-This is ../../gmp/doc/gmp.info, produced by makeinfo version 4.8 from
-../../gmp/doc/gmp.texi.
+This is gmp.info, produced by makeinfo version 6.7 from gmp.texi.
- This manual describes how to install and use the GNU multiple
-precision arithmetic library, version 5.0.1.
+This manual describes how to install and use the GNU multiple precision
+arithmetic library, version 6.2.1.
- Copyright 1991, 1993, 1994, 1995, 1996, 1997, 1998, 1999, 2000,
-2001, 2002, 2003, 2004, 2005, 2006, 2007, 2008, 2009, 2010 Free
-Software Foundation, Inc.
-
- 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::.
+ Copyright 1991, 1993-2016, 2018-2020 Free Software Foundation, Inc.
+ 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
* gmp: (gmp). GNU Multiple Precision Arithmetic Library.
GNU MP
******
- This manual describes how to install and use the GNU multiple
-precision arithmetic library, version 5.0.1.
-
- Copyright 1991, 1993, 1994, 1995, 1996, 1997, 1998, 1999, 2000,
-2001, 2002, 2003, 2004, 2005, 2006, 2007, 2008, 2009, 2010 Free
-Software Foundation, Inc.
+This manual describes how to install and use the GNU multiple precision
+arithmetic library, version 6.2.1.
- 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::.
+ Copyright 1991, 1993-2016, 2018-2020 Free Software Foundation, Inc.
+ 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:
* 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.
+* Formatted Output:: 'printf' style output.
+* Formatted Input:: 'scanf' style input.
* C++ Class Interface:: Class wrappers around GMP types.
-* BSD Compatible Functions:: All functions found in BSD MP.
* Custom Allocation:: How to customize the internal allocation.
* Language Bindings:: Using GMP from other languages.
* Algorithms:: What happens behind the scenes.
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.
+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 GNU MP 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.
+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 GNU MP 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 GNU MP library are
-found in the Lesser General Public License version 3 that accompanies
-the source code, see `COPYING.LIB'. Certain demonstration programs are
-provided under the terms of the plain General Public License version 3,
-see `COPYING'.
+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.
+
+ More precisely, the GNU MP library is dual licensed, under the
+conditions of the GNU Lesser General Public License version 3 (see
+'COPYING.LESSERv3'), or the GNU General Public License version 2 (see
+'COPYINGv2'). This is the recipient's choice, and the recipient also
+has the additional option of applying later versions of these licenses.
+(The reason for this dual licensing is to make it possible to use the
+library with programs which are licensed under GPL version 2, but which
+for historical or other reasons do not allow use under later versions of
+the GPL).
+
+ Programs which are not part of the library itself, such as
+demonstration programs and the GMP testsuite, are licensed under the
+terms of the GNU General Public License version 3 (see 'COPYINGv3'), or
+any later version.
\1f
File: gmp.info, Node: Introduction to GMP, Next: Installing GMP, Prev: Copying, Up: Top
Many applications use just a few hundred bits of precision; but some
applications may need thousands or even millions of bits. GMP 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.
+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 GMP is achieved by using fullwords as the basic
arithmetic type, by using sophisticated algorithms, by including
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 29000, AMD K6, K6-2, Athlon, and Athlon64, Hitachi
-SuperH and SH-2, HPPA 1.0, 1.1 and 2.0, Intel Pentium, Pentium
-Pro/II/III, Pentium 4, generic x86, Intel IA-64, i960, Motorola
-MC68000, MC68020, MC88100, and MC88110, Motorola/IBM PowerPC 32 and 64,
-National NS32000, IBM POWER, MIPS R3000, R4000, SPARCv7, SuperSPARC,
-generic SPARCv8, UltraSPARC, DEC VAX, and Zilog Z8000. Some
-optimizations also for Cray vector systems, Clipper, IBM ROMP (RT), and
-Pyramid AP/XP.
+ There is assembly code for these CPUs: ARM Cortex-A9, Cortex-A15, and
+generic ARM, DEC Alpha 21064, 21164, and 21264, AMD K8 and K10 (sold
+under many brands, e.g. Athlon64, Phenom, Opteron) Bulldozer, and
+Bobcat, Intel Pentium, Pentium Pro/II/III, Pentium 4, Core2, Nehalem,
+Sandy bridge, Haswell, generic x86, Intel IA-64, Motorola/IBM PowerPC 32
+and 64 such as POWER970, POWER5, POWER6, and POWER7, MIPS 32-bit and
+64-bit, SPARC 32-bit ad 64-bit with special support for all UltraSPARC
+models. There is also assembly code for many obsolete CPUs.
For up-to-date information on GMP, please see the GMP web pages at
- `http://gmplib.org/'
+ <https://gmplib.org/>
The latest version of the library is available at
- `ftp://ftp.gnu.org/gnu/gmp/'
+ <https://ftp.gnu.org/gnu/gmp/>
- Many sites around the world mirror `ftp.gnu.org', please use a mirror
-near you, see `http://www.gnu.org/order/ftp.html' for a full list.
+ Many sites around the world mirror 'ftp.gnu.org', please use a mirror
+near you, see <https://www.gnu.org/order/ftp.html> for a full list.
There are three public mailing lists of interest. One for release
announcements, one for general questions and discussions about usage of
the GMP library and one for bug reports. For more information, see
- `http://gmplib.org/mailman/listinfo/'.
+ <https://gmplib.org/mailman/listinfo/>.
- The proper place for bug reports is <gmp-bugs@gmplib.org>. See
-*Note Reporting Bugs:: for information about reporting bugs.
+ The proper place for bug reports is <gmp-bugs@gmplib.org>. See *note
+Reporting Bugs:: for information about reporting bugs.
1.1 How to use this Manual
==========================
-Everyone should read *Note GMP Basics::. If you need to install the
-library yourself, then read *Note Installing GMP::. If you have a
-system with multiple ABIs, then read *Note ABI and ISA::, for the
+Everyone should read *note GMP Basics::. If you need to install the
+library yourself, then read *note Installing GMP::. 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
make check
-And you can install (under `/usr/local' by default) with
+And you can install (under '/usr/local' by default) with
make install
If you experience problems, please report them to
-<gmp-bugs@gmplib.org>. See *Note Reporting Bugs::, for information on
+<gmp-bugs@gmplib.org>. See *note Reporting Bugs::, for information on
what to include in useful bug reports.
* Menu:
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
+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
+ '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',
+ 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,
+ To compile in a separate build directory, 'cd' to that directory,
and prefix the configure command with the path to the GMP source
directory. For example
cd /my/build/dir
- /my/sources/gmp-5.0.1/configure
+ /my/sources/gmp-6.2.1/configure
- Not all `make' programs have the necessary features (`VPATH') to
- support this. In particular, SunOS and Slowaris `make' have bugs
+ Not all 'make' programs have the necessary features ('VPATH') to
+ support this. In particular, SunOS and Slowaris '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 GMP
- to install under a particular tree. The default is `/usr/local'.
-
- `--exec-prefix' can be used to direct architecture-dependent files
- like `libgmp.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 `gmp.h' and
- `mp.h' are architecture-dependent since they encode certain
- aspects of `libgmp', so it will be necessary to ensure both
- `$prefix/include' and `$exec_prefix/include' are available to the
- compiler.
-
-`--disable-shared', `--disable-static'
+ 'make' instead.
+
+'--prefix' and '--exec-prefix'
+ The '--prefix' option can be used in the normal way to direct GMP
+ to install under a particular tree. The default is '/usr/local'.
+
+ '--exec-prefix' can be used to direct architecture-dependent files
+ like 'libgmp.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 'gmp.h' is
+ architecture-dependent since it encodes certain aspects of
+ 'libgmp', so it will be necessary to ensure both '$prefix/include'
+ and '$exec_prefix/include' are available to the compiler.
+
+'--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'
+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
+ '--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
+ 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'
+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
+ '--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.
+ 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.
+ './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 `libgmp', `./configure' uses
- linking tests for various purposes, such as determining what
- functions are available on the host 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 'libgmp', './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
+ 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'.
+ guess the build system type if the 'PATH' has only a
+ cross-compiling 'cc'.
- Note that the `--target' option is not appropriate for GMP. It's
- for use when building compiler tools, with `--host' being where
- they will run, and `--target' what they'll produce code for.
+ Note that the '--target' option is not appropriate for GMP. 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 GMP are only interested in the
- `--host' part, being where they'll run. (Some past versions of
- GMP used `--target' incorrectly.)
+ '--host' part, being where they'll run. (Some past versions of GMP
+ used '--target' incorrectly.)
CPU types
In general, if you want a library that runs as fast as possible,
you should configure GMP 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 GMP for the exact machine type
- you intend to run it on.
+ The best idea is always to build GMP for the exact machine type you
+ intend to run it on.
- The following CPUs have specific support. See `configure.in' for
+ The following CPUs have specific support. See 'configure.ac' for
details of what code and compiler options they select.
* Alpha: alpha, alphaev5, alphaev56, alphapca56, alphapca57,
pentium2, pentium3, pentium4, k6, k62, k63, athlon, amd64,
viac3, viac32
- * Other: a29k, arm, clipper, i960, ns32k, pyramid, sh, sh2, vax,
- z8k
+ * Other: arm, sh, sh2, vax,
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
+ desired, the generic C code can be selected with the configure
+ '--disable-assembly'.
- 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.
+ 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, where
+Fat binary, '--enable-fat'
+ Using '--enable-fat' selects a "fat binary" build on x86, where
optimized low level subroutines are chosen at runtime according to
- the CPU detected. This means more code, but gives good
- performance on all x86 chips. (This option might become available
- for more architectures in the future.)
+ the CPU detected. This means more code, but gives good performance
+ on all x86 chips. (This option might become available for more
+ architectures in the future.)
-`ABI'
+'ABI'
On some systems GMP supports multiple ABIs (application binary
interfaces), meaning data type sizes and calling conventions. By
default GMP chooses the best ABI available, but a particular ABI
./configure --host=mips64-sgi-irix6 ABI=n32
- See *Note ABI and ISA::, for the available choices on relevant
+ See *note ABI and ISA::, for the available choices on relevant
CPUs, and what applications need to do.
-`CC', `CFLAGS'
+'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
+ 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
+ CPU and compiler. The usual 'CFLAGS="-whatever"' can be passed to
+ './configure' to use something different or to set good flags for
systems GMP 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
+ 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
+ Note that when 'CC' and 'CFLAGS' are specified on a system
supporting multiple ABIs it's important to give an explicit
- `ABI=whatever', since GMP can't determine the ABI just from the
+ 'ABI=whatever', since GMP can't determine the ABI just from the
flags and won't be able to select the correct assembly code.
- If just `CC' is selected then normal default `CFLAGS' for that
- compiler will be used (if GMP recognises it). For example
- `CC=gcc' can be used to force the use of GCC, with default flags
- (and default ABI).
+ If just 'CC' is selected then normal default 'CFLAGS' for that
+ compiler will be used (if GMP 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
+'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.
+ Preprocessing is done separately in some configure tests.
-`CC_FOR_BUILD'
+'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
+ 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.
+ 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
+ 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 GMP can be enabled with `--enable-cxx', in which
+C++ Support, '--enable-cxx'
+ C++ support in GMP 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
+ '--enable-cxx=detect' can be used to enable C++ support only if a
compiler can be found. The C++ support consists of a library
- `libgmpxx.la' and header file `gmpxx.h' (*note Headers and
+ 'libgmpxx.la' and header file 'gmpxx.h' (*note Headers and
Libraries::).
- A separate `libgmpxx.la' has been adopted rather than having C++
- objects within `libgmp.la' in order to ensure dynamic linked C
+ A separate 'libgmpxx.la' has been adopted rather than having C++
+ objects within 'libgmp.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.
- `libgmpxx.la' will use certain internals from `libgmp.la' and can
- only be expected to work with `libgmp.la' from the same GMP
+ 'libgmpxx.la' will use certain internals from 'libgmp.la' and can
+ only be expected to work with 'libgmp.la' from the same GMP
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 `libgmpxx.la' will be usable only with the C++ compiler
+ In general 'libgmpxx.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'
+'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++'.
+ 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
+ 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
+ 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>'
+Temporary Memory, '--enable-alloca=<choice>'
GMP 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.
+ '--enable-alloca=malloc-reentrant'.
- * `malloc-notreentrant' - the heap, with global variables.
+ * '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'.
+ '--disable-alloca' is the same as 'no'.
- * `yes' - a synonym for `alloca'.
+ * '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'.
- * `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
+ '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
+ '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
+ 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
+ An additional choice '--enable-alloca=debug' is available, to help
when debugging memory related problems (*note Debugging::).
-FFT Multiplication, `--disable-fft'
+FFT Multiplication, '--disable-fft'
By default multiplications are done using Karatsuba, 3-way Toom,
- and Fermat FFT. The FFT is only used on large to very large
- operands and can be disabled to save code size if desired.
+ higher degree Toom, and Fermat FFT. The FFT is only used on large
+ to very large operands and can be disabled to save code size if
+ desired.
-Berkeley MP, `--enable-mpbsd'
- The Berkeley MP compatibility library (`libmp') and header file
- (`mp.h') are built and installed only if `--enable-mpbsd' is used.
- *Note BSD Compatible Functions::.
-
-Assertion Checking, `--enable-assert'
+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'
+Execution Profiling, '--enable-profiling=prof/gprof/instrument'
Enable profiling support, in one of various styles, *note
Profiling::.
-`MPN_PATH'
+'MPN_PATH'
Various assembly versions of each 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
+ 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"
is completely unnecessary.
Documentation
- The source for the document you're now reading is `doc/gmp.texi',
- in Texinfo format, see *Note Texinfo: (texinfo)Top.
+ The source for the document you're now reading is 'doc/gmp.texi',
+ in Texinfo format, see *note Texinfo: (texinfo)Top.
- Info format `doc/gmp.info' is included in the distribution. The
+ Info format 'doc/gmp.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.
+ 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'
+ Some supplementary notes can also be found in the 'doc'
subdirectory.
-
\1f
File: gmp.info, Node: ABI and ISA, Next: Notes for Package Builds, Prev: Build Options, Up: Installing GMP
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.
-GMP supports some CPUs like this in both ABIs. In fact within GMP
-`ABI' means a combination of chip ABI, plus how GMP chooses to use it.
-For example in some 32-bit ABIs, GMP may support a limb as either a
-32-bit `long' or a 64-bit `long long'.
+ 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. GMP
+supports some CPUs like this in both ABIs. In fact within GMP 'ABI'
+means a combination of chip ABI, plus how GMP chooses to use it. For
+example in some 32-bit ABIs, GMP may support a limb as either a 32-bit
+'long' or a 64-bit 'long long'.
- By default GMP 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 GMP compatible with other libraries, or
+ By default GMP 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 GMP 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 `gmp.h'. This is convenient
-for applications, but it does mean that `gmp.h' will vary, and can't be
-just copied around. `gmp.h' remains compiler independent though, since
-all compilers for a particular ABI will be expected to use the same
-limb type.
+ Usually a limb is implemented as a 'long'. When a 'long long' limb
+is used this is encoded in the generated 'gmp.h'. This is convenient
+for applications, but it does mean that 'gmp.h' will vary, and can't be
+just copied around. 'gmp.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 GMP,
-and it might be as simple as configuring with a special `libdir', or it
+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 `./configure' and `make' each.
+to done separately, with a fresh './configure' and 'make' each.
-AMD64 (`x86_64')
+AMD64 ('x86_64')
On AMD64 systems supporting both 32-bit and 64-bit modes for
applications, the following ABI choices are available.
- `ABI=64'
+ '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
+ Applications will usually not need special compiler flags, but
+ for reference the option is
gcc -m64
- `ABI=32'
+ '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
+ 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
+ (In GCC 2.95 and earlier there's no '-m32' option, it's the
only mode.)
+ 'ABI=x32'
+ The x32 ABI uses 64-bit limbs but 32-bit pointers. Like the
+ 64-bit ABI, it makes full use of the chip's arithmetic
+ capabilities. This ABI is not supported by all operating
+ systems.
-HPPA 2.0 (`hppa2.0*', `hppa64')
+ gcc -mx32
- `ABI=2.0w'
+
+HPPA 2.0 ('hppa2.0*', 'hppa64')
+ 'ABI=2.0w'
The 2.0w ABI uses 64-bit limbs and pointers and is available
on HP-UX 11 or up. Applications must be compiled with
gcc [built for 2.0w]
cc +DD64
- `ABI=2.0n'
+ 'ABI=2.0n'
The 2.0n ABI means the 32-bit HPPA 1.0 ABI and all its normal
calling conventions, but with 64-bit instructions permitted
- within functions. GMP uses a 64-bit `long long' for a limb.
+ within functions. GMP uses a 64-bit 'long long' for a limb.
This ABI is available on hppa64 GNU/Linux and on HP-UX 10 or
higher. Applications must be compiled with
cc +DA2.0 +e
Note that current versions of GCC (eg. 3.2) don't generate
- 64-bit instructions for `long long' operations and so may be
+ 64-bit instructions for 'long long' operations and so may be
slower than for 2.0w. (The GMP assembly code is the same
though.)
- `ABI=1.0'
+ 'ABI=1.0'
HPPA 2.0 CPUs can run all HPPA 1.0 and 1.1 code in the 32-bit
HPPA 1.0 ABI. No special compiler options are needed for
applications.
- All three ABIs are available for CPU types `hppa2.0w', `hppa2.0'
- and `hppa64', but for CPU type `hppa2.0n' only 2.0n or 1.0 are
+ All three ABIs are available for CPU types 'hppa2.0w', 'hppa2.0'
+ and 'hppa64', but for CPU type 'hppa2.0n' only 2.0n or 1.0 are
considered.
Note that GCC on HP-UX has no options to choose between 2.0n and
- 2.0w modes, unlike HP `cc'. Instead it must be built for one or
+ 2.0w modes, unlike HP 'cc'. Instead it must be built for one or
the other ABI. GMP will detect how it was built, and skip to the
- corresponding `ABI'.
+ corresponding 'ABI'.
-IA-64 under HP-UX (`ia64*-*-hpux*', `itanium*-*-hpux*')
+IA-64 under HP-UX ('ia64*-*-hpux*', 'itanium*-*-hpux*')
HP-UX supports two ABIs for IA-64. GMP performance is the same in
both.
- `ABI=32'
- In the 32-bit ABI, pointers, `int's and `long's are 32 bits
- and GMP uses a 64 bit `long long' for a limb. Applications
+ 'ABI=32'
+ In the 32-bit ABI, pointers, 'int's and 'long's are 32 bits
+ and GMP 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 GMP
- uses a `long' for a limb. Applications must be compiled with
+ 'ABI=64'
+ In the 64-bit ABI, 'long's and pointers are 64 bits and GMP
+ 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
+ On other IA-64 systems, GNU/Linux for instance, 'ABI=64' is the
only choice.
-MIPS under IRIX 6 (`mips*-*-irix[6789]')
+MIPS under IRIX 6 ('mips*-*-irix[6789]')
IRIX 6 always has a 64-bit MIPS 3 or better CPU, and supports ABIs
o32, n32, and 64. n32 or 64 are recommended, and GMP performance
will be the same in each. The default is n32.
- `ABI=o32'
+ 'ABI=o32'
The o32 ABI is 32-bit pointers and integers, and no 64-bit
- operations. GMP will be slower than in n32 or 64, this
- option only exists to support old compilers, eg. GCC 2.7.2.
+ operations. GMP will be slower than in n32 or 64, this option
+ only exists to support old compilers, eg. GCC 2.7.2.
Applications can be compiled with no special flags on an old
compiler, or on a newer compiler with
gcc -mabi=32
cc -32
- `ABI=n32'
- The n32 ABI is 32-bit pointers and integers, but with a
- 64-bit limb using a `long long'. Applications must be
- compiled with
+ 'ABI=n32'
+ The n32 ABI is 32-bit pointers and integers, but with a 64-bit
+ limb using a 'long long'. Applications must be compiled with
gcc -mabi=n32
cc -n32
- `ABI=64'
+ 'ABI=64'
The 64-bit ABI is 64-bit pointers and integers. Applications
must be compiled with
and the MIPS 2 code.
-PowerPC 64 (`powerpc64', `powerpc620', `powerpc630', `powerpc970', `power4', `power5')
-
- `ABI=aix64'
+PowerPC 64 ('powerpc64', 'powerpc620', 'powerpc630', 'powerpc970', 'power4', 'power5')
+ 'ABI=mode64'
The AIX 64 ABI uses 64-bit limbs and pointers and is the
- default on PowerPC 64 `*-*-aix*' systems. Applications must
+ default on PowerPC 64 '*-*-aix*' systems. Applications must
be compiled with
gcc -maix64
xlc -q64
- `ABI=mode64'
- The `mode64' ABI uses 64-bit limbs and pointers, and is the
- default on 64-bit GNU/Linux, BSD, and Mac OS X/Darwin
- systems. Applications must be compiled with
+ On 64-bit GNU/Linux, BSD, and Mac OS X/Darwin systems, the
+ applications must be compiled with
gcc -m64
- `ABI=mode32'
- The `mode32' ABI uses a 64-bit `long long' limb but with the
+ '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 for systems where the
- true 64-bit ABIs are unavailable. No special compiler
- options are needed for applications.
+ conventions. This is the default for systems where the true
+ 64-bit ABI is unavailable. No special compiler options are
+ typically needed for applications. This ABI is not available
+ under AIX.
- `ABI=32'
+ 'ABI=32'
This is the basic 32-bit PowerPC ABI, with a 32-bit limb. No
special compiler options are needed for applications.
- GMP 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.
-
+ GMP's speed is greatest for the 'mode64' ABI, the 'mode32' ABI is
+ 2nd best. In 'ABI=32' only the 32-bit ISA is used and this doesn't
+ make full use of a 64-bit chip.
