big>]</big><var></var>)</td></tr></table></dt>
-<dd>
-Solves
-<BR><P></P>
-<DIV ALIGN="CENTER" CLASS="mathdisplay">
-<!-- MATH
- \begin{displaymath}
-A X = B
-\end{displaymath}
- -->
-
-<IMG
- WIDTH="60" HEIGHT="24" BORDER="0"
- SRC="img47.gif"
- ALT="\begin{displaymath}
-AX=B
-\end{displaymath}">
-</DIV>
-<BR CLEAR="ALL">
-<P></P>
-where <I>A</I> is a real symmetric or complex Hermitian of order <I>n</I>.
-On exit, <var>B</var> is replaced by the solution.
-The matrices <var>A</var> and <var>B</var> must have the same type (<code>'d'</code> or
-<code>'z'</code>).
-The optional argument <var>ipiv</var> is an integer matrix of length at
-least <var>n</var>.
-If <var>ipiv</var> is provided, then <tt class="function">hesv()</tt> solves the system and
-returns the factorization in <var>A</var> and <var>ipiv</var>.
-If <var>ipiv</var> is not specified, then <tt class="function">hesv()</tt> solves the
-system but does not return the factorization and does not modify
-<var>A</var>.
-Raises an <code>ArithmeticError</code> if the matrix is singular.
-</dl>
-
-<P>
-<dl><dt><table cellpadding="0" cellspacing="0"><tr valign="baseline">
- <td><nobr><b><tt id='l2h-87' xml:id='l2h-87' class="function">hetrf</tt></b>(</nobr></td>
- <td><var>A, ipiv</var><big>[</big><var>, uplo='L'</var><big>]</big><var></var>)</td></tr></table></dt>
-<dd>
-LDL<SPAN CLASS="MATH"><IMG
- WIDTH="14" HEIGHT="18" ALIGN="BOTTOM" BORDER="0"
- SRC="img51.gif"
- ALT="${}\mathrm{^H}$"></SPAN> factorization
-<BR><P></P>
-<DIV ALIGN="CENTER" CLASS="mathdisplay">
-<!-- MATH
- \begin{displaymath}
-PAP^T = LDL^H
-\end{displaymath}
- -->
-
-<IMG
- WIDTH="115" HEIGHT="24" BORDER="0"
- SRC="img54.gif"
- ALT="\begin{displaymath}
-PAP^T = LDL^H
-\end{displaymath}">
-</DIV>
-<BR CLEAR="ALL">
-<P></P>
-of a real symmetric or complex Hermitian matrix of order <I>n</I>.
-<var>ipiv</var> is an <code>'i'</code> matrix of length at least <var>n</var>.
-On exit, <var>A</var> and <var>ipiv</var> contain the factorization.
-Raises an <code>ArithmeticError</code> if the matrix is singular.
-</dl>
-
-<P>
-<dl><dt><table cellpadding="0" cellspacing="0"><tr valign="baseline">
- <td><nobr><b><tt id='l2h-88' xml:id='l2h-88' class="function">hetrs</tt></b>(</nobr></td>
- <td><var>A, ipiv, B</var><big>[</big><var>, uplo='L'</var><big>]</big><var></var>)</td></tr></table></dt>
-<dd>
-Solves
-<BR><P></P>
-<DIV ALIGN="CENTER" CLASS="mathdisplay">
-<!-- MATH
- \begin{displaymath}
-A X = B
-\end{displaymath}
- -->
-
-<IMG
- WIDTH="60" HEIGHT="24" BORDER="0"
- SRC="img47.gif"
- ALT="\begin{displaymath}
-AX=B
-\end{displaymath}">
-</DIV>
-<BR CLEAR="ALL">
-<P></P>
-given the LDL<SPAN CLASS="MATH"><IMG
- WIDTH="14" HEIGHT="18" ALIGN="BOTTOM" BORDER="0"
- SRC="img51.gif"
- ALT="${}\mathrm{^H}$"></SPAN> factorization computed by
-<tt class="function">hetrf()</tt> or <tt class="function">hesv()</tt>.
-</dl>
-
-<P>
-<dl><dt><table cellpadding="0" cellspacing="0"><tr valign="baseline">
- <td><nobr><b><tt id='l2h-89' xml:id='l2h-89' class="function">hetri</tt></b>(</nobr></td>
- <td><var>A, ipiv</var><big>[</big><var>, uplo='L'</var><big>]</big><var></var>)</td></tr></table></dt>
-<dd>
-Computes the inverse of a real symmetric or complex Hermitian matrix.
-On entry, <var>A</var> and <var>ipiv</var> contain the LDL<SPAN CLASS="MATH"><IMG
- WIDTH="14" HEIGHT="18" ALIGN="BOTTOM" BORDER="0"
- SRC="img51.gif"
- ALT="${}\mathrm{^H}$"></SPAN>
-factorization computed by <tt class="function">hetrf()</tt> or <tt class="function">hesv()</tt>.
-On exit, <var>A</var> contains the inverse.
-</dl>
-
-<P>
-As an example we solve the KKT system (<A href="e-kkt-example.html#e-kkt-example">4.1</A>).
-<div class="verbatim"><pre>
->>> from cvxopt.lapack import sysv
->>> K = matrix(0.0, (m+n,m+n))
->>> K[: (m+n)*m : m+n+1] = -d**2
->>> K[:m, m:] = A
->>> x = matrix(0.0, (m+n,1))
->>> x[:m], x[m:] = b1, b2
->>> sysv(K, x, uplo='U')
-</pre></div>
-
-<P>
-
-<DIV CLASS="navigation">
-<div class='online-navigation'>
-<p></p><hr />
-<table align="center" width="100%" cellpadding="0" cellspacing="2">
-<tr>
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- href="e-kkt-example.html"><img src='previous.gif'
- border='0' height='32' alt='Previous Page' width='32' /></A></td>
-<td class='online-navigation'><a rel="parent" title="4. The LAPACK Interface"
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- href="genindex.html"><img src='index.gif'
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-</DIV>
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diff --git a/doc/cvxopt/node23.html b/doc/cvxopt/node23.html
deleted file mode 100644
index 8a0d840..0000000
--- a/doc/cvxopt/node23.html
+++ /dev/null
@@ -1,182 +0,0 @@
-<!DOCTYPE html PUBLIC "-//W3C//DTD HTML 4.0 Transitional//EN">
-<html>
-<head>
-<link rel="STYLESHEET" href="cvxopt.css" type='text/css' />
-<link rel="first" href="cvxopt.html" title='CVXOPT: A Python Package for Convex Optimization' />
-<link rel='contents' href='contents.html' title="Contents" />
-<link rel='index' href='genindex.html' title='Index' />
-<link rel='last' href='about.html' title='About this document...' />
-<link rel='help' href='about.html' title='About this document...' />
-<link rel="next" href="node24.html" />
-<link rel="prev" href="node22.html" />
-<link rel="parent" href="node19.html" />
-<link rel="next" href="node24.html" />
-<meta name='aesop' content='information' />
-<title>4.4 Triangular Linear Equations</title>
-</head>
-<body>
-<DIV CLASS="navigation">
-<div id='top-navigation-panel' xml:id='top-navigation-panel'>
-<table align="center" width="100%" cellpadding="0" cellspacing="2">
-<tr>
-<td class='online-navigation'><a rel="prev" title="4.3 Symmetric and Hermitian"
- href="node22.html"><img src='previous.gif'
- border='0' height='32' alt='Previous Page' width='32' /></A></td>
-<td class='online-navigation'><a rel="parent" title="4. The LAPACK Interface"
- href="node19.html"><img src='up.gif'
- border='0' height='32' alt='Up One Level' width='32' /></A></td>
-<td class='online-navigation'><a rel="next" title="4.5 Least-Squares and Least-Norm"
- href="node24.html"><img src='next.gif'
- border='0' height='32' alt='Next Page' width='32' /></A></td>
-<td align="center" width="100%">CVXOPT: A Python Package for Convex Optimization</td>
-<td class='online-navigation'><a rel="contents" title="Table of Contents"
- href="contents.html"><img src='contents.gif'
- border='0' height='32' alt='Contents' width='32' /></A></td>
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-<td class='online-navigation'><a rel="index" title="Index"
- href="genindex.html"><img src='index.gif'
- border='0' height='32' alt='Index' width='32' /></A></td>
-</tr></table>
-<div class='online-navigation'>
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-<b class="navlabel">Next:</b>
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-</div>
-<hr /></div>
-</DIV>
-<!--End of Navigation Panel-->
-
-<H1><A NAME="SECTION006400000000000000000">
-4.4 Triangular Linear Equations</A>
-</H1>
-
-<P>
-<dl><dt><table cellpadding="0" cellspacing="0"><tr valign="baseline">
- <td><nobr><b><tt id='l2h-90' xml:id='l2h-90' class="function">trtrs</tt></b>(</nobr></td>
- <td><var>A, B</var><big>[</big><var>, uplo='L'</var><big>[</big><var>,
-trans='N'</var><big>[</big><var>, diag='N'</var><big>]</big><var></var><big>]</big><var></var><big>]</big><var></var>)</td></tr></table></dt>
-<dd>
-Solves a triangular set of equations
-<BR><P></P>
-<DIV ALIGN="CENTER" CLASS="mathdisplay">
-<!-- MATH
- \begin{displaymath}
-AX=B \quad (\mathrm{trans} = \mathrm{'N'}), \qquad
- A^TX=B \quad (\mathrm{trans} = \mathrm{'T'}), \qquad
- A^HX=B \quad (\mathrm{trans} = \mathrm{'C'}),
-\end{displaymath}
- -->
-
-<IMG
- WIDTH="601" HEIGHT="28" BORDER="0"
- SRC="img43.gif"
- ALT="\begin{displaymath}
-AX=B \quad (\mathrm{trans} = \mathrm{'N'}), \qquad
-A^TX=B ...
-...'T'}), \qquad
-A^HX=B \quad (\mathrm{trans} = \mathrm{'C'}),
-\end{displaymath}">
-</DIV>
-<BR CLEAR="ALL">
-<P></P>
-where <I>A</I> is real or complex and triangular of order <I>n</I>,
-and <var>B</var> is a matrix with <I>n</I> rows.
-<var>A</var> and <var>B</var> are matrices with the same type (<code>'d'</code> or <code>'z'</code>).
-<tt class="function">trtrs()</tt> is similar to <tt class="function">blas.trsm()</tt>, except
-that it raises an <code>ArithmeticError</code> if a diagonal element of
-<var>A</var> is zero (whereas <tt class="function">blas.trsm()</tt> returns <code>inf</code>
-values).
-</dl>
-
-<P>
-<dl><dt><table cellpadding="0" cellspacing="0"><tr valign="baseline">
- <td><nobr><b><tt id='l2h-91' xml:id='l2h-91' class="function">trtri</tt></b>(</nobr></td>
- <td><var>A</var><big>[</big><var>, uplo='L'</var><big>[</big><var>, diag='N'</var><big>]</big><var></var><big>]</big><var></var>)</td></tr></table></dt>
-<dd>
-Computes the inverse of a real or complex triangular matrix <I>A</I>.
-On exit, <var>A</var> contains the inverse.
-</dl>
-
-<P>
-<dl><dt><table cellpadding="0" cellspacing="0"><tr valign="baseline">
- <td><nobr><b><tt id='l2h-92' xml:id='l2h-92' class="function">tbtrs</tt></b>(</nobr></td>
- <td><var>A, B</var><big>[</big><var>, uplo='L'</var><big>[</big><var>,
-trans='T'</var><big>[</big><var>,diag='N'</var><big>]</big><var></var><big>]</big><var></var><big>]</big><var></var>)</td></tr></table></dt>
-<dd>
-Solves a triangular set of equations
-<BR><P></P>
-<DIV ALIGN="CENTER" CLASS="mathdisplay">
-<!-- MATH
- \begin{displaymath}
-AX=B \quad (\mathrm{trans} = \mathrm{'N'}), \qquad
- A^TX=B \quad (\mathrm{trans} = \mathrm{'T'}), \qquad
- A^HX=B \quad (\mathrm{trans} = \mathrm{'C'}),
-\end{displaymath}
- -->
-
-<IMG
- WIDTH="601" HEIGHT="28" BORDER="0"
- SRC="img43.gif"
- ALT="\begin{displaymath}
-AX=B \quad (\mathrm{trans} = \mathrm{'N'}), \qquad
-A^TX=B ...
-...'T'}), \qquad
-A^HX=B \quad (\mathrm{trans} = \mathrm{'C'}),
-\end{displaymath}">
-</DIV>
-<BR CLEAR="ALL">
-<P></P>
-where <I>A</I> is real or complex triangular band matrix of order <I>n</I>,
-and <var>B</var> is a matrix with <I>n</I> rows.
-The diagonals of <I>A</I> are stored in <var>A</var> using the BLAS conventions
-for triangular band matrices.
-<var>A</var> and <var>B</var> are matrices with the same type (<code>'d'</code> or <code>'z'</code>).
-On exit, <var>B</var> is replaced by the solution <I>X</I>.
-</dl>
-
-<P>
-
-<DIV CLASS="navigation">
-<div class='online-navigation'>
-<p></p><hr />
-<table align="center" width="100%" cellpadding="0" cellspacing="2">
-<tr>
-<td class='online-navigation'><a rel="prev" title="4.3 Symmetric and Hermitian"
- href="node22.html"><img src='previous.gif'
- border='0' height='32' alt='Previous Page' width='32' /></A></td>
-<td class='online-navigation'><a rel="parent" title="4. The LAPACK Interface"
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-<td class='online-navigation'><a rel="next" title="4.5 Least-Squares and Least-Norm"
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-<td align="center" width="100%">CVXOPT: A Python Package for Convex Optimization</td>
-<td class='online-navigation'><a rel="contents" title="Table of Contents"
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- border='0' height='32' alt='Contents' width='32' /></A></td>
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- border='0' height='32' alt='Index' width='32' /></A></td>
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-</div>
-<hr />
-<span class="release-info">Release 0.8.2, documentation updated on February 6, 2007.</span>
-</DIV>
-<!--End of Navigation Panel-->
-
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-</HTML>
diff --git a/doc/cvxopt/node24.html b/doc/cvxopt/node24.html
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--- a/doc/cvxopt/node24.html
+++ /dev/null
@@ -1,402 +0,0 @@
-<!DOCTYPE html PUBLIC "-//W3C//DTD HTML 4.0 Transitional//EN">
-<html>
-<head>
-<link rel="STYLESHEET" href="cvxopt.css" type='text/css' />
-<link rel="first" href="cvxopt.html" title='CVXOPT: A Python Package for Convex Optimization' />
-<link rel='contents' href='contents.html' title="Contents" />
-<link rel='index' href='genindex.html' title='Index' />
-<link rel='last' href='about.html' title='About this document...' />
-<link rel='help' href='about.html' title='About this document...' />
-<link rel="next" href="node25.html" />
-<link rel="prev" href="node23.html" />
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-<meta name='aesop' content='information' />
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-</head>
-<body>
-<DIV CLASS="navigation">
-<div id='top-navigation-panel' xml:id='top-navigation-panel'>
-<table align="center" width="100%" cellpadding="0" cellspacing="2">
-<tr>
-<td class='online-navigation'><a rel="prev" title="4.4 Triangular Linear Equations"
- href="node23.html"><img src='previous.gif'
- border='0' height='32' alt='Previous Page' width='32' /></A></td>
-<td class='online-navigation'><a rel="parent" title="4. The LAPACK Interface"
- href="node19.html"><img src='up.gif'
- border='0' height='32' alt='Up One Level' width='32' /></A></td>
-<td class='online-navigation'><a rel="next" title="4.6 Symmetric and Hermitian"
- href="node25.html"><img src='next.gif'
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-<td align="center" width="100%">CVXOPT: A Python Package for Convex Optimization</td>
-<td class='online-navigation'><a rel="contents" title="Table of Contents"
- href="contents.html"><img src='contents.gif'
- border='0' height='32' alt='Contents' width='32' /></A></td>
-<td class='online-navigation'><img src='blank.gif'
- border='0' height='32' alt='' width='32' /></td>
-<td class='online-navigation'><a rel="index" title="Index"
- href="genindex.html"><img src='index.gif'
- border='0' height='32' alt='Index' width='32' /></A></td>
-</tr></table>
-<div class='online-navigation'>
-<b class="navlabel">Previous:</b>
-<a class="sectref" rel="prev" href="node23.html">4.4 Triangular Linear Equations</A>
-<b class="navlabel">Up:</b>
-<a class="sectref" rel="parent" href="node19.html">4. The LAPACK Interface</A>
-<b class="navlabel">Next:</b>
-<a class="sectref" rel="next" href="node25.html">4.6 Symmetric and Hermitian</A>
-</div>
-<hr /></div>
-</DIV>
-<!--End of Navigation Panel-->
-
-<H1><A NAME="SECTION006500000000000000000">
-4.5 Least-Squares and Least-Norm Problems</A>
-</H1>
-<dl><dt><table cellpadding="0" cellspacing="0"><tr valign="baseline">
- <td><nobr><b><tt id='l2h-93' xml:id='l2h-93' class="function">gels</tt></b>(</nobr></td>
- <td><var>A, B</var><big>[</big><var>, trans='N'</var><big>]</big><var></var>)</td></tr></table></dt>
-<dd>
-Solves least-squares and least-norm problems with a full rank
-<I>m</I> by <I>n</I> matrix <I>A</I>.
-
-<P>
-
-<OL>
-<LI><var>trans</var> is <code>'N'</code>. If <I>m</I> is greater than or equal
-to <I>n</I>, <tt class="function">gels()</tt> solves the least-squares problem
-<BR><P></P>
-<DIV ALIGN="CENTER" CLASS="mathdisplay">
-<!-- MATH
- \begin{displaymath}
-\begin{array}{ll}
- \mbox{minimize} & \|AX-B\|_F.
- \end{array}
-\end{displaymath}
- -->
-
-<IMG
- WIDTH="165" HEIGHT="30" BORDER="0"
- SRC="img55.gif"
- ALT="\begin{displaymath}
-\begin{array}{ll}
-\mbox{minimize} & \Vert AX-B\Vert _F.
-\end{array}
-\end{displaymath}">
-</DIV>
-<BR CLEAR="ALL">
-<P></P>
-If <I>m</I> is less than or equal to <I>n</I>, <tt class="function">gels()</tt> solves
-the least-norm problem
-<BR><P></P>
-<DIV ALIGN="CENTER" CLASS="mathdisplay">
-<!-- MATH
- \begin{displaymath}
-\begin{array}{ll}
- \mbox{minimize} & \|X\|_F \\
- \mbox{subject to} & AX = B.
- \end{array}
-\end{displaymath}
- -->
-
-<IMG
- WIDTH="141" HEIGHT="45" BORDER="0"
- SRC="img56.gif"
- ALT="\begin{displaymath}
-\begin{array}{ll}
-\mbox{minimize} & \Vert X\Vert _F \\
-\mbox{subject to} & AX = B.
-\end{array}\end{displaymath}">
-</DIV>
-<BR CLEAR="ALL">
-<P></P>
-
-<P>
-</LI>
-<LI><var>trans</var> is <code>'T'</code> or <code>'C'</code> and <var>A</var> and <var>B</var>
-are real. If <I>m</I> is greater than or equal to <I>n</I>,
-<tt class="function">gels()</tt> solves the least-norm problem
-<BR><P></P>
-<DIV ALIGN="CENTER" CLASS="mathdisplay">
-<!-- MATH
- \begin{displaymath}
-\begin{array}{ll}
- \mbox{minimize} & \|X\|_F \\
- \mbox{subject to} & A^TX=B.
- \end{array}
-\end{displaymath}
- -->
-
-<IMG
- WIDTH="151" HEIGHT="45" BORDER="0"
- SRC="img57.gif"
- ALT="\begin{displaymath}
-\begin{array}{ll}
-\mbox{minimize} & \Vert X\Vert _F \\
-\mbox{subject to} & A^TX=B.
-\end{array}\end{displaymath}">
-</DIV>
-<BR CLEAR="ALL">
-<P></P>
-If <I>m</I> is less than or equal to <I>n</I>, <tt class="function">gels()</tt> solves
-the least-squares problem
-<BR><P></P>
-<DIV ALIGN="CENTER" CLASS="mathdisplay">
-<!-- MATH
- \begin{displaymath}
-\begin{array}{ll}
- \mbox{minimize} & \|A^TX-B\|_F.
- \end{array}
-\end{displaymath}
- -->
-
-<IMG
- WIDTH="175" HEIGHT="30" BORDER="0"
- SRC="img58.gif"
- ALT="\begin{displaymath}
-\begin{array}{ll}
-\mbox{minimize} & \Vert A^TX-B\Vert _F.
-\end{array}\end{displaymath}">
-</DIV>
-<BR CLEAR="ALL">
-<P></P>
-
-<P>
-</LI>
-<LI><var>trans</var> is <code>'C'</code> and <var>A</var> and <var>B</var>
-are complex. If <I>m</I> is greater than or equal to <I>n</I>,
-<tt class="function">gels()</tt> solves the least-norm problem
-<BR><P></P>
-<DIV ALIGN="CENTER" CLASS="mathdisplay">
-<!-- MATH
- \begin{displaymath}
-\begin{array}{ll}
- \mbox{minimize} & \|X\|_F \\
- \mbox{subject to} & A^HX=B.
- \end{array}
-\end{displaymath}
- -->
-
-<IMG
- WIDTH="153" HEIGHT="45" BORDER="0"
- SRC="img59.gif"
- ALT="\begin{displaymath}
-\begin{array}{ll}
-\mbox{minimize} & \Vert X\Vert _F \\
-\mbox{subject to} & A^HX=B.
-\end{array}\end{displaymath}">
-</DIV>
-<BR CLEAR="ALL">
-<P></P>
-If <I>m</I> is less than or equal to <I>n</I>, <tt class="function">gels()</tt> solves
-the least-squares problem
-<BR><P></P>
-<DIV ALIGN="CENTER" CLASS="mathdisplay">
-<!-- MATH
- \begin{displaymath}
-\begin{array}{ll}
- \mbox{minimize} & \|A^HX-B\|_F.
- \end{array}
-\end{displaymath}
- -->
-
-<IMG
- WIDTH="177" HEIGHT="30" BORDER="0"
- SRC="img60.gif"
- ALT="\begin{displaymath}
-\begin{array}{ll}
-\mbox{minimize} & \Vert A^HX-B\Vert _F.
-\end{array}\end{displaymath}">
-</DIV>
-<BR CLEAR="ALL">
-<P></P>
-</LI>
-</OL>
-<var>A</var> and <var>B</var> must have the same typecode (<code>'d'</code> or <code>'z'</code>).
-<var>trans</var> = <code>'T'</code> is not allowed if <var>A</var> is complex.
-On exit, the solution <I>X</I> is stored as the leading submatrix
-of <var>B</var>.
-The array <var>A</var> is overwritten with details of the QR or the LQ
-factorization of <I>A</I>.
-Note that <tt class="function">gels()</tt> does not check whether <SPAN CLASS="MATH"><IMG
- WIDTH="16" HEIGHT="14" ALIGN="BOTTOM" BORDER="0"
- SRC="img53.gif"
- ALT="$A$"></SPAN> is full rank.
-</dl>
-
-<P>
-<dl><dt><table cellpadding="0" cellspacing="0"><tr valign="baseline">
- <td><nobr><b><tt id='l2h-94' xml:id='l2h-94' class="function">geqrf</tt></b>(</nobr></td>
- <td><var>A, tau</var>)</td></tr></table></dt>
-<dd>
-QR factorization of a real or complex matrix <var>A</var>:
-<BR><P></P>
-<DIV ALIGN="CENTER" CLASS="mathdisplay">
-<!-- MATH
- \begin{displaymath}
-A = Q R.
-\end{displaymath}
- -->
-
-<IMG
- WIDTH="61" HEIGHT="27" BORDER="0"
- SRC="img61.gif"
- ALT="\begin{displaymath}
-A = Q R.
-\end{displaymath}">
-</DIV>
-<BR CLEAR="ALL">
-<P></P>
-If <var>A</var> is <I>m</I> by <I>n</I>, then <I>Q</I> is <I>m</I> by <I>m</I>
-and orthogonal/unitary, and <var>R</var> is <I>m</I> by <I>n</I>
-and upper triangular (if <var>m</var> is greater than or equal to <var>n</var>),
-or upper trapezoidal (if <var>m</var> is less than or equal to <var>n</var>).
-<var>tau</var> is a matrix of the same type as <var>A</var> and of length at
-least min{<var>m</var>, <var>n</var>}.
-On exit, <I>R</I> is stored in the upper triangular part of <var>A</var>.
-The matrix <I>Q</I> is stored as a product of min{<var>m</var>, <var>n</var>}
-elementary reflectors in the first min{<var>m</var>, <var>n</var>} columns
-of <var>A</var> and in <var>tau</var>.
-</dl>
-
-<P>
-<dl><dt><table cellpadding="0" cellspacing="0"><tr valign="baseline">
- <td><nobr><b><tt id='l2h-95' xml:id='l2h-95' class="function">ormqr</tt></b>(</nobr></td>
- <td><var>A, tau, C</var><big>[</big><var>, side='L'</var><big>[</big><var>,
-trans='N'</var><big>]</big><var></var><big>]</big><var></var>)</td></tr></table></dt>
-<dd>
-Product with a real orthogonal matrix:
-<BR><P></P>
-<DIV ALIGN="CENTER" CLASS="mathdisplay">
-<!-- MATH
- \begin{displaymath}
-C := \mathop{\mathrm{op}}(Q)C \quad (\mathrm{side} = \mathrm{'L'}), \qquad
- C := C\mathop{\mathrm{op}}(Q) \quad (\mathrm{side} = \mathrm{'R'}), \qquad
- \mathop{\mathrm{op}}(Q) = \left\{ \begin{array}{ll}
- Q & \mathrm{trans} = \mathrm{'N'} \\
- Q^T & \mathrm{trans} = \mathrm{'T'},
-\end{array}\right.
-\end{displaymath}
- -->
-
-<IMG
- WIDTH="658" HEIGHT="45" BORDER="0"
- SRC="img62.gif"
- ALT="\begin{displaymath}
-C := \mathop{\mathrm{op}}(Q)C \quad (\mathrm{side} = \mathr...
-...} \\
-Q^T & \mathrm{trans} = \mathrm{'T'},
-\end{array}\right.
-\end{displaymath}">
-</DIV>
-<BR CLEAR="ALL">
-<P></P>
-where <I>Q</I> is square and orthogonal.
-<I>Q</I> is stored in <var>A</var> and <var>tau</var> as a product
-of min{<var>A</var>.<tt class="member">size</tt>[0], <var>A</var>.<tt class="member">size</tt>[1]}
-elementary reflectors, as computed by <tt class="function">geqrf()</tt>.
-</dl>
-
-<P>
-<dl><dt><table cellpadding="0" cellspacing="0"><tr valign="baseline">
- <td><nobr><b><tt id='l2h-96' xml:id='l2h-96' class="function">unmqr</tt></b>(</nobr></td>
- <td><var>A, tau, C</var><big>[</big><var>, side='L'</var><big>[</big><var>,
-trans='N'</var><big>]</big><var></var><big>]</big><var></var>)</td></tr></table></dt>
-<dd>
-Product with a real orthogonal or complex unitary matrix:
-<BR><P></P>
-<DIV ALIGN="CENTER" CLASS="mathdisplay">
-<!-- MATH
- \begin{displaymath}
-C := \mathop{\mathrm{op}}(Q)C \quad (\mathrm{side} = \mathrm{'L'}), \qquad
- C := C\mathop{\mathrm{op}}(Q) \quad (\mathrm{side} = \mathrm{'R'}), \qquad
- \mathop{\mathrm{op}}(Q) = \left\{ \begin{array}{ll}
- Q & \mathrm{trans} = \mathrm{'N'} \\
- Q^T & \mathrm{trans} = \mathrm{'T'} \\
- Q^H & \mathrm{trans} = \mathrm{'C'},
-\end{array}\right.
-\end{displaymath}
- -->
-
-<IMG
- WIDTH="662" HEIGHT="64" BORDER="0"
- SRC="img63.gif"
- ALT="\begin{displaymath}
-C := \mathop{\mathrm{op}}(Q)C \quad (\mathrm{side} = \mathr...
-...} \\
-Q^H & \mathrm{trans} = \mathrm{'C'},
-\end{array}\right.
-\end{displaymath}">
-</DIV>
-<BR CLEAR="ALL">
-<P></P>
-<I>Q</I> is square and orthogonal or unitary.
-<I>Q</I> is stored in <var>A</var> and <var>tau</var> as a product of
-min{<var>A</var>.<tt class="member">size</tt>[0], <var>A</var>.<tt class="member">size</tt>[1]}
-elementary reflectors, as computed by <tt class="function">geqrf()</tt>.
-The arrays <var>A</var>, <var>tau</var> and <var>C</var> must have the same type.
-<code><var>trans</var> = 'T'</code> is only allowed if the typecode is <code>'d'</code>.
-</dl>
-
-<P>
-In the following example, we solve a least-squares problem
-by a direct call to <tt class="function">gels()</tt>, and by separate calls to
-<tt class="function">geqrf()</tt>, <tt class="function">ormqr()</tt>, and <tt class="function">trtrs()</tt>.
-<div class="verbatim"><pre>
->>> from cvxopt import random, blas, lapack
->>> from cvxopt.base import matrix
->>> m, n = 10, 5
->>> A, b = random.normal(m,n), random.normal(m,1)
->>> x1 = +b
->>> lapack.gels(+A, x1) # x1[:n] minimizes ||A*x1[:n] - b||_2
->>> tau = matrix(0.0, (n,1))
->>> lapack.geqrf(A, tau) # A = [Q1, Q2] * [R1; 0]
->>> x2 = +b
->>> lapack.ormqr(A, tau, x2, trans='T') # x2 := [Q1, Q2]' * b
->>> lapack.trtrs(A[:n,:], x2, uplo='U') # x2[:n] := R1^{-1}*x2[:n]
->>> blas.nrm2(x1[:n] - x2[:n])
-3.0050798580569307e-16
-</pre></div>
-
-<P>
-
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+++ /dev/null
@@ -1,257 +0,0 @@
-<!DOCTYPE html PUBLIC "-//W3C//DTD HTML 4.0 Transitional//EN">
-<html>
-<head>
-<link rel="STYLESHEET" href="cvxopt.css" type='text/css' />
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-
-<H1><A NAME="SECTION006600000000000000000">
-4.6 Symmetric and Hermitian Eigenvalue Decomposition</A>
-</H1>
-The first four routines compute all or selected eigenvalues and
-eigenvectors of a real symmetric matrix <SPAN CLASS="MATH"><IMG
- WIDTH="16" HEIGHT="14" ALIGN="BOTTOM" BORDER="0"
- SRC="img53.gif"
- ALT="$A$"></SPAN>:
-<BR><P></P>
-<DIV ALIGN="CENTER" CLASS="mathdisplay">
-<!-- MATH
- \begin{displaymath}
-A = V\mbox{\bf diag}\,(\lambda)V^T,\qquad V^TV = I.
-\end{displaymath}
- -->
-
-<IMG
- WIDTH="229" HEIGHT="28" BORDER="0"
- SRC="img64.gif"
- ALT="\begin{displaymath}
-A = V\mbox{\bf diag}\,(\lambda)V^T,\qquad V^TV = I.
-\end{displaymath}">
-</DIV>
-<BR CLEAR="ALL">
-<P></P>
-
-<P>
-<dl><dt><table cellpadding="0" cellspacing="0"><tr valign="baseline">
- <td><nobr><b><tt id='l2h-97' xml:id='l2h-97' class="function">syev</tt></b>(</nobr></td>
- <td><var>A, W</var><big>[</big><var>, jobz='N'</var><big>[</big><var>, uplo='L'</var><big>]</big><var></var><big>]</big><var></var>)</td></tr></table></dt>
-<dd>
-Eigenvalue decomposition of a real symmetric matrix of order <I>n</I>.
-<var>W</var> is a real matrix of length at least <I>n</I>.
-On exit, <var>W</var> contains the eigenvalues in ascending order.
-If <var>jobz</var> is <code>'V'</code>, the eigenvectors are also computed
-and returned in <var>A</var>.
-If <var>jobz</var> is <code>'N'</code>, the eigenvectors are not returned and the
-contents of <var>A</var> are destroyed.
-Raises an <code>ArithmeticError</code> if the eigenvalue decomposition fails.
-</dl>
-
-<P>
-<dl><dt><table cellpadding="0" cellspacing="0"><tr valign="baseline">
- <td><nobr><b><tt id='l2h-98' xml:id='l2h-98' class="function">syevd</tt></b>(</nobr></td>
- <td><var>A, W</var><big>[</big><var>, jobz='N'</var><big>[</big><var>, uplo='L'</var><big>]</big><var></var><big>]</big><var></var>)</td></tr></table></dt>
-<dd>
-This is an alternative to <tt class="function">syev()</tt>, based on a different
-algorithm. It is faster on large problems, but also uses more memory.
-</dl>
-
-<P>
-<dl><dt><table cellpadding="0" cellspacing="0"><tr valign="baseline">
- <td><nobr><b><tt id='l2h-99' xml:id='l2h-99' class="function">syevx</tt></b>(</nobr></td>
- <td><var>A, W</var><big>[</big><var>, jobz='N'</var><big>[</big><var>,
-range='A'</var><big>[</big><var>, uplo='L'</var><big>[</big><var>, vl=0.0, vu=0.0</var><big>[</big><var>,
-il=1, iu=1</var><big>[</big><var>, Z=<code>None</code></var><big>]</big><var></var><big>]</big><var></var><big>]</big><var></var><big>]</big><var></var><big>]</big><var></var><big>]</big><var></var>)</td></tr></table></dt>
-<dd>
-Computes selected eigenvalues and eigenvectors of a real symmetric
-matrix <var>A</var> of order <I>n</I>.
-
-<P>
-<var>W</var> is a real matrix of length at least <var>n</var>.
-On exit, <var>W</var> contains the eigenvalues in ascending order.
-If <var>range</var> is <code>'A'</code>, all the eigenvalues are computed.
-If <var>range</var> is <code>'I'</code>, eigenvalues <var>il</var> through <var>iu</var>
-are computed, where <var>1</var> <code><=</code> <var>il</var> <code><=</code> <var>iu</var>
-<code><=</code> <var>n</var>.
-If <var>range</var> is <code>'V'</code>, the eigenvalues in the interval
-<code>(<var>vl</var>,<var>vu</var>]</code> are computed.
-
-<P>
-If <var>jobz</var> is <code>'V'</code>, the (normalized) eigenvectors are
-computed, and returned in <var>Z</var>. If <var>jobz</var> is <code>'N'</code>, the
-eigenvectors are not computed. In both cases, the contents of <var>A</var>
-are destroyed on exit.
-<var>Z</var> is optional (and not referenced) if <var>jobz</var> is <code>'N'</code>.
-It is required if <var>jobz</var> is <code>'V'</code> and must have at least
-<var>n</var> columns if <var>range</var> is <code>'A'</code> or <code>'V'</code> and at
-least <code><var>iu</var>-<var>il</var>+1</code> columns if <var>range</var> is <code>'I'</code>.
-
-<P>
-<tt class="function">syevx()</tt> returns the number of computed eigenvalues.
-</dl>
-
-<P>
-<dl><dt><table cellpadding="0" cellspacing="0"><tr valign="baseline">
- <td><nobr><b><tt id='l2h-100' xml:id='l2h-100' class="function">syevr</tt></b>(</nobr></td>
- <td><var>A, W</var><big>[</big><var>, jobz='N'</var><big>[</big><var>,
-range='A'</var><big>[</big><var>, uplo='L'</var><big>[</big><var>, vl=0.0, vu=0.0</var><big>[</big><var>,
-il=1, iu=n</var><big>[</big><var>, Z=<code>None</code></var><big>]</big><var></var><big>]</big><var></var><big>]</big><var></var><big>]</big><var></var><big>]</big><var></var><big>]</big><var></var>)</td></tr></table></dt>
-<dd>
-This is an alternative to <tt class="function">syevx()</tt>.
-<tt class="function">syevr()</tt> is the most recent LAPACK routine for symmetric
-eigenvalue problems, and expected to supersede the three other
-routines in future releases.
-</dl>
-
-<P>
-The next four routines can be used to compute eigenvalues and
-eigenvectors for complex Hermitian matrices:
-<BR><P></P>
-<DIV ALIGN="CENTER" CLASS="mathdisplay">
-<!-- MATH
- \begin{displaymath}
-A = V\mbox{\bf diag}\,(\lambda)V^H,\qquad V^HV = I.
-\end{displaymath}
- -->
-
-<IMG
- WIDTH="233" HEIGHT="28" BORDER="0"
- SRC="img65.gif"
- ALT="\begin{displaymath}
-A = V\mbox{\bf diag}\,(\lambda)V^H,\qquad V^HV = I.
-\end{displaymath}">
-</DIV>
-<BR CLEAR="ALL">
-<P></P>
-For real symmetric matrices they are identical to the corresponding
-<tt class="function">syev_()</tt> routines.
-
-<P>
-<dl><dt><table cellpadding="0" cellspacing="0"><tr valign="baseline">
- <td><nobr><b><tt id='l2h-101' xml:id='l2h-101' class="function">heev</tt></b>(</nobr></td>
- <td><var>A, W</var><big>[</big><var>, jobz='N'</var><big>[</big><var>, uplo='L'</var><big>]</big><var></var><big>]</big><var></var>)</td></tr></table></dt>
-<dd>
-Eigenvalue decomposition of a real symmetric or complex Hermitian
-matrix of order <I>n</I>.
-The calling sequence is identical to <tt class="function">syev()</tt>, except that
-<var>A</var> can be real or complex.
-</dl>
-
-<P>
-<dl><dt><table cellpadding="0" cellspacing="0"><tr valign="baseline">
- <td><nobr><b><tt id='l2h-102' xml:id='l2h-102' class="function">heevd</tt></b>(</nobr></td>
- <td><var>A, W</var><big>[</big><var>, jobz='N'</var><big>[</big><var>, uplo='L'</var><big>]</big><var></var><big>]</big><var></var>)</td></tr></table></dt>
-<dd>
-This is an alternative to <tt class="function">heev()</tt>.
-</dl>
-
-<P>
-<dl><dt><table cellpadding="0" cellspacing="0"><tr valign="baseline">
- <td><nobr><b><tt id='l2h-103' xml:id='l2h-103' class="function">heevx</tt></b>(</nobr></td>
- <td><var>A, W</var><big>[</big><var>, jobz='N'</var><big>[</big><var>,
-range='A'</var><big>[</big><var>, uplo='L'</var><big>[</big><var>, vl=0.0, vu=0.0 </var><big>[</big><var>,
-il=1, iu=n</var><big>[</big><var>, Z=<code>None</code></var><big>]</big><var></var><big>]</big><var></var><big>]</big><var></var><big>]</big><var></var><big>]</big><var></var><big>]</big><var></var>)</td></tr></table></dt>
-<dd>
-Computes selected eigenvalues and eigenvectors of a real symmetric
-or complex Hermitian matrix of order <I>n</I>.
-The calling sequence is identical to <tt class="function">syevx()</tt>,
-except that <var>A</var> can be real or complex.
-<var>Z</var> must have the same type as <var>A</var>.
-</dl>
-
-<P>
-<dl><dt><table cellpadding="0" cellspacing="0"><tr valign="baseline">
- <td><nobr><b><tt id='l2h-104' xml:id='l2h-104' class="function">heevr</tt></b>(</nobr></td>
- <td><var>A, W</var><big>[</big><var>, jobz='N'</var><big>[</big><var>,
-range='A'</var><big>[</big><var>, uplo='L'</var><big>[</big><var>, vl=0.0, vu=0.0</var><big>[</big><var>,
-il=1, iu=n</var><big>[</big><var>, Z=<code>None</code></var><big>]</big><var></var><big>]</big><var></var><big>]</big><var></var><big>]</big><var></var><big>]</big><var></var><big>]</big><var></var>)</td></tr></table></dt>
-<dd>
-This is an alternative to <tt class="function">heevx()</tt>.
-</dl>
-
-<P>
-
-<DIV CLASS="navigation">
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-
-<H1><A NAME="SECTION006800000000000000000">
-4.8 Singular Value Decomposition</A>
-</H1>
-
-<P>
-<dl><dt><table cellpadding="0" cellspacing="0"><tr valign="baseline">
- <td><nobr><b><tt id='l2h-107' xml:id='l2h-107' class="function">gesvd</tt></b>(</nobr></td>
- <td><var>A, S</var><big>[</big><var>, jobu='N'</var><big>[</big><var>,
-jobvt='N'</var><big>[</big><var>, U=<code>None</code></var><big>[</big><var>, Vt=<code>None</code></var><big>]</big><var></var><big>]</big><var></var><big>]</big><var></var><big>]</big><var></var>)</td></tr></table></dt>
-<dd>
-Singular value decomposition
-<BR><P></P>
-<DIV ALIGN="CENTER" CLASS="mathdisplay">
-<!-- MATH
- \begin{displaymath}
-A = U \Sigma V^T, \qquad A = U \Sigma V^H
-\end{displaymath}
- -->
-
-<IMG
- WIDTH="200" HEIGHT="27" BORDER="0"
- SRC="img69.gif"
- ALT="\begin{displaymath}
-A = U \Sigma V^T, \qquad A = U \Sigma V^H
-\end{displaymath}">
-</DIV>
-<BR CLEAR="ALL">
-<P></P>
-of a real or complex <I>m</I> by <I>n</I> matrix <var>A</var>.
-
-<P>
-<var>S</var> is a real matrix of length at least min{<I>m</I>, <I>n</I>}.
-On exit, its first min{<I>m</I>, <I>n</I>} elements are the
-singular values in descending order.
-
-<P>
-The argument <var>jobu</var> controls how many left singular vectors are
-computed. The possible values are <code>'N'</code>, <code>'A'</code>, <code>'S'</code>
-and <code>'O'</code>.
-If <var>jobu</var> is <code>'N'</code>, no left singular vectors are
-computed.
-If <var>jobu</var> is <code>'A'</code>, all left singular vectors are computed
-and returned as columns of <var>U</var>.
-If <var>jobu</var> is <code>'S'</code>, the first min{<I>m</I>,<I>n</I>} left
-singular vectors are computed and returned as columns of <var>U</var>.
-If <var>jobu</var> is <code>'O'</code>, the first min{<I>m</I>,<I>n</I>} left
-singular vectors are computed and returned as columns of <var>A</var>.
-The argument <var>U</var> is <code>None</code> (if <var>jobu</var> is <code>'N'</code>
-or <code>'A'</code>) or a matrix of the same type as <var>A</var>.
-
-<P>
-The argument <var>jobvt</var> controls how many right singular vectors are
-computed. The possible values are <code>'N'</code>, <code>'A'</code>, <code>'S'</code>
-and <code>'O'</code>.
-If <var>jobvt</var> is <code>'N'</code>, no right singular vectors are
-computed. If <var>jobvt</var> is <code>'A'</code>, all right singular vectors
-are computed and returned as rows of <var>Vt</var>.
-If <var>jobvt</var> is <code>'S'</code>, the first min{<I>m</I>,<I>n</I>} right
-singular vectors are computed and their (conjugate) transposes are
-returned as rows of <var>Vt</var>.
-If <var>jobvt</var> is <code>'O'</code>, the first min{<I>m</I>, <I>n</I>} right
-singular vectors are computed and their (conjugate) transposes
-are returned as rows of <var>A</var>.
-Note that the (conjugate) transposes of the right singular vectors
-(<EM>i.e.</EM>, the matrix <SPAN CLASS="MATH"><IMG
- WIDTH="29" HEIGHT="16" ALIGN="BOTTOM" BORDER="0"
- SRC="img70.gif"
- ALT="$V^H$"></SPAN>) are returned in <var>Vt</var> or <var>A</var>.
-The argument <var>Vt</var> can be <code>None</code> (if <var>jobvt</var> is <code>'N'</code>
-or <code>'A'</code>) or a matrix of the same type as <var>A</var>.
-
-<P>
-On exit, the contents of <var>A</var> are destroyed.
-</dl>
-
-<P>
-<dl><dt><table cellpadding="0" cellspacing="0"><tr valign="baseline">
- <td><nobr><b><tt id='l2h-108' xml:id='l2h-108' class="function">gesdd</tt></b>(</nobr></td>
- <td><var>A, S</var><big>[</big><var>, jobz='N'</var><big>[</big><var>,
-U=<code>None</code></var><big>[</big><var>, Vt=<code>None</code></var><big>]</big><var></var><big>]</big><var></var><big>]</big><var></var>)</td></tr></table></dt>
-<dd>
-Singular value decomposition of a real or complex <I>m</I> by <I>n</I>
-matrix <var>A</var>. This function is based on a divide-and-conquer
-algorithm and is faster than <tt class="function">gesvd()</tt>.
-
-<P>
-<var>S</var> is a real matrix of length at least min{<I>m</I>, <I>n</I>}.
-On exit, its first min{<I>m</I>, <I>n</I>} elements are the
-singular values in descending order.
-
-<P>
-The argument <var>jobz</var> controls how many singular vectors are
-computed. The possible values are <code>'N'</code>, <code>'A'</code>, <code>'S'</code>
-and <code>'O'</code>.
-If <var>jobz</var> is <code>'N'</code>, no singular vectors are computed.
-If <var>jobz</var> is <code>'A'</code>, all <I>m</I> left singular vectors are
-computed and returned as columns of <var>U</var> and all <I>n</I> right
-singular vectors are computed and returned as rows of <var>Vt</var>.
-If <var>jobz</var> is <code>'S'</code>, the first min{<I>m</I>, <I>n</I>} left
-and right singular vectors are computed and returned as columns of
-<var>U</var> and rows of <var>Vt</var>.
-If <var>jobz</var> is <code>'O'</code> and <I>m</I> is greater than or equal
-to <I>n</I>, the first <I>n</I> left singular vectors are returned as
-columns of <var>A</var> and the <I>n</I> right singular vectors are returned
-as rows of <var>Vt</var>. If <var>jobz</var> is <code>'O'</code> and <I>m</I> is less
-than <I>n</I>, the <I>m</I> left singular vectors are returned as columns
-of <var>U</var> and the first <I>m</I> right singular vectors are returned
-as rows of <var>A</var>.
-Note that the (conjugate) transposes of the right singular vectors
-are returned in <var>Vt</var> or <var>A</var>.
-
-<P>
-The argument <var>U</var> can be <code>None</code> (if <var>jobz</var> is <code>'N'</code>
-or <code>'A'</code> of <var>jobz</var> is <code>'O'</code> and <I>m</I> is greater than
-or equal to <I>n</I>) or a matrix of the same type as <var>A</var>.
-The argument <var>Vt</var> can be <code>None</code> (if <var>jobz</var> is <code>'N'</code>
-or <code>'A'</code> or <var>jobz</var> is <code>'O'</code> and <I>m</I> is less than
-<I>n</I>) or a matrix of the same type as <var>A</var>.
-
-<P>
-On exit, the contents of <var>A</var> are destroyed.
-</dl>
-
-<P>
-
-<DIV CLASS="navigation">
-<div class='online-navigation'>
-<p></p><hr />
-<table align="center" width="100%" cellpadding="0" cellspacing="2">
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@@ -1,197 +0,0 @@
-<!DOCTYPE html PUBLIC "-//W3C//DTD HTML 4.0 Transitional//EN">
-<html>
-<head>
-<link rel="STYLESHEET" href="cvxopt.css" type='text/css' />
-<link rel="first" href="cvxopt.html" title='CVXOPT: A Python Package for Convex Optimization' />
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-
-<H1><A NAME="SECTION006900000000000000000">
-4.9 Example: Analytic Centering</A>
-</H1>
-The analytic centering problem is defined as
-<BR><P></P>
-<DIV ALIGN="CENTER" CLASS="mathdisplay">
-<!-- MATH
- \begin{displaymath}
-\begin{array}{ll}
- \mbox{minimize} & -\sum_{i=1}^m \log(b_i-a_i^Tx).
- \end{array}
-\end{displaymath}
- -->
-
-<IMG
- WIDTH="227" HEIGHT="30" BORDER="0"
- SRC="img71.gif"
- ALT="\begin{displaymath}
-\begin{array}{ll}
-\mbox{minimize} & -\sum_{i=1}^m \log(b_i-a_i^Tx).
-\end{array}\end{displaymath}">
-</DIV>
-<BR CLEAR="ALL">
-<P></P>
-In the code below we solve the problem using Newton's method.
-At each iteration the Newton direction is computed by solving
-a positive definite set of linear equations
-<BR><P></P>
-<DIV ALIGN="CENTER" CLASS="mathdisplay">
-<!-- MATH
- \begin{displaymath}
-A^T \mbox{\bf diag}\,(b-Ax)^{-2} A v = -\mbox{\bf diag}\,(b-Ax)^{-1}{\bf 1}
-\end{displaymath}
- -->
-
-<IMG
- WIDTH="301" HEIGHT="28" BORDER="0"
- SRC="img72.gif"
- ALT="\begin{displaymath}
-A^T \mbox{\bf diag}\,(b-Ax)^{-2} A v = -\mbox{\bf diag}\,(b-Ax)^{-1}{\bf 1}
-\end{displaymath}">
-</DIV>
-<BR CLEAR="ALL">
-<P></P>
-(where <I>A</I> has rows <SPAN CLASS="MATH"><IMG
- WIDTH="22" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
- SRC="img73.gif"
- ALT="$a_i^T$"></SPAN>), and a suitable step size is determined
-by a backtracking line search.
-
-<P>
-We use the level-3 BLAS function <tt class="function">syrk()</tt> to form the Hessian
-matrix and the LAPACK function <tt class="function">posv()</tt> to solving the Newton
-system. The code can be further optimized by replacing the
-matrix-vector products with the level-2 BLAS function <tt class="function">gemv()</tt>.
-
-<P>
-<div class="verbatim"><pre>
-from cvxopt.base import matrix, log, mul, div
-from cvxopt import blas, lapack, random
-from math import sqrt
-
-def acent(A,b):
- """
- Returns the analytic center of A*x <= b.
- We assume that b > 0 and the feasible set is bounded.
- """
-
- MAXITERS = 100
- ALPHA = 0.01
- BETA = 0.5
- TOL = 1e-8
-
- m, n = A.size
- x = matrix(0.0, (n,1))
- H = matrix(0.0, (n,n))
- g = matrix(0.0, (n,1))
-
- for iter in xrange(MAXITERS):
-
- # Gradient is g = A^T * (1./(b-A*x)).
- d = (b-A*x)**-1
- g = A.T * d
-
- # Hessian is H = A^T * diag(d)^2 * A.
- Asc = mul( d[:,n*[0]], A )
- blas.syrk(Asc, H, trans='T')
-
- # Newton step is v = -H^-1 * g.
- v = -g
- lapack.posv(H, v)
-
- # Terminate if Newton decrement is less than TOL.
- lam = blas.dot(g, v)
- if sqrt(-lam) < TOL: return x
-
- # Backtracking line search.
- y = mul(A*v, d)
- step = 1.0
- while 1-step*max(y) < 0: step *= BETA
- while True:
- if -sum(log(1-step*y)) < ALPHA*step*lam: break
- step *= BETA
- x += step*v
-</pre></div>
-
-<DIV CLASS="navigation">
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-<!DOCTYPE html PUBLIC "-//W3C//DTD HTML 4.0 Transitional//EN">
-<html>
-<head>
-<link rel="STYLESHEET" href="cvxopt.css" type='text/css' />
-<link rel="first" href="cvxopt.html" title='CVXOPT: A Python Package for Convex Optimization' />
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-<title>1. Introduction</title>
-</head>
-<body>
-<DIV CLASS="navigation">
-<div id='top-navigation-panel' xml:id='top-navigation-panel'>
-<table align="center" width="100%" cellpadding="0" cellspacing="2">
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-<td align="center" width="100%">CVXOPT: A Python Package for Convex Optimization</td>
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-
-<H1><A NAME="SECTION003000000000000000000"></A><A NAME="chap:intro"></A>
-<BR>
-1. Introduction
-</H1>
-
-<P>
-CVXOPT is a free software package for convex optimization based on
-the Python programming language.
-It can be used with the interactive Python interpreter,
-on the command line by executing Python scripts, or integrated in
-other software via Python extension modules.
-Its main purpose is to make the development of software for convex
-optimization applications straightforward by building on Python's
-extensive standard library and on the strengths of Python as a
-high-level programming language.
-
-<P>
-Release 0.8.2 of CVXOPT includes routines for basic linear algebra
-calculations,
-interfaces to efficient libraries for solving dense and sparse linear
-equations,
-convex optimization solvers written in Python,
-interfaces to a few other optimization libraries,
-and a modeling tool for piecewise-linear convex optimization problems.
-These components are organized in different modules.
-<DL>
-<DT><STRONG><tt class="module">cvxopt.base</tt></STRONG></DT>
-<DD>This module defines a Python type
- <tt class="class">matrix</tt> for storing and manipulating dense matrices,
- a Python type <tt class="class">spmatrix</tt> for storing and manipulating sparse
- matrices, and routines for sparse matrix-vector and matrix-matrix
- multiplication (see chapters <A HREF="node4.html#chap:matrix">2</A>
- and <A HREF="node33.html#chap:spmatrix">6</A>).
-</DD>
-<DT><STRONG><tt class="module">cvxopt.random</tt></STRONG></DT>
-<DD>Routines for generating random matrices
- with uniformly or normally distributed entries
- (see section <A href="s-random.html#s-random">2.7</A>).
-</DD>
-<DT><STRONG><tt class="module">cvxopt.blas</tt></STRONG></DT>
-<DD>Interface to most of the double-precision
- real and complex BLAS (chapter <A HREF="node14.html#chap:blas">3</A>).
-</DD>
-<DT><STRONG><tt class="module">cvxopt.lapack</tt></STRONG></DT>
-<DD>Interface to the dense double-precision
- real and complex linear equation solvers and eigenvalue routines
- from LAPACK (chapter <A HREF="node19.html#chap:lapack">4</A>).
-</DD>
-<DT><STRONG><tt class="module">cvxopt.fftw</tt></STRONG></DT>
-<DD>An optional interface to the
- discrete transform routines from FFTW (section <A href="c-fftw.html#c-fftw">5</A>).
-</DD>
-<DT><STRONG><tt class="module">cvxopt.amd</tt></STRONG></DT>
-<DD>Interface to the approximate minimum degree
- ordering routine from AMD (chapter <A href="s-orderings.html#s-orderings">7.1</A>).
-</DD>
-<DT><STRONG><tt class="module">cvxopt.umfpack</tt></STRONG></DT>
-<DD>Interface to the sparse LU solver
- from UMFPACK (section <A href="s-umfpack.html#s-umfpack">7.2</A>).
-</DD>
-<DT><STRONG><tt class="module">cvxopt.cholmod</tt></STRONG></DT>
-<DD>Interface to the sparse Cholesky
- solver from CHOLMOD (section <A href="s-cholmod.html#s-cholmod">7.3</A>).
-</DD>
-<DT><STRONG><tt class="module">cvxopt.solvers</tt></STRONG></DT>
-<DD>Convex optimization routines
- and optional interfaces to solvers from GLPK, MOSEK and DSDP5
- (chapter <A HREF="node45.html#chap:solvers">8</A>).
-</DD>
-<DT><STRONG><tt class="module">cvxopt.modeling</tt></STRONG></DT>
-<DD>Routines for specifying and solving
- linear programs and convex optimization problems with piecewise-linear
- cost and constraint functions (chapter <A HREF="node55.html#chap:modeling">9</A>).
-</DD>
-<DT><STRONG><tt class="module">cvxopt.info</tt></STRONG></DT>
-<DD>Defines a string <code>version</code> with
- the version number of the CVXOPT installation and a function
- <tt class="function">license()</tt> that prints the CVXOPT license.
-</DD>
-</DL>
-The modules are described in detail in this manual and in the
-on-line Python help facility <b class="program">pydoc</b>. Several example scripts
-are included in the distribution.
-
-<DIV CLASS="navigation">
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-<!DOCTYPE html PUBLIC "-//W3C//DTD HTML 4.0 Transitional//EN">
-<html>
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-5.1 Discrete Fourier Transform</A>
-</H1>
-<dl><dt><table cellpadding="0" cellspacing="0"><tr valign="baseline">
- <td><nobr><b><tt id='l2h-109' xml:id='l2h-109' class="function">dft</tt></b>(</nobr></td>
- <td><var>X</var>)</td></tr></table></dt>
-<dd>
-Replaces the columns of a dense complex matrix with their discrete
-Fourier transforms: if <var>X</var> has <I>n</I> rows,
-<BR><P></P>
-<DIV ALIGN="CENTER" CLASS="mathdisplay">
-<!-- MATH
- \begin{displaymath}
-X[k,:] := \sum_{j=0}^{n-1} e^{-2\pi j k \sqrt{-1}/n} X[j,:],
- \qquad k=0,\ldots,n-1.
-\end{displaymath}
- -->
-
-<IMG
- WIDTH="375" HEIGHT="58" BORDER="0"
- SRC="img74.gif"
- ALT="\begin{displaymath}
-X[k,:] := \sum_{j=0}^{n-1} e^{-2\pi j k \sqrt{-1}/n} X[j,:],
-\qquad k=0,\ldots,n-1.
-\end{displaymath}">
-</DIV>
-<BR CLEAR="ALL">
-<P></P>
-</dl>
-
-<P>
-<dl><dt><table cellpadding="0" cellspacing="0"><tr valign="baseline">
- <td><nobr><b><tt id='l2h-110' xml:id='l2h-110' class="function">idft</tt></b>(</nobr></td>
- <td><var>X</var>)</td></tr></table></dt>
-<dd>
-Replaces the columns of a dense complex matrix with their inverse
-discrete Fourier transforms: if <var>X</var> has <I>n</I> rows,
-<BR><P></P>
-<DIV ALIGN="CENTER" CLASS="mathdisplay">
-<!-- MATH
- \begin{displaymath}
-X[k,:] :=
- \frac{1}{n} \sum_{j=0}^{n-1} e^{2\pi j k \sqrt{-1}/n} X[j,:],
- \qquad k=0,\ldots,n-1.
-\end{displaymath}
- -->
-
-<IMG
- WIDTH="382" HEIGHT="58" BORDER="0"
- SRC="img75.gif"
- ALT="\begin{displaymath}
-X[k,:] :=
-\frac{1}{n} \sum_{j=0}^{n-1} e^{2\pi j k \sqrt{-1}/n} X[j,:],
-\qquad k=0,\ldots,n-1.
-\end{displaymath}">
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-<H1><A NAME="SECTION007200000000000000000">
-5.2 Discrete Cosine Transform</A>
-</H1>
-<dl><dt><table cellpadding="0" cellspacing="0"><tr valign="baseline">
- <td><nobr><b><tt id='l2h-111' xml:id='l2h-111' class="function">dct</tt></b>(</nobr></td>
- <td><var>X</var><big>[</big><var>, type=2</var><big>]</big><var></var>)</td></tr></table></dt>
-<dd>
-Replaces the columns of a dense real matrix with their discrete
-cosine transforms. The second argument, an integer between 1 and 4,
-denotes the type of transform (DCT-I, DCT-II, DCT-III, DCT-IV).
-The DCT-I transform requires that the row dimension of <var>X</var> is at
-least 2.
-These transforms are defined as follows
-(for a matrix with <I>n</I> rows).
-<P></P>
-<DIV ALIGN="CENTER" CLASS="mathdisplay">
-<!-- MATH
- \begin{eqnarray*}
-\mbox{DCT-I:} \qquad
- X[k,:] & := & X[0,:] + (-1)^k X[n-1,:] +
- 2 \sum_{j=1}^{n-2} X[j,:] \cos(\pi j k /(n-1)),
- \qquad k=0,\ldots,n-1.\\
-\mbox{DCT-II:} \qquad
- X[k,:] & := & 2 \sum_{j=0}^{n-1} X[j,:] \cos(\pi(j+1/2)k/n),
- \qquad k=0,\ldots,n-1.\\
-\mbox{DCT-III:} \qquad
- X[k,:] & := & X[0,:] + 2 \sum_{j=1}^{n-1} X[j,:] \cos(\pi j(k+1/2)/n),
- \qquad k=0,\ldots,n-1.\\
-\mbox{DCT-IV:} \qquad
- X[k,:] & := & 2 \sum_{j=0}^{n-1} X[j,:] \cos(\pi (j+1/2)(k+1/2)/n),
- \qquad k=0,\ldots,n-1.
-\end{eqnarray*}
- -->
-<IMG
- WIDTH="724" HEIGHT="230" BORDER="0"
- SRC="img76.gif"
- ALT="\begin{eqnarray*}
-\mbox{DCT-I:} \qquad
-X[k,:] & := & X[0,:] + (-1)^k X[n-1,:] +...
-...n-1} X[j,:] \cos(\pi (j+1/2)(k+1/2)/n),
-\qquad k=0,\ldots,n-1.
-\end{eqnarray*}"></DIV>
-<BR CLEAR="ALL"><P></P>
-<BR CLEAR="ALL"><P></P>
-</dl>
-
-<P>
-<dl><dt><table cellpadding="0" cellspacing="0"><tr valign="baseline">
- <td><nobr><b><tt id='l2h-112' xml:id='l2h-112' class="function">idct</tt></b>(</nobr></td>
- <td><var>X</var><big>[</big><var>, type=2</var><big>]</big><var></var>)</td></tr></table></dt>
-<dd>
-Replaces the columns of a dense real matrix with the inverses
-of the discrete cosine transforms defined above.
-</dl>
-
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-
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-</H1>
-<dl><dt><table cellpadding="0" cellspacing="0"><tr valign="baseline">
- <td><nobr><b><tt id='l2h-113' xml:id='l2h-113' class="function">dst</tt></b>(</nobr></td>
- <td><var>X</var><big>[</big><var>, type=1</var><big>]</big><var></var>)</td></tr></table></dt>
-<dd>
-Replaces the columns of a dense real matrix with their discrete
-sine transforms. The second argument, an integer between 1 and 4,
-denotes the type of transform (DST-I, DST-II, DST-III, DST-IV).
-These transforms are defined as follows
-(for a matrix with <I>n</I> rows).
-<P></P>
-<DIV ALIGN="CENTER" CLASS="mathdisplay">
-<!-- MATH
- \begin{eqnarray*}
-\mbox{DST-I:} \qquad
- X[k,:] & := &
- 2 \sum_{j=0}^{n-1} X[j,:] \sin(\pi(j+1)(k+1)/(n+1)),
- \qquad k=0,\ldots,n-1.\\
-\mbox{DST-II:} \qquad
- X[k,:] & := & 2 \sum_{j=0}^{n-1} X[j,:] \sin(\pi(j+1/2)(k+1)/n),
- \qquad k=0,\ldots,n-1.\\
-\mbox{DST-III:} \qquad
- X[k,:] & := & (-1)^k X[n-1,:] + 2 \sum_{j=0}^{n-2}
- X[j,:] \sin(\pi(j+1)(k+1/2)/n),
- \qquad k=0,\ldots,n-1. \\
-\mbox{DST-IV:} \qquad
- X[k,:] & := & 2 \sum_{j=0}^{n-1} X[j,:] \sin(\pi (j+1/2)(k+1/2)/n),
- \qquad k=0,\ldots,n-1.
-\end{eqnarray*}
- -->
-<IMG
- WIDTH="714" HEIGHT="230" BORDER="0"
- SRC="img77.gif"
- ALT="\begin{eqnarray*}
-\mbox{DST-I:} \qquad
-X[k,:] & := &
-2 \sum_{j=0}^{n-1} X[j,:...
-...n-1} X[j,:] \sin(\pi (j+1/2)(k+1/2)/n),
-\qquad k=0,\ldots,n-1.
-\end{eqnarray*}"></DIV>
-<BR CLEAR="ALL"><P></P>
-<BR CLEAR="ALL"><P></P>
-</dl>
-
-<P>
-<dl><dt><table cellpadding="0" cellspacing="0"><tr valign="baseline">
- <td><nobr><b><tt id='l2h-114' xml:id='l2h-114' class="function">idst</tt></b>(</nobr></td>
- <td><var>X</var><big>[</big><var>, type=1</var><big>]</big><var></var>)</td></tr></table></dt>
-<dd>
-Replaces the columns of a dense real matrix with the inverses
-of the discrete sine transforms defined above.
-</dl>
-
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-<p></p><hr />
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+++ /dev/null
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-<!DOCTYPE html PUBLIC "-//W3C//DTD HTML 4.0 Transitional//EN">
-<html>
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-
-<H1><A NAME="SECTION008000000000000000000"></A><A NAME="chap:spmatrix"></A>
-<BR>
-6. Sparse Matrices (<tt class="module">cvxopt.base</tt>)
-</H1>
-
-<P>
-In this chapter we discuss the <tt class="class">spmatrix</tt> object defined in
-<tt class="module">cvxopt.base</tt>.
-
-<P>
-
-<p><br /></p><hr class='online-navigation' />
-<div class='online-navigation'>
-<!--Table of Child-Links-->
-<A NAME="CHILD_LINKS"><STRONG>Subsections</STRONG></a>
-
-<UL CLASS="ChildLinks">
-<LI><A href="s-creating-spmatrix.html">6.1 Creating Sparse Matrices</a>
-<LI><A href="e-spA-example.html">6.2 Attributes and Methods</a>
-<LI><A href="s-spmatrix-arith.html">6.3 Arithmetic Operations</a>
-<LI><A href="node37.html">6.4 Indexing and Slicing</a>
-<LI><A href="node38.html">6.5 Built-In Functions</a>
-<LI><A href="node39.html">6.6 Sparse BLAS Functions</a>
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-<!DOCTYPE html PUBLIC "-//W3C//DTD HTML 4.0 Transitional//EN">
-<html>
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-
-<H1><A NAME="SECTION008400000000000000000">
-6.4 Indexing and Slicing</A>
-</H1>
-Sparse matrices can be indexed the same way as dense matrices
-(see section <A href="s-indexing.html#s-indexing">2.4</A> ).
-
-<P>
-<div class="verbatim"><pre>
->>> from cvxopt.base import spmatrix
->>> A = spmatrix([0,2,-1,2,-2,1], [0,1,2,0,2,1], [0,0,0,1,1,2])
->>> print A[:,[0,1]]
-SIZE: (3,2)
-(0, 0) 0.0000e+00
-(1, 0) 2.0000e+00
-(2, 0) -1.0000e+00
-(0, 1) 2.0000e+00
-(2, 1) -2.0000e+00
->>> B = spmatrix([0,2*1j,0,-2], [1,2,1,2], [0,0,1,1,])
->>> print B[-2:,-2:]
-SIZE: (2,2)
-(0, 0) 0.0000e+00-j0.0000e+00
-(1, 0) 2.0000e+00-j0.0000e+00
-(0, 1) 0.0000e+00-j0.0000e+00
-(1, 1) 0.0000e+00-j2.0000e+00
-</pre></div>
-
-<P>
-An indexed sparse matrix <code><var>A</var>[<var>I</var>]</code> or
-<code><var>A</var>[<var>I</var>,<var>J</var>]</code> can also be the target of an
-assignment. The righthand side of the assignment can be a scalar
-(a Python integer, float, or complex, or a 1 by 1 dense matrix),
-a sequence of numbers, or a sparse or dense matrix of
-compatible dimensions.
-If the righthand side is a scalar, it is treated as a dense matrix of
-the same size as the lefthand side and with all its entries equal to
-the scalar.
-If the righthand side is a sequence of numbers, they are treated as
-the elements of a dense matrix in column-major order.
-
-<P>
-We continue the example above.
-<div class="verbatim"><pre>
->>> C = spmatrix([10,-20,30], [0,2,1], [0,0,1])
->>> A[:,0] = C[:,0]
->>> print A
-SIZE: (3,3)
-(0, 0) 1.0000e+01
-(2, 0) -2.0000e+01
-(0, 1) 2.0000e+00
-(2, 1) -2.0000e+00
-(1, 2) 1.0000e+00
->>> D = matrix(range(6), (3,2))
->>> A[:,0] = D[:,0]
->>> print A
-SIZE: (3,3)
-(0, 0) 0.0000e+00
-(1, 0) 1.0000e+00
-(2, 0) 2.0000e+00
-(0, 1) 2.0000e+00
-(2, 1) -2.0000e+00
-(1, 2) 1.0000e+00
->>> A[:,0] = 1
->>> print A
-SIZE: (3,3)
-(0, 0) 1.0000e+00
-(1, 0) 1.0000e+00
-(2, 0) 1.0000e+00
-(0, 1) 2.0000e+00
-(2, 1) -2.0000e+00
-(1, 2) 1.0000e+00
->>> A[:,0] = 0
->>> print A
-TYPE: (3,3)
-(0, 0) 0.0000e+00
-(1, 0) 0.0000e+00
-(2, 0) 0.0000e+00
-(0, 1) 2.0000e+00
-(2, 1) -2.0000e+00
-(1, 2) 1.0000e+00
-</pre></div>
-
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-<!DOCTYPE html PUBLIC "-//W3C//DTD HTML 4.0 Transitional//EN">
-<html>
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-<H1><A NAME="SECTION008500000000000000000">
-6.5 Built-In Functions</A>
-</H1>
-
-<P>
-The functions described in the table of section <A href="s-builtinfuncs.html#s-builtinfuncs">2.5</A>
-also work with sparse matrix arguments. The difference is that for a
-sparse matrix only the nonzero entries are considered.
-
-<P>
-<dl><dt><table cellpadding="0" cellspacing="0"><tr valign="baseline">
- <td><nobr><b><tt id='l2h-123' xml:id='l2h-123' class="function">len</tt></b>(</nobr></td>
- <td><var>x</var>)</td></tr></table></dt>
-<dd>
-If <var>x</var> is a <tt class="class">spmatrix</tt>, returns the number of nonzero entries in
-<var>x</var>.
-</dl>
-
-<P>
-<dl><dt><table cellpadding="0" cellspacing="0"><tr valign="baseline">
- <td><nobr><b><tt id='l2h-124' xml:id='l2h-124' class="function">bool</tt></b>(</nobr></td>
- <td><var></var><big>[</big><var>x</var><big>]</big><var></var>)</td></tr></table></dt>
-<dd>
-If <var>x</var> is a <tt class="class">spmatrix</tt>, returns <code>False</code> if <var>x</var> has at least one
-nonzero entry; <code>False</code> otherwise.
-</dl>
-
-<P>
-<dl><dt><table cellpadding="0" cellspacing="0"><tr valign="baseline">
- <td><nobr><b><tt id='l2h-125' xml:id='l2h-125' class="function">max</tt></b>(</nobr></td>
- <td><var>x</var>)</td></tr></table></dt>
-<dd>
-If <var>x</var> is a <tt class="class">spmatrix</tt>, returns the maximum nonzero entry of <var>x</var>.
-</dl>
-
-<P>
-<dl><dt><table cellpadding="0" cellspacing="0"><tr valign="baseline">
- <td><nobr><b><tt id='l2h-126' xml:id='l2h-126' class="function">min</tt></b>(</nobr></td>
- <td><var>x</var>)</td></tr></table></dt>
-<dd>
-If <var>x</var> is a <tt class="class">spmatrix</tt>, returns the minimum nonzero entry of <var>x</var>.
-</dl>
-
-<P>
-<dl><dt><table cellpadding="0" cellspacing="0"><tr valign="baseline">
- <td><nobr><b><tt id='l2h-127' xml:id='l2h-127' class="function">abs</tt></b>(</nobr></td>
- <td><var>x</var>)</td></tr></table></dt>
-<dd>
-If <var>x</var> is a <tt class="class">spmatrix</tt>, returns a sparse matrix with the absolute
-value of the elements of <var>x</var> and the same sparsity pattern.
-</dl>
-
-<P>
-<dl><dt><table cellpadding="0" cellspacing="0"><tr valign="baseline">
- <td><nobr><b><tt id='l2h-128' xml:id='l2h-128' class="function">sum</tt></b>(</nobr></td>
- <td><var>x</var><big>[</big><var>, start=0.0</var><big>]</big><var></var>)</td></tr></table></dt>
-<dd>
-If <var>x</var> is a <tt class="class">spmatrix</tt>, returns the sum of <var>start</var> and the
-elements of <var>x</var>.
-</dl>
-
-<P>
-The functions <tt class="function">list()</tt>, <tt class="function">tuple()</tt>, <tt class="function">zip()</tt>,
-<tt class="function">map()</tt>, <tt class="function">filter()</tt> also take sparse matrix arguments.
-They work as for dense matrices, again with the difference that only
-the nonzero entries are considered.
-
-<P>
-In the following example we square the entries of the
-matrix (<A href="e-spA-example.html#e-spA-example">6.2</A>).
-
-<P>
-<div class="verbatim"><pre>
->>> A = spmatrix([2,1,2,2,1,3,4], [1,2,0,2,3,0,2], [0,0,1,1,2,3,3])
->>> B = spmatrix(map(lambda x: x**2, A), A.I, A.J)
->>> print B
-SIZE: (4,4)
-(1, 0) 4.0000e+00
-(2, 0) 1.0000e+00
-(0, 1) 4.0000e+00
-(2, 1) 4.0000e+00
-(3, 2) 1.0000e+00
-(0, 3) 9.0000e+00
-(2, 3) 1.6000e+01
-</pre></div>
-
-<P>
-The expression "<tt class="samp"><var>x</var> in <var>A</var></tt>" returns <code>True</code> if a nonzero
-entry of <var>A</var> is equal to <var>x</var> and <code>False</code> otherwise.
-
-<P>
-
-<DIV CLASS="navigation">
-<div class='online-navigation'>
-<p></p><hr />
-<table align="center" width="100%" cellpadding="0" cellspacing="2">
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+++ /dev/null
@@ -1,375 +0,0 @@
-<!DOCTYPE html PUBLIC "-//W3C//DTD HTML 4.0 Transitional//EN">
-<html>
-<head>
-<link rel="STYLESHEET" href="cvxopt.css" type='text/css' />
-<link rel="first" href="cvxopt.html" title='CVXOPT: A Python Package for Convex Optimization' />
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-<td class='online-navigation'><a rel="next" title="7. Sparse Linear Equation"
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-
-<H1><A NAME="SECTION008600000000000000000">
-6.6 Sparse BLAS Functions</A>
-</H1>
-The <tt class="module">cvxopt.base</tt> module includes a few arithmetic functions
-that extend functions from <tt class="module">cvxopt.blas</tt> to sparse matrices.
-These functions are faster than the corresponding operations
-implemented using the overloaded arithmetic described
-in section <A href="s-spmatrix-arith.html#s-spmatrix-arith">6.3</A>.
-They also work in-place, <EM>i.e.</EM>, they modify their arguments without
-creating new objects.
-
-<P>
-<dl><dt><table cellpadding="0" cellspacing="0"><tr valign="baseline">
- <td><nobr><b><tt id='l2h-129' xml:id='l2h-129' class="function">gemv</tt></b>(</nobr></td>
- <td><var>A, x, y</var><big>[</big><var>, trans='N'</var><big>[</big><var>,
- alpha=1.0</var><big>[</big><var>, beta=0.0</var><big>]</big><var></var><big>]</big><var></var><big>]</big><var></var>)</td></tr></table></dt>
-<dd>
-Matrix-vector product with a general dense or sparse matrix:
-<BR><P></P>
-<DIV ALIGN="CENTER" CLASS="mathdisplay">
-<!-- MATH
- \begin{displaymath}
-y := \alpha Ax + \beta y \quad (\mathrm{trans} = \mathrm{'N'}),
- \qquad
-y := \alpha A^T x + \beta y \quad (\mathrm{trans} = \mathrm{'T'}),
- \qquad
-y := \alpha A^H x + \beta y \quad (\mathrm{trans} = \mathrm{'C'}).
-\end{displaymath}
- -->
-
-<IMG
- WIDTH="707" HEIGHT="28" BORDER="0"
- SRC="img17.gif"
- ALT="\begin{displaymath}
-y := \alpha Ax + \beta y \quad (\mathrm{trans} = \mathrm{'N'...
-...\alpha A^H x + \beta y \quad (\mathrm{trans} = \mathrm{'C'}).
-\end{displaymath}">
-</DIV>
-<BR CLEAR="ALL">
-<P></P>
-If <var>A</var> is a dense matrix, this is identical to
-<tt class="function">blas.gemv()</tt>. If <var>A</var> is sparse, the result is the same
-as when <tt class="function">blas.gemv()</tt> is called with <code>matrix(A)</code> as
-argument, however, without explicitly converting <var>A</var> to dense.
-</dl>
-
-<P>
-<dl><dt><table cellpadding="0" cellspacing="0"><tr valign="baseline">
- <td><nobr><b><tt id='l2h-130' xml:id='l2h-130' class="function">symv</tt></b>(</nobr></td>
- <td><var>A, x, y</var><big>[</big><var>, uplo='L'</var><big>[</big><var>,
-alpha=1.0</var><big>[</big><var>, beta=0.0</var><big>]</big><var></var><big>]</big><var></var><big>]</big><var></var>)</td></tr></table></dt>
-<dd>
-Matrix-vector product with a dense or sparse real symmetric matrix:
-<BR><P></P>
-<DIV ALIGN="CENTER" CLASS="mathdisplay">
-<!-- MATH
- \begin{displaymath}
-y := \alpha A x + \beta y.
-\end{displaymath}
- -->
-
-<IMG
- WIDTH="106" HEIGHT="27" BORDER="0"
- SRC="img82.gif"
- ALT="\begin{displaymath}
-y := \alpha A x + \beta y.
-\end{displaymath}">
-</DIV>
-<BR CLEAR="ALL">
-<P></P>
-If <var>A</var> is a dense matrix, this is identical to
-<tt class="function">blas.symv()</tt>. If <var>A</var> is sparse, the result is the same
-as when <tt class="function">blas.symv()</tt> is called with <code>matrix(A)</code> as
-argument, however, without explicitly converting <var>A</var> to dense.
-</dl>
-
-<P>
-<dl><dt><table cellpadding="0" cellspacing="0"><tr valign="baseline">
- <td><nobr><b><tt id='l2h-131' xml:id='l2h-131' class="function">gemm</tt></b>(</nobr></td>
- <td><var>A, B, C</var><big>[</big><var>, transA='N'</var><big>[</big><var>,
-transB='N'</var><big>[</big><var>, alpha=1.0</var><big>[</big><var>, beta=0.0</var><big>[</big><var>,
-partial=False</var><big>]</big><var></var><big>]</big><var></var><big>]</big><var></var><big>]</big><var></var><big>]</big><var></var>)</td></tr></table></dt>
-<dd>
-Matrix-matrix product of two general sparse or dense matrices:
-<BR><P></P>
-<DIV ALIGN="CENTER" CLASS="mathdisplay">
-<!-- MATH
- \begin{displaymath}
-C := \alpha \mathop{\mathrm{op}}(A) \mathop{\mathrm{op}}(B) + \beta C
-\end{displaymath}
- -->
-
-<IMG
- WIDTH="177" HEIGHT="28" BORDER="0"
- SRC="img31.gif"
- ALT="\begin{displaymath}
-C := \alpha \mathop{\mathrm{op}}(A) \mathop{\mathrm{op}}(B) + \beta C
-\end{displaymath}">
-</DIV>
-<BR CLEAR="ALL">
-<P></P>
-where
-<BR><P></P>
-<DIV ALIGN="CENTER" CLASS="mathdisplay">
-<!-- MATH
- \begin{displaymath}
-\mathop{\mathrm{op}}(A) = \left\{ \begin{array}{ll}
- A & \mathrm{transA} = \mathrm{'N'} \\
- A^T & \mathrm{transA} = \mathrm{'T'} \\
- A^H & \mathrm{transA} = \mathrm{'C'} \end{array} \right.
-\qquad
-\mathop{\mathrm{op}}(B) = \left\{ \begin{array}{ll}
- B & \mathrm{transB} = \mathrm{'N'} \\
- B^T & \mathrm{transB} = \mathrm{'T'} \\
- B^H & \mathrm{transB} = \mathrm{'C'}. \end{array} \right.
-\end{displaymath}
- -->
-
-<IMG
- WIDTH="474" HEIGHT="64" BORDER="0"
- SRC="img32.gif"
- ALT="\begin{displaymath}
-\mathop{\mathrm{op}}(A) = \left\{ \begin{array}{ll}
-A & \ma...
-...\\
-B^H & \mathrm{transB} = \mathrm{'C'}. \end{array} \right.
-\end{displaymath}">
-</DIV>
-<BR CLEAR="ALL">
-<P></P>
-If <var>A</var>, <var>B</var> and <var>C</var> are dense matrices, this is
-identical to <tt class="function">blas.gemm()</tt>, described in section <A href="s-blas3.html#s-blas3">3.4</A>,
-and the argument <var>partial</var> is ignored.
-
-<P>
-If <var>A</var> and/or <var>B</var> are sparse and <var>C</var> is dense, the result
-is the same as when <tt class="function">blas.gemm()</tt> is called with
-<code>matrix(A)</code> and <code>matrix(B)</code> as arguments, without explicitly
-converting <var>A</var> and <var>B</var> to dense.
-The argument <var>partial</var> is ignored.
-
-<P>
-If <var>C</var> is a sparse matrix, the matrix-matrix product in the
-definition of <tt class="function">blas.gemm()</tt> is computed, but as a sparse
-matrix.
-If <var>partial</var> is <code>False</code>, the result is stored in <var>C</var>,
-and the sparsity pattern of <var>C</var> is modified if necessary.
-If <var>partial</var> is <code>True</code>, the operation only updates the nonzero
-elements in <var>C</var>, even if the sparsity pattern of <var>C</var> differs
-from that of the matrix product.
-</dl>
-
-<P>
-<dl><dt><table cellpadding="0" cellspacing="0"><tr valign="baseline">
- <td><nobr><b><tt id='l2h-132' xml:id='l2h-132' class="function">syrk</tt></b>(</nobr></td>
- <td><var>A, C</var><big>[</big><var>, uplo='L'</var><big>[</big><var>,
-trans='N'</var><big>[</big><var>, alpha=1.0</var><big>[</big><var>, beta=0.0</var><big>[</big><var>,
-partial=False</var><big>]</big><var></var><big>]</big><var></var><big>]</big><var></var><big>]</big><var></var><big>]</big><var></var>)</td></tr></table></dt>
-<dd>
-Rank-<I>k</I> update of a sparse or dense real or complex symmetric
-matrix:
-<BR><P></P>
-<DIV ALIGN="CENTER" CLASS="mathdisplay">
-<!-- MATH
- \begin{displaymath}
-C := \alpha AA^T + \beta C \quad (\mathrm{trans} = \mathrm{'N'}),
- \qquad
- C := \alpha A^TA + \beta C \quad (\mathrm{trans} = \mathrm{'T'}),
-\end{displaymath}
- -->
-
-<IMG
- WIDTH="501" HEIGHT="28" BORDER="0"
- SRC="img36.gif"
- ALT="\begin{displaymath}
-C := \alpha AA^T + \beta C \quad (\mathrm{trans} = \mathrm{...
-... \alpha A^TA + \beta C \quad (\mathrm{trans} = \mathrm{'T'}),
-\end{displaymath}">
-</DIV>
-<BR CLEAR="ALL">
-<P></P>
-If <var>A</var> and <var>C</var> are dense, this is identical to
-<tt class="function">blas.syrk()</tt>, described in section <A href="s-blas3.html#s-blas3">3.4</A>,
-and the argument <var>partial</var> is ignored.
-
-<P>
-If <var>A</var> is sparse and <var>C</var> is dense, the result is the same as
-when <tt class="function">blas.syrk()</tt> is called with <code>matrix(A)</code> as
-argument, without explicitly converting <var>A</var> to dense.
-The argument <var>partial</var> is ignored.
-
-<P>
-If <var>C</var> is sparse, the product in the definition of
-<tt class="function">blas.syrk()</tt> is computed, but as a sparse matrix.
-If <var>partial</var> is <code>False</code>, the result is stored in <var>C</var>,
-and the sparsity pattern of <var>C</var> is modified if necessary.
-If <var>partial</var> is <code>True</code>, the operation only updates the nonzero
-elements in <var>C</var>, even if the sparsity pattern of <var>C</var> differs
-from that of the matrix product.
-</dl>
-
-<P>
-In the following example, we first compute
-<BR><P></P>
-<DIV ALIGN="CENTER" CLASS="mathdisplay">
-<!-- MATH
- \begin{displaymath}
-C = A^TB, \qquad
-A = \left[ \begin{array}{ccc}
-0 & 1 & 0 \\1 & 0 & 1 \\0 & 1 & 0 \\1 & 0 & 0 \end{array}\right],
-\qquad
-B = \left[ \begin{array}{ccc}
- 0 & -1 & 0 \\2 & 0 & 2 \\0 & 3 & 0 \\2 & 0 & 0
- \end{array}\right].
-\end{displaymath}
- -->
-
-<IMG
- WIDTH="418" HEIGHT="83" BORDER="0"
- SRC="img83.gif"
- ALT="\begin{displaymath}
-C = A^TB, \qquad
-A = \left[ \begin{array}{ccc}
-0 & 1 & 0 \\ ...
-...0 \\ 2 & 0 & 2 \\ 0 & 3 & 0 \\ 2 & 0 & 0
-\end{array}\right].
-\end{displaymath}">
-</DIV>
-<BR CLEAR="ALL">
-<P></P>
-<div class="verbatim"><pre>
->>> from cvxopt.base import spmatrix, gemm
->>> A = spmatrix(1, [1,3,0,2,1], [0,0,1,1,2])
->>> B = spmatrix([2,2,-1,3,2], [1,3,0,2,1], [0,0,1,1,2])
->>> C = spmatrix([], [], [], size=(3,3))
->>> gemm(A, B, C, transA='T')
->>> print C
-SIZE: (3,3)
-(0, 0) 4.0000e+00
-(2, 0) 2.0000e+00
-(1, 1) 2.0000e+00
-(0, 2) 2.0000e+00
-(2, 2) 2.0000e+00
-</pre></div>
-Now suppose we want to replace <I>C</I>
-<BR><P></P>
-<DIV ALIGN="CENTER" CLASS="mathdisplay">
-<!-- MATH
- \begin{displaymath}
-C = A^TD, \qquad
-D = \left[ \begin{array}{ccc}
- 0 & 1 & 0 \\3 & 0 & -2 \\0 & 1 & 0 \\4 & 0 & 0
- \end{array}\right].
-\end{displaymath}
- -->
-
-<IMG
- WIDTH="252" HEIGHT="83" BORDER="0"
- SRC="img84.gif"
- ALT="\begin{displaymath}
-C = A^TD, \qquad
-D = \left[ \begin{array}{ccc}
-0 & 1 & 0 \\ 3 & 0 & -2 \\ 0 & 1 & 0 \\ 4 & 0 & 0
-\end{array}\right].
-\end{displaymath}">
-</DIV>
-<BR CLEAR="ALL">
-<P></P>
-The new matrix has the same sparsity pattern as <var>C</var>, so we can
-use <tt class="function">gemm()</tt> with the <code>partial=True</code> option.
-This saves time in large sparse matrix multiplications when the
-sparsity pattern of the result is known beforehand.
-<div class="verbatim"><pre>
->>> D = spmatrix([3,4,1,1,-2], [1,3,0,2,1], [0,0,1,1,2])
->>> gemm(A, D, C, transA='T', partial=True)
->>> print C
-SIZE: (3,3)
-(0, 0) 7.0000e+00
-(2, 0) 3.0000e+00
-(1, 1) 2.0000e+00
-(0, 2) -2.0000e+00
-(2, 2) -2.0000e+00
-</pre></div>
-
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-<BR>
-2. Dense Matrices (<tt class="module">cvxopt.base</tt>)
-</H1>
-
-<P>
-The <tt class="module">cvxopt.base</tt> module defines two new Python types:
-<tt class="class">matrix</tt> objects, used for dense matrix computations, and
-<tt class="class">spmatrix</tt> objects, used for sparse matrix computations.
-In this chapter we describe the dense <tt class="class">matrix</tt> object.
-
-<P>
-
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-<LI><A href="node6.html">2.2 Attributes and Methods</a>
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-8. Optimization Routines (<tt class="module">cvxopt.solvers</tt>)
-</H1>
-
-<P>
-
-<p><br /></p><hr class='online-navigation' />
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-<LI><A href="node51.html">8.6 Exploiting Structure in LPs and SDPs</a>
-<LI><A href="node52.html">8.7 Exploiting Structure in Nonlinear Convex Programs</a>
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-<H1><A NAME="SECTION0010200000000000000000">
-8.2 Quadratic Programming</A>
-</H1>
-<dl><dt><table cellpadding="0" cellspacing="0"><tr valign="baseline">
- <td><nobr><b><tt id='l2h-146' xml:id='l2h-146' class="function">qp</tt></b>(</nobr></td>
- <td><var>P, q, </var><big>[</big><var>, G, h </var><big>[</big><var>, A, b</var><big>[</big><var>,
-solver</var><big>]</big><var></var><big>]</big><var></var><big>]</big><var></var>)</td></tr></table></dt>
-<dd>
-Solves a convex quadratic program
-<BR><P></P>
-<DIV ALIGN="CENTER" CLASS="mathdisplay">
-<!-- MATH
- \begin{displaymath}
-\begin{array}{ll}
-\mbox{minimize} & (1/2) x^TPx + q^T x \\
-\mbox{subject to} & Gx \preceq h \\& Ax = b.
-\end{array}
-\end{displaymath}
- -->
-
-<IMG
- WIDTH="199" HEIGHT="64" BORDER="0"
- SRC="img116.gif"
- ALT="\begin{displaymath}
-\begin{array}{ll}
-\mbox{minimize} & (1/2) x^TPx + q^T x \\
-\mbox{subject to} & Gx \preceq h \\ & Ax = b.
-\end{array}\end{displaymath}">
-</DIV>
-<BR CLEAR="ALL">
-<P></P>
-
-<P>
-<var>P</var> is a square dense or sparse real matrix, representing a
-symmetric matrix in <code>'L'</code> storage, <EM>i.e.</EM>, only the lower
-triangular part of <var>P</var> is referenced.
-<var>G</var> and <var>A</var> are dense or sparse real matrices.
-Their default values are sparse matrices with zero columns.
-<var>q</var>, <var>h</var> and <var>b</var> are single-column real dense matrices.
-The default values of <var>h</var> and <var>b</var> are matrices of size (0,1).
-
-<P>
-The default CVXOPT solver is used when the <var>solver</var> argument
-is absent or <code>None</code>. The MOSEK solver (if installed) can be
-selected by setting <var>solver</var>=<code>'mosek'</code>.
-
-<P>
-<tt class="function">qp()</tt> returns a dictionary with keys
-<code>'status'</code>, <code>'x'</code>, <code>'s'</code>, <code>'y'</code>, <code>'z'</code>.
-The possible values of the <code>'status'</code> key are as follows.
-<DL>
-<DT><STRONG><code>'optimal'</code></STRONG></DT>
-<DD>In this case the
-<code>'x'</code> entry is the primal optimal solution,
-the <code>'s'</code> entry is the corresponding slack in the inequality
-constraints, the <code>'z'</code> and <code>'y'</code> entries are the optimal
-values of the dual variables associated with the linear inequality
-and linear equality constraints.
-These values (approximately) satisfy the optimality conditions
-<BR><P></P>
-<DIV ALIGN="CENTER" CLASS="mathdisplay">
-<!-- MATH
- \begin{displaymath}
-Px + q + G^T z + A^T y = 0, \qquad Gx + s = h, \qquad
- Ax = b, \qquad s \succeq 0, \qquad z \succeq 0, \qquad s^T z = 0.
-\end{displaymath}
- -->
-
-<IMG
- WIDTH="603" HEIGHT="27" BORDER="0"
- SRC="img117.gif"
- ALT="\begin{displaymath}
-Px + q + G^T z + A^T y = 0, \qquad Gx + s = h, \qquad
-Ax = b, \qquad s \succeq 0, \qquad z \succeq 0, \qquad s^T z = 0.
-\end{displaymath}">
-</DIV>
-<BR CLEAR="ALL">
-<P></P>
-
-<P>
-</DD>
-<DT><STRONG><code>'primal infeasible'</code></STRONG></DT>
-<DD>This only applies when
-<var>solver</var> is <code>'mosek'</code>, and means that a certificate of
-primal infeasibility has been found. The <code>'x'</code> and <code>'s'</code>
-entries are <code>None</code>, and the
-<code>'z'</code> and <code>'y'</code> entries are vectors that approximately satisfy
-<BR><P></P>
-<DIV ALIGN="CENTER" CLASS="mathdisplay">
-<!-- MATH
- \begin{displaymath}
-G^Tz + A^T y = 0, \qquad h^Tz + b^Ty = -1, \qquad z \succeq 0.
-\end{displaymath}
- -->
-
-<IMG
- WIDTH="340" HEIGHT="27" BORDER="0"
- SRC="img113.gif"
- ALT="\begin{displaymath}
-G^T z + A^T y = 0, \qquad h^T z + b^T y = -1, \qquad z \succeq 0.
-\end{displaymath}">
-</DIV>
-<BR CLEAR="ALL">
-<P></P>
-
-<P>
-</DD>
-<DT><STRONG><code>'dual infeasible'</code></STRONG></DT>
-<DD>This only applies when
-<var>solver</var> is <code>'mosek'</code>, and means that a certificate of
-dual infeasibility has been found. The <code>'z'</code> and <code>'y'</code>
-entries are <code>None</code>, and the <code>'x'</code> and <code>'s'</code> entries are
-vectors that approximately satisfy
-<BR><P></P>
-<DIV ALIGN="CENTER" CLASS="mathdisplay">
-<!-- MATH
- \begin{displaymath}
-Px = 0, \qquad q^Tx = -1, \qquad Gx + s = 0, \qquad Ax=0, \qquad
- s \succeq 0.
-\end{displaymath}
- -->
-
-<IMG
- WIDTH="442" HEIGHT="27" BORDER="0"
- SRC="img118.gif"
- ALT="\begin{displaymath}
-Px = 0, \qquad q^Tx = -1, \qquad Gx + s = 0, \qquad Ax=0, \qquad
-s \succeq 0.
-\end{displaymath}">
-</DIV>
-<BR CLEAR="ALL">
-<P></P>
-
-<P>
-</DD>
-<DT><STRONG><code>'unknown'</code></STRONG></DT>
-<DD>This means that the algorithm reached
-the maximum number of iterations before a solution was found.
-The <code>'x'</code>, <code>'s'</code>, <code>'y'</code>, <code>'z'</code> entries are <code>None</code>.
-</DD>
-</DL>
-</dl>
-
-<P>
-As an example we compute the trade-off curve on page 187
-of the book <em class="citetitle"><a
- href="http://www.stanford.edu/~boyd/cvxbook"
- title="Convex
-Optimization"
- >Convex
-Optimization</a></em>, by solving the quadratic program
-<BR><P></P>
-<DIV ALIGN="CENTER" CLASS="mathdisplay">
-<!-- MATH
- \begin{displaymath}
-\begin{array}{ll}
-\mbox{minimize} & -\bar p^T x + \mu x^T S x \\
-\mbox{subject to} & {\bf 1}^T x = 1, \quad x \succeq 0
-\end{array}
-\end{displaymath}
- -->
-
-<IMG
- WIDTH="195" HEIGHT="45" BORDER="0"
- SRC="img119.gif"
- ALT="\begin{displaymath}
-\begin{array}{ll}
-\mbox{minimize} & -\bar p^T x + \mu x^T S ...
-...ox{subject to} & {\bf 1}^T x = 1, \quad x \succeq 0
-\end{array}\end{displaymath}">
-</DIV>
-<BR CLEAR="ALL">
-<P></P>
-for a sequence of positive values of <I>mu</I>.
-The code below computes the trade-off curve and produces two figures
-using the <a class="ulink" href="http://matplotlib.sourceforge.net"
- >Matplotlib</a> package.
-<DIV ALIGN="CENTER">
-<IMG
- WIDTH="453" HEIGHT="340" ALIGN="BOTTOM" BORDER="0"
- SRC="img120.gif"
- ALT="\includegraphics[width=10cm]{figures/portfolio1.eps}">
-
-<IMG
- WIDTH="453" HEIGHT="340" ALIGN="BOTTOM" BORDER="0"
- SRC="img121.gif"
- ALT="\includegraphics[width=10cm]{figures/portfolio2.eps}">
-
-</DIV>
-
-<P>
-<div class="verbatim"><pre>
-from math import sqrt
-from cvxopt.base import matrix
-from cvxopt.blas import dot
-from cvxopt.solvers import qp
-import pylab
-
-# Problem data.
-n = 4
-S = matrix([[ 4e-2, 6e-3, -4e-3, 0.0 ],
- [ 6e-3, 1e-2, 0.0, 0.0 ],
- [-4e-3, 0.0, 2.5e-3, 0.0 ],
- [ 0.0, 0.0, 0.0, 0.0 ]])
-pbar = matrix([.12, .10, .07, .03])
-G = matrix(0.0, (n,n))
-G[::n+1] = -1.0
-h = matrix(0.0, (n,1))
-A = matrix(1.0, (1,n))
-b = matrix(1.0)
-
-# Compute trade-off.
-N = 100
-mus = [ 10**(5.0*t/N-1.0) for t in xrange(N) ]
-portfolios = [ qp(mu*S, -pbar, G, h, A, b)['x'] for mu in mus ]
-returns = [ dot(pbar,x) for x in portfolios ]
-risks = [ sqrt(dot(x, S*x)) for x in portfolios ]
-
-# Plot trade-off curve and optimal allocations.
-pylab.figure(1, facecolor='w')
-pylab.plot(risks, returns)
-pylab.xlabel('standard deviation')
-pylab.ylabel('expected return')
-pylab.axis([0, 0.2, 0, 0.15])
-pylab.title('Risk-return trade-off curve (fig 4.12)')
-pylab.yticks([0.00, 0.05, 0.10, 0.15])
-
-pylab.figure(2, facecolor='w')
-c1 = [ x[0] for x in portfolios ]
-c2 = [ x[0] + x[1] for x in portfolios ]
-c3 = [ x[0] + x[1] + x[2] for x in portfolios ]
-c4 = [ x[0] + x[1] + x[2] + x[3] for x in portfolios ]
-pylab.fill(risks + [.20], c1 + [0.0], '#F0F0F0')
-pylab.fill(risks[-1::-1] + risks, c2[-1::-1] + c1, '#D0D0D0')
-pylab.fill(risks[-1::-1] + risks, c3[-1::-1] + c2, '#F0F0F0')
-pylab.fill(risks[-1::-1] + risks, c4[-1::-1] + c3, '#D0D0D0')
-pylab.axis([0.0, 0.2, 0.0, 1.0])
-pylab.xlabel('standard deviation')
-pylab.ylabel('allocation')
-pylab.text(.15,.5,'x1')
-pylab.text(.10,.7,'x2')
-pylab.text(.05,.7,'x3')
-pylab.text(.01,.7,'x4')
-pylab.title('Optimal allocations (fig 4.12)')
-pylab.show()
-</pre></div>
-
-<P>
-
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-<p></p><hr />
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-<!DOCTYPE html PUBLIC "-//W3C//DTD HTML 4.0 Transitional//EN">
-<html>
-<head>
-<link rel="STYLESHEET" href="cvxopt.css" type='text/css' />
-<link rel="first" href="cvxopt.html" title='CVXOPT: A Python Package for Convex Optimization' />
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-
-<H1><A NAME="SECTION0010300000000000000000">
-8.3 Geometric Programming</A>
-</H1>
-<dl><dt><table cellpadding="0" cellspacing="0"><tr valign="baseline">
- <td><nobr><b><tt id='l2h-147' xml:id='l2h-147' class="function">gp</tt></b>(</nobr></td>
- <td><var>K, F, g </var><big>[</big><var>, G, h </var><big>[</big><var>, A, b</var><big>]</big><var></var><big>]</big><var></var>)</td></tr></table></dt>
-<dd>
-Solves a geometric program in convex form
-<BR><P></P>
-<DIV ALIGN="CENTER" CLASS="mathdisplay">
-<!-- MATH
- \begin{displaymath}
-\begin{array}{ll}
-\mbox{minimize} & f_0(x) = \mathop{\bf lse}(F_0x+g_0) \\
-\mbox{subject to} & f_i(x) = \mathop{\bf lse}(F_ix+g_i) \leq 0,\quad i=1,\ldots,m \\
- & Gx \preceq h \\
- & Ax=b
-\end{array}
-\end{displaymath}
- -->
-
-<IMG
- WIDTH="359" HEIGHT="83" BORDER="0"
- SRC="img122.gif"
- ALT="\begin{displaymath}
-\begin{array}{ll}
-\mbox{minimize} & f_0(x) = \mathop{\bf lse...
-...,\quad i=1,\ldots,m \\
-& Gx \preceq h \\
-& Ax=b
-\end{array}\end{displaymath}">
-</DIV>
-<BR CLEAR="ALL">
-<P></P>
-where
-<BR><P></P>
-<DIV ALIGN="CENTER" CLASS="mathdisplay">
-<!-- MATH
- \begin{displaymath}
-\mathop{\bf lse}(u) = \log \sum_k \exp(u_k), \qquad
- F = \left[ \begin{array}{cccc}
- F_0^T & F_1^T & \cdots & F_m^T \end{array}\right]^T, \qquad
- g = \left[ \begin{array}{cccc}
- g_0^T & g_1^T & \cdots & g_m^T \end{array}\right]^T.
-\end{displaymath}
- -->
-
-<IMG
- WIDTH="649" HEIGHT="45" BORDER="0"
- SRC="img123.gif"
- ALT="\begin{displaymath}
-\mathop{\bf lse}(u) = \log \sum_k \exp(u_k), \qquad
-F = \l...
-...}{cccc}
-g_0^T & g_1^T & \cdots & g_m^T \end{array}\right]^T.
-\end{displaymath}">
-</DIV>
-<BR CLEAR="ALL">
-<P></P>
-<var>K</var> is a list of <var>m</var>+1 positive integers with
-<code><var>K</var>[<var>i</var>]</code>
-equal to the number of rows in <I>Fi</I>.
-<var>F</var> is a dense or sparse real matrix of size
-<code>(sum(<var>K</var>),<var>n</var>)</code>.
-<var>g</var> is a dense real matrix of size <code>(sum(<var>K</var>),1)</code>.
-<var>G</var> and <var>A</var> are dense or sparse real matrices.
-Their default values are sparse matrices with zero rows.
-<var>h</var> and <var>b</var> are dense real matrices with one column.
-Their default values are matrices of size (0,1).
-
-<P>
-<tt class="function">gp()</tt> returns a dictionary with keys
-<code>'status'</code>, <code>'x'</code>, <code>'snl'</code>, <code>'sl'</code>,
-<code>'y'</code>, <code>'znl'</code> and <code>'zl'</code>.
-The possible values of the <code>'status'</code> key are:
-<DL>
-<DT><STRONG><code>'optimal'</code></STRONG></DT>
-<DD>In this case the
-<code>'x'</code> entry is the primal optimal solution,
-the <code>'snl'</code> and <code>'sl'</code> entries are the corresponding slacks
-in the nonlinear and linear inequality constraints.
-The <code>'znl'</code>, <code>'zl'</code> and <code>'y'</code> entries are the optimal
-values of the dual variables associated with the nonlinear and linear
-inequality constraints and the linear equality constraints.
-These values approximately satisfy
-<BR><P></P>
-<DIV ALIGN="CENTER" CLASS="mathdisplay">
-<!-- MATH
- \begin{displaymath}
-\nabla f_0(x) + \sum_{k=1}^m z_{\mathrm{nl},k}
- \nabla f_k(x) + G^T z_\mathrm{l} + A^T y = 0, \qquad
- f_k(x) + s_{\mathrm{nl},k} = 0, \quad k=1,\ldots,m, \qquad
- Gx + s_\mathrm{l} = h, \qquad Ax=b
-\end{displaymath}
- -->
-
-<IMG
- WIDTH="747" HEIGHT="53" BORDER="0"
- SRC="img124.gif"
- ALT="\begin{displaymath}
-\nabla f_0(x) + \sum_{k=1}^m z_{\mathrm{nl},k}
-\nabla f_k...
-...quad k=1,\ldots,m, \qquad
-Gx + s_\mathrm{l} = h, \qquad Ax=b
-\end{displaymath}">
-</DIV>
-<BR CLEAR="ALL">
-<P></P>
-and
-<BR><P></P>
-<DIV ALIGN="CENTER" CLASS="mathdisplay">
-<!-- MATH
- \begin{displaymath}
-s_\mathrm{nl}\succeq 0, \qquad s_\mathrm{l}\succeq 0, \qquad
-z_\mathrm{nl} \succeq 0, \qquad z_\mathrm{l} \succeq 0, \qquad
-s_\mathrm{nl}^T z_\mathrm{nl} + s_\mathrm{l}^T z_\mathrm{l} = 0.
-\end{displaymath}
- -->
-
-<IMG
- WIDTH="453" HEIGHT="28" BORDER="0"
- SRC="img125.gif"
- ALT="\begin{displaymath}
-s_\mathrm{nl}\succeq 0, \qquad s_\mathrm{l}\succeq 0, \qquad...
-...\mathrm{nl}^T z_\mathrm{nl} + s_\mathrm{l}^T z_\mathrm{l} = 0.
-\end{displaymath}">
-</DIV>
-<BR CLEAR="ALL">
-<P></P>
-
-<P>
-</DD>
-<DT><STRONG><code>'unknown'</code></STRONG></DT>
-<DD>This means that the algorithm reached
-the maximum number of iterations before a solution was found.
-The <code>'x'</code>, <code>'snl'</code>, <code>'sl'</code>, <code>'y'</code>, <code>'znl'</code>
-and <code>'zl'</code> entries are <code>None</code>.
-</DD>
-</DL>
-</dl>
-
-<P>
-As an example, we solve the small GP on page 8 of the paper
-<em class="citetitle"><a
- href="http://www.stanford.edu/~boyd/gp_tutorial"
- title="A Tutorial on Geometric Programming"
- >A Tutorial on Geometric Programming</a></em>.
-The posynomial form of the problem is
-<BR><P></P>
-<DIV ALIGN="CENTER" CLASS="mathdisplay">
-<!-- MATH
- \begin{displaymath}
-\begin{array}{ll}
- \mbox{minimize} & w^{-1} h^{-1} d^{-1} \\
- \mbox{subject to}
- & (2/A_\mathrm{wall}) hw + (2/A_\mathrm{wall})hd \leq 1 \\
- & (1/A_\mathrm{flr}) wd \leq 1 \\
- & \alpha wh^{-1} \leq 1 \\
- & (1/\beta) hw^{-1} \leq 1 \\
- & \gamma wd^{-1} \leq 1 \\
- & (1/\delta)dw^{-1} \leq 1
- \end{array}
-\end{displaymath}
- -->
-
-<IMG
- WIDTH="288" HEIGHT="140" BORDER="0"
- SRC="img126.gif"
- ALT="\begin{displaymath}
-\begin{array}{ll}
-\mbox{minimize} & w^{-1} h^{-1} d^{-1} \...
-...mma wd^{-1} \leq 1 \\
-& (1/\delta)dw^{-1} \leq 1
-\end{array}\end{displaymath}">
-</DIV>
-<BR CLEAR="ALL">
-<P></P>
-with variables <I>h</I>, <I>w</I>, <I>d</I>.
-
-<P>
-<div class="verbatim"><pre>
-from cvxopt.base import matrix, log, exp
-from cvxopt import solvers
-
-Aflr = 1000.0
-Awall = 100.0
-alpha = 0.5
-beta = 2.0
-gamma = 0.5
-delta = 2.0
-
-F = matrix( [[-1., 1., 1., 0., -1., 1., 0., 0.],
- [-1., 1., 0., 1., 1., -1., 1., -1.],
- [-1., 0., 1., 1., 0., 0., -1., 1.]])
-g = log( matrix( [1.0, 2/Awall, 2/Awall, 1/Aflr, alpha, 1/beta, gamma, 1/delta]) )
-K = [1, 2, 1, 1, 1, 1, 1]
-h, w, d = exp( solvers.gp(K, F, g)['x'] )
-</pre></div>
-
-<P>
-
-<DIV CLASS="navigation">
-<div class='online-navigation'>
-<p></p><hr />
-<table align="center" width="100%" cellpadding="0" cellspacing="2">
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-<!DOCTYPE html PUBLIC "-//W3C//DTD HTML 4.0 Transitional//EN">
-<html>
-<head>
-<link rel="STYLESHEET" href="cvxopt.css" type='text/css' />
-<link rel="first" href="cvxopt.html" title='CVXOPT: A Python Package for Convex Optimization' />
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-</head>
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-<td align="center" width="100%">CVXOPT: A Python Package for Convex Optimization</td>
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- border='0' height='32' alt='Index' width='32' /></A></td>
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-<div class='online-navigation'>
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-<a class="sectref" rel="prev" href="e-nlcp.html">8.5 Nonlinear Convex Programming</A>
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-</div>
-<hr /></div>
-</DIV>
-<!--End of Navigation Panel-->
-
-<H1><A NAME="SECTION0010600000000000000000">
-8.6 Exploiting Structure in LPs and SDPs</A>
-</H1>
-The solvers <tt class="function">lp()</tt> and <tt class="function">sdp()</tt> are interfaces to
-a common function <tt class="function">conelp()</tt>, which can also be called
-directly. When calling <tt class="function">conelp()</tt>, the user must provide
-functions for evaluating the constraint functions and for
-solving the linear equations (KKT equations) that are solved in
-each iteration of the algorithm.
-This is useful for LPs and SDPs that possess some interesting
-structure that makes it possible to solve the KKT equations fast.
-
-<P>
-<dl><dt><table cellpadding="0" cellspacing="0"><tr valign="baseline">
- <td><nobr><b><tt id='l2h-150' xml:id='l2h-150' class="function">conelp</tt></b>(</nobr></td>
- <td><var>c, kktsolver</var><big>[</big><var>, Gl, hl</var><big>[</big><var>,
-Gs, hs</var><big>[</big><var>, A, b</var><big>[</big><var>, primalstart</var><big>[</big><var>, dualstart</var><big>]</big><var></var><big>]</big><var></var><big>]</big><var></var><big>]</big><var></var><big>]</big><var></var>)</td></tr></table></dt>
-<dd>
-Solves the pair of primal and dual SDPs (<A href="s-sdpsolver.html#e-sdp">8.2</A>).
-The arguments <var>c</var>, <var>hl</var>, <var>hs</var>, <var>b</var>,
-<var>primalstart</var>, <var>dualstart</var> have the same meaning as in
-<tt class="function">sdp()</tt>.
-The arguments <var>kktsolver</var>, <var>Gl</var>, <var>Gs</var>, <var>A</var> are
-functions that must handle the following calling sequences.
-
-<P>
-
-<UL>
-<LI><tt class="function">kktsolver</tt>(<var>d</var>, <var>R</var>) with <var>d</var> a positive
-dense real matrix of size (<I>ml</I>,<I>1</I>), and
-<var>R</var> a list of <I>N</I> square dense real matrices
-<code><var>R</var>[k]</code> of order <I>m_k</I>, returns a function for
-solving the equation
-<P></P>
-<DIV ALIGN="CENTER" CLASS="mathdisplay">
-<!-- MATH
- \begin{eqnarray*}
-A^T u_y + G_\mathrm{l}^T u_{z_\mathrm{l}} +
- G_\mathrm{s}^T(u_{z_\mathrm{s}}) & = & b_x \\
- A u_x & = & b_y \\
-G_\mathrm{l}u_x - \mbox{\bf diag}\,(d)^{-2} u_{z_\mathrm{l}} & = &
- b_{z_\mathrm{l}} \\
-G_\mathrm{s}(u_x) - R^{-T} R^{-1} u_{z_\mathrm{s}} R^{-T} R^{-1}
- & = & b_{z_\mathrm{s}}.
-\end{eqnarray*}
- -->
-<IMG
- WIDTH="276" HEIGHT="100" BORDER="0"
- SRC="img148.gif"
- ALT="\begin{eqnarray*}
-A^T u_y + G_\mathrm{l}^T u_{z_\mathrm{l}} +
-G_\mathrm{s}^T(...
-...} R^{-1} u_{z_\mathrm{s}} R^{-T} R^{-1}
-& = & b_{z_\mathrm{s}}.
-\end{eqnarray*}"></DIV>
-<BR CLEAR="ALL"><P></P>
-<BR CLEAR="ALL"><P></P>
-The function created by "<tt class="samp">f = kktsolver(d, R)</tt>" will be
-called as "<tt class="samp">f(bx, by, bzl, bzs)</tt>".
-On entry, <var>bx</var>, <var>by</var>, <var>bzl</var> and <var>bzs</var> contain the
-righthand side. On exit, they should contain the solution of the KKT
-system, with <I>uzl</I> and <I>uzs</I> scaled:
-<BR><P></P>
-<DIV ALIGN="CENTER" CLASS="mathdisplay">
-<!-- MATH
- \begin{displaymath}
-b_x := u_x, \qquad
- b_y := u_y, \qquad
- b_{z_\mathrm{l}} := \mbox{\bf diag}\,(d)^{-1} u_{z_\mathrm{l}}, \qquad
- b_{z_\mathrm{s}} := R^{-1} u_{z_\mathrm{s}} R^{-T}.
-\end{displaymath}
- -->
-
-<IMG
- WIDTH="494" HEIGHT="29" BORDER="0"
- SRC="img149.gif"
- ALT="\begin{displaymath}
-b_x := u_x, \qquad
-b_y := u_y, \qquad
-b_{z_\mathrm{l}} :=...
-...}, \qquad
-b_{z_\mathrm{s}} := R^{-1} u_{z_\mathrm{s}} R^{-T}.
-\end{displaymath}">
-</DIV>
-<BR CLEAR="ALL">
-<P></P>
-
-<P>
-</LI>
-<LI><tt class="function">Gl</tt>(<var>x</var>, <var>y</var><big>[</big>,
-<var>alpha</var>=1.0<big>[</big>, <var>beta</var>=0.0<big>[</big>,
-<var>trans</var>=<code>'N'</code><big>]</big><big>]</big><big>]</big>) evaluates the matrix-vector products
-<BR><P></P>
-<DIV ALIGN="CENTER" CLASS="mathdisplay">
-<!-- MATH
- \begin{displaymath}
-y := \alpha G_\mathrm{l}x + \beta y \quad
- (\mathrm{trans} = \mathrm{'N'}), \qquad
-y := \alpha G_\mathrm{l}^T x + \beta y \quad
- (\mathrm{trans} = \mathrm{'T'}).
-\end{displaymath}
- -->
-
-<IMG
- WIDTH="475" HEIGHT="28" BORDER="0"
- SRC="img150.gif"
- ALT="\begin{displaymath}
-y := \alpha G_\mathrm{l}x + \beta y \quad
-(\mathrm{trans} ...
-...thrm{l}^T x + \beta y \quad
-(\mathrm{trans} = \mathrm{'T'}).
-\end{displaymath}">
-</DIV>
-<BR CLEAR="ALL">
-<P></P>
-
-<P>
-</LI>
-<LI><tt class="function">Gs</tt>(<var>x</var>, <var>y</var><big>[</big>,
-<var>alpha</var>=1.0<big>[</big>, <var>beta</var>=0.0<big>[</big>,
-<var>trans</var>=<code>'N'</code><big>]</big><big>]</big><big>]</big>)
-evaluates the linear mappings
-<BR><P></P>
-<DIV ALIGN="CENTER" CLASS="mathdisplay">
-<!-- MATH
- \begin{displaymath}
-y := \alpha G_\mathrm{s}(x) + \beta y \quad
- (\mathrm{trans} = \mathrm{'N'}), \qquad
-y := \alpha G_\mathrm{s}^T(x) + \beta y \quad
- (\mathrm{trans} = \mathrm{'T'}).
-\end{displaymath}
- -->
-
-<IMG
- WIDTH="501" HEIGHT="28" BORDER="0"
- SRC="img151.gif"
- ALT="\begin{displaymath}
-y := \alpha G_\mathrm{s}(x) + \beta y \quad
-(\mathrm{trans...
-...hrm{s}^T(x) + \beta y \quad
-(\mathrm{trans} = \mathrm{'T'}).
-\end{displaymath}">
-</DIV>
-<BR CLEAR="ALL">
-<P></P>
-
-<P>
-</LI>
-<LI><tt class="function">A</tt>(<var>x</var>, <var>y</var><big>[</big>,
-<var>alpha</var>=1.0<big>[</big>, <var>beta</var>=0.0<big>[</big>,
-<var>trans</var>=<code>'N'</code><big>]</big><big>]</big><big>]</big>)
-evaluates the matrix vector products
-<BR><P></P>
-<DIV ALIGN="CENTER" CLASS="mathdisplay">
-<!-- MATH
- \begin{displaymath}
-y := \alpha Ax + \beta y \quad
- (\mathrm{trans} = \mathrm{'N'}), \qquad
-y := \alpha A^Tx + \beta y \quad
- (\mathrm{trans} = \mathrm{'T'}).
-\end{displaymath}
- -->
-
-<IMG
- WIDTH="469" HEIGHT="28" BORDER="0"
- SRC="img152.gif"
- ALT="\begin{displaymath}
-y := \alpha Ax + \beta y \quad
-(\mathrm{trans} = \mathrm{'...
-...\alpha A^Tx + \beta y \quad
-(\mathrm{trans} = \mathrm{'T'}).
-\end{displaymath}">
-</DIV>
-<BR CLEAR="ALL">
-<P></P>
-</LI>
-</UL>
-</dl>
-
-<P>
-<DL>
-<DT><STRONG>Example: 1-norm approximation</STRONG></DT>
-<DD><P>
-The optimization problem
-<BR><P></P>
-<DIV ALIGN="CENTER" CLASS="mathdisplay">
-<!-- MATH
- \begin{displaymath}
-\begin{array}{ll}
- \mbox{minimize} & \|Pu-q\|_1
- \end{array}
-\end{displaymath}
- -->
-
-<IMG
- WIDTH="147" HEIGHT="30" BORDER="0"
- SRC="img153.gif"
- ALT="\begin{displaymath}
-\begin{array}{ll}
-\mbox{minimize} & \Vert Pu-q\Vert _1
-\end{array}\end{displaymath}">
-</DIV>
-<BR CLEAR="ALL">
-<P></P>
-can be formulated as an LP
-<BR><P></P>
-<DIV ALIGN="CENTER" CLASS="mathdisplay">
-<!-- MATH
- \begin{displaymath}
-\begin{array}{ll}
- \mbox{minimize} & {\bf 1}^T v \\
- \mbox{subject to} & -v \preceq Pu - q \preceq v.
- \end{array}
-\end{displaymath}
- -->
-
-<IMG
- WIDTH="201" HEIGHT="45" BORDER="0"
- SRC="img154.gif"
- ALT="\begin{displaymath}
-\begin{array}{ll}
-\mbox{minimize} & {\bf 1}^T v \\
-\mbox{subject to} & -v \preceq Pu - q \preceq v.
-\end{array}\end{displaymath}">
-</DIV>
-<BR CLEAR="ALL">
-<P></P>
-By exploiting the structure in the inequalities, the cost of
-an iteration of an interior-point method can be reduced
-to the cost of least-squares problem of the same dimensions.
-(See section 11.8.2 in the book
-<em class="citetitle"><a
- href="http://www.ee.ucla.edu/~vandenbe/cvxbook"
- title="Convex Optimization"
- >Convex Optimization</a></em>.)
-The code belows taks advantage of this fact.
-
-<P>
-<div class="verbatim"><pre>
-from cvxopt import base, blas, lapack, solvers
-from cvxopt.base import matrix, spmatrix, mul, div
-
-def l1(P, q):
- """
-
- Returns the solution u, w of the l1 approximation problem
-
- (primal) minimize ||P*u - q||_1
-
- (dual) maximize q'*w
- subject to P'*w = 0
- ||w||_infty <= 1.
- """
-
- m, n = P.size
-
- # Solve equivalent LP
- #
- # minimize [0; 1]' * [u; v]
- # subject to [P, -I; -P, -I] * [u; v] <= [q; -q]
- #
- # maximize -[q; -q]' * z
- # subject to [P', -P']*z = 0
- # [-I, -I]*z + 1 = 0
- # z >= 0
-
- c = matrix(n*[0.0] + m*[1.0])
- h = matrix([q, -q])
-
- def Fi(x, y, alpha=1.0, beta=0.0, trans='N'):
- if trans=='N':
- # y := alpha * [P, -I; -P, -I] * x + beta*y
- u = P*x[:n]
- y[:m] = alpha * ( u - x[n:]) + beta*y[:m]
- y[m:] = alpha * (-u - x[n:]) + beta*y[m:]
-
- else:
- # y := alpha * [P', -P'; -I, -I] * x + beta*y
- y[:n] = alpha * P.T * (x[:m] - x[m:]) + beta*y[:n]
- y[n:] = -alpha * (x[:m] + x[m:]) + beta*y[n:]
-
-
- def kktsolver(d, R):
-
- # Returns a function f(x,y,zl,zs) that solves
- #
- # [ 0 0 P' -P' ] [ x[:n] ] [ bx[:n] ]
- # [ 0 0 -I -I ] [ x[n:] ] [ bx[n:] ]
- # [ P -I -D1^{-1} 0 ] [ zl[:m]] = [ bzl[:m] ]
- # [-P -I 0 -D2^{-1} ] [ zl[m:]] [ bzl[m:] ]
- #
- # where D1 = diag(d[:m])^2, D2 = diag(d[m:])^2.
- #
- # On entry bx, bzl are stored in x, zl.
- # On exit x, zl contain the solution, with zl scaled: zl./d is
- # returned instead of zl.
-
- # Factor A = 4*P'*D*P where D = d1.*d2 ./(d1+d2) and
- # d1 = d[:m].^2, d2 = d[m:].^2.
-
- d1, d2 = d[:m]**2, d[m:]**2
- D = div( mul(d1,d2), d1+d2 )
- A = P.T * spmatrix(4*D, range(m), range(m)) * P
- lapack.potrf(A)
-
- def f(x, y, zl, zs):
-
- # Solve for x[:n]:
- #
- # A*x[:n] = bx[:n] + P' * ( ((D1-D2)*(D1+D2)^{-1})*bx[n:]
- # + (2*D1*D2*(D1+D2)^{-1}) * (bzl[:m] - bzl[m:]) ).
- x[:n] += P.T * ( mul(div(d1-d2, d1+d2), x[n:]) + mul(2*D, zl[:m]-zl[m:]) )
- lapack.potrs(A, x)
-
- # x[n:] := (D1+D2)^{-1} * (bx[n:] - D1*bzl[:m] - D2*bzl[m:] + (D1-D2)*P*x[:n])
- u = P*x[:n]
- x[n:] = div(x[n:] - mul(d1, zl[:m]) - mul(d2, zl[m:]) + mul(d1-d2, u), d1+d2)
-
- # z[:m] := d1[:m] .* ( P*x[:n] - x[n:] - bzl[:m])
- # z[m:] := d2[m:] .* (-P*x[:n] - x[n:] - bzl[m:])
- zl[:m] = mul(d[:m], u-x[n:]-zl[:m])
- zl[m:] = mul(d[m:], -u-x[n:]-zl[m:])
-
- return f
-
- sol = solvers.conelp(c, kktsolver, Gl=Fi, hl=h)
- return sol['x'][:n], sol['zl'][m:] - sol['zl'][:m]
-</pre></div>
-
-<P>
-</DD>
-<DT><STRONG>Example: SDP with diagonal linear term</STRONG></DT>
-<DD><P>
-The SDP
-<BR><P></P>
-<DIV ALIGN="CENTER" CLASS="mathdisplay">
-<!-- MATH
- \begin{displaymath}
-\begin{array}{ll}
- \mbox{minimize} & {\bf 1}^T x \\
- \mbox{subject to} & W + \mbox{\bf diag}\,(x) \succeq 0
- \end{array}
-\end{displaymath}
- -->
-
-<IMG
- WIDTH="195" HEIGHT="45" BORDER="0"
- SRC="img155.gif"
- ALT="\begin{displaymath}
-\begin{array}{ll}
-\mbox{minimize} & {\bf 1}^T x \\
-\mbox{subject to} & W + \mbox{\bf diag}\,(x) \succeq 0
-\end{array}
-\end{displaymath}">
-</DIV>
-<BR CLEAR="ALL">
-<P></P>
-can be solved efficiently by exploiting properties of the diag operator.
-
-<P>
-<div class="verbatim"><pre>
-from cvxopt import base, blas, lapack, solvers
-from cvxopt.base import matrix
-
-def mcsdp(w):
- """
- Returns solution x, z to
-
- (primal) minimize sum(x)
- subject to w + diag(x) >= 0
-
- (dual) maximize -tr(w*z)
- subject to diag(z) = 1
- z >= 0.
- """
-
- n = w.size[0]
-
- def Fs(x, y, alpha=1.0, beta=0.0, trans='N'):
- """
- y := alpha*(-diag(x)) + beta*y.
- """
- if trans=='N':
- # x is a vector; y[0] is a matrix.
- y[0] *= beta
- y[0][::n+1] -= alpha * x
- else:
- # x[0] is a matrix; y is a vector.
- y *= beta
- y -= alpha * x[::n+1]
-
-
- def cngrnc(r, x, alpha=1.0):
- """
- Congruence transformation
-
- x := alpha * r'*x*r.
-
- r and x are square 'd' matrices.
- """
-
- # Scale diagonal of x by 1/2.
- x[::n+1] *= 0.5
-
- # a := tril(x)*r
- a = +r
- blas.trmm(x, a, side='L')
-
- # x := alpha*(a*r' + r*a')
- blas.syr2k(r, a, x, trans='T', alpha=alpha)
-
-
- def kktsolver(d, r):
-
- # t = r*r' as a nonsymmetric matrix.
- t = matrix(0.0, (n,n))
- blas.gemm(r[0], r[0], t, transB='T')
-
- # Cholesky factorization of tsq = t.*t.
- tsq = t**2
- lapack.potrf(tsq)
-
- def f(x, y, zl, zs):
- """
- Solve
- -diag(zs) = bx
- -diag(x) - inv(r*r')*zs*inv(r*r') = bs.
-
- On entry, x and zs contain bx and bs.
- On exit, they contain the solution, with zs scaled
- (inv(r)'*zs*inv(r) is returned instead of zs).
-
- We solve
-
- ((r*r') .* (r*r')) * x = bx - diag(t*bs*t)
-
- and take zs = -r' * (diag(x) + bs) * r.
- """
-
- # tbst := t * zs * t = t * bs * t
- tbst = +zs[0]
- cngrnc(t, tbst)
-
- # x := x - diag(tbst) = bx - diag(r*r' * bs * r*r')
- x -= tbst[::n+1]
-
- # x := (t.*t)^{-1} * x = (t.*t)^{-1} * (bx - diag(t*bs*t))
- lapack.potrs(tsq, x)
-
- # zs := zs + diag(x) = bs + diag(x)
- zs[0][::n+1] += x
-
- # zs := -r' * zs * r = -r' * (diag(x) + bs) * r
- cngrnc(r[0], zs[0], alpha=-1.0)
-
- return f
-
- c = matrix(1.0, (n,1))
- sol = solvers.conelp(c, kktsolver, Gs=Fs, hs=[w])
- return sol['x'], sol['zs'][0]
-</pre></div>
-</DD>
-</DL>
-
-<P>
-
-<DIV CLASS="navigation">
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-<p></p><hr />
-<table align="center" width="100%" cellpadding="0" cellspacing="2">
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-
-<H1><A NAME="SECTION0010700000000000000000">
-8.7 Exploiting Structure in Nonlinear Convex Programs</A>
-</H1>
-The solvers <tt class="function">gp()</tt>, <tt class="function">gp()</tt> and <tt class="function">cp()</tt> are
-interfaces to <tt class="function">nlcp()</tt>, which can also be called directly
-but requires user-provided functions for evaluating the constraint
-and for solving the KKT equations.
-
-<P>
-<dl><dt><table cellpadding="0" cellspacing="0"><tr valign="baseline">
- <td><nobr><b><tt id='l2h-151' xml:id='l2h-151' class="function">nlcp</tt></b>(</nobr></td>
- <td><var>kktsolver, F</var><big>[</big><var>, G, h</var><big>[</big><var>, A, b</var><big>]</big><var></var><big>]</big><var></var>)</td></tr></table></dt>
-<dd>
-Solves the nonlinear convex optimization problem (<A href="e-nlcp.html#e-nlcp">8.3</A>).
-
-<P>
-The meaning of the arguments <var>h</var> and <var>b</var> is the same
-as for <tt class="function">cp()</tt>.
-The arguments <var>kktsolver</var>, <var>F</var>, <var>G</var> and <var>A</var> are
-functions that must handle the following calling sequences.
-
-<UL>
-<LI><tt class="function">kktsolver</tt>(<var>x</var>, <var>z</var>, <var>dnl</var>, <var>dl</var>),
-returns a function for solving KKT systems
-<P></P>
-<DIV ALIGN="CENTER" CLASS="mathdisplay">
-<!-- MATH
- \begin{eqnarray*}
-\sum_{k=0}^m z_k \nabla^2 f_k(x)u_x + A^T u_y +
- D \tilde f(x)^T u_{z_\mathrm{nl}} +
- G_\mathrm{l}^T u_{z_\mathrm{l}} & = & b_x \\
- A x & = & b_y \\
- D\tilde f(x) x - \mbox{\bf diag}\,(d_\mathrm{nl})^{-2} z_\mathrm{nl} & = &
- b_{z_\mathrm{nl}} \\
- G_\mathrm{l}x - \mbox{\bf diag}\,(d_\mathrm{l})^{-2} z_\mathrm{l} & = &
- b_{z_\mathrm{l}}
-\end{eqnarray*}
- -->
-<IMG
- WIDTH="402" HEIGHT="125" BORDER="0"
- SRC="img156.gif"
- ALT="\begin{eqnarray*}
-\sum_{k=0}^m z_k \nabla^2 f_k(x)u_x + A^T u_y +
-D \tilde f(...
-...diag}\,(d_\mathrm{l})^{-2} z_\mathrm{l} & = &
-b_{z_\mathrm{l}}
-\end{eqnarray*}"></DIV>
-<BR CLEAR="ALL"><P></P>
-<BR CLEAR="ALL"><P></P>
-where <!-- MATH
- $\tilde f = (f_1, \ldots, f_m)$
- -->
-<SPAN CLASS="MATH"><IMG
- WIDTH="117" HEIGHT="38" ALIGN="MIDDLE" BORDER="0"
- SRC="img142.gif"
- ALT="$\tilde f = (f_1,\ldots, f_m)$"></SPAN>.
-The arguments are single-column real dense matrices. <var>x</var> is in the
-domain of the objective and constraint functions. <var>z</var>, <var>dnl</var>
-and <var>dl</var> are positive vectors.
-
-<P>
-The function <var>f</var> created by
-"<tt class="samp">f = kktsolver(bx, by, bznl, bzl)</tt>" will be
-called as "<tt class="samp">f(bx, by, bznl, bzl)</tt>".
-On entry, the arguments contain the righthand sides. On exit, they
-should be replaced by the solution.
-
-<P>
-</LI>
-<LI>Called with no arguments, <code><tt class="function">F</tt>()</code> returns a tuple
-(<var>m</var>, <var>x0</var>), where <var>m</var> is the number of nonlinear
-inequality constraints) and <var>x0</var> is a point in the domain
-of <I>f</I>).
-
-<P>
-Called with one argument, <tt class="function">F(<var>x</var>)</tt> returns a tuple
-(<var>f</var>, <var>Df</var>). <var>f</var> is a dense matrix of size (<I>m</I>+1,1)
-with the function values of the objective function and the
-nonlinear constraint functions at <var>x</var>.
-<var>Df</var> is a dense or sparse real matrix of size (<var>m</var>+1,<var>n</var>)
-with <code><var>Df</var>[<var>k</var>,:]</code> equal to the transpose of the gradient
-of <I>f_k</I> at <I>x</I>.
-Alternatively, <var>Df</var> can be given as a function. In that case
-the function call <tt class="function">Df</tt>(<var>u</var>,<var>v</var>),
-where <var>u</var> and <var>v</var> are dense column vectors, should evaluate
-<BR><P></P>
-<DIV ALIGN="CENTER" CLASS="mathdisplay">
-<!-- MATH
- \begin{displaymath}
-v := \sum_{k=0}^m u_k \nabla f_k(x) + v.
-\end{displaymath}
- -->
-
-<IMG
- WIDTH="157" HEIGHT="53" BORDER="0"
- SRC="img157.gif"
- ALT="\begin{displaymath}
-v := \sum_{k=0}^m u_k \nabla f_k(x) + v.
-\end{displaymath}">
-</DIV>
-<BR CLEAR="ALL">
-<P></P>
-
-<P>
-If <var>x</var> is not in the domain of <I>f</I>, <tt class="function">F</tt>(<var>x</var>)
-returns <code>None</code> or (<code>None</code>,<code>None</code>).
-
-<P>
-</LI>
-<LI><tt class="function">G</tt>(<var>x</var>, <var>y</var><big>[</big>,
-<var>alpha</var>=1.0<big>[</big>, <var>beta</var>=0.0<big>[</big>,
-<var>trans</var>=<code>'N'</code><big>]</big><big>]</big><big>]</big>) evaluates the matrix-vector products
-<BR><P></P>
-<DIV ALIGN="CENTER" CLASS="mathdisplay">
-<!-- MATH
- \begin{displaymath}
-y := \alpha G x + \beta y \quad (\mathrm{trans} = \mathrm{'N'}), \qquad
-y := \alpha G^T x + \beta y \quad (\mathrm{trans} = \mathrm{'T'}).
-\end{displaymath}
- -->
-
-<IMG
- WIDTH="471" HEIGHT="28" BORDER="0"
- SRC="img158.gif"
- ALT="\begin{displaymath}
-y := \alpha G x + \beta y \quad (\mathrm{trans} = \mathrm{'N...
-... \alpha G^T x + \beta y \quad (\mathrm{trans} = \mathrm{'T'}).
-\end{displaymath}">
-</DIV>
-<BR CLEAR="ALL">
-<P></P>
-Alternatively, <var>G</var> can be specified as a real sparse or dense
-matrix.
-
-<P>
-</LI>
-<LI><tt class="function">A</tt>(<var>x</var>, <var>y</var><big>[</big>,
-<var>alpha</var>=1.0<big>[</big>, <var>beta</var>=0.0<big>[</big>,
-<var>trans</var>=<code>'N'</code><big>]</big><big>]</big><big>]</big>) evaluates the matrix vector products
-<BR><P></P>
-<DIV ALIGN="CENTER" CLASS="mathdisplay">
-<!-- MATH
- \begin{displaymath}
-y := \alpha Ax + \beta y \quad
- (\mathrm{trans} = \mathrm{'N'}), \qquad
-y := \alpha A^Tx + \beta y \quad
- (\mathrm{trans} = \mathrm{'T'}).
-\end{displaymath}
- -->
-
-<IMG
- WIDTH="469" HEIGHT="28" BORDER="0"
- SRC="img152.gif"
- ALT="\begin{displaymath}
-y := \alpha Ax + \beta y \quad
-(\mathrm{trans} = \mathrm{'...
-...\alpha A^Tx + \beta y \quad
-(\mathrm{trans} = \mathrm{'T'}).
-\end{displaymath}">
-</DIV>
-<BR CLEAR="ALL">
-<P></P>
-Alternatively, <var>A</var> can be specified as a real sparse or dense
-matrix.
-</LI>
-</UL>
-</dl>
-
-<P>
-As an example, we consider the 1-norm regularized least-squares problem
-<BR><P></P>
-<DIV ALIGN="CENTER" CLASS="mathdisplay">
-<!-- MATH
- \begin{displaymath}
-\begin{array}{ll}
-\mbox{minimize} & \|Ax - y\|_2^2 + \|x\|_1
-\end{array}
-\end{displaymath}
- -->
-
-<IMG
- WIDTH="199" HEIGHT="30" BORDER="0"
- SRC="img159.gif"
- ALT="\begin{displaymath}
-\begin{array}{ll}
-\mbox{minimize} & \Vert Ax - y\Vert _2^2 + \Vert x\Vert _1
-\end{array}\end{displaymath}">
-</DIV>
-<BR CLEAR="ALL">
-<P></P>
-with variable <I>x</I>. The problem is equivalent to the quadratic
-program
-<BR><P></P>
-<DIV ALIGN="CENTER" CLASS="mathdisplay">
-<!-- MATH
- \begin{displaymath}
-\begin{array}{ll}
- \mbox{minimize} & \|Ax - y\|_2^2 + {\bf 1}^T u \\
- \mbox{subject to} & -u \preceq x \preceq u
- \end{array}
-\end{displaymath}
- -->
-
-<IMG
- WIDTH="197" HEIGHT="45" BORDER="0"
- SRC="img160.gif"
- ALT="\begin{displaymath}
-\begin{array}{ll}
-\mbox{minimize} & \Vert Ax - y\Vert _2^2...
-... u \\
-\mbox{subject to} & -u \preceq x \preceq u
-\end{array}\end{displaymath}">
-</DIV>
-<BR CLEAR="ALL">
-<P></P>
-with variables <I>x</I> and <I>u</I>. The implementation below is
-efficient when <I>A</I> has many more columns than rows.
-
-<P>
-<div class="verbatim"><pre>
-from cvxopt.base import matrix, spmatrix, mul, div
-from cvxopt import blas, lapack, solvers
-
-m, n = A.size
-def F(x=None):
- """
- Function and gradient evaluation of
-
- f = || A*x[:n] - y ||_2^2 + sum(x[n:])
- """
-
- nvars = 2*n
- if x is None: return 0, matrix(0.0, (nvars,1))
- r = A*x[:n] - y
- f = blas.nrm2(r)**2 + sum(x[n:])
- gradf = matrix(1.0, (1,2*n))
- blas.gemv(A, r, gradf, alpha=2.0, trans='T')
- return f, +gradf
-
-
-def G(u, v, alpha=1.0, beta=0.0, trans='N'):
- """
- v := alpha*[I, -I; -I, -I] * u + beta * v (trans = 'N' or 'T')
- """
-
- v *= beta
- v[:n] += alpha*(u[:n] - u[n:])
- v[n:] += alpha*(-u[:n] - u[n:])
-
-h = matrix(0.0, (2*n,1))
-
-
-# Customized solver for the KKT system
-#
-# [ 2.0*z[0]*A'*A 0 I -I ] [x[:n] ] [bx[:n] ]
-# [ 0 0 -I -I ] [x[n:] ] = [bx[n:] ].
-# [ I -I -D1^-1 0 ] [zl[:n]] [bzl[:n]]
-# [ -I -I 0 -D2^-1 ] [zl[n:]] [bzl[n:]]
-#
-#
-# We first eliminate zl and x[n:]:
-#
-# ( 2*z[0]*A'*A + 4*D1*D2*(D1+D2)^-1 ) * x[:n] = bx[:n] - (D2-D1)*(D1+D2)^-1 * bx[n:]
-# + D1 * ( I + (D2-D1)*(D1+D2)^-1 ) * bzl[:n] - D2 * ( I - (D2-D1)*(D1+D2)^-1 ) * bzl[n:]
-#
-# x[n:] = (D1+D2)^-1 * ( bx[n:] - D1*bzl[:n] - D2*bzl[n:] ) - (D2-D1)*(D1+D2)^-1 * x[:n]
-# zl[:n] = D1 * ( x[:n] - x[n:] - bzl[:n] )
-# zl[n:] = D2 * (-x[:n] - x[n:] - bzl[n:] ).
-#
-# The first equation has the form
-#
-# (z[0]*A'*A + D)*x[:n] = rhs
-#
-# and is equivalent to
-#
-# [ D A' ] [ x:n] ] = [ rhs ]
-# [ A -1/z[0]*I ] [ v ] [ 0 ].
-#
-# It can be solved as
-#
-# ( A*D^-1*A' + 1/z[0]*I ) * v = A * D^-1 * rhs
-# x[:n] = D^-1 * ( rhs - A'*v ).
-
-S = matrix(0.0, (m,m))
-Asc = matrix(0.0, (m,n))
-v = matrix(0.0, (m,1))
-def kktsolver(x, z, dnl, dl):
-
- # Factor
- #
- # S = A*D^-1*A' + 1/z[0]*I
- #
- # where D = 2*D1*D2*(D1+D2)^-1, D1 = dl[:n]**2, D2 = dl[n:]**2.
-
- d1, d2 = dl[:n]**2, dl[n:]**2 # d1 = diag(D1), d2 = diag(D2)
- # ds is square root of diagonal of D
- ds = sqrt(2.0) * div( mul(dl[:n], dl[n:]), sqrt(d1+d2) )
- d3 = div(d2 - d1, d1 + d2)
-
- # Asc = A*diag(d)^-1/2
- Asc = A * spmatrix( ds**-1, range(n), range(n))
-
- # S = 1/z[0]*I + A * D^-1 * A'
- blas.syrk(Asc, S)
- S[::m+1] += 1.0 / z[0]
- lapack.potrf(S)
-
- def g(x, y, znl, zl):
-
- x[:n] = 0.5 * ( x[:n] - mul(d3, x[n:]) + mul(d1, zl[:n] + mul(d3, zl[:n])) - mul(d2, zl[n:] - mul(d3, zl[n:])) )
- x[:n] = div( x[:n], ds)
-
- # Solve
- #
- # S * v = 0.5 * A * D^-1 * ( bx[:n] - (D2-D1)*(D1+D2)^-1 * bx[n:]
- # + D1 * ( I + (D2-D1)*(D1+D2)^-1 ) * bzl[:n] - D2 * ( I - (D2-D1)*(D1+D2)^-1 ) * bzl[n:] )
-
- blas.gemv(Asc, x, v)
- lapack.potrs(S, v)
-
- # x[:n] = D^-1 * ( rhs - A'*v ).
- blas.gemv(Asc, v, x, alpha=-1.0, beta=1.0, trans='T')
- x[:n] = div(x[:n], ds)
-
- # x[n:] = (D1+D2)^-1 * ( bx[n:] - D1*bzl[:n] - D2*bzl[n:] ) - (D2-D1)*(D1+D2)^-1 * x[:n]
- x[n:] = div( x[n:] - mul(d1, zl[:n]) - mul(d2, zl[n:]), d1+d2 ) - mul( d3, x[:n] )
-
- # zl[:n] = D1 * ( x[:n] - x[n:] - bzl[:n] )
- # zl[n:] = D2 * ( -x[:n] - x[n:] - bzl[n:] ).
- zl[:n] = mul( d1, x[:n] - x[n:] - zl[:n] )
- zl[n:] = mul( d2, -x[:n] - x[n:] - zl[n:] )
-
- return g
-
-x = solvers.nlcp(kktsolver, F, G, h)['x'][:n]
-</pre></div>
-
-<P>
-
-<DIV CLASS="navigation">
-<div class='online-navigation'>
-<p></p><hr />
-<table align="center" width="100%" cellpadding="0" cellspacing="2">
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+++ /dev/null
@@ -1,127 +0,0 @@
-<!DOCTYPE html PUBLIC "-//W3C//DTD HTML 4.0 Transitional//EN">
-<html>
-<head>
-<link rel="STYLESHEET" href="cvxopt.css" type='text/css' />
-<link rel="first" href="cvxopt.html" title='CVXOPT: A Python Package for Convex Optimization' />
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-</DIV>
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-
-<H1><A NAME="SECTION0011000000000000000000"></A>
-<A NAME="chap:modeling"></A>
-<BR>
-9. Modeling (<tt class="module">cvxopt.modeling</tt>)
-</H1>
-The module <tt class="module">cvxopt.modeling</tt> can be used to specify and solve
-optimization problems with convex piecewise-linear objective and
-constraint functions.
-
-<P>
-To specify an optimization problem one first defines
-the optimization variables (see section <A href="s-variables.html#s-variables">9.1</A>),
-and then defines the objective and constraint functions
-using linear operations (vector addition and subtraction,
-matrix-vector multiplication, indexing and slicing)
-and nested evaluations of <tt class="function">max()</tt>, <tt class="function">min()</tt>,
-<tt class="function">abs()</tt> and <tt class="function">sum()</tt> (see section <A href="s-functions.html#s-functions">9.2</A>).
-
-<P>
-
-<p><br /></p><hr class='online-navigation' />
-<div class='online-navigation'>
-<!--Table of Child-Links-->
-<A NAME="CHILD_LINKS"><STRONG>Subsections</STRONG></a>
-
-<UL CLASS="ChildLinks">
-<LI><A href="s-variables.html">9.1 Variables</a>
-<LI><A href="s-functions.html">9.2 Functions</a>
-<LI><A href="node58.html">9.3 Constraints</a>
-<LI><A href="s-lp.html">9.4 Optimization Problems</a>
-<LI><A href="node60.html">9.5 Examples</a>
-</ul>
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-</div>
-
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-<div class='online-navigation'>
-<p></p><hr />
-<table align="center" width="100%" cellpadding="0" cellspacing="2">
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-<!DOCTYPE html PUBLIC "-//W3C//DTD HTML 4.0 Transitional//EN">
-<html>
-<head>
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-
-<H1><A NAME="SECTION0011300000000000000000">
-9.3 Constraints</A>
-</H1>
-Linear equality and inequality constraints of the form
-<BR><P></P>
-<DIV ALIGN="CENTER" CLASS="mathdisplay">
-<!-- MATH
- \begin{displaymath}
-f(x_1,\ldots,x_n) = 0, \qquad f(x_1,\ldots,x_n) \preceq 0,
-\end{displaymath}
- -->
-
-<IMG
- WIDTH="283" HEIGHT="28" BORDER="0"
- SRC="img174.gif"
- ALT="\begin{displaymath}
-f(x_1,\ldots,x_n) = 0, \qquad f(x_1,\ldots,x_n) \preceq 0,
-\end{displaymath}">
-</DIV>
-<BR CLEAR="ALL">
-<P></P>
-where <SPAN CLASS="MATH"><IMG
- WIDTH="13" HEIGHT="30" ALIGN="MIDDLE" BORDER="0"
- SRC="img175.gif"
- ALT="$f$"></SPAN> is a convex function, are represented by constraint
-objects. Equality constraints are created by expressions of the form
-<BLOCKQUOTE>
-<code><var>f1</var> == <var>f2</var></code>.
-
-</BLOCKQUOTE>
-Here <var>f1</var> and <var>f2</var> can be any objects for which the
-difference <code><var>f1</var>-<var>f2</var></code> yields an affine function.
-Inequality constraints are created by expressions of the form
-<BLOCKQUOTE>
-<code><var>f1</var> <= <var>f2</var></code>, <code><var>f2</var> >= <var>f1</var></code>,
-
-</BLOCKQUOTE>
-where <var>f1</var> and <var>f2</var> can be any objects for which the difference
-<code><var>f1</var>-<var>f2</var></code> yields a convex piecewise-linear function.
-The comparison operators first convert the expressions to
-<code><var>f1</var>-<var>f2</var> == 0</code>, resp. <code><var>f1</var>-<var>f2</var> <= 0</code>,
-and then return a new constraint object with constraint
-function
-<code><var>f1</var>-<var>f2</var></code>.
-
-<P>
-In the following example we create three constraints
-<BR><P></P>
-<DIV ALIGN="CENTER" CLASS="mathdisplay">
-<!-- MATH
- \begin{displaymath}
-0 \preceq x \preceq {\bf 1}, \qquad {\bf 1}^T x = 2,
-\end{displaymath}
- -->
-
-<IMG
- WIDTH="168" HEIGHT="27" BORDER="0"
- SRC="img176.gif"
- ALT="\begin{displaymath}
-0 \preceq x \preceq {\bf 1}, \qquad {\bf 1}^T x = 2,
-\end{displaymath}">
-</DIV>
-<BR CLEAR="ALL">
-<P></P>
-for a variable of length 5.
-<div class="verbatim"><pre>
->>> x = variable(5,'x')
->>> c1 = (x <= 1)
->>> c2 = (x >= 0)
->>> c3 = (sum(x) == 2)
-</pre></div>
-
-<P>
-The built-in fucntion <tt class="function">len()</tt> returns the dimension of the
-constraint function.
-
-<P>
-Constraints have four public attributes.
-<dl><dt><table cellpadding="0" cellspacing="0"><tr valign="baseline">
- <td><nobr><b><tt id='l2h-159' xml:id='l2h-159' class="method">type</tt></b>(</nobr></td>
- <td><var></var>)</td></tr></table></dt>
-<dd>
-Returns <code>'='</code> if the constraint is an equality constraint,
-and <code>'<'</code> if the constraint is an inequality constraint.
-</dl>
-
-<P>
-<dl><dt><table cellpadding="0" cellspacing="0"><tr valign="baseline">
- <td><nobr><b><tt id='l2h-160' xml:id='l2h-160' class="method">value</tt></b>(</nobr></td>
- <td><var></var>)</td></tr></table></dt>
-<dd>
-Returns the value of the constraint function.
-</dl>
-
-<P>
-<dl><dt><b><tt id='l2h-161' xml:id='l2h-161' class="member">multiplier</tt></b></dt>
-<dd>
-For a constraint <var>c</var>, <var>c</var>.<tt class="member">multiplier</tt> is a
-variable object of dimension <code>len(<var>c</var>)</code>.
-It is used to represent the Lagrange multiplier or dual variable
-associated with the constraint.
-Its value is initialized as <code>None</code>, and can be modified
-by making an assignment to <var>c</var>.<tt class="member">multiplier</tt>.<tt class="member">value</tt>.
-</dl>
-
-<P>
-<dl><dt><b><tt id='l2h-162' xml:id='l2h-162' class="member">name</tt></b></dt>
-<dd>
-The name of the constraint. Changing the name of a constraint
-also changes the name of the multiplier of <var>c</var>.
-For example, the command <code><var>c</var>.<tt class="member">name</tt> = 'newname'</code> also
-changes
-<var>c</var>.<tt class="member">multiplier</tt>.<tt class="member">name</tt> to <code>'newname_mul'</code>.
-</dl>
-
-<P>
-
-<DIV CLASS="navigation">
-<div class='online-navigation'>
-<p></p><hr />
-<table align="center" width="100%" cellpadding="0" cellspacing="2">
-<tr>
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- href="s-functions.html"><img src='previous.gif'
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-<td class='online-navigation'><a rel="parent" title="9. Modeling (cvxopt.modeling)"
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-<td class='online-navigation'><a rel="next" title="9.4 Optimization Problems"
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-<td align="center" width="100%">CVXOPT: A Python Package for Convex Optimization</td>
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diff --git a/doc/cvxopt/node6.html b/doc/cvxopt/node6.html
deleted file mode 100644
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--- a/doc/cvxopt/node6.html
+++ /dev/null
@@ -1,200 +0,0 @@
-<!DOCTYPE html PUBLIC "-//W3C//DTD HTML 4.0 Transitional//EN">
-<html>
-<head>
-<link rel="STYLESHEET" href="cvxopt.css" type='text/css' />
-<link rel="first" href="cvxopt.html" title='CVXOPT: A Python Package for Convex Optimization' />
-<link rel='contents' href='contents.html' title="Contents" />
-<link rel='index' href='genindex.html' title='Index' />
-<link rel='last' href='about.html' title='About this document...' />
-<link rel='help' href='about.html' title='About this document...' />
-<link rel="next" href="s-arithmetic.html" />
-<link rel="prev" href="s-creating-matrices.html" />
-<link rel="parent" href="node4.html" />
-<link rel="next" href="s-arithmetic.html" />
-<meta name='aesop' content='information' />
-<title>2.2 Attributes and Methods</title>
-</head>
-<body>
-<DIV CLASS="navigation">
-<div id='top-navigation-panel' xml:id='top-navigation-panel'>
-<table align="center" width="100%" cellpadding="0" cellspacing="2">
-<tr>
-<td class='online-navigation'><a rel="prev" title="2.1 Creating Matrices"
- href="s-creating-matrices.html"><img src='previous.gif'
- border='0' height='32' alt='Previous Page' width='32' /></A></td>
-<td class='online-navigation'><a rel="parent" title="2. Dense Matrices (cvxopt.base)"
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-<td class='online-navigation'><a rel="next" title="2.3 Arithmetic Operations"
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-<td align="center" width="100%">CVXOPT: A Python Package for Convex Optimization</td>
-<td class='online-navigation'><a rel="contents" title="Table of Contents"
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-</div>
-<hr /></div>
-</DIV>
-<!--End of Navigation Panel-->
-
-<H1><A NAME="SECTION004200000000000000000">
-2.2 Attributes and Methods</A>
-</H1>
-A <tt class="class">matrix</tt> has the following attributes.
-
-<P>
-<dl><dt><b><tt id='l2h-2' xml:id='l2h-2' class="member">size</tt></b></dt>
-<dd>
-A tuple with the dimensions of the matrix. This is a read-only
-attribute; operations that change the size of a matrix are not
-permitted.
-</dl>
-
-<P>
-<dl><dt><b><tt id='l2h-3' xml:id='l2h-3' class="member">typecode</tt></b></dt>
-<dd>
-A <tt class="ctype">char</tt>, either <code>'i'</code>, <code>'d'</code>, or <code>'z'</code>, for integer, real
-and complex matrices, respectively. A read-only attribute.
-</dl>
-
-<P>
-<dl><dt><table cellpadding="0" cellspacing="0"><tr valign="baseline">
- <td><nobr><b><tt id='l2h-4' xml:id='l2h-4' class="method">trans</tt></b>(</nobr></td>
- <td><var></var>)</td></tr></table></dt>
-<dd>
-Returns the transpose of the matrix as a new matrix.
-One can also use <code>A.T</code> instead of <code>A.trans()</code>.
-</dl>
-
-<P>
-<dl><dt><table cellpadding="0" cellspacing="0"><tr valign="baseline">
- <td><nobr><b><tt id='l2h-5' xml:id='l2h-5' class="method">ctrans</tt></b>(</nobr></td>
- <td><var></var>)</td></tr></table></dt>
-<dd>
-Returns the conjugate transpose of the matrix as a new matrix.
-One can also use <code>A.H</code> instead of <code>A.ctrans()</code>.
-</dl>
-
-<P>
-<dl><dt><table cellpadding="0" cellspacing="0"><tr valign="baseline">
- <td><nobr><b><tt id='l2h-6' xml:id='l2h-6' class="method">real</tt></b>(</nobr></td>
- <td><var></var>)</td></tr></table></dt>
-<dd>
-For complex matrices, returns the real part as a real matrix.
-For integer and real matrices, returns a copy of the matrix.
-</dl>
-
-<P>
-<dl><dt><table cellpadding="0" cellspacing="0"><tr valign="baseline">
- <td><nobr><b><tt id='l2h-7' xml:id='l2h-7' class="method">imag</tt></b>(</nobr></td>
- <td><var></var>)</td></tr></table></dt>
-<dd>
-For complex matrices, returns the imaginary part as a real matrix.
-For integer and real matrices, returns an integer or real zero matrix.
-</dl>
-
-<P>
-<dl><dt><b><tt id='l2h-8' xml:id='l2h-8' class="member">__array_struct__</tt></b></dt>
-<dd>
-A PyCObject implementing the NumPy array interface
-(see section <A href="s-array-interface.html#s-array-interface">2.8</A> for details).
-</dl>
-
-<P>
-<dl><dt><table cellpadding="0" cellspacing="0"><tr valign="baseline">
- <td><nobr><b><tt id='l2h-9' xml:id='l2h-9' class="method">tofile</tt></b>(</nobr></td>
- <td><var>f</var>)</td></tr></table></dt>
-<dd>
-Writes the elements of the matrix in column-major order to a binary
-file <var>f</var>.
-</dl>
-
-<P>
-<dl><dt><table cellpadding="0" cellspacing="0"><tr valign="baseline">
- <td><nobr><b><tt id='l2h-10' xml:id='l2h-10' class="method">fromfile</tt></b>(</nobr></td>
- <td><var>f</var>)</td></tr></table></dt>
-<dd>
-Reads the contents of a binary file <var>f</var> into the matrix object.
-</dl>
-
-<P>
-The last two methods are illustrated in the following example.
-<div class="verbatim"><pre>
->>> from cvxopt.base import matrix
->>> A = matrix([[1.,2.,3.], [4.,5.,6.]])
->>> print A
- 1.0000e+00 4.0000e+00
- 2.0000e+00 5.0000e+00
- 3.0000e+00 6.0000e+00
->>> f = open('mat.bin','w')
->>> A.tofile(f)
->>> f.close()
->>> B = matrix(0.0, (2,3))
->>> f = open('mat.bin','r')
->>> B.fromfile(f)
->>> f.close()
->>> print B
- 1.0000e+00 3.0000e+00 5.0000e+00
- 2.0000e+00 4.0000e+00 6.0000e+00
-</pre></div>
-
-<P>
-Matrices can also be written to or read from files using the
-<tt class="function">dump()</tt> and <tt class="function">load()</tt> functions in the
-<tt class="module">pickle</tt> module.
