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Copyright 1991, 1993, 1994, 1995, 1996, 1997, 1998, 1999, 2000, 2001, 2002 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 1.1 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 C. GNU Free Documentation License.
GNU MP Copying Conditions GMP Copying Conditions (LGPL). 1. Introduction to GNU MP Brief introduction to GNU MP. 2. Installing GMP How to configure and compile the GMP library. 3. GMP Basics What every GMP user should know. 4. Reporting Bugs How to usefully report bugs. 5. Integer Functions Functions for arithmetic on signed integers. 6. Rational Number Functions Functions for arithmetic on rational numbers. 7. Floating-point Functions Functions for arithmetic on floats. 8. Low-level Functions Fast functions for natural numbers. 9. Random Number Functions Functions for generating random numbers. 10. Formatted Output printfstyle output.11. Formatted Input scanfstyle input.12. C++ Class Interface Class wrappers around GMP types. 13. Berkeley MP Compatible Functions All functions found in BSD MP. 14. Custom Allocation How to customize the internal allocation. 15. Language Bindings Using GMP from other languages. 16. Algorithms What happens behind the scenes. 17. Internals How values are represented behind the scenes.
A. Contributors Who brings your this library? B. References Some useful papers and books to read. C. GNU Free Documentation License Concept Index Function and Type Index
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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.
The precise conditions of the license for the GNU MP library are found in the Lesser General Public License version 2.1 that accompanies the source code, see `COPYING.LIB'. Certain demonstration programs are provided under the terms of the plain General Public License version 2, see `COPYING'.
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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 carefully optimized assembly code for these CPUs: ARM, DEC Alpha 21064, 21164, and 21264, AMD 29000, AMD K6, K6-2 and Athlon, Hitachi SuperH and SH-2, HPPA 1.0, 1.1 and 2.0, Intel Pentium, Pentium Pro/II/III, Pentium 4, generic x86, Intel IA-64, i960, Motorola MC68000, MC68020, MC88100, and MC88110, Motorola/IBM PowerPC 32 and 64, National NS32000, IBM POWER, MIPS R3000, R4000, SPARCv7, SuperSPARC, generic SPARCv8, UltraSPARC, DEC VAX, and Zilog Z8000. Some optimizations also for Cray vector systems, Clipper, IBM ROMP (RT), and Pyramid AP/XP.
There is a mailing list for GMP users. To join it, send a mail to gmp-request@swox.com with the word `subscribe' in the message body (not in the subject line).
For up-to-date information on GMP, please see the GMP web pages at
http://swox.com/gmp/ |
The latest version of the library is available at
ftp://ftp.gnu.org/gnu/gmp |
Many sites around the world mirror `ftp.gnu.org', please use a mirror near you, see http://www.gnu.org/order/ftp.html for a full list.
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Everyone should read 3. GMP Basics. If you need to install the library yourself, then read 2. Installing GMP. If you have a system with multiple ABIs, then read 2.2 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.
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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 bug-gmp@gnu.org. See 4. Reporting Bugs, for information on what to include in useful bug reports.
2.1 Build Options 2.2 ABI and ISA 2.3 Notes for Package Builds 2.4 Notes for Particular Systems 2.5 Known Build Problems
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All the usual autoconf configure options are available, run `./configure --help' for a summary. The file `INSTALL.autoconf' has some generic installation information too.
`configure' requires various Unix-like tools. On an MS-DOS system Cygwin, DJGPP or MINGW can be used. See
http://www.cygnus.com/cygwin http://www.delorie.com/djgpp http://www.mingw.org |
The `macos' directory contains an unsupported port to MacOS 9 on Power Macintosh. Note that MacOS X "Darwin" can use the normal `./configure'.
It might be possible to build without the help of `configure', certainly all the code is there, but unfortunately you'll be on your own.
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-4.0.1/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.
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.
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.
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 checked for, 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 with 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.
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.)
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.in' for details of what code and compiler options they select.
CPUs not listed will use generic C code.
If some of the assembly code causes problems, or if otherwise desired, the generic C code can be selected with CPU `none'. For example,
./configure --build=none-unknown-freebsd3.5 |
Note that this will run quite slowly, but it should be portable and should at least make it possible to get something running if all else fails.
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 --build=mips64-sgi-irix6 ABI=n32 |
See 2.2 ABI and ISA, for the available choices on relevant CPUs, and what applications need to do.
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 some 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 assembler 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).