-Sparc V9 (`sparc64', `sparcv9', `ultrasparc*')
- `ABI=64'
+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
+ higher, or Sun 'cc' is required. On GNU/Linux, depending on
+ the default 'gcc' mode, applications must be compiled with
gcc -m64
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, GMP 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
+ 'ABI=32'
+ For the basic 32-bit ABI, GMP 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'
+ recommended. ('gcc' 2.8 and earlier only support '-mv8'
though.)
gcc -mv8plus
cc -xarch=v8plus
- GMP speed is greatest in `ABI=64', so it's the default where
+ GMP 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.
+ 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
+ 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
+ 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
GMP 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
+ 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 will be upwardly binary compatible in each release
application depends on a new enough GMP.
An auxiliary mechanism may also be needed to express that
-`libgmpxx.la' (from `--enable-cxx', *note Build Options::) requires
-`libgmp.la' from the same GMP 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.
+'libgmpxx.la' (from '--enable-cxx', *note Build Options::) requires
+'libgmp.la' from the same GMP 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.
+'--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,
+ 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 GMP 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.
+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').
+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
+'/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 `gmp.h' is a generated file, and will be architecture and
+ Note that 'gmp.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 `gmp.h'
+will be important that an application compile gets the correct 'gmp.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/gmp.h'
-which tests preprocessor symbols and chooses the correct actual `gmp.h'.
+options then it might be necessary to create a '/usr/include/gmp.h'
+which tests preprocessor symbols and chooses the correct actual 'gmp.h'.
\1f
File: gmp.info, Node: Notes for Particular Systems, Next: Known Build Problems, Prev: Notes for Package Builds, Up: Installing GMP
================================
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
+ 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 `libgmp.a' a symlink to `libgmp.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.
+ Note that the '--disable-static' is necessary because in a shared
+ build libtool makes 'libgmp.a' a symlink to 'libgmp.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
+ 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. GMP `./configure' will demand GCC 2.95.4 or later.
+ operands. GMP './configure' will demand GCC 2.95.4 or later.
Compaq C++
- Compaq C++ on OSF 5.1 has two flavours of `iostream', a standard
- one and an old pre-standard one (see `man iostream_intro'). GMP
+ Compaq C++ on OSF 5.1 has two flavours of 'iostream', a standard
+ one and an old pre-standard one (see 'man iostream_intro'). GMP
can only use the standard one, which unfortunately is not the
- default but must be selected by defining `__USE_STD_IOSTREAM'.
+ default but must be selected by defining '__USE_STD_IOSTREAM'.
Configure with for instance
./configure --enable-cxx CPPFLAGS=-D__USE_STD_IOSTREAM
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 GMP 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 GMP.
+ floating point). The GMP 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 GMP.
+
+FreeBSD 7.x, 8.x, 9.0, 9.1, 9.2
+ 'm4' in these releases of FreeBSD has an eval function which
+ ignores its 2nd and 3rd arguments, which 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 FreeBSD 9.3 and 10.0, so either upgrade or use GNU m4. Note
+ that the FreeBSD package system installs GNU m4 under the name
+ 'gm4', which GMP cannot guess.
+
+FreeBSD 7.x, 8.x, 9.x
+ GMP releases starting with 6.0 do not support 'ABI=32' on
+ FreeBSD/amd64 prior to release 10.0 of the system. The cause is a
+ broken 'limits.h', which GMP no longer works around.
MS-DOS and MS Windows
On an MS-DOS system DJGPP can be used to build GMP, and on an MS
Windows system Cygwin, DJGPP and MINGW can be used. All three are
excellent ports of GCC and the various GNU tools.
- `http://www.cygwin.com/'
- `http://www.delorie.com/djgpp/'
- `http://www.mingw.org/'
+ <https://www.cygwin.com/>
+ <http://www.delorie.com/djgpp/>
+ <http://www.mingw.org/>
Microsoft also publishes an Interix "Services for Unix" which can
- be used to build GMP on Windows (with a normal `./configure'), but
+ be used to build GMP on Windows (with a normal './configure'), but
it's not free software.
MS Windows DLLs
- On systems `*-*-cygwin*', `*-*-mingw*' and `*-*-pw32*' by default
+ On systems '*-*-cygwin*', '*-*-mingw*' and '*-*-pw32*' by default
GMP 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 `gmp.h' must be different.
+ directives in 'gmp.h' must be different.
A MINGW DLL build of GMP can be used with Microsoft C. 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 `libmp' and `libgmpxx'.
+ 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 'libmp' and 'libgmpxx'.
cd .libs
lib /def:libgmp-3.dll.def /out:libgmp-3.lib
- MINGW uses the C runtime library `msvcrt.dll' for I/O, so
+ MINGW uses the C runtime library 'msvcrt.dll' for I/O, so
applications wanting to use the GMP I/O routines must be compiled
- with `cl /MD' to do the same. If one of the other C runtime
+ 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 GMP string functions and confine I/O to the application.
Motorola 68k CPU Types
- `m68k' is taken to mean 68000. `m68020' or higher will give a
- performance boost on applicable CPUs. `m68360' can be used for
- CPU32 series chips. `m68302' can be used for "Dragonball" series
- chips, though this is merely a synonym for `m68000'.
+ 'm68k' is taken to mean 68000. 'm68020' or higher will give a
+ performance boost on applicable CPUs. 'm68360' can be used for
+ CPU32 series chips. 'm68302' can be used for "Dragonball" series
+ chips, though this is merely a synonym for 'm68000'.
+
+NetBSD 5.x
+ 'm4' in these releases of NetBSD has an eval function which ignores
+ its 2nd and 3rd arguments, which 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
+ NetBSD 6, so either upgrade or use GNU m4. Note that the NetBSD
+ package system installs GNU m4 under the name 'gm4', which GMP
+ cannot guess.
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'.
+ '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.
Power CPU Types
- In GMP, CPU types `power*' and `powerpc*' will each use
+ In GMP, CPU types 'power*' and 'powerpc*' will each use
instructions not available on the other, so it's important to
choose the right one for the CPU that will be used. Currently GMP
has no assembly code support for using just the common instruction
subset. To get executables that run on both, the current
- suggestion is to use the generic C code (CPU `none'), possibly
- with appropriate compiler options (like `-mcpu=common' for `gcc').
- CPU `rs6000' (which is not a CPU but a family of workstations) is
- accepted by `config.sub', but is currently equivalent to `none'.
+ suggestion is to use the generic C code ('--disable-assembly'),
+ possibly with appropriate compiler options (like '-mcpu=common' for
+ 'gcc'). CPU 'rs6000' (which is not a CPU but a family of
+ workstations) is accepted by 'config.sub', but is currently
+ equivalent to '--disable-assembly'.
Sparc CPU Types
- `sparcv8' or `supersparc' on relevant systems will give a
+ 'sparcv8' or 'supersparc' on relevant systems will give a
significant performance increase over the V7 code selected by plain
- `sparc'.
+ 'sparc'.
Sparc App Regs
The GMP assembly 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
+ "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),
+ '-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 GMP with CPU `none' to avoid the assembly code.
+ purposes. In these cases the only suggestion currently is to build
+ GMP with '--disable-assembly' to avoid the assembly code.
SunOS 4
- `/usr/bin/m4' lacks various features needed to process `.asm'
- files, and instead `./configure' will automatically use
- `/usr/5bin/m4', which we believe is always available (if not then
+ '/usr/bin/m4' lacks various features needed to process '.asm'
+ files, and instead './configure' will automatically use
+ '/usr/5bin/m4', which we believe is always available (if not then
use GNU m4).
x86 CPU Types
- `i586', `pentium' or `pentiummmx' code is good for its intended P5
+ '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
+ (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.
+ 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
+ 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
+ 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.
\1f
2.5 Known Build Problems
========================
-You might find more up-to-date information at `http://gmplib.org/'.
+You might find more up-to-date information at <https://gmplib.org/>.
Compiler link options
The version of libtool currently in use rather aggressively strips
./configure CC=gcc-with-my-options
-DJGPP (`*-*-msdosdjgpp*')
- The DJGPP port of `bash' 2.03 is unable to run the `configure'
+DJGPP ('*-*-msdosdjgpp*')
+ The DJGPP port of 'bash' 2.03 is unable to run the 'configure'
script, it exits silently, having died writing a preamble to
- `config.log'. Use `bash' 2.04 or higher.
+ 'config.log'. Use 'bash' 2.04 or higher.
- `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
+ '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.
-GNU binutils `strip' prior to 2.12
- `strip' from GNU binutils 2.11 and earlier should not be used on
- the static libraries `libgmp.a' and `libmp.a' since it will
- discard all but the last of multiple archive members with the same
- name, like the three versions of `init.o' in `libgmp.a'. Binutils
- 2.12 or higher can be used successfully.
+GNU binutils 'strip' prior to 2.12
+ 'strip' from GNU binutils 2.11 and earlier should not be used on
+ the static libraries 'libgmp.a' and 'libmp.a' since it will discard
+ all but the last of multiple archive members with the same name,
+ like the three versions of 'init.o' in 'libgmp.a'. Binutils 2.12
+ or higher can be used successfully.
- The shared libraries `libgmp.so' and `libmp.so' are not affected by
- this and any version of `strip' can be used on them.
+ The shared libraries 'libgmp.so' and 'libmp.so' are not affected by
+ this and any version of 'strip' can be used on them.
-`make' syntax error
+'make' syntax error
On certain versions of SCO OpenServer 5 and IRIX 6.5 the native
- `make' is unable to handle the long dependencies list for
- `libgmp.la'. The symptom is a "syntax error" on the following
- line of the top-level `Makefile'.
+ 'make' is unable to handle the long dependencies list for
+ 'libgmp.la'. The symptom is a "syntax error" on the following line
+ of the top-level 'Makefile'.
libgmp.la: $(libgmp_la_OBJECTS) $(libgmp_la_DEPENDENCIES)
Either use GNU Make, or as a workaround remove
- `$(libgmp_la_DEPENDENCIES)' from that line (which will make the
- initial build work, but if any recompiling is done `libgmp.la'
+ '$(libgmp_la_DEPENDENCIES)' from that line (which will make the
+ initial build work, but if any recompiling is done 'libgmp.la'
might not be rebuilt).
-MacOS X (`*-*-darwin*')
+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
+ MacOS X using the native 'cc' (which is a modified GCC), not a
plain GCC. A static-only build should work though
- (`--disable-shared').
+ ('--disable-shared').
NeXT prior to 3.3
- The system compiler on old versions of NeXT was a massacred and
- old GCC, even if it called itself `cc'. This compiler cannot be
- used to build GMP, you need to get a real GCC, and install that.
- (NeXT may have fixed this in release 3.3 of their system.)
+ The system compiler on old versions of NeXT was a massacred and old
+ GCC, even if it called itself 'cc'. This compiler cannot be used
+ to build GMP, you need to get a real GCC, and install that. (NeXT
+ may have fixed this in release 3.3 of their system.)
POWER and PowerPC
Bugs in GCC 2.7.2 (and 2.6.3) mean it can't be used to compile GMP
latter has serious bugs.
Solaris 2.6
- The system `sed' prints an error "Output line too long" when
- libtool builds `libgmp.la'. This doesn't seem to cause any
- obvious ill effects, but GNU `sed' is recommended, to avoid any
- doubt.
+ The system 'sed' prints an error "Output line too long" when
+ libtool builds 'libgmp.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 GMP 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.
+Sparc Solaris 2.7 with gcc 2.95.2 in 'ABI=32'
+ A shared library build of GMP 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.
\1f
File: gmp.info, Node: Performance optimization, Prev: Known Build Problems, Up: Installing GMP
============================
For optimal performance, build GMP for the exact CPU type of the target
-computer, see *Note Build Options::.
+computer, see *note Build Options::.
Unlike what is the case for most other programs, the compiler
typically doesn't matter much, since GMP uses assembly language for the
most critical operation.
In particular for long-running GMP applications, and applications
-demanding extremely large numbers, building and running the `tuneup'
-program in the `tune' subdirectory, can be important. For example,
+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.
+ will generate better contents for the 'gmp-mparam.h' parameter file.
- To use the results, put the output in the file file indicated in the
-`Parameters for ...' header. Then recompile from scratch.
+ 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
+ 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 GMP for extremely large numbers, you may want to
-run `tuneup' with a large NNN value.
+you're going to use GMP for extremely large numbers, you may want to run
+'tuneup' with a large NNN value.
\1f
File: gmp.info, Node: GMP Basics, Next: Reporting Bugs, Prev: Installing GMP, Up: Top
3 GMP 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 GMP.*
+*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 GMP.*
* Menu:
=========================
All declarations needed to use GMP are collected in the include file
-`gmp.h'. It is designed to work with both C and C++ compilers.
+'gmp.h'. It is designed to work with both C and C++ compilers.
#include <gmp.h>
- Note however that prototypes for GMP functions with `FILE *'
-parameters are only provided if `<stdio.h>' is included too.
+ Note however that prototypes for GMP functions with 'FILE *'
+parameters are only provided if '<stdio.h>' is included too.
#include <stdio.h>
#include <gmp.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.
+ Likewise '<stdarg.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 GMP must link against the `libgmp' library. On a
-typical Unix-like system this can be done with `-lgmp', for example
+ All programs using GMP must link against the 'libgmp' library. On a
+typical Unix-like system this can be done with '-lgmp', for example
gcc myprogram.c -lgmp
- GMP C++ functions are in a separate `libgmpxx' library. This is
+ GMP C++ functions are in a separate 'libgmpxx' library. This is
built and installed if C++ support has been enabled (*note Build
Options::). For example,
g++ mycxxprog.cc -lgmpxx -lgmp
- GMP is built using Libtool and an application can use that to link
-if desired, *note GNU Libtool: (libtool)Top.
+ GMP is built using Libtool and an application can use that to link if
+desired, *note GNU Libtool: (libtool)Top.
If GMP 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
+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.
\1f
In this manual, "integer" usually means a multiple precision integer, as
defined by the GMP library. The C data type for such integers is
-`mpz_t'. Here are some examples of how to declare such integers:
+'mpz_t'. Here are some examples of how to declare such integers:
mpz_t sum;
mpz_t vec[20];
"Rational number" means a multiple precision fraction. The C data
-type for these fractions is `mpq_t'. For example:
+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:
+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.
+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'.
+is 'mp_limb_t'.
Counts of limbs of a multi-precision number represented in the C type
-`mp_size_t'. Currently this is normally a `long', but on some systems
-it's an `int' for efficiency, and on some systems it will be `long
-long' in the future.
+'mp_size_t'. Currently this is normally a 'long', but on some systems
+it's an 'int' for efficiency, and on some systems it will be 'long long'
+in the future.
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 .
+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:
+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.
+ Also, in general 'mp_bitcnt_t' is used for bit counts and ranges, and
+'size_t' is used for byte or character counts.
\1f
File: gmp.info, Node: Function Classes, Next: Variable Conventions, Prev: Nomenclature and Types, Up: GMP Basics
There are six classes of functions in the GMP library:
1. Functions for signed integer arithmetic, with names beginning with
- `mpz_'. The associated type is `mpz_t'. There are about 150
+ '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
+ 'mpq_'. The associated type is 'mpq_t'. There are about 35
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
+ 'mpf_'. The associated type is 'mpf_t'. There are about 70
functions is this class. (*note Floating-point Functions::)
- 4. Functions compatible with Berkeley MP, such as `itom', `madd', and
- `mult'. The associated type is `MINT'. (*note BSD Compatible
- 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 60 (hard-to-use) functions in this
+ class. (*note Low-level Functions::)
- 5. 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::)
-
- 6. Miscellaneous functions. Functions for setting up custom
+ 5. Miscellaneous functions. Functions for setting up custom
allocation and functions for generating random numbers. (*note
Custom Allocation::, and *note Random Number Functions::)
========================
GMP functions generally have output arguments before input arguments.
-This notation is by analogy with the assignment operator. The BSD MP
-compatibility functions are exceptions, having the output arguments
-last.
+This notation is by analogy with the assignment operator.
GMP 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', can be used to square 'x' and put the result back in 'x' with
mpz_mul (x, x, x);
- Before you can assign to a GMP 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
+ Before you can assign to a GMP 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
mpz_clear (n);
}
+ GMP types like 'mpz_t' are implemented as one-element arrays of
+certain structures. Declaring a variable creates an object with the
+fields GMP needs, but variables are normally manipulated by using the
+pointer to the object. For both behavior and efficiency reasons, it is
+discouraged to make copies of the GMP object itself (either directly or
+via aggregate objects containing such GMP objects). If copies are done,
+all of them must be used read-only; using a copy as the output of some
+function will invalidate all the other copies. 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 GMP
+releases.
+
\1f
File: gmp.info, Node: Parameter Conventions, Next: Memory Management, Prev: Variable Conventions, Up: GMP Basics
=========================
When a GMP 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.
+call-by-reference, meaning that when 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 a GMP 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.
+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
+ Here's an example accepting an 'mpz_t' parameter, doing a
calculation, and storing the result to the indicated parameter.
void
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.
+ Our function 'foo' works even if its caller 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 GMP 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 GMP 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 GMP
-releases.
+ Since GMP types are implemented as one-element arrays, using a GMP
+variable as a parameter passes a pointer to the object. Hence the
+call-by-reference.
\1f
File: gmp.info, Node: Memory Management, Next: Reentrancy, Prev: Parameter Conventions, Up: GMP Basics
3.6 Memory Management
=====================
-The GMP types like `mpz_t' are small, containing only a couple of sizes,
+The GMP types like 'mpz_t' are small, containing only a couple of sizes,
and pointers to allocated data. Once a variable is initialized, GMP
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.
+ '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.
+point can use 'mpz_realloc2', or clear variables no longer needed.
- `mpf_t' variables, in the current implementation, use a fixed amount
+ '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::.
+ 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::.
\1f
File: gmp.info, Node: Reentrancy, Next: Useful Macros and Constants, Prev: Memory Management, Up: GMP Basics
GMP 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),
+ * If configured with '--enable-alloca=malloc-notreentrant' (or with
+ '--enable-alloca=notreentrant' when 'alloca' is not available),
then naturally GMP 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.
+ * '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.
- * `mpz_random' and the other old random number functions use a global
+ * 'mpz_random' and the other old random number functions use a global
random state and are hence not reentrant. The newer random number
- functions that accept a `gmp_randstate_t' parameter can be used
+ functions that accept a 'gmp_randstate_t' parameter can be used
instead.
- * `gmp_randinit' (obsolete) returns an error indication through a
+ * 'gmp_randinit' (obsolete) returns an error indication through a
global variable, which is not thread safe. Applications are
- advised to use `gmp_randinit_default' or `gmp_randinit_lc_2exp'
+ advised to use 'gmp_randinit_default' or 'gmp_randinit_lc_2exp'
instead.
- * `mp_set_memory_functions' uses global variables to store the
+ * '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
+ 'mp_set_memory_functions' (or 'malloc' and friends by default) are
not reentrant, then GMP will not be reentrant either.
- * If the standard I/O functions such as `fwrite' are not reentrant
+ * If the standard I/O functions such as 'fwrite' are not reentrant
then the GMP I/O functions using them will not be reentrant either.
* It's safe for two threads to read from the same GMP 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.
+ simultaneously, but it's not safe for one to read while 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.
\1f
File: gmp.info, Node: Useful Macros and Constants, Next: Compatibility with older versions, Prev: Reentrancy, Up: GMP Basics
-- Global Constant: const char * const gmp_version
The GMP version number, as a null-terminated string, in the form
- "i.j.k". This release is "5.0.1". Note that the format "i.j" was
- used when k was zero was used before version 4.3.0.
+ "i.j.k". This release is "6.2.1". Note that the format "i.j" was
+ used, before version 4.3.0, when k was zero.
-- Macro: __GMP_CC
-- Macro: __GMP_CFLAGS
3.9 Compatibility with older versions
=====================================
-This version of GMP is upwardly binary compatible with all 4.x and 3.x
-versions, and upwardly compatible at the source level with all 2.x
+This version of GMP is upwardly binary compatible with all 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.
+ * '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
+ * '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.
- 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. Please see the GMP 2 manual for details.
+ * 'mpn_bdivmod', documented as preliminary in GMP 4, has been
+ removed.
- The Berkeley MP compatibility library (*note BSD Compatible
-Functions::) is source and binary compatible with the standard `libmp'.
+ 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 5.
+Please see the GMP 2 manual for details.
\1f
File: gmp.info, Node: Demonstration Programs, Next: Efficiency, Prev: Compatibility with older versions, Up: GMP Basics
3.10 Demonstration programs
===========================
-The `demos' subdirectory has some sample programs using GMP. These
-aren't built or installed, but there's a `Makefile' with rules for them.
+The 'demos' subdirectory has some sample programs using GMP. These
+aren't built or installed, but there's a 'Makefile' with rules for them.
For instance,
make pexpr
The following programs are provided
- * `pexpr' is an expression evaluator, the program used on the GMP
- web page.
-
- * The `calc' subdirectory has a similar but simpler evaluator using
- `lex' and `yacc'.
-
- * The `expr' subdirectory is yet another expression evaluator, a
+ * 'pexpr' is an expression evaluator, the program used on the GMP web
+ page.
+ * The 'calc' subdirectory has a similar but simpler evaluator using
+ 'lex' and 'yacc'.
+ * The 'expr' subdirectory is yet another expression evaluator, a
library designed for ease of use within a C program. See
- `demos/expr/README' for more information.
-
- * `factorize' is a Pollard-Rho factorization program.
-
- * `isprime' is a command-line interface to the `mpz_probab_prime_p'
+ 'demos/expr/README' for more information.
+ * 'factorize' is a Pollard-Rho factorization program.
+ * 'isprime' is a command-line interface to the 'mpz_probab_prime_p'
function.
-
- * `primes' counts or lists primes in an interval, using a sieve.
-
- * `qcn' is an example use of `mpz_kronecker_ui' to estimate quadratic
+ * 'primes' counts or lists primes in an interval, using a sieve.
+ * 'qcn' is an example use of 'mpz_kronecker_ui' to estimate quadratic
class numbers.
-
- * The `perl' subdirectory is a comprehensive perl interface to GMP.
- See `demos/perl/INSTALL' for more information. Documentation is
- in POD format in `demos/perl/GMP.pm'.
+ * The 'perl' subdirectory is a comprehensive perl interface to GMP.