-
-<P>
-
-<DIV CLASS="navigation">
-<div class='online-navigation'>
-<p></p><hr />
-<table align="center" width="100%" cellpadding="0" cellspacing="2">
-<tr>
-<td class='online-navigation'><a rel="prev" title="2.1 Creating Matrices"
- href="s-creating-matrices.html"><img src='previous.gif'
- border='0' height='32' alt='Previous Page' width='32' /></A></td>
-<td class='online-navigation'><a rel="parent" title="2. Dense Matrices (cvxopt.base)"
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-<td align="center" width="100%">CVXOPT: A Python Package for Convex Optimization</td>
-<td class='online-navigation'><a rel="contents" title="Table of Contents"
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-</DIV>
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--- a/doc/cvxopt/node60.html
+++ /dev/null
@@ -1,358 +0,0 @@
-<!DOCTYPE html PUBLIC "-//W3C//DTD HTML 4.0 Transitional//EN">
-<html>
-<head>
-<link rel="STYLESHEET" href="cvxopt.css" type='text/css' />
-<link rel="first" href="cvxopt.html" title='CVXOPT: A Python Package for Convex Optimization' />
-<link rel='contents' href='contents.html' title="Contents" />
-<link rel='index' href='genindex.html' title='Index' />
-<link rel='last' href='about.html' title='About this document...' />
-<link rel='help' href='about.html' title='About this document...' />
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-<link rel="parent" href="node55.html" />
-<link rel="next" href="node61.html" />
-<meta name='aesop' content='information' />
-<title>9.5 Examples</title>
-</head>
-<body>
-<DIV CLASS="navigation">
-<div id='top-navigation-panel' xml:id='top-navigation-panel'>
-<table align="center" width="100%" cellpadding="0" cellspacing="2">
-<tr>
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- href="s-lp.html"><img src='previous.gif'
- border='0' height='32' alt='Previous Page' width='32' /></A></td>
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- href="node55.html"><img src='up.gif'
- border='0' height='32' alt='Up One Level' width='32' /></A></td>
-<td class='online-navigation'><a rel="next" title="10. C API"
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-<td align="center" width="100%">CVXOPT: A Python Package for Convex Optimization</td>
-<td class='online-navigation'><a rel="contents" title="Table of Contents"
- href="contents.html"><img src='contents.gif'
- border='0' height='32' alt='Contents' width='32' /></A></td>
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-</div>
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-
-<H1><A NAME="SECTION0011500000000000000000">
-9.5 Examples</A>
-</H1>
-
-<P>
-<DL>
-<DT><STRONG>Norm and Penalty Approximation</STRONG></DT>
-<DD><P>
-In the first example we solve the norm approximation problems
-<BR><P></P>
-<DIV ALIGN="CENTER" CLASS="mathdisplay">
-<!-- MATH
- \begin{displaymath}
-\begin{array}{ll}
- \mbox{minimize} & \|Ax - b\|_\infty,
- \end{array} \qquad
- \begin{array}{ll}
- \mbox{minimize} & \|Ax - b\|_1
- \end{array},
-\end{displaymath}
- -->
-
-<IMG
- WIDTH="364" HEIGHT="30" BORDER="0"
- SRC="img178.gif"
- ALT="\begin{displaymath}
-\begin{array}{ll}
-\mbox{minimize} & \Vert Ax - b\Vert _\i...
-...ay}{ll}
-\mbox{minimize} & \Vert Ax - b\Vert _1
-\end{array},
-\end{displaymath}">
-</DIV>
-<BR CLEAR="ALL">
-<P></P>
-and the penalty approximation problem
-<BR><P></P>
-<DIV ALIGN="CENTER" CLASS="mathdisplay">
-<!-- MATH
- \begin{displaymath}
-\begin{array}{ll}
- \mbox{minimize} & \sum_k \phi((Ax-b)_k),
- \end{array} \qquad
- \phi(u) = \left\{\begin{array}{ll}
- 0 & |u| \leq 3/4 \\
- |u|-3/4 & 3/4 \leq |u| \leq 3/2 \\
- 2|u|-9/4 & |u| \geq 3/2.\end{array}\right.
-\end{displaymath}
- -->
-
-<IMG
- WIDTH="503" HEIGHT="64" BORDER="0"
- SRC="img179.gif"
- ALT="\begin{displaymath}
-\begin{array}{ll}
-\mbox{minimize} & \sum_k \phi((Ax-b)_k)...
-...
-2\vert u\vert-9/4 & \vert u\vert \geq 3/2.\end{array}\right.
-\end{displaymath}">
-</DIV>
-<BR CLEAR="ALL">
-<P></P>
-We use randomly generated data.
-
-<P>
-The code uses the <a class="ulink" href="http://matplotlib.sourceforge.net"
- >Matplotlib</a>
-package for plotting the histograms of the residual vectors for the
-two solutions. It generates the figure shown below.
-
-<P>
-<div class="verbatim"><pre>
-from cvxopt.random import normal
-from cvxopt.modeling import variable, op, max, sum
-import pylab
-
-m, n = 500, 100
-A = normal(m,n)
-b = normal(m)
-
-x1 = variable(n)
-op(max(abs(A*x1-b))).solve()
-
-x2 = variable(n)
-op(sum(abs(A*x2-b))).solve()
-
-x3 = variable(n)
-op(sum(max(0, abs(A*x3-b)-0.75, 2*abs(A*x3-b)-2.25))).solve()
-
-pylab.subplot(311)
-pylab.hist(A*x1.value-b, m/5)
-pylab.subplot(312)
-pylab.hist(A*x2.value-b, m/5)
-pylab.subplot(313)
-pylab.hist(A*x3.value-b, m/5)
-pylab.show()
-</pre></div>
-
-<P>
-<DIV ALIGN="CENTER">
-<IMG
- WIDTH="556" HEIGHT="418" ALIGN="BOTTOM" BORDER="0"
- SRC="img180.gif"
- ALT="\includegraphics[width=\linewidth]{figures/normappr.eps}">
-
-</DIV>
-
-<P>
-Equivalently, we can formulate and solve the problems as LPs.
-<div class="verbatim"><pre>
-t = variable()
-x1 = variable(n)
-op(t, [-t <= A*x1-b, A*x1-b<=t]).solve()
-
-u = variable(m)
-x2 = variable(n)
-op(sum(u), [-u <= A*x2+b, A*x2+b <= u]).solve()
-
-v = variable(m)
-x3 = variable(n)
-op(sum(v), [v >= 0, v >= A*x3+b-0.75, v >= -(A*x3+b)-0.75, v >= 2*(A*x3-b)-2.25, v >= -2*(A*x3-b)-2.25]).solve()
-</pre></div>
-
-<P>
-</DD>
-<DT><STRONG>Robust Linear Programming</STRONG></DT>
-<DD>The robust LP
-<BR><P></P>
-<DIV ALIGN="CENTER" CLASS="mathdisplay">
-<!-- MATH
- \begin{displaymath}
-\begin{array}{ll}
- \mbox{minimize} & c^T x \\
- \mbox{subject to} & \sup_{\|v\|_\infty \leq 1}
- (a_i+v)^T x \leq b_i, \qquad i=1,\ldots,m
- \end{array}
-\end{displaymath}
- -->
-
-<IMG
- WIDTH="379" HEIGHT="46" BORDER="0"
- SRC="img181.gif"
- ALT="\begin{displaymath}
-\begin{array}{ll}
-\mbox{minimize} & c^T x \\
-\mbox{subje...
-...leq 1}
-(a_i+v)^T x \leq b_i, \qquad i=1,\ldots,m
-\end{array}\end{displaymath}">
-</DIV>
-<BR CLEAR="ALL">
-<P></P>
-is equivalent to the problem
-<BR><P></P>
-<DIV ALIGN="CENTER" CLASS="mathdisplay">
-<!-- MATH
- \begin{displaymath}
-\begin{array}{ll}
- \mbox{minimize} & c^Tx \\
- \mbox{subject to} & a_i^Tx + \|x\|_1 \leq b_i, \qquad i=1,\ldots,m.
- \end{array}
-\end{displaymath}
- -->
-
-<IMG
- WIDTH="315" HEIGHT="45" BORDER="0"
- SRC="img182.gif"
- ALT="\begin{displaymath}
-\begin{array}{ll}
-\mbox{minimize} & c^Tx \\
-\mbox{subjec...
-...x + \Vert x\Vert _1 \leq b_i, \qquad i=1,\ldots,m.
-\end{array}\end{displaymath}">
-</DIV>
-<BR CLEAR="ALL">
-<P></P>
-The following code computes the solution and the solution of
-the equivalent LP
-<BR><P></P>
-<DIV ALIGN="CENTER" CLASS="mathdisplay">
-<!-- MATH
- \begin{displaymath}
-\begin{array}{ll}
- \mbox{minimize} & c^Tx \\
- \mbox{subject to} & a_i^Tx + {\bf 1}^Ty \leq b_i, \qquad i=1,\ldots,m \\
-& -y \preceq x \preceq y
-\end{array}
-\end{displaymath}
- -->
-
-<IMG
- WIDTH="308" HEIGHT="64" BORDER="0"
- SRC="img183.gif"
- ALT="\begin{displaymath}
-\begin{array}{ll}
-\mbox{minimize} & c^Tx \\
-\mbox{subjec...
-...i, \qquad i=1,\ldots,m \\
-& -y \preceq x \preceq y
-\end{array}\end{displaymath}">
-</DIV>
-<BR CLEAR="ALL">
-<P></P>
-for randomly generated data.
-
-<P>
-<div class="verbatim"><pre>
-from cvxopt.random import normal, uniform
-from cvxopt.modeling import variable, dot, op, sum
-from cvxopt.blas import nrm2
-
-m, n = 500, 100
-A = normal(m,n)
-b = uniform(m)
-c = normal(n)
-
-x = variable(n)
-op(dot(c,x), A*x+sum(abs(x)) <= b).solve()
-
-x2 = variable(n)
-y = variable(n)
-op(dot(c,x2), [A*x2+sum(y) <= b, -y <= x2, x2 <= y]).solve()
-</pre></div>
-
-<P>
-</DD>
-<DT><STRONG>1-Norm Support Vector Classifier</STRONG></DT>
-<DD><P>
-The following problem arises in classification:
-<BR><P></P>
-<DIV ALIGN="CENTER" CLASS="mathdisplay">
-<!-- MATH
- \begin{displaymath}
-\begin{array}{ll}
-\mbox{minimize} & \|x\|_1 + {\bf 1}^Tu \\
-\mbox{subject to} & Ax \succeq {\bf 1}-u \\
-& u \succeq 0.
-\end{array}
-\end{displaymath}
- -->
-
-<IMG
- WIDTH="157" HEIGHT="64" BORDER="0"
- SRC="img184.gif"
- ALT="\begin{displaymath}
-\begin{array}{ll}
-\mbox{minimize} & \Vert x\Vert _1 + {\bf 1...
-...bject to} & Ax \succeq {\bf 1}-u \\
-& u \succeq 0.
-\end{array}\end{displaymath}">
-</DIV>
-<BR CLEAR="ALL">
-<P></P>
-It can be solved as follows.
-<div class="verbatim"><pre>
-x = variable(A.size[1],'x')
-u = variable(A.size[0],'u')
-op(sum(abs(x)) + sum(u), [A*x >= 1-u, u >= 0]).solve()
-</pre></div>
-An equivalent unconstrained formulation is
-<div class="verbatim"><pre>
-x = variable(A.size[1],'x')
-op(sum(abs(x)) + sum(max(0,1-A*x))).solve()
-</pre></div>
-</DD>
-</DL>
-
-<DIV CLASS="navigation">
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-<p></p><hr />
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--- a/doc/cvxopt/node61.html
+++ /dev/null
@@ -1,135 +0,0 @@
-<!DOCTYPE html PUBLIC "-//W3C//DTD HTML 4.0 Transitional//EN">
-<html>
-<head>
-<link rel="STYLESHEET" href="cvxopt.css" type='text/css' />
-<link rel="first" href="cvxopt.html" title='CVXOPT: A Python Package for Convex Optimization' />
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-<link rel='index' href='genindex.html' title='Index' />
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-</head>
-<body>
-<DIV CLASS="navigation">
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-<table align="center" width="100%" cellpadding="0" cellspacing="2">
-<tr>
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-<hr /></div>
-</DIV>
-<!--End of Navigation Panel-->
-
-<H1><A NAME="SECTION0012000000000000000000"></A><A NAME="chap:c-api"></A>
-<BR>
-10. C API
-</H1>
-The API can be used to extend CVXOPT with interfaces to
-external C routines and libraries.
-A C program that creates or manipulates the dense or sparse matrix
-objects defined in <tt class="module">cvxopt.base</tt> must include the
-<span class="file">cvxopt.h</span> header file in the <span class="file">src</span> directory of the
-distribution.
-
-<P>
-Before the C-API can be used in an extension module it must be
-initialized by calling the macro <tt class="cfunction">import_cvxopt</tt>.
-As an example
-we show the module initialization from the <tt class="module">cvxopt.blas</tt> module,
-which itself uses the API:
-<div class="verbatim"><pre>
-PyMODINIT_FUNC initblas(void)
-{
- PyObject *m;
-
- m = Py_InitModule3("cvxopt.blas", blas_functions, blas__doc__);
-
- if (import_cvxopt() < 0)
- return;
-}
-</pre></div>
-
-<P>
-
-<p><br /></p><hr class='online-navigation' />
-<div class='online-navigation'>
-<!--Table of Child-Links-->
-<A NAME="CHILD_LINKS"><STRONG>Subsections</STRONG></a>
-
-<UL CLASS="ChildLinks">
-<LI><A href="node62.html">10.1 Dense Matrices</a>
-<LI><A href="node63.html">10.2 Sparse Matrices</a>
-</ul>
-<!--End of Table of Child-Links-->
-</div>
-
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-<p></p><hr />
-<table align="center" width="100%" cellpadding="0" cellspacing="2">
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+++ /dev/null
@@ -1,162 +0,0 @@
-<!DOCTYPE html PUBLIC "-//W3C//DTD HTML 4.0 Transitional//EN">
-<html>
-<head>
-<link rel="STYLESHEET" href="cvxopt.css" type='text/css' />
-<link rel="first" href="cvxopt.html" title='CVXOPT: A Python Package for Convex Optimization' />
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-
-<H1><A NAME="SECTION0012100000000000000000">
-10.1 Dense Matrices</A>
-</H1>
-As can be seen from the header file <span class="file">cvxopt.h</span>, a <tt class="class">matrix</tt> is
-essentially a structure with four fields.
-The fields <tt class="cdata">nrows</tt> and <tt class="cdata">ncols</tt> are two integers that
-specify the dimensions.
-The <tt class="cdata">id</tt> field controls the type of the matrix and can have
-values <tt class="constant">DOUBLE</tt>, <tt class="constant">INT</tt> and <tt class="constant">COMPLEX</tt>.
-The <tt class="cdata">buffer</tt> field is an array that contains the matrix elements
-stored contiguously in column-major order.
-
-<P>
-The following C functions can be used to create matrices.
-<dl><dt><table cellpadding="0" cellspacing="0"><tr valign="baseline"><td><nobr>matrix* <b><tt id='l2h-174' xml:id='l2h-174' class="cfunction">Matrix_New</tt></b>(</nobr></td><td>int <var>nrows</var>, int <var>ncols</var>, int <var>id</var>)</td></tr></table></dt>
-<dd>
-Returns a <tt class="class">matrix</tt> object of type <var>id</var> with <var>nrows</var> rows
-and <var>ncols</var> columns. The elements of the matrix are uninitialized.
-</dd></dl>
-
-<P>
-<dl><dt><table cellpadding="0" cellspacing="0"><tr valign="baseline"><td><nobr>matrix* <b><tt id='l2h-175' xml:id='l2h-175' class="cfunction">Matrix_NewFromMatrix</tt></b>(</nobr></td><td>matrix *<var>src</var>, int <var>id</var>)</td></tr></table></dt>
-<dd>
-Returns a copy of the matrix <var>src</var> converted to type <var>id</var>.
-The following type conversions are allowed: <code>'i'</code> to <code>'d'</code>,
-<code>'i'</code> to <code>'z'</code> and <code>'d'</code> to <code>'z'</code>.
-</dd></dl>
-
-<P>
-<dl><dt><table cellpadding="0" cellspacing="0"><tr valign="baseline"><td><nobr>matrix* <b><tt id='l2h-176' xml:id='l2h-176' class="cfunction">Matrix_NewFromSequence</tt></b>(</nobr></td><td>PyListObject *<var>x</var>, int <var>id</var>)</td></tr></table></dt>
-<dd>
-Creates a matrix of type <var>id</var> from the Python sequence type <var>x</var>. The
-returned matrix has size <code>(len(<var>x</var>),1)</code>.
-The size can be changed by modifying the <tt class="member">nrows</tt> and
-<tt class="member">ncols</tt> fields of the returned matrix.
-</dd></dl>
-
-<P>
-To illustrate the creation and manipulation of dense matrices (as well
-as the Python C API), we show the code for the <tt class="function">uniform()</tt>
-function from <tt class="module">cvxopt.random</tt> described in
-section <A href="s-random.html#s-random">2.7</A>.
-<div class="verbatim"><pre>
-PyObject * uniform(PyObject *self, PyObject *args, PyObject *kwrds)
-{
- matrix *obj;
- int i, nrows, ncols = 1;
- double a = 0, b = 1;
- char *kwlist[] = {"nrows", "ncols", "a", "b", NULL};
-
- if (!PyArg_ParseTupleAndKeywords(args, kwrds, "i|idd", kwlist,
- &nrows, &ncols, &a, &b)) return NULL;
-
- if ((nrows<0) || (ncols<0)) {
- PyErr_SetString(PyExc_TypeError, "dimensions must be non-negative");
- return NULL;
- }
-
- if (!(obj = Matrix_New(nrows, ncols, DOUBLE)))
- return PyErr_NoMemory();
-
- for (i = 0; i < nrows*ncols; i++)
- MAT_BUFD(obj)[i] = Uniform(a,b);
-
- return (PyObject *)obj;
-}
-</pre></div>
-
-<P>
-
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-<H1><A NAME="SECTION0012200000000000000000">
-10.2 Sparse Matrices</A>
-</H1>
-Sparse matrices are stored in compressed column storage (CCS)
-format. For a general <var>nrows</var> by <var>ncols</var> sparse matrix with
-<var>nnz</var> nonzero entries this means the following. The sparsity
-pattern and the nonzero values are stored in three fields:
-<DL>
-<DT><STRONG>values:</STRONG></DT>
-<DD>A <code>'d'</code> or <code>'z'</code> matrix of size <code>(<var>nnz</var>,1)</code>
- with the nonzero entries of the matrix stored columnwise.
-</DD>
-<DT><STRONG>rowind:</STRONG></DT>
-<DD>An array of integers of length <var>nnz</var> containing the
-row indices of the nonzero entries, stored in the same order as
-<var>values</var>.
-</DD>
-<DT><STRONG>colptr:</STRONG></DT>
-<DD>An array of integers of length <code><var>ncols</var>+1</code> with
-for each column of the matrix the index of the first element in
-<var>values</var> from that column. More precisely,
-<code><var>colptr</var>[0]</code> is <code>0</code>, and for
-<code><var>k</var> = 0, 1, ..., <var>ncols</var>-1</code>,
-<code><var>colptr</var>[k+1]</code> is equal to <code><var>colptr</var>[k]</code> plus the
-number of nonzeros in column <var>k</var> of the matrix.
-Thus, <code><var>colptr</var>[<var>ncols</var>]</code> is equal to <var>nnz</var>, the
-number of nonzero entries.
-</DD>
-</DL>
-
-<P>
-For example, for the matrix
-<BR><P></P>
-<DIV ALIGN="CENTER" CLASS="mathdisplay">
-<!-- MATH
- \begin{displaymath}
-A=\left [\begin{array}{cccc}
- 1 & 0 & 0 & 5\\
- 2 & 0 & 4 & 0\\
- 0 & 0 & 0 & 6\\
- 3 & 0 & 0 & 0
-\end{array}\right]
-\end{displaymath}
- -->
-
-<IMG
- WIDTH="144" HEIGHT="83" BORDER="0"
- SRC="img185.gif"
- ALT="\begin{displaymath}
-A=\left [\begin{array}{cccc}
-1 & 0 & 0 & 5\\
-2 & 0 & 4 & 0\\
-0 & 0 & 0 & 6\\
-3 & 0 & 0 & 0
-\end{array}\right]
-\end{displaymath}">
-</DIV>
-<BR CLEAR="ALL">
-<P></P>
-the elements of <var>values</var>, <var>rowind</var> and <var>colptr</var> are:
-<BLOCKQUOTE>
-<var>values</var>: 1.0, 2.0, 3.0, 4.0, 5.0, 6.0,
- <var>rowind</var>: 0, 1,3, 1, 0, 2,
- <var>colptr</var>: 0, 3, 3, 4, 6.
-
-</BLOCKQUOTE>
-It is crucial that for each column the row indices in <var>rowind</var> are
-sorted; the equivalent representation
-<BLOCKQUOTE>
-<var>values</var>: 3.0, 2.0, 1.0, 4.0, 5.0, 6.0,
- <var>rowind</var>: 3, 1, 0, 1, 0, 2,
- <var>colptr</var>: 0, 3, 3, 4, 6
-
-</BLOCKQUOTE>
-is not allowed (and will likely cause the program to crash).
-
-<P>
-The <tt class="cdata">nzmax</tt> field specifies the number of non-zero elements the
-matrix can store. It is equal to the length of <var>rowind</var> and
-<var>values</var>; this number can be larger that
-<code><var>colptr</var>[<var>nrows</var>]</code>, but never less.
-This field makes it possible to preallocate a certain amount of
-memory to avoid reallocations if the matrix is constructed
-sequentially by filling in elements.
-In general the <var>nzmax</var> field can safely be ignored, however, since
-it will always be adjusted automatically as the number of non-zero
-elements grows.
-
-<P>
-The <tt class="cdata">id</tt> field controls the type of the matrix and can have
-values <tt class="constant">DOUBLE</tt> and <tt class="constant">COMPLEX</tt>.
-
-<P>
-Sparse matrices are created using the following functions from the API.
-<dl><dt><table cellpadding="0" cellspacing="0"><tr valign="baseline"><td><nobr>spmatrix* <b><tt id='l2h-177' xml:id='l2h-177' class="cfunction">SpMatrix_New</tt></b>(</nobr></td><td>int <var>nrows</var>, int <var>ncols</var>, int
- <var>nzmax</var>, int <var>id</var>)</td></tr></table></dt>
-<dd>
- Returns a sparse zero matrix with <var>nrows</var> rows and
- <var>ncols</var> columns. <var>nzmax</var> is the number of elements that will
- be allocated (the length of the <var>values</var> and <var>rowind</var>
- fields).
-</dd></dl>
-
-<P>
-<dl><dt><table cellpadding="0" cellspacing="0"><tr valign="baseline"><td><nobr>spmatrix* <b><tt id='l2h-178' xml:id='l2h-178' class="cfunction">SpMatrix_NewFromMatrix</tt></b>(</nobr></td><td>spmatrix *<var>src</var>, int
- <var>id</var>)</td></tr></table></dt>
-<dd>
- Returns a copy the sparse matrix <var>src</var>.
-</dd></dl>
-
-<P>
-<dl><dt><table cellpadding="0" cellspacing="0"><tr valign="baseline"><td><nobr>spmatrix* <b><tt id='l2h-179' xml:id='l2h-179' class="cfunction">SpMatrix_NewFromIJV</tt></b>(</nobr></td><td>matrix *I, matrix *J,
- matrix *V, int <var>nrows</var>, int <var>ncols</var>, int <var>nzmax</var>, int <var>id</var>)</td></tr></table></dt>
-<dd>
- Creates a sparse matrix with <var>nrows</var> rows and <var>ncols</var>
- columns from a triplet description. <var>I</var> and <var>J</var>
- must be integer matrices and <var>V</var> either a double or complex matrix,
- or <tt class="constant">NULL</tt>. If <var>V</var> is <tt class="constant">NULL</tt> the values of the
- entries in the matrix are undefined, otherwise they are
- specified by <var>V</var>. Repeated entries in <var>V</var> are summed.
- The number of allocated elements is given by <var>nzmax</var>, which is
- adjusted if it is smaller than the required amount.
-</dd></dl>
-
-<P>
-We illustrate use of the sparse matrix class by listing the source
-code for the <tt class="method">real()</tt> method, which returns the real part of
-a sparse matrix:
-
-<P>
-<div class="verbatim"><pre>
-static PyObject * spmatrix_real(spmatrix *self) {
-
- if (SP_ID(self) != COMPLEX)
- return (PyObject *)SpMatrix_NewFromMatrix(self, 0, SP_ID(self));
-
- spmatrix *ret = SpMatrix_New(SP_NROWS(self), SP_NCOLS(self),
- SP_NNZ(self), DOUBLE);
- if (!ret) return PyErr_NoMemory();
-
- int i;
- for (i=0; i < SP_NNZ(self); i++)
- SP_VALD(ret)[i] = creal(SP_VALZ(self)[i]);
-
- memcpy(SP_COL(ret), SP_COL(self), (SP_NCOLS(self)+1)*sizeof(int_t));
- memcpy(SP_ROW(ret), SP_ROW(self), SP_NNZ(self)*sizeof(int_t));
- return (PyObject *)ret;
-}
-</pre></div>
-
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-<!DOCTYPE html PUBLIC "-//W3C//DTD HTML 4.0 Transitional//EN">
-<html>
-<head>
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-
-<H1><A NAME="SECTION004300000000000000000"></A> <A NAME="s-arithmetic"></A>
-<BR>
-2.3 Arithmetic Operations
-</H1>
-The following table lists the arithmetic operations defined for
-dense matrices. In this table <var>A</var> and <var>B</var> are dense matrices
-with compatible dimensions, <var>c</var> is a scalar (a Python number
-or a dense 1 by 1 matrix), and <var>d</var> is a Python number.
-
-<P>
-<DIV ALIGN="CENTER">
-<TABLE CELLPADDING=3 BORDER="1">
-<TR><TD ALIGN="LEFT">Unary plus/minus</TD>
-<TD ALIGN="LEFT"><code>+<var>A</var></code>, <code>-<var>A</var></code></TD>
-</TR>
-<TR><TD ALIGN="LEFT">Addition</TD>
-<TD ALIGN="LEFT"><code><var>A</var>+<var>B</var></code>, <code><var>A</var>+<var>c</var></code>,
- <code><var>c</var>+<var>A</var></code></TD>
-</TR>
-<TR><TD ALIGN="LEFT">Subtraction</TD>
-<TD ALIGN="LEFT"><code><var>A</var>-<var>B</var></code>, <code><var>A</var>-<var>c</var></code>,
- <code><var>c</var>-<var>A</var></code></TD>
-</TR>
-<TR><TD ALIGN="LEFT">Matrix multiplication</TD>
-<TD ALIGN="LEFT"><var>A</var>*<var>B</var></TD>
-</TR>
-<TR><TD ALIGN="LEFT">Scalar multiplication and division</TD>
-<TD ALIGN="LEFT"><code><var>c</var>*<var>A</var></code>,
- <code><var>A</var>*<var>c</var></code>, <code><var>A</var>/<var>c</var></code></TD>
-</TR>
-<TR><TD ALIGN="LEFT">Remainder after division</TD>
-<TD ALIGN="LEFT"><code><var>A</var>%<var>c</var></code></TD>
-</TR>
-<TR><TD ALIGN="LEFT">Elementwise exponentiation</TD>
-<TD ALIGN="LEFT"><code><var>A</var>**<var>d</var></code></TD>
-</TR>
-</TABLE>
-</DIV>
-
-<P>
-If <var>c</var> in the expressions <code><var>A</var>+<var>c</var></code>,
-<code><var>c</var>+<var>A</var></code>, <code><var>A</var>-<var>c</var></code>,
-<code><var>c</var>-<var>A</var></code> is a number, then it is interpreted as a
-matrix with the same dimensions as <var>A</var>, type given by the type
-of <var>c</var>, and all entries equal to <var>c</var>.
-If <var>c</var> is a 1 by 1 matrix and <var>A</var> is not 1 by 1, then <var>c</var>
-is interpreted as a matrix with the same size of <var>A</var> and all
-entries equal to <code><var>c</var>[0]</code>.
-
-<P>
-Postmultiplying a matrix with a number <var>c</var> means the same as
-premultiplying, <EM>i.e.</EM>, scalar multiplication. Dividing a matrix by
-<var>c</var> means dividing all entries by <var>c</var>.
-If <var>c</var> is a 1 by 1 matrix and the product
-<code><var>c</var>*<var>A</var></code> or <code><var>A</var>*<var>c</var></code> cannot be
-interpreted as a matrix-matrix product, then it is interpreted as
-<code><var>c</var>[0]*<var>A</var></code>.
-The division <code><var>A</var>/<var>c</var></code> and remainder
-<code><var>A</var>%<var>c</var></code> with <var>c</var> a 1 by 1 matrix are always
-interpreted as <code><var>A</var>/<var>c</var>[0]</code>, resp.,
-<code><var>A</var>%<var>c</var>[0]</code>.
-
-<P>
-If one of the operands in the arithmetic operations is integer
-(a scalar integer or a matrix of type <code>'i'</code>) and the other operand
-is double (a scalar float or a matrix of type <code>'d'</code>), then the
-integer operand is converted to double, and the result is a matrix of
-type <code>'d'</code>.
-If one of the operands is integer or double, and the other operand is
-complex (a scalar complex or a matrix of type <code>'z'</code>),
-then the first operand is converted to complex, and the result is
-a matrix of type <code>'z'</code>.
-
-<P>
-The result of <code><var>A</var>**<var>d</var></code> is a complex matrix
-if <var>A</var> or <var>d</var> are complex, and real otherwise.
-
-<P>
-Note that Python rounds the result of an integer division towards minus
-infinity.
-
-<P>
-The following in-place operations are also defined, but only if
-they do not change the type or the size of the matrix <var>A</var>:
-<DIV ALIGN="CENTER">
-<TABLE CELLPADDING=3 BORDER="1">
-<TR><TD ALIGN="LEFT">In-place addition</TD>
-<TD ALIGN="LEFT"><code><var>A</var>+=<var>B</var></code>, <code><var>A</var>+=<var>c</var></code></TD>
-</TR>
-<TR><TD ALIGN="LEFT">In-place subtraction</TD>
-<TD ALIGN="LEFT"><code><var>A</var>-=<var>B</var></code>, <code><var>A</var>-=<var>c</var></code></TD>
-</TR>
-<TR><TD ALIGN="LEFT">In-place scalar multiplication and division</TD>
-<TD ALIGN="LEFT"><code><var>A</var>*=<var>c</var></code>, <code><var>A</var>/=<var>c</var></code></TD>
-</TR>
-<TR><TD ALIGN="LEFT">In-place remainder</TD>
-<TD ALIGN="LEFT"><code><var>A</var>%=<var>c</var></code></TD>
-</TR>
-</TABLE>
-</DIV>
-
-<P>
-For example, if <var>A</var> has type <code>'i'</code>, then <code><var>A</var>+=<var>B</var></code>
-is allowed if <var>B</var> has type <code>'i'</code>.
-It is not allowed if <var>B</var> has type <code>'d'</code>or <code>'z'</code>because the
-addition <code><var>A</var>+<var>B</var></code> results in a matrix of
-type <code>'d'</code>or <code>'z'</code>and therefore cannot be assigned to <var>A</var>
-without changing its type.
-
-<P>
-In-place matrix-matrix products are not allowed. (Except when
-<var>c</var> is a 1 by 1 matrix, in which case <code><var>A</var>*=<var>c</var></code> is
-interpreted as the scalar product <code><var>A</var>*=<var>c</var>[0]</code>.)
-
-<P>
-It is important to know when a matrix operation creates
-a new object. The following rules apply.
-
-<UL>
-<LI>A simple assignment ("<tt class="samp">A = B</tt>") is given the standard
-Python interpretation, <EM>i.e.</EM>, it assigns to the variable <var>A</var> a
-reference (or pointer) to the object referenced by <var>B</var>.
-<div class="verbatim"><pre>
->>> B = matrix([[1.,2.], [3.,4.]])
->>> print B
- 1.0000e+00 3.0000e+00
- 2.0000e+00 4.0000e+00
->>> A = B
->>> A[0,0] = -1
->>> print B # modifying A[0,0] also modified B[0,0]
--1.0000e+00 3.0000e+00
- 2.0000e+00 4.0000e+00
-</pre></div>
-
-<P>
-</LI>
-<LI>The regular (<EM>i.e.</EM>, not in-place) arithmetic operations always
-return new objects. Hence "<tt class="samp">A = +B</tt>" is equivalent to
-"<tt class="samp">A = matrix(B)</tt>".
-<div class="verbatim"><pre>
->>> B = matrix([[1.,2.], [3.,4.]])
->>> A = +B
->>> A[0,0] = -1
->>> print B # modifying A[0,0] does not modify B[0,0]
- 1.0000e+00 3.0000e+00
- 2.0000e+00 4.0000e+00
-</pre></div>
-
-<P>
-</LI>
-<LI>The in-place operations directly modify the
- coefficients of the existing matrix object and do not create a new
- object.
-<div class="verbatim"><pre>
->>> B = matrix([[1.,2.], [3.,4.]])
->>> A = B
->>> A *= 2
->>> print B # in-place operation also changed B
- 2.0000e+00 6.0000e+00
- 4.0000e+00 8.0000e+00
->>> A = 2*A
->>> print B # regular operation creates a new A, so does not change B
- 2.0000e+00 6.0000e+00
- 4.0000e+00 8.0000e+00
-</pre></div>
-</LI>
-</UL>
-
-<P>
-The restrictions on in-place operations follow the principle that once
-a matrix object is created, its size and type cannot be modified.
-The only matrix attributes that can be changed are the values of the
-elements. The values can be changed by in-place operations or by an
-indexed assignment, as explained in the next section.
-
-<P>
-
-<DIV CLASS="navigation">
-<div class='online-navigation'>
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-<H1><A NAME="SECTION004800000000000000000"></A> <A NAME="s-array-interface"></A>
-<BR>
-2.8 The NumPy Array Interface
-</H1>
-
-<P>
-The CVXOPT <tt class="class">matrix</tt> object is compatible with the NumPy Array Interface,
-which allows Python objects that represent multidimensional
-arrays to exchange data using information stored in the
-attribute <code>__array_struct__</code>.
-
-<P>
-<div class="seealso">
- <p class="heading">See Also:</p>
-
-<dl compact="compact" class="seeurl">
- <dt><a href='http://numpy.scipy.org/array_interface.shtml'
- >NumPy Array Interface Specification</a></dt>
- <dd></dd>
- </dl>
-
-<P>
-<dl compact="compact" class="seeurl">
- <dt><a href='http://numpy.scipy.org'
- >NumPy home page</a></dt>
- <dd></dd>
- </dl>
-</div>
-
-<P>
-As already mentioned in section <A href="s-creating-matrices.html#s-creating-matrices">2.1</A>,
-a two-dimensional array object (for example, a 2D <tt class="module">numarray</tt>
-array) can be converted to a <tt class="class">matrix</tt> object by using the
-<tt class="function">matrix()</tt> constructor.
-Conversely, CVXOPT matrices can be used as array-like objects
-in <tt class="module">numarray</tt>. The following example illustrates the
-compatibility of CVXOPT matrices and <tt class="module">numarray</tt> arrays.
-<div class="verbatim"><pre>
->>> from cvxopt import matrix
->>> a = matrix(range(6), (2,3), 'd')
->>> print a
- 0.0000e+00 2.0000e+00 4.0000e+00
- 1.0000e+00 3.0000e+00 5.0000e+00
->>> from numarray import array
->>> b = array(a)
->>> print b
-[[ 0. 2. 4.]
- [ 1. 3. 5.]]
->>> print a*b
-[[ 0. 4. 16.]
- [ 1. 9. 25.]]
-</pre></div>
-In the last expression <code>a*b</code> is interpreted as a <tt class="module">numarray</tt>
-multiplication (<EM>i.e.</EM>, componentwise) even though one operand is a
-<tt class="class">matrix</tt> object.
-
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deleted file mode 100644
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--- a/doc/cvxopt/s-blas1.html
+++ /dev/null
@@ -1,344 +0,0 @@
-<!DOCTYPE html PUBLIC "-//W3C//DTD HTML 4.0 Transitional//EN">
-<html>
-<head>
-<link rel="STYLESHEET" href="cvxopt.css" type='text/css' />
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-<title>3.2 Level 1 BLAS</title>
-</head>
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-<DIV CLASS="navigation">
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-<table align="center" width="100%" cellpadding="0" cellspacing="2">
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-<H1><A NAME="SECTION005200000000000000000"></A> <A NAME="s-blas1"></A>
-<BR>
-3.2 Level 1 BLAS
-</H1>
-The level 1 functions implement vector operations.
-
-<P>
-<dl><dt><table cellpadding="0" cellspacing="0"><tr valign="baseline">
- <td><nobr><b><tt id='l2h-28' xml:id='l2h-28' class="function">scal</tt></b>(</nobr></td>
- <td><var>alpha, x</var>)</td></tr></table></dt>
-<dd>
-Scales a vector by a constant:
-<BR><P></P>
-<DIV ALIGN="CENTER" CLASS="mathdisplay">
-<!-- MATH
- \begin{displaymath}
-x := \alpha x.
-\end{displaymath}
- -->
-
-<IMG
- WIDTH="56" HEIGHT="24" BORDER="0"
- SRC="img8.gif"
- ALT="\begin{displaymath}
-x := \alpha x.
-\end{displaymath}">
-</DIV>
-<BR CLEAR="ALL">
-<P></P>
-If <var>x</var> is a real <tt class="class">matrix</tt>, the scalar argument <var>alpha</var> must be a
-Python integer or float. If <var>x</var> is complex, <var>alpha</var> can be an
-integer, float, or complex.
-</dl>
-
-<P>
-<dl><dt><table cellpadding="0" cellspacing="0"><tr valign="baseline">
- <td><nobr><b><tt id='l2h-29' xml:id='l2h-29' class="function">nrm2</tt></b>(</nobr></td>
- <td><var>x</var>)</td></tr></table></dt>
-<dd>
-Euclidean norm of a vector: returns
-<BR><P></P>
-<DIV ALIGN="CENTER" CLASS="mathdisplay">
-<!-- MATH
- \begin{displaymath}
-\|x\|_2.
-\end{displaymath}
- -->
-
-<IMG
- WIDTH="33" HEIGHT="28" BORDER="0"
- SRC="img9.gif"
- ALT="\begin{displaymath}
-\Vert x\Vert _2.
-\end{displaymath}">
-</DIV>
-<BR CLEAR="ALL">
-<P></P>
-</dl>
-
-<P>
-<dl><dt><table cellpadding="0" cellspacing="0"><tr valign="baseline">
- <td><nobr><b><tt id='l2h-30' xml:id='l2h-30' class="function">asum</tt></b>(</nobr></td>
- <td><var>x</var>)</td></tr></table></dt>
-<dd>
-L-1 norm of a vector: returns
-<BR><P></P>
-<DIV ALIGN="CENTER" CLASS="mathdisplay">
-<!-- MATH
- \begin{displaymath}
-\|x\|_1 \quad \mbox{($x$\ real)}, \qquad
-\|\Re x\|_1 + \|\Im x\|_1 \quad \mbox{($x$\ complex)}.
-\end{displaymath}
- -->
-
-<IMG
- WIDTH="337" HEIGHT="28" BORDER="0"
- SRC="img10.gif"
- ALT="\begin{displaymath}
-\Vert x\Vert _1 \quad \mbox{($x$\ real)}, \qquad
-\Vert\Re x\Vert _1 + \Vert\Im x\Vert _1 \quad \mbox{($x$\ complex)}.
-\end{displaymath}">
-</DIV>
-<BR CLEAR="ALL">
-<P></P>
-</dl>
-
-<P>
-<dl><dt><table cellpadding="0" cellspacing="0"><tr valign="baseline">
- <td><nobr><b><tt id='l2h-31' xml:id='l2h-31' class="function">iamax</tt></b>(</nobr></td>
- <td><var>x</var>)</td></tr></table></dt>
-<dd>
-Returns
-<BR><P></P>
-<DIV ALIGN="CENTER" CLASS="mathdisplay">
-<!-- MATH
- \begin{displaymath}
-\mathop{\rm argmax}_{k=0,\ldots,n-1} |x_k| \quad \mbox{($x$\ real)}, \qquad
- \mathop{\rm argmax}_{k=0,\ldots,n-1} |\Re x_k| + |\Im x_k| \quad
- \mbox{($x$\ complex)}.
-\end{displaymath}
- -->
-
-<IMG
- WIDTH="457" HEIGHT="41" BORDER="0"
- SRC="img11.gif"
- ALT="\begin{displaymath}
-\mathop{\rm argmax}_{k=0,\ldots,n-1} \vert x_k\vert \quad \...
-...e x_k\vert + \vert\Im x_k\vert \quad
-\mbox{($x$\ complex)}.
-\end{displaymath}">
-</DIV>
-<BR CLEAR="ALL">
-<P></P>
-If more than one coefficient achieves the maximum, the index of the
-first <I>k</I> is returned.
-</dl>
-
-<P>
-<dl><dt><table cellpadding="0" cellspacing="0"><tr valign="baseline">
- <td><nobr><b><tt id='l2h-32' xml:id='l2h-32' class="function">swap</tt></b>(</nobr></td>
- <td><var>x, y</var>)</td></tr></table></dt>
-<dd>
-Interchanges two vectors:
-<BR><P></P>
-<DIV ALIGN="CENTER" CLASS="mathdisplay">
-<!-- MATH
- \begin{displaymath}
-x \leftrightarrow y.
-\end{displaymath}
- -->
-
-<IMG
- WIDTH="44" HEIGHT="27" BORDER="0"
- SRC="img12.gif"
- ALT="\begin{displaymath}
-x \leftrightarrow y.
-\end{displaymath}">
-</DIV>
-<BR CLEAR="ALL">
-<P></P>
-<var>x</var> and <var>y</var> are matrices of the same type (<code>'d'</code> or <code>'z'</code>).
-</dl>
-
-<P>
-<dl><dt><table cellpadding="0" cellspacing="0"><tr valign="baseline">
- <td><nobr><b><tt id='l2h-33' xml:id='l2h-33' class="function">copy</tt></b>(</nobr></td>
- <td><var>x, y</var>)</td></tr></table></dt>
-<dd>
-Copies a vector to another vector:
-<BR><P></P>
-<DIV ALIGN="CENTER" CLASS="mathdisplay">
-<!-- MATH
- \begin{displaymath}
-y := x.
-\end{displaymath}
- -->
-
-<IMG
- WIDTH="47" HEIGHT="27" BORDER="0"
- SRC="img13.gif"
- ALT="\begin{displaymath}
-y := x.
-\end{displaymath}">
-</DIV>
-<BR CLEAR="ALL">
-<P></P>
-<var>x</var> and <var>y</var> are matrices of the same type (<code>'d'</code> or <code>'z'</code>).
-</dl>
-
-<P>
-<dl><dt><table cellpadding="0" cellspacing="0"><tr valign="baseline">
- <td><nobr><b><tt id='l2h-34' xml:id='l2h-34' class="function">axpy</tt></b>(</nobr></td>
- <td><var>x, y</var><big>[</big><var>,alpha=1.0</var><big>]</big><var></var>)</td></tr></table></dt>
-<dd>
-Constant times a vector plus a vector:
-<BR><P></P>
-<DIV ALIGN="CENTER" CLASS="mathdisplay">
-<!-- MATH
- \begin{displaymath}
-y := \alpha x + y.
-\end{displaymath}
- -->
-
-<IMG
- WIDTH="85" HEIGHT="27" BORDER="0"
- SRC="img14.gif"
- ALT="\begin{displaymath}
-y := \alpha x + y.
-\end{displaymath}">
-</DIV>
-<BR CLEAR="ALL">
-<P></P>
-<var>x</var> and <var>y</var> are matrices of the same type (<code>'d'</code> or <code>'z'</code>).
-If <var>x</var> is real, the scalar argument <var>alpha</var> must be a Python
-integer or float. If <var>x</var> is complex, <var>alpha</var> can be an integer,
-float, or complex.
-</dl>
-
-<P>
-<dl><dt><table cellpadding="0" cellspacing="0"><tr valign="baseline">
- <td><nobr><b><tt id='l2h-35' xml:id='l2h-35' class="function">dot</tt></b>(</nobr></td>
- <td><var>x, y</var>)</td></tr></table></dt>
-<dd>
-Returns
-<BR><P></P>
-<DIV ALIGN="CENTER" CLASS="mathdisplay">
-<!-- MATH
- \begin{displaymath}
-x^Hy.
-\end{displaymath}
- -->
-
-<IMG
- WIDTH="32" HEIGHT="27" BORDER="0"
- SRC="img15.gif"
- ALT="\begin{displaymath}
-x^Hy.
-\end{displaymath}">
-</DIV>
-<BR CLEAR="ALL">
-<P></P>
-<var>x</var> and <var>y</var> are matrices of the same type (<code>'d'</code> or <code>'z'</code>).
-</dl>
-
-<P>
-<dl><dt><table cellpadding="0" cellspacing="0"><tr valign="baseline">
- <td><nobr><b><tt id='l2h-36' xml:id='l2h-36' class="function">dotu</tt></b>(</nobr></td>
- <td><var>x, y</var>)</td></tr></table></dt>
-<dd>
-Returns
-<BR><P></P>
-<DIV ALIGN="CENTER" CLASS="mathdisplay">
-<!-- MATH
- \begin{displaymath}
-x^Ty.
-\end{displaymath}
- -->
-
-<IMG
- WIDTH="30" HEIGHT="27" BORDER="0"
- SRC="img16.gif"
- ALT="\begin{displaymath}
-x^Ty.
-\end{displaymath}">
-</DIV>
-<BR CLEAR="ALL">
-<P></P>
-<var>x</var> and <var>y</var> are matrices of the same type (<code>'d'</code> or <code>'z'</code>).
-</dl>
-
-<P>
-
-<DIV CLASS="navigation">
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-<BR>
-3.3 Level 2 BLAS
-</H1>
-The level 2 functions implement matrix-vector products and
-rank-1 and rank-2 matrix updates.
-Different types of matrix structure can be exploited using the
-conventions of section <A href="s-conventions.html#s-conventions">3.1</A>.
-
-<P>
-<dl><dt><table cellpadding="0" cellspacing="0"><tr valign="baseline">
- <td><nobr><b><tt id='l2h-37' xml:id='l2h-37' class="function">gemv</tt></b>(</nobr></td>
- <td><var>A, x, y</var><big>[</big><var>, trans='N'</var><big>[</big><var>,
-alpha=1.0</var><big>[</big><var>, beta=0.0</var><big>]</big><var></var><big>]</big><var></var><big>]</big><var></var>)</td></tr></table></dt>
-<dd>
-Matrix-vector product with a general matrix:
-<BR><P></P>
-<DIV ALIGN="CENTER" CLASS="mathdisplay">
-<!-- MATH
- \begin{displaymath}
-y := \alpha Ax + \beta y \quad (\mathrm{trans} = \mathrm{'N'}),
- \qquad
-y := \alpha A^T x + \beta y \quad (\mathrm{trans} = \mathrm{'T'}),
- \qquad
-y := \alpha A^H x + \beta y \quad (\mathrm{trans} = \mathrm{'C'}).
-\end{displaymath}
- -->
-
-<IMG
- WIDTH="707" HEIGHT="28" BORDER="0"
- SRC="img17.gif"
- ALT="\begin{displaymath}
-y := \alpha Ax + \beta y \quad (\mathrm{trans} = \mathrm{'N'...
-...\alpha A^H x + \beta y \quad (\mathrm{trans} = \mathrm{'C'}).
-\end{displaymath}">
-</DIV>
-<BR CLEAR="ALL">
-<P></P>
-The arguments <var>A</var>, <var>x</var> and <var>y</var> must have the same type
-(<code>'d'</code> or <code>'z'</code>). Complex values of <var>alpha</var> and <var>beta</var> are only
-allowed if <var>A</var> is complex.
-</dl>
-
-<P>
-<dl><dt><table cellpadding="0" cellspacing="0"><tr valign="baseline">
- <td><nobr><b><tt id='l2h-38' xml:id='l2h-38' class="function">symv</tt></b>(</nobr></td>
- <td><var>A, x, y</var><big>[</big><var>, uplo='L'</var><big>[</big><var>,
-alpha=1.0</var><big>[</big><var>, beta=0.0</var><big>]</big><var></var><big>]</big><var></var><big>]</big><var></var>)</td></tr></table></dt>
-<dd>
-Matrix-vector product with a real symmetric matrix:
-<BR><P></P>
-<DIV ALIGN="CENTER" CLASS="mathdisplay">
-<!-- MATH
- \begin{displaymath}
-y := \alpha A x + \beta y,
-\end{displaymath}
- -->
-
-<IMG
- WIDTH="107" HEIGHT="27" BORDER="0"
- SRC="img18.gif"
- ALT="\begin{displaymath}
-y := \alpha A x + \beta y,
-\end{displaymath}">
-</DIV>
-<BR CLEAR="ALL">
-<P></P>
-where <I>A</I> is a real symmetric matrix.
-The arguments <var>A</var>, <var>x</var> and <var>y</var> must have
-type <code>'d'</code> and <var>alpha</var> and <var>beta</var> must be real.
-</dl>
-
-<P>
-<dl><dt><table cellpadding="0" cellspacing="0"><tr valign="baseline">
- <td><nobr><b><tt id='l2h-39' xml:id='l2h-39' class="function">hemv</tt></b>(</nobr></td>
- <td><var>A, x, y</var><big>[</big><var>, uplo='L'</var><big>[</big><var>,
-alpha=1.0</var><big>[</big><var>, beta=0.0</var><big>]</big><var></var><big>]</big><var></var><big>]</big><var></var>)</td></tr></table></dt>
-<dd>
-Matrix-vector product with a real symmetric or complex Hermitian
-matrix:
-<BR><P></P>
-<DIV ALIGN="CENTER" CLASS="mathdisplay">
-<!-- MATH
- \begin{displaymath}
-y := \alpha A x + \beta y,
-\end{displaymath}
- -->
-
-<IMG
- WIDTH="107" HEIGHT="27" BORDER="0"
- SRC="img18.gif"
- ALT="\begin{displaymath}
-y := \alpha A x + \beta y,
-\end{displaymath}">
-</DIV>
-<BR CLEAR="ALL">
-<P></P>
-where <I>A</I> is real symmetric or complex Hermitian.
-The arguments <var>A</var>, <var>x</var> and <var>y</var> must have the same
-type (<code>'d'</code> or <code>'z'</code>).
-Complex values of <var>alpha</var> and <var>beta</var> are only
-allowed if <var>A</var> is complex.
-</dl>
-
-<P>
-<dl><dt><table cellpadding="0" cellspacing="0"><tr valign="baseline">
- <td><nobr><b><tt id='l2h-40' xml:id='l2h-40' class="function">trmv</tt></b>(</nobr></td>
- <td><var>A, x</var><big>[</big><var>, uplo='L'</var><big>[</big><var>,
-trans='N'</var><big>[</big><var>, diag='N'</var><big>]</big><var></var><big>]</big><var></var><big>]</big><var></var>)</td></tr></table></dt>
-<dd>
-Matrix-vector product with a triangular matrix:
-<BR><P></P>
-<DIV ALIGN="CENTER" CLASS="mathdisplay">
-<!-- MATH
- \begin{displaymath}
-x := Ax \quad (\mathrm{trans} = \mathrm{'N'}), \qquad
-x := A^T x \quad (\mathrm{trans} = \mathrm{'T'}), \qquad
-x := A^H x \quad (\mathrm{trans} = \mathrm{'C'}),
-\end{displaymath}
- -->
-
-<IMG
- WIDTH="586" HEIGHT="28" BORDER="0"
- SRC="img19.gif"
- ALT="\begin{displaymath}
-x := Ax \quad (\mathrm{trans} = \mathrm{'N'}), \qquad
-x := A...
-...'}), \qquad
-x := A^H x \quad (\mathrm{trans} = \mathrm{'C'}),
-\end{displaymath}">
-</DIV>
-<BR CLEAR="ALL">
-<P></P>
-where <I>A</I> is square and triangular.
-The arguments <var>A</var> and <var>x</var> must have the same type
-(<code>'d'</code> or <code>'z'</code>).
-</dl>
-
-<P>
-<dl><dt><table cellpadding="0" cellspacing="0"><tr valign="baseline">
- <td><nobr><b><tt id='l2h-41' xml:id='l2h-41' class="function">trsv</tt></b>(</nobr></td>
- <td><var>A, x</var><big>[</big><var>, uplo='L'</var><big>[</big><var>,
-trans='N'</var><big>[</big><var>, diag='N'</var><big>]</big><var></var><big>]</big><var></var><big>]</big><var></var>)</td></tr></table></dt>
-<dd>
-Solution of a nonsingular triangular set of linear equations:
-<BR><P></P>
-<DIV ALIGN="CENTER" CLASS="mathdisplay">
-<!-- MATH
- \begin{displaymath}
-x := A^{-1}x \quad (\mathrm{trans} = \mathrm{'N'}), \qquad
-x := A^{-T}x \quad (\mathrm{trans} = \mathrm{'T'}), \qquad
-x := A^{-H}x \quad (\mathrm{trans} = \mathrm{'C'}),
-\end{displaymath}
- -->
-
-<IMG
- WIDTH="623" HEIGHT="28" BORDER="0"
- SRC="img20.gif"
- ALT="\begin{displaymath}
-x := A^{-1}x \quad (\mathrm{trans} = \mathrm{'N'}), \qquad
-x...
-..., \qquad
-x := A^{-H}x \quad (\mathrm{trans} = \mathrm{'C'}),
-\end{displaymath}">
-</DIV>
-<BR CLEAR="ALL">
-<P></P>
-where <I>A</I> is square and triangular with nonzero diagonal
-elements. The arguments <var>A</var> and <var>x</var> must have the same type
-(<code>'d'</code> or <code>'z'</code>).
-</dl>
-
-<P>
-<dl><dt><table cellpadding="0" cellspacing="0"><tr valign="baseline">
- <td><nobr><b><tt id='l2h-42' xml:id='l2h-42' class="function">gbmv</tt></b>(</nobr></td>
- <td><var>A, m, kl, x, y</var><big>[</big><var>, trans='N'
-</var><big>[</big><var>, alpha=1.0</var><big>[</big><var>, beta=0.0</var><big>]</big><var></var><big>]</big><var></var><big>]</big><var></var>)</td></tr></table></dt>
-<dd>
-Matrix-vector product with a general band matrix:
-<BR><P></P>
-<DIV ALIGN="CENTER" CLASS="mathdisplay">
-<!-- MATH
- \begin{displaymath}
-y := \alpha Ax + \beta y \quad (\mathrm{trans} = \mathrm{'N'}), \qquad
-y := \alpha A^T x + \beta y \quad (\mathrm{trans} = \mathrm{'T'}),
-\qquad
-y := \alpha A^H x + \beta y \quad (\mathrm{trans} = \mathrm{'C'}),
-\end{displaymath}
- -->
-
-<IMG
- WIDTH="708" HEIGHT="28" BORDER="0"
- SRC="img21.gif"
- ALT="\begin{displaymath}
-y := \alpha Ax + \beta y \quad (\mathrm{trans} = \mathrm{'N'...
-... \alpha A^H x + \beta y \quad (\mathrm{trans} = \mathrm{'C'}),
-\end{displaymath}">
-</DIV>
-<BR CLEAR="ALL">
-<P></P>
-where <I>A</I> is a rectangular band matrix with <var>m</var> rows and
-<var>kl</var> subdiagonals.
-The arguments <var>A</var>, <var>x</var> and <var>y</var> must have the same
-type (<code>'d'</code> or <code>'z'</code>).
-Complex values of <var>alpha</var> and <var>beta</var> are only allowed if <var>A</var> is
- complex.
-</dl>
-
-<P>
-<dl><dt><table cellpadding="0" cellspacing="0"><tr valign="baseline">
- <td><nobr><b><tt id='l2h-43' xml:id='l2h-43' class="function">sbmv</tt></b>(</nobr></td>
- <td><var>A, x, y</var><big>[</big><var>, uplo='L'</var><big>[</big><var>,
-alpha=1.0</var><big>[</big><var>, beta=0.0</var><big>]</big><var></var><big>]</big><var></var><big>]</big><var></var>)</td></tr></table></dt>
-<dd>
-Matrix-vector product with a real symmetric band matrix:
-<BR><P></P>
-<DIV ALIGN="CENTER" CLASS="mathdisplay">
-<!-- MATH
- \begin{displaymath}
-y := \alpha Ax + \beta y,
-\end{displaymath}
- -->
-
-<IMG
- WIDTH="107" HEIGHT="27" BORDER="0"
- SRC="img18.gif"
- ALT="\begin{displaymath}
-y := \alpha A x + \beta y,
-\end{displaymath}">
-</DIV>
-<BR CLEAR="ALL">
-<P></P>
-where <I>A</I> is a real symmetric band matrix.
-The arguments <var>A</var>, <var>x</var> and <var>y</var> must have type <code>'d'</code> and
-<var>alpha</var> and <var>beta</var> must be real.
-</dl>
-
-<P>
-<dl><dt><table cellpadding="0" cellspacing="0"><tr valign="baseline">
- <td><nobr><b><tt id='l2h-44' xml:id='l2h-44' class="function">hbmv</tt></b>(</nobr></td>
- <td><var>A, x, y</var><big>[</big><var>, uplo='L'</var><big>[</big><var>,
-alpha=1.0</var><big>[</big><var>, beta=0.0</var><big>]</big><var></var><big>]</big><var></var><big>]</big><var></var>)</td></tr></table></dt>
-<dd>
-Matrix-vector product with a real symmetric or complex Hermitian
-band matrix:
-<BR><P></P>
-<DIV ALIGN="CENTER" CLASS="mathdisplay">
-<!-- MATH
- \begin{displaymath}
-y := \alpha Ax + \beta y,
-\end{displaymath}
- -->
-
-<IMG
- WIDTH="107" HEIGHT="27" BORDER="0"
- SRC="img18.gif"
- ALT="\begin{displaymath}
-y := \alpha A x + \beta y,
-\end{displaymath}">
-</DIV>
-<BR CLEAR="ALL">
-<P></P>
-where <I>A</I> is a real symmetric or complex Hermitian band matrix.
-The arguments <var>A</var>, <var>x</var> and <var>y</var> must have the same type
-(<code>'d'</code> or <code>'z'</code>).
-Complex values of <var>alpha</var> and <var>beta</var> are only allowed if
-<var>A</var> is complex.
-</dl>
-
-<P>
-<dl><dt><table cellpadding="0" cellspacing="0"><tr valign="baseline">
- <td><nobr><b><tt id='l2h-45' xml:id='l2h-45' class="function">tbmv</tt></b>(</nobr></td>
- <td><var>A, x</var><big>[</big><var>, uplo='L'</var><big>[</big><var>,
-trans</var><big>[</big><var>, diag</var><big>]</big><var></var><big>]</big><var></var><big>]</big><var></var>)</td></tr></table></dt>
-<dd>
-Matrix-vector product with a triangular band matrix:
-<BR><P></P>
-<DIV ALIGN="CENTER" CLASS="mathdisplay">
-<!-- MATH
- \begin{displaymath}
-x := Ax \quad (\mathrm{trans} = \mathrm{'N'}), \qquad
-x := A^T x \quad (\mathrm{trans} = \mathrm{'T'}), \qquad
-x := A^H x \quad (\mathrm{trans} = \mathrm{'C'}).
-\end{displaymath}
- -->
-
-<IMG
- WIDTH="585" HEIGHT="28" BORDER="0"
- SRC="img22.gif"
- ALT="\begin{displaymath}
-x := Ax \quad (\mathrm{trans} = \mathrm{'N'}), \qquad
-x := A...
-...}), \qquad
-x := A^H x \quad (\mathrm{trans} = \mathrm{'C'}).
-\end{displaymath}">
-</DIV>
-<BR CLEAR="ALL">
-<P></P>
-The arguments <var>A</var> and <var>x</var> must have the same type
-(<code>'d'</code> or <code>'z'</code>).
-</dl>
-
-<P>
-<dl><dt><table cellpadding="0" cellspacing="0"><tr valign="baseline">
- <td><nobr><b><tt id='l2h-46' xml:id='l2h-46' class="function">tbsv</tt></b>(</nobr></td>
- <td><var>A, x</var><big>[</big><var>, uplo='L'</var><big>[</big><var>,
-trans</var><big>[</big><var>, diag</var><big>]</big><var></var><big>]</big><var></var><big>]</big><var></var>)</td></tr></table></dt>
-<dd>
-Solution of a triangular banded set of linear equations:
-<BR><P></P>
-<DIV ALIGN="CENTER" CLASS="mathdisplay">
-<!-- MATH
- \begin{displaymath}
-x := A^{-1}x \quad (\mathrm{trans} = \mathrm{'N'}), \qquad
-x := A^{-T} x \quad (\mathrm{trans} = \mathrm{'T'}), \qquad
-x := A^{-H} x \quad (\mathrm{trans} = \mathrm{'T'}),
-\end{displaymath}
- -->
-
-<IMG
- WIDTH="623" HEIGHT="28" BORDER="0"
- SRC="img23.gif"
- ALT="\begin{displaymath}
-x := A^{-1}x \quad (\mathrm{trans} = \mathrm{'N'}), \qquad
-x...
-..., \qquad
-x := A^{-H} x \quad (\mathrm{trans} = \mathrm{'T'}),
-\end{displaymath}">
-</DIV>
-<BR CLEAR="ALL">
-<P></P>
-where <I>A</I> is a triangular band matrix of with nonzero diagonal
-elements.
-The arguments <var>A</var> and <var>x</var> must have the same type
-(<code>'d'</code> or <code>'z'</code>).
-</dl>
-
-<P>
-<dl><dt><table cellpadding="0" cellspacing="0"><tr valign="baseline">
- <td><nobr><b><tt id='l2h-47' xml:id='l2h-47' class="function">ger</tt></b>(</nobr></td>
- <td><var>x, y, A</var><big>[</big><var>, alpha=1.0</var><big>]</big><var></var>)</td></tr></table></dt>
-<dd>
-General rank-1 update:
-<BR><P></P>
-<DIV ALIGN="CENTER" CLASS="mathdisplay">
-<!-- MATH
- \begin{displaymath}
-A := A + \alpha x y^H,
-\end{displaymath}
- -->
-
-<IMG
- WIDTH="112" HEIGHT="27" BORDER="0"
- SRC="img24.gif"
- ALT="\begin{displaymath}
-A := A + \alpha x y^H,
-\end{displaymath}">
-</DIV>
-<BR CLEAR="ALL">
-<P></P>
-where <I>A</I> is a general matrix.
-The arguments <var>A</var>, <var>x</var> and <var>y</var> must have the same type
-(<code>'d'</code> or <code>'z'</code>).
-Complex values of <var>alpha</var> are only allowed if <var>A</var> is complex.
-</dl>
-
-<P>
-<dl><dt><table cellpadding="0" cellspacing="0"><tr valign="baseline">
- <td><nobr><b><tt id='l2h-48' xml:id='l2h-48' class="function">geru</tt></b>(</nobr></td>
- <td><var>x, y, A</var><big>[</big><var>, alpha=1.0</var><big>]</big><var></var>)</td></tr></table></dt>
-<dd>
-General rank-1 update:
-<BR><P></P>
-<DIV ALIGN="CENTER" CLASS="mathdisplay">
-<!-- MATH
- \begin{displaymath}
-A := A + \alpha x y^T,
-\end{displaymath}
- -->
-
-<IMG
- WIDTH="110" HEIGHT="27" BORDER="0"
- SRC="img25.gif"
- ALT="\begin{displaymath}
-A := A + \alpha x y^T,
-\end{displaymath}">
-</DIV>
-<BR CLEAR="ALL">
-<P></P>
-where <I>A</I> is a general matrix.
-The arguments <var>A</var>, <var>x</var> and <var>y</var> must have the same type
-(<code>'d'</code> or <code>'z'</code>).
-Complex values of <var>alpha</var> are only allowed if <var>A</var> is complex.
-</dl>
-
-<P>
-<dl><dt><table cellpadding="0" cellspacing="0"><tr valign="baseline">
- <td><nobr><b><tt id='l2h-49' xml:id='l2h-49' class="function">syr</tt></b>(</nobr></td>
- <td><var>x, A</var><big>[</big><var>, uplo='L'</var><big>[</big><var>, alpha=1.0</var><big>]</big><var></var><big>]</big><var></var>)</td></tr></table></dt>
-<dd>
-Symmetric rank-1 update:
-<BR><P></P>
-<DIV ALIGN="CENTER" CLASS="mathdisplay">
-<!-- MATH
- \begin{displaymath}
-A := A + \alpha xx^T,
-\end{displaymath}
- -->
-
-<IMG
- WIDTH="112" HEIGHT="27" BORDER="0"
- SRC="img26.gif"
- ALT="\begin{displaymath}
-A := A + \alpha xx^T,
-\end{displaymath}">
-</DIV>
-<BR CLEAR="ALL">
-<P></P>
-where <I>A</I> is a real symmetric matrix.
-The arguments <var>A</var> and <var>x</var> must have type <code>'d'</code>.
-<var>alpha</var> must be a real number.
-</dl>
-
-<P>
-<dl><dt><table cellpadding="0" cellspacing="0"><tr valign="baseline">
- <td><nobr><b><tt id='l2h-50' xml:id='l2h-50' class="function">her</tt></b>(</nobr></td>
- <td><var>x, A</var><big>[</big><var>, uplo='L'</var><big>[</big><var>, alpha=1.0</var><big>]</big><var></var><big>]</big><var></var>)</td></tr></table></dt>
-<dd>
-Hermitian rank-1 update:
-<BR><P></P>
-<DIV ALIGN="CENTER" CLASS="mathdisplay">
-<!-- MATH
- \begin{displaymath}
-A := A + \alpha xx^H,
-\end{displaymath}
- -->
-
-<IMG
- WIDTH="114" HEIGHT="27" BORDER="0"
- SRC="img27.gif"
- ALT="\begin{displaymath}
-A := A + \alpha xx^H,
-\end{displaymath}">
-</DIV>
-<BR CLEAR="ALL">
-<P></P>
-where <I>A</I> is a real symmetric or complex Hermitian matrix.
-The arguments <var>A</var> and <var>x</var> must have the same type
-(<code>'d'</code> or <code>'z'</code>).
-<var>alpha</var> must be a real number.
-</dl>
-
-<P>
-<dl><dt><table cellpadding="0" cellspacing="0"><tr valign="baseline">
- <td><nobr><b><tt id='l2h-51' xml:id='l2h-51' class="function">syr2</tt></b>(</nobr></td>
- <td><var>x, y, A</var><big>[</big><var>, uplo='L'</var><big>[</big><var>,
-alpha=1.0</var><big>]</big><var></var><big>]</big><var></var>)</td></tr></table></dt>
-<dd>
-Symmetric rank-2 update:
-<BR><P></P>
-<DIV ALIGN="CENTER" CLASS="mathdisplay">
-<!-- MATH
- \begin{displaymath}
-A := A + \alpha (xy^T + yx^T),
-\end{displaymath}
- -->
-
-<IMG
- WIDTH="170" HEIGHT="28" BORDER="0"
- SRC="img28.gif"
- ALT="\begin{displaymath}
-A := A + \alpha (xy^T + yx^T),
-\end{displaymath}">
-</DIV>
-<BR CLEAR="ALL">
-<P></P>
-where <I>A</I> is a real symmetric matrix.
-The arguments <var>A</var>, <var>x</var> and <var>y</var> must have type <code>'d'</code>.
-<var>alpha</var> must be real.
-</dl>
-
-<P>
-<dl><dt><table cellpadding="0" cellspacing="0"><tr valign="baseline">
- <td><nobr><b><tt id='l2h-52' xml:id='l2h-52' class="function">her2</tt></b>(</nobr></td>
- <td><var>x, y, A</var><big>[</big><var>, uplo='L'</var><big>[</big><var>,
-alpha=1.0</var><big>]</big><var></var><big>]</big><var></var>)</td></tr></table></dt>
-<dd>
-Symmetric rank-2 update:
-<BR><P></P>
-<DIV ALIGN="CENTER" CLASS="mathdisplay">
-<!-- MATH
- \begin{displaymath}
-A := A + \alpha xy^H + \bar \alpha yx^H,
-\end{displaymath}
- -->
-
-<IMG
- WIDTH="172" HEIGHT="27" BORDER="0"
- SRC="img29.gif"
- ALT="\begin{displaymath}
-A := A + \alpha xy^H + \bar \alpha yx^H,
-\end{displaymath}">
-</DIV>
-<BR CLEAR="ALL">
-<P></P>
-where <I>A</I> is a a real symmetric or complex Hermitian matrix.
-The arguments <var>A</var>, <var>x</var> and <var>y</var> must have the same type
-(<code>'d'</code> or <code>'z'</code>).
-Complex values of <var>alpha</var> are only allowed if <var>A</var> is complex.
-</dl>
-
-<P>
-As an example, the following code multiplies the tridiagonal matrix
-<BR><P></P>
-<DIV ALIGN="CENTER" CLASS="mathdisplay">
-<!-- MATH
- \begin{displaymath}
-A = \left[\begin{array}{rrrr}
- 1 & 6 & 0 & 0 \\
- 2 & -4 & 3 & 0 \\
- 0 & -3 & -1 & 1
- \end{array}\right]
-\end{displaymath}
- -->
-
-<IMG
- WIDTH="169" HEIGHT="64" BORDER="0"
- SRC="img30.gif"
- ALT="\begin{displaymath}
-A = \left[\begin{array}{rrrr}
-1 & 6 & 0 & 0 \\
-2 & -4 & 3 & 0 \\
-0 & -3 & -1 & 1
-\end{array}\right]
-\end{displaymath}">
-</DIV>
-<BR CLEAR="ALL">
-<P></P>
-with the vector <I>x</I> = (1,-1,2,-2).
-<div class="verbatim"><pre>
->>> from cvxopt.base import matrix
->>> from cvxopt.blas import gbmv
->>> A = matrix([[0., 1., 2.], [6., -4., -3.], [3., -1., 0.], [1., 0., 0.]])
->>> x = matrix([1., -1., 2., -2.])
->>> y = matrix(0., (3,1))
->>> gbmv(A, 3, 1, x, y)
->>> print y
--5.0000e+00
- 1.2000e+01
--1.0000e+00
-</pre></div>
-
-<P>
-The following example illustrates the use of <tt class="function">tbsv()</tt>.
-<div class="verbatim"><pre>
->>> from cvxopt.base import matrix
->>> from cvxopt.blas import tbsv
->>> A = matrix([-6., 5., -1., 2.], (1,4))
->>> x = matrix(1.0, (4,1))
->>> tbsv(A, x) # x := diag(A)^{-1}*x
->>> print x
--1.6667e-01
- 2.0000e-01
--1.0000e+00
- 5.0000e-01
-</pre></div>
-
-<P>
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-<!DOCTYPE html PUBLIC "-//W3C//DTD HTML 4.0 Transitional//EN">
-<html>
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-
-<H1><A NAME="SECTION005400000000000000000"></A> <A NAME="s-blas3"></A>
-<BR>
-3.4 Level 3 BLAS
-</H1>
-The level 3 BLAS include functions for matrix-matrix multiplication.
-
-<P>
-<dl><dt><table cellpadding="0" cellspacing="0"><tr valign="baseline">
- <td><nobr><b><tt id='l2h-53' xml:id='l2h-53' class="function">gemm</tt></b>(</nobr></td>
- <td><var>A, B, C</var><big>[</big><var>, transA='N'</var><big>[</big><var>,
-transB='N'</var><big>[</big><var>, alpha=1.0</var><big>[</big><var>, beta=0.0</var><big>]</big><var></var><big>]</big><var></var><big>]</big><var></var><big>]</big><var></var>)</td></tr></table></dt>
-<dd>
-Matrix-matrix product of two general matrices:
-<BR><P></P>
-<DIV ALIGN="CENTER" CLASS="mathdisplay">
-<!-- MATH
- \begin{displaymath}
-C := \alpha \mathop{\mathrm{op}}(A) \mathop{\mathrm{op}}(B) + \beta C
-\end{displaymath}
- -->
-
-<IMG
- WIDTH="177" HEIGHT="28" BORDER="0"
- SRC="img31.gif"
- ALT="\begin{displaymath}
-C := \alpha \mathop{\mathrm{op}}(A) \mathop{\mathrm{op}}(B) + \beta C
-\end{displaymath}">
-</DIV>
-<BR CLEAR="ALL">
-<P></P>
-where
-<BR><P></P>
-<DIV ALIGN="CENTER" CLASS="mathdisplay">
-<!-- MATH
- \begin{displaymath}
-\mathop{\mathrm{op}}(A) = \left\{ \begin{array}{ll}
- A & \mathrm{transA} = \mathrm{'N'} \\
- A^T & \mathrm{transA} = \mathrm{'T'} \\
- A^H & \mathrm{transA} = \mathrm{'C'} \end{array} \right.
-\qquad
-\mathop{\mathrm{op}}(B) = \left\{ \begin{array}{ll}
- B & \mathrm{transB} = \mathrm{'N'} \\
- B^T & \mathrm{transB} = \mathrm{'T'} \\
- B^H & \mathrm{transB} = \mathrm{'C'}. \end{array} \right.
-\end{displaymath}
- -->
-
-<IMG
- WIDTH="474" HEIGHT="64" BORDER="0"
- SRC="img32.gif"
- ALT="\begin{displaymath}
-\mathop{\mathrm{op}}(A) = \left\{ \begin{array}{ll}
-A & \ma...
-...\\
-B^H & \mathrm{transB} = \mathrm{'C'}. \end{array} \right.
-\end{displaymath}">
-</DIV>
-<BR CLEAR="ALL">
-<P></P>
-The arguments <var>A</var>, <var>B</var> and <var>C</var> must have the same type
-(<code>'d'</code> or <code>'z'</code>).
-Complex values of <var>alpha</var> and <var>beta</var> are only allowed
-if <var>A</var> is complex.
-</dl>
-
-<P>
-<dl><dt><table cellpadding="0" cellspacing="0"><tr valign="baseline">
- <td><nobr><b><tt id='l2h-54' xml:id='l2h-54' class="function">symm</tt></b>(</nobr></td>
- <td><var>A, B, C</var><big>[</big><var>, side='L'</var><big>[</big><var>,
-uplo='L'</var><big>[</big><var>, alpha=1.0</var><big>[</big><var>, beta=0.0</var><big>]</big><var></var><big>]</big><var></var><big>]</big><var></var><big>]</big><var></var>)</td></tr></table></dt>
-<dd>
-Product of a real or complex symmetric matrix <I>A</I> and a general
-matrix <I>B</I>:
-<BR><P></P>
-<DIV ALIGN="CENTER" CLASS="mathdisplay">
-<!-- MATH
- \begin{displaymath}
-C := \alpha AB + \beta C \quad (\mathrm{side} = \mathrm{'L'}), \qquad
- C := \alpha BA + \beta C \quad (\mathrm{side} = \mathrm{'R'}).
-\end{displaymath}
- -->
-
-<IMG
- WIDTH="462" HEIGHT="28" BORDER="0"
- SRC="img33.gif"
- ALT="\begin{displaymath}
-C := \alpha AB + \beta C \quad (\mathrm{side} = \mathrm{'L'...
-... := \alpha BA + \beta C \quad (\mathrm{side} = \mathrm{'R'}).
-\end{displaymath}">
-</DIV>
-<BR CLEAR="ALL">
-<P></P>
-The arguments <var>A</var>, <var>B</var> and <var>C</var> must have the same type
-(<code>'d'</code> or <code>'z'</code>). Complex values of <var>alpha</var> and <var>beta</var> are
-only allowed if <var>A</var> is complex.
-</dl>
-
-<P>
-<dl><dt><table cellpadding="0" cellspacing="0"><tr valign="baseline">
- <td><nobr><b><tt id='l2h-55' xml:id='l2h-55' class="function">hemm</tt></b>(</nobr></td>
- <td><var>A, B, C</var><big>[</big><var>, side='L'</var><big>[</big><var>,
-uplo='L'</var><big>[</big><var>, alpha=1.0</var><big>[</big><var>, beta=0.0</var><big>]</big><var></var><big>]</big><var></var><big>]</big><var></var><big>]</big><var></var>)</td></tr></table></dt>
-<dd>
-Product of a real symmetric or complex Hermitian matrix <I>A</I> and a
-general matrix <I>B</I>:
-<BR><P></P>
-<DIV ALIGN="CENTER" CLASS="mathdisplay">
-<!-- MATH
- \begin{displaymath}
-C := \alpha AB + \beta C \quad (\mathrm{side} = \mathrm{'L'}), \qquad
- C := \alpha BA + \beta C \quad (\mathrm{side} = \mathrm{'R'}).
-\end{displaymath}
- -->
-
-<IMG
- WIDTH="462" HEIGHT="28" BORDER="0"
- SRC="img33.gif"
- ALT="\begin{displaymath}
-C := \alpha AB + \beta C \quad (\mathrm{side} = \mathrm{'L'...
-... := \alpha BA + \beta C \quad (\mathrm{side} = \mathrm{'R'}).
-\end{displaymath}">
-</DIV>
-<BR CLEAR="ALL">
-<P></P>
-The arguments <var>A</var>, <var>B</var> and <var>C</var> must have the same type
-(<code>'d'</code> or <code>'z'</code>).
-Complex values of <var>alpha</var> and <var>beta</var> are only allowed if
-<var>A</var> is complex.
-</dl>
-
-<P>
-<dl><dt><table cellpadding="0" cellspacing="0"><tr valign="baseline">
- <td><nobr><b><tt id='l2h-56' xml:id='l2h-56' class="function">trmm</tt></b>(</nobr></td>
- <td><var>A, B</var><big>[</big><var>, side='L'</var><big>[</big><var>,
-uplo='L'</var><big>[</big><var>, transA='N'</var><big>[</big><var>, diag='N'</var><big>[</big><var>,
-alpha=1.0</var><big>]</big><var></var><big>]</big><var></var><big>]</big><var></var><big>]</big><var></var><big>]</big><var></var>)</td></tr></table></dt>
-<dd>
-Product of a triangular matrix <I>A</I> and a general matrix <I>B</I>:
-<BR><P></P>
-<DIV ALIGN="CENTER" CLASS="mathdisplay">
-<!-- MATH
- \begin{displaymath}
-B := \alpha\mathop{\mathrm{op}}(A)B \quad (\mathrm{side} = \mathrm{'L'}), \qquad
- B := \alpha B\mathop{\mathrm{op}}(A) \quad (\mathrm{side} = \mathrm{'R'}), \qquad
- \mathop{\mathrm{op}}(A) = \left\{ \begin{array}{ll}
- A & \mathrm{transA} = \mathrm{'N'} \\
- A^T & \mathrm{transA} = \mathrm{'T'} \\
- A^H & \mathrm{transA} = \mathrm{'C'}. \end{array} \right.
-\end{displaymath}
- -->
-
-<IMG
- WIDTH="696" HEIGHT="64" BORDER="0"
- SRC="img34.gif"
- ALT="\begin{displaymath}
-B := \alpha\mathop{\mathrm{op}}(A)B \quad (\mathrm{side} = ...
-...\\
-A^H & \mathrm{transA} = \mathrm{'C'}. \end{array} \right.
-\end{displaymath}">
-</DIV>
-<BR CLEAR="ALL">
-<P></P>
-The arguments <var>A</var> and <var>B</var> must have the same type (<code>'d'</code> or
-<code>'z'</code>). Complex values of <var>alpha</var> are only allowed if <var>A</var> is
-complex.
-</dl>
-
-<P>
-<dl><dt><table cellpadding="0" cellspacing="0"><tr valign="baseline">
- <td><nobr><b><tt id='l2h-57' xml:id='l2h-57' class="function">trsm</tt></b>(</nobr></td>
- <td><var>A, B</var><big>[</big><var>, side='L'</var><big>[</big><var>,
-uplo='L'</var><big>[</big><var>, transA='N'</var><big>[</big><var>, diag='N'</var><big>[</big><var>,
-alpha=1.0</var><big>]</big><var></var><big>]</big><var></var><big>]</big><var></var><big>]</big><var></var><big>]</big><var></var>)</td></tr></table></dt>
-<dd>
-Solution of a nonsingular triangular system of equations:
-<BR><P></P>
-<DIV ALIGN="CENTER" CLASS="mathdisplay">
-<!-- MATH
- \begin{displaymath}
-B := \alpha \mathop{\mathrm{op}}(A)^{-1}B \quad (\mathrm{side} = \mathrm{'L'}), \qquad
- B := \alpha B\mathop{\mathrm{op}}(A)^{-1} \quad (\mathrm{side} = \mathrm{'R'}), \qquad
- \mathop{\mathrm{op}}(A) = \left\{ \begin{array}{ll}
- A & \mathrm{transA} = \mathrm{'N'} \\
- A^T & \mathrm{transA} = \mathrm{'T'} \\
- A^H & \mathrm{transA} = \mathrm{'C'}, \end{array} \right.
-\end{displaymath}
- -->
-
-<IMG
- WIDTH="731" HEIGHT="64" BORDER="0"
- SRC="img35.gif"
- ALT="\begin{displaymath}
-B := \alpha \mathop{\mathrm{op}}(A)^{-1}B \quad (\mathrm{si...
-...\\
-A^H & \mathrm{transA} = \mathrm{'C'}, \end{array} \right.
-\end{displaymath}">
-</DIV>
-<BR CLEAR="ALL">
-<P></P>
-where <I>A</I> is triangular and <I>B</I> is a general matrix.
-The arguments <var>A</var> and <var>B</var> must have the same type (<code>'d'</code> or
-<code>'z'</code>). Complex values of <var>alpha</var> are only allowed if <var>A</var> is
-complex.
-</dl>
-
-<P>
-<dl><dt><table cellpadding="0" cellspacing="0"><tr valign="baseline">
- <td><nobr><b><tt id='l2h-58' xml:id='l2h-58' class="function">syrk</tt></b>(</nobr></td>
- <td><var>A, C</var><big>[</big><var>, uplo='L'</var><big>[</big><var>,
-trans='N'</var><big>[</big><var>, alpha=1.0</var><big>[</big><var>, beta=0.0</var><big>]</big><var></var><big>]</big><var></var><big>]</big><var></var><big>]</big><var></var>)</td></tr></table></dt>
-<dd>
-Rank-<I>k</I> update of a real or complex symmetric matrix <I>C</I>:
-<BR><P></P>
-<DIV ALIGN="CENTER" CLASS="mathdisplay">
-<!-- MATH
- \begin{displaymath}
-C := \alpha AA^T + \beta C \quad (\mathrm{trans} = \mathrm{'N'}),
- \qquad
- C := \alpha A^TA + \beta C \quad (\mathrm{trans} = \mathrm{'T'}),
-\end{displaymath}
- -->
-
-<IMG
- WIDTH="501" HEIGHT="28" BORDER="0"
- SRC="img36.gif"
- ALT="\begin{displaymath}
-C := \alpha AA^T + \beta C \quad (\mathrm{trans} = \mathrm{...
-... \alpha A^TA + \beta C \quad (\mathrm{trans} = \mathrm{'T'}),
-\end{displaymath}">
-</DIV>
-<BR CLEAR="ALL">
-<P></P>
-where <I>A</I> is a general matrix.
-The arguments <var>A</var> and <var>C</var> must have the same type (<code>'d'</code> or
-<code>'z'</code>). Complex values of <var>alpha</var> and <var>beta</var> are only allowed
-if <var>A</var> is complex.
-</dl>
-
-<P>
-<dl><dt><table cellpadding="0" cellspacing="0"><tr valign="baseline">
- <td><nobr><b><tt id='l2h-59' xml:id='l2h-59' class="function">herk</tt></b>(</nobr></td>
- <td><var>A, C</var><big>[</big><var>, uplo='L'</var><big>[</big><var>,
-trans='N'</var><big>[</big><var>, alpha=1.0</var><big>[</big><var>, beta=0.0</var><big>]</big><var></var><big>]</big><var></var><big>]</big><var></var><big>]</big><var></var>)</td></tr></table></dt>
-<dd>
-Rank-<I>k</I> update of a real symmetric or complex Hermitian matrix
-<I>C</I>:
-<BR><P></P>
-<DIV ALIGN="CENTER" CLASS="mathdisplay">
-<!-- MATH
- \begin{displaymath}
-C := \alpha AA^H + \beta C \quad (\mathrm{trans} = \mathrm{'N'}),
- \qquad
- C := \alpha A^HA + \beta C \quad (\mathrm{trans} = \mathrm{'C'}),
-\end{displaymath}
- -->
-
-<IMG
- WIDTH="505" HEIGHT="28" BORDER="0"
- SRC="img37.gif"
- ALT="\begin{displaymath}
-C := \alpha AA^H + \beta C \quad (\mathrm{trans} = \mathrm{...
-...= \alpha A^HA + \beta C \quad (\mathrm{trans} = \mathrm{'C'}),
-\end{displaymath}">
-</DIV>
-<BR CLEAR="ALL">
-<P></P>
-where <I>A</I> is a general matrix.
-The arguments <var>A</var> and <var>C</var> must have the same type (<code>'d'</code> or
-<code>'z'</code>). <var>alpha</var> and <var>beta</var> must be real.
-</dl>
-
-<P>
-<dl><dt><table cellpadding="0" cellspacing="0"><tr valign="baseline">
- <td><nobr><b><tt id='l2h-60' xml:id='l2h-60' class="function">syr2k</tt></b>(</nobr></td>
- <td><var>A, B, C</var><big>[</big><var>, uplo='L'</var><big>[</big><var>,
-trans='N'</var><big>[</big><var>, alpha=1.0</var><big>[</big><var>, beta=0.0</var><big>]</big><var></var><big>]</big><var></var><big>]</big><var></var><big>]</big><var></var>)</td></tr></table></dt>
-<dd>
-Rank-<I>2k</I> update of a real or complex symmetric matrix <I>C</I>:
-<BR><P></P>
-<DIV ALIGN="CENTER" CLASS="mathdisplay">
-<!-- MATH
- \begin{displaymath}
-C := \alpha (AB^T + BA^T) + \beta C \quad
- (\mathrm{trans} = \mathrm{'N'}), \qquad
- C := \alpha (A^TB + B^TA) + \beta C \quad
- (\mathrm{trans} = \mathrm{'T'}).
-\end{displaymath}
- -->
-
-<IMG
- WIDTH="607" HEIGHT="28" BORDER="0"
- SRC="img38.gif"
- ALT="\begin{displaymath}
-C := \alpha (AB^T + BA^T) + \beta C \quad
-(\mathrm{trans}...
-...TB + B^TA) + \beta C \quad
-(\mathrm{trans} = \mathrm{'T'}).
-\end{displaymath}">
-</DIV>
-<BR CLEAR="ALL">
-<P></P>
-<I>A</I> and <I>B</I> are general real or complex matrices.
-The arguments <var>A</var>, <var>B</var> and <var>C</var> must have the same
-type. Complex values of <var>alpha</var> and <var>beta</var> are only
-allowed if <var>A</var> is complex.
-</dl>
-
-<P>
-<dl><dt><table cellpadding="0" cellspacing="0"><tr valign="baseline">
- <td><nobr><b><tt id='l2h-61' xml:id='l2h-61' class="function">her2k</tt></b>(</nobr></td>
- <td><var>A, B, C</var><big>[</big><var>, uplo='L'</var><big>[</big><var>,
-trans='N'</var><big>[</big><var>, alpha=1.0</var><big>[</big><var> beta=0.0</var><big>]</big><var></var><big>]</big><var></var><big>]</big><var></var><big>]</big><var></var>)</td></tr></table></dt>
-<dd>
-Rank-<I>2k</I> update of a real symmetric or complex Hermitian matrix
-<I>C</I>:
-<BR><P></P>
-<DIV ALIGN="CENTER" CLASS="mathdisplay">
-<!-- MATH
- \begin{displaymath}
-C := \alpha AB^H + \bar \alpha BA^H + \beta C \quad
- (\mathrm{trans} = \mathrm{'N'}), \qquad
- C := \alpha A^HB + \bar\alpha B^HA + \beta C \quad
- (\mathrm{trans} = \mathrm{'C'}),
-\end{displaymath}
- -->
-
-<IMG
- WIDTH="612" HEIGHT="28" BORDER="0"
- SRC="img39.gif"
- ALT="\begin{displaymath}
-C := \alpha AB^H + \bar \alpha BA^H + \beta C \quad
-(\mat...
-...alpha B^HA + \beta C \quad
-(\mathrm{trans} = \mathrm{'C'}),
-\end{displaymath}">
-</DIV>
-<BR CLEAR="ALL">
-<P></P>
-where <I>A</I> and <I>B</I> are general matrices.
-The arguments <var>A</var>, <var>B</var> and <var>C</var> must have the same type
-(<code>'d'</code> or <code>'z'</code>). Complex values of <var>alpha</var> are only allowed if
-<var>A</var> is complex. <var>beta</var> must be real.
-</dl>
-
-<DIV CLASS="navigation">
-<div class='online-navigation'>
-<p></p><hr />
-<table align="center" width="100%" cellpadding="0" cellspacing="2">
-<tr>
-<td class='online-navigation'><a rel="prev" title="3.3 Level 2 BLAS"
- href="s-blas2.html"><img src='previous.gif'
- border='0' height='32' alt='Previous Page' width='32' /></A></td>
-<td class='online-navigation'><a rel="parent" title="3. The BLAS Interface"
- href="node14.html"><img src='up.gif'
- border='0' height='32' alt='Up One Level' width='32' /></A></td>
-<td class='online-navigation'><a rel="next" title="4. The LAPACK Interface"
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-<b class="navlabel">Previous:</b>
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--- a/doc/cvxopt/s-builtinfuncs.html
+++ /dev/null
@@ -1,219 +0,0 @@
-<!DOCTYPE html PUBLIC "-//W3C//DTD HTML 4.0 Transitional//EN">
-<html>
-<head>
-<link rel="STYLESHEET" href="cvxopt.css" type='text/css' />
-<link rel="first" href="cvxopt.html" title='CVXOPT: A Python Package for Convex Optimization' />
-<link rel='contents' href='contents.html' title="Contents" />
-<link rel='index' href='genindex.html' title='Index' />
-<link rel='last' href='about.html' title='About this document...' />
-<link rel='help' href='about.html' title='About this document...' />
-<link rel="next" href="s-otherfuncs.html" />
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-<link rel="parent" href="node4.html" />
-<link rel="next" href="s-otherfuncs.html" />
-<meta name='aesop' content='information' />
-<title>2.5 Built-in Functions</title>
-</head>
-<body>
-<DIV CLASS="navigation">
-<div id='top-navigation-panel' xml:id='top-navigation-panel'>
-<table align="center" width="100%" cellpadding="0" cellspacing="2">
-<tr>
-<td class='online-navigation'><a rel="prev" title="2.4 Indexing and Slicing"
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-<td class='online-navigation'><a rel="parent" title="2. Dense Matrices (cvxopt.base)"
- href="node4.html"><img src='up.gif'
- border='0' height='32' alt='Up One Level' width='32' /></A></td>
-<td class='online-navigation'><a rel="next" title="2.6 Other Matrix Functions"
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-<td align="center" width="100%">CVXOPT: A Python Package for Convex Optimization</td>
-<td class='online-navigation'><a rel="contents" title="Table of Contents"
- href="contents.html"><img src='contents.gif'
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-<td class='online-navigation'><a rel="index" title="Index"
- href="genindex.html"><img src='index.gif'
- border='0' height='32' alt='Index' width='32' /></A></td>
-</tr></table>
-<div class='online-navigation'>
-<b class="navlabel">Previous:</b>
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-<b class="navlabel">Next:</b>
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-</div>
-<hr /></div>
-</DIV>
-<!--End of Navigation Panel-->
-
-<H1><A NAME="SECTION004500000000000000000"></A> <A NAME="s-builtinfuncs"></A>
-<BR>
-2.5 Built-in Functions
-</H1>
-Many Python built-in functions and operations can be used with matrix
-arguments. We list some useful examples.
-
-<P>
-<dl><dt><table cellpadding="0" cellspacing="0"><tr valign="baseline">
- <td><nobr><b><tt id='l2h-11' xml:id='l2h-11' class="function">len</tt></b>(</nobr></td>
- <td><var>x</var>)</td></tr></table></dt>
-<dd>
-Returns the product of the number of rows and the number of columns.
-</dl>
-
-<P>
-<dl><dt><table cellpadding="0" cellspacing="0"><tr valign="baseline">
- <td><nobr><b><tt id='l2h-12' xml:id='l2h-12' class="function">bool</tt></b>(</nobr></td>
- <td><var></var><big>[</big><var>x</var><big>]</big><var></var>)</td></tr></table></dt>
-<dd>
-Returns <code>False</code> if <var>x</var> is empty (<EM>i.e.</EM>, <code>len(<var>x</var>)</code> is zero)
-and <code>True</code> otherwise.
-</dl>
-
-<P>
-<dl><dt><table cellpadding="0" cellspacing="0"><tr valign="baseline">
- <td><nobr><b><tt id='l2h-13' xml:id='l2h-13' class="function">max</tt></b>(</nobr></td>
- <td><var>x</var>)</td></tr></table></dt>
-<dd>
-Returns the maximum element of <var>x</var>.
-</dl>
-
-<P>
-<dl><dt><table cellpadding="0" cellspacing="0"><tr valign="baseline">
- <td><nobr><b><tt id='l2h-14' xml:id='l2h-14' class="function">min</tt></b>(</nobr></td>
- <td><var>x</var>)</td></tr></table></dt>
-<dd>
-Returns the minimum element of <var>x</var>.
-</dl>
-
-<P>
-<dl><dt><table cellpadding="0" cellspacing="0"><tr valign="baseline">
- <td><nobr><b><tt id='l2h-15' xml:id='l2h-15' class="function">abs</tt></b>(</nobr></td>
- <td><var>x</var>)</td></tr></table></dt>
-<dd>
-Returns a matrix with the absolute values of the elements of <var>x</var>.
-</dl>
-
-<P>
-<dl><dt><table cellpadding="0" cellspacing="0"><tr valign="baseline">
- <td><nobr><b><tt id='l2h-16' xml:id='l2h-16' class="function">sum</tt></b>(</nobr></td>
- <td><var>x</var><big>[</big><var>, start=0.0</var><big>]</big><var></var>)</td></tr></table></dt>
-<dd>
-Returns the sum of <var>start</var> and the elements of <var>x</var>.
-</dl>
-
-<P>
-Matrices can be used as arguments to the <tt class="function">list()</tt>,
-<tt class="function">tuple()</tt>, <tt class="function">zip()</tt>, <tt class="function">map()</tt>, and
-<tt class="function">filter()</tt> functions described in section 2.1 of the Python
-Library Reference. <code>list(<var>A</var>)</code> and <code>tuple(<var>A</var>)</code>
-construct a list, respectively a tuple, from the elements of <var>A</var>.
-<code>zip(<var>A</var>,<var>B</var>,...)</code> returns a list of tuples,
-with the <I>i</I>th tuple containing the <I>i</I>th elements of <var>A</var>,
-<var>B</var>, ....
-
-<P>
-<div class="verbatim"><pre>
->>> from cvxopt.base import matrix
->>> A = matrix([[-11., -5., -20.], [-6., -0., 7.]])
->>> B = matrix(range(6), (3,2))
->>> list(A)
-[-11.0, -5.0, -20.0, -6.0, 0.0, 7.0]
->>> tuple(B)
-(0, 1, 2, 3, 4, 5)
->>> zip(A,B)
-[(-11.0, 0), (-5.0, 1), (-20.0, 2), (-6.0, 3), (0.0, 4), (7.0, 5)]
-</pre></div>
-
-<P>
-<code>map(<var>f</var>,<var>A</var>)</code>, where <var>f</var> is a function and <var>A</var>
-is a matrix, returns a list constructed by applying <var>f</var> to each
-element of <var>A</var>. Multiple arguments can be provided, for example,
-as in <code>map(<var>f</var>,<var>A</var>,<var>B</var>)</code>, if <var>f</var> is a function
-with two arguments.
-<div class="verbatim"><pre>
->>> A = matrix([[5, -4, 10, -7], [-1, -5, -6, 2], [6, 1, 5, 2], [-1, 2, -3, -7]])
->>> B = matrix([[4,-15, 9, -14], [-4, -12, 1, -22], [-10, -9, 9, 12], [-9, -7,-11, -6]])
->>> print matrix(map(max, A, B), (4,4)) # takes componentwise maximum
- 5 -1 6 -1
- -4 -5 1 2
- 10 1 9 -3
- -7 2 12 -6
-</pre></div>
-
-<P>
-<code>filter(<var>f</var>,<var>A</var>)</code>, where <var>f</var> is a function and
-<var>A</var> is a matrix, returns a list containing the elements of <var>A</var>
-for which <var>f</var> is true.
-<div class="verbatim"><pre>
->>> print filter(lambda x: x%2, A) # list of odd elements in A
-[5, -7, -1, -5, 1, 5, -1, -3, -7]
->>> print filter(lambda x: -2 < x < 3, A) # list of elements between -2 and 3
-[-1, 2, 1, 2, -1, 2]
-</pre></div>
-
-<P>
-It is also possible to iterate over matrix elements, as illustrated in
-the following example.
-<div class="verbatim"><pre>
->>> A = matrix([[5, -3], [9, 11]])
->>> for x in A: print max(x,0)
-...
-5
-0
-9
-11
->>> [max(x,0) for x in A]
-[5, 0, 9, 11]
-</pre></div>
-
-<P>
-The expression "<tt class="samp"><var>x</var> in <var>A</var></tt>" returns <code>True</code> if an element
-of <var>A</var> is equal to <var>x</var> and <code>False</code> otherwise.
-
-<P>
-
-<DIV CLASS="navigation">
-<div class='online-navigation'>
-<p></p><hr />
-<table align="center" width="100%" cellpadding="0" cellspacing="2">
-<tr>
-<td class='online-navigation'><a rel="prev" title="2.4 Indexing and Slicing"
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- border='0' height='32' alt='Previous Page' width='32' /></A></td>
-<td class='online-navigation'><a rel="parent" title="2. Dense Matrices (cvxopt.base)"
- href="node4.html"><img src='up.gif'
- border='0' height='32' alt='Up One Level' width='32' /></A></td>
-<td class='online-navigation'><a rel="next" title="2.6 Other Matrix Functions"
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-<td align="center" width="100%">CVXOPT: A Python Package for Convex Optimization</td>
-<td class='online-navigation'><a rel="contents" title="Table of Contents"
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-<td class='online-navigation'><img src='blank.gif'
- border='0' height='32' alt='' width='32' /></td>
-<td class='online-navigation'><a rel="index" title="Index"
- href="genindex.html"><img src='index.gif'
- border='0' height='32' alt='Index' width='32' /></A></td>
-</tr></table>
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-<b class="navlabel">Previous:</b>
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-</div>
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diff --git a/doc/cvxopt/s-cholmod.html b/doc/cvxopt/s-cholmod.html
deleted file mode 100644
index 2980c4a..0000000
--- a/doc/cvxopt/s-cholmod.html
+++ /dev/null
@@ -1,514 +0,0 @@
-<!DOCTYPE html PUBLIC "-//W3C//DTD HTML 4.0 Transitional//EN">
-<html>
-<head>
-<link rel="STYLESHEET" href="cvxopt.css" type='text/css' />
-<link rel="first" href="cvxopt.html" title='CVXOPT: A Python Package for Convex Optimization' />
-<link rel='contents' href='contents.html' title="Contents" />
-<link rel='index' href='genindex.html' title='Index' />
-<link rel='last' href='about.html' title='About this document...' />
-<link rel='help' href='about.html' title='About this document...' />
-<link rel="next" href="e-covsel.html" />
-<link rel="prev" href="s-umfpack.html" />
-<link rel="parent" href="c-spsolvers.html" />
-<link rel="next" href="e-covsel.html" />
-<meta name='aesop' content='information' />
-<title>7.3 Positive Definite Linear Equations (cvxopt.cholmod)</title>
-</head>
-<body>
-<DIV CLASS="navigation">
-<div id='top-navigation-panel' xml:id='top-navigation-panel'>
-<table align="center" width="100%" cellpadding="0" cellspacing="2">
-<tr>
-<td class='online-navigation'><a rel="prev" title="7.2 General Linear Equations"
- href="s-umfpack.html"><img src='previous.gif'
- border='0' height='32' alt='Previous Page' width='32' /></A></td>
-<td class='online-navigation'><a rel="parent" title="7. Sparse Linear Equation"
- href="c-spsolvers.html"><img src='up.gif'
- border='0' height='32' alt='Up One Level' width='32' /></A></td>
-<td class='online-navigation'><a rel="next" title="7.4 Example: Covariance Selection"
- href="e-covsel.html"><img src='next.gif'
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-<td align="center" width="100%">CVXOPT: A Python Package for Convex Optimization</td>
-<td class='online-navigation'><a rel="contents" title="Table of Contents"
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- border='0' height='32' alt='Contents' width='32' /></A></td>
-<td class='online-navigation'><img src='blank.gif'
- border='0' height='32' alt='' width='32' /></td>
-<td class='online-navigation'><a rel="index" title="Index"
- href="genindex.html"><img src='index.gif'
- border='0' height='32' alt='Index' width='32' /></A></td>
-</tr></table>
-<div class='online-navigation'>
-<b class="navlabel">Previous:</b>
-<a class="sectref" rel="prev" href="s-umfpack.html">7.2 General Linear Equations</A>
-<b class="navlabel">Up:</b>
-<a class="sectref" rel="parent" href="c-spsolvers.html">7. Sparse Linear Equation</A>
-<b class="navlabel">Next:</b>
-<a class="sectref" rel="next" href="e-covsel.html">7.4 Example: Covariance Selection</A>
-</div>
-<hr /></div>
-</DIV>
-<!--End of Navigation Panel-->
-
-<H1><A NAME="SECTION009300000000000000000"></A>
-<A NAME="s-cholmod"></A>
-<BR>
-7.3 Positive Definite Linear Equations (<tt class="module">cvxopt.cholmod</tt>)
-</H1>
-
-<P>
-<tt class="module">cvxopt.cholmod</tt> is an interface to the Cholesky factorization
-routines of the CHOLMOD package.
-It includes functions for Cholesky factorization of sparse positive
-definite matrices, and for solving sparse sets of linear equations with
-positive definite matrices.
-The routines can also be used for computing LDL<SPAN CLASS="MATH"><IMG
- WIDTH="14" HEIGHT="18" ALIGN="BOTTOM" BORDER="0"
- SRC="img40.gif"
- ALT="$\mathrm{{}^T}$"></SPAN>
-(or LDL<SPAN CLASS="MATH"><IMG
- WIDTH="14" HEIGHT="18" ALIGN="BOTTOM" BORDER="0"
- SRC="img89.gif"
- ALT="$\mathrm{{}^H}$"></SPAN>) factorizations of symmetric indefinite matrices
-(with L unit lower-triangular and D diagonal and nonsingular) if
-such a factorization exists.
-
-<P>
-<div class="seealso">
- <p class="heading">See Also:</p>
-
-<dl compact="compact" class="seeurl">
- <dt><a href='http://www.cise.ufl.edu/research/sparse/cholmod'
- >CHOLMOD code, documentation, copyright and license.</a></dt>
- <dd></dd>
- </dl>
-</div>
-
-<P>
-<dl><dt><table cellpadding="0" cellspacing="0"><tr valign="baseline">
- <td><nobr><b><tt id='l2h-138' xml:id='l2h-138' class="function">linsolve</tt></b>(</nobr></td>
- <td><var>A, B</var><big>[</big><var>, p=<code>None</code></var><big>[</big><var>,
-uplo='L'</var><big>]</big><var></var><big>]</big><var></var>)</td></tr></table></dt>
-<dd>
-Solves
-<BR><P></P>
-<DIV ALIGN="CENTER" CLASS="mathdisplay">
-<!-- MATH
- \begin{displaymath}
-AX = B
-\end{displaymath}
- -->
-
-<IMG
- WIDTH="60" HEIGHT="24" BORDER="0"
- SRC="img47.gif"
- ALT="\begin{displaymath}
-AX=B
-\end{displaymath}">
-</DIV>
-<BR CLEAR="ALL">
-<P></P>
-with <I>A</I> sparse and real symmetric or complex Hermitian.
-<var>B</var> is a dense matrix of the same type as <var>A</var>. On exit it
-is overwritten with the solution.
-The argument <var>p</var> is an integer matrix with length equal to the
-order of <var>A</var>, and specifies an optional reordering of <var>A</var>.
-If <var>p</var> is not specified, CHOLMOD used a reordering from the
-AMD library.
-Raises an <code>ArithmeticError</code> if the factorization does not exist.
-</dl>
-
-<P>
-As an example, we solve
-<BR>
-<DIV ALIGN="RIGHT" CLASS="mathdisplay">
-
-<!-- MATH
- \begin{equation}
-\left[ \begin{array}{rrrr}
- 10 & 0 & 3 & 0 \\
- 0 & 5 & 0 & -2 \\
- 3 & 0 & 5 & 0 \\
- 0 & -2 & 0 & 2
- \end{array}\right] X = \left[ \begin{array}{cc}
- 0 & 4 \\1 & 5 \\2 & 6 \\3 & 7\end{array} \right].
-\end{equation}
- -->
-<A NAME="e-A-pd"></A>
-<TABLE WIDTH="100%" ALIGN="CENTER">
-<TR VALIGN="MIDDLE"><TD></TD><TD ALIGN="CENTER" NOWRAP><A NAME="e-A-pd"></A><IMG
- WIDTH="258" HEIGHT="83" BORDER="0"
- SRC="img90.gif"
- ALT="\begin{displaymath}
-\left[ \begin{array}{rrrr}
-10 & 0 & 3 & 0 \\
-0 & 5 & 0 & ...
-...ray}{cc}
-0 & 4 \\ 1 & 5 \\ 2 & 6 \\ 3 & 7\end{array} \right].
-\end{displaymath}"></TD>
-<TD CLASS="eqno" WIDTH=10 ALIGN="RIGHT">
-(7.2)</TD></TR>
-</TABLE>
-<BR CLEAR="ALL"></DIV><P></P>
-<div class="verbatim"><pre>
->>> from cvxopt.base import matrix, spmatrix
->>> from cvxopt import cholmod
->>> A = spmatrix([10,3, 5,-2, 5, 2], [0,2, 1,3, 2, 3], [0,0, 1,1, 2, 3])
->>> X = matrix(range(8), (4,2), 'd')
->>> cholmod.linsolve(A,X)
->>> print X
--1.4634e-01 4.8780e-02
- 1.3333e+00 4.0000e+00
- 4.8780e-01 1.1707e+00
- 2.8333e+00 7.5000e+00
-</pre></div>
-
-<P>
-<dl><dt><table cellpadding="0" cellspacing="0"><tr valign="baseline">
- <td><nobr><b><tt id='l2h-139' xml:id='l2h-139' class="function">splinsolve</tt></b>(</nobr></td>
- <td><var>A, B</var><big>[</big><var>, p=<code>None</code></var><big>[</big><var>,
-uplo='L'</var><big>]</big><var></var><big>]</big><var></var>)</td></tr></table></dt>
-<dd>
-Similar to <tt class="function">linsolve()</tt> except that <var>B</var> is a sparse
-matrix and that the solution is returned as an output
-argument (as a new sparse matrix). <var>B</var> is not modified.
-</dl>
-
-<P>
-The following code computes the inverse of the coefficient matrix
-in (<A HREF="#e-A-pd">7.2</A>) as a sparse matrix.
-<div class="verbatim"><pre>
->>> X = cholmod.splinsolve(A, spmatrix(1.0,range(4),range(4)))
->>> print X
-SIZE: (4,4)
-(0, 0) 1.2195e-01
-(2, 0) -7.3171e-02
-(1, 1) 3.3333e-01
-(3, 1) 3.3333e-01
-(0, 2) -7.3171e-02
-(2, 2) 2.4390e-01
-(1, 3) 3.3333e-01
-(3, 3) 8.3333e-01
-</pre></div>
-
-<P>
-The functions <tt class="function">linsolve()</tt> and <tt class="function">splinsolve()</tt> are
-equivalent to <tt class="function">symbolic()</tt> and <tt class="function">numeric()</tt> called in
-sequence, followed by <tt class="function">solve()</tt>, respectively,
-<tt class="function">spsolve()</tt>.
-
-<P>
-<dl><dt><table cellpadding="0" cellspacing="0"><tr valign="baseline">
- <td><nobr><b><tt id='l2h-140' xml:id='l2h-140' class="function">symbolic</tt></b>(</nobr></td>
- <td><var>A</var><big>[</big><var>, p=<code>None</code></var><big>[</big><var>, uplo='L'</var><big>]</big><var></var><big>]</big><var></var>)</td></tr></table></dt>
-<dd>
-Performs a symbolic analysis of a sparse real symmetric or
-complex Hermitian matrix <var>A</var> for one of the two factorizations:
-<BR>
-<DIV ALIGN="RIGHT" CLASS="mathdisplay">
-
-<!-- MATH
- \begin{equation}
-PAP^T = LL^T, \qquad PAP^T = LL^H,
-\end{equation}
- -->
-<A NAME="e-chol-ll"></A>
-<TABLE WIDTH="100%" ALIGN="CENTER">
-<TR VALIGN="MIDDLE"><TD></TD><TD ALIGN="CENTER" NOWRAP><A NAME="e-chol-ll"></A><IMG
- WIDTH="244" HEIGHT="27" BORDER="0"
- SRC="img91.gif"
- ALT="\begin{displaymath}
-PAP^T = LL^T, \qquad PAP^T = LL^H,
-\end{displaymath}"></TD>
-<TD CLASS="eqno" WIDTH=10 ALIGN="RIGHT">
-(7.3)</TD></TR>
-</TABLE>
-<BR CLEAR="ALL"></DIV><P></P>
-and
-<BR>
-<DIV ALIGN="RIGHT" CLASS="mathdisplay">
-
-<!-- MATH
- \begin{equation}
-PAP^T = LDL^T, \qquad PAP^T = LDL^H,
-\end{equation}
- -->
-<A NAME="e-chol-ldl"></A>
-<TABLE WIDTH="100%" ALIGN="CENTER">
-<TR VALIGN="MIDDLE"><TD></TD><TD ALIGN="CENTER" NOWRAP><A NAME="e-chol-ldl"></A><IMG
- WIDTH="272" HEIGHT="27" BORDER="0"
- SRC="img92.gif"
- ALT="\begin{displaymath}
-PAP^T = LDL^T, \qquad PAP^T = LDL^H,
-\end{displaymath}"></TD>
-<TD CLASS="eqno" WIDTH=10 ALIGN="RIGHT">
-(7.4)</TD></TR>
-</TABLE>
-<BR CLEAR="ALL"></DIV><P></P>
-where <I>P</I> is a permutation matrix, <I>L</I> is lower triangular
-(unit lower triangular in the second factorization), and
-<I>D</I> is nonsingular diagonal. The type of factorization depends
-on the value of <code>options['supernodal']</code>.
-
-<P>
-If <var>uplo</var> is <code>'L'</code>, only the lower triangular part of <var>A</var>
-is accessed and the upper triangular part is ignored.
-If <var>uplo</var> is <code>'U'</code>, only the upper triangular part of <var>A</var>
-is accessed and the lower triangular part is ignored.
-
-<P>
-The symbolic factorization is returned as an opaque C object that
-can be passed to <tt class="function">cholmod.numeric()</tt>.
-</dl>
-
-<P>
-<dl><dt><table cellpadding="0" cellspacing="0"><tr valign="baseline">
- <td><nobr><b><tt id='l2h-141' xml:id='l2h-141' class="function">numeric</tt></b>(</nobr></td>
- <td><var>A, F</var>)</td></tr></table></dt>
-<dd>
-Performs a numeric factorization of a sparse symmetric matrix
-as (<A HREF="#e-chol-ll">7.3</A>) or (<A HREF="#e-chol-ldl">7.4</A>).
-The argument <var>F</var> is the symbolic factorization
-computed by <tt class="function">cholmod.symbolic()</tt> applied to the matrix <var>A</var>,
-or to another sparse matrix with the same sparsity pattern and
-typecode, or by
-<tt class="function">cholmod.numeric()</tt> applied to a matrix with the same
-sparsity pattern and typecode as <var>A</var>.
-
-<P>
-If <var>F</var> was created by a <tt class="function">cholmod.symbolic</tt> with
-<var>uplo</var> equal to <code>'L'</code>, then only the lower triangular part
-of <var>A</var> is accessed and the upper triangular part is ignored.
-If it was created with <var>uplo</var> is <code>'U'</code>, then only the upper
-triangular part of <var>A</var> is accessed and the lower triangular part
-is ignored.
-
-<P>
-On successful exit, the factorization is stored in <var>F</var>.
-Raises an <code>ArithmeticError</code> if the factorization does not
-exist.
-</dl>
-
-<P>
-<dl><dt><table cellpadding="0" cellspacing="0"><tr valign="baseline">
- <td><nobr><b><tt id='l2h-142' xml:id='l2h-142' class="function">solve</tt></b>(</nobr></td>
- <td><var>F, B</var><big>[</big><var>, sys=0</var><big>]</big><var></var>)</td></tr></table></dt>
-<dd>
-Solves one of the following linear equations where <var>B</var> is a dense
-matrix and <var>F</var> is the numeric
-factorization (<A HREF="#e-chol-ll">7.3</A>) or (<A HREF="#e-chol-ldl">7.4</A>) computed by
-<tt class="function">cholmod_numeric()</tt>.
-<var>sys</var> is an integer with values between 0 and 8.
-
-<P>
-<DIV ALIGN="CENTER">
-<TABLE CELLPADDING=3 BORDER="1">
-<TR><TD ALIGN="CENTER"><var>sys</var></TD>
-<TD ALIGN="CENTER">equation</TD>
-</TR>
-<TR><TD ALIGN="CENTER">0</TD>
-<TD ALIGN="CENTER"><SPAN CLASS="MATH"><IMG
- WIDTH="64" HEIGHT="14" ALIGN="BOTTOM" BORDER="0"
- SRC="img93.gif"
- ALT="$AX=B$"></SPAN></TD>
-</TR>
-<TR><TD ALIGN="CENTER">1</TD>
-<TD ALIGN="CENTER"><SPAN CLASS="MATH"><IMG
- WIDTH="98" HEIGHT="16" ALIGN="BOTTOM" BORDER="0"
- SRC="img94.gif"
- ALT="$LDL^TX=B$"></SPAN></TD>
-</TR>
-<TR><TD ALIGN="CENTER">2</TD>
-<TD ALIGN="CENTER"><SPAN CLASS="MATH"><IMG
- WIDTH="88" HEIGHT="14" ALIGN="BOTTOM" BORDER="0"
- SRC="img95.gif"
- ALT="$LDLX=B$"></SPAN></TD>
-</TR>
-<TR><TD ALIGN="CENTER">3</TD>
-<TD ALIGN="CENTER"><SPAN CLASS="MATH"><IMG
- WIDTH="87" HEIGHT="16" ALIGN="BOTTOM" BORDER="0"
- SRC="img96.gif"
- ALT="$DL^TX=B$"></SPAN></TD>
-</TR>
-<TR><TD ALIGN="CENTER">4</TD>
-<TD ALIGN="CENTER"><SPAN CLASS="MATH"><IMG
- WIDTH="63" HEIGHT="14" ALIGN="BOTTOM" BORDER="0"
- SRC="img97.gif"
- ALT="$LX=B$"></SPAN></TD>
-</TR>
-<TR><TD ALIGN="CENTER">5</TD>
-<TD ALIGN="CENTER"><SPAN CLASS="MATH"><IMG
- WIDTH="73" HEIGHT="16" ALIGN="BOTTOM" BORDER="0"
- SRC="img98.gif"
- ALT="$L^TX=B$"></SPAN></TD>
-</TR>
-<TR><TD ALIGN="CENTER">6</TD>
-<TD ALIGN="CENTER"><SPAN CLASS="MATH"><IMG
- WIDTH="66" HEIGHT="14" ALIGN="BOTTOM" BORDER="0"
- SRC="img99.gif"
- ALT="$DX=B$"></SPAN></TD>
-</TR>
-<TR><TD ALIGN="CENTER">7</TD>
-<TD ALIGN="CENTER"><SPAN CLASS="MATH"><IMG
- WIDTH="75" HEIGHT="16" ALIGN="BOTTOM" BORDER="0"
- SRC="img100.gif"
- ALT="$P^TX=B$"></SPAN></TD>
-</TR>
-<TR><TD ALIGN="CENTER">8</TD>
-<TD ALIGN="CENTER"><SPAN CLASS="MATH"><IMG
- WIDTH="65" HEIGHT="14" ALIGN="BOTTOM" BORDER="0"
- SRC="img101.gif"
- ALT="$PX=B$"></SPAN></TD>
-</TR>
-</TABLE>
-</DIV>
-
-<P>
-(If <var>F</var> is a Cholesky factorization of the form (<A HREF="#e-chol-ll">7.3</A>),
-<I>D</I> is an identity matrix in this table.
-If <var>A</var> is complex, <SPAN CLASS="MATH"><IMG
- WIDTH="25" HEIGHT="16" ALIGN="BOTTOM" BORDER="0"
- SRC="img102.gif"
- ALT="$L^T$"></SPAN> should be replaced by <SPAN CLASS="MATH"><IMG
- WIDTH="27" HEIGHT="16" ALIGN="BOTTOM" BORDER="0"
- SRC="img103.gif"
- ALT="$L^H$"></SPAN>.)
-
-<P>
-The matrix <var>B</var> is a dense <code>'d'</code> or <code>'z'</code> matrix, with the same type
-as <var>A</var>. On exit it is overwritten by the solution.
-</dl>
-
-<P>
-<dl><dt><table cellpadding="0" cellspacing="0"><tr valign="baseline">
- <td><nobr><b><tt id='l2h-143' xml:id='l2h-143' class="function">spsolve</tt></b>(</nobr></td>
- <td><var>F, B</var><big>[</big><var>, sys=0</var><big>]</big><var></var>)</td></tr></table></dt>
-<dd>
-Similar to <tt class="function">solve()</tt>, except that <var>B</var> is a
-sparse matrix, and the solution is returned as an output argument
-(as a sparse matrix). <var>B</var> must have the same typecode as <var>A</var>.
-</dl>
-
-<P>
-For the same example as above:
-<div class="verbatim"><pre>
->>> X = matrix(range(8), (4,2), 'd')
->>> F = cholmod.symbolic(A)
->>> cholmod.numeric(A,F)
->>> cholmod.solve(F,X)
->>> print X
--1.4634e-01 4.8780e-02
- 1.3333e+00 4.0000e+00
- 4.8780e-01 1.1707e+00
- 2.8333e+00 7.5000e+00
-</pre></div>
-
-<P>
-<dl><dt><table cellpadding="0" cellspacing="0"><tr valign="baseline">
- <td><nobr><b><tt id='l2h-144' xml:id='l2h-144' class="function">diag</tt></b>(</nobr></td>
- <td><var>F</var>)</td></tr></table></dt>
-<dd>
-Returns the diagonal elements of the Cholesky factor <I>L</I>
-in (<A HREF="#e-chol-ll">7.3</A>), as a dense matrix of the same type as <var>A</var>.
-Note that this only applies to Cholesky factorizations.
-The matrix <I>D</I> in an LDL<SPAN CLASS="MATH"><IMG
- WIDTH="14" HEIGHT="18" ALIGN="BOTTOM" BORDER="0"
- SRC="img40.gif"
- ALT="$\mathrm{{}^T}$"></SPAN> factorization can be
-retrieved via <tt class="function">cholmod.solve()</tt> with <var>sys</var> equal to 6.
-</dl>
-
-<P>
-In the functions listed above, the default values of the control
-parameters described in the CHOLMOD user guide are used, except for
-<tt class="ctype">Common->print</tt> which is set to 0 instead of 3
-and
-<tt class="ctype">Common->supernodal</tt> which is set to 2 instead of 1.
-These parameters (and a few others) can be modified by making an
-entry in the dictionary <tt class="member">cholmod.options</tt>.
-The meaning of these parameters is as follows.
-<DL>
-<DT><STRONG><code>options['supernodal']</code></STRONG></DT>
-<DD>If equal to 0, a
-factorization (<A HREF="#e-chol-ldl">7.4</A>) is computed using a simplicial
-algorithm.
-If equal to 2, a factorization (<A HREF="#e-chol-ll">7.3</A>) is
-computed using a supernodal algorithm.
-If equal to 1, the most efficient of the two factorizations is
-selected, based on the sparsity pattern. Default: 2.
-
-<P>
-</DD>
-<DT><STRONG><code>options['print']</code></STRONG></DT>
-<DD>A nonnegative integer that controls the
-amount of output printed to the screen.
-Default: 0 (no output).
-</DD>
-</DL>
-
-<P>
-As an example that illustrates <tt class="function">diag()</tt> and the use of
-<tt class="member">cholmod.options</tt>, we compute the logarithm of the determinant
-of the coefficient matrix in (<A HREF="#e-A-pd">7.2</A>) by two methods.
-<div class="verbatim"><pre>
->>> import math
->>> from cvxopt.cholmod import options
->>> from cvxopt.base import log
->>> options['supernodal'] = 2
->>> F = cholmod.symbolic(A)
->>> cholmod.numeric(A,F)
->>> print 2.0 * sum(log(cholmod.diag(F)))
-5.50533153593
-</pre></div>
-<div class="verbatim"><pre>
->>> options['supernodal'] = 0
->>> F = cholmod.symbolic(A)
->>> cholmod.numeric(A,F)
->>> Di = matrix(1.0, (4,1))
->>> cholmod.solve(F,Di,sys=6)
->>> print -sum(log(Di))
-5.50533153593
-</pre></div>
-
-<P>
-
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+++ /dev/null
@@ -1,470 +0,0 @@
-<!DOCTYPE html PUBLIC "-//W3C//DTD HTML 4.0 Transitional//EN">
-<html>
-<head>
-<link rel="STYLESHEET" href="cvxopt.css" type='text/css' />
-<link rel="first" href="cvxopt.html" title='CVXOPT: A Python Package for Convex Optimization' />
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-</head>
-<body>
-<DIV CLASS="navigation">
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-
-<H1><A NAME="SECTION005100000000000000000"></A> <A NAME="s-conventions"></A>
-<BR>
-3.1 Matrix Classes
-</H1>
-
-<P>
-The BLAS exploit several types of matrix structure: symmetric,
-Hermitian, triangular, and banded. We represent all these matrix
-classes by dense real or complex <tt class="class">matrix</tt> objects, with additional
-arguments that specify the structure.
-
-<P>
-<DL>
-<DT><STRONG>Vector</STRONG></DT>
-<DD>A real or complex <I>n</I>-vector is represented by a <tt class="class">matrix</tt> of type
-<code>'d'</code> or <code>'z'</code> and length <I>n</I>, with the entries of the vector
-stored in column-major order.
-
-<P>
-</DD>
-<DT><STRONG>General matrix</STRONG></DT>
-<DD>A general real or complex <I>m</I> by <I>n</I> matrix is represented by
-a real or complex <tt class="class">matrix</tt> of size (<I>m</I>, <I>n</I>).
-
-<P>
-</DD>
-<DT><STRONG>Symmetric matrix</STRONG></DT>
-<DD>A real or complex symmetric matrix of order <I>n</I> is represented
-by a real or complex <tt class="class">matrix</tt> of size (<I>n</I>, <I>n</I>), and a character
-argument <var>uplo</var> with two possible values:
-<code>'L'</code> and <code>'U'</code>.
-If <var>uplo</var> is <code>'L'</code>, the lower triangular part of the
-symmetric matrix is stored; if <var>uplo</var> is <code>'U'</code>, the upper
-triangular part is stored. A square <tt class="class">matrix</tt> <var>X</var> of size
-(<I>n</I>, <I>n</I>) can therefore be used to represent the symmetric
-matrices
-<P></P>
-<DIV ALIGN="CENTER" CLASS="mathdisplay">
-<!-- MATH
- \begin{eqnarray*}
-& \left[\begin{array}{ccccc}
-X[0,0] & X[1,0] & X[2,0] & \cdots & X[n-1,0] \\
-X[1,0] & X[1,1] & X[2,1] & \cdots & X[n-1,1] \\
-X[2,0] & X[2,1] & X[2,2] & \cdots & X[n-1,2] \\
-\vdots & \vdots & \vdots & \ddots & \vdots \\
-X[n-1,0] & X[n-1,1] & X[n-1,2] & \cdots & X[n-1,n-1]
-\end{array}\right] & \mbox{if uplo = 'L'}, \\
-& \left[\begin{array}{ccccc}
-X[0,0] & X[0,1] & X[0,2] & \cdots & X[0,n-1] \\
-X[0,1] & X[1,1] & X[1,2] & \cdots & X[1,n-1] \\
-X[0,2] & X[1,2] & X[2,2] & \cdots & X[2,n-1] \\
-\vdots & \vdots & \vdots & \ddots & \vdots \\
-X[0,n-1] & X[1,n-1] & X[2,n-1] & \cdots & X[n-1,n-1]
-\end{array}\right] & \mbox{if uplo = U'}.
-\end{eqnarray*}
- -->
-<IMG
- WIDTH="542" HEIGHT="225" BORDER="0"
- SRC="img1.gif"
- ALT="\begin{eqnarray*}
-& \left[\begin{array}{ccccc}
-X[0,0] & X[1,0] & X[2,0] & \cdots...
-...& \cdots & X[n-1,n-1]
-\end{array}\right] & \mbox{if uplo = U'}.
-\end{eqnarray*}"></DIV>
-<BR CLEAR="ALL"><P></P>
-<BR CLEAR="ALL"><P></P>
-
-<P>
-</DD>
-<DT><STRONG>Complex Hermitian matrix</STRONG></DT>
-<DD>A complex Hermitian matrix of order <I>n</I> is represented
-by a <tt class="class">matrix</tt> of type <code>'z'</code> and size (<I>n</I>, <I>n</I>), and
-a character argument <var>uplo</var> with the ame meaning as for symmetric
-matrices.
-A complex <tt class="class">matrix</tt> <var>X</var> of size (<I>n</I>, <I>n</I>) can
-represent the Hermitian matrices
-<P></P>
-<DIV ALIGN="CENTER" CLASS="mathdisplay">
-<!-- MATH
- \begin{eqnarray*}
-&
-\left[\begin{array}{ccccc}
-\Re X[0,0] & \bar X[1,0] & \bar X[2,0] & \cdots & \bar X[n-1,0] \\
-X[1,0] & \Re X[1,1] & \bar X[2,1] & \cdots & \bar X[n-1,1] \\
-X[2,0] & X[2,1] & \Re X[2,2] & \cdots & \bar X[n-1,2] \\
-\vdots & \vdots & \vdots & \ddots & \vdots \\
-X[n-1,0] & X[n-1,1] & X[n-1,2] & \cdots & \Re X[n-1,n-1]
-\end{array}\right] & \mbox{if uplo = 'L'}, \\
-& \left[\begin{array}{ccccc}
-\Re X[0,0] & X[0,1] & X[0,2] & \cdots & X[0,n-1] \\
-\bar X[0,1] & \Re X[1,1] & X[1,2] & \cdots & X[1,n-1] \\
-\bar X[0,2] & \bar X[1,2] & \Re X[2,2] & \cdots & X[2,n-1] \\
-\vdots & \vdots & \vdots & \ddots & \vdots \\
-\bar X[0,n-1] & \bar X[1,n-1] & \bar X[2,n-1] & \cdots & \Re X[n-1,n-1]
-\end{array}\right] & \mbox{if uplo = 'U'}.
-\end{eqnarray*}
- -->
-<IMG
- WIDTH="556" HEIGHT="225" BORDER="0"
- SRC="img2.gif"
- ALT="\begin{eqnarray*}
-&
-\left[\begin{array}{ccccc}
-\Re X[0,0] & \bar X[1,0] & \bar X...
-...dots & \Re X[n-1,n-1]
-\end{array}\right] & \mbox{if uplo = 'U'}.
-\end{eqnarray*}"></DIV>
-<BR CLEAR="ALL"><P></P>
-<BR CLEAR="ALL"><P></P>
-
-<P>
-</DD>
-<DT><STRONG>Triangular matrix</STRONG></DT>
-<DD>A real or complex triangular matrix of order <I>n</I> is represented
-by a real or complex <tt class="class">matrix</tt> of size (<I>n</I>, <I>n</I>), and two
-character arguments: an argument <var>uplo</var> with possible values
-<code>'L'</code> and <code>'U'</code> to distinguish between lower and upper
-triangular matrices, and an argument <var>diag</var> with possible values
-<code>'U'</code> and <code>'N'</code> to distinguish between unit and non-unit
-triangular matrices. A square <tt class="class">matrix</tt> <var>X</var> of size
-(<I>n</I>, <I>n</I>) can represent the triangular matrices
-<P></P>
-<DIV ALIGN="CENTER" CLASS="mathdisplay">
-<!-- MATH
- \begin{eqnarray*}
-& \left[\begin{array}{ccccc}
-X[0,0] & 0 & 0 & \cdots & 0 \\
-X[1,0] & X[1,1] & 0 & \cdots & 0 \\
-X[2,0] & X[2,1] & X[2,2] & \cdots & 0 \\
-\vdots & \vdots & \vdots & \ddots & \vdots \\
-X[n-1,0] & X[n-1,1] & X[n-1,2] & \cdots & X[n-1,n-1]
-\end{array}\right] & \mbox{if uplo = 'L' and diag = 'N'}, \\
-& \left[\begin{array}{ccccc}
-1 & 0 & 0 & \cdots & 0 \\
-X[1,0] & 1 & 0 & \cdots & 0 \\
-X[2,0] & X[2,1] & 1 & \cdots & 0 \\
-\vdots & \vdots & \vdots & \ddots & \vdots \\
-X[n-1,0] & X[n-1,1] & X[n-1,2] & \cdots & 1
-\end{array}\right] & \mbox{if uplo = 'L' and diag = 'U'}, \\
-& \left[\begin{array}{ccccc}
-X[0,0] & X[0,1] & X[0,2] & \cdots & X[0,n-1] \\
-0 & X[1,1] & X[1,2] & \cdots & X[1,n-1] \\
-0 & 0 & X[2,2] & \cdots & X[2,n-1] \\
-\vdots & \vdots & \vdots & \ddots & \vdots \\
-0 & 0 & 0 & \cdots & X[n-1,n-1]
-\end{array}\right] & \mbox{if uplo = 'U' and diag = 'N'}, \\
-& \left[\begin{array}{ccccc}
-1 & X[0,1] & X[0,2] & \cdots & X[0,n-1] \\
-0 & 1 & X[1,2] & \cdots & X[1,n-1] \\
-0 & 0 & 1 & \cdots & X[2,n-1] \\
-\vdots & \vdots & \vdots & \ddots & \vdots \\
-0 & 0 & 0 & \cdots & 1
-\end{array}\right] & \mbox{if uplo = 'U' and diag = 'U'}.
-\end{eqnarray*}
- -->
-<IMG
- WIDTH="643" HEIGHT="450" BORDER="0"
- SRC="img3.gif"
- ALT="\begin{eqnarray*}
-& \left[\begin{array}{ccccc}
-X[0,0] & 0 & 0 & \cdots & 0 \\
-X...
-...ts & 1
-\end{array}\right] & \mbox{if uplo = 'U' and diag = 'U'}.
-\end{eqnarray*}"></DIV>
-<BR CLEAR="ALL"><P></P>
-<BR CLEAR="ALL"><P></P>
-
-<P>
-</DD>
-<DT><STRONG>General band matrix</STRONG></DT>
-<DD>A general real or complex <I>m</I> by <I>n</I> band matrix with <I>kl</I>
-subdiagonals and <I>ku</I> superdiagonals is represented by a real or
-complex <tt class="class">matrix</tt> <var>X</var> of size (<I>kl+ku+1</I>, <I>n</I>), and the two
-integers <I>m</I> and <I>kl</I>.
-The diagonals of the band matrix are stored in the rows of <var>X</var>,
-starting at the top diagonal, and shifted horizontally so that the
-entries of the
-<I>k</I>th column of the band matrix are stored in column <I>k</I> of
-<var>X</var>. A <tt class="class">matrix</tt> <var>X</var> of size (<I>kl+ku+1</I>, <I>n</I>) therefore
-represents the <I>m</I> by <I>n</I> band matrix
-<BR><P></P>
-<DIV ALIGN="CENTER" CLASS="mathdisplay">
-<!-- MATH
- \begin{displaymath}
-\left[ \begin{array}{ccccccc}
-X[k_u,0] & X[k_u-1,1] & X[k_u-2,2] & \cdots & X[0,k_u] & 0 & \cdots \\
-X[k_u+1,0] & X[k_u,1] & X[k_u-1,2] & \cdots & X[1,k_u] & X[0,k_u+1] & \cdots \\
-X[k_u+2,0] & X[k_u+1,1] & X[k_u,2] & \cdots & X[2,k_u] & X[1,k_u+1] & \cdots \\
- \vdots & \vdots & \vdots & \ddots & \vdots & \vdots & \ddots \\
-X[k_u+k_l,0] & X[k_u+k_l-1,1] & X[k_u+k_l-2,2] & \cdots & & & \\
-0 & X[k_u+k_l,1] & X[k_u+k_l-1,2] & \cdots & & & \\
-\vdots & \vdots & \vdots & \ddots & & &
-\end{array}\right].
-\end{displaymath}
- -->
-
-<IMG
- WIDTH="624" HEIGHT="161" BORDER="0"
- SRC="img4.gif"
- ALT="\begin{displaymath}
-\left[ \begin{array}{ccccccc}
-X[k_u,0] & X[k_u-1,1] & X[k_u-...
-...
-\vdots & \vdots & \vdots & \ddots & & &
-\end{array}\right].
-\end{displaymath}">
-</DIV>
-<BR CLEAR="ALL">
-<P></P>
-
-<P>
-</DD>
-<DT><STRONG>Symmetric band matrix</STRONG></DT>
-<DD>A real or complex symmetric band matrix of order <I>n</I> with <I>k</I>
-subdiagonals, is represented by a real or complex matrix <var>X</var> of
-size (<I>k</I>+1, <I>n</I>), and an argument <I>uplo</I> to indicate
-whether the subdiagonals (<I>uplo</I> is <code>'L'</code>) or superdiagonals
-(<I>uplo</I> is <code>'U'</code>) are stored.
-The <I>k</I>+1 diagonals are stored as rows of <var>X</var>, starting at the top
-diagonal (<EM>i.e.</EM>, the main diagonal if <I>uplo</I> is <code>'L'</code>, or
-the <I>k</I>th superdiagonal if <I>uplo</I> is <code>'U'</code>) and shifted
-horizontally so that the entries of the
-<I>k</I>th column of the band matrix are stored in column <I>k</I> of
-<var>X</var>. A <tt class="class">matrix</tt> <I>X</I> of size (<I>k</I>+1, <I>n</I>) can therefore
-represent the band matrices
-<P></P>
-<DIV ALIGN="CENTER" CLASS="mathdisplay">
-<!-- MATH
- \begin{eqnarray*}
-& \left[ \begin{array}{ccccccc}
-X[0,0] & X[1,0] & X[2,0] & \cdots & X[k,0] & 0 & \cdots \\
-X[1,0] & X[0,1] & X[1,1] & \cdots & X[k-1,1] & X[k,1] & \cdots \\
-X[2,0] & X[1,1] & X[0,2] & \cdots & X[k-2,2] & X[k-1,2] & \cdots \\
-\vdots & \vdots & \vdots & \ddots & \vdots & \vdots & \ddots \\
-X[k,0] & X[k-1,1] & X[k-2,2] & \cdots & & & \\
-0 & X[k,1] & X[k-1,2] & \cdots & & & \\
-\vdots & \vdots & \vdots & \ddots & & &
-\end{array}\right] & \mbox{if uplo = 'L'}, \\
-&
-\left[ \begin{array}{ccccccc}
-X[k,0] & X[k-1,1] & X[k-2,2] & \cdots & X[0,k] & 0 & \cdots \\
-X[k-1,1] & X[k,1] & X[k-1,2] & \cdots & X[1,k] & X[0,k+1] & \cdots \\
-X[k-2,2] & X[k-1,2] & X[k,2] & \cdots & X[2,k] & X[1,k+1] & \cdots \\
-\vdots & \vdots & \vdots & \ddots & \vdots & \vdots & \ddots \\
-X[0,k] & X[1,k] & X[2,k] & \cdots & & & \\
-0 & X[0,k+1] & X[1,k+1] & \cdots & & & \\
-\vdots & \vdots & \vdots & \ddots & & &
-\end{array}\right] & \mbox{if uplo='U'}.
-\end{eqnarray*}
- -->
-<IMG
- WIDTH="608" HEIGHT="322" BORDER="0"
- SRC="img5.gif"
- ALT="\begin{eqnarray*}
-& \left[ \begin{array}{ccccccc}
-X[0,0] & X[1,0] & X[2,0] & \cd...
-... \vdots & \ddots & & &
-\end{array}\right] & \mbox{if uplo='U'}.
-\end{eqnarray*}"></DIV>
-<BR CLEAR="ALL"><P></P>
-<BR CLEAR="ALL"><P></P>
-
-<P>
-</DD>
-<DT><STRONG>Hermitian band matrix</STRONG></DT>
-<DD>A complex Hermitian band matrix of order <I>n</I> with <I>k</I>
-subdiagonals is represented by a complex matrix of size
-(<I>k+1</I>, <I>n</I>) and an argument <var>uplo</var>.
-A <tt class="class">matrix</tt> <I>X</I> of size (<I>k</I>+1, <I>n</I>) can represent the band
-matrices
-<P></P>
-<DIV ALIGN="CENTER" CLASS="mathdisplay">
-<!-- MATH
- \begin{eqnarray*}
-& \left[ \begin{array}{ccccccc}
-\Re X[0,0] & \bar X[1,0] & \bar X[2,0] & \cdots & \bar X[k,0] & 0 & \cdots \\
-X[1,0] & \Re X[0,1] & \bar X[1,1] & \cdots & \bar X[k-1,1] & \bar X[k,1] & \cdots \\
-X[2,0] & X[1,1] & \Re X[0,2] & \cdots & \bar X[k-2,2] & \bar X[k-1,2] & \cdots \\
-\vdots & \vdots & \vdots & \ddots & \vdots & \vdots & \ddots \\
-X[k,0] & X[k-1,1] & X[k-2,2] & \cdots & & & \\
-0 & X[k,1] & X[k-1,2] & \cdots & & & \\
-\vdots & \vdots & \vdots & \ddots & & &
-\end{array}\right] & \mbox{if uplo = 'L'}, \\
-&
-\left[ \begin{array}{ccccccc}
-\Re X[k,0] & X[k-1,1] & X[k-2,2] & \cdots & X[0,k] & 0 & \cdots \\
-\bar X[k-1,1] & \Re X[k,1] & X[k-1,2] & \cdots & X[1,k] & X[0,k+1] & \cdots \\
-\bar X[k-2,2] & \bar X[k-1,2] & \Re X[k,2] & \cdots & X[2,k] & X[1,k+1] & \cdots \\
-\vdots & \vdots & \vdots & \ddots & \vdots & \vdots & \ddots \\
-\bar X[0,k] & \bar X[1,k] & \bar X[2,k] & \cdots & & & \\
-0 & \bar X[0,k+1] & \bar X[1,k+1] & \cdots & & & \\
-\vdots & \vdots & \vdots & \ddots & & &
-\end{array}\right] & \mbox{if uplo='U'}.
-\end{eqnarray*}
- -->
-<IMG
- WIDTH="619" HEIGHT="322" BORDER="0"
- SRC="img6.gif"
- ALT="\begin{eqnarray*}
-& \left[ \begin{array}{ccccccc}
-\Re X[0,0] & \bar X[1,0] & \ba...
-... \vdots & \ddots & & &
-\end{array}\right] & \mbox{if uplo='U'}.
-\end{eqnarray*}"></DIV>
-<BR CLEAR="ALL"><P></P>
-<BR CLEAR="ALL"><P></P>
-
-<P>
-</DD>
-<DT><STRONG>Triangular band matrix</STRONG></DT>
-<DD>A triangular band matrix of order <I>n</I> with <I>k</I> subdiagonals or
-superdiagonals is represented by a real complex matrix of size
-(<I>k+1</I>, <I>n</I>) and two character arguments <var>uplo</var> and
-<var>diag</var>.
-A <tt class="class">matrix</tt> <I>X</I> of size (<I>k</I>+1, <I>n</I>) can represent the band
-matrices
-<P></P>
-<DIV ALIGN="CENTER" CLASS="mathdisplay">
-<!-- MATH
- \begin{eqnarray*}
-& \left[ \begin{array}{cccc}
-X[0,0] & 0 & 0 & \cdots \\
-X[1,0] & X[0,1] & 0 & \cdots \\
-X[2,0] & X[1,1] & X[0,2] & \cdots \\
-\vdots & \vdots & \vdots & \ddots \\
-X[k,0] & X[k-1,1] & X[k-2,2] & \cdots \\
-0 & X[k,1] & X[k-1,1] & \cdots \\
-\vdots & \vdots & \vdots & \ddots
-\end{array}\right] & \mbox{if uplo = 'L' and diag = 'N'}, \\
-& \left[ \begin{array}{cccc}
-1 & 0 & 0 & \cdots \\
-X[1,0] & 1 & 0 & \cdots \\
-X[2,0] & X[1,1] & 1 & \cdots \\
-\vdots & \vdots & \vdots & \ddots \\
-X[k,0] & X[k-1,1] & X[k-2,2] & \cdots \\
-0 & X[k,1] & X[k-1,2] & \cdots \\
-\vdots & \vdots & \vdots & \ddots
-\end{array}\right] & \mbox{if uplo = 'L' and diag = 'U'}, \\
-& \left[ \begin{array}{ccccccc}
-X[k,0] & X[k-1,1] & X[k-2,3] & \cdots & X[0,k] & 0 & \cdots\\
-0 & X[k,1] & X[k-1,2] & \cdots & X[1,k] & X[0,k+1] & \cdots \\
-0 & 0 & X[k,2] & \cdots & X[2,k] & X[1,k+1] & \cdots \\
-\vdots & \vdots & \vdots & \ddots & \vdots & \vdots & \ddots
-\end{array}\right] & \mbox{if uplo = 'U' and diag = 'N'}, \\
-& \left[ \begin{array}{ccccccc}
-1 & X[k-1,1] & X[k-2,3] & \cdots & X[0,k] & 0 & \cdots\\
-0 & 1 & X[k-1,2] & \cdots & X[1,k] & X[0,k+1] & \cdots \\
-0 & 0 & 1 & \cdots & X[2,k] & X[1,k+1] & \cdots \\
-\vdots & \vdots & \vdots & \ddots & \vdots & \vdots & \ddots
-\end{array}\right] & \mbox{if uplo = 'U' and diag = 'U'}.
-\end{eqnarray*}
- -->
-<IMG
- WIDTH="681" HEIGHT="509" BORDER="0"
- SRC="img7.gif"
- ALT="\begin{eqnarray*}
-& \left[ \begin{array}{cccc}
-X[0,0] & 0 & 0 & \cdots \\
-X[1,0...
-...ddots
-\end{array}\right] & \mbox{if uplo = 'U' and diag = 'U'}.
-\end{eqnarray*}"></DIV>
-<BR CLEAR="ALL"><P></P>
-<BR CLEAR="ALL"><P></P>
-</DD>
-</DL>
-
-<P>
-When discussing BLAS functions in the following sections we will
-omit several less important optional arguments that can
-be used to select submatrices for in-place operations.
-The complete specification is documented in the docstrings of the
-source code and the <b class="program">pydoc</b> help program.
-
-<P>
-
-<DIV CLASS="navigation">
-<div class='online-navigation'>
-<p></p><hr />
-<table align="center" width="100%" cellpadding="0" cellspacing="2">
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+++ /dev/null
@@ -1,261 +0,0 @@
-<!DOCTYPE html PUBLIC "-//W3C//DTD HTML 4.0 Transitional//EN">
-<html>
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-
-<H1><A NAME="SECTION004100000000000000000"></A><A NAME="s-creating-matrices"></A>
-<BR>
-2.1 Creating Matrices
-</H1>
-A <tt class="class">matrix</tt> object is created by calling the function <tt class="function">matrix()</tt>.
-The arguments specify the values of the coefficients, the
-dimensions, and the type (integer, double or complex) of the matrix.
-
-<P>
-<dl><dt><table cellpadding="0" cellspacing="0"><tr valign="baseline">
- <td><nobr><b><tt id='l2h-1' xml:id='l2h-1' class="function">matrix</tt></b>(</nobr></td>
- <td><var>x</var><big>[</big><var>, size</var><big>[</big><var>, tc</var><big>]</big><var></var><big>]</big><var></var>)</td></tr></table></dt>
-<dd>
-<var>size</var> is a tuple of length two with the matrix dimensions.
-The number of rows and/or the number of columns can be zero.
-
-<P>
-<var>tc</var> stands for typecode. The possible values are <code>'i'</code>, <code>'d'</code> and
-<code>'z'</code>, for integer, real (double) and complex matrices, respectively.
-
-<P>
-<var>x</var> can be a number, a sequence of numbers, a dense or sparse
-matrix, a two-dimensional <tt class="module">numarray</tt> array, or a list of
-lists of matrices and numbers.
-
-<UL>
-<LI>If <var>x</var> is a number (Python integer, float or complex), a matrix
-is created with the dimensions specified by <var>size</var> and with all the
-coefficients equal to <var>x</var>.
-The default value of <var>size</var> is (1,1), and the default value
-of <var>tc</var> is the type of <var>x</var>.
-If necessary, the type of <var>x</var> is converted (from integer to double
-when used to create a matrix of type <code>'d'</code>, and from integer or
-double to complex when used to create a matrix of type <code>'z'</code>).
-
-<P>
-<div class="verbatim"><pre>
->>> from cvxopt.base import matrix
->>> A = matrix(1, (1,4))
->>> print A
- 1 1 1 1
->>> A = matrix(1.0, (1,4))
->>> print A
- 1.0000e+00 1.0000e+00 1.0000e+00 1.0000e+00
->>> A = matrix(1+1j)
->>> print A
- 1.0000e+00+j1.0000e+00
-</pre></div>
-
-<P>
-</LI>
-<LI>If <var>x</var> is a sequence of numbers (list, tuple, <tt class="module">array</tt>
-array, xrange object, one-dimensional <tt class="module">numarray</tt> array, ...),
-then the numbers are interpreted as the coefficients of a matrix in
-column-major order. The length of <var>x</var> must be equal to the product
-of <var>size</var>[0] and <var>size</var>[1].
-If <var>size</var> is not specified, a matrix with one column is created.
-If <var>tc</var> is not specified, it is determined from the elements of
-<var>x</var> (and if that is impossible, for example because <var>x</var> is
-an empty list, a value <code>'i'</code> is used).
-Type conversion takes place as for scalar <var>x</var>.
-
-<P>
-The following example shows several ways to define the same integer
-matrix.
-<div class="verbatim"><pre>
->>> A = matrix([0, 1, 2, 3], (2,2))
->>> A = matrix((0, 1, 2, 3), (2,2))
->>> A = matrix(xrange(4), (2,2))
->>> from array import array
->>> A = matrix(array('i', [0,1,2,3]), (2,2))
->>> print A
- 0 2
- 1 3
-</pre></div>
-
-<P>
-</LI>
-<LI>If <var>x</var> is a dense or sparse matrix (a <tt class="class">matrix</tt> or a <tt class="class">spmatrix</tt>
-object), or a two-dimensional <tt class="module">numarray</tt> array of type <code>'i'</code>,
-<code>'d'</code> or <code>'z'</code>, then the coefficients of <var>x</var> are copied, in
-column-major order, to a new matrix of the given size.
-The total number of elements in the new matrix (the product of
-<var>size</var>[0] and <var>size</var>[1]) must be the same as the product of
-the dimensions of <var>x</var>. If <var>size</var> is not specified, the
-dimensions of <var>x</var> are used.
-The default value of <var>tc</var> is the type of <var>x</var>.
-Type conversion takes place when the type of <var>x</var> differs from
-<var>tc</var>, in a similar way as for scalar <var>x</var>.
-
-<P>
-<div class="verbatim"><pre>
->>> A = matrix([1., 2., 3., 4., 5., 6.], (2,3))
->>> print A
- 1.0000e+00 3.0000e+00 5.0000e+00
- 2.0000e+00 4.0000e+00 6.0000e+00
->>> B = matrix(A, (3,2))
->>> print B
- 1.0000e+00 4.0000e+00
- 2.0000e+00 5.0000e+00
- 3.0000e+00 6.0000e+00
->>> C = matrix(B, tc='z')
->>> print C
- 1.0000e+00-j0.0000e+00 4.0000e+00-j0.0000e+00
- 2.0000e+00-j0.0000e+00 5.0000e+00-j0.0000e+00
- 3.0000e+00-j0.0000e+00 6.0000e+00-j0.0000e+00
->>> from numarray import array
->>> x = array([1., 2., 3., 4., 5., 6.], shape=(2,3))
->>> print x
-[[ 1. 2. 3.]
- [ 4. 5. 6.]]
->>> y = matrix(x)
->>> print y
- 1.0000e+00 2.0000e+00 3.0000e+00
- 4.0000e+00 5.0000e+00 6.0000e+00
-</pre></div>
-
-<P>
-</LI>
-<LI>If <var>x</var> is a list of lists of matrices
-(<tt class="class">matrix</tt> or <tt class="class">spmatrix</tt> objects) or numbers (Python integer, float or
-complex), then each element of <var>x</var> is interpreted as a
-block-column stored in column-major order.
-If <var>size</var> is not specified, the block-columns are juxtaposed
-to obtain a matrix with <code>len(<var>x</var>)</code> block-columns.
-If <var>size</var> is specified, then the matrix with <code>len(<var>x</var>)</code>
-block-columns is resized by copying its elements in column-major order
-into a matrix of the dimensions given by <var>size</var>.
-If <var>tc</var> is not specified, it is determined from the elements of
-<var>x</var> (and if that is impossible, for example because <var>x</var> is
-a list of empty lists, a value <code>'i'</code> is used).
-The same rules for type conversion apply as for scalar <var>x</var>.
-
-<P>
-<div class="verbatim"><pre>
->>> A = matrix([[1., 2.], [3., 4.], [5., 6.]])
->>> print A
- 1.0000e+00 3.0000e+00 5.0000e+00
- 2.0000e+00 4.0000e+00 6.0000e+00
->>> A1 = matrix([1, 2], (2,1))
->>> B1 = matrix([6, 7, 8, 9, 10, 11], (2,3))
->>> B2 = matrix([12, 13, 14, 15, 16, 17], (2,3))
->>> B3 = matrix([18, 19, 20], (1,3))
->>> print matrix([[A1, 3.0, 4.0, 5.0], [B1, B2, B3]])
- 1.0000e+00 6.0000e+00 8.0000e+00 1.0000e+01
- 2.0000e+00 7.0000e+00 9.0000e+00 1.1000e+01
- 3.0000e+00 1.2000e+01 1.4000e+01 1.6000e+01
- 4.0000e+00 1.3000e+01 1.5000e+01 1.7000e+01
- 5.0000e+00 1.8000e+01 1.9000e+01 2.0000e+01
-</pre></div>
-
-<P>
-A matrix with a single block-column can be represented by a single
-list (<EM>i.e.</EM>, when the length of <var>x</var> is one, it can be replaced with
-<code><var>x</var>[0]</code>).
-<div class="verbatim"><pre>
->>> print matrix([B1, B2, B3])
- 6 8 10
- 7 9 11
- 12 14 16
- 13 15 17
- 18 19 20
-</pre></div>
-</LI>
-</UL>
-</dl>
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-<!DOCTYPE html PUBLIC "-//W3C//DTD HTML 4.0 Transitional//EN">
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-<H1><A NAME="SECTION008100000000000000000"></A> <A NAME="s-creating-spmatrix"></A>
-<BR>
-6.1 Creating Sparse Matrices
-</H1>
-A general <tt class="class">spmatrix</tt> object can be thought of as a <em>triplet
-description</em> of a sparse matrix, <EM>i.e.</EM>, a list of entries of the matrix,
-with for each entry the value, row index, and column index.
-Entries that are not included in the list are assumed to be zero.
-For example, the sparse matrix
-<BR>
-<DIV ALIGN="RIGHT" CLASS="mathdisplay">
-
-<!-- MATH
- \begin{equation}
-A = \left[ \begin{array}{rrrrr}
- 0 & 2 & 0 & 0 & 3 \\
- 2 & 0 & 0 & 0 & 0 \\
- -1 & -2 & 0 & 4 & 0 \\
- 0 & 0 & 1 & 0 & 0 \end{array} \right]
-\end{equation}
- -->
-<A NAME="e-sparse-A"></A>
-<TABLE WIDTH="100%" ALIGN="CENTER">
-<TR VALIGN="MIDDLE"><TD></TD><TD ALIGN="CENTER" NOWRAP><A NAME="e-sparse-A"></A><IMG
- WIDTH="193" HEIGHT="83" BORDER="0"
- SRC="img78.gif"
- ALT="\begin{displaymath}
-A = \left[ \begin{array}{rrrrr}
-0 & 2 & 0 & 0 & 3 \\
-2 &...
-...-1 & -2 & 0 & 4 & 0 \\
-0 & 0 & 1 & 0 & 0 \end{array} \right]
-\end{displaymath}"></TD>
-<TD CLASS="eqno" WIDTH=10 ALIGN="RIGHT">
-(6.1)</TD></TR>
-</TABLE>
-<BR CLEAR="ALL"></DIV><P></P>
-has the triplet description
-<BR><P></P>
-<DIV ALIGN="CENTER" CLASS="mathdisplay">
-<!-- MATH
- \begin{displaymath}
-(2,1,0), \qquad (-1,2,0), \qquad (2,0,1), \qquad (-2,2,1), \qquad
-(1,3,2), \qquad (4,2,3), \qquad (3,0,4).
-\end{displaymath}
- -->
-
-<IMG
- WIDTH="611" HEIGHT="28" BORDER="0"
- SRC="img79.gif"
- ALT="\begin{displaymath}
-(2,1,0), \qquad (-1,2,0), \qquad (2,0,1), \qquad (-2,2,1), \qquad
-(1,3,2), \qquad (4,2,3), \qquad (3,0,4).
-\end{displaymath}">
-</DIV>
-<BR CLEAR="ALL">
-<P></P>
-The list may include entries with a zero value, so triplet
-descriptions are not necessarily unique.
-The list
-<BR><P></P>
-<DIV ALIGN="CENTER" CLASS="mathdisplay">
-<!-- MATH
- \begin{displaymath}
-(2,1,0), \qquad (-1,2,0), \qquad (0,3,0), \qquad (2,0,1), \qquad
- (-2,2,1), \qquad (1,3,2), \qquad (4,2,3), \qquad (3,0,4)
-\end{displaymath}
- -->
-
-<IMG
- WIDTH="696" HEIGHT="28" BORDER="0"
- SRC="img80.gif"
- ALT="\begin{displaymath}
-(2,1,0), \qquad (-1,2,0), \qquad (0,3,0), \qquad (2,0,1), \qquad
-(-2,2,1), \qquad (1,3,2), \qquad (4,2,3), \qquad (3,0,4)
-\end{displaymath}">
-</DIV>
-<BR CLEAR="ALL">
-<P></P>
-is another triplet description of the same matrix.
-
-<P>
-An <tt class="class">spmatrix</tt> object corresponds to a particular triplet
-description of a sparse matrix. We will refer to the entries in
-the triplet description as the <em>nonzero entries</em> of the object,
-even though they may have a numerical value zero.
-
-<P>
-Two functions are provided to create sparse matrices.
-The first, <tt class="function">spmatrix()</tt>, constructs a sparse matrix from
-a triplet description.
-<dl><dt><table cellpadding="0" cellspacing="0"><tr valign="baseline">
- <td><nobr><b><tt id='l2h-115' xml:id='l2h-115' class="function">spmatrix</tt></b>(</nobr></td>
- <td><var>x, I, J</var><big>[</big><var>, size</var><big>[</big><var>, tc</var><big>]</big><var></var><big>]</big><var></var>)</td></tr></table></dt>
-<dd>
-
-<P>
-<var>I</var> and <var>J</var> are sequences of integers (lists, tuples,
-<tt class="module">array</tt> arrays, xrange objects, ...) or integer matrices
-(<tt class="class">matrix</tt> objects with typecode <code>'i'</code>), containing the row and column
-indices of the nonzero entries.
-The lengths of <var>I</var> and <var>J</var> must be equal. If they
-are matrices, they are treated as lists of indices stored in
-column-major order, <EM>i.e.</EM>, as lists <code>list(<var>I</var>)</code>, respectively,
-<code>list(<var>J</var>)</code>.
-
-<P>
-<var>size</var> is a tuple of nonnegative integers with the row and column
-dimensions of the matrix.
-The <var>size</var> argument is only needed when creating a matrix with
-a zero last row or last column. If <var>size</var> is not specified, it
-is determined from <var>I</var> and <var>J</var>:
-the default value for <code><var>size</var>[0]</code> is <code>max(<var>I</var>)+1</code>
-if <var>I</var> is nonempty and zero otherwise.
-The default value for <code><var>size</var>[1]</code> is
-<code>max(<var>J</var>)+1</code> if <var>J</var> is nonempty and zero otherwise.
