[mpir] 05/09: Remove non-DFSG documentation files.
Doug Torrance
dtorrance-guest at moszumanska.debian.org
Fri Mar 13 11:41:53 UTC 2015
This is an automated email from the git hooks/post-receive script.
dtorrance-guest pushed a commit to branch master
in repository mpir.
commit 0b181c510cd5e0695803e6e8d25b64b352b77cbe
Author: Doug Torrance <dtorrance at monmouthcollege.edu>
Date: Thu Mar 12 19:34:32 2015 -0500
Remove non-DFSG documentation files.
We use uscan to repack the orig tarball and append +dfsg1 to the version
number. We need a patch to allow the package to build without these files.
Also take the opportunity to add the ITP bug number to debian/changelog and
set the priority to the new default (medium).
---
debian/README.source | 14 +
debian/changelog | 6 +-
debian/copyright | 1 +
debian/patches/no_texinfo.patch | 14 +
debian/patches/series | 1 +
debian/watch | 3 +-
doc/fdl.texi | 506 --
doc/mpir.info | 177 -
doc/mpir.info-1 | 7061 --------------------------
doc/mpir.info-2 | 3427 -------------
doc/mpir.texi | 10370 --------------------------------------
doc/version.texi | 4 -
12 files changed, 35 insertions(+), 21549 deletions(-)
diff --git a/debian/README.source b/debian/README.source
new file mode 100644
index 0000000..d49b6b9
--- /dev/null
+++ b/debian/README.source
@@ -0,0 +1,14 @@
+We repack the MPIR source code for Debian.
+
+In particular, the Texinfo documentation files were released under the GNU Free
+Documentation License, which does not meet the Debian Free Software Guidelines
+[1].
+
+This repacking is accomplished using uscan. In particular, we declare the
+files to be removed using the Files-Excluded field in debian/copyright and add
+the +dfsgN suffix to the source tarball using the repacksuffix option in
+debian/watch.
+
+[1] https://wiki.debian.org/DFSGLicenses#GNU_Free_Documentation_License_.28GFDL.29
+
+ -- Doug Torrance <dtorrance at monmouthcollege.edu>, Thu, 12 Mar 2015 20:37:49 -0500
diff --git a/debian/changelog b/debian/changelog
index 6e2facd..5626aed 100644
--- a/debian/changelog
+++ b/debian/changelog
@@ -1,5 +1,5 @@
-mpir (2.6.0-1) UNRELEASED; urgency=low
+mpir (2.6.0+dfsg1-1) UNRELEASED; urgency=medium
- * Initial release.
+ * Initial release (Closes: #708391).
- -- Felix Salfelder <felix at salfelder.org> Wed, 15 May 2013 21:49:56 +0200
+ -- Doug Torrance <dtorrance at monmouthcollege.edu> Thu, 12 Mar 2015 19:21:27 -0500
diff --git a/debian/copyright b/debian/copyright
index b0c2e54..3bec66a 100644
--- a/debian/copyright
+++ b/debian/copyright
@@ -1,6 +1,7 @@
Format: http://www.debian.org/doc/packaging-manuals/copyright-format/1.0/
Upstream-Name: mpir
Source: http://mpir.org/
+Files-Excluded: doc/*.texi doc/mpir.info*
Files: *
Copyright: 1991, 1996, 1999, 2000 Free Software Foundation, Inc.
diff --git a/debian/patches/no_texinfo.patch b/debian/patches/no_texinfo.patch
new file mode 100644
index 0000000..20a3dae
--- /dev/null
+++ b/debian/patches/no_texinfo.patch
@@ -0,0 +1,14 @@
+Description: Don't try and build texinfo files.
+ We removed them since they are released under a non-DFSG license.
+Author: Doug Torrance <dtorrance at monmouthcollege.edu>
+Last-Update: 2015-03-09
+
+--- a/doc/Makefile.am
++++ b/doc/Makefile.am
+@@ -22,6 +22,3 @@
+
+
+ EXTRA_DIST = isa_abi_headache
+-
+-info_TEXINFOS = mpir.texi
+-gmp_TEXINFOS = fdl.texi
diff --git a/debian/patches/series b/debian/patches/series
index cd90e64..24ed556 100644
--- a/debian/patches/series
+++ b/debian/patches/series
@@ -1,3 +1,4 @@
no_lzma.patch
fix_self_reference.patch
fix_t-scan_segfault.patch
+no_texinfo.patch
diff --git a/debian/watch b/debian/watch
index e0eb43e..94025f1 100644
--- a/debian/watch
+++ b/debian/watch
@@ -1,3 +1,4 @@
version=3
-opts=uversionmangle=s/(\d)[_\.\-\+]?((RC|rc|pre|dev|beta|alpha)\d*)$/$1~$2/ \
+opts=uversionmangle=s/(\d)[_\.\-\+]?((RC|rc|pre|dev|beta|alpha)\d*)$/$1~$2/,\
+dversionmangle=s/\+dfsg1//,repacksuffix=+dfsg1 \
http://mpir.org/mpir-(.+)\.(?:zip|tgz|tbz|txz|(?:tar\.(?:gz|bz2|xz|lz|lzma)))
diff --git a/doc/fdl.texi b/doc/fdl.texi
deleted file mode 100644
index 8805f1a..0000000
--- a/doc/fdl.texi
+++ /dev/null
@@ -1,506 +0,0 @@
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diff --git a/doc/mpir.info b/doc/mpir.info
deleted file mode 100644
index 2c03079..0000000
--- a/doc/mpir.info
+++ /dev/null
@@ -1,177 +0,0 @@
-This is mpir.info, produced by makeinfo version 4.13 from mpir.texi.
-
-This manual describes how to install and use MPIR, the Multiple
-Precision Integers and Rationals library, version 2.6.0.
-
- Copyright 1991, 1993, 1994, 1995, 1996, 1997, 1998, 1999, 2000,
-2001, 2002, 2003, 2004, 2005, 2006, 2007, 2008, 2009, 2010 Free
-Software Foundation, Inc.
-
- Copyright 2008, 2009, 2010 William Hart
-
- Permission is granted to copy, distribute and/or modify this
-document under the terms of the GNU Free Documentation License, Version
-1.3 or any later version published by the Free Software Foundation;
-with no Invariant Sections, with the Front-Cover Texts being "A GNU
-Manual", and with the Back-Cover Texts being "You have freedom to copy
-and modify this GNU Manual, like GNU software". A copy of the license
-is included in *note GNU Free Documentation License::.
-
-INFO-DIR-SECTION GNU libraries
-START-INFO-DIR-ENTRY
-* mpir: (mpir). MPIR Multiple Precision Integers and Rationals Library.
-END-INFO-DIR-ENTRY
-
-
-Indirect:
-mpir.info-1: 1026
-mpir.info-2: 299211
-
-Tag Table:
-(Indirect)
-Node: Top1026
-Node: Copying3245
-Node: Introduction to MPIR4965
-Node: Installing MPIR7135
-Node: Build Options8573
-Node: ABI and ISA24632
-Node: Notes for Package Builds31401
-Node: Notes for Particular Systems34839
-Node: Known Build Problems42271
-Node: Performance optimization43959
-Node: MPIR Basics45095
-Node: Headers and Libraries45746
-Node: Nomenclature and Types47201
-Node: MPIR on Windows x6449122
-Node: Function Classes51049
-Node: Variable Conventions52584
-Node: Parameter Conventions54113
-Node: Memory Management56165
-Node: Reentrancy57297
-Node: Useful Macros and Constants58734
-Node: Compatibility with older versions59925
-Node: Efficiency60962
-Node: Debugging68641
-Node: Profiling75041
-Node: Autoconf79075
-Node: Emacs80949
-Node: Reporting Bugs81558
-Node: Integer Functions84125
-Node: Initializing Integers84904
-Node: Assigning Integers86838
-Node: Simultaneous Integer Init & Assign88617
-Node: Converting Integers90445
-Node: Integer Arithmetic94061
-Node: Integer Division95527
-Node: Integer Exponentiation101432
-Node: Integer Roots102255
-Node: Number Theoretic Functions104042
-Node: Integer Comparisons112099
-Node: Integer Logic and Bit Fiddling113450
-Node: I/O of Integers116010
-Node: Integer Random Numbers118897
-Node: Integer Import and Export120661
-Node: Miscellaneous Integer Functions124648
-Node: Integer Special Functions126494
-Node: Rational Number Functions129665
-Node: Initializing Rationals130860
-Node: Rational Conversions133316
-Node: Rational Arithmetic135048
-Node: Comparing Rationals136353
-Node: Applying Integer Functions137670
-Node: I/O of Rationals139154
-Node: Floating-point Functions141017
-Node: Initializing Floats143908
-Node: Assigning Floats147996
-Node: Simultaneous Float Init & Assign150546
-Node: Converting Floats152058
-Node: Float Arithmetic155281
-Node: Float Comparison157145
-Node: I/O of Floats158473
-Node: Miscellaneous Float Functions161044
-Node: Low-level Functions163677
-Node: Random Number Functions189839
-Node: Random State Initialization190909
-Node: Random State Seeding192962
-Node: Random State Miscellaneous194332
-Node: Formatted Output194924
-Node: Formatted Output Strings195170
-Node: Formatted Output Functions200410
-Node: C++ Formatted Output204487
-Node: Formatted Input207174
-Node: Formatted Input Strings207411
-Node: Formatted Input Functions212073
-Node: C++ Formatted Input215046
-Node: C++ Class Interface216954
-Node: C++ Interface General217952
-Node: C++ Interface Integers221032
-Node: C++ Interface Rationals224418
-Node: C++ Interface Floats228064
-Node: C++ Interface Random Numbers233291
-Node: C++ Interface Limitations235399
-Node: Custom Allocation238245
-Node: Language Bindings242891
-Node: Algorithms246921
-Node: Multiplication Algorithms247626
-Node: Basecase Multiplication248838
-Node: Karatsuba Multiplication250750
-Node: Toom 3-Way Multiplication254380
-Node: Toom 4-Way Multiplication260796
-Node: FFT Multiplication262170
-Node: Other Multiplication267596
-Node: Unbalanced Multiplication270278
-Node: Division Algorithms271174
-Node: Single Limb Division271522
-Node: Basecase Division274442
-Node: Divide and Conquer Division275645
-Node: Exact Division277915
-Node: Exact Remainder281115
-Node: Small Quotient Division283408
-Node: Greatest Common Divisor Algorithms285007
-Node: Binary GCD285299
-Node: Lehmer's GCD287975
-Node: Subquadratic GCD290179
-Node: Extended GCD292633
-Node: Jacobi Symbol293192
-Node: Powering Algorithms294109
-Node: Normal Powering Algorithm294373
-Node: Modular Powering Algorithm294902
-Node: Root Extraction Algorithms295966
-Node: Square Root Algorithm296282
-Node: Nth Root Algorithm298425
-Node: Perfect Square Algorithm299211
-Node: Perfect Power Algorithm301298
-Node: Radix Conversion Algorithms301920
-Node: Binary to Radix302297
-Node: Radix to Binary306228
-Node: Other Algorithms308193
-Node: Prime Testing Algorithm308547
-Node: Factorial Algorithm309774
-Node: Binomial Coefficients Algorithm311211
-Node: Fibonacci Numbers Algorithm312106
-Node: Lucas Numbers Algorithm314582
-Node: Random Number Algorithms315304
-Node: Assembler Coding317428
-Node: Assembler Code Organisation318405
-Node: Assembler Basics319378
-Node: Assembler Carry Propagation320537
-Node: Assembler Cache Handling322375
-Node: Assembler Functional Units324543
-Node: Assembler Floating Point326164
-Node: Assembler SIMD Instructions329947
-Node: Assembler Software Pipelining330936
-Node: Assembler Loop Unrolling332004
-Node: Assembler Writing Guide334224
-Node: Internals336994
-Node: Integer Internals337508
-Node: Rational Internals339765
-Node: Float Internals341005
-Node: Raw Output Internals348333
-Node: C++ Interface Internals349528
-Node: Contributors352815
-Node: References361810
-Node: GNU Free Documentation License369557
-Node: Concept Index394727
-Node: Function Index437105
-
-End Tag Table
diff --git a/doc/mpir.info-1 b/doc/mpir.info-1
deleted file mode 100644
index c03ceff..0000000
--- a/doc/mpir.info-1
+++ /dev/null
@@ -1,7061 +0,0 @@
-This is mpir.info, produced by makeinfo version 4.13 from mpir.texi.
-
-This manual describes how to install and use MPIR, the Multiple
-Precision Integers and Rationals library, version 2.6.0.
-
- Copyright 1991, 1993, 1994, 1995, 1996, 1997, 1998, 1999, 2000,
-2001, 2002, 2003, 2004, 2005, 2006, 2007, 2008, 2009, 2010 Free
-Software Foundation, Inc.
-
- Copyright 2008, 2009, 2010 William Hart
-
- Permission is granted to copy, distribute and/or modify this
-document under the terms of the GNU Free Documentation License, Version
-1.3 or any later version published by the Free Software Foundation;
-with no Invariant Sections, with the Front-Cover Texts being "A GNU
-Manual", and with the Back-Cover Texts being "You have freedom to copy
-and modify this GNU Manual, like GNU software". A copy of the license
-is included in *note GNU Free Documentation License::.
-
-INFO-DIR-SECTION GNU libraries
-START-INFO-DIR-ENTRY
-* mpir: (mpir). MPIR Multiple Precision Integers and Rationals Library.
-END-INFO-DIR-ENTRY
-
-
-File: mpir.info, Node: Top, Next: Copying, Prev: (dir), Up: (dir)
-
-MPIR
-****
-
- This manual describes how to install and use MPIR, the Multiple
-Precision Integers and Rationals library, version 2.6.0.
-
- Copyright 1991, 1993, 1994, 1995, 1996, 1997, 1998, 1999, 2000,
-2001, 2002, 2003, 2004, 2005, 2006, 2007, 2008, 2009, 2010 Free
-Software Foundation, Inc.
-
- Copyright 2008, 2009, 2010 William Hart
-
- Permission is granted to copy, distribute and/or modify this
-document under the terms of the GNU Free Documentation License, Version
-1.3 or any later version published by the Free Software Foundation;
-with no Invariant Sections, with the Front-Cover Texts being "A GNU
-Manual", and with the Back-Cover Texts being "You have freedom to copy
-and modify this GNU Manual, like GNU software". A copy of the license
-is included in *note GNU Free Documentation License::.
-
-
-* Menu:
-
-* Copying:: MPIR Copying Conditions (LGPL).
-* Introduction to MPIR:: Brief introduction to MPIR.
-* Installing MPIR:: How to configure and compile the MPIR library.
-* MPIR Basics:: What every MPIR user should know.
-* Reporting Bugs:: How to usefully report bugs.
-* Integer Functions:: Functions for arithmetic on signed integers.
-* Rational Number Functions:: Functions for arithmetic on rational numbers.
-* Floating-point Functions:: Functions for arithmetic on floats.
-* Low-level Functions:: Fast functions for natural numbers.
-* Random Number Functions:: Functions for generating random numbers.
-* Formatted Output:: `printf' style output.
-* Formatted Input:: `scanf' style input.
-* C++ Class Interface:: Class wrappers around MPIR types.
-* Custom Allocation:: How to customize the internal allocation.
-* Language Bindings:: Using MPIR from other languages.
-* Algorithms:: What happens behind the scenes.
-* Internals:: How values are represented behind the scenes.
-
-* Contributors:: Who brings you this library?
-* References:: Some useful papers and books to read.
-* GNU Free Documentation License::
-* Concept Index::
-* Function Index::
-
-
-File: mpir.info, Node: Copying, Next: Introduction to MPIR, Prev: Top, Up: Top
-
-MPIR Copying Conditions
-***********************
-
-This library is "free"; this means that everyone is free to use it and
-free to redistribute it on a free basis. The library is not in the
-public domain; it is copyrighted and there are restrictions on its
-distribution, but these restrictions are designed to permit everything
-that a good cooperating citizen would want to do. What is not allowed
-is to try to prevent others from further sharing any version of this
-library that they might get from you.
-
- Specifically, we want to make sure that you have the right to give
-away copies of the library, that you receive source code or else can
-get it if you want it, that you can change this library or use pieces
-of it in new free programs, and that you know you can do these things.
-
- To make sure that everyone has such rights, we have to forbid you to
-deprive anyone else of these rights. For example, if you distribute
-copies of the MPIR library, you must give the recipients all the rights
-that you have. You must make sure that they, too, receive or can get
-the source code. And you must tell them their rights.
-
- Also, for our own protection, we must make certain that everyone
-finds out that there is no warranty for the MPIR library. If it is
-modified by someone else and passed on, we want their recipients to
-know that what they have is not what we distributed, so that any
-problems introduced by others will not reflect on our reputation.
-
- The precise conditions of the license for the MPIR library are found
-in the Lesser General Public License version 3 that accompanies the
-source code, see `COPYING.LIB'.
-
-
-File: mpir.info, Node: Introduction to MPIR, Next: Installing MPIR, Prev: Copying, Up: Top
-
-1 Introduction to MPIR
-**********************
-
-MPIR is a portable library written in C for arbitrary precision
-arithmetic on integers, rational numbers, and floating-point numbers.
-It aims to provide the fastest possible arithmetic for all applications
-that need higher precision than is directly supported by the basic C
-types.
-
- Many applications use just a few hundred bits of precision; but some
-applications may need thousands or even millions of bits. MPIR is
-designed to give good performance for both, by choosing algorithms
-based on the sizes of the operands, and by carefully keeping the
-overhead at a minimum.
-
- The speed of MPIR is achieved by using fullwords as the basic
-arithmetic type, by using sophisticated algorithms, by including
-carefully optimized assembly code for the most common inner loops for
-many different CPUs, and by a general emphasis on speed (as opposed to
-simplicity or elegance).
-
- There is assembly code for these CPUs: ARM, DEC Alpha 21064, 21164,
-and 21264, AMD K6, K6-2, Athlon, K8 and K10, Intel Pentium, Pentium
-Pro/II/III, Pentium 4, generic x86, Intel IA-64, Core 2, i7, Atom,
-Motorola/IBM PowerPC 32 and 64, MIPS R3000, R4000, SPARCv7, SuperSPARC,
-generic SPARCv8, UltraSPARC,
-
-For up-to-date information on, and latest version of, MPIR, please see
-the MPIR web pages at
-
- `http://www.mpir.org/'
-
- There are a number of public mailing lists of interest. The
-development list is
-
- `http://groups.google.com/group/mpir-devel/'.
-
- The proper place for bug reports is
-`http://groups.google.com/group/mpir-devel'. See *note Reporting
-Bugs:: for information about reporting bugs.
-
-
-1.1 How to use this Manual
-==========================
-
-Everyone should read *note MPIR Basics::. If you need to install the
-library yourself, then read *note Installing MPIR::. If you have a
-system with multiple ABIs, then read *note ABI and ISA::, for the
-compiler options that must be used on applications.
-
- The rest of the manual can be used for later reference, although it
-is probably a good idea to glance through it.
-
-
-File: mpir.info, Node: Installing MPIR, Next: MPIR Basics, Prev: Introduction to MPIR, Up: Top
-
-2 Installing MPIR
-*****************
-
-MPIR has an autoconf/automake/libtool based configuration system. On a
-Unix-like system a basic build can be done with
-
- ./configure
- make
-
-Some self-tests can be run with
-
- make check
-
-And you can install (under `/usr/local' by default) with
-
- make install
-
- Important note: by default MPIR produces libraries named libmpir,
-etc., and the header file mpir.h. If you wish to have MPIR to build a
-library named libgmp as well, etc., and a gmp.h header file, so that
-you can use mpir with programs designed to only work with GMP, then use
-the `--enable-gmpcompat' option when invoking configure:
-
- ./configure --enable-gmpcompat
-
- Note gmp.h is only created upon running make install.
-
- MPIR is compatible with GMP when the `--enable-gmpcompat' option is
-used, except that the GMP secure cryptographic functions are not
-available.
-
- Some deprecated GMP functionality may be unavailable if this option
-is not selected.
-
- If you experience problems, please report them to
-
- `http://groups.google.com/group/mpir-devel'.
-
- See *note Reporting Bugs::, for information on what to include in
-useful bug reports.
-
-* Menu:
-
-* Build Options::
-* ABI and ISA::
-* Notes for Package Builds::
-* Notes for Particular Systems::
-* Known Build Problems::
-* Performance optimization::
-
-
-File: mpir.info, Node: Build Options, Next: ABI and ISA, Prev: Installing MPIR, Up: Installing MPIR
-
-2.1 Build Options
-=================
-
-All the usual autoconf configure options are available, run `./configure
---help' for a summary. The file `INSTALL.autoconf' has some generic
-installation information too.
-
-Tools
- `configure' requires various Unix-like tools. See *note Notes for
- Particular Systems::, for some options on non-Unix systems.
-
- It might be possible to build without the help of `configure',
- certainly all the code is there, but unfortunately you'll be on
- your own.
-
-Build Directory
- To compile in a separate build directory, `cd' to that directory,
- and prefix the configure command with the path to the MPIR source
- directory. For example
-
- cd /my/build/dir
- /my/sources/mpir-2.6.0/configure
-
- Not all `make' programs have the necessary features (`VPATH') to
- support this. In particular, SunOS and Solaris `make' have bugs
- that make them unable to build in a separate directory. Use GNU
- `make' instead.
-
-`--prefix' and `--exec-prefix'
- The `--prefix' option can be used in the normal way to direct MPIR
- to install under a particular tree. The default is `/usr/local'.
-
- `--exec-prefix' can be used to direct architecture-dependent files
- like `libmpir.a' to a different location. This can be used to
- share architecture-independent parts like the documentation, but
- separate the dependent parts. Note however that `mpir.h' and
- `mp.h' are architecture-dependent since they encode certain
- aspects of `libmpir', so it will be necessary to ensure both
- `$prefix/include' and `$exec_prefix/include' are available to the
- compiler.
-
-`--enable-gmpcompat'
- By default make builds libmpir library files (and libmpirxx if C++
- headers are requested) and the mpir.h header file. This option
- allows you to specify that you want additional libraries created
- called libgmp (and libgmpxx), etc., for libraries and gmp.h for
- compatibility with GMP (except for GMP secure cryptograhic
- functions, which are not available in MPIR).
-
-`--disable-shared', `--disable-static'
- By default both shared and static libraries are built (where
- possible), but one or other can be disabled. Shared libraries
- result in smaller executables and permit code sharing between
- separate running processes, but on some CPUs are slightly slower,
- having a small cost on each function call.
-
-Native Compilation, `--build=CPU-VENDOR-OS'
- For normal native compilation, the system can be specified with
- `--build'. By default `./configure' uses the output from running
- `./config.guess'. On some systems `./config.guess' can determine
- the exact CPU type, on others it will be necessary to give it
- explicitly. For example,
-
- ./configure --build=ultrasparc-sun-solaris2.7
-
- In all cases the `OS' part is important, since it controls how
- libtool generates shared libraries. Running `./config.guess' is
- the simplest way to see what it should be, if you don't know
- already.
-
-Cross Compilation, `--host=CPU-VENDOR-OS'
- When cross-compiling, the system used for compiling is given by
- `--build' and the system where the library will run is given by
- `--host'. For example when using a FreeBSD Athlon system to build
- GNU/Linux m68k binaries,
-
- ./configure --build=athlon-pc-freebsd3.5 --host=m68k-mac-linux-gnu
-
- Compiler tools are sought first with the host system type as a
- prefix. For example `m68k-mac-linux-gnu-ranlib' is tried, then
- plain `ranlib'. This makes it possible for a set of
- cross-compiling tools to co-exist with native tools. The prefix
- is the argument to `--host', and this can be an alias, such as
- `m68k-linux'. But note that tools don't have to be setup this
- way, it's enough to just have a `PATH' with a suitable
- cross-compiling `cc' etc.
-
- Compiling for a different CPU in the same family as the build
- system is a form of cross-compilation, though very possibly this
- would merely be special options on a native compiler. In any case
- `./configure' avoids depending on being able to run code on the
- build system, which is important when creating binaries for a
- newer CPU since they very possibly won't run on the build system.
-
- In all cases the compiler must be able to produce an executable
- (of whatever format) from a standard C `main'. Although only
- object files will go to make up `libmpir', `./configure' uses
- linking tests for various purposes, such as determining what
- functions are available on the host system.
-
- Currently a warning is given unless an explicit `--build' is used
- when cross-compiling, because it may not be possible to correctly
- guess the build system type if the `PATH' has only a
- cross-compiling `cc'.
-
- Note that the `--target' option is not appropriate for MPIR. It's
- for use when building compiler tools, with `--host' being where
- they will run, and `--target' what they'll produce code for.
- Ordinary programs or libraries like MPIR are only interested in
- the `--host' part, being where they'll run.
-
-CPU types
- In general, if you want a library that runs as fast as possible,
- you should configure MPIR for the exact CPU type your system uses.
- However, this may mean the binaries won't run on older members of
- the family, and might run slower on other members, older or newer.
- The best idea is always to build MPIR for the exact machine type
- you intend to run it on.
-
- The following CPUs have specific support. See `configure.in' for
- details of what code and compiler options they select.
-
- * Alpha: alpha, alphaev5, alphaev56, alphapca56, alphapca57,
- alphaev6, alphaev67, alphaev68 alphaev7
-
- * IA-64: ia64, itanium, itanium2
-
- * MIPS: mips, mips3, mips64
-
- * PowerPC: powerpc, powerpc64, powerpc401, powerpc403,
- powerpc405, powerpc505, powerpc601, powerpc602, powerpc603,
- powerpc603e, powerpc604, powerpc604e, powerpc620, powerpc630,
- powerpc740, powerpc7400, powerpc7450, powerpc750, powerpc801,
- powerpc821, powerpc823, powerpc860, powerpc970
-
- * SPARC: sparc, sparcv8, microsparc, supersparc, sparcv9,
- ultrasparc, ultrasparc2, ultrasparc2i, ultrasparc3, sparc64
-
- * x86 family: pentium, pentiummmx, pentiumpro, pentium2,
- pentium3, pentium4, netburst, netburstlahf, prescott, core,
- core2, penryn, nehalem, nano atom, k5, k6, k62, k63, k7, k8,
- k10 k102 viac3, viac32
-
- * Other: arm,
-
- CPUs not listed will use generic C code.
-
-Generic C Build
- If some of the assembly code causes problems, or if otherwise
- desired, the generic C code can be selected with CPU `none'. For
- example,
-
- ./configure --host=none-unknown-freebsd3.5
-
- Note that this will run quite slowly, but it should be portable
- and should at least make it possible to get something running if
- all else fails.
-
-Fat binary, `--enable-fat'
- Using `--enable-fat' selects a "fat binary" build on x86 or x86_64
- systems, where optimized low level subroutines are chosen at
- runtime according to the CPU detected. This means more code, but
- gives reasonable performance from a single binary for all x86
- chips, or similarly for all x86_64 chips. (This option might
- become available for more architectures in the future.)
-
-`ABI'
- On some systems MPIR supports multiple ABIs (application binary
- interfaces), meaning data type sizes and calling conventions. By
- default MPIR chooses the best ABI available, but a particular ABI
- can be selected. For example
-
- ./configure --host=mips64-sgi-irix6 ABI=n32
-
- See *note ABI and ISA::, for the available choices on relevant
- CPUs, and what applications need to do.
-
-`CC', `CFLAGS'
- By default the C compiler used is chosen from among some likely
- candidates, with `gcc' normally preferred if it's present. The
- usual `CC=whatever' can be passed to `./configure' to choose
- something different.
-
- For various systems, default compiler flags are set based on the
- CPU and compiler. The usual `CFLAGS="-whatever"' can be passed to
- `./configure' to use something different or to set good flags for
- systems MPIR doesn't otherwise know.
-
- The `CC' and `CFLAGS' used are printed during `./configure', and
- can be found in each generated `Makefile'. This is the easiest way
- to check the defaults when considering changing or adding
- something.
-
- Note that when `CC' and `CFLAGS' are specified on a system
- supporting multiple ABIs it's important to give an explicit
- `ABI=whatever', since MPIR can't determine the ABI just from the
- flags and won't be able to select the correct assembler code.
-
- If just `CC' is selected then normal default `CFLAGS' for that
- compiler will be used (if MPIR recognises it). For example
- `CC=gcc' can be used to force the use of GCC, with default flags
- (and default ABI).
-
-`CPPFLAGS'
- Any flags like `-D' defines or `-I' includes required by the
- preprocessor should be set in `CPPFLAGS' rather than `CFLAGS'.
- Compiling is done with both `CPPFLAGS' and `CFLAGS', but
- preprocessing uses just `CPPFLAGS'. This distinction is because
- most preprocessors won't accept all the flags the compiler does.
- Preprocessing is done separately in some configure tests, and in
- the `ansi2knr' support for K&R compilers.
-
-`CC_FOR_BUILD'
- Some build-time programs are compiled and run to generate
- host-specific data tables. `CC_FOR_BUILD' is the compiler used
- for this. It doesn't need to be in any particular ABI or mode, it
- merely needs to generate executables that can run. The default is
- to try the selected `CC' and some likely candidates such as `cc'
- and `gcc', looking for something that works.
-
- No flags are used with `CC_FOR_BUILD' because a simple invocation
- like `cc foo.c' should be enough. If some particular options are
- required they can be included as for instance `CC_FOR_BUILD="cc
- -whatever"'.
-
-C++ Support, `--enable-cxx'
- C++ support in MPIR can be enabled with `--enable-cxx', in which
- case a C++ compiler will be required. As a convenience
- `--enable-cxx=detect' can be used to enable C++ support only if a
- compiler can be found. The C++ support consists of a library
- `libmpirxx.la' and header file `mpirxx.h' (*note Headers and
- Libraries::).
-
- A separate `libmpirxx.la' has been adopted rather than having C++
- objects within `libmpir.la' in order to ensure dynamic linked C
- programs aren't bloated by a dependency on the C++ standard
- library, and to avoid any chance that the C++ compiler could be
- required when linking plain C programs.
-
- `libmpirxx.la' will use certain internals from `libmpir.la' and can
- only be expected to work with `libmpir.la' from the same MPIR
- version. Future changes to the relevant internals will be
- accompanied by renaming, so a mismatch will cause unresolved
- symbols rather than perhaps mysterious misbehaviour.
-
- In general `libmpirxx.la' will be usable only with the C++
- compiler that built it, since name mangling and runtime support
- are usually incompatible between different compilers.
-
-`CXX', `CXXFLAGS'
- When C++ support is enabled, the C++ compiler and its flags can be
- set with variables `CXX' and `CXXFLAGS' in the usual way. The
- default for `CXX' is the first compiler that works from a list of
- likely candidates, with `g++' normally preferred when available.
- The default for `CXXFLAGS' is to try `CFLAGS', `CFLAGS' without
- `-g', then for `g++' either `-g -O2' or `-O2', or for other
- compilers `-g' or nothing. Trying `CFLAGS' this way is convenient
- when using `gcc' and `g++' together, since the flags for `gcc' will
- usually suit `g++'.
-
- It's important that the C and C++ compilers match, meaning their
- startup and runtime support routines are compatible and that they
- generate code in the same ABI (if there's a choice of ABIs on the
- system). `./configure' isn't currently able to check these things
- very well itself, so for that reason `--disable-cxx' is the
- default, to avoid a build failure due to a compiler mismatch.
- Perhaps this will change in the future.
-
- Incidentally, it's normally not good enough to set `CXX' to the
- same as `CC'. Although `gcc' for instance recognises `foo.cc' as
- C++ code, only `g++' will invoke the linker the right way when
- building an executable or shared library from C++ object files.
-
-Temporary Memory, `--enable-alloca=<choice>'
- MPIR allocates temporary workspace using one of the following
- three methods, which can be selected with for instance
- `--enable-alloca=malloc-reentrant'.
-
- * `alloca' - C library or compiler builtin.
-
- * `malloc-reentrant' - the heap, in a re-entrant fashion.
-
- * `malloc-notreentrant' - the heap, with global variables.
-
- For convenience, the following choices are also available.
- `--disable-alloca' is the same as `no'.
-
- * `yes' - a synonym for `alloca'.
-
- * `no' - a synonym for `malloc-reentrant'.
-
- * `reentrant' - `alloca' if available, otherwise
- `malloc-reentrant'. This is the default.
-
- * `notreentrant' - `alloca' if available, otherwise
- `malloc-notreentrant'.
-
- `alloca' is reentrant and fast, and is recommended. It actually
- allocates just small blocks on the stack; larger ones use
- malloc-reentrant.
-
- `malloc-reentrant' is, as the name suggests, reentrant and thread
- safe, but `malloc-notreentrant' is faster and should be used if
- reentrancy is not required.
-
- The two malloc methods in fact use the memory allocation functions
- selected by `mp_set_memory_functions', these being `malloc' and
- friends by default. *Note Custom Allocation::.
-
- An additional choice `--enable-alloca=debug' is available, to help
- when debugging memory related problems (*note Debugging::).
-
-FFT Multiplication, `--disable-fft'
- By default multiplications are done using Karatsuba, Toom, and FFT
- algorithms. The FFT is only used on large to very large operands
- and can be disabled to save code size if desired.
-
-Assertion Checking, `--enable-assert'
- This option enables some consistency checking within the library.
- This can be of use while debugging, *note Debugging::.
-
-Execution Profiling, `--enable-profiling=prof/gprof/instrument'
- Enable profiling support, in one of various styles, *note
- Profiling::.
-
-`MPN_PATH'
- Various assembler versions of mpn subroutines are provided. For a
- given CPU, a search is made though a path to choose a version of
- each. For example `sparcv8' has
-
- MPN_PATH="sparc32/v8 sparc32 generic"
-
- which means look first for v8 code, then plain sparc32 (which is
- v7), and finally fall back on generic C. Knowledgeable users with
- special requirements can specify a different path. Normally this
- is completely unnecessary.
-
-Documentation
- The source for the document you're now reading is `doc/mpir.texi',
- in Texinfo format, see *note Texinfo: (texinfo)Top.
-
- Info format `doc/mpir.info' is included in the distribution. The
- usual automake targets are available to make PostScript, DVI, PDF
- and HTML (these will require various TeX and Texinfo tools).
-
- DocBook and XML can be generated by the Texinfo `makeinfo' program
- too, see *note Options for `makeinfo': (texinfo)makeinfo options.
-
- Some supplementary notes can also be found in the `doc'
- subdirectory.
-
-
-
-File: mpir.info, Node: ABI and ISA, Next: Notes for Package Builds, Prev: Build Options, Up: Installing MPIR
-
-2.2 ABI and ISA
-===============
-
-ABI (Application Binary Interface) refers to the calling conventions
-between functions, meaning what registers are used and what sizes the
-various C data types are. ISA (Instruction Set Architecture) refers to
-the instructions and registers a CPU has available.
-
- Some 64-bit ISA CPUs have both a 64-bit ABI and a 32-bit ABI
-defined, the latter for compatibility with older CPUs in the family.
-MPIR supports some CPUs like this in both ABIs. In fact within MPIR
-`ABI' means a combination of chip ABI, plus how MPIR chooses to use it.
-For example in some 32-bit ABIs, MPIR may support a limb as either a
-32-bit `long' or a 64-bit `long long'.
-
- By default MPIR chooses the best ABI available for a given system,
-and this generally gives significantly greater speed. But an ABI can
-be chosen explicitly to make MPIR compatible with other libraries, or
-particular application requirements. For example,
-
- ./configure ABI=32
-
- In all cases it's vital that all object code used in a given program
-is compiled for the same ABI.
-
- Usually a limb is implemented as a `long'. When a `long long' limb
-is used this is encoded in the generated `mpir.h'. This is convenient
-for applications, but it does mean that `mpir.h' will vary, and can't
-be just copied around. `mpir.h' remains compiler independent though,
-since all compilers for a particular ABI will be expected to use the
-same limb type.
-
- Currently no attempt is made to follow whatever conventions a system
-has for installing library or header files built for a particular ABI.
-This will probably only matter when installing multiple builds of MPIR,
-and it might be as simple as configuring with a special `libdir', or it
-might require more than that. Note that builds for different ABIs need
-to done separately, with a fresh (`make distclean'), `./configure' and
-`make'.
-
-
-AMD64 (`x86_64')
- On AMD64 systems supporting both 32-bit and 64-bit modes for
- applications, the following ABI choices are available.
-
- `ABI=64'
- The 64-bit ABI uses 64-bit limbs and pointers and makes full
- use of the chip architecture. This is the default.
- Applications will usually not need special compiler flags,
- but for reference the option is
-
- gcc -m64
-
- `ABI=32'
- The 32-bit ABI is the usual i386 conventions. This will be
- slower, and is not recommended except for inter-operating
- with other code not yet 64-bit capable. Applications must be
- compiled with
-
- gcc -m32
-
- (In GCC 2.95 and earlier there's no `-m32' option, it's the
- only mode.)
-
-
-IA-64 under HP-UX (`ia64*-*-hpux*', `itanium*-*-hpux*')
- HP-UX supports two ABIs for IA-64. MPIR performance is the same
- in both.
-
- `ABI=32'
- In the 32-bit ABI, pointers, `int's and `long's are 32 bits
- and MPIR uses a 64 bit `long long' for a limb. Applications
- can be compiled without any special flags since this ABI is
- the default in both HP C and GCC, but for reference the flags
- are
-
- gcc -milp32
- cc +DD32
-
- `ABI=64'
- In the 64-bit ABI, `long's and pointers are 64 bits and MPIR
- uses a `long' for a limb. Applications must be compiled with
-
- gcc -mlp64
- cc +DD64
-
- On other IA-64 systems, GNU/Linux for instance, `ABI=64' is the
- only choice.
-
-
-PowerPC 64 (`powerpc64', `powerpc620', `powerpc630', `powerpc970')
-
- `ABI=aix64'
- The AIX 64 ABI uses 64-bit limbs and pointers and is the
- default on PowerPC 64 `*-*-aix*' systems. Applications must
- be compiled with
-
- gcc -maix64
- xlc -q64
-
- `ABI=mode32'
- The `mode32' ABI uses a 64-bit `long long' limb but with the
- chip still in 32-bit mode and using 32-bit calling
- conventions. This is the default on PowerPC 64 `*-*-darwin*'
- systems. No special compiler options are needed for
- applications.
-
- `ABI=32'
- This is the basic 32-bit PowerPC ABI, with a 32-bit limb. No
- special compiler options are needed for applications.
-
- MPIR speed is greatest in `aix64' and `mode32'. In `ABI=32' only
- the 32-bit ISA is used and this doesn't make full use of a 64-bit
- chip. On a suitable system we could perhaps use more of the ISA,
- but there are no plans to do so.
-
-
-Sparc V9 (`sparc64', `sparcv9', `ultrasparc*')
-
- `ABI=64'
- The 64-bit V9 ABI is available on the various BSD sparc64
- ports, recent versions of Sparc64 GNU/Linux, and Solaris 2.7
- and up (when the kernel is in 64-bit mode). GCC 3.2 or
- higher, or Sun `cc' is required. On GNU/Linux, depending on
- the default `gcc' mode, applications must be compiled with
-
- gcc -m64
-
- On Solaris applications must be compiled with
-
- gcc -m64 -mptr64 -Wa,-xarch=v9 -mcpu=v9
- cc -xarch=v9
-
- On the BSD sparc64 systems no special options are required,
- since 64-bits is the only ABI available.
-
- `ABI=32'
- For the basic 32-bit ABI, MPIR still uses as much of the V9
- ISA as it can. In the Sun documentation this combination is
- known as "v8plus". On GNU/Linux, depending on the default
- `gcc' mode, applications may need to be compiled with
-
- gcc -m32
-
- On Solaris, no special compiler options are required for
- applications, though using something like the following is
- recommended. (`gcc' 2.8 and earlier only support `-mv8'
- though.)
-
- gcc -mv8plus
- cc -xarch=v8plus
-
- MPIR speed is greatest in `ABI=64', so it's the default where
- available. The speed is partly because there are extra registers
- available and partly because 64-bits is considered the more
- important case and has therefore had better code written for it.
-
- Don't be confused by the names of the `-m' and `-x' compiler
- options, they're called `arch' but effectively control both ABI
- and ISA.
-
- On Solaris 2.6 and earlier, only `ABI=32' is available since the
- kernel doesn't save all registers.
-
- On Solaris 2.7 with the kernel in 32-bit mode, a normal native
- build will reject `ABI=64' because the resulting executables won't
- run. `ABI=64' can still be built if desired by making it look
- like a cross-compile, for example
-
- ./configure --build=none --host=sparcv9-sun-solaris2.7 ABI=64
-
-
-File: mpir.info, Node: Notes for Package Builds, Next: Notes for Particular Systems, Prev: ABI and ISA, Up: Installing MPIR
-
-2.3 Notes for Package Builds
-============================
-
-MPIR should present no great difficulties for packaging in a binary
-distribution.
-
- Libtool is used to build the library and `-version-info' is set
-appropriately, having started from `3:0:0' in GMP 3.0 (*note Library
-interface versions: (libtool)Versioning.).
-
- The GMP 4 series and MPIR 1 series will be upwardly binary
-compatible in each release and will be upwardly binary compatible with
-all of the GMP 3 series. Additional function interfaces may be added
-in each release, so on systems where libtool versioning is not fully
-checked by the loader an auxiliary mechanism may be needed to express
-that a dynamic linked application depends on a new enough MPIR.
-
- From MPIR 2.0.0 binary compatibility with the GMP 5 series will be
-maintained with the exception of the availability of secure functions
-for cryptography, which will not be supported in MPIR. For full GMP
-compatibility, including deprecated functionality, the
-`--enable-gmpcompat' configuration option must be used.
-
- An auxiliary mechanism may also be needed to express that
-`libmpirxx.la' (from `--enable-cxx', *note Build Options::) requires
-`libmpir.la' from the same MPIR version, since this is not done by the
-libtool versioning, nor otherwise. A mismatch will result in
-unresolved symbols from the linker, or perhaps the loader.
-
- When building a package for a CPU family, care should be taken to use
-`--host' (or `--build') to choose the least common denominator among
-the CPUs which might use the package. For example this might mean plain
-`sparc' (meaning V7) for SPARCs.
-
- For x86s, `--enable-fat' sets things up for a fat binary build,
-making a runtime selection of optimized low level routines. This is a
-good choice for packaging to run on a range of x86 chips.
-
- Users who care about speed will want MPIR built for their exact CPU
-type, to make best use of the available optimizations. Providing a way
-to suitably rebuild a package may be useful. This could be as simple
-as making it possible for a user to omit `--build' (and `--host') so
-`./config.guess' will detect the CPU. But a way to manually specify a
-`--build' will be wanted for systems where `./config.guess' is inexact.
-
- On systems with multiple ABIs, a packaged build will need to decide
-which among the choices is to be provided, see *note ABI and ISA::. A
-given run of `./configure' etc will only build one ABI. If a second
-ABI is also required then a second run of `./configure' etc must be
-made, starting from a clean directory tree (`make distclean').
-
- As noted under "ABI and ISA", currently no attempt is made to follow
-system conventions for install locations that vary with ABI, such as
-`/usr/lib/sparcv9' for `ABI=64' as opposed to `/usr/lib' for `ABI=32'.
-A package build can override `libdir' and other standard variables as
-necessary.
-
- Note that `mpir.h' is a generated file, and will be architecture and
-ABI dependent. When attempting to install two ABIs simultaneously it
-will be important that an application compile gets the correct `mpir.h'
-for its desired ABI. If compiler include paths don't vary with ABI
-options then it might be necessary to create a `/usr/include/mpir.h'
-which tests preprocessor symbols and chooses the correct actual
-`mpir.h'.
-
-
-File: mpir.info, Node: Notes for Particular Systems, Next: Known Build Problems, Prev: Notes for Package Builds, Up: Installing MPIR
-
-2.4 Notes for Particular Systems
-================================
-
-AIX 3 and 4
- On systems `*-*-aix[34]*' shared libraries are disabled by
- default, since some versions of the native `ar' fail on the
- convenience libraries used. A shared build can be attempted with
-
- ./configure --enable-shared --disable-static
-
- Note that the `--disable-static' is necessary because in a shared
- build libtool makes `libmpir.a' a symlink to `libmpir.so',
- apparently for the benefit of old versions of `ld' which only
- recognise `.a', but unfortunately this is done even if a fully
- functional `ld' is available.
-
-ARM
- On systems `arm*-*-*', versions of GCC up to and including 2.95.3
- have a bug in unsigned division, giving wrong results for some
- operands. MPIR `./configure' will demand GCC 2.95.4 or later.
-
-Floating Point Mode
- On some systems, the hardware floating point has a control mode
- which can set all operations to be done in a particular precision,
- for instance single, double or extended on x86 systems (x87
- floating point). The MPIR functions involving a `double' cannot
- be expected to operate to their full precision when the hardware
- is in single precision mode. Of course this affects all code,
- including application code, not just MPIR.
-
-MS-DOS and MS Windows
- On an MS Windows system Cygwin and MINGW can be used , they are
- ports of GCC and the various GNU tools.
-
- `http://www.cygwin.com/'
- `http://www.mingw.org/'
-
- Cygwin is a 32 bit build only but mingw is 32 or 64 bit build.
- Depending on how the mingw tools are installed will determine the
- best procedure for building , because of the large number of ways
- this can be achieved it is best to search the MPIR devel mailing
- list or the mingw mailing list.
-
- For building with MSVC we provide a number of ways.
-
- In addition, project files for MSVC are provided, allowing MPIR to
- build on Microsoft's compiler. For Visual Studio 2010 see the
- readme.txt file in the build.vc10 directory. The MSVC projects
- provides full assembler support and for `x86_64' CPU's this will
- produce far superior results. These project files can also be
- accessed via the command line with the batch files `configure.bat'
- and `make.bat' which have a `unix like' interface , however they
- are not very well tested and are due to be replaced.
-
- An another alternative is `configure' and `make' in the `win'
- directory , these again have a `unix like' syntax , these are
- tested regularly and also have the advantage of working with
- VS2005 and up (including the free/express versions). There is some
- auto detection of the compiler , but it's probably best to set it
- explicity using the usual `call
- "%VS90COMNTOOLS%\..\..\VC\vcvarsall.bat" amd64' in the command
- window. The program `YASM' is also required and should be in path
- or the `%YASMPATH%' varible set. If `configure' guesses wrong ,
- close the window and try again changing the `ABI=...' selection
- and or the `vcvarsall.bat' options. `make' supports the usual
- `clean' and `check' options . The `only' bug is that shared
- library builds `dll's' fail the make check in the C++ parts for
- `istream' and `ostream' with some unresolved symbols.
-
-MS Windows DLLs
- On systems `*-*-cygwin*' and `*-*-mingw*' by default MPIR builds
- only a static library, but a DLL can be built instead using
-
- ./configure --disable-static --enable-shared
-
- Static and DLL libraries can't both be built, since certain export
- directives in `mpir.h' must be different.
-
- Libtool doesn't install a `.lib' format import library, but it can
- be created with MS `lib' as follows, and copied to the install
- directory. Similarly for `libmpir' and `libmpirxx'.
-
- cd .libs
- lib /def:libgmp-3.dll.def /out:libgmp-3.lib
-
- MINGW uses the C runtime library `msvcrt.dll' for I/O, so
- applications wanting to use the MPIR I/O routines must be compiled
- with `cl /MD' to do the same. If one of the other C runtime
- library choices provided by MS C is desired then the suggestion is
- to use the MPIR string functions and confine I/O to the
- application.
-
-OpenBSD 2.6
- `m4' in this release of OpenBSD has a bug in `eval' that makes it
- unsuitable for `.asm' file processing. `./configure' will detect
- the problem and either abort or choose another m4 in the `PATH'.
- The bug is fixed in OpenBSD 2.7, so either upgrade or use GNU m4.
-
-Sparc CPU Types
- `sparcv8' or `supersparc' on relevant systems will give a
- significant performance increase over the V7 code selected by plain
- `sparc'.
-
-Sparc App Regs
- The MPIR assembler code for both 32-bit and 64-bit Sparc clobbers
- the "application registers" `g2', `g3' and `g4', the same way that
- the GCC default `-mapp-regs' does (*note SPARC Options: (gcc)SPARC
- Options.).
-
- This makes that code unsuitable for use with the special V9
- `-mcmodel=embmedany' (which uses `g4' as a data segment pointer),
- and for applications wanting to use those registers for special
- purposes. In these cases the only suggestion currently is to
- build MPIR with CPU `none' to avoid the assembler code.
-
-SPARC Solaris
- Building applications against MPIR on SPARC Solaris (including
- `make check') requires the `LD_LIBRARY_PATH' to be set
- appropriately. In particular if one is building with `ABI=64' the
- linker needs to know where to find `libgcc' (often often
- `/usr/lib/sparcv9' or `/usr/local/lib/sparcv9' or `/lib/sparcv9').
-
- It is not enough to specify the location in `LD_LIBRARY_PATH_64'
- unless `LD_LIBRARY_PATH_64' is added to `LD_LIBRARY_PATH'.
- Specifically the 64 bit `libgcc' path needs to be in
- `LD_LIBRARY_PATH'.
-
- The linker is able to automatically distinguish 32 and 64 bit
- libraries, so it is safe to include paths to both the 32 and 64
- bit libraries in the `LD_LIBRARY_PATH'.
-
-Solaris 10 First Release on SPARC
- MPIR fails to build with Solaris 10 first release. Patch 123647-01
- for SPARC, released by Sun in August 2006 fixes this problem.
-
-x86 CPU Types
- `i586', `pentium' or `pentiummmx' code is good for its intended P5
- Pentium chips, but quite slow when run on Intel P6 class chips
- (PPro, P-II, P-III). `i386' is a better choice when making
- binaries that must run on both.
-
-x86 MMX and SSE2 Code
- If the CPU selected has MMX code but the assembler doesn't support
- it, a warning is given and non-MMX code is used instead. This
- will be an inferior build, since the MMX code that's present is
- there because it's faster than the corresponding plain integer
- code. The same applies to SSE2.
-
- Old versions of `gas' don't support MMX instructions, in particular
- version 1.92.3 that comes with FreeBSD 2.2.8 or the more recent
- OpenBSD 3.1 doesn't.
-
- Solaris 2.6 and 2.7 `as' generate incorrect object code for
- register to register `movq' instructions, and so can't be used for
- MMX code. Install a recent `gas' if MMX code is wanted on these
- systems.
-
-
-File: mpir.info, Node: Known Build Problems, Next: Performance optimization, Prev: Notes for Particular Systems, Up: Installing MPIR
-
-2.5 Known Build Problems
-========================
-
-You might find more up-to-date information at `http://www.mpir.org/'.
-
-Compiler link options
- The version of libtool currently in use rather aggressively strips
- compiler options when linking a shared library. This will
- hopefully be relaxed in the future, but for now if this is a
- problem the suggestion is to create a little script to hide them,
- and for instance configure with
-
- ./configure CC=gcc-with-my-options
-
- `make all' was found to run out of memory during the final
- `libgmp.la' link on one system tested, despite having 64Mb
- available. Running `make libgmp.la' directly helped, perhaps
- recursing into the various subdirectories uses up memory.
-
-MacOS X (`*-*-darwin*')
- Libtool currently only knows how to create shared libraries on
- MacOS X using the native `cc' (which is a modified GCC), not a
- plain GCC. A static-only build should work though
- (`--disable-shared').
-
-Solaris 2.6
- The system `sed' prints an error "Output line too long" when
- libtool builds `libmpir.la'. This doesn't seem to cause any
- obvious ill effects, but GNU `sed' is recommended, to avoid any
- doubt.
-
-Sparc Solaris 2.7 with gcc 2.95.2 in `ABI=32'
- A shared library build of MPIR seems to fail in this combination,
- it builds but then fails the tests, apparently due to some
- incorrect data relocations within `gmp_randinit_lc_2exp_size'.
- The exact cause is unknown, `--disable-shared' is recommended.
-
-
-File: mpir.info, Node: Performance optimization, Prev: Known Build Problems, Up: Installing MPIR
-
-2.6 Performance optimization
-============================
-
-For optimal performance, build MPIR for the exact CPU type of the target
-computer, see *note Build Options::.
-
- Unlike what is the case for most other programs, the compiler
-typically doesn't matter much, since MPIR uses assembly language for
-the most critical operations.
-
- In particular for long-running MPIR applications, and applications
-demanding extremely large numbers, building and running the `tuneup'
-program in the `tune' subdirectory, can be important. For example,
-
- cd tune
- make tuneup
- ./tuneup
-
- will generate better contents for the `gmp-mparam.h' parameter file.
-
- To use the results, put the output in the file indicated in the
-`Parameters for ...' header. Then recompile from scratch.
-
- The `tuneup' program takes one useful parameter, `-f NNN', which
-instructs the program how long to check FFT multiply parameters. If
-you're going to use MPIR for extremely large numbers, you may want to
-run `tuneup' with a large NNN value.
-
-
-File: mpir.info, Node: MPIR Basics, Next: Reporting Bugs, Prev: Installing MPIR, Up: Top
-
-3 MPIR Basics
-*************
-
-*Using functions, macros, data types, etc. not documented in this
-manual is strongly discouraged. If you do so your application is
-guaranteed to be incompatible with future versions of MPIR.*
-
-* Menu:
-
-* Headers and Libraries::
-* Nomenclature and Types::
-* MPIR on Windows x64::
-* Function Classes::
-* Variable Conventions::
-* Parameter Conventions::
-* Memory Management::
-* Reentrancy::
-* Useful Macros and Constants::
-* Compatibility with older versions::
-* Efficiency::
-* Debugging::
-* Profiling::
-* Autoconf::
-* Emacs::
-
-
-File: mpir.info, Node: Headers and Libraries, Next: Nomenclature and Types, Prev: MPIR Basics, Up: MPIR Basics
-
-3.1 Headers and Libraries
-=========================
-
-All declarations needed to use MPIR are collected in the include file
-`mpir.h'. It is designed to work with both C and C++ compilers.
-
- #include <mpir.h>
-
- Note however that prototypes for MPIR functions with `FILE *'
-parameters are only provided if `<stdio.h>' is included too.
-
- #include <stdio.h>
- #include <mpir.h>
-
- Likewise `<stdarg.h>' (or `<varargs.h>') is required for prototypes
-with `va_list' parameters, such as `gmp_vprintf'. And `<obstack.h>'
-for prototypes with `struct obstack' parameters, such as
-`gmp_obstack_printf', when available.
-
- All programs using MPIR must link against the `libmpir' library. On
-a typical Unix-like system this can be done with `-lmpir' respectively,
-for example
-
- gcc myprogram.c -lmpir
-
- MPIR C++ functions are in a separate `libmpirxx' library. This is
-built and installed if C++ support has been enabled (*note Build
-Options::). For example,
-
- g++ mycxxprog.cc -lmpirxx -lmpir
-
- MPIR is built using Libtool and an application can use that to link
-if desired, *note GNU Libtool: (libtool)Top.
-
- If MPIR has been installed to a non-standard location then it may be
-necessary to use `-I' and `-L' compiler options to point to the right
-directories, and some sort of run-time path for a shared library.
-
-
-File: mpir.info, Node: Nomenclature and Types, Next: MPIR on Windows x64, Prev: Headers and Libraries, Up: MPIR Basics
-
-3.2 Nomenclature and Types
-==========================
-
-In this manual, "integer" usually means a multiple precision integer, as
-defined by the MPIR library. The C data type for such integers is
-`mpz_t'. Here are some examples of how to declare such integers:
-
- mpz_t sum;
-
- struct foo { mpz_t x, y; };
-
- mpz_t vec[20];
-
- "Rational number" means a multiple precision fraction. The C data
-type for these fractions is `mpq_t'. For example:
-
- mpq_t quotient;
-
- "Floating point number" or "Float" for short, is an arbitrary
-precision mantissa with a limited precision exponent. The C data type
-for such objects is `mpf_t'. For example:
-
- mpf_t fp;
-
- The floating point functions accept and return exponents in the C
-type `mp_exp_t'. Currently this is usually a `long', but on some
-systems it's an `int' for efficiency.
-
- A "limb" means the part of a multi-precision number that fits in a
-single machine word. (We chose this word because a limb of the human
-body is analogous to a digit, only larger, and containing several
-digits.) Normally a limb is 32 or 64 bits. The C data type for a limb
-is `mp_limb_t'.
-
- Counts of limbs are represented in the C type `mp_size_t'. Currently
-this is normally a `long', but on some systems it's an `int' for
-efficiency.
-
- Counts of bits of a multi-precision number are represented in the C
-type `mp_bitcnt_t'. Currently this is always an `unsigned long', but on
-some systems it will be an `unsigned long long' in the future .
-
- "Random state" means an algorithm selection and current state data.
-The C data type for such objects is `gmp_randstate_t'. For example:
-
- gmp_randstate_t rstate;
-
- Also, in general `mp_bitcnt_t' is used for bit counts and ranges, and
-`size_t' is used for byte or character counts.
-
-
-File: mpir.info, Node: MPIR on Windows x64, Next: Function Classes, Prev: Nomenclature and Types, Up: MPIR Basics
-
-3.3 MPIR on Windows x64
-=======================
-
-Although Windows x64 is a 64-bit operating system, Microsoft has
-decided to make long integers 32-bits, which is inconsistent when
-compared with almost all other 64-bit operating systems. This has
-caused many subtle bugs when open source code is ported to Windows x64
-because many developers reasonably expect to find that long integers on
-a 64-bit operating system will be 64 bits long.
-
- MPIR contains functions with suffixes of `_ui' and `_si' that are
-used to input unsigned and signed integers into and convert them for use
-with MPIR's multiple precision integers (mpz types). For example, the
-following functions set an `mpz_t' integer from `unsigned' and `signed
-long' integers respectively.
-
- -- Function: void mpz_set_ui (mpz_t, unsigned long int)
-
- -- Function: void mpz_set_si (mpz_t, signed long int)
-
-
- Also, the following functions obtain `unsigned' and `signed long
-int' values from an MPIR multiple precision integer (`mpz_t').
-
- -- Function: unsigned long int mpz_get_ui (mpz_t)
-
- -- Function: signed long int mpz_get_si (mpz_t)
-
-
- To bring MPIR on Windows x64 into line with other 64-bit operating
-systems two new types have been introduced throughout MPIR:
-
- * `mpir_ui' defined as `unsigned long int' on all but Windows x64,
- defined as `unsigned long long int' on Windows x64
-
- * `mpir_si' defined as `signed long int' on all but Windows x64,
- defined as `signed long long int' on Windows x64
-
-
- The above prototypes in MPIR 2.6.0 are changed to:
-
- -- Function: void mpz_set_ui (mpz_t, mpir_ui)
-
- -- Function: void mpz_set_si (mpz_t, mpir_si)
-
- -- Function: mpir_ui mpz_get_ui (mpz_t)
-
- -- Function: mpir_si mpz_get_si (mpz_t)
-
-
- These changes are applied to all MPIR functions with `_ui' and `_si'
-suffixes.
-
-
-File: mpir.info, Node: Function Classes, Next: Variable Conventions, Prev: MPIR on Windows x64, Up: MPIR Basics
-
-3.4 Function Classes
-====================
-
-There are five classes of functions in the MPIR library:
-
- 1. Functions for signed integer arithmetic, with names beginning with
- `mpz_'. The associated type is `mpz_t'. There are about 150
- functions in this class. (*note Integer Functions::)
-
- 2. Functions for rational number arithmetic, with names beginning with
- `mpq_'. The associated type is `mpq_t'. There are about 40
- functions in this class, but the integer functions can be used for
- arithmetic on the numerator and denominator separately. (*note
- Rational Number Functions::)
-
- 3. Functions for floating-point arithmetic, with names beginning with
- `mpf_'. The associated type is `mpf_t'. There are about 60
- functions is this class. (*note Floating-point Functions::)
-
- 4. Fast low-level functions that operate on natural numbers. These
- are used by the functions in the preceding groups, and you can
- also call them directly from very time-critical user programs.
- These functions' names begin with `mpn_'. The associated type is
- array of `mp_limb_t'. There are about 30 (hard-to-use) functions
- in this class. (*note Low-level Functions::)
-
- 5. Miscellaneous functions. Functions for setting up custom
- allocation and functions for generating random numbers. (*note
- Custom Allocation::, and *note Random Number Functions::)
-
-
-File: mpir.info, Node: Variable Conventions, Next: Parameter Conventions, Prev: Function Classes, Up: MPIR Basics
-
-3.5 Variable Conventions
-========================
-
-MPIR functions generally have output arguments before input arguments.
-This notation is by analogy with the assignment operator.
-
- MPIR lets you use the same variable for both input and output in one
-call. For example, the main function for integer multiplication,
-`mpz_mul', can be used to square `x' and put the result back in `x' with
-
- mpz_mul (x, x, x);
-
- Before you can assign to an MPIR variable, you need to initialize it
-by calling one of the special initialization functions. When you're
-done with a variable, you need to clear it out, using one of the
-functions for that purpose. Which function to use depends on the type
-of variable. See the chapters on integer functions, rational number
-functions, and floating-point functions for details.
-
- A variable should only be initialized once, or at least cleared
-between each initialization. After a variable has been initialized, it
-may be assigned to any number of times.
-
- For efficiency reasons, avoid excessive initializing and clearing.
-In general, initialize near the start of a function and clear near the
-end. For example,
-
- void
- foo (void)
- {
- mpz_t n;
- int i;
- mpz_init (n);
- for (i = 1; i < 100; i++)
- {
- mpz_mul (n, ...);
- mpz_fdiv_q (n, ...);
- ...
- }
- mpz_clear (n);
- }
-
-
-File: mpir.info, Node: Parameter Conventions, Next: Memory Management, Prev: Variable Conventions, Up: MPIR Basics
-
-3.6 Parameter Conventions
-=========================
-
-When an MPIR variable is used as a function parameter, it's effectively
-a call-by-reference, meaning if the function stores a value there it
-will change the original in the caller. Parameters which are
-input-only can be designated `const' to provoke a compiler error or
-warning on attempting to modify them.
-
- When a function is going to return an MPIR result, it should
-designate a parameter that it sets, like the library functions do.
-More than one value can be returned by having more than one output
-parameter, again like the library functions. A `return' of an `mpz_t'
-etc doesn't return the object, only a pointer, and this is almost
-certainly not what's wanted.
-
- Here's an example accepting an `mpz_t' parameter, doing a
-calculation, and storing the result to the indicated parameter.
-
- void
- foo (mpz_t result, const mpz_t param, mpir_ui n)
- {
- mpir_ui i;
- mpz_mul_ui (result, param, n);
- for (i = 1; i < n; i++)
- mpz_add_ui (result, result, i*7);
- }
-
- int
- main (void)
- {
- mpz_t r, n;
- mpz_init (r);
- mpz_init_set_str (n, "123456", 0);
- foo (r, n, 20L);
- gmp_printf ("%Zd\n", r);
- return 0;
- }
-
- `foo' works even if the mainline passes the same variable for
-`param' and `result', just like the library functions. But sometimes
-it's tricky to make that work, and an application might not want to
-bother supporting that sort of thing.
-
- For interest, the MPIR types `mpz_t' etc are implemented as
-one-element arrays of certain structures. This is why declaring a
-variable creates an object with the fields MPIR needs, but then using
-it as a parameter passes a pointer to the object. Note that the actual
-fields in each `mpz_t' etc are for internal use only and should not be
-accessed directly by code that expects to be compatible with future
-MPIR releases.
-
-
-File: mpir.info, Node: Memory Management, Next: Reentrancy, Prev: Parameter Conventions, Up: MPIR Basics
-
-3.7 Memory Management
-=====================
-
-The MPIR types like `mpz_t' are small, containing only a couple of
-sizes, and pointers to allocated data. Once a variable is initialized,
-MPIR takes care of all space allocation. Additional space is allocated
-whenever a variable doesn't have enough.
-
- `mpz_t' and `mpq_t' variables never reduce their allocated space.
-Normally this is the best policy, since it avoids frequent reallocation.
-Applications that need to return memory to the heap at some particular
-point can use `mpz_realloc2', or clear variables no longer needed.
-
- `mpf_t' variables, in the current implementation, use a fixed amount
-of space, determined by the chosen precision and allocated at
-initialization, so their size doesn't change.
-
- All memory is allocated using `malloc' and friends by default, but
-this can be changed, see *note Custom Allocation::. Temporary memory
-on the stack is also used (via `alloca'), but this can be changed at
-build-time if desired, see *note Build Options::.
-
-
-File: mpir.info, Node: Reentrancy, Next: Useful Macros and Constants, Prev: Memory Management, Up: MPIR Basics
-
-3.8 Reentrancy
-==============
-
-MPIR is reentrant and thread-safe, with some exceptions:
-
- * If configured with `--enable-alloca=malloc-notreentrant' (or with
- `--enable-alloca=notreentrant' when `alloca' is not available),
- then naturally MPIR is not reentrant.
-
- * `mpf_set_default_prec' and `mpf_init' use a global variable for the
- selected precision. `mpf_init2' can be used instead, and in the
- C++ interface an explicit precision to the `mpf_class' constructor.
-
- * `mp_set_memory_functions' uses global variables to store the
- selected memory allocation functions.
-
- * If the memory allocation functions set by a call to
- `mp_set_memory_functions' (or `malloc' and friends by default) are
- not reentrant, then MPIR will not be reentrant either.
-
- * If the standard I/O functions such as `fwrite' are not reentrant
- then the MPIR I/O functions using them will not be reentrant
- either.
-
- * It's safe for two threads to read from the same MPIR variable
- simultaneously, but it's not safe for one to read while the
- another might be writing, nor for two threads to write
- simultaneously. It's not safe for two threads to generate a
- random number from the same `gmp_randstate_t' simultaneously,
- since this involves an update of that variable.
-
-
-File: mpir.info, Node: Useful Macros and Constants, Next: Compatibility with older versions, Prev: Reentrancy, Up: MPIR Basics
-
-3.9 Useful Macros and Constants
-===============================
-
- -- Global Constant: const int mp_bits_per_limb
- The number of bits per limb.
-
- -- Macro: __GNU_MP_VERSION
- -- Macro: __GNU_MP_VERSION_MINOR
- -- Macro: __GNU_MP_VERSION_PATCHLEVEL
- The major and minor GMP version, and patch level, respectively, as
- integers. For GMP i.j.k, these numbers will be i, j, and k,
- respectively. These numbers represent the version of GMP fully
- supported by this version of MPIR.
-
- -- Macro: __MPIR_VERSION
- -- Macro: __MPIR_VERSION_MINOR
- -- Macro: __MPIR_VERSION_PATCHLEVEL
- The major and minor MPIR version, and patch level, respectively,
- as integers. For MPIR i.j.k, these numbers will be i, j, and k,
- respectively.
-
- -- Global Constant: const char * const gmp_version
- The GNU MP version number, as a null-terminated string, in the form
- "i.j.k".
-
- -- Global Constant: const char * const mpir_version
- The MPIR version number, as a null-terminated string, in the form
- "i.j.k". This release is "2.6.0".
-
-
-File: mpir.info, Node: Compatibility with older versions, Next: Efficiency, Prev: Useful Macros and Constants, Up: MPIR Basics
-
-3.10 Compatibility with older versions
-======================================
-
-This version of MPIR is upwardly binary compatible with all GMP 5.x,
-4.x and 3.x versions, and upwardly compatible at the source level with
-all 2.x versions, with the following exceptions.
-
- * `mpn_gcd' had its source arguments swapped as of GMP 3.0, for
- consistency with other `mpn' functions.
-
- * `mpf_get_prec' counted precision slightly differently in GMP 3.0
- and 3.0.1, but in 3.1 reverted to the 2.x style.
-
- * MPIR does not support the secure cryptographic functions provided
- by GMP.
-
- * Full GMP compatibility is only available when the
- `--enable-gmpcompat' configure option is used.
-
- There are a number of compatibility issues between GMP 1 and GMP 2
-that of course also apply when porting applications from GMP 1 to GMP 4
-and MPIR 1 and 2. Please see the GMP 2 manual for details.
-
-
-File: mpir.info, Node: Efficiency, Next: Debugging, Prev: Compatibility with older versions, Up: MPIR Basics
-
-3.11 Efficiency
-===============
-
-Small Operands
- On small operands, the time for function call overheads and memory
- allocation can be significant in comparison to actual calculation.
- This is unavoidable in a general purpose variable precision
- library, although MPIR attempts to be as efficient as it can on
- both large and small operands.
-
-Static Linking
- On some CPUs, in particular the x86s, the static `libmpir.a'
- should be used for maximum speed, since the PIC code in the shared
- `libmpir.so' will have a small overhead on each function call and
- global data address. For many programs this will be
- insignificant, but for long calculations there's a gain to be had.
-
-Initializing and Clearing
- Avoid excessive initializing and clearing of variables, since this
- can be quite time consuming, especially in comparison to otherwise
- fast operations like addition.
-
- A language interpreter might want to keep a free list or stack of
- initialized variables ready for use. It should be possible to
- integrate something like that with a garbage collector too.
-
-Reallocations
- An `mpz_t' or `mpq_t' variable used to hold successively increasing
- values will have its memory repeatedly `realloc'ed, which could be
- quite slow or could fragment memory, depending on the C library.
- If an application can estimate the final size then `mpz_init2' or
- `mpz_realloc2' can be called to allocate the necessary space from
- the beginning (*note Initializing Integers::).
-
- It doesn't matter if a size set with `mpz_init2' or `mpz_realloc2'
- is too small, since all functions will do a further reallocation
- if necessary. Badly overestimating memory required will waste
- space though.
-
-`2exp' Functions
- It's up to an application to call functions like `mpz_mul_2exp'
- when appropriate. General purpose functions like `mpz_mul' make
- no attempt to identify powers of two or other special forms,
- because such inputs will usually be very rare and testing every
- time would be wasteful.
-
-`ui' and `si' Functions
- The `ui' functions and the small number of `si' functions exist for
- convenience and should be used where applicable. But if for
- example an `mpz_t' contains a value that fits in an `unsigned long'
- (`unsigned long long' on Windows x64) there's no need extract it
- and call a `ui' function, just use the regular `mpz' function.
-
-In-Place Operations
- `mpz_abs', `mpq_abs', `mpf_abs', `mpz_neg', `mpq_neg' and
- `mpf_neg' are fast when used for in-place operations like
- `mpz_abs(x,x)', since in the current implementation only a single
- field of `x' needs changing. On suitable compilers (GCC for
- instance) this is inlined too.
-
- `mpz_add_ui', `mpz_sub_ui', `mpf_add_ui' and `mpf_sub_ui' benefit
- from an in-place operation like `mpz_add_ui(x,x,y)', since usually
- only one or two limbs of `x' will need to be changed. The same
- applies to the full precision `mpz_add' etc if `y' is small. If
- `y' is big then cache locality may be helped, but that's all.
-
- `mpz_mul' is currently the opposite, a separate destination is
- slightly better. A call like `mpz_mul(x,x,y)' will, unless `y' is
- only one limb, make a temporary copy of `x' before forming the
- result. Normally that copying will only be a tiny fraction of the
- time for the multiply, so this is not a particularly important
- consideration.
-
- `mpz_set', `mpq_set', `mpq_set_num', `mpf_set', etc, make no
- attempt to recognise a copy of something to itself, so a call like
- `mpz_set(x,x)' will be wasteful. Naturally that would never be
- written deliberately, but if it might arise from two pointers to
- the same object then a test to avoid it might be desirable.
-
- if (x != y)
- mpz_set (x, y);
-
- Note that it's never worth introducing extra `mpz_set' calls just
- to get in-place operations. If a result should go to a particular
- variable then just direct it there and let MPIR take care of data
- movement.
-
-Divisibility Testing (Small Integers)
- `mpz_divisible_ui_p' and `mpz_congruent_ui_p' are the best
- functions for testing whether an `mpz_t' is divisible by an
- individual small integer. They use an algorithm which is faster
- than `mpz_tdiv_ui', but which gives no useful information about
- the actual remainder, only whether it's zero (or a particular
- value).
-
- However when testing divisibility by several small integers, it's
- best to take a remainder modulo their product, to save
- multi-precision operations. For instance to test whether a number
- is divisible by any of 23, 29 or 31 take a remainder modulo
- 23*29*31 = 20677 and then test that.
-
- The division functions like `mpz_tdiv_q_ui' which give a quotient
- as well as a remainder are generally a little slower than the
- remainder-only functions like `mpz_tdiv_ui'. If the quotient is
- only rarely wanted then it's probably best to just take a
- remainder and then go back and calculate the quotient if and when
- it's wanted (`mpz_divexact_ui' can be used if the remainder is
- zero).
-
-Rational Arithmetic
- The `mpq' functions operate on `mpq_t' values with no common
- factors in the numerator and denominator. Common factors are
- checked-for and cast out as necessary. In general, cancelling
- factors every time is the best approach since it minimizes the
- sizes for subsequent operations.
-
- However, applications that know something about the factorization
- of the values they're working with might be able to avoid some of
- the GCDs used for canonicalization, or swap them for divisions.
- For example when multiplying by a prime it's enough to check for
- factors of it in the denominator instead of doing a full GCD. Or
- when forming a big product it might be known that very little
- cancellation will be possible, and so canonicalization can be left
- to the end.
-
- The `mpq_numref' and `mpq_denref' macros give access to the
- numerator and denominator to do things outside the scope of the
- supplied `mpq' functions. *Note Applying Integer Functions::.
-
- The canonical form for rationals allows mixed-type `mpq_t' and
- integer additions or subtractions to be done directly with
- multiples of the denominator. This will be somewhat faster than
- `mpq_add'. For example,
-
- /* mpq increment */
- mpz_add (mpq_numref(q), mpq_numref(q), mpq_denref(q));
-
- /* mpq += unsigned long */
- mpz_addmul_ui (mpq_numref(q), mpq_denref(q), 123UL);
-
- /* mpq -= mpz */
- mpz_submul (mpq_numref(q), mpq_denref(q), z);
-
-Number Sequences
- Functions like `mpz_fac_ui', `mpz_fib_ui' and `mpz_bin_uiui' are
- designed for calculating isolated values. If a range of values is
- wanted it's probably best to call to get a starting point and
- iterate from there.
-
-Text Input/Output
- Hexadecimal or octal are suggested for input or output in text
- form. Power-of-2 bases like these can be converted much more
- efficiently than other bases, like decimal. For big numbers
- there's usually nothing of particular interest to be seen in the
- digits, so the base doesn't matter much.
-
- Maybe we can hope octal will one day become the normal base for
- everyday use, as proposed by King Charles XII of Sweden and later
- reformers.
-
-
-File: mpir.info, Node: Debugging, Next: Profiling, Prev: Efficiency, Up: MPIR Basics
-
-3.12 Debugging
-==============
-
-Stack Overflow
- Depending on the system, a segmentation violation or bus error
- might be the only indication of stack overflow. See
- `--enable-alloca' choices in *note Build Options::, for how to
- address this.
-
- In new enough versions of GCC, `-fstack-check' may be able to
- ensure an overflow is recognised by the system before too much
- damage is done, or `-fstack-limit-symbol' or
- `-fstack-limit-register' may be able to add checking if the system
- itself doesn't do any (*note Options for Code Generation:
- (gcc)Code Gen Options.). These options must be added to the
- `CFLAGS' used in the MPIR build (*note Build Options::), adding
- them just to an application will have no effect. Note also
- they're a slowdown, adding overhead to each function call and each
- stack allocation.
-
-Heap Problems
- The most likely cause of application problems with MPIR is heap
- corruption. Failing to `init' MPIR variables will have
- unpredictable effects, and corruption arising elsewhere in a
- program may well affect MPIR. Initializing MPIR variables more
- than once or failing to clear them will cause memory leaks.
-
- In all such cases a `malloc' debugger is recommended. On a GNU or
- BSD system the standard C library `malloc' has some diagnostic
- facilities, see *note Allocation Debugging: (libc)Allocation
- Debugging, or `man 3 malloc'. Other possibilities, in no
- particular order, include
-
- `http://dmalloc.com/'
- `http://www.perens.com/FreeSoftware/' (electric fence)
- `http://www.gnupdate.org/components/leakbug/'
- `http://wwww.gnome.org/projects/memprof'
-
- The MPIR default allocation routines in `memory.c' also have a
- simple sentinel scheme which can be enabled with `#define DEBUG'
- in that file. This is mainly designed for detecting buffer
- overruns during MPIR development, but might find other uses.
-
-Stack Backtraces
- On some systems the compiler options MPIR uses by default can
- interfere with debugging. In particular on x86 and 68k systems
- `-fomit-frame-pointer' is used and this generally inhibits stack
- backtracing. Recompiling without such options may help while
- debugging, though the usual caveats about it potentially moving a
- memory problem or hiding a compiler bug will apply.
-
-GDB, the GNU Debugger
- A sample `.gdbinit' is included in the distribution, showing how
- to call some undocumented dump functions to print MPIR variables
- from within GDB. Note that these functions shouldn't be used in
- final application code since they're undocumented and may be
- subject to incompatible changes in future versions of MPIR.
-
-Source File Paths
- MPIR has multiple source files with the same name, in different
- directories. For example `mpz', `mpq' and `mpf' each have an
- `init.c'. If the debugger can't already determine the right one
- it may help to build with absolute paths on each C file. One way
- to do that is to use a separate object directory with an absolute
- path to the source directory.
-
- cd /my/build/dir
- /my/source/dir/gmp-2.6.0/configure
-
- This works via `VPATH', and might require GNU `make'. Alternately
- it might be possible to change the `.c.lo' rules appropriately.
-
-Assertion Checking
- The build option `--enable-assert' is available to add some
- consistency checks to the library (see *note Build Options::).
- These are likely to be of limited value to most applications.
- Assertion failures are just as likely to indicate memory
- corruption as a library or compiler bug.
-
- Applications using the low-level `mpn' functions, however, will
- benefit from `--enable-assert' since it adds checks on the
- parameters of most such functions, many of which have subtle
- restrictions on their usage. Note however that only the generic C
- code has checks, not the assembler code, so CPU `none' should be
- used for maximum checking.
-
-Temporary Memory Checking
- The build option `--enable-alloca=debug' arranges that each block
- of temporary memory in MPIR is allocated with a separate call to
- `malloc' (or the allocation function set with
- `mp_set_memory_functions').
-
- This can help a malloc debugger detect accesses outside the
- intended bounds, or detect memory not released. In a normal
- build, on the other hand, temporary memory is allocated in blocks
- which MPIR divides up for its own use, or may be allocated with a
- compiler builtin `alloca' which will go nowhere near any malloc
- debugger hooks.
-
-Maximum Debuggability
- To summarize the above, an MPIR build for maximum debuggability
- would be
-
- ./configure --disable-shared --enable-assert \
- --enable-alloca=debug --host=none CFLAGS=-g
-
- For C++, add `--enable-cxx CXXFLAGS=-g'.
-
-Checker
- The GCC checker (`http://savannah.gnu.org/projects/checker/') can
- be used with MPIR. It contains a stub library which means MPIR
- applications compiled with checker can use a normal MPIR build.
-
- A build of MPIR with checking within MPIR itself can be made.
- This will run very very slowly. On GNU/Linux for example,
-
- ./configure --host=none-pc-linux-gnu CC=checkergcc
-
- `--host=none' must be used, since the MPIR assembler code doesn't
- support the checking scheme. The MPIR C++ features cannot be
- used, since current versions of checker (0.9.9.1) don't yet
- support the standard C++ library.
-
-Valgrind
- The valgrind program (`http://valgrind.org/') is a memory checker
- for x86s. It translates and emulates machine instructions to do
- strong checks for uninitialized data (at the level of individual
- bits), memory accesses through bad pointers, and memory leaks.
-
- Recent versions of Valgrind are getting support for MMX and
- SSE/SSE2 instructions, for past versions MPIR will need to be
- configured not to use those, ie. for an x86 without them (for
- instance plain `i486').
-
-Other Problems
- Any suspected bug in MPIR itself should be isolated to make sure
- it's not an application problem, see *note Reporting Bugs::.
-
-
-File: mpir.info, Node: Profiling, Next: Autoconf, Prev: Debugging, Up: MPIR Basics
-
-3.13 Profiling
-==============
-
-Running a program under a profiler is a good way to find where it's
-spending most time and where improvements can be best sought. The
-profiling choices for a MPIR build are as follows.
-
-`--disable-profiling'
- The default is to add nothing special for profiling.
-
- It should be possible to just compile the mainline of a program
- with `-p' and use `prof' to get a profile consisting of
- timer-based sampling of the program counter. Most of the MPIR
- assembler code has the necessary symbol information.
-
- This approach has the advantage of minimizing interference with
- normal program operation, but on most systems the resolution of
- the sampling is quite low (10 milliseconds for instance),
- requiring long runs to get accurate information.
-
-`--enable-profiling=prof'
- Build with support for the system `prof', which means `-p' added
- to the `CFLAGS'.
-
- This provides call counting in addition to program counter
- sampling, which allows the most frequently called routines to be
- identified, and an average time spent in each routine to be
- determined.
-
- The x86 assembler code has support for this option, but on other
- processors the assembler routines will be as if compiled without
- `-p' and therefore won't appear in the call counts.
-
- On some systems, such as GNU/Linux, `-p' in fact means `-pg' and in
- this case `--enable-profiling=gprof' described below should be used
- instead.
-
-`--enable-profiling=gprof'
- Build with support for `gprof' (*note GNU gprof: (gprof)Top.),
- which means `-pg' added to the `CFLAGS'.
-
- This provides call graph construction in addition to call counting
- and program counter sampling, which makes it possible to count
- calls coming from different locations. For example the number of
- calls to `mpn_mul' from `mpz_mul' versus the number from
- `mpf_mul'. The program counter sampling is still flat though, so
- only a total time in `mpn_mul' would be accumulated, not a
- separate amount for each call site.
-
- The x86 assembler code has support for this option, but on other
- processors the assembler routines will be as if compiled without
- `-pg' and therefore not be included in the call counts.
-
- On x86 and m68k systems `-pg' and `-fomit-frame-pointer' are
- incompatible, so the latter is omitted from the default flags in
- that case, which might result in poorer code generation.
-
- Incidentally, it should be possible to use the `gprof' program
- with a plain `--enable-profiling=prof' build. But in that case
- only the `gprof -p' flat profile and call counts can be expected
- to be valid, not the `gprof -q' call graph.
-
-`--enable-profiling=instrument'
- Build with the GCC option `-finstrument-functions' added to the
- `CFLAGS' (*note Options for Code Generation: (gcc)Code Gen
- Options.).
-
- This inserts special instrumenting calls at the start and end of
- each function, allowing exact timing and full call graph
- construction.
-
- This instrumenting is not normally a standard system feature and
- will require support from an external library, such as
-
- `http://sourceforge.net/projects/fnccheck/'
-
- This should be included in `LIBS' during the MPIR configure so
- that test programs will link. For example,
-
- ./configure --enable-profiling=instrument LIBS=-lfc
-
- On a GNU system the C library provides dummy instrumenting
- functions, so programs compiled with this option will link. In
- this case it's only necessary to ensure the correct library is
- added when linking an application.
-
- The x86 assembler code supports this option, but on other
- processors the assembler routines will be as if compiled without
- `-finstrument-functions' meaning time spent in them will
- effectively be attributed to their caller.
-
-
-File: mpir.info, Node: Autoconf, Next: Emacs, Prev: Profiling, Up: MPIR Basics
-
-3.14 Autoconf
-=============
-
-Autoconf based applications can easily check whether MPIR is installed.
-The only thing to be noted is that GMP/MPIR library symbols from
-version 3 of GMP and version 1 of MPIR onwards have prefixes like
-`__gmpz'. The following therefore would be a simple test,
-
- AC_CHECK_LIB(mpir, __gmpz_init)
-
- This just uses the default `AC_CHECK_LIB' actions for found or not
-found, but an application that must have MPIR would want to generate an
-error if not found. For example,
-
- AC_CHECK_LIB(mpir, __gmpz_init, ,
- [AC_MSG_ERROR([MPIR not found, see http://www.mpir.org/])])
-
- If functions added in some particular version of GMP/MPIR are
-required, then one of those can be used when checking. For example
-`mpz_mul_si' was added in GMP 3.1,
-
- AC_CHECK_LIB(mpir, __gmpz_mul_si, ,
- [AC_MSG_ERROR(
- [GMP/MPIR not found, or not GMP 3.1 or up or MPIR 1.0 or up, see http://www.mpir.org/])])
-
- An alternative would be to test the version number in `mpir.h' using
-say `AC_EGREP_CPP'. That would make it possible to test the exact
-version, if some particular sub-minor release is known to be necessary.
-
- In general it's recommended that applications should simply demand a
-new enough MPIR rather than trying to provide supplements for features
-not available in past versions.
-
- Occasionally an application will need or want to know the size of a
-type at configuration or preprocessing time, not just with `sizeof' in
-the code. This can be done in the normal way with `mp_limb_t' etc, but
-GMP 4.0 or up and MPIR 1.0 and up is best for this, since prior
-versions needed certain `-D' defines on systems using a `long long'
-limb. The following would suit Autoconf 2.50 or up,
-
- AC_CHECK_SIZEOF(mp_limb_t, , [#include <mpir.h>])
-
-
-File: mpir.info, Node: Emacs, Prev: Autoconf, Up: MPIR Basics
-
-3.15 Emacs
-==========
-
-<C-h C-i> (`info-lookup-symbol') is a good way to find documentation on
-C functions while editing (*note Info Documentation Lookup: (emacs)Info
-Lookup.).
-
- The MPIR manual can be included in such lookups by putting the
-following in your `.emacs',
-
- (eval-after-load "info-look"
- '(let ((mode-value (assoc 'c-mode (assoc 'symbol info-lookup-alist))))
- (setcar (nthcdr 3 mode-value)
- (cons '("(gmp)Function Index" nil "^ -.* " "\\>")
- (nth 3 mode-value)))))
-
-
-File: mpir.info, Node: Reporting Bugs, Next: Integer Functions, Prev: MPIR Basics, Up: Top
-
-4 Reporting Bugs
-****************
-
-If you think you have found a bug in the MPIR library, please
-investigate it and report it. We have made this library available to
-you, and it is not too much to ask you to report the bugs you find.
-
- Before you report a bug, check it's not already addressed in *note
-Known Build Problems::, or perhaps *note Notes for Particular
-Systems::. You may also want to check `http://www.mpir.org/' for
-patches for this release.
-
- Please include the following in any report,
-
- * The MPIR version number, and if pre-packaged or patched then say
- so.
-
- * A test program that makes it possible for us to reproduce the bug.
- Include instructions on how to run the program.
-
- * A description of what is wrong. If the results are incorrect, in
- what way. If you get a crash, say so.
-
- * If you get a crash, include a stack backtrace from the debugger if
- it's informative (`where' in `gdb', or `$C' in `adb').
-
- * Please do not send core dumps, executables or `strace's.
-
- * The configuration options you used when building MPIR, if any.
-
- * The name of the compiler and its version. For `gcc', get the
- version with `gcc -v', otherwise perhaps `what `which cc`', or
- similar.
-
- * The output from running `uname -a'.
-
- * The output from running `./config.guess', and from running
- `./configfsf.guess' (might be the same).
-
- * If the bug is related to `configure', then the contents of
- `config.log'.
-
- * If the bug is related to an `asm' file not assembling, then the
- contents of `config.m4' and the offending line or lines from the
- temporary `mpn/tmp-<file>.s'.
-
- Please make an effort to produce a self-contained report, with
-something definite that can be tested or debugged. Vague queries or
-piecemeal messages are difficult to act on and don't help the
-development effort.
-
- It is not uncommon that an observed problem is actually due to a bug
-in the compiler; the MPIR code tends to explore interesting corners in
-compilers.
-
- If your bug report is good, we will do our best to help you get a
-corrected version of the library; if the bug report is poor, we won't
-do anything about it (except maybe ask you to send a better report).
-
- Send your report to: `http://groups.google.com/group/mpir-devel'.
-
- If you think something in this manual is unclear, or downright
-incorrect, or if the language needs to be improved, please send a note
-to the same address.
-
-
-File: mpir.info, Node: Integer Functions, Next: Rational Number Functions, Prev: Reporting Bugs, Up: Top
-
-5 Integer Functions
-*******************
-
-This chapter describes the MPIR functions for performing integer
-arithmetic. These functions start with the prefix `mpz_'.
-
- MPIR integers are stored in objects of type `mpz_t'.
-
-* Menu:
-
-* Initializing Integers::
-* Assigning Integers::
-* Simultaneous Integer Init & Assign::
-* Converting Integers::
-* Integer Arithmetic::
-* Integer Division::
-* Integer Exponentiation::
-* Integer Roots::
-* Number Theoretic Functions::
-* Integer Comparisons::
-* Integer Logic and Bit Fiddling::
-* I/O of Integers::
-* Integer Random Numbers::
-* Integer Import and Export::
-* Miscellaneous Integer Functions::
-* Integer Special Functions::
-
-
-File: mpir.info, Node: Initializing Integers, Next: Assigning Integers, Prev: Integer Functions, Up: Integer Functions
-
-5.1 Initialization Functions
-============================
-
-The functions for integer arithmetic assume that all integer objects are
-initialized. You do that by calling the function `mpz_init'. For
-example,
-
- {
- mpz_t integ;
- mpz_init (integ);
- ...
- mpz_add (integ, ...);
- ...
- mpz_sub (integ, ...);
-
- /* Unless the program is about to exit, do ... */
- mpz_clear (integ);
- }
-
- As you can see, you can store new values any number of times, once an
-object is initialized.
-
- -- Function: void mpz_init (mpz_t INTEGER)
- Initialize INTEGER, and set its value to 0.
-
- -- Function: void mpz_inits (mpz_t X, ...)
- Initialize a NULL-terminated list of `mpz_t' variables, and set
- their values to 0.
-
- -- Function: void mpz_init2 (mpz_t INTEGER, mp_bitcnt_t N)
- Initialize INTEGER, with space for N bits, and set its value to 0.
-
- N is only the initial space, INTEGER will grow automatically in
- the normal way, if necessary, for subsequent values stored.
- `mpz_init2' makes it possible to avoid such reallocations if a
- maximum size is known in advance.
-
- -- Function: void mpz_clear (mpz_t INTEGER)
- Free the space occupied by INTEGER. Call this function for all
- `mpz_t' variables when you are done with them.
-
- -- Function: void mpz_clears (mpz_t X, ...)
- Free the space occupied by a NULL-terminated list of `mpz_t'
- variables.
-
- -- Function: void mpz_realloc2 (mpz_t INTEGER, mp_bitcnt_t N)
- Change the space allocated for INTEGER to N bits. The value in
- INTEGER is preserved if it fits, or is set to 0 if not.
-
- This function can be used to increase the space for a variable in
- order to avoid repeated automatic reallocations, or to decrease it
- to give memory back to the heap.
-
-
-File: mpir.info, Node: Assigning Integers, Next: Simultaneous Integer Init & Assign, Prev: Initializing Integers, Up: Integer Functions
-
-5.2 Assignment Functions
-========================
-
-These functions assign new values to already initialized integers
-(*note Initializing Integers::).
-
- -- Function: void mpz_set (mpz_t ROP, mpz_t OP)
- -- Function: void mpz_set_ui (mpz_t ROP, mpir_ui OP)
- -- Function: void mpz_set_si (mpz_t ROP, mpir_si OP)
- -- Function: void mpz_set_ux (mpz_t ROP, uintmax_t OP)
- -- Function: void mpz_set_sx (mpz_t ROP, intmax_t OP)
- -- Function: void mpz_set_d (mpz_t ROP, double OP)
- -- Function: void mpz_set_q (mpz_t ROP, mpq_t OP)
- -- Function: void mpz_set_f (mpz_t ROP, mpf_t OP)
- Set the value of ROP from OP. Note the intmax versions are only
- available if you have stdint.h header file on your system.
-
- `mpz_set_d', `mpz_set_q' and `mpz_set_f' truncate OP to make it an
- integer.
-
- -- Function: int mpz_set_str (mpz_t ROP, char *STR, int BASE)
- Set the value of ROP from STR, a null-terminated C string in base
- BASE. White space is allowed in the string, and is simply ignored.
-
- The BASE may vary from 2 to 62, or if BASE is 0, then the leading
- characters are used: `0x' and `0X' for hexadecimal, `0b' and `0B'
- for binary, `0' for octal, or decimal otherwise.
-
- For bases up to 36, case is ignored; upper-case and lower-case
- letters have the same value. For bases 37 to 62, upper-case
- letter represent the usual 10..35 while lower-case letter
- represent 36..61.
-
- This function returns 0 if the entire string is a valid number in
- base BASE. Otherwise it returns -1.
-
- -- Function: void mpz_swap (mpz_t ROP1, mpz_t ROP2)
- Swap the values ROP1 and ROP2 efficiently.
-
-
-File: mpir.info, Node: Simultaneous Integer Init & Assign, Next: Converting Integers, Prev: Assigning Integers, Up: Integer Functions
-
-5.3 Combined Initialization and Assignment Functions
-====================================================
-
-For convenience, MPIR provides a parallel series of initialize-and-set
-functions which initialize the output and then store the value there.
-These functions' names have the form `mpz_init_set...'
-
- Here is an example of using one:
-
- {
- mpz_t pie;
- mpz_init_set_str (pie, "3141592653589793238462643383279502884", 10);
- ...
- mpz_sub (pie, ...);
- ...
- mpz_clear (pie);
- }
-
-Once the integer has been initialized by any of the `mpz_init_set...'
-functions, it can be used as the source or destination operand for the
-ordinary integer functions. Don't use an initialize-and-set function
-on a variable already initialized!
-
- -- Function: void mpz_init_set (mpz_t ROP, mpz_t OP)
- -- Function: void mpz_init_set_ui (mpz_t ROP, mpir_ui OP)
- -- Function: void mpz_init_set_si (mpz_t ROP, mpir_si OP)
- -- Function: void mpz_init_set_ux (mpz_t ROP, uintmax_t OP)
- -- Function: void mpz_init_set_sx (mpz_t ROP, intmax_t OP)
- -- Function: void mpz_init_set_d (mpz_t ROP, double OP)
- Initialize ROP with limb space and set the initial numeric value
- from OP. Note the intmax versions are only available if you have
- stdint.h header file on your system.
-
- -- Function: int mpz_init_set_str (mpz_t ROP, char *STR, int BASE)
- Initialize ROP and set its value like `mpz_set_str' (see its
- documentation above for details).
-
- If the string is a correct base BASE number, the function returns
- 0; if an error occurs it returns -1. ROP is initialized even if
- an error occurs. (I.e., you have to call `mpz_clear' for it.)
-
-
-File: mpir.info, Node: Converting Integers, Next: Integer Arithmetic, Prev: Simultaneous Integer Init & Assign, Up: Integer Functions
-
-5.4 Conversion Functions
-========================
-
-This section describes functions for converting MPIR integers to
-standard C types. Functions for converting _to_ MPIR integers are
-described in *note Assigning Integers:: and *note I/O of Integers::.
-
- -- Function: mpir_ui mpz_get_ui (mpz_t OP)
- Return the value of OP as an `mpir_ui'.
-
- If OP is too big to fit an `mpir_ui' then just the least
- significant bits that do fit are returned. The sign of OP is
- ignored, only the absolute value is used.
-
- -- Function: mpir_si mpz_get_si (mpz_t OP)
- If OP fits into a `mpir_si' return the value of OP. Otherwise
- return the least significant part of OP, with the same sign as OP.
-
- If OP is too big to fit in a `mpir_si', the returned result is
- probably not very useful. To find out if the value will fit, use
- the function `mpz_fits_slong_p'.
-
- -- Function: uintmax_t mpz_get_ux (mpz_t OP)
- Return the value of OP as an `uintmax_t'.
-
- If OP is too big to fit an `uintmax_t' then just the least
- significant bits that do fit are returned. The sign of OP is
- ignored, only the absolute value is used. Note the intmax versions
- are only available if you have stdint.h header file on your system.
-
- -- Function: intmax_t mpz_get_sx (mpz_t OP)
- If OP fits into a `intmax_t' return the value of OP. Otherwise
- return the least significant part of OP, with the same sign as OP.
-
- If OP is too big to fit in a `intmax_t', the returned result is
- probably not very useful. Note the intmax versions are only
- available if you have stdint.h header file on your system.
-
- -- Function: double mpz_get_d (mpz_t OP)
- Convert OP to a `double', truncating if necessary (ie. rounding
- towards zero).
-
- If the exponent from the conversion is too big, the result is
- system dependent. An infinity is returned where available. A
- hardware overflow trap may or may not occur.
-
- -- Function: double mpz_get_d_2exp (mpir_si *EXP, mpz_t OP)
- Convert OP to a `double', truncating if necessary (ie. rounding
- towards zero), and returning the exponent separately.
-
- The return value is in the range 0.5<=abs(D)<1 and the exponent is
- stored to `*EXP'. D * 2^EXP is the (truncated) OP value. If OP
- is zero, the return is 0.0 and 0 is stored to `*EXP'.
-
- This is similar to the standard C `frexp' function (*note
- Normalization Functions: (libc)Normalization Functions.).
-
- -- Function: char * mpz_get_str (char *STR, int BASE, mpz_t OP)
- Convert OP to a string of digits in base BASE. The base may vary
- from 2 to 36 or from -2 to -36.
-
- For BASE in the range 2..36, digits and lower-case letters are
- used; for -2..-36, digits and upper-case letters are used; for
- 37..62, digits, upper-case letters, and lower-case letters (in
- that significance order) are used.
-
- If STR is `NULL', the result string is allocated using the current
- allocation function (*note Custom Allocation::). The block will be
- `strlen(str)+1' bytes, that being exactly enough for the string and
- null-terminator.
-
- If STR is not `NULL', it should point to a block of storage large
- enough for the result, that being `mpz_sizeinbase (OP, BASE) + 2'.
- The two extra bytes are for a possible minus sign, and the
- null-terminator.
-
- A pointer to the result string is returned, being either the
- allocated block, or the given STR.
-
-
-File: mpir.info, Node: Integer Arithmetic, Next: Integer Division, Prev: Converting Integers, Up: Integer Functions
-
-5.5 Arithmetic Functions
-========================
-
- -- Function: void mpz_add (mpz_t ROP, mpz_t OP1, mpz_t OP2)
- -- Function: void mpz_add_ui (mpz_t ROP, mpz_t OP1, mpir_ui OP2)
- Set ROP to OP1 + OP2.
-
- -- Function: void mpz_sub (mpz_t ROP, mpz_t OP1, mpz_t OP2)
- -- Function: void mpz_sub_ui (mpz_t ROP, mpz_t OP1, mpir_ui OP2)
- -- Function: void mpz_ui_sub (mpz_t ROP, mpir_ui OP1, mpz_t OP2)
- Set ROP to OP1 - OP2.
-
- -- Function: void mpz_mul (mpz_t ROP, mpz_t OP1, mpz_t OP2)
- -- Function: void mpz_mul_si (mpz_t ROP, mpz_t OP1, mpir_si OP2)
- -- Function: void mpz_mul_ui (mpz_t ROP, mpz_t OP1, mpir_ui OP2)
- Set ROP to OP1 times OP2.
-
- -- Function: void mpz_addmul (mpz_t ROP, mpz_t OP1, mpz_t OP2)
- -- Function: void mpz_addmul_ui (mpz_t ROP, mpz_t OP1, mpir_ui OP2)
- Set ROP to ROP + OP1 times OP2.
-
- -- Function: void mpz_submul (mpz_t ROP, mpz_t OP1, mpz_t OP2)
- -- Function: void mpz_submul_ui (mpz_t ROP, mpz_t OP1, mpir_ui OP2)
- Set ROP to ROP - OP1 times OP2.
-
- -- Function: void mpz_mul_2exp (mpz_t ROP, mpz_t OP1, mp_bitcnt_t OP2)
- Set ROP to OP1 times 2 raised to OP2. This operation can also be
- defined as a left shift by OP2 bits.
-
- -- Function: void mpz_neg (mpz_t ROP, mpz_t OP)
- Set ROP to -OP.
-
- -- Function: void mpz_abs (mpz_t ROP, mpz_t OP)
- Set ROP to the absolute value of OP.
-
-
-File: mpir.info, Node: Integer Division, Next: Integer Exponentiation, Prev: Integer Arithmetic, Up: Integer Functions
-
-5.6 Division Functions
-======================
-
-Division is undefined if the divisor is zero. Passing a zero divisor
-to the division or modulo functions (including the modular powering
-functions `mpz_powm' and `mpz_powm_ui'), will cause an intentional
-division by zero. This lets a program handle arithmetic exceptions in
-these functions the same way as for normal C `int' arithmetic.
-
- -- Function: void mpz_cdiv_q (mpz_t Q, mpz_t N, mpz_t D)
- -- Function: void mpz_cdiv_r (mpz_t R, mpz_t N, mpz_t D)
- -- Function: void mpz_cdiv_qr (mpz_t Q, mpz_t R, mpz_t N, mpz_t D)
- -- Function: mpir_ui mpz_cdiv_q_ui (mpz_t Q, mpz_t N, mpir_ui D)
- -- Function: mpir_ui mpz_cdiv_r_ui (mpz_t R, mpz_t N, mpir_ui D)
- -- Function: mpir_ui mpz_cdiv_qr_ui (mpz_t Q, mpz_t R, mpz_t N,
- mpir_ui D)
- -- Function: mpir_ui mpz_cdiv_ui (mpz_t N, mpir_ui D)
- -- Function: void mpz_cdiv_q_2exp (mpz_t Q, mpz_t N, mp_bitcnt_t B)
- -- Function: void mpz_cdiv_r_2exp (mpz_t R, mpz_t N, mp_bitcnt_t B)
-
- -- Function: void mpz_fdiv_q (mpz_t Q, mpz_t N, mpz_t D)
- -- Function: void mpz_fdiv_r (mpz_t R, mpz_t N, mpz_t D)
- -- Function: void mpz_fdiv_qr (mpz_t Q, mpz_t R, mpz_t N, mpz_t D)
- -- Function: mpir_ui mpz_fdiv_q_ui (mpz_t Q, mpz_t N, mpir_ui D)
- -- Function: mpir_ui mpz_fdiv_r_ui (mpz_t R, mpz_t N, mpir_ui D)
- -- Function: mpir_ui mpz_fdiv_qr_ui (mpz_t Q, mpz_t R, mpz_t N,
- mpir_ui D)
- -- Function: mpir_ui mpz_fdiv_ui (mpz_t N, mpir_ui D)
- -- Function: void mpz_fdiv_q_2exp (mpz_t Q, mpz_t N, mp_bitcnt_t B)
- -- Function: void mpz_fdiv_r_2exp (mpz_t R, mpz_t N, mp_bitcnt_t B)
-
- -- Function: void mpz_tdiv_q (mpz_t Q, mpz_t N, mpz_t D)
- -- Function: void mpz_tdiv_r (mpz_t R, mpz_t N, mpz_t D)
- -- Function: void mpz_tdiv_qr (mpz_t Q, mpz_t R, mpz_t N, mpz_t D)
- -- Function: mpir_ui mpz_tdiv_q_ui (mpz_t Q, mpz_t N, mpir_ui D)
- -- Function: mpir_ui mpz_tdiv_r_ui (mpz_t R, mpz_t N, mpir_ui D)
- -- Function: mpir_ui mpz_tdiv_qr_ui (mpz_t Q, mpz_t R, mpz_t N,
- mpir_ui D)
- -- Function: mpir_ui mpz_tdiv_ui (mpz_t N, mpir_ui D)
- -- Function: void mpz_tdiv_q_2exp (mpz_t Q, mpz_t N, mp_bitcnt_t B)
- -- Function: void mpz_tdiv_r_2exp (mpz_t R, mpz_t N, mp_bitcnt_t B)
-
- Divide N by D, forming a quotient Q and/or remainder R. For the
- `2exp' functions, D=2^B. The rounding is in three styles, each
- suiting different applications.
-
- * `cdiv' rounds Q up towards +infinity, and R will have the
- opposite sign to D. The `c' stands for "ceil".
-
- * `fdiv' rounds Q down towards -infinity, and R will have the
- same sign as D. The `f' stands for "floor".
-
- * `tdiv' rounds Q towards zero, and R will have the same sign
- as N. The `t' stands for "truncate".
-
- In all cases Q and R will satisfy N=Q*D+R, and R will satisfy
- 0<=abs(R)<abs(D).
-
- The `q' functions calculate only the quotient, the `r' functions
- only the remainder, and the `qr' functions calculate both. Note
- that for `qr' the same variable cannot be passed for both Q and R,
- or results will be unpredictable.
-
- For the `ui' variants the return value is the remainder, and in
- fact returning the remainder is all the `div_ui' functions do. For
- `tdiv' and `cdiv' the remainder can be negative, so for those the
- return value is the absolute value of the remainder.
-
- For the `2exp' variants the divisor is 2^B. These functions are
- implemented as right shifts and bit masks, but of course they
- round the same as the other functions.
-
- For positive N both `mpz_fdiv_q_2exp' and `mpz_tdiv_q_2exp' are
- simple bitwise right shifts. For negative N, `mpz_fdiv_q_2exp' is
- effectively an arithmetic right shift treating N as twos complement
- the same as the bitwise logical functions do, whereas
- `mpz_tdiv_q_2exp' effectively treats N as sign and magnitude.
-
- -- Function: void mpz_mod (mpz_t R, mpz_t N, mpz_t D)
- -- Function: mpir_ui mpz_mod_ui (mpz_t R, mpz_t N, mpir_ui D)
- Set R to N `mod' D. The sign of the divisor is ignored; the
- result is always non-negative.
-
- `mpz_mod_ui' is identical to `mpz_fdiv_r_ui' above, returning the
- remainder as well as setting R. See `mpz_fdiv_ui' above if only
- the return value is wanted.
-
- -- Function: void mpz_divexact (mpz_t Q, mpz_t N, mpz_t D)
- -- Function: void mpz_divexact_ui (mpz_t Q, mpz_t N, mpir_ui D)
- Set Q to N/D. These functions produce correct results only when
- it is known in advance that D divides N.
-
- These routines are much faster than the other division functions,
- and are the best choice when exact division is known to occur, for
- example reducing a rational to lowest terms.
-
- -- Function: int mpz_divisible_p (mpz_t N, mpz_t D)
- -- Function: int mpz_divisible_ui_p (mpz_t N, mpir_ui D)
- -- Function: int mpz_divisible_2exp_p (mpz_t N, mp_bitcnt_t B)
- Return non-zero if N is exactly divisible by D, or in the case of
- `mpz_divisible_2exp_p' by 2^B.
-
- N is divisible by D if there exists an integer Q satisfying N =
- Q*D. Unlike the other division functions, D=0 is accepted and
- following the rule it can be seen that only 0 is considered
- divisible by 0.
-
- -- Function: int mpz_congruent_p (mpz_t N, mpz_t C, mpz_t D)
- -- Function: int mpz_congruent_ui_p (mpz_t N, mpir_ui C, mpir_ui D)
- -- Function: int mpz_congruent_2exp_p (mpz_t N, mpz_t C, mp_bitcnt_t B)
- Return non-zero if N is congruent to C modulo D, or in the case of
- `mpz_congruent_2exp_p' modulo 2^B.
-
- N is congruent to C mod D if there exists an integer Q satisfying
- N = C + Q*D. Unlike the other division functions, D=0 is accepted
- and following the rule it can be seen that N and C are considered
- congruent mod 0 only when exactly equal.
-
-
-File: mpir.info, Node: Integer Exponentiation, Next: Integer Roots, Prev: Integer Division, Up: Integer Functions
-
-5.7 Exponentiation Functions
-============================
-
- -- Function: void mpz_powm (mpz_t ROP, mpz_t BASE, mpz_t EXP, mpz_t
- MOD)
- -- Function: void mpz_powm_ui (mpz_t ROP, mpz_t BASE, mpir_ui EXP,
- mpz_t MOD)
- Set ROP to (BASE raised to EXP) modulo MOD.
-
- A negative EXP is supported in `mpz_powm' if an inverse BASE^-1
- mod MOD exists (see `mpz_invert' in *note Number Theoretic
- Functions::). If an inverse doesn't exist then a divide by zero
- is raised.
-
- -- Function: void mpz_pow_ui (mpz_t ROP, mpz_t BASE, mpir_ui EXP)
- -- Function: void mpz_ui_pow_ui (mpz_t ROP, mpir_ui BASE, mpir_ui EXP)
- Set ROP to BASE raised to EXP. The case 0^0 yields 1.
-
-
-File: mpir.info, Node: Integer Roots, Next: Number Theoretic Functions, Prev: Integer Exponentiation, Up: Integer Functions
-
-5.8 Root Extraction Functions
-=============================
-
- -- Function: int mpz_root (mpz_t ROP, mpz_t OP, mpir_ui N)
- Set ROP to the truncated integer part of the Nth root of OP.
- Return non-zero if the computation was exact, i.e., if OP is ROP
- to the Nth power.
-
- -- Function: void mpz_nthroot (mpz_t ROP, mpz_t OP, mpir_ui N)
- Set ROP to the truncated integer part of the Nth root of OP.
-
- -- Function: void mpz_rootrem (mpz_t ROOT, mpz_t REM, mpz_t U, mpir_ui
- N)
- Set ROOT to the truncated integer part of the Nth root of U. Set
- REM to the remainder, U-ROOT**N.
-
- -- Function: void mpz_sqrt (mpz_t ROP, mpz_t OP)
- Set ROP to the truncated integer part of the square root of OP.
-
- -- Function: void mpz_sqrtrem (mpz_t ROP1, mpz_t ROP2, mpz_t OP)
- Set ROP1 to the truncated integer part of the square root of OP,
- like `mpz_sqrt'. Set ROP2 to the remainder OP-ROP1*ROP1, which
- will be zero if OP is a perfect square.
-
- If ROP1 and ROP2 are the same variable, the results are undefined.
-
- -- Function: int mpz_perfect_power_p (mpz_t OP)
- Return non-zero if OP is a perfect power, i.e., if there exist
- integers A and B, with B>1, such that OP equals A raised to the
- power B.
-
- Under this definition both 0 and 1 are considered to be perfect
- powers. Negative values of OP are accepted, but of course can
- only be odd perfect powers.
-
- -- Function: int mpz_perfect_square_p (mpz_t OP)
- Return non-zero if OP is a perfect square, i.e., if the square
- root of OP is an integer. Under this definition both 0 and 1 are
- considered to be perfect squares.
-
-
-File: mpir.info, Node: Number Theoretic Functions, Next: Integer Comparisons, Prev: Integer Roots, Up: Integer Functions
-
-5.9 Number Theoretic Functions
-==============================
-
- -- Function: int mpz_probable_prime_p (mpz_t N, gmp_randstate_t STATE,
- int PROB, mpir_ui DIV)
- Determine whether N is a probable prime with the chance of error
- being at most 1 in 2^prob. return value is 1 if N is probably
- prime, or 0 if N is definitely composite.
-
- This function does some trial divisions to speed up the average
- case, then some probabilistic primality tests to achieve the
- desired level of error.
-
- DIV can be used to inform the function that trial division up to
- DIV has already been performed on N and so N has NO divisors <=
- DIV.Use 0 to inform the function that no trial division has been
- done.
-
- *This function interface is preliminary and may change in the
- future.*
-
- -- Function: int mpz_likely_prime_p (mpz_t N, gmp_randstate_t STATE,
- mpir_ui DIV)
- Determine whether N is likely a prime, i.e. you can consider it a
- prime for practical purposes. return value is 1 if N can be
- considered prime, or 0 if N is definitely composite.
-
- This function does some trial divisions to speed up the average
- case, then some probabilistic primality tests. The term "likely"
- refers to the fact that the number will not have small factors.
-
- DIV can be used to inform the function that trial division up to
- DIV has already been performed on N and so N has NO divisors <= DIV
-
- *This function interface is preliminary and may change in the
- future.*
-
- -- Function: int mpz_probab_prime_p (mpz_t N, int REPS)
- Determine whether N is prime. Return 2 if N is definitely prime,
- return 1 if N is probably prime (without being certain), or return
- 0 if N is definitely composite.
-
- This function does some trial divisions, then some Miller-Rabin
- probabilistic primality tests. REPS controls how many such tests
- are done, 5 to 10 is a reasonable number, more will reduce the
- chances of a composite being returned as "probably prime".
-
- Miller-Rabin and similar tests can be more properly called
- compositeness tests. Numbers which fail are known to be composite
- but those which pass might be prime or might be composite. Only a
- few composites pass, hence those which pass are considered
- probably prime.
-
- *This function is obsolete. It will disappear from future MPIR
- releases.*
-
- -- Function: void mpz_nextprime (mpz_t ROP, mpz_t OP)
- Set ROP to the next prime greater than OP.
