This is ../../gmp/doc/gmp.info, produced by makeinfo version 4.8 from
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../../gmp/doc/gmp.texi.
|
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This manual describes how to install and use the GNU multiple
|
precision arithmetic library, version 6.1.0.
|
|
Copyright 1991, 1993-2015 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::.
|
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INFO-DIR-SECTION GNU libraries
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START-INFO-DIR-ENTRY
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* gmp: (gmp). GNU Multiple Precision Arithmetic Library.
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END-INFO-DIR-ENTRY
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File: gmp.info, Node: Top, Next: Copying, Prev: (dir), Up: (dir)
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GNU MP
|
******
|
|
This manual describes how to install and use the GNU multiple
|
precision arithmetic library, version 6.1.0.
|
|
Copyright 1991, 1993-2015 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:
|
|
* Copying:: GMP Copying Conditions (LGPL).
|
* Introduction to GMP:: Brief introduction to GNU MP.
|
* Installing GMP:: How to configure and compile the GMP library.
|
* GMP Basics:: What every GMP user should know.
|
* Reporting Bugs:: How to usefully report bugs.
|
* Integer Functions:: Functions for arithmetic on signed integers.
|
* Rational Number Functions:: Functions for arithmetic on rational numbers.
|
* Floating-point Functions:: Functions for arithmetic on floats.
|
* Low-level Functions:: Fast functions for natural numbers.
|
* Random Number Functions:: Functions for generating random numbers.
|
* Formatted Output:: `printf' style output.
|
* Formatted Input:: `scanf' style input.
|
* C++ Class Interface:: Class wrappers around GMP types.
|
* Custom Allocation:: How to customize the internal allocation.
|
* Language Bindings:: Using GMP from other languages.
|
* Algorithms:: What happens behind the scenes.
|
* Internals:: How values are represented behind the scenes.
|
|
* Contributors:: Who brings you this library?
|
* References:: Some useful papers and books to read.
|
* GNU Free Documentation License::
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* Concept Index::
|
* Function Index::
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|
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File: gmp.info, Node: Copying, Next: Introduction to GMP, Prev: Top, Up: Top
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GNU MP Copying Conditions
|
*************************
|
|
This library is "free"; this means that everyone is free to use it and
|
free to redistribute it on a free basis. The library is not in the
|
public domain; it is copyrighted and there are restrictions on its
|
distribution, but these restrictions are designed to permit everything
|
that a good cooperating citizen would want to do. What is not allowed
|
is to try to prevent others from further sharing any version of this
|
library that they might get from you.
|
|
Specifically, we want to make sure that you have the right to give
|
away copies of the library, that you receive source code or else can
|
get it if you want it, that you can change this library or use pieces
|
of it in new free programs, and that you know you can do these things.
|
|
To make sure that everyone has such rights, we have to forbid you to
|
deprive anyone else of these rights. For example, if you distribute
|
copies of the 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.
|
|
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.
|
|
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.
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|
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File: gmp.info, Node: Introduction to GMP, Next: Installing GMP, Prev: Copying, Up: Top
|
|
1 Introduction to GNU MP
|
************************
|
|
GNU MP is a portable library written in C for arbitrary precision
|
arithmetic on integers, rational numbers, and floating-point numbers.
|
It aims to provide the fastest possible arithmetic for all applications
|
that need higher precision than is directly supported by the basic C
|
types.
|
|
Many applications use just a few hundred bits of precision; but some
|
applications may need thousands or even millions of bits. 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.
|
|
The speed of GMP is achieved by using fullwords as the basic
|
arithmetic type, by using sophisticated algorithms, by including
|
carefully optimized assembly code for the most common inner loops for
|
many different CPUs, and by a general emphasis on speed (as opposed to
|
simplicity or elegance).
|
|
There is assembly code for these CPUs: ARM 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
|
|
`https://gmplib.org/'
|
|
The latest version of the library is available at
|
|
`https://ftp.gnu.org/gnu/gmp/'
|
|
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
|
|
`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.
|
|
|
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
|
compiler options that must be used on applications.
|
|
The rest of the manual can be used for later reference, although it
|
is probably a good idea to glance through it.
|
|
|
File: gmp.info, Node: Installing GMP, Next: GMP Basics, Prev: Introduction to GMP, Up: Top
|
|
2 Installing GMP
|
****************
|
|
GMP has an autoconf/automake/libtool based configuration system. On a
|
Unix-like system a basic build can be done with
|
|
./configure
|
make
|
|
Some self-tests can be run with
|
|
make check
|
|
And you can install (under `/usr/local' by default) with
|
|
make install
|
|
If you experience problems, please report them to
|
<gmp-bugs@gmplib.org>. See *Note Reporting Bugs::, for information on
|
what to include in useful bug reports.
|
|
* Menu:
|
|
* Build Options::
|
* ABI and ISA::
|
* Notes for Package Builds::
|
* Notes for Particular Systems::
|
* Known Build Problems::
|
* Performance optimization::
|
|
|
File: gmp.info, Node: Build Options, Next: ABI and ISA, Prev: Installing GMP, Up: Installing GMP
|
|
2.1 Build Options
|
=================
|
|
All the usual autoconf configure options are available, run `./configure
|
--help' for a summary. The file `INSTALL.autoconf' has some generic
|
installation information too.
|
|
Tools
|
`configure' requires various Unix-like tools. See *Note Notes for
|
Particular Systems::, for some options on non-Unix systems.
|
|
It might be possible to build without the help of `configure',
|
certainly all the code is there, but unfortunately you'll be on
|
your own.
|
|
Build Directory
|
To compile in a separate build directory, `cd' to that directory,
|
and prefix the configure command with the path to the GMP source
|
directory. For example
|
|
cd /my/build/dir
|
/my/sources/gmp-6.1.0/configure
|
|
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'
|
By default both shared and static libraries are built (where
|
possible), but one or other can be disabled. Shared libraries
|
result in smaller executables and permit code sharing between
|
separate running processes, but on some CPUs are slightly slower,
|
having a small cost on each function call.
|
|
Native Compilation, `--build=CPU-VENDOR-OS'
|
For normal native compilation, the system can be specified with
|
`--build'. By default `./configure' uses the output from running
|
`./config.guess'. On some systems `./config.guess' can determine
|
the exact CPU type, on others it will be necessary to give it
|
explicitly. For example,
|
|
./configure --build=ultrasparc-sun-solaris2.7
|
|
In all cases the `OS' part is important, since it controls how
|
libtool generates shared libraries. Running `./config.guess' is
|
the simplest way to see what it should be, if you don't know
|
already.
|
|
Cross Compilation, `--host=CPU-VENDOR-OS'
|
When cross-compiling, the system used for compiling is given by
|
`--build' and the system where the library will run is given by
|
`--host'. For example when using a FreeBSD Athlon system to build
|
GNU/Linux m68k binaries,
|
|
./configure --build=athlon-pc-freebsd3.5 --host=m68k-mac-linux-gnu
|
|
Compiler tools are sought first with the host system type as a
|
prefix. For example `m68k-mac-linux-gnu-ranlib' is tried, then
|
plain `ranlib'. This makes it possible for a set of
|
cross-compiling tools to co-exist with native tools. The prefix
|
is the argument to `--host', and this can be an alias, such as
|
`m68k-linux'. But note that tools don't have to be setup this
|
way, it's enough to just have a `PATH' with a suitable
|
cross-compiling `cc' etc.
|
|
Compiling for a different CPU in the same family as the build
|
system is a form of cross-compilation, though very possibly this
|
would merely be special options on a native compiler. In any case
|
`./configure' avoids depending on being able to run code on the
|
build system, which is important when creating binaries for a
|
newer CPU since they very possibly won't run on the build system.
|
|
In all cases the compiler must be able to produce an executable
|
(of whatever format) from a standard C `main'. Although only
|
object files will go to make up `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
|
when cross-compiling, because it may not be possible to correctly
|
guess the build system type if the `PATH' has only a
|
cross-compiling `cc'.
|
|
Note that the `--target' option is not appropriate for 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.)
|
|
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 following CPUs have specific support. See `configure.ac' for
|
details of what code and compiler options they select.
|
|
* Alpha: alpha, alphaev5, alphaev56, alphapca56, alphapca57,
|
alphaev6, alphaev67, alphaev68 alphaev7
|
|
* Cray: c90, j90, t90, sv1
|
|
* HPPA: hppa1.0, hppa1.1, hppa2.0, hppa2.0n, hppa2.0w, hppa64
|
|
* IA-64: ia64, itanium, itanium2
|
|
* MIPS: mips, mips3, mips64
|
|
* Motorola: m68k, m68000, m68010, m68020, m68030, m68040,
|
m68060, m68302, m68360, m88k, m88110
|
|
* POWER: power, power1, power2, power2sc
|
|
* PowerPC: powerpc, powerpc64, powerpc401, powerpc403,
|
powerpc405, powerpc505, powerpc601, powerpc602, powerpc603,
|
powerpc603e, powerpc604, powerpc604e, powerpc620, powerpc630,
|
powerpc740, powerpc7400, powerpc7450, powerpc750, powerpc801,
|
powerpc821, powerpc823, powerpc860, powerpc970
|
|
* SPARC: sparc, sparcv8, microsparc, supersparc, sparcv9,
|
ultrasparc, ultrasparc2, ultrasparc2i, ultrasparc3, sparc64
|
|
* x86 family: i386, i486, i586, pentium, pentiummmx, pentiumpro,
|
pentium2, pentium3, pentium4, k6, k62, k63, athlon, amd64,
|
viac3, viac32
|
|
* 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 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.
|
|
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.)
|
|
`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
|
can be selected. For example
|
|
./configure --host=mips64-sgi-irix6 ABI=n32
|
|
See *Note ABI and ISA::, for the available choices on relevant
|
CPUs, and what applications need to do.
|
|
`CC', `CFLAGS'
|
By default the C compiler used is chosen from among some likely
|
candidates, with `gcc' normally preferred if it's present. The
|
usual `CC=whatever' can be passed to `./configure' to choose
|
something different.
|
|
For various systems, default compiler flags are set based on the
|
CPU and compiler. The usual `CFLAGS="-whatever"' can be passed to
|
`./configure' to use something different or to set good flags for
|
systems 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
|
to check the defaults when considering changing or adding
|
something.
|
|
Note that when `CC' and `CFLAGS' are specified on a system
|
supporting multiple ABIs it's important to give an explicit
|
`ABI=whatever', since 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).
|
|
`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.
|
|
`CC_FOR_BUILD'
|
Some build-time programs are compiled and run to generate
|
host-specific data tables. `CC_FOR_BUILD' is the compiler used
|
for this. It doesn't need to be in any particular ABI or mode, it
|
merely needs to generate executables that can run. The default is
|
to try the selected `CC' and some likely candidates such as `cc'
|
and `gcc', looking for something that works.
|
|
No flags are used with `CC_FOR_BUILD' because a simple invocation
|
like `cc foo.c' should be enough. If some particular options are
|
required they can be included as for instance `CC_FOR_BUILD="cc
|
-whatever"'.
|
|
C++ Support, `--enable-cxx'
|
C++ support in 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
|
compiler can be found. The C++ support consists of a library
|
`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
|
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
|
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
|
that built it, since name mangling and runtime support are usually
|
incompatible between different compilers.
|
|
`CXX', `CXXFLAGS'
|
When C++ support is enabled, the C++ compiler and its flags can be
|
set with variables `CXX' and `CXXFLAGS' in the usual way. The
|
default for `CXX' is the first compiler that works from a list of
|
likely candidates, with `g++' normally preferred when available.
|
The default for `CXXFLAGS' is to try `CFLAGS', `CFLAGS' without
|
`-g', then for `g++' either `-g -O2' or `-O2', or for other
|
compilers `-g' or nothing. Trying `CFLAGS' this way is convenient
|
when using `gcc' and `g++' together, since the flags for `gcc' will
|
usually suit `g++'.
|
|
It's important that the C and C++ compilers match, meaning their
|
startup and runtime support routines are compatible and that they
|
generate code in the same ABI (if there's a choice of ABIs on the
|
system). `./configure' isn't currently able to check these things
|
very well itself, so for that reason `--disable-cxx' is the
|
default, to avoid a build failure due to a compiler mismatch.
|
Perhaps this will change in the future.
|
|
Incidentally, it's normally not good enough to set `CXX' to the
|
same as `CC'. Although `gcc' for instance recognises `foo.cc' as
|
C++ code, only `g++' will invoke the linker the right way when
|
building an executable or shared library from C++ object files.
|
|
Temporary Memory, `--enable-alloca=<choice>'
|
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.
|
|
* `malloc-notreentrant' - the heap, with global variables.
|
|
For convenience, the following choices are also available.
|
`--disable-alloca' is the same as `no'.
|
|
* `yes' - a synonym for `alloca'.
|
|
* `no' - a synonym for `malloc-reentrant'.
|
|
* `reentrant' - `alloca' if available, otherwise
|
`malloc-reentrant'. This is the default.
|
|
* `notreentrant' - `alloca' if available, otherwise
|
`malloc-notreentrant'.
|
|
`alloca' is reentrant and fast, and is recommended. It actually
|
allocates just small blocks on the stack; larger ones use
|
malloc-reentrant.
|
|
`malloc-reentrant' is, as the name suggests, reentrant and thread
|
safe, but `malloc-notreentrant' is faster and should be used if
|
reentrancy is not required.
|
|
The two malloc methods in fact use the memory allocation functions
|
selected by `mp_set_memory_functions', these being `malloc' and
|
friends by default. *Note Custom Allocation::.
|
|
An additional choice `--enable-alloca=debug' is available, to help
|
when debugging memory related problems (*note Debugging::).
|
|
FFT Multiplication, `--disable-fft'
|
By default multiplications are done using Karatsuba, 3-way Toom,
|
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.
|
|
Assertion Checking, `--enable-assert'
|
This option enables some consistency checking within the library.
|
This can be of use while debugging, *note Debugging::.
|
|
Execution Profiling, `--enable-profiling=prof/gprof/instrument'
|
Enable profiling support, in one of various styles, *note
|
Profiling::.
|
|
`MPN_PATH'
|
Various 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
|
|
MPN_PATH="sparc32/v8 sparc32 generic"
|
|
which means look first for v8 code, then plain sparc32 (which is
|
v7), and finally fall back on generic C. Knowledgeable users with
|
special requirements can specify a different path. Normally this
|
is completely unnecessary.
|
|
Documentation
|
The source for the document you're now reading is `doc/gmp.texi',
|
in Texinfo format, see *Note Texinfo: (texinfo)Top.
|
|
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.
|
|
Some supplementary notes can also be found in the `doc'
|
subdirectory.
|
|
|
|
File: gmp.info, Node: ABI and ISA, Next: Notes for Package Builds, Prev: Build Options, Up: Installing GMP
|
|
2.2 ABI and ISA
|
===============
|
|
ABI (Application Binary Interface) refers to the calling conventions
|
between functions, meaning what registers are used and what sizes the
|
various C data types are. ISA (Instruction Set Architecture) refers to
|
the instructions and registers a CPU has available.
|
|
Some 64-bit ISA CPUs have both a 64-bit ABI and a 32-bit ABI
|
defined, the latter for compatibility with older CPUs in the family.
|
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
|
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.
|
|
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
|
might require more than that. Note that builds for different ABIs need
|
to done separately, with a fresh `./configure' and `make' each.
|
|
|
AMD64 (`x86_64')
|
On AMD64 systems supporting both 32-bit and 64-bit modes for
|
applications, the following ABI choices are available.
|
|
`ABI=64'
|
The 64-bit ABI uses 64-bit limbs and pointers and makes full
|
use of the chip architecture. This is the default.
|
Applications will usually not need special compiler flags,
|
but for reference the option is
|
|
gcc -m64
|
|
`ABI=32'
|
The 32-bit ABI is the usual i386 conventions. This will be
|
slower, and is not recommended except for inter-operating
|
with other code not yet 64-bit capable. Applications must be
|
compiled with
|
|
gcc -m32
|
|
(In GCC 2.95 and earlier there's no `-m32' option, it's the
|
only mode.)
|
|
`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.
|
|
gcc -mx32
|
|
|
|
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'
|
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.
|
This ABI is available on hppa64 GNU/Linux and on HP-UX 10 or
|
higher. Applications must be compiled with
|
|
gcc [built for 2.0n]
|
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
|
slower than for 2.0w. (The GMP assembly code is the same
|
though.)
|
|
`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
|
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
|
the other ABI. GMP will detect how it was built, and skip to the
|
corresponding `ABI'.
|
|
|
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
|
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
|
|
gcc -mlp64
|
cc +DD64
|
|
On other IA-64 systems, GNU/Linux for instance, `ABI=64' is the
|
only choice.
|
|
|
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'
|
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.
|
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
|
|
gcc -mabi=n32
|
cc -n32
|
|
`ABI=64'
|
The 64-bit ABI is 64-bit pointers and integers. Applications
|
must be compiled with
|
|
gcc -mabi=64
|
cc -64
|
|
Note that MIPS GNU/Linux, as of kernel version 2.2, doesn't have
|
the necessary support for n32 or 64 and so only gets a 32-bit limb
|
and the MIPS 2 code.
|
|
|
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
|
be compiled with
|
|
gcc -maix64
|
xlc -q64
|
|
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
|
chip still in 32-bit mode and using 32-bit calling
|
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'
|
This is the basic 32-bit PowerPC ABI, with a 32-bit limb. No
|
special compiler options are needed for applications.
|
|
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'
|
The 64-bit V9 ABI is available on the various BSD sparc64
|
ports, recent versions of Sparc64 GNU/Linux, and Solaris 2.7
|
and up (when the kernel is in 64-bit mode). GCC 3.2 or
|
higher, or Sun `cc' is required. On GNU/Linux, depending on
|
the default `gcc' mode, applications must be compiled with
|
|
gcc -m64
|
|
On Solaris applications must be compiled with
|
|
gcc -m64 -mptr64 -Wa,-xarch=v9 -mcpu=v9
|
cc -xarch=v9
|
|
On the BSD sparc64 systems no special options are required,
|
since 64-bits is the only ABI available.
|
|
`ABI=32'
|
For the basic 32-bit ABI, 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'
|
though.)
|
|
gcc -mv8plus
|
cc -xarch=v8plus
|
|
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.
|
|
On Solaris 2.6 and earlier, only `ABI=32' is available since the
|
kernel doesn't save all registers.
|
|
On Solaris 2.7 with the kernel in 32-bit mode, a normal native
|
build will reject `ABI=64' because the resulting executables won't
|
run. `ABI=64' can still be built if desired by making it look
|
like a cross-compile, for example
|
|
./configure --build=none --host=sparcv9-sun-solaris2.7 ABI=64
|
|
|
File: gmp.info, Node: Notes for Package Builds, Next: Notes for Particular Systems, Prev: ABI and ISA, Up: Installing GMP
|
|
2.3 Notes for Package Builds
|
============================
|
|
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
|
interface versions: (libtool)Versioning.).
|
|
The GMP 4 series will be upwardly binary compatible in each release
|
and will be upwardly binary compatible with all of the GMP 3 series.
|
Additional function interfaces may be added in each release, so on
|
systems where libtool versioning is not fully checked by the loader an
|
auxiliary mechanism may be needed to express that a dynamic linked
|
application depends on a new enough 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.
|
|
When building a package for a CPU family, care should be taken to use
|
`--host' (or `--build') to choose the least common denominator among
|
the CPUs which might use the package. For example this might mean plain
|
`sparc' (meaning V7) for SPARCs.
|
|
For x86s, `--enable-fat' sets things up for a fat binary build,
|
making a runtime selection of optimized low level routines. This is a
|
good choice for packaging to run on a range of x86 chips.
|
|
Users who care about speed will want 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.
|
|
On systems with multiple ABIs, a packaged build will need to decide
|
which among the choices is to be provided, see *Note ABI and ISA::. A
|
given run of `./configure' etc will only build one ABI. If a second
|
ABI is also required then a second run of `./configure' etc must be
|
made, starting from a clean directory tree (`make distclean').
|
|
As noted under "ABI and ISA", currently no attempt is made to follow
|
system conventions for install locations that vary with ABI, such as
|
`/usr/lib/sparcv9' for `ABI=64' as opposed to `/usr/lib' for `ABI=32'.
|
A package build can override `libdir' and other standard variables as
|
necessary.
|
|
Note that `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'
|
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'.
|
|
|
File: gmp.info, Node: Notes for Particular Systems, Next: Known Build Problems, Prev: Notes for Package Builds, Up: Installing GMP
|
|
2.4 Notes for Particular Systems
|
================================
|
|
AIX 3 and 4
|
On systems `*-*-aix[34]*' shared libraries are disabled by
|
default, since some versions of the native `ar' fail on the
|
convenience libraries used. A shared build can be attempted with
|
|
./configure --enable-shared --disable-static
|
|
Note that the `--disable-static' is necessary because in a shared
|
build libtool makes `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
|
have a bug in unsigned division, giving wrong results for some
|
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
|
can only use the standard one, which unfortunately is not the
|
default but must be selected by defining `__USE_STD_IOSTREAM'.
|
Configure with for instance
|
|
./configure --enable-cxx CPPFLAGS=-D__USE_STD_IOSTREAM
|
|
Floating Point Mode
|
On some systems, the hardware floating point has a control mode
|
which can set all operations to be done in a particular precision,
|
for instance single, double or extended on x86 systems (x87
|
floating point). The 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/'
|
|
Microsoft also publishes an Interix "Services for Unix" which can
|
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
|
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.
|
|
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'.