-
-<P>
-<var>tc</var> is the typecode, <code>'d'</code> or <code>'z'</code>, for double and complex
-matrices, respectively. Integer sparse matrices are not implemented.
-
-<P>
-<var>x</var> can be a number, a sequence of numbers, or a dense matrix.
-This argument specifies the numerical values of the nonzero entries.
-
-<UL>
-<LI>If <var>x</var> is a number (Python integer, float or complex),
-a matrix is created with the sparsity pattern defined by <var>I</var> and
-<var>J</var>, and nonzero entries initialized to the value of <var>x</var>.
-The default value of <var>tc</var> is <code>'d'</code> if <var>x</var> is integer or float,
-and <code>'z'</code> if <var>x</var> is complex.
-
-<P>
-The following code creates a 4 by 4 sparse identity matrix.
-<div class="verbatim"><pre>
->>> from cvxopt.base import spmatrix
->>> A = spmatrix(1.0, range(4), range(4))
->>> print A
-SIZE: (4,4)
-(0, 0) 1.0000e+00
-(1, 1) 1.0000e+00
-(2, 2) 1.0000e+00
-(3, 3) 1.0000e+00
-</pre></div>
-
-<P>
-</LI>
-<LI>If <var>x</var> is a sequence of numbers, a sparse matrix is created
-with the entries of <var>x</var> copied to the entries indexed by <var>I</var>
-and <var>J</var>. The list <var>x</var> must have the same length as <var>I</var> and
-<var>J</var>.
-The default value of <var>tc</var> is determined from the elements of
-<var>x</var>:
-<code>'d'</code> if <var>x</var> contains integers and floating-point numbers or
-if <var>x</var> is an empty list,
-and <code>'z'</code> if <var>x</var> contains at least one complex number.
-
-<P>
-As an example, the matrix (<A HREF="#e-sparse-A">6.1</A>) can be created as follows.
-<div class="verbatim"><pre>
->>> A = spmatrix([2,-1,2,-2,1,4,3], [1,2,0,2,3,2,0], [0,0,1,1,2,3,4])
->>> print A
-SIZE: (4,5)
-(1, 0) 2.0000e+00
-(2, 0) -1.0000e+00
-(0, 1) 2.0000e+00
-(2, 1) -2.0000e+00
-(3, 2) 1.0000e+00
-(2, 3) 4.0000e+00
-(0, 4) 3.0000e+00
-</pre></div>
-
-<P>
-</LI>
-<LI>If <var>x</var> is a dense matrix, a sparse matrix is created with
-all the entries of <var>x</var> copied, in column-major order, to the
-entries indexed by <var>I</var> and <var>J</var>.
-The matrix <var>x</var> must have the same length as <var>I</var> and <var>J</var>.
-The default value of <var>tc</var> is <code>'d'</code> if <var>x</var> is an <code>'i'</code> or <code>'d'</code> matrix, and <code>'z'</code> otherwise.
-</LI>
-</UL>
-
-<P>
-If <var>I</var> and <var>J</var> contain repeated entries, the corresponding
-values of the coefficients are added.
-</dl>
-
-<P>
-The function <tt class="function">sparse()</tt> constructs a sparse matrix from
-a block-matrix description.
-
-<P>
-<dl><dt><table cellpadding="0" cellspacing="0"><tr valign="baseline">
- <td><nobr><b><tt id='l2h-116' xml:id='l2h-116' class="function">sparse</tt></b>(</nobr></td>
- <td><var>x</var><big>[</big><var>, tc</var><big>]</big><var></var>)</td></tr></table></dt>
-<dd>
-<var>tc</var> is the typecode, <code>'d'</code> or <code>'z'</code>, for double and complex
-matrices, respectively.
-
-<P>
-<var>x</var> can be a <tt class="class">matrix</tt>, <tt class="class">spmatrix</tt>, or a list of lists of matrices
-(<tt class="class">matrix</tt> or <tt class="class">spmatrix</tt> objects) and numbers (Python integer, float or
-complex).
-
-<UL>
-<LI>If <var>x</var> is a <tt class="class">matrix</tt> or <tt class="class">spmatrix</tt> object, then a sparse matrix
-of the same size and the same numerical value is created.
-Numerical zeros in <var>x</var> are treated as structural zeros and removed
-from the triplet description of the new sparse matrix.
-
-<P>
-</LI>
-<LI>If <var>x</var> is a list of lists of matrices (<tt class="class">matrix</tt> or <tt class="class">spmatrix</tt>)
-and numbers (Python integer, float or complex) then each element of
-<var>x</var> is interpreted as a (block-)column matrix stored in
-colum-major order, and a block-matrix is constructed by juxtaposing
-the <code>len(<var>x</var>)</code> block-columns
-(as in <tt class="function">matrix()</tt>, see section <A href="s-creating-matrices.html#s-creating-matrices">2.1</A>).
-Numerical zeros are removed from the triplet description of the new
-matrix.
-
-<P>
-The following example shows how to construct a sparse block-matrix.
-<div class="verbatim"><pre>
->>> from cvxopt.base import matrix, spmatrix, sparse
->>> A = matrix([[1, 2, 0], [2, 1, 2], [0, 2, 1]])
->>> B = spmatrix([], [], [], (3,3))
->>> C = spmatrix([3, 4, 5], [0, 1, 2], [0, 1, 2])
->>> print sparse([[A, B], [B, C]])
-SIZE: (6,6)
-(0, 0) 1.0000e+00
-(1, 0) 2.0000e+00
-(0, 1) 2.0000e+00
-(1, 1) 1.0000e+00
-(2, 1) 2.0000e+00
-(1, 2) 2.0000e+00
-(2, 2) 1.0000e+00
-(3, 3) 3.0000e+00
-(4, 4) 4.0000e+00
-(5, 5) 5.0000e+00
-</pre></div>
-
-<P>
-A matrix with a single block-column can be represented by a single
-list.
-<div class="verbatim"><pre>
->>> print sparse([A, C])
-SIZE: (6,3)
-(0, 0) 1.0000e+00
-(1, 0) 2.0000e+00
-(3, 0) 3.0000e+00
-(0, 1) 2.0000e+00
-(1, 1) 1.0000e+00
-(2, 1) 2.0000e+00
-(4, 1) 4.0000e+00
-(1, 2) 2.0000e+00
-(2, 2) 1.0000e+00
-(5, 2) 5.0000e+00
-</pre></div>
-</LI>
-</UL>
-
-<P>
-</dl>
-
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-<H1><A NAME="SECTION0010800000000000000000"></A> <A NAME="s-external"></A>
-<BR>
-8.8 Optional Solvers
-</H1>
-CVXOPT includes optional interfaces to several other optimization
-libraries.
-
-<P>
-<DL>
-<DT><STRONG>GLPK</STRONG></DT>
-<DD><tt class="function">lp()</tt> with the <code>solver='glpk'</code> option uses
-the the simplex algorithm in
-<a class="ulink" href="http://www.gnu.org/software/glpk/glpk.html"
- >GLPK (GNU Linear Programming Kit)</a>.
-
-<P>
-</DD>
-<DT><STRONG>MOSEK</STRONG></DT>
-<DD><tt class="function">lp()</tt> and <tt class="function">qp()</tt> with the
-<code>solver='mosek'</code> option call routines from
-<a class="ulink" href="http://www.mosek.com"
- >MOSEK</a> version 4.
-
-<P>
-</DD>
-<DT><STRONG>DSDP</STRONG></DT>
-<DD><tt class="function">sdp()</tt> with the <code>solver='dsdp'</code> option uses
-the
-<a class="ulink" href="http://www-unix.mcs.anl.gov/DSDP"
- >DSDP5.8</a> solver.
-</DD>
-</DL>
-GLPK, MOSEK and DSDP are not included in the CVXOPT distribution and
-need to be installed separately.
-
-<P>
-
-<DIV CLASS="navigation">
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-<!DOCTYPE html PUBLIC "-//W3C//DTD HTML 4.0 Transitional//EN">
-<html>
-<head>
-<link rel="STYLESHEET" href="cvxopt.css" type='text/css' />
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-
-<H1><A NAME="SECTION0011200000000000000000"></A> <A NAME="s-functions"></A>
-<BR>
-9.2 Functions
-</H1>
-Objective and constraint functions can be defined via overloaded
-operations on variables and other functions. A function <var>f</var> is
-interpreted as a column vector, with length <code>len(<var>f</var>)</code>
-and with a value that depends on the values of its variables.
-Functions have two public attributes.
-<dl><dt><table cellpadding="0" cellspacing="0"><tr valign="baseline">
- <td><nobr><b><tt id='l2h-155' xml:id='l2h-155' class="method">variables</tt></b>(</nobr></td>
- <td><var></var>)</td></tr></table></dt>
-<dd>
-Returns a copy of the list of variables of the function.
-</dl>
-
-<P>
-<dl><dt><table cellpadding="0" cellspacing="0"><tr valign="baseline">
- <td><nobr><b><tt id='l2h-156' xml:id='l2h-156' class="method">value</tt></b>(</nobr></td>
- <td><var></var>)</td></tr></table></dt>
-<dd>
-The function value. If any of the variables of <var>f</var> has value
-<code>None</code>, then <var>f</var>.<tt class="method">value()</tt> returns <code>None</code>.
-Otherwise, it returns a dense <code>'d'</code> matrix of size
-(<code>len(<var>f</var>)</code>,1) with the function value computed from the
-<tt class="member">value</tt> attributes of the variables of <var>f</var>.
-</dl>
-
-<P>
-Three types of functions are supported: affine, convex
-piecewise-linear and concave piecewise-linear.
-
-<P>
-<b>Affine functions</b> represent vector valued functions of the form
-<BR><P></P>
-<DIV ALIGN="CENTER" CLASS="mathdisplay">
-<!-- MATH
- \begin{displaymath}
-f(x_1,\ldots,x_n) = A_1 x_1 + \cdots + A_n x_n + b.
-\end{displaymath}
- -->
-
-<IMG
- WIDTH="272" HEIGHT="28" BORDER="0"
- SRC="img169.gif"
- ALT="\begin{displaymath}
-f(x_1,\ldots,x_n) = A_1 x_1 + \cdots + A_n x_n + b.
-\end{displaymath}">
-</DIV>
-<BR CLEAR="ALL">
-<P></P>
-The coefficients can be scalars or dense or sparse matrices. The
-constant term is a scalar or a column vector.
-
-<P>
-Affine functions result from the following operations.
-<DL>
-<DT><STRONG>Unary operations</STRONG></DT>
-<DD>For a variable <var>x</var>, the unary operation <code>+<var>x</var></code> results in
-an affine function with <var>x</var> as variable, coefficient 1.0, and
-constant term 0.0. The unary operation <code>-<var>x</var></code> returns an
-affine function with <var>x</var> as variable, coefficient -1.0, and
-constant term 0.0. For an affine function <var>f</var>, <code>+<var>f</var></code> is
-a copy of <var>f</var>, and <code>-<var>f</var></code> is a copy of <var>f</var> with the
-signs of its coefficients and constant term reversed.
-
-<P>
-</DD>
-<DT><STRONG>Addition and subtraction</STRONG></DT>
-<DD>Sums and differences of affine functions, variables and constants
-result in new affine functions.
-The constant terms in the sum can be of type integer or float, or
-dense or sparse <code>'d'</code> matrices with one column.
-
-<P>
-The rules for addition and subtraction follow the conventions for
-matrix addition and subtraction in sections <A href="s-arithmetic.html#s-arithmetic">2.3</A>
-and <A href="s-spmatrix-arith.html#s-spmatrix-arith">6.3</A>, with variables and affine functions
-interpreted as dense <code>'d'</code> matrices with one column.
-In particular, a scalar term (integer, float, 1 by 1 dense <code>'d'</code> matrix,
-variable of length 1, or affine function of length 1) can be added
-to an affine function or variable of length greater than 1.
-
-<P>
-</DD>
-<DT><STRONG>Multiplication</STRONG></DT>
-<DD>Suppose <var>v</var> is an affine function or a variable, and <var>a</var> is
-an integer, float, sparse or dense <code>'d'</code> matrix. The products
-<code><var>a</var>*<var>v</var></code> and <code><var>v</var>*<var>a</var></code> are
-valid affine functions whenever the product is allowed under the rules
-for matrix and scalar multiplication of sections <A href="s-arithmetic.html#s-arithmetic">2.3</A>
-and <A href="s-spmatrix-arith.html#s-spmatrix-arith">6.3</A>, with <var>v</var> interpreted as a <code>'d'</code> matrix
-with one column.
-In particular, the product <code><var>a</var>*<var>v</var></code> is defined if
-<var>a</var> is a scalar (integer, float or 1 by 1 dense <code>'d'</code> matrix),
-or a matrix (dense or sparse) with
-<code><var>a</var>.<tt class="member">size</tt>[1] = len(<var>v</var>)</code>.
-The operation <code><var>v</var>*<var>a</var></code> is defined if <var>a</var> is scalar,
-or if <code>len(<var>v</var>)</code> = 1 and <var>a</var> is a matrix with one
-column.
-
-<P>
-</DD>
-<DT><STRONG>Inner products</STRONG></DT>
-<DD>The following two functions return scalar affine functions defined
-as inner products of a constant vector with a variable or affine
-function.
-<dl><dt><table cellpadding="0" cellspacing="0"><tr valign="baseline">
- <td><nobr><b><tt id='l2h-157' xml:id='l2h-157' class="function">sum</tt></b>(</nobr></td>
- <td><var>v</var>)</td></tr></table></dt>
-<dd>
-The argument is an affine function or a variable. The result is an
-affine function of length 1, with the sum of the components of the
-argument <var>v</var>.
-</dl>
-
-<P>
-<dl><dt><table cellpadding="0" cellspacing="0"><tr valign="baseline">
- <td><nobr><b><tt id='l2h-158' xml:id='l2h-158' class="function">dot</tt></b>(</nobr></td>
- <td><var>u,v</var>)</td></tr></table></dt>
-<dd>
-If <var>v</var> is a variable or affine function and <var>u</var> is a <code>'d'</code> matrix of size (<code>len(<var>v</var>)</code>,1), then
-<code>dot(<var>u</var>,<var>v</var>)</code> and <code>dot(<var>v</var>,<var>u</var>)</code> are
-equivalent to <code><var>u</var>.<tt class="method">trans()</tt>*<var>v</var></code>.
-
-<P>
-If <var>u</var> and <var>v</var> are dense matrices, then
-<code>dot(<var>u</var>,<var>v</var>)</code>
-is equivalent to the function <code>blas.dot(<var>u</var>,<var>v</var>)</code>
-defined in section <A href="s-blas1.html#s-blas1">3.2</A>, <EM>i.e.</EM>, it returns the inner product of
-the two matrices.
-</dl>
-</DD>
-</DL>
-
-<P>
-In the following example, the variable <var>x</var> has length 1 and
-<var>y</var> has length 2.
-The functions <var>f</var> and <var>g</var> are given by
-<P></P>
-<DIV ALIGN="CENTER" CLASS="mathdisplay">
-<!-- MATH
- \begin{eqnarray*}
-f(x,y) & = & \left[ \begin{array}{c} 2 \\2 \end{array}\right] x
- + y + \left[ \begin{array}{c} 3 \\3 \end{array}\right], \\
- g(x,y) & = &
- \left[ \begin{array}{cc} 1 & 3 \\2 & 4 \end{array}\right] f(x,y)
- + \left[ \begin{array}{cc} 1 & 1 \\1 & 1 \end{array} \right] y +
- \left[ \begin{array}{c} 1 \\-1 \end{array} \right] \\
- & = & \left[ \begin{array}{c} 8 \\12 \end{array}\right] x
- + \left[ \begin{array}{cc} 2 & 4 \\3 & 5 \end{array}\right] y
- + \left[ \begin{array}{c} 13 \\17\end{array}\right].
-\end{eqnarray*}
- -->
-<IMG
- WIDTH="366" HEIGHT="134" BORDER="0"
- SRC="img170.gif"
- ALT="\begin{eqnarray*}
-f(x,y) & = & \left[ \begin{array}{c} 2 \\ 2 \end{array}\right...
-...\right] y
-+ \left[ \begin{array}{c} 13 \\ 17\end{array}\right].
-\end{eqnarray*}"></DIV>
-<BR CLEAR="ALL"><P></P>
-<BR CLEAR="ALL"><P></P>
-<div class="verbatim"><pre>
->>> from cvxopt.modeling import variable
->>> x = variable(1,'x')
->>> y = variable(2,'y')
->>> f = 2*x + y + 3
->>> A = matrix([[1., 2.], [3.,4.]])
->>> b = matrix([1.,-1.])
->>> g = A*f + sum(y) + b
->>> print g
-affine function of length 2
-constant term:
- 1.3000e+01
- 1.7000e+01
-linear term: linear function of length 2
-coefficient of variable(2,'y'):
- 2.0000e+00 4.0000e+00
- 3.0000e+00 5.0000e+00
-coefficient of variable(1,'x'):
- 8.0000e+00
- 1.2000e+01
-</pre></div>
-
-<P>
-<DL>
-<DT><STRONG>In-place operations</STRONG></DT>
-<DD>For an affine function <var>f</var> the operations <code><var>f</var> += <var>u</var></code>
-and <code><var>f</var> -= <var>u</var></code>, with <var>u</var> a constant, a variable or
-an affine function, are allowed if they do not change the length of
-<var>f</var>, <EM>i.e.</EM>, if <var>u</var> has length <code>len(<var>f</var>)</code> or length 1.
-In-place multiplication <code><var>f</var> *= <var>u</var></code> and division
-<code><var>f</var> /= <var>u</var></code> are allowed if <var>u</var> is an integer, float, or
-1 by 1 matrix.
-</DD>
-</DL>
-
-<P>
-<DL>
-<DT><STRONG>Indexing and slicing</STRONG></DT>
-<DD>Variables and affine functions admit
-single-argument indexing of the four types described in
-section <A href="s-indexing.html#s-indexing">2.4</A>. The result of an indexing or slicing
-operation is an affine function.
-</DD>
-</DL>
-
-<P>
-<div class="verbatim"><pre>
->>> x = variable(4,'x')
->>> f = x[::2]
->>> print f
->>> linear function of length 2
-linear term: linear function of length 2
-coefficient of variable(4,'x'):
-TYPE: general
-SIZE: (2,4)
-(0, 0) 1.0000e+00
-(1, 2) 1.0000e+00
->>> y = variable(3,'x')
->>> g = matrix(range(12),(3,4),'d')*x - 3*y + 1
->>> print g[0] + g[2]
-affine function of length 1
-constant term:
- 2.0000e+00
-linear term: linear function of length 1
-coefficient of variable(4,'x'):
- 2.0000e+00 8.0000e+00 1.4000e+01 2.0000e+01
-coefficient of variable(3,'y'):
-TYPE: general
-SIZE: (1,3)
-(0, 0) -3.0000e+00
-(0, 2) -3.0000e+00
-</pre></div>
-
-<P>
-The general expression of a <b>convex piecewise-linear</b> function is
-<BR><P></P>
-<DIV ALIGN="CENTER" CLASS="mathdisplay">
-<!-- MATH
- \begin{displaymath}
-f(x_1,\ldots,x_n) = b + A_1 x_1 + \cdots + A_n x_n +
- \sum_{k=1}^K \max (y_1, y_2, \ldots, y_{m_k}).
-\end{displaymath}
- -->
-
-<IMG
- WIDTH="458" HEIGHT="56" BORDER="0"
- SRC="img171.gif"
- ALT="\begin{displaymath}
-f(x_1,\ldots,x_n) = b + A_1 x_1 + \cdots + A_n x_n +
-\sum_{k=1}^K \max (y_1, y_2, \ldots, y_{m_k}).
-\end{displaymath}">
-</DIV>
-<BR CLEAR="ALL">
-<P></P>
-The maximum in this expression is a componentwise maximum of its
-vector arguments, which can be constant vectors, variables, affine
-functions or convex piecewise-linear functions.
-The general expression for a <b>concave piecewise-linear</b> function
-is
-<BR><P></P>
-<DIV ALIGN="CENTER" CLASS="mathdisplay">
-<!-- MATH
- \begin{displaymath}
-f(x_1,\ldots,x_n) = b + A_1 x_1 + \cdots + A_n x_n +
- \sum_{k=1}^K \min (y_1, y_2, \ldots, y_{m_k}).
-\end{displaymath}
- -->
-
-<IMG
- WIDTH="456" HEIGHT="56" BORDER="0"
- SRC="img172.gif"
- ALT="\begin{displaymath}
-f(x_1,\ldots,x_n) = b + A_1 x_1 + \cdots + A_n x_n +
-\sum_{k=1}^K \min (y_1, y_2, \ldots, y_{m_k}).
-\end{displaymath}">
-</DIV>
-<BR CLEAR="ALL">
-<P></P>
-Here the arguments of the <tt class="function">min()</tt> can be constants, variables,
-affine functions or concave piecewise-linear functions.
-
-<P>
-Piecewise-linear functions can be created using the following
-operations.
-<DL>
-<DT><STRONG><tt class="function">max</tt></STRONG></DT>
-<DD>If the arguments in
-<code><var>f</var> = max(<var>y1</var>,<var>y2</var>, ...)</code>
-do not include any variables or functions, then the Python built-in
-<tt class="function">max()</tt> is evaluated.
-
-<P>
-If one or more of the arguments are variables or functions,
-<tt class="function">max()</tt> returns a piecewise-linear function defined as the
-elementwise maximum of its arguments.
-In other words, <var>f</var>[<var>k</var>] =
-<code>max(<var>y1</var>[<var>k</var>],<var>y2</var>[<var>k</var>], ...)</code>
-for <var>k</var>=0, ..., <code>len(<var>f</var>)</code>-1.
-The length of <var>f</var> is equal to the maximum of the lengths of the
-arguments. Each argument must have length equal to
-<code>len(<var>f</var>)</code> or length one.
-Arguments with length one are interpreted as vectors of
-length <code>len(<var>f</var>)</code> with identical entries.
-
-<P>
-The arguments can be scalars of type integer or float,
-dense <code>'d'</code> matrices with one column, variables, affine functions or
-convex piecewise-linear functions.
-
-<P>
-With one argument, <code><var>f</var> = max(<var>u</var>)</code> is interpreted as
-<code><var>f</var> = max(<var>u</var>[0],<var>u</var>[1],...,
-<var>u</var>[<tt class="function">len</tt>(<var>u</var>)-1])</code>.
-
-<P>
-</DD>
-<DT><STRONG><tt class="function">min</tt></STRONG></DT>
-<DD>Similar to <tt class="function">max()</tt> but returns a concave
-piecewise-linear function. The arguments can be scalars of type
-integer or float, dense <code>'d'</code> matrices with one column,
-variables, affine functions or concave piecewise-linear functions.
-
-<P>
-</DD>
-<DT><STRONG><tt class="function">abs</tt></STRONG></DT>
-<DD>If <var>u</var> is a variable or affine function then
-<code><var>f</var> = abs(<var>u</var>)</code> returns the convex piecewise-linear
-function <code>max(<var>u</var>,-<var>u</var>)</code>.
-
-<P>
-</DD>
-<DT><STRONG>Unary plus and minus</STRONG></DT>
-<DD><code>+<var>f</var></code> creates a copy of <var>f</var>.
-<code>-<var>f</var></code> is a concave piecewise-linear function if <var>f</var> is
-convex and a convex piecewise-linear function if <var>f</var> is concave.
-
-<P>
-</DD>
-<DT><STRONG>Addition and subtraction</STRONG></DT>
-<DD>Sums and differences involving
-piecewise-linear functions are allowed if they result in convex
-or concave functions. For example, one can add two convex or two
-concave functions, but not a convex and a concave function.
-The command <code>sum(<var>f</var>)</code> is equivalent
-to <code><var>f</var>[0] + <var>f</var>[1] + ...+ <var>f</var>[len(<var>f</var>)-1]</code>.
-
-<P>
-</DD>
-<DT><STRONG>Multiplication</STRONG></DT>
-<DD>Scalar multiplication <code><var>a</var>*<var>f</var></code> of a
-piecewise-linear function <var>f</var> is defined if <var>a</var>
-is an integer, float, 1 by 1 <code>'d'</code> matrix.
-Matrix-matrix multiplications <code><var>a</var>*<var>f</var></code> or
-<code><var>f</var>*<var>a</var></code> are only defined if <var>a</var> is a dense or
-sparse 1 by 1 matrix.
-
-<P>
-</DD>
-<DT><STRONG>Indexing and slicing</STRONG></DT>
-<DD>Piecewise-linear functions admit
-single-argument indexing of the four types described in
-section <A href="s-indexing.html#s-indexing">2.4</A>. The result of an indexing or slicing
-operation is a new piecewise-linear function.
-</DD>
-</DL>
-
-<P>
-In the following example, <var>f</var> is the 1-norm of a vector
-variable <var>x</var> of length 10, <var>g</var> is its infinity-norm and
-<var>h</var> is the function
-<BR><P></P>
-<DIV ALIGN="CENTER" CLASS="mathdisplay">
-<!-- MATH
- \begin{displaymath}
-h(x) = \sum_k \phi(x[k]),\qquad
- \phi(u) = \left\{\begin{array}{ll}
- 0 & |u| \leq 1 \\
- |u|-1 & 1 \leq |u| \leq 2 \\
- 2|u|-3 & |u| \geq 2. \end{array}\right.
-\end{displaymath}
- -->
-
-<IMG
- WIDTH="383" HEIGHT="64" BORDER="0"
- SRC="img173.gif"
- ALT="\begin{displaymath}
-h(x) = \sum_k \phi(x[k]),\qquad
-\phi(u) = \left\{\begin{ar...
-...\\
-2\vert u\vert-3 & \vert u\vert \geq 2. \end{array}\right.
-\end{displaymath}">
-</DIV>
-<BR CLEAR="ALL">
-<P></P>
-<div class="verbatim"><pre>
->>> f = sum(abs(x))
->>> g = max(abs(x))
->>> h = sum(max(0, abs(x)-1, 2*abs(x)-3))
-</pre></div>
-
-<P>
-<DL>
-<DT><STRONG>In-place operations</STRONG></DT>
-<DD>If <var>f</var> is piecewise-linear then the in-place operations
-<code><var>f</var> += <var>u</var></code>, <code><var>f</var> -= <var>u</var></code>,
-<code><var>f</var> *= <var>u</var></code>, <code><var>f</var> /= <var>u</var></code> are defined if the
-corresponding expanded operations <code><var>f</var> = <var>f</var>+<var>u</var></code>,
-<code><var>f</var> = <var>f</var>-<var>u</var></code>, <code><var>f</var> = <var>f</var>*<var>u</var></code>
-and <code><var>f</var> = <var>f</var>/<var>u</var></code> are defined and if they do not
-change the length of <var>f</var>.
-</DD>
-</DL>
-
-<P>
-
-<DIV CLASS="navigation">
-<div class='online-navigation'>
-<p></p><hr />
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+++ /dev/null
@@ -1,263 +0,0 @@
-<!DOCTYPE html PUBLIC "-//W3C//DTD HTML 4.0 Transitional//EN">
-<html>
-<head>
-<link rel="STYLESHEET" href="cvxopt.css" type='text/css' />
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-
-<H1><A NAME="SECTION004400000000000000000"></A> <A NAME="s-indexing"></A>
-<BR>
-2.4 Indexing and Slicing
-</H1>
-
-<P>
-Matrices can be indexed using one or two arguments. In single-argument
-indexing of a matrix <var>A</var>, the index runs from
-<code>-len(<var>A</var>)</code> to <code>len(<var>A</var>)-1</code>, and is interpreted as an
-index in the one-dimensional array of coefficients of <var>A</var>
-in column-major order. Negative indices have the standard Python
-interpretation: for negative <var>k</var>, <code><var>A</var>[<var>k</var>]</code> is the
-same element as <code><var>A</var>[len(<var>A</var>)+<var>k</var>]</code>.
-
-<P>
-Four different types of one-argument indexing are implemented.
-
-<OL>
-<LI>The index can be a single integer. This returns a
-number, <EM>e.g.</EM>, <code><var>A</var>[0]</code> is the first element of <var>A</var>.
-
-<P>
-</LI>
-<LI>The index can be an integer matrix. This returns a
-column matrix: the command "<tt class="samp">A[matrix([0,1,2,3])]</tt>"
-returns the 4 by 1 matrix consisting of the first four elements of
-<var>A</var>. The size of the index matrix is ignored:
-"<tt class="samp">A[matrix([0,1,2,3], (2,2))]</tt>" returns the same 4 by 1 matrix.
-
-<P>
-</LI>
-<LI>The index can be a list of integers. This returns a column
-matrix, <EM>e.g.</EM>, <code><var>A</var>[[0,1,2,3]]</code> is the 4 by 1 matrix consisting
-of elements 0, 1, 2, 3 of <var>A</var>.
-
-<P>
-</LI>
-<LI>The index can be a Python slice. This returns a matrix with one
-column (possibly 0 by 1, or 1 by 1). For example, <code><var>A</var>[::2]</code>
-is the column matrix defined by taking every other element of <var>A</var>,
-stored in column-major order.
-<code><var>A</var>[0:0]</code> is a matrix with size (0,1).
-</LI>
-</OL>
-Thus, single-argument indexing returns a scalar (if the index is an
-integer), or a matrix with one column. This is consistent with the
-interpretation that single-argument indexing accesses the matrix in
-column-major order.
-
-<P>
-Note that an index list or an index matrix are equivalent,
-but they are both useful, especially when we perform operations on
-index sets. For example, if <var>I</var> and <var>J</var> are lists then
-<code><var>I</var>+<var>J</var></code> is the concatenated list, and <code>2*<var>I</var></code>
-is <var>I</var> repeated twice. If they are matrices, these operations are
-interpreted as arithmetic operations.
-For large index sets, indexing with integer matrices is also faster
-than indexing with lists.
-
-<P>
-The following example illustrates one-argument indexing.
-<div class="verbatim"><pre>
->>> from cvxopt.base import matrix
->>> A = matrix(range(16), (4,4), 'd')
->>> print A
- 0.0000e+00 4.0000e+00 8.0000e+00 1.2000e+01
- 1.0000e+00 5.0000e+00 9.0000e+00 1.3000e+01
- 2.0000e+00 6.0000e+00 1.0000e+01 1.4000e+01
- 3.0000e+00 7.0000e+00 1.1000e+01 1.5000e+01
->>> A[4]
-4.0
->>> I = matrix([0, 5, 10, 15])
->>> print A[I] # the diagonal
- 0.0000e+00
- 5.0000e+00
- 1.0000e+01
- 1.5000e+01
->>> I = [0,2]; J = [1,3]
->>> print A[2*I+J] # duplicate I and append J
- 0.0000e+00
- 2.0000e+00
- 0.0000e+00
- 2.0000e+00
- 1.0000e+00
- 3.0000e+00
->>> I = matrix([0, 2]); J = matrix([1, 3])
->>> print A[2*I+J] # multiply I by 2 and add J
- 1.0000e+00
- 7.0000e+00
->>> print A[4::4] # get every fourth element skipping the first four
- 4.0000e+00
- 8.0000e+00
- 1.2000e+01
-</pre></div>
-
-<P>
-In two-argument indexing the arguments can be any combinations of the
-four types listed above. The first argument indexes the rows of
-the matrix and the second argument indexes the columns. If both
-indices are scalars, then a scalar is returned. In all other cases,
-a matrix is returned. We continue the example.
-<div class="verbatim"><pre>
->>> print A[:,1]
- 4.0000e+00
- 5.0000e+00
- 6.0000e+00
- 7.0000e+00
->>> J = matrix([0, 2])
->>> print A[J,J]
- 0.0000e+00 8.0000e+00
- 2.0000e+00 1.0000e+01
->>> print A[:2, -2:]
- 8.0000e+00 1.2000e+01
- 9.0000e+00 1.3000e+01
-</pre></div>
-
-<P>
-Expressions of the form <code><var>A</var>[<var>I</var>]</code> or
-<code><var>A</var>[<var>I</var>,<var>J</var>]</code> can also appear on the lefthand side
-of an assignment.
-The righthand side must be a scalar (<EM>i.e.</EM>, a number or a 1 by 1 dense
-matrix), a sequence of numbers, or a dense or sparse matrix.
-If the righthand side is a scalar, it is interpreted as a
-matrix with identical entries and the dimensions of the lefthand side.
-If the righthand side is a sequence of numbers (list, tuple,
-<tt class="module">array</tt> array, xrange object, ...) its values are
-interpreted as the coefficients of the lefthand side in column-major
-order. If the righthand side is a matrix (<tt class="class">matrix</tt> or <tt class="class">spmatrix</tt>), it must
-have the same size as the lefthand side. Sparse matrices are
-converted to dense in the assignment.
-
-<P>
-Indexed assignments are only allowed if they do not change the type of
-the matrix. For example, if <var>A</var> is a matrix with type <code>'d'</code>, then
-<code><var>A</var>[<var>I</var>] = <var>B</var></code> is only permitted if <var>B</var> is
-an integer, a float, or a matrix of type <code>'i'</code> or <code>'d'</code>.
-If <var>A</var> is an integer matrix, then <code><var>A</var>[<var>I</var>] = <var>B</var></code>
-is only permitted if <var>B</var> is an integer or an integer matrix.
-
-<P>
-The following example illlustrates indexed assignment.
-<div class="verbatim"><pre>
->>> A = matrix(range(16), (4,4))
->>> A[::2,::2] = matrix([[-1, -2], [-3, -4]])
->>> print A
- -1 4 -3 12
- 1 5 9 13
- -2 6 -4 14
- 3 7 11 15
->>> A[::5] += 1
->>> print A
- 0 4 -3 12
- 1 6 9 13
- -2 6 -3 14
- 3 7 11 16
->>> A[0,:] = -1, 1, -1, 1
->>> print A
- -1 1 -1 1
- 1 6 9 13
- -2 6 -3 14
- 3 7 11 16
->>> A[2:,2:] = xrange(4)
->>> print A
- -1 1 -1 1
- 1 6 9 13
- -2 6 0 2
- 3 7 1 3
-</pre></div>
-
-<P>
-
-<DIV CLASS="navigation">
-<div class='online-navigation'>
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+++ /dev/null
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-<!DOCTYPE html PUBLIC "-//W3C//DTD HTML 4.0 Transitional//EN">
-<html>
-<head>
-<link rel="STYLESHEET" href="cvxopt.css" type='text/css' />
-<link rel="first" href="cvxopt.html" title='CVXOPT: A Python Package for Convex Optimization' />
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-<H1><A NAME="SECTION0011400000000000000000"></A> <A NAME="s-lp"></A>
-<BR>
-9.4 Optimization Problems
-</H1>
-
-<P>
-Optimization problems are be constructed by calling the following
-function.
-<dl><dt><table cellpadding="0" cellspacing="0"><tr valign="baseline">
- <td><nobr><b><span class="typelabel">class</span> <tt id='l2h-163' xml:id='l2h-163' class="class">op</tt></b>(</nobr></td>
- <td><var></var><big>[</big><var>objective</var><big>[</big><var>, constraints</var><big>[</big><var>, name</var><big>]</big><var></var><big>]</big><var></var><big>]</big><var></var>)</td></tr></table></dt>
-<dd>
-The first argument specifies the objective function to be minimized.
-It can be an affine or convex piecewise-linear function with length 1,
-a variable with length 1, or a scalar constant
-(integer, float or 1 by 1 dense <code>'d'</code> matrix). The default value is
-<code>0.0</code>.
-
-<P>
-The second argument is a single constraint, or a list of
-constraint objects. The default value is an empty list.
-
-<P>
-The third argument is a string with a name for the problem.
-The default value is the empty string.
-</dl>
-
-<P>
-The following attributes and methods are useful for examining
-and modifying optimization problems.
-
-<P>
-<dl><dt><b><tt id='l2h-164' xml:id='l2h-164' class="member">objective</tt></b></dt>
-<dd>
-The objective or cost function. One can write to this
-attribute to change the objective of an existing problem.
-</dl>
-
-<P>
-<dl><dt><table cellpadding="0" cellspacing="0"><tr valign="baseline">
- <td><nobr><b><tt id='l2h-165' xml:id='l2h-165' class="method">variables</tt></b>(</nobr></td>
- <td><var></var>)</td></tr></table></dt>
-<dd>
-Returns a list of the variables of the problem.
-</dl>
-
-<P>
-<dl><dt><table cellpadding="0" cellspacing="0"><tr valign="baseline">
- <td><nobr><b><tt id='l2h-166' xml:id='l2h-166' class="method">constraints</tt></b>(</nobr></td>
- <td><var></var>)</td></tr></table></dt>
-<dd>
-Returns a list of the constraints.
-</dl>
-
-<P>
-<dl><dt><table cellpadding="0" cellspacing="0"><tr valign="baseline">
- <td><nobr><b><tt id='l2h-167' xml:id='l2h-167' class="method">inequalities</tt></b>(</nobr></td>
- <td><var></var>)</td></tr></table></dt>
-<dd>
-Returns a list of the inequality constraints.
-</dl>
-
-<P>
-<dl><dt><table cellpadding="0" cellspacing="0"><tr valign="baseline">
- <td><nobr><b><tt id='l2h-168' xml:id='l2h-168' class="method">equalities</tt></b>(</nobr></td>
- <td><var></var>)</td></tr></table></dt>
-<dd>
-Returns a list of the equality constraints.
-</dl>
-
-<P>
-<dl><dt><table cellpadding="0" cellspacing="0"><tr valign="baseline">
- <td><nobr><b><tt id='l2h-169' xml:id='l2h-169' class="method">delconstraint</tt></b>(</nobr></td>
- <td><var>c</var>)</td></tr></table></dt>
-<dd>
-Deletes constraint <TT>c</TT> from the problem.
-</dl>
-
-<P>
-<dl><dt><table cellpadding="0" cellspacing="0"><tr valign="baseline">
- <td><nobr><b><tt id='l2h-170' xml:id='l2h-170' class="method">addconstraint</tt></b>(</nobr></td>
- <td><var>c</var>)</td></tr></table></dt>
-<dd>
-Adds constraint <TT>c</TT> to the problem.
-</dl>
-
-<P>
-An optimization problem with convex piecewise-linear objective and
-constraints can be solved by calling the method <tt class="function">solve()</tt>.
-
-<P>
-<dl><dt><table cellpadding="0" cellspacing="0"><tr valign="baseline">
- <td><nobr><b><tt id='l2h-171' xml:id='l2h-171' class="method">solve</tt></b>(</nobr></td>
- <td><var></var><big>[</big><var>format</var><big>[</big><var>, solver</var><big>]</big><var></var><big>]</big><var></var>)</td></tr></table></dt>
-<dd>
-This function converts the optimization problem to a linear program
-in matrix form and then solves it using the solver described in
-section <A href="s-lpsolver.html#s-lpsolver">8.1</A>.
-
-<P>
-The first argument is either <code>'dense'</code> or <code>'sparse'</code>, and
-denotes the matrix types used in the matrix representation of the LP.
-The default value is <code>'dense'</code>.
-
-<P>
-The second argument is either <code>None</code>. <code>'glpk'</code> or <code>'mosek'</code>,
-and selects one of three available LP solvers: a default solver
-written in Python, the GLPK solver (if installed) or the
-MOSEK LP solver (if installed); see section <A href="s-lpsolver.html#s-lpsolver">8.1</A>.
-The default value is <code>None</code>.
-
-<P>
-The solver reports the outcome of optimization by setting
-the attribute <var>self</var>.<tt class="member">status</tt> and by modifying
-the <tt class="member">value</tt> attributes of the variables and the constraint
-multipliers of the problem.
-
-<UL>
-<LI>If the problem is solved to optimality,
-<var>self</var>.<tt class="member">status</tt> is set to <code>'optimal'</code>.
-The <tt class="member">value</tt> attributes of the variables in the problem are set
-to their computed solutions, and the <tt class="member">value</tt> attributes of the
-multipliers of the constraints of the problem are set to the computed
-dual optimal solution.
-
-<P>
-</LI>
-<LI>If it is determined that the problem is infeasible,
-<var>self</var>.<tt class="member">status</tt> is set to <code>'primal infeasible'</code>.
-The <tt class="member">value</tt> attributes of the variables are set to <code>None</code>.
-The <tt class="member">value</tt> attributes of the
-multipliers of the constraints of the problem are set to a certificate
-of primal infeasibility.
-With the <code>'glpk'</code> option, <tt class="function">solve()</tt>
-does not provide certificates of infeasibility.
-
-<P>
-</LI>
-<LI>If it is determined that the problem is dual infeasible,
-<var>self</var>.<tt class="member">status</tt> is set to <code>'dual infeasible'</code>.
-The <tt class="member">value</tt> attributes of the multipliers of the constraints of
-the problem are set to <code>None</code>.
-The <tt class="member">value</tt> attributes of the
-variables are set to a certificate of dual infeasibility.
-With the <code>'glpk'</code> option, <tt class="function">solve()</tt> does not provide
-certificates of infeasibility.
-
-<P>
-</LI>
-<LI>If the problem was not solved successfully,
-<var>self</var>.<tt class="member">status</tt> is set to <code>'unknown'</code>.
-The <tt class="member">value</tt> attributes of the variables and the constraint
-multipliers are set to <code>None</code>.
-</LI>
-</UL>
-</dl>
-We refer to section <A href="s-lpsolver.html#s-lpsolver">8.1</A> for details on the algorithms and
-the different solver options.
-
-<P>
-As an example we solve the LP
-<BR><P></P>
-<DIV ALIGN="CENTER" CLASS="mathdisplay">
-<!-- MATH
- \begin{displaymath}
-\begin{array}{ll}
- \mbox{minimize} & -4x - 5y \\
- \mbox{subject to} & 2x +y \leq 3 \\
- & x +2y \leq 3 \\
- & x \geq 0, \quad y \geq 0.
- \end{array}
-\end{displaymath}
- -->
-
-<IMG
- WIDTH="179" HEIGHT="83" BORDER="0"
- SRC="img177.gif"
- ALT="\begin{displaymath}
-\begin{array}{ll}
-\mbox{minimize} & -4x - 5y \\
-\mbox{su...
-...
-& x +2y \leq 3 \\
-& x \geq 0, \quad y \geq 0.
-\end{array}\end{displaymath}">
-</DIV>
-<BR CLEAR="ALL">
-<P></P>
-<div class="verbatim"><pre>
->>> x = variable()
->>> y = variable()
->>> c1 = ( 2*x+y <= 3 )
->>> c2 = ( x+2*y <= 3 )
->>> c3 = ( x >= 0 )
->>> c4 = ( y >= 0 )
->>> lp1 = op(-4*x-5*y, [c1,c2,c3,c4])
->>> lp1.solve()
->>> print lp1.status
-optimal
->>> print lp1.objective.value()
--9.0000e+00
->>> print x.value
- 1.0000e-00
->>> print y.value
- 1.0000e-00
->>> print c1.multiplier.value
- 1.0000e-00
->>> print c2.multiplier.value
- 2.0000e-00
->>> print c3.multiplier.value
- 8.8912e-09
->>> print c4.multiplier.value
- 9.8567e-09
-</pre></div>
-
-<P>
-We can solve the same LP in matrix form as follows.
-<div class="verbatim"><pre>
->>> x = variable(2)
->>> A = matrix([[2.,1.,-1.,0.], [1.,2.,0.,-1.]])
->>> b = matrix([3.,3.,0.,0.])
->>> c = matrix([-4.,-5.])
->>> ineq = ( A*x <= b )
->>> lp2 = op(dot(c,x), ineq)
->>> lp2.solve()
->>> print lp2.objective.value()
--9.0000e+00
->>> print x.value
- 1.0000e-00
- 1.0000e-00
->>> print ineq.multiplier.value
- 1.0000e+00
- 2.0000e+00
- 8.8912e-09
- 9.8567e-09
-</pre></div>
-
-<P>
-The op class also includes two methods for writing and reading
-files in
-<a class="ulink" href="http://www-fp.mcs.anl.gov/otc/Guide/OptWeb/continuous/constrained/linearprog/mps.html"
- >MPS format</a>.
-
-<P>
-<dl><dt><table cellpadding="0" cellspacing="0"><tr valign="baseline">
- <td><nobr><b><tt id='l2h-172' xml:id='l2h-172' class="method">tofile</tt></b>(</nobr></td>
- <td><var>filename</var>)</td></tr></table></dt>
-<dd>
-If the problem is an LP, writes it to the file <code>'filename'</code> using
-the MPS format. Row and column labels are assigned based on the
-variable and constraint names in the LP.
-</dl>
-
-<P>
-<dl><dt><table cellpadding="0" cellspacing="0"><tr valign="baseline">
- <td><nobr><b><tt id='l2h-173' xml:id='l2h-173' class="method">fromfile</tt></b>(</nobr></td>
- <td><var>filename</var>)</td></tr></table></dt>
-<dd>
-Reads the LP from the file <code>'filename'</code>. The file must be
-a fixed-format MPS file. Some features of the MPS format are not
-supported: comments beginning with dollar signs,
-the row types 'DE', 'DL', 'DG', and 'DN', and the capability of
-reading multiple righthand side, bound or range vectors.
-</dl>
-
-<P>
-
-<DIV CLASS="navigation">
-<div class='online-navigation'>
-<p></p><hr />
-<table align="center" width="100%" cellpadding="0" cellspacing="2">
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-<!DOCTYPE html PUBLIC "-//W3C//DTD HTML 4.0 Transitional//EN">
-<html>
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-
-<H1><A NAME="SECTION0010100000000000000000"></A> <A NAME="s-lpsolver"></A>
-<BR>
-8.1 Linear Programming
-</H1>
-
-<P>
-<dl><dt><table cellpadding="0" cellspacing="0"><tr valign="baseline">
- <td><nobr><b><tt id='l2h-145' xml:id='l2h-145' class="function">lp</tt></b>(</nobr></td>
- <td><var>c, G, h</var><big>[</big><var>, A, b</var><big>[</big><var>,
-solver</var><big>[</big><var>, primalstart</var><big>[</big><var>, dualstart</var><big>]</big><var></var><big>]</big><var></var><big>]</big><var></var><big>]</big><var></var>)</td></tr></table></dt>
-<dd>
-Solves the pair of primal and dual LPs
-<BR>
-<DIV ALIGN="RIGHT" CLASS="mathdisplay">
-
-<!-- MATH
- \begin{equation}
-\mbox{Primal:}\quad \begin{array}[t]{ll}
-\mbox{minimize} & c^Tx \\
-\mbox{subject to} & Gx + s = h \\& Ax=b \\& s \succeq 0
-\end{array} \qquad\qquad
-\mbox{Dual:}\quad
-\begin{array}[t]{ll}
-\mbox{maximize} & -h^T z - b^T y \\
-\mbox{subject to} & G^Tz + A^T y + c = 0 \\& z \succeq 0.
-\end{array}
-\end{equation}
- -->
-<A NAME="e-lp"></A>
-<TABLE WIDTH="100%" ALIGN="CENTER">
-<TR VALIGN="MIDDLE"><TD></TD><TD ALIGN="CENTER" NOWRAP><A NAME="e-lp"></A><IMG
- WIDTH="571" HEIGHT="87" BORDER="0"
- SRC="img111.gif"
- ALT="\begin{displaymath}
-\mbox{Primal:}\quad \begin{array}[t]{ll}
-\mbox{minimize} & c...
-...ubject to} & G^Tz + A^T y + c = 0 \\ & z \succeq 0.
-\end{array}\end{displaymath}"></TD>
-<TD CLASS="eqno" WIDTH=10 ALIGN="RIGHT">
-(8.1)</TD></TR>
-</TABLE>
-<BR CLEAR="ALL"></DIV><P></P>
-<var>c</var>, <var>h</var> and <var>b</var> are real single-column dense matrices.
-<var>G</var> and <var>A</var> are real dense or sparse matrices.
-The default values for <var>A</var> and <var>b</var> are sparse matrices with
-zero rows, meaning that there are no equality constraints.
-
-<P>
-The <var>solver</var> argument is used to choose among three solvers.
-When it is omitted or <code>None</code>, the default CVXOPT solver is used.
-The external solvers GLPK and MOSEK (if installed) can be selected by
-setting <code><var>solver</var>='glpk'</code> or
-<code><var>solver</var>='mosek'</code>; see section <A href="s-external.html#s-external">8.8</A>.
-
-<P>
-The optional arguments <var>primalstart</var> and <var>dualstart</var> are only
-referenced by the default solver.
-<var>primalstart</var> is a dictionary with keys <code>'x'</code> and <code>'s'</code>,
-used as an optional primal starting point.
-<var>dualstart</var> is a dictionary with keys <code>'y'</code> and <code>'z'</code>,
-used as an optional dual starting point.
-
-<P>
-<tt class="function">lp()</tt> returns a dictionary with keys <code>'status'</code>,
-<code>'x'</code>, <code>'s'</code>, <code>'y'</code>, <code>'z'</code>.
-The possible values of the <code>'status'</code> item are as follows.
-<DL>
-<DT><STRONG><code>'optimal'.</code></STRONG></DT>
-<DD>In this case the <code>'x'</code>, <code>'s'</code>,
-<code>'y'</code> and <code>'z'</code> entries contain the primal and dual solutions,
-which approximately satisfy
-<BR><P></P>
-<DIV ALIGN="CENTER" CLASS="mathdisplay">
-<!-- MATH
- \begin{displaymath}
-Gx + s = h, \qquad Ax=b, \qquad G^T z + A^T y + c = 0, \qquad
- s \succeq 0, \qquad z \succeq 0, \qquad s^T z =0.
-\end{displaymath}
- -->
-
-<IMG
- WIDTH="567" HEIGHT="27" BORDER="0"
- SRC="img112.gif"
- ALT="\begin{displaymath}
-Gx + s = h, \qquad Ax=b, \qquad G^T z + A^T y + c = 0, \qquad
-s \succeq 0, \qquad z \succeq 0, \qquad s^T z =0.
-\end{displaymath}">
-</DIV>
-<BR CLEAR="ALL">
-<P></P>
-
-<P>
-</DD>
-<DT><STRONG><code>'primal infeasible'.</code></STRONG></DT>
-<DD>The <code>'x'</code> and <code>'s'</code> entries are <code>None</code>, and the <code>'y'</code>,
-<code>'z'</code> entries provide an approximate certificate of
-infeasibility, <EM>i.e.</EM>, vectors that approximately satisfy
-<BR><P></P>
-<DIV ALIGN="CENTER" CLASS="mathdisplay">
-<!-- MATH
- \begin{displaymath}
-G^T z + A^T y = 0, \qquad h^T z + b^T y = -1, \qquad z \succeq 0.
-\end{displaymath}
- -->
-
-<IMG
- WIDTH="340" HEIGHT="27" BORDER="0"
- SRC="img113.gif"
- ALT="\begin{displaymath}
-G^T z + A^T y = 0, \qquad h^T z + b^T y = -1, \qquad z \succeq 0.
-\end{displaymath}">
-</DIV>
-<BR CLEAR="ALL">
-<P></P>
-With the <code>'glpk'</code> option, no proof of infeasibility is returned
-(all entries of the dictionary are <code>None</code>).
-
-<P>
-</DD>
-<DT><STRONG><code>'dual infeasible'.</code></STRONG></DT>
-<DD>The LP is dual infeasible.
-The <code>'y'</code> and <code>'z'</code> entries are <code>None</code>, and the <code>'x'</code>
-and <code>'s'</code> entries contain an approximate certificate of dual
-infeasibility
-<BR><P></P>
-<DIV ALIGN="CENTER" CLASS="mathdisplay">
-<!-- MATH
- \begin{displaymath}
-Gx + s = 0, \qquad Ax=0, \qquad c^T x = -1, \qquad s \succeq 0.
-\end{displaymath}
- -->
-
-<IMG
- WIDTH="350" HEIGHT="27" BORDER="0"
- SRC="img114.gif"
- ALT="\begin{displaymath}
-Gx + s = 0, \qquad Ax=0, \qquad c^T x = -1, \qquad s \succeq 0.
-\end{displaymath}">
-</DIV>
-<BR CLEAR="ALL">
-<P></P>
-With the <code>'glpk'</code> option, no proof of dual infeasibility is
-returned.
-
-<P>
-</DD>
-<DT><STRONG><code>'unknown'</code>.</STRONG></DT>
-<DD>The <code>'x'</code>, <code>'s'</code>, <code>'y'</code>,
-<code>'z'</code> entries are <code>None</code>.
-</DD>
-</DL>
-
-<P>
-</dl>
-
-<P>
-As a simple example we solve the LP
-<BR><P></P>
-<DIV ALIGN="CENTER" CLASS="mathdisplay">
-<!-- MATH
- \begin{displaymath}
-\begin{array}[t]{ll}
- \mbox{minimize} & -4x_1 - 5x_2 \\
- \mbox{subject to} & 2x_1 + x_2 \leq 3 \\
- & x_1 + 2x_2 \leq 3 \\
- & x_1 \geq 0, \quad x_2 \geq 0.
- \end{array}
-\end{displaymath}
- -->
-
-<IMG
- WIDTH="195" HEIGHT="87" BORDER="0"
- SRC="img115.gif"
- ALT="\begin{displaymath}
-\begin{array}[t]{ll}
-\mbox{minimize} & -4x_1 - 5x_2 \\
-\...
-...2x_2 \leq 3 \\
-& x_1 \geq 0, \quad x_2 \geq 0.
-\end{array}
-\end{displaymath}">
-</DIV>
-<BR CLEAR="ALL">
-<P></P>
-<div class="verbatim"><pre>
->>> from cvxopt.base import matrix
->>> from cvxopt import solvers
->>> c = matrix([-4., -5.])
->>> G = matrix([[2., 1., -1., 0.], [1., 2., 0., -1.]])
->>> h = matrix([3., 3., 0., 0.])
->>> sol = solvers.lp(c,G,h)
->>> print sol['x']
- 1.0000e-00
- 1.0000e-00
-</pre></div>
-
-<P>
-
-<DIV CLASS="navigation">
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-
-<H1><A NAME="SECTION009100000000000000000"></A> <A NAME="s-orderings"></A>
-<BR>
-7.1 Matrix Orderings (<tt class="module">cvxopt.amd</tt>)
-</H1>
-
-<P>
-CVXOPT includes an interface to the AMD library for computing
-approximate minimum degree orderings of sparse matrices.
-
-<P>
-<div class="seealso">
- <p class="heading">See Also:</p>
-
-<dl compact="compact" class="seeurl">
- <dt><a href='http://www.cise.ufl.edu/research/sparse/amd'
- >AMD code,
-documentation, copyright and license.</a></dt>
- <dd></dd>
- </dl>
-<div class="seetext"><p>P. R. Amestoy, T. A. Davis, I. S. Duff,
-Algorithm 837: AMD, An Approximate Minimum Degree Ordering Algorithm,
-ACM Transactions on Mathematical Software, 30(3), 381-388, 2004.</p></div>
-</div>
-
-<P>
-<dl><dt><table cellpadding="0" cellspacing="0"><tr valign="baseline">
- <td><nobr><b><tt id='l2h-133' xml:id='l2h-133' class="function">order</tt></b>(</nobr></td>
- <td><var>A</var><big>[</big><var>, uplo='L'</var><big>]</big><var></var>)</td></tr></table></dt>
-<dd>
-Computes the approximate mimimum degree ordering of a symmetric sparse
-matrix <I>A</I>.
-The ordering is returned as an integer dense matrix with length equal
-to the order of <var>A</var>. Its entries specify a permutation that
-reduces fill-in during the Cholesky factorization.
-More precisely, if <code><var>p</var> = order(<var>A</var>)</code>, then
-<code><var>A</var>[<var>p</var>,<var>p</var>]</code> has sparser Cholesky factors
-than <code><var>A</var></code>.
-</dl>
-
-<P>
-As an example we consider the matrix
-<BR><P></P>
-<DIV ALIGN="CENTER" CLASS="mathdisplay">
-<!-- MATH
- \begin{displaymath}
-\left[ \begin{array}{rrrr}
- 10 & 0 & 3 & 0 \\
- 0 & 5 & 0 & -2 \\
- 3 & 0 & 5 & 0 \\
- 0 & -2 & 0 & 2
-\end{array}\right].
-\end{displaymath}
- -->
-
-<IMG
- WIDTH="150" HEIGHT="83" BORDER="0"
- SRC="img85.gif"
- ALT="\begin{displaymath}
-\left[ \begin{array}{rrrr}
-10 & 0 & 3 & 0 \\
-0 & 5 & 0 & -2 \\
-3 & 0 & 5 & 0 \\
-0 & -2 & 0 & 2
-\end{array}\right].
-\end{displaymath}">
-</DIV>
-<BR CLEAR="ALL">
-<P></P>
-<div class="verbatim"><pre>
->>> from cvxopt.base import spmatrix
->>> from cvxopt import amd
->>> A = spmatrix([10,3,5,-2,5,2], [0,2,1,2,2,3], [0,0,1,1,2,3])
->>> P = amd.order(A)
->>> print P
- 1
- 0
- 2
- 3
-</pre></div>
-
-<P>
-
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+++ /dev/null
@@ -1,171 +0,0 @@
-<!DOCTYPE html PUBLIC "-//W3C//DTD HTML 4.0 Transitional//EN">
-<html>
-<head>
-<link rel="STYLESHEET" href="cvxopt.css" type='text/css' />
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-<title>2.6 Other Matrix Functions</title>
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-<H1><A NAME="SECTION004600000000000000000"></A> <A NAME="s-otherfuncs"></A>
-<BR>
-2.6 Other Matrix Functions
-</H1>
-The following functions of dense matrices can be imported from
-<tt class="module">cvxopt.base</tt>.
-<dl><dt><table cellpadding="0" cellspacing="0"><tr valign="baseline">
- <td><nobr><b><tt id='l2h-17' xml:id='l2h-17' class="function">sqrt</tt></b>(</nobr></td>
- <td><var>x</var>)</td></tr></table></dt>
-<dd>
-The elementwise square root of <var>x</var>. The result is returned
-as a real matrix if <var>x</var> is an integer or real matrix and
-as a complex matrix if <var>x</var> is a complex matrix. Raises an
-exception when <var>x</var> is an integer or real matrix with negative
-elements.
-</dl>
-
-<P>
-<dl><dt><table cellpadding="0" cellspacing="0"><tr valign="baseline">
- <td><nobr><b><tt id='l2h-18' xml:id='l2h-18' class="function">sin</tt></b>(</nobr></td>
- <td><var>x</var>)</td></tr></table></dt>
-<dd>
-The sine function applied elementwise to <var>x</var>.
-The result is returned as a real matrix if <var>x</var> is an integer
-or real matrix and as a complex matrix otherwise.
-</dl>
-
-<P>
-<dl><dt><table cellpadding="0" cellspacing="0"><tr valign="baseline">
- <td><nobr><b><tt id='l2h-19' xml:id='l2h-19' class="function">cos</tt></b>(</nobr></td>
- <td><var>x</var>)</td></tr></table></dt>
-<dd>
-The cosine function applied elementwise to <var>x</var>.
-The result is returned as a real matrix if <var>x</var> is an integer
-or real matrix and as a complex matrix otherwise.
-</dl>
-
-<P>
-<dl><dt><table cellpadding="0" cellspacing="0"><tr valign="baseline">
- <td><nobr><b><tt id='l2h-20' xml:id='l2h-20' class="function">exp</tt></b>(</nobr></td>
- <td><var>x</var>)</td></tr></table></dt>
-<dd>
-The exponential function applied elementwise to <var>x</var>.
-The result is returned as a real matrix if <var>x</var> is an integer
-or real matrix and as a complex matrix otherwise.
-</dl>
-
-<P>
-<dl><dt><table cellpadding="0" cellspacing="0"><tr valign="baseline">
- <td><nobr><b><tt id='l2h-21' xml:id='l2h-21' class="function">log</tt></b>(</nobr></td>
- <td><var>x</var>)</td></tr></table></dt>
-<dd>
-The natural logarithm applied elementwise to <var>x</var>.
-The result is returned as a real matrix if <var>x</var> is an integer
-or real matrix and as a complex matrix otherwise.
-Raises an exception when <var>x</var> is an integer or real matrix with
-nonnegative elements, or a complex matrix with zero elements.
-</dl>
-
-<P>
-<dl><dt><table cellpadding="0" cellspacing="0"><tr valign="baseline">
- <td><nobr><b><tt id='l2h-22' xml:id='l2h-22' class="function">mul</tt></b>(</nobr></td>
- <td><var>x, y</var>)</td></tr></table></dt>
-<dd>
-The elementwise product of <var>x</var> and <var>y</var>.
-The two matrices must have the same size and type.
-</dl>
-
-<P>
-<dl><dt><table cellpadding="0" cellspacing="0"><tr valign="baseline">
- <td><nobr><b><tt id='l2h-23' xml:id='l2h-23' class="function">div</tt></b>(</nobr></td>
- <td><var>x, y</var>)</td></tr></table></dt>
-<dd>
-The elementwise division of <var>x</var> by <var>y</var>.
-The two matrices must have the same size and type.
-</dl>
-
-<P>
-
-<DIV CLASS="navigation">
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+++ /dev/null
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-<!DOCTYPE html PUBLIC "-//W3C//DTD HTML 4.0 Transitional//EN">
-<html>
-<head>
-<link rel="STYLESHEET" href="cvxopt.css" type='text/css' />
-<link rel="first" href="cvxopt.html" title='CVXOPT: A Python Package for Convex Optimization' />
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-</head>
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-
-<H1><A NAME="SECTION0010900000000000000000"></A> <A NAME="s-parameters"></A>
-<BR>
-8.9 Algorithm Parameters
-</H1>
-In this section we list some algorithm control parameters that can
-be modified without editing the source code.
-These control parameters are accessible via the dictionary
-<tt class="member">solvers.options</tt>. By default the dictionary
-is empty and the default values of the parameters are used.
-
-<P>
-One can change the parameters in the <b>default</b> solvers by
-adding entries with the following key values.
-<DL>
-<DT><STRONG><code>'show_progress'</code></STRONG></DT>
-<DD><code>True</code> or <code>False</code>; turns the output to the screen on or off
-(default: <code>True</code>).
-</DD>
-<DT><STRONG><code>'maxiters'</code></STRONG></DT>
-<DD>maximum number of iterations (default: 100).
-</DD>
-<DT><STRONG><code>'abstol'</code></STRONG></DT>
-<DD>absolute accuracy (default: <code>1e-7</code>).
-</DD>
-<DT><STRONG><code>'reltol'</code></STRONG></DT>
-<DD>relative accuracy (default: <code>1e-7</code>).
-</DD>
-<DT><STRONG><code>'feastol'</code></STRONG></DT>
-<DD>tolerance for feasibility conditions (default:
-<code>1e-7</code>).
-</DD>
-<DT><STRONG><code>'refinement'</code></STRONG></DT>
-<DD><code>True</code> or <code>False</code>. If <code>True</code>,
-one step of iterative refinement is applied after solving KKT equations
-in <tt class="function">conelp()</tt>, <tt class="function">lp()</tt>, and <tt class="function">sdp()</tt>
-(default: <code>True</code>).
-</DD>
-</DL>
-For example the command
-<div class="verbatim"><pre>
->>> from cvxopt import solvers
->>> solvers.options['show_progress'] = False
-</pre></div>
-turns off the screen output during calls to the solvers.
-The tolerances <var>abstol</var>, <var>reltol</var> and <var>feastol</var> have the
-following meaning. <tt class="function">conelp()</tt> terminates with
-status <code>'optimal'</code>, if
-<BR><P></P>
-<DIV ALIGN="CENTER" CLASS="mathdisplay">
-<!-- MATH
- \begin{displaymath}
-s_\mathrm{l} \succeq 0, \qquad S_\mathrm{s} \succeq 0,
-\qquad
- \frac{\|G_\mathrm{l}x+s_\mathrm{l}-h_\mathrm{l}\|_2}
- {\max\{1,\|h_\mathrm{l}\|_2\}} \leq \epsilon_\mathrm{feas},
-\qquad
-\frac{\|G_\mathrm{s}(x)+S_\mathrm{s}-H_\mathrm{s}\|_F}
-{\max\{1,\|H_\mathrm{s}\|_F\}} \leq \epsilon_\mathrm{feas},
-\qquad
-\frac{\|Ax-b\|_2}{\max\{1,\|b\|_2\}} \leq \epsilon_\mathrm{feas},
-\end{displaymath}
- -->
-
-<IMG
- WIDTH="748" HEIGHT="44" BORDER="0"
- SRC="img161.gif"
- ALT="\begin{displaymath}
-s_\mathrm{l} \succeq 0, \qquad S_\mathrm{s} \succeq 0,
-\qqu...
-...t _2}{\max\{1,\Vert b\Vert _2\}} \leq \epsilon_\mathrm{feas},
-\end{displaymath}">
-</DIV>
-<BR CLEAR="ALL">
-<P></P>
-<BR><P></P>
-<DIV ALIGN="CENTER" CLASS="mathdisplay">
-<!-- MATH
- \begin{displaymath}
-z_\mathrm{l} \succeq 0, \qquad Z_\mathrm{s} \succeq 0, \qquad
-\frac{\|G_\mathrm{l}^Tz_\mathrm{l}+
-G_\mathrm{s}^T(Z_\mathrm{s}) + A^Ty+c\|_2}{\max\{1,\|c\|_2\}}
-\leq \epsilon_\mathrm{feas},
-\end{displaymath}
- -->
-
-<IMG
- WIDTH="426" HEIGHT="45" BORDER="0"
- SRC="img162.gif"
- ALT="\begin{displaymath}
-z_\mathrm{l} \succeq 0, \qquad Z_\mathrm{s} \succeq 0, \qqua...
-... _2}{\max\{1,\Vert c\Vert _2\}}
-\leq \epsilon_\mathrm{feas},
-\end{displaymath}">
-</DIV>
-<BR CLEAR="ALL">
-<P></P>
-and
-<BR><P></P>
-<DIV ALIGN="CENTER" CLASS="mathdisplay">
-<!-- MATH
- \begin{displaymath}
-s_\mathrm{l}^T z_\mathrm{l} + \mathop{\bf tr}(S_\mathrm{s}Z_\mathrm{s}) \leq
- \epsilon_\mathrm{abs} \qquad \mbox{or} \qquad
-\left( \min\left\{c^Tx,
- h_\mathrm{l}^T z_\mathrm{l} + \mathop{\bf tr}(H_\mathrm{s} Z_\mathrm{s})
- + b^Ty \right\} < 0, \quad
- \frac{s_\mathrm{l}^Tz_\mathrm{l} + \mathop{\bf tr}(S_\mathrm{s}Z_\mathrm{s})}
- {-\min\{c^Tx, h_\mathrm{l}^Tz_\mathrm{l} +
- \mathop{\bf tr}(H_\mathrm{s} Z_\mathrm{s}) + b^T y\}} \leq \epsilon_\mathrm{rel}
-\right).
-\end{displaymath}
- -->
-
-<IMG
- WIDTH="842" HEIGHT="47" BORDER="0"
- SRC="img163.gif"
- ALT="\begin{displaymath}
-s_\mathrm{l}^T z_\mathrm{l} + \mathop{\bf tr}(S_\mathrm{s}Z...
-...} Z_\mathrm{s}) + b^T y\}} \leq \epsilon_\mathrm{rel}
-\right).
-\end{displaymath}">
-</DIV>
-<BR CLEAR="ALL">
-<P></P>
-It returns with status <code>'primal infeasible'</code> if
-<BR><P></P>
-<DIV ALIGN="CENTER" CLASS="mathdisplay">
-<!-- MATH
- \begin{displaymath}
-z_\mathrm{l} \succeq 0, \qquad
-Z_\mathrm{s} \succeq 0, \qquad
-\qquad \|G_\mathrm{l}^Tz_\mathrm{l} + G_\mathrm{s}^T(Z_\mathrm{s})
- +A^Ty\|_2 \leq \epsilon_\mathrm{feas},
- \qquad h_\mathrm{l}^Tz_\mathrm{l} + \mathop{\bf tr}(H_\mathrm{s} Z_\mathrm{s})
- +b^Ty = -1.
-\end{displaymath}
- -->
-
-<IMG
- WIDTH="637" HEIGHT="28" BORDER="0"
- SRC="img164.gif"
- ALT="\begin{displaymath}
-z_\mathrm{l} \succeq 0, \qquad
-Z_\mathrm{s} \succeq 0, \qqua...
-...{l} + \mathop{\bf tr}(H_\mathrm{s} Z_\mathrm{s})
-+b^Ty = -1.
-\end{displaymath}">
-</DIV>
-<BR CLEAR="ALL">
-<P></P>
-It returns with status <code>'dual infeasible'</code> if
-<BR><P></P>
-<DIV ALIGN="CENTER" CLASS="mathdisplay">
-<!-- MATH
- \begin{displaymath}
-s_\mathrm{l} \succeq 0, \qquad
-S_\mathrm{s} \succeq 0, \qquad
-\qquad
-\|G_\mathrm{l}x+s_\mathrm{l}\|_2 \leq \epsilon_\mathrm{feas}, \qquad
-\|G_\mathrm{s}(x)+S_\mathrm{s}\|_F \leq \epsilon_\mathrm{feas}, \qquad
-\|Ax\|_2 \leq \epsilon_\mathrm{feas}, \qquad
-c^Tx = -1.
-\end{displaymath}
- -->
-
-<IMG
- WIDTH="742" HEIGHT="28" BORDER="0"
- SRC="img165.gif"
- ALT="\begin{displaymath}
-s_\mathrm{l} \succeq 0, \qquad
-S_\mathrm{s} \succeq 0, \qqua...
-...Vert Ax\Vert _2 \leq \epsilon_\mathrm{feas}, \qquad
-c^Tx = -1.
-\end{displaymath}">
-</DIV>
-<BR CLEAR="ALL">
-<P></P>
-The functions <tt class="function">lp()</tt> and <tt class="function">sdp()</tt> call
-<tt class="function">conelp()</tt> and hence use the same stopping criteria.
-
-<P>
-<tt class="function">nlcp()</tt> returns with status <code>'optimal'</code> if
-<BR><P></P>
-<DIV ALIGN="CENTER" CLASS="mathdisplay">
-<!-- MATH
- \begin{displaymath}
-\frac{\| \nabla f_0(x) + D\tilde f(x)^Tz_\mathrm{nl} +
- G^Tz_\mathrm{l} + A^T y \|_2 }
-{\max\{ 1,
-\| \nabla f_0(x_0) + D\tilde f(x_0)^T{\bf 1}+ G^T{\bf 1}\|_2 \}}
-\leq \epsilon_\mathrm{feas}, \qquad
-\frac{\| ( \tilde f(x) + s_{\mathrm{nl}}, Gx + s_\mathrm{l} - h,
- Ax-b ) \|_2}
-{\max\{1, \| ( \tilde f(x_0) + {\bf 1},
-Gx_0 + {\bf 1}-h, Ax_0-b) \|_2 \}} \leq \epsilon_\mathrm{feas}
-\end{displaymath}
- -->
-
-<IMG
- WIDTH="742" HEIGHT="49" BORDER="0"
- SRC="img166.gif"
- ALT="\begin{displaymath}
-\frac{\Vert \nabla f_0(x) + D\tilde f(x)^Tz_\mathrm{nl} +
-...
-...+ {\bf 1}-h, Ax_0-b) \Vert _2 \}} \leq \epsilon_\mathrm{feas}
-\end{displaymath}">
-</DIV>
-<BR CLEAR="ALL">
-<P></P>
-where <I>x0</I> is the point returned by <code>F()</code>, and
-<BR><P></P>
-<DIV ALIGN="CENTER" CLASS="mathdisplay">
-<!-- MATH
- \begin{displaymath}
-\mathrm{gap} \leq \epsilon_\mathrm{abs}
-\qquad \mbox{or} \qquad \left( f_0(x) < 0, \quad
-\frac{\mathrm{gap}} {-f_0(x)} \leq \epsilon_\mathrm{rel} \right)
-\qquad \mbox{or} \qquad
-\left( L(x,y,z) > 0, \quad \frac{\mathrm{gap}}
-{L(x,y,z)} \leq \epsilon_\mathrm{rel} \right)
-\end{displaymath}
- -->
-
-<IMG
- WIDTH="684" HEIGHT="45" BORDER="0"
- SRC="img167.gif"
- ALT="\begin{displaymath}
-\mathrm{gap} \leq \epsilon_\mathrm{abs}
-\qquad \mbox{or} \qq...
-...ac{\mathrm{gap}}
-{L(x,y,z)} \leq \epsilon_\mathrm{rel} \right)
-\end{displaymath}">
-</DIV>
-<BR CLEAR="ALL">
-<P></P>
-where
-<BR><P></P>
-<DIV ALIGN="CENTER" CLASS="mathdisplay">
-<!-- MATH
- \begin{displaymath}
-\mathrm{gap} =
-\left[\begin{array}{c} s_\mathrm{nl} \\s_\mathrm{l}
-\end{array}\right]^T
-\left[\begin{array}{c} z_\mathrm{nl} \\z_\mathrm{l}
-\end{array}\right],
-\qquad
-L(x,y,z) = f_0(x) + z_\mathrm{nl}^T \tilde f(x) +
- z_\mathrm{l}^T (Gx-h) + y^T(Ax-b).
-\end{displaymath}
- -->
-
-<IMG
- WIDTH="558" HEIGHT="48" BORDER="0"
- SRC="img168.gif"
- ALT="\begin{displaymath}
-\mathrm{gap} =
-\left[\begin{array}{c} s_\mathrm{nl} \\ s_\...
-...athrm{nl}^T \tilde f(x) +
-z_\mathrm{l}^T (Gx-h) + y^T(Ax-b).
-\end{displaymath}">
-</DIV>
-<BR CLEAR="ALL">
-<P></P>
-The functions <tt class="function">qp()</tt>, <tt class="function">gp()</tt> and <tt class="function">cp()</tt>
-call <tt class="function">nlcp()</tt> and hence use the same stopping criteria
-(with <I>x0</I>=0 for <tt class="function">qp()</tt> and <tt class="function">gp()</tt>).
-
-<P>
-The control parameters listed in the <b>GLPK</b> documentation are
-set to their default values and can also be customized by making
-an entry in <tt class="member">solvers.options</tt>.
-The keys in the dictionary are strings with the name of the GLPK
-parameter. The command
-<div class="verbatim"><pre>
->>> from cvxopt import solvers
->>> solvers.options['LPX_K_MSGLEV'] = 0
-</pre></div>
-turns off the screen output subsequent calls <tt class="function">lp()</tt> with
-the <code>'glpk'</code> option.
-
-<P>
-The <b>MOSEK</b> <a class="ulink" href="http://www.mosek.com/products/3/tools/doc/html/tools/node22.html"
- >control parameters</a>
-are set to their default values.
-The corresponding keys in <code>solvers.options</code> are strings with the
-name of the MOSEK parameter. For example the command
-<div class="verbatim"><pre>
->>> from cvxopt import solvers
->>> solvers.options['MSK_IPAR_LOG'] = 0
-</pre></div>
-turns off the screen output during calls of <tt class="function">lp()</tt>
-and <tt class="function">qp()</tt> with the <code>'mosek'</code> option.
-
-<P>
-The following control parameters affect the <b>DSDP</b> algorithm:
-<DL>
-<DT><STRONG><code>'DSDP_Monitor'</code></STRONG></DT>
-<DD>the interval (in number of iterations)
- at which output is printed to the screen
-(default: 0).
-</DD>
-<DT><STRONG><code>'DSDP_MaxIts'</code></STRONG></DT>
-<DD>maximum number of iterations.
-</DD>
-<DT><STRONG><code>'DSDP_GapTolerance'</code></STRONG></DT>
-<DD>relative accuracy (default:
-<code>1e-5</code>).
-</DD>
-</DL>
-
-<DIV CLASS="navigation">
-<div class='online-navigation'>
-<p></p><hr />
-<table align="center" width="100%" cellpadding="0" cellspacing="2">
-<tr>
-<td class='online-navigation'><a rel="prev" title="8.8 Optional Solvers"
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- href="genindex.html"><img src='index.gif'
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-<hr />
-<span class="release-info">Release 0.8.2, documentation updated on February 6, 2007.</span>
-</DIV>
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diff --git a/doc/cvxopt/s-random.html b/doc/cvxopt/s-random.html
deleted file mode 100644
index 47e0f57..0000000
--- a/doc/cvxopt/s-random.html
+++ /dev/null
@@ -1,167 +0,0 @@
-<!DOCTYPE html PUBLIC "-//W3C//DTD HTML 4.0 Transitional//EN">
-<html>
-<head>
-<link rel="STYLESHEET" href="cvxopt.css" type='text/css' />
-<link rel="first" href="cvxopt.html" title='CVXOPT: A Python Package for Convex Optimization' />
-<link rel='contents' href='contents.html' title="Contents" />
-<link rel='index' href='genindex.html' title='Index' />
-<link rel='last' href='about.html' title='About this document...' />
-<link rel='help' href='about.html' title='About this document...' />
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-<link rel="parent" href="node4.html" />
-<link rel="next" href="s-array-interface.html" />
-<meta name='aesop' content='information' />
-<title>2.7 Randomly Generated Matrices</title>
-</head>
-<body>
-<DIV CLASS="navigation">
-<div id='top-navigation-panel' xml:id='top-navigation-panel'>
-<table align="center" width="100%" cellpadding="0" cellspacing="2">
-<tr>
-<td class='online-navigation'><a rel="prev" title="2.6 Other Matrix Functions"
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-<td class='online-navigation'><a rel="next" title="2.8 The NumPy Array"
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-<td class='online-navigation'><a rel="contents" title="Table of Contents"
- href="contents.html"><img src='contents.gif'
- border='0' height='32' alt='Contents' width='32' /></A></td>
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-<td class='online-navigation'><a rel="index" title="Index"
- href="genindex.html"><img src='index.gif'
- border='0' height='32' alt='Index' width='32' /></A></td>
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-<div class='online-navigation'>
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-<hr /></div>
-</DIV>
-<!--End of Navigation Panel-->
-
-<H1><A NAME="SECTION004700000000000000000"></A> <A NAME="s-random"></A>
-<BR>
-2.7 Randomly Generated Matrices
-</H1>
-The module <tt class="module">cvxopt.random</tt> provides functions for generating
-random matrices. Two types of random matrices are defined:
-matrices with normally distributed entries and matrices with uniformly
-distributed entries.
-
-<P>
-The pseudo-random number generators used to
-generate the random matrices are from the package described in the
-references below.
-<div class="seealso">
- <p class="heading">See Also:</p>
-
-<dl compact="compact" class="seeurl">
- <dt><a href='http://www.cs.wm.edu/~va/software/park/park.html'
- >S. Park, Random Number Generators.</a></dt>
- <dd></dd>
- </dl>
-
-<P>
-S. Park, D. Geyer, Random Number Generators: Good Ones Are
-Hard To Find,
-Communications of the ACM, October 1988.
-</div>
-
-<P>
-<dl><dt><table cellpadding="0" cellspacing="0"><tr valign="baseline">
- <td><nobr><b><tt id='l2h-24' xml:id='l2h-24' class="function">normal</tt></b>(</nobr></td>
- <td><var>nrows</var><big>[</big><var>, ncols</var><big>[</big><var>,
- mean</var><big>[</big><var>, std</var><big>]</big><var></var><big>]</big><var></var><big>]</big><var></var>)</td></tr></table></dt>
-<dd>
-Returns a type <code>'d'</code> matrix of size <var>nrows</var> by
-<var>ncols</var> with random elements chosen from a normal distribution
-with mean <var>mean</var> and standard deviation <var>std</var>.
-The default values for the optional arguments are
-<var>ncols</var>=1, <var>mean</var>=0.0, <var>std</var>=1.0.
-</dl>
-
-<P>
-<dl><dt><table cellpadding="0" cellspacing="0"><tr valign="baseline">
- <td><nobr><b><tt id='l2h-25' xml:id='l2h-25' class="function">uniform</tt></b>(</nobr></td>
- <td><var>nrows</var><big>[</big><var>, ncols</var><big>[</big><var>,
- a</var><big>[</big><var>, b</var><big>]</big><var></var><big>]</big><var></var><big>]</big><var></var>)</td></tr></table></dt>
-<dd>
-Returns a type <code>'d'</code> matrix of size <var>nrows</var> by
-<var>ncols</var> matrix with random elements, uniformly distributed
-between <var>a</var> and <var>b</var>.
-The default values for the optional arguments are
-<var>ncols</var>=1, <var>a</var>=0.0, <var>b</var>=1.0.
-</dl>
-
-<P>
-<dl><dt><table cellpadding="0" cellspacing="0"><tr valign="baseline">
- <td><nobr><b><tt id='l2h-26' xml:id='l2h-26' class="function">getseed</tt></b>(</nobr></td>
- <td><var></var>)</td></tr></table></dt>
-<dd>
-Returns the current seed value (the state of the random number
-generator).
-</dl>
-
-<P>
-<dl><dt><table cellpadding="0" cellspacing="0"><tr valign="baseline">
- <td><nobr><b><tt id='l2h-27' xml:id='l2h-27' class="function">setseed</tt></b>(</nobr></td>
- <td><var></var><big>[</big><var>value</var><big>]</big><var></var>)</td></tr></table></dt>
-<dd>
-Sets the seed value. <var>value</var> must be a nonnegative integer.
-If <var>value</var> is absent or equal to zero, the seed value is taken
-from the system clock.
-</dl>
-
-<P>
-
-<DIV CLASS="navigation">
-<div class='online-navigation'>
-<p></p><hr />
-<table align="center" width="100%" cellpadding="0" cellspacing="2">
-<tr>
-<td class='online-navigation'><a rel="prev" title="2.6 Other Matrix Functions"
- href="s-otherfuncs.html"><img src='previous.gif'
- border='0' height='32' alt='Previous Page' width='32' /></A></td>
-<td class='online-navigation'><a rel="parent" title="2. Dense Matrices (cvxopt.base)"
- href="node4.html"><img src='up.gif'
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-<td class='online-navigation'><a rel="next" title="2.8 The NumPy Array"
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-<td align="center" width="100%">CVXOPT: A Python Package for Convex Optimization</td>
-<td class='online-navigation'><a rel="contents" title="Table of Contents"
- href="contents.html"><img src='contents.gif'
- border='0' height='32' alt='Contents' width='32' /></A></td>
-<td class='online-navigation'><img src='blank.gif'
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-<td class='online-navigation'><a rel="index" title="Index"
- href="genindex.html"><img src='index.gif'
- border='0' height='32' alt='Index' width='32' /></A></td>
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-<!DOCTYPE html PUBLIC "-//W3C//DTD HTML 4.0 Transitional//EN">
-<html>
-<head>
-<link rel="STYLESHEET" href="cvxopt.css" type='text/css' />
-<link rel="first" href="cvxopt.html" title='CVXOPT: A Python Package for Convex Optimization' />
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-<link rel='help' href='about.html' title='About this document...' />
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-<title>8.4 Semidefinite Programming</title>
-</head>
-<body>
-<DIV CLASS="navigation">
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-<table align="center" width="100%" cellpadding="0" cellspacing="2">
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-
-<H1><A NAME="SECTION0010400000000000000000"></A> <A NAME="s-sdpsolver"></A>
-<BR>
-8.4 Semidefinite Programming
-</H1>
-We use the following notation for a pair of primal and dual
-semidefinite programs (SDPs):
-<BR>
-<DIV ALIGN="RIGHT" CLASS="mathdisplay">
-
-<!-- MATH
- \begin{equation}
-\mbox{Primal:}\quad \begin{array}[t]{ll}
-\mbox{minimize} & c^Tx \\
-\mbox{subject to} & G_\mathrm{l}x + s_\mathrm{l} = h_\mathrm{l} \\
- & G_\mathrm{s}(x) + S_\mathrm{s} = H_\mathrm{s} \\
- & Ax=b \\& s_\mathrm{l} \succeq 0, \quad S_\mathrm{s} \succeq 0
-\end{array} \qquad\qquad
-\mbox{Dual:}\quad
-\begin{array}[t]{ll}
-\mbox{maximize} & -h_\mathrm{l}^T z_\mathrm{l} -
- \mathop{\bf tr}(H_\mathrm{s} Z_\mathrm{s}) - b^T y \\
-\mbox{subject to} & G_\mathrm{l}^Tz_\mathrm{l} +
- G_\mathrm{s}^T(Z_\mathrm{s}) + A^T y + c = 0 \\
- & z_\mathrm{l} \succeq 0, \quad Z_\mathrm{s} \succeq 0.
-\end{array}
-\end{equation}
- -->
-<A NAME="e-sdp"></A>
-<TABLE WIDTH="100%" ALIGN="CENTER">
-<TR VALIGN="MIDDLE"><TD></TD><TD ALIGN="CENTER" NOWRAP><A NAME="e-sdp"></A><IMG
- WIDTH="682" HEIGHT="106" BORDER="0"
- SRC="img127.gif"
- ALT="\begin{displaymath}
-\mbox{Primal:}\quad \begin{array}[t]{ll}
-\mbox{minimize} & c...
-...\mathrm{l} \succeq 0, \quad Z_\mathrm{s} \succeq 0.
-\end{array}\end{displaymath}"></TD>
-<TD CLASS="eqno" WIDTH=10 ALIGN="RIGHT">
-(8.2)</TD></TR>
-</TABLE>
-<BR CLEAR="ALL"></DIV><P></P>
-The dimensions of the primal and dual variables are
-<BR><P></P>
-<DIV ALIGN="CENTER" CLASS="mathdisplay">
-<!-- MATH
- \begin{displaymath}
-x\in {\mbox{\bf R}}^n, \qquad s_\mathrm{l} \in {\mbox{\bf R}}^m,
-\qquad S_\mathrm{s} \in {\mbox{\bf S}}^{m_0} \times \cdots \times
-{\mbox{\bf S}}^{m_{N-1}}, \qquad
- y \in{\mbox{\bf R}}^p, \qquad z_\mathrm{l}\in {\mbox{\bf R}}^m,
-\qquad Z_\mathrm{s} \in {\mbox{\bf S}}^{m_0} \times \cdots \times
-{\mbox{\bf S}}^{m_{N-1}},
-\end{displaymath}
- -->
-
-<IMG
- WIDTH="705" HEIGHT="27" BORDER="0"
- SRC="img128.gif"
- ALT="\begin{displaymath}
-x\in {\mbox{\bf R}}^n, \qquad s_\mathrm{l} \in {\mbox{\bf R...
-...{\bf S}}^{m_0} \times \cdots \times
-{\mbox{\bf S}}^{m_{N-1}},
-\end{displaymath}">
-</DIV>
-<BR CLEAR="ALL">
-<P></P>
-where <!-- MATH
- ${\mbox{\bf S}}^n$
- -->
-<SPAN CLASS="MATH"><IMG
- WIDTH="21" HEIGHT="32" ALIGN="MIDDLE" BORDER="0"
- SRC="img129.gif"
- ALT="${\mbox{\bf S}}^n$"></SPAN> is the set of real symmetric matrices
-of order <I>n</I>. The problem data are the matrices
-<BR><P></P>
-<DIV ALIGN="CENTER" CLASS="mathdisplay">
-<!-- MATH
- \begin{displaymath}
-c\in{\mbox{\bf R}}^n, \qquad G_\mathrm{l} \in{\mbox{\bf R}}^{m\times n},
-\qquad h_\mathrm{l} \in {\mbox{\bf R}}^{m}, \qquad
- H_\mathrm{s} \in {\mbox{\bf S}}^{m_0} \times \cdots \times {\mbox{\bf S}}^{m_{N-1}},
-\qquad
-A \in {\mbox{\bf R}}^{p\times n}, \qquad b \in {\mbox{\bf R}}^{p},
-\end{displaymath}
- -->
-
-<IMG
- WIDTH="645" HEIGHT="27" BORDER="0"
- SRC="img130.gif"
- ALT="\begin{displaymath}
-c\in{\mbox{\bf R}}^n, \qquad G_\mathrm{l} \in{\mbox{\bf R}}^...
-... {\mbox{\bf R}}^{p\times n}, \qquad b \in {\mbox{\bf R}}^{p},
-\end{displaymath}">
-</DIV>
-<BR CLEAR="ALL">
-<P></P>
-and the linear mapping
-<!-- MATH
- $G_\mathrm{s} : {\mbox{\bf R}}^n \rightarrow {\mbox{\bf S}}^{m_1} \times \cdots \times
-{\mbox{\bf S}}^{m_N}$
- -->
-<SPAN CLASS="MATH"><IMG
- WIDTH="196" HEIGHT="32" ALIGN="MIDDLE" BORDER="0"
- SRC="img131.gif"
- ALT="$G_\mathrm{s} : {\mbox{\bf R}}^n \rightarrow {\mbox{\bf S}}^{m_1} \times \cdots \times
-{\mbox{\bf S}}^{m_N}$"></SPAN> and its adjoint <!-- MATH
- $G_\mathrm{s}^T$
- -->
-<SPAN CLASS="MATH"><IMG
- WIDTH="26" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
- SRC="img132.gif"
- ALT="$G_\mathrm{s}^T$"></SPAN>.
-As for LPs we store vector variables as dense real matrices with one
-column.
-Block-diagonal symmetric matrices are stored as lists of square
-dense real matrices, with the lower triangular part of each matrix
-representing the lower triangular part of a diagonal block. Entries
-above the diagonal are not referenced.
-
-<P>
-<dl><dt><table cellpadding="0" cellspacing="0"><tr valign="baseline">
- <td><nobr><b><tt id='l2h-148' xml:id='l2h-148' class="function">sdp</tt></b>(</nobr></td>
- <td><var>c</var><big>[</big><var>, Gl, hl</var><big>[</big><var>,
- Gs, hs</var><big>[</big><var>, A, b</var><big>[</big><var>, solver</var><big>[</big><var>,
- primalstart</var><big>[</big><var>, dualstart</var><big>]</big><var></var><big>]</big><var></var><big>]</big><var></var><big>]</big><var></var><big>]</big><var></var><big>]</big><var></var>)</td></tr></table></dt>
-<dd>
-
-<P>
-Solves the pair of primal and dual SDPs (<A HREF="#e-sdp">8.2</A>).
-
-<P>
-<var>c</var> is a dense real matrix with one column.
-<var>Gl</var> and <var>A</var> are dense or sparse real matrices.
-<var>hl</var> and <var>b</var> are dense real matrices with one column.
-The default values for <var>Gl</var>, <var>hl</var>, <var>A</var> and <var>b</var> are
-empty matrices, <EM>i.e.</EM>, matrices with zero rows.
-
-<P>
-<var>Gs</var> and <var>hs</var> are lists of length <I>N</I> that specify the
-linear matrix inequality constraints.
-<var>hs</var> is a list of square dense real matrices <code><var>hs</var>[k]</code> of
-order <I>m_k</I>.
-<var>Gs</var> is a list of dense or sparse real matrices
-<code><var>Gs</var>[k]</code> with <I>m_k</I>*<I>m_k</I> rows and <I>n</I> columns,
-such that the product <code><var>Gs</var>[k]*<var>x</var></code>
-is the <I>k</I>th diagonal block of <I>Gs</I>(<var>x</var>), stored
-columnwise.
-
-<P>
-The <var>solver</var> argument is used to choose between two
-solvers: the default solver (used when <var>solver</var> is absent or
-equal to <code>None</code>) and the external solver DSDP5
-(<code><var>solver</var>='dsdp'</code>); see section <A href="s-external.html#s-external">8.8</A>.
-With the <code>'dsdp'</code> option the code does not accept problems with
-equality constraints.
-
-<P>
-The optional argument <var>primalstart</var> is a dictionary with keys
-<code>'x'</code>, <code>'sl'</code>, and <code>'ss'</code>, used as an optional primal
-starting point.
-<var>dualstart</var> is a dictionary with keys <code>'y'</code>, <code>'zl'</code>,
-<code>'zs'</code>, used as an optional dual starting point.
-These two arguments are ignored when the DSDP solver is used.
-
-<P>
-<tt class="function">sdp()</tt> returns a dictionary with keys <code>'status'</code>,
-<code>'x'</code>, <code>'sl'</code>, <code>'ss'</code>, <code>'y'</code>, <code>'zl'</code>,
-<code>'ss'</code>.
-The possible values of the <code>'status'</code> item are as follows.
-<DL>
-<DT><STRONG><code>'optimal'.</code></STRONG></DT>
-<DD>In this case the <code>'x'</code>, <code>'sl'</code>,
-<code>'ss'</code>, <code>'y'</code>, <code>'zl'</code>, <code>'zs'</code> entries contain
-primal and dual optimal solutions, which approximately satisfy
-<BR><P></P>
-<DIV ALIGN="CENTER" CLASS="mathdisplay">
-<!-- MATH
- \begin{displaymath}
-G_\mathrm{l}x+s_\mathrm{l} = h_\mathrm{l},
-\qquad
-G_\mathrm{s}(x)+S_\mathrm{s} = H_\mathrm{s},
-\qquad
-Ax=b, \qquad
-G_\mathrm{l}^Tz_\mathrm{l} + G_\mathrm{s}^T(Z_\mathrm{s}) + A^Ty+c = 0
-\end{displaymath}
- -->
-
-<IMG
- WIDTH="553" HEIGHT="28" BORDER="0"
- SRC="img133.gif"
- ALT="\begin{displaymath}
-G_\mathrm{l}x+s_\mathrm{l} = h_\mathrm{l},
-\qquad
-G_\mathr...
-...m{l}^Tz_\mathrm{l} + G_\mathrm{s}^T(Z_\mathrm{s}) + A^Ty+c = 0
-\end{displaymath}">
-</DIV>
-<BR CLEAR="ALL">
-<P></P>
-and
-<BR><P></P>
-<DIV ALIGN="CENTER" CLASS="mathdisplay">
-<!-- MATH
- \begin{displaymath}
-s_\mathrm{l} \succeq 0, \qquad S_\mathrm{s} \succeq 0, \qquad
- z_\mathrm{l} \succeq 0, \qquad Z_\mathrm{s} \succeq 0, \qquad
- s_\mathrm{l}^T z_\mathrm{l} + \mathop{\bf tr}(S_\mathrm{s}Z_\mathrm{s}) = 0.
-\end{displaymath}
- -->
-
-<IMG
- WIDTH="469" HEIGHT="28" BORDER="0"
- SRC="img134.gif"
- ALT="\begin{displaymath}
-s_\mathrm{l} \succeq 0, \qquad S_\mathrm{s} \succeq 0, \qqu...
-... z_\mathrm{l} + \mathop{\bf tr}(S_\mathrm{s}Z_\mathrm{s}) = 0.
-\end{displaymath}">
-</DIV>
-<BR CLEAR="ALL">
-<P></P>
-
-<P>
-</DD>
-<DT><STRONG><code>'primal infeasible'.</code></STRONG></DT>
-<DD>The <code>'x'</code>, <code>'sl'</code> and <code>'ss'</code> entries are <code>None</code>,
-and the <code>'y'</code>, <code>'zl'</code>, <code>'zs'</code> entries provide an
-approximate certificate of infeasibility:
-<BR><P></P>
-<DIV ALIGN="CENTER" CLASS="mathdisplay">
-<!-- MATH
- \begin{displaymath}
-G_\mathrm{l}^Tz_\mathrm{l} + G_\mathrm{s}^T(Z_\mathrm{s}) +A^Ty = 0
-\qquad h_\mathrm{l}^Tz_\mathrm{l} + \mathop{\bf tr}(H_\mathrm{s} Z_\mathrm{s})
- +b^Ty = -1, \qquad
-z_\mathrm{l} \succeq 0, \qquad Z_\mathrm{s} \succeq 0.
-\end{displaymath}
- -->
-
-<IMG
- WIDTH="553" HEIGHT="28" BORDER="0"
- SRC="img135.gif"
- ALT="\begin{displaymath}
-G_\mathrm{l}^Tz_\mathrm{l} + G_\mathrm{s}^T(Z_\mathrm{s}) +A...
-...\qquad
-z_\mathrm{l} \succeq 0, \qquad Z_\mathrm{s} \succeq 0.
-\end{displaymath}">
-</DIV>
-<BR CLEAR="ALL">
-<P></P>
-
-<P>
-</DD>
-<DT><STRONG><code>'dual infeasible'.</code></STRONG></DT>
-<DD>The SDP is dual infeasible.
-The <code>'y'</code>, <code>'zl'</code> and <code>'zs'</code> entries are <code>None</code>,
-and the <code>'x'</code>, <code>'sl'</code>, <code>'ss'</code> entries contain an
-approximate certificate of dual infeasibility:
-<BR><P></P>
-<DIV ALIGN="CENTER" CLASS="mathdisplay">
-<!-- MATH
- \begin{displaymath}
-G_\mathrm{l}x+s_\mathrm{l} = 0, \qquad
-G_\mathrm{s}(x)+S_\mathrm{s} = 0, \qquad
-Ax = 0, \qquad c^Tx = -1, \qquad
-s_\mathrm{l} \succeq 0, \qquad S_\mathrm{s} \succeq 0.
-\end{displaymath}
- -->
-
-<IMG
- WIDTH="576" HEIGHT="28" BORDER="0"
- SRC="img136.gif"
- ALT="\begin{displaymath}
-G_\mathrm{l}x+s_\mathrm{l} = 0, \qquad
-G_\mathrm{s}(x)+S_\ma...
-...\qquad
-s_\mathrm{l} \succeq 0, \qquad S_\mathrm{s} \succeq 0.
-\end{displaymath}">
-</DIV>
-<BR CLEAR="ALL">
-<P></P>
-
-<P>
-</DD>
-<DT><STRONG><code>'unknown'</code>.</STRONG></DT>
-<DD>The <code>'x'</code>, <code>'sl'</code>, <code>'ss'</code>,
-<code>'y'</code>, <code>'zl'</code> and <code>'zs'</code> entries are <code>None</code>.
-</DD>
-</DL>
-</dl>
-
-<P>
-We illustrate the calling sequence with a small example.
-<BR><P></P>
-<DIV ALIGN="CENTER" CLASS="mathdisplay">
-<!-- MATH
- \begin{displaymath}
-\begin{array}{ll}
-\mbox{minimize} & x_1 - x_2 + x_3 \\
-\mbox{subject to} &
- x_1 \left[ \begin{array}{cc} -7 & -11 \\-11 & 3
- \end{array}\right] +
- x_2 \left[ \begin{array}{cc}
- 7 & -18 \\-18 & 8 \end{array}\right] +
- x_3 \left[ \begin{array}{cc}
- -2 & -8 \\-8 & 1
- \end{array}\right] \preceq
- \left[ \begin{array}{cc}
- 33 & -9 \\-9 & 26 \end{array}\right] \\*[1ex]
-& x_1 \left[ \begin{array}{ccc}
- -21 & -11 & 0 \\-11 & 10 & 8 \\0 & 8 & 5
- \end{array}\right] +
- x_2 \left[ \begin{array}{ccc}
- 0 & 10 & 16 \\
-10 & -10 & -10 \\
-16 & -10 & 3
- \end{array}\right] +
- x_3 \left[ \begin{array}{ccc}
- -5 & 2 & -17 \\
- 2 & -6 & -7 \\
- -17 & 8 & 6
- \end{array}\right]
-\preceq \left[ \begin{array}{ccc}
- 14 & 9 & 40 \\
- 9 & 91 & 10 \\
- 40 & 10 & 15
-\end{array} \right]
-\end{array}
-\end{displaymath}
- -->
-
-<IMG
- WIDTH="726" HEIGHT="121" BORDER="0"
- SRC="img137.gif"
- ALT="\begin{displaymath}
-\begin{array}{ll}
-\mbox{minimize} & x_1 - x_2 + x_3 \\
-\mbo...
-...
-9 & 91 & 10 \\
-40 & 10 & 15
-\end{array} \right]
-\end{array}\end{displaymath}">
-</DIV>
-<BR CLEAR="ALL">
-<P></P>
-<div class="verbatim"><pre>
->>> from cvxopt.base import matrix
->>> from cvxopt import solvers
->>> c = matrix([1.,-1.,1.])
->>> G = [ matrix([[-7., -11., -11., 3.],
- [ 7., -18., -18., 8.],
- [-2., -8., -8., 1.]]) ]
->>> G += [ matrix([[-21., -11., 0., -11., 10., 8., 0., 8., 5.],
- [ 0., 10., 16., 10., -10., -10., 16., -10., 3.],
- [ -5., 2., -17., 2., -6., 8., -17., -7., 6.]]) ]
->>> h = [ matrix([[33., -9.], [-9., 26.]]) ]
->>> h += [ matrix([[14., 9., 40.], [9., 91., 10.], [40., 10., 15.]]) ]
->>> sol = solvers.sdp(c, Gs=G, hs=h)
->>> print sol['x']
- -3.6775e-01
- 1.8983e+00
- -8.8747e-01
->>> print sol['zs'][0]
- 3.9613e-03 0.0000e+00
- -4.3390e-03 4.7526e-03
->>> print sol['zs'][1]
- 5.5803e-02 0.0000e+00 0.0000e+00
- -2.4103e-03 1.0411e-04 0.0000e+00
- 2.4214e-02 -1.0459e-03 1.0507e-02
-</pre></div>
-Note that only the lower triangular parts of the dual variables are
-returned (in the example the returned values of the upper triangular
-elements happen to be zero, but this is not necessarily the case).
-
-<P>
-Only the entries in <var>Gs</var> and <var>hs</var> that correspond to lower
-triangular entries need to be provided, so in the example <var>h</var> and
-<var>G</var> can also be defined as follows.
-<div class="verbatim"><pre>
->>> G = [ matrix([[-7., -11., 0., 3.],
- [ 7., -18., 0., 8.],
- [-2., -8., 0., 1.]]) ]
->>> G += [ matrix([[-21., -11., 0., 0., 10., 8., 0., 0., 5.],
- [ 0., 10., 16., 0., -10., -10., 0., 0., 3.],
- [ -5., 2., -17., 0., -6., 8., 0., 0., 6.]]) ]
->>> h = [ matrix([[33., -9.], [0., 26.]]) ]
->>> h += [ matrix([[14., 9., 40.], [0., 91., 10.], [0., 0., 15.]]) ]
-</pre></div>
-
-<P>
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+++ /dev/null
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-<!DOCTYPE html PUBLIC "-//W3C//DTD HTML 4.0 Transitional//EN">
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-
-<H1><A NAME="SECTION008300000000000000000"></A> <A NAME="s-spmatrix-arith"></A>
-<BR>
-6.3 Arithmetic Operations
-</H1>
-Most of the operations defined for dense <code>'d'</code> and <code>'z'</code> matrices
-(section <A href="s-arithmetic.html#s-arithmetic">2.3</A>) are also defined for sparse matrices.
-In the following table, <var>A</var> is a sparse matrix,
-<var>B</var> is sparse or dense, and <var>c</var> is a scalar, defined as a
-Python number or a 1 by 1 dense matrix.
-
-<P>
-<DIV ALIGN="CENTER">
-<TABLE CELLPADDING=3 BORDER="1">
-<TR><TD ALIGN="LEFT">Unary plus/minus</TD>
-<TD ALIGN="LEFT"><code>+<var>A</var></code>, <code>-<var>A</var></code></TD>
-</TR>
-<TR><TD ALIGN="LEFT">Addition</TD>
-<TD ALIGN="LEFT"><code><var>A</var>+<var>B</var></code>, <code><var>B</var>+<var>A</var></code>,
- <code><var>A</var>+<var>c</var></code>, <code><var>c</var>+<var>A</var></code></TD>
-</TR>
-<TR><TD ALIGN="LEFT">Subtraction</TD>
-<TD ALIGN="LEFT"><code><var>A</var>-<var>B</var></code>, <code><var>B</var>-<var>A</var></code>,
- <code><var>A</var>-<var>c</var></code>, <code><var>c</var>-<var>A</var></code></TD>
-</TR>
-<TR><TD ALIGN="LEFT">Matrix multiplication</TD>
-<TD ALIGN="LEFT"><code><var>A</var>*<var>B</var></code>,
- <code><var>B</var>*<var>A</var></code></TD>
-</TR>
-<TR><TD ALIGN="LEFT">Scalar multiplication and division</TD>
-<TD ALIGN="LEFT"><code><var>c</var>*<var>A</var></code>,
- <code><var>A</var>*<var>c</var></code>, <code><var>A</var>/<var>c</var></code></TD>
-</TR>
-</TABLE>
-</DIV>
-
-<P>
-If <var>B</var> is a dense matrix, then the result of
-<code><var>A</var>+<var>B</var></code>, <code><var>B</var>+<var>A</var></code>, <code><var>A</var>-<var>B</var></code>,
-<code><var>B</var>-<var>A</var></code> is a dense matrix.
-The typecode of the result is <code>'d'</code> if <var>A</var> has typcode <code>'d'</code> and
-<var>B</var> has typecode <code>'i'</code> or <code>'d'</code>,
-and it is <code>'z'</code> if <var>A</var> and/or <var>B</var> have typecode <code>'z'</code>.
-
-<P>
-If <var>B</var> is a sparse matrix, then the result of
-<code><var>A</var>+<var>B</var></code>, <code><var>B</var>+<var>A</var></code>,
-<code><var>A</var>-<var>B</var></code>, <code><var>B</var>-<var>A</var></code> is a sparse
-matrix. The typecode of the result is <code>'d'</code> if <var>A</var> and <var>B</var>
-have typecode <code>'d'</code>, and <code>'z'</code> otherwise.
-
-<P>
-If <var>c</var> in <code><var>A</var>+<var>c</var></code>, <code><var>A</var>-<var>c</var></code>,
-<code><var>c</var>+<var>A</var></code>, <code><var>c</var>-<var>A</var></code> is a number,
-then it is interpreted as a dense matrix with the same size as
-<var>A</var>, typecode given by the type of <var>c</var>, and all entries equal
-to <var>c</var>.
-If <var>c</var> is a 1 by 1 dense matrix and the size of <var>A</var> is not 1
-by 1, then <var>c</var>
-is interpreted as a dense matrix of the same size as <var>A</var>,
-typecode given by the typecode of <var>c</var>, and all entries equal to
-<code><var>c</var>[0]</code>.
-
-<P>
-The result of a matrix-matrix product <code><var>A</var>*<var>B</var></code> or
-<code><var>B</var>*<var>A</var></code> is a dense matrix if <var>B</var> is dense, and sparse
-if <var>B</var> is sparse. The matrix-matrix product is not allowed if
-<var>B</var> is a dense <code>'i'</code> matrix.
-
-<P>
-If <var>c</var> is a number (Python integer float or complex), then the
-operations <code><var>c</var>*<var>A</var></code> and <code><var>A</var>*<var>c</var></code> define
-scalar multiplication and return a sparse matrix.
-
-<P>
-If <var>c</var> is a 1 by 1 dense matrix, then, if possible, the products
-<code><var>c</var>*<var>A</var></code> and <code><var>A</var>*<var>c</var></code> are interpreted as
-matrix-matrix products and a dense matrix is returned.
-If the product cannot be interpreted as a matrix-matrix product
-(either because the dimensions of <var>A</var> are incompatible or because
-<var>c</var> has typecode <code>'i'</code>), then the product is interpreted as the
-scalar multiplication with <code><var>c</var>[0]</code> and a sparse matrix is
-returned.
-
-<P>
-The division <code><var>A</var>/<var>c</var></code> is interpreted as scalar
-multiplication with <code>1.0/<var>c</var></code> if <var>c</var> is a number,
-or with <code>1.0/<var>c</var>[0]</code> if <var>c</var> is a 1 by 1 dense matrix.
-
-<P>
-The following in-place operations are defined for a sparse matrix
-<var>A</var> if they do not change the dimensions or type of <var>A</var>.
-<DIV ALIGN="CENTER">
-<TABLE CELLPADDING=3 BORDER="1">
-<TR><TD ALIGN="LEFT">In-place addition</TD>
-<TD ALIGN="LEFT"><code><var>A</var>+=<var>B</var></code>,
- <code><var>A</var>+=<var>c</var></code></TD>
-</TR>
-<TR><TD ALIGN="LEFT">In-place subtraction</TD>
-<TD ALIGN="LEFT"><code><var>A</var>-=<var>B</var></code>,
- <code><var>A</var>-=<var>c</var></code></TD>
-</TR>
-<TR><TD ALIGN="LEFT">In-place scalar multiplication and division</TD>
-<TD ALIGN="LEFT"><code><var>A</var>*=<var>c</var></code>, <code><var>A</var>/=<var>c</var></code></TD>
-</TR>
-</TABLE>
-</DIV>
-
-<P>
-For example, "<tt class="samp"><var>A</var> += 1.0</tt>" is not allowed because the
-operation "<tt class="samp"><var>A</var> = <var>A</var> + 1.0</tt>" results in a dense matrix,
-so it cannot be assigned to <var>A</var> without changing its type.
-
-<P>
-In-place matrix-matrix products are not allowed. (Except when
-<var>c</var> is a 1 by 1 dense matrix, in which case <code><var>A</var>*=<var>c</var></code>
-is interpreted as a scalar product <code><var>A</var>*=<var>c</var>[0]</code>.)
-
-<P>
-As for dense operations, the in-place sparse operations do not return
-a new matrix but modify the existing object <var>A</var>.
-The restrictions on in-place operations follow the principle that once
-a sparse matrix is created, its size and type cannot be modified.
-The only attributes that can be modified are the sparsity pattern and
-the numerical values of the nonzero elements.
-These attributes can be modified by in-place operations or by indexed
-assignments.
-
-<P>
-
-<DIV CLASS="navigation">
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-<p></p><hr />
-<table align="center" width="100%" cellpadding="0" cellspacing="2">
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+++ /dev/null
@@ -1,346 +0,0 @@
-<!DOCTYPE html PUBLIC "-//W3C//DTD HTML 4.0 Transitional//EN">
-<html>
-<head>
-<link rel="STYLESHEET" href="cvxopt.css" type='text/css' />
-<link rel="first" href="cvxopt.html" title='CVXOPT: A Python Package for Convex Optimization' />
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-
-<H1><A NAME="SECTION009200000000000000000"></A>
-<A NAME="s-umfpack"></A>
-<BR>
-7.2 General Linear Equations (<tt class="module">cvxopt.umfpack</tt>)
-</H1>
-The module <tt class="module">cvxopt.umfpack</tt> includes four functions for solving
-sparse non-symmetric sets of linear equations.
-They call routines from the UMFPACK library, with all control options
-set to the default values described in the UMFPACK user guide.
-
-<P>
-<div class="seealso">
- <p class="heading">See Also:</p>
-
-<dl compact="compact" class="seeurl">
- <dt><a href='http://www.cise.ufl.edu/research/sparse/umfpack'
- >UMFPACK code, documentation, copyright and license.</a></dt>
- <dd></dd>
- </dl>
-<div class="seetext"><p>T. A. Davis,
-Algorithm 832: UMFPACK - an unsymmetric-pattern multifrontal method
-with a column pre-ordering strategy,
-ACM Transactions on Mathematical Software, 30(2), 196-199, 2004.</p></div>
-</div>
-
-<P>
-<dl><dt><table cellpadding="0" cellspacing="0"><tr valign="baseline">
- <td><nobr><b><tt id='l2h-134' xml:id='l2h-134' class="function">linsolve</tt></b>(</nobr></td>
- <td><var>A, B</var><big>[</big><var>, trans='N'</var><big>]</big><var></var>)</td></tr></table></dt>
-<dd>
-Solves a sparse set of linear equations
-<BR><P></P>
-<DIV ALIGN="CENTER" CLASS="mathdisplay">
-<!-- MATH
- \begin{displaymath}
-AX = B \quad (\mathrm{trans} = \mathrm{'N'}), \qquad
- A^TX = B \quad (\mathrm{trans} = \mathrm{'T'}), \qquad
- A^HX = B \quad (\mathrm{trans} = \mathrm{'C'}),
-\end{displaymath}
- -->
-
-<IMG
- WIDTH="601" HEIGHT="28" BORDER="0"
- SRC="img43.gif"
- ALT="\begin{displaymath}
-AX=B \quad (\mathrm{trans} = \mathrm{'N'}), \qquad
-A^TX=B ...
-...'T'}), \qquad
-A^HX=B \quad (\mathrm{trans} = \mathrm{'C'}),
-\end{displaymath}">
-</DIV>
-<BR CLEAR="ALL">
-<P></P>
-where <var>A</var> is a sparse matrix and <var>B</var> is a dense matrix of
-the same type (<code>'d'</code> or <code>'z'</code>) as <var>A</var>. On exit <var>B</var> contains
-the solution.
-Raises an <code>ArithmeticError</code> exception if the coefficient matrix
-is singular.
-</dl>
-In the following example we solve an equation with
-coefficient matrix
-<BR>
-<DIV ALIGN="RIGHT" CLASS="mathdisplay">
-
-<!-- MATH
- \begin{equation}
-A = \left[\begin{array}{rrrrr}
- 2 & 3 & 0 & 0 & 0 \\
- 3 & 0 & 4 & 0 & 6 \\
- 0 &-1 &-3 & 2 & 0 \\
- 0 & 0 & 1 & 0 & 0 \\
- 0 & 4 & 2 & 0 & 1
- \end{array}\right].
-\end{equation}
- -->
-<A NAME="e-sp-Adef"></A>
-<TABLE WIDTH="100%" ALIGN="CENTER">
-<TR VALIGN="MIDDLE"><TD></TD><TD ALIGN="CENTER" NOWRAP><A NAME="e-sp-Adef"></A><IMG
- WIDTH="205" HEIGHT="102" BORDER="0"
- SRC="img86.gif"
- ALT="\begin{displaymath}
-A = \left[\begin{array}{rrrrr}
-2 & 3 & 0 & 0 & 0 \\
-3 & 0...
-...0 & 0 & 1 & 0 & 0 \\
-0 & 4 & 2 & 0 & 1
-\end{array}\right].
-\end{displaymath}"></TD>
-<TD CLASS="eqno" WIDTH=10 ALIGN="RIGHT">
-(7.1)</TD></TR>
-</TABLE>
-<BR CLEAR="ALL"></DIV><P></P>
-<div class="verbatim"><pre>
->>> from cvxopt.base import spmatrix, matrix
->>> from cvxopt import umfpack
->>> V = [2,3, 3,-1,4, 4,-3,1,2, 2, 6,1]
->>> I = [0,1, 0, 2,4, 1, 2,3,4, 2, 1,4]
->>> J = [0,0, 1, 1,1, 2, 2,2,2, 3, 4,4]
->>> A = spmatrix(V,I,J)
->>> B = matrix(1.0, (5,1))
->>> umfpack.linsolve(A,B)
->>> print B
- 5.7895e-01
--5.2632e-02
- 1.0000e+00
- 1.9737e+00
--7.8947e-01
-</pre></div>
-The function <tt class="function">umfpack.linsolve()</tt> is equivalent to the
-following three functions called in sequence.
-<dl><dt><table cellpadding="0" cellspacing="0"><tr valign="baseline">
- <td><nobr><b><tt id='l2h-135' xml:id='l2h-135' class="function">symbolic</tt></b>(</nobr></td>
- <td><var>A</var>)</td></tr></table></dt>
-<dd>
-Reorders the columns of <var>A</var> to reduce fill-in and performs a
-symbolic LU factorization. <var>A</var> is a sparse, possibly rectangular,
-matrix.
-Returns the symbolic factorization as an opaque C object that can be
-passed on to <tt class="function">umfpack.numeric()</tt>.
-</dl>
-
-<P>
-<dl><dt><table cellpadding="0" cellspacing="0"><tr valign="baseline">
- <td><nobr><b><tt id='l2h-136' xml:id='l2h-136' class="function">numeric</tt></b>(</nobr></td>
- <td><var>A, F</var>)</td></tr></table></dt>
-<dd>
-Performs a numeric LU factorization of a sparse, possibly rectangular,
-matrix <var>A</var>. The argument <var>F</var> is the symbolic factorization
-computed by <tt class="function">umfpack.symbolic()</tt> applied to the matrix <var>A</var>,
-or another sparse matrix with the same sparsity pattern, dimensions,
-and type. The numeric factorization is returned as an opaque C object
-that that can be passed on to <tt class="function">umfpack.solve()</tt>. Raises an
-<code>ArithmeticError</code> if the matrix is singular.
-</dl>
-
-<P>
-<dl><dt><table cellpadding="0" cellspacing="0"><tr valign="baseline">
- <td><nobr><b><tt id='l2h-137' xml:id='l2h-137' class="function">solve</tt></b>(</nobr></td>
- <td><var>A, F, B</var><big>[</big><var>, trans='N'</var><big>]</big><var></var>)</td></tr></table></dt>
-<dd>
-Solves a set of linear equations
-<BR><P></P>
-<DIV ALIGN="CENTER" CLASS="mathdisplay">
-<!-- MATH
- \begin{displaymath}
-AX = B \quad (\mathrm{trans} = \mathrm{'N'}), \qquad
- A^TX = B \quad (\mathrm{trans} = \mathrm{'T'}), \qquad
- A^HX = B \quad (\mathrm{trans} = \mathrm{'C'}),
-\end{displaymath}
- -->
-
-<IMG
- WIDTH="601" HEIGHT="28" BORDER="0"
- SRC="img43.gif"
- ALT="\begin{displaymath}
-AX=B \quad (\mathrm{trans} = \mathrm{'N'}), \qquad
-A^TX=B ...
-...'T'}), \qquad
-A^HX=B \quad (\mathrm{trans} = \mathrm{'C'}),
-\end{displaymath}">
-</DIV>
-<BR CLEAR="ALL">
-<P></P>
-where <var>A</var> is a sparse matrix and <var>B</var> is a dense matrix of
-the same type as <var>A</var>.
-The argument <var>F</var> is a numeric factorization computed by
-<tt class="function">umfpack.numeric()</tt>.
-On exit <var>B</var> is overwritten by the solution.
-</dl>
-These separate functions are useful for solving several sets
-of linear equations with the same coefficient matrix and different
-righthand sides, or with coefficient matrices that share the same
-sparsity pattern.
-The symbolic factorization depends only on the sparsity pattern of
-the matrix, and not on the numerical values of the nonzero
-coefficients.
-The numerical factorization on the other hand depends on the sparsity
-pattern of the matrix and on its the numerical values.
-
-<P>
-As an example, suppose <I>A</I> is the matrix (<A HREF="#e-sp-Adef">7.1</A>) and
-<BR><P></P>
-<DIV ALIGN="CENTER" CLASS="mathdisplay">
-<!-- MATH
- \begin{displaymath}
-B = \left[\begin{array}{rrrrr}
- 4 & 3 & 0 & 0 & 0 \\
- 3 & 0 & 4 & 0 & 6 \\
- 0 &-1 &-3 & 2 & 0 \\
- 0 & 0 & 1 & 0 & 0 \\
- 0 & 4 & 2 & 0 & 2
- \end{array}\right],
-\end{displaymath}
- -->
-
-<IMG
- WIDTH="206" HEIGHT="102" BORDER="0"
- SRC="img87.gif"
- ALT="\begin{displaymath}
-B = \left[\begin{array}{rrrrr}
-4 & 3 & 0 & 0 & 0 \\
-3 & 0...
-...0 & 0 & 1 & 0 & 0 \\
-0 & 4 & 2 & 0 & 2
-\end{array}\right],
-\end{displaymath}">
-</DIV>
-<BR CLEAR="ALL">
-<P></P>
-which differs from <I>A</I> in its first and last entries.
-The following code computes
-<BR><P></P>
-<DIV ALIGN="CENTER" CLASS="mathdisplay">
-<!-- MATH
- \begin{displaymath}
-x = A^{-T}B^{-1}A^{-1}{\bf 1}.
-\end{displaymath}
- -->
-
-<IMG
- WIDTH="132" HEIGHT="24" BORDER="0"
- SRC="img88.gif"
- ALT="\begin{displaymath}
-x = A^{-T}B^{-1}A^{-1}{\bf 1}.
-\end{displaymath}">
-</DIV>
-<BR CLEAR="ALL">
-<P></P>
-<div class="verbatim"><pre>
->>> from cvxopt.base import spmatrix, matrix
->>> from cvxopt import umfpack
->>> VA = [2,3, 3,-1,4, 4,-3,1,2, 2, 6,1]
->>> VB = [4,3, 3,-1,4, 4,-3,1,2, 2, 6,2]
->>> I = [0,1, 0, 2,4, 1, 2,3,4, 2, 1,4]
->>> J = [0,0, 1, 1,1, 2, 2,2,2, 3, 4,4]
->>> A = spmatrix(VA, I, J)
->>> B = spmatrix(VB, I, J)
->>> x = matrix(1.0, (5,1))
->>> Fs = umfpack.symbolic(A)
->>> FA = umfpack.numeric(A, Fs)
->>> FB = umfpack.numeric(B, Fs)
->>> umfpack.solve(A, FA, x)
->>> umfpack.solve(B, FB, x)
->>> umfpack.solve(A, FA, x, trans='T')
->>> print x
- 5.8065e-01
--2.3660e-01
- 1.6280e+00
- 8.0656e+00
--1.3075e-01
-</pre></div>
-
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-<!--End of Navigation Panel-->
-
-<H1><A NAME="SECTION0011100000000000000000"></A> <A NAME="s-variables"></A>
-<BR>
-9.1 Variables
-</H1>
-Optimization variables are represented by variable objects.
-
-<P>
-<dl><dt><table cellpadding="0" cellspacing="0"><tr valign="baseline">
- <td><nobr><b><span class="typelabel">class</span> <tt id='l2h-152' xml:id='l2h-152' class="class">variable</tt></b>(</nobr></td>
- <td><var></var><big>[</big><var>size</var><big>[</big><var>, name</var><big>]</big><var></var><big>]</big><var></var>)</td></tr></table></dt>
-<dd>
-A vector variable. The first argument is the dimension of the
-vector (a positive integer with default value 1).
-The second argument is a string with a name for the variable.
-The name is optional and has default value <code>""</code>. It is only used
-when displaying variables (or objects that depend on variables, such
-as functions or constraints)
-using <tt class="function">print</tt> statements, when calling the built-in functions
-<tt class="function">repr()</tt> or <tt class="function">str()</tt>, or when writing linear programs
-to MPS files.
-</dl>
-The function <tt class="function">len()</tt> returns the length of a
-variable. A variable <var>x</var> has two attributes.
-<dl><dt><b><tt id='l2h-153' xml:id='l2h-153' class="member">name</tt></b></dt>
-<dd>
-The name of the variable.
-</dl>
-
-<P>
-<dl><dt><b><tt id='l2h-154' xml:id='l2h-154' class="member">value</tt></b></dt>
-<dd>
-Either <code>None</code> or a dense <code>'d'</code> matrix of size <code>len(<var>x</var>)</code> by 1.
-
-<P>
-The attribute <var>x</var>.<tt class="member">value</tt> is set to <code>None</code> when the
-variable <var>x</var> is created. It can be given a numerical value later,
-typically by solving an LP that has <var>x</var> as one of its variables.
-One can also make an explicit assignment
-<code><var>x</var>.<tt class="member">value</tt> = <var>y</var></code>. The
-assigned value <var>y</var> must be an integer or float, or a
-dense <code>'d'</code> matrix of size (<code>len(<var>x</var>)</code>,1).
-If <var>y</var> is an integer or float all the elements of
-<var>x</var>.<tt class="member">value</tt> are set to the value of <var>y</var>.
-</dl>
-
-<P>
-<div class="verbatim"><pre>
->>> from cvxopt.base import matrix
->>> from cvxopt.modeling import variable
->>> x = variable(3,'a')
->>> len(x)
-3
->>> print x.name
-a
->>> print x.value
-None
->>> x.value = matrix([1.,2.,3.])
->>> print x.value
- 1.0000e+00
- 2.0000e+00
- 3.0000e+00
->>> x.value = 1
->>> print x.value
- 1.0000e+00
- 1.0000e+00
- 1.0000e+00
-</pre></div>
-
-<P>
-
-<DIV CLASS="navigation">
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-<td align="center" width="100%">CVXOPT: A Python Package for Convex Optimization</td>
-<td class='online-navigation'><a rel="contents" title="Table of Contents"
- href="contents.html"><img src='contents.gif'
- border='0' height='32' alt='Contents' width='32' /></A></td>
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-<td class='online-navigation'><a rel="index" title="Index"
- href="genindex.html"><img src='index.gif'
- border='0' height='32' alt='Index' width='32' /></A></td>
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-</div>
-</div>
-<hr />
-<span class="release-info">Release 0.8.2, documentation updated on February 6, 2007.</span>
-</DIV>
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-
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-</HTML>
diff --git a/doc/cvxopt/up.gif b/doc/cvxopt/up.gif
deleted file mode 100644
index a9d3e13..0000000
Binary files a/doc/cvxopt/up.gif and /dev/null differ
diff --git a/doc/intro.tex b/doc/intro.tex
index f9f510c..985b446 100644
--- a/doc/intro.tex
+++ b/doc/intro.tex
@@ -10,7 +10,7 @@ optimization applications straightforward by building on Python's
extensive standard library and on the strengths of Python as a
high-level programming language.
-Release 0.8.2 of CVXOPT includes routines for basic linear algebra
+Release 0.9 of CVXOPT includes routines for basic linear algebra
calculations,
interfaces to efficient libraries for solving dense and sparse linear
equations,
@@ -43,7 +43,7 @@ These components are organized in different modules.
solver from CHOLMOD (section~\ref{s-cholmod}).
\item[\module{cvxopt.solvers}] Convex optimization routines
and optional interfaces to solvers from GLPK, MOSEK and DSDP5
- (chapter~\ref{chap:solvers}).
+ (chapters~\ref{chap:coneprog} and~\ref{chap:solvers}).
\item[\module{cvxopt.modeling}] Routines for specifying and solving
linear programs and convex optimization problems with piecewise-linear
cost and constraint functions (chapter~\ref{chap:modeling}).
diff --git a/doc/lapack.tex b/doc/lapack.tex
index 15f4433..4d7df49 100644
--- a/doc/lapack.tex
+++ b/doc/lapack.tex
@@ -216,7 +216,7 @@ by \function{gbsv()}.
1.4286e-01
-2.3810e-02
\end{verbatim}
-An alternative method uses \function{getrf()} for the factorization.
+An alternative method uses \function{gbtrf()} for the factorization.
\begin{verbatim}
>>> Ac[kl:,:] = A
>>> gbtrf(Ac, n, kl, ipiv)
@@ -421,10 +421,10 @@ Solves
\[
A X = B,
\]
-where {\it A} is an {\it n} by {\it n} real symmetric or complex
-Hermitian tridiagonal matrix, with diagonal \var{d} (a \dtc\ matrix of
-length {\it n}) and subdiagonal \var{e} (a \dtc\ or \ztc\ matrix of length
-{\it n}-1).
+where {\it A} is an {\it n} by {\it n} positive definite real symmetric
+or complex Hermitian tridiagonal matrix, with diagonal
+\var{d} (a \dtc\ matrix of length {\it n}) and subdiagonal \var{e}
+(a \dtc\ or \ztc\ matrix of length {\it n}-1).
The arguments \var{e} and \var{B} must have the same type.
On exit \var{d} contains the diagonal elements of {\it D} in
the LDL${}\mathrm{^T}$ or LDL${}\mathrm{^H}$ factorization
@@ -436,7 +436,8 @@ Raises an \code{ArithmeticError} if the matrix is singular.
\begin{funcdesc}{pttrf}{d, e}
LDL${}\mathrm{^T}$ or LDL${}\mathrm{^H}$ factorization of an {\it n} by
-{\it n} real symmetric or complex Hermitian tridiagonal matrix {\it A}.
+{\it n} positive definite real symmetric or complex Hermitian tridiagonal
+matrix {\it A}.
On entry, the argument \var{d} is a \dtc\ matrix with the diagonal elements
of {\it A}. The argument \var{e} is \dtc\ or \ztc\ matrix with
the subdiagonal elements of {\it A}.
@@ -450,12 +451,12 @@ Solves a set of linear equations
\[
AX=B
\]
-where {\it A} is an {\it n} by {\it n} real symmetric or complex
-Hermitian tridiagonal matrix, given its LDL${}\mathrm{^T}$ or
+where {\it A} is an {\it n} by {\it n} positive definite real symmetric
+or complex Hermitian tridiagonal matrix, given its LDL${}\mathrm{^T}$ or
LDL${}\mathrm{^H}$ factorization.
The argument \var{d} is the diagonal of the diagonal matrix {\it D}.
The argument \var{uplo} only matters for complex matrices.
-If \var{uplo} is \code{'L'}, then \var{e} contains the subdiagonal
+If \var{uplo} is \code{'L'}, then on exit \var{e} contains the subdiagonal
elements of the unit bidiagonal matrix {\it L}.
If \var{uplo} is \code{'U'}, then \var{e} contains the complex
conjugates of the elements of the unit bidiagonal matrix {\it L}.
diff --git a/doc/modeling.tex b/doc/modeling.tex
index e380dd5..e31e924 100644
--- a/doc/modeling.tex
+++ b/doc/modeling.tex
@@ -28,11 +28,11 @@ to MPS files.
\end{classdesc}
The function \function{len()} returns the length of a
\pytype{variable}. A \pytype{variable} \var{x} has two attributes.
-\begin{memberdesc}[variable]{name}
+\begin{memberdesc}{variable}{name}
The name of the variable.
\end{memberdesc}
-\begin{memberdesc}[variable]{value}
+\begin{memberdesc}{variable}{value}
Either \None\ or a dense \dtc\ matrix of size \code{len(\var{x})} by 1.
The attribute \var{x}.\member{value} is set to \None\ when the
@@ -75,15 +75,15 @@ operations on variables and other functions. A function \var{f} is
interpreted as a column vector, with length \code{len(\var{f})}
and with a value that depends on the values of its variables.
Functions have two public attributes.
-\begin{methoddesc}{variables}{}
+\begin{methoddesc}{funciton}{variables}
Returns a copy of the list of variables of the function.
\end{methoddesc}
-\begin{methoddesc}{value}{}
+\begin{methoddesc}{function}{value}
The function value. If any of the variables of \var{f} has value
-\None, then \var{f}.\method{value()} returns \None.
+\None, then \var{f}.\function{value()} returns \None.
Otherwise, it returns a dense \dtc\ matrix of size
-(\code{len(\var{f})},1) with the function value computed from the
+(\code{len(\var{f}),1}) with the function value computed from the
\member{value} attributes of the variables of \var{f}.
\end{methoddesc}
@@ -152,7 +152,7 @@ argument \var{v}.
If \var{v} is a variable or affine function and \var{u} is a \dtc\
matrix of size (\code{len(\var{v})},1), then
\code{dot(\var{u},\var{v})} and \code{dot(\var{v},\var{u})} are
-equivalent to \code{\var{u}.\method{trans()}*\var{v}}.
+equivalent to \code{\var{u}.\function{trans()}*\var{v}}.
If \var{u} and \var{v} are dense matrices, then
\code{dot(\var{u},\var{v})}
@@ -272,9 +272,9 @@ do not include any variables or functions, then the Python built-in
If one or more of the arguments are variables or functions,
\function{max()} returns a piecewise-linear function defined as the
elementwise maximum of its arguments.
-In other words, \var{f}[\var{k}] =
-\code{max(\var{y1}[\var{k}],\var{y2}[\var{k}], \ldots)}
-for \var{k}=0, \ldots, \code{len(\var{f})}-1.
+In other words, \code{\var{f}[\var{k}] =
+max(\var{y1}[\var{k}],\var{y2}[\var{k}], \ldots)}
+for \var{k}=0, \ldots, \code{len(\var{f})-1}.
The length of \var{f} is equal to the maximum of the lengths of the
arguments. Each argument must have length equal to
\code{len(\var{f})} or length one.
@@ -334,6 +334,8 @@ variable \var{x} of length 10, \var{g} is its infinity-norm and
2|u|-3 & |u| \geq 2. \end{array}\right.
\]
\begin{verbatim}
+>>> from cvxopt.modeling import variable, max
+>>> x = variable(10, 'x')
>>> f = sum(abs(x))
>>> g = max(abs(x))
>>> h = sum(max(0, abs(x)-1, 2*abs(x)-3))
@@ -390,16 +392,16 @@ The built-in fucntion \function{len()} returns the dimension of the
constraint function.
Constraints have four public attributes.
-\begin{methoddesc}[constraint]{type}{}
+\begin{methoddesc}{constraint}{type}{}
Returns \code{'='} if the constraint is an equality constraint,
and \code{'<'} if the constraint is an inequality constraint.
\end{methoddesc}
-\begin{methoddesc}[constraint]{value}{}
+\begin{methoddesc}{constraint}{value}{}
Returns the value of the constraint function.
\end{methoddesc}
-\begin{memberdesc}[constraint]{multiplier}
+\begin{memberdesc}{constraint}{multiplier}
For a constraint \var{c}, \var{c}.\member{multiplier} is a
\pytype{variable} object of dimension \code{len(\var{c})}.
It is used to represent the Lagrange multiplier or dual variable
@@ -408,7 +410,7 @@ Its value is initialized as \None, and can be modified
by making an assignment to \var{c}.\member{multiplier}.\member{value}.
\end{memberdesc}
-\begin{memberdesc}[constraint]{name}
+\begin{memberdesc}{constraint}{name}
The name of the constraint. Changing the name of a constraint
also changes the name of the multiplier of \var{c}.
For example, the command \code{\var{c}.\member{name} = 'newname'} also
@@ -438,39 +440,39 @@ The default value is the empty string.
The following attributes and methods are useful for examining
and modifying optimization problems.
-\begin{memberdesc}[op]{objective}
+\begin{memberdesc}{op}{objective}
The objective or cost function. One can write to this
attribute to change the objective of an existing problem.
\end{memberdesc}
-\begin{methoddesc}[lp]{variables}{}
+\begin{methoddesc}{op}{variables}
Returns a list of the variables of the problem.
\end{methoddesc}
-\begin{methoddesc}[lp]{constraints}{}
+\begin{methoddesc}{op}{constraints}
Returns a list of the constraints.
\end{methoddesc}
-\begin{methoddesc}[lp]{inequalities}{}
+\begin{methoddesc}{op}{inequalities}
Returns a list of the inequality constraints.
\end{methoddesc}
-\begin{methoddesc}[lp]{equalities}{}
+\begin{methoddesc}{op}{equalities}
Returns a list of the equality constraints.
\end{methoddesc}
-\begin{methoddesc}[lp]{delconstraint}{c}
+\begin{funcdesc}{delconstraint}{c}
Deletes constraint {\tt c} from the problem.
-\end{methoddesc}
+\end{funcdesc}
-\begin{methoddesc}[lp]{addconstraint}{c}
+\begin{funcdesc}{addconstraint}{c}
Adds constraint {\tt c} to the problem.
-\end{methoddesc}
+\end{funcdesc}
An optimization problem with convex piecewise-linear objective and
constraints can be solved by calling the method \function{solve()}.
-\begin{methoddesc}[op]{solve}{\optional{format\optional{, solver}}}
+\begin{funcdesc}{solve}{\optional{format\optional{, solver}}}
This function converts the optimization problem to a linear program
in matrix form and then solves it using the solver described in
section~\ref{s-lpsolver}.
@@ -520,7 +522,7 @@ certificates of infeasibility.
The \member{value} attributes of the variables and the constraint
multipliers are set to \None.
\EIT
-\end{methoddesc}
+\end{funcdesc}
We refer to section~\ref{s-lpsolver} for details on the algorithms and
the different solver options.
@@ -585,19 +587,19 @@ The \pytype{op} class also includes two methods for writing and reading
files in
\ulink{MPS format}{http://www-fp.mcs.anl.gov/otc/Guide/OptWeb/continuous/constrained/linearprog/mps.html}.
-\begin{methoddesc}[op]{tofile}{filename}
+\begin{funcdesc}{tofile}{filename}
If the problem is an LP, writes it to the file \code{'filename'} using
the MPS format. Row and column labels are assigned based on the
variable and constraint names in the LP.
-\end{methoddesc}
+\end{funcdesc}
-\begin{methoddesc}[lp]{fromfile}{filename}
+\begin{funcdesc}{fromfile}{filename}
Reads the LP from the file \code{'filename'}. The file must be
a fixed-format MPS file. Some features of the MPS format are not
supported: comments beginning with dollar signs,
the row types 'DE', 'DL', 'DG', and 'DN', and the capability of
reading multiple righthand side, bound or range vectors.
-\end{methoddesc}
+\end{funcdesc}
\section{Examples}
diff --git a/doc/python.sty b/doc/python.sty
deleted file mode 100644
index 3fdb8b7..0000000
--- a/doc/python.sty
+++ /dev/null
@@ -1,1330 +0,0 @@
-%
-% python.sty for the Python docummentation [works only with Latex2e]
-%
-
-\NeedsTeXFormat{LaTeX2e}[1995/12/01]
-\ProvidesPackage{python}
- [1998/01/11 LaTeX package (Python markup)]
-
-\RequirePackage{longtable}
-
-% Uncomment these two lines to ignore the paper size and make the page
-% size more like a typical published manual.
-%\renewcommand{\paperheight}{9in}
-%\renewcommand{\paperwidth}{8.5in} % typical squarish manual
-%\renewcommand{\paperwidth}{7in} % O'Reilly ``Programmming Python''
-
-% These packages can be used to add marginal annotations which indicate
-% index entries and labels; useful for reviewing this messy documentation!
-%
-%\RequirePackage{showkeys}
-%\RequirePackage{showidx}
-
-% If we ever want to indent paragraphs, this needs to be changed.
-% This is used inside the macros defined here instead of coding
-% \noindent directly.
-\let\py at parindent=\noindent
-
-% for PDF output, use maximal compression & a lot of other stuff
-% (test for PDF recommended by Tanmoy Bhattacharya <tanmoy at qcd.lanl.gov>)
-%
-\newif\ifpy at doing@page at targets
-\py at doing@page at targetsfalse
-
-\newif\ifpdf\pdffalse
-\ifx\pdfoutput\undefined\else\ifcase\pdfoutput
-\else
- \pdftrue
- \input{pdfcolor}
- \let\py at LinkColor=\NavyBlue
- \let\py at NormalColor=\Black
- \pdfcompresslevel=9
- \pdfpagewidth=\paperwidth % page width of PDF output
- \pdfpageheight=\paperheight % page height of PDF output
- %
- % Pad the number with '0' to 3 digits wide so no page name is a prefix
- % of any other.
- %
- \newcommand{\py at targetno}[1]{\ifnum#1<100 0\fi\ifnum#1<10 0\fi#1}
- \newcommand{\py at pageno}{\py at targetno\thepage}
- %
- % This definition allows the entries in the page-view of the ToC to be
- % active links. Some work, some don't.
- %
- \let\py at OldContentsline=\contentsline
- %
- % Backward compatibility hack: pdfTeX 0.13 defined \pdfannotlink,
- % but it changed to \pdfstartlink in 0.14. This let's us use either
- % version and still get useful behavior.
- %
- \@ifundefined{pdfstartlink}{
- \let\pdfstartlink=\pdfannotlink
- }{}
- %
- % The \py at parindent here is a hack -- we're forcing pdfTeX into
- % horizontal mode since \pdfstartlink requires that.
- \def\py at pdfstartlink{%
- \ifvmode\py at parindent\fi%
- \pdfstartlink%
- }
- %
- % Macro that takes two args: the name to link to and the content of
- % the link. This takes care of the PDF magic, getting the colors
- % the same for each link, and avoids having lots of garbage all over
- % this style file.
- \newcommand{\py at linkToName}[2]{%
- \py at pdfstartlink attr{/Border [0 0 0]} goto name{#1}%
- \py at LinkColor#2\py at NormalColor%
- \pdfendlink%
- }
- % Compute the padded page number separately since we end up with a pair of
- % \relax tokens; this gets the right string computed and works.
- \renewcommand{\contentsline}[3]{%
- \def\my at pageno{\py at targetno{#3}}%
- \py at OldContentsline{#1}{\py at linkToName{page\my at pageno}{#2}}{#3}%
- }
- \AtEndDocument{
- \def\_{\string_}
- \InputIfFileExists{\jobname.bkm}{\pdfcatalog{/PageMode /UseOutlines}}{}
- }
- \newcommand{\py at target}[1]{%
- \ifpy at doing@page at targets%
- {\pdfdest name{#1} xyz}%
- \fi%
- }
- \let\py at OldLabel=\label
- \renewcommand{\label}[1]{%
- \py at OldLabel{#1}%
- \py at target{label-#1}%
- }
- % This stuff adds a page# destination to every PDF page, where # is three
- % digits wide, padded with leading zeros. This doesn't really help with
- % the frontmatter, but does fine with the body.
- %
- % This is *heavily* based on the hyperref package.
- %
- \def\@begindvi{%
- \unvbox \@begindvibox
- \@hyperfixhead
- }
- \def\@hyperfixhead{%
- \let\H at old@thehead\@thehead
- \global\def\@foo{\py at target{page\py at pageno}}%
- \expandafter\ifx\expandafter\@empty\H at old@thehead
- \def\H at old@thehead{\hfil}\fi
- \def\@thehead{\@foo\relax\H at old@thehead}%
- }
-\fi\fi
-
-% Increase printable page size (copied from fullpage.sty)
-\topmargin 0pt
-\advance \topmargin by -\headheight
-\advance \topmargin by -\headsep
-
-% attempt to work a little better for A4 users
-\textheight \paperheight
-\advance\textheight by -2in
-
-\oddsidemargin 0pt
-\evensidemargin 0pt
-%\evensidemargin -.25in % for ``manual size'' documents
-\marginparwidth 0.5in
-
-\textwidth \paperwidth
-\advance\textwidth by -2in
-
-
-% Style parameters and macros used by most documents here
-\raggedbottom
-\sloppy
-\parindent = 0mm
-\parskip = 2mm
-\hbadness = 5000 % don't print trivial gripes
-
-\pagestyle{empty} % start this way; change for
-\pagenumbering{roman} % ToC & chapters
-
-% Use this to set the font family for headers and other decor:
-\newcommand{\py at HeaderFamily}{\sffamily}
-
-% Set up abstract ways to get the normal and smaller font sizes that
-% work even in footnote context.
-\newif\ifpy at infootnote \py at infootnotefalse
-\let\py at oldmakefntext\@makefntext
-\def\@makefntext#1{%
- \bgroup%
- \py at infootnotetrue
- \py at oldmakefntext{#1}%
- \egroup%
-}
-\def\py at defaultsize{%
- \ifpy at infootnote\footnotesize\else\normalsize\fi%
-}
-\def\py at smallsize{%
- \ifpy at infootnote\scriptsize\else\small\fi%
-}
-
-% Redefine the 'normal' header/footer style when using "fancyhdr" package:
-\@ifundefined{fancyhf}{}{
- % Use \pagestyle{normal} as the primary pagestyle for text.
- \fancypagestyle{normal}{
- \fancyhf{}
- \fancyfoot[LE,RO]{{\py at HeaderFamily\thepage}}
- \fancyfoot[LO]{{\py at HeaderFamily\nouppercase{\rightmark}}}
- \fancyfoot[RE]{{\py at HeaderFamily\nouppercase{\leftmark}}}
- \renewcommand{\headrulewidth}{0pt}
- \renewcommand{\footrulewidth}{0.4pt}
- }
- % Update the plain style so we get the page number & footer line,
- % but not a chapter or section title. This is to keep the first
- % page of a chapter and the blank page between chapters `clean.'
- \fancypagestyle{plain}{
- \fancyhf{}
- \fancyfoot[LE,RO]{{\py at HeaderFamily\thepage}}
- \renewcommand{\headrulewidth}{0pt}
- \renewcommand{\footrulewidth}{0.4pt}
- }
- % Redefine \cleardoublepage so that the blank page between chapters
- % gets the plain style and not the fancy style. This is described
- % in the documentation for the fancyhdr package by Piet von Oostrum.
- \@ifundefined{chapter}{}{
- \renewcommand{\cleardoublepage}{
- \clearpage\if at openright \ifodd\c at page\else
- \hbox{}
- \thispagestyle{plain}
- \newpage
- \if at twocolumn\hbox{}\newpage\fi\fi\fi
- }
- }
-}
-
-% This sets up the {verbatim} environment to be indented and a minipage,
-% and to have all the other mostly nice properties that we want for
-% code samples.
-
-\let\py at OldVerbatim=\verbatim
-\let\py at OldEndVerbatim=\endverbatim
-\RequirePackage{verbatim}
-\let\py at OldVerbatimInput=\verbatiminput
-
-% Variable used by begin code command
-\newlength{\py at codewidth}
-
-\renewcommand{\verbatim}{%
- \setlength{\parindent}{1cm}%
- % Calculate the text width for the minipage:
- \setlength{\py at codewidth}{\linewidth}%
- \addtolength{\py at codewidth}{-\parindent}%
- %
- \par\indent%
- \begin{minipage}[t]{\py at codewidth}%
- \small%
- \py at OldVerbatim%
-}
-\renewcommand{\endverbatim}{%
- \py at OldEndVerbatim%
- \end{minipage}%
-}
-\renewcommand{\verbatiminput}[1]{%
- {\setlength{\parindent}{1cm}%
- % Calculate the text width for the minipage:
- \setlength{\py at codewidth}{\linewidth}%
- \addtolength{\py at codewidth}{-\parindent}%
- %
- \small%
- \begin{list}{}{\setlength{\leftmargin}{1cm}}
- \item%
- \py at OldVerbatimInput{#1}%
- \end{list}
- }%
-}
-
-% This does a similar thing for the {alltt} environment:
-\RequirePackage{alltt}
-\let\py at OldAllTT=\alltt
-\let\py at OldEndAllTT=\endalltt
-
-\renewcommand{\alltt}{%
- \setlength{\parindent}{1cm}%
- % Calculate the text width for the minipage:
- \setlength{\py at codewidth}{\linewidth}%
- \addtolength{\py at codewidth}{-\parindent}%
- \let\e=\textbackslash%
- %
- \par\indent%
- \begin{minipage}[t]{\py at codewidth}%
- \small%
- \py at OldAllTT%
-}
-\renewcommand{\endalltt}{%
- \py at OldEndAllTT%
- \end{minipage}%
-}
-
-
-\newcommand{\py at modulebadkey}{{--just-some-junk--}}
-
-
-%% Lots of index-entry generation support.
-
-% Command to wrap around stuff that refers to function / module /
-% attribute names in the index. Default behavior: like \code{}. To
-% just keep the index entries in the roman font, uncomment the second
-% definition; it matches O'Reilly style more.
-%
-\newcommand{\py at idxcode}[1]{\texttt{#1}}
-%\renewcommand{\py at idxcode}[1]{#1}
-
-% Command to generate two index entries (using subentries)
-\newcommand{\indexii}[2]{\index{#1!#2}\index{#2!#1}}
-
-% And three entries (using only one level of subentries)
-\newcommand{\indexiii}[3]{\index{#1!#2 #3}\index{#2!#3, #1}\index{#3!#1 #2}}
-
-% And four (again, using only one level of subentries)
-\newcommand{\indexiv}[4]{
-\index{#1!#2 #3 #4}
-\index{#2!#3 #4, #1}
-\index{#3!#4, #1 #2}
-\index{#4!#1 #2 #3}
-}
-
-% Command to generate a reference to a function, statement, keyword,
-% operator.
-\newcommand{\kwindex}[1]{\indexii{keyword}{#1@{\py at idxcode{#1}}}}
-\newcommand{\stindex}[1]{\indexii{statement}{#1@{\py at idxcode{#1}}}}
-\newcommand{\opindex}[1]{\indexii{operator}{#1@{\py at idxcode{#1}}}}
-\newcommand{\exindex}[1]{\indexii{exception}{#1@{\py at idxcode{#1}}}}
-\newcommand{\obindex}[1]{\indexii{object}{#1}}
-\newcommand{\bifuncindex}[1]{%
- \index{#1@{\py at idxcode{#1()}} (built-in function)}}
-
-% Add an index entry for a module
-\newcommand{\py at refmodule}[2]{\index{#1@{\py at idxcode{#1}} (#2module)}}
-\newcommand{\refmodindex}[1]{\py at refmodule{#1}{}}
-\newcommand{\refbimodindex}[1]{\py at refmodule{#1}{built-in }}
-\newcommand{\refexmodindex}[1]{\py at refmodule{#1}{extension }}
-\newcommand{\refstmodindex}[1]{\py at refmodule{#1}{standard }}
-
-% Refer to a module's documentation using a hyperlink of the module's
-% name, at least if we're building PDF:
-\ifpdf
- \newcommand{\refmodule}[2][\py at modulebadkey]{%
- \ifx\py at modulebadkey#1\def\py at modulekey{#2}\else\def\py at modulekey{#1}\fi%
- \py at linkToName{label-module-\py at modulekey}{\module{#2}}%
- }
-\else
- \newcommand{\refmodule}[2][\py at modulebadkey]{\module{#2}}
-\fi
-
-% support for the module index
-\newif\ifpy at UseModuleIndex
-\py at UseModuleIndexfalse
-
-\newcommand{\makemodindex}{
- \newwrite\modindexfile
- \openout\modindexfile=mod\jobname.idx
- \py at UseModuleIndextrue
-}
-
-% Add the defining entry for a module
-\newcommand{\py at modindex}[2]{%
- \renewcommand{\py at thismodule}{#1}
- \setindexsubitem{(in module #1)}%
- \index{#1@{\py at idxcode{#1}} (#2module)|textbf}%
- \ifpy at UseModuleIndex%
- \@ifundefined{py at modplat@\py at thismodulekey}{
- \write\modindexfile{\protect\indexentry{#1@{\texttt{#1}}}{\thepage}}%
- }{\write\modindexfile{\protect\indexentry{#1@{\texttt{#1} %
- \emph{(\py at platformof[\py at thismodulekey]{})}}}{\thepage}}%
- }
- \fi%
-}
-
-% *** XXX *** THE NEXT FOUR MACROS ARE NOW OBSOLETE !!! ***
-
-% built-in & Python modules in the main distribution
-\newcommand{\bimodindex}[1]{\py at modindex{#1}{built-in }%
- \typeout{*** MACRO bimodindex IS OBSOLETE -- USE declaremodule INSTEAD!}}
-\newcommand{\stmodindex}[1]{\py at modindex{#1}{standard }%
- \typeout{*** MACRO stmodindex IS OBSOLETE -- USE declaremodule INSTEAD!}}
-
-% Python & extension modules outside the main distribution
-\newcommand{\modindex}[1]{\py at modindex{#1}{}%
- \typeout{*** MACRO modindex IS OBSOLETE -- USE declaremodule INSTEAD!}}
-\newcommand{\exmodindex}[1]{\py at modindex{#1}{extension }%
- \typeout{*** MACRO exmodindex IS OBSOLETE -- USE declaremodule INSTEAD!}}
-
-% Additional string for an index entry
-\newif\ifpy at usingsubitem\py at usingsubitemfalse
-\newcommand{\py at indexsubitem}{}
-\newcommand{\setindexsubitem}[1]{\renewcommand{\py at indexsubitem}{ #1}%
- \py at usingsubitemtrue}
-\newcommand{\ttindex}[1]{%
- \ifpy at usingsubitem
- \index{#1@{\py at idxcode{#1}}\py at indexsubitem}%
- \else%
- \index{#1@{\py at idxcode{#1}}}%
- \fi%
-}
-\newcommand{\withsubitem}[2]{%
- \begingroup%
- \def\ttindex##1{\index{##1@{\py at idxcode{##1}} #1}}%
- #2%
- \endgroup%
-}
-
-
-% Module synopsis processing -----------------------------------------------
-%
-\newcommand{\py at thisclass}{}
-\newcommand{\py at thismodule}{}
-\newcommand{\py at thismodulekey}{}
-\newcommand{\py at thismoduletype}{}
-
-\newcommand{\py at standardIndexModule}[1]{\py at modindex{#1}{standard }}
-\newcommand{\py at builtinIndexModule}[1]{\py at modindex{#1}{built-in }}
-\newcommand{\py at extensionIndexModule}[1]{\py at modindex{#1}{extension }}
-\newcommand{\py at IndexModule}[1]{\py at modindex{#1}{}}
-
-\newif\ifpy at HaveModSynopsis \py at HaveModSynopsisfalse
-\newif\ifpy at ModSynopsisFileIsOpen \py at ModSynopsisFileIsOpenfalse
-\newif\ifpy at HaveModPlatform \py at HaveModPlatformfalse
-
-% \declaremodule[key]{type}{name}
-\newcommand{\declaremodule}[3][\py at modulebadkey]{
- \py at openModSynopsisFile
- \renewcommand{\py at thismoduletype}{#2}
- \ifx\py at modulebadkey#1
- \renewcommand{\py at thismodulekey}{#3}
- \else
- \renewcommand{\py at thismodulekey}{#1}
- \fi
- \@ifundefined{py@#2IndexModule}{%
- \typeout{*** MACRO declaremodule called with unknown module type: `#2'}
- \py at IndexModule{#3}%
- }{%
- \csname py@#2IndexModule\endcsname{#3}%
- }
- \label{module-\py at thismodulekey}
-}
-\newif\ifpy at ModPlatformFileIsOpen \py at ModPlatformFileIsOpenfalse
-\newcommand{\py at ModPlatformFilename}{\jobname.pla}
-\newcommand{\platform}[1]{
- \ifpy at ModPlatformFileIsOpen\else
- \newwrite\py at ModPlatformFile
- \openout\py at ModPlatformFile=\py at ModPlatformFilename
- \py at ModPlatformFileIsOpentrue
- \fi
-}
-\InputIfFileExists{\jobname.pla}{}{}
-\newcommand{\py at platformof}[2][\py at modulebadkey]{%
- \ifx\py at modulebadkey#1 \def\py at key{#2}%
- \else \def\py at key{#1}%
- \fi%
- \csname py at modplat@\py at key\endcsname%
-}
-\newcommand{\ignorePlatformAnnotation}[1]{}
-
-% \moduleauthor{name}{email}
-\newcommand{\moduleauthor}[2]{}
-
-% \sectionauthor{name}{email}
-\newcommand{\sectionauthor}[2]{}
-
-
-\newcommand{\py at defsynopsis}{Module has no synopsis.}
-\newcommand{\py at modulesynopsis}{\py at defsynopsis}
-\newcommand{\modulesynopsis}[1]{
- \py at HaveModSynopsistrue
- \renewcommand{\py at modulesynopsis}{#1}
-}
-
-% define the file
-\newwrite\py at ModSynopsisFile
-
-% hacked from \addtocontents from latex.ltx:
-\long\def\py at writeModSynopsisFile#1{%
- \protected at write\py at ModSynopsisFile%
- {\let\label\@gobble \let\index\@gobble \let\glossary\@gobble}%
- {\string#1}%
-}
-\newcommand{\py at closeModSynopsisFile}{
- \ifpy at ModSynopsisFileIsOpen
- \closeout\py at ModSynopsisFile
- \py at ModSynopsisFileIsOpenfalse
- \fi
-}
-\newcommand{\py at openModSynopsisFile}{
- \ifpy at ModSynopsisFileIsOpen\else
- \openout\py at ModSynopsisFile=\py at ModSynopsisFilename
- \py at ModSynopsisFileIsOpentrue
- \fi
-}
-
-\newcommand{\py at ProcessModSynopsis}{
- \ifpy at HaveModSynopsis
- \py at writeModSynopsisFile{\modulesynopsis%
- {\py at thismodulekey}{\py at thismodule}%
- {\py at thismoduletype}{\py at modulesynopsis}}%
- \py at HaveModSynopsisfalse
- \fi
- \renewcommand{\py at modulesynopsis}{\py at defsynopsis}
-}
-\AtEndDocument{\py at ProcessModSynopsis\py at closeModSynopsisFile}
-
-
-\long\def\py at writeModPlatformFile#1{%
- \protected at write\py at ModPlatformFile%
- {\let\label\@gobble \let\index\@gobble \let\glossary\@gobble}%
- {\string#1}%
-}
-
-
-\newcommand{\localmoduletable}{
- \IfFileExists{\py at ModSynopsisFilename}{
- \begin{synopsistable}
- \input{\py at ModSynopsisFilename}
- \end{synopsistable}
- }{}
-}
-
-\ifpdf
- \newcommand{\py at ModSynopsisSummary}[4]{%
- \py at linkToName{label-module-#1}{\bfcode{#2}} & #4\\
- }
-\else
- \newcommand{\py at ModSynopsisSummary}[4]{\bfcode{#2} & #4\\}
-\fi
-\newenvironment{synopsistable}{
- % key, name, type, synopsis
- \let\modulesynopsis=\py at ModSynopsisSummary
- \begin{tabular}{ll}
-}{
- \end{tabular}
-}
-%
-% --------------------------------------------------------------------------
-
-
-\newcommand{\py at reset}{
- \py at usingsubitemfalse
- \py at ProcessModSynopsis
- \renewcommand{\py at thisclass}{}
- \renewcommand{\py at thismodule}{}
- \renewcommand{\py at thismodulekey}{}
- \renewcommand{\py at thismoduletype}{}
-}
-
-% Augment the sectioning commands used to get our own font family in place,
-% and reset some internal data items:
-\renewcommand{\section}{\py at reset%
- \@startsection{section}{1}{\z@}%
- {-3.5ex \@plus -1ex \@minus -.2ex}%
- {2.3ex \@plus.2ex}%
- {\reset at font\Large\py at HeaderFamily}}
-\renewcommand{\subsection}{\@startsection{subsection}{2}{\z@}%
- {-3.25ex\@plus -1ex \@minus -.2ex}%
- {1.5ex \@plus .2ex}%
- {\reset at font\large\py at HeaderFamily}}
-\renewcommand{\subsubsection}{\@startsection{subsubsection}{3}{\z@}%
- {-3.25ex\@plus -1ex \@minus -.2ex}%
- {1.5ex \@plus .2ex}%
- {\reset at font\normalsize\py at HeaderFamily}}
-\renewcommand{\paragraph}{\@startsection{paragraph}{4}{\z@}%
- {3.25ex \@plus1ex \@minus.2ex}%
- {-1em}%
- {\reset at font\normalsize\py at HeaderFamily}}
-\renewcommand{\subparagraph}{\@startsection{subparagraph}{5}{\parindent}%
- {3.25ex \@plus1ex \@minus .2ex}%
- {-1em}%
- {\reset at font\normalsize\py at HeaderFamily}}
-
-
-% This gets the underscores closer to the right width; the only change
-% from standard LaTeX is the width specified.