-
- This function uses a probabilistic algorithm to identify primes.
- For practical purposes it's adequate, the chance of a composite
- passing will be extremely small.
-
- *This function is obsolete. It will disappear from future MPIR
- releases.*
-
- -- Function: void mpz_next_likely_prime (mpz_t ROP, mpz_t OP,
- gmp_randstate_t STATE)
- Set ROP to the next likely prime greater than OP.
-
- This function uses a probabilistic algorithm to identify primes.
- For practical purposes it's adequate, the chance of a composite
- passing will be extremely small.
-
- The variable STATE must be initialized by calling one of the
- `gmp_randinit' functions (*note Random State Initialization::)
- before invoking this function.
-
- -- Function: void mpz_gcd (mpz_t ROP, mpz_t OP1, mpz_t OP2)
- Set ROP to the greatest common divisor of OP1 and OP2. The result
- is always positive even if one or both input operands are negative.
-
- -- Function: mpir_ui mpz_gcd_ui (mpz_t ROP, mpz_t OP1, mpir_ui OP2)
- Compute the greatest common divisor of OP1 and OP2. If ROP is not
- `NULL', store the result there.
-
- If the result is small enough to fit in an `mpir_ui', it is
- returned. If the result does not fit, 0 is returned, and the
- result is equal to the argument OP1. Note that the result will
- always fit if OP2 is non-zero.
-
- -- Function: void mpz_gcdext (mpz_t G, mpz_t S, mpz_t T, mpz_t A,
- mpz_t B)
- Set G to the greatest common divisor of A and B, and in addition
- set S and T to coefficients satisfying A*S + B*T = G. G is always
- positive, even if one or both of A and B are negative.
-
- If T is `NULL' then that value is not computed.
-
- -- Function: void mpz_lcm (mpz_t ROP, mpz_t OP1, mpz_t OP2)
- -- Function: void mpz_lcm_ui (mpz_t ROP, mpz_t OP1, mpir_ui OP2)
- Set ROP to the least common multiple of OP1 and OP2. ROP is
- always positive, irrespective of the signs of OP1 and OP2. ROP
- will be zero if either OP1 or OP2 is zero.
-
- -- Function: int mpz_invert (mpz_t ROP, mpz_t OP1, mpz_t OP2)
- Compute the inverse of OP1 modulo OP2 and put the result in ROP.
- If the inverse exists, the return value is non-zero and ROP will
- satisfy 0 <= ROP < OP2. If an inverse doesn't exist the return
- value is zero and ROP is undefined.
-
- -- Function: int mpz_jacobi (mpz_t A, mpz_t B)
- Calculate the Jacobi symbol (A/B). This is defined only for B odd.
-
- -- Function: int mpz_legendre (mpz_t A, mpz_t P)
- Calculate the Legendre symbol (A/P). This is defined only for P
- an odd positive prime, and for such P it's identical to the Jacobi
- symbol.
-
- -- Function: int mpz_kronecker (mpz_t A, mpz_t B)
- -- Function: int mpz_kronecker_si (mpz_t A, mpir_si B)
- -- Function: int mpz_kronecker_ui (mpz_t A, mpir_ui B)
- -- Function: int mpz_si_kronecker (mpir_si A, mpz_t B)
- -- Function: int mpz_ui_kronecker (mpir_ui A, mpz_t B)
- Calculate the Jacobi symbol (A/B) with the Kronecker extension
- (a/2)=(2/a) when a odd, or (a/2)=0 when a even.
-
- When B is odd the Jacobi symbol and Kronecker symbol are
- identical, so `mpz_kronecker_ui' etc can be used for mixed
- precision Jacobi symbols too.
-
- For more information see Henri Cohen section 1.4.2 (*note
- References::), or any number theory textbook. See also the
- example program `demos/qcn.c' which uses `mpz_kronecker_ui' on the
- MPIR website.
-
- -- Function: mp_bitcnt_t mpz_remove (mpz_t ROP, mpz_t OP, mpz_t F)
- Remove all occurrences of the factor F from OP and store the
- result in ROP. The return value is how many such occurrences were
- removed.
-
- -- Function: void mpz_fac_ui (mpz_t ROP, mpir_ui OP)
- Set ROP to OP!, the factorial of OP.
-
- -- Function: void mpz_bin_ui (mpz_t ROP, mpz_t N, mpir_ui K)
- -- Function: void mpz_bin_uiui (mpz_t ROP, mpir_ui N, mpir_ui K)
- Compute the binomial coefficient N over K and store the result in
- ROP. Negative values of N are supported by `mpz_bin_ui', using
- the identity bin(-n,k) = (-1)^k * bin(n+k-1,k), see Knuth volume 1
- section 1.2.6 part G.
-
- -- Function: void mpz_fib_ui (mpz_t FN, mpir_ui N)
- -- Function: void mpz_fib2_ui (mpz_t FN, mpz_t FNSUB1, mpir_ui N)
- `mpz_fib_ui' sets FN to to F[n], the N'th Fibonacci number.
- `mpz_fib2_ui' sets FN to F[n], and FNSUB1 to F[n-1].
-
- These functions are designed for calculating isolated Fibonacci
- numbers. When a sequence of values is wanted it's best to start
- with `mpz_fib2_ui' and iterate the defining F[n+1]=F[n]+F[n-1] or
- similar.
-
- -- Function: void mpz_lucnum_ui (mpz_t LN, mpir_ui N)
- -- Function: void mpz_lucnum2_ui (mpz_t LN, mpz_t LNSUB1, mpir_ui N)
- `mpz_lucnum_ui' sets LN to to L[n], the N'th Lucas number.
- `mpz_lucnum2_ui' sets LN to L[n], and LNSUB1 to L[n-1].
-
- These functions are designed for calculating isolated Lucas
- numbers. When a sequence of values is wanted it's best to start
- with `mpz_lucnum2_ui' and iterate the defining L[n+1]=L[n]+L[n-1]
- or similar.
-
- The Fibonacci numbers and Lucas numbers are related sequences, so
- it's never necessary to call both `mpz_fib2_ui' and
- `mpz_lucnum2_ui'. The formulas for going from Fibonacci to Lucas
- can be found in *note Lucas Numbers Algorithm::, the reverse is
- straightforward too.
-
-
-File: mpir.info, Node: Integer Comparisons, Next: Integer Logic and Bit Fiddling, Prev: Number Theoretic Functions, Up: Integer Functions
-
-5.10 Comparison Functions
-=========================
-
- -- Function: int mpz_cmp (mpz_t OP1, mpz_t OP2)
- -- Function: int mpz_cmp_d (mpz_t OP1, double OP2)
- -- Macro: int mpz_cmp_si (mpz_t OP1, mpir_si OP2)
- -- Macro: int mpz_cmp_ui (mpz_t OP1, mpir_ui OP2)
- Compare OP1 and OP2. Return a positive value if OP1 > OP2, zero
- if OP1 = OP2, or a negative value if OP1 < OP2.
-
- `mpz_cmp_ui' and `mpz_cmp_si' are macros and will evaluate their
- arguments more than once. `mpz_cmp_d' can be called with an
- infinity, but results are undefined for a NaN.
-
- -- Function: int mpz_cmpabs (mpz_t OP1, mpz_t OP2)
- -- Function: int mpz_cmpabs_d (mpz_t OP1, double OP2)
- -- Function: int mpz_cmpabs_ui (mpz_t OP1, mpir_ui OP2)
- Compare the absolute values of OP1 and OP2. Return a positive
- value if abs(OP1) > abs(OP2), zero if abs(OP1) = abs(OP2), or a
- negative value if abs(OP1) < abs(OP2).
-
- `mpz_cmpabs_d' can be called with an infinity, but results are
- undefined for a NaN.
-
- -- Macro: int mpz_sgn (mpz_t OP)
- Return +1 if OP > 0, 0 if OP = 0, and -1 if OP < 0.
-
- This function is actually implemented as a macro. It evaluates
- its argument multiple times.
-
-
-File: mpir.info, Node: Integer Logic and Bit Fiddling, Next: I/O of Integers, Prev: Integer Comparisons, Up: Integer Functions
-
-5.11 Logical and Bit Manipulation Functions
-===========================================
-
-These functions behave as if twos complement arithmetic were used
-(although sign-magnitude is the actual implementation). The least
-significant bit is number 0.
-
- -- Function: void mpz_and (mpz_t ROP, mpz_t OP1, mpz_t OP2)
- Set ROP to OP1 bitwise-and OP2.
-
- -- Function: void mpz_ior (mpz_t ROP, mpz_t OP1, mpz_t OP2)
- Set ROP to OP1 bitwise inclusive-or OP2.
-
- -- Function: void mpz_xor (mpz_t ROP, mpz_t OP1, mpz_t OP2)
- Set ROP to OP1 bitwise exclusive-or OP2.
-
- -- Function: void mpz_com (mpz_t ROP, mpz_t OP)
- Set ROP to the one's complement of OP.
-
- -- Function: mp_bitcnt_t mpz_popcount (mpz_t OP)
- If OP>=0, return the population count of OP, which is the number
- of 1 bits in the binary representation. If OP<0, the number of 1s
- is infinite, and the return value is ULONG_MAX, the largest
- possible `mp_bitcnt_t'.
-
- -- Function: mp_bitcnt_t mpz_hamdist (mpz_t OP1, mpz_t OP2)
- If OP1 and OP2 are both >=0 or both <0, return the hamming
- distance between the two operands, which is the number of bit
- positions where OP1 and OP2 have different bit values. If one
- operand is >=0 and the other <0 then the number of bits different
- is infinite, and the return value is the largest possible
- `imp_bitcnt_t'.
-
- -- Function: mp_bitcnt_t mpz_scan0 (mpz_t OP, mp_bitcnt_t STARTING_BIT)
- -- Function: mp_bitcnt_t mpz_scan1 (mpz_t OP, mp_bitcnt_t STARTING_BIT)
- Scan OP, starting from bit STARTING_BIT, towards more significant
- bits, until the first 0 or 1 bit (respectively) is found. Return
- the index of the found bit.
-
- If the bit at STARTING_BIT is already what's sought, then
- STARTING_BIT is returned.
-
- If there's no bit found, then the largest possible `mp_bitcnt_t' is
- returned. This will happen in `mpz_scan0' past the end of a
- positive number, or `mpz_scan1' past the end of a nonnegative
- number.
-
- -- Function: void mpz_setbit (mpz_t ROP, mp_bitcnt_t BIT_INDEX)
- Set bit BIT_INDEX in ROP.
-
- -- Function: void mpz_clrbit (mpz_t ROP, mp_bitcnt_t BIT_INDEX)
- Clear bit BIT_INDEX in ROP.
-
- -- Function: void mpz_combit (mpz_t ROP, mp_bitcnt_t BIT_INDEX)
- Complement bit BIT_INDEX in ROP.
-
- -- Function: int mpz_tstbit (mpz_t OP, mp_bitcnt_t BIT_INDEX)
- Test bit BIT_INDEX in OP and return 0 or 1 accordingly.
-
-
-File: mpir.info, Node: I/O of Integers, Next: Integer Random Numbers, Prev: Integer Logic and Bit Fiddling, Up: Integer Functions
-
-5.12 Input and Output Functions
-===============================
-
-Functions that perform input from a stdio stream, and functions that
-output to a stdio stream. Passing a `NULL' pointer for a STREAM
-argument to any of these functions will make them read from `stdin' and
-write to `stdout', respectively.
-
- When using any of these functions, it is a good idea to include
-`stdio.h' before `mpir.h', since that will allow `mpir.h' to define
-prototypes for these functions.
-
- -- Function: size_t mpz_out_str (FILE *STREAM, int BASE, mpz_t OP)
- Output OP on stdio stream STREAM, as a string of digits in base
- BASE. The base argument may vary from 2 to 62 or from -2 to -36.
-
- For BASE in the range 2..36, digits and lower-case letters are
- used; for -2..-36, digits and upper-case letters are used; for
- 37..62, digits, upper-case letters, and lower-case letters (in
- that significance order) are used.
-
- Return the number of bytes written, or if an error occurred,
- return 0.
-
- -- Function: size_t mpz_inp_str (mpz_t ROP, FILE *STREAM, int BASE)
- Input a possibly white-space preceded string in base BASE from
- stdio stream STREAM, and put the read integer in ROP.
-
- The BASE may vary from 2 to 62, or if BASE is 0, then the leading
- characters are used: `0x' and `0X' for hexadecimal, `0b' and `0B'
- for binary, `0' for octal, or decimal otherwise.
-
- For bases up to 36, case is ignored; upper-case and lower-case
- letters have the same value. For bases 37 to 62, upper-case
- letter represent the usual 10..35 while lower-case letter
- represent 36..61.
-
- Return the number of bytes read, or if an error occurred, return 0.
-
- -- Function: size_t mpz_out_raw (FILE *STREAM, mpz_t OP)
- Output OP on stdio stream STREAM, in raw binary format. The
- integer is written in a portable format, with 4 bytes of size
- information, and that many bytes of limbs. Both the size and the
- limbs are written in decreasing significance order (i.e., in
- big-endian).
-
- The output can be read with `mpz_inp_raw'.
-
- Return the number of bytes written, or if an error occurred,
- return 0.
-
- The output of this can not be read by `mpz_inp_raw' from GMP 1,
- because of changes necessary for compatibility between 32-bit and
- 64-bit machines.
-
- -- Function: size_t mpz_inp_raw (mpz_t ROP, FILE *STREAM)
- Input from stdio stream STREAM in the format written by
- `mpz_out_raw', and put the result in ROP. Return the number of
- bytes read, or if an error occurred, return 0.
-
- This routine can read the output from `mpz_out_raw' also from GMP
- 1, in spite of changes necessary for compatibility between 32-bit
- and 64-bit machines.
-
-
-File: mpir.info, Node: Integer Random Numbers, Next: Integer Import and Export, Prev: I/O of Integers, Up: Integer Functions
-
-5.13 Random Number Functions
-============================
-
-The random number functions of MPIR come in two groups; older function
-that rely on a global state, and newer functions that accept a state
-parameter that is read and modified. Please see the *note Random
-Number Functions:: for more information on how to use and not to use
-random number functions.
-
- -- Function: void mpz_urandomb (mpz_t ROP, gmp_randstate_t STATE,
- mp_bitcnt_t N)
- Generate a uniformly distributed random integer in the range 0 to
- 2^N-1, inclusive.
-
- The variable STATE must be initialized by calling one of the
- `gmp_randinit' functions (*note Random State Initialization::)
- before invoking this function.
-
- -- Function: void mpz_urandomm (mpz_t ROP, gmp_randstate_t STATE,
- mpz_t N)
- Generate a uniform random integer in the range 0 to N-1, inclusive.
-
- The variable STATE must be initialized by calling one of the
- `gmp_randinit' functions (*note Random State Initialization::)
- before invoking this function.
-
- -- Function: void mpz_rrandomb (mpz_t ROP, gmp_randstate_t STATE,
- mp_bitcnt_t N)
- Generate a random integer with long strings of zeros and ones in
- the binary representation. Useful for testing functions and
- algorithms, since this kind of random numbers have proven to be
- more likely to trigger corner-case bugs. The random number will
- be in the range 0 to 2^N-1, inclusive.
-
- The variable STATE must be initialized by calling one of the
- `gmp_randinit' functions (*note Random State Initialization::)
- before invoking this function.
-
-
-File: mpir.info, Node: Integer Import and Export, Next: Miscellaneous Integer Functions, Prev: Integer Random Numbers, Up: Integer Functions
-
-5.14 Integer Import and Export
-==============================
-
-`mpz_t' variables can be converted to and from arbitrary words of binary
-data with the following functions.
-
- -- Function: void mpz_import (mpz_t ROP, size_t COUNT, int ORDER,
- size_t SIZE, int ENDIAN, size_t NAILS, const void *OP)
- Set ROP from an array of word data at OP.
-
- The parameters specify the format of the data. COUNT many words
- are read, each SIZE bytes. ORDER can be 1 for most significant
- word first or -1 for least significant first. Within each word
- ENDIAN can be 1 for most significant byte first, -1 for least
- significant first, or 0 for the native endianness of the host CPU.
- The most significant NAILS bits of each word are skipped, this can
- be 0 to use the full words.
-
- There is no sign taken from the data, ROP will simply be a positive
- integer. An application can handle any sign itself, and apply it
- for instance with `mpz_neg'.
-
- There are no data alignment restrictions on OP, any address is
- allowed.
-
- Here's an example converting an array of `mpir_ui' data, most
- significant element first, and host byte order within each value.
-
- mpir_ui a[20];
- mpz_t z;
- mpz_import (z, 20, 1, sizeof(a[0]), 0, 0, a);
-
- This example assumes the full `sizeof' bytes are used for data in
- the given type, which is usually true, and certainly true for
- `mpir_ui' everywhere we know of. However on Cray vector systems
- it may be noted that `short' and `int' are always stored in 8
- bytes (and with `sizeof' indicating that) but use only 32 or 46
- bits. The NAILS feature can account for this, by passing for
- instance `8*sizeof(int)-INT_BIT'.
-
- -- Function: void * mpz_export (void *ROP, size_t *COUNTP, int ORDER,
- size_t SIZE, int ENDIAN, size_t NAILS, mpz_t OP)
- Fill ROP with word data from OP.
-
- The parameters specify the format of the data produced. Each word
- will be SIZE bytes and ORDER can be 1 for most significant word
- first or -1 for least significant first. Within each word ENDIAN
- can be 1 for most significant byte first, -1 for least significant
- first, or 0 for the native endianness of the host CPU. The most
- significant NAILS bits of each word are unused and set to zero,
- this can be 0 to produce full words.
-
- The number of words produced is written to `*COUNTP', or COUNTP
- can be `NULL' to discard the count. ROP must have enough space
- for the data, or if ROP is `NULL' then a result array of the
- necessary size is allocated using the current MPIR allocation
- function (*note Custom Allocation::). In either case the return
- value is the destination used, either ROP or the allocated block.
-
- If OP is non-zero then the most significant word produced will be
- non-zero. If OP is zero then the count returned will be zero and
- nothing written to ROP. If ROP is `NULL' in this case, no block
- is allocated, just `NULL' is returned.
-
- The sign of OP is ignored, just the absolute value is exported. An
- application can use `mpz_sgn' to get the sign and handle it as
- desired. (*note Integer Comparisons::)
-
- There are no data alignment restrictions on ROP, any address is
- allowed.
-
- When an application is allocating space itself the required size
- can be determined with a calculation like the following. Since
- `mpz_sizeinbase' always returns at least 1, `count' here will be
- at least one, which avoids any portability problems with
- `malloc(0)', though if `z' is zero no space at all is actually
- needed (or written).
-
- numb = 8*size - nail;
- count = (mpz_sizeinbase (z, 2) + numb-1) / numb;
- p = malloc (count * size);
-
-
-File: mpir.info, Node: Miscellaneous Integer Functions, Next: Integer Special Functions, Prev: Integer Import and Export, Up: Integer Functions
-
-5.15 Miscellaneous Functions
-============================
-
- -- Function: int mpz_fits_ulong_p (mpz_t OP)
- -- Function: int mpz_fits_slong_p (mpz_t OP)
- -- Function: int mpz_fits_uint_p (mpz_t OP)
- -- Function: int mpz_fits_sint_p (mpz_t OP)
- -- Function: int mpz_fits_ushort_p (mpz_t OP)
- -- Function: int mpz_fits_sshort_p (mpz_t OP)
- Return non-zero iff the value of OP fits in an `unsigned long',
- `long', `unsigned int', `signed int', `unsigned short int', or
- `signed short int', respectively. Otherwise, return zero.
-
- -- Macro: int mpz_odd_p (mpz_t OP)
- -- Macro: int mpz_even_p (mpz_t OP)
- Determine whether OP is odd or even, respectively. Return
- non-zero if yes, zero if no. These macros evaluate their argument
- more than once.
-
- -- Function: size_t mpz_sizeinbase (mpz_t OP, int BASE)
- Return the size of OP measured in number of digits in the given
- BASE. BASE can vary from 2 to 36. The sign of OP is ignored,
- just the absolute value is used. The result will be either exact
- or 1 too big. If BASE is a power of 2, the result is always
- exact. If OP is zero the return value is always 1.
-
- This function can be used to determine the space required when
- converting OP to a string. The right amount of allocation is
- normally two more than the value returned by `mpz_sizeinbase', one
- extra for a minus sign and one for the null-terminator.
-
- It will be noted that `mpz_sizeinbase(OP,2)' can be used to locate
- the most significant 1 bit in OP, counting from 1. (Unlike the
- bitwise functions which start from 0, *Note Logical and Bit
- Manipulation Functions: Integer Logic and Bit Fiddling.)
-
-
-File: mpir.info, Node: Integer Special Functions, Prev: Miscellaneous Integer Functions, Up: Integer Functions
-
-5.16 Special Functions
-======================
-
-The functions in this section are for various special purposes. Most
-applications will not need them.
-
- -- Function: void mpz_array_init (mpz_t INTEGER_ARRAY, size_t
- ARRAY_SIZE, mp_size_t FIXED_NUM_BITS)
- This is a special type of initialization. *Fixed* space of
- FIXED_NUM_BITS is allocated to each of the ARRAY_SIZE integers in
- INTEGER_ARRAY. There is no way to free the storage allocated by
- this function. Don't call `mpz_clear'!
-
- The INTEGER_ARRAY parameter is the first `mpz_t' in the array. For
- example,
-
- mpz_t arr[20000];
- mpz_array_init (arr[0], 20000, 512);
-
- This function is only intended for programs that create a large
- number of integers and need to reduce memory usage by avoiding the
- overheads of allocating and reallocating lots of small blocks. In
- normal programs this function is not recommended.
-
- The space allocated to each integer by this function will not be
- automatically increased, unlike the normal `mpz_init', so an
- application must ensure it is sufficient for any value stored.
- The following space requirements apply to various routines,
-
- * `mpz_abs', `mpz_neg', `mpz_set', `mpz_set_si' and
- `mpz_set_ui' need room for the value they store.
-
- * `mpz_add', `mpz_add_ui', `mpz_sub' and `mpz_sub_ui' need room
- for the larger of the two operands, plus an extra
- `mp_bits_per_limb'.
-
- * `mpz_mul', `mpz_mul_ui' and `mpz_mul_ui' need room for the sum
- of the number of bits in their operands, but each rounded up
- to a multiple of `mp_bits_per_limb'.
-
- * `mpz_swap' can be used between two array variables, but not
- between an array and a normal variable.
-
- For other functions, or if in doubt, the suggestion is to
- calculate in a regular `mpz_init' variable and copy the result to
- an array variable with `mpz_set'.
-
- *This function is obsolete. It will disappear from future MPIR
- releases.*
-
- -- Function: void * _mpz_realloc (mpz_t INTEGER, mp_size_t NEW_ALLOC)
- Change the space for INTEGER to NEW_ALLOC limbs. The value in
- INTEGER is preserved if it fits, or is set to 0 if not. The return
- value is not useful to applications and should be ignored.
-
- `mpz_realloc2' is the preferred way to accomplish allocation
- changes like this. `mpz_realloc2' and `_mpz_realloc' are the same
- except that `_mpz_realloc' takes its size in limbs.
-
- -- Function: mp_limb_t mpz_getlimbn (mpz_t OP, mp_size_t N)
- Return limb number N from OP. The sign of OP is ignored, just the
- absolute value is used. The least significant limb is number 0.
-
- `mpz_size' can be used to find how many limbs make up OP.
- `mpz_getlimbn' returns zero if N is outside the range 0 to
- `mpz_size(OP)-1'.
-
- -- Function: size_t mpz_size (mpz_t OP)
- Return the size of OP measured in number of limbs. If OP is zero,
- the returned value will be zero.
-
-
-File: mpir.info, Node: Rational Number Functions, Next: Floating-point Functions, Prev: Integer Functions, Up: Top
-
-6 Rational Number Functions
-***************************
-
-This chapter describes the MPIR functions for performing arithmetic on
-rational numbers. These functions start with the prefix `mpq_'.
-
- Rational numbers are stored in objects of type `mpq_t'.
-
- All rational arithmetic functions assume operands have a canonical
-form, and canonicalize their result. The canonical from means that the
-denominator and the numerator have no common factors, and that the
-denominator is positive. Zero has the unique representation 0/1.
-
- Pure assignment functions do not canonicalize the assigned variable.
-It is the responsibility of the user to canonicalize the assigned
-variable before any arithmetic operations are performed on that
-variable.
-
- -- Function: void mpq_canonicalize (mpq_t OP)
- Remove any factors that are common to the numerator and
- denominator of OP, and make the denominator positive.
-
-* Menu:
-
-* Initializing Rationals::
-* Rational Conversions::
-* Rational Arithmetic::
-* Comparing Rationals::
-* Applying Integer Functions::
-* I/O of Rationals::
-
-
-File: mpir.info, Node: Initializing Rationals, Next: Rational Conversions, Prev: Rational Number Functions, Up: Rational Number Functions
-
-6.1 Initialization and Assignment Functions
-===========================================
-
- -- Function: void mpq_init (mpq_t DEST_RATIONAL)
- Initialize DEST_RATIONAL and set it to 0/1. Each variable should
- normally only be initialized once, or at least cleared out (using
- the function `mpq_clear') between each initialization.
-
- -- Function: void mpq_inits (mpq_t X, ...)
- Initialize a NULL-terminated list of `mpq_t' variables, and set
- their values to 0/1.
-
- -- Function: void mpq_clear (mpq_t RATIONAL_NUMBER)
- Free the space occupied by RATIONAL_NUMBER. Make sure to call this
- function for all `mpq_t' variables when you are done with them.
-
- -- Function: void mpq_clears (mpq_t X, ...)
- Free the space occupied by a NULL-terminated list of `mpq_t'
- variables.
-
- -- Function: void mpq_set (mpq_t ROP, mpq_t OP)
- -- Function: void mpq_set_z (mpq_t ROP, mpz_t OP)
- Assign ROP from OP.
-
- -- Function: void mpq_set_ui (mpq_t ROP, mpir_ui OP1, mpir_ui OP2)
- -- Function: void mpq_set_si (mpq_t ROP, mpir_si OP1, mpir_ui OP2)
- Set the value of ROP to OP1/OP2. Note that if OP1 and OP2 have
- common factors, ROP has to be passed to `mpq_canonicalize' before
- any operations are performed on ROP.
-
- -- Function: int mpq_set_str (mpq_t ROP, char *STR, int BASE)
- Set ROP from a null-terminated string STR in the given BASE.
-
- The string can be an integer like "41" or a fraction like
- "41/152". The fraction must be in canonical form (*note Rational
- Number Functions::), or if not then `mpq_canonicalize' must be
- called.
-
- The numerator and optional denominator are parsed the same as in
- `mpz_set_str' (*note Assigning Integers::). White space is
- allowed in the string, and is simply ignored. The BASE can vary
- from 2 to 62, or if BASE is 0 then the leading characters are
- used: `0x' or `0X' for hex, `0b' or `0B' for binary, `0' for
- octal, or decimal otherwise. Note that this is done separately
- for the numerator and denominator, so for instance `0xEF/100' is
- 239/100, whereas `0xEF/0x100' is 239/256.
-
- The return value is 0 if the entire string is a valid number, or
- -1 if not.
-
- -- Function: void mpq_swap (mpq_t ROP1, mpq_t ROP2)
- Swap the values ROP1 and ROP2 efficiently.
-
-
-File: mpir.info, Node: Rational Conversions, Next: Rational Arithmetic, Prev: Initializing Rationals, Up: Rational Number Functions
-
-6.2 Conversion Functions
-========================
-
- -- Function: double mpq_get_d (mpq_t OP)
- Convert OP to a `double', truncating if necessary (ie. rounding
- towards zero).
-
- If the exponent from the conversion is too big or too small to fit
- a `double' then the result is system dependent. For too big an
- infinity is returned when available. For too small 0.0 is
- normally returned. Hardware overflow, underflow and denorm traps
- may or may not occur.
-
- -- Function: void mpq_set_d (mpq_t ROP, double OP)
- -- Function: void mpq_set_f (mpq_t ROP, mpf_t OP)
- Set ROP to the value of OP. There is no rounding, this conversion
- is exact.
-
- -- Function: char * mpq_get_str (char *STR, int BASE, mpq_t OP)
- Convert OP to a string of digits in base BASE. The base may vary
- from 2 to 36. The string will be of the form `num/den', or if the
- denominator is 1 then just `num'.
-
- If STR is `NULL', the result string is allocated using the current
- allocation function (*note Custom Allocation::). The block will be
- `strlen(str)+1' bytes, that being exactly enough for the string and
- null-terminator.
-
- If STR is not `NULL', it should point to a block of storage large
- enough for the result, that being
-
- mpz_sizeinbase (mpq_numref(OP), BASE)
- + mpz_sizeinbase (mpq_denref(OP), BASE) + 3
-
- The three extra bytes are for a possible minus sign, possible
- slash, and the null-terminator.
-
- A pointer to the result string is returned, being either the
- allocated block, or the given STR.
-
-
-File: mpir.info, Node: Rational Arithmetic, Next: Comparing Rationals, Prev: Rational Conversions, Up: Rational Number Functions
-
-6.3 Arithmetic Functions
-========================
-
- -- Function: void mpq_add (mpq_t SUM, mpq_t ADDEND1, mpq_t ADDEND2)
- Set SUM to ADDEND1 + ADDEND2.
-
- -- Function: void mpq_sub (mpq_t DIFFERENCE, mpq_t MINUEND, mpq_t
- SUBTRAHEND)
- Set DIFFERENCE to MINUEND - SUBTRAHEND.
-
- -- Function: void mpq_mul (mpq_t PRODUCT, mpq_t MULTIPLIER, mpq_t
- MULTIPLICAND)
- Set PRODUCT to MULTIPLIER times MULTIPLICAND.
-
- -- Function: void mpq_mul_2exp (mpq_t ROP, mpq_t OP1, mp_bitcnt_t OP2)
- Set ROP to OP1 times 2 raised to OP2.
-
- -- Function: void mpq_div (mpq_t QUOTIENT, mpq_t DIVIDEND, mpq_t
- DIVISOR)
- Set QUOTIENT to DIVIDEND/DIVISOR.
-
- -- Function: void mpq_div_2exp (mpq_t ROP, mpq_t OP1, mp_bitcnt_t OP2)
- Set ROP to OP1 divided by 2 raised to OP2.
-
- -- Function: void mpq_neg (mpq_t NEGATED_OPERAND, mpq_t OPERAND)
- Set NEGATED_OPERAND to -OPERAND.
-
- -- Function: void mpq_abs (mpq_t ROP, mpq_t OP)
- Set ROP to the absolute value of OP.
-
- -- Function: void mpq_inv (mpq_t INVERTED_NUMBER, mpq_t NUMBER)
- Set INVERTED_NUMBER to 1/NUMBER. If the new denominator is zero,
- this routine will divide by zero.
-
-
-File: mpir.info, Node: Comparing Rationals, Next: Applying Integer Functions, Prev: Rational Arithmetic, Up: Rational Number Functions
-
-6.4 Comparison Functions
-========================
-
- -- Function: int mpq_cmp (mpq_t OP1, mpq_t OP2)
- Compare OP1 and OP2. Return a positive value if OP1 > OP2, zero
- if OP1 = OP2, and a negative value if OP1 < OP2.
-
- To determine if two rationals are equal, `mpq_equal' is faster than
- `mpq_cmp'.
-
- -- Macro: int mpq_cmp_ui (mpq_t OP1, mpir_ui NUM2, mpir_ui DEN2)
- -- Macro: int mpq_cmp_si (mpq_t OP1, mpir_si NUM2, mpir_ui DEN2)
- Compare OP1 and NUM2/DEN2. Return a positive value if OP1 >
- NUM2/DEN2, zero if OP1 = NUM2/DEN2, and a negative value if OP1 <
- NUM2/DEN2.
-
- NUM2 and DEN2 are allowed to have common factors.
-
- These functions are implemented as a macros and evaluate their
- arguments multiple times.
-
- -- Macro: int mpq_sgn (mpq_t OP)
- Return +1 if OP > 0, 0 if OP = 0, and -1 if OP < 0.
-
- This function is actually implemented as a macro. It evaluates its
- arguments multiple times.
-
- -- Function: int mpq_equal (mpq_t OP1, mpq_t OP2)
- Return non-zero if OP1 and OP2 are equal, zero if they are
- non-equal. Although `mpq_cmp' can be used for the same purpose,
- this function is much faster.
-
-
-File: mpir.info, Node: Applying Integer Functions, Next: I/O of Rationals, Prev: Comparing Rationals, Up: Rational Number Functions
-
-6.5 Applying Integer Functions to Rationals
-===========================================
-
-The set of `mpq' functions is quite small. In particular, there are few
-functions for either input or output. The following functions give
-direct access to the numerator and denominator of an `mpq_t'.
-
- Note that if an assignment to the numerator and/or denominator could
-take an `mpq_t' out of the canonical form described at the start of
-this chapter (*note Rational Number Functions::) then
-`mpq_canonicalize' must be called before any other `mpq' functions are
-applied to that `mpq_t'.
-
- -- Macro: mpz_t mpq_numref (mpq_t OP)
- -- Macro: mpz_t mpq_denref (mpq_t OP)
- Return a reference to the numerator and denominator of OP,
- respectively. The `mpz' functions can be used on the result of
- these macros.
-
- -- Function: void mpq_get_num (mpz_t NUMERATOR, mpq_t RATIONAL)
- -- Function: void mpq_get_den (mpz_t DENOMINATOR, mpq_t RATIONAL)
- -- Function: void mpq_set_num (mpq_t RATIONAL, mpz_t NUMERATOR)
- -- Function: void mpq_set_den (mpq_t RATIONAL, mpz_t DENOMINATOR)
- Get or set the numerator or denominator of a rational. These
- functions are equivalent to calling `mpz_set' with an appropriate
- `mpq_numref' or `mpq_denref'. Direct use of `mpq_numref' or
- `mpq_denref' is recommended instead of these functions.
-
-
-File: mpir.info, Node: I/O of Rationals, Prev: Applying Integer Functions, Up: Rational Number Functions
-
-6.6 Input and Output Functions
-==============================
-
-When using any of these functions, it's a good idea to include `stdio.h'
-before `mpir.h', since that will allow `mpir.h' to define prototypes
-for these functions.
-
- Passing a `NULL' pointer for a STREAM argument to any of these
-functions will make them read from `stdin' and write to `stdout',
-respectively.
-
- -- Function: size_t mpq_out_str (FILE *STREAM, int BASE, mpq_t OP)
- Output OP on stdio stream STREAM, as a string of digits in base
- BASE. The base may vary from 2 to 36. Output is in the form
- `num/den' or if the denominator is 1 then just `num'.
-
- Return the number of bytes written, or if an error occurred,
- return 0.
-
- -- Function: size_t mpq_inp_str (mpq_t ROP, FILE *STREAM, int BASE)
- Read a string of digits from STREAM and convert them to a rational
- in ROP. Any initial white-space characters are read and
- discarded. Return the number of characters read (including white
- space), or 0 if a rational could not be read.
-
- The input can be a fraction like `17/63' or just an integer like
- `123'. Reading stops at the first character not in this form, and
- white space is not permitted within the string. If the input
- might not be in canonical form, then `mpq_canonicalize' must be
- called (*note Rational Number Functions::).
-
- The BASE can be between 2 and 36, or can be 0 in which case the
- leading characters of the string determine the base, `0x' or `0X'
- for hexadecimal, `0' for octal, or decimal otherwise. The leading
- characters are examined separately for the numerator and
- denominator of a fraction, so for instance `0x10/11' is 16/11,
- whereas `0x10/0x11' is 16/17.
-
-
-File: mpir.info, Node: Floating-point Functions, Next: Low-level Functions, Prev: Rational Number Functions, Up: Top
-
-7 Floating-point Functions
-**************************
-
-MPIR floating point numbers are stored in objects of type `mpf_t' and
-functions operating on them have an `mpf_' prefix.
-
- The mantissa of each float has a user-selectable precision, limited
-only by available memory. Each variable has its own precision, and
-that can be increased or decreased at any time.
-
- The exponent of each float is a fixed precision, one machine word on
-most systems. In the current implementation the exponent is a count of
-limbs, so for example on a 32-bit system this means a range of roughly
-2^-68719476768 to 2^68719476736, or on a 64-bit system this will be
-greater. Note however `mpf_get_str' can only return an exponent which
-fits an `mp_exp_t' and currently `mpf_set_str' doesn't accept exponents
-bigger than a `mpir_si'.
-
- Each variable keeps a size for the mantissa data actually in use.
-This means that if a float is exactly represented in only a few bits
-then only those bits will be used in a calculation, even if the
-selected precision is high.
-
- All calculations are performed to the precision of the destination
-variable. Each function is defined to calculate with "infinite
-precision" followed by a truncation to the destination precision, but
-of course the work done is only what's needed to determine a result
-under that definition.
-
- The precision selected for a variable is a minimum value, MPIR may
-increase it a little to facilitate efficient calculation. Currently
-this means rounding up to a whole limb, and then sometimes having a
-further partial limb, depending on the high limb of the mantissa. But
-applications shouldn't be concerned by such details.
-
- The mantissa in stored in binary, as might be imagined from the fact
-precisions are expressed in bits. One consequence of this is that
-decimal fractions like 0.1 cannot be represented exactly. The same is
-true of plain IEEE `double' floats. This makes both highly unsuitable
-for calculations involving money or other values that should be exact
-decimal fractions. (Suitably scaled integers, or perhaps rationals,
-are better choices.)
-
- `mpf' functions and variables have no special notion of infinity or
-not-a-number, and applications must take care not to overflow the
-exponent or results will be unpredictable. This might change in a
-future release.
-
- Note that the `mpf' functions are _not_ intended as a smooth
-extension to IEEE P754 arithmetic. In particular results obtained on
-one computer often differ from the results on a computer with a
-different word size.
-
-* Menu:
-
-* Initializing Floats::
-* Assigning Floats::
-* Simultaneous Float Init & Assign::
-* Converting Floats::
-* Float Arithmetic::
-* Float Comparison::
-* I/O of Floats::
-* Miscellaneous Float Functions::
-
-
-File: mpir.info, Node: Initializing Floats, Next: Assigning Floats, Prev: Floating-point Functions, Up: Floating-point Functions
-
-7.1 Initialization Functions
-============================
-
- -- Function: void mpf_set_default_prec (mp_bitcnt_t PREC)
- Set the default precision to be *at least* PREC bits. All
- subsequent calls to `mpf_init' will use this precision, but
- previously initialized variables are unaffected.
-
- -- Function: mp_bitcnt_t mpf_get_default_prec (void)
- Return the default precision actually used.
-
- An `mpf_t' object must be initialized before storing the first value
-in it. The functions `mpf_init' and `mpf_init2' are used for that
-purpose.
-
- -- Function: void mpf_init (mpf_t X)
- Initialize X to 0. Normally, a variable should be initialized
- once only or at least be cleared, using `mpf_clear', between
- initializations. The precision of X is undefined unless a default
- precision has already been established by a call to
- `mpf_set_default_prec'.
-
- -- Function: void mpf_init2 (mpf_t X, mp_bitcnt_t PREC)
- Initialize X to 0 and set its precision to be *at least* PREC
- bits. Normally, a variable should be initialized once only or at
- least be cleared, using `mpf_clear', between initializations.
-
- -- Function: void mpf_inits (mpf_t X, ...)
- Initialize a NULL-terminated list of `mpf_t' variables, and set
- their values to 0. The precision of the initialized variables is
- undefined unless a default precision has already been established
- by a call to `mpf_set_default_prec'.
-
- -- Function: void mpf_clear (mpf_t X)
- Free the space occupied by X. Make sure to call this function for
- all `mpf_t' variables when you are done with them.
-
- -- Function: void mpf_clears (mpf_t X, ...)
- Free the space occupied by a NULL-terminated list of `mpf_t'
- variables.
-
- Here is an example on how to initialize floating-point variables:
- {
- mpf_t x, y;
- mpf_init (x); /* use default precision */
- mpf_init2 (y, 256); /* precision _at least_ 256 bits */
- ...
- /* Unless the program is about to exit, do ... */
- mpf_clear (x);
- mpf_clear (y);
- }
-
- The following three functions are useful for changing the precision
-during a calculation. A typical use would be for adjusting the
-precision gradually in iterative algorithms like Newton-Raphson, making
-the computation precision closely match the actual accurate part of the
-numbers.
-
- -- Function: mp_bitcnt_t mpf_get_prec (mpf_t OP)
- Return the current precision of OP, in bits.
-
- -- Function: void mpf_set_prec (mpf_t ROP, mp_bitcnt_t PREC)
- Set the precision of ROP to be *at least* PREC bits. The value in
- ROP will be truncated to the new precision.
-
- This function requires a call to `realloc', and so should not be
- used in a tight loop.
-
- -- Function: void mpf_set_prec_raw (mpf_t ROP, mp_bitcnt_t PREC)
- Set the precision of ROP to be *at least* PREC bits, without
- changing the memory allocated.
-
- PREC must be no more than the allocated precision for ROP, that
- being the precision when ROP was initialized, or in the most recent
- `mpf_set_prec'.
-
- The value in ROP is unchanged, and in particular if it had a higher
- precision than PREC it will retain that higher precision. New
- values written to ROP will use the new PREC.
-
- Before calling `mpf_clear' or the full `mpf_set_prec', another
- `mpf_set_prec_raw' call must be made to restore ROP to its original
- allocated precision. Failing to do so will have unpredictable
- results.
-
- `mpf_get_prec' can be used before `mpf_set_prec_raw' to get the
- original allocated precision. After `mpf_set_prec_raw' it
- reflects the PREC value set.
-
- `mpf_set_prec_raw' is an efficient way to use an `mpf_t' variable
- at different precisions during a calculation, perhaps to gradually
- increase precision in an iteration, or just to use various
- different precisions for different purposes during a calculation.
-
-
-File: mpir.info, Node: Assigning Floats, Next: Simultaneous Float Init & Assign, Prev: Initializing Floats, Up: Floating-point Functions
-
-7.2 Assignment Functions
-========================
-
-These functions assign new values to already initialized floats (*note
-Initializing Floats::).
-
- -- Function: void mpf_set (mpf_t ROP, mpf_t OP)
- -- Function: void mpf_set_ui (mpf_t ROP, mpir_ui OP)
- -- Function: void mpf_set_si (mpf_t ROP, mpir_si OP)
- -- Function: void mpf_set_d (mpf_t ROP, double OP)
- -- Function: void mpf_set_z (mpf_t ROP, mpz_t OP)
- -- Function: void mpf_set_q (mpf_t ROP, mpq_t OP)
- Set the value of ROP from OP.
-
- -- Function: int mpf_set_str (mpf_t ROP, char *STR, int BASE)
- Set the value of ROP from the string in STR. The string is of the
- form `M at N' or, if the base is 10 or less, alternatively `MeN'.
- `M' is the mantissa and `N' is the exponent. The mantissa is
- always in the specified base. The exponent is either in the
- specified base or, if BASE is negative, in decimal. The decimal
- point expected is taken from the current locale, on systems
- providing `localeconv'.
-
- The argument BASE may be in the ranges 2 to 62, or -62 to -2.
- Negative values are used to specify that the exponent is in
- decimal.
-
- For bases up to 36, case is ignored; upper-case and lower-case
- letters have the same value; for bases 37 to 62, upper-case letter
- represent the usual 10..35 while lower-case letter represent
- 36..61.
-
- Unlike the corresponding `mpz' function, the base will not be
- determined from the leading characters of the string if BASE is 0.
- This is so that numbers like `0.23' are not interpreted as octal.
-
- White space is allowed in the string, and is simply ignored.
- [This is not really true; white-space is ignored in the beginning
- of the string and within the mantissa, but not in other places,
- such as after a minus sign or in the exponent. We are considering
- changing the definition of this function, making it fail when
- there is any white-space in the input, since that makes a lot of
- sense. Please tell us your opinion about this change. Do you
- really want it to accept "3 14" as meaning 314 as it does now?]
-
- This function returns 0 if the entire string is a valid number in
- base BASE. Otherwise it returns -1.
-
- -- Function: void mpf_swap (mpf_t ROP1, mpf_t ROP2)
- Swap ROP1 and ROP2 efficiently. Both the values and the
- precisions of the two variables are swapped.
-
-
-File: mpir.info, Node: Simultaneous Float Init & Assign, Next: Converting Floats, Prev: Assigning Floats, Up: Floating-point Functions
-
-7.3 Combined Initialization and Assignment Functions
-====================================================
-
-For convenience, MPIR provides a parallel series of initialize-and-set
-functions which initialize the output and then store the value there.
-These functions' names have the form `mpf_init_set...'
-
- Once the float has been initialized by any of the `mpf_init_set...'
-functions, it can be used as the source or destination operand for the
-ordinary float functions. Don't use an initialize-and-set function on
-a variable already initialized!
-
- -- Function: void mpf_init_set (mpf_t ROP, mpf_t OP)
- -- Function: void mpf_init_set_ui (mpf_t ROP, mpir_ui OP)
- -- Function: void mpf_init_set_si (mpf_t ROP, mpir_si OP)
- -- Function: void mpf_init_set_d (mpf_t ROP, double OP)
- Initialize ROP and set its value from OP.
-
- The precision of ROP will be taken from the active default
- precision, as set by `mpf_set_default_prec'.
-
- -- Function: int mpf_init_set_str (mpf_t ROP, char *STR, int BASE)
- Initialize ROP and set its value from the string in STR. See
- `mpf_set_str' above for details on the assignment operation.
-
- Note that ROP is initialized even if an error occurs. (I.e., you
- have to call `mpf_clear' for it.)
-
- The precision of ROP will be taken from the active default
- precision, as set by `mpf_set_default_prec'.
-
-
-File: mpir.info, Node: Converting Floats, Next: Float Arithmetic, Prev: Simultaneous Float Init & Assign, Up: Floating-point Functions
-
-7.4 Conversion Functions
-========================
-
- -- Function: double mpf_get_d (mpf_t OP)
- Convert OP to a `double', truncating if necessary (ie. rounding
- towards zero).
-
- If the exponent in OP is too big or too small to fit a `double'
- then the result is system dependent. For too big an infinity is
- returned when available. For too small 0.0 is normally returned.
- Hardware overflow, underflow and denorm traps may or may not occur.
-
- -- Function: double mpf_get_d_2exp (mpir_si *EXP, mpf_t OP)
- Convert OP to a `double', truncating if necessary (ie. rounding
- towards zero), and with an exponent returned separately.
-
- The return value is in the range 0.5<=abs(D)<1 and the exponent is
- stored to `*EXP'. D * 2^EXP is the (truncated) OP value. If OP
- is zero, the return is 0.0 and 0 is stored to `*EXP'.
-
- This is similar to the standard C `frexp' function (*note
- Normalization Functions: (libc)Normalization Functions.).
-
- -- Function: mpir_si mpf_get_si (mpf_t OP)
- -- Function: mpir_ui mpf_get_ui (mpf_t OP)
- Convert OP to a `mpir_si' or `mpir_ui', truncating any fraction
- part. If OP is too big for the return type, the result is
- undefined.
-
- See also `mpf_fits_slong_p' and `mpf_fits_ulong_p' (*note
- Miscellaneous Float Functions::).
-
- -- Function: char * mpf_get_str (char *STR, mp_exp_t *EXPPTR, int
- BASE, size_t N_DIGITS, mpf_t OP)
- Convert OP to a string of digits in base BASE. BASE can vary from
- 2 to 362 or from -2 to -36. Up to N_DIGITS digits will be
- generated. Trailing zeros are not returned. No more digits than
- can be accurately represented by OP are ever generated. If
- N_DIGITS is 0 then that accurate maximum number of digits are
- generated.
-
- For BASE in the range 2..36, digits and lower-case letters are
- used; for -2..-36, digits and upper-case letters are used; for
- 37..62, digits, upper-case letters, and lower-case letters (in
- that significance order) are used.
-
- If STR is `NULL', the result string is allocated using the current
- allocation function (*note Custom Allocation::). The block will be
- `strlen(str)+1' bytes, that being exactly enough for the string and
- null-terminator.
-
- If STR is not `NULL', it should point to a block of N_DIGITS + 2
- bytes, that being enough for the mantissa, a possible minus sign,
- and a null-terminator. When N_DIGITS is 0 to get all significant
- digits, an application won't be able to know the space required,
- and STR should be `NULL' in that case.
-
- The generated string is a fraction, with an implicit radix point
- immediately to the left of the first digit. The applicable
- exponent is written through the EXPPTR pointer. For example, the
- number 3.1416 would be returned as string "31416" and exponent 1.
-
- When OP is zero, an empty string is produced and the exponent
- returned is 0.
-
- A pointer to the result string is returned, being either the
- allocated block or the given STR.
-
-
-File: mpir.info, Node: Float Arithmetic, Next: Float Comparison, Prev: Converting Floats, Up: Floating-point Functions
-
-7.5 Arithmetic Functions
-========================
-
- -- Function: void mpf_add (mpf_t ROP, mpf_t OP1, mpf_t OP2)
- -- Function: void mpf_add_ui (mpf_t ROP, mpf_t OP1, mpir_ui OP2)
- Set ROP to OP1 + OP2.
-
- -- Function: void mpf_sub (mpf_t ROP, mpf_t OP1, mpf_t OP2)
- -- Function: void mpf_ui_sub (mpf_t ROP, mpir_ui OP1, mpf_t OP2)
- -- Function: void mpf_sub_ui (mpf_t ROP, mpf_t OP1, mpir_ui OP2)
- Set ROP to OP1 - OP2.
-
- -- Function: void mpf_mul (mpf_t ROP, mpf_t OP1, mpf_t OP2)
- -- Function: void mpf_mul_ui (mpf_t ROP, mpf_t OP1, mpir_ui OP2)
- Set ROP to OP1 times OP2.
-
- Division is undefined if the divisor is zero, and passing a zero
-divisor to the divide functions will make these functions intentionally
-divide by zero. This lets the user handle arithmetic exceptions in
-these functions in the same manner as other arithmetic exceptions.
-
- -- Function: void mpf_div (mpf_t ROP, mpf_t OP1, mpf_t OP2)
- -- Function: void mpf_ui_div (mpf_t ROP, mpir_ui OP1, mpf_t OP2)
- -- Function: void mpf_div_ui (mpf_t ROP, mpf_t OP1, mpir_ui OP2)
- Set ROP to OP1/OP2.
-
- -- Function: void mpf_sqrt (mpf_t ROP, mpf_t OP)
- -- Function: void mpf_sqrt_ui (mpf_t ROP, mpir_ui OP)
- Set ROP to the square root of OP.
-
- -- Function: void mpf_pow_ui (mpf_t ROP, mpf_t OP1, mpir_ui OP2)
- Set ROP to OP1 raised to the power OP2.
-
- -- Function: void mpf_neg (mpf_t ROP, mpf_t OP)
- Set ROP to -OP.
-
- -- Function: void mpf_abs (mpf_t ROP, mpf_t OP)
- Set ROP to the absolute value of OP.
-
- -- Function: void mpf_mul_2exp (mpf_t ROP, mpf_t OP1, mp_bitcnt_t OP2)
- Set ROP to OP1 times 2 raised to OP2.
-
- -- Function: void mpf_div_2exp (mpf_t ROP, mpf_t OP1, mp_bitcnt_t OP2)
- Set ROP to OP1 divided by 2 raised to OP2.
-
-
-File: mpir.info, Node: Float Comparison, Next: I/O of Floats, Prev: Float Arithmetic, Up: Floating-point Functions
-
-7.6 Comparison Functions
-========================
-
- -- Function: int mpf_cmp (mpf_t OP1, mpf_t OP2)
- -- Function: int mpf_cmp_d (mpf_t OP1, double OP2)
- -- Function: int mpf_cmp_ui (mpf_t OP1, mpir_ui OP2)
- -- Function: int mpf_cmp_si (mpf_t OP1, mpir_si OP2)
- Compare OP1 and OP2. Return a positive value if OP1 > OP2, zero
- if OP1 = OP2, and a negative value if OP1 < OP2.
-
- `mpf_cmp_d' can be called with an infinity, but results are
- undefined for a NaN.
-
- -- Function: int mpf_eq (mpf_t OP1, mpf_t OP2, mp_bitcnt_t op3)
- Return non-zero if the first OP3 bits of OP1 and OP2 are equal,
- zero otherwise. I.e., test if OP1 and OP2 are approximately equal.
-
- In the future values like 1000 and 0111 may be considered the same
- to 3 bits (on the basis that their difference is that small).
-
- -- Function: void mpf_reldiff (mpf_t ROP, mpf_t OP1, mpf_t OP2)
- Compute the relative difference between OP1 and OP2 and store the
- result in ROP. This is abs(OP1-OP2)/OP1.
-
- -- Macro: int mpf_sgn (mpf_t OP)
- Return +1 if OP > 0, 0 if OP = 0, and -1 if OP < 0.
-
- This function is actually implemented as a macro. It evaluates
- its arguments multiple times.
-
-
-File: mpir.info, Node: I/O of Floats, Next: Miscellaneous Float Functions, Prev: Float Comparison, Up: Floating-point Functions
-
-7.7 Input and Output Functions
-==============================
-
-Functions that perform input from a stdio stream, and functions that
-output to a stdio stream. Passing a `NULL' pointer for a STREAM
-argument to any of these functions will make them read from `stdin' and
-write to `stdout', respectively.
-
- When using any of these functions, it is a good idea to include
-`stdio.h' before `mpir.h', since that will allow `mpir.h' to define
-prototypes for these functions.
-
- -- Function: size_t mpf_out_str (FILE *STREAM, int BASE, size_t
- N_DIGITS, mpf_t OP)
- Print OP to STREAM, as a string of digits. Return the number of
- bytes written, or if an error occurred, return 0.
-
- The mantissa is prefixed with an `0.' and is in the given BASE,
- which may vary from 2 to 36. An exponent then printed, separated
- by an `e', or if BASE is greater than 10 then by an `@'. The
- exponent is always in decimal. The decimal point follows the
- current locale, on systems providing `localeconv'.
-
- For BASE in the range 2..36, digits and lower-case letters are
- used; for -2..-36, digits and upper-case letters are used; for
- 37..62, digits, upper-case letters, and lower-case letters (in
- that significance order) are used.
-
- Up to N_DIGITS will be printed from the mantissa, except that no
- more digits than are accurately representable by OP will be
- printed. N_DIGITS can be 0 to select that accurate maximum.
-
- -- Function: size_t mpf_inp_str (mpf_t ROP, FILE *STREAM, int BASE)
- Read a string in base BASE from STREAM, and put the read float in
- ROP. The string is of the form `M at N' or, if the base is 10 or
- less, alternatively `MeN'. `M' is the mantissa and `N' is the
- exponent. The mantissa is always in the specified base. The
- exponent is either in the specified base or, if BASE is negative,
- in decimal. The decimal point expected is taken from the current
- locale, on systems providing `localeconv'.
-
- The argument BASE may be in the ranges 2 to 36, or -36 to -2.
- Negative values are used to specify that the exponent is in
- decimal.
-
- Unlike the corresponding `mpz' function, the base will not be
- determined from the leading characters of the string if BASE is 0.
- This is so that numbers like `0.23' are not interpreted as octal.
-
- Return the number of bytes read, or if an error occurred, return 0.
-
-
-File: mpir.info, Node: Miscellaneous Float Functions, Prev: I/O of Floats, Up: Floating-point Functions
-
-7.8 Miscellaneous Functions
-===========================
-
- -- Function: void mpf_ceil (mpf_t ROP, mpf_t OP)
- -- Function: void mpf_floor (mpf_t ROP, mpf_t OP)
- -- Function: void mpf_trunc (mpf_t ROP, mpf_t OP)
- Set ROP to OP rounded to an integer. `mpf_ceil' rounds to the
- next higher integer, `mpf_floor' to the next lower, and `mpf_trunc'
- to the integer towards zero.
-
- -- Function: int mpf_integer_p (mpf_t OP)
- Return non-zero if OP is an integer.
-
- -- Function: int mpf_fits_ulong_p (mpf_t OP)
- -- Function: int mpf_fits_slong_p (mpf_t OP)
- -- Function: int mpf_fits_uint_p (mpf_t OP)
- -- Function: int mpf_fits_sint_p (mpf_t OP)
- -- Function: int mpf_fits_ushort_p (mpf_t OP)
- -- Function: int mpf_fits_sshort_p (mpf_t OP)
- Return non-zero if OP would fit in the respective C data type, when
- truncated to an integer.
-
- -- Function: void mpf_urandomb (mpf_t ROP, gmp_randstate_t STATE,
- mp_bitcnt_t NBITS)
- Generate a uniformly distributed random float in ROP, such that 0
- <= ROP < 1, with NBITS significant bits in the mantissa.
-
- The variable STATE must be initialized by calling one of the
- `gmp_randinit' functions (*note Random State Initialization::)
- before invoking this function.
-
- -- Function: void mpf_rrandomb (mpf_t ROP, gmp_randstate_t STATE,
- mp_size_t MAX_SIZE, mp_exp_t EXP)
- Generate a random float of at most MAX_SIZE limbs, with long
- strings of zeros and ones in the binary representation. The
- exponent of the number is in the interval -EXP to EXP (in limbs).
- This function is useful for testing functions and algorithms,
- since these kind of random numbers have proven to be more likely
- to trigger corner-case bugs. Negative random numbers are
- generated when MAX_SIZE is negative.
-
- *This interface is preliminary. It might change incompatibly in
- future revisions.*
-
- -- Function: void mpf_random2 (mpf_t ROP, mp_size_t MAX_SIZE, mp_exp_t
- EXP)
- Generate a random float of at most MAX_SIZE limbs, with long
- strings of zeros and ones in the binary representation. The
- exponent of the number is in the interval -EXP to EXP (in limbs).
- This function is useful for testing functions and algorithms,
- since these kind of random numbers have proven to be more likely
- to trigger corner-case bugs. Negative random numbers are
- generated when MAX_SIZE is negative.
-
- *This function is obsolete. It will disappear from future MPIR
- releases.*
-
-
-File: mpir.info, Node: Low-level Functions, Next: Random Number Functions, Prev: Floating-point Functions, Up: Top
-
-8 Low-level Functions
-*********************
-
-This chapter describes low-level MPIR functions, used to implement the
-high-level MPIR functions, but also intended for time-critical user
-code.
-
- These functions start with the prefix `mpn_'.
-
- The `mpn' functions are designed to be as fast as possible, *not* to
-provide a coherent calling interface. The different functions have
-somewhat similar interfaces, but there are variations that make them
-hard to use. These functions do as little as possible apart from the
-real multiple precision computation, so that no time is spent on things
-that not all callers need.
-
- A source operand is specified by a pointer to the least significant
-limb and a limb count. A destination operand is specified by just a
-pointer. It is the responsibility of the caller to ensure that the
-destination has enough space for storing the result.
-
- With this way of specifying operands, it is possible to perform
-computations on subranges of an argument, and store the result into a
-subrange of a destination.
-
- A common requirement for all functions is that each source area
-needs at least one limb. No size argument may be zero. Unless
-otherwise stated, in-place operations are allowed where source and
-destination are the same, but not where they only partly overlap.
-
- The `mpn' functions are the base for the implementation of the
-`mpz_', `mpf_', and `mpq_' functions.
-
- This example adds the number beginning at S1P and the number
-beginning at S2P and writes the sum at DESTP. All areas have N limbs.
-
- cy = mpn_add_n (destp, s1p, s2p, n)
-
- It should be noted that the `mpn' functions make no attempt to
-identify high or low zero limbs on their operands, or other special
-forms. On random data such cases will be unlikely and it'd be wasteful
-for every function to check every time. An application knowing
-something about its data can take steps to trim or perhaps split its
-calculations.
-
-
-In the notation used below, a source operand is identified by the
-pointer to the least significant limb, and the limb count in braces.
-For example, {S1P, S1N}.
-
- -- Function: mp_limb_t mpn_add_n (mp_limb_t *RP, const mp_limb_t *S1P,
- const mp_limb_t *S2P, mp_size_t N)
- Add {S1P, N} and {S2P, N}, and write the N least significant limbs
- of the result to RP. Return carry, either 0 or 1.
-
- This is the lowest-level function for addition. It is the
- preferred function for addition, since it is written in assembly
- for most CPUs. For addition of a variable to itself (i.e., S1P
- equals S2P, use `mpn_lshift' with a count of 1 for optimal speed.
-
- -- Function: mp_limb_t mpn_add_1 (mp_limb_t *RP, const mp_limb_t *S1P,
- mp_size_t N, mp_limb_t S2LIMB)
- Add {S1P, N} and S2LIMB, and write the N least significant limbs
- of the result to RP. Return carry, either 0 or 1.
-
- -- Function: mp_limb_t mpn_add (mp_limb_t *RP, const mp_limb_t *S1P,
- mp_size_t S1N, const mp_limb_t *S2P, mp_size_t S2N)
- Add {S1P, S1N} and {S2P, S2N}, and write the S1N least significant
- limbs of the result to RP. Return carry, either 0 or 1.
-
- This function requires that S1N is greater than or equal to S2N.
-
- -- Function: mp_limb_t mpn_sub_n (mp_limb_t *RP, const mp_limb_t *S1P,
- const mp_limb_t *S2P, mp_size_t N)
- Subtract {S2P, N} from {S1P, N}, and write the N least significant
- limbs of the result to RP. Return borrow, either 0 or 1.
-
- This is the lowest-level function for subtraction. It is the
- preferred function for subtraction, since it is written in
- assembly for most CPUs.
-
- -- Function: mp_limb_t mpn_sub_1 (mp_limb_t *RP, const mp_limb_t *S1P,
- mp_size_t N, mp_limb_t S2LIMB)
- Subtract S2LIMB from {S1P, N}, and write the N least significant
- limbs of the result to RP. Return borrow, either 0 or 1.
-
- -- Function: mp_limb_t mpn_sub (mp_limb_t *RP, const mp_limb_t *S1P,
- mp_size_t S1N, const mp_limb_t *S2P, mp_size_t S2N)
- Subtract {S2P, S2N} from {S1P, S1N}, and write the S1N least
- significant limbs of the result to RP. Return borrow, either 0 or
- 1.
-
- This function requires that S1N is greater than or equal to S2N.
-
- -- Function: void mpn_neg (mp_limb_t *RP, const mp_limb_t *SP,
- mp_size_t N)
- Perform the negation of {SP, N}, and write the result to {RP, N}.
- Return carry-out.
-
- -- Function: void mpn_mul_n (mp_limb_t *RP, const mp_limb_t *S1P,
- const mp_limb_t *S2P, mp_size_t N)
- Multiply {S1P, N} and {S2P, N}, and write the 2*N-limb result to
- RP.
-
- The destination has to have space for 2*N limbs, even if the
- product's most significant limb is zero. No overlap is permitted
- between the destination and either source.
-
- If the input operands are the same, `mpn_sqr' will generally be
- faster.
-
- -- Function: mp_limb_t mpn_mul_1 (mp_limb_t *RP, const mp_limb_t *S1P,
- mp_size_t N, mp_limb_t S2LIMB)
- Multiply {S1P, N} by S2LIMB, and write the N least significant
- limbs of the product to RP. Return the most significant limb of
- the product. {S1P, N} and {RP, N} are allowed to overlap provided
- RP <= S1P.
-
- This is a low-level function that is a building block for general
- multiplication as well as other operations in MPIR. It is written
- in assembly for most CPUs.
-
- Don't call this function if S2LIMB is a power of 2; use
- `mpn_lshift' with a count equal to the logarithm of S2LIMB
- instead, for optimal speed.
-
- -- Function: mp_limb_t mpn_addmul_1 (mp_limb_t *RP, const mp_limb_t
- *S1P, mp_size_t N, mp_limb_t S2LIMB)
- Multiply {S1P, N} and S2LIMB, and add the N least significant
- limbs of the product to {RP, N} and write the result to RP.
- Return the most significant limb of the product, plus carry-out
- from the addition.
-
- This is a low-level function that is a building block for general
- multiplication as well as other operations in MPIR. It is written
- in assembly for most CPUs.
-
- -- Function: mp_limb_t mpn_submul_1 (mp_limb_t *RP, const mp_limb_t
- *S1P, mp_size_t N, mp_limb_t S2LIMB)
- Multiply {S1P, N} and S2LIMB, and subtract the N least significant
- limbs of the product from {RP, N} and write the result to RP.
- Return the most significant limb of the product, minus borrow-out
- from the subtraction.
-
- This is a low-level function that is a building block for general
- multiplication and division as well as other operations in MPIR.
- It is written in assembly for most CPUs.
-
- -- Function: mp_limb_t mpn_mul (mp_limb_t *RP, const mp_limb_t *S1P,
- mp_size_t S1N, const mp_limb_t *S2P, mp_size_t S2N)
- Multiply {S1P, S1N} and {S2P, S2N}, and write the result to RP.
- Return the most significant limb of the result.
-
- The destination has to have space for S1N + S2N limbs, even if the
- result might be one limb smaller.
-
- This function requires that S1N is greater than or equal to S2N.
- The destination must be distinct from both input operands.
-
- -- Function: void mpn_sqr (mp_limb_t *RP, const mp_limb_t *S1P,
- mp_size_t N)
- Compute the square of {S1P, N} and write the 2*N-limb result to RP.
-
- The destination has to have space for 2*N limbs, even if the
- result's most significant limb is zero. No overlap is permitted
- between the destination and the source.
-
- -- Function: void mpn_tdiv_qr (mp_limb_t *QP, mp_limb_t *RP, mp_size_t
- QXN, const mp_limb_t *NP, mp_size_t NN, const mp_limb_t *DP,
- mp_size_t DN)
- Divide {NP, NN} by {DP, DN} and put the quotient at {QP, NN-DN+1}
- and the remainder at {RP, DN}. The quotient is rounded towards 0.
-
- No overlap is permitted between arguments. NN must be greater
- than or equal to DN. The most significant limb of DP must be
- non-zero. The QXN operand must be zero.
-
- -- Function: mp_limb_t mpn_divrem (mp_limb_t *R1P, mp_size_t QXN,
- mp_limb_t *RS2P, mp_size_t RS2N, const mp_limb_t *S3P,
- mp_size_t S3N)
- [This function is obsolete. Please call `mpn_tdiv_qr' instead for
- best performance.]
-
- Divide {RS2P, RS2N} by {S3P, S3N}, and write the quotient at R1P,
- with the exception of the most significant limb, which is
- returned. The remainder replaces the dividend at RS2P; it will be
- S3N limbs long (i.e., as many limbs as the divisor).
-
- In addition to an integer quotient, QXN fraction limbs are
- developed, and stored after the integral limbs. For most usages,
- QXN will be zero.
-
- It is required that RS2N is greater than or equal to S3N. It is
- required that the most significant bit of the divisor is set.
-
- If the quotient is not needed, pass RS2P + S3N as R1P. Aside from
- that special case, no overlap between arguments is permitted.
-
- Return the most significant limb of the quotient, either 0 or 1.
-
- The area at R1P needs to be RS2N - S3N + QXN limbs large.
-
- -- Function: mp_limb_t mpn_divrem_1 (mp_limb_t *R1P, mp_size_t QXN,
- mp_limb_t *S2P, mp_size_t S2N, mp_limb_t S3LIMB)
- -- Macro: mp_limb_t mpn_divmod_1 (mp_limb_t *R1P, mp_limb_t *S2P,
- mp_size_t S2N, mp_limb_t S3LIMB)
- Divide {S2P, S2N} by S3LIMB, and write the quotient at R1P.
- Return the remainder.
-
- The integer quotient is written to {R1P+QXN, S2N} and in addition
- QXN fraction limbs are developed and written to {R1P, QXN}.
- Either or both S2N and QXN can be zero. For most usages, QXN will
- be zero.
-
- `mpn_divmod_1' exists for upward source compatibility and is
- simply a macro calling `mpn_divrem_1' with a QXN of 0.
-
- The areas at R1P and S2P have to be identical or completely
- separate, not partially overlapping.
-
- -- Macro: mp_limb_t mpn_divexact_by3 (mp_limb_t *RP, mp_limb_t *SP,
- mp_size_t N)
- -- Function: mp_limb_t mpn_divexact_by3c (mp_limb_t *RP, mp_limb_t
- *SP, mp_size_t N, mp_limb_t CARRY)
- Divide {SP, N} by 3, expecting it to divide exactly, and writing
- the result to {RP, N}. If 3 divides exactly, the return value is
- zero and the result is the quotient. If not, the return value is
- non-zero and the result won't be anything useful.
-
- `mpn_divexact_by3c' takes an initial carry parameter, which can be
- the return value from a previous call, so a large calculation can
- be done piece by piece from low to high. `mpn_divexact_by3' is
- simply a macro calling `mpn_divexact_by3c' with a 0 carry
- parameter.
-
- These routines use a multiply-by-inverse and will be faster than
- `mpn_divrem_1' on CPUs with fast multiplication but slow division.
-
- The source a, result q, size n, initial carry i, and return value
- c satisfy c*b^n + a-i = 3*q, where b=2^GMP_NUMB_BITS. The return
- c is always 0, 1 or 2, and the initial carry i must also be 0, 1
- or 2 (these are both borrows really). When c=0 clearly q=(a-i)/3.
- When c!=0, the remainder (a-i) mod 3 is given by 3-c, because b ==
- 1 mod 3 (when `mp_bits_per_limb' is even, which is always so
- currently).
-
- -- Function: mp_limb_t mpn_mod_1 (mp_limb_t *S1P, mp_size_t S1N,
- mp_limb_t S2LIMB)
- Divide {S1P, S1N} by S2LIMB, and return the remainder. S1N can be
- zero.
-
- -- Function: mp_limb_t mpn_bdivmod (mp_limb_t *RP, mp_limb_t *S1P,
- mp_size_t S1N, const mp_limb_t *S2P, mp_size_t S2N, mpir_ui D)
- This function puts the low floor(D/mp_bits_per_limb) limbs of Q =
- {S1P, S1N}/{S2P, S2N} mod 2^D at RP, and returns the high D mod
- `mp_bits_per_limb' bits of Q.
-
- {S1P, S1N} - Q * {S2P, S2N} mod 2^(S1N*mp_bits_per_limb) is placed
- at S1P. Since the low floor(D/mp_bits_per_limb) limbs of this
- difference are zero, it is possible to overwrite the low limbs at
- S1P with this difference, provided RP <= S1P.
-
- This function requires that S1N * mp_bits_per_limb >= D, and that
- {S2P, S2N} is odd.
-
- *This interface is preliminary. It might change incompatibly in
- future revisions.*
-
- -- Function: mp_limb_t mpn_lshift (mp_limb_t *RP, const mp_limb_t *SP,
- mp_size_t N, unsigned int COUNT)
- Shift {SP, N} left by COUNT bits, and write the result to {RP, N}.
- The bits shifted out at the left are returned in the least
- significant COUNT bits of the return value (the rest of the return
- value is zero).
-
- COUNT must be in the range 1 to mp_bits_per_limb-1. The regions
- {SP, N} and {RP, N} may overlap, provided RP >= SP.
-
- This function is written in assembly for most CPUs.
-
- -- Function: mp_limb_t mpn_rshift (mp_limb_t *RP, const mp_limb_t *SP,
- mp_size_t N, unsigned int COUNT)
- Shift {SP, N} right by COUNT bits, and write the result to {RP,
- N}. The bits shifted out at the right are returned in the most
- significant COUNT bits of the return value (the rest of the return
- value is zero).
-
- COUNT must be in the range 1 to mp_bits_per_limb-1. The regions
- {SP, N} and {RP, N} may overlap, provided RP <= SP.
-
- This function is written in assembly for most CPUs.
-
- -- Function: int mpn_cmp (const mp_limb_t *S1P, const mp_limb_t *S2P,
- mp_size_t N)
- Compare {S1P, N} and {S2P, N} and return a positive value if S1 >
- S2, 0 if they are equal, or a negative value if S1 < S2.
-
- -- Function: mp_size_t mpn_gcd (mp_limb_t *RP, mp_limb_t *S1P,
- mp_size_t S1N, mp_limb_t *S2P, mp_size_t S2N)
- Set {RP, RETVAL} to the greatest common divisor of {S1P, S1N} and
- {S2P, S2N}. The result can be up to S2N limbs, the return value
- is the actual number produced. Both source operands are destroyed.
-
- {S1P, S1N} must have at least as many bits as {S2P, S2N}. {S2P,
- S2N} must be odd. Both operands must have non-zero most
- significant limbs. No overlap is permitted between {S1P, S1N} and
- {S2P, S2N}.
-
- -- Function: mp_limb_t mpn_gcd_1 (const mp_limb_t *S1P, mp_size_t S1N,
- mp_limb_t S2LIMB)
- Return the greatest common divisor of {S1P, S1N} and S2LIMB. Both
- operands must be non-zero.
-
- -- Function: mp_size_t mpn_gcdext (mp_limb_t *GP, mp_limb_t *SP,
- mp_size_t *SN, mp_limb_t *XP, mp_size_t XN, mp_limb_t *YP,
- mp_size_t YN)
- Let U be defined by {XP, XN} and let V be defined by {YP, YN}.
-
- Compute the greatest common divisor G of U and V. Compute a
- cofactor S such that G = US + VT. The second cofactor T is not
- computed but can easily be obtained from (G - U*S) / V (the
- division will be exact). It is required that U >= V > 0.
-
- S satisfies S = 1 or abs(S) < V / (2 G). S = 0 if and only if V
- divides U (i.e., G = V).
-
- Store G at GP and let the return value define its limb count.
- Store S at SP and let |*SN| define its limb count. S can be
- negative; when this happens *SN will be negative. The areas at GP
- and SP should each have room for XN+1 limbs.
-
- The areas {XP, XN+1} and {YP, YN+1} are destroyed (i.e. the input
- operands plus an extra limb past the end of each).
-
- Compatibility note: MPIR versions 1.3,2.0 and GMP versions
- 4.3.0,4.3.1 defined S less strictly. Earlier as well as later GMP
- releases define S as described here.
-
- -- Function: mp_size_t mpn_sqrtrem (mp_limb_t *R1P, mp_limb_t *R2P,
- const mp_limb_t *SP, mp_size_t N)
- Compute the square root of {SP, N} and put the result at {R1P,
- ceil(N/2)} and the remainder at {R2P, RETVAL}. R2P needs space
- for N limbs, but the return value indicates how many are produced.
-
- The most significant limb of {SP, N} must be non-zero. The areas
- {R1P, ceil(N/2)} and {SP, N} must be completely separate. The
- areas {R2P, N} and {SP, N} must be either identical or completely
- separate.
-
- If the remainder is not wanted then R2P can be `NULL', and in this
- case the return value is zero or non-zero according to whether the
- remainder would have been zero or non-zero.
-
- A return value of zero indicates a perfect square. See also
- `mpz_perfect_square_p'.
-
- -- Function: mp_size_t mpn_get_str (unsigned char *STR, int BASE,
- mp_limb_t *S1P, mp_size_t S1N)
- Convert {S1P, S1N} to a raw unsigned char array at STR in base
- BASE, and return the number of characters produced. There may be
- leading zeros in the string. The string is not in ASCII; to
- convert it to printable format, add the ASCII codes for `0' or
- `A', depending on the base and range. BASE can vary from 2 to 256.
-
- The most significant limb of the input {S1P, S1N} must be
- non-zero. The input {S1P, S1N} is clobbered, except when BASE is
- a power of 2, in which case it's unchanged.
-
- The area at STR has to have space for the largest possible number
- represented by a S1N long limb array, plus one extra character.
-
- -- Function: mp_size_t mpn_set_str (mp_limb_t *RP, const unsigned char
- *STR, size_t STRSIZE, int BASE)
- Convert bytes {STR,STRSIZE} in the given BASE to limbs at RP.
-
- STR[0] is the most significant byte and STR[STRSIZE-1] is the
- least significant. Each byte should be a value in the range 0 to
- BASE-1, not an ASCII character. BASE can vary from 2 to 256.
-
- The return value is the number of limbs written to RP. If the most
- significant input byte is non-zero then the high limb at RP will be
- non-zero, and only that exact number of limbs will be required
- there.
-
- If the most significant input byte is zero then there may be high
- zero limbs written to RP and included in the return value.
-
- STRSIZE must be at least 1, and no overlap is permitted between
- {STR,STRSIZE} and the result at RP.
-
- -- Function: mp_bitcnt_t mpn_scan0 (const mp_limb_t *S1P, imp_bitcnt_t
- BIT)
- Scan S1P from bit position BIT for the next clear bit.
-
- It is required that there be a clear bit within the area at S1P at
- or beyond bit position BIT, so that the function has something to
- return.
-
- -- Function: mp_bitcnt_t mpn_scan1 (const mp_limb_t *S1P, mp_bitcnt_t
- BIT)
- Scan S1P from bit position BIT for the next set bit.
-
- It is required that there be a set bit within the area at S1P at or
- beyond bit position BIT, so that the function has something to
- return.
-
- -- Function: void mpn_random (mp_limb_t *R1P, mp_size_t R1N)
- -- Function: void mpn_random2 (mp_limb_t *R1P, mp_size_t R1N)
- Generate a random number of length R1N and store it at R1P. The
- most significant limb is always non-zero. `mpn_random' generates
- uniformly distributed limb data, `mpn_random2' generates long
- strings of zeros and ones in the binary representation.
-
- `mpn_random2' is intended for testing the correctness of the `mpn'
- routines.
-
- *These functions are obsolete. They will disappear from future
- MPIR releases.*
-
- -- Function: void mpn_urandomb (mp_limb_t *RP, gmp_randstate_t STATE,
- mpir_ui N)
- Generate a uniform random number of length N bits and store it at
- RP.
-
- *This function interface is preliminary and may change in the
- future.*
-
- -- Function: void mpn_urandomm (mp_limb_t *RP, gmp_randstate_t STATE,
- const mp_limb_t *MP, mp_size_t N)
- Generate a uniform random number modulo (MP,N) of length N limbs
- and store it at RP.
-
- *This function interface is preliminary and may change in the
- future.*
-
- -- Function: void mpn_randomb (mp_limb_t *RP, gmp_randstate_t STATE,
- mp_size_t N)
- Generate a random number of length N limbs and store it at RP.
- The most significant limb is always non-zero.
-
- *This function interface is preliminary and may change in the
- future.*
-
- -- Function: void mpn_rrandom (mp_limb_t *RP, gmp_randstate_t STATE,
- mp_size_t N)
- Generate a random number of length N limbs and store it at RP.
- The most significant limb is always non-zero. Generates long
- strings of zeros and ones in the binary representation and is
- intended for testing the correctness of the `mpn' routines.
-
- *This function interface is preliminary and may change in the
- future.*
-
- -- Function: mp_bitcnt_t mpn_popcount (const mp_limb_t *S1P, mp_size_t
- N)
- Count the number of set bits in {S1P, N}.
-
- -- Function: mp_bitcnt_t mpn_hamdist (const mp_limb_t *S1P, const
- mp_limb_t *S2P, mp_size_t N)
- Compute the hamming distance between {S1P, N} and {S2P, N}, which
- is the number of bit positions where the two operands have
- different bit values.
-
- -- Function: int mpn_perfect_square_p (const mp_limb_t *S1P, mp_size_t
- N)
- Return non-zero iff {S1P, N} is a perfect square.
-
- -- Function: void mpn_and_n (mp_limb_t *RP, const mp_limb_t *S1P,
- const mp_limb_t *S2P, mp_size_t N)
- Perform the bitwise logical and of {S1P, N} and {S2P, N}, and
- write the result to {RP, N}.
-
- -- Function: void mpn_ior_n (mp_limb_t *RP, const mp_limb_t *S1P,
- const mp_limb_t *S2P, mp_size_t N)
- Perform the bitwise logical inclusive or of {S1P, N} and {S2P, N},
- and write the result to {RP, N}.
-
- -- Function: void mpn_xor_n (mp_limb_t *RP, const mp_limb_t *S1P,
- const mp_limb_t *S2P, mp_size_t N)
- Perform the bitwise logical exclusive or of {S1P, N} and {S2P, N},
- and write the result to {RP, N}.
-
- -- Function: void mpn_andn_n (mp_limb_t *RP, const mp_limb_t *S1P,
- const mp_limb_t *S2P, mp_size_t N)
- Perform the bitwise logical and of {S1P, N} and the bitwise
- complement of {S2P, N}, and write the result to {RP, N}.
-
- -- Function: void mpn_iorn_n (mp_limb_t *RP, const mp_limb_t *S1P,
- const mp_limb_t *S2P, mp_size_t N)
- Perform the bitwise logical inclusive or of {S1P, N} and the
- bitwise complement of {S2P, N}, and write the result to {RP, N}.
-
- -- Function: void mpn_nand_n (mp_limb_t *RP, const mp_limb_t *S1P,
- const mp_limb_t *S2P, mp_size_t N)
- Perform the bitwise logical and of {S1P, N} and {S2P, N}, and
- write the bitwise complement of the result to {RP, N}.
-
- -- Function: void mpn_nior_n (mp_limb_t *RP, const mp_limb_t *S1P,
- const mp_limb_t *S2P, mp_size_t N)
- Perform the bitwise logical inclusive or of {S1P, N} and {S2P, N},
- and write the bitwise complement of the result to {RP, N}.
-
- -- Function: void mpn_xnor_n (mp_limb_t *RP, const mp_limb_t *S1P,
- const mp_limb_t *S2P, mp_size_t N)
- Perform the bitwise logical exclusive or of {S1P, N} and {S2P, N},
- and write the bitwise complement of the result to {RP, N}.
-
- -- Function: void mpn_com (mp_limb_t *RP, const mp_limb_t *SP,
- mp_size_t N)
- Perform the bitwise complement of {SP, N}, and write the result to
- {RP, N}.
-
- -- Function: void mpn_copyi (mp_limb_t *RP, const mp_limb_t *S1P,
- mp_size_t N)
- Copy from {S1P, N} to {RP, N}, increasingly.
-
- -- Function: void mpn_copyd (mp_limb_t *RP, const mp_limb_t *S1P,
- mp_size_t N)
- Copy from {S1P, N} to {RP, N}, decreasingly.
-
- -- Function: void mpn_zero (mp_limb_t *RP, mp_size_t N)
- Zero {RP, N}.
-
-
-8.1 Nails
-=========
-
-*Everything in this section is highly experimental and may disappear or
-be subject to incompatible changes in a future version of MPIR.*
-
- N.B: Nails are currently disabled and not supported in MPIR. They
-may or may not return in a future version of MPIR.
-
- Nails are an experimental feature whereby a few bits are left unused
-at the top of each `mp_limb_t'. This can significantly improve carry
-handling on some processors.
-
- All the `mpn' functions accepting limb data will expect the nail
-bits to be zero on entry, and will return data with the nails similarly
-all zero. This applies both to limb vectors and to single limb
-arguments.
-
- Nails can be enabled by configuring with `--enable-nails'. By
-default the number of bits will be chosen according to what suits the
-host processor, but a particular number can be selected with
-`--enable-nails=N'.
-
- At the mpn level, a nail build is neither source nor binary
-compatible with a non-nail build, strictly speaking. But programs
-acting on limbs only through the mpn functions are likely to work
-equally well with either build, and judicious use of the definitions
-below should make any program compatible with either build, at the
-source level.
-
- For the higher level routines, meaning `mpz' etc, a nail build
-should be fully source and binary compatible with a non-nail build.
-
- -- Macro: GMP_NAIL_BITS
- -- Macro: GMP_NUMB_BITS
- -- Macro: GMP_LIMB_BITS
- `GMP_NAIL_BITS' is the number of nail bits, or 0 when nails are
- not in use. `GMP_NUMB_BITS' is the number of data bits in a limb.
- `GMP_LIMB_BITS' is the total number of bits in an `mp_limb_t'. In
- all cases
-
- GMP_LIMB_BITS == GMP_NAIL_BITS + GMP_NUMB_BITS
-
- -- Macro: GMP_NAIL_MASK
- -- Macro: GMP_NUMB_MASK
- Bit masks for the nail and number parts of a limb.
- `GMP_NAIL_MASK' is 0 when nails are not in use.
-
- `GMP_NAIL_MASK' is not often needed, since the nail part can be
- obtained with `x >> GMP_NUMB_BITS', and that means one less large
- constant, which can help various RISC chips.
-
- -- Macro: GMP_NUMB_MAX
- The maximum value that can be stored in the number part of a limb.
- This is the same as `GMP_NUMB_MASK', but can be used for clarity
- when doing comparisons rather than bit-wise operations.
-
- The term "nails" comes from finger or toe nails, which are at the
-ends of a limb (arm or leg). "numb" is short for number, but is also
-how the developers felt after trying for a long time to come up with
-sensible names for these things.
-
- In the future (the distant future most likely) a non-zero nail might
-be permitted, giving non-unique representations for numbers in a limb
-vector. This would help vector processors since carries would only
-ever need to propagate one or two limbs.
-
-
-File: mpir.info, Node: Random Number Functions, Next: Formatted Output, Prev: Low-level Functions, Up: Top
-
-9 Random Number Functions
-*************************
-
-Sequences of pseudo-random numbers in MPIR are generated using a
-variable of type `gmp_randstate_t', which holds an algorithm selection
-and a current state. Such a variable must be initialized by a call to
-one of the `gmp_randinit' functions, and can be seeded with one of the
-`gmp_randseed' functions.
-
- The functions actually generating random numbers are described in
-*note Integer Random Numbers::, and *note Miscellaneous Float
-Functions::.
-
- The older style random number functions don't accept a
-`gmp_randstate_t' parameter but instead share a global variable of that
-type. They use a default algorithm and are currently not seeded
-(though perhaps that will change in the future). The new functions
-accepting a `gmp_randstate_t' are recommended for applications that
-care about randomness.
-
-* Menu:
-
-* Random State Initialization::
-* Random State Seeding::
-* Random State Miscellaneous::
-
-
-File: mpir.info, Node: Random State Initialization, Next: Random State Seeding, Prev: Random Number Functions, Up: Random Number Functions
-
-9.1 Random State Initialization
-===============================
-
- -- Function: void gmp_randinit_default (gmp_randstate_t STATE)
- Initialize STATE with a default algorithm. This will be a
- compromise between speed and randomness, and is recommended for
- applications with no special requirements. Currently this is
- `gmp_randinit_mt'.
-
- -- Function: void gmp_randinit_mt (gmp_randstate_t STATE)
- Initialize STATE for a Mersenne Twister algorithm. This algorithm
- is fast and has good randomness properties.
-
- -- Function: void gmp_randinit_lc_2exp (gmp_randstate_t STATE, mpz_t
- A, mpir_ui C, mp_bitcnt_t M2EXP)
- Initialize STATE with a linear congruential algorithm X = (A*X +
- C) mod 2^M2EXP.
-
- The low bits of X in this algorithm are not very random. The least
- significant bit will have a period no more than 2, and the second
- bit no more than 4, etc. For this reason only the high half of
- each X is actually used.
-
- When a random number of more than M2EXP/2 bits is to be generated,
- multiple iterations of the recurrence are used and the results
- concatenated.
-
- -- Function: int gmp_randinit_lc_2exp_size (gmp_randstate_t STATE,
- mp_bitcnt_t SIZE)
- Initialize STATE for a linear congruential algorithm as per
- `gmp_randinit_lc_2exp'. A, C and M2EXP are selected from a table,
- chosen so that SIZE bits (or more) of each X will be used, ie.
- M2EXP/2 >= SIZE.
-
- If successful the return value is non-zero. If SIZE is bigger
- than the table data provides then the return value is zero. The
- maximum SIZE currently supported is 128.
-
- -- Function: int gmp_randinit_set (gmp_randstate_t ROP,
- gmp_randstate_t OP)
- Initialize ROP with a copy of the algorithm and state from OP.
-
- -- Function: void gmp_randclear (gmp_randstate_t STATE)
- Free all memory occupied by STATE.
-
-
-File: mpir.info, Node: Random State Seeding, Next: Random State Miscellaneous, Prev: Random State Initialization, Up: Random Number Functions
-
-9.2 Random State Seeding
-========================
-
- -- Function: void gmp_randseed (gmp_randstate_t STATE, mpz_t SEED)
- -- Function: void gmp_randseed_ui (gmp_randstate_t STATE, mpir_ui SEED)
- Set an initial seed value into STATE.
-
- The size of a seed determines how many different sequences of
- random numbers that it's possible to generate. The "quality" of
- the seed is the randomness of a given seed compared to the
- previous seed used, and this affects the randomness of separate
- number sequences. The method for choosing a seed is critical if
- the generated numbers are to be used for important applications,
- such as generating cryptographic keys.
-
- Traditionally the system time has been used to seed, but care
- needs to be taken with this. If an application seeds often and
- the resolution of the system clock is low, then the same sequence
- of numbers might be repeated. Also, the system time is quite easy
- to guess, so if unpredictability is required then it should
- definitely not be the only source for the seed value. On some
- systems there's a special device `/dev/random' which provides
- random data better suited for use as a seed.
-
-
-File: mpir.info, Node: Random State Miscellaneous, Prev: Random State Seeding, Up: Random Number Functions
-
-9.3 Random State Miscellaneous
-==============================
-
- -- Function: mpir_ui gmp_urandomb_ui (gmp_randstate_t STATE, mpir_ui N)
- Return a uniformly distributed random number of N bits, ie. in the
- range 0 to 2^N-1 inclusive. N must be less than or equal to the
- number of bits in an `mpir_ui'.
-
- -- Function: mpir_ui gmp_urandomm_ui (gmp_randstate_t STATE, mpir_ui N)
- Return a uniformly distributed random number in the range 0 to
- N-1, inclusive.
-
-
-File: mpir.info, Node: Formatted Output, Next: Formatted Input, Prev: Random Number Functions, Up: Top
-
-10 Formatted Output
-*******************
-
-* Menu:
-
-* Formatted Output Strings::
-* Formatted Output Functions::
-* C++ Formatted Output::
-
-
-File: mpir.info, Node: Formatted Output Strings, Next: Formatted Output Functions, Prev: Formatted Output, Up: Formatted Output
-
-10.1 Format Strings
-===================
-
-`gmp_printf' and friends accept format strings similar to the standard C
-`printf' (*note Formatted Output: (libc)Formatted Output.). A format
-specification is of the form
-
- % [flags] [width] [.[precision]] [type] conv
-
- MPIR adds types `Z', `Q' and `F' for `mpz_t', `mpq_t' and `mpf_t'
-respectively, `M' for `mp_limb_t', and `N' for an `mp_limb_t' array.
-`Z', `Q', `M' and `N' behave like integers. `Q' will print a `/' and a
-denominator, if needed. `F' behaves like a float. For example,
-
- mpz_t z;
- gmp_printf ("%s is an mpz %Zd\n", "here", z);
-
- mpq_t q;
- gmp_printf ("a hex rational: %#40Qx\n", q);
-
- mpf_t f;
- int n;
- gmp_printf ("fixed point mpf %.*Ff with %d digits\n", n, f, n);
-
- mp_limb_t l;
- gmp_printf ("limb %Mu\n", limb);
-
- const mp_limb_t *ptr;
- mp_size_t size;
- gmp_printf ("limb array %Nx\n", ptr, size);
-
- For `N' the limbs are expected least significant first, as per the
-`mpn' functions (*note Low-level Functions::). A negative size can be
-given to print the value as a negative.
-
- All the standard C `printf' types behave the same as the C library
-`printf', and can be freely intermixed with the MPIR extensions. In the
-current implementation the standard parts of the format string are
-simply handed to `printf' and only the MPIR extensions handled directly.
-
- The flags accepted are as follows. GLIBC style ' is only for the
-standard C types (not the MPIR types), and only if the C library
-supports it.
-
- 0 pad with zeros (rather than spaces)
- # show the base with `0x', `0X' or `0'
- + always show a sign
- (space) show a space or a `-' sign
- ' group digits, GLIBC style (not MPIR
- types)
-
- The optional width and precision can be given as a number within the
-format string, or as a `*' to take an extra parameter of type `int', the
-same as the standard `printf'.
-
- The standard types accepted are as follows. `h' and `l' are
-portable, the rest will depend on the compiler (or include files) for
-the type and the C library for the output.
-
- h short
- hh char
- j intmax_t or uintmax_t
- l long or wchar_t
- ll long long
- L long double
- q quad_t or u_quad_t
- t ptrdiff_t
- z size_t
-
-The MPIR types are
-
- F mpf_t, float conversions
- Q mpq_t, integer conversions
- M mp_limb_t, integer conversions
- N mp_limb_t array, integer conversions
- Z mpz_t, integer conversions
-
- The conversions accepted are as follows. `a' and `A' are always
-supported for `mpf_t' but depend on the C library for standard C float
-types. `m' and `p' depend on the C library.
-
- a A hex floats, C99 style
- c character
- d decimal integer
- e E scientific format float
- f fixed point float
- i same as d
- g G fixed or scientific float
- m `strerror' string, GLIBC style
- n store characters written so far
- o octal integer
- p pointer
- s string
- u unsigned integer
- x X hex integer
-
- `o', `x' and `X' are unsigned for the standard C types, but for
-types `Z', `Q' and `N' they are signed. `u' is not meaningful for `Z',
-`Q' and `N'.
-
- `M' is a proxy for the C library `l' or `L', according to the size
-of `mp_limb_t'. Unsigned conversions will be usual, but a signed
-conversion can be used and will interpret the value as a twos complement
-negative.
-
- `n' can be used with any type, even the MPIR types.
-
- Other types or conversions that might be accepted by the C library
-`printf' cannot be used through `gmp_printf', this includes for
-instance extensions registered with GLIBC `register_printf_function'.
-Also currently there's no support for POSIX `$' style numbered arguments
-(perhaps this will be added in the future).
-
- The precision field has it's usual meaning for integer `Z' and float
-`F' types, but is currently undefined for `Q' and should not be used
-with that.
-
- `mpf_t' conversions only ever generate as many digits as can be
-accurately represented by the operand, the same as `mpf_get_str' does.
-Zeros will be used if necessary to pad to the requested precision. This
-happens even for an `f' conversion of an `mpf_t' which is an integer,
-for instance 2^1024 in an `mpf_t' of 128 bits precision will only
-produce about 40 digits, then pad with zeros to the decimal point. An
-empty precision field like `%.Fe' or `%.Ff' can be used to specifically
-request just the significant digits.
-
- The decimal point character (or string) is taken from the current
-locale settings on systems which provide `localeconv' (*note Locales
-and Internationalization: (libc)Locales.). The C library will normally
-do the same for standard float output.
-
- The format string is only interpreted as plain `char's, multibyte
-characters are not recognised. Perhaps this will change in the future.
-
-
-File: mpir.info, Node: Formatted Output Functions, Next: C++ Formatted Output, Prev: Formatted Output Strings, Up: Formatted Output
-
-10.2 Functions
-==============
-
-Each of the following functions is similar to the corresponding C
-library function. The basic `printf' forms take a variable argument
-list. The `vprintf' forms take an argument pointer, see *note Variadic
-Functions: (libc)Variadic Functions, or `man 3 va_start'.
-
- It should be emphasised that if a format string is invalid, or the
-arguments don't match what the format specifies, then the behaviour of
-any of these functions will be unpredictable. GCC format string
-checking is not available, since it doesn't recognise the MPIR
-extensions.
-
- The file based functions `gmp_printf' and `gmp_fprintf' will return
--1 to indicate a write error. Output is not "atomic", so partial
-output may be produced if a write error occurs. All the functions can
-return -1 if the C library `printf' variant in use returns -1, but this
-shouldn't normally occur.
-
- -- Function: int gmp_printf (const char *FMT, ...)
- -- Function: int gmp_vprintf (const char *FMT, va_list AP)
- Print to the standard output `stdout'. Return the number of
- characters written, or -1 if an error occurred.
-
- -- Function: int gmp_fprintf (FILE *FP, const char *FMT, ...)
- -- Function: int gmp_vfprintf (FILE *FP, const char *FMT, va_list AP)
- Print to the stream FP. Return the number of characters written,
- or -1 if an error occurred.
-
- -- Function: int gmp_sprintf (char *BUF, const char *FMT, ...)
- -- Function: int gmp_vsprintf (char *BUF, const char *FMT, va_list AP)
- Form a null-terminated string in BUF. Return the number of
- characters written, excluding the terminating null.
-
- No overlap is permitted between the space at BUF and the string
- FMT.
-
- These functions are not recommended, since there's no protection
- against exceeding the space available at BUF.
-
- -- Function: int gmp_snprintf (char *BUF, size_t SIZE, const char
- *FMT, ...)
- -- Function: int gmp_vsnprintf (char *BUF, size_t SIZE, const char
- *FMT, va_list AP)
- Form a null-terminated string in BUF. No more than SIZE bytes
- will be written. To get the full output, SIZE must be enough for
- the string and null-terminator.
-
- The return value is the total number of characters which ought to
- have been produced, excluding the terminating null. If RETVAL >=
- SIZE then the actual output has been truncated to the first SIZE-1
- characters, and a null appended.
-
- No overlap is permitted between the region {BUF,SIZE} and the FMT
- string.
-
- Notice the return value is in ISO C99 `snprintf' style. This is
- so even if the C library `vsnprintf' is the older GLIBC 2.0.x
- style.
-
- -- Function: int gmp_asprintf (char **PP, const char *FMT, ...)
- -- Function: int gmp_vasprintf (char **PP, const char *FMT, va_list AP)
- Form a null-terminated string in a block of memory obtained from
- the current memory allocation function (*note Custom
- Allocation::). The block will be the size of the string and
- null-terminator. The address of the block in stored to *PP. The
- return value is the number of characters produced, excluding the
- null-terminator.
-
- Unlike the C library `asprintf', `gmp_asprintf' doesn't return -1
- if there's no more memory available, it lets the current allocation
- function handle that.
-
- -- Function: int gmp_obstack_printf (struct obstack *OB, const char
- *FMT, ...)
- -- Function: int gmp_obstack_vprintf (struct obstack *OB, const char
- *FMT, va_list AP)
- Append to the current object in OB. The return value is the
- number of characters written. A null-terminator is not written.
-
- FMT cannot be within the current object in OB, since that object
- might move as it grows.
-
- These functions are available only when the C library provides the
- obstack feature, which probably means only on GNU systems, see
- *note Obstacks: (libc)Obstacks.
-
-
-File: mpir.info, Node: C++ Formatted Output, Prev: Formatted Output Functions, Up: Formatted Output
-
-10.3 C++ Formatted Output
-=========================
-
-The following functions are provided in `libmpirxx' (*note Headers and
-Libraries::), which is built if C++ support is enabled (*note Build
-Options::). Prototypes are available from `<mpir.h>'.
-
- -- Function: ostream& operator<< (ostream& STREAM, mpz_t OP)
- Print OP to STREAM, using its `ios' formatting settings.
- `ios::width' is reset to 0 after output, the same as the standard
- `ostream operator<<' routines do.
-
- In hex or octal, OP is printed as a signed number, the same as for
- decimal. This is unlike the standard `operator<<' routines on
- `int' etc, which instead give twos complement.
-
- -- Function: ostream& operator<< (ostream& STREAM, mpq_t OP)
- Print OP to STREAM, using its `ios' formatting settings.
- `ios::width' is reset to 0 after output, the same as the standard
- `ostream operator<<' routines do.
-
- Output will be a fraction like `5/9', or if the denominator is 1
- then just a plain integer like `123'.
-
- In hex or octal, OP is printed as a signed value, the same as for
- decimal. If `ios::showbase' is set then a base indicator is shown
- on both the numerator and denominator (if the denominator is
- required).
-
- -- Function: ostream& operator<< (ostream& STREAM, mpf_t OP)
- Print OP to STREAM, using its `ios' formatting settings.
- `ios::width' is reset to 0 after output, the same as the standard
- `ostream operator<<' routines do.
-
- The decimal point follows the standard library float `operator<<',
- which on recent systems means the `std::locale' imbued on STREAM.
-
- Hex and octal are supported, unlike the standard `operator<<' on
- `double'. The mantissa will be in hex or octal, the exponent will
- be in decimal. For hex the exponent delimiter is an `@'. This is
- as per `mpf_out_str'.
-
- `ios::showbase' is supported, and will put a base on the mantissa,
- for example hex `0x1.8' or `0x0.8', or octal `01.4' or `00.4'.
- This last form is slightly strange, but at least differentiates
- itself from decimal.
-
- These operators mean that MPIR types can be printed in the usual C++
-way, for example,
-
- mpz_t z;
- int n;
- ...
- cout << "iteration " << n << " value " << z << "\n";
-
- But note that `ostream' output (and `istream' input, *note C++
-Formatted Input::) is the only overloading available for the MPIR types
-and that for instance using `+' with an `mpz_t' will have unpredictable
-results. For classes with overloading, see *note C++ Class Interface::.
-
-
-File: mpir.info, Node: Formatted Input, Next: C++ Class Interface, Prev: Formatted Output, Up: Top
-
-11 Formatted Input
-******************
-
-* Menu:
-
-* Formatted Input Strings::
-* Formatted Input Functions::
-* C++ Formatted Input::
-
-
-File: mpir.info, Node: Formatted Input Strings, Next: Formatted Input Functions, Prev: Formatted Input, Up: Formatted Input
-
-11.1 Formatted Input Strings
-============================
-
-`gmp_scanf' and friends accept format strings similar to the standard C
-`scanf' (*note Formatted Input: (libc)Formatted Input.). A format
-specification is of the form
-
- % [flags] [width] [type] conv
-
- MPIR adds types `Z', `Q' and `F' for `mpz_t', `mpq_t' and `mpf_t'
-respectively. `Z' and `Q' behave like integers. `Q' will read a `/'
-and a denominator, if present. `F' behaves like a float.
-
- MPIR variables don't require an `&' when passed to `gmp_scanf', since
-they're already "call-by-reference". For example,
-
- /* to read say "a(5) = 1234" */
- int n;
- mpz_t z;
- gmp_scanf ("a(%d) = %Zd\n", &n, z);
-
- mpq_t q1, q2;
- gmp_sscanf ("0377 + 0x10/0x11", "%Qi + %Qi", q1, q2);
-
- /* to read say "topleft (1.55,-2.66)" */
- mpf_t x, y;
- char buf[32];
- gmp_scanf ("%31s (%Ff,%Ff)", buf, x, y);
-
- All the standard C `scanf' types behave the same as in the C library
-`scanf', and can be freely intermixed with the MPIR extensions. In the
-current implementation the standard parts of the format string are
-simply handed to `scanf' and only the MPIR extensions handled directly.
-
- The flags accepted are as follows. `a' and `'' will depend on
-support from the C library, and `'' cannot be used with MPIR types.
-
- * read but don't store
- a allocate a buffer (string conversions)
- ' grouped digits, GLIBC style (not MPIR
- types)
-
- The standard types accepted are as follows. `h' and `l' are
-portable, the rest will depend on the compiler (or include files) for
-the type and the C library for the input.
-
- h short
- hh char
- j intmax_t or uintmax_t
- l long int, double or wchar_t
- ll long long
- L long double
- q quad_t or u_quad_t
- t ptrdiff_t
- z size_t
-
-The MPIR types are
-
- F mpf_t, float conversions
- Q mpq_t, integer conversions
- Z mpz_t, integer conversions
-
- The conversions accepted are as follows. `p' and `[' will depend on
-support from the C library, the rest are standard.
-
- c character or characters
- d decimal integer
- e E f g G float
- i integer with base indicator
- n characters read so far
- o octal integer
- p pointer
- s string of non-whitespace characters
- u decimal integer
- x X hex integer
- [ string of characters in a set
-
- `e', `E', `f', `g' and `G' are identical, they all read either fixed
-point or scientific format, and either upper or lower case `e' for the
-exponent in scientific format.
-
- C99 style hex float format (`printf %a', *note Formatted Output
-Strings::) is always accepted for `mpf_t', but for the standard float
-types it will depend on the C library.
-
- `x' and `X' are identical, both accept both upper and lower case
-hexadecimal.
-
- `o', `u', `x' and `X' all read positive or negative values. For the
-standard C types these are described as "unsigned" conversions, but
-that merely affects certain overflow handling, negatives are still
-allowed (per `strtoul', *note Parsing of Integers: (libc)Parsing of
-Integers.). For MPIR types there are no overflows, so `d' and `u' are
-identical.
-
- `Q' type reads the numerator and (optional) denominator as given.
-If the value might not be in canonical form then `mpq_canonicalize'
-must be called before using it in any calculations (*note Rational
-Number Functions::).
-
- `Qi' will read a base specification separately for the numerator and
-denominator. For example `0x10/11' would be 16/11, whereas `0x10/0x11'
-would be 16/17.
-
- `n' can be used with any of the types above, even the MPIR types.
-`*' to suppress assignment is allowed, though in that case it would do
-nothing at all.
-
- Other conversions or types that might be accepted by the C library
-`scanf' cannot be used through `gmp_scanf'.
-
- Whitespace is read and discarded before a field, except for `c' and
-`[' conversions.
-
- For float conversions, the decimal point character (or string)
-expected is taken from the current locale settings on systems which
-provide `localeconv' (*note Locales and Internationalization:
-(libc)Locales.). The C library will normally do the same for standard
-float input.
-
- The format string is only interpreted as plain `char's, multibyte
-characters are not recognised. Perhaps this will change in the future.
-
-
-File: mpir.info, Node: Formatted Input Functions, Next: C++ Formatted Input, Prev: Formatted Input Strings, Up: Formatted Input
-
-11.2 Formatted Input Functions
-==============================
-
-Each of the following functions is similar to the corresponding C
-library function. The plain `scanf' forms take a variable argument
-list. The `vscanf' forms take an argument pointer, see *note Variadic
-Functions: (libc)Variadic Functions, or `man 3 va_start'.
-
- It should be emphasised that if a format string is invalid, or the
-arguments don't match what the format specifies, then the behaviour of
-any of these functions will be unpredictable. GCC format string
-checking is not available, since it doesn't recognise the MPIR
-extensions.
-
- No overlap is permitted between the FMT string and any of the results
-produced.
-
- -- Function: int gmp_scanf (const char *FMT, ...)
- -- Function: int gmp_vscanf (const char *FMT, va_list AP)
- Read from the standard input `stdin'.
-
- -- Function: int gmp_fscanf (FILE *FP, const char *FMT, ...)
- -- Function: int gmp_vfscanf (FILE *FP, const char *FMT, va_list AP)
- Read from the stream FP.
-
- -- Function: int gmp_sscanf (const char *S, const char *FMT, ...)
- -- Function: int gmp_vsscanf (const char *S, const char *FMT, va_list
- AP)
- Read from a null-terminated string S.
-
- The return value from each of these functions is the same as the
-standard C99 `scanf', namely the number of fields successfully parsed
-and stored. `%n' fields and fields read but suppressed by `*' don't
-count towards the return value.
-
- If end of input (or a file error) is reached before a character for
-a field or a literal, and if no previous non-suppressed fields have
-matched, then the return value is `EOF' instead of 0. A whitespace
-character in the format string is only an optional match and doesn't
-induce an `EOF' in this fashion. Leading whitespace read and discarded
-for a field don't count as characters for that field.
-
- For the MPIR types, input parsing follows C99 rules, namely one
-character of lookahead is used and characters are read while they
-continue to meet the format requirements. If this doesn't provide a
-complete number then the function terminates, with that field not
-stored nor counted towards the return value. For instance with `mpf_t'
-an input `1.23e-XYZ' would be read up to the `X' and that character
-pushed back since it's not a digit. The string `1.23e-' would then be
-considered invalid since an `e' must be followed by at least one digit.
-
- For the standard C types, in the current implementation MPIR calls
-the C library `scanf' functions, which might have looser rules about
-what constitutes a valid input.
-
- Note that `gmp_sscanf' is the same as `gmp_fscanf' and only does one
-character of lookahead when parsing. Although clearly it could look at
-its entire input, it is deliberately made identical to `gmp_fscanf',
-the same way C99 `sscanf' is the same as `fscanf'.
-
-
-File: mpir.info, Node: C++ Formatted Input, Prev: Formatted Input Functions, Up: Formatted Input
-
-11.3 C++ Formatted Input
-========================
-
-The following functions are provided in `libmpirxx' (*note Headers and
-Libraries::), which is built only if C++ support is enabled (*note
-Build Options::). Prototypes are available from `<mpir.h>'.
-
- -- Function: istream& operator>> (istream& STREAM, mpz_t ROP)
- Read ROP from STREAM, using its `ios' formatting settings.
-
- -- Function: istream& operator>> (istream& STREAM, mpq_t ROP)
- An integer like `123' will be read, or a fraction like `5/9'. No
- whitespace is allowed around the `/'. If the fraction is not in
- canonical form then `mpq_canonicalize' must be called (*note
- Rational Number Functions::) before operating on it.
-
- As per integer input, an `0' or `0x' base indicator is read when
- none of `ios::dec', `ios::oct' or `ios::hex' are set. This is
- done separately for numerator and denominator, so that for instance
- `0x10/11' is 16/11 and `0x10/0x11' is 16/17.
-
- -- Function: istream& operator>> (istream& STREAM, mpf_t ROP)
- Read ROP from STREAM, using its `ios' formatting settings.
-
- Hex or octal floats are not supported, but might be in the future,
- or perhaps it's best to accept only what the standard float
- `operator>>' does.
-
- Note that digit grouping specified by the `istream' locale is
-currently not accepted. Perhaps this will change in the future.
-
-
- These operators mean that MPIR types can be read in the usual C++
-way, for example,
-
- mpz_t z;
- ...
- cin >> z;
-
- But note that `istream' input (and `ostream' output, *note C++
-Formatted Output::) is the only overloading available for the MPIR
-types and that for instance using `+' with an `mpz_t' will have
-unpredictable results. For classes with overloading, see *note C++
-Class Interface::.
-
-
-File: mpir.info, Node: C++ Class Interface, Next: Custom Allocation, Prev: Formatted Input, Up: Top
-
-12 C++ Class Interface
-**********************
-
-This chapter describes the C++ class based interface to MPIR.
-
- All MPIR C language types and functions can be used in C++ programs,
-since `mpir.h' has `extern "C"' qualifiers, but the class interface
-offers overloaded functions and operators which may be more convenient.
-
- Due to the implementation of this interface, a reasonably recent C++
-compiler is required, one supporting namespaces, partial specialization
-of templates and member templates. For GCC this means version 2.91 or
-later.
-
- *Everything described in this chapter is to be considered preliminary
-and might be subject to incompatible changes if some unforeseen
-difficulty reveals itself.*
-
-* Menu:
-
-* C++ Interface General::
-* C++ Interface Integers::
-* C++ Interface Rationals::
-* C++ Interface Floats::
-* C++ Interface Random Numbers::
-* C++ Interface Limitations::
-
-
-File: mpir.info, Node: C++ Interface General, Next: C++ Interface Integers, Prev: C++ Class Interface, Up: C++ Class Interface
-
-12.1 C++ Interface General
-==========================
-
-All the C++ classes and functions are available with
-
- #include <mpirxx.h>
-
- Programs should be linked with the `libmpirxx' and `libmpir'
-libraries. For example,
-
- g++ mycxxprog.cc -lmpirxx -lmpir
-
-The classes defined are
-
- -- Class: mpz_class
- -- Class: mpq_class
- -- Class: mpf_class
-
- The standard operators and various standard functions are overloaded
-to allow arithmetic with these classes. For example,
-
- int
- main (void)
- {
- mpz_class a, b, c;
-
- a = 1234;
- b = "-5678";
- c = a+b;
- cout << "sum is " << c << "\n";
- cout << "absolute value is " << abs(c) << "\n";
-
- return 0;
- }
-
- An important feature of the implementation is that an expression like
-`a=b+c' results in a single call to the corresponding `mpz_add',
-without using a temporary for the `b+c' part. Expressions which by
-their nature imply intermediate values, like `a=b*c+d*e', still use
-temporaries though.
-
- The classes can be freely intermixed in expressions, as can the
-classes and the standard types `mpir_si', `mpir_ui' and `double'.
-Smaller types like `int' or `float' can also be intermixed, since C++
-will promote them.
-
- Note that `bool' is not accepted directly, but must be explicitly
-cast to an `int' first. This is because C++ will automatically convert
-any pointer to a `bool', so if MPIR accepted `bool' it would make all
-sorts of invalid class and pointer combinations compile but almost
-certainly not do anything sensible.
-
- Conversions back from the classes to standard C++ types aren't done
-automatically, instead member functions like `get_si' are provided (see
-the following sections for details).