|
|
cd .libs
|
lib /def:libgmp-3.dll.def /out:libgmp-3.lib
|
|
MINGW uses the C runtime library `msvcrt.dll' for I/O, so
|
applications wanting to use the GMP I/O routines must be compiled
|
with `cl /MD' to do the same. If one of the other C runtime
|
library choices provided by MS C is desired then the suggestion is
|
to use the 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'.
|
|
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'.
|
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
|
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 (`--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
|
significant performance increase over the V7 code selected by plain
|
`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
|
Options.).
|
|
This makes that code unsuitable for use with the special V9
|
`-mcmodel=embmedany' (which uses `g4' as a data segment pointer),
|
and for applications wanting to use those registers for special
|
purposes. In these cases the only suggestion currently is to
|
build 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
|
use GNU m4).
|
|
x86 CPU Types
|
`i586', `pentium' or `pentiummmx' code is good for its intended P5
|
Pentium chips, but quite slow when run on Intel P6 class chips
|
(PPro, P-II, P-III). `i386' is a better choice when making
|
binaries that must run on both.
|
|
x86 MMX and SSE2 Code
|
If the CPU selected has MMX code but the assembler doesn't support
|
it, a warning is given and non-MMX code is used instead. This
|
will be an inferior build, since the MMX code that's present is
|
there because it's faster than the corresponding plain integer
|
code. The same applies to SSE2.
|
|
Old versions of `gas' don't support MMX instructions, in particular
|
version 1.92.3 that comes with FreeBSD 2.2.8 or the more recent
|
OpenBSD 3.1 doesn't.
|
|
Solaris 2.6 and 2.7 `as' generate incorrect object code for
|
register to register `movq' instructions, and so can't be used for
|
MMX code. Install a recent `gas' if MMX code is wanted on these
|
systems.
|
|
|
File: gmp.info, Node: Known Build Problems, Next: Performance optimization, Prev: Notes for Particular Systems, Up: Installing GMP
|
|
2.5 Known Build Problems
|
========================
|
|
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
|
compiler options when linking a shared library. This will
|
hopefully be relaxed in the future, but for now if this is a
|
problem the suggestion is to create a little script to hide them,
|
and for instance configure with
|
|
./configure CC=gcc-with-my-options
|
|
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.
|
|
`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.
|
|
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
|
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'.
|
|
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'
|
might not be rebuilt).
|
|
MacOS X (`*-*-darwin*')
|
Libtool currently only knows how to create shared libraries on
|
MacOS X using the native `cc' (which is a modified GCC), not a
|
plain GCC. A static-only build should work though
|
(`--disable-shared').
|
|
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.)
|
|
POWER and PowerPC
|
Bugs in GCC 2.7.2 (and 2.6.3) mean it can't be used to compile GMP
|
on POWER or PowerPC. If you want to use GCC for these machines,
|
get GCC 2.7.2.1 (or later).
|
|
Sequent Symmetry
|
Use the GNU assembler instead of the system assembler, since the
|
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.
|
|
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.
|
|
|
File: gmp.info, Node: Performance optimization, Prev: Known Build Problems, Up: Installing GMP
|
|
2.6 Performance optimization
|
============================
|
|
For optimal performance, build GMP for the exact CPU type of the target
|
computer, see *Note Build Options::.
|
|
Unlike what is the case for most other programs, the compiler
|
typically doesn't matter much, since 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,
|
|
cd tune
|
make tuneup
|
./tuneup
|
|
will generate better contents for the `gmp-mparam.h' parameter file.
|
|
To use the results, put the output in the file indicated in the
|
`Parameters for ...' header. Then recompile from scratch.
|
|
The `tuneup' program takes one useful parameter, `-f NNN', which
|
instructs the program how long to check FFT multiply parameters. If
|
you're going to use GMP for extremely large numbers, you may want to
|
run `tuneup' with a large NNN value.
|
|
|
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.*
|
|
* Menu:
|
|
* Headers and Libraries::
|
* Nomenclature and Types::
|
* Function Classes::
|
* Variable Conventions::
|
* Parameter Conventions::
|
* Memory Management::
|
* Reentrancy::
|
* Useful Macros and Constants::
|
* Compatibility with older versions::
|
* Demonstration Programs::
|
* Efficiency::
|
* Debugging::
|
* Profiling::
|
* Autoconf::
|
* Emacs::
|
|
|
File: gmp.info, Node: Headers and Libraries, Next: Nomenclature and Types, Prev: GMP Basics, Up: GMP Basics
|
|
3.1 Headers and Libraries
|
=========================
|
|
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.
|
|
#include <gmp.h>
|
|
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>' 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
|
|
gcc myprogram.c -lgmp
|
|
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.
|
|
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
|
directories, and some sort of run-time path for a shared library.
|
|
|
File: gmp.info, Node: Nomenclature and Types, Next: Function Classes, Prev: Headers and Libraries, Up: GMP Basics
|
|
3.2 Nomenclature and Types
|
==========================
|
|
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 sum;
|
|
struct foo { mpz_t x, y; };
|
|
mpz_t vec[20];
|
|
"Rational number" means a multiple precision fraction. The C data
|
type for these fractions is `mpq_t'. For example:
|
|
mpq_t quotient;
|
|
"Floating point number" or "Float" for short, is an arbitrary
|
precision mantissa with a limited precision exponent. The C data type
|
for such objects is `mpf_t'. For example:
|
|
mpf_t fp;
|
|
The floating point functions accept and return exponents in the C
|
type `mp_exp_t'. Currently this is usually a `long', but on some
|
systems it's an `int' for efficiency.
|
|
A "limb" means the part of a multi-precision number that fits in a
|
single machine word. (We chose this word because a limb of the human
|
body is analogous to a digit, only larger, and containing several
|
digits.) Normally a limb is 32 or 64 bits. The C data type for a limb
|
is `mp_limb_t'.
|
|
Counts of limbs 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.
|
|
Counts of bits of a multi-precision number are represented in the C
|
type `mp_bitcnt_t'. Currently this is always an `unsigned long', but on
|
some systems it will be an `unsigned long long' in the future.
|
|
"Random state" means an algorithm selection and current state data.
|
The C data type for such objects is `gmp_randstate_t'. For example:
|
|
gmp_randstate_t rstate;
|
|
Also, in general `mp_bitcnt_t' is used for bit counts and ranges, and
|
`size_t' is used for byte or character counts.
|
|
|
File: gmp.info, Node: Function Classes, Next: Variable Conventions, Prev: Nomenclature and Types, Up: GMP Basics
|
|
3.3 Function Classes
|
====================
|
|
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
|
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 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 70
|
functions is this class. (*note Floating-point Functions::)
|
|
4. Fast low-level functions that operate on natural numbers. These
|
are used by the functions in the preceding groups, and you can
|
also call them directly from very time-critical user programs.
|
These functions' names begin with `mpn_'. The associated type is
|
array of `mp_limb_t'. There are about 60 (hard-to-use) functions
|
in this class. (*note Low-level Functions::)
|
|
5. Miscellaneous functions. Functions for setting up custom
|
allocation and functions for generating random numbers. (*note
|
Custom Allocation::, and *note Random Number Functions::)
|
|
|
File: gmp.info, Node: Variable Conventions, Next: Parameter Conventions, Prev: Function Classes, Up: GMP Basics
|
|
3.4 Variable Conventions
|
========================
|
|
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.
|
|
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 (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
|
functions, and floating-point functions for details.
|
|
A variable should only be initialized once, or at least cleared
|
between each initialization. After a variable has been initialized, it
|
may be assigned to any number of times.
|
|
For efficiency reasons, avoid excessive initializing and clearing.
|
In general, initialize near the start of a function and clear near the
|
end. For example,
|
|
void
|
foo (void)
|
{
|
mpz_t n;
|
int i;
|
mpz_init (n);
|
for (i = 1; i < 100; i++)
|
{
|
mpz_mul (n, ...);
|
mpz_fdiv_q (n, ...);
|
...
|
}
|
mpz_clear (n);
|
}
|
|
|
File: gmp.info, Node: Parameter Conventions, Next: Memory Management, Prev: Variable Conventions, Up: GMP Basics
|
|
3.5 Parameter Conventions
|
=========================
|
|
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.
|
|
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.
|
|
Here's an example accepting an `mpz_t' parameter, doing a
|
calculation, and storing the result to the indicated parameter.
|
|
void
|
foo (mpz_t result, const mpz_t param, unsigned long n)
|
{
|
unsigned long i;
|
mpz_mul_ui (result, param, n);
|
for (i = 1; i < n; i++)
|
mpz_add_ui (result, result, i*7);
|
}
|
|
int
|
main (void)
|
{
|
mpz_t r, n;
|
mpz_init (r);
|
mpz_init_set_str (n, "123456", 0);
|
foo (r, n, 20L);
|
gmp_printf ("%Zd\n", r);
|
return 0;
|
}
|
|
`foo' works even if the mainline passes the same variable for
|
`param' and `result', just like the library functions. But sometimes
|
it's tricky to make that work, and an application might not want to
|
bother supporting that sort of thing.
|
|
For interest, the 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.
|
|
|
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,
|
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.
|
Normally this is the best policy, since it avoids frequent reallocation.
|
Applications that need to return memory to the heap at some particular
|
point can use `mpz_realloc2', or clear variables no longer needed.
|
|
`mpf_t' variables, in the current implementation, use a fixed amount
|
of space, determined by the chosen precision and allocated at
|
initialization, so their size doesn't change.
|
|
All memory is allocated using `malloc' and friends by default, but
|
this can be changed, see *Note Custom Allocation::. Temporary memory
|
on the stack is also used (via `alloca'), but this can be changed at
|
build-time if desired, see *Note Build Options::.
|
|
|
File: gmp.info, Node: Reentrancy, Next: Useful Macros and Constants, Prev: Memory Management, Up: GMP Basics
|
|
3.7 Reentrancy
|
==============
|
|
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),
|
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.
|
|
* `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
|
instead.
|
|
* `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'
|
instead.
|
|
* `mp_set_memory_functions' uses global variables to store the
|
selected memory allocation functions.
|
|
* If the memory allocation functions set by a call to
|
`mp_set_memory_functions' (or `malloc' and friends by default) are
|
not reentrant, then GMP will not be reentrant either.
|
|
* 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 another
|
might be writing, nor for two threads to write simultaneously.
|
It's not safe for two threads to generate a random number from the
|
same `gmp_randstate_t' simultaneously, since this involves an
|
update of that variable.
|
|
|
File: gmp.info, Node: Useful Macros and Constants, Next: Compatibility with older versions, Prev: Reentrancy, Up: GMP Basics
|
|
3.8 Useful Macros and Constants
|
===============================
|
|
-- Global Constant: const int mp_bits_per_limb
|
The number of bits per limb.
|
|
-- Macro: __GNU_MP_VERSION
|
-- Macro: __GNU_MP_VERSION_MINOR
|
-- Macro: __GNU_MP_VERSION_PATCHLEVEL
|
The major and minor GMP version, and patch level, respectively, as
|
integers. For GMP i.j, these numbers will be i, j, and 0,
|
respectively. For GMP i.j.k, these numbers will be i, j, and k,
|
respectively.
|
|
-- 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 "6.1.0". Note that the format "i.j" was
|
used, before version 4.3.0, when k was zero.
|
|
-- Macro: __GMP_CC
|
-- Macro: __GMP_CFLAGS
|
The compiler and compiler flags, respectively, used when compiling
|
GMP, as strings.
|
|
|
File: gmp.info, Node: Compatibility with older versions, Next: Demonstration Programs, Prev: Useful Macros and Constants, Up: GMP Basics
|
|
3.9 Compatibility with older versions
|
=====================================
|
|
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.
|
|
* `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.
|
|
* `mpn_bdivmod', documented as preliminary in GMP 4, has been
|
removed.
|
|
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.
|
|
|
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.
|
For instance,
|
|
make pexpr
|
./pexpr 68^975+10
|
|
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
|
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'
|
function.
|
|
* `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'.
|
|
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.
|
|
|
File: gmp.info, Node: Efficiency, Next: Debugging, Prev: Demonstration Programs, Up: GMP Basics
|
|
3.11 Efficiency
|
===============
|
|
Small Operands
|
On small operands, the time for function call overheads and memory
|
allocation can be significant in comparison to actual calculation.
|
This is unavoidable in a general purpose variable precision
|
library, although 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
|
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.
|
|
Initializing and Clearing
|
Avoid excessive initializing and clearing of variables, since this
|
can be quite time consuming, especially in comparison to otherwise
|
fast operations like addition.
|
|
A language interpreter might want to keep a free list or stack of
|
initialized variables ready for use. It should be possible to
|
integrate something like that with a garbage collector too.
|
|
Reallocations
|
An `mpz_t' or `mpq_t' variable used to hold successively increasing
|
values will have its memory repeatedly `realloc'ed, which could be
|
quite slow or could fragment memory, depending on the C library.
|
If an application can estimate the final size then `mpz_init2' or
|
`mpz_realloc2' can be called to allocate the necessary space from
|
the beginning (*note Initializing Integers::).
|
|
It doesn't matter if a size set with `mpz_init2' or `mpz_realloc2'
|
is too small, since all functions will do a further reallocation
|
if necessary. Badly overestimating memory required will waste
|
space though.
|
|
`2exp' Functions
|
It's up to an application to call functions like `mpz_mul_2exp'
|
when appropriate. General purpose functions like `mpz_mul' make
|
no attempt to identify powers of two or other special forms,
|
because such inputs will usually be very rare and testing every
|
time would be wasteful.
|
|
`ui' and `si' Functions
|
The `ui' functions and the small number of `si' functions exist for
|
convenience and should be used where applicable. But if for
|
example an `mpz_t' contains a value that fits in an `unsigned
|
long' there's no need extract it and call a `ui' function, just
|
use the regular `mpz' function.
|
|
In-Place Operations
|
`mpz_abs', `mpq_abs', `mpf_abs', `mpz_neg', `mpq_neg' and
|
`mpf_neg' are fast when used for in-place operations like
|
`mpz_abs(x,x)', since in the current implementation only a single
|
field of `x' needs changing. On suitable compilers (GCC for
|
instance) this is inlined too.
|
|
`mpz_add_ui', `mpz_sub_ui', `mpf_add_ui' and `mpf_sub_ui' benefit
|
from an in-place operation like `mpz_add_ui(x,x,y)', since usually
|
only one or two limbs of `x' will need to be changed. The same
|
applies to the full precision `mpz_add' etc if `y' is small. If
|
`y' is big then cache locality may be helped, but that's all.
|
|
`mpz_mul' is currently the opposite, a separate destination is
|
slightly better. A call like `mpz_mul(x,x,y)' will, unless `y' is
|
only one limb, make a temporary copy of `x' before forming the
|
result. Normally that copying will only be a tiny fraction of the
|
time for the multiply, so this is not a particularly important
|
consideration.
|
|
`mpz_set', `mpq_set', `mpq_set_num', `mpf_set', etc, make no
|
attempt to recognise a copy of something to itself, so a call like
|
`mpz_set(x,x)' will be wasteful. Naturally that would never be
|
written deliberately, but if it might arise from two pointers to
|
the same object then a test to avoid it might be desirable.
|
|
if (x != y)
|
mpz_set (x, y);
|
|
Note that it's never worth introducing extra `mpz_set' calls just
|
to get in-place operations. If a result should go to a particular
|
variable then just direct it there and let 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
|
individual small integer. They use an algorithm which is faster
|
than `mpz_tdiv_ui', but which gives no useful information about
|
the actual remainder, only whether it's zero (or a particular
|
value).
|
|
However when testing divisibility by several small integers, it's
|
best to take a remainder modulo their product, to save
|
multi-precision operations. For instance to test whether a number
|
is divisible by any of 23, 29 or 31 take a remainder modulo
|
23*29*31 = 20677 and then test that.
|
|
The division functions like `mpz_tdiv_q_ui' which give a quotient
|
as well as a remainder are generally a little slower than the
|
remainder-only functions like `mpz_tdiv_ui'. If the quotient is
|
only rarely wanted then it's probably best to just take a
|
remainder and then go back and calculate the quotient if and when
|
it's wanted (`mpz_divexact_ui' can be used if the remainder is
|
zero).
|
|
Rational Arithmetic
|
The `mpq' functions operate on `mpq_t' values with no common
|
factors in the numerator and denominator. Common factors are
|
checked-for and cast out as necessary. In general, cancelling
|
factors every time is the best approach since it minimizes the
|
sizes for subsequent operations.
|
|
However, applications that know something about the factorization
|
of the values they're working with might be able to avoid some of
|
the GCDs used for canonicalization, or swap them for divisions.
|
For example when multiplying by a prime it's enough to check for
|
factors of it in the denominator instead of doing a full GCD. Or
|
when forming a big product it might be known that very little
|
cancellation will be possible, and so canonicalization can be left
|
to the end.
|
|
The `mpq_numref' and `mpq_denref' macros give access to the
|
numerator and denominator to do things outside the scope of the
|
supplied `mpq' functions. *Note Applying Integer Functions::.
|
|
The canonical form for rationals allows mixed-type `mpq_t' and
|
integer additions or subtractions to be done directly with
|
multiples of the denominator. This will be somewhat faster than
|
`mpq_add'. For example,
|
|
/* mpq increment */
|
mpz_add (mpq_numref(q), mpq_numref(q), mpq_denref(q));
|
|
/* mpq += unsigned long */
|
mpz_addmul_ui (mpq_numref(q), mpq_denref(q), 123UL);
|
|
/* mpq -= mpz */
|
mpz_submul (mpq_numref(q), mpq_denref(q), z);
|
|
Number Sequences
|
Functions like `mpz_fac_ui', `mpz_fib_ui' and `mpz_bin_uiui' are
|
designed for calculating isolated values. If a range of values is
|
wanted it's probably best to call to get a starting point and
|
iterate from there.
|
|
Text Input/Output
|
Hexadecimal or octal are suggested for input or output in text
|
form. Power-of-2 bases like these can be converted much more
|
efficiently than other bases, like decimal. For big numbers
|
there's usually nothing of particular interest to be seen in the
|
digits, so the base doesn't matter much.
|
|
Maybe we can hope octal will one day become the normal base for
|
everyday use, as proposed by King Charles XII of Sweden and later
|
reformers.
|
|
|
File: gmp.info, Node: Debugging, Next: Profiling, Prev: Efficiency, Up: GMP Basics
|
|
3.12 Debugging
|
==============
|
|
Stack Overflow
|
Depending on the system, a segmentation violation or bus error
|
might be the only indication of stack overflow. See
|
`--enable-alloca' choices in *Note Build Options::, for how to
|
address this.
|
|
In new enough versions of GCC, `-fstack-check' may be able to
|
ensure an overflow is recognised by the system before too much
|
damage is done, or `-fstack-limit-symbol' or
|
`-fstack-limit-register' may be able to add checking if the system
|
itself doesn't do any (*note Options for Code Generation:
|
(gcc)Code Gen Options.). These options must be added to the
|
`CFLAGS' used in the 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
|
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
|
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/'
|
|
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
|
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.
|
|
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.
|
|
cd /my/build/dir
|
/my/source/dir/gmp-6.1.0/configure
|
|
This works via `VPATH', and might require GNU `make'. Alternately
|
it might be possible to change the `.c.lo' rules appropriately.
|
|
Assertion Checking
|
The build option `--enable-assert' is available to add some
|
consistency checks to the library (see *Note Build Options::).
|
These are likely to be of limited value to most applications.
|
Assertion failures are just as likely to indicate memory
|
corruption as a library or compiler bug.
|
|
Applications using the low-level `mpn' functions, however, will
|
benefit from `--enable-assert' since it adds checks on the
|
parameters of most such functions, many of which have subtle
|
restrictions on their usage. Note however that only the generic C
|
code has checks, not the assembly code, so `--disable-assembly'
|
should be used for maximum checking.
|
|
Temporary Memory Checking
|
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').
|
|
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.
|
|
Maximum Debuggability
|
To summarize the above, a GMP build for maximum debuggability
|
would be
|
|
./configure --disable-shared --enable-assert \
|
--enable-alloca=debug --disable-assembly CFLAGS=-g
|
|
For C++, add `--enable-cxx CXXFLAGS=-g'.
|
|
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 --disable-assembly CC=checkergcc
|
|
`--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
|
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::.
|
|
|
File: gmp.info, Node: Profiling, Next: Autoconf, Prev: Debugging, Up: GMP Basics
|
|
3.13 Profiling
|
==============
|
|
Running a program under a profiler is a good way to find where it's
|
spending most time and where improvements can be best sought. The
|
profiling choices for a GMP build are as follows.
|
|
`--disable-profiling'
|
The default is to add nothing special for profiling.
|
|
It should be possible to just compile the mainline of a program
|
with `-p' and use `prof' to get a profile consisting of
|
timer-based sampling of the program counter. Most of the 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.
|
|
`--enable-profiling=prof'
|
Build with support for the system `prof', which means `-p' added
|
to the `CFLAGS'.
|
|
This provides call counting in addition to program counter
|
sampling, which allows the most frequently called routines to be
|
identified, and an average time spent in each routine to be
|
determined.
|
|
The x86 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.
|
|
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'.
|
|
This provides call graph construction in addition to call counting
|
and program counter sampling, which makes it possible to count
|
calls coming from different locations. For example the number of
|
calls to `mpn_mul' from `mpz_mul' versus the number from
|
`mpf_mul'. The program counter sampling is still flat though, so
|
only a total time in `mpn_mul' would be accumulated, not a
|
separate amount for each call site.
|
|
The x86 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.
|
|
On x86 and m68k systems `-pg' and `-fomit-frame-pointer' are
|
incompatible, so the latter is omitted from the default flags in
|
that case, which might result in poorer code generation.