-
-\DeclareTextCommandDefault{\textunderscore}{%
- \leavevmode \kern.06em\vbox{\hrule\@width.55em}}
-
-% Underscore hack (only act like subscript operator if in math mode)
-%
-% The following is due to Mark Wooding (the old version didn't work with
-% Latex 2e.
-
-%%\DeclareRobustCommand\hackscore{%
-%% \ifmmode_\else\textunderscore\fi%
-%%}
-%%\begingroup
-%%\catcode`\_\active
-%%\def\next{%
-%% \AtBeginDocument{\catcode`\_\active\def_{\hackscore{}}}%
-%%}
-%%\expandafter\endgroup\next
-
-
-% Now for a lot of semantically-loaded environments that do a ton of magical
-% things to get the right formatting and index entries for the stuff in
-% Python modules and C API.
-
-
-% {fulllineitems} is used in one place in libregex.tex, but is really for
-% internal use in this file.
-%
-\newcommand{\py at itemnewline}[1]{%
- \@tempdima\linewidth%
- \advance\@tempdima \leftmargin\makebox[\@tempdima][l]{#1}%
-}
-
-\newenvironment{fulllineitems}{
- \begin{list}{}{\labelwidth \leftmargin \labelsep 0pt
- \rightmargin 0pt \topsep -\parskip \partopsep \parskip
- \itemsep -\parsep
- \let\makelabel=\py at itemnewline}
-}{\end{list}}
-
-% \optional is mostly for use in the arguments parameters to the various
-% {*desc} environments defined below, but may be used elsewhere. Known to
-% be used in the debugger chapter.
-%
-% Typical usage:
-%
-% \begin{funcdesc}{myfunc}{reqparm\optional{, optparm}}
-% ^^^ ^^^
-% No space here No space here
-%
-% When a function has multiple optional parameters, \optional should be
-% nested, not chained. This is right:
-%
-% \begin{funcdesc}{myfunc}{\optional{parm1\optional{, parm2}}}
-%
-\let\py at badkey=\@undefined
-
-\newcommand{\optional}[1]{%
- {\textnormal{\Large[}}{#1}\hspace{0.5mm}{\textnormal{\Large]}}}
-
-% This can be used when a function or method accepts an varying number
-% of arguments, such as by using the *args syntax in the parameter list.
-\newcommand{\py at moreargs}{...}
-
-% This can be used when you don't want to document the parameters to a
-% function or method, but simply state that it's an alias for
-% something else.
-\newcommand{\py at unspecified}{...}
-
-
-\newlength{\py at argswidth}
-\newcommand{\py at sigparams}[1]{%
- \parbox[t]{\py at argswidth}{\py at varvars{#1}\code{)}}}
-\newcommand{\py at sigline}[2]{%
- \settowidth{\py at argswidth}{#1\code{(}}%
- \addtolength{\py at argswidth}{-2\py at argswidth}%
- \addtolength{\py at argswidth}{\textwidth}%
- \item[#1\code{(}\py at sigparams{#2}]}
-
-% C functions ------------------------------------------------------------
-% \begin{cfuncdesc}[refcount]{type}{name}{arglist}
-% Note that the [refcount] slot should only be filled in by
-% tools/anno-api.py; it pulls the value from the refcounts database.
-\newcommand{\cfuncline}[3]{
- \py at sigline{\code{#1 \bfcode{#2}}}{#3}%
- \index{#2@{\py at idxcode{#2()}}}
-}
-\newenvironment{cfuncdesc}[4][\py at badkey]{
- \begin{fulllineitems}
- \cfuncline{#2}{#3}{#4}
- \ifx#1\@undefined\else%
- \emph{Return value: \textbf{#1}.}\\
- \fi
-}{\end{fulllineitems}}
-
-% C variables ------------------------------------------------------------
-% \begin{cvardesc}{type}{name}
-\newenvironment{cvardesc}[2]{
- \begin{fulllineitems}
- \item[\code{#1 \bfcode{#2}}\index{#2@{\py at idxcode{#2}}}]
-}{\end{fulllineitems}}
-
-% C data types -----------------------------------------------------------
-% \begin{ctypedesc}[index name]{typedef name}
-\newenvironment{ctypedesc}[2][\py at badkey]{
- \begin{fulllineitems}
- \item[\bfcode{#2}%
- \ifx#1\@undefined%
- \index{#2@{\py at idxcode{#2}} (C type)}
- \else%
- \index{#2@{\py at idxcode{#1}} (C type)}
- \fi]
-}{\end{fulllineitems}}
-
-% C type fields ----------------------------------------------------------
-% \begin{cmemberdesc}{container type}{ctype}{membername}
-\newcommand{\cmemberline}[3]{
- \item[\code{#2 \bfcode{#3}}]
- \index{#3@{\py at idxcode{#3}} (#1 member)}
-}
-\newenvironment{cmemberdesc}[3]{
- \begin{fulllineitems}
- \cmemberline{#1}{#2}{#3}
-}{\end{fulllineitems}}
-
-% Funky macros -----------------------------------------------------------
-% \begin{csimplemacrodesc}{name}
-% -- "simple" because it has no args; NOT for constant definitions!
-\newenvironment{csimplemacrodesc}[1]{
- \begin{fulllineitems}
- \item[\bfcode{#1}\index{#1@{\py at idxcode{#1}} (macro)}]
-}{\end{fulllineitems}}
-
-% simple functions (not methods) -----------------------------------------
-% \begin{funcdesc}{name}{args}
-\newcommand{\funcline}[2]{%
- \funclineni{#1}{#2}%
- \index{#1@{\py at idxcode{#1()}} (in module \py at thismodule)}}
-\newenvironment{funcdesc}[2]{
- \begin{fulllineitems}
- \funcline{#1}{#2}
-}{\end{fulllineitems}}
-
-% similar to {funcdesc}, but doesn't add to the index
-\newcommand{\funclineni}[2]{%
- \py at sigline{\bfcode{#1}}{#2}}
-\newenvironment{funcdescni}[2]{
- \begin{fulllineitems}
- \funclineni{#1}{#2}
-}{\end{fulllineitems}}
-
-% classes ----------------------------------------------------------------
-% \begin{classdesc}{name}{constructor args}
-\newenvironment{classdesc}[2]{
- % Using \renewcommand doesn't work for this, for unknown reasons:
- \global\def\py at thisclass{#1}
- \begin{fulllineitems}
- \py at sigline{\strong{class }\bfcode{#1}}{#2}%
- \index{#1@{\py at idxcode{#1}} (class in \py at thismodule)}
-}{\end{fulllineitems}}
-
-% \begin{classdesc*}{name}
-\newenvironment{classdesc*}[1]{
- % Using \renewcommand doesn't work for this, for unknown reasons:
- \global\def\py at thisclass{#1}
- \begin{fulllineitems}
- \item[\strong{class }\code{\bfcode{#1}}%
- \index{#1@{\py at idxcode{#1}} (class in \py at thismodule)}]
-}{\end{fulllineitems}}
-
-% \begin{excclassdesc}{name}{constructor args}
-% but indexes as an exception
-\newenvironment{excclassdesc}[2]{
- % Using \renewcommand doesn't work for this, for unknown reasons:
- \global\def\py at thisclass{#1}
- \begin{fulllineitems}
- \py at sigline{\strong{exception }\bfcode{#1}}{#2}%
- \index{#1@{\py at idxcode{#1}} (exception in \py at thismodule)}
-}{\end{fulllineitems}}
-
-% There is no corresponding {excclassdesc*} environment. To describe
-% a class exception without parameters, use the {excdesc} environment.
-
-
-\let\py at classbadkey=\@undefined
-
-% object method ----------------------------------------------------------
-% \begin{methoddesc}[classname]{methodname}{args}
-\newcommand{\methodline}[3][\@undefined]{
- \methodlineni{#2}{#3}
- \ifx#1\@undefined
- \index{#2@{\py at idxcode{#2()}} (\py at thisclass\ method)}
- \else
- \index{#2@{\py at idxcode{#2()}} (#1 method)}
- \fi
-}
-\newenvironment{methoddesc}[3][\@undefined]{
- \begin{fulllineitems}
- \ifx#1\@undefined
- \methodline{#2}{#3}
- \else
- \def\py at thisclass{#1}
- \methodline{#2}{#3}
- \fi
-}{\end{fulllineitems}}
-
-% similar to {methoddesc}, but doesn't add to the index
-% (never actually uses the optional argument)
-\newcommand{\methodlineni}[3][\py at classbadkey]{%
- \py at sigline{\bfcode{#2}}{#3}}
-\newenvironment{methoddescni}[3][\py at classbadkey]{
- \begin{fulllineitems}
- \methodlineni{#2}{#3}
-}{\end{fulllineitems}}
-
-% object data attribute --------------------------------------------------
-% \begin{memberdesc}[classname]{membername}
-\newcommand{\memberline}[2][\py at classbadkey]{%
- \ifx#1\@undefined
- \memberlineni{#2}
- \index{#2@{\py at idxcode{#2}} (\py at thisclass\ attribute)}
- \else
- \memberlineni{#2}
- \index{#2@{\py at idxcode{#2}} (#1 attribute)}
- \fi
-}
-\newenvironment{memberdesc}[2][\py at classbadkey]{
- \begin{fulllineitems}
- \ifx#1\@undefined
- \memberline{#2}
- \else
- \def\py at thisclass{#1}
- \memberline{#2}
- \fi
-}{\end{fulllineitems}}
-
-% similar to {memberdesc}, but doesn't add to the index
-% (never actually uses the optional argument)
-\newcommand{\memberlineni}[2][\py at classbadkey]{\item[\bfcode{#2}]}
-\newenvironment{memberdescni}[2][\py at classbadkey]{
- \begin{fulllineitems}
- \memberlineni{#2}
-}{\end{fulllineitems}}
-
-% For exceptions: --------------------------------------------------------
-% \begin{excdesc}{name}
-% -- for constructor information, use excclassdesc instead
-\newenvironment{excdesc}[1]{
- \begin{fulllineitems}
- \item[\strong{exception }\bfcode{#1}%
- \index{#1@{\py at idxcode{#1}} (exception in \py at thismodule)}]
-}{\end{fulllineitems}}
-
-% Module data or constants: ----------------------------------------------
-% \begin{datadesc}{name}
-\newcommand{\dataline}[1]{%
- \datalineni{#1}\index{#1@{\py at idxcode{#1}} (data in \py at thismodule)}}
-\newenvironment{datadesc}[1]{
- \begin{fulllineitems}
- \dataline{#1}
-}{\end{fulllineitems}}
-
-% similar to {datadesc}, but doesn't add to the index
-\newcommand{\datalineni}[1]{\item[\bfcode{#1}]\nopagebreak}
-\newenvironment{datadescni}[1]{
- \begin{fulllineitems}
- \datalineni{#1}
-}{\end{fulllineitems}}
-
-% bytecode instruction ---------------------------------------------------
-% \begin{opcodedesc}{name}{var}
-% -- {var} may be {}
-\newenvironment{opcodedesc}[2]{
- \begin{fulllineitems}
- \item[\bfcode{#1}\quad\var{#2}]
-}{\end{fulllineitems}}
-
-
-\newcommand{\nodename}[1]{\label{#1}}
-
-% For these commands, use \command{} to get the typography right, not
-% {\command}. This works better with the texinfo translation.
-\newcommand{\ABC}{{\sc abc}}
-\newcommand{\UNIX}{{\sc Unix}}
-\newcommand{\POSIX}{POSIX}
-\newcommand{\ASCII}{{\sc ascii}}
-\newcommand{\Cpp}{C\protect\raisebox{.18ex}{++}}
-\newcommand{\C}{C}
-\newcommand{\EOF}{{\sc eof}}
-\newcommand{\NULL}{\constant{NULL}}
-\newcommand{\infinity}{\ensuremath{\infty}}
-\newcommand{\plusminus}{\ensuremath{\pm}}
-
-% \guilabel{Start}
-\newcommand{\guilabel}[1]{\textsf{#1}}
-% \menuselection{Start \sub Programs \sub Python}
-\newcommand{\menuselection}[1]{\guilabel{{\def\sub{ \ensuremath{>} }#1}}}
-
-% Also for consistency: spell Python "Python", not "python"!
-
-% code is the most difficult one...
-\newcommand{\code}[1]{\textrm{\@vobeyspaces\@noligs\def\{{\char`\{}\def\}{\char`\}}\def\~{\char`\~}\def\^{\char`\^}\def\e{\char`\\}\def\${\char`\$}\def\#{\char`\#}\def\&{\char`\&}\def\%{\char`\%}%
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-
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-\renewcommand{\emph}[1]{{\em #1}}
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-\newcommand{\strong}[1]{{\bf #1}}
-% let's experiment with a new font:
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-
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-\ifpdf
- \newcommand{\url}[1]{{%
- \py at pdfstartlink attr{/Border [0 0 0]} user{/S /URI /URI (#1)}%
- \py at LinkColor% color of the link text
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-
-\newcommand{\py at varvars}[1]{{%
- {\let\unspecified=\py at unspecified%
- \let\moreargs=\py at moreargs%
- \var{#1}}}}
-
-% I'd really like to get rid of this!
-\newif\iftexi\texifalse
-
-% This is used to get l2h to put the copyright and abstract on
-% a separate HTML page.
-\newif\ifhtml\htmlfalse
-
-
-% These should be used for all references to identifiers which are
-% used to refer to instances of specific language constructs. See the
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-% For now, don't do anything really fancy with them; just use them as
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-
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-
-% constants defined in Python modules or C headers, not language constants:
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-
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-\newcommand{\rfc}[1]{RFC #1\index{RFC!RFC #1}}
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-\newcommand{\programopt}[1]{\strong{#1}}
-% Note that \longprogramopt provides the '--'!
-\newcommand{\longprogramopt}[1]{\strong{-{}-#1}}
-
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-\ifpdf
- \newcommand{\ulink}[2]{{%
- % For PDF, we *should* only generate a link when the URL is absolute.
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- \py at LinkColor% color of the link text
- #1%
- \py at NormalColor% Turn it back off; these are declarative
- \pdfendlink}% and don't appear bound to the current
- }% formatting "box".
-\else
- \newcommand{\ulink}[2]{#1}
-\fi
-
-% cited titles: \citetitle{Title of Work}
-% online: \citetitle[url-to-resource]{Title of Work}
-\ifpdf
- \newcommand{\citetitle}[2][\py at modulebadkey]{%
- \ifx\py at modulebadkey#1\emph{#2}\else\ulink{\emph{#2}}{#1}\fi%
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- \newcommand{\citetitle}[2][URL]{\emph{#2}}
-\fi
-
-
-
-% This version is being checked in for the historical record; it shows
-% how I've managed to get some aspects of this to work. It will not
-% be used in practice, so a subsequent revision will change things
-% again. This version has problems, but shows how to do something
-% that proved more tedious than I'd expected, so I don't want to lose
-% the example completely.
-%
-\newcommand{\grammartoken}[1]{\texttt{#1}}
-\newenvironment{productionlist}[1][\py at badkey]{
- \def\optional##1{{\Large[}##1{\Large]}}
- \def\production##1##2{\code{##1}&::=&\code{##2}\\}
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- \def\token##1{##1}
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- \endSbox
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-\newcommand{\py at noticeend@note}{}
-
-% a 'warning' gets more visible distinction:
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-\newcommand{\py at noticestart@warning}{\py at heavybox}
-\newcommand{\py at noticeend@warning}{\py at endheavybox}
-
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-\newcommand{\note}[1]{\strong{\py at noticelabel@note} #1}
-\newcommand{\warning}[1]{\strong{\py at noticelabel@warning} #1}
-
-% Deprecation stuff.
-% Should be extended to allow an index / list of deprecated stuff. But
-% there's a lot of stuff that needs to be done to make that automatable.
-%
-% First parameter is the release number that deprecates the feature, the
-% second is the action the should be taken by users of the feature.
-%
-% Example:
-% \deprecated{1.5.1}{Use \method{frobnicate()} instead.}
-%
-\newcommand{\deprecated}[2]{%
- \strong{Deprecated since release #1.} #2\par}
-
-% New stuff.
-% This should be used to mark things which have been added to the
-% development tree but that aren't in the release, but are documented.
-% This allows release of documentation that already includes updated
-% descriptions. Place at end of descriptor environment.
-%
-% Example:
-% \versionadded{1.5.2}
-% \versionchanged[short explanation]{2.0}
-%
-\newcommand{\versionadded}[2][\py at badkey]{%
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-}{%
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-}
-
-\newenvironment{longtablev}[7]{%
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- \\%
- \hline\endhead%
-}{%
- \end{longtable}%
- \end{center}%
-}
-
-% XXX Don't think we can use this yet, though it cleans up some
-% tedious markup. There's no equivalent for the HTML transform yet,
-% and that needs to exist. I don't know how to write it.
-%
-% This should really have something that makes it easier to bind a
-% table's ``Notes'' column and an associated tablenotes environment,
-% and generates the right magic for getting the numbers right in the
-% table.
-%
-% So this is quite incomplete.
-%
-\newcounter{py at tablenotescounter}
-\newenvironment{tablenotes}{%
- \noindent Notes:
- \par
- \setcounter{py at tablenotescounter}{0}
- \begin{list}{(\arabic{py at tablenotescounter})}%
- {\usecounter{py at tablenotescounter}}
-}{\end{list}}
-
-
-% Cross-referencing (AMK, new impl. FLD)
-% Sample usage:
-% \begin{seealso}
-% \seemodule{rand}{Uniform random number generator.}; % Module xref
-% \seetext{\emph{Encyclopedia Britannica}}. % Ref to a book
-%
-% % A funky case: module name contains '_'; have to supply an optional key
-% \seemodule[copyreg]{copy_reg}{Interface constructor registration for
-% \module{pickle}.}
-% \end{seealso}
-%
-% Note that the last parameter for \seemodule and \seetext should be complete
-% sentences and be terminated with the proper punctuation.
-
-\ifpdf
- \newcommand{\py at seemodule}[3][\py at modulebadkey]{%
- \par%
- \ifx\py at modulebadkey#1\def\py at modulekey{#2}\else\def\py at modulekey{#1}\fi%
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- (section \ref{module-\py at modulekey}):]
- #3
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- #3
- \end{fulllineitems}
- }
-\fi
-
-% \seelink{url}{link text}{why it's interesting}
-\newcommand{\py at seelink}[3]{%
- \par
- \begin{fulllineitems}
- \item[\ulink{#2}{#1}]
- #3
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-% \seetitle[url]{title}{why it's interesting}
-\newcommand{\py at seetitle}[3][\py at modulebadkey]{%
- \par
- \begin{fulllineitems}
- \item[\citetitle{#2}]
- \ifx\py at modulebadkey#1\else
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-\newcommand{\py at seepep}[3]{%
- \par%
- \begin{fulllineitems}
- \item[\pep{#1}, ``\emph{#2}'']
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-\newcommand{\py at seerfc}[3]{%
- \par%
- \begin{fulllineitems}
- \item[\rfc{#1}, ``\emph{#2}'']
- #3
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-% \seeurl{url}{why it's interesting}
-\newcommand{\py at seeurl}[2]{%
- \par%
- \begin{fulllineitems}
- \item[\url{#1}]
- #2
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- \par
- \def\seetext##1{\par{##1}}
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- \let\seeurl=\py at seeurl
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-}{\par}
-\newenvironment{seealso}{
- \par
- \strong{See Also:}
- \par
- \def\seetext##1{\par{##1}}
- \let\seemodule=\py at seemodule
- \let\seepep=\py at seepep
- \let\seerfc=\py at seerfc
- \let\seetitle=\py at seetitle
- \let\seeurl=\py at seeurl
- \let\seelink=\py at seelink
-}{\par}
-
-% Allow the Python release number to be specified independently of the
-% \date{}. This allows the date to reflect the document's date and
-% release to specify the Python release that is documented.
-%
-\newcommand{\py at release}{}
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-\newcommand{\shortversion}{}
-\newcommand{\releaseinfo}{}
-\newcommand{\releasename}{Release}
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-\newcommand{\setreleaseinfo}[1]{%
- \renewcommand{\releaseinfo}{#1}}
-
-% Allow specification of the author's address separately from the
-% author's name. This can be used to format them differently, which
-% is a good thing.
-%
-\newcommand{\py at authoraddress}{}
-\newcommand{\authoraddress}[1]{\renewcommand{\py at authoraddress}{#1}}
-\let\developersaddress=\authoraddress
-\let\developer=\author
-\let\developers=\author
-
-% This sets up the fancy chapter headings that make the documents look
-% at least a little better than the usual LaTeX output.
-%
-\@ifundefined{ChTitleVar}{}{
- \ChNameVar{\raggedleft\normalsize\py at HeaderFamily}
- \ChNumVar{\raggedleft \bfseries\Large\py at HeaderFamily}
- \ChTitleVar{\raggedleft \rm\Huge\py at HeaderFamily}
- % This creates chapter heads without the leading \vspace*{}:
- \def\@makechapterhead#1{%
- {\parindent \z@ \raggedright \normalfont
- \ifnum \c at secnumdepth >\m at ne
- \DOCH
- \fi
- \interlinepenalty\@M
- \DOTI{#1}
- }
- }
-}
-
-
-% Definition lists; requested by AMK for HOWTO documents. Probably useful
-% elsewhere as well, so keep in in the general style support.
-%
-\newenvironment{definitions}{%
- \begin{description}%
- \def\term##1{\item[##1]\mbox{}\\*[0mm]}
-}{%
- \end{description}%
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-
-% Tell TeX about pathological hyphenation cases:
-\hyphenation{Base-HTTP-Re-quest-Hand-ler}
diff --git a/doc/solvers.tex b/doc/solvers.tex
index 90c014f..00579e8 100644
--- a/doc/solvers.tex
+++ b/doc/solvers.tex
@@ -1,558 +1,8 @@
-\chapter{Optimization Routines (\module{cvxopt.solvers})}
+\chapter{Nonlinear Convex Programming (\module{cvxopt.solvers})}
\label{chap:solvers}
-\section{Linear Programming} \label{s-lpsolver}
-
-\begin{funcdesc}{lp}{c, G, h\optional{, A, b\optional{,
-solver\optional{, primalstart\optional{, dualstart}}}}}
-Solves the pair of primal and dual LPs
-\BEQ \label{e-lp}
-\mbox{Primal:}\quad \begin{array}[t]{ll}
-\mbox{minimize} & c^Tx \\
-\mbox{subject to} & Gx + s = h \\ & Ax=b \\ & s \succeq 0
-\end{array} \qquad\qquad
-\mbox{Dual:}\quad
-\begin{array}[t]{ll}
-\mbox{maximize} & -h^T z - b^T y \\
-\mbox{subject to} & G^Tz + A^T y + c = 0 \\ & z \succeq 0.
-\end{array}
-\EEQ
-\var{c}, \var{h} and \var{b} are real single-column dense matrices.
-\var{G} and \var{A} are real dense or sparse matrices.
-The default values for \var{A} and \var{b} are sparse matrices with
-zero rows, meaning that there are no equality constraints.
-
-The \var{solver} argument is used to choose among three solvers.
-When it is omitted or \None, the default CVXOPT solver is used.
-The external solvers GLPK and MOSEK (if installed) can be selected by
-setting \code{\var{solver}='glpk'} or
-\code{\var{solver}='mosek'}; see section~\ref{s-external}.
-
-The optional arguments \var{primalstart} and \var{dualstart} are only
-referenced by the default solver.
-\var{primalstart} is a dictionary with keys \code{'x'} and \code{'s'},
-used as an optional primal starting point.
-\var{dualstart} is a dictionary with keys \code{'y'} and \code{'z'},
-used as an optional dual starting point.
-
-\function{lp()} returns a dictionary with keys \code{'status'},
-\code{'x'}, \code{'s'}, \code{'y'}, \code{'z'}.
-The possible values of the \code{'status'} item are as follows.
-\begin{description}
-\item[\code{'optimal'.}] In this case the \code{'x'}, \code{'s'},
-\code{'y'} and \code{'z'} entries contain the primal and dual solutions,
-which approximately satisfy
-\[
- Gx + s = h, \qquad Ax=b, \qquad G^T z + A^T y + c = 0, \qquad
- s \succeq 0, \qquad z \succeq 0, \qquad s^T z =0.
-\]
-
-\item[\code{'primal infeasible'.}]
-The \code{'x'} and \code{'s'} entries are \None, and the \code{'y'},
-\code{'z'} entries provide an approximate certificate of
-infeasibility, \ie, vectors that approximately satisfy
-\[
- G^T z + A^T y = 0, \qquad h^T z + b^T y = -1, \qquad z \succeq 0.
-\]
-With the \code{'glpk'} option, no proof of infeasibility is returned
-(all entries of the dictionary are \None).
-
-\item[\code{'dual infeasible'.}] The LP is dual infeasible.
-The \code{'y'} and \code{'z'} entries are \None, and the \code{'x'}
-and \code{'s'} entries contain an approximate certificate of dual
-infeasibility
-\[
- Gx + s = 0, \qquad Ax=0, \qquad c^T x = -1, \qquad s \succeq 0.
-\]
-With the \code{'glpk'} option, no proof of dual infeasibility is
-returned.
-
-\item[\code{'unknown'}.] The \code{'x'}, \code{'s'}, \code{'y'},
-\code{'z'} entries are \None.
-\end{description}
-
-%The default solver requires that
-%\[
-% \Rank(A) = p, \qquad \Rank (\left[ \begin{array}{c} G \\ A\end{array}
-% \right]) = n
-%\]
-%where \var{p} is the number of equality constraints and \var{n}
-%is the number of primal variables (dimension of \var{x}).
-\end{funcdesc}
-
-As a simple example we solve the LP
-\[
- \begin{array}[t]{ll}
- \mbox{minimize} & -4x_1 - 5x_2 \\
- \mbox{subject to} & 2x_1 + x_2 \leq 3 \\
- & x_1 + 2x_2 \leq 3 \\
- & x_1 \geq 0, \quad x_2 \geq 0.
- \end{array}
-\]
-\begin{verbatim}
->>> from cvxopt.base import matrix
->>> from cvxopt import solvers
->>> c = matrix([-4., -5.])
->>> G = matrix([[2., 1., -1., 0.], [1., 2., 0., -1.]])
->>> h = matrix([3., 3., 0., 0.])
->>> sol = solvers.lp(c,G,h)
->>> print sol['x']
- 1.0000e-00
- 1.0000e-00
-\end{verbatim}
-
-\section{Quadratic Programming}
-\begin{funcdesc}{qp}{P, q, \optional{, G, h \optional{, A, b\optional{,
-solver}}}}
-Solves a convex quadratic program
-\[
-\begin{array}{ll}
-\mbox{minimize} & (1/2) x^TPx + q^T x \\
-\mbox{subject to} & Gx \preceq h \\ & Ax = b.
-\end{array}
-\]
-
-\var{P} is a square dense or sparse real matrix, representing a
-symmetric matrix in \code{'L'} storage, \ie, only the lower
-triangular part of \var{P} is referenced.
-\var{G} and \var{A} are dense or sparse real matrices.
-Their default values are sparse matrices with zero columns.
-\var{q}, \var{h} and \var{b} are single-column real dense matrices.
-The default values of \var{h} and \var{b} are matrices of size (0,1).
-
-The default CVXOPT solver is used when the \var{solver} argument
-is absent or \None. The MOSEK solver (if installed) can be
-selected by setting \var{solver}=\code{'mosek'}.
-
-\function{qp()} returns a dictionary with keys
-\code{'status'}, \code{'x'}, \code{'s'}, \code{'y'}, \code{'z'}.
-The possible values of the \code{'status'} key are as follows.
-\begin{description}
-\item[\code{'optimal'}] In this case the
-\code{'x'} entry is the primal optimal solution,
-the \code{'s'} entry is the corresponding slack in the inequality
-constraints, the \code{'z'} and \code{'y'} entries are the optimal
-values of the dual variables associated with the linear inequality
-and linear equality constraints.
-These values (approximately) satisfy the optimality conditions
-\[
- Px + q + G^T z + A^T y = 0, \qquad Gx + s = h, \qquad
- Ax = b, \qquad s \succeq 0, \qquad z \succeq 0, \qquad s^T z = 0.
-\]
-
-\item [\code{'primal infeasible'}] This only applies when
-\var{solver} is \code{'mosek'}, and means that a certificate of
-primal infeasibility has been found. The \code{'x'} and \code{'s'}
-entries are \None, and the
-\code{'z'} and \code{'y'} entries are vectors that approximately satisfy
-\[
- G^Tz + A^T y = 0, \qquad h^Tz + b^Ty = -1, \qquad z \succeq 0.
-\]
-
-\item [\code{'dual infeasible'}] This only applies when
-\var{solver} is \code{'mosek'}, and means that a certificate of
-dual infeasibility has been found. The \code{'z'} and \code{'y'}
-entries are \None, and the \code{'x'} and \code{'s'} entries are
-vectors that approximately satisfy
-\[
- Px = 0, \qquad q^Tx = -1, \qquad Gx + s = 0, \qquad Ax=0, \qquad
- s \succeq 0.
-\]
-
-\item[\code{'unknown'}] This means that the algorithm reached
-the maximum number of iterations before a solution was found.
-The \code{'x'}, \code{'s'}, \code{'y'}, \code{'z'} entries are \None.
-\end{description}
-\end{funcdesc}
-
-As an example we compute the trade-off curve on page 187
-of the book \citetitle[http://www.stanford.edu/\~{}boyd/cvxbook]{Convex
-Optimization}, by solving the quadratic program
-\[
-\begin{array}{ll}
-\mbox{minimize} & -\bar p^T x + \mu x^T S x \\
-\mbox{subject to} & \ones^T x = 1, \quad x \succeq 0
-\end{array}
-\]
-for a sequence of positive values of {\it mu}.
-The code below computes the trade-off curve and produces two figures
-using the \ulink{Matplotlib}{http://matplotlib.sourceforge.net} package.
-\begin{center}
-\includegraphics[width=10cm]{figures/portfolio1.eps}
-\hspace*{\fill}
-\includegraphics[width=10cm]{figures/portfolio2.eps}
-\end{center}
-
-\begin{verbatim}
-from math import sqrt
-from cvxopt.base import matrix
-from cvxopt.blas import dot
-from cvxopt.solvers import qp
-import pylab
-
-# Problem data.
-n = 4
-S = matrix([[ 4e-2, 6e-3, -4e-3, 0.0 ],
- [ 6e-3, 1e-2, 0.0, 0.0 ],
- [-4e-3, 0.0, 2.5e-3, 0.0 ],
- [ 0.0, 0.0, 0.0, 0.0 ]])
-pbar = matrix([.12, .10, .07, .03])
-G = matrix(0.0, (n,n))
-G[::n+1] = -1.0
-h = matrix(0.0, (n,1))
-A = matrix(1.0, (1,n))
-b = matrix(1.0)
-
-# Compute trade-off.
-N = 100
-mus = [ 10**(5.0*t/N-1.0) for t in xrange(N) ]
-portfolios = [ qp(mu*S, -pbar, G, h, A, b)['x'] for mu in mus ]
-returns = [ dot(pbar,x) for x in portfolios ]
-risks = [ sqrt(dot(x, S*x)) for x in portfolios ]
-
-# Plot trade-off curve and optimal allocations.
-pylab.figure(1, facecolor='w')
-pylab.plot(risks, returns)
-pylab.xlabel('standard deviation')
-pylab.ylabel('expected return')
-pylab.axis([0, 0.2, 0, 0.15])
-pylab.title('Risk-return trade-off curve (fig 4.12)')
-pylab.yticks([0.00, 0.05, 0.10, 0.15])
-
-pylab.figure(2, facecolor='w')
-c1 = [ x[0] for x in portfolios ]
-c2 = [ x[0] + x[1] for x in portfolios ]
-c3 = [ x[0] + x[1] + x[2] for x in portfolios ]
-c4 = [ x[0] + x[1] + x[2] + x[3] for x in portfolios ]
-pylab.fill(risks + [.20], c1 + [0.0], '#F0F0F0')
-pylab.fill(risks[-1::-1] + risks, c2[-1::-1] + c1, '#D0D0D0')
-pylab.fill(risks[-1::-1] + risks, c3[-1::-1] + c2, '#F0F0F0')
-pylab.fill(risks[-1::-1] + risks, c4[-1::-1] + c3, '#D0D0D0')
-pylab.axis([0.0, 0.2, 0.0, 1.0])
-pylab.xlabel('standard deviation')
-pylab.ylabel('allocation')
-pylab.text(.15,.5,'x1')
-pylab.text(.10,.7,'x2')
-pylab.text(.05,.7,'x3')
-pylab.text(.01,.7,'x4')
-pylab.title('Optimal allocations (fig 4.12)')
-pylab.show()
-\end{verbatim}
-
-
-\section{Geometric Programming}
-\begin{funcdesc}{gp}{K, F, g \optional{, G, h \optional{, A, b}}}
-Solves a geometric program in convex form
-\[
-\begin{array}{ll}
-\mbox{minimize} & f_0(x) = \lse(F_0x+g_0) \\
-\mbox{subject to} & f_i(x) = \lse(F_ix+g_i) \leq 0,\quad i=1,\ldots,m \\
- & Gx \preceq h \\
- & Ax=b
-\end{array}
-\]
-where
-\[
- \lse(u) = \log \sum_k \exp(u_k), \qquad
- F = \left[ \begin{array}{cccc}
- F_0^T & F_1^T & \cdots & F_m^T \end{array}\right]^T, \qquad
- g = \left[ \begin{array}{cccc}
- g_0^T & g_1^T & \cdots & g_m^T \end{array}\right]^T.
-\]
-\var{K} is a list of \var{m}+1 positive integers with
-\code{\var{K}[\var i]}
-equal to the number of rows in {\it Fi}.
-\var{F} is a dense or sparse real matrix of size
-\code{(sum(\var K),\var n)}.
-\var{g} is a dense real matrix of size \code{(sum(\var K),1)}.
-\var{G} and \var{A} are dense or sparse real matrices.
-Their default values are sparse matrices with zero rows.
-\var{h} and \var{b} are dense real matrices with one column.
-Their default values are matrices of size (0,1).
-
-\function{gp()} returns a dictionary with keys
-\code{'status'}, \code{'x'}, \code{'snl'}, \code{'sl'},
-\code{'y'}, \code{'znl'} and \code{'zl'}.
-The possible values of the \code{'status'} key are:
-\begin{description}
-\item[\code{'optimal'}] In this case the
-\code{'x'} entry is the primal optimal solution,
-the \code{'snl'} and \code{'sl'} entries are the corresponding slacks
-in the nonlinear and linear inequality constraints.
-The \code{'znl'}, \code{'zl'} and \code{'y'} entries are the optimal
-values of the dual variables associated with the nonlinear and linear
-inequality constraints and the linear equality constraints.
-These values approximately satisfy
-\[
- \nabla f_0(x) + \sum_{k=1}^m z_{\mathrm{nl},k}
- \nabla f_k(x) + G^T z_\mathrm{l} + A^T y = 0, \qquad
- f_k(x) + s_{\mathrm{nl},k} = 0, \quad k=1,\ldots,m, \qquad
- Gx + s_\mathrm{l} = h, \qquad Ax=b
-\]
-and
-\[
-s_\mathrm{nl}\succeq 0, \qquad s_\mathrm{l}\succeq 0, \qquad
-z_\mathrm{nl} \succeq 0, \qquad z_\mathrm{l} \succeq 0, \qquad
-s_\mathrm{nl}^T z_\mathrm{nl} + s_\mathrm{l}^T z_\mathrm{l} = 0.
-\]
-
-\item[\code{'unknown'}] This means that the algorithm reached
-the maximum number of iterations before a solution was found.
-The \code{'x'}, \code{'snl'}, \code{'sl'}, \code{'y'}, \code{'znl'}
-and \code{'zl'} entries are \None.
-\end{description}
-\end{funcdesc}
-
-As an example, we solve the small GP on page~8 of the paper
-\citetitle[http://www.stanford.edu/\~{}boyd/gp\_tutorial]{A Tutorial on Geometric Programming}.
-The posynomial form of the problem is
-\[
- \begin{array}{ll}
- \mbox{minimize} & w^{-1} h^{-1} d^{-1} \\
- \mbox{subject to}
- & (2/A_\mathrm{wall}) hw + (2/A_\mathrm{wall})hd \leq 1 \\
- & (1/A_\mathrm{flr}) wd \leq 1 \\
- & \alpha wh^{-1} \leq 1 \\
- & (1/\beta) hw^{-1} \leq 1 \\
- & \gamma wd^{-1} \leq 1 \\
- & (1/\delta)dw^{-1} \leq 1
- \end{array}
-\]
-with variables {\it h}, {\it w}, {\it d}.
-
-\begin{verbatim}
-from cvxopt.base import matrix, log, exp
-from cvxopt import solvers
-
-Aflr = 1000.0
-Awall = 100.0
-alpha = 0.5
-beta = 2.0
-gamma = 0.5
-delta = 2.0
-
-F = matrix( [[-1., 1., 1., 0., -1., 1., 0., 0.],
- [-1., 1., 0., 1., 1., -1., 1., -1.],
- [-1., 0., 1., 1., 0., 0., -1., 1.]])
-g = log( matrix( [1.0, 2/Awall, 2/Awall, 1/Aflr, alpha, 1/beta, gamma, 1/delta]) )
-K = [1, 2, 1, 1, 1, 1, 1]
-h, w, d = exp( solvers.gp(K, F, g)['x'] )
-\end{verbatim}
-
-\section{Semidefinite Programming} \label{s-sdpsolver}
-We use the following notation for a pair of primal and dual
-semidefinite programs (SDPs):
-\BEQ \label{e-sdp}
-\mbox{Primal:}\quad \begin{array}[t]{ll}
-\mbox{minimize} & c^Tx \\
-\mbox{subject to} & G_\mathrm{l}x + s_\mathrm{l} = h_\mathrm{l} \\
- & G_\mathrm{s}(x) + S_\mathrm{s} = H_\mathrm{s} \\
- & Ax=b \\ & s_\mathrm{l} \succeq 0, \quad S_\mathrm{s} \succeq 0
-\end{array} \qquad\qquad
-\mbox{Dual:}\quad
-\begin{array}[t]{ll}
-\mbox{maximize} & -h_\mathrm{l}^T z_\mathrm{l} -
- \Tr(H_\mathrm{s} Z_\mathrm{s}) - b^T y \\
-\mbox{subject to} & G_\mathrm{l}^Tz_\mathrm{l} +
- G_\mathrm{s}^T(Z_\mathrm{s}) + A^T y + c = 0 \\
- & z_\mathrm{l} \succeq 0, \quad Z_\mathrm{s} \succeq 0.
-\end{array}
-\EEQ
-The dimensions of the primal and dual variables are
-\[
- x\in \reals^n, \qquad s_\mathrm{l} \in \reals^m,
-\qquad S_\mathrm{s} \in \symm^{m_0} \times \cdots \times
-\symm^{m_{N-1}}, \qquad
- y \in\reals^p, \qquad z_\mathrm{l}\in \reals^m,
-\qquad Z_\mathrm{s} \in \symm^{m_0} \times \cdots \times
-\symm^{m_{N-1}},
-\]
-where $\symm^n$ is the set of real symmetric matrices
-of order {\it n}. The problem data are the matrices
-\[
-c\in\reals^n, \qquad G_\mathrm{l} \in\reals^{m\times n},
-\qquad h_\mathrm{l} \in \reals^{m}, \qquad
- H_\mathrm{s} \in \symm^{m_0} \times \cdots \times \symm^{m_{N-1}},
-\qquad
-A \in \reals^{p\times n}, \qquad b \in \reals^{p},
-\]
-and the linear mapping
-$G_\mathrm{s} : \reals^n \rightarrow \symm^{m_1} \times \cdots \times
-\symm^{m_N}$ and its adjoint $G_\mathrm{s}^T$.
-As for LPs we store vector variables as dense real matrices with one
-column.
-Block-diagonal symmetric matrices are stored as lists of square
-dense real matrices, with the lower triangular part of each matrix
-representing the lower triangular part of a diagonal block. Entries
-above the diagonal are not referenced.
-
-\begin{funcdesc}{sdp}{c\optional{, Gl, hl\optional{,
- Gs, hs\optional{, A, b\optional{, solver\optional{,
- primalstart\optional{, dualstart}}}}}}}
-
-Solves the pair of primal and dual SDPs~(\ref{e-sdp}).
-
-\var{c} is a dense real matrix with one column.
-\var{Gl} and \var{A} are dense or sparse real matrices.
-\var{hl} and \var{b} are dense real matrices with one column.
-The default values for \var{Gl}, \var{hl}, \var{A} and \var{b} are
-empty matrices, \ie, matrices with zero rows.
-
-\var{Gs} and \var{hs} are lists of length {\it N} that specify the
-linear matrix inequality constraints.
-\var{hs} is a list of square dense real matrices \code{\var{hs}[k]} of
-order {\it m\_k}.
-\var{Gs} is a list of dense or sparse real matrices
-\code{\var{Gs}[k]} with {\it m\_k}*{\it m\_k} rows and {\it n} columns,
-such that the product \code{\var{Gs}[k]*\var{x}}
-is the {\it k}th diagonal block of {\it Gs}(\var{x}), stored
-columnwise.
-
-The \var{solver} argument is used to choose between two
-solvers: the default solver (used when \var{solver} is absent or
-equal to \None) and the external solver DSDP5
-(\code{\var{solver}='dsdp'}); see section~\ref{s-external}.
-With the \code{'dsdp'} option the code does not accept problems with
-equality constraints.
-
-The optional argument \var{primalstart} is a dictionary with keys
-\code{'x'}, \code{'sl'}, and \code{'ss'}, used as an optional primal
-starting point.
-\var{dualstart} is a dictionary with keys \code{'y'}, \code{'zl'},
-\code{'zs'}, used as an optional dual starting point.
-These two arguments are ignored when the DSDP solver is used.
-
-\function{sdp()} returns a dictionary with keys \code{'status'},
-\code{'x'}, \code{'sl'}, \code{'ss'}, \code{'y'}, \code{'zl'},
-\code{'ss'}.
-The possible values of the \code{'status'} item are as follows.
-\begin{description}
-\item[\code{'optimal'.}] In this case the \code{'x'}, \code{'sl'},
-\code{'ss'}, \code{'y'}, \code{'zl'}, \code{'zs'} entries contain
-primal and dual optimal solutions, which approximately satisfy
-\[
- G_\mathrm{l}x+s_\mathrm{l} = h_\mathrm{l},
-\qquad
-G_\mathrm{s}(x)+S_\mathrm{s} = H_\mathrm{s},
-\qquad
-Ax=b, \qquad
-G_\mathrm{l}^Tz_\mathrm{l} + G_\mathrm{s}^T(Z_\mathrm{s}) + A^Ty+c = 0
-\]
-and
-\[
- s_\mathrm{l} \succeq 0, \qquad S_\mathrm{s} \succeq 0, \qquad
- z_\mathrm{l} \succeq 0, \qquad Z_\mathrm{s} \succeq 0, \qquad
- s_\mathrm{l}^T z_\mathrm{l} + \Tr(S_\mathrm{s}Z_\mathrm{s}) = 0.
-\]
-
-\item[\code{'primal infeasible'.}]
-The \code{'x'}, \code{'sl'} and \code{'ss'} entries are \None,
-and the \code{'y'}, \code{'zl'}, \code{'zs'} entries provide an
-approximate certificate of infeasibility:
-\[
-G_\mathrm{l}^Tz_\mathrm{l} + G_\mathrm{s}^T(Z_\mathrm{s}) +A^Ty = 0
-\qquad h_\mathrm{l}^Tz_\mathrm{l} + \Tr(H_\mathrm{s} Z_\mathrm{s})
- +b^Ty = -1, \qquad
-z_\mathrm{l} \succeq 0, \qquad Z_\mathrm{s} \succeq 0.
-\]
-
-\item[\code{'dual infeasible'.}] The SDP is dual infeasible.
-The \code{'y'}, \code{'zl'} and \code{'zs'} entries are \None,
-and the \code{'x'}, \code{'sl'}, \code{'ss'} entries contain an
-approximate certificate of dual infeasibility:
-\[
-G_\mathrm{l}x+s_\mathrm{l} = 0, \qquad
-G_\mathrm{s}(x)+S_\mathrm{s} = 0, \qquad
-Ax = 0, \qquad c^Tx = -1, \qquad
-s_\mathrm{l} \succeq 0, \qquad S_\mathrm{s} \succeq 0.
-\]
-
-\item[\code{'unknown'}.] The \code{'x'}, \code{'sl'}, \code{'ss'},
-\code{'y'}, \code{'zl'} and \code{'zs'} entries are \None.
-\end{description}
-\end{funcdesc}
-
-We illustrate the calling sequence with a small example.
-\[
-\begin{array}{ll}
-\mbox{minimize} & x_1 - x_2 + x_3 \\
-\mbox{subject to} &
- x_1 \left[ \begin{array}{cc} -7 & -11 \\ -11 & 3
- \end{array}\right] +
- x_2 \left[ \begin{array}{cc}
- 7 & -18 \\ -18 & 8 \end{array}\right] +
- x_3 \left[ \begin{array}{cc}
- -2 & -8 \\ -8 & 1
- \end{array}\right] \preceq
- \left[ \begin{array}{cc}
- 33 & -9 \\ -9 & 26 \end{array}\right] \\*[1ex]
-& x_1 \left[ \begin{array}{ccc}
- -21 & -11 & 0 \\ -11 & 10 & 8 \\ 0 & 8 & 5
- \end{array}\right] +
- x_2 \left[ \begin{array}{ccc}
- 0 & 10 & 16 \\
-10 & -10 & -10 \\
-16 & -10 & 3
- \end{array}\right] +
- x_3 \left[ \begin{array}{ccc}
- -5 & 2 & -17 \\
- 2 & -6 & -7 \\
- -17 & 8 & 6
- \end{array}\right]
-\preceq \left[ \begin{array}{ccc}
- 14 & 9 & 40 \\
- 9 & 91 & 10 \\
- 40 & 10 & 15
-\end{array} \right]
-\end{array}
-\]
-\begin{verbatim}
->>> from cvxopt.base import matrix
->>> from cvxopt import solvers
->>> c = matrix([1.,-1.,1.])
->>> G = [ matrix([[-7., -11., -11., 3.],
- [ 7., -18., -18., 8.],
- [-2., -8., -8., 1.]]) ]
->>> G += [ matrix([[-21., -11., 0., -11., 10., 8., 0., 8., 5.],
- [ 0., 10., 16., 10., -10., -10., 16., -10., 3.],
- [ -5., 2., -17., 2., -6., 8., -17., -7., 6.]]) ]
->>> h = [ matrix([[33., -9.], [-9., 26.]]) ]
->>> h += [ matrix([[14., 9., 40.], [9., 91., 10.], [40., 10., 15.]]) ]
->>> sol = solvers.sdp(c, Gs=G, hs=h)
->>> print sol['x']
- -3.6775e-01
- 1.8983e+00
- -8.8747e-01
->>> print sol['zs'][0]
- 3.9613e-03 0.0000e+00
- -4.3390e-03 4.7526e-03
->>> print sol['zs'][1]
- 5.5803e-02 0.0000e+00 0.0000e+00
- -2.4103e-03 1.0411e-04 0.0000e+00
- 2.4214e-02 -1.0459e-03 1.0507e-02
-\end{verbatim}
-Note that only the lower triangular parts of the dual variables are
-returned (in the example the returned values of the upper triangular
-elements happen to be zero, but this is not necessarily the case).
-
-Only the entries in \var{Gs} and \var{hs} that correspond to lower
-triangular entries need to be provided, so in the example \var{h} and
-\var{G} can also be defined as follows.
-\begin{verbatim}
->>> G = [ matrix([[-7., -11., 0., 3.],
- [ 7., -18., 0., 8.],
- [-2., -8., 0., 1.]]) ]
->>> G += [ matrix([[-21., -11., 0., 0., 10., 8., 0., 0., 5.],
- [ 0., 10., 16., 0., -10., -10., 0., 0., 3.],
- [ -5., 2., -17., 0., -6., 8., 0., 0., 6.]]) ]
->>> h = [ matrix([[33., -9.], [0., 26.]]) ]
->>> h += [ matrix([[14., 9., 40.], [0., 91., 10.], [0., 0., 15.]]) ]
-\end{verbatim}
-
-
-\section{Nonlinear Convex Programming}
-\begin{funcdesc}{cp}{F\optional{, G, h\optional{, A, b}}}
-Solves an optimization problem
+The functions in this chapter are intended for nonlinear convex
+optimization problems in the format
\BEQ \label{e-nlcp}
\begin{array}{ll}
\mbox{minimize} & f_0(x) \\
@@ -562,7 +12,22 @@ Solves an optimization problem
\end{array}
\EEQ
with $f=(f_0,\ldots,f_m)$ convex and twice differentiable.
-
+The inequalities are componentwise vector inequalities.
+
+The most important function in this chapter is \function{solvers.cp()},
+described in section~\ref{s-cp}. There are also functions for two
+special problem classes: quadratic programming (section~\ref{s-qp}) and
+geometric programming~(section~\ref{s-gp}).
+These solvers are all interfaces to a more general function
+\function{nlcp()}, which can also be called directly but requires
+user-provided functions for evaluating the constraints and for solving
+KKT equations. This allows the user to exploit certain types of
+problem structure (section~\ref{s-nlcp}).
+
+\section{General Solver} \label{s-cp}
+\begin{funcdesc}{cp}{F\optional{, G, h\optional{, A, b}}}
+Solves an optimization problem~(\ref{e-nlcp})
+with $f=(f_0,\ldots,f_m)$ convex and twice differentiable.
\var{F} is a function that evaluates the objective and nonlinear
constraint functions. It must handle the following calling sequences.
@@ -570,22 +35,22 @@ constraint functions. It must handle the following calling sequences.
\item \code{F()} returns a tuple (\var{m}, \var{x0}), where \var{m} is
the number of nonlinear constraints and \var{x0} is a point in the
domain of {\it f}. \var{x0} is a dense real matrix of size
- (\var{n},1).
+ ({\it n}, 1).
\item \code{F(x)}, with \var{x} a dense real matrix of size
- (\var{n},1), returns a tuple (\var{f}, \var{Df}).
- \var{f} is a dense real matrix of size (\var{m}+1,1), with
+ ({\it n},1), returns a tuple (\var{f}, \var{Df}).
+ \var{f} is a dense real matrix of size ({\it m}+1,1), with
\code{\var{f}[\var{k}]} equal to {\it f\_k(x)}.
(If {\it m} is zero, \var{f} can also be returned as a number.)
- \var{Df} is a dense or sparse real matrix of size (\var{m}+1,\var{n})
+ \var{Df} is a dense or sparse real matrix of size ({\it m}+1, {\it n})
with \code{\var{Df}[\var{k},:]} equal to the transpose of the gradient
of {\it f\_k} at {\it x}.
If \var{x} is not in the domain of {\it f}, \code{F(x)} returns
\None\ or a tuple (\None,\None).
\item \code{F(x,z)}, with \var{x} a dense real matrix of size
- (\var{n},1) and \var{z} a positive dense real matrix of size
- (\var{m}+1,1) returns a tuple (\var{f}, \var{Df}, \var{H}).
+ ({\it n},1) and \var{z} a positive dense real matrix of size
+ ({\it m}+1,1) returns a tuple (\var{f}, \var{Df}, \var{H}).
\var{f} and \var{Df} are defined as above.
\var{H} is a square dense or sparse real matrix of size
({\it n}, {\it n}), whose lower triangular part contains the lower
@@ -597,8 +62,8 @@ constraint functions. It must handle the following calling sequences.
\var{x} is in the domain of {\it f}.
\end{itemize}
-\var{G} and \var{A} are dense or sparse real matrices with \var{n}
-columns. Their default values are matrices of size (0,\var{n}).
+\var{G} and \var{A} are dense or sparse real matrices with {\it n}
+columns. Their default values are matrices of size (0, {\it n}).
\var{h} and \var{b} are dense real matrices with one column, and the
same number of rows as \var{G} and \var A, respectively.
Their default values are matrices of size (0,1).
@@ -637,17 +102,17 @@ The \code{'x'}, \code{'snl'}, \code{'sl'},
\code{'y'}, \code{'znl'} and \code{'zl'} entries are \None.
\end{description}
-%\function{cp()} requires that the problem is solvable and that the
-%Karush-Kuhn-Tucker matrix
-%\[
-%\left[\begin{array}{cccc}
-%\sum_{k=0}^m z_k \nabla^2 f_k(x) & D\tilde f(x)^T & G^T & A^T\\
-%D\tilde f(x) & -\diag(d_1) & 0 & 0 \\
-% G & 0 & -\diag(d_2) & 0 \\
-% A & 0 & 0 & 0 \end{array}\right]
-%%\]
-%is nonsingular for all {\it x}, all nonnegative {\it z}, and all
-%positive {\it d\_1}, {\it d\_2}.
+\function{cp()} requires that the problem is solvable and that the
+Karush-Kuhn-Tucker matrix
+\[
+\left[\begin{array}{cccc}
+\sum_{k=0}^m z_k \nabla^2 f_k(x) & D\tilde f(x)^T & G^T & A^T\\
+D\tilde f(x) & -\diag(d_1) & 0 & 0 \\
+ G & 0 & -\diag(d_2) & 0 \\
+ A & 0 & 0 & 0 \end{array}\right]
+\]
+is nonsingular for all {\it x}, all nonnegative {\it z}, and all positive
+$d_1$, $d_2$.
\end{funcdesc}
@@ -713,7 +178,7 @@ def robls(A, b, rho):
\item[Example: floor planning]
This example is the floor planning problem of section 8.8.2 in the book
-\citetitle[http://www.stanford.edu/\~{}boyd/cvxbook]{Convex Optimization}:
+\citetitle{http://www.stanford.edu/\~{}boyd/cvxbook}{Convex Optimization}:
\[
\begin{array}{ll}
\mbox{minimize} & W + H \\
@@ -877,317 +342,247 @@ pylab.show()
\end{center}
\end{description}
-\section{Exploiting Structure in LPs and SDPs}
-The solvers \function{lp()} and \function{sdp()} are interfaces to
-a common function \function{conelp()}, which can also be called
-directly. When calling \function{conelp()}, the user must provide
-functions for evaluating the constraint functions and for
-solving the linear equations (KKT equations) that are solved in
-each iteration of the algorithm.
-This is useful for LPs and SDPs that possess some interesting
-structure that makes it possible to solve the KKT equations fast.
-
-\begin{funcdesc}{conelp}{c, kktsolver\optional{, Gl, hl\optional{,
-Gs, hs\optional{, A, b\optional{, primalstart\optional{, dualstart}}}}}}
-Solves the pair of primal and dual SDPs~(\ref{e-sdp}).
-The arguments \var{c}, \var{hl}, \var{hs}, \var{b},
-\var{primalstart}, \var{dualstart} have the same meaning as in
-\function{sdp()}.
-The arguments \var{kktsolver}, \var{Gl}, \var{Gs}, \var{A} are
-functions that must handle the following calling sequences.
-
-\begin{itemize}
-\item \function{kktsolver}(\var{d}, \var{R}) with \var{d} a positive
-dense real matrix of size ({\it ml},{\it 1}), and
-\var{R} a list of {\it N} square dense real matrices
-\code{\var{R}[k]} of order {\it m\_k}, returns a function for
-solving the equation
-\BEAS
- A^T u_y + G_\mathrm{l}^T u_{z_\mathrm{l}} +
- G_\mathrm{s}^T(u_{z_\mathrm{s}}) & = & b_x \\
- A u_x & = & b_y \\
-G_\mathrm{l}u_x - \diag(d)^{-2} u_{z_\mathrm{l}} & = &
- b_{z_\mathrm{l}} \\
-G_\mathrm{s}(u_x) - R^{-T} R^{-1} u_{z_\mathrm{s}} R^{-T} R^{-1}
- & = & b_{z_\mathrm{s}}.
-\EEAS
-The function created by \samp{f = kktsolver(d, R)} will be
-called as \samp{f(bx, by, bzl, bzs)}.
-On entry, \var{bx}, \var{by}, \var{bzl} and \var{bzs} contain the
-righthand side. On exit, they should contain the solution of the KKT
-system, with {\it uzl} and {\it uzs} scaled:
+\section{Quadratic Programming} \label{s-qp}
+\begin{funcdesc}{qp}{P, q, \optional{, G, h \optional{, A, b\optional{,
+solver}}}}
+Solves a convex quadratic program
\[
- b_x := u_x, \qquad
- b_y := u_y, \qquad
- b_{z_\mathrm{l}} := \diag(d)^{-1} u_{z_\mathrm{l}}, \qquad
- b_{z_\mathrm{s}} := R^{-1} u_{z_\mathrm{s}} R^{-T}.
+\begin{array}{ll}
+\mbox{minimize} & (1/2) x^TPx + q^T x \\
+\mbox{subject to} & Gx \preceq h \\ & Ax = b.
+\end{array}
\]
-\item\function{Gl}(\var{x}, \var{y}\optional{,
-\var{alpha}=1.0\optional{, \var{beta}=0.0\optional{,
-\var{trans}=\code{'N'}}}}) evaluates the matrix-vector products
+\var{P} is a square dense or sparse real matrix, representing a
+symmetric matrix in \code{'L'} storage, \ie, only the lower
+triangular part of \var{P} is referenced.
+\var{G} and \var{A} are dense or sparse real matrices.
+Their default values are sparse matrices with zero columns.
+\var{q}, \var{h} and \var{b} are single-column real dense matrices.
+The default values of \var{h} and \var{b} are matrices of size (0,1).
+
+The default CVXOPT solver is used when the \var{solver} argument
+is absent or \None. The MOSEK solver (if installed) can be
+selected by setting \var{solver}=\code{'mosek'}.
+
+\function{qp()} returns a dictionary with keys
+\code{'status'}, \code{'x'}, \code{'s'}, \code{'y'}, \code{'z'}.
+The possible values of the \code{'status'} key are as follows.
+\begin{description}
+\item[\code{'optimal'}] In this case the
+\code{'x'} entry is the primal optimal solution,
+the \code{'s'} entry is the corresponding slack in the inequality
+constraints, the \code{'z'} and \code{'y'} entries are the optimal
+values of the dual variables associated with the linear inequality
+and linear equality constraints.
+These values (approximately) satisfy the optimality conditions
\[
-y := \alpha G_\mathrm{l}x + \beta y \quad
- (\mathrm{trans} = \mathrm{'N'}), \qquad
-y := \alpha G_\mathrm{l}^T x + \beta y \quad
- (\mathrm{trans} = \mathrm{'T'}).
+ Px + q + G^T z + A^T y = 0, \qquad Gx + s = h, \qquad
+ Ax = b, \qquad s \succeq 0, \qquad z \succeq 0, \qquad s^T z = 0.
\]
-\item \function{Gs}(\var{x}, \var{y}\optional{,
-\var{alpha}=1.0\optional{, \var{beta}=0.0\optional{,
-\var{trans}=\code{'N'}}}})
-evaluates the linear mappings
+\item [\code{'primal infeasible'}] This only applies when
+\var{solver} is \code{'mosek'}, and means that a certificate of
+primal infeasibility has been found. The \code{'x'} and \code{'s'}
+entries are \None, and the
+\code{'z'} and \code{'y'} entries are vectors that approximately satisfy
\[
-y := \alpha G_\mathrm{s}(x) + \beta y \quad
- (\mathrm{trans} = \mathrm{'N'}), \qquad
-y := \alpha G_\mathrm{s}^T(x) + \beta y \quad
- (\mathrm{trans} = \mathrm{'T'}).
+ G^Tz + A^T y = 0, \qquad h^Tz + b^Ty = -1, \qquad z \succeq 0.
\]
-\item \function{A}(\var{x}, \var{y}\optional{,
-\var{alpha}=1.0\optional{, \var{beta}=0.0\optional{,
-\var{trans}=\code{'N'}}}})
-evaluates the matrix vector products
+\item [\code{'dual infeasible'}] This only applies when
+\var{solver} is \code{'mosek'}, and means that a certificate of
+dual infeasibility has been found. The \code{'z'} and \code{'y'}
+entries are \None, and the \code{'x'} and \code{'s'} entries are
+vectors that approximately satisfy
\[
-y := \alpha Ax + \beta y \quad
- (\mathrm{trans} = \mathrm{'N'}), \qquad
-y := \alpha A^Tx + \beta y \quad
- (\mathrm{trans} = \mathrm{'T'}).
+ Px = 0, \qquad q^Tx = -1, \qquad Gx + s = 0, \qquad Ax=0, \qquad
+ s \succeq 0.
\]
-\end{itemize}
-\end{funcdesc}
-\begin{description}
-\item[Example: 1-norm approximation]
+\item[\code{'unknown'}] This means that the algorithm reached
+the maximum number of iterations before a solution was found.
+The \code{'x'}, \code{'s'}, \code{'y'}, \code{'z'} entries are \None.
+\end{description}
+\end{funcdesc}
-The optimization problem
-\[
- \begin{array}{ll}
- \mbox{minimize} & \|Pu-q\|_1
- \end{array}
-\]
-can be formulated as an LP
+As an example we compute the trade-off curve on page 187
+of the book \citetitle{http://www.stanford.edu/\~{}boyd/cvxbook}{Convex
+Optimization}, by solving the quadratic program
\[
- \begin{array}{ll}
- \mbox{minimize} & \ones^T v \\
- \mbox{subject to} & -v \preceq Pu - q \preceq v.
- \end{array}
+\begin{array}{ll}
+\mbox{minimize} & -\bar p^T x + \mu x^T S x \\
+\mbox{subject to} & \ones^T x = 1, \quad x \succeq 0
+\end{array}
\]
-By exploiting the structure in the inequalities, the cost of
-an iteration of an interior-point method can be reduced
-to the cost of least-squares problem of the same dimensions.
-(See section 11.8.2 in the book
-\citetitle[http://www.ee.ucla.edu/\~{}vandenbe/cvxbook]{Convex Optimization}.)
-The code belows taks advantage of this fact.
+for a sequence of positive values of {\it mu}.
+The code below computes the trade-off curve and produces two figures
+using the \ulink{Matplotlib}{http://matplotlib.sourceforge.net} package.
+\begin{center}
+\includegraphics[width=10cm]{figures/portfolio1.eps}
+\hspace*{\fill}
+\includegraphics[width=10cm]{figures/portfolio2.eps}
+\end{center}
\begin{verbatim}
-from cvxopt import base, blas, lapack, solvers
-from cvxopt.base import matrix, spmatrix, mul, div
-
-def l1(P, q):
- """
-
- Returns the solution u, w of the l1 approximation problem
-
- (primal) minimize ||P*u - q||_1
-
- (dual) maximize q'*w
- subject to P'*w = 0
- ||w||_infty <= 1.
- """
-
- m, n = P.size
-
- # Solve equivalent LP
- #
- # minimize [0; 1]' * [u; v]
- # subject to [P, -I; -P, -I] * [u; v] <= [q; -q]
- #
- # maximize -[q; -q]' * z
- # subject to [P', -P']*z = 0
- # [-I, -I]*z + 1 = 0
- # z >= 0
-
- c = matrix(n*[0.0] + m*[1.0])
- h = matrix([q, -q])
-
- def Fi(x, y, alpha=1.0, beta=0.0, trans='N'):
- if trans=='N':
- # y := alpha * [P, -I; -P, -I] * x + beta*y
- u = P*x[:n]
- y[:m] = alpha * ( u - x[n:]) + beta*y[:m]
- y[m:] = alpha * (-u - x[n:]) + beta*y[m:]
-
- else:
- # y := alpha * [P', -P'; -I, -I] * x + beta*y
- y[:n] = alpha * P.T * (x[:m] - x[m:]) + beta*y[:n]
- y[n:] = -alpha * (x[:m] + x[m:]) + beta*y[n:]
-
-
- def kktsolver(d, R):
-
- # Returns a function f(x,y,zl,zs) that solves
- #
- # [ 0 0 P' -P' ] [ x[:n] ] [ bx[:n] ]
- # [ 0 0 -I -I ] [ x[n:] ] [ bx[n:] ]
- # [ P -I -D1^{-1} 0 ] [ zl[:m]] = [ bzl[:m] ]
- # [-P -I 0 -D2^{-1} ] [ zl[m:]] [ bzl[m:] ]
- #
- # where D1 = diag(d[:m])^2, D2 = diag(d[m:])^2.
- #
- # On entry bx, bzl are stored in x, zl.
- # On exit x, zl contain the solution, with zl scaled: zl./d is
- # returned instead of zl.
-
- # Factor A = 4*P'*D*P where D = d1.*d2 ./(d1+d2) and
- # d1 = d[:m].^2, d2 = d[m:].^2.
-
- d1, d2 = d[:m]**2, d[m:]**2
- D = div( mul(d1,d2), d1+d2 )
- A = P.T * spmatrix(4*D, range(m), range(m)) * P
- lapack.potrf(A)
+from math import sqrt
+from cvxopt.base import matrix
+from cvxopt.blas import dot
+from cvxopt.solvers import qp
+import pylab
- def f(x, y, zl, zs):
+# Problem data.
+n = 4
+S = matrix([[ 4e-2, 6e-3, -4e-3, 0.0 ],
+ [ 6e-3, 1e-2, 0.0, 0.0 ],
+ [-4e-3, 0.0, 2.5e-3, 0.0 ],
+ [ 0.0, 0.0, 0.0, 0.0 ]])
+pbar = matrix([.12, .10, .07, .03])
+G = matrix(0.0, (n,n))
+G[::n+1] = -1.0
+h = matrix(0.0, (n,1))
+A = matrix(1.0, (1,n))
+b = matrix(1.0)
- # Solve for x[:n]:
- #
- # A*x[:n] = bx[:n] + P' * ( ((D1-D2)*(D1+D2)^{-1})*bx[n:]
- # + (2*D1*D2*(D1+D2)^{-1}) * (bzl[:m] - bzl[m:]) ).
- x[:n] += P.T * ( mul(div(d1-d2, d1+d2), x[n:]) + mul(2*D, zl[:m]-zl[m:]) )
- lapack.potrs(A, x)
+# Compute trade-off.
+N = 100
+mus = [ 10**(5.0*t/N-1.0) for t in xrange(N) ]
+portfolios = [ qp(mu*S, -pbar, G, h, A, b)['x'] for mu in mus ]
+returns = [ dot(pbar,x) for x in portfolios ]
+risks = [ sqrt(dot(x, S*x)) for x in portfolios ]
- # x[n:] := (D1+D2)^{-1} * (bx[n:] - D1*bzl[:m] - D2*bzl[m:] + (D1-D2)*P*x[:n])
- u = P*x[:n]
- x[n:] = div(x[n:] - mul(d1, zl[:m]) - mul(d2, zl[m:]) + mul(d1-d2, u), d1+d2)
+# Plot trade-off curve and optimal allocations.
+pylab.figure(1, facecolor='w')
+pylab.plot(risks, returns)
+pylab.xlabel('standard deviation')
+pylab.ylabel('expected return')
+pylab.axis([0, 0.2, 0, 0.15])
+pylab.title('Risk-return trade-off curve (fig 4.12)')
+pylab.yticks([0.00, 0.05, 0.10, 0.15])
- # z[:m] := d1[:m] .* ( P*x[:n] - x[n:] - bzl[:m])
- # z[m:] := d2[m:] .* (-P*x[:n] - x[n:] - bzl[m:])
- zl[:m] = mul(d[:m], u-x[n:]-zl[:m])
- zl[m:] = mul(d[m:], -u-x[n:]-zl[m:])
+pylab.figure(2, facecolor='w')
+c1 = [ x[0] for x in portfolios ]
+c2 = [ x[0] + x[1] for x in portfolios ]
+c3 = [ x[0] + x[1] + x[2] for x in portfolios ]
+c4 = [ x[0] + x[1] + x[2] + x[3] for x in portfolios ]
+pylab.fill(risks + [.20], c1 + [0.0], '#F0F0F0')
+pylab.fill(risks[-1::-1] + risks, c2[-1::-1] + c1, '#D0D0D0')
+pylab.fill(risks[-1::-1] + risks, c3[-1::-1] + c2, '#F0F0F0')
+pylab.fill(risks[-1::-1] + risks, c4[-1::-1] + c3, '#D0D0D0')
+pylab.axis([0.0, 0.2, 0.0, 1.0])
+pylab.xlabel('standard deviation')
+pylab.ylabel('allocation')
+pylab.text(.15,.5,'x1')
+pylab.text(.10,.7,'x2')
+pylab.text(.05,.7,'x3')
+pylab.text(.01,.7,'x4')
+pylab.title('Optimal allocations (fig 4.12)')
+pylab.show()
+\end{verbatim}
- return f
+\section{Geometric Programming} \label{s-gp}
+\begin{funcdesc}{gp}{K, F, g \optional{, G, h \optional{, A, b}}}
+Solves a geometric program in convex form
+\[
+\begin{array}{ll}
+\mbox{minimize} & f_0(x) = \lse(F_0x+g_0) \\
+\mbox{subject to} & f_i(x) = \lse(F_ix+g_i) \leq 0,\quad i=1,\ldots,m \\
+ & Gx \preceq h \\
+ & Ax=b
+\end{array}
+\]
+where
+\[
+ \lse(u) = \log \sum_k \exp(u_k), \qquad
+ F = \left[ \begin{array}{cccc}
+ F_0^T & F_1^T & \cdots & F_m^T \end{array}\right]^T, \qquad
+ g = \left[ \begin{array}{cccc}
+ g_0^T & g_1^T & \cdots & g_m^T \end{array}\right]^T.
+\]
+\var{K} is a list of {\it m}+1 positive integers with
+\code{\var{K}[\var i]}
+equal to the number of rows in {\it F\_i}.
+\var{F} is a dense or sparse real matrix of size
+\code{(sum(\var K),\var n)}.
+\var{g} is a dense real matrix with one column and the same number of
+rows as \var{F}.
+\var{G} and \var{A} are dense or sparse real matrices.
+Their default values are sparse matrices with zero rows.
+\var{h} and \var{b} are dense real matrices with one column.
+Their default values are matrices of size (0,1).
- sol = solvers.conelp(c, kktsolver, Gl=Fi, hl=h)
- return sol['x'][:n], sol['zl'][m:] - sol['zl'][:m]
-\end{verbatim}
+\function{gp()} returns a dictionary with keys
+\code{'status'}, \code{'x'}, \code{'snl'}, \code{'sl'},
+\code{'y'}, \code{'znl'} and \code{'zl'}.
+The possible values of the \code{'status'} key are:
+\begin{description}
+\item[\code{'optimal'}] In this case the
+\code{'x'} entry is the primal optimal solution,
+the \code{'snl'} and \code{'sl'} entries are the corresponding slacks
+in the nonlinear and linear inequality constraints.
+The \code{'znl'}, \code{'zl'} and \code{'y'} entries are the optimal
+values of the dual variables associated with the nonlinear and linear
+inequality constraints and the linear equality constraints.
+These values approximately satisfy
+\[
+ \nabla f_0(x) + \sum_{k=1}^m z_{\mathrm{nl},k}
+ \nabla f_k(x) + G^T z_\mathrm{l} + A^T y = 0, \qquad
+ f_k(x) + s_{\mathrm{nl},k} = 0, \quad k=1,\ldots,m, \qquad
+ Gx + s_\mathrm{l} = h, \qquad Ax=b
+\]
+and
+\[
+s_\mathrm{nl}\succeq 0, \qquad s_\mathrm{l}\succeq 0, \qquad
+z_\mathrm{nl} \succeq 0, \qquad z_\mathrm{l} \succeq 0, \qquad
+s_\mathrm{nl}^T z_\mathrm{nl} + s_\mathrm{l}^T z_\mathrm{l} = 0.
+\]
-\item[Example: SDP with diagonal linear term]
+\item[\code{'unknown'}] This means that the algorithm reached
+the maximum number of iterations before a solution was found.
+The \code{'x'}, \code{'snl'}, \code{'sl'}, \code{'y'}, \code{'znl'}
+and \code{'zl'} entries are \None.
+\end{description}
+\end{funcdesc}
-The SDP
+As an example, we solve the small GP of section 2.4 of the paper
+\citetitle{http://www.stanford.edu/\~{}boyd/gp\_tutorial}{A Tutorial on Geometric Programming}.
+The posynomial form of the problem is
\[
\begin{array}{ll}
- \mbox{minimize} & \ones^T x \\
- \mbox{subject to} & W + \diag(x) \succeq 0
- \end{array}
+ \mbox{minimize} & w^{-1} h^{-1} d^{-1} \\
+ \mbox{subject to}
+ & (2/A_\mathrm{wall}) hw + (2/A_\mathrm{wall})hd \leq 1 \\
+ & (1/A_\mathrm{flr}) wd \leq 1 \\
+ & \alpha wh^{-1} \leq 1 \\
+ & (1/\beta) hw^{-1} \leq 1 \\
+ & \gamma wd^{-1} \leq 1 \\
+ & (1/\delta)dw^{-1} \leq 1
+ \end{array}
\]
-can be solved efficiently by exploiting properties of the diag operator.
+with variables {\it h}, {\it w}, {\it d}.
\begin{verbatim}
-from cvxopt import base, blas, lapack, solvers
-from cvxopt.base import matrix
-
-def mcsdp(w):
- """
- Returns solution x, z to
-
- (primal) minimize sum(x)
- subject to w + diag(x) >= 0
-
- (dual) maximize -tr(w*z)
- subject to diag(z) = 1
- z >= 0.
- """
-
- n = w.size[0]
-
- def Fs(x, y, alpha=1.0, beta=0.0, trans='N'):
- """
- y := alpha*(-diag(x)) + beta*y.
- """
- if trans=='N':
- # x is a vector; y[0] is a matrix.
- y[0] *= beta
- y[0][::n+1] -= alpha * x
- else:
- # x[0] is a matrix; y is a vector.
- y *= beta
- y -= alpha * x[::n+1]
-
-
- def cngrnc(r, x, alpha=1.0):
- """
- Congruence transformation
-
- x := alpha * r'*x*r.
-
- r and x are square 'd' matrices.
- """
-
- # Scale diagonal of x by 1/2.
- x[::n+1] *= 0.5
-
- # a := tril(x)*r
- a = +r
- blas.trmm(x, a, side='L')
-
- # x := alpha*(a*r' + r*a')
- blas.syr2k(r, a, x, trans='T', alpha=alpha)
-
-
- def kktsolver(d, r):
-
- # t = r*r' as a nonsymmetric matrix.
- t = matrix(0.0, (n,n))
- blas.gemm(r[0], r[0], t, transB='T')
-
- # Cholesky factorization of tsq = t.*t.
- tsq = t**2
- lapack.potrf(tsq)
-
- def f(x, y, zl, zs):
- """
- Solve
- -diag(zs) = bx
- -diag(x) - inv(r*r')*zs*inv(r*r') = bs.
-
- On entry, x and zs contain bx and bs.
- On exit, they contain the solution, with zs scaled
- (inv(r)'*zs*inv(r) is returned instead of zs).
-
- We solve
-
- ((r*r') .* (r*r')) * x = bx - diag(t*bs*t)
-
- and take zs = -r' * (diag(x) + bs) * r.
- """
-
- # tbst := t * zs * t = t * bs * t
- tbst = +zs[0]
- cngrnc(t, tbst)
-
- # x := x - diag(tbst) = bx - diag(r*r' * bs * r*r')
- x -= tbst[::n+1]
-
- # x := (t.*t)^{-1} * x = (t.*t)^{-1} * (bx - diag(t*bs*t))
- lapack.potrs(tsq, x)
-
- # zs := zs + diag(x) = bs + diag(x)
- zs[0][::n+1] += x
-
- # zs := -r' * zs * r = -r' * (diag(x) + bs) * r
- cngrnc(r[0], zs[0], alpha=-1.0)
+from cvxopt.base import matrix, log, exp
+from cvxopt import solvers
- return f
+Aflr = 1000.0
+Awall = 100.0
+alpha = 0.5
+beta = 2.0
+gamma = 0.5
+delta = 2.0
- c = matrix(1.0, (n,1))
- sol = solvers.conelp(c, kktsolver, Gs=Fs, hs=[w])
- return sol['x'], sol['zs'][0]
+F = matrix( [[-1., 1., 1., 0., -1., 1., 0., 0.],
+ [-1., 1., 0., 1., 1., -1., 1., -1.],
+ [-1., 0., 1., 1., 0., 0., -1., 1.]])
+g = log( matrix( [1.0, 2/Awall, 2/Awall, 1/Aflr, alpha, 1/beta, gamma, 1/delta]) )
+K = [1, 2, 1, 1, 1, 1, 1]
+h, w, d = exp( solvers.gp(K, F, g)['x'] )
\end{verbatim}
-\end{description}
-\section{Exploiting Structure in Nonlinear Convex Programs}
-The solvers \function{gp()}, \function{gp()} and \function{cp()} are
+\section{Exploiting Structure} \label{s-nlcp}
+The solvers \function{cp()}, \function{qp()} and \function{gp()} are
interfaces to \function{nlcp()}, which can also be called directly
but requires user-provided functions for evaluating the constraint
and for solving the KKT equations.
@@ -1405,34 +800,13 @@ def kktsolver(x, z, dnl, dl):
x = solvers.nlcp(kktsolver, F, G, h)['x'][:n]
\end{verbatim}
-\section{Optional Solvers} \label{s-external}
-CVXOPT includes optional interfaces to several other optimization
-libraries.
-
-\begin{description}
-\item[GLPK] \function{lp()} with the \code{solver='glpk'} option uses
-the the simplex algorithm in
-\ulink{GLPK (GNU Linear Programming Kit)}{http://www.gnu.org/software/glpk/glpk.html}.
-
-\item[MOSEK] \function{lp()} and \function{qp()} with the
-\code{solver='mosek'} option call routines from
-\ulink{MOSEK}{http://www.mosek.com} version 4.
-
-\item[DSDP] \function{sdp()} with the \code{solver='dsdp'} option uses
-the
-\ulink{DSDP5.8}{http://www-unix.mcs.anl.gov/DSDP} solver.
-\end{description}
-GLPK, MOSEK and DSDP are not included in the CVXOPT distribution and
-need to be installed separately.
\section{Algorithm Parameters} \label{s-parameters}
-In this section we list some algorithm control parameters that can
-be modified without editing the source code.
-These control parameters are accessible via the dictionary
-\member{solvers.options}. By default the dictionary
+The following algorithm control parameters are accessible via the
+dictionary \member{solvers.options}. By default the dictionary
is empty and the default values of the parameters are used.
-One can change the parameters in the \textbf{default} solvers by
+One can change the parameters in the default solvers by
adding entries with the following key values.
\begin{description}
\item[\code{'show\_progress'}]
@@ -1443,10 +817,6 @@ adding entries with the following key values.
\item[\code{'reltol'}] relative accuracy (default: \code{1e-7}).
\item[\code{'feastol'}] tolerance for feasibility conditions (default:
\code{1e-7}).
-\item[\code{'refinement'}] \True\ or \False. If \True,
-one step of iterative refinement is applied after solving KKT equations
-in \function{conelp()}, \function{lp()}, and \function{sdp()}
-(default: \True).
\end{description}
For example the command
\begin{verbatim}
@@ -1455,58 +825,7 @@ For example the command
\end{verbatim}
turns off the screen output during calls to the solvers.
The tolerances \var{abstol}, \var{reltol} and \var{feastol} have the
-following meaning. \function{conelp()} terminates with
-status \code{'optimal'}, if
-\[
-s_\mathrm{l} \succeq 0, \qquad S_\mathrm{s} \succeq 0,
-\qquad
- \frac{\|G_\mathrm{l}x+s_\mathrm{l}-h_\mathrm{l}\|_2}
- {\max\{1,\|h_\mathrm{l}\|_2\}} \leq \epsilon_\mathrm{feas},
-\qquad
-\frac{\|G_\mathrm{s}(x)+S_\mathrm{s}-H_\mathrm{s}\|_F}
-{\max\{1,\|H_\mathrm{s}\|_F\}} \leq \epsilon_\mathrm{feas},
-\qquad
-\frac{\|Ax-b\|_2}{\max\{1,\|b\|_2\}} \leq \epsilon_\mathrm{feas},
-\]
-\[
-z_\mathrm{l} \succeq 0, \qquad Z_\mathrm{s} \succeq 0, \qquad
-\frac{\|G_\mathrm{l}^Tz_\mathrm{l}+
-G_\mathrm{s}^T(Z_\mathrm{s}) + A^Ty+c\|_2}{\max\{1,\|c\|_2\}}
-\leq \epsilon_\mathrm{feas},
-\]
-and
-\[
- s_\mathrm{l}^T z_\mathrm{l} + \Tr(S_\mathrm{s}Z_\mathrm{s}) \leq
- \epsilon_\mathrm{abs} \qquad \mbox{or} \qquad
-\left( \min\left\{c^Tx,
- h_\mathrm{l}^T z_\mathrm{l} + \Tr(H_\mathrm{s} Z_\mathrm{s})
- + b^Ty \right\} < 0, \quad
- \frac{s_\mathrm{l}^Tz_\mathrm{l} + \Tr(S_\mathrm{s}Z_\mathrm{s})}
- {-\min\{c^Tx, h_\mathrm{l}^Tz_\mathrm{l} +
- \Tr(H_\mathrm{s} Z_\mathrm{s}) + b^T y\}} \leq \epsilon_\mathrm{rel}
-\right).
-\]
-It returns with status \code{'primal infeasible'} if
-\[
-z_\mathrm{l} \succeq 0, \qquad
-Z_\mathrm{s} \succeq 0, \qquad
-\qquad \|G_\mathrm{l}^Tz_\mathrm{l} + G_\mathrm{s}^T(Z_\mathrm{s})
- +A^Ty\|_2 \leq \epsilon_\mathrm{feas},
- \qquad h_\mathrm{l}^Tz_\mathrm{l} + \Tr(H_\mathrm{s} Z_\mathrm{s})
- +b^Ty = -1.
-\]
-It returns with status \code{'dual infeasible'} if
-\[
-s_\mathrm{l} \succeq 0, \qquad
-S_\mathrm{s} \succeq 0, \qquad
-\qquad
-\|G_\mathrm{l}x+s_\mathrm{l}\|_2 \leq \epsilon_\mathrm{feas}, \qquad
-\|G_\mathrm{s}(x)+S_\mathrm{s}\|_F \leq \epsilon_\mathrm{feas}, \qquad
-\|Ax\|_2 \leq \epsilon_\mathrm{feas}, \qquad
-c^Tx = -1.
-\]
-The functions \function{lp()} and \function{sdp()} call
-\function{conelp()} and hence use the same stopping criteria.
+following meaning in \function{nlcp()}.
\function{nlcp()} returns with status \code{'optimal'} if
\[
@@ -1544,19 +863,7 @@ The functions \function{qp()}, \function{gp()} and \function{cp()}
call \function{nlcp()} and hence use the same stopping criteria
(with {\it x0}=0 for \function{qp()} and \function{gp()}).
-The control parameters listed in the \textbf{GLPK} documentation are
-set to their default values and can also be customized by making
-an entry in \member{solvers.options}.
-The keys in the dictionary are strings with the name of the GLPK
-parameter. The command
-\begin{verbatim}
->>> from cvxopt import solvers
->>> solvers.options['LPX_K_MSGLEV'] = 0
-\end{verbatim}
-turns off the screen output subsequent calls \function{lp()} with
-the \code{'glpk'} option.
-
-The \textbf{MOSEK} \ulink{control parameters}{http://www.mosek.com/products/3/tools/doc/html/tools/node22.html}
+The MOSEK \ulink{control parameters}{http://www.mosek.com/fileadmin/products/3/tools/doc/html/tools/node22.html}
are set to their default values.
The corresponding keys in \code{solvers.options} are strings with the
name of the MOSEK parameter. For example the command
@@ -1564,15 +871,5 @@ name of the MOSEK parameter. For example the command
>>> from cvxopt import solvers
>>> solvers.options['MSK_IPAR_LOG'] = 0
\end{verbatim}
-turns off the screen output during calls of \function{lp()}
-and \function{qp()} with the \code{'mosek'} option.
-
-The following control parameters affect the \textbf{DSDP} algorithm:
-\begin{description}
-\item[\code{'DSDP\_Monitor'}] the interval (in number of iterations)
- at which output is printed to the screen
-(default: 0).
-\item[\code{'DSDP\_MaxIts'}] maximum number of iterations.
-\item[\code{'DSDP\_GapTolerance'}] relative accuracy (default:
-\code{1e-5}).
-\end{description}
+turns off the screen output during calls of \function{qp()} with the
+\code{'mosek'} option.
diff --git a/doc/spsolvers.tex b/doc/spsolvers.tex
index 56b556f..827998a 100644
--- a/doc/spsolvers.tex
+++ b/doc/spsolvers.tex
@@ -210,8 +210,8 @@ routines of the CHOLMOD package.
It includes functions for Cholesky factorization of sparse positive
definite matrices, and for solving sparse sets of linear equations with
positive definite matrices.
-The routines can also be used for computing LDL$\mathrm{{}^T}$
-(or LDL$\mathrm{{}^H}$) factorizations of symmetric indefinite matrices
+The routines can also be used for computing {LDL$\mathrm{{}^T}$}
+(or {LDL$\mathrm{{}^H}$}) factorizations of symmetric indefinite matrices
(with L unit lower-triangular and D diagonal and nonsingular) if
such a factorization exists.
@@ -300,7 +300,7 @@ and
where {\it P} is a permutation matrix, {\it L} is lower triangular
(unit lower triangular in the second factorization), and
{\it D} is nonsingular diagonal. The type of factorization depends
-on the value of \code{options['supernodal']}.
+on the value of \code{options['supernodal']} (see below).
If \var{uplo} is \code{'L'}, only the lower triangular part of \var{A}
is accessed and the upper triangular part is ignored.
@@ -413,7 +413,6 @@ of the coefficient matrix in~(\ref{e-A-pd}) by two methods.
>>> import math
>>> from cvxopt.cholmod import options
>>> from cvxopt.base import log
->>> options['supernodal'] = 2
>>> F = cholmod.symbolic(A)
>>> cholmod.numeric(A,F)
>>> print 2.0 * sum(log(cholmod.diag(F)))
diff --git a/examples/book/README b/examples/book/README
index 6ca767c..9068854 100644
--- a/examples/book/README
+++ b/examples/book/README
@@ -4,4 +4,4 @@ www.stanford.edu/~boyd/cvxbook).
The scripts require the plotting library Matplotlib, available at
matplotlib.sourceforge.net. They were tested with Matplotlib
-version 0.83.2-2.
+version 0.87.7-0.3.
diff --git a/examples/book/chap4/rls b/examples/book/chap4/rls
index 50ddc05..b9a1e35 100755
--- a/examples/book/chap4/rls
+++ b/examples/book/chap4/rls
@@ -10,7 +10,7 @@ from cvxopt.base import matrix
from cvxopt import solvers
solvers.options['show_progress'] = 0
-data = load(open("rls.pic",'r'))
+data = load(open("rls.bin",'r'))
A, b = data['A'], data['b']
m, n = A.size
@@ -61,14 +61,14 @@ for alpha in alpha2[1:]:
ubnds += [ blas.dot(c, solvers.sdp(c, Gs=[G], hs=[-h])['x']) ]
pylab.figure(1, facecolor='w')
-pylab.plot(lbnds, alpha1, '-', ubnds, alpha2, '-')
+pylab.plot(lbnds, alpha1, 'b-', ubnds, alpha2, 'b-')
kmax = max([ k for k in xrange(len(alpha1)) if alpha1[k] <
blas.nrm2(xls)**2 ])
pylab.plot( [ blas.nrm2(b)**2 ] + lbnds[:kmax] +
[ blas.nrm2(A*xls-b)**2 ], [0.0] + alpha1[:kmax] +
[ blas.nrm2(xls)**2 ], '-', linewidth=2)
pylab.plot([ blas.nrm2(b)**2, blas.nrm2(A*xls-b)**2 ],
- [0.0, blas.nrm2(xls)**2], 'o')
+ [0.0, blas.nrm2(xls)**2], 'bo')
pylab.fill(lbnds[-1::-1] + ubnds + [ubnds[-1]],
alpha1[-1::-1] + alpha2+ [alpha1[-1]], '#D0D0D0')
pylab.axis([0, 15, -1.0, 15])
diff --git a/examples/book/chap4/rls.pic b/examples/book/chap4/rls.bin
similarity index 100%
rename from examples/book/chap4/rls.pic
rename to examples/book/chap4/rls.bin
diff --git a/examples/book/chap6/consumerpref b/examples/book/chap6/consumerpref
index fc404b4..988f63d 100755
--- a/examples/book/chap6/consumerpref
+++ b/examples/book/chap6/consumerpref
@@ -59,15 +59,15 @@ m = B.size[1]
nopts = 200
a = (1.0/nopts)*matrix(range(nopts), tc='d')
X, Y = a[:,nopts*[0]].T, a[:,nopts*[0]]
-pylab.figure(1, facecolor='w')
-pylab.plot(B[0,:], B[1,:], 'wo', markeredgecolor='b')
-pylab.contour(pylab.array(X), pylab.array(Y), pylab.array(utility(X,Y)),
- [.1*(k+1) for k in xrange(9)], colors='k')
-pylab.xlabel('x1')
-pylab.ylabel('x2')
-pylab.title('Goods baskets and utility function (fig. 6.25)')
-print "Close figure to start analysis."
-pylab.show()
+#pylab.figure(1, facecolor='w')
+#pylab.plot(B[0,:], B[1,:], 'wo', markeredgecolor='b')
+#pylab.contour(pylab.array(X), pylab.array(Y), pylab.array(utility(X,Y)),
+# [.1*(k+1) for k in xrange(9)], colors='k')
+#pylab.xlabel('x1')
+#pylab.ylabel('x2')
+#pylab.title('Goods baskets and utility function (fig. 6.25)')
+#print "Close figure to start analysis."
+#pylab.show()
# P are basket indices in order of increasing preference
diff --git a/examples/book/chap6/cvxfit b/examples/book/chap6/cvxfit
index f649289..6220be2 100755
--- a/examples/book/chap6/cvxfit
+++ b/examples/book/chap6/cvxfit
@@ -9,7 +9,7 @@ from cvxopt.base import matrix, spmatrix, mul
from pickle import load
solvers.options['show_progress'] = 0
-data = load(open('cvxfit.pic','r'))
+data = load(open('cvxfit.bin','r'))
u, y = data['u'], data['y']
m = len(u)
diff --git a/examples/book/chap6/cvxfit.pic b/examples/book/chap6/cvxfit.bin
similarity index 100%
rename from examples/book/chap6/cvxfit.pic
rename to examples/book/chap6/cvxfit.bin
diff --git a/examples/book/chap6/huber b/examples/book/chap6/huber
index 14d0b3b..d9a2daf 100755
--- a/examples/book/chap6/huber
+++ b/examples/book/chap6/huber
@@ -9,7 +9,7 @@ import pylab
from pickle import load
solvers.options['show_progress'] = 0
-data = load(open('huber.pic','r'))
+data = load(open('huber.bin','r'))
u, v = data['u'], data['v']
m, n = len(u), 2
diff --git a/examples/book/chap6/huber.pic b/examples/book/chap6/huber.bin
similarity index 100%
rename from examples/book/chap6/huber.pic
rename to examples/book/chap6/huber.bin
diff --git a/examples/book/chap6/old/README b/examples/book/chap6/old/README
new file mode 100644
index 0000000..7df549e
--- /dev/null
+++ b/examples/book/chap6/old/README
@@ -0,0 +1,2 @@
+tv and smoothrec use cholmod
+tv_new and smoothrec_new use lapack band solvers
diff --git a/examples/book/chap6/smoothrec b/examples/book/chap6/old/smoothrec
similarity index 80%
copy from examples/book/chap6/smoothrec
copy to examples/book/chap6/old/smoothrec
index ddca231..18b4b9a 100755
--- a/examples/book/chap6/smoothrec
+++ b/examples/book/chap6/old/smoothrec
@@ -4,7 +4,7 @@
# Quadratic smoothing.
from math import pi
-from cvxopt import random, blas, lapack
+from cvxopt import random, blas, lapack, cholmod
from cvxopt.base import matrix, spmatrix, sin, mul
import pylab
@@ -27,28 +27,35 @@ pylab.xlabel('i')
# A = D'*D is an n by n tridiagonal matrix with -1.0 on the
# upper/lower diagonal and 1, 2, 2, ..., 2, 2, 1 on the diagonal.
-Ad = matrix([1.0] + (n-2)*[2.0] + [1.0])
-As = matrix(-1.0, (n-1,1))
+A = spmatrix( (n-1)*[-1.0] + [1.0] + (n-2)*[2.0] + [1.0],
+ range(1,n) + [0] + range(1,n-1) + [n-1],
+ range(n-1) + [0] + range(1,n-1) + [n-1])
+I = spmatrix(1.0, range(n), range(n))
+F = cholmod.symbolic(A)
nopts = 50
deltas = -10.0 + 20.0/(nopts-1) * matrix(range(nopts))
cost1, cost2 = [], []
for delta in deltas:
+ cholmod.numeric(I + 10**delta * A, F )
xr = +corr
- lapack.ptsv(1.0 + 10**delta * Ad, 10**delta *As, xr)
+ cholmod.solve(F, xr)
cost1 += [blas.nrm2(xr - corr)]
cost2 += [blas.nrm2(xr[1:] - xr[:-1])]
# Find solutions with ||xhat - xcorr || roughly equal to 8.0, 3.1, 1.0.
mv1, k1 = min(zip([abs(c - 8.0) for c in cost1], range(nopts)))
+cholmod.numeric(I + 10**deltas[k1] * A, F )
xr1 = +corr
-lapack.ptsv(1.0 + 10**deltas[k1] * Ad, 10**deltas[k1] *As, xr1)
+cholmod.solve(F, xr1)
mv2, k2 = min(zip([abs(c - 3.1) for c in cost1], range(nopts)))
+cholmod.numeric(I + 10**deltas[k2] * A, F )
xr2 = +corr
-lapack.ptsv(1.0 + 10**deltas[k2] * Ad, 10**deltas[k2] *As, xr2)
+cholmod.solve(F, xr2)
mv3, k3 = min(zip([abs(c - 1.0) for c in cost1], range(nopts)))
+cholmod.numeric(I + 10**deltas[k3] * A, F )
xr3 = +corr
-lapack.ptsv(1.0 + 10**deltas[k3] * Ad, 10**deltas[k3] *As, xr3)
+cholmod.solve(F, xr3)
pylab.figure(2, facecolor='w')
pylab.plot(cost1, cost2, [blas.nrm2(corr)], [0], 'o',
diff --git a/examples/book/chap6/smoothrec b/examples/book/chap6/old/smoothrec_new
similarity index 71%
copy from examples/book/chap6/smoothrec
copy to examples/book/chap6/old/smoothrec_new
index ddca231..87a9213 100755
--- a/examples/book/chap6/smoothrec
+++ b/examples/book/chap6/old/smoothrec_new
@@ -3,6 +3,8 @@
# Figures 6.8-10, pages 313-314
# Quadratic smoothing.
+### Same as smoothrec, but uses LAPACK solvers for band matrices.
+
from math import pi
from cvxopt import random, blas, lapack
from cvxopt.base import matrix, spmatrix, sin, mul
@@ -27,28 +29,44 @@ pylab.xlabel('i')
# A = D'*D is an n by n tridiagonal matrix with -1.0 on the
# upper/lower diagonal and 1, 2, 2, ..., 2, 2, 1 on the diagonal.
-Ad = matrix([1.0] + (n-2)*[2.0] + [1.0])
-As = matrix(-1.0, (n-1,1))
+## A = spmatrix( (n-1)*[-1.0] + [1.0] + (n-2)*[2.0] + [1.0],
+## range(1,n) + [0] + range(1,n-1) + [n-1],
+## range(n-1) + [0] + range(1,n-1) + [n-1])
+## I = spmatrix(1.0, range(n), range(n))
+## F = cholmod.symbolic(A)
+
+A = matrix(0.0, (2,n))
+A[0,:] = [1.0] + (n-2)*[2.0] + [1.0]
+A[1,:n-1] = -1.0
+I = matrix(n*[[1.0, 0.0]])
nopts = 50
deltas = -10.0 + 20.0/(nopts-1) * matrix(range(nopts))
cost1, cost2 = [], []
for delta in deltas:
+## cholmod.numeric(I + 10**delta * A, F )
xr = +corr
- lapack.ptsv(1.0 + 10**delta * Ad, 10**delta *As, xr)
+## cholmod.solve(F, xr)
+ lapack.pbsv(I + 10**delta * A, xr)
cost1 += [blas.nrm2(xr - corr)]
cost2 += [blas.nrm2(xr[1:] - xr[:-1])]
# Find solutions with ||xhat - xcorr || roughly equal to 8.0, 3.1, 1.0.
mv1, k1 = min(zip([abs(c - 8.0) for c in cost1], range(nopts)))
+## cholmod.numeric(I + 10**deltas[k1] * A, F )
xr1 = +corr
-lapack.ptsv(1.0 + 10**deltas[k1] * Ad, 10**deltas[k1] *As, xr1)
+## cholmod.solve(F, xr1)
+lapack.pbsv(I + 10**deltas[k1] * A, xr1)
mv2, k2 = min(zip([abs(c - 3.1) for c in cost1], range(nopts)))
+## cholmod.numeric(I + 10**deltas[k2] * A, F )
xr2 = +corr
-lapack.ptsv(1.0 + 10**deltas[k2] * Ad, 10**deltas[k2] *As, xr2)
+## cholmod.solve(F, xr2)
+lapack.pbsv(I + 10**deltas[k2] * A, xr2)
mv3, k3 = min(zip([abs(c - 1.0) for c in cost1], range(nopts)))
+## cholmod.numeric(I + 10**deltas[k3] * A, F )
xr3 = +corr
-lapack.ptsv(1.0 + 10**deltas[k3] * Ad, 10**deltas[k3] *As, xr3)
+## cholmod.solve(F, xr3)
+lapack.pbsv(I + 10**deltas[k3] * A, xr3)
pylab.figure(2, facecolor='w')
pylab.plot(cost1, cost2, [blas.nrm2(corr)], [0], 'o',
diff --git a/examples/book/chap6/tv b/examples/book/chap6/old/tv
similarity index 89%
copy from examples/book/chap6/tv
copy to examples/book/chap6/old/tv
index 0b4948f..335a177 100755
--- a/examples/book/chap6/tv
+++ b/examples/book/chap6/old/tv
@@ -4,7 +4,7 @@
# Total variation reconstruction.
from math import pi
-from cvxopt import random, blas, lapack, solvers
+from cvxopt import random, blas, lapack, cholmod, solvers
from cvxopt.base import matrix, spmatrix, sin, mul, div
solvers.options['show_progress'] = 0
import pylab
@@ -32,29 +32,36 @@ pylab.axis([0, 2000, -2, 2])
# Quadratic smoothing.
# A = D'*D is an n by n tridiagonal matrix with -1.0 on the
# upper/lower diagonal and 1, 2, 2, ..., 2, 2, 1 on the diagonal.
-Ad = matrix([1.0] + (n-2)*[2.0] + [1.0])
-As = matrix(-1.0, (n-1,1))
+A = spmatrix( (n-1)*[-1.0] + [1.0] + (n-2)*[2.0] + [1.0],
+ range(1,n) + [0] + range(1,n-1) + [n-1],
+ range(n-1) + [0] + range(1,n-1) + [n-1])
+I = spmatrix(1.0, range(n), range(n))
+F = cholmod.symbolic(A)
nopts = 100
deltas = -10.0 + 20.0/(nopts-1) * matrix(range(nopts))
cost1, cost2 = [], []
for delta in deltas:
+ cholmod.numeric(I + 10**delta * A, F )
xr = +corr
- lapack.ptsv(1.0 + 10**delta * Ad, 10**delta * As, xr)
+ cholmod.solve(F, xr)
cost1 += [blas.nrm2(xr - corr)]
cost2 += [blas.nrm2(xr[1:] - xr[:-1])]
# Find solutions with ||xhat - xcorr || roughly equal to 4, 7, 10.
mv1, k1 = min(zip([abs(c - 10.0) for c in cost1], range(nopts)))
+cholmod.numeric(I + 10**deltas[k1] * A, F )
xr1 = +corr
-lapack.ptsv(1.0 + 10**deltas[k1] * Ad, 10**deltas[k1] * As, xr1)
+cholmod.solve(F, xr1)
mv2, k2 = min(zip([abs(c - 7.0) for c in cost1], range(nopts)))
+cholmod.numeric(I + 10**deltas[k2] * A, F )
xr2 = +corr
-lapack.ptsv(1.0 + 10**deltas[k2] * Ad, 10**deltas[k2] * As, xr2)
+cholmod.solve(F, xr2)
mv3, k3 = min(zip([abs(c - 4.0) for c in cost1], range(nopts)))
+cholmod.numeric(I + 10**deltas[k3] * A, F )
xr3 = +corr
-lapack.ptsv(1.0 + 10**deltas[k3] * Ad, 10**deltas[k3] * As, xr3)
+cholmod.solve(F, xr3)
pylab.figure(2, facecolor='w')
pylab.plot(cost1, cost2, [blas.nrm2(corr)], [0], 'o',
@@ -152,9 +159,10 @@ def tv(delta):
# [ D -I -D1 0 ]
# [ -D -I 0 -D2 ].
- # Diagonal and subdiagonal.
- Sd = matrix(0.0, (n,1))
- Se = matrix(0.0, (n-1,1))
+ # First do a symbolic factorization of tridiagonal matrix.
+ S = spmatrix((2*n-1)*[1.0], range(1,n)+range(n),
+ range(n-1)+range(n))
+ fact = cholmod.symbolic(S)
def kktsolver(x, z, dnl, dl):
"""
@@ -169,11 +177,11 @@ def tv(delta):
d1 = dl[:n-1]**2
d2 = dl[n-1:]**2
d = 4.0*div( mul(d1,d2), d1+d2)
- Sd[:] = z[0]
- Sd[:n-1] += d
- Sd[1:] += d
- Se[:] = -d
- lapack.pttrf(Sd, Se)
+ S[::n+1] = z[0]
+ S[n+1::n+1] = S[n+1::n+1] + d
+ S[:(n**2-1):n+1] = S[:(n**2-1):n+1] + d
+ S[1::n+1] = -d
+ cholmod.numeric(S, fact)
def g(x, y, znl, zl):
"""
@@ -207,7 +215,7 @@ def tv(delta):
x[1:n] += y
# x[:n] := S^-1 * x[:n]
- lapack.pttrs(Sd, Se, x)
+ cholmod.solve(fact, x)
# u = D*x[:n]
u = x[1:n] - x[0:n-1]
@@ -226,7 +234,6 @@ def tv(delta):
return solvers.nlcp(kktsolver, F, G, h)['x'][:n]
-
nopts = 15
deltas = -3.0 + (3.0-(-3.0))/(nopts-1) * matrix(range(nopts))
cost1, cost2 = [], []
diff --git a/examples/book/chap6/tv b/examples/book/chap6/old/tv_new
similarity index 93%
copy from examples/book/chap6/tv
copy to examples/book/chap6/old/tv_new
index 0b4948f..201d46d 100755
--- a/examples/book/chap6/tv
+++ b/examples/book/chap6/old/tv_new
@@ -3,6 +3,8 @@
# Figures 6.11-14, pages 315-317.
# Total variation reconstruction.
+### same as tv, but uses LAPACK solvers for band systems
+
from math import pi
from cvxopt import random, blas, lapack, solvers
from cvxopt.base import matrix, spmatrix, sin, mul, div
@@ -32,8 +34,11 @@ pylab.axis([0, 2000, -2, 2])
# Quadratic smoothing.
# A = D'*D is an n by n tridiagonal matrix with -1.0 on the
# upper/lower diagonal and 1, 2, 2, ..., 2, 2, 1 on the diagonal.
-Ad = matrix([1.0] + (n-2)*[2.0] + [1.0])
-As = matrix(-1.0, (n-1,1))
+
+A = matrix(0.0, (2,n))
+A[0,:] = [1.0] + (n-2)*[2.0] + [1.0]
+A[1,:n-1] = -1.0
+I = matrix(n*[[1.0, 0.0]])
nopts = 100
deltas = -10.0 + 20.0/(nopts-1) * matrix(range(nopts))
@@ -41,20 +46,20 @@ deltas = -10.0 + 20.0/(nopts-1) * matrix(range(nopts))
cost1, cost2 = [], []
for delta in deltas:
xr = +corr
- lapack.ptsv(1.0 + 10**delta * Ad, 10**delta * As, xr)
+ lapack.pbsv(I + 10**delta * A, xr)
cost1 += [blas.nrm2(xr - corr)]
cost2 += [blas.nrm2(xr[1:] - xr[:-1])]
# Find solutions with ||xhat - xcorr || roughly equal to 4, 7, 10.
mv1, k1 = min(zip([abs(c - 10.0) for c in cost1], range(nopts)))
xr1 = +corr
-lapack.ptsv(1.0 + 10**deltas[k1] * Ad, 10**deltas[k1] * As, xr1)
+lapack.pbsv(I + 10**deltas[k1] * A, xr1)
mv2, k2 = min(zip([abs(c - 7.0) for c in cost1], range(nopts)))
xr2 = +corr
-lapack.ptsv(1.0 + 10**deltas[k2] * Ad, 10**deltas[k2] * As, xr2)
+lapack.pbsv(I + 10**deltas[k2] * A, xr2)
mv3, k3 = min(zip([abs(c - 4.0) for c in cost1], range(nopts)))
xr3 = +corr
-lapack.ptsv(1.0 + 10**deltas[k3] * Ad, 10**deltas[k3] * As, xr3)
+lapack.pbsv(I + 10**deltas[k3] * A, xr3)
pylab.figure(2, facecolor='w')
pylab.plot(cost1, cost2, [blas.nrm2(corr)], [0], 'o',
@@ -152,9 +157,8 @@ def tv(delta):
# [ D -I -D1 0 ]
# [ -D -I 0 -D2 ].
- # Diagonal and subdiagonal.
- Sd = matrix(0.0, (n,1))
- Se = matrix(0.0, (n-1,1))
+ # First do a symbolic factorization of tridiagonal matrix.
+ S = matrix(0.0, (2,n))
def kktsolver(x, z, dnl, dl):
"""
@@ -169,11 +173,11 @@ def tv(delta):
d1 = dl[:n-1]**2
d2 = dl[n-1:]**2
d = 4.0*div( mul(d1,d2), d1+d2)
- Sd[:] = z[0]
- Sd[:n-1] += d
- Sd[1:] += d
- Se[:] = -d
- lapack.pttrf(Sd, Se)
+ S[0,:] = z[0]
+ S[0,:n-1] += d.T
+ S[0,1:] += d.T
+ S[1,:n-1] = -d.T
+ lapack.pbtrf(S)
def g(x, y, znl, zl):
"""
@@ -207,7 +211,7 @@ def tv(delta):
x[1:n] += y
# x[:n] := S^-1 * x[:n]
- lapack.pttrs(Sd, Se, x)
+ lapack.pbtrs(S, x)
# u = D*x[:n]
u = x[1:n] - x[0:n-1]
@@ -241,6 +245,7 @@ xtv2 = tv(10**deltas[k2])
mv3, k3 = min(zip([abs(c - 5.0) for c in cost2], range(nopts)))
xtv3 = tv(10**deltas[k3])
+
pylab.figure(1, facecolor='w', figsize=(8,5))
pylab.subplot(211)
pylab.plot(t, ex)
diff --git a/examples/book/chap6/polapprox b/examples/book/chap6/polapprox
index 1d9524b..0f2c294 100755
--- a/examples/book/chap6/polapprox
+++ b/examples/book/chap6/polapprox
@@ -10,7 +10,7 @@ import pylab
from pickle import load
solvers.options['show_progress'] = 0
-data = load(open('polapprox.pic','r'))
+data = load(open('polapprox.bin','r'))
t, y = data['t'], data['y']
m = len(t)
diff --git a/examples/book/chap6/polapprox.pic b/examples/book/chap6/polapprox.bin
similarity index 100%
rename from examples/book/chap6/polapprox.pic
rename to examples/book/chap6/polapprox.bin
diff --git a/examples/book/chap6/regsel b/examples/book/chap6/regsel
index d583672..2de8c57 100755
--- a/examples/book/chap6/regsel
+++ b/examples/book/chap6/regsel
@@ -11,7 +11,7 @@ from cvxopt.base import matrix, spmatrix, mul
from pickle import load
solvers.options['show_progress'] = 0
-data = load(open('regsel.pic','r'))
+data = load(open('regsel.bin','r'))
A, b = data['A'], data['b']
m, n = A.size
@@ -67,14 +67,14 @@ for c in xrange(m+1):
res2 += [ min(r), min(r) ]
card2 += [ c, c+1 ]
-pylab.figure(1, facecolor='w')
-pylab.plot( res2[::2], card2[::2], 'o')
-pylab.plot( res2, card2, '-')
-pylab.xlabel('||A*x-b||_2')
-pylab.ylabel('card(x)')
-pylab.title('Sparse regressor selection (fig 6.7)')
-print "Close figure to start exhaustive search."
-pylab.show()
+#pylab.figure(1, facecolor='w')
+#pylab.plot( res2[::2], card2[::2], 'o')
+#pylab.plot( res2, card2, '-')
+#pylab.xlabel('||A*x-b||_2')
+#pylab.ylabel('card(x)')
+#pylab.title('Sparse regressor selection (fig 6.7)')
+#print "Close figure to start exhaustive search."
+#pylab.show()
# Exhaustive search.
diff --git a/examples/book/chap6/regsel.pic b/examples/book/chap6/regsel.bin
similarity index 100%
rename from examples/book/chap6/regsel.pic
rename to examples/book/chap6/regsel.bin
diff --git a/examples/book/chap6/robls b/examples/book/chap6/robls
index 6efcbf7..ffe497f 100755
--- a/examples/book/chap6/robls
+++ b/examples/book/chap6/robls
@@ -55,7 +55,7 @@ def wcls(A, Ap, b):
# Figure 6.15
-data = load(open('robls.pic','r'))['6.15']
+data = load(open('robls.bin','r'))['6.15']
A, b, B = data['A'], data['b'], data['B']
m, n = A.size
@@ -102,7 +102,7 @@ pylab.title('Robust least-squares (fig.6.15)')
# Figure 6.16
-data = load(open('robls.pic','r'))['6.16']
+data = load(open('robls.bin','r'))['6.16']
A, Ap, b = data['A0'], [data['A1'], data['A2']], data['b']
(m, n), p = A.size, len(Ap)
diff --git a/examples/book/chap6/robls.pic b/examples/book/chap6/robls.bin
similarity index 100%
rename from examples/book/chap6/robls.pic
rename to examples/book/chap6/robls.bin
diff --git a/examples/book/chap6/smoothrec b/examples/book/chap6/smoothrec
index ddca231..219661a 100755
--- a/examples/book/chap6/smoothrec
+++ b/examples/book/chap6/smoothrec
@@ -51,10 +51,10 @@ xr3 = +corr
lapack.ptsv(1.0 + 10**deltas[k3] * Ad, 10**deltas[k3] *As, xr3)
pylab.figure(2, facecolor='w')
-pylab.plot(cost1, cost2, [blas.nrm2(corr)], [0], 'o',
- [0], [blas.nrm2(corr[1:] - corr[:-1])], 'o')
-pylab.plot([cost1[k1]], [cost2[k1]], 'o', [cost1[k2]], [cost2[k2]], 'o',
- [cost1[k3]], [cost2[k3]], 'o')
+pylab.plot(cost1, cost2, [blas.nrm2(corr)], [0], 'bo',
+ [0], [blas.nrm2(corr[1:] - corr[:-1])], 'bo')
+pylab.plot([cost1[k1]], [cost2[k1]], 'bo', [cost1[k2]], [cost2[k2]], 'bo',
+ [cost1[k3]], [cost2[k3]], 'bo')
pylab.text(cost1[k1], cost2[k1],'1')
pylab.text(cost1[k2], cost2[k2],'2')
pylab.text(cost1[k3], cost2[k3],'3')
diff --git a/examples/book/chap6/tv b/examples/book/chap6/tv
index 0b4948f..fc5d35d 100755
--- a/examples/book/chap6/tv
+++ b/examples/book/chap6/tv
@@ -57,10 +57,10 @@ xr3 = +corr
lapack.ptsv(1.0 + 10**deltas[k3] * Ad, 10**deltas[k3] * As, xr3)
pylab.figure(2, facecolor='w')
-pylab.plot(cost1, cost2, [blas.nrm2(corr)], [0], 'o',
- [0], [blas.nrm2(corr[1:] - corr[:-1])], 'o')
-pylab.plot([cost1[k1]], [cost2[k1]], 'o',
- [cost1[k2]], [cost2[k2]], 'o', [cost1[k3]], [cost2[k3]], 'o')
+pylab.plot(cost1, cost2, [blas.nrm2(corr)], [0], 'bo',
+ [0], [blas.nrm2(corr[1:] - corr[:-1])], 'bo')
+pylab.plot([cost1[k1]], [cost2[k1]], 'bo',
+ [cost1[k2]], [cost2[k2]], 'bo', [cost1[k3]], [cost2[k3]], 'bo')
pylab.text(cost1[k1], cost2[k1], '1')
pylab.text(cost1[k2], cost2[k2], '2')
pylab.text(cost1[k3], cost2[k3], '3')
@@ -85,8 +85,8 @@ pylab.plot(t, xr3)
pylab.axis([0, 2000, -2.0, 2.0])
pylab.ylabel('xhat3[i]')
pylab.xlabel('i')
-print "Close figures to start total variation reconstruction."
-pylab.show()
+#print "Close figures to start total variation reconstruction."
+#pylab.show()
# Total variation smoothing.
@@ -241,7 +241,7 @@ xtv2 = tv(10**deltas[k2])
mv3, k3 = min(zip([abs(c - 5.0) for c in cost2], range(nopts)))
xtv3 = tv(10**deltas[k3])
-pylab.figure(1, facecolor='w', figsize=(8,5))
+pylab.figure(4, facecolor='w', figsize=(8,5))
pylab.subplot(211)
pylab.plot(t, ex)
pylab.ylabel('x[i]')
@@ -254,11 +254,11 @@ pylab.ylabel('xcor[i]')
pylab.xlabel('i')
pylab.axis([0, 2000, -2, 2])
-pylab.figure(2, facecolor='w') #figsize=(8,7.5))
-pylab.plot(cost1, cost2, [blas.nrm2(corr)], [0], 'o',
- [0], [blas.asum(corr[1:] - corr[:-1])], 'o')
-pylab.plot([cost1[k1]], [cost2[k1]], 'o', [cost1[k2]], [cost2[k2]], 'o',
- [cost1[k3]], [cost2[k3]], 'o')
+pylab.figure(5, facecolor='w') #figsize=(8,7.5))
+pylab.plot(cost1, cost2, [blas.nrm2(corr)], [0], 'bo',
+ [0], [blas.asum(corr[1:] - corr[:-1])], 'bo')
+pylab.plot([cost1[k1]], [cost2[k1]], 'bo', [cost1[k2]], [cost2[k2]], 'bo',
+ [cost1[k3]], [cost2[k3]], 'bo')
pylab.text(cost1[k1], cost2[k1],'1')
pylab.text(cost1[k2], cost2[k2],'2')
pylab.text(cost1[k3], cost2[k3],'3')
@@ -268,7 +268,7 @@ pylab.xlabel('||xhat-xcor||_2')
pylab.ylabel('||D*xhat||_1')
pylab.title('Optimal trade-off curve (fig. 6.13)')
-pylab.figure(3, facecolor='w', figsize=(8,7.5))
+pylab.figure(6, facecolor='w', figsize=(8,7.5))
pylab.subplot(311)
pylab.plot(t, xtv1)
pylab.axis([0, 2000, -2.0, 2.0])
diff --git a/examples/book/chap7/logreg b/examples/book/chap7/logreg
index 17c2d43..61fb736 100755
--- a/examples/book/chap7/logreg
+++ b/examples/book/chap7/logreg
@@ -8,7 +8,7 @@ from cvxopt import solvers
from cvxopt.base import matrix, spmatrix, log, exp, div
solvers.options['show_progress'] = False
-data = pickle.load(open("logreg.pic"))
+data = pickle.load(open("logreg.bin"))
u, y = data['u'], data['y']
# minimize sum_{y_k = 1} (a*uk + b) + sum log (1 + exp(a*u + b))
diff --git a/examples/book/chap7/logreg.pic b/examples/book/chap7/logreg.bin
similarity index 100%
rename from examples/book/chap7/logreg.pic
rename to examples/book/chap7/logreg.bin
diff --git a/examples/book/chap7/probbounds b/examples/book/chap7/probbounds
index 959ef62..844385c 100755
--- a/examples/book/chap7/probbounds
+++ b/examples/book/chap7/probbounds
@@ -147,8 +147,8 @@ def makefig1():
pylab.title('Geometrical interpretation of Chebyshev bound (fig. 7.7)')
pylab.axis('off')
makefig1()
-print "Close figure to continue."
-pylab.show()
+#print "Close figure to continue."
+#pylab.show()
# Compute bounds for s0 with sigma in [0,2.5]
@@ -166,8 +166,8 @@ pylab.title('Chebyshev lower bounds (fig 7.6)')
pylab.text(sigmas[nosigmas/2], bnds0[nosigmas/2], 's0')
pylab.xlabel('sigma')
pylab.ylabel('Probability of correct detection')
-print "Close figure to continue."
-pylab.show()
+#print "Close figure to continue."
+#pylab.show()
# Bounds for s1.
b1 -= A1*C[1] # put s1 at the origin
@@ -182,8 +182,8 @@ pylab.text(sigmas[nosigmas/2], bnds0[nosigmas/2], 's0')
pylab.text(sigmas[nosigmas/2], bnds1[nosigmas/2], 's1')
pylab.xlabel('sigma')
pylab.ylabel('Probability of correct detection')
-print "Close figure to continue."
-pylab.show()
+#print "Close figure to continue."
+#pylab.show()
# Bounds for s2.
b2 -= A2*C[2] # put s2 at the origin
diff --git a/examples/book/chap8/linsep b/examples/book/chap8/linsep
index 15f1c89..4b94ca2 100755
--- a/examples/book/chap8/linsep
+++ b/examples/book/chap8/linsep
@@ -10,7 +10,7 @@ from cvxopt.base import matrix, spmatrix, log, exp, div
from cvxopt.modeling import variable, op
solvers.options['show_progress'] = False
-data = pickle.load(open("linsep.pic"))
+data = pickle.load(open("linsep.bin"))
X, Y = data['X'], data['Y']
n, N, M = X.size[0], X.size[1], Y.size[1]
diff --git a/examples/book/chap8/linsep.pic b/examples/book/chap8/linsep.bin
similarity index 100%
rename from examples/book/chap8/linsep.pic
rename to examples/book/chap8/linsep.bin
diff --git a/examples/book/chap8/placement b/examples/book/chap8/placement
index 7f95fc2..e9f7657 100755
--- a/examples/book/chap8/placement
+++ b/examples/book/chap8/placement
@@ -11,7 +11,7 @@ from cvxopt.base import matrix, spmatrix, sqrt, mul
from cvxopt.modeling import variable, op
solvers.options['show_progress'] = False
-data = pickle.load(open("placement.pic", "r"))
+data = pickle.load(open("placement.bin", "r"))
Xf = data['X'] # M by n matrix with coordinates of M fixed nodes
M = Xf.size[0]
E = data['E'] # list of edges
diff --git a/examples/book/chap8/placement.pic b/examples/book/chap8/placement.bin
similarity index 100%
rename from examples/book/chap8/placement.pic
rename to examples/book/chap8/placement.bin
diff --git a/examples/doc/acent b/examples/doc/acent
index 336ec1f..10df856 100755
--- a/examples/doc/acent
+++ b/examples/doc/acent
@@ -1,6 +1,6 @@
#!/usr/bin/python
-# The analytic centering example of section 4.8.
+# The analytic centering example of section 4.9.
from cvxopt.base import matrix, log, mul, div
from cvxopt import blas, lapack, random
diff --git a/examples/doc/covsel b/examples/doc/covsel
index ff45f43..67d218b 100755
--- a/examples/doc/covsel
+++ b/examples/doc/covsel
@@ -85,6 +85,6 @@ def covsel(Y):
return K
-Y = load(open("covsel.pic","r"))
+Y = load(open("covsel.bin","r"))
print "%d rows/columns, %d nonzeros\n" %(Y.size[0], len(Y))
covsel(Y)
diff --git a/examples/doc/covsel.pic b/examples/doc/covsel.bin
similarity index 100%
rename from examples/doc/covsel.pic
rename to examples/doc/covsel.bin
diff --git a/examples/doc/l1 b/examples/doc/l1
index 92ab0f3..9091890 100755
--- a/examples/doc/l1
+++ b/examples/doc/l1
@@ -1,6 +1,6 @@
#!/usr/bin/python
-# The 1-norm approximation example of section 8.6.
+# The 1-norm approximation example of section 8.5.
from cvxopt import base, random, blas, lapack, solvers
from cvxopt.base import matrix, spmatrix, mul, div
@@ -34,8 +34,8 @@ def l1(P, q):
c = matrix(n*[0.0] + m*[1.0])
h = matrix([q, -q])
- def Fi(x, y, alpha=1.0, beta=0.0, trans='N'):
- if trans=='N':
+ def Fi(x, y, alpha = 1.0, beta = 0.0, trans = 'N'):
+ if trans == 'N':
# y := alpha * [P, -I; -P, -I] * x + beta*y
u = P*x[:n]
y[:m] = alpha * ( u - x[n:]) + beta*y[:m]
@@ -47,52 +47,53 @@ def l1(P, q):
y[n:] = -alpha * (x[:m] + x[m:]) + beta*y[n:]
- def kktsolver(d, R):
+ def Fkkt(W):
- # Returns a function f(x,y,zl,zs) that solves
+ # Returns a function f(x, y, z) that solves
#
- # [ 0 0 P' -P' ] [ x[:n] ] [ bx[:n] ]
- # [ 0 0 -I -I ] [ x[n:] ] [ bx[n:] ]
- # [ P -I -D1^{-1} 0 ] [ zl[:m]] = [ bzl[:m] ]
- # [-P -I 0 -D2^{-1} ] [ zl[m:]] [ bzl[m:] ]
+ # [ 0 0 P' -P' ] [ x[:n] ] [ bx[:n] ]
+ # [ 0 0 -I -I ] [ x[n:] ] [ bx[n:] ]
+ # [ P -I -D1^{-1} 0 ] [ z[:m] ] = [ bz[:m] ]
+ # [-P -I 0 -D2^{-1} ] [ z[m:] ] [ bz[m:] ]
#
- # where D1 = diag(d[:m])^2, D2 = diag(d[m:])^2.
+ # where D1 = diag(di[:m])^2, D2 = diag(di[m:])^2 and di = W['di'].
#
- # On entry bx, bzl are stored in x, zl.
- # On exit x, zl contain the solution, with zl scaled (d.*zl is
- # returned instead of zl).
+ # On entry bx, bz are stored in x, z.
+ # On exit x, z contain the solution, with z scaled (di .* z is
+ # returned instead of z).
# Factor A = 4*P'*D*P where D = d1.*d2 ./(d1+d2) and
# d1 = d[:m].^2, d2 = d[m:].^2.
- d1, d2 = d[:m]**2, d[m:]**2
+ di = W['di']
+ d1, d2 = di[:m]**2, di[m:]**2
D = div( mul(d1,d2), d1+d2 )
A = P.T * spmatrix(4*D, range(m), range(m)) * P
lapack.potrf(A)
- def f(x, y, zl, zs):
+ def f(x, y, z):
# Solve for x[:n]:
#
# A*x[:n] = bx[:n] + P' * ( ((D1-D2)*(D1+D2)^{-1})*bx[n:]
- # + (2*D1*D2*(D1+D2)^{-1}) * (bzl[:m] - bzl[m:]) ).
+ # + (2*D1*D2*(D1+D2)^{-1}) * (bz[:m] - bz[m:]) ).
x[:n] += P.T * ( mul( div(d1-d2, d1+d2), x[n:]) +
- mul( 2*D, zl[:m]-zl[m:] ) )
+ mul( 2*D, z[:m]-z[m:] ) )
lapack.potrs(A, x)
- # x[n:] := (D1+D2)^{-1} * (bx[n:] - D1*bzl[:m] - D2*bzl[m:]
+ # x[n:] := (D1+D2)^{-1} * (bx[n:] - D1*bz[:m] - D2*bz[m:]
# + (D1-D2)*P*x[:n])
u = P*x[:n]
- x[n:] = div( x[n:] - mul(d1, zl[:m]) - mul(d2, zl[m:]) +
+ x[n:] = div( x[n:] - mul(d1, z[:m]) - mul(d2, z[m:]) +
mul(d1-d2, u), d1+d2 )
- # z[:m] := d1[:m] .* ( P*x[:n] - x[n:] - bzl[:m])
- # z[m:] := d2[m:] .* (-P*x[:n] - x[n:] - bzl[m:])
+ # z[:m] := d1[:m] .* ( P*x[:n] - x[n:] - bz[:m])
+ # z[m:] := d2[m:] .* (-P*x[:n] - x[n:] - bz[m:])
- zl[:m] = mul(d[:m], u-x[n:]-zl[:m])
- zl[m:] = mul(d[m:], -u-x[n:]-zl[m:])
+ z[:m] = mul(di[:m], u-x[n:]-z[:m])
+ z[m:] = mul(di[m:], -u-x[n:]-z[m:])
return f
@@ -117,9 +118,10 @@ def l1(P, q):
w = matrix(0.0, (m,1))
z0 = matrix([.5*(1+w), .5*(1-w)])
- sol = solvers.conelp(c, kktsolver, Gl=Fi, hl=h,
- primalstart={'x': x0, 'sl':s0}, dualstart={'zl': z0})
- return sol['x'][:n], sol['zl'][m:] - sol['zl'][:m]
+ dims = {'l': 2*m, 'q': [], 's': []}
+ sol = solvers.conelp(c, Fi, h, dims, kktsolver = Fkkt,
+ primalstart={'x': x0, 's': s0}, dualstart={'z': z0})
+ return sol['x'][:n], sol['z'][m:] - sol['z'][:m]
random.setseed()
m, n = 500, 100
diff --git a/examples/doc/mcsdp b/examples/doc/mcsdp
index 43b4c06..2408946 100755
--- a/examples/doc/mcsdp
+++ b/examples/doc/mcsdp
@@ -1,6 +1,6 @@
#!/usr/bin/python
-# The SDP example of section 8.6.
+# The SDP example of section 8.5.
from cvxopt import base, blas, lapack, random, solvers
from cvxopt.base import matrix
@@ -19,22 +19,22 @@ def mcsdp(w):
n = w.size[0]
- def Fs(x, y, alpha=1.0, beta=0.0, trans='N'):
+ def Fs(x, y, alpha = 1.0, beta = 0.0, trans = 'N'):
"""
y := alpha*(-diag(x)) + beta*y.
"""
if trans=='N':
- # x is a vector; y[0] is a matrix.
- blas.scal(beta, y[0])
- blas.axpy(x, y[0], alpha=-alpha, incy=n+1)
+ # x is a vector; y is a matrix.
+ blas.scal(beta, y)
+ blas.axpy(x, y, alpha = -alpha, incy = n+1)
else:
- # x[0] is a matrix; y is a vector.
+ # x is a matrix; y is a vector.
blas.scal(beta, y)
- blas.axpy(x[0], y, alpha=-alpha, incx=n+1)
+ blas.axpy(x, y, alpha = -alpha, incx = n+1)
- def cngrnc(r, x, alpha=1.0):
+ def cngrnc(r, x, alpha = 1.0):
"""
Congruence transformation
@@ -48,54 +48,58 @@ def mcsdp(w):
# a := tril(x)*r
a = +r
- blas.trmm(x, a, side='L')
+ tx = matrix(x, (n,n))
+ blas.trmm(tx, a, side='L')
# x := alpha*(a*r' + r*a')
- blas.syr2k(r, a, x, trans='T', alpha=alpha)
-
+ blas.syr2k(r, a, tx, trans = 'T', alpha = alpha)
+ x[:] = tx[:]
+
- def kktsolver(d, r):
+ def Fkkt(W):
+
+ rti = W['rti'][0]
- # t = r*r' as a nonsymmetric matrix.
+ # t = rti*rti' as a nonsymmetric matrix.
t = matrix(0.0, (n,n))
- blas.gemm(r[0], r[0], t, transB='T')
+ blas.gemm(rti, rti, t, transB = 'T')
# Cholesky factorization of tsq = t.*t.
tsq = t**2
lapack.potrf(tsq)
- def f(x, y, zl, zs):
+ def f(x, y, z):
"""
Solve
- -diag(zs) = bx
- -diag(x) - inv(r*r')*zs*inv(r*r') = bs
+ -diag(z) = bx
+ -diag(x) - inv(rti*rti') * z * inv(rti*rti') = bs
- On entry, x and zs contain bx and bs.
- On exit, they contain the solution, with zs scaled
- (inv(r)'*zs*inv(r) is returned instead of zs).
+ On entry, x and z contain bx and bs.
+ On exit, they contain the solution, with z scaled
+ (inv(rti)'*z*inv(rti) is returned instead of z).
We first solve
- ((r*r') .* (r*r')) * x = bx - diag(t*bs*t)
+ ((rti*rti') .* (rti*rti')) * x = bx - diag(t*bs*t)
- and take zs = -r' * (diag(x) + bs) * r.
+ and take z = -rti' * (diag(x) + bs) * rti.
"""
# tbst := t * zs * t = t * bs * t
- tbst = +zs[0]
+ tbst = matrix(z, (n,n))
cngrnc(t, tbst)
- # x := x - diag(tbst) = bx - diag(r*r' * bs * r*r')
+ # x := x - diag(tbst) = bx - diag(rti*rti' * bs * rti*rti')
x -= tbst[::n+1]
# x := (t.*t)^{-1} * x = (t.*t)^{-1} * (bx - diag(t*bs*t))
lapack.potrs(tsq, x)
- # zs := zs + diag(x) = bs + diag(x)
- zs[0][::n+1] += x
+ # z := z + diag(x) = bs + diag(x)
+ z[::n+1] += x
- # zs := -r' * zs * r = -r' * (diag(x) + bs) * r
- cngrnc(r[0], zs[0], alpha=-1.0)
+ # z := -rti' * z * rti = -rti' * (diag(x) + bs) * rti
+ cngrnc(rti, z, alpha = -1.0)
return f
@@ -112,9 +116,10 @@ def mcsdp(w):
z0 = matrix(0.0, (n,n))
z0[::n+1] = 1.0
- sol = solvers.conelp(c, kktsolver, Gs=Fs, hs=[w],
- primalstart={'x': x0, 'ss': [s0]}, dualstart={'zs': [z0]})
- return sol['x'], sol['zs'][0]
+ dims = {'l': 0, 'q': [], 's': [n]}
+ sol = solvers.conelp(c, Fs, w[:], dims, kktsolver = Fkkt,
+ primalstart = {'x': x0, 's': s0[:]}, dualstart = {'z': z0[:]})
+ return sol['x'], matrix(sol['z'], (n,n))
n = 100
w = random.normal(n,n)
diff --git a/examples/doc/qcl1 b/examples/doc/qcl1
new file mode 100755
index 0000000..d3bf32c
--- /dev/null
+++ b/examples/doc/qcl1
@@ -0,0 +1,136 @@
+#!/usr/bin/python
+
+# The quadratically constrained 1-norm minimization example of section 8.5.
+
+from cvxopt import random, blas, lapack, solvers
+from cvxopt.base import matrix, mul, div
+from math import sqrt
+
+def qcl1(A, b):
+
+ """
+ Returns the solution u, z of
+
+ (primal) minimize || u ||_1
+ subject to || A * u - b ||_2 <= 1
+
+ (dual) maximize b^T z - ||z||_2
+ subject to || A'*z ||_inf <= 1.
+
+ Exploits structure, assuming A is m by n with m >= n.
+ """
+
+ m, n = A.size
+
+ # Solve equivalent cone LP with variables x = [u; v]:
+ #
+ # minimize [0; 1]' * x
+ # subject to [ I -I ] * x <= [ 0 ] (componentwise)
+ # [-I -I ] * x <= [ 0 ] (componentwise)
+ # [ 0 0 ] * x <= [ 1 ] (SOC)
+ # [-A 0 ] [ -b ].
+ #
+ # maximize -t + b' * w
+ # subject to z1 - z2 = A'*w
+ # z1 + z2 = 1
+ # z1 >= 0, z2 >=0, ||w||_2 <= t.
+
+ c = matrix(n*[0.0] + n*[1.0])
+ h = matrix( 0.0, (2*n + m + 1, 1))
+ h[2*n] = 1.0
+ h[2*n+1:] = -b
+
+ def G(x, y, alpha = 1.0, beta = 0.0, trans = 'N'):
+ y *= beta
+ if trans=='N':
+ # y += alpha * G * x
+ y[:n] += alpha * (x[:n] - x[n:2*n])
+ y[n:2*n] += alpha * (-x[:n] - x[n:2*n])
+ y[2*n+1:] -= alpha * A*x[:n]
+
+ else:
+ # y += alpha * G'*x
+ y[:n] += alpha * (x[:n] - x[n:2*n] - A.T * x[-m:])
+ y[n:] -= alpha * (x[:n] + x[n:2*n])
+
+
+ def Fkkt(W):
+
+ # Returns a function f(x, y, z) that solves
+ #
+ # [ 0 G' ] [ x ] = [ bx ]
+ # [ G -W'*W ] [ z ] [ bz ].
+
+ # First factor
+ #
+ # S = G' * W**-1 * W**-T * G
+ # = [0; -A]' * W3^-2 * [0; -A] + 4 * (W1**2 + W2**2)**-1
+ #
+ # where
+ #
+ # W1 = diag(d1) with d1 = W['d'][:n] = 1 ./ W['di'][:n]
+ # W2 = diag(d2) with d2 = W['d'][n:] = 1 ./ W['di'][n:]
+ # W3 = beta * (2*v*v' - J), W3^-1 = 1/beta * (2*J*v*v'*J - J)
+ # with beta = W['beta'][0], v = W['v'][0], J = [1, 0; 0, -I].
+
+ # As = W3^-1 * [ 0 ; -A ] = 1/beta * ( 2*J*v * v' - I ) * [0; A]
+ beta, v = W['beta'][0], W['v'][0]
+ As = 2 * v * (v[1:].T * A)
+ As[1:,:] *= -1.0
+ As[1:,:] -= A
+ As /= beta
+
+ # S = As'*As + 4 * (W1**2 + W2**2)**-1
+ S = As.T * As
+ d1, d2 = W['d'][:n], W['d'][n:]
+ d = 4.0 * (d1**2 + d2**2)**-1
+ S[::n+1] += d
+ lapack.potrf(S)
+
+ def f(x, y, z):
+
+ # z := - W**-T * z
+ z[:n] = -div( z[:n], d1 )
+ z[n:2*n] = -div( z[n:2*n], d2 )
+ z[2*n:] -= 2.0*v*( v[0]*z[2*n] - blas.dot(v[1:], z[2*n+1:]) )
+ z[2*n+1:] *= -1.0
+ z[2*n:] /= beta
+
+ # x := x - G' * W**-1 * z
+ x[:n] -= div(z[:n], d1) - div(z[n:2*n], d2) + As.T * z[-(m+1):]
+ x[n:] += div(z[:n], d1) + div(z[n:2*n], d2)
+
+ # Solve for x[:n]:
+ #
+ # S*x[:n] = x[:n] - (W1**2 - W2**2)(W1**2 + W2**2)^-1 * x[n:]
+
+ x[:n] -= mul( div(d1**2 - d2**2, d1**2 + d2**2), x[n:])
+ lapack.potrs(S, x)
+
+ # Solve for x[n:]:
+ #
+ # (d1**-2 + d2**-2) * x[n:] = x[n:] + (d1**-2 - d2**-2)*x[:n]
+
+ x[n:] += mul( d1**-2 - d2**-2, x[:n])
+ x[n:] = div( x[n:], d1**-2 + d2**-2)
+
+ # z := z + W^-T * G*x
+ z[:n] += div( x[:n] - x[n:2*n], d1)
+ z[n:2*n] += div( -x[:n] - x[n:2*n], d2)
+ z[2*n:] += As*x[:n]
+
+ return f
+
+ dims = {'l': 2*n, 'q': [m+1], 's': []}
+ sol = solvers.conelp(c, G, h, dims, kktsolver = Fkkt)
+ if sol['status'] == 'optimal':
+ return sol['x'][:n], sol['z'][-m:]
+ else:
+ return None, None
+
+random.setseed()
+m, n = 100, 100
+A, b = random.normal(m,n), random.normal(m,1)
+
+x, z = qcl1(A, b)
+if x is None: print "infeasible"
diff --git a/src/C/SuiteSparse/AMD/Doc/AMD_UserGuide.pdf b/src/C/SuiteSparse/AMD/Doc/AMD_UserGuide.pdf
deleted file mode 100644
index 0fcffe6..0000000
Binary files a/src/C/SuiteSparse/AMD/Doc/AMD_UserGuide.pdf and /dev/null differ
diff --git a/src/C/SuiteSparse/AMD/Doc/AMD_UserGuide.tex b/src/C/SuiteSparse/AMD/Doc/AMD_UserGuide.tex
index 2ea79e0..9381142 100644
--- a/src/C/SuiteSparse/AMD/Doc/AMD_UserGuide.tex
+++ b/src/C/SuiteSparse/AMD/Doc/AMD_UserGuide.tex
@@ -13,7 +13,7 @@
\begin{document}
%------------------------------------------------------------------------------
-\title{AMD Version 2.0 User Guide}
+\title{AMD Version 2.2 User Guide}
\author{Patrick R. Amestoy\thanks{ENSEEIHT-IRIT,
2 rue Camichel 31017 Toulouse, France.
email: amestoy at enseeiht.fr. http://www.enseeiht.fr/$\sim$amestoy.}
@@ -34,7 +34,7 @@ http://www.numerical.rl.ac.uk/people/isd/isd.html.
This work was supported by the EPSRC under grant GR/R46441.
}}
-\date{Dec 12, 2006}
+\date{May 31, 2007}
\maketitle
%------------------------------------------------------------------------------
@@ -48,9 +48,9 @@ A MATLAB interface is included.
%------------------------------------------------------------------------------
Technical report TR-04-002 (revised), CISE Department, University of Florida,
-Gainesville, FL, 2006.
+Gainesville, FL, 2007.
-AMD Version 2.0, Copyright\copyright 2006 by Timothy A.
+AMD Version 2.2, Copyright\copyright 2007 by Timothy A.
Davis, Patrick R. Amestoy, and Iain S. Duff. All Rights Reserved.
AMD is available under alternate licences; contact T. Davis for details.
@@ -173,8 +173,8 @@ see \cite{GeorgeLiu89}.
\section{Availability}
%------------------------------------------------------------------------------
-In addition to appearing as a Collected Algorithm of the ACM,
-AMD Version 2.0 is available at http://www.cise.ufl.edu/research/sparse.
+In addition to appearing as a Collected Algorithm of the ACM, \newline
+AMD is available at http://www.cise.ufl.edu/research/sparse.
The Fortran version is available as the routine {\tt MC47} in HSL
(formerly the Harwell Subroutine Library) \cite{hsl:2002}.
@@ -183,12 +183,8 @@ The Fortran version is available as the routine {\tt MC47} in HSL
%------------------------------------------------------------------------------
The MATLAB function {\tt amd} is now a built-in function in MATLAB 7.3
-(R2006b). Compiling the AMD mexFunction, here, causes a name conflict
-(it is MATLAB that is conflict with AMD, not the other way around ...).
-
-To avoid this issue, either use the built-in {\tt amd}, or use the
-{\tt amd2} function provided here. {\tt amd}
-and {\tt amd2} differ in how the optional parameters are passed
+(R2006b). The built-in {\tt amd} and the {\tt amd2} function provided here
+differ in how the optional parameters are passed
(the 2nd input parameter).
To use AMD2 in MATLAB, you must first compile the AMD2 mexFunction.
@@ -735,6 +731,11 @@ This works on any system with MATLAB, including Windows.
Alternately, type {\tt make} in the {\tt AMD/MATLAB} directory,
or just use the built-in {\tt amd} in MATLAB 7.3 or later.
+If you have MATLAB 7.2 or earlier, you must first edit UFconfig/UFconfig.h to
+remove the "-largeArrayDims" option from the MEX command, prior to
+{\tt make mex} or {\tt make} in the MATLAB directory
+(or just use {\tt amd\_make.m} inside MATLAB.
+
If you are including AMD as a subset of a larger library and do not want
to link the C standard I/O library, or if you simply do not need to use
them, you can safely remove the {\tt amd\_control.c} and {\tt amd\_info.c}
@@ -776,12 +777,13 @@ and gives details on their use.
{\footnotesize
\begin{verbatim}
+
/* ========================================================================= */
/* === AMD: approximate minimum degree ordering =========================== */
/* ========================================================================= */
/* ------------------------------------------------------------------------- */
-/* AMD Version 2.0, Copyright (c) 2006 by Timothy A. Davis, */
+/* AMD Version 2.2, Copyright (c) 2007 by Timothy A. Davis, */
/* Patrick R. Amestoy, and Iain S. Duff. See ../README.txt for License. */
/* email: davis at cise.ufl.edu CISE Department, Univ. of Florida. */
/* web: http://www.cise.ufl.edu/research/sparse/amd */
@@ -1176,11 +1178,11 @@ void amd_l_info (double Info [ ]) ;
* Versions 1.1 and earlier of AMD do not include a #define'd version number.
*/
-#define AMD_DATE "Dec 12, 2006"
+#define AMD_DATE "May 31, 2007"
#define AMD_VERSION_CODE(main,sub) ((main) * 1000 + (sub))
#define AMD_MAIN_VERSION 2
-#define AMD_SUB_VERSION 0
-#define AMD_SUBSUB_VERSION 4
+#define AMD_SUB_VERSION 2
+#define AMD_SUBSUB_VERSION 0
#define AMD_VERSION AMD_VERSION_CODE(AMD_MAIN_VERSION,AMD_SUB_VERSION)
#ifdef __cplusplus
@@ -1188,6 +1190,7 @@ void amd_l_info (double Info [ ]) ;
#endif
#endif
+#endif
\end{verbatim}
}
diff --git a/src/C/SuiteSparse/AMD/Doc/ChangeLog b/src/C/SuiteSparse/AMD/Doc/ChangeLog
index da57054..9b98e38 100644
--- a/src/C/SuiteSparse/AMD/Doc/ChangeLog
+++ b/src/C/SuiteSparse/AMD/Doc/ChangeLog
@@ -1,3 +1,11 @@
+May 31, 2007: version 2.2.0
+
+ * port to 64-bit MATLAB
+
+ * Makefile moved from Source/ to Lib/
+
+ * minor changes to printing routines (amd_control.c, amd_info.c)
+
Dec 12, 2006, version 2.0.4
* minor MATLAB code cleanup
diff --git a/src/C/SuiteSparse/AMD/Doc/License b/src/C/SuiteSparse/AMD/Doc/License
index 52e5087..b01008a 100644
--- a/src/C/SuiteSparse/AMD/Doc/License
+++ b/src/C/SuiteSparse/AMD/Doc/License
@@ -1,4 +1,4 @@
-AMD Version 2.0, Copyright (c) 2006 by Timothy A. Davis,
+AMD, Copyright (c) by Timothy A. Davis,
Patrick R. Amestoy, and Iain S. Duff. All Rights Reserved.
AMD is available under alternate licenses, contact T. Davis for details.
diff --git a/src/C/SuiteSparse/AMD/Include/amd.h b/src/C/SuiteSparse/AMD/Include/amd.h
index 4c73b42..e4fd47e 100644
--- a/src/C/SuiteSparse/AMD/Include/amd.h
+++ b/src/C/SuiteSparse/AMD/Include/amd.h
@@ -3,7 +3,7 @@
/* ========================================================================= */
/* ------------------------------------------------------------------------- */
-/* AMD Version 2.0, Copyright (c) 2006 by Timothy A. Davis, */
+/* AMD Version 2.2, Copyright (c) 2007 by Timothy A. Davis, */
/* Patrick R. Amestoy, and Iain S. Duff. See ../README.txt for License. */
/* email: davis at cise.ufl.edu CISE Department, Univ. of Florida. */
/* web: http://www.cise.ufl.edu/research/sparse/amd */
@@ -398,11 +398,11 @@ void amd_l_info (double Info [ ]) ;
* Versions 1.1 and earlier of AMD do not include a #define'd version number.
*/
-#define AMD_DATE "Dec 12, 2006"
+#define AMD_DATE "May 31, 2007"
#define AMD_VERSION_CODE(main,sub) ((main) * 1000 + (sub))
#define AMD_MAIN_VERSION 2
-#define AMD_SUB_VERSION 0
-#define AMD_SUBSUB_VERSION 4
+#define AMD_SUB_VERSION 2
+#define AMD_SUBSUB_VERSION 0
#define AMD_VERSION AMD_VERSION_CODE(AMD_MAIN_VERSION,AMD_SUB_VERSION)
#ifdef __cplusplus
diff --git a/src/C/SuiteSparse/AMD/Include/amd_internal.h b/src/C/SuiteSparse/AMD/Include/amd_internal.h
index 68b6e4f..d515b64 100644
--- a/src/C/SuiteSparse/AMD/Include/amd_internal.h
+++ b/src/C/SuiteSparse/AMD/Include/amd_internal.h
@@ -3,7 +3,7 @@
/* ========================================================================= */
/* ------------------------------------------------------------------------- */
-/* AMD Version 2.0, Copyright (c) 2006 by Timothy A. Davis, */
+/* AMD, Copyright (c) Timothy A. Davis, */
/* Patrick R. Amestoy, and Iain S. Duff. See ../README.txt for License. */
/* email: davis at cise.ufl.edu CISE Department, Univ. of Florida. */
/* web: http://www.cise.ufl.edu/research/sparse/amd */
diff --git a/src/C/SuiteSparse/AMD/Source/GNUmakefile b/src/C/SuiteSparse/AMD/Lib/GNUmakefile
similarity index 88%
rename from src/C/SuiteSparse/AMD/Source/GNUmakefile
rename to src/C/SuiteSparse/AMD/Lib/GNUmakefile
index 571ad1c..f59b1d6 100644
--- a/src/C/SuiteSparse/AMD/Source/GNUmakefile
+++ b/src/C/SuiteSparse/AMD/Lib/GNUmakefile
@@ -30,13 +30,13 @@ AMDL = $(addsuffix .o, $(subst amd_,amd_l_,$(AMD)))
# compile each int and long routine (with no real/complex version)
#-------------------------------------------------------------------------------
-amd_global.o: amd_global.c $(INC)
+amd_global.o: ../Source/amd_global.c $(INC)
$(C) -c $< -o $@
-amd_i_%.o: amd_%.c $(INC)
+amd_i_%.o: ../Source/amd_%.c $(INC)
$(C) -DDINT -c $< -o $@
-amd_l_%.o: amd_%.c $(INC)
+amd_l_%.o: ../Source/amd_%.c $(INC)
$(C) -DDLONG -c $< -o $@
#-------------------------------------------------------------------------------
@@ -55,11 +55,11 @@ fortran: ../Lib/libamdf77.a
AMDF77 = amd.o amdbar.o
-amd.o: amd.f
- $(F77) $(F77FLAGS) -c amd.f -o amd.o
+amd.o: ../Source/amd.f
+ $(F77) $(F77FLAGS) -c ../Source/amd.f -o amd.o
-amdbar.o: amdbar.f
- $(F77) $(F77FLAGS) -c amdbar.f -o amdbar.o
+amdbar.o: ../Source/amdbar.f
+ $(F77) $(F77FLAGS) -c ../Source/amdbar.f -o amdbar.o
../Lib/libamdf77.a: $(AMDF77)
$(AR) ../Lib/libamdf77.a $^
@@ -75,5 +75,4 @@ clean:
purge: distclean
distclean: clean
- - $(RM) ../Lib/libamd.a
- - $(RM) ../Lib/libamdf77.a
+ - $(RM) ../Lib/libamd.a ../Lib/libamdf77.a
diff --git a/src/C/SuiteSparse/AMD/Source/Makefile b/src/C/SuiteSparse/AMD/Lib/Makefile
similarity index 50%
rename from src/C/SuiteSparse/AMD/Source/Makefile
rename to src/C/SuiteSparse/AMD/Lib/Makefile
index 409046a..8e0f197 100644
--- a/src/C/SuiteSparse/AMD/Source/Makefile
+++ b/src/C/SuiteSparse/AMD/Lib/Makefile
@@ -13,31 +13,31 @@ include ../../UFconfig/UFconfig.mk
C = $(CC) $(CFLAGS) $(CONFIG) -I../Include -I../../UFconfig
everything:
- $(C) -DDINT -c amd_aat.c -o amd_i_aat.o
- $(C) -DDINT -c amd_1.c -o amd_i_1.o
- $(C) -DDINT -c amd_2.c -o amd_i_2.o
- $(C) -DDINT -c amd_dump.c -o amd_i_dump.o
- $(C) -DDINT -c amd_postorder.c -o amd_i_postorder.o
- $(C) -DDINT -c amd_post_tree.c -o amd_i_post_tree.o
- $(C) -DDINT -c amd_defaults.c -o amd_i_defaults.o
- $(C) -DDINT -c amd_order.c -o amd_i_order.o
- $(C) -DDINT -c amd_control.c -o amd_i_control.o
- $(C) -DDINT -c amd_info.c -o amd_i_info.o
- $(C) -DDINT -c amd_valid.c -o amd_i_valid.o
- $(C) -DDINT -c amd_preprocess.c -o amd_i_preprocess.o
- $(C) -DDLONG -c amd_aat.c -o amd_l_aat.o
- $(C) -DDLONG -c amd_1.c -o amd_l_1.o
- $(C) -DDLONG -c amd_2.c -o amd_l_2.o
- $(C) -DDLONG -c amd_dump.c -o amd_l_dump.o
- $(C) -DDLONG -c amd_postorder.c -o amd_l_postorder.o
- $(C) -DDLONG -c amd_post_tree.c -o amd_l_post_tree.o
- $(C) -DDLONG -c amd_defaults.c -o amd_l_defaults.o
- $(C) -DDLONG -c amd_order.c -o amd_l_order.o
- $(C) -DDLONG -c amd_control.c -o amd_l_control.o
- $(C) -DDLONG -c amd_info.c -o amd_l_info.o
- $(C) -DDLONG -c amd_valid.c -o amd_l_valid.o
- $(C) -DDLONG -c amd_preprocess.c -o amd_l_preprocess.o
- $(C) -c amd_global.c
+ $(C) -DDINT -c ../Source/amd_aat.c -o amd_i_aat.o
+ $(C) -DDINT -c ../Source/amd_1.c -o amd_i_1.o
+ $(C) -DDINT -c ../Source/amd_2.c -o amd_i_2.o
+ $(C) -DDINT -c ../Source/amd_dump.c -o amd_i_dump.o
+ $(C) -DDINT -c ../Source/amd_postorder.c -o amd_i_postorder.o
+ $(C) -DDINT -c ../Source/amd_post_tree.c -o amd_i_post_tree.o
+ $(C) -DDINT -c ../Source/amd_defaults.c -o amd_i_defaults.o
+ $(C) -DDINT -c ../Source/amd_order.c -o amd_i_order.o
+ $(C) -DDINT -c ../Source/amd_control.c -o amd_i_control.o
+ $(C) -DDINT -c ../Source/amd_info.c -o amd_i_info.o
+ $(C) -DDINT -c ../Source/amd_valid.c -o amd_i_valid.o
+ $(C) -DDINT -c ../Source/amd_preprocess.c -o amd_i_preprocess.o
+ $(C) -DDLONG -c ../Source/amd_aat.c -o amd_l_aat.o
+ $(C) -DDLONG -c ../Source/amd_1.c -o ../Source/amd_l_1.o
+ $(C) -DDLONG -c ../Source/amd_2.c -o amd_l_2.o
+ $(C) -DDLONG -c ../Source/amd_dump.c -o amd_l_dump.o
+ $(C) -DDLONG -c ../Source/amd_postorder.c -o amd_l_postorder.o
+ $(C) -DDLONG -c ../Source/amd_post_tree.c -o amd_l_post_tree.o
+ $(C) -DDLONG -c ../Source/amd_defaults.c -o amd_l_defaults.o
+ $(C) -DDLONG -c ../Source/amd_order.c -o amd_l_order.o
+ $(C) -DDLONG -c ../Source/amd_control.c -o amd_l_control.o
+ $(C) -DDLONG -c ../Source/amd_info.c -o amd_l_info.o
+ $(C) -DDLONG -c ../Source/amd_valid.c -o amd_l_valid.o
+ $(C) -DDLONG -c ../Source/amd_preprocess.c -o amd_l_preprocess.o
+ $(C) -c ../Source/amd_global.c
$(AR) ../Lib/libamd.a amd_i_aat.o amd_i_1.o amd_i_2.o amd_i_dump.o \
amd_i_postorder.o amd_i_post_tree.o amd_i_defaults.o amd_i_order.o \
amd_i_control.o amd_i_info.o amd_i_valid.o amd_l_aat.o amd_l_1.o \
@@ -51,8 +51,8 @@ everything:
#-------------------------------------------------------------------------------
fortran:
- $(F77) $(F77FLAGS) -c amd.f -o amd.o
- $(F77) $(F77FLAGS) -c amdbar.f -o amdbar.o
+ $(F77) $(F77FLAGS) -c ../Source/amd.f -o amd.o
+ $(F77) $(F77FLAGS) -c ../Source/amdbar.f -o amdbar.o
$(AR) ../Lib/libamdf77.a amd.o amdbar.o
- $(RANLIB) ../Lib/libamdf77.a
@@ -66,5 +66,4 @@ clean:
purge: distclean
distclean: clean
- - $(RM) ../Lib/libamd.a
- - $(RM) ../Lib/libamdf77.a
+ - $(RM) ../Lib/libamd.a ../Lib/libamdf77.a
diff --git a/src/C/SuiteSparse/AMD/Makefile b/src/C/SuiteSparse/AMD/Makefile
index 07c7aef..ce727aa 100644
--- a/src/C/SuiteSparse/AMD/Makefile
+++ b/src/C/SuiteSparse/AMD/Makefile
@@ -6,26 +6,26 @@ default: demo
include ../UFconfig/UFconfig.mk
-# Compile all C code, including the C-callable routine and the mexFunctions.
+# Compile all C code, including the C-callable routines.
# Do not compile the FORTRAN versions, or MATLAB interface.
demo:
- ( cd Source ; $(MAKE) )
+ ( cd Lib ; $(MAKE) )
( cd Demo ; $(MAKE) )
# Compile all C code, including the C-callable routine and the mexFunctions.
# Do not compile the FORTRAN versions.
all:
- ( cd Source ; $(MAKE) )
+ ( cd Lib ; $(MAKE) )
( cd Demo ; $(MAKE) )
( cd MATLAB ; $(MAKE) )
# compile just the C-callable libraries (not mexFunctions or Demos)
library:
- ( cd Source ; $(MAKE) )
+ ( cd Lib ; $(MAKE) )
# compile the FORTRAN libraries and demo programs (not compiled by "make all")
fortran:
- ( cd Source ; $(MAKE) fortran )
+ ( cd Lib ; $(MAKE) fortran )
( cd Demo ; $(MAKE) fortran )
# compile a FORTRAN demo program that calls the C version of AMD
@@ -35,14 +35,14 @@ cross:
# remove object files, but keep the compiled programs and library archives
clean:
- ( cd Source ; $(MAKE) clean )
+ ( cd Lib ; $(MAKE) clean )
( cd Demo ; $(MAKE) clean )
( cd MATLAB ; $(MAKE) clean )
( cd Doc ; $(MAKE) clean )
# clean, and then remove compiled programs and library archives
purge:
- ( cd Source ; $(MAKE) purge )
+ ( cd Lib ; $(MAKE) purge )
( cd Demo ; $(MAKE) purge )
( cd MATLAB ; $(MAKE) purge )
( cd Doc ; $(MAKE) purge )
diff --git a/src/C/SuiteSparse/AMD/README.txt b/src/C/SuiteSparse/AMD/README.txt
index ea6b415..c6fd2e0 100644
--- a/src/C/SuiteSparse/AMD/README.txt
+++ b/src/C/SuiteSparse/AMD/README.txt
@@ -1,3 +1,7 @@
+AMD Version 2.2, Copyright (c) 2007 by Timothy A.