-
- Also there are no automatic conversions from the classes to the
-corresponding MPIR C types, instead a reference to the underlying C
-object can be obtained with the following functions,
-
- -- Function: mpz_t mpz_class::get_mpz_t ()
- -- Function: mpq_t mpq_class::get_mpq_t ()
- -- Function: mpf_t mpf_class::get_mpf_t ()
-
- These can be used to call a C function which doesn't have a C++ class
-interface. For example to set `a' to the GCD of `b' and `c',
-
- mpz_class a, b, c;
- ...
- mpz_gcd (a.get_mpz_t(), b.get_mpz_t(), c.get_mpz_t());
-
- In the other direction, a class can be initialized from the
-corresponding MPIR C type, or assigned to if an explicit constructor is
-used. In both cases this makes a copy of the value, it doesn't create
-any sort of association. For example,
-
- mpz_t z;
- // ... init and calculate z ...
- mpz_class x(z);
- mpz_class y;
- y = mpz_class (z);
-
- There are no namespace setups in `mpirxx.h', all types and functions
-are simply put into the global namespace. This is what `mpir.h' has
-done in the past, and continues to do for compatibility. The extras
-provided by `mpirxx.h' follow MPIR naming conventions and are unlikely
-to clash with anything.
-
-
-File: mpir.info, Node: C++ Interface Integers, Next: C++ Interface Rationals, Prev: C++ Interface General, Up: C++ Class Interface
-
-12.2 C++ Interface Integers
-===========================
-
- -- Function: void mpz_class::mpz_class (type N)
- Construct an `mpz_class'. All the standard C++ types may be used
- `long double', and all the MPIR C++ classes can be used. Any
- necessary conversion follows the corresponding C function, for
- example `double' follows `mpz_set_d' (*note Assigning Integers::).
-
- -- Function: void mpz_class::mpz_class (mpz_t Z)
- Construct an `mpz_class' from an `mpz_t'. The value in Z is
- copied into the new `mpz_class', there won't be any permanent
- association between it and Z.
-
- -- Function: void mpz_class::mpz_class (const char *S)
- -- Function: void mpz_class::mpz_class (const char *S, int BASE = 0)
- -- Function: void mpz_class::mpz_class (const string& S)
- -- Function: void mpz_class::mpz_class (const string& S, int BASE = 0)
- Construct an `mpz_class' converted from a string using
- `mpz_set_str' (*note Assigning Integers::).
-
- If the string is not a valid integer, an `std::invalid_argument'
- exception is thrown. The same applies to `operator='.
-
- -- Function: mpz_class operator/ (mpz_class A, mpz_class D)
- -- Function: mpz_class operator% (mpz_class A, mpz_class D)
- Divisions involving `mpz_class' round towards zero, as per the
- `mpz_tdiv_q' and `mpz_tdiv_r' functions (*note Integer Division::).
- This is the same as the C99 `/' and `%' operators.
-
- The `mpz_fdiv...' or `mpz_cdiv...' functions can always be called
- directly if desired. For example,
-
- mpz_class q, a, d;
- ...
- mpz_fdiv_q (q.get_mpz_t(), a.get_mpz_t(), d.get_mpz_t());
-
- -- Function: mpz_class abs (mpz_class OP1)
- -- Function: int cmp (mpz_class OP1, type OP2)
- -- Function: int cmp (type OP1, mpz_class OP2)
- -- Function: bool mpz_class::fits_sint_p (void)
- -- Function: bool mpz_class::fits_slong_p (void)
- -- Function: bool mpz_class::fits_sshort_p (void)
- -- Function: bool mpz_class::fits_uint_p (void)
- -- Function: bool mpz_class::fits_ulong_p (void)
- -- Function: bool mpz_class::fits_ushort_p (void)
- -- Function: double mpz_class::get_d (void)
- -- Function: mpir_si mpz_class::get_si (void)
- -- Function: string mpz_class::get_str (int BASE = 10)
- -- Function: mpir_ui mpz_class::get_ui (void)
- -- Function: int mpz_class::set_str (const char *STR, int BASE)
- -- Function: int mpz_class::set_str (const string& STR, int BASE)
- -- Function: int sgn (mpz_class OP)
- -- Function: mpz_class sqrt (mpz_class OP)
- These functions provide a C++ class interface to the corresponding
- MPIR C routines.
-
- `cmp' can be used with any of the classes or the standard C++
- types, except `long double'.
-
-
- Overloaded operators for combinations of `mpz_class' and `double'
-are provided for completeness, but it should be noted that if the given
-`double' is not an integer then the way any rounding is done is
-currently unspecified. The rounding might take place at the start, in
-the middle, or at the end of the operation, and it might change in the
-future.
-
- Conversions between `mpz_class' and `double', however, are defined
-to follow the corresponding C functions `mpz_get_d' and `mpz_set_d'.
-And comparisons are always made exactly, as per `mpz_cmp_d'.
-
-
-File: mpir.info, Node: C++ Interface Rationals, Next: C++ Interface Floats, Prev: C++ Interface Integers, Up: C++ Class Interface
-
-12.3 C++ Interface Rationals
-============================
-
-In all the following constructors, if a fraction is given then it
-should be in canonical form, or if not then `mpq_class::canonicalize'
-called.
-
- -- Function: void mpq_class::mpq_class (type OP)
- -- Function: void mpq_class::mpq_class (integer NUM, integer DEN)
- Construct an `mpq_class'. The initial value can be a single value
- of any type, or a pair of integers (`mpz_class' or standard C++
- integer types) representing a fraction, except that `long double'
- is not supported. For example,
-
- mpq_class q (99);
- mpq_class q (1.75);
- mpq_class q (1, 3);
-
- -- Function: void mpq_class::mpq_class (mpq_t Q)
- Construct an `mpq_class' from an `mpq_t'. The value in Q is
- copied into the new `mpq_class', there won't be any permanent
- association between it and Q.
-
- -- Function: void mpq_class::mpq_class (const char *S)
- -- Function: void mpq_class::mpq_class (const char *S, int BASE = 0)
- -- Function: void mpq_class::mpq_class (const string& S)
- -- Function: void mpq_class::mpq_class (const string& S, int BASE = 0)
- Construct an `mpq_class' converted from a string using
- `mpq_set_str' (*note Initializing Rationals::).
-
- If the string is not a valid rational, an `std::invalid_argument'
- exception is thrown. The same applies to `operator='.
-
- -- Function: void mpq_class::canonicalize ()
- Put an `mpq_class' into canonical form, as per *note Rational
- Number Functions::. All arithmetic operators require their
- operands in canonical form, and will return results in canonical
- form.
-
- -- Function: mpq_class abs (mpq_class OP)
- -- Function: int cmp (mpq_class OP1, type OP2)
- -- Function: int cmp (type OP1, mpq_class OP2)
- -- Function: double mpq_class::get_d (void)
- -- Function: string mpq_class::get_str (int BASE = 10)
- -- Function: int mpq_class::set_str (const char *STR, int BASE)
- -- Function: int mpq_class::set_str (const string& STR, int BASE)
- -- Function: int sgn (mpq_class OP)
- These functions provide a C++ class interface to the corresponding
- MPIR C routines.
-
- `cmp' can be used with any of the classes or the standard C++
- types, except `long double'.
-
- -- Function: mpz_class& mpq_class::get_num ()
- -- Function: mpz_class& mpq_class::get_den ()
- Get a reference to an `mpz_class' which is the numerator or
- denominator of an `mpq_class'. This can be used both for read and
- write access. If the object returned is modified, it modifies the
- original `mpq_class'.
-
- If direct manipulation might produce a non-canonical value, then
- `mpq_class::canonicalize' must be called before further operations.
-
- -- Function: mpz_t mpq_class::get_num_mpz_t ()
- -- Function: mpz_t mpq_class::get_den_mpz_t ()
- Get a reference to the underlying `mpz_t' numerator or denominator
- of an `mpq_class'. This can be passed to C functions expecting an
- `mpz_t'. Any modifications made to the `mpz_t' will modify the
- original `mpq_class'.
-
- If direct manipulation might produce a non-canonical value, then
- `mpq_class::canonicalize' must be called before further operations.
-
- -- Function: istream& operator>> (istream& STREAM, mpq_class& ROP);
- Read ROP from STREAM, using its `ios' formatting settings, the
- same as `mpq_t operator>>' (*note C++ Formatted Input::).
-
- If the ROP read might not be in canonical form then
- `mpq_class::canonicalize' must be called.
-
-
-File: mpir.info, Node: C++ Interface Floats, Next: C++ Interface Random Numbers, Prev: C++ Interface Rationals, Up: C++ Class Interface
-
-12.4 C++ Interface Floats
-=========================
-
-When an expression requires the use of temporary intermediate
-`mpf_class' values, like `f=g*h+x*y', those temporaries will have the
-same precision as the destination `f'. Explicit constructors can be
-used if this doesn't suit.
-
- -- Function: mpf_class::mpf_class (type OP)
- -- Function: mpf_class::mpf_class (type OP, mpir_ui PREC)
- Construct an `mpf_class'. Any standard C++ type can be used,
- except `long double', and any of the MPIR C++ classes can be used.
-
- If PREC is given, the initial precision is that value, in bits. If
- PREC is not given, then the initial precision is determined by the
- type of OP given. An `mpz_class', `mpq_class', or C++ builtin
- type will give the default `mpf' precision (*note Initializing
- Floats::). An `mpf_class' or expression will give the precision
- of that value. The precision of a binary expression is the higher
- of the two operands.
-
- mpf_class f(1.5); // default precision
- mpf_class f(1.5, 500); // 500 bits (at least)
- mpf_class f(x); // precision of x
- mpf_class f(abs(x)); // precision of x
- mpf_class f(-g, 1000); // 1000 bits (at least)
- mpf_class f(x+y); // greater of precisions of x and y
-
- -- Function: void mpf_class::mpf_class (const char *S)
- -- Function: void mpf_class::mpf_class (const char *S, mpir_ui PREC,
- int BASE = 0)
- -- Function: void mpf_class::mpf_class (const string& S)
- -- Function: void mpf_class::mpf_class (const string& S, mpir_ui PREC,
- int BASE = 0)
- Construct an `mpf_class' converted from a string using
- `mpf_set_str' (*note Assigning Floats::). If PREC is given, the
- initial precision is that value, in bits. If not, the default
- `mpf' precision (*note Initializing Floats::) is used.
-
- If the string is not a valid float, an `std::invalid_argument'
- exception is thrown. The same applies to `operator='.
-
- -- Function: mpf_class& mpf_class::operator= (type OP)
- Convert and store the given OP value to an `mpf_class' object. The
- same types are accepted as for the constructors above.
-
- Note that `operator=' only stores a new value, it doesn't copy or
- change the precision of the destination, instead the value is
- truncated if necessary. This is the same as `mpf_set' etc. Note
- in particular this means for `mpf_class' a copy constructor is not
- the same as a default constructor plus assignment.
-
- mpf_class x (y); // x created with precision of y
-
- mpf_class x; // x created with default precision
- x = y; // value truncated to that precision
-
- Applications using templated code may need to be careful about the
- assumptions the code makes in this area, when working with
- `mpf_class' values of various different or non-default precisions.
- For instance implementations of the standard `complex' template
- have been seen in both styles above, though of course `complex' is
- normally only actually specified for use with the builtin float
- types.
-
- -- Function: mpf_class abs (mpf_class OP)
- -- Function: mpf_class ceil (mpf_class OP)
- -- Function: int cmp (mpf_class OP1, type OP2)
- -- Function: int cmp (type OP1, mpf_class OP2)
- -- Function: bool mpf_class::fits_sint_p (void)
- -- Function: bool mpf_class::fits_slong_p (void)
- -- Function: bool mpf_class::fits_sshort_p (void)
- -- Function: bool mpf_class::fits_uint_p (void)
- -- Function: bool mpf_class::fits_ulong_p (void)
- -- Function: bool mpf_class::fits_ushort_p (void)
- -- Function: mpf_class floor (mpf_class OP)
- -- Function: mpf_class hypot (mpf_class OP1, mpf_class OP2)
- -- Function: double mpf_class::get_d (void)
- -- Function: mpir_si mpf_class::get_si (void)
- -- Function: string mpf_class::get_str (mp_exp_t& EXP, int BASE = 10,
- size_t DIGITS = 0)
- -- Function: mpir_ui mpf_class::get_ui (void)
- -- Function: int mpf_class::set_str (const char *STR, int BASE)
- -- Function: int mpf_class::set_str (const string& STR, int BASE)
- -- Function: int sgn (mpf_class OP)
- -- Function: mpf_class sqrt (mpf_class OP)
- -- Function: mpf_class trunc (mpf_class OP)
- These functions provide a C++ class interface to the corresponding
- MPIR C routines.
-
- `cmp' can be used with any of the classes or the standard C++
- types, except `long double'.
-
- The accuracy provided by `hypot' is not currently guaranteed.
-
- -- Function: mp_bitcnt_t mpf_class::get_prec ()
- -- Function: void mpf_class::set_prec (mp_bitcnt_t PREC)
- -- Function: void mpf_class::set_prec_raw (mp_bitcnt_t PREC)
- Get or set the current precision of an `mpf_class'.
-
- The restrictions described for `mpf_set_prec_raw' (*note
- Initializing Floats::) apply to `mpf_class::set_prec_raw'. Note
- in particular that the `mpf_class' must be restored to it's
- allocated precision before being destroyed. This must be done by
- application code, there's no automatic mechanism for it.
-
-
-File: mpir.info, Node: C++ Interface Random Numbers, Next: C++ Interface Limitations, Prev: C++ Interface Floats, Up: C++ Class Interface
-
-12.5 C++ Interface Random Numbers
-=================================
-
- -- Class: gmp_randclass
- The C++ class interface to the MPIR random number functions uses
- `gmp_randclass' to hold an algorithm selection and current state,
- as per `gmp_randstate_t'.
-
- -- Function: gmp_randclass::gmp_randclass (void (*RANDINIT)
- (gmp_randstate_t, ...), ...)
- Construct a `gmp_randclass', using a call to the given RANDINIT
- function (*note Random State Initialization::). The arguments
- expected are the same as RANDINIT, but with `mpz_class' instead of
- `mpz_t'. For example,
-
- gmp_randclass r1 (gmp_randinit_default);
- gmp_randclass r2 (gmp_randinit_lc_2exp_size, 32);
- gmp_randclass r3 (gmp_randinit_lc_2exp, a, c, m2exp);
- gmp_randclass r4 (gmp_randinit_mt);
-
- `gmp_randinit_lc_2exp_size' will fail if the size requested is too
- big, an `std::length_error' exception is thrown in that case.
-
- -- Function: void gmp_randclass::seed (mpir_ui S)
- -- Function: void gmp_randclass::seed (mpz_class S)
- Seed a random number generator. See *note Random Number
- Functions::, for how to choose a good seed.
-
- -- Function: mpz_class gmp_randclass::get_z_bits (mpir_ui BITS)
- -- Function: mpz_class gmp_randclass::get_z_bits (mpz_class BITS)
- Generate a random integer with a specified number of bits.
-
- -- Function: mpz_class gmp_randclass::get_z_range (mpz_class N)
- Generate a random integer in the range 0 to N-1 inclusive.
-
- -- Function: mpf_class gmp_randclass::get_f ()
- -- Function: mpf_class gmp_randclass::get_f (mpir_ui PREC)
- Generate a random float F in the range 0 <= F < 1. F will be to
- PREC bits precision, or if PREC is not given then to the precision
- of the destination. For example,
-
- gmp_randclass r;
- ...
- mpf_class f (0, 512); // 512 bits precision
- f = r.get_f(); // random number, 512 bits
-
-
-File: mpir.info, Node: C++ Interface Limitations, Prev: C++ Interface Random Numbers, Up: C++ Class Interface
-
-12.6 C++ Interface Limitations
-==============================
-
-`mpq_class' and Templated Reading
- A generic piece of template code probably won't know that
- `mpq_class' requires a `canonicalize' call if inputs read with
- `operator>>' might be non-canonical. This can lead to incorrect
- results.
-
- `operator>>' behaves as it does for reasons of efficiency. A
- canonicalize can be quite time consuming on large operands, and is
- best avoided if it's not necessary.
-
- But this potential difficulty reduces the usefulness of
- `mpq_class'. Perhaps a mechanism to tell `operator>>' what to do
- will be adopted in the future, maybe a preprocessor define, a
- global flag, or an `ios' flag pressed into service. Or maybe, at
- the risk of inconsistency, the `mpq_class' `operator>>' could
- canonicalize and leave `mpq_t' `operator>>' not doing so, for use
- on those occasions when that's acceptable. Send feedback or
- alternate ideas to `http://groups.google.com/group/mpir-devel'.
-
-Subclassing
- Subclassing the MPIR C++ classes works, but is not currently
- recommended.
-
- Expressions involving subclasses resolve correctly (or seem to),
- but in normal C++ fashion the subclass doesn't inherit
- constructors and assignments. There's many of those in the MPIR
- classes, and a good way to reestablish them in a subclass is not
- yet provided.
-
-Templated Expressions
- A subtle difficulty exists when using expressions together with
- application-defined template functions. Consider the following,
- with `T' intended to be some numeric type,
-
- template <class T>
- T fun (const T &, const T &);
-
- When used with, say, plain `mpz_class' variables, it works fine:
- `T' is resolved as `mpz_class'.
-
- mpz_class f(1), g(2);
- fun (f, g); // Good
-
- But when one of the arguments is an expression, it doesn't work.
-
- mpz_class f(1), g(2), h(3);
- fun (f, g+h); // Bad
-
- This is because `g+h' ends up being a certain expression template
- type internal to `mpirxx.h', which the C++ template resolution
- rules are unable to automatically convert to `mpz_class'. The
- workaround is simply to add an explicit cast.
-
- mpz_class f(1), g(2), h(3);
- fun (f, mpz_class(g+h)); // Good
-
- Similarly, within `fun' it may be necessary to cast an expression
- to type `T' when calling a templated `fun2'.
-
- template <class T>
- void fun (T f, T g)
- {
- fun2 (f, f+g); // Bad
- }
-
- template <class T>
- void fun (T f, T g)
- {
- fun2 (f, T(f+g)); // Good
- }
-
-
-File: mpir.info, Node: Custom Allocation, Next: Language Bindings, Prev: C++ Class Interface, Up: Top
-
-13 Custom Allocation
-********************
-
-By default MPIR uses `malloc', `realloc' and `free' for memory
-allocation, and if they fail MPIR prints a message to the standard
-error output and terminates the program.
-
- Alternate functions can be specified, to allocate memory in a
-different way or to have a different error action on running out of
-memory.
-
- -- Function: void mp_set_memory_functions (
- void *(*ALLOC_FUNC_PTR) (size_t),
- void *(*REALLOC_FUNC_PTR) (void *, size_t, size_t),
- void (*FREE_FUNC_PTR) (void *, size_t))
- Replace the current allocation functions from the arguments. If
- an argument is `NULL', the corresponding default function is used.
-
- These functions will be used for all memory allocation done by
- MPIR, apart from temporary space from `alloca' if that function is
- available and MPIR is configured to use it (*note Build Options::).
-
- *Be sure to call `mp_set_memory_functions' only when there are no
- active MPIR objects allocated using the previous memory functions!
- Usually that means calling it before any other MPIR function.*
-
- The functions supplied should fit the following declarations:
-
- -- Function: void * allocate_function (size_t ALLOC_SIZE)
- Return a pointer to newly allocated space with at least ALLOC_SIZE
- bytes.
-
- -- Function: void * reallocate_function (void *PTR, size_t OLD_SIZE,
- size_t NEW_SIZE)
- Resize a previously allocated block PTR of OLD_SIZE bytes to be
- NEW_SIZE bytes.
-
- The block may be moved if necessary or if desired, and in that
- case the smaller of OLD_SIZE and NEW_SIZE bytes must be copied to
- the new location. The return value is a pointer to the resized
- block, that being the new location if moved or just PTR if not.
-
- PTR is never `NULL', it's always a previously allocated block.
- NEW_SIZE may be bigger or smaller than OLD_SIZE.
-
- -- Function: void free_function (void *PTR, size_t SIZE)
- De-allocate the space pointed to by PTR.
-
- PTR is never `NULL', it's always a previously allocated block of
- SIZE bytes.
-
- A "byte" here means the unit used by the `sizeof' operator.
-
- The OLD_SIZE parameters to REALLOCATE_FUNCTION and FREE_FUNCTION are
-passed for convenience, but of course can be ignored if not needed.
-The default functions using `malloc' and friends for instance don't use
-them.
-
- No error return is allowed from any of these functions, if they
-return then they must have performed the specified operation. In
-particular note that ALLOCATE_FUNCTION or REALLOCATE_FUNCTION mustn't
-return `NULL'.
-
- Getting a different fatal error action is a good use for custom
-allocation functions, for example giving a graphical dialog rather than
-the default print to `stderr'. How much is possible when genuinely out
-of memory is another question though.
-
- There's currently no defined way for the allocation functions to
-recover from an error such as out of memory, they must terminate
-program execution. A `longjmp' or throwing a C++ exception will have
-undefined results. This may change in the future.
-
- MPIR may use allocated blocks to hold pointers to other allocated
-blocks. This will limit the assumptions a conservative garbage
-collection scheme can make.
-
- Any custom allocation functions must align pointers to limb
-boundaries. Thus if a limb is eight bytes (e.g. on x86_64), then all
-blocks must be aligned to eight byte boundaries. Check the
-configuration options for the custom allocation library in use. It is
-not necessary to align blocks to SSE boundaries even when SSE code is
-used. All MPIR assembly routines assume limb boundary alignment only
-(which is the default for most standard memory managers).
-
- Since the default MPIR allocation uses `malloc' and friends, those
-functions will be linked in even if the first thing a program does is an
-`mp_set_memory_functions'. It's necessary to change the MPIR sources if
-this is a problem.
-
-
- -- Function: void mp_get_memory_functions (
- void *(**ALLOC_FUNC_PTR) (size_t),
- void *(**REALLOC_FUNC_PTR) (void *, size_t, size_t),
- void (**FREE_FUNC_PTR) (void *, size_t))
- Get the current allocation functions, storing function pointers to
- the locations given by the arguments. If an argument is `NULL',
- that function pointer is not stored.
-
- For example, to get just the current free function,
-
- void (*freefunc) (void *, size_t);
-
- mp_get_memory_functions (NULL, NULL, &freefunc);
-
-
-File: mpir.info, Node: Language Bindings, Next: Algorithms, Prev: Custom Allocation, Up: Top
-
-14 Language Bindings
-********************
-
-The following packages and projects offer access to MPIR from languages
-other than C, though perhaps with varying levels of functionality and
-efficiency.
-
-
-C++
- * MPIR C++ class interface, *note C++ Class Interface::
- Straightforward interface, expression templates to eliminate
- temporaries.
-
- * ALP `http://www-sop.inria.fr/saga/logiciels/ALP/'
- Linear algebra and polynomials using templates.
-
- * CLN `http://www.ginac.de/CLN/'
- High level classes for arithmetic.
-
- * LiDIA `http://www.informatik.tu-darmstadt.de/TI/LiDIA/'
- A C++ library for computational number theory.
-
- * Linbox `http://www.linalg.org/'
- Sparse vectors and matrices.
-
- * NTL `http://www.shoup.net/ntl/'
- A C++ number theory library.
-
-Eiffel
- * Eiffel Interface `http://www.eiffelroom.org/node/407'
- An Eiffel Interface to MPFR, MPC and MPIR by Chris Saunders.
-
-Fortran
- * Omni F77 `http://phase.hpcc.jp/Omni/home.html'
- Arbitrary precision floats.
-
-Haskell
- * Glasgow Haskell Compiler `http://www.haskell.org/ghc/'
-
-Java
- * Kaffe `http://www.kaffe.org/'
-
-Lisp
- * Embeddable Common Lisp
- `http://ecls.sourceforge.net/download.html'
-
- * GNU Common Lisp `http://www.gnu.org/software/gcl/gcl.html'
-
- * Librep `http://librep.sourceforge.net/'
-
- * XEmacs (21.5.18 beta and up) `http://www.xemacs.org'
- Optional big integers, rationals and floats using MPIR.
-
-M4
- * GNU m4 betas `http://www.seindal.dk/rene/gnu/'
- Optionally provides an arbitrary precision `mpeval'.
-
-ML
- * MLton compiler `http://mlton.org/'
-
-Objective Caml
- * Numerix `http://pauillac.inria.fr/~quercia/'
- Optionally using GMP.
-
-Oz
- * Mozart `http://www.mozart-oz.org/'
-
-Pascal
- * GNU Pascal Compiler `http://www.gnu-pascal.de/'
- GMP unit.
-
- * Numerix `http://pauillac.inria.fr/~quercia/'
- For Free Pascal, optionally using GMP.
-
-Perl
- * GMP module, see `demos/perl' on the MPIR website.
-
- * Math::GMP `http://www.cpan.org/'
- Compatible with Math::BigInt, but not as many functions as
- the GMP module above.
-
- * Math::BigInt::GMP `http://www.cpan.org/'
- Plug Math::GMP into normal Math::BigInt operations.
-
-PHP
- * mpz module in the main distribution, `http://php.net/'
-
-Pike
- * mpz module in the standard distribution,
- `http://pike.ida.liu.se/'
-
-Prolog
- * SWI Prolog `http://www.swi-prolog.org/'
- Arbitrary precision floats.
-
-Python
- * mpz module in the standard distribution,
- `http://www.python.org/'
-
- * GMPY `http://gmpy.sourceforge.net/'
-
-Scheme
- * GNU Guile (upcoming 1.8)
- `http://www.gnu.org/software/guile/guile.html'
-
- * RScheme `http://www.rscheme.org/'
-
-Smalltalk
- * GNU Smalltalk
- `http://www.smalltalk.org/versions/GNUSmalltalk.html'
-
-Other
- * ALGLIB `http://www.alglib.net/'
- Numerical analysis and data processing library.
-
- * Axiom `http://savannah.nongnu.org/projects/axiom'
- Computer algebra using GCL.
-
- * GiNaC `http://www.ginac.de/'
- C++ computer algebra using CLN.
-
- * GOO `http://www.googoogaga.org/'
- Dynamic object oriented language.
-
- * Maxima `http://www.ma.utexas.edu/users/wfs/maxima.html'
- Macsyma computer algebra using GCL.
-
- * Q `http://q-lang.sourceforge.net/'
- Equational programming system.
-
- * Regina `http://regina.sourceforge.net/'
- Topological calculator.
-
- * Sage `http://www.sagemath.org/'
- Computer Algebra System written in Python and Cython.
-
- * Yacas `http://yacas.sourceforge.net/homepage.html'
- Yet another computer algebra system.
-
-
-
-File: mpir.info, Node: Algorithms, Next: Internals, Prev: Language Bindings, Up: Top
-
-15 Algorithms
-*************
-
-This chapter is an introduction to some of the algorithms used for
-various MPIR operations. The code is likely to be hard to understand
-without knowing something about the algorithms.
-
- Some MPIR internals are mentioned, but applications that expect to be
-compatible with future MPIR releases should take care to use only the
-documented functions.
-
-* Menu:
-
-* Multiplication Algorithms::
-* Division Algorithms::
-* Greatest Common Divisor Algorithms::
-* Powering Algorithms::
-* Root Extraction Algorithms::
-* Radix Conversion Algorithms::
-* Other Algorithms::
-* Assembler Coding::
-
-
-File: mpir.info, Node: Multiplication Algorithms, Next: Division Algorithms, Prev: Algorithms, Up: Algorithms
-
-15.1 Multiplication
-===================
-
-NxN limb multiplications and squares are done using one of six
-algorithms, as the size N increases.
-
- Algorithm Mul Threshold
- Basecase (none)
- Karatsuba `MUL_KARATSUBA_THRESHOLD'
- Toom-3 `MUL_TOOM3_THRESHOLD'
- Toom-4 `MUL_TOOM4_THRESHOLD'
- Toom-8(.5) `MUL_TOOM8H_THRESHOLD'
- FFT `MUL_FFT_FULL_THRESHOLD'
-
- Algorithm Sqr Threshold
- Basecase (none)
- Karatsuba `SQR_KARATSUBA_THRESHOLD'
- Toom-3 `SQR_TOOM3_THRESHOLD'
- Toom-4 `SQR_TOOM4_THRESHOLD'
- Toom-8 `SQR_TOOM8_THRESHOLD'
- FFT `SQR_FFT_FULL_THRESHOLD'
-
- NxM multiplications of operands with different sizes above
-`MUL_KARATSUBA_THRESHOLD' are done using unbalanced Toom algorithms or
-with the FFT. See (*note Unbalanced Multiplication::).
-
-* Menu:
-
-* Basecase Multiplication::
-* Karatsuba Multiplication::
-* Toom 3-Way Multiplication::
-* Toom 4-Way Multiplication::
-* FFT Multiplication::
-* Other Multiplication::
-* Unbalanced Multiplication::
-
-
-File: mpir.info, Node: Basecase Multiplication, Next: Karatsuba Multiplication, Prev: Multiplication Algorithms, Up: Multiplication Algorithms
-
-15.1.1 Basecase Multiplication
-------------------------------
-
-Basecase NxM multiplication is a straightforward rectangular set of
-cross-products, the same as long multiplication done by hand and for
-that reason sometimes known as the schoolbook or grammar school method.
-This is an O(N*M) algorithm. See Knuth section 4.3.1 algorithm M
-(*note References::), and the `mpn/generic/mul_basecase.c' code.
-
- Assembler implementations of `mpn_mul_basecase' are essentially the
-same as the generic C code, but have all the usual assembler tricks and
-obscurities introduced for speed.
-
- A square can be done in roughly half the time of a multiply, by
-using the fact that the cross products above and below the diagonal are
-the same. A triangle of products below the diagonal is formed, doubled
-(left shift by one bit), and then the products on the diagonal added.
-This can be seen in `mpn/generic/sqr_basecase.c'. Again the assembler
-implementations take essentially the same approach.
-
- u0 u1 u2 u3 u4
- +---+---+---+---+---+
- u0 | d | | | | |
- +---+---+---+---+---+
- u1 | | d | | | |
- +---+---+---+---+---+
- u2 | | | d | | |
- +---+---+---+---+---+
- u3 | | | | d | |
- +---+---+---+---+---+
- u4 | | | | | d |
- +---+---+---+---+---+
-
- In practice squaring isn't a full 2x faster than multiplying, it's
-usually around 1.5x. Less than 1.5x probably indicates
-`mpn_sqr_basecase' wants improving on that CPU.
-
- On some CPUs `mpn_mul_basecase' can be faster than the generic C
-`mpn_sqr_basecase' on some small sizes. `SQR_BASECASE_THRESHOLD' is
-the size at which to use `mpn_sqr_basecase', this will be zero if that
-routine should be used always.
-
-
-File: mpir.info, Node: Karatsuba Multiplication, Next: Toom 3-Way Multiplication, Prev: Basecase Multiplication, Up: Multiplication Algorithms
-
-15.1.2 Karatsuba Multiplication
--------------------------------
-
-The Karatsuba multiplication algorithm is described in Knuth section
-4.3.3 part A, and various other textbooks. A brief description is
-given here.
-
- The inputs x and y are treated as each split into two parts of equal
-length (or the most significant part one limb shorter if N is odd).
-
- high low
- +----------+----------+
- | x1 | x0 |
- +----------+----------+
-
- +----------+----------+
- | y1 | y0 |
- +----------+----------+
-
- Let b be the power of 2 where the split occurs, ie. if x0 is k limbs
-(y0 the same) then b=2^(k*mp_bits_per_limb). With that x=x1*b+x0 and
-y=y1*b+y0, and the following holds,
-
- x*y = (b^2+b)*x1*y1 - b*(x1-x0)*(y1-y0) + (b+1)*x0*y0
-
- This formula means doing only three multiplies of (N/2)x(N/2) limbs,
-whereas a basecase multiply of NxN limbs is equivalent to four
-multiplies of (N/2)x(N/2). The factors (b^2+b) etc represent the
-positions where the three products must be added.
-
- high low
- +--------+--------+ +--------+--------+
- | x1*y1 | | x0*y0 |
- +--------+--------+ +--------+--------+
- +--------+--------+
- add | x1*y1 |
- +--------+--------+
- +--------+--------+
- add | x0*y0 |
- +--------+--------+
- +--------+--------+
- sub | (x1-x0)*(y1-y0) |
- +--------+--------+
-
- The term (x1-x0)*(y1-y0) is best calculated as an absolute value,
-and the sign used to choose to add or subtract. Notice the sum
-high(x0*y0)+low(x1*y1) occurs twice, so it's possible to do 5*k limb
-additions, rather than 6*k, but in MPIR extra function call overheads
-outweigh the saving.
-
- Squaring is similar to multiplying, but with x=y the formula reduces
-to an equivalent with three squares,
-
- x^2 = (b^2+b)*x1^2 - b*(x1-x0)^2 + (b+1)*x0^2
-
- The final result is accumulated from those three squares the same
-way as for the three multiplies above. The middle term (x1-x0)^2 is now
-always positive.
-
- A similar formula for both multiplying and squaring can be
-constructed with a middle term (x1+x0)*(y1+y0). But those sums can
-exceed k limbs, leading to more carry handling and additions than the
-form above.
-
- Karatsuba multiplication is asymptotically an O(N^1.585) algorithm,
-the exponent being log(3)/log(2), representing 3 multiplies each 1/2
-the size of the inputs. This is a big improvement over the basecase
-multiply at O(N^2) and the advantage soon overcomes the extra additions
-Karatsuba performs. `MUL_KARATSUBA_THRESHOLD' can be as little as 10
-limbs. The `SQR' threshold is usually about twice the `MUL'.
-
- The basecase algorithm will take a time of the form M(N) = a*N^2 +
-b*N + c and the Karatsuba algorithm K(N) = 3*M(N/2) + d*N + e, which
-expands to K(N) = 3/4*a*N^2 + 3/2*b*N + 3*c + d*N + e. The factor 3/4
-for a means per-crossproduct speedups in the basecase code will
-increase the threshold since they benefit M(N) more than K(N). And
-conversely the 3/2 for b means linear style speedups of b will increase
-the threshold since they benefit K(N) more than M(N). The latter can
-be seen for instance when adding an optimized `mpn_sqr_diagonal' to
-`mpn_sqr_basecase'. Of course all speedups reduce total time, and in
-that sense the algorithm thresholds are merely of academic interest.
-
-
-File: mpir.info, Node: Toom 3-Way Multiplication, Next: Toom 4-Way Multiplication, Prev: Karatsuba Multiplication, Up: Multiplication Algorithms
-
-15.1.3 Toom 3-Way Multiplication
---------------------------------
-
-The Karatsuba formula is the simplest case of a general approach to
-splitting inputs that leads to both Toom and FFT algorithms. A
-description of Toom can be found in Knuth section 4.3.3, with an
-example 3-way calculation after Theorem A. The 3-way form used in MPIR
-is described here.
-
- The operands are each considered split into 3 pieces of equal length
-(or the most significant part 1 or 2 limbs shorter than the other two).
-
- high low
- +----------+----------+----------+
- | x2 | x1 | x0 |
- +----------+----------+----------+
-
- +----------+----------+----------+
- | y2 | y1 | y0 |
- +----------+----------+----------+
-
-These parts are treated as the coefficients of two polynomials
-
- X(t) = x2*t^2 + x1*t + x0
- Y(t) = y2*t^2 + y1*t + y0
-
- Let b equal the power of 2 which is the size of the x0, x1, y0 and
-y1 pieces, ie. if they're k limbs each then b=2^(k*mp_bits_per_limb).
-With this x=X(b) and y=Y(b).
-
- Let a polynomial W(t)=X(t)*Y(t) and suppose its coefficients are
-
- W(t) = w4*t^4 + w3*t^3 + w2*t^2 + w1*t + w0
-
- The w[i] are going to be determined, and when they are they'll give
-the final result using w=W(b), since x*y=X(b)*Y(b)=W(b). The
-coefficients will be roughly b^2 each, and the final W(b) will be an
-addition like,
-
- high low
- +-------+-------+
- | w4 |
- +-------+-------+
- +--------+-------+
- | w3 |
- +--------+-------+
- +--------+-------+
- | w2 |
- +--------+-------+
- +--------+-------+
- | w1 |
- +--------+-------+
- +-------+-------+
- | w0 |
- +-------+-------+
-
- The w[i] coefficients could be formed by a simple set of cross
-products, like w4=x2*y2, w3=x2*y1+x1*y2, w2=x2*y0+x1*y1+x0*y2 etc, but
-this would need all nine x[i]*y[j] for i,j=0,1,2, and would be
-equivalent merely to a basecase multiply. Instead the following
-approach is used.
-
- X(t) and Y(t) are evaluated and multiplied at 5 points, giving
-values of W(t) at those points. In MPIR the following points are used,
-
- Point Value
- t=0 x0 * y0, which gives w0 immediately
- t=1 (x2+x1+x0) * (y2+y1+y0)
- t=-1 (x2-x1+x0) * (y2-y1+y0)
- t=2 (4*x2+2*x1+x0) * (4*y2+2*y1+y0)
- t=inf x2 * y2, which gives w4 immediately
-
- At t=-1 the values can be negative and that's handled using the
-absolute values and tracking the sign separately. At t=inf the value
-is actually X(t)*Y(t)/t^4 in the limit as t approaches infinity, but
-it's much easier to think of as simply x2*y2 giving w4 immediately
-(much like x0*y0 at t=0 gives w0 immediately).
-
- Each of the points substituted into W(t)=w4*t^4+...+w0 gives a
-linear combination of the w[i] coefficients, and the value of those
-combinations has just been calculated.
-
- W(0) = w0
- W(1) = w4 + w3 + w2 + w1 + w0
- W(-1) = w4 - w3 + w2 - w1 + w0
- W(2) = 16*w4 + 8*w3 + 4*w2 + 2*w1 + w0
- W(inf) = w4
-
- This is a set of five equations in five unknowns, and some
-elementary linear algebra quickly isolates each w[i]. This involves
-adding or subtracting one W(t) value from another, and a couple of
-divisions by powers of 2 and one division by 3, the latter using the
-special `mpn_divexact_by3' (*note Exact Division::).
-
- The conversion of W(t) values to the coefficients is interpolation.
-A polynomial of degree 4 like W(t) is uniquely determined by values
-known at 5 different points. The points are arbitrary and can be
-chosen to make the linear equations come out with a convenient set of
-steps for quickly isolating the w[i].
-
- Squaring follows the same procedure as multiplication, but there's
-only one X(t) and it's evaluated at the 5 points, and those values
-squared to give values of W(t). The interpolation is then identical,
-and in fact the same `toom3_interpolate' subroutine is used for both
-squaring and multiplying.
-
- Toom-3 is asymptotically O(N^1.465), the exponent being
-log(5)/log(3), representing 5 recursive multiplies of 1/3 the original
-size each. This is an improvement over Karatsuba at O(N^1.585), though
-Toom does more work in the evaluation and interpolation and so it only
-realizes its advantage above a certain size.
-
- Near the crossover between Toom-3 and Karatsuba there's generally a
-range of sizes where the difference between the two is small.
-`MUL_TOOM3_THRESHOLD' is a somewhat arbitrary point in that range and
-successive runs of the tune program can give different values due to
-small variations in measuring. A graph of time versus size for the two
-shows the effect, see `tune/README'.
-
- At the fairly small sizes where the Toom-3 thresholds occur it's
-worth remembering that the asymptotic behaviour for Karatsuba and
-Toom-3 can't be expected to make accurate predictions, due of course to
-the big influence of all sorts of overheads, and the fact that only a
-few recursions of each are being performed. Even at large sizes
-there's a good chance machine dependent effects like cache architecture
-will mean actual performance deviates from what might be predicted.
-
- The formula given for the Karatsuba algorithm (*note Karatsuba
-Multiplication::) has an equivalent for Toom-3 involving only five
-multiplies, but this would be complicated and unenlightening.
-
- An alternate view of Toom-3 can be found in Zuras (*note
-References::), using a vector to represent the x and y splits and a
-matrix multiplication for the evaluation and interpolation stages. The
-matrix inverses are not meant to be actually used, and they have
-elements with values much greater than in fact arise in the
-interpolation steps. The diagram shown for the 3-way is attractive,
-but again doesn't have to be implemented that way and for example with
-a bit of rearrangement just one division by 6 can be done.
-
-
-File: mpir.info, Node: Toom 4-Way Multiplication, Next: FFT Multiplication, Prev: Toom 3-Way Multiplication, Up: Multiplication Algorithms
-
-15.1.4 Toom 4-Way Multiplication
---------------------------------
-
-Karatsuba and Toom-3 split the operands into 2 and 3 coefficients,
-respectively. Toom-4 analogously splits the operands into 4
-coefficients. Using the notation from the section on Toom-3
-multiplication, we form two polynomials:
-
- X(t) = x3*t^3 + x2*t^2 + x1*t + x0
- Y(t) = y3*t^3 + y2*t^2 + y1*t + y0
-
- X(t) and Y(t) are evaluated and multiplied at 7 points, giving
-values of W(t) at those points. In MPIR the following points are used,
-
- Point Value
- t=0 x0 * y0, which gives w0 immediately
- t=1/2 (x3+2*x2+4*x1+8*x0) * (y3+2*y2+4*y1+8*y0)
- t=-1/2 (-x3+2*x2-4*x1+8*x0) * (-y3+2*y2-4*y1+8*y0)
- t=1 (x3+x2+x1+x0) * (y3+y2+y1+y0)
- t=-1 (-x3+x2-x1+x0) * (-y3+y2-y1+y0)
- t=2 (8*x3+4*x2+2*x1+x0) * (8*y3+4*y2+2*y1+y0)
- t=inf x3 * y3, which gives w6 immediately
-
- The number of additions and subtractions for Toom-4 is much larger
-than for Toom-3. But several subexpressions occur multiple times, for
-example x2+x0, occurs for both t=1 and t=-1.
-
- Toom-4 is asymptotically O(N^1.404), the exponent being
-log(7)/log(4), representing 7 recursive multiplies of 1/4 the original
-size each.
-
-
-File: mpir.info, Node: FFT Multiplication, Next: Other Multiplication, Prev: Toom 4-Way Multiplication, Up: Multiplication Algorithms
-
-15.1.5 FFT Multiplication
--------------------------
-
-This section is out-of-date and will be updated when the new FFT is
-added.
-
- At large to very large sizes a Fermat style FFT multiplication is
-used, following Scho"nhage and Strassen (*note References::).
-Descriptions of FFTs in various forms can be found in many textbooks,
-for instance Knuth section 4.3.3 part C or Lipson chapter IX. A brief
-description of the form used in MPIR is given here.
-
- The multiplication done is x*y mod 2^N+1, for a given N. A full
-product x*y is obtained by choosing N>=bits(x)+bits(y) and padding x
-and y with high zero limbs. The modular product is the native form for
-the algorithm, so padding to get a full product is unavoidable.
-
- The algorithm follows a split, evaluate, pointwise multiply,
-interpolate and combine similar to that described above for Karatsuba
-and Toom-3. A k parameter controls the split, with an FFT-k splitting
-into 2^k pieces of M=N/2^k bits each. N must be a multiple of
-(2^k)*mp_bits_per_limb so the split falls on limb boundaries, avoiding
-bit shifts in the split and combine stages.
-
- The evaluations, pointwise multiplications, and interpolation, are
-all done modulo 2^N'+1 where N' is 2M+k+3 rounded up to a multiple of
-2^k and of `mp_bits_per_limb'. The results of interpolation will be
-the following negacyclic convolution of the input pieces, and the
-choice of N' ensures these sums aren't truncated.
-
- ---
- \ b
- w[n] = / (-1) * x[i] * y[j]
- ---
- i+j==b*2^k+n
- b=0,1
-
- The points used for the evaluation are g^i for i=0 to 2^k-1 where
-g=2^(2N'/2^k). g is a 2^k'th root of unity mod 2^N'+1, which produces
-necessary cancellations at the interpolation stage, and it's also a
-power of 2 so the fast fourier transforms used for the evaluation and
-interpolation do only shifts, adds and negations.
-
- The pointwise multiplications are done modulo 2^N'+1 and either
-recurse into a further FFT or use a plain multiplication (Toom-3,
-Karatsuba or basecase), whichever is optimal at the size N'. The
-interpolation is an inverse fast fourier transform. The resulting set
-of sums of x[i]*y[j] are added at appropriate offsets to give the final
-result.
-
- Squaring is the same, but x is the only input so it's one transform
-at the evaluate stage and the pointwise multiplies are squares. The
-interpolation is the same.
-
- For a mod 2^N+1 product, an FFT-k is an O(N^(k/(k-1))) algorithm,
-the exponent representing 2^k recursed modular multiplies each
-1/2^(k-1) the size of the original. Each successive k is an asymptotic
-improvement, but overheads mean each is only faster at bigger and
-bigger sizes. In the code, `MUL_FFT_TABLE' and `SQR_FFT_TABLE' are the
-thresholds where each k is used. Each new k effectively swaps some
-multiplying for some shifts, adds and overheads.
-
- A mod 2^N+1 product can be formed with a normal NxN->2N bit multiply
-plus a subtraction, so an FFT and Toom-3 etc can be compared directly.
-A k=4 FFT at O(N^1.333) can be expected to be the first faster than
-Toom-3 at O(N^1.465). In practice this is what's found, with
-`MUL_FFT_MODF_THRESHOLD' and `SQR_FFT_MODF_THRESHOLD' being between 300
-and 1000 limbs, depending on the CPU. So far it's been found that only
-very large FFTs recurse into pointwise multiplies above these sizes.
-
- When an FFT is to give a full product, the change of N to 2N doesn't
-alter the theoretical complexity for a given k, but for the purposes of
-considering where an FFT might be first used it can be assumed that the
-FFT is recursing into a normal multiply and that on that basis it's
-doing 2^k recursed multiplies each 1/2^(k-2) the size of the inputs,
-making it O(N^(k/(k-2))). This would mean k=7 at O(N^1.4) would be the
-first FFT faster than Toom-3. In practice `MUL_FFT_FULL_THRESHOLD' and
-`SQR_FFT_FULL_THRESHOLD' have been found to be in the k=8 range,
-somewhere between 3000 and 10000 limbs.
-
- The way N is split into 2^k pieces and then 2M+k+3 is rounded up to
-a multiple of 2^k and `mp_bits_per_limb' means that when
-2^k>=mp_bits_per_limb the effective N is a multiple of 2^(2k-1) bits.
-The +k+3 means some values of N just under such a multiple will be
-rounded to the next. The complexity calculations above assume that a
-favourable size is used, meaning one which isn't padded through
-rounding, and it's also assumed that the extra +k+3 bits are negligible
-at typical FFT sizes.
-
- The practical effect of the 2^(2k-1) constraint is to introduce a
-step-effect into measured speeds. For example k=8 will round N up to a
-multiple of 32768 bits, so for a 32-bit limb there'll be 512 limb
-groups of sizes for which `mpn_mul_n' runs at the same speed. Or for
-k=9 groups of 2048 limbs, k=10 groups of 8192 limbs, etc. In practice
-it's been found each k is used at quite small multiples of its size
-constraint and so the step effect is quite noticeable in a time versus
-size graph.
-
- The threshold determinations currently measure at the mid-points of
-size steps, but this is sub-optimal since at the start of a new step it
-can happen that it's better to go back to the previous k for a while.
-Something more sophisticated for `MUL_FFT_TABLE' and `SQR_FFT_TABLE'
-will be needed.
-
-
-File: mpir.info, Node: Other Multiplication, Next: Unbalanced Multiplication, Prev: FFT Multiplication, Up: Multiplication Algorithms
-
-15.1.6 Other Multiplication
----------------------------
-
-The Toom algorithms described above (*note Toom 3-Way Multiplication::),
-*note Toom 4-Way Multiplication::) generalize to split into an arbitrary
-number of pieces, as per Knuth section 4.3.3 algorithm C. MPIR currently
-implements Toom 8 routines.
-
- These are generated automatically via a technique due to Bodrato
-(*note References::) which mixes evaluation, pointwise multiplication
-and interpolation phases. The routine used is called Toom 8.5. See
-Bodrato's paper.
-
- For general Toom-n a split into r+1 pieces is made, and evaluations
-and pointwise multiplications done at 2*r+1 points. A 4-way split does
-7 pointwise multiplies, 5-way does 9, etc. Asymptotically an (r+1)-way
-algorithm is O(N^(log(2*r+1)/log(r+1))). Only the pointwise
-multiplications count towards big-O complexity, but the time spent in
-the evaluate and interpolate stages grows with r and has a significant
-practical impact, with the asymptotic advantage of each r realized only
-at bigger and bigger sizes. The overheads grow as O(N*r), whereas in
-an r=2^k FFT they grow only as O(N*log(r)).
-
- Knuth algorithm C evaluates at points 0,1,2,...,2*r, but exercise 4
-uses -r,...,0,...,r and the latter saves some small multiplies in the
-evaluate stage (or rather trades them for additions), and has a further
-saving of nearly half the interpolate steps. The idea is to separate
-odd and even final coefficients and then perform algorithm C steps C7
-and C8 on them separately. The divisors at step C7 become j^2 and the
-multipliers at C8 become 2*t*j-j^2.
-
- Splitting odd and even parts through positive and negative points
-can be thought of as using -1 as a square root of unity. If a 4th root
-of unity was available then a further split and speedup would be
-possible, but no such root exists for plain integers. Going to complex
-integers with i=sqrt(-1) doesn't help, essentially because in cartesian
-form it takes three real multiplies to do a complex multiply. The
-existence of 2^k'th roots of unity in a suitable ring or field lets the
-fast fourier transform keep splitting and get to O(N*log(r)).
-
- Floating point FFTs use complex numbers approximating Nth roots of
-unity. Some processors have special support for such FFTs. But these
-are not used in MPIR since it's very difficult to guarantee an exact
-result (to some number of bits). An occasional difference of 1 in the
-last bit might not matter to a typical signal processing algorithm, but
-is of course of vital importance to MPIR.
-
-
-File: mpir.info, Node: Unbalanced Multiplication, Prev: Other Multiplication, Up: Multiplication Algorithms
-
-15.1.7 Unbalanced Multiplication
---------------------------------
-
-Multiplication of operands with different sizes, both below
-`MUL_KARATSUBA_THRESHOLD' are done with plain schoolbook multiplication
-(*note Basecase Multiplication::).
-
- For really large operands, we invoke the FFT directly.
-
- For operands between these sizes, we use Toom inspired algorithms
-suggested by Alberto Zanoni and Marco Bodrato. The idea is to split
-the operands into polynomials of different degree. These algorithms are
-denoted ToomMN where the first input is broken into M components and
-the second operand is broken into N components. MPIR currently
-implements Toom32, Toom33, Toom44, Toom53 and Toom8h which deals with a
-variety of sizes where the product polynomial will have length 15 or 16.
-
-
-File: mpir.info, Node: Division Algorithms, Next: Greatest Common Divisor Algorithms, Prev: Multiplication Algorithms, Up: Algorithms
-
-15.2 Division Algorithms
-========================
-
-* Menu:
-
-* Single Limb Division::
-* Basecase Division::
-* Divide and Conquer Division::
-* Exact Division::
-* Exact Remainder::
-* Small Quotient Division::
-
-
-File: mpir.info, Node: Single Limb Division, Next: Basecase Division, Prev: Division Algorithms, Up: Division Algorithms
-
-15.2.1 Single Limb Division
----------------------------
-
-Nx1 division is implemented using repeated 2x1 divisions from high to
-low, either with a hardware divide instruction or a multiplication by
-inverse, whichever is best on a given CPU.
-
- The multiply by inverse follows section 8 of "Division by Invariant
-Integers using Multiplication" by Granlund and Montgomery (*note
-References::) and is implemented as `udiv_qrnnd_preinv' in
-`gmp-impl.h'. The idea is to have a fixed-point approximation to 1/d
-(see `invert_limb') and then multiply by the high limb (plus one bit)
-of the dividend to get a quotient q. With d normalized (high bit set),
-q is no more than 1 too small. Subtracting q*d from the dividend gives
-a remainder, and reveals whether q or q-1 is correct.
-
- The result is a division done with two multiplications and four or
-five arithmetic operations. On CPUs with low latency multipliers this
-can be much faster than a hardware divide, though the cost of
-calculating the inverse at the start may mean it's only better on
-inputs bigger than say 4 or 5 limbs.
-
- When a divisor must be normalized, either for the generic C
-`__udiv_qrnnd_c' or the multiply by inverse, the division performed is
-actually a*2^k by d*2^k where a is the dividend and k is the power
-necessary to have the high bit of d*2^k set. The bit shifts for the
-dividend are usually accomplished "on the fly" meaning by extracting
-the appropriate bits at each step. Done this way the quotient limbs
-come out aligned ready to store. When only the remainder is wanted, an
-alternative is to take the dividend limbs unshifted and calculate r = a
-mod d*2^k followed by an extra final step r*2^k mod d*2^k. This can
-help on CPUs with poor bit shifts or few registers.
-
- The multiply by inverse can be done two limbs at a time. The
-calculation is basically the same, but the inverse is two limbs and the
-divisor treated as if padded with a low zero limb. This means more
-work, since the inverse will need a 2x2 multiply, but the four 1x1s to
-do that are independent and can therefore be done partly or wholly in
-parallel. Likewise for a 2x1 calculating q*d. The net effect is to
-process two limbs with roughly the same two multiplies worth of latency
-that one limb at a time gives. This extends to 3 or 4 limbs at a time,
-though the extra work to apply the inverse will almost certainly soon
-reach the limits of multiplier throughput.
-
- A similar approach in reverse can be taken to process just half a
-limb at a time if the divisor is only a half limb. In this case the
-1x1 multiply for the inverse effectively becomes two (1/2)x1 for each
-limb, which can be a saving on CPUs with a fast half limb multiply, or
-in fact if the only multiply is a half limb, and especially if it's not
-pipelined.
-
-
-File: mpir.info, Node: Basecase Division, Next: Divide and Conquer Division, Prev: Single Limb Division, Up: Division Algorithms
-
-15.2.2 Basecase Division
-------------------------
-
-This section is out-of-date.
-
- Basecase NxM division is like long division done by hand, but in base
-2^mp_bits_per_limb. See Knuth section 4.3.1 algorithm D.
-
- Briefly stated, while the dividend remains larger than the divisor,
-a high quotient limb is formed and the Nx1 product q*d subtracted at
-the top end of the dividend. With a normalized divisor (most
-significant bit set), each quotient limb can be formed with a 2x1
-division and a 1x1 multiplication plus some subtractions. The 2x1
-division is by the high limb of the divisor and is done either with a
-hardware divide or a multiply by inverse (the same as in *note Single
-Limb Division::) whichever is faster. Such a quotient is sometimes one
-too big, requiring an addback of the divisor, but that happens rarely.
-
- With Q=N-M being the number of quotient limbs, this is an O(Q*M)
-algorithm and will run at a speed similar to a basecase QxM
-multiplication, differing in fact only in the extra multiply and divide
-for each of the Q quotient limbs.
-
-
-File: mpir.info, Node: Divide and Conquer Division, Next: Exact Division, Prev: Basecase Division, Up: Division Algorithms
-
-15.2.3 Divide and Conquer Division
-----------------------------------
-
-This section is out-of-date
-
- For divisors larger than `DIV_DC_THRESHOLD', division is done by
-dividing. Or to be precise by a recursive divide and conquer algorithm
-based on work by Moenck and Borodin, Jebelean, and Burnikel and Ziegler
-(*note References::).
-
- The algorithm consists essentially of recognising that a 2NxN
-division can be done with the basecase division algorithm (*note
-Basecase Division::), but using N/2 limbs as a base, not just a single
-limb. This way the multiplications that arise are (N/2)x(N/2) and can
-take advantage of Karatsuba and higher multiplication algorithms (*note
-Multiplication Algorithms::). The two "digits" of the quotient are
-formed by recursive Nx(N/2) divisions.
-
- If the (N/2)x(N/2) multiplies are done with a basecase multiplication
-then the work is about the same as a basecase division, but with more
-function call overheads and with some subtractions separated from the
-multiplies. These overheads mean that it's only when N/2 is above
-`MUL_KARATSUBA_THRESHOLD' that divide and conquer is of use.
-
- `DIV_DC_THRESHOLD' is based on the divisor size N, so it will be
-somewhere above twice `MUL_KARATSUBA_THRESHOLD', but how much above
-depends on the CPU. An optimized `mpn_mul_basecase' can lower
-`DIV_DC_THRESHOLD' a little by offering a ready-made advantage over
-repeated `mpn_submul_1' calls.
-
- Divide and conquer is asymptotically O(M(N)*log(N)) where M(N) is
-the time for an NxN multiplication done with FFTs. The actual time is
-a sum over multiplications of the recursed sizes, as can be seen near
-the end of section 2.2 of Burnikel and Ziegler. For example, within
-the Toom-3 range, divide and conquer is 2.63*M(N). With higher
-algorithms the M(N) term improves and the multiplier tends to log(N).
-In practice, at moderate to large sizes, a 2NxN division is about 2 to
-4 times slower than an NxN multiplication.
-
- Newton's method used for division is asymptotically O(M(N)) and
-should therefore be superior to divide and conquer, but it's believed
-this would only be for large to very large N.
-
-
-File: mpir.info, Node: Exact Division, Next: Exact Remainder, Prev: Divide and Conquer Division, Up: Division Algorithms
-
-15.2.4 Exact Division
----------------------
-
-This section is out-of-date
-
- A so-called exact division is when the dividend is known to be an
-exact multiple of the divisor. Jebelean's exact division algorithm
-uses this knowledge to make some significant optimizations (*note
-References::).
-
- The idea can be illustrated in decimal for example with 368154
-divided by 543. Because the low digit of the dividend is 4, the low
-digit of the quotient must be 8. This is arrived at from 4*7 mod 10,
-using the fact 7 is the modular inverse of 3 (the low digit of the
-divisor), since 3*7 == 1 mod 10. So 8*543=4344 can be subtracted from
-the dividend leaving 363810. Notice the low digit has become zero.
-
- The procedure is repeated at the second digit, with the next
-quotient digit 7 (7 == 1*7 mod 10), subtracting 7*543=3801, leaving
-325800. And finally at the third digit with quotient digit 6 (8*7 mod
-10), subtracting 6*543=3258 leaving 0. So the quotient is 678.
-
- Notice however that the multiplies and subtractions don't need to
-extend past the low three digits of the dividend, since that's enough
-to determine the three quotient digits. For the last quotient digit no
-subtraction is needed at all. On a 2NxN division like this one, only
-about half the work of a normal basecase division is necessary.
-
- For an NxM exact division producing Q=N-M quotient limbs, the saving
-over a normal basecase division is in two parts. Firstly, each of the
-Q quotient limbs needs only one multiply, not a 2x1 divide and
-multiply. Secondly, the crossproducts are reduced when Q>M to
-Q*M-M*(M+1)/2, or when Q<=M to Q*(Q-1)/2. Notice the savings are
-complementary. If Q is big then many divisions are saved, or if Q is
-small then the crossproducts reduce to a small number.
-
- The modular inverse used is calculated efficiently by
-`modlimb_invert' in `gmp-impl.h'. This does four multiplies for a
-32-bit limb, or six for a 64-bit limb. `tune/modlinv.c' has some
-alternate implementations that might suit processors better at bit
-twiddling than multiplying.
-
- The sub-quadratic exact division described by Jebelean in "Exact
-Division with Karatsuba Complexity" is not currently implemented. It
-uses a rearrangement similar to the divide and conquer for normal
-division (*note Divide and Conquer Division::), but operating from low
-to high. A further possibility not currently implemented is
-"Bidirectional Exact Integer Division" by Krandick and Jebelean which
-forms quotient limbs from both the high and low ends of the dividend,
-and can halve once more the number of crossproducts needed in a 2NxN
-division.
-
- A special case exact division by 3 exists in `mpn_divexact_by3',
-supporting Toom-3 multiplication and `mpq' canonicalizations. It forms
-quotient digits with a multiply by the modular inverse of 3 (which is
-`0xAA..AAB') and uses two comparisons to determine a borrow for the next
-limb. The multiplications don't need to be on the dependent chain, as
-long as the effect of the borrows is applied, which can help chips with
-pipelined multipliers.
-
-
-File: mpir.info, Node: Exact Remainder, Next: Small Quotient Division, Prev: Exact Division, Up: Division Algorithms
-
-15.2.5 Exact Remainder
-----------------------
-
-If the exact division algorithm is done with a full subtraction at each
-stage and the dividend isn't a multiple of the divisor, then low zero
-limbs are produced but with a remainder in the high limbs. For
-dividend a, divisor d, quotient q, and b = 2^mp_bits_per_limb, this
-remainder r is of the form
-
- a = q*d + r*b^n
-
- n represents the number of zero limbs produced by the subtractions,
-that being the number of limbs produced for q. r will be in the range
-0<=r<d and can be viewed as a remainder, but one shifted up by a factor
-of b^n.
-
- Carrying out full subtractions at each stage means the same number
-of cross products must be done as a normal division, but there's still
-some single limb divisions saved. When d is a single limb some
-simplifications arise, providing good speedups on a number of
-processors.
-
- `mpn_bdivmod', `mpn_divexact_by3', `mpn_modexact_1_odd' and the
-`redc' function in `mpz_powm' differ subtly in how they return r,
-leading to some negations in the above formula, but all are essentially
-the same.
-
- Clearly r is zero when a is a multiple of d, and this leads to
-divisibility or congruence tests which are potentially more efficient
-than a normal division.
-
- The factor of b^n on r can be ignored in a GCD when d is odd, hence
-the use of `mpn_bdivmod' in `mpn_gcd', and the use of
-`mpn_modexact_1_odd' by `mpn_gcd_1' and `mpz_kronecker_ui' etc (*note
-Greatest Common Divisor Algorithms::).
-
- Montgomery's REDC method for modular multiplications uses operands
-of the form of x*b^-n and y*b^-n and on calculating (x*b^-n)*(y*b^-n)
-uses the factor of b^n in the exact remainder to reach a product in the
-same form (x*y)*b^-n (*note Modular Powering Algorithm::).
-
- Notice that r generally gives no useful information about the
-ordinary remainder a mod d since b^n mod d could be anything. If
-however b^n == 1 mod d, then r is the negative of the ordinary
-remainder. This occurs whenever d is a factor of b^n-1, as for example
-with 3 in `mpn_divexact_by3'. For a 32 or 64 bit limb other such
-factors include 5, 17 and 257, but no particular use has been found for
-this.
-
-
-File: mpir.info, Node: Small Quotient Division, Prev: Exact Remainder, Up: Division Algorithms
-
-15.2.6 Small Quotient Division
-------------------------------
-
-An NxM division where the number of quotient limbs Q=N-M is small can
-be optimized somewhat.
-
- An ordinary basecase division normalizes the divisor by shifting it
-to make the high bit set, shifting the dividend accordingly, and
-shifting the remainder back down at the end of the calculation. This
-is wasteful if only a few quotient limbs are to be formed. Instead a
-division of just the top 2*Q limbs of the dividend by the top Q limbs
-of the divisor can be used to form a trial quotient. This requires
-only those limbs normalized, not the whole of the divisor and dividend.
-
- A multiply and subtract then applies the trial quotient to the M-Q
-unused limbs of the divisor and N-Q dividend limbs (which includes Q
-limbs remaining from the trial quotient division). The starting trial
-quotient can be 1 or 2 too big, but all cases of 2 too big and most
-cases of 1 too big are detected by first comparing the most significant
-limbs that will arise from the subtraction. An addback is done if the
-quotient still turns out to be 1 too big.
-
- This whole procedure is essentially the same as one step of the
-basecase algorithm done in a Q limb base, though with the trial
-quotient test done only with the high limbs, not an entire Q limb
-"digit" product. The correctness of this weaker test can be
-established by following the argument of Knuth section 4.3.1 exercise
-20 but with the v2*q>b*r+u2 condition appropriately relaxed.
-
-
-File: mpir.info, Node: Greatest Common Divisor Algorithms, Next: Powering Algorithms, Prev: Division Algorithms, Up: Algorithms
-
-15.3 Greatest Common Divisor
-============================
-
-* Menu:
-
-* Binary GCD::
-* Lehmer's GCD::
-* Subquadratic GCD::
-* Extended GCD::
-* Jacobi Symbol::
-
-
-File: mpir.info, Node: Binary GCD, Next: Lehmer's GCD, Prev: Greatest Common Divisor Algorithms, Up: Greatest Common Divisor Algorithms
-
-15.3.1 Binary GCD
------------------
-
-At small sizes MPIR uses an O(N^2) binary style GCD. This is described
-in many textbooks, for example Knuth section 4.5.2 algorithm B. It
-simply consists of successively reducing odd operands a and b using
-
- a,b = abs(a-b),min(a,b)
- strip factors of 2 from a
-
- The Euclidean GCD algorithm, as per Knuth algorithms E and A,
-reduces using a mod b but this has so far been found to be slower
-everywhere. One reason the binary method does well is that the implied
-quotient at each step is usually small, so often only one or two
-subtractions are needed to get the same effect as a division.
-Quotients 1, 2 and 3 for example occur 67.7% of the time, see Knuth
-section 4.5.3 Theorem E.
-
- When the implied quotient is large, meaning b is much smaller than
-a, then a division is worthwhile. This is the basis for the initial a
-mod b reductions in `mpn_gcd' and `mpn_gcd_1' (the latter for both Nx1
-and 1x1 cases). But after that initial reduction, big quotients occur
-too rarely to make it worth checking for them.
-
-
- The final 1x1 GCD in `mpn_gcd_1' is done in the generic C code as
-described above. For two N-bit operands, the algorithm takes about
-0.68 iterations per bit. For optimum performance some attention needs
-to be paid to the way the factors of 2 are stripped from a.
-
- Firstly it may be noted that in twos complement the number of low
-zero bits on a-b is the same as b-a, so counting or testing can begin on
-a-b without waiting for abs(a-b) to be determined.
-
- A loop stripping low zero bits tends not to branch predict well,
-since the condition is data dependent. But on average there's only a
-few low zeros, so an option is to strip one or two bits arithmetically
-then loop for more (as done for AMD K6). Or use a lookup table to get
-a count for several bits then loop for more (as done for AMD K7). An
-alternative approach is to keep just one of a or b odd and iterate
-
- a,b = abs(a-b), min(a,b)
- a = a/2 if even
- b = b/2 if even
-
- This requires about 1.25 iterations per bit, but stripping of a
-single bit at each step avoids any branching. Repeating the bit strip
-reduces to about 0.9 iterations per bit, which may be a worthwhile
-tradeoff.
-
- Generally with the above approaches a speed of perhaps 6 cycles per
-bit can be achieved, which is still not terribly fast with for instance
-a 64-bit GCD taking nearly 400 cycles. It's this sort of time which
-means it's not usually advantageous to combine a set of divisibility
-tests into a GCD.
-
-
-File: mpir.info, Node: Lehmer's GCD, Next: Subquadratic GCD, Prev: Binary GCD, Up: Greatest Common Divisor Algorithms
-
-15.3.2 Lehmer's GCD
--------------------
-
-Lehmer's improvement of the Euclidean algorithms is based on the
-observation that the initial part of the quotient sequence depends only
-on the most significant parts of the inputs. The variant of Lehmer's
-algorithm used in MPIR splits off the most significant two limbs, as
-suggested, e.g., in "A Double-Digit Lehmer-Euclid Algorithm" by
-Jebelean (*note References::). The quotients of two double-limb inputs
-are collected as a 2 by 2 matrix with single-limb elements. This is
-done by the function `mpn_hgcd2'. The resulting matrix is applied to
-the inputs using `mpn_mul_1' and `mpn_submul_1'. Each iteration usually
-reduces the inputs by almost one limb. In the rare case of a large
-quotient, no progress can be made by examining just the most
-significant two limbs, and the quotient is computing using plain
-division.
-
- The resulting algorithm is asymptotically O(N^2), just as the
-Euclidean algorithm and the binary algorithm. The quadratic part of the
-work are the calls to `mpn_mul_1' and `mpn_submul_1'. For small sizes,
-the linear work is also significant. There are roughly N calls to the
-`mpn_hgcd2' function. This function uses a couple of important
-optimizations:
-
- * It uses the same relaxed notion of correctness as `mpn_hgcd' (see
- next section). This means that when called with the most
- significant two limbs of two large numbers, the returned matrix
- does not always correspond exactly to the initial quotient
- sequence for the two large numbers; the final quotient may
- sometimes be one off.
-
- * It takes advantage of the fact the quotients are usually small.
- The division operator is not used, since the corresponding
- assembler instruction is very slow on most architectures. (This
- code could probably be improved further, it uses many branches
- that are unfriendly to prediction).
-
- * It switches from double-limb calculations to single-limb
- calculations half-way through, when the input numbers have been
- reduced in size from two limbs to one and a half.
-
-
-
-File: mpir.info, Node: Subquadratic GCD, Next: Extended GCD, Prev: Lehmer's GCD, Up: Greatest Common Divisor Algorithms
-
-15.3.3 Subquadratic GCD
------------------------
-
-For inputs larger than `GCD_DC_THRESHOLD', GCD is computed via the HGCD
-(Half GCD) function, as a generalization to Lehmer's algorithm.
-
- Let the inputs a,b be of size N limbs each. Put S = floor(N/2) + 1.
-Then HGCD(a,b) returns a transformation matrix T with non-negative
-elements, and reduced numbers (c;d) = T^-1 (a;b). The reduced numbers
-c,d must be larger than S limbs, while their difference abs(c-d) must
-fit in S limbs. The matrix elements will also be of size roughly N/2.
-
- The HGCD base case uses Lehmer's algorithm, but with the above stop
-condition that returns reduced numbers and the corresponding
-transformation matrix half-way through. For inputs larger than
-`HGCD_THRESHOLD', HGCD is computed recursively, using the divide and
-conquer algorithm in "On Scho"nhage's algorithm and subquadratic
-integer GCD computation" by Mo"ller (*note References::). The recursive
-algorithm consists of these main steps.
-
- * Call HGCD recursively, on the most significant N/2 limbs. Apply the
- resulting matrix T_1 to the full numbers, reducing them to a size
- just above 3N/2.
-
- * Perform a small number of division or subtraction steps to reduce
- the numbers to size below 3N/2. This is essential mainly for the
- unlikely case of large quotients.
-
- * Call HGCD recursively, on the most significant N/2 limbs of the
- reduced numbers. Apply the resulting matrix T_2 to the full
- numbers, reducing them to a size just above N/2.
-
- * Compute T = T_1 T_2.
-
- * Perform a small number of division and subtraction steps to
- satisfy the requirements, and return.
-
- GCD is then implemented as a loop around HGCD, similarly to Lehmer's
-algorithm. Where Lehmer repeatedly chops off the top two limbs, calls
-`mpn_hgcd2', and applies the resulting matrix to the full numbers, the
-subquadratic GCD chops off the most significant third of the limbs (the
-proportion is a tuning parameter, and 1/3 seems to be more efficient
-than, e.g, 1/2), calls `mpn_hgcd', and applies the resulting matrix.
-Once the input numbers are reduced to size below `GCD_DC_THRESHOLD',
-Lehmer's algorithm is used for the rest of the work.
-
- The asymptotic running time of both HGCD and GCD is O(M(N)*log(N)),
-where M(N) is the time for multiplying two N-limb numbers.
-
-
-File: mpir.info, Node: Extended GCD, Next: Jacobi Symbol, Prev: Subquadratic GCD, Up: Greatest Common Divisor Algorithms
-
-15.3.4 Extended GCD
--------------------
-
-The extended GCD function, or gcdext, calculates gcd(a,b) and also one
-of the cofactors x and y satisfying a*x+b*y=gcd(a,b). The algorithms
-used for plain GCD are extended to handle this case.
-
- Lehmer's algorithm is used for sizes up to `GCDEXT_DC_THRESHOLD'.
-Above this threshold, GCDEXT is implemented as a loop around HGCD, but
-with more book-keeping to keep track of the cofactors.
-
-
-File: mpir.info, Node: Jacobi Symbol, Prev: Extended GCD, Up: Greatest Common Divisor Algorithms
-
-15.3.5 Jacobi Symbol
---------------------
-
-`mpz_jacobi' and `mpz_kronecker' are currently implemented with a
-simple binary algorithm similar to that described for the GCDs (*note
-Binary GCD::). They're not very fast when both inputs are large.
-Lehmer's multi-step improvement or a binary based multi-step algorithm
-is likely to be better.
-
- When one operand fits a single limb, and that includes
-`mpz_kronecker_ui' and friends, an initial reduction is done with
-either `mpn_mod_1' or `mpn_modexact_1_odd', followed by the binary
-algorithm on a single limb. The binary algorithm is well suited to a
-single limb, and the whole calculation in this case is quite efficient.
-
- In all the routines sign changes for the result are accumulated
-using some bit twiddling, avoiding table lookups or conditional jumps.
-
-
-File: mpir.info, Node: Powering Algorithms, Next: Root Extraction Algorithms, Prev: Greatest Common Divisor Algorithms, Up: Algorithms
-
-15.4 Powering Algorithms
-========================
-
-* Menu:
-
-* Normal Powering Algorithm::
-* Modular Powering Algorithm::
-
-
-File: mpir.info, Node: Normal Powering Algorithm, Next: Modular Powering Algorithm, Prev: Powering Algorithms, Up: Powering Algorithms
-
-15.4.1 Normal Powering
-----------------------
-
-Normal `mpz' or `mpf' powering uses a simple binary algorithm,
-successively squaring and then multiplying by the base when a 1 bit is
-seen in the exponent, as per Knuth section 4.6.3. The "left to right"
-variant described there is used rather than algorithm A, since it's
-just as easy and can be done with somewhat less temporary memory.
-
-
-File: mpir.info, Node: Modular Powering Algorithm, Prev: Normal Powering Algorithm, Up: Powering Algorithms
-
-15.4.2 Modular Powering
------------------------
-
-Modular powering is implemented using a 2^k-ary sliding window
-algorithm, as per "Handbook of Applied Cryptography" algorithm 14.85
-(*note References::). k is chosen according to the size of the
-exponent. Larger exponents use larger values of k, the choice being
-made to minimize the average number of multiplications that must
-supplement the squaring.
-
- The modular multiplies and squares use either a simple division or
-the REDC method by Montgomery (*note References::). REDC is a little
-faster, essentially saving N single limb divisions in a fashion similar
-to an exact remainder (*note Exact Remainder::). The current REDC has
-some limitations. It's only O(N^2) so above `POWM_THRESHOLD' division
-becomes faster and is used. It doesn't attempt to detect small bases,
-but rather always uses a REDC form, which is usually a full size
-operand. And lastly it's only applied to odd moduli.
-
-
-File: mpir.info, Node: Root Extraction Algorithms, Next: Radix Conversion Algorithms, Prev: Powering Algorithms, Up: Algorithms
-
-15.5 Root Extraction Algorithms
-===============================
-
-* Menu:
-
-* Square Root Algorithm::
-* Nth Root Algorithm::
-* Perfect Square Algorithm::
-* Perfect Power Algorithm::
-
-
-File: mpir.info, Node: Square Root Algorithm, Next: Nth Root Algorithm, Prev: Root Extraction Algorithms, Up: Root Extraction Algorithms
-
-15.5.1 Square Root
-------------------
-
-Square roots are taken using the "Karatsuba Square Root" algorithm by
-Paul Zimmermann (*note References::).
-
- An input n is split into four parts of k bits each, so with b=2^k we
-have n = a3*b^3 + a2*b^2 + a1*b + a0. Part a3 must be "normalized" so
-that either the high or second highest bit is set. In MPIR, k is kept
-on a limb boundary and the input is left shifted (by an even number of
-bits) to normalize.
-
- The square root of the high two parts is taken, by recursive
-application of the algorithm (bottoming out in a one-limb Newton's
-method),
-
- s1,r1 = sqrtrem (a3*b + a2)
-
- This is an approximation to the desired root and is extended by a
-division to give s,r,
-
- q,u = divrem (r1*b + a1, 2*s1)
- s = s1*b + q
- r = u*b + a0 - q^2
-
- The normalization requirement on a3 means at this point s is either
-correct or 1 too big. r is negative in the latter case, so
-
- if r < 0 then
- r = r + 2*s - 1
- s = s - 1
-
- The algorithm is expressed in a divide and conquer form, but as
-noted in the paper it can also be viewed as a discrete variant of
-Newton's method, or as a variation on the schoolboy method (no longer
-taught) for square roots two digits at a time.
-
- If the remainder r is not required then usually only a few high limbs
-of r and u need to be calculated to determine whether an adjustment to
-s is required. This optimization is not currently implemented.
-
- In the Karatsuba multiplication range this algorithm is
-O(1.5*M(N/2)), where M(n) is the time to multiply two numbers of n
-limbs. In the FFT multiplication range this grows to a bound of
-O(6*M(N/2)). In practice a factor of about 1.5 to 1.8 is found in the
-Karatsuba and Toom-3 ranges, growing to 2 or 3 in the FFT range.
-
- The algorithm does all its calculations in integers and the resulting
-`mpn_sqrtrem' is used for both `mpz_sqrt' and `mpf_sqrt'. The extended
-precision given by `mpf_sqrt_ui' is obtained by padding with zero limbs.
-
-
-File: mpir.info, Node: Nth Root Algorithm, Next: Perfect Square Algorithm, Prev: Square Root Algorithm, Up: Root Extraction Algorithms
-
-15.5.2 Nth Root
----------------
-
-Integer Nth roots are taken using Newton's method with the following
-iteration, where A is the input and n is the root to be taken.
-
- 1 A
- a[i+1] = - * ( --------- + (n-1)*a[i] )
- n a[i]^(n-1)
-
- The initial approximation a[1] is generated bitwise by successively
-powering a trial root with or without new 1 bits, aiming to be just
-above the true root. The iteration converges quadratically when
-started from a good approximation. When n is large more initial bits
-are needed to get good convergence. The current implementation is not
-particularly well optimized.
-
diff --git a/doc/mpir.info-2 b/doc/mpir.info-2
deleted file mode 100644
index 1d3ae70..0000000
--- a/doc/mpir.info-2
+++ /dev/null
@@ -1,3427 +0,0 @@
-This is mpir.info, produced by makeinfo version 4.13 from mpir.texi.
-
-This manual describes how to install and use MPIR, the Multiple
-Precision Integers and Rationals library, version 2.6.0.
-
- Copyright 1991, 1993, 1994, 1995, 1996, 1997, 1998, 1999, 2000,
-2001, 2002, 2003, 2004, 2005, 2006, 2007, 2008, 2009, 2010 Free
-Software Foundation, Inc.
-
- Copyright 2008, 2009, 2010 William Hart
-
- Permission is granted to copy, distribute and/or modify this
-document under the terms of the GNU Free Documentation License, Version
-1.3 or any later version published by the Free Software Foundation;
-with no Invariant Sections, with the Front-Cover Texts being "A GNU
-Manual", and with the Back-Cover Texts being "You have freedom to copy
-and modify this GNU Manual, like GNU software". A copy of the license
-is included in *note GNU Free Documentation License::.
-
-INFO-DIR-SECTION GNU libraries
-START-INFO-DIR-ENTRY
-* mpir: (mpir). MPIR Multiple Precision Integers and Rationals Library.
-END-INFO-DIR-ENTRY
-
-
-File: mpir.info, Node: Perfect Square Algorithm, Next: Perfect Power Algorithm, Prev: Nth Root Algorithm, Up: Root Extraction Algorithms
-
-15.5.3 Perfect Square
----------------------
-
-A significant fraction of non-squares can be quickly identified by
-checking whether the input is a quadratic residue modulo small integers.
-
- `mpz_perfect_square_p' first tests the input mod 256, which means
-just examining the low byte. Only 44 different values occur for
-squares mod 256, so 82.8% of inputs can be immediately identified as
-non-squares.
-
- On a 32-bit system similar tests are done mod 9, 5, 7, 13 and 17,
-for a total 99.25% of inputs identified as non-squares. On a 64-bit
-system 97 is tested too, for a total 99.62%.
-
- These moduli are chosen because they're factors of 2^24-1 (or 2^48-1
-for 64-bits), and such a remainder can be quickly taken just using
-additions (see `mpn_mod_34lsub1').
-
- When nails are in use moduli are instead selected by the `gen-psqr.c'
-program and applied with an `mpn_mod_1'. The same 2^24-1 or 2^48-1
-could be done with nails using some extra bit shifts, but this is not
-currently implemented.
-
- In any case each modulus is applied to the `mpn_mod_34lsub1' or
-`mpn_mod_1' remainder and a table lookup identifies non-squares. By
-using a "modexact" style calculation, and suitably permuted tables,
-just one multiply each is required, see the code for details. Moduli
-are also combined to save operations, so long as the lookup tables
-don't become too big. `gen-psqr.c' does all the pre-calculations.
-
- A square root must still be taken for any value that passes these
-tests, to verify it's really a square and not one of the small fraction
-of non-squares that get through (ie. a pseudo-square to all the tested
-bases).
-
- Clearly more residue tests could be done, `mpz_perfect_square_p' only
-uses a compact and efficient set. Big inputs would probably benefit
-from more residue testing, small inputs might be better off with less.
-The assumed distribution of squares versus non-squares in the input
-would affect such considerations.
-
-
-File: mpir.info, Node: Perfect Power Algorithm, Prev: Perfect Square Algorithm, Up: Root Extraction Algorithms
-
-15.5.4 Perfect Power
---------------------
-
-Detecting perfect powers is required by some factorization algorithms.
-Currently `mpz_perfect_power_p' is implemented using repeated Nth root
-extractions, though naturally only prime roots need to be considered.
-(*Note Nth Root Algorithm::.)
-
- If a prime divisor p with multiplicity e can be found, then only
-roots which are divisors of e need to be considered, much reducing the
-work necessary. To this end divisibility by a set of small primes is
-checked.
-
-
-File: mpir.info, Node: Radix Conversion Algorithms, Next: Other Algorithms, Prev: Root Extraction Algorithms, Up: Algorithms
-
-15.6 Radix Conversion
-=====================
-
-Radix conversions are less important than other algorithms. A program
-dominated by conversions should probably use a different data
-representation.
-
-* Menu:
-
-* Binary to Radix::
-* Radix to Binary::
-
-
-File: mpir.info, Node: Binary to Radix, Next: Radix to Binary, Prev: Radix Conversion Algorithms, Up: Radix Conversion Algorithms
-
-15.6.1 Binary to Radix
-----------------------
-
-Conversions from binary to a power-of-2 radix use a simple and fast
-O(N) bit extraction algorithm.
-
- Conversions from binary to other radices use one of two algorithms.
-Sizes below `GET_STR_PRECOMPUTE_THRESHOLD' use a basic O(N^2) method.
-Repeated divisions by b^n are made, where b is the radix and n is the
-biggest power that fits in a limb. But instead of simply using the
-remainder r from such divisions, an extra divide step is done to give a
-fractional limb representing r/b^n. The digits of r can then be
-extracted using multiplications by b rather than divisions. Special
-case code is provided for decimal, allowing multiplications by 10 to
-optimize to shifts and adds.
-
- Above `GET_STR_PRECOMPUTE_THRESHOLD' a sub-quadratic algorithm is
-used. For an input t, powers b^(n*2^i) of the radix are calculated,
-until a power between t and sqrt(t) is reached. t is then divided by
-that largest power, giving a quotient which is the digits above that
-power, and a remainder which is those below. These two parts are in
-turn divided by the second highest power, and so on recursively. When
-a piece has been divided down to less than `GET_STR_DC_THRESHOLD'
-limbs, the basecase algorithm described above is used.
-
- The advantage of this algorithm is that big divisions can make use
-of the sub-quadratic divide and conquer division (*note Divide and
-Conquer Division::), and big divisions tend to have less overheads than
-lots of separate single limb divisions anyway. But in any case the
-cost of calculating the powers b^(n*2^i) must first be overcome.
-
- `GET_STR_PRECOMPUTE_THRESHOLD' and `GET_STR_DC_THRESHOLD' represent
-the same basic thing, the point where it becomes worth doing a big
-division to cut the input in half. `GET_STR_PRECOMPUTE_THRESHOLD'
-includes the cost of calculating the radix power required, whereas
-`GET_STR_DC_THRESHOLD' assumes that's already available, which is the
-case when recursing.
-
- Since the base case produces digits from least to most significant
-but they want to be stored from most to least, it's necessary to
-calculate in advance how many digits there will be, or at least be sure
-not to underestimate that. For MPIR the number of input bits is
-multiplied by `chars_per_bit_exactly' from `mp_bases', rounding up.
-The result is either correct or one too big.
-
- Examining some of the high bits of the input could increase the
-chance of getting the exact number of digits, but an exact result every
-time would not be practical, since in general the difference between
-numbers 100... and 99... is only in the last few bits and the work to
-identify 99... might well be almost as much as a full conversion.
-
- `mpf_get_str' doesn't currently use the algorithm described here, it
-multiplies or divides by a power of b to move the radix point to the
-just above the highest non-zero digit (or at worst one above that
-location), then multiplies by b^n to bring out digits. This is O(N^2)
-and is certainly not optimal.
-
- The r/b^n scheme described above for using multiplications to bring
-out digits might be useful for more than a single limb. Some brief
-experiments with it on the base case when recursing didn't give a
-noticeable improvement, but perhaps that was only due to the
-implementation. Something similar would work for the sub-quadratic
-divisions too, though there would be the cost of calculating a bigger
-radix power.
-
- Another possible improvement for the sub-quadratic part would be to
-arrange for radix powers that balanced the sizes of quotient and
-remainder produced, ie. the highest power would be an b^(n*k)
-approximately equal to sqrt(t), not restricted to a 2^i factor. That
-ought to smooth out a graph of times against sizes, but may or may not
-be a net speedup.
-
-
-File: mpir.info, Node: Radix to Binary, Prev: Binary to Radix, Up: Radix Conversion Algorithms
-
-15.6.2 Radix to Binary
-----------------------
-
-This section is out-of-date.
-
- Conversions from a power-of-2 radix into binary use a simple and fast
-O(N) bitwise concatenation algorithm.
-
- Conversions from other radices use one of two algorithms. Sizes
-below `SET_STR_THRESHOLD' use a basic O(N^2) method. Groups of n
-digits are converted to limbs, where n is the biggest power of the base
-b which will fit in a limb, then those groups are accumulated into the
-result by multiplying by b^n and adding. This saves multi-precision
-operations, as per Knuth section 4.4 part E (*note References::). Some
-special case code is provided for decimal, giving the compiler a chance
-to optimize multiplications by 10.
-
- Above `SET_STR_THRESHOLD' a sub-quadratic algorithm is used. First
-groups of n digits are converted into limbs. Then adjacent limbs are
-combined into limb pairs with x*b^n+y, where x and y are the limbs.
-Adjacent limb pairs are combined into quads similarly with x*b^(2n)+y.
-This continues until a single block remains, that being the result.
-
- The advantage of this method is that the multiplications for each x
-are big blocks, allowing Karatsuba and higher algorithms to be used.
-But the cost of calculating the powers b^(n*2^i) must be overcome.
-`SET_STR_THRESHOLD' usually ends up quite big, around 5000 digits, and
-on some processors much bigger still.
-
- `SET_STR_THRESHOLD' is based on the input digits (and tuned for
-decimal), though it might be better based on a limb count, so as to be
-independent of the base. But that sort of count isn't used by the base
-case and so would need some sort of initial calculation or estimate.
-
- The main reason `SET_STR_THRESHOLD' is so much bigger than the
-corresponding `GET_STR_PRECOMPUTE_THRESHOLD' is that `mpn_mul_1' is
-much faster than `mpn_divrem_1' (often by a factor of 10, or more).
-
-
-File: mpir.info, Node: Other Algorithms, Next: Assembler Coding, Prev: Radix Conversion Algorithms, Up: Algorithms
-
-15.7 Other Algorithms
-=====================
-
-* Menu:
-
-* Prime Testing Algorithm::
-* Factorial Algorithm::
-* Binomial Coefficients Algorithm::
-* Fibonacci Numbers Algorithm::
-* Lucas Numbers Algorithm::
-* Random Number Algorithms::
-
-
-File: mpir.info, Node: Prime Testing Algorithm, Next: Factorial Algorithm, Prev: Other Algorithms, Up: Other Algorithms
-
-15.7.1 Prime Testing
---------------------
-
-This section is somewhat out-of-date.
-
- The primality testing in `mpz_probab_prime_p' (*note Number
-Theoretic Functions::) first does some trial division by small factors
-and then uses the Miller-Rabin probabilistic primality testing
-algorithm, as described in Knuth section 4.5.4 algorithm P (*note
-References::).
-
- For an odd input n, and with n = q*2^k+1 where q is odd, this
-algorithm selects a random base x and tests whether x^q mod n is 1 or
--1, or an x^(q*2^j) mod n is 1, for 1<=j<=k. If so then n is probably
-prime, if not then n is definitely composite.
-
- Any prime n will pass the test, but some composites do too. Such
-composites are known as strong pseudoprimes to base x. No n is a
-strong pseudoprime to more than 1/4 of all bases (see Knuth exercise
-22), hence with x chosen at random there's no more than a 1/4 chance a
-"probable prime" will in fact be composite.
-
- In fact strong pseudoprimes are quite rare, making the test much more
-powerful than this analysis would suggest, but 1/4 is all that's proven
-for an arbitrary n.
-
-
-File: mpir.info, Node: Factorial Algorithm, Next: Binomial Coefficients Algorithm, Prev: Prime Testing Algorithm, Up: Other Algorithms
-
-15.7.2 Factorial
-----------------
-
-This section is out-of-date.
-
- Factorials are calculated by a combination of removal of twos,
-powering, and binary splitting. The procedure can be best illustrated
-with an example,
-
- 23! = 1.2.3.4.5.6.7.8.9.10.11.12.13.14.15.16.17.18.19.20.21.22.23
-
-has factors of two removed,
-
- 23! = 2^19.1.1.3.1.5.3.7.1.9.5.11.3.13.7.15.1.17.9.19.5.21.11.23
-
-and the resulting terms collected up according to their multiplicity,
-
- 23! = 2^19.(3.5)^3.(7.9.11)^2.(13.15.17.19.21.23)
-
- Each sequence such as 13.15.17.19.21.23 is evaluated by splitting
-into every second term, as for instance (13.17.21).(15.19.23), and the
-same recursively on each half. This is implemented iteratively using
-some bit twiddling.
-
- Such splitting is more efficient than repeated Nx1 multiplies since
-it forms big multiplies, allowing Karatsuba and higher algorithms to be
-used. And even below the Karatsuba threshold a big block of work can
-be more efficient for the basecase algorithm.
-
- Splitting into subsequences of every second term keeps the resulting
-products more nearly equal in size than would the simpler approach of
-say taking the first half and second half of the sequence. Nearly
-equal products are more efficient for the current multiply
-implementation.
-
-
-File: mpir.info, Node: Binomial Coefficients Algorithm, Next: Fibonacci Numbers Algorithm, Prev: Factorial Algorithm, Up: Other Algorithms
-
-15.7.3 Binomial Coefficients
-----------------------------
-
-Binomial coefficients C(n,k) are calculated by first arranging k <= n/2
-using C(n,k) = C(n,n-k) if necessary, and then evaluating the following
-product simply from i=2 to i=k.
-
- k (n-k+i)
- C(n,k) = (n-k+1) * prod -------
- i=2 i
-
- It's easy to show that each denominator i will divide the product so
-far, so the exact division algorithm is used (*note Exact Division::).
-
- The numerators n-k+i and denominators i are first accumulated into
-as many fit a limb, to save multi-precision operations, though for
-`mpz_bin_ui' this applies only to the divisors, since n is an `mpz_t'
-and n-k+i in general won't fit in a limb at all.
-
-
-File: mpir.info, Node: Fibonacci Numbers Algorithm, Next: Lucas Numbers Algorithm, Prev: Binomial Coefficients Algorithm, Up: Other Algorithms
-
-15.7.4 Fibonacci Numbers
-------------------------
-
-The Fibonacci functions `mpz_fib_ui' and `mpz_fib2_ui' are designed for
-calculating isolated F[n] or F[n],F[n-1] values efficiently.
-
- For small n, a table of single limb values in `__gmp_fib_table' is
-used. On a 32-bit limb this goes up to F[47], or on a 64-bit limb up
-to F[93]. For convenience the table starts at F[-1].
-
- Beyond the table, values are generated with a binary powering
-algorithm, calculating a pair F[n] and F[n-1] working from high to low
-across the bits of n. The formulas used are
-
- F[2k+1] = 4*F[k]^2 - F[k-1]^2 + 2*(-1)^k
- F[2k-1] = F[k]^2 + F[k-1]^2
-
- F[2k] = F[2k+1] - F[2k-1]
-
- At each step, k is the high b bits of n. If the next bit of n is 0
-then F[2k],F[2k-1] is used, or if it's a 1 then F[2k+1],F[2k] is used,
-and the process repeated until all bits of n are incorporated. Notice
-these formulas require just two squares per bit of n.
-
- It'd be possible to handle the first few n above the single limb
-table with simple additions, using the defining Fibonacci recurrence
-F[k+1]=F[k]+F[k-1], but this is not done since it usually turns out to
-be faster for only about 10 or 20 values of n, and including a block of
-code for just those doesn't seem worthwhile. If they really mattered
-it'd be better to extend the data table.
-
- Using a table avoids lots of calculations on small numbers, and
-makes small n go fast. A bigger table would make more small n go fast,
-it's just a question of balancing size against desired speed. For MPIR
-the code is kept compact, with the emphasis primarily on a good
-powering algorithm.
-
- `mpz_fib2_ui' returns both F[n] and F[n-1], but `mpz_fib_ui' is only
-interested in F[n]. In this case the last step of the algorithm can
-become one multiply instead of two squares. One of the following two
-formulas is used, according as n is odd or even.
-
- F[2k] = F[k]*(F[k]+2F[k-1])
-
- F[2k+1] = (2F[k]+F[k-1])*(2F[k]-F[k-1]) + 2*(-1)^k
-
- F[2k+1] here is the same as above, just rearranged to be a multiply.
-For interest, the 2*(-1)^k term both here and above can be applied just
-to the low limb of the calculation, without a carry or borrow into
-further limbs, which saves some code size. See comments with
-`mpz_fib_ui' and the internal `mpn_fib2_ui' for how this is done.
-
-
-File: mpir.info, Node: Lucas Numbers Algorithm, Next: Random Number Algorithms, Prev: Fibonacci Numbers Algorithm, Up: Other Algorithms
-
-15.7.5 Lucas Numbers
---------------------
-
-`mpz_lucnum2_ui' derives a pair of Lucas numbers from a pair of
-Fibonacci numbers with the following simple formulas.
-
- L[k] = F[k] + 2*F[k-1]
- L[k-1] = 2*F[k] - F[k-1]
-
- `mpz_lucnum_ui' is only interested in L[n], and some work can be
-saved. Trailing zero bits on n can be handled with a single square
-each.
-
- L[2k] = L[k]^2 - 2*(-1)^k
-
- And the lowest 1 bit can be handled with one multiply of a pair of
-Fibonacci numbers, similar to what `mpz_fib_ui' does.
-
- L[2k+1] = 5*F[k-1]*(2*F[k]+F[k-1]) - 4*(-1)^k
-
-
-File: mpir.info, Node: Random Number Algorithms, Prev: Lucas Numbers Algorithm, Up: Other Algorithms
-
-15.7.6 Random Numbers
----------------------
-
-For the `urandomb' functions, random numbers are generated simply by
-concatenating bits produced by the generator. As long as the generator
-has good randomness properties this will produce well-distributed N bit
-numbers.
-
- For the `urandomm' functions, random numbers in a range 0<=R<N are
-generated by taking values R of ceil(log2(N)) bits each until one
-satisfies R<N. This will normally require only one or two attempts,
-but the attempts are limited in case the generator is somehow
-degenerate and produces only 1 bits or similar.
-
- The Mersenne Twister generator is by Matsumoto and Nishimura (*note
-References::). It has a non-repeating period of 2^19937-1, which is a
-Mersenne prime, hence the name of the generator. The state is 624
-words of 32-bits each, which is iterated with one XOR and shift for each
-32-bit word generated, making the algorithm very fast. Randomness
-properties are also very good and this is the default algorithm used by
-MPIR.
-
- Linear congruential generators are described in many text books, for
-instance Knuth volume 2 (*note References::). With a modulus M and
-parameters A and C, a integer state S is iterated by the formula S <-
-A*S+C mod M. At each step the new state is a linear function of the
-previous, mod M, hence the name of the generator.
-
- In MPIR only moduli of the form 2^N are supported, and the current
-implementation is not as well optimized as it could be. Overheads are
-significant when N is small, and when N is large clearly the multiply
-at each step will become slow. This is not a big concern, since the
-Mersenne Twister generator is better in every respect and is therefore
-recommended for all normal applications.
-
- For both generators the current state can be deduced by observing
-enough output and applying some linear algebra (over GF(2) in the case
-of the Mersenne Twister). This generally means raw output is
-unsuitable for cryptographic applications without further hashing or
-the like.
-
-
-File: mpir.info, Node: Assembler Coding, Prev: Other Algorithms, Up: Algorithms
-
-15.8 Assembler Coding
-=====================
-
-The assembler subroutines in MPIR are the most significant source of
-speed at small to moderate sizes. At larger sizes algorithm selection
-becomes more important, but of course speedups in low level routines
-will still speed up everything proportionally.
-
- Carry handling and widening multiplies that are important for MPIR
-can't be easily expressed in C. GCC `asm' blocks help a lot and are
-provided in `longlong.h', but hand coding low level routines invariably
-offers a speedup over generic C by a factor of anything from 2 to 10.
-
-* Menu:
-
-* Assembler Code Organisation::
-* Assembler Basics::
-* Assembler Carry Propagation::
-* Assembler Cache Handling::
-* Assembler Functional Units::
-* Assembler Floating Point::
-* Assembler SIMD Instructions::
-* Assembler Software Pipelining::
-* Assembler Loop Unrolling::
-* Assembler Writing Guide::
-
-
-File: mpir.info, Node: Assembler Code Organisation, Next: Assembler Basics, Prev: Assembler Coding, Up: Assembler Coding
-
-15.8.1 Code Organisation
-------------------------
-
-The various `mpn' subdirectories contain machine-dependent code, written
-in C or assembler. The `mpn/generic' subdirectory contains default
-code, used when there's no machine-specific version of a particular
-file.
-
- Each `mpn' subdirectory is for an ISA family. Generally 32-bit and
-64-bit variants in a family cannot share code and have separate
-directories. Within a family further subdirectories may exist for CPU
-variants.
-
- In each directory a `nails' subdirectory may exist, holding code with
-nails support for that CPU variant. A `NAILS_SUPPORT' directive in each
-file indicates the nails values the code handles. Nails code only
-exists where it's faster, or promises to be faster, than plain code.
-There's no effort put into nails if they're not going to enhance a
-given CPU.
-
-
-File: mpir.info, Node: Assembler Basics, Next: Assembler Carry Propagation, Prev: Assembler Code Organisation, Up: Assembler Coding
-
-15.8.2 Assembler Basics
------------------------
-
-`mpn_addmul_1' and `mpn_submul_1' are the most important routines for
-overall MPIR performance. All multiplications and divisions come down
-to repeated calls to these. `mpn_add_n', `mpn_sub_n', `mpn_lshift' and
-`mpn_rshift' are next most important.
-
- On some CPUs assembler versions of the internal functions
-`mpn_mul_basecase' and `mpn_sqr_basecase' give significant speedups,
-mainly through avoiding function call overheads. They can also
-potentially make better use of a wide superscalar processor, as can
-bigger primitives like `mpn_addmul_2' or `mpn_addmul_4'.
-
- The restrictions on overlaps between sources and destinations (*note
-Low-level Functions::) are designed to facilitate a variety of
-implementations. For example, knowing `mpn_add_n' won't have partly
-overlapping sources and destination means reading can be done far ahead
-of writing on superscalar processors, and loops can be vectorized on a
-vector processor, depending on the carry handling.
-
-
-File: mpir.info, Node: Assembler Carry Propagation, Next: Assembler Cache Handling, Prev: Assembler Basics, Up: Assembler Coding
-
-15.8.3 Carry Propagation
-------------------------
-
-The problem that presents most challenges in MPIR is propagating
-carries from one limb to the next. In functions like `mpn_addmul_1' and
-`mpn_add_n', carries are the only dependencies between limb operations.
-
- On processors with carry flags, a straightforward CISC style `adc' is
-generally best. AMD K6 `mpn_addmul_1' however is an example of an
-unusual set of circumstances where a branch works out better.
-
- On RISC processors generally an add and compare for overflow is
-used. This sort of thing can be seen in `mpn/generic/aors_n.c'. Some
-carry propagation schemes require 4 instructions, meaning at least 4
-cycles per limb, but other schemes may use just 1 or 2. On wide
-superscalar processors performance may be completely determined by the
-number of dependent instructions between carry-in and carry-out for
-each limb.
-
- On vector processors good use can be made of the fact that a carry
-bit only very rarely propagates more than one limb. When adding a
-single bit to a limb, there's only a carry out if that limb was
-`0xFF...FF' which on random data will be only 1 in 2^mp_bits_per_limb.
-`mpn/cray/add_n.c' is an example of this, it adds all limbs in
-parallel, adds one set of carry bits in parallel and then only rarely
-needs to fall through to a loop propagating further carries.
-
- On the x86s, GCC (as of version 2.95.2) doesn't generate
-particularly good code for the RISC style idioms that are necessary to
-handle carry bits in C. Often conditional jumps are generated where
-`adc' or `sbb' forms would be better. And so unfortunately almost any
-loop involving carry bits needs to be coded in assembler for best
-results.
-
-
-File: mpir.info, Node: Assembler Cache Handling, Next: Assembler Functional Units, Prev: Assembler Carry Propagation, Up: Assembler Coding
-
-15.8.4 Cache Handling
----------------------
-
-MPIR aims to perform well both on operands that fit entirely in L1
-cache and those which don't.
-
- Basic routines like `mpn_add_n' or `mpn_lshift' are often used on
-large operands, so L2 and main memory performance is important for them.
-`mpn_mul_1' and `mpn_addmul_1' are mostly used for multiply and square
-basecases, so L1 performance matters most for them, unless assembler
-versions of `mpn_mul_basecase' and `mpn_sqr_basecase' exist, in which
-case the remaining uses are mostly for larger operands.
-
- For L2 or main memory operands, memory access times will almost
-certainly be more than the calculation time. The aim therefore is to
-maximize memory throughput, by starting a load of the next cache line
-while processing the contents of the previous one. Clearly this is
-only possible if the chip has a lock-up free cache or some sort of
-prefetch instruction. Most current chips have both these features.
-
- Prefetching sources combines well with loop unrolling, since a
-prefetch can be initiated once per unrolled loop (or more than once if
-the loop covers more than one cache line).
-
- On CPUs without write-allocate caches, prefetching destinations will
-ensure individual stores don't go further down the cache hierarchy,
-limiting bandwidth. Of course for calculations which are slow anyway,
-like `mpn_divrem_1', write-throughs might be fine.
-
- The distance ahead to prefetch will be determined by memory latency
-versus throughput. The aim of course is to have data arriving
-continuously, at peak throughput. Some CPUs have limits on the number
-of fetches or prefetches in progress.
-
- If a special prefetch instruction doesn't exist then a plain load
-can be used, but in that case care must be taken not to attempt to read
-past the end of an operand, since that might produce a segmentation
-violation.
-
- Some CPUs or systems have hardware that detects sequential memory
-accesses and initiates suitable cache movements automatically, making
-life easy.
-
-
-File: mpir.info, Node: Assembler Functional Units, Next: Assembler Floating Point, Prev: Assembler Cache Handling, Up: Assembler Coding
-
-15.8.5 Functional Units
------------------------
-
-When choosing an approach for an assembler loop, consideration is given
-to what operations can execute simultaneously and what throughput can
-thereby be achieved. In some cases an algorithm can be tweaked to
-accommodate available resources.
-
- Loop control will generally require a counter and pointer updates,
-costing as much as 5 instructions, plus any delays a branch introduces.
-CPU addressing modes might reduce pointer updates, perhaps by allowing
-just one updating pointer and others expressed as offsets from it, or
-on CISC chips with all addressing done with the loop counter as a
-scaled index.
-
- The final loop control cost can be amortised by processing several
-limbs in each iteration (*note Assembler Loop Unrolling::). This at
-least ensures loop control isn't a big fraction the work done.
-
- Memory throughput is always a limit. If perhaps only one load or
-one store can be done per cycle then 3 cycles/limb will the top speed
-for "binary" operations like `mpn_add_n', and any code achieving that
-is optimal.
-
- Integer resources can be freed up by having the loop counter in a
-float register, or by pressing the float units into use for some
-multiplying, perhaps doing every second limb on the float side (*note
-Assembler Floating Point::).
-
- Float resources can be freed up by doing carry propagation on the
-integer side, or even by doing integer to float conversions in integers
-using bit twiddling.
-
-
-File: mpir.info, Node: Assembler Floating Point, Next: Assembler SIMD Instructions, Prev: Assembler Functional Units, Up: Assembler Coding
-
-15.8.6 Floating Point
----------------------
-
-Floating point arithmetic is used in MPIR for multiplications on CPUs
-with poor integer multipliers. It's mostly useful for `mpn_mul_1',
-`mpn_addmul_1' and `mpn_submul_1' on 64-bit machines, and
-`mpn_mul_basecase' on both 32-bit and 64-bit machines.
-
- With IEEE 53-bit double precision floats, integer multiplications
-producing up to 53 bits will give exact results. Breaking a 64x64
-multiplication into eight 16x32->48 bit pieces is convenient. With
-some care though six 21x32->53 bit products can be used, if one of the
-lower two 21-bit pieces also uses the sign bit.
-
- For the `mpn_mul_1' family of functions on a 64-bit machine, the
-invariant single limb is split at the start, into 3 or 4 pieces.
-Inside the loop, the bignum operand is split into 32-bit pieces. Fast
-conversion of these unsigned 32-bit pieces to floating point is highly
-machine-dependent. In some cases, reading the data into the integer
-unit, zero-extending to 64-bits, then transferring to the floating
-point unit back via memory is the only option.
-
- Converting partial products back to 64-bit limbs is usually best
-done as a signed conversion. Since all values are smaller than 2^53,
-signed and unsigned are the same, but most processors lack unsigned
-conversions.
-
-
-
- Here is a diagram showing 16x32 bit products for an `mpn_mul_1' or
-`mpn_addmul_1' with a 64-bit limb. The single limb operand V is split
-into four 16-bit parts. The multi-limb operand U is split in the loop
-into two 32-bit parts.
-
- +---+---+---+---+
- |v48|v32|v16|v00| V operand
- +---+---+---+---+
-
- +-------+---+---+
- x | u32 | u00 | U operand (one limb)
- +---------------+
-
- ---------------------------------
-
- +-----------+
- | u00 x v00 | p00 48-bit products
- +-----------+
- +-----------+
- | u00 x v16 | p16
- +-----------+
- +-----------+
- | u00 x v32 | p32
- +-----------+
- +-----------+
- | u00 x v48 | p48
- +-----------+
- +-----------+
- | u32 x v00 | r32
- +-----------+
- +-----------+
- | u32 x v16 | r48
- +-----------+
- +-----------+
- | u32 x v32 | r64
- +-----------+
- +-----------+
- | u32 x v48 | r80
- +-----------+
-
- p32 and r32 can be summed using floating-point addition, and
-likewise p48 and r48. p00 and p16 can be summed with r64 and r80 from
-the previous iteration.
-
- For each loop then, four 49-bit quantities are transfered to the
-integer unit, aligned as follows,
-
- |-----64bits----|-----64bits----|
- +------------+
- | p00 + r64' | i00
- +------------+
- +------------+
- | p16 + r80' | i16
- +------------+
- +------------+
- | p32 + r32 | i32
- +------------+
- +------------+
- | p48 + r48 | i48
- +------------+
-
- The challenge then is to sum these efficiently and add in a carry
-limb, generating a low 64-bit result limb and a high 33-bit carry limb
-(i48 extends 33 bits into the high half).
-
-
-File: mpir.info, Node: Assembler SIMD Instructions, Next: Assembler Software Pipelining, Prev: Assembler Floating Point, Up: Assembler Coding
-
-15.8.7 SIMD Instructions
-------------------------
-
-The single-instruction multiple-data support in current microprocessors
-is aimed at signal processing algorithms where each data point can be
-treated more or less independently. There's generally not much support
-for propagating the sort of carries that arise in MPIR.
-
- SIMD multiplications of say four 16x16 bit multiplies only do as much
-work as one 32x32 from MPIR's point of view, and need some shifts and
-adds besides. But of course if say the SIMD form is fully pipelined
-and uses less instruction decoding then it may still be worthwhile.
-
- On the x86 chips, MMX has so far found a use in `mpn_rshift' and
-`mpn_lshift', and is used in a special case for 16-bit multipliers in
-the P55 `mpn_mul_1'. SSE2 is used for Pentium 4 `mpn_mul_1',
-`mpn_addmul_1', and `mpn_submul_1'.
-
-
-File: mpir.info, Node: Assembler Software Pipelining, Next: Assembler Loop Unrolling, Prev: Assembler SIMD Instructions, Up: Assembler Coding
-
-15.8.8 Software Pipelining
---------------------------
-
-Software pipelining consists of scheduling instructions around the
-branch point in a loop. For example a loop might issue a load not for
-use in the present iteration but the next, thereby allowing extra
-cycles for the data to arrive from memory.
-
- Naturally this is wanted only when doing things like loads or
-multiplies that take several cycles to complete, and only where a CPU
-has multiple functional units so that other work can be done in the
-meantime.
-
- A pipeline with several stages will have a data value in progress at
-each stage and each loop iteration moves them along one stage. This is
-like juggling.
-
- If the latency of some instruction is greater than the loop time
-then it will be necessary to unroll, so one register has a result ready
-to use while another (or multiple others) are still in progress.
-(*note Assembler Loop Unrolling::).
-
-
-File: mpir.info, Node: Assembler Loop Unrolling, Next: Assembler Writing Guide, Prev: Assembler Software Pipelining, Up: Assembler Coding
-
-15.8.9 Loop Unrolling
----------------------
-
-Loop unrolling consists of replicating code so that several limbs are
-processed in each loop. At a minimum this reduces loop overheads by a
-corresponding factor, but it can also allow better register usage, for
-example alternately using one register combination and then another.
-Judicious use of `m4' macros can help avoid lots of duplication in the
-source code.
-
- Any amount of unrolling can be handled with a loop counter that's
-decremented by N each time, stopping when the remaining count is less
-than the further N the loop will process. Or by subtracting N at the
-start, the termination condition becomes when the counter C is less
-than 0 (and the count of remaining limbs is C+N).
-
- Alternately for a power of 2 unroll the loop count and remainder can
-be established with a shift and mask. This is convenient if also
-making a computed jump into the middle of a large loop.
-
- The limbs not a multiple of the unrolling can be handled in various
-ways, for example
-
- * A simple loop at the end (or the start) to process the excess.
- Care will be wanted that it isn't too much slower than the
- unrolled part.
-
- * A set of binary tests, for example after an 8-limb unrolling, test
- for 4 more limbs to process, then a further 2 more or not, and
- finally 1 more or not. This will probably take more code space
- than a simple loop.
-
- * A `switch' statement, providing separate code for each possible
- excess, for example an 8-limb unrolling would have separate code
- for 0 remaining, 1 remaining, etc, up to 7 remaining. This might
- take a lot of code, but may be the best way to optimize all cases
- in combination with a deep pipelined loop.
-
- * A computed jump into the middle of the loop, thus making the first
- iteration handle the excess. This should make times smoothly
- increase with size, which is attractive, but setups for the jump
- and adjustments for pointers can be tricky and could become quite
- difficult in combination with deep pipelining.
-
-
-File: mpir.info, Node: Assembler Writing Guide, Prev: Assembler Loop Unrolling, Up: Assembler Coding
-
-15.8.10 Writing Guide
----------------------
-
-This is a guide to writing software pipelined loops for processing limb
-vectors in assembler.
-
- First determine the algorithm and which instructions are needed.
-Code it without unrolling or scheduling, to make sure it works. On a
-3-operand CPU try to write each new value to a new register, this will
-greatly simplify later steps.
-
- Then note for each instruction the functional unit and/or issue port
-requirements. If an instruction can use either of two units, like U0
-or U1 then make a category "U0/U1". Count the total using each unit
-(or combined unit), and count all instructions.
-
- Figure out from those counts the best possible loop time. The goal
-will be to find a perfect schedule where instruction latencies are
-completely hidden. The total instruction count might be the limiting
-factor, or perhaps a particular functional unit. It might be possible
-to tweak the instructions to help the limiting factor.