|
|
Incidentally, it should be possible to use the `gprof' program
|
with a plain `--enable-profiling=prof' build. But in that case
|
only the `gprof -p' flat profile and call counts can be expected
|
to be valid, not the `gprof -q' call graph.
|
|
`--enable-profiling=instrument'
|
Build with the GCC option `-finstrument-functions' added to the
|
`CFLAGS' (*note Options for Code Generation: (gcc)Code Gen
|
Options.).
|
|
This inserts special instrumenting calls at the start and end of
|
each function, allowing exact timing and full call graph
|
construction.
|
|
This instrumenting is not normally a standard system feature and
|
will require support from an external library, such as
|
|
`http://sourceforge.net/projects/fnccheck/'
|
|
This should be included in `LIBS' during the GMP configure so that
|
test programs will link. For example,
|
|
./configure --enable-profiling=instrument LIBS=-lfc
|
|
On a GNU system the C library provides dummy instrumenting
|
functions, so programs compiled with this option will link. In
|
this case it's only necessary to ensure the correct library is
|
added when linking an application.
|
|
The x86 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.
|
|
|
File: gmp.info, Node: Autoconf, Next: Emacs, Prev: Profiling, Up: GMP Basics
|
|
3.14 Autoconf
|
=============
|
|
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,
|
|
AC_CHECK_LIB(gmp, __gmpz_init)
|
|
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 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'
|
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 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
|
version, if some particular sub-minor release is known to be necessary.
|
|
In general it's recommended that applications should simply demand a
|
new enough GMP rather than trying to provide supplements for features
|
not available in past versions.
|
|
Occasionally an application will need or want to know the size of a
|
type at configuration or preprocessing time, not just with `sizeof' in
|
the code. This can be done in the normal way with `mp_limb_t' etc, but
|
GMP 4.0 or up 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>])
|
|
|
File: gmp.info, Node: Emacs, Prev: Autoconf, Up: GMP Basics
|
|
3.15 Emacs
|
==========
|
|
<C-h C-i> (`info-lookup-symbol') is a good way to find documentation on
|
C functions while editing (*note Info Documentation Lookup: (emacs)Info
|
Lookup.).
|
|
The GMP manual can be included in such lookups by putting the
|
following in your `.emacs',
|
|
(eval-after-load "info-look"
|
'(let ((mode-value (assoc 'c-mode (assoc 'symbol info-lookup-alist))))
|
(setcar (nthcdr 3 mode-value)
|
(cons '("(gmp)Function Index" nil "^ -.* " "\\>")
|
(nth 3 mode-value)))))
|
|
|
File: gmp.info, Node: Reporting Bugs, Next: Integer Functions, Prev: GMP Basics, Up: Top
|
|
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.
|
|
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,
|
|
* The GMP version number, and if pre-packaged or patched then say so.
|
|
* A test program that makes it possible for us to reproduce the bug.
|
Include instructions on how to run the program.
|
|
* A description of what is wrong. If the results are incorrect, in
|
what way. If you get a crash, say so.
|
|
* If you get a crash, include a stack backtrace from the debugger if
|
it's informative (`where' in `gdb', or `$C' in `adb').
|
|
* Please do not send core dumps, executables or `strace's.
|
|
* The `configure' 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
|
similar.
|
|
* The output from running `uname -a'.
|
|
* The output from running `./config.guess', and from running
|
`./configfsf.guess' (might be the same).
|
|
* If the bug is related to `configure', then the 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'.
|
|
Please make an effort to produce a self-contained report, with
|
something definite that can be tested or debugged. Vague queries or
|
piecemeal messages are difficult to act on and don't help the
|
development effort.
|
|
It is not uncommon that an observed problem is actually due to a bug
|
in the compiler; the GMP code tends to explore interesting corners in
|
compilers.
|
|
If your bug report is good, we will do our best to help you get a
|
corrected version of the library; if the bug report is poor, we won't
|
do anything about it (except maybe ask you to send a better report).
|
|
Send your report to: <gmp-bugs@gmplib.org>.
|
|
If you think something in this manual is unclear, or downright
|
incorrect, or if the language needs to be improved, please send a note
|
to the same address.
|
|
|
File: gmp.info, Node: Integer Functions, Next: Rational Number Functions, Prev: Reporting Bugs, Up: Top
|
|
5 Integer Functions
|
*******************
|
|
This chapter describes the GMP functions for performing integer
|
arithmetic. These functions start with the prefix `mpz_'.
|
|
GMP integers are stored in objects of type `mpz_t'.
|
|
* Menu:
|
|
* Initializing Integers::
|
* Assigning Integers::
|
* Simultaneous Integer Init & Assign::
|
* Converting Integers::
|
* Integer Arithmetic::
|
* Integer Division::
|
* Integer Exponentiation::
|
* Integer Roots::
|
* Number Theoretic Functions::
|
* Integer Comparisons::
|
* Integer Logic and Bit Fiddling::
|
* I/O of Integers::
|
* Integer Random Numbers::
|
* Integer Import and Export::
|
* Miscellaneous Integer Functions::
|
* Integer Special Functions::
|
|
|
File: gmp.info, Node: Initializing Integers, Next: Assigning Integers, Prev: Integer Functions, Up: Integer Functions
|
|
5.1 Initialization Functions
|
============================
|
|
The functions for integer arithmetic assume that all integer objects are
|
initialized. You do that by calling the function `mpz_init'. For
|
example,
|
|
{
|
mpz_t integ;
|
mpz_init (integ);
|
...
|
mpz_add (integ, ...);
|
...
|
mpz_sub (integ, ...);
|
|
/* Unless the program is about to exit, do ... */
|
mpz_clear (integ);
|
}
|
|
As you can see, you can store new values any number of times, once an
|
object is initialized.
|
|
-- Function: void mpz_init (mpz_t X)
|
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
|
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
|
necessary; reallocation is handled automatically by GMP when
|
needed.
|
|
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
|
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'
|
variables when you are done with them.
|
|
-- Function: void mpz_clears (mpz_t X, ...)
|
Free the space occupied by a NULL-terminated list of `mpz_t'
|
variables.
|
|
-- Function: void mpz_realloc2 (mpz_t X, mp_bitcnt_t N)
|
Change the space allocated for X to N bits. The value in X is
|
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
|
automatic reallocations, or to decrease it to give memory back to
|
the heap.
|
|
|
File: gmp.info, Node: Assigning Integers, Next: Simultaneous Integer Init & Assign, Prev: Initializing Integers, Up: Integer Functions
|
|
5.2 Assignment Functions
|
========================
|
|
These functions assign new values to already initialized integers
|
(*note Initializing Integers::).
|
|
-- Function: void mpz_set (mpz_t ROP, 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, 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
|
integer.
|
|
-- 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.
|
|
For bases up to 36, case is ignored; upper-case and lower-case
|
letters have the same value. For bases 37 to 62, upper-case
|
letter represent the usual 10..35 while lower-case letter
|
represent 36..61.
|
|
This function returns 0 if the entire string is a valid number in
|
base BASE. Otherwise it returns -1.
|
|
-- Function: void mpz_swap (mpz_t ROP1, mpz_t ROP2)
|
Swap the values ROP1 and ROP2 efficiently.
|
|
|
File: gmp.info, Node: Simultaneous Integer Init & Assign, Next: Converting Integers, Prev: Assigning Integers, Up: Integer Functions
|
|
5.3 Combined Initialization and Assignment 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 `mpz_init_set...'
|
|
Here is an example of using one:
|
|
{
|
mpz_t pie;
|
mpz_init_set_str (pie, "3141592653589793238462643383279502884", 10);
|
...
|
mpz_sub (pie, ...);
|
...
|
mpz_clear (pie);
|
}
|
|
Once the integer has been initialized by any of the `mpz_init_set...'
|
functions, it can be used as the source or destination operand for the
|
ordinary integer functions. Don't use an initialize-and-set function
|
on a variable already initialized!
|
|
-- Function: void mpz_init_set (mpz_t ROP, 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, 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.)
|
|
|
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::.
|
|
-- 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
|
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 (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'.
|
|
-- 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, 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'.
|
|
This is similar to the standard C `frexp' function (*note
|
Normalization Functions: (libc)Normalization Functions.).
|
|
-- 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.
|
|
If STR is `NULL', the result string is allocated using the current
|
allocation function (*note Custom Allocation::). The block will be
|
`strlen(str)+1' bytes, that being exactly enough for the string and
|
null-terminator.
|
|
If STR is not `NULL', it should point to a block of storage large
|
enough for the result, that being `mpz_sizeinbase (OP, BASE) + 2'.
|
The two extra bytes are for a possible minus sign, and the
|
null-terminator.
|
|
A pointer to the result string is returned, being either the
|
allocated block, or the given STR.
|
|
|
File: gmp.info, Node: Integer Arithmetic, Next: Integer Division, Prev: Converting Integers, Up: Integer Functions
|
|
5.5 Arithmetic Functions
|
========================
|
|
-- 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, 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, 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, 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, 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, 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, const mpz_t OP)
|
Set ROP to -OP.
|
|
-- Function: void mpz_abs (mpz_t ROP, const mpz_t OP)
|
Set ROP to the absolute value of OP.
|
|
|
File: gmp.info, Node: Integer Division, Next: Integer Exponentiation, Prev: Integer Arithmetic, Up: Integer Functions
|
|
5.6 Division Functions
|
======================
|
|
Division is undefined if the divisor is zero. Passing a zero divisor
|
to the division or modulo functions (including the modular powering
|
functions `mpz_powm' and `mpz_powm_ui'), will cause an intentional
|
division by zero. This lets a program handle arithmetic exceptions in
|
these functions the same way as for normal C `int' arithmetic.
|
|
-- Function: void mpz_cdiv_q (mpz_t Q, 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, const mpz_t N,
|
unsigned long int D)
|
-- Function: unsigned long int mpz_cdiv_qr_ui (mpz_t Q, mpz_t R,
|
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, 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, const mpz_t N,
|
unsigned long int D)
|
-- Function: unsigned long int mpz_fdiv_qr_ui (mpz_t Q, mpz_t R,
|
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, 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, const mpz_t N,
|
unsigned long int D)
|
-- Function: unsigned long int mpz_tdiv_qr_ui (mpz_t Q, mpz_t R,
|
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, 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
|
suiting different applications.
|
|
* `cdiv' rounds Q up towards +infinity, and R will have the
|
opposite sign to D. The `c' stands for "ceil".
|
|
* `fdiv' rounds Q down towards -infinity, and R will have the
|
same sign as D. The `f' stands for "floor".
|
|
* `tdiv' rounds Q towards zero, and R will have the same sign
|
as N. The `t' stands for "truncate".
|
|
In all cases Q and R will satisfy N=Q*D+R, and R will satisfy
|
0<=abs(R)<abs(D).
|
|
The `q' functions calculate only the quotient, the `r' functions
|
only the remainder, and the `qr' functions calculate both. Note
|
that for `qr' the same variable cannot be passed for both Q and R,
|
or results will be unpredictable.
|
|
For the `ui' variants the return value is the remainder, and in
|
fact returning the remainder is all the `div_ui' functions do. For
|
`tdiv' and `cdiv' the remainder can be negative, so for those the
|
return value is the absolute value of the remainder.
|
|
For the `2exp' variants the divisor is 2^B. These functions are
|
implemented as right shifts and bit masks, but of course they
|
round the same as the other functions.
|
|
For positive N both `mpz_fdiv_q_2exp' and `mpz_tdiv_q_2exp' are
|
simple bitwise right shifts. For negative N, `mpz_fdiv_q_2exp' is
|
effectively an arithmetic right shift treating N as twos complement
|
the same as the bitwise logical functions do, whereas
|
`mpz_tdiv_q_2exp' effectively treats N as sign and magnitude.
|
|
-- Function: void mpz_mod (mpz_t R, 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.
|
|
`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, 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 (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.
|
|
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 (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.
|
|
N is congruent to C mod D if there exists an integer Q satisfying
|
N = C + Q*D. Unlike the other division functions, D=0 is accepted
|
and following the rule it can be seen that N and C are considered
|
congruent mod 0 only when exactly equal.
|
|
|
File: gmp.info, Node: Integer Exponentiation, Next: Integer Roots, Prev: Integer Division, Up: Integer Functions
|
|
5.7 Exponentiation Functions
|
============================
|
|
-- 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
|
inverse doesn't exist then a divide by zero is raised.
|
|
-- 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.
|
|
This function is designed to take the same time 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. This function
|
is intended for cryptographic purposes, where resilience to
|
side-channel attacks is desired.
|
|
-- 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.
|
|
|
File: gmp.info, Node: Integer Roots, Next: Number Theoretic Functions, Prev: Integer Exponentiation, Up: Integer Functions
|
|
5.8 Root Extraction Functions
|
=============================
|
|
-- Function: int mpz_root (mpz_t ROP, 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, const mpz_t U,
|
unsigned long int N)
|
Set ROOT to the truncated integer part of the Nth root of U. Set
|
REM to the remainder, U-ROOT**N.
|
|
-- Function: void mpz_sqrt (mpz_t ROP, 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, 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
|
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 (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.
|
|
-- 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.
|
|
|
File: gmp.info, Node: Number Theoretic Functions, Next: Integer Comparisons, Prev: Integer Roots, Up: Integer Functions
|
|
5.9 Number Theoretic Functions
|
==============================
|
|
-- 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 non-prime.
|
|
This function performs some trial divisions, then REPS 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 a probability
|
of less than 4^(-REPS). Reasonable values of REPS are between 15
|
and 50.
|
|
-- 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, 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, 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.
|
|
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, 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 (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 abs(A) = abs(B), then S = 0, T = sgn(B).
|
|
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 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, 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 < 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 (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 (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 (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.
|
|
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'.
|
|
-- 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 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, 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
|
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].
|
|
These functions are designed for calculating isolated Fibonacci
|
numbers. When a sequence of values is wanted it's best to start
|
with `mpz_fib2_ui' and iterate the defining F[n+1]=F[n]+F[n-1] or
|
similar.
|
|
-- Function: void mpz_lucnum_ui (mpz_t LN, 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].
|
|
These functions are designed for calculating isolated Lucas
|
numbers. When a sequence of values is wanted it's best to start
|
with `mpz_lucnum2_ui' and iterate the defining L[n+1]=L[n]+L[n-1]
|
or similar.
|
|
The Fibonacci numbers and Lucas numbers are related sequences, so
|
it's never necessary to call both `mpz_fib2_ui' and
|
`mpz_lucnum2_ui'. The formulas for going from Fibonacci to Lucas
|
can be found in *Note Lucas Numbers Algorithm::, the reverse is
|
straightforward too.
|
|
|
File: gmp.info, Node: Integer Comparisons, Next: Integer Logic and Bit Fiddling, Prev: Number Theoretic Functions, Up: Integer Functions
|
|
5.10 Comparison Functions
|
=========================
|
|
-- Function: int mpz_cmp (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
|
infinity, but results are undefined for a NaN.
|
|
-- 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
|
undefined for a NaN.
|
|
-- 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.
|
|
|
File: gmp.info, Node: Integer Logic and Bit Fiddling, Next: I/O of Integers, Prev: Integer Comparisons, Up: Integer Functions
|
|
5.11 Logical and Bit Manipulation Functions
|
===========================================
|
|
These functions behave as if twos complement arithmetic were used
|
(although sign-magnitude is the actual implementation). The least
|
significant bit is number 0.
|
|
-- Function: void mpz_and (mpz_t ROP, const mpz_t OP1, const mpz_t OP2)
|
Set ROP to OP1 bitwise-and 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, const mpz_t OP1, const mpz_t OP2)
|
Set ROP to OP1 bitwise exclusive-or OP2.
|
|
-- 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 (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
|
number.
|
|
-- Function: void mpz_setbit (mpz_t ROP, mp_bitcnt_t BIT_INDEX)
|
Set bit BIT_INDEX in ROP.
|
|
-- Function: void mpz_clrbit (mpz_t ROP, mp_bitcnt_t BIT_INDEX)
|
Clear bit BIT_INDEX in ROP.
|
|
-- Function: void mpz_combit (mpz_t ROP, mp_bitcnt_t BIT_INDEX)
|
Complement bit BIT_INDEX in ROP.
|
|
-- Function: int mpz_tstbit (const mpz_t OP, mp_bitcnt_t BIT_INDEX)
|
Test bit BIT_INDEX in OP and return 0 or 1 accordingly.
|
|
|
File: gmp.info, Node: I/O of Integers, Next: Integer Random Numbers, Prev: Integer Logic and Bit Fiddling, Up: Integer Functions
|
|
5.12 Input and Output Functions
|
===============================
|
|
Functions that perform input from a stdio stream, and functions that
|
output to a stdio stream, 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
|
prototypes for these functions.
|
|
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.
|
|
Return the number of bytes written, or if an error occurred,
|
return 0.
|
|
-- Function: size_t mpz_inp_str (mpz_t ROP, FILE *STREAM, int BASE)
|
Input a possibly white-space preceded string in base BASE from
|
stdio stream STREAM, and put the read integer in ROP.
|
|
The BASE may vary from 2 to 62, or if BASE is 0, then the leading
|
characters are used: `0x' and `0X' for hexadecimal, `0b' and `0B'
|
for binary, `0' for octal, or decimal otherwise.
|
|
For bases up to 36, case is ignored; upper-case and lower-case
|
letters have the same value. For bases 37 to 62, upper-case
|
letter represent the usual 10..35 while lower-case letter
|
represent 36..61.
|
|
Return the number of bytes read, or if an error occurred, return 0.
|
|
-- Function: size_t mpz_out_raw (FILE *STREAM, 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'.
|
|
Return the number of bytes written, or if an error occurred,
|
return 0.
|
|
The output of this can not be read by `mpz_inp_raw' from GMP 1,
|
because of changes necessary for compatibility between 32-bit and
|
64-bit machines.
|
|
-- Function: size_t mpz_inp_raw (mpz_t ROP, FILE *STREAM)
|
Input from stdio stream STREAM in the format written by
|
`mpz_out_raw', and put the result in ROP. Return the number of
|
bytes read, or if an error occurred, return 0.
|
|
This routine can read the output from `mpz_out_raw' also from GMP
|
1, in spite of changes necessary for compatibility between 32-bit
|
and 64-bit machines.
|
|
|
File: gmp.info, Node: Integer Random Numbers, Next: Integer Import and Export, Prev: I/O of Integers, Up: Integer Functions
|
|
5.13 Random Number Functions
|
============================
|
|
The random number functions of 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.
|
|
-- Function: void mpz_urandomb (mpz_t ROP, gmp_randstate_t STATE,
|
mp_bitcnt_t N)
|
Generate a uniformly distributed random integer in the range 0 to
|
2^N-1, inclusive.
|
|
The variable STATE must be initialized by calling one of the
|
`gmp_randinit' functions (*Note Random State Initialization::)
|
before invoking this function.
|
|
-- Function: void mpz_urandomm (mpz_t ROP, gmp_randstate_t STATE,
|
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::)
|
before invoking this function.
|
|
-- Function: void mpz_rrandomb (mpz_t ROP, gmp_randstate_t STATE,
|
mp_bitcnt_t N)
|
Generate a random integer with long strings of zeros and ones in
|
the binary representation. Useful for testing functions and
|
algorithms, since this kind of random numbers have proven to be
|
more likely to trigger corner-case bugs. The random number will
|
be in the range 0 to 2^N-1, inclusive.
|
|
The variable STATE must be initialized by calling one of the
|
`gmp_randinit' functions (*Note Random State Initialization::)
|
before invoking this function.
|
|
-- 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.
|
|
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.
|
Negative random numbers are generated when MAX_SIZE is negative.
|
|
This function is obsolete. Use `mpz_rrandomb' instead.
|
|
|
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
|
data with the following functions.
|
|
-- Function: void mpz_import (mpz_t ROP, size_t COUNT, int ORDER,
|
size_t SIZE, int ENDIAN, size_t NAILS, const void *OP)
|
Set ROP from an array of word data at OP.
|
|
The parameters specify the format of the data. COUNT many words
|
are read, each SIZE bytes. ORDER can be 1 for most significant
|
word first or -1 for least significant first. Within each word
|
ENDIAN can be 1 for most significant byte first, -1 for least
|
significant first, or 0 for the native endianness of the host CPU.
|
The most significant NAILS bits of each word are skipped, this
|
can be 0 to use the full words.
|
|
There is no sign taken from the data, ROP will simply be a positive
|
integer. An application can handle any sign itself, and apply it
|
for instance with `mpz_neg'.
|
|
There are no data alignment restrictions on OP, any address is
|
allowed.
|
|
Here's an example converting an array of `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
|
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'.
|
|
-- Function: void * mpz_export (void *ROP, size_t *COUNTP, int ORDER,
|
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
|
will be SIZE bytes and ORDER can be 1 for most significant word
|
first or -1 for least significant first. Within each word ENDIAN
|
can be 1 for most significant byte first, -1 for least significant
|
first, or 0 for the native endianness of the host CPU. The most
|
significant NAILS bits of each word are unused and set to zero,
|
this can be 0 to produce full words.
|
|
The number of words produced is written to `*COUNTP', or COUNTP
|
can be `NULL' to discard the count. ROP must have enough space
|
for the data, or if ROP is `NULL' then a result array of the
|
necessary size is allocated using the current 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.
|
|
The sign of OP is ignored, just the absolute value is exported. An
|
application can use `mpz_sgn' to get the sign and handle it as
|
desired. (*note Integer Comparisons::)
|
|
There are no data alignment restrictions on ROP, any address is
|
allowed.