+Davis, Patrick R. Amestoy, and Iain S. Duff. All Rights Reserved.
+AMD is available under alternate licences; contact T. Davis for details.
+
AMD: a set of routines for permuting sparse matrices prior to
factorization. Includes a version in C, a version in Fortran, and a MATLAB
mexFunction.
@@ -19,15 +23,12 @@ Quick start (for MATLAB users);
To compile, test, and install the AMD mexFunction, cd to the
AMD/MATLAB directory and type amd_make at the MATLAB prompt.
- NOTE: DO NOT ATTEMPT TO USE THIS CODE IN 64-BIT MATLAB (v7.3).
- It is not yet ported to that version of MATLAB.
+If you have MATLAB 7.2 or earlier and use "make mex", you must first edit
+UFconfig/UFconfig.h to remove the "-largeArrayDims" option from the MEX command
+(or just use amd_make.m inside MATLAB).
-------------------------------------------------------------------------------
-AMD Version 2.0, Copyright (c) 2006 by Timothy A.
-Davis, Patrick R. Amestoy, and Iain S. Duff. All Rights Reserved.
-AMD is available under alternate licences; contact T. Davis for details.
-
AMD License:
Your use or distribution of AMD or any modified version of
@@ -132,9 +133,6 @@ Files and directories in the AMD distribution:
Source directory:
---------------------------------------------------------------------------
- GNUmakefile a nice Makefile, for GNU make
- Makefile an ugly Unix Makefile (for older make's)
-
amd_order.c user-callable, primary AMD ordering routine
amd_control.c user-callable, prints the control parameters
amd_defaults.c user-callable, sets default control parameters
@@ -200,6 +198,7 @@ Files and directories in the AMD distribution:
amd2.m MATLAB help file for AMD
amd_make.m MATLAB m-file for compiling AMD mexFunction
+ amd_install.m compile and install the AMD mexFunction
amd_mex.c AMD mexFunction for MATLAB
@@ -211,4 +210,6 @@ Files and directories in the AMD distribution:
Lib directory: libamd.a and libamdf77.a libraries placed here
---------------------------------------------------------------------------
+ GNUmakefile a nice Makefile, for GNU make
+ Makefile an ugly Unix Makefile (for older make's)
libamd.def AMD definitions for Windows
diff --git a/src/C/SuiteSparse/AMD/Source/amd_1.c b/src/C/SuiteSparse/AMD/Source/amd_1.c
index a85c184..30cb277 100644
--- a/src/C/SuiteSparse/AMD/Source/amd_1.c
+++ b/src/C/SuiteSparse/AMD/Source/amd_1.c
@@ -3,7 +3,7 @@
/* ========================================================================= */
/* ------------------------------------------------------------------------- */
-/* AMD Version 2.0, Copyright (c) 2006 by Timothy A. Davis, */
+/* AMD, Copyright (c) Timothy A. Davis, */
/* Patrick R. Amestoy, and Iain S. Duff. See ../README.txt for License. */
/* email: davis at cise.ufl.edu CISE Department, Univ. of Florida. */
/* web: http://www.cise.ufl.edu/research/sparse/amd */
diff --git a/src/C/SuiteSparse/AMD/Source/amd_2.c b/src/C/SuiteSparse/AMD/Source/amd_2.c
index a760877..97b4f7a 100644
--- a/src/C/SuiteSparse/AMD/Source/amd_2.c
+++ b/src/C/SuiteSparse/AMD/Source/amd_2.c
@@ -3,7 +3,7 @@
/* ========================================================================= */
/* ------------------------------------------------------------------------- */
-/* AMD Version 2.0, Copyright (c) 2006 by Timothy A. Davis, */
+/* AMD, Copyright (c) Timothy A. Davis, */
/* Patrick R. Amestoy, and Iain S. Duff. See ../README.txt for License. */
/* email: davis at cise.ufl.edu CISE Department, Univ. of Florida. */
/* web: http://www.cise.ufl.edu/research/sparse/amd */
diff --git a/src/C/SuiteSparse/AMD/Source/amd_aat.c b/src/C/SuiteSparse/AMD/Source/amd_aat.c
index 5c92931..4f02b75 100644
--- a/src/C/SuiteSparse/AMD/Source/amd_aat.c
+++ b/src/C/SuiteSparse/AMD/Source/amd_aat.c
@@ -3,7 +3,7 @@
/* ========================================================================= */
/* ------------------------------------------------------------------------- */
-/* AMD Version 2.0, Copyright (c) 2006 by Timothy A. Davis, */
+/* AMD, Copyright (c) Timothy A. Davis, */
/* Patrick R. Amestoy, and Iain S. Duff. See ../README.txt for License. */
/* email: davis at cise.ufl.edu CISE Department, Univ. of Florida. */
/* web: http://www.cise.ufl.edu/research/sparse/amd */
diff --git a/src/C/SuiteSparse/AMD/Source/amd_control.c b/src/C/SuiteSparse/AMD/Source/amd_control.c
index a38eb4d..c2aec9f 100644
--- a/src/C/SuiteSparse/AMD/Source/amd_control.c
+++ b/src/C/SuiteSparse/AMD/Source/amd_control.c
@@ -3,7 +3,7 @@
/* ========================================================================= */
/* ------------------------------------------------------------------------- */
-/* AMD Version 2.0, Copyright (c) 2006 by Timothy A. Davis, */
+/* AMD, Copyright (c) Timothy A. Davis, */
/* Patrick R. Amestoy, and Iain S. Duff. See ../README.txt for License. */
/* email: davis at cise.ufl.edu CISE Department, Univ. of Florida. */
/* web: http://www.cise.ufl.edu/research/sparse/amd */
@@ -35,9 +35,9 @@ GLOBAL void AMD_control
aggressive = AMD_DEFAULT_AGGRESSIVE ;
}
- PRINTF (("\namd version %d.%d, %s: approximate minimum degree ordering:\n"
+ PRINTF (("\nAMD version %d.%d.%d, %s: approximate minimum degree ordering\n"
" dense row parameter: %g\n", AMD_MAIN_VERSION, AMD_SUB_VERSION,
- AMD_DATE, alpha)) ;
+ AMD_SUBSUB_VERSION, AMD_DATE, alpha)) ;
if (alpha < 0)
{
@@ -53,10 +53,12 @@ GLOBAL void AMD_control
if (aggressive)
{
- PRINTF ((" aggressive absorption: yes\n\n")) ;
+ PRINTF ((" aggressive absorption: yes\n")) ;
}
else
{
- PRINTF ((" aggressive absorption: no\n\n")) ;
+ PRINTF ((" aggressive absorption: no\n")) ;
}
+
+ PRINTF ((" size of AMD integer: %d\n\n", sizeof (Int))) ;
}
diff --git a/src/C/SuiteSparse/AMD/Source/amd_defaults.c b/src/C/SuiteSparse/AMD/Source/amd_defaults.c
index 59c10e4..ffe3f4b 100644
--- a/src/C/SuiteSparse/AMD/Source/amd_defaults.c
+++ b/src/C/SuiteSparse/AMD/Source/amd_defaults.c
@@ -3,7 +3,7 @@
/* ========================================================================= */
/* ------------------------------------------------------------------------- */
-/* AMD Version 2.0, Copyright (c) 2006 by Timothy A. Davis, */
+/* AMD, Copyright (c) Timothy A. Davis, */
/* Patrick R. Amestoy, and Iain S. Duff. See ../README.txt for License. */
/* email: davis at cise.ufl.edu CISE Department, Univ. of Florida. */
/* web: http://www.cise.ufl.edu/research/sparse/amd */
@@ -25,6 +25,7 @@ GLOBAL void AMD_defaults
)
{
Int i ;
+
if (Control != (double *) NULL)
{
for (i = 0 ; i < AMD_CONTROL ; i++)
diff --git a/src/C/SuiteSparse/AMD/Source/amd_dump.c b/src/C/SuiteSparse/AMD/Source/amd_dump.c
index cac782e..89d67b8 100644
--- a/src/C/SuiteSparse/AMD/Source/amd_dump.c
+++ b/src/C/SuiteSparse/AMD/Source/amd_dump.c
@@ -3,7 +3,7 @@
/* ========================================================================= */
/* ------------------------------------------------------------------------- */
-/* AMD Version 2.0, Copyright (c) 2006 by Timothy A. Davis, */
+/* AMD, Copyright (c) Timothy A. Davis, */
/* Patrick R. Amestoy, and Iain S. Duff. See ../README.txt for License. */
/* email: davis at cise.ufl.edu CISE Department, Univ. of Florida. */
/* web: http://www.cise.ufl.edu/research/sparse/amd */
@@ -40,7 +40,10 @@ GLOBAL void AMD_debug_init ( char *s )
fscanf (f, ID, &AMD_debug) ;
fclose (f) ;
}
- if (AMD_debug >= 0) printf ("%s: AMD_debug_init, D= "ID"\n", s, AMD_debug);
+ if (AMD_debug >= 0)
+ {
+ printf ("%s: AMD_debug_init, D= "ID"\n", s, AMD_debug) ;
+ }
}
/* ========================================================================= */
diff --git a/src/C/SuiteSparse/AMD/Source/amd_global.c b/src/C/SuiteSparse/AMD/Source/amd_global.c
index f306593..93f2b45 100644
--- a/src/C/SuiteSparse/AMD/Source/amd_global.c
+++ b/src/C/SuiteSparse/AMD/Source/amd_global.c
@@ -3,7 +3,7 @@
/* ========================================================================= */
/* ------------------------------------------------------------------------- */
-/* AMD Version 2.0, Copyright (c) 2006 by Timothy A. Davis, */
+/* AMD, Copyright (c) Timothy A. Davis, */
/* Patrick R. Amestoy, and Iain S. Duff. See ../README.txt for License. */
/* email: davis at cise.ufl.edu CISE Department, Univ. of Florida. */
/* web: http://www.cise.ufl.edu/research/sparse/amd */
diff --git a/src/C/SuiteSparse/AMD/Source/amd_info.c b/src/C/SuiteSparse/AMD/Source/amd_info.c
index 9cce8a9..0a842ad 100644
--- a/src/C/SuiteSparse/AMD/Source/amd_info.c
+++ b/src/C/SuiteSparse/AMD/Source/amd_info.c
@@ -3,7 +3,7 @@
/* ========================================================================= */
/* ------------------------------------------------------------------------- */
-/* AMD Version 2.0, Copyright (c) 2006 by Timothy A. Davis, */
+/* AMD, Copyright (c) Timothy A. Davis, */
/* Patrick R. Amestoy, and Iain S. Duff. See ../README.txt for License. */
/* email: davis at cise.ufl.edu CISE Department, Univ. of Florida. */
/* web: http://www.cise.ufl.edu/research/sparse/amd */
@@ -24,6 +24,9 @@ GLOBAL void AMD_info
{
double n, ndiv, nmultsubs_ldl, nmultsubs_lu, lnz, lnzd ;
+ PRINTF (("\nAMD version %d.%d.%d, %s, results:\n",
+ AMD_MAIN_VERSION, AMD_SUB_VERSION, AMD_SUBSUB_VERSION, AMD_DATE)) ;
+
if (!Info)
{
return ;
@@ -37,9 +40,7 @@ GLOBAL void AMD_info
lnzd = (n >= 0 && lnz >= 0) ? (n + lnz) : (-1) ;
/* AMD return status */
- PRINTF ((
- "\namd: approximate minimum degree ordering, results:\n"
- " status: ")) ;
+ PRINTF ((" status: ")) ;
if (Info [AMD_STATUS] == AMD_OK)
{
PRINTF (("OK\n")) ;
diff --git a/src/C/SuiteSparse/AMD/Source/amd_order.c b/src/C/SuiteSparse/AMD/Source/amd_order.c
index 438d3b7..d3f6853 100644
--- a/src/C/SuiteSparse/AMD/Source/amd_order.c
+++ b/src/C/SuiteSparse/AMD/Source/amd_order.c
@@ -3,7 +3,7 @@
/* ========================================================================= */
/* ------------------------------------------------------------------------- */
-/* AMD Version 2.0, Copyright (c) 2006 by Timothy A. Davis, */
+/* AMD, Copyright (c) Timothy A. Davis, */
/* Patrick R. Amestoy, and Iain S. Duff. See ../README.txt for License. */
/* email: davis at cise.ufl.edu CISE Department, Univ. of Florida. */
/* web: http://www.cise.ufl.edu/research/sparse/amd */
diff --git a/src/C/SuiteSparse/AMD/Source/amd_post_tree.c b/src/C/SuiteSparse/AMD/Source/amd_post_tree.c
index 9695b18..b4e063d 100644
--- a/src/C/SuiteSparse/AMD/Source/amd_post_tree.c
+++ b/src/C/SuiteSparse/AMD/Source/amd_post_tree.c
@@ -3,7 +3,7 @@
/* ========================================================================= */
/* ------------------------------------------------------------------------- */
-/* AMD Version 2.0, Copyright (c) 2006 by Timothy A. Davis, */
+/* AMD, Copyright (c) Timothy A. Davis, */
/* Patrick R. Amestoy, and Iain S. Duff. See ../README.txt for License. */
/* email: davis at cise.ufl.edu CISE Department, Univ. of Florida. */
/* web: http://www.cise.ufl.edu/research/sparse/amd */
diff --git a/src/C/SuiteSparse/AMD/Source/amd_postorder.c b/src/C/SuiteSparse/AMD/Source/amd_postorder.c
index 57c6ef2..4adcea3 100644
--- a/src/C/SuiteSparse/AMD/Source/amd_postorder.c
+++ b/src/C/SuiteSparse/AMD/Source/amd_postorder.c
@@ -3,7 +3,7 @@
/* ========================================================================= */
/* ------------------------------------------------------------------------- */
-/* AMD Version 2.0, Copyright (c) 2006 by Timothy A. Davis, */
+/* AMD, Copyright (c) Timothy A. Davis, */
/* Patrick R. Amestoy, and Iain S. Duff. See ../README.txt for License. */
/* email: davis at cise.ufl.edu CISE Department, Univ. of Florida. */
/* web: http://www.cise.ufl.edu/research/sparse/amd */
diff --git a/src/C/SuiteSparse/AMD/Source/amd_preprocess.c b/src/C/SuiteSparse/AMD/Source/amd_preprocess.c
index 6b8bd66..86ea07f 100644
--- a/src/C/SuiteSparse/AMD/Source/amd_preprocess.c
+++ b/src/C/SuiteSparse/AMD/Source/amd_preprocess.c
@@ -3,7 +3,7 @@
/* ========================================================================= */
/* ------------------------------------------------------------------------- */
-/* AMD Version 2.0, Copyright (c) 2006 by Timothy A. Davis, */
+/* AMD, Copyright (c) Timothy A. Davis, */
/* Patrick R. Amestoy, and Iain S. Duff. See ../README.txt for License. */
/* email: davis at cise.ufl.edu CISE Department, Univ. of Florida. */
/* web: http://www.cise.ufl.edu/research/sparse/amd */
diff --git a/src/C/SuiteSparse/AMD/Source/amd_valid.c b/src/C/SuiteSparse/AMD/Source/amd_valid.c
index cd91e21..4d05925 100644
--- a/src/C/SuiteSparse/AMD/Source/amd_valid.c
+++ b/src/C/SuiteSparse/AMD/Source/amd_valid.c
@@ -3,7 +3,7 @@
/* ========================================================================= */
/* ------------------------------------------------------------------------- */
-/* AMD Version 2.0, Copyright (c) 2006 by Timothy A. Davis, */
+/* AMD, Copyright (c) Timothy A. Davis, */
/* Patrick R. Amestoy, and Iain S. Duff. See ../README.txt for License. */
/* email: davis at cise.ufl.edu CISE Department, Univ. of Florida. */
/* web: http://www.cise.ufl.edu/research/sparse/amd */
@@ -46,6 +46,7 @@ GLOBAL Int AMD_valid
)
{
Int nz, j, p1, p2, ilast, i, p, result = AMD_OK ;
+
if (n_row < 0 || n_col < 0 || Ap == NULL || Ai == NULL)
{
return (AMD_INVALID) ;
diff --git a/src/C/SuiteSparse/CHOLMOD/Check/cholmod_check.c b/src/C/SuiteSparse/CHOLMOD/Check/cholmod_check.c
index 49a638b..13a092d 100644
--- a/src/C/SuiteSparse/CHOLMOD/Check/cholmod_check.c
+++ b/src/C/SuiteSparse/CHOLMOD/Check/cholmod_check.c
@@ -562,6 +562,7 @@ static int check_common
/* workspace and parameters are valid */
P3 ("%s", " OK\n") ;
+ P4 ("%s", "\n") ;
return (TRUE) ;
}
@@ -842,6 +843,7 @@ static UF_long check_sparse
/* matrix is valid */
P4 (" nnz on diagonal: "ID"\n", dnz) ;
P3 ("%s", " OK\n") ;
+ P4 ("%s", "\n") ;
*nnzdiag = dnz ;
return (TRUE) ;
}
@@ -980,6 +982,7 @@ static int check_dense
/* dense is valid */
P3 ("%s", " OK\n") ;
+ P4 ("%s", "\n") ;
return (TRUE) ;
}
@@ -1068,6 +1071,7 @@ static int check_subset
if (len <= 0 || S == NULL)
{
P3 ("%s", " OK\n") ;
+ P4 ("%s", "\n") ;
return (TRUE) ;
}
@@ -1098,6 +1102,7 @@ static int check_subset
}
}
P3 ("%s", " OK\n") ;
+ P4 ("%s", "\n") ;
return (TRUE) ;
}
@@ -1317,6 +1322,7 @@ int CHOLMOD(print_perm)
if (ok)
{
P3 ("%s", " OK\n") ;
+ P4 ("%s", "\n") ;
}
return (ok) ;
}
@@ -1381,6 +1387,7 @@ static int check_parent
}
}
P3 ("%s", " OK\n") ;
+ P4 ("%s", "\n") ;
return (TRUE) ;
}
@@ -1933,6 +1940,7 @@ static int check_factor
/* factor is valid */
P3 (" nz "ID"", lnz) ;
P3 ("%s", " OK\n") ;
+ P4 ("%s", "\n") ;
return (TRUE) ;
}
@@ -2118,6 +2126,7 @@ static int check_triplet
/* triplet matrix is valid */
P3 ("%s", " OK\n") ;
+ P4 ("%s", "\n") ;
return (TRUE) ;
}
diff --git a/src/C/SuiteSparse/CHOLMOD/Check/cholmod_read.c b/src/C/SuiteSparse/CHOLMOD/Check/cholmod_read.c
index afdbddf..089f524 100644
--- a/src/C/SuiteSparse/CHOLMOD/Check/cholmod_read.c
+++ b/src/C/SuiteSparse/CHOLMOD/Check/cholmod_read.c
@@ -518,7 +518,7 @@ static cholmod_triplet *read_triplet
cholmod_triplet *T ;
double l1, l2 ;
Int nitems, xtype, unknown, k, nshould, is_lower, is_upper, one_based, i, j,
- imax, jmax, ignore, skew_symmetric, p, complex_symmetric ;
+ imax, jmax, skew_symmetric, p, complex_symmetric ;
size_t s, nnz2, extra ;
int ok = TRUE ;
@@ -895,9 +895,9 @@ static cholmod_dense *read_dense
)
{
double x, z ;
- double *Xx ;
+ double *Xx = NULL ;
cholmod_dense *X ;
- Int nitems, xtype, nshould, i, j, k, kup, first ;
+ Int nitems, xtype = -1, nshould = 0, i, j, k, kup, first ;
/* ---------------------------------------------------------------------- */
/* quick return for empty matrix */
@@ -1286,7 +1286,8 @@ void *CHOLMOD(read_matrix)
if (*mtype == CHOLMOD_TRIPLET)
{
- /* read in the triplet matrix */
+ /* read in the triplet matrix, converting to unsymmetric format if
+ * prefer == 1 */
T = read_triplet (f, nrow, ncol, nnz, stype, prefer == 1, buf, Common) ;
if (prefer == 0)
{
diff --git a/src/C/SuiteSparse/CHOLMOD/Cholesky/cholmod_solve.c b/src/C/SuiteSparse/CHOLMOD/Cholesky/cholmod_solve.c
index f8ab992..b51fc80 100644
--- a/src/C/SuiteSparse/CHOLMOD/Cholesky/cholmod_solve.c
+++ b/src/C/SuiteSparse/CHOLMOD/Cholesky/cholmod_solve.c
@@ -1062,7 +1062,7 @@ cholmod_dense *CHOLMOD(solve)
/* ------------------------------------------------------------------ */
#ifndef NSUPERNODAL
- Int ok ;
+ Int blas_ok = TRUE ;
/* allocate workspace */
cholmod_dense *E ;
@@ -1085,24 +1085,31 @@ cholmod_dense *CHOLMOD(solve)
if (sys == CHOLMOD_A || sys == CHOLMOD_LDLt)
{
- ok = CHOLMOD(super_lsolve) (L, Y, E, Common) ; /* Y = L\Y */
- ok = ok && CHOLMOD(super_ltsolve) (L, Y, E, Common) ; /* Y = L'\Y*/
+ blas_ok = CHOLMOD(super_lsolve) (L, Y, E, Common) ; /* Y = L\Y */
+ blas_ok = blas_ok &&
+ CHOLMOD(super_ltsolve) (L, Y, E, Common) ; /* Y = L'\Y*/
}
else if (sys == CHOLMOD_L || sys == CHOLMOD_LD)
{
- ok = CHOLMOD(super_lsolve) (L, Y, E, Common) ; /* Y = L\Y */
+ blas_ok = CHOLMOD(super_lsolve) (L, Y, E, Common) ; /* Y = L\Y */
}
else if (sys == CHOLMOD_Lt || sys == CHOLMOD_DLt)
{
- ok = CHOLMOD(super_ltsolve) (L, Y, E, Common) ; /* Y = L'\Y*/
+ blas_ok = CHOLMOD(super_ltsolve) (L, Y, E, Common) ; /* Y = L'\Y*/
}
CHOLMOD(free_dense) (&E, Common) ;
iperm (Y, Perm, 0, nrhs, X) ; /* X = P'*Y */
- if (CHECK_BLAS_INT && !ok)
+ if (CHECK_BLAS_INT && !blas_ok)
{
- /* integer overflow in the BLAS */
+ /* Integer overflow in the BLAS. This is probably impossible,
+ * since the BLAS were used to create the supernodal factorization.
+ * It might be possible for the calls to the BLAS to differ between
+ * factorization and forward/backsolves, however. This statement
+ * is untested; it does not appear in the compiled code if
+ * CHECK_BLAS_INT is true (when the same integer is used in CHOLMOD
+ * and the BLAS. */
CHOLMOD(free_dense) (&X, Common) ;
}
diff --git a/src/C/SuiteSparse/CHOLMOD/Doc/ChangeLog b/src/C/SuiteSparse/CHOLMOD/Doc/ChangeLog
index f16584d..6cc80d3 100644
--- a/src/C/SuiteSparse/CHOLMOD/Doc/ChangeLog
+++ b/src/C/SuiteSparse/CHOLMOD/Doc/ChangeLog
@@ -1,3 +1,22 @@
+May 31, 2007, version 1.5.0
+
+ * 64-bit MATLAB interface
+
+ * MATLAB interface back-ported to MATLAB 6.1.
+
+ * bug fix: solving Dx=b using a supernodal factorization, in
+ cholmod_l_solve, when sizeof(UF_long) > sizeof(BLAS integer)
+
+ * changes to Makefiles to reflect directory changes in COLAMD and CCOLAMD
+ v2.7.0 directory structure (CHOLMOD v1.5 requires v2.7.0 of those
+ two packages)
+
+ * update to Modify/cholmod_updown.c, to allow input vector R to be packed
+ or unpacked.
+
+ * bug fix to Tcov/huge.c test code, for 64-bit case (this has no effect
+ on the CHOLMOD library itself, just the test code)
+
Dec 12, 2006, version 1.4.0
* added support for large files (larger than 2GB)
diff --git a/src/C/SuiteSparse/CHOLMOD/Doc/Makefile b/src/C/SuiteSparse/CHOLMOD/Doc/Makefile
index e661e91..6945903 100644
--- a/src/C/SuiteSparse/CHOLMOD/Doc/Makefile
+++ b/src/C/SuiteSparse/CHOLMOD/Doc/Makefile
@@ -36,7 +36,6 @@ M = \
../MATLAB/ldlsolve.m \
../MATLAB/ldlsplit.m \
../MATLAB/ldlupdate.m \
- ../MATLAB/lu_normest.m \
../MATLAB/metis.m \
../MATLAB/nesdis.m \
../MATLAB/resymbol.m \
@@ -62,7 +61,6 @@ UserGuide.pdf: UserGuide.tex UserGuide.bib $(I) $(C) $(M) Makefile getproto rule
./getmproto ../MATLAB/ldlsolve.m > _ldlsolve_m.tex
./getmproto ../MATLAB/ldlsplit.m > _ldlsplit_m.tex
./getmproto ../MATLAB/ldlupdate.m > _ldlupdate_m.tex
- ./getmproto ../MATLAB/lu_normest.m > _lu_normest_m.tex
./getmproto ../MATLAB/metis.m > _metis_m.tex
./getmproto ../MATLAB/mread.m > _mread_m.tex
./getmproto ../MATLAB/spsym.m > _spsym_m.tex
diff --git a/src/C/SuiteSparse/CHOLMOD/Doc/UserGuide.pdf b/src/C/SuiteSparse/CHOLMOD/Doc/UserGuide.pdf
deleted file mode 100644
index d25561f..0000000
Binary files a/src/C/SuiteSparse/CHOLMOD/Doc/UserGuide.pdf and /dev/null differ
diff --git a/src/C/SuiteSparse/CHOLMOD/Doc/UserGuide.tex b/src/C/SuiteSparse/CHOLMOD/Doc/UserGuide.tex
index c3fa5a9..1f656fd 100644
--- a/src/C/SuiteSparse/CHOLMOD/Doc/UserGuide.tex
+++ b/src/C/SuiteSparse/CHOLMOD/Doc/UserGuide.tex
@@ -21,7 +21,7 @@ Dept. of Computer and Information Science and Engineering \\
Univ. of Florida, Gainesville, FL}
\title{User Guide for CHOLMOD: a sparse Cholesky factorization and
modification package}
-\date{Version 1.4, Dec 12, 2006}
+\date{Version 1.5, May 31, 2007}
\maketitle
%-------------------------------------------------------------------------------
@@ -374,7 +374,6 @@ directory. The following functions are provided:
{\tt ldlsolve} & {\tt x = L'}$\backslash${\tt (D}$\backslash${\tt (L}$\backslash${\tt b))} \\
{\tt ldlsplit} & split the output of {\tt ldlchol} into {\tt L} and {\tt D} \\
{\tt ldlupdate} & update/downdate an {\tt L*D*L'} factorization \\
-{\tt lu\_normest} & estimate {\tt norm(A-L*U')} \\
{\tt metis} & interface to {\tt METIS\_NodeND} ordering \\
{\tt mread} & read a sparse or dense Matrix Market file \\
{\tt mwrite} & write a sparse or dense Matrix Market file \\
@@ -406,14 +405,11 @@ Each function is described in the next sections.
\subsection{{\tt lchol}: $\m{LL}\tr$ factorization} \input{_lchol_m.tex}
\subsection{{\tt ldlchol}: $\m{LDL}\tr$ factorization} \input{_ldlchol_m.tex}
\newpage
-\subsection{{\tt ldl\_normest}: estimate $||\m{A}-\m{LDL}\tr||$} \input{_ldl_normest_m.tex}
\subsection{{\tt ldlsolve}: solve using an $\m{LDL}\tr$ factorization} \input{_ldlsolve_m.tex}
\subsection{{\tt ldlsplit}: split an $\m{LDL}\tr$ factorization} \input{_ldlsplit_m.tex}
\newpage
\subsection{{\tt ldlupdate}: update/downdate an $\m{LDL}\tr$ factorization} \input{_ldlupdate_m.tex}
\newpage
-\subsection{{\tt lu\_normest}: estimate $||\m{A}-\m{LU}||$} \input{_lu_normest_m.tex}
-\newpage
\subsection{{\tt mread}: read a sparse or dense matrix from a Matrix Market file}\input{_mread_m.tex}
\subsection{{\tt mwrite}: write a sparse or densematrix to a Matrix Market file} \input{_mwrite_m.tex}
\newpage
@@ -488,6 +484,13 @@ Also ensure {\tt -fexceptions} is in the {\tt CFLAGS} option in the
If you do not make these modifications, the CHOLMOD mexFunctions
will terminate MATLAB if they encounter an error.
+If you have MATLAB 7.2 or earlier and use {\tt make mex} in the
+{\tt CHOLMOD} directory (equivalently, {\tt make} in {\tt CHOLMOD/MATLAB}),
+you must first edit
+{\tt UFconfig/UFconfig.h} to remove the {\tt -largeArrayDims}
+option from the MEX command
+(or just use {\tt cholmod\_make.m} inside MATLAB).
+
Next, compile your METIS 4.0.1 library by typing {\tt make} in the
{\tt metis-4.0} directory. Then type {\tt make} in the {\tt CHOLMOD/MATLAB}
directory. This will compile the C-callable libraries for
diff --git a/src/C/SuiteSparse/CHOLMOD/Include/cholmod_core.h b/src/C/SuiteSparse/CHOLMOD/Include/cholmod_core.h
index 65cf0c0..e282f9d 100644
--- a/src/C/SuiteSparse/CHOLMOD/Include/cholmod_core.h
+++ b/src/C/SuiteSparse/CHOLMOD/Include/cholmod_core.h
@@ -244,10 +244,10 @@
* #endif
*/
-#define CHOLMOD_DATE "Dec 12, 2006"
+#define CHOLMOD_DATE "May 31, 2007"
#define CHOLMOD_VER_CODE(main,sub) ((main) * 1000 + (sub))
#define CHOLMOD_MAIN_VERSION 1
-#define CHOLMOD_SUB_VERSION 4
+#define CHOLMOD_SUB_VERSION 5
#define CHOLMOD_SUBSUB_VERSION 0
#define CHOLMOD_VERSION \
CHOLMOD_VER_CODE(CHOLMOD_MAIN_VERSION,CHOLMOD_SUB_VERSION)
diff --git a/src/C/SuiteSparse/CHOLMOD/Include/cholmod_internal.h b/src/C/SuiteSparse/CHOLMOD/Include/cholmod_internal.h
index 0cabd5a..89160d6 100644
--- a/src/C/SuiteSparse/CHOLMOD/Include/cholmod_internal.h
+++ b/src/C/SuiteSparse/CHOLMOD/Include/cholmod_internal.h
@@ -212,7 +212,7 @@ size_t cholmod_l_add_size_t (size_t a, size_t b, int *ok) ;
size_t cholmod_l_mult_size_t (size_t a, size_t k, int *ok) ;
/* -------------------------------------------------------------------------- */
-/* double, UF_long */
+/* double (also complex double), UF_long */
/* -------------------------------------------------------------------------- */
#ifdef DLONG
@@ -255,7 +255,7 @@ size_t cholmod_l_mult_size_t (size_t a, size_t k, int *ok) ;
#error "single-precision not yet supported"
/* -------------------------------------------------------------------------- */
-/* double, int: this is the default */
+/* double (also complex double), int: this is the default */
/* -------------------------------------------------------------------------- */
#else
diff --git a/src/C/SuiteSparse/CHOLMOD/Include/cholmod_matrixops.h b/src/C/SuiteSparse/CHOLMOD/Include/cholmod_matrixops.h
index 59f19da..b4d090b 100644
--- a/src/C/SuiteSparse/CHOLMOD/Include/cholmod_matrixops.h
+++ b/src/C/SuiteSparse/CHOLMOD/Include/cholmod_matrixops.h
@@ -133,7 +133,7 @@ int cholmod_sdmult
(
/* ---- input ---- */
cholmod_sparse *A, /* sparse matrix to multiply */
- int transpose, /* use A if 0, or A' if 1, or A.' if -1 */
+ int transpose, /* use A if 0, or A' otherwise */
double alpha [2], /* scale factor for A */
double beta [2], /* scale factor for Y */
cholmod_dense *X, /* dense matrix to multiply */
diff --git a/src/C/SuiteSparse/CHOLMOD/Lib/Makefile b/src/C/SuiteSparse/CHOLMOD/Lib/Makefile
index 117f869..7d736d3 100644
--- a/src/C/SuiteSparse/CHOLMOD/Lib/Makefile
+++ b/src/C/SuiteSparse/CHOLMOD/Lib/Makefile
@@ -115,10 +115,9 @@ libcholmod.a: $(OBJ)
$(OBJ): $(INC)
-I = -I../../AMD/Include -I../../AMD/Source -I../../COLAMD \
- -I$(METIS_PATH)/Lib -I../../CCOLAMD -I../../CAMD/Include -I../Include \
- -I../../UFconfig
-
+I = -I../../AMD/Include -I../../AMD/Source -I../../COLAMD/Include \
+ -I$(METIS_PATH)/Lib -I../../CCOLAMD/Include -I../../CAMD/Include \
+ -I../Include -I../../UFconfig
#-------------------------------------------------------------------------------
# Check Module:
diff --git a/src/C/SuiteSparse/CHOLMOD/README.txt b/src/C/SuiteSparse/CHOLMOD/README.txt
index 61f0131..985a190 100644
--- a/src/C/SuiteSparse/CHOLMOD/README.txt
+++ b/src/C/SuiteSparse/CHOLMOD/README.txt
@@ -1,4 +1,5 @@
CHOLMOD: a sparse CHOLesky MODification package
+Version 1.5, May 31, 2007. Copyright (c) 2005-2007.
-----------------------------------------------
CHOLMOD is a set of routines for factorizing sparse symmetric positive
@@ -12,7 +13,6 @@ CHOLMOD: a sparse CHOLesky MODification package
C and MATLAB interfaces. This code works on Microsoft Windows and many
versions of Unix and Linux.
-Version 1.4, Dec 12, 2006. Copyright (c) 2005-2006.
Some Modules of CHOLMOD are copyrighted by the University of Florida (the
Core and Partition Modules). The rest are copyrighted by the authors:
@@ -54,8 +54,9 @@ also ensures your mexFunctions are compiled with -fexceptions, so that
exceptions are handled properly (when hitting control-C in the MATLAB command
window, for example).
- NOTE: DO NOT ATTEMPT TO USE THIS CODE IN 64-BIT MATLAB (v7.3).
- It is not yet ported to that version of MATLAB.
+If you have MATLAB 7.2 or earlier and use "make mex", you must first edit
+UFconfig/UFconfig.h to remove the "-largeArrayDims" option from the MEX command
+(or just use cholmod_make.m inside MATLAB).
On the Pentium, do NOT use the Intel MKL BLAS prior to MKL Version 8.0 with
CHOLMOD. Older versions (prior to 8.0) have a bug in dgemm when computing
diff --git a/src/C/SuiteSparse/CHOLMOD/Supernodal/cholmod_super_solve.c b/src/C/SuiteSparse/CHOLMOD/Supernodal/cholmod_super_solve.c
index 58191f0..2d81222 100644
--- a/src/C/SuiteSparse/CHOLMOD/Supernodal/cholmod_super_solve.c
+++ b/src/C/SuiteSparse/CHOLMOD/Supernodal/cholmod_super_solve.c
@@ -42,7 +42,7 @@
* workspace: none
*/
-int CHOLMOD(super_lsolve)
+int CHOLMOD(super_lsolve) /* TRUE if OK, FALSE if BLAS overflow occured */
(
/* ---- input ---- */
cholmod_factor *L, /* factor to use for the forward solve */
@@ -54,7 +54,7 @@ int CHOLMOD(super_lsolve)
cholmod_common *Common
)
{
- int ok = TRUE ;
+ int blas_ok = TRUE ;
/* ---------------------------------------------------------------------- */
/* check inputs */
@@ -108,19 +108,19 @@ int CHOLMOD(super_lsolve)
{
case CHOLMOD_REAL:
- ok = r_cholmod_super_lsolve (L, X, E, Common) ;
+ blas_ok = r_cholmod_super_lsolve (L, X, E, Common) ;
break ;
case CHOLMOD_COMPLEX:
- ok = c_cholmod_super_lsolve (L, X, E, Common) ;
+ blas_ok = c_cholmod_super_lsolve (L, X, E, Common) ;
break ;
}
- if (CHECK_BLAS_INT && !ok)
+ if (CHECK_BLAS_INT && !blas_ok)
{
ERROR (CHOLMOD_TOO_LARGE, "problem too large for the BLAS") ;
}
- return (ok) ;
+ return (blas_ok) ;
}
@@ -137,7 +137,7 @@ int CHOLMOD(super_lsolve)
* workspace: none
*/
-int CHOLMOD(super_ltsolve)
+int CHOLMOD(super_ltsolve) /* TRUE if OK, FALSE if BLAS overflow occured */
(
/* ---- input ---- */
cholmod_factor *L, /* factor to use for the backsolve */
@@ -149,7 +149,7 @@ int CHOLMOD(super_ltsolve)
cholmod_common *Common
)
{
- int ok = TRUE ;
+ int blas_ok = TRUE ;
/* ---------------------------------------------------------------------- */
/* check inputs */
@@ -203,19 +203,19 @@ int CHOLMOD(super_ltsolve)
{
case CHOLMOD_REAL:
- ok = r_cholmod_super_ltsolve (L, X, E, Common) ;
+ blas_ok = r_cholmod_super_ltsolve (L, X, E, Common) ;
break ;
case CHOLMOD_COMPLEX:
- ok = c_cholmod_super_ltsolve (L, X, E, Common) ;
+ blas_ok = c_cholmod_super_ltsolve (L, X, E, Common) ;
break ;
}
- if (CHECK_BLAS_INT && !ok)
+ if (CHECK_BLAS_INT && !blas_ok)
{
ERROR (CHOLMOD_TOO_LARGE, "problem too large for the BLAS") ;
}
- return (ok) ;
+ return (blas_ok) ;
}
#endif
diff --git a/src/C/SuiteSparse/COLAMD/ChangeLog b/src/C/SuiteSparse/COLAMD/ChangeLog
deleted file mode 100644
index e16e946..0000000
--- a/src/C/SuiteSparse/COLAMD/ChangeLog
+++ /dev/null
@@ -1,123 +0,0 @@
-Dec 12, 2006, version 2.5.2
-
- * minor MATLAB cleanup. MATLAB functions renamed colamd2 and symamd2,
- so that they do not conflict with the built-in versions. Note that
- the MATLAB built-in functions colamd and symamd are identical to
- the colamd and symamd functions here.
-
-Aug 31, 2006: Version 2.5.1
-
- * minor change to colamd.m and symamd.m, to use etree instead
- of sparsfun.
-
-Apr. 30, 2006: Version 2.5
-
- * colamd_recommended modified, to do more careful integer overflow
- checking. It now returns size_t, not int. colamd_l_recommended
- also returns size_t. A zero is returned if an error occurs. A
- postive return value denotes success. In v2.4 and earlier,
- -1 was returned on error (an int or long).
-
- * long replaced with UF_long integer, which is long except on WIN64.
-
-Nov 15, 2005:
-
- * minor editting of comments; version number (2.4) unchanged.
-
-Changes from Version 2.3 to 2.4 (Aug 30, 2005)
-
- * Makefile now relies on ../UFconfig/UFconfig.mk
-
- * changed the dense row/col detection. The meaning of the knobs
- has thus changed.
-
- * added an option to turn off aggressive absorption. It was
- always on in versions 2.3 and earlier.
-
- * added a #define'd version number
-
- * added a function pointer (colamd_printf) for COLAMD's printing.
-
- * added a -DNPRINT option, to turn off printing at compile-time.
-
- * added a check for integer overflow in colamd_recommended
-
- * minor changes to allow for more simpler 100% test coverage
-
- * bug fix. If symamd v2.3 fails to allocate its copy of the input
- matrix, then it erroneously frees a calloc'd workspace twice.
- This bug has no effect on the MATLAB symamd mexFunction, since
- mxCalloc terminates the mexFunction if it fails to allocate
- memory. Similarly, UMFPACK is not affected because it does not
- use symamd. The bug has no effect on the colamd ordering
- routine in v2.3.
-
-Changes from Version 2.2 to 2.3 (Sept. 8, 2003)
-
- * removed the call to the MATLAB spparms ('spumoni') function.
- This can take a lot of time if you are ordering many small
- matrices. Only affects the MATLAB interface (colamdmex.c,
- symamdmex.c, colamdtestmex.c, and symamdtestmex.c). The
- usage of the optional 2nd argument to the colamd and symamd
- mexFunctions was changed accordingly.
-
-Changes from Version 2.1 to 2.2 (Sept. 23, 2002)
-
- * extensive testing routines added (colamd_test.m, colamdtestmex.c,
- and symamdtestmex.c), and the Makefile modified accordingly.
-
- * a few typos in the comments corrected
-
- * use of the MATLAB "flops" command removed from colamd_demo, and an
- m-file routine luflops.m added.
-
- * an explicit typecast from unsigned to int added, for COLAMD_C and
- COLAMD_R in colamd.h.
-
- * #include <stdio.h> added to colamd_example.c
-
-
-Changes from Version 2.0 to 2.1 (May 4, 2001)
-
- * TRUE and FALSE are predefined on some systems, so they are defined
- here only if not already defined.
-
- * web site changed
-
- * UNIX Makefile modified, to handle the case if "." is not in your path.
-
-
-Changes from Version 1.0 to 2.0 (January 31, 2000)
-
- No bugs were found in version 1.1. These changes merely add new
- functionality.
-
- * added the COLAMD_RECOMMENDED (nnz, n_row, n_col) macro.
-
- * moved the output statistics, from A, to a separate output argument.
- The arguments changed for the C-callable routines.
-
- * added colamd_report and symamd_report.
-
- * added a C-callable symamd routine. Formerly, symamd was only
- available as a mexFunction from MATLAB.
-
- * added error-checking to symamd. Formerly, it assumed its input
- was error-free.
-
- * added the optional stats and knobs arguments to the symamd mexFunction
-
- * deleted colamd_help. A help message is still available from
- "help colamd" and "help symamd" in MATLAB.
-
- * deleted colamdtree.m and symamdtree.m. Now, colamd.m and symamd.m
- also do the elimination tree post-ordering. The Version 1.1
- colamd and symamd mexFunctions, which do not do the post-
- ordering, are now visible as colamdmex and symamdmex from
- MATLAB. Essentialy, the post-ordering is now the default
- behavior of colamd.m and symamd.m, to match the behavior of
- colmmd and symmmd. The post-ordering is only available in the
- MATLAB interface, not the C-callable interface.
-
- * made a slight change to the dense row/column detection in symamd,
- to match the stated specifications.
diff --git a/src/C/SuiteSparse/COLAMD/Demo/Makefile b/src/C/SuiteSparse/COLAMD/Demo/Makefile
new file mode 100644
index 0000000..f08df2a
--- /dev/null
+++ b/src/C/SuiteSparse/COLAMD/Demo/Makefile
@@ -0,0 +1,43 @@
+#-----------------------------------------------------------------------------
+# compile the COLAMD demo
+#-----------------------------------------------------------------------------
+
+default: colamd_example colamd_l_example
+
+include ../../UFconfig/UFconfig.mk
+
+I = -I../Include -I../../UFconfig
+
+C = $(CC) $(CFLAGS) $(I)
+
+library:
+ ( cd ../Lib ; $(MAKE) )
+
+#------------------------------------------------------------------------------
+# Create the demo program, run it, and compare the output
+#------------------------------------------------------------------------------
+
+dist:
+
+colamd_example: colamd_example.c library
+ $(C) -o colamd_example colamd_example.c ../Lib/libcolamd.a -lm
+ - ./colamd_example > my_colamd_example.out
+ - diff colamd_example.out my_colamd_example.out
+
+colamd_l_example: colamd_l_example.c library
+ $(C) -o colamd_l_example colamd_l_example.c ../Lib/libcolamd.a -lm
+ - ./colamd_l_example > my_colamd_l_example.out
+ - diff colamd_example.out my_colamd_example.out
+
+#------------------------------------------------------------------------------
+# Remove all but the files in the original distribution
+#------------------------------------------------------------------------------
+
+clean:
+ - $(RM) $(CLEAN)
+
+purge: distclean
+
+distclean: clean
+ - $(RM) colamd_example colamd_l_example
+ - $(RM) my_colamd_example.out my_colamd_l_example.out
diff --git a/src/C/SuiteSparse/COLAMD/colamd_example.c b/src/C/SuiteSparse/COLAMD/Demo/colamd_example.c
similarity index 100%
rename from src/C/SuiteSparse/COLAMD/colamd_example.c
rename to src/C/SuiteSparse/COLAMD/Demo/colamd_example.c
diff --git a/src/C/SuiteSparse/COLAMD/colamd_example.out b/src/C/SuiteSparse/COLAMD/Demo/colamd_example.out
similarity index 92%
rename from src/C/SuiteSparse/COLAMD/colamd_example.out
rename to src/C/SuiteSparse/COLAMD/Demo/colamd_example.out
index 45377ca..0c4026c 100644
--- a/src/C/SuiteSparse/COLAMD/colamd_example.out
+++ b/src/C/SuiteSparse/COLAMD/Demo/colamd_example.out
@@ -15,7 +15,7 @@ Column 3, with 2 entries:
row 1
row 3
-colamd version 2.6, Dec 12, 2006: OK.
+colamd version 2.7, May 31, 2007: OK.
colamd: number of dense or empty rows ignored: 0
colamd: number of dense or empty columns ignored: 0
colamd: number of garbage collections performed: 0
@@ -38,7 +38,7 @@ Column 3, with 1 entries:
row 4
Column 4, with 0 entries:
-symamd version 2.6, Dec 12, 2006: OK.
+symamd version 2.7, May 31, 2007: OK.
symamd: number of dense or empty rows ignored: 0
symamd: number of dense or empty columns ignored: 0
symamd: number of garbage collections performed: 0
diff --git a/src/C/SuiteSparse/COLAMD/colamd_l_example.c b/src/C/SuiteSparse/COLAMD/Demo/colamd_l_example.c
similarity index 100%
rename from src/C/SuiteSparse/COLAMD/colamd_l_example.c
rename to src/C/SuiteSparse/COLAMD/Demo/colamd_l_example.c
diff --git a/src/C/SuiteSparse/COLAMD/colamd_l_example.out b/src/C/SuiteSparse/COLAMD/Demo/colamd_l_example.out
similarity index 92%
rename from src/C/SuiteSparse/COLAMD/colamd_l_example.out
rename to src/C/SuiteSparse/COLAMD/Demo/colamd_l_example.out
index f088b27..bbf8901 100644
--- a/src/C/SuiteSparse/COLAMD/colamd_l_example.out
+++ b/src/C/SuiteSparse/COLAMD/Demo/colamd_l_example.out
@@ -15,7 +15,7 @@ Column 3, with 2 entries:
row 1
row 3
-colamd version 2.6, Dec 12, 2006: OK.
+colamd version 2.7, May 31, 2007: OK.
colamd: number of dense or empty rows ignored: 0
colamd: number of dense or empty columns ignored: 0
colamd: number of garbage collections performed: 0
@@ -38,7 +38,7 @@ Column 3, with 1 entries:
row 4
Column 4, with 0 entries:
-symamd version 2.6, Dec 12, 2006: OK.
+symamd version 2.7, May 31, 2007: OK.
symamd: number of dense or empty rows ignored: 0
symamd: number of dense or empty columns ignored: 0
symamd: number of garbage collections performed: 0
diff --git a/src/C/SuiteSparse/COLAMD/Doc/ChangeLog b/src/C/SuiteSparse/COLAMD/Doc/ChangeLog
new file mode 100644
index 0000000..29308e9
--- /dev/null
+++ b/src/C/SuiteSparse/COLAMD/Doc/ChangeLog
@@ -0,0 +1,129 @@
+May 31, 2007: version 2.7.0
+
+ * ported to 64-bit MATLAB
+
+ * subdirectories added (Source/, Include/, Lib/, Doc/, MATLAB/, Demo/)
+
+Dec 12, 2006, version 2.5.2
+
+ * minor MATLAB cleanup. MATLAB functions renamed colamd2 and symamd2,
+ so that they do not conflict with the built-in versions. Note that
+ the MATLAB built-in functions colamd and symamd are identical to
+ the colamd and symamd functions here.
+
+Aug 31, 2006: Version 2.5.1
+
+ * minor change to colamd.m and symamd.m, to use etree instead
+ of sparsfun.
+
+Apr. 30, 2006: Version 2.5
+
+ * colamd_recommended modified, to do more careful integer overflow
+ checking. It now returns size_t, not int. colamd_l_recommended
+ also returns size_t. A zero is returned if an error occurs. A
+ postive return value denotes success. In v2.4 and earlier,
+ -1 was returned on error (an int or long).
+
+ * long replaced with UF_long integer, which is long except on WIN64.
+
+Nov 15, 2005:
+
+ * minor editting of comments; version number (2.4) unchanged.
+
+Changes from Version 2.3 to 2.4 (Aug 30, 2005)
+
+ * Makefile now relies on ../UFconfig/UFconfig.mk
+
+ * changed the dense row/col detection. The meaning of the knobs
+ has thus changed.
+
+ * added an option to turn off aggressive absorption. It was
+ always on in versions 2.3 and earlier.
+
+ * added a #define'd version number
+
+ * added a function pointer (colamd_printf) for COLAMD's printing.
+
+ * added a -DNPRINT option, to turn off printing at compile-time.
+
+ * added a check for integer overflow in colamd_recommended
+
+ * minor changes to allow for more simpler 100% test coverage
+
+ * bug fix. If symamd v2.3 fails to allocate its copy of the input
+ matrix, then it erroneously frees a calloc'd workspace twice.
+ This bug has no effect on the MATLAB symamd mexFunction, since
+ mxCalloc terminates the mexFunction if it fails to allocate
+ memory. Similarly, UMFPACK is not affected because it does not
+ use symamd. The bug has no effect on the colamd ordering
+ routine in v2.3.
+
+Changes from Version 2.2 to 2.3 (Sept. 8, 2003)
+
+ * removed the call to the MATLAB spparms ('spumoni') function.
+ This can take a lot of time if you are ordering many small
+ matrices. Only affects the MATLAB interface (colamdmex.c,
+ symamdmex.c, colamdtestmex.c, and symamdtestmex.c). The
+ usage of the optional 2nd argument to the colamd and symamd
+ mexFunctions was changed accordingly.
+
+Changes from Version 2.1 to 2.2 (Sept. 23, 2002)
+
+ * extensive testing routines added (colamd_test.m, colamdtestmex.c,
+ and symamdtestmex.c), and the Makefile modified accordingly.
+
+ * a few typos in the comments corrected
+
+ * use of the MATLAB "flops" command removed from colamd_demo, and an
+ m-file routine luflops.m added.
+
+ * an explicit typecast from unsigned to int added, for COLAMD_C and
+ COLAMD_R in colamd.h.
+
+ * #include <stdio.h> added to colamd_example.c
+
+
+Changes from Version 2.0 to 2.1 (May 4, 2001)
+
+ * TRUE and FALSE are predefined on some systems, so they are defined
+ here only if not already defined.
+
+ * web site changed
+
+ * UNIX Makefile modified, to handle the case if "." is not in your path.
+
+
+Changes from Version 1.0 to 2.0 (January 31, 2000)
+
+ No bugs were found in version 1.1. These changes merely add new
+ functionality.
+
+ * added the COLAMD_RECOMMENDED (nnz, n_row, n_col) macro.
+
+ * moved the output statistics, from A, to a separate output argument.
+ The arguments changed for the C-callable routines.
+
+ * added colamd_report and symamd_report.
+
+ * added a C-callable symamd routine. Formerly, symamd was only
+ available as a mexFunction from MATLAB.
+
+ * added error-checking to symamd. Formerly, it assumed its input
+ was error-free.
+
+ * added the optional stats and knobs arguments to the symamd mexFunction
+
+ * deleted colamd_help. A help message is still available from
+ "help colamd" and "help symamd" in MATLAB.
+
+ * deleted colamdtree.m and symamdtree.m. Now, colamd.m and symamd.m
+ also do the elimination tree post-ordering. The Version 1.1
+ colamd and symamd mexFunctions, which do not do the post-
+ ordering, are now visible as colamdmex and symamdmex from
+ MATLAB. Essentialy, the post-ordering is now the default
+ behavior of colamd.m and symamd.m, to match the behavior of
+ colmmd and symmmd. The post-ordering is only available in the
+ MATLAB interface, not the C-callable interface.
+
+ * made a slight change to the dense row/column detection in symamd,
+ to match the stated specifications.
diff --git a/src/C/SuiteSparse/COLAMD/lesser.txt b/src/C/SuiteSparse/COLAMD/Doc/lesser.txt
similarity index 100%
rename from src/C/SuiteSparse/COLAMD/lesser.txt
rename to src/C/SuiteSparse/COLAMD/Doc/lesser.txt
diff --git a/src/C/SuiteSparse/COLAMD/colamd.h b/src/C/SuiteSparse/COLAMD/Include/colamd.h
similarity index 98%
rename from src/C/SuiteSparse/COLAMD/colamd.h
rename to src/C/SuiteSparse/COLAMD/Include/colamd.h
index 7af4213..dc53ad5 100644
--- a/src/C/SuiteSparse/COLAMD/colamd.h
+++ b/src/C/SuiteSparse/COLAMD/Include/colamd.h
@@ -21,7 +21,7 @@
Notice:
- Copyright (c) 1998-2006, Timothy A. Davis, All Rights Reserved.
+ Copyright (c) 1998-2007, Timothy A. Davis, All Rights Reserved.
THIS MATERIAL IS PROVIDED AS IS, WITH ABSOLUTELY NO WARRANTY
EXPRESSED OR IMPLIED. ANY USE IS AT YOUR OWN RISK.
@@ -81,10 +81,11 @@ extern "C" {
* Versions 2.3 and earlier of COLAMD do not include a #define'd version number.
*/
-#define COLAMD_DATE "Dec 12, 2006"
+#define COLAMD_DATE "May 31, 2007"
#define COLAMD_VERSION_CODE(main,sub) ((main) * 1000 + (sub))
#define COLAMD_MAIN_VERSION 2
-#define COLAMD_SUB_VERSION 6
+#define COLAMD_SUB_VERSION 7
+#define COLAMD_SUBSUB_VERSION 0
#define COLAMD_VERSION \
COLAMD_VERSION_CODE(COLAMD_MAIN_VERSION,COLAMD_SUB_VERSION)
diff --git a/src/C/SuiteSparse/COLAMD/Lib/Makefile b/src/C/SuiteSparse/COLAMD/Lib/Makefile
new file mode 100644
index 0000000..b1571ef
--- /dev/null
+++ b/src/C/SuiteSparse/COLAMD/Lib/Makefile
@@ -0,0 +1,32 @@
+#-------------------------------------------------------------------------------
+# COLAMD Makefile
+#-------------------------------------------------------------------------------
+
+default: libcolamd.a
+
+include ../../UFconfig/UFconfig.mk
+
+I = -I../Include -I../../UFconfig
+
+INC = ../Include/colamd.h ../../UFconfig/UFconfig.h
+
+SRC = ../Source/colamd.c ../Source/colamd_global.c
+
+# creates libcolamd.a, a C-callable COLAMD library
+libcolamd.a: $(SRC) $(INC)
+ $(CC) $(CFLAGS) $(I) -c ../Source/colamd_global.c
+ $(CC) $(CFLAGS) $(I) -c ../Source/colamd.c
+ $(CC) $(CFLAGS) $(I) -c ../Source/colamd.c -DDLONG -o colamd_l.o
+ $(AR) libcolamd.a colamd.o colamd_l.o colamd_global.o
+
+ccode: libcolamd.a
+
+library: libcolamd.a
+
+clean:
+ - $(RM) $(CLEAN)
+
+purge: distclean
+
+distclean: clean
+ - $(RM) libcolamd.a
diff --git a/src/C/SuiteSparse/COLAMD/Contents.m b/src/C/SuiteSparse/COLAMD/MATLAB/Contents.m
similarity index 73%
rename from src/C/SuiteSparse/COLAMD/Contents.m
rename to src/C/SuiteSparse/COLAMD/MATLAB/Contents.m
index 5639549..960fd14 100644
--- a/src/C/SuiteSparse/COLAMD/Contents.m
+++ b/src/C/SuiteSparse/COLAMD/MATLAB/Contents.m
@@ -7,6 +7,7 @@
% helper and test functions:
% colamd_demo - demo for colamd, column approx minimum degree ordering algorithm
% colamd_make - compiles COLAMD2 and SYMAMD2 for MATLAB
+% colamd_make - compiles and installs COLAMD2 and SYMAMD2 for MATLAB
% colamd_test - test colamd2 and symamd2
% luflops - compute the flop count for sparse LU factorization
%
@@ -14,5 +15,5 @@
% p = colamd2 (A)
%
-% Copyright 2006, Timothy A. Davis
-
+% Copyright 1998-2007, Timothy A. Davis, and Stefan Larimore
+% Developed in collaboration with J. Gilbert and E. Ng.
diff --git a/src/C/SuiteSparse/COLAMD/MATLAB/Makefile b/src/C/SuiteSparse/COLAMD/MATLAB/Makefile
new file mode 100644
index 0000000..ee9415f
--- /dev/null
+++ b/src/C/SuiteSparse/COLAMD/MATLAB/Makefile
@@ -0,0 +1,35 @@
+# COLAMD Makefile for MATLAB mexFunctions
+
+default: colamd2 symamd2
+
+include ../../UFconfig/UFconfig.mk
+
+I = -I../../UFconfig -I../Include
+
+INC = ../Include/colamd.h ../../UFconfig/UFconfig.h
+
+SRC = ../Source/colamd.c ../Source/colamd_global.c
+
+MX = $(MEX) -DDLONG $(I)
+
+# Compiles the MATLAB-callable routines
+mex: colamd2 symamd2
+
+symamd2: symamdmex.c $(INC) $(SRC)
+ $(MX) -output symamd2mex symamdmex.c $(SRC)
+
+colamd2: colamdmex.c $(INC) $(SRC)
+ $(MX) -output colamd2mex colamdmex.c $(SRC)
+
+# Compiles the extensive test code
+test: mex colamdtestmex.c symamdtestmex.c $(INC) $(SRC)
+ $(MX) colamdtestmex.c $(SRC)
+ $(MX) symamdtestmex.c $(SRC)
+
+clean:
+ - $(RM) $(CLEAN)
+
+purge: distclean
+
+distclean: clean
+ - $(RM) *.mex* *.dll
diff --git a/src/C/SuiteSparse/COLAMD/colamd2.m b/src/C/SuiteSparse/COLAMD/MATLAB/colamd2.m
similarity index 82%
rename from src/C/SuiteSparse/COLAMD/colamd2.m
rename to src/C/SuiteSparse/COLAMD/MATLAB/colamd2.m
index 69348c3..9916cc7 100644
--- a/src/C/SuiteSparse/COLAMD/colamd2.m
+++ b/src/C/SuiteSparse/COLAMD/MATLAB/colamd2.m
@@ -24,43 +24,27 @@ function [p,stats] = colamd2 (S, knobs)
% are < 0, respectively. If knobs(3) is nonzero, stats and knobs are
% printed. The default is knobs = [10 10 0]. Note that knobs differs from
% earlier versions of colamd.
-%
-% Type the command "type colamd2" for a description of the optional stats
-% output and for the copyright information.
-%
-% Authors: S. Larimore and T. Davis, University of Florida. Developed in
-% collaboration with J. Gilbert and E. Ng.
+
+% Copyright 1998-2007, Timothy A. Davis, and Stefan Larimore
+% Developed in collaboration with J. Gilbert and E. Ng.
%
% Acknowledgements: This work was supported by the National Science
% Foundation, under grants DMS-9504974 and DMS-9803599.
-% Notice:
-%
-% Copyright 1998-2006, Timothy A. Davis, All Rights Reserved.
-%
-% See http://www.cise.ufl.edu/research/sparse/colamd (the colamd.c
-% file) for the License.
-%
-% Availability:
-%
-% colamd, symamd, amd, ccolamd, and csymamd are available at
-% http://www.cise.ufl.edu/research/sparse
-
%-------------------------------------------------------------------------------
% Perform the colamd ordering:
%-------------------------------------------------------------------------------
-if (nargout <= 1 && nargin == 1)
+if (nargout <= 1 & nargin == 1) %#ok
p = colamd2mex (S) ;
-elseif (nargout <= 1 && nargin == 2)
+elseif (nargout <= 1 & nargin == 2) %#ok
p = colamd2mex (S, knobs) ;
-elseif (nargout == 2 && nargin == 1)
+elseif (nargout == 2 & nargin == 1) %#ok
[p, stats] = colamd2mex (S) ;
-elseif (nargout == 2 && nargin == 2)
+elseif (nargout == 2 & nargin == 2) %#ok
[p, stats] = colamd2mex (S, knobs) ;
else
- error ('MATLAB:colamd:WrongInputOrOutputNumber',...
- 'colamd: incorrect number of input and/or output arguments') ;
+ error ('colamd: incorrect number of input and/or output arguments') ;
end
%-------------------------------------------------------------------------------
diff --git a/src/C/SuiteSparse/COLAMD/colamd_demo.m b/src/C/SuiteSparse/COLAMD/MATLAB/colamd_demo.m
similarity index 93%
rename from src/C/SuiteSparse/COLAMD/colamd_demo.m
rename to src/C/SuiteSparse/COLAMD/MATLAB/colamd_demo.m
index 265daf4..f2fd442 100644
--- a/src/C/SuiteSparse/COLAMD/colamd_demo.m
+++ b/src/C/SuiteSparse/COLAMD/MATLAB/colamd_demo.m
@@ -7,13 +7,13 @@
% ordering methods for MATLAB. They are typically faster (sometimes much
% faster) and typically provide better orderings than their MATLAB counterparts:
%
-% colamd a replacement for colmmd.
+% colamd a replacement for colmmd.
%
-% Typical usage: p = colamd (A) ;
+% Typical usage: p = colamd (A) ;
%
-% symamd a replacement for symmmd. Based on colamd.
+% symamd a replacement for symmmd. Based on colamd.
%
-% Typical usage: p = symamd (A) ;
+% Typical usage: p = symamd (A) ;
%
% For a description of the methods used, see the colamd.c file.
%
@@ -24,7 +24,8 @@
% Minor changes: in MATLAB 7, symmmd and colmmd are flagged as "obsolete".
% This demo checks if they exist, so it should still work when they are removed.
-% Copyright 2006, Timothy A. Davis, University of Florida
+% Copyright 1998-2007, Timothy A. Davis, and Stefan Larimore
+% Developed in collaboration with J. Gilbert and E. Ng.
%-------------------------------------------------------------------------------
% Print the introduction, the help info, and compile the mexFunctions
@@ -86,7 +87,7 @@ try
fprintf (1, 'residual: %e\n', ...
norm (A*x-b)) ;
catch
- fprintf (1, 'colmmd is obsolete\n') ;
+ fprintf (1, 'colmmd is obsolete; test skipped\n') ;
end
fprintf (1, '\n\nSolving via lu (PA = LU), without regard for sparsity:\n') ;
@@ -136,7 +137,7 @@ try
fprintf (1, 'flop count for Cholesky of A(:,p)''A(:,p): %d\n', ...
sum (lnz.^2)) ;
catch
- fprintf (1, 'colmmd is obsolete\n') ;
+ fprintf (1, 'colmmd is obsolete; test skipped\n') ;
end
%-------------------------------------------------------------------------------
diff --git a/src/C/SuiteSparse/COLAMD/MATLAB/colamd_install.m b/src/C/SuiteSparse/COLAMD/MATLAB/colamd_install.m
new file mode 100644
index 0000000..7ac0110
--- /dev/null
+++ b/src/C/SuiteSparse/COLAMD/MATLAB/colamd_install.m
@@ -0,0 +1,18 @@
+function colamd_install
+%COLAMD_MAKE to compile and install the colamd2 and symamd2 mexFunction.
+% Your current directory must be COLAMD/MATLAB for this function to work.
+%
+% Example:
+% colamd_install
+%
+% See also colamd2, symamd2.
+
+% Copyright 1998-2007, Timothy A. Davis, and Stefan Larimore
+% Developed in collaboration with J. Gilbert and E. Ng.
+
+colamd_make
+addpath (pwd)
+fprintf ('\nThe following path has been added. You may wish to add it\n') ;
+fprintf ('permanently, using the MATLAB pathtool command.\n') ;
+fprintf ('%s\n\n', pwd) ;
+colamd_demo
diff --git a/src/C/SuiteSparse/COLAMD/MATLAB/colamd_make.m b/src/C/SuiteSparse/COLAMD/MATLAB/colamd_make.m
new file mode 100644
index 0000000..1748f9d
--- /dev/null
+++ b/src/C/SuiteSparse/COLAMD/MATLAB/colamd_make.m
@@ -0,0 +1,29 @@
+function colamd_make
+%COLAMD_MAKE compiles COLAMD2 and SYMAMD2 for MATLAB
+%
+% Example:
+% colamd_make
+%
+% See also colamd, symamd
+
+% Copyright 1998-2007, Timothy A. Davis, and Stefan Larimore
+% Developed in collaboration with J. Gilbert and E. Ng.
+
+details = 0 ; % 1 if details of each command are to be printed
+d = '' ;
+if (~isempty (strfind (computer, '64')))
+ d = '-largeArrayDims' ;
+end
+src = '../Source/colamd.c ../Source/colamd_global.c' ;
+cmd = sprintf ('mex -DDLONG -O %s -I../../UFconfig -I../Include -output ', d) ;
+s = [cmd 'colamd2mex colamdmex.c ' src] ;
+if (details)
+ fprintf ('%s\n', s) ;
+end
+eval (s) ;
+s = [cmd 'symamd2mex symamdmex.c ' src] ;
+if (details)
+ fprintf ('%s\n', s) ;
+end
+eval (s) ;
+fprintf ('COLAMD2 and SYMAMD2 successfully compiled.\n') ;
diff --git a/src/C/SuiteSparse/COLAMD/colamd_test.m b/src/C/SuiteSparse/COLAMD/MATLAB/colamd_test.m
similarity index 89%
rename from src/C/SuiteSparse/COLAMD/colamd_test.m
rename to src/C/SuiteSparse/COLAMD/MATLAB/colamd_test.m
index cd909b6..7de7e81 100644
--- a/src/C/SuiteSparse/COLAMD/colamd_test.m
+++ b/src/C/SuiteSparse/COLAMD/MATLAB/colamd_test.m
@@ -12,21 +12,19 @@ function colamd_test
% the test mexFunctions are to be compiled.
%
% See also colamd2, symamd2
-%
-% Tim Davis
-% http://www.cise.ufl.edu/research/sparse/colamd/
-% Copyright 2006, Timothy A. Davis, University of Florida
+% Copyright 1998-2007, Timothy A. Davis, and Stefan Larimore
+% Developed in collaboration with J. Gilbert and E. Ng.
help colamd_test
s = input (...
-'Compile colamd2, symand, and the test codes? (y/n, default is yes): ', 's') ;
+'Compile colamd2, symand2, and the test codes? (y/n, default is yes): ', 's') ;
do_compile = 1 ;
if (~isempty (s))
- if (s (1) == 'n' || s (1) == 'N')
+ if (s (1) == 'n' | s (1) == 'N') %#ok
do_compile = 0 ;
end
end
@@ -34,20 +32,26 @@ end
if (do_compile)
fprintf ('Compiling colamd2, symamd2, and test mexFunctions.\n') ;
colamd_make ;
- cmd = 'mex -O -I../UFconfig colamdtestmex.c colamd.c colamd_global.c' ;
- fprintf ('%s\n', cmd) ;
- eval (cmd) ;
- cmd = 'mex -O -I../UFconfig symamdtestmex.c colamd.c colamd_global.c' ;
- fprintf ('%s\n', cmd) ;
- eval (cmd) ;
+
+ d = '' ;
+ if (~isempty (strfind (computer, '64')))
+ d = '-largeArrayDims' ;
+ end
+ cmd = sprintf ('mex -DDLONG -O %s -I../../UFconfig -I../Include ', d) ;
+ src = '../Source/colamd.c ../Source/colamd_global.c' ;
+ eval ([cmd 'colamdtestmex.c ' src]) ;
+ eval ([cmd 'symamdtestmex.c ' src]) ;
fprintf ('Done compiling.\n') ;
+
end
fprintf ('\nThe following codes will be tested:\n') ;
which colamd2
which symamd2
-which colamdmex
-which symamdmex
+which colamd2mex
+which symamd2mex
+which colamdtestmex
+which symamdtestmex
fprintf ('\nStarting the tests. Please be patient.\n') ;
@@ -256,7 +260,7 @@ for trial = 1:400
Alo = tril (A, -1) ;
nnull = length (find (sum (Alo') == 0 & sum (Alo) == 0)) ; %#ok
- if (stats (2) ~= nnull || nnull < 5)
+ if (stats (2) ~= nnull | nnull < 5) %#ok
error ('symamd2: wrong number of null columns') ;
end
if (any (null_col ~= p ((n-4):n)))
@@ -295,7 +299,7 @@ end
fprintf (' OK\n') ;
-fprintf ('\ncolamd and symamd2: all tests passed\n\n') ;
+fprintf ('\ncolamd2 and symamd2: all tests passed\n\n') ;
%-------------------------------------------------------------------------------
@@ -361,7 +365,7 @@ end
function check_perm (p, A)
% check_perm: check for a valid permutation vector
-if (isempty (A) && isempty (p)) %#ok
+if (isempty (A) & isempty (p)) %#ok
% empty permutation vectors of empty matrices are OK
return
end
@@ -465,13 +469,13 @@ A = spones (A) ;
function [p,stats] = Tcolamd (S, knobs)
% Tcolamd: run colamd2 in a testing mode
-if (nargout <= 1 && nargin == 1)
+if (nargout <= 1 & nargin == 1) %#ok
p = colamdtestmex (S) ;
-elseif (nargout <= 1 && nargin == 2)
+elseif (nargout <= 1 & nargin == 2) %#ok
p = colamdtestmex (S, knobs) ;
-elseif (nargout == 2 && nargin == 1)
+elseif (nargout == 2 & nargin == 1) %#ok
[p, stats] = colamdtestmex (S) ;
-elseif (nargout == 2 && nargin == 2)
+elseif (nargout == 2 & nargin == 2) %#ok
[p, stats] = colamdtestmex (S, knobs) ;
else
error ('colamd2: incorrect number of input and/or output arguments') ;
@@ -488,13 +492,13 @@ end
function [p, stats] = Tsymamd (S, knobs)
% Tsymamd: run symamd2 in a testing mode
-if (nargout <= 1 && nargin == 1)
+if (nargout <= 1 & nargin == 1) %#ok
p = symamdtestmex (S) ;
-elseif (nargout <= 1 && nargin == 2)
+elseif (nargout <= 1 & nargin == 2) %#ok
p = symamdtestmex (S, knobs) ;
-elseif (nargout == 2 && nargin == 1)
+elseif (nargout == 2 & nargin == 1) %#ok
[p, stats] = symamdtestmex (S) ;
-elseif (nargout == 2 && nargin == 2)
+elseif (nargout == 2 & nargin == 2) %#ok
[p, stats] = symamdtestmex (S, knobs) ;
else
error ('symamd2: incorrect number of input and/or output arguments') ;
diff --git a/src/C/SuiteSparse/COLAMD/colamdmex.c b/src/C/SuiteSparse/COLAMD/MATLAB/colamdmex.c
similarity index 83%
rename from src/C/SuiteSparse/COLAMD/colamdmex.c
rename to src/C/SuiteSparse/COLAMD/MATLAB/colamdmex.c
index d5ae3fd..93f16d5 100644
--- a/src/C/SuiteSparse/COLAMD/colamdmex.c
+++ b/src/C/SuiteSparse/COLAMD/MATLAB/colamdmex.c
@@ -23,7 +23,7 @@
Notice:
- Copyright (c) 1998-2006, Timothy A. Davis, All Rights Reserved.
+ Copyright (c) 1998-2007, Timothy A. Davis, All Rights Reserved.
See http://www.cise.ufl.edu/research/sparse/colamd (the colamd.c
file) for the License.
@@ -48,6 +48,7 @@
#include "matrix.h"
#include <stdlib.h>
#include <string.h>
+#include "UFconfig.h"
/* ========================================================================== */
/* === colamd mexFunction =================================================== */
@@ -65,21 +66,21 @@ void mexFunction
{
/* === Local variables ================================================== */
- int *A ; /* colamd's copy of the matrix, and workspace */
- int *p ; /* colamd's copy of the column pointers */
- int Alen ; /* size of A */
- int n_col ; /* number of columns of A */
- int n_row ; /* number of rows of A */
- int nnz ; /* number of entries in A */
- int full ; /* TRUE if input matrix full, FALSE if sparse */
+ UF_long *A ; /* colamd's copy of the matrix, and workspace */
+ UF_long *p ; /* colamd's copy of the column pointers */
+ UF_long Alen ; /* size of A */
+ UF_long n_col ; /* number of columns of A */
+ UF_long n_row ; /* number of rows of A */
+ UF_long nnz ; /* number of entries in A */
+ UF_long full ; /* TRUE if input matrix full, FALSE if sparse */
double knobs [COLAMD_KNOBS] ; /* colamd user-controllable parameters */
double *out_perm ; /* output permutation vector */
double *out_stats ; /* output stats vector */
double *in_knobs ; /* input knobs vector */
- int i ; /* loop counter */
+ UF_long i ; /* loop counter */
mxArray *Ainput ; /* input matrix handle */
- int spumoni ; /* verbosity variable */
- int stats [COLAMD_STATS] ; /* stats for colamd */
+ UF_long spumoni ; /* verbosity variable */
+ UF_long stats [COLAMD_STATS] ; /* stats for colamd */
colamd_printf = mexPrintf ; /* COLAMD printf routine */
@@ -93,7 +94,7 @@ void mexFunction
/* === Get knobs ======================================================== */
- colamd_set_defaults (knobs) ;
+ colamd_l_set_defaults (knobs) ;
spumoni = 0 ;
/* check for user-passed knobs */
@@ -103,7 +104,7 @@ void mexFunction
i = mxGetNumberOfElements (prhs [1]) ;
if (i > 0) knobs [COLAMD_DENSE_ROW] = in_knobs [0] ;
if (i > 1) knobs [COLAMD_DENSE_COL] = in_knobs [1] ;
- if (i > 2) spumoni = (int) (in_knobs [2] != 0) ;
+ if (i > 2) spumoni = (UF_long) (in_knobs [2] != 0) ;
}
/* print knob settings if spumoni is set */
@@ -155,10 +156,10 @@ void mexFunction
n_col = mxGetN (Ainput) ;
/* get column pointer vector so we can find nnz */
- p = (int *) mxCalloc (n_col+1, sizeof (int)) ;
- (void) memcpy (p, mxGetJc (Ainput), (n_col+1)*sizeof (int)) ;
+ p = (UF_long *) mxCalloc (n_col+1, sizeof (UF_long)) ;
+ (void) memcpy (p, mxGetJc (Ainput), (n_col+1)*sizeof (UF_long)) ;
nnz = p [n_col] ;
- Alen = (int) colamd_recommended (nnz, n_row, n_col) ;
+ Alen = (UF_long) colamd_l_recommended (nnz, n_row, n_col) ;
if (Alen == 0)
{
mexErrMsgTxt ("colamd: problem too large") ;
@@ -166,8 +167,8 @@ void mexFunction
/* === Copy input matrix into workspace ================================= */
- A = (int *) mxCalloc (Alen, sizeof (int)) ;
- (void) memcpy (A, mxGetIr (Ainput), nnz*sizeof (int)) ;
+ A = (UF_long *) mxCalloc (Alen, sizeof (UF_long)) ;
+ (void) memcpy (A, mxGetIr (Ainput), nnz*sizeof (UF_long)) ;
if (full)
{
@@ -176,9 +177,9 @@ void mexFunction
/* === Order the columns (destroys A) =================================== */
- if (!colamd (n_row, n_col, Alen, A, p, knobs, stats))
+ if (!colamd_l (n_row, n_col, Alen, A, p, knobs, stats))
{
- colamd_report (stats) ;
+ colamd_l_report (stats) ;
mexErrMsgTxt ("colamd error!") ;
}
mxFree (A) ;
@@ -199,7 +200,7 @@ void mexFunction
/* print stats if spumoni is set */
if (spumoni)
{
- colamd_report (stats) ;
+ colamd_l_report (stats) ;
}
if (nlhs == 2)
diff --git a/src/C/SuiteSparse/COLAMD/colamdtestmex.c b/src/C/SuiteSparse/COLAMD/MATLAB/colamdtestmex.c
similarity index 87%
rename from src/C/SuiteSparse/COLAMD/colamdtestmex.c
rename to src/C/SuiteSparse/COLAMD/MATLAB/colamdtestmex.c
index fe3a4c4..9b90a46 100644
--- a/src/C/SuiteSparse/COLAMD/colamdtestmex.c
+++ b/src/C/SuiteSparse/COLAMD/MATLAB/colamdtestmex.c
@@ -37,7 +37,7 @@
Notice:
- Copyright (c) 1998-2006, Timothy A. Davis, All Rights Reserved.
+ Copyright (c) 1998-2007, Timothy A. Davis, All Rights Reserved.
See http://www.cise.ufl.edu/research/sparse/colamd (the colamd.c
file) for the License.
@@ -63,15 +63,16 @@
#include "matrix.h"
#include <stdlib.h>
#include <string.h>
+#include "UFconfig.h"
static void dump_matrix
(
- int A [ ],
- int p [ ],
- int n_row,
- int n_col,
- int Alen,
- int limit
+ UF_long A [ ],
+ UF_long p [ ],
+ UF_long n_row,
+ UF_long n_col,
+ UF_long Alen,
+ UF_long limit
) ;
/* ========================================================================== */
@@ -90,24 +91,24 @@ void mexFunction
{
/* === Local variables ================================================== */
- int *A ; /* colamd's copy of the matrix, and workspace */
- int *p ; /* colamd's copy of the column pointers */
- int Alen ; /* size of A */
- int n_col ; /* number of columns of A */
- int n_row ; /* number of rows of A */
- int nnz ; /* number of entries in A */
- int full ; /* TRUE if input matrix full, FALSE if sparse */
+ UF_long *A ; /* colamd's copy of the matrix, and workspace */
+ UF_long *p ; /* colamd's copy of the column pointers */
+ UF_long Alen ; /* size of A */
+ UF_long n_col ; /* number of columns of A */
+ UF_long n_row ; /* number of rows of A */
+ UF_long nnz ; /* number of entries in A */
+ UF_long full ; /* TRUE if input matrix full, FALSE if sparse */
double knobs [COLAMD_KNOBS] ; /* colamd user-controllable parameters */
double *out_perm ; /* output permutation vector */
double *out_stats ; /* output stats vector */
double *in_knobs ; /* input knobs vector */
- int i ; /* loop counter */
+ UF_long i ; /* loop counter */
mxArray *Ainput ; /* input matrix handle */
- int spumoni ; /* verbosity variable */
- int stats2 [COLAMD_STATS] ; /* stats for colamd */
+ UF_long spumoni ; /* verbosity variable */
+ UF_long stats2 [COLAMD_STATS] ; /* stats for colamd */
- int *cp, *cp_end, result, col, length ;
- int *stats ;
+ UF_long *cp, *cp_end, result, col, length ;
+ UF_long *stats ;
stats = stats2 ;
colamd_printf = mexPrintf ; /* COLAMD printf routine */
@@ -132,7 +133,7 @@ void mexFunction
/* === Get knobs ======================================================== */
- colamd_set_defaults (knobs) ;
+ colamd_l_set_defaults (knobs) ;
spumoni = 0 ;
/* check for user-passed knobs */
@@ -142,7 +143,7 @@ void mexFunction
i = mxGetNumberOfElements (prhs [1]) ;
if (i > 0) knobs [COLAMD_DENSE_ROW] = in_knobs [0] ;
if (i > 1) knobs [COLAMD_DENSE_COL] = in_knobs [1] ;
- if (i > 2) spumoni = (int) in_knobs [2] ;
+ if (i > 2) spumoni = (UF_long) in_knobs [2] ;
}
/* print knob settings if spumoni is set */
@@ -194,10 +195,10 @@ void mexFunction
n_col = mxGetN (Ainput) ;
/* get column pointer vector so we can find nnz */
- p = (int *) mxCalloc (n_col+1, sizeof (int)) ;
- (void) memcpy (p, mxGetJc (Ainput), (n_col+1)*sizeof (int)) ;
+ p = (UF_long *) mxCalloc (n_col+1, sizeof (UF_long)) ;
+ (void) memcpy (p, mxGetJc (Ainput), (n_col+1)*sizeof (UF_long)) ;
nnz = p [n_col] ;
- Alen = (int) colamd_recommended (nnz, n_row, n_col) ;
+ Alen = (UF_long) colamd_l_recommended (nnz, n_row, n_col) ;
if (Alen == 0)
{
mexErrMsgTxt ("colamd: problem too large") ;
@@ -215,8 +216,8 @@ void mexFunction
/* Here only for testing */
/* size of the Col and Row structures */
-#define COLAMD_C(n_col) (((n_col) + 1) * 24 / sizeof (int))
-#define COLAMD_R(n_row) (((n_row) + 1) * 16 / sizeof (int))
+#define COLAMD_C(n_col) (((n_col) + 1) * 24 / sizeof (UF_long))
+#define COLAMD_R(n_row) (((n_row) + 1) * 16 / sizeof (UF_long))
#ifdef MIN
#undef MIN
#endif
@@ -238,8 +239,8 @@ void mexFunction
/* === Copy input matrix into workspace ================================= */
- A = (int *) mxCalloc (Alen, sizeof (int)) ;
- (void) memcpy (A, mxGetIr (Ainput), nnz*sizeof (int)) ;
+ A = (UF_long *) mxCalloc (Alen, sizeof (UF_long)) ;
+ (void) memcpy (A, mxGetIr (Ainput), nnz*sizeof (UF_long)) ;
if (full)
{
@@ -269,7 +270,7 @@ void mexFunction
*/
/* jumble appropriately */
- switch ((int) in_knobs [4])
+ switch ((UF_long) in_knobs [4])
{
case 0 :
@@ -367,7 +368,7 @@ void mexFunction
mexPrintf ("colamdtest: A not present\n") ;
}
result = 0 ; /* A not present */
- A = (int *) NULL ;
+ A = (UF_long *) NULL ;
break ;
case 8 :
@@ -376,7 +377,7 @@ void mexFunction
mexPrintf ("colamdtest: p not present\n") ;
}
result = 0 ; /* p not present */
- p = (int *) NULL ;
+ p = (UF_long *) NULL ;
break ;
case 9 :
@@ -464,7 +465,7 @@ void mexFunction
mexPrintf ("colamdtest: stats not present\n") ;
}
result = 0 ; /* stats not present */
- stats = (int *) NULL ;
+ stats = (UF_long *) NULL ;
break ;
case 13 :
@@ -481,7 +482,7 @@ void mexFunction
/* === Order the columns (destroys A) =================================== */
- if (!colamd (n_row, n_col, Alen, A, p, knobs, stats))
+ if (!colamd_l (n_row, n_col, Alen, A, p, knobs, stats))
{
/* return p = -1 if colamd failed */
@@ -493,7 +494,7 @@ void mexFunction
if (spumoni > 0 || result)
{
- colamd_report (stats) ;
+ colamd_l_report (stats) ;
}
if (result)
@@ -507,7 +508,7 @@ void mexFunction
if (!result)
{
- colamd_report (stats) ;
+ colamd_l_report (stats) ;
mexErrMsgTxt ("colamd should have returned FALSE\n") ;
}
mxFree (A) ;
@@ -528,7 +529,7 @@ void mexFunction
/* print stats if spumoni > 0 */
if (spumoni > 0)
{
- colamd_report (stats) ;
+ colamd_l_report (stats) ;
}
if (nlhs == 2)
@@ -550,15 +551,15 @@ void mexFunction
static void dump_matrix
(
- int A [ ],
- int p [ ],
- int n_row,
- int n_col,
- int Alen,
- int limit
+ UF_long A [ ],
+ UF_long p [ ],
+ UF_long n_row,
+ UF_long n_col,
+ UF_long Alen,
+ UF_long limit
)
{
- int col, k, row ;
+ UF_long col, k, row ;
mexPrintf ("dump matrix: nrow %d ncol %d Alen %d\n", n_row, n_col, Alen) ;
diff --git a/src/C/SuiteSparse/COLAMD/luflops.m b/src/C/SuiteSparse/COLAMD/MATLAB/luflops.m
similarity index 96%
rename from src/C/SuiteSparse/COLAMD/luflops.m
rename to src/C/SuiteSparse/COLAMD/MATLAB/luflops.m
index 0f6f46d..f3c9e45 100644
--- a/src/C/SuiteSparse/COLAMD/luflops.m
+++ b/src/C/SuiteSparse/COLAMD/MATLAB/luflops.m
@@ -24,9 +24,8 @@ function fl = luflops (L, U)
% See NA Digest, Vol 00, #50, Tuesday, Dec. 5, 2000
%
% See also symbfact
-%
-% Copyright 2006, Timothy A. Davis, University of Florida
+% Copyright 1998-2007, Timothy A. Davis
Lnz = full (sum (spones (L))) - 1 ; % off diagonal nz in cols of L
diff --git a/src/C/SuiteSparse/COLAMD/symamd2.m b/src/C/SuiteSparse/COLAMD/MATLAB/symamd2.m
similarity index 82%
rename from src/C/SuiteSparse/COLAMD/symamd2.m
rename to src/C/SuiteSparse/COLAMD/MATLAB/symamd2.m
index 2ec705c..ecae450 100644
--- a/src/C/SuiteSparse/COLAMD/symamd2.m
+++ b/src/C/SuiteSparse/COLAMD/MATLAB/symamd2.m
@@ -24,43 +24,26 @@ function [p, stats] = symamd2 (S, knobs)
% No rows/columns are removed if knobs(1)<0. If knobs(2) is nonzero, stats
% and knobs are printed. The default is knobs = [10 0]. Note that knobs
% differs from earlier versions of symamd.
-%
-% Type the command "type symamd2" for a description of the optional stats
-% output and for the copyright information.
-%
-% Authors: S. Larimore and T. Davis, University of Florida. Developed in
-% collaboration with J. Gilbert and E. Ng.
-%
+
+% Copyright 1998-2007, Timothy A. Davis, and Stefan Larimore
+% Developed in collaboration with J. Gilbert and E. Ng.
% Acknowledgements: This work was supported by the National Science
% Foundation, under grants DMS-9504974 and DMS-9803599.
-% Notice:
-%
-% Copyright 1998-2006, Timothy A. Davis, All Rights Reserved.
-%
-% See http://www.cise.ufl.edu/research/sparse/colamd (the colamd.c
-% file) for the License.
-%
-% Availability:
-%
-% colamd, symamd, amd, ccolamd, and csymamd are available at
-% http://www.cise.ufl.edu/research/sparse
-
%-------------------------------------------------------------------------------
% perform the symamd ordering:
%-------------------------------------------------------------------------------
-if (nargout <= 1 && nargin == 1)
+if (nargout <= 1 & nargin == 1) %#ok
p = symamd2mex (S) ;
-elseif (nargout <= 1 && nargin == 2)
+elseif (nargout <= 1 & nargin == 2) %#ok
p = symamd2mex (S, knobs) ;
-elseif (nargout == 2 && nargin == 1)
+elseif (nargout == 2 & nargin == 1) %#ok
[p, stats] = symamd2mex (S) ;
-elseif (nargout == 2 && nargin == 2)
+elseif (nargout == 2 & nargin == 2) %#ok
[p, stats] = symamd2mex (S, knobs) ;
else
- error('MATLAB:symamd:WrongInputOrOutputNumber',...
- 'symamd: incorrect number of input and/or output arguments.') ;
+ error('symamd: incorrect number of input and/or output arguments.') ;
end
%-------------------------------------------------------------------------------
diff --git a/src/C/SuiteSparse/COLAMD/symamdmex.c b/src/C/SuiteSparse/COLAMD/MATLAB/symamdmex.c
similarity index 84%
rename from src/C/SuiteSparse/COLAMD/symamdmex.c
rename to src/C/SuiteSparse/COLAMD/MATLAB/symamdmex.c
index 74ed936..253366d 100644
--- a/src/C/SuiteSparse/COLAMD/symamdmex.c
+++ b/src/C/SuiteSparse/COLAMD/MATLAB/symamdmex.c
@@ -25,7 +25,7 @@
Notice:
- Copyright (c) 1998-2006, Timothy A. Davis. All Rights Reserved.
+ Copyright (c) 1998-2007, Timothy A. Davis. All Rights Reserved.
See http://www.cise.ufl.edu/research/sparse/colamd (the colamd.c
file) for the License.
@@ -49,6 +49,7 @@
#include "mex.h"
#include "matrix.h"
#include <stdlib.h>
+#include "UFconfig.h"
/* ========================================================================== */
/* === symamd mexFunction =================================================== */
@@ -66,20 +67,20 @@ void mexFunction
{
/* === Local variables ================================================== */
- int *perm ; /* column ordering of M and ordering of A */
- int *A ; /* row indices of input matrix A */
- int *p ; /* column pointers of input matrix A */
- int n_col ; /* number of columns of A */
- int n_row ; /* number of rows of A */
- int full ; /* TRUE if input matrix full, FALSE if sparse */
+ UF_long *perm ; /* column ordering of M and ordering of A */
+ UF_long *A ; /* row indices of input matrix A */
+ UF_long *p ; /* column pointers of input matrix A */
+ UF_long n_col ; /* number of columns of A */
+ UF_long n_row ; /* number of rows of A */
+ UF_long full ; /* TRUE if input matrix full, FALSE if sparse */
double knobs [COLAMD_KNOBS] ; /* colamd user-controllable parameters */
double *out_perm ; /* output permutation vector */
double *out_stats ; /* output stats vector */
double *in_knobs ; /* input knobs vector */
- int i ; /* loop counter */
+ UF_long i ; /* loop counter */
mxArray *Ainput ; /* input matrix handle */
- int spumoni ; /* verbosity variable */
- int stats [COLAMD_STATS] ; /* stats for symamd */
+ UF_long spumoni ; /* verbosity variable */
+ UF_long stats [COLAMD_STATS] ; /* stats for symamd */
colamd_printf = mexPrintf ; /* COLAMD printf routine */
@@ -93,7 +94,7 @@ void mexFunction
/* === Get knobs ======================================================== */
- colamd_set_defaults (knobs) ;
+ colamd_l_set_defaults (knobs) ;
spumoni = 0 ;
/* check for user-passed knobs */
@@ -102,7 +103,7 @@ void mexFunction
in_knobs = mxGetPr (prhs [1]) ;
i = mxGetNumberOfElements (prhs [1]) ;
if (i > 0) knobs [COLAMD_DENSE_ROW] = in_knobs [0] ;
- if (i > 1) spumoni = (int) (in_knobs [1] != 0) ;
+ if (i > 1) spumoni = (UF_long) (in_knobs [1] != 0) ;
}
/* print knob settings if spumoni is set */
@@ -147,15 +148,15 @@ void mexFunction
mexErrMsgTxt ("symamd: matrix must be square.") ;
}
- A = mxGetIr (Ainput) ;
- p = mxGetJc (Ainput) ;
- perm = (int *) mxCalloc (n_col+1, sizeof (int)) ;
+ A = (UF_long *) mxGetIr (Ainput) ;
+ p = (UF_long *) mxGetJc (Ainput) ;
+ perm = (UF_long *) mxCalloc (n_col+1, sizeof (UF_long)) ;
/* === Order the rows and columns of A (does not destroy A) ============= */
- if (!symamd (n_col, A, p, perm, knobs, stats, &mxCalloc, &mxFree))
+ if (!symamd_l (n_col, A, p, perm, knobs, stats, &mxCalloc, &mxFree))
{
- symamd_report (stats) ;
+ symamd_l_report (stats) ;
mexErrMsgTxt ("symamd error!") ;
}
@@ -180,7 +181,7 @@ void mexFunction
/* print stats if spumoni is set */
if (spumoni)
{
- symamd_report (stats) ;
+ symamd_l_report (stats) ;
}
if (nlhs == 2)
diff --git a/src/C/SuiteSparse/COLAMD/symamdtestmex.c b/src/C/SuiteSparse/COLAMD/MATLAB/symamdtestmex.c
similarity index 87%
rename from src/C/SuiteSparse/COLAMD/symamdtestmex.c
rename to src/C/SuiteSparse/COLAMD/MATLAB/symamdtestmex.c
index 75c946e..c41101f 100644
--- a/src/C/SuiteSparse/COLAMD/symamdtestmex.c
+++ b/src/C/SuiteSparse/COLAMD/MATLAB/symamdtestmex.c
@@ -33,7 +33,7 @@
Notice:
- Copyright (c) 1998-2006, Timothy A. Davis. All Rights Reserved.
+ Copyright (c) 1998-2007, Timothy A. Davis. All Rights Reserved.
See http://www.cise.ufl.edu/research/sparse/colamd (the colamd.c
file) for the License.
@@ -59,15 +59,16 @@
#include "matrix.h"
#include <stdlib.h>
#include <string.h>
+#include "UFconfig.h"
static void dump_matrix
(
- int A [ ],
- int p [ ],
- int n_row,
- int n_col,
- int Alen,
- int limit
+ UF_long A [ ],
+ UF_long p [ ],
+ UF_long n_row,
+ UF_long n_col,
+ UF_long Alen,
+ UF_long limit
) ;
/* ========================================================================== */
@@ -86,23 +87,23 @@ void mexFunction
{
/* === Local variables ================================================== */
- int *perm ; /* column ordering of M and ordering of A */
- int *A ; /* row indices of input matrix A */
- int *p ; /* column pointers of input matrix A */
- int n_col ; /* number of columns of A */
- int n_row ; /* number of rows of A */
- int full ; /* TRUE if input matrix full, FALSE if sparse */
+ UF_long *perm ; /* column ordering of M and ordering of A */
+ UF_long *A ; /* row indices of input matrix A */
+ UF_long *p ; /* column pointers of input matrix A */
+ UF_long n_col ; /* number of columns of A */
+ UF_long n_row ; /* number of rows of A */
+ UF_long full ; /* TRUE if input matrix full, FALSE if sparse */
double knobs [COLAMD_KNOBS] ; /* colamd user-controllable parameters */
double *out_perm ; /* output permutation vector */
double *out_stats ; /* output stats vector */
double *in_knobs ; /* input knobs vector */
- int i ; /* loop counter */
+ UF_long i ; /* loop counter */
mxArray *Ainput ; /* input matrix handle */
- int spumoni ; /* verbosity variable */
- int stats2 [COLAMD_STATS] ; /* stats for symamd */
+ UF_long spumoni ; /* verbosity variable */
+ UF_long stats2 [COLAMD_STATS] ; /* stats for symamd */
- int *cp, *cp_end, result, nnz, col, length ;
- int *stats ;
+ UF_long *cp, *cp_end, result, nnz, col, length ;
+ UF_long *stats ;
stats = stats2 ;
colamd_printf = mexPrintf ; /* COLAMD printf routine */
@@ -127,7 +128,7 @@ void mexFunction
/* === Get knobs ======================================================== */
- colamd_set_defaults (knobs) ;
+ colamd_l_set_defaults (knobs) ;
spumoni = 0 ;
/* check for user-passed knobs */
@@ -136,7 +137,7 @@ void mexFunction
in_knobs = mxGetPr (prhs [1]) ;
i = mxGetNumberOfElements (prhs [1]) ;
if (i > 0) knobs [COLAMD_DENSE_ROW] = in_knobs [0] ;
- if (i > 1) spumoni = (int) in_knobs [1] ;
+ if (i > 1) spumoni = (UF_long) in_knobs [1] ;
}
/* print knob settings if spumoni is set */
@@ -183,8 +184,8 @@ void mexFunction
}
/* p = mxGetJc (Ainput) ; */
- p = (int *) mxCalloc (n_col+1, sizeof (int)) ;
- (void) memcpy (p, mxGetJc (Ainput), (n_col+1)*sizeof (int)) ;
+ p = (UF_long *) mxCalloc (n_col+1, sizeof (UF_long)) ;
+ (void) memcpy (p, mxGetJc (Ainput), (n_col+1)*sizeof (UF_long)) ;
nnz = p [n_col] ;
if (spumoni > 0)
@@ -193,10 +194,10 @@ void mexFunction
}
/* A = mxGetIr (Ainput) ; */
- A = (int *) mxCalloc (nnz+1, sizeof (int)) ;
- (void) memcpy (A, mxGetIr (Ainput), nnz*sizeof (int)) ;
+ A = (UF_long *) mxCalloc (nnz+1, sizeof (UF_long)) ;
+ (void) memcpy (A, mxGetIr (Ainput), nnz*sizeof (UF_long)) ;
- perm = (int *) mxCalloc (n_col+1, sizeof (int)) ;
+ perm = (UF_long *) mxCalloc (n_col+1, sizeof (UF_long)) ;
/* === Jumble matrix ======================================================== */
@@ -221,7 +222,7 @@ void mexFunction
*/
/* jumble appropriately */
- switch ((int) in_knobs [2])
+ switch ((UF_long) in_knobs [2])
{
case 0 :
@@ -312,7 +313,7 @@ void mexFunction
mexPrintf ("symamdtest: A not present\n") ;
}
result = 0 ; /* A not present */
- A = (int *) NULL ;
+ A = (UF_long *) NULL ;
break ;
case 8 :
@@ -321,7 +322,7 @@ void mexFunction
mexPrintf ("symamdtest: p not present\n") ;
}
result = 0 ; /* p not present */
- p = (int *) NULL ;
+ p = (UF_long *) NULL ;
break ;
case 9 :
@@ -409,7 +410,7 @@ void mexFunction
mexPrintf ("symamdtest: stats not present\n") ;
}
result = 0 ; /* stats not present */
- stats = (int *) NULL ;
+ stats = (UF_long *) NULL ;
break ;
case 13 :
@@ -425,7 +426,7 @@ void mexFunction
/* === Order the rows and columns of A (does not destroy A) ============= */
- if (!symamd (n_col, A, p, perm, knobs, stats, &mxCalloc, &mxFree))
+ if (!symamd_l (n_col, A, p, perm, knobs, stats, &mxCalloc, &mxFree))
{
/* return p = -1 if colamd failed */
@@ -437,7 +438,7 @@ void mexFunction
if (spumoni > 0 || result)
{
- symamd_report (stats) ;
+ symamd_l_report (stats) ;
}
if (result)
@@ -451,7 +452,7 @@ void mexFunction
if (!result)
{
- symamd_report (stats) ;
+ symamd_l_report (stats) ;
mexErrMsgTxt ("symamd should have returned FALSE\n") ;
}
@@ -476,7 +477,7 @@ void mexFunction
/* print stats if spumoni > 0 */
if (spumoni > 0)
{
- symamd_report (stats) ;
+ symamd_l_report (stats) ;
}
if (nlhs == 2)
@@ -504,15 +505,15 @@ void mexFunction
static void dump_matrix
(
- int A [ ],
- int p [ ],
- int n_row,
- int n_col,
- int Alen,
- int limit
+ UF_long A [ ],
+ UF_long p [ ],
+ UF_long n_row,
+ UF_long n_col,
+ UF_long Alen,
+ UF_long limit
)
{
- int col, k, row ;
+ UF_long col, k, row ;
mexPrintf ("dump matrix: nrow %d ncol %d Alen %d\n", n_row, n_col, Alen) ;
diff --git a/src/C/SuiteSparse/COLAMD/Makefile b/src/C/SuiteSparse/COLAMD/Makefile
index 07a2518..fd1397b 100644
--- a/src/C/SuiteSparse/COLAMD/Makefile
+++ b/src/C/SuiteSparse/COLAMD/Makefile
@@ -1,51 +1,49 @@
+#------------------------------------------------------------------------------
+# COLAMD Makefile
+#------------------------------------------------------------------------------
-default: libcolamd.a colamd_example colamd_l_example
+default: demo
include ../UFconfig/UFconfig.mk
-I = -I../UFconfig
+# Compile all C code, including the C-callable routine and the mexFunctions.
+# Do not the MATLAB interface.
+demo:
+ ( cd Lib ; $(MAKE) )
+ ( cd Demo ; $(MAKE) )
-colamd_example: colamd_example.c libcolamd.a
- $(CC) $(CFLAGS) $(I) -o colamd_example colamd_example.c libcolamd.a -lm
- - ./colamd_example > my_colamd_example.out
- - diff colamd_example.out my_colamd_example.out
+# Compile all C code, including the C-callable routine and the mexFunctions.
+all:
+ ( cd Lib ; $(MAKE) )
+ ( cd Demo ; $(MAKE) )
+ ( cd MATLAB ; $(MAKE) )
-colamd_l_example: colamd_l_example.c libcolamd.a
- $(CC) $(CFLAGS) $(I) -o colamd_l_example colamd_l_example.c libcolamd.a -lm
- - ./colamd_l_example > my_colamd_l_example.out
- - diff colamd_example.out my_colamd_example.out
+# compile just the C-callable libraries (not mexFunctions or Demos)
+library:
+ ( cd Lib ; $(MAKE) )
-purge: distclean
-
-distclean: clean2
- - $(RM) libcolamd.a
-
-clean2: clean
- - $(RM) *.o *.dll colamd_example colamd_l_example
- - $(RM) colamd2mex.mex* symamd2mex.mex*
- - $(RM) colamdtestmex.mex* symamdtestmex.mex*
- - $(RM) my_colamd_example.out my_colamd_l_example.out
+# remove object files, but keep the compiled programs and library archives
+clean:
+ ( cd Lib ; $(MAKE) clean )
+ ( cd Demo ; $(MAKE) clean )
+ ( cd MATLAB ; $(MAKE) clean )
-# Compiles the MATLAB-callable routines
-mex: colamdmex.c symamdmex.c libcolamd.a
- $(MEX) -output colamd2mex $(I) colamdmex.c libcolamd.a
- $(MEX) -output symamd2mex $(I) symamdmex.c libcolamd.a
+# clean, and then remove compiled programs and library archives
+purge:
+ ( cd Lib ; $(MAKE) purge )
+ ( cd Demo ; $(MAKE) purge )
+ ( cd MATLAB ; $(MAKE) purge )
-# Compiles the extensive test code
-test: mex colamdtestmex.c symamdtestmex.c libcolamd.a
- $(MEX) $(I) colamdtestmex.c libcolamd.a
- $(MEX) $(I) symamdtestmex.c libcolamd.a
+distclean: purge
-# creates libcolamd.a, a C-callable COLAMD library
-libcolamd.a: colamd.c colamd_global.c colamd.h
- $(CC) $(CFLAGS) $(I) -c colamd_global.c
- $(CC) $(CFLAGS) $(I) -c colamd.c
- $(CC) $(CFLAGS) $(I) -c colamd.c -DDLONG -o colamd_l.o
- $(AR) libcolamd.a colamd.o colamd_l.o colamd_global.o
+# get ready for distribution
+dist: purge
+ ( cd Demo ; $(MAKE) dist )
-ccode: libcolamd.a
+ccode: library
-library: libcolamd.a
+lib: library
-clean:
- - $(RM) $(CLEAN)
+# compile the MATLAB mexFunction
+mex:
+ ( cd MATLAB ; $(MAKE) )
diff --git a/src/C/SuiteSparse/COLAMD/README.txt b/src/C/SuiteSparse/COLAMD/README.txt
index 5561ec3..5ed81c7 100644
--- a/src/C/SuiteSparse/COLAMD/README.txt
+++ b/src/C/SuiteSparse/COLAMD/README.txt
@@ -1,4 +1,4 @@
-The COLAMD ordering method - Version 2.6
+The COLAMD ordering method - Version 2.7
-------------------------------------------------------------------------------
The COLAMD column approximate minimum degree ordering algorithm computes
@@ -18,18 +18,25 @@ and symamd. This will work on any computer with MATLAB (Unix, PC, or Mac).
Alternatively, type "make" (in Unix) to compile and run a simple example C
code, without using MATLAB.
+To compile and install the colamd m-files and mexFunctions, just cd to
+COLAMD/MATLAB and type colamd_install in the MATLAB command window.
+A short demo will run. Optionally, type colamd_test to run an extensive tests.
+Type "make" in Unix in the COLAMD directory to compile the C-callable
+library and to run a short demo.
+
+If you have MATLAB 7.2 or earlier, you must first edit UFconfig/UFconfig.h to
+remove the "-largeArrayDims" option from the MEX command (or just use
+colamd_make.m inside MATLAB).
+
Colamd is a built-in routine in MATLAB, available from The
-Mathworks, Inc. Under most cases, the compiled COLAMD from Versions 2.0
-through 2.6 do not differ. Colamd Versions 2.2 and 2.3 differ only in their
+Mathworks, Inc. Under most cases, the compiled COLAMD from Versions 2.0 to the
+current version do not differ. Colamd Versions 2.2 and 2.3 differ only in their
mexFunction interaces to MATLAB. v2.4 fixes a bug in the symamd routine in
v2.3. The bug (in v2.3 and earlier) has no effect on the MATLAB symamd
mexFunction. v2.5 adds additional checks for integer overflow, so that
the "int" version can be safely used with 64-bit pointers. Refer to the
ChangeLog for more details.
- NOTE: DO NOT ATTEMPT TO USE THIS CODE IN 64-BIT MATLAB (v7.3).
- It is not yet ported to that version of MATLAB.
-
To use colamd and symamd within an application written in C, all you need are
colamd.c, colamd_global.c, and colamd.h, which are the C-callable
colamd/symamd codes. See colamd.c for more information on how to call
@@ -37,7 +44,7 @@ colamd from a C program.
Requires UFconfig, in the ../UFconfig directory relative to this directory.
- Copyright (c) 1998-2006, Timothy A. Davis, All Rights Reserved.
+ Copyright (c) 1998-2007, Timothy A. Davis, All Rights Reserved.
See http://www.cise.ufl.edu/research/sparse/colamd (the colamd.c
file) for the License.
@@ -72,37 +79,49 @@ in collaboration with John Gilbert, Xerox PARC (now at UC Santa Barbara),
and Esmong Ng, Lawrence Berkeley National Laboratory (much of this work
he did while at Oak Ridge National Laboratory).
-COLAMD files
-
- colamd.c: the primary colamd computational kernel.
-
- colamd.h: include file for colamd/symamd library.
-
- colamd.m: the MATLAB interface to colamd.
-
- colamd_demo.m: MATLAB demo file for colamd and symamd
- (also compiles the colamdmex and symamdmex mexFunctions).
-
- colamdmex.c: colamd mexFunction for use in MATLAB.
-
- colamd_example.c: example C main program that calls colamd and symamd.
-
- colamd_example.out: output of colamd_example.c.
-
- Makefile: Makefile for colamd_example.c
-
- symamd.m: the MATLAB interface to symamd.
-
- symamdmex.c: symamd mexFunction for use in MATLAB.
-
- README: this file
-
- ChangeLog: a log of changes since Version 1.0.
-
- colamd_test.m: test code
-
- colamdtestmex.c: test code
-
- luflops.m: test code
-
- symamdtestmex.c: test code
+COLAMD files:
+
+ Demo simple demo
+ Doc additional documentation (see colamd.c for more)
+ Include include file
+ Lib compiled C-callable library
+ Makefile primary Unix Makefile
+ MATLAB MATLAB functions
+ README.txt this file
+ Source C source code
+
+ ./Demo:
+ colamd_example.c simple example
+ colamd_example.out output of colamd_example.c
+ colamd_l_example.c simple example, long integers
+ colamd_l_example.out output of colamd_l_example.c
+ Makefile Makefile for C demos
+
+ ./Doc:
+ ChangeLog change log
+ lesser.txt license
+
+ ./Include:
+ colamd.h include file
+
+ ./Lib:
+ Makefile Makefile for C-callable library
+
+ ./MATLAB:
+ colamd2.m MATLAB interface for colamd2
+ colamd_demo.m simple demo
+ colamd_install.m compile and install colamd2 and symamd2
+ colamd_make.m compile colamd2 and symamd2
+ colamdmex.ca MATLAB mexFunction for colamd2
+ colamd_test.m extensive test
+ colamdtestmex.c test function for colamd
+ Contents.m contents of the MATLAB directory
+ luflops.m test code
+ Makefile Makefile for MATLAB functions
+ symamd2.m MATLAB interface for symamd2
+ symamdmex.c MATLAB mexFunction for symamd2
+ symamdtestmex.c test function for symamd
+
+ ./Source:
+ colamd.c primary source code
+ colamd_global.c globally defined function pointers (malloc, free, ...)
diff --git a/src/C/SuiteSparse/COLAMD/colamd.c b/src/C/SuiteSparse/COLAMD/Source/colamd.c
similarity index 99%
rename from src/C/SuiteSparse/COLAMD/colamd.c
rename to src/C/SuiteSparse/COLAMD/Source/colamd.c
index abf67e7..5fe20d6 100644
--- a/src/C/SuiteSparse/COLAMD/colamd.c
+++ b/src/C/SuiteSparse/COLAMD/Source/colamd.c
@@ -46,7 +46,7 @@
Copyright and License:
- Copyright (c) 1998-2006, Timothy A. Davis, All Rights Reserved.
+ Copyright (c) 1998-2007, Timothy A. Davis, All Rights Reserved.
COLAMD is also available under alternate licenses, contact T. Davis
for details.
diff --git a/src/C/SuiteSparse/COLAMD/colamd_global.c b/src/C/SuiteSparse/COLAMD/Source/colamd_global.c
similarity index 94%
rename from src/C/SuiteSparse/COLAMD/colamd_global.c
rename to src/C/SuiteSparse/COLAMD/Source/colamd_global.c
index 6c9dfde..4d1ae22 100644
--- a/src/C/SuiteSparse/COLAMD/colamd_global.c
+++ b/src/C/SuiteSparse/COLAMD/Source/colamd_global.c
@@ -3,7 +3,7 @@
/* ========================================================================== */
/* ----------------------------------------------------------------------------
- * COLAMD, Copyright (C) 2006, Timothy A. Davis.
+ * COLAMD, Copyright (C) 2007, Timothy A. Davis.
* See License.txt for the Version 2.1 of the GNU Lesser General Public License
* http://www.cise.ufl.edu/research/sparse
* -------------------------------------------------------------------------- */
diff --git a/src/C/SuiteSparse/COLAMD/colamd_make.m b/src/C/SuiteSparse/COLAMD/colamd_make.m
deleted file mode 100644
index 48d8b98..0000000
--- a/src/C/SuiteSparse/COLAMD/colamd_make.m
+++ /dev/null
@@ -1,18 +0,0 @@
-function colamd_make
-%COLAMD_MAKE compiles COLAMD2 and SYMAMD2 for MATLAB
-%
-% Example:
-% colamd_make
-%
-% See also colamd, symamd
-
-% Copyright 2006, Timothy A. Davis, University of Florida
-
-
-if (~isempty (strfind (computer, '64')))
- error ('64-bit version not yet supported') ;
-end
-
-mex -output colamd2mex -O -I../UFconfig colamdmex.c colamd.c colamd_global.c
-mex -output symamd2mex -O -I../UFconfig symamdmex.c colamd.c colamd_global.c
-fprintf ('COLAMD2 and SYMAMD2 successfully compiled.\n') ;
diff --git a/src/C/SuiteSparse/Contents.m b/src/C/SuiteSparse/Contents.m
index b1c7cce..6ea5659 100644
--- a/src/C/SuiteSparse/Contents.m
+++ b/src/C/SuiteSparse/Contents.m
@@ -3,25 +3,25 @@
% Only the primary MATLAB functions are listed below.
%
% Example:
-% SuiteSparse_install - compiles and installs all of SuiteSparse, and runs
-% several demos and tests.
+% SuiteSparse_install
+% compiles and installs all of SuiteSparse, and runs several demos and tests.
%
-%-------------------
+%-------------------------------------------------------------------------------
% Ordering methods:
-%-------------------
+%-------------------------------------------------------------------------------
%
-% amd - approximate minimum degree ordering.
-% colamd - column approximate minimum degree ordering.
-% symamd - symmetrix approximate min degree ordering based on colamd.
+% amd2 - approximate minimum degree ordering.
+% colamd2 - column approximate minimum degree ordering.
+% symamd2 - symmetrix approximate min degree ordering based on colamd.
% camd - constrained amd.
% ccolamd - constrained colamd.
% csymamd - constrained symamd.
%
-%---------------------------------------------------------------
+%-------------------------------------------------------------------------------
% CHOLMOD: a sparse supernodal Cholesky update/downdate package:
-%---------------------------------------------------------------
+%-------------------------------------------------------------------------------
%
-% cholmod - computes x=A\b when A is symmetric and positive definite.
+% cholmod2 - computes x=A\b when A is symmetric and positive definite.
% chol2 - same as MATLAB chol(sparse(A)), just faster.
% lchol - computes an LL' factorization.
% ldlchol - computes an LDL' factorization.
@@ -44,12 +44,13 @@
% mwrite - write a sparse matrix in Matrix Market format
% spsym - determine the symmetry of a sparse matrix
%
-%------------------------------------------
-% CSPARSE: a Concise Sparse matrix package:
-%------------------------------------------
+%-------------------------------------------------------------------------------
+% CSPARSE / CXSPARSE: a Concise Sparse matrix package:
+%-------------------------------------------------------------------------------
%
-% Matrices used in CSparse must in general be either sparse and real,
-% or dense vectors. Ordering methods can accept any sparse matrix.
+% Matrices used in CSparse must in general be either sparse and real, or
+% dense vectors. Ordering methods can accept any sparse matrix. CXSparse
+% supports complex matrices and 64-bit MATLAB; it is installed by default.
%
% cs_add - sparse matrix addition.
% cs_amd - approximate minimum degree ordering.
@@ -84,25 +85,25 @@
% cs_sparse - convert a triplet form into a sparse matrix.
% cs_sqr - symbolic sparse QR factorization.
% cs_symperm - symmetric permutation of a symmetric matrix.
-% cs_transpose - transpose a real sparse matrix.
+% cs_transpose - transpose a sparse matrix.
% cs_updown - rank-1 update/downdate of a sparse Cholesky factorization.
% cs_usolve - solve a sparse upper triangular system U*x=b.
% cs_utsolve - solve a sparse lower triangular system U'*x=b.
% cspy - plot a sparse matrix in color.
-% ccspy - plot the connected components of a matrix.
+% ccspy - plot the connected components of a matrix.
%
-%-------------------------------
+%-------------------------------------------------------------------------------
% LDL: Sparse LDL factorization:
-%-------------------------------
+%-------------------------------------------------------------------------------
%
% ldlsparse - LDL' factorization of a real, sparse, symmetric matrix.
% ldlrow - an m-file description of the algorithm used by LDL.
%
-%-----------------------------------------------
+%-------------------------------------------------------------------------------
% UMFPACK: the Unsymmetric MultiFrontal Package:
-%-----------------------------------------------
+%-------------------------------------------------------------------------------
%
-% umfpack - computes x=A\b, x=A/b, or lu (A) for a sparse matrix A
+% umfpack2 - computes x=A\b, x=A/b, or lu (A) for a sparse matrix A
% umfpack_details - details on all the options for using umfpack in MATLAB
% umfpack_report - prints optional control settings and statistics
% umfpack_btf - factorize A using a block triangular form
@@ -111,24 +112,15 @@
% (duplicate of CHOLMOD/lu_normest, for completeness)
% luflop - given L and U, computes # of flops required
%
-%------------------------------------------------------
-% RBio: read/write matrices in Rutherford/Boeing format
-%------------------------------------------------------
-%
-% RBread - read a sparse matrix from a Rutherford/Boeing file
-% RBreade - read a symmetric finite-element matrix from a R/B file
-% RBtype - determine the Rutherford/Boeing type of a sparse matrix
-% RBwrite - write a sparse matrix to a Rutherford/Boeing file
-%
+%-------------------------------------------------------------------------------
+% UFGET: MATLAB interface to the UF Sparse Matrix Collection
%-------------------------------------------------------------------------------
%
% For help on compiling SuiteSparse or the demos, testing functions, etc.,
-% please see the help for each individual package.
-%
-% NOTE: None of the packages above have yet been ported to 64-bit MATLAB.
-% Do not attempt to use these in 64-bit MATLAB.
+% please see the help for each individual package. UFcollection and RBio
+% are two additional toolboxes, for managing the UF Sparse Matrix Collection.
%
-% Copyright 2006, Timothy A. Davis
+% Copyright 2007, Timothy A. Davis
% http://www.cise.ufl.edu/research/sparse
help SuiteSparse
diff --git a/src/C/SuiteSparse/Makefile b/src/C/SuiteSparse/Makefile
index dafd591..da7dc78 100644
--- a/src/C/SuiteSparse/Makefile
+++ b/src/C/SuiteSparse/Makefile
@@ -24,7 +24,7 @@ default:
library: default
-# Compile the MATLAB mexFunctions
+# Compile the MATLAB mexFunctions (except RBio and UFcollection)
mex:
( cd AMD ; $(MAKE) mex )
( cd CAMD ; $(MAKE) mex )
@@ -35,9 +35,8 @@ mex:
( cd CCOLAMD ; $(MAKE) mex )
( cd CHOLMOD ; $(MAKE) mex )
( cd UMFPACK ; $(MAKE) mex )
+ ( cd CXSparse ; $(MAKE) mex )
( cd CSparse ; $(MAKE) mex )
- ( cd RBio ; $(MAKE) )
- ( cd UFcollection ; $(MAKE) )
# Remove all files not in the original distribution
purge:
@@ -54,8 +53,8 @@ purge:
( cd CHOLMOD ; $(MAKE) purge )
( cd CSparse ; $(MAKE) purge )
( cd CXSparse ; $(MAKE) purge )
- ( cd RBio ; $(MAKE) purge )
- ( cd UFcollection ; $(MAKE) purge )
+ ( cd RBio ; $(RM) *.mex* )
+ ( cd UFcollection ; $(RM) *.mex* )
# ( cd LPDASA ; $(MAKE) purge )
# ( cd PARAKLETE ; $(MAKE) purge )
@@ -74,8 +73,6 @@ clean:
( cd CHOLMOD ; $(MAKE) clean )
( cd CSparse ; $(MAKE) clean )
( cd CXSparse ; $(MAKE) clean )
- ( cd RBio ; $(MAKE) clean )
- ( cd UFcollection ; $(MAKE) clean )
# ( cd LPDASA ; $(MAKE) clean )
# ( cd PARAKLETE ; $(MAKE) clean )
diff --git a/src/C/SuiteSparse/README.txt b/src/C/SuiteSparse/README.txt
index a87e1ea..924c514 100644
--- a/src/C/SuiteSparse/README.txt
+++ b/src/C/SuiteSparse/README.txt
@@ -11,39 +11,32 @@ SuiteSparse_install. All packages will be compiled, and several demos will be
run.
================================================================================
-All codes, below, are stable except KLU and BTF. KLU and BTF are "beta", but
-robust enough for production use. They merely are still in development, and
-are missing a few minor features to make them "1.0". Thus the "beta". They
-are bug-free as far as I know, and are in use in commercial circuit simulation
-packages.
-
-Nov, 2006. SuiteSparse version 2.3. Note that SuiteSparse is now given
-its own version number, rather than merely a date of release.
+May 31, 2007. SuiteSparse version 3.0.
UF suite of sparse matrix algorithms:
AMD approximate minimum degree ordering
- CAMD constrained column approximate minimum degree ordering
+ CAMD constrained approximate minimum degree ordering
COLAMD column approximate minimum degree ordering
CCOLAMD constrained column approximate minimum degree ordering
- BTF permutation to block triangular form (beta)
+ BTF permutation to block triangular form
KLU sparse LU factorization, primarily for circuit simulation.
- Requires AMD, COLAMD, and BTF (beta). Optionally
- uses CHOLMOD, CCOLAMD, and METIS.
+ Requires AMD, COLAMD, and BTF. Optionally uses CHOLMOD,
+ CAMD, CCOLAMD, and METIS.
UMFPACK sparse LU factorization. Requires AMD and the BLAS.
CHOLMOD sparse Cholesky factorization. Requires AMD, COLAMD, CCOLAMD,
- METIS, the BLAS, and LAPACK.
+ the BLAS, and LAPACK. Optionally uses METIS.
UFconfig configuration file for all the above packages. The
UFconfig/UFconfig.mk is included in the Makefile's of all
- packages. CSparse and CXSparse do not use UFconfig.
+ packages. CSparse and RBio do not use UFconfig.
CSparse a concise sparse matrix package, developed for my upcoming
book, "Direct Methods for Sparse Linear Systems", to be
@@ -59,62 +52,52 @@ UF suite of sparse matrix algorithms:
LPDASA LP dual active set algorithm (to appear)
-See http://www.netlib.org/blas for the Fortran reference BLAS (slow, but they
-work). See http://www.tacc.utexas.edu/~kgoto/ or
-http://www.cs.utexas.edu/users/flame/goto/ for an optimized BLAS.
-See http://www.netlib.org/lapack for LAPACK. The UFconfig/UFconfig.mk
-file assumes the Goto BLAS; change -lgoto to -l(your BLAS library here),
-if you have another BLAS (-lblas, for example).
-
-CHOLMOD requires METIS 4.0.1 (http://www-users.cs.umn.edu/~karypis/metis)
-by default. Place a copy of the metis-4.0 directory in the same directory
-(SuiteSparse) containing this README file. cd to metis-4.0 and edit the
-Makefile.in file. I recommend making these changes to metis-4.0/Makefile.in:
-
-CC = gcc
-OPTFLAGS = -O3
-COPTIONS = -fexceptions -D_FILE_OFFSET_BITS=64 -D_LARGEFILE64_SOURCE
-
-then type "make". You can now compile CHOLMOD. First,
-edit the UFconfig/UFconfig.mk file (see that file for instructions), if
-necessary. Next, type "make" in this directory to compile all packages in
-this distribution. CHOLMOD can be compiled without METIS (use -DNPARTITION).
+CHOLMOD and KLU optionally use METIS 4.0.1
+(http://www-users.cs.umn.edu/~karypis/metis). Place a copy of the metis-4.0
+directory in the same directory (SuiteSparse) containing this README file.
Refer to each package for license, copyright, and author information. All
codes are authored or co-authored by Timothy A. Davis, CISE Dept., Univ. of
Florida. email: my last name @ cise dot ufl dot edu.
-To compile each package, cd to the top-level directory (AMD, COLAMD, etc)
-and type "make". Type "make clean" in the same directory to remove all but
-the compiled libraries. Type "make distclean" to remove all files not in
-the original distribution. Alternatively, just type "make" in this directory.
+================================================================================
+If you use SuiteSparse_install in MATLAB, stop reading here.
+================================================================================
+
+
-If you intend on compiling the MATLAB mexFunction interfaces, UFconfig.mk
-should use
+----------------------------
+To use "make" in Unix/Linux:
+----------------------------
- CFLAGS = -O3 -fexceptions
+ See http://www.netlib.org/blas for the Fortran reference BLAS (slow, but
+ they work). See http://www.tacc.utexas.edu/~kgoto/ or
+ http://www.cs.utexas.edu/users/flame/goto/ for an optimized BLAS. See
+ http://www.netlib.org/lapack for LAPACK. The UFconfig/UFconfig.mk file
+ assumes the vanilla BLAS (-lblas). You should use an optimized BLAS;
+ otherwise UMFPACK and CHOLMOD will be slow. Change -lblas to -l(your BLAS
+ library here).
-(for Linux), to ensure that exceptions are properly caught. See your
-default MATLAB mexopts.sh file for how to do this for other systems
-(type the command "mex -v"). Alternatively, you can use the various M-files
-in each package to compile them from within the MATLAB Command Window, or
-just type "SuiteSparse_install" in the MATLAB Command Window when MATLAB's
-working directory is this one. That is the only way to compile these packages
-for Windows, unless you have Cygwin or wish to write your own MS Visual Studio
-scripts.
+ cd to metis-4.0 and edit the Makefile.in file. I recommend making these
+ changes to metis-4.0/Makefile.in:
---------------------------------------------------------------------------------
+ CC = gcc
+ OPTFLAGS = -O3
+ COPTIONS = -fexceptions -D_FILE_OFFSET_BITS=64 -D_LARGEFILE64_SOURCE
-To turn on debugging (for development only, not needed by the typical user):
+ then type "make". Now compile CHOLMOD.
-SuiteSparse/UFconfig/UFconfig.mk
- change CFLAGS = -O to CFLAGS = -g
+ To compile all the C-callable libraries in SuiteSparse: First, edit the
+ UFconfig/UFconfig.mk file (see that file for instructions), if necessary.
+ Next, type "make" in this directory to compile all packages in this
+ distribution. CHOLMOD can be compiled without METIS (use -DNPARTITION);
+ this option is handled by SuiteSparse_install.m in MATLAB automatically, if
+ the metis-4.0 directory does not appear (in the same directory as CHOLMOD,
+ AMD, UMFPACK, etc).
-To turn on debugging, add the line "#undef NDEBUG" in the following files.
-To turn off debugging, remove that line.
+ To compile each package, cd to the top-level directory (AMD, COLAMD, etc)
+ and type "make". Type "make clean" in the same directory to remove all but
+ the compiled libraries. Type "make distclean" to remove all files not in
+ the original distribution. Alternatively, just type "make" in this
+ directory.
- SuiteSparse/CHOLMOD/Include/cholmod_internal.h
- SuiteSparse/AMD/Source/amd_internal.h
- SuiteSparse/CAMD/Source/camd_internal.h
- SuiteSparse/CCOLAMD/ccolmod.c
- SuiteSparse/COLAMD/colamd.c
diff --git a/src/C/SuiteSparse/README_cvxopt b/src/C/SuiteSparse/README_cvxopt
index 83bc799..6e4c6ec 100644
--- a/src/C/SuiteSparse/README_cvxopt
+++ b/src/C/SuiteSparse/README_cvxopt
@@ -1,5 +1,5 @@
-This is the January 30, 2007 (version 2.4.0) distribution of the
-SuiteSparse package, with the following files and directories removed.
+This is the May 31, 2007 (version 3.0) distribution of the SuiteSparse
+package, with the following files and directories removed.
AMD/Demo
AMD/MATLAB
diff --git a/src/C/SuiteSparse/SuiteSparse_demo.m b/src/C/SuiteSparse/SuiteSparse_demo.m
new file mode 100644
index 0000000..1247014
--- /dev/null
+++ b/src/C/SuiteSparse/SuiteSparse_demo.m
@@ -0,0 +1,113 @@
+function SuiteSparse_demo (matrixpath)
+%SUITESPARSE_DEMO a demo of all packages in SuiteSparse
+%
+% Example:
+% SuiteSparse_demo
+%
+% See also umfpack, cholmod, amd, camd, colamd, ccolamd, btf, klu,
+% CSparse, CXSparse, ldlsparse
+
+% Copyright (c) Timothy A. Davis, Univ. of Florida
+
+if (nargin < 1)
+ try
+ % older versions of MATLAB do not have an input argument to mfilename
+ p = mfilename ('fullpath') ;
+ t = strfind (p, filesep) ;
+ matrixpath = [ p(1:t(end)) 'CXSparse/Matrix' ] ;
+ catch
+ % mfilename failed, assume we're in the SuiteSparse directory
+ matrixpath = 'CXSparse/Matrix' ;
+ end
+end
+
+input ('Hit enter to run the CXSparse demo: ') ;
+try
+ cs_demo (0, matrixpath)
+catch
+ fprintf ('\nIf you have an older version of MATLAB, you must run the\n') ;
+ fprintf ('SuiteSparse_demo while in the SuiteSparse directory.\n\n') ;
+ fprintf ('CXSparse demo failed\n' )
+end
+
+input ('Hit enter to run the UMFPACK demo: ') ;
+try
+ umfpack_demo (1)
+catch
+ disp (lasterr) ;
+ fprintf ('UMFPACK demo failed\n' )
+end
+
+input ('Hit enter to run the CHOLMOD demo: ') ;
+try
+ cholmod_demo
+catch
+ disp (lasterr) ;
+ fprintf ('CHOLMOD demo failed\n' )
+end
+
+input ('Hit enter to run the CHOLMOD graph partitioning demo: ') ;
+try
+ graph_demo
+catch
+ disp (lasterr) ;
+ fprintf ('graph_demo failed, probably because METIS not installed\n') ;
+end
+
+input ('Hit enter to run the AMD demo: ') ;
+try
+ amd_demo
+catch
+ disp (lasterr) ;
+ fprintf ('AMD demo failed\n' )
+end
+
+input ('Hit enter to run the CAMD demo: ') ;
+try
+ camd_demo
+catch
+ disp (lasterr) ;
+ fprintf ('CAMD demo failed\n' )
+end
+
+input ('Hit enter to run the COLAMD demo: ') ;
+try
+ colamd_demo
+catch
+ disp (lasterr) ;
+ fprintf ('COLAMD demo failed\n' )
+end
+
+input ('Hit enter to run the CCOLAMD demo: ') ;
+try
+ ccolamd_demo
+catch
+ disp (lasterr) ;
+ fprintf ('CCOLAMD demo failed\n' )
+end
+
+input ('Hit enter to run the BTF demo: ') ;
+try
+ btf_demo
+catch
+ disp (lasterr) ;
+ fprintf ('BTF demo failed\n' )
+end
+
+input ('Hit enter to run the KLU demo: ') ;
+try
+ klu_demo
+catch
+ disp (lasterr) ;
+ fprintf ('KLU demo failed\n' )
+end
+
+input ('Hit enter to run the LDL demo: ') ;
+try
+ ldldemo
+catch
+ disp (lasterr) ;
+ fprintf ('LDL demo failed\n' )
+end
+
+fprintf ('\n\n---- SuiteSparse demos complete\n') ;
diff --git a/src/C/SuiteSparse/UFconfig/README.txt b/src/C/SuiteSparse/UFconfig/README.txt
index b505181..0e165a4 100644
--- a/src/C/SuiteSparse/UFconfig/README.txt
+++ b/src/C/SuiteSparse/UFconfig/README.txt
@@ -1,25 +1,30 @@
-This file contains configuration settings for
-all many of the software packages that I develop or
-co-author:
+UFconfig contains configuration settings for all many of the software packages
+that I develop or co-author. Note that older versions of some of these packages
+do not require UFconfig.
- Package Version Description
- ------- ------- -----------
- AMD 1.2 or later approximate minimum degree ordering
- CAMD any
- COLAMD 2.4 or later column approximate minimum degree ordering
- CCOLAMD any constrained approximate minimum degree ordering
- UMFPACK 4.5 or later sparse LU factorization, with the BLAS
- CXSparse any
- CHOLMOD any sparse Cholesky factorization, update/downdate
- KLU 1.0 or later sparse LU factorization, BLAS-free
- BTF 1.0 or later permutation to block triangular form
- LDL any concise sparse LDL'
- LPDASA any LP Dual Active Set Algorithm
+ Package Description
+ ------- -----------
+ AMD approximate minimum degree ordering
+ CAMD constrained AMD
+ COLAMD column approximate minimum degree ordering
+ CCOLAMD constrained approximate minimum degree ordering
+ UMFPACK sparse LU factorization, with the BLAS
+ CXSparse int/long/real/complex version of CSparse
+ CHOLMOD sparse Cholesky factorization, update/downdate
+ KLU sparse LU factorization, BLAS-free
+ BTF permutation to block triangular form
+ LDL concise sparse LDL'
+ LPDASA LP Dual Active Set Algorithm
-In addition, the xerbla/ directory contains Fortan and C versions
-of the BLAS/LAPACK xerbla routine, which is called when an invalid
-input is passed to the BLAS or LAPACK. The xerbla provided here
-does not print any message, so the entire Fortran I/O library does
-not need to be linked into a C application. Most versions of the
-BLAS contain xerbla, but those from K. Goto do not. Use this if
-you need too.
+UFconfig is not required by:
+
+ CSparse a Concise Sparse matrix package
+ RBio read/write files in Rutherford/Boeing format
+ UFcollection tools for managing the UF Sparse Matrix Collection
+
+In addition, the xerbla/ directory contains Fortan and C versions of the
+BLAS/LAPACK xerbla routine, which is called when an invalid input is passed to
+the BLAS or LAPACK. The xerbla provided here does not print any message, so
+the entire Fortran I/O library does not need to be linked into a C application.
+Most versions of the BLAS contain xerbla, but those from K. Goto do not. Use
+this if you need too.
diff --git a/src/C/SuiteSparse/UFconfig/UFconfig.h b/src/C/SuiteSparse/UFconfig/UFconfig.h
index 051a40f..91bbeac 100644
--- a/src/C/SuiteSparse/UFconfig/UFconfig.h
+++ b/src/C/SuiteSparse/UFconfig/UFconfig.h
@@ -30,7 +30,7 @@
*
* This file also defines the SUITESPARSE_VERSION and related definitions.
*
- * Copyright (c) 2006, University of Florida. No licensing restrictions
+ * Copyright (c) 2007, University of Florida. No licensing restrictions
* apply to this file or to the UFconfig directory. Author: Timothy A. Davis.
*/
@@ -76,33 +76,33 @@ extern "C" {
* version of SuiteSparse, with another package from another version of
* SuiteSparse, may or may not work.
*
- * SuiteSparse Version 2.4 contains the following packages:
+ * SuiteSparse Version 3.0.0 contains the following packages:
*
- * AMD version 2.0.4
- * CAMD version 2.1.3
- * COLAMD version 2.6.0
- * CCOLAMD version 2.5.2
- * CHOLMOD version 1.4.0
- * CSparse version 2.0.7
- * CXSparse version 2.0.7
- * KLU version 0.11
- * BTF version 0.11
- * LDL version 1.3.4
+ * AMD version 2.2.0
+ * CAMD version 2.2.0
+ * COLAMD version 2.7.0
+ * CCOLAMD version 2.7.0
+ * CHOLMOD version 1.5.0
+ * CSparse version 2.2.0
+ * CXSparse version 2.2.0
+ * KLU version 1.0.0
+ * BTF version 1.0.0
+ * LDL version 2.0.0
* UFconfig version number is the same as SuiteSparse
- * UMFPACK version 5.0.3
- * RBio version 1.0.0
- * UFcollection version 1.0.1
+ * UMFPACK version 5.1.0
+ * RBio version 1.1.0
+ * UFcollection version 1.1.0
*
* Other package dependencies:
* BLAS required by CHOLMOD and UMFPACK
* LAPACK required by CHOLMOD
- * METIS 4.0.1 required by CHOLMOD (optional)
+ * METIS 4.0.1 required by CHOLMOD (optional) and KLU (optional)
*/
-#define SUITESPARSE_DATE "Dec 13, 2006"
+#define SUITESPARSE_DATE "May 31, 2007"
#define SUITESPARSE_VER_CODE(main,sub) ((main) * 1000 + (sub))
-#define SUITESPARSE_MAIN_VERSION 2
-#define SUITESPARSE_SUB_VERSION 4
+#define SUITESPARSE_MAIN_VERSION 3
+#define SUITESPARSE_SUB_VERSION 0
#define SUITESPARSE_SUBSUB_VERSION 0
#define SUITESPARSE_VERSION \
SUITESPARSE_VER_CODE(SUITESPARSE_MAIN_VERSION,SUITESPARSE_SUB_VERSION)
@@ -110,6 +110,4 @@ extern "C" {
#ifdef __cplusplus
}
#endif
-
-
#endif
diff --git a/src/C/SuiteSparse/UFconfig/UFconfig.mk b/src/C/SuiteSparse/UFconfig/UFconfig.mk
index 61dcbee..8d33eab 100644
--- a/src/C/SuiteSparse/UFconfig/UFconfig.mk
+++ b/src/C/SuiteSparse/UFconfig/UFconfig.mk
@@ -17,6 +17,7 @@
# BTF 0.8 or later permutation to block triangular form
# LDL 1.2 or later concise sparse LDL'
# LPDASA any linear program solve (dual active set algorithm)
+# CXSparse any extended version of CSparse (int/long, real/complex)
#
# The UFconfig directory and the above packages should all appear in a single
# directory, in order for the Makefile's within each package to find this file.
@@ -50,8 +51,15 @@ F77LIB =
# C and Fortran libraries
LIB = -lm
-# For compiling MATLAB mexFunctions
-MEX = mex -O
+# For compiling MATLAB mexFunctions (MATLAB 7.4 or later)
+MEX = mex -O -largeArrayDims -lmwlapack
+
+# For compiling MATLAB mexFunctions (MATLAB 7.3, but the 7.4 might work OK)
+# MEX = mex -O -largeArrayDims
+
+# For MATLAB 7.2 or earlier, you must use one of these options:
+# MEX = mex -O -lmwlapack
+# MEX = mex -O
# Which version of MAKE you are using (default is "make")
# MAKE = make
@@ -71,9 +79,13 @@ MEX = mex -O
# on a 2.5Ghz dual-core AMD Opteron.
# These settings will probably not work, since there is no fixed convention for
-# naming the BLAS and LAPACK library (*.a or *.so) files. Assume the Goto
-# BLAS are available.
-BLAS = -lgoto -lgfortran -lgfortranbegin
+# naming the BLAS and LAPACK library (*.a or *.so) files.
+
+# Using the Goto BLAS:
+# BLAS = -lgoto -lgfortran -lgfortranbegin
+
+# This is probably slow ... it might connect to the Standard Reference BLAS:
+BLAS = -lblas -lgfortran -lgfortranbegin
LAPACK = -llapack
# The BLAS might not contain xerbla, an error-handling routine for LAPACK and
diff --git a/src/C/SuiteSparse/UMFPACK/Doc/ChangeLog b/src/C/SuiteSparse/UMFPACK/Doc/ChangeLog
index 4ecbfd7..c2e258b 100644
--- a/src/C/SuiteSparse/UMFPACK/Doc/ChangeLog
+++ b/src/C/SuiteSparse/UMFPACK/Doc/ChangeLog
@@ -1,3 +1,12 @@
+May 31, 2007, version 5.1.0
+
+ * port to 64-bit MATLAB
+
+ * Makefiles updated to reflect directory changes to AMD (UMFPACK v5.1.0
+ requires AMD v2.2.0)
+
+ * Source/Makefile and GNUMakefile moved to Lib/
+
Dec 12, 2006: version 5.0.3
* minor MATLAB cleanup. Renamed umfpack mexFunction to umfpack2, to avoid
diff --git a/src/C/SuiteSparse/UMFPACK/Doc/QuickStart.pdf b/src/C/SuiteSparse/UMFPACK/Doc/QuickStart.pdf
deleted file mode 100644
index 986f282..0000000
Binary files a/src/C/SuiteSparse/UMFPACK/Doc/QuickStart.pdf and /dev/null differ
diff --git a/src/C/SuiteSparse/UMFPACK/Doc/QuickStart.tex b/src/C/SuiteSparse/UMFPACK/Doc/QuickStart.tex
index 9958477..61846a2 100644
--- a/src/C/SuiteSparse/UMFPACK/Doc/QuickStart.tex
+++ b/src/C/SuiteSparse/UMFPACK/Doc/QuickStart.tex
@@ -18,8 +18,8 @@
\author{Timothy A. Davis \\
Dept. of Computer and Information Science and Engineering \\
Univ. of Florida, Gainesville, FL}
-\title{UMFPACK Version 5.0 Quick Start Guide}
-\date{May 5, 2006}
+\title{UMFPACK Version 5.1 Quick Start Guide}
+\date{May 31, 2007}
\maketitle
%-------------------------------------------------------------------------------
@@ -35,7 +35,7 @@ Univ. of Florida, Gainesville, FL}
\end{abstract}
%-------------------------------------------------------------------------------
-UMFPACK Version 5.0, Copyright\copyright 1995-2006 by Timothy A. Davis.
+UMFPACK Version 5.1, Copyright\copyright 1995-2006 by Timothy A. Davis.
All Rights Reserved. Refer to the UMFPACK User Guide
for the License. See \newline
http://www.cise.ufl.edu/research/sparse/umfpack
@@ -234,8 +234,8 @@ for more details.
\label{Install}
%-------------------------------------------------------------------------------
-You will need to install both UMFPACK v5.0 and AMD v2.0 to use UMFPACK.
-Note that UMFPACK v5.0 cannot use AMD v1.2 or earlier.
+You will need to install both UMFPACK v5.1 and AMD v2.2 to use UMFPACK.
+Note that UMFPACK v5.1 cannot use AMD v1.2 or earlier.
The {\tt UMFPACK} and {\tt AMD} subdirectories must be placed side-by-side
within the same parent directory. AMD is a stand-alone package that
is required by UMFPACK. UMFPACK can be compiled without the
@@ -243,8 +243,8 @@ BLAS but your performance will be much less than what it should be.
System-dependent configurations are in the {\tt UFconfig/UFconfig.mk}
file. The default
-settings will work on most systems, except that UMFPACK will be compiled so
-that it does not use the BLAS. Sample configurations are provided
+settings will work on most systems, except for the BLAS definition.
+Sample configurations are provided
for Linux, Sun Solaris, SGI IRIX, IBM AIX, and the DEC/Compaq Alpha.
To compile and install both packages,
@@ -256,9 +256,15 @@ compiled and tested in the {\tt UMFPACK/Demo} directory.
The outputs of these demo programs will then be compared with output
files in the distribution. Expect to see a few differences, such as
residual norms, compile-time control settings, and perhaps memory usage
-differences. The AMD and MATLAB mexFunctions for
-use in MATLAB will also be compiled. If you do not have MATLAB,
-type {\tt make lib} instead.
+differences.
+
+To use {\tt make} to compile the MATLAB mexFunctions for MATLAB
+and AMD, you can either type {\tt make mex} in the UMFPACK directory.
+You may first need to edit the {\tt UFconfig/UFconfig.mk} file to
+modify the definition of the {\tt MEX}, if you have a version of MATLAB
+older than Version 7.2. Remove the {\tt -largeArrayDims} definition.
+If you use the MATLAB command {\tt umfpack\_make} in the MATLAB directory,
+then this case is handled for you automatically.
If you compile UMFPACK and AMD and then later change the
{\tt UFconfig/UFconfig.mk} file
diff --git a/src/C/SuiteSparse/UMFPACK/Doc/UserGuide.pdf b/src/C/SuiteSparse/UMFPACK/Doc/UserGuide.pdf
deleted file mode 100644
index a6087e1..0000000
Binary files a/src/C/SuiteSparse/UMFPACK/Doc/UserGuide.pdf and /dev/null differ
diff --git a/src/C/SuiteSparse/UMFPACK/Doc/UserGuide.stex b/src/C/SuiteSparse/UMFPACK/Doc/UserGuide.stex
index 601857d..8cf79f1 100644
--- a/src/C/SuiteSparse/UMFPACK/Doc/UserGuide.stex
+++ b/src/C/SuiteSparse/UMFPACK/Doc/UserGuide.stex
@@ -20,8 +20,8 @@
\author{Timothy A. Davis \\
Dept. of Computer and Information Science and Engineering \\
Univ. of Florida, Gainesville, FL}
-\title{UMFPACK Version 5.0 User Guide}
-\date{May 5, 2006}
+\title{UMFPACK Version 5.1 User Guide}
+\date{May 31, 2007}
\maketitle
%-------------------------------------------------------------------------------
@@ -38,7 +38,7 @@ Univ. of Florida, Gainesville, FL}
Technical Report TR-04-003 (revised)
-UMFPACK Version 5.0, Copyright\copyright 1995-2006 by Timothy A. Davis.
+UMFPACK Version 5.1, Copyright\copyright 1995-2006 by Timothy A. Davis.
All Rights Reserved.
UMFPACK is available under alternate licences; contact T. Davis for details.
@@ -297,6 +297,12 @@ Subroutine Library.
A detailed list of changes is in the {\tt ChangeLog} file.
%-------------------------------------------------------------------------------
+\subsection{Version 5.1.0}
+%-------------------------------------------------------------------------------
+
+Port of MATLAB interface to 64-bit MATLAB.
+
+%-------------------------------------------------------------------------------
\subsection{Version 5.0.3}
%-------------------------------------------------------------------------------
@@ -1500,7 +1506,7 @@ In this case, no {\tt Symbolic} or {\tt Numeric} object was created.
\item {\tt UMFPACK\_ERROR\_internal\_error}, (-911):
An internal error has occurred, of unknown cause. This is either a bug
in UMFPACK, or the result of a memory overrun from your program.
- Try modifying the file {\tt AMD/Source/amd\_internal.h} and adding
+ Try modifying the file {\tt AMD/Include/amd\_internal.h} and adding
the statement {\tt \#undef NDEBUG}, to enable the debugging mode.
Recompile UMFPACK and rerun your program.
A failed assertion might occur which
@@ -2052,14 +2058,20 @@ compiled and tested in the {\tt UMFPACK/Demo} directory.
The outputs of these demo programs will then be compared with output
files in the distribution. Expect to see a few differences, such as
residual norms, compile-time control settings, and perhaps memory usage
-differences. The AMD and UMFPACK mexFunctions for
-use in MATLAB will also be compiled. If you do not have MATLAB 6.0 or
-later, type {\tt make lib} instead.
+differences.
+
+To use {\tt make} to compile the MATLAB mexFunctions for MATLAB
+and AMD, you can either type {\tt make mex} in the UMFPACK directory.
+You may first need to edit the {\tt UFconfig/UFconfig.mk} file to
+modify the definition of the {\tt MEX}, if you have a version of MATLAB
+older than Version 7.2. Remove the {\tt -largeArrayDims} definition.
+If you use the MATLAB command {\tt umfpack\_make} in the MATLAB directory,
+then this case is handled for you automatically.
-If you have the GNU version of {\tt make}, the {\tt Source/GNUmakefile} and
+If you have the GNU version of {\tt make}, the {\tt Lib/GNUmakefile} and
{\tt MATLAB/GNUmakefile} files are used. These are much more concise than
what the ``old'' version of {\tt make} can handle. If you do not have
-GNU {\tt make}, the {\tt Source/Makefile} and {\tt MATLAB/Makefile} files
+GNU {\tt make}, the {\tt Lib/Makefile} and {\tt MATLAB/Makefile} files
are used instead. Each UMFPACK source file is compiled into four
versions ({\tt double} / complex, and {\tt int} / {\tt UF\_long}). A proper
old-style {\tt Makefile} is cumbersome in this case, so these two
@@ -2200,13 +2212,13 @@ to compile the stand-alone AMD mexFunction (this is not required to
compile the UMFPACK mexFunction). This works on any computer with MATLAB,
including Windows.
-You will be prompted to select several configuration options, including
-whether or not to use the BLAS.
-MATLAB 5.3 (or earlier) does not include the BLAS, so you either have to
-compile UMFPACK without the BLAS (UMFPACK will be slow), or modify your
-{\tt <matlab>/bin/mexopts.sh} by adding your BLAS library
-to the {\tt CLIBS} string,
-where {\tt <matlab>} is the directory in which MATLAB is installed.
+% You will be prompted to select several configuration options, including
+% whether or not to use the BLAS.
+% MATLAB 5.3 (or earlier) does not include the BLAS, so you either have to
+% compile UMFPACK without the BLAS (UMFPACK will be slow), or modify your
+% {\tt <matlab>/bin/mexopts.sh} by adding your BLAS library
+% to the {\tt CLIBS} string,
+% where {\tt <matlab>} is the directory in which MATLAB is installed.
If you are using Windows and the {\tt lcc} compiler bundled with
MATLAB 6.1, then you may need to copy the
diff --git a/src/C/SuiteSparse/UMFPACK/Include/umfpack.h b/src/C/SuiteSparse/UMFPACK/Include/umfpack.h
index f128f43..d0eb203 100644
--- a/src/C/SuiteSparse/UMFPACK/Include/umfpack.h
+++ b/src/C/SuiteSparse/UMFPACK/Include/umfpack.h
@@ -3,7 +3,7 @@
/* ========================================================================== */
/* -------------------------------------------------------------------------- */
-/* UMFPACK Version 5.0, Copyright (c) 1995-2006 by Timothy A. Davis. CISE, */
+/* UMFPACK Copyright (c) Timothy A. Davis, CISE, */
/* Univ. of Florida. All Rights Reserved. See ../Doc/License for License. */
/* web: http://www.cise.ufl.edu/research/sparse/umfpack */
/* -------------------------------------------------------------------------- */
@@ -97,7 +97,7 @@ extern "C" {
/* Version, copyright, and license */
/* -------------------------------------------------------------------------- */
-#define UMFPACK_VERSION "UMFPACK V5.0.3 (Dec 12, 2006)"
+#define UMFPACK_VERSION "UMFPACK V5.1.0 (May 31, 2007)"
#define UMFPACK_COPYRIGHT \
"UMFPACK: Copyright (c) 2005-2006 by Timothy A. Davis. All Rights Reserved.\n"
@@ -163,11 +163,11 @@ extern "C" {
* above.