+ See 'demos/perl/INSTALL' for more information. Documentation is in
+ POD format in 'demos/perl/GMP.pm'.
As an aside, consideration has been given at various times to some
-sort of expression evaluation within the main GMP library. Going
-beyond something minimal quickly leads to matters like user-defined
-functions, looping, fixnums for control variables, etc, which are
-considered outside the scope of GMP (much closer to language
-interpreters or compilers, *Note Language Bindings::.) Something
-simple for program input convenience may yet be a possibility, a
-combination of the `expr' demo and the `pexpr' tree back-end perhaps.
-But for now the above evaluators are offered as illustrations.
+sort of expression evaluation within the main GMP library. Going beyond
+something minimal quickly leads to matters like user-defined functions,
+looping, fixnums for control variables, etc, which are considered
+outside the scope of GMP (much closer to language interpreters or
+compilers, *Note Language Bindings::.) Something simple for program
+input convenience may yet be a possibility, a combination of the 'expr'
+demo and the 'pexpr' tree back-end perhaps. But for now the above
+evaluators are offered as illustrations.
\1f
File: gmp.info, Node: Efficiency, Next: Debugging, Prev: Demonstration Programs, Up: GMP Basics
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 GMP attempts to be as efficient as it can on
- both large and small operands.
+ library, although GMP 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 `libgmp.a' should
+ On some CPUs, in particular the x86s, the static 'libgmp.a' should
be used for maximum speed, since the PIC code in the shared
- `libgmp.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.
+ 'libgmp.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
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
+ 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
+ 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.
+ 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.
+'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
+'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' there's no need extract it and call a `ui' function, just
- use the regular `mpz' function.
+ example an 'mpz_t' contains a value that fits in an 'unsigned long'
+ 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
+ '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
+ '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
+ '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
+ 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 GMP 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
+ '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).
+ 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
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
+ 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).
+ 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
+ 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
cancellation will be possible, and so canonicalization can be left
to the end.
- The `mpq_numref' and `mpq_denref' macros give access to the
+ 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::.
+ supplied 'mpq' functions. *Note Applying Integer Functions::.
- The canonical form for rationals allows mixed-type `mpq_t' and
+ 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_add'. For example,
/* mpq increment */
mpz_add (mpq_numref(q), mpq_numref(q), mpq_denref(q));
mpz_submul (mpq_numref(q), mpq_denref(q), z);
Number Sequences
- Functions like `mpz_fac_ui', `mpz_fib_ui' and `mpz_bin_uiui' are
+ 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.
+ wanted it's probably best to get a starting point and iterate from
+ there.
Text Input/Output
Hexadecimal or octal are suggested for input or output in text
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
+ '--enable-alloca' choices in *note Build Options::, for how to
address this.
- In new enough versions of GCC, `-fstack-check' may be able to
+ 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 GMP 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.
+ 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 GMP 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 GMP is heap
- corruption. Failing to `init' GMP variables will have
+ corruption. Failing to 'init' GMP variables will have
unpredictable effects, and corruption arising elsewhere in a
program may well affect GMP. Initializing GMP 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
+ 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://www.inf.ethz.ch/personal/biere/projects/ccmalloc/'
- `http://dmalloc.com/'
- `http://www.perens.com/FreeSoftware/' (electric fence)
- `http://packages.debian.org/stable/devel/fda'
- `http://www.gnupdate.org/components/leakbug/'
- `http://people.redhat.com/~otaylor/memprof/'
- `http://www.cbmamiga.demon.co.uk/mpatrol/'
+ <http://cs.ecs.baylor.edu/~donahoo/tools/ccmalloc/>
+ <http://dmalloc.com/>
+ <https://wiki.gnome.org/Apps/MemProf>
- The GMP 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 GMP development, but might find other uses.
+ The GMP 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 GMP development, but might find other uses.
Stack Backtraces
On some systems the compiler options GMP uses by default can
interfere with debugging. In particular on x86 and 68k systems
- `-fomit-frame-pointer' is used and this generally inhibits stack
+ '-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 GMP 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 GMP.
+ A sample '.gdbinit' is included in the distribution, showing how to
+ call some undocumented dump functions to print GMP 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 GMP.
Source File Paths
GMP 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.
+ 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-5.0.1/configure
+ /my/source/dir/gmp-6.2.1/configure
- This works via `VPATH', and might require GNU `make'. Alternately
- it might be possible to change the `.c.lo' rules appropriately.
+ 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::).
+ 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.
+ 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
+ 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 assembly code, so CPU `none' should be
- used for maximum checking.
+ code has checks, not the assembly code, so '--disable-assembly'
+ should be used for maximum checking.
Temporary Memory Checking
- The build option `--enable-alloca=debug' arranges that each block
+ The build option '--enable-alloca=debug' arranges that each block
of temporary memory in GMP is allocated with a separate call to
- `malloc' (or the allocation function set with
- `mp_set_memory_functions').
+ '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 GMP divides up for its own use, or may be allocated with a
- compiler builtin `alloca' which will go nowhere near any malloc
- debugger hooks.
+ intended bounds, or detect memory not released. In a normal build,
+ on the other hand, temporary memory is allocated in blocks which
+ GMP 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, a GMP build for maximum debuggability
- would be
+ To summarize the above, a GMP build for maximum debuggability would
+ be
./configure --disable-shared --enable-assert \
- --enable-alloca=debug --host=none CFLAGS=-g
+ --enable-alloca=debug --disable-assembly CFLAGS=-g
- For C++, add `--enable-cxx CXXFLAGS=-g'.
+ For C++, add '--enable-cxx CXXFLAGS=-g'.
Checker
- The GCC checker (`http://savannah.nongnu.org/projects/checker/')
+ The GCC checker (<https://savannah.nongnu.org/projects/checker/>)
can be used with GMP. It contains a stub library which means GMP
applications compiled with checker can use a normal GMP build.
A build of GMP with checking within GMP itself can be made. This
will run very very slowly. On GNU/Linux for example,
- ./configure --host=none-pc-linux-gnu CC=checkergcc
+ ./configure --disable-assembly CC=checkergcc
- `--host=none' must be used, since the GMP assembly code doesn't
- support the checking scheme. The GMP C++ features cannot be used,
- since current versions of checker (0.9.9.1) don't yet support the
- standard C++ library.
+ '--disable-assembly' must be used, since the GMP assembly code
+ doesn't support the checking scheme. The GMP 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 GMP will need to be
- configured not to use those, ie. for an x86 without them (for
- instance plain `i486').
+ Valgrind (<http://valgrind.org/>) is a memory checker for x86, ARM,
+ MIPS, PowerPC, and S/390. 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.
+
+ Valgrind does not always support every possible instruction, in
+ particular ones recently added to an ISA. Valgrind might therefore
+ be incompatible with a recent GMP or even a less recent GMP which
+ is compiled using a recent GCC.
+
+ GMP's assembly code sometimes promotes a read of the limbs to some
+ larger size, for efficiency. GMP will do this even at the start
+ and end of a multilimb operand, using naturally aligned operations
+ on the larger type. This may lead to benign reads outside of
+ allocated areas, triggering complaints from Valgrind. Valgrind's
+ option '--partial-loads-ok=yes' should help.
Other Problems
Any suspected bug in GMP itself should be isolated to make sure
- it's not an application problem, see *Note Reporting Bugs::.
+ it's not an application problem, see *note Reporting Bugs::.
\1f
File: gmp.info, Node: Profiling, Next: Autoconf, Prev: Debugging, Up: GMP Basics
spending most time and where improvements can be best sought. The
profiling choices for a GMP build are as follows.
-`--disable-profiling'
+'--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 GMP
- assembly code has the necessary symbol information.
+ with '-p' and use 'prof' to get a profile consisting of timer-based
+ sampling of the program counter. Most of the GMP assembly 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.
+ 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'.
+'--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
The x86 assembly code has support for this option, but on other
processors the assembly routines will be as if compiled without
- `-p' and therefore won't appear in the call counts.
+ '-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
+ 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', which means `-pg' added to the
- `CFLAGS'.
+'--enable-profiling=gprof'
+ Build with support for 'gprof', 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.
+ 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 assembly code has support for this option, but on other
processors the assembly routines will be as if compiled without
- `-pg' and therefore not be included in the call counts.
+ '-pg' and therefore not be included in the call counts.
- On x86 and m68k systems `-pg' and `-fomit-frame-pointer' are
+ 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.
+ 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
+'--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
This instrumenting is not normally a standard system feature and
will require support from an external library, such as
- `http://sourceforge.net/projects/fnccheck/'
+ <https://sourceforge.net/projects/fnccheck/>
- This should be included in `LIBS' during the GMP configure so that
+ This should be included in 'LIBS' during the GMP configure so that
test programs will link. For example,
./configure --enable-profiling=instrument LIBS=-lfc
this case it's only necessary to ensure the correct library is
added when linking an application.
- The x86 assembly code supports this option, but on other
- processors the assembly routines will be as if compiled without
- `-finstrument-functions' meaning time spent in them will
+ The x86 assembly code supports this option, but on other processors
+ the assembly routines will be as if compiled without
+ '-finstrument-functions' meaning time spent in them will
effectively be attributed to their caller.
\1f
Autoconf based applications can easily check whether GMP is installed.
The only thing to be noted is that GMP library symbols from version 3
-onwards have prefixes like `__gmpz'. The following therefore would be
-a simple test,
+onwards have prefixes like '__gmpz'. The following therefore would be a
+simple test,
AC_CHECK_LIB(gmp, __gmpz_init)
- This just uses the default `AC_CHECK_LIB' actions for found or not
+ This just uses the default 'AC_CHECK_LIB' actions for found or not
found, but an application that must have GMP would want to generate an
error if not found. For example,
AC_CHECK_LIB(gmp, __gmpz_init, ,
- [AC_MSG_ERROR([GNU MP not found, see http://gmplib.org/])])
+ [AC_MSG_ERROR([GNU MP not found, see https://gmplib.org/])])
If functions added in some particular version of GMP are required,
-then one of those can be used when checking. For example `mpz_mul_si'
+then one of those can be used when checking. For example 'mpz_mul_si'
was added in GMP 3.1,
AC_CHECK_LIB(gmp, __gmpz_mul_si, ,
[AC_MSG_ERROR(
- [GNU MP not found, or not 3.1 or up, see http://gmplib.org/])])
+ [GNU MP not found, or not 3.1 or up, see https://gmplib.org/])])
- An alternative would be to test the version number in `gmp.h' using
-say `AC_EGREP_CPP'. That would make it possible to test the exact
+ An alternative would be to test the version number in 'gmp.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
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 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,
+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 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 <gmp.h>])
3.15 Emacs
==========
-<C-h C-i> (`info-lookup-symbol') is a good way to find documentation on
+<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 GMP manual can be included in such lookups by putting the
-following in your `.emacs',
+following in your '.emacs',
(eval-after-load "info-look"
'(let ((mode-value (assoc 'c-mode (assoc 'symbol info-lookup-alist))))
4 Reporting Bugs
****************
-If you think you have found a bug in the GMP 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.
+If you think you have found a bug in the GMP 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://gmplib.org/' for patches
-for this release.
+ 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 <https://gmplib.org/> for patches for this
+release.
Please include the following in any report,
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').
+ it's informative ('where' in 'gdb', or '$C' in 'adb').
+
+ * Please do not send core dumps, executables or 'strace's.
- * Please do not send core dumps, executables or `strace's.
+ * The 'configure' options you used when building GMP, if any.
- * The configuration options you used when building GMP, if any.
+ * The output from 'configure', as printed to stdout, with any options
+ used.
- * The name of the compiler and its version. For `gcc', get the
- version with `gcc -v', otherwise perhaps `what `which cc`', or
+ * 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 'uname -a'.
- * The output from running `./config.guess', and from running
- `./configfsf.guess' (might be the same).
+ * The output from running './config.guess', and from running
+ './configfsf.guess' (might be the same).
- * If the bug is related to `configure', then the compressed contents
- of `config.log'.
+ * If the bug is related to 'configure', then the compressed 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'.
+ * 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
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).
+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: <gmp-bugs@gmplib.org>.
*******************
This chapter describes the GMP functions for performing integer
-arithmetic. These functions start with the prefix `mpz_'.
+arithmetic. These functions start with the prefix 'mpz_'.
- GMP integers are stored in objects of type `mpz_t'.
+ GMP integers are stored in objects of type 'mpz_t'.
* Menu:
============================
The functions for integer arithmetic assume that all integer objects are
-initialized. You do that by calling the function `mpz_init'. For
+initialized. You do that by calling the function 'mpz_init'. For
example,
{
Initialize X, and set its value to 0.
-- Function: void mpz_inits (mpz_t X, ...)
- Initialize a NULL-terminated list of `mpz_t' variables, and set
+ Initialize a NULL-terminated list of 'mpz_t' variables, and set
their values to 0.
-- Function: void mpz_init2 (mpz_t X, mp_bitcnt_t N)
Initialize X, with space for N-bit numbers, and set its value to 0.
- Calling this function instead of `mpz_init' or `mpz_inits' is never
+ Calling this function instead of 'mpz_init' or 'mpz_inits' is never
necessary; reallocation is handled automatically by GMP when
needed.
- N is only the initial space, X will grow automatically in the
+ While N defines the initial space, X will grow automatically in the
normal way, if necessary, for subsequent values stored.
- `mpz_init2' makes it possible to avoid such reallocations if a
+ 'mpz_init2' makes it possible to avoid such reallocations if a
maximum size is known in advance.
+ In preparation for an operation, GMP often allocates one limb more
+ than ultimately needed. To make sure GMP will not perform
+ reallocation for X, you need to add the number of bits in
+ 'mp_limb_t' to N.
+
-- Function: void mpz_clear (mpz_t X)
- Free the space occupied by X. Call this function for all `mpz_t'
+ Free the space occupied by X. 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'
+ Free the space occupied by a NULL-terminated list of 'mpz_t'
variables.
-- Function: void mpz_realloc2 (mpz_t X, mp_bitcnt_t N)
preserved if it fits, or is set to 0 if not.
Calling this function is never necessary; reallocation is handled
- automatically by GMP when needed. But this function can be used
- to increase the space for a variable in order to avoid repeated
+ automatically by GMP when needed. But 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.
5.2 Assignment Functions
========================
-These functions assign new values to already initialized integers
-(*note Initializing Integers::).
+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 (mpz_t ROP, const mpz_t OP)
-- Function: void mpz_set_ui (mpz_t ROP, unsigned long int OP)
-- Function: void mpz_set_si (mpz_t ROP, signed long int 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)
+ -- Function: void mpz_set_q (mpz_t ROP, const mpq_t OP)
+ -- Function: void mpz_set_f (mpz_t ROP, const mpf_t OP)
Set the value of ROP from OP.
- `mpz_set_d', `mpz_set_q' and `mpz_set_f' truncate OP to make it an
+ '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)
+ -- Function: int mpz_set_str (mpz_t ROP, const 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.
+ 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.
+ 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.
For convenience, GMP 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...'
+These functions' names have the form 'mpz_init_set...'
Here is an example of using one:
mpz_clear (pie);
}
-Once the integer has been initialized by any of the `mpz_init_set...'
+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!
+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 (mpz_t ROP, const mpz_t OP)
-- Function: void mpz_init_set_ui (mpz_t ROP, unsigned long int OP)
-- Function: void mpz_init_set_si (mpz_t ROP, signed long int 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.
- -- 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
+ -- Function: int mpz_init_set_str (mpz_t ROP, const 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.)
+ 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.)
\1f
File: gmp.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 GMP integers to
-standard C types. Functions for converting _to_ GMP integers are
-described in *Note Assigning Integers:: and *Note I/O of Integers::.
+This section describes functions for converting GMP integers to standard
+C types. Functions for converting _to_ GMP integers are described in
+*note Assigning Integers:: and *note I/O of Integers::.
- -- Function: unsigned long int mpz_get_ui (mpz_t OP)
- Return the value of OP as an `unsigned long'.
+ -- Function: unsigned long int mpz_get_ui (const mpz_t OP)
+ Return the value of OP as an 'unsigned long'.
- If OP is too big to fit an `unsigned long' then just the least
+ If OP is too big to fit an 'unsigned long' then just the least
significant bits that do fit are returned. The sign of OP is
ignored, only the absolute value is used.
- -- Function: signed long int mpz_get_si (mpz_t OP)
- If OP fits into a `signed long int' return the value of OP.
+ -- Function: signed long int mpz_get_si (const mpz_t OP)
+ If OP fits into a 'signed long int' 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 `signed long int', the returned
- result is probably not very useful. To find out if the value will
- fit, use the function `mpz_fits_slong_p'.
+ If OP is too big to fit in a 'signed long int', the returned result
+ is probably not very useful. To find out if the value will fit,
+ use the function 'mpz_fits_slong_p'.
- -- Function: double mpz_get_d (mpz_t OP)
- Convert OP to a `double', truncating if necessary (ie. rounding
+ -- Function: double mpz_get_d (const mpz_t OP)
+ Convert OP to a 'double', truncating if necessary (i.e. 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 (signed long int *EXP, mpz_t OP)
- Convert OP to a `double', truncating if necessary (ie. rounding
+ -- Function: double mpz_get_d_2exp (signed long int *EXP, const mpz_t
+ OP)
+ Convert OP to a 'double', truncating if necessary (i.e. 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'.
+ 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.).
+ This is similar to the standard C 'frexp' function (*note
+ (libc)Normalization Functions::).
- -- Function: char * mpz_get_str (char *STR, int BASE, mpz_t OP)
+ -- Function: char * mpz_get_str (char *STR, int BASE, const mpz_t OP)
Convert OP to 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.
+ 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
+ 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
+ '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'.
+ 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.
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, unsigned long int
- OP2)
+ -- Function: void mpz_add (mpz_t ROP, const mpz_t OP1, const mpz_t OP2)
+ -- Function: void mpz_add_ui (mpz_t ROP, const mpz_t OP1, unsigned long
+ int 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, unsigned long int
- OP2)
- -- Function: void mpz_ui_sub (mpz_t ROP, unsigned long int OP1, mpz_t
- OP2)
+ -- Function: void mpz_sub (mpz_t ROP, const mpz_t OP1, const mpz_t OP2)
+ -- Function: void mpz_sub_ui (mpz_t ROP, const mpz_t OP1, unsigned long
+ int OP2)
+ -- Function: void mpz_ui_sub (mpz_t ROP, unsigned long int OP1, const
+ 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, long int OP2)
- -- Function: void mpz_mul_ui (mpz_t ROP, mpz_t OP1, unsigned long int
- OP2)
+ -- Function: void mpz_mul (mpz_t ROP, const mpz_t OP1, const mpz_t OP2)
+ -- Function: void mpz_mul_si (mpz_t ROP, const mpz_t OP1, long int OP2)
+ -- Function: void mpz_mul_ui (mpz_t ROP, const mpz_t OP1, unsigned long
+ int 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, unsigned long
- int OP2)
+ -- Function: void mpz_addmul (mpz_t ROP, const mpz_t OP1, const mpz_t
+ OP2)
+ -- Function: void mpz_addmul_ui (mpz_t ROP, const mpz_t OP1, unsigned
+ long int 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, unsigned long
- int OP2)
+ -- Function: void mpz_submul (mpz_t ROP, const mpz_t OP1, const mpz_t
+ OP2)
+ -- Function: void mpz_submul_ui (mpz_t ROP, const mpz_t OP1, unsigned
+ long int OP2)
Set ROP to ROP - OP1 times OP2.
- -- Function: void mpz_mul_2exp (mpz_t ROP, mpz_t OP1, mp_bitcnt_t OP2)
+ -- Function: void mpz_mul_2exp (mpz_t ROP, const 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)
+ -- Function: void mpz_neg (mpz_t ROP, const mpz_t OP)
Set ROP to -OP.
- -- Function: void mpz_abs (mpz_t ROP, mpz_t OP)
+ -- Function: void mpz_abs (mpz_t ROP, const mpz_t OP)
Set ROP to the absolute value of OP.
\1f
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 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.