Any flags like `-D' defines or `-I' includes required by the preprocessor should be set in `CPPFLAGS' rather than `CFLAGS'. Compiling is done with both `CPPFLAGS' and `CFLAGS', but preprocessing uses just `CPPFLAGS'. This distinction is because most preprocessors won't accept all the flags the compiler does. Preprocessing is done separately in some configure tests, and in the `ansi2knr' support for K&R compilers.
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.
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 object files.
GMP allocates temporary workspace using one of the following three methods, which can be selected with for instance `--enable-alloca=malloc-reentrant'.
For convenience, the following choices are also available. `--disable-alloca' is the same as `--enable-alloca=no'.
alloca if available, otherwise
`malloc-reentrant'. This is the default.
alloca if available, otherwise
`malloc-notreentrant'.
alloca is reentrant and fast, and is recommended, but when working with
large numbers it can overflow the available stack space, in which case one of
the two malloc methods will need to be used. Alternately it might be possible
to increase available stack with limit, ulimit or
setrlimit, or under DJGPP with stubedit or
_stklen. Note that depending on the system the only indication of
stack overflow might be a segmentation violation.
`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. See section 14. Custom Allocation.
An additional choice `--enable-alloca=debug' is available, to help when debugging memory related problems (see section 3.10 Debugging).
By default multiplications are done using Karatsuba, 3-way Toom-Cook, and Fermat FFT. The FFT is only used on large to very large operands and can be disabled to save code size if desired.
The Berkeley MP compatibility library (`libmp') and header file (`mp.h') are built and installed only if `--enable-mpbsd' is used. See section 13. Berkeley MP Compatible Functions.
The optional MPFR functions are built and installed only if `--enable-mpfr' is used. These are in a separate library `libmpfr.a' and are documented separately too (see section `Introduction to MPFR' in MPFR).
This option enables some consistency checking within the library. This can be of use while debugging, see section 3.10 Debugging.
Profiling support can be enabled either for prof or gprof.
This adds `-p' or `-pg' respectively to `CFLAGS', and for some
systems adds corresponding mcount calls to the assembler code.
See section 3.11 Profiling.
Various assembler versions of mpn subroutines are provided, and, for a given CPU, a search is made though a path to choose a version of each. For example `sparcv8' has path `sparc32/v8 sparc32 generic', which means it looks first for v8 code, then plain sparc32, and finally falls back on generic C. Knowledgeable users with special requirements can specify a path with `MPN_PATH="dir list"'. This will normally be unnecessary because all sensible paths should be available under one or other CPU.
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 document you're now reading is `gmp.texi'. The usual automake targets are available to make PostScript `gmp.ps' and/or DVI `gmp.dvi'.
HTML can be produced with `makeinfo --html', see section `Generating HTML' in Texinfo. Or alternately `texi2html', see section `About' in Texinfo To HTML.
PDF can be produced with `texi2dvi --pdf' (see section `PDF Output' in Texinfo) or with `pdftex'.
Some supplementary notes can be found in the `doc' subdirectory.
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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.
The 2.0w ABI uses 64-bit limbs and pointers and is available on HP-UX 11 or up
when using cc. gcc support for this is in progress.
Applications must be compiled with
cc +DD64 |
The 2.0n ABI means the 32-bit HPPA 1.0 ABI but with a 64-bit limb using
long long. This is available on HP-UX 10 or up when using
cc. No gcc support is planned for this. Applications
must be compiled with
cc +DA2.0 +e |
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 CPUs `hppa2.0w' and `hppa2.0', but for CPU `hppa2.0n' only 2.0n or 1.0 are allowed.
IRIX 6 supports the n32 and 64 ABIs and always has a 64-bit MIPS 3 or better
CPU. In both these ABIs GMP uses a 64-bit limb. A new enough gcc
is required (2.95 for instance).
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 |
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.
The AIX 64 ABI uses 64-bit limbs and pointers and is available on systems `powerpc64*-*-aix*'. Applications must be compiled (and linked) with
gcc -maix64 xlc -q64 |
This uses the 32-bit ABI but a 64-bit limb using GCC long long in
64-bit registers. Applications must be compiled with
gcc -mpowerpc64 |
This is the basic 32-bit PowerPC ABI. No special compiler options are needed for applications.