*/
-#define UMFPACK_DATE "Dec 12, 2006"
+#define UMFPACK_DATE "May 31, 2007"
#define UMFPACK_VER_CODE(main,sub) ((main) * 1000 + (sub))
#define UMFPACK_MAIN_VERSION 5
-#define UMFPACK_SUB_VERSION 0
-#define UMFPACK_SUBSUB_VERSION 3
+#define UMFPACK_SUB_VERSION 1
+#define UMFPACK_SUBSUB_VERSION 0
#define UMFPACK_VER UMFPACK_VER_CODE(UMFPACK_MAIN_VERSION,UMFPACK_SUB_VERSION)
/* -------------------------------------------------------------------------- */
diff --git a/src/C/SuiteSparse/UMFPACK/Include/umfpack_col_to_triplet.h b/src/C/SuiteSparse/UMFPACK/Include/umfpack_col_to_triplet.h
index 427571e..948a65e 100644
--- a/src/C/SuiteSparse/UMFPACK/Include/umfpack_col_to_triplet.h
+++ b/src/C/SuiteSparse/UMFPACK/Include/umfpack_col_to_triplet.h
@@ -3,7 +3,7 @@
/* ========================================================================== */
/* -------------------------------------------------------------------------- */
-/* UMFPACK Version 5.0, Copyright (c) 1995-2006 by Timothy A. Davis. CISE, */
+/* UMFPACK Copyright (c) Timothy A. Davis, CISE, */
/* Univ. of Florida. All Rights Reserved. See ../Doc/License for License. */
/* web: http://www.cise.ufl.edu/research/sparse/umfpack */
/* -------------------------------------------------------------------------- */
diff --git a/src/C/SuiteSparse/UMFPACK/Include/umfpack_defaults.h b/src/C/SuiteSparse/UMFPACK/Include/umfpack_defaults.h
index bc40313..a588d4f 100644
--- a/src/C/SuiteSparse/UMFPACK/Include/umfpack_defaults.h
+++ b/src/C/SuiteSparse/UMFPACK/Include/umfpack_defaults.h
@@ -3,7 +3,7 @@
/* ========================================================================== */
/* -------------------------------------------------------------------------- */
-/* UMFPACK Version 5.0, Copyright (c) 1995-2006 by Timothy A. Davis. CISE, */
+/* UMFPACK Copyright (c) Timothy A. Davis, CISE, */
/* Univ. of Florida. All Rights Reserved. See ../Doc/License for License. */
/* web: http://www.cise.ufl.edu/research/sparse/umfpack */
/* -------------------------------------------------------------------------- */
diff --git a/src/C/SuiteSparse/UMFPACK/Include/umfpack_free_numeric.h b/src/C/SuiteSparse/UMFPACK/Include/umfpack_free_numeric.h
index 94bbf98..986a6ab 100644
--- a/src/C/SuiteSparse/UMFPACK/Include/umfpack_free_numeric.h
+++ b/src/C/SuiteSparse/UMFPACK/Include/umfpack_free_numeric.h
@@ -3,7 +3,7 @@
/* ========================================================================== */
/* -------------------------------------------------------------------------- */
-/* UMFPACK Version 5.0, Copyright (c) 1995-2006 by Timothy A. Davis. CISE, */
+/* UMFPACK Copyright (c) Timothy A. Davis, CISE, */
/* Univ. of Florida. All Rights Reserved. See ../Doc/License for License. */
/* web: http://www.cise.ufl.edu/research/sparse/umfpack */
/* -------------------------------------------------------------------------- */
diff --git a/src/C/SuiteSparse/UMFPACK/Include/umfpack_free_symbolic.h b/src/C/SuiteSparse/UMFPACK/Include/umfpack_free_symbolic.h
index cf9262e..ad3aeb6 100644
--- a/src/C/SuiteSparse/UMFPACK/Include/umfpack_free_symbolic.h
+++ b/src/C/SuiteSparse/UMFPACK/Include/umfpack_free_symbolic.h
@@ -3,7 +3,7 @@
/* ========================================================================== */
/* -------------------------------------------------------------------------- */
-/* UMFPACK Version 5.0, Copyright (c) 1995-2006 by Timothy A. Davis. CISE, */
+/* UMFPACK Copyright (c) Timothy A. Davis, CISE, */
/* Univ. of Florida. All Rights Reserved. See ../Doc/License for License. */
/* web: http://www.cise.ufl.edu/research/sparse/umfpack */
/* -------------------------------------------------------------------------- */
diff --git a/src/C/SuiteSparse/UMFPACK/Include/umfpack_get_determinant.h b/src/C/SuiteSparse/UMFPACK/Include/umfpack_get_determinant.h
index 71b4158..702c1f1 100644
--- a/src/C/SuiteSparse/UMFPACK/Include/umfpack_get_determinant.h
+++ b/src/C/SuiteSparse/UMFPACK/Include/umfpack_get_determinant.h
@@ -3,7 +3,7 @@
/* ========================================================================== */
/* -------------------------------------------------------------------------- */
-/* UMFPACK Version 5.0, Copyright (c) 1995-2006 by Timothy A. Davis. CISE, */
+/* UMFPACK Copyright (c) Timothy A. Davis, CISE, */
/* Univ. of Florida. All Rights Reserved. See ../Doc/License for License. */
/* web: http://www.cise.ufl.edu/research/sparse/umfpack */
/* UMFPACK_get_determinant contributed by David Bateman, Motorola, Paris. */
diff --git a/src/C/SuiteSparse/UMFPACK/Include/umfpack_get_lunz.h b/src/C/SuiteSparse/UMFPACK/Include/umfpack_get_lunz.h
index 8e53520..512cffa 100644
--- a/src/C/SuiteSparse/UMFPACK/Include/umfpack_get_lunz.h
+++ b/src/C/SuiteSparse/UMFPACK/Include/umfpack_get_lunz.h
@@ -3,7 +3,7 @@
/* ========================================================================== */
/* -------------------------------------------------------------------------- */
-/* UMFPACK Version 5.0, Copyright (c) 1995-2006 by Timothy A. Davis. CISE, */
+/* UMFPACK Copyright (c) Timothy A. Davis, CISE, */
/* Univ. of Florida. All Rights Reserved. See ../Doc/License for License. */
/* web: http://www.cise.ufl.edu/research/sparse/umfpack */
/* -------------------------------------------------------------------------- */
diff --git a/src/C/SuiteSparse/UMFPACK/Include/umfpack_get_numeric.h b/src/C/SuiteSparse/UMFPACK/Include/umfpack_get_numeric.h
index 07c8252..45ba0e6 100644
--- a/src/C/SuiteSparse/UMFPACK/Include/umfpack_get_numeric.h
+++ b/src/C/SuiteSparse/UMFPACK/Include/umfpack_get_numeric.h
@@ -3,7 +3,7 @@
/* ========================================================================== */
/* -------------------------------------------------------------------------- */
-/* UMFPACK Version 5.0, Copyright (c) 1995-2006 by Timothy A. Davis. CISE, */
+/* UMFPACK Copyright (c) Timothy A. Davis, CISE, */
/* Univ. of Florida. All Rights Reserved. See ../Doc/License for License. */
/* web: http://www.cise.ufl.edu/research/sparse/umfpack */
/* -------------------------------------------------------------------------- */
diff --git a/src/C/SuiteSparse/UMFPACK/Include/umfpack_get_symbolic.h b/src/C/SuiteSparse/UMFPACK/Include/umfpack_get_symbolic.h
index 545825e..21b7cc2 100644
--- a/src/C/SuiteSparse/UMFPACK/Include/umfpack_get_symbolic.h
+++ b/src/C/SuiteSparse/UMFPACK/Include/umfpack_get_symbolic.h
@@ -3,7 +3,7 @@
/* ========================================================================== */
/* -------------------------------------------------------------------------- */
-/* UMFPACK Version 5.0, Copyright (c) 1995-2006 by Timothy A. Davis. CISE, */
+/* UMFPACK Copyright (c) Timothy A. Davis, CISE, */
/* Univ. of Florida. All Rights Reserved. See ../Doc/License for License. */
/* web: http://www.cise.ufl.edu/research/sparse/umfpack */
/* -------------------------------------------------------------------------- */
diff --git a/src/C/SuiteSparse/UMFPACK/Include/umfpack_global.h b/src/C/SuiteSparse/UMFPACK/Include/umfpack_global.h
index c2c946b..51c64dd 100644
--- a/src/C/SuiteSparse/UMFPACK/Include/umfpack_global.h
+++ b/src/C/SuiteSparse/UMFPACK/Include/umfpack_global.h
@@ -3,7 +3,7 @@
/* ========================================================================== */
/* -------------------------------------------------------------------------- */
-/* UMFPACK Version 5.0, Copyright (c) 1995-2006 by Timothy A. Davis. CISE, */
+/* UMFPACK Copyright (c) Timothy A. Davis, CISE, */
/* Univ. of Florida. All Rights Reserved. See ../Doc/License for License. */
/* web: http://www.cise.ufl.edu/research/sparse/umfpack */
/* -------------------------------------------------------------------------- */
diff --git a/src/C/SuiteSparse/UMFPACK/Include/umfpack_load_numeric.h b/src/C/SuiteSparse/UMFPACK/Include/umfpack_load_numeric.h
index 4473821..0b84c99 100644
--- a/src/C/SuiteSparse/UMFPACK/Include/umfpack_load_numeric.h
+++ b/src/C/SuiteSparse/UMFPACK/Include/umfpack_load_numeric.h
@@ -3,7 +3,7 @@
/* ========================================================================== */
/* -------------------------------------------------------------------------- */
-/* UMFPACK Version 5.0, Copyright (c) 1995-2006 by Timothy A. Davis. CISE, */
+/* UMFPACK Copyright (c) Timothy A. Davis, CISE, */
/* Univ. of Florida. All Rights Reserved. See ../Doc/License for License. */
/* web: http://www.cise.ufl.edu/research/sparse/umfpack */
/* -------------------------------------------------------------------------- */
diff --git a/src/C/SuiteSparse/UMFPACK/Include/umfpack_load_symbolic.h b/src/C/SuiteSparse/UMFPACK/Include/umfpack_load_symbolic.h
index 8952445..8bd5728 100644
--- a/src/C/SuiteSparse/UMFPACK/Include/umfpack_load_symbolic.h
+++ b/src/C/SuiteSparse/UMFPACK/Include/umfpack_load_symbolic.h
@@ -3,7 +3,7 @@
/* ========================================================================== */
/* -------------------------------------------------------------------------- */
-/* UMFPACK Version 5.0, Copyright (c) 1995-2006 by Timothy A. Davis. CISE, */
+/* UMFPACK Copyright (c) Timothy A. Davis, CISE, */
/* Univ. of Florida. All Rights Reserved. See ../Doc/License for License. */
/* web: http://www.cise.ufl.edu/research/sparse/umfpack */
/* -------------------------------------------------------------------------- */
diff --git a/src/C/SuiteSparse/UMFPACK/Include/umfpack_numeric.h b/src/C/SuiteSparse/UMFPACK/Include/umfpack_numeric.h
index 0f437b9..d871ef1 100644
--- a/src/C/SuiteSparse/UMFPACK/Include/umfpack_numeric.h
+++ b/src/C/SuiteSparse/UMFPACK/Include/umfpack_numeric.h
@@ -3,7 +3,7 @@
/* ========================================================================== */
/* -------------------------------------------------------------------------- */
-/* UMFPACK Version 5.0, Copyright (c) 1995-2006 by Timothy A. Davis. CISE, */
+/* UMFPACK Copyright (c) Timothy A. Davis, CISE, */
/* Univ. of Florida. All Rights Reserved. See ../Doc/License for License. */
/* web: http://www.cise.ufl.edu/research/sparse/umfpack */
/* -------------------------------------------------------------------------- */
diff --git a/src/C/SuiteSparse/UMFPACK/Include/umfpack_qsymbolic.h b/src/C/SuiteSparse/UMFPACK/Include/umfpack_qsymbolic.h
index 28362f2..00f9cd4 100644
--- a/src/C/SuiteSparse/UMFPACK/Include/umfpack_qsymbolic.h
+++ b/src/C/SuiteSparse/UMFPACK/Include/umfpack_qsymbolic.h
@@ -3,7 +3,7 @@
/* ========================================================================== */
/* -------------------------------------------------------------------------- */
-/* UMFPACK Version 5.0, Copyright (c) 1995-2006 by Timothy A. Davis. CISE, */
+/* UMFPACK Copyright (c) Timothy A. Davis, CISE, */
/* Univ. of Florida. All Rights Reserved. See ../Doc/License for License. */
/* web: http://www.cise.ufl.edu/research/sparse/umfpack */
/* -------------------------------------------------------------------------- */
diff --git a/src/C/SuiteSparse/UMFPACK/Include/umfpack_report_control.h b/src/C/SuiteSparse/UMFPACK/Include/umfpack_report_control.h
index 2084409..361a353 100644
--- a/src/C/SuiteSparse/UMFPACK/Include/umfpack_report_control.h
+++ b/src/C/SuiteSparse/UMFPACK/Include/umfpack_report_control.h
@@ -3,7 +3,7 @@
/* ========================================================================== */
/* -------------------------------------------------------------------------- */
-/* UMFPACK Version 5.0, Copyright (c) 1995-2006 by Timothy A. Davis. CISE, */
+/* UMFPACK Copyright (c) Timothy A. Davis, CISE, */
/* Univ. of Florida. All Rights Reserved. See ../Doc/License for License. */
/* web: http://www.cise.ufl.edu/research/sparse/umfpack */
/* -------------------------------------------------------------------------- */
diff --git a/src/C/SuiteSparse/UMFPACK/Include/umfpack_report_info.h b/src/C/SuiteSparse/UMFPACK/Include/umfpack_report_info.h
index e78ae44..5822721 100644
--- a/src/C/SuiteSparse/UMFPACK/Include/umfpack_report_info.h
+++ b/src/C/SuiteSparse/UMFPACK/Include/umfpack_report_info.h
@@ -3,7 +3,7 @@
/* ========================================================================== */
/* -------------------------------------------------------------------------- */
-/* UMFPACK Version 5.0, Copyright (c) 1995-2006 by Timothy A. Davis. CISE, */
+/* UMFPACK Copyright (c) Timothy A. Davis, CISE, */
/* Univ. of Florida. All Rights Reserved. See ../Doc/License for License. */
/* web: http://www.cise.ufl.edu/research/sparse/umfpack */
/* -------------------------------------------------------------------------- */
diff --git a/src/C/SuiteSparse/UMFPACK/Include/umfpack_report_matrix.h b/src/C/SuiteSparse/UMFPACK/Include/umfpack_report_matrix.h
index 7148896..0d17e2d 100644
--- a/src/C/SuiteSparse/UMFPACK/Include/umfpack_report_matrix.h
+++ b/src/C/SuiteSparse/UMFPACK/Include/umfpack_report_matrix.h
@@ -3,7 +3,7 @@
/* ========================================================================== */
/* -------------------------------------------------------------------------- */
-/* UMFPACK Version 5.0, Copyright (c) 1995-2006 by Timothy A. Davis. CISE, */
+/* UMFPACK Copyright (c) Timothy A. Davis, CISE, */
/* Univ. of Florida. All Rights Reserved. See ../Doc/License for License. */
/* web: http://www.cise.ufl.edu/research/sparse/umfpack */
/* -------------------------------------------------------------------------- */
diff --git a/src/C/SuiteSparse/UMFPACK/Include/umfpack_report_numeric.h b/src/C/SuiteSparse/UMFPACK/Include/umfpack_report_numeric.h
index 635f8cd..ff4eb11 100644
--- a/src/C/SuiteSparse/UMFPACK/Include/umfpack_report_numeric.h
+++ b/src/C/SuiteSparse/UMFPACK/Include/umfpack_report_numeric.h
@@ -3,7 +3,7 @@
/* ========================================================================== */
/* -------------------------------------------------------------------------- */
-/* UMFPACK Version 5.0, Copyright (c) 1995-2006 by Timothy A. Davis. CISE, */
+/* UMFPACK Copyright (c) Timothy A. Davis, CISE, */
/* Univ. of Florida. All Rights Reserved. See ../Doc/License for License. */
/* web: http://www.cise.ufl.edu/research/sparse/umfpack */
/* -------------------------------------------------------------------------- */
diff --git a/src/C/SuiteSparse/UMFPACK/Include/umfpack_report_perm.h b/src/C/SuiteSparse/UMFPACK/Include/umfpack_report_perm.h
index 0bf0865..4810e32 100644
--- a/src/C/SuiteSparse/UMFPACK/Include/umfpack_report_perm.h
+++ b/src/C/SuiteSparse/UMFPACK/Include/umfpack_report_perm.h
@@ -3,7 +3,7 @@
/* ========================================================================== */
/* -------------------------------------------------------------------------- */
-/* UMFPACK Version 5.0, Copyright (c) 1995-2006 by Timothy A. Davis. CISE, */
+/* UMFPACK Copyright (c) Timothy A. Davis, CISE, */
/* Univ. of Florida. All Rights Reserved. See ../Doc/License for License. */
/* web: http://www.cise.ufl.edu/research/sparse/umfpack */
/* -------------------------------------------------------------------------- */
diff --git a/src/C/SuiteSparse/UMFPACK/Include/umfpack_report_status.h b/src/C/SuiteSparse/UMFPACK/Include/umfpack_report_status.h
index 5acf73c..24376ae 100644
--- a/src/C/SuiteSparse/UMFPACK/Include/umfpack_report_status.h
+++ b/src/C/SuiteSparse/UMFPACK/Include/umfpack_report_status.h
@@ -3,7 +3,7 @@
/* ========================================================================== */
/* -------------------------------------------------------------------------- */
-/* UMFPACK Version 5.0, Copyright (c) 1995-2006 by Timothy A. Davis. CISE, */
+/* UMFPACK Copyright (c) Timothy A. Davis, CISE, */
/* Univ. of Florida. All Rights Reserved. See ../Doc/License for License. */
/* web: http://www.cise.ufl.edu/research/sparse/umfpack */
/* -------------------------------------------------------------------------- */
diff --git a/src/C/SuiteSparse/UMFPACK/Include/umfpack_report_symbolic.h b/src/C/SuiteSparse/UMFPACK/Include/umfpack_report_symbolic.h
index a3a9413..eb61182 100644
--- a/src/C/SuiteSparse/UMFPACK/Include/umfpack_report_symbolic.h
+++ b/src/C/SuiteSparse/UMFPACK/Include/umfpack_report_symbolic.h
@@ -3,7 +3,7 @@
/* ========================================================================== */
/* -------------------------------------------------------------------------- */
-/* UMFPACK Version 5.0, Copyright (c) 1995-2006 by Timothy A. Davis. CISE, */
+/* UMFPACK Copyright (c) Timothy A. Davis, CISE, */
/* Univ. of Florida. All Rights Reserved. See ../Doc/License for License. */
/* web: http://www.cise.ufl.edu/research/sparse/umfpack */
/* -------------------------------------------------------------------------- */
diff --git a/src/C/SuiteSparse/UMFPACK/Include/umfpack_report_triplet.h b/src/C/SuiteSparse/UMFPACK/Include/umfpack_report_triplet.h
index 0f4f60a..4e7a605 100644
--- a/src/C/SuiteSparse/UMFPACK/Include/umfpack_report_triplet.h
+++ b/src/C/SuiteSparse/UMFPACK/Include/umfpack_report_triplet.h
@@ -3,7 +3,7 @@
/* ========================================================================== */
/* -------------------------------------------------------------------------- */
-/* UMFPACK Version 5.0, Copyright (c) 1995-2006 by Timothy A. Davis. CISE, */
+/* UMFPACK Copyright (c) Timothy A. Davis, CISE, */
/* Univ. of Florida. All Rights Reserved. See ../Doc/License for License. */
/* web: http://www.cise.ufl.edu/research/sparse/umfpack */
/* -------------------------------------------------------------------------- */
diff --git a/src/C/SuiteSparse/UMFPACK/Include/umfpack_report_vector.h b/src/C/SuiteSparse/UMFPACK/Include/umfpack_report_vector.h
index d930b1e..2c4c386 100644
--- a/src/C/SuiteSparse/UMFPACK/Include/umfpack_report_vector.h
+++ b/src/C/SuiteSparse/UMFPACK/Include/umfpack_report_vector.h
@@ -3,7 +3,7 @@
/* ========================================================================== */
/* -------------------------------------------------------------------------- */
-/* UMFPACK Version 5.0, Copyright (c) 1995-2006 by Timothy A. Davis. CISE, */
+/* UMFPACK Copyright (c) Timothy A. Davis, CISE, */
/* Univ. of Florida. All Rights Reserved. See ../Doc/License for License. */
/* web: http://www.cise.ufl.edu/research/sparse/umfpack */
/* -------------------------------------------------------------------------- */
diff --git a/src/C/SuiteSparse/UMFPACK/Include/umfpack_save_numeric.h b/src/C/SuiteSparse/UMFPACK/Include/umfpack_save_numeric.h
index 72eb3b1..708e581 100644
--- a/src/C/SuiteSparse/UMFPACK/Include/umfpack_save_numeric.h
+++ b/src/C/SuiteSparse/UMFPACK/Include/umfpack_save_numeric.h
@@ -3,7 +3,7 @@
/* ========================================================================== */
/* -------------------------------------------------------------------------- */
-/* UMFPACK Version 5.0, Copyright (c) 1995-2006 by Timothy A. Davis. CISE, */
+/* UMFPACK Copyright (c) Timothy A. Davis, CISE, */
/* Univ. of Florida. All Rights Reserved. See ../Doc/License for License. */
/* web: http://www.cise.ufl.edu/research/sparse/umfpack */
/* -------------------------------------------------------------------------- */
diff --git a/src/C/SuiteSparse/UMFPACK/Include/umfpack_save_symbolic.h b/src/C/SuiteSparse/UMFPACK/Include/umfpack_save_symbolic.h
index 3e64a26..2dffdd6 100644
--- a/src/C/SuiteSparse/UMFPACK/Include/umfpack_save_symbolic.h
+++ b/src/C/SuiteSparse/UMFPACK/Include/umfpack_save_symbolic.h
@@ -3,7 +3,7 @@
/* ========================================================================== */
/* -------------------------------------------------------------------------- */
-/* UMFPACK Version 5.0, Copyright (c) 1995-2006 by Timothy A. Davis. CISE, */
+/* UMFPACK Copyright (c) Timothy A. Davis, CISE, */
/* Univ. of Florida. All Rights Reserved. See ../Doc/License for License. */
/* web: http://www.cise.ufl.edu/research/sparse/umfpack */
/* -------------------------------------------------------------------------- */
diff --git a/src/C/SuiteSparse/UMFPACK/Include/umfpack_scale.h b/src/C/SuiteSparse/UMFPACK/Include/umfpack_scale.h
index 6f698b1..62fc32a 100644
--- a/src/C/SuiteSparse/UMFPACK/Include/umfpack_scale.h
+++ b/src/C/SuiteSparse/UMFPACK/Include/umfpack_scale.h
@@ -3,7 +3,7 @@
/* ========================================================================== */
/* -------------------------------------------------------------------------- */
-/* UMFPACK Version 5.0, Copyright (c) 1995-2006 by Timothy A. Davis. CISE, */
+/* UMFPACK Copyright (c) Timothy A. Davis, CISE, */
/* Univ. of Florida. All Rights Reserved. See ../Doc/License for License. */
/* web: http://www.cise.ufl.edu/research/sparse/umfpack */
/* -------------------------------------------------------------------------- */
diff --git a/src/C/SuiteSparse/UMFPACK/Include/umfpack_solve.h b/src/C/SuiteSparse/UMFPACK/Include/umfpack_solve.h
index a4b984b..0932091 100644
--- a/src/C/SuiteSparse/UMFPACK/Include/umfpack_solve.h
+++ b/src/C/SuiteSparse/UMFPACK/Include/umfpack_solve.h
@@ -3,7 +3,7 @@
/* ========================================================================== */
/* -------------------------------------------------------------------------- */
-/* UMFPACK Version 5.0, Copyright (c) 1995-2006 by Timothy A. Davis. CISE, */
+/* UMFPACK Copyright (c) Timothy A. Davis, CISE, */
/* Univ. of Florida. All Rights Reserved. See ../Doc/License for License. */
/* web: http://www.cise.ufl.edu/research/sparse/umfpack */
/* -------------------------------------------------------------------------- */
diff --git a/src/C/SuiteSparse/UMFPACK/Include/umfpack_symbolic.h b/src/C/SuiteSparse/UMFPACK/Include/umfpack_symbolic.h
index 52ed167..1deacc2 100644
--- a/src/C/SuiteSparse/UMFPACK/Include/umfpack_symbolic.h
+++ b/src/C/SuiteSparse/UMFPACK/Include/umfpack_symbolic.h
@@ -3,7 +3,7 @@
/* ========================================================================== */
/* -------------------------------------------------------------------------- */
-/* UMFPACK Version 5.0, Copyright (c) 1995-2006 by Timothy A. Davis. CISE, */
+/* UMFPACK Copyright (c) Timothy A. Davis, CISE, */
/* Univ. of Florida. All Rights Reserved. See ../Doc/License for License. */
/* web: http://www.cise.ufl.edu/research/sparse/umfpack */
/* -------------------------------------------------------------------------- */
diff --git a/src/C/SuiteSparse/UMFPACK/Include/umfpack_tictoc.h b/src/C/SuiteSparse/UMFPACK/Include/umfpack_tictoc.h
index c2d7e57..da10426 100644
--- a/src/C/SuiteSparse/UMFPACK/Include/umfpack_tictoc.h
+++ b/src/C/SuiteSparse/UMFPACK/Include/umfpack_tictoc.h
@@ -3,7 +3,7 @@
/* ========================================================================== */
/* -------------------------------------------------------------------------- */
-/* UMFPACK Version 5.0, Copyright (c) 1995-2006 by Timothy A. Davis. CISE, */
+/* UMFPACK Copyright (c) Timothy A. Davis, CISE, */
/* Univ. of Florida. All Rights Reserved. See ../Doc/License for License. */
/* web: http://www.cise.ufl.edu/research/sparse/umfpack */
/* -------------------------------------------------------------------------- */
diff --git a/src/C/SuiteSparse/UMFPACK/Include/umfpack_timer.h b/src/C/SuiteSparse/UMFPACK/Include/umfpack_timer.h
index 7284a08..413a177 100644
--- a/src/C/SuiteSparse/UMFPACK/Include/umfpack_timer.h
+++ b/src/C/SuiteSparse/UMFPACK/Include/umfpack_timer.h
@@ -3,7 +3,7 @@
/* ========================================================================== */
/* -------------------------------------------------------------------------- */
-/* UMFPACK Version 5.0, Copyright (c) 1995-2006 by Timothy A. Davis. CISE, */
+/* UMFPACK Copyright (c) Timothy A. Davis, CISE, */
/* Univ. of Florida. All Rights Reserved. See ../Doc/License for License. */
/* web: http://www.cise.ufl.edu/research/sparse/umfpack */
/* -------------------------------------------------------------------------- */
diff --git a/src/C/SuiteSparse/UMFPACK/Include/umfpack_transpose.h b/src/C/SuiteSparse/UMFPACK/Include/umfpack_transpose.h
index 15aa60d..b12bb0b 100644
--- a/src/C/SuiteSparse/UMFPACK/Include/umfpack_transpose.h
+++ b/src/C/SuiteSparse/UMFPACK/Include/umfpack_transpose.h
@@ -3,7 +3,7 @@
/* ========================================================================== */
/* -------------------------------------------------------------------------- */
-/* UMFPACK Version 5.0, Copyright (c) 1995-2006 by Timothy A. Davis. CISE, */
+/* UMFPACK Copyright (c) Timothy A. Davis, CISE, */
/* Univ. of Florida. All Rights Reserved. See ../Doc/License for License. */
/* web: http://www.cise.ufl.edu/research/sparse/umfpack */
/* -------------------------------------------------------------------------- */
diff --git a/src/C/SuiteSparse/UMFPACK/Include/umfpack_triplet_to_col.h b/src/C/SuiteSparse/UMFPACK/Include/umfpack_triplet_to_col.h
index b519d86..cb45278 100644
--- a/src/C/SuiteSparse/UMFPACK/Include/umfpack_triplet_to_col.h
+++ b/src/C/SuiteSparse/UMFPACK/Include/umfpack_triplet_to_col.h
@@ -3,7 +3,7 @@
/* ========================================================================== */
/* -------------------------------------------------------------------------- */
-/* UMFPACK Version 5.0, Copyright (c) 1995-2006 by Timothy A. Davis. CISE, */
+/* UMFPACK Copyright (c) Timothy A. Davis, CISE, */
/* Univ. of Florida. All Rights Reserved. See ../Doc/License for License. */
/* web: http://www.cise.ufl.edu/research/sparse/umfpack */
/* -------------------------------------------------------------------------- */
diff --git a/src/C/SuiteSparse/UMFPACK/Include/umfpack_wsolve.h b/src/C/SuiteSparse/UMFPACK/Include/umfpack_wsolve.h
index c2ed033..38d15ba 100644
--- a/src/C/SuiteSparse/UMFPACK/Include/umfpack_wsolve.h
+++ b/src/C/SuiteSparse/UMFPACK/Include/umfpack_wsolve.h
@@ -3,7 +3,7 @@
/* ========================================================================== */
/* -------------------------------------------------------------------------- */
-/* UMFPACK Version 5.0, Copyright (c) 1995-2006 by Timothy A. Davis. CISE, */
+/* UMFPACK Copyright (c) Timothy A. Davis, CISE, */
/* Univ. of Florida. All Rights Reserved. See ../Doc/License for License. */
/* web: http://www.cise.ufl.edu/research/sparse/umfpack */
/* -------------------------------------------------------------------------- */
diff --git a/src/C/SuiteSparse/UMFPACK/Source/GNUmakefile b/src/C/SuiteSparse/UMFPACK/Lib/GNUmakefile
similarity index 74%
rename from src/C/SuiteSparse/UMFPACK/Source/GNUmakefile
rename to src/C/SuiteSparse/UMFPACK/Lib/GNUmakefile
index 1326653..559ed99 100644
--- a/src/C/SuiteSparse/UMFPACK/Source/GNUmakefile
+++ b/src/C/SuiteSparse/UMFPACK/Lib/GNUmakefile
@@ -7,7 +7,7 @@ default: ../Lib/libumfpack.a
include ../../UFconfig/UFconfig.mk
C = $(CC) $(CFLAGS) $(UMFPACK_CONFIG) \
- -I../Include -I../../AMD/Include -I../../UFconfig
+ -I../Include -I../Source -I../../AMD/Include -I../../UFconfig
#-------------------------------------------------------------------------------
# source files
@@ -66,9 +66,10 @@ GENERIC = umfpack_timer umfpack_tictoc umfpack_global
#-------------------------------------------------------------------------------
INC = ../Include/umfpack.h ../../UFconfig/UFconfig.h \
- umf_config.h umf_version.h umf_internal.h umf_triplet.h \
- $(addsuffix .h,$(UMFCH)) \
- $(addsuffix .h,$(UMFINT)) \
+ ../Source/umf_config.h ../Source/umf_version.h \
+ ../Source/umf_internal.h ../Source/umf_triplet.h \
+ $(addprefix ../Source/, $(addsuffix .h,$(UMFCH))) \
+ $(addprefix ../Source/, $(addsuffix .h,$(UMFINT))) \
$(addprefix ../Include/, $(addsuffix .h,$(USER))) \
$(addprefix ../Include/, $(addsuffix .h,$(GENERIC))) \
../../AMD/Include/amd_internal.h ../../AMD/Include/amd.h
@@ -89,157 +90,157 @@ GN = $(addsuffix .o, $(subst umfpack_,umfpack_gn_,$(GENERIC)))
# compile each int and UF_long routine (with no real/complex version)
#-------------------------------------------------------------------------------
-umf_i_%.o: umf_%.c $(INC)
+umf_i_%.o: ../Source/umf_%.c $(INC)
$(C) -DDINT -c $< -o $@
-umf_l_%.o: umf_%.c $(INC)
+umf_l_%.o: ../Source/umf_%.c $(INC)
$(C) -DDLONG -c $< -o $@
#-------------------------------------------------------------------------------
# compile each routine in the DI version
#-------------------------------------------------------------------------------
-umf_di_%.o: umf_%.c $(INC)
+umf_di_%.o: ../Source/umf_%.c $(INC)
$(C) -DDINT -c $< -o $@
-umf_di_%hsolve.o: umf_%tsolve.c $(INC)
+umf_di_%hsolve.o: ../Source/umf_%tsolve.c $(INC)
$(C) -DDINT -DCONJUGATE_SOLVE -c $< -o $@
-umf_di_triplet_map_x.o: umf_triplet.c $(INC)
+umf_di_triplet_map_x.o: ../Source/umf_triplet.c $(INC)
$(C) -DDINT -DDO_MAP -DDO_VALUES -c $< -o $@
-umf_di_triplet_map_nox.o: umf_triplet.c $(INC)
+umf_di_triplet_map_nox.o: ../Source/umf_triplet.c $(INC)
$(C) -DDINT -DDO_MAP -c $< -o $@
-umf_di_triplet_nomap_x.o: umf_triplet.c $(INC)
+umf_di_triplet_nomap_x.o: ../Source/umf_triplet.c $(INC)
$(C) -DDINT -DDO_VALUES -c $< -o $@
-umf_di_triplet_nomap_nox.o: umf_triplet.c $(INC)
+umf_di_triplet_nomap_nox.o: ../Source/umf_triplet.c $(INC)
$(C) -DDINT -c $< -o $@
-umf_di_assemble_fixq.o: umf_assemble.c $(INC)
+umf_di_assemble_fixq.o: ../Source/umf_assemble.c $(INC)
$(C) -DDINT -DFIXQ -c $< -o $@
-umf_di_store_lu_drop.o: umf_store_lu.c $(INC)
+umf_di_store_lu_drop.o: ../Source/umf_store_lu.c $(INC)
$(C) -DDINT -DDROP -c $< -o $@
-umfpack_di_wsolve.o: umfpack_solve.c $(INC)
+umfpack_di_wsolve.o: ../Source/umfpack_solve.c $(INC)
$(C) -DDINT -DWSOLVE -c $< -o $@
-umfpack_di_%.o: umfpack_%.c $(INC)
+umfpack_di_%.o: ../Source/umfpack_%.c $(INC)
$(C) -DDINT -c $< -o $@
#-------------------------------------------------------------------------------
# compile each routine in the DL version
#-------------------------------------------------------------------------------
-umf_dl_%.o: umf_%.c $(INC)
+umf_dl_%.o: ../Source/umf_%.c $(INC)
$(C) -DDLONG -c $< -o $@
-umf_dl_%hsolve.o: umf_%tsolve.c $(INC)
+umf_dl_%hsolve.o: ../Source/umf_%tsolve.c $(INC)
$(C) -DDLONG -DCONJUGATE_SOLVE -c $< -o $@
-umf_dl_triplet_map_x.o: umf_triplet.c $(INC)
+umf_dl_triplet_map_x.o: ../Source/umf_triplet.c $(INC)
$(C) -DDLONG -DDO_MAP -DDO_VALUES -c $< -o $@
-umf_dl_triplet_map_nox.o: umf_triplet.c $(INC)
+umf_dl_triplet_map_nox.o: ../Source/umf_triplet.c $(INC)
$(C) -DDLONG -DDO_MAP -c $< -o $@
-umf_dl_triplet_nomap_x.o: umf_triplet.c $(INC)
+umf_dl_triplet_nomap_x.o: ../Source/umf_triplet.c $(INC)
$(C) -DDLONG -DDO_VALUES -c $< -o $@
-umf_dl_triplet_nomap_nox.o: umf_triplet.c $(INC)
+umf_dl_triplet_nomap_nox.o: ../Source/umf_triplet.c $(INC)
$(C) -DDLONG -c $< -o $@
-umf_dl_assemble_fixq.o: umf_assemble.c $(INC)
+umf_dl_assemble_fixq.o: ../Source/umf_assemble.c $(INC)
$(C) -DDLONG -DFIXQ -c $< -o $@
-umf_dl_store_lu_drop.o: umf_store_lu.c $(INC)
+umf_dl_store_lu_drop.o: ../Source/umf_store_lu.c $(INC)
$(C) -DDLONG -DDROP -c $< -o $@
-umfpack_dl_wsolve.o: umfpack_solve.c $(INC)
+umfpack_dl_wsolve.o: ../Source/umfpack_solve.c $(INC)
$(C) -DDLONG -DWSOLVE -c $< -o $@
-umfpack_dl_%.o: umfpack_%.c $(INC)
+umfpack_dl_%.o: ../Source/umfpack_%.c $(INC)
$(C) -DDLONG -c $< -o $@
#-------------------------------------------------------------------------------
# compile each routine in the ZI version
#-------------------------------------------------------------------------------
-umf_zi_%.o: umf_%.c $(INC)
+umf_zi_%.o: ../Source/umf_%.c $(INC)
$(C) -DZINT -c $< -o $@
-umf_zi_%hsolve.o: umf_%tsolve.c $(INC)
+umf_zi_%hsolve.o: ../Source/umf_%tsolve.c $(INC)
$(C) -DZINT -DCONJUGATE_SOLVE -c $< -o $@
-umf_zi_triplet_map_x.o: umf_triplet.c $(INC)
+umf_zi_triplet_map_x.o: ../Source/umf_triplet.c $(INC)
$(C) -DZINT -DDO_MAP -DDO_VALUES -c $< -o $@
-umf_zi_triplet_map_nox.o: umf_triplet.c $(INC)
+umf_zi_triplet_map_nox.o: ../Source/umf_triplet.c $(INC)
$(C) -DZINT -DDO_MAP -c $< -o $@
-umf_zi_triplet_nomap_x.o: umf_triplet.c $(INC)
+umf_zi_triplet_nomap_x.o: ../Source/umf_triplet.c $(INC)
$(C) -DZINT -DDO_VALUES -c $< -o $@
-umf_zi_triplet_nomap_nox.o: umf_triplet.c $(INC)
+umf_zi_triplet_nomap_nox.o: ../Source/umf_triplet.c $(INC)
$(C) -DZINT -c $< -o $@
-umf_zi_assemble_fixq.o: umf_assemble.c $(INC)
+umf_zi_assemble_fixq.o: ../Source/umf_assemble.c $(INC)
$(C) -DZINT -DFIXQ -c $< -o $@
-umf_zi_store_lu_drop.o: umf_store_lu.c $(INC)
+umf_zi_store_lu_drop.o: ../Source/umf_store_lu.c $(INC)
$(C) -DZINT -DDROP -c $< -o $@
-umfpack_zi_wsolve.o: umfpack_solve.c $(INC)
+umfpack_zi_wsolve.o: ../Source/umfpack_solve.c $(INC)
$(C) -DZINT -DWSOLVE -c $< -o $@
-umfpack_zi_%.o: umfpack_%.c $(INC)
+umfpack_zi_%.o: ../Source/umfpack_%.c $(INC)
$(C) -DZINT -c $< -o $@
#-------------------------------------------------------------------------------
# compile each routine in the ZL version
#-------------------------------------------------------------------------------
-umf_zl_%.o: umf_%.c $(INC)
+umf_zl_%.o: ../Source/umf_%.c $(INC)
$(C) -DZLONG -c $< -o $@
-umf_zl_%hsolve.o: umf_%tsolve.c $(INC)
+umf_zl_%hsolve.o: ../Source/umf_%tsolve.c $(INC)
$(C) -DZLONG -DCONJUGATE_SOLVE -c $< -o $@
-umf_zl_triplet_map_x.o: umf_triplet.c $(INC)
+umf_zl_triplet_map_x.o: ../Source/umf_triplet.c $(INC)
$(C) -DZLONG -DDO_MAP -DDO_VALUES -c $< -o $@
-umf_zl_triplet_map_nox.o: umf_triplet.c $(INC)
+umf_zl_triplet_map_nox.o: ../Source/umf_triplet.c $(INC)
$(C) -DZLONG -DDO_MAP -c $< -o $@
-umf_zl_triplet_nomap_x.o: umf_triplet.c $(INC)
+umf_zl_triplet_nomap_x.o: ../Source/umf_triplet.c $(INC)
$(C) -DZLONG -DDO_VALUES -c $< -o $@
-umf_zl_triplet_nomap_nox.o: umf_triplet.c $(INC)
+umf_zl_triplet_nomap_nox.o: ../Source/umf_triplet.c $(INC)
$(C) -DZLONG -c $< -o $@
-umf_zl_assemble_fixq.o: umf_assemble.c $(INC)
+umf_zl_assemble_fixq.o: ../Source/umf_assemble.c $(INC)
$(C) -DZLONG -DFIXQ -c $< -o $@
-umf_zl_store_lu_drop.o: umf_store_lu.c $(INC)
+umf_zl_store_lu_drop.o: ../Source/umf_store_lu.c $(INC)
$(C) -DZLONG -DDROP -c $< -o $@
-umfpack_zl_wsolve.o: umfpack_solve.c $(INC)
+umfpack_zl_wsolve.o: ../Source/umfpack_solve.c $(INC)
$(C) -DZLONG -DWSOLVE -c $< -o $@
-umfpack_zl_%.o: umfpack_%.c $(INC)
+umfpack_zl_%.o: ../Source/umfpack_%.c $(INC)
$(C) -DZLONG -c $< -o $@
#-------------------------------------------------------------------------------
# Create the generic routines (GN) using a generic rule
#-------------------------------------------------------------------------------
-umfpack_gn_%.o: umfpack_%.c $(INC)
+umfpack_gn_%.o: ../Source/umfpack_%.c $(INC)
$(C) -c $< -o $@
#-------------------------------------------------------------------------------
-# Create the libumfpack.a library
+# Create the ../Lib/libumfpack.a library
#-------------------------------------------------------------------------------
../Lib/libumfpack.a: $(II) $(LL) $(GN) $(DI) $(DL) $(ZI) $(ZL)
diff --git a/src/C/SuiteSparse/UMFPACK/Lib/Makefile b/src/C/SuiteSparse/UMFPACK/Lib/Makefile
new file mode 100644
index 0000000..8444719
--- /dev/null
+++ b/src/C/SuiteSparse/UMFPACK/Lib/Makefile
@@ -0,0 +1,483 @@
+#-------------------------------------------------------------------------------
+# UMFPACK Makefile for compiling on Unix systems (for original make only)
+#-------------------------------------------------------------------------------
+
+# This is a very ugly Makefile, and is only provided for those who do not
+# have GNU make. Note that it is not used if you have GNU make. It ignores
+# dependency checking and just compiles everything.
+
+default: everything
+
+include ../../UFconfig/UFconfig.mk
+
+C = $(CC) $(CFLAGS) $(UMFPACK_CONFIG) -I../Include -I../../AMD/Include \
+ -I../Source
+
+everything:
+ $(C) -c ../Source/umfpack_global.c -o umfpack_gn_global.o
+ $(C) -DDINT -c ../Source/umf_analyze.c -o umf_i_analyze.o
+ $(C) -DDINT -c ../Source/umf_apply_order.c -o umf_i_apply_order.o
+ $(C) -DDINT -c ../Source/umf_colamd.c -o umf_i_colamd.o
+ $(C) -DDINT -c ../Source/umf_free.c -o umf_i_free.o
+ $(C) -DDINT -c ../Source/umf_fsize.c -o umf_i_fsize.o
+ $(C) -DDINT -c ../Source/umf_is_permutation.c -o umf_i_is_permutation.o
+ $(C) -DDINT -c ../Source/umf_malloc.c -o umf_i_malloc.o
+ $(C) -DDINT -c ../Source/umf_realloc.c -o umf_i_realloc.o
+ $(C) -DDINT -c ../Source/umf_report_perm.c -o umf_i_report_perm.o
+ $(C) -DDINT -c ../Source/umf_singletons.c -o umf_i_singletons.o
+ $(C) -DDLONG -c ../Source/umf_analyze.c -o umf_l_analyze.o
+ $(C) -DDLONG -c ../Source/umf_apply_order.c -o umf_l_apply_order.o
+ $(C) -DDLONG -c ../Source/umf_colamd.c -o umf_l_colamd.o
+ $(C) -DDLONG -c ../Source/umf_free.c -o umf_l_free.o
+ $(C) -DDLONG -c ../Source/umf_fsize.c -o umf_l_fsize.o
+ $(C) -DDLONG -c ../Source/umf_is_permutation.c -o umf_l_is_permutation.o
+ $(C) -DDLONG -c ../Source/umf_malloc.c -o umf_l_malloc.o
+ $(C) -DDLONG -c ../Source/umf_realloc.c -o umf_l_realloc.o
+ $(C) -DDLONG -c ../Source/umf_report_perm.c -o umf_l_report_perm.o
+ $(C) -DDLONG -c ../Source/umf_singletons.c -o umf_l_singletons.o
+ $(C) -c ../Source/umfpack_timer.c -o umfpack_gn_timer.o
+ $(C) -c ../Source/umfpack_tictoc.c -o umfpack_gn_tictoc.o
+ $(C) -DDINT -DCONJUGATE_SOLVE -c ../Source/umf_ltsolve.c -o umf_di_lhsolve.o
+ $(C) -DDINT -DCONJUGATE_SOLVE -c ../Source/umf_utsolve.c -o umf_di_uhsolve.o
+ $(C) -DDINT -DDO_MAP -c ../Source/umf_triplet.c -o umf_di_triplet_map_nox.o
+ $(C) -DDINT -DDO_VALUES -c ../Source/umf_triplet.c -o umf_di_triplet_nomap_x.o
+ $(C) -DDINT -c ../Source/umf_triplet.c -o umf_di_triplet_nomap_nox.o
+ $(C) -DDINT -DDO_MAP -DDO_VALUES -c ../Source/umf_triplet.c -o umf_di_triplet_map_x.o
+ $(C) -DDINT -DFIXQ -c ../Source/umf_assemble.c -o umf_di_assemble_fixq.o
+ $(C) -DDINT -DDROP -c ../Source/umf_store_lu.c -o umf_di_store_lu_drop.o
+ $(C) -DDINT -c ../Source/umf_assemble.c -o umf_di_assemble.o
+ $(C) -DDINT -c ../Source/umf_blas3_update.c -o umf_di_blas3_update.o
+ $(C) -DDINT -c ../Source/umf_build_tuples.c -o umf_di_build_tuples.o
+ $(C) -DDINT -c ../Source/umf_create_element.c -o umf_di_create_element.o
+ $(C) -DDINT -c ../Source/umf_dump.c -o umf_di_dump.o
+ $(C) -DDINT -c ../Source/umf_extend_front.c -o umf_di_extend_front.o
+ $(C) -DDINT -c ../Source/umf_garbage_collection.c -o umf_di_garbage_collection.o
+ $(C) -DDINT -c ../Source/umf_get_memory.c -o umf_di_get_memory.o
+ $(C) -DDINT -c ../Source/umf_init_front.c -o umf_di_init_front.o
+ $(C) -DDINT -c ../Source/umf_kernel.c -o umf_di_kernel.o
+ $(C) -DDINT -c ../Source/umf_kernel_init.c -o umf_di_kernel_init.o
+ $(C) -DDINT -c ../Source/umf_kernel_wrapup.c -o umf_di_kernel_wrapup.o
+ $(C) -DDINT -c ../Source/umf_local_search.c -o umf_di_local_search.o
+ $(C) -DDINT -c ../Source/umf_lsolve.c -o umf_di_lsolve.o
+ $(C) -DDINT -c ../Source/umf_ltsolve.c -o umf_di_ltsolve.o
+ $(C) -DDINT -c ../Source/umf_mem_alloc_element.c -o umf_di_mem_alloc_element.o
+ $(C) -DDINT -c ../Source/umf_mem_alloc_head_block.c -o umf_di_mem_alloc_head_block.o
+ $(C) -DDINT -c ../Source/umf_mem_alloc_tail_block.c -o umf_di_mem_alloc_tail_block.o
+ $(C) -DDINT -c ../Source/umf_mem_free_tail_block.c -o umf_di_mem_free_tail_block.o
+ $(C) -DDINT -c ../Source/umf_mem_init_memoryspace.c -o umf_di_mem_init_memoryspace.o
+ $(C) -DDINT -c ../Source/umf_report_vector.c -o umf_di_report_vector.o
+ $(C) -DDINT -c ../Source/umf_row_search.c -o umf_di_row_search.o
+ $(C) -DDINT -c ../Source/umf_scale_column.c -o umf_di_scale_column.o
+ $(C) -DDINT -c ../Source/umf_set_stats.c -o umf_di_set_stats.o
+ $(C) -DDINT -c ../Source/umf_solve.c -o umf_di_solve.o
+ $(C) -DDINT -c ../Source/umf_symbolic_usage.c -o umf_di_symbolic_usage.o
+ $(C) -DDINT -c ../Source/umf_transpose.c -o umf_di_transpose.o
+ $(C) -DDINT -c ../Source/umf_tuple_lengths.c -o umf_di_tuple_lengths.o
+ $(C) -DDINT -c ../Source/umf_usolve.c -o umf_di_usolve.o
+ $(C) -DDINT -c ../Source/umf_utsolve.c -o umf_di_utsolve.o
+ $(C) -DDINT -c ../Source/umf_valid_numeric.c -o umf_di_valid_numeric.o
+ $(C) -DDINT -c ../Source/umf_valid_symbolic.c -o umf_di_valid_symbolic.o
+ $(C) -DDINT -c ../Source/umf_grow_front.c -o umf_di_grow_front.o
+ $(C) -DDINT -c ../Source/umf_start_front.c -o umf_di_start_front.o
+ $(C) -DDINT -c ../Source/umf_2by2.c -o umf_di_2by2.o
+ $(C) -DDINT -c ../Source/umf_store_lu.c -o umf_di_store_lu.o
+ $(C) -DDINT -c ../Source/umf_scale.c -o umf_di_scale.o
+ $(C) -DDINT -DWSOLVE -c ../Source/umfpack_solve.c -o umfpack_di_wsolve.o
+ $(C) -DDINT -c ../Source/umfpack_col_to_triplet.c -o umfpack_di_col_to_triplet.o
+ $(C) -DDINT -c ../Source/umfpack_defaults.c -o umfpack_di_defaults.o
+ $(C) -DDINT -c ../Source/umfpack_free_numeric.c -o umfpack_di_free_numeric.o
+ $(C) -DDINT -c ../Source/umfpack_free_symbolic.c -o umfpack_di_free_symbolic.o
+ $(C) -DDINT -c ../Source/umfpack_get_numeric.c -o umfpack_di_get_numeric.o
+ $(C) -DDINT -c ../Source/umfpack_get_lunz.c -o umfpack_di_get_lunz.o
+ $(C) -DDINT -c ../Source/umfpack_get_symbolic.c -o umfpack_di_get_symbolic.o
+ $(C) -DDINT -c ../Source/umfpack_get_determinant.c -o umfpack_di_get_determinant.o
+ $(C) -DDINT -c ../Source/umfpack_numeric.c -o umfpack_di_numeric.o
+ $(C) -DDINT -c ../Source/umfpack_qsymbolic.c -o umfpack_di_qsymbolic.o
+ $(C) -DDINT -c ../Source/umfpack_report_control.c -o umfpack_di_report_control.o
+ $(C) -DDINT -c ../Source/umfpack_report_info.c -o umfpack_di_report_info.o
+ $(C) -DDINT -c ../Source/umfpack_report_matrix.c -o umfpack_di_report_matrix.o
+ $(C) -DDINT -c ../Source/umfpack_report_numeric.c -o umfpack_di_report_numeric.o
+ $(C) -DDINT -c ../Source/umfpack_report_perm.c -o umfpack_di_report_perm.o
+ $(C) -DDINT -c ../Source/umfpack_report_status.c -o umfpack_di_report_status.o
+ $(C) -DDINT -c ../Source/umfpack_report_symbolic.c -o umfpack_di_report_symbolic.o
+ $(C) -DDINT -c ../Source/umfpack_report_triplet.c -o umfpack_di_report_triplet.o
+ $(C) -DDINT -c ../Source/umfpack_report_vector.c -o umfpack_di_report_vector.o
+ $(C) -DDINT -c ../Source/umfpack_solve.c -o umfpack_di_solve.o
+ $(C) -DDINT -c ../Source/umfpack_symbolic.c -o umfpack_di_symbolic.o
+ $(C) -DDINT -c ../Source/umfpack_transpose.c -o umfpack_di_transpose.o
+ $(C) -DDINT -c ../Source/umfpack_triplet_to_col.c -o umfpack_di_triplet_to_col.o
+ $(C) -DDINT -c ../Source/umfpack_scale.c -o umfpack_di_scale.o
+ $(C) -DDINT -c ../Source/umfpack_load_numeric.c -o umfpack_di_load_numeric.o
+ $(C) -DDINT -c ../Source/umfpack_save_numeric.c -o umfpack_di_save_numeric.o
+ $(C) -DDINT -c ../Source/umfpack_load_symbolic.c -o umfpack_di_load_symbolic.o
+ $(C) -DDINT -c ../Source/umfpack_save_symbolic.c -o umfpack_di_save_symbolic.o
+ $(C) -DDLONG -DCONJUGATE_SOLVE -c ../Source/umf_ltsolve.c -o umf_dl_lhsolve.o
+ $(C) -DDLONG -DCONJUGATE_SOLVE -c ../Source/umf_utsolve.c -o umf_dl_uhsolve.o
+ $(C) -DDLONG -DDO_MAP -c ../Source/umf_triplet.c -o umf_dl_triplet_map_nox.o
+ $(C) -DDLONG -DDO_VALUES -c ../Source/umf_triplet.c -o umf_dl_triplet_nomap_x.o
+ $(C) -DDLONG -c ../Source/umf_triplet.c -o umf_dl_triplet_nomap_nox.o
+ $(C) -DDLONG -DDO_MAP -DDO_VALUES -c ../Source/umf_triplet.c -o umf_dl_triplet_map_x.o
+ $(C) -DDLONG -DFIXQ -c ../Source/umf_assemble.c -o umf_dl_assemble_fixq.o
+ $(C) -DDLONG -DDROP -c ../Source/umf_store_lu.c -o umf_dl_store_lu_drop.o
+ $(C) -DDLONG -c ../Source/umf_assemble.c -o umf_dl_assemble.o
+ $(C) -DDLONG -c ../Source/umf_blas3_update.c -o umf_dl_blas3_update.o
+ $(C) -DDLONG -c ../Source/umf_build_tuples.c -o umf_dl_build_tuples.o
+ $(C) -DDLONG -c ../Source/umf_create_element.c -o umf_dl_create_element.o
+ $(C) -DDLONG -c ../Source/umf_dump.c -o umf_dl_dump.o
+ $(C) -DDLONG -c ../Source/umf_extend_front.c -o umf_dl_extend_front.o
+ $(C) -DDLONG -c ../Source/umf_garbage_collection.c -o umf_dl_garbage_collection.o
+ $(C) -DDLONG -c ../Source/umf_get_memory.c -o umf_dl_get_memory.o
+ $(C) -DDLONG -c ../Source/umf_init_front.c -o umf_dl_init_front.o
+ $(C) -DDLONG -c ../Source/umf_kernel.c -o umf_dl_kernel.o
+ $(C) -DDLONG -c ../Source/umf_kernel_init.c -o umf_dl_kernel_init.o
+ $(C) -DDLONG -c ../Source/umf_kernel_wrapup.c -o umf_dl_kernel_wrapup.o
+ $(C) -DDLONG -c ../Source/umf_local_search.c -o umf_dl_local_search.o
+ $(C) -DDLONG -c ../Source/umf_lsolve.c -o umf_dl_lsolve.o
+ $(C) -DDLONG -c ../Source/umf_ltsolve.c -o umf_dl_ltsolve.o
+ $(C) -DDLONG -c ../Source/umf_mem_alloc_element.c -o umf_dl_mem_alloc_element.o
+ $(C) -DDLONG -c ../Source/umf_mem_alloc_head_block.c -o umf_dl_mem_alloc_head_block.o
+ $(C) -DDLONG -c ../Source/umf_mem_alloc_tail_block.c -o umf_dl_mem_alloc_tail_block.o
+ $(C) -DDLONG -c ../Source/umf_mem_free_tail_block.c -o umf_dl_mem_free_tail_block.o
+ $(C) -DDLONG -c ../Source/umf_mem_init_memoryspace.c -o umf_dl_mem_init_memoryspace.o
+ $(C) -DDLONG -c ../Source/umf_report_vector.c -o umf_dl_report_vector.o
+ $(C) -DDLONG -c ../Source/umf_row_search.c -o umf_dl_row_search.o
+ $(C) -DDLONG -c ../Source/umf_scale_column.c -o umf_dl_scale_column.o
+ $(C) -DDLONG -c ../Source/umf_set_stats.c -o umf_dl_set_stats.o
+ $(C) -DDLONG -c ../Source/umf_solve.c -o umf_dl_solve.o
+ $(C) -DDLONG -c ../Source/umf_symbolic_usage.c -o umf_dl_symbolic_usage.o
+ $(C) -DDLONG -c ../Source/umf_transpose.c -o umf_dl_transpose.o
+ $(C) -DDLONG -c ../Source/umf_tuple_lengths.c -o umf_dl_tuple_lengths.o
+ $(C) -DDLONG -c ../Source/umf_usolve.c -o umf_dl_usolve.o
+ $(C) -DDLONG -c ../Source/umf_utsolve.c -o umf_dl_utsolve.o
+ $(C) -DDLONG -c ../Source/umf_valid_numeric.c -o umf_dl_valid_numeric.o
+ $(C) -DDLONG -c ../Source/umf_valid_symbolic.c -o umf_dl_valid_symbolic.o
+ $(C) -DDLONG -c ../Source/umf_grow_front.c -o umf_dl_grow_front.o
+ $(C) -DDLONG -c ../Source/umf_start_front.c -o umf_dl_start_front.o
+ $(C) -DDLONG -c ../Source/umf_2by2.c -o umf_dl_2by2.o
+ $(C) -DDLONG -c ../Source/umf_store_lu.c -o umf_dl_store_lu.o
+ $(C) -DDLONG -c ../Source/umf_scale.c -o umf_dl_scale.o
+ $(C) -DDLONG -DWSOLVE -c ../Source/umfpack_solve.c -o umfpack_dl_wsolve.o
+ $(C) -DDLONG -c ../Source/umfpack_col_to_triplet.c -o umfpack_dl_col_to_triplet.o
+ $(C) -DDLONG -c ../Source/umfpack_defaults.c -o umfpack_dl_defaults.o
+ $(C) -DDLONG -c ../Source/umfpack_free_numeric.c -o umfpack_dl_free_numeric.o
+ $(C) -DDLONG -c ../Source/umfpack_free_symbolic.c -o umfpack_dl_free_symbolic.o
+ $(C) -DDLONG -c ../Source/umfpack_get_numeric.c -o umfpack_dl_get_numeric.o
+ $(C) -DDLONG -c ../Source/umfpack_get_lunz.c -o umfpack_dl_get_lunz.o
+ $(C) -DDLONG -c ../Source/umfpack_get_symbolic.c -o umfpack_dl_get_symbolic.o
+ $(C) -DDLONG -c ../Source/umfpack_get_determinant.c -o umfpack_dl_get_determinant.o
+ $(C) -DDLONG -c ../Source/umfpack_numeric.c -o umfpack_dl_numeric.o
+ $(C) -DDLONG -c ../Source/umfpack_qsymbolic.c -o umfpack_dl_qsymbolic.o
+ $(C) -DDLONG -c ../Source/umfpack_report_control.c -o umfpack_dl_report_control.o
+ $(C) -DDLONG -c ../Source/umfpack_report_info.c -o umfpack_dl_report_info.o
+ $(C) -DDLONG -c ../Source/umfpack_report_matrix.c -o umfpack_dl_report_matrix.o
+ $(C) -DDLONG -c ../Source/umfpack_report_numeric.c -o umfpack_dl_report_numeric.o
+ $(C) -DDLONG -c ../Source/umfpack_report_perm.c -o umfpack_dl_report_perm.o
+ $(C) -DDLONG -c ../Source/umfpack_report_status.c -o umfpack_dl_report_status.o
+ $(C) -DDLONG -c ../Source/umfpack_report_symbolic.c -o umfpack_dl_report_symbolic.o
+ $(C) -DDLONG -c ../Source/umfpack_report_triplet.c -o umfpack_dl_report_triplet.o
+ $(C) -DDLONG -c ../Source/umfpack_report_vector.c -o umfpack_dl_report_vector.o
+ $(C) -DDLONG -c ../Source/umfpack_solve.c -o umfpack_dl_solve.o
+ $(C) -DDLONG -c ../Source/umfpack_symbolic.c -o umfpack_dl_symbolic.o
+ $(C) -DDLONG -c ../Source/umfpack_transpose.c -o umfpack_dl_transpose.o
+ $(C) -DDLONG -c ../Source/umfpack_triplet_to_col.c -o umfpack_dl_triplet_to_col.o
+ $(C) -DDLONG -c ../Source/umfpack_scale.c -o umfpack_dl_scale.o
+ $(C) -DDLONG -c ../Source/umfpack_load_numeric.c -o umfpack_dl_load_numeric.o
+ $(C) -DDLONG -c ../Source/umfpack_save_numeric.c -o umfpack_dl_save_numeric.o
+ $(C) -DDLONG -c ../Source/umfpack_load_symbolic.c -o umfpack_dl_load_symbolic.o
+ $(C) -DDLONG -c ../Source/umfpack_save_symbolic.c -o umfpack_dl_save_symbolic.o
+ $(C) -DZINT -DCONJUGATE_SOLVE -c ../Source/umf_ltsolve.c -o umf_zi_lhsolve.o
+ $(C) -DZINT -DCONJUGATE_SOLVE -c ../Source/umf_utsolve.c -o umf_zi_uhsolve.o
+ $(C) -DZINT -DDO_MAP -c ../Source/umf_triplet.c -o umf_zi_triplet_map_nox.o
+ $(C) -DZINT -DDO_VALUES -c ../Source/umf_triplet.c -o umf_zi_triplet_nomap_x.o
+ $(C) -DZINT -c ../Source/umf_triplet.c -o umf_zi_triplet_nomap_nox.o
+ $(C) -DZINT -DDO_MAP -DDO_VALUES -c ../Source/umf_triplet.c -o umf_zi_triplet_map_x.o
+ $(C) -DZINT -DFIXQ -c ../Source/umf_assemble.c -o umf_zi_assemble_fixq.o
+ $(C) -DZINT -DDROP -c ../Source/umf_store_lu.c -o umf_zi_store_lu_drop.o
+ $(C) -DZINT -c ../Source/umf_assemble.c -o umf_zi_assemble.o
+ $(C) -DZINT -c ../Source/umf_blas3_update.c -o umf_zi_blas3_update.o
+ $(C) -DZINT -c ../Source/umf_build_tuples.c -o umf_zi_build_tuples.o
+ $(C) -DZINT -c ../Source/umf_create_element.c -o umf_zi_create_element.o
+ $(C) -DZINT -c ../Source/umf_dump.c -o umf_zi_dump.o
+ $(C) -DZINT -c ../Source/umf_extend_front.c -o umf_zi_extend_front.o
+ $(C) -DZINT -c ../Source/umf_garbage_collection.c -o umf_zi_garbage_collection.o
+ $(C) -DZINT -c ../Source/umf_get_memory.c -o umf_zi_get_memory.o
+ $(C) -DZINT -c ../Source/umf_init_front.c -o umf_zi_init_front.o
+ $(C) -DZINT -c ../Source/umf_kernel.c -o umf_zi_kernel.o
+ $(C) -DZINT -c ../Source/umf_kernel_init.c -o umf_zi_kernel_init.o
+ $(C) -DZINT -c ../Source/umf_kernel_wrapup.c -o umf_zi_kernel_wrapup.o
+ $(C) -DZINT -c ../Source/umf_local_search.c -o umf_zi_local_search.o
+ $(C) -DZINT -c ../Source/umf_lsolve.c -o umf_zi_lsolve.o
+ $(C) -DZINT -c ../Source/umf_ltsolve.c -o umf_zi_ltsolve.o
+ $(C) -DZINT -c ../Source/umf_mem_alloc_element.c -o umf_zi_mem_alloc_element.o
+ $(C) -DZINT -c ../Source/umf_mem_alloc_head_block.c -o umf_zi_mem_alloc_head_block.o
+ $(C) -DZINT -c ../Source/umf_mem_alloc_tail_block.c -o umf_zi_mem_alloc_tail_block.o
+ $(C) -DZINT -c ../Source/umf_mem_free_tail_block.c -o umf_zi_mem_free_tail_block.o
+ $(C) -DZINT -c ../Source/umf_mem_init_memoryspace.c -o umf_zi_mem_init_memoryspace.o
+ $(C) -DZINT -c ../Source/umf_report_vector.c -o umf_zi_report_vector.o
+ $(C) -DZINT -c ../Source/umf_row_search.c -o umf_zi_row_search.o
+ $(C) -DZINT -c ../Source/umf_scale_column.c -o umf_zi_scale_column.o
+ $(C) -DZINT -c ../Source/umf_set_stats.c -o umf_zi_set_stats.o
+ $(C) -DZINT -c ../Source/umf_solve.c -o umf_zi_solve.o
+ $(C) -DZINT -c ../Source/umf_symbolic_usage.c -o umf_zi_symbolic_usage.o
+ $(C) -DZINT -c ../Source/umf_transpose.c -o umf_zi_transpose.o
+ $(C) -DZINT -c ../Source/umf_tuple_lengths.c -o umf_zi_tuple_lengths.o
+ $(C) -DZINT -c ../Source/umf_usolve.c -o umf_zi_usolve.o
+ $(C) -DZINT -c ../Source/umf_utsolve.c -o umf_zi_utsolve.o
+ $(C) -DZINT -c ../Source/umf_valid_numeric.c -o umf_zi_valid_numeric.o
+ $(C) -DZINT -c ../Source/umf_valid_symbolic.c -o umf_zi_valid_symbolic.o
+ $(C) -DZINT -c ../Source/umf_grow_front.c -o umf_zi_grow_front.o
+ $(C) -DZINT -c ../Source/umf_start_front.c -o umf_zi_start_front.o
+ $(C) -DZINT -c ../Source/umf_2by2.c -o umf_zi_2by2.o
+ $(C) -DZINT -c ../Source/umf_store_lu.c -o umf_zi_store_lu.o
+ $(C) -DZINT -c ../Source/umf_scale.c -o umf_zi_scale.o
+ $(C) -DZINT -DWSOLVE -c ../Source/umfpack_solve.c -o umfpack_zi_wsolve.o
+ $(C) -DZINT -c ../Source/umfpack_col_to_triplet.c -o umfpack_zi_col_to_triplet.o
+ $(C) -DZINT -c ../Source/umfpack_defaults.c -o umfpack_zi_defaults.o
+ $(C) -DZINT -c ../Source/umfpack_free_numeric.c -o umfpack_zi_free_numeric.o
+ $(C) -DZINT -c ../Source/umfpack_free_symbolic.c -o umfpack_zi_free_symbolic.o
+ $(C) -DZINT -c ../Source/umfpack_get_numeric.c -o umfpack_zi_get_numeric.o
+ $(C) -DZINT -c ../Source/umfpack_get_lunz.c -o umfpack_zi_get_lunz.o
+ $(C) -DZINT -c ../Source/umfpack_get_symbolic.c -o umfpack_zi_get_symbolic.o
+ $(C) -DZINT -c ../Source/umfpack_get_determinant.c -o umfpack_zi_get_determinant.o
+ $(C) -DZINT -c ../Source/umfpack_numeric.c -o umfpack_zi_numeric.o
+ $(C) -DZINT -c ../Source/umfpack_qsymbolic.c -o umfpack_zi_qsymbolic.o
+ $(C) -DZINT -c ../Source/umfpack_report_control.c -o umfpack_zi_report_control.o
+ $(C) -DZINT -c ../Source/umfpack_report_info.c -o umfpack_zi_report_info.o
+ $(C) -DZINT -c ../Source/umfpack_report_matrix.c -o umfpack_zi_report_matrix.o
+ $(C) -DZINT -c ../Source/umfpack_report_numeric.c -o umfpack_zi_report_numeric.o
+ $(C) -DZINT -c ../Source/umfpack_report_perm.c -o umfpack_zi_report_perm.o
+ $(C) -DZINT -c ../Source/umfpack_report_status.c -o umfpack_zi_report_status.o
+ $(C) -DZINT -c ../Source/umfpack_report_symbolic.c -o umfpack_zi_report_symbolic.o
+ $(C) -DZINT -c ../Source/umfpack_report_triplet.c -o umfpack_zi_report_triplet.o
+ $(C) -DZINT -c ../Source/umfpack_report_vector.c -o umfpack_zi_report_vector.o
+ $(C) -DZINT -c ../Source/umfpack_solve.c -o umfpack_zi_solve.o
+ $(C) -DZINT -c ../Source/umfpack_symbolic.c -o umfpack_zi_symbolic.o
+ $(C) -DZINT -c ../Source/umfpack_transpose.c -o umfpack_zi_transpose.o
+ $(C) -DZINT -c ../Source/umfpack_triplet_to_col.c -o umfpack_zi_triplet_to_col.o
+ $(C) -DZINT -c ../Source/umfpack_scale.c -o umfpack_zi_scale.o
+ $(C) -DZINT -c ../Source/umfpack_load_numeric.c -o umfpack_zi_load_numeric.o
+ $(C) -DZINT -c ../Source/umfpack_save_numeric.c -o umfpack_zi_save_numeric.o
+ $(C) -DZINT -c ../Source/umfpack_load_symbolic.c -o umfpack_zi_load_symbolic.o
+ $(C) -DZINT -c ../Source/umfpack_save_symbolic.c -o umfpack_zi_save_symbolic.o
+ $(C) -DZLONG -DCONJUGATE_SOLVE -c ../Source/umf_ltsolve.c -o umf_zl_lhsolve.o
+ $(C) -DZLONG -DCONJUGATE_SOLVE -c ../Source/umf_utsolve.c -o umf_zl_uhsolve.o
+ $(C) -DZLONG -DDO_MAP -c ../Source/umf_triplet.c -o umf_zl_triplet_map_nox.o
+ $(C) -DZLONG -DDO_VALUES -c ../Source/umf_triplet.c -o umf_zl_triplet_nomap_x.o
+ $(C) -DZLONG -c ../Source/umf_triplet.c -o umf_zl_triplet_nomap_nox.o
+ $(C) -DZLONG -DDO_MAP -DDO_VALUES -c ../Source/umf_triplet.c -o umf_zl_triplet_map_x.o
+ $(C) -DZLONG -DFIXQ -c ../Source/umf_assemble.c -o umf_zl_assemble_fixq.o
+ $(C) -DZLONG -DDROP -c ../Source/umf_store_lu.c -o umf_zl_store_lu_drop.o
+ $(C) -DZLONG -c ../Source/umf_assemble.c -o umf_zl_assemble.o
+ $(C) -DZLONG -c ../Source/umf_blas3_update.c -o umf_zl_blas3_update.o
+ $(C) -DZLONG -c ../Source/umf_build_tuples.c -o umf_zl_build_tuples.o
+ $(C) -DZLONG -c ../Source/umf_create_element.c -o umf_zl_create_element.o
+ $(C) -DZLONG -c ../Source/umf_dump.c -o umf_zl_dump.o
+ $(C) -DZLONG -c ../Source/umf_extend_front.c -o umf_zl_extend_front.o
+ $(C) -DZLONG -c ../Source/umf_garbage_collection.c -o umf_zl_garbage_collection.o
+ $(C) -DZLONG -c ../Source/umf_get_memory.c -o umf_zl_get_memory.o
+ $(C) -DZLONG -c ../Source/umf_init_front.c -o umf_zl_init_front.o
+ $(C) -DZLONG -c ../Source/umf_kernel.c -o umf_zl_kernel.o
+ $(C) -DZLONG -c ../Source/umf_kernel_init.c -o umf_zl_kernel_init.o
+ $(C) -DZLONG -c ../Source/umf_kernel_wrapup.c -o umf_zl_kernel_wrapup.o
+ $(C) -DZLONG -c ../Source/umf_local_search.c -o umf_zl_local_search.o
+ $(C) -DZLONG -c ../Source/umf_lsolve.c -o umf_zl_lsolve.o
+ $(C) -DZLONG -c ../Source/umf_ltsolve.c -o umf_zl_ltsolve.o
+ $(C) -DZLONG -c ../Source/umf_mem_alloc_element.c -o umf_zl_mem_alloc_element.o
+ $(C) -DZLONG -c ../Source/umf_mem_alloc_head_block.c -o umf_zl_mem_alloc_head_block.o
+ $(C) -DZLONG -c ../Source/umf_mem_alloc_tail_block.c -o umf_zl_mem_alloc_tail_block.o
+ $(C) -DZLONG -c ../Source/umf_mem_free_tail_block.c -o umf_zl_mem_free_tail_block.o
+ $(C) -DZLONG -c ../Source/umf_mem_init_memoryspace.c -o umf_zl_mem_init_memoryspace.o
+ $(C) -DZLONG -c ../Source/umf_report_vector.c -o umf_zl_report_vector.o
+ $(C) -DZLONG -c ../Source/umf_row_search.c -o umf_zl_row_search.o
+ $(C) -DZLONG -c ../Source/umf_scale_column.c -o umf_zl_scale_column.o
+ $(C) -DZLONG -c ../Source/umf_set_stats.c -o umf_zl_set_stats.o
+ $(C) -DZLONG -c ../Source/umf_solve.c -o umf_zl_solve.o
+ $(C) -DZLONG -c ../Source/umf_symbolic_usage.c -o umf_zl_symbolic_usage.o
+ $(C) -DZLONG -c ../Source/umf_transpose.c -o umf_zl_transpose.o
+ $(C) -DZLONG -c ../Source/umf_tuple_lengths.c -o umf_zl_tuple_lengths.o
+ $(C) -DZLONG -c ../Source/umf_usolve.c -o umf_zl_usolve.o
+ $(C) -DZLONG -c ../Source/umf_utsolve.c -o umf_zl_utsolve.o
+ $(C) -DZLONG -c ../Source/umf_valid_numeric.c -o umf_zl_valid_numeric.o
+ $(C) -DZLONG -c ../Source/umf_valid_symbolic.c -o umf_zl_valid_symbolic.o
+ $(C) -DZLONG -c ../Source/umf_grow_front.c -o umf_zl_grow_front.o
+ $(C) -DZLONG -c ../Source/umf_start_front.c -o umf_zl_start_front.o
+ $(C) -DZLONG -c ../Source/umf_2by2.c -o umf_zl_2by2.o
+ $(C) -DZLONG -c ../Source/umf_store_lu.c -o umf_zl_store_lu.o
+ $(C) -DZLONG -c ../Source/umf_scale.c -o umf_zl_scale.o
+ $(C) -DZLONG -DWSOLVE -c ../Source/umfpack_solve.c -o umfpack_zl_wsolve.o
+ $(C) -DZLONG -c ../Source/umfpack_col_to_triplet.c -o umfpack_zl_col_to_triplet.o
+ $(C) -DZLONG -c ../Source/umfpack_defaults.c -o umfpack_zl_defaults.o
+ $(C) -DZLONG -c ../Source/umfpack_free_numeric.c -o umfpack_zl_free_numeric.o
+ $(C) -DZLONG -c ../Source/umfpack_free_symbolic.c -o umfpack_zl_free_symbolic.o
+ $(C) -DZLONG -c ../Source/umfpack_get_numeric.c -o umfpack_zl_get_numeric.o
+ $(C) -DZLONG -c ../Source/umfpack_get_lunz.c -o umfpack_zl_get_lunz.o
+ $(C) -DZLONG -c ../Source/umfpack_get_symbolic.c -o umfpack_zl_get_symbolic.o
+ $(C) -DZLONG -c ../Source/umfpack_get_determinant.c -o umfpack_zl_get_determinant.o
+ $(C) -DZLONG -c ../Source/umfpack_numeric.c -o umfpack_zl_numeric.o
+ $(C) -DZLONG -c ../Source/umfpack_qsymbolic.c -o umfpack_zl_qsymbolic.o
+ $(C) -DZLONG -c ../Source/umfpack_report_control.c -o umfpack_zl_report_control.o
+ $(C) -DZLONG -c ../Source/umfpack_report_info.c -o umfpack_zl_report_info.o
+ $(C) -DZLONG -c ../Source/umfpack_report_matrix.c -o umfpack_zl_report_matrix.o
+ $(C) -DZLONG -c ../Source/umfpack_report_numeric.c -o umfpack_zl_report_numeric.o
+ $(C) -DZLONG -c ../Source/umfpack_report_perm.c -o umfpack_zl_report_perm.o
+ $(C) -DZLONG -c ../Source/umfpack_report_status.c -o umfpack_zl_report_status.o
+ $(C) -DZLONG -c ../Source/umfpack_report_symbolic.c -o umfpack_zl_report_symbolic.o
+ $(C) -DZLONG -c ../Source/umfpack_report_triplet.c -o umfpack_zl_report_triplet.o
+ $(C) -DZLONG -c ../Source/umfpack_report_vector.c -o umfpack_zl_report_vector.o
+ $(C) -DZLONG -c ../Source/umfpack_solve.c -o umfpack_zl_solve.o
+ $(C) -DZLONG -c ../Source/umfpack_symbolic.c -o umfpack_zl_symbolic.o
+ $(C) -DZLONG -c ../Source/umfpack_transpose.c -o umfpack_zl_transpose.o
+ $(C) -DZLONG -c ../Source/umfpack_triplet_to_col.c -o umfpack_zl_triplet_to_col.o
+ $(C) -DZLONG -c ../Source/umfpack_scale.c -o umfpack_zl_scale.o
+ $(C) -DZLONG -c ../Source/umfpack_load_numeric.c -o umfpack_zl_load_numeric.o
+ $(C) -DZLONG -c ../Source/umfpack_save_numeric.c -o umfpack_zl_save_numeric.o
+ $(C) -DZLONG -c ../Source/umfpack_load_symbolic.c -o umfpack_zl_load_symbolic.o
+ $(C) -DZLONG -c ../Source/umfpack_save_symbolic.c -o umfpack_zl_save_symbolic.o
+ $(AR) ../Lib/libumfpack.a \
+ umfpack_gn_global.o \
+ umf_i_analyze.o umf_i_apply_order.o umf_i_colamd.o umf_i_free.o \
+ umf_i_fsize.o umf_i_is_permutation.o umf_i_malloc.o umf_i_realloc.o \
+ umf_i_report_perm.o umf_i_singletons.o \
+ umf_l_analyze.o umf_l_apply_order.o umf_l_colamd.o umf_l_free.o \
+ umf_l_fsize.o umf_l_is_permutation.o umf_l_malloc.o umf_l_realloc.o \
+ umf_l_report_perm.o umf_l_singletons.o \
+ umfpack_gn_timer.o umfpack_gn_tictoc.o \
+ umf_di_lhsolve.o \
+ umf_di_uhsolve.o umf_di_triplet_map_nox.o umf_di_triplet_nomap_x.o \
+ umf_di_triplet_nomap_nox.o umf_di_triplet_map_x.o \
+ umf_di_assemble_fixq.o umf_di_store_lu_drop.o umf_di_assemble.o \
+ umf_di_blas3_update.o umf_di_build_tuples.o \
+ umf_di_create_element.o umf_di_dump.o umf_di_extend_front.o \
+ umf_di_garbage_collection.o umf_di_get_memory.o \
+ umf_di_init_front.o umf_di_kernel.o umf_di_kernel_init.o \
+ umf_di_kernel_wrapup.o umf_di_local_search.o umf_di_lsolve.o \
+ umf_di_ltsolve.o umf_di_mem_alloc_element.o \
+ umf_di_mem_alloc_head_block.o umf_di_mem_alloc_tail_block.o \
+ umf_di_mem_free_tail_block.o umf_di_mem_init_memoryspace.o \
+ umf_di_report_vector.o umf_di_row_search.o umf_di_scale_column.o \
+ umf_di_set_stats.o umf_di_solve.o umf_di_symbolic_usage.o \
+ umf_di_transpose.o umf_di_tuple_lengths.o umf_di_usolve.o \
+ umf_di_utsolve.o umf_di_valid_numeric.o umf_di_valid_symbolic.o \
+ umf_di_grow_front.o umf_di_start_front.o umf_di_2by2.o \
+ umf_di_store_lu.o umf_di_scale.o umfpack_di_wsolve.o \
+ umfpack_di_col_to_triplet.o umfpack_di_defaults.o \
+ umfpack_di_free_numeric.o umfpack_di_free_symbolic.o \
+ umfpack_di_get_numeric.o umfpack_di_get_lunz.o \
+ umfpack_di_get_symbolic.o umfpack_di_get_determinant.o \
+ umfpack_di_numeric.o \
+ umfpack_di_qsymbolic.o umfpack_di_report_control.o \
+ umfpack_di_report_info.o umfpack_di_report_matrix.o \
+ umfpack_di_report_numeric.o umfpack_di_report_perm.o \
+ umfpack_di_report_status.o umfpack_di_report_symbolic.o \
+ umfpack_di_report_triplet.o umfpack_di_report_vector.o \
+ umfpack_di_solve.o umfpack_di_symbolic.o umfpack_di_transpose.o \
+ umfpack_di_triplet_to_col.o umfpack_di_scale.o \
+ umfpack_di_load_numeric.o umfpack_di_save_numeric.o \
+ umfpack_di_load_symbolic.o umfpack_di_save_symbolic.o \
+ umf_dl_lhsolve.o \
+ umf_dl_uhsolve.o umf_dl_triplet_map_nox.o umf_dl_triplet_nomap_x.o \
+ umf_dl_triplet_nomap_nox.o umf_dl_triplet_map_x.o \
+ umf_dl_assemble_fixq.o umf_dl_store_lu_drop.o umf_dl_assemble.o \
+ umf_dl_blas3_update.o umf_dl_build_tuples.o \
+ umf_dl_create_element.o umf_dl_dump.o umf_dl_extend_front.o \
+ umf_dl_garbage_collection.o umf_dl_get_memory.o \
+ umf_dl_init_front.o umf_dl_kernel.o umf_dl_kernel_init.o \
+ umf_dl_kernel_wrapup.o umf_dl_local_search.o umf_dl_lsolve.o \
+ umf_dl_ltsolve.o umf_dl_mem_alloc_element.o \
+ umf_dl_mem_alloc_head_block.o umf_dl_mem_alloc_tail_block.o \
+ umf_dl_mem_free_tail_block.o umf_dl_mem_init_memoryspace.o \
+ umf_dl_report_vector.o umf_dl_row_search.o umf_dl_scale_column.o \
+ umf_dl_set_stats.o umf_dl_solve.o umf_dl_symbolic_usage.o \
+ umf_dl_transpose.o umf_dl_tuple_lengths.o umf_dl_usolve.o \
+ umf_dl_utsolve.o umf_dl_valid_numeric.o umf_dl_valid_symbolic.o \
+ umf_dl_grow_front.o umf_dl_start_front.o umf_dl_2by2.o \
+ umf_dl_store_lu.o umf_dl_scale.o umfpack_dl_wsolve.o \
+ umfpack_dl_col_to_triplet.o umfpack_dl_defaults.o \
+ umfpack_dl_free_numeric.o umfpack_dl_free_symbolic.o \
+ umfpack_dl_get_numeric.o umfpack_dl_get_lunz.o \
+ umfpack_dl_get_symbolic.o umfpack_dl_get_determinant.o \
+ umfpack_dl_numeric.o \
+ umfpack_dl_qsymbolic.o umfpack_dl_report_control.o \
+ umfpack_dl_report_info.o umfpack_dl_report_matrix.o \
+ umfpack_dl_report_numeric.o umfpack_dl_report_perm.o \
+ umfpack_dl_report_status.o umfpack_dl_report_symbolic.o \
+ umfpack_dl_report_triplet.o umfpack_dl_report_vector.o \
+ umfpack_dl_solve.o umfpack_dl_symbolic.o umfpack_dl_transpose.o \
+ umfpack_dl_triplet_to_col.o umfpack_dl_scale.o \
+ umfpack_dl_load_numeric.o umfpack_dl_save_numeric.o \
+ umfpack_dl_load_symbolic.o umfpack_dl_save_symbolic.o \
+ umf_zi_lhsolve.o \
+ umf_zi_uhsolve.o umf_zi_triplet_map_nox.o umf_zi_triplet_nomap_x.o \
+ umf_zi_triplet_nomap_nox.o umf_zi_triplet_map_x.o \
+ umf_zi_assemble_fixq.o umf_zi_store_lu_drop.o umf_zi_assemble.o \
+ umf_zi_blas3_update.o umf_zi_build_tuples.o \
+ umf_zi_create_element.o umf_zi_dump.o umf_zi_extend_front.o \
+ umf_zi_garbage_collection.o umf_zi_get_memory.o \
+ umf_zi_init_front.o umf_zi_kernel.o umf_zi_kernel_init.o \
+ umf_zi_kernel_wrapup.o umf_zi_local_search.o umf_zi_lsolve.o \
+ umf_zi_ltsolve.o umf_zi_mem_alloc_element.o \
+ umf_zi_mem_alloc_head_block.o umf_zi_mem_alloc_tail_block.o \
+ umf_zi_mem_free_tail_block.o umf_zi_mem_init_memoryspace.o \
+ umf_zi_report_vector.o umf_zi_row_search.o umf_zi_scale_column.o \
+ umf_zi_set_stats.o umf_zi_solve.o umf_zi_symbolic_usage.o \
+ umf_zi_transpose.o umf_zi_tuple_lengths.o umf_zi_usolve.o \
+ umf_zi_utsolve.o umf_zi_valid_numeric.o umf_zi_valid_symbolic.o \
+ umf_zi_grow_front.o umf_zi_start_front.o umf_zi_2by2.o \
+ umf_zi_store_lu.o umf_zi_scale.o umfpack_zi_wsolve.o \
+ umfpack_zi_col_to_triplet.o umfpack_zi_defaults.o \
+ umfpack_zi_free_numeric.o umfpack_zi_free_symbolic.o \
+ umfpack_zi_get_numeric.o umfpack_zi_get_lunz.o \
+ umfpack_zi_get_symbolic.o umfpack_zi_get_determinant.o \
+ umfpack_zi_numeric.o \
+ umfpack_zi_qsymbolic.o umfpack_zi_report_control.o \
+ umfpack_zi_report_info.o umfpack_zi_report_matrix.o \
+ umfpack_zi_report_numeric.o umfpack_zi_report_perm.o \
+ umfpack_zi_report_status.o umfpack_zi_report_symbolic.o \
+ umfpack_zi_report_triplet.o umfpack_zi_report_vector.o \
+ umfpack_zi_solve.o umfpack_zi_symbolic.o umfpack_zi_transpose.o \
+ umfpack_zi_triplet_to_col.o umfpack_zi_scale.o \
+ umfpack_zi_load_numeric.o umfpack_zi_save_numeric.o \
+ umfpack_zi_load_symbolic.o umfpack_zi_save_symbolic.o \
+ umf_zl_lhsolve.o \
+ umf_zl_uhsolve.o umf_zl_triplet_map_nox.o umf_zl_triplet_nomap_x.o \
+ umf_zl_triplet_nomap_nox.o umf_zl_triplet_map_x.o \
+ umf_zl_assemble_fixq.o umf_zl_store_lu_drop.o umf_zl_assemble.o \
+ umf_zl_blas3_update.o umf_zl_build_tuples.o \
+ umf_zl_create_element.o umf_zl_dump.o umf_zl_extend_front.o \
+ umf_zl_garbage_collection.o umf_zl_get_memory.o \
+ umf_zl_init_front.o umf_zl_kernel.o umf_zl_kernel_init.o \
+ umf_zl_kernel_wrapup.o umf_zl_local_search.o umf_zl_lsolve.o \
+ umf_zl_ltsolve.o umf_zl_mem_alloc_element.o \
+ umf_zl_mem_alloc_head_block.o umf_zl_mem_alloc_tail_block.o \
+ umf_zl_mem_free_tail_block.o umf_zl_mem_init_memoryspace.o \
+ umf_zl_report_vector.o umf_zl_row_search.o umf_zl_scale_column.o \
+ umf_zl_set_stats.o umf_zl_solve.o umf_zl_symbolic_usage.o \
+ umf_zl_transpose.o umf_zl_tuple_lengths.o umf_zl_usolve.o \
+ umf_zl_utsolve.o umf_zl_valid_numeric.o umf_zl_valid_symbolic.o \
+ umf_zl_grow_front.o umf_zl_start_front.o umf_zl_2by2.o \
+ umf_zl_store_lu.o umf_zl_scale.o umfpack_zl_wsolve.o \
+ umfpack_zl_col_to_triplet.o umfpack_zl_defaults.o \
+ umfpack_zl_free_numeric.o umfpack_zl_free_symbolic.o \
+ umfpack_zl_get_numeric.o umfpack_zl_get_lunz.o \
+ umfpack_zl_get_symbolic.o umfpack_zl_get_determinant.o \
+ umfpack_zl_numeric.o \
+ umfpack_zl_qsymbolic.o umfpack_zl_report_control.o \
+ umfpack_zl_report_info.o umfpack_zl_report_matrix.o \
+ umfpack_zl_report_numeric.o umfpack_zl_report_perm.o \
+ umfpack_zl_report_status.o umfpack_zl_report_symbolic.o \
+ umfpack_zl_report_triplet.o umfpack_zl_report_vector.o \
+ umfpack_zl_solve.o umfpack_zl_symbolic.o umfpack_zl_transpose.o \
+ umfpack_zl_triplet_to_col.o umfpack_zl_scale.o \
+ umfpack_zl_load_numeric.o umfpack_zl_save_numeric.o \
+ umfpack_zl_load_symbolic.o umfpack_zl_save_symbolic.o
+ - $(RANLIB) ../Lib/libumfpack.a
+
+#-------------------------------------------------------------------------------
+# Remove all but the files in the original distribution
+#-------------------------------------------------------------------------------
+
+purge: clean
+ - $(RM) ../Lib/libumfpack.a
+
+clean:
+ - $(RM) $(CLEAN)
diff --git a/src/C/SuiteSparse/UMFPACK/Makefile b/src/C/SuiteSparse/UMFPACK/Makefile
index 07afd04..80f2f73 100644
--- a/src/C/SuiteSparse/UMFPACK/Makefile
+++ b/src/C/SuiteSparse/UMFPACK/Makefile
@@ -11,17 +11,17 @@ include ../UFconfig/UFconfig.mk
# compile all C code (except hb, fortran, and fortran64), including AMD and the
# MATLAB mexFunctions
all:
- ( cd ../AMD ; $(MAKE) )
- ( cd ../AMD/MATLAB ; $(MAKE) )
- ( cd Source ; $(MAKE) )
+ ( cd ../AMD ; $(MAKE) library )
+ ( cd ../AMD ; $(MAKE) mex )
+ ( cd Lib ; $(MAKE) )
( cd Demo ; $(MAKE) )
( cd MATLAB ; $(MAKE) )
- cat Doc/License
# compile just the C-callable libraries and demo programs (not mexFunctions)
library:
- ( cd ../AMD ; $(MAKE) )
- ( cd Source ; $(MAKE) )
+ ( cd ../AMD ; $(MAKE) library )
+ ( cd Lib ; $(MAKE) )
( cd Demo ; $(MAKE) )
- cat Doc/License
@@ -40,7 +40,7 @@ hb:
# remove object files, but keep the compiled programs and library archives
clean:
( cd ../AMD ; $(MAKE) clean )
- ( cd Source ; $(MAKE) clean )
+ ( cd Lib ; $(MAKE) clean )
( cd Demo ; $(MAKE) clean )
( cd MATLAB ; $(MAKE) clean )
( cd Doc ; $(MAKE) clean )
@@ -48,7 +48,7 @@ clean:
# clean, and then remove compiled programs and library archives
purge:
( cd ../AMD ; $(MAKE) purge )
- ( cd Source ; $(MAKE) purge )
+ ( cd Lib ; $(MAKE) purge )
( cd Demo ; $(MAKE) purge )
( cd MATLAB ; $(MAKE) purge )
( cd Doc ; $(MAKE) purge )
diff --git a/src/C/SuiteSparse/UMFPACK/README.txt b/src/C/SuiteSparse/UMFPACK/README.txt
index 61ed73c..eec0f74 100644
--- a/src/C/SuiteSparse/UMFPACK/README.txt
+++ b/src/C/SuiteSparse/UMFPACK/README.txt
@@ -189,9 +189,6 @@ Files and directories in the UMFPACK distribution:
Source directory:
----------------------------------------------------------------------------
- GNUmakefile a nice Makefile, for GNU make
- Makefile an ugly Unix Makefile (for older make's)
-
cholmod_blas.h an exact copy of CHOLMOD/Include/cholmod_blas.h
umfpack_col_to_triplet.c convert col form to triplet
@@ -370,8 +367,6 @@ Files and directories in the UMFPACK distribution:
version of UMFPACK (useful prototype for
Microsoft Visual Studio project)
- Opteron64/ demo output files on an AMD Opteron
-
----------------------------------------------------------------------------
MATLAB directory:
----------------------------------------------------------------------------
@@ -396,7 +391,6 @@ Files and directories in the UMFPACK distribution:
west0067.mat sparse matrix for umfpack_demo.m
umfpack_demo.m.out output of umfpack_demo.m
- umfpack_demo.m.out_verbose ditto, but with print level 2, on AMD Opteron
umfpack_simple.m.out output of umfpack_simple
lcc_lib/lapacksyms.def LAPACK definitions for lcc compiler (Windows)
@@ -406,4 +400,6 @@ Files and directories in the UMFPACK distribution:
Lib directory: libumfpack.a library placed here
----------------------------------------------------------------------------
+ GNUmakefile a nice Makefile, for GNU make
+ Makefile an ugly Unix Makefile (for older make's)
libumfpack.def UMPFACK definitions for Windows
diff --git a/src/C/SuiteSparse/UMFPACK/Source/Makefile b/src/C/SuiteSparse/UMFPACK/Source/Makefile
deleted file mode 100644
index 56ab48b..0000000
--- a/src/C/SuiteSparse/UMFPACK/Source/Makefile
+++ /dev/null
@@ -1,482 +0,0 @@
-#-------------------------------------------------------------------------------
-# UMFPACK Makefile for compiling on Unix systems (for original make only)
-#-------------------------------------------------------------------------------
-
-# This is a very ugly Makefile, and is only provided for those who do not
-# have GNU make. Note that it is not used if you have GNU make. It ignores
-# dependency checking and just compiles everything.
-
-default: everything
-
-include ../../UFconfig/UFconfig.mk
-
-C = $(CC) $(CFLAGS) $(UMFPACK_CONFIG) -I../Include -I../../AMD/Include
-
-everything:
- $(C) -c umfpack_global.c -o umfpack_gn_global.o
- $(C) -DDINT -c umf_analyze.c -o umf_i_analyze.o
- $(C) -DDINT -c umf_apply_order.c -o umf_i_apply_order.o
- $(C) -DDINT -c umf_colamd.c -o umf_i_colamd.o
- $(C) -DDINT -c umf_free.c -o umf_i_free.o
- $(C) -DDINT -c umf_fsize.c -o umf_i_fsize.o
- $(C) -DDINT -c umf_is_permutation.c -o umf_i_is_permutation.o
- $(C) -DDINT -c umf_malloc.c -o umf_i_malloc.o
- $(C) -DDINT -c umf_realloc.c -o umf_i_realloc.o
- $(C) -DDINT -c umf_report_perm.c -o umf_i_report_perm.o
- $(C) -DDINT -c umf_singletons.c -o umf_i_singletons.o
- $(C) -DDLONG -c umf_analyze.c -o umf_l_analyze.o
- $(C) -DDLONG -c umf_apply_order.c -o umf_l_apply_order.o
- $(C) -DDLONG -c umf_colamd.c -o umf_l_colamd.o
- $(C) -DDLONG -c umf_free.c -o umf_l_free.o
- $(C) -DDLONG -c umf_fsize.c -o umf_l_fsize.o
- $(C) -DDLONG -c umf_is_permutation.c -o umf_l_is_permutation.o
- $(C) -DDLONG -c umf_malloc.c -o umf_l_malloc.o
- $(C) -DDLONG -c umf_realloc.c -o umf_l_realloc.o
- $(C) -DDLONG -c umf_report_perm.c -o umf_l_report_perm.o
- $(C) -DDLONG -c umf_singletons.c -o umf_l_singletons.o
- $(C) -c umfpack_timer.c -o umfpack_gn_timer.o
- $(C) -c umfpack_tictoc.c -o umfpack_gn_tictoc.o
- $(C) -DDINT -DCONJUGATE_SOLVE -c umf_ltsolve.c -o umf_di_lhsolve.o
- $(C) -DDINT -DCONJUGATE_SOLVE -c umf_utsolve.c -o umf_di_uhsolve.o
- $(C) -DDINT -DDO_MAP -c umf_triplet.c -o umf_di_triplet_map_nox.o
- $(C) -DDINT -DDO_VALUES -c umf_triplet.c -o umf_di_triplet_nomap_x.o
- $(C) -DDINT -c umf_triplet.c -o umf_di_triplet_nomap_nox.o
- $(C) -DDINT -DDO_MAP -DDO_VALUES -c umf_triplet.c -o umf_di_triplet_map_x.o
- $(C) -DDINT -DFIXQ -c umf_assemble.c -o umf_di_assemble_fixq.o
- $(C) -DDINT -DDROP -c umf_store_lu.c -o umf_di_store_lu_drop.o
- $(C) -DDINT -c umf_assemble.c -o umf_di_assemble.o
- $(C) -DDINT -c umf_blas3_update.c -o umf_di_blas3_update.o
- $(C) -DDINT -c umf_build_tuples.c -o umf_di_build_tuples.o
- $(C) -DDINT -c umf_create_element.c -o umf_di_create_element.o
- $(C) -DDINT -c umf_dump.c -o umf_di_dump.o
- $(C) -DDINT -c umf_extend_front.c -o umf_di_extend_front.o
- $(C) -DDINT -c umf_garbage_collection.c -o umf_di_garbage_collection.o
- $(C) -DDINT -c umf_get_memory.c -o umf_di_get_memory.o
- $(C) -DDINT -c umf_init_front.c -o umf_di_init_front.o
- $(C) -DDINT -c umf_kernel.c -o umf_di_kernel.o
- $(C) -DDINT -c umf_kernel_init.c -o umf_di_kernel_init.o
- $(C) -DDINT -c umf_kernel_wrapup.c -o umf_di_kernel_wrapup.o
- $(C) -DDINT -c umf_local_search.c -o umf_di_local_search.o
- $(C) -DDINT -c umf_lsolve.c -o umf_di_lsolve.o
- $(C) -DDINT -c umf_ltsolve.c -o umf_di_ltsolve.o
- $(C) -DDINT -c umf_mem_alloc_element.c -o umf_di_mem_alloc_element.o
- $(C) -DDINT -c umf_mem_alloc_head_block.c -o umf_di_mem_alloc_head_block.o
- $(C) -DDINT -c umf_mem_alloc_tail_block.c -o umf_di_mem_alloc_tail_block.o
- $(C) -DDINT -c umf_mem_free_tail_block.c -o umf_di_mem_free_tail_block.o
- $(C) -DDINT -c umf_mem_init_memoryspace.c -o umf_di_mem_init_memoryspace.o
- $(C) -DDINT -c umf_report_vector.c -o umf_di_report_vector.o
- $(C) -DDINT -c umf_row_search.c -o umf_di_row_search.o
- $(C) -DDINT -c umf_scale_column.c -o umf_di_scale_column.o
- $(C) -DDINT -c umf_set_stats.c -o umf_di_set_stats.o
- $(C) -DDINT -c umf_solve.c -o umf_di_solve.o
- $(C) -DDINT -c umf_symbolic_usage.c -o umf_di_symbolic_usage.o
- $(C) -DDINT -c umf_transpose.c -o umf_di_transpose.o
- $(C) -DDINT -c umf_tuple_lengths.c -o umf_di_tuple_lengths.o
- $(C) -DDINT -c umf_usolve.c -o umf_di_usolve.o
- $(C) -DDINT -c umf_utsolve.c -o umf_di_utsolve.o
- $(C) -DDINT -c umf_valid_numeric.c -o umf_di_valid_numeric.o
- $(C) -DDINT -c umf_valid_symbolic.c -o umf_di_valid_symbolic.o
- $(C) -DDINT -c umf_grow_front.c -o umf_di_grow_front.o
- $(C) -DDINT -c umf_start_front.c -o umf_di_start_front.o
- $(C) -DDINT -c umf_2by2.c -o umf_di_2by2.o
- $(C) -DDINT -c umf_store_lu.c -o umf_di_store_lu.o
- $(C) -DDINT -c umf_scale.c -o umf_di_scale.o
- $(C) -DDINT -DWSOLVE -c umfpack_solve.c -o umfpack_di_wsolve.o
- $(C) -DDINT -c umfpack_col_to_triplet.c -o umfpack_di_col_to_triplet.o
- $(C) -DDINT -c umfpack_defaults.c -o umfpack_di_defaults.o
- $(C) -DDINT -c umfpack_free_numeric.c -o umfpack_di_free_numeric.o
- $(C) -DDINT -c umfpack_free_symbolic.c -o umfpack_di_free_symbolic.o
- $(C) -DDINT -c umfpack_get_numeric.c -o umfpack_di_get_numeric.o
- $(C) -DDINT -c umfpack_get_lunz.c -o umfpack_di_get_lunz.o
- $(C) -DDINT -c umfpack_get_symbolic.c -o umfpack_di_get_symbolic.o
- $(C) -DDINT -c umfpack_get_determinant.c -o umfpack_di_get_determinant.o
- $(C) -DDINT -c umfpack_numeric.c -o umfpack_di_numeric.o
- $(C) -DDINT -c umfpack_qsymbolic.c -o umfpack_di_qsymbolic.o
- $(C) -DDINT -c umfpack_report_control.c -o umfpack_di_report_control.o
- $(C) -DDINT -c umfpack_report_info.c -o umfpack_di_report_info.o
- $(C) -DDINT -c umfpack_report_matrix.c -o umfpack_di_report_matrix.o
- $(C) -DDINT -c umfpack_report_numeric.c -o umfpack_di_report_numeric.o
- $(C) -DDINT -c umfpack_report_perm.c -o umfpack_di_report_perm.o
- $(C) -DDINT -c umfpack_report_status.c -o umfpack_di_report_status.o
- $(C) -DDINT -c umfpack_report_symbolic.c -o umfpack_di_report_symbolic.o
- $(C) -DDINT -c umfpack_report_triplet.c -o umfpack_di_report_triplet.o
- $(C) -DDINT -c umfpack_report_vector.c -o umfpack_di_report_vector.o
- $(C) -DDINT -c umfpack_solve.c -o umfpack_di_solve.o
- $(C) -DDINT -c umfpack_symbolic.c -o umfpack_di_symbolic.o
- $(C) -DDINT -c umfpack_transpose.c -o umfpack_di_transpose.o
- $(C) -DDINT -c umfpack_triplet_to_col.c -o umfpack_di_triplet_to_col.o
- $(C) -DDINT -c umfpack_scale.c -o umfpack_di_scale.o
- $(C) -DDINT -c umfpack_load_numeric.c -o umfpack_di_load_numeric.o
- $(C) -DDINT -c umfpack_save_numeric.c -o umfpack_di_save_numeric.o
- $(C) -DDINT -c umfpack_load_symbolic.c -o umfpack_di_load_symbolic.o
- $(C) -DDINT -c umfpack_save_symbolic.c -o umfpack_di_save_symbolic.o
- $(C) -DDLONG -DCONJUGATE_SOLVE -c umf_ltsolve.c -o umf_dl_lhsolve.o
- $(C) -DDLONG -DCONJUGATE_SOLVE -c umf_utsolve.c -o umf_dl_uhsolve.o
- $(C) -DDLONG -DDO_MAP -c umf_triplet.c -o umf_dl_triplet_map_nox.o
- $(C) -DDLONG -DDO_VALUES -c umf_triplet.c -o umf_dl_triplet_nomap_x.o
- $(C) -DDLONG -c umf_triplet.c -o umf_dl_triplet_nomap_nox.o
- $(C) -DDLONG -DDO_MAP -DDO_VALUES -c umf_triplet.c -o umf_dl_triplet_map_x.o
- $(C) -DDLONG -DFIXQ -c umf_assemble.c -o umf_dl_assemble_fixq.o
- $(C) -DDLONG -DDROP -c umf_store_lu.c -o umf_dl_store_lu_drop.o
- $(C) -DDLONG -c umf_assemble.c -o umf_dl_assemble.o
- $(C) -DDLONG -c umf_blas3_update.c -o umf_dl_blas3_update.o
- $(C) -DDLONG -c umf_build_tuples.c -o umf_dl_build_tuples.o
- $(C) -DDLONG -c umf_create_element.c -o umf_dl_create_element.o
- $(C) -DDLONG -c umf_dump.c -o umf_dl_dump.o
- $(C) -DDLONG -c umf_extend_front.c -o umf_dl_extend_front.o
- $(C) -DDLONG -c umf_garbage_collection.c -o umf_dl_garbage_collection.o
- $(C) -DDLONG -c umf_get_memory.c -o umf_dl_get_memory.o
- $(C) -DDLONG -c umf_init_front.c -o umf_dl_init_front.o
- $(C) -DDLONG -c umf_kernel.c -o umf_dl_kernel.o
- $(C) -DDLONG -c umf_kernel_init.c -o umf_dl_kernel_init.o
- $(C) -DDLONG -c umf_kernel_wrapup.c -o umf_dl_kernel_wrapup.o
- $(C) -DDLONG -c umf_local_search.c -o umf_dl_local_search.o
- $(C) -DDLONG -c umf_lsolve.c -o umf_dl_lsolve.o
- $(C) -DDLONG -c umf_ltsolve.c -o umf_dl_ltsolve.o
- $(C) -DDLONG -c umf_mem_alloc_element.c -o umf_dl_mem_alloc_element.o
- $(C) -DDLONG -c umf_mem_alloc_head_block.c -o umf_dl_mem_alloc_head_block.o
- $(C) -DDLONG -c umf_mem_alloc_tail_block.c -o umf_dl_mem_alloc_tail_block.o
- $(C) -DDLONG -c umf_mem_free_tail_block.c -o umf_dl_mem_free_tail_block.o
- $(C) -DDLONG -c umf_mem_init_memoryspace.c -o umf_dl_mem_init_memoryspace.o
- $(C) -DDLONG -c umf_report_vector.c -o umf_dl_report_vector.o
- $(C) -DDLONG -c umf_row_search.c -o umf_dl_row_search.o
- $(C) -DDLONG -c umf_scale_column.c -o umf_dl_scale_column.o
- $(C) -DDLONG -c umf_set_stats.c -o umf_dl_set_stats.o
- $(C) -DDLONG -c umf_solve.c -o umf_dl_solve.o
- $(C) -DDLONG -c umf_symbolic_usage.c -o umf_dl_symbolic_usage.o
- $(C) -DDLONG -c umf_transpose.c -o umf_dl_transpose.o
- $(C) -DDLONG -c umf_tuple_lengths.c -o umf_dl_tuple_lengths.o
- $(C) -DDLONG -c umf_usolve.c -o umf_dl_usolve.o
- $(C) -DDLONG -c umf_utsolve.c -o umf_dl_utsolve.o
- $(C) -DDLONG -c umf_valid_numeric.c -o umf_dl_valid_numeric.o
- $(C) -DDLONG -c umf_valid_symbolic.c -o umf_dl_valid_symbolic.o
- $(C) -DDLONG -c umf_grow_front.c -o umf_dl_grow_front.o
- $(C) -DDLONG -c umf_start_front.c -o umf_dl_start_front.o
- $(C) -DDLONG -c umf_2by2.c -o umf_dl_2by2.o
- $(C) -DDLONG -c umf_store_lu.c -o umf_dl_store_lu.o
- $(C) -DDLONG -c umf_scale.c -o umf_dl_scale.o
- $(C) -DDLONG -DWSOLVE -c umfpack_solve.c -o umfpack_dl_wsolve.o
- $(C) -DDLONG -c umfpack_col_to_triplet.c -o umfpack_dl_col_to_triplet.o
- $(C) -DDLONG -c umfpack_defaults.c -o umfpack_dl_defaults.o
- $(C) -DDLONG -c umfpack_free_numeric.c -o umfpack_dl_free_numeric.o
- $(C) -DDLONG -c umfpack_free_symbolic.c -o umfpack_dl_free_symbolic.o
- $(C) -DDLONG -c umfpack_get_numeric.c -o umfpack_dl_get_numeric.o
- $(C) -DDLONG -c umfpack_get_lunz.c -o umfpack_dl_get_lunz.o
- $(C) -DDLONG -c umfpack_get_symbolic.c -o umfpack_dl_get_symbolic.o
- $(C) -DDLONG -c umfpack_get_determinant.c -o umfpack_dl_get_determinant.o
- $(C) -DDLONG -c umfpack_numeric.c -o umfpack_dl_numeric.o
- $(C) -DDLONG -c umfpack_qsymbolic.c -o umfpack_dl_qsymbolic.o
- $(C) -DDLONG -c umfpack_report_control.c -o umfpack_dl_report_control.o
- $(C) -DDLONG -c umfpack_report_info.c -o umfpack_dl_report_info.o
- $(C) -DDLONG -c umfpack_report_matrix.c -o umfpack_dl_report_matrix.o
- $(C) -DDLONG -c umfpack_report_numeric.c -o umfpack_dl_report_numeric.o
- $(C) -DDLONG -c umfpack_report_perm.c -o umfpack_dl_report_perm.o
- $(C) -DDLONG -c umfpack_report_status.c -o umfpack_dl_report_status.o
- $(C) -DDLONG -c umfpack_report_symbolic.c -o umfpack_dl_report_symbolic.o
- $(C) -DDLONG -c umfpack_report_triplet.c -o umfpack_dl_report_triplet.o
- $(C) -DDLONG -c umfpack_report_vector.c -o umfpack_dl_report_vector.o
- $(C) -DDLONG -c umfpack_solve.c -o umfpack_dl_solve.o
- $(C) -DDLONG -c umfpack_symbolic.c -o umfpack_dl_symbolic.o
- $(C) -DDLONG -c umfpack_transpose.c -o umfpack_dl_transpose.o
- $(C) -DDLONG -c umfpack_triplet_to_col.c -o umfpack_dl_triplet_to_col.o
- $(C) -DDLONG -c umfpack_scale.c -o umfpack_dl_scale.o
- $(C) -DDLONG -c umfpack_load_numeric.c -o umfpack_dl_load_numeric.o
- $(C) -DDLONG -c umfpack_save_numeric.c -o umfpack_dl_save_numeric.o
- $(C) -DDLONG -c umfpack_load_symbolic.c -o umfpack_dl_load_symbolic.o
- $(C) -DDLONG -c umfpack_save_symbolic.c -o umfpack_dl_save_symbolic.o
- $(C) -DZINT -DCONJUGATE_SOLVE -c umf_ltsolve.c -o umf_zi_lhsolve.o
- $(C) -DZINT -DCONJUGATE_SOLVE -c umf_utsolve.c -o umf_zi_uhsolve.o
- $(C) -DZINT -DDO_MAP -c umf_triplet.c -o umf_zi_triplet_map_nox.o
- $(C) -DZINT -DDO_VALUES -c umf_triplet.c -o umf_zi_triplet_nomap_x.o
- $(C) -DZINT -c umf_triplet.c -o umf_zi_triplet_nomap_nox.o
- $(C) -DZINT -DDO_MAP -DDO_VALUES -c umf_triplet.c -o umf_zi_triplet_map_x.o
- $(C) -DZINT -DFIXQ -c umf_assemble.c -o umf_zi_assemble_fixq.o
- $(C) -DZINT -DDROP -c umf_store_lu.c -o umf_zi_store_lu_drop.o
- $(C) -DZINT -c umf_assemble.c -o umf_zi_assemble.o
- $(C) -DZINT -c umf_blas3_update.c -o umf_zi_blas3_update.o
- $(C) -DZINT -c umf_build_tuples.c -o umf_zi_build_tuples.o
- $(C) -DZINT -c umf_create_element.c -o umf_zi_create_element.o
- $(C) -DZINT -c umf_dump.c -o umf_zi_dump.o
- $(C) -DZINT -c umf_extend_front.c -o umf_zi_extend_front.o
- $(C) -DZINT -c umf_garbage_collection.c -o umf_zi_garbage_collection.o
- $(C) -DZINT -c umf_get_memory.c -o umf_zi_get_memory.o
- $(C) -DZINT -c umf_init_front.c -o umf_zi_init_front.o
- $(C) -DZINT -c umf_kernel.c -o umf_zi_kernel.o
- $(C) -DZINT -c umf_kernel_init.c -o umf_zi_kernel_init.o
- $(C) -DZINT -c umf_kernel_wrapup.c -o umf_zi_kernel_wrapup.o
- $(C) -DZINT -c umf_local_search.c -o umf_zi_local_search.o
- $(C) -DZINT -c umf_lsolve.c -o umf_zi_lsolve.o
- $(C) -DZINT -c umf_ltsolve.c -o umf_zi_ltsolve.o
- $(C) -DZINT -c umf_mem_alloc_element.c -o umf_zi_mem_alloc_element.o
- $(C) -DZINT -c umf_mem_alloc_head_block.c -o umf_zi_mem_alloc_head_block.o
- $(C) -DZINT -c umf_mem_alloc_tail_block.c -o umf_zi_mem_alloc_tail_block.o
- $(C) -DZINT -c umf_mem_free_tail_block.c -o umf_zi_mem_free_tail_block.o
- $(C) -DZINT -c umf_mem_init_memoryspace.c -o umf_zi_mem_init_memoryspace.o
- $(C) -DZINT -c umf_report_vector.c -o umf_zi_report_vector.o
- $(C) -DZINT -c umf_row_search.c -o umf_zi_row_search.o
- $(C) -DZINT -c umf_scale_column.c -o umf_zi_scale_column.o
- $(C) -DZINT -c umf_set_stats.c -o umf_zi_set_stats.o
- $(C) -DZINT -c umf_solve.c -o umf_zi_solve.o
- $(C) -DZINT -c umf_symbolic_usage.c -o umf_zi_symbolic_usage.o
- $(C) -DZINT -c umf_transpose.c -o umf_zi_transpose.o
- $(C) -DZINT -c umf_tuple_lengths.c -o umf_zi_tuple_lengths.o
- $(C) -DZINT -c umf_usolve.c -o umf_zi_usolve.o
- $(C) -DZINT -c umf_utsolve.c -o umf_zi_utsolve.o
- $(C) -DZINT -c umf_valid_numeric.c -o umf_zi_valid_numeric.o
- $(C) -DZINT -c umf_valid_symbolic.c -o umf_zi_valid_symbolic.o
- $(C) -DZINT -c umf_grow_front.c -o umf_zi_grow_front.o
- $(C) -DZINT -c umf_start_front.c -o umf_zi_start_front.o
- $(C) -DZINT -c umf_2by2.c -o umf_zi_2by2.o
- $(C) -DZINT -c umf_store_lu.c -o umf_zi_store_lu.o
- $(C) -DZINT -c umf_scale.c -o umf_zi_scale.o
- $(C) -DZINT -DWSOLVE -c umfpack_solve.c -o umfpack_zi_wsolve.o
- $(C) -DZINT -c umfpack_col_to_triplet.c -o umfpack_zi_col_to_triplet.o
- $(C) -DZINT -c umfpack_defaults.c -o umfpack_zi_defaults.o
- $(C) -DZINT -c umfpack_free_numeric.c -o umfpack_zi_free_numeric.o
- $(C) -DZINT -c umfpack_free_symbolic.c -o umfpack_zi_free_symbolic.o
- $(C) -DZINT -c umfpack_get_numeric.c -o umfpack_zi_get_numeric.o
- $(C) -DZINT -c umfpack_get_lunz.c -o umfpack_zi_get_lunz.o
- $(C) -DZINT -c umfpack_get_symbolic.c -o umfpack_zi_get_symbolic.o
- $(C) -DZINT -c umfpack_get_determinant.c -o umfpack_zi_get_determinant.o
- $(C) -DZINT -c umfpack_numeric.c -o umfpack_zi_numeric.o
- $(C) -DZINT -c umfpack_qsymbolic.c -o umfpack_zi_qsymbolic.o
- $(C) -DZINT -c umfpack_report_control.c -o umfpack_zi_report_control.o
- $(C) -DZINT -c umfpack_report_info.c -o umfpack_zi_report_info.o
- $(C) -DZINT -c umfpack_report_matrix.c -o umfpack_zi_report_matrix.o
- $(C) -DZINT -c umfpack_report_numeric.c -o umfpack_zi_report_numeric.o
- $(C) -DZINT -c umfpack_report_perm.c -o umfpack_zi_report_perm.o
- $(C) -DZINT -c umfpack_report_status.c -o umfpack_zi_report_status.o
- $(C) -DZINT -c umfpack_report_symbolic.c -o umfpack_zi_report_symbolic.o
- $(C) -DZINT -c umfpack_report_triplet.c -o umfpack_zi_report_triplet.o
- $(C) -DZINT -c umfpack_report_vector.c -o umfpack_zi_report_vector.o
- $(C) -DZINT -c umfpack_solve.c -o umfpack_zi_solve.o
- $(C) -DZINT -c umfpack_symbolic.c -o umfpack_zi_symbolic.o
- $(C) -DZINT -c umfpack_transpose.c -o umfpack_zi_transpose.o
- $(C) -DZINT -c umfpack_triplet_to_col.c -o umfpack_zi_triplet_to_col.o
- $(C) -DZINT -c umfpack_scale.c -o umfpack_zi_scale.o
- $(C) -DZINT -c umfpack_load_numeric.c -o umfpack_zi_load_numeric.o
- $(C) -DZINT -c umfpack_save_numeric.c -o umfpack_zi_save_numeric.o
- $(C) -DZINT -c umfpack_load_symbolic.c -o umfpack_zi_load_symbolic.o
- $(C) -DZINT -c umfpack_save_symbolic.c -o umfpack_zi_save_symbolic.o
- $(C) -DZLONG -DCONJUGATE_SOLVE -c umf_ltsolve.c -o umf_zl_lhsolve.o
- $(C) -DZLONG -DCONJUGATE_SOLVE -c umf_utsolve.c -o umf_zl_uhsolve.o
- $(C) -DZLONG -DDO_MAP -c umf_triplet.c -o umf_zl_triplet_map_nox.o
- $(C) -DZLONG -DDO_VALUES -c umf_triplet.c -o umf_zl_triplet_nomap_x.o
- $(C) -DZLONG -c umf_triplet.c -o umf_zl_triplet_nomap_nox.o
- $(C) -DZLONG -DDO_MAP -DDO_VALUES -c umf_triplet.c -o umf_zl_triplet_map_x.o
- $(C) -DZLONG -DFIXQ -c umf_assemble.c -o umf_zl_assemble_fixq.o
- $(C) -DZLONG -DDROP -c umf_store_lu.c -o umf_zl_store_lu_drop.o
- $(C) -DZLONG -c umf_assemble.c -o umf_zl_assemble.o
- $(C) -DZLONG -c umf_blas3_update.c -o umf_zl_blas3_update.o
- $(C) -DZLONG -c umf_build_tuples.c -o umf_zl_build_tuples.o
- $(C) -DZLONG -c umf_create_element.c -o umf_zl_create_element.o
- $(C) -DZLONG -c umf_dump.c -o umf_zl_dump.o
- $(C) -DZLONG -c umf_extend_front.c -o umf_zl_extend_front.o
- $(C) -DZLONG -c umf_garbage_collection.c -o umf_zl_garbage_collection.o
- $(C) -DZLONG -c umf_get_memory.c -o umf_zl_get_memory.o
- $(C) -DZLONG -c umf_init_front.c -o umf_zl_init_front.o
- $(C) -DZLONG -c umf_kernel.c -o umf_zl_kernel.o
- $(C) -DZLONG -c umf_kernel_init.c -o umf_zl_kernel_init.o
- $(C) -DZLONG -c umf_kernel_wrapup.c -o umf_zl_kernel_wrapup.o
- $(C) -DZLONG -c umf_local_search.c -o umf_zl_local_search.o
- $(C) -DZLONG -c umf_lsolve.c -o umf_zl_lsolve.o
- $(C) -DZLONG -c umf_ltsolve.c -o umf_zl_ltsolve.o
- $(C) -DZLONG -c umf_mem_alloc_element.c -o umf_zl_mem_alloc_element.o
- $(C) -DZLONG -c umf_mem_alloc_head_block.c -o umf_zl_mem_alloc_head_block.o
- $(C) -DZLONG -c umf_mem_alloc_tail_block.c -o umf_zl_mem_alloc_tail_block.o
- $(C) -DZLONG -c umf_mem_free_tail_block.c -o umf_zl_mem_free_tail_block.o
- $(C) -DZLONG -c umf_mem_init_memoryspace.c -o umf_zl_mem_init_memoryspace.o
- $(C) -DZLONG -c umf_report_vector.c -o umf_zl_report_vector.o
- $(C) -DZLONG -c umf_row_search.c -o umf_zl_row_search.o
- $(C) -DZLONG -c umf_scale_column.c -o umf_zl_scale_column.o
- $(C) -DZLONG -c umf_set_stats.c -o umf_zl_set_stats.o
- $(C) -DZLONG -c umf_solve.c -o umf_zl_solve.o
- $(C) -DZLONG -c umf_symbolic_usage.c -o umf_zl_symbolic_usage.o
- $(C) -DZLONG -c umf_transpose.c -o umf_zl_transpose.o
- $(C) -DZLONG -c umf_tuple_lengths.c -o umf_zl_tuple_lengths.o
- $(C) -DZLONG -c umf_usolve.c -o umf_zl_usolve.o
- $(C) -DZLONG -c umf_utsolve.c -o umf_zl_utsolve.o
- $(C) -DZLONG -c umf_valid_numeric.c -o umf_zl_valid_numeric.o
- $(C) -DZLONG -c umf_valid_symbolic.c -o umf_zl_valid_symbolic.o
- $(C) -DZLONG -c umf_grow_front.c -o umf_zl_grow_front.o
- $(C) -DZLONG -c umf_start_front.c -o umf_zl_start_front.o
- $(C) -DZLONG -c umf_2by2.c -o umf_zl_2by2.o
- $(C) -DZLONG -c umf_store_lu.c -o umf_zl_store_lu.o
- $(C) -DZLONG -c umf_scale.c -o umf_zl_scale.o
- $(C) -DZLONG -DWSOLVE -c umfpack_solve.c -o umfpack_zl_wsolve.o
- $(C) -DZLONG -c umfpack_col_to_triplet.c -o umfpack_zl_col_to_triplet.o
- $(C) -DZLONG -c umfpack_defaults.c -o umfpack_zl_defaults.o
- $(C) -DZLONG -c umfpack_free_numeric.c -o umfpack_zl_free_numeric.o
- $(C) -DZLONG -c umfpack_free_symbolic.c -o umfpack_zl_free_symbolic.o
- $(C) -DZLONG -c umfpack_get_numeric.c -o umfpack_zl_get_numeric.o
- $(C) -DZLONG -c umfpack_get_lunz.c -o umfpack_zl_get_lunz.o
- $(C) -DZLONG -c umfpack_get_symbolic.c -o umfpack_zl_get_symbolic.o
- $(C) -DZLONG -c umfpack_get_determinant.c -o umfpack_zl_get_determinant.o
- $(C) -DZLONG -c umfpack_numeric.c -o umfpack_zl_numeric.o
- $(C) -DZLONG -c umfpack_qsymbolic.c -o umfpack_zl_qsymbolic.o
- $(C) -DZLONG -c umfpack_report_control.c -o umfpack_zl_report_control.o
- $(C) -DZLONG -c umfpack_report_info.c -o umfpack_zl_report_info.o
- $(C) -DZLONG -c umfpack_report_matrix.c -o umfpack_zl_report_matrix.o
- $(C) -DZLONG -c umfpack_report_numeric.c -o umfpack_zl_report_numeric.o
- $(C) -DZLONG -c umfpack_report_perm.c -o umfpack_zl_report_perm.o
- $(C) -DZLONG -c umfpack_report_status.c -o umfpack_zl_report_status.o
- $(C) -DZLONG -c umfpack_report_symbolic.c -o umfpack_zl_report_symbolic.o
- $(C) -DZLONG -c umfpack_report_triplet.c -o umfpack_zl_report_triplet.o
- $(C) -DZLONG -c umfpack_report_vector.c -o umfpack_zl_report_vector.o
- $(C) -DZLONG -c umfpack_solve.c -o umfpack_zl_solve.o
- $(C) -DZLONG -c umfpack_symbolic.c -o umfpack_zl_symbolic.o
- $(C) -DZLONG -c umfpack_transpose.c -o umfpack_zl_transpose.o
- $(C) -DZLONG -c umfpack_triplet_to_col.c -o umfpack_zl_triplet_to_col.o
- $(C) -DZLONG -c umfpack_scale.c -o umfpack_zl_scale.o
- $(C) -DZLONG -c umfpack_load_numeric.c -o umfpack_zl_load_numeric.o
- $(C) -DZLONG -c umfpack_save_numeric.c -o umfpack_zl_save_numeric.o
- $(C) -DZLONG -c umfpack_load_symbolic.c -o umfpack_zl_load_symbolic.o
- $(C) -DZLONG -c umfpack_save_symbolic.c -o umfpack_zl_save_symbolic.o
- $(AR) ../Lib/libumfpack.a \
- umfpack_gn_global.o \
- umf_i_analyze.o umf_i_apply_order.o umf_i_colamd.o umf_i_free.o \
- umf_i_fsize.o umf_i_is_permutation.o umf_i_malloc.o umf_i_realloc.o \
- umf_i_report_perm.o umf_i_singletons.o \
- umf_l_analyze.o umf_l_apply_order.o umf_l_colamd.o umf_l_free.o \
- umf_l_fsize.o umf_l_is_permutation.o umf_l_malloc.o umf_l_realloc.o \
- umf_l_report_perm.o umf_l_singletons.o \
- umfpack_gn_timer.o umfpack_gn_tictoc.o \
- umf_di_lhsolve.o \
- umf_di_uhsolve.o umf_di_triplet_map_nox.o umf_di_triplet_nomap_x.o \
- umf_di_triplet_nomap_nox.o umf_di_triplet_map_x.o \
- umf_di_assemble_fixq.o umf_di_store_lu_drop.o umf_di_assemble.o \
- umf_di_blas3_update.o umf_di_build_tuples.o \
- umf_di_create_element.o umf_di_dump.o umf_di_extend_front.o \
- umf_di_garbage_collection.o umf_di_get_memory.o \
- umf_di_init_front.o umf_di_kernel.o umf_di_kernel_init.o \
- umf_di_kernel_wrapup.o umf_di_local_search.o umf_di_lsolve.o \
- umf_di_ltsolve.o umf_di_mem_alloc_element.o \
- umf_di_mem_alloc_head_block.o umf_di_mem_alloc_tail_block.o \
- umf_di_mem_free_tail_block.o umf_di_mem_init_memoryspace.o \
- umf_di_report_vector.o umf_di_row_search.o umf_di_scale_column.o \
- umf_di_set_stats.o umf_di_solve.o umf_di_symbolic_usage.o \
- umf_di_transpose.o umf_di_tuple_lengths.o umf_di_usolve.o \
- umf_di_utsolve.o umf_di_valid_numeric.o umf_di_valid_symbolic.o \
- umf_di_grow_front.o umf_di_start_front.o umf_di_2by2.o \
- umf_di_store_lu.o umf_di_scale.o umfpack_di_wsolve.o \
- umfpack_di_col_to_triplet.o umfpack_di_defaults.o \
- umfpack_di_free_numeric.o umfpack_di_free_symbolic.o \
- umfpack_di_get_numeric.o umfpack_di_get_lunz.o \
- umfpack_di_get_symbolic.o umfpack_di_get_determinant.o \
- umfpack_di_numeric.o \
- umfpack_di_qsymbolic.o umfpack_di_report_control.o \
- umfpack_di_report_info.o umfpack_di_report_matrix.o \
- umfpack_di_report_numeric.o umfpack_di_report_perm.o \
- umfpack_di_report_status.o umfpack_di_report_symbolic.o \
- umfpack_di_report_triplet.o umfpack_di_report_vector.o \
- umfpack_di_solve.o umfpack_di_symbolic.o umfpack_di_transpose.o \
- umfpack_di_triplet_to_col.o umfpack_di_scale.o \
- umfpack_di_load_numeric.o umfpack_di_save_numeric.o \
- umfpack_di_load_symbolic.o umfpack_di_save_symbolic.o \
- umf_dl_lhsolve.o \
- umf_dl_uhsolve.o umf_dl_triplet_map_nox.o umf_dl_triplet_nomap_x.o \
- umf_dl_triplet_nomap_nox.o umf_dl_triplet_map_x.o \
- umf_dl_assemble_fixq.o umf_dl_store_lu_drop.o umf_dl_assemble.o \
- umf_dl_blas3_update.o umf_dl_build_tuples.o \
- umf_dl_create_element.o umf_dl_dump.o umf_dl_extend_front.o \
- umf_dl_garbage_collection.o umf_dl_get_memory.o \
- umf_dl_init_front.o umf_dl_kernel.o umf_dl_kernel_init.o \
- umf_dl_kernel_wrapup.o umf_dl_local_search.o umf_dl_lsolve.o \
- umf_dl_ltsolve.o umf_dl_mem_alloc_element.o \
- umf_dl_mem_alloc_head_block.o umf_dl_mem_alloc_tail_block.o \
- umf_dl_mem_free_tail_block.o umf_dl_mem_init_memoryspace.o \
- umf_dl_report_vector.o umf_dl_row_search.o umf_dl_scale_column.o \
- umf_dl_set_stats.o umf_dl_solve.o umf_dl_symbolic_usage.o \
- umf_dl_transpose.o umf_dl_tuple_lengths.o umf_dl_usolve.o \
- umf_dl_utsolve.o umf_dl_valid_numeric.o umf_dl_valid_symbolic.o \
- umf_dl_grow_front.o umf_dl_start_front.o umf_dl_2by2.o \
- umf_dl_store_lu.o umf_dl_scale.o umfpack_dl_wsolve.o \
- umfpack_dl_col_to_triplet.o umfpack_dl_defaults.o \
- umfpack_dl_free_numeric.o umfpack_dl_free_symbolic.o \
- umfpack_dl_get_numeric.o umfpack_dl_get_lunz.o \
- umfpack_dl_get_symbolic.o umfpack_dl_get_determinant.o \
- umfpack_dl_numeric.o \
- umfpack_dl_qsymbolic.o umfpack_dl_report_control.o \
- umfpack_dl_report_info.o umfpack_dl_report_matrix.o \
- umfpack_dl_report_numeric.o umfpack_dl_report_perm.o \
- umfpack_dl_report_status.o umfpack_dl_report_symbolic.o \
- umfpack_dl_report_triplet.o umfpack_dl_report_vector.o \
- umfpack_dl_solve.o umfpack_dl_symbolic.o umfpack_dl_transpose.o \
- umfpack_dl_triplet_to_col.o umfpack_dl_scale.o \
- umfpack_dl_load_numeric.o umfpack_dl_save_numeric.o \
- umfpack_dl_load_symbolic.o umfpack_dl_save_symbolic.o \
- umf_zi_lhsolve.o \
- umf_zi_uhsolve.o umf_zi_triplet_map_nox.o umf_zi_triplet_nomap_x.o \
- umf_zi_triplet_nomap_nox.o umf_zi_triplet_map_x.o \
- umf_zi_assemble_fixq.o umf_zi_store_lu_drop.o umf_zi_assemble.o \
- umf_zi_blas3_update.o umf_zi_build_tuples.o \
- umf_zi_create_element.o umf_zi_dump.o umf_zi_extend_front.o \
- umf_zi_garbage_collection.o umf_zi_get_memory.o \
- umf_zi_init_front.o umf_zi_kernel.o umf_zi_kernel_init.o \
- umf_zi_kernel_wrapup.o umf_zi_local_search.o umf_zi_lsolve.o \
- umf_zi_ltsolve.o umf_zi_mem_alloc_element.o \
- umf_zi_mem_alloc_head_block.o umf_zi_mem_alloc_tail_block.o \
- umf_zi_mem_free_tail_block.o umf_zi_mem_init_memoryspace.o \
- umf_zi_report_vector.o umf_zi_row_search.o umf_zi_scale_column.o \
- umf_zi_set_stats.o umf_zi_solve.o umf_zi_symbolic_usage.o \
- umf_zi_transpose.o umf_zi_tuple_lengths.o umf_zi_usolve.o \
- umf_zi_utsolve.o umf_zi_valid_numeric.o umf_zi_valid_symbolic.o \
- umf_zi_grow_front.o umf_zi_start_front.o umf_zi_2by2.o \
- umf_zi_store_lu.o umf_zi_scale.o umfpack_zi_wsolve.o \
- umfpack_zi_col_to_triplet.o umfpack_zi_defaults.o \
- umfpack_zi_free_numeric.o umfpack_zi_free_symbolic.o \
- umfpack_zi_get_numeric.o umfpack_zi_get_lunz.o \
- umfpack_zi_get_symbolic.o umfpack_zi_get_determinant.o \
- umfpack_zi_numeric.o \
- umfpack_zi_qsymbolic.o umfpack_zi_report_control.o \
- umfpack_zi_report_info.o umfpack_zi_report_matrix.o \
- umfpack_zi_report_numeric.o umfpack_zi_report_perm.o \
- umfpack_zi_report_status.o umfpack_zi_report_symbolic.o \
- umfpack_zi_report_triplet.o umfpack_zi_report_vector.o \
- umfpack_zi_solve.o umfpack_zi_symbolic.o umfpack_zi_transpose.o \
- umfpack_zi_triplet_to_col.o umfpack_zi_scale.o \
- umfpack_zi_load_numeric.o umfpack_zi_save_numeric.o \
- umfpack_zi_load_symbolic.o umfpack_zi_save_symbolic.o \
- umf_zl_lhsolve.o \
- umf_zl_uhsolve.o umf_zl_triplet_map_nox.o umf_zl_triplet_nomap_x.o \
- umf_zl_triplet_nomap_nox.o umf_zl_triplet_map_x.o \
- umf_zl_assemble_fixq.o umf_zl_store_lu_drop.o umf_zl_assemble.o \
- umf_zl_blas3_update.o umf_zl_build_tuples.o \
- umf_zl_create_element.o umf_zl_dump.o umf_zl_extend_front.o \
- umf_zl_garbage_collection.o umf_zl_get_memory.o \
- umf_zl_init_front.o umf_zl_kernel.o umf_zl_kernel_init.o \
- umf_zl_kernel_wrapup.o umf_zl_local_search.o umf_zl_lsolve.o \
- umf_zl_ltsolve.o umf_zl_mem_alloc_element.o \
- umf_zl_mem_alloc_head_block.o umf_zl_mem_alloc_tail_block.o \
- umf_zl_mem_free_tail_block.o umf_zl_mem_init_memoryspace.o \
- umf_zl_report_vector.o umf_zl_row_search.o umf_zl_scale_column.o \
- umf_zl_set_stats.o umf_zl_solve.o umf_zl_symbolic_usage.o \
- umf_zl_transpose.o umf_zl_tuple_lengths.o umf_zl_usolve.o \
- umf_zl_utsolve.o umf_zl_valid_numeric.o umf_zl_valid_symbolic.o \
- umf_zl_grow_front.o umf_zl_start_front.o umf_zl_2by2.o \
- umf_zl_store_lu.o umf_zl_scale.o umfpack_zl_wsolve.o \
- umfpack_zl_col_to_triplet.o umfpack_zl_defaults.o \
- umfpack_zl_free_numeric.o umfpack_zl_free_symbolic.o \
- umfpack_zl_get_numeric.o umfpack_zl_get_lunz.o \
- umfpack_zl_get_symbolic.o umfpack_zl_get_determinant.o \
- umfpack_zl_numeric.o \
- umfpack_zl_qsymbolic.o umfpack_zl_report_control.o \
- umfpack_zl_report_info.o umfpack_zl_report_matrix.o \
- umfpack_zl_report_numeric.o umfpack_zl_report_perm.o \
- umfpack_zl_report_status.o umfpack_zl_report_symbolic.o \
- umfpack_zl_report_triplet.o umfpack_zl_report_vector.o \
- umfpack_zl_solve.o umfpack_zl_symbolic.o umfpack_zl_transpose.o \
- umfpack_zl_triplet_to_col.o umfpack_zl_scale.o \
- umfpack_zl_load_numeric.o umfpack_zl_save_numeric.o \
- umfpack_zl_load_symbolic.o umfpack_zl_save_symbolic.o
- - $(RANLIB) ../Lib/libumfpack.a
-
-#-------------------------------------------------------------------------------
-# Remove all but the files in the original distribution
-#-------------------------------------------------------------------------------
-
-purge: clean
- - $(RM) ../Lib/libumfpack.a
-
-clean:
- - $(RM) $(CLEAN)
diff --git a/src/C/SuiteSparse/UMFPACK/Source/cholmod_blas.h b/src/C/SuiteSparse/UMFPACK/Source/cholmod_blas.h
index 9619804..709f2d0 100644
--- a/src/C/SuiteSparse/UMFPACK/Source/cholmod_blas.h
+++ b/src/C/SuiteSparse/UMFPACK/Source/cholmod_blas.h
@@ -3,8 +3,8 @@
/* ========================================================================== */
/* -----------------------------------------------------------------------------
- * CHOLMOD/Include/cholmod_blas.h. Version 1.2.
- * Copyright (C) 2005-2006, Univ. of Florida. Author: Timothy A. Davis
+ * CHOLMOD/Include/cholmod_blas.h.
+ * Copyright (C) Univ. of Florida. Author: Timothy A. Davis
* CHOLMOD/Include/cholmod_blas.h is licensed under Version 2.1 of the GNU
* Lesser General Public License. See lesser.txt for a text of the license.
* CHOLMOD is also available under other licenses; contact authors for details.