+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: unsigned long int mpz_cdiv_q_ui (mpz_t Q, mpz_t N,
+ -- Function: void mpz_cdiv_q (mpz_t Q, const mpz_t N, const mpz_t D)
+ -- Function: void mpz_cdiv_r (mpz_t R, const mpz_t N, const mpz_t D)
+ -- Function: void mpz_cdiv_qr (mpz_t Q, mpz_t R, const mpz_t N, const
+ mpz_t D)
+
+ -- Function: unsigned long int mpz_cdiv_q_ui (mpz_t Q, const mpz_t N,
unsigned long int D)
- -- Function: unsigned long int mpz_cdiv_r_ui (mpz_t R, mpz_t N,
+ -- Function: unsigned long int mpz_cdiv_r_ui (mpz_t R, const mpz_t N,
unsigned long int D)
-- Function: unsigned long int mpz_cdiv_qr_ui (mpz_t Q, mpz_t R,
- mpz_t N, unsigned long int D)
- -- Function: unsigned long int mpz_cdiv_ui (mpz_t N,
+ const mpz_t N, unsigned long int D)
+ -- Function: unsigned long int mpz_cdiv_ui (const mpz_t N,
unsigned long int 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: unsigned long int mpz_fdiv_q_ui (mpz_t Q, mpz_t N,
+ -- Function: void mpz_cdiv_q_2exp (mpz_t Q, const mpz_t N,
+ mp_bitcnt_t B)
+ -- Function: void mpz_cdiv_r_2exp (mpz_t R, const mpz_t N,
+ mp_bitcnt_t B)
+
+ -- Function: void mpz_fdiv_q (mpz_t Q, const mpz_t N, const mpz_t D)
+ -- Function: void mpz_fdiv_r (mpz_t R, const mpz_t N, const mpz_t D)
+ -- Function: void mpz_fdiv_qr (mpz_t Q, mpz_t R, const mpz_t N, const
+ mpz_t D)
+
+ -- Function: unsigned long int mpz_fdiv_q_ui (mpz_t Q, const mpz_t N,
unsigned long int D)
- -- Function: unsigned long int mpz_fdiv_r_ui (mpz_t R, mpz_t N,
+ -- Function: unsigned long int mpz_fdiv_r_ui (mpz_t R, const mpz_t N,
unsigned long int D)
-- Function: unsigned long int mpz_fdiv_qr_ui (mpz_t Q, mpz_t R,
- mpz_t N, unsigned long int D)
- -- Function: unsigned long int mpz_fdiv_ui (mpz_t N,
+ const mpz_t N, unsigned long int D)
+ -- Function: unsigned long int mpz_fdiv_ui (const mpz_t N,
unsigned long int 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: unsigned long int mpz_tdiv_q_ui (mpz_t Q, mpz_t N,
+ -- Function: void mpz_fdiv_q_2exp (mpz_t Q, const mpz_t N,
+ mp_bitcnt_t B)
+ -- Function: void mpz_fdiv_r_2exp (mpz_t R, const mpz_t N,
+ mp_bitcnt_t B)
+
+ -- Function: void mpz_tdiv_q (mpz_t Q, const mpz_t N, const mpz_t D)
+ -- Function: void mpz_tdiv_r (mpz_t R, const mpz_t N, const mpz_t D)
+ -- Function: void mpz_tdiv_qr (mpz_t Q, mpz_t R, const mpz_t N, const
+ mpz_t D)
+
+ -- Function: unsigned long int mpz_tdiv_q_ui (mpz_t Q, const mpz_t N,
unsigned long int D)
- -- Function: unsigned long int mpz_tdiv_r_ui (mpz_t R, mpz_t N,
+ -- Function: unsigned long int mpz_tdiv_r_ui (mpz_t R, const mpz_t N,
unsigned long int D)
-- Function: unsigned long int mpz_tdiv_qr_ui (mpz_t Q, mpz_t R,
- mpz_t N, unsigned long int D)
- -- Function: unsigned long int mpz_tdiv_ui (mpz_t N,
+ const mpz_t N, unsigned long int D)
+ -- Function: unsigned long int mpz_tdiv_ui (const mpz_t N,
unsigned long int 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)
+
+ -- Function: void mpz_tdiv_q_2exp (mpz_t Q, const mpz_t N,
+ mp_bitcnt_t B)
+ -- Function: void mpz_tdiv_r_2exp (mpz_t R, const 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
+ '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".
+ * '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".
+ * '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".
+ * '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,
+ 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
+ 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 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
+ 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.
+ '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: unsigned long int mpz_mod_ui (mpz_t R, mpz_t N,
+ -- Function: void mpz_mod (mpz_t R, const mpz_t N, const mpz_t D)
+ -- Function: unsigned long int mpz_mod_ui (mpz_t R, const mpz_t N,
unsigned long int D)
- Set R to N `mod' D. The sign of the divisor is ignored; the
- result is always non-negative.
+ 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
+ '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, unsigned long D)
- Set Q to N/D. These functions produce correct results only when
- it is known in advance that D divides N.
+ -- Function: void mpz_divexact (mpz_t Q, const mpz_t N, const mpz_t D)
+ -- Function: void mpz_divexact_ui (mpz_t Q, const mpz_t N, unsigned
+ long 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, unsigned long int D)
- -- Function: int mpz_divisible_2exp_p (mpz_t N, mp_bitcnt_t B)
+ -- Function: int mpz_divisible_p (const mpz_t N, const mpz_t D)
+ -- Function: int mpz_divisible_ui_p (const mpz_t N, unsigned long int
+ D)
+ -- Function: int mpz_divisible_2exp_p (const 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.
+ '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, unsigned long int C,
- unsigned long int D)
- -- Function: int mpz_congruent_2exp_p (mpz_t N, mpz_t C, mp_bitcnt_t B)
+ -- Function: int mpz_congruent_p (const mpz_t N, const mpz_t C, const
+ mpz_t D)
+ -- Function: int mpz_congruent_ui_p (const mpz_t N, unsigned long int
+ C, unsigned long int D)
+ -- Function: int mpz_congruent_2exp_p (const mpz_t N, const 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.
+ '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
+ 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.
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, unsigned long
- int EXP, mpz_t MOD)
+ -- Function: void mpz_powm (mpz_t ROP, const mpz_t BASE, const mpz_t
+ EXP, const mpz_t MOD)
+ -- Function: void mpz_powm_ui (mpz_t ROP, const mpz_t BASE, unsigned
+ long int EXP, const mpz_t MOD)
Set ROP to (BASE raised to EXP) modulo MOD.
- Negative EXP is supported if an inverse BASE^-1 mod MOD exists
- (see `mpz_invert' in *Note Number Theoretic Functions::). If an
+ Negative EXP is supported if the 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_powm_sec (mpz_t ROP, mpz_t BASE, mpz_t EXP,
- mpz_t MOD)
+ -- Function: void mpz_powm_sec (mpz_t ROP, const mpz_t BASE, const
+ mpz_t EXP, const mpz_t MOD)
Set ROP to (BASE raised to EXP) modulo MOD.
It is required that EXP > 0 and that MOD is odd.
is intended for cryptographic purposes, where resilience to
side-channel attacks is desired.
- -- Function: void mpz_pow_ui (mpz_t ROP, mpz_t BASE, unsigned long int
- EXP)
+ -- Function: void mpz_pow_ui (mpz_t ROP, const mpz_t BASE, unsigned
+ long int EXP)
-- Function: void mpz_ui_pow_ui (mpz_t ROP, unsigned long int BASE,
unsigned long int EXP)
Set ROP to BASE raised to EXP. The case 0^0 yields 1.
5.8 Root Extraction Functions
=============================
- -- Function: int mpz_root (mpz_t ROP, mpz_t OP, unsigned long int 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: int mpz_root (mpz_t ROP, const mpz_t OP, unsigned long int
+ 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_rootrem (mpz_t ROOT, mpz_t REM, mpz_t U,
+ -- Function: void mpz_rootrem (mpz_t ROOT, mpz_t REM, const mpz_t U,
unsigned long int N)
- Set ROOT to the truncated integer part of the Nth root of U. Set
+ 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_sqrt (mpz_t ROP, const 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)
+ -- Function: void mpz_sqrtrem (mpz_t ROP1, mpz_t ROP2, const 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
+ 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)
+ -- Function: int mpz_perfect_power_p (const 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.
+ 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
+ -- Function: int mpz_perfect_square_p (const 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.
\1f
5.9 Number Theoretic Functions
==============================
- -- Function: int mpz_probab_prime_p (mpz_t N, int REPS)
+ -- Function: int mpz_probab_prime_p (const 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.
+ 0 if N is definitely non-prime.
- 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".
+ This function performs some trial divisions, a Baillie-PSW probable
+ prime test, then REPS-24 Miller-Rabin probabilistic primality
+ tests. A higher REPS value will reduce the chances of a non-prime
+ being identified as "probably prime". A composite number will be
+ identified as a prime with an asymptotic probability of less than
+ 4^(-REPS). Reasonable values of REPS are between 15 and 50.
- 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.
+ GMP versions up to and including 6.1.2 did not use the Baillie-PSW
+ primality test. In those older versions of GMP, this function
+ performed REPS Miller-Rabin tests.
- -- Function: void mpz_nextprime (mpz_t ROP, mpz_t OP)
+ -- Function: void mpz_nextprime (mpz_t ROP, const 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.
- -- Function: void mpz_gcd (mpz_t ROP, mpz_t OP1, mpz_t OP2)
+ -- Function: void mpz_gcd (mpz_t ROP, const mpz_t OP1, const 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.
+ Except if both inputs are zero; then this function defines gcd(0,0)
+ = 0.
- -- Function: unsigned long int mpz_gcd_ui (mpz_t ROP, mpz_t OP1,
+ -- Function: unsigned long int mpz_gcd_ui (mpz_t ROP, const mpz_t OP1,
unsigned long int OP2)
Compute the greatest common divisor of OP1 and OP2. If ROP is not
- `NULL', store the result there.
+ 'NULL', store the result there.
- If the result is small enough to fit in an `unsigned long int', it
+ If the result is small enough to fit in an 'unsigned long int', 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)
+ -- Function: void mpz_gcdext (mpz_t G, mpz_t S, mpz_t T, const mpz_t A,
+ const 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. The value
- in G is always positive, even if one or both of A and B are
- negative. The values in S and T are chosen such that abs(S) <=
- abs(B) and abs(T) <= abs(A).
+ set S and T to coefficients satisfying A*S + B*T = G. The value in
+ G is always positive, even if one or both of A and B are negative
+ (or zero if both inputs are zero). The values in S and T are
+ chosen such that normally, abs(S) < abs(B) / (2 G) and abs(T) <
+ abs(A) / (2 G), and these relations define S and T uniquely. There
+ are a few exceptional cases:
- If T is `NULL' then that value is not computed.
+ If abs(A) = abs(B), then S = 0, T = sgn(B).
- -- Function: void mpz_lcm (mpz_t ROP, mpz_t OP1, mpz_t OP2)
- -- Function: void mpz_lcm_ui (mpz_t ROP, mpz_t OP1, unsigned long 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.
+ Otherwise, S = sgn(A) if B = 0 or abs(B) = 2 G, and T = sgn(B) if A
+ = 0 or abs(A) = 2 G.
+
+ In all cases, S = 0 if and only if G = abs(B), i.e., if B divides A
+ or A = B = 0.
+
+ If T or G is 'NULL' then that value is not computed.
+
+ -- Function: void mpz_lcm (mpz_t ROP, const mpz_t OP1, const mpz_t OP2)
+ -- Function: void mpz_lcm_ui (mpz_t ROP, const mpz_t OP1, unsigned long
+ 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)
+ -- Function: int mpz_invert (mpz_t ROP, const mpz_t OP1, const 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.
+ satisfy 0 <= ROP < abs(OP2) (with ROP = 0 possible only when
+ abs(OP2) = 1, i.e., in the somewhat degenerate zero ring). If an
+ inverse doesn't exist the return value is zero and ROP is
+ undefined. The behaviour of this function is undefined when OP2 is
+ zero.
- -- Function: int mpz_jacobi (mpz_t A, mpz_t B)
+ -- Function: int mpz_jacobi (const mpz_t A, const 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
+ -- Function: int mpz_legendre (const mpz_t A, const 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, long B)
- -- Function: int mpz_kronecker_ui (mpz_t A, unsigned long B)
- -- Function: int mpz_si_kronecker (long A, mpz_t B)
- -- Function: int mpz_ui_kronecker (unsigned long A, mpz_t B)
+ -- Function: int mpz_kronecker (const mpz_t A, const mpz_t B)
+ -- Function: int mpz_kronecker_si (const mpz_t A, long B)
+ -- Function: int mpz_kronecker_ui (const mpz_t A, unsigned long B)
+ -- Function: int mpz_si_kronecker (long A, const mpz_t B)
+ -- Function: int mpz_ui_kronecker (unsigned long A, const 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.
+ 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'.
+ References::), or any number theory textbook. See also the example
+ program 'demos/qcn.c' which uses 'mpz_kronecker_ui'.
- -- 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
+ -- Function: mp_bitcnt_t mpz_remove (mpz_t ROP, const mpz_t OP, const
+ 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, unsigned long int OP)
- Set ROP to OP!, the factorial of OP.
+ -- Function: void mpz_fac_ui (mpz_t ROP, unsigned long int N)
+ -- Function: void mpz_2fac_ui (mpz_t ROP, unsigned long int N)
+ -- Function: void mpz_mfac_uiui (mpz_t ROP, unsigned long int N,
+ unsigned long int M)
+ Set ROP to the factorial of N: 'mpz_fac_ui' computes the plain
+ factorial N!, 'mpz_2fac_ui' computes the double-factorial N!!, and
+ 'mpz_mfac_uiui' the M-multi-factorial N!^(M).
+
+ -- Function: void mpz_primorial_ui (mpz_t ROP, unsigned long int N)
+ Set ROP to the primorial of N, i.e. the product of all positive
+ prime numbers <=N.
- -- Function: void mpz_bin_ui (mpz_t ROP, mpz_t N, unsigned long int K)
+ -- Function: void mpz_bin_ui (mpz_t ROP, const mpz_t N, unsigned long
+ int K)
-- Function: void mpz_bin_uiui (mpz_t ROP, unsigned long int N,
unsigned long int 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
+ 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, unsigned long int N)
-- Function: void mpz_fib2_ui (mpz_t FN, mpz_t FNSUB1, unsigned long
int 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].
+ '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
+ 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, unsigned long int N)
- -- Function: void mpz_lucnum2_ui (mpz_t LN, mpz_t LNSUB1, unsigned
- long int 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].
+ -- Function: void mpz_lucnum2_ui (mpz_t LN, mpz_t LNSUB1, unsigned long
+ int 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]
+ 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
+ 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.
\1f
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, signed long int OP2)
- -- Macro: int mpz_cmp_ui (mpz_t OP1, unsigned long int OP2)
- Compare OP1 and OP2. Return a positive value if OP1 > OP2, zero
- if OP1 = OP2, or a negative value if OP1 < OP2.
+ -- Function: int mpz_cmp (const mpz_t OP1, const mpz_t OP2)
+ -- Function: int mpz_cmp_d (const mpz_t OP1, double OP2)
+ -- Macro: int mpz_cmp_si (const mpz_t OP1, signed long int OP2)
+ -- Macro: int mpz_cmp_ui (const mpz_t OP1, unsigned long int 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
+ '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, unsigned long int OP2)
+ -- Function: int mpz_cmpabs (const mpz_t OP1, const mpz_t OP2)
+ -- Function: int mpz_cmpabs_d (const mpz_t OP1, double OP2)
+ -- Function: int mpz_cmpabs_ui (const mpz_t OP1, unsigned long int 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
+ 'mpz_cmpabs_d' can be called with an infinity, but results are
undefined for a NaN.
- -- Macro: int mpz_sgn (mpz_t OP)
+ -- Macro: int mpz_sgn (const 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.
+ This function is actually implemented as a macro. It evaluates its
+ argument multiple times.
\1f
File: gmp.info, Node: Integer Logic and Bit Fiddling, Next: I/O of Integers, Prev: Integer Comparisons, Up: Integer Functions
(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)
+ -- Function: void mpz_and (mpz_t ROP, const mpz_t OP1, const mpz_t OP2)
Set ROP to OP1 bitwise-and OP2.
- -- Function: void mpz_ior (mpz_t ROP, mpz_t OP1, mpz_t OP2)
+ -- Function: void mpz_ior (mpz_t ROP, const mpz_t OP1, const mpz_t OP2)
Set ROP to OP1 bitwise inclusive-or OP2.
- -- Function: void mpz_xor (mpz_t ROP, mpz_t OP1, mpz_t OP2)
+ -- Function: void mpz_xor (mpz_t ROP, const mpz_t OP1, const mpz_t OP2)
Set ROP to OP1 bitwise exclusive-or OP2.
- -- Function: void mpz_com (mpz_t ROP, mpz_t OP)
+ -- Function: void mpz_com (mpz_t ROP, const 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 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
- `mp_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)
+ -- Function: mp_bitcnt_t mpz_popcount (const 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 the largest possible
+ 'mp_bitcnt_t'.
+
+ -- Function: mp_bitcnt_t mpz_hamdist (const mpz_t OP1, const 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 'mp_bitcnt_t'.
+
+ -- Function: mp_bitcnt_t mpz_scan0 (const mpz_t OP, mp_bitcnt_t
+ STARTING_BIT)
+ -- Function: mp_bitcnt_t mpz_scan1 (const 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
- negative number, or `mpz_scan1' past the end of a nonnegative
+ 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
+ negative number, or 'mpz_scan1' past the end of a nonnegative
number.
-- Function: void mpz_setbit (mpz_t ROP, mp_bitcnt_t BIT_INDEX)
-- 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)
+ -- Function: int mpz_tstbit (const mpz_t OP, mp_bitcnt_t BIT_INDEX)
Test bit BIT_INDEX in OP and return 0 or 1 accordingly.
\1f
===============================
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.
+output to a stdio stream, of 'mpz' numbers. 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 `gmp.h', since that will allow `gmp.h' to define
+'stdio.h' before 'gmp.h', since that will allow 'gmp.h' to define
prototypes for these functions.
- -- Function: size_t mpz_out_str (FILE *STREAM, int BASE, mpz_t OP)
+ See also *note Formatted Output:: and *note Formatted Input::.
+
+ -- Function: size_t mpz_out_str (FILE *STREAM, int BASE, const 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.
+ 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.
+ 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.
+ 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.
+ 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)
+ -- Function: size_t mpz_out_raw (FILE *STREAM, const 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'.
+ The output can be read with 'mpz_inp_raw'.
- Return the number of bytes written, or if an error occurred,
- return 0.
+ 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,
+ 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
+ '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
+ 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.
The random number functions of GMP 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.
+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)
2^N-1, inclusive.
The variable STATE must be initialized by calling one of the
- `gmp_randinit' functions (*Note Random State Initialization::)
+ 'gmp_randinit' functions (*note Random State Initialization::)
before invoking this function.
- -- Function: void mpz_urandomm (mpz_t ROP, gmp_randstate_t STATE,
+ -- Function: void mpz_urandomm (mpz_t ROP, gmp_randstate_t STATE, const
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::)
+ 'gmp_randinit' functions (*note Random State Initialization::)
before invoking this function.
-- Function: void mpz_rrandomb (mpz_t ROP, gmp_randstate_t STATE,
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.
+ more likely to trigger corner-case bugs. The random number will be
+ in the range 2^(N-1) to 2^N-1, inclusive.
The variable STATE must be initialized by calling one of the
- `gmp_randinit' functions (*Note Random State Initialization::)
+ 'gmp_randinit' functions (*note Random State Initialization::)
before invoking this function.
-- Function: void mpz_random (mpz_t ROP, mp_size_t MAX_SIZE)
Generate a random integer of at most MAX_SIZE limbs. The generated
random number doesn't satisfy any particular requirements of
- randomness. Negative random numbers are generated when MAX_SIZE
- is negative.
+ randomness. Negative random numbers are generated when MAX_SIZE is
+ negative.
- This function is obsolete. Use `mpz_urandomb' or `mpz_urandomm'
+ This function is obsolete. Use 'mpz_urandomb' or 'mpz_urandomm'
instead.
-- Function: void mpz_random2 (mpz_t ROP, mp_size_t MAX_SIZE)
Generate a random integer of at most MAX_SIZE limbs, 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.
+ 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.
Negative random numbers are generated when MAX_SIZE is negative.
- This function is obsolete. Use `mpz_rrandomb' instead.
+ This function is obsolete. Use 'mpz_rrandomb' instead.
\1f
File: gmp.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
+'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,
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.
+ 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'.
+ 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 `unsigned long' data, most
+ Here's an example converting an array of 'unsigned long' data, most
significant element first, and host byte order within each value.
unsigned long a[20];
/* Initialize Z and A */
mpz_import (z, 20, 1, sizeof(a[0]), 0, 0, a);
- This example assumes the full `sizeof' bytes are used for data in
+ This example assumes the full 'sizeof' bytes are used for data in
the given type, which is usually true, and certainly true for
- `unsigned long' 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'.
+ 'unsigned long' 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)
+ size_t SIZE, int ENDIAN, size_t NAILS, const mpz_t OP)
Fill ROP with word data from OP.
The parameters specify the format of the data produced. Each word
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 GMP allocation
- function (*note Custom Allocation::). In either case the return
- value is the destination used, either ROP or the allocated block.
+ 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 GMP 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.
+ 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
+ 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
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).
+ '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;
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 int',
- `signed long int', `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)
+ -- Function: int mpz_fits_ulong_p (const mpz_t OP)
+ -- Function: int mpz_fits_slong_p (const mpz_t OP)
+ -- Function: int mpz_fits_uint_p (const mpz_t OP)
+ -- Function: int mpz_fits_sint_p (const mpz_t OP)
+ -- Function: int mpz_fits_ushort_p (const mpz_t OP)
+ -- Function: int mpz_fits_sshort_p (const mpz_t OP)
+ Return non-zero iff the value of OP fits in an 'unsigned long int',
+ 'signed long int', 'unsigned int', 'signed int', 'unsigned short
+ int', or 'signed short int', respectively. Otherwise, return zero.
+
+ -- Macro: int mpz_odd_p (const mpz_t OP)
+ -- Macro: int mpz_even_p (const 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 (const 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 62. 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.
+ BASE. BASE can vary from 2 to 62. 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
+ 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
+ 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.)
-- Function: void mpz_array_init (mpz_t INTEGER_ARRAY, mp_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 is an obsolete function. Do not use it.*
-- 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.
+ '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)
+ -- Function: mp_limb_t mpz_getlimbn (const 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'.
+ '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)
+ -- Function: size_t mpz_size (const mpz_t OP)
Return the size of OP measured in number of limbs. If OP is zero,
the returned value will be zero.
+ -- Function: const mp_limb_t * mpz_limbs_read (const mpz_t X)
+ Return a pointer to the limb array representing the absolute value
+ of X. The size of the array is 'mpz_size(X)'. Intended for read
+ access only.
+
+ -- Function: mp_limb_t * mpz_limbs_write (mpz_t X, mp_size_t N)
+ -- Function: mp_limb_t * mpz_limbs_modify (mpz_t X, mp_size_t N)
+ Return a pointer to the limb array, intended for write access. The
+ array is reallocated as needed, to make room for N limbs. Requires
+ N > 0. The 'mpz_limbs_modify' function returns an array that holds
+ the old absolute value of X, while 'mpz_limbs_write' may destroy
+ the old value and return an array with unspecified contents.
+
+ -- Function: void mpz_limbs_finish (mpz_t X, mp_size_t S)
+ Updates the internal size field of X. Used after writing to the
+ limb array pointer returned by 'mpz_limbs_write' or
+ 'mpz_limbs_modify' is completed. The array should contain abs(S)
+ valid limbs, representing the new absolute value for X, and the
+ sign of X is taken from the sign of S. This function never
+ reallocates X, so the limb pointer remains valid.