-
- Suppose the loop time is N, then make N issue buckets, with the
-final loop branch at the end of the last. Now fill the buckets with
-dummy instructions using the functional units desired. Run this to
-make sure the intended speed is reached.
-
- Now replace the dummy instructions with the real instructions from
-the slow but correct loop you started with. The first will typically
-be a load instruction. Then the instruction using that value is placed
-in a bucket an appropriate distance down. Run the loop again, to check
-it still runs at target speed.
-
- Keep placing instructions, frequently measuring the loop. After a
-few you will need to wrap around from the last bucket back to the top
-of the loop. If you used the new-register for new-value strategy above
-then there will be no register conflicts. If not then take care not to
-clobber something already in use. Changing registers at this time is
-very error prone.
-
- The loop will overlap two or more of the original loop iterations,
-and the computation of one vector element result will be started in one
-iteration of the new loop, and completed one or several iterations
-later.
-
- The final step is to create feed-in and wind-down code for the loop.
-A good way to do this is to make a copy (or copies) of the loop at the
-start and delete those instructions which don't have valid antecedents,
-and at the end replicate and delete those whose results are unwanted
-(including any further loads).
-
- The loop will have a minimum number of limbs loaded and processed,
-so the feed-in code must test if the request size is smaller and skip
-either to a suitable part of the wind-down or to special code for small
-sizes.
-
-
-File: mpir.info, Node: Internals, Next: Contributors, Prev: Algorithms, Up: Top
-
-16 Internals
-************
-
-*This chapter is provided only for informational purposes and the
-various internals described here may change in future MPIR releases.
-Applications expecting to be compatible with future releases should use
-only the documented interfaces described in previous chapters.*
-
-* Menu:
-
-* Integer Internals::
-* Rational Internals::
-* Float Internals::
-* Raw Output Internals::
-* C++ Interface Internals::
-
-
-File: mpir.info, Node: Integer Internals, Next: Rational Internals, Prev: Internals, Up: Internals
-
-16.1 Integer Internals
-======================
-
-`mpz_t' variables represent integers using sign and magnitude, in space
-dynamically allocated and reallocated. The fields are as follows.
-
-`_mp_size'
- The number of limbs, or the negative of that when representing a
- negative integer. Zero is represented by `_mp_size' set to zero,
- in which case the `_mp_d' data is unused.
-
-`_mp_d'
- A pointer to an array of limbs which is the magnitude. These are
- stored "little endian" as per the `mpn' functions, so `_mp_d[0]'
- is the least significant limb and `_mp_d[ABS(_mp_size)-1]' is the
- most significant. Whenever `_mp_size' is non-zero, the most
- significant limb is non-zero.
-
- Currently there's always at least one limb allocated, so for
- instance `mpz_set_ui' never needs to reallocate, and `mpz_get_ui'
- can fetch `_mp_d[0]' unconditionally (though its value is then
- only wanted if `_mp_size' is non-zero).
-
-`_mp_alloc'
- `_mp_alloc' is the number of limbs currently allocated at `_mp_d',
- and naturally `_mp_alloc >= ABS(_mp_size)'. When an `mpz' routine
- is about to (or might be about to) increase `_mp_size', it checks
- `_mp_alloc' to see whether there's enough space, and reallocates
- if not. `MPZ_REALLOC' is generally used for this.
-
- The various bitwise logical functions like `mpz_and' behave as if
-negative values were twos complement. But sign and magnitude is always
-used internally, and necessary adjustments are made during the
-calculations. Sometimes this isn't pretty, but sign and magnitude are
-best for other routines.
-
- Some internal temporary variables are setup with `MPZ_TMP_INIT' and
-these have `_mp_d' space obtained from `TMP_ALLOC' rather than the
-memory allocation functions. Care is taken to ensure that these are
-big enough that no reallocation is necessary (since it would have
-unpredictable consequences).
-
- `_mp_size' and `_mp_alloc' are `int', although `mp_size_t' is
-usually a `long'. This is done to make the fields just 32 bits on some
-64 bits systems, thereby saving a few bytes of data space but still
-providing plenty of range.
-
-
-File: mpir.info, Node: Rational Internals, Next: Float Internals, Prev: Integer Internals, Up: Internals
-
-16.2 Rational Internals
-=======================
-
-`mpq_t' variables represent rationals using an `mpz_t' numerator and
-denominator (*note Integer Internals::).
-
- The canonical form adopted is denominator positive (and non-zero),
-no common factors between numerator and denominator, and zero uniquely
-represented as 0/1.
-
- It's believed that casting out common factors at each stage of a
-calculation is best in general. A GCD is an O(N^2) operation so it's
-better to do a few small ones immediately than to delay and have to do
-a big one later. Knowing the numerator and denominator have no common
-factors can be used for example in `mpq_mul' to make only two cross
-GCDs necessary, not four.
-
- This general approach to common factors is badly sub-optimal in the
-presence of simple factorizations or little prospect for cancellation,
-but MPIR has no way to know when this will occur. As per *note
-Efficiency::, that's left to applications. The `mpq_t' framework might
-still suit, with `mpq_numref' and `mpq_denref' for direct access to the
-numerator and denominator, or of course `mpz_t' variables can be used
-directly.
-
-
-File: mpir.info, Node: Float Internals, Next: Raw Output Internals, Prev: Rational Internals, Up: Internals
-
-16.3 Float Internals
-====================
-
-Efficient calculation is the primary aim of MPIR floats and the use of
-whole limbs and simple rounding facilitates this.
-
- `mpf_t' floats have a variable precision mantissa and a single
-machine word signed exponent. The mantissa is represented using sign
-and magnitude.
-
- most least
- significant significant
- limb limb
-
- _mp_d
- |---- _mp_exp ---> |
- _____ _____ _____ _____ _____
- |_____|_____|_____|_____|_____|
- . <------------ radix point
-
- <-------- _mp_size --------->
-
-The fields are as follows.
-
-`_mp_size'
- The number of limbs currently in use, or the negative of that when
- representing a negative value. Zero is represented by `_mp_size'
- and `_mp_exp' both set to zero, and in that case the `_mp_d' data
- is unused. (In the future `_mp_exp' might be undefined when
- representing zero.)
-
-`_mp_prec'
- The precision of the mantissa, in limbs. In any calculation the
- aim is to produce `_mp_prec' limbs of result (the most significant
- being non-zero).
-
-`_mp_d'
- A pointer to the array of limbs which is the absolute value of the
- mantissa. These are stored "little endian" as per the `mpn'
- functions, so `_mp_d[0]' is the least significant limb and
- `_mp_d[ABS(_mp_size)-1]' the most significant.
-
- The most significant limb is always non-zero, but there are no
- other restrictions on its value, in particular the highest 1 bit
- can be anywhere within the limb.
-
- `_mp_prec+1' limbs are allocated to `_mp_d', the extra limb being
- for convenience (see below). There are no reallocations during a
- calculation, only in a change of precision with `mpf_set_prec'.
-
-`_mp_exp'
- The exponent, in limbs, determining the location of the implied
- radix point. Zero means the radix point is just above the most
- significant limb. Positive values mean a radix point offset
- towards the lower limbs and hence a value >= 1, as for example in
- the diagram above. Negative exponents mean a radix point further
- above the highest limb.
-
- Naturally the exponent can be any value, it doesn't have to fall
- within the limbs as the diagram shows, it can be a long way above
- or a long way below. Limbs other than those included in the
- `{_mp_d,_mp_size}' data are treated as zero.
-
- `_mp_size' and `_mp_prec' are `int', although `mp_size_t' is usually
-a `long'. This is done to make the fields just 32 bits on some 64 bits
-systems, thereby saving a few bytes of data space but still providing
-plenty of range.
-
-
-The following various points should be noted.
-
-Low Zeros
- The least significant limbs `_mp_d[0]' etc can be zero, though
- such low zeros can always be ignored. Routines likely to produce
- low zeros check and avoid them to save time in subsequent
- calculations, but for most routines they're quite unlikely and
- aren't checked.
-
-Mantissa Size Range
- The `_mp_size' count of limbs in use can be less than `_mp_prec' if
- the value can be represented in less. This means low precision
- values or small integers stored in a high precision `mpf_t' can
- still be operated on efficiently.
-
- `_mp_size' can also be greater than `_mp_prec'. Firstly a value is
- allowed to use all of the `_mp_prec+1' limbs available at `_mp_d',
- and secondly when `mpf_set_prec_raw' lowers `_mp_prec' it leaves
- `_mp_size' unchanged and so the size can be arbitrarily bigger than
- `_mp_prec'.
-
-Rounding
- All rounding is done on limb boundaries. Calculating `_mp_prec'
- limbs with the high non-zero will ensure the application requested
- minimum precision is obtained.
-
- The use of simple "trunc" rounding towards zero is efficient,
- since there's no need to examine extra limbs and increment or
- decrement.
-
-Bit Shifts
- Since the exponent is in limbs, there are no bit shifts in basic
- operations like `mpf_add' and `mpf_mul'. When differing exponents
- are encountered all that's needed is to adjust pointers to line up
- the relevant limbs.
-
- Of course `mpf_mul_2exp' and `mpf_div_2exp' will require bit
- shifts, but the choice is between an exponent in limbs which
- requires shifts there, or one in bits which requires them almost
- everywhere else.
-
-Use of `_mp_prec+1' Limbs
- The extra limb on `_mp_d' (`_mp_prec+1' rather than just
- `_mp_prec') helps when an `mpf' routine might get a carry from its
- operation. `mpf_add' for instance will do an `mpn_add' of
- `_mp_prec' limbs. If there's no carry then that's the result, but
- if there is a carry then it's stored in the extra limb of space and
- `_mp_size' becomes `_mp_prec+1'.
-
- Whenever `_mp_prec+1' limbs are held in a variable, the low limb
- is not needed for the intended precision, only the `_mp_prec' high
- limbs. But zeroing it out or moving the rest down is unnecessary.
- Subsequent routines reading the value will simply take the high
- limbs they need, and this will be `_mp_prec' if their target has
- that same precision. This is no more than a pointer adjustment,
- and must be checked anyway since the destination precision can be
- different from the sources.
-
- Copy functions like `mpf_set' will retain a full `_mp_prec+1' limbs
- if available. This ensures that a variable which has `_mp_size'
- equal to `_mp_prec+1' will get its full exact value copied.
- Strictly speaking this is unnecessary since only `_mp_prec' limbs
- are needed for the application's requested precision, but it's
- considered that an `mpf_set' from one variable into another of the
- same precision ought to produce an exact copy.
-
-Application Precisions
- `__GMPF_BITS_TO_PREC' converts an application requested precision
- to an `_mp_prec'. The value in bits is rounded up to a whole limb
- then an extra limb is added since the most significant limb of
- `_mp_d' is only non-zero and therefore might contain only one bit.
-
- `__GMPF_PREC_TO_BITS' does the reverse conversion, and removes the
- extra limb from `_mp_prec' before converting to bits. The net
- effect of reading back with `mpf_get_prec' is simply the precision
- rounded up to a multiple of `mp_bits_per_limb'.
-
- Note that the extra limb added here for the high only being
- non-zero is in addition to the extra limb allocated to `_mp_d'.
- For example with a 32-bit limb, an application request for 250
- bits will be rounded up to 8 limbs, then an extra added for the
- high being only non-zero, giving an `_mp_prec' of 9. `_mp_d' then
- gets 10 limbs allocated. Reading back with `mpf_get_prec' will
- take `_mp_prec' subtract 1 limb and multiply by 32, giving 256
- bits.
-
- Strictly speaking, the fact the high limb has at least one bit
- means that a float with, say, 3 limbs of 32-bits each will be
- holding at least 65 bits, but for the purposes of `mpf_t' it's
- considered simply to be 64 bits, a nice multiple of the limb size.
-
-
-File: mpir.info, Node: Raw Output Internals, Next: C++ Interface Internals, Prev: Float Internals, Up: Internals
-
-16.4 Raw Output Internals
-=========================
-
-`mpz_out_raw' uses the following format.
-
- +------+------------------------+
- | size | data bytes |
- +------+------------------------+
-
- The size is 4 bytes written most significant byte first, being the
-number of subsequent data bytes, or the twos complement negative of
-that when a negative integer is represented. The data bytes are the
-absolute value of the integer, written most significant byte first.
-
- The most significant data byte is always non-zero, so the output is
-the same on all systems, irrespective of limb size.
-
- In GMP 1, leading zero bytes were written to pad the data bytes to a
-multiple of the limb size. `mpz_inp_raw' will still accept this, for
-compatibility.
-
- The use of "big endian" for both the size and data fields is
-deliberate, it makes the data easy to read in a hex dump of a file.
-Unfortunately it also means that the limb data must be reversed when
-reading or writing, so neither a big endian nor little endian system
-can just read and write `_mp_d'.
-
-
-File: mpir.info, Node: C++ Interface Internals, Prev: Raw Output Internals, Up: Internals
-
-16.5 C++ Interface Internals
-============================
-
-A system of expression templates is used to ensure something like
-`a=b+c' turns into a simple call to `mpz_add' etc. For `mpf_class' the
-scheme also ensures the precision of the final destination is used for
-any temporaries within a statement like `f=w*x+y*z'. These are
-important features which a naive implementation cannot provide.
-
- A simplified description of the scheme follows. The true scheme is
-complicated by the fact that expressions have different return types.
-For detailed information, refer to the source code.
-
- To perform an operation, say, addition, we first define a "function
-object" evaluating it,
-
- struct __gmp_binary_plus
- {
- static void eval(mpf_t f, mpf_t g, mpf_t h) { mpf_add(f, g, h); }
- };
-
-And an "additive expression" object,
-
- __gmp_expr<__gmp_binary_expr<mpf_class, mpf_class, __gmp_binary_plus> >
- operator+(const mpf_class &f, const mpf_class &g)
- {
- return __gmp_expr
- <__gmp_binary_expr<mpf_class, mpf_class, __gmp_binary_plus> >(f, g);
- }
-
- The seemingly redundant `__gmp_expr<__gmp_binary_expr<...>>' is used
-to encapsulate any possible kind of expression into a single template
-type. In fact even `mpf_class' etc are `typedef' specializations of
-`__gmp_expr'.
-
- Next we define assignment of `__gmp_expr' to `mpf_class'.
-
- template <class T>
- mpf_class & mpf_class::operator=(const __gmp_expr<T> &expr)
- {
- expr.eval(this->get_mpf_t(), this->precision());
- return *this;
- }
-
- template <class Op>
- void __gmp_expr<__gmp_binary_expr<mpf_class, mpf_class, Op> >::eval
- (mpf_t f, mp_bitcnt_t precision)
- {
- Op::eval(f, expr.val1.get_mpf_t(), expr.val2.get_mpf_t());
- }
-
- where `expr.val1' and `expr.val2' are references to the expression's
-operands (here `expr' is the `__gmp_binary_expr' stored within the
-`__gmp_expr').
-
- This way, the expression is actually evaluated only at the time of
-assignment, when the required precision (that of `f') is known.
-Furthermore the target `mpf_t' is now available, thus we can call
-`mpf_add' directly with `f' as the output argument.
-
- Compound expressions are handled by defining operators taking
-subexpressions as their arguments, like this:
-
- template <class T, class U>
- __gmp_expr
- <__gmp_binary_expr<__gmp_expr<T>, __gmp_expr<U>, __gmp_binary_plus> >
- operator+(const __gmp_expr<T> &expr1, const __gmp_expr<U> &expr2)
- {
- return __gmp_expr
- <__gmp_binary_expr<__gmp_expr<T>, __gmp_expr<U>, __gmp_binary_plus> >
- (expr1, expr2);
- }
-
- And the corresponding specializations of `__gmp_expr::eval':
-
- template <class T, class U, class Op>
- void __gmp_expr
- <__gmp_binary_expr<__gmp_expr<T>, __gmp_expr<U>, Op> >::eval
- (mpf_t f, mp_bitcnt_t precision)
- {
- // declare two temporaries
- mpf_class temp1(expr.val1, precision), temp2(expr.val2, precision);
- Op::eval(f, temp1.get_mpf_t(), temp2.get_mpf_t());
- }
-
- The expression is thus recursively evaluated to any level of
-complexity and all subexpressions are evaluated to the precision of `f'.
-
-
-File: mpir.info, Node: Contributors, Next: References, Prev: Internals, Up: Top
-
-Appendix A Contributors
-***********************
-
-Torbjorn Granlund wrote the original GMP library and is still
-developing and maintaining it. Several other individuals and
-organizations have contributed to GMP in various ways. Here is a list
-in chronological order:
-
- Gunnar Sjoedin and Hans Riesel helped with mathematical problems in
-early versions of the library.
-
- Richard Stallman contributed to the interface design and revised the
-first version of this manual.
-
- Brian Beuning and Doug Lea helped with testing of early versions of
-the library and made creative suggestions.
-
- John Amanatides of York University in Canada contributed the function
-`mpz_probab_prime_p'.
-
- Paul Zimmermann of Inria sparked the development of GMP 2, with his
-comparisons between bignum packages.
-
- Ken Weber (Kent State University, Universidade Federal do Rio Grande
-do Sul) contributed `mpz_gcd', `mpz_divexact', `mpn_gcd', and
-`mpn_bdivmod', partially supported by CNPq (Brazil) grant 301314194-2.
-
- Per Bothner of Cygnus Support helped to set up GMP to use Cygnus'
-configure. He has also made valuable suggestions and tested numerous
-intermediary releases.
-
- Joachim Hollman was involved in the design of the `mpf' interface,
-and in the `mpz' design revisions for version 2.
-
- Bennet Yee contributed the initial versions of `mpz_jacobi' and
-`mpz_legendre'.
-
- Andreas Schwab contributed the files `mpn/m68k/lshift.S' and
-`mpn/m68k/rshift.S' (now in `.asm' form).
-
- The development of floating point functions of GNU MP 2, were
-supported in part by the ESPRIT-BRA (Basic Research Activities) 6846
-project POSSO (POlynomial System SOlving).
-
- GNU MP 2 was finished and released by SWOX AB, SWEDEN, in
-cooperation with the IDA Center for Computing Sciences, USA.
-
- Robert Harley of Inria, France and David Seal of ARM, England,
-suggested clever improvements for population count.
-
- Robert Harley also wrote highly optimized Karatsuba and 3-way Toom
-multiplication functions for GMP 3. He also contributed the ARM
-assembly code.
-
- Torsten Ekedahl of the Mathematical department of Stockholm
-University provided significant inspiration during several phases of
-the GMP development. His mathematical expertise helped improve several
-algorithms.
-
- Paul Zimmermann wrote the Divide and Conquer division code, the REDC
-code, the REDC-based mpz_powm code, the FFT multiply code, and the
-Karatsuba square root code. He also rewrote the Toom3 code for GMP
-4.2. The ECMNET project Paul is organizing was a driving force behind
-many of the optimizations in GMP 3.
-
- Linus Nordberg wrote the new configure system based on autoconf and
-implemented the new random functions.
-
- Kent Boortz made the Mac OS 9 port.
-
- Kevin Ryde worked on a number of things: optimized x86 code, m4 asm
-macros, parameter tuning, speed measuring, the configure system,
-function inlining, divisibility tests, bit scanning, Jacobi symbols,
-Fibonacci and Lucas number functions, printf and scanf functions, perl
-interface, demo expression parser, the algorithms chapter in the
-manual, `gmpasm-mode.el', and various miscellaneous improvements
-elsewhere.
-
- Steve Root helped write the optimized alpha 21264 assembly code.
-
- Gerardo Ballabio wrote the `gmpxx.h' C++ class interface and the C++
-`istream' input routines.
-
- GNU MP 4 was finished and released by Torbjorn Granlund and Kevin
-Ryde. Torbjorn's work was partially funded by the IDA Center for
-Computing Sciences, USA.
-
- Jason Moxham rewrote `mpz_fac_ui'.
-
- Pedro Gimeno implemented the Mersenne Twister and made other random
-number improvements.
-
- (This list is chronological, not ordered after significance. If you
-have contributed to GMP/MPIR but are not listed above, please tell
-`http://groups.google.com/group/mpir-devel' about the omission!)
-
- Thanks go to Hans Thorsen for donating an SGI system for the GMP
-test system environment.
-
- In 2008 GMP was forked and gave rise to the MPIR (Multiple Precision
-Integers and Rationals) project. In 2010 version 2.0.0 of MPIR switched
-to LGPL v3+ and much code from GMP was again incorporated into MPIR.
-
- The MPIR project has largely been a collaboration of William Hart,
-Brian Gladman and Jason Moxham. MPIR code not obtained from GMP and not
-specifically mentioned elsewhere below is likely written by one of
-these three.
-
- William Hart did much of the early MPIR coding including build
-system fixes. His contributions also include Toom 4 and 7 code and
-variants, extended GCD based on Niels Mollers ngcd work, asymptotically
-fast division code. He does much of the release management work.
-
- Brian Gladman wrote and maintains MSVC project files. He has also
-done much of the conversion of assembly code to yasm format. He rewrote
-the benchmark program and developed MSVC ports of tune, speed, try and
-the benchmark code. He helped with many aspects of the merging of GMP
-code into MPIR after the switch to LGPL v3+.
-
- Jason Moxham has contributed a great deal of x86 assembly code. He
-has also contributed improved root code and mulhi and mullo routines
-and implemented Peter Montgomery's single limb remainder algorithm. He
-has also contributed a command line build system for Windows and
-numerous build system fixes.
-
- The following people have either contributed directly to the MPIR
-project, made code available on their websites or contributed code to
-the official GNU project which has been used in MPIR.
-
- Pierrick Gaudry wrote some fast assembly support for AMD 64.
-
- Jason Martin wrote some fast assembly patches for Core 2 and
-converted them to intel format. He also did the initial merge of Niels
-Moller's fast GCD patches. He wrote fast addmul functions for Itanium.
-
- Gonzalo Tornaria helped patch config.guess and associated files to
-distinguish modern processors. He also patched mpirbench.
-
- Michael Abshoff helped resolve some build issues on various
-platforms. He served for a while as release manager for the MPIR
-project.
-
- Mariah Lennox contributed patches to mpirbench and various build
-failure reports. She has also reported gcc bugs found during MPIR
-development.
-
- Niels Moller wrote the fast ngcd code for computing integer GCD, the
-quadratic Hensel division code and precomputed inverse code for
-Euclidean division. He also made contributions to the Toom multiply
-code, especially helper functions to simplify Toom evaluations.
-
- Pierrick Gaudry provided initial AMD 64 assembly support and revised
-the FFT code.
-
- Paul Zimmermann provided an mpz implementation of Toom 4, wrote much
-of the FFT code, wrote some of the rootrem code and contributed
-invert.c for computing precomputed inverses.
-
- Alexander Kruppa revised the FFT code.
-
- Torbjorn Granlund revised the FFT code and wrote a lot of division
-code, including the quadratic Euclidean division code, many parts of
-the divide and conquer division code, both Hensel and Euclidean, and
-his code was also reused for parts of the asymptotically fast division
-code. He also helped write the root code and wrote much of the Itanium
-assembly code and a couple of Core 2 assembly functions and part of the
-basecase middle product assembly code for x86 64 bit. He also wrote the
-improved string input and output code and made improvements to the GCD
-and extended GCD code. Torbjorn is also responsible for numerous other
-bits and pieces that have been used from the GNU project.
-
- Marco Bodrato and Alberto Zanoni suggested the unbalanced multiply
-strategy and found optimal Toom multiplication sequences.
-
- Marco Bodrato wrote an mpz implementation of the Toom 7 code and
-wrote most of the Toom 8.5 multiply and squaring code. He also helped
-write the divide and conquer Euclidean division code.
-
- Robert Gerbicz contributed fast factorial code.
-
- David Harvey wrote fast middle product code and divide and conquer
-approximate quotient code for both Euclidean and Hensel division and
-contributed to the quadratic Hensel code.
-
- T. R. Nicely wrote primality tests used in the benchmark code.
-
- Jeff Gilchrist assisted with the porting of T. R. Nicely's primality
-code to MPIR and helped with tuning.
-
- Peter Shrimpton wrote the BPSW primality test used up to
-GMP_LIMB_BITS.
-
- Thanks to Microsoft for supporting Jason Moxham to work on a command
-line build system for Windows and some assembly improvements for
-Windows.
-
- Thanks to the Free Software Foundation France for giving us access
-to their build farm.
-
- Thanks to William Stein for giving us access to his sage.math
-machines for testing and for hosting the MPIR website, and for
-supporting us in inumerably many other ways.
-
- Minh Van Nguyen served as release manager for MPIR 2.1.0.
-
- Case Vanhorsen helped with release testing.
-
- David Cleaver filed a bug report.
-
- Julien Puydt provided tuning values.
-
- Leif Lionhardy provided tuning values.
-
- Jean-Pierre Flori provided tuning values.
-
-
-File: mpir.info, Node: References, Next: GNU Free Documentation License, Prev: Contributors, Up: Top
-
-Appendix B References
-*********************
-
-B.1 Books
-=========
-
- * Jonathan M. Borwein and Peter B. Borwein, "Pi and the AGM: A Study
- in Analytic Number Theory and Computational Complexity", Wiley,
- 1998.
-
- * Henri Cohen, "A Course in Computational Algebraic Number Theory",
- Graduate Texts in Mathematics number 138, Springer-Verlag, 1993.
- `http://www.math.u-bordeaux.fr/~cohen/'
-
- * Richard Crandall, Carl Pomerance, "Prime Numbers: A Computational
- Perspective" 2nd edition, Springer, 2005.
-
- * Donald E. Knuth, "The Art of Computer Programming", volume 2,
- "Seminumerical Algorithms", 3rd edition, Addison-Wesley, 1998.
- `http://www-cs-faculty.stanford.edu/~knuth/taocp.html'
-
- * John D. Lipson, "Elements of Algebra and Algebraic Computing", The
- Benjamin Cummings Publishing Company Inc, 1981.
-
- * Alfred J. Menezes, Paul C. van Oorschot and Scott A. Vanstone,
- "Handbook of Applied Cryptography",
- `http://www.cacr.math.uwaterloo.ca/hac/'
-
- * Richard M. Stallman, "Using and Porting GCC", Free Software
- Foundation, 1999, available online
- `http://gcc.gnu.org/onlinedocs/', and in the GCC package
- `ftp://ftp.gnu.org/gnu/gcc/'
-
-B.2 Papers
-==========
-
- * Dan Bernstein, "Detecting perfect powers in essentially linear
- time", Math. Comp. (67) pp. 1253-1283, 1998.
-
- * Yves Bertot, Nicolas Magaud and Paul Zimmermann, "A Proof of GMP
- Square Root", Journal of Automated Reasoning, volume 29, 2002, pp.
- 225-252. Also available online as INRIA Research Report 4475,
- June 2001, `http://www.inria.fr/rrrt/rr-4475.html'
-
- * Marco Bodrato, Alberto Zanoni, "Integer and Polynomial
- Multiplication: Towards optimal Toom-Cook Matrices", ISAAC 2007
- Proceedings, Ontario, Canada, July 29 - August 1, 2007, ACM Press.
- Available online at `http://ln.bodrato.it/issac2007_pdf'
-
- * Marco Bodrato, "High degree Toom`n'half for balanced and
- unbalanced multiplication", E. Antelo, D. Hough and P. Ienne,
- editors, Proceedings of the 20th IEEE Symposium on Computer
- Arithmetic, IEEE, Tubingen, Germany, July 25-27, 2011, pp. 15-222.
- See `http://bodrato.it/papers'
-
- * Richard Brent and Paul Zimmermann, "Modern Computer Arithmetic",
- version 0.4, November 2009,
- `http://www.loria.fr/~zimmerma/mca/mca-0.4.pdf'
-
- * Christoph Burnikel and Joachim Ziegler, "Fast Recursive Division",
- Max-Planck-Institut fuer Informatik Research Report MPI-I-98-1-022,
- `http://data.mpi-sb.mpg.de/internet/reports.nsf/NumberView/1998-1-022'
-
- * Agner Fog, "Software optimization resources", online at
- `http://www.agner.org/optimize/'
-
- * Pierrick Gaudry, Alexander Kruppa, Paul Zimmermann, "A GMP-based
- implementation of Schoenhage-Strassen's large integer
- multiplication algorithm", ISAAC 2007 Proceedings, Ontario,
- Canada, July 29 - August 1, 2007, pp. 167-174, ACM Press. Full
- text available at
- `http://hal.inria.fr/docs/00/14/86/20/PDF/fft.final.pdf'
-
- * Torbjorn Granlund and Peter L. Montgomery, "Division by Invariant
- Integers using Multiplication", in Proceedings of the SIGPLAN
- PLDI'94 Conference, June 1994. Also available
- `ftp://ftp.cwi.nl/pub/pmontgom/divcnst.psa4.gz' (and .psl.gz).
-
- * Niels Mo"ller and Torbjo"rn Granlund, "Improved division by
- invariant integers", to appear.
-
- * Torbjo"rn Granlund and Niels Mo"ller, "Division of integers large
- and small", to appear.
-
- * David Harvey, "The Karatsuba middle product for integers",
- (preprint), 2009. Available at
- `http://www.cims.nyu.edu/~harvey/mulmid/mulmid.pdf'
-
- * Tudor Jebelean, "An algorithm for exact division", Journal of
- Symbolic Computation, volume 15, 1993, pp. 169-180. Research
- report version available
- `ftp://ftp.risc.uni-linz.ac.at/pub/techreports/1992/92-35.ps.gz'
-
- * Tudor Jebelean, "Exact Division with Karatsuba Complexity -
- Extended Abstract", RISC-Linz technical report 96-31,
- `ftp://ftp.risc.uni-linz.ac.at/pub/techreports/1996/96-31.ps.gz'
-
- * Tudor Jebelean, "Practical Integer Division with Karatsuba
- Complexity", ISSAC 97, pp. 339-341. Technical report available
- `ftp://ftp.risc.uni-linz.ac.at/pub/techreports/1996/96-29.ps.gz'
-
- * Tudor Jebelean, "A Generalization of the Binary GCD Algorithm",
- ISSAC 93, pp. 111-116. Technical report version available
- `ftp://ftp.risc.uni-linz.ac.at/pub/techreports/1993/93-01.ps.gz'
-
- * Tudor Jebelean, "A Double-Digit Lehmer-Euclid Algorithm for
- Finding the GCD of Long Integers", Journal of Symbolic
- Computation, volume 19, 1995, pp. 145-157. Technical report
- version also available
- `ftp://ftp.risc.uni-linz.ac.at/pub/techreports/1992/92-69.ps.gz'
-
- * Werner Krandick, Jeremy R. Johnson, "Efficient Multiprecision
- Floating Point Multiplication with Exact Rounding", Technical
- Report, RISC Linz, 1993, available at
- `ftp://ftp.risc.uni-linz.ac.at/pub/techreports/1993/93-76.ps.gz'
-
- * Werner Krandick and Tudor Jebelean, "Bidirectional Exact Integer
- Division", Journal of Symbolic Computation, volume 21, 1996, pp.
- 441-455. Early technical report version also available
- `ftp://ftp.risc.uni-linz.ac.at/pub/techreports/1994/94-50.ps.gz'
-
- * Makoto Matsumoto and Takuji Nishimura, "Mersenne Twister: A
- 623-dimensionally equidistributed uniform pseudorandom number
- generator", ACM Transactions on Modelling and Computer Simulation,
- volume 8, January 1998, pp. 3-30. Available online
- `http://www.math.keio.ac.jp/~nisimura/random/doc/mt.ps.gz' (or
- .pdf)
-
- * R. Moenck and A. Borodin, "Fast Modular Transforms via Division",
- Proceedings of the 13th Annual IEEE Symposium on Switching and
- Automata Theory, October 1972, pp. 90-96. Reprinted as "Fast
- Modular Transforms", Journal of Computer and System Sciences,
- volume 8, number 3, June 1974, pp. 366-386.
-
- * Niels Mo"ller, "On Schoenhage's algorithm and subquadratic integer
- GCD computation", Math. Comp. 2007. Available online at
- `http://www.lysator.liu.se/~nisse/archive/S0025-5718-07-02017-0.pdf'
-
- * Peter L. Montgomery, "Modular Multiplication Without Trial
- Division", in Mathematics of Computation, volume 44, number 170,
- April 1985.
-
- * Thom Mulders, "On short multiplications and divisions", Appl.
- Algebra Engrg. Comm. Comput. 11 (2000), no. 1, pp. 69-88. Tech.
- report No. 276, Dept. of Comp. Sci., ETH Zurich, Nov 1997,
- available online at
- `ftp://ftp.inf.ethz.ch/pub/publications/tech-reports/2xx/276.pdf'
-
- * Arnold Scho"nhage and Volker Strassen, "Schnelle Multiplikation
- grosser Zahlen", Computing 7, 1971, pp. 281-292.
-
- * A. Scho"nhage, A. F. W. Grotefeld and E. Vetter, "Fast Algorithms,
- A Multitape Turing Machine Implementation" BI
- Wissenschafts-Verlag, Mannheim, 1994.
-
- * Kenneth Weber, "The accelerated integer GCD algorithm", ACM
- Transactions on Mathematical Software, volume 21, number 1, March
- 1995, pp. 111-122.
-
- * Paul Zimmermann, "Karatsuba Square Root", INRIA Research Report
- 3805, November 1999, `http://www.inria.fr/rrrt/rr-3805.html'
-
- * Paul Zimmermann, "A Proof of GMP Fast Division and Square Root
- Implementations",
- `http://www.loria.fr/~zimmerma/papers/proof-div-sqrt.ps.gz'
-
- * Dan Zuras, "On Squaring and Multiplying Large Integers", ARITH-11:
- IEEE Symposium on Computer Arithmetic, 1993, pp. 260 to 271.
- Reprinted as "More on Multiplying and Squaring Large Integers",
- IEEE Transactions on Computers, volume 43, number 8, August 1994,
- pp. 899-908.
-
-
-File: mpir.info, Node: GNU Free Documentation License, Next: Concept Index, Prev: References, Up: Top
-
-Appendix C GNU Free Documentation License
-*****************************************
-
- Version 1.3, 3 November 2008
-
- Copyright (C) 2000, 2001, 2002, 2007, 2008 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.
-
- 0. PREAMBLE
-
- The purpose of this License is to make a manual, textbook, or other
- functional and useful document "free" in the sense of freedom: to
- assure everyone the effective freedom to copy and redistribute it,
- with or without modifying it, either commercially or
- noncommercially. Secondarily, this License preserves for the
- author and publisher a way to get credit for their work, while not
- being considered responsible for modifications made by others.
-
- This License is a kind of "copyleft", which means that derivative
- works of the document must themselves be free in the same sense.
- It complements the GNU General Public License, which is a copyleft
- license designed for free software.
-
- We have designed this License in order to use it for manuals for
- free software, because free software needs free documentation: a
- free program should come with manuals providing the same freedoms
- that the software does. But this License is not limited to
- software manuals; it can be used for any textual work, regardless
- of subject matter or whether it is published as a printed book.
- We recommend this License principally for works whose purpose is
- instruction or reference.
-
- 1. APPLICABILITY AND DEFINITIONS
-
- This License applies to any manual or other work, in any medium,
- that contains a notice placed by the copyright holder saying it
- can be distributed under the terms of this License. Such a notice
- grants a world-wide, royalty-free license, unlimited in duration,
- to use that work under the conditions stated herein. The
- "Document", below, refers to any such manual or work. Any member
- of the public is a licensee, and is addressed as "you". You
- accept the license if you copy, modify or distribute the work in a
- way requiring permission under copyright law.
-
- A "Modified Version" of the Document means any work containing the
- Document or a portion of it, either copied verbatim, or with
- modifications and/or translated into another language.
-
- A "Secondary Section" is a named appendix or a front-matter section
- of the Document that deals exclusively with the relationship of the
- publishers or authors of the Document to the Document's overall
- subject (or to related matters) and contains nothing that could
- fall directly within that overall subject. (Thus, if the Document
- is in part a textbook of mathematics, a Secondary Section may not
- explain any mathematics.) The relationship could be a matter of
- historical connection with the subject or with related matters, or
- of legal, commercial, philosophical, ethical or political position
- regarding them.
-
- The "Invariant Sections" are certain Secondary Sections whose
- titles are designated, as being those of Invariant Sections, in
- the notice that says that the Document is released under this
- License. If a section does not fit the above definition of
- Secondary then it is not allowed to be designated as Invariant.
- The Document may contain zero Invariant Sections. If the Document
- does not identify any Invariant Sections then there are none.
-
- The "Cover Texts" are certain short passages of text that are
- listed, as Front-Cover Texts or Back-Cover Texts, in the notice
- that says that the Document is released under this License. A
- Front-Cover Text may be at most 5 words, and a Back-Cover Text may
- be at most 25 words.
-
- A "Transparent" copy of the Document means a machine-readable copy,
- represented in a format whose specification is available to the
- general public, that is suitable for revising the document
- straightforwardly with generic text editors or (for images
- composed of pixels) generic paint programs or (for drawings) some
- widely available drawing editor, and that is suitable for input to
- text formatters or for automatic translation to a variety of
- formats suitable for input to text formatters. A copy made in an
- otherwise Transparent file format whose markup, or absence of
- markup, has been arranged to thwart or discourage subsequent
- modification by readers is not Transparent. An image format is
- not Transparent if used for any substantial amount of text. A
- copy that is not "Transparent" is called "Opaque".
-
- Examples of suitable formats for Transparent copies include plain
- ASCII without markup, Texinfo input format, LaTeX input format,
- SGML or XML using a publicly available DTD, and
- standard-conforming simple HTML, PostScript or PDF designed for
- human modification. Examples of transparent image formats include
- PNG, XCF and JPG. Opaque formats include proprietary formats that
- can be read and edited only by proprietary word processors, SGML or
- XML for which the DTD and/or processing tools are not generally
- available, and the machine-generated HTML, PostScript or PDF
- produced by some word processors for output purposes only.
-
- The "Title Page" means, for a printed book, the title page itself,
- plus such following pages as are needed to hold, legibly, the
- material this License requires to appear in the title page. For
- works in formats which do not have any title page as such, "Title
- Page" means the text near the most prominent appearance of the
- work's title, preceding the beginning of the body of the text.
-
- The "publisher" means any person or entity that distributes copies
- of the Document to the public.
-
- A section "Entitled XYZ" means a named subunit of the Document
- whose title either is precisely XYZ or contains XYZ in parentheses
- following text that translates XYZ in another language. (Here XYZ
- stands for a specific section name mentioned below, such as
- "Acknowledgements", "Dedications", "Endorsements", or "History".)
- To "Preserve the Title" of such a section when you modify the
- Document means that it remains a section "Entitled XYZ" according
- to this definition.
-
- The Document may include Warranty Disclaimers next to the notice
- which states that this License applies to the Document. These
- Warranty Disclaimers are considered to be included by reference in
- this License, but only as regards disclaiming warranties: any other
- implication that these Warranty Disclaimers may have is void and
- has no effect on the meaning of this License.
-
- 2. VERBATIM COPYING
-
- You may copy and distribute the Document in any medium, either
- commercially or noncommercially, provided that this License, the
- copyright notices, and the license notice saying this License
- applies to the Document are reproduced in all copies, and that you
- add no other conditions whatsoever to those of this License. You
- may not use technical measures to obstruct or control the reading
- or further copying of the copies you make or distribute. However,
- you may accept compensation in exchange for copies. If you
- distribute a large enough number of copies you must also follow
- the conditions in section 3.
-
- You may also lend copies, under the same conditions stated above,
- and you may publicly display copies.
-
- 3. COPYING IN QUANTITY
-
- If you publish printed copies (or copies in media that commonly
- have printed covers) of the Document, numbering more than 100, and
- the Document's license notice requires Cover Texts, you must
- enclose the copies in covers that carry, clearly and legibly, all
- these Cover Texts: Front-Cover Texts on the front cover, and
- Back-Cover Texts on the back cover. Both covers must also clearly
- and legibly identify you as the publisher of these copies. The
- front cover must present the full title with all words of the
- title equally prominent and visible. You may add other material
- on the covers in addition. Copying with changes limited to the
- covers, as long as they preserve the title of the Document and
- satisfy these conditions, can be treated as verbatim copying in
- other respects.
-
- If the required texts for either cover are too voluminous to fit
- legibly, you should put the first ones listed (as many as fit
- reasonably) on the actual cover, and continue the rest onto
- adjacent pages.
-
- If you publish or distribute Opaque copies of the Document
- numbering more than 100, you must either include a
- machine-readable Transparent copy along with each Opaque copy, or
- state in or with each Opaque copy a computer-network location from
- which the general network-using public has access to download
- using public-standard network protocols a complete Transparent
- copy of the Document, free of added material. If you use the
- latter option, you must take reasonably prudent steps, when you
- begin distribution of Opaque copies in quantity, to ensure that
- this Transparent copy will remain thus accessible at the stated
- location until at least one year after the last time you
- distribute an Opaque copy (directly or through your agents or
- retailers) of that edition to the public.
-
- It is requested, but not required, that you contact the authors of
- the Document well before redistributing any large number of
- copies, to give them a chance to provide you with an updated
- version of the Document.
-
- 4. MODIFICATIONS
-
- You may copy and distribute a Modified Version of the Document
- under the conditions of sections 2 and 3 above, provided that you
- release the Modified Version under precisely this License, with
- the Modified Version filling the role of the Document, thus
- licensing distribution and modification of the Modified Version to
- whoever possesses a copy of it. In addition, you must do these
- things in the Modified Version:
-
- A. Use in the Title Page (and on the covers, if any) a title
- distinct from that of the Document, and from those of
- previous versions (which should, if there were any, be listed
- in the History section of the Document). You may use the
- same title as a previous version if the original publisher of
- that version gives permission.
-
- B. List on the Title Page, as authors, one or more persons or
- entities responsible for authorship of the modifications in
- the Modified Version, together with at least five of the
- principal authors of the Document (all of its principal
- authors, if it has fewer than five), unless they release you
- from this requirement.
-
- C. State on the Title page the name of the publisher of the
- Modified Version, as the publisher.
-
- D. Preserve all the copyright notices of the Document.
-
- E. Add an appropriate copyright notice for your modifications
- adjacent to the other copyright notices.
-
- F. Include, immediately after the copyright notices, a license
- notice giving the public permission to use the Modified
- Version under the terms of this License, in the form shown in
- the Addendum below.
-
- G. Preserve in that license notice the full lists of Invariant
- Sections and required Cover Texts given in the Document's
- license notice.
-
- H. Include an unaltered copy of this License.
-
- I. Preserve the section Entitled "History", Preserve its Title,
- and add to it an item stating at least the title, year, new
- authors, and publisher of the Modified Version as given on
- the Title Page. If there is no section Entitled "History" in
- the Document, create one stating the title, year, authors,
- and publisher of the Document as given on its Title Page,
- then add an item describing the Modified Version as stated in
- the previous sentence.
-
- J. Preserve the network location, if any, given in the Document
- for public access to a Transparent copy of the Document, and
- likewise the network locations given in the Document for
- previous versions it was based on. These may be placed in
- the "History" section. You may omit a network location for a
- work that was published at least four years before the
- Document itself, or if the original publisher of the version
- it refers to gives permission.
-
- K. For any section Entitled "Acknowledgements" or "Dedications",
- Preserve the Title of the section, and preserve in the
- section all the substance and tone of each of the contributor
- acknowledgements and/or dedications given therein.
-
- L. Preserve all the Invariant Sections of the Document,
- unaltered in their text and in their titles. Section numbers
- or the equivalent are not considered part of the section
- titles.
-
- M. Delete any section Entitled "Endorsements". Such a section
- may not be included in the Modified Version.
-
- N. Do not retitle any existing section to be Entitled
- "Endorsements" or to conflict in title with any Invariant
- Section.
-
- O. Preserve any Warranty Disclaimers.
-
- If the Modified Version includes new front-matter sections or
- appendices that qualify as Secondary Sections and contain no
- material copied from the Document, you may at your option
- designate some or all of these sections as invariant. To do this,
- add their titles to the list of Invariant Sections in the Modified
- Version's license notice. These titles must be distinct from any
- other section titles.
-
- You may add a section Entitled "Endorsements", provided it contains
- nothing but endorsements of your Modified Version by various
- parties--for example, statements of peer review or that the text
- has been approved by an organization as the authoritative
- definition of a standard.
-
- You may add a passage of up to five words as a Front-Cover Text,
- and a passage of up to 25 words as a Back-Cover Text, to the end
- of the list of Cover Texts in the Modified Version. Only one
- passage of Front-Cover Text and one of Back-Cover Text may be
- added by (or through arrangements made by) any one entity. If the
- Document already includes a cover text for the same cover,
- previously added by you or by arrangement made by the same entity
- you are acting on behalf of, you may not add another; but you may
- replace the old one, on explicit permission from the previous
- publisher that added the old one.
-
- The author(s) and publisher(s) of the Document do not by this
- License give permission to use their names for publicity for or to
- assert or imply endorsement of any Modified Version.
-
- 5. COMBINING DOCUMENTS
-
- You may combine the Document with other documents released under
- this License, under the terms defined in section 4 above for
- modified versions, provided that you include in the combination
- all of the Invariant Sections of all of the original documents,
- unmodified, and list them all as Invariant Sections of your
- combined work in its license notice, and that you preserve all
- their Warranty Disclaimers.
-
- The combined work need only contain one copy of this License, and
- multiple identical Invariant Sections may be replaced with a single
- copy. If there are multiple Invariant Sections with the same name
- but different contents, make the title of each such section unique
- by adding at the end of it, in parentheses, the name of the
- original author or publisher of that section if known, or else a
- unique number. Make the same adjustment to the section titles in
- the list of Invariant Sections in the license notice of the
- combined work.
-
- In the combination, you must combine any sections Entitled
- "History" in the various original documents, forming one section
- Entitled "History"; likewise combine any sections Entitled
- "Acknowledgements", and any sections Entitled "Dedications". You
- must delete all sections Entitled "Endorsements."
-
- 6. COLLECTIONS OF DOCUMENTS
-
- You may make a collection consisting of the Document and other
- documents released under this License, and replace the individual
- copies of this License in the various documents with a single copy
- that is included in the collection, provided that you follow the
- rules of this License for verbatim copying of each of the
- documents in all other respects.
-
- You may extract a single document from such a collection, and
- distribute it individually under this License, provided you insert
- a copy of this License into the extracted document, and follow
- this License in all other respects regarding verbatim copying of
- that document.
-
- 7. AGGREGATION WITH INDEPENDENT WORKS
-
- A compilation of the Document or its derivatives with other
- separate and independent documents or works, in or on a volume of
- a storage or distribution medium, is called an "aggregate" if the
- copyright resulting from the compilation is not used to limit the
- legal rights of the compilation's users beyond what the individual
- works permit. When the Document is included in an aggregate, this
- License does not apply to the other works in the aggregate which
- are not themselves derivative works of the Document.
-
- If the Cover Text requirement of section 3 is applicable to these
- copies of the Document, then if the Document is less than one half
- of the entire aggregate, the Document's Cover Texts may be placed
- on covers that bracket the Document within the aggregate, or the
- electronic equivalent of covers if the Document is in electronic
- form. Otherwise they must appear on printed covers that bracket
- the whole aggregate.
-
- 8. TRANSLATION
-
- Translation is considered a kind of modification, so you may
- distribute translations of the Document under the terms of section
- 4. Replacing Invariant Sections with translations requires special
- permission from their copyright holders, but you may include
- translations of some or all Invariant Sections in addition to the
- original versions of these Invariant Sections. You may include a
- translation of this License, and all the license notices in the
- Document, and any Warranty Disclaimers, provided that you also
- include the original English version of this License and the
- original versions of those notices and disclaimers. In case of a
- disagreement between the translation and the original version of
- this License or a notice or disclaimer, the original version will
- prevail.
-
- If a section in the Document is Entitled "Acknowledgements",
- "Dedications", or "History", the requirement (section 4) to
- Preserve its Title (section 1) will typically require changing the
- actual title.
-
- 9. TERMINATION
-
- You may not copy, modify, sublicense, or distribute the Document
- except as expressly provided under this License. Any attempt
- otherwise to copy, modify, sublicense, or distribute it is void,
- and will automatically terminate your rights under this License.
-
- However, if you cease all violation of this License, then your
- license from a particular copyright holder is reinstated (a)
- provisionally, unless and until the copyright holder explicitly
- and finally terminates your license, and (b) permanently, if the
- copyright holder fails to notify you of the violation by some
- reasonable means prior to 60 days after the cessation.
-
- Moreover, your license from a particular copyright holder is
- reinstated permanently if the copyright holder notifies you of the
- violation by some reasonable means, this is the first time you have
- received notice of violation of this License (for any work) from
- that copyright holder, and you cure the violation prior to 30 days
- after your receipt of the notice.
-
- Termination of your rights under this section does not terminate
- the licenses of parties who have received copies or rights from
- you under this License. If your rights have been terminated and
- not permanently reinstated, receipt of a copy of some or all of
- the same material does not give you any rights to use it.
-
- 10. FUTURE REVISIONS OF THIS LICENSE
-
- The Free Software Foundation may publish new, revised versions of
- the GNU Free Documentation License from time to time. Such new
- versions will be similar in spirit to the present version, but may
- differ in detail to address new problems or concerns. See
- `http://www.gnu.org/copyleft/'.
-
- Each version of the License is given a distinguishing version
- number. If the Document specifies that a particular numbered
- version of this License "or any later version" applies to it, you
- have the option of following the terms and conditions either of
- that specified version or of any later version that has been
- published (not as a draft) by the Free Software Foundation. If
- the Document does not specify a version number of this License,
- you may choose any version ever published (not as a draft) by the
- Free Software Foundation. If the Document specifies that a proxy
- can decide which future versions of this License can be used, that
- proxy's public statement of acceptance of a version permanently
- authorizes you to choose that version for the Document.
-
- 11. RELICENSING
-
- "Massive Multiauthor Collaboration Site" (or "MMC Site") means any
- World Wide Web server that publishes copyrightable works and also
- provides prominent facilities for anybody to edit those works. A
- public wiki that anybody can edit is an example of such a server.
- A "Massive Multiauthor Collaboration" (or "MMC") contained in the
- site means any set of copyrightable works thus published on the MMC
- site.
-
- "CC-BY-SA" means the Creative Commons Attribution-Share Alike 3.0
- license published by Creative Commons Corporation, a not-for-profit
- corporation with a principal place of business in San Francisco,
- California, as well as future copyleft versions of that license
- published by that same organization.
-
- "Incorporate" means to publish or republish a Document, in whole or
- in part, as part of another Document.
-
- An MMC is "eligible for relicensing" if it is licensed under this
- License, and if all works that were first published under this
- License somewhere other than this MMC, and subsequently
- incorporated in whole or in part into the MMC, (1) had no cover
- texts or invariant sections, and (2) were thus incorporated prior
- to November 1, 2008.
-
- The operator of an MMC Site may republish an MMC contained in the
- site under CC-BY-SA on the same site at any time before August 1,
- 2009, provided the MMC is eligible for relicensing.
-
-
-ADDENDUM: How to use this License for your documents
-====================================================
-
-To use this License in a document you have written, include a copy of
-the License in the document and put the following copyright and license
-notices just after the title page:
-
- Copyright (C) YEAR YOUR NAME.
- Permission is granted to copy, distribute and/or modify this document
- under the terms of the GNU Free Documentation License, Version 1.3
- or any later version published by the Free Software Foundation;
- with no Invariant Sections, no Front-Cover Texts, and no Back-Cover
- Texts. A copy of the license is included in the section entitled ``GNU
- Free Documentation License''.
-
- If you have Invariant Sections, Front-Cover Texts and Back-Cover
-Texts, replace the "with...Texts." line with this:
-
- with the Invariant Sections being LIST THEIR TITLES, with
- the Front-Cover Texts being LIST, and with the Back-Cover Texts
- being LIST.
-
- If you have Invariant Sections without Cover Texts, or some other
-combination of the three, merge those two alternatives to suit the
-situation.
-
- If your document contains nontrivial examples of program code, we
-recommend releasing these examples in parallel under your choice of
-free software license, such as the GNU General Public License, to
-permit their use in free software.
-
-
-File: mpir.info, Node: Concept Index, Next: Function Index, Prev: GNU Free Documentation License, Up: Top
-
-Concept Index
-*************
-
- [index ]
-* Menu:
-
-* #include: Headers and Libraries.
- (line 6)
-* --build: Build Options. (line 60)
-* --disable-fft: Build Options. (line 316)
-* --disable-shared: Build Options. (line 53)
-* --disable-static: Build Options. (line 53)
-* --enable-alloca: Build Options. (line 277)
-* --enable-assert: Build Options. (line 321)
-* --enable-cxx: Build Options. (line 229)
-* --enable-fat: Build Options. (line 162)
-* --enable-gmpcompat: Build Options. (line 45)
-* --enable-profiling <1>: Profiling. (line 6)
-* --enable-profiling: Build Options. (line 325)
-* --exec-prefix: Build Options. (line 32)
-* --host: Build Options. (line 74)
-* --prefix: Build Options. (line 32)
-* -finstrument-functions: Profiling. (line 66)
-* 2exp functions: Efficiency. (line 43)
-* 80x86: Notes for Particular Systems.
- (line 139)
-* ABI <1>: ABI and ISA. (line 6)
-* ABI: Build Options. (line 170)
-* About this manual: Introduction to MPIR.
- (line 48)
-* AC_CHECK_LIB: Autoconf. (line 11)
-* AIX <1>: Notes for Particular Systems.
- (line 7)
-* AIX: ABI and ISA. (line 96)
-* Algorithms: Algorithms. (line 6)
-* alloca: Build Options. (line 277)
-* Allocation of memory: Custom Allocation. (line 6)
-* AMD64: ABI and ISA. (line 45)
-* Application Binary Interface: ABI and ISA. (line 6)
-* Arithmetic functions <1>: Float Arithmetic. (line 6)
-* Arithmetic functions <2>: Rational Arithmetic. (line 6)
-* Arithmetic functions: Integer Arithmetic. (line 6)
-* ARM: Notes for Particular Systems.
- (line 20)
-* Assembler cache handling: Assembler Cache Handling.
- (line 6)
-* Assembler carry propagation: Assembler Carry Propagation.
- (line 6)
-* Assembler code organisation: Assembler Code Organisation.
- (line 6)
-* Assembler coding: Assembler Coding. (line 6)
-* Assembler floating Point: Assembler Floating Point.
- (line 6)
-* Assembler loop unrolling: Assembler Loop Unrolling.
- (line 6)
-* Assembler SIMD: Assembler SIMD Instructions.
- (line 6)
-* Assembler software pipelining: Assembler Software Pipelining.
- (line 6)
-* Assembler writing guide: Assembler Writing Guide.
- (line 6)
-* Assertion checking <1>: Debugging. (line 76)
-* Assertion checking: Build Options. (line 321)
-* Assignment functions <1>: Simultaneous Float Init & Assign.
- (line 6)
-* Assignment functions <2>: Assigning Floats. (line 6)
-* Assignment functions <3>: Initializing Rationals.
- (line 6)
-* Assignment functions <4>: Simultaneous Integer Init & Assign.
- (line 6)
-* Assignment functions: Assigning Integers. (line 6)
-* Autoconf: Autoconf. (line 6)
-* Basics: MPIR Basics. (line 6)
-* Binomial coefficient algorithm: Binomial Coefficients Algorithm.
- (line 6)
-* Binomial coefficient functions: Number Theoretic Functions.
- (line 149)
-* Bit manipulation functions: Integer Logic and Bit Fiddling.
- (line 6)
-* Bit scanning functions: Integer Logic and Bit Fiddling.
- (line 38)
-* Bit shift left: Integer Arithmetic. (line 29)
-* Bit shift right: Integer Division. (line 44)
-* Bits per limb: Useful Macros and Constants.
- (line 7)
-* Bug reporting: Reporting Bugs. (line 6)
-* Build directory: Build Options. (line 19)
-* Build notes for binary packaging: Notes for Package Builds.
- (line 6)
-* Build notes for particular systems: Notes for Particular Systems.
- (line 6)
-* Build options: Build Options. (line 6)
-* Build problems known: Known Build Problems.
- (line 6)
-* Build system: Build Options. (line 60)
-* Building MPIR: Installing MPIR. (line 6)
-* Bus error: Debugging. (line 7)
-* C compiler: Build Options. (line 181)
-* C++ compiler: Build Options. (line 253)
-* C++ interface: C++ Class Interface. (line 6)
-* C++ interface internals: C++ Interface Internals.
- (line 6)
-* C++ istream input: C++ Formatted Input. (line 6)
-* C++ ostream output: C++ Formatted Output.
- (line 6)
-* C++ support: Build Options. (line 229)
-* CC: Build Options. (line 181)
-* CC_FOR_BUILD: Build Options. (line 216)
-* CFLAGS: Build Options. (line 181)
-* Checker: Debugging. (line 112)
-* checkergcc: Debugging. (line 119)
-* Code organisation: Assembler Code Organisation.
- (line 6)
-* Comparison functions <1>: Float Comparison. (line 6)
-* Comparison functions <2>: Comparing Rationals. (line 6)
-* Comparison functions: Integer Comparisons. (line 6)
-* Compatibility with older versions: Compatibility with older versions.
- (line 6)
-* Conditions for copying MPIR: Copying. (line 6)
-* Configuring MPIR: Installing MPIR. (line 6)
-* Congruence algorithm: Exact Remainder. (line 30)
-* Congruence functions: Integer Division. (line 113)
-* Constants: Useful Macros and Constants.
- (line 6)
-* Contributors: Contributors. (line 6)
-* Conventions for parameters: Parameter Conventions.
- (line 6)
-* Conventions for variables: Variable Conventions.
- (line 6)
-* Conversion functions <1>: Converting Floats. (line 6)
-* Conversion functions <2>: Rational Conversions.
- (line 6)
-* Conversion functions: Converting Integers. (line 6)
-* Copying conditions: Copying. (line 6)
-* CPPFLAGS: Build Options. (line 207)
-* CPU types <1>: Build Options. (line 115)
-* CPU types: Introduction to MPIR.
- (line 24)
-* Cross compiling: Build Options. (line 74)
-* Custom allocation: Custom Allocation. (line 6)
-* CXX: Build Options. (line 253)
-* CXXFLAGS: Build Options. (line 253)
-* Cygwin: Notes for Particular Systems.
- (line 34)
-* Darwin: Known Build Problems.
- (line 23)
-* Debugging: Debugging. (line 6)
-* Digits in an integer: Miscellaneous Integer Functions.
- (line 23)
-* Divisibility algorithm: Exact Remainder. (line 30)
-* Divisibility functions: Integer Division. (line 102)
-* Divisibility testing: Efficiency. (line 91)
-* Division algorithms: Division Algorithms. (line 6)
-* Division functions <1>: Float Arithmetic. (line 27)
-* Division functions <2>: Rational Arithmetic. (line 22)
-* Division functions: Integer Division. (line 6)
-* DLLs: Notes for Particular Systems.
- (line 73)
-* DocBook: Build Options. (line 348)
-* Documentation formats: Build Options. (line 341)
-* Documentation license: GNU Free Documentation License.
- (line 6)
-* DVI: Build Options. (line 344)
-* Efficiency: Efficiency. (line 6)
-* Emacs: Emacs. (line 6)
-* Exact division functions: Integer Division. (line 92)
-* Exact remainder: Exact Remainder. (line 6)
-* Exec prefix: Build Options. (line 32)
-* Execution profiling <1>: Profiling. (line 6)
-* Execution profiling: Build Options. (line 325)
-* Exponentiation functions <1>: Float Arithmetic. (line 34)
-* Exponentiation functions: Integer Exponentiation.
- (line 6)
-* Export: Integer Import and Export.
- (line 45)
-* Extended GCD: Number Theoretic Functions.
- (line 96)
-* Factor removal functions: Number Theoretic Functions.
- (line 140)
-* Factorial algorithm: Factorial Algorithm. (line 6)
-* Factorial functions: Number Theoretic Functions.
- (line 145)
-* Fast Fourier Transform: FFT Multiplication. (line 6)
-* Fat binary: Build Options. (line 162)
-* FFT multiplication <1>: FFT Multiplication. (line 6)
-* FFT multiplication: Build Options. (line 316)
-* Fibonacci number algorithm: Fibonacci Numbers Algorithm.
- (line 6)
-* Fibonacci sequence functions: Number Theoretic Functions.
- (line 156)
-* Float arithmetic functions: Float Arithmetic. (line 6)
-* Float assignment functions <1>: Simultaneous Float Init & Assign.
- (line 6)
-* Float assignment functions: Assigning Floats. (line 6)
-* Float comparison functions: Float Comparison. (line 6)
-* Float conversion functions: Converting Floats. (line 6)
-* Float functions: Floating-point Functions.
- (line 6)
-* Float initialization functions <1>: Simultaneous Float Init & Assign.
- (line 6)
-* Float initialization functions: Initializing Floats. (line 6)
-* Float input and output functions: I/O of Floats. (line 6)
-* Float internals: Float Internals. (line 6)
-* Float miscellaneous functions: Miscellaneous Float Functions.
- (line 6)
-* Float random number functions: Miscellaneous Float Functions.
- (line 27)
-* Float rounding functions: Miscellaneous Float Functions.
- (line 9)
-* Float sign tests: Float Comparison. (line 28)
-* Floating point mode: Notes for Particular Systems.
- (line 25)
-* Floating-point functions: Floating-point Functions.
- (line 6)
-* Floating-point number: Nomenclature and Types.
- (line 21)
-* fnccheck: Profiling. (line 77)
-* Formatted input: Formatted Input. (line 6)
-* Formatted output: Formatted Output. (line 6)
-* Free Documentation License: GNU Free Documentation License.
- (line 6)
-* frexp <1>: Converting Floats. (line 23)
-* frexp: Converting Integers. (line 57)
-* Function classes: Function Classes. (line 6)
-* FunctionCheck: Profiling. (line 77)
-* GCC Checker: Debugging. (line 112)
-* GCD algorithms: Greatest Common Divisor Algorithms.
- (line 6)
-* GCD extended: Number Theoretic Functions.
- (line 96)
-* GCD functions: Number Theoretic Functions.
- (line 82)
-* GDB: Debugging. (line 55)
-* Generic C: Build Options. (line 151)
-* GNU Debugger: Debugging. (line 55)
-* GNU Free Documentation License: GNU Free Documentation License.
- (line 6)
-* gprof: Profiling. (line 41)
-* Greatest common divisor algorithms: Greatest Common Divisor Algorithms.
- (line 6)
-* Greatest common divisor functions: Number Theoretic Functions.
- (line 82)
-* Hardware floating point mode: Notes for Particular Systems.
- (line 25)
-* Headers: Headers and Libraries.
- (line 6)
-* Heap problems: Debugging. (line 24)
-* Home page: Introduction to MPIR.
- (line 30)
-* Host system: Build Options. (line 74)
-* HP-UX: ABI and ISA. (line 69)
-* I/O functions <1>: I/O of Floats. (line 6)
-* I/O functions <2>: I/O of Rationals. (line 6)
-* I/O functions: I/O of Integers. (line 6)
-* i386: Notes for Particular Systems.
- (line 139)
-* IA-64: ABI and ISA. (line 69)
-* Import: Integer Import and Export.
- (line 11)
-* In-place operations: Efficiency. (line 57)
-* Include files: Headers and Libraries.
- (line 6)
-* info-lookup-symbol: Emacs. (line 6)
-* Initialization functions <1>: Random State Initialization.
- (line 6)
-* Initialization functions <2>: Simultaneous Float Init & Assign.
- (line 6)
-* Initialization functions <3>: Initializing Floats. (line 6)
-* Initialization functions <4>: Initializing Rationals.
- (line 6)
-* Initialization functions <5>: Simultaneous Integer Init & Assign.
- (line 6)
-* Initialization functions: Initializing Integers.
- (line 6)
-* Initializing and clearing: Efficiency. (line 21)
-* Input functions <1>: Formatted Input Functions.
- (line 6)
-* Input functions <2>: I/O of Floats. (line 6)
-* Input functions <3>: I/O of Rationals. (line 6)
-* Input functions: I/O of Integers. (line 6)
-* Install prefix: Build Options. (line 32)
-* Installing MPIR: Installing MPIR. (line 6)
-* Instruction Set Architecture: ABI and ISA. (line 6)
-* instrument-functions: Profiling. (line 66)
-* Integer: Nomenclature and Types.
- (line 6)
-* Integer arithmetic functions: Integer Arithmetic. (line 6)
-* Integer assignment functions <1>: Simultaneous Integer Init & Assign.
- (line 6)
-* Integer assignment functions: Assigning Integers. (line 6)
-* Integer bit manipulation functions: Integer Logic and Bit Fiddling.
- (line 6)
-* Integer comparison functions: Integer Comparisons. (line 6)
-* Integer conversion functions: Converting Integers. (line 6)
-* Integer division functions: Integer Division. (line 6)
-* Integer exponentiation functions: Integer Exponentiation.
- (line 6)
-* Integer export: Integer Import and Export.
- (line 45)
-* Integer functions: Integer Functions. (line 6)
-* Integer import: Integer Import and Export.
- (line 11)
-* Integer initialization functions <1>: Simultaneous Integer Init & Assign.
- (line 6)
-* Integer initialization functions: Initializing Integers.
- (line 6)
-* Integer input and output functions: I/O of Integers. (line 6)
-* Integer internals: Integer Internals. (line 6)
-* Integer logical functions: Integer Logic and Bit Fiddling.
- (line 6)
-* Integer miscellaneous functions: Miscellaneous Integer Functions.
- (line 6)
-* Integer random number functions: Integer Random Numbers.
- (line 6)
-* Integer root functions: Integer Roots. (line 6)
-* Integer sign tests: Integer Comparisons. (line 28)
-* Integer special functions: Integer Special Functions.
- (line 6)
-* Internals: Internals. (line 6)
-* Introduction: Introduction to MPIR.
- (line 6)
-* Inverse modulo functions: Number Theoretic Functions.
- (line 109)
-* ISA: ABI and ISA. (line 6)
-* istream input: C++ Formatted Input. (line 6)
-* Jacobi symbol algorithm: Jacobi Symbol. (line 6)
-* Jacobi symbol functions: Number Theoretic Functions.
- (line 115)
-* Karatsuba multiplication: Karatsuba Multiplication.
- (line 6)
-* Karatsuba square root algorithm: Square Root Algorithm.
- (line 6)
-* Kronecker symbol functions: Number Theoretic Functions.
- (line 127)
-* Language bindings: Language Bindings. (line 6)
-* LCM functions: Number Theoretic Functions.
- (line 104)
-* Least common multiple functions: Number Theoretic Functions.
- (line 104)
-* Legendre symbol functions: Number Theoretic Functions.
- (line 118)
-* libmpir: Headers and Libraries.
- (line 22)
-* libmpirxx: Headers and Libraries.
- (line 28)
-* Libraries: Headers and Libraries.
- (line 22)
-* Libtool: Headers and Libraries.
- (line 34)
-* Libtool versioning: Notes for Package Builds.
- (line 9)
-* License conditions: Copying. (line 6)
-* Limb: Nomenclature and Types.
- (line 31)
-* Limb size: Useful Macros and Constants.
- (line 7)
-* Linear congruential algorithm: Random Number Algorithms.
- (line 25)
-* Linear congruential random numbers: Random State Initialization.
- (line 18)
-* Linking: Headers and Libraries.
- (line 22)
-* Logical functions: Integer Logic and Bit Fiddling.
- (line 6)
-* Low-level functions: Low-level Functions. (line 6)
-* Lucas number algorithm: Lucas Numbers Algorithm.
- (line 6)
-* Lucas number functions: Number Theoretic Functions.
- (line 166)
-* MacOS X: Known Build Problems.
- (line 23)
-* Mailing lists: Introduction to MPIR.
- (line 35)
-* Malloc debugger: Debugging. (line 30)
-* Malloc problems: Debugging. (line 24)
-* Memory allocation: Custom Allocation. (line 6)
-* Memory management: Memory Management. (line 6)
-* Mersenne twister algorithm: Random Number Algorithms.
- (line 17)
-* Mersenne twister random numbers: Random State Initialization.
- (line 13)
-* MINGW: Notes for Particular Systems.
- (line 34)
-* Miscellaneous float functions: Miscellaneous Float Functions.
- (line 6)
-* Miscellaneous integer functions: Miscellaneous Integer Functions.
- (line 6)
-* MMX: Notes for Particular Systems.
- (line 145)
-* Modular inverse functions: Number Theoretic Functions.
- (line 109)
-* Most significant bit: Miscellaneous Integer Functions.
- (line 34)
-* MPIR version number: Useful Macros and Constants.
- (line 12)
-* mpir.h: Headers and Libraries.
- (line 6)
-* mpirxx.h: C++ Interface General.
- (line 8)
-* MPN_PATH: Build Options. (line 329)
-* MS Windows: Notes for Particular Systems.
- (line 34)
-* MS-DOS: Notes for Particular Systems.
- (line 34)
-* Multi-threading: Reentrancy. (line 6)
-* Multiplication algorithms: Multiplication Algorithms.
- (line 6)
-* Nails: Low-level Functions. (line 530)
-* Native compilation: Build Options. (line 60)
-* Next likely prime function: Number Theoretic Functions.
- (line 71)
-* Next prime function: Number Theoretic Functions.
- (line 60)
-* Nomenclature: Nomenclature and Types.
- (line 6)
-* Non-Unix systems: Build Options. (line 11)
-* Nth root algorithm: Nth Root Algorithm. (line 6)
-* Number sequences: Efficiency. (line 147)
-* Number theoretic functions: Number Theoretic Functions.
- (line 6)
-* Numerator and denominator: Applying Integer Functions.
- (line 6)
-* obstack output: Formatted Output Functions.
- (line 81)
-* OpenBSD: Notes for Particular Systems.
- (line 96)
-* Optimizing performance: Performance optimization.
- (line 6)
-* ostream output: C++ Formatted Output.
- (line 6)
-* Other languages: Language Bindings. (line 6)
-* Output functions <1>: Formatted Output Functions.
- (line 6)
-* Output functions <2>: I/O of Floats. (line 6)
-* Output functions <3>: I/O of Rationals. (line 6)
-* Output functions: I/O of Integers. (line 6)
-* Packaged builds: Notes for Package Builds.
- (line 6)
-* Parameter conventions: Parameter Conventions.
- (line 6)
-* Particular systems: Notes for Particular Systems.
- (line 6)
-* Past GMP/MPIR versions: Compatibility with older versions.
- (line 6)
-* PDF: Build Options. (line 344)
-* Perfect power algorithm: Perfect Power Algorithm.
- (line 6)
-* Perfect power functions: Integer Roots. (line 30)
-* Perfect square algorithm: Perfect Square Algorithm.
- (line 6)
-* Perfect square functions: Integer Roots. (line 39)
-* Postscript: Build Options. (line 344)
-* Powering algorithms: Powering Algorithms. (line 6)
-* Powering functions <1>: Float Arithmetic. (line 34)
-* Powering functions: Integer Exponentiation.
- (line 6)
-* PowerPC: ABI and ISA. (line 94)
-* Precision of floats: Floating-point Functions.
- (line 6)
-* Precision of hardware floating point: Notes for Particular Systems.
- (line 25)
-* Prefix: Build Options. (line 32)
-* Prime testing algorithms: Prime Testing Algorithm.
- (line 6)
-* Prime testing functions: Number Theoretic Functions.
- (line 8)
-* printf formatted output: Formatted Output. (line 6)
-* Probable prime testing functions: Number Theoretic Functions.
- (line 8)
-* prof: Profiling. (line 24)
-* Profiling: Profiling. (line 6)
-* Radix conversion algorithms: Radix Conversion Algorithms.
- (line 6)
-* Random number algorithms: Random Number Algorithms.
- (line 6)
-* Random number functions <1>: Random Number Functions.
- (line 6)
-* Random number functions <2>: Miscellaneous Float Functions.
- (line 27)
-* Random number functions: Integer Random Numbers.
- (line 6)
-* Random number seeding: Random State Seeding.
- (line 6)
-* Random number state: Random State Initialization.
- (line 6)
-* Random state: Nomenclature and Types.
- (line 45)
-* Rational arithmetic: Efficiency. (line 113)
-* Rational arithmetic functions: Rational Arithmetic. (line 6)
-* Rational assignment functions: Initializing Rationals.
- (line 6)
-* Rational comparison functions: Comparing Rationals. (line 6)
-* Rational conversion functions: Rational Conversions.
- (line 6)
-* Rational initialization functions: Initializing Rationals.
- (line 6)
-* Rational input and output functions: I/O of Rationals. (line 6)
-* Rational internals: Rational Internals. (line 6)
-* Rational number: Nomenclature and Types.
- (line 16)
-* Rational number functions: Rational Number Functions.
- (line 6)
-* Rational numerator and denominator: Applying Integer Functions.
- (line 6)
-* Rational sign tests: Comparing Rationals. (line 25)
-* Raw output internals: Raw Output Internals.
- (line 6)
-* Reallocations: Efficiency. (line 30)
-* Reentrancy: Reentrancy. (line 6)
-* References: References. (line 6)
-* Remove factor functions: Number Theoretic Functions.
- (line 140)
-* Reporting bugs: Reporting Bugs. (line 6)
-* Root extraction algorithm: Nth Root Algorithm. (line 6)
-* Root extraction algorithms: Root Extraction Algorithms.
- (line 6)
-* Root extraction functions <1>: Float Arithmetic. (line 31)
-* Root extraction functions: Integer Roots. (line 6)
-* Root testing functions: Integer Roots. (line 30)
-* Rounding functions: Miscellaneous Float Functions.
- (line 9)
-* Scan bit functions: Integer Logic and Bit Fiddling.
- (line 38)
-* scanf formatted input: Formatted Input. (line 6)
-* Seeding random numbers: Random State Seeding.
- (line 6)
-* Segmentation violation: Debugging. (line 7)
-* Shared library versioning: Notes for Package Builds.
- (line 9)
-* Sign tests <1>: Float Comparison. (line 28)
-* Sign tests <2>: Comparing Rationals. (line 25)
-* Sign tests: Integer Comparisons. (line 28)
-* Size in digits: Miscellaneous Integer Functions.
- (line 23)
-* Small operands: Efficiency. (line 7)
-* Solaris <1>: Known Build Problems.
- (line 29)
-* Solaris <2>: Notes for Particular Systems.
- (line 135)
-* Solaris: ABI and ISA. (line 121)
-* Sparc: Notes for Particular Systems.
- (line 102)
-* Sparc V9: ABI and ISA. (line 121)
-* Special integer functions: Integer Special Functions.
- (line 6)
-* Square root algorithm: Square Root Algorithm.
- (line 6)
-* SSE2: Notes for Particular Systems.
- (line 145)
-* Stack backtrace: Debugging. (line 47)
-* Stack overflow <1>: Debugging. (line 7)
-* Stack overflow: Build Options. (line 277)
-* Static linking: Efficiency. (line 14)
-* stdarg.h: Headers and Libraries.
- (line 17)
-* stdio.h: Headers and Libraries.
- (line 11)
-* Sun: ABI and ISA. (line 121)
-* Systems: Notes for Particular Systems.
- (line 6)
-* Temporary memory: Build Options. (line 277)
-* Texinfo: Build Options. (line 341)
-* Text input/output: Efficiency. (line 153)
-* Thread safety: Reentrancy. (line 6)
-* Toom multiplication <1>: Other Multiplication.
- (line 6)
-* Toom multiplication <2>: Toom 4-Way Multiplication.
- (line 6)
-* Toom multiplication: Toom 3-Way Multiplication.
- (line 6)
-* Types: Nomenclature and Types.
- (line 6)
-* ui and si functions: Efficiency. (line 50)
-* Unbalanced multiplication: Unbalanced Multiplication.
- (line 6)
-* Upward compatibility: Compatibility with older versions.
- (line 6)
-* Useful macros and constants: Useful Macros and Constants.
- (line 6)
-* User-defined precision: Floating-point Functions.
- (line 6)
-* Valgrind: Debugging. (line 127)
-* Variable conventions: Variable Conventions.
- (line 6)
-* Version number: Useful Macros and Constants.
- (line 12)
-* Web page: Introduction to MPIR.
- (line 30)
-* Windows <1>: MPIR on Windows x64. (line 6)
-* Windows: Notes for Particular Systems.
- (line 34)
-* x86: Notes for Particular Systems.
- (line 139)
-* x87: Notes for Particular Systems.
- (line 25)
-* XML: Build Options. (line 348)
-
-
-File: mpir.info, Node: Function Index, Prev: Concept Index, Up: Top
-
-Function and Type Index
-***********************
-
- [index ]
-* Menu:
-
-* __GNU_MP_VERSION: Useful Macros and Constants.
- (line 10)
-* __GNU_MP_VERSION_MINOR: Useful Macros and Constants.
- (line 11)
-* __GNU_MP_VERSION_PATCHLEVEL: Useful Macros and Constants.
- (line 12)
-* __MPIR_VERSION: Useful Macros and Constants.
- (line 18)
-* __MPIR_VERSION_MINOR: Useful Macros and Constants.
- (line 19)
-* __MPIR_VERSION_PATCHLEVEL: Useful Macros and Constants.
- (line 20)
-* _mpz_realloc: Integer Special Functions.
- (line 54)
-* abs <1>: C++ Interface Floats.
- (line 69)
-* abs <2>: C++ Interface Rationals.
- (line 43)
-* abs: C++ Interface Integers.
- (line 41)
-* ceil: C++ Interface Floats.
- (line 70)
-* cmp <1>: C++ Interface Floats.
- (line 71)
-* cmp <2>: C++ Interface Rationals.
- (line 44)
-* cmp: C++ Interface Integers.
- (line 42)
-* floor: C++ Interface Floats.
- (line 79)
-* gmp_asprintf: Formatted Output Functions.
- (line 65)
-* gmp_fprintf: Formatted Output Functions.
- (line 29)
-* gmp_fscanf: Formatted Input Functions.
- (line 25)
-* GMP_LIMB_BITS: Low-level Functions. (line 563)
-* GMP_NAIL_BITS: Low-level Functions. (line 561)
-* GMP_NAIL_MASK: Low-level Functions. (line 571)
-* GMP_NUMB_BITS: Low-level Functions. (line 562)
-* GMP_NUMB_MASK: Low-level Functions. (line 572)
-* GMP_NUMB_MAX: Low-level Functions. (line 580)
-* gmp_obstack_printf: Formatted Output Functions.
- (line 79)
-* gmp_obstack_vprintf: Formatted Output Functions.
- (line 81)
-* gmp_printf: Formatted Output Functions.
- (line 24)
-* gmp_randclass: C++ Interface Random Numbers.
- (line 7)
-* gmp_randclass::get_f: C++ Interface Random Numbers.
- (line 39)
-* gmp_randclass::get_z_bits: C++ Interface Random Numbers.
- (line 32)
-* gmp_randclass::get_z_range: C++ Interface Random Numbers.
- (line 36)
-* gmp_randclass::gmp_randclass: C++ Interface Random Numbers.
- (line 13)
-* gmp_randclass::seed: C++ Interface Random Numbers.
- (line 27)
-* gmp_randclear: Random State Initialization.
- (line 46)
-* gmp_randinit_default: Random State Initialization.
- (line 7)
-* gmp_randinit_lc_2exp: Random State Initialization.
- (line 18)
-* gmp_randinit_lc_2exp_size: Random State Initialization.
- (line 32)
-* gmp_randinit_mt: Random State Initialization.
- (line 13)
-* gmp_randinit_set: Random State Initialization.
- (line 43)
-* gmp_randseed: Random State Seeding.
- (line 7)
-* gmp_randseed_ui: Random State Seeding.
- (line 8)
-* gmp_randstate_t: Nomenclature and Types.
- (line 45)
-* gmp_scanf: Formatted Input Functions.
- (line 21)
-* gmp_snprintf: Formatted Output Functions.
- (line 46)
-* gmp_sprintf: Formatted Output Functions.
- (line 34)
-* gmp_sscanf: Formatted Input Functions.
- (line 29)
-* gmp_urandomb_ui: Random State Miscellaneous.
- (line 7)
-* gmp_urandomm_ui: Random State Miscellaneous.
- (line 12)
-* gmp_vasprintf: Formatted Output Functions.
- (line 66)
-* gmp_version: Useful Macros and Constants.
- (line 25)
-* gmp_vfprintf: Formatted Output Functions.
- (line 30)
-* gmp_vfscanf: Formatted Input Functions.
- (line 26)
-* gmp_vprintf: Formatted Output Functions.
- (line 25)
-* gmp_vscanf: Formatted Input Functions.
- (line 22)
-* gmp_vsnprintf: Formatted Output Functions.
- (line 48)
-* gmp_vsprintf: Formatted Output Functions.
- (line 35)
-* gmp_vsscanf: Formatted Input Functions.
- (line 31)
-* hypot: C++ Interface Floats.
- (line 80)
-* long: MPIR on Windows x64. (line 28)
-* mp_bitcnt_t: Nomenclature and Types.
- (line 41)
-* mp_bits_per_limb: Useful Macros and Constants.
- (line 7)
-* mp_exp_t: Nomenclature and Types.
- (line 27)
-* mp_get_memory_functions: Custom Allocation. (line 98)
-* mp_limb_t: Nomenclature and Types.
- (line 31)
-* mp_set_memory_functions: Custom Allocation. (line 18)
-* mp_size_t: Nomenclature and Types.
- (line 37)
-* mpf_abs: Float Arithmetic. (line 40)
-* mpf_add: Float Arithmetic. (line 7)
-* mpf_add_ui: Float Arithmetic. (line 8)
-* mpf_ceil: Miscellaneous Float Functions.
- (line 7)
-* mpf_class: C++ Interface General.
- (line 20)
-* mpf_class::fits_sint_p: C++ Interface Floats.
- (line 73)
-* mpf_class::fits_slong_p: C++ Interface Floats.
- (line 74)
-* mpf_class::fits_sshort_p: C++ Interface Floats.
- (line 75)
-* mpf_class::fits_uint_p: C++ Interface Floats.
- (line 76)
-* mpf_class::fits_ulong_p: C++ Interface Floats.
- (line 77)
-* mpf_class::fits_ushort_p: C++ Interface Floats.
- (line 78)
-* mpf_class::get_d: C++ Interface Floats.
- (line 81)
-* mpf_class::get_mpf_t: C++ Interface General.
- (line 66)
-* mpf_class::get_prec: C++ Interface Floats.
- (line 99)
-* mpf_class::get_si: C++ Interface Floats.
- (line 82)
-* mpf_class::get_str: C++ Interface Floats.
- (line 84)
-* mpf_class::get_ui: C++ Interface Floats.
- (line 85)
-* mpf_class::mpf_class: C++ Interface Floats.
- (line 12)
-* mpf_class::operator=: C++ Interface Floats.
- (line 46)
-* mpf_class::set_prec: C++ Interface Floats.
- (line 100)
-* mpf_class::set_prec_raw: C++ Interface Floats.
- (line 101)
-* mpf_class::set_str: C++ Interface Floats.
- (line 86)
-* mpf_clear: Initializing Floats. (line 37)
-* mpf_clears: Initializing Floats. (line 41)
-* mpf_cmp: Float Comparison. (line 7)
-* mpf_cmp_d: Float Comparison. (line 8)
-* mpf_cmp_si: Float Comparison. (line 10)
-* mpf_cmp_ui: Float Comparison. (line 9)
-* mpf_div: Float Arithmetic. (line 25)
-* mpf_div_2exp: Float Arithmetic. (line 46)
-* mpf_div_ui: Float Arithmetic. (line 27)
-* mpf_eq: Float Comparison. (line 17)
-* mpf_fits_sint_p: Miscellaneous Float Functions.
- (line 20)
-* mpf_fits_slong_p: Miscellaneous Float Functions.
- (line 18)
-* mpf_fits_sshort_p: Miscellaneous Float Functions.
- (line 22)
-* mpf_fits_uint_p: Miscellaneous Float Functions.
- (line 19)
-* mpf_fits_ulong_p: Miscellaneous Float Functions.
- (line 17)
-* mpf_fits_ushort_p: Miscellaneous Float Functions.
- (line 21)
-* mpf_floor: Miscellaneous Float Functions.
- (line 8)
-* mpf_get_d: Converting Floats. (line 7)
-* mpf_get_d_2exp: Converting Floats. (line 16)
-* mpf_get_default_prec: Initializing Floats. (line 12)
-* mpf_get_prec: Initializing Floats. (line 62)
-* mpf_get_si: Converting Floats. (line 27)
-* mpf_get_str: Converting Floats. (line 37)
-* mpf_get_ui: Converting Floats. (line 28)
-* mpf_init: Initializing Floats. (line 19)
-* mpf_init2: Initializing Floats. (line 26)
-* mpf_init_set: Simultaneous Float Init & Assign.
- (line 16)
-* mpf_init_set_d: Simultaneous Float Init & Assign.
- (line 19)
-* mpf_init_set_si: Simultaneous Float Init & Assign.
- (line 18)
-* mpf_init_set_str: Simultaneous Float Init & Assign.
- (line 25)
-* mpf_init_set_ui: Simultaneous Float Init & Assign.
- (line 17)
-* mpf_inits: Initializing Floats. (line 31)
-* mpf_inp_str: I/O of Floats. (line 36)
-* mpf_integer_p: Miscellaneous Float Functions.
- (line 14)
-* mpf_mul: Float Arithmetic. (line 16)
-* mpf_mul_2exp: Float Arithmetic. (line 43)
-* mpf_mul_ui: Float Arithmetic. (line 17)
-* mpf_neg: Float Arithmetic. (line 37)
-* mpf_out_str: I/O of Floats. (line 17)
-* mpf_pow_ui: Float Arithmetic. (line 34)
-* mpf_random2: Miscellaneous Float Functions.
- (line 49)
-* mpf_reldiff: Float Comparison. (line 24)
-* mpf_rrandomb: Miscellaneous Float Functions.
- (line 36)
-* mpf_set: Assigning Floats. (line 10)
-* mpf_set_d: Assigning Floats. (line 13)
-* mpf_set_default_prec: Initializing Floats. (line 7)
-* mpf_set_prec: Initializing Floats. (line 65)
-* mpf_set_prec_raw: Initializing Floats. (line 72)
-* mpf_set_q: Assigning Floats. (line 15)
-* mpf_set_si: Assigning Floats. (line 12)
-* mpf_set_str: Assigning Floats. (line 18)
-* mpf_set_ui: Assigning Floats. (line 11)
-* mpf_set_z: Assigning Floats. (line 14)
-* mpf_sgn: Float Comparison. (line 28)
-* mpf_sqrt: Float Arithmetic. (line 30)
-* mpf_sqrt_ui: Float Arithmetic. (line 31)
-* mpf_sub: Float Arithmetic. (line 11)
-* mpf_sub_ui: Float Arithmetic. (line 13)
-* mpf_swap: Assigning Floats. (line 52)
-* mpf_t: Nomenclature and Types.
- (line 21)
-* mpf_trunc: Miscellaneous Float Functions.
- (line 9)
-* mpf_ui_div: Float Arithmetic. (line 26)
-* mpf_ui_sub: Float Arithmetic. (line 12)
-* mpf_urandomb: Miscellaneous Float Functions.
- (line 27)
-* mpir_version: Useful Macros and Constants.
- (line 29)
-* mpn_add: Low-level Functions. (line 70)
-* mpn_add_1: Low-level Functions. (line 65)
-* mpn_add_n: Low-level Functions. (line 55)
-* mpn_addmul_1: Low-level Functions. (line 131)
-* mpn_and_n: Low-level Functions. (line 472)
-* mpn_andn_n: Low-level Functions. (line 487)
-* mpn_bdivmod: Low-level Functions. (line 256)
-* mpn_cmp: Low-level Functions. (line 297)
-* mpn_com: Low-level Functions. (line 512)
-* mpn_copyd: Low-level Functions. (line 521)
-* mpn_copyi: Low-level Functions. (line 517)
-* mpn_divexact_by3: Low-level Functions. (line 225)
-* mpn_divexact_by3c: Low-level Functions. (line 227)
-* mpn_divmod_1: Low-level Functions. (line 209)
-* mpn_divrem: Low-level Functions. (line 183)
-* mpn_divrem_1: Low-level Functions. (line 207)
-* mpn_gcd: Low-level Functions. (line 302)
-* mpn_gcd_1: Low-level Functions. (line 313)
-* mpn_gcdext: Low-level Functions. (line 319)
-* mpn_get_str: Low-level Functions. (line 361)
-* mpn_hamdist: Low-level Functions. (line 462)
-* mpn_ior_n: Low-level Functions. (line 477)
-* mpn_iorn_n: Low-level Functions. (line 492)
-* mpn_lshift: Low-level Functions. (line 273)
-* mpn_mod_1: Low-level Functions. (line 251)
-* mpn_mul: Low-level Functions. (line 153)
-* mpn_mul_1: Low-level Functions. (line 116)
-* mpn_mul_n: Low-level Functions. (line 104)
-* mpn_nand_n: Low-level Functions. (line 497)
-* mpn_neg: Low-level Functions. (line 99)
-* mpn_nior_n: Low-level Functions. (line 502)
-* mpn_perfect_square_p: Low-level Functions. (line 468)
-* mpn_popcount: Low-level Functions. (line 458)
-* mpn_random: Low-level Functions. (line 410)
-* mpn_random2: Low-level Functions. (line 411)
-* mpn_randomb: Low-level Functions. (line 440)
-* mpn_rrandom: Low-level Functions. (line 448)
-* mpn_rshift: Low-level Functions. (line 285)
-* mpn_scan0: Low-level Functions. (line 395)
-* mpn_scan1: Low-level Functions. (line 403)
-* mpn_set_str: Low-level Functions. (line 376)
-* mpn_sqr: Low-level Functions. (line 164)
-* mpn_sqrtrem: Low-level Functions. (line 343)
-* mpn_sub: Low-level Functions. (line 91)
-* mpn_sub_1: Low-level Functions. (line 86)
-* mpn_sub_n: Low-level Functions. (line 77)
-* mpn_submul_1: Low-level Functions. (line 142)
-* mpn_tdiv_qr: Low-level Functions. (line 173)
-* mpn_urandomb: Low-level Functions. (line 424)
-* mpn_urandomm: Low-level Functions. (line 432)
-* mpn_xnor_n: Low-level Functions. (line 507)
-* mpn_xor_n: Low-level Functions. (line 482)
-* mpn_zero: Low-level Functions. (line 524)
-* mpq_abs: Rational Arithmetic. (line 31)
-* mpq_add: Rational Arithmetic. (line 7)
-* mpq_canonicalize: Rational Number Functions.
- (line 22)
-* mpq_class: C++ Interface General.
- (line 19)
-* mpq_class::canonicalize: C++ Interface Rationals.
- (line 37)
-* mpq_class::get_d: C++ Interface Rationals.
- (line 46)
-* mpq_class::get_den: C++ Interface Rationals.
- (line 58)
-* mpq_class::get_den_mpz_t: C++ Interface Rationals.
- (line 68)
-* mpq_class::get_mpq_t: C++ Interface General.
- (line 65)
-* mpq_class::get_num: C++ Interface Rationals.
- (line 57)
-* mpq_class::get_num_mpz_t: C++ Interface Rationals.
- (line 67)
-* mpq_class::get_str: C++ Interface Rationals.
- (line 47)
-* mpq_class::mpq_class: C++ Interface Rationals.
- (line 11)
-* mpq_class::set_str: C++ Interface Rationals.
- (line 48)
-* mpq_clear: Initializing Rationals.
- (line 16)
-* mpq_clears: Initializing Rationals.
- (line 20)
-* mpq_cmp: Comparing Rationals. (line 7)
-* mpq_cmp_si: Comparing Rationals. (line 15)
-* mpq_cmp_ui: Comparing Rationals. (line 14)
-* mpq_denref: Applying Integer Functions.
- (line 18)
-* mpq_div: Rational Arithmetic. (line 22)
-* mpq_div_2exp: Rational Arithmetic. (line 25)
-* mpq_equal: Comparing Rationals. (line 31)
-* mpq_get_d: Rational Conversions.
- (line 7)
-* mpq_get_den: Applying Integer Functions.
- (line 24)
-* mpq_get_num: Applying Integer Functions.
- (line 23)
-* mpq_get_str: Rational Conversions.
- (line 22)
-* mpq_init: Initializing Rationals.
- (line 7)
-* mpq_inits: Initializing Rationals.
- (line 12)
-* mpq_inp_str: I/O of Rationals. (line 23)
-* mpq_inv: Rational Arithmetic. (line 34)
-* mpq_mul: Rational Arithmetic. (line 15)
-* mpq_mul_2exp: Rational Arithmetic. (line 18)
-* mpq_neg: Rational Arithmetic. (line 28)
-* mpq_numref: Applying Integer Functions.
- (line 17)
-* mpq_out_str: I/O of Rationals. (line 15)
-* mpq_set: Initializing Rationals.
- (line 24)
-* mpq_set_d: Rational Conversions.
- (line 17)
-* mpq_set_den: Applying Integer Functions.
- (line 26)
-* mpq_set_f: Rational Conversions.
- (line 18)
-* mpq_set_num: Applying Integer Functions.
- (line 25)
-* mpq_set_si: Initializing Rationals.
- (line 29)
-* mpq_set_str: Initializing Rationals.
- (line 34)
-* mpq_set_ui: Initializing Rationals.
- (line 28)
-* mpq_set_z: Initializing Rationals.
- (line 25)
-* mpq_sgn: Comparing Rationals. (line 25)
-* mpq_sub: Rational Arithmetic. (line 11)
-* mpq_swap: Initializing Rationals.
- (line 54)
-* mpq_t: Nomenclature and Types.
- (line 16)
-* mpz_abs: Integer Arithmetic. (line 36)
-* mpz_add: Integer Arithmetic. (line 7)
-* mpz_add_ui: Integer Arithmetic. (line 8)
-* mpz_addmul: Integer Arithmetic. (line 21)
-* mpz_addmul_ui: Integer Arithmetic. (line 22)
-* mpz_and: Integer Logic and Bit Fiddling.
- (line 11)
-* mpz_array_init: Integer Special Functions.
- (line 11)
-* mpz_bin_ui: Number Theoretic Functions.
- (line 148)
-* mpz_bin_uiui: Number Theoretic Functions.
- (line 149)
-* mpz_cdiv_q: Integer Division. (line 13)
-* mpz_cdiv_q_2exp: Integer Division. (line 21)
-* mpz_cdiv_q_ui: Integer Division. (line 16)
-* mpz_cdiv_qr: Integer Division. (line 15)
-* mpz_cdiv_qr_ui: Integer Division. (line 19)
-* mpz_cdiv_r: Integer Division. (line 14)
-* mpz_cdiv_r_2exp: Integer Division. (line 22)
-* mpz_cdiv_r_ui: Integer Division. (line 17)
-* mpz_cdiv_ui: Integer Division. (line 20)
-* mpz_class: C++ Interface General.
- (line 18)
-* mpz_class::fits_sint_p: C++ Interface Integers.
- (line 44)
-* mpz_class::fits_slong_p: C++ Interface Integers.
- (line 45)
-* mpz_class::fits_sshort_p: C++ Interface Integers.
- (line 46)
-* mpz_class::fits_uint_p: C++ Interface Integers.
- (line 47)
-* mpz_class::fits_ulong_p: C++ Interface Integers.
- (line 48)
-* mpz_class::fits_ushort_p: C++ Interface Integers.
- (line 49)
-* mpz_class::get_d: C++ Interface Integers.
- (line 50)
-* mpz_class::get_mpz_t: C++ Interface General.
- (line 64)
-* mpz_class::get_si: C++ Interface Integers.
- (line 51)
-* mpz_class::get_str: C++ Interface Integers.
- (line 52)
-* mpz_class::get_ui: C++ Interface Integers.
- (line 53)
-* mpz_class::mpz_class: C++ Interface Integers.
- (line 7)
-* mpz_class::set_str: C++ Interface Integers.
- (line 54)
-* mpz_clear: Initializing Integers.
- (line 41)
-* mpz_clears: Initializing Integers.
- (line 45)
-* mpz_clrbit: Integer Logic and Bit Fiddling.
- (line 54)
-* mpz_cmp: Integer Comparisons. (line 7)
-* mpz_cmp_d: Integer Comparisons. (line 8)
-* mpz_cmp_si: Integer Comparisons. (line 9)
-* mpz_cmp_ui: Integer Comparisons. (line 10)
-* mpz_cmpabs: Integer Comparisons. (line 18)
-* mpz_cmpabs_d: Integer Comparisons. (line 19)
-* mpz_cmpabs_ui: Integer Comparisons. (line 20)
-* mpz_com: Integer Logic and Bit Fiddling.
- (line 20)
-* mpz_combit: Integer Logic and Bit Fiddling.
- (line 57)
-* mpz_congruent_2exp_p: Integer Division. (line 113)
-* mpz_congruent_p: Integer Division. (line 111)
-* mpz_congruent_ui_p: Integer Division. (line 112)
-* mpz_divexact: Integer Division. (line 91)
-* mpz_divexact_ui: Integer Division. (line 92)
-* mpz_divisible_2exp_p: Integer Division. (line 102)
-* mpz_divisible_p: Integer Division. (line 100)
-* mpz_divisible_ui_p: Integer Division. (line 101)
-* mpz_even_p: Miscellaneous Integer Functions.
- (line 18)
-* mpz_export: Integer Import and Export.
- (line 45)
-* mpz_fac_ui: Number Theoretic Functions.
- (line 145)
-* mpz_fdiv_q: Integer Division. (line 24)
-* mpz_fdiv_q_2exp: Integer Division. (line 32)
-* mpz_fdiv_q_ui: Integer Division. (line 27)
-* mpz_fdiv_qr: Integer Division. (line 26)
-* mpz_fdiv_qr_ui: Integer Division. (line 30)
-* mpz_fdiv_r: Integer Division. (line 25)
-* mpz_fdiv_r_2exp: Integer Division. (line 33)
-* mpz_fdiv_r_ui: Integer Division. (line 28)
-* mpz_fdiv_ui: Integer Division. (line 31)
-* mpz_fib2_ui: Number Theoretic Functions.
- (line 156)
-* mpz_fib_ui: Number Theoretic Functions.
- (line 155)
-* mpz_fits_sint_p: Miscellaneous Integer Functions.
- (line 10)
-* mpz_fits_slong_p: Miscellaneous Integer Functions.
- (line 8)
-* mpz_fits_sshort_p: Miscellaneous Integer Functions.
- (line 12)
-* mpz_fits_uint_p: Miscellaneous Integer Functions.
- (line 9)
-* mpz_fits_ulong_p: Miscellaneous Integer Functions.
- (line 7)
-* mpz_fits_ushort_p: Miscellaneous Integer Functions.
- (line 11)
-* mpz_gcd: Number Theoretic Functions.
- (line 82)
-* mpz_gcd_ui: Number Theoretic Functions.
- (line 86)
-* mpz_gcdext: Number Theoretic Functions.
- (line 96)
-* mpz_get_d: Converting Integers. (line 42)
-* mpz_get_d_2exp: Converting Integers. (line 50)
-* mpz_get_si <1>: Converting Integers. (line 18)
-* mpz_get_si: MPIR on Windows x64. (line 51)
-* mpz_get_str: Converting Integers. (line 61)
-* mpz_get_sx: Converting Integers. (line 34)
-* mpz_get_ui <1>: Converting Integers. (line 11)
-* mpz_get_ui: MPIR on Windows x64. (line 49)
-* mpz_get_ux: Converting Integers. (line 26)
-* mpz_getlimbn: Integer Special Functions.
- (line 63)
-* mpz_hamdist: Integer Logic and Bit Fiddling.
- (line 29)
-* mpz_import: Integer Import and Export.
- (line 11)
-* mpz_init: Initializing Integers.
- (line 26)
-* mpz_init2: Initializing Integers.
- (line 33)
-* mpz_init_set: Simultaneous Integer Init & Assign.
- (line 27)
-* mpz_init_set_d: Simultaneous Integer Init & Assign.
- (line 32)
-* mpz_init_set_si: Simultaneous Integer Init & Assign.
- (line 29)
-* mpz_init_set_str: Simultaneous Integer Init & Assign.
- (line 37)
-* mpz_init_set_sx: Simultaneous Integer Init & Assign.
- (line 31)
-* mpz_init_set_ui: Simultaneous Integer Init & Assign.
- (line 28)
-* mpz_init_set_ux: Simultaneous Integer Init & Assign.
- (line 30)
-* mpz_inits: Initializing Integers.
- (line 29)
-* mpz_inp_raw: I/O of Integers. (line 59)
-* mpz_inp_str: I/O of Integers. (line 28)
-* mpz_invert: Number Theoretic Functions.
- (line 109)
-* mpz_ior: Integer Logic and Bit Fiddling.
- (line 14)
-* mpz_jacobi: Number Theoretic Functions.
- (line 115)
-* mpz_kronecker: Number Theoretic Functions.
- (line 123)
-* mpz_kronecker_si: Number Theoretic Functions.
- (line 124)
-* mpz_kronecker_ui: Number Theoretic Functions.
- (line 125)
-* mpz_lcm: Number Theoretic Functions.
- (line 103)
-* mpz_lcm_ui: Number Theoretic Functions.
- (line 104)
-* mpz_legendre: Number Theoretic Functions.
- (line 118)
-* mpz_likely_prime_p: Number Theoretic Functions.
- (line 26)
-* mpz_lucnum2_ui: Number Theoretic Functions.
- (line 166)
-* mpz_lucnum_ui: Number Theoretic Functions.
- (line 165)
-* mpz_mod: Integer Division. (line 82)
-* mpz_mod_ui: Integer Division. (line 83)
-* mpz_mul: Integer Arithmetic. (line 16)
-* mpz_mul_2exp: Integer Arithmetic. (line 29)
-* mpz_mul_si: Integer Arithmetic. (line 17)
-* mpz_mul_ui: Integer Arithmetic. (line 18)
-* mpz_neg: Integer Arithmetic. (line 33)
-* mpz_next_likely_prime: Number Theoretic Functions.
- (line 71)
-* mpz_nextprime: Number Theoretic Functions.
- (line 60)
-* mpz_nthroot: Integer Roots. (line 12)
-* mpz_odd_p: Miscellaneous Integer Functions.
- (line 17)
-* mpz_out_raw: I/O of Integers. (line 43)
-* mpz_out_str: I/O of Integers. (line 16)
-* mpz_perfect_power_p: Integer Roots. (line 30)
-* mpz_perfect_square_p: Integer Roots. (line 39)
-* mpz_popcount: Integer Logic and Bit Fiddling.
- (line 23)
-* mpz_pow_ui: Integer Exponentiation.
- (line 18)
-* mpz_powm: Integer Exponentiation.
- (line 8)
-* mpz_powm_ui: Integer Exponentiation.
- (line 10)
-* mpz_probab_prime_p: Number Theoretic Functions.
- (line 41)
-* mpz_probable_prime_p: Number Theoretic Functions.
- (line 8)
-* mpz_realloc2: Initializing Integers.
- (line 49)
-* mpz_remove: Number Theoretic Functions.
- (line 140)
-* mpz_root: Integer Roots. (line 7)
-* mpz_rootrem: Integer Roots. (line 16)
-* mpz_rrandomb: Integer Random Numbers.
- (line 31)
-* mpz_scan0: Integer Logic and Bit Fiddling.
- (line 37)
-* mpz_scan1: Integer Logic and Bit Fiddling.
- (line 38)
-* mpz_set: Assigning Integers. (line 10)
-* mpz_set_d: Assigning Integers. (line 15)
-* mpz_set_f: Assigning Integers. (line 17)
-* mpz_set_q: Assigning Integers. (line 16)
-* mpz_set_si <1>: Assigning Integers. (line 12)
-* mpz_set_si: MPIR on Windows x64. (line 22)
-* mpz_set_str: Assigning Integers. (line 24)
-* mpz_set_sx: Assigning Integers. (line 14)
-* mpz_set_ui <1>: Assigning Integers. (line 11)
-* mpz_set_ui: MPIR on Windows x64. (line 20)
-* mpz_set_ux: Assigning Integers. (line 13)
-* mpz_setbit: Integer Logic and Bit Fiddling.
- (line 51)
-* mpz_sgn: Integer Comparisons. (line 28)
-* mpz_si_kronecker: Number Theoretic Functions.
- (line 126)
-* mpz_size: Integer Special Functions.
- (line 71)
-* mpz_sizeinbase: Miscellaneous Integer Functions.
- (line 23)
-* mpz_sqrt: Integer Roots. (line 20)
-* mpz_sqrtrem: Integer Roots. (line 23)
-* mpz_sub: Integer Arithmetic. (line 11)
-* mpz_sub_ui: Integer Arithmetic. (line 12)
-* mpz_submul: Integer Arithmetic. (line 25)
-* mpz_submul_ui: Integer Arithmetic. (line 26)
-* mpz_swap: Assigning Integers. (line 40)
-* mpz_t: Nomenclature and Types.
- (line 6)
-* mpz_tdiv_q: Integer Division. (line 35)
-* mpz_tdiv_q_2exp: Integer Division. (line 43)
-* mpz_tdiv_q_ui: Integer Division. (line 38)
-* mpz_tdiv_qr: Integer Division. (line 37)
-* mpz_tdiv_qr_ui: Integer Division. (line 41)
-* mpz_tdiv_r: Integer Division. (line 36)
-* mpz_tdiv_r_2exp: Integer Division. (line 44)
-* mpz_tdiv_r_ui: Integer Division. (line 39)
-* mpz_tdiv_ui: Integer Division. (line 42)
-* mpz_tstbit: Integer Logic and Bit Fiddling.
- (line 60)
-* mpz_ui_kronecker: Number Theoretic Functions.
- (line 127)
-* mpz_ui_pow_ui: Integer Exponentiation.
- (line 19)
-* mpz_ui_sub: Integer Arithmetic. (line 13)
-* mpz_urandomb: Integer Random Numbers.
- (line 14)
-* mpz_urandomm: Integer Random Numbers.
- (line 23)
-* mpz_xor: Integer Logic and Bit Fiddling.
- (line 17)
-* operator%: C++ Interface Integers.
- (line 29)
-* operator/: C++ Interface Integers.
- (line 28)
-* operator<<: C++ Formatted Output.
- (line 11)
-* operator>> <1>: C++ Interface Rationals.
- (line 77)
-* operator>>: C++ Formatted Input. (line 11)
-* sgn <1>: C++ Interface Floats.
- (line 88)
-* sgn <2>: C++ Interface Rationals.
- (line 50)
-* sgn: C++ Interface Integers.
- (line 56)
-* sqrt <1>: C++ Interface Floats.
- (line 89)
-* sqrt: C++ Interface Integers.
- (line 57)
-* trunc: C++ Interface Floats.
- (line 90)
-
-
diff --git a/doc/mpir.texi b/doc/mpir.texi
deleted file mode 100644
index 3aa54d1..0000000
--- a/doc/mpir.texi
+++ /dev/null
@@ -1,10370 +0,0 @@
-\input texinfo @c -*-texinfo-*-
- at c %**start of header
- at setfilename mpir.info
- at include version.texi
- at settitle MPIR @value{VERSION}
- at synindex tp fn
- at iftex
- at afourpaper
- at end iftex
- at comment %**end of header
-
- at copying
-This manual describes how to install and use MPIR, the Multiple Precision Integers and Rationals
-library, version @value{VERSION}.
-
-Copyright 1991, 1993, 1994, 1995, 1996, 1997, 1998, 1999, 2000, 2001, 2002,
-2003, 2004, 2005, 2006, 2007, 2008, 2009, 2010 Free Software Foundation, Inc.
-
-Copyright 2008, 2009, 2010 William Hart
-
-Permission is granted to copy, distribute and/or modify this document under
-the terms of the GNU Free Documentation License, Version 1.3 or any later
-version published by the Free Software Foundation; with no Invariant Sections,
-with the Front-Cover Texts being ``A GNU Manual'', and with the Back-Cover
-Texts being ``You have freedom to copy and modify this GNU Manual, like GNU
-software''. A copy of the license is included in
- at ref{GNU Free Documentation License}.
- at end copying
- at c Note the @ref above must be on one line, a line break in an @ref within
- at c @copying will bomb in recent texinfo.tex (eg. 2004-04-07.08 which comes
- at c with texinfo 4.7), with messages about missing @endcsname.
-
-
- at c Texinfo version 4.2 or up will be needed to process this file.
- at c
- at c The version number and edition number are taken from version.texi provided
- at c by automake (note that it's regenerated only if you configure with
- at c --enable-maintainer-mode).
- at c
- at c Notes discussing the present version number of GMP/MPIR in relation to previous
- at c ones (for instance in the "Compatibility" section) must be updated
- at c manually though.
- at c
- at c @cindex entries have been made for function categories and programming
- at c topics. The "mpn" section is not included in this, because a beginner
- at c looking for "GCD" or something is only going to be confused by pointers to
- at c low level routines.
- at c
- at c @cindex entries are present for processors and systems when there's
- at c particular notes concerning them, but not just for everything MPIR
- at c supports.
- at c
- at c Index entries for files use @code rather than @file, @samp or @option,
- at c since the latter come out with quotes in TeX, which are nice in the text
- at c but don't look so good in index columns.
- at c
- at c Tex:
- at c
- at c A suitable texinfo.tex is supplied, a newer one should work equally well.
- at c
- at c HTML:
- at c
- at c Nothing special is done for links to external manuals, they just come out
- at c in the usual makeinfo style, eg. "../libc/Locales.html". If you have
- at c local copies of such manuals then this is a good thing, if not then you
- at c may want to search-and-replace to some online source.
- at c
-
- at dircategory GNU libraries
- at direntry
-* mpir: (mpir). MPIR Multiple Precision Integers and Rationals Library.
- at end direntry
-
- at c html <meta name="description" content="...">
- at documentdescription
-How to install and use the MPIR multiple precision arithmetic library, version @value{VERSION}.
- at end documentdescription
-
- at c smallbook
- at finalout
- at setchapternewpage on
-
- at ifnottex
- at node Top, Copying, (dir), (dir)
- at top MPIR
- at end ifnottex
-
- at iftex
- at titlepage
- at title MPIR
- at subtitle The Multiple Precision Integers and Rationals Library
- at subtitle Edition @value{EDITION}
- at subtitle @value{UPDATED}
-
- at author Original Authors: Torbjorn Granlund and the GMP Development Team
- at author Subsequent modifications: William Hart and the MPIR Team
- at c @email{goodwillhart at gmail.com}
-
- at c Include the Distribution inside the titlepage so
- at c that headings are turned off.
-
- at tex
-\global\parindent=0pt
-\global\parskip=8pt
-\global\baselineskip=13pt
- at end tex
-
- at page
- at vskip 0pt plus 1filll
- at end iftex
-
- at insertcopying
- at ifnottex
- at sp 1
- at end ifnottex
-
- at iftex
- at end titlepage
- at headings double
- at end iftex
-
- at c Don't bother with contents for html, the menus seem adequate.
- at ifnothtml
- at contents
- at end ifnothtml
-
- at menu
-* Copying:: MPIR Copying Conditions (LGPL).
-* Introduction to MPIR:: Brief introduction to MPIR.
-* Installing MPIR:: How to configure and compile the MPIR library.
-* MPIR Basics:: What every MPIR user should know.
-* Reporting Bugs:: How to usefully report bugs.
-* Integer Functions:: Functions for arithmetic on signed integers.
-* Rational Number Functions:: Functions for arithmetic on rational numbers.
-* Floating-point Functions:: Functions for arithmetic on floats.
-* Low-level Functions:: Fast functions for natural numbers.
-* Random Number Functions:: Functions for generating random numbers.
-* Formatted Output:: @code{printf} style output.
-* Formatted Input:: @code{scanf} style input.
-* C++ Class Interface:: Class wrappers around MPIR types.
-* Custom Allocation:: How to customize the internal allocation.
-* Language Bindings:: Using MPIR from other languages.
-* Algorithms:: What happens behind the scenes.
-* Internals:: How values are represented behind the scenes.
-
-* Contributors:: Who brings you this library?
-* References:: Some useful papers and books to read.
-* GNU Free Documentation License::
-* Concept Index::
-* Function Index::
- at end menu
-
-
- at c @m{T,N} is $T$ in tex or @math{N} otherwise. This is an easy way to give
- at c different forms for math in tex and info. Commas in N or T don't work,
- at c but @C{} can be used instead. \, works in info but not in tex.
- at iftex
- at macro m {T,N}
- at tex$\T\$@end tex
- at end macro
- at end iftex
- at ifnottex
- at macro m {T,N}
- at math{\N\}
- at end macro
- at end ifnottex
-
- at macro C {}
-,
- at end macro
-
- at c @ms{V,N} is $V_N$ in tex or just vn otherwise. This suits simple
- at c subscripts like @ms{x,0}.