The 64-bit V9 ABI is available on Solaris 2.7 and up and GNU/Linux. GCC 2.95
or up, or Sun cc is required. Applications must be compiled with
gcc -m64 -mptr64 -Wa,-xarch=v9 -mcpu=v9 cc -xarch=v9 |
On Solaris 2.6 and earlier, and on Solaris 2.7 with the kernel in 32-bit mode, only the plain V8 32-bit ABI can be used, since the kernel doesn't save all registers. GMP still uses as much of the V9 ISA as it can in these circumstances. No special compiler options are required for applications, though using something like the following requesting V9 code within the V8 ABI is recommended.
gcc -mv8plus cc -xarch=v8plus |
gcc 2.8 and earlier only supports `-mv8' though.
Don't be confused by the names of these sparc `-m' and `-x' options, they're called `arch' but they effectively control the ABI.
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 |
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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. 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', see section 2.1 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 necessitate `i386' for x86s, or plain `sparc' (meaning V7) for SPARCs.
Users who care about speed will want GMP built for their exact CPU type, to make 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.
Note that `gmp.h' is a generated file, and will be architecture and ABI dependent.
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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.
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.
./configure --disable-static --enable-shared |
Static and DLL libraries can't both be built, since certain export directives in `gmp.h' must be different. `--enable-cxx' cannot be used when building a DLL, since libtool doesn't currently support C++ DLLs. This might change in the future.
GCC is recommended for compiling GMP, but the resulting DLL can be used with any compiler. On mingw only the standard Windows libraries will be needed, on Cygwin the usual cygwin runtime will be required.
`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'.
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.
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 assembler code support for using
just the common instruction subset. To get executables that run on both, the
current suggestion is to use the generic C code (CPU `none'), possibly
with appropriate compiler options (like `-mcpu=common' for
gcc). CPU `rs6000' (which is not a CPU but a family of
workstations) is accepted by `config.sub', but is currently equivalent to
`none'.
`sparcv8' or `supersparc' on relevant systems will give a significant performance increase over the V7 code.
/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).
`i386' selects generic code which will run reasonably well on all x86 chips.
`i586', `pentium' or `pentiummmx' code is good for the 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.
`pentium4' and an SSE2 capable assembler are important for best results on Pentium 4. The specific code is for instance roughly a 2x{} to 3x 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 doesn't (and unfortunately there's no newer assembler for that system).
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.
GCC 2.95.2 and 2.95.3 miscompiled some versions of `mpz/powm.c' when
`-march=pentiumpro' was used, so for relevant CPUs that option is only in
the default CFLAGS for GCC 2.95.4 and up.
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You might find more up-to-date information at http://swox.com/gmp/.
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. A separate `make libgmp.la' helped, perhaps recursing into the various subdirectories uses up memory.
strip
GNU binutils strip should not be used on the static libraries
`libgmp.a' and `libmp.a', neither directly nor via `make
install-strip'. It can be used on the shared libraries `libgmp.so' and
`libmp.so' though.
Currently (binutils 2.10.0), strip unpacks an archive then operates
on the files, but GMP contains multiple object files of the same name
(eg. three versions of `init.o'), and they overwrite each other, leaving
only the one that happens to be last.
If stripped static libraries are wanted, the suggested workaround is to build normally, strip the separate object files, and do another `make all' to rebuild. Alternately `CFLAGS' with `-g' omitted can always be used if it's just debugging which is unwanted.
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.)
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).
Use the GNU assembler instead of the system assembler, since the latter has serious bugs.
The system sed prints an error "Output line too long" when libtool
builds `libgmp.la'. This doesn't seem cause any obvious ill effects, but
GNU sed is recommended, to avoid any doubt.
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.
When creating a DLL version of `libgmp', libtool creates wrapper scripts
like `t-mul' for programs that would normally be `t-mul.exe', in
order to setup the right library paths etc. This works fine, but the absence
of `t-mul.exe' etc causes make to think they need recompiling
every time, which is an annoyance when re-running a `make check'.
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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> |
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.
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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.
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.
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There are six classes of functions in the GMP library:
mpz_. The associated type is mpz_t. There are about 150
functions in this class.
mpq_. The associated type is mpq_t. There are about 40
functions in this class, but the integer functions can be used for arithmetic
on the numerator and denominator separately.
mpf_. The associated type is mpf_t. There are about 60
functions is this class.
itom, madd, and
mult. The associated type is MINT.
mpn_. The associated type is array of mp_limb_t. There are
about 30 (hard-to-use) functions in this class.
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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);
}
|
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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.