+
+ void foo (mpz_t x)
+ {
+ mp_size_t n, i;
+ mp_limb_t *xp;
+
+ n = mpz_size (x);
+ xp = mpz_limbs_modify (x, 2*n);
+ for (i = 0; i < n; i++)
+ xp[n+i] = xp[n-1-i];
+ mpz_limbs_finish (x, mpz_sgn (x) < 0 ? - 2*n : 2*n);
+ }
+
+ -- Function: mpz_srcptr mpz_roinit_n (mpz_t X, const mp_limb_t *XP,
+ mp_size_t XS)
+ Special initialization of X, using the given limb array and size.
+ X should be treated as read-only: it can be passed safely as input
+ to any mpz function, but not as an output. The array XP must point
+ to at least a readable limb, its size is abs(XS), and the sign of X
+ is the sign of XS. For convenience, the function returns X, but
+ cast to a const pointer type.
+
+ void foo (mpz_t x)
+ {
+ static const mp_limb_t y[3] = { 0x1, 0x2, 0x3 };
+ mpz_t tmp;
+ mpz_add (x, x, mpz_roinit_n (tmp, y, 3));
+ }
+
+ -- Macro: mpz_t MPZ_ROINIT_N (mp_limb_t *XP, mp_size_t XS)
+ This macro expands to an initializer which can be assigned to an
+ mpz_t variable. The limb array XP must point to at least a
+ readable limb, moreover, unlike the 'mpz_roinit_n' function, the
+ array must be normalized: if XS is non-zero, then 'XP[abs(XS)-1]'
+ must be non-zero. Intended primarily for constant values. Using
+ it for non-constant values requires a C compiler supporting C99.
+
+ void foo (mpz_t x)
+ {
+ static const mp_limb_t ya[3] = { 0x1, 0x2, 0x3 };
+ static const mpz_t y = MPZ_ROINIT_N ((mp_limb_t *) ya, 3);
+
+ mpz_add (x, x, y);
+ }
+
\1f
File: gmp.info, Node: Rational Number Functions, Next: Floating-point Functions, Prev: Integer Functions, Up: Top
***************************
This chapter describes the GMP functions for performing arithmetic on
-rational numbers. These functions start with the prefix `mpq_'.
+rational numbers. These functions start with the prefix 'mpq_'.
- Rational numbers are stored in objects of type `mpq_t'.
+ 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
+form, and canonicalize their result. The canonical form means that the
denominator and the numerator have no common factors, and that the
denominator is positive. Zero has the unique representation 0/1.
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.
+ Remove any factors that are common to the numerator and denominator
+ of OP, and make the denominator positive.
* Menu:
===========================================
-- Function: void mpq_init (mpq_t X)
- Initialize X 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.
+ Initialize X 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
+ Initialize a NULL-terminated list of 'mpq_t' variables, and set
their values to 0/1.
-- Function: void mpq_clear (mpq_t X)
Free the space occupied by X. Make sure to call this function for
- all `mpq_t' variables when you are done with them.
+ 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'
+ 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)
+ -- Function: void mpq_set (mpq_t ROP, const mpq_t OP)
+ -- Function: void mpq_set_z (mpq_t ROP, const mpz_t OP)
Assign ROP from OP.
-- Function: void mpq_set_ui (mpq_t ROP, unsigned long int OP1,
-- Function: void mpq_set_si (mpq_t ROP, signed long int OP1, unsigned
long int 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
+ 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)
+ -- Function: int mpq_set_str (mpq_t ROP, const 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 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.
+ '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.
+ 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.
6.2 Conversion Functions
========================
- -- Function: double mpq_get_d (mpq_t OP)
- Convert OP to a `double', truncating if necessary (ie. rounding
+ -- Function: double mpq_get_d (const mpq_t OP)
+ Convert OP to a 'double', truncating if necessary (i.e. 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.
+ 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)
+ -- Function: void mpq_set_f (mpq_t ROP, const 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'.
+ -- Function: char * mpq_get_str (char *STR, int BASE, const mpq_t OP)
+ Convert OP to a string of digits in base BASE. The base argument
+ may vary from 2 to 62 or from -2 to -36. The string will be of the
+ form 'num/den', or if the denominator is 1 then just 'num'.
+
+ 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
+ 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
+ '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
+ 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)
6.3 Arithmetic Functions
========================
- -- Function: void mpq_add (mpq_t SUM, mpq_t ADDEND1, mpq_t ADDEND2)
+ -- Function: void mpq_add (mpq_t SUM, const mpq_t ADDEND1, const mpq_t
+ ADDEND2)
Set SUM to ADDEND1 + ADDEND2.
- -- Function: void mpq_sub (mpq_t DIFFERENCE, mpq_t MINUEND, mpq_t
- SUBTRAHEND)
+ -- Function: void mpq_sub (mpq_t DIFFERENCE, const mpq_t MINUEND, const
+ mpq_t SUBTRAHEND)
Set DIFFERENCE to MINUEND - SUBTRAHEND.
- -- Function: void mpq_mul (mpq_t PRODUCT, mpq_t MULTIPLIER, mpq_t
- MULTIPLICAND)
+ -- Function: void mpq_mul (mpq_t PRODUCT, const mpq_t MULTIPLIER, const
+ mpq_t MULTIPLICAND)
Set PRODUCT to MULTIPLIER times MULTIPLICAND.
- -- Function: void mpq_mul_2exp (mpq_t ROP, mpq_t OP1, mp_bitcnt_t OP2)
+ -- Function: void mpq_mul_2exp (mpq_t ROP, const 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)
+ -- Function: void mpq_div (mpq_t QUOTIENT, const mpq_t DIVIDEND, const
+ mpq_t DIVISOR)
Set QUOTIENT to DIVIDEND/DIVISOR.
- -- Function: void mpq_div_2exp (mpq_t ROP, mpq_t OP1, mp_bitcnt_t OP2)
+ -- Function: void mpq_div_2exp (mpq_t ROP, const 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)
+ -- Function: void mpq_neg (mpq_t NEGATED_OPERAND, const mpq_t OPERAND)
Set NEGATED_OPERAND to -OPERAND.
- -- Function: void mpq_abs (mpq_t ROP, mpq_t OP)
+ -- Function: void mpq_abs (mpq_t ROP, const mpq_t OP)
Set ROP to the absolute value of OP.
- -- Function: void mpq_inv (mpq_t INVERTED_NUMBER, mpq_t NUMBER)
+ -- Function: void mpq_inv (mpq_t INVERTED_NUMBER, const mpq_t NUMBER)
Set INVERTED_NUMBER to 1/NUMBER. If the new denominator is zero,
this routine will divide by zero.
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.
+ -- Function: int mpq_cmp (const mpq_t OP1, const mpq_t OP2)
+ -- Function: int mpq_cmp_z (const mpq_t OP1, const mpz_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'.
+ To determine if two rationals are equal, 'mpq_equal' is faster than
+ 'mpq_cmp'.
- -- Macro: int mpq_cmp_ui (mpq_t OP1, unsigned long int NUM2, unsigned
- long int DEN2)
- -- Macro: int mpq_cmp_si (mpq_t OP1, long int NUM2, unsigned long int
- DEN2)
+ -- Macro: int mpq_cmp_ui (const mpq_t OP1, unsigned long int NUM2,
+ unsigned long int DEN2)
+ -- Macro: int mpq_cmp_si (const mpq_t OP1, long int NUM2, unsigned long
+ int 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.
These functions are implemented as a macros and evaluate their
arguments multiple times.
- -- Macro: int mpq_sgn (mpq_t OP)
+ -- Macro: int mpq_sgn (const 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.
+ argument multiple times.
- -- Function: int mpq_equal (mpq_t OP1, mpq_t OP2)
+ -- Function: int mpq_equal (const mpq_t OP1, const 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,
+ non-equal. Although 'mpq_cmp' can be used for the same purpose,
this function is much faster.
\1f
6.5 Applying Integer Functions to Rationals
===========================================
-The set of `mpq' functions is quite small. In particular, there are few
+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'.
+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'.
+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)
+ -- Macro: mpz_t mpq_numref (const mpq_t OP)
+ -- Macro: mpz_t mpq_denref (const mpq_t OP)
Return a reference to the numerator and denominator of OP,
- respectively. The `mpz' functions can be used on the result of
+ 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)
+ -- Function: void mpq_get_num (mpz_t NUMERATOR, const mpq_t RATIONAL)
+ -- Function: void mpq_get_den (mpz_t DENOMINATOR, const mpq_t RATIONAL)
+ -- Function: void mpq_set_num (mpq_t RATIONAL, const mpz_t NUMERATOR)
+ -- Function: void mpq_set_den (mpq_t RATIONAL, const 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.
+ 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.
\1f
File: gmp.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 `gmp.h', since that will allow `gmp.h' to define prototypes for
-these functions.
+Functions that perform input from a stdio stream, and functions that
+output to a stdio stream, of 'mpq' numbers. 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 'gmp.h', since that will allow 'gmp.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.
+ See also *note Formatted Output:: and *note Formatted Input::.
- -- Function: size_t mpq_out_str (FILE *STREAM, int BASE, mpq_t OP)
+ -- Function: size_t mpq_out_str (FILE *STREAM, int BASE, const 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'.
+ BASE. The base argument may vary from 2 to 62 or from -2 to -36.
+ Output is in the form 'num/den' or if the denominator is 1 then
+ just 'num'.
+
+ 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.
+ 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.
+ 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 62, or can be 0 in which case the
+ leading characters of the string determine the base, '0x' or '0X'
+ for hexadecimal, '0b' and '0B' for binary, '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.
\1f
File: gmp.info, Node: Floating-point Functions, Next: Low-level Functions, Prev: Rational Number Functions, Up: Top
7 Floating-point Functions
**************************
-GMP floating point numbers are stored in objects of type `mpf_t' and
-functions operating on them have an `mpf_' prefix.
+GMP 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 mantissa of each float has a user-selectable precision, in
+practice only limited by available memory. Each variable has its own
+precision, and that can be increased or decreased at any time. This
+selectable precision is a minimum value, GMP rounds it up to a whole
+limb.
- The exponent of each float is a fixed precision, one machine word on
+ The accuracy of a calculation is determined by the priorly set
+precision of the destination variable and the numeric values of the
+input variables. Input variables' set precisions do not affect
+calculations (except indirectly as their values might have been affected
+when they were assigned).
+
+ The exponent of each float has 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 `long'.
-
- 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, GMP 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
+2^-68719476768 to 2^68719476736, or on a 64-bit system this will be much
+greater. Note however that '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 'long'.
+
+ Each variable keeps track of 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 variable's
+selected precision is high. This is a performance optimization; it does
+not affect the numeric results.
+
+ Internally, GMP sometimes calculates with higher precision than that
+of the destination variable in order to limit errors. Final results are
+always truncated to the destination variable's precision.
+
+ The mantissa is stored in binary. 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
+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.)
+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.
+ The '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.
- Note that the `mpf' functions are _not_ intended as a smooth
+ 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.
+ New projects should consider using the GMP extension library MPFR
+(<http://mpfr.org>) instead. MPFR provides well-defined precision and
+accurate rounding, and thereby naturally extends IEEE P754.
+
* Menu:
* Initializing Floats::
-- 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
+ 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
+ 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
+ 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'.
+ '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.
+ 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
+ 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'.
+ 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.
+ 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'
+ Free the space occupied by a NULL-terminated list of 'mpf_t'
variables.
Here is an example on how to initialize floating-point variables:
the computation precision closely match the actual accurate part of the
numbers.
- -- Function: mp_bitcnt_t mpf_get_prec (mpf_t OP)
+ -- Function: mp_bitcnt_t mpf_get_prec (const 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
+ 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)
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'.
+ '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
+ 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_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
+ '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.
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 (mpf_t ROP, const mpf_t OP)
-- Function: void mpf_set_ui (mpf_t ROP, unsigned long int OP)
-- Function: void mpf_set_si (mpf_t ROP, signed long int 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)
+ -- Function: void mpf_set_z (mpf_t ROP, const mpz_t OP)
+ -- Function: void mpf_set_q (mpf_t ROP, const mpq_t OP)
Set the value of ROP from OP.
- -- Function: int mpf_set_str (mpf_t ROP, char *STR, int BASE)
+ -- Function: int mpf_set_str (mpf_t ROP, const char *STR, int BASE)
Set the value of ROP from the string in STR. The string is of the
- form `M@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'.
+ form 'M@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
represent the usual 10..35 while lower-case letter represent
36..61.
- Unlike the corresponding `mpz' function, the base will not be
+ 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.
+ 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?]
+ 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.
+ Swap ROP1 and ROP2 efficiently. Both the values and the precisions
+ of the two variables are swapped.
\1f
File: gmp.info, Node: Simultaneous Float Init & Assign, Next: Converting Floats, Prev: Assigning Floats, Up: Floating-point Functions
For convenience, GMP 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...'
+These functions' names have the form 'mpf_init_set...'
- Once the float has been initialized by any of the `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!
+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 (mpf_t ROP, const mpf_t OP)
-- Function: void mpf_init_set_ui (mpf_t ROP, unsigned long int OP)
-- Function: void mpf_init_set_si (mpf_t ROP, signed long int 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'.
+ precision, as set by 'mpf_set_default_prec'.
- -- Function: int mpf_init_set_str (mpf_t ROP, char *STR, int BASE)
+ -- Function: int mpf_init_set_str (mpf_t ROP, const 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.
+ '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.)
+ 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'.
+ precision, as set by 'mpf_set_default_prec'.
\1f
File: gmp.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
+ -- Function: double mpf_get_d (const mpf_t OP)
+ Convert OP to a 'double', truncating if necessary (i.e. rounding
towards zero).
- If the exponent in OP is too big or too small to fit a `double'
+ 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 (signed long int *EXP, mpf_t OP)
- Convert OP to a `double', truncating if necessary (ie. rounding
+ -- Function: double mpf_get_d_2exp (signed long int *EXP, const mpf_t
+ OP)
+ Convert OP to a 'double', truncating if necessary (i.e. 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'.
+ 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.).
+ This is similar to the standard C 'frexp' function (*note
+ (libc)Normalization Functions::).
- -- Function: long mpf_get_si (mpf_t OP)
- -- Function: unsigned long mpf_get_ui (mpf_t OP)
- Convert OP to a `long' or `unsigned long', truncating any fraction
+ -- Function: long mpf_get_si (const mpf_t OP)
+ -- Function: unsigned long mpf_get_ui (const mpf_t OP)
+ Convert OP to a 'long' or 'unsigned long', 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
+ 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)
+ -- Function: char * mpf_get_str (char *STR, mp_exp_t *EXPPTR, int BASE,
+ size_t N_DIGITS, const mpf_t OP)
Convert OP to a string of digits in base BASE. The base argument
may vary from 2 to 62 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.
+ 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.
+ 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
+ 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
+ '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
+ 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.
+ 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
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, unsigned long int
- OP2)
+ -- Function: void mpf_add (mpf_t ROP, const mpf_t OP1, const mpf_t OP2)
+ -- Function: void mpf_add_ui (mpf_t ROP, const mpf_t OP1, unsigned long
+ int 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, unsigned long int OP1, mpf_t
- OP2)
- -- Function: void mpf_sub_ui (mpf_t ROP, mpf_t OP1, unsigned long int
- OP2)
+ -- Function: void mpf_sub (mpf_t ROP, const mpf_t OP1, const mpf_t OP2)
+ -- Function: void mpf_ui_sub (mpf_t ROP, unsigned long int OP1, const
+ mpf_t OP2)
+ -- Function: void mpf_sub_ui (mpf_t ROP, const mpf_t OP1, unsigned long
+ int 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, unsigned long int
- OP2)
+ -- Function: void mpf_mul (mpf_t ROP, const mpf_t OP1, const mpf_t OP2)
+ -- Function: void mpf_mul_ui (mpf_t ROP, const mpf_t OP1, unsigned long
+ int OP2)
Set ROP to OP1 times OP2.
Division is undefined if the divisor is zero, and passing a zero
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, unsigned long int OP1, mpf_t
- OP2)
- -- Function: void mpf_div_ui (mpf_t ROP, mpf_t OP1, unsigned long int
- OP2)
+ -- Function: void mpf_div (mpf_t ROP, const mpf_t OP1, const mpf_t OP2)
+ -- Function: void mpf_ui_div (mpf_t ROP, unsigned long int OP1, const
+ mpf_t OP2)
+ -- Function: void mpf_div_ui (mpf_t ROP, const mpf_t OP1, unsigned long
+ int OP2)
Set ROP to OP1/OP2.
- -- Function: void mpf_sqrt (mpf_t ROP, mpf_t OP)
+ -- Function: void mpf_sqrt (mpf_t ROP, const mpf_t OP)
-- Function: void mpf_sqrt_ui (mpf_t ROP, unsigned long int OP)
Set ROP to the square root of OP.
- -- Function: void mpf_pow_ui (mpf_t ROP, mpf_t OP1, unsigned long int
- OP2)
+ -- Function: void mpf_pow_ui (mpf_t ROP, const mpf_t OP1, unsigned long
+ int OP2)
Set ROP to OP1 raised to the power OP2.
- -- Function: void mpf_neg (mpf_t ROP, mpf_t OP)
+ -- Function: void mpf_neg (mpf_t ROP, const mpf_t OP)
Set ROP to -OP.
- -- Function: void mpf_abs (mpf_t ROP, mpf_t OP)
+ -- Function: void mpf_abs (mpf_t ROP, const 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)
+ -- Function: void mpf_mul_2exp (mpf_t ROP, const 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)
+ -- Function: void mpf_div_2exp (mpf_t ROP, const mpf_t OP1, mp_bitcnt_t
+ OP2)
Set ROP to OP1 divided by 2 raised to OP2.
\1f
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, unsigned long int OP2)
- -- Function: int mpf_cmp_si (mpf_t OP1, signed long int OP2)
- Compare OP1 and OP2. Return a positive value if OP1 > OP2, zero
- if OP1 = OP2, and a negative value if OP1 < OP2.
+ -- Function: int mpf_cmp (const mpf_t OP1, const mpf_t OP2)
+ -- Function: int mpf_cmp_z (const mpf_t OP1, const mpz_t OP2)
+ -- Function: int mpf_cmp_d (const mpf_t OP1, double OP2)
+ -- Function: int mpf_cmp_ui (const mpf_t OP1, unsigned long int OP2)
+ -- Function: int mpf_cmp_si (const mpf_t OP1, signed long int 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
+ '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.
-
- Caution 1: All version of GMP up to version 4.2.4 compared just
- whole limbs, meaning sometimes more than OP3 bits, sometimes fewer.
+ -- Function: int mpf_eq (const mpf_t OP1, const mpf_t OP2, mp_bitcnt_t
+ op3)
+ *This function is mathematically ill-defined and should not be
+ used.*
- Caution 2: This function will consider XXX11...111 and XX100...000
- different, even if ... is replaced by a semi-infinite number of
- bits. Such numbers are really just one ulp off, and should be
- considered equal.
+ Return non-zero if the first OP3 bits of OP1 and OP2 are equal,
+ zero otherwise. Note that numbers like e.g., 256 (binary
+ 100000000) and 255 (binary 11111111) will never be equal by this
+ function's measure, and furthermore that 0 will only be equal to
+ itself.
- -- Function: void mpf_reldiff (mpf_t ROP, mpf_t OP1, mpf_t OP2)
+ -- Function: void mpf_reldiff (mpf_t ROP, const mpf_t OP1, const 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)
+ -- Macro: int mpf_sgn (const 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.
+ This function is actually implemented as a macro. It evaluates its
+ argument multiple times.
\1f
File: gmp.info, Node: I/O of Floats, Next: Miscellaneous Float Functions, Prev: Float Comparison, Up: Floating-point 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.
+output to a stdio stream, of 'mpf' numbers. 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 `gmp.h', since that will allow `gmp.h' to define
+'stdio.h' before 'gmp.h', since that will allow 'gmp.h' to define
prototypes for these functions.
+ See also *note Formatted Output:: and *note Formatted Input::.
+
-- Function: size_t mpf_out_str (FILE *STREAM, int BASE, size_t
- N_DIGITS, mpf_t OP)
+ N_DIGITS, const 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 62 or from -2 to -36. An exponent is
- then printed, separated by an `e', or if the 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'.
+ The mantissa is prefixed with an '0.' and is in the given BASE,
+ which may vary from 2 to 62 or from -2 to -36. An exponent is then
+ printed, separated by an 'e', or if the 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.
+ 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
-- 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@N' or, if the base is 10 or
- less, alternatively `MeN'. `M' is the mantissa and `N' is the
+ ROP. The string is of the form 'M@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'.
+ 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
+ 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.
+ 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.
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: void mpf_ceil (mpf_t ROP, const mpf_t OP)
+ -- Function: void mpf_floor (mpf_t ROP, const mpf_t OP)
+ -- Function: void mpf_trunc (mpf_t ROP, const 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)
+ -- Function: int mpf_integer_p (const 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)
+ -- Function: int mpf_fits_ulong_p (const mpf_t OP)
+ -- Function: int mpf_fits_slong_p (const mpf_t OP)
+ -- Function: int mpf_fits_uint_p (const mpf_t OP)
+ -- Function: int mpf_fits_sint_p (const mpf_t OP)
+ -- Function: int mpf_fits_ushort_p (const mpf_t OP)
+ -- Function: int mpf_fits_sshort_p (const 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.
+ <= ROP < 1, with NBITS significant bits in the mantissa or less if
+ the precision of ROP is smaller.
The variable STATE must be initialized by calling one of the
- `gmp_randinit' functions (*Note Random State Initialization::)
+ 'gmp_randinit' functions (*note Random State Initialization::)
before invoking this function.
-- Function: void mpf_random2 (mpf_t ROP, mp_size_t MAX_SIZE, mp_exp_t
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 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.
\1f
File: gmp.info, Node: Low-level Functions, Next: Random Number Functions, Prev: Floating-point Functions, Up: Top
This chapter describes low-level GMP functions, used to implement the
high-level GMP functions, but also intended for time-critical user code.
- These functions start with the prefix `mpn_'.
+ These functions start with the prefix 'mpn_'.
- The `mpn' functions are designed to be as fast as possible, *not* to
+ 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
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.
+ 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.