- at iftex
- at macro ms {V,N}
- at tex$\V\_{\N\}$@end tex
- at end macro
- at end iftex
- at ifnottex
- at macro ms {V,N}
-\V\\N\
- at end macro
- at end ifnottex
-
- at c @nicode{S} is plain S in info, or @code{S} elsewhere. This can be used
- at c when the quotes that @code{} gives in info aren't wanted, but the
- at c fontification in tex or html is wanted. Doesn't work as @nicode{'\\0'}
- at c though (gives two backslashes in tex).
- at ifinfo
- at macro nicode {S}
-\S\
- at end macro
- at end ifinfo
- at ifnotinfo
- at macro nicode {S}
- at code{\S\}
- at end macro
- at end ifnotinfo
-
- at c @nisamp{S} is plain S in info, or @samp{S} elsewhere. This can be used
- at c when the quotes that @samp{} gives in info aren't wanted, but the
- at c fontification in tex or html is wanted.
- at ifinfo
- at macro nisamp {S}
-\S\
- at end macro
- at end ifinfo
- at ifnotinfo
- at macro nisamp {S}
- at samp{\S\}
- at end macro
- at end ifnotinfo
-
- at c Usage: @GMPtimes{}
- at c Give either \times or the word "times".
- at tex
-\gdef\GMPtimes{\times}
- at end tex
- at ifnottex
- at macro GMPtimes
-times
- at end macro
- at end ifnottex
-
- at c Usage: @GMPmultiply{}
- at c Give * in info, or nothing in tex.
- at tex
-\gdef\GMPmultiply{}
- at end tex
- at ifnottex
- at macro GMPmultiply
-*
- at end macro
- at end ifnottex
-
- at c Usage: @GMPabs{x}
- at c Give either |x| in tex, or abs(x) in info or html.
- at tex
-\gdef\GMPabs#1{|#1|}
- at end tex
- at ifnottex
- at macro GMPabs {X}
- at abs{}(\X\)
- at end macro
- at end ifnottex
-
- at c Usage: @GMPfloor{x}
- at c Give either \lfloor x\rfloor in tex, or floor(x) in info or html.
- at tex
-\gdef\GMPfloor#1{\lfloor #1\rfloor}
- at end tex
- at ifnottex
- at macro GMPfloor {X}
-floor(\X\)
- at end macro
- at end ifnottex
-
- at c Usage: @GMPceil{x}
- at c Give either \lceil x\rceil in tex, or ceil(x) in info or html.
- at tex
-\gdef\GMPceil#1{\lceil #1 \rceil}
- at end tex
- at ifnottex
- at macro GMPceil {X}
-ceil(\X\)
- at end macro
- at end ifnottex
-
- at c Math operators already available in tex, made available in info too.
- at c For example @bmod{} can be used in both tex and info.
- at ifnottex
- at macro bmod
-mod
- at end macro
- at macro gcd
-gcd
- at end macro
- at macro ge
->=
- at end macro
- at macro le
-<=
- at end macro
- at macro log
-log
- at end macro
- at macro min
-min
- at end macro
- at macro leftarrow
-<-
- at end macro
- at macro rightarrow
-->
- at end macro
- at end ifnottex
-
- at c New math operators.
- at c @abs{} can be used in both tex and info, or just \abs in tex.
- at tex
-\gdef\abs{\mathop{\rm abs}}
- at end tex
- at ifnottex
- at macro abs
-abs
- at end macro
- at end ifnottex
-
- at c @cross{} is a \times symbol in tex, or an "x" in info. In tex it works
- at c inside or outside $ $.
- at tex
-\gdef\cross{\ifmmode\times\else$\times$\fi}
- at end tex
- at ifnottex
- at macro cross
-x
- at end macro
- at end ifnottex
-
- at c @times{} made available as a "*" in info and html (already works in tex).
- at ifnottex
- at macro times
-*
- at end macro
- at end ifnottex
-
- at c Usage: @W{text}
- at c Like @w{} but working in math mode too.
- at tex
-\gdef\W#1{\ifmmode{#1}\else\w{#1}\fi}
- at end tex
- at ifnottex
- at macro W {S}
- at w{\S\}
- at end macro
- at end ifnottex
-
- at c Usage: \GMPdisplay{text}
- at c Put the given text in an @display style indent, but without turning off
- at c paragraph reflow etc.
- at tex
-\gdef\GMPdisplay#1{%
-\noindent
-\advance\leftskip by \lispnarrowing
-#1\par}
- at end tex
-
- at c Usage: \GMPhat
- at c A new \hat that will work in math mode, unlike the texinfo redefined
- at c version.
- at tex
-\gdef\GMPhat{\mathaccent"705E}
- at end tex
-
- at c Usage: \GMPraise{text}
- at c For use in a $ $ math expression as an alternative to "^". This is good
- at c for @code{} in an exponent, since there seems to be no superscript font
- at c for that.
- at tex
-\gdef\GMPraise#1{\mskip0.5\thinmuskip\hbox{\raise0.8ex\hbox{#1}}}
- at end tex
-
- at c Usage: @texlinebreak{}
- at c A line break as per @*, but only in tex.
- at iftex
- at macro texlinebreak
-@*
- at end macro
- at end iftex
- at ifnottex
- at macro texlinebreak
- at end macro
- at end ifnottex
-
- at c Usage: @maybepagebreak
- at c Allow tex to insert a page break, if it feels the urge.
- at c Normally blocks of @deftypefun/funx are kept together, which can lead to
- at c some poor page break positioning if it's a big block, like the sets of
- at c division functions etc.
- at tex
-\gdef\maybepagebreak{\penalty0}
- at end tex
- at ifnottex
- at macro maybepagebreak
- at end macro
- at end ifnottex
-
- at c Usage: @GMPreftop{info,title}
- at c Usage: @GMPpxreftop{info,title}
- at c
- at c Like @ref{} and @pxref{}, but designed for a reference to the top of a
- at c document, not a particular section. The TeX output for plain @ref insists
- at c on printing a particular section, GMPreftop gives just the title.
- at c
- at c The texinfo manual recommends putting a likely section name in references
- at c like this, eg. "Introduction", but it seems better to just give the title.
- at c
- at iftex
- at macro GMPreftop{info,title}
- at i{\title\}
- at end macro
- at macro GMPpxreftop{info,title}
-see @i{\title\}
- at end macro
- at end iftex
- at c
- at ifnottex
- at macro GMPreftop{info,title}
- at ref{Top,\title\,\title\,\info\,\title\}
- at end macro
- at macro GMPpxreftop{info,title}
- at pxref{Top,\title\,\title\,\info\,\title\}
- at end macro
- at end ifnottex
-
-
- at node Copying, Introduction to MPIR, Top, Top
- at comment node-name, next, previous, up
- at unnumbered MPIR Copying Conditions
- at cindex Copying conditions
- at cindex Conditions for copying MPIR
- at cindex License conditions
-
-This library is @dfn{free}; this means that everyone is free to use it and
-free to redistribute it on a free basis. The library is not in the public
-domain; it is copyrighted and there are restrictions on its distribution, but
-these restrictions are designed to permit everything that a good cooperating
-citizen would want to do. What is not allowed is to try to prevent others
-from further sharing any version of this library that they might get from
-you. at refill
-
-Specifically, we want to make sure that you have the right to give away copies
-of the library, that you receive source code or else can get it if you want
-it, that you can change this library or use pieces of it in new free programs,
-and that you know you can do these things. at refill
-
-To make sure that everyone has such rights, we have to forbid you to deprive
-anyone else of these rights. For example, if you distribute copies of the MPIR
-library, you must give the recipients all the rights that you have. You
-must make sure that they, too, receive or can get the source code. And you
-must tell them their rights. at refill
-
-Also, for our own protection, we must make certain that everyone finds out
-that there is no warranty for the MPIR library. If it is modified by
-someone else and passed on, we want their recipients to know that what they
-have is not what we distributed, so that any problems introduced by others
-will not reflect on our reputation. at refill
-
-The precise conditions of the license for the MPIR library are found in the
-Lesser General Public License version 3 that accompanies the source code,
-see @file{COPYING.LIB}.
-
-
- at node Introduction to MPIR, Installing MPIR, Copying, Top
- at comment node-name, next, previous, up
- at chapter Introduction to MPIR
- at cindex Introduction
-
-MPIR is a portable library written in C for arbitrary precision arithmetic
-on integers, rational numbers, and floating-point numbers. It aims to provide
-the fastest possible arithmetic for all applications that need higher
-precision than is directly supported by the basic C types.
-
-Many applications use just a few hundred bits of precision; but some
-applications may need thousands or even millions of bits. MPIR is designed to
-give good performance for both, by choosing algorithms based on the sizes of
-the operands, and by carefully keeping the overhead at a minimum.
-
-The speed of MPIR is achieved by using fullwords as the basic arithmetic type,
-by using sophisticated algorithms, by including carefully optimized assembly
-code for the most common inner loops for many different CPUs, and by a general
-emphasis on speed (as opposed to simplicity or elegance).
-
-There is assembly code for these CPUs:
- at cindex CPU types
-ARM,
-DEC Alpha 21064, 21164, and 21264,
-AMD K6, K6-2, Athlon, K8 and K10,
-Intel Pentium, Pentium Pro/II/III, Pentium 4, generic x86,
-Intel IA-64, Core 2, i7, Atom,
-Motorola/IBM PowerPC 32 and 64,
-MIPS R3000, R4000,
-SPARCv7, SuperSPARC, generic SPARCv8, UltraSPARC,
-
- at cindex Home page
- at cindex Web page
- at noindent
-For up-to-date information on, and latest version of, MPIR, please see the MPIR web pages at
-
- at display
- at uref{http://www.mpir.org/}
- at end display
-
- at cindex Mailing lists
-There are a number of public mailing lists of interest. The development list is
-
- at display
- at uref{http://groups.google.com/group/mpir-devel/}.
- at end display
-
-The proper place for bug reports is @uref{http://groups.google.com/group/mpir-devel}. See
- at ref{Reporting Bugs} for information about reporting bugs.
-
- at sp 1
- at section How to use this Manual
- at cindex About this manual
-
-Everyone should read @ref{MPIR Basics}. If you need to install the library
-yourself, then read @ref{Installing MPIR}. If you have a system with multiple
-ABIs, then read @ref{ABI and ISA}, for the compiler options that must be used
-on applications.
-
-The rest of the manual can be used for later reference, although it is
-probably a good idea to glance through it.
-
-
- at node Installing MPIR, MPIR Basics, Introduction to MPIR, Top
- at comment node-name, next, previous, up
- at chapter Installing MPIR
- at cindex Installing MPIR
- at cindex Configuring MPIR
- at cindex Building MPIR
-
-MPIR has an autoconf/automake/libtool based configuration system. On a
-Unix-like system a basic build can be done with
-
- at example
-./configure
-make
- at end example
-
- at noindent
-Some self-tests can be run with
-
- at example
-make check
- at end example
-
- at noindent
-And you can install (under @file{/usr/local} by default) with
-
- at example
-make install
- at end example
-
-Important note: by default MPIR produces libraries named libmpir, etc., and the header file
-mpir.h. If you wish to have MPIR to build a library named libgmp as well, etc., and a
-gmp.h header file, so that you can use mpir with programs designed to only work with GMP,
-then use the @samp{--enable-gmpcompat} option when invoking configure:
-
- at example
-./configure --enable-gmpcompat
- at end example
-
-Note gmp.h is only created upon running make install.
-
-MPIR is compatible with GMP when the @samp{--enable-gmpcompat} option is used, except that the GMP secure cryptographic functions are not available.
-
-Some deprecated GMP functionality may be unavailable if this option is not selected.
-
-If you experience problems, please report them to
-
- at uref{http://groups.google.com/group/mpir-devel}.
-
-See @ref{Reporting Bugs}, for information on what to include in useful bug
-reports.
-
- at menu
-* Build Options::
-* ABI and ISA::
-* Notes for Package Builds::
-* Notes for Particular Systems::
-* Known Build Problems::
-* Performance optimization::
- at end menu
-
-
- at node Build Options, ABI and ISA, Installing MPIR, Installing MPIR
- at section Build Options
- at cindex Build options
-
-All the usual autoconf configure options are available, run @samp{./configure
---help} for a summary. The file @file{INSTALL.autoconf} has some generic
-installation information too.
-
- at table @asis
- at item Tools
- at cindex Non-Unix systems
- at samp{configure} requires various Unix-like tools. See @ref{Notes for
-Particular Systems}, for some options on non-Unix systems.
-
-It might be possible to build without the help of @samp{configure}, certainly
-all the code is there, but unfortunately you'll be on your own.
-
- at item Build Directory
- at cindex Build directory
-To compile in a separate build directory, @command{cd} to that directory, and
-prefix the configure command with the path to the MPIR source directory. For
-example
-
- at example
-cd /my/build/dir
-/my/sources/mpir- at value{VERSION}/configure
- at end example
-
-Not all @samp{make} programs have the necessary features (@code{VPATH}) to
-support this. In particular, SunOS and Solaris @command{make} have bugs that
-make them unable to build in a separate directory. Use GNU @command{make}
-instead.
-
- at item @option{--prefix} and @option{--exec-prefix}
- at cindex Prefix
- at cindex Exec prefix
- at cindex Install prefix
- at cindex @code{--prefix}
- at cindex @code{--exec-prefix}
-The @option{--prefix} option can be used in the normal way to direct MPIR to
-install under a particular tree. The default is @samp{/usr/local}.
-
- at option{--exec-prefix} can be used to direct architecture-dependent files like
- at file{libmpir.a} to a different location. This can be used to share
-architecture-independent parts like the documentation, but separate the
-dependent parts. Note however that @file{mpir.h} and @file{mp.h} are
-architecture-dependent since they encode certain aspects of @file{libmpir}, so
-it will be necessary to ensure both @file{$prefix/include} and
- at file{$exec_prefix/include} are available to the compiler.
-
- at item @option{--enable-gmpcompat}
- at cindex @code{--enable-gmpcompat}
-By default make builds libmpir library files (and libmpirxx if C++ headers are requested) and the mpir.h header file. This option allows you to specify that you want additional libraries created called libgmp (and libgmpxx), etc., for libraries and gmp.h for compatibility with GMP (except for GMP secure cryptograhic functions, which are not available in MPIR).
-
- at item @option{--disable-shared}, @option{--disable-static}
- at cindex @code{--disable-shared}
- at cindex @code{--disable-static}
-By default both shared and static libraries are built (where possible), but
-one or other can be disabled. Shared libraries result in smaller executables
-and permit code sharing between separate running processes, but on some CPUs
-are slightly slower, having a small cost on each function call.
-
- at item Native Compilation, @option{--build=CPU-VENDOR-OS}
- at cindex Native compilation
- at cindex Build system
- at cindex @code{--build}
-For normal native compilation, the system can be specified with
- at samp{--build}. By default @samp{./configure} uses the output from running
- at samp{./config.guess}. On some systems @samp{./config.guess} can determine
-the exact CPU type, on others it will be necessary to give it explicitly. For
-example,
-
- at example
-./configure --build=ultrasparc-sun-solaris2.7
- at end example
-
-In all cases the @samp{OS} part is important, since it controls how libtool
-generates shared libraries. Running @samp{./config.guess} is the simplest way
-to see what it should be, if you don't know already.
-
- at item Cross Compilation, @option{--host=CPU-VENDOR-OS}
- at cindex Cross compiling
- at cindex Host system
- at cindex @code{--host}
-When cross-compiling, the system used for compiling is given by @samp{--build}
-and the system where the library will run is given by @samp{--host}. For
-example when using a FreeBSD Athlon system to build GNU/Linux m68k binaries,
-
- at example
-./configure --build=athlon-pc-freebsd3.5 --host=m68k-mac-linux-gnu
- at end example
-
-Compiler tools are sought first with the host system type as a prefix. For
-example @command{m68k-mac-linux-gnu-ranlib} is tried, then plain
- at command{ranlib}. This makes it possible for a set of cross-compiling tools
-to co-exist with native tools. The prefix is the argument to @samp{--host},
-and this can be an alias, such as @samp{m68k-linux}. But note that tools
-don't have to be setup this way, it's enough to just have a @env{PATH} with a
-suitable cross-compiling @command{cc} etc.
-
-Compiling for a different CPU in the same family as the build system is a form
-of cross-compilation, though very possibly this would merely be special
-options on a native compiler. In any case @samp{./configure} avoids depending
-on being able to run code on the build system, which is important when
-creating binaries for a newer CPU since they very possibly won't run on the
-build system.
-
-In all cases the compiler must be able to produce an executable (of whatever
-format) from a standard C @code{main}. Although only object files will go to
-make up @file{libmpir}, @samp{./configure} uses linking tests for various
-purposes, such as determining what functions are available on the host system.
-
-Currently a warning is given unless an explicit @samp{--build} is used when
-cross-compiling, because it may not be possible to correctly guess the build
-system type if the @env{PATH} has only a cross-compiling @command{cc}.
-
-Note that the @samp{--target} option is not appropriate for MPIR at . It's for use
-when building compiler tools, with @samp{--host} being where they will run,
-and @samp{--target} what they'll produce code for. Ordinary programs or
-libraries like MPIR are only interested in the @samp{--host} part, being where
-they'll run.
-
- at item CPU types
- at cindex CPU types
-In general, if you want a library that runs as fast as possible, you should
-configure MPIR for the exact CPU type your system uses. However, this may mean
-the binaries won't run on older members of the family, and might run slower on
-other members, older or newer. The best idea is always to build MPIR for the
-exact machine type you intend to run it on.
-
-The following CPUs have specific support. See @file{configure.in} for details
-of what code and compiler options they select.
-
- at itemize @bullet
-
- at c Keep this formatting, it's easy to read and it can be grepped to
- at c automatically test that CPUs listed get through ./config.sub
-
- at item
-Alpha:
- at nisamp{alpha},
- at nisamp{alphaev5},
- at nisamp{alphaev56},
- at nisamp{alphapca56},
- at nisamp{alphapca57},
- at nisamp{alphaev6},
- at nisamp{alphaev67},
- at nisamp{alphaev68}
- at nisamp{alphaev7}
-
- at item
-IA-64:
- at nisamp{ia64},
- at nisamp{itanium},
- at nisamp{itanium2}
-
- at item
-MIPS:
- at nisamp{mips},
- at nisamp{mips3},
- at nisamp{mips64}
-
- at item
-PowerPC:
- at nisamp{powerpc},
- at nisamp{powerpc64},
- at nisamp{powerpc401},
- at nisamp{powerpc403},
- at nisamp{powerpc405},
- at nisamp{powerpc505},
- at nisamp{powerpc601},
- at nisamp{powerpc602},
- at nisamp{powerpc603},
- at nisamp{powerpc603e},
- at nisamp{powerpc604},
- at nisamp{powerpc604e},
- at nisamp{powerpc620},
- at nisamp{powerpc630},
- at nisamp{powerpc740},
- at nisamp{powerpc7400},
- at nisamp{powerpc7450},
- at nisamp{powerpc750},
- at nisamp{powerpc801},
- at nisamp{powerpc821},
- at nisamp{powerpc823},
- at nisamp{powerpc860},
- at nisamp{powerpc970}
-
- at item
-SPARC:
- at nisamp{sparc},
- at nisamp{sparcv8},
- at nisamp{microsparc},
- at nisamp{supersparc},
- at nisamp{sparcv9},
- at nisamp{ultrasparc},
- at nisamp{ultrasparc2},
- at nisamp{ultrasparc2i},
- at nisamp{ultrasparc3},
- at nisamp{sparc64}
-
- at item
-x86 family:
- at nisamp{pentium},
- at nisamp{pentiummmx},
- at nisamp{pentiumpro},
- at nisamp{pentium2},
- at nisamp{pentium3},
- at nisamp{pentium4},
- at nisamp{netburst},
- at nisamp{netburstlahf},
- at nisamp{prescott},
- at nisamp{core},
- at nisamp{core2},
- at nisamp{penryn},
- at nisamp{nehalem},
- at nisamp{nano}
- at nisamp{atom},
- at nisamp{k5},
- at nisamp{k6},
- at nisamp{k62},
- at nisamp{k63},
- at nisamp{k7},
- at nisamp{k8},
- at nisamp{k10}
- at nisamp{k102}
- at nisamp{viac3},
- at nisamp{viac32}
-
- at item
-Other:
- at nisamp{arm},
- at end itemize
-
-CPUs not listed will use generic C code.
-
- at item Generic C Build
- at cindex Generic C
-If some of the assembly code causes problems, or if otherwise desired, the
-generic C code can be selected with CPU @samp{none}. For example,
-
- at example
-./configure --host=none-unknown-freebsd3.5
- at end example
-
-Note that this will run quite slowly, but it should be portable and should at
-least make it possible to get something running if all else fails.
-
- at item Fat binary, @option{--enable-fat}
- at cindex Fat binary
- at cindex @option{--enable-fat}
-Using @option{--enable-fat} selects a ``fat binary'' build on x86 or x86_64
-systems, where optimized low level subroutines are chosen at runtime according
-to the CPU detected. This means more code, but gives reasonable performance
-from a single binary for all x86 chips, or similarly for all x86_64 chips.
-(This option might become available for more architectures in the future.)
-
- at item @option{ABI}
- at cindex ABI
-On some systems MPIR supports multiple ABIs (application binary interfaces),
-meaning data type sizes and calling conventions. By default MPIR chooses the
-best ABI available, but a particular ABI can be selected. For example
-
- at example
-./configure --host=mips64-sgi-irix6 ABI=n32
- at end example
-
-See @ref{ABI and ISA}, for the available choices on relevant CPUs, and what
-applications need to do.
-
- at item @option{CC}, @option{CFLAGS}
- at cindex C compiler
- at cindex @code{CC}
- at cindex @code{CFLAGS}
-By default the C compiler used is chosen from among some likely candidates,
-with @command{gcc} normally preferred if it's present. The usual
- at samp{CC=whatever} can be passed to @samp{./configure} to choose something
-different.
-
-For various systems, default compiler flags are set based on the CPU and
-compiler. The usual @samp{CFLAGS="-whatever"} can be passed to
- at samp{./configure} to use something different or to set good flags for systems
-MPIR doesn't otherwise know.
-
-The @samp{CC} and @samp{CFLAGS} used are printed during @samp{./configure},
-and can be found in each generated @file{Makefile}. This is the easiest way
-to check the defaults when considering changing or adding something.
-
-Note that when @samp{CC} and @samp{CFLAGS} are specified on a system
-supporting multiple ABIs it's important to give an explicit
- at samp{ABI=whatever}, since MPIR can't determine the ABI just from the flags and
-won't be able to select the correct assembler code.
-
-If just @samp{CC} is selected then normal default @samp{CFLAGS} for that
-compiler will be used (if MPIR recognises it). For example @samp{CC=gcc} can
-be used to force the use of GCC, with default flags (and default ABI).
-
- at item @option{CPPFLAGS}
- at cindex @code{CPPFLAGS}
-Any flags like @samp{-D} defines or @samp{-I} includes required by the
-preprocessor should be set in @samp{CPPFLAGS} rather than @samp{CFLAGS}.
-Compiling is done with both @samp{CPPFLAGS} and @samp{CFLAGS}, but
-preprocessing uses just @samp{CPPFLAGS}. This distinction is because most
-preprocessors won't accept all the flags the compiler does. Preprocessing is
-done separately in some configure tests, and in the @samp{ansi2knr} support
-for K&R compilers.
-
- at item @option{CC_FOR_BUILD}
- at cindex @code{CC_FOR_BUILD}
-Some build-time programs are compiled and run to generate host-specific data
-tables. @samp{CC_FOR_BUILD} is the compiler used for this. It doesn't need
-to be in any particular ABI or mode, it merely needs to generate executables
-that can run. The default is to try the selected @samp{CC} and some likely
-candidates such as @samp{cc} and @samp{gcc}, looking for something that works.
-
-No flags are used with @samp{CC_FOR_BUILD} because a simple invocation like
- at samp{cc foo.c} should be enough. If some particular options are required
-they can be included as for instance @samp{CC_FOR_BUILD="cc -whatever"}.
-
- at item C++ Support, @option{--enable-cxx}
- at cindex C++ support
- at cindex @code{--enable-cxx}
-C++ support in MPIR can be enabled with @samp{--enable-cxx}, in which case a
-C++ compiler will be required. As a convenience @samp{--enable-cxx=detect}
-can be used to enable C++ support only if a compiler can be found. The C++
-support consists of a library @file{libmpirxx.la} and header file
- at file{mpirxx.h} (@pxref{Headers and Libraries}).
-
-A separate @file{libmpirxx.la} has been adopted rather than having C++ objects
-within @file{libmpir.la} in order to ensure dynamic linked C programs aren't
-bloated by a dependency on the C++ standard library, and to avoid any chance
-that the C++ compiler could be required when linking plain C programs.
-
- at file{libmpirxx.la} will use certain internals from @file{libmpir.la} and can
-only be expected to work with @file{libmpir.la} from the same MPIR version.
-Future changes to the relevant internals will be accompanied by renaming, so a
-mismatch will cause unresolved symbols rather than perhaps mysterious
-misbehaviour.
-
-In general @file{libmpirxx.la} will be usable only with the C++ compiler that
-built it, since name mangling and runtime support are usually incompatible
-between different compilers.
-
- at item @option{CXX}, @option{CXXFLAGS}
- at cindex C++ compiler
- at cindex @code{CXX}
- at cindex @code{CXXFLAGS}
-When C++ support is enabled, the C++ compiler and its flags can be set with
-variables @samp{CXX} and @samp{CXXFLAGS} in the usual way. The default for
- at samp{CXX} is the first compiler that works from a list of likely candidates,
-with @command{g++} normally preferred when available. The default for
- at samp{CXXFLAGS} is to try @samp{CFLAGS}, @samp{CFLAGS} without @samp{-g}, then
-for @command{g++} either @samp{-g -O2} or @samp{-O2}, or for other compilers
- at samp{-g} or nothing. Trying @samp{CFLAGS} this way is convenient when using
- at samp{gcc} and @samp{g++} together, since the flags for @samp{gcc} will
-usually suit @samp{g++}.
-
-It's important that the C and C++ compilers match, meaning their startup and
-runtime support routines are compatible and that they generate code in the
-same ABI (if there's a choice of ABIs on the system). @samp{./configure}
-isn't currently able to check these things very well itself, so for that
-reason @samp{--disable-cxx} is the default, to avoid a build failure due to a
-compiler mismatch. Perhaps this will change in the future.
-
-Incidentally, it's normally not good enough to set @samp{CXX} to the same as
- at samp{CC}. Although @command{gcc} for instance recognises @file{foo.cc} as
-C++ code, only @command{g++} will invoke the linker the right way when
-building an executable or shared library from C++ object files.
-
- at item Temporary Memory, @option{--enable-alloca=<choice>}
- at cindex Temporary memory
- at cindex Stack overflow
- at cindex @code{alloca}
- at cindex @code{--enable-alloca}
-MPIR allocates temporary workspace using one of the following three methods,
-which can be selected with for instance
- at samp{--enable-alloca=malloc-reentrant}.
-
- at itemize @bullet
- at item
- at samp{alloca} - C library or compiler builtin.
- at item
- at samp{malloc-reentrant} - the heap, in a re-entrant fashion.
- at item
- at samp{malloc-notreentrant} - the heap, with global variables.
- at end itemize
-
-For convenience, the following choices are also available.
- at samp{--disable-alloca} is the same as @samp{no}.
-
- at itemize @bullet
- at item
- at samp{yes} - a synonym for @samp{alloca}.
- at item
- at samp{no} - a synonym for @samp{malloc-reentrant}.
- at item
- at samp{reentrant} - @code{alloca} if available, otherwise
- at samp{malloc-reentrant}. This is the default.
- at item
- at samp{notreentrant} - @code{alloca} if available, otherwise
- at samp{malloc-notreentrant}.
- at end itemize
-
- at code{alloca} is reentrant and fast, and is recommended. It actually allocates
-just small blocks on the stack; larger ones use malloc-reentrant.
-
- at samp{malloc-reentrant} is, as the name suggests, reentrant and thread safe,
-but @samp{malloc-notreentrant} is faster and should be used if reentrancy is
-not required.
-
-The two malloc methods in fact use the memory allocation functions selected by
- at code{mp_set_memory_functions}, these being @code{malloc} and friends by
-default. @xref{Custom Allocation}.
-
-An additional choice @samp{--enable-alloca=debug} is available, to help when
-debugging memory related problems (@pxref{Debugging}).
-
- at item FFT Multiplication, @option{--disable-fft}
- at cindex FFT multiplication
- at cindex @code{--disable-fft}
-By default multiplications are done using Karatsuba, Toom, and
-FFT algorithms at . The FFT is only used on large to very large operands and
-can be disabled to save code size if desired.
-
- at item Assertion Checking, @option{--enable-assert}
- at cindex Assertion checking
- at cindex @code{--enable-assert}
-This option enables some consistency checking within the library. This can be
-of use while debugging, @pxref{Debugging}.
-
- at item Execution Profiling, @option{--enable-profiling=prof/gprof/instrument}
- at cindex Execution profiling
- at cindex @code{--enable-profiling}
-Enable profiling support, in one of various styles, @pxref{Profiling}.
-
- at item @option{MPN_PATH}
- at cindex @code{MPN_PATH}
-Various assembler versions of mpn subroutines are provided. For a given
-CPU, a search is made though a path to choose a version of each. For example
- at samp{sparcv8} has
-
- at example
-MPN_PATH="sparc32/v8 sparc32 generic"
- at end example
-
-which means look first for v8 code, then plain sparc32 (which is v7), and
-finally fall back on generic C at . Knowledgeable users with special requirements
-can specify a different path. Normally this is completely unnecessary.
-
- at item Documentation
- at cindex Documentation formats
- at cindex Texinfo
-The source for the document you're now reading is @file{doc/mpir.texi}, in
-Texinfo format, see @GMPreftop{texinfo, Texinfo}.
-
- at cindex Postscript
- at cindex DVI
- at cindex PDF
-Info format @samp{doc/mpir.info} is included in the distribution. The usual
-automake targets are available to make PostScript, DVI, PDF and HTML (these
-will require various @TeX{} and Texinfo tools).
-
- at cindex DocBook
- at cindex XML
-DocBook and XML can be generated by the Texinfo @command{makeinfo} program
-too, see @ref{makeinfo options,, Options for @command{makeinfo}, texinfo,
-Texinfo}.
-
-Some supplementary notes can also be found in the @file{doc} subdirectory.
-
- at end table
-
-
- at need 2000
- at node ABI and ISA, Notes for Package Builds, Build Options, Installing MPIR
- at section ABI and ISA
- at cindex ABI
- at cindex Application Binary Interface
- at cindex ISA
- at cindex Instruction Set Architecture
-
-ABI (Application Binary Interface) refers to the calling conventions between
-functions, meaning what registers are used and what sizes the various C data
-types are. ISA (Instruction Set Architecture) refers to the instructions and
-registers a CPU has available.
-
-Some 64-bit ISA CPUs have both a 64-bit ABI and a 32-bit ABI defined, the
-latter for compatibility with older CPUs in the family. MPIR supports some
-CPUs like this in both ABIs. In fact within MPIR @samp{ABI} means a
-combination of chip ABI, plus how MPIR chooses to use it. For example in some
-32-bit ABIs, MPIR may support a limb as either a 32-bit @code{long} or a 64-bit
- at code{long long}.
-
-By default MPIR chooses the best ABI available for a given system, and this
-generally gives significantly greater speed. But an ABI can be chosen
-explicitly to make MPIR compatible with other libraries, or particular
-application requirements. For example,
-
- at example
-./configure ABI=32
- at end example
-
-In all cases it's vital that all object code used in a given program is
-compiled for the same ABI.
-
-Usually a limb is implemented as a @code{long}. When a @code{long long} limb
-is used this is encoded in the generated @file{mpir.h}. This is convenient for
-applications, but it does mean that @file{mpir.h} will vary, and can't be just
-copied around. @file{mpir.h} remains compiler independent though, since all
-compilers for a particular ABI will be expected to use the same limb type.
-
-Currently no attempt is made to follow whatever conventions a system has for
-installing library or header files built for a particular ABI at . This will
-probably only matter when installing multiple builds of MPIR, and it might be
-as simple as configuring with a special @samp{libdir}, or it might require
-more than that. Note that builds for different ABIs need to done separately,
-with a fresh (@command{make distclean}), @command{./configure} and @command{make}.
-
- at sp 1
- at table @asis
- at need 1000
- at item AMD64 (@samp{x86_64})
- at cindex AMD64
-On AMD64 systems supporting both 32-bit and 64-bit modes for applications, the
-following ABI choices are available.
-
- at table @asis
- at item @samp{ABI=64}
-The 64-bit ABI uses 64-bit limbs and pointers and makes full use of the chip
-architecture. This is the default. Applications will usually not need
-special compiler flags, but for reference the option is
-
- at example
-gcc -m64
- at end example
-
- at item @samp{ABI=32}
-The 32-bit ABI is the usual i386 conventions. This will be slower, and is not
-recommended except for inter-operating with other code not yet 64-bit capable.
-Applications must be compiled with
-
- at example
-gcc -m32
- at end example
-
-(In GCC 2.95 and earlier there's no @samp{-m32} option, it's the only mode.)
- at end table
-
- at sp 1
- at need 1500
- at item IA-64 under HP-UX (@samp{ia64*-*-hpux*}, @samp{itanium*-*-hpux*})
- at cindex IA-64
- at cindex HP-UX
-HP-UX supports two ABIs for IA-64. MPIR performance is the same in both.
-
- at table @asis
- at item @samp{ABI=32}
-In the 32-bit ABI, pointers, @code{int}s and @code{long}s are 32 bits and MPIR
-uses a 64 bit @code{long long} for a limb. Applications can be compiled
-without any special flags since this ABI is the default in both HP C and GCC,
-but for reference the flags are
-
- at example
-gcc -milp32
-cc +DD32
- at end example
-
- at item @samp{ABI=64}
-In the 64-bit ABI, @code{long}s and pointers are 64 bits and MPIR uses a
- at code{long} for a limb. Applications must be compiled with
-
- at example
-gcc -mlp64
-cc +DD64
- at end example
- at end table
-
-On other IA-64 systems, GNU/Linux for instance, @samp{ABI=64} is the only
-choice.
-
- at sp 1
- at need 1000
- at item PowerPC 64 (@samp{powerpc64}, @samp{powerpc620}, @samp{powerpc630}, @samp{powerpc970})
- at cindex PowerPC
- at table @asis
- at item @samp{ABI=aix64}
- at cindex AIX
-The AIX 64 ABI uses 64-bit limbs and pointers and is the default on PowerPC 64
- at samp{*-*-aix*} systems. Applications must be compiled with
-
- at example
-gcc -maix64
-xlc -q64
- at end example
-
- at item @samp{ABI=mode32}
- at cindex AIX
-The @samp{mode32} ABI uses a 64-bit @code{long long} limb but with the chip
-still in 32-bit mode and using 32-bit calling conventions. This is the
-default on PowerPC 64 @samp{*-*-darwin*} systems. No special compiler options
-are needed for applications.
-
- at item @samp{ABI=32}
-This is the basic 32-bit PowerPC ABI, with a 32-bit limb. No special compiler
-options are needed for applications.
- at end table
-
-MPIR speed is greatest in @samp{aix64} and @samp{mode32}. In @samp{ABI=32}
-only the 32-bit ISA is used and this doesn't make full use of a 64-bit chip.
-On a suitable system we could perhaps use more of the ISA, but there are no
-plans to do so.
-
- at sp 1
- at need 1000
- at item Sparc V9 (@samp{sparc64}, @samp{sparcv9}, @samp{ultrasparc*})
- at cindex Sparc V9
- at cindex Solaris
- at cindex Sun
- at table @asis
- at item @samp{ABI=64}
-The 64-bit V9 ABI is available on the various BSD sparc64 ports, recent
-versions of Sparc64 GNU/Linux, and Solaris 2.7 and up (when the kernel is in
-64-bit mode). GCC 3.2 or higher, or Sun @command{cc} is required. On
-GNU/Linux, depending on the default @command{gcc} mode, applications must be
-compiled with
-
- at example
-gcc -m64
- at end example
-
-On Solaris applications must be compiled with
-
- at example
-gcc -m64 -mptr64 -Wa,-xarch=v9 -mcpu=v9
-cc -xarch=v9
- at end example
-
-On the BSD sparc64 systems no special options are required, since 64-bits is
-the only ABI available.
-
- at item @samp{ABI=32}
-For the basic 32-bit ABI, MPIR still uses as much of the V9 ISA as it can. In
-the Sun documentation this combination is known as ``v8plus''. On GNU/Linux,
-depending on the default @command{gcc} mode, applications may need to be
-compiled with
-
- at example
-gcc -m32
- at end example
-
-On Solaris, no special compiler options are required for applications, though
-using something like the following is recommended. (@command{gcc} 2.8 and
-earlier only support @samp{-mv8} though.)
-
- at example
-gcc -mv8plus
-cc -xarch=v8plus
- at end example
- at end table
-
-MPIR speed is greatest in @samp{ABI=64}, so it's the default where available.
-The speed is partly because there are extra registers available and partly
-because 64-bits is considered the more important case and has therefore had
-better code written for it.
-
-Don't be confused by the names of the @samp{-m} and @samp{-x} compiler
-options, they're called @samp{arch} but effectively control both ABI and ISA at .
-
-On Solaris 2.6 and earlier, only @samp{ABI=32} is available since the kernel
-doesn't save all registers.
-
-On Solaris 2.7 with the kernel in 32-bit mode, a normal native build will
-reject @samp{ABI=64} because the resulting executables won't run.
- at samp{ABI=64} can still be built if desired by making it look like a
-cross-compile, for example
-
- at example
-./configure --build=none --host=sparcv9-sun-solaris2.7 ABI=64
- at end example
- at end table
-
-
- at need 2000
- at node Notes for Package Builds, Notes for Particular Systems, ABI and ISA, Installing MPIR
- at section Notes for Package Builds
- at cindex Build notes for binary packaging
- at cindex Packaged builds
-
-MPIR should present no great difficulties for packaging in a binary
-distribution.
-
- at cindex Libtool versioning
- at cindex Shared library versioning
-Libtool is used to build the library and @samp{-version-info} is set
-appropriately, having started from @samp{3:0:0} in GMP 3.0 (@pxref{Versioning,
-Library interface versions, Library interface versions, libtool, GNU
-Libtool}).
-
-The GMP 4 series and MPIR 1 series will be upwardly binary compatible in each
-release and will be upwardly binary compatible with all of the GMP 3 series.
-Additional function interfaces may be added in each release, so on systems where
-libtool versioning is not fully checked by the loader an auxiliary mechanism may be
-needed to express that a dynamic linked application depends on a new enough
-MPIR.
-
-From MPIR 2.0.0 binary compatibility with the GMP 5 series will be maintained
-with the exception of the availability of secure functions for cryptography,
-which will not be supported in MPIR. For full GMP compatibility, including
-deprecated functionality, the @samp{--enable-gmpcompat} configuration option
-must be used.
-
-An auxiliary mechanism may also be needed to express that @file{libmpirxx.la}
-(from @option{--enable-cxx}, @pxref{Build Options}) requires @file{libmpir.la}
-from the same MPIR version, since this is not done by the libtool versioning,
-nor otherwise. A mismatch will result in unresolved symbols from the linker,
-or perhaps the loader.
-
-When building a package for a CPU family, care should be taken to use
- at samp{--host} (or @samp{--build}) to choose the least common denominator among
-the CPUs which might use the package. For example this might mean plain
- at samp{sparc} (meaning V7) for SPARCs.
-
-For x86s, @option{--enable-fat} sets things up for a fat binary build, making a
-runtime selection of optimized low level routines. This is a good choice for
-packaging to run on a range of x86 chips.
-
-Users who care about speed will want MPIR built for their exact CPU type, to
-make best use of the available optimizations. Providing a way to suitably
-rebuild a package may be useful. This could be as simple as making it
-possible for a user to omit @samp{--build} (and @samp{--host}) so
- at samp{./config.guess} will detect the CPU at . But a way to manually specify a
- at samp{--build} will be wanted for systems where @samp{./config.guess} is
-inexact.
-
-On systems with multiple ABIs, a packaged build will need to decide which
-among the choices is to be provided, see @ref{ABI and ISA}. A given run of
- at samp{./configure} etc will only build one ABI at . If a second ABI is also
-required then a second run of @samp{./configure} etc must be made, starting
-from a clean directory tree (@samp{make distclean}).
-
-As noted under ``ABI and ISA'', currently no attempt is made to follow system
-conventions for install locations that vary with ABI, such as
- at file{/usr/lib/sparcv9} for @samp{ABI=64} as opposed to @file{/usr/lib} for
- at samp{ABI=32}. A package build can override @samp{libdir} and other standard
-variables as necessary.
-
-Note that @file{mpir.h} is a generated file, and will be architecture and ABI
-dependent. When attempting to install two ABIs simultaneously it will be
-important that an application compile gets the correct @file{mpir.h} for its
-desired ABI at . If compiler include paths don't vary with ABI options then it
-might be necessary to create a @file{/usr/include/mpir.h} which tests
-preprocessor symbols and chooses the correct actual @file{mpir.h}.
-
-
- at need 2000
- at node Notes for Particular Systems, Known Build Problems, Notes for Package Builds, Installing MPIR
- at section Notes for Particular Systems
- at cindex Build notes for particular systems
- at cindex Particular systems
- at cindex Systems
- at table @asis
-
- at c This section is more or less meant for notes about performance or about
- at c build problems that have been worked around but might leave a user
- at c scratching their head. Fun with different ABIs on a system belongs in the
- at c above section.
-
- at item AIX 3 and 4
- at cindex AIX
-On systems @samp{*-*-aix[34]*} shared libraries are disabled by default, since
-some versions of the native @command{ar} fail on the convenience libraries
-used. A shared build can be attempted with
-
- at example
-./configure --enable-shared --disable-static
- at end example
-
-Note that the @samp{--disable-static} is necessary because in a shared build
-libtool makes @file{libmpir.a} a symlink to @file{libmpir.so}, apparently for
-the benefit of old versions of @command{ld} which only recognise @file{.a},
-but unfortunately this is done even if a fully functional @command{ld} is
-available.
-
- at item ARM
- at cindex ARM
-On systems @samp{arm*-*-*}, versions of GCC up to and including 2.95.3 have a
-bug in unsigned division, giving wrong results for some operands. MPIR
- at samp{./configure} will demand GCC 2.95.4 or later.
-
- at item Floating Point Mode
- at cindex Floating point mode
- at cindex Hardware floating point mode
- at cindex Precision of hardware floating point
- at cindex x87
-On some systems, the hardware floating point has a control mode which can set
-all operations to be done in a particular precision, for instance single,
-double or extended on x86 systems (x87 floating point). The MPIR functions
-involving a @code{double} cannot be expected to operate to their full
-precision when the hardware is in single precision mode. Of course this
-affects all code, including application code, not just MPIR.
-
- at item MS-DOS and MS Windows
- at cindex MS-DOS
- at cindex MS Windows
- at cindex Windows
- at cindex Cygwin
- at cindex MINGW
-On an MS Windows system Cygwin and MINGW can be used , they
-are ports of GCC and the various GNU tools.
-
- at display
- at uref{http://www.cygwin.com/}
- at uref{http://www.mingw.org/}
- at end display
-
-Cygwin is a 32 bit build only but mingw is 32 or 64 bit build. Depending on how
-the mingw tools are installed will determine the best procedure for building ,
-because of the large number of ways this can be achieved it is best to search
-the MPIR devel mailing list or the mingw mailing list.
-
-For building with MSVC we provide a number of ways.
-
-In addition, project files for MSVC are provided, allowing MPIR to
-build on Microsoft's compiler. For Visual Studio 2010 see the readme.txt
-file in the build.vc10
-directory. The MSVC projects provides full assembler support and for
- at samp{x86_64} CPU's this will produce far superior results. These project
-files can also be accessed via the command line with the batch files
- at samp{configure.bat} and @samp{make.bat} which have a @samp{unix like}
-interface , however they are not very well tested and are due to be replaced.
-
-An another alternative is @samp{configure} and @samp{make} in the @samp{win}
-directory , these again have a @samp{unix like} syntax , these are tested
-regularly and also have the advantage of working with VS2005 and up (including
-the free/express versions). There is some auto detection of the compiler , but it's
-probably best to set it explicity using the usual
- at samp{call "%VS90COMNTOOLS%\..\..\VC\vcvarsall.bat" amd64} in the command window.
-The program @samp{YASM} is also required and should be in path or the @samp{%YASMPATH%}
-varible set.
-If @samp{configure} guesses wrong , close the window and try again changing the @samp{ABI=...}
-selection and or the @samp{vcvarsall.bat} options.
- at samp{make} supports the usual @samp{clean} and @samp{check} options .
-The @samp{only} bug is that shared library builds @samp{dll's} fail the make check
-in the C++ parts for @samp{istream} and @samp{ostream} with some unresolved
-symbols.
-
-
- at item MS Windows DLLs
- at cindex DLLs
- at cindex MS Windows
- at cindex Windows
-On systems @samp{*-*-cygwin*} and @samp{*-*-mingw*} by
-default MPIR builds only a static library, but a DLL can be built instead using
-
- at example
-./configure --disable-static --enable-shared
- at end example
-
-Static and DLL libraries can't both be built, since certain export directives
-in @file{mpir.h} must be different.
-
-Libtool doesn't install a @file{.lib} format import library, but it can be
-created with MS @command{lib} as follows, and copied to the install directory.
-Similarly for @file{libmpir} and @file{libmpirxx}.
-
- at example
-cd .libs
-lib /def:libgmp-3.dll.def /out:libgmp-3.lib
- at end example
-
-MINGW uses the C runtime library @samp{msvcrt.dll} for I/O, so applications
-wanting to use the MPIR I/O routines must be compiled with @samp{cl /MD} to do
-the same. If one of the other C runtime library choices provided by MS C is
-desired then the suggestion is to use the MPIR string functions and confine I/O
-to the application.
-
- at item OpenBSD 2.6
- at cindex OpenBSD
- at command{m4} in this release of OpenBSD has a bug in @code{eval} that makes it
-unsuitable for @file{.asm} file processing. @samp{./configure} will detect
-the problem and either abort or choose another m4 in the @env{PATH}. The bug
-is fixed in OpenBSD 2.7, so either upgrade or use GNU m4.
-
- at item Sparc CPU Types
- at cindex Sparc
- at samp{sparcv8} or @samp{supersparc} on relevant systems will give a
-significant performance increase over the V7 code selected by plain
- at samp{sparc}.
-
- at item Sparc App Regs
- at cindex Sparc
-The MPIR assembler code for both 32-bit and 64-bit Sparc clobbers the
-``application registers'' @code{g2}, @code{g3} and @code{g4}, the same way
-that the GCC default @samp{-mapp-regs} does (@pxref{SPARC Options,, SPARC
-Options, gcc, Using the GNU Compiler Collection (GCC)}).
-
-This makes that code unsuitable for use with the special V9
- at samp{-mcmodel=embmedany} (which uses @code{g4} as a data segment pointer),
-and for applications wanting to use those registers for special purposes. In
-these cases the only suggestion currently is to build MPIR with CPU @samp{none}
-to avoid the assembler code.
-
- at item SPARC Solaris
- at cindex Sparc
-Building applications against MPIR on SPARC Solaris (including @code{make
-check}) requires the @code{LD_LIBRARY_PATH} to be set appropriately. In
-particular if one is building with @code{ABI=64} the linker needs to know
-where to find @code{libgcc} (often often @code{/usr/lib/sparcv9}
-or @code{/usr/local/lib/sparcv9} or @code{/lib/sparcv9}).
-
-It is not enough to specify the location in @code{LD_LIBRARY_PATH_64} unless
- at code{LD_LIBRARY_PATH_64} is added to @code{LD_LIBRARY_PATH}. Specifically
-the 64 bit @code{libgcc} path needs to be in @code{LD_LIBRARY_PATH}.
-
-The linker is able to automatically distinguish 32 and 64 bit libraries,
-so it is safe to include paths to both the 32 and 64 bit libraries in the
- at code{LD_LIBRARY_PATH}.
-
- at item Solaris 10 First Release on SPARC
- at cindex Solaris
-MPIR fails to build with Solaris 10 first release. Patch 123647-01 for SPARC, released by Sun in August 2006 fixes this problem.
-
- at item x86 CPU Types
- at cindex x86
- at cindex 80x86
- at cindex i386
- at samp{i586}, @samp{pentium} or @samp{pentiummmx} code is good for its intended
-P5 Pentium chips, but quite slow when run on Intel P6 class chips (PPro, P-II,
-P-III)@. @samp{i386} is a better choice when making binaries that must run on
-both.
-
- at item x86 MMX and SSE2 Code
- at cindex MMX
- at cindex SSE2
-If the CPU selected has MMX code but the assembler doesn't support it, a
-warning is given and non-MMX code is used instead. This will be an inferior
-build, since the MMX code that's present is there because it's faster than the
-corresponding plain integer code. The same applies to SSE2.
-
-Old versions of @samp{gas} don't support MMX instructions, in particular
-version 1.92.3 that comes with FreeBSD 2.2.8 or the more recent OpenBSD 3.1
-doesn't.
-
-Solaris 2.6 and 2.7 @command{as} generate incorrect object code for register
-to register @code{movq} instructions, and so can't be used for MMX code.
-Install a recent @command{gas} if MMX code is wanted on these systems.
- at end table
-
-
- at need 2000
- at node Known Build Problems, Performance optimization, Notes for Particular Systems, Installing MPIR
- at section Known Build Problems
- at cindex Build problems known
-
- at c This section is more or less meant for known build problems that are not
- at c otherwise worked around and require some sort of manual intervention.
-
-You might find more up-to-date information at @uref{http://www.mpir.org/}.
-
- at table @asis
- at item Compiler link options
-The version of libtool currently in use rather aggressively strips compiler
-options when linking a shared library. This will hopefully be relaxed in the
-future, but for now if this is a problem the suggestion is to create a little
-script to hide them, and for instance configure with
-
- at example
-./configure CC=gcc-with-my-options
- at end example
-
- at samp{make all} was found to run out of memory during the final
- at file{libgmp.la} link on one system tested, despite having 64Mb available.
-Running @samp{make libgmp.la} directly helped, perhaps recursing into the
-various subdirectories uses up memory.
-
- at item MacOS X (@samp{*-*-darwin*})
- at cindex MacOS X
- at cindex Darwin
-Libtool currently only knows how to create shared libraries on MacOS X using
-the native @command{cc} (which is a modified GCC), not a plain GCC at . A
-static-only build should work though (@samp{--disable-shared}).
-
- at item Solaris 2.6
- at cindex Solaris
-The system @command{sed} prints an error ``Output line too long'' when libtool
-builds @file{libmpir.la}. This doesn't seem to cause any obvious ill effects,
-but GNU @command{sed} is recommended, to avoid any doubt.
-
- at item Sparc Solaris 2.7 with gcc 2.95.2 in @samp{ABI=32}
- at cindex Solaris
-A shared library build of MPIR seems to fail in this combination, it builds but
-then fails the tests, apparently due to some incorrect data relocations within
- at code{gmp_randinit_lc_2exp_size}. The exact cause is unknown,
- at samp{--disable-shared} is recommended.
- at end table
-
-
- at need 2000
- at node Performance optimization, , Known Build Problems, Installing MPIR
- at section Performance optimization
- at cindex Optimizing performance
-
- at c At some point, this should perhaps move to a separate chapter on optimizing
- at c performance.
-
-For optimal performance, build MPIR for the exact CPU type of the target
-computer, see @ref{Build Options}.
-
-Unlike what is the case for most other programs, the compiler typically
-doesn't matter much, since MPIR uses assembly language for the most critical
-operations.
-
-In particular for long-running MPIR applications, and applications demanding
-extremely large numbers, building and running the @code{tuneup} program in the
- at file{tune} subdirectory, can be important. For example,
-
- at example
-cd tune
-make tuneup
-./tuneup
- at end example
-
-will generate better contents for the @file{gmp-mparam.h} parameter file.
-
-To use the results, put the output in the file indicated in the
- at samp{Parameters for ...} header. Then recompile from scratch.
-
-The @code{tuneup} program takes one useful parameter, @samp{-f NNN}, which
-instructs the program how long to check FFT multiply parameters. If you're
-going to use MPIR for extremely large numbers, you may want to run @code{tuneup}
-with a large NNN value.
-
-
- at node MPIR Basics, Reporting Bugs, Installing MPIR, Top
- at comment node-name, next, previous, up
- at chapter MPIR Basics
- at cindex Basics
-
- at strong{Using functions, macros, data types, etc.@: not documented in this
-manual is strongly discouraged. If you do so your application is guaranteed
-to be incompatible with future versions of MPIR.}
-
- at menu
-* Headers and Libraries::
-* Nomenclature and Types::
-* MPIR on Windows x64::
-* Function Classes::
-* Variable Conventions::
-* Parameter Conventions::
-* Memory Management::
-* Reentrancy::
-* Useful Macros and Constants::
-* Compatibility with older versions::
-* Efficiency::
-* Debugging::
-* Profiling::
-* Autoconf::
-* Emacs::
- at end menu
-
- at node Headers and Libraries, Nomenclature and Types, MPIR Basics, MPIR Basics
- at section Headers and Libraries
- at cindex Headers
-
- at cindex @file{mpir.h}
- at cindex Include files
- at cindex @code{#include}
-All declarations needed to use MPIR are collected in the include file
- at file{mpir.h}. It is designed to work with both C and C++ compilers.
-
- at example
-#include <mpir.h>
- at end example
-
- at cindex @code{stdio.h}
-Note however that prototypes for MPIR functions with @code{FILE *} parameters
-are only provided if @code{<stdio.h>} is included too.
-
- at example
-#include <stdio.h>
-#include <mpir.h>
- at end example
-
- at cindex @code{stdarg.h}
-Likewise @code{<stdarg.h>} (or @code{<varargs.h>}) is required for prototypes
-with @code{va_list} parameters, such as @code{gmp_vprintf}. And
- at code{<obstack.h>} for prototypes with @code{struct obstack} parameters, such
-as @code{gmp_obstack_printf}, when available.
-
- at cindex Libraries
- at cindex Linking
- at cindex @code{libmpir}
-All programs using MPIR must link against the @file{libmpir} library. On a
-typical Unix-like system this can be done with @samp{-lmpir} respectively, for example
-
- at example
-gcc myprogram.c -lmpir
- at end example
-
- at cindex @code{libmpirxx}
-MPIR C++ functions are in a separate @file{libmpirxx} library. This is built
-and installed if C++ support has been enabled (@pxref{Build Options}). For
-example,
-
- at example
-g++ mycxxprog.cc -lmpirxx -lmpir
- at end example
-
- at cindex Libtool
-MPIR is built using Libtool and an application can use that to link if desired,
- at GMPpxreftop{libtool, GNU Libtool}
-
-If MPIR has been installed to a non-standard location then it may be necessary
-to use @samp{-I} and @samp{-L} compiler options to point to the right
-directories, and some sort of run-time path for a shared library.
-
-
- at node Nomenclature and Types, MPIR on Windows x64, Headers and Libraries, MPIR Basics
- at section Nomenclature and Types
- at cindex Nomenclature
- at cindex Types
-
- at cindex Integer
- at tindex @code{mpz_t}
-In this manual, @dfn{integer} usually means a multiple precision integer, as
-defined by the MPIR library. The C data type for such integers is @code{mpz_t}.
-Here are some examples of how to declare such integers:
-
- at example
-mpz_t sum;
-
-struct foo @{ mpz_t x, y; @};
-
-mpz_t vec[20];
- at end example
-
- at cindex Rational number
- at tindex @code{mpq_t}
- at dfn{Rational number} means a multiple precision fraction. The C data type
-for these fractions is @code{mpq_t}. For example:
-
- at example
-mpq_t quotient;
- at end example
-
- at cindex Floating-point number
- at tindex @code{mpf_t}
- at dfn{Floating point number} or @dfn{Float} for short, is an arbitrary precision
-mantissa with a limited precision exponent. The C data type for such objects
-is @code{mpf_t}. For example:
-
- at example
-mpf_t fp;
- at end example
-
- at tindex @code{mp_exp_t}
-The floating point functions accept and return exponents in the C type
- at code{mp_exp_t}. Currently this is usually a @code{long}, but on some systems
-it's an @code{int} for efficiency.
-
- at cindex Limb
- at tindex @code{mp_limb_t}
-A @dfn{limb} means the part of a multi-precision number that fits in a single
-machine word. (We chose this word because a limb of the human body is
-analogous to a digit, only larger, and containing several digits.) Normally a
-limb is 32 or 64 bits. The C data type for a limb is @code{mp_limb_t}.
-
- at tindex @code{mp_size_t}
-Counts of limbs are represented in the C type @code{mp_size_t}. Currently
-this is normally a @code{long}, but on some systems it's an @code{int} for
-efficiency.
-
- at tindex @code{mp_bitcnt_t}
-Counts of bits of a multi-precision number are represented in the C type
- at code{mp_bitcnt_t}. Currently this is always an @code{unsigned long}, but on
-some systems it will be an @code{unsigned long long} in the future .
-
- at cindex Random state
- at tindex @code{gmp_randstate_t}
- at dfn{Random state} means an algorithm selection and current state data. The C
-data type for such objects is @code{gmp_randstate_t}. For example:
-
- at example
-gmp_randstate_t rstate;
- at end example
-
-Also, in general @code{mp_bitcnt_t} is used for bit counts and ranges, and
- at code{size_t} is used for byte or character counts.
-
- at node MPIR on Windows x64, Function Classes, Nomenclature and Types, MPIR Basics
- at section MPIR on Windows x64
- at cindex Windows
-
-Although Windows x64 is a 64-bit operating system, Microsoft has decided to
-make long integers 32-bits, which is inconsistent when compared with almost
-all other 64-bit operating systems. This has caused many subtle bugs when
-open source code is ported to Windows x64 because many developers reasonably
-expect to find that long integers on a 64-bit operating system will be 64
-bits long.
-
-MPIR contains functions with suffixes of @code{_ui} and @code{_si} that are
-used to input unsigned and signed integers into and convert them for use
-with MPIR's multiple precision integers (mpz types). For example, the
-following functions set an @code{mpz_t} integer from @code{unsigned} and
- at code{signed long} integers respectively.
-
- at deftypefun void mpz_set_ui (mpz_t, unsigned long int)
- at end deftypefun
-
- at deftypefun void mpz_set_si (mpz_t, signed long int)
- at end deftypefun
-
- at sp 1
-Also, the following functions obtain @code{unsigned} and
- at code{signed long int} values from an MPIR multiple precision integer
-(@code{mpz_t}).
-
- at deftypefun unsigned long int mpz_get_ui (mpz_t)
- at end deftypefun
-
- at deftypefun signed long int mpz_get_si (mpz_t)
- at end deftypefun
-
- at sp 1
-To bring MPIR on Windows x64 into line with other 64-bit operating systems
-two new types have been introduced throughout MPIR:
-
- at itemize @bullet
- at item @code{mpir_ui} defined as @code{unsigned long int} on all but Windows x64, defined as @code{unsigned long long int} on Windows x64
-
- at item @code{mpir_si} defined as @code{signed long int} on all but Windows x64, defined as @code{signed long long int} on Windows x64
- at end itemize
-
- at sp 1
-The above prototypes in MPIR 2.6.0 are changed to:
-
- at deftypefun void mpz_set_ui (mpz_t, mpir_ui)
- at end deftypefun
-
- at deftypefun void mpz_set_si (mpz_t, mpir_si)
- at end deftypefun
-
- at deftypefun mpir_ui mpz_get_ui (mpz_t)
- at end deftypefun
-
- at deftypefun mpir_si mpz_get_si (mpz_t)
- at end deftypefun
-
- at sp 1
-These changes are applied to all MPIR functions with @code{_ui} and
- at code{_si} suffixes.
-
- at node Function Classes, Variable Conventions, MPIR on Windows x64, MPIR Basics
- at section Function Classes
- at cindex Function classes
-
-There are five classes of functions in the MPIR library:
-
- at enumerate
- at item
-Functions for signed integer arithmetic, with names beginning with
- at code{mpz_}. The associated type is @code{mpz_t}. There are about 150
-functions in this class. (@pxref{Integer Functions})
-
- at item
-Functions for rational number arithmetic, with names beginning with
- at code{mpq_}. The associated type is @code{mpq_t}. There are about 40
-functions in this class, but the integer functions can be used for arithmetic
-on the numerator and denominator separately. (@pxref{Rational Number
-Functions})
-
- at item
-Functions for floating-point arithmetic, with names beginning with
- at code{mpf_}. The associated type is @code{mpf_t}. There are about 60
-functions is this class. (@pxref{Floating-point Functions})
-
- at item
-Fast low-level functions that operate on natural numbers. These are used by
-the functions in the preceding groups, and you can also call them directly
-from very time-critical user programs. These functions' names begin with
- at code{mpn_}. The associated type is array of @code{mp_limb_t}. There are
-about 30 (hard-to-use) functions in this class. (@pxref{Low-level Functions})
-
- at item
-Miscellaneous functions. Functions for setting up custom allocation and
-functions for generating random numbers. (@pxref{Custom Allocation}, and
- at pxref{Random Number Functions})
- at end enumerate
-
-
- at node Variable Conventions, Parameter Conventions, Function Classes, MPIR Basics
- at section Variable Conventions
- at cindex Variable conventions
- at cindex Conventions for variables
-
-MPIR functions generally have output arguments before input arguments. This
-notation is by analogy with the assignment operator.
-
-MPIR lets you use the same variable for both input and output in one call. For
-example, the main function for integer multiplication, @code{mpz_mul}, can be
-used to square @code{x} and put the result back in @code{x} with
-
- at example
-mpz_mul (x, x, x);
- at end example
-
-Before you can assign to an MPIR variable, you need to initialize it by calling
-one of the special initialization functions. When you're done with a
-variable, you need to clear it out, using one of the functions for that
-purpose. Which function to use depends on the type of variable. See the
-chapters on integer functions, rational number functions, and floating-point
-functions for details.
-
-A variable should only be initialized once, or at least cleared between each
-initialization. After a variable has been initialized, it may be assigned to
-any number of times.
-
-For efficiency reasons, avoid excessive initializing and clearing. In
-general, initialize near the start of a function and clear near the end. For
-example,
-
- at example
-void
-foo (void)
-@{
- mpz_t n;
- int i;
- mpz_init (n);
- for (i = 1; i < 100; i++)
- @{
- mpz_mul (n, @dots{});
- mpz_fdiv_q (n, @dots{});
- @dots{}
- @}
- mpz_clear (n);
-@}
- at end example
-
-
- at node Parameter Conventions, Memory Management, Variable Conventions, MPIR Basics
- at section Parameter Conventions
- at cindex Parameter conventions
- at cindex Conventions for parameters
-
-When an MPIR variable is used as a function parameter, it's effectively a
-call-by-reference, meaning if the function stores a value there it will change
-the original in the caller. Parameters which are input-only can be designated
- at code{const} to provoke a compiler error or warning on attempting to modify
-them.
-
-When a function is going to return an MPIR result, it should designate a
-parameter that it sets, like the library functions do. More than one value
-can be returned by having more than one output parameter, again like the
-library functions. A @code{return} of an @code{mpz_t} etc doesn't return the
-object, only a pointer, and this is almost certainly not what's wanted.
-
-Here's an example accepting an @code{mpz_t} parameter, doing a calculation,
-and storing the result to the indicated parameter.
-
- at example
-void
-foo (mpz_t result, const mpz_t param, mpir_ui n)
-@{
- mpir_ui i;
- mpz_mul_ui (result, param, n);
- for (i = 1; i < n; i++)
- mpz_add_ui (result, result, i*7);
-@}
-
-int
-main (void)
-@{
- mpz_t r, n;
- mpz_init (r);
- mpz_init_set_str (n, "123456", 0);
- foo (r, n, 20L);
- gmp_printf ("%Zd\n", r);
- return 0;
-@}
- at end example
-
- at code{foo} works even if the mainline passes the same variable for
- at code{param} and @code{result}, just like the library functions. But
-sometimes it's tricky to make that work, and an application might not want to
-bother supporting that sort of thing.
-
-For interest, the MPIR types @code{mpz_t} etc are implemented as one-element
-arrays of certain structures. This is why declaring a variable creates an
-object with the fields MPIR needs, but then using it as a parameter passes a
-pointer to the object. Note that the actual fields in each @code{mpz_t} etc
-are for internal use only and should not be accessed directly by code that
-expects to be compatible with future MPIR releases.
-
-
- at need 1000
- at node Memory Management, Reentrancy, Parameter Conventions, MPIR Basics
- at section Memory Management
- at cindex Memory management
-
-The MPIR types like @code{mpz_t} are small, containing only a couple of sizes,
-and pointers to allocated data. Once a variable is initialized, MPIR takes
-care of all space allocation. Additional space is allocated whenever a
-variable doesn't have enough.
-
- at code{mpz_t} and @code{mpq_t} variables never reduce their allocated space.
-Normally this is the best policy, since it avoids frequent reallocation.
-Applications that need to return memory to the heap at some particular point
-can use @code{mpz_realloc2}, or clear variables no longer needed.
-
- at code{mpf_t} variables, in the current implementation, use a fixed amount of
-space, determined by the chosen precision and allocated at initialization, so
-their size doesn't change.
-
-All memory is allocated using @code{malloc} and friends by default, but this
-can be changed, see @ref{Custom Allocation}. Temporary memory on the stack is
-also used (via @code{alloca}), but this can be changed at build-time if
-desired, see @ref{Build Options}.
-
-
- at node Reentrancy, Useful Macros and Constants, Memory Management, MPIR Basics
- at section Reentrancy
- at cindex Reentrancy
- at cindex Thread safety
- at cindex Multi-threading
-
- at noindent
-MPIR is reentrant and thread-safe, with some exceptions:
-
- at itemize @bullet
- at item
-If configured with @option{--enable-alloca=malloc-notreentrant} (or with
- at option{--enable-alloca=notreentrant} when @code{alloca} is not available),
-then naturally MPIR is not reentrant.
-
- at item
- at code{mpf_set_default_prec} and @code{mpf_init} use a global variable for the
-selected precision. @code{mpf_init2} can be used instead, and in the C++
-interface an explicit precision to the @code{mpf_class} constructor.
-
- at item
- at code{mp_set_memory_functions} uses global variables to store the selected
-memory allocation functions.
-
- at item
-If the memory allocation functions set by a call to
- at code{mp_set_memory_functions} (or @code{malloc} and friends by default) are
-not reentrant, then MPIR will not be reentrant either.
-
- at item
-If the standard I/O functions such as @code{fwrite} are not reentrant then the
-MPIR I/O functions using them will not be reentrant either.
-
- at item
-It's safe for two threads to read from the same MPIR variable simultaneously,
-but it's not safe for one to read while the another might be writing, nor for
-two threads to write simultaneously. It's not safe for two threads to
-generate a random number from the same @code{gmp_randstate_t} simultaneously,
-since this involves an update of that variable.
- at end itemize
-
-
- at need 2000
- at node Useful Macros and Constants, Compatibility with older versions, Reentrancy, MPIR Basics
- at section Useful Macros and Constants
- at cindex Useful macros and constants
- at cindex Constants
-
- at deftypevr {Global Constant} {const int} mp_bits_per_limb
- at findex mp_bits_per_limb
- at cindex Bits per limb
- at cindex Limb size
-The number of bits per limb.
- at end deftypevr
-
- at defmac __GNU_MP_VERSION
- at defmacx __GNU_MP_VERSION_MINOR
- at defmacx __GNU_MP_VERSION_PATCHLEVEL
- at cindex Version number
- at cindex MPIR version number
-The major and minor GMP version, and patch level, respectively, as integers.
-For GMP i.j.k, these numbers will be i, j, and k, respectively.
-These numbers represent the version of GMP fully supported by this version of MPIR.
- at end defmac
-
- at defmac __MPIR_VERSION
- at defmacx __MPIR_VERSION_MINOR
- at defmacx __MPIR_VERSION_PATCHLEVEL
- at cindex Version number
- at cindex MPIR version number
-The major and minor MPIR version, and patch level, respectively, as integers.
-For MPIR i.j.k, these numbers will be i, j, and k, respectively.
- at end defmac
-
- at deftypevr {Global Constant} {const char * const} gmp_version
- at findex gmp_version
-The GNU MP version number, as a null-terminated string, in the form
-``i.j.k''.
- at end deftypevr
-
- at deftypevr {Global Constant} {const char * const} mpir_version
- at findex mpir_version
-The MPIR version number, as a null-terminated string, in the form
-``i.j.k''. This release is @nicode{"@value{VERSION}"}.
- at end deftypevr
-
- at node Compatibility with older versions, Efficiency, Useful Macros and Constants, MPIR Basics
- at section Compatibility with older versions
- at cindex Compatibility with older versions
- at cindex Past GMP/MPIR versions
- at cindex Upward compatibility
-
-This version of MPIR is upwardly binary compatible with all GMP 5.x, 4.x and 3.x
-versions, and upwardly compatible at the source level with all 2.x versions,
-with the following exceptions.
-
- at itemize @bullet
- at item
- at code{mpn_gcd} had its source arguments swapped as of GMP 3.0, for consistency
-with other @code{mpn} functions.
-
- at item
- at code{mpf_get_prec} counted precision slightly differently in GMP 3.0 and
-3.0.1, but in 3.1 reverted to the 2.x style.
-
- at item
-MPIR does not support the secure cryptographic functions provided by GMP.
-
- at item
-Full GMP compatibility is only available when the @samp{--enable-gmpcompat}
-configure option is used.
- at end itemize
-
-There are a number of compatibility issues between GMP 1 and GMP 2 that of
-course also apply when porting applications from GMP 1 to GMP 4 and MPIR 1
-and 2. Please see the GMP 2 manual for details.
-
- at need 1000
- at node Efficiency, Debugging, Compatibility with older versions, MPIR Basics
- at section Efficiency
- at cindex Efficiency
-
- at table @asis
- at item Small Operands
- at cindex Small operands
-On small operands, the time for function call overheads and memory allocation
-can be significant in comparison to actual calculation. This is unavoidable
-in a general purpose variable precision library, although MPIR attempts to be
-as efficient as it can on both large and small operands.
-
- at item Static Linking
- at cindex Static linking
-On some CPUs, in particular the x86s, the static @file{libmpir.a} should be
-used for maximum speed, since the PIC code in the shared @file{libmpir.so} will
-have a small overhead on each function call and global data address. For many
-programs this will be insignificant, but for long calculations there's a gain
-to be had.
-
- at item Initializing and Clearing
- at cindex Initializing and clearing
-Avoid excessive initializing and clearing of variables, since this can be
-quite time consuming, especially in comparison to otherwise fast operations
-like addition.
-
-A language interpreter might want to keep a free list or stack of
-initialized variables ready for use. It should be possible to integrate
-something like that with a garbage collector too.
-
- at item Reallocations
- at cindex Reallocations
-An @code{mpz_t} or @code{mpq_t} variable used to hold successively increasing
-values will have its memory repeatedly @code{realloc}ed, which could be quite
-slow or could fragment memory, depending on the C library. If an application
-can estimate the final size then @code{mpz_init2} or @code{mpz_realloc2} can
-be called to allocate the necessary space from the beginning
-(@pxref{Initializing Integers}).
-
-It doesn't matter if a size set with @code{mpz_init2} or @code{mpz_realloc2}
-is too small, since all functions will do a further reallocation if necessary.
-Badly overestimating memory required will waste space though.
-
- at item @code{2exp} Functions
- at cindex @code{2exp} functions
-It's up to an application to call functions like @code{mpz_mul_2exp} when
-appropriate. General purpose functions like @code{mpz_mul} make no attempt to
-identify powers of two or other special forms, because such inputs will
-usually be very rare and testing every time would be wasteful.
-
- at item @code{ui} and @code{si} Functions
- at cindex @code{ui} and @code{si} functions
-The @code{ui} functions and the small number of @code{si} functions exist for
-convenience and should be used where applicable. But if for example an
- at code{mpz_t} contains a value that fits in an @code{unsigned long}
-(@code{unsigned long long} on Windows x64) there's no need extract it and
-call a @code{ui} function, just use the regular @code{mpz} function.
-
- at item In-Place Operations
- at cindex In-place operations
- at code{mpz_abs}, @code{mpq_abs}, @code{mpf_abs}, @code{mpz_neg}, @code{mpq_neg}
-and @code{mpf_neg} are fast when used for in-place operations like
- at code{mpz_abs(x,x)}, since in the current implementation only a single field
-of @code{x} needs changing. On suitable compilers (GCC for instance) this is
-inlined too.
-
- at code{mpz_add_ui}, @code{mpz_sub_ui}, @code{mpf_add_ui} and @code{mpf_sub_ui}
-benefit from an in-place operation like @code{mpz_add_ui(x,x,y)}, since
-usually only one or two limbs of @code{x} will need to be changed. The same
-applies to the full precision @code{mpz_add} etc if @code{y} is small. If
- at code{y} is big then cache locality may be helped, but that's all.
-
- at code{mpz_mul} is currently the opposite, a separate destination is slightly
-better. A call like @code{mpz_mul(x,x,y)} will, unless @code{y} is only one
-limb, make a temporary copy of @code{x} before forming the result. Normally
-that copying will only be a tiny fraction of the time for the multiply, so
-this is not a particularly important consideration.
-
- at code{mpz_set}, @code{mpq_set}, @code{mpq_set_num}, @code{mpf_set}, etc, make
-no attempt to recognise a copy of something to itself, so a call like
- at code{mpz_set(x,x)} will be wasteful. Naturally that would never be written
-deliberately, but if it might arise from two pointers to the same object then
-a test to avoid it might be desirable.
-
- at example
-if (x != y)
- mpz_set (x, y);
- at end example
-
-Note that it's never worth introducing extra @code{mpz_set} calls just to get
-in-place operations. If a result should go to a particular variable then just
-direct it there and let MPIR take care of data movement.
-
- at item Divisibility Testing (Small Integers)
- at cindex Divisibility testing
- at code{mpz_divisible_ui_p} and @code{mpz_congruent_ui_p} are the best functions
-for testing whether an @code{mpz_t} is divisible by an individual small
-integer. They use an algorithm which is faster than @code{mpz_tdiv_ui}, but
-which gives no useful information about the actual remainder, only whether
-it's zero (or a particular value).
-
-However when testing divisibility by several small integers, it's best to take
-a remainder modulo their product, to save multi-precision operations. For
-instance to test whether a number is divisible by any of 23, 29 or 31 take a
-remainder modulo @math{23 at times{}29 at times{}31 = 20677} and then test that.
-
-The division functions like @code{mpz_tdiv_q_ui} which give a quotient as well
-as a remainder are generally a little slower than the remainder-only functions
-like @code{mpz_tdiv_ui}. If the quotient is only rarely wanted then it's
-probably best to just take a remainder and then go back and calculate the
-quotient if and when it's wanted (@code{mpz_divexact_ui} can be used if the
-remainder is zero).
-
- at item Rational Arithmetic
- at cindex Rational arithmetic
-The @code{mpq} functions operate on @code{mpq_t} values with no common factors
-in the numerator and denominator. Common factors are checked-for and cast out
-as necessary. In general, cancelling factors every time is the best approach
-since it minimizes the sizes for subsequent operations.
-
-However, applications that know something about the factorization of the
-values they're working with might be able to avoid some of the GCDs used for
-canonicalization, or swap them for divisions. For example when multiplying by
-a prime it's enough to check for factors of it in the denominator instead of
-doing a full GCD at . Or when forming a big product it might be known that very
-little cancellation will be possible, and so canonicalization can be left to
-the end.
-
-The @code{mpq_numref} and @code{mpq_denref} macros give access to the
-numerator and denominator to do things outside the scope of the supplied
- at code{mpq} functions. @xref{Applying Integer Functions}.
-
-The canonical form for rationals allows mixed-type @code{mpq_t} and integer
-additions or subtractions to be done directly with multiples of the
-denominator. This will be somewhat faster than @code{mpq_add}. For example,
-
- at example
-/* mpq increment */
-mpz_add (mpq_numref(q), mpq_numref(q), mpq_denref(q));
-
-/* mpq += unsigned long */
-mpz_addmul_ui (mpq_numref(q), mpq_denref(q), 123UL);
-
-/* mpq -= mpz */
-mpz_submul (mpq_numref(q), mpq_denref(q), z);
- at end example
-
- at item Number Sequences
- at cindex Number sequences
-Functions like @code{mpz_fac_ui}, @code{mpz_fib_ui} and @code{mpz_bin_uiui}
-are designed for calculating isolated values. If a range of values is wanted
-it's probably best to call to get a starting point and iterate from there.
-
- at item Text Input/Output
- at cindex Text input/output
-Hexadecimal or octal are suggested for input or output in text form.
-Power-of-2 bases like these can be converted much more efficiently than other
-bases, like decimal. For big numbers there's usually nothing of particular
-interest to be seen in the digits, so the base doesn't matter much.
-
-Maybe we can hope octal will one day become the normal base for everyday use,
-as proposed by King Charles XII of Sweden and later reformers.
- at c Reference: Knuth volume 2 section 4.1, page 184 of second edition. :-)
- at end table
-
-
- at node Debugging, Profiling, Efficiency, MPIR Basics
- at section Debugging
- at cindex Debugging
-
- at table @asis
- at item Stack Overflow
- at cindex Stack overflow
- at cindex Segmentation violation
- at cindex Bus error
-Depending on the system, a segmentation violation or bus error might be the
-only indication of stack overflow. See @samp{--enable-alloca} choices in
- at ref{Build Options}, for how to address this.
-
-In new enough versions of GCC, @samp{-fstack-check} may be able to ensure an
-overflow is recognised by the system before too much damage is done, or
- at samp{-fstack-limit-symbol} or @samp{-fstack-limit-register} may be able to
-add checking if the system itself doesn't do any (@pxref{Code Gen Options,,
-Options for Code Generation, gcc, Using the GNU Compiler Collection (GCC)}).
-These options must be added to the @samp{CFLAGS} used in the MPIR build
-(@pxref{Build Options}), adding them just to an application will have no
-effect. Note also they're a slowdown, adding overhead to each function call
-and each stack allocation.
-
- at item Heap Problems
- at cindex Heap problems
- at cindex Malloc problems
-The most likely cause of application problems with MPIR is heap corruption.
-Failing to @code{init} MPIR variables will have unpredictable effects, and
-corruption arising elsewhere in a program may well affect MPIR at . Initializing
-MPIR variables more than once or failing to clear them will cause memory leaks.
-
- at cindex Malloc debugger
-In all such cases a @code{malloc} debugger is recommended. On a GNU or BSD
-system the standard C library @code{malloc} has some diagnostic facilities,
-see @ref{Allocation Debugging,, Allocation Debugging, libc, The GNU C Library
-Reference Manual}, or @samp{man 3 malloc}. Other possibilities, in no
-particular order, include
-
- at display
- at uref{http://dmalloc.com/}
- at uref{http://www.perens.com/FreeSoftware/} @ (electric fence)
- at uref{http://www.gnupdate.org/components/leakbug/}
- at uref{http://wwww.gnome.org/projects/memprof}
- at end display
-
-The MPIR default allocation routines in @file{memory.c} also have a simple
-sentinel scheme which can be enabled with @code{#define DEBUG} in that file.
-This is mainly designed for detecting buffer overruns during MPIR development,
-but might find other uses.
-
- at item Stack Backtraces
- at cindex Stack backtrace
-On some systems the compiler options MPIR uses by default can interfere with
-debugging. In particular on x86 and 68k systems @samp{-fomit-frame-pointer}
-is used and this generally inhibits stack backtracing. Recompiling without
-such options may help while debugging, though the usual caveats about it
-potentially moving a memory problem or hiding a compiler bug will apply.
-
- at item GDB, the GNU Debugger
- at cindex GDB
- at cindex GNU Debugger
-A sample @file{.gdbinit} is included in the distribution, showing how to call
-some undocumented dump functions to print MPIR variables from within GDB at . Note
-that these functions shouldn't be used in final application code since they're
-undocumented and may be subject to incompatible changes in future versions of
-MPIR.
-
- at item Source File Paths
-MPIR has multiple source files with the same name, in different directories.
-For example @file{mpz}, @file{mpq} and @file{mpf} each have an
- at file{init.c}. If the debugger can't already determine the right one it may
-help to build with absolute paths on each C file. One way to do that is to
-use a separate object directory with an absolute path to the source directory.
-
- at example
-cd /my/build/dir
-/my/source/dir/gmp- at value{VERSION}/configure
- at end example
-
-This works via @code{VPATH}, and might require GNU @command{make}.
-Alternately it might be possible to change the @code{.c.lo} rules
-appropriately.
-
- at item Assertion Checking
- at cindex Assertion checking
-The build option @option{--enable-assert} is available to add some consistency
-checks to the library (see @ref{Build Options}). These are likely to be of
-limited value to most applications. Assertion failures are just as likely to
-indicate memory corruption as a library or compiler bug.
-
-Applications using the low-level @code{mpn} functions, however, will benefit
-from @option{--enable-assert} since it adds checks on the parameters of most
-such functions, many of which have subtle restrictions on their usage. Note
-however that only the generic C code has checks, not the assembler code, so
-CPU @samp{none} should be used for maximum checking.
-
- at item Temporary Memory Checking
-The build option @option{--enable-alloca=debug} arranges that each block of
-temporary memory in MPIR is allocated with a separate call to @code{malloc} (or
-the allocation function set with @code{mp_set_memory_functions}).
-
-This can help a malloc debugger detect accesses outside the intended bounds,
-or detect memory not released. In a normal build, on the other hand,
-temporary memory is allocated in blocks which MPIR divides up for its own use,
-or may be allocated with a compiler builtin @code{alloca} which will go
-nowhere near any malloc debugger hooks.
-
- at item Maximum Debuggability
-To summarize the above, an MPIR build for maximum debuggability would be
-
- at example
-./configure --disable-shared --enable-assert \
- --enable-alloca=debug --host=none CFLAGS=-g
- at end example
-
-For C++, add @samp{--enable-cxx CXXFLAGS=-g}.
-
- at item Checker
- at cindex Checker
- at cindex GCC Checker
-The GCC checker (@uref{http://savannah.gnu.org/projects/checker/}) can be used
-with MPIR at . It contains a stub library which means MPIR applications compiled
-with checker can use a normal MPIR build.
-
-A build of MPIR with checking within MPIR itself can be made. This will run
-very very slowly. On GNU/Linux for example,
-
- at cindex @command{checkergcc}
- at example
-./configure --host=none-pc-linux-gnu CC=checkergcc
- at end example
-
- at samp{--host=none} must be used, since the MPIR assembler code doesn't support
-the checking scheme. The MPIR C++ features cannot be used, since current
-versions of checker (0.9.9.1) don't yet support the standard C++ library.
-
- at item Valgrind
- at cindex Valgrind
-The valgrind program (@uref{http://valgrind.org/}) is a memory
-checker for x86s. It translates and emulates machine instructions to do
-strong checks for uninitialized data (at the level of individual bits), memory
-accesses through bad pointers, and memory leaks.
-
-Recent versions of Valgrind are getting support for MMX and SSE/SSE2
-instructions, for past versions MPIR will need to be configured not to use
-those, ie.@: for an x86 without them (for instance plain @samp{i486}).
-
- at item Other Problems
-Any suspected bug in MPIR itself should be isolated to make sure it's not an
-application problem, see @ref{Reporting Bugs}.
- at end table
-
-
- at node Profiling, Autoconf, Debugging, MPIR Basics
- at section Profiling
- at cindex Profiling
- at cindex Execution profiling
- at cindex @code{--enable-profiling}
-
-Running a program under a profiler is a good way to find where it's spending
-most time and where improvements can be best sought. The profiling choices
-for a MPIR build are as follows.
-
- at table @asis
- at item @samp{--disable-profiling}
-The default is to add nothing special for profiling.
-
-It should be possible to just compile the mainline of a program with @code{-p}
-and use @command{prof} to get a profile consisting of timer-based sampling of
-the program counter. Most of the MPIR assembler code has the necessary symbol
-information.
-
-This approach has the advantage of minimizing interference with normal program
-operation, but on most systems the resolution of the sampling is quite low (10
-milliseconds for instance), requiring long runs to get accurate information.
-
- at item @samp{--enable-profiling=prof}
- at cindex @code{prof}
-Build with support for the system @command{prof}, which means @samp{-p} added
-to the @samp{CFLAGS}.
-
-This provides call counting in addition to program counter sampling, which
-allows the most frequently called routines to be identified, and an average
-time spent in each routine to be determined.
-
-The x86 assembler code has support for this option, but on other processors
-the assembler routines will be as if compiled without @samp{-p} and therefore
-won't appear in the call counts.
-
-On some systems, such as GNU/Linux, @samp{-p} in fact means @samp{-pg} and in
-this case @samp{--enable-profiling=gprof} described below should be used
-instead.
-
- at item @samp{--enable-profiling=gprof}
- at cindex @code{gprof}
-Build with support for @command{gprof} (@GMPpxreftop{gprof, GNU gprof}), which
-means @samp{-pg} added to the @samp{CFLAGS}.
-
-This provides call graph construction in addition to call counting and program
-counter sampling, which makes it possible to count calls coming from different
-locations. For example the number of calls to @code{mpn_mul} from
- at code{mpz_mul} versus the number from @code{mpf_mul}. The program counter
-sampling is still flat though, so only a total time in @code{mpn_mul} would be
-accumulated, not a separate amount for each call site.
-
-The x86 assembler code has support for this option, but on other processors
-the assembler routines will be as if compiled without @samp{-pg} and therefore
-not be included in the call counts.
-
-On x86 and m68k systems @samp{-pg} and @samp{-fomit-frame-pointer} are
-incompatible, so the latter is omitted from the default flags in that case,
-which might result in poorer code generation.
-
-Incidentally, it should be possible to use the @command{gprof} program with a
-plain @samp{--enable-profiling=prof} build. But in that case only the
- at samp{gprof -p} flat profile and call counts can be expected to be valid, not
-the @samp{gprof -q} call graph.
-
- at item @samp{--enable-profiling=instrument}
- at cindex @code{-finstrument-functions}
- at cindex @code{instrument-functions}
-Build with the GCC option @samp{-finstrument-functions} added to the
- at samp{CFLAGS} (@pxref{Code Gen Options,, Options for Code Generation, gcc,
-Using the GNU Compiler Collection (GCC)}).
-
-This inserts special instrumenting calls at the start and end of each
-function, allowing exact timing and full call graph construction.
-
-This instrumenting is not normally a standard system feature and will require
-support from an external library, such as
-
- at cindex FunctionCheck
- at cindex fnccheck
- at display
- at uref{http://sourceforge.net/projects/fnccheck/}
- at end display
-
-This should be included in @samp{LIBS} during the MPIR configure so that test
-programs will link. For example,
-
- at example
-./configure --enable-profiling=instrument LIBS=-lfc
- at end example
-
-On a GNU system the C library provides dummy instrumenting functions, so
-programs compiled with this option will link. In this case it's only
-necessary to ensure the correct library is added when linking an application.
-
-The x86 assembler code supports this option, but on other processors the
-assembler routines will be as if compiled without
- at samp{-finstrument-functions} meaning time spent in them will effectively be
-attributed to their caller.
- at end table
-
-
- at node Autoconf, Emacs, Profiling, MPIR Basics
- at section Autoconf
- at cindex Autoconf
-
-Autoconf based applications can easily check whether MPIR is installed. The
-only thing to be noted is that GMP/MPIR library symbols from version 3 of GMP
-and version 1 of MPIR onwards have prefixes like @code{__gmpz}. The following
-therefore would be a simple test,
-
- at cindex @code{AC_CHECK_LIB}
- at example
-AC_CHECK_LIB(mpir, __gmpz_init)
- at end example
-
-This just uses the default @code{AC_CHECK_LIB} actions for found or not found,
-but an application that must have MPIR would want to generate an error if not
-found. For example,
-
- at example
-AC_CHECK_LIB(mpir, __gmpz_init, ,
- [AC_MSG_ERROR([MPIR not found, see http://www.mpir.org/])])
- at end example
-
-If functions added in some particular version of GMP/MPIR are required, then one of
-those can be used when checking. For example @code{mpz_mul_si} was added in
-GMP 3.1,
-
- at example
-AC_CHECK_LIB(mpir, __gmpz_mul_si, ,
- [AC_MSG_ERROR(
- [GMP/MPIR not found, or not GMP 3.1 or up or MPIR 1.0 or up, see http://www.mpir.org/])])
- at end example
-
-An alternative would be to test the version number in @file{mpir.h} using say
- at code{AC_EGREP_CPP}. That would make it possible to test the exact version,
-if some particular sub-minor release is known to be necessary.
-
-In general it's recommended that applications should simply demand a new
-enough MPIR rather than trying to provide supplements for features not
-available in past versions.
-
-Occasionally an application will need or want to know the size of a type at
-configuration or preprocessing time, not just with @code{sizeof} in the code.
-This can be done in the normal way with @code{mp_limb_t} etc, but GMP 4.0 or
-up and MPIR 1.0 and up is best for this, since prior versions needed certain
- at samp{-D} defines on systems using a @code{long long} limb. The following
-would suit Autoconf 2.50 or up,
-
- at example
-AC_CHECK_SIZEOF(mp_limb_t, , [#include <mpir.h>])
- at end example
-
-
- at node Emacs, , Autoconf, MPIR Basics
- at section Emacs
- at cindex Emacs
- at cindex @code{info-lookup-symbol}
-
- at key{C-h C-i} (@code{info-lookup-symbol}) is a good way to find documentation
-on C functions while editing (@pxref{Info Lookup, , Info Documentation Lookup,
-emacs, The Emacs Editor}).
-
-The MPIR manual can be included in such lookups by putting the following in
-your @file{.emacs},
-
- at c This isn't pretty, but there doesn't seem to be a better way (in emacs
- at c 21.2 at least). info-lookup->mode-value could be used for the "assoc"s,
- at c but that function isn't documented, whereas info-lookup-alist is.
- at c
- at example
-(eval-after-load "info-look"
- '(let ((mode-value (assoc 'c-mode (assoc 'symbol info-lookup-alist))))
- (setcar (nthcdr 3 mode-value)
- (cons '("(gmp)Function Index" nil "^ -.* " "\\>")
- (nth 3 mode-value)))))
- at end example
-
-
- at node Reporting Bugs, Integer Functions, MPIR Basics, Top
- at comment node-name, next, previous, up
- at chapter Reporting Bugs
- at cindex Reporting bugs
- at cindex Bug reporting
-
-If you think you have found a bug in the MPIR library, please investigate it
-and report it. We have made this library available to you, and it is not too
-much to ask you to report the bugs you find.
-
-Before you report a bug, check it's not already addressed in @ref{Known Build
-Problems}, or perhaps @ref{Notes for Particular Systems}. You may also want
-to check @uref{http://www.mpir.org/} for patches for this release.
-
-Please include the following in any report,
-
- at itemize @bullet
- at item
-The MPIR version number, and if pre-packaged or patched then say so.
-
- at item
-A test program that makes it possible for us to reproduce the bug. Include
-instructions on how to run the program.
-
- at item
-A description of what is wrong. If the results are incorrect, in what way.
-If you get a crash, say so.
-
- at item
-If you get a crash, include a stack backtrace from the debugger if it's
-informative (@samp{where} in @command{gdb}, or @samp{$C} in @command{adb}).
-
- at item
-Please do not send core dumps, executables or @command{strace}s.
-
- at item
-The configuration options you used when building MPIR, if any.
-
- at item
-The name of the compiler and its version. For @command{gcc}, get the version
-with @samp{gcc -v}, otherwise perhaps @samp{what `which cc`}, or similar.
-
- at item
-The output from running @samp{uname -a}.
-
- at item
-The output from running @samp{./config.guess}, and from running
- at samp{./configfsf.guess} (might be the same).
-
- at item
-If the bug is related to @samp{configure}, then the contents of
- at file{config.log}.
-
- at item
-If the bug is related to an @file{asm} file not assembling, then the contents
-of @file{config.m4} and the offending line or lines from the temporary
- at file{mpn/tmp-<file>.s}.
- at end itemize
-
-Please make an effort to produce a self-contained report, with something
-definite that can be tested or debugged. Vague queries or piecemeal messages
-are difficult to act on and don't help the development effort.
-
-It is not uncommon that an observed problem is actually due to a bug in the
-compiler; the MPIR code tends to explore interesting corners in compilers.
-
-If your bug report is good, we will do our best to help you get a corrected
-version of the library; if the bug report is poor, we won't do anything about
-it (except maybe ask you to send a better report).
-
-Send your report to: @uref{http://groups.google.com/group/mpir-devel}.
-
-If you think something in this manual is unclear, or downright incorrect, or if
-the language needs to be improved, please send a note to the same address.
-
-
- at node Integer Functions, Rational Number Functions, Reporting Bugs, Top
- at comment node-name, next, previous, up
- at chapter Integer Functions
- at cindex Integer functions
-
-This chapter describes the MPIR functions for performing integer arithmetic.
-These functions start with the prefix @code{mpz_}.
-
-MPIR integers are stored in objects of type @code{mpz_t}.
-
- at menu
-* Initializing Integers::
-* Assigning Integers::
-* Simultaneous Integer Init & Assign::
-* Converting Integers::
-* Integer Arithmetic::
-* Integer Division::
-* Integer Exponentiation::
-* Integer Roots::
-* Number Theoretic Functions::
-* Integer Comparisons::
-* Integer Logic and Bit Fiddling::
-* I/O of Integers::
-* Integer Random Numbers::
-* Integer Import and Export::
-* Miscellaneous Integer Functions::
-* Integer Special Functions::
- at end menu
-
- at node Initializing Integers, Assigning Integers, Integer Functions, Integer Functions
- at comment node-name, next, previous, up
- at section Initialization Functions
- at cindex Integer initialization functions
- at cindex Initialization functions
-
-The functions for integer arithmetic assume that all integer objects are
-initialized. You do that by calling the function @code{mpz_init}. For
-example,
-
- at example
-@{
- mpz_t integ;
- mpz_init (integ);
- @dots{}
- mpz_add (integ, @dots{});
- @dots{}
- mpz_sub (integ, @dots{});
-
- /* Unless the program is about to exit, do ... */
- mpz_clear (integ);
-@}
- at end example
-
-As you can see, you can store new values any number of times, once an
-object is initialized.
-
- at deftypefun void mpz_init (mpz_t @var{integer})
-Initialize @var{integer}, and set its value to 0.
- at end deftypefun
-
- at deftypefun void mpz_inits (mpz_t @var{x}, ...)
-Initialize a NULL-terminated list of @code{mpz_t} variables, and set their
-values to 0.
- at end deftypefun
-
- at deftypefun void mpz_init2 (mpz_t @var{integer}, mp_bitcnt_t @var{n})
-Initialize @var{integer}, with space for @var{n} bits, and set its value to 0.
-
- at var{n} is only the initial space, @var{integer} will grow automatically in
-the normal way, if necessary, for subsequent values stored. @code{mpz_init2}
-makes it possible to avoid such reallocations if a maximum size is known in
-advance.
- at end deftypefun
-
- at deftypefun void mpz_clear (mpz_t @var{integer})
-Free the space occupied by @var{integer}. Call this function for all
- at code{mpz_t} variables when you are done with them.
- at end deftypefun
-
- at deftypefun void mpz_clears (mpz_t @var{x}, ...)
-Free the space occupied by a NULL-terminated list of @code{mpz_t} variables.
- at end deftypefun
-
- at deftypefun void mpz_realloc2 (mpz_t @var{integer}, mp_bitcnt_t @var{n})
-Change the space allocated for @var{integer} to @var{n} bits. The value in
- at var{integer} is preserved if it fits, or is set to 0 if not.
-
-This function can be used to increase the space for a variable in order to
-avoid repeated automatic reallocations, or to decrease it to give memory back
-to the heap.
- at end deftypefun
-
-
- at node Assigning Integers, Simultaneous Integer Init & Assign, Initializing Integers, Integer Functions
- at comment node-name, next, previous, up
- at section Assignment Functions
- at cindex Integer assignment functions
- at cindex Assignment functions
-
-These functions assign new values to already initialized integers
-(@pxref{Initializing Integers}).
-
- at deftypefun void mpz_set (mpz_t @var{rop}, mpz_t @var{op})
- at deftypefunx void mpz_set_ui (mpz_t @var{rop}, mpir_ui @var{op})
- at deftypefunx void mpz_set_si (mpz_t @var{rop}, mpir_si @var{op})
- at deftypefunx void mpz_set_ux (mpz_t @var{rop}, uintmax_t @var{op})
- at deftypefunx void mpz_set_sx (mpz_t @var{rop}, intmax_t @var{op})
- at deftypefunx void mpz_set_d (mpz_t @var{rop}, double @var{op})
- at deftypefunx void mpz_set_q (mpz_t @var{rop}, mpq_t @var{op})
- at deftypefunx void mpz_set_f (mpz_t @var{rop}, mpf_t @var{op})
-Set the value of @var{rop} from @var{op}. Note the intmax versions are only available
-if you have stdint.h header file on your system.
-
- at code{mpz_set_d}, @code{mpz_set_q} and @code{mpz_set_f} truncate @var{op} to
-make it an integer.
- at end deftypefun
-
- at deftypefun int mpz_set_str (mpz_t @var{rop}, char *@var{str}, int @var{base})
-Set the value of @var{rop} from @var{str}, a null-terminated C string in base
- at var{base}. White space is allowed in the string, and is simply ignored.
-
-The @var{base} may vary from 2 to 62, or if @var{base} is 0, then the leading
-characters are used: @code{0x} and @code{0X} for hexadecimal, @code{0b} and
- at code{0B} for binary, @code{0} for octal, or decimal otherwise.
-
-For bases up to 36, case is ignored; upper-case and lower-case letters have
-the same value. For bases 37 to 62, upper-case letter represent the usual
-10..35 while lower-case letter represent 36..61.
-
-This function returns 0 if the entire string is a valid number in base
- at var{base}. Otherwise it returns @minus{}1.
- at c
- at c It turns out that it is not entirely true that this function ignores
- at c white-space. It does ignore it between digits, but not after a minus sign
- at c or within or after ``0x''. Some thought was given to disallowing all
- at c whitespace, but that would be an incompatible change, whitespace has been
- at c documented as ignored ever since GMP 1.
- at c
- at end deftypefun
-
- at deftypefun void mpz_swap (mpz_t @var{rop1}, mpz_t @var{rop2})
-Swap the values @var{rop1} and @var{rop2} efficiently.
- at end deftypefun
-
-
- at node Simultaneous Integer Init & Assign, Converting Integers, Assigning Integers, Integer Functions
- at comment node-name, next, previous, up
- at section Combined Initialization and Assignment Functions
- at cindex Integer assignment functions
- at cindex Assignment functions
- at cindex Integer initialization functions
- at cindex Initialization functions
-
-For convenience, MPIR provides a parallel series of initialize-and-set functions
-which initialize the output and then store the value there. These functions'
-names have the form @code{mpz_init_set at dots{}}
-
-Here is an example of using one:
-
- at example
-@{
- mpz_t pie;
- mpz_init_set_str (pie, "3141592653589793238462643383279502884", 10);
- @dots{}
- mpz_sub (pie, @dots{});
- @dots{}
- mpz_clear (pie);
-@}
- at end example
-
- at noindent
-Once the integer has been initialized by any of the @code{mpz_init_set at dots{}}
-functions, it can be used as the source or destination operand for the ordinary
-integer functions. Don't use an initialize-and-set function on a variable
-already initialized!
-
- at deftypefun void mpz_init_set (mpz_t @var{rop}, mpz_t @var{op})
- at deftypefunx void mpz_init_set_ui (mpz_t @var{rop}, mpir_ui @var{op})
- at deftypefunx void mpz_init_set_si (mpz_t @var{rop}, mpir_si @var{op})
- at deftypefunx void mpz_init_set_ux (mpz_t @var{rop}, uintmax_t @var{op})
- at deftypefunx void mpz_init_set_sx (mpz_t @var{rop}, intmax_t @var{op})
- at deftypefunx void mpz_init_set_d (mpz_t @var{rop}, double @var{op})
-Initialize @var{rop} with limb space and set the initial numeric value from
- at var{op}. Note the intmax versions are only available
-if you have stdint.h header file on your system.
- at end deftypefun
-
- at deftypefun int mpz_init_set_str (mpz_t @var{rop}, char *@var{str}, int @var{base})
-Initialize @var{rop} and set its value like @code{mpz_set_str} (see its
-documentation above for details).
-
-If the string is a correct base @var{base} number, the function returns 0;
-if an error occurs it returns @minus{}1. @var{rop} is initialized even if
-an error occurs. (I.e., you have to call @code{mpz_clear} for it.)
- at end deftypefun
-
-
- at node Converting Integers, Integer Arithmetic, Simultaneous Integer Init & Assign, Integer Functions
- at comment node-name, next, previous, up
- at section Conversion Functions
- at cindex Integer conversion functions
- at cindex Conversion functions
-
-This section describes functions for converting MPIR integers to standard C
-types. Functions for converting @emph{to} MPIR integers are described in
- at ref{Assigning Integers} and @ref{I/O of Integers}.
-
- at deftypefun mpir_ui mpz_get_ui (mpz_t @var{op})
-Return the value of @var{op} as an @code{mpir_ui}.
-
-If @var{op} is too big to fit an @code{mpir_ui} then just the least
-significant bits that do fit are returned. The sign of @var{op} is ignored,
-only the absolute value is used.
- at end deftypefun
-
- at deftypefun mpir_si mpz_get_si (mpz_t @var{op})
-If @var{op} fits into a @code{mpir_si} return the value of @var{op}.
-Otherwise return the least significant part of @var{op}, with the same sign
-as @var{op}.
-
-If @var{op} is too big to fit in a @code{mpir_si}, the returned
-result is probably not very useful. To find out if the value will fit, use
-the function @code{mpz_fits_slong_p}.
- at end deftypefun
-
- at deftypefun {uintmax_t} mpz_get_ux (mpz_t @var{op})
-Return the value of @var{op} as an @code{uintmax_t}.
-
-If @var{op} is too big to fit an @code{uintmax_t} then just the least
-significant bits that do fit are returned. The sign of @var{op} is ignored,
-only the absolute value is used. Note the intmax versions are only available
-if you have stdint.h header file on your system.
- at end deftypefun
-
- at deftypefun {intmax_t} mpz_get_sx (mpz_t @var{op})
-If @var{op} fits into a @code{intmax_t} return the value of @var{op}.
-Otherwise return the least significant part of @var{op}, with the same sign
-as @var{op}.
-
-If @var{op} is too big to fit in a @code{intmax_t}, the returned
-result is probably not very useful. Note the intmax versions are only available
-if you have stdint.h header file on your system.
- at end deftypefun
-
- at deftypefun double mpz_get_d (mpz_t @var{op})
-Convert @var{op} to a @code{double}, truncating if necessary (ie.@: rounding
-towards zero).
-
-If the exponent from the conversion is too big, the result is system
-dependent. An infinity is returned where available. A hardware overflow trap
-may or may not occur.
- at end deftypefun
-
- at deftypefun double mpz_get_d_2exp (mpir_si *@var{exp}, mpz_t @var{op})
-Convert @var{op} to a @code{double}, truncating if necessary (ie.@: rounding
-towards zero), and returning the exponent separately.
-
-The return value is in the range @math{0.5 at le{}@GMPabs{@var{d}}<1} and the
-exponent is stored to @code{*@var{exp}}. @m{@var{d} * 2^{exp}, @var{d} *
-2^@var{exp}} is the (truncated) @var{op} value. If @var{op} is zero, the
-return is @math{0.0} and 0 is stored to @code{*@var{exp}}.
-
- at cindex @code{frexp}
-This is similar to the standard C @code{frexp} function (@pxref{Normalization
-Functions,,, libc, The GNU C Library Reference Manual}).
- at end deftypefun
-
- at deftypefun {char *} mpz_get_str (char *@var{str}, int @var{base}, mpz_t @var{op})
-Convert @var{op} to a string of digits in base @var{base}. The base may vary
-from 2 to 36 or from @minus{}2 to @minus{}36.
-
-For @var{base} in the range 2..36, digits and lower-case letters are used; for
- at minus{}2.. at minus{}36, digits and upper-case letters are used; for 37..62,
-digits, upper-case letters, and lower-case letters (in that significance order)
-are used.
-
-If @var{str} is @code{NULL}, the result string is allocated using the current
-allocation function (@pxref{Custom Allocation}). The block will be
- at code{strlen(str)+1} bytes, that being exactly enough for the string and
-null-terminator.
-
-If @var{str} is not @code{NULL}, it should point to a block of storage large
-enough for the result, that being @code{mpz_sizeinbase (@var{op}, @var{base})
-+ 2}. The two extra bytes are for a possible minus sign, and the
-null-terminator.
-
-A pointer to the result string is returned, being either the allocated block,
-or the given @var{str}.
- at end deftypefun
-
-
- at need 2000
- at node Integer Arithmetic, Integer Division, Converting Integers, Integer Functions
- at comment node-name, next, previous, up
- at section Arithmetic Functions
- at cindex Integer arithmetic functions
- at cindex Arithmetic functions
-
- at deftypefun void mpz_add (mpz_t @var{rop}, mpz_t @var{op1}, mpz_t @var{op2})
- at deftypefunx void mpz_add_ui (mpz_t @var{rop}, mpz_t @var{op1}, mpir_ui @var{op2})
-Set @var{rop} to @math{@var{op1} + @var{op2}}.
- at end deftypefun
-
- at deftypefun void mpz_sub (mpz_t @var{rop}, mpz_t @var{op1}, mpz_t @var{op2})
- at deftypefunx void mpz_sub_ui (mpz_t @var{rop}, mpz_t @var{op1}, mpir_ui @var{op2})
- at deftypefunx void mpz_ui_sub (mpz_t @var{rop}, mpir_ui @var{op1}, mpz_t @var{op2})
-Set @var{rop} to @var{op1} @minus{} @var{op2}.
- at end deftypefun
-
- at deftypefun void mpz_mul (mpz_t @var{rop}, mpz_t @var{op1}, mpz_t @var{op2})
- at deftypefunx void mpz_mul_si (mpz_t @var{rop}, mpz_t @var{op1}, mpir_si @var{op2})
- at deftypefunx void mpz_mul_ui (mpz_t @var{rop}, mpz_t @var{op1}, mpir_ui @var{op2})
-Set @var{rop} to @math{@var{op1} @GMPtimes{} @var{op2}}.
- at end deftypefun
-
- at deftypefun void mpz_addmul (mpz_t @var{rop}, mpz_t @var{op1}, mpz_t @var{op2})
- at deftypefunx void mpz_addmul_ui (mpz_t @var{rop}, mpz_t @var{op1}, mpir_ui @var{op2})
-Set @var{rop} to @math{@var{rop} + @var{op1} @GMPtimes{} @var{op2}}.
- at end deftypefun
-
- at deftypefun void mpz_submul (mpz_t @var{rop}, mpz_t @var{op1}, mpz_t @var{op2})
- at deftypefunx void mpz_submul_ui (mpz_t @var{rop}, mpz_t @var{op1}, mpir_ui @var{op2})
-Set @var{rop} to @math{@var{rop} - @var{op1} @GMPtimes{} @var{op2}}.
- at end deftypefun
-
- at deftypefun void mpz_mul_2exp (mpz_t @var{rop}, mpz_t @var{op1}, mp_bitcnt_t @var{op2})
- at cindex Bit shift left
-Set @var{rop} to @m{@var{op1} \times 2^{op2}, @var{op1} times 2 raised to
- at var{op2}}. This operation can also be defined as a left shift by @var{op2}
-bits.
- at end deftypefun
-
- at deftypefun void mpz_neg (mpz_t @var{rop}, mpz_t @var{op})
-Set @var{rop} to @minus{}@var{op}.
- at end deftypefun
-
- at deftypefun void mpz_abs (mpz_t @var{rop}, mpz_t @var{op})
-Set @var{rop} to the absolute value of @var{op}.
- at end deftypefun
-
-
- at need 2000
- at node Integer Division, Integer Exponentiation, Integer Arithmetic, Integer Functions
- at section Division Functions
- at cindex Integer division functions
- at cindex Division functions
-
-Division is undefined if the divisor is zero. Passing a zero divisor to the
-division or modulo functions (including the modular powering functions
- at code{mpz_powm} and @code{mpz_powm_ui}), will cause an intentional division by
-zero. This lets a program handle arithmetic exceptions in these functions the
-same way as for normal C @code{int} arithmetic.
-
- at c Separate deftypefun groups for cdiv, fdiv and tdiv produce a blank line
- at c between each, and seem to let tex do a better job of page breaks than an
- at c @sp 1 in the middle of one big set.
-
- at deftypefun void mpz_cdiv_q (mpz_t @var{q}, mpz_t @var{n}, mpz_t @var{d})
- at deftypefunx void mpz_cdiv_r (mpz_t @var{r}, mpz_t @var{n}, mpz_t @var{d})
- at deftypefunx void mpz_cdiv_qr (mpz_t @var{q}, mpz_t @var{r}, mpz_t @var{n}, mpz_t @var{d})
- at maybepagebreak
- at deftypefunx mpir_ui mpz_cdiv_q_ui (mpz_t @var{q}, mpz_t @var{n}, mpir_ui @var{d})
- at deftypefunx mpir_ui mpz_cdiv_r_ui (mpz_t @var{r}, mpz_t @var{n}, mpir_ui @var{d})
- at deftypefunx mpir_ui mpz_cdiv_qr_ui (mpz_t @var{q}, mpz_t @var{r}, @w{mpz_t @var{n}}, mpir_ui @var{d})
- at deftypefunx mpir_ui mpz_cdiv_ui (mpz_t @var{n}, mpir_ui @var{d})
- at maybepagebreak
- at deftypefunx void mpz_cdiv_q_2exp (mpz_t @var{q}, mpz_t @var{n}, @w{mp_bitcnt_t @var{b}})
- at deftypefunx void mpz_cdiv_r_2exp (mpz_t @var{r}, mpz_t @var{n}, @w{mp_bitcnt_t @var{b}})
- at end deftypefun
-
- at deftypefun void mpz_fdiv_q (mpz_t @var{q}, mpz_t @var{n}, mpz_t @var{d})
- at deftypefunx void mpz_fdiv_r (mpz_t @var{r}, mpz_t @var{n}, mpz_t @var{d})
- at deftypefunx void mpz_fdiv_qr (mpz_t @var{q}, mpz_t @var{r}, mpz_t @var{n}, mpz_t @var{d})
- at maybepagebreak
- at deftypefunx mpir_ui mpz_fdiv_q_ui (mpz_t @var{q}, mpz_t @var{n}, mpir_ui @var{d})
- at deftypefunx mpir_ui mpz_fdiv_r_ui (mpz_t @var{r}, mpz_t @var{n}, mpir_ui @var{d})
- at deftypefunx mpir_ui mpz_fdiv_qr_ui (mpz_t @var{q}, mpz_t @var{r}, @w{mpz_t @var{n}}, mpir_ui @var{d})
- at deftypefunx mpir_ui mpz_fdiv_ui (mpz_t @var{n}, mpir_ui @var{d})
- at maybepagebreak
- at deftypefunx void mpz_fdiv_q_2exp (mpz_t @var{q}, mpz_t @var{n}, @w{mp_bitcnt_t @var{b}})
- at deftypefunx void mpz_fdiv_r_2exp (mpz_t @var{r}, mpz_t @var{n}, @w{mp_bitcnt_t @var{b}})
- at end deftypefun
-
- at deftypefun void mpz_tdiv_q (mpz_t @var{q}, mpz_t @var{n}, mpz_t @var{d})
- at deftypefunx void mpz_tdiv_r (mpz_t @var{r}, mpz_t @var{n}, mpz_t @var{d})
- at deftypefunx void mpz_tdiv_qr (mpz_t @var{q}, mpz_t @var{r}, mpz_t @var{n}, mpz_t @var{d})
- at maybepagebreak
- at deftypefunx mpir_ui mpz_tdiv_q_ui (mpz_t @var{q}, mpz_t @var{n}, mpir_ui @var{d})
- at deftypefunx mpir_ui mpz_tdiv_r_ui (mpz_t @var{r}, mpz_t @var{n}, mpir_ui @var{d})
- at deftypefunx mpir_ui mpz_tdiv_qr_ui (mpz_t @var{q}, mpz_t @var{r}, @w{mpz_t @var{n}}, mpir_ui @var{d})
- at deftypefunx mpir_ui mpz_tdiv_ui (mpz_t @var{n}, mpir_ui @var{d})
- at maybepagebreak
- at deftypefunx void mpz_tdiv_q_2exp (mpz_t @var{q}, mpz_t @var{n}, @w{mp_bitcnt_t @var{b}})
- at deftypefunx void mpz_tdiv_r_2exp (mpz_t @var{r}, mpz_t @var{n}, @w{mp_bitcnt_t @var{b}})
- at cindex Bit shift right
-
- at sp 1
-Divide @var{n} by @var{d}, forming a quotient @var{q} and/or remainder
- at var{r}. For the @code{2exp} functions, @m{@var{d}=2^b, @var{d}=2^@var{b}}.