|
|
When an application is allocating space itself the required size
|
can be determined with a calculation like the following. Since
|
`mpz_sizeinbase' always returns at least 1, `count' here will be
|
at least one, which avoids any portability problems with
|
`malloc(0)', though if `z' is zero no space at all is actually
|
needed (or written).
|
|
numb = 8*size - nail;
|
count = (mpz_sizeinbase (z, 2) + numb-1) / numb;
|
p = malloc (count * size);
|
|
|
File: gmp.info, Node: Miscellaneous Integer Functions, Next: Integer Special Functions, Prev: Integer Import and Export, Up: Integer Functions
|
|
5.15 Miscellaneous Functions
|
============================
|
|
-- Function: int mpz_fits_ulong_p (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.
|
|
This function can be used to determine the space required when
|
converting OP to a string. The right amount of allocation is
|
normally two more than the value returned by `mpz_sizeinbase', one
|
extra for a minus sign and one for the null-terminator.
|
|
It will be noted that `mpz_sizeinbase(OP,2)' can be used to locate
|
the most significant 1 bit in OP, counting from 1. (Unlike the
|
bitwise functions which start from 0, *Note Logical and Bit
|
Manipulation Functions: Integer Logic and Bit Fiddling.)
|
|
|
File: gmp.info, Node: Integer Special Functions, Prev: Miscellaneous Integer Functions, Up: Integer Functions
|
|
5.16 Special Functions
|
======================
|
|
The functions in this section are for various special purposes. Most
|
applications will not need them.
|
|
-- Function: void mpz_array_init (mpz_t INTEGER_ARRAY, mp_size_t
|
ARRAY_SIZE, mp_size_t FIXED_NUM_BITS)
|
*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.
|
|
-- 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'.
|
|
-- 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);
|
}
|
|
|
File: gmp.info, Node: Rational Number Functions, Next: Floating-point Functions, Prev: Integer Functions, Up: Top
|
|
6 Rational Number Functions
|
***************************
|
|
This chapter describes the GMP functions for performing arithmetic on
|
rational numbers. These functions start with the prefix `mpq_'.
|
|
Rational numbers are stored in objects of type `mpq_t'.
|
|
All rational arithmetic functions assume operands have a canonical
|
form, and canonicalize their result. The canonical from means that the
|
denominator and the numerator have no common factors, and that the
|
denominator is positive. Zero has the unique representation 0/1.
|
|
Pure assignment functions do not canonicalize the assigned variable.
|
It is the responsibility of the user to canonicalize the assigned
|
variable before any arithmetic operations are performed on that
|
variable.
|
|
-- Function: void mpq_canonicalize (mpq_t OP)
|
Remove any factors that are common to the numerator and
|
denominator of OP, and make the denominator positive.
|
|
* Menu:
|
|
* Initializing Rationals::
|
* Rational Conversions::
|
* Rational Arithmetic::
|
* Comparing Rationals::
|
* Applying Integer Functions::
|
* I/O of Rationals::
|
|
|
File: gmp.info, Node: Initializing Rationals, Next: Rational Conversions, Prev: Rational Number Functions, Up: Rational Number Functions
|
|
6.1 Initialization and Assignment Functions
|
===========================================
|
|
-- Function: void mpq_init (mpq_t 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.
|
|
-- Function: void mpq_inits (mpq_t X, ...)
|
Initialize a NULL-terminated list of `mpq_t' variables, and set
|
their values to 0/1.
|
|
-- Function: void mpq_clear (mpq_t X)
|
Free the space occupied by X. Make sure to call this function for
|
all `mpq_t' variables when you are done with them.
|
|
-- Function: void mpq_clears (mpq_t X, ...)
|
Free the space occupied by a NULL-terminated list of `mpq_t'
|
variables.
|
|
-- Function: void mpq_set (mpq_t ROP, 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,
|
unsigned long int OP2)
|
-- 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
|
any operations are performed on ROP.
|
|
-- 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 numerator and optional denominator are parsed the same as in
|
`mpz_set_str' (*note Assigning Integers::). White space is
|
allowed in the string, and is simply ignored. The BASE can vary
|
from 2 to 62, or if BASE is 0 then the leading characters are
|
used: `0x' or `0X' for hex, `0b' or `0B' for binary, `0' for
|
octal, or decimal otherwise. Note that this is done separately
|
for the numerator and denominator, so for instance `0xEF/100' is
|
239/100, whereas `0xEF/0x100' is 239/256.
|
|
The return value is 0 if the entire string is a valid number, or
|
-1 if not.
|
|
-- Function: void mpq_swap (mpq_t ROP1, mpq_t ROP2)
|
Swap the values ROP1 and ROP2 efficiently.
|
|
|
File: gmp.info, Node: Rational Conversions, Next: Rational Arithmetic, Prev: Initializing Rationals, Up: Rational Number Functions
|
|
6.2 Conversion Functions
|
========================
|
|
-- 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.
|
|
-- Function: void mpq_set_d (mpq_t ROP, double 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, const mpq_t OP)
|
Convert OP to a string of digits in base BASE. The base may vary
|
from 2 to 36. The string will be of the form `num/den', or if the
|
denominator is 1 then just `num'.
|
|
If STR is `NULL', the result string is allocated using the current
|
allocation function (*note Custom Allocation::). The block will be
|
`strlen(str)+1' bytes, that being exactly enough for the string and
|
null-terminator.
|
|
If STR is not `NULL', it should point to a block of storage large
|
enough for the result, that being
|
|
mpz_sizeinbase (mpq_numref(OP), BASE)
|
+ mpz_sizeinbase (mpq_denref(OP), BASE) + 3
|
|
The three extra bytes are for a possible minus sign, possible
|
slash, and the null-terminator.
|
|
A pointer to the result string is returned, being either the
|
allocated block, or the given STR.
|
|
|
File: gmp.info, Node: Rational Arithmetic, Next: Comparing Rationals, Prev: Rational Conversions, Up: Rational Number Functions
|
|
6.3 Arithmetic Functions
|
========================
|
|
-- Function: void mpq_add (mpq_t SUM, const mpq_t ADDEND1, const mpq_t
|
ADDEND2)
|
Set SUM to ADDEND1 + ADDEND2.
|
|
-- 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, const mpq_t MULTIPLIER,
|
const mpq_t MULTIPLICAND)
|
Set PRODUCT to MULTIPLIER times MULTIPLICAND.
|
|
-- 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, const mpq_t DIVIDEND, const
|
mpq_t DIVISOR)
|
Set QUOTIENT to DIVIDEND/DIVISOR.
|
|
-- 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, const mpq_t OPERAND)
|
Set NEGATED_OPERAND to -OPERAND.
|
|
-- 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, const mpq_t NUMBER)
|
Set INVERTED_NUMBER to 1/NUMBER. If the new denominator is zero,
|
this routine will divide by zero.
|
|
|
File: gmp.info, Node: Comparing Rationals, Next: Applying Integer Functions, Prev: Rational Arithmetic, Up: Rational Number Functions
|
|
6.4 Comparison Functions
|
========================
|
|
-- 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'.
|
|
-- 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.
|
|
NUM2 and DEN2 are allowed to have common factors.
|
|
These functions are implemented as a macros and evaluate their
|
arguments multiple times.
|
|
-- Macro: int mpq_sgn (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
|
argument multiple times.
|
|
-- 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,
|
this function is much faster.
|
|
|
File: gmp.info, Node: Applying Integer Functions, Next: I/O of Rationals, Prev: Comparing Rationals, Up: Rational Number Functions
|
|
6.5 Applying Integer Functions to Rationals
|
===========================================
|
|
The set of `mpq' functions is quite small. In particular, there are few
|
functions for either input or output. The following functions give
|
direct access to the numerator and denominator of an `mpq_t'.
|
|
Note that if an assignment to the numerator and/or denominator could
|
take an `mpq_t' out of the canonical form described at the start of
|
this chapter (*note Rational Number Functions::) then
|
`mpq_canonicalize' must be called before any other `mpq' functions are
|
applied to that `mpq_t'.
|
|
-- Macro: mpz_t mpq_numref (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
|
these macros.
|
|
-- 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.
|
|
|
File: gmp.info, Node: I/O of Rationals, Prev: Applying Integer Functions, Up: Rational Number Functions
|
|
6.6 Input and Output 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.
|
|
See also *Note Formatted Output:: and *Note Formatted Input::.
|
|
-- 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'.
|
|
Return the number of bytes written, or if an error occurred,
|
return 0.
|
|
-- Function: size_t mpq_inp_str (mpq_t ROP, FILE *STREAM, int BASE)
|
Read a string of digits from STREAM and convert them to a rational
|
in ROP. Any initial white-space characters are read and
|
discarded. Return the number of characters read (including white
|
space), or 0 if a rational could not be read.
|
|
The input can be a fraction like `17/63' or just an integer like
|
`123'. Reading stops at the first character not in this form, and
|
white space is not permitted within the string. If the input
|
might not be in canonical form, then `mpq_canonicalize' must be
|
called (*note Rational Number Functions::).
|
|
The BASE can be between 2 and 36, or can be 0 in which case the
|
leading characters of the string determine the base, `0x' or `0X'
|
for hexadecimal, `0' for octal, or decimal otherwise. The leading
|
characters are examined separately for the numerator and
|
denominator of a fraction, so for instance `0x10/11' is 16/11,
|
whereas `0x10/0x11' is 16/17.
|
|
|
File: 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.
|
|
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 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
|
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
|
for calculations involving money or other values that should be exact
|
decimal fractions. (Suitably scaled integers, or perhaps rationals,
|
are better choices.)
|
|
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
|
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::
|
* Assigning Floats::
|
* Simultaneous Float Init & Assign::
|
* Converting Floats::
|
* Float Arithmetic::
|
* Float Comparison::
|
* I/O of Floats::
|
* Miscellaneous Float Functions::
|
|
|
File: gmp.info, Node: Initializing Floats, Next: Assigning Floats, Prev: Floating-point Functions, Up: Floating-point Functions
|
|
7.1 Initialization Functions
|
============================
|
|
-- Function: void mpf_set_default_prec (mp_bitcnt_t PREC)
|
Set the default precision to be *at least* PREC bits. All
|
subsequent calls to `mpf_init' will use this precision, but
|
previously initialized variables are unaffected.
|
|
-- Function: mp_bitcnt_t mpf_get_default_prec (void)
|
Return the default precision actually used.
|
|
An `mpf_t' object must be initialized before storing the first value
|
in it. The functions `mpf_init' and `mpf_init2' are used for that
|
purpose.
|
|
-- Function: void mpf_init (mpf_t X)
|
Initialize X to 0. Normally, a variable should be initialized
|
once only or at least be cleared, using `mpf_clear', between
|
initializations. The precision of X is undefined unless a default
|
precision has already been established by a call to
|
`mpf_set_default_prec'.
|
|
-- Function: void mpf_init2 (mpf_t X, mp_bitcnt_t PREC)
|
Initialize X to 0 and set its precision to be *at least* PREC
|
bits. Normally, a variable should be initialized once only or at
|
least be cleared, using `mpf_clear', between initializations.
|
|
-- Function: void mpf_inits (mpf_t X, ...)
|
Initialize a NULL-terminated list of `mpf_t' variables, and set
|
their values to 0. The precision of the initialized variables is
|
undefined unless a default precision has already been established
|
by a call to `mpf_set_default_prec'.
|
|
-- Function: void mpf_clear (mpf_t X)
|
Free the space occupied by X. Make sure to call this function for
|
all `mpf_t' variables when you are done with them.
|
|
-- Function: void mpf_clears (mpf_t X, ...)
|
Free the space occupied by a NULL-terminated list of `mpf_t'
|
variables.
|
|
Here is an example on how to initialize floating-point variables:
|
{
|
mpf_t x, y;
|
mpf_init (x); /* use default precision */
|
mpf_init2 (y, 256); /* precision _at least_ 256 bits */
|
...
|
/* Unless the program is about to exit, do ... */
|
mpf_clear (x);
|
mpf_clear (y);
|
}
|
|
The following three functions are useful for changing the precision
|
during a calculation. A typical use would be for adjusting the
|
precision gradually in iterative algorithms like Newton-Raphson, making
|
the computation precision closely match the actual accurate part of the
|
numbers.
|
|
-- Function: mp_bitcnt_t mpf_get_prec (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
|
used in a tight loop.
|
|
-- Function: void mpf_set_prec_raw (mpf_t ROP, mp_bitcnt_t PREC)
|
Set the precision of ROP to be *at least* PREC bits, without
|
changing the memory allocated.
|
|
PREC must be no more than the allocated precision for ROP, that
|
being the precision when ROP was initialized, or in the most recent
|
`mpf_set_prec'.
|
|
The value in ROP is unchanged, and in particular if it had a higher
|
precision than PREC it will retain that higher precision. New
|
values written to ROP will use the new PREC.
|
|
Before calling `mpf_clear' or the full `mpf_set_prec', another
|
`mpf_set_prec_raw' call must be made to restore ROP to its original
|
allocated precision. Failing to do so will have unpredictable
|
results.
|
|
`mpf_get_prec' can be used before `mpf_set_prec_raw' to get the
|
original allocated precision. After `mpf_set_prec_raw' it
|
reflects the PREC value set.
|
|
`mpf_set_prec_raw' is an efficient way to use an `mpf_t' variable
|
at different precisions during a calculation, perhaps to gradually
|
increase precision in an iteration, or just to use various
|
different precisions for different purposes during a calculation.
|
|
|
File: gmp.info, Node: Assigning Floats, Next: Simultaneous Float Init & Assign, Prev: Initializing Floats, Up: Floating-point Functions
|
|
7.2 Assignment Functions
|
========================
|
|
These functions assign new values to already initialized floats (*note
|
Initializing Floats::).
|
|
-- Function: void mpf_set (mpf_t ROP, 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, 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, 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'.
|
|
The argument BASE may be in the ranges 2 to 62, or -62 to -2.
|
Negative values are used to specify that the exponent is in
|
decimal.
|
|
For bases up to 36, case is ignored; upper-case and lower-case
|
letters have the same value; for bases 37 to 62, upper-case letter
|
represent the usual 10..35 while lower-case letter represent
|
36..61.
|
|
Unlike the corresponding `mpz' function, the base will not be
|
determined from the leading characters of the string if BASE is 0.
|
This is so that numbers like `0.23' are not interpreted as octal.
|
|
White space is allowed in the string, and is simply ignored.
|
[This is not really true; white-space is ignored in the beginning
|
of the string and within the mantissa, but not in other places,
|
such as after a minus sign or in the exponent. We are considering
|
changing the definition of this function, making it fail when
|
there is any white-space in the input, since that makes a lot of
|
sense. Please tell us your opinion about this change. Do you
|
really want it to accept "3 14" as meaning 314 as it does now?]
|
|
This function returns 0 if the entire string is a valid number in
|
base BASE. Otherwise it returns -1.
|
|
-- Function: void mpf_swap (mpf_t ROP1, mpf_t ROP2)
|
Swap ROP1 and ROP2 efficiently. Both the values and the
|
precisions of the two variables are swapped.
|
|
|
File: gmp.info, Node: Simultaneous Float Init & Assign, Next: Converting Floats, Prev: Assigning Floats, Up: Floating-point Functions
|
|
7.3 Combined Initialization and Assignment Functions
|
====================================================
|
|
For convenience, 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...'
|
|
Once the float has been initialized by any of the `mpf_init_set...'
|
functions, it can be used as the source or destination operand for the
|
ordinary float functions. Don't use an initialize-and-set function on
|
a variable already initialized!
|
|
-- Function: void mpf_init_set (mpf_t ROP, 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'.
|
|
-- 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.
|
|
Note that ROP is initialized even if an error occurs. (I.e., you
|
have to call `mpf_clear' for it.)
|
|
The precision of ROP will be taken from the active default
|
precision, as set by `mpf_set_default_prec'.
|
|
|
File: 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 (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'
|
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, 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'.
|
|
This is similar to the standard C `frexp' function (*note
|
Normalization Functions: (libc)Normalization Functions.).
|
|
-- 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
|
Miscellaneous Float Functions::).
|
|
-- 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.
|
|
For BASE in the range 2..36, digits and lower-case letters are
|
used; for -2..-36, digits and upper-case letters are used; for
|
37..62, digits, upper-case letters, and lower-case letters (in
|
that significance order) are used.
|
|
If STR is `NULL', the result string is allocated using the current
|
allocation function (*note Custom Allocation::). The block will be
|
`strlen(str)+1' bytes, that being exactly enough for the string and
|
null-terminator.
|
|
If STR is not `NULL', it should point to a block of N_DIGITS + 2
|
bytes, that being enough for the mantissa, a possible minus sign,
|
and a null-terminator. When N_DIGITS is 0 to get all significant
|
digits, an application won't be able to know the space required,
|
and STR should be `NULL' in that case.
|
|
The generated string is a fraction, with an implicit radix point
|
immediately to the left of the first digit. The applicable
|
exponent is written through the EXPPTR pointer. For example, the
|
number 3.1416 would be returned as string "31416" and exponent 1.
|
|
When OP is zero, an empty string is produced and the exponent
|
returned is 0.
|
|
A pointer to the result string is returned, being either the
|
allocated block or the given STR.
|
|
|
File: gmp.info, Node: Float Arithmetic, Next: Float Comparison, Prev: Converting Floats, Up: Floating-point Functions
|
|
7.5 Arithmetic Functions
|
========================
|
|
-- Function: void mpf_add (mpf_t ROP, 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, 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, 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
|
divisor to the divide functions will make these functions intentionally
|
divide by zero. This lets the user handle arithmetic exceptions in
|
these functions in the same manner as other arithmetic exceptions.
|
|
-- Function: void mpf_div (mpf_t ROP, 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, 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, const mpf_t OP1, unsigned
|
long int OP2)
|
Set ROP to OP1 raised to the power OP2.
|
|
-- Function: void mpf_neg (mpf_t ROP, const mpf_t OP)
|
Set ROP to -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, 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, const mpf_t OP1,
|
mp_bitcnt_t OP2)
|
Set ROP to OP1 divided by 2 raised to OP2.
|
|
|
File: gmp.info, Node: Float Comparison, Next: I/O of Floats, Prev: Float Arithmetic, Up: Floating-point Functions
|
|
7.6 Comparison Functions
|
========================
|
|
-- 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
|
undefined for a NaN.
|
|
-- 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.*
|
|
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, 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 (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 argument multiple times.
|
|
|
File: gmp.info, Node: I/O of Floats, Next: Miscellaneous Float Functions, Prev: Float Comparison, Up: Floating-point Functions
|
|
7.7 Input and Output Functions
|
==============================
|
|
Functions that perform input from a stdio stream, and functions that
|
output to a stdio stream, 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
|
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, 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'.
|
|
For BASE in the range 2..36, digits and lower-case letters are
|
used; for -2..-36, digits and upper-case letters are used; for
|
37..62, digits, upper-case letters, and lower-case letters (in
|
that significance order) are used.
|
|
Up to N_DIGITS will be printed from the mantissa, except that no
|
more digits than are accurately representable by OP will be
|
printed. N_DIGITS can be 0 to select that accurate maximum.
|
|
-- Function: size_t mpf_inp_str (mpf_t ROP, FILE *STREAM, int BASE)
|
Read a string in base BASE from STREAM, and put the read float in
|
ROP. The string is of the form `M@N' or, if the base is 10 or
|
less, alternatively `MeN'. `M' is the mantissa and `N' is the
|
exponent. The mantissa is always in the specified base. The
|
exponent is either in the specified base or, if BASE is negative,
|
in decimal. The decimal point expected is taken from the current
|
locale, on systems providing `localeconv'.
|
|
The argument BASE may be in the ranges 2 to 36, or -36 to -2.
|
Negative values are used to specify that the exponent is in
|
decimal.
|
|
Unlike the corresponding `mpz' function, the base will not be
|
determined from the leading characters of the string if BASE is 0.
|
This is so that numbers like `0.23' are not interpreted as octal.
|
|
Return the number of bytes read, or if an error occurred, return 0.
|
|
|
File: gmp.info, Node: Miscellaneous Float Functions, Prev: I/O of Floats, Up: Floating-point Functions
|
|
7.8 Miscellaneous Functions
|
===========================
|
|
-- Function: void mpf_ceil (mpf_t ROP, 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 (const mpf_t OP)
|
Return non-zero if OP is an integer.
|
|
-- 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 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::)
|
before invoking this function.
|
|
-- Function: void mpf_random2 (mpf_t ROP, mp_size_t MAX_SIZE, mp_exp_t
|
EXP)
|
Generate a random float of at most MAX_SIZE limbs, with long
|
strings of zeros and ones in the binary representation. The
|
exponent of the number is in the interval -EXP to EXP (in limbs).
|
This function is useful for testing functions and algorithms,
|
since these kind of random numbers have proven to be more likely
|
to trigger corner-case bugs. Negative random numbers are
|
generated when MAX_SIZE is negative.
|
|
|
File: gmp.info, Node: Low-level Functions, Next: Random Number Functions, Prev: Floating-point Functions, Up: Top
|
|
8 Low-level Functions
|
*********************
|
|
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_'.
|
|
The `mpn' functions are designed to be as fast as possible, *not* to
|
provide a coherent calling interface. The different functions have
|
somewhat similar interfaces, but there are variations that make them
|
hard to use. These functions do as little as possible apart from the
|
real multiple precision computation, so that no time is spent on things
|
that not all callers need.
|
|
A source operand is specified by a pointer to the least significant
|
limb and a limb count. A destination operand is specified by just a
|
pointer. It is the responsibility of the caller to ensure that the
|
destination has enough space for storing the result.