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, 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 as both
param and result, just like the library functions. But
sometimes this is tricky to arrange, 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.
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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 14. Custom Allocation. Temporary memory on the stack is
also used (via alloca), but this can be changed at build-time if
desired, see 2.1 Build Options.
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GMP is reentrant and thread-safe, with some exceptions:
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.
mp_set_memory_functions uses global variables to store the selected
memory allocation functions.
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.
mp_set_memory_functions (or malloc and friends by default) are
not reentrant, then GMP will not be reentrant either.
fwrite are not reentrant then the
GMP I/O functions using them will not be reentrant either.
gmp_randstate_t simultaneously,
since this involves an update of that variable.
<ctype.h> macros use per-file static
variables and may not be reentrant, depending whether the compiler optimizes
away fetches from them. The GMP text-based input functions are affected.
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"4.0.1".
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This version of GMP is upwardly binary compatible with all 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.
There are a number of compatibility issues between GMP 1 and GMP 2 that of course also apply when porting applications from GMP 1 to GMP 4. Please see the GMP 2 manual for details.
The Berkeley MP compatibility library (see section 13. Berkeley MP Compatible Functions) is source and binary compatible with the standard `libmp'.
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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.
mpz_t or mpq_t variable used to hold successively increasing
values will have its memory repeatedly realloced, 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
(see section 5.1 Initialization Functions).
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
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
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.
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.
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*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).
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. See section 6.5 Applying Integer Functions to Rationals.
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); |
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.
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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
section `Allocation Debugging' in The GNU C Library Reference Manual, or
`man 3 malloc'. Other possibilities, in no particular order, include
cd /my/build/dir /my/source/dir/gmp-4.0.1/configure |
This works via VPATH, and might require GNU make.
Alternately it might be possible to change the .c.lo rules
appropriately.
Applications using the low-level mpn functions, however, will benefit
from `--enable-assert' since it adds checks on the parameters of most
such functions, many of which have subtle restrictions on their usage. Note
however that only the generic C code has checks, not the assembler code, so
CPU `none' should be used for maximum checking.
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.
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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.
Depending on the system, it may be possible to get a flat profile, meaning simple timer sampling of the program counter, with no special GMP build options, just a `-p' when compiling the mainline. This is a good way to ensure minimum interference with normal operation. The necessary symbol type and size information exists in most of the GMP assembler code.
The `--enable-profiling' build option can be used to add suitable
compiler flags, either for prof (`-p') or gprof
(`-pg'), see 2.1 Build Options. Which of the two is available and what
they do will depend on the system, and possibly on support available in
`libc'. For some systems appropriate corresponding mcount calls
are added to the assembler code too.
On x86 systems prof gives call counting, so that average time spent
in a function can be determined. gprof, where supported, adds call
graph construction, so for instance calls to mpn_add_n from
mpz_add and from mpz_mul can be differentiated.
On x86 and 68k systems `-pg' and `-fomit-frame-pointer' are
incompatible, so the latter is not used when gprof profiling is
selected, which may result in poorer code generation. If prof
profiling is selected instead it should still be possible to use
gprof, but only the `gprof -p' flat profile and call counts can
be expected to be valid, not the `gprof -q' call graph.
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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 http://swox.com/gmp])]) |
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 http://swox.com/gmp])]) |
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.
An application that can use either GMP 2 or 3 will need to test for
__gmpz_init (GMP 3 and up) or mpz_init (GMP 2), and it's also
worth checking for `libgmp2' since Debian GNU/Linux systems used that
name in the past. For example,
AC_CHECK_LIB(gmp, __gmpz_init, ,
[AC_CHECK_LIB(gmp, mpz_init, ,
[AC_CHECK_LIB(gmp2, mpz_init)])])
|
In general it's suggested 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>]) |
The optional mpfr functions are provided in a separate
`libmpfr.a', and this might be from GMP with `--enable-mpfr' or
from MPFR installed separately. Either way `libmpfr' depends on
`libgmp', it doesn't stand alone. Currently only a static
`libmpfr.a' will be available, not a shared library, since upward binary
compatibility is not guaranteed.
AC_CHECK_LIB(mpfr, mpfr_add, , [AC_MSG_ERROR( [Need MPFR either from GNU MP 4 or separate MPFR package. See http://www.mpfr.org or http://swox.com/gmp]) |
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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 2.5 Known Build Problems, or perhaps 2.4 Notes for Particular Systems. You may also want to check http://swox.com/gmp/ for patches for this release.