diff --git a/src/C/SuiteSparse/UMFPACK/Source/umf_2by2.c b/src/C/SuiteSparse/UMFPACK/Source/umf_2by2.c
index 7a4e3e1..42616dc 100644
--- a/src/C/SuiteSparse/UMFPACK/Source/umf_2by2.c
+++ b/src/C/SuiteSparse/UMFPACK/Source/umf_2by2.c
@@ -3,7 +3,7 @@
/* ========================================================================== */
/* -------------------------------------------------------------------------- */
-/* UMFPACK Version 5.0, Copyright (c) 1995-2006 by Timothy A. Davis. CISE, */
+/* UMFPACK Copyright (c) Timothy A. Davis, CISE, */
/* Univ. of Florida. All Rights Reserved. See ../Doc/License for License. */
/* web: http://www.cise.ufl.edu/research/sparse/umfpack */
/* -------------------------------------------------------------------------- */
diff --git a/src/C/SuiteSparse/UMFPACK/Source/umf_2by2.h b/src/C/SuiteSparse/UMFPACK/Source/umf_2by2.h
index 078625e..3cb8b45 100644
--- a/src/C/SuiteSparse/UMFPACK/Source/umf_2by2.h
+++ b/src/C/SuiteSparse/UMFPACK/Source/umf_2by2.h
@@ -1,5 +1,5 @@
/* -------------------------------------------------------------------------- */
-/* UMFPACK Version 5.0, Copyright (c) 1995-2006 by Timothy A. Davis. CISE, */
+/* UMFPACK Copyright (c) Timothy A. Davis, CISE, */
/* Univ. of Florida. All Rights Reserved. See ../Doc/License for License. */
/* web: http://www.cise.ufl.edu/research/sparse/umfpack */
/* -------------------------------------------------------------------------- */
diff --git a/src/C/SuiteSparse/UMFPACK/Source/umf_analyze.c b/src/C/SuiteSparse/UMFPACK/Source/umf_analyze.c
index fa8aecd..7744635 100644
--- a/src/C/SuiteSparse/UMFPACK/Source/umf_analyze.c
+++ b/src/C/SuiteSparse/UMFPACK/Source/umf_analyze.c
@@ -3,7 +3,7 @@
/* ========================================================================== */
/* -------------------------------------------------------------------------- */
-/* UMFPACK Version 5.0, Copyright (c) 1995-2006 by Timothy A. Davis. CISE, */
+/* UMFPACK Copyright (c) Timothy A. Davis, CISE, */
/* Univ. of Florida. All Rights Reserved. See ../Doc/License for License. */
/* web: http://www.cise.ufl.edu/research/sparse/umfpack */
/* -------------------------------------------------------------------------- */
diff --git a/src/C/SuiteSparse/UMFPACK/Source/umf_analyze.h b/src/C/SuiteSparse/UMFPACK/Source/umf_analyze.h
index c0592a6..733ba03 100644
--- a/src/C/SuiteSparse/UMFPACK/Source/umf_analyze.h
+++ b/src/C/SuiteSparse/UMFPACK/Source/umf_analyze.h
@@ -1,5 +1,5 @@
/* -------------------------------------------------------------------------- */
-/* UMFPACK Version 5.0, Copyright (c) 1995-2006 by Timothy A. Davis. CISE, */
+/* UMFPACK Copyright (c) Timothy A. Davis, CISE, */
/* Univ. of Florida. All Rights Reserved. See ../Doc/License for License. */
/* web: http://www.cise.ufl.edu/research/sparse/umfpack */
/* -------------------------------------------------------------------------- */
diff --git a/src/C/SuiteSparse/UMFPACK/Source/umf_apply_order.c b/src/C/SuiteSparse/UMFPACK/Source/umf_apply_order.c
index 996f2e3..8e90d3a 100644
--- a/src/C/SuiteSparse/UMFPACK/Source/umf_apply_order.c
+++ b/src/C/SuiteSparse/UMFPACK/Source/umf_apply_order.c
@@ -3,7 +3,7 @@
/* ========================================================================== */
/* -------------------------------------------------------------------------- */
-/* UMFPACK Version 5.0, Copyright (c) 1995-2006 by Timothy A. Davis. CISE, */
+/* UMFPACK Copyright (c) Timothy A. Davis, CISE, */
/* Univ. of Florida. All Rights Reserved. See ../Doc/License for License. */
/* web: http://www.cise.ufl.edu/research/sparse/umfpack */
/* -------------------------------------------------------------------------- */
diff --git a/src/C/SuiteSparse/UMFPACK/Source/umf_apply_order.h b/src/C/SuiteSparse/UMFPACK/Source/umf_apply_order.h
index 8363a96..dd7e682 100644
--- a/src/C/SuiteSparse/UMFPACK/Source/umf_apply_order.h
+++ b/src/C/SuiteSparse/UMFPACK/Source/umf_apply_order.h
@@ -1,5 +1,5 @@
/* -------------------------------------------------------------------------- */
-/* UMFPACK Version 5.0, Copyright (c) 1995-2006 by Timothy A. Davis. CISE, */
+/* UMFPACK Copyright (c) Timothy A. Davis, CISE, */
/* Univ. of Florida. All Rights Reserved. See ../Doc/License for License. */
/* web: http://www.cise.ufl.edu/research/sparse/umfpack */
/* -------------------------------------------------------------------------- */
diff --git a/src/C/SuiteSparse/UMFPACK/Source/umf_assemble.c b/src/C/SuiteSparse/UMFPACK/Source/umf_assemble.c
index 0bc175f..8618be4 100644
--- a/src/C/SuiteSparse/UMFPACK/Source/umf_assemble.c
+++ b/src/C/SuiteSparse/UMFPACK/Source/umf_assemble.c
@@ -3,7 +3,7 @@
/* ========================================================================== */
/* -------------------------------------------------------------------------- */
-/* UMFPACK Version 5.0, Copyright (c) 1995-2006 by Timothy A. Davis. CISE, */
+/* UMFPACK Copyright (c) Timothy A. Davis, CISE, */
/* Univ. of Florida. All Rights Reserved. See ../Doc/License for License. */
/* web: http://www.cise.ufl.edu/research/sparse/umfpack */
/* -------------------------------------------------------------------------- */
diff --git a/src/C/SuiteSparse/UMFPACK/Source/umf_assemble.h b/src/C/SuiteSparse/UMFPACK/Source/umf_assemble.h
index d219e2c..e976b48 100644
--- a/src/C/SuiteSparse/UMFPACK/Source/umf_assemble.h
+++ b/src/C/SuiteSparse/UMFPACK/Source/umf_assemble.h
@@ -1,5 +1,5 @@
/* -------------------------------------------------------------------------- */
-/* UMFPACK Version 5.0, Copyright (c) 1995-2006 by Timothy A. Davis. CISE, */
+/* UMFPACK Copyright (c) Timothy A. Davis, CISE, */
/* Univ. of Florida. All Rights Reserved. See ../Doc/License for License. */
/* web: http://www.cise.ufl.edu/research/sparse/umfpack */
/* -------------------------------------------------------------------------- */
diff --git a/src/C/SuiteSparse/UMFPACK/Source/umf_blas3_update.c b/src/C/SuiteSparse/UMFPACK/Source/umf_blas3_update.c
index f11fb7b..8ff3bc4 100644
--- a/src/C/SuiteSparse/UMFPACK/Source/umf_blas3_update.c
+++ b/src/C/SuiteSparse/UMFPACK/Source/umf_blas3_update.c
@@ -3,7 +3,7 @@
/* ========================================================================== */
/* -------------------------------------------------------------------------- */
-/* UMFPACK Version 5.0, Copyright (c) 1995-2006 by Timothy A. Davis. CISE, */
+/* UMFPACK Copyright (c) Timothy A. Davis, CISE, */
/* Univ. of Florida. All Rights Reserved. See ../Doc/License for License. */
/* web: http://www.cise.ufl.edu/research/sparse/umfpack */
/* -------------------------------------------------------------------------- */
diff --git a/src/C/SuiteSparse/UMFPACK/Source/umf_blas3_update.h b/src/C/SuiteSparse/UMFPACK/Source/umf_blas3_update.h
index 62cea48..c77b9ac 100644
--- a/src/C/SuiteSparse/UMFPACK/Source/umf_blas3_update.h
+++ b/src/C/SuiteSparse/UMFPACK/Source/umf_blas3_update.h
@@ -1,5 +1,5 @@
/* -------------------------------------------------------------------------- */
-/* UMFPACK Version 5.0, Copyright (c) 1995-2006 by Timothy A. Davis. CISE, */
+/* UMFPACK Copyright (c) Timothy A. Davis, CISE, */
/* Univ. of Florida. All Rights Reserved. See ../Doc/License for License. */
/* web: http://www.cise.ufl.edu/research/sparse/umfpack */
/* -------------------------------------------------------------------------- */
diff --git a/src/C/SuiteSparse/UMFPACK/Source/umf_build_tuples.c b/src/C/SuiteSparse/UMFPACK/Source/umf_build_tuples.c
index 4895089..36feed6 100644
--- a/src/C/SuiteSparse/UMFPACK/Source/umf_build_tuples.c
+++ b/src/C/SuiteSparse/UMFPACK/Source/umf_build_tuples.c
@@ -3,7 +3,7 @@
/* ========================================================================== */
/* -------------------------------------------------------------------------- */
-/* UMFPACK Version 5.0, Copyright (c) 1995-2006 by Timothy A. Davis. CISE, */
+/* UMFPACK Copyright (c) Timothy A. Davis, CISE, */
/* Univ. of Florida. All Rights Reserved. See ../Doc/License for License. */
/* web: http://www.cise.ufl.edu/research/sparse/umfpack */
/* -------------------------------------------------------------------------- */
diff --git a/src/C/SuiteSparse/UMFPACK/Source/umf_build_tuples.h b/src/C/SuiteSparse/UMFPACK/Source/umf_build_tuples.h
index 8ee8052..0b0410c 100644
--- a/src/C/SuiteSparse/UMFPACK/Source/umf_build_tuples.h
+++ b/src/C/SuiteSparse/UMFPACK/Source/umf_build_tuples.h
@@ -1,5 +1,5 @@
/* -------------------------------------------------------------------------- */
-/* UMFPACK Version 5.0, Copyright (c) 1995-2006 by Timothy A. Davis. CISE, */
+/* UMFPACK Copyright (c) Timothy A. Davis, CISE, */
/* Univ. of Florida. All Rights Reserved. See ../Doc/License for License. */
/* web: http://www.cise.ufl.edu/research/sparse/umfpack */
/* -------------------------------------------------------------------------- */
diff --git a/src/C/SuiteSparse/UMFPACK/Source/umf_colamd.c b/src/C/SuiteSparse/UMFPACK/Source/umf_colamd.c
index edbbad0..5a9b168 100644
--- a/src/C/SuiteSparse/UMFPACK/Source/umf_colamd.c
+++ b/src/C/SuiteSparse/UMFPACK/Source/umf_colamd.c
@@ -3,7 +3,7 @@
/* ========================================================================== */
/* -------------------------------------------------------------------------- */
-/* UMFPACK Version 5.0, Copyright (c) 1995-2006 by Timothy A. Davis. CISE, */
+/* UMFPACK Copyright (c) Timothy A. Davis, CISE, */
/* Univ. of Florida. All Rights Reserved. See ../Doc/License for License. */
/* web: http://www.cise.ufl.edu/research/sparse/umfpack */
/* -------------------------------------------------------------------------- */
diff --git a/src/C/SuiteSparse/UMFPACK/Source/umf_colamd.h b/src/C/SuiteSparse/UMFPACK/Source/umf_colamd.h
index c7d9167..51675a5 100644
--- a/src/C/SuiteSparse/UMFPACK/Source/umf_colamd.h
+++ b/src/C/SuiteSparse/UMFPACK/Source/umf_colamd.h
@@ -3,7 +3,7 @@
/* ========================================================================== */
/* -------------------------------------------------------------------------- */
-/* UMFPACK Version 5.0, Copyright (c) 1995-2006 by Timothy A. Davis. CISE, */
+/* UMFPACK Copyright (c) Timothy A. Davis, CISE, */
/* Univ. of Florida. All Rights Reserved. See ../Doc/License for License. */
/* web: http://www.cise.ufl.edu/research/sparse/umfpack */
/* -------------------------------------------------------------------------- */
diff --git a/src/C/SuiteSparse/UMFPACK/Source/umf_config.h b/src/C/SuiteSparse/UMFPACK/Source/umf_config.h
index 224cf52..79e9f5d 100644
--- a/src/C/SuiteSparse/UMFPACK/Source/umf_config.h
+++ b/src/C/SuiteSparse/UMFPACK/Source/umf_config.h
@@ -3,7 +3,7 @@
/* ========================================================================== */
/* -------------------------------------------------------------------------- */
-/* UMFPACK Version 5.0, Copyright (c) 1995-2006 by Timothy A. Davis. CISE, */
+/* UMFPACK Copyright (c) Timothy A. Davis, CISE, */
/* Univ. of Florida. All Rights Reserved. See ../Doc/License for License. */
/* web: http://www.cise.ufl.edu/research/sparse/umfpack */
/* -------------------------------------------------------------------------- */
diff --git a/src/C/SuiteSparse/UMFPACK/Source/umf_create_element.c b/src/C/SuiteSparse/UMFPACK/Source/umf_create_element.c
index 2588a64..581ba62 100644
--- a/src/C/SuiteSparse/UMFPACK/Source/umf_create_element.c
+++ b/src/C/SuiteSparse/UMFPACK/Source/umf_create_element.c
@@ -3,7 +3,7 @@
/* ========================================================================== */
/* -------------------------------------------------------------------------- */
-/* UMFPACK Version 5.0, Copyright (c) 1995-2006 by Timothy A. Davis. CISE, */
+/* UMFPACK Copyright (c) Timothy A. Davis, CISE, */
/* Univ. of Florida. All Rights Reserved. See ../Doc/License for License. */
/* web: http://www.cise.ufl.edu/research/sparse/umfpack */
/* -------------------------------------------------------------------------- */
diff --git a/src/C/SuiteSparse/UMFPACK/Source/umf_create_element.h b/src/C/SuiteSparse/UMFPACK/Source/umf_create_element.h
index f2beda6..b6025bb 100644
--- a/src/C/SuiteSparse/UMFPACK/Source/umf_create_element.h
+++ b/src/C/SuiteSparse/UMFPACK/Source/umf_create_element.h
@@ -1,5 +1,5 @@
/* -------------------------------------------------------------------------- */
-/* UMFPACK Version 5.0, Copyright (c) 1995-2006 by Timothy A. Davis. CISE, */
+/* UMFPACK Copyright (c) Timothy A. Davis, CISE, */
/* Univ. of Florida. All Rights Reserved. See ../Doc/License for License. */
/* web: http://www.cise.ufl.edu/research/sparse/umfpack */
/* -------------------------------------------------------------------------- */
diff --git a/src/C/SuiteSparse/UMFPACK/Source/umf_dump.c b/src/C/SuiteSparse/UMFPACK/Source/umf_dump.c
index 962ad9c..888a316 100644
--- a/src/C/SuiteSparse/UMFPACK/Source/umf_dump.c
+++ b/src/C/SuiteSparse/UMFPACK/Source/umf_dump.c
@@ -3,7 +3,7 @@
/* ========================================================================== */
/* -------------------------------------------------------------------------- */
-/* UMFPACK Version 5.0, Copyright (c) 1995-2006 by Timothy A. Davis. CISE, */
+/* UMFPACK Copyright (c) Timothy A. Davis, CISE, */
/* Univ. of Florida. All Rights Reserved. See ../Doc/License for License. */
/* web: http://www.cise.ufl.edu/research/sparse/umfpack */
/* -------------------------------------------------------------------------- */
diff --git a/src/C/SuiteSparse/UMFPACK/Source/umf_dump.h b/src/C/SuiteSparse/UMFPACK/Source/umf_dump.h
index 2e0e512..71ec745 100644
--- a/src/C/SuiteSparse/UMFPACK/Source/umf_dump.h
+++ b/src/C/SuiteSparse/UMFPACK/Source/umf_dump.h
@@ -1,5 +1,5 @@
/* -------------------------------------------------------------------------- */
-/* UMFPACK Version 5.0, Copyright (c) 1995-2006 by Timothy A. Davis. CISE, */
+/* UMFPACK Copyright (c) Timothy A. Davis, CISE, */
/* Univ. of Florida. All Rights Reserved. See ../Doc/License for License. */
/* web: http://www.cise.ufl.edu/research/sparse/umfpack */
/* -------------------------------------------------------------------------- */
diff --git a/src/C/SuiteSparse/UMFPACK/Source/umf_extend_front.c b/src/C/SuiteSparse/UMFPACK/Source/umf_extend_front.c
index ee17d7f..08cdc38 100644
--- a/src/C/SuiteSparse/UMFPACK/Source/umf_extend_front.c
+++ b/src/C/SuiteSparse/UMFPACK/Source/umf_extend_front.c
@@ -3,7 +3,7 @@
/* ========================================================================== */
/* -------------------------------------------------------------------------- */
-/* UMFPACK Version 5.0, Copyright (c) 1995-2006 by Timothy A. Davis. CISE, */
+/* UMFPACK Copyright (c) Timothy A. Davis, CISE, */
/* Univ. of Florida. All Rights Reserved. See ../Doc/License for License. */
/* web: http://www.cise.ufl.edu/research/sparse/umfpack */
/* -------------------------------------------------------------------------- */
diff --git a/src/C/SuiteSparse/UMFPACK/Source/umf_extend_front.h b/src/C/SuiteSparse/UMFPACK/Source/umf_extend_front.h
index cfb284e..6c51986 100644
--- a/src/C/SuiteSparse/UMFPACK/Source/umf_extend_front.h
+++ b/src/C/SuiteSparse/UMFPACK/Source/umf_extend_front.h
@@ -1,5 +1,5 @@
/* -------------------------------------------------------------------------- */
-/* UMFPACK Version 5.0, Copyright (c) 1995-2006 by Timothy A. Davis. CISE, */
+/* UMFPACK Copyright (c) Timothy A. Davis, CISE, */
/* Univ. of Florida. All Rights Reserved. See ../Doc/License for License. */
/* web: http://www.cise.ufl.edu/research/sparse/umfpack */
/* -------------------------------------------------------------------------- */
diff --git a/src/C/SuiteSparse/UMFPACK/Source/umf_free.c b/src/C/SuiteSparse/UMFPACK/Source/umf_free.c
index 864d3d6..b5321f7 100644
--- a/src/C/SuiteSparse/UMFPACK/Source/umf_free.c
+++ b/src/C/SuiteSparse/UMFPACK/Source/umf_free.c
@@ -3,7 +3,7 @@
/* ========================================================================== */
/* -------------------------------------------------------------------------- */
-/* UMFPACK Version 5.0, Copyright (c) 1995-2006 by Timothy A. Davis. CISE, */
+/* UMFPACK Copyright (c) Timothy A. Davis, CISE, */
/* Univ. of Florida. All Rights Reserved. See ../Doc/License for License. */
/* web: http://www.cise.ufl.edu/research/sparse/umfpack */
/* -------------------------------------------------------------------------- */
diff --git a/src/C/SuiteSparse/UMFPACK/Source/umf_free.h b/src/C/SuiteSparse/UMFPACK/Source/umf_free.h
index 045e40b..cacf331 100644
--- a/src/C/SuiteSparse/UMFPACK/Source/umf_free.h
+++ b/src/C/SuiteSparse/UMFPACK/Source/umf_free.h
@@ -1,5 +1,5 @@
/* -------------------------------------------------------------------------- */
-/* UMFPACK Version 5.0, Copyright (c) 1995-2006 by Timothy A. Davis. CISE, */
+/* UMFPACK Copyright (c) Timothy A. Davis, CISE, */
/* Univ. of Florida. All Rights Reserved. See ../Doc/License for License. */
/* web: http://www.cise.ufl.edu/research/sparse/umfpack */
/* -------------------------------------------------------------------------- */
diff --git a/src/C/SuiteSparse/UMFPACK/Source/umf_fsize.c b/src/C/SuiteSparse/UMFPACK/Source/umf_fsize.c
index 6411bf9..d464071 100644
--- a/src/C/SuiteSparse/UMFPACK/Source/umf_fsize.c
+++ b/src/C/SuiteSparse/UMFPACK/Source/umf_fsize.c
@@ -3,7 +3,7 @@
/* ========================================================================== */
/* -------------------------------------------------------------------------- */
-/* UMFPACK Version 5.0, Copyright (c) 1995-2006 by Timothy A. Davis. CISE, */
+/* UMFPACK Copyright (c) Timothy A. Davis, CISE, */
/* Univ. of Florida. All Rights Reserved. See ../Doc/License for License. */
/* web: http://www.cise.ufl.edu/research/sparse/umfpack */
/* -------------------------------------------------------------------------- */
diff --git a/src/C/SuiteSparse/UMFPACK/Source/umf_fsize.h b/src/C/SuiteSparse/UMFPACK/Source/umf_fsize.h
index 27601be..1bd83d4 100644
--- a/src/C/SuiteSparse/UMFPACK/Source/umf_fsize.h
+++ b/src/C/SuiteSparse/UMFPACK/Source/umf_fsize.h
@@ -1,5 +1,5 @@
/* -------------------------------------------------------------------------- */
-/* UMFPACK Version 5.0, Copyright (c) 1995-2006 by Timothy A. Davis. CISE, */
+/* UMFPACK Copyright (c) Timothy A. Davis, CISE, */
/* Univ. of Florida. All Rights Reserved. See ../Doc/License for License. */
/* web: http://www.cise.ufl.edu/research/sparse/umfpack */
/* -------------------------------------------------------------------------- */
diff --git a/src/C/SuiteSparse/UMFPACK/Source/umf_garbage_collection.c b/src/C/SuiteSparse/UMFPACK/Source/umf_garbage_collection.c
index 15e7163..73c40f1 100644
--- a/src/C/SuiteSparse/UMFPACK/Source/umf_garbage_collection.c
+++ b/src/C/SuiteSparse/UMFPACK/Source/umf_garbage_collection.c
@@ -3,7 +3,7 @@
/* ========================================================================== */
/* -------------------------------------------------------------------------- */
-/* UMFPACK Version 5.0, Copyright (c) 1995-2006 by Timothy A. Davis. CISE, */
+/* UMFPACK Copyright (c) Timothy A. Davis, CISE, */
/* Univ. of Florida. All Rights Reserved. See ../Doc/License for License. */
/* web: http://www.cise.ufl.edu/research/sparse/umfpack */
/* -------------------------------------------------------------------------- */
diff --git a/src/C/SuiteSparse/UMFPACK/Source/umf_garbage_collection.h b/src/C/SuiteSparse/UMFPACK/Source/umf_garbage_collection.h
index 69d25d9..9d2d8b7 100644
--- a/src/C/SuiteSparse/UMFPACK/Source/umf_garbage_collection.h
+++ b/src/C/SuiteSparse/UMFPACK/Source/umf_garbage_collection.h
@@ -1,5 +1,5 @@
/* -------------------------------------------------------------------------- */
-/* UMFPACK Version 5.0, Copyright (c) 1995-2006 by Timothy A. Davis. CISE, */
+/* UMFPACK Copyright (c) Timothy A. Davis, CISE, */
/* Univ. of Florida. All Rights Reserved. See ../Doc/License for License. */
/* web: http://www.cise.ufl.edu/research/sparse/umfpack */
/* -------------------------------------------------------------------------- */
diff --git a/src/C/SuiteSparse/UMFPACK/Source/umf_get_memory.c b/src/C/SuiteSparse/UMFPACK/Source/umf_get_memory.c
index bbef6be..01bc940 100644
--- a/src/C/SuiteSparse/UMFPACK/Source/umf_get_memory.c
+++ b/src/C/SuiteSparse/UMFPACK/Source/umf_get_memory.c
@@ -3,7 +3,7 @@
/* ========================================================================== */
/* -------------------------------------------------------------------------- */
-/* UMFPACK Version 5.0, Copyright (c) 1995-2006 by Timothy A. Davis. CISE, */
+/* UMFPACK Copyright (c) Timothy A. Davis, CISE, */
/* Univ. of Florida. All Rights Reserved. See ../Doc/License for License. */
/* web: http://www.cise.ufl.edu/research/sparse/umfpack */
/* -------------------------------------------------------------------------- */
diff --git a/src/C/SuiteSparse/UMFPACK/Source/umf_get_memory.h b/src/C/SuiteSparse/UMFPACK/Source/umf_get_memory.h
index 6131478..317d97a 100644
--- a/src/C/SuiteSparse/UMFPACK/Source/umf_get_memory.h
+++ b/src/C/SuiteSparse/UMFPACK/Source/umf_get_memory.h
@@ -1,5 +1,5 @@
/* -------------------------------------------------------------------------- */
-/* UMFPACK Version 5.0, Copyright (c) 1995-2006 by Timothy A. Davis. CISE, */
+/* UMFPACK Copyright (c) Timothy A. Davis, CISE, */
/* Univ. of Florida. All Rights Reserved. See ../Doc/License for License. */
/* web: http://www.cise.ufl.edu/research/sparse/umfpack */
/* -------------------------------------------------------------------------- */
diff --git a/src/C/SuiteSparse/UMFPACK/Source/umf_grow_front.c b/src/C/SuiteSparse/UMFPACK/Source/umf_grow_front.c
index 9afa6d6..566b2b7 100644
--- a/src/C/SuiteSparse/UMFPACK/Source/umf_grow_front.c
+++ b/src/C/SuiteSparse/UMFPACK/Source/umf_grow_front.c
@@ -3,7 +3,7 @@
/* ========================================================================== */
/* -------------------------------------------------------------------------- */
-/* UMFPACK Version 5.0, Copyright (c) 1995-2006 by Timothy A. Davis. CISE, */
+/* UMFPACK Copyright (c) Timothy A. Davis, CISE, */
/* Univ. of Florida. All Rights Reserved. See ../Doc/License for License. */
/* web: http://www.cise.ufl.edu/research/sparse/umfpack */
/* -------------------------------------------------------------------------- */
diff --git a/src/C/SuiteSparse/UMFPACK/Source/umf_grow_front.h b/src/C/SuiteSparse/UMFPACK/Source/umf_grow_front.h
index e63acd3..8078897 100644
--- a/src/C/SuiteSparse/UMFPACK/Source/umf_grow_front.h
+++ b/src/C/SuiteSparse/UMFPACK/Source/umf_grow_front.h
@@ -1,5 +1,5 @@
/* -------------------------------------------------------------------------- */
-/* UMFPACK Version 5.0, Copyright (c) 1995-2006 by Timothy A. Davis. CISE, */
+/* UMFPACK Copyright (c) Timothy A. Davis, CISE, */
/* Univ. of Florida. All Rights Reserved. See ../Doc/License for License. */
/* web: http://www.cise.ufl.edu/research/sparse/umfpack */
/* -------------------------------------------------------------------------- */
diff --git a/src/C/SuiteSparse/UMFPACK/Source/umf_init_front.c b/src/C/SuiteSparse/UMFPACK/Source/umf_init_front.c
index b82be9f..c8344db 100644
--- a/src/C/SuiteSparse/UMFPACK/Source/umf_init_front.c
+++ b/src/C/SuiteSparse/UMFPACK/Source/umf_init_front.c
@@ -3,7 +3,7 @@
/* ========================================================================== */
/* -------------------------------------------------------------------------- */
-/* UMFPACK Version 5.0, Copyright (c) 1995-2006 by Timothy A. Davis. CISE, */
+/* UMFPACK Copyright (c) Timothy A. Davis, CISE, */
/* Univ. of Florida. All Rights Reserved. See ../Doc/License for License. */
/* web: http://www.cise.ufl.edu/research/sparse/umfpack */
/* -------------------------------------------------------------------------- */
diff --git a/src/C/SuiteSparse/UMFPACK/Source/umf_init_front.h b/src/C/SuiteSparse/UMFPACK/Source/umf_init_front.h
index b7115fb..9801b72 100644
--- a/src/C/SuiteSparse/UMFPACK/Source/umf_init_front.h
+++ b/src/C/SuiteSparse/UMFPACK/Source/umf_init_front.h
@@ -1,5 +1,5 @@
/* -------------------------------------------------------------------------- */
-/* UMFPACK Version 5.0, Copyright (c) 1995-2006 by Timothy A. Davis. CISE, */
+/* UMFPACK Copyright (c) Timothy A. Davis, CISE, */
/* Univ. of Florida. All Rights Reserved. See ../Doc/License for License. */
/* web: http://www.cise.ufl.edu/research/sparse/umfpack */
/* -------------------------------------------------------------------------- */
diff --git a/src/C/SuiteSparse/UMFPACK/Source/umf_internal.h b/src/C/SuiteSparse/UMFPACK/Source/umf_internal.h
index 7c231e8..cd46b03 100644
--- a/src/C/SuiteSparse/UMFPACK/Source/umf_internal.h
+++ b/src/C/SuiteSparse/UMFPACK/Source/umf_internal.h
@@ -3,7 +3,7 @@
/* ========================================================================== */
/* -------------------------------------------------------------------------- */
-/* UMFPACK Version 5.0, Copyright (c) 1995-2006 by Timothy A. Davis. CISE, */
+/* UMFPACK Copyright (c) Timothy A. Davis, CISE, */
/* Univ. of Florida. All Rights Reserved. See ../Doc/License for License. */
/* web: http://www.cise.ufl.edu/research/sparse/umfpack */
/* -------------------------------------------------------------------------- */
diff --git a/src/C/SuiteSparse/UMFPACK/Source/umf_is_permutation.c b/src/C/SuiteSparse/UMFPACK/Source/umf_is_permutation.c
index 05ff965..7120736 100644
--- a/src/C/SuiteSparse/UMFPACK/Source/umf_is_permutation.c
+++ b/src/C/SuiteSparse/UMFPACK/Source/umf_is_permutation.c
@@ -3,7 +3,7 @@
/* ========================================================================== */
/* -------------------------------------------------------------------------- */
-/* UMFPACK Version 5.0, Copyright (c) 1995-2006 by Timothy A. Davis. CISE, */
+/* UMFPACK Copyright (c) Timothy A. Davis, CISE, */
/* Univ. of Florida. All Rights Reserved. See ../Doc/License for License. */
/* web: http://www.cise.ufl.edu/research/sparse/umfpack */
/* -------------------------------------------------------------------------- */
diff --git a/src/C/SuiteSparse/UMFPACK/Source/umf_is_permutation.h b/src/C/SuiteSparse/UMFPACK/Source/umf_is_permutation.h
index 89256bd..2468545 100644
--- a/src/C/SuiteSparse/UMFPACK/Source/umf_is_permutation.h
+++ b/src/C/SuiteSparse/UMFPACK/Source/umf_is_permutation.h
@@ -1,5 +1,5 @@
/* -------------------------------------------------------------------------- */
-/* UMFPACK Version 5.0, Copyright (c) 1995-2006 by Timothy A. Davis. CISE, */
+/* UMFPACK Copyright (c) Timothy A. Davis, CISE, */
/* Univ. of Florida. All Rights Reserved. See ../Doc/License for License. */
/* web: http://www.cise.ufl.edu/research/sparse/umfpack */
/* -------------------------------------------------------------------------- */
diff --git a/src/C/SuiteSparse/UMFPACK/Source/umf_kernel.c b/src/C/SuiteSparse/UMFPACK/Source/umf_kernel.c
index e94522c..54e756f 100644
--- a/src/C/SuiteSparse/UMFPACK/Source/umf_kernel.c
+++ b/src/C/SuiteSparse/UMFPACK/Source/umf_kernel.c
@@ -3,7 +3,7 @@
/* ========================================================================== */
/* -------------------------------------------------------------------------- */
-/* UMFPACK Version 5.0, Copyright (c) 1995-2006 by Timothy A. Davis. CISE, */
+/* UMFPACK Copyright (c) Timothy A. Davis, CISE, */
/* Univ. of Florida. All Rights Reserved. See ../Doc/License for License. */
/* web: http://www.cise.ufl.edu/research/sparse/umfpack */
/* -------------------------------------------------------------------------- */
diff --git a/src/C/SuiteSparse/UMFPACK/Source/umf_kernel.h b/src/C/SuiteSparse/UMFPACK/Source/umf_kernel.h
index 4acd2ae..0f5d70d 100644
--- a/src/C/SuiteSparse/UMFPACK/Source/umf_kernel.h
+++ b/src/C/SuiteSparse/UMFPACK/Source/umf_kernel.h
@@ -1,5 +1,5 @@
/* -------------------------------------------------------------------------- */
-/* UMFPACK Version 5.0, Copyright (c) 1995-2006 by Timothy A. Davis. CISE, */
+/* UMFPACK Copyright (c) Timothy A. Davis, CISE, */
/* Univ. of Florida. All Rights Reserved. See ../Doc/License for License. */
/* web: http://www.cise.ufl.edu/research/sparse/umfpack */
/* -------------------------------------------------------------------------- */
diff --git a/src/C/SuiteSparse/UMFPACK/Source/umf_kernel_init.c b/src/C/SuiteSparse/UMFPACK/Source/umf_kernel_init.c
index b15db9c..7ec3ed6 100644
--- a/src/C/SuiteSparse/UMFPACK/Source/umf_kernel_init.c
+++ b/src/C/SuiteSparse/UMFPACK/Source/umf_kernel_init.c
@@ -3,7 +3,7 @@
/* ========================================================================== */
/* -------------------------------------------------------------------------- */
-/* UMFPACK Version 5.0, Copyright (c) 1995-2006 by Timothy A. Davis. CISE, */
+/* UMFPACK Copyright (c) Timothy A. Davis, CISE, */
/* Univ. of Florida. All Rights Reserved. See ../Doc/License for License. */
/* web: http://www.cise.ufl.edu/research/sparse/umfpack */
/* -------------------------------------------------------------------------- */
diff --git a/src/C/SuiteSparse/UMFPACK/Source/umf_kernel_init.h b/src/C/SuiteSparse/UMFPACK/Source/umf_kernel_init.h
index 312e8e8..dc86d04 100644
--- a/src/C/SuiteSparse/UMFPACK/Source/umf_kernel_init.h
+++ b/src/C/SuiteSparse/UMFPACK/Source/umf_kernel_init.h
@@ -1,5 +1,5 @@
/* -------------------------------------------------------------------------- */
-/* UMFPACK Version 5.0, Copyright (c) 1995-2006 by Timothy A. Davis. CISE, */
+/* UMFPACK Copyright (c) Timothy A. Davis, CISE, */
/* Univ. of Florida. All Rights Reserved. See ../Doc/License for License. */
/* web: http://www.cise.ufl.edu/research/sparse/umfpack */
/* -------------------------------------------------------------------------- */
diff --git a/src/C/SuiteSparse/UMFPACK/Source/umf_kernel_wrapup.c b/src/C/SuiteSparse/UMFPACK/Source/umf_kernel_wrapup.c
index 98f2821..1609c2c 100644
--- a/src/C/SuiteSparse/UMFPACK/Source/umf_kernel_wrapup.c
+++ b/src/C/SuiteSparse/UMFPACK/Source/umf_kernel_wrapup.c
@@ -3,7 +3,7 @@
/* ========================================================================== */
/* -------------------------------------------------------------------------- */
-/* UMFPACK Version 5.0, Copyright (c) 1995-2006 by Timothy A. Davis. CISE, */
+/* UMFPACK Copyright (c) Timothy A. Davis, CISE, */
/* Univ. of Florida. All Rights Reserved. See ../Doc/License for License. */
/* web: http://www.cise.ufl.edu/research/sparse/umfpack */
/* -------------------------------------------------------------------------- */
diff --git a/src/C/SuiteSparse/UMFPACK/Source/umf_kernel_wrapup.h b/src/C/SuiteSparse/UMFPACK/Source/umf_kernel_wrapup.h
index 2cee302..e3e8fbd 100644
--- a/src/C/SuiteSparse/UMFPACK/Source/umf_kernel_wrapup.h
+++ b/src/C/SuiteSparse/UMFPACK/Source/umf_kernel_wrapup.h
@@ -1,5 +1,5 @@
/* -------------------------------------------------------------------------- */
-/* UMFPACK Version 5.0, Copyright (c) 1995-2006 by Timothy A. Davis. CISE, */
+/* UMFPACK Copyright (c) Timothy A. Davis, CISE, */
/* Univ. of Florida. All Rights Reserved. See ../Doc/License for License. */
/* web: http://www.cise.ufl.edu/research/sparse/umfpack */
/* -------------------------------------------------------------------------- */
diff --git a/src/C/SuiteSparse/UMFPACK/Source/umf_local_search.c b/src/C/SuiteSparse/UMFPACK/Source/umf_local_search.c
index 6bc59a0..d81491b 100644
--- a/src/C/SuiteSparse/UMFPACK/Source/umf_local_search.c
+++ b/src/C/SuiteSparse/UMFPACK/Source/umf_local_search.c
@@ -3,7 +3,7 @@
/* ========================================================================== */
/* -------------------------------------------------------------------------- */
-/* UMFPACK Version 5.0, Copyright (c) 1995-2006 by Timothy A. Davis. CISE, */
+/* UMFPACK Copyright (c) Timothy A. Davis, CISE, */
/* Univ. of Florida. All Rights Reserved. See ../Doc/License for License. */
/* web: http://www.cise.ufl.edu/research/sparse/umfpack */
/* -------------------------------------------------------------------------- */
diff --git a/src/C/SuiteSparse/UMFPACK/Source/umf_local_search.h b/src/C/SuiteSparse/UMFPACK/Source/umf_local_search.h
index bd039f4..434d66f 100644
--- a/src/C/SuiteSparse/UMFPACK/Source/umf_local_search.h
+++ b/src/C/SuiteSparse/UMFPACK/Source/umf_local_search.h
@@ -1,5 +1,5 @@
/* -------------------------------------------------------------------------- */
-/* UMFPACK Version 5.0, Copyright (c) 1995-2006 by Timothy A. Davis. CISE, */
+/* UMFPACK Copyright (c) Timothy A. Davis, CISE, */
/* Univ. of Florida. All Rights Reserved. See ../Doc/License for License. */
/* web: http://www.cise.ufl.edu/research/sparse/umfpack */
/* -------------------------------------------------------------------------- */
diff --git a/src/C/SuiteSparse/UMFPACK/Source/umf_lsolve.c b/src/C/SuiteSparse/UMFPACK/Source/umf_lsolve.c
index a8c0c2d..cf3bf9f 100644
--- a/src/C/SuiteSparse/UMFPACK/Source/umf_lsolve.c
+++ b/src/C/SuiteSparse/UMFPACK/Source/umf_lsolve.c
@@ -3,7 +3,7 @@
/* ========================================================================== */
/* -------------------------------------------------------------------------- */
-/* UMFPACK Version 5.0, Copyright (c) 1995-2006 by Timothy A. Davis. CISE, */
+/* UMFPACK Copyright (c) Timothy A. Davis, CISE, */
/* Univ. of Florida. All Rights Reserved. See ../Doc/License for License. */
/* web: http://www.cise.ufl.edu/research/sparse/umfpack */
/* -------------------------------------------------------------------------- */
diff --git a/src/C/SuiteSparse/UMFPACK/Source/umf_lsolve.h b/src/C/SuiteSparse/UMFPACK/Source/umf_lsolve.h
index 187b950..211adff 100644
--- a/src/C/SuiteSparse/UMFPACK/Source/umf_lsolve.h
+++ b/src/C/SuiteSparse/UMFPACK/Source/umf_lsolve.h
@@ -1,5 +1,5 @@
/* -------------------------------------------------------------------------- */
-/* UMFPACK Version 5.0, Copyright (c) 1995-2006 by Timothy A. Davis. CISE, */
+/* UMFPACK Copyright (c) Timothy A. Davis, CISE, */
/* Univ. of Florida. All Rights Reserved. See ../Doc/License for License. */
/* web: http://www.cise.ufl.edu/research/sparse/umfpack */
/* -------------------------------------------------------------------------- */
diff --git a/src/C/SuiteSparse/UMFPACK/Source/umf_ltsolve.c b/src/C/SuiteSparse/UMFPACK/Source/umf_ltsolve.c
index 383cf9b..b259d27 100644
--- a/src/C/SuiteSparse/UMFPACK/Source/umf_ltsolve.c
+++ b/src/C/SuiteSparse/UMFPACK/Source/umf_ltsolve.c
@@ -3,7 +3,7 @@
/* ========================================================================== */
/* -------------------------------------------------------------------------- */
-/* UMFPACK Version 5.0, Copyright (c) 1995-2006 by Timothy A. Davis. CISE, */
+/* UMFPACK Copyright (c) Timothy A. Davis, CISE, */
/* Univ. of Florida. All Rights Reserved. See ../Doc/License for License. */
/* web: http://www.cise.ufl.edu/research/sparse/umfpack */
/* -------------------------------------------------------------------------- */
diff --git a/src/C/SuiteSparse/UMFPACK/Source/umf_ltsolve.h b/src/C/SuiteSparse/UMFPACK/Source/umf_ltsolve.h
index ba5995a..02f7734 100644
--- a/src/C/SuiteSparse/UMFPACK/Source/umf_ltsolve.h
+++ b/src/C/SuiteSparse/UMFPACK/Source/umf_ltsolve.h
@@ -1,5 +1,5 @@
/* -------------------------------------------------------------------------- */
-/* UMFPACK Version 5.0, Copyright (c) 1995-2006 by Timothy A. Davis. CISE, */
+/* UMFPACK Copyright (c) Timothy A. Davis, CISE, */
/* Univ. of Florida. All Rights Reserved. See ../Doc/License for License. */
/* web: http://www.cise.ufl.edu/research/sparse/umfpack */
/* -------------------------------------------------------------------------- */
diff --git a/src/C/SuiteSparse/UMFPACK/Source/umf_malloc.c b/src/C/SuiteSparse/UMFPACK/Source/umf_malloc.c
index 97c43fd..09ef770 100644
--- a/src/C/SuiteSparse/UMFPACK/Source/umf_malloc.c
+++ b/src/C/SuiteSparse/UMFPACK/Source/umf_malloc.c
@@ -3,7 +3,7 @@
/* ========================================================================== */
/* -------------------------------------------------------------------------- */
-/* UMFPACK Version 5.0, Copyright (c) 1995-2006 by Timothy A. Davis. CISE, */
+/* UMFPACK Copyright (c) Timothy A. Davis, CISE, */
/* Univ. of Florida. All Rights Reserved. See ../Doc/License for License. */
/* web: http://www.cise.ufl.edu/research/sparse/umfpack */
/* -------------------------------------------------------------------------- */
diff --git a/src/C/SuiteSparse/UMFPACK/Source/umf_malloc.h b/src/C/SuiteSparse/UMFPACK/Source/umf_malloc.h
index 486a491..8ee9934 100644
--- a/src/C/SuiteSparse/UMFPACK/Source/umf_malloc.h
+++ b/src/C/SuiteSparse/UMFPACK/Source/umf_malloc.h
@@ -1,5 +1,5 @@
/* -------------------------------------------------------------------------- */
-/* UMFPACK Version 5.0, Copyright (c) 1995-2006 by Timothy A. Davis. CISE, */
+/* UMFPACK Copyright (c) Timothy A. Davis, CISE, */
/* Univ. of Florida. All Rights Reserved. See ../Doc/License for License. */
/* web: http://www.cise.ufl.edu/research/sparse/umfpack */
/* -------------------------------------------------------------------------- */
diff --git a/src/C/SuiteSparse/UMFPACK/Source/umf_mem_alloc_element.c b/src/C/SuiteSparse/UMFPACK/Source/umf_mem_alloc_element.c
index 73553bd..cef3ba7 100644
--- a/src/C/SuiteSparse/UMFPACK/Source/umf_mem_alloc_element.c
+++ b/src/C/SuiteSparse/UMFPACK/Source/umf_mem_alloc_element.c
@@ -3,7 +3,7 @@
/* ========================================================================== */
/* -------------------------------------------------------------------------- */
-/* UMFPACK Version 5.0, Copyright (c) 1995-2006 by Timothy A. Davis. CISE, */
+/* UMFPACK Copyright (c) Timothy A. Davis, CISE, */
/* Univ. of Florida. All Rights Reserved. See ../Doc/License for License. */
/* web: http://www.cise.ufl.edu/research/sparse/umfpack */
/* -------------------------------------------------------------------------- */
diff --git a/src/C/SuiteSparse/UMFPACK/Source/umf_mem_alloc_element.h b/src/C/SuiteSparse/UMFPACK/Source/umf_mem_alloc_element.h
index 4463071..e0b259f 100644
--- a/src/C/SuiteSparse/UMFPACK/Source/umf_mem_alloc_element.h
+++ b/src/C/SuiteSparse/UMFPACK/Source/umf_mem_alloc_element.h
@@ -1,5 +1,5 @@
/* -------------------------------------------------------------------------- */
-/* UMFPACK Version 5.0, Copyright (c) 1995-2006 by Timothy A. Davis. CISE, */
+/* UMFPACK Copyright (c) Timothy A. Davis, CISE, */
/* Univ. of Florida. All Rights Reserved. See ../Doc/License for License. */
/* web: http://www.cise.ufl.edu/research/sparse/umfpack */
/* -------------------------------------------------------------------------- */
diff --git a/src/C/SuiteSparse/UMFPACK/Source/umf_mem_alloc_head_block.c b/src/C/SuiteSparse/UMFPACK/Source/umf_mem_alloc_head_block.c
index bc9cd22..2600182 100644
--- a/src/C/SuiteSparse/UMFPACK/Source/umf_mem_alloc_head_block.c
+++ b/src/C/SuiteSparse/UMFPACK/Source/umf_mem_alloc_head_block.c
@@ -3,7 +3,7 @@
/* ========================================================================== */
/* -------------------------------------------------------------------------- */
-/* UMFPACK Version 5.0, Copyright (c) 1995-2006 by Timothy A. Davis. CISE, */
+/* UMFPACK Copyright (c) Timothy A. Davis, CISE, */
/* Univ. of Florida. All Rights Reserved. See ../Doc/License for License. */
/* web: http://www.cise.ufl.edu/research/sparse/umfpack */
/* -------------------------------------------------------------------------- */
diff --git a/src/C/SuiteSparse/UMFPACK/Source/umf_mem_alloc_head_block.h b/src/C/SuiteSparse/UMFPACK/Source/umf_mem_alloc_head_block.h
index d861330..510a6fa 100644
--- a/src/C/SuiteSparse/UMFPACK/Source/umf_mem_alloc_head_block.h
+++ b/src/C/SuiteSparse/UMFPACK/Source/umf_mem_alloc_head_block.h
@@ -1,5 +1,5 @@
/* -------------------------------------------------------------------------- */
-/* UMFPACK Version 5.0, Copyright (c) 1995-2006 by Timothy A. Davis. CISE, */
+/* UMFPACK Copyright (c) Timothy A. Davis, CISE, */
/* Univ. of Florida. All Rights Reserved. See ../Doc/License for License. */
/* web: http://www.cise.ufl.edu/research/sparse/umfpack */
/* -------------------------------------------------------------------------- */
diff --git a/src/C/SuiteSparse/UMFPACK/Source/umf_mem_alloc_tail_block.c b/src/C/SuiteSparse/UMFPACK/Source/umf_mem_alloc_tail_block.c
index afe8ee1..8499644 100644
--- a/src/C/SuiteSparse/UMFPACK/Source/umf_mem_alloc_tail_block.c
+++ b/src/C/SuiteSparse/UMFPACK/Source/umf_mem_alloc_tail_block.c
@@ -3,7 +3,7 @@
/* ========================================================================== */
/* -------------------------------------------------------------------------- */
-/* UMFPACK Version 5.0, Copyright (c) 1995-2006 by Timothy A. Davis. CISE, */
+/* UMFPACK Copyright (c) Timothy A. Davis, CISE, */
/* Univ. of Florida. All Rights Reserved. See ../Doc/License for License. */
/* web: http://www.cise.ufl.edu/research/sparse/umfpack */
/* -------------------------------------------------------------------------- */
diff --git a/src/C/SuiteSparse/UMFPACK/Source/umf_mem_alloc_tail_block.h b/src/C/SuiteSparse/UMFPACK/Source/umf_mem_alloc_tail_block.h
index 91ae285..d072a73 100644
--- a/src/C/SuiteSparse/UMFPACK/Source/umf_mem_alloc_tail_block.h
+++ b/src/C/SuiteSparse/UMFPACK/Source/umf_mem_alloc_tail_block.h
@@ -1,5 +1,5 @@
/* -------------------------------------------------------------------------- */
-/* UMFPACK Version 5.0, Copyright (c) 1995-2006 by Timothy A. Davis. CISE, */
+/* UMFPACK Copyright (c) Timothy A. Davis, CISE, */
/* Univ. of Florida. All Rights Reserved. See ../Doc/License for License. */
/* web: http://www.cise.ufl.edu/research/sparse/umfpack */
/* -------------------------------------------------------------------------- */
diff --git a/src/C/SuiteSparse/UMFPACK/Source/umf_mem_free_tail_block.c b/src/C/SuiteSparse/UMFPACK/Source/umf_mem_free_tail_block.c
index 6aea23a..05a6313 100644
--- a/src/C/SuiteSparse/UMFPACK/Source/umf_mem_free_tail_block.c
+++ b/src/C/SuiteSparse/UMFPACK/Source/umf_mem_free_tail_block.c
@@ -3,7 +3,7 @@
/* ========================================================================== */
/* -------------------------------------------------------------------------- */
-/* UMFPACK Version 5.0, Copyright (c) 1995-2006 by Timothy A. Davis. CISE, */
+/* UMFPACK Copyright (c) Timothy A. Davis, CISE, */
/* Univ. of Florida. All Rights Reserved. See ../Doc/License for License. */
/* web: http://www.cise.ufl.edu/research/sparse/umfpack */
/* -------------------------------------------------------------------------- */
diff --git a/src/C/SuiteSparse/UMFPACK/Source/umf_mem_free_tail_block.h b/src/C/SuiteSparse/UMFPACK/Source/umf_mem_free_tail_block.h
index 15c4c3b..14c7324 100644
--- a/src/C/SuiteSparse/UMFPACK/Source/umf_mem_free_tail_block.h
+++ b/src/C/SuiteSparse/UMFPACK/Source/umf_mem_free_tail_block.h
@@ -1,5 +1,5 @@
/* -------------------------------------------------------------------------- */
-/* UMFPACK Version 5.0, Copyright (c) 1995-2006 by Timothy A. Davis. CISE, */
+/* UMFPACK Copyright (c) Timothy A. Davis, CISE, */
/* Univ. of Florida. All Rights Reserved. See ../Doc/License for License. */
/* web: http://www.cise.ufl.edu/research/sparse/umfpack */
/* -------------------------------------------------------------------------- */
diff --git a/src/C/SuiteSparse/UMFPACK/Source/umf_mem_init_memoryspace.c b/src/C/SuiteSparse/UMFPACK/Source/umf_mem_init_memoryspace.c
index 3b4e0c0..8981dd2 100644
--- a/src/C/SuiteSparse/UMFPACK/Source/umf_mem_init_memoryspace.c
+++ b/src/C/SuiteSparse/UMFPACK/Source/umf_mem_init_memoryspace.c
@@ -3,7 +3,7 @@
/* ========================================================================== */
/* -------------------------------------------------------------------------- */
-/* UMFPACK Version 5.0, Copyright (c) 1995-2006 by Timothy A. Davis. CISE, */
+/* UMFPACK Copyright (c) Timothy A. Davis, CISE, */
/* Univ. of Florida. All Rights Reserved. See ../Doc/License for License. */
/* web: http://www.cise.ufl.edu/research/sparse/umfpack */
/* -------------------------------------------------------------------------- */
diff --git a/src/C/SuiteSparse/UMFPACK/Source/umf_mem_init_memoryspace.h b/src/C/SuiteSparse/UMFPACK/Source/umf_mem_init_memoryspace.h
index 30a09b8..45d552f 100644
--- a/src/C/SuiteSparse/UMFPACK/Source/umf_mem_init_memoryspace.h
+++ b/src/C/SuiteSparse/UMFPACK/Source/umf_mem_init_memoryspace.h
@@ -1,5 +1,5 @@
/* -------------------------------------------------------------------------- */
-/* UMFPACK Version 5.0, Copyright (c) 1995-2006 by Timothy A. Davis. CISE, */
+/* UMFPACK Copyright (c) Timothy A. Davis, CISE, */
/* Univ. of Florida. All Rights Reserved. See ../Doc/License for License. */
/* web: http://www.cise.ufl.edu/research/sparse/umfpack */
/* -------------------------------------------------------------------------- */
diff --git a/src/C/SuiteSparse/UMFPACK/Source/umf_multicompile.c b/src/C/SuiteSparse/UMFPACK/Source/umf_multicompile.c
index ffafef3..0603560 100644
--- a/src/C/SuiteSparse/UMFPACK/Source/umf_multicompile.c
+++ b/src/C/SuiteSparse/UMFPACK/Source/umf_multicompile.c
@@ -3,7 +3,7 @@
/* ========================================================================== */
/* -------------------------------------------------------------------------- */
-/* UMFPACK Version 5.0, Copyright (c) 1995-2006 by Timothy A. Davis. CISE, */
+/* UMFPACK Copyright (c) Timothy A. Davis, CISE, */
/* Univ. of Florida. All Rights Reserved. See ../Doc/License for License. */
/* web: http://www.cise.ufl.edu/research/sparse/umfpack */
/* -------------------------------------------------------------------------- */
diff --git a/src/C/SuiteSparse/UMFPACK/Source/umf_realloc.c b/src/C/SuiteSparse/UMFPACK/Source/umf_realloc.c
index 8504d0b..ee0f70d 100644
--- a/src/C/SuiteSparse/UMFPACK/Source/umf_realloc.c
+++ b/src/C/SuiteSparse/UMFPACK/Source/umf_realloc.c
@@ -3,7 +3,7 @@
/* ========================================================================== */
/* -------------------------------------------------------------------------- */
-/* UMFPACK Version 5.0, Copyright (c) 1995-2006 by Timothy A. Davis. CISE, */
+/* UMFPACK Copyright (c) Timothy A. Davis, CISE, */
/* Univ. of Florida. All Rights Reserved. See ../Doc/License for License. */
/* web: http://www.cise.ufl.edu/research/sparse/umfpack */
/* -------------------------------------------------------------------------- */
diff --git a/src/C/SuiteSparse/UMFPACK/Source/umf_realloc.h b/src/C/SuiteSparse/UMFPACK/Source/umf_realloc.h
index b55c30e..5b2f1c2 100644
--- a/src/C/SuiteSparse/UMFPACK/Source/umf_realloc.h
+++ b/src/C/SuiteSparse/UMFPACK/Source/umf_realloc.h
@@ -1,5 +1,5 @@
/* -------------------------------------------------------------------------- */
-/* UMFPACK Version 5.0, Copyright (c) 1995-2006 by Timothy A. Davis. CISE, */
+/* UMFPACK Copyright (c) Timothy A. Davis, CISE, */
/* Univ. of Florida. All Rights Reserved. See ../Doc/License for License. */
/* web: http://www.cise.ufl.edu/research/sparse/umfpack */
/* -------------------------------------------------------------------------- */
diff --git a/src/C/SuiteSparse/UMFPACK/Source/umf_report_perm.c b/src/C/SuiteSparse/UMFPACK/Source/umf_report_perm.c
index bc25faa..c97052c 100644
--- a/src/C/SuiteSparse/UMFPACK/Source/umf_report_perm.c
+++ b/src/C/SuiteSparse/UMFPACK/Source/umf_report_perm.c
@@ -3,7 +3,7 @@
/* ========================================================================== */
/* -------------------------------------------------------------------------- */
-/* UMFPACK Version 5.0, Copyright (c) 1995-2006 by Timothy A. Davis. CISE, */
+/* UMFPACK Copyright (c) Timothy A. Davis, CISE, */
/* Univ. of Florida. All Rights Reserved. See ../Doc/License for License. */
/* web: http://www.cise.ufl.edu/research/sparse/umfpack */
/* -------------------------------------------------------------------------- */
diff --git a/src/C/SuiteSparse/UMFPACK/Source/umf_report_perm.h b/src/C/SuiteSparse/UMFPACK/Source/umf_report_perm.h
index dd81936..6b18d85 100644
--- a/src/C/SuiteSparse/UMFPACK/Source/umf_report_perm.h
+++ b/src/C/SuiteSparse/UMFPACK/Source/umf_report_perm.h
@@ -1,5 +1,5 @@
/* -------------------------------------------------------------------------- */
-/* UMFPACK Version 5.0, Copyright (c) 1995-2006 by Timothy A. Davis. CISE, */
+/* UMFPACK Copyright (c) Timothy A. Davis, CISE, */
/* Univ. of Florida. All Rights Reserved. See ../Doc/License for License. */
/* web: http://www.cise.ufl.edu/research/sparse/umfpack */
/* -------------------------------------------------------------------------- */
diff --git a/src/C/SuiteSparse/UMFPACK/Source/umf_report_vector.c b/src/C/SuiteSparse/UMFPACK/Source/umf_report_vector.c
index 7587204..4b602d1 100644
--- a/src/C/SuiteSparse/UMFPACK/Source/umf_report_vector.c
+++ b/src/C/SuiteSparse/UMFPACK/Source/umf_report_vector.c
@@ -3,7 +3,7 @@
/* ========================================================================== */
/* -------------------------------------------------------------------------- */
-/* UMFPACK Version 5.0, Copyright (c) 1995-2006 by Timothy A. Davis. CISE, */
+/* UMFPACK Copyright (c) Timothy A. Davis, CISE, */
/* Univ. of Florida. All Rights Reserved. See ../Doc/License for License. */
/* web: http://www.cise.ufl.edu/research/sparse/umfpack */
/* -------------------------------------------------------------------------- */
diff --git a/src/C/SuiteSparse/UMFPACK/Source/umf_report_vector.h b/src/C/SuiteSparse/UMFPACK/Source/umf_report_vector.h
index 665a8d4..6a758d7 100644
--- a/src/C/SuiteSparse/UMFPACK/Source/umf_report_vector.h
+++ b/src/C/SuiteSparse/UMFPACK/Source/umf_report_vector.h
@@ -1,5 +1,5 @@
/* -------------------------------------------------------------------------- */
-/* UMFPACK Version 5.0, Copyright (c) 1995-2006 by Timothy A. Davis. CISE, */
+/* UMFPACK Copyright (c) Timothy A. Davis, CISE, */
/* Univ. of Florida. All Rights Reserved. See ../Doc/License for License. */
/* web: http://www.cise.ufl.edu/research/sparse/umfpack */
/* -------------------------------------------------------------------------- */
diff --git a/src/C/SuiteSparse/UMFPACK/Source/umf_row_search.c b/src/C/SuiteSparse/UMFPACK/Source/umf_row_search.c
index aefb36d..f20ca0c 100644
--- a/src/C/SuiteSparse/UMFPACK/Source/umf_row_search.c
+++ b/src/C/SuiteSparse/UMFPACK/Source/umf_row_search.c
@@ -3,7 +3,7 @@
/* ========================================================================== */
/* -------------------------------------------------------------------------- */
-/* UMFPACK Version 5.0, Copyright (c) 1995-2006 by Timothy A. Davis. CISE, */
+/* UMFPACK Copyright (c) Timothy A. Davis, CISE, */
/* Univ. of Florida. All Rights Reserved. See ../Doc/License for License. */
/* web: http://www.cise.ufl.edu/research/sparse/umfpack */
/* -------------------------------------------------------------------------- */
diff --git a/src/C/SuiteSparse/UMFPACK/Source/umf_row_search.h b/src/C/SuiteSparse/UMFPACK/Source/umf_row_search.h
index 8a9dac0..e0171ee 100644
--- a/src/C/SuiteSparse/UMFPACK/Source/umf_row_search.h
+++ b/src/C/SuiteSparse/UMFPACK/Source/umf_row_search.h
@@ -1,5 +1,5 @@
/* -------------------------------------------------------------------------- */
-/* UMFPACK Version 5.0, Copyright (c) 1995-2006 by Timothy A. Davis. CISE, */
+/* UMFPACK Copyright (c) Timothy A. Davis, CISE, */
/* Univ. of Florida. All Rights Reserved. See ../Doc/License for License. */
/* web: http://www.cise.ufl.edu/research/sparse/umfpack */
/* -------------------------------------------------------------------------- */
diff --git a/src/C/SuiteSparse/UMFPACK/Source/umf_scale.c b/src/C/SuiteSparse/UMFPACK/Source/umf_scale.c
index c716bee..98fb3ba 100644
--- a/src/C/SuiteSparse/UMFPACK/Source/umf_scale.c
+++ b/src/C/SuiteSparse/UMFPACK/Source/umf_scale.c
@@ -3,7 +3,7 @@
/* ========================================================================== */
/* -------------------------------------------------------------------------- */
-/* UMFPACK Version 5.0, Copyright (c) 1995-2006 by Timothy A. Davis. CISE, */
+/* UMFPACK Copyright (c) Timothy A. Davis, CISE, */
/* Univ. of Florida. All Rights Reserved. See ../Doc/License for License. */
/* web: http://www.cise.ufl.edu/research/sparse/umfpack */
/* -------------------------------------------------------------------------- */
diff --git a/src/C/SuiteSparse/UMFPACK/Source/umf_scale.h b/src/C/SuiteSparse/UMFPACK/Source/umf_scale.h
index 5b24afb..99799f5 100644
--- a/src/C/SuiteSparse/UMFPACK/Source/umf_scale.h
+++ b/src/C/SuiteSparse/UMFPACK/Source/umf_scale.h
@@ -1,5 +1,5 @@
/* -------------------------------------------------------------------------- */
-/* UMFPACK Version 5.0, Copyright (c) 1995-2006 by Timothy A. Davis. CISE, */
+/* UMFPACK Copyright (c) Timothy A. Davis, CISE, */
/* Univ. of Florida. All Rights Reserved. See ../Doc/License for License. */
/* web: http://www.cise.ufl.edu/research/sparse/umfpack */
/* -------------------------------------------------------------------------- */
diff --git a/src/C/SuiteSparse/UMFPACK/Source/umf_scale_column.c b/src/C/SuiteSparse/UMFPACK/Source/umf_scale_column.c
index ace38bc..782b716 100644
--- a/src/C/SuiteSparse/UMFPACK/Source/umf_scale_column.c
+++ b/src/C/SuiteSparse/UMFPACK/Source/umf_scale_column.c
@@ -3,7 +3,7 @@
/* ========================================================================== */
/* -------------------------------------------------------------------------- */
-/* UMFPACK Version 5.0, Copyright (c) 1995-2006 by Timothy A. Davis. CISE, */
+/* UMFPACK Copyright (c) Timothy A. Davis, CISE, */
/* Univ. of Florida. All Rights Reserved. See ../Doc/License for License. */
/* web: http://www.cise.ufl.edu/research/sparse/umfpack */
/* -------------------------------------------------------------------------- */
diff --git a/src/C/SuiteSparse/UMFPACK/Source/umf_scale_column.h b/src/C/SuiteSparse/UMFPACK/Source/umf_scale_column.h
index 166a4e5..2306787 100644
--- a/src/C/SuiteSparse/UMFPACK/Source/umf_scale_column.h
+++ b/src/C/SuiteSparse/UMFPACK/Source/umf_scale_column.h
@@ -1,5 +1,5 @@
/* -------------------------------------------------------------------------- */
-/* UMFPACK Version 5.0, Copyright (c) 1995-2006 by Timothy A. Davis. CISE, */
+/* UMFPACK Copyright (c) Timothy A. Davis, CISE, */
/* Univ. of Florida. All Rights Reserved. See ../Doc/License for License. */
/* web: http://www.cise.ufl.edu/research/sparse/umfpack */
/* -------------------------------------------------------------------------- */
diff --git a/src/C/SuiteSparse/UMFPACK/Source/umf_set_stats.c b/src/C/SuiteSparse/UMFPACK/Source/umf_set_stats.c
index 1458763..8dfe2ca 100644
--- a/src/C/SuiteSparse/UMFPACK/Source/umf_set_stats.c
+++ b/src/C/SuiteSparse/UMFPACK/Source/umf_set_stats.c
@@ -3,7 +3,7 @@
/* ========================================================================== */
/* -------------------------------------------------------------------------- */
-/* UMFPACK Version 5.0, Copyright (c) 1995-2006 by Timothy A. Davis. CISE, */
+/* UMFPACK Copyright (c) Timothy A. Davis, CISE, */
/* Univ. of Florida. All Rights Reserved. See ../Doc/License for License. */
/* web: http://www.cise.ufl.edu/research/sparse/umfpack */
/* -------------------------------------------------------------------------- */
diff --git a/src/C/SuiteSparse/UMFPACK/Source/umf_set_stats.h b/src/C/SuiteSparse/UMFPACK/Source/umf_set_stats.h
index deb654d..f4342ce 100644
--- a/src/C/SuiteSparse/UMFPACK/Source/umf_set_stats.h
+++ b/src/C/SuiteSparse/UMFPACK/Source/umf_set_stats.h
@@ -1,5 +1,5 @@
/* -------------------------------------------------------------------------- */
-/* UMFPACK Version 5.0, Copyright (c) 1995-2006 by Timothy A. Davis. CISE, */
+/* UMFPACK Copyright (c) Timothy A. Davis, CISE, */
/* Univ. of Florida. All Rights Reserved. See ../Doc/License for License. */
/* web: http://www.cise.ufl.edu/research/sparse/umfpack */
/* -------------------------------------------------------------------------- */
diff --git a/src/C/SuiteSparse/UMFPACK/Source/umf_singletons.c b/src/C/SuiteSparse/UMFPACK/Source/umf_singletons.c
index 88b12db..9d1617d 100644
--- a/src/C/SuiteSparse/UMFPACK/Source/umf_singletons.c
+++ b/src/C/SuiteSparse/UMFPACK/Source/umf_singletons.c
@@ -3,7 +3,7 @@
/* ========================================================================== */
/* -------------------------------------------------------------------------- */
-/* UMFPACK Version 5.0, Copyright (c) 1995-2006 by Timothy A. Davis. CISE, */
+/* UMFPACK Copyright (c) Timothy A. Davis, CISE, */
/* Univ. of Florida. All Rights Reserved. See ../Doc/License for License. */
/* web: http://www.cise.ufl.edu/research/sparse/umfpack */
/* -------------------------------------------------------------------------- */
@@ -13,27 +13,28 @@
* that the matrix is in the following form:
*
* x x x x x x x x x
- * 0 x x x x x x x x
- * 0 0 x x x x x x x
- * 0 0 0 x 0 0 0 0 0
- * 0 0 0 x x 0 0 0 0
- * 0 0 0 x x s s s s
- * 0 0 0 x x s s s s
- * 0 0 0 x x s s s s
- * 0 0 0 x x s s s s
+ * . x x x x x x x x
+ * . . x x x x x x x
+ * . . . x . . . . .
+ * . . . x x . . . .
+ * . . . x x s s s s
+ * . . . x x s s s s
+ * . . . x x s s s s
+ * . . . x x s s s s
*
* The above example has 3 column singletons (the first three columns and
* their corresponding pivot rows) and 2 row singletons. The singletons are
* ordered first, because they have zero Markowitz cost. The LU factorization
* for these first five rows and columns is free - there is no work to do
* (except to scale the pivot columns for the 2 row singletons), and no
- * fill-in occurs. * The remaining * submatrix (4-by-4 in the above example)
+ * fill-in occurs. The remaining submatrix (4-by-4 in the above example)
* has no rows or columns with degree one. It may have empty rows or columns.
*
* This algorithm does not perform a full permutation to block triangular
* form. If there are one or more singletons, then the matrix can be
* permuted to block triangular form, but UMFPACK does not perform the full
- * BTF permutation (see also "dmperm" in MATLAB).
+ * BTF permutation (see also "dmperm" in MATLAB, CSparse cs_dmperm,
+ * and SuiteSparse/BTF).
*/
#include "umf_internal.h"
diff --git a/src/C/SuiteSparse/UMFPACK/Source/umf_singletons.h b/src/C/SuiteSparse/UMFPACK/Source/umf_singletons.h
index 1d3316f..5b2e498 100644
--- a/src/C/SuiteSparse/UMFPACK/Source/umf_singletons.h
+++ b/src/C/SuiteSparse/UMFPACK/Source/umf_singletons.h
@@ -1,5 +1,5 @@
/* -------------------------------------------------------------------------- */
-/* UMFPACK Version 5.0, Copyright (c) 1995-2006 by Timothy A. Davis. CISE, */
+/* UMFPACK Copyright (c) Timothy A. Davis, CISE, */
/* Univ. of Florida. All Rights Reserved. See ../Doc/License for License. */
/* web: http://www.cise.ufl.edu/research/sparse/umfpack */
/* -------------------------------------------------------------------------- */
diff --git a/src/C/SuiteSparse/UMFPACK/Source/umf_solve.c b/src/C/SuiteSparse/UMFPACK/Source/umf_solve.c
index 8f37ec0..fc78586 100644
--- a/src/C/SuiteSparse/UMFPACK/Source/umf_solve.c
+++ b/src/C/SuiteSparse/UMFPACK/Source/umf_solve.c
@@ -3,7 +3,7 @@
/* ========================================================================== */
/* -------------------------------------------------------------------------- */
-/* UMFPACK Version 5.0, Copyright (c) 1995-2006 by Timothy A. Davis. CISE, */
+/* UMFPACK Copyright (c) Timothy A. Davis, CISE, */
/* Univ. of Florida. All Rights Reserved. See ../Doc/License for License. */
/* web: http://www.cise.ufl.edu/research/sparse/umfpack */
/* -------------------------------------------------------------------------- */
diff --git a/src/C/SuiteSparse/UMFPACK/Source/umf_solve.h b/src/C/SuiteSparse/UMFPACK/Source/umf_solve.h
index 6eb1cf0..80a796c 100644
--- a/src/C/SuiteSparse/UMFPACK/Source/umf_solve.h
+++ b/src/C/SuiteSparse/UMFPACK/Source/umf_solve.h
@@ -1,5 +1,5 @@
/* -------------------------------------------------------------------------- */
-/* UMFPACK Version 5.0, Copyright (c) 1995-2006 by Timothy A. Davis. CISE, */
+/* UMFPACK Copyright (c) Timothy A. Davis, CISE, */
/* Univ. of Florida. All Rights Reserved. See ../Doc/License for License. */
/* web: http://www.cise.ufl.edu/research/sparse/umfpack */
/* -------------------------------------------------------------------------- */
diff --git a/src/C/SuiteSparse/UMFPACK/Source/umf_start_front.c b/src/C/SuiteSparse/UMFPACK/Source/umf_start_front.c
index cff6ad4..aa581d4 100644
--- a/src/C/SuiteSparse/UMFPACK/Source/umf_start_front.c
+++ b/src/C/SuiteSparse/UMFPACK/Source/umf_start_front.c
@@ -3,7 +3,7 @@
/* ========================================================================== */
/* -------------------------------------------------------------------------- */
-/* UMFPACK Version 5.0, Copyright (c) 1995-2006 by Timothy A. Davis. CISE, */
+/* UMFPACK Copyright (c) Timothy A. Davis, CISE, */
/* Univ. of Florida. All Rights Reserved. See ../Doc/License for License. */
/* web: http://www.cise.ufl.edu/research/sparse/umfpack */
/* -------------------------------------------------------------------------- */
diff --git a/src/C/SuiteSparse/UMFPACK/Source/umf_start_front.h b/src/C/SuiteSparse/UMFPACK/Source/umf_start_front.h
index d6d63b3..1a1c4f2 100644
--- a/src/C/SuiteSparse/UMFPACK/Source/umf_start_front.h
+++ b/src/C/SuiteSparse/UMFPACK/Source/umf_start_front.h
@@ -1,5 +1,5 @@
/* -------------------------------------------------------------------------- */
-/* UMFPACK Version 5.0, Copyright (c) 1995-2006 by Timothy A. Davis. CISE, */
+/* UMFPACK Copyright (c) Timothy A. Davis, CISE, */
/* Univ. of Florida. All Rights Reserved. See ../Doc/License for License. */
/* web: http://www.cise.ufl.edu/research/sparse/umfpack */
/* -------------------------------------------------------------------------- */
diff --git a/src/C/SuiteSparse/UMFPACK/Source/umf_store_lu.c b/src/C/SuiteSparse/UMFPACK/Source/umf_store_lu.c
index 4235917..0d0ea09 100644
--- a/src/C/SuiteSparse/UMFPACK/Source/umf_store_lu.c
+++ b/src/C/SuiteSparse/UMFPACK/Source/umf_store_lu.c
@@ -3,7 +3,7 @@
/* ========================================================================== */
/* -------------------------------------------------------------------------- */
-/* UMFPACK Version 5.0, Copyright (c) 1995-2006 by Timothy A. Davis. CISE, */
+/* UMFPACK Copyright (c) Timothy A. Davis, CISE, */
/* Univ. of Florida. All Rights Reserved. See ../Doc/License for License. */
/* web: http://www.cise.ufl.edu/research/sparse/umfpack */
/* -------------------------------------------------------------------------- */
diff --git a/src/C/SuiteSparse/UMFPACK/Source/umf_store_lu.h b/src/C/SuiteSparse/UMFPACK/Source/umf_store_lu.h
index b78528f..82627ad 100644
--- a/src/C/SuiteSparse/UMFPACK/Source/umf_store_lu.h
+++ b/src/C/SuiteSparse/UMFPACK/Source/umf_store_lu.h
@@ -1,5 +1,5 @@
/* -------------------------------------------------------------------------- */
-/* UMFPACK Version 5.0, Copyright (c) 1995-2006 by Timothy A. Davis. CISE, */
+/* UMFPACK Copyright (c) Timothy A. Davis, CISE, */
/* Univ. of Florida. All Rights Reserved. See ../Doc/License for License. */
/* web: http://www.cise.ufl.edu/research/sparse/umfpack */
/* -------------------------------------------------------------------------- */
diff --git a/src/C/SuiteSparse/UMFPACK/Source/umf_symbolic_usage.c b/src/C/SuiteSparse/UMFPACK/Source/umf_symbolic_usage.c
index dff210e..8c417cd 100644
--- a/src/C/SuiteSparse/UMFPACK/Source/umf_symbolic_usage.c
+++ b/src/C/SuiteSparse/UMFPACK/Source/umf_symbolic_usage.c
@@ -3,7 +3,7 @@
/* ========================================================================== */
/* -------------------------------------------------------------------------- */
-/* UMFPACK Version 5.0, Copyright (c) 1995-2006 by Timothy A. Davis. CISE, */
+/* UMFPACK Copyright (c) Timothy A. Davis, CISE, */
/* Univ. of Florida. All Rights Reserved. See ../Doc/License for License. */
/* web: http://www.cise.ufl.edu/research/sparse/umfpack */
/* -------------------------------------------------------------------------- */
diff --git a/src/C/SuiteSparse/UMFPACK/Source/umf_symbolic_usage.h b/src/C/SuiteSparse/UMFPACK/Source/umf_symbolic_usage.h
index fbe979a..3fad24a 100644
--- a/src/C/SuiteSparse/UMFPACK/Source/umf_symbolic_usage.h
+++ b/src/C/SuiteSparse/UMFPACK/Source/umf_symbolic_usage.h
@@ -1,5 +1,5 @@
/* -------------------------------------------------------------------------- */
-/* UMFPACK Version 5.0, Copyright (c) 1995-2006 by Timothy A. Davis. CISE, */
+/* UMFPACK Copyright (c) Timothy A. Davis, CISE, */
/* Univ. of Florida. All Rights Reserved. See ../Doc/License for License. */
/* web: http://www.cise.ufl.edu/research/sparse/umfpack */
/* -------------------------------------------------------------------------- */
diff --git a/src/C/SuiteSparse/UMFPACK/Source/umf_transpose.c b/src/C/SuiteSparse/UMFPACK/Source/umf_transpose.c
index 85cd34d..061c97b 100644
--- a/src/C/SuiteSparse/UMFPACK/Source/umf_transpose.c
+++ b/src/C/SuiteSparse/UMFPACK/Source/umf_transpose.c
@@ -3,7 +3,7 @@
/* ========================================================================== */
/* -------------------------------------------------------------------------- */
-/* UMFPACK Version 5.0, Copyright (c) 1995-2006 by Timothy A. Davis. CISE, */
+/* UMFPACK Copyright (c) Timothy A. Davis, CISE, */
/* Univ. of Florida. All Rights Reserved. See ../Doc/License for License. */
/* web: http://www.cise.ufl.edu/research/sparse/umfpack */
/* -------------------------------------------------------------------------- */
diff --git a/src/C/SuiteSparse/UMFPACK/Source/umf_transpose.h b/src/C/SuiteSparse/UMFPACK/Source/umf_transpose.h
index 5d44106..4c4dee9 100644
--- a/src/C/SuiteSparse/UMFPACK/Source/umf_transpose.h
+++ b/src/C/SuiteSparse/UMFPACK/Source/umf_transpose.h
@@ -1,5 +1,5 @@
/* -------------------------------------------------------------------------- */
-/* UMFPACK Version 5.0, Copyright (c) 1995-2006 by Timothy A. Davis. CISE, */
+/* UMFPACK Copyright (c) Timothy A. Davis, CISE, */
/* Univ. of Florida. All Rights Reserved. See ../Doc/License for License. */
/* web: http://www.cise.ufl.edu/research/sparse/umfpack */
/* -------------------------------------------------------------------------- */
diff --git a/src/C/SuiteSparse/UMFPACK/Source/umf_triplet.c b/src/C/SuiteSparse/UMFPACK/Source/umf_triplet.c
index 363e7a4..b1f0b30 100644
--- a/src/C/SuiteSparse/UMFPACK/Source/umf_triplet.c
+++ b/src/C/SuiteSparse/UMFPACK/Source/umf_triplet.c
@@ -3,7 +3,7 @@
/* ========================================================================== */
/* -------------------------------------------------------------------------- */
-/* UMFPACK Version 5.0, Copyright (c) 1995-2006 by Timothy A. Davis. CISE, */
+/* UMFPACK Copyright (c) Timothy A. Davis, CISE, */
/* Univ. of Florida. All Rights Reserved. See ../Doc/License for License. */
/* web: http://www.cise.ufl.edu/research/sparse/umfpack */
/* -------------------------------------------------------------------------- */
diff --git a/src/C/SuiteSparse/UMFPACK/Source/umf_triplet.h b/src/C/SuiteSparse/UMFPACK/Source/umf_triplet.h
index 5a3a9d9..62f9729 100644
--- a/src/C/SuiteSparse/UMFPACK/Source/umf_triplet.h
+++ b/src/C/SuiteSparse/UMFPACK/Source/umf_triplet.h
@@ -1,5 +1,5 @@
/* -------------------------------------------------------------------------- */
-/* UMFPACK Version 5.0, Copyright (c) 1995-2006 by Timothy A. Davis. CISE, */
+/* UMFPACK Copyright (c) Timothy A. Davis, CISE, */
/* Univ. of Florida. All Rights Reserved. See ../Doc/License for License. */
/* web: http://www.cise.ufl.edu/research/sparse/umfpack */
/* -------------------------------------------------------------------------- */
diff --git a/src/C/SuiteSparse/UMFPACK/Source/umf_tuple_lengths.c b/src/C/SuiteSparse/UMFPACK/Source/umf_tuple_lengths.c
index cd8f9ea..c0680de 100644
--- a/src/C/SuiteSparse/UMFPACK/Source/umf_tuple_lengths.c
+++ b/src/C/SuiteSparse/UMFPACK/Source/umf_tuple_lengths.c
@@ -3,7 +3,7 @@
/* ========================================================================== */
/* -------------------------------------------------------------------------- */
-/* UMFPACK Version 5.0, Copyright (c) 1995-2006 by Timothy A. Davis. CISE, */
+/* UMFPACK Copyright (c) Timothy A. Davis, CISE, */
/* Univ. of Florida. All Rights Reserved. See ../Doc/License for License. */
/* web: http://www.cise.ufl.edu/research/sparse/umfpack */
/* -------------------------------------------------------------------------- */
diff --git a/src/C/SuiteSparse/UMFPACK/Source/umf_tuple_lengths.h b/src/C/SuiteSparse/UMFPACK/Source/umf_tuple_lengths.h
index e859092..a31ccb9 100644
--- a/src/C/SuiteSparse/UMFPACK/Source/umf_tuple_lengths.h
+++ b/src/C/SuiteSparse/UMFPACK/Source/umf_tuple_lengths.h
@@ -1,5 +1,5 @@
/* -------------------------------------------------------------------------- */
-/* UMFPACK Version 5.0, Copyright (c) 1995-2006 by Timothy A. Davis. CISE, */
+/* UMFPACK Copyright (c) Timothy A. Davis, CISE, */
/* Univ. of Florida. All Rights Reserved. See ../Doc/License for License. */
/* web: http://www.cise.ufl.edu/research/sparse/umfpack */
/* -------------------------------------------------------------------------- */
diff --git a/src/C/SuiteSparse/UMFPACK/Source/umf_usolve.c b/src/C/SuiteSparse/UMFPACK/Source/umf_usolve.c
index 043ebc4..ca2e4ad 100644
--- a/src/C/SuiteSparse/UMFPACK/Source/umf_usolve.c
+++ b/src/C/SuiteSparse/UMFPACK/Source/umf_usolve.c
@@ -3,7 +3,7 @@
/* ========================================================================== */
/* -------------------------------------------------------------------------- */
-/* UMFPACK Version 5.0, Copyright (c) 1995-2006 by Timothy A. Davis. CISE, */
+/* UMFPACK Copyright (c) Timothy A. Davis, CISE, */
/* Univ. of Florida. All Rights Reserved. See ../Doc/License for License. */
/* web: http://www.cise.ufl.edu/research/sparse/umfpack */
/* -------------------------------------------------------------------------- */
diff --git a/src/C/SuiteSparse/UMFPACK/Source/umf_usolve.h b/src/C/SuiteSparse/UMFPACK/Source/umf_usolve.h
index 5055278..3b950d3 100644
--- a/src/C/SuiteSparse/UMFPACK/Source/umf_usolve.h
+++ b/src/C/SuiteSparse/UMFPACK/Source/umf_usolve.h
@@ -1,5 +1,5 @@
/* -------------------------------------------------------------------------- */
-/* UMFPACK Version 5.0, Copyright (c) 1995-2006 by Timothy A. Davis. CISE, */
+/* UMFPACK Copyright (c) Timothy A. Davis, CISE, */
/* Univ. of Florida. All Rights Reserved. See ../Doc/License for License. */
/* web: http://www.cise.ufl.edu/research/sparse/umfpack */
/* -------------------------------------------------------------------------- */
diff --git a/src/C/SuiteSparse/UMFPACK/Source/umf_utsolve.c b/src/C/SuiteSparse/UMFPACK/Source/umf_utsolve.c
index fb3f4a8..ba78364 100644
--- a/src/C/SuiteSparse/UMFPACK/Source/umf_utsolve.c
+++ b/src/C/SuiteSparse/UMFPACK/Source/umf_utsolve.c
@@ -3,7 +3,7 @@
/* ========================================================================== */
/* -------------------------------------------------------------------------- */
-/* UMFPACK Version 5.0, Copyright (c) 1995-2006 by Timothy A. Davis. CISE, */
+/* UMFPACK Copyright (c) Timothy A. Davis, CISE, */
/* Univ. of Florida. All Rights Reserved. See ../Doc/License for License. */
/* web: http://www.cise.ufl.edu/research/sparse/umfpack */
/* -------------------------------------------------------------------------- */
diff --git a/src/C/SuiteSparse/UMFPACK/Source/umf_utsolve.h b/src/C/SuiteSparse/UMFPACK/Source/umf_utsolve.h
index 59c0685..dae3143 100644
--- a/src/C/SuiteSparse/UMFPACK/Source/umf_utsolve.h
+++ b/src/C/SuiteSparse/UMFPACK/Source/umf_utsolve.h
@@ -1,5 +1,5 @@
/* -------------------------------------------------------------------------- */
-/* UMFPACK Version 5.0, Copyright (c) 1995-2006 by Timothy A. Davis. CISE, */
+/* UMFPACK Copyright (c) Timothy A. Davis, CISE, */
/* Univ. of Florida. All Rights Reserved. See ../Doc/License for License. */
/* web: http://www.cise.ufl.edu/research/sparse/umfpack */
/* -------------------------------------------------------------------------- */
diff --git a/src/C/SuiteSparse/UMFPACK/Source/umf_valid_numeric.c b/src/C/SuiteSparse/UMFPACK/Source/umf_valid_numeric.c
index e43ebab..ea3bb07 100644
--- a/src/C/SuiteSparse/UMFPACK/Source/umf_valid_numeric.c
+++ b/src/C/SuiteSparse/UMFPACK/Source/umf_valid_numeric.c
@@ -3,7 +3,7 @@
/* ========================================================================== */
/* -------------------------------------------------------------------------- */
-/* UMFPACK Version 5.0, Copyright (c) 1995-2006 by Timothy A. Davis. CISE, */
+/* UMFPACK Copyright (c) Timothy A. Davis, CISE, */
/* Univ. of Florida. All Rights Reserved. See ../Doc/License for License. */
/* web: http://www.cise.ufl.edu/research/sparse/umfpack */
/* -------------------------------------------------------------------------- */
diff --git a/src/C/SuiteSparse/UMFPACK/Source/umf_valid_numeric.h b/src/C/SuiteSparse/UMFPACK/Source/umf_valid_numeric.h
index 6d99218..0b258e4 100644
--- a/src/C/SuiteSparse/UMFPACK/Source/umf_valid_numeric.h
+++ b/src/C/SuiteSparse/UMFPACK/Source/umf_valid_numeric.h
@@ -1,5 +1,5 @@
/* -------------------------------------------------------------------------- */
-/* UMFPACK Version 5.0, Copyright (c) 1995-2006 by Timothy A. Davis. CISE, */
+/* UMFPACK Copyright (c) Timothy A. Davis, CISE, */
/* Univ. of Florida. All Rights Reserved. See ../Doc/License for License. */
/* web: http://www.cise.ufl.edu/research/sparse/umfpack */
/* -------------------------------------------------------------------------- */
diff --git a/src/C/SuiteSparse/UMFPACK/Source/umf_valid_symbolic.c b/src/C/SuiteSparse/UMFPACK/Source/umf_valid_symbolic.c
index 396d548..2f65a0c 100644
--- a/src/C/SuiteSparse/UMFPACK/Source/umf_valid_symbolic.c
+++ b/src/C/SuiteSparse/UMFPACK/Source/umf_valid_symbolic.c
@@ -3,7 +3,7 @@
/* ========================================================================== */
/* -------------------------------------------------------------------------- */
-/* UMFPACK Version 5.0, Copyright (c) 1995-2006 by Timothy A. Davis. CISE, */
+/* UMFPACK Copyright (c) Timothy A. Davis, CISE, */
/* Univ. of Florida. All Rights Reserved. See ../Doc/License for License. */
/* web: http://www.cise.ufl.edu/research/sparse/umfpack */
/* -------------------------------------------------------------------------- */
diff --git a/src/C/SuiteSparse/UMFPACK/Source/umf_valid_symbolic.h b/src/C/SuiteSparse/UMFPACK/Source/umf_valid_symbolic.h
index 906f70c..741a799 100644
--- a/src/C/SuiteSparse/UMFPACK/Source/umf_valid_symbolic.h
+++ b/src/C/SuiteSparse/UMFPACK/Source/umf_valid_symbolic.h
@@ -1,5 +1,5 @@
/* -------------------------------------------------------------------------- */
-/* UMFPACK Version 5.0, Copyright (c) 1995-2006 by Timothy A. Davis. CISE, */
+/* UMFPACK Copyright (c) Timothy A. Davis, CISE, */
/* Univ. of Florida. All Rights Reserved. See ../Doc/License for License. */
/* web: http://www.cise.ufl.edu/research/sparse/umfpack */
/* -------------------------------------------------------------------------- */
diff --git a/src/C/SuiteSparse/UMFPACK/Source/umf_version.h b/src/C/SuiteSparse/UMFPACK/Source/umf_version.h
index d5fe456..98d765f 100644
--- a/src/C/SuiteSparse/UMFPACK/Source/umf_version.h
+++ b/src/C/SuiteSparse/UMFPACK/Source/umf_version.h
@@ -3,7 +3,7 @@
/* ========================================================================== */
/* -------------------------------------------------------------------------- */
-/* UMFPACK Version 5.0, Copyright (c) 1995-2006 by Timothy A. Davis. CISE, */
+/* UMFPACK Copyright (c) Timothy A. Davis, CISE, */
/* Univ. of Florida. All Rights Reserved. See ../Doc/License for License. */
/* web: http://www.cise.ufl.edu/research/sparse/umfpack */
/* -------------------------------------------------------------------------- */
diff --git a/src/C/SuiteSparse/UMFPACK/Source/umfpack_col_to_triplet.c b/src/C/SuiteSparse/UMFPACK/Source/umfpack_col_to_triplet.c
index 3075e2a..f49a13d 100644
--- a/src/C/SuiteSparse/UMFPACK/Source/umfpack_col_to_triplet.c
+++ b/src/C/SuiteSparse/UMFPACK/Source/umfpack_col_to_triplet.c
@@ -3,7 +3,7 @@
/* ========================================================================== */
/* -------------------------------------------------------------------------- */
-/* UMFPACK Version 5.0, Copyright (c) 1995-2006 by Timothy A. Davis. CISE, */
+/* UMFPACK Copyright (c) Timothy A. Davis, CISE, */
/* Univ. of Florida. All Rights Reserved. See ../Doc/License for License. */
/* web: http://www.cise.ufl.edu/research/sparse/umfpack */
/* -------------------------------------------------------------------------- */
diff --git a/src/C/SuiteSparse/UMFPACK/Source/umfpack_defaults.c b/src/C/SuiteSparse/UMFPACK/Source/umfpack_defaults.c
index b8205b7..761ecd1 100644
--- a/src/C/SuiteSparse/UMFPACK/Source/umfpack_defaults.c
+++ b/src/C/SuiteSparse/UMFPACK/Source/umfpack_defaults.c
@@ -3,7 +3,7 @@
/* ========================================================================== */
/* -------------------------------------------------------------------------- */
-/* UMFPACK Version 5.0, Copyright (c) 1995-2006 by Timothy A. Davis. CISE, */
+/* UMFPACK Copyright (c) Timothy A. Davis, CISE, */
/* Univ. of Florida. All Rights Reserved. See ../Doc/License for License. */
/* web: http://www.cise.ufl.edu/research/sparse/umfpack */
/* -------------------------------------------------------------------------- */
diff --git a/src/C/SuiteSparse/UMFPACK/Source/umfpack_free_numeric.c b/src/C/SuiteSparse/UMFPACK/Source/umfpack_free_numeric.c
index 36a6af1..7e4e531 100644
--- a/src/C/SuiteSparse/UMFPACK/Source/umfpack_free_numeric.c
+++ b/src/C/SuiteSparse/UMFPACK/Source/umfpack_free_numeric.c
@@ -3,7 +3,7 @@
/* ========================================================================== */
/* -------------------------------------------------------------------------- */
-/* UMFPACK Version 5.0, Copyright (c) 1995-2006 by Timothy A. Davis. CISE, */
+/* UMFPACK Copyright (c) Timothy A. Davis, CISE, */
/* Univ. of Florida. All Rights Reserved. See ../Doc/License for License. */
/* web: http://www.cise.ufl.edu/research/sparse/umfpack */
/* -------------------------------------------------------------------------- */
diff --git a/src/C/SuiteSparse/UMFPACK/Source/umfpack_free_symbolic.c b/src/C/SuiteSparse/UMFPACK/Source/umfpack_free_symbolic.c
index ccd4834..b865c43 100644
--- a/src/C/SuiteSparse/UMFPACK/Source/umfpack_free_symbolic.c
+++ b/src/C/SuiteSparse/UMFPACK/Source/umfpack_free_symbolic.c
@@ -3,7 +3,7 @@
/* ========================================================================== */
/* -------------------------------------------------------------------------- */
-/* UMFPACK Version 5.0, Copyright (c) 1995-2006 by Timothy A. Davis. CISE, */
+/* UMFPACK Copyright (c) Timothy A. Davis, CISE, */
/* Univ. of Florida. All Rights Reserved. See ../Doc/License for License. */
/* web: http://www.cise.ufl.edu/research/sparse/umfpack */
/* -------------------------------------------------------------------------- */
diff --git a/src/C/SuiteSparse/UMFPACK/Source/umfpack_get_determinant.c b/src/C/SuiteSparse/UMFPACK/Source/umfpack_get_determinant.c
index 0f47e79..bfeacaa 100644
--- a/src/C/SuiteSparse/UMFPACK/Source/umfpack_get_determinant.c
+++ b/src/C/SuiteSparse/UMFPACK/Source/umfpack_get_determinant.c
@@ -3,7 +3,7 @@
/* ========================================================================== */
/* -------------------------------------------------------------------------- */
-/* UMFPACK Version 5.0, Copyright (c) 1995-2006 by Timothy A. Davis. CISE, */
+/* UMFPACK Copyright (c) Timothy A. Davis, CISE, */
/* Univ. of Florida. All Rights Reserved. See ../Doc/License for License. */
/* web: http://www.cise.ufl.edu/research/sparse/umfpack */
/* UMFPACK_get_determinant contributed by David Bateman, Motorola, Paris. */
diff --git a/src/C/SuiteSparse/UMFPACK/Source/umfpack_get_lunz.c b/src/C/SuiteSparse/UMFPACK/Source/umfpack_get_lunz.c
index 2bfc714..21752d3 100644
--- a/src/C/SuiteSparse/UMFPACK/Source/umfpack_get_lunz.c
+++ b/src/C/SuiteSparse/UMFPACK/Source/umfpack_get_lunz.c
@@ -3,7 +3,7 @@
/* ========================================================================== */
/* -------------------------------------------------------------------------- */
-/* UMFPACK Version 5.0, Copyright (c) 1995-2006 by Timothy A. Davis. CISE, */
+/* UMFPACK Copyright (c) Timothy A. Davis, CISE, */
/* Univ. of Florida. All Rights Reserved. See ../Doc/License for License. */
/* web: http://www.cise.ufl.edu/research/sparse/umfpack */
/* -------------------------------------------------------------------------- */
diff --git a/src/C/SuiteSparse/UMFPACK/Source/umfpack_get_numeric.c b/src/C/SuiteSparse/UMFPACK/Source/umfpack_get_numeric.c
index 8028a86..9dd2208 100644
--- a/src/C/SuiteSparse/UMFPACK/Source/umfpack_get_numeric.c
+++ b/src/C/SuiteSparse/UMFPACK/Source/umfpack_get_numeric.c
@@ -3,7 +3,7 @@
/* ========================================================================== */
/* -------------------------------------------------------------------------- */
-/* UMFPACK Version 5.0, Copyright (c) 1995-2006 by Timothy A. Davis. CISE, */
+/* UMFPACK Copyright (c) Timothy A. Davis, CISE, */
/* Univ. of Florida. All Rights Reserved. See ../Doc/License for License. */
/* web: http://www.cise.ufl.edu/research/sparse/umfpack */
/* -------------------------------------------------------------------------- */
diff --git a/src/C/SuiteSparse/UMFPACK/Source/umfpack_get_symbolic.c b/src/C/SuiteSparse/UMFPACK/Source/umfpack_get_symbolic.c
index 9363534..560d38c 100644
--- a/src/C/SuiteSparse/UMFPACK/Source/umfpack_get_symbolic.c
+++ b/src/C/SuiteSparse/UMFPACK/Source/umfpack_get_symbolic.c
@@ -3,7 +3,7 @@
/* ========================================================================== */
/* -------------------------------------------------------------------------- */
-/* UMFPACK Version 5.0, Copyright (c) 1995-2006 by Timothy A. Davis. CISE, */
+/* UMFPACK Copyright (c) Timothy A. Davis, CISE, */
/* Univ. of Florida. All Rights Reserved. See ../Doc/License for License. */
/* web: http://www.cise.ufl.edu/research/sparse/umfpack */
/* -------------------------------------------------------------------------- */
diff --git a/src/C/SuiteSparse/UMFPACK/Source/umfpack_global.c b/src/C/SuiteSparse/UMFPACK/Source/umfpack_global.c
index b612c1e..b227573 100644
--- a/src/C/SuiteSparse/UMFPACK/Source/umfpack_global.c
+++ b/src/C/SuiteSparse/UMFPACK/Source/umfpack_global.c
@@ -3,7 +3,7 @@
/* ========================================================================== */
/* -------------------------------------------------------------------------- */
-/* UMFPACK Version 5.0, Copyright (c) 1995-2006 by Timothy A. Davis. CISE, */
+/* UMFPACK Copyright (c) Timothy A. Davis, CISE, */
/* Univ. of Florida. All Rights Reserved. See ../Doc/License for License. */
/* web: http://www.cise.ufl.edu/research/sparse/umfpack */
/* -------------------------------------------------------------------------- */
diff --git a/src/C/SuiteSparse/UMFPACK/Source/umfpack_load_numeric.c b/src/C/SuiteSparse/UMFPACK/Source/umfpack_load_numeric.c
index 18dfcfe..ddd6447 100644
--- a/src/C/SuiteSparse/UMFPACK/Source/umfpack_load_numeric.c
+++ b/src/C/SuiteSparse/UMFPACK/Source/umfpack_load_numeric.c
@@ -3,7 +3,7 @@
/* ========================================================================== */
/* -------------------------------------------------------------------------- */
-/* UMFPACK Version 5.0, Copyright (c) 1995-2006 by Timothy A. Davis. CISE, */
+/* UMFPACK Copyright (c) Timothy A. Davis, CISE, */
/* Univ. of Florida. All Rights Reserved. See ../Doc/License for License. */
/* web: http://www.cise.ufl.edu/research/sparse/umfpack */
/* -------------------------------------------------------------------------- */
diff --git a/src/C/SuiteSparse/UMFPACK/Source/umfpack_load_symbolic.c b/src/C/SuiteSparse/UMFPACK/Source/umfpack_load_symbolic.c
index 6401645..ac3076d 100644
--- a/src/C/SuiteSparse/UMFPACK/Source/umfpack_load_symbolic.c
+++ b/src/C/SuiteSparse/UMFPACK/Source/umfpack_load_symbolic.c
@@ -3,7 +3,7 @@
/* ========================================================================== */
/* -------------------------------------------------------------------------- */
-/* UMFPACK Version 5.0, Copyright (c) 1995-2006 by Timothy A. Davis. CISE, */
+/* UMFPACK Copyright (c) Timothy A. Davis, CISE, */
/* Univ. of Florida. All Rights Reserved. See ../Doc/License for License. */
/* web: http://www.cise.ufl.edu/research/sparse/umfpack */
/* -------------------------------------------------------------------------- */
diff --git a/src/C/SuiteSparse/UMFPACK/Source/umfpack_numeric.c b/src/C/SuiteSparse/UMFPACK/Source/umfpack_numeric.c
index 0a8d67f..a96a9fd 100644
--- a/src/C/SuiteSparse/UMFPACK/Source/umfpack_numeric.c
+++ b/src/C/SuiteSparse/UMFPACK/Source/umfpack_numeric.c
@@ -3,7 +3,7 @@
/* ========================================================================== */
/* -------------------------------------------------------------------------- */
-/* UMFPACK Version 5.0, Copyright (c) 1995-2006 by Timothy A. Davis. CISE, */
+/* UMFPACK Copyright (c) Timothy A. Davis, CISE, */
/* Univ. of Florida. All Rights Reserved. See ../Doc/License for License. */
/* web: http://www.cise.ufl.edu/research/sparse/umfpack */
/* -------------------------------------------------------------------------- */
diff --git a/src/C/SuiteSparse/UMFPACK/Source/umfpack_qsymbolic.c b/src/C/SuiteSparse/UMFPACK/Source/umfpack_qsymbolic.c
index fc7f267..3e2674f 100644
--- a/src/C/SuiteSparse/UMFPACK/Source/umfpack_qsymbolic.c
+++ b/src/C/SuiteSparse/UMFPACK/Source/umfpack_qsymbolic.c
@@ -3,7 +3,7 @@
/* ========================================================================== */
/* -------------------------------------------------------------------------- */
-/* UMFPACK Version 5.0, Copyright (c) 1995-2006 by Timothy A. Davis. CISE, */
+/* UMFPACK Copyright (c) Timothy A. Davis, CISE, */
/* Univ. of Florida. All Rights Reserved. See ../Doc/License for License. */
/* web: http://www.cise.ufl.edu/research/sparse/umfpack */
/* -------------------------------------------------------------------------- */
diff --git a/src/C/SuiteSparse/UMFPACK/Source/umfpack_report_control.c b/src/C/SuiteSparse/UMFPACK/Source/umfpack_report_control.c
index 74a2ed7..30a51b9 100644
--- a/src/C/SuiteSparse/UMFPACK/Source/umfpack_report_control.c
+++ b/src/C/SuiteSparse/UMFPACK/Source/umfpack_report_control.c
@@ -3,7 +3,7 @@
/* ========================================================================== */
/* -------------------------------------------------------------------------- */
-/* UMFPACK Version 5.0, Copyright (c) 1995-2006 by Timothy A. Davis. CISE, */
+/* UMFPACK Copyright (c) Timothy A. Davis, CISE, */
/* Univ. of Florida. All Rights Reserved. See ../Doc/License for License. */
/* web: http://www.cise.ufl.edu/research/sparse/umfpack */
/* -------------------------------------------------------------------------- */
diff --git a/src/C/SuiteSparse/UMFPACK/Source/umfpack_report_info.c b/src/C/SuiteSparse/UMFPACK/Source/umfpack_report_info.c
index ff7818b..ac55466 100644
--- a/src/C/SuiteSparse/UMFPACK/Source/umfpack_report_info.c
+++ b/src/C/SuiteSparse/UMFPACK/Source/umfpack_report_info.c
@@ -3,7 +3,7 @@
/* ========================================================================== */
/* -------------------------------------------------------------------------- */
-/* UMFPACK Version 5.0, Copyright (c) 1995-2006 by Timothy A. Davis. CISE, */
+/* UMFPACK Copyright (c) Timothy A. Davis, CISE, */
/* Univ. of Florida. All Rights Reserved. See ../Doc/License for License. */
/* web: http://www.cise.ufl.edu/research/sparse/umfpack */
/* -------------------------------------------------------------------------- */
diff --git a/src/C/SuiteSparse/UMFPACK/Source/umfpack_report_matrix.c b/src/C/SuiteSparse/UMFPACK/Source/umfpack_report_matrix.c
index 6add888..6074ea4 100644
--- a/src/C/SuiteSparse/UMFPACK/Source/umfpack_report_matrix.c
+++ b/src/C/SuiteSparse/UMFPACK/Source/umfpack_report_matrix.c
@@ -3,7 +3,7 @@
/* ========================================================================== */
/* -------------------------------------------------------------------------- */
-/* UMFPACK Version 5.0, Copyright (c) 1995-2006 by Timothy A. Davis. CISE, */
+/* UMFPACK Copyright (c) Timothy A. Davis, CISE, */
/* Univ. of Florida. All Rights Reserved. See ../Doc/License for License. */
/* web: http://www.cise.ufl.edu/research/sparse/umfpack */
/* -------------------------------------------------------------------------- */
diff --git a/src/C/SuiteSparse/UMFPACK/Source/umfpack_report_numeric.c b/src/C/SuiteSparse/UMFPACK/Source/umfpack_report_numeric.c
index 73a5d24..f70555f 100644
--- a/src/C/SuiteSparse/UMFPACK/Source/umfpack_report_numeric.c
+++ b/src/C/SuiteSparse/UMFPACK/Source/umfpack_report_numeric.c
@@ -3,7 +3,7 @@
/* ========================================================================== */
/* -------------------------------------------------------------------------- */
-/* UMFPACK Version 5.0, Copyright (c) 1995-2006 by Timothy A. Davis. CISE, */
+/* UMFPACK Copyright (c) Timothy A. Davis, CISE, */
/* Univ. of Florida. All Rights Reserved. See ../Doc/License for License. */
/* web: http://www.cise.ufl.edu/research/sparse/umfpack */
/* -------------------------------------------------------------------------- */
diff --git a/src/C/SuiteSparse/UMFPACK/Source/umfpack_report_perm.c b/src/C/SuiteSparse/UMFPACK/Source/umfpack_report_perm.c
index 591122b..5f3baab 100644
--- a/src/C/SuiteSparse/UMFPACK/Source/umfpack_report_perm.c
+++ b/src/C/SuiteSparse/UMFPACK/Source/umfpack_report_perm.c
@@ -3,7 +3,7 @@
/* ========================================================================== */
/* -------------------------------------------------------------------------- */
-/* UMFPACK Version 5.0, Copyright (c) 1995-2006 by Timothy A. Davis. CISE, */
+/* UMFPACK Copyright (c) Timothy A. Davis, CISE, */
/* Univ. of Florida. All Rights Reserved. See ../Doc/License for License. */
/* web: http://www.cise.ufl.edu/research/sparse/umfpack */
/* -------------------------------------------------------------------------- */
diff --git a/src/C/SuiteSparse/UMFPACK/Source/umfpack_report_status.c b/src/C/SuiteSparse/UMFPACK/Source/umfpack_report_status.c
index 068bb7c..5e4e142 100644
--- a/src/C/SuiteSparse/UMFPACK/Source/umfpack_report_status.c
+++ b/src/C/SuiteSparse/UMFPACK/Source/umfpack_report_status.c
@@ -3,7 +3,7 @@
/* ========================================================================== */
/* -------------------------------------------------------------------------- */
-/* UMFPACK Version 5.0, Copyright (c) 1995-2006 by Timothy A. Davis. CISE, */
+/* UMFPACK Copyright (c) Timothy A. Davis, CISE, */
/* Univ. of Florida. All Rights Reserved. See ../Doc/License for License. */
/* web: http://www.cise.ufl.edu/research/sparse/umfpack */
/* -------------------------------------------------------------------------- */
diff --git a/src/C/SuiteSparse/UMFPACK/Source/umfpack_report_symbolic.c b/src/C/SuiteSparse/UMFPACK/Source/umfpack_report_symbolic.c
index 4142834..57e134e 100644
--- a/src/C/SuiteSparse/UMFPACK/Source/umfpack_report_symbolic.c
+++ b/src/C/SuiteSparse/UMFPACK/Source/umfpack_report_symbolic.c
@@ -3,7 +3,7 @@
/* ========================================================================== */
/* -------------------------------------------------------------------------- */
-/* UMFPACK Version 5.0, Copyright (c) 1995-2006 by Timothy A. Davis. CISE, */
+/* UMFPACK Copyright (c) Timothy A. Davis, CISE, */
/* Univ. of Florida. All Rights Reserved. See ../Doc/License for License. */
/* web: http://www.cise.ufl.edu/research/sparse/umfpack */
/* -------------------------------------------------------------------------- */
diff --git a/src/C/SuiteSparse/UMFPACK/Source/umfpack_report_triplet.c b/src/C/SuiteSparse/UMFPACK/Source/umfpack_report_triplet.c
index 6bb6fb8..ba11a69 100644
--- a/src/C/SuiteSparse/UMFPACK/Source/umfpack_report_triplet.c
+++ b/src/C/SuiteSparse/UMFPACK/Source/umfpack_report_triplet.c
@@ -3,7 +3,7 @@
/* ========================================================================== */
/* -------------------------------------------------------------------------- */
-/* UMFPACK Version 5.0, Copyright (c) 1995-2006 by Timothy A. Davis. CISE, */
+/* UMFPACK Copyright (c) Timothy A. Davis, CISE, */
/* Univ. of Florida. All Rights Reserved. See ../Doc/License for License. */
/* web: http://www.cise.ufl.edu/research/sparse/umfpack */
/* -------------------------------------------------------------------------- */
diff --git a/src/C/SuiteSparse/UMFPACK/Source/umfpack_report_vector.c b/src/C/SuiteSparse/UMFPACK/Source/umfpack_report_vector.c
index 3b27d3f..a88804c 100644
--- a/src/C/SuiteSparse/UMFPACK/Source/umfpack_report_vector.c
+++ b/src/C/SuiteSparse/UMFPACK/Source/umfpack_report_vector.c
@@ -3,7 +3,7 @@
/* ========================================================================== */
/* -------------------------------------------------------------------------- */
-/* UMFPACK Version 5.0, Copyright (c) 1995-2006 by Timothy A. Davis. CISE, */
+/* UMFPACK Copyright (c) Timothy A. Davis, CISE, */
/* Univ. of Florida. All Rights Reserved. See ../Doc/License for License. */
/* web: http://www.cise.ufl.edu/research/sparse/umfpack */
/* -------------------------------------------------------------------------- */
diff --git a/src/C/SuiteSparse/UMFPACK/Source/umfpack_save_numeric.c b/src/C/SuiteSparse/UMFPACK/Source/umfpack_save_numeric.c
index 2d8a73f..dc2ae32 100644
--- a/src/C/SuiteSparse/UMFPACK/Source/umfpack_save_numeric.c
+++ b/src/C/SuiteSparse/UMFPACK/Source/umfpack_save_numeric.c
@@ -3,7 +3,7 @@
/* ========================================================================== */
/* -------------------------------------------------------------------------- */
-/* UMFPACK Version 5.0, Copyright (c) 1995-2006 by Timothy A. Davis. CISE, */
+/* UMFPACK Copyright (c) Timothy A. Davis, CISE, */
/* Univ. of Florida. All Rights Reserved. See ../Doc/License for License. */
/* web: http://www.cise.ufl.edu/research/sparse/umfpack */
/* -------------------------------------------------------------------------- */
diff --git a/src/C/SuiteSparse/UMFPACK/Source/umfpack_save_symbolic.c b/src/C/SuiteSparse/UMFPACK/Source/umfpack_save_symbolic.c
index c326b28..7469e04 100644
--- a/src/C/SuiteSparse/UMFPACK/Source/umfpack_save_symbolic.c
+++ b/src/C/SuiteSparse/UMFPACK/Source/umfpack_save_symbolic.c
@@ -3,7 +3,7 @@
/* ========================================================================== */
/* -------------------------------------------------------------------------- */
-/* UMFPACK Version 5.0, Copyright (c) 1995-2006 by Timothy A. Davis. CISE, */
+/* UMFPACK Copyright (c) Timothy A. Davis, CISE, */
/* Univ. of Florida. All Rights Reserved. See ../Doc/License for License. */
/* web: http://www.cise.ufl.edu/research/sparse/umfpack */
/* -------------------------------------------------------------------------- */
diff --git a/src/C/SuiteSparse/UMFPACK/Source/umfpack_scale.c b/src/C/SuiteSparse/UMFPACK/Source/umfpack_scale.c
index 5a08616..49467a5 100644
--- a/src/C/SuiteSparse/UMFPACK/Source/umfpack_scale.c
+++ b/src/C/SuiteSparse/UMFPACK/Source/umfpack_scale.c
@@ -3,7 +3,7 @@
/* ========================================================================== */
/* -------------------------------------------------------------------------- */
-/* UMFPACK Version 5.0, Copyright (c) 1995-2006 by Timothy A. Davis. CISE, */
+/* UMFPACK Copyright (c) Timothy A. Davis, CISE, */
/* Univ. of Florida. All Rights Reserved. See ../Doc/License for License. */
/* web: http://www.cise.ufl.edu/research/sparse/umfpack */
/* -------------------------------------------------------------------------- */
diff --git a/src/C/SuiteSparse/UMFPACK/Source/umfpack_solve.c b/src/C/SuiteSparse/UMFPACK/Source/umfpack_solve.c
index fcdebe2..c4e8263 100644
--- a/src/C/SuiteSparse/UMFPACK/Source/umfpack_solve.c
+++ b/src/C/SuiteSparse/UMFPACK/Source/umfpack_solve.c
@@ -3,7 +3,7 @@
/* ========================================================================== */
/* -------------------------------------------------------------------------- */
-/* UMFPACK Version 5.0, Copyright (c) 1995-2006 by Timothy A. Davis. CISE, */
+/* UMFPACK Copyright (c) Timothy A. Davis, CISE, */
/* Univ. of Florida. All Rights Reserved. See ../Doc/License for License. */
/* web: http://www.cise.ufl.edu/research/sparse/umfpack */
/* -------------------------------------------------------------------------- */
diff --git a/src/C/SuiteSparse/UMFPACK/Source/umfpack_symbolic.c b/src/C/SuiteSparse/UMFPACK/Source/umfpack_symbolic.c
index 39ee106..41be72b 100644
--- a/src/C/SuiteSparse/UMFPACK/Source/umfpack_symbolic.c
+++ b/src/C/SuiteSparse/UMFPACK/Source/umfpack_symbolic.c
@@ -3,7 +3,7 @@
/* ========================================================================== */
/* -------------------------------------------------------------------------- */
-/* UMFPACK Version 5.0, Copyright (c) 1995-2006 by Timothy A. Davis. CISE, */
+/* UMFPACK Copyright (c) Timothy A. Davis, CISE, */
/* Univ. of Florida. All Rights Reserved. See ../Doc/License for License. */
/* web: http://www.cise.ufl.edu/research/sparse/umfpack */
/* -------------------------------------------------------------------------- */
diff --git a/src/C/SuiteSparse/UMFPACK/Source/umfpack_tictoc.c b/src/C/SuiteSparse/UMFPACK/Source/umfpack_tictoc.c
index 605f789..7fc5fba 100644
--- a/src/C/SuiteSparse/UMFPACK/Source/umfpack_tictoc.c
+++ b/src/C/SuiteSparse/UMFPACK/Source/umfpack_tictoc.c
@@ -3,7 +3,7 @@
/* ========================================================================== */
/* -------------------------------------------------------------------------- */
-/* UMFPACK Version 5.0, Copyright (c) 1995-2006 by Timothy A. Davis. CISE, */
+/* UMFPACK Copyright (c) Timothy A. Davis, CISE, */
/* Univ. of Florida. All Rights Reserved. See ../Doc/License for License. */
/* web: http://www.cise.ufl.edu/research/sparse/umfpack */
/* -------------------------------------------------------------------------- */
diff --git a/src/C/SuiteSparse/UMFPACK/Source/umfpack_timer.c b/src/C/SuiteSparse/UMFPACK/Source/umfpack_timer.c
index 7e8d20b..f4f245c 100644
--- a/src/C/SuiteSparse/UMFPACK/Source/umfpack_timer.c
+++ b/src/C/SuiteSparse/UMFPACK/Source/umfpack_timer.c
@@ -3,7 +3,7 @@
/* ========================================================================== */
/* -------------------------------------------------------------------------- */
-/* UMFPACK Version 5.0, Copyright (c) 1995-2006 by Timothy A. Davis. CISE, */
+/* UMFPACK Copyright (c) Timothy A. Davis, CISE, */
/* Univ. of Florida. All Rights Reserved. See ../Doc/License for License. */
/* web: http://www.cise.ufl.edu/research/sparse/umfpack */
/* -------------------------------------------------------------------------- */
diff --git a/src/C/SuiteSparse/UMFPACK/Source/umfpack_transpose.c b/src/C/SuiteSparse/UMFPACK/Source/umfpack_transpose.c
index 4a6b9ba..25ea211 100644
--- a/src/C/SuiteSparse/UMFPACK/Source/umfpack_transpose.c
+++ b/src/C/SuiteSparse/UMFPACK/Source/umfpack_transpose.c
@@ -3,7 +3,7 @@
/* ========================================================================== */
/* -------------------------------------------------------------------------- */
-/* UMFPACK Version 5.0, Copyright (c) 1995-2006 by Timothy A. Davis. CISE, */
+/* UMFPACK Copyright (c) Timothy A. Davis, CISE, */
/* Univ. of Florida. All Rights Reserved. See ../Doc/License for License. */
/* web: http://www.cise.ufl.edu/research/sparse/umfpack */
/* -------------------------------------------------------------------------- */
diff --git a/src/C/SuiteSparse/UMFPACK/Source/umfpack_triplet_to_col.c b/src/C/SuiteSparse/UMFPACK/Source/umfpack_triplet_to_col.c
index 92f5a8d..97750d1 100644
--- a/src/C/SuiteSparse/UMFPACK/Source/umfpack_triplet_to_col.c
+++ b/src/C/SuiteSparse/UMFPACK/Source/umfpack_triplet_to_col.c
@@ -3,7 +3,7 @@
/* ========================================================================== */
/* -------------------------------------------------------------------------- */
-/* UMFPACK Version 5.0, Copyright (c) 1995-2006 by Timothy A. Davis. CISE, */
+/* UMFPACK Copyright (c) Timothy A. Davis, CISE, */
/* Univ. of Florida. All Rights Reserved. See ../Doc/License for License. */
/* web: http://www.cise.ufl.edu/research/sparse/umfpack */
/* -------------------------------------------------------------------------- */
diff --git a/src/C/amd.c b/src/C/amd.c
index 3d1f41d..848c56f 100644
--- a/src/C/amd.c
+++ b/src/C/amd.c
@@ -1,6 +1,20 @@
/*
- * This file is part of CVXOPT version 0.8.2.
* Copyright 2004-2007 J. Dahl and L. Vandenberghe.
+ *
+ * This file is part of CVXOPT version 0.9.
+ *
+ * CVXOPT is free software; you can redistribute it and/or modify
+ * it under the terms of the GNU General Public License as published by
+ * the Free Software Foundation; either version 3 of the License, or
+ * (at your option) any later version.
+ *
+ * CVXOPT is distributed in the hope that it will be useful,
+ * but WITHOUT ANY WARRANTY; without even the implied warranty of
+ * MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the
+ * GNU General Public License for more details.
+ *
+ * You should have received a copy of the GNU General Public License
+ * along with this program. If not, see <http://www.gnu.org/licenses/>.
*/
#include "cvxopt.h"
diff --git a/src/C/base.c b/src/C/base.c
index 7834906..d03e6d2 100644
--- a/src/C/base.c
+++ b/src/C/base.c
@@ -1,6 +1,20 @@
/*
- * This file is part of CVXOPT version 0.8.2.
* Copyright 2004-2007 J. Dahl and L. Vandenberghe.
+ *
+ * This file is part of CVXOPT version 0.9.
+ *
+ * CVXOPT is free software; you can redistribute it and/or modify
+ * it under the terms of the GNU General Public License as published by
+ * the Free Software Foundation; either version 3 of the License, or
+ * (at your option) any later version.
+ *
+ * CVXOPT is distributed in the hope that it will be useful,
+ * but WITHOUT ANY WARRANTY; without even the implied warranty of
+ * MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the
+ * GNU General Public License for more details.
+ *
+ * You should have received a copy of the GNU General Public License
+ * along with this program. If not, see <http://www.gnu.org/licenses/>.
*/
#define BASE_MODULE
diff --git a/src/C/blas.c b/src/C/blas.c
index 825b48c..d4792c7 100644
--- a/src/C/blas.c
+++ b/src/C/blas.c
@@ -1,6 +1,20 @@
/*
- * This file is part of CVXOPT version 0.8.2.
* Copyright 2004-2007 J. Dahl and L. Vandenberghe.
+ *
+ * This file is part of CVXOPT version 0.9.
+ *
+ * CVXOPT is free software; you can redistribute it and/or modify
+ * it under the terms of the GNU General Public License as published by
+ * the Free Software Foundation; either version 3 of the License, or
+ * (at your option) any later version.
+ *
+ * CVXOPT is distributed in the hope that it will be useful,
+ * but WITHOUT ANY WARRANTY; without even the implied warranty of
+ * MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the
+ * GNU General Public License for more details.
+ *
+ * You should have received a copy of the GNU General Public License
+ * along with this program. If not, see <http://www.gnu.org/licenses/>.
*/
#include "Python.h"
@@ -1700,7 +1714,7 @@ static PyObject* ger(PyObject *self, PyObject *args, PyObject *kwrds)
number a;
int m=-1, n=-1, ldA=0, ix=1, iy=1, oA=0, ox=0, oy=0;
char *kwlist[] = {"x", "y", "A", "alpha", "m", "n", "incx", "incy",
- "ldA", "offsetx", "offsety", "offsetZ", NULL};
+ "ldA", "offsetx", "offsety", "offsetA", NULL};
if (!PyArg_ParseTupleAndKeywords(args, kwrds, "OOO|Oiiiiiiii",
kwlist, &x, &y, &A, &ao, &m, &n, &ix, &iy, &ldA, &ox, &oy, &oA))
@@ -1783,7 +1797,7 @@ static PyObject* geru(PyObject *self, PyObject *args, PyObject *kwrds)
number a;
int m=-1, n=-1, ldA=0, ix=1, iy=1, oA=0, ox=0, oy=0;
char *kwlist[] = {"x", "y", "A", "alpha", "m", "n", "incx", "incy",
- "ldA", "offsetx", "offsety", "offsetZ", NULL};
+ "ldA", "offsetx", "offsety", "offsetA", NULL};
if (!PyArg_ParseTupleAndKeywords(args, kwrds, "OOO|Oiiiiiiii",
kwlist, &x, &y, &A, &ao, &m, &n, &ix, &iy, &ldA, &ox, &oy, &oA))
@@ -2185,7 +2199,9 @@ static char doc_gemm[] =
"C := alpha*A*B^H + beta*C if transA = 'N' and transB = 'C'.\n"
"C := alpha*A^T*B^H + beta*C if transA = 'T' and transB = 'C'.\n"
"C := alpha*A^H*B^H + beta*C if transA = 'C' and transB = 'C'.\n"
- "If k=0, this reduces to C := beta*C.\n\n"
+ "The number of rows of the matrix product is m. The number of \n"
+ "columns is n. The inner dimension is k. If k=0, this reduces \n"
+ "to C := beta*C.\n\n"
"ARGUMENTS\n\n"
"A 'd' or 'z' matrix\n\n"
"B 'd' or 'z' matrix. Must have the same type as A.\n\n"
diff --git a/src/C/cholmod.c b/src/C/cholmod.c
index 5a2c8a1..eab380d 100644
--- a/src/C/cholmod.c
+++ b/src/C/cholmod.c
@@ -1,6 +1,20 @@
/*
- * This file is part of CVXOPT version 0.8.2.
* Copyright 2004-2007 J. Dahl and L. Vandenberghe.
+ *
+ * This file is part of CVXOPT version 0.9.
+ *
+ * CVXOPT is free software; you can redistribute it and/or modify
+ * it under the terms of the GNU General Public License as published by
+ * the Free Software Foundation; either version 3 of the License, or
+ * (at your option) any later version.
+ *
+ * CVXOPT is distributed in the hope that it will be useful,
+ * but WITHOUT ANY WARRANTY; without even the implied warranty of
+ * MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the
+ * GNU General Public License for more details.
+ *
+ * You should have received a copy of the GNU General Public License
+ * along with this program. If not, see <http://www.gnu.org/licenses/>.
*/
#define NO_ANSI99_COMPLEX
diff --git a/src/C/cvxopt.h b/src/C/cvxopt.h
index f37f7a8..ebed7ee 100644
--- a/src/C/cvxopt.h
+++ b/src/C/cvxopt.h
@@ -1,6 +1,20 @@
/*
- * This file is part of CVXOPT version 0.8.2.
* Copyright 2004-2007 J. Dahl and L. Vandenberghe.
+ *
+ * This file is part of CVXOPT version 0.9.
+ *
+ * CVXOPT is free software; you can redistribute it and/or modify
+ * it under the terms of the GNU General Public License as published by
+ * the Free Software Foundation; either version 3 of the License, or
+ * (at your option) any later version.
+ *
+ * CVXOPT is distributed in the hope that it will be useful,
+ * but WITHOUT ANY WARRANTY; without even the implied warranty of
+ * MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the
+ * GNU General Public License for more details.
+ *
+ * You should have received a copy of the GNU General Public License
+ * along with this program. If not, see <http://www.gnu.org/licenses/>.
*/
#include "Python.h"
diff --git a/src/C/dense.c b/src/C/dense.c
index 882cbc4..d4e12eb 100644
--- a/src/C/dense.c
+++ b/src/C/dense.c
@@ -1,6 +1,20 @@
/*
- * This file is part of CVXOPT version 0.8.2.
* Copyright 2004-2007 J. Dahl and L. Vandenberghe.
+ *
+ * This file is part of CVXOPT version 0.9.
+ *
+ * CVXOPT is free software; you can redistribute it and/or modify
+ * it under the terms of the GNU General Public License as published by
+ * the Free Software Foundation; either version 3 of the License, or
+ * (at your option) any later version.
+ *
+ * CVXOPT is distributed in the hope that it will be useful,
+ * but WITHOUT ANY WARRANTY; without even the implied warranty of
+ * MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the
+ * GNU General Public License for more details.
+ *
+ * You should have received a copy of the GNU General Public License
+ * along with this program. If not, see <http://www.gnu.org/licenses/>.
*/
#define BASE_MODULE
@@ -199,10 +213,9 @@ matrix *Matrix_NewFromArrayStruct(PyObject *obj, int id, int_t *ndim)
}
/* XXX: revise flags check */
- //if (src->flags != 0x701) {
- if (!(src->flags & 0x001)) {
+ if (!(src->flags & 0x001) && !(src->flags & 0x002)) {
Py_DECREF(cobj);
- PY_ERR_TYPE("array not contiguous)");
+ PY_ERR_TYPE("error converting array");
}
*ndim = src->nd;
diff --git a/src/C/dsdp.c b/src/C/dsdp.c
index 0f120d9..c0a9120 100644
--- a/src/C/dsdp.c
+++ b/src/C/dsdp.c
@@ -1,6 +1,20 @@
/*
- * This file is part of CVXOPT version 0.8.2.
* Copyright 2004-2007 J. Dahl and L. Vandenberghe.
+ *
+ * This file is part of CVXOPT version 0.9.
+ *
+ * CVXOPT is free software; you can redistribute it and/or modify
+ * it under the terms of the GNU General Public License as published by
+ * the Free Software Foundation; either version 3 of the License, or
+ * (at your option) any later version.
+ *
+ * CVXOPT is distributed in the hope that it will be useful,
+ * but WITHOUT ANY WARRANTY; without even the implied warranty of
+ * MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the
+ * GNU General Public License for more details.
+ *
+ * You should have received a copy of the GNU General Public License
+ * along with this program. If not, see <http://www.gnu.org/licenses/>.
*/
#include "cvxopt.h"
diff --git a/src/C/fftw.c b/src/C/fftw.c
index 65feadd..4b0585a 100644
--- a/src/C/fftw.c
+++ b/src/C/fftw.c
@@ -1,6 +1,20 @@
/*
- * This file is part of CVXOPT version 0.8.2.
* Copyright 2004-2007 J. Dahl and L. Vandenberghe.
+ *
+ * This file is part of CVXOPT version 0.9.
+ *
+ * CVXOPT is free software; you can redistribute it and/or modify
+ * it under the terms of the GNU General Public License as published by
+ * the Free Software Foundation; either version 3 of the License, or
+ * (at your option) any later version.
+ *
+ * CVXOPT is distributed in the hope that it will be useful,
+ * but WITHOUT ANY WARRANTY; without even the implied warranty of
+ * MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the
+ * GNU General Public License for more details.
+ *
+ * You should have received a copy of the GNU General Public License
+ * along with this program. If not, see <http://www.gnu.org/licenses/>.
*/
#include "cvxopt.h"
diff --git a/src/C/glpk.c b/src/C/glpk.c
index d05dd31..3b2a470 100644
--- a/src/C/glpk.c
+++ b/src/C/glpk.c
@@ -1,6 +1,20 @@
/*
- * This file is part of CVXOPT version 0.8.2.
* Copyright 2004-2007 J. Dahl and L. Vandenberghe.
+ *
+ * This file is part of CVXOPT version 0.9.
+ *
+ * CVXOPT is free software; you can redistribute it and/or modify
+ * it under the terms of the GNU General Public License as published by
+ * the Free Software Foundation; either version 3 of the License, or
+ * (at your option) any later version.
+ *
+ * CVXOPT is distributed in the hope that it will be useful,
+ * but WITHOUT ANY WARRANTY; without even the implied warranty of
+ * MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the
+ * GNU General Public License for more details.
+ *
+ * You should have received a copy of the GNU General Public License
+ * along with this program. If not, see <http://www.gnu.org/licenses/>.
*/
#include "cvxopt.h"
diff --git a/src/C/lapack.c b/src/C/lapack.c
index 469f694..4bd3f10 100644
--- a/src/C/lapack.c
+++ b/src/C/lapack.c
@@ -1,6 +1,20 @@
/*
- * This file is part of CVXOPT version 0.8.2.
* Copyright 2004-2007 J. Dahl and L. Vandenberghe.
+ *
+ * This file is part of CVXOPT version 0.9.
+ *
+ * CVXOPT is free software; you can redistribute it and/or modify
+ * it under the terms of the GNU General Public License as published by
+ * the Free Software Foundation; either version 3 of the License, or
+ * (at your option) any later version.
+ *
+ * CVXOPT is distributed in the hope that it will be useful,
+ * but WITHOUT ANY WARRANTY; without even the implied warranty of
+ * MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the
+ * GNU General Public License for more details.
+ *
+ * You should have received a copy of the GNU General Public License
+ * along with this program. If not, see <http://www.gnu.org/licenses/>.
*/
#include "Python.h"
diff --git a/src/C/misc.h b/src/C/misc.h
index fb8a9fb..4072847 100644
--- a/src/C/misc.h
+++ b/src/C/misc.h
@@ -1,6 +1,20 @@
/*
- * This file is part of CVXOPT version 0.8.2.
* Copyright 2004-2007 J. Dahl and L. Vandenberghe.
+ *
+ * This file is part of CVXOPT version 0.9.
+ *
+ * CVXOPT is free software; you can redistribute it and/or modify
+ * it under the terms of the GNU General Public License as published by
+ * the Free Software Foundation; either version 3 of the License, or
+ * (at your option) any later version.
+ *
+ * CVXOPT is distributed in the hope that it will be useful,
+ * but WITHOUT ANY WARRANTY; without even the implied warranty of
+ * MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the
+ * GNU General Public License for more details.
+ *
+ * You should have received a copy of the GNU General Public License
+ * along with this program. If not, see <http://www.gnu.org/licenses/>.
*/
#ifndef __MISC__
diff --git a/src/C/mosek.c b/src/C/mosek.c
index 6900da3..b0882ec 100644
--- a/src/C/mosek.c
+++ b/src/C/mosek.c
@@ -1,6 +1,20 @@
/*
- * This file is part of CVXOPT version 0.8.2.
* Copyright 2004-2007 J. Dahl and L. Vandenberghe.
+ *
+ * This file is part of CVXOPT version 0.9.
+ *
+ * CVXOPT is free software; you can redistribute it and/or modify
+ * it under the terms of the GNU General Public License as published by
+ * the Free Software Foundation; either version 3 of the License, or
+ * (at your option) any later version.
+ *
+ * CVXOPT is distributed in the hope that it will be useful,
+ * but WITHOUT ANY WARRANTY; without even the implied warranty of
+ * MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the
+ * GNU General Public License for more details.
+ *
+ * You should have received a copy of the GNU General Public License
+ * along with this program. If not, see <http://www.gnu.org/licenses/>.
*/
diff --git a/src/C/random.c b/src/C/random.c
index 49c4bca..ce1e714 100644
--- a/src/C/random.c
+++ b/src/C/random.c
@@ -1,6 +1,20 @@
/*
- * This file is part of CVXOPT version 0.8.2.
* Copyright 2004-2007 J. Dahl and L. Vandenberghe.
+ *
+ * This file is part of CVXOPT version 0.9.
+ *
+ * CVXOPT is free software; you can redistribute it and/or modify
+ * it under the terms of the GNU General Public License as published by
+ * the Free Software Foundation; either version 3 of the License, or
+ * (at your option) any later version.
+ *
+ * CVXOPT is distributed in the hope that it will be useful,
+ * but WITHOUT ANY WARRANTY; without even the implied warranty of
+ * MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the
+ * GNU General Public License for more details.
+ *
+ * You should have received a copy of the GNU General Public License
+ * along with this program. If not, see <http://www.gnu.org/licenses/>.
*/
#include "cvxopt.h"
diff --git a/src/C/sparse.c b/src/C/sparse.c
index 0785ba0..9541cd9 100644
--- a/src/C/sparse.c
+++ b/src/C/sparse.c
@@ -1,6 +1,20 @@
/*
- * This file is part of CVXOPT version 0.8.2.
* Copyright 2004-2007 J. Dahl and L. Vandenberghe.
+ *
+ * This file is part of CVXOPT version 0.9.
+ *
+ * CVXOPT is free software; you can redistribute it and/or modify
+ * it under the terms of the GNU General Public License as published by
+ * the Free Software Foundation; either version 3 of the License, or
+ * (at your option) any later version.
+ *
+ * CVXOPT is distributed in the hope that it will be useful,
+ * but WITHOUT ANY WARRANTY; without even the implied warranty of
+ * MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the
+ * GNU General Public License for more details.
+ *
+ * You should have received a copy of the GNU General Public License
+ * along with this program. If not, see <http://www.gnu.org/licenses/>.
*/
#define BASE_MODULE
diff --git a/src/C/umfpack.c b/src/C/umfpack.c
index e79468c..4908204 100644
--- a/src/C/umfpack.c
+++ b/src/C/umfpack.c
@@ -1,6 +1,20 @@
/*
- * This file is part of CVXOPT version 0.8.2.
* Copyright 2004-2007 J. Dahl and L. Vandenberghe.
+ *
+ * This file is part of CVXOPT version 0.9.
+ *
+ * CVXOPT is free software; you can redistribute it and/or modify
+ * it under the terms of the GNU General Public License as published by
+ * the Free Software Foundation; either version 3 of the License, or
+ * (at your option) any later version.
+ *
+ * CVXOPT is distributed in the hope that it will be useful,
+ * but WITHOUT ANY WARRANTY; without even the implied warranty of
+ * MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the
+ * GNU General Public License for more details.
+ *
+ * You should have received a copy of the GNU General Public License
+ * along with this program. If not, see <http://www.gnu.org/licenses/>.
*/
#include "cvxopt.h"
diff --git a/src/python/coneprog.py b/src/python/coneprog.py
index 50b0ce2..04c9a44 100644
--- a/src/python/coneprog.py
+++ b/src/python/coneprog.py
@@ -1,123 +1,755 @@
"""
-Linear and semidefinite programming solver.
+Solver for linear, second-order cone and semidefinite programming.
"""
-# This file is part of CVXOPT version 0.8.2.
# Copyright 2004-2007 J. Dahl and L. Vandenberghe.
+#
+# This file is part of CVXOPT version 0.9.
+#
+# CVXOPT is free software; you can redistribute it and/or modify
+# it under the terms of the GNU General Public License as published by
+# the Free Software Foundation; either version 3 of the License, or
+# (at your option) any later version.
+#
+# CVXOPT is distributed in the hope that it will be useful,
+# but WITHOUT ANY WARRANTY; without even the implied warranty of
+# MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the
+# GNU General Public License for more details.
+#
+# You should have received a copy of the GNU General Public License
+# along with this program. If not, see <http://www.gnu.org/licenses/>.
+
import math
from cvxopt import base, blas, lapack, cholmod, misc
from cvxopt.base import matrix, spmatrix
-cholmod.options['supernodal'] = 2
__all__ = []
options = {}
-def conelp(c, kktsolver, Gl=None, hl=None, Gs=None, hs=None, A=None,
- b=None, primalstart=None, dualstart=None, xnewcopy=matrix,
- xdot=blas.dot, xaxpy=blas.axpy, xscal=blas.scal, ynewcopy=matrix,
- ydot=blas.dot, yaxpy=blas.axpy, yscal=blas.scal):
- """
-
- Solves a pair of primal and dual cone programs
- minimize <c,x>
- subject to Gl(x) + sl = hl
- Gs(x) + ss = hs
- A(x) = b
- sl >= 0, ss >= 0
+def scale(x, W, trans = 'N', inverse = 'N'):
- maximize -<hl,zl> - <hs,zs> - <b,y>
- subject to Gl'(zl) + Gs'(zs) + A'(y) + c = 0
- zl >= 0, zs >= 0
+ # Computes
+ #
+ # x := W*x (trans is 'N', inverse = 'N')
+ # x := W^T*x (trans is 'T', inverse = 'N')
+ # x := W^{-1}*x (trans is 'N', inverse = 'I')
+ # x := W^{-T}*x (trans is 'T', inverse = 'I').
+ #
+ # x is a dense 'd' matrix.
+ #
+ # W is a dictionary with entries:
+ #
+ # - W['d']: positive vector
+ # - W['di']: componentwise inverse of W['d']
+ # - W['v']: lists of second order cone vectors with unit hyperbolic
+ # norms
+ # - W['beta']: list of positive numbers
+ # - W['r']: list of square matrices
+ # - W['rti']: list of square matrices. rti[k] is the inverse
+ # transpose of r[k].
- Gl: V -> reals^ml, Gs: V -> S^m1 x ... x S^mN, and A: V -> W are
- linear mappings, where V and W are real vector spaces.
- Input arguments:
+ # Scaling for 'l' component xk is xk := d .* xk; inverse scaling is
+ # xk ./ d = di .* xk, where d = W['d'], di = W['di'].
- c is a matrix, list of matrices, dictionary of matrices, ...
- representing a vector in the vector space V.
+ if inverse == 'N': w = W['d']
+ else: w = W['di']
+ m = w.size[0]
+ for k in xrange(x.size[1]):
+ blas.tbmv(w, x, n = m, k = 0, ldA = 1, offsetx = k*x.size[0])
- Gl is a function Gl(x, y, alpha=1.0, beta=0.0, trans='N'):
- y := alpha*Gl(x) + beta*y if trans is 'N'
- y := alpha*Gl'(x) + beta*y if trans is 'T'.
+ # Scaling for 'q' component is
+ #
+ # xk := beta * (2*v*v' - J) * xk
+ # = beta * (2*v*(xk'*v)' - J*xk)
+ #
+ # where beta = W['beta'][k], v = W['v'][k], J = [1, 0; 0, -I].
+ #
+ # Inverse scaling is
+ #
+ # xk := 1/beta * (2*J*v*v'*J - J) * xk
+ # = 1/beta * (-J) * (2*v*((-J*xk)'*v)' + xk).
+
+ w = matrix(0.0, (x.size[1], 1))
+ ind = m
+ for k in xrange(len(W['v'])):
+ v = W['v'][k]
+ m = v.size[0]
+ if inverse == 'I':
+ blas.scal(-1.0, x, offset = ind, inc = x.size[0])
+ blas.gemv(x, v, w, trans = 'T', m = m, n = x.size[1], offsetA =
+ ind, ldA = x.size[0])
+ blas.scal(-1.0, x, offset = ind, inc = x.size[0])
+ blas.ger(v, w, x, alpha = 2.0, m = m, n = x.size[1], ldA =
+ x.size[0], offsetA = ind)
+ if inverse == 'I':
+ blas.scal(-1.0, x, offset = ind, inc = x.size[0])
+ a = 1.0 / W['beta'][k]
+ else:
+ a = W['beta'][k]
+ for i in xrange(x.size[1]):
+ blas.scal(a, x, n = m, offset = ind + i*x.size[0])
+ ind += m
- If trans is 'N', x is in V and y is in R^ml.
- If trans is 'T', x is in R^ml and y is in V.
- hl is a dense 'd' matrix of size (ml,1).
+ # Scaling for 's' component xk is
+ #
+ # xk := vec( r' * mat(xk) * r ) if trans = 'N'
+ # xk := vec( r * mat(xk) * r' ) if trans = 'T'.
+ #
+ # r is kth element of W['r'].
+ #
+ # Inverse scaling is
+ #
+ # xk := vec( rti * mat(xk) * rti' ) if trans = 'N'
+ # xk := vec( rti' * mat(xk) * rti ) if trans = 'T'.
+ #
+ # rti is kth element of W['rti'].
- Gs is a function Gs(x, y, alpha=1.0, beta=0.0, trans='N'):
+ maxn = max( [0] + [ r.size[0] for r in W['r'] ] )
+ a = matrix(0.0, (maxn, maxn))
+ for k in xrange(len(W['r'])):
- y := alpha*Gs(x) + beta*y if trans is 'N'
- y := alpha*Gs'(x) + beta*y if trans is 'T'.
+ if inverse == 'N':
+ r = W['r'][k]
+ if trans == 'N': t = 'T'
+ else: t = 'N'
+ else:
+ r = W['rti'][k]
+ t = trans
+
+ n = r.size[0]
+ for i in xrange(x.size[1]):
+
+ # scale diagonal of xk by 0.5
+ blas.scal(0.5, x, offset = ind + i*x.size[0], inc = n+1, n = n)
+
+ # a = r*tril(x) (t is 'N') or a = tril(x)*r (t is 'T')
+ blas.copy(r, a)
+ if t == 'N':
+ blas.trmm(x, a, side = 'R', m = n, n = n, ldA = n, ldB = n,
+ offsetA = ind + i*x.size[0])
+ else:
+ blas.trmm(x, a, side = 'L', m = n, n = n, ldA = n, ldB = n,
+ offsetA = ind + i*x.size[0])
+
+ # x := (r*a' + a*r') if t is 'N'
+ # x := (r'*a + a'*r) if t is 'T'
+ blas.syr2k(r, a, x, trans = t, n = n, k = n, ldB = n, ldC = n,
+ offsetC = ind + i*x.size[0])
+
+ ind += n**2
- If trans is 'N', x is in V and y is in S^m1 x ... x S^mN.
- If trans is 'T', x is in S^m1 x ... x S^mN and y is in V.
- hs is a list of N square dense 'd' matrices, representing a
- symmetric block diagonal matrix in 'L' storage.
+def scale2(lmbda, x, dims, inverse = 'N'):
- A is a function A(x, y, alpha, beta, trans='N'):
+ # x := H(lambda^{1/2}) * x (inverse is 'N')
+ # x := H(lambda^{-1/2}) * x (inverse is 'I')
+ #
+ # H is the Hessian of the logarithmic barrier.
+
- y := alpha*A(x) + beta*y if trans is 'N'
- y := alpha*A'(x) + beta*y if trans is 'T'.
+ # For the 'l' block,
+ #
+ # xk := xk ./ l (inverse is 'N')
+ # xk := xk .* l (inverse is 'I')
+ #
+ # where l is lmbda[:dims['l']].
- If trans is 'N', x is in V and y is in W
- If trans is 'T', x is in S^m1 x ... x S^mN and y is in V.
+ if inverse == 'N':
+ blas.tbsv(lmbda, x, n = dims['l'], k = 0, ldA = 1)
+ else:
+ blas.tbmv(lmbda, x, n = dims['l'], k = 0, ldA = 1)
+
+
+ # For 'q' blocks, if inverse is 'N',
+ #
+ # xk := 1/a * [ l'*J*xk;
+ # xk[1:] - (xk[0] + l'*J*xk) / (l[0] + 1) * l[1:] ].
+ #
+ # If inverse is 'I',
+ #
+ # xk := a * [ l'*xk;
+ # xk[1:] + (xk[0] + l'*xk) / (l[0] + 1) * l[1:] ].
+ #
+ # a = sqrt(lambda_k' * J * lambda_k), l = lambda_k / a.
- b is a matrix, list of matrices, dictionary of matrices, ...
- representing a vector in the vector space W.
+ ind = dims['l']
+ for m in dims['q']:
+ a = jnrm2(lmbda, n = m, offset = ind)
+ if inverse == 'N':
+ lx = jdot(lmbda, x, n = m, offsetx = ind, offsety = ind)/a
+ else:
+ lx = blas.dot(lmbda, x, n = m, offsetx = ind, offsety = ind)/a
+ x0 = x[ind]
+ x[ind] = lx
+ c = (lx + x0) / (lmbda[ind]/a + 1) / a
+ if inverse == 'N': c *= -1.0
+ blas.axpy(lmbda, x, alpha = c, n = m-1, offsetx = ind+1, offsety =
+ ind+1)
+ if inverse == 'N': a = 1.0/a
+ blas.scal(a, x, offset = ind, n = m)
+ ind += m
- kktsolver is a function that returns a function for solving
- KKT systems
+
+ # For the 's' blocks, if inverse is 'N',
+ #
+ # xk := vec( diag(l)^{-1/2} * mat(xk) * diag(k)^{-1/2}).
+ #
+ # If inverse is 'I',
+ #
+ # xk := vec( diag(l)^{1/2} * mat(xk) * diag(k)^{1/2}).
+ #
+ # where l is kth block of lambda.
+ #
+ # We scale upper and lower triangular part of mat(xk) because the
+ # inverse operation will be applied to nonsymmetric matrices.
+
+ ind2 = ind
+ for k in xrange(len(dims['s'])):
+ m = dims['s'][k]
+ for j in xrange(m):
+ c = math.sqrt(lmbda[ind2+j]) * base.sqrt(lmbda[ind2:ind2+m])
+ if inverse == 'N':
+ blas.tbsv(c, x, n = m, k = 0, ldA = 1, offsetx = ind + j*m)
+ else:
+ blas.tbmv(c, x, n = m, k = 0, ldA = 1, offsetx = ind + j*m)
+ ind += m*m
+ ind2 += m
+
+
+def pack(x, y, dims, offsetx = 0, offsety = 0):
+
+ # The vector x is an element of S, with the 's' components stored
+ # in unpacked storage. On return, x is copied to y with the 's'
+ # components matrices stored in packed storage and the off-diagonal
+ # entries scaled by sqrt(2).
+
+ nlq = dims['l'] + sum(dims['q'])
+ np = sum([ n*(n+1)/2 for n in dims['s'] ])
+ blas.copy(x, y, n = nlq, offsetx = offsetx, offsety = offsety)
+ iu, ip = offsetx + nlq, offsety + nlq
+ for n in dims['s']:
+ for k in xrange(n):
+ blas.copy(x, y, n = n-k, offsetx = iu + k*(n+1), offsety = ip)
+ y[ip] /= math.sqrt(2)
+ ip += n-k
+ iu += n**2
+ blas.scal(math.sqrt(2.0), y, n = np, offset = offsety+nlq)
+
+
+def unpack(x, y, dims, offsetx = 0, offsety = 0):
+
+ # The vector x is an element of S, with the 's' components stored
+ # in unpacked storage and off-diagonal entries scaled by sqrt(2).
+ # On return, x is copied to y with the 's' components stored in
+ # unpacked storage and off-diagonal entries scaled by sqrt(2).
+
+ nlq = dims['l'] + sum(dims['q'])
+ nu = sum([ n**2 for n in dims['s'] ])
+ blas.copy(x, y, n = nlq, offsetx = offsetx, offsety = offsety)
+ iu, ip = offsety+nlq, offsetx+nlq
+ for n in dims['s']:
+ for k in xrange(n):
+ blas.copy(x, y, n = n-k, offsetx = ip, offsety = iu+k*(n+1))
+ y[iu+k*(n+1)] *= math.sqrt(2)
+ ip += n-k
+ iu += n**2
+ blas.scal(1.0/math.sqrt(2.0), y, n = nu, offset = offsety+nlq)
+
+
+def sdot(x, y, dims):
+
+ # Returns the inner product of two vectors in S
+
+ ind = dims['l'] + sum(dims['q'])
+ a = blas.dot(x, y, n = ind)
+ for m in dims['s']:
+ a += blas.dot(x, y, offsetx = ind, offsety = ind, incx = m+1,
+ incy = m+1, n = m)
+ for j in xrange(1, m):
+ a += 2.0 * blas.dot(x, y, incx = m+1, incy = m+1,
+ offsetx = ind+j, offsety = ind+j, n = m-j)
+ ind += m**2
+ return a
+
+
+def snrm2(x, dims):
+
+ # Returns the norm of a vector in S
+
+ return math.sqrt(sdot(x, x, dims))
+
+
+def sgemv(A, x, y, dims, trans = 'N', alpha = 1.0, beta = 0.0, m = None,
+ n = None, offsetA = 0, offsety = 0):
+
+ # A is a matrix or spmatrix of size (N, n) where
+ #
+ # N = dims['l'] + sum(dims['q']) + sum( k**2 for k in dims['s'] ).
+ #
+ # If trans is 'N':
+ #
+ # y := alpha*A*x + beta * y (trans = 'N').
+ #
+ # x is a vector of length n. y is a vector of length N.
+ #
+ # If trans is 'T':
+ #
+ # y := alpha*A'*x + beta * y (trans = 'T').
+ #
+ # x is a vector of length N. y is a vector of length n.
+ #
+ # The 's' components in S are stored in unpacked 'L' storage.
+
+ if m is None: m = A.size[0]
+ if n is None: n = A.size[1]
+
+ if trans == 'T' and alpha:
+ ind = dims['l'] + sum(dims['q'])
+ for mk in dims['s']:
+ # Set upper triangular part of x to zero and scale strict
+ # lower triangular part by 2.
+ for j in xrange(1, mk):
+ blas.scal(0.0, x, n = mk-j, inc = mk, offset =
+ ind + j*(mk + 1) - 1)
+ blas.scal(2.0, x, offset = ind + mk*(j-1) + j, n = mk-j)
+ ind += mk**2
+
+ base.gemv(A, x, y, trans = trans, alpha = alpha, beta = beta, m = m,
+ n = n, offsetA = offsetA, offsety = offsety)
+
+ if trans == 'T' and alpha:
+ ind = dims['l'] + sum(dims['q'])
+ for mk in dims['s']:
+ # Scale strict lower triangular part of x by 0.5.
+ for j in xrange(1, mk):
+ blas.scal(0.5, x, offset = ind + mk*(j-1) + j, n = mk-j)
+ ind += mk**2
+
+
+def jdot(x, y, n = None, offsetx = 0, offsety = 0):
+
+ # Returns x' * J * y, where J = [1, 0; 0, -I].
+
+ if n is None:
+ if len(x) != len(y): raise ValueError, "x and y must have the "\
+ "same length"
+ n = len(x)
+ return x[offsetx] * y[offsety] - blas.dot(x, y, n = n-1,
+ offsetx = offsetx + 1, offsety = offsety + 1)
+
+
+def jnrm2(x, n = None, offset = 0):
+
+ # Returns sqrt(x' * J * x) where J = [1, 0; 0, -I], for a vector
+ # x in a second order cone.
+
+ if n is None: n = len(x)
+ a = blas.nrm2(x, n = n-1, offset = offset+1)
+ return math.sqrt(x[offset] - a) * math.sqrt(x[offset] + a)
+
+
+def symm(x, n, offset = 0):
+
+ # Fills in the upper triangular part of the symmetric matrix stored in
+ # x[offset : offset+n*n] using 'L' storage.
+
+ if n <= 1: pass
+ for i in xrange(n-1):
+ blas.copy(x, x, offsetx = offset + i*(n+1) + 1, offsety =
+ offset + (i+1)*(n+1) - 1, incy = n, n = n-i-1)
+
+
+def sprod(x, y, dims, diag = 'N'):
+
+ # The product x := (y o x). If diag is 'D', the 's' part of y is
+ # diagonal and only the diagonal is stored.
+
+
+ # For the 'l' block:
+ #
+ # yk o xk = yk .* xk.
+
+ blas.tbmv(y, x, n = dims['l'], k = 0, ldA = 1)
+
+
+ # For 'q' blocks:
+ #
+ # [ lo l1' ]
+ # yk o xk = [ ] * xk
+ # [ lo l0*I ]
+ #
+ # where yk = (l0, l1).
+
+ ind = dims['l']
+ for m in dims['q']:
+ dd = blas.dot(x, y, offsetx = ind, offsety = ind, n = m)
+ blas.scal(y[ind], x, offset = ind+1, n = m-1)
+ blas.axpy(y, x, alpha = x[ind], n = m-1, offsetx = ind+1, offsety
+ = ind+1)
+ x[ind] = dd
+ ind += m
+
+
+ # For the 's' blocks:
+ #
+ # yk o sk = .5 * ( Yk * mat(xk) + mat(xk) * Yk )
+ #
+ # where Yk = mat(yk) if diag is 'N' and Yk = diag(yk) if diag is 'D'.
+
+ if diag is 'N':
+ maxm = max([0] + dims['s'])
+ A = matrix(0.0, (maxm, maxm))
+
+ for m in dims['s']:
+ blas.copy(x, A, offsetx = ind, n = m*m)
+
+ # Write upper triangular part of A and yk.
+ for i in xrange(m-1):
+ symm(A, m)
+ symm(y, m, offset = ind)
+
+ # xk = 0.5 * (A*yk + yk*A)
+ blas.syr2k(A, y, x, alpha = 0.5, n = m, k = m, ldA = m, ldB =
+ m, ldC = m, offsetB = ind, offsetC = ind)
+
+ ind += m*m
+
+ else:
+ ind2 = ind
+ for m in dims['s']:
+ for j in xrange(m):
+ u = 0.5 * ( y[ind2+j:ind2+m] + y[ind2+j] )
+ blas.tbmv(u, x, n = m-j, k = 0, ldA = 1, offsetx =
+ ind + j*(m+1))
+ ind += m*m
+ ind2 += m
+
+
+def sinv(x, y, dims):
+
+ # The inverse product x := (y o\ x), when the 's' components of y are
+ # diagonal.
+
+ # For the 'l' block:
+ #
+ # yk o\ xk = yk .\ xk.
+
+ blas.tbsv(y, x, n = dims['l'], k = 0, ldA = 1)
+
+
+ # For the 'q' blocks:
+ #
+ # [ l0 -l1' ]
+ # yk o\ xk = 1/a^2 * [ ] * xk
+ # [ -l1 (a*I + l1*l1')/l0 ]
+ #
+ # where yk = (l0, l1) and a = l0^2 - l1'*l1.
+
+ ind = dims['l']
+ for m in dims['q']:
+ aa = jnrm2(y, n = m, offset = ind)**2
+ cc = x[ind]
+ dd = blas.dot(y, x, offsetx = ind+1, offsety = ind+1, n = m-1)
+ x[ind] = cc * y[ind] - dd
+ blas.scal(aa / y[ind], x, n = m-1, offset = ind+1)
+ blas.axpy(y, x, alpha = dd/y[ind] - cc, n = m-1, offsetx = ind+1,
+ offsety = ind+1)
+ blas.scal(1.0/aa, x, n = m, offset = ind)
+ ind += m
+
+
+ # For the 's' blocks:
+ #
+ # yk o\ xk = xk ./ gamma
+ #
+ # where gammaij = .5 * (yk_i + yk_j).
+
+ ind2 = ind
+ for m in dims['s']:
+ for j in xrange(m):
+ u = 0.5 * ( y[ind2+j:ind2+m] + y[ind2+j] )
+ blas.tbsv(u, x, n = m-j, k = 0, ldA = 1, offsetx = ind +
+ j*(m+1))
+ ind += m*m
+ ind2 += m
- [ 0 A' Gl' Gs' ] [ x ] [ bx ]
- [ A 0 0 0 ] [ y ] = [ by ].
- [ Gl 0 -diag(d)^2 0 ] [ zl ] [ bzl ]
- [ Gs 0 0 -r*r'*()*r*r' ] [ zs ] [ bzs ]
+
+def max_step(x, dims, sigma = None):
+
+ # Returns min {t | x + t*e >= 0}.
+ # When called with the argument sigma, also returns the eigenvalues
+ # (in sigma) and the eigenvectors (in x) of the 's' components of x.
+
+ t = []
+ ind = dims['l']
+ if ind: t += [ -min(x[:ind]) ]
+ for m in dims['q']:
+ if m: t += [ blas.nrm2(x, offset = ind+1, n = m-1) - x[ind] ]
+ ind += m
+ if sigma is None and dims['s']:
+ Q = matrix(0.0, (max(dims['s']), max(dims['s'])))
+ w = matrix(0.0, (max(dims['s']),1))
+ ind2 = 0
+ for m in dims['s']:
+ if sigma is None:
+ blas.copy(x, Q, offsetx = ind, n = m**2)
+ lapack.syevr(Q, w, range = 'I', il = 1, iu = 1, n = m, ldA = m)
+ if m: t += [ -w[0] ]
+ else:
+ lapack.syevd(x, sigma, jobz = 'V', n = m, ldA = m, offsetA =
+ ind, offsetW = ind2)
+ if m: t += [ -sigma[ind2] ]
+ ind += m*m
+ ind2 += m
+ if t: return max(t)
+ else: return 0.0
+
+
+
+def conelp(c, G, h, dims, A = None, b = None, primalstart = None,
+ dualstart = None, kktsolver = None, xnewcopy = matrix, xdot =
+ blas.dot, xaxpy = blas.axpy, xscal = blas.scal, ynewcopy = matrix,
+ ydot = blas.dot, yaxpy = blas.axpy, yscal = blas.scal):
+
+ """
+ Solves a pair of primal and dual cone programs
+
+ minimize c'*x maximize -h'*z - b'*y
+ subject to G*x + s = h subject to G'*z + A'*y + c = 0
+ A*x = b z >= 0.
+ s >= 0
+
+ The inequalities are with respect to a cone C defined as the Cartesian
+ product of N + M + 1 cones:
+
+ C = C_0 x C_1 x .... x C_N x C_{N+1} x ... x C_{N+M}.
+
+ The first cone C_0 is the nonnegative orthant of dimension ml.
+ The next N cones are second order cones of dimension mq[0], ...,
+ mq[N-1]. The second order cone of dimension m is defined as
+
+ { (u0, u1) in R x R^{m-1} | u0 >= ||u1||_2 }.
+
+ The next M cones are positive semidefinite cones of order ms[0], ...,
+ ms[M-1] >= 0.
+
+
+ Input arguments (basic usage).
+
+ c is a dense 'd' matrix of size (n,1), where n is the dimension of
+ the primal variable x.
+
+ dims is a dictionary with the dimensions of the components of C.
+ It has three fields.
+ - dims['l'] = ml, the dimension of the nonnegative orthant C_0.
+ (ml >= 0.)
+ - dims['q'] = mq = [ mq[0], mq[1], ..., mq[N-1] ], a list of N
+ integers with the dimensions of the second order cones C_1, ...,
+ C_N. (N >= 0 and mq[k] >= 1.)
+ - dims['s'] = ms = [ ms[0], ms[1], ..., ms[M-1] ], a list of M
+ integers with the orders of the semidefinite cones C_{N+1}, ...,
+ C_{N+M}. (M >= 0 and ms[k] >= 0.)
+
+ G is a dense or sparse 'd' matrix of size (K,n), where
+
+ K = ml + mq[0] + ... + mq[N-1] + ms[0]**2 + ... + ms[M-1]**2.
+
+ Each column of G describes a vector
+
+ v = ( v_0, v_1, ..., v_N, vec(v_{N+1}), ..., vec(v_{N+M}) )
+
+ in V = R^ml x R^mq[0] x ... x R^mq[N-1] x S^ms[0] x ... x S^ms[M-1]
+ stored as a column vector
+
+ [ v_0; v_1; ...; v_N; vec(v_{N+1}); ...; vec(v_{N+M}) ].
+
+ Here, if u is a symmetric matrix of order m, then vec(u) is the
+ matrix u stored in column major order as a vector of length m**2.
+ We use BLAS unpacked 'L' storage, i.e., the entries in vec(u)
+ corresponding to the strictly upper triangular entries of u are
+ not referenced.
+
+ h is a dense 'd' matrix of size (K,1), representing a vector in V,
+ in the same format as the columns of G.
+
+ A is a dense or sparse 'd' matrix of size (p,n). The default
+ value is a sparse 'd' matrix of size (0,n).
+
+ b is a dense 'd' matrix of size (p,1). The default value is a
+ dense 'd' matrix of size (0,1).
+
+ The argument primalstart is a dictionary with keys 'x', 's'. It
+ specifies an optional primal starting point.
+ - primalstart['x'] is a dense 'd' matrix of size (n,1).
+ - primalstart['s'] is a dense 'd' matrix of size (K,1),
+ representing a vector that is strictly positive with respect
+ to the cone C.
+
+ The argument dualstart is a dictionary with keys 'y', 'z'. It
+ specifies an optional dual starting point.
+ - dualstart['y'] is a dense 'd' matrix of size (p,1).
+ - dualstart['z'] is a dense 'd' matrix of size (K,1), representing
+ a vector that is strictly positive with respect to the cone C.
+
+ The other arguments are normally not needed. They allow one to
+ exploit certain types of structure in cone LPs, as described below.
+
+
+ Output.
+
+ conelp() returns a dictionary with keys 'status', 'x', 's', 'z',
+ 'y'.
+
+ If status is 'optimal', x, s, y, z are approximate primal and
+ dual optimal solutions.
+
+ If status is 'primal infeasible', x = s = None, and z, y are an
+ approximate proof of infeasibility:
+
+ h'*z + b'*y = -1, G'*z + A'*y = 0, z >= 0.
+
+ If status is 'dual infeasible', z = y = None, and x, s are an
+ approximate proof of dual infeasibility:
+
+ c'*x = -1, G*x + s = 0, A*x = 0, s >= 0.
+
+ If status is 'unknown', x, y, s, z are None.
+
+
+
+ Exploiting problem structure.
+
+ Three mechanisms are provided to express problem structure in
+ cone LPs.
+
+ First, instead of matrices, G and A are allowed to be Python
+ functions that evaluate the linear mappings G*x, A*x and their
+ adjoints. If G is a function, the call G(x, y, alpha, beta, trans)
+ should evaluate the matrix-vector products
+
+ y := alpha * G * x + beta * y if trans is 'N'
+ y := alpha * G' * x + beta * y if trans is 'T'.
+
+ The arguments x and y are required. The other arguments have
+ default values alpha = 1.0, beta = 0.0, trans = 'N'.
+
+ If A is a function, the call A(x, y, alpha, beta, trans) should
+ evaluate the matrix-vectors products
+
+ y := alpha * A * x + beta * y if trans is 'N'
+ y := alpha * A' * x + beta * y if trans is 'T'.
+
+ The arguments x and y are required. The other arguments
+ have default values alpha = 1.0, beta = 0.0, trans = 'N'.
+
+ If G and/or A are functions, then the argument kktsolver is
+ required.
+
+
+ Second, the user can provide a customized routine for solving the
+ linear equations (`KKT systems')
- Called as f = kktsolver(di, rti) with di = 1.0./d (a dense 'd'
- matrix of size (ml,1)) and rti = (r')^{-1} (a list of square
- dense 'd' matrices of size (mk,mk)).
- The KKT system is solved by f(x,y,zl,zs). On entry, x, y, zl,
- zs contain the righthand side. On exit, they contain the
- solution, with zl, zs scaled: d.*zl and r'*zs*r are returned
- instead of zl and zs.
-
- xnewcopy, xdot, xscal, xaxpy, ynewcopy, ydot, yscal, yaxpy are
- Python functions.
- xnewcopy(x), ynewcopy(x) create new copies of vectors in V and
- W, respectively.
- xdot(x,y), ydot(x,y) return the inner product of vectors in V
- and W, respectively.
- xscal(alpha, x), yscal(alpha,x) compute x := alpha*x for x in V
- and W, respectively.
- xaxpy(x, y, alpha=1, beta=0), yaxpy(x, y, alpha=1, beta=0)
- compute y := alpha*x + y for vectors in V and W, respectively.
-
- conelp() returns a dictionary with keys 'status', 'x', 'sl',
- 'ss', 'y', 'zl', 'zs'.
-
- The control parameters can be modified by adding an entry to the
- dictionary options.
+ [ 0 A' G' ] [ x ] [ bx ]
+ [ A 0 0 ] [ y ] = [ by ]
+ [ G 0 -W'*W ] [ z ] [ bz ]
- options['show_progress'] True/False (default: True)
- options['maxiters'] positive integer (default: 100)
- options['abstol'] scalar (default: 1e-7)
- options['reltol'] scalar (default: 1e-7)
- options['feastol'] scalar (default: 1e-7)
- options['refinement'] True/False (default: True)
+ that form the most expensive step of the algorithm. Here W is a
+ scaling matrix, a block diagonal mapping
+
+ W*z = ( W0*z_0, ..., W_{N+M}*z_{N+M} )
+
+ from V to V, defined as follows.
+
+ - For the 'l' block (W_0):
+
+ W_0 = diag(d),
+
+ with d a positive vector of length ml.
+
+ - For the 'q' blocks (W_{k+1}, k = 0, ..., N-1):
+
+ W_{k+1} = beta_k * ( 2 * v_k * v_k' - J )
+
+ where beta_k is a positive scalar, v_k is a vector in R^mq[k]
+ with v_k[0] > 0 and v_k'*J*v_k = 1, and J = [1, 0; 0, -I].
+
+ - For the 's' blocks (W_{k+N}, k = 0, ..., M-1):
+
+ W_k * x = vec(r_k' * mat(x) * r_k)
+
+ where r_k is a nonsingular matrix of order ms[k], and mat(x) is
+ the inverse of the vec operation.
+
+ The optional argument kktsolver is a Python function that will be
+ called as f = kktsolver(W), where W is a dictionary that contains
+ the parameters of the scaling:
+
+ - W['d'] is a positive 'd' matrix of size (ml, 1).
+ - W['di'] is a positive 'd' matrix with the elementwise inverse of
+ W['d'].
+ - W['beta'] is a list [ beta_0, ..., beta_{N-1} ]
+ - W['v'] is a list [ v_0, ..., v_{N-1} ]
+ - W['r'] is a list [ r_0, ..., r_{M-1} ]
+ - W['rti'] is a list [ rti_0, ..., rti_{M-1} ], with rti_k the
+ inverse of the transpose of r_k.
+
+ The call f = kktsolver(W) should return a function f for solving
+ the KKT system. The KKT system is solved by f(x, y, z). On
+ entry, x, y, z contain the righthand side bx, by, bz. On exit,
+ they contain the solution, with z scaled: W*z is returned instead
+ of z.
+
+
+ Finally, instead of using the default representation of the primal
+ variable x and the dual variable y as one-column 'd' matrices,
+ we can represent these variables (and the corresponding parameters
+ c and b) by arbitrary Python objects (matrices, lists,
+ dictionaries, etc), provided the user supplies the functions
+ xnewcopy, xdot, xscal, xaxpy, ynewcopy, ydot, yscal, yaxpy.
+
+ If X is the vector space of primal variables x, then:
+ - xnewcopy(u) creates a new copy of the vector u in X.
+ - xdot(u, v) returns the inner product of two vectors u and v in X.
+ - xscal(alpha, u) computes u := alpha*u, where alpha is a scalar
+ and u is a vector in X.
+ - xaxpy(u, v, alpha = 1.0, beta = 0.0) computes v := alpha*u + v
+ for a scalar alpha and two vectors u and v in X.
+ If this option is used, the argument c must be in the same format
+ as x, the arguments G and A must be Python functions, and the
+ argument kktsolver is required.
+
+ If Y is the vector space of primal variables y:
+ - ynewcopy(u) creates a new copy of the vector u in Y.
+ - ydot(u, v) returns the inner product of two vectors u and v in Y.
+ - yscal(alpha, u) computes u := alpha*u, where alpha is a scalar
+ and u is a vector in Y.
+ - yaxpy(u, v, alpha = 1.0, beta = 0.0) computes v := alpha*u + v
+ for a scalar alpha and two vectors u and v in Y.
+ If this option is used, the argument b must be in the same format
+ as y, the argument A must be a Python function, and the argument
+ kktsolver is required.
+
+
+ Control parameters.
+
+ The following control parameters can be modified by adding an
+ entry to the dictionary options.
+
+ options['show_progress'] True/False (default: True)
+ options['maxiters'] positive integer (default: 100)
+ options['abstol'] scalar (default: 1e-7)
+ options['reltol'] scalar (default: 1e-6)
+ options['feastol'] scalar (default: 1e-7).
"""
EXPON = 3
STEP = 0.99
- try: DEBUG = options['debug']
- except KeyError: DEBUG = False
-
try: MAXITERS = options['maxiters']
except KeyError: MAXITERS = 100
else:
@@ -132,7 +764,7 @@ def conelp(c, kktsolver, Gl=None, hl=None, Gs=None, hs=None, A=None,
raise ValueError, "options['abstol'] must be a scalar"
try: RELTOL = options['reltol']
- except KeyError: RELTOL = 1e-7
+ except KeyError: RELTOL = 1e-6
else:
if type(RELTOL) is not float and type(RELTOL) is not int:
raise ValueError, "options['reltol'] must be a scalar"
@@ -146,250 +778,813 @@ def conelp(c, kktsolver, Gl=None, hl=None, Gs=None, hs=None, A=None,
try: show_progress = options['show_progress']
except KeyError: show_progress = True
+ if type(dims['l']) is not int or dims['l'] < 0:
+ raise TypeError, "'dims['l']' must be a nonnegative integer"
+ if [ k for k in dims['q'] if type(k) is not int or k < 1 ]:
+ raise TypeError, "'dims['q']' must be a list of positive integers"
+ if [ k for k in dims['s'] if type(k) is not int or k < 0 ]:
+ raise TypeError, "'dims['s']' must be a list of nonnegative " \
+ "integers"
+
try: refinement = options['refinement']
except KeyError: refinement = True
- def f0(x, y, alpha=1.0, beta=0.0, trans='N'):
- # y := alpha*F(x) + beta*y (trans = 'N')
- # y := alpha*F'(x) + beta*y (trans = 'T')
- # when F : R^n -> a 0-dimensional space
- if trans == 'N': pass
- else: xscal(beta, y)
- if Gl is None: Gl = f0
- if Gs is None: Gs = f0
- if A is None: A = f0
- if hl is None: hl = matrix(0.0, (0,1))
- if hs is None: hs = []
+ # Number of second-order and positive semidefinite cones.
+ Nq, Ns = len(dims['q']), len(dims['s'])
+
+ # Logarithmic degree of the product cone.
+ cdeg = dims['l'] + Nq + sum(dims['s'])
+
+ # Dimension of the product cone, with 's' components unpacked.
+ cdim = dims['l'] + sum(dims['q']) + sum([k**2 for k in dims['s']])
+
+ # Dimension of the product cone, with 's' components packed.
+ cdim_pckd = dims['l'] + sum(dims['q']) + sum([k*(k+1)/2 for k
+ in dims['s']])
+
+ # Dimension of the product cone, with diagonal 's' components.
+ cdim_diag = dims['l'] + sum(dims['q']) + sum(dims['s'])
+
+ # Data for kth 'q' constraint are found in rows indq[k]:indq[k+1] of G.
+ indq = [ dims['l'] ]
+ for k in dims['q']: indq = indq + [ indq[-1] + k ]
+
+ # Data for kth 's' constraint are found in rows inds[k]:inds[k+1] of G.
+ inds = [ indq[-1] ]
+ for k in dims['s']: inds = inds + [ inds[-1] + k**2 ]
+
+ if type(h) is not matrix or h.typecode != 'd' or h.size[1] != 1:
+ raise TypeError, "'h' must be a 'd' matrix with 1 column"
+ if type(G) is matrix or type(G) is spmatrix:
+ n = G.size[1]
+ if G.typecode != 'd' or G.size[0] != cdim:
+ raise TypeError, "'G' must be a 'd' matrix with %d rows " %cdim
+ if h.size[0] != cdim:
+ raise TypeError, "'h' must have %d rows" %cdim
+ def Gf(x, y, trans = 'N', alpha = 1.0, beta = 0.0):
+ sgemv(G, x, y, dims, trans = trans, alpha = alpha, beta = beta)
+ else:
+ if kktsolver is None:
+ raise ValueError, "argument 'kktsolver' must be provided if "\
+ "'G' is a function"
+ Gf = G
+
if b is None: b = matrix(0.0, (0,1))
- ml, ms, p = hl.size[0], [ h_k.size[0] for h_k in hs ], b.size[0]
- maxm = max( [0] + ms )
- m = ml + sum(ms)
-
- def xcopy(x, y): xscal(0, y); xaxpy(x, y)
- def ycopy(x, y): yscal(0, y); yaxpy(x, y)
-
- x = xnewcopy(c); xscal(0,x)
- sl = matrix(1.0, (ml,1))
- ss = [ matrix(0.0, (m_k,m_k)) for m_k in ms ]
- for sk in ss: sk[::sk.size[0]+1] = 1.0
- y = ynewcopy(b); yscal(0,y)
- zl = matrix(1.0, (ml,1))
- zs = [ matrix(0.0, (m_k,m_k)) for m_k in ms ]
- for zk in zs: zk[::zk.size[0]+1] = 1.0
+ if type(b) is not matrix or b.typecode != 'd' or b.size[1] != 1:
+ raise TypeError, "'b' must be a 'd' matrix with 1 column"
+ if A is None:
+ if type(G) is matrix or type(G) is spmatrix:
+ A = spmatrix([], [], [], (0,n))
+ else:
+ def A(x, y, trans = 'N', alpha = 1.0, beta = 0.0):
+ if trans == 'N': pass
+ else: xscal(beta, y)
+ if type(A) is matrix or type(A) is spmatrix:
+ p = A.size[0]
+ if A.typecode != 'd' or A.size[1] != n:
+ raise TypeError, "'A' must be a 'd' matrix with %d columns " %n
+ if b.size[0] != p:
+ raise TypeError, "'b' must have %d rows" %p
+ def Af(x, y, trans = 'N', alpha = 1.0, beta = 0.0):
+ base.gemv(A, x, y, trans = trans, alpha = alpha, beta = beta)
+ else:
+ if kktsolver is None:
+ raise ValueError, "argument 'kktsolver' must be provided if "\
+ "'A' is a function"
+ Af = A
+
+ def xcopy(x, y):
+ xscal(0.0, y)
+ xaxpy(x, y)
+
+ def ycopy(x, y):
+ yscal(0.0, y)
+ yaxpy(x, y)
+
+ if kktsolver is None:
+ if dims['q'] or dims['s']:
+ kktsolver = 'qr'
+ else:
+ kktsolver = 'chol'
+ if kktsolver in ('qr', 'ldl', 'ldl2', 'chol'):
+ if type(A) is not matrix and type(A) is not spmatrix:
+ raise TypeError, "A must be a matrix or spmatrix if " \
+ "kktsolver is '" + kktsolver + "'"
+ if type(G) is not matrix and type(G) is not spmatrix:
+ raise TypeError, "G must be a matrix or spmatrix if " \
+ "kktsolver is '" + kktsolver + "'"
+ if p > n or cdim_pckd + p < n:
+ raise ValueError, "Rank(A) < p or Rank([G; A]) < n"
+
+ if kktsolver == 'qr':
+
+ # The default kktsolver, except for LPs.
+ #
+ # Two QR factorizations
+ #
+ # A' = [Q1, Q2] * [R1; 0], W^{-T} * G * Q1 = Q3*R3
+ #
+ # (with columns of W^{-T}*G in packed storage).
+
+ # A' = [Q1, Q2] * [R1; 0]
+ if type(A) is matrix:
+ QA = +A.T
+ else:
+ QA = matrix(A.T)
+ tauA = matrix(0.0, (p,1))
+ lapack.geqrf(QA, tauA)
+
+ Gs = matrix(0.0, (cdim, n))
+ tauG = matrix(0.0, (n-p,1))
+ g = matrix(0.0, (cdim, 1))
+ u = matrix(0.0, (cdim_pckd, 1))
+ vv = matrix(0.0, (n,1))
+ w = matrix(0.0, (cdim_pckd, 1))
+
+ def kktsolver(W):
+
+ # Gs = W^{-T}*G, in packed storage.
+ Gs[:,:] = G
+ scale(Gs, W, trans = 'T', inverse = 'I')
+ for k in xrange(n):
+ g[:] = Gs[:, k]
+ pack(g, Gs, dims, offsety = k*Gs.size[0])
+
+ # Gs := [ Gs1, Gs2 ]
+ # = Gs * [ Q1, Q2 ]
+ lapack.ormqr(QA, tauA, Gs, side = 'R', m = cdim_pckd)
+
+ # QR factorization Gs2 := [ Q3, Q4 ] * [ R3; 0 ]
+ lapack.geqrf(Gs, tauG, n = n-p, m = cdim_pckd, offsetA =
+ Gs.size[0]*p)
+
+ def solve_kkt(x, y, z):
+
+ # On entry, x, y, z contain bx, by, bz. On exit, they
+ # contain the solution x, y, W*z of
+ #
+ # [ 0 A' G'*W^{-1} ] [ x ] [bx ]
+ # [ A 0 0 ] * [ y ] = [by ].
+ # [ W^{-T}*G 0 -I ] [ W*z ] [W^{-T}*bz]
+ #
+ # The system is solved in five steps:
+ #
+ # w := W^{-T}*bz - Gs1*R1^{-T}*by
+ # u := R3^{-T}*Q2'*bx + Q3'*w
+ # W*z := Q3*u - w
+ # y := R1^{-1} * (Q1'*bx - Gs1'*(W*z))
+ # x := [ Q1, Q2 ] * [ R1^{-T}*by; R3^{-1}*u ]
+
+ # w := W^{-T} * bz in packed storage
+ scale(z, W, trans = 'T', inverse = 'I')
+ pack(z, w, dims)
+
+ # vv := [ Q1'*bx; R3^{-T}*Q2'*bx ]
+ blas.copy(x, vv)
+ lapack.ormqr(QA, tauA, vv, trans='T')
+ lapack.trtrs(Gs, vv, uplo = 'U', trans = 'T', n = n-p,
+ offsetA = Gs.size[0]*p, offsetB = p)
+
+ # x[:p] := R1^{-T} * by
+ blas.copy(y, x)
+ lapack.trtrs(QA, x, uplo = 'U', trans = 'T', n = p)
+
+ # w := w - Gs1 * x[:p]
+ # = W^{-T}*bz - Gs1*by
+ blas.gemv(Gs, x, w, alpha = -1.0, beta = 1.0, n = p,
+ m = cdim_pckd)
+
+ # u := [ Q3'*w + v[p:]; 0 ]
+ # = [ Q3'*w + R3^{-T}*Q2'*bx; 0 ]
+ blas.copy(w, u)
+ lapack.ormqr(Gs, tauG, u, trans = 'T', k = n-p, offsetA =
+ Gs.size[0]*p, m = cdim_pckd)
+ blas.axpy(vv, u, offsetx = p, n = n-p)
+ blas.scal(0.0, u, offset = n-p)
+
+ # x[p:] := R3^{-1} * u[:n-p]
+ blas.copy(u, x, offsety = p, n = n-p)
+ lapack.trtrs(Gs, x, uplo='U', n = n-p, offsetA =
+ Gs.size[0]*p, offsetB = p)
+
+ # x is now [ R1^{-T}*by; R3^{-1}*u[:n-p] ]
+ # x := [Q1 Q2]*x
+ lapack.ormqr(QA, tauA, x)
+
+ # u := [Q3, Q4] * u - w
+ # = Q3 * u[:n-p] - w
+ lapack.ormqr(Gs, tauG, u, k = n-p, m = cdim_pckd,
+ offsetA = Gs.size[0]*p)
+ blas.axpy(w, u, alpha = -1.0)
+
+ # y := R1^{-1} * ( v[:p] - Gs1'*u )
+ # = R1^{-1} * ( Q1'*bx - Gs1'*u )
+ blas.copy(vv, y, n = p)
+ blas.gemv(Gs, u, y, m = cdim_pckd, n = p, trans = 'T',
+ alpha = -1.0, beta = 1.0)
+ lapack.trtrs(QA, y, uplo = 'U', n=p)
+
+ unpack(u, z, dims)
+
+ return solve_kkt
+
+
+ elif kktsolver == 'ldl':
+
+ # LDL factorization of
+ #
+ # [ 0 A' G'*W^{-1} ]
+ # K = [ A 0 0 ].
+ # [ W^{-T}*G 0 -I ]
+
+ ldK = n + p + cdim_pckd
+ K = matrix(0.0, (ldK, ldK))
+ ipiv = matrix(0, (ldK, 1))
+ u = matrix(0.0, (ldK, 1))
+ g = matrix(0.0, (G.size[0], 1))
+
+ def kktsolver(W):
+
+ blas.scal(0.0, K)
+ K[(ldK+1)*(p+n) :: ldK+1] = -1.0
+ K[n:n+p, :n] = A
+ for k in xrange(n):
+ g[:] = G[:,k]
+ scale(g, W, trans = 'T', inverse = 'I')
+ pack(g, K, dims, offsety = k*ldK+n+p)
+ lapack.sytrf(K, ipiv)
+
+ def solve_kkt(x, y, z):
+
+ # Solve
+ #
+ # [ x ] [ bx ]
+ # K * [ y ] = [ by ]
+ # [ W*z ] [ W^{-T}*bz ]
+ #
+ # and return x, y, W*z.
+ #
+ # On entry, x, y, z contain bx, by, bz. On exit, they
+ # contain the solution.
+
+ blas.copy(x, u)
+ blas.copy(y, u, offsety=n)
+ scale(z, W, trans='T', inverse='I')
+ pack(z, u, dims, offsety = n+p)
+ lapack.sytrs(K, ipiv, u)
+ blas.copy(u, x, n=n)
+ blas.copy(u, y, offsetx = n, n = p)
+ unpack(u, z, dims, offsetx = n+p)
+
+ return solve_kkt
+
+
+ elif kktsolver == 'ldl2':
+
+ # LDL or Cholesky factorization of
+ #
+ # [ G' * W^{-1} * W^{-T} * G A' ]
+ # K = [ ]
+ # [ A 0 ].
+
+ ldK = n + p
+ K = matrix(0.0, (ldK, ldK))
+ if p: ipiv = matrix(0, (ldK, 1))
+ g = matrix(0.0, (G.size[0], 1))
+ u = matrix(0.0, (ldK, 1))
+
+ def kktsolver(W):
+
+ blas.scal(0.0, K)
+ K[n:,:n] = A
+ for k in xrange(n):
+ g[:] = G[:,k]
+ scale(g, W, trans = 'T', inverse = 'I')
+ scale(g, W, inverse = 'I')
+ sgemv(G, g, K, dims, trans = 'T', beta = 1.0, n = n-k,
+ offsetA = G.size[0]*k, offsety = (ldK + 1)*k)
+ if p: lapack.sytrf(K, ipiv)
+ else: lapack.potrf(K)
+
+ def solve_kkt(x, y, z):
+
+ # Solve
+ #
+ # [ x ] [ bx + G' * W^{-1} * W^{-T} * bz ]
+ # K * [ ] = [ ]
+ # [ y ] [ by ]
+ #
+ # and return x, y, W*z = W^{-T} * (G*x - bz).
+
+ blas.copy(z, g)
+ scale(g, W, trans = 'T', inverse = 'I')
+ scale(g, W, inverse = 'I')
+ sgemv(G, g, u, dims, trans = 'T')
+ blas.axpy(x, u)
+ blas.copy(y, u, offsety = n)
+ if p: lapack.sytrs(K, ipiv, u)
+ else: lapack.potrs(K, u)
+ blas.copy(u, x, n = n)
+ blas.copy(u, y, offsetx = n, n = p)
+ sgemv(G, x, z, dims, alpha = 1.0, beta = -1.0)
+ scale(z, W, trans = 'T', inverse = 'I')
+
+ return solve_kkt
+
+ elif kktsolver == 'chol' and (dims['q'] or dims['s']):
+
+ # Dense Cholesky factorizations of
+ #
+ # S = G' * W^{-1} * W^{-T} * G + A' * A
+ # K = A * S^{-1} * A'.
+
+ S, K = matrix(0.0, (n,n)), matrix(0.0, (p,p))
+ g = matrix(0.0, (G.size[0], 1))
+
+ def kktsolver(W):
+
+ # S = G' * W^{-1} * W^{-T} * G + A' * A
+ blas.scal(0.0, S)
+ for k in xrange(n):
+ g[:] = G[:,k]
+ scale(g, W, trans = 'T', inverse = 'I')
+ scale(g, W, inverse = 'I')
+ sgemv(G, g, S, dims, trans = 'T', beta = 1.0, n = n-k,
+ offsetA = G.size[0]*k, offsety = (n+1)*k)
+ base.syrk(A, S, trans='T', beta=1.0)
+ lapack.potrf(S)
+
+ # Asct := L^{-1}*A'. Factor K = Asct'*Asct.
+ if type(A) is matrix:
+ Asct = A.T
+ else:
+ Asct = matrix(A.T)
+ blas.trsm(S, Asct)
+ blas.syrk(Asct, K, trans='T')
+ lapack.potrf(K)
+
+ def solve_kkt(x, y, z):
+
+ # Solve for y, x:
+ #
+ # K * y = A * S^{-1} * ( bx + G'*W^{-1}*W^{-T}*bz +
+ # A'*by ) - by
+ # S*x = bx + G'*W^{-1}*W^{-T}*bz + A'*by - A'*y.
+ #
+ # Wz = W^{-T} * ( G*x - bz ).
+
+ # x := L^{-1} * ( bx + G'*W^{-1}*W^{-T}*bz + A' * by ).
+ blas.copy(z, g)
+ scale(g, W, trans = 'T', inverse = 'I')
+ scale(g, W, inverse = 'I')
+ sgemv(G, g, x, dims, beta = 1.0, trans = 'T')
+ base.gemv(A, y, x, trans='T', beta=1.0)
+ blas.trsv(S, x)
+
+ # y := K^{-1} * (A * L^{-T} * x - by)
+ base.gemv(Asct, x, y, trans = 'T', beta = -1.0)
+ lapack.potrs(K, y)
+
+ # x := L^{-T} * (x - Asc'*y)
+ base.gemv(Asct, y, x, alpha = -1.0, beta = 1.0)
+ blas.trsv(S, x, trans='T')
+
+ # Wz = W^{-T} * ( G*x - bz ).