-The rounding is in three styles, each suiting different applications.
-
- at itemize @bullet
- at item
- at code{cdiv} rounds @var{q} up towards @m{+\infty, +infinity}, and @var{r} will
-have the opposite sign to @var{d}. The @code{c} stands for ``ceil''.
-
- at item
- at code{fdiv} rounds @var{q} down towards @m{-\infty, @minus{}infinity}, and
- at var{r} will have the same sign as @var{d}. The @code{f} stands for
-``floor''.
-
- at item
- at code{tdiv} rounds @var{q} towards zero, and @var{r} will have the same sign
-as @var{n}. The @code{t} stands for ``truncate''.
- at end itemize
-
-In all cases @var{q} and @var{r} will satisfy
- at m{@var{n}=@var{q}@var{d}+ at var{r}, @var{n}=@var{q}*@var{d}+ at var{r}}, and
- at var{r} will satisfy @math{0 at le{}@GMPabs{@var{r}}<@GMPabs{@var{d}}}.
-
-The @code{q} functions calculate only the quotient, the @code{r} functions
-only the remainder, and the @code{qr} functions calculate both. Note that for
- at code{qr} the same variable cannot be passed for both @var{q} and @var{r}, or
-results will be unpredictable.
-
-For the @code{ui} variants the return value is the remainder, and in fact
-returning the remainder is all the @code{div_ui} functions do. For
- at code{tdiv} and @code{cdiv} the remainder can be negative, so for those the
-return value is the absolute value of the remainder.
-
-For the @code{2exp} variants the divisor is @m{2^b,2^@var{b}}. These
-functions are implemented as right shifts and bit masks, but of course they
-round the same as the other functions.
-
-For positive @var{n} both @code{mpz_fdiv_q_2exp} and @code{mpz_tdiv_q_2exp}
-are simple bitwise right shifts. For negative @var{n}, @code{mpz_fdiv_q_2exp}
-is effectively an arithmetic right shift treating @var{n} as twos complement
-the same as the bitwise logical functions do, whereas @code{mpz_tdiv_q_2exp}
-effectively treats @var{n} as sign and magnitude.
- at end deftypefun
-
- at deftypefun void mpz_mod (mpz_t @var{r}, mpz_t @var{n}, mpz_t @var{d})
- at deftypefunx mpir_ui mpz_mod_ui (mpz_t @var{r}, mpz_t @var{n}, mpir_ui @var{d})
-Set @var{r} to @var{n} @code{mod} @var{d}. The sign of the divisor is
-ignored; the result is always non-negative.
-
- at code{mpz_mod_ui} is identical to @code{mpz_fdiv_r_ui} above, returning the
-remainder as well as setting @var{r}. See @code{mpz_fdiv_ui} above if only
-the return value is wanted.
- at end deftypefun
-
- at deftypefun void mpz_divexact (mpz_t @var{q}, mpz_t @var{n}, mpz_t @var{d})
- at deftypefunx void mpz_divexact_ui (mpz_t @var{q}, mpz_t @var{n}, mpir_ui @var{d})
- at cindex Exact division functions
-Set @var{q} to @var{n}/@var{d}. These functions produce correct results only
-when it is known in advance that @var{d} divides @var{n}.
-
-These routines are much faster than the other division functions, and are the
-best choice when exact division is known to occur, for example reducing a
-rational to lowest terms.
- at end deftypefun
-
- at deftypefun int mpz_divisible_p (mpz_t @var{n}, mpz_t @var{d})
- at deftypefunx int mpz_divisible_ui_p (mpz_t @var{n}, mpir_ui @var{d})
- at deftypefunx int mpz_divisible_2exp_p (mpz_t @var{n}, mp_bitcnt_t @var{b})
- at cindex Divisibility functions
-Return non-zero if @var{n} is exactly divisible by @var{d}, or in the case of
- at code{mpz_divisible_2exp_p} by @m{2^b,2^@var{b}}.
-
- at var{n} is divisible by @var{d} if there exists an integer @var{q} satisfying
- at math{@var{n} = @var{q}@GMPmultiply{}@var{d}}. Unlike the other division
-functions, @math{@var{d}=0} is accepted and following the rule it can be seen
-that only 0 is considered divisible by 0.
- at end deftypefun
-
- at deftypefun int mpz_congruent_p (mpz_t @var{n}, mpz_t @var{c}, mpz_t @var{d})
- at deftypefunx int mpz_congruent_ui_p (mpz_t @var{n}, mpir_ui @var{c}, mpir_ui @var{d})
- at deftypefunx int mpz_congruent_2exp_p (mpz_t @var{n}, mpz_t @var{c}, mp_bitcnt_t @var{b})
- at cindex Divisibility functions
- at cindex Congruence functions
-Return non-zero if @var{n} is congruent to @var{c} modulo @var{d}, or in the
-case of @code{mpz_congruent_2exp_p} modulo @m{2^b,2^@var{b}}.
-
- at var{n} is congruent to @var{c} mod @var{d} if there exists an integer @var{q}
-satisfying @math{@var{n} = @var{c} + @var{q}@GMPmultiply{}@var{d}}. Unlike
-the other division functions, @math{@var{d}=0} is accepted and following the
-rule it can be seen that @var{n} and @var{c} are considered congruent mod 0
-only when exactly equal.
- at end deftypefun
-
-
- at need 2000
- at node Integer Exponentiation, Integer Roots, Integer Division, Integer Functions
- at section Exponentiation Functions
- at cindex Integer exponentiation functions
- at cindex Exponentiation functions
- at cindex Powering functions
-
- at deftypefun void mpz_powm (mpz_t @var{rop}, mpz_t @var{base}, mpz_t @var{exp}, mpz_t @var{mod})
- at deftypefunx void mpz_powm_ui (mpz_t @var{rop}, mpz_t @var{base}, mpir_ui @var{exp}, mpz_t @var{mod})
-Set @var{rop} to @m{base^{exp} \bmod mod, (@var{base} raised to @var{exp})
-modulo @var{mod}}.
-
-A negative @var{exp} is supported in @code{mpz_powm} if an inverse
- at math{@var{base}^@W{-1} @bmod @var{mod}} exists (see @code{mpz_invert} in
- at ref{Number Theoretic Functions}).
-If an inverse doesn't exist then a divide by zero is raised.
- at end deftypefun
-
- at deftypefun void mpz_pow_ui (mpz_t @var{rop}, mpz_t @var{base}, mpir_ui @var{exp})
- at deftypefunx void mpz_ui_pow_ui (mpz_t @var{rop}, mpir_ui @var{base}, mpir_ui @var{exp})
-Set @var{rop} to @m{base^{exp}, @var{base} raised to @var{exp}}. The case
- at math{0^0} yields 1.
- at end deftypefun
-
-
- at need 2000
- at node Integer Roots, Number Theoretic Functions, Integer Exponentiation, Integer Functions
- at section Root Extraction Functions
- at cindex Integer root functions
- at cindex Root extraction functions
-
- at deftypefun int mpz_root (mpz_t @var{rop}, mpz_t @var{op}, mpir_ui @var{n})
-Set @var{rop} to @m{\lfloor\root n \of {op}\rfloor at C{},} the truncated integer
-part of the @var{n}th root of @var{op}. Return non-zero if the computation
-was exact, i.e., if @var{op} is @var{rop} to the @var{n}th power.
- at end deftypefun
-
- at deftypefun void mpz_nthroot (mpz_t @var{rop}, mpz_t @var{op}, mpir_ui @var{n})
-Set @var{rop} to @m{\lfloor\root n \of {op}\rfloor at C{},} the truncated integer
-part of the @var{n}th root of @var{op}.
- at end deftypefun
-
- at deftypefun void mpz_rootrem (mpz_t @var{root}, mpz_t @var{rem}, mpz_t @var{u}, mpir_ui @var{n})
-Set @var{root} to @m{\lfloor\root n \of {u}\rfloor at C{},} the truncated
-integer part of the @var{n}th root of @var{u}. Set @var{rem} to the
-remainder, @m{(@var{u} - @var{root}^n),
- at var{u}@minus{}@var{root}**@var{n}}.
- at end deftypefun
-
- at deftypefun void mpz_sqrt (mpz_t @var{rop}, mpz_t @var{op})
-Set @var{rop} to @m{\lfloor\sqrt{@var{op}}\rfloor at C{},} the truncated
-integer part of the square root of @var{op}.
- at end deftypefun
-
- at deftypefun void mpz_sqrtrem (mpz_t @var{rop1}, mpz_t @var{rop2}, mpz_t @var{op})
-Set @var{rop1} to @m{\lfloor\sqrt{@var{op}}\rfloor, the truncated integer part
-of the square root of @var{op}}, like @code{mpz_sqrt}. Set @var{rop2} to the
-remainder @m{(@var{op} - @var{rop1}^2),
- at var{op}@minus{}@var{rop1}*@var{rop1}}, which will be zero if @var{op} is a
-perfect square.
-
-If @var{rop1} and @var{rop2} are the same variable, the results are
-undefined.
- at end deftypefun
-
- at deftypefun int mpz_perfect_power_p (mpz_t @var{op})
- at cindex Perfect power functions
- at cindex Root testing functions
-Return non-zero if @var{op} is a perfect power, i.e., if there exist integers
- at m{a, at var{a}} and @m{b, at var{b}}, with @m{b>1, @var{b}>1}, such that
- at m{@var{op}=a^b, @var{op} equals @var{a} raised to the power @var{b}}.
-
-Under this definition both 0 and 1 are considered to be perfect powers.
-Negative values of @var{op} are accepted, but of course can only be odd
-perfect powers.
- at end deftypefun
-
- at deftypefun int mpz_perfect_square_p (mpz_t @var{op})
- at cindex Perfect square functions
- at cindex Root testing functions
-Return non-zero if @var{op} is a perfect square, i.e., if the square root of
- at var{op} is an integer. Under this definition both 0 and 1 are considered to
-be perfect squares.
- at end deftypefun
-
-
- at need 2000
- at node Number Theoretic Functions, Integer Comparisons, Integer Roots, Integer Functions
- at section Number Theoretic Functions
- at cindex Number theoretic functions
-
- at deftypefun int mpz_probable_prime_p (mpz_t @var{n}, gmp_randstate_t @var{state}, int @var{prob}, mpir_ui @var{div})
- at cindex Prime testing functions
- at cindex Probable prime testing functions
-Determine whether @var{n} is a probable prime with the chance of error being at most 1 in 2^prob.
-return value is 1 if @var{n} is probably prime, or 0 if
- at var{n} is definitely composite.
-
-This function does some trial divisions to speed up the average case, then some probabilistic
-primality tests to achieve the desired level of error.
-
- at var{div} can be used to inform the function that trial division up to @var{div} has
-already been performed on @var{n} and so @var{n} has NO divisors <= @var{div}.Use 0 to
-inform the function that no trial division has been done.
-
- at strong{This function interface is preliminary and may change in the future.}
- at end deftypefun
-
- at deftypefun int mpz_likely_prime_p (mpz_t @var{n}, gmp_randstate_t @var{state}, mpir_ui @var{div})
- at cindex Prime testing functions
- at cindex Probable prime testing functions
-Determine whether @var{n} is likely a prime, i.e. you can consider it a prime for practical purposes.
-return value is 1 if @var{n} can be considered prime, or 0 if
- at var{n} is definitely composite.
-
-This function does some trial divisions to speed up the average case, then some probabilistic
-primality tests. The term ``likely'' refers to the fact that the number will not have small factors.
-
- at var{div} can be used to inform the function that trial division up to @var{div} has
-already been performed on @var{n} and so @var{n} has NO divisors <= @var{div}
-
- at strong{This function interface is preliminary and may change in the future.}
- at end deftypefun
-
- at deftypefun int mpz_probab_prime_p (mpz_t @var{n}, int @var{reps})
- at cindex Prime testing functions
- at cindex Probable prime testing functions
-Determine whether @var{n} is prime. Return 2 if @var{n} is definitely prime,
-return 1 if @var{n} is probably prime (without being certain), or return 0 if
- at var{n} is definitely composite.
-
-This function does some trial divisions, then some Miller-Rabin probabilistic
-primality tests. @var{reps} controls how many such tests are done, 5 to 10 is
-a reasonable number, more will reduce the chances of a composite being
-returned as ``probably prime''.
-
-Miller-Rabin and similar tests can be more properly called compositeness
-tests. Numbers which fail are known to be composite but those which pass
-might be prime or might be composite. Only a few composites pass, hence those
-which pass are considered probably prime.
-
- at strong{This function is obsolete. It will disappear from future MPIR releases.}
- at end deftypefun
-
- at deftypefun void mpz_nextprime (mpz_t @var{rop}, mpz_t @var{op})
- at cindex Next prime function
-Set @var{rop} to the next prime greater than @var{op}.
-
-This function uses a probabilistic algorithm to identify primes. For
-practical purposes it's adequate, the chance of a composite passing will be
-extremely small.
-
- at strong{This function is obsolete. It will disappear from future MPIR releases.}
- at end deftypefun
-
- at deftypefun void mpz_next_likely_prime (mpz_t @var{rop}, mpz_t @var{op}, gmp_randstate_t @var{state})
- at cindex Next likely prime function
-Set @var{rop} to the next likely prime greater than @var{op}.
-
-This function uses a probabilistic algorithm to identify primes. For
-practical purposes it's adequate, the chance of a composite passing will be
-extremely small.
-
-The variable @var{state} must be initialized by calling one of the
- at code{gmp_randinit} functions (@ref{Random State Initialization})
-before invoking this function.
- at end deftypefun
-
- at deftypefun void mpz_gcd (mpz_t @var{rop}, mpz_t @var{op1}, mpz_t @var{op2})
- at cindex Greatest common divisor functions
- at cindex GCD functions
-Set @var{rop} to the greatest common divisor of @var{op1} and @var{op2}.
-The result is always positive even if one or both input operands
-are negative.
- at end deftypefun
-
- at deftypefun mpir_ui mpz_gcd_ui (mpz_t @var{rop}, mpz_t @var{op1}, mpir_ui @var{op2})
-Compute the greatest common divisor of @var{op1} and @var{op2}. If
- at var{rop} is not @code{NULL}, store the result there.
-
-If the result is small enough to fit in an @code{mpir_ui}, it is
-returned. If the result does not fit, 0 is returned, and the result is equal
-to the argument @var{op1}. Note that the result will always fit if @var{op2}
-is non-zero.
- at end deftypefun
-
- at deftypefun void mpz_gcdext (mpz_t @var{g}, mpz_t @var{s}, mpz_t @var{t}, mpz_t @var{a}, mpz_t @var{b})
- at cindex Extended GCD
- at cindex GCD extended
-Set @var{g} to the greatest common divisor of @var{a} and @var{b}, and in
-addition set @var{s} and @var{t} to coefficients satisfying
- at math{@var{a}@GMPmultiply{}@var{s} + @var{b}@GMPmultiply{}@var{t} = @var{g}}.
- at var{g} is always positive, even if one or both of @var{a} and @var{b} are
-negative.
-
-If @var{t} is @code{NULL} then that value is not computed.
- at end deftypefun
-
- at deftypefun void mpz_lcm (mpz_t @var{rop}, mpz_t @var{op1}, mpz_t @var{op2})
- at deftypefunx void mpz_lcm_ui (mpz_t @var{rop}, mpz_t @var{op1}, mpir_ui @var{op2})
- at cindex Least common multiple functions
- at cindex LCM functions
-Set @var{rop} to the least common multiple of @var{op1} and @var{op2}.
- at var{rop} is always positive, irrespective of the signs of @var{op1} and
- at var{op2}. @var{rop} will be zero if either @var{op1} or @var{op2} is zero.
- at end deftypefun
-
- at deftypefun int mpz_invert (mpz_t @var{rop}, mpz_t @var{op1}, mpz_t @var{op2})
- at cindex Modular inverse functions
- at cindex Inverse modulo functions
-Compute the inverse of @var{op1} modulo @var{op2} and put the result in
- at var{rop}. If the inverse exists, the return value is non-zero and @var{rop}
-will satisfy @math{0 @le{} @var{rop} < @var{op2}}. If an inverse doesn't exist
-the return value is zero and @var{rop} is undefined.
- at end deftypefun
-
- at deftypefun int mpz_jacobi (mpz_t @var{a}, mpz_t @var{b})
- at cindex Jacobi symbol functions
-Calculate the Jacobi symbol @m{\left(a \over b\right),
-(@var{a}/@var{b})}. This is defined only for @var{b} odd.
- at end deftypefun
-
- at deftypefun int mpz_legendre (mpz_t @var{a}, mpz_t @var{p})
- at cindex Legendre symbol functions
-Calculate the Legendre symbol @m{\left(a \over p\right),
-(@var{a}/@var{p})}. This is defined only for @var{p} an odd positive
-prime, and for such @var{p} it's identical to the Jacobi symbol.
- at end deftypefun
-
- at deftypefun int mpz_kronecker (mpz_t @var{a}, mpz_t @var{b})
- at deftypefunx int mpz_kronecker_si (mpz_t @var{a}, mpir_si @var{b})
- at deftypefunx int mpz_kronecker_ui (mpz_t @var{a}, mpir_ui @var{b})
- at deftypefunx int mpz_si_kronecker (mpir_si @var{a}, mpz_t @var{b})
- at deftypefunx int mpz_ui_kronecker (mpir_ui @var{a}, mpz_t @var{b})
- at cindex Kronecker symbol functions
-Calculate the Jacobi symbol @m{\left(a \over b\right),
-(@var{a}/@var{b})} with the Kronecker extension @m{\left(a \over
-2\right) = \left(2 \over a\right), (a/2)=(2/a)} when @math{a} odd, or
- at m{\left(a \over 2\right) = 0, (a/2)=0} when @math{a} even.
-
-When @var{b} is odd the Jacobi symbol and Kronecker symbol are
-identical, so @code{mpz_kronecker_ui} etc can be used for mixed
-precision Jacobi symbols too.
-
-For more information see Henri Cohen section 1.4.2 (@pxref{References}),
-or any number theory textbook. See also the example program
- at file{demos/qcn.c} which uses @code{mpz_kronecker_ui} on the MPIR website.
- at end deftypefun
-
- at deftypefun {mp_bitcnt_t} mpz_remove (mpz_t @var{rop}, mpz_t @var{op}, mpz_t @var{f})
- at cindex Remove factor functions
- at cindex Factor removal functions
-Remove all occurrences of the factor @var{f} from @var{op} and store the
-result in @var{rop}. The return value is how many such occurrences were
-removed.
- at end deftypefun
-
- at deftypefun void mpz_fac_ui (mpz_t @var{rop}, mpir_ui @var{op})
- at cindex Factorial functions
-Set @var{rop} to @var{op}!, the factorial of @var{op}.
- at end deftypefun
-
- at deftypefun void mpz_bin_ui (mpz_t @var{rop}, mpz_t @var{n}, mpir_ui @var{k})
- at deftypefunx void mpz_bin_uiui (mpz_t @var{rop}, mpir_ui @var{n}, mpir_ui @var{k})
- at cindex Binomial coefficient functions
-Compute the binomial coefficient @m{\left({n}\atop{k}\right), @var{n} over
- at var{k}} and store the result in @var{rop}. Negative values of @var{n} are
-supported by @code{mpz_bin_ui}, using the identity
- at m{\left({-n}\atop{k}\right) = (-1)^k \left({n+k-1}\atop{k}\right),
-bin(-n at C{}k) = (-1)^k * bin(n+k-1 at C{}k)}, see Knuth volume 1 section 1.2.6
-part G.
- at end deftypefun
-
- at deftypefun void mpz_fib_ui (mpz_t @var{fn}, mpir_ui @var{n})
- at deftypefunx void mpz_fib2_ui (mpz_t @var{fn}, mpz_t @var{fnsub1}, mpir_ui @var{n})
- at cindex Fibonacci sequence functions
- at code{mpz_fib_ui} sets @var{fn} to to @m{F_n,F[n]}, the @var{n}'th Fibonacci
-number. @code{mpz_fib2_ui} sets @var{fn} to @m{F_n,F[n]}, and @var{fnsub1} to
- at m{F_{n-1},F[n-1]}.
-
-These functions are designed for calculating isolated Fibonacci numbers. When
-a sequence of values is wanted it's best to start with @code{mpz_fib2_ui} and
-iterate the defining @m{F_{n+1} = F_n + F_{n-1}, F[n+1]=F[n]+F[n-1]} or
-similar.
- at end deftypefun
-
- at deftypefun void mpz_lucnum_ui (mpz_t @var{ln}, mpir_ui @var{n})
- at deftypefunx void mpz_lucnum2_ui (mpz_t @var{ln}, mpz_t @var{lnsub1}, mpir_ui @var{n})
- at cindex Lucas number functions
- at code{mpz_lucnum_ui} sets @var{ln} to to @m{L_n,L[n]}, the @var{n}'th Lucas
-number. @code{mpz_lucnum2_ui} sets @var{ln} to @m{L_n,L[n]}, and @var{lnsub1}
-to @m{L_{n-1},L[n-1]}.
-
-These functions are designed for calculating isolated Lucas numbers. When a
-sequence of values is wanted it's best to start with @code{mpz_lucnum2_ui} and
-iterate the defining @m{L_{n+1} = L_n + L_{n-1}, L[n+1]=L[n]+L[n-1]} or
-similar.
-
-The Fibonacci numbers and Lucas numbers are related sequences, so it's never
-necessary to call both @code{mpz_fib2_ui} and @code{mpz_lucnum2_ui}. The
-formulas for going from Fibonacci to Lucas can be found in @ref{Lucas Numbers
-Algorithm}, the reverse is straightforward too.
- at end deftypefun
-
-
- at node Integer Comparisons, Integer Logic and Bit Fiddling, Number Theoretic Functions, Integer Functions
- at comment node-name, next, previous, up
- at section Comparison Functions
- at cindex Integer comparison functions
- at cindex Comparison functions
-
- at deftypefn Function int mpz_cmp (mpz_t @var{op1}, mpz_t @var{op2})
- at deftypefnx Function int mpz_cmp_d (mpz_t @var{op1}, double @var{op2})
- at deftypefnx Macro int mpz_cmp_si (mpz_t @var{op1}, mpir_si @var{op2})
- at deftypefnx Macro int mpz_cmp_ui (mpz_t @var{op1}, mpir_ui @var{op2})
-Compare @var{op1} and @var{op2}. Return a positive value if @math{@var{op1} >
- at var{op2}}, zero if @math{@var{op1} = @var{op2}}, or a negative value if
- at math{@var{op1} < @var{op2}}.
-
- at code{mpz_cmp_ui} and @code{mpz_cmp_si} are macros and will evaluate their
-arguments more than once. @code{mpz_cmp_d} can be called with an infinity,
-but results are undefined for a NaN.
- at end deftypefn
-
- at deftypefn Function int mpz_cmpabs (mpz_t @var{op1}, mpz_t @var{op2})
- at deftypefnx Function int mpz_cmpabs_d (mpz_t @var{op1}, double @var{op2})
- at deftypefnx Function int mpz_cmpabs_ui (mpz_t @var{op1}, mpir_ui @var{op2})
-Compare the absolute values of @var{op1} and @var{op2}. Return a positive
-value if @math{@GMPabs{@var{op1}} > @GMPabs{@var{op2}}}, zero if
- at math{@GMPabs{@var{op1}} = @GMPabs{@var{op2}}}, or a negative value if
- at math{@GMPabs{@var{op1}} < @GMPabs{@var{op2}}}.
-
- at code{mpz_cmpabs_d} can be called with an infinity, but results are undefined
-for a NaN.
- at end deftypefn
-
- at deftypefn Macro int mpz_sgn (mpz_t @var{op})
- at cindex Sign tests
- at cindex Integer sign tests
-Return @math{+1} if @math{@var{op} > 0}, 0 if @math{@var{op} = 0}, and
- at math{-1} if @math{@var{op} < 0}.
-
-This function is actually implemented as a macro. It evaluates its argument
-multiple times.
- at end deftypefn
-
-
- at node Integer Logic and Bit Fiddling, I/O of Integers, Integer Comparisons, Integer Functions
- at comment node-name, next, previous, up
- at section Logical and Bit Manipulation Functions
- at cindex Logical functions
- at cindex Bit manipulation functions
- at cindex Integer logical functions
- at cindex Integer bit manipulation functions
-
-These functions behave as if twos complement arithmetic were used (although
-sign-magnitude is the actual implementation). The least significant bit is
-number 0.
-
- at deftypefun void mpz_and (mpz_t @var{rop}, mpz_t @var{op1}, mpz_t @var{op2})
-Set @var{rop} to @var{op1} bitwise-and @var{op2}.
- at end deftypefun
-
- at deftypefun void mpz_ior (mpz_t @var{rop}, mpz_t @var{op1}, mpz_t @var{op2})
-Set @var{rop} to @var{op1} bitwise inclusive-or @var{op2}.
- at end deftypefun
-
- at deftypefun void mpz_xor (mpz_t @var{rop}, mpz_t @var{op1}, mpz_t @var{op2})
-Set @var{rop} to @var{op1} bitwise exclusive-or @var{op2}.
- at end deftypefun
-
- at deftypefun void mpz_com (mpz_t @var{rop}, mpz_t @var{op})
-Set @var{rop} to the one's complement of @var{op}.
- at end deftypefun
-
- at deftypefun {mp_bitcnt_t} mpz_popcount (mpz_t @var{op})
-If @math{@var{op}@ge{}0}, return the population count of @var{op}, which is
-the number of 1 bits in the binary representation. If @math{@var{op}<0}, the
-number of 1s is infinite, and the return value is @var{ULONG_MAX}, the largest
-possible @code{mp_bitcnt_t}.
- at end deftypefun
-
- at deftypefun {mp_bitcnt_t} mpz_hamdist (mpz_t @var{op1}, mpz_t @var{op2})
-If @var{op1} and @var{op2} are both @math{@ge{}0} or both @math{<0}, return
-the hamming distance between the two operands, which is the number of bit
-positions where @var{op1} and @var{op2} have different bit values. If one
-operand is @math{@ge{}0} and the other @math{<0} then the number of bits
-different is infinite, and the return value is the largest
-possible @code{imp_bitcnt_t}.
- at end deftypefun
-
- at deftypefun {mp_bitcnt_t} mpz_scan0 (mpz_t @var{op}, mp_bitcnt_t @var{starting_bit})
- at deftypefunx mp_bitcnt_t mpz_scan1 (mpz_t @var{op}, mp_bitcnt_t @var{starting_bit})
- at cindex Bit scanning functions
- at cindex Scan bit functions
-Scan @var{op}, starting from bit @var{starting_bit}, towards more significant
-bits, until the first 0 or 1 bit (respectively) is found. Return the index of
-the found bit.
-
-If the bit at @var{starting_bit} is already what's sought, then
- at var{starting_bit} is returned.
-
-If there's no bit found, then the largest possible @code{mp_bitcnt_t} is
-returned. This will happen in @code{mpz_scan0} past the end of a positive
-number, or @code{mpz_scan1} past the end of a nonnegative number.
- at end deftypefun
-
- at deftypefun void mpz_setbit (mpz_t @var{rop}, mp_bitcnt_t @var{bit_index})
-Set bit @var{bit_index} in @var{rop}.
- at end deftypefun
-
- at deftypefun void mpz_clrbit (mpz_t @var{rop}, mp_bitcnt_t @var{bit_index})
-Clear bit @var{bit_index} in @var{rop}.
- at end deftypefun
-
- at deftypefun void mpz_combit (mpz_t @var{rop}, mp_bitcnt_t @var{bit_index})
-Complement bit @var{bit_index} in @var{rop}.
- at end deftypefun
-
- at deftypefun int mpz_tstbit (mpz_t @var{op}, mp_bitcnt_t @var{bit_index})
-Test bit @var{bit_index} in @var{op} and return 0 or 1 accordingly.
- at end deftypefun
-
- at node I/O of Integers, Integer Random Numbers, Integer Logic and Bit Fiddling, Integer Functions
- at comment node-name, next, previous, up
- at section Input and Output Functions
- at cindex Integer input and output functions
- at cindex Input functions
- at cindex Output functions
- at cindex I/O functions
-
-Functions that perform input from a stdio stream, and functions that output to
-a stdio stream. Passing a @code{NULL} pointer for a @var{stream} argument to any of
-these functions will make them read from @code{stdin} and write to
- at code{stdout}, respectively.
-
-When using any of these functions, it is a good idea to include @file{stdio.h}
-before @file{mpir.h}, since that will allow @file{mpir.h} to define prototypes
-for these functions.
-
- at deftypefun size_t mpz_out_str (FILE *@var{stream}, int @var{base}, mpz_t @var{op})
-Output @var{op} on stdio stream @var{stream}, as a string of digits in base
- at var{base}. The base argument may vary from 2 to 62 or from @minus{}2 to
- at minus{}36.
-
-For @var{base} in the range 2..36, digits and lower-case letters are used; for
- at minus{}2.. at minus{}36, digits and upper-case letters are used; for 37..62,
-digits, upper-case letters, and lower-case letters (in that significance order)
-are used.
-
-Return the number of bytes written, or if an error occurred, return 0.
- at end deftypefun
-
- at deftypefun size_t mpz_inp_str (mpz_t @var{rop}, FILE *@var{stream}, int @var{base})
-Input a possibly white-space preceded string in base @var{base} from stdio
-stream @var{stream}, and put the read integer in @var{rop}.
-
-The @var{base} may vary from 2 to 62, or if @var{base} is 0, then the leading
-characters are used: @code{0x} and @code{0X} for hexadecimal, @code{0b} and
- at code{0B} for binary, @code{0} for octal, or decimal otherwise.
-
-For bases up to 36, case is ignored; upper-case and lower-case letters have
-the same value. For bases 37 to 62, upper-case letter represent the usual
-10..35 while lower-case letter represent 36..61.
-
-Return the number of bytes read, or if an error occurred, return 0.
- at end deftypefun
-
- at deftypefun size_t mpz_out_raw (FILE *@var{stream}, mpz_t @var{op})
-Output @var{op} on stdio stream @var{stream}, in raw binary format. The
-integer is written in a portable format, with 4 bytes of size information, and
-that many bytes of limbs. Both the size and the limbs are written in
-decreasing significance order (i.e., in big-endian).
-
-The output can be read with @code{mpz_inp_raw}.
-
-Return the number of bytes written, or if an error occurred, return 0.
-
-The output of this can not be read by @code{mpz_inp_raw} from GMP 1, because
-of changes necessary for compatibility between 32-bit and 64-bit machines.
- at end deftypefun
-
- at deftypefun size_t mpz_inp_raw (mpz_t @var{rop}, FILE *@var{stream})
-Input from stdio stream @var{stream} in the format written by
- at code{mpz_out_raw}, and put the result in @var{rop}. Return the number of
-bytes read, or if an error occurred, return 0.
-
-This routine can read the output from @code{mpz_out_raw} also from GMP 1, in
-spite of changes necessary for compatibility between 32-bit and 64-bit
-machines.
- at end deftypefun
-
-
- at need 2000
- at node Integer Random Numbers, Integer Import and Export, I/O of Integers, Integer Functions
- at comment node-name, next, previous, up
- at section Random Number Functions
- at cindex Integer random number functions
- at cindex Random number functions
-
-The random number functions of MPIR come in two groups; older function
-that rely on a global state, and newer functions that accept a state
-parameter that is read and modified. Please see the @ref{Random Number
-Functions} for more information on how to use and not to use random
-number functions.
-
- at deftypefun void mpz_urandomb (mpz_t @var{rop}, gmp_randstate_t @var{state}, mp_bitcnt_t @var{n})
-Generate a uniformly distributed random integer in the range 0 to @m{2^n-1,
-2^@var{n}@minus{}1}, inclusive.
-
-The variable @var{state} must be initialized by calling one of the
- at code{gmp_randinit} functions (@ref{Random State Initialization})
-before invoking this function.
- at end deftypefun
-
- at deftypefun void mpz_urandomm (mpz_t @var{rop}, gmp_randstate_t @var{state}, mpz_t @var{n})
-Generate a uniform random integer in the range 0 to @math{@var{n}-1},
-inclusive.
-
-The variable @var{state} must be initialized by calling one of the
- at code{gmp_randinit} functions (@ref{Random State Initialization})
-before invoking this function.
- at end deftypefun
-
- at deftypefun void mpz_rrandomb (mpz_t @var{rop}, gmp_randstate_t @var{state}, mp_bitcnt_t @var{n})
-Generate a random integer with long strings of zeros and ones in the
-binary representation. Useful for testing functions and algorithms,
-since this kind of random numbers have proven to be more likely to
-trigger corner-case bugs. The random number will be in the range
-0 to @m{2^n-1, 2^@var{n}@minus{}1}, inclusive.
-
-The variable @var{state} must be initialized by calling one of the
- at code{gmp_randinit} functions (@ref{Random State Initialization})
-before invoking this function.
- at end deftypefun
-
- at node Integer Import and Export, Miscellaneous Integer Functions, Integer Random Numbers, Integer Functions
- at section Integer Import and Export
-
- at code{mpz_t} variables can be converted to and from arbitrary words of binary
-data with the following functions.
-
- at deftypefun void mpz_import (mpz_t @var{rop}, size_t @var{count}, int @var{order}, size_t @var{size}, int @var{endian}, size_t @var{nails}, const void *@var{op})
- at cindex Integer import
- at cindex Import
-Set @var{rop} from an array of word data at @var{op}.
-
-The parameters specify the format of the data. @var{count} many words are
-read, each @var{size} bytes. @var{order} can be 1 for most significant word
-first or -1 for least significant first. Within each word @var{endian} can be
-1 for most significant byte first, -1 for least significant first, or 0 for
-the native endianness of the host CPU at . The most significant @var{nails} bits
-of each word are skipped, this can be 0 to use the full words.
-
-There is no sign taken from the data, @var{rop} will simply be a positive
-integer. An application can handle any sign itself, and apply it for instance
-with @code{mpz_neg}.
-
-There are no data alignment restrictions on @var{op}, any address is allowed.
-
-Here's an example converting an array of @code{mpir_ui} data, most
-significant element first, and host byte order within each value.
-
- at example
-mpir_ui a[20];
-mpz_t z;
-mpz_import (z, 20, 1, sizeof(a[0]), 0, 0, a);
- at end example
-
-This example assumes the full @code{sizeof} bytes are used for data in the
-given type, which is usually true, and certainly true for @code{mpir_ui}
-everywhere we know of. However on Cray vector systems it may be noted that
- at code{short} and @code{int} are always stored in 8 bytes (and with
- at code{sizeof} indicating that) but use only 32 or 46 bits. The @var{nails}
-feature can account for this, by passing for instance
- at code{8*sizeof(int)-INT_BIT}.
- at end deftypefun
-
- at deftypefun {void *} mpz_export (void *@var{rop}, size_t *@var{countp}, int @var{order}, size_t @var{size}, int @var{endian}, size_t @var{nails}, mpz_t @var{op})
- at cindex Integer export
- at cindex Export
-Fill @var{rop} with word data from @var{op}.
-
-The parameters specify the format of the data produced. Each word will be
- at var{size} bytes and @var{order} can be 1 for most significant word first or
--1 for least significant first. Within each word @var{endian} can be 1 for
-most significant byte first, -1 for least significant first, or 0 for the
-native endianness of the host CPU at . The most significant @var{nails} bits of
-each word are unused and set to zero, this can be 0 to produce full words.
-
-The number of words produced is written to @code{*@var{countp}}, or
- at var{countp} can be @code{NULL} to discard the count. @var{rop} must have
-enough space for the data, or if @var{rop} is @code{NULL} then a result array
-of the necessary size is allocated using the current MPIR allocation function
-(@pxref{Custom Allocation}). In either case the return value is the
-destination used, either @var{rop} or the allocated block.
-
-If @var{op} is non-zero then the most significant word produced will be
-non-zero. If @var{op} is zero then the count returned will be zero and
-nothing written to @var{rop}. If @var{rop} is @code{NULL} in this case, no
-block is allocated, just @code{NULL} is returned.
-
-The sign of @var{op} is ignored, just the absolute value is exported. An
-application can use @code{mpz_sgn} to get the sign and handle it as desired.
-(@pxref{Integer Comparisons})
-
-There are no data alignment restrictions on @var{rop}, any address is allowed.
-
-When an application is allocating space itself the required size can be
-determined with a calculation like the following. Since @code{mpz_sizeinbase}
-always returns at least 1, @code{count} here will be at least one, which
-avoids any portability problems with @code{malloc(0)}, though if @code{z} is
-zero no space at all is actually needed (or written).
-
- at example
-numb = 8*size - nail;
-count = (mpz_sizeinbase (z, 2) + numb-1) / numb;
-p = malloc (count * size);
- at end example
- at end deftypefun
-
-
- at need 2000
- at node Miscellaneous Integer Functions, Integer Special Functions, Integer Import and Export, Integer Functions
- at comment node-name, next, previous, up
- at section Miscellaneous Functions
- at cindex Miscellaneous integer functions
- at cindex Integer miscellaneous functions
-
- at deftypefun int mpz_fits_ulong_p (mpz_t @var{op})
- at deftypefunx int mpz_fits_slong_p (mpz_t @var{op})
- at deftypefunx int mpz_fits_uint_p (mpz_t @var{op})
- at deftypefunx int mpz_fits_sint_p (mpz_t @var{op})
- at deftypefunx int mpz_fits_ushort_p (mpz_t @var{op})
- at deftypefunx int mpz_fits_sshort_p (mpz_t @var{op})
-Return non-zero iff the value of @var{op} fits in an @code{unsigned long},
- at code{long}, @code{unsigned int}, @code{signed int}, @code{unsigned
-short int}, or @code{signed short int}, respectively. Otherwise, return zero.
- at end deftypefun
-
- at deftypefn Macro int mpz_odd_p (mpz_t @var{op})
- at deftypefnx Macro int mpz_even_p (mpz_t @var{op})
-Determine whether @var{op} is odd or even, respectively. Return non-zero if
-yes, zero if no. These macros evaluate their argument more than once.
- at end deftypefn
-
- at deftypefun size_t mpz_sizeinbase (mpz_t @var{op}, int @var{base})
- at cindex Size in digits
- at cindex Digits in an integer
-Return the size of @var{op} measured in number of digits in the given
- at var{base}. @var{base} can vary from 2 to 36. The sign of @var{op} is
-ignored, just the absolute value is used. The result will be either exact or
-1 too big. If @var{base} is a power of 2, the result is always exact. If
- at var{op} is zero the return value is always 1.
-
-This function can be used to determine the space required when converting
- at var{op} to a string. The right amount of allocation is normally two more
-than the value returned by @code{mpz_sizeinbase}, one extra for a minus sign
-and one for the null-terminator.
-
- at cindex Most significant bit
-It will be noted that @code{mpz_sizeinbase(@var{op},2)} can be used to locate
-the most significant 1 bit in @var{op}, counting from 1. (Unlike the bitwise
-functions which start from 0, @xref{Integer Logic and Bit Fiddling,, Logical
-and Bit Manipulation Functions}.)
- at end deftypefun
-
-
- at node Integer Special Functions, , Miscellaneous Integer Functions, Integer Functions
- at section Special Functions
- at cindex Special integer functions
- at cindex Integer special functions
-
-The functions in this section are for various special purposes. Most
-applications will not need them.
-
- at deftypefun void mpz_array_init (mpz_t @var{integer_array}, size_t @var{array_size}, @w{mp_size_t @var{fixed_num_bits}})
-This is a special type of initialization. @strong{Fixed} space of
- at var{fixed_num_bits} is allocated to each of the @var{array_size} integers in
- at var{integer_array}. There is no way to free the storage allocated by this
-function. Don't call @code{mpz_clear}!
-
-The @var{integer_array} parameter is the first @code{mpz_t} in the array. For
-example,
-
- at example
-mpz_t arr[20000];
-mpz_array_init (arr[0], 20000, 512);
- at end example
-
- at c In case anyone's wondering, yes this parameter style is a bit anomalous,
- at c it'd probably be nicer if it was "arr" instead of "arr[0]". Obviously the
- at c two differ only in the declaration, not the pointer value, but changing is
- at c not possible since it'd provoke warnings or errors in existing sources.
-
-This function is only intended for programs that create a large number
-of integers and need to reduce memory usage by avoiding the overheads of
-allocating and reallocating lots of small blocks. In normal programs this
-function is not recommended.
-
-The space allocated to each integer by this function will not be automatically
-increased, unlike the normal @code{mpz_init}, so an application must ensure it
-is sufficient for any value stored. The following space requirements apply to
-various routines,
-
- at itemize @bullet
- at item
- at code{mpz_abs}, @code{mpz_neg}, @code{mpz_set}, @code{mpz_set_si} and
- at code{mpz_set_ui} need room for the value they store.
-
- at item
- at code{mpz_add}, @code{mpz_add_ui}, @code{mpz_sub} and @code{mpz_sub_ui} need
-room for the larger of the two operands, plus an extra
- at code{mp_bits_per_limb}.
-
- at item
- at code{mpz_mul}, @code{mpz_mul_ui} and @code{mpz_mul_ui} need room for the sum
-of the number of bits in their operands, but each rounded up to a multiple of
- at code{mp_bits_per_limb}.
-
- at item
- at code{mpz_swap} can be used between two array variables, but not between an
-array and a normal variable.
- at end itemize
-
-For other functions, or if in doubt, the suggestion is to calculate in a
-regular @code{mpz_init} variable and copy the result to an array variable with
- at code{mpz_set}.
-
- at strong{This function is obsolete. It will disappear from future MPIR releases.}
- at end deftypefun
-
- at deftypefun {void *} _mpz_realloc (mpz_t @var{integer}, mp_size_t @var{new_alloc})
-Change the space for @var{integer} to @var{new_alloc} limbs. The value in
- at var{integer} is preserved if it fits, or is set to 0 if not. The return
-value is not useful to applications and should be ignored.
-
- at code{mpz_realloc2} is the preferred way to accomplish allocation changes like
-this. @code{mpz_realloc2} and @code{_mpz_realloc} are the same except that
- at code{_mpz_realloc} takes its size in limbs.
- at end deftypefun
-
- at deftypefun mp_limb_t mpz_getlimbn (mpz_t @var{op}, mp_size_t @var{n})
-Return limb number @var{n} from @var{op}. The sign of @var{op} is ignored,
-just the absolute value is used. The least significant limb is number 0.
-
- at code{mpz_size} can be used to find how many limbs make up @var{op}.
- at code{mpz_getlimbn} returns zero if @var{n} is outside the range 0 to
- at code{mpz_size(@var{op})-1}.
- at end deftypefun
-
- at deftypefun size_t mpz_size (mpz_t @var{op})
-Return the size of @var{op} measured in number of limbs. If @var{op} is zero,
-the returned value will be zero.
- at c (@xref{Nomenclature}, for an explanation of the concept @dfn{limb}.)
- at end deftypefun
-
-
-
- at node Rational Number Functions, Floating-point Functions, Integer Functions, Top
- at comment node-name, next, previous, up
- at chapter Rational Number Functions
- at cindex Rational number functions
-
-This chapter describes the MPIR functions for performing arithmetic on rational
-numbers. These functions start with the prefix @code{mpq_}.
-
-Rational numbers are stored in objects of type @code{mpq_t}.
-
-All rational arithmetic functions assume operands have a canonical form, and
-canonicalize their result. The canonical from means that the denominator and
-the numerator have no common factors, and that the denominator is positive.
-Zero has the unique representation 0/1.
-
-Pure assignment functions do not canonicalize the assigned variable. It is
-the responsibility of the user to canonicalize the assigned variable before
-any arithmetic operations are performed on that variable.
-
- at deftypefun void mpq_canonicalize (mpq_t @var{op})
-Remove any factors that are common to the numerator and denominator of
- at var{op}, and make the denominator positive.
- at end deftypefun
-
- at menu
-* Initializing Rationals::
-* Rational Conversions::
-* Rational Arithmetic::
-* Comparing Rationals::
-* Applying Integer Functions::
-* I/O of Rationals::
- at end menu
-
- at node Initializing Rationals, Rational Conversions, Rational Number Functions, Rational Number Functions
- at comment node-name, next, previous, up
- at section Initialization and Assignment Functions
- at cindex Rational assignment functions
- at cindex Assignment functions
- at cindex Rational initialization functions
- at cindex Initialization functions
-
- at deftypefun void mpq_init (mpq_t @var{dest_rational})
-Initialize @var{dest_rational} and set it to 0/1. Each variable should
-normally only be initialized once, or at least cleared out (using the function
- at code{mpq_clear}) between each initialization.
- at end deftypefun
-
- at deftypefun void mpq_inits (mpq_t @var{x}, ...)
-Initialize a NULL-terminated list of @code{mpq_t} variables, and set their
-values to 0/1.
- at end deftypefun
-
- at deftypefun void mpq_clear (mpq_t @var{rational_number})
-Free the space occupied by @var{rational_number}. Make sure to call this
-function for all @code{mpq_t} variables when you are done with them.
- at end deftypefun
-
- at deftypefun void mpq_clears (mpq_t @var{x}, ...)
-Free the space occupied by a NULL-terminated list of @code{mpq_t} variables.
- at end deftypefun
-
- at deftypefun void mpq_set (mpq_t @var{rop}, mpq_t @var{op})
- at deftypefunx void mpq_set_z (mpq_t @var{rop}, mpz_t @var{op})
-Assign @var{rop} from @var{op}.
- at end deftypefun
-
- at deftypefun void mpq_set_ui (mpq_t @var{rop}, mpir_ui @var{op1}, mpir_ui @var{op2})
- at deftypefunx void mpq_set_si (mpq_t @var{rop}, mpir_si @var{op1}, mpir_ui @var{op2})
-Set the value of @var{rop} to @var{op1}/@var{op2}. Note that if @var{op1} and
- at var{op2} have common factors, @var{rop} has to be passed to
- at code{mpq_canonicalize} before any operations are performed on @var{rop}.
- at end deftypefun
-
- at deftypefun int mpq_set_str (mpq_t @var{rop}, char *@var{str}, int @var{base})
-Set @var{rop} from a null-terminated string @var{str} in the given @var{base}.
-
-The string can be an integer like ``41'' or a fraction like ``41/152''. The
-fraction must be in canonical form (@pxref{Rational Number Functions}), or if
-not then @code{mpq_canonicalize} must be called.
-
-The numerator and optional denominator are parsed the same as in
- at code{mpz_set_str} (@pxref{Assigning Integers}). White space is allowed in
-the string, and is simply ignored. The @var{base} can vary from 2 to 62, or
-if @var{base} is 0 then the leading characters are used: @code{0x} or @code{0X} for hex,
- at code{0b} or @code{0B} for binary,
- at code{0} for octal, or decimal otherwise. Note that this is done separately
-for the numerator and denominator, so for instance @code{0xEF/100} is 239/100,
-whereas @code{0xEF/0x100} is 239/256.
-
-The return value is 0 if the entire string is a valid number, or @minus{}1 if
-not.
- at end deftypefun
-
- at deftypefun void mpq_swap (mpq_t @var{rop1}, mpq_t @var{rop2})
-Swap the values @var{rop1} and @var{rop2} efficiently.
- at end deftypefun
-
-
- at need 2000
- at node Rational Conversions, Rational Arithmetic, Initializing Rationals, Rational Number Functions
- at comment node-name, next, previous, up
- at section Conversion Functions
- at cindex Rational conversion functions
- at cindex Conversion functions
-
- at deftypefun double mpq_get_d (mpq_t @var{op})
-Convert @var{op} to a @code{double}, truncating if necessary (ie.@: rounding
-towards zero).
-
-If the exponent from the conversion is too big or too small to fit a
- at code{double} then the result is system dependent. For too big an infinity is
-returned when available. For too small @math{0.0} is normally returned.
-Hardware overflow, underflow and denorm traps may or may not occur.
- at end deftypefun
-
- at deftypefun void mpq_set_d (mpq_t @var{rop}, double @var{op})
- at deftypefunx void mpq_set_f (mpq_t @var{rop}, mpf_t @var{op})
-Set @var{rop} to the value of @var{op}. There is no rounding, this conversion
-is exact.
- at end deftypefun
-
- at deftypefun {char *} mpq_get_str (char *@var{str}, int @var{base}, mpq_t @var{op})
-Convert @var{op} to a string of digits in base @var{base}. The base may vary
-from 2 to 36. The string will be of the form @samp{num/den}, or if the
-denominator is 1 then just @samp{num}.
-
-If @var{str} is @code{NULL}, the result string is allocated using the current
-allocation function (@pxref{Custom Allocation}). The block will be
- at code{strlen(str)+1} bytes, that being exactly enough for the string and
-null-terminator.
-
-If @var{str} is not @code{NULL}, it should point to a block of storage large
-enough for the result, that being
-
- at example
-mpz_sizeinbase (mpq_numref(@var{op}), @var{base})
-+ mpz_sizeinbase (mpq_denref(@var{op}), @var{base}) + 3
- at end example
-
-The three extra bytes are for a possible minus sign, possible slash, and the
-null-terminator.
-
-A pointer to the result string is returned, being either the allocated block,
-or the given @var{str}.
- at end deftypefun
-
-
- at node Rational Arithmetic, Comparing Rationals, Rational Conversions, Rational Number Functions
- at comment node-name, next, previous, up
- at section Arithmetic Functions
- at cindex Rational arithmetic functions
- at cindex Arithmetic functions
-
- at deftypefun void mpq_add (mpq_t @var{sum}, mpq_t @var{addend1}, mpq_t @var{addend2})
-Set @var{sum} to @var{addend1} + @var{addend2}.
- at end deftypefun
-
- at deftypefun void mpq_sub (mpq_t @var{difference}, mpq_t @var{minuend}, mpq_t @var{subtrahend})
-Set @var{difference} to @var{minuend} @minus{} @var{subtrahend}.
- at end deftypefun
-
- at deftypefun void mpq_mul (mpq_t @var{product}, mpq_t @var{multiplier}, mpq_t @var{multiplicand})
-Set @var{product} to @math{@var{multiplier} @GMPtimes{} @var{multiplicand}}.
- at end deftypefun
-
- at deftypefun void mpq_mul_2exp (mpq_t @var{rop}, mpq_t @var{op1}, mp_bitcnt_t @var{op2})
-Set @var{rop} to @m{@var{op1} \times 2^{op2}, @var{op1} times 2 raised to
- at var{op2}}.
- at end deftypefun
-
- at deftypefun void mpq_div (mpq_t @var{quotient}, mpq_t @var{dividend}, mpq_t @var{divisor})
- at cindex Division functions
-Set @var{quotient} to @var{dividend}/@var{divisor}.
- at end deftypefun
-
- at deftypefun void mpq_div_2exp (mpq_t @var{rop}, mpq_t @var{op1}, mp_bitcnt_t @var{op2})
-Set @var{rop} to @m{@var{op1}/2^{op2}, @var{op1} divided by 2 raised to
- at var{op2}}.
- at end deftypefun
-
- at deftypefun void mpq_neg (mpq_t @var{negated_operand}, mpq_t @var{operand})
-Set @var{negated_operand} to @minus{}@var{operand}.
- at end deftypefun
-
- at deftypefun void mpq_abs (mpq_t @var{rop}, mpq_t @var{op})
-Set @var{rop} to the absolute value of @var{op}.
- at end deftypefun
-
- at deftypefun void mpq_inv (mpq_t @var{inverted_number}, mpq_t @var{number})
-Set @var{inverted_number} to 1/@var{number}. If the new denominator is
-zero, this routine will divide by zero.
- at end deftypefun
-
- at node Comparing Rationals, Applying Integer Functions, Rational Arithmetic, Rational Number Functions
- at comment node-name, next, previous, up
- at section Comparison Functions
- at cindex Rational comparison functions
- at cindex Comparison functions
-
- at deftypefun int mpq_cmp (mpq_t @var{op1}, mpq_t @var{op2})
-Compare @var{op1} and @var{op2}. Return a positive value if @math{@var{op1} >
- at var{op2}}, zero if @math{@var{op1} = @var{op2}}, and a negative value if
- at math{@var{op1} < @var{op2}}.
-
-To determine if two rationals are equal, @code{mpq_equal} is faster than
- at code{mpq_cmp}.
- at end deftypefun
-
- at deftypefn Macro int mpq_cmp_ui (mpq_t @var{op1}, mpir_ui @var{num2}, mpir_ui @var{den2})
- at deftypefnx Macro int mpq_cmp_si (mpq_t @var{op1}, mpir_si @var{num2}, mpir_ui @var{den2})
-Compare @var{op1} and @var{num2}/@var{den2}. Return a positive value if
- at math{@var{op1} > @var{num2}/@var{den2}}, zero if @math{@var{op1} =
- at var{num2}/@var{den2}}, and a negative value if @math{@var{op1} <
- at var{num2}/@var{den2}}.
-
- at var{num2} and @var{den2} are allowed to have common factors.
-
-These functions are implemented as a macros and evaluate their arguments
-multiple times.
- at end deftypefn
-
- at deftypefn Macro int mpq_sgn (mpq_t @var{op})
- at cindex Sign tests
- at cindex Rational sign tests
-Return @math{+1} if @math{@var{op} > 0}, 0 if @math{@var{op} = 0}, and
- at math{-1} if @math{@var{op} < 0}.
-
-This function is actually implemented as a macro. It evaluates its
-arguments multiple times.
- at end deftypefn
-
- at deftypefun int mpq_equal (mpq_t @var{op1}, mpq_t @var{op2})
-Return non-zero if @var{op1} and @var{op2} are equal, zero if they are
-non-equal. Although @code{mpq_cmp} can be used for the same purpose, this
-function is much faster.
- at end deftypefun
-
- at node Applying Integer Functions, I/O of Rationals, Comparing Rationals, Rational Number Functions
- at comment node-name, next, previous, up
- at section Applying Integer Functions to Rationals
- at cindex Rational numerator and denominator
- at cindex Numerator and denominator
-
-The set of @code{mpq} functions is quite small. In particular, there are few
-functions for either input or output. The following functions give direct
-access to the numerator and denominator of an @code{mpq_t}.
-
-Note that if an assignment to the numerator and/or denominator could take an
- at code{mpq_t} out of the canonical form described at the start of this chapter
-(@pxref{Rational Number Functions}) then @code{mpq_canonicalize} must be
-called before any other @code{mpq} functions are applied to that @code{mpq_t}.
-
- at deftypefn Macro mpz_t mpq_numref (mpq_t @var{op})
- at deftypefnx Macro mpz_t mpq_denref (mpq_t @var{op})
-Return a reference to the numerator and denominator of @var{op}, respectively.
-The @code{mpz} functions can be used on the result of these macros.
- at end deftypefn
-
- at deftypefun void mpq_get_num (mpz_t @var{numerator}, mpq_t @var{rational})
- at deftypefunx void mpq_get_den (mpz_t @var{denominator}, mpq_t @var{rational})
- at deftypefunx void mpq_set_num (mpq_t @var{rational}, mpz_t @var{numerator})
- at deftypefunx void mpq_set_den (mpq_t @var{rational}, mpz_t @var{denominator})
-Get or set the numerator or denominator of a rational. These functions are
-equivalent to calling @code{mpz_set} with an appropriate @code{mpq_numref} or
- at code{mpq_denref}. Direct use of @code{mpq_numref} or @code{mpq_denref} is
-recommended instead of these functions.
- at end deftypefun
-
-
- at need 2000
- at node I/O of Rationals, , Applying Integer Functions, Rational Number Functions
- at comment node-name, next, previous, up
- at section Input and Output Functions
- at cindex Rational input and output functions
- at cindex Input functions
- at cindex Output functions
- at cindex I/O functions
-
-When using any of these functions, it's a good idea to include @file{stdio.h}
-before @file{mpir.h}, since that will allow @file{mpir.h} to define prototypes
-for these functions.
-
-Passing a @code{NULL} pointer for a @var{stream} argument to any of these
-functions will make them read from @code{stdin} and write to @code{stdout},
-respectively.
-
- at deftypefun size_t mpq_out_str (FILE *@var{stream}, int @var{base}, mpq_t @var{op})
-Output @var{op} on stdio stream @var{stream}, as a string of digits in base
- at var{base}. The base may vary from 2 to 36. Output is in the form
- at samp{num/den} or if the denominator is 1 then just @samp{num}.
-
-Return the number of bytes written, or if an error occurred, return 0.
- at end deftypefun
-
- at deftypefun size_t mpq_inp_str (mpq_t @var{rop}, FILE *@var{stream}, int @var{base})
-Read a string of digits from @var{stream} and convert them to a rational in
- at var{rop}. Any initial white-space characters are read and discarded. Return
-the number of characters read (including white space), or 0 if a rational
-could not be read.
-
-The input can be a fraction like @samp{17/63} or just an integer like
- at samp{123}. Reading stops at the first character not in this form, and white
-space is not permitted within the string. If the input might not be in
-canonical form, then @code{mpq_canonicalize} must be called (@pxref{Rational
-Number Functions}).
-
-The @var{base} can be between 2 and 36, or can be 0 in which case the leading
-characters of the string determine the base, @samp{0x} or @samp{0X} for
-hexadecimal, @samp{0} for octal, or decimal otherwise. The leading characters
-are examined separately for the numerator and denominator of a fraction, so
-for instance @samp{0x10/11} is @math{16/11}, whereas @samp{0x10/0x11} is
- at math{16/17}.
- at end deftypefun
-
-
- at node Floating-point Functions, Low-level Functions, Rational Number Functions, Top
- at comment node-name, next, previous, up
- at chapter Floating-point Functions
- at cindex Floating-point functions
- at cindex Float functions
- at cindex User-defined precision
- at cindex Precision of floats
-
-MPIR floating point numbers are stored in objects of type @code{mpf_t} and
-functions operating on them have an @code{mpf_} prefix.
-
-The mantissa of each float has a user-selectable precision, limited only by
-available memory. Each variable has its own precision, and that can be
-increased or decreased at any time.
-
-The exponent of each float is a fixed precision, one machine word on most
-systems. In the current implementation the exponent is a count of limbs, so
-for example on a 32-bit system this means a range of roughly
- at math{2^@W{-68719476768}} to @math{2^@W{68719476736}}, or on a 64-bit system
-this will be greater. Note however @code{mpf_get_str} can only return an
-exponent which fits an @code{mp_exp_t} and currently @code{mpf_set_str}
-doesn't accept exponents bigger than a @code{mpir_si}.
-
-Each variable keeps a size for the mantissa data actually in use. This means
-that if a float is exactly represented in only a few bits then only those bits
-will be used in a calculation, even if the selected precision is high.
-
-All calculations are performed to the precision of the destination variable.
-Each function is defined to calculate with ``infinite precision'' followed by
-a truncation to the destination precision, but of course the work done is only
-what's needed to determine a result under that definition.
-
-The precision selected for a variable is a minimum value, MPIR may increase it
-a little to facilitate efficient calculation. Currently this means rounding
-up to a whole limb, and then sometimes having a further partial limb,
-depending on the high limb of the mantissa. But applications shouldn't be
-concerned by such details.
-
-The mantissa in stored in binary, as might be imagined from the fact
-precisions are expressed in bits. One consequence of this is that decimal
-fractions like @math{0.1} cannot be represented exactly. The same is true of
-plain IEEE @code{double} floats. This makes both highly unsuitable for
-calculations involving money or other values that should be exact decimal
-fractions. (Suitably scaled integers, or perhaps rationals, are better
-choices.)
-
- at code{mpf} functions and variables have no special notion of infinity or
-not-a-number, and applications must take care not to overflow the exponent or
-results will be unpredictable. This might change in a future release.
-
-Note that the @code{mpf} functions are @emph{not} intended as a smooth
-extension to IEEE P754 arithmetic. In particular results obtained on one
-computer often differ from the results on a computer with a different word
-size.
-
- at menu
-* Initializing Floats::
-* Assigning Floats::
-* Simultaneous Float Init & Assign::
-* Converting Floats::
-* Float Arithmetic::
-* Float Comparison::
-* I/O of Floats::
-* Miscellaneous Float Functions::
- at end menu
-
- at node Initializing Floats, Assigning Floats, Floating-point Functions, Floating-point Functions
- at comment node-name, next, previous, up
- at section Initialization Functions
- at cindex Float initialization functions
- at cindex Initialization functions
-
- at deftypefun void mpf_set_default_prec (mp_bitcnt_t @var{prec})
-Set the default precision to be @strong{at least} @var{prec} bits. All
-subsequent calls to @code{mpf_init} will use this precision, but previously
-initialized variables are unaffected.
- at end deftypefun
-
- at deftypefun {mp_bitcnt_t} mpf_get_default_prec (void)
-Return the default precision actually used.
- at end deftypefun
-
-An @code{mpf_t} object must be initialized before storing the first value in
-it. The functions @code{mpf_init} and @code{mpf_init2} are used for that
-purpose.
-
- at deftypefun void mpf_init (mpf_t @var{x})
-Initialize @var{x} to 0. Normally, a variable should be initialized once only
-or at least be cleared, using @code{mpf_clear}, between initializations. The
-precision of @var{x} is undefined unless a default precision has already been
-established by a call to @code{mpf_set_default_prec}.
- at end deftypefun
-
- at deftypefun void mpf_init2 (mpf_t @var{x}, mp_bitcnt_t @var{prec})
-Initialize @var{x} to 0 and set its precision to be @strong{at least}
- at var{prec} bits. Normally, a variable should be initialized once only or at
-least be cleared, using @code{mpf_clear}, between initializations.
- at end deftypefun
-
- at deftypefun void mpf_inits (mpf_t @var{x}, ...)
-Initialize a NULL-terminated list of @code{mpf_t} variables, and set their
-values to 0. The precision of the initialized variables is undefined unless a
-default precision has already been established by a call to
- at code{mpf_set_default_prec}.
- at end deftypefun
-
- at deftypefun void mpf_clear (mpf_t @var{x})
-Free the space occupied by @var{x}. Make sure to call this function for all
- at code{mpf_t} variables when you are done with them.
- at end deftypefun
-
- at deftypefun void mpf_clears (mpf_t @var{x}, ...)
-Free the space occupied by a NULL-terminated list of @code{mpf_t} variables.
- at end deftypefun
-
- at need 2000
-Here is an example on how to initialize floating-point variables:
- at example
-@{
- mpf_t x, y;
- mpf_init (x); /* use default precision */
- mpf_init2 (y, 256); /* precision @emph{at least} 256 bits */
- @dots{}
- /* Unless the program is about to exit, do ... */
- mpf_clear (x);
- mpf_clear (y);
-@}
- at end example
-
-The following three functions are useful for changing the precision during a
-calculation. A typical use would be for adjusting the precision gradually in
-iterative algorithms like Newton-Raphson, making the computation precision
-closely match the actual accurate part of the numbers.
-
- at deftypefun {mp_bitcnt_t} mpf_get_prec (mpf_t @var{op})
-Return the current precision of @var{op}, in bits.
- at end deftypefun
-
- at deftypefun void mpf_set_prec (mpf_t @var{rop}, mp_bitcnt_t @var{prec})
-Set the precision of @var{rop} to be @strong{at least} @var{prec} bits. The
-value in @var{rop} will be truncated to the new precision.
-
-This function requires a call to @code{realloc}, and so should not be used in
-a tight loop.
- at end deftypefun
-
- at deftypefun void mpf_set_prec_raw (mpf_t @var{rop}, mp_bitcnt_t @var{prec})
-Set the precision of @var{rop} to be @strong{at least} @var{prec} bits,
-without changing the memory allocated.
-
- at var{prec} must be no more than the allocated precision for @var{rop}, that
-being the precision when @var{rop} was initialized, or in the most recent
- at code{mpf_set_prec}.
-
-The value in @var{rop} is unchanged, and in particular if it had a higher
-precision than @var{prec} it will retain that higher precision. New values
-written to @var{rop} will use the new @var{prec}.
-
-Before calling @code{mpf_clear} or the full @code{mpf_set_prec}, another
- at code{mpf_set_prec_raw} call must be made to restore @var{rop} to its original
-allocated precision. Failing to do so will have unpredictable results.
-
- at code{mpf_get_prec} can be used before @code{mpf_set_prec_raw} to get the
-original allocated precision. After @code{mpf_set_prec_raw} it reflects the
- at var{prec} value set.
-
- at code{mpf_set_prec_raw} is an efficient way to use an @code{mpf_t} variable at
-different precisions during a calculation, perhaps to gradually increase
-precision in an iteration, or just to use various different precisions for
-different purposes during a calculation.
- at end deftypefun
-
-
- at need 2000
- at node Assigning Floats, Simultaneous Float Init & Assign, Initializing Floats, Floating-point Functions
- at comment node-name, next, previous, up
- at section Assignment Functions
- at cindex Float assignment functions
- at cindex Assignment functions
-
-These functions assign new values to already initialized floats
-(@pxref{Initializing Floats}).
-
- at deftypefun void mpf_set (mpf_t @var{rop}, mpf_t @var{op})
- at deftypefunx void mpf_set_ui (mpf_t @var{rop}, mpir_ui @var{op})
- at deftypefunx void mpf_set_si (mpf_t @var{rop}, mpir_si @var{op})
- at deftypefunx void mpf_set_d (mpf_t @var{rop}, double @var{op})
- at deftypefunx void mpf_set_z (mpf_t @var{rop}, mpz_t @var{op})
- at deftypefunx void mpf_set_q (mpf_t @var{rop}, mpq_t @var{op})
-Set the value of @var{rop} from @var{op}.
- at end deftypefun
-
- at deftypefun int mpf_set_str (mpf_t @var{rop}, char *@var{str}, int @var{base})
-Set the value of @var{rop} from the string in @var{str}. The string is of the
-form @samp{M@@N} or, if the base is 10 or less, alternatively @samp{MeN}.
- at samp{M} is the mantissa and @samp{N} is the exponent. The mantissa is always
-in the specified base. The exponent is either in the specified base or, if
- at var{base} is negative, in decimal. The decimal point expected is taken from
-the current locale, on systems providing @code{localeconv}.
-
-The argument @var{base} may be in the ranges 2 to 62, or @minus{}62 to
- at minus{}2. Negative values are used to specify that the exponent is in
-decimal.
-
-For bases up to 36, case is ignored; upper-case and lower-case letters have
-the same value; for bases 37 to 62, upper-case letter represent the usual
-10..35 while lower-case letter represent 36..61.
-
-Unlike the corresponding @code{mpz} function, the base will not be determined
-from the leading characters of the string if @var{base} is 0. This is so that
-numbers like @samp{0.23} are not interpreted as octal.
-
-White space is allowed in the string, and is simply ignored. [This is not
-really true; white-space is ignored in the beginning of the string and within
-the mantissa, but not in other places, such as after a minus sign or in the
-exponent. We are considering changing the definition of this function, making
-it fail when there is any white-space in the input, since that makes a lot of
-sense. Please tell us your opinion about this change. Do you really want it
-to accept @nicode{"3 14"} as meaning 314 as it does now?]
-
-This function returns 0 if the entire string is a valid number in base
- at var{base}. Otherwise it returns @minus{}1.
- at end deftypefun
-
- at deftypefun void mpf_swap (mpf_t @var{rop1}, mpf_t @var{rop2})
-Swap @var{rop1} and @var{rop2} efficiently. Both the values and the
-precisions of the two variables are swapped.
- at end deftypefun
-
-
- at node Simultaneous Float Init & Assign, Converting Floats, Assigning Floats, Floating-point Functions
- at comment node-name, next, previous, up
- at section Combined Initialization and Assignment Functions
- at cindex Float assignment functions
- at cindex Assignment functions
- at cindex Float initialization functions
- at cindex Initialization functions
-
-For convenience, MPIR provides a parallel series of initialize-and-set functions
-which initialize the output and then store the value there. These functions'
-names have the form @code{mpf_init_set at dots{}}
-
-Once the float has been initialized by any of the @code{mpf_init_set at dots{}}
-functions, it can be used as the source or destination operand for the ordinary
-float functions. Don't use an initialize-and-set function on a variable
-already initialized!
-
- at deftypefun void mpf_init_set (mpf_t @var{rop}, mpf_t @var{op})
- at deftypefunx void mpf_init_set_ui (mpf_t @var{rop}, mpir_ui @var{op})
- at deftypefunx void mpf_init_set_si (mpf_t @var{rop}, mpir_si @var{op})
- at deftypefunx void mpf_init_set_d (mpf_t @var{rop}, double @var{op})
-Initialize @var{rop} and set its value from @var{op}.
-
-The precision of @var{rop} will be taken from the active default precision, as
-set by @code{mpf_set_default_prec}.
- at end deftypefun
-
- at deftypefun int mpf_init_set_str (mpf_t @var{rop}, char *@var{str}, int @var{base})
-Initialize @var{rop} and set its value from the string in @var{str}. See
- at code{mpf_set_str} above for details on the assignment operation.
-
-Note that @var{rop} is initialized even if an error occurs. (I.e., you have to
-call @code{mpf_clear} for it.)
-
-The precision of @var{rop} will be taken from the active default precision, as
-set by @code{mpf_set_default_prec}.
- at end deftypefun
-
-
- at node Converting Floats, Float Arithmetic, Simultaneous Float Init & Assign, Floating-point Functions
- at comment node-name, next, previous, up
- at section Conversion Functions
- at cindex Float conversion functions
- at cindex Conversion functions
-
- at deftypefun double mpf_get_d (mpf_t @var{op})
-Convert @var{op} to a @code{double}, truncating if necessary (ie.@: rounding
-towards zero).
-
-If the exponent in @var{op} is too big or too small to fit a @code{double}
-then the result is system dependent. For too big an infinity is returned when
-available. For too small @math{0.0} is normally returned. Hardware overflow,
-underflow and denorm traps may or may not occur.
- at end deftypefun
-
- at deftypefun double mpf_get_d_2exp (mpir_si *@var{exp}, mpf_t @var{op})
-Convert @var{op} to a @code{double}, truncating if necessary (ie.@: rounding
-towards zero), and with an exponent returned separately.
-
-The return value is in the range @math{0.5 at le{}@GMPabs{@var{d}}<1} and the
-exponent is stored to @code{*@var{exp}}. @m{@var{d} * 2^{exp}, @var{d} *
-2^@var{exp}} is the (truncated) @var{op} value. If @var{op} is zero, the
-return is @math{0.0} and 0 is stored to @code{*@var{exp}}.
-
- at cindex @code{frexp}
-This is similar to the standard C @code{frexp} function (@pxref{Normalization
-Functions,,, libc, The GNU C Library Reference Manual}).
- at end deftypefun
-
- at deftypefun mpir_si mpf_get_si (mpf_t @var{op})
- at deftypefunx mpir_ui mpf_get_ui (mpf_t @var{op})
-Convert @var{op} to a @code{mpir_si} or @code{mpir_ui}, truncating any
-fraction part. If @var{op} is too big for the return type, the result is
-undefined.
-
-See also @code{mpf_fits_slong_p} and @code{mpf_fits_ulong_p}
-(@pxref{Miscellaneous Float Functions}).
- at end deftypefun
-
- at deftypefun {char *} mpf_get_str (char *@var{str}, mp_exp_t *@var{expptr}, int @var{base}, size_t @var{n_digits}, mpf_t @var{op})
-Convert @var{op} to a string of digits in base @var{base}. @var{base} can vary
-from 2 to 362 or from @minus{}2 to @minus{}36. Up to @var{n_digits} digits
-will be generated. Trailing zeros are not returned. No more digits than can
-be accurately represented by @var{op} are ever generated. If @var{n_digits}
-is 0 then that accurate maximum number of digits are generated.
-
-For @var{base} in the range 2..36, digits and lower-case letters are used; for
- at minus{}2.. at minus{}36, digits and upper-case letters are used; for 37..62,
-digits, upper-case letters, and lower-case letters (in that significance order)
-are used.
-
-If @var{str} is @code{NULL}, the result string is allocated using the current
-allocation function (@pxref{Custom Allocation}). The block will be
- at code{strlen(str)+1} bytes, that being exactly enough for the string and
-null-terminator.
-
-If @var{str} is not @code{NULL}, it should point to a block of
- at math{@var{n_digits} + 2} bytes, that being enough for the mantissa, a
-possible minus sign, and a null-terminator. When @var{n_digits} is 0 to get
-all significant digits, an application won't be able to know the space
-required, and @var{str} should be @code{NULL} in that case.
-
-The generated string is a fraction, with an implicit radix point immediately
-to the left of the first digit. The applicable exponent is written through
-the @var{expptr} pointer. For example, the number 3.1416 would be returned as
-string @nicode{"31416"} and exponent 1.
-
-When @var{op} is zero, an empty string is produced and the exponent returned
-is 0.
-
-A pointer to the result string is returned, being either the allocated block
-or the given @var{str}.
- at end deftypefun
-
-
- at node Float Arithmetic, Float Comparison, Converting Floats, Floating-point Functions
- at comment node-name, next, previous, up
- at section Arithmetic Functions
- at cindex Float arithmetic functions
- at cindex Arithmetic functions
-
- at deftypefun void mpf_add (mpf_t @var{rop}, mpf_t @var{op1}, mpf_t @var{op2})
- at deftypefunx void mpf_add_ui (mpf_t @var{rop}, mpf_t @var{op1}, mpir_ui @var{op2})
-Set @var{rop} to @math{@var{op1} + @var{op2}}.
- at end deftypefun
-
- at deftypefun void mpf_sub (mpf_t @var{rop}, mpf_t @var{op1}, mpf_t @var{op2})
- at deftypefunx void mpf_ui_sub (mpf_t @var{rop}, mpir_ui @var{op1}, mpf_t @var{op2})
- at deftypefunx void mpf_sub_ui (mpf_t @var{rop}, mpf_t @var{op1}, mpir_ui @var{op2})
-Set @var{rop} to @var{op1} @minus{} @var{op2}.
- at end deftypefun
-
- at deftypefun void mpf_mul (mpf_t @var{rop}, mpf_t @var{op1}, mpf_t @var{op2})
- at deftypefunx void mpf_mul_ui (mpf_t @var{rop}, mpf_t @var{op1}, mpir_ui @var{op2})
-Set @var{rop} to @math{@var{op1} @GMPtimes{} @var{op2}}.
- at end deftypefun
-
-Division is undefined if the divisor is zero, and passing a zero divisor to the
-divide functions will make these functions intentionally divide by zero. This
-lets the user handle arithmetic exceptions in these functions in the same
-manner as other arithmetic exceptions.
-
- at deftypefun void mpf_div (mpf_t @var{rop}, mpf_t @var{op1}, mpf_t @var{op2})
- at deftypefunx void mpf_ui_div (mpf_t @var{rop}, mpir_ui @var{op1}, mpf_t @var{op2})
- at deftypefunx void mpf_div_ui (mpf_t @var{rop}, mpf_t @var{op1}, mpir_ui @var{op2})
- at cindex Division functions
-Set @var{rop} to @var{op1}/@var{op2}.
- at end deftypefun
-
- at deftypefun void mpf_sqrt (mpf_t @var{rop}, mpf_t @var{op})
- at deftypefunx void mpf_sqrt_ui (mpf_t @var{rop}, mpir_ui @var{op})
- at cindex Root extraction functions
-Set @var{rop} to @m{\sqrt{@var{op}}, the square root of @var{op}}.
- at end deftypefun
-
- at deftypefun void mpf_pow_ui (mpf_t @var{rop}, mpf_t @var{op1}, mpir_ui @var{op2})
- at cindex Exponentiation functions
- at cindex Powering functions
-Set @var{rop} to @m{@var{op1}^{op2}, @var{op1} raised to the power @var{op2}}.
- at end deftypefun
-
- at deftypefun void mpf_neg (mpf_t @var{rop}, mpf_t @var{op})
-Set @var{rop} to @minus{}@var{op}.
- at end deftypefun
-
- at deftypefun void mpf_abs (mpf_t @var{rop}, mpf_t @var{op})
-Set @var{rop} to the absolute value of @var{op}.
- at end deftypefun
-
- at deftypefun void mpf_mul_2exp (mpf_t @var{rop}, mpf_t @var{op1}, mp_bitcnt_t @var{op2})
-Set @var{rop} to @m{@var{op1} \times 2^{op2}, @var{op1} times 2 raised to
- at var{op2}}.
- at end deftypefun
-
- at deftypefun void mpf_div_2exp (mpf_t @var{rop}, mpf_t @var{op1}, mp_bitcnt_t @var{op2})
-Set @var{rop} to @m{@var{op1}/2^{op2}, @var{op1} divided by 2 raised to
- at var{op2}}.
- at end deftypefun
-
- at node Float Comparison, I/O of Floats, Float Arithmetic, Floating-point Functions
- at comment node-name, next, previous, up
- at section Comparison Functions
- at cindex Float comparison functions
- at cindex Comparison functions
-
- at deftypefun int mpf_cmp (mpf_t @var{op1}, mpf_t @var{op2})
- at deftypefunx int mpf_cmp_d (mpf_t @var{op1}, double @var{op2})
- at deftypefunx int mpf_cmp_ui (mpf_t @var{op1}, mpir_ui @var{op2})
- at deftypefunx int mpf_cmp_si (mpf_t @var{op1}, mpir_si @var{op2})
-Compare @var{op1} and @var{op2}. Return a positive value if @math{@var{op1} >
- at var{op2}}, zero if @math{@var{op1} = @var{op2}}, and a negative value if
- at math{@var{op1} < @var{op2}}.
-
- at code{mpf_cmp_d} can be called with an infinity, but results are undefined for
-a NaN.
- at end deftypefun
-
- at deftypefun int mpf_eq (mpf_t @var{op1}, mpf_t @var{op2}, mp_bitcnt_t op3)
-Return non-zero if the first @var{op3} bits of @var{op1} and @var{op2} are
-equal, zero otherwise. I.e., test if @var{op1} and @var{op2} are approximately
-equal.
-
-In the future values like 1000 and 0111 may be considered the same
-to 3 bits (on the basis that their difference is that small).
- at end deftypefun
-
- at deftypefun void mpf_reldiff (mpf_t @var{rop}, mpf_t @var{op1}, mpf_t @var{op2})
-Compute the relative difference between @var{op1} and @var{op2} and store the
-result in @var{rop}. This is @math{@GMPabs{@var{op1}- at var{op2}}/@var{op1}}.
- at end deftypefun
-
- at deftypefn Macro int mpf_sgn (mpf_t @var{op})
- at cindex Sign tests
- at cindex Float sign tests
-Return @math{+1} if @math{@var{op} > 0}, 0 if @math{@var{op} = 0}, and
- at math{-1} if @math{@var{op} < 0}.
-
-This function is actually implemented as a macro. It evaluates its arguments
-multiple times.
- at end deftypefn
-
- at node I/O of Floats, Miscellaneous Float Functions, Float Comparison, Floating-point Functions
- at comment node-name, next, previous, up
- at section Input and Output Functions
- at cindex Float input and output functions
- at cindex Input functions
- at cindex Output functions
- at cindex I/O functions
-
-Functions that perform input from a stdio stream, and functions that output to
-a stdio stream. Passing a @code{NULL} pointer for a @var{stream} argument to
-any of these functions will make them read from @code{stdin} and write to
- at code{stdout}, respectively.
-
-When using any of these functions, it is a good idea to include @file{stdio.h}
-before @file{mpir.h}, since that will allow @file{mpir.h} to define prototypes
-for these functions.
-
- at deftypefun size_t mpf_out_str (FILE *@var{stream}, int @var{base}, size_t @var{n_digits}, mpf_t @var{op})
-Print @var{op} to @var{stream}, as a string of digits. Return the number of
-bytes written, or if an error occurred, return 0.
-
-The mantissa is prefixed with an @samp{0.} and is in the given @var{base},
-which may vary from 2 to 36. An exponent then printed, separated by an
- at samp{e}, or if @var{base} is greater than 10 then by an @samp{@@}. The
-exponent is always in decimal. The decimal point follows the current locale,
-on systems providing @code{localeconv}.
-
-For @var{base} in the range 2..36, digits and lower-case letters are used; for
- at minus{}2.. at minus{}36, digits and upper-case letters are used; for 37..62,
-digits, upper-case letters, and lower-case letters (in that significance order)
-are used.
-
-Up to @var{n_digits} will be printed from the mantissa, except that no more
-digits than are accurately representable by @var{op} will be printed.
- at var{n_digits} can be 0 to select that accurate maximum.
- at end deftypefun
-
- at deftypefun size_t mpf_inp_str (mpf_t @var{rop}, FILE *@var{stream}, int @var{base})
-Read a string in base @var{base} from @var{stream}, and put the read float in
- at var{rop}. The string is of the form @samp{M@@N} or, if the base is 10 or
-less, alternatively @samp{MeN}. @samp{M} is the mantissa and @samp{N} is the
-exponent. The mantissa is always in the specified base. The exponent is
-either in the specified base or, if @var{base} is negative, in decimal. The
-decimal point expected is taken from the current locale, on systems providing
- at code{localeconv}.
-
-The argument @var{base} may be in the ranges 2 to 36, or @minus{}36 to
- at minus{}2. Negative values are used to specify that the exponent is in
-decimal.
-
-Unlike the corresponding @code{mpz} function, the base will not be determined
-from the leading characters of the string if @var{base} is 0. This is so that
-numbers like @samp{0.23} are not interpreted as octal.
-
-Return the number of bytes read, or if an error occurred, return 0.
- at end deftypefun
-
- at c @deftypefun void mpf_out_raw (FILE *@var{stream}, mpf_t @var{float})
- at c Output @var{float} on stdio stream @var{stream}, in raw binary
- at c format. The float is written in a portable format, with 4 bytes of
- at c size information, and that many bytes of limbs. Both the size and the
- at c limbs are written in decreasing significance order.
- at c @end deftypefun
-
- at c @deftypefun void mpf_inp_raw (mpf_t @var{float}, FILE *@var{stream})
- at c Input from stdio stream @var{stream} in the format written by
- at c @code{mpf_out_raw}, and put the result in @var{float}.
- at c @end deftypefun
-
-
- at node Miscellaneous Float Functions, , I/O of Floats, Floating-point Functions
- at comment node-name, next, previous, up
- at section Miscellaneous Functions
- at cindex Miscellaneous float functions
- at cindex Float miscellaneous functions
-
- at deftypefun void mpf_ceil (mpf_t @var{rop}, mpf_t @var{op})
- at deftypefunx void mpf_floor (mpf_t @var{rop}, mpf_t @var{op})
- at deftypefunx void mpf_trunc (mpf_t @var{rop}, mpf_t @var{op})
- at cindex Rounding functions
- at cindex Float rounding functions
-Set @var{rop} to @var{op} rounded to an integer. @code{mpf_ceil} rounds to the
-next higher integer, @code{mpf_floor} to the next lower, and @code{mpf_trunc}
-to the integer towards zero.
- at end deftypefun
-
- at deftypefun int mpf_integer_p (mpf_t @var{op})
-Return non-zero if @var{op} is an integer.
- at end deftypefun
-
- at deftypefun int mpf_fits_ulong_p (mpf_t @var{op})
- at deftypefunx int mpf_fits_slong_p (mpf_t @var{op})
- at deftypefunx int mpf_fits_uint_p (mpf_t @var{op})
- at deftypefunx int mpf_fits_sint_p (mpf_t @var{op})
- at deftypefunx int mpf_fits_ushort_p (mpf_t @var{op})
- at deftypefunx int mpf_fits_sshort_p (mpf_t @var{op})
-Return non-zero if @var{op} would fit in the respective C data type, when
-truncated to an integer.
- at end deftypefun
-
- at deftypefun void mpf_urandomb (mpf_t @var{rop}, gmp_randstate_t @var{state}, mp_bitcnt_t @var{nbits})
- at cindex Random number functions
- at cindex Float random number functions
-Generate a uniformly distributed random float in @var{rop}, such that @math{0
- at le{} @var{rop} < 1}, with @var{nbits} significant bits in the mantissa.
-
-The variable @var{state} must be initialized by calling one of the
- at code{gmp_randinit} functions (@ref{Random State Initialization}) before
-invoking this function.
- at end deftypefun
-
- at deftypefun void mpf_rrandomb (mpf_t @var{rop}, gmp_randstate_t @var{state}, mp_size_t @var{max_size}, mp_exp_t @var{exp})
-Generate a random float of at most @var{max_size} limbs, with long strings of
-zeros and ones in the binary representation. The exponent of the number is in
-the interval @minus{}@var{exp} to @var{exp} (in limbs). This function is
-useful for testing functions and algorithms, since these kind of random
-numbers have proven to be more likely to trigger corner-case bugs. Negative
-random numbers are generated when @var{max_size} is negative.
-
- at strong{This interface is preliminary. It might change incompatibly in future revisions.}
- at end deftypefun
-
- at deftypefun void mpf_random2 (mpf_t @var{rop}, mp_size_t @var{max_size}, mp_exp_t @var{exp})
-Generate a random float of at most @var{max_size} limbs, with long strings of
-zeros and ones in the binary representation. The exponent of the number is in
-the interval @minus{}@var{exp} to @var{exp} (in limbs). This function is
-useful for testing functions and algorithms, since these kind of random
-numbers have proven to be more likely to trigger corner-case bugs. Negative
-random numbers are generated when @var{max_size} is negative.
-
- at strong{This function is obsolete. It will disappear from future MPIR releases.}
- at end deftypefun
-
- at node Low-level Functions, Random Number Functions, Floating-point Functions, Top
- at comment node-name, next, previous, up
- at chapter Low-level Functions
- at cindex Low-level functions
-
-This chapter describes low-level MPIR functions, used to implement the
-high-level MPIR functions, but also intended for time-critical user code.
-
-These functions start with the prefix @code{mpn_}.
-
- at c 1. Some of these function clobber input operands.
- at c
-
-The @code{mpn} functions are designed to be as fast as possible, @strong{not}
-to provide a coherent calling interface. The different functions have somewhat
-similar interfaces, but there are variations that make them hard to use. These
-functions do as little as possible apart from the real multiple precision
-computation, so that no time is spent on things that not all callers need.
-
-A source operand is specified by a pointer to the least significant limb and a
-limb count. A destination operand is specified by just a pointer. It is the
-responsibility of the caller to ensure that the destination has enough space
-for storing the result.
-
-With this way of specifying operands, it is possible to perform computations on
-subranges of an argument, and store the result into a subrange of a
-destination.
-
-A common requirement for all functions is that each source area needs at least
-one limb. No size argument may be zero. Unless otherwise stated, in-place
-operations are allowed where source and destination are the same, but not where
-they only partly overlap.
-
-The @code{mpn} functions are the base for the implementation of the
- at code{mpz_}, @code{mpf_}, and @code{mpq_} functions.
-
-This example adds the number beginning at @var{s1p} and the number beginning at
- at var{s2p} and writes the sum at @var{destp}. All areas have @var{n} limbs.
-
- at example
-cy = mpn_add_n (destp, s1p, s2p, n)
- at end example
-
-It should be noted that the @code{mpn} functions make no attempt to identify
-high or low zero limbs on their operands, or other special forms. On random
-data such cases will be unlikely and it'd be wasteful for every function to
-check every time. An application knowing something about its data can take
-steps to trim or perhaps split its calculations.
- at c
- at c For reference, within gmp mpz_t operands never have high zero limbs, and
- at c we rate low zero limbs as unlikely too (or something an application should
- at c handle). This is a prime motivation for not stripping zero limbs in say
- at c mpn_mul_n etc.
- at c
- at c Other applications doing variable-length calculations will quite likely do
- at c something similar to mpz. And even if not then it's highly likely zero
- at c limb stripping can be done at just a few judicious points, which will be
- at c more efficient than having lots of mpn functions checking every time.
-
- at sp 1
- at noindent
-In the notation used below, a source operand is identified by the pointer to
-the least significant limb, and the limb count in braces. For example,
-@{@var{s1p}, @var{s1n}@}.
-
- at deftypefun mp_limb_t mpn_add_n (mp_limb_t *@var{rp}, const mp_limb_t *@var{s1p}, const mp_limb_t *@var{s2p}, mp_size_t @var{n})
-Add @{@var{s1p}, @var{n}@} and @{@var{s2p}, @var{n}@}, and write the @var{n}
-least significant limbs of the result to @var{rp}. Return carry, either 0 or
-1.
-
-This is the lowest-level function for addition. It is the preferred function
-for addition, since it is written in assembly for most CPUs. For addition of
-a variable to itself (i.e., @var{s1p} equals @var{s2p}, use @code{mpn_lshift}
-with a count of 1 for optimal speed.
- at end deftypefun
-
- at deftypefun mp_limb_t mpn_add_1 (mp_limb_t *@var{rp}, const mp_limb_t *@var{s1p}, mp_size_t @var{n}, mp_limb_t @var{s2limb})
-Add @{@var{s1p}, @var{n}@} and @var{s2limb}, and write the @var{n} least
-significant limbs of the result to @var{rp}. Return carry, either 0 or 1.
- at end deftypefun
-
- at deftypefun mp_limb_t mpn_add (mp_limb_t *@var{rp}, const mp_limb_t *@var{s1p}, mp_size_t @var{s1n}, const mp_limb_t *@var{s2p}, mp_size_t @var{s2n})
-Add @{@var{s1p}, @var{s1n}@} and @{@var{s2p}, @var{s2n}@}, and write the
- at var{s1n} least significant limbs of the result to @var{rp}. Return carry,
-either 0 or 1.
-
-This function requires that @var{s1n} is greater than or equal to @var{s2n}.
- at end deftypefun
-
- at deftypefun mp_limb_t mpn_sub_n (mp_limb_t *@var{rp}, const mp_limb_t *@var{s1p}, const mp_limb_t *@var{s2p}, mp_size_t @var{n})
-Subtract @{@var{s2p}, @var{n}@} from @{@var{s1p}, @var{n}@}, and write the
- at var{n} least significant limbs of the result to @var{rp}. Return borrow,
-either 0 or 1.
-
-This is the lowest-level function for subtraction. It is the preferred
-function for subtraction, since it is written in assembly for most CPUs.
- at end deftypefun
-
- at deftypefun mp_limb_t mpn_sub_1 (mp_limb_t *@var{rp}, const mp_limb_t *@var{s1p}, mp_size_t @var{n}, mp_limb_t @var{s2limb})
-Subtract @var{s2limb} from @{@var{s1p}, @var{n}@}, and write the @var{n} least
-significant limbs of the result to @var{rp}. Return borrow, either 0 or 1.
- at end deftypefun
-
- at deftypefun mp_limb_t mpn_sub (mp_limb_t *@var{rp}, const mp_limb_t *@var{s1p}, mp_size_t @var{s1n}, const mp_limb_t *@var{s2p}, mp_size_t @var{s2n})
-Subtract @{@var{s2p}, @var{s2n}@} from @{@var{s1p}, @var{s1n}@}, and write the
- at var{s1n} least significant limbs of the result to @var{rp}. Return borrow,
-either 0 or 1.
-
-This function requires that @var{s1n} is greater than or equal to
- at var{s2n}.
- at end deftypefun
-
- at deftypefun void mpn_neg (mp_limb_t *@var{rp}, const mp_limb_t *@var{sp}, mp_size_t @var{n})
-Perform the negation of @{@var{sp}, @var{n}@}, and write the result to
-@{@var{rp}, @var{n}@}. Return carry-out.
- at end deftypefun
-
- at deftypefun void mpn_mul_n (mp_limb_t *@var{rp}, const mp_limb_t *@var{s1p}, const mp_limb_t *@var{s2p}, mp_size_t @var{n})
-Multiply @{@var{s1p}, @var{n}@} and @{@var{s2p}, @var{n}@}, and write the
-2*@var{n}-limb result to @var{rp}.
-
-The destination has to have space for 2*@var{n} limbs, even if the product's
-most significant limb is zero. No overlap is permitted between the
-destination and either source.
-
-If the input operands are the same, @code{mpn_sqr} will generally be faster.
- at end deftypefun
-
- at deftypefun mp_limb_t mpn_mul_1 (mp_limb_t *@var{rp}, const mp_limb_t *@var{s1p}, mp_size_t @var{n}, mp_limb_t @var{s2limb})
-Multiply @{@var{s1p}, @var{n}@} by @var{s2limb}, and write the @var{n} least
-significant limbs of the product to @var{rp}. Return the most significant
-limb of the product. @{@var{s1p}, @var{n}@} and @{@var{rp}, @var{n}@} are
-allowed to overlap provided @math{@var{rp} @le{} @var{s1p}}.
-
-This is a low-level function that is a building block for general
-multiplication as well as other operations in MPIR at . It is written in assembly
-for most CPUs.
-
-Don't call this function if @var{s2limb} is a power of 2; use @code{mpn_lshift}
-with a count equal to the logarithm of @var{s2limb} instead, for optimal speed.
- at end deftypefun
-
- at deftypefun mp_limb_t mpn_addmul_1 (mp_limb_t *@var{rp}, const mp_limb_t *@var{s1p}, mp_size_t @var{n}, mp_limb_t @var{s2limb})
-Multiply @{@var{s1p}, @var{n}@} and @var{s2limb}, and add the @var{n} least
-significant limbs of the product to @{@var{rp}, @var{n}@} and write the result
-to @var{rp}. Return the most significant limb of the product, plus carry-out
-from the addition.
-
-This is a low-level function that is a building block for general
-multiplication as well as other operations in MPIR at . It is written in assembly
-for most CPUs.
- at end deftypefun
-
- at deftypefun mp_limb_t mpn_submul_1 (mp_limb_t *@var{rp}, const mp_limb_t *@var{s1p}, mp_size_t @var{n}, mp_limb_t @var{s2limb})
-Multiply @{@var{s1p}, @var{n}@} and @var{s2limb}, and subtract the @var{n}
-least significant limbs of the product from @{@var{rp}, @var{n}@} and write the
-result to @var{rp}. Return the most significant limb of the product, minus
-borrow-out from the subtraction.
-
-This is a low-level function that is a building block for general
-multiplication and division as well as other operations in MPIR at . It is written
-in assembly for most CPUs.
- at end deftypefun
-
- at deftypefun mp_limb_t mpn_mul (mp_limb_t *@var{rp}, const mp_limb_t *@var{s1p}, mp_size_t @var{s1n}, const mp_limb_t *@var{s2p}, mp_size_t @var{s2n})
-Multiply @{@var{s1p}, @var{s1n}@} and @{@var{s2p}, @var{s2n}@}, and write the
-result to @var{rp}. Return the most significant limb of the result.
-
-The destination has to have space for @var{s1n} + @var{s2n} limbs, even if the
-result might be one limb smaller.
-
-This function requires that @var{s1n} is greater than or equal to
- at var{s2n}. The destination must be distinct from both input operands.
- at end deftypefun
-
- at deftypefun void mpn_sqr (mp_limb_t *@var{rp}, const mp_limb_t *@var{s1p}, mp_size_t @var{n})
-Compute the square of @{@var{s1p}, @var{n}@} and write the 2*@var{n}-limb
-result to @var{rp}.
-
-The destination has to have space for 2*@var{n} limbs, even if the result's
-most significant limb is zero. No overlap is permitted between the
-destination and the source.
- at end deftypefun
-
- at deftypefun void mpn_tdiv_qr (mp_limb_t *@var{qp}, mp_limb_t *@var{rp}, mp_size_t @var{qxn}, const mp_limb_t *@var{np}, mp_size_t @var{nn}, const mp_limb_t *@var{dp}, mp_size_t @var{dn})
-Divide @{@var{np}, @var{nn}@} by @{@var{dp}, @var{dn}@} and put the quotient
-at @{@var{qp}, @var{nn}@minus{}@var{dn}+1@} and the remainder at @{@var{rp},
- at var{dn}@}. The quotient is rounded towards 0.
-
-No overlap is permitted between arguments. @var{nn} must be greater than or
-equal to @var{dn}. The most significant limb of @var{dp} must be non-zero.
-The @var{qxn} operand must be zero.
- at comment FIXME: Relax overlap requirements!
- at end deftypefun
-
- at deftypefun mp_limb_t mpn_divrem (mp_limb_t *@var{r1p}, mp_size_t @var{qxn}, mp_limb_t *@var{rs2p}, mp_size_t @var{rs2n}, const mp_limb_t *@var{s3p}, mp_size_t @var{s3n})
-[This function is obsolete. Please call @code{mpn_tdiv_qr} instead for best
-performance.]
-
-Divide @{@var{rs2p}, @var{rs2n}@} by @{@var{s3p}, @var{s3n}@}, and write the
-quotient at @var{r1p}, with the exception of the most significant limb, which
-is returned. The remainder replaces the dividend at @var{rs2p}; it will be
- at var{s3n} limbs long (i.e., as many limbs as the divisor).
-
-In addition to an integer quotient, @var{qxn} fraction limbs are developed, and
-stored after the integral limbs. For most usages, @var{qxn} will be zero.
-
-It is required that @var{rs2n} is greater than or equal to @var{s3n}. It is
-required that the most significant bit of the divisor is set.
-
-If the quotient is not needed, pass @var{rs2p} + @var{s3n} as @var{r1p}. Aside
-from that special case, no overlap between arguments is permitted.
-
-Return the most significant limb of the quotient, either 0 or 1.
-
-The area at @var{r1p} needs to be @var{rs2n} @minus{} @var{s3n} + @var{qxn}
-limbs large.
- at end deftypefun
-
-
- at deftypefn Function mp_limb_t mpn_divrem_1 (mp_limb_t *@var{r1p}, mp_size_t @var{qxn}, @w{mp_limb_t *@var{s2p}}, mp_size_t @var{s2n}, mp_limb_t @var{s3limb})
- at deftypefnx Macro mp_limb_t mpn_divmod_1 (mp_limb_t *@var{r1p}, mp_limb_t *@var{s2p}, @w{mp_size_t @var{s2n}}, @w{mp_limb_t @var{s3limb}})
-Divide @{@var{s2p}, @var{s2n}@} by @var{s3limb}, and write the quotient at
- at var{r1p}. Return the remainder.
-
-The integer quotient is written to @{@var{r1p}+ at var{qxn}, @var{s2n}@} and in
-addition @var{qxn} fraction limbs are developed and written to @{@var{r1p},
- at var{qxn}@}. Either or both @var{s2n} and @var{qxn} can be zero. For most
-usages, @var{qxn} will be zero.
-
- at code{mpn_divmod_1} exists for upward source compatibility and is simply a
-macro calling @code{mpn_divrem_1} with a @var{qxn} of 0.
-
-The areas at @var{r1p} and @var{s2p} have to be identical or completely
-separate, not partially overlapping.
- at end deftypefn
-
- at deftypefn Macro mp_limb_t mpn_divexact_by3 (mp_limb_t *@var{rp}, mp_limb_t *@var{sp}, @w{mp_size_t @var{n}})
- at deftypefnx Function mp_limb_t mpn_divexact_by3c (mp_limb_t *@var{rp}, mp_limb_t *@var{sp}, @w{mp_size_t @var{n}}, mp_limb_t @var{carry})
-Divide @{@var{sp}, @var{n}@} by 3, expecting it to divide exactly, and writing
-the result to @{@var{rp}, @var{n}@}. If 3 divides exactly, the return value is
-zero and the result is the quotient. If not, the return value is non-zero and
-the result won't be anything useful.
-
- at code{mpn_divexact_by3c} takes an initial carry parameter, which can be the
-return value from a previous call, so a large calculation can be done piece by
-piece from low to high. @code{mpn_divexact_by3} is simply a macro calling
- at code{mpn_divexact_by3c} with a 0 carry parameter.
-
-These routines use a multiply-by-inverse and will be faster than
- at code{mpn_divrem_1} on CPUs with fast multiplication but slow division.
-
-The source @math{a}, result @math{q}, size @math{n}, initial carry @math{i},
-and return value @math{c} satisfy @m{cb^n+a-i=3q, c*b^n + a-i = 3*q}, where
- at m{b=2\GMPraise{@code{GMP\_NUMB\_BITS}}, b=2^GMP_NUMB_BITS}. The
-return @math{c} is always 0, 1 or 2, and the initial carry @math{i} must also
-be 0, 1 or 2 (these are both borrows really). When @math{c=0} clearly
- at math{q=(a-i)/3}. When @m{c \neq 0, c!=0}, the remainder @math{(a-i) @bmod{}
-3} is given by @math{3-c}, because @math{b @equiv{} 1 @bmod{} 3} (when
- at code{mp_bits_per_limb} is even, which is always so currently).
- at end deftypefn
-
- at deftypefun mp_limb_t mpn_mod_1 (mp_limb_t *@var{s1p}, mp_size_t @var{s1n}, mp_limb_t @var{s2limb})
-Divide @{@var{s1p}, @var{s1n}@} by @var{s2limb}, and return the remainder.
- at var{s1n} can be zero.
- at end deftypefun
-
- at deftypefun mp_limb_t mpn_bdivmod (mp_limb_t *@var{rp}, mp_limb_t *@var{s1p}, mp_size_t @var{s1n}, const mp_limb_t *@var{s2p}, mp_size_t @var{s2n}, mpir_ui @var{d})
-This function puts the low
- at math{@GMPfloor{@var{d}/@nicode{mp\_bits\_per\_limb}}} limbs of @var{q} =
-@{@var{s1p}, @var{s1n}@}/@{@var{s2p}, @var{s2n}@} mod @m{2^d,2^@var{d}} at
- at var{rp}, and returns the high @var{d} mod @code{mp_bits_per_limb} bits of
- at var{q}.
-
-@{@var{s1p}, @var{s1n}@} - @var{q} * @{@var{s2p}, @var{s2n}@} mod @m{2
-\GMPraise{@var{s1n}*@code{mp\_bits\_per\_limb}},
-2^(@var{s1n}*@nicode{mp\_bits\_per\_limb})} is placed at @var{s1p}. Since the
-low @math{@GMPfloor{@var{d}/@nicode{mp\_bits\_per\_limb}}} limbs of this
-difference are zero, it is possible to overwrite the low limbs at @var{s1p}
-with this difference, provided @math{@var{rp} @le{} @var{s1p}}.
-
-This function requires that @math{@var{s1n} * @nicode{mp\_bits\_per\_limb}
- at ge{} @var{D}}, and that @{@var{s2p}, @var{s2n}@} is odd.
-
- at strong{This interface is preliminary. It might change incompatibly in future revisions.}
- at end deftypefun
-
- at deftypefun mp_limb_t mpn_lshift (mp_limb_t *@var{rp}, const mp_limb_t *@var{sp}, mp_size_t @var{n}, unsigned int @var{count})
-Shift @{@var{sp}, @var{n}@} left by @var{count} bits, and write the result to
-@{@var{rp}, @var{n}@}. The bits shifted out at the left are returned in the
-least significant @var{count} bits of the return value (the rest of the return
-value is zero).
-
- at var{count} must be in the range 1 to @nicode{mp_bits_per_limb}@minus{}1. The
-regions @{@var{sp}, @var{n}@} and @{@var{rp}, @var{n}@} may overlap, provided
- at math{@var{rp} @ge{} @var{sp}}.
-
-This function is written in assembly for most CPUs.
- at end deftypefun
-
- at deftypefun mp_limb_t mpn_rshift (mp_limb_t *@var{rp}, const mp_limb_t *@var{sp}, mp_size_t @var{n}, unsigned int @var{count})
-Shift @{@var{sp}, @var{n}@} right by @var{count} bits, and write the result to
-@{@var{rp}, @var{n}@}. The bits shifted out at the right are returned in the
-most significant @var{count} bits of the return value (the rest of the return
-value is zero).
-
- at var{count} must be in the range 1 to @nicode{mp_bits_per_limb}@minus{}1. The
-regions @{@var{sp}, @var{n}@} and @{@var{rp}, @var{n}@} may overlap, provided
- at math{@var{rp} @le{} @var{sp}}.
-
-This function is written in assembly for most CPUs.
- at end deftypefun
-
- at deftypefun int mpn_cmp (const mp_limb_t *@var{s1p}, const mp_limb_t *@var{s2p}, mp_size_t @var{n})
-Compare @{@var{s1p}, @var{n}@} and @{@var{s2p}, @var{n}@} and return a
-positive value if @math{@var{s1} > @var{s2}}, 0 if they are equal, or a
-negative value if @math{@var{s1} < @var{s2}}.
- at end deftypefun
-
- at deftypefun mp_size_t mpn_gcd (mp_limb_t *@var{rp}, mp_limb_t *@var{s1p}, mp_size_t @var{s1n}, mp_limb_t *@var{s2p}, mp_size_t @var{s2n})
-Set @{@var{rp}, @var{retval}@} to the greatest common divisor of @{@var{s1p},
- at var{s1n}@} and @{@var{s2p}, @var{s2n}@}. The result can be up to @var{s2n}
-limbs, the return value is the actual number produced. Both source operands
-are destroyed.
-
-@{@var{s1p}, @var{s1n}@} must have at least as many bits as @{@var{s2p},
- at var{s2n}@}. @{@var{s2p}, @var{s2n}@} must be odd. Both operands must have
-non-zero most significant limbs. No overlap is permitted between @{@var{s1p},
- at var{s1n}@} and @{@var{s2p}, @var{s2n}@}.
- at end deftypefun
-
- at deftypefun mp_limb_t mpn_gcd_1 (const mp_limb_t *@var{s1p}, mp_size_t @var{s1n}, mp_limb_t @var{s2limb})
-Return the greatest common divisor of @{@var{s1p}, @var{s1n}@} and
- at var{s2limb}. Both operands must be non-zero.
- at end deftypefun
-
- at deftypefun mp_size_t mpn_gcdext (mp_limb_t *@var{gp}, mp_limb_t *@var{sp}, mp_size_t *@var{sn}, mp_limb_t *@var{xp}, mp_size_t @var{xn}, mp_limb_t *@var{yp}, mp_size_t @var{yn})
-Let @m{U, at var{U}} be defined by @{@var{xp}, @var{xn}@} and let @m{V, at var{V}} be
-defined by @{@var{yp}, @var{yn}@}.
-
-Compute the greatest common divisor @math{G} of @math{U} and @math{V}. Compute
-a cofactor @math{S} such that @math{G = US + VT}. The second cofactor @var{T}
-is not computed but can easily be obtained from @m{(G - US) / V, (@var{G} -
- at var{U}*@var{S}) / @var{V}} (the division will be exact). It is required that
- at math{U @ge V > 0}.
-
- at math{S} satisfies @math{S = 1} or @math{@GMPabs{S} < V / (2 G)}. @math{S =
-0} if and only if @math{V} divides @math{U} (i.e., @math{G = V}).
-
-Store @math{G} at @var{gp} and let the return value define its limb count.
-Store @math{S} at @var{sp} and let |*@var{sn}| define its limb count. @math{S}
-can be negative; when this happens *@var{sn} will be negative. The areas at
- at var{gp} and @var{sp} should each have room for @math{@var{xn}+1} limbs.
-
-The areas @{@var{xp}, @math{@var{xn}+1}@} and @{@var{yp}, @math{@var{yn}+1}@}
-are destroyed (i.e.@: the input operands plus an extra limb past the end of
-each).
-
-Compatibility note: MPIR versions 1.3,2.0 and GMP versions 4.3.0,4.3.1 defined @math{S} less strictly.
-Earlier as well as later GMP releases define @math{S} as described here.
- at end deftypefun
-
- at deftypefun mp_size_t mpn_sqrtrem (mp_limb_t *@var{r1p}, mp_limb_t *@var{r2p}, const mp_limb_t *@var{sp}, mp_size_t @var{n})
-Compute the square root of @{@var{sp}, @var{n}@} and put the result at
-@{@var{r1p}, @math{@GMPceil{@var{n}/2}}@} and the remainder at @{@var{r2p},
- at var{retval}@}. @var{r2p} needs space for @var{n} limbs, but the return value
-indicates how many are produced.
-
-The most significant limb of @{@var{sp}, @var{n}@} must be non-zero. The
-areas @{@var{r1p}, @math{@GMPceil{@var{n}/2}}@} and @{@var{sp}, @var{n}@} must
-be completely separate. The areas @{@var{r2p}, @var{n}@} and @{@var{sp},
- at var{n}@} must be either identical or completely separate.
-
-If the remainder is not wanted then @var{r2p} can be @code{NULL}, and in this
-case the return value is zero or non-zero according to whether the remainder
-would have been zero or non-zero.
-
-A return value of zero indicates a perfect square. See also
- at code{mpz_perfect_square_p}.
- at end deftypefun
-
- at deftypefun mp_size_t mpn_get_str (unsigned char *@var{str}, int @var{base}, mp_limb_t *@var{s1p}, mp_size_t @var{s1n})
-Convert @{@var{s1p}, @var{s1n}@} to a raw unsigned char array at @var{str} in
-base @var{base}, and return the number of characters produced. There may be
-leading zeros in the string. The string is not in ASCII; to convert it to
-printable format, add the ASCII codes for @samp{0} or @samp{A}, depending on
-the base and range. @var{base} can vary from 2 to 256.
-
-The most significant limb of the input @{@var{s1p}, @var{s1n}@} must be
-non-zero. The input @{@var{s1p}, @var{s1n}@} is clobbered, except when
- at var{base} is a power of 2, in which case it's unchanged.
-
-The area at @var{str} has to have space for the largest possible number
-represented by a @var{s1n} long limb array, plus one extra character.
- at end deftypefun
-
- at deftypefun mp_size_t mpn_set_str (mp_limb_t *@var{rp}, const unsigned char *@var{str}, size_t @var{strsize}, int @var{base})
-Convert bytes @{@var{str}, at var{strsize}@} in the given @var{base} to limbs at
- at var{rp}.
-
- at math{@var{str}[0]} is the most significant byte and
- at math{@var{str}[@var{strsize}-1]} is the least significant. Each byte should
-be a value in the range 0 to @math{@var{base}-1}, not an ASCII character.
- at var{base} can vary from 2 to 256.
-
-The return value is the number of limbs written to @var{rp}. If the most
-significant input byte is non-zero then the high limb at @var{rp} will be
-non-zero, and only that exact number of limbs will be required there.
-
-If the most significant input byte is zero then there may be high zero limbs
-written to @var{rp} and included in the return value.
-
- at var{strsize} must be at least 1, and no overlap is permitted between
-@{@var{str}, at var{strsize}@} and the result at @var{rp}.
- at end deftypefun
-
- at deftypefun {mp_bitcnt_t} mpn_scan0 (const mp_limb_t *@var{s1p}, imp_bitcnt_t @var{bit})
-Scan @var{s1p} from bit position @var{bit} for the next clear bit.
-
-It is required that there be a clear bit within the area at @var{s1p} at or
-beyond bit position @var{bit}, so that the function has something to return.
- at end deftypefun
-
- at deftypefun {mp_bitcnt_t} mpn_scan1 (const mp_limb_t *@var{s1p}, mp_bitcnt_t @var{bit})
-Scan @var{s1p} from bit position @var{bit} for the next set bit.
-
-It is required that there be a set bit within the area at @var{s1p} at or
-beyond bit position @var{bit}, so that the function has something to return.
- at end deftypefun
-
- at deftypefun void mpn_random (mp_limb_t *@var{r1p}, mp_size_t @var{r1n})
- at deftypefunx void mpn_random2 (mp_limb_t *@var{r1p}, mp_size_t @var{r1n})
-Generate a random number of length @var{r1n} and store it at @var{r1p}. The
-most significant limb is always non-zero. @code{mpn_random} generates
-uniformly distributed limb data, @code{mpn_random2} generates long strings of
-zeros and ones in the binary representation.
-
- at code{mpn_random2} is intended for testing the correctness of the @code{mpn}
-routines.
-
- at strong{These functions are obsolete. They will disappear from future MPIR releases.}
- at end deftypefun
-
- at deftypefun void mpn_urandomb (mp_limb_t *@var{rp}, gmp_randstate_t @var{state}, mpir_ui @var{n})
-Generate a uniform random number of length @var{n} bits and store it at @var{rp}.
-
- at strong{This function interface is preliminary and may change in the future.}
- at end deftypefun
-
- at deftypefun void mpn_urandomm (mp_limb_t *@var{rp}, gmp_randstate_t @var{state}, const mp_limb_t *@var{mp}, mp_size_t @var{n})
-Generate a uniform random number modulo (@var{mp}, at var{n}) of length @var{n} limbs and store it at @var{rp}.