+ 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
+ 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
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.
+ 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.
+ 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)
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.
+ 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)
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,
+ -- Function: mp_limb_t 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.
+ This is equivalent to calling 'mpn_sub_n' with a N-limb zero
+ minuend and passing {SP, N} as subtrahend. Return borrow, either 0
+ or 1.
- -- Function: void mpn_mul_n (mp_limb_t *RP, const mp_limb_t *S1P,
- const mp_limb_t *S2P, mp_size_t N)
+ -- 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.
product's most significant limb is zero. No overlap is permitted
between the destination and either source.
- If the two input operands are the same, use `mpn_sqr'.
+ If the two input operands are the same, use 'mpn_sqr'.
-- 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)
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
+ The destination has to have space for 2N limbs, even if the
result's most significant limb is zero. No overlap is permitted
between the destination and the source.
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.
+ '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.
+ 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. {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 GMP. It is written
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, plus borrow-out
- from the subtraction.
+ from the subtraction. {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 and division as well as other operations in GMP.
- It is written in assembly for most CPUs.
+ multiplication and division as well as other operations in GMP. It
+ is written in assembly for most CPUs.
-- 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,
-- 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
+ [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).
+ 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,
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.
+ 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.
+ 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.
+ '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.
-- Function: mp_limb_t mpn_divmod (mp_limb_t *R1P, 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
+ [This function is obsolete. Please call 'mpn_tdiv_qr' instead for
best performance.]
+ -- Function: void mpn_divexact_1 (mp_limb_t * RP, const mp_limb_t * SP,
+ mp_size_t N, mp_limb_t D)
+ Divide {SP, N} by D, expecting it to divide exactly, and writing
+ the result to {RP, N}. If D doesn't divide exactly, the value
+ written to {RP, N} is undefined. The areas at RP and SP 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)
+ -- 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
+ '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
+ 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.
+ '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).
+ 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,
+ -- Function: mp_limb_t mpn_mod_1 (const 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_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
+ 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).
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 *XP,
- mp_size_t XN, mp_limb_t *YP, mp_size_t YN)
+ -- Function: int mpn_zero_p (const mp_limb_t *SP, mp_size_t N)
+ Test {SP, N} and return 1 if the operand is zero, 0 otherwise.
+
+ -- Function: mp_size_t mpn_gcd (mp_limb_t *RP, mp_limb_t *XP, mp_size_t
+ XN, mp_limb_t *YP, mp_size_t YN)
Set {RP, RETVAL} to the greatest common divisor of {XP, XN} and
{YP, YN}. The result can be up to YN limbs, the return value is
the actual number produced. Both source operands are destroyed.
- {XP, XN} must have at least as many bits as {YP, YN}. {YP, YN}
- must be odd. Both operands must have non-zero most significant
- limbs. No overlap is permitted between {XP, XN} and {YP, YN}.
+ It is required that XN >= YN > 0, the most significant limb of {YP,
+ YN} must be non-zero, and at least one of the two operands must be
+ odd. No overlap is permitted between {XP, XN} and {YP, YN}.
-- Function: mp_limb_t mpn_gcd_1 (const mp_limb_t *XP, mp_size_t XN,
mp_limb_t YLIMB)
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}.
+ mp_size_t *SN, mp_limb_t *UP, mp_size_t UN, mp_limb_t *VP,
+ mp_size_t VN)
+ Let U be defined by {UP, UN} and let V be defined by {VP, VN}.
- 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
+ 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.
+ division will be exact). It is required that UN >= VN > 0, and the
+ most significant limb of {VP, VN} must be non-zero.
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.
+ negative; when this happens *SN will be negative. The area at GP
+ should have room for VN limbs and the area at SP should have room
+ for VN+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).
+ Both source operands are destroyed.
- Compatibility note: GMP 4.3.0 and 4.3.1 defined S less strictly.
+ Compatibility notes: GMP 4.3.0 and 4.3.1 defined S less strictly.
Earlier as well as later GMP releases define S as described here.
+ GMP releases before GMP 4.3.0 required additional space for both
+ input and output areas. More precisely, the areas {UP, UN+1} and
+ {VP, VN+1} were destroyed (i.e. the operands plus an extra limb
+ past the end of each), and the areas pointed to by GP and SP should
+ each have room for UN+1 limbs.
-- 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.
+ 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
+ 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'.
+ 'mpn_perfect_square_p'.
+
+ -- Function: size_t mpn_sizeinbase (const mp_limb_t *XP, mp_size_t N,
+ int BASE)
+ Return the size of {XP,N} measured in number of digits in the given
+ BASE. BASE can vary from 2 to 62. Requires N > 0 and XP[N-1] > 0.
+ The result will be either exact or 1 too big. If BASE is a power
+ of 2, the result is always exact.
-- 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.
+ 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 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.
*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.
+ STR[0] is the most significant input byte and STR[STRSIZE-1] is the
+ least significant input byte. 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.
+ The converted value is {RP,RN} where RN is the return value. If
+ the most significant input byte STR[0] is non-zero, then RP[RN-1]
+ will be non-zero, else RP[RN-1] and some number of subsequent limbs
+ may be zero.
- If the most significant input byte is zero then there may be high
- zero limbs written to RP and included in the return value.
+ The area at RP has to have space for the largest possible number
+ with STRSIZE digits in the chosen base, plus one extra limb.
- STRSIZE must be at least 1, and no overlap is permitted between
- {STR,STRSIZE} and the result at RP.
+ The input must have at least one byte, and no overlap is permitted
+ between {STR,STRSIZE} and the result at RP.
-- Function: mp_bitcnt_t mpn_scan0 (const mp_limb_t *S1P, mp_bitcnt_t
BIT)
-- 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
+ 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'
+ 'mpn_random2' is intended for testing the correctness of the 'mpn'
routines.
-- Function: mp_bitcnt_t mpn_popcount (const mp_limb_t *S1P, mp_size_t
-- 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.
+ Return non-zero iff {S1P, N} is a perfect square. The most
+ significant limb of the input {S1P, N} must be non-zero.
- -- 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_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)
+ -- 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)
+ -- 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_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}.
+ 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)
Zero {RP, N}.
-8.1 Nails
+8.1 Low-level functions for cryptography
+========================================
+
+The functions prefixed with 'mpn_sec_' and 'mpn_cnd_' are designed to
+perform the exact same low-level operations and have the same cache
+access patterns for any two same-size arguments, assuming that function
+arguments are placed at the same position and that the machine state is
+identical upon function entry. These functions are intended for
+cryptographic purposes, where resilience to side-channel attacks is
+desired.
+
+ These functions are less efficient than their "leaky" counterparts;
+their performance for operands of the sizes typically used for
+cryptographic applications is between 15% and 100% worse. For larger
+operands, these functions might be inadequate, since they rely on
+asymptotically elementary algorithms.
+
+ These functions do not make any explicit allocations. Those of these
+functions that need scratch space accept a scratch space operand. This
+convention allows callers to keep sensitive data in designated memory
+areas. Note however that compilers may choose to spill scalar values
+used within these functions to their stack frame and that such scalars
+may contain sensitive data.
+
+ In addition to these specially crafted functions, the following 'mpn'
+functions are naturally side-channel resistant: 'mpn_add_n',
+'mpn_sub_n', 'mpn_lshift', 'mpn_rshift', 'mpn_zero', 'mpn_copyi',
+'mpn_copyd', 'mpn_com', and the logical function ('mpn_and_n', etc).
+
+ There are some exceptions from the side-channel resilience: (1) Some
+assembly implementations of 'mpn_lshift' identify shift-by-one as a
+special case. This is a problem iff the shift count is a function of
+sensitive data. (2) Alpha ev6 and Pentium4 using 64-bit limbs have
+leaky 'mpn_add_n' and 'mpn_sub_n'. (3) Alpha ev6 has a leaky
+'mpn_mul_1' which also makes 'mpn_sec_mul' on those systems unsafe.
+
+ -- Function: mp_limb_t mpn_cnd_add_n (mp_limb_t CND, mp_limb_t *RP,
+ const mp_limb_t *S1P, const mp_limb_t *S2P, mp_size_t N)
+ -- Function: mp_limb_t mpn_cnd_sub_n (mp_limb_t CND, mp_limb_t *RP,
+ const mp_limb_t *S1P, const mp_limb_t *S2P, mp_size_t N)
+ These functions do conditional addition and subtraction. If CND is
+ non-zero, they produce the same result as a regular 'mpn_add_n' or
+ 'mpn_sub_n', and if CND is zero, they copy {S1P,N} to the result
+ area and return zero. The functions are designed to have timing
+ and memory access patterns depending only on size and location of
+ the data areas, but independent of the condition CND. Like for
+ 'mpn_add_n' and 'mpn_sub_n', on most machines, the timing will also
+ be independent of the actual limb values.
+
+ -- Function: mp_limb_t mpn_sec_add_1 (mp_limb_t *RP, const mp_limb_t
+ *AP, mp_size_t N, mp_limb_t B, mp_limb_t *TP)
+ -- Function: mp_limb_t mpn_sec_sub_1 (mp_limb_t *RP, const mp_limb_t
+ *AP, mp_size_t N, mp_limb_t B, mp_limb_t *TP)
+ Set R to A + B or A - B, respectively, where R = {RP,N}, A =
+ {AP,N}, and B is a single limb. Returns carry.
+
+ These functions take O(N) time, unlike the leaky functions
+ 'mpn_add_1' which are O(1) on average. They require scratch space
+ of 'mpn_sec_add_1_itch(N)' and 'mpn_sec_sub_1_itch(N)' limbs,
+ respectively, to be passed in the TP parameter. The scratch space
+ requirements are guaranteed to be at most N limbs, and increase
+ monotonously in the operand size.
+
+ -- Function: void mpn_cnd_swap (mp_limb_t CND, volatile mp_limb_t *AP,
+ volatile mp_limb_t *BP, mp_size_t N)
+ If CND is non-zero, swaps the contents of the areas {AP,N} and
+ {BP,N}. Otherwise, the areas are left unmodified. Implemented
+ using logical operations on the limbs, with the same memory
+ accesses independent of the value of CND.
+
+ -- Function: void mpn_sec_mul (mp_limb_t *RP, const mp_limb_t *AP,
+ mp_size_t AN, const mp_limb_t *BP, mp_size_t BN, mp_limb_t
+ *TP)
+ -- Function: mp_size_t mpn_sec_mul_itch (mp_size_t AN, mp_size_t BN)
+ Set R to A * B, where A = {AP,AN}, B = {BP,BN}, and R = {RP,AN+BN}.
+
+ It is required that AN >= BN > 0.
+
+ No overlapping between R and the input operands is allowed. For A
+ = B, use 'mpn_sec_sqr' for optimal performance.
+
+ This function requires scratch space of 'mpn_sec_mul_itch(AN, BN)'
+ limbs to be passed in the TP parameter. The scratch space
+ requirements are guaranteed to increase monotonously in the operand
+ sizes.
+
+ -- Function: void mpn_sec_sqr (mp_limb_t *RP, const mp_limb_t *AP,
+ mp_size_t AN, mp_limb_t *TP)
+ -- Function: mp_size_t mpn_sec_sqr_itch (mp_size_t AN)
+ Set R to A^2, where A = {AP,AN}, and R = {RP,2AN}.
+
+ It is required that AN > 0.
+
+ No overlapping between R and the input operands is allowed.
+
+ This function requires scratch space of 'mpn_sec_sqr_itch(AN)'
+ limbs to be passed in the TP parameter. The scratch space
+ requirements are guaranteed to increase monotonously in the operand
+ size.
+
+ -- Function: void mpn_sec_powm (mp_limb_t *RP, const mp_limb_t *BP,
+ mp_size_t BN, const mp_limb_t *EP, mp_bitcnt_t ENB, const
+ mp_limb_t *MP, mp_size_t N, mp_limb_t *TP)
+ -- Function: mp_size_t mpn_sec_powm_itch (mp_size_t BN, mp_bitcnt_t
+ ENB, size_t N)
+ Set R to (B raised to E) modulo M, where R = {RP,N}, M = {MP,N},
+ and E = {EP,ceil(ENB / 'GMP\_NUMB\_BITS')}.
+
+ It is required that B > 0, that M > 0 is odd, and that E < 2^ENB,
+ with ENB > 0.
+
+ No overlapping between R and the input operands is allowed.
+
+ This function requires scratch space of 'mpn_sec_powm_itch(BN, ENB,
+ N)' limbs to be passed in the TP parameter. The scratch space
+ requirements are guaranteed to increase monotonously in the operand
+ sizes.
+
+ -- Function: void mpn_sec_tabselect (mp_limb_t *RP, const mp_limb_t
+ *TAB, mp_size_t N, mp_size_t NENTS, mp_size_t WHICH)
+ Select entry WHICH from table TAB, which has NENTS entries, each N
+ limbs. Store the selected entry at RP.
+
+ This function reads the entire table to avoid side-channel
+ information leaks.
+
+ -- Function: mp_limb_t mpn_sec_div_qr (mp_limb_t *QP, mp_limb_t *NP,
+ mp_size_t NN, const mp_limb_t *DP, mp_size_t DN, mp_limb_t
+ *TP)
+ -- Function: mp_size_t mpn_sec_div_qr_itch (mp_size_t NN, mp_size_t DN)
+
+ Set Q to the truncated quotient N / D and R to N modulo D, where N
+ = {NP,NN}, D = {DP,DN}, Q's most significant limb is the function
+ return value and the remaining limbs are {QP,NN-DN}, and R =
+ {NP,DN}.
+
+ It is required that NN >= DN >= 1, and that DP[DN-1] != 0. This
+ does not imply that N >= D since N might be zero-padded.
+
+ Note the overlapping between N and R. No other operand overlapping
+ is allowed. The entire space occupied by N is overwritten.
+
+ This function requires scratch space of 'mpn_sec_div_qr_itch(NN,
+ DN)' limbs to be passed in the TP parameter.
+
+ -- Function: void mpn_sec_div_r (mp_limb_t *NP, mp_size_t NN, const
+ mp_limb_t *DP, mp_size_t DN, mp_limb_t *TP)
+ -- Function: mp_size_t mpn_sec_div_r_itch (mp_size_t NN, mp_size_t DN)
+
+ Set R to N modulo D, where N = {NP,NN}, D = {DP,DN}, and R =
+ {NP,DN}.
+
+ It is required that NN >= DN >= 1, and that DP[DN-1] != 0. This
+ does not imply that N >= D since N might be zero-padded.
+
+ Note the overlapping between N and R. No other operand overlapping
+ is allowed. The entire space occupied by N is overwritten.
+
+ This function requires scratch space of 'mpn_sec_div_r_itch(NN,
+ DN)' limbs to be passed in the TP parameter.
+
+ -- Function: int mpn_sec_invert (mp_limb_t *RP, mp_limb_t *AP, const
+ mp_limb_t *MP, mp_size_t N, mp_bitcnt_t NBCNT, mp_limb_t *TP)
+ -- Function: mp_size_t mpn_sec_invert_itch (mp_size_t N)
+ Set R to the inverse of A modulo M, where R = {RP,N}, A = {AP,N},
+ and M = {MP,N}. *This function's interface is preliminary.*
+
+ If an inverse exists, return 1, otherwise return 0 and leave R
+ undefined. In either case, the input A is destroyed.
+
+ It is required that M is odd, and that NBCNT >= ceil(\log(A+1)) +
+ ceil(\log(M+1)). A safe choice is NBCNT = 2 * N * GMP_NUMB_BITS,
+ but a smaller value might improve performance if M or A are known
+ to have leading zero bits.
+
+ This function requires scratch space of 'mpn_sec_invert_itch(N)'
+ limbs to be passed in the TP parameter.
+
+
+8.2 Nails
=========
*Everything in this section is highly experimental and may disappear or
be subject to incompatible changes in a future version of GMP.*
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
+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.
+ 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
+ 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'.
+'--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
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.
+ 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
+ '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.
+ 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
+ '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
+ 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
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.
+vector. This would help vector processors since carries would only ever
+need to propagate one or two limbs.
\1f
File: gmp.info, Node: Random Number Functions, Next: Formatted Output, Prev: Low-level Functions, Up: Top
9 Random Number Functions
*************************
-Sequences of pseudo-random numbers in GMP 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.
+Sequences of pseudo-random numbers in GMP 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
+*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.
+'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:
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'.
+ '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, unsigned long C, mp_bitcnt_t M2EXP)
- Initialize STATE with a linear congruential algorithm X = (A*X +
- C) mod 2^M2EXP.
+ -- Function: void gmp_randinit_lc_2exp (gmp_randstate_t STATE, const
+ mpz_t A, unsigned long 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
-- 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.
+ '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, i.e.
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.
+ 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: void gmp_randinit_set (gmp_randstate_t ROP,
gmp_randstate_t OP)
*This function is obsolete.*
Initialize STATE with an algorithm selected by ALG. The only
- choice is `GMP_RAND_ALG_LC', which is `gmp_randinit_lc_2exp_size'
- described above. A third parameter of type `unsigned long' is
+ choice is 'GMP_RAND_ALG_LC', which is 'gmp_randinit_lc_2exp_size'
+ described above. A third parameter of type 'unsigned long' is
required, this is the SIZE for that function.
- `GMP_RAND_ALG_DEFAULT' or 0 are the same as `GMP_RAND_ALG_LC'.
+ 'GMP_RAND_ALG_DEFAULT' or 0 are the same as 'GMP_RAND_ALG_LC'.
- `gmp_randinit' sets bits in the global variable `gmp_errno' to
- indicate an error. `GMP_ERROR_UNSUPPORTED_ARGUMENT' if ALG is
- unsupported, or `GMP_ERROR_INVALID_ARGUMENT' if the SIZE parameter
+ 'gmp_randinit' sets bits in the global variable 'gmp_errno' to
+ indicate an error. 'GMP_ERROR_UNSUPPORTED_ARGUMENT' if ALG is
+ unsupported, or 'GMP_ERROR_INVALID_ARGUMENT' if the SIZE parameter
is too big. It may be noted this error reporting is not thread
- safe (a good reason to use `gmp_randinit_lc_2exp_size' instead).
+ safe (a good reason to use 'gmp_randinit_lc_2exp_size' instead).
-- Function: void gmp_randclear (gmp_randstate_t STATE)
Free all memory occupied by STATE.
9.2 Random State Seeding
========================
- -- Function: void gmp_randseed (gmp_randstate_t STATE, mpz_t SEED)
+ -- Function: void gmp_randseed (gmp_randstate_t STATE, const mpz_t
+ SEED)
-- Function: void gmp_randseed_ui (gmp_randstate_t STATE,
unsigned long int 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.
+ 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.
\1f
File: gmp.info, Node: Random State Miscellaneous, Prev: Random State Seeding, Up: Random Number Functions
-- Function: unsigned long gmp_urandomb_ui (gmp_randstate_t STATE,
unsigned long N)
- Return a uniformly distributed random number of N bits, ie. in the
+ Return a uniformly distributed random number of N bits, i.e. in the
range 0 to 2^N-1 inclusive. N must be less than or equal to the
- number of bits in an `unsigned long'.
+ number of bits in an 'unsigned long'.
-- Function: unsigned long gmp_urandomm_ui (gmp_randstate_t STATE,
unsigned long N)
- Return a uniformly distributed random number in the range 0 to
- N-1, inclusive.
+ Return a uniformly distributed random number in the range 0 to N-1,
+ inclusive.
\1f
File: gmp.info, Node: Formatted Output, Next: Formatted Input, Prev: Random Number Functions, Up: Top
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
+'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
- GMP 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,
+ GMP 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);
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
+ 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 GMP extensions. In the
+ All the standard C 'printf' types behave the same as the C library
+'printf', and can be freely intermixed with the GMP extensions. In the
current implementation the standard parts of the format string are
-simply handed to `printf' and only the GMP extensions handled directly.
+simply handed to 'printf' and only the GMP extensions handled directly.
The flags accepted are as follows. GLIBC style ' is only for the
-standard C types (not the GMP types), and only if the C library
-supports it.
+standard C types (not the GMP types), and only if the C library supports
+it.
0 pad with zeros (rather than spaces)
- # show the base with `0x', `0X' or `0'
+ # show the base with '0x', '0X' or '0'
+ always show a sign
- (space) show a space or a `-' sign
- ' group digits, GLIBC style (not GMP types)
+ (space) show a space or a '-' sign
+ ' group digits, GLIBC style (not GMP
+ 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'.
+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
+ 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.
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.
+ 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
f fixed point float
i same as d
g G fixed or scientific float
- m `strerror' string, GLIBC style
+ m 'strerror' string, GLIBC style
n store characters written so far
o octal integer
p pointer
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'.
+ '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
+ '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 GMP types.
+ 'n' can be used with any type, even the GMP 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
+'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
+ The precision field has its 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.
+ '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
+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.
+empty precision field like '%.Fe' or '%.Ff' can be used to specifically
+request just the significant digits. Without any dot and thus no
+precision field, a precision value of 6 will be used. Note that these
+rules mean that '%Ff', '%.Ff', and '%.0Ff' will all be different.
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.
+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
+ The format string is only interpreted as plain 'char's, multibyte
characters are not recognised. Perhaps this will change in the future.
\1f
==============
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'.
+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
checking is not available, since it doesn't recognise the GMP
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
+ 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
+ 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, ...)
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_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.
+ 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 >=
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.
+ 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.
+ 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
+ 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.
*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.
+ 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.
+ *note Obstacks: (libc)Obstacks.
\1f
File: gmp.info, Node: C++ Formatted Output, Prev: Formatted Output Functions, Up: Formatted Output
10.3 C++ Formatted Output
=========================
-The following functions are provided in `libgmpxx' (*note Headers and
+The following functions are provided in 'libgmpxx' (*note Headers and
Libraries::), which is built if C++ support is enabled (*note Build
-Options::). Prototypes are available from `<gmp.h>'.
+Options::). Prototypes are available from '<gmp.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.
+ -- Function: ostream& operator<< (ostream& STREAM, const 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.
+ 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.
+ -- Function: ostream& operator<< (ostream& STREAM, const 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'.