Please include the following in any report,
gdb, or `$C' in adb).
straces.
gcc, get the version
with `gcc -v', otherwise perhaps `what `which cc`', or similar.
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: bug-gmp@gnu.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.
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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.
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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.
n is only the initial space, integer will grow automatically in
the normal way, if necessary, for subsequent values stored. mpz_init2
makes it possible to avoid such reallocations if a maximum size is known in
advance.
mpz_t variables when you are done with them.
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.
The space will not be automatically increased, unlike the normal
mpz_init, but instead an application must ensure it's sufficient for
any value stored. The following space requirements apply to various
functions,
mpz_abs, mpz_neg, mpz_set, mpz_set_si and
mpz_set_ui need room for the value they store.
mpz_add, mpz_add_ui, mpz_sub and mpz_sub_ui need
room for the larger of the two operands, plus an extra
mp_bits_per_limb.
mpz_mul, mpz_mul_ui and mpz_mul_ui need room for the sum
of the number of bits in their operands, but each rounded up to a multiple of
mp_bits_per_limb.
mpz_swap can be used between two array variables, but not between an
array and a normal variable.
For other functions, or if in doubt, the suggestion is to calculate in a
regular mpz_init variable and copy the result to an array variable with
mpz_set.
mpz_array_init can reduce memory usage in algorithms that need large
arrays of integers, since it avoids allocating and reallocating lots of small
memory blocks. There is no way to free the storage allocated by this
function. Don't call mpz_clear!
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 the new size in limbs.
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These functions assign new values to already initialized integers (see section 5.1 Initialization Functions).
mpz_set_d, mpz_set_q and mpz_set_f truncate op to
make it an integer.
This function returns 0 if the entire string is a valid number in base base. Otherwise it returns -1.
[It turns out that it is not entirely true that this function ignores
white-space. It does ignore it between digits, but not after a minus sign or
within or after "0x". 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. Send your opinion of this change to
bug-gmp@gnu.org. Do you really want it to accept "3 14" as
meaning 314 as it does now?]
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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!
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.)
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This section describes functions for converting GMP integers to standard C types. Functions for converting to GMP integers are described in 5.2 Assignment Functions and 5.12 Input and Output Functions.
mpz_tdiv_q_2exp(..., op, CHAR_BIT*sizeof(unsigned long
int)) can be used to decompose an integer into unsigned longs.
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 large 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.
double.
If str is NULL, the result string is allocated using the current
allocation function (see section 14. 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.
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.
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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.
Divide n by d, forming a quotient q and/or remainder
r. For the 2exp functions, \N\=2^b, 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\=qd+r, n=q*d+r}, and r will satisfy 0<=<abs}.
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.
The 2exp functions are right shifts and bit masks, but of course
rounding 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.
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.
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.
mpz_divisible_2exp_p by 2^b.
mpz_congruent_2exp_p modulo 2^b.
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Negative exp is supported if an inverse base^-1 mod
mod exists (see mpz_invert in 5.9 Number Theoretic Functions).
If an inverse doesn't exist then a divide by zero is raised.
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mpz_sqrt. Set rop2 to the
remainder \N\ - rop1^2),
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.
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.
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This function does some trial divisions, then some Miller-Rabin probabilistic primality tests. reps controls how many such tests are done, 5 to 10 is a reasonable number, more will reduce the chances of a composite being returned as "probably prime".
Miller-Rabin and similar tests can be more properly called compositeness tests. Numbers which fail are known to be composite but those which pass might be prime or might be composite. Only a few composites pass, hence those which pass are considered probably prime.
This function uses a probabilistic algorithm to identify primes. For practical purposes it's adequate, the chance of a composite passing will be extremely small.
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.
NULL, that argument is
not computed.
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 (see section B. References),
or any number theory textbook. See also the example program
`demos/qcn.c' which uses mpz_kronecker_ui.
mpz_bin_ui, using the identity
\N\\atop{k}\right) = (-1)^k \left({n+k-1}\atop{k}\right),
bin(-n,k)}, see Knuth volume 1 section 1.2.6
part G.
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
\N\,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 \N\ = F_n + F_{n-1}, F[n+1]=F[n]+F[n-1]} or
similar.
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 \N\,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 \N\ = L_n + L_{n-1}, L[n+1]=L[n]+L[n-1]} or
similar.
The Fibonacci numbers