-
- at strong{This function interface is preliminary and may change in the future.}
- at end deftypefun
-
- at deftypefun void mpn_randomb (mp_limb_t *@var{rp}, gmp_randstate_t @var{state}, mp_size_t @var{n})
-Generate a random number of length @var{n} limbs and store it at @var{rp}.
-The most significant limb is always non-zero.
-
- at strong{This function interface is preliminary and may change in the future.}
- at end deftypefun
-
- at deftypefun void mpn_rrandom (mp_limb_t *@var{rp}, gmp_randstate_t @var{state}, mp_size_t @var{n})
-Generate a random number of length @var{n} limbs and store it at @var{rp}.
-The most significant limb is always non-zero. Generates long strings of
-zeros and ones in the binary representation and is intended for testing the correctness of the @code{mpn}
-routines.
-
- at strong{This function interface is preliminary and may change in the future.}
- at end deftypefun
-
-
- at deftypefun {mp_bitcnt_t} mpn_popcount (const mp_limb_t *@var{s1p}, mp_size_t @var{n})
-Count the number of set bits in @{@var{s1p}, @var{n}@}.
- at end deftypefun
-
- at deftypefun {mp_bitcnt_t} mpn_hamdist (const mp_limb_t *@var{s1p}, const mp_limb_t *@var{s2p}, mp_size_t @var{n})
-Compute the hamming distance between @{@var{s1p}, @var{n}@} and @{@var{s2p},
- at var{n}@}, which is the number of bit positions where the two operands have
-different bit values.
- at end deftypefun
-
- at deftypefun int mpn_perfect_square_p (const mp_limb_t *@var{s1p}, mp_size_t @var{n})
-Return non-zero iff @{@var{s1p}, @var{n}@} is a perfect square.
- at end deftypefun
-
- at deftypefun void mpn_and_n (mp_limb_t *@var{rp}, const mp_limb_t *@var{s1p}, const mp_limb_t *@var{s2p}, mp_size_t @var{n})
-Perform the bitwise logical and of @{@var{s1p}, @var{n}@} and @{@var{s2p},
- at var{n}@}, and write the result to @{@var{rp}, @var{n}@}.
- at end deftypefun
-
- at deftypefun void mpn_ior_n (mp_limb_t *@var{rp}, const mp_limb_t *@var{s1p}, const mp_limb_t *@var{s2p}, mp_size_t @var{n})
-Perform the bitwise logical inclusive or of @{@var{s1p}, @var{n}@} and
-@{@var{s2p}, @var{n}@}, and write the result to @{@var{rp}, @var{n}@}.
- at end deftypefun
-
- at deftypefun void mpn_xor_n (mp_limb_t *@var{rp}, const mp_limb_t *@var{s1p}, const mp_limb_t *@var{s2p}, mp_size_t @var{n})
-Perform the bitwise logical exclusive or of @{@var{s1p}, @var{n}@} and
-@{@var{s2p}, @var{n}@}, and write the result to @{@var{rp}, @var{n}@}.
- at end deftypefun
-
- at deftypefun void mpn_andn_n (mp_limb_t *@var{rp}, const mp_limb_t *@var{s1p}, const mp_limb_t *@var{s2p}, mp_size_t @var{n})
-Perform the bitwise logical and of @{@var{s1p}, @var{n}@} and the bitwise
-complement of @{@var{s2p}, @var{n}@}, and write the result to @{@var{rp}, @var{n}@}.
- at end deftypefun
-
- at deftypefun void mpn_iorn_n (mp_limb_t *@var{rp}, const mp_limb_t *@var{s1p}, const mp_limb_t *@var{s2p}, mp_size_t @var{n})
-Perform the bitwise logical inclusive or of @{@var{s1p}, @var{n}@} and the bitwise
-complement of @{@var{s2p}, @var{n}@}, and write the result to @{@var{rp}, @var{n}@}.
- at end deftypefun
-
- at deftypefun void mpn_nand_n (mp_limb_t *@var{rp}, const mp_limb_t *@var{s1p}, const mp_limb_t *@var{s2p}, mp_size_t @var{n})
-Perform the bitwise logical and of @{@var{s1p}, @var{n}@} and @{@var{s2p},
- at var{n}@}, and write the bitwise complement of the result to @{@var{rp}, @var{n}@}.
- at end deftypefun
-
- at deftypefun void mpn_nior_n (mp_limb_t *@var{rp}, const mp_limb_t *@var{s1p}, const mp_limb_t *@var{s2p}, mp_size_t @var{n})
-Perform the bitwise logical inclusive or of @{@var{s1p}, @var{n}@} and
-@{@var{s2p}, @var{n}@}, and write the bitwise complement of the result to
-@{@var{rp}, @var{n}@}.
- at end deftypefun
-
- at deftypefun void mpn_xnor_n (mp_limb_t *@var{rp}, const mp_limb_t *@var{s1p}, const mp_limb_t *@var{s2p}, mp_size_t @var{n})
-Perform the bitwise logical exclusive or of @{@var{s1p}, @var{n}@} and
-@{@var{s2p}, @var{n}@}, and write the bitwise complement of the result to
-@{@var{rp}, @var{n}@}.
- at end deftypefun
-
- at deftypefun void mpn_com (mp_limb_t *@var{rp}, const mp_limb_t *@var{sp}, mp_size_t @var{n})
-Perform the bitwise complement of @{@var{sp}, @var{n}@}, and write the result
-to @{@var{rp}, @var{n}@}.
- at end deftypefun
-
- at deftypefun void mpn_copyi (mp_limb_t *@var{rp}, const mp_limb_t *@var{s1p}, mp_size_t @var{n})
-Copy from @{@var{s1p}, @var{n}@} to @{@var{rp}, @var{n}@}, increasingly.
- at end deftypefun
-
- at deftypefun void mpn_copyd (mp_limb_t *@var{rp}, const mp_limb_t *@var{s1p}, mp_size_t @var{n})
-Copy from @{@var{s1p}, @var{n}@} to @{@var{rp}, @var{n}@}, decreasingly.
- at end deftypefun
-
- at deftypefun void mpn_zero (mp_limb_t *@var{rp}, mp_size_t @var{n})
-Zero @{@var{rp}, @var{n}@}.
- at end deftypefun
-
- at sp 1
- at section Nails
- at cindex Nails
-
- at strong{Everything in this section is highly experimental and may disappear or
-be subject to incompatible changes in a future version of MPIR.}
-
-N.B: Nails are currently disabled and not supported in MPIR. They may or may not return in a future version of MPIR.
-
-Nails are an experimental feature whereby a few bits are left unused at the
-top of each @code{mp_limb_t}. This can significantly improve carry handling
-on some processors.
-
-All the @code{mpn} functions accepting limb data will expect the nail bits to
-be zero on entry, and will return data with the nails similarly all zero.
-This applies both to limb vectors and to single limb arguments.
-
-Nails can be enabled by configuring with @samp{--enable-nails}. By default
-the number of bits will be chosen according to what suits the host processor,
-but a particular number can be selected with @samp{--enable-nails=N}.
-
-At the mpn level, a nail build is neither source nor binary compatible with a
-non-nail build, strictly speaking. But programs acting on limbs only through
-the mpn functions are likely to work equally well with either build, and
-judicious use of the definitions below should make any program compatible with
-either build, at the source level.
-
-For the higher level routines, meaning @code{mpz} etc, a nail build should be
-fully source and binary compatible with a non-nail build.
-
- at defmac GMP_NAIL_BITS
- at defmacx GMP_NUMB_BITS
- at defmacx GMP_LIMB_BITS
- at code{GMP_NAIL_BITS} is the number of nail bits, or 0 when nails are not in
-use. @code{GMP_NUMB_BITS} is the number of data bits in a limb.
- at code{GMP_LIMB_BITS} is the total number of bits in an @code{mp_limb_t}. In
-all cases
-
- at example
-GMP_LIMB_BITS == GMP_NAIL_BITS + GMP_NUMB_BITS
- at end example
- at end defmac
-
- at defmac GMP_NAIL_MASK
- at defmacx GMP_NUMB_MASK
-Bit masks for the nail and number parts of a limb. @code{GMP_NAIL_MASK} is 0
-when nails are not in use.
-
- at code{GMP_NAIL_MASK} is not often needed, since the nail part can be obtained
-with @code{x >> GMP_NUMB_BITS}, and that means one less large constant, which
-can help various RISC chips.
- at end defmac
-
- at defmac GMP_NUMB_MAX
-The maximum value that can be stored in the number part of a limb. This is
-the same as @code{GMP_NUMB_MASK}, but can be used for clarity when doing
-comparisons rather than bit-wise operations.
- at end defmac
-
-The term ``nails'' comes from finger or toe nails, which are at the ends of a
-limb (arm or leg). ``numb'' is short for number, but is also how the
-developers felt after trying for a long time to come up with sensible names
-for these things.
-
-In the future (the distant future most likely) a non-zero nail might be
-permitted, giving non-unique representations for numbers in a limb vector.
-This would help vector processors since carries would only ever need to
-propagate one or two limbs.
-
-
- at node Random Number Functions, Formatted Output, Low-level Functions, Top
- at chapter Random Number Functions
- at cindex Random number functions
-
-Sequences of pseudo-random numbers in MPIR are generated using a variable of
-type @code{gmp_randstate_t}, which holds an algorithm selection and a current
-state. Such a variable must be initialized by a call to one of the
- at code{gmp_randinit} functions, and can be seeded with one of the
- at code{gmp_randseed} functions.
-
-The functions actually generating random numbers are described in @ref{Integer
-Random Numbers}, and @ref{Miscellaneous Float Functions}.
-
-The older style random number functions don't accept a @code{gmp_randstate_t}
-parameter but instead share a global variable of that type. They use a
-default algorithm and are currently not seeded (though perhaps that will
-change in the future). The new functions accepting a @code{gmp_randstate_t}
-are recommended for applications that care about randomness.
-
- at menu
-* Random State Initialization::
-* Random State Seeding::
-* Random State Miscellaneous::
- at end menu
-
- at node Random State Initialization, Random State Seeding, Random Number Functions, Random Number Functions
- at section Random State Initialization
- at cindex Random number state
- at cindex Initialization functions
-
- at deftypefun void gmp_randinit_default (gmp_randstate_t @var{state})
-Initialize @var{state} with a default algorithm. This will be a compromise
-between speed and randomness, and is recommended for applications with no
-special requirements. Currently this is @code{gmp_randinit_mt}.
- at end deftypefun
-
- at deftypefun void gmp_randinit_mt (gmp_randstate_t @var{state})
- at cindex Mersenne twister random numbers
-Initialize @var{state} for a Mersenne Twister algorithm. This algorithm is
-fast and has good randomness properties.
- at end deftypefun
-
- at deftypefun void gmp_randinit_lc_2exp (gmp_randstate_t @var{state}, mpz_t @var{a}, @w{mpir_ui @var{c}}, @w{mp_bitcnt_t @var{m2exp}})
- at cindex Linear congruential random numbers
-Initialize @var{state} with a linear congruential algorithm @m{X = (@var{a}X +
- at var{c}) @bmod 2^{m2exp}, X = (@var{a}*X + @var{c}) mod 2^@var{m2exp}}.
-
-The low bits of @math{X} in this algorithm are not very random. The least
-significant bit will have a period no more than 2, and the second bit no more
-than 4, etc. For this reason only the high half of each @math{X} is actually
-used.
-
-When a random number of more than @math{@var{m2exp}/2} bits is to be
-generated, multiple iterations of the recurrence are used and the results
-concatenated.
- at end deftypefun
-
- at deftypefun int gmp_randinit_lc_2exp_size (gmp_randstate_t @var{state}, mp_bitcnt_t @var{size})
- at cindex Linear congruential random numbers
-Initialize @var{state} for a linear congruential algorithm as per
- at code{gmp_randinit_lc_2exp}. @var{a}, @var{c} and @var{m2exp} are selected
-from a table, chosen so that @var{size} bits (or more) of each @math{X} will
-be used, ie.@: @math{@var{m2exp}/2 @ge{} @var{size}}.
-
-If successful the return value is non-zero. If @var{size} is bigger than the
-table data provides then the return value is zero. The maximum @var{size}
-currently supported is 128.
- at end deftypefun
-
- at deftypefun int gmp_randinit_set (gmp_randstate_t @var{rop}, gmp_randstate_t @var{op})
-Initialize @var{rop} with a copy of the algorithm and state from @var{op}.
- at end deftypefun
-
- at deftypefun void gmp_randclear (gmp_randstate_t @var{state})
-Free all memory occupied by @var{state}.
- at end deftypefun
-
-
- at node Random State Seeding, Random State Miscellaneous, Random State Initialization, Random Number Functions
- at section Random State Seeding
- at cindex Random number seeding
- at cindex Seeding random numbers
-
- at deftypefun void gmp_randseed (gmp_randstate_t @var{state}, mpz_t @var{seed})
- at deftypefunx void gmp_randseed_ui (gmp_randstate_t @var{state}, mpir_ui @var{seed})
-Set an initial seed value into @var{state}.
-
-The size of a seed determines how many different sequences of random numbers
-that it's possible to generate. The ``quality'' of the seed is the randomness
-of a given seed compared to the previous seed used, and this affects the
-randomness of separate number sequences. The method for choosing a seed is
-critical if the generated numbers are to be used for important applications,
-such as generating cryptographic keys.
-
-Traditionally the system time has been used to seed, but care needs to be
-taken with this. If an application seeds often and the resolution of the
-system clock is low, then the same sequence of numbers might be repeated.
-Also, the system time is quite easy to guess, so if unpredictability is
-required then it should definitely not be the only source for the seed value.
-On some systems there's a special device @file{/dev/random} which provides
-random data better suited for use as a seed.
- at end deftypefun
-
-
- at node Random State Miscellaneous, , Random State Seeding, Random Number Functions
- at section Random State Miscellaneous
-
- at deftypefun mpir_ui gmp_urandomb_ui (gmp_randstate_t @var{state}, mpir_ui @var{n})
-Return a uniformly distributed random number of @var{n} bits, ie.@: in the
-range 0 to @m{2^n-1,2^@var{n}-1} inclusive. @var{n} must be less than or
-equal to the number of bits in an @code{mpir_ui}.
- at end deftypefun
-
- at deftypefun mpir_ui gmp_urandomm_ui (gmp_randstate_t @var{state}, mpir_ui @var{n})
-Return a uniformly distributed random number in the range 0 to
- at math{@var{n}-1}, inclusive.
- at end deftypefun
-
-
- at node Formatted Output, Formatted Input, Random Number Functions, Top
- at chapter Formatted Output
- at cindex Formatted output
- at cindex @code{printf} formatted output
-
- at menu
-* Formatted Output Strings::
-* Formatted Output Functions::
-* C++ Formatted Output::
- at end menu
-
- at node Formatted Output Strings, Formatted Output Functions, Formatted Output, Formatted Output
- at section Format Strings
-
- at code{gmp_printf} and friends accept format strings similar to the standard C
- at code{printf} (@pxref{Formatted Output,, Formatted Output, libc, The GNU C
-Library Reference Manual}). A format specification is of the form
-
- at example
-% [flags] [width] [.[precision]] [type] conv
- at end example
-
-MPIR adds types @samp{Z}, @samp{Q} and @samp{F} for @code{mpz_t}, @code{mpq_t}
-and @code{mpf_t} respectively, @samp{M} for @code{mp_limb_t}, and @samp{N} for
-an @code{mp_limb_t} array. @samp{Z}, @samp{Q}, @samp{M} and @samp{N} behave
-like integers. @samp{Q} will print a @samp{/} and a denominator, if needed.
- at samp{F} behaves like a float. For example,
-
- at example
-mpz_t z;
-gmp_printf ("%s is an mpz %Zd\n", "here", z);
-
-mpq_t q;
-gmp_printf ("a hex rational: %#40Qx\n", q);
-
-mpf_t f;
-int n;
-gmp_printf ("fixed point mpf %.*Ff with %d digits\n", n, f, n);
-
-mp_limb_t l;
-gmp_printf ("limb %Mu\n", limb);
-
-const mp_limb_t *ptr;
-mp_size_t size;
-gmp_printf ("limb array %Nx\n", ptr, size);
- at end example
-
-For @samp{N} the limbs are expected least significant first, as per the
- at code{mpn} functions (@pxref{Low-level Functions}). A negative size can be
-given to print the value as a negative.
-
-All the standard C @code{printf} types behave the same as the C library
- at code{printf}, and can be freely intermixed with the MPIR extensions. In the
-current implementation the standard parts of the format string are simply
-handed to @code{printf} and only the MPIR extensions handled directly.
-
-The flags accepted are as follows. GLIBC style @nisamp{'} is only for the
-standard C types (not the MPIR types), and only if the C library supports it.
-
- at quotation
- at multitable {(space)} {MMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMM}
- at item @nicode{0} @tab pad with zeros (rather than spaces)
- at item @nicode{#} @tab show the base with @samp{0x}, @samp{0X} or @samp{0}
- at item @nicode{+} @tab always show a sign
- at item (space) @tab show a space or a @samp{-} sign
- at item @nicode{'} @tab group digits, GLIBC style (not MPIR types)
- at end multitable
- at end quotation
-
-The optional width and precision can be given as a number within the format
-string, or as a @samp{*} to take an extra parameter of type @code{int}, the
-same as the standard @code{printf}.
-
-The standard types accepted are as follows. @samp{h} and @samp{l} are
-portable, the rest will depend on the compiler (or include files) for the type
-and the C library for the output.
-
- at quotation
- at multitable {(space)} {MMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMM}
- at item @nicode{h} @tab @nicode{short}
- at item @nicode{hh} @tab @nicode{char}
- at item @nicode{j} @tab @nicode{intmax_t} or @nicode{uintmax_t}
- at item @nicode{l} @tab @nicode{long} or @nicode{wchar_t}
- at item @nicode{ll} @tab @nicode{long long}
- at item @nicode{L} @tab @nicode{long double}
- at item @nicode{q} @tab @nicode{quad_t} or @nicode{u_quad_t}
- at item @nicode{t} @tab @nicode{ptrdiff_t}
- at item @nicode{z} @tab @nicode{size_t}
- at end multitable
- at end quotation
-
- at noindent
-The MPIR types are
-
- at quotation
- at multitable {(space)} {MMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMM}
- at item @nicode{F} @tab @nicode{mpf_t}, float conversions
- at item @nicode{Q} @tab @nicode{mpq_t}, integer conversions
- at item @nicode{M} @tab @nicode{mp_limb_t}, integer conversions
- at item @nicode{N} @tab @nicode{mp_limb_t} array, integer conversions
- at item @nicode{Z} @tab @nicode{mpz_t}, integer conversions
- at end multitable
- at end quotation
-
-The conversions accepted are as follows. @samp{a} and @samp{A} are always
-supported for @code{mpf_t} but depend on the C library for standard C float
-types. @samp{m} and @samp{p} depend on the C library.
-
- at quotation
- at multitable {(space)} {MMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMM}
- at item @nicode{a} @nicode{A} @tab hex floats, C99 style
- at item @nicode{c} @tab character
- at item @nicode{d} @tab decimal integer
- at item @nicode{e} @nicode{E} @tab scientific format float
- at item @nicode{f} @tab fixed point float
- at item @nicode{i} @tab same as @nicode{d}
- at item @nicode{g} @nicode{G} @tab fixed or scientific float
- at item @nicode{m} @tab @code{strerror} string, GLIBC style
- at item @nicode{n} @tab store characters written so far
- at item @nicode{o} @tab octal integer
- at item @nicode{p} @tab pointer
- at item @nicode{s} @tab string
- at item @nicode{u} @tab unsigned integer
- at item @nicode{x} @nicode{X} @tab hex integer
- at end multitable
- at end quotation
-
- at samp{o}, @samp{x} and @samp{X} are unsigned for the standard C types, but for
-types @samp{Z}, @samp{Q} and @samp{N} they are signed. @samp{u} is not
-meaningful for @samp{Z}, @samp{Q} and @samp{N}.
-
- at samp{M} is a proxy for the C library @samp{l} or @samp{L}, according to the
-size of @code{mp_limb_t}. Unsigned conversions will be usual, but a signed
-conversion can be used and will interpret the value as a twos complement
-negative.
-
- at samp{n} can be used with any type, even the MPIR types.
-
-Other types or conversions that might be accepted by the C library
- at code{printf} cannot be used through @code{gmp_printf}, this includes for
-instance extensions registered with GLIBC @code{register_printf_function}.
-Also currently there's no support for POSIX @samp{$} style numbered arguments
-(perhaps this will be added in the future).
-
-The precision field has it's usual meaning for integer @samp{Z} and float
- at samp{F} types, but is currently undefined for @samp{Q} and should not be used
-with that.
-
- at code{mpf_t} conversions only ever generate as many digits as can be
-accurately represented by the operand, the same as @code{mpf_get_str} does.
-Zeros will be used if necessary to pad to the requested precision. This
-happens even for an @samp{f} conversion of an @code{mpf_t} which is an
-integer, for instance @math{2^@W{1024}} in an @code{mpf_t} of 128 bits
-precision will only produce about 40 digits, then pad with zeros to the
-decimal point. An empty precision field like @samp{%.Fe} or @samp{%.Ff} can
-be used to specifically request just the significant digits.
-
-The decimal point character (or string) is taken from the current locale
-settings on systems which provide @code{localeconv} (@pxref{Locales,, Locales
-and Internationalization, libc, The GNU C Library Reference Manual}). The C
-library will normally do the same for standard float output.
-
-The format string is only interpreted as plain @code{char}s, multibyte
-characters are not recognised. Perhaps this will change in the future.
-
-
- at node Formatted Output Functions, C++ Formatted Output, Formatted Output Strings, Formatted Output
- at section Functions
- at cindex Output functions
-
-Each of the following functions is similar to the corresponding C library
-function. The basic @code{printf} forms take a variable argument list. The
- at code{vprintf} forms take an argument pointer, see @ref{Variadic Functions,,
-Variadic Functions, libc, The GNU C Library Reference Manual}, or @samp{man 3
-va_start}.
-
-It should be emphasised that if a format string is invalid, or the arguments
-don't match what the format specifies, then the behaviour of any of these
-functions will be unpredictable. GCC format string checking is not available,
-since it doesn't recognise the MPIR extensions.
-
-The file based functions @code{gmp_printf} and @code{gmp_fprintf} will return
- at math{-1} to indicate a write error. Output is not ``atomic'', so partial
-output may be produced if a write error occurs. All the functions can return
- at math{-1} if the C library @code{printf} variant in use returns @math{-1}, but
-this shouldn't normally occur.
-
- at deftypefun int gmp_printf (const char *@var{fmt}, @dots{})
- at deftypefunx int gmp_vprintf (const char *@var{fmt}, va_list @var{ap})
-Print to the standard output @code{stdout}. Return the number of characters
-written, or @math{-1} if an error occurred.
- at end deftypefun
-
- at deftypefun int gmp_fprintf (FILE *@var{fp}, const char *@var{fmt}, @dots{})
- at deftypefunx int gmp_vfprintf (FILE *@var{fp}, const char *@var{fmt}, va_list @var{ap})
-Print to the stream @var{fp}. Return the number of characters written, or
- at math{-1} if an error occurred.
- at end deftypefun
-
- at deftypefun int gmp_sprintf (char *@var{buf}, const char *@var{fmt}, @dots{})
- at deftypefunx int gmp_vsprintf (char *@var{buf}, const char *@var{fmt}, va_list @var{ap})
-Form a null-terminated string in @var{buf}. Return the number of characters
-written, excluding the terminating null.
-
-No overlap is permitted between the space at @var{buf} and the string
- at var{fmt}.
-
-These functions are not recommended, since there's no protection against
-exceeding the space available at @var{buf}.
- at end deftypefun
-
- at deftypefun int gmp_snprintf (char *@var{buf}, size_t @var{size}, const char *@var{fmt}, @dots{})
- at deftypefunx int gmp_vsnprintf (char *@var{buf}, size_t @var{size}, const char *@var{fmt}, va_list @var{ap})
-Form a null-terminated string in @var{buf}. No more than @var{size} bytes
-will be written. To get the full output, @var{size} must be enough for the
-string and null-terminator.
-
-The return value is the total number of characters which ought to have been
-produced, excluding the terminating null. If @math{@var{retval} @ge{}
- at var{size}} then the actual output has been truncated to the first
- at math{@var{size}-1} characters, and a null appended.
-
-No overlap is permitted between the region @{@var{buf}, at var{size}@} and the
- at var{fmt} string.
-
-Notice the return value is in ISO C99 @code{snprintf} style. This is so even
-if the C library @code{vsnprintf} is the older GLIBC 2.0.x style.
- at end deftypefun
-
- at deftypefun int gmp_asprintf (char **@var{pp}, const char *@var{fmt}, @dots{})
- at deftypefunx int gmp_vasprintf (char **@var{pp}, const char *@var{fmt}, va_list @var{ap})
-Form a null-terminated string in a block of memory obtained from the current
-memory allocation function (@pxref{Custom Allocation}). The block will be the
-size of the string and null-terminator. The address of the block in stored to
-*@var{pp}. The return value is the number of characters produced, excluding
-the null-terminator.
-
-Unlike the C library @code{asprintf}, @code{gmp_asprintf} doesn't return
- at math{-1} if there's no more memory available, it lets the current allocation
-function handle that.
- at end deftypefun
-
- at deftypefun int gmp_obstack_printf (struct obstack *@var{ob}, const char *@var{fmt}, @dots{})
- at deftypefunx int gmp_obstack_vprintf (struct obstack *@var{ob}, const char *@var{fmt}, va_list @var{ap})
- at cindex @code{obstack} output
-Append to the current object in @var{ob}. The return value is the number of
-characters written. A null-terminator is not written.
-
- at var{fmt} cannot be within the current object in @var{ob}, since that object
-might move as it grows.
-
-These functions are available only when the C library provides the obstack
-feature, which probably means only on GNU systems, see @ref{Obstacks,,
-Obstacks, libc, The GNU C Library Reference Manual}.
- at end deftypefun
-
-
- at node C++ Formatted Output, , Formatted Output Functions, Formatted Output
- at section C++ Formatted Output
- at cindex C++ @code{ostream} output
- at cindex @code{ostream} output
-
-The following functions are provided in @file{libmpirxx} (@pxref{Headers and
-Libraries}), which is built if C++ support is enabled (@pxref{Build Options}).
-Prototypes are available from @code{<mpir.h>}.
-
- at deftypefun ostream& operator<< (ostream& @var{stream}, mpz_t @var{op})
-Print @var{op} to @var{stream}, using its @code{ios} formatting settings.
- at code{ios::width} is reset to 0 after output, the same as the standard
- at code{ostream operator<<} routines do.
-
-In hex or octal, @var{op} is printed as a signed number, the same as for
-decimal. This is unlike the standard @code{operator<<} routines on @code{int}
-etc, which instead give twos complement.
- at end deftypefun
-
- at deftypefun ostream& operator<< (ostream& @var{stream}, mpq_t @var{op})
-Print @var{op} to @var{stream}, using its @code{ios} formatting settings.
- at code{ios::width} is reset to 0 after output, the same as the standard
- at code{ostream operator<<} routines do.
-
-Output will be a fraction like @samp{5/9}, or if the denominator is 1 then
-just a plain integer like @samp{123}.
-
-In hex or octal, @var{op} is printed as a signed value, the same as for
-decimal. If @code{ios::showbase} is set then a base indicator is shown on
-both the numerator and denominator (if the denominator is required).
- at end deftypefun
-
- at deftypefun ostream& operator<< (ostream& @var{stream}, mpf_t @var{op})
-Print @var{op} to @var{stream}, using its @code{ios} formatting settings.
- at code{ios::width} is reset to 0 after output, the same as the standard
- at code{ostream operator<<} routines do.
-
-The decimal point follows the standard library float @code{operator<<}, which
-on recent systems means the @code{std::locale} imbued on @var{stream}.
-
-Hex and octal are supported, unlike the standard @code{operator<<} on
- at code{double}. The mantissa will be in hex or octal, the exponent will be in
-decimal. For hex the exponent delimiter is an @samp{@@}. This is as per
- at code{mpf_out_str}.
-
- at code{ios::showbase} is supported, and will put a base on the mantissa, for
-example hex @samp{0x1.8} or @samp{0x0.8}, or octal @samp{01.4} or @samp{00.4}.
-This last form is slightly strange, but at least differentiates itself from
-decimal.
- at end deftypefun
-
-These operators mean that MPIR types can be printed in the usual C++ way, for
-example,
-
- at example
-mpz_t z;
-int n;
-...
-cout << "iteration " << n << " value " << z << "\n";
- at end example
-
-But note that @code{ostream} output (and @code{istream} input, @pxref{C++
-Formatted Input}) is the only overloading available for the MPIR types and that
-for instance using @code{+} with an @code{mpz_t} will have unpredictable
-results. For classes with overloading, see @ref{C++ Class Interface}.
-
-
- at node Formatted Input, C++ Class Interface, Formatted Output, Top
- at chapter Formatted Input
- at cindex Formatted input
- at cindex @code{scanf} formatted input
-
- at menu
-* Formatted Input Strings::
-* Formatted Input Functions::
-* C++ Formatted Input::
- at end menu
-
-
- at node Formatted Input Strings, Formatted Input Functions, Formatted Input, Formatted Input
- at section Formatted Input Strings
-
- at code{gmp_scanf} and friends accept format strings similar to the standard C
- at code{scanf} (@pxref{Formatted Input,, Formatted Input, libc, The GNU C
-Library Reference Manual}). A format specification is of the form
-
- at example
-% [flags] [width] [type] conv
- at end example
-
-MPIR adds types @samp{Z}, @samp{Q} and @samp{F} for @code{mpz_t}, @code{mpq_t}
-and @code{mpf_t} respectively. @samp{Z} and @samp{Q} behave like integers.
- at samp{Q} will read a @samp{/} and a denominator, if present. @samp{F} behaves
-like a float.
-
-MPIR variables don't require an @code{&} when passed to @code{gmp_scanf}, since
-they're already ``call-by-reference''. For example,
-
- at example
-/* to read say "a(5) = 1234" */
-int n;
-mpz_t z;
-gmp_scanf ("a(%d) = %Zd\n", &n, z);
-
-mpq_t q1, q2;
-gmp_sscanf ("0377 + 0x10/0x11", "%Qi + %Qi", q1, q2);
-
-/* to read say "topleft (1.55,-2.66)" */
-mpf_t x, y;
-char buf[32];
-gmp_scanf ("%31s (%Ff,%Ff)", buf, x, y);
- at end example
-
-All the standard C @code{scanf} types behave the same as in the C library
- at code{scanf}, and can be freely intermixed with the MPIR extensions. In the
-current implementation the standard parts of the format string are simply
-handed to @code{scanf} and only the MPIR extensions handled directly.
-
-The flags accepted are as follows. @samp{a} and @samp{'} will depend on
-support from the C library, and @samp{'} cannot be used with MPIR types.
-
- at quotation
- at multitable {(space)} {MMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMM}
- at item @nicode{*} @tab read but don't store
- at item @nicode{a} @tab allocate a buffer (string conversions)
- at item @nicode{'} @tab grouped digits, GLIBC style (not MPIR types)
- at end multitable
- at end quotation
-
-The standard types accepted are as follows. @samp{h} and @samp{l} are
-portable, the rest will depend on the compiler (or include files) for the type
-and the C library for the input.
-
- at quotation
- at multitable {(space)} {MMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMM}
- at item @nicode{h} @tab @nicode{short}
- at item @nicode{hh} @tab @nicode{char}
- at item @nicode{j} @tab @nicode{intmax_t} or @nicode{uintmax_t}
- at item @nicode{l} @tab @nicode{long int}, @nicode{double} or @nicode{wchar_t}
- at item @nicode{ll} @tab @nicode{long long}
- at item @nicode{L} @tab @nicode{long double}
- at item @nicode{q} @tab @nicode{quad_t} or @nicode{u_quad_t}
- at item @nicode{t} @tab @nicode{ptrdiff_t}
- at item @nicode{z} @tab @nicode{size_t}
- at end multitable
- at end quotation
-
- at noindent
-The MPIR types are
-
- at quotation
- at multitable {(space)} {MMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMM}
- at item @nicode{F} @tab @nicode{mpf_t}, float conversions
- at item @nicode{Q} @tab @nicode{mpq_t}, integer conversions
- at item @nicode{Z} @tab @nicode{mpz_t}, integer conversions
- at end multitable
- at end quotation
-
-The conversions accepted are as follows. @samp{p} and @samp{[} will depend on
-support from the C library, the rest are standard.
-
- at quotation
- at multitable {(space)} {MMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMM}
- at item @nicode{c} @tab character or characters
- at item @nicode{d} @tab decimal integer
- at item @nicode{e} @nicode{E} @nicode{f} @nicode{g} @nicode{G}
- @tab float
- at item @nicode{i} @tab integer with base indicator
- at item @nicode{n} @tab characters read so far
- at item @nicode{o} @tab octal integer
- at item @nicode{p} @tab pointer
- at item @nicode{s} @tab string of non-whitespace characters
- at item @nicode{u} @tab decimal integer
- at item @nicode{x} @nicode{X} @tab hex integer
- at item @nicode{[} @tab string of characters in a set
- at end multitable
- at end quotation
-
- at samp{e}, @samp{E}, @samp{f}, @samp{g} and @samp{G} are identical, they all
-read either fixed point or scientific format, and either upper or lower case
- at samp{e} for the exponent in scientific format.
-
-C99 style hex float format (@code{printf %a}, @pxref{Formatted Output
-Strings}) is always accepted for @code{mpf_t}, but for the standard float
-types it will depend on the C library.
-
- at samp{x} and @samp{X} are identical, both accept both upper and lower case
-hexadecimal.
-
- at samp{o}, @samp{u}, @samp{x} and @samp{X} all read positive or negative
-values. For the standard C types these are described as ``unsigned''
-conversions, but that merely affects certain overflow handling, negatives are
-still allowed (per @code{strtoul}, @pxref{Parsing of Integers,, Parsing of
-Integers, libc, The GNU C Library Reference Manual}). For MPIR types there are
-no overflows, so @samp{d} and @samp{u} are identical.
-
- at samp{Q} type reads the numerator and (optional) denominator as given. If the
-value might not be in canonical form then @code{mpq_canonicalize} must be
-called before using it in any calculations (@pxref{Rational Number
-Functions}).
-
- at samp{Qi} will read a base specification separately for the numerator and
-denominator. For example @samp{0x10/11} would be 16/11, whereas
- at samp{0x10/0x11} would be 16/17.
-
- at samp{n} can be used with any of the types above, even the MPIR types.
- at samp{*} to suppress assignment is allowed, though in that case it would do
-nothing at all.
-
-Other conversions or types that might be accepted by the C library
- at code{scanf} cannot be used through @code{gmp_scanf}.
-
-Whitespace is read and discarded before a field, except for @samp{c} and
- at samp{[} conversions.
-
-For float conversions, the decimal point character (or string) expected is
-taken from the current locale settings on systems which provide
- at code{localeconv} (@pxref{Locales,, Locales and Internationalization, libc,
-The GNU C Library Reference Manual}). The C library will normally do the same
-for standard float input.
-
-The format string is only interpreted as plain @code{char}s, multibyte
-characters are not recognised. Perhaps this will change in the future.
-
-
- at node Formatted Input Functions, C++ Formatted Input, Formatted Input Strings, Formatted Input
- at section Formatted Input Functions
- at cindex Input functions
-
-Each of the following functions is similar to the corresponding C library
-function. The plain @code{scanf} forms take a variable argument list. The
- at code{vscanf} forms take an argument pointer, see @ref{Variadic Functions,,
-Variadic Functions, libc, The GNU C Library Reference Manual}, or @samp{man 3
-va_start}.
-
-It should be emphasised that if a format string is invalid, or the arguments
-don't match what the format specifies, then the behaviour of any of these
-functions will be unpredictable. GCC format string checking is not available,
-since it doesn't recognise the MPIR extensions.
-
-No overlap is permitted between the @var{fmt} string and any of the results
-produced.
-
- at deftypefun int gmp_scanf (const char *@var{fmt}, @dots{})
- at deftypefunx int gmp_vscanf (const char *@var{fmt}, va_list @var{ap})
-Read from the standard input @code{stdin}.
- at end deftypefun
-
- at deftypefun int gmp_fscanf (FILE *@var{fp}, const char *@var{fmt}, @dots{})
- at deftypefunx int gmp_vfscanf (FILE *@var{fp}, const char *@var{fmt}, va_list @var{ap})
-Read from the stream @var{fp}.
- at end deftypefun
-
- at deftypefun int gmp_sscanf (const char *@var{s}, const char *@var{fmt}, @dots{})
- at deftypefunx int gmp_vsscanf (const char *@var{s}, const char *@var{fmt}, va_list @var{ap})
-Read from a null-terminated string @var{s}.
- at end deftypefun
-
-The return value from each of these functions is the same as the standard C99
- at code{scanf}, namely the number of fields successfully parsed and stored.
- at samp{%n} fields and fields read but suppressed by @samp{*} don't count
-towards the return value.
-
-If end of input (or a file error) is reached before a character for a field or
-a literal, and if no previous non-suppressed fields have matched, then the
-return value is @code{EOF} instead of 0. A whitespace character in the format
-string is only an optional match and doesn't induce an @code{EOF} in this
-fashion. Leading whitespace read and discarded for a field don't count as
-characters for that field.
-
-For the MPIR types, input parsing follows C99 rules, namely one character of
-lookahead is used and characters are read while they continue to meet the
-format requirements. If this doesn't provide a complete number then the
-function terminates, with that field not stored nor counted towards the return
-value. For instance with @code{mpf_t} an input @samp{1.23e-XYZ} would be read
-up to the @samp{X} and that character pushed back since it's not a digit. The
-string @samp{1.23e-} would then be considered invalid since an @samp{e} must
-be followed by at least one digit.
-
-For the standard C types, in the current implementation MPIR calls the C
-library @code{scanf} functions, which might have looser rules about what
-constitutes a valid input.
-
-Note that @code{gmp_sscanf} is the same as @code{gmp_fscanf} and only does one
-character of lookahead when parsing. Although clearly it could look at its
-entire input, it is deliberately made identical to @code{gmp_fscanf}, the same
-way C99 @code{sscanf} is the same as @code{fscanf}.
-
-
- at node C++ Formatted Input, , Formatted Input Functions, Formatted Input
- at section C++ Formatted Input
- at cindex C++ @code{istream} input
- at cindex @code{istream} input
-
-The following functions are provided in @file{libmpirxx} (@pxref{Headers and
-Libraries}), which is built only if C++ support is enabled (@pxref{Build
-Options}). Prototypes are available from @code{<mpir.h>}.
-
- at deftypefun istream& operator>> (istream& @var{stream}, mpz_t @var{rop})
-Read @var{rop} from @var{stream}, using its @code{ios} formatting settings.
- at end deftypefun
-
- at deftypefun istream& operator>> (istream& @var{stream}, mpq_t @var{rop})
-An integer like @samp{123} will be read, or a fraction like @samp{5/9}. No
-whitespace is allowed around the @samp{/}. If the fraction is not in
-canonical form then @code{mpq_canonicalize} must be called (@pxref{Rational
-Number Functions}) before operating on it.
-
-As per integer input, an @samp{0} or @samp{0x} base indicator is read when
-none of @code{ios::dec}, @code{ios::oct} or @code{ios::hex} are set. This is
-done separately for numerator and denominator, so that for instance
- at samp{0x10/11} is @math{16/11} and @samp{0x10/0x11} is @math{16/17}.
- at end deftypefun
-
- at deftypefun istream& operator>> (istream& @var{stream}, mpf_t @var{rop})
-Read @var{rop} from @var{stream}, using its @code{ios} formatting settings.
-
-Hex or octal floats are not supported, but might be in the future, or perhaps
-it's best to accept only what the standard float @code{operator>>} does.
- at end deftypefun
-
-Note that digit grouping specified by the @code{istream} locale is currently
-not accepted. Perhaps this will change in the future.
-
- at sp 1
-These operators mean that MPIR types can be read in the usual C++ way, for
-example,
-
- at example
-mpz_t z;
-...
-cin >> z;
- at end example
-
-But note that @code{istream} input (and @code{ostream} output, @pxref{C++
-Formatted Output}) is the only overloading available for the MPIR types and
-that for instance using @code{+} with an @code{mpz_t} will have unpredictable
-results. For classes with overloading, see @ref{C++ Class Interface}.
-
-
-
- at node C++ Class Interface, Custom Allocation, Formatted Input, Top
- at chapter C++ Class Interface
- at cindex C++ interface
-
-This chapter describes the C++ class based interface to MPIR.
-
-All MPIR C language types and functions can be used in C++ programs, since
- at file{mpir.h} has @code{extern "C"} qualifiers, but the class interface offers
-overloaded functions and operators which may be more convenient.
-
-Due to the implementation of this interface, a reasonably recent C++ compiler
-is required, one supporting namespaces, partial specialization of templates
-and member templates. For GCC this means version 2.91 or later.
-
- at strong{Everything described in this chapter is to be considered preliminary
-and might be subject to incompatible changes if some unforeseen difficulty
-reveals itself.}
-
- at menu
-* C++ Interface General::
-* C++ Interface Integers::
-* C++ Interface Rationals::
-* C++ Interface Floats::
-* C++ Interface Random Numbers::
-* C++ Interface Limitations::
- at end menu
-
-
- at node C++ Interface General, C++ Interface Integers, C++ Class Interface, C++ Class Interface
- at section C++ Interface General
-
- at noindent
-All the C++ classes and functions are available with
-
- at cindex @code{mpirxx.h}
- at example
-#include <mpirxx.h>
- at end example
-
-Programs should be linked with the @file{libmpirxx} and @file{libmpir}
-libraries. For example,
-
- at example
-g++ mycxxprog.cc -lmpirxx -lmpir
- at end example
-
- at noindent
-The classes defined are
-
- at deftp Class mpz_class
- at deftpx Class mpq_class
- at deftpx Class mpf_class
- at end deftp
-
-The standard operators and various standard functions are overloaded to allow
-arithmetic with these classes. For example,
-
- at example
-int
-main (void)
-@{
- mpz_class a, b, c;
-
- a = 1234;
- b = "-5678";
- c = a+b;
- cout << "sum is " << c << "\n";
- cout << "absolute value is " << abs(c) << "\n";
-
- return 0;
-@}
- at end example
-
-An important feature of the implementation is that an expression like
- at code{a=b+c} results in a single call to the corresponding @code{mpz_add},
-without using a temporary for the @code{b+c} part. Expressions which by their
-nature imply intermediate values, like @code{a=b*c+d*e}, still use temporaries
-though.
-
-The classes can be freely intermixed in expressions, as can the classes and
-the standard types @code{mpir_si}, @code{mpir_ui} and @code{double}.
-Smaller types like @code{int} or @code{float} can also be intermixed, since
-C++ will promote them.
-
-Note that @code{bool} is not accepted directly, but must be explicitly cast to
-an @code{int} first. This is because C++ will automatically convert any
-pointer to a @code{bool}, so if MPIR accepted @code{bool} it would make all
-sorts of invalid class and pointer combinations compile but almost certainly
-not do anything sensible.
-
-Conversions back from the classes to standard C++ types aren't done
-automatically, instead member functions like @code{get_si} are provided (see
-the following sections for details).
-
-Also there are no automatic conversions from the classes to the corresponding
-MPIR C types, instead a reference to the underlying C object can be obtained
-with the following functions,
-
- at deftypefun mpz_t mpz_class::get_mpz_t ()
- at deftypefunx mpq_t mpq_class::get_mpq_t ()
- at deftypefunx mpf_t mpf_class::get_mpf_t ()
- at end deftypefun
-
-These can be used to call a C function which doesn't have a C++ class
-interface. For example to set @code{a} to the GCD of @code{b} and @code{c},
-
- at example
-mpz_class a, b, c;
-...
-mpz_gcd (a.get_mpz_t(), b.get_mpz_t(), c.get_mpz_t());
- at end example
-
-In the other direction, a class can be initialized from the corresponding MPIR
-C type, or assigned to if an explicit constructor is used. In both cases this
-makes a copy of the value, it doesn't create any sort of association. For
-example,
-
- at example
-mpz_t z;
-// ... init and calculate z ...
-mpz_class x(z);
-mpz_class y;
-y = mpz_class (z);
- at end example
-
-There are no namespace setups in @file{mpirxx.h}, all types and functions are
-simply put into the global namespace. This is what @file{mpir.h} has done in
-the past, and continues to do for compatibility. The extras provided by
- at file{mpirxx.h} follow MPIR naming conventions and are unlikely to clash with
-anything.
-
-
- at node C++ Interface Integers, C++ Interface Rationals, C++ Interface General, C++ Class Interface
- at section C++ Interface Integers
-
- at deftypefun void mpz_class::mpz_class (type @var{n})
-Construct an @code{mpz_class}. All the standard C++ types may be used
- at code{long double}, and all the MPIR C++ classes can be
-used. Any necessary conversion follows the corresponding C function, for
-example @code{double} follows @code{mpz_set_d} (@pxref{Assigning Integers}).
- at end deftypefun
-
- at deftypefun void mpz_class::mpz_class (mpz_t @var{z})
-Construct an @code{mpz_class} from an @code{mpz_t}. The value in @var{z} is
-copied into the new @code{mpz_class}, there won't be any permanent association
-between it and @var{z}.
- at end deftypefun
-
- at deftypefun void mpz_class::mpz_class (const char *@var{s})
- at deftypefunx void mpz_class::mpz_class (const char *@var{s}, int @var{base} = 0)
- at deftypefunx void mpz_class::mpz_class (const string& @var{s})
- at deftypefunx void mpz_class::mpz_class (const string& @var{s}, int @var{base} = 0)
-Construct an @code{mpz_class} converted from a string using @code{mpz_set_str}
-(@pxref{Assigning Integers}).
-
-If the string is not a valid integer, an @code{std::invalid_argument}
-exception is thrown. The same applies to @code{operator=}.
- at end deftypefun
-
- at deftypefun mpz_class operator/ (mpz_class @var{a}, mpz_class @var{d})
- at deftypefunx mpz_class operator% (mpz_class @var{a}, mpz_class @var{d})
-Divisions involving @code{mpz_class} round towards zero, as per the
- at code{mpz_tdiv_q} and @code{mpz_tdiv_r} functions (@pxref{Integer Division}).
-This is the same as the C99 @code{/} and @code{%} operators.
-
-The @code{mpz_fdiv at dots{}} or @code{mpz_cdiv at dots{}} functions can always be called
-directly if desired. For example,
-
- at example
-mpz_class q, a, d;
-...
-mpz_fdiv_q (q.get_mpz_t(), a.get_mpz_t(), d.get_mpz_t());
- at end example
- at end deftypefun
-
- at deftypefun mpz_class abs (mpz_class @var{op1})
- at deftypefunx int cmp (mpz_class @var{op1}, type @var{op2})
- at deftypefunx int cmp (type @var{op1}, mpz_class @var{op2})
- at maybepagebreak
- at deftypefunx bool mpz_class::fits_sint_p (void)
- at deftypefunx bool mpz_class::fits_slong_p (void)
- at deftypefunx bool mpz_class::fits_sshort_p (void)
- at maybepagebreak
- at deftypefunx bool mpz_class::fits_uint_p (void)
- at deftypefunx bool mpz_class::fits_ulong_p (void)
- at deftypefunx bool mpz_class::fits_ushort_p (void)
- at maybepagebreak
- at deftypefunx double mpz_class::get_d (void)
- at deftypefunx mpir_si mpz_class::get_si (void)
- at deftypefunx string mpz_class::get_str (int @var{base} = 10)
- at deftypefunx mpir_ui mpz_class::get_ui (void)
- at maybepagebreak
- at deftypefunx int mpz_class::set_str (const char *@var{str}, int @var{base})
- at deftypefunx int mpz_class::set_str (const string& @var{str}, int @var{base})
- at deftypefunx int sgn (mpz_class @var{op})
- at deftypefunx mpz_class sqrt (mpz_class @var{op})
-These functions provide a C++ class interface to the corresponding MPIR C
-routines.
-
- at code{cmp} can be used with any of the classes or the standard C++ types,
-except @code{long double}.
- at end deftypefun
-
- at sp 1
-Overloaded operators for combinations of @code{mpz_class} and @code{double}
-are provided for completeness, but it should be noted that if the given
- at code{double} is not an integer then the way any rounding is done is currently
-unspecified. The rounding might take place at the start, in the middle, or at
-the end of the operation, and it might change in the future.
-
-Conversions between @code{mpz_class} and @code{double}, however, are defined
-to follow the corresponding C functions @code{mpz_get_d} and @code{mpz_set_d}.
-And comparisons are always made exactly, as per @code{mpz_cmp_d}.
-
-
- at node C++ Interface Rationals, C++ Interface Floats, C++ Interface Integers, C++ Class Interface
- at section C++ Interface Rationals
-
-In all the following constructors, if a fraction is given then it should be in
-canonical form, or if not then @code{mpq_class::canonicalize} called.
-
- at deftypefun void mpq_class::mpq_class (type @var{op})
- at deftypefunx void mpq_class::mpq_class (integer @var{num}, integer @var{den})
-Construct an @code{mpq_class}. The initial value can be a single value of any
-type, or a pair of integers (@code{mpz_class} or standard C++ integer types)
-representing a fraction, except that @code{long double}
-is not supported. For example,
-
- at example
-mpq_class q (99);
-mpq_class q (1.75);
-mpq_class q (1, 3);
- at end example
- at end deftypefun
-
- at deftypefun void mpq_class::mpq_class (mpq_t @var{q})
-Construct an @code{mpq_class} from an @code{mpq_t}. The value in @var{q} is
-copied into the new @code{mpq_class}, there won't be any permanent association
-between it and @var{q}.
- at end deftypefun
-
- at deftypefun void mpq_class::mpq_class (const char *@var{s})
- at deftypefunx void mpq_class::mpq_class (const char *@var{s}, int @var{base} = 0)
- at deftypefunx void mpq_class::mpq_class (const string& @var{s})
- at deftypefunx void mpq_class::mpq_class (const string& @var{s}, int @var{base} = 0)
-Construct an @code{mpq_class} converted from a string using @code{mpq_set_str}
-(@pxref{Initializing Rationals}).
-
-If the string is not a valid rational, an @code{std::invalid_argument}
-exception is thrown. The same applies to @code{operator=}.
- at end deftypefun
-
- at deftypefun void mpq_class::canonicalize ()
-Put an @code{mpq_class} into canonical form, as per @ref{Rational Number
-Functions}. All arithmetic operators require their operands in canonical
-form, and will return results in canonical form.
- at end deftypefun
-
- at deftypefun mpq_class abs (mpq_class @var{op})
- at deftypefunx int cmp (mpq_class @var{op1}, type @var{op2})
- at deftypefunx int cmp (type @var{op1}, mpq_class @var{op2})
- at maybepagebreak
- at deftypefunx double mpq_class::get_d (void)
- at deftypefunx string mpq_class::get_str (int @var{base} = 10)
- at maybepagebreak
- at deftypefunx int mpq_class::set_str (const char *@var{str}, int @var{base})
- at deftypefunx int mpq_class::set_str (const string& @var{str}, int @var{base})
- at deftypefunx int sgn (mpq_class @var{op})
-These functions provide a C++ class interface to the corresponding MPIR C
-routines.
-
- at code{cmp} can be used with any of the classes or the standard C++ types,
-except @code{long double}.
- at end deftypefun
-
- at deftypefun {mpz_class&} mpq_class::get_num ()
- at deftypefunx {mpz_class&} mpq_class::get_den ()
-Get a reference to an @code{mpz_class} which is the numerator or denominator
-of an @code{mpq_class}. This can be used both for read and write access. If
-the object returned is modified, it modifies the original @code{mpq_class}.
-
-If direct manipulation might produce a non-canonical value, then
- at code{mpq_class::canonicalize} must be called before further operations.
- at end deftypefun
-
- at deftypefun mpz_t mpq_class::get_num_mpz_t ()
- at deftypefunx mpz_t mpq_class::get_den_mpz_t ()
-Get a reference to the underlying @code{mpz_t} numerator or denominator of an
- at code{mpq_class}. This can be passed to C functions expecting an
- at code{mpz_t}. Any modifications made to the @code{mpz_t} will modify the
-original @code{mpq_class}.
-
-If direct manipulation might produce a non-canonical value, then
- at code{mpq_class::canonicalize} must be called before further operations.
- at end deftypefun
-
- at deftypefun istream& operator>> (istream& @var{stream}, mpq_class& @var{rop});
-Read @var{rop} from @var{stream}, using its @code{ios} formatting settings,
-the same as @code{mpq_t operator>>} (@pxref{C++ Formatted Input}).
-
-If the @var{rop} read might not be in canonical form then
- at code{mpq_class::canonicalize} must be called.
- at end deftypefun
-
-
- at node C++ Interface Floats, C++ Interface Random Numbers, C++ Interface Rationals, C++ Class Interface
- at section C++ Interface Floats
-
-When an expression requires the use of temporary intermediate @code{mpf_class}
-values, like @code{f=g*h+x*y}, those temporaries will have the same precision
-as the destination @code{f}. Explicit constructors can be used if this
-doesn't suit.
-
- at deftypefun {} mpf_class::mpf_class (type @var{op})
- at deftypefunx {} mpf_class::mpf_class (type @var{op}, mpir_ui @var{prec})
-Construct an @code{mpf_class}. Any standard C++ type can be used, except
- at code{long double}, and any of the MPIR C++ classes can be
-used.
-
-If @var{prec} is given, the initial precision is that value, in bits. If
- at var{prec} is not given, then the initial precision is determined by the type
-of @var{op} given. An @code{mpz_class}, @code{mpq_class}, or C++
-builtin type will give the default @code{mpf} precision (@pxref{Initializing
-Floats}). An @code{mpf_class} or expression will give the precision of that
-value. The precision of a binary expression is the higher of the two
-operands.
-
- at example
-mpf_class f(1.5); // default precision
-mpf_class f(1.5, 500); // 500 bits (at least)
-mpf_class f(x); // precision of x
-mpf_class f(abs(x)); // precision of x
-mpf_class f(-g, 1000); // 1000 bits (at least)
-mpf_class f(x+y); // greater of precisions of x and y
- at end example
- at end deftypefun
-
- at deftypefun void mpf_class::mpf_class (const char *@var{s})
- at deftypefunx void mpf_class::mpf_class (const char *@var{s}, mpir_ui @var{prec}, int @var{base} = 0)
- at deftypefunx void mpf_class::mpf_class (const string& @var{s})
- at deftypefunx void mpf_class::mpf_class (const string& @var{s}, mpir_ui @var{prec}, int @var{base} = 0)
-Construct an @code{mpf_class} converted from a string using @code{mpf_set_str}
-(@pxref{Assigning Floats}). If @var{prec} is given, the initial precision is
-that value, in bits. If not, the default @code{mpf} precision
-(@pxref{Initializing Floats}) is used.
-
-If the string is not a valid float, an @code{std::invalid_argument} exception
-is thrown. The same applies to @code{operator=}.
- at end deftypefun
-
- at deftypefun {mpf_class&} mpf_class::operator= (type @var{op})
-Convert and store the given @var{op} value to an @code{mpf_class} object. The
-same types are accepted as for the constructors above.
-
-Note that @code{operator=} only stores a new value, it doesn't copy or change
-the precision of the destination, instead the value is truncated if necessary.
-This is the same as @code{mpf_set} etc. Note in particular this means for
- at code{mpf_class} a copy constructor is not the same as a default constructor
-plus assignment.
-
- at example
-mpf_class x (y); // x created with precision of y
-
-mpf_class x; // x created with default precision
-x = y; // value truncated to that precision
- at end example
-
-Applications using templated code may need to be careful about the assumptions
-the code makes in this area, when working with @code{mpf_class} values of
-various different or non-default precisions. For instance implementations of
-the standard @code{complex} template have been seen in both styles above,
-though of course @code{complex} is normally only actually specified for use
-with the builtin float types.
- at end deftypefun
-
- at deftypefun mpf_class abs (mpf_class @var{op})
- at deftypefunx mpf_class ceil (mpf_class @var{op})
- at deftypefunx int cmp (mpf_class @var{op1}, type @var{op2})
- at deftypefunx int cmp (type @var{op1}, mpf_class @var{op2})
- at maybepagebreak
- at deftypefunx bool mpf_class::fits_sint_p (void)
- at deftypefunx bool mpf_class::fits_slong_p (void)
- at deftypefunx bool mpf_class::fits_sshort_p (void)
- at maybepagebreak
- at deftypefunx bool mpf_class::fits_uint_p (void)
- at deftypefunx bool mpf_class::fits_ulong_p (void)
- at deftypefunx bool mpf_class::fits_ushort_p (void)
- at maybepagebreak
- at deftypefunx mpf_class floor (mpf_class @var{op})
- at deftypefunx mpf_class hypot (mpf_class @var{op1}, mpf_class @var{op2})
- at maybepagebreak
- at deftypefunx double mpf_class::get_d (void)
- at deftypefunx mpir_si mpf_class::get_si (void)
- at deftypefunx string mpf_class::get_str (mp_exp_t& @var{exp}, int @var{base} = 10, size_t @var{digits} = 0)
- at deftypefunx mpir_ui mpf_class::get_ui (void)
- at maybepagebreak
- at deftypefunx int mpf_class::set_str (const char *@var{str}, int @var{base})
- at deftypefunx int mpf_class::set_str (const string& @var{str}, int @var{base})
- at deftypefunx int sgn (mpf_class @var{op})
- at deftypefunx mpf_class sqrt (mpf_class @var{op})
- at deftypefunx mpf_class trunc (mpf_class @var{op})
-These functions provide a C++ class interface to the corresponding MPIR C
-routines.
-
- at code{cmp} can be used with any of the classes or the standard C++ types,
-except @code{long double}.
-
-The accuracy provided by @code{hypot} is not currently guaranteed.
- at end deftypefun
-
- at deftypefun {mp_bitcnt_t} mpf_class::get_prec ()
- at deftypefunx void mpf_class::set_prec (mp_bitcnt_t @var{prec})
- at deftypefunx void mpf_class::set_prec_raw (mp_bitcnt_t @var{prec})
-Get or set the current precision of an @code{mpf_class}.
-
-The restrictions described for @code{mpf_set_prec_raw} (@pxref{Initializing
-Floats}) apply to @code{mpf_class::set_prec_raw}. Note in particular that the
- at code{mpf_class} must be restored to it's allocated precision before being
-destroyed. This must be done by application code, there's no automatic
-mechanism for it.
- at end deftypefun
-
-
- at node C++ Interface Random Numbers, C++ Interface Limitations, C++ Interface Floats, C++ Class Interface
- at section C++ Interface Random Numbers
-
- at deftp Class gmp_randclass
-The C++ class interface to the MPIR random number functions uses
- at code{gmp_randclass} to hold an algorithm selection and current state, as per
- at code{gmp_randstate_t}.
- at end deftp
-
- at deftypefun {} gmp_randclass::gmp_randclass (void (*@var{randinit}) (gmp_randstate_t, @dots{}), @dots{})
-Construct a @code{gmp_randclass}, using a call to the given @var{randinit}
-function (@pxref{Random State Initialization}). The arguments expected are
-the same as @var{randinit}, but with @code{mpz_class} instead of @code{mpz_t}.
-For example,
-
- at example
-gmp_randclass r1 (gmp_randinit_default);
-gmp_randclass r2 (gmp_randinit_lc_2exp_size, 32);
-gmp_randclass r3 (gmp_randinit_lc_2exp, a, c, m2exp);
-gmp_randclass r4 (gmp_randinit_mt);
- at end example
-
- at code{gmp_randinit_lc_2exp_size} will fail if the size requested is too big,
-an @code{std::length_error} exception is thrown in that case.
- at end deftypefun
-
- at deftypefun void gmp_randclass::seed (mpir_ui @var{s})
- at deftypefunx void gmp_randclass::seed (mpz_class @var{s})
-Seed a random number generator. See @pxref{Random Number Functions}, for how
-to choose a good seed.
- at end deftypefun
-
- at deftypefun mpz_class gmp_randclass::get_z_bits (mpir_ui @var{bits})
- at deftypefunx mpz_class gmp_randclass::get_z_bits (mpz_class @var{bits})
-Generate a random integer with a specified number of bits.
- at end deftypefun
-
- at deftypefun mpz_class gmp_randclass::get_z_range (mpz_class @var{n})
-Generate a random integer in the range 0 to @math{@var{n}-1} inclusive.
- at end deftypefun
-
- at deftypefun mpf_class gmp_randclass::get_f ()
- at deftypefunx mpf_class gmp_randclass::get_f (mpir_ui @var{prec})
-Generate a random float @var{f} in the range @math{0 <= @var{f} < 1}. @var{f}
-will be to @var{prec} bits precision, or if @var{prec} is not given then to
-the precision of the destination. For example,
-
- at example
-gmp_randclass r;
-...
-mpf_class f (0, 512); // 512 bits precision
-f = r.get_f(); // random number, 512 bits
- at end example
- at end deftypefun
-
-
-
- at node C++ Interface Limitations, , C++ Interface Random Numbers, C++ Class Interface
- at section C++ Interface Limitations
-
- at table @asis
- at item @code{mpq_class} and Templated Reading
-A generic piece of template code probably won't know that @code{mpq_class}
-requires a @code{canonicalize} call if inputs read with @code{operator>>}
-might be non-canonical. This can lead to incorrect results.
-
- at code{operator>>} behaves as it does for reasons of efficiency. A
-canonicalize can be quite time consuming on large operands, and is best
-avoided if it's not necessary.
-
-But this potential difficulty reduces the usefulness of @code{mpq_class}.
-Perhaps a mechanism to tell @code{operator>>} what to do will be adopted in
-the future, maybe a preprocessor define, a global flag, or an @code{ios} flag
-pressed into service. Or maybe, at the risk of inconsistency, the
- at code{mpq_class} @code{operator>>} could canonicalize and leave @code{mpq_t}
- at code{operator>>} not doing so, for use on those occasions when that's
-acceptable. Send feedback or alternate ideas to @uref{http://groups.google.com/group/mpir-devel}.
-
- at item Subclassing
-Subclassing the MPIR C++ classes works, but is not currently recommended.
-
-Expressions involving subclasses resolve correctly (or seem to), but in normal
-C++ fashion the subclass doesn't inherit constructors and assignments.
-There's many of those in the MPIR classes, and a good way to reestablish them
-in a subclass is not yet provided.
-
- at item Templated Expressions
-A subtle difficulty exists when using expressions together with
-application-defined template functions. Consider the following, with @code{T}
-intended to be some numeric type,
-
- at example
-template <class T>
-T fun (const T &, const T &);
- at end example
-
- at noindent
-When used with, say, plain @code{mpz_class} variables, it works fine: @code{T}
-is resolved as @code{mpz_class}.
-
- at example
-mpz_class f(1), g(2);
-fun (f, g); // Good
- at end example
-
- at noindent
-But when one of the arguments is an expression, it doesn't work.
-
- at example
-mpz_class f(1), g(2), h(3);
-fun (f, g+h); // Bad
- at end example
-
-This is because @code{g+h} ends up being a certain expression template type
-internal to @code{mpirxx.h}, which the C++ template resolution rules are unable
-to automatically convert to @code{mpz_class}. The workaround is simply to add
-an explicit cast.
-
- at example
-mpz_class f(1), g(2), h(3);
-fun (f, mpz_class(g+h)); // Good
- at end example
-
-Similarly, within @code{fun} it may be necessary to cast an expression to type
- at code{T} when calling a templated @code{fun2}.
-
- at example
-template <class T>
-void fun (T f, T g)
-@{
- fun2 (f, f+g); // Bad
-@}
-
-template <class T>
-void fun (T f, T g)
-@{
- fun2 (f, T(f+g)); // Good
-@}
- at end example
- at end table
-
- at node Custom Allocation, Language Bindings, C++ Class Interface, Top
- at comment node-name, next, previous, up
- at chapter Custom Allocation
- at cindex Custom allocation
- at cindex Memory allocation
- at cindex Allocation of memory
-
-By default MPIR uses @code{malloc}, @code{realloc} and @code{free} for memory
-allocation, and if they fail MPIR prints a message to the standard error output
-and terminates the program.
-
-Alternate functions can be specified, to allocate memory in a different way or
-to have a different error action on running out of memory.
-
- at deftypefun void mp_set_memory_functions (@* void *(*@var{alloc_func_ptr}) (size_t), @* void *(*@var{realloc_func_ptr}) (void *, size_t, size_t), @* void (*@var{free_func_ptr}) (void *, size_t))
-Replace the current allocation functions from the arguments. If an argument
-is @code{NULL}, the corresponding default function is used.
-
-These functions will be used for all memory allocation done by MPIR, apart from
-temporary space from @code{alloca} if that function is available and MPIR is
-configured to use it (@pxref{Build Options}).
-
- at strong{Be sure to call @code{mp_set_memory_functions} only when there are no
-active MPIR objects allocated using the previous memory functions! Usually
-that means calling it before any other MPIR function.}
- at end deftypefun
-
-The functions supplied should fit the following declarations:
-
- at deftypevr Function {void *} allocate_function (size_t @var{alloc_size})
-Return a pointer to newly allocated space with at least @var{alloc_size}
-bytes.
- at end deftypevr
-
- at deftypevr Function {void *} reallocate_function (void *@var{ptr}, size_t @var{old_size}, size_t @var{new_size})
-Resize a previously allocated block @var{ptr} of @var{old_size} bytes to be
- at var{new_size} bytes.
-
-The block may be moved if necessary or if desired, and in that case the
-smaller of @var{old_size} and @var{new_size} bytes must be copied to the new
-location. The return value is a pointer to the resized block, that being the
-new location if moved or just @var{ptr} if not.
-
- at var{ptr} is never @code{NULL}, it's always a previously allocated block.
- at var{new_size} may be bigger or smaller than @var{old_size}.
- at end deftypevr
-
- at deftypevr Function void free_function (void *@var{ptr}, size_t @var{size})
-De-allocate the space pointed to by @var{ptr}.
-
- at var{ptr} is never @code{NULL}, it's always a previously allocated block of
- at var{size} bytes.
- at end deftypevr
-
-A @dfn{byte} here means the unit used by the @code{sizeof} operator.
-
-The @var{old_size} parameters to @var{reallocate_function} and
- at var{free_function} are passed for convenience, but of course can be ignored
-if not needed. The default functions using @code{malloc} and friends for
-instance don't use them.
-
-No error return is allowed from any of these functions, if they return then
-they must have performed the specified operation. In particular note that
- at var{allocate_function} or @var{reallocate_function} mustn't return
- at code{NULL}.
-
-Getting a different fatal error action is a good use for custom allocation
-functions, for example giving a graphical dialog rather than the default print
-to @code{stderr}. How much is possible when genuinely out of memory is
-another question though.
-
-There's currently no defined way for the allocation functions to recover from
-an error such as out of memory, they must terminate program execution. A
- at code{longjmp} or throwing a C++ exception will have undefined results. This
-may change in the future.
-
-MPIR may use allocated blocks to hold pointers to other allocated blocks. This
-will limit the assumptions a conservative garbage collection scheme can make.
-
-Any custom allocation functions must align pointers to limb boundaries. Thus if a limb is eight bytes (e.g. on x86_64), then all blocks must be aligned to eight byte boundaries. Check the configuration options for the custom allocation library in use. It is not necessary to align blocks to SSE boundaries even when SSE code is used. All MPIR assembly routines assume limb boundary alignment only (which is the default for most standard memory managers).
-
-Since the default MPIR allocation uses @code{malloc} and friends, those
-functions will be linked in even if the first thing a program does is an
- at code{mp_set_memory_functions}. It's necessary to change the MPIR sources if
-this is a problem.
-
- at sp 1
- at deftypefun void mp_get_memory_functions (@* void *(**@var{alloc_func_ptr}) (size_t), @* void *(**@var{realloc_func_ptr}) (void *, size_t, size_t), @* void (**@var{free_func_ptr}) (void *, size_t))
-Get the current allocation functions, storing function pointers to the
-locations given by the arguments. If an argument is @code{NULL}, that
-function pointer is not stored.
-
- at need 1000
-For example, to get just the current free function,
-
- at example
-void (*freefunc) (void *, size_t);
-
-mp_get_memory_functions (NULL, NULL, &freefunc);
- at end example
- at end deftypefun
-
- at node Language Bindings, Algorithms, Custom Allocation, Top
- at chapter Language Bindings
- at cindex Language bindings
- at cindex Other languages
-
-The following packages and projects offer access to MPIR from languages other
-than C, though perhaps with varying levels of functionality and efficiency.
-
- at c @spaceuref{U} is the same as @uref{U}, but with a couple of extra spaces
- at c in tex, just to separate the URL from the preceding text a bit.
- at iftex
- at macro spaceuref {U}
-@ @ @uref{\U\}
- at end macro
- at end iftex
- at ifnottex
- at macro spaceuref {U}
- at uref{\U\}
- at end macro
- at end ifnottex
-
- at sp 1
- at table @asis
- at item C++
- at itemize @bullet
- at item
-MPIR C++ class interface, @pxref{C++ Class Interface} @* Straightforward
-interface, expression templates to eliminate temporaries.
- at item
-ALP @spaceuref{http://www-sop.inria.fr/saga/logiciels/ALP/} @* Linear algebra and
-polynomials using templates.
- at item
-CLN @spaceuref{http://www.ginac.de/CLN/} @* High level classes for arithmetic.
- at item
-LiDIA @spaceuref{http://www.informatik.tu-darmstadt.de/TI/LiDIA/} @* A C++
-library for computational number theory.
- at item
-Linbox @spaceuref{http://www.linalg.org/} @* Sparse vectors and matrices.
- at item
-NTL @spaceuref{http://www.shoup.net/ntl/} @* A C++ number theory library.
- at end itemize
-
- at item Eiffel
- at itemize @bullet
- at item
-Eiffel Interface @spaceuref{http://www.eiffelroom.org/node/407} @* An Eiffel Interface to MPFR, MPC and MPIR by Chris Saunders.
- at end itemize
-
- at item Fortran
- at itemize @bullet
- at item
-Omni F77 @spaceuref{http://phase.hpcc.jp/Omni/home.html} @* Arbitrary
-precision floats.
- at end itemize
-
- at item Haskell
- at itemize @bullet
- at item
-Glasgow Haskell Compiler @spaceuref{http://www.haskell.org/ghc/}
- at end itemize
-
- at item Java
- at itemize @bullet
- at item
-Kaffe @spaceuref{http://www.kaffe.org/}
- at end itemize
-
- at item Lisp
- at itemize @bullet
- at item
-Embeddable Common Lisp @spaceuref{http://ecls.sourceforge.net/download.html}
- at item
-GNU Common Lisp @spaceuref{http://www.gnu.org/software/gcl/gcl.html}
- at item
-Librep @spaceuref{http://librep.sourceforge.net/}
- at item
- at c FIXME: When there's a stable release with gmp support, just refer to it
- at c rather than bothering to talk about betas.
-XEmacs (21.5.18 beta and up) @spaceuref{http://www.xemacs.org} @* Optional
-big integers, rationals and floats using MPIR.
- at end itemize
-
- at item M4
- at itemize @bullet
- at item
- at c FIXME: When there's a stable release with gmp support, just refer to it
- at c rather than bothering to talk about betas.
-GNU m4 betas @spaceuref{http://www.seindal.dk/rene/gnu/} @* Optionally provides
-an arbitrary precision @code{mpeval}.
- at end itemize
-
- at item ML
- at itemize @bullet
- at item
-MLton compiler @spaceuref{http://mlton.org/}
- at end itemize
-
- at item Objective Caml
- at itemize @bullet
- at item
-Numerix @spaceuref{http://pauillac.inria.fr/~quercia/} @* Optionally using
-GMP.
- at end itemize
-
- at item Oz
- at itemize @bullet
- at item
-Mozart @spaceuref{http://www.mozart-oz.org/}
- at end itemize
-
- at item Pascal
- at itemize @bullet
- at item
-GNU Pascal Compiler @spaceuref{http://www.gnu-pascal.de/} @* GMP unit.
- at item
-Numerix @spaceuref{http://pauillac.inria.fr/~quercia/} @* For Free Pascal,
-optionally using GMP.
- at end itemize
-
- at item Perl
- at itemize @bullet
- at item
-GMP module, see @file{demos/perl} on the MPIR website.
- at item
-Math::GMP @spaceuref{http://www.cpan.org/} @* Compatible with Math::BigInt, but
-not as many functions as the GMP module above.
- at item
-Math::BigInt::GMP @spaceuref{http://www.cpan.org/} @* Plug Math::GMP into
-normal Math::BigInt operations.
- at end itemize
-
- at item PHP
- at itemize @bullet
- at item
-mpz module in the main distribution, @uref{http://php.net/}
- at end itemize
-
- at need 1000
- at item Pike
- at itemize @bullet
- at item
-mpz module in the standard distribution, @uref{http://pike.ida.liu.se/}
- at end itemize
-
- at need 500
- at item Prolog
- at itemize @bullet
- at item
-SWI Prolog @spaceuref{http://www.swi-prolog.org/} @*
-Arbitrary precision floats.
- at end itemize
-
- at item Python
- at itemize @bullet
- at item
-mpz module in the standard distribution, @uref{http://www.python.org/}
- at item
-GMPY @uref{http://gmpy.sourceforge.net/}
- at end itemize
-
- at item Scheme
- at itemize @bullet
- at item
-GNU Guile (upcoming 1.8) @spaceuref{http://www.gnu.org/software/guile/guile.html}
- at item
-RScheme @spaceuref{http://www.rscheme.org/}
- at c
- at c For reference, MzScheme uses some of gmp, but (as of version 205) it only
- at c has copies of some of the generic C code, and we don't consider that a
- at c language binding to gmp.
- at c
- at end itemize
-
- at item Smalltalk
- at itemize @bullet
- at item
-GNU Smalltalk @spaceuref{http://www.smalltalk.org/versions/GNUSmalltalk.html}
- at end itemize
-
- at item Other
- at itemize @bullet
- at item
-ALGLIB @uref{http://www.alglib.net/} @* Numerical analysis and data processing
-library.
- at item
-Axiom @uref{http://savannah.nongnu.org/projects/axiom} @* Computer algebra
-using GCL.
- at item
-GiNaC @spaceuref{http://www.ginac.de/} @* C++ computer algebra using CLN.
- at item
-GOO @spaceuref{http://www.googoogaga.org/} @* Dynamic object oriented
-language.
- at item
-Maxima @uref{http://www.ma.utexas.edu/users/wfs/maxima.html} @* Macsyma
-computer algebra using GCL.
- at item
-Q @spaceuref{http://q-lang.sourceforge.net/} @* Equational programming system.
- at item
-Regina @spaceuref{http://regina.sourceforge.net/} @* Topological calculator.
- at item
-Sage @spaceuref{http://www.sagemath.org/} @* Computer Algebra System written
-in Python and Cython.
- at item
-Yacas @spaceuref{http://yacas.sourceforge.net/homepage.html} @* Yet another
-computer algebra system.
- at end itemize
-
- at end table
-
-
- at node Algorithms, Internals, Language Bindings, Top
- at chapter Algorithms
- at cindex Algorithms
-
-This chapter is an introduction to some of the algorithms used for various MPIR
-operations. The code is likely to be hard to understand without knowing
-something about the algorithms.
-
-Some MPIR internals are mentioned, but applications that expect to be
-compatible with future MPIR releases should take care to use only the
-documented functions.
-
- at menu
-* Multiplication Algorithms::
-* Division Algorithms::
-* Greatest Common Divisor Algorithms::
-* Powering Algorithms::
-* Root Extraction Algorithms::
-* Radix Conversion Algorithms::
-* Other Algorithms::
-* Assembler Coding::
- at end menu
-
-
- at node Multiplication Algorithms, Division Algorithms, Algorithms, Algorithms
- at section Multiplication
- at cindex Multiplication algorithms
-
-N at cross{}N limb multiplications and squares are done using one of six
-algorithms, as the size N increases.
-
- at quotation
- at multitable {KaratsubaMMM} {MMMMMMMMMMMMMMMMMMMMMMMMMMMMMMM}
- at item Algorithm @tab Mul Threshold
- at item Basecase @tab (none)
- at item Karatsuba @tab @code{MUL_KARATSUBA_THRESHOLD}
- at item Toom-3 @tab @code{MUL_TOOM3_THRESHOLD}
- at item Toom-4 @tab @code{MUL_TOOM4_THRESHOLD}
- at item Toom-8(.5) @tab @code{MUL_TOOM8H_THRESHOLD}
- at item FFT @tab @code{MUL_FFT_FULL_THRESHOLD}
- at end multitable
- at end quotation
-
- at quotation
- at multitable {KaratsubaMMM} {MMMMMMMMMMMMMMMMMMMMMMMMMMMMMMM}
- at item Algorithm @tab Sqr Threshold
- at item Basecase @tab (none)
- at item Karatsuba @tab @code{SQR_KARATSUBA_THRESHOLD}
- at item Toom-3 @tab @code{SQR_TOOM3_THRESHOLD}
- at item Toom-4 @tab @code{SQR_TOOM4_THRESHOLD}
- at item Toom-8 @tab @code{SQR_TOOM8_THRESHOLD}
- at item FFT @tab @code{SQR_FFT_FULL_THRESHOLD}
- at end multitable
- at end quotation
-
-N at cross{}M multiplications of operands with different sizes above
- at code{MUL_KARATSUBA_THRESHOLD} are done using unbalanced Toom algorithms or
-with the FFT. See (@pxref{Unbalanced Multiplication}).
-
- at menu
-* Basecase Multiplication::
-* Karatsuba Multiplication::
-* Toom 3-Way Multiplication::
-* Toom 4-Way Multiplication::
-* FFT Multiplication::
-* Other Multiplication::
-* Unbalanced Multiplication::
- at end menu
-
-
- at node Basecase Multiplication, Karatsuba Multiplication, Multiplication Algorithms, Multiplication Algorithms
- at subsection Basecase Multiplication
-
-Basecase N at cross{}M multiplication is a straightforward rectangular set of
-cross-products, the same as long multiplication done by hand and for that
-reason sometimes known as the schoolbook or grammar school method. This is an
- at m{O(NM),O(N*M)} algorithm. See Knuth section 4.3.1 algorithm M
-(@pxref{References}), and the @file{mpn/generic/mul_basecase.c} code.
-
-Assembler implementations of @code{mpn_mul_basecase} are essentially the same
-as the generic C code, but have all the usual assembler tricks and
-obscurities introduced for speed.
-
-A square can be done in roughly half the time of a multiply, by using the fact
-that the cross products above and below the diagonal are the same. A triangle
-of products below the diagonal is formed, doubled (left shift by one bit), and
-then the products on the diagonal added. This can be seen in
- at file{mpn/generic/sqr_basecase.c}. Again the assembler implementations take
-essentially the same approach.
-
- at tex
-\def\GMPline#1#2#3#4#5#6{%
- \hbox {%
- \vrule height 2.5ex depth 1ex
- \hbox to 2em {\hfil{#2}\hfil}%
- \vrule \hbox to 2em {\hfil{#3}\hfil}%
- \vrule \hbox to 2em {\hfil{#4}\hfil}%
- \vrule \hbox to 2em {\hfil{#5}\hfil}%
- \vrule \hbox to 2em {\hfil{#6}\hfil}%
- \vrule}}
-\GMPdisplay{
- \hbox{%
- \vbox{%
- \hbox to 1.5em {\vrule height 2.5ex depth 1ex width 0pt}%
- \hbox {\vrule height 2.5ex depth 1ex width 0pt u0\hfil}%
- \hbox {\vrule height 2.5ex depth 1ex width 0pt u1\hfil}%
- \hbox {\vrule height 2.5ex depth 1ex width 0pt u2\hfil}%
- \hbox {\vrule height 2.5ex depth 1ex width 0pt u3\hfil}%
- \hbox {\vrule height 2.5ex depth 1ex width 0pt u4\hfil}%
- \vfill}%
- \vbox{%
- \hbox{%
- \hbox to 2em {\hfil u0\hfil}%
- \hbox to 2em {\hfil u1\hfil}%
- \hbox to 2em {\hfil u2\hfil}%
- \hbox to 2em {\hfil u3\hfil}%
- \hbox to 2em {\hfil u4\hfil}}%
- \vskip 0.7ex
- \hrule
- \GMPline{u0}{d}{}{}{}{}%
- \hrule
- \GMPline{u1}{}{d}{}{}{}%
- \hrule
- \GMPline{u2}{}{}{d}{}{}%
- \hrule
- \GMPline{u3}{}{}{}{d}{}%
- \hrule
- \GMPline{u4}{}{}{}{}{d}%
- \hrule}}}
- at end tex
- at ifnottex
- at example
- at group
- u0 u1 u2 u3 u4
- +---+---+---+---+---+
-u0 | d | | | | |
- +---+---+---+---+---+
-u1 | | d | | | |
- +---+---+---+---+---+
-u2 | | | d | | |
- +---+---+---+---+---+
-u3 | | | | d | |
- +---+---+---+---+---+
-u4 | | | | | d |
- +---+---+---+---+---+
- at end group
- at end example
- at end ifnottex
-
-In practice squaring isn't a full 2 at cross{} faster than multiplying, it's
-usually around 1.5 at cross{}. Less than 1.5 at cross{} probably indicates
- at code{mpn_sqr_basecase} wants improving on that CPU.
-
-On some CPUs @code{mpn_mul_basecase} can be faster than the generic C
- at code{mpn_sqr_basecase} on some small sizes. @code{SQR_BASECASE_THRESHOLD} is
-the size at which to use @code{mpn_sqr_basecase}, this will be zero if that
-routine should be used always.
-
-
- at node Karatsuba Multiplication, Toom 3-Way Multiplication, Basecase Multiplication, Multiplication Algorithms
- at subsection Karatsuba Multiplication
- at cindex Karatsuba multiplication
-
-The Karatsuba multiplication algorithm is described in Knuth section 4.3.3
-part A, and various other textbooks. A brief description is given here.
-
-The inputs @math{x} and @math{y} are treated as each split into two parts of
-equal length (or the most significant part one limb shorter if N is odd).
-
- at tex
-% GMPboxwidth used for all the multiplication pictures
-\global\newdimen\GMPboxwidth \global\GMPboxwidth=5em
-% GMPboxdepth and GMPboxheight are also used for the float pictures
-\global\newdimen\GMPboxdepth \global\GMPboxdepth=1ex
-\global\newdimen\GMPboxheight \global\GMPboxheight=2ex
-\gdef\GMPvrule{\vrule height \GMPboxheight depth \GMPboxdepth}
-\def\GMPbox#1#2{%
- \vbox {%
- \hrule
- \hbox to 2\GMPboxwidth{%
- \GMPvrule \hfil $#1$\hfil \vrule \hfil $#2$\hfil \vrule}%
- \hrule}}
-\GMPdisplay{%
-\vbox{%
- \hbox to 2\GMPboxwidth {high \hfil low}
- \vskip 0.7ex
- \GMPbox{x_1}{x_0}
- \vskip 0.5ex
- \GMPbox{y_1}{y_0}
-}}
- at end tex
- at ifnottex
- at example
- at group
- high low
-+----------+----------+
-| x1 | x0 |
-+----------+----------+
-
-+----------+----------+
-| y1 | y0 |
-+----------+----------+
- at end group
- at end example
- at end ifnottex
-
-Let @math{b} be the power of 2 where the split occurs, ie.@: if @ms{x,0} is
- at math{k} limbs (@ms{y,0} the same) then
- at m{b=2\GMPraise{$k*$@code{mp\_bits\_per\_limb}}, b=2^(k*mp_bits_per_limb)}.
-With that @m{x=x_1b+x_0,x=x1*b+x0} and @m{y=y_1b+y_0,y=y1*b+y0}, and the
-following holds,
-
- at display
- at m{xy = (b^2+b)x_1y_1 - b(x_1-x_0)(y_1-y_0) + (b+1)x_0y_0,
- x*y = (b^2+b)*x1*y1 - b*(x1-x0)*(y1-y0) + (b+1)*x0*y0}
- at end display
-
-This formula means doing only three multiplies of (N/2)@cross{}(N/2) limbs,
-whereas a basecase multiply of N at cross{}N limbs is equivalent to four
-multiplies of (N/2)@cross{}(N/2). The factors @math{(b^2+b)} etc represent
-the positions where the three products must be added.
-
- at tex
-\def\GMPboxA#1#2{%
- \vbox{%
- \hrule
- \hbox{%
- \GMPvrule
- \hbox to 2\GMPboxwidth {\hfil\hbox{$#1$}\hfil}%
- \vrule
- \hbox to 2\GMPboxwidth {\hfil\hbox{$#2$}\hfil}%
- \vrule}
- \hrule}}
-\def\GMPboxB#1#2{%
- \hbox{%
- \raise \GMPboxdepth \hbox to \GMPboxwidth {\hfil #1\hskip 0.5em}%
- \vbox{%
- \hrule
- \hbox{%
- \GMPvrule
- \hbox to 2\GMPboxwidth {\hfil\hbox{$#2$}\hfil}%
- \vrule}%
- \hrule}}}
-\GMPdisplay{%
-\vbox{%
- \hbox to 4\GMPboxwidth {high \hfil low}
- \vskip 0.7ex
- \GMPboxA{x_1y_1}{x_0y_0}
- \vskip 0.5ex
- \GMPboxB{$+$}{x_1y_1}
- \vskip 0.5ex
- \GMPboxB{$+$}{x_0y_0}
- \vskip 0.5ex
- \GMPboxB{$-$}{(x_1-x_0)(y_1-y_0)}
-}}
- at end tex
- at ifnottex
- at example
- at group
- high low
-+--------+--------+ +--------+--------+
-| x1*y1 | | x0*y0 |
-+--------+--------+ +--------+--------+
- +--------+--------+
- add | x1*y1 |
- +--------+--------+
- +--------+--------+
- add | x0*y0 |
- +--------+--------+
- +--------+--------+
- sub | (x1-x0)*(y1-y0) |
- +--------+--------+
- at end group
- at end example
- at end ifnottex
-
-The term @m{(x_1-x_0)(y_1-y_0),(x1-x0)*(y1-y0)} is best calculated as an
-absolute value, and the sign used to choose to add or subtract. Notice the
-sum @m{\mathop{\rm high}(x_0y_0)+\mathop{\rm low}(x_1y_1),
-high(x0*y0)+low(x1*y1)} occurs twice, so it's possible to do @m{5k,5*k} limb
-additions, rather than @m{6k,6*k}, but in MPIR extra function call overheads
-outweigh the saving.
-
-Squaring is similar to multiplying, but with @math{x=y} the formula reduces to
-an equivalent with three squares,
-
- at display
- at m{x^2 = (b^2+b)x_1^2 - b(x_1-x_0)^2 + (b+1)x_0^2,
- x^2 = (b^2+b)*x1^2 - b*(x1-x0)^2 + (b+1)*x0^2}
- at end display
-
-The final result is accumulated from those three squares the same way as for
-the three multiplies above. The middle term @m{(x_1-x_0)^2,(x1-x0)^2} is now
-always positive.
-
-A similar formula for both multiplying and squaring can be constructed with a
-middle term @m{(x_1+x_0)(y_1+y_0),(x1+x0)*(y1+y0)}. But those sums can exceed
- at math{k} limbs, leading to more carry handling and additions than the form
-above.
-
-Karatsuba multiplication is asymptotically an @math{O(N^@W{1.585})} algorithm,
-the exponent being @m{\log3/\log2,log(3)/log(2)}, representing 3 multiplies
-each @math{1/2} the size of the inputs. This is a big improvement over the
-basecase multiply at @math{O(N^2)} and the advantage soon overcomes the extra
-additions Karatsuba performs. @code{MUL_KARATSUBA_THRESHOLD} can be as little
-as 10 limbs. The @code{SQR} threshold is usually about twice the @code{MUL}.
-
-The basecase algorithm will take a time of the form @m{M(N) = aN^2 + bN + c,
-M(N) = a*N^2 + b*N + c} and the Karatsuba algorithm @m{K(N) = 3M(N/2) + dN +
-e, K(N) = 3*M(N/2) + d*N + e}, which expands to @m{K(N) = {3\over4} aN^2 +
-{3\over2} bN + 3c + dN + e, K(N) = 3/4*a*N^2 + 3/2*b*N + 3*c + d*N + e}. The
-factor @m{3\over4, 3/4} for @math{a} means per-crossproduct speedups in the
-basecase code will increase the threshold since they benefit @math{M(N)} more
-than @math{K(N)}. And conversely the @m{3\over2, 3/2} for @math{b} means
-linear style speedups of @math{b} will increase the threshold since they
-benefit @math{K(N)} more than @math{M(N)}. The latter can be seen for
-instance when adding an optimized @code{mpn_sqr_diagonal} to
- at code{mpn_sqr_basecase}. Of course all speedups reduce total time, and in
-that sense the algorithm thresholds are merely of academic interest.
-
-
- at node Toom 3-Way Multiplication, Toom 4-Way Multiplication, Karatsuba Multiplication, Multiplication Algorithms
- at subsection Toom 3-Way Multiplication
- at cindex Toom multiplication
-
-The Karatsuba formula is the simplest case of a general approach to splitting
-inputs that leads to both Toom and FFT algorithms. A description of
-Toom can be found in Knuth section 4.3.3, with an example 3-way
-calculation after Theorem A at . The 3-way form used in MPIR is described here.
-
-The operands are each considered split into 3 pieces of equal length (or the
-most significant part 1 or 2 limbs shorter than the other two).
-
- at tex
-\def\GMPbox#1#2#3{%
- \vbox{%
- \hrule \vfil
- \hbox to 3\GMPboxwidth {%
- \GMPvrule
- \hfil$#1$\hfil
- \vrule
- \hfil$#2$\hfil
- \vrule
- \hfil$#3$\hfil
- \vrule}%
- \vfil \hrule
-}}
-\GMPdisplay{%
-\vbox{%
- \hbox to 3\GMPboxwidth {high \hfil low}
- \vskip 0.7ex
- \GMPbox{x_2}{x_1}{x_0}
- \vskip 0.5ex
- \GMPbox{y_2}{y_1}{y_0}
- \vskip 0.5ex
-}}
- at end tex
- at ifnottex
- at example
- at group
- high low
-+----------+----------+----------+
-| x2 | x1 | x0 |
-+----------+----------+----------+
-
-+----------+----------+----------+
-| y2 | y1 | y0 |
-+----------+----------+----------+
- at end group
- at end example
- at end ifnottex
-
- at noindent
-These parts are treated as the coefficients of two polynomials
-
- at display
- at group
- at m{X(t) = x_2t^2 + x_1t + x_0,
- X(t) = x2*t^2 + x1*t + x0}
- at m{Y(t) = y_2t^2 + y_1t + y_0,
- Y(t) = y2*t^2 + y1*t + y0}
- at end group
- at end display
-
-Let @math{b} equal the power of 2 which is the size of the @ms{x,0}, @ms{x,1},
- at ms{y,0} and @ms{y,1} pieces, ie.@: if they're @math{k} limbs each then
- at m{b=2\GMPraise{$k*$@code{mp\_bits\_per\_limb}}, b=2^(k*mp_bits_per_limb)}.
-With this @math{x=X(b)} and @math{y=Y(b)}.
-
-Let a polynomial @m{W(t)=X(t)Y(t),W(t)=X(t)*Y(t)} and suppose its coefficients
-are
-
- at display
- at m{W(t) = w_4t^4 + w_3t^3 + w_2t^2 + w_1t + w_0,
- W(t) = w4*t^4 + w3*t^3 + w2*t^2 + w1*t + w0}
- at end display
-
-The @m{w_i,w[i]} are going to be determined, and when they are they'll give
-the final result using @math{w=W(b)}, since
- at m{xy=X(b)Y(b),x*y=X(b)*Y(b)=W(b)}. The coefficients will be roughly
- at math{b^2} each, and the final @math{W(b)} will be an addition like,
-
- at tex
-\def\GMPbox#1#2{%
- \moveright #1\GMPboxwidth
- \vbox{%
- \hrule
- \hbox{%
- \GMPvrule
- \hbox to 2\GMPboxwidth {\hfil$#2$\hfil}%
- \vrule}%
- \hrule
-}}
-\GMPdisplay{%
-\vbox{%
- \hbox to 6\GMPboxwidth {high \hfil low}%
- \vskip 0.7ex
- \GMPbox{0}{w_4}
- \vskip 0.5ex
- \GMPbox{1}{w_3}
- \vskip 0.5ex
- \GMPbox{2}{w_2}
- \vskip 0.5ex
- \GMPbox{3}{w_1}
- \vskip 0.5ex
- \GMPbox{4}{w_1}
-}}
- at end tex
- at ifnottex
- at example
- at group
- high low
-+-------+-------+
-| w4 |
-+-------+-------+
- +--------+-------+
- | w3 |
- +--------+-------+
- +--------+-------+
- | w2 |
- +--------+-------+
- +--------+-------+
- | w1 |
- +--------+-------+
- +-------+-------+
- | w0 |
- +-------+-------+
- at end group
- at end example
- at end ifnottex
-
-The @m{w_i,w[i]} coefficients could be formed by a simple set of cross
-products, like @m{w_4=x_2y_2,w4=x2*y2}, @m{w_3=x_2y_1+x_1y_2,w3=x2*y1+x1*y2},
- at m{w_2=x_2y_0+x_1y_1+x_0y_2,w2=x2*y0+x1*y1+x0*y2} etc, but this would need all
-nine @m{x_iy_j,x[i]*y[j]} for @math{i,j=0,1,2}, and would be equivalent merely
-to a basecase multiply. Instead the following approach is used.
-
- at math{X(t)} and @math{Y(t)} are evaluated and multiplied at 5 points, giving
-values of @math{W(t)} at those points. In MPIR the following points are used,
-
- at quotation
- at multitable {@m{t=\infty,t=inf}M} {MMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMM}
- at item Point @tab Value
- at item @math{t=0} @tab @m{x_0y_0,x0 * y0}, which gives @ms{w,0} immediately
- at item @math{t=1} @tab @m{(x_2+x_1+x_0)(y_2+y_1+y_0),(x2+x1+x0) * (y2+y1+y0)}
- at item @math{t=-1} @tab @m{(x_2-x_1+x_0)(y_2-y_1+y_0),(x2-x1+x0) * (y2-y1+y0)}
- at item @math{t=2} @tab @m{(4x_2+2x_1+x_0)(4y_2+2y_1+y_0),(4*x2+2*x1+x0) * (4*y2+2*y1+y0)}
- at item @m{t=\infty,t=inf} @tab @m{x_2y_2,x2 * y2}, which gives @ms{w,4} immediately
- at end multitable
- at end quotation
-
-At @math{t=-1} the values can be negative and that's handled using the
-absolute values and tracking the sign separately. At @m{t=\infty,t=inf} the
-value is actually @m{\lim_{t\to\infty} {X(t)Y(t)\over t^4}, X(t)*Y(t)/t^4 in
-the limit as t approaches infinity}, but it's much easier to think of as
-simply @m{x_2y_2,x2*y2} giving @ms{w,4} immediately (much like
- at m{x_0y_0,x0*y0} at @math{t=0} gives @ms{w,0} immediately).
-
-Each of the points substituted into
- at m{W(t)=w_4t^4+\cdots+w_0,W(t)=w4*t^4+ at dots{}+w0} gives a linear combination
-of the @m{w_i,w[i]} coefficients, and the value of those combinations has just
-been calculated.
-
- at tex
-\GMPdisplay{%
-$\matrix{%
-W(0) & = & & & & & & & & & w_0 \cr
-W(1) & = & w_4 & + & w_3 & + & w_2 & + & w_1 & + & w_0 \cr
-W(-1) & = & w_4 & - & w_3 & + & w_2 & - & w_1 & + & w_0 \cr
-W(2) & = & 16w_4 & + & 8w_3 & + & 4w_2 & + & 2w_1 & + & w_0 \cr
-W(\infty) & = & w_4 \cr
-}$}
- at end tex
- at ifnottex
- at example
- at group
-W(0) = w0
-W(1) = w4 + w3 + w2 + w1 + w0
-W(-1) = w4 - w3 + w2 - w1 + w0
-W(2) = 16*w4 + 8*w3 + 4*w2 + 2*w1 + w0
-W(inf) = w4
- at end group
- at end example
- at end ifnottex
-
-This is a set of five equations in five unknowns, and some elementary linear
-algebra quickly isolates each @m{w_i,w[i]}. This involves adding or
-subtracting one @math{W(t)} value from another, and a couple of divisions by
-powers of 2 and one division by 3, the latter using the special
- at code{mpn_divexact_by3} (@pxref{Exact Division}).
-
-The conversion of @math{W(t)} values to the coefficients is interpolation. A
-polynomial of degree 4 like @math{W(t)} is uniquely determined by values known
-at 5 different points. The points are arbitrary and can be chosen to make the
-linear equations come out with a convenient set of steps for quickly isolating
-the @m{w_i,w[i]}.
-
-Squaring follows the same procedure as multiplication, but there's only one
- at math{X(t)} and it's evaluated at the 5 points, and those values squared to
-give values of @math{W(t)}. The interpolation is then identical, and in fact
-the same @code{toom3_interpolate} subroutine is used for both squaring and
-multiplying.
-
-Toom-3 is asymptotically @math{O(N^@W{1.465})}, the exponent being
- at m{\log5/\log3,log(5)/log(3)}, representing 5 recursive multiplies of 1/3 the
-original size each. This is an improvement over Karatsuba at
- at math{O(N^@W{1.585})}, though Toom does more work in the evaluation and
-interpolation and so it only realizes its advantage above a certain size.
-
-Near the crossover between Toom-3 and Karatsuba there's generally a range of
-sizes where the difference between the two is small.
- at code{MUL_TOOM3_THRESHOLD} is a somewhat arbitrary point in that range and
-successive runs of the tune program can give different values due to small
-variations in measuring. A graph of time versus size for the two shows the
-effect, see @file{tune/README}.
-
-At the fairly small sizes where the Toom-3 thresholds occur it's worth
-remembering that the asymptotic behaviour for Karatsuba and Toom-3 can't be
-expected to make accurate predictions, due of course to the big influence of
-all sorts of overheads, and the fact that only a few recursions of each are
-being performed. Even at large sizes there's a good chance machine dependent
-effects like cache architecture will mean actual performance deviates from
-what might be predicted.
-
-The formula given for the Karatsuba algorithm (@pxref{Karatsuba
-Multiplication}) has an equivalent for Toom-3 involving only five multiplies,
-but this would be complicated and unenlightening.
-
-An alternate view of Toom-3 can be found in Zuras (@pxref{References}), using
-a vector to represent the @math{x} and @math{y} splits and a matrix
-multiplication for the evaluation and interpolation stages. The matrix
-inverses are not meant to be actually used, and they have elements with values
-much greater than in fact arise in the interpolation steps. The diagram shown
-for the 3-way is attractive, but again doesn't have to be implemented that way
-and for example with a bit of rearrangement just one division by 6 can be
-done.
-
- at node Toom 4-Way Multiplication, FFT Multiplication, Toom 3-Way Multiplication, Multiplication Algorithms
- at subsection Toom 4-Way Multiplication
- at cindex Toom multiplication
-
-Karatsuba and Toom-3 split the operands into 2 and 3 coefficients,
-respectively. Toom-4 analogously splits the operands into 4 coefficients.
-Using the notation from the section on Toom-3 multiplication, we form two
-polynomials:
-
- at display
- at group
- at m{X(t) = x_3t^3 + x_2t^2 + x_1t + x_0,
- X(t) = x3*t^3 + x2*t^2 + x1*t + x0}
- at m{Y(t) = y_3t^3 + y_2t^2 + y_1t + y_0,
- Y(t) = y3*t^3 + y2*t^2 + y1*t + y0}
- at end group
- at end display
-
- at math{X(t)} and @math{Y(t)} are evaluated and multiplied at 7 points, giving
-values of @math{W(t)} at those points. In MPIR the following points are used,
-
- at quotation
- at multitable {@m{t=-1/2,t=inf}M} {MMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMM}
- at item Point @tab Value
- at item @math{t=0} @tab @m{x_0y_0,x0 * y0}, which gives @ms{w,0} immediately
- at item @math{t=1/2} @tab @m{(x_3+2x_2+4x_1+8x_0)(y_3+2y_2+4y_1+8y_0),(x3+2*x2+4*x1+8*x0) * (y3+2*y2+4*y1+8*y0)}
- at item @math{t=-1/2} @tab @m{(-x_3+2x_2-4x_1+8x_0)(-y_3+2y_2-4y_1+8y_0),(-x3+2*x2-4*x1+8*x0) * (-y3+2*y2-4*y1+8*y0)}
- at item @math{t=1} @tab @m{(x_3+x_2+x_1+x_0)(y_3+y_2+y_1+y_0),(x3+x2+x1+x0) * (y3+y2+y1+y0)}
- at item @math{t=-1} @tab @m{(-x_3+x_2-x_1+x_0)(-y_3+y_2-y_1+y_0),(-x3+x2-x1+x0) * (-y3+y2-y1+y0)}
- at item @math{t=2} @tab @m{(8x_3+4x_2+2x_1+x_0)(8y_3+4y_2+2y_1+y_0),(8*x3+4*x2+2*x1+x0) * (8*y3+4*y2+2*y1+y0)}
- at item @m{t=\infty,t=inf} @tab @m{x_3y_3,x3 * y3}, which gives @ms{w,6} immediately
- at end multitable
- at end quotation
-
-The number of additions and subtractions for Toom-4 is much larger than for Toom-3.
-But several subexpressions occur multiple times, for example @m{x_2+x_0,x2+x0}, occurs
-for both @math{t=1} and @math{t=-1}.
-
-Toom-4 is asymptotically @math{O(N^@W{1.404})}, the exponent being
- at m{\log7/\log4,log(7)/log(4)}, representing 7 recursive multiplies of 1/4 the
-original size each.
-
- at node FFT Multiplication, Other Multiplication, Toom 4-Way Multiplication, Multiplication Algorithms
- at subsection FFT Multiplication
- at cindex FFT multiplication
- at cindex Fast Fourier Transform
-
-This section is out-of-date and will be updated when the new FFT is added.
-
-At large to very large sizes a Fermat style FFT multiplication is used,
-following Sch@"onhage and Strassen (@pxref{References}). Descriptions of FFTs
-in various forms can be found in many textbooks, for instance Knuth section
-4.3.3 part C or Lipson chapter IX at . A brief description of the form used in
-MPIR is given here.
-
-The multiplication done is @m{xy \bmod 2^N+1, x*y mod 2^N+1}, for a given
- at math{N}. A full product @m{xy,x*y} is obtained by choosing @m{N \ge
-\mathop{\rm bits}(x)+\mathop{\rm bits}(y), N>=bits(x)+bits(y)} and padding
- at math{x} and @math{y} with high zero limbs. The modular product is the native
-form for the algorithm, so padding to get a full product is unavoidable.
-
-The algorithm follows a split, evaluate, pointwise multiply, interpolate and
-combine similar to that described above for Karatsuba and Toom-3. A @math{k}
-parameter controls the split, with an FFT- at math{k} splitting into @math{2^k}
-pieces of @math{M=N/2^k} bits each. @math{N} must be a multiple of
- at m{2^k\times at code{mp\_bits\_per\_limb}, (2^k)*@nicode{mp_bits_per_limb}} so
-the split falls on limb boundaries, avoiding bit shifts in the split and
-combine stages.
-
-The evaluations, pointwise multiplications, and interpolation, are all done
-modulo @m{2^{N'}+1, 2^N'+1} where @math{N'} is @math{2M+k+3} rounded up to a
-multiple of @math{2^k} and of @code{mp_bits_per_limb}. The results of
-interpolation will be the following negacyclic convolution of the input
-pieces, and the choice of @math{N'} ensures these sums aren't truncated.
- at tex
-$$ w_n = \sum_{{i+j = b2^k+n}\atop{b=0,1}} (-1)^b x_i y_j $$
- at end tex
- at ifnottex
-
- at example
- ---
- \ b
-w[n] = / (-1) * x[i] * y[j]
- ---
- i+j==b*2^k+n
- b=0,1
- at end example
-
- at end ifnottex
-The points used for the evaluation are @math{g^i} for @math{i=0} to
- at math{2^k-1} where @m{g=2^{2N'/2^k}, g=2^(2N'/2^k)}. @math{g} is a
- at m{2^k,2^k'}th root of unity mod @m{2^{N'}+1,2^N'+1}, which produces necessary
-cancellations at the interpolation stage, and it's also a power of 2 so the
-fast fourier transforms used for the evaluation and interpolation do only
-shifts, adds and negations.
-
-The pointwise multiplications are done modulo @m{2^{N'}+1, 2^N'+1} and either
-recurse into a further FFT or use a plain multiplication (Toom-3, Karatsuba or
-basecase), whichever is optimal at the size @math{N'}. The interpolation is
-an inverse fast fourier transform. The resulting set of sums of @m{x_iy_j,
-x[i]*y[j]} are added at appropriate offsets to give the final result.
-
-Squaring is the same, but @math{x} is the only input so it's one transform at
-the evaluate stage and the pointwise multiplies are squares. The
-interpolation is the same.
-
-For a mod @math{2^N+1} product, an FFT- at math{k} is an @m{O(N^{k/(k-1)}),
-O(N^(k/(k-1)))} algorithm, the exponent representing @math{2^k} recursed
-modular multiplies each @m{1/2^{k-1},1/2^(k-1)} the size of the original.
-Each successive @math{k} is an asymptotic improvement, but overheads mean each
-is only faster at bigger and bigger sizes. In the code, @code{MUL_FFT_TABLE}
-and @code{SQR_FFT_TABLE} are the thresholds where each @math{k} is used. Each
-new @math{k} effectively swaps some multiplying for some shifts, adds and
-overheads.
-
-A mod @math{2^N+1} product can be formed with a normal
- at math{N at cross{}N at rightarrow{}2N} bit multiply plus a subtraction, so an FFT
-and Toom-3 etc can be compared directly. A @math{k=4} FFT at
- at math{O(N^@W{1.333})} can be expected to be the first faster than Toom-3 at
- at math{O(N^@W{1.465})}. In practice this is what's found, with
- at code{MUL_FFT_MODF_THRESHOLD} and @code{SQR_FFT_MODF_THRESHOLD} being between
-300 and 1000 limbs, depending on the CPU at . So far it's been found that only
-very large FFTs recurse into pointwise multiplies above these sizes.
-
-When an FFT is to give a full product, the change of @math{N} to @math{2N}
-doesn't alter the theoretical complexity for a given @math{k}, but for the
-purposes of considering where an FFT might be first used it can be assumed
-that the FFT is recursing into a normal multiply and that on that basis it's
-doing @math{2^k} recursed multiplies each @m{1/2^{k-2},1/2^(k-2)} the size of
-the inputs, making it @m{O(N^{k/(k-2)}), O(N^(k/(k-2)))}. This would mean
- at math{k=7} at @math{O(N^@W{1.4})} would be the first FFT faster than Toom-3.
-In practice @code{MUL_FFT_FULL_THRESHOLD} and @code{SQR_FFT_FULL_THRESHOLD}
-have been found to be in the @math{k=8} range, somewhere between 3000 and
-10000 limbs.
-
-The way @math{N} is split into @math{2^k} pieces and then @math{2M+k+3} is
-rounded up to a multiple of @math{2^k} and @code{mp_bits_per_limb} means that
-when @math{2^k at ge{}@nicode{mp\_bits\_per\_limb}} the effective @math{N} is a
-multiple of @m{2^{2k-1},2^(2k-1)} bits. The @math{+k+3} means some values of
- at math{N} just under such a multiple will be rounded to the next. The
-complexity calculations above assume that a favourable size is used, meaning
-one which isn't padded through rounding, and it's also assumed that the extra
- at math{+k+3} bits are negligible at typical FFT sizes.
-
-The practical effect of the @m{2^{2k-1},2^(2k-1)} constraint is to introduce a
-step-effect into measured speeds. For example @math{k=8} will round @math{N}
-up to a multiple of 32768 bits, so for a 32-bit limb there'll be 512 limb
-groups of sizes for which @code{mpn_mul_n} runs at the same speed. Or for
- at math{k=9} groups of 2048 limbs, @math{k=10} groups of 8192 limbs, etc. In
-practice it's been found each @math{k} is used at quite small multiples of its
-size constraint and so the step effect is quite noticeable in a time versus
-size graph.
-
-The threshold determinations currently measure at the mid-points of size
-steps, but this is sub-optimal since at the start of a new step it can happen
-that it's better to go back to the previous @math{k} for a while. Something
-more sophisticated for @code{MUL_FFT_TABLE} and @code{SQR_FFT_TABLE} will be
-needed.
-
-
- at node Other Multiplication, Unbalanced Multiplication, FFT Multiplication, Multiplication Algorithms
- at subsection Other Multiplication
- at cindex Toom multiplication
-
-The Toom algorithms described above (@pxref{Toom 3-Way Multiplication}),
- at pxref{Toom 4-Way Multiplication}) generalize to split into an arbitrary
-number of pieces, as per Knuth section 4.3.3 algorithm C at . MPIR currently
-implements Toom 8 routines.
-
-These are generated automatically via a technique due to Bodrato
-(@pxref{References}) which mixes evaluation, pointwise multiplication and
-interpolation phases. The routine used is called Toom 8.5. See Bodrato's
-paper.
-
-For general Toom-n a split into @math{r+1} pieces is made, and evaluations and
-pointwise multiplications done at @m{2r+1,2*r+1} points. A 4-way split does 7
-pointwise multiplies, 5-way does 9, etc. Asymptotically an @math{(r+1)}-way
-algorithm is @m{O(N^{log(2r+1)/log(r+1)}, O(N^(log(2*r+1)/log(r+1)))}. Only
-the pointwise multiplications count towards big- at math{O} complexity, but the
-time spent in the evaluate and interpolate stages grows with @math{r} and has
-a significant practical impact, with the asymptotic advantage of each @math{r}
-realized only at bigger and bigger sizes. The overheads grow as
- at m{O(Nr),O(N*r)}, whereas in an @math{r=2^k} FFT they grow only as @m{O(N \log
-r), O(N*log(r))}.
-
-Knuth algorithm C evaluates at points 0,1,2, at dots{}, at m{2r,2*r}, but exercise 4
-uses @math{-r}, at dots{},0, at dots{}, at math{r} and the latter saves some small
-multiplies in the evaluate stage (or rather trades them for additions), and
-has a further saving of nearly half the interpolate steps. The idea is to
-separate odd and even final coefficients and then perform algorithm C steps C7
-and C8 on them separately. The divisors at step C7 become @math{j^2} and the
-multipliers at C8 become @m{2tj-j^2,2*t*j-j^2}.
-
-Splitting odd and even parts through positive and negative points can be
-thought of as using @math{-1} as a square root of unity. If a 4th root of
-unity was available then a further split and speedup would be possible, but no
-such root exists for plain integers. Going to complex integers with
- at m{i=\sqrt{-1}, i=sqrt(-1)} doesn't help, essentially because in cartesian
-form it takes three real multiplies to do a complex multiply. The existence
-of @m{2^k,2^k'}th roots of unity in a suitable ring or field lets the fast
-fourier transform keep splitting and get to @m{O(N \log r), O(N*log(r))}.
-
-Floating point FFTs use complex numbers approximating Nth roots of unity.
-Some processors have special support for such FFTs. But these are not used in
-MPIR since it's very difficult to guarantee an exact result (to some number of
-bits). An occasional difference of 1 in the last bit might not matter to a
-typical signal processing algorithm, but is of course of vital importance to
-MPIR.
-
- at node Unbalanced Multiplication, , Other Multiplication, Multiplication Algorithms
- at subsection Unbalanced Multiplication
- at cindex Unbalanced multiplication
-
-Multiplication of operands with different sizes, both below
- at code{MUL_KARATSUBA_THRESHOLD} are done with plain schoolbook multiplication
-(@pxref{Basecase Multiplication}).
-
-For really large operands, we invoke the FFT directly.
-
-For operands between these sizes, we use Toom inspired algorithms suggested by
-Alberto Zanoni and Marco Bodrato. The idea is to split the operands into
-polynomials of different degree. These algorithms are denoted ToomMN where
-the first input is broken into M components and the second operand is broken
-into N components. MPIR currently implements Toom32, Toom33, Toom44, Toom53 and
-Toom8h which deals with a variety of sizes where the product polynomial will
-have length 15 or 16.