+ 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
+ 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.
+ -- Function: ostream& operator<< (ostream& STREAM, const 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.
+ 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'.
+ 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'.
+ '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.
...
cout << "iteration " << n << " value " << z << "\n";
- But note that `ostream' output (and `istream' input, *note C++
+ But note that 'ostream' output (and 'istream' input, *note C++
Formatted Input::) is the only overloading available for the GMP types
-and that for instance using `+' with an `mpz_t' will have unpredictable
-results. For classes with overloading, see *Note C++ Class Interface::.
+and that for instance using '+' with an 'mpz_t' will have unpredictable
+results. For classes with overloading, see *note C++ Class Interface::.
\1f
File: gmp.info, Node: Formatted Input, Next: C++ Class Interface, Prev: Formatted Output, Up: Top
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
+'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
- GMP 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.
+ GMP 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.
- GMP variables don't require an `&' when passed to `gmp_scanf', since
+ GMP 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" */
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 GMP extensions. In the
+ All the standard C 'scanf' types behave the same as in the C library
+'scanf', and can be freely intermixed with the GMP extensions. In the
current implementation the standard parts of the format string are
-simply handed to `scanf' and only the GMP extensions handled directly.
+simply handed to 'scanf' and only the GMP extensions handled directly.
- The flags accepted are as follows. `a' and `'' will depend on
-support from the C library, and `'' cannot be used with GMP types.
+ The flags accepted are as follows. 'a' and ''' will depend on
+support from the C library, and ''' cannot be used with GMP types.
* read but don't store
a allocate a buffer (string conversions)
' grouped digits, GLIBC style (not GMP
types)
- The standard types accepted are as follows. `h' and `l' are
+ 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.
Q mpq_t, integer conversions
Z mpz_t, integer conversions
- The conversions accepted are as follows. `p' and `[' will depend on
+ 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
+ e E f g float
+ G
i integer with base indicator
n characters read so far
o octal 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
+ '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
+ 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
+ '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 GMP types there are no overflows, so `d' and `u' are
-identical.
+ '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 GMP 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::).
+ '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'
+ '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 GMP types.
-`*' to suppress assignment is allowed, though in that case it would do
+ 'n' can be used with any of the types above, even the GMP 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'.
+'scanf' cannot be used through 'gmp_scanf'.
- Whitespace is read and discarded before a field, except for `c' and
-`[' conversions.
+ 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:
+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
+ The format string is only interpreted as plain 'char's, multibyte
characters are not recognised. Perhaps this will change in the future.
\1f
==============================
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'.
+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
-- Function: int gmp_scanf (const char *FMT, ...)
-- Function: int gmp_vscanf (const char *FMT, va_list AP)
- Read from the standard input `stdin'.
+ 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 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
+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
+ 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
+induce an 'EOF' in this fashion. Leading whitespace read and discarded
for a field don't count as characters for that field.
For the GMP 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.
+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 GMP calls
-the C library `scanf' functions, which might have looser rules about
-what constitutes a valid input.
+ For the standard C types, in the current implementation GMP 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
+ 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'.
+its entire input, it is deliberately made identical to 'gmp_fscanf', the
+same way C99 'sscanf' is the same as 'fscanf'.
\1f
File: gmp.info, Node: C++ Formatted Input, Prev: Formatted Input Functions, Up: Formatted Input
11.3 C++ Formatted Input
========================
-The following functions are provided in `libgmpxx' (*note Headers and
-Libraries::), which is built only if C++ support is enabled (*note
-Build Options::). Prototypes are available from `<gmp.h>'.
+The following functions are provided in 'libgmpxx' (*note Headers and
+Libraries::), which is built only if C++ support is enabled (*note Build
+Options::). Prototypes are available from '<gmp.h>'.
-- Function: istream& operator>> (istream& STREAM, mpz_t ROP)
- Read ROP from STREAM, using its `ios' formatting settings.
+ 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
+ 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.
+ 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.
+ 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.
+ 'operator>>' does.
- Note that digit grouping specified by the `istream' locale is
+ Note that digit grouping specified by the 'istream' locale is
currently not accepted. Perhaps this will change in the future.
- These operators mean that GMP types can be read in the usual C++
-way, for example,
+ These operators mean that GMP 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++
+ But note that 'istream' input (and 'ostream' output, *note C++
Formatted Output::) is the only overloading available for the GMP types
-and that for instance using `+' with an `mpz_t' will have unpredictable
-results. For classes with overloading, see *Note C++ Class Interface::.
+and that for instance using '+' with an 'mpz_t' will have unpredictable
+results. For classes with overloading, see *note C++ Class Interface::.
\1f
-File: gmp.info, Node: C++ Class Interface, Next: BSD Compatible Functions, Prev: Formatted Input, Up: Top
+File: gmp.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 GMP.
All GMP C language types and functions can be used in C++ programs,
-since `gmp.h' has `extern "C"' qualifiers, but the class interface
+since 'gmp.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.
+of templates and member templates.
*Everything described in this chapter is to be considered preliminary
and might be subject to incompatible changes if some unforeseen
#include <gmpxx.h>
- Programs should be linked with the `libgmpxx' and `libgmp'
-libraries. For example,
+ Programs should be linked with the 'libgmpxx' and 'libgmp' libraries.
+For example,
g++ mycxxprog.cc -lgmpxx -lgmp
}
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.
+'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 `long', `unsigned long' and `double'.
-Smaller types like `int' or `float' can also be intermixed, since C++
+classes and the standard types 'long', 'unsigned long' 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 GMP accepted `bool' it would make all
+ 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 GMP 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
+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
-- 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',
+interface. For example to set 'a' to the GCD of 'b' and 'c',
mpz_class a, b, c;
...
mpz_class y;
y = mpz_class (z);
- There are no namespace setups in `gmpxx.h', all types and functions
-are simply put into the global namespace. This is what `gmp.h' has
-done in the past, and continues to do for compatibility. The extras
-provided by `gmpxx.h' follow GMP naming conventions and are unlikely to
-clash with anything.
+ There are no namespace setups in 'gmpxx.h', all types and functions
+are simply put into the global namespace. This is what 'gmp.h' has done
+in the past, and continues to do for compatibility. The extras provided
+by 'gmpxx.h' follow GMP naming conventions and are unlikely to clash
+with anything.
\1f
File: gmp.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,
- except `long long' and `long double', and all the GMP C++ classes
- can be used. Any necessary conversion follows the corresponding C
- function, for example `double' follows `mpz_set_d' (*note
+ -- Function: mpz_class::mpz_class (type N)
+ Construct an 'mpz_class'. All the standard C++ types may be used,
+ except 'long long' and 'long double', and all the GMP C++ classes
+ can be used, although conversions from 'mpq_class' and 'mpf_class'
+ are 'explicit'. 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: explicit mpz_class::mpz_class (const 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::).
+ -- Function: explicit mpz_class::mpz_class (const char *S, int BASE =
+ 0)
+ -- Function: explicit 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='.
+ 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 (const char *STR)
+ With C++11 compilers, integers can be constructed with the syntax
+ '123_mpz' which is equivalent to 'mpz_class("123")'.
-- 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.
+ 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
+ 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: mpz_class abs (mpz_class OP)
-- 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: long mpz_class::get_si (void)
-- Function: string mpz_class::get_str (int BASE = 10)
-- Function: unsigned long 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)
+
+ -- Function: mpz_class gcd (mpz_class OP1, mpz_class OP2)
+ -- Function: mpz_class lcm (mpz_class OP1, mpz_class OP2)
+ -- Function: mpz_class mpz_class::factorial (type OP)
+ -- Function: mpz_class factorial (mpz_class OP)
+ -- Function: mpz_class mpz_class::primorial (type OP)
+ -- Function: mpz_class primorial (mpz_class OP)
+ -- Function: mpz_class mpz_class::fibonacci (type OP)
+ -- Function: mpz_class fibonacci (mpz_class OP)
+
+ -- Function: void mpz_class::swap (mpz_class& OP)
+ -- Function: void swap (mpz_class& OP1, mpz_class& OP2)
These functions provide a C++ class interface to the corresponding
- GMP C routines.
+ GMP C routines. Calling 'factorial' or 'primorial' on a negative
+ number is undefined.
- `cmp' can be used with any of the classes or the standard C++
- types, except `long long' and `long double'.
+ 'cmp' can be used with any of the classes or the standard C++
+ types, except 'long long' and '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
+ 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'.
+ 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'.
\1f
File: gmp.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.
+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 long'
- and `long double' are not supported. For example,
+ -- Function: mpq_class::mpq_class (type OP)
+ -- Function: mpq_class::mpq_class (integer NUM, integer DEN)
+ Construct an 'mpq_class'. The initial value can be a single value
+ of any type (conversion from 'mpf_class' is 'explicit'), or a pair
+ of integers ('mpz_class' or standard C++ integer types)
+ representing a fraction, except that 'long long' and 'long double'
+ are 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: explicit mpq_class::mpq_class (const 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: explicit mpq_class::mpq_class (const char *S, int BASE =
+ 0)
+ -- Function: explicit 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::).
- -- 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='.
- If the string is not a valid rational, an `std::invalid_argument'
- exception is thrown. The same applies to `operator='.
+ -- Function: mpq_class operator"" _mpq (const char *STR)
+ With C++11 compilers, integral rationals can be constructed with
+ the syntax '123_mpq' which is equivalent to 'mpq_class(123_mpz)'.
+ Other rationals can be built as '-1_mpq/2' or '0xb_mpq/123456_mpz'.
-- Function: void mpq_class::canonicalize ()
- Put an `mpq_class' into canonical form, as per *Note Rational
+ 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)
+
+ -- Function: void mpq_class::swap (mpq_class& OP)
+ -- Function: void swap (mpq_class& OP1, mpq_class& OP2)
These functions provide a C++ class interface to the corresponding
GMP C routines.
- `cmp' can be used with any of the classes or the standard C++
- types, except `long long' and `long double'.
+ 'cmp' can be used with any of the classes or the standard C++
+ types, except 'long long' and '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
+ 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'.
+ original 'mpq_class'.
If direct manipulation might produce a non-canonical value, then
- `mpq_class::canonicalize' must be called before further operations.
+ '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'.
+ 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.
+ '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::).
+ 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.
+ 'mpq_class::canonicalize' must be called.
\1f
File: gmp.info, Node: C++ Interface Floats, Next: C++ Interface Random Numbers, Prev: C++ Interface Rationals, Up: C++ Class Interface
=========================
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
+'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, unsigned long PREC)
- Construct an `mpf_class'. Any standard C++ type can be used,
- except `long long' and `long double', and any of the GMP C++
+ -- Function: mpf_class::mpf_class (type OP)
+ -- Function: mpf_class::mpf_class (type OP, mp_bitcnt_t PREC)
+ Construct an 'mpf_class'. Any standard C++ type can be used,
+ except 'long long' and 'long double', and any of the GMP 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.
+ 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(-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, unsigned long
- PREC, int BASE = 0)
- -- Function: void mpf_class::mpf_class (const string& S)
- -- Function: void mpf_class::mpf_class (const string& S, unsigned long
- PREC, int BASE = 0)
- Construct an `mpf_class' converted from a string using
- `mpf_set_str' (*note Assigning Floats::). If PREC is given, the
+ -- Function: explicit mpf_class::mpf_class (const mpf_t F)
+ -- Function: mpf_class::mpf_class (const mpf_t F, mp_bitcnt_t PREC)
+ Construct an 'mpf_class' from an 'mpf_t'. The value in F is copied
+ into the new 'mpf_class', there won't be any permanent association
+ between it and F.
+
+ If PREC is given, the initial precision is that value, in bits. If
+ PREC is not given, then the initial precision is that of F.
+
+ -- Function: explicit mpf_class::mpf_class (const char *S)
+ -- Function: mpf_class::mpf_class (const char *S, mp_bitcnt_t PREC, int
+ BASE = 0)
+ -- Function: explicit mpf_class::mpf_class (const string& S)
+ -- Function: mpf_class::mpf_class (const string& S, mp_bitcnt_t 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.
+ '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='.
+ If the string is not a valid float, an 'std::invalid_argument'
+ exception is thrown. The same applies to 'operator='.
+
+ -- Function: mpf_class operator"" _mpf (const char *STR)
+ With C++11 compilers, floats can be constructed with the syntax
+ '1.23e-1_mpf' which is equivalent to 'mpf_class("1.23e-1")'.
-- Function: mpf_class& mpf_class::operator= (type OP)
- Convert and store the given OP value to an `mpf_class' object. The
+ 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
+ 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
+ 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
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
+ '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 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: long mpf_class::get_si (void)
-- Function: string mpf_class::get_str (mp_exp_t& EXP, int BASE = 10,
size_t DIGITS = 0)
-- Function: unsigned long 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: void mpf_class::swap (mpf_class& OP)
+ -- Function: void swap (mpf_class& OP1, mpf_class& OP2)
-- Function: mpf_class trunc (mpf_class OP)
These functions provide a C++ class interface to the corresponding
GMP C routines.
- `cmp' can be used with any of the classes or the standard C++
- types, except `long long' and `long double'.
+ 'cmp' can be used with any of the classes or the standard C++
+ types, except 'long long' and 'long double'.
- The accuracy provided by `hypot' is not currently guaranteed.
+ 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'.
+ 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.
+ 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.
\1f
File: gmp.info, Node: C++ Interface Random Numbers, Next: C++ Interface Limitations, Prev: C++ Interface Floats, Up: C++ Class Interface
-- Class: gmp_randclass
The C++ class interface to the GMP random number functions uses
- `gmp_randclass' to hold an algorithm selection and current state,
- as per `gmp_randstate_t'.
+ 'gmp_randclass' to hold an algorithm selection and current state,
+ as per 'gmp_randstate_t'.
- -- Function: gmp_randclass::gmp_randclass (void (*RANDINIT)
+ -- Function: gmp_randclass::gmp_randclass (void (*RANDINIT)
(gmp_randstate_t, ...), ...)
- Construct a `gmp_randclass', using a call to the given RANDINIT
+ 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,
+ 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.
+ '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: gmp_randclass::gmp_randclass (gmp_randalg_t ALG, ...)
- Construct a `gmp_randclass' using the same parameters as
- `gmp_randinit' (*note Random State Initialization::). This
+ -- Function: gmp_randclass::gmp_randclass (gmp_randalg_t ALG, ...)
+ Construct a 'gmp_randclass' using the same parameters as
+ 'gmp_randinit' (*note Random State Initialization::). This
function is obsolete and the above RANDINIT style should be
preferred.
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 (unsigned long BITS)
+ -- Function: mpz_class gmp_randclass::get_z_bits (mp_bitcnt_t BITS)
-- Function: mpz_class gmp_randclass::get_z_bits (mpz_class BITS)
Generate a random integer with a specified number of bits.
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 (unsigned long PREC)
+ -- Function: mpf_class gmp_randclass::get_f (mp_bitcnt_t 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,
12.6 C++ Interface Limitations
==============================
-`mpq_class' and Templated Reading
+'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
+ '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
+ '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
+ '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
+ 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 <gmp-bugs@gmplib.org>.
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 GMP
- classes, and a good way to reestablish them in a subclass is not
- yet provided.
+ but in normal C++ fashion the subclass doesn't inherit constructors
+ and assignments. There's many of those in the GMP 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,
+ 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'.
+ 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
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 `gmpxx.h', which the C++ template resolution
- rules are unable to automatically convert to `mpz_class'. The
- workaround is simply to add an explicit cast.
+ This is because 'g+h' ends up being a certain expression template
+ type internal to 'gmpxx.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'.
+ 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, T(f+g)); // Good
}
-\1f
-File: gmp.info, Node: BSD Compatible Functions, Next: Custom Allocation, Prev: C++ Class Interface, Up: Top
-
-13 Berkeley MP Compatible Functions
-***********************************
-
-These functions are intended to be fully compatible with the Berkeley MP
-library which is available on many BSD derived U*ix systems. The
-`--enable-mpbsd' option must be used when building GNU MP to make these
-available (*note Installing GMP::).
-
- The original Berkeley MP library has a usage restriction: you cannot
-use the same variable as both source and destination in a single
-function call. The compatible functions in GNU MP do not share this
-restriction--inputs and outputs may overlap.
-
- It is not recommended that new programs are written using these
-functions. Apart from the incomplete set of functions, the interface
-for initializing `MINT' objects is more error prone, and the `pow'
-function collides with `pow' in `libm.a'.
-
- Include the header `mp.h' to get the definition of the necessary
-types and functions. If you are on a BSD derived system, make sure to
-include GNU `mp.h' if you are going to link the GNU `libmp.a' to your
-program. This means that you probably need to give the `-I<dir>'
-option to the compiler, where `<dir>' is the directory where you have
-GNU `mp.h'.
-
- -- Function: MINT * itom (signed short int INITIAL_VALUE)
- Allocate an integer consisting of a `MINT' object and dynamic limb
- space. Initialize the integer to INITIAL_VALUE. Return a pointer
- to the `MINT' object.
-
- -- Function: MINT * xtom (char *INITIAL_VALUE)
- Allocate an integer consisting of a `MINT' object and dynamic limb
- space. Initialize the integer from INITIAL_VALUE, a hexadecimal,
- null-terminated C string. Return a pointer to the `MINT' object.
-
- -- Function: void move (MINT *SRC, MINT *DEST)
- Set DEST to SRC by copying. Both variables must be previously
- initialized.
-
- -- Function: void madd (MINT *SRC_1, MINT *SRC_2, MINT *DESTINATION)
- Add SRC_1 and SRC_2 and put the sum in DESTINATION.
-
- -- Function: void msub (MINT *SRC_1, MINT *SRC_2, MINT *DESTINATION)
- Subtract SRC_2 from SRC_1 and put the difference in DESTINATION.
-
- -- Function: void mult (MINT *SRC_1, MINT *SRC_2, MINT *DESTINATION)
- Multiply SRC_1 and SRC_2 and put the product in DESTINATION.
-
- -- Function: void mdiv (MINT *DIVIDEND, MINT *DIVISOR, MINT *QUOTIENT,
- MINT *REMAINDER)
- -- Function: void sdiv (MINT *DIVIDEND, signed short int DIVISOR, MINT
- *QUOTIENT, signed short int *REMAINDER)
- Set QUOTIENT to DIVIDEND/DIVISOR, and REMAINDER to DIVIDEND mod
- DIVISOR. The quotient is rounded towards zero; the remainder has
- the same sign as the dividend unless it is zero.
-
- Some implementations of these functions work differently--or not
- at all--for negative arguments.
-
- -- Function: void msqrt (MINT *OP, MINT *ROOT, MINT *REMAINDER)
- Set ROOT to the truncated integer part of the square root of OP,
- like `mpz_sqrt'. Set REMAINDER to OP-ROOT*ROOT, i.e. zero if OP
- is a perfect square.
-
- If ROOT and REMAINDER are the same variable, the results are
- undefined.
-
- -- Function: void pow (MINT *BASE, MINT *EXP, MINT *MOD, MINT *DEST)
- Set DEST to (BASE raised to EXP) modulo MOD.
-
- Note that the name `pow' clashes with `pow' from the standard C
- math library (*note Exponentiation and Logarithms: (libc)Exponents
- and Logarithms.). An application will only be able to use one or
- the other.
-
- -- Function: void rpow (MINT *BASE, signed short int EXP, MINT *DEST)
- Set DEST to BASE raised to EXP.
+C++11
+ C++11 provides several new ways in which types can be inferred:
+ 'auto', 'decltype', etc. While they can be very convenient, they
+ don't mix well with expression templates. In this example, the
+ addition is performed twice, as if we had defined 'sum' as a macro.
- -- Function: void gcd (MINT *OP1, MINT *OP2, MINT *RES)
- Set RES to the greatest common divisor of OP1 and OP2.
+ mpz_class z = 33;
+ auto sum = z + z;
+ mpz_class prod = sum * sum;
- -- Function: int mcmp (MINT *OP1, MINT *OP2)
- Compare OP1 and OP2. Return a positive value if OP1 > OP2, zero
- if OP1 = OP2, and a negative value if OP1 < OP2.
+ This other example may crash, though some compilers might make it
+ look like it is working, because the expression 'z+z' goes out of
+ scope before it is evaluated.
- -- Function: void min (MINT *DEST)
- Input a decimal string from `stdin', and put the read integer in
- DEST. SPC and TAB are allowed in the number string, and are
- ignored.
+ mpz_class z = 33;
+ auto sum = z + z + z;
+ mpz_class prod = sum * 2;
- -- Function: void mout (MINT *SRC)
- Output SRC to `stdout', as a decimal string. Also output a
- newline.
-
- -- Function: char * mtox (MINT *OP)
- Convert OP to a hexadecimal string, and return a pointer to the
- string. The returned string is allocated using the default memory
- allocation function, `malloc' by default. It will be
- `strlen(str)+1' bytes, that being exactly enough for the string
- and null-terminator.
-
- -- Function: void mfree (MINT *OP)
- De-allocate, the space used by OP. *This function should only be
- passed a value returned by `itom' or `xtom'.*
+ It is thus strongly recommended to avoid 'auto' anywhere a GMP C++
+ expression may appear.
\1f
-File: gmp.info, Node: Custom Allocation, Next: Language Bindings, Prev: BSD Compatible Functions, Up: Top
+File: gmp.info, Node: Custom Allocation, Next: Language Bindings, Prev: C++ Class Interface, Up: Top
-14 Custom Allocation
+13 Custom Allocation
********************
-By default GMP uses `malloc', `realloc' and `free' for memory
+By default GMP uses 'malloc', 'realloc' and 'free' for memory
allocation, and if they fail GMP prints a message to the standard error
output and terminates the program.
different way or to have a different error action on running out of
memory.
- This feature is available in the Berkeley compatibility library
-(*note BSD Compatible Functions::) as well as the main GMP library.
-
-- 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.
+ 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
- GMP, apart from temporary space from `alloca' if that function is
+ These functions will be used for all memory allocation done by GMP,
+ apart from temporary space from 'alloca' if that function is
available and GMP is configured to use it (*note Build Options::).