|
|
With this way of specifying operands, it is possible to perform
|
computations on subranges of an argument, and store the result into a
|
subrange of a destination.
|
|
A common requirement for all functions is that each source area
|
needs at least one limb. No size argument may be zero. Unless
|
otherwise stated, in-place operations are allowed where source and
|
destination are the same, but not where they only partly overlap.
|
|
The `mpn' functions are the base for the implementation of the
|
`mpz_', `mpf_', and `mpq_' functions.
|
|
This example adds the number beginning at S1P and the number
|
beginning at S2P and writes the sum at DESTP. All areas have N limbs.
|
|
cy = mpn_add_n (destp, s1p, s2p, n)
|
|
It should be noted that the `mpn' functions make no attempt to
|
identify high or low zero limbs on their operands, or other special
|
forms. On random data such cases will be unlikely and it'd be wasteful
|
for every function to check every time. An application knowing
|
something about its data can take steps to trim or perhaps split its
|
calculations.
|
|
|
In the notation used below, a source operand is identified by the
|
pointer to the least significant limb, and the limb count in braces.
|
For example, {S1P, S1N}.
|
|
-- Function: mp_limb_t mpn_add_n (mp_limb_t *RP, const mp_limb_t *S1P,
|
const mp_limb_t *S2P, mp_size_t N)
|
Add {S1P, N} and {S2P, N}, and write the N least significant limbs
|
of the result to RP. Return carry, either 0 or 1.
|
|
This is the lowest-level function for addition. It is the
|
preferred function for addition, since it is written in assembly
|
for most CPUs. For addition of a variable to itself (i.e., S1P
|
equals S2P) use `mpn_lshift' with a count of 1 for optimal speed.
|
|
-- Function: mp_limb_t mpn_add_1 (mp_limb_t *RP, const mp_limb_t *S1P,
|
mp_size_t N, mp_limb_t S2LIMB)
|
Add {S1P, N} and S2LIMB, and write the N least significant limbs
|
of the result to RP. Return carry, either 0 or 1.
|
|
-- Function: mp_limb_t mpn_add (mp_limb_t *RP, const mp_limb_t *S1P,
|
mp_size_t S1N, const mp_limb_t *S2P, mp_size_t S2N)
|
Add {S1P, S1N} and {S2P, S2N}, and write the S1N least significant
|
limbs of the result to RP. Return carry, either 0 or 1.
|
|
This function requires that S1N is greater than or equal to S2N.
|
|
-- Function: mp_limb_t mpn_sub_n (mp_limb_t *RP, const mp_limb_t *S1P,
|
const mp_limb_t *S2P, mp_size_t N)
|
Subtract {S2P, N} from {S1P, N}, and write the N least significant
|
limbs of the result to RP. Return borrow, either 0 or 1.
|
|
This is the lowest-level function for subtraction. It is the
|
preferred function for subtraction, since it is written in
|
assembly for most CPUs.
|
|
-- Function: mp_limb_t mpn_sub_1 (mp_limb_t *RP, const mp_limb_t *S1P,
|
mp_size_t N, mp_limb_t S2LIMB)
|
Subtract S2LIMB from {S1P, N}, and write the N least significant
|
limbs of the result to RP. Return borrow, either 0 or 1.
|
|
-- Function: mp_limb_t mpn_sub (mp_limb_t *RP, const mp_limb_t *S1P,
|
mp_size_t S1N, const mp_limb_t *S2P, mp_size_t S2N)
|
Subtract {S2P, S2N} from {S1P, S1N}, and write the S1N least
|
significant limbs of the result to RP. Return borrow, either 0 or
|
1.
|
|
This function requires that S1N is greater than or equal to S2N.
|
|
-- Function: 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}.
|
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)
|
Multiply {S1P, N} and {S2P, N}, and write the 2*N-limb result to
|
RP.
|
|
The destination has to have space for 2*N limbs, even if the
|
product's most significant limb is zero. No overlap is permitted
|
between the destination and either source.
|
|
If the 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)
|
Multiply {S1P, S1N} and {S2P, S2N}, and write the (S1N+S2N)-limb
|
result to RP. Return the most significant limb of the result.
|
|
The destination has to have space for S1N + S2N limbs, even if the
|
product's most significant limb is zero. No overlap is permitted
|
between the destination and either source.
|
|
This function requires that S1N is greater than or equal to S2N.
|
|
-- Function: void mpn_sqr (mp_limb_t *RP, const mp_limb_t *S1P,
|
mp_size_t N)
|
Compute the square of {S1P, N} and write the 2*N-limb result to RP.
|
|
The destination has to have space for 2N limbs, even if the
|
result's most significant limb is zero. No overlap is permitted
|
between the destination and the source.
|
|
-- Function: mp_limb_t mpn_mul_1 (mp_limb_t *RP, const mp_limb_t *S1P,
|
mp_size_t N, mp_limb_t S2LIMB)
|
Multiply {S1P, N} by S2LIMB, and write the N least significant
|
limbs of the product to RP. Return the most significant limb of
|
the product. {S1P, N} and {RP, N} are allowed to overlap provided
|
RP <= S1P.
|
|
This is a low-level function that is a building block for general
|
multiplication as well as other operations in GMP. It is written
|
in assembly for most CPUs.
|
|
Don't call this function if S2LIMB is a power of 2; use
|
`mpn_lshift' with a count equal to the logarithm of S2LIMB
|
instead, for optimal speed.
|
|
-- Function: mp_limb_t mpn_addmul_1 (mp_limb_t *RP, const mp_limb_t
|
*S1P, mp_size_t N, mp_limb_t S2LIMB)
|
Multiply {S1P, N} and S2LIMB, and add the N least significant
|
limbs of the product to {RP, N} and write the result to RP.
|
Return the most significant limb of the product, plus carry-out
|
from the addition. {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
|
in assembly for most CPUs.
|
|
-- Function: mp_limb_t mpn_submul_1 (mp_limb_t *RP, const mp_limb_t
|
*S1P, mp_size_t N, mp_limb_t S2LIMB)
|
Multiply {S1P, N} and S2LIMB, and subtract the N least significant
|
limbs of the product from {RP, N} and write the result to RP.
|
Return the most significant limb of the product, plus borrow-out
|
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.
|
|
-- Function: void mpn_tdiv_qr (mp_limb_t *QP, mp_limb_t *RP, mp_size_t
|
QXN, const mp_limb_t *NP, mp_size_t NN, const mp_limb_t *DP,
|
mp_size_t DN)
|
Divide {NP, NN} by {DP, DN} and put the quotient at {QP, NN-DN+1}
|
and the remainder at {RP, DN}. The quotient is rounded towards 0.
|
|
No overlap is permitted between arguments, except that NP might
|
equal RP. The dividend size NN must be greater than or equal to
|
divisor size DN. The most significant limb of the divisor must be
|
non-zero. The QXN operand must be zero.
|
|
-- Function: mp_limb_t mpn_divrem (mp_limb_t *R1P, mp_size_t QXN,
|
mp_limb_t *RS2P, mp_size_t RS2N, const mp_limb_t *S3P,
|
mp_size_t S3N)
|
[This function is obsolete. Please call `mpn_tdiv_qr' instead for
|
best performance.]
|
|
Divide {RS2P, RS2N} by {S3P, S3N}, and write the quotient at R1P,
|
with the exception of the most significant limb, which is
|
returned. The remainder replaces the dividend at RS2P; it will be
|
S3N limbs long (i.e., as many limbs as the divisor).
|
|
In addition to an integer quotient, QXN fraction limbs are
|
developed, and stored after the integral limbs. For most usages,
|
QXN will be zero.
|
|
It is required that RS2N is greater than or equal to S3N. It is
|
required that the most significant bit of the divisor is set.
|
|
If the quotient is not needed, pass RS2P + S3N as R1P. Aside from
|
that special case, no overlap between arguments is permitted.
|
|
Return the most significant limb of the quotient, either 0 or 1.
|
|
The area at R1P needs to be RS2N - S3N + QXN limbs large.
|
|
-- Function: mp_limb_t mpn_divrem_1 (mp_limb_t *R1P, mp_size_t QXN,
|
mp_limb_t *S2P, mp_size_t S2N, mp_limb_t S3LIMB)
|
-- Macro: mp_limb_t mpn_divmod_1 (mp_limb_t *R1P, mp_limb_t *S2P,
|
mp_size_t S2N, mp_limb_t S3LIMB)
|
Divide {S2P, S2N} by S3LIMB, and write the quotient at R1P.
|
Return the remainder.
|
|
The integer quotient is written to {R1P+QXN, S2N} and in addition
|
QXN fraction limbs are developed and written to {R1P, QXN}.
|
Either or both S2N and QXN can be zero. For most usages, QXN will
|
be zero.
|
|
`mpn_divmod_1' exists for upward source compatibility and is
|
simply a macro calling `mpn_divrem_1' with a QXN of 0.
|
|
The areas at R1P and S2P have to be identical or completely
|
separate, not partially overlapping.
|
|
-- 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
|
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)
|
Divide {SP, N} by 3, expecting it to divide exactly, and writing
|
the result to {RP, N}. If 3 divides exactly, the return value is
|
zero and the result is the quotient. If not, the return value is
|
non-zero and the result won't be anything useful.
|
|
`mpn_divexact_by3c' takes an initial carry parameter, which can be
|
the return value from a previous call, so a large calculation can
|
be done piece by piece from low to high. `mpn_divexact_by3' is
|
simply a macro calling `mpn_divexact_by3c' with a 0 carry
|
parameter.
|
|
These routines use a multiply-by-inverse and will be faster than
|
`mpn_divrem_1' on CPUs with fast multiplication but slow division.
|
|
The source a, result q, size n, initial carry i, and return value
|
c satisfy c*b^n + a-i = 3*q, where b=2^GMP_NUMB_BITS. The return
|
c is always 0, 1 or 2, and the initial carry i must also be 0, 1
|
or 2 (these are both borrows really). When c=0 clearly q=(a-i)/3.
|
When c!=0, the remainder (a-i) mod 3 is given by 3-c, because b
|
== 1 mod 3 (when `mp_bits_per_limb' is even, which is always so
|
currently).
|
|
-- Function: mp_limb_t mpn_mod_1 (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_lshift (mp_limb_t *RP, const mp_limb_t *SP,
|
mp_size_t N, unsigned int COUNT)
|
Shift {SP, N} left by COUNT bits, and write the result to {RP, N}.
|
The bits shifted out at the left are returned in the least
|
significant COUNT bits of the return value (the rest of the return
|
value is zero).
|
|
COUNT must be in the range 1 to mp_bits_per_limb-1. The regions
|
{SP, N} and {RP, N} may overlap, provided RP >= SP.
|
|
This function is written in assembly for most CPUs.
|
|
-- Function: mp_limb_t mpn_rshift (mp_limb_t *RP, const mp_limb_t *SP,
|
mp_size_t N, unsigned int COUNT)
|
Shift {SP, N} right by COUNT bits, and write the result to {RP,
|
N}. The bits shifted out at the right are returned in the most
|
significant COUNT bits of the return value (the rest of the return
|
value is zero).
|
|
COUNT must be in the range 1 to mp_bits_per_limb-1. The regions
|
{SP, N} and {RP, N} may overlap, provided RP <= SP.
|
|
This function is written in assembly for most CPUs.
|
|
-- Function: int mpn_cmp (const mp_limb_t *S1P, const mp_limb_t *S2P,
|
mp_size_t N)
|
Compare {S1P, N} and {S2P, N} and return a positive value if S1 >
|
S2, 0 if they are equal, or a negative value if S1 < S2.
|
|
-- Function: 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.
|
|
It is required that XN >= YN > 0, and the most significant limb of
|
{YP, YN} must be non-zero. 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)
|
Return the greatest common divisor of {XP, XN} and YLIMB. Both
|
operands must be non-zero.
|
|
-- Function: mp_size_t mpn_gcdext (mp_limb_t *GP, mp_limb_t *SP,
|
mp_size_t *SN, mp_limb_t *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
|
computed but can easily be obtained from (G - U*S) / V (the
|
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 area at GP
|
should have room for VN limbs and the area at SP should have room
|
for VN+1 limbs.
|
|
Both source operands are destroyed.
|
|
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.
|
|
The most significant limb of {SP, N} must be non-zero. The areas
|
{R1P, ceil(N/2)} and {SP, N} must be completely separate. The
|
areas {R2P, N} and {SP, N} must be either identical or completely
|
separate.
|
|
If the remainder is not wanted then R2P can be `NULL', and in this
|
case the return value is zero or non-zero according to whether the
|
remainder would have been zero or non-zero.
|
|
A return value of zero indicates a perfect square. See also
|
`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.
|
|
The most significant limb of the input {S1P, S1N} must be
|
non-zero. The input {S1P, S1N} is clobbered, except when BASE is
|
a power of 2, in which case it's unchanged.
|
|
The area at STR has to have space for the largest possible number
|
represented by a S1N long limb array, plus one extra character.
|
|
-- Function: mp_size_t mpn_set_str (mp_limb_t *RP, const unsigned char
|
*STR, size_t STRSIZE, int BASE)
|
Convert bytes {STR,STRSIZE} in the given BASE to limbs at RP.
|
|
STR[0] is the most significant 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 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.
|
|
The area at RP has to have space for the largest possible number
|
with STRSIZE digits in the chosen base, plus one extra limb.
|
|
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)
|
Scan S1P from bit position BIT for the next clear bit.
|
|
It is required that there be a clear bit within the area at S1P at
|
or beyond bit position BIT, so that the function has something to
|
return.
|
|
-- Function: mp_bitcnt_t mpn_scan1 (const mp_limb_t *S1P, mp_bitcnt_t
|
BIT)
|
Scan S1P from bit position BIT for the next set bit.
|
|
It is required that there be a set bit within the area at S1P at or
|
beyond bit position BIT, so that the function has something to
|
return.
|
|
-- Function: void mpn_random (mp_limb_t *R1P, mp_size_t R1N)
|
-- Function: void mpn_random2 (mp_limb_t *R1P, mp_size_t R1N)
|
Generate a random number of length R1N and store it at R1P. The
|
most significant limb is always non-zero. `mpn_random' generates
|
uniformly distributed limb data, `mpn_random2' generates long
|
strings of zeros and ones in the binary representation.
|
|
`mpn_random2' is intended for testing the correctness of the `mpn'
|
routines.
|
|
-- Function: mp_bitcnt_t mpn_popcount (const mp_limb_t *S1P, mp_size_t
|
N)
|
Count the number of set bits in {S1P, N}.
|
|
-- Function: mp_bitcnt_t mpn_hamdist (const mp_limb_t *S1P, const
|
mp_limb_t *S2P, mp_size_t N)
|
Compute the hamming distance between {S1P, N} and {S2P, N}, which
|
is the number of bit positions where the two operands have
|
different bit values.
|
|
-- Function: int mpn_perfect_square_p (const mp_limb_t *S1P, mp_size_t
|
N)
|
Return non-zero iff {S1P, N} is a perfect square. 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_ior_n (mp_limb_t *RP, const mp_limb_t *S1P,
|
const mp_limb_t *S2P, mp_size_t N)
|
Perform the bitwise logical inclusive or of {S1P, N} and {S2P, N},
|
and write the result to {RP, N}.
|
|
-- Function: void mpn_xor_n (mp_limb_t *RP, const mp_limb_t *S1P,
|
const mp_limb_t *S2P, mp_size_t N)
|
Perform the bitwise logical exclusive or of {S1P, N} and {S2P, N},
|
and write the result to {RP, N}.
|
|
-- Function: void mpn_andn_n (mp_limb_t *RP, const mp_limb_t *S1P,
|
const mp_limb_t *S2P, mp_size_t N)
|
Perform the bitwise logical and of {S1P, N} and the bitwise
|
complement of {S2P, N}, and write the result to {RP, N}.
|
|
-- Function: void mpn_iorn_n (mp_limb_t *RP, const mp_limb_t *S1P,
|
const mp_limb_t *S2P, mp_size_t N)
|
Perform the bitwise logical inclusive or of {S1P, N} and the
|
bitwise complement of {S2P, N}, and write the result to {RP, N}.
|
|
-- Function: void mpn_nand_n (mp_limb_t *RP, const mp_limb_t *S1P,
|
const mp_limb_t *S2P, mp_size_t N)
|
Perform the bitwise logical and of {S1P, N} and {S2P, N}, and
|
write the bitwise complement of the result to {RP, N}.
|
|
-- Function: void mpn_nior_n (mp_limb_t *RP, const mp_limb_t *S1P,
|
const mp_limb_t *S2P, mp_size_t N)
|
Perform the bitwise logical inclusive or of {S1P, N} and {S2P, N},
|
and write the bitwise complement of the result to {RP, N}.
|
|
-- Function: void mpn_xnor_n (mp_limb_t *RP, const mp_limb_t *S1P,
|
const mp_limb_t *S2P, mp_size_t N)
|
Perform the bitwise logical exclusive or of {S1P, N} and {S2P, N},
|
and write the bitwise complement of the result to {RP, N}.
|
|
-- Function: void mpn_com (mp_limb_t *RP, const mp_limb_t *SP,
|
mp_size_t N)
|
Perform the bitwise complement of {SP, N}, and write the result to
|
{RP, N}.
|
|
-- Function: void mpn_copyi (mp_limb_t *RP, const mp_limb_t *S1P,
|
mp_size_t N)
|
Copy from {S1P, N} to {RP, N}, increasingly.
|
|
-- Function: void mpn_copyd (mp_limb_t *RP, const mp_limb_t *S1P,
|
mp_size_t N)
|
Copy from {S1P, N} to {RP, N}, decreasingly.
|
|
-- Function: void mpn_zero (mp_limb_t *RP, mp_size_t N)
|
Zero {RP, N}.
|
|
|
8.1 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.
|
|
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
|
handling on some processors.
|
|
All the `mpn' functions accepting limb data will expect the nail
|
bits to be zero on entry, and will return data with the nails similarly
|
all zero. This applies both to limb vectors and to single limb
|
arguments.
|
|
Nails can be enabled by configuring with `--enable-nails'. By
|
default the number of bits will be chosen according to what suits the
|
host processor, but a particular number can be selected with
|
`--enable-nails=N'.
|
|
At the mpn level, a nail build is neither source nor binary
|
compatible with a non-nail build, strictly speaking. But programs
|
acting on limbs only through the mpn functions are likely to work
|
equally well with either build, and judicious use of the definitions
|
below should make any program compatible with either build, at the
|
source level.
|
|
For the higher level routines, meaning `mpz' etc, a nail build
|
should be fully source and binary compatible with a non-nail build.
|
|
-- Macro: GMP_NAIL_BITS
|
-- Macro: GMP_NUMB_BITS
|
-- Macro: GMP_LIMB_BITS
|
`GMP_NAIL_BITS' is the number of nail bits, or 0 when nails are
|
not in use. `GMP_NUMB_BITS' is the number of data bits in a limb.
|
`GMP_LIMB_BITS' is the total number of bits in an `mp_limb_t'. In
|
all cases
|
|
GMP_LIMB_BITS == GMP_NAIL_BITS + GMP_NUMB_BITS
|
|
-- Macro: GMP_NAIL_MASK
|
-- Macro: GMP_NUMB_MASK
|
Bit masks for the nail and number parts of a limb.
|
`GMP_NAIL_MASK' is 0 when nails are not in use.
|
|
`GMP_NAIL_MASK' is not often needed, since the nail part can be
|
obtained with `x >> GMP_NUMB_BITS', and that means one less large
|
constant, which can help various RISC chips.
|
|
-- Macro: GMP_NUMB_MAX
|
The maximum value that can be stored in the number part of a limb.
|
This is the same as `GMP_NUMB_MASK', but can be used for clarity
|
when doing comparisons rather than bit-wise operations.
|
|
The term "nails" comes from finger or toe nails, which are at the
|
ends of a limb (arm or leg). "numb" is short for number, but is also
|
how the developers felt after trying for a long time to come up with
|
sensible names for these things.
|
|
In the future (the distant future most likely) a non-zero nail might
|
be permitted, giving non-unique representations for numbers in a limb
|
vector. This would help vector processors since carries would only
|
ever need to propagate one or two limbs.
|
|
|
File: 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.
|
|
The functions actually generating random numbers are described in
|
*Note Integer Random Numbers::, and *Note Miscellaneous Float
|
Functions::.
|
|
The older style random number functions don't accept a
|
`gmp_randstate_t' parameter but instead share a global variable of that
|
type. They use a default algorithm and are currently not seeded
|
(though perhaps that will change in the future). The new functions
|
accepting a `gmp_randstate_t' are recommended for applications that
|
care about randomness.
|
|
* Menu:
|
|
* Random State Initialization::
|
* Random State Seeding::
|
* Random State Miscellaneous::
|
|
|
File: gmp.info, Node: Random State Initialization, Next: Random State Seeding, Prev: Random Number Functions, Up: Random Number Functions
|
|
9.1 Random State Initialization
|
===============================
|
|
-- Function: void gmp_randinit_default (gmp_randstate_t STATE)
|
Initialize STATE with a default algorithm. This will be a
|
compromise between speed and randomness, and is recommended for
|
applications with no special requirements. Currently this is
|
`gmp_randinit_mt'.
|
|
-- Function: void gmp_randinit_mt (gmp_randstate_t STATE)
|
Initialize STATE for a Mersenne Twister algorithm. This algorithm
|
is fast and has good randomness properties.
|
|
-- Function: void gmp_randinit_lc_2exp (gmp_randstate_t STATE, 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
|
bit no more than 4, etc. For this reason only the high half of
|
each X is actually used.