-
- at node Division Algorithms, Greatest Common Divisor Algorithms, Multiplication Algorithms, Algorithms
- at section Division Algorithms
- at cindex Division algorithms
-
- at menu
-* Single Limb Division::
-* Basecase Division::
-* Divide and Conquer Division::
-* Exact Division::
-* Exact Remainder::
-* Small Quotient Division::
- at end menu
-
-
- at node Single Limb Division, Basecase Division, Division Algorithms, Division Algorithms
- at subsection Single Limb Division
-
-N at cross{}1 division is implemented using repeated 2 at cross{}1 divisions from
-high to low, either with a hardware divide instruction or a multiplication by
-inverse, whichever is best on a given CPU.
-
-The multiply by inverse follows section 8 of ``Division by Invariant Integers
-using Multiplication'' by Granlund and Montgomery (@pxref{References}) and is
-implemented as @code{udiv_qrnnd_preinv} in @file{gmp-impl.h}. The idea is to
-have a fixed-point approximation to @math{1/d} (see @code{invert_limb}) and
-then multiply by the high limb (plus one bit) of the dividend to get a
-quotient @math{q}. With @math{d} normalized (high bit set), @math{q} is no
-more than 1 too small. Subtracting @m{qd,q*d} from the dividend gives a
-remainder, and reveals whether @math{q} or @math{q-1} is correct.
-
-The result is a division done with two multiplications and four or five
-arithmetic operations. On CPUs with low latency multipliers this can be much
-faster than a hardware divide, though the cost of calculating the inverse at
-the start may mean it's only better on inputs bigger than say 4 or 5 limbs.
-
-When a divisor must be normalized, either for the generic C
- at code{__udiv_qrnnd_c} or the multiply by inverse, the division performed is
-actually @m{a2^k,a*2^k} by @m{d2^k,d*2^k} where @math{a} is the dividend and
- at math{k} is the power necessary to have the high bit of @m{d2^k,d*2^k} set.
-The bit shifts for the dividend are usually accomplished ``on the fly''
-meaning by extracting the appropriate bits at each step. Done this way the
-quotient limbs come out aligned ready to store. When only the remainder is
-wanted, an alternative is to take the dividend limbs unshifted and calculate
- at m{r = a \bmod d2^k, r = a mod d*2^k} followed by an extra final step @m{r2^k
-\bmod d2^k, r*2^k mod d*2^k}. This can help on CPUs with poor bit shifts or
-few registers.
-
-The multiply by inverse can be done two limbs at a time. The calculation is
-basically the same, but the inverse is two limbs and the divisor treated as if
-padded with a low zero limb. This means more work, since the inverse will
-need a 2 at cross{}2 multiply, but the four 1 at cross{}1s to do that are
-independent and can therefore be done partly or wholly in parallel. Likewise
-for a 2 at cross{}1 calculating @m{qd,q*d}. The net effect is to process two
-limbs with roughly the same two multiplies worth of latency that one limb at a
-time gives. This extends to 3 or 4 limbs at a time, though the extra work to
-apply the inverse will almost certainly soon reach the limits of multiplier
-throughput.
-
-A similar approach in reverse can be taken to process just half a limb at a
-time if the divisor is only a half limb. In this case the 1 at cross{}1 multiply
-for the inverse effectively becomes two @m{{1\over2}\times1, (1/2)x1} for each
-limb, which can be a saving on CPUs with a fast half limb multiply, or in fact
-if the only multiply is a half limb, and especially if it's not pipelined.
-
-
- at node Basecase Division, Divide and Conquer Division, Single Limb Division, Division Algorithms
- at subsection Basecase Division
-
-This section is out-of-date.
-
-Basecase N at cross{}M division is like long division done by hand, but in base
- at m{2\GMPraise{@code{mp\_bits\_per\_limb}}, 2^mp_bits_per_limb}. See Knuth
-section 4.3.1 algorithm D.
-
-Briefly stated, while the dividend remains larger than the divisor, a high
-quotient limb is formed and the N at cross{}1 product @m{qd,q*d} subtracted at
-the top end of the dividend. With a normalized divisor (most significant bit
-set), each quotient limb can be formed with a 2 at cross{}1 division and a
-1 at cross{}1 multiplication plus some subtractions. The 2 at cross{}1 division is
-by the high limb of the divisor and is done either with a hardware divide or a
-multiply by inverse (the same as in @ref{Single Limb Division}) whichever is
-faster. Such a quotient is sometimes one too big, requiring an addback of the
-divisor, but that happens rarely.
-
-With Q=N at minus{}M being the number of quotient limbs, this is an
- at m{O(QM),O(Q*M)} algorithm and will run at a speed similar to a basecase
-Q at cross{}M multiplication, differing in fact only in the extra multiply and
-divide for each of the Q quotient limbs.
-
-
- at node Divide and Conquer Division, Exact Division, Basecase Division, Division Algorithms
- at subsection Divide and Conquer Division
-
-This section is out-of-date
-
-For divisors larger than @code{DIV_DC_THRESHOLD}, division is done by dividing.
-Or to be precise by a recursive divide and conquer algorithm based on work by
-Moenck and Borodin, Jebelean, and Burnikel and Ziegler (@pxref{References}).
-
-The algorithm consists essentially of recognising that a 2N at cross{}N division
-can be done with the basecase division algorithm (@pxref{Basecase Division}),
-but using N/2 limbs as a base, not just a single limb. This way the
-multiplications that arise are (N/2)@cross{}(N/2) and can take advantage of
-Karatsuba and higher multiplication algorithms (@pxref{Multiplication
-Algorithms}). The two ``digits'' of the quotient are formed by recursive
-N at cross{}(N/2) divisions.
-
-If the (N/2)@cross{}(N/2) multiplies are done with a basecase multiplication
-then the work is about the same as a basecase division, but with more function
-call overheads and with some subtractions separated from the multiplies.
-These overheads mean that it's only when N/2 is above
- at code{MUL_KARATSUBA_THRESHOLD} that divide and conquer is of use.
-
- at code{DIV_DC_THRESHOLD} is based on the divisor size N, so it will be somewhere
-above twice @code{MUL_KARATSUBA_THRESHOLD}, but how much above depends on the
-CPU at . An optimized @code{mpn_mul_basecase} can lower @code{DIV_DC_THRESHOLD} a
-little by offering a ready-made advantage over repeated @code{mpn_submul_1}
-calls.
-
-Divide and conquer is asymptotically @m{O(M(N)\log N),O(M(N)*log(N))} where
- at math{M(N)} is the time for an N at cross{}N multiplication done with FFTs. The
-actual time is a sum over multiplications of the recursed sizes, as can be
-seen near the end of section 2.2 of Burnikel and Ziegler. For example, within
-the Toom-3 range, divide and conquer is @m{2.63M(N), 2.63*M(N)}. With higher
-algorithms the @math{M(N)} term improves and the multiplier tends to @m{\log
-N, log(N)}. In practice, at moderate to large sizes, a 2N at cross{}N division
-is about 2 to 4 times slower than an N at cross{}N multiplication.
-
-Newton's method used for division is asymptotically @math{O(M(N))} and should
-therefore be superior to divide and conquer, but it's believed this would only
-be for large to very large N.
-
-
- at node Exact Division, Exact Remainder, Divide and Conquer Division, Division Algorithms
- at subsection Exact Division
-
-This section is out-of-date
-
-A so-called exact division is when the dividend is known to be an exact
-multiple of the divisor. Jebelean's exact division algorithm uses this
-knowledge to make some significant optimizations (@pxref{References}).
-
-The idea can be illustrated in decimal for example with 368154 divided by
-543. Because the low digit of the dividend is 4, the low digit of the
-quotient must be 8. This is arrived at from @m{4 \mathord{\times} 7 \bmod 10,
-4*7 mod 10}, using the fact 7 is the modular inverse of 3 (the low digit of
-the divisor), since @m{3 \mathord{\times} 7 \mathop{\equiv} 1 \bmod 10, 3*7
- at equiv{} 1 mod 10}. So @m{8\mathord{\times}543 = 4344,8*543=4344} can be
-subtracted from the dividend leaving 363810. Notice the low digit has become
-zero.
-
-The procedure is repeated at the second digit, with the next quotient digit 7
-(@m{1 \mathord{\times} 7 \bmod 10, 7 @equiv{} 1*7 mod 10}), subtracting
- at m{7\mathord{\times}543 = 3801,7*543=3801}, leaving 325800. And finally at
-the third digit with quotient digit 6 (@m{8 \mathord{\times} 7 \bmod 10, 8*7
-mod 10}), subtracting @m{6\mathord{\times}543 = 3258,6*543=3258} leaving 0.
-So the quotient is 678.
-
-Notice however that the multiplies and subtractions don't need to extend past
-the low three digits of the dividend, since that's enough to determine the
-three quotient digits. For the last quotient digit no subtraction is needed
-at all. On a 2N at cross{}N division like this one, only about half the work of
-a normal basecase division is necessary.
-
-For an N at cross{}M exact division producing Q=N at minus{}M quotient limbs, the
-saving over a normal basecase division is in two parts. Firstly, each of the
-Q quotient limbs needs only one multiply, not a 2 at cross{}1 divide and
-multiply. Secondly, the crossproducts are reduced when @math{Q>M} to
- at m{QM-M(M+1)/2,Q*M-M*(M+1)/2}, or when @math{Q at le{}M} to @m{Q(Q-1)/2,
-Q*(Q-1)/2}. Notice the savings are complementary. If Q is big then many
-divisions are saved, or if Q is small then the crossproducts reduce to a small
-number.
-
-The modular inverse used is calculated efficiently by @code{modlimb_invert} in
- at file{gmp-impl.h}. This does four multiplies for a 32-bit limb, or six for a
-64-bit limb. @file{tune/modlinv.c} has some alternate implementations that
-might suit processors better at bit twiddling than multiplying.
-
-The sub-quadratic exact division described by Jebelean in ``Exact Division
-with Karatsuba Complexity'' is not currently implemented. It uses a
-rearrangement similar to the divide and conquer for normal division
-(@pxref{Divide and Conquer Division}), but operating from low to high. A
-further possibility not currently implemented is ``Bidirectional Exact Integer
-Division'' by Krandick and Jebelean which forms quotient limbs from both the
-high and low ends of the dividend, and can halve once more the number of
-crossproducts needed in a 2N at cross{}N division.
-
-A special case exact division by 3 exists in @code{mpn_divexact_by3},
-supporting Toom-3 multiplication and @code{mpq} canonicalizations. It forms
-quotient digits with a multiply by the modular inverse of 3 (which is
- at code{0xAA..AAB}) and uses two comparisons to determine a borrow for the next
-limb. The multiplications don't need to be on the dependent chain, as long as
-the effect of the borrows is applied, which can help chips with pipelined
-multipliers.
-
-
- at node Exact Remainder, Small Quotient Division, Exact Division, Division Algorithms
- at subsection Exact Remainder
- at cindex Exact remainder
-
-If the exact division algorithm is done with a full subtraction at each stage
-and the dividend isn't a multiple of the divisor, then low zero limbs are
-produced but with a remainder in the high limbs. For dividend @math{a},
-divisor @math{d}, quotient @math{q}, and @m{b = 2
-\GMPraise{@code{mp\_bits\_per\_limb}}, b = 2^mp_bits_per_limb}, this remainder
- at math{r} is of the form
- at tex
-$$ a = qd + r b^n $$
- at end tex
- at ifnottex
-
- at example
-a = q*d + r*b^n
- at end example
-
- at end ifnottex
- at math{n} represents the number of zero limbs produced by the subtractions,
-that being the number of limbs produced for @math{q}. @math{r} will be in the
-range @math{0 at le{}r<d} and can be viewed as a remainder, but one shifted up by
-a factor of @math{b^n}.
-
-Carrying out full subtractions at each stage means the same number of cross
-products must be done as a normal division, but there's still some single limb
-divisions saved. When @math{d} is a single limb some simplifications arise,
-providing good speedups on a number of processors.
-
- at code{mpn_bdivmod}, @code{mpn_divexact_by3}, @code{mpn_modexact_1_odd} and the
- at code{redc} function in @code{mpz_powm} differ subtly in how they return
- at math{r}, leading to some negations in the above formula, but all are
-essentially the same.
-
- at cindex Divisibility algorithm
- at cindex Congruence algorithm
-Clearly @math{r} is zero when @math{a} is a multiple of @math{d}, and this
-leads to divisibility or congruence tests which are potentially more efficient
-than a normal division.
-
-The factor of @math{b^n} on @math{r} can be ignored in a GCD when @math{d} is
-odd, hence the use of @code{mpn_bdivmod} in @code{mpn_gcd}, and the use of
- at code{mpn_modexact_1_odd} by @code{mpn_gcd_1} and @code{mpz_kronecker_ui} etc
-(@pxref{Greatest Common Divisor Algorithms}).
-
-Montgomery's REDC method for modular multiplications uses operands of the form
-of @m{xb^{-n}, x*b^-n} and @m{yb^{-n}, y*b^-n} and on calculating @m{(xb^{-n})
-(yb^{-n}), (x*b^-n)*(y*b^-n)} uses the factor of @math{b^n} in the exact
-remainder to reach a product in the same form @m{(xy)b^{-n}, (x*y)*b^-n}
-(@pxref{Modular Powering Algorithm}).
-
-Notice that @math{r} generally gives no useful information about the ordinary
-remainder @math{a @bmod d} since @math{b^n @bmod d} could be anything. If
-however @math{b^n @equiv{} 1 @bmod d}, then @math{r} is the negative of the
-ordinary remainder. This occurs whenever @math{d} is a factor of
- at math{b^n-1}, as for example with 3 in @code{mpn_divexact_by3}. For a 32 or
-64 bit limb other such factors include 5, 17 and 257, but no particular use
-has been found for this.
-
-
- at node Small Quotient Division, , Exact Remainder, Division Algorithms
- at subsection Small Quotient Division
-
-An N at cross{}M division where the number of quotient limbs Q=N at minus{}M is
-small can be optimized somewhat.
-
-An ordinary basecase division normalizes the divisor by shifting it to make
-the high bit set, shifting the dividend accordingly, and shifting the
-remainder back down at the end of the calculation. This is wasteful if only a
-few quotient limbs are to be formed. Instead a division of just the top
- at m{\rm2Q,2*Q} limbs of the dividend by the top Q limbs of the divisor can be
-used to form a trial quotient. This requires only those limbs normalized, not
-the whole of the divisor and dividend.
-
-A multiply and subtract then applies the trial quotient to the M at minus{}Q
-unused limbs of the divisor and N at minus{}Q dividend limbs (which includes Q
-limbs remaining from the trial quotient division). The starting trial
-quotient can be 1 or 2 too big, but all cases of 2 too big and most cases of 1
-too big are detected by first comparing the most significant limbs that will
-arise from the subtraction. An addback is done if the quotient still turns
-out to be 1 too big.
-
-This whole procedure is essentially the same as one step of the basecase
-algorithm done in a Q limb base, though with the trial quotient test done only
-with the high limbs, not an entire Q limb ``digit'' product. The correctness
-of this weaker test can be established by following the argument of Knuth
-section 4.3.1 exercise 20 but with the @m{v_2 \GMPhat q > b \GMPhat r
-+ u_2, v2*q>b*r+u2} condition appropriately relaxed.
-
-
- at need 1000
- at node Greatest Common Divisor Algorithms, Powering Algorithms, Division Algorithms, Algorithms
- at section Greatest Common Divisor
- at cindex Greatest common divisor algorithms
- at cindex GCD algorithms
-
- at menu
-* Binary GCD::
-* Lehmer's GCD::
-* Subquadratic GCD::
-* Extended GCD::
-* Jacobi Symbol::
- at end menu
-
-
- at node Binary GCD, Lehmer's GCD, Greatest Common Divisor Algorithms, Greatest Common Divisor Algorithms
- at subsection Binary GCD
-
-At small sizes MPIR uses an @math{O(N^2)} binary style GCD at . This is described
-in many textbooks, for example Knuth section 4.5.2 algorithm B at . It simply
-consists of successively reducing odd operands @math{a} and @math{b} using
-
- at quotation
- at math{a,b = @abs{}(a-b), at min{}(a,b)} @*
-strip factors of 2 from @math{a}
- at end quotation
-
-The Euclidean GCD algorithm, as per Knuth algorithms E and A, reduces using
- at math{a @bmod b} but this has so far been found to be slower everywhere. One
-reason the binary method does well is that the implied quotient at each step
-is usually small, so often only one or two subtractions are needed to get the
-same effect as a division. Quotients 1, 2 and 3 for example occur 67.7% of
-the time, see Knuth section 4.5.3 Theorem E.
-
-When the implied quotient is large, meaning @math{b} is much smaller than
- at math{a}, then a division is worthwhile. This is the basis for the initial
- at math{a @bmod b} reductions in @code{mpn_gcd} and @code{mpn_gcd_1} (the latter
-for both N at cross{}1 and 1 at cross{}1 cases). But after that initial reduction,
-big quotients occur too rarely to make it worth checking for them.
-
- at sp 1
-The final @math{1 at cross{}1} GCD in @code{mpn_gcd_1} is done in the generic C
-code as described above. For two N-bit operands, the algorithm takes about
-0.68 iterations per bit. For optimum performance some attention needs to be
-paid to the way the factors of 2 are stripped from @math{a}.
-
-Firstly it may be noted that in twos complement the number of low zero bits on
- at math{a-b} is the same as @math{b-a}, so counting or testing can begin on
- at math{a-b} without waiting for @math{@abs{}(a-b)} to be determined.
-
-A loop stripping low zero bits tends not to branch predict well, since the
-condition is data dependent. But on average there's only a few low zeros, so
-an option is to strip one or two bits arithmetically then loop for more (as
-done for AMD K6). Or use a lookup table to get a count for several bits then
-loop for more (as done for AMD K7). An alternative approach is to keep just
-one of @math{a} or @math{b} odd and iterate
-
- at quotation
- at math{a,b = @abs{}(a-b), @min{}(a,b)} @*
- at math{a = a/2} if even @*
- at math{b = b/2} if even
- at end quotation
-
-This requires about 1.25 iterations per bit, but stripping of a single bit at
-each step avoids any branching. Repeating the bit strip reduces to about 0.9
-iterations per bit, which may be a worthwhile tradeoff.
-
-Generally with the above approaches a speed of perhaps 6 cycles per bit can be
-achieved, which is still not terribly fast with for instance a 64-bit GCD
-taking nearly 400 cycles. It's this sort of time which means it's not usually
-advantageous to combine a set of divisibility tests into a GCD.
-
-
- at node Lehmer's GCD, Subquadratic GCD, Binary GCD, Greatest Common Divisor Algorithms
- at subsection Lehmer's GCD
-
-Lehmer's improvement of the Euclidean algorithms is based on the observation
-that the initial part of the quotient sequence depends only on the most
-significant parts of the inputs. The variant of Lehmer's algorithm used in MPIR
-splits off the most significant two limbs, as suggested, e.g., in ``A
-Double-Digit Lehmer-Euclid Algorithm'' by Jebelean (@pxref{References}). The
-quotients of two double-limb inputs are collected as a 2 by 2 matrix with
-single-limb elements. This is done by the function @code{mpn_hgcd2}. The
-resulting matrix is applied to the inputs using @code{mpn_mul_1} and
- at code{mpn_submul_1}. Each iteration usually reduces the inputs by almost one
-limb. In the rare case of a large quotient, no progress can be made by
-examining just the most significant two limbs, and the quotient is computing
-using plain division.
-
-The resulting algorithm is asymptotically @math{O(N^2)}, just as the Euclidean
-algorithm and the binary algorithm. The quadratic part of the work are
-the calls to @code{mpn_mul_1} and @code{mpn_submul_1}. For small sizes, the
-linear work is also significant. There are roughly @math{N} calls to the
- at code{mpn_hgcd2} function. This function uses a couple of important
-optimizations:
-
- at itemize
- at item
-It uses the same relaxed notion of correctness as @code{mpn_hgcd} (see next
-section). This means that when called with the most significant two limbs of
-two large numbers, the returned matrix does not always correspond exactly to
-the initial quotient sequence for the two large numbers; the final quotient
-may sometimes be one off.
-
- at item
-It takes advantage of the fact the quotients are usually small. The division
-operator is not used, since the corresponding assembler instruction is very
-slow on most architectures. (This code could probably be improved further, it
-uses many branches that are unfriendly to prediction).
-
- at item
-It switches from double-limb calculations to single-limb calculations half-way
-through, when the input numbers have been reduced in size from two limbs to
-one and a half.
-
- at end itemize
-
- at node Subquadratic GCD, Extended GCD, Lehmer's GCD, Greatest Common Divisor Algorithms
- at subsection Subquadratic GCD
-
-For inputs larger than @code{GCD_DC_THRESHOLD}, GCD is computed via the HGCD
-(Half GCD) function, as a generalization to Lehmer's algorithm.
-
-Let the inputs @math{a,b} be of size @math{N} limbs each. Put @m{S=\lfloor N/2
-\rfloor + 1, S = floor(N/2) + 1}. Then HGCD(a,b) returns a transformation
-matrix @math{T} with non-negative elements, and reduced numbers @math{(c;d) =
-T^{-1} (a;b)}. The reduced numbers @math{c,d} must be larger than @math{S}
-limbs, while their difference @math{abs(c-d)} must fit in @math{S} limbs. The
-matrix elements will also be of size roughly @math{N/2}.
-
-The HGCD base case uses Lehmer's algorithm, but with the above stop condition
-that returns reduced numbers and the corresponding transformation matrix
-half-way through. For inputs larger than @code{HGCD_THRESHOLD}, HGCD is
-computed recursively, using the divide and conquer algorithm in ``On
-Sch@"onhage's algorithm and subquadratic integer GCD computation'' by M@"oller
-(@pxref{References}). The recursive algorithm consists of these main
-steps.
-
- at itemize
-
- at item
-Call HGCD recursively, on the most significant @math{N/2} limbs. Apply the
-resulting matrix @math{T_1} to the full numbers, reducing them to a size just
-above @math{3N/2}.
-
- at item
-Perform a small number of division or subtraction steps to reduce the numbers
-to size below @math{3N/2}. This is essential mainly for the unlikely case of
-large quotients.
-
- at item
-Call HGCD recursively, on the most significant @math{N/2} limbs of the reduced
-numbers. Apply the resulting matrix @math{T_2} to the full numbers, reducing
-them to a size just above @math{N/2}.
-
- at item
-Compute @math{T = T_1 T_2}.
-
- at item
-Perform a small number of division and subtraction steps to satisfy the
-requirements, and return.
- at end itemize
-
-GCD is then implemented as a loop around HGCD, similarly to Lehmer's
-algorithm. Where Lehmer repeatedly chops off the top two limbs, calls
- at code{mpn_hgcd2}, and applies the resulting matrix to the full numbers, the
-subquadratic GCD chops off the most significant third of the limbs (the
-proportion is a tuning parameter, and @math{1/3} seems to be more efficient
-than, e.g, @math{1/2}), calls @code{mpn_hgcd}, and applies the resulting
-matrix. Once the input numbers are reduced to size below
- at code{GCD_DC_THRESHOLD}, Lehmer's algorithm is used for the rest of the work.
-
-The asymptotic running time of both HGCD and GCD is @m{O(M(N)\log N),O(M(N)*log(N))},
-where @math{M(N)} is the time for multiplying two @math{N}-limb numbers.
-
- at node Extended GCD, Jacobi Symbol, Subquadratic GCD, Greatest Common Divisor Algorithms
- at subsection Extended GCD
-
-The extended GCD function, or gcdext, calculates @math{@gcd{}(a,b)} and also
-one of the cofactors @math{x} and @math{y} satisfying @m{ax+by=\gcd(a at C{}b),
-a*x+b*y=gcd(a at C{}b)}. The algorithms used for plain GCD are extended to
-handle this case.
-
-Lehmer's algorithm is used for sizes up to @code{GCDEXT_DC_THRESHOLD}. Above
-this threshold, GCDEXT is implemented as a loop around HGCD, but with more
-book-keeping to keep track of the cofactors.
-
- at node Jacobi Symbol, , Extended GCD, Greatest Common Divisor Algorithms
- at subsection Jacobi Symbol
- at cindex Jacobi symbol algorithm
-
- at code{mpz_jacobi} and @code{mpz_kronecker} are currently implemented with a
-simple binary algorithm similar to that described for the GCDs (@pxref{Binary
-GCD}). They're not very fast when both inputs are large. Lehmer's multi-step
-improvement or a binary based multi-step algorithm is likely to be better.
-
-When one operand fits a single limb, and that includes @code{mpz_kronecker_ui}
-and friends, an initial reduction is done with either @code{mpn_mod_1} or
- at code{mpn_modexact_1_odd}, followed by the binary algorithm on a single limb.
-The binary algorithm is well suited to a single limb, and the whole
-calculation in this case is quite efficient.
-
-In all the routines sign changes for the result are accumulated using some bit
-twiddling, avoiding table lookups or conditional jumps.
-
-
- at need 1000
- at node Powering Algorithms, Root Extraction Algorithms, Greatest Common Divisor Algorithms, Algorithms
- at section Powering Algorithms
- at cindex Powering algorithms
-
- at menu
-* Normal Powering Algorithm::
-* Modular Powering Algorithm::
- at end menu
-
-
- at node Normal Powering Algorithm, Modular Powering Algorithm, Powering Algorithms, Powering Algorithms
- at subsection Normal Powering
-
-Normal @code{mpz} or @code{mpf} powering uses a simple binary algorithm,
-successively squaring and then multiplying by the base when a 1 bit is seen in
-the exponent, as per Knuth section 4.6.3. The ``left to right''
-variant described there is used rather than algorithm A, since it's just as
-easy and can be done with somewhat less temporary memory.
-
-
- at node Modular Powering Algorithm, , Normal Powering Algorithm, Powering Algorithms
- at subsection Modular Powering
-
-Modular powering is implemented using a @math{2^k}-ary sliding window
-algorithm, as per ``Handbook of Applied Cryptography'' algorithm 14.85
-(@pxref{References}). @math{k} is chosen according to the size of the
-exponent. Larger exponents use larger values of @math{k}, the choice being
-made to minimize the average number of multiplications that must supplement
-the squaring.
-
-The modular multiplies and squares use either a simple division or the REDC
-method by Montgomery (@pxref{References}). REDC is a little faster,
-essentially saving N single limb divisions in a fashion similar to an exact
-remainder (@pxref{Exact Remainder}). The current REDC has some limitations.
-It's only @math{O(N^2)} so above @code{POWM_THRESHOLD} division becomes faster
-and is used. It doesn't attempt to detect small bases, but rather always uses
-a REDC form, which is usually a full size operand. And lastly it's only
-applied to odd moduli.
-
-
- at node Root Extraction Algorithms, Radix Conversion Algorithms, Powering Algorithms, Algorithms
- at section Root Extraction Algorithms
- at cindex Root extraction algorithms
-
- at menu
-* Square Root Algorithm::
-* Nth Root Algorithm::
-* Perfect Square Algorithm::
-* Perfect Power Algorithm::
- at end menu
-
-
- at node Square Root Algorithm, Nth Root Algorithm, Root Extraction Algorithms, Root Extraction Algorithms
- at subsection Square Root
- at cindex Square root algorithm
- at cindex Karatsuba square root algorithm
-
-Square roots are taken using the ``Karatsuba Square Root'' algorithm by Paul
-Zimmermann (@pxref{References}).
-
-An input @math{n} is split into four parts of @math{k} bits each, so with
- at math{b=2^k} we have @m{n = a_3b^3 + a_2b^2 + a_1b + a_0, n = a3*b^3 + a2*b^2
-+ a1*b + a0}. Part @ms{a,3} must be ``normalized'' so that either the high or
-second highest bit is set. In MPIR, @math{k} is kept on a limb boundary and
-the input is left shifted (by an even number of bits) to normalize.
-
-The square root of the high two parts is taken, by recursive application of
-the algorithm (bottoming out in a one-limb Newton's method),
- at tex
-$$ s',r' = \mathop{\rm sqrtrem} \> (a_3b + a_2) $$
- at end tex
- at ifnottex
-
- at example
-s1,r1 = sqrtrem (a3*b + a2)
- at end example
-
- at end ifnottex
-This is an approximation to the desired root and is extended by a division to
-give @math{s}, at math{r},
- at tex
-$$\eqalign{
-q,u &= \mathop{\rm divrem} \> (r'b + a_1, 2s') \cr
-s &= s'b + q \cr
-r &= ub + a_0 - q^2
-}$$
- at end tex
- at ifnottex
-
- at example
-q,u = divrem (r1*b + a1, 2*s1)
-s = s1*b + q
-r = u*b + a0 - q^2
- at end example
-
- at end ifnottex
-The normalization requirement on @ms{a,3} means at this point @math{s} is
-either correct or 1 too big. @math{r} is negative in the latter case, so
- at tex
-$$\eqalign{
-\mathop{\rm if} \; r &< 0 \; \mathop{\rm then} \cr
-r &\leftarrow r + 2s - 1 \cr
-s &\leftarrow s - 1
-}$$
- at end tex
- at ifnottex
-
- at example
-if r < 0 then
- r = r + 2*s - 1
- s = s - 1
- at end example
-
- at end ifnottex
-The algorithm is expressed in a divide and conquer form, but as noted in the
-paper it can also be viewed as a discrete variant of Newton's method, or as a
-variation on the schoolboy method (no longer taught) for square roots two
-digits at a time.
-
-If the remainder @math{r} is not required then usually only a few high limbs
-of @math{r} and @math{u} need to be calculated to determine whether an
-adjustment to @math{s} is required. This optimization is not currently
-implemented.
-
-In the Karatsuba multiplication range this algorithm is @m{O({3\over2}
-M(N/2)),O(1.5*M(N/2))}, where @math{M(n)} is the time to multiply two numbers
-of @math{n} limbs. In the FFT multiplication range this grows to a bound of
- at m{O(6 M(N/2)),O(6*M(N/2))}. In practice a factor of about 1.5 to 1.8 is
-found in the Karatsuba and Toom-3 ranges, growing to 2 or 3 in the FFT range.
-
-The algorithm does all its calculations in integers and the resulting
- at code{mpn_sqrtrem} is used for both @code{mpz_sqrt} and @code{mpf_sqrt}.
-The extended precision given by @code{mpf_sqrt_ui} is obtained by
-padding with zero limbs.
-
-
- at node Nth Root Algorithm, Perfect Square Algorithm, Square Root Algorithm, Root Extraction Algorithms
- at subsection Nth Root
- at cindex Root extraction algorithm
- at cindex Nth root algorithm
-
-Integer Nth roots are taken using Newton's method with the following
-iteration, where @math{A} is the input and @math{n} is the root to be taken.
- at tex
-$$a_{i+1} = {1\over n} \left({A \over a_i^{n-1}} + (n-1)a_i \right)$$
- at end tex
- at ifnottex
-
- at example
- 1 A
-a[i+1] = - * ( --------- + (n-1)*a[i] )
- n a[i]^(n-1)
- at end example
-
- at end ifnottex
-The initial approximation @m{a_1,a[1]} is generated bitwise by successively
-powering a trial root with or without new 1 bits, aiming to be just above the
-true root. The iteration converges quadratically when started from a good
-approximation. When @math{n} is large more initial bits are needed to get
-good convergence. The current implementation is not particularly well
-optimized.
-
-
- at node Perfect Square Algorithm, Perfect Power Algorithm, Nth Root Algorithm, Root Extraction Algorithms
- at subsection Perfect Square
- at cindex Perfect square algorithm
-
-A significant fraction of non-squares can be quickly identified by checking
-whether the input is a quadratic residue modulo small integers.
-
- at code{mpz_perfect_square_p} first tests the input mod 256, which means just
-examining the low byte. Only 44 different values occur for squares mod 256,
-so 82.8% of inputs can be immediately identified as non-squares.
-
-On a 32-bit system similar tests are done mod 9, 5, 7, 13 and 17, for a total
-99.25% of inputs identified as non-squares. On a 64-bit system 97 is tested
-too, for a total 99.62%.
-
-These moduli are chosen because they're factors of @math{2^@W{24}-1} (or
- at math{2^@W{48}-1} for 64-bits), and such a remainder can be quickly taken just
-using additions (see @code{mpn_mod_34lsub1}).
-
-When nails are in use moduli are instead selected by the @file{gen-psqr.c}
-program and applied with an @code{mpn_mod_1}. The same @math{2^@W{24}-1} or
- at math{2^@W{48}-1} could be done with nails using some extra bit shifts, but
-this is not currently implemented.
-
-In any case each modulus is applied to the @code{mpn_mod_34lsub1} or
- at code{mpn_mod_1} remainder and a table lookup identifies non-squares. By
-using a ``modexact'' style calculation, and suitably permuted tables, just one
-multiply each is required, see the code for details. Moduli are also combined
-to save operations, so long as the lookup tables don't become too big.
- at file{gen-psqr.c} does all the pre-calculations.
-
-A square root must still be taken for any value that passes these tests, to
-verify it's really a square and not one of the small fraction of non-squares
-that get through (ie.@: a pseudo-square to all the tested bases).
-
-Clearly more residue tests could be done, @code{mpz_perfect_square_p} only
-uses a compact and efficient set. Big inputs would probably benefit from more
-residue testing, small inputs might be better off with less. The assumed
-distribution of squares versus non-squares in the input would affect such
-considerations.
-
-
- at node Perfect Power Algorithm, , Perfect Square Algorithm, Root Extraction Algorithms
- at subsection Perfect Power
- at cindex Perfect power algorithm
-
-Detecting perfect powers is required by some factorization algorithms.
-Currently @code{mpz_perfect_power_p} is implemented using repeated Nth root
-extractions, though naturally only prime roots need to be considered.
-(@xref{Nth Root Algorithm}.)
-
-If a prime divisor @math{p} with multiplicity @math{e} can be found, then only
-roots which are divisors of @math{e} need to be considered, much reducing the
-work necessary. To this end divisibility by a set of small primes is checked.
-
-
- at node Radix Conversion Algorithms, Other Algorithms, Root Extraction Algorithms, Algorithms
- at section Radix Conversion
- at cindex Radix conversion algorithms
-
-Radix conversions are less important than other algorithms. A program
-dominated by conversions should probably use a different data representation.
-
- at menu
-* Binary to Radix::
-* Radix to Binary::
- at end menu
-
-
- at node Binary to Radix, Radix to Binary, Radix Conversion Algorithms, Radix Conversion Algorithms
- at subsection Binary to Radix
-
-Conversions from binary to a power-of-2 radix use a simple and fast
- at math{O(N)} bit extraction algorithm.
-
-Conversions from binary to other radices use one of two algorithms. Sizes
-below @code{GET_STR_PRECOMPUTE_THRESHOLD} use a basic @math{O(N^2)} method.
-Repeated divisions by @math{b^n} are made, where @math{b} is the radix and
- at math{n} is the biggest power that fits in a limb. But instead of simply
-using the remainder @math{r} from such divisions, an extra divide step is done
-to give a fractional limb representing @math{r/b^n}. The digits of @math{r}
-can then be extracted using multiplications by @math{b} rather than divisions.
-Special case code is provided for decimal, allowing multiplications by 10 to
-optimize to shifts and adds.
-
-Above @code{GET_STR_PRECOMPUTE_THRESHOLD} a sub-quadratic algorithm is used.
-For an input @math{t}, powers @m{b^{n2^i},b^(n*2^i)} of the radix are
-calculated, until a power between @math{t} and @m{\sqrt{t},sqrt(t)} is
-reached. @math{t} is then divided by that largest power, giving a quotient
-which is the digits above that power, and a remainder which is those below.
-These two parts are in turn divided by the second highest power, and so on
-recursively. When a piece has been divided down to less than
- at code{GET_STR_DC_THRESHOLD} limbs, the basecase algorithm described above is
-used.
-
-The advantage of this algorithm is that big divisions can make use of the
-sub-quadratic divide and conquer division (@pxref{Divide and Conquer
-Division}), and big divisions tend to have less overheads than lots of
-separate single limb divisions anyway. But in any case the cost of
-calculating the powers @m{b^{n2^i},b^(n*2^i)} must first be overcome.
-
- at code{GET_STR_PRECOMPUTE_THRESHOLD} and @code{GET_STR_DC_THRESHOLD} represent
-the same basic thing, the point where it becomes worth doing a big division to
-cut the input in half. @code{GET_STR_PRECOMPUTE_THRESHOLD} includes the cost
-of calculating the radix power required, whereas @code{GET_STR_DC_THRESHOLD}
-assumes that's already available, which is the case when recursing.
-
-Since the base case produces digits from least to most significant but they
-want to be stored from most to least, it's necessary to calculate in advance
-how many digits there will be, or at least be sure not to underestimate that.
-For MPIR the number of input bits is multiplied by @code{chars_per_bit_exactly}
-from @code{mp_bases}, rounding up. The result is either correct or one too
-big.
-
-Examining some of the high bits of the input could increase the chance of
-getting the exact number of digits, but an exact result every time would not
-be practical, since in general the difference between numbers 100 at dots{} and
-99 at dots{} is only in the last few bits and the work to identify 99 at dots{}
-might well be almost as much as a full conversion.
-
- at code{mpf_get_str} doesn't currently use the algorithm described here, it
-multiplies or divides by a power of @math{b} to move the radix point to the
-just above the highest non-zero digit (or at worst one above that location),
-then multiplies by @math{b^n} to bring out digits. This is @math{O(N^2)} and
-is certainly not optimal.
-
-The @math{r/b^n} scheme described above for using multiplications to bring out
-digits might be useful for more than a single limb. Some brief experiments
-with it on the base case when recursing didn't give a noticeable improvement,
-but perhaps that was only due to the implementation. Something similar would
-work for the sub-quadratic divisions too, though there would be the cost of
-calculating a bigger radix power.
-
-Another possible improvement for the sub-quadratic part would be to arrange
-for radix powers that balanced the sizes of quotient and remainder produced,
-ie.@: the highest power would be an @m{b^{nk},b^(n*k)} approximately equal to
- at m{\sqrt{t},sqrt(t)}, not restricted to a @math{2^i} factor. That ought to
-smooth out a graph of times against sizes, but may or may not be a net
-speedup.
-
-
- at node Radix to Binary, , Binary to Radix, Radix Conversion Algorithms
- at subsection Radix to Binary
-
-This section is out-of-date.
-
-Conversions from a power-of-2 radix into binary use a simple and fast
- at math{O(N)} bitwise concatenation algorithm.
-
-Conversions from other radices use one of two algorithms. Sizes below
- at code{SET_STR_THRESHOLD} use a basic @math{O(N^2)} method. Groups of @math{n}
-digits are converted to limbs, where @math{n} is the biggest power of the base
- at math{b} which will fit in a limb, then those groups are accumulated into the
-result by multiplying by @math{b^n} and adding. This saves multi-precision
-operations, as per Knuth section 4.4 part E (@pxref{References}). Some
-special case code is provided for decimal, giving the compiler a chance to
-optimize multiplications by 10.
-
-Above @code{SET_STR_THRESHOLD} a sub-quadratic algorithm is used. First
-groups of @math{n} digits are converted into limbs. Then adjacent limbs are
-combined into limb pairs with @m{xb^n+y,x*b^n+y}, where @math{x} and @math{y}
-are the limbs. Adjacent limb pairs are combined into quads similarly with
- at m{xb^{2n}+y,x*b^(2n)+y}. This continues until a single block remains, that
-being the result.
-
-The advantage of this method is that the multiplications for each @math{x} are
-big blocks, allowing Karatsuba and higher algorithms to be used. But the cost
-of calculating the powers @m{b^{n2^i},b^(n*2^i)} must be overcome.
- at code{SET_STR_THRESHOLD} usually ends up quite big, around 5000 digits, and on
-some processors much bigger still.
-
- at code{SET_STR_THRESHOLD} is based on the input digits (and tuned for decimal),
-though it might be better based on a limb count, so as to be independent of
-the base. But that sort of count isn't used by the base case and so would
-need some sort of initial calculation or estimate.
-
-The main reason @code{SET_STR_THRESHOLD} is so much bigger than the
-corresponding @code{GET_STR_PRECOMPUTE_THRESHOLD} is that @code{mpn_mul_1} is
-much faster than @code{mpn_divrem_1} (often by a factor of 10, or more).
-
-
- at need 1000
- at node Other Algorithms, Assembler Coding, Radix Conversion Algorithms, Algorithms
- at section Other Algorithms
-
- at menu
-* Prime Testing Algorithm::
-* Factorial Algorithm::
-* Binomial Coefficients Algorithm::
-* Fibonacci Numbers Algorithm::
-* Lucas Numbers Algorithm::
-* Random Number Algorithms::
- at end menu
-
-
- at node Prime Testing Algorithm, Factorial Algorithm, Other Algorithms, Other Algorithms
- at subsection Prime Testing
- at cindex Prime testing algorithms
-
-This section is somewhat out-of-date.
-
-The primality testing in @code{mpz_probab_prime_p} (@pxref{Number Theoretic
-Functions}) first does some trial division by small factors and then uses the
-Miller-Rabin probabilistic primality testing algorithm, as described in Knuth
-section 4.5.4 algorithm P (@pxref{References}).
-
-For an odd input @math{n}, and with @math{n = q at GMPmultiply{}2^k+1} where
- at math{q} is odd, this algorithm selects a random base @math{x} and tests
-whether @math{x^q @bmod{} n} is 1 or @math{-1}, or an @m{x^{q2^j} \bmod n,
-x^(q*2^j) mod n} is @math{1}, for @math{1 at le{}j at le{}k}. If so then @math{n}
-is probably prime, if not then @math{n} is definitely composite.
-
-Any prime @math{n} will pass the test, but some composites do too. Such
-composites are known as strong pseudoprimes to base @math{x}. No @math{n} is
-a strong pseudoprime to more than @math{1/4} of all bases (see Knuth exercise
-22), hence with @math{x} chosen at random there's no more than a @math{1/4}
-chance a ``probable prime'' will in fact be composite.
-
-In fact strong pseudoprimes are quite rare, making the test much more
-powerful than this analysis would suggest, but @math{1/4} is all that's proven
-for an arbitrary @math{n}.
-
-
- at node Factorial Algorithm, Binomial Coefficients Algorithm, Prime Testing Algorithm, Other Algorithms
- at subsection Factorial
- at cindex Factorial algorithm
-
-This section is out-of-date.
-
-Factorials are calculated by a combination of removal of twos, powering, and
-binary splitting. The procedure can be best illustrated with an example,
-
- at quotation
- at math{23! = 1.2.3.4.5.6.7.8.9.10.11.12.13.14.15.16.17.18.19.20.21.22.23}
- at end quotation
-
- at noindent
-has factors of two removed,
-
- at quotation
- at math{23! = 2^{19}.1.1.3.1.5.3.7.1.9.5.11.3.13.7.15.1.17.9.19.5.21.11.23}
- at end quotation
-
- at noindent
-and the resulting terms collected up according to their multiplicity,
-
- at quotation
- at math{23! = 2^{19}.(3.5)^3.(7.9.11)^2.(13.15.17.19.21.23)}
- at end quotation
-
-Each sequence such as @math{13.15.17.19.21.23} is evaluated by splitting into
-every second term, as for instance @math{(13.17.21).(15.19.23)}, and the same
-recursively on each half. This is implemented iteratively using some bit
-twiddling.
-
-Such splitting is more efficient than repeated N at cross{}1 multiplies since it
-forms big multiplies, allowing Karatsuba and higher algorithms to be used.
-And even below the Karatsuba threshold a big block of work can be more
-efficient for the basecase algorithm.
-
-Splitting into subsequences of every second term keeps the resulting products
-more nearly equal in size than would the simpler approach of say taking the
-first half and second half of the sequence. Nearly equal products are more
-efficient for the current multiply implementation.
-
-
- at node Binomial Coefficients Algorithm, Fibonacci Numbers Algorithm, Factorial Algorithm, Other Algorithms
- at subsection Binomial Coefficients
- at cindex Binomial coefficient algorithm
-
-Binomial coefficients @m{\left({n}\atop{k}\right), C(n at C{}k)} are calculated
-by first arranging @math{k @le{} n/2} using @m{\left({n}\atop{k}\right) =
-\left({n}\atop{n-k}\right), C(n at C{}k) = C(n at C{}n-k)} if necessary, and then
-evaluating the following product simply from @math{i=2} to @math{i=k}.
- at tex
-$$ \left({n}\atop{k}\right) = (n-k+1) \prod_{i=2}^{k} {{n-k+i} \over i} $$
- at end tex
- at ifnottex
-
- at example
- k (n-k+i)
-C(n,k) = (n-k+1) * prod -------
- i=2 i
- at end example
-
- at end ifnottex
-It's easy to show that each denominator @math{i} will divide the product so
-far, so the exact division algorithm is used (@pxref{Exact Division}).
-
-The numerators @math{n-k+i} and denominators @math{i} are first accumulated
-into as many fit a limb, to save multi-precision operations, though for
- at code{mpz_bin_ui} this applies only to the divisors, since @math{n} is an
- at code{mpz_t} and @math{n-k+i} in general won't fit in a limb at all.
-
-
- at node Fibonacci Numbers Algorithm, Lucas Numbers Algorithm, Binomial Coefficients Algorithm, Other Algorithms
- at subsection Fibonacci Numbers
- at cindex Fibonacci number algorithm
-
-The Fibonacci functions @code{mpz_fib_ui} and @code{mpz_fib2_ui} are designed
-for calculating isolated @m{F_n,F[n]} or @m{F_n,F[n]}, at m{F_{n-1},F[n-1]}
-values efficiently.
-
-For small @math{n}, a table of single limb values in @code{__gmp_fib_table} is
-used. On a 32-bit limb this goes up to @m{F_{47},F[47]}, or on a 64-bit limb
-up to @m{F_{93},F[93]}. For convenience the table starts at @m{F_{-1},F[-1]}.
-
-Beyond the table, values are generated with a binary powering algorithm,
-calculating a pair @m{F_n,F[n]} and @m{F_{n-1},F[n-1]} working from high to
-low across the bits of @math{n}. The formulas used are
- at tex
-$$\eqalign{
- F_{2k+1} &= 4F_k^2 - F_{k-1}^2 + 2(-1)^k \cr
- F_{2k-1} &= F_k^2 + F_{k-1}^2 \cr
- F_{2k} &= F_{2k+1} - F_{2k-1}
-}$$
- at end tex
- at ifnottex
-
- at example
-F[2k+1] = 4*F[k]^2 - F[k-1]^2 + 2*(-1)^k
-F[2k-1] = F[k]^2 + F[k-1]^2
-
-F[2k] = F[2k+1] - F[2k-1]
- at end example
-
- at end ifnottex
-At each step, @math{k} is the high @math{b} bits of @math{n}. If the next bit
-of @math{n} is 0 then @m{F_{2k},F[2k]}, at m{F_{2k-1},F[2k-1]} is used, or if
-it's a 1 then @m{F_{2k+1},F[2k+1]}, at m{F_{2k},F[2k]} is used, and the process
-repeated until all bits of @math{n} are incorporated. Notice these formulas
-require just two squares per bit of @math{n}.
-
-It'd be possible to handle the first few @math{n} above the single limb table
-with simple additions, using the defining Fibonacci recurrence @m{F_{k+1} =
-F_k + F_{k-1}, F[k+1]=F[k]+F[k-1]}, but this is not done since it usually
-turns out to be faster for only about 10 or 20 values of @math{n}, and
-including a block of code for just those doesn't seem worthwhile. If they
-really mattered it'd be better to extend the data table.
-
-Using a table avoids lots of calculations on small numbers, and makes small
- at math{n} go fast. A bigger table would make more small @math{n} go fast, it's
-just a question of balancing size against desired speed. For MPIR the code is
-kept compact, with the emphasis primarily on a good powering algorithm.
-
- at code{mpz_fib2_ui} returns both @m{F_n,F[n]} and @m{F_{n-1},F[n-1]}, but
- at code{mpz_fib_ui} is only interested in @m{F_n,F[n]}. In this case the last
-step of the algorithm can become one multiply instead of two squares. One of
-the following two formulas is used, according as @math{n} is odd or even.
- at tex
-$$\eqalign{
- F_{2k} &= F_k (F_k + 2F_{k-1}) \cr
- F_{2k+1} &= (2F_k + F_{k-1}) (2F_k - F_{k-1}) + 2(-1)^k
-}$$
- at end tex
- at ifnottex
-
- at example
-F[2k] = F[k]*(F[k]+2F[k-1])
-
-F[2k+1] = (2F[k]+F[k-1])*(2F[k]-F[k-1]) + 2*(-1)^k
- at end example
-
- at end ifnottex
- at m{F_{2k+1},F[2k+1]} here is the same as above, just rearranged to be a
-multiply. For interest, the @m{2(-1)^k, 2*(-1)^k} term both here and above
-can be applied just to the low limb of the calculation, without a carry or
-borrow into further limbs, which saves some code size. See comments with
- at code{mpz_fib_ui} and the internal @code{mpn_fib2_ui} for how this is done.
-
-
- at node Lucas Numbers Algorithm, Random Number Algorithms, Fibonacci Numbers Algorithm, Other Algorithms
- at subsection Lucas Numbers
- at cindex Lucas number algorithm
-
- at code{mpz_lucnum2_ui} derives a pair of Lucas numbers from a pair of Fibonacci
-numbers with the following simple formulas.
- at tex
-$$\eqalign{
- L_k &= F_k + 2F_{k-1} \cr
- L_{k-1} &= 2F_k - F_{k-1}
-}$$
- at end tex
- at ifnottex
-
- at example
-L[k] = F[k] + 2*F[k-1]
-L[k-1] = 2*F[k] - F[k-1]
- at end example
-
- at end ifnottex
- at code{mpz_lucnum_ui} is only interested in @m{L_n,L[n]}, and some work can be
-saved. Trailing zero bits on @math{n} can be handled with a single square
-each.
- at tex
-$$ L_{2k} = L_k^2 - 2(-1)^k $$
- at end tex
- at ifnottex
-
- at example
-L[2k] = L[k]^2 - 2*(-1)^k
- at end example
-
- at end ifnottex
-And the lowest 1 bit can be handled with one multiply of a pair of Fibonacci
-numbers, similar to what @code{mpz_fib_ui} does.
- at tex
-$$ L_{2k+1} = 5F_{k-1} (2F_k + F_{k-1}) - 4(-1)^k $$
- at end tex
- at ifnottex
-
- at example
-L[2k+1] = 5*F[k-1]*(2*F[k]+F[k-1]) - 4*(-1)^k
- at end example
-
- at end ifnottex
-
-
- at node Random Number Algorithms, , Lucas Numbers Algorithm, Other Algorithms
- at subsection Random Numbers
- at cindex Random number algorithms
-
-For the @code{urandomb} functions, random numbers are generated simply by
-concatenating bits produced by the generator. As long as the generator has
-good randomness properties this will produce well-distributed @math{N} bit
-numbers.
-
-For the @code{urandomm} functions, random numbers in a range @math{0 at le{}R<N}
-are generated by taking values @math{R} of @m{\lceil \log_2 N \rceil,
-ceil(log2(N))} bits each until one satisfies @math{R<N}. This will normally
-require only one or two attempts, but the attempts are limited in case the
-generator is somehow degenerate and produces only 1 bits or similar.
-
- at cindex Mersenne twister algorithm
-The Mersenne Twister generator is by Matsumoto and Nishimura
-(@pxref{References}). It has a non-repeating period of @math{2^@W{19937}-1},
-which is a Mersenne prime, hence the name of the generator. The state is 624
-words of 32-bits each, which is iterated with one XOR and shift for each
-32-bit word generated, making the algorithm very fast. Randomness properties
-are also very good and this is the default algorithm used by MPIR.
-
- at cindex Linear congruential algorithm
-Linear congruential generators are described in many text books, for instance
-Knuth volume 2 (@pxref{References}). With a modulus @math{M} and parameters
- at math{A} and @math{C}, a integer state @math{S} is iterated by the formula
- at math{S @leftarrow{} A at GMPmultiply{}S+C @bmod{} M}. At each step the new
-state is a linear function of the previous, mod @math{M}, hence the name of
-the generator.
-
-In MPIR only moduli of the form @math{2^N} are supported, and the current
-implementation is not as well optimized as it could be. Overheads are
-significant when @math{N} is small, and when @math{N} is large clearly the
-multiply at each step will become slow. This is not a big concern, since the
-Mersenne Twister generator is better in every respect and is therefore
-recommended for all normal applications.
-
-For both generators the current state can be deduced by observing enough
-output and applying some linear algebra (over GF(2) in the case of the
-Mersenne Twister). This generally means raw output is unsuitable for
-cryptographic applications without further hashing or the like.
-
-
- at node Assembler Coding, , Other Algorithms, Algorithms
- at section Assembler Coding
- at cindex Assembler coding
-
-The assembler subroutines in MPIR are the most significant source of speed at
-small to moderate sizes. At larger sizes algorithm selection becomes more
-important, but of course speedups in low level routines will still speed up
-everything proportionally.
-
-Carry handling and widening multiplies that are important for MPIR can't be
-easily expressed in C at . GCC @code{asm} blocks help a lot and are provided in
- at file{longlong.h}, but hand coding low level routines invariably offers a
-speedup over generic C by a factor of anything from 2 to 10.
-
- at menu
-* Assembler Code Organisation::
-* Assembler Basics::
-* Assembler Carry Propagation::
-* Assembler Cache Handling::
-* Assembler Functional Units::
-* Assembler Floating Point::
-* Assembler SIMD Instructions::
-* Assembler Software Pipelining::
-* Assembler Loop Unrolling::
-* Assembler Writing Guide::
- at end menu
-
-
- at node Assembler Code Organisation, Assembler Basics, Assembler Coding, Assembler Coding
- at subsection Code Organisation
- at cindex Assembler code organisation
- at cindex Code organisation
-
-The various @file{mpn} subdirectories contain machine-dependent code, written
-in C or assembler. The @file{mpn/generic} subdirectory contains default code,
-used when there's no machine-specific version of a particular file.
-
-Each @file{mpn} subdirectory is for an ISA family. Generally 32-bit and
-64-bit variants in a family cannot share code and have separate directories.
-Within a family further subdirectories may exist for CPU variants.
-
-In each directory a @file{nails} subdirectory may exist, holding code with
-nails support for that CPU variant. A @code{NAILS_SUPPORT} directive in each
-file indicates the nails values the code handles. Nails code only exists
-where it's faster, or promises to be faster, than plain code. There's no
-effort put into nails if they're not going to enhance a given CPU.
-
-
- at node Assembler Basics, Assembler Carry Propagation, Assembler Code Organisation, Assembler Coding
- at subsection Assembler Basics
-
- at code{mpn_addmul_1} and @code{mpn_submul_1} are the most important routines
-for overall MPIR performance. All multiplications and divisions come down to
-repeated calls to these. @code{mpn_add_n}, @code{mpn_sub_n},
- at code{mpn_lshift} and @code{mpn_rshift} are next most important.
-
-On some CPUs assembler versions of the internal functions
- at code{mpn_mul_basecase} and @code{mpn_sqr_basecase} give significant speedups,
-mainly through avoiding function call overheads. They can also potentially
-make better use of a wide superscalar processor, as can bigger primitives like
- at code{mpn_addmul_2} or @code{mpn_addmul_4}.
-
-The restrictions on overlaps between sources and destinations
-(@pxref{Low-level Functions}) are designed to facilitate a variety of
-implementations. For example, knowing @code{mpn_add_n} won't have partly
-overlapping sources and destination means reading can be done far ahead of
-writing on superscalar processors, and loops can be vectorized on a vector
-processor, depending on the carry handling.
-
-
- at node Assembler Carry Propagation, Assembler Cache Handling, Assembler Basics, Assembler Coding
- at subsection Carry Propagation
- at cindex Assembler carry propagation
-
-The problem that presents most challenges in MPIR is propagating carries from
-one limb to the next. In functions like @code{mpn_addmul_1} and
- at code{mpn_add_n}, carries are the only dependencies between limb operations.
-
-On processors with carry flags, a straightforward CISC style @code{adc} is
-generally best. AMD K6 @code{mpn_addmul_1} however is an example of an
-unusual set of circumstances where a branch works out better.
-
-On RISC processors generally an add and compare for overflow is used. This
-sort of thing can be seen in @file{mpn/generic/aors_n.c}. Some carry
-propagation schemes require 4 instructions, meaning at least 4 cycles per
-limb, but other schemes may use just 1 or 2. On wide superscalar processors
-performance may be completely determined by the number of dependent
-instructions between carry-in and carry-out for each limb.
-
-On vector processors good use can be made of the fact that a carry bit only
-very rarely propagates more than one limb. When adding a single bit to a
-limb, there's only a carry out if that limb was @code{0xFF at dots{}FF} which on
-random data will be only 1 in @m{2\GMPraise{@code{mp\_bits\_per\_limb}},
-2^mp_bits_per_limb}. @file{mpn/cray/add_n.c} is an example of this, it adds
-all limbs in parallel, adds one set of carry bits in parallel and then only
-rarely needs to fall through to a loop propagating further carries.
-
-On the x86s, GCC (as of version 2.95.2) doesn't generate particularly good code
-for the RISC style idioms that are necessary to handle carry bits in
-C at . Often conditional jumps are generated where @code{adc} or @code{sbb} forms
-would be better. And so unfortunately almost any loop involving carry bits
-needs to be coded in assembler for best results.
-
-
- at node Assembler Cache Handling, Assembler Functional Units, Assembler Carry Propagation, Assembler Coding
- at subsection Cache Handling
- at cindex Assembler cache handling
-
-MPIR aims to perform well both on operands that fit entirely in L1 cache and
-those which don't.
-
-Basic routines like @code{mpn_add_n} or @code{mpn_lshift} are often used on
-large operands, so L2 and main memory performance is important for them.
- at code{mpn_mul_1} and @code{mpn_addmul_1} are mostly used for multiply and
-square basecases, so L1 performance matters most for them, unless assembler
-versions of @code{mpn_mul_basecase} and @code{mpn_sqr_basecase} exist, in
-which case the remaining uses are mostly for larger operands.
-
-For L2 or main memory operands, memory access times will almost certainly be
-more than the calculation time. The aim therefore is to maximize memory
-throughput, by starting a load of the next cache line while processing the
-contents of the previous one. Clearly this is only possible if the chip has a
-lock-up free cache or some sort of prefetch instruction. Most current chips
-have both these features.
-
-Prefetching sources combines well with loop unrolling, since a prefetch can be
-initiated once per unrolled loop (or more than once if the loop covers more
-than one cache line).
-
-On CPUs without write-allocate caches, prefetching destinations will ensure
-individual stores don't go further down the cache hierarchy, limiting
-bandwidth. Of course for calculations which are slow anyway, like
- at code{mpn_divrem_1}, write-throughs might be fine.
-
-The distance ahead to prefetch will be determined by memory latency versus
-throughput. The aim of course is to have data arriving continuously, at peak
-throughput. Some CPUs have limits on the number of fetches or prefetches in
-progress.
-
-If a special prefetch instruction doesn't exist then a plain load can be used,
-but in that case care must be taken not to attempt to read past the end of an
-operand, since that might produce a segmentation violation.
-
-Some CPUs or systems have hardware that detects sequential memory accesses and
-initiates suitable cache movements automatically, making life easy.
-
-
- at node Assembler Functional Units, Assembler Floating Point, Assembler Cache Handling, Assembler Coding
- at subsection Functional Units
-
-When choosing an approach for an assembler loop, consideration is given to
-what operations can execute simultaneously and what throughput can thereby be
-achieved. In some cases an algorithm can be tweaked to accommodate available
-resources.
-
-Loop control will generally require a counter and pointer updates, costing as
-much as 5 instructions, plus any delays a branch introduces. CPU addressing
-modes might reduce pointer updates, perhaps by allowing just one updating
-pointer and others expressed as offsets from it, or on CISC chips with all
-addressing done with the loop counter as a scaled index.
-
-The final loop control cost can be amortised by processing several limbs in
-each iteration (@pxref{Assembler Loop Unrolling}). This at least ensures loop
-control isn't a big fraction the work done.
-
-Memory throughput is always a limit. If perhaps only one load or one store
-can be done per cycle then 3 cycles/limb will the top speed for ``binary''
-operations like @code{mpn_add_n}, and any code achieving that is optimal.
-
-Integer resources can be freed up by having the loop counter in a float
-register, or by pressing the float units into use for some multiplying,
-perhaps doing every second limb on the float side (@pxref{Assembler Floating
-Point}).
-
-Float resources can be freed up by doing carry propagation on the integer
-side, or even by doing integer to float conversions in integers using bit
-twiddling.
-
-
- at node Assembler Floating Point, Assembler SIMD Instructions, Assembler Functional Units, Assembler Coding
- at subsection Floating Point
- at cindex Assembler floating Point
-
-Floating point arithmetic is used in MPIR for multiplications on CPUs with poor
-integer multipliers. It's mostly useful for @code{mpn_mul_1},
- at code{mpn_addmul_1} and @code{mpn_submul_1} on 64-bit machines, and
- at code{mpn_mul_basecase} on both 32-bit and 64-bit machines.
-
-With IEEE 53-bit double precision floats, integer multiplications producing up
-to 53 bits will give exact results. Breaking a 64 at cross{}64 multiplication
-into eight 16 at cross{}@math{32 at rightarrow{}48} bit pieces is convenient. With
-some care though six 21 at cross{}@math{32 at rightarrow{}53} bit products can be
-used, if one of the lower two 21-bit pieces also uses the sign bit.
-
-For the @code{mpn_mul_1} family of functions on a 64-bit machine, the
-invariant single limb is split at the start, into 3 or 4 pieces. Inside the
-loop, the bignum operand is split into 32-bit pieces. Fast conversion of
-these unsigned 32-bit pieces to floating point is highly machine-dependent.
-In some cases, reading the data into the integer unit, zero-extending to
-64-bits, then transferring to the floating point unit back via memory is the
-only option.
-
-Converting partial products back to 64-bit limbs is usually best done as a
-signed conversion. Since all values are smaller than @m{2^{53},2^53}, signed
-and unsigned are the same, but most processors lack unsigned conversions.
-
- at sp 2
-
-Here is a diagram showing 16 at cross{}32 bit products for an @code{mpn_mul_1} or
- at code{mpn_addmul_1} with a 64-bit limb. The single limb operand V is split
-into four 16-bit parts. The multi-limb operand U is split in the loop into
-two 32-bit parts.
-
- at tex
-\global\newdimen\GMPbits \global\GMPbits=0.18em
-\def\GMPbox#1#2#3{%
- \hbox{%
- \hbox to 128\GMPbits{\hfil
- \vbox{%
- \hrule
- \hbox to 48\GMPbits {\GMPvrule \hfil$#2$\hfil \vrule}%
- \hrule}%
- \hskip #1\GMPbits}%
- \raise \GMPboxdepth \hbox{\hskip 2em #3}}}
-%
-\GMPdisplay{%
- \vbox{%
- \hbox{%
- \hbox to 128\GMPbits {\hfil
- \vbox{%
- \hrule
- \hbox to 64\GMPbits{%
- \GMPvrule \hfil$v48$\hfil
- \vrule \hfil$v32$\hfil
- \vrule \hfil$v16$\hfil
- \vrule \hfil$v00$\hfil
- \vrule}
- \hrule}}%
- \raise \GMPboxdepth \hbox{\hskip 2em V Operand}}
- \vskip 0.5ex
- \hbox{%
- \hbox to 128\GMPbits {\hfil
- \raise \GMPboxdepth \hbox{$\times$\hskip 1.5em}%
- \vbox{%
- \hrule
- \hbox to 64\GMPbits {%
- \GMPvrule \hfil$u32$\hfil
- \vrule \hfil$u00$\hfil
- \vrule}%
- \hrule}}%
- \raise \GMPboxdepth \hbox{\hskip 2em U Operand (one limb)}}%
- \vskip 0.5ex
- \hbox{\vbox to 2ex{\hrule width 128\GMPbits}}%
- \GMPbox{0}{u00 \times v00}{$p00$\hskip 1.5em 48-bit products}%
- \vskip 0.5ex
- \GMPbox{16}{u00 \times v16}{$p16$}
- \vskip 0.5ex
- \GMPbox{32}{u00 \times v32}{$p32$}
- \vskip 0.5ex
- \GMPbox{48}{u00 \times v48}{$p48$}
- \vskip 0.5ex
- \GMPbox{32}{u32 \times v00}{$r32$}
- \vskip 0.5ex
- \GMPbox{48}{u32 \times v16}{$r48$}
- \vskip 0.5ex
- \GMPbox{64}{u32 \times v32}{$r64$}
- \vskip 0.5ex
- \GMPbox{80}{u32 \times v48}{$r80$}
-}}
- at end tex
- at ifnottex
- at example
- at group
- +---+---+---+---+
- |v48|v32|v16|v00| V operand
- +---+---+---+---+
-
- +-------+---+---+
- x | u32 | u00 | U operand (one limb)
- +---------------+
-
----------------------------------
-
- +-----------+
- | u00 x v00 | p00 48-bit products
- +-----------+
- +-----------+
- | u00 x v16 | p16
- +-----------+
- +-----------+
- | u00 x v32 | p32
- +-----------+
- +-----------+
- | u00 x v48 | p48
- +-----------+
- +-----------+
- | u32 x v00 | r32
- +-----------+
- +-----------+
- | u32 x v16 | r48
- +-----------+
- +-----------+
- | u32 x v32 | r64
- +-----------+
-+-----------+
-| u32 x v48 | r80
-+-----------+
- at end group
- at end example
- at end ifnottex
-
- at math{p32} and @math{r32} can be summed using floating-point addition, and
-likewise @math{p48} and @math{r48}. @math{p00} and @math{p16} can be summed
-with @math{r64} and @math{r80} from the previous iteration.
-
-For each loop then, four 49-bit quantities are transfered to the integer unit,
-aligned as follows,
-
- at tex
-% GMPbox here should be 49 bits wide, but use 51 to better show p16+r80'
-% crossing into the upper 64 bits.
-\def\GMPbox#1#2#3{%
- \hbox{%
- \hbox to 128\GMPbits {%
- \hfil
- \vbox{%
- \hrule
- \hbox to 51\GMPbits {\GMPvrule \hfil$#2$\hfil \vrule}%
- \hrule}%
- \hskip #1\GMPbits}%
- \raise \GMPboxdepth \hbox{\hskip 1.5em $#3$\hfil}%
-}}
-\newbox\b \setbox\b\hbox{64 bits}%
-\newdimen\bw \bw=\wd\b \advance\bw by 2em
-\newdimen\x \x=128\GMPbits
-\advance\x by -2\bw
-\divide\x by4
-\GMPdisplay{%
- \vbox{%
- \hbox to 128\GMPbits {%
- \GMPvrule
- \raise 0.5ex \vbox{\hrule \hbox to \x {}}%
- \hfil 64 bits\hfil
- \raise 0.5ex \vbox{\hrule \hbox to \x {}}%
- \vrule
- \raise 0.5ex \vbox{\hrule \hbox to \x {}}%
- \hfil 64 bits\hfil
- \raise 0.5ex \vbox{\hrule \hbox to \x {}}%
- \vrule}%
- \vskip 0.7ex
- \GMPbox{0}{p00+r64'}{i00}
- \vskip 0.5ex
- \GMPbox{16}{p16+r80'}{i16}
- \vskip 0.5ex
- \GMPbox{32}{p32+r32}{i32}
- \vskip 0.5ex
- \GMPbox{48}{p48+r48}{i48}
-}}
- at end tex
- at ifnottex
- at example
- at group
-|-----64bits----|-----64bits----|
- +------------+
- | p00 + r64' | i00
- +------------+
- +------------+
- | p16 + r80' | i16
- +------------+
- +------------+
- | p32 + r32 | i32
- +------------+
- +------------+
- | p48 + r48 | i48
- +------------+
- at end group
- at end example
- at end ifnottex
-
-The challenge then is to sum these efficiently and add in a carry limb,
-generating a low 64-bit result limb and a high 33-bit carry limb (@math{i48}
-extends 33 bits into the high half).
-
-
- at node Assembler SIMD Instructions, Assembler Software Pipelining, Assembler Floating Point, Assembler Coding
- at subsection SIMD Instructions
- at cindex Assembler SIMD
-
-The single-instruction multiple-data support in current microprocessors is
-aimed at signal processing algorithms where each data point can be treated
-more or less independently. There's generally not much support for
-propagating the sort of carries that arise in MPIR.
-
-SIMD multiplications of say four 16 at cross{}16 bit multiplies only do as much
-work as one 32 at cross{}32 from MPIR's point of view, and need some shifts and
-adds besides. But of course if say the SIMD form is fully pipelined and uses
-less instruction decoding then it may still be worthwhile.
-
-On the x86 chips, MMX has so far found a use in @code{mpn_rshift} and
- at code{mpn_lshift}, and is used in a special case for 16-bit multipliers in the
-P55 @code{mpn_mul_1}. SSE2 is used for Pentium 4 @code{mpn_mul_1},
- at code{mpn_addmul_1}, and @code{mpn_submul_1}.
-
-
- at node Assembler Software Pipelining, Assembler Loop Unrolling, Assembler SIMD Instructions, Assembler Coding
- at subsection Software Pipelining
- at cindex Assembler software pipelining
-
-Software pipelining consists of scheduling instructions around the branch
-point in a loop. For example a loop might issue a load not for use in the
-present iteration but the next, thereby allowing extra cycles for the data to
-arrive from memory.
-
-Naturally this is wanted only when doing things like loads or multiplies that
-take several cycles to complete, and only where a CPU has multiple functional
-units so that other work can be done in the meantime.
-
-A pipeline with several stages will have a data value in progress at each
-stage and each loop iteration moves them along one stage. This is like
-juggling.
-
-If the latency of some instruction is greater than the loop time then it will
-be necessary to unroll, so one register has a result ready to use while
-another (or multiple others) are still in progress. (@pxref{Assembler Loop
-Unrolling}).
-
-
- at node Assembler Loop Unrolling, Assembler Writing Guide, Assembler Software Pipelining, Assembler Coding
- at subsection Loop Unrolling
- at cindex Assembler loop unrolling
-
-Loop unrolling consists of replicating code so that several limbs are
-processed in each loop. At a minimum this reduces loop overheads by a
-corresponding factor, but it can also allow better register usage, for example
-alternately using one register combination and then another. Judicious use of
- at command{m4} macros can help avoid lots of duplication in the source code.
-
-Any amount of unrolling can be handled with a loop counter that's decremented
-by @math{N} each time, stopping when the remaining count is less than the
-further @math{N} the loop will process. Or by subtracting @math{N} at the
-start, the termination condition becomes when the counter @math{C} is less
-than 0 (and the count of remaining limbs is @math{C+N}).
-
-Alternately for a power of 2 unroll the loop count and remainder can be
-established with a shift and mask. This is convenient if also making a
-computed jump into the middle of a large loop.
-
-The limbs not a multiple of the unrolling can be handled in various ways, for
-example
-
- at itemize @bullet
- at item
-A simple loop at the end (or the start) to process the excess. Care will be
-wanted that it isn't too much slower than the unrolled part.
-
- at item
-A set of binary tests, for example after an 8-limb unrolling, test for 4 more
-limbs to process, then a further 2 more or not, and finally 1 more or not.
-This will probably take more code space than a simple loop.
-
- at item
-A @code{switch} statement, providing separate code for each possible excess,
-for example an 8-limb unrolling would have separate code for 0 remaining, 1
-remaining, etc, up to 7 remaining. This might take a lot of code, but may be
-the best way to optimize all cases in combination with a deep pipelined loop.
-
- at item
-A computed jump into the middle of the loop, thus making the first iteration
-handle the excess. This should make times smoothly increase with size, which
-is attractive, but setups for the jump and adjustments for pointers can be
-tricky and could become quite difficult in combination with deep pipelining.
- at end itemize
-
-
- at node Assembler Writing Guide, , Assembler Loop Unrolling, Assembler Coding
- at subsection Writing Guide
- at cindex Assembler writing guide
-
-This is a guide to writing software pipelined loops for processing limb
-vectors in assembler.
-
-First determine the algorithm and which instructions are needed. Code it
-without unrolling or scheduling, to make sure it works. On a 3-operand CPU
-try to write each new value to a new register, this will greatly simplify later
-steps.
-
-Then note for each instruction the functional unit and/or issue port
-requirements. If an instruction can use either of two units, like U0 or U1
-then make a category ``U0/U1''. Count the total using each unit (or combined
-unit), and count all instructions.
-
-Figure out from those counts the best possible loop time. The goal will be to
-find a perfect schedule where instruction latencies are completely hidden.
-The total instruction count might be the limiting factor, or perhaps a
-particular functional unit. It might be possible to tweak the instructions to
-help the limiting factor.
-
-Suppose the loop time is @math{N}, then make @math{N} issue buckets, with the
-final loop branch at the end of the last. Now fill the buckets with dummy
-instructions using the functional units desired. Run this to make sure the
-intended speed is reached.
-
-Now replace the dummy instructions with the real instructions from the slow
-but correct loop you started with. The first will typically be a load
-instruction. Then the instruction using that value is placed in a bucket an
-appropriate distance down. Run the loop again, to check it still runs at
-target speed.
-
-Keep placing instructions, frequently measuring the loop. After a few you
-will need to wrap around from the last bucket back to the top of the loop. If
-you used the new-register for new-value strategy above then there will be no
-register conflicts. If not then take care not to clobber something already in
-use. Changing registers at this time is very error prone.
-
-The loop will overlap two or more of the original loop iterations, and the
-computation of one vector element result will be started in one iteration of
-the new loop, and completed one or several iterations later.
-
-The final step is to create feed-in and wind-down code for the loop. A good
-way to do this is to make a copy (or copies) of the loop at the start and
-delete those instructions which don't have valid antecedents, and at the end
-replicate and delete those whose results are unwanted (including any further
-loads).
-
-The loop will have a minimum number of limbs loaded and processed, so the
-feed-in code must test if the request size is smaller and skip either to a
-suitable part of the wind-down or to special code for small sizes.
-
-
- at node Internals, Contributors, Algorithms, Top
- at chapter Internals
- at cindex Internals
-
- at strong{This chapter is provided only for informational purposes and the
-various internals described here may change in future MPIR releases.
-Applications expecting to be compatible with future releases should use only
-the documented interfaces described in previous chapters.}
-
- at menu
-* Integer Internals::
-* Rational Internals::
-* Float Internals::
-* Raw Output Internals::
-* C++ Interface Internals::
- at end menu
-
- at node Integer Internals, Rational Internals, Internals, Internals
- at section Integer Internals
- at cindex Integer internals
-
- at code{mpz_t} variables represent integers using sign and magnitude, in space
-dynamically allocated and reallocated. The fields are as follows.
-
- at table @asis
- at item @code{_mp_size}
-The number of limbs, or the negative of that when representing a negative
-integer. Zero is represented by @code{_mp_size} set to zero, in which case
-the @code{_mp_d} data is unused.
-
- at item @code{_mp_d}
-A pointer to an array of limbs which is the magnitude. These are stored
-``little endian'' as per the @code{mpn} functions, so @code{_mp_d[0]} is the
-least significant limb and @code{_mp_d[ABS(_mp_size)-1]} is the most
-significant. Whenever @code{_mp_size} is non-zero, the most significant limb
-is non-zero.
-
-Currently there's always at least one limb allocated, so for instance
- at code{mpz_set_ui} never needs to reallocate, and @code{mpz_get_ui} can fetch
- at code{_mp_d[0]} unconditionally (though its value is then only wanted if
- at code{_mp_size} is non-zero).
-
- at item @code{_mp_alloc}
- at code{_mp_alloc} is the number of limbs currently allocated at @code{_mp_d},
-and naturally @code{_mp_alloc >= ABS(_mp_size)}. When an @code{mpz} routine
-is about to (or might be about to) increase @code{_mp_size}, it checks
- at code{_mp_alloc} to see whether there's enough space, and reallocates if not.
- at code{MPZ_REALLOC} is generally used for this.
- at end table
-
-The various bitwise logical functions like @code{mpz_and} behave as if
-negative values were twos complement. But sign and magnitude is always used
-internally, and necessary adjustments are made during the calculations.
-Sometimes this isn't pretty, but sign and magnitude are best for other
-routines.
-
-Some internal temporary variables are setup with @code{MPZ_TMP_INIT} and these
-have @code{_mp_d} space obtained from @code{TMP_ALLOC} rather than the memory
-allocation functions. Care is taken to ensure that these are big enough that
-no reallocation is necessary (since it would have unpredictable consequences).
-
- at code{_mp_size} and @code{_mp_alloc} are @code{int}, although @code{mp_size_t}
-is usually a @code{long}. This is done to make the fields just 32 bits on
-some 64 bits systems, thereby saving a few bytes of data space but still
-providing plenty of range.
-
-
- at node Rational Internals, Float Internals, Integer Internals, Internals
- at section Rational Internals
- at cindex Rational internals
-
- at code{mpq_t} variables represent rationals using an @code{mpz_t} numerator and
-denominator (@pxref{Integer Internals}).
-
-The canonical form adopted is denominator positive (and non-zero), no common
-factors between numerator and denominator, and zero uniquely represented as
-0/1.
-
-It's believed that casting out common factors at each stage of a calculation
-is best in general. A GCD is an @math{O(N^2)} operation so it's better to do
-a few small ones immediately than to delay and have to do a big one later.
-Knowing the numerator and denominator have no common factors can be used for
-example in @code{mpq_mul} to make only two cross GCDs necessary, not four.
-
-This general approach to common factors is badly sub-optimal in the presence
-of simple factorizations or little prospect for cancellation, but MPIR has no
-way to know when this will occur. As per @ref{Efficiency}, that's left to
-applications. The @code{mpq_t} framework might still suit, with
- at code{mpq_numref} and @code{mpq_denref} for direct access to the numerator and
-denominator, or of course @code{mpz_t} variables can be used directly.
-
-
- at node Float Internals, Raw Output Internals, Rational Internals, Internals
- at section Float Internals
- at cindex Float internals
-
-Efficient calculation is the primary aim of MPIR floats and the use of whole
-limbs and simple rounding facilitates this.
-
- at code{mpf_t} floats have a variable precision mantissa and a single machine
-word signed exponent. The mantissa is represented using sign and magnitude.
-
- at c FIXME: The arrow heads don't join to the lines exactly.
- at tex
-\global\newdimen\GMPboxwidth \GMPboxwidth=5em
-\global\newdimen\GMPboxheight \GMPboxheight=3ex
-\def\centreline{\hbox{\raise 0.8ex \vbox{\hrule \hbox{\hfil}}}}
-\GMPdisplay{%
-\vbox{%
- \hbox to 5\GMPboxwidth {most significant limb \hfil least significant limb}
- \vskip 0.7ex
- \def\GMPcentreline#1{\hbox{\raise 0.5 ex \vbox{\hrule \hbox to #1 {}}}}
- \hbox {
- \hbox to 3\GMPboxwidth {%
- \setbox 0 = \hbox{@code{\_mp\_exp}}%
- \dimen0=3\GMPboxwidth
- \advance\dimen0 by -\wd0
- \divide\dimen0 by 2
- \advance\dimen0 by -1em
- \setbox1 = \hbox{$\rightarrow$}%
- \dimen1=\dimen0
- \advance\dimen1 by -\wd1
- \GMPcentreline{\dimen0}%
- \hfil
- \box0%
- \hfil
- \GMPcentreline{\dimen1{}}%
- \box1}
- \hbox to 2\GMPboxwidth {\hfil @code{\_mp\_d}}}
- \vskip 0.5ex
- \vbox {%
- \hrule
- \hbox{%
- \vrule height 2ex depth 1ex
- \hbox to \GMPboxwidth {}%
- \vrule
- \hbox to \GMPboxwidth {}%
- \vrule
- \hbox to \GMPboxwidth {}%
- \vrule
- \hbox to \GMPboxwidth {}%
- \vrule
- \hbox to \GMPboxwidth {}%
- \vrule}
- \hrule
- }
- \hbox {%
- \hbox to 0.8 pt {}
- \hbox to 3\GMPboxwidth {%
- \hfil $\cdot$} \hbox {$\leftarrow$ radix point\hfil}}
- \hbox to 5\GMPboxwidth{%
- \setbox 0 = \hbox{@code{\_mp\_size}}%
- \dimen0 = 5\GMPboxwidth
- \advance\dimen0 by -\wd0
- \divide\dimen0 by 2
- \advance\dimen0 by -1em
- \dimen1 = \dimen0
- \setbox1 = \hbox{$\leftarrow$}%
- \setbox2 = \hbox{$\rightarrow$}%
- \advance\dimen0 by -\wd1
- \advance\dimen1 by -\wd2
- \hbox to 0.3 em {}%
- \box1
- \GMPcentreline{\dimen0}%
- \hfil
- \box0
- \hfil
- \GMPcentreline{\dimen1}%
- \box2}
-}}
- at end tex
- at ifnottex
- at example
- most least
-significant significant
- limb limb
-
- _mp_d
- |---- _mp_exp ---> |
- _____ _____ _____ _____ _____
- |_____|_____|_____|_____|_____|
- . <------------ radix point
-
- <-------- _mp_size --------->
- at sp 1
- at end example
- at end ifnottex
-
- at noindent
-The fields are as follows.
-
- at table @asis
- at item @code{_mp_size}
-The number of limbs currently in use, or the negative of that when
-representing a negative value. Zero is represented by @code{_mp_size} and
- at code{_mp_exp} both set to zero, and in that case the @code{_mp_d} data is
-unused. (In the future @code{_mp_exp} might be undefined when representing
-zero.)
-
- at item @code{_mp_prec}
-The precision of the mantissa, in limbs. In any calculation the aim is to
-produce @code{_mp_prec} limbs of result (the most significant being non-zero).
-
- at item @code{_mp_d}
-A pointer to the array of limbs which is the absolute value of the mantissa.
-These are stored ``little endian'' as per the @code{mpn} functions, so
- at code{_mp_d[0]} is the least significant limb and
- at code{_mp_d[ABS(_mp_size)-1]} the most significant.
-
-The most significant limb is always non-zero, but there are no other
-restrictions on its value, in particular the highest 1 bit can be anywhere
-within the limb.
-
- at code{_mp_prec+1} limbs are allocated to @code{_mp_d}, the extra limb being
-for convenience (see below). There are no reallocations during a calculation,
-only in a change of precision with @code{mpf_set_prec}.
-
- at item @code{_mp_exp}
-The exponent, in limbs, determining the location of the implied radix point.
-Zero means the radix point is just above the most significant limb. Positive
-values mean a radix point offset towards the lower limbs and hence a value
- at math{@ge{} 1}, as for example in the diagram above. Negative exponents mean
-a radix point further above the highest limb.
-
-Naturally the exponent can be any value, it doesn't have to fall within the
-limbs as the diagram shows, it can be a long way above or a long way below.
-Limbs other than those included in the @code{@{_mp_d,_mp_size@}} data
-are treated as zero.
- at end table
-
- at code{_mp_size} and @code{_mp_prec} are @code{int}, although @code{mp_size_t}
-is usually a @code{long}. This is done to make the fields just 32 bits on
-some 64 bits systems, thereby saving a few bytes of data space but still
-providing plenty of range.
-
-
- at sp 1
- at noindent
-The following various points should be noted.
-
- at table @asis
- at item Low Zeros
-The least significant limbs @code{_mp_d[0]} etc can be zero, though such low
-zeros can always be ignored. Routines likely to produce low zeros check and
-avoid them to save time in subsequent calculations, but for most routines
-they're quite unlikely and aren't checked.
-
- at item Mantissa Size Range
-The @code{_mp_size} count of limbs in use can be less than @code{_mp_prec} if
-the value can be represented in less. This means low precision values or
-small integers stored in a high precision @code{mpf_t} can still be operated
-on efficiently.
-
- at code{_mp_size} can also be greater than @code{_mp_prec}. Firstly a value is
-allowed to use all of the @code{_mp_prec+1} limbs available at @code{_mp_d},
-and secondly when @code{mpf_set_prec_raw} lowers @code{_mp_prec} it leaves
- at code{_mp_size} unchanged and so the size can be arbitrarily bigger than
- at code{_mp_prec}.
-
- at item Rounding
-All rounding is done on limb boundaries. Calculating @code{_mp_prec} limbs
-with the high non-zero will ensure the application requested minimum precision
-is obtained.
-
-The use of simple ``trunc'' rounding towards zero is efficient, since there's
-no need to examine extra limbs and increment or decrement.
-
- at item Bit Shifts
-Since the exponent is in limbs, there are no bit shifts in basic operations
-like @code{mpf_add} and @code{mpf_mul}. When differing exponents are
-encountered all that's needed is to adjust pointers to line up the relevant
-limbs.
-
-Of course @code{mpf_mul_2exp} and @code{mpf_div_2exp} will require bit shifts,
-but the choice is between an exponent in limbs which requires shifts there, or
-one in bits which requires them almost everywhere else.
-
- at item Use of @code{_mp_prec+1} Limbs
-The extra limb on @code{_mp_d} (@code{_mp_prec+1} rather than just
- at code{_mp_prec}) helps when an @code{mpf} routine might get a carry from its
-operation. @code{mpf_add} for instance will do an @code{mpn_add} of
- at code{_mp_prec} limbs. If there's no carry then that's the result, but if
-there is a carry then it's stored in the extra limb of space and
- at code{_mp_size} becomes @code{_mp_prec+1}.
-
-Whenever @code{_mp_prec+1} limbs are held in a variable, the low limb is not
-needed for the intended precision, only the @code{_mp_prec} high limbs. But
-zeroing it out or moving the rest down is unnecessary. Subsequent routines
-reading the value will simply take the high limbs they need, and this will be
- at code{_mp_prec} if their target has that same precision. This is no more than
-a pointer adjustment, and must be checked anyway since the destination
-precision can be different from the sources.
-
-Copy functions like @code{mpf_set} will retain a full @code{_mp_prec+1} limbs
-if available. This ensures that a variable which has @code{_mp_size} equal to
- at code{_mp_prec+1} will get its full exact value copied. Strictly speaking
-this is unnecessary since only @code{_mp_prec} limbs are needed for the
-application's requested precision, but it's considered that an @code{mpf_set}
-from one variable into another of the same precision ought to produce an exact
-copy.
-
- at item Application Precisions
- at code{__GMPF_BITS_TO_PREC} converts an application requested precision to an
- at code{_mp_prec}. The value in bits is rounded up to a whole limb then an
-extra limb is added since the most significant limb of @code{_mp_d} is only
-non-zero and therefore might contain only one bit.
-
- at code{__GMPF_PREC_TO_BITS} does the reverse conversion, and removes the extra
-limb from @code{_mp_prec} before converting to bits. The net effect of
-reading back with @code{mpf_get_prec} is simply the precision rounded up to a
-multiple of @code{mp_bits_per_limb}.
-
-Note that the extra limb added here for the high only being non-zero is in
-addition to the extra limb allocated to @code{_mp_d}. For example with a
-32-bit limb, an application request for 250 bits will be rounded up to 8
-limbs, then an extra added for the high being only non-zero, giving an
- at code{_mp_prec} of 9. @code{_mp_d} then gets 10 limbs allocated. Reading
-back with @code{mpf_get_prec} will take @code{_mp_prec} subtract 1 limb and
-multiply by 32, giving 256 bits.
-
-Strictly speaking, the fact the high limb has at least one bit means that a
-float with, say, 3 limbs of 32-bits each will be holding at least 65 bits, but
-for the purposes of @code{mpf_t} it's considered simply to be 64 bits, a nice
-multiple of the limb size.
- at end table
-
-
- at node Raw Output Internals, C++ Interface Internals, Float Internals, Internals
- at section Raw Output Internals
- at cindex Raw output internals
-
- at noindent
- at code{mpz_out_raw} uses the following format.
-
- at tex
-\global\newdimen\GMPboxwidth \GMPboxwidth=5em
-\global\newdimen\GMPboxheight \GMPboxheight=3ex
-\def\centreline{\hbox{\raise 0.8ex \vbox{\hrule \hbox{\hfil}}}}
-\GMPdisplay{%
-\vbox{%
- \def\GMPcentreline#1{\hbox{\raise 0.5 ex \vbox{\hrule \hbox to #1 {}}}}
- \vbox {%
- \hrule
- \hbox{%
- \vrule height 2.5ex depth 1.5ex
- \hbox to \GMPboxwidth {\hfil size\hfil}%
- \vrule
- \hbox to 3\GMPboxwidth {\hfil data bytes\hfil}%
- \vrule}
- \hrule}
-}}
- at end tex
- at ifnottex
- at example
-+------+------------------------+
-| size | data bytes |
-+------+------------------------+
- at end example
- at end ifnottex
-
-The size is 4 bytes written most significant byte first, being the number of
-subsequent data bytes, or the twos complement negative of that when a negative
-integer is represented. The data bytes are the absolute value of the integer,
-written most significant byte first.
-
-The most significant data byte is always non-zero, so the output is the same
-on all systems, irrespective of limb size.
-
-In GMP 1, leading zero bytes were written to pad the data bytes to a multiple
-of the limb size. @code{mpz_inp_raw} will still accept this, for
-compatibility.
-
-The use of ``big endian'' for both the size and data fields is deliberate, it
-makes the data easy to read in a hex dump of a file. Unfortunately it also
-means that the limb data must be reversed when reading or writing, so neither
-a big endian nor little endian system can just read and write @code{_mp_d}.
-
-
- at node C++ Interface Internals, , Raw Output Internals, Internals
- at section C++ Interface Internals
- at cindex C++ interface internals
-
-A system of expression templates is used to ensure something like @code{a=b+c}
-turns into a simple call to @code{mpz_add} etc. For @code{mpf_class}
-the scheme also ensures the precision of the final
-destination is used for any temporaries within a statement like
- at code{f=w*x+y*z}. These are important features which a naive implementation
-cannot provide.
-
-A simplified description of the scheme follows. The true scheme is
-complicated by the fact that expressions have different return types. For
-detailed information, refer to the source code.
-
-To perform an operation, say, addition, we first define a ``function object''
-evaluating it,
-
- at example
-struct __gmp_binary_plus
-@{
- static void eval(mpf_t f, mpf_t g, mpf_t h) @{ mpf_add(f, g, h); @}
-@};
- at end example
-
- at noindent
-And an ``additive expression'' object,
-
- at example
-__gmp_expr<__gmp_binary_expr<mpf_class, mpf_class, __gmp_binary_plus> >
-operator+(const mpf_class &f, const mpf_class &g)
-@{
- return __gmp_expr
- <__gmp_binary_expr<mpf_class, mpf_class, __gmp_binary_plus> >(f, g);
-@}
- at end example
-
-The seemingly redundant @code{__gmp_expr<__gmp_binary_expr<@dots{}>>} is used to
-encapsulate any possible kind of expression into a single template type. In
-fact even @code{mpf_class} etc are @code{typedef} specializations of
- at code{__gmp_expr}.
-
-Next we define assignment of @code{__gmp_expr} to @code{mpf_class}.
-
- at example
-template <class T>
-mpf_class & mpf_class::operator=(const __gmp_expr<T> &expr)
-@{
- expr.eval(this->get_mpf_t(), this->precision());
- return *this;
-@}
-
-template <class Op>
-void __gmp_expr<__gmp_binary_expr<mpf_class, mpf_class, Op> >::eval
-(mpf_t f, mp_bitcnt_t precision)
-@{
- Op::eval(f, expr.val1.get_mpf_t(), expr.val2.get_mpf_t());
-@}
- at end example
-
-where @code{expr.val1} and @code{expr.val2} are references to the expression's
-operands (here @code{expr} is the @code{__gmp_binary_expr} stored within the
- at code{__gmp_expr}).
-
-This way, the expression is actually evaluated only at the time of assignment,
-when the required precision (that of @code{f}) is known. Furthermore the
-target @code{mpf_t} is now available, thus we can call @code{mpf_add} directly
-with @code{f} as the output argument.
-
-Compound expressions are handled by defining operators taking subexpressions
-as their arguments, like this:
-
- at example
-template <class T, class U>
-__gmp_expr
-<__gmp_binary_expr<__gmp_expr<T>, __gmp_expr<U>, __gmp_binary_plus> >
-operator+(const __gmp_expr<T> &expr1, const __gmp_expr<U> &expr2)
-@{
- return __gmp_expr
- <__gmp_binary_expr<__gmp_expr<T>, __gmp_expr<U>, __gmp_binary_plus> >
- (expr1, expr2);
-@}
- at end example
-
-And the corresponding specializations of @code{__gmp_expr::eval}:
-
- at example
-template <class T, class U, class Op>
-void __gmp_expr
-<__gmp_binary_expr<__gmp_expr<T>, __gmp_expr<U>, Op> >::eval
-(mpf_t f, mp_bitcnt_t precision)
-@{
- // declare two temporaries
- mpf_class temp1(expr.val1, precision), temp2(expr.val2, precision);
- Op::eval(f, temp1.get_mpf_t(), temp2.get_mpf_t());
-@}
- at end example
-
-The expression is thus recursively evaluated to any level of complexity and
-all subexpressions are evaluated to the precision of @code{f}.
-
-
- at node Contributors, References, Internals, Top
- at comment node-name, next, previous, up
- at appendix Contributors
- at cindex Contributors
-
-Torbjorn Granlund wrote the original GMP library and is still developing and
-maintaining it. Several other individuals and organizations have contributed
-to GMP in various ways. Here is a list in chronological order:
-
-Gunnar Sjoedin and Hans Riesel helped with mathematical problems in early
-versions of the library.
-
-Richard Stallman contributed to the interface design and revised the first
-version of this manual.
-
-Brian Beuning and Doug Lea helped with testing of early versions of the
-library and made creative suggestions.
-
-John Amanatides of York University in Canada contributed the function
- at code{mpz_probab_prime_p}.
-
-Paul Zimmermann of Inria sparked the development of GMP 2, with his
-comparisons between bignum packages.
-
-Ken Weber (Kent State University, Universidade Federal do Rio Grande do Sul)
-contributed @code{mpz_gcd}, @code{mpz_divexact}, @code{mpn_gcd}, and
- at code{mpn_bdivmod}, partially supported by CNPq (Brazil) grant 301314194-2.
-
-Per Bothner of Cygnus Support helped to set up GMP to use Cygnus' configure.
-He has also made valuable suggestions and tested numerous intermediary
-releases.
-
-Joachim Hollman was involved in the design of the @code{mpf} interface, and in
-the @code{mpz} design revisions for version 2.
-
-Bennet Yee contributed the initial versions of @code{mpz_jacobi} and
- at code{mpz_legendre}.
-
-Andreas Schwab contributed the files @file{mpn/m68k/lshift.S} and
- at file{mpn/m68k/rshift.S} (now in @file{.asm} form).
-
-The development of floating point functions of GNU MP 2, were supported in part
-by the ESPRIT-BRA (Basic Research Activities) 6846 project POSSO (POlynomial
-System SOlving).
-
-GNU MP 2 was finished and released by SWOX AB, SWEDEN, in cooperation with the
-IDA Center for Computing Sciences, USA.
-
-Robert Harley of Inria, France and David Seal of ARM, England, suggested clever
-improvements for population count.
-
-Robert Harley also wrote highly optimized Karatsuba and 3-way Toom
-multiplication functions for GMP 3. He also contributed the ARM assembly
-code.
-
-Torsten Ekedahl of the Mathematical department of Stockholm University provided
-significant inspiration during several phases of the GMP development. His
-mathematical expertise helped improve several algorithms.
-
-Paul Zimmermann wrote the Divide and Conquer division code, the REDC code, the
-REDC-based mpz_powm code, the FFT multiply code, and the Karatsuba square root
-code. He also rewrote the Toom3 code for GMP 4.2. The ECMNET project Paul is
-organizing was a driving force behind many of the optimizations in GMP 3.
-
-Linus Nordberg wrote the new configure system based on autoconf and
-implemented the new random functions.
-
-Kent Boortz made the Mac OS 9 port.
-
-Kevin Ryde worked on a number of things: optimized x86 code, m4 asm macros,
-parameter tuning, speed measuring, the configure system, function inlining,
-divisibility tests, bit scanning, Jacobi symbols, Fibonacci and Lucas number
-functions, printf and scanf functions, perl interface, demo expression parser,
-the algorithms chapter in the manual, @file{gmpasm-mode.el}, and various
-miscellaneous improvements elsewhere.
-
-Steve Root helped write the optimized alpha 21264 assembly code.
-
-Gerardo Ballabio wrote the @file{gmpxx.h} C++ class interface and the C++
- at code{istream} input routines.
-
-GNU MP 4 was finished and released by Torbjorn Granlund and Kevin Ryde.
-Torbjorn's work was partially funded by the IDA Center for Computing Sciences,
-USA.
-
-Jason Moxham rewrote @code{mpz_fac_ui}.
-
-Pedro Gimeno implemented the Mersenne Twister and made other random number
-improvements.
-
-(This list is chronological, not ordered after significance. If you have
-contributed to GMP/MPIR but are not listed above, please tell
- at uref{http://groups.google.com/group/mpir-devel} about the omission!)
-
-Thanks go to Hans Thorsen for donating an SGI system for the GMP test system
-environment.
-
-In 2008 GMP was forked and gave rise to the MPIR (Multiple Precision Integers
-and Rationals) project. In 2010 version 2.0.0 of MPIR switched to LGPL v3+
-and much code from GMP was again incorporated into MPIR.
-
-The MPIR project has largely been a collaboration of William Hart, Brian
-Gladman and Jason Moxham. MPIR code not obtained from GMP and not specifically
-mentioned elsewhere below is likely written by one of these three.
-
-William Hart did much of the early MPIR coding including build system fixes.
-His contributions also include Toom 4 and 7 code and variants, extended GCD
-based on Niels Mollers ngcd work, asymptotically fast division code. He does
-much of the release management work.
-
-Brian Gladman wrote and maintains MSVC project files. He has also done much of
-the conversion of assembly code to yasm format. He rewrote the benchmark
-program and developed MSVC ports of tune, speed, try and the benchmark code.
-He helped with many aspects of the merging of GMP code into MPIR after the
-switch to LGPL v3+.
-
-Jason Moxham has contributed a great deal of x86 assembly code. He has also
-contributed improved root code and mulhi and mullo routines and implemented
-Peter Montgomery's single limb remainder algorithm. He has also contributed
-a command line build system for Windows and numerous build system fixes.
-
-The following people have either contributed directly to the MPIR project,
-made code available on their websites or contributed code to the official
-GNU project which has been used in MPIR.
-
-Pierrick Gaudry wrote some fast assembly support for AMD 64.
-
-Jason Martin wrote some fast assembly patches for Core 2 and converted them to
-intel format. He also did the initial merge of Niels Moller's fast GCD patches.
-He wrote fast addmul functions for Itanium.
-
-Gonzalo Tornaria helped patch config.guess and associated files to distinguish
-modern processors. He also patched mpirbench.
-
-Michael Abshoff helped resolve some build issues on various platforms. He served for a while as release manager for the MPIR project.
-
-Mariah Lennox contributed patches to mpirbench and various build failure reports. She has also reported gcc bugs found during MPIR development.
-
-Niels Moller wrote the fast ngcd code for computing integer GCD, the quadratic
-Hensel division code and precomputed inverse code for Euclidean division.
-He also made contributions to the Toom multiply code,
-especially helper functions to simplify Toom evaluations.
-
-Pierrick Gaudry provided initial AMD 64 assembly support and revised the FFT code.
-
-Paul Zimmermann provided an mpz implementation of Toom 4, wrote much of the FFT code, wrote some of the rootrem code and contributed invert.c for computing
-precomputed inverses.
-
-Alexander Kruppa revised the FFT code.
-
-Torbjorn Granlund revised the FFT code and wrote a lot of division code,
-including the quadratic Euclidean division code, many parts of the divide
-and conquer division code, both Hensel and Euclidean, and his code was also
-reused for parts of the asymptotically fast division code. He also helped
-write the root code and wrote much of the Itanium assembly code and a couple
-of Core 2 assembly functions and part of the basecase middle product assembly
-code for x86 64 bit. He also wrote the improved string input and output code
-and made improvements to the GCD and extended GCD code. Torbjorn is also
-responsible for numerous other bits and pieces that have been used from
-the GNU project.
-
-Marco Bodrato and Alberto Zanoni suggested the unbalanced multiply strategy
-and found optimal Toom multiplication sequences.
-
-Marco Bodrato wrote an mpz implementation of the Toom 7 code and wrote most of
-the Toom 8.5 multiply and squaring code. He also helped write the divide and conquer Euclidean division code.
-
-Robert Gerbicz contributed fast factorial code.
-
-David Harvey wrote fast middle product code and divide and conquer approximate
-quotient code for both Euclidean and Hensel division and contributed to the
-quadratic Hensel code.
-
-T. R. Nicely wrote primality tests used in the benchmark code.
-
-Jeff Gilchrist assisted with the porting of T. R. Nicely's primality code to MPIR and helped with tuning.
-
-Peter Shrimpton wrote the BPSW primality test used up to GMP_LIMB_BITS.
-
-Thanks to Microsoft for supporting Jason Moxham to work on a command line
-build system for Windows and some assembly improvements for Windows.
-
-Thanks to the Free Software Foundation France for giving us access to their
-build farm.
-
-Thanks to William Stein for giving us access to his sage.math machines for
-testing and for hosting the MPIR website, and for supporting us in inumerably
-many other ways.
-
-Minh Van Nguyen served as release manager for MPIR 2.1.0.
-
-Case Vanhorsen helped with release testing.
-
-David Cleaver filed a bug report.
-
-Julien Puydt provided tuning values.
-
-Leif Lionhardy provided tuning values.
-
-Jean-Pierre Flori provided tuning values.
-
- at node References, GNU Free Documentation License, Contributors, Top
- at comment node-name, next, previous, up
- at appendix References
- at cindex References
-
- at c FIXME: In tex, the @uref's are unhyphenated, which is good for clarity,
- at c but being long words they upset paragraph formatting (the preceding line
- at c can get badly stretched). Would like an conditional @* style line break
- at c if the uref is too long to fit on the last line of the paragraph, but it's
- at c not clear how to do that. For now explicit @texlinebreak{}s are used on
- at c paragraphs that come out bad.
-
- at section Books
-
- at itemize @bullet
- at item
-Jonathan M. Borwein and Peter B. Borwein, ``Pi and the AGM: A Study in
-Analytic Number Theory and Computational Complexity'', Wiley, 1998.
-
- at item
-Henri Cohen, ``A Course in Computational Algebraic Number Theory'', Graduate
-Texts in Mathematics number 138, Springer-Verlag, 1993.
- at texlinebreak{} @uref{http://www.math.u-bordeaux.fr/~cohen/}
-
- at item
-Richard Crandall, Carl Pomerance, ``Prime Numbers: A Computational Perspective'' 2nd edition, Springer, 2005.
-
- at item
-Donald E. Knuth, ``The Art of Computer Programming'', volume 2,
-``Seminumerical Algorithms'', 3rd edition, Addison-Wesley, 1998.
- at texlinebreak{} @uref{http://www-cs-faculty.stanford.edu/~knuth/taocp.html}
-
- at item
-John D. Lipson, ``Elements of Algebra and Algebraic Computing'',
-The Benjamin Cummings Publishing Company Inc, 1981.
-
- at item
-Alfred J. Menezes, Paul C. van Oorschot and Scott A. Vanstone, ``Handbook of
-Applied Cryptography'', @uref{http://www.cacr.math.uwaterloo.ca/hac/}
-
- at item
-Richard M. Stallman, ``Using and Porting GCC'', Free Software Foundation, 1999,
-available online @uref{http://gcc.gnu.org/onlinedocs/}, and in
-the GCC package @uref{ftp://ftp.gnu.org/gnu/gcc/}
- at end itemize
-
- at section Papers
-
- at itemize @bullet
- at item
-Dan Bernstein, ``Detecting perfect powers in essentially linear time'', Math. Comp. (67) pp.@: 1253-1283, 1998.
-
- at item
-Yves Bertot, Nicolas Magaud and Paul Zimmermann, ``A Proof of GMP Square
-Root'', Journal of Automated Reasoning, volume 29, 2002, pp.@: 225-252. Also
-available online as INRIA Research Report 4475, June 2001,
- at uref{http://www.inria.fr/rrrt/rr-4475.html}
-
- at item
-Marco Bodrato, Alberto Zanoni, ``Integer and Polynomial Multiplication: Towards optimal Toom-Cook Matrices'', ISAAC 2007 Proceedings, Ontario, Canada, July 29 - August 1, 2007, ACM Press. Available online at @uref{http://ln.bodrato.it/issac2007_pdf}
-
- at item
-Marco Bodrato, ``High degree Toom`n'half for balanced and unbalanced multiplication'', E. Antelo, D. Hough and P. Ienne, editors, Proceedings of the 20th IEEE Symposium on Computer Arithmetic, IEEE, Tubingen, Germany, July 25-27, 2011, pp. 15--222. See @uref{http://bodrato.it/papers}
-
- at item
-Richard Brent and Paul Zimmermann, ``Modern Computer Arithmetic'',
-version 0.4, November 2009, @uref{http://www.loria.fr/~zimmerma/mca/mca-0.4.pdf}
-
- at item
-Christoph Burnikel and Joachim Ziegler, ``Fast Recursive Division'',
-Max-Planck-Institut fuer Informatik Research Report MPI-I-98-1-022,
- at texlinebreak{} @uref{http://data.mpi-sb.mpg.de/internet/reports.nsf/NumberView/1998-1-022}
-
- at item
-Agner Fog, ``Software optimization resources'', online at @uref{http://www.agner.org/optimize/}
-
- at item
-Pierrick Gaudry, Alexander Kruppa, Paul Zimmermann, ``A GMP-based implementation of Schoenhage-Strassen's large integer multiplication algorithm'', ISAAC 2007 Proceedings, Ontario, Canada, July 29 - August 1, 2007, pp.@: 167-174, ACM Press. Full text available at @uref{http://hal.inria.fr/docs/00/14/86/20/PDF/fft.final.pdf}
-
- at item
-Torbjorn Granlund and Peter L. Montgomery, ``Division by Invariant Integers
-using Multiplication'', in Proceedings of the SIGPLAN PLDI'94 Conference, June
-1994. Also available @uref{ftp://ftp.cwi.nl/pub/pmontgom/divcnst.psa4.gz}
-(and .psl.gz).
-
- at item
-Niels M@"oller and Torbj@"orn Granlund, ``Improved division by invariant
-integers'', to appear.
-
- at item
-Torbj@"orn Granlund and Niels M@"oller, ``Division of integers large and
-small'', to appear.
-
- at item
-David Harvey, ``The Karatsuba middle product for integers'', (preprint), 2009. Available at @uref{http://www.cims.nyu.edu/~harvey/mulmid/mulmid.pdf}
-
- at item
-Tudor Jebelean,
-``An algorithm for exact division'',
-Journal of Symbolic Computation,
-volume 15, 1993, pp.@: 169-180.
-Research report version available @texlinebreak{}
- at uref{ftp://ftp.risc.uni-linz.ac.at/pub/techreports/1992/92-35.ps.gz}
-
- at item
-Tudor Jebelean, ``Exact Division with Karatsuba Complexity - Extended
-Abstract'', RISC-Linz technical report 96-31, @texlinebreak{}
- at uref{ftp://ftp.risc.uni-linz.ac.at/pub/techreports/1996/96-31.ps.gz}
-
- at item
-Tudor Jebelean, ``Practical Integer Division with Karatsuba Complexity'',
-ISSAC 97, pp.@: 339-341. Technical report available @texlinebreak{}
- at uref{ftp://ftp.risc.uni-linz.ac.at/pub/techreports/1996/96-29.ps.gz}
-
- at item
-Tudor Jebelean, ``A Generalization of the Binary GCD Algorithm'', ISSAC 93,
-pp.@: 111-116. Technical report version available @texlinebreak{}
- at uref{ftp://ftp.risc.uni-linz.ac.at/pub/techreports/1993/93-01.ps.gz}
-
- at item
-Tudor Jebelean, ``A Double-Digit Lehmer-Euclid Algorithm for Finding the GCD
-of Long Integers'', Journal of Symbolic Computation, volume 19, 1995,
-pp.@: 145-157. Technical report version also available @texlinebreak{}
- at uref{ftp://ftp.risc.uni-linz.ac.at/pub/techreports/1992/92-69.ps.gz}
-
- at item
-Werner Krandick, Jeremy R. Johnson, ``Efficient Multiprecision Floating Point Multiplication with Exact Rounding'', Technical Report, RISC Linz, 1993, available at @uref{ftp://ftp.risc.uni-linz.ac.at/pub/techreports/1993/93-76.ps.gz}
-
- at item
-Werner Krandick and Tudor Jebelean, ``Bidirectional Exact Integer Division'',
-Journal of Symbolic Computation, volume 21, 1996, pp.@: 441-455. Early
-technical report version also available
- at uref{ftp://ftp.risc.uni-linz.ac.at/pub/techreports/1994/94-50.ps.gz}
-
- at item
-Makoto Matsumoto and Takuji Nishimura, ``Mersenne Twister: A 623-dimensionally
-equidistributed uniform pseudorandom number generator'', ACM Transactions on
-Modelling and Computer Simulation, volume 8, January 1998, pp.@: 3-30.
-Available online @texlinebreak{}
- at uref{http://www.math.keio.ac.jp/~nisimura/random/doc/mt.ps.gz} (or .pdf)
-
- at item
-R. Moenck and A. Borodin, ``Fast Modular Transforms via Division'',
-Proceedings of the 13th Annual IEEE Symposium on Switching and Automata
-Theory, October 1972, pp.@: 90-96. Reprinted as ``Fast Modular Transforms'',
-Journal of Computer and System Sciences, volume 8, number 3, June 1974,
-pp.@: 366-386.
-
- at item
-Niels M@"oller, ``On Schoenhage's algorithm and subquadratic integer GCD computation'', Math. Comp. 2007. Available online at @uref{http://www.lysator.liu.se/~nisse/archive/S0025-5718-07-02017-0.pdf}
-
- at item
-Peter L. Montgomery, ``Modular Multiplication Without Trial Division'', in
-Mathematics of Computation, volume 44, number 170, April 1985.
-
- at item
-Thom Mulders, ``On short multiplications and divisions'', Appl. Algebra Engrg. Comm. Comput. 11 (2000), no. 1, pp.@: 69-88. Tech. report No. 276, Dept. of Comp. Sci., ETH Zurich, Nov 1997, available online at @uref{ftp://ftp.inf.ethz.ch/pub/publications/tech-reports/2xx/276.pdf}
-
- at item
-Arnold Sch@"onhage and Volker Strassen, ``Schnelle Multiplikation grosser
-Zahlen'', Computing 7, 1971, pp.@: 281-292.
-
- at item
-A. Sch@"onhage, A. F. W. Grotefeld and E. Vetter, "Fast Algorithms, A Multitape Turing Machine Implementation" BI Wissenschafts-Verlag, Mannheim, 1994.
-
- at item
-Kenneth Weber, ``The accelerated integer GCD algorithm'',
-ACM Transactions on Mathematical Software,
-volume 21, number 1, March 1995, pp.@: 111-122.
-
- at item
-Paul Zimmermann, ``Karatsuba Square Root'', INRIA Research Report 3805,
-November 1999, @uref{http://www.inria.fr/rrrt/rr-3805.html}
-
- at item
-Paul Zimmermann, ``A Proof of GMP Fast Division and Square Root
-Implementations'', @texlinebreak{}
- at uref{http://www.loria.fr/~zimmerma/papers/proof-div-sqrt.ps.gz}
-
- at item
-Dan Zuras, ``On Squaring and Multiplying Large Integers'', ARITH-11: IEEE
-Symposium on Computer Arithmetic, 1993, pp.@: 260 to 271. Reprinted as ``More
-on Multiplying and Squaring Large Integers'', IEEE Transactions on Computers,
-volume 43, number 8, August 1994, pp.@: 899-908.
- at end itemize
-
-
- at node GNU Free Documentation License, Concept Index, References, Top
- at appendix GNU Free Documentation License
- at cindex GNU Free Documentation License
- at cindex Free Documentation License
- at cindex Documentation license
- at include fdl.texi
-
-
- at node Concept Index, Function Index, GNU Free Documentation License, Top
- at comment node-name, next, previous, up
- at unnumbered Concept Index
- at printindex cp
-
- at node Function Index, , Concept Index, Top
- at comment node-name, next, previous, up
- at unnumbered Function and Type Index
- at printindex fn
-
- at bye
-
- at c Local variables:
- at c fill-column: 78
- at c compile-command: "make mpir.info"
- at c End:
diff --git a/doc/version.texi b/doc/version.texi
deleted file mode 100644
index 352f744..0000000
--- a/doc/version.texi
+++ /dev/null
@@ -1,4 +0,0 @@
- at set UPDATED 8 November 2012
- at set UPDATED-MONTH November 2012
- at set EDITION 2.6.0
- at set VERSION 2.6.0
--
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