- *Be sure to call `mp_set_memory_functions' only when there are no
+ *Be sure to call 'mp_set_memory_functions' only when there are no
active GMP objects allocated using the previous memory functions!
Usually that means calling it before any other GMP function.*
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.
+ 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.
+ 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
+ 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.
+ 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.
+ The REALLOCATE_FUNCTION parameter OLD_SIZE and the FREE_FUNCTION
+parameter SIZE are passed for convenience, but of course they can be
+ignored if not needed by an implementation. 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'.
+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
+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.
+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.
GMP may use allocated blocks to hold pointers to other allocated
blocks. This will limit the assumptions a conservative garbage
collection scheme can make.
- Since the default GMP allocation uses `malloc' and friends, those
+ Since the default GMP 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 GMP sources if
+'mp_set_memory_functions'. It's necessary to change the GMP sources if
this is a problem.
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',
+ 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,
\1f
File: gmp.info, Node: Language Bindings, Next: Algorithms, Prev: Custom Allocation, Up: Top
-15 Language Bindings
+14 Language Bindings
********************
The following packages and projects offer access to GMP from languages
* GMP C++ class interface, *note C++ Class Interface::
Straightforward interface, expression templates to eliminate
temporaries.
-
- * ALP `http://www-sop.inria.fr/saga/logiciels/ALP/'
+ * ALP <https://www-sop.inria.fr/saga/logiciels/ALP/>
Linear algebra and polynomials using templates.
-
- * Arithmos `http://www.win.ua.ac.be/~cant/arithmos/'
- Rationals with infinities and square roots.
-
- * CLN `http://www.ginac.de/CLN/'
+ * CLN <https://www.ginac.de/CLN/>
High level classes for arithmetic.
-
- * LiDIA `http://www.cdc.informatik.tu-darmstadt.de/TI/LiDIA/'
- A C++ library for computational number theory.
-
- * Linbox `http://www.linalg.org/'
+ * Linbox <http://www.linalg.org/>
Sparse vectors and matrices.
-
- * NTL `http://www.shoup.net/ntl/'
+ * NTL <http://www.shoup.net/ntl/>
A C++ number theory library.
-Fortran
- * Omni F77 `http://phase.hpcc.jp/Omni/home.html'
- Arbitrary precision floats.
+Eiffel
+ * Eiffelroom <http://www.eiffelroom.org/node/442>
Haskell
- * Glasgow Haskell Compiler `http://www.haskell.org/ghc/'
+ * Glasgow Haskell Compiler <https://www.haskell.org/ghc/>
Java
- * Kaffe `http://www.kaffe.org/'
-
- * Kissme `http://kissme.sourceforge.net/'
+ * Kaffe <https://github.com/kaffe/kaffe>
Lisp
- * 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'
+ * GNU Common Lisp <https://www.gnu.org/software/gcl/gcl.html>
+ * Librep <http://librep.sourceforge.net/>
+ * XEmacs (21.5.18 beta and up) <https://www.xemacs.org>
Optional big integers, rationals and floats using GMP.
-M4
- * GNU m4 betas `http://www.seindal.dk/rene/gnu/'
- Optionally provides an arbitrary precision `mpeval'.
-
ML
- * MLton compiler `http://mlton.org/'
+ * MLton compiler <http://mlton.org/>
Objective Caml
- * MLGMP `http://www.di.ens.fr/~monniaux/programmes.html.en'
-
- * Numerix `http://pauillac.inria.fr/~quercia/'
+ * MLGMP <https://opam.ocaml.org/packages/mlgmp/>
+ * Numerix <http://pauillac.inria.fr/~quercia/>
Optionally using GMP.
Oz
- * Mozart `http://www.mozart-oz.org/'
+ * Mozart <https://mozart.github.io/>
Pascal
- * GNU Pascal Compiler `http://www.gnu-pascal.de/'
+ * GNU Pascal Compiler <http://www.gnu-pascal.de/>
GMP unit.
-
- * Numerix `http://pauillac.inria.fr/~quercia/'
+ * Numerix <http://pauillac.inria.fr/~quercia/>
For Free Pascal, optionally using GMP.
Perl
- * GMP module, see `demos/perl' in the GMP sources (*note
+ * GMP module, see 'demos/perl' in the GMP sources (*note
Demonstration Programs::).
-
- * 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/'
+ * Math::GMP <https://www.cpan.org/>
+ Compatible with Math::BigInt, but not as many functions as the
+ GMP module above.
+ * Math::BigInt::GMP <https://www.cpan.org/>
Plug Math::GMP into normal Math::BigInt operations.
Pike
- * mpz module in the standard distribution,
- `http://pike.ida.liu.se/'
+ * pikempz module in the standard distribution,
+ <https://pike.lysator.liu.se/>
Prolog
- * SWI Prolog `http://www.swi-prolog.org/'
+ * SWI Prolog <http://www.swi-prolog.org/>
Arbitrary precision floats.
Python
- * mpz module in the standard distribution,
- `http://www.python.org/'
+ * GMPY <https://code.google.com/p/gmpy/>
- * GMPY `http://gmpy.sourceforge.net/'
+Ruby
+ * <https://rubygems.org/gems/gmp>
Scheme
- * GNU Guile (upcoming 1.8)
- `http://www.gnu.org/software/guile/guile.html'
-
- * RScheme `http://www.rscheme.org/'
-
- * STklos `http://www.stklos.org/'
+ * GNU Guile <https://www.gnu.org/software/guile/guile.html>
+ * RScheme <https://www.rscheme.org/>
+ * STklos <http://www.stklos.net/>
Smalltalk
- * GNU Smalltalk
- `http://www.smalltalk.org/versions/GNUSmalltalk.html'
+ * GNU Smalltalk <http://smalltalk.gnu.org/>
Other
- * Axiom `http://savannah.nongnu.org/projects/axiom'
+ * Axiom <https://savannah.nongnu.org/projects/axiom>
Computer algebra using GCL.
-
- * DrGenius `http://drgenius.seul.org/'
+ * DrGenius <http://drgenius.seul.org/>
Geometry system and mathematical programming language.
-
- * GiNaC `http://www.ginac.de/'
+ * GiNaC <httsp://www.ginac.de/>
C++ computer algebra using CLN.
-
- * GOO `http://www.googoogaga.org/'
+ * GOO <https://www.eecs.berkeley.edu/~jrb/goo/>
Dynamic object oriented language.
-
- * Maxima `http://www.ma.utexas.edu/users/wfs/maxima.html'
+ * Maxima <https://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/'
+ * Regina <http://regina.sourceforge.net/>
Topological calculator.
-
- * Yacas `http://www.xs4all.nl/~apinkus/yacas.html'
+ * Yacas <http://yacas.sourceforge.net>
Yet another computer algebra system.
-
\1f
File: gmp.info, Node: Algorithms, Next: Internals, Prev: Language Bindings, Up: Top
-16 Algorithms
+15 Algorithms
*************
This chapter is an introduction to some of the algorithms used for
\1f
File: gmp.info, Node: Multiplication Algorithms, Next: Division Algorithms, Prev: Algorithms, Up: Algorithms
-16.1 Multiplication
+15.1 Multiplication
===================
-NxN limb multiplications and squares are done using one of five
+NxN limb multiplications and squares are done using one of seven
algorithms, as the size N increases.
Algorithm Threshold
Basecase (none)
- Karatsuba `MUL_TOOM22_THRESHOLD'
- Toom-3 `MUL_TOOM33_THRESHOLD'
- Toom-4 `MUL_TOOM44_THRESHOLD'
- FFT `MUL_FFT_THRESHOLD'
+ Karatsuba 'MUL_TOOM22_THRESHOLD'
+ Toom-3 'MUL_TOOM33_THRESHOLD'
+ Toom-4 'MUL_TOOM44_THRESHOLD'
+ Toom-6.5 'MUL_TOOM6H_THRESHOLD'
+ Toom-8.5 'MUL_TOOM8H_THRESHOLD'
+ FFT 'MUL_FFT_THRESHOLD'
- Similarly for squaring, with the `SQR' thresholds.
+ Similarly for squaring, with the 'SQR' thresholds.
NxM multiplications of operands with different sizes above
-`MUL_TOOM22_THRESHOLD' are currently done by special Toom-inspired
+'MUL_TOOM22_THRESHOLD' are currently done by special Toom-inspired
algorithms or directly with FFT, depending on operand size (*note
Unbalanced Multiplication::).
* Karatsuba Multiplication::
* Toom 3-Way Multiplication::
* Toom 4-Way Multiplication::
+* Higher degree Toom'n'half::
* FFT Multiplication::
* Other Multiplication::
* Unbalanced Multiplication::
\1f
File: gmp.info, Node: Basecase Multiplication, Next: Karatsuba Multiplication, Prev: Multiplication Algorithms, Up: Multiplication Algorithms
-16.1.1 Basecase Multiplication
+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.
+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.
- Assembly implementations of `mpn_mul_basecase' are essentially the
+ Assembly implementations of 'mpn_mul_basecase' are essentially the
same as the generic C code, but have all the usual assembly 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
+ 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 assembly
+This can be seen in 'mpn/generic/sqr_basecase.c'. Again the assembly
implementations take essentially the same approach.
u0 u1 u2 u3 u4
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.
+'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
+ 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.
\1f
File: gmp.info, Node: Karatsuba Multiplication, Next: Toom 3-Way Multiplication, Prev: Basecase Multiplication, Up: Multiplication Algorithms
-16.1.2 Karatsuba Multiplication
+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.
+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).
| y1 | y0 |
+----------+----------+
- Let b be the power of 2 where the split occurs, ie. if x0 is k limbs
+ Let b be the power of 2 where the split occurs, i.e. 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,
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
+ 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 GMP extra function call overheads
outweigh the saving.
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
+ 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
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
+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_TOOM22_THRESHOLD' can be as little as 10
-limbs. The `SQR' threshold is usually about twice the `MUL'.
+Karatsuba performs. 'MUL_TOOM22_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
+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.
\1f
File: gmp.info, Node: Toom 3-Way Multiplication, Next: Toom 4-Way Multiplication, Prev: Karatsuba Multiplication, Up: Multiplication Algorithms
-16.1.3 Toom 3-Way Multiplication
+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 GMP
-is described here.
+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 GMP 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).
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).
+ Let b equal the power of 2 which is the size of the x0, x1, y0 and y1
+pieces, i.e. 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
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 GMP the following points are used,
+ X(t) and Y(t) are evaluated and multiplied at 5 points, giving values
+of W(t) at those points. In GMP the following points are used,
Point Value
t=0 x0 * y0, which gives w0 immediately
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).
+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
+ 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(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::).
+ 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].
+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
+and in fact the same 'toom_interpolate_5pts' subroutine is used for both
squaring and multiplying.
Toom-3 is asymptotically O(N^1.465), the exponent being
Near the crossover between Toom-3 and Karatsuba there's generally a
range of sizes where the difference between the two is small.
-`MUL_TOOM33_THRESHOLD' is a somewhat arbitrary point in that range and
+'MUL_TOOM33_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'.
+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.
+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
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.
+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.
\1f
-File: gmp.info, Node: Toom 4-Way Multiplication, Next: FFT Multiplication, Prev: Toom 3-Way Multiplication, Up: Multiplication Algorithms
+File: gmp.info, Node: Toom 4-Way Multiplication, Next: Higher degree Toom'n'half, Prev: Toom 3-Way Multiplication, Up: Multiplication Algorithms
-16.1.4 Toom 4-Way Multiplication
+15.1.4 Toom 4-Way Multiplication
--------------------------------
Karatsuba and Toom-3 split the operands into 2 and 3 coefficients,
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 GMP the following points are used,
+ X(t) and Y(t) are evaluated and multiplied at 7 points, giving values
+of W(t) at those points. In GMP the following points are used,
Point Value
t=0 x0 * y0, which gives w0 immediately
size each.
\1f
-File: gmp.info, Node: FFT Multiplication, Next: Other Multiplication, Prev: Toom 4-Way Multiplication, Up: Multiplication Algorithms
+File: gmp.info, Node: Higher degree Toom'n'half, Next: FFT Multiplication, Prev: Toom 4-Way Multiplication, Up: Multiplication Algorithms
+
+15.1.5 Higher degree Toom'n'half
+--------------------------------
+
+The Toom algorithms described above (*note Toom 3-Way Multiplication::,
+*note Toom 4-Way Multiplication::) generalizes to split into an
+arbitrary number of pieces. In general a split of two equally long
+operands into r pieces leads to evaluations and pointwise
+multiplications done at 2*r-1 points. To fully exploit symmetries it
+would be better to have a multiple of 4 points, that's why for higher
+degree Toom'n'half is used.
+
+ Toom'n'half means that the existence of one more piece is considered
+for a single operand. It can be virtual, i.e. zero, or real, when the
+two operand are not exactly balanced. By choosing an even r, Toom-r+1/2
+requires 2r points, a multiple of four.
+
+ The quadruplets of points include 0, inf, +1, -1 and +-2^i, +-2^-i .
+Each of them giving shortcuts for the evaluation phase and for some
+steps in the interpolation phase. Further tricks are used to reduce the
+memory footprint of the whole multiplication algorithm to a memory
+buffer equal in size to the result of the product.
-16.1.5 FFT Multiplication
+ Current GMP uses both Toom-6'n'half and Toom-8'n'half.
+
+\1f
+File: gmp.info, Node: FFT Multiplication, Next: Other Multiplication, Prev: Higher degree Toom'n'half, Up: Multiplication Algorithms
+
+15.1.6 FFT Multiplication
-------------------------
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 GMP is given here.
+following Schönhage 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 GMP 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 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
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.
+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
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
+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.
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.
+ 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
+'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.
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_THRESHOLD' and
-`SQR_FFT_THRESHOLD' have been found to be in the k=8 range, somewhere
+first FFT faster than Toom-3. In practice 'MUL_FFT_THRESHOLD' and
+'SQR_FFT_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 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
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
+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'
+Something more sophisticated for 'MUL_FFT_TABLE' and 'SQR_FFT_TABLE'
will be needed.
\1f
File: gmp.info, Node: Other Multiplication, Next: Unbalanced Multiplication, Prev: FFT Multiplication, Up: Multiplication Algorithms
-16.1.6 Other Multiplication
+15.1.7 Other Multiplication
---------------------------
The Toom algorithms described above (*note Toom 3-Way Multiplication::,
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)).
+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
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)).
+ 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
\1f
File: gmp.info, Node: Unbalanced Multiplication, Prev: Other Multiplication, Up: Multiplication Algorithms
-16.1.7 Unbalanced Multiplication
+15.1.8 Unbalanced Multiplication
--------------------------------
Multiplication of operands with different sizes, both below
-`MUL_TOOM22_THRESHOLD' are done with plain schoolbook multiplication
+'MUL_TOOM22_THRESHOLD' are done with plain schoolbook multiplication
(*note Basecase Multiplication::).
For really large operands, we invoke 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. GMP currently
-splits the smaller operand onto 2 coefficients, i.e., a polynomial of
-degree 1, but the larger operand can be split into 2, 3, or 4
-coefficients, i.e., a polynomial of degree 1 to 3.
+suggested by Alberto Zanoni and Marco Bodrato. The idea is to split the
+operands into polynomials of different degree. GMP currently splits the
+smaller operand onto 2 coefficients, i.e., a polynomial of degree 1, but
+the larger operand can be split into 2, 3, or 4 coefficients, i.e., a
+polynomial of degree 1 to 3.
\1f
File: gmp.info, Node: Division Algorithms, Next: Greatest Common Divisor Algorithms, Prev: Multiplication Algorithms, Up: Algorithms
-16.2 Division Algorithms
+15.2 Division Algorithms
========================
* Menu:
\1f
File: gmp.info, Node: Single Limb Division, Next: Basecase Division, Prev: Division Algorithms, Up: Division Algorithms
-16.2.1 Single Limb Division
+15.2.1 Single Limb Division
---------------------------
Nx1 division is implemented using repeated 2x1 divisions from high to
inverse, whichever is best on a given CPU.
The multiply by inverse follows "Improved division by invariant
-integers" by Mo"ller and Granlund (*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.
+integers" by Möller and Granlund (*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.
+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
+'__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
+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.
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
+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.
\1f
File: gmp.info, Node: Basecase Division, Next: Divide and Conquer Division, Prev: Single Limb Division, Up: Division Algorithms
-16.2.2 Basecase Division
+15.2.2 Basecase Division
------------------------
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, and
-`mpn/generic/sb_divrem_mn.c'.
-
- 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.
+'mpn/generic/sb_divrem_mn.c'.
+
+ 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
\1f
File: gmp.info, Node: Divide and Conquer Division, Next: Block-Wise Barrett Division, Prev: Basecase Division, Up: Division Algorithms
-16.2.3 Divide and Conquer Division
+15.2.3 Divide and Conquer Division
----------------------------------
-For divisors larger than `DC_DIV_QR_THRESHOLD', division is done by
+For divisors larger than 'DC_DIV_QR_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::).
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_TOOM22_THRESHOLD' that divide and conquer is of use.
-
- `DC_DIV_QR_THRESHOLD' is based on the divisor size N, so it will be
-somewhere above twice `MUL_TOOM22_THRESHOLD', but how much above
-depends on the CPU. An optimized `mpn_mul_basecase' can lower
-`DC_DIV_QR_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.
+'MUL_TOOM22_THRESHOLD' that divide and conquer is of use.
+
+ 'DC_DIV_QR_THRESHOLD' is based on the divisor size N, so it will be
+somewhere above twice 'MUL_TOOM22_THRESHOLD', but how much above depends
+on the CPU. An optimized 'mpn_mul_basecase' can lower
+'DC_DIV_QR_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.
\1f
File: gmp.info, Node: Block-Wise Barrett Division, Next: Exact Division, Prev: Divide and Conquer Division, Up: Division Algorithms
-16.2.4 Block-Wise Barrett Division
+15.2.4 Block-Wise Barrett Division
----------------------------------
For the largest divisions, a block-wise Barrett division algorithm is
used. Here, the divisor is inverted to a precision determined by the
-relative size of the dividend and divisor. Blocks of quotient limbs
-are then generated by multiplying blocks from the dividend by the
-inverse.
+relative size of the dividend and divisor. Blocks of quotient limbs are
+then generated by multiplying blocks from the dividend by the inverse.
- Our block-wise algorithm computes a smaller inverse than in the
-plain Barrett algorithm. For a 2n/n division, the inverse will be just
+ Our block-wise algorithm computes a smaller inverse than in the plain
+Barrett algorithm. For a 2n/n division, the inverse will be just
ceil(n/2) limbs.
\1f
File: gmp.info, Node: Exact Division, Next: Exact Remainder, Prev: Block-Wise Barrett Division, Up: Division Algorithms
-16.2.5 Exact Division
+15.2.5 Exact Division
---------------------
A so-called exact division is when the dividend is known to be an exact
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.
+ 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
+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 `binvert_limb'
-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
+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 'binvert_limb'
+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
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
+ 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
+'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.
-\1f
-File: gmp.info, Node: Exact Remainder, Next: Small Quotient Division, Prev: Exact Division, Up: Division Algorithms
-
-16.2.6 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_divexact_by3', `mpn_modexact_1_odd' and the `mpn_redc_X'
-functions 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_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.
-
-\1f
-File: gmp.info, Node: Small Quotient Division, Prev: Exact Remainder, Up: Division Algorithms
-
-16.2.7 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.
-
-\1f
-File: gmp.info, Node: Greatest Common Divisor Algorithms, Next: Powering Algorithms, Prev: Division Algorithms, Up: Algorithms
-
-16.3 Greatest Common Divisor
-============================
-
-* Menu:
-
-* Binary GCD::
-* Lehmer's Algorithm::
-* Subquadratic GCD::
-* Extended GCD::
-* Jacobi Symbol::
-
-\1f
-File: gmp.info, Node: Binary GCD, Next: Lehmer's Algorithm, Prev: Greatest Common Divisor Algorithms, Up: Greatest Common Divisor Algorithms
-
-16.3.1 Binary GCD
------------------
-
-At small sizes GMP 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,
-repeatedly computes the quotient q = floor(a/b) and replaces a,b by v,
-u - q v. The binary algorithm has so far been found to be faster than
-the Euclidean algorithm 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.
-
- Currently, the binary algorithm is used for GCD only when N < 3.
-
-\1f
-File: gmp.info, Node: Lehmer's Algorithm, Next: Subquadratic GCD, Prev: Binary GCD, Up: Greatest Common Divisor Algorithms
-
-16.3.2 Lehmer's algorithm
--------------------------
-
-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 GMP 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.
-
-
-\1f
-File: gmp.info, Node: Subquadratic GCD, Next: Extended GCD, Prev: Lehmer's Algorithm, Up: Greatest Common Divisor Algorithms
-
-16.3.3 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.
-
-\1f
-File: gmp.info, Node: Extended GCD, Next: Jacobi Symbol, Prev: Subquadratic GCD, Up: Greatest Common Divisor Algorithms
-
-16.3.4 Extended GCD
--------------------
-
-The extended GCD function, or GCDEXT, calculates gcd(a,b) and also
-cofactors x and y satisfying a*x+b*y=gcd(a,b). All the algorithms used
-for plain GCD are extended to handle this case. The binary algorithm is
-used only for single-limb GCDEXT. 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. This gives the same asymptotic running time as
-for GCD and HGCD, O(M(N)*log(N))
-
- One difference to plain GCD is that while the inputs a and b are
-reduced as the algorithm proceeds, the cofactors x and y grow in size.
-This makes the tuning of the chopping-point more difficult. The current
-code chops off the most significant half of the inputs for the call to
-HGCD in the first iteration, and the most significant two thirds for
-the remaining calls. This strategy could surely be improved. Also the
-stop condition for the loop, where Lehmer's algorithm is invoked once
-the inputs are reduced below `GCDEXT_DC_THRESHOLD', could maybe be
-improved by taking into account the current size of the cofactors.
-
-\1f
-File: gmp.info, Node: Jacobi Symbol, Prev: Extended GCD, Up: Greatest Common Divisor Algorithms
-
-16.3.5 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.
-
projects / xonotic / xonotic.git / blobdiff