|
|
When a random number of more than M2EXP/2 bits is to be generated,
|
multiple iterations of the recurrence are used and the results
|
concatenated.
|
|
-- Function: int gmp_randinit_lc_2exp_size (gmp_randstate_t STATE,
|
mp_bitcnt_t SIZE)
|
Initialize STATE for a linear congruential algorithm as per
|
`gmp_randinit_lc_2exp'. A, C and M2EXP are selected from a table,
|
chosen so that SIZE bits (or more) of each X will be used, 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.
|
|
-- Function: void gmp_randinit_set (gmp_randstate_t ROP,
|
gmp_randstate_t OP)
|
Initialize ROP with a copy of the algorithm and state from OP.
|
|
-- Function: void gmp_randinit (gmp_randstate_t STATE,
|
gmp_randalg_t ALG, ...)
|
*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
|
required, this is the SIZE for that function.
|
`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
|
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).
|
|
-- Function: void gmp_randclear (gmp_randstate_t STATE)
|
Free all memory occupied by STATE.
|
|
|
File: gmp.info, Node: Random State Seeding, Next: Random State Miscellaneous, Prev: Random State Initialization, Up: Random Number Functions
|
|
9.2 Random State Seeding
|
========================
|
|
-- Function: void gmp_randseed (gmp_randstate_t STATE, 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.
|
|
|
File: gmp.info, Node: Random State Miscellaneous, Prev: Random State Seeding, Up: Random Number Functions
|
|
9.3 Random State Miscellaneous
|
==============================
|
|
-- Function: unsigned long gmp_urandomb_ui (gmp_randstate_t STATE,
|
unsigned long N)
|
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'.
|
|
-- 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.
|
|
|
File: gmp.info, Node: Formatted Output, Next: Formatted Input, Prev: Random Number Functions, Up: Top
|
|
10 Formatted Output
|
*******************
|
|
* Menu:
|
|
* Formatted Output Strings::
|
* Formatted Output Functions::
|
* C++ Formatted Output::
|
|
|
File: gmp.info, Node: Formatted Output Strings, Next: Formatted Output Functions, Prev: Formatted Output, Up: Formatted Output
|
|
10.1 Format Strings
|
===================
|
|
`gmp_printf' and friends accept format strings similar to the standard C
|
`printf' (*note Formatted Output: (libc)Formatted Output.). A format
|
specification is of the form
|
|
% [flags] [width] [.[precision]] [type] conv
|
|
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);
|
|
mpq_t q;
|
gmp_printf ("a hex rational: %#40Qx\n", q);
|
|
mpf_t f;
|
int n;
|
gmp_printf ("fixed point mpf %.*Ff with %d digits\n", n, f, n);
|
|
mp_limb_t l;
|
gmp_printf ("limb %Mu\n", l);
|
|
const mp_limb_t *ptr;
|
mp_size_t size;
|
gmp_printf ("limb array %Nx\n", ptr, size);
|
|
For `N' the limbs are expected least significant first, as per the
|
`mpn' functions (*note Low-level Functions::). A negative size can be
|
given to print the value as a negative.
|
|
All the standard C `printf' types behave the same as the C library
|
`printf', and can be freely intermixed with the 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.
|
|
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.
|
|
0 pad with zeros (rather than spaces)
|
# show the base with `0x', `0X' or `0'
|
+ always show a sign
|
(space) show a space or a `-' sign
|
' group digits, GLIBC style (not 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'.
|
|
The standard types accepted are as follows. `h' and `l' are
|
portable, the rest will depend on the compiler (or include files) for
|
the type and the C library for the output.
|
|
h short
|
hh char
|
j intmax_t or uintmax_t
|
l long or wchar_t
|
ll long long
|
L long double
|
q quad_t or u_quad_t
|
t ptrdiff_t
|
z size_t
|
|
The GMP types are
|
|
F mpf_t, float conversions
|
Q mpq_t, integer conversions
|
M mp_limb_t, integer conversions
|
N mp_limb_t array, integer conversions
|
Z mpz_t, integer conversions
|
|
The conversions accepted are as follows. `a' and `A' are always
|
supported for `mpf_t' but depend on the C library for standard C float
|
types. `m' and `p' depend on the C library.
|
|
a A hex floats, C99 style
|
c character
|
d decimal integer
|
e E scientific format float
|
f fixed point float
|
i same as d
|
g G fixed or scientific float
|
m `strerror' string, GLIBC style
|
n store characters written so far
|
o octal integer
|
p pointer
|
s string
|
u unsigned integer
|
x X hex integer
|
|
`o', `x' and `X' are unsigned for the standard C types, but for
|
types `Z', `Q' and `N' they are signed. `u' is not meaningful for `Z',
|
`Q' and `N'.
|
|
`M' is a proxy for the C library `l' or `L', according to the size
|
of `mp_limb_t'. Unsigned conversions will be usual, but a signed
|
conversion can be used and will interpret the value as a twos complement
|
negative.
|
|
`n' can be used with any type, even the 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
|
(perhaps this will be added in the future).
|
|
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.
|
Zeros will be used if necessary to pad to the requested precision. This
|
happens even for an `f' conversion of an `mpf_t' which is an integer,
|
for instance 2^1024 in an `mpf_t' of 128 bits precision will only
|
produce about 40 digits, then pad with zeros to the decimal point. An
|
empty precision field like `%.Fe' or `%.Ff' can be used to specifically
|
request just the significant digits. 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.
|
|
The format string is only interpreted as plain `char's, multibyte
|
characters are not recognised. Perhaps this will change in the future.
|
|
|
File: gmp.info, Node: Formatted Output Functions, Next: C++ Formatted Output, Prev: Formatted Output Strings, Up: Formatted Output
|
|
10.2 Functions
|
==============
|
|
Each of the following functions is similar to the corresponding C
|
library function. The basic `printf' forms take a variable argument
|
list. The `vprintf' forms take an argument pointer, see *Note Variadic
|
Functions: (libc)Variadic Functions, or `man 3 va_start'.
|
|
It should be emphasised that if a format string is invalid, or the
|
arguments don't match what the format specifies, then the behaviour of
|
any of these functions will be unpredictable. GCC format string
|
checking is not available, since it doesn't recognise the 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
|
shouldn't normally occur.
|
|
-- Function: int gmp_printf (const char *FMT, ...)
|
-- Function: int gmp_vprintf (const char *FMT, va_list AP)
|
Print to the standard output `stdout'. Return the number of
|
characters written, or -1 if an error occurred.
|
|
-- Function: int gmp_fprintf (FILE *FP, const char *FMT, ...)
|
-- Function: int gmp_vfprintf (FILE *FP, const char *FMT, va_list AP)
|
Print to the stream FP. Return the number of characters written,
|
or -1 if an error occurred.
|
|
-- Function: int gmp_sprintf (char *BUF, const char *FMT, ...)
|
-- Function: int gmp_vsprintf (char *BUF, const char *FMT, va_list AP)
|
Form a null-terminated string in BUF. Return the number of
|
characters written, excluding the terminating null.
|
|
No overlap is permitted between the space at BUF and the string
|
FMT.
|
|
These functions are not recommended, since there's no protection
|
against exceeding the space available at BUF.
|
|
-- Function: int gmp_snprintf (char *BUF, size_t SIZE, const char
|
*FMT, ...)
|
-- Function: int gmp_vsnprintf (char *BUF, size_t SIZE, const char
|
*FMT, va_list AP)
|
Form a null-terminated string in BUF. No more than SIZE bytes
|
will be written. To get the full output, SIZE must be enough for
|
the string and null-terminator.
|
|
The return value is the total number of characters which ought to
|
have been produced, excluding the terminating null. If RETVAL >=
|
SIZE then the actual output has been truncated to the first SIZE-1
|
characters, and a null appended.
|
|
No overlap is permitted between the region {BUF,SIZE} and the FMT
|
string.
|
|
Notice the return value is in ISO C99 `snprintf' style. This is
|
so even if the C library `vsnprintf' is the older GLIBC 2.0.x
|
style.
|
|
-- Function: int gmp_asprintf (char **PP, const char *FMT, ...)
|
-- Function: int gmp_vasprintf (char **PP, const char *FMT, va_list AP)
|
Form a null-terminated string in a block of memory obtained from
|
the current memory allocation function (*note Custom
|
Allocation::). The block will be the size of the string and
|
null-terminator. The address of the block in stored to *PP. The
|
return value is the number of characters produced, excluding the
|
null-terminator.
|
|
Unlike the C library `asprintf', `gmp_asprintf' doesn't return -1
|
if there's no more memory available, it lets the current allocation
|
function handle that.
|
|
-- Function: int gmp_obstack_printf (struct obstack *OB, const char
|
*FMT, ...)
|
-- Function: int gmp_obstack_vprintf (struct obstack *OB, const char
|
*FMT, va_list AP)
|
Append to the current object in OB. The return value is the
|
number of characters written. A null-terminator is not written.
|
|
FMT cannot be within the current object in OB, since that object
|
might move as it grows.
|
|
These functions are available only when the C library provides the
|
obstack feature, which probably means only on GNU systems, see
|
*Note Obstacks: (libc)Obstacks.
|
|
|
File: 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
|
Libraries::), which is built if C++ support is enabled (*note Build
|
Options::). Prototypes are available from `<gmp.h>'.
|
|
-- 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.
|
|
-- 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'.
|
|
In hex or octal, OP is printed as a signed value, the same as for
|
decimal. If `ios::showbase' is set then a base indicator is shown
|
on both the numerator and denominator (if the denominator is
|
required).
|
|
-- Function: ostream& operator<< (ostream& STREAM, 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.
|
|
Hex and octal are supported, unlike the standard `operator<<' on
|
`double'. The mantissa will be in hex or octal, the exponent will
|
be in decimal. For hex the exponent delimiter is an `@'. This is
|
as per `mpf_out_str'.
|
|
`ios::showbase' is supported, and will put a base on the mantissa,
|
for example hex `0x1.8' or `0x0.8', or octal `01.4' or `00.4'.
|
This last form is slightly strange, but at least differentiates
|
itself from decimal.
|
|
These operators mean that GMP types can be printed in the usual C++
|
way, for example,
|
|
mpz_t z;
|
int n;
|
...
|
cout << "iteration " << n << " value " << z << "\n";
|
|
But note that `ostream' output (and `istream' input, *note C++
|
Formatted Input::) is the only overloading available for the GMP types
|
and that for instance using `+' with an `mpz_t' will have unpredictable
|
results. For classes with overloading, see *Note C++ Class Interface::.
|
|
|
File: gmp.info, Node: Formatted Input, Next: C++ Class Interface, Prev: Formatted Output, Up: Top
|
|
11 Formatted Input
|
******************
|
|
* Menu:
|
|
* Formatted Input Strings::
|
* Formatted Input Functions::
|
* C++ Formatted Input::
|
|
|
File: gmp.info, Node: Formatted Input Strings, Next: Formatted Input Functions, Prev: Formatted Input, Up: Formatted Input
|
|
11.1 Formatted Input Strings
|
============================
|
|
`gmp_scanf' and friends accept format strings similar to the standard C
|
`scanf' (*note Formatted Input: (libc)Formatted Input.). A format
|
specification is of the form
|
|
% [flags] [width] [type] conv
|
|
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
|
they're already "call-by-reference". For example,
|
|
/* to read say "a(5) = 1234" */
|
int n;
|
mpz_t z;
|
gmp_scanf ("a(%d) = %Zd\n", &n, z);
|
|
mpq_t q1, q2;
|
gmp_sscanf ("0377 + 0x10/0x11", "%Qi + %Qi", q1, q2);
|
|
/* to read say "topleft (1.55,-2.66)" */
|
mpf_t x, y;
|
char buf[32];
|
gmp_scanf ("%31s (%Ff,%Ff)", buf, x, y);
|
|
All the standard C `scanf' types behave the same as in the C library
|
`scanf', and can be freely intermixed with the 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.
|
|
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
|
portable, the rest will depend on the compiler (or include files) for
|
the type and the C library for the input.
|
|
h short
|
hh char
|
j intmax_t or uintmax_t
|
l long int, double or wchar_t
|
ll long long
|
L long double
|
q quad_t or u_quad_t
|
t ptrdiff_t
|
z size_t
|
|
The GMP types are
|
|
F mpf_t, float conversions
|
Q mpq_t, integer conversions
|
Z mpz_t, integer conversions
|
|
The conversions accepted are as follows. `p' and `[' will depend on
|
support from the C library, the rest are standard.
|
|
c character or characters
|
d decimal integer
|
e E f g G float
|
i integer with base indicator
|
n characters read so far
|
o octal integer
|
p pointer
|
s string of non-whitespace characters
|
u decimal integer
|
x X hex integer
|
[ string of characters in a set
|
|
`e', `E', `f', `g' and `G' are identical, they all read either fixed
|
point or scientific format, and either upper or lower case `e' for the
|
exponent in scientific format.
|
|
C99 style hex float format (`printf %a', *note Formatted Output
|
Strings::) is always accepted for `mpf_t', but for the standard float
|
types it will depend on the C library.
|
|
`x' and `X' are identical, both accept both upper and lower case
|
hexadecimal.
|
|
`o', `u', `x' and `X' all read positive or negative values. For the
|
standard C types these are described as "unsigned" conversions, but
|
that merely affects certain overflow handling, negatives are still
|
allowed (per `strtoul', *note Parsing of Integers: (libc)Parsing of
|
Integers.). For 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::).
|
|
`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
|
nothing at all.
|
|
Other conversions or types that might be accepted by the C library
|
`scanf' cannot be used through `gmp_scanf'.
|
|
Whitespace is read and discarded before a field, except for `c' and
|
`[' conversions.
|
|
For float conversions, the decimal point character (or string)
|
expected is taken from the current locale settings on systems which
|
provide `localeconv' (*note Locales and Internationalization:
|
(libc)Locales.). The C library will normally do the same for standard
|
float input.
|
|
The format string is only interpreted as plain `char's, multibyte
|
characters are not recognised. Perhaps this will change in the future.
|
|
|
File: gmp.info, Node: Formatted Input Functions, Next: C++ Formatted Input, Prev: Formatted Input Strings, Up: Formatted Input
|
|
11.2 Formatted Input Functions
|
==============================
|
|
Each of the following functions is similar to the corresponding C
|
library function. The plain `scanf' forms take a variable argument
|
list. The `vscanf' forms take an argument pointer, see *Note Variadic
|
Functions: (libc)Variadic Functions, or `man 3 va_start'.
|
|
It should be emphasised that if a format string is invalid, or the
|
arguments don't match what the format specifies, then the behaviour of
|
any of these functions will be unpredictable. GCC format string
|
checking is not available, since it doesn't recognise the GMP
|
extensions.
|
|
No overlap is permitted between the FMT string and any of the results
|
produced.
|
|
-- Function: int gmp_scanf (const char *FMT, ...)
|
-- Function: int gmp_vscanf (const char *FMT, va_list AP)
|
Read from the standard input `stdin'.
|
|
-- Function: int gmp_fscanf (FILE *FP, const char *FMT, ...)
|
-- Function: int gmp_vfscanf (FILE *FP, const char *FMT, va_list AP)
|
Read from the stream FP.
|
|
-- Function: int gmp_sscanf (const char *S, const char *FMT, ...)
|
-- Function: int gmp_vsscanf (const char *S, const char *FMT, va_list
|
AP)
|
Read from a null-terminated string S.
|
|
The return value from each of these functions is the same as the
|
standard C99 `scanf', namely the number of fields successfully parsed
|
and stored. `%n' fields and fields read but suppressed by `*' don't
|
count towards the return value.
|
|
If end of input (or a file error) is reached before a character for
|
a field or a literal, and if no previous non-suppressed fields have
|
matched, then the return value is `EOF' instead of 0. A whitespace
|
character in the format string is only an optional match and doesn't
|
induce an `EOF' in this fashion. Leading whitespace read and discarded
|
for a field don't count as characters for that field.
|
|
For the 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.
|
|
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
|
character of lookahead when parsing. Although clearly it could look at
|
its entire input, it is deliberately made identical to `gmp_fscanf',
|
the same way C99 `sscanf' is the same as `fscanf'.
|
|
|
File: 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>'.
|
|
-- Function: istream& operator>> (istream& STREAM, mpz_t ROP)
|
Read ROP from STREAM, using its `ios' formatting settings.
|
|
-- Function: istream& operator>> (istream& STREAM, mpq_t ROP)
|
An integer like `123' will be read, or a fraction like `5/9'. No
|
whitespace is allowed around the `/'. If the fraction is not in
|
canonical form then `mpq_canonicalize' must be called (*note
|
Rational Number Functions::) before operating on it.
|
|
As per integer input, an `0' or `0x' base indicator is read when
|
none of `ios::dec', `ios::oct' or `ios::hex' are set. This is
|
done separately for numerator and denominator, so that for instance
|
`0x10/11' is 16/11 and `0x10/0x11' is 16/17.
|
|
-- Function: istream& operator>> (istream& STREAM, mpf_t ROP)
|
Read ROP from STREAM, using its `ios' formatting settings.
|
|
Hex or octal floats are not supported, but might be in the future,
|
or perhaps it's best to accept only what the standard float
|
`operator>>' does.
|
|
Note that digit grouping specified by the `istream' locale is
|
currently not accepted. Perhaps this will change in the future.
|
|
|
These operators mean that 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++
|
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::.
|
|
|
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
|
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.
|
|
*Everything described in this chapter is to be considered preliminary
|
and might be subject to incompatible changes if some unforeseen
|
difficulty reveals itself.*
|
|
* Menu:
|
|
* C++ Interface General::
|
* C++ Interface Integers::
|
* C++ Interface Rationals::
|
* C++ Interface Floats::
|
* C++ Interface Random Numbers::
|
* C++ Interface Limitations::
|
|
|
File: gmp.info, Node: C++ Interface General, Next: C++ Interface Integers, Prev: C++ Class Interface, Up: C++ Class Interface
|
|
12.1 C++ Interface General
|
==========================
|
|
All the C++ classes and functions are available with
|
|
#include <gmpxx.h>
|
|
Programs should be linked with the `libgmpxx' and `libgmp'
|
libraries. For example,
|
|
g++ mycxxprog.cc -lgmpxx -lgmp
|
|
The classes defined are
|
|
-- Class: mpz_class
|
-- Class: mpq_class
|
-- Class: mpf_class
|
|
The standard operators and various standard functions are overloaded
|
to allow arithmetic with these classes. For example,
|
|
int
|
main (void)
|
{
|
mpz_class a, b, c;
|
|
a = 1234;
|
b = "-5678";
|
c = a+b;
|
cout << "sum is " << c << "\n";
|
cout << "absolute value is " << abs(c) << "\n";
|
|
return 0;
|
}
|
|
An important feature of the implementation is that an expression like
|
`a=b+c' results in a single call to the corresponding `mpz_add',
|
without using a temporary for the `b+c' part. Expressions which by
|
their nature imply intermediate values, like `a=b*c+d*e', still use
|
temporaries though.
|
|
The classes can be freely intermixed in expressions, as can the
|
classes and the standard types `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
|
sorts of invalid class and pointer combinations compile but almost
|
certainly not do anything sensible.
|
|
Conversions back from the classes to standard C++ types aren't done
|
automatically, instead member functions like `get_si' are provided (see
|
the following sections for details).
|
|
Also there are no automatic conversions from the classes to the
|
corresponding GMP C types, instead a reference to the underlying C
|
object can be obtained with the following functions,
|
|
-- Function: mpz_t mpz_class::get_mpz_t ()
|
-- Function: mpq_t mpq_class::get_mpq_t ()
|
-- Function: mpf_t mpf_class::get_mpf_t ()
|
|
These can be used to call a C function which doesn't have a C++ class
|
interface. For example to set `a' to the GCD of `b' and `c',
|
|
mpz_class a, b, c;
|
...
|
mpz_gcd (a.get_mpz_t(), b.get_mpz_t(), c.get_mpz_t());
|
|
In the other direction, a class can be initialized from the
|
corresponding GMP C type, or assigned to if an explicit constructor is
|
used. In both cases this makes a copy of the value, it doesn't create
|
any sort of association. For example,
|
|
mpz_t z;
|
// ... init and calculate z ...
|
mpz_class x(z);
|
mpz_class y;
|
y = mpz_class (z);
|
|
There are no namespace setups in `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.
|
|
|
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: 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: 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: 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='.
|
|
-- 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.
|
|
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 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: 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.
|
|
`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
|
currently unspecified. The rounding might take place at the start, in
|
the middle, or at the end of the operation, and it might change in the
|
future.
|
|
Conversions between `mpz_class' and `double', however, are defined
|
to follow the corresponding C functions `mpz_get_d' and `mpz_set_d'.
|
And comparisons are always made exactly, as per `mpz_cmp_d'.
|
|
|
File: 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.
|
|
-- 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: 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::).
|
|
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
|
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'.
|
|
-- Function: mpz_class& mpq_class::get_num ()
|
-- Function: mpz_class& mpq_class::get_den ()
|
Get a reference to an `mpz_class' which is the numerator or
|
denominator of an `mpq_class'. This can be used both for read and
|
write access. If the object returned is modified, it modifies the
|
original `mpq_class'.
|
|
If direct manipulation might produce a non-canonical value, then
|
`mpq_class::canonicalize' must be called before further operations.
|
|
-- Function: mpz_t mpq_class::get_num_mpz_t ()
|
-- Function: mpz_t mpq_class::get_den_mpz_t ()
|
Get a reference to the underlying `mpz_t' numerator or denominator
|
of an `mpq_class'. This can be passed to C functions expecting an
|
`mpz_t'. Any modifications made to the `mpz_t' will modify the
|
original `mpq_class'.
|
|
If direct manipulation might produce a non-canonical value, then
|
`mpq_class::canonicalize' must be called before further operations.
|
|
-- Function: istream& operator>> (istream& STREAM, mpq_class& ROP);
|
Read ROP from STREAM, using its `ios' formatting settings, the
|
same as `mpq_t operator>>' (*note C++ Formatted Input::).
|
|
If the ROP read might not be in canonical form then
|
`mpq_class::canonicalize' must be called.
|
|
|
File: gmp.info, Node: C++ Interface Floats, Next: C++ Interface Random Numbers, Prev: C++ Interface Rationals, Up: C++ Class Interface
|
|
12.4 C++ Interface Floats
|
=========================
|
|
When an expression requires the use of temporary intermediate
|
`mpf_class' values, like `f=g*h+x*y', those temporaries will have the
|
same precision as the destination `f'. Explicit constructors can be
|
used if this doesn't suit.
|
|
-- Function: mpf_class::mpf_class (type OP)
|
-- Function: mpf_class::mpf_class (type OP, 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.
|
|
mpf_class f(1.5); // default precision
|
mpf_class f(1.5, 500); // 500 bits (at least)
|
mpf_class f(x); // precision of x
|
mpf_class f(abs(x)); // precision of x
|
mpf_class f(-g, 1000); // 1000 bits (at least)
|
mpf_class f(x+y); // greater of precisions of x and y
|
|
-- Function: 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.
|
|
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
|
same types are accepted as for the constructors above.
|
|
Note that `operator=' only stores a new value, it doesn't copy or
|
change the precision of the destination, instead the value is
|
truncated if necessary. This is the same as `mpf_set' etc. Note
|
in particular this means for `mpf_class' a copy constructor is not
|
the same as a default constructor plus assignment.
|
|
mpf_class x (y); // x created with precision of y
|
|
mpf_class x; // x created with default precision
|
x = y; // value truncated to that precision
|
|
Applications using templated code may need to be careful about the
|
assumptions the code makes in this area, when working with
|
`mpf_class' values of various different or non-default precisions.
|
For instance implementations of the standard `complex' template
|
have been seen in both styles above, though of course `complex' is
|
normally only actually specified for use with the builtin float
|
types.
|
|
-- Function: mpf_class abs (mpf_class OP)
|
-- Function: mpf_class ceil (mpf_class OP)
|
-- Function: int cmp (mpf_class OP1, type OP2)
|
-- Function: int cmp (type OP1, mpf_class OP2)
|
-- Function: bool mpf_class::fits_sint_p (void)
|
-- Function: bool mpf_class::fits_slong_p (void)
|
-- Function: bool mpf_class::fits_sshort_p (void)
|
-- Function: bool mpf_class::fits_uint_p (void)
|
-- Function: bool mpf_class::fits_ulong_p (void)
|
-- Function: bool mpf_class::fits_ushort_p (void)
|
-- Function: mpf_class floor (mpf_class OP)
|
-- Function: mpf_class hypot (mpf_class OP1, mpf_class OP2)
|
-- Function: double mpf_class::get_d (void)
|
-- Function: 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'.
|
|
The accuracy provided by `hypot' is not currently guaranteed.
|
|
-- Function: mp_bitcnt_t mpf_class::get_prec ()
|
-- Function: void mpf_class::set_prec (mp_bitcnt_t PREC)
|
-- Function: void mpf_class::set_prec_raw (mp_bitcnt_t PREC)
|
Get or set the current precision of an `mpf_class'.
|
|
The restrictions described for `mpf_set_prec_raw' (*note
|
Initializing Floats::) apply to `mpf_class::set_prec_raw'. Note
|
in particular that the `mpf_class' must be restored to it's
|
allocated precision before being destroyed. This must be done by
|
application code, there's no automatic mechanism for it.
|
|
|
File: gmp.info, Node: C++ Interface Random Numbers, Next: C++ Interface Limitations, Prev: C++ Interface Floats, Up: C++ Class Interface
|
|
12.5 C++ Interface Random Numbers
|
=================================
|
|
-- Class: gmp_randclass
|
The C++ class interface to the GMP random number functions uses
|
`gmp_randclass' to hold an algorithm selection and current state,
|
as per `gmp_randstate_t'.
|
|
-- Function: gmp_randclass::gmp_randclass (void (*RANDINIT)
|
(gmp_randstate_t, ...), ...)
|
Construct a `gmp_randclass', using a call to the given RANDINIT
|
function (*note Random State Initialization::). The arguments
|
expected are the same as RANDINIT, but with `mpz_class' instead of
|
`mpz_t'. For example,
|
|
gmp_randclass r1 (gmp_randinit_default);
|
gmp_randclass r2 (gmp_randinit_lc_2exp_size, 32);
|
gmp_randclass r3 (gmp_randinit_lc_2exp, a, c, m2exp);
|
gmp_randclass r4 (gmp_randinit_mt);
|
|
`gmp_randinit_lc_2exp_size' will fail if the size requested is too
|
big, an `std::length_error' exception is thrown in that case.
|
|
-- Function: 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.
|
|
-- Function: void gmp_randclass::seed (unsigned long int S)
|
-- Function: void gmp_randclass::seed (mpz_class S)
|
Seed a random number generator. See *note Random Number
|
Functions::, for how to choose a good seed.
|
|
-- Function: mpz_class gmp_randclass::get_z_bits (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.
|
|
-- Function: mpz_class gmp_randclass::get_z_range (mpz_class N)
|
Generate a random integer in the range 0 to N-1 inclusive.
|
|
-- Function: mpf_class gmp_randclass::get_f ()
|
-- Function: mpf_class gmp_randclass::get_f (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,
|
|
gmp_randclass r;
|
...
|
mpf_class f (0, 512); // 512 bits precision
|
f = r.get_f(); // random number, 512 bits
|
|
|
File: gmp.info, Node: C++ Interface Limitations, Prev: C++ Interface Random Numbers, Up: C++ Class Interface
|
|
12.6 C++ Interface Limitations
|
==============================
|
|
`mpq_class' and Templated Reading
|
A generic piece of template code probably won't know that
|
`mpq_class' requires a `canonicalize' call if inputs read with
|
`operator>>' might be non-canonical. This can lead to incorrect
|
results.
|
|
`operator>>' behaves as it does for reasons of efficiency. A
|
canonicalize can be quite time consuming on large operands, and is
|
best avoided if it's not necessary.
|
|
But this potential difficulty reduces the usefulness of
|
`mpq_class'. Perhaps a mechanism to tell `operator>>' what to do
|
will be adopted in the future, maybe a preprocessor define, a
|
global flag, or an `ios' flag pressed into service. Or maybe, at
|
the risk of inconsistency, the `mpq_class' `operator>>' could
|
canonicalize and leave `mpq_t' `operator>>' not doing so, for use
|
on those occasions when that's acceptable. Send feedback or
|
alternate ideas to <gmp-bugs@gmplib.org>.
|
|
Subclassing
|
Subclassing the GMP C++ classes works, but is not currently
|
recommended.
|
|
Expressions involving subclasses resolve correctly (or seem to),
|
but in normal C++ fashion the subclass doesn't inherit
|
constructors and assignments. There's many of those in the 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,
|
|
template <class T>
|
T fun (const T &, const T &);
|
|
When used with, say, plain `mpz_class' variables, it works fine:
|
`T' is resolved as `mpz_class'.
|
|
mpz_class f(1), g(2);
|
fun (f, g); // Good
|
|
But when one of the arguments is an expression, it doesn't work.
|
|
mpz_class f(1), g(2), h(3);
|
fun (f, g+h); // Bad
|
|
This is because `g+h' ends up being a certain expression template
|
type internal to `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'.
|
|
template <class T>
|
void fun (T f, T g)
|
{
|
fun2 (f, f+g); // Bad
|
}
|
|
template <class T>
|
void fun (T f, T g)
|
{
|
fun2 (f, T(f+g)); // Good
|
}
|
|
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.
|
|
mpz_class z = 33;
|
auto sum = z + z;
|
mpz_class prod = sum * sum;
|
|
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.
|
|
mpz_class z = 33;
|
auto sum = z + z + z;
|
mpz_class prod = sum * 2;
|
|
It is thus strongly recommended to avoid `auto' anywhere a GMP C++
|
expression may appear.
|
|
|
File: gmp.info, Node: Custom Allocation, Next: Language Bindings, Prev: C++ Class Interface, Up: Top
|
|
13 Custom Allocation
|
********************
|
|
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.
|
|
Alternate functions can be specified, to allocate memory in a
|
different way or to have a different error action on running out of
|
memory.
|
|
-- Function: void mp_set_memory_functions (
|
void *(*ALLOC_FUNC_PTR) (size_t),
|
void *(*REALLOC_FUNC_PTR) (void *, size_t, size_t),
|
void (*FREE_FUNC_PTR) (void *, size_t))
|
Replace the current allocation functions from the arguments. If
|
an argument is `NULL', the corresponding default function is used.
|
|
These functions will be used for all memory allocation done by
|
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
|
active GMP objects allocated using the previous memory functions!
|
Usually that means calling it before any other GMP function.*
|
|
The functions supplied should fit the following declarations:
|
|
-- Function: void * allocate_function (size_t ALLOC_SIZE)
|
Return a pointer to newly allocated space with at least ALLOC_SIZE
|
bytes.
|
|
-- Function: void * reallocate_function (void *PTR, size_t OLD_SIZE,
|
size_t NEW_SIZE)
|
Resize a previously allocated block PTR of OLD_SIZE bytes to be
|
NEW_SIZE bytes.
|
|
The block may be moved if necessary or if desired, and in that
|
case the smaller of OLD_SIZE and NEW_SIZE bytes must be copied to
|
the new location. The return value is a pointer to the resized
|
block, that being the new location if moved or just PTR if not.
|
|
PTR is never `NULL', it's always a previously allocated block.
|
NEW_SIZE may be bigger or smaller than OLD_SIZE.
|
|
-- Function: void free_function (void *PTR, size_t SIZE)
|
De-allocate the space pointed to by PTR.
|
|
PTR is never `NULL', it's always a previously allocated block of
|
SIZE bytes.
|
|
A "byte" here means the unit used by the `sizeof' operator.
|
|
The 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'.
|
|
Getting a different fatal error action is a good use for custom
|
allocation functions, for example giving a graphical dialog rather than
|
the default print to `stderr'. How much is possible when genuinely out
|
of memory is another question though.
|
|
There's currently no defined way for the allocation functions to
|
recover from an error such as out of memory, they must terminate
|
program execution. A `longjmp' or throwing a C++ exception will have
|
undefined results. This may change in the future.
|
|
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
|
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
|
this is a problem.
|
|
|
-- Function: void mp_get_memory_functions (
|
void *(**ALLOC_FUNC_PTR) (size_t),
|
void *(**REALLOC_FUNC_PTR) (void *, size_t, size_t),
|
void (**FREE_FUNC_PTR) (void *, size_t))
|
Get the current allocation functions, storing function pointers to
|
the locations given by the arguments. If an argument is `NULL',
|
that function pointer is not stored.
|
|
For example, to get just the current free function,
|
|
void (*freefunc) (void *, size_t);
|
|
mp_get_memory_functions (NULL, NULL, &freefunc);
|
|
|
File: gmp.info, Node: Language Bindings, Next: Algorithms, Prev: Custom Allocation, Up: Top
|
|
14 Language Bindings
|
********************
|
|
The following packages and projects offer access to GMP from languages
|
other than C, though perhaps with varying levels of functionality and
|
efficiency.
|
|
|
C++
|
* GMP C++ class interface, *note C++ Class Interface::
|
Straightforward interface, expression templates to eliminate
|
temporaries.
|
|
* ALP `https://www-sop.inria.fr/saga/logiciels/ALP/'
|
Linear algebra and polynomials using templates.
|
|
* Arithmos `http://cant.ua.ac.be/old/arithmos/'
|
Rationals with infinities and square roots.
|
|
* CLN `http://www.ginac.de/CLN/'
|
High level classes for arithmetic.
|
|
* Linbox `http://www.linalg.org/'
|
Sparse vectors and matrices.
|
|
* NTL `http://www.shoup.net/ntl/'
|
A C++ number theory library.
|
|
Eiffel
|
* Eiffelroom `http://www.eiffelroom.org/node/442'
|
|
Haskell
|
* Glasgow Haskell Compiler `https://www.haskell.org/ghc/'
|
|
Java
|
* Kaffe `https://github.com/kaffe/kaffe'
|
|
Lisp
|
* GNU Common Lisp `https://www.gnu.org/software/gcl/gcl.html'
|
|
* Librep `http://librep.sourceforge.net/'
|
|
* XEmacs (21.5.18 beta and up) `http://www.xemacs.org'
|
Optional big integers, rationals and floats using GMP.
|
|
M4
|
* GNU m4 betas `http://www.seindal.dk/rene/gnu/'
|
Optionally provides an arbitrary precision `mpeval'.
|
|
ML
|
* MLton compiler `http://mlton.org/'
|
|
Objective Caml
|
* MLGMP `http://opam.ocamlpro.com/pkg/mlgmp.20120224.html'
|
|
* Numerix `http://pauillac.inria.fr/~quercia/'
|
Optionally using GMP.
|
|
Oz
|
* Mozart `http://mozart.github.io/'
|
|
Pascal
|
* GNU Pascal Compiler `http://www.gnu-pascal.de/'
|
GMP unit.
|
|
* Numerix `http://pauillac.inria.fr/~quercia/'
|
For Free Pascal, optionally using GMP.
|
|
Perl
|
* GMP module, see `demos/perl' 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/'
|
Plug Math::GMP into normal Math::BigInt operations.
|
|
Pike
|
* mpz module in the standard distribution,
|
`http://pike.ida.liu.se/'
|
|
Prolog
|
* SWI Prolog `http://www.swi-prolog.org/'
|
Arbitrary precision floats.
|
|
Python
|
* GMPY `https://code.google.com/p/gmpy/'
|
|
Ruby
|
* http://rubygems.org/gems/gmp
|
|
Scheme
|
* GNU Guile `https://www.gnu.org/software/guile/guile.html'
|
|
* RScheme `http://www.rscheme.org/'
|
|
* STklos `http://www.stklos.net/'
|
|
Smalltalk
|
* GNU Smalltalk
|
`http://www.smalltalk.org/versions/GNUSmalltalk.html'
|
|
Other
|
* Axiom `https://savannah.nongnu.org/projects/axiom'
|
Computer algebra using GCL.
|
|
* DrGenius `http://drgenius.seul.org/'
|
Geometry system and mathematical programming language.
|
|
* GiNaC `http://www.ginac.de/'
|
C++ computer algebra using CLN.
|
|
* GOO `https://www.eecs.berkeley.edu/~jrb/goo/'
|
Dynamic object oriented language.
|
|
* Maxima `https://www.ma.utexas.edu/users/wfs/maxima.html'
|
Macsyma computer algebra using GCL.
|
|
* Regina `http://regina.sourceforge.net/'
|
Topological calculator.
|
|
* Yacas `http://yacas.sourceforge.net'
|
Yet another computer algebra system.
|
|
|
|
File: gmp.info, Node: Algorithms, Next: Internals, Prev: Language Bindings, Up: Top
|
|
15 Algorithms
|
*************
|
|
This chapter is an introduction to some of the algorithms used for
|
various GMP operations. The code is likely to be hard to understand
|
without knowing something about the algorithms.
|
|
Some GMP internals are mentioned, but applications that expect to be
|
compatible with future GMP releases should take care to use only the
|
documented functions.
|
|
* Menu:
|
|
* Multiplication Algorithms::
|
* Division Algorithms::
|
* Greatest Common Divisor Algorithms::
|
* Powering Algorithms::
|
* Root Extraction Algorithms::
|
* Radix Conversion Algorithms::
|
* Other Algorithms::
|
* Assembly Coding::
|
|
|
File: gmp.info, Node: Multiplication Algorithms, Next: Division Algorithms, Prev: Algorithms, Up: Algorithms
|
|
15.1 Multiplication
|
===================
|
|
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'
|
Toom-6.5 `MUL_TOOM6H_THRESHOLD'
|
Toom-8.5 `MUL_TOOM8H_THRESHOLD'
|
FFT `MUL_FFT_THRESHOLD'
|
|
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
|
algorithms or directly with FFT, depending on operand size (*note
|
Unbalanced Multiplication::).
|
|
* Menu:
|
|
* Basecase Multiplication::
|
* Karatsuba Multiplication::
|
* Toom 3-Way Multiplication::
|
* Toom 4-Way Multiplication::
|
* Higher degree Toom'n'half::
|
* FFT Multiplication::
|
* Other Multiplication::
|
* Unbalanced Multiplication::
|
|
|
File: gmp.info, Node: Basecase Multiplication, Next: Karatsuba Multiplication, Prev: Multiplication Algorithms, Up: Multiplication Algorithms
|
|
15.1.1 Basecase Multiplication
|
------------------------------
|
|
Basecase NxM multiplication is a straightforward rectangular set of
|
cross-products, the same as long multiplication done by hand and for
|
that reason sometimes known as the schoolbook or grammar school method.
|
This is an O(N*M) algorithm. See Knuth section 4.3.1 algorithm M
|
(*note References::), and the `mpn/generic/mul_basecase.c' code.
|
|
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
|
(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
|
implementations take essentially the same approach.
|
|
u0 u1 u2 u3 u4
|
+---+---+---+---+---+
|
u0 | d | | | | |
|
+---+---+---+---+---+
|
u1 | | d | | | |
|
+---+---+---+---+---+
|
u2 | | | d | | |
|
+---+---+---+---+---+
|
u3 | | | | d | |
|
+---+---+---+---+---+
|
u4 | | | | | d |
|
+---+---+---+---+---+
|
|
In practice squaring isn't a full 2x faster than multiplying, it's
|
usually around 1.5x. Less than 1.5x probably indicates
|
`mpn_sqr_basecase' wants improving on that CPU.
|
|
On some CPUs `mpn_mul_basecase' can be faster than the generic C
|
`mpn_sqr_basecase' on some small sizes. `SQR_BASECASE_THRESHOLD' is
|
the size at which to use `mpn_sqr_basecase', this will be zero if that
|
routine should be used always.
|
|
|
File: gmp.info, Node: Karatsuba Multiplication, Next: Toom 3-Way Multiplication, Prev: Basecase Multiplication, Up: Multiplication Algorithms
|
|
15.1.2 Karatsuba Multiplication
|
-------------------------------
|
|
The Karatsuba multiplication algorithm is described in Knuth section
|
4.3.3 part A, and various other textbooks. A brief description is
|
given here.
|
|
The inputs x and y are treated as each split into two parts of equal
|
length (or the most significant part one limb shorter if N is odd).
|
|
high low
|
+----------+----------+
|
| x1 | x0 |
|
+----------+----------+
|
|
+----------+----------+
|
| y1 | y0 |
|
+----------+----------+
|
|
Let b be the power of 2 where the split occurs, 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,
|
|
x*y = (b^2+b)*x1*y1 - b*(x1-x0)*(y1-y0) + (b+1)*x0*y0
|
|
This formula means doing only three multiplies of (N/2)x(N/2) limbs,
|
whereas a basecase multiply of NxN limbs is equivalent to four
|
multiplies of (N/2)x(N/2). The factors (b^2+b) etc represent the
|
positions where the three products must be added.
|
|
high low
|
+--------+--------+ +--------+--------+
|
| x1*y1 | | x0*y0 |
|
+--------+--------+ +--------+--------+
|
+--------+--------+
|
add | x1*y1 |
|
+--------+--------+
|
+--------+--------+
|
add | x0*y0 |
|
+--------+--------+
|
+--------+--------+
|
sub | (x1-x0)*(y1-y0) |
|
+--------+--------+
|
|
The term (x1-x0)*(y1-y0) is best calculated as an absolute value,
|
and the sign used to choose to add or subtract. Notice the sum
|
high(x0*y0)+low(x1*y1) occurs twice, so it's possible to do 5*k limb
|
additions, rather than 6*k, but in GMP extra function call overheads
|
outweigh the saving.
|
|
Squaring is similar to multiplying, but with x=y the formula reduces
|
to an equivalent with three squares,
|
|
x^2 = (b^2+b)*x1^2 - b*(x1-x0)^2 + (b+1)*x0^2
|
|
The final result is accumulated from those three squares the same
|
way as for the three multiplies above. The middle term (x1-x0)^2 is now
|
always positive.
|
|
A similar formula for both multiplying and squaring can be
|
constructed with a middle term (x1+x0)*(y1+y0). But those sums can
|
exceed k limbs, leading to more carry handling and additions than the
|
form above.
|
|
Karatsuba multiplication is asymptotically an O(N^1.585) algorithm,
|
the exponent being log(3)/log(2), representing 3 multiplies each 1/2
|
the size of the inputs. This is a big improvement over the basecase
|
multiply at O(N^2) and the advantage soon overcomes the extra additions
|
Karatsuba performs. `MUL_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
|
that sense the algorithm thresholds are merely of academic interest.
|
|
|
File: gmp.info, Node: Toom 3-Way Multiplication, Next: Toom 4-Way Multiplication, Prev: Karatsuba Multiplication, Up: Multiplication Algorithms
|
|
15.1.3 Toom 3-Way Multiplication
|
--------------------------------
|
|
The Karatsuba formula is the simplest case of a general approach to
|
splitting inputs that leads to both Toom and FFT algorithms. A
|
description of Toom can be found in Knuth section 4.3.3, with an
|
example 3-way calculation after Theorem A. The 3-way form used in 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).
|
|
high low
|
+----------+----------+----------+
|
| x2 | x1 | x0 |
|
+----------+----------+----------+
|
|
+----------+----------+----------+
|
| y2 | y1 | y0 |
|
+----------+----------+----------+
|
|
These parts are treated as the coefficients of two polynomials
|
|
X(t) = x2*t^2 + x1*t + x0
|
Y(t) = y2*t^2 + y1*t + y0
|
|
Let b equal the power of 2 which is the size of the x0, x1, y0 and
|
y1 pieces, 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
|
|
W(t) = w4*t^4 + w3*t^3 + w2*t^2 + w1*t + w0
|
|
The w[i] are going to be determined, and when they are they'll give
|
the final result using w=W(b), since x*y=X(b)*Y(b)=W(b). The
|
coefficients will be roughly b^2 each, and the final W(b) will be an
|
addition like,
|
|
high low
|
+-------+-------+
|
| w4 |
|
+-------+-------+
|
+--------+-------+
|
| w3 |
|
+--------+-------+
|
+--------+-------+
|
| w2 |
|
+--------+-------+
|
+--------+-------+
|
| w1 |
|
+--------+-------+
|
+-------+-------+
|
| w0 |
|
+-------+-------+
|
|
The w[i] coefficients could be formed by a simple set of cross
|
products, like w4=x2*y2, w3=x2*y1+x1*y2, w2=x2*y0+x1*y1+x0*y2 etc, but
|
this would need all nine x[i]*y[j] for i,j=0,1,2, and would be
|
equivalent merely to a basecase multiply. Instead the following
|
approach is used.
|
|
X(t) and Y(t) are evaluated and multiplied at 5 points, giving
|
values of W(t) at those points. In GMP the following points are used,
|
|
Point Value
|
t=0 x0 * y0, which gives w0 immediately
|
t=1 (x2+x1+x0) * (y2+y1+y0)
|
t=-1 (x2-x1+x0) * (y2-y1+y0)
|
t=2 (4*x2+2*x1+x0) * (4*y2+2*y1+y0)
|
t=inf x2 * y2, which gives w4 immediately
|
|
At t=-1 the values can be negative and that's handled using the
|
absolute values and tracking the sign separately. At t=inf the value
|
is actually X(t)*Y(t)/t^4 in the limit as t approaches infinity, but
|
it's much easier to think of as simply x2*y2 giving w4 immediately
|
(much like x0*y0 at t=0 gives w0 immediately).
|
|
Each of the points substituted into W(t)=w4*t^4+...+w0 gives a
|
linear combination of the w[i] coefficients, and the value of those
|
combinations has just been calculated.
|
|
W(0) = w0
|
W(1) = w4 + w3 + w2 + w1 + w0
|
W(-1) = w4 - w3 + w2 - w1 + w0
|
W(2) = 16*w4 + 8*w3 + 4*w2 + 2*w1 + w0
|
W(inf) = w4
|
|
This is a set of five equations in five unknowns, and some
|
elementary linear algebra quickly isolates each w[i]. This involves
|
adding or subtracting one W(t) value from another, and a couple of
|
divisions by powers of 2 and one division by 3, the latter using the
|
special `mpn_divexact_by3' (*note Exact Division::).
|
|
The conversion of W(t) values to the coefficients is interpolation.
|
A polynomial of degree 4 like W(t) is uniquely determined by values
|
known at 5 different points. The points are arbitrary and can be
|
chosen to make the linear equations come out with a convenient set of
|
steps for quickly isolating the w[i].
|
|
Squaring follows the same procedure as multiplication, but there's
|
only one X(t) and it's evaluated at the 5 points, and those values
|
squared to give values of W(t). The interpolation is then identical,
|
and in fact the same `toom_interpolate_5pts' subroutine is used for
|
both squaring and multiplying.
|
|
Toom-3 is asymptotically O(N^1.465), the exponent being
|
log(5)/log(3), representing 5 recursive multiplies of 1/3 the original
|
size each. This is an improvement over Karatsuba at O(N^1.585), though
|
Toom does more work in the evaluation and interpolation and so it only
|
realizes its advantage above a certain size.
|
|
Near the crossover between Toom-3 and Karatsuba there's generally a
|
range of sizes where the difference between the two is small.
|
`MUL_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'.
|
|
At the fairly small sizes where the Toom-3 thresholds occur it's
|
worth remembering that the asymptotic behaviour for Karatsuba and
|
Toom-3 can't be expected to make accurate predictions, due of course to
|
the big influence of all sorts of overheads, and the fact that only a
|
few recursions of each are being performed. Even at large sizes
|
there's a good chance machine dependent effects like cache architecture
|
will mean actual performance deviates from what might be predicted.
|
|
The formula given for the Karatsuba algorithm (*note Karatsuba
|
Multiplication::) has an equivalent for Toom-3 involving only five
|
multiplies, but this would be complicated and unenlightening.
|
|
An alternate view of Toom-3 can be found in Zuras (*note
|
References::), using a vector to represent the x and y splits and a
|
matrix multiplication for the evaluation and interpolation stages. The
|
matrix inverses are not meant to be actually used, and they have
|
elements with values much greater than in fact arise in the
|
interpolation steps. The diagram shown for the 3-way is attractive,
|
but again doesn't have to be implemented that way and for example with
|
a bit of rearrangement just one division by 6 can be done.
|
|
|
File: gmp.info, Node: Toom 4-Way Multiplication, Next: Higher degree Toom'n'half, Prev: Toom 3-Way Multiplication, Up: Multiplication Algorithms
|
|
15.1.4 Toom 4-Way Multiplication
|
--------------------------------
|
|
Karatsuba and Toom-3 split the operands into 2 and 3 coefficients,
|
respectively. Toom-4 analogously splits the operands into 4
|
coefficients. Using the notation from the section on Toom-3
|
multiplication, we form two polynomials:
|
|
X(t) = x3*t^3 + x2*t^2 + x1*t + x0
|
Y(t) = y3*t^3 + y2*t^2 + y1*t + y0
|
|
X(t) and Y(t) are evaluated and multiplied at 7 points, giving
|
values of W(t) at those points. In GMP the following points are used,
|
|
Point Value
|
t=0 x0 * y0, which gives w0 immediately
|
t=1/2 (x3+2*x2+4*x1+8*x0) * (y3+2*y2+4*y1+8*y0)
|
t=-1/2 (-x3+2*x2-4*x1+8*x0) * (-y3+2*y2-4*y1+8*y0)
|
t=1 (x3+x2+x1+x0) * (y3+y2+y1+y0)
|
t=-1 (-x3+x2-x1+x0) * (-y3+y2-y1+y0)
|
t=2 (8*x3+4*x2+2*x1+x0) * (8*y3+4*y2+2*y1+y0)
|
t=inf x3 * y3, which gives w6 immediately
|
|
The number of additions and subtractions for Toom-4 is much larger
|
than for Toom-3. But several subexpressions occur multiple times, for
|
example x2+x0, occurs for both t=1 and t=-1.
|
|
Toom-4 is asymptotically O(N^1.404), the exponent being
|
log(7)/log(4), representing 7 recursive multiplies of 1/4 the original
|
size each.
|
|
|
File: 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 four-plets 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 equanl in size to the result of the product.
|
|
Current GMP uses both Toom-6'n'half and Toom-8'n'half.
|
|
|
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.
|
|
The multiplication done is x*y mod 2^N+1, for a given N. A full
|
product x*y is obtained by choosing N>=bits(x)+bits(y) and padding x
|
and y with high zero limbs. The modular product is the native form for
|
the algorithm, so padding to get a full product is unavoidable.
|
|
The algorithm follows a split, evaluate, pointwise multiply,
|
interpolate and combine similar to that described above for Karatsuba
|
and Toom-3. A k parameter controls the split, with an FFT-k splitting
|
into 2^k pieces of M=N/2^k bits each. N must be a multiple of
|
(2^k)*mp_bits_per_limb so the split falls on limb boundaries, avoiding
|
bit shifts in the split and combine stages.
|
|
The evaluations, pointwise multiplications, and interpolation, are
|
all done modulo 2^N'+1 where N' is 2M+k+3 rounded up to a multiple of
|
2^k and of `mp_bits_per_limb'. The results of interpolation will be
|
the following negacyclic convolution of the input pieces, and the
|
choice of N' ensures these sums aren't truncated.
|
|
---
|
\ b
|
w[n] = / (-1) * x[i] * y[j]
|
---
|
i+j==b*2^k+n
|
b=0,1
|
|
The points used for the evaluation are g^i for i=0 to 2^k-1 where
|
g=2^(2N'/2^k). g is a 2^k'th root of unity mod 2^N'+1, which produces
|
necessary cancellations at the interpolation stage, and it's also a
|
power of 2 so the fast Fourier transforms used for the evaluation and
|
interpolation do only shifts, adds and negations.
|
|
The pointwise multiplications are done modulo 2^N'+1 and either
|
recurse into a further FFT or use a plain multiplication (Toom-3,
|
Karatsuba or basecase), whichever is optimal at the size N'. The
|
interpolation is an inverse fast Fourier transform. The resulting set
|
of sums of x[i]*y[j] are added at appropriate offsets to give the final
|
result.
|
|
Squaring is the same, but x is the only input so it's one transform
|
at the evaluate stage and the pointwise multiplies are squares. The
|
interpolation is the same.
|
|
For a mod 2^N+1 product, an FFT-k is an O(N^(k/(k-1))) algorithm,
|
the exponent representing 2^k recursed modular multiplies each
|
1/2^(k-1) the size of the original. Each successive k is an asymptotic
|
improvement, but overheads mean each is only faster at bigger and
|
bigger sizes. In the code, `MUL_FFT_TABLE' and `SQR_FFT_TABLE' are the
|
thresholds where each k is used. Each new k effectively swaps some
|
multiplying for some shifts, adds and overheads.
|
|
A mod 2^N+1 product can be formed with a normal NxN->2N bit multiply
|
plus a subtraction, so an FFT and Toom-3 etc can be compared directly.
|
A k=4 FFT at O(N^1.333) can be expected to be the first faster than
|
Toom-3 at O(N^1.465). In practice this is what's found, with
|
`MUL_FFT_MODF_THRESHOLD' and `SQR_FFT_MODF_THRESHOLD' being between 300
|
and 1000 limbs, depending on the CPU. So far it's been found that only
|
very large FFTs recurse into pointwise multiplies above these sizes.
|
|
When an FFT is to give a full product, the change of N to 2N doesn't
|
alter the theoretical complexity for a given k, but for the purposes of
|
considering where an FFT might be first used it can be assumed that the
|
FFT is recursing into a normal multiply and that on that basis it's
|
doing 2^k recursed multiplies each 1/2^(k-2) the size of the inputs,
|
making it O(N^(k/(k-2))). This would mean k=7 at O(N^1.4) would be the
|
first FFT faster than Toom-3. In practice `MUL_FFT_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 +k+3 means some values of N just under such a multiple will be
|
rounded to the next. The complexity calculations above assume that a
|
favourable size is used, meaning one which isn't padded through
|
rounding, and it's also assumed that the extra +k+3 bits are negligible
|
at typical FFT sizes.
|
|
The practical effect of the 2^(2k-1) constraint is to introduce a
|
step-effect into measured speeds. For example k=8 will round N up to a
|
multiple of 32768 bits, so for a 32-bit limb there'll be 512 limb
|
groups of sizes for which `mpn_mul_n' runs at the same speed. Or for
|
k=9 groups of 2048 limbs, k=10 groups of 8192 limbs, etc. In practice
|
it's been found each k is used at quite small multiples of its size
|
constraint and so the step effect is quite noticeable in a time versus
|
size graph.
|
|
The threshold determinations currently measure at the mid-points of
|
size steps, but this is sub-optimal since at the start of a new step it
|
can happen that it's better to go back to the previous k for a while.
|
Something more sophisticated for `MUL_FFT_TABLE' and `SQR_FFT_TABLE'
|
will be needed.
|
|
|
File: gmp.info, Node: Other Multiplication, Next: Unbalanced Multiplication, Prev: FFT Multiplication, Up: Multiplication Algorithms
|
|
15.1.7 Other Multiplication
|
---------------------------
|
|
The Toom algorithms described above (*note Toom 3-Way Multiplication::,
|
*note Toom 4-Way Multiplication::) generalizes to split into an
|
arbitrary number of pieces, as per Knuth section 4.3.3 algorithm C.
|
This is not currently used. The notes here are merely for interest.
|
|
In general a split into r+1 pieces is made, and evaluations and
|
pointwise multiplications done at 2*r+1 points. A 4-way split does 7
|
pointwise multiplies, 5-way does 9, etc. Asymptotically an (r+1)-way
|
algorithm is O(N^(log(2*r+1)/log(r+1))). Only the pointwise
|
multiplications count towards big-O complexity, but the time spent in
|
the evaluate and interpolate stages grows with r and has a significant
|
practical impact, with the asymptotic advantage of each r realized only
|
at bigger and bigger sizes. The overheads grow as O(N*r), whereas in
|
an r=2^k FFT they grow only as O(N*log(r)).
|
|
Knuth algorithm C evaluates at points 0,1,2,...,2*r, but exercise 4
|
uses -r,...,0,...,r and the latter saves some small multiplies in the
|
evaluate stage (or rather trades them for additions), and has a further
|
saving of nearly half the interpolate steps. The idea is to separate
|
odd and even final coefficients and then perform algorithm C steps C7
|
and C8 on them separately. The divisors at step C7 become j^2 and the
|
multipliers at C8 become 2*t*j-j^2.
|
|
Splitting odd and even parts through positive and negative points
|
can be thought of as using -1 as a square root of unity. If a 4th root
|
of unity was available then a further split and speedup would be
|
possible, but no such root exists for plain integers. Going to complex
|
integers with i=sqrt(-1) doesn't help, essentially because in Cartesian
|
form it takes three real multiplies to do a complex multiply. The
|
existence of 2^k'th roots of unity in a suitable ring or field lets the
|
fast Fourier transform keep splitting and get to O(N*log(r)).
|
|
Floating point FFTs use complex numbers approximating Nth roots of
|
unity. Some processors have special support for such FFTs. But these
|
are not used in GMP since it's very difficult to guarantee an exact
|
result (to some number of bits). An occasional difference of 1 in the
|
last bit might not matter to a typical signal processing algorithm, but
|
is of course of vital importance to GMP.
|
|
|
File: gmp.info, Node: Unbalanced Multiplication, Prev: Other Multiplication, Up: Multiplication Algorithms
|
|
15.1.8 Unbalanced Multiplication
|
--------------------------------
|
|
Multiplication of operands with different sizes, both below
|
`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.
|
|
|
File: gmp.info, Node: Division Algorithms, Next: Greatest Common Divisor Algorithms, Prev: Multiplication Algorithms, Up: Algorithms
|
|
15.2 Division Algorithms
|
========================
|
|
* Menu:
|
|
* Single Limb Division::
|
* Basecase Division::
|
* Divide and Conquer Division::
|
* Block-Wise Barrett Division::
|
* Exact Division::
|
* Exact Remainder::
|
* Small Quotient Division::
|
|
|
File: gmp.info, Node: Single Limb Division, Next: Basecase Division, Prev: Division Algorithms, Up: Division Algorithms
|
|
15.2.1 Single Limb Division
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---------------------------
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Nx1 division is implemented using repeated 2x1 divisions from high to
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low, either with a hardware divide instruction or a multiplication by
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inverse, whichever is best on a given CPU.
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The multiply by inverse follows "Improved division by invariant
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integers" by Mo"ller and Granlund (*note References::) and is
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implemented as `udiv_qrnnd_preinv' in `gmp-impl.h'. The idea is to
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have a fixed-point approximation to 1/d (see `invert_limb') and then
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multiply by the high limb (plus one bit) of the dividend to get a
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quotient q. With d normalized (high bit set), q is no more than 1 too
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small. Subtracting q*d from the dividend gives a remainder, and
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reveals whether q or q-1 is correct.
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The result is a division done with two multiplications and four or
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five arithmetic operations. On CPUs with low latency multipliers this
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can be much faster than a hardware divide, though the cost of
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calculating the inverse at the start may mean it's only better on
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inputs bigger than say 4 or 5 limbs.
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When a divisor must be normalized, either for the generic C
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`__udiv_qrnnd_c' or the multiply by inverse, the division performed is
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actually a*2^k by d*2^k where a is the dividend and k is the power
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necessary to have the high bit of d*2^k set. The bit shifts for the
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dividend are usually accomplished "on the fly" meaning by extracting
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the appropriate bits at each step. Done this way the quotient limbs
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come out aligned ready to store. When only the remainder is wanted, an
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alternative is to take the dividend limbs unshifted and calculate r = a
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mod d*2^k followed by an extra final step r*2^k mod d*2^k. This can
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help on CPUs with poor bit shifts or few registers.
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The multiply by inverse can be done two limbs at a time. The
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calculation is basically the same, but the inverse is two limbs and the
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divisor treated as if padded with a low zero limb. This means more
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work, since the inverse will need a 2x2 multiply, but the four 1x1s to
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do that are independent and can therefore be done partly or wholly in
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parallel. Likewise for a 2x1 calculating q*d. The net effect is to
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process two limbs with roughly the same two multiplies worth of latency
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that one limb at a time gives. This extends to 3 or 4 limbs at a time,
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though the extra work to apply the inverse will almost certainly soon
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reach the limits of multiplier throughput.
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A similar approach in reverse can be taken to process just half a
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limb at a time if the divisor is only a half limb. In this case the
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1x1 multiply for the inverse effectively becomes two (1/2)x1 for each
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limb, which can be a saving on CPUs with a fast half limb multiply, or
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in fact if the only multiply is a half limb, and especially if it's not
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pipelined.
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File: gmp.info, Node: Basecase Division, Next: Divide and Conquer Division, Prev: Single Limb Division, Up: Division Algorithms
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15.2.2 Basecase Division
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------------------------
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Basecase NxM division is like long division done by hand, but in base
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2^mp_bits_per_limb. See Knuth section 4.3.1 algorithm D, and
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`mpn/generic/sb_divrem_mn.c'.
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Briefly stated, while the dividend remains larger than the divisor,
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a high quotient limb is formed and the Nx1 product q*d subtracted at
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the top end of the dividend. With a normalized divisor (most
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significant bit set), each quotient limb can be formed with a 2x1
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division and a 1x1 multiplication plus some subtractions. The 2x1
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division is by the high limb of the divisor and is done either with a
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hardware divide or a multiply by inverse (the same as in *Note Single
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Limb Division::) whichever is faster. Such a quotient is sometimes one
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too big, requiring an addback of the divisor, but that happens rarely.
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With Q=N-M being the number of quotient limbs, this is an O(Q*M)
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algorithm and will run at a speed similar to a basecase QxM
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multiplication, differing in fact only in the extra multiply and divide
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for each of the Q quotient limbs.
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File: gmp.info, Node: Divide and Conquer Division, Next: Block-Wise Barrett Division, Prev: Basecase Division, Up: Division Algorithms
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15.2.3 Divide and Conquer Division
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----------------------------------
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For divisors larger than `DC_DIV_QR_THRESHOLD', division is done by
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dividing. Or to be precise by a recursive divide and conquer algorithm
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based on work by Moenck and Borodin, Jebelean, and Burnikel and Ziegler
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(*note References::).
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The algorithm consists essentially of recognising that a 2NxN
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division can be done with the basecase division algorithm (*note
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Basecase Division::), but using N/2 limbs as a base, not just a single
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limb. This way the multiplications that arise are (N/2)x(N/2) and can
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take advantage of Karatsuba and higher multiplication algorithms (*note
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Multiplication Algorithms::). The two "digits" of the quotient are
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formed by recursive Nx(N/2) divisions.
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If the (N/2)x(N/2) multiplies are done with a basecase multiplication
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then the work is about the same as a basecase division, but with more
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function call overheads and with some subtractions separated from the
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multiplies. These overheads mean that it's only when N/2 is above
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`MUL_TOOM22_THRESHOLD' that divide and conquer is of use.
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`DC_DIV_QR_THRESHOLD' is based on the divisor size N, so it will be
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somewhere above twice `MUL_TOOM22_THRESHOLD', but how much above
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depends on the CPU. An optimized `mpn_mul_basecase' can lower
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`DC_DIV_QR_THRESHOLD' a little by offering a ready-made advantage over
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repeated `mpn_submul_1' calls.
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Divide and conquer is asymptotically O(M(N)*log(N)) where M(N) is
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the time for an NxN multiplication done with FFTs. The actual time is
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a sum over multiplications of the recursed sizes, as can be seen near
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the end of section 2.2 of Burnikel and Ziegler. For example, within
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the Toom-3 range, divide and conquer is 2.63*M(N). With higher
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algorithms the M(N) term improves and the multiplier tends to log(N).
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In practice, at moderate to large sizes, a 2NxN division is about 2 to
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4 times slower than an NxN multiplication.
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File: gmp.info, Node: Block-Wise Barrett Division, Next: Exact Division, Prev: Divide and Conquer Division, Up: Division Algorithms
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15.2.4 Block-Wise Barrett Division
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----------------------------------
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For the largest divisions, a block-wise Barrett division algorithm is
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used. Here, the divisor is inverted to a precision determined by the
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relative size of the dividend and divisor. Blocks of quotient limbs
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are then generated by multiplying blocks from the dividend by the
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inverse.
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Our block-wise algorithm computes a smaller inverse than in the
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plain Barrett algorithm. For a 2n/n division, the inverse will be just
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ceil(n/2) limbs.
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