+ sgemv(G, x, z, dims, alpha = 1.0, beta = -1.0)
+ scale(z, W, trans = 'T', inverse = 'I')
+
+ return solve_kkt
+
+
+ elif kktsolver == 'chol' and not dims['q'] and not dims['s']:
+
+ # This is the default kktsolver for LPs. It exploits sparsity to
+ # some extent.
+ #
+ # Factor
+ #
+ # S = G' * W^{-2} * G where W = diag( W['di'] )^{-1}
+ # K = A * S^{-1} * A',
+ #
+ # using dense (L*L') or sparse (P'*L*L'*P) Cholesky factorizations.
+ # If S turns out to be singular in the first factorization, then
+ # switch to factoring
+ #
+ # S = G' * W^{-2} * G + A' * A
+ # K = A * S^{-1} * A'.
+
+ F = {'firstcall': True, 'singular': False}
+ if type(G) is matrix:
+ Gs = matrix(0.0, G.size)
+ F['S'] = matrix(0.0, (n,n))
+ K = matrix(0.0, (p,p))
+ else:
+ Gs = spmatrix(0.0, G.I, G.J, G.size)
+ F['S'] = spmatrix([], [], [], (n,n), 'd')
+ F['Sf'] = None
+ if type(A) is matrix:
+ K = matrix(0.0, (p,p))
+ else:
+ K = spmatrix([], [], [], (p,p), 'd')
+ m = dims['l']
+
+ def kktsolver(W):
+
+ # Gs = W^{-1} * G
+ base.gemm( spmatrix(W['di'], range(m), range(m)), G, Gs,
+ partial = True)
+
+ if F['firstcall']:
+ F['firstcall'] = False
+ base.syrk(Gs, F['S'], trans = 'T')
+ try:
+ if type(F['S']) is matrix:
+ lapack.potrf(F['S'])
+ else:
+ F['Sf'] = cholmod.symbolic(F['S'])
+ cholmod.numeric(F['S'], F['Sf'])
+ except ArithmeticError:
+ F['singular'] = True
+ if type(A) is matrix and type(F['S']) is spmatrix:
+ F['S'] = matrix(0.0, (n,n))
+ base.syrk(Gs, F['S'], trans = 'T')
+ base.syrk(A, F['S'], trans = 'T', beta = 1.0)
+ if type(F['S']) is matrix:
+ lapack.potrf(F['S'])
+ else:
+ F['Sf'] = cholmod.symbolic(F['S'])
+ cholmod.numeric(F['S'], F['Sf'])
+
+ else:
+ base.syrk(Gs, F['S'], trans = 'T', partial = True)
+ if F['singular']:
+ base.syrk(A, F['S'], trans = 'T', beta = 1.0, partial
+ = True)
+ if type(F['S']) is matrix:
+ lapack.potrf(F['S'])
+ else:
+ cholmod.numeric(F['S'], F['Sf'])
+
+ if type(F['S']) is matrix:
+ # Asct := L^{-1}*A'. Factor K = Asct'*Asct.
+ if type(A) is matrix:
+ Asct = A.T
+ else:
+ Asct = matrix(A.T)
+ blas.trsm(F['S'], Asct)
+ blas.syrk(Asct, K, trans='T')
+ lapack.potrf(K)
+
+ else:
+ # Asct := L^{-1}*P*A'. Factor K = Asct'*Asct.
+ if type(A) is matrix:
+ Asct = A.T
+ cholmod.solve(F['Sf'], Asct, sys = 7)
+ cholmod.solve(F['Sf'], Asct, sys = 4)
+ blas.syrk(Asct, K, trans = 'T')
+ lapack.potrf(K)
+ else:
+ Asct = cholmod.spsolve(F['Sf'], A.T, sys = 7)
+ Asct = cholmod.spsolve(F['Sf'], Asct, sys = 4)
+ base.syrk(Asct, K, trans = 'T')
+ Kf = cholmod.symbolic(K)
+ cholmod.numeric(K, Kf)
+
+ def solve_kkt(x, y, z):
+
+ # If not F['singular']:
+ #
+ # K*y = A * S^{-1} * ( bx + G'*W^{-2}*bz ) - by
+ # S*x = bx + G'*W^{-2}*bz - A'*y
+ # W*z = W^{-1} * ( G*x - bz ).
+ #
+ # If F['singular']:
+ #
+ # K*y = A * S^{-1} * ( bx + G'*W^{-1}*W^{-T}*bz +
+ # A'*by ) - by
+ # S*x = bx + G'*W^{-1}*W^{-T}*bz + A'*by - A'*y.
+ # W*z = W^{-T} * ( G*x - bz ).
+
+
+ # z := W^{-1} * z = W^{-1} * bz
+ blas.tbmv(W['di'], z, n = m, k = 0, ldA = 1)
+
+ # If not F['singular']:
+ # x := L^{-1} * P * (x + Gs'*z)
+ # = L^{-1} * P * (x + G'*W^{-2}*bz)
+ #
+ # If F['singular']:
+ # x := L^{-1} * P * (x + Gs'*z + A'*y))
+ # = L^{-1} * P * (x + G'*W^{-2}*bz + A'*y)
+
+ base.gemv(Gs, z, x, trans = 'T', beta = 1.0)
+ if F['singular']:
+ base.gemv(A, y, x, trans = 'T', beta = 1.0)
+ if type(F['S']) is matrix:
+ blas.trsv(F['S'], x)
+ else:
+ cholmod.solve(F['Sf'], x, sys = 7)
+ cholmod.solve(F['Sf'], x, sys = 4)
+
+ # y := K^{-1} * (Asc*x - y)
+ # = K^{-1} * (A * S^{-1} * (bx + G'*W^{-2}*bz) - by)
+ # (if not F['singular'])
+ # = K^{-1} * (A * S^{-1} * (bx + G'*W^{-2}*bz + A'*by)
+ # - by)
+ # (if F['singular']).
+
+ base.gemv(Asct, x, y, trans = 'T', beta = -1.0)
+ if type(K) is matrix:
+ lapack.potrs(K, y)
+ else:
+ cholmod.solve(Kf, y)
+
+ # x := P' * L^{-T} * (x - Asc'*y)
+ # = S^{-1} * (bx + G'*W^{-2}*bz - A'*y)
+ # (if not F['singular'])
+ # = S^{-1} * (bx + G'*W^{-2}*bz + A'*by - A'*y)
+ # (if F['singular'])
+
+ base.gemv(Asct, y, x, alpha = -1.0, beta = 1.0)
+ if type(F['S']) is matrix:
+ blas.trsv(F['S'], x, trans='T')
+ else:
+ cholmod.solve(F['Sf'], x, sys = 5)
+ cholmod.solve(F['Sf'], x, sys = 8)
+
+ # W*z := Gs*x - z = W^{-1} * (G*x - bz)
+ base.gemv(Gs, x, z, beta = -1.0)
+
+ return solve_kkt
+
+ else:
+ # User provided kktsolver.
+
+ pass
+
+ x = xnewcopy(c); xscal(0.0, x)
+ y = ynewcopy(b); yscal(0.0, y)
+ s, z = matrix(0.0, (cdim,1)), matrix(0.0, (cdim,1))
resx, hresx = xnewcopy(c), xnewcopy(c)
- reszl, hreszl = matrix(0.0, (ml,1)), matrix(0.0, (ml,1))
- reszs = [ matrix(0.0, (m_k,m_k)) for m_k in ms ]
- hreszs = [ matrix(0.0, (m_k,m_k)) for m_k in ms ]
resy, hresy = ynewcopy(b), ynewcopy(b)
- dx = xnewcopy(c)
- dsl = matrix(0.0, (ml,1))
- dss = [ matrix(0.0, (m_k,m_k)) for m_k in ms ]
- dy = ynewcopy(b)
- dzl = matrix(0.0, (ml,1))
- dzs = [ matrix(0.0, (m_k,m_k)) for m_k in ms ]
+ resz, hresz = matrix(0.0, (cdim,1)), matrix(0.0, (cdim,1))
+ dx, dy = xnewcopy(c), ynewcopy(b)
+ ds, dz = matrix(0.0, (cdim,1)), matrix(0.0, (cdim,1))
+ dx1, dy1 = xnewcopy(c), ynewcopy(b)
+ ds1, dz1 = matrix(0.0, (cdim,1)), matrix(0.0, (cdim,1))
dkappa, dtau = matrix(0.0, (1,1)), matrix(0.0, (1,1))
- dx1 = xnewcopy(c)
- dy1 = ynewcopy(b)
- dsl1 = matrix(0.0, (ml,1))
- dss1 = [ matrix(0.0, (m_k,m_k)) for m_k in ms ]
- dzl1 = matrix(0.0, (ml,1))
- dzs1 = [ matrix(0.0, (m_k,m_k)) for m_k in ms ]
- if refinement:
- dx2 = xnewcopy(c)
- dy2 = ynewcopy(b)
- dsl2 = matrix(0.0, (ml,1))
- dss2 = [ matrix(0.0, (m_k,m_k)) for m_k in ms ]
- dzl2 = matrix(0.0, (ml,1))
- dzs2 = [ matrix(0.0, (m_k,m_k)) for m_k in ms ]
- dkappa2, dtau2 = matrix(0.0, (1,1)), matrix(0.0, (1,1))
- r = [ matrix(0.0, (m_k,m_k)) for m_k in ms ]
- rti = [ matrix(0.0, (m_k,m_k)) for m_k in ms ]
- lmbda = matrix(0.0, (m+1,1))
- gamma = [ matrix(0.0, (m_k,m_k)) for m_k in ms ]
- thl = matrix(0.0, (ml,1))
- ths = [ matrix(0.0, (m_k, m_k)) for m_k in ms ]
- sigs, sigz = matrix(0.0, (m+1,1)), matrix(0.0, (m+1,1))
- ones = matrix(1.0, (max(maxm,ml),1))
- work, work2 = matrix(0.0, (maxm,maxm)), matrix(0.0, (maxm,maxm))
-
-
- # Initial points more or less as in Mehrotra's paper.
+ dx2, dy2 = xnewcopy(c), ynewcopy(b)
+ ds2, dz2 = matrix(0.0, (cdim,1)), matrix(0.0, (cdim,1))
+ dkappa2, dtau2 = matrix(0.0, (1,1)), matrix(0.0, (1,1))
+ th = matrix(0.0, (cdim,1))
+ sigs = matrix(0.0, (sum(dims['s']), 1))
+ sigz = matrix(0.0, (sum(dims['s']), 1))
+ lmbda = matrix(0.0, (cdim_diag + 1, 1))
+ lmbdasq = matrix(0.0, (cdim_diag + 1, 1))
+ work = matrix(0.0, (max( [0] + dims['s'] )**2, 1))
+
+
+ # Select initial points.
if primalstart is None or dualstart is None:
# Factor
#
- # [ 0 A' Gl' Gs' ]
- # [ A 0 0 0 ]
- # [ Gl 0 -I 0 ]
- # [ Gs 0 0 -I ]
-
- dli = matrix(1.0, (ml,1))
- for e in rti:
- blas.scal(0.0, e)
- e[::e.size[0]+1] = 1.0
- try:
- f = kktsolver(dli, rti)
+ # [ 0 A' G' ]
+ # [ A 0 0 ].
+ # [ G 0 -I ]
+
+ W = {}
+ W['d'] = matrix(1.0, (dims['l'], 1))
+ W['di'] = matrix(1.0, (dims['l'], 1))
+ W['v'] = [ matrix(0.0, (m,1)) for m in dims['q'] ]
+ W['beta'] = Nq * [ 1.0 ]
+ for v in W['v']: v[0] = 1.0
+ W['r'] = [ matrix(0.0, (m,m)) for m in dims['s'] ]
+ W['rti'] = [ matrix(0.0, (m,m)) for m in dims['s'] ]
+ for r in W['r']: r[::r.size[0]+1 ] = 1.0
+ for rti in W['rti']: rti[::rti.size[0]+1 ] = 1.0
+ try: f = kktsolver(W)
except ArithmeticError:
- raise ValueError, "Rank(A) < p or Rank([Gl; Gs; A]) < n"
+ raise ValueError, "Rank(A) < p or Rank([G; A]) < n"
if primalstart is None:
- # minimize || Gl(x) - hl ||^2
- # subject to A(x) = b
- #
- # by solving
- #
- # [ 0 A' Gl' Gs' ] [x ] [0 ]
- # [ A 0 0 0 ] [dy ] = [b ]
- # [ Gl 0 -I 0 ] [dzl] [hl]
- # [ Gs 0 0 -I ] [dzs] [hs]
- #
- # (sl, ss) = (hl-Gl*x, hs-Gs(x)) and add a multiple of the
- # identity to make (sl,ss) >= 0.
+ # minimize || G * x - h ||^2
+ # subject to A * x = b
+ #
+ # by solving
+ #
+ # [ 0 A' G' ] [ x ] [ 0 ]
+ # [ A 0 0 ] * [ dy ] = [ b ].
+ # [ G 0 -I ] [ -s ] [ h ]
xscal(0.0, x)
ycopy(b, dy)
- blas.copy(hl, dzl)
- misc.scopy(hs, dzs)
- f(x, dy, dzl, dzs)
- blas.scal(0.0, sl); blas.axpy(dzl, sl, alpha=-1)
- misc.sscal(0.0, ss); misc.saxpy(dzs, ss, alpha=-1)
- blas.copy(sl, lmbda)
- if sl: ap = min(sl)
- else: ap = 0.0
- for e in ss:
- if e.size[0]:
- lapack.syevr(+e, lmbda, range='I', il=1, iu=1)
- ap = min(ap, lmbda[0])
- if ap < 0.0:
- sl += -1.5*ap
- for e in ss: e[::e.size[0]+1] -= 1.5*ap
+ blas.copy(h, s)
+ try: f(x, dy, s)
+ except ArithmeticError:
+ raise ValueError, "Rank(A) < p or Rank([G; A]) < n"
+ blas.scal(-1.0, s)
else:
xcopy(primalstart['x'], x)
- if ml: blas.copy(primalstart['sl'], sl)
- if ms: misc.scopy(primalstart['ss'], ss)
+ blas.copy(primalstart['s'], s)
+
+ # ts = min{ t | s + t*e >= 0 }
+ ts = max_step(s, dims)
+ if ts >= 0 and primalstart:
+ raise ValueError, "initial s is not positive"
if dualstart is None:
- # minimize || zl ||^2
- # subject to Gl'(zl) + A'(y) + c = 0
+ # minimize || z ||^2
+ # subject to G'*z + A'*y + c = 0
#
# by solving
#
- # [ 0 A' Gl' Gs' ] [dx] [-c]
- # [ A 0 0 0 ] [y ] = [ 0]
- # [ Gl 0 -I 0 ] [zl] [ 0]
- # [ Gs 0 0 -I ] [zs] [ 0]
+ # [ 0 A' G' ] [ dx ] [ -c ]
+ # [ A 0 0 ] [ y ] = [ 0 ].
+ # [ G 0 -I ] [ z ] [ 0 ]
- xcopy(c, dx); xscal(-1.0, dx)
+ xcopy(c, dx);
+ xscal(-1.0, dx)
yscal(0.0, y)
- blas.scal(0.0, zl)
- misc.sscal(0.0, zs)
- f(dx, y, zl, zs)
- blas.copy(zl, lmbda)
- if zl: ad = min(zl)
- else: ad = 0.0
- for e in zs:
- if e.size[0]:
- lapack.syevr(+e, lmbda, range='I', il=1, iu=1)
- ad = min(ad, lmbda[0])
- if ad <= 0.0:
- zl += -1.5*ad
- for e in zs: e[::e.size[0]+1] -= 1.5*ad
+ blas.scal(0.0, z)
+ try: f(dx, y, z)
+ except ArithmeticError:
+ raise ValueError, "Rank(A) < p or Rank([G; A]) < n"
else:
- if p: ycopy(dualstart['y'], y)
- if ml: blas.copy(dualstart['zl'], zl)
- if ms: misc.scopy(dualstart['zs'], zs)
-
- gap = blas.dot(sl,zl) + misc.sdot(ss, zs)
-
- if primalstart is None and dualstart is None and gap < 1e-10:
- # If gap is zero, z, s must have zero components, and this means
- # that ad and ap were zero. Therefore the initial point
- # happens to be feasible and optimal. So it does not matter
- # if (sl,ss) or (zl,zs) are not strictly positive and we can
- # skip to the computation of the residuals.
- pass
-
- else:
- if primalstart is None:
- # (sl, ss) >= could be non-strictly positive so we need to
- # make it positive definite. Note that sum(zl) + tr(zs)
- # cannot be zero, because (zl,zs)=0 means dualstart is None
- # and gap = 0, and that was handled above.
- trz = blas.asum(zl) + sum([ blas.asum(e, inc=e.size[0]+1)
- for e in zs ])
- a = max(gap, 1e-8) / trz
- sl += a
- for e in ss: e[::e.size[0]+1] += a
-
- if dualstart is None:
- # Idem for (zl, zs).
- trs = blas.asum(sl) + sum([ blas.asum(e, inc=e.size[0]+1)
- for e in ss ])
- a = max(gap, 1e-8) / trs
- zl += a
- for e in zs: e[::e.size[0]+1] += a
+ if 'y' in dualstart: ycopy(dualstart['y'], y)
+ blas.copy(dualstart['z'], z)
+
+ # tz = min{ t | z + t*e >= 0 }
+ tz = max_step(z, dims)
+ if tz >= 0 and dualstart:
+ raise ValueError, "initial z is not positive"
+
+ trz = sum(z[:dims['l']]) + sum(z[indq[:-1]]) + sum([
+ sum(z[inds[k] : inds[k+1] : dims['s'][k]+1]) for k in
+ xrange(len(dims['s'])) ])
+ trs = sum(s[:dims['l']]) + sum(s[indq[:-1]]) + sum([
+ sum(s[inds[k] : inds[k+1] : dims['s'][k]+1]) for k in
+ xrange(len(dims['s'])) ])
+ nrms = snrm2(s, dims)
+ nrmz = snrm2(z, dims)
+
+ if primalstart is None and dualstart is None:
+
+ gap = sdot(s, z, dims)
+ pcost = xdot(c,x)
+ dcost = -ydot(b,y) - sdot(h,z,dims)
+
+ if ts <= 0 and tz <= 0 and (gap <= ABSTOL or ( pcost < 0.0 and
+ gap / -pcost <= RELTOL ) or (dcost > 0.0 and gap / dcost
+ <= RELTOL)):
+
+ # The initial points we constructed happen to be feasible and
+ # optimal.
+ ind = dims['l'] + sum(dims['q'])
+ for m in dims['s']:
+ symm(s, m, ind)
+ symm(z, m, ind)
+ ind += m**2
+ return {'status': 'optimal', 'x': x, 'y': y, 's': s, 'z': z}
+
+ if ts >= -1e-8 * max(nrms, 1.0):
+ a = 1.0 + ts
+ s[:dims['l']] += a
+ s[indq[:-1]] += a
+ ind = dims['l'] + sum(dims['q'])
+ for m in dims['s']:
+ s[ind : ind+m*m : m+1] += a
+ ind += m**2
+
+ if tz >= -1e-8 * max(nrmz, 1.0):
+ a = 1.0 + tz
+ z[:dims['l']] += a
+ z[indq[:-1]] += a
+ ind = dims['l'] + sum(dims['q'])
+ for m in dims['s']:
+ z[ind : ind+m*m : m+1] += a
+ ind += m**2
+
+ elif primalstart is None and dualstart is not None:
+
+ if ts >= -1e-8 * max(nrms, 1.0):
+ a = 1.0 + ts
+ s[:dims['l']] += a
+ s[indq[:-1]] += a
+ ind = dims['l'] + sum(dims['q'])
+ for m in dims['s']:
+ s[ind : ind+m*m : m+1] += a
+ ind += m**2
+
+ elif primalstart is not None and dualstart is None:
+
+ if tz >= -1e-8 * max(nrmz, 1.0):
+ a = 1.0 + tz
+ z[:dims['l']] += a
+ z[indq[:-1]] += a
+ ind = dims['l'] + sum(dims['q'])
+ for m in dims['s']:
+ z[ind : ind+m*m : m+1] += a
+ ind += m**2
+
+
+ if 0: # random starting points for debugging
+
+ from cvxopt import random
+ n = len(c)
+ x = random.normal(n,1)
+ y = random.normal(p,1)
+ s, z = matrix(0.0, (cdim,1)), matrix(0.0, (cdim,1))
+ s[:dims['l']] = random.uniform(dims['l'], 1)
+ z[:dims['l']] = random.uniform(dims['l'], 1)
+ for k in xrange(Nq):
+ mk = dims['q'][k]
+ sk = random.normal(mk, 1)
+ sk[0] = blas.nrm2(sk[1:]) + random.uniform(1)
+ s[indq[k]:indq[k+1]] = sk
+ zk = random.normal(mk, 1)
+ zk[0] = blas.nrm2(zk[1:]) + random.uniform(1)
+ z[indq[k]:indq[k+1]] = zk
+ for k in xrange(Ns):
+ mk = dims['s'][k]
+ sk = random.normal(mk, mk)
+ sk = sk*sk.T
+ s[inds[k]:inds[k+1]] = sk[:]
+ zk = random.normal(mk, mk)
+ zk = zk*zk.T
+ z[inds[k]:inds[k+1]] = zk[:]
+
+ if 0: # starting point = e for debugging
+
+ from cvxopt import random
+ n = len(c)
+ x = matrix(0.0, (n,1))
+ y = matrix(0.0, (p,1))
+ s, z = matrix(0.0, (cdim,1)), matrix(0.0, (cdim,1))
+ s[:dims['l']] = 1.0
+ s[indq[:-1]] = 1.0
+ ind = dims['l'] + sum(dims['q'])
+ for m in dims['s']:
+ s[ind : ind+m*m : m+1] = 1.0
+ ind += m**2
+ z[:dims['l']] = 1.0
+ z[indq[:-1]] = 1.0
+ ind = dims['l'] + sum(dims['q'])
+ for m in dims['s']:
+ z[ind : ind+m*m : m+1] = 1.0
+ ind += m**2
tau, kappa = 1.0, 1.0
- gap = blas.dot(sl, zl) + misc.sdot(ss, zs)
+ gap = sdot(s, z, dims)
for iters in xrange(MAXITERS):
- # hresx = -A'(y) - Gl'(zl) - Gs'(zs)
- A(y, hresx, alpha=-1.0, trans='T')
- Gl(zl, hresx, alpha=-1.0, beta=1.0, trans='T')
- Gs(zs, hresx, alpha=-1.0, beta=1.0, trans='T')
+ # hresx = -A'(y) - G'(z)
+ Af(y, hresx, alpha = -1.0, trans = 'T')
+ Gf(z, hresx, alpha = -1.0, beta = 1.0, trans = 'T')
- # resx = hresx - c*tau = -A'(y) - Gl'(zl) - Gs'(zs) - c*tau
+ # resx = hresx - c*tau
+ # = -A'(y) - G'(z) - c*tau
xcopy(hresx, resx)
- xaxpy(c, resx, alpha=-tau)
+ xaxpy(c, resx, alpha = -tau)
# hresy = A(x)
- A(x, hresy)
+ Af(x, hresy)
- # resy = hresy - b*tau = A(x) - b*tau
+ # resy = hresy - b*tau
+ # = A(x) - b*tau
ycopy(hresy, resy)
- yaxpy(b, resy, alpha=-tau)
-
- # hreszl = sl + Gl(x)
- Gl(x, hreszl)
- blas.axpy(sl, hreszl)
+ yaxpy(b, resy, alpha = -tau)
- # reszl = hreszl - hl*tau = sl + Gl(x) - hl*tau
- blas.scal(0, reszl)
- blas.axpy(hreszl, reszl)
- blas.axpy(hl, reszl, alpha=-tau)
+ # hresz = s + G(x)
+ Gf(x, hresz)
+ blas.axpy(s, hresz)
- # hreszs = ss + Gs(x)
- Gs(x, hreszs)
- misc.saxpy(ss, hreszs)
+ # resz = hresz - h*tau
+ # = s + G(x) - h*tau
+ blas.scal(0, resz)
+ blas.axpy(hresz, resz)
+ blas.axpy(h, resz, alpha = -tau)
- # reszs = hreszs - hs*tau = ss + Gs(x) - hs*tau
- misc.sscal(0, reszs)
- misc.saxpy(hreszs, reszs)
- misc.saxpy(hs, reszs, alpha=-tau)
-
- # rest = kappa + <c,x> + <b,y> + <hl,zl> + <hs,zs>
- cx, by = xdot(c,x), ydot(b,y)
- hlzl, hszs = blas.dot(hl,zl), misc.sdot(hs,zs)
- rest = kappa + cx + by + hlzl + hszs
+ # rest = kappa + <c,x> + <b,y> + <h,z>
+ cx, by, hz = xdot(c,x), ydot(b,y), sdot(h, z, dims)
+ rest = kappa + cx + by + hz
# stopping criteria
- pcost, dcost = cx/tau, -(by + hlzl + hszs) / tau
- nrmhresx = math.sqrt(xdot(hresx, hresx))
- nrmresx = math.sqrt(xdot(resx, resx)) / tau
- nrmhresy = math.sqrt(ydot(hresy, hresy))
- nrmresy = math.sqrt(ydot(resy, resy)) / tau
- nrmhreszl = blas.nrm2(hreszl)
- nrmreszl = blas.nrm2(reszl) / tau
- nrmhreszs = misc.snrm2(hreszs)
- nrmreszs = misc.snrm2(reszs) / tau
+ pcost, dcost = cx/tau, -(by + hz) / tau
+ nrmhresx = math.sqrt( xdot(hresx, hresx) )
+ nrmresx = math.sqrt( xdot(resx, resx) ) / tau
+ nrmhresy = math.sqrt( ydot(hresy, hresy) )
+ nrmresy = math.sqrt( ydot(resy, resy) ) / tau
+ nrmhresz = snrm2(hresz, dims)
+ nrmresz = snrm2(resz, dims) / tau
if pcost < 0.0:
relgap = gap / -pcost
elif dcost > 0.0:
@@ -400,255 +1595,301 @@ def conelp(c, kktsolver, Gl=None, hl=None, Gs=None, hs=None, A=None,
if iters == 0:
nrmresx0 = max(1.0, math.sqrt(xdot(c,c)))
nrmresy0 = max(1.0, math.sqrt(ydot(b,b)))
- nrmreszl0 = max(1.0, blas.nrm2(hl))
- nrmreszs0 = max(1.0, misc.snrm2(hs))
+ nrmresz0 = max(1.0, snrm2(h, dims))
if show_progress:
if iters==0:
print "% 10s% 12s% 10s% 8s% 7s % 5s" %("pcost", "dcost",
"gap", "pres", "dres", "k/t")
print "%2d: % 8.4e % 8.4e % 4.0e% 7.0e% 7.0e% 7.0e" \
- %(iters, pcost, dcost, gap, max(nrmreszl/nrmreszl0,
- nrmreszs/nrmreszs0, nrmresy/nrmresy0), nrmresx/nrmresx0,
- kappa/tau)
-
- if max(nrmreszl/nrmreszl0, nrmreszs/nrmreszs0,
- nrmresy/nrmresy0) <= FEASTOL and \
- nrmresx/nrmresx0 <= FEASTOL and (gap <= ABSTOL or
- (relgap is not None and relgap <= RELTOL)):
+ %(iters, pcost, dcost, gap, max(nrmresz/nrmresz0,
+ nrmresy/nrmresy0), nrmresx/nrmresx0, kappa/tau)
+
+ if max(nrmresz/nrmresz0, nrmresy/nrmresy0) <= FEASTOL and \
+ nrmresx/nrmresx0 <= FEASTOL and ( gap <= ABSTOL or
+ (relgap is not None and relgap <= RELTOL) ):
xscal(1.0/tau, x)
yscal(1.0/tau, y)
- blas.scal(1.0/tau, sl)
- misc.sscal(1.0/tau, ss)
- blas.scal(1.0/tau, zl)
- misc.sscal(1.0/tau, zs)
- return {'status': 'optimal', 'x': x, 'y': y, 'sl': sl,
- 'ss': ss, 'zl': zl, 'zs': zs}
-
- elif hlzl + hszs + by < 0.0 and nrmhresx / (-hlzl - hszs - by) \
- <= FEASTOL:
- yscal(1.0/(-hlzl - hszs - by), y)
- blas.scal(1.0/(-hlzl - hszs - by), zl)
- misc.sscal(1.0/(-hlzl - hszs - by), zs)
- return {'status': 'primal infeasible', 'x': None,
- 'sl': None, 'ss': None, 'y': y, 'zl': zl, 'zs': zs}
-
- elif cx < 0.0 and max(nrmhresy, nrmhreszl, nrmhreszs) / (-cx) \
- <= FEASTOL:
+ blas.scal(1.0/tau, s)
+ blas.scal(1.0/tau, z)
+ ind = dims['l'] + sum(dims['q'])
+ for m in dims['s']:
+ symm(s, m, ind)
+ symm(z, m, ind)
+ ind += m**2
+ return {'status': 'optimal', 'x': x, 'y': y, 's': s, 'z': z}
+
+ elif hz + by < 0.0 and nrmhresx/nrmresx0 / (-hz - by) <= FEASTOL:
+ yscal(1.0/(-hz - by), y)
+ blas.scal(1.0/(-hz - by), z)
+ ind = dims['l'] + sum(dims['q'])
+ for m in dims['s']:
+ symm(z, m, ind)
+ ind += m**2
+ return {'status': 'primal infeasible', 'x': None, 's': None,
+ 'y': y, 'z': z }
+
+ elif cx < 0.0 and max(nrmhresy/nrmresy0, nrmhresz/nrmresz0) \
+ / (-cx) <= FEASTOL:
xscal(1.0/(-cx), x)
- blas.scal(1.0/(-cx), sl)
- misc.sscal(1.0/(-cx), ss)
- return {'status': 'dual infeasible', 'x': x, 'sl': sl,
- 'ss': ss, 'y': None, 'zl': None, 'zs': None}
+ blas.scal(1.0/(-cx), s)
+ ind = dims['l'] + sum(dims['q'])
+ for m in dims['s']:
+ symm(s, m, ind)
+ ind += m**2
+ return {'status': 'dual infeasible', 'x': x, 's': s, 'y': None,
+ 'z': None}
if iters == 0:
- # Compute initial scaling.
+ # Compute initial scaling W:
+ #
+ # W * z = W^{-T} * s = lambda
+ # dg * tau = 1/dg * kappa = lambdag.
+
+ W = {}
+
+
+ # For kappa, tau block:
#
- # For the linear inequalities:
+ # dg = sqrt( kappa / tau )
+ # dgi = sqrt( tau / kappa )
+ # lambda_g = sqrt( tau * kappa )
+ #
+ # lambda_g is stored in the last position of lmbda.
+
+ dg = math.sqrt( kappa / tau )
+ dgi = math.sqrt( tau / kappa )
+ lmbda[-1] = math.sqrt( tau * kappa )
+
+
+ # For the 'l' block:
#
- # dl = sqrt(sl./zl), dli = sqrt(zl./sl)
- # lmbdl = sqrt(sl.*zl).
+ # W['d'] = sqrt( sk ./ zk )
+ # W['di'] = sqrt( zk ./ sk )
+ # lambdak = sqrt( sk .* zk )
#
- # Store lmbdl in lmbda[:ml].
+ # where sk and zk are the first dims['l'] entries of s and z.
+ # lambda_k is stored in the first dims['l'] positions of lmbda.
+
+ m = dims['l']
+ W['d'] = base.sqrt( base.div( s[:m], z[:m] ))
+ W['di'] = W['d']**-1
+ lmbda[:m] = base.sqrt( base.mul( s[:m], z[:m] ) )
+
+
+ # For the 'q' blocks, compute lists 'v', 'beta' of length Nq.
#
- # Matrix inequalities: lists of square matrices r, rti with
+ # The vector v[k] has unit hyperbolic norm:
+ #
+ # (sqrt( v[k]' * J * v[k] ) = 1 with J = [1, 0; 0, -I]).
+ #
+ # beta[k] is a positive scalar.
#
- # r[k]'*ss[k]^{-1}*r[k] = diag(lmbds[k])^{-1}
- # r[k]'*zs[k]*r[k] = diag(lmbds[k])
+ # The hyperbolic Householder matrix H = 2*v[k]*v[k]' - J
+ # defined by v[k] satisfies
+ #
+ # (beta[k] * H) * zk = (beta[k] * H) \ sk = lambda_k
#
- # rti[k] is the inverse of r[k]', so that
+ # where sk = s[indq[k]:indq[k+1]], zk = z[indq[k]:indq[k+1]].
#
- # rti[k]'*ss[k]*rti[k] = diag(lmbds[k])^{-1}
- # rti[k]'*zs[k]^{-1}*rti[k] = diag(lmbds[k]).
+ # lambda_k is stored in lmbda[indq[k]:indq[k+1]].
+
+ ind = dims['l']
+ W['v'] = [ matrix(0.0, (k,1)) for k in dims['q'] ]
+ W['beta'] = Nq * [ 0.0 ]
+
+ for k in xrange(Nq):
+ m = dims['q'][k]
+ v = W['v'][k]
+
+ # a = sqrt( sk' * J * sk ) where J = [1, 0; 0, -I]
+ aa = jnrm2(s, offset = ind, n = m)
+
+ # b = sqrt( zk' * J * zk )
+ bb = jnrm2(z, offset = ind, n = m)
+
+ # beta[k] = ( a / b )**1/2
+ W['beta'][k] = math.sqrt( aa / bb )
+
+ # c = sqrt( (sk/a)' * (zk/b) + 1 ) / sqrt(2)
+ cc = math.sqrt( ( blas.dot(s, z, n = m, offsetx = ind,
+ offsety = ind) / aa / bb + 1.0 ) / 2.0 )
+
+ # vk = 1/(2*c) * ( (sk/a) + J * (zk/b) )
+ blas.copy(z, v, offsetx = ind, n = m)
+ blas.scal(-1.0/bb, v)
+ v[0] *= -1.0
+ blas.axpy(s, v, 1.0/aa, offsetx = ind, n = m)
+ blas.scal(1.0/2.0/cc, v)
+
+ # v[k] = 1/sqrt(2*(vk0 + 1)) * ( vk + e ), e = [1; 0]
+ v[0] += 1.0
+ blas.scal(1.0/math.sqrt(2.0 * v[0]), v)
+
+
+ # To get the scaled variable lambda_k
+ #
+ # d = sk0/a + zk0/b + 2*c
+ # lambda_k = [ c; (c + zk0/b)/d * sk1/a +
+ # (c + sk0/a)/d * zk1/b ]
+ # lambda_k *= sqrt(a * b)
+
+ lmbda[ind] = cc
+ dd = 2*cc + s[ind]/aa + z[ind]/bb
+ blas.copy(s, lmbda, offsetx = ind+1, offsety = ind+1,
+ n = m-1)
+ blas.scal((cc + z[ind]/bb)/dd/aa, lmbda, n = m-1, offset
+ = ind+1)
+ blas.axpy(z, lmbda, (cc + s[ind]/aa)/dd/bb, n = m-1,
+ offsetx = ind+1, offsety = ind+1)
+ blas.scal(math.sqrt(aa*bb), lmbda, offset = ind, n = m)
+
+ ind += m
+
+
+ # For the 's' blocks: compute two lists 'r' and 'rti' of
+ # length Ns.
#
- # Store (lmbds[0], ..., lmbds[N-1]) in lmbda[ml:ml+sum(ms)].
+ # r[k]' * sk^{-1} * r[k] = diag(lambda_k)^{-1}
+ # r[k]' * zk * r[k] = diag(lambda_k)
#
- # Also compute gamma: gamma[k] has entries
+ # where sk and zk are the entries inds[k] : inds[k+1] of
+ # s and z, reshaped into symmetric matrices.
#
- # gamma[k][i,j] = 0.5*(lmbds[k][i]+lmbds[k][j])
+ # rti[k] is the inverse of r[k]', so
#
- # For kappa, tau:
+ # rti[k]' * sk * rti[k] = diag(lambda_k)^{-1}
+ # rti[k]' * zk^{-1} * rti[k] = diag(lambda_k).
#
- # dg = sqrt(kappa/tau), dgi = sqrt(tau/kappa)
- # lmbdg = sqrt(tau*kappa)
+ # The vectors lambda_k are stored in
#
- # Store lmbdg in lmbda[-1].
-
- dli = base.sqrt(misc.ihad(zl, sl))
- lmbda[:ml] = base.sqrt(misc.had(sl,zl))
+ # lmbda[ dims['l'] + sum(dims['q']) : -1 ]
+
+ W['r'] = [ matrix(0.0, (m,m)) for m in dims['s'] ]
+ W['rti'] = [ matrix(0.0, (m,m)) for m in dims['s'] ]
- dgi = math.sqrt(tau/kappa)
- lmbda[-1] = math.sqrt(tau*kappa)
+ ind2 = ind
+ for k in xrange(Ns):
+ m = dims['s'][k]
+ r, rti = W['r'][k], W['rti'][k]
- ind = ml
- for k in xrange(len(ms)):
+ # Factor sk = L1*L1'; store L1 in ds[inds[k]:inds[k+1]].
+ blas.copy(s, ds, offsetx = ind2, offsety = ind2, n = m**2)
+ lapack.potrf(ds, n = m, ldA = m, offsetA = ind2)
- # Factor ss = L1*L1', zs = L2*L2'. Store L1 and L2 in
- # dss and dzs.
- blas.copy(ss[k], dss[k]); lapack.potrf(dss[k])
- blas.copy(zs[k], dzs[k]); lapack.potrf(dzs[k])
+ # Factor zs[k] = L2*L2'; store L2 in dz[inds[k]:inds[k+1]].
+ blas.copy(z, dz, offsetx = ind2, offsety = ind2, n = m**2)
+ lapack.potrf(dz, n = m, ldA = m, offsetA = ind2)
- # SVD of L2'*L1 = U*diag(lambda)*V'. Keep U in work.
- for i in xrange(ms[k]): dss[k][i,i+1:] = 0
- blas.copy(dss[k], work)
- blas.trmm(dzs[k], work, transA='T', ldB=ms[k], n=ms[k],
- m=ms[k])
- lapack.gesvd(work, lmbda, jobu='O', ldA=ms[k], m=ms[k],
- n=ms[k], offsetS=ind)
+ # SVD L2'*L1 = U*diag(lambda_k)*V'. Keep U in work.
+ for i in xrange(m):
+ blas.scal(0.0, ds, offset = ind2 + i*m, n = i)
+ blas.copy(ds, work, offsetx = ind2, n = m**2)
+ blas.trmm(dz, work, transA = 'T', ldA = m, ldB = m, n = m,
+ m = m, offsetA = ind2)
+ lapack.gesvd(work, lmbda, jobu = 'O', ldA = m, m = m,
+ n = m, offsetS = ind)
- # r = L2^{-T}*U*diag(sqrt(lambda))
- # = L1*V*diag(1./sqrt(lambda))
- # rti = L2*U*diag(1./sqrt(lambda))
- # = L1^{-T}*V*diag(sqrt(lambda))
- blas.copy(work, r[k], n=ms[k]*ms[k])
- blas.trsm(dzs[k], r[k], transA='T')
- blas.copy(work, rti[k], n=ms[k]*ms[k])
- blas.trmm(dzs[k], rti[k])
- for i in xrange(ms[k]):
- a = math.sqrt(lmbda[ind+i])
- blas.scal(a, r[k], offset=ms[k]*i, n=ms[k])
- blas.scal(1/a, rti[k], offset=ms[k]*i, n=ms[k])
-
- # gamma[k][i,j] = .5 * (lambda_i + lambda_j)
- # Use blas.ger to make sure there are no zeros or nan's
- # in upper triangular part.
- blas.scal(0.0, gamma[k])
- blas.ger(lmbda, ones, gamma[k], alpha=.5, offsetx=ind)
- blas.ger(ones, lmbda, gamma[k], alpha=.5, offsety=ind)
-
- ind += ms[k]
+ # r = L2^{-T} * U
+ blas.copy(work, r, n = m*m)
+ blas.trsm(dz, r, transA = 'T', m = m, n = m, ldA = m,
+ offsetA = ind2)
+
+ # rti = L2 * U
+ blas.copy(work, rti, n = m*m)
+ blas.trmm(dz, rti, m = m, n = m, ldA = m, offsetA = ind2)
+
+ # r := r * diag(sqrt(lambda_k))
+ # rti := rti * diag(1 ./ sqrt(lambda_k))
+ for i in xrange(m):
+ a = math.sqrt( lmbda[ind+i] )
+ blas.scal(a, r, offset = m*i, n = m)
+ blas.scal(1.0/a, rti, offset = m*i, n = m)
+
+ ind += m
+ ind2 += m*m
+
-
# Define a function
#
- # solve_newton(dx, dy, dzl, dzs, dtau, dsl, dss, dkappa)
+ # solve_newton(dx, dy, dz, dtau, ds, dkappa)
#
# for solving
#
- # [0 ] [ 0 A' Gl' Gs' 0 ] [dx ] [rhsx ]
- # [0 ] [-A 0 0 0 b ] [dy ] [rhsy ]
- # [dsl ] - [-Gl 0 0 0 hl] [dzl ] = -[rhszl ]
- # [dsk ] [-Gs 0 0 0 hs] [dzs ] [rhszs ]
- # [dkappa] [-c' -b' -hl' -hs' 0 ] [dtau] [rhstau]
- #
- # sl.*dzl + zl.*dsl = -rhssl
- # Hc(dzs*ss + zs*dss) = -rhsss
+ # [ 0 ] [ 0 A' G' c ] [ dx ] [ rhsx ]
+ # [ 0 ] [ -A 0 0 b ] [ dy ] [ rhsy ]
+ # [ ds ] - [ -G 0 0 h ] [ dz ] = -[ rhsz ]
+ # [ dkappa ] [ -c' -b' -h' 0 ] [ dtau ] [ rhstau ]
+ #
+ # s o dz + z o dz = -rhss
# kappa*dtau + tau*dkappa = -rhskappa.
- #
- # where Hc(x) = 0.5 * (r'*x*(r')^{-1} + r^{-1}*x'*r).
-
- try: f = kktsolver(dli, rti)
+
+ try: f = kktsolver(W)
except ArithmeticError:
if iters == 0 and primalstart and dualstart:
- raise ValueError, "Rank(A) < p or Rank([Gl; Gs; A]) < n"
+ raise ValueError, "Rank(A) < p or Rank([G; A]) < n"
else:
raise ArithmeticError, "singular KKT matrix"
- # thl := dli .* hl; ths := rti'*hs*rti
- blas.copy(hl, thl)
- misc.had2(thl, dli)
- misc.cngrnc(rti, hs, ths, trans='T')
+ # th = W^{-T} * h
+ blas.copy(h, th)
+ scale(th, W, trans = 'T', inverse = 'I')
# Solve
#
- # [ 0 A' Gl' Gs' ] [dx1 ] [c ]
- # [-A 0 0 0 ] [dy1 ] = -dgi * [b ]
- # [-Gl 0 diag(dl)^2 0 ] [dzl1] [hl]
- # [-Gs 0 0 r*r'*()*r*r'] [dzs1] [hs]
+ # [ 0 A' G' ] [ dx1 ] [ c ]
+ # [-A 0 0 ]*[ dy1 ] = -dgi * [ b ].
+ # [-G 0 W'*W ] [ dzl ] [ h ]
xcopy(c, dx1); xscal(-1, dx1)
ycopy(b, dy1)
- blas.copy(hl, dzl1)
- misc.scopy(hs, dzs1)
- f(dx1, dy1, dzl1, dzs1)
+ blas.copy(h, dz1)
+ try: f(dx1, dy1, dz1)
+ except ArithmeticError:
+ if iters == 0 and primalstart and dualstart:
+ raise ValueError, "Rank(A) < p or Rank([G; A]) < n"
+ else:
+ raise ArithmeticError, "singular KKT matrix"
xscal(dgi, dx1)
yscal(dgi, dy1)
- blas.scal(dgi, dzl1)
- misc.sscal(dgi, dzs1)
-
- if DEBUG:
- print "first 4x4 system"
-
- # unscaled variables
- duzl1 = misc.had(dzl1, dli)
- duzs1 = [ +z_k for z_k in dzs1 ]
- misc.cngrnc(rti, duzs1, duzs1)
-
- rxdb = xnewcopy(c)
- A(dy1, rxdb, alpha=1, beta=dgi, trans='T')
- Gl(duzl1, rxdb, alpha=1, beta=1, trans='T')
- Gs(duzs1, rxdb, alpha=1, beta=1, trans='T')
- print " norm(rxdb) = %e" %math.sqrt(xdot(rxdb,rxdb))
-
- rydb = ynewcopy(b)
- A(dx1, rydb, alpha=-1.0, beta=dgi)
- print " norm(rydb) = %e" %math.sqrt(ydot(rydb,rydb))
+ blas.scal(dgi, dz1)
- rzldb = +hl
- Gl(dx1, rzldb, alpha=-1.0, beta=dgi)
- rzldb += misc.ihad(duzl1, misc.had(dli,dli))
- print " norm(rzldb) = %e" %blas.nrm2(rzldb)
-
- rzsdb = [ +hs_k for hs_k in hs ]
- Gs(dx1, rzsdb, alpha=-1.0, beta=dgi)
- misc.cngrnc(r, duzs1, duzs1, trans='T')
- misc.cngrnc(r, duzs1, rzsdb, alpha=1.0, beta=1.0)
- print " norm(rzsdb) = %e\n" %misc.snrm2(rzsdb)
-
-
- def solve_newton1(dx, dy, dzl, dzs, dtau, dsl, dss, dkappa):
+ def solve_newton1(dx, dy, dz, dtau, ds, dkappa):
# Solve without refinement
#
- # [0 ] [ 0 A' Gl' Gs' 0 ] [dx ] [rhsx ]
- # [0 ] [-A 0 0 0 b ] [dy ] [rhsy ]
- # [dsl ] - [-Gl 0 0 0 hl] [dzl ] = -[rhszl ]
- # [dsk ] [-Gs 0 0 0 hs] [dzs ] [rhszs ]
- # [dkappa] [-c' -b' -hl' -hs' 0 ] [dtau] [rhstau]
- #
- # sl.*dzl + zl.*dsl = -rhssl
- # Hc(dzs*ss + zs*dss) = -rhsss
- # kappa*dtau + tau*dkappa = -rhskappa
+ # [0 ] [ 0 A' G' 0 ] [dx ] [rhsx ]
+ # [0 ] [-A 0 0 b ] [dy ] [rhsy ]
+ # [ds ] - [-G 0 0 h ] [dz ] = -[rhsz ]
+ # [dkappa] [-c' -b' -h' 0 ] [dtau] [rhstau]
+ #
+ # s o dz + z o dz = -rhss
+ # kappa*dtau + tau*dkappa = -rhskappa.
#
- # Last three equations in scaled variables:
+ # Last two equations in scaled variables:
#
- # lmbdl .* (dzl + dsl) = -rhssl
- # gamma .* (dzs + dss) = -rhsss
- # lmbdg * (dtau + dkappa) = -rhskappa.
+ # lmbda o (W*dz + W^{-T}*ds) = -rhss
+ # lmbdg * (w*dtau + dkappa/w) = -rhskappa.
#
- # On entry, the righthand sides are stored in dx, dy, dzl,
- # dzs, dtau, dssl, dsss, dkappa.
- # On exit, scaled quantities are returned for dsl, dzl,
- # dss, dzs.
-
- if DEBUG:
- rxdb = xnewcopy(dx)
- rydb = ynewcopy(dy)
- rzldb = +dzl
- rzsdb = [ +dzs_k for dzs_k in dzs ]
- rtdb = dtau[0]
- rsldb = +dsl
- rssdb = [ +dss_k for dss_k in dss ]
- rkdb = dkappa[0]
+ # On entry, the righthand sides are stored in dx, dy, dz, dtau,
+ # ds, dkappa. On exit, scaled quantities are returned for ds,
+ # dz, dtau, dkappa.
# dy := -dy = -rhsy
yscal(-1.0, dy)
- # dsl := -dsl ./ lmbdl (replace with btmv)
- misc.ihad2(dsl, lmbda, n=ml); blas.scal(-1,dsl)
-
- # dzl := -(dzl + dsl ./ dli) = -(rhszl - rhssl./zl)
- blas.axpy(misc.ihad(dsl,dli), dzl); blas.scal(-1, dzl)
+ # ds := -lambda o\ ds (\o is inverse of o)
+ sinv(ds, lmbda, dims)
+ blas.scal(-1.0, ds)
- # dss := -dss ./ gamma
- for num, den in zip(dss, gamma): misc.ihad2(num, den)
- misc.sscal(-1.0, dss)
+ # dz := -(dz + W'*ds) = -rhsz + W'*(lambda o\ rhss)
+ blas.copy(ds, ds1)
+ scale(ds1, W, trans = 'T')
+ blas.axpy(ds1, dz)
+ blas.scal(-1.0, dz)
- # dzs := -(dzs + r*dss*r') = -(rhszs - r*(rhss./gamma)*r')
- misc.cngrnc(r, dss, dzs, alpha=-1.0, beta=-1.0)
-
# dkappa[0] := -dkappa[0]/lmbd[-1]
# = -rhskappa / sqrt(kappa*tau)
dkappa[0] = -dkappa[0] / lmbda[-1]
@@ -660,479 +1901,519 @@ def conelp(c, kktsolver, Gl=None, hl=None, Gs=None, hs=None, A=None,
# Solve
#
- # [ 0 A' Gl' Gs' 0 ][dx ] [rhsx ]
- # [-A 0 0 0 b ][dy ] [rhsy ]
- # [-Gl 0 Dl 0 hl ][dzl ] = [rhszl-rhssl./zl ]
- # [-Gs 0 0 Ds hs ][dzs ] [rhszs-r*(rhszs./ga)*r']
- # [-c' -b' -hl' -hs' k/t][dtau] [rhst-rhsk/tau ]
- #
- # Dl = diag(dl)^2, Ds = r*r'*()*r*r'.
-
- if DEBUG:
- rxdb2 = xnewcopy(dx)
- rydb2 = ynewcopy(dy); xscal(-1, rydb2)
- rzldb2 = -dzl
- rzsdb2 = [-dzs_k for dzs_k in dzs]
-
- rxdb3 = xnewcopy(dx)
- rydb3 = ynewcopy(dy); xscal(-1, rydb3)
- rzldb3 = -dzl
- rzsdb3 = [-dzs_k for dzs_k in dzs]
-
- f(dx, dy, dzl, dzs)
-
- if DEBUG:
- print "second 4x4 system"
-
- # unscaled variables
- duzl = misc.had(dzl, dli)
- duzs = [ +z_k for z_k in dzs ]
- misc.cngrnc(rti, duzs, duzs)
-
- A(dy, rxdb2, alpha=-1.0, beta=1.0, trans='T')
- Gl(duzl, rxdb2, alpha=-1.0, beta=1.0, trans='T')
- Gs(duzs, rxdb2, alpha=-1.0, beta=1.0, trans='T')
- print " norm(rxdb2) = %e" \
- %math.sqrt(xdot(rxdb2,rxdb2))
-
- A(dx, rydb2, alpha=1.0, beta=1.0)
- print " norm(rydb2) = %e" \
- %math.sqrt(ydot(rydb2,rydb2))
-
- Gl(dx, rzldb2, alpha=1.0, beta=1.0)
- rzldb2 -= misc.ihad(duzl, misc.had(dli,dli))
- print " norm(rzldb2) = %e" %blas.nrm2(rzldb2)
-
- Gs(dx, rzsdb2, alpha=1.0, beta=1.0)
- misc.cngrnc(r, duzs, duzs, trans='T')
- misc.cngrnc(r, duzs, rzsdb2, alpha=-1.0, beta=1.0)
- print " norm(rzsdb2) = %e\n" %misc.snrm2(rzsdb2)
-
- dtau[0] = dgi * ( dtau[0] + xdot(c,dx) + ydot(b,dy) +
- blas.dot(thl, dzl) + misc.sdot(ths, dzs) ) / \
- ( 1.0 + blas.dot(dzl1, dzl1) + misc.sdot(dzs1, dzs1) )
- xaxpy(dx1, dx, alpha=dtau[0])
- yaxpy(dy1, dy, alpha=dtau[0])
- blas.axpy(dzl1, dzl, alpha=dtau[0])
- misc.saxpy(dzs1, dzs, alpha=dtau[0])
+ # [ 0 A' G' 0 ] [dx ] [ rhsx ]
+ # [ -A 0 0 b ] [dy ] [ rhsy ]
+ # [ -G 0 W'*W h ] [dz ] = [ rhsz - lambda o\ rhss ]
+ # [ -c' -b' -h' k/t ] [dtau] [ rhst - rhsk/tau ].
+
+ f(dx, dy, dz)
+
+ dtau[0] = dgi * ( dtau[0] + xdot(c, dx) + ydot(b, dy) +
+ sdot(th, dz, dims) ) / ( 1.0 + sdot(dz1, dz1, dims) )
+ xaxpy(dx1, dx, alpha = dtau[0])
+ yaxpy(dy1, dy, alpha = dtau[0])
+ blas.axpy(dz1, dz, alpha = dtau[0])
- # dsl := dsl - dzl = -rhsl ./ lmbdl - dtzl
- blas.axpy(dzl, dsl, alpha=-1)
-
- # dss := dss - dzs = -rhss ./ Gamma - dzs
- misc.saxpy(dzs, dss, alpha=-1)
+ # ds := ds - dz = - lambda o\ rhs - dz
+ blas.axpy(dz, ds, alpha = -1)
dkappa[0] -= dtau[0]
- if DEBUG:
- print "solvenewton errors"
-
- # unscaled variables
- duzl = misc.had(dzl, dli)
- dusl = misc.ihad(dsl, dli)
- duzs = [ +z_k for z_k in dzs ]
- misc.cngrnc(rti, duzs, duzs)
- duss = [ +s_k for s_k in dss ]
- misc.cngrnc(r, duss, duss)
- dutau = dtau[0] * dgi
- dukappa = dkappa[0] / dgi
-
- print " norm(rsldb) = %e" %blas.nrm2(rsldb +
- misc.had(lmbda[:ml], dzl+dsl))
-
- for k in xrange(len(ms)):
- for j in xrange(ms[k]):
- rssdb[k][j,j+1:] = rssdb[k][j+1:,j].trans()
- rssdb[k] += misc.had(dzs[k] + dss[k], gamma[k])
- print " norm(rssdb) = %e" %misc.snrm2(rssdb)
- print " abs(rtdb) = %e" %abs(rtdb + dukappa +
- xdot(c,dx) + ydot(b,dy) + blas.dot(hl,duzl) +
- misc.sdot(hs,duzs))
-
- A(dy, rxdb, alpha=-1.0, beta=1.0, trans='T')
- Gl(duzl, rxdb, alpha=-1.0, beta=1.0, trans='T')
- Gs(duzs, rxdb, alpha=-1.0, beta=1.0, trans='T')
- xaxpy(c, rxdb, alpha=-dutau)
- print " norm(rxdb) = %e" %math.sqrt(xdot(rxdb,rxdb))
-
- A(dx, rydb, alpha=1.0, beta=1.0)
- yaxpy(b, rydb, alpha=-dutau)
- print " norm(rydb) = %e" %math.sqrt(ydot(rydb,rydb))
-
- rzldb += dusl - hl*dutau
- Gl(dx, rzldb, alpha=1.0, beta=1.0)
- print " norm(rzldb) = %e" %blas.nrm2(rzldb)
-
- misc.saxpy(duss, rzsdb)
- misc.saxpy(hs, rzsdb, alpha=-dutau)
- Gs(dx, rzsdb, alpha=1.0, beta=1.0)
- print " norm(rzsdb) = %e\n" %misc.snrm2(rzsdb)
-
- if refinement:
- def solve_newton(dx, dy, dzl, dzs, dtau, dsl, dss, dkappa):
+ if refinement:
+ def solve_newton(dx, dy, dz, dtau, ds, dkappa):
- if DEBUG:
- rxdb2 = xnewcopy(dx)
- rydb2 = ynewcopy(dy)
- rzldb2 = +dzl
- rzsdb2 = [+dzs_k for dzs_k in dzs]
- rtdb2 = dtau[0]
- rsldb2 = +dsl
- rssdb2 = [+dss_k for dss_k in dss]
- rkdb2 = dkappa[0]
-
# copy righthand sides to dx2 etc
xcopy(dx, dx2)
ycopy(dy, dy2)
- blas.copy(dzl, dzl2)
- misc.scopy(dzs, dzs2)
+ blas.copy(dz, dz2)
dtau2[0] = dtau[0]
- blas.copy(dsl, dsl2)
- misc.scopy(dss, dss2)
+ blas.copy(ds, ds2)
dkappa2[0] = dkappa[0]
- solve_newton1(dx, dy, dzl, dzs, dtau, dsl, dss, dkappa)
+ solve_newton1(dx, dy, dz, dtau, ds, dkappa)
# store residuals in dx2, dy2, etc
#
- # [0 ] [ 0 A' Gl' Gs' c ] [dx ] [rhsx ]
- # [0 ] [-A 0 0 0 b ] [dy ] [rhsy ]
- # [dsl ] - [-Gl 0 0 0 hl] [dzl ] + [rhszl ]
- # [dss ] [-Gs 0 0 0 hs] [dzs ] [rhszs ]
- # [dkappa] [-c' -b' -hl' -hs' 0 ] [dtau] [rhstau]
+ # [0 ] [ 0 A' G' c ] [dx ] [rhsx ]
+ # [0 ] [-A 0 0 b ] [dy ] [rhsy ]
+ # [ds ] - [-Gl 0 0 h ] [dz ] + [rhsz ]
+ # [dkappa] [-c' -b' -h' 0 ] [dtau] [rhstau]
#
- # sl.*dzl + zl.*dsl + rhssl
- # Hc(dzs*ss + zs*dss) + rhsss
- # kappa*dtau + tau*dkappa + rhskappa
+ # s o dz + z o dz = -rhss
+ # kappa*dtau + tau*dkappa = -rhskappa.
#
- # Last 3 equations with scaled variables:
- #
- # lmbdl .* (dzl + dsl) + rhssl
- # 0.5 * (diag(lmbds)*dzs + dzs*diag(lmbds) +
- # diag(lmbds)*dss + dss*diag(lmbds)) + rhsss
- # lmbdg * (dtau + dkappa) + rhskappa.
+ # Last two equations in scaled variables:
+ #
+ # lmbda o (W*dz + W^{-T}*ds) = -rhss
+ # lmbdg * (w*dtau + dkappa/w) = -rhskappa.
- # Store unscaled steps in sl, zl, ss, zs.
- blas.copy(dzl, zl)
- misc.had2(zl, dli)
- blas.copy(dsl, sl)
- misc.ihad2(sl, dli)
- misc.scopy(dzs, zs)
- misc.cngrnc(rti, zs, zs)
- misc.scopy(dss, ss)
- misc.cngrnc(r, ss, ss)
+ # Store unscaled steps in s, z.
+ blas.copy(dz, z)
+ scale(z, W, inverse = 'I')
+ blas.copy(ds, s)
+ scale(s, W, trans = 'T')
dutau = dtau[0] * dgi
dukappa = dkappa[0] / dgi
- A(dy, dx2, alpha=-1.0, beta=1.0, trans='T')
- Gl(zl, dx2, alpha=-1.0, beta=1.0, trans='T')
- Gs(zs, dx2, alpha=-1.0, beta=1.0, trans='T')
- xaxpy(c, dx2, alpha=-dutau)
+ Af(dy, dx2, alpha = -1.0, beta = 1.0, trans = 'T')
+ Gf(z, dx2, alpha = -1.0, beta = 1.0, trans = 'T')
+ xaxpy(c, dx2, alpha =- dutau)
+
+ Af(dx, dy2, alpha = 1.0, beta = 1.0)
+ yaxpy(b, dy2, alpha = -dutau)
- A(dx, dy2, alpha=1.0, beta=1.0)
- yaxpy(b, dy2, alpha=-dutau)
+ Gf(dx, dz2, alpha = 1.0, beta = 1.0)
+ blas.axpy(h, dz2, alpha = -dutau)
+ blas.axpy(s, dz2)
- Gl(dx, dzl2, alpha=1.0, beta=1.0)
- blas.axpy(hl, dzl2, alpha=-dutau)
- blas.axpy(sl, dzl2)
+ dtau2[0] += dukappa + xdot(c,dx) + ydot(b,dy) + sdot(h, z,
+ dims)
- Gs(dx, dzs2, alpha=1.0, beta=1.0)
- misc.saxpy(hs, dzs2, alpha=-dutau)
- misc.saxpy(ss, dzs2)
+ # s := lmbda o (W*dz + W^{-T}*ds)
+ blas.copy(ds, s)
+ blas.axpy(dz, s)
+ sprod(s, lmbda, dims, diag = 'D')
+
+ blas.axpy(s, ds2)
- dtau2[0] += dukappa + xdot(c,dx) + ydot(b,dy) + \
- blas.dot(hl, zl) + misc.sdot(hs, zs)
+ dkappa2[0] += lmbda[-1] * (dtau[0] + dkappa[0])
+
+ solve_newton1(dx2, dy2, dz2, dtau2, ds2, dkappa2)
+
+ xaxpy(dx2, dx)
+ yaxpy(dy2, dy)
+ blas.axpy(dz2, dz)
+ dtau[0] += dtau2[0]
+ blas.axpy(ds2, ds)
+ dkappa[0] += dkappa2[0]
+
+ else: # no iterative refinement
+
+ solve_newton = solve_newton1
+
+ mu = blas.nrm2(lmbda)**2 / (cdeg + 1)
+
+
+ # Affine scaling step is solution with right hand side
+ #
+ # rhsx = resx
+ # rhsy = resy
+ # rhsz = resz
+ # rhstau = rest
+ # rhss = lambda o lambda
+ # rhskappa = kappa * tau.
+
+ # dx := resx, dy := resy, dz := resz, dtau := rest
+ xcopy(resx, dx)
+ ycopy(resy, dy)
+ blas.copy(resz, dz)
+ dtau[0] = rest
+
+ # lmbdasq := lmbda o lmbda
+ blas.copy(lmbda, lmbdasq)
+ blas.tbmv(lmbda, lmbdasq, n = dims['l'], k = 0, ldA = 1)
+ ind = dims['l']
+ for m in dims['q']:
+ lmbdasq[ind] = blas.nrm2(lmbda, offset = ind, n = m)**2
+ blas.scal(2.0*lmbda[ind], lmbdasq, n = m-1, offset = ind+1)
+ ind += m
+ # Diagonal symmetric matrices are stored as vectors in lmbdasq.
+ blas.tbmv(lmbda, lmbdasq, n = sum(dims['s']) + 1, k = 0, ldA = 1,
+ offsetA = ind, offsetx = ind)
+
+ # ds := lambda o lambda
+ blas.copy(lmbdasq, ds, n = dims['l'] + sum(dims['q']))
+ ind = dims['l'] + sum(dims['q'])
+ ind2 = ind
+ blas.scal(0.0, ds, offset = ind)
+ for m in dims['s']:
+ blas.copy(lmbdasq, ds, n = m, offsetx = ind2, offsety = ind,
+ incy = m+1)
+ ind += m*m
+ ind2 += m
+
+ # dkappa = kappa*tau
+ dkappa[0] = lmbdasq[-1]
+
+ solve_newton(dx, dy, dz, dtau, ds, dkappa)
+
+ # Maximum step to boundary
+ blas.copy(ds, ds2)
+ scale2(lmbda, ds2, dims)
+ ts = max_step(ds2, dims)
+ blas.copy(dz, dz2)
+ scale2(lmbda, dz2, dims)
+ tz = max_step(dz2, dims)
+ tt = -dtau[0]/lmbda[-1]
+ tk = -dkappa[0]/lmbda[-1]
+ step = min(1.0, 1.0 / max( [ 0.0, ts, tz, tt, tk ] ))
+ sigma = (1.0 - step)**EXPON
+
+
+ # Centering-corrector step is solution with right hand side
+ #
+ # rhsx = (1 - sigma) * resx
+ # rhsy = (1 - sigma) * resy
+ # rhsz = (1 - sigma) * resz
+ # rhstau = (1 - sigma) * rest
+ # rhss = lambda o lambda + ds o dz - sigma*mu*e
+ # rhskappa = lambdag**2 + dkappa * dtau - sigma*mu.
+
+ # ds := dz o ds + lambda o lambda - sigma*mu*e
+ sprod(ds, dz, dims)
+ blas.axpy(lmbdasq, ds, n = dims['l'] + sum(dims['q']))
+ ds[:dims['l']] -= sigma*mu
+ ds[indq[:-1]] -= sigma*mu
+ ind = dims['l'] + sum(dims['q'])
+ ind2 = ind
+ for m in dims['s']:
+ blas.axpy(lmbdasq, ds, n = m, offsetx = ind2, offsety = ind,
+ incy = m+1)
+ ds[ind : ind+m*m : m+1] -= sigma*mu
+ ind += m*m
+ ind2 += m
+ dkappa[0] = lmbdasq[-1] + dkappa[0]*dtau[0] - sigma*mu
+ xcopy(resx, dx); xscal(1.0 - sigma, dx)
+ ycopy(resy, dy); yscal(1.0 - sigma, dy)
+ blas.copy(resz, dz); blas.scal(1.0 - sigma, dz)
+ dtau[0] = (1.0 - sigma) * rest
+
+ solve_newton(dx, dy, dz, dtau, ds, dkappa)
+
+ # Maximum step to boundary.
+ # Compute eigenvalue decomposition of symmetric matrices in ds, dz.
+ # The eigenvectors of dsk, dzk are stored in dsk, dzk.
+ # The eigenvalues are stored in sigs, sigz.
+
+ scale2(lmbda, ds, dims)
+ ts = max_step(ds, dims, sigs)
+ scale2(lmbda, dz, dims)
+ tz = max_step(dz, dims, sigz)
+ tt = -dtau[0]/lmbda[-1]
+ tk = -dkappa[0]/lmbda[-1]
+ step = min(1.0, STEP / max([ 0.0, ts, tz, tt, tk ]))
+
+ # ds := e + step*ds for 'l' and 'q' blocks.
+ blas.scal(step, ds, n = dims['l'] + sum(dims['q']))
+ ds[:dims['l']] += 1.0
+ ds[indq[:-1]] += 1.0
+
+ # sigs := e + step*sigs for 's' blocks.
+ blas.scal(step, sigs)
+ sigs += 1.0
+
+ # dz := e + step*dz for 'l' and 'q' blocks.
+ blas.scal(step, dz, n = dims['l'] + sum(dims['q']))
+ dz[:dims['l']] += 1.0
+ dz[indq[:-1]] += 1.0
+
+ # sigz := e + step*sigz for 's' blocks.
+ blas.scal(step, sigz)
+ sigz += 1.0
+
+ # ds := H(lambda)^{-1/2} * ds and dz := H(lambda)^{-1/2} * dz.
+ scale2(lmbda, ds, dims, inverse = 'I')
+ scale2(lmbda, dz, dims, inverse = 'I')
+
+ # The 'l' and 'q' components of ds and dz now contain the updated
+ # variables in the current scaling. The 's' components of ds
+ # and dz contain
+ #
+ # Lambda^1/2 * Qs * Lambda^1/2
+ # Lambda^1/2 * Qz * Lambda^1/2
+ #
+ # where Lambda^1/2 * (Qs * diag(sigs) * Qs') * Lambda^1/2 and
+ # Lambda^1/2 * (Qz * diag(sigs) * Qz') * Lambda^1/2 are the
+ # updated variablaes in the current scaling.
+
+
+ # Update lambda and scaling.
+
+ # For kappa, tau block:
+ #
+ # dg := sqrt( (kappa + step*dkappa) / (tau + step*dtau) )
+ # = dg * sqrt( (1 - step*tk) / (1 - step*tt) )
+ #
+ # lmbda[-1] := sqrt((tau + step*dtau) * (kappa + step*dkappa))
+ # = lmbda[-1] * sqrt(( 1 - step*tt) * (1 - step*tk))
+
+ dg *= math.sqrt(1.0 - step*tk) / math.sqrt(1.0 - step*tt)
+ dgi = 1.0 / dg
+ lmbda[-1] *= math.sqrt(1.0 - step*tt) * math.sqrt(1.0 - step*tk)
+
+
+ # 'l' blocks
+ #
+ # d := d .* sqrt( ds ./ dz )
+ # lmbda := lmbda .* sqrt(ds) .* sqrt(dz)
+
+ m = dims['l']
+ ds[:m] = base.sqrt( ds[:m] )
+ dz[:m] = base.sqrt( dz[:m] )
+
+ # d := d .* ds .* dz
+ blas.tbmv(ds, W['d'], n = m, k = 0, ldA = 1)
+ blas.tbsv(dz, W['d'], n = m, k = 0, ldA = 1)
+ W['di'][:m] = W['d'][:m] ** -1
+
+ # lmbda := ds .* dz
+ blas.copy(ds, lmbda, n = m)
+ blas.tbmv(dz, lmbda, n = m, k = 0, ldA = 1)
+
+
+ # 'q' blocks.
+ #
+ # Let st and zt be the new variables in the old scaling:
+ #
+ # st = ds_k, zt = dz_k
+ #
+ # and a = sqrt(st' * J * st), b = sqrt(zt' * J * zt).
+ #
+ # 1. Compute the hyperbolic Householder transformation 2*q*q' - J
+ # that maps st/a to zt/b.
+ #
+ # c = sqrt( (1 + st'*zt/(a*b)) / 2 )
+ # q = (st/a + J*zt/b) / (2*c).
+ #
+ # The new scaling point is
+ #
+ # wk := betak * sqrt(a/b) * (2*v[k]*v[k]' - J) * q
+ #
+ # with betak = W['beta'][k].
+ #
+ # 3. The scaled variable:
+ #
+ # lambda_k0 = sqrt(a*b) * c
+ # lambda_k1 = sqrt(a*b) * ( (2vk*vk' - J) * (-d*q + u/2) )_1
+ #
+ # where
+ #
+ # u = st/a - J*zt/b
+ # d = ( vk0 * (vk'*u) + u0/2 ) / (2*vk0 *(vk'*q) - q0 + 1).
+ #
+ # 4. Update scaling
+ #
+ # v[k] := wk^1/2
+ # = 1 / sqrt(2*(wk0 + 1)) * (wk + e).
+ # beta[k] *= sqrt(a/b)
- dsl2[:] += misc.had(dzl+dsl, lmbda[:ml])
- for k in xrange(len(ms)):
- dss2[k] += misc.had(dzs[k]+dss[k], gamma[k])
+ ind = dims['l']
+ for k in xrange(Nq):
- dkappa2[0] += lmbda[-1] * (dtau[0] + dkappa[0])
+ m = dims['q'][k]
+ v = W['v'][k]
- solve_newton1(dx2, dy2, dzl2, dzs2, dtau2, dsl2, dss2,
- dkappa2)
+ # ln = sqrt( lambda_k' * J * lambda_k )
+ ln = jnrm2(lmbda, n = m, offset = ind)
- xaxpy(dx2, dx)
- yaxpy(dy2, dy)
- blas.axpy(dzl2, dzl)
- misc.saxpy(dzs2, dzs)
- dtau[0] += dtau2[0]
- blas.axpy(dsl2, dsl)
- misc.saxpy(dss2, dss)
- dkappa[0] += dkappa2[0]
+ # a = sqrt( dsk' * J * dsk ) = sqrt( st' * J * st )
+ # ds := ds / a = st / a
+ aa = jnrm2(ds, offset = ind, n = m)
+ blas.scal(1.0/aa, ds, offset = ind, n = m)
- if DEBUG:
- print "solvenewton2 errors"
-
- # unscaled variables
- duzl = misc.had(dzl, dli)
- dusl = misc.ihad(dsl, dli)
- duzs = [ +z_k for z_k in dzs ]
- misc.cngrnc(rti, duzs, duzs)
- duss = [ +s_k for s_k in dss ]
- misc.cngrnc(r, duss, duss)
- dutau = dtau[0]*dgi
- dukappa = dkappa[0]/dgi
-
- print " norm(rsldb2) = %e" %blas.nrm2(rsldb2 +
- misc.had(lmbda[:ml],dzl+dsl))
-
- for k in xrange(len(ms)):
- for j in xrange(ms[k]): rssdb2[k][j,j+1:] = \
- rssdb2[k][j+1:,j].trans()
- rssdb2[k] += misc.had(dzs[k] + dss[k], gamma[k])
- print " norm(rssdb2) = %e" %misc.snrm2(rssdb2)
-
- print " abs(rtdb2) = %e" %abs(rtdb2 + dukappa
- + xdot(c,dx) + ydot(b,dy) + blas.dot(hl,duzl)
- + misc.sdot(hs,duzs))
-
- A(dy, rxdb2, alpha=-1.0, beta=1.0, trans='T')
- Gl(duzl, rxdb2, alpha=-1.0, beta=1.0, trans='T')
- Gs(duzs, rxdb2, alpha=-1.0, beta=1.0, trans='T')
- xaxpy(c, rxdb2, alpha=-dutau)
- print " norm(rxdb2) = %e" \
- %math.sqrt(xdot(rxdb2,rxdb2))
-
- A(dx, rydb2, alpha=1.0, beta=1.0)
- yaxpy(b, rydb2, alpha=-dutau)
- print " norm(rydb2) = %e" \
- %math.sqrt(ydot(rydb2,rydb2))
- rzldb2 += dusl - hl*dutau
- Gl(dx, rzldb2, alpha=1.0, beta=1.0)
- print " norm(rzldb2) = %e" %blas.nrm2(rzldb2)
-
- misc.saxpy(duss, rzsdb2)
- misc.saxpy(hs, rzsdb2, alpha=-dutau)
- Gs(dx, rzsdb2, alpha=1.0, beta=1.0)
- print " norm(rzsdb2) = %e\n" %misc.snrm2(rzsdb2)
+ # b = sqrt( dzk' * J * dzk ) = sqrt( zt' * J * zt )
+ # dz := dz / a = zt / b
+ bb = jnrm2(dz, offset = ind, n = m)
+ blas.scal(1.0/bb, dz, offset = ind, n = m)
- else:
- solve_newton = solve_newton1
+ # c = sqrt( ( 1 + (st'*zt) / (a*b) ) / 2 )
+ cc = math.sqrt( ( 1.0 + blas.dot(ds, dz, offsetx = ind,
+ offsety = ind, n = m) ) / 2.0 )
- mu = blas.dot(lmbda,lmbda) / (m+1)
+ # vs = v' * st / a
+ vs = blas.dot(v, ds, offsety = ind, n = m)
- # Affine scaling step
- blas.copy(lmbda, dsl, n=ml)
- misc.had2(dsl,lmbda, n=ml)
- misc.sscal(0.0, dss)
- ind = ml
- for e, mk in zip(dss, ms):
- blas.copy(lmbda, e, n=mk, offsetx=ind, incy=mk+1)
- blas.tbmv(lmbda, e, n=mk, k=0, ldA=1, offsetA=ind,
- incx=mk+1)
- ind += mk
- dkappa[0] = kappa*tau
- xcopy(resx, dx)
- ycopy(resy, dy)
- blas.copy(reszl, dzl)
- misc.scopy(reszs, dzs)
- dtau[0] = rest
- solve_newton(dx, dy, dzl, dzs, dtau, dsl, dss, dkappa)
+ # vz = v' * J *zt / b
+ vz = jdot(v, dz, offsety = ind, n = m)
- # Maximum step to boundary
- blas.copy(dzl, sigz)
- misc.ihad2(sigz, lmbda, n=ml)
- blas.copy(dsl, sigs)
- misc.ihad2(sigs, lmbda, n=ml)
- misc.scopy(dss, ss)
- misc.scopy(dzs, zs)
- ind = ml
- for k in xrange(len(ms)):
- for i in xrange(ms[k]):
- a = 1.0 / math.sqrt(lmbda[ind+i])
- blas.scal(a, ss[k], offset=i, n=i+1, inc=ms[k])
- blas.scal(a, ss[k], offset=i*ms[k]+i, n=ms[k]-i)
- blas.scal(a, zs[k], offset=i, n=i+1, inc=ms[k])
- blas.scal(a, zs[k], offset=i*ms[k]+i, n=ms[k]-i)
- lapack.syevd(ss[k], sigs, offsetW=ind, jobz='V')
- lapack.syevd(zs[k], sigz, offsetW=ind, jobz='V')
- ind += ms[k]
- sigz[-1], sigs[-1] = dtau[0]/lmbda[-1], dkappa[0]/lmbda[-1]
- step = min(1.0, STEP / max(-min(sigz), -min(sigs)))
-
- newmu = ( (1-step) * blas.dot(lmbda,lmbda) + step**2 *
- (blas.dot(dsl, dzl) + misc.sdot(dss, dzs) +
- dkappa[0]*dtau[0]) ) / ( m+1)
- sigma = min(1.0, (newmu/mu)**EXPON)
-
- # Centering-corrector step
- misc.had2(dsl, dzl)
- blas.axpy(misc.had(lmbda[:ml], lmbda[:ml]), dsl)
- blas.axpy(ones, dsl, alpha=-sigma*mu, n=ml)
- ind = ml
- for k in xrange(len(ms)):
- # work := dss[k] with upper triangular part
- for i in xrange(ms[k]-1):
- dss[k][i,i+1:] = dss[k][i+1:,i].trans()
- dzs[k][i,i+1:] = dzs[k][i+1:,i].trans()
- blas.copy(dss[k], work)
-
- # dss[k] := 0.5 * (work*dzs[k] + dzs[k]*work)
- # = 0.5 * (dss[k]*dzs[k] + dzs[k]*dss[k])
- blas.syr2k(work, dzs[k], dss[k], n=ms[k], k=ms[k],
- ldA=ms[k], alpha=0.5)
-
- # dss[k] := dss[k] + diag(lambda)^2 - sigma*mu*I
- blas.copy(lmbda, work, n=ms[k], offsetx=ind)
- blas.tbmv(lmbda, work, n=ms[k], k=0, ldA=1, offsetA=ind)
- blas.axpy(work, dss[k], incy=ms[k]+1, n=ms[k])
- blas.axpy(ones, dss[k], alpha=-sigma*mu, incy=ms[k]+1,
- n=ms[k])
-
- ind += ms[k]
-
- dkappa[0] = lmbda[-1]**2 + dkappa[0]*dtau[0] - sigma*mu
- xcopy(resx, dx); xscal(1.0-sigma, dx)
- ycopy(resy,dy); yscal(1.0-sigma, dy)
- blas.copy(reszl, dzl); blas.scal(1.0-sigma, dzl)
- misc.scopy(reszs, dzs); misc.sscal(1.0-sigma, dzs)
- dtau[0] = (1.0-sigma)*rest
- solve_newton(dx, dy, dzl, dzs, dtau, dsl, dss, dkappa)
+ # vq = v' * q where q = (st/a + J * zt/b) / (2 * c)
+ vq = (vs + vz ) / 2.0 / cc
- # Maximum step to boundary
- blas.copy(dzl, sigz)
- misc.ihad2(sigz, lmbda, n=ml)
- blas.copy(dsl, sigs)
- misc.ihad2(sigs, lmbda, n=ml)
- ind = ml
- for k in xrange(len(ms)):
- for i in xrange(ms[k]):
- a = 1.0 / math.sqrt(lmbda[ind+i])
- blas.scal(a, dss[k], offset=i, n=i+1, inc=ms[k])
- blas.scal(a, dss[k], offset=i*(ms[k]+1), n=ms[k]-i)
- blas.scal(a, dzs[k], offset=i, n=i+1, inc=ms[k])
- blas.scal(a, dzs[k], offset=i*(ms[k]+1), n=ms[k]-i)
- lapack.syevd(dss[k], sigs, offsetW=ind, jobz='V')
- lapack.syevd(dzs[k], sigz, offsetW=ind, jobz='V')
- ind += ms[k]
- sigz[-1], sigs[-1] = dtau[0]/lmbda[-1], dkappa[0]/lmbda[-1]
- step = min(1.0, STEP / max(-min(sigz), -min(sigs)))
-
- # Update lmbdl, dli
- dli = misc.ihad(dli, base.sqrt(misc.ihad(lmbda[:ml] + step*dsl,
- lmbda[:ml] + step*dzl)))
- lmbda[:ml] = base.sqrt(misc.had(lmbda[:ml]+step*dsl,
- lmbda[:ml]+step*dzl))
-
- dgi = dgi * math.sqrt( (lmbda[-1] + step*dtau[0]) /
- (lmbda[-1] + step*dkappa[0]) )
- lmbda[-1] = math.sqrt( (lmbda[-1] + step*dtau[0]) *
- (lmbda[-1] + step*dkappa[0]) )
-
- # Update scaling matrices, lmbdas, gamma.
- ind = ml
- for k in xrange(len(ms)):
- for i in xrange(ms[k]):
- # dss[k] := L1
- # = lmbdas[k]^{1/2} * dss[k] * (1+step*sigs)^{1/2}
- # dzs[k] := L2
- # = lmbdas[k]^{1/2} * dzs[k] * (1+step*sigz)^{1/2}
- a = math.sqrt(lmbda[ind+i])
- blas.scal(a, dss[k], offset=i, inc=ms[k], n=ms[k])
- blas.scal(a, dzs[k], offset=i, inc=ms[k], n=ms[k])
- blas.scal(math.sqrt(1+step*sigs[ind+i]), dss[k],
- offset=ms[k]*i, n=ms[k])
- blas.scal(math.sqrt(1+step*sigz[ind+i]), dzs[k],
- offset=ms[k]*i, n=ms[k])
-
- # r[k] := r[k]*dss[k] = r[k]*L1[k]
- # rti[k] := rti[k]*dzs[k] = rti[k]*L2[k]
- blas.gemm(r[k], dss[k], work, ldC=ms[k])
- blas.copy(work, r[k], n=ms[k]**2)
- blas.gemm(rti[k], dzs[k], work, ldC=ms[k])
- blas.copy(work, rti[k], n=ms[k]**2)
-
- # SVD of L2'*L1 = U*lmbds^+*V', store U in dss[k] and V' in
- # dzs[k]
- blas.gemm(dzs[k], dss[k], work, transA='T', ldC=ms[k])
- lapack.gesvd(work, lmbda, jobu='A', jobvt='A', m=ms[k],
- n=ms[k], ldA=ms[k], U=dss[k], Vt=dzs[k], offsetS=ind)
-
- # r[k] := r[k]*V, rti[k] := rti[k]*U
- blas.gemm(r[k], dzs[k], work, transB='T', ldC=ms[k])
- blas.copy(work, r[k], n=ms[k]**2)
- blas.gemm(rti[k], dss[k], work, ldC=ms[k])
- blas.copy(work, rti[k], n=ms[k]**2)
-
- # r[k] := r[k]*lambda^{-1/2}; rti[k] := rti[k]*lambda^{-1/2}
- for i in xrange(ms[k]):
- a = 1.0 / math.sqrt(lmbda[ind+i])
- blas.scal(a, r[k], offset=ms[k]*i, n=ms[k])
- blas.scal(a, rti[k], offset=ms[k]*i, n=ms[k])
+ # vu = v' * u where u = st/a - J * zt/b
+ vu = vs - vz
- # gamma[k][i,j] = .5 * (lambda_i + lambda_j)
- # Use blas.ger to make sure there are no zeros or nan's in
- # upper triangular part.
+ # lambda_k0 = c
+ lmbda[ind] = cc
- blas.scal(0.0, gamma[k])
- blas.ger(lmbda, ones, gamma[k], alpha=.5, offsetx=ind)
- blas.ger(ones, lmbda, gamma[k], alpha=.5, offsety=ind)
+ # wk0 = 2 * vk0 * (vk' * q) - q0
+ wk0 = 2 * v[0] * vq - ( ds[ind] + dz[ind] ) / 2.0 / cc
- ind += ms[k]
+ # d = (v[0] * (vk' * u) - u0/2) / (wk0 + 1)
+ dd = (v[0] * vu - ds[ind]/2.0 + dz[ind]/2.0) / (wk0 + 1.0)
+ # lambda_k1 = 2 * v_k1 * vk' * (-d*q + u/2) - d*q1 + u1/2
+ blas.copy(v, lmbda, offsetx = 1, offsety = ind+1, n = m-1)
+ blas.scal(2.0 * (-dd * vq + 0.5 * vu), lmbda, offset = ind+1,
+ n = m-1)
+ blas.axpy(ds, lmbda, 0.5 * (1.0 - dd/cc), offsetx = ind+1,
+ offsety = ind+1, n = m-1)
+ blas.axpy(dz, lmbda, 0.5 * (1.0 + dd/cc), offsetx = ind+1,
+ offsety = ind+1, n = m-1)
- # Compute new sl, zl, ss, zs, tau, kappa (used to evaluate the
- # residuals).
+ # Scale so that sqrt(lambda_k' * J * lambda_k) = sqrt(aa*bb).
+ blas.scal(math.sqrt(aa*bb), lmbda, offset = ind, n = m)
+
+ # v := (2*v*v' - J) * q
+ # = 2 * (v'*q) * v' - (J* st/a + zt/b) / (2*c)
+ blas.scal(2.0 * vq, v)
+ v[0] -= ds[ind] / 2.0 / cc
+ blas.axpy(ds, v, 0.5/cc, offsetx = ind+1, offsety = 1,
+ n = m-1)
+ blas.axpy(dz, v, -0.5/cc, offsetx = ind, n = m)
+
+ # v := v^{1/2} = 1/sqrt(2 * (v0 + 1)) * (v + e)
+ v[0] += 1.0
+ blas.scal(1.0 / math.sqrt(2.0 * v[0]), v)
+
+ # beta[k] *= ( aa / bb )**1/2
+ W['beta'][k] *= math.sqrt( aa / bb )
+
+ ind += m
- # sl := lmbdl ./ dli
- blas.copy(lmbda, sl, n=ml)
- misc.ihad2(sl, dli)
- # zl = lmbdl .* dli
- blas.copy(lmbda, zl, n=ml)
- misc.had2(zl, dli)
+ # 's' blocks
+ #
+ # Let st, zt be the updated variables in the old scaling:
+ #
+ # st = ds * diag(sigs ./ lambda) * ds'
+ # zt = dz * diag(sigs ./ lambda) * dz'.
+ #
+ # 1. Compute
+ #
+ # L1 = dsk * diag(sigs_k ./ lambda_k)^{1/2}
+ # L2 = dzk * diag(sigz_k ./ lambda_k)^{1/2}.
+ #
+ # We have
+ #
+ # L1 * L1' = st, L2 * L2' = zt.
+ #
+ # 2. SVD L2'*L1 = Uk * lambda_k^+ * Vk'.
+ #
+ # 3. New scaling is
+ #
+ # r[k] := r[k] * L1 * Vk * diag(lambda_k^+)^{-1/2}
+ # rti[k] := r[k] * L2 * Uk * diag(lambda_k^+)^{-1/2}.
+
+ ind = dims['l'] + sum(dims['q'])
+ ind2, ind3 = ind, 0
+
+ # sigs := sigs./lambda. sigz := sigz./lambda.
+ blas.tbsv(lmbda, sigs, n = sum(dims['s']), k = 0, ldA = 1,
+ offsetA = ind)
+ blas.tbsv(lmbda, sigz, n = sum(dims['s']), k = 0, ldA = 1,
+ offsetA = ind)
+
+ for k in xrange(Ns):
+ m = dims['s'][k]
+ r, rti = W['r'][k], W['rti'][k]
+
+ # dsk := L1 = dsk * sqrt(sigs). dzk := L2 = dzk * sqrt(sigz).
+ for i in xrange(m):
+ blas.scal(math.sqrt(sigs[ind3+i]), ds, offset = ind2 + m*i,
+ n = m)
+ blas.scal(math.sqrt(sigz[ind3+i]), dz, offset = ind2 + m*i,
+ n = m)
+
+ # r := r*dsk = r*L1
+ blas.gemm(r, ds, work, m = m, n = m, k = m, ldB = m, ldC = m,
+ offsetB = ind2)
+ blas.copy(work, r, n = m**2)
+
+ # rti := rti*dzk = rti*L2
+ blas.gemm(rti, dz, work, m = m, n = m, k = m, ldB = m, ldC = m,
+ offsetB = ind2)
+ blas.copy(work, rti, n = m**2)
+
+ # SVD L2'*L1 = U * lmbds^+ * V'; store U in dsk and V' in dzk.
+ blas.gemm(dz, ds, work, transA = 'T', m = m, n = m, k = m,
+ ldA = m, ldB = m, ldC = m, offsetA = ind2, offsetB = ind2)
+ lapack.gesvd(work, lmbda, jobu = 'A', jobvt = 'A', m = m,
+ n = m, ldA = m, U = ds, Vt = dz, ldU = m, ldVt = m,
+ offsetS = ind, offsetU = ind2, offsetVt = ind2)
+
+ # r := r*V
+ blas.gemm(r, dz, work, transB = 'T', m = m, n = m, k = m,
+ ldB = m, ldC = m, offsetB = ind2)
+ blas.copy(work, r, n = m**2)
+
+ # rti := rti*U
+ blas.gemm(rti, ds, work, n = m, m = m, k = m, ldB = m, ldC = m,
+ offsetB = ind2)
+ blas.copy(work, rti, n = m**2)
+
+ # r := r*lambda^{-1/2}; rti := rti*lambda^{-1/2}
+ for i in xrange(m):
+ a = 1.0 / math.sqrt(lmbda[ind+i])
+ blas.scal(a, r, offset = m*i, n = m)
+ blas.scal(a, rti, offset = m*i, n = m)
+
+ ind += m
+ ind2 += m*m
+ ind3 += m
+
+ xaxpy(dx, x, alpha = step)
+ yaxpy(dy, y, alpha = step)
+
+
+ # Unscale s, z, tau, kappa (unscaled variables are used only to
+ # compute feasibility residuals).
+
+ blas.copy(lmbda, s, n = dims['l'] + sum(dims['q']))
+ ind = dims['l'] + sum(dims['q'])
+ ind2 = ind
+ for m in dims['s']:
+ blas.scal(0.0, s, offset = ind2)
+ blas.copy(lmbda, s, offsetx = ind, offsety = ind2, n = m,
+ incy = m+1)
+ ind += m
+ ind2 += m*m
+ scale(s, W, trans = 'T')
+
+ blas.copy(lmbda, z, n = dims['l'] + sum(dims['q']))
+ ind = dims['l'] + sum(dims['q'])
+ ind2 = ind
+ for m in dims['s']:
+ blas.scal(0.0, z, offset = ind2)
+ blas.copy(lmbda, z, offsetx = ind, offsety = ind2, n = m,
+ incy = m+1)
+ ind += m
+ ind2 += m*m
+ scale(z, W, inverse = 'I')
kappa, tau = lmbda[-1]/dgi, lmbda[-1]*dgi
+ gap = blas.dot(lmbda, lmbda, n = cdeg) / tau**2
- # ss = r*lambda*r', zs = rti*lambda*rti'
- ind = ml
- for k in xrange(len(ms)):
- blas.copy(r[k], work)
- blas.copy(rti[k], work2)
- for i in xrange(ms[k]):
- a = math.sqrt(lmbda[ind+i])
- blas.scal(a, work, offset=ms[k]*i, n=ms[k])
- blas.scal(a, work2, offset=ms[k]*i, n=ms[k])
- blas.syrk(work, ss[k], ldA=ms[k], n=ms[k], k=ms[k])
- blas.syrk(work2, zs[k], ldA=ms[k], n=ms[k], k=ms[k])
- ind += ms[k]
-
- xaxpy(dx, x, alpha=step)
- yaxpy(dy, y, alpha=step)
-
- gap = blas.dot(lmbda, lmbda, n=m-1) / tau**2
+ return {'status': 'unknown', 'x': None, 'y': None, 's': None,
+ 'z': None}
- return {'status': 'unknown', 'x': None, 'y': None, 'sl': None,
- 'ss': None, 'zl': None, 'zs': None}
-
-def lp(c, G, h, A=None, b=None, solver=None, primalstart=None,
- dualstart=None):
+def lp(c, G, h, A = None, b = None, solver = None, primalstart = None,
+ dualstart = None):
"""
Solves a pair of primal and dual LPs
minimize c'*x maximize -h'*z - b'*y
subject to G*x + s = h subject to G'*z + A'*y + c = 0
- A*x = b z >= 0
+ A*x = b z >= 0.
s >= 0
- Input arguments:
- G is mxn, h is mx1, A is pxn, b is px1. G and A must be dense
- or sparse 'd' matrices. h and b are dense 'd' matrices with
- one column. The default values for A and b are empty matrices
- with zero rows.
+ Input arguments
+
+ G is m x n, h is m x 1, A is p x n, b is p x 1. G and A must be
+ dense or sparse 'd' matrices. h and b are dense 'd' matrices
+ with one column. The default values for A and b are empty
+ matrices with zero rows.
solver is None, 'glpk' or 'mosek'. The default solver (None)
- uses an LP solver implemented in Python. The 'glpk' solver is
- the simplex LP solver from GLPK. The 'mosek' solver is the LP
- solver from MOSEK.
+ uses the cvxopt conelp() function. The 'glpk' solver is the
+ simplex LP solver from GLPK. The 'mosek' solver is the LP solver
+ from MOSEK.
The arguments primalstart and dualstart are ignored when solver
is 'glpk' or 'mosek' and are optional when solver is None.
@@ -1185,9 +2466,8 @@ def lp(c, G, h, A=None, b=None, solver=None, primalstart=None,
options['show_progress'] True/False (default: True)
options['maxiters'] positive integer (default: 100)
options['abstol'] scalar (default: 1e-7)
- options['reltol'] scalar (default: 1e-7)
- options['feastol'] scalar (default: 1e-7)
- options['refinement'] True/False (default: True)
+ options['reltol'] scalar (default: 1e-6)
+ options['feastol'] scalar (default: 1e-7).
The control parameter names for GLPK and MOSEK can be found in the
GLPK and MOSEK documentation. Options that are not recognized are
@@ -1258,354 +2538,313 @@ def lp(c, G, h, A=None, b=None, solver=None, primalstart=None,
x, s, y, z = None, None, None, None
return {'status': status, 'x': x, 's': s, 'y': y, 'z': z}
- if p > n or n-p > m:
- raise ValueError, "Rank(A) < p or Rank([G; A]) < n"
+ return conelp(c, G, h, {'l': m, 'q': [], 's': []}, A, b, primalstart
+ = primalstart, dualstart = dualstart)
- def Fi(x, y, alpha=1.0, beta=0.0, trans='N'):
- base.gemv(G, x, y, alpha=alpha, beta=beta, trans=trans)
- def Fe(x, y, alpha=1.0, beta=0.0, trans='N'):
- base.gemv(A, x, y, alpha=alpha, beta=beta, trans=trans)
- def Fs(x, y, alpha=1.0, beta=0.0, trans='N'):
- pass
+def socp(c, Gl = None, hl = None, Gq = None, hq = None, A = None, b = None,
+ primalstart = None, dualstart = None):
+
+ """
+ Solves a pair of primal and dual SOCPs
+
+ minimize c'*x
+ subject to Gl*x + sl = hl
+ Gq[k]*x + sq[k] = hq[k], k = 0, ..., N-1
+ A*x = b
+ sl >= 0,
+ sq[k] >= 0, k = 0, ..., N-1
+
+ maximize -hl'*z - sum_k hq[k]'*zq[k] - b'*y
+ subject to Gl'*zl + sum_k Gq[k]'*zq[k] + A'*y + c = 0
+ zl >= 0, zq[k] >= 0, k = 0, ..., N-1.
+
+ The inequalities sl >= 0 and zl >= 0 are elementwise vector
+ inequalities. The inequalities sq[k] >= 0, zq[k] >= 0 are second
+ order cone inequalities, i.e., equivalent to
+ sq[k][0] >= || sq[k][1:] ||_2, zq[k][0] >= || zq[k][1:] ||_2.
- # kktsolver(di, rti) returns a function for solving
- #
- # [ 0 A' G' ] [x] [bx]
- # [ A 0 0 ] [y] = [by].
- # [ G 0 -diag(di)^{-2} ] [z] [bz]
- #
- # di is a positive mx1 'd' matrix; the argument rti is ignored.
- #
- # We return the scaled z: z./di instead of z.
- global kktmethod
- try: kktmethod = options['kktmethod']
- except KeyError: kktmethod = 3
- else:
- if type(kktmethod) is not int or kktmethod not in [1,2,3]:
- raise ValueError, "options['kktmethod'] must be 1, 2, or 3"
+ Input arguments
+
+ Gl is a dense or sparse 'd' matrix of size (ml, n). hl is a
+ dense 'd' matrix of size (ml, 1). The default values of Gl and hl
+ are matrices with zero rows.
+
+ The argument Gq is a list of N dense or sparse 'd' matrices of
+ size (m[k] n), k = 0, ..., N-1, where m[k] >= 1. hq is a list
+ of N dense 'd' matrices of size (m[k], 1), k = 0, ..., N-1.
+ The default values of Gq and hq are empty lists.
- if kktmethod == 1:
+ A is a dense or sparse 'd' matrix of size (p,1). b is a dense 'd'
+ matrix of size (p,1). The default values of A and b are matrices
+ with zero rows.
- # The entire matrix is factored using a dense LDL factorization.
+ The argument primalstart is a dictionary with keys 'x', 'sl', 'sq',
+ and specifies an optional primal starting point.
+ primalstart['x'] is a dense 'd' matrix of size (n,1).
+ primalstart['sl'] is a positive dense 'd' matrix of size (ml,1).
+ primalstart['sq'] is a list of matrices of size (m[k],1), positive
+ with respect to the second order cone of order m[k].
- K = matrix(0.0, (n+p+m,n+p+m))
- ipiv = matrix(0, (n+p+m,1))
- u = matrix(0.0, (n+p+m,1))
- def kktsolver(di, rti):
- blas.scal(0, K)
- K[n:n+p,:n], K[n+p:,:n] = A, G
- K[(m+p+n+1)*(p+n)::n+m+p+1] = -misc.ihad(1.0,
- misc.had(di, di))
- lapack.sytrf(K, ipiv)
- def solve_kkt(x, y, z, zs):
- blas.copy(x, u)
- blas.copy(y, u, offsety=n)
- blas.copy(z, u, offsety=n+p)
- lapack.sytrs(K, ipiv, u)
- blas.copy(u, x, n=n)
- blas.copy(u, y, offsetx=n, n=p)
- blas.copy(u, z, offsetx=n+p, n=m)
- misc.ihad2(z, di)
- return solve_kkt
+ The argument dualstart is a dictionary with keys 'y', 'zl', 'zq',
+ and specifies an optional dual starting point.
+ dualstart['y'] is a dense 'd' matrix of size (p,1).
+ dualstart['zl'] is a positive dense 'd' matrix of size (ml,1).
+ dualstart['sq'] is a list of matrices of size (m[k],1), positive
+ with respect to the second order cone of order m[k].
- else:
- if type(G) is matrix:
- Gsc = matrix(0.0, (m,n))
- else:
- Gsc = spmatrix(0.0, G.I, G.J, (m,n))
- def GDG(Gsc, S, di, pattern_S_is_correct=True):
+ Returns a dictionary with keys 'status', 'x', 'sl', 'sq', 'y', 'zl',
+ 'zq'.
- # Compute
- #
- # Gsc := diag(di)*G and
- # S[:n,:n] := G' * diag(di)^2 * G.
- #
- # If G is dense, Gsc and S must be dense. If G is sparse,
- # Gsc is sparse and S may be dense or sparse.
+ If status is 'optimal', x, sl, sq, y, zl, zq are approximate
+ primal and dual optimal solutions.
- base.gemm(spmatrix(di, range(m), range(m), tc='d'), G,
- Gsc, partial=True)
- base.syrk(Gsc, S, trans='T', partial=pattern_S_is_correct)
+ If status is 'primal infeasible', x = sl = sq = None, and y, zl,
+ zq are a proof of infeasibility:
- if kktmethod == 2:
+ hl'*zl + sum_k hq[k]' * zq[k] + b'*y = -1,
+ Gl'*zl + sum_k Gq[k]' * zq[k] + A'*y = 0,
+ zl >= 0, zq[k] >= 0, k = 0, ..., N-1.
- # We compute x, y, z by solving
- #
- # [G'*diag(di)^2*G A'] [x] [bx + G'*(di.^2).*bz)]
- # [ ]*[ ] = [ ]
- # [A 0 ] [y] [by ]
- #
- # z = (di.^2) .* (G*x - bz).
- #
- # We return the scaled z: z = di .* (G*x - bz).
- #
- # The coefficient matrix is factored using a dense LDL
- # (if p>0) or dense Cholesky factorization (if p=0).
-
- K = matrix(0.0, (n+p,n+p))
- if p: ipiv = matrix(0, (n+p,1))
- u = matrix(0.0, (n+p,1))
- def kktsolver(di, rti):
- blas.scal(0.0, K)
- K[n:,:n] = A
- GDG(Gsc, K, di)
- if p: lapack.sytrf(K, ipiv)
- else: lapack.potrf(K)
- def solve_kkt(x, y, z, zs):
- blas.copy(x, u)
- blas.copy(y, u, offsety=n)
- misc.had2(z, di)
- base.gemv(Gsc, z, u, trans='T', beta=1.0)
- if p: lapack.sytrs(K, ipiv, u)
- else: lapack.potrs(K, u)
- blas.copy(u, x, n=n)
- blas.copy(u, y, n=p, offsetx=n)
- base.gemv(Gsc, x, z, beta=-1.0)
- return solve_kkt
+ If status is 'dual infeasible', zl = zq = y = None, and x, sl, sq
+ are a proof of dual infeasibility:
- else:
+ c'*x = -1, Gl*x + sl = 0, Gq[k]*x + sq[k] = 0, A*x = 0,
+ sl >= 0, sq[k] >= 0, k = 0, ..., N-1.
- # Solve
- #
- # K*y = A * S^{-1} * (bx + G'*((di.^2).*bz)) - by
- # S*x = bx + G'*((di.^2).*bz) - A'*y
- # z = (di.^2) .* (G*x - bz).
- #
- # where K = A*S^{-1}*A', S = G'*diag(di)^2*G.
- #
- # If in the first call to kktsolver, S is singular, switch
- # to kktmethod 4, i.e., solve
- #
- # K*y = A * S^{-1} * (bx + G'*((di.^2).*bz) + A'*by) - by
- # S*x = bx + G'*((di.^2).*bz) + A'*by - A'*y
- # z = (di.^2) .* (G*x - bz).
- #
- # where K = A'*S^{-1}*A, S = G'*diag(di)^2*G + A'*A
- #
- # The matrices K and S are factored using dense or sparse
- # Cholesky factorizations.
- #
- # We return the scaled z: z = di .* (G*x - bz).
+ If status is 'unknown', x, y, sl, sq, zl, zq are None.
- global S, Sf, firstcall
- firstcall = True
- if type(G) is matrix:
- S = matrix(0.0, (n,n))
- K = matrix(0.0, (p,p))
- else:
- S = spmatrix([], [], [], (n,n), 'd')
- if type(A) is matrix: K = matrix(0.0, (p,p))
- else: K = spmatrix([], [], [], (p,p), tc='d')
-
- def factorS(di,S):
-
- # Factor S = G'*diag(di)^2*G if kktmethod = 3.
- # Factor S = G'*diag(di)^2*G + A'*A if kktmethod = 4.
- # If S is sparse, store factorization in Sf.
-
- global Sf
- GDG(Gsc, S, di, not firstcall)
- if kktmethod == 4:
- base.syrk(A, S, trans='T', beta=1.0, partial=not
- firstcall)
- if type(S) is matrix:
- lapack.potrf(S)
- else:
- if firstcall: Sf = cholmod.symbolic(S)
- cholmod.numeric(S, Sf)
-
- def kktsolver(di, rti):
- global firstcall, kktmethod, S
- if kktmethod == 3:
- try: factorS(di,S)
- except ArithmeticError:
- if firstcall:
- kktmethod = 4
- if type(S) is spmatrix and type(A) is \
- matrix: S = matrix(0.0, (n,n))
- factorS(di,S)
- else: raise ArithmeticError
- else:
- factorS(di,S)
- firstcall = False
+ The following parameters control the execution of the default
+ solver.
- if type(S) is matrix:
- # Asct := L^{-1}*A', factor K = Asct'*Asct
- if type(A) is matrix:
- Asct = A.trans()
- else:
- Asct = matrix(A.trans())
- blas.trsm(S, Asct)
- blas.syrk(Asct, K, trans='T')
- lapack.potrf(K)
+ options['show_progress'] True/False (default: True)
+ options['maxiters'] positive integer (default: 100)
+ options['abstol'] scalar (default: 1e-7)
+ options['reltol'] scalar (default: 1e-6)
+ options['feastol'] scalar (default: 1e-7).
+ """
- else:
- # Asct := L^{-1}*P*A' and factor K = Asct'*Asct.
- if type(A) is matrix:
- Asct = A.trans()
- cholmod.solve(Sf, Asct, sys=7)
- cholmod.solve(Sf, Asct, sys=4)
- blas.syrk(Asct, K, trans='T')
- lapack.potrf(K)
- else:
- Asct = cholmod.spsolve(Sf, A.trans(), sys=7)
- Asct = cholmod.spsolve(Sf, Asct, sys=4)
- base.syrk(Asct, K, trans='T')
- Kf = cholmod.symbolic(K)
- cholmod.numeric(K, Kf)
-
- def solve_kkt(x, y, z, zs):
-
- # If kktmethod is 3:
- # x := L^{-1}*P * (x + G'*((di.^2).*z))
- # = L^{-1}*P * (bx + G'*((di.^2).*bz))
- #
- # If kktmethod is 4:
- # x := L^{-1}*P * (x + G'*((di.^2).*z) + A'*y)
- # = L^{-1}*P * (bx + G'*((di.^2).*bz) + A'*by)
- misc.had2(z, di)
- base.gemv(Gsc, z, x, trans='T', beta=1.0)
- if kktmethod == 4:
- base.gemv(A, y, x, trans='T', beta=1.0)
- if type(S) is matrix:
- blas.trsv(S, x)
- else:
- cholmod.solve(Sf, x, sys=7)
- cholmod.solve(Sf, x, sys=4)
-
- # y := K^{-1} * (Asc*x - y)
- # = K^{-1} * (A*S^{-1}* (bx + G'*((di.^2).*bz))
- # - by) (kktmethod = 3)
- # = K^{-1} * (A*S^{-1}* (bx + G'*((di.^2).*bz))
- # + A'*by) - by) (kktmethod = 4)
- base.gemv(Asct, x, y, trans='T', beta=-1.0)
- if type(K) is matrix:
- lapack.potrs(K, y)
- else:
- cholmod.solve(Kf, y)
-
- # x := P' * L^{-T} * (x - Asc'*y)
- # = S^{-1}* (bx + G'*((di.^2).*bz) - A'*y)
- # (kktmethod = 3)
- # = S^{-1}*(bx + G'*((di.^2).*bz) + A'*by -
- # A'*y) (kktmethod = 4)
- base.gemv(Asct, y, x, alpha=-1.0, beta=1.0)
- if type(S) is matrix:
- blas.trsv(S, x, trans='T')
- else:
- cholmod.solve(Sf, x, sys=5)
- cholmod.solve(Sf, x, sys=8)
+ if type(c) is not matrix or c.typecode != 'd' or c.size[1] != 1:
+ raise TypeError, "'c' must be a dense column matrix"
+ n = c.size[0]
+ if n < 1: raise ValueError, "number of variables must be at least 1"
+
+ if Gl is None: Gl = spmatrix([], [], [], (0,n), tc='d')
+ if (type(Gl) is not matrix and type(Gl) is not spmatrix) or \
+ Gl.typecode != 'd' or Gl.size[1] != n:
+ raise TypeError, "'Gl' must be a dense or sparse 'd' matrix "\
+ "with %d columns" %n
+ ml = Gl.size[0]
+ if hl is None: hl = matrix(0.0, (0,1))
+ if type(hl) is not matrix or hl.typecode != 'd' or \
+ hl.size != (ml,1):
+ raise TypeError, "'hl' must be a dense 'd' matrix of " \
+ "size (%d,1)" %ml
+
+ if Gq is None: Gq = []
+ if type(Gq) is not list or [ G for G in Gq if (type(G) is not matrix
+ and type(G) is not spmatrix) or G.typecode != 'd' or
+ G.size[1] != n ]:
+ raise TypeError, "'Gq' must be a list of sparse or dense 'd' "\
+ "matrices with %d columns" %n
+ mq = [ G.size[0] for G in Gq ]
+ a = [ k for k in xrange(len(mq)) if mq[k] == 0 ]
+ if a: raise TypeError, "the number of rows of Gq[%d] is zero" %a[0]
+ if hq is None: hq = []
+ if type(hq) is not list or len(hq) != len(mq) or [ h for h in hq if
+ (type(h) is not matrix and type(h) is not spmatrix) or
+ h.typecode != 'd' ]:
+ raise TypeError, "'hq' must be a list of %d dense or sparse "\
+ "'d' matrices" %len(mq)
+ a = [ k for k in xrange(len(mq)) if hq[k].size != (mq[k], 1) ]
+ if a:
+ k = a[0]
+ raise TypeError, "'hq[%d]' has size (%d,%d). Expected size "\
+ "is (%d,1)." %(k, hq[k].size[0], hq[k].size[1], mq[k])
- # z := Gsc*x - z = di .* (G*x - bz)
- base.gemv(Gsc, x, z, beta=-1.0)
+ if A is None: A = spmatrix([], [], [], (0,n), 'd')
+ if (type(A) is not matrix and type(A) is not spmatrix) or \
+ A.typecode != 'd' or A.size[1] != n:
+ raise TypeError, "'A' must be a dense or sparse 'd' matrix "\
+ "with %d columns" %n
+ p = A.size[0]
+ if b is None: b = matrix(0.0, (0,1))
+ if type(b) is not matrix or b.typecode != 'd' or b.size != (p,1):
+ raise TypeError, "'b' must be a dense matrix of size (%d,1)" %p
- return solve_kkt
+ dims = {'l': ml, 'q': mq, 's': []}
+ N = ml + sum(mq)
+ h = matrix(0.0, (N,1))
+ if type(Gl) is matrix or [ Gk for Gk in Gq if type(Gk) is matrix ]:
+ G = matrix(0.0, (N, n))
+ else:
+ G = spmatrix([], [], [], (N, n), 'd')
+ h[:ml] = hl
+ G[:ml,:] = Gl
+ ind = ml
+ for k in xrange(len(mq)):
+ h[ind : ind + mq[k]] = hq[k]
+ G[ind : ind + mq[k], :] = Gq[k]
+ ind += mq[k]
if primalstart:
- pst = {'x': primalstart['x'], 'sl': primalstart['s'], 'ss': []}
+ ps = {}
+ ps['x'] = primalstart['x']
+ ps['s'] = matrix(0.0, (N,1))
+ if ml: ps['s'][:ml] = primalstart['sl']
+ if mq:
+ ind = ml
+ for k in xrange(len(mq)):
+ ps['s'][ind : ind + mq[k]] = primalstart['sq'][k][:]
+ ind += mq[k]
else:
- pst = None
+ ps = None
+
if dualstart:
- dst = {'y': dualstart['y'], 'zl': dualstart['z'], 'zs': []}
+ ds = {}
+ if p: ds['y'] = dualstart['y']
+ ds['z'] = matrix(0.0, (N,1))
+ if ml: ds['z'][:ml] = dualstart['zl']
+ if mq:
+ ind = ml
+ for k in xrange(len(mq)):
+ ds['z'][ind : ind + mq[k]] = dualstart['zq'][k][:]
+ ind += mq[k]
+ else:
+ ds = None
+
+ sol = conelp(c, G, h, dims, A = A, b = b, primalstart = ps, dualstart
+ = ds)
+ val = {'status': sol['status'], 'x': sol['x'], 'y': sol['y']}
+ if sol['s'] is None:
+ val['sl'] = None
+ val['sq'] = None
else:
- dst = None
- sol = conelp(c, kktsolver, Gl=Fi, hl=h, A=Fe, b=b, primalstart=pst,
- dualstart=dst)
- return {'status': sol['status'], 'x': sol['x'], 'y': sol['y'],
- 'z': sol['zl'], 's': sol['sl']}
+ val['sl'] = sol['s'][:ml]
+ val['sq'] = [ matrix(0.0, (m,1)) for m in mq ]
+ ind = ml
+ for k in xrange(len(mq)):
+ val['sq'][k][:] = sol['s'][ind : ind+mq[k]]
+ ind += mq[k]
+ if sol['z'] is None:
+ val['zl'] = None
+ val['zq'] = None
+ else:
+ val['zl'] = sol['z'][:ml]
+ val['zq'] = [ matrix(0.0, (m,1)) for m in mq]
+ ind = ml
+ for k in xrange(len(mq)):
+ val['zq'][k][:] = sol['z'][ind : ind+mq[k]]
+ ind += mq[k]
+ return val
-def sdp(c, Gl=None, hl=None, Gs=None, hs=None, A=None, b=None,
- solver=None, primalstart=None, dualstart=None):
+
+def sdp(c, Gl = None, hl = None, Gs = None, hs = None, A = None, b = None,
+ solver = None, primalstart = None, dualstart = None):
"""
Solves a pair of primal and dual SDPs
minimize c'*x
subject to Gl*x + sl = hl
- Gs(x) + ss = hs
+ mat(Gs[k]*x) + ss[k] = hs[k], k = 0, ..., N-1
A*x = b
- sl >= 0, ss >= 0
-
- maximize -hl'*z - tr(hs,zs) - b'*y
- subject to Gl'*zl + Gs'(zs) + A'*y + c = 0
- zl >= 0, zs >= 0
+ sl >= 0, ss[k] >= 0, k = 0, ..., N-1
+
+ maximize -hl'*z - sum_k trace(hs[k]*zs[k]) - b'*y
+ subject to Gl'*zl + sum_k Gs[k]'*vec(zs[k]) + A'*y + c = 0
+ zl >= 0, zs[k] >= 0, k = 0, ..., N-1.
+
+ The inequalities sl >= 0 and zl >= 0 are elementwise vector
+ inequalities. The inequalities ss[k] >= 0, zs[k] >= 0 are matrix
+ inequalities, i.e., the symmetric matrices ss[k] and zs[k] must be
+ positive semidefinite. mat(Gs[k]*x) is the symmetric matrix X with
+ X[:] = Gs[k]*x. For a symmetric matrix, zs[k], vec(zs[k]) is the
+ vector zs[k][:].
+
- Input arguments:
+ Input arguments
- Gl is an mlxn dense or sparse 'd' matrix.
- hl is an mlx1 dense 'd' matrix. The default values of Gl and
- hl are matrices with zero rows.
+ Gl is a dense or sparse 'd' matrix of size (ml, n). hl is a
+ dense 'd' matrix of size (ml, 1). The default values of Gl and hl
+ are matrices with zero rows.
The argument Gs is a list of N dense or sparse 'd' matrices of
- size (ms[k]**2, n), k=1,...,N.
- The matrix-vector products Gs[k]*x, k=1,...,N, represent a
- block-diagonal symmetric matrix with N diagonal blocks of size
- (ms[k],ms[k]), stored columnwise as vectors.
- hs is a list of dense 'd' matrices of size (ms[k],ms[k]).
- Only the lower-triangular part of hs[k] is accessed.
-
- A is a pxn dense or sparse 'd' matrix.
- b is a px1 dense 'd' matrix. The default values of A and b are
- matrices with zero rows.
-
- solver is None or 'dsdp'. The default solver (None) uses an
- SDP solver implemented in Python. The 'dsdp' solver uses an
- interface to DSDP5. It does not accept problems with equality
+ size (m[k]**2, n), k = 0, ..., N-1. The columns of Gs[k]
+ represent symmetric matrices stored as vectors in column major
+ order. hs is a list of N dense 'd' matrices of size (m[k], m[k]),
+ k = 0, ..., N-1. The columns of Gs[k] and the matrices hs[k]
+ represent symmetric matrices in 'L' storage, i.e., only the lower
+ triangular elements are accessed. The default values of Gs and
+ hs are empty lists.
+
+ A is a dense or sparse 'd' matrix of size (p,n). b is a dense 'd'
+ matrix of size (p,1). The default values of A and b are matrices
+ with zero rows.
+
+ solver is None or 'dsdp'. The default solver (None) calls
+ cvxopt.conelp(). The 'dsdp' solver uses an interface to DSDP5.
+ The 'dsdp' solver does not accept problems with equality
constraints (A and b must have zero rows, or be absent).
The argument primalstart is a dictionary with keys 'x', 'sl',
- 'ss', and specifies a primal starting point. primalstart['x']
- must be a dense 'd' matrix of length n; primalstart['sl'] must
- be a positive dense 'd' matrix of length ml; primalstart['ss']
- must be a list of positive definite matrices of size
- ms[k]xms[k].
- The argument dualstart is a dictionary with keys 'zl', 'zs',
- 'y' and specifies a dual starting point. primalstart['y']
- must be a dense 'd' matrix of length p; primalstart['zl']
- must be a positive dense 'd' matrix of length ml;
- primalstart['zs'] must be a list of positive definite matrices
- of size (ms[k],ms[k]).
+ 'ss', and specifies an optional primal starting point.
+ primalstart['x'] is a dense 'd' matrix of length n;
+ primalstart['sl'] is a positive dense 'd' matrix of length ml;
+ primalstart['ss'] is a list of positive definite matrices of
+ size (ms[k], ms[k]). Only the lower triangular parts of these
+ matrices will be accessed.
+
+ The argument dualstart is a dictionary with keys 'zl', 'zs', 'y'
+ and specifies an optional dual starting point.
+ dualstart['y'] is a dense 'd' matrix of length p;
+ dualstart['zl'] must be a positive dense 'd' matrix of length ml;
+ dualstart['zs'] is a list of positive definite matrices of
+ size (ms[k], ms[k]). Only the lower triangular parts of these
+ matrices will be accessed.
+
The arguments primalstart and dualstart are ignored when solver
is 'dsdp'.
+
Returns a dictionary with keys 'status', 'x', 'sl', 'ss', 'y', 'zl',
'zs'.
- If status is 'optimal', x, sl, ss, y, zl, zs are the primal and
- dual optimal solutions.
+ If status is 'optimal', x, sl, ss, y, zl, zs are approximate
+ primal and dual optimal solutions.
If status is 'primal infeasible', x = sl = ss = None, and zl,
zs, y are a proof of infeasibility:
- hl'*zl + tr(hs,zs) + b'*y = -1,
- Gl'*zl + Gs'(zs) + A'*y = 0,
- zl >= 0, zs >= 0.
+ hl'*zl + sum_k tr(hs[k]*zs[k]) + b'*y = -1,
+ Gl'*zl + sum_k Gs[k]'*vec(zs[k]) + A'*y = 0,
+ zl >= 0, zs[k] >= 0, k = 0, ..., N-1.
If status is 'dual infeasible', zl = zs = y = None, and x, sl,
ss are a proof of dual infeasibility:
- c'*x = -1, Gl*x + sl = 0, Gs(x) + ss = 0, A*x = 0,
- sl >= 0, ss >= 0.
+ c'*x = -1,
+ Gl*x + sl = 0, mat(Gs[k]*x] + ss[k] = 0, k = 0, ..., N-1
+ A*x = 0, sl >= 0, ss[k] >= 0, k = 0, ..., N-1.
If status is 'unknown', x, y, sl, ss, zl, zs are None.
+
The following parameters control the execution of the default
solver.
options['show_progress'] True/False (default: True)
options['maxiters'] positive integer (default: 100)
options['abstol'] scalar (default: 1e-7)
- options['reltol'] scalar (default: 1e-7)
- options['feastol'] scalar (default: 1e-7)
- options['refinement'] True/False (default: True).
+ options['reltol'] scalar (default: 1e-6)
+ options['feastol'] scalar (default: 1e-7).
The execution of the 'dsdp' solver is controlled by:
@@ -1623,7 +2862,7 @@ def sdp(c, Gl=None, hl=None, Gs=None, hs=None, A=None, b=None,
if (type(Gl) is not matrix and type(Gl) is not spmatrix) or \
Gl.typecode != 'd' or Gl.size[1] != n:
raise TypeError, "'Gl' must be a dense or sparse 'd' matrix "\
- "with %d columns" %n
+ "with %d columns" %n
ml = Gl.size[0]
if hl is None: hl = matrix(0.0, (0,1))
if type(hl) is not matrix or hl.typecode != 'd' or \
@@ -1631,22 +2870,26 @@ def sdp(c, Gl=None, hl=None, Gs=None, hs=None, A=None, b=None,
raise TypeError, "'hl' must be a 'd' matrix of size (%d,1)" %ml
if Gs is None: Gs = []
- if type(Gs) is not list or [ Gs_k for Gs_k in Gs if (type(Gs_k) is
- not matrix and type(Gs_k) is not spmatrix) or
- Gs_k.typecode != 'd' or Gs_k.size[1] != n ]:
+ if type(Gs) is not list or [ G for G in Gs if (type(G) is not matrix
+ and type(G) is not spmatrix) or G.typecode != 'd' or
+ G.size[1] != n ]:
raise TypeError, "'Gs' must be a list of sparse or dense 'd' "\
"matrices with %d columns" %n
- ms = [ int(math.sqrt(Gs_k.size[0])) for Gs_k in Gs ]
- maxm, N = max([0] + ms), len(ms)
- a = [ k for k in xrange(len(Gs)) if ms[k]**2 != Gs[k].size[0] ]
+ ms = [ int(math.sqrt(G.size[0])) for G in Gs ]
+ a = [ k for k in xrange(len(ms)) if ms[k]**2 != Gs[k].size[0] ]
if a: raise TypeError, "the squareroot of the number of rows in "\
"'Gs[%d]' is not an integer" %k
if hs is None: hs = []
- if type(hs) is not list or len(hs) != N or [ k for k in xrange(N)
- if type(hs[k]) not in (matrix, spmatrix) or
- hs[k].typecode != 'd' or hs[k].size != (ms[k],ms[k]) ]:
+ if type(hs) is not list or len(hs) != len(ms) or [ h for h in hs if
+ (type(h) is not matrix and type(h) is not spmatrix) or
+ h.typecode != 'd' ]:
raise TypeError, "'hs' must be a list of %d dense or sparse "\
- "'d' matrices of size (%d,%d)" %(N,ms[k],ms[k])
+ "'d' matrices" %len(ms)
+ a = [ k for k in xrange(len(ms)) if hs[k].size != (ms[k],ms[k]) ]
+ if a:
+ k = a[0]
+ raise TypeError, "hs[%d] has size (%d,%d). Expected size is "\
+ "(%d,%d)." %(k,hs[k].size[0], hs[k].size[1], ms[k], ms[k])
if A is None: A = spmatrix([], [], [], (0,n), 'd')
if (type(A) is not matrix and type(A) is not spmatrix) or \
@@ -1658,39 +2901,20 @@ def sdp(c, Gl=None, hl=None, Gs=None, hs=None, A=None, b=None,
if type(b) is not matrix or b.typecode != 'd' or b.size != (p,1):
raise TypeError, "'b' must be a dense matrix of size (%d,1)" %p
- if p > n or n-p > ml+ sum([mk*(mk+1)/2 for mk in ms]):
- raise ValueError, "Rank(A) < p or Rank([Gl; Gs; A]) < n"
-
- if not sum(ms): # call lp()
- if primalstart is not None:
- ps = primalstart.copy()
- if ps.has_key('sl'): ps['s'] = ps['sl']
- else:
- ps = None
- if dualstart is not None:
- ds = dualstart.copy()
- if ds.has_key('zl'): ds['z'] = ds['zl']
- else:
- ds = None
- sol = lp(c, Gl, hl, A, b, solver=solver, primalstart=ps,
- dualstart=ds)
- sol['sl'] = sol['s']; del sol['s']
- sol['ss'] = [ matrix(0.0, (0,0)) for k in xrange(N)]
- sol['zl'] = sol['z']; del sol['z']
- sol['zs'] = [ matrix(0.0, (0,0)) for k in xrange(N)];
- return sol
-
if solver == 'dsdp':
try: from cvxopt import dsdp
except ImportError: raise ValueError, "invalid option "\
"(solver = 'dsdp'): cvxopt.dsdp is not installed"
dsdp.options = options
- if p: raise ValueError, "sdp() with the 'solver=dsdp' option "\
+ if p: raise ValueError, "sdp() with the solver = 'dsdp' option "\
"does not handle problems with equality constraints"
dsdpstatus, x, r, zl, zs = dsdp.sdp(c, Gl, hl, Gs, hs)
sl = hl - Gl*x
- ss = [ hs[k] - matrix(Gs[k]*x, hs[k].size) for k in
- xrange(len(hs)) ]
+ ss = [ hs[k] - matrix(Gs[k]*x, (ms[k], ms[k])) for k in
+ xrange(len(ms)) ]
+ for k in xrange(len(ms)):
+ symm(ss[k], ms[k])
+ symm(zs[k], ms[k])
if dsdpstatus == 'DSDP_PDFEASIBLE':
y = matrix(0.0, (0,1))
status = 'optimal'
@@ -1715,437 +2939,73 @@ def sdp(c, Gl=None, hl=None, Gs=None, hs=None, A=None, b=None,
return {'status': status, 'x': x, 'y': y, 'sl': sl, 'ss': ss,
'zl': zl, 'zs': zs}
+ dims = {'l': ml, 'q': [], 's': ms}
+ N = ml + sum([ m**2 for m in ms ])
+ h = matrix(0.0, (N,1))
+ if type(Gl) is matrix or [ Gk for Gk in Gs if type(Gk) is matrix ]:
+ G = matrix(0.0, (N, n))
+ else:
+ G = spmatrix([], [], [], (N, n), 'd')
+ h[:ml] = hl
+ G[:ml,:] = Gl
+ ind = ml
+ for k in xrange(len(ms)):
+ m = ms[k]
+ h[ind : ind + m*m] = hs[k][:]
+ G[ind : ind + m*m, :] = Gs[k]
+ ind += m**2
- def Fi(x, y, alpha=1.0, beta=0.0, trans='N'):
- base.gemv(Gl, x, y, alpha=alpha, beta=beta, trans=trans)
-
- def Fs(x, y, alpha=1.0, beta=0.0, trans='N'):
- if trans == 'N':
- for k in xrange(N):
- # y[k] := alpha*mat(Gs[k]*x) + beta*y[k]
- base.gemv(Gs[k], x, y[k], alpha=alpha, beta=beta)
-
- else:
- # y[i] := alpha*sum_k tr(mat(Gs[k][:,i])*x[k]) + beta*y[i]
- # for i=0,...,n-1.
-
- for k in xrange(N):
- # Scale diagonal of x[k] by 1/2 and set the upper
- # triangular part to zero, so that the matrix product
- # of Gs[k]' and vec(x[k]) is equal to the inner
- # product of mat(Gs[k][:j]) with x[k]. Then make the
- # matrix-vector product.
- blas.scal(0.5, x[k], inc=ms[k]+1)
- for j in xrange(ms[k]):
- blas.scal(0.0, x[k], offset=j+ms[k]*(j+1),
- inc=ms[k])
- if k == 0:
- base.gemv(Gs[k], x[k], y, alpha=2*alpha, beta=beta,
- trans='T')
- else:
- base.gemv(Gs[k], x[k], y, alpha=2*alpha, beta=1.0,
- trans='T')
- blas.scal(2.0, x[k], inc=ms[k]+1)
-
- def Fe(x, y, alpha=1.0, beta=0.0, trans='N'):
- base.gemv(A, x, y, alpha=alpha, beta=beta, trans=trans)
-
-
- # kktsolver(di, rti) returns a function for solving
- #
- # [ 0 A' Gl' Gs()' ] [x ] [bx ]
- # [ A 0 0 0 ] [y ] = [by ].
- # [ Gl 0 -diag(d)^2 0 ] [zl] [bzl]
- # [ Gs 0 0 -r*r'*()*r*r' ] [zs] [bzs]
- #
- # di = 1./d is a positive mx1 'd' matrix;
- # rti = inv(r') is a list of square matrices.
- #
- # The scaled zl and zs are returned: d.*zl instead of zl, r'*zs*r
- # instead of zs.
-
- global kktmethod
- try: kktmethod = options['kktmethod']
- except KeyError: kktmethod = 5
+ if primalstart:
+ ps = {}
+ ps['x'] = primalstart['x']
+ ps['s'] = matrix(0.0, (N,1))
+ if ml: ps['s'][:ml] = primalstart['sl']
+ if ms:
+ ind = ml
+ for k in xrange(len(ms)):
+ m = ms[k]
+ ps['s'][ind : ind + m*m] = primalstart['ss'][k][:]
+ ind += m**2
else:
- if type(kktmethod) is not int or kktmethod not in [1,4,5]:
- raise ValueError, "options['kktmethod'] must be 1, 4, or 5"
-
- if kktmethod == 1:
-
- # The matrix
- #
- # [ 0 A' Glsc' Gssc()' ]
- # [ A 0 0 0 ]
- # [ Glsc 0 -I 0 ]
- # [ Gssc 0 0 -(1/2)*I ]
- #
- # is factored as LDL, with:
- #
- # Glsc = diag(d)^{-1}*G = diag(di)*G
- # Gssc() = rti'*Gs()*rti in packed storage with diagonal
- # elements scaled by 1/sqrt(2).
-
- pms = [ mk*(mk+1)/2 for mk in ms ]
- K = matrix(0.0, (n+p+ml+sum(pms), n+p+ml+sum(pms)))
- ipiv = matrix(0, (K.size[0],1))
- Gssc = [ matrix(0.0, (m_k,m_k)) for m_k in ms ]
- u = matrix(0.0, (K.size[0],1))
-
- def kktsolver(di, rti):
- blas.scal(0.0, K)
-
- # 3,3-block of K is -I.
- K[ (K.size[0]+1)*(p+n) : (K.size[0]+1)*(p+n+ml):
- K.size[0]+1 ] = -1.0
-
- # 4,4-block of K is -I/2.
- K[ (K.size[0]+1)*(p+n+ml) :: K.size[0]+1 ] = -0.5
-
- # 2,1-block of K is A.
- K[n:n+p,:n] = A
-
- # 3,1-block of K is diag(di)*Gl.
- K[n+p:n+p+ml, :n] = Gl
- for k in xrange(ml):
- blas.scal(di[k], K, n=n, offset=n+p+k, inc=K.size[0])
-
- # 4,1-block of K is rti'*G()*rti in packed storage with
- # diagonal elements scaled by 1/sqrt(2).
- for j in xrange(n):
- # Gssc = rti' * mat(Gs[:,j]) * rti.
- for k in xrange(N): Gssc[k][:] = Gs[k][:,j]
- misc.cngrnc(rti, Gssc, Gssc, trans='T')
-
- # copy Gssc in packed storage to column j of K
- ind = n+p+ml + j*K.size[0]
- for k in xrange(N):
- blas.scal(1.0/math.sqrt(2), Gssc[k], inc=ms[k]+1)
- for i in xrange(ms[k]):
- blas.copy(Gssc[k], K, n=ms[k]-i,
- offsetx=(ms[k]+1)*i, offsety=ind)
- ind += ms[k]-i
-
- lapack.sytrf(K, ipiv)
-
- def solve_kkt(x, y, zl, zs):
-
- # Solve
- #
- # [0 A' Glsc' Gssc()'] [x ] [bx ]
- # [A 0 0 0 ] [y ] = [by ]
- # [Glsc 0 -I 0 ] [d.*zl ] [di.*zl ]
- # [Gssc 0 0 -(1/2)*I] [2*r'*zs*r] [rti'*zs*rti]
- #
- # and return x, y, d.*zl, r'*zs*r.
-
- blas.copy(x, u)
- blas.copy(y, u, offsety=n)
- misc.had2(zl, di)
- blas.copy(zl, u, offsety=n+p)
- misc.cngrnc(rti, zs, zs, trans='T')
- ind = n+p+ml
- for k in xrange(N):
- blas.scal(1.0/math.sqrt(2), zs[k], inc=ms[k]+1)
- for i in xrange(ms[k]):
- blas.copy(zs[k], u, n=ms[k]-i,
- offsetx=(ms[k]+1)*i, offsety=ind)
- ind += ms[k]-i
- lapack.sytrs(K, ipiv, u)
- blas.copy(u, x, n=n)
- blas.copy(u, y, offsetx=n, n=p)
- blas.copy(u, zl, offsetx=n+p, n=ml)
- ind = n+p+ml
- for k in xrange(N):
- for i in xrange(ms[k]):
- blas.copy(u, zs[k], offsetx=ind,
- offsety=i*(ms[k]+1), n=ms[k]-i)
- zs[k][i,i] *= math.sqrt(2)
- ind += ms[k]-i
- misc.sscal(0.5, zs)
-
- return solve_kkt
-
- elif kktmethod == 2:
-
- pass # not implemented
-
- elif kktmethod == 3:
-
- pass # not implemented
-
- elif kktmethod == 4:
-
- # Factor the matrices
- #
- # S = Gl'*diag(di)^2*Gl + Gs'(rti*rti'*Gs(x)*rti*rti')
- # + A'*A
- # K = A*S^{-1}*A'.
- #
- # Solve for y, x:
- #
- # K*y = A * S^{-1} * (bx + Gl'*diag(di)^2*bzl
- # + Gs'(rti*rti'*bzs*rti*rti') + A'*by) - by
- # S*x = bx + Gl'*diag(di)^2*bzl + Gs'(rti*rti'*bzs*rti*rti')
- # + A'*by - A'*y.
- #
- # Compute scaled zl, zs:
- #
- # zl = di .* (Gl*x - bzl)
- # zs = rti' * (Gs(x) - bzs) * rti.
-
- S, K = matrix(0.0, (n,n)), matrix(0.0, (p,p))
- if type(Gl) is matrix:
- Glsc = matrix(0.0, (ml,n))
- else:
- Glsc = spmatrix(0.0, Gl.I, Gl.J, (ml,n))
- T = [ matrix(0.0, (m_k,m_k)) for m_k in ms ]
- Gssc = [ matrix(0.0, (m_k,m_k)) for m_k in ms ]
- zsc = [ matrix(0.0, (m_k,m_k)) for m_k in ms ]
-
- def kktsolver(di, rti):
-
- # S = Gl'*diag(di)^2*Gl
- base.gemm(spmatrix(di, range(ml), range(ml), tc='d'), Gl,
- Glsc, partial=True)
- base.syrk(Glsc, S, trans='T')
-
- # T = rti*rti' as a nonsymmetric matrix
- for k in xrange(N):
- blas.gemm(rti[k], rti[k], T[k], transB='T')
-
- for j in xrange(n):
- # Gssc[k] = T[k] * mat(Gs[k][:,j]) * T[k]
- for k in xrange(N): Gssc[k][:] = Gs[k][:,j]
- misc.cngrnc(T, Gssc, Gssc, trans='T')
-
- # N matrix vector products to get lower triangular part
- # of column j of S.
- for k in xrange(N):
- blas.scal(0.5, Gssc[k], inc=ms[k]+1)
- for i in xrange(ms[k]):
- blas.scal(0.0, Gssc[k], offset=i+ms[k]*(i+1),
- inc=ms[k])
- base.gemv(Gs[k], Gssc[k], S, trans='T', n=n-j,
- alpha=2.0, beta=1.0, offsetA=j*(ms[k]**2),
- offsety=j*(n+1))
-
- # S += A'*A
- base.syrk(A, S, trans='T', beta=1.0)
-
- # factor S = L*L'
- lapack.potrf(S)
-
- # Asct := L^{-1}*A', factor K = Asct'*Asct
- if type(A) is matrix:
- Asct = A.trans()
- else:
- Asct = matrix(A.trans())
- blas.trsm(S, Asct)
- blas.syrk(Asct, K, trans='T')
- lapack.potrf(K)
-
- def solve_kkt(x, y, zl, zs):
-
- # zl := di.*zl = di.*bzl
- # zsc := T*zs*T = T*bzs*T
- misc.had2(zl, di)
- misc.cngrnc(T, zs, zsc, trans='T')
-
- # x := x + Glsc'*zl + Gs'(zsc) + A'*y
- # = bx + Gl'*diag(di)^2*bzl) + Gs'(T*bzs*T) + A'*by
- base.gemv(Glsc, zl, x, trans='T', beta=1.0)
- Fs(zsc, x, alpha=1.0, beta=1.0, trans='T')
- base.gemv(A, y, x, trans='T', beta=1.0)
-
- # x := L^{-1}*x
- # = L^{-1} * (bx + Gl'*diag(di)^2*bzl + Gs'(T*bzs*T)
- # + A'*by)
- blas.trsv(S, x)
-
- # y := Asc*x - y
- # = A*S^{-1}* (bx + Gl'*diag(di)^2.*bzl +
- # Gs'(T*mat(bzs)*T) + A'*by) - by
- base.gemv(Asct, x, y, trans='T', beta=-1.0)
-
- # y := K^{-1} * y
- # = K^{-1} * (A*S^{-1}* (bx + Gl'*diag(di)^2*bzl +
- # Gs'(T*mat(bzs)*T) + A'*by) - by)
- lapack.potrs(K, y)
-
- # x := x - Asc'*y
- base.gemv(Asct, y, x, alpha=-1.0, beta=1.0)
-
- # x := L^{-T} * x
- # = S^{-1} * (bx + Gl'*diag(di)^2*bzl +
- # Gs'*(T*bzs*T) + A'*by - A'*y)
- blas.trsv(S, x, trans='T')
-
- # zl := Glsc*x - zl
- # = di .* (Gl*x - bzl)
- base.gemv(Glsc, x, zl, beta=-1.0)
-
- # zs = rti' * (Gs(x) - bzs) * rti
- Fs(x, zs, alpha=1.0, beta=-1.0)
- misc.cngrnc(rti, zs, zs, trans='T')
-
- return solve_kkt
-
- elif kktmethod == 5:
-
- # Solve by eliminating the equality constraints and using
- # QR factorizations
- #
- # A' = Q1*R1, G*Q1 = Q3*R3
- #
- # where G = [Glsc; Gssc], Glsc = diag(di)*G, and
- # Gssc = rti'*Gs()*rti (in packed storage with off-diagonal
- # elements scaled by sqrt(2)).
-
- # A' = [Q1, Q2] * [R1; 0]
- if type(A) is matrix:
- QA = +A.trans()
- else:
- QA = matrix(A.trans())
- tauA, tauG = matrix(0.0, (p,1)), matrix(0.0, (n-p,1))
- lapack.geqrf(QA, tauA)
-
- pms = [ mk*(mk+1)/2 for mk in ms ]
- Gssc = [ matrix(0.0, (m_k,m_k)) for m_k in ms ]
- G = matrix(0.0, (ml+sum(pms), n))
- z = matrix(0.0, (ml+sum(pms),1))
- w = matrix(0.0, (ml+sum(pms),1))
- v = matrix(0.0, (n,1))
- def kktsolver(di, rti):
-
- # G = [Glsc; Gssc] where , Glsc = diag(di)*G, and
- # Gssc = rti'*Gs()*rti (in packed storage with off-diagonal
- # elements scaled by sqrt(2)).
-
- G[:ml,:] = Gl
- for k in xrange(ml):
- blas.scal(di[k], G, n=n, offset=k, inc=G.size[0])
-
- for j in xrange(n):
- # Gssc = sqrt(2) * rti' * mat(Gs[:,j]) * rti.
- for k in xrange(N): Gssc[k][:] = Gs[k][:,j]
- misc.cngrnc(rti, Gssc, Gssc, trans='T',
- alpha=math.sqrt(2))
- # copy Gssc in packed storage to column j of G and
- # scale diagonal elements by 1/sqrt(2).
- ind = ml + j*G.size[0]
- for k in xrange(N):
- blas.scal(1.0/math.sqrt(2), Gssc[k], inc=ms[k]+1)
- for i in xrange(ms[k]):
- blas.copy(Gssc[k], G, n=ms[k]-i,
- offsetx=(ms[k]+1)*i, offsety=ind)
- ind += ms[k]-i
-
- # G = [G1, G2] := G * [Q1, Q2]
- lapack.ormqr(QA, tauA, G, side='R')
-
- # QR factorization G2 := [Q3, Q4] * [R3; 0].
- lapack.geqrf(G, tauG, n=n-p, offsetA=G.size[0]*p)
-
- def solve_kkt(x, y, zl, zs):
-
- # Solve
- #
- # [0 A' G'] [x ] [bx]
- # [A 0 0 ] [y ] = [by]
- # [G 0 -I ] [z ] [bz]
- #
- # where
- # z = [d.*zl; r'*zs*r] (with r'*zs*r in packed
- # storage with off-diagonal elements scaled by
- # sqrt(2))
- # bz = [di.*bzl; rti'*bzs*rti] (with rti'*bzs*rti in
- # packed storage with off-diagonal elements
- # scaled by sqrt(2))
- #
- # If we eliminate x = [Q1 Q2] * [ R1^{-T}*by; v ] the
- # equations reduce to
- #
- # G2'*z = Q2'*bx
- # G2*v - z = bz - G1*R1^{-T}*by
- # R1*y = Q1'*bx - G1'*z.
- #
- # This can be solved in 5 steps:
- #
- # w := bz - G1*R1^{-T}*by
- # u := Q3'*w + R3^{-T}*Q2'*bx
- # z := Q3*u - w
- # x := [Q1, Q2] * [ R1^{-T}*by; R3^{-1}*u]
- # y := R1^{-1} * (Q1'*bx - G1'*z).
-
- # w := bz = [bzlc; bzsc]
- misc.had2(zl, di)
- blas.copy(zl, w)
- misc.cngrnc(rti, zs, zs, trans='T', alpha=math.sqrt(2))
- ind = ml
- for k in xrange(N):
- blas.scal(1.0/math.sqrt(2), zs[k], inc=ms[k]+1)
- for i in xrange(ms[k]):
- blas.copy(zs[k], w, n=ms[k]-i,
- offsetx=(ms[k]+1)*i, offsety=ind)
- ind += ms[k]-i
-
- # v := [Q1'*bx; R3^{-T}*Q2'*bx]
- blas.copy(x, v)
- lapack.ormqr(QA, tauA, v, trans='T')
- lapack.trtrs(G, v, uplo='U', trans='T', n=n-p,
- offsetA=G.size[0]*p, offsetB=p)
-
- # x[:p] := R1^{-T}*by
- blas.copy(y, x)
- lapack.trtrs(QA, x, uplo='U', trans='T', n=p)
-
- # w := w - G1*x[:p]
- # = bz - G1*R1^{-T}*by
- blas.gemv(G, x, w, alpha=-1.0, beta=1.0, n=p)
-
- # z := [Q3'*w + v[p:]; 0]
- # = [Q3'*w + R3^{-T}*Q2'*bx; 0]
- blas.copy(w, z)
- lapack.ormqr(G, tauG, z, trans='T', k=n-p,
- offsetA=G.size[0]*p)
- blas.axpy(v, z, offsetx=p, n=n-p)
- blas.scal(0.0, z, offset=n-p)
-
- # x[p:] := R3^{-1}*z[:n-p]
- blas.copy(z, x, offsety=p, n=n-p)
- lapack.trtrs(G, x, uplo='U', n=n-p, offsetA=G.size[0]*p,
- offsetB=p)
-
- # x is now [ R1^{-T}*by; Q3'*w + R3^{-T}*Q2'*bx ]
- # x := [Q1 Q2]*x
- lapack.ormqr(QA, tauA, x)
-
- # z := [Q3, Q4] * z - w
- lapack.ormqr(G, tauG, z, k=n-p, offsetA=G.size[0]*p)
- blas.axpy(w, z, alpha=-1.0)
-
- # y := R1^{-1} * ( v[:p] - G1'*z )
- # = R1^{-1} * ( Q1'*bx - G1'*z )
- blas.copy(v, y, n=p)
- blas.gemv(G, z, y, n=p, trans='T', alpha=-1.0, beta=1.0)
- lapack.trtrs(QA, y, uplo='U', n=p)
-
- # zl = z[:ml]
- blas.copy(z, zl, n=ml)
-
- # zs = z[ml:] unpacked and with offdiagonal elts
- # scaled by 1/sqrt(2)
- ind = ml
- for k in xrange(N):
- for i in xrange(ms[k]):
- blas.copy(z, zs[k], offsetx=ind,
- offsety=i*(ms[k]+1), n=ms[k]-i)
- ind += ms[k]-i
- blas.scal(math.sqrt(2.0), zs[k], inc=ms[k]+1)
- misc.sscal(1.0/math.sqrt(2.0), zs)
+ ps = None
- return solve_kkt
-
- return conelp(c, kktsolver, Gl=Fi, hl=hl, Gs=Fs, hs=hs, A=Fe, b=b,
- primalstart=primalstart, dualstart=dualstart)
+ if dualstart:
+ ds = {}
+ if p: ds['y'] = dualstart['y']
+ ds['z'] = matrix(0.0, (N,1))
+ if ml: ds['z'][:ml] = dualstart['zl']
+ if ms:
+ ind = ml
+ for k in xrange(len(ms)):
+ m = ms[k]
+ ds['z'][ind : ind + m*m] = dualstart['zs'][k][:]
+ ind += m**2
+ else:
+ ds = None
+
+ sol = conelp(c, G, h, dims, A = A, b = b, primalstart = ps, dualstart
+ = ds)
+ val = {'status': sol['status'], 'x': sol['x'], 'y': sol['y']}
+ if sol['s'] is None:
+ val['sl'] = None
+ val['ss'] = None
+ else:
+ val['sl'] = sol['s'][:ml]
+ val['ss'] = [ matrix(0.0, (mk, mk)) for mk in ms ]
+ ind = ml
+ for k in xrange(len(ms)):
+ m = ms[k]
+ val['ss'][k][:] = sol['s'][ind:ind+m*m]
+ ind += m**2
+ if sol['z'] is None:
+ val['zl'] = None
+ val['zs'] = None
+ else:
+ val['zl'] = sol['z'][:ml]
+ val['zs'] = [ matrix(0.0, (mk, mk)) for mk in ms ]
+ ind = ml
+ for k in xrange(len(ms)):
+ m = ms[k]
+ val['zs'][k][:] = sol['z'][ind:ind+m*m]
+ ind += m**2
+ return val
diff --git a/src/python/cvxprog.py b/src/python/cvxprog.py
index 59658d3..d014f56 100644
--- a/src/python/cvxprog.py
+++ b/src/python/cvxprog.py
@@ -6,8 +6,22 @@ for quadratic and geometric programming. Also includes an interface
to the quadratic programming solver from MOSEK.
"""
-# This file is part of CVXOPT version 0.8.2.
# Copyright 2004-2007 J. Dahl and L. Vandenberghe.
+#
+# This file is part of CVXOPT version 0.9.
+#
+# CVXOPT is free software; you can redistribute it and/or modify
+# it under the terms of the GNU General Public License as published by
+# the Free Software Foundation; either version 3 of the License, or
+# (at your option) any later version.
+#
+# CVXOPT is distributed in the hope that it will be useful,
+# but WITHOUT ANY WARRANTY; without even the implied warranty of
+# MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the
+# GNU General Public License for more details.
+#
+# You should have received a copy of the GNU General Public License
+# along with this program. If not, see <http://www.gnu.org/licenses/>.
import math
from cvxopt import base, blas, lapack, misc
diff --git a/src/python/info.py b/src/python/info.py
index 43dd9db..3911d7f 100644
--- a/src/python/info.py
+++ b/src/python/info.py
@@ -1,41 +1,45 @@
-version = '0.8.2'
+version = '0.9'
def license(): print(
-"""CVXOPT version 0.8.2. Copyright (c) 2004-2007 J. Dahl and L. Vandenberghe.
+"""
+CVXOPT version 0.9. Copyright (c) 2004-2007 J. Dahl and L. Vandenberghe.
-This program is free software; you can redistribute it and/or modify it under
-the terms of the GNU General Public License as published by the Free Software
-Foundation; either version 2 of the License, or (at your option) any later
-version.
+This program is free software; you can redistribute it and/or modify
+it under the terms of the GNU General Public License as published by
+the Free Software Foundation; either version 3 of the License, or
+(at your option) any later version.
-This program is distributed in the hope that it will be useful, but WITHOUT
-ANY WARRANTY; without even the implied warranty of MERCHANTABILITY or FITNESS
-FOR A PARTICULAR PURPOSE. See the GNU General Public License for more details.
+This program is distributed in the hope that it will be useful,
+but WITHOUT ANY WARRANTY; without even the implied warranty of
+MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the
+GNU General Public License for more details.
-The full text of the GNU General Public License can be found at
-www.gnu.org/copyleft/gpl.html or in the LICENSE file distributed with CVXOPT.
+You should have received a copy of the GNU General Public License
+along with this program. If not, see <http://www.gnu.org/licenses/>.
The CVXOPT distribution includes source code for the following software
-libraries. The copyright of these libraries is owned by their authors.
+libraries.
1. Part of the SuiteSparse suite of sparse matrix algorithms, including:
- - AMD Version 2.0. Copyright (c) 2006 by Timothy A. Davis,
- Patrick R. Amestoy, and Iain S. Duff.
- - CHOLMOD Version 1.4. Copyright (c) 2005-2006 by University of Florida,
- Timothy A. Davis and W. Hager.
- - COLAMD version 2.6. Copyright (c) 1998-2006 by Timothy A. Davis.
- - UMFPACK Version 5.0.2. Copyright (c) 1995-2006 by Timothy A. Davis.
-
- These packages are licensed under the terms of the GNU Lesser General Public
- License (UMFPACK, parts of CHOLMOD, AMD, COLAMD) and the GNU General Public
- License (parts of CHOLMOD). For details, consult the README files in the
- source directories or the website listed below.
+ - AMD Version 2.1. Copyright (c) 2007 by Timothy A. Davis, Patrick R.
+ Amestoy, and Iain S. Duff.
+ - CHOLMOD Version 1.5. Copyright (c) 2005-2007 by University of Florida,
+ Timothy A. Davis and W. Hager.
+ - COLAMD version 2.7. Copyright (c) 1998-2007 by Timothy A. Davis.
+ - UMFPACK Version 5.0.2. Copyright (c) 1995-2006 by Timothy A. Davis.
+
+ These packages are licensed under the terms of the GNU Lesser General
+ Public License, version 2.1 or higher (UMFPACK, parts of CHOLMOD, AMD,
+ COLAMD) and the GNU General Public License, version 2 or higher
+ (the Supernodal module of CHOLMOD). For copyright and license details,
+ consult the README files in the source directories or the website
+ listed below.
Availability: www.cise.ufl.edu/research/sparse.
-2. RNGS Random Number Generation -- Multiple Streams (Sep. 22, 1998) by Steve
- Park and Dave Geyer.
+2. RNGS Random Number Generation -- Multiple Streams (Sep. 22, 1998) by
+ Steve Park & Dave Geyer.
- Availability: www.cs.wm.edu/~va/software/park/park.html.
+ Availability: www.cs.wm.edu/~va/software/park/park.html.
"""
)
diff --git a/src/python/misc.py b/src/python/misc.py
index df2f8f2..7dc0ca1 100644
--- a/src/python/misc.py
+++ b/src/python/misc.py
@@ -2,8 +2,22 @@
Miscellaneous functions used by the CVXOPT solvers.
"""
-# This file is part of CVXOPT version 0.8.2.
# Copyright 2004-2007 J. Dahl and L. Vandenberghe.
+#
+# This file is part of CVXOPT version 0.9.
+#
+# CVXOPT is free software; you can redistribute it and/or modify
+# it under the terms of the GNU General Public License as published by
+# the Free Software Foundation; either version 3 of the License, or
+# (at your option) any later version.
+#
+# CVXOPT is distributed in the hope that it will be useful,
+# but WITHOUT ANY WARRANTY; without even the implied warranty of
+# MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the
+# GNU General Public License for more details.
+#
+# You should have received a copy of the GNU General Public License
+# along with this program. If not, see <http://www.gnu.org/licenses/>.
import math
from cvxopt import base, blas, lapack, cholmod
diff --git a/src/python/modeling.py b/src/python/modeling.py
index 8b8cb70..bc3c04b 100644
--- a/src/python/modeling.py
+++ b/src/python/modeling.py
@@ -5,8 +5,22 @@ Routines for specifying and solving convex optimization problems with
piecewise-linear objective and constraint functions.
"""
-# This file is part of CVXOPT version 0.8.2.
# Copyright 2004-2007 J. Dahl and L. Vandenberghe.
+#
+# This file is part of CVXOPT version 0.9.
+#
+# CVXOPT is free software; you can redistribute it and/or modify
+# it under the terms of the GNU General Public License as published by
+# the Free Software Foundation; either version 3 of the License, or
+# (at your option) any later version.
+#
+# CVXOPT is distributed in the hope that it will be useful,
+# but WITHOUT ANY WARRANTY; without even the implied warranty of
+# MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the
+# GNU General Public License for more details.
+#
+# You should have received a copy of the GNU General Public License
+# along with this program. If not, see <http://www.gnu.org/licenses/>.
from string import strip
from cvxopt.base import matrix, spmatrix
diff --git a/src/python/solvers.py b/src/python/solvers.py
index 5184dab..b8ab0da 100644
--- a/src/python/solvers.py
+++ b/src/python/solvers.py
@@ -1,24 +1,39 @@
"""
Convex optimization solvers.
-conelp: solves semidefinite programs specified via functions.
+conelp: solves cone programs.
cp: solves nonlinear convex problems.
lp: solves linear programs.
gp: solves geometric programs.
nlcp: solves nonlinear convex problems specified via functions.
qp: solves quadratic programs.
sdp: solves semidefinite programs.
+socp: solves second-order cone programs.
options: dictionary with customizable algorithm parameters.
"""
-# This file is part of CVXOPT version 0.8.2.
# Copyright 2004-2007 J. Dahl and L. Vandenberghe.
+#
+# This file is part of CVXOPT version 0.9.
+#
+# CVXOPT is free software; you can redistribute it and/or modify
+# it under the terms of the GNU General Public License as published by
+# the Free Software Foundation; either version 3 of the License, or
+# (at your option) any later version.
+#
+# CVXOPT is distributed in the hope that it will be useful,
+# but WITHOUT ANY WARRANTY; without even the implied warranty of
+# MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the
+# GNU General Public License for more details.
+#
+# You should have received a copy of the GNU General Public License
+# along with this program. If not, see <http://www.gnu.org/licenses/>.
import cvxopt
from cvxopt.cvxprog import cp, qp, gp, nlcp
-from cvxopt.coneprog import conelp, lp, sdp
+from cvxopt.coneprog import conelp, lp, sdp, socp
solvecp, solveqp, solvegp, solvelp, solvesdp = cp, qp, gp, lp, sdp
options = {}
cvxopt.cvxprog.options = options
cvxopt.coneprog.options = options
-__all__ = ['conelp', 'cp', 'lp', 'gp', 'nlcp', 'qp', 'sdp']
+__all__ = ['conelp', 'cp', 'lp', 'gp', 'nlcp', 'qp', 'sdp', 'socp']
diff --git a/src/setup.py b/src/setup.py
index d3fb1dd..96ccf9a 100644
--- a/src/setup.py
+++ b/src/setup.py
@@ -119,7 +119,6 @@ umfpack = Extension('umfpack',
['C/SuiteSparse_cvxopt_extra/umfpack/' + s for s in
listdir('C/SuiteSparse_cvxopt_extra/umfpack')])
-
# Build for int or long?
import sys
if sys.maxint > 2**31: MACROS += [('DLONG','')]
@@ -129,12 +128,12 @@ cholmod = Extension('cholmod',
libraries = ['lapack', 'blas', 'g2c'],
include_dirs = [ 'C/SuiteSparse/CHOLMOD/Include',
'C/SuiteSparse/COLAMD', 'C/SuiteSparse/AMD/Include',
- 'C/SuiteSparse/UFconfig' ],
+ 'C/SuiteSparse/UFconfig', 'C/SuiteSparse/COLAMD/Include' ],
define_macros = MACROS + [('NPARTITION', '1')],
sources = [ 'C/cholmod.c' ] +
['C/SuiteSparse/AMD/Source/' + s for s in ['amd_global.c',
'amd_postorder.c', 'amd_post_tree.c', 'amd_2.c']] +
- ['C/SuiteSparse/COLAMD/' + s for s in ['colamd.c',
+ ['C/SuiteSparse/COLAMD/Source/' + s for s in ['colamd.c',
'colamd_global.c']] +
['C/SuiteSparse/CHOLMOD/Core/' + s for s in
listdir('C/SuiteSparse/CHOLMOD/Core') if s[-2:] == '.c' and
@@ -158,7 +157,20 @@ extmods += [base, blas, lapack, random, umfpack, cholmod, amd]
setup (name = 'cvxopt',
description = 'Convex optimization package',
- version = '0.8.2',
+ version = '0.9',
+ long_description = '''
+CVXOPT is a free software package for convex optimization based on the
+Python programming language. It can be used with the interactive Python
+interpreter, on the command line by executing Python scripts, or
+integrated in other software via Python extension modules. Its main
+purpose is to make the development of software for convex optimization
+applications straightforward by building on Python's extensive standard
+library and on the strengths of Python as a high-level programming
+language.''',
+ author='J. Dahl and L. Vandenberghe',
+ author_email='joachim at es.aau.dk, vandenbe at ee.ucla.edu',
+ url='http://abel.ee.ucla.edu/cvxopt',
+ license='GNU GPL version 3',
ext_package = "cvxopt",
ext_modules = extmods,
package_dir = {"cvxopt": "python"},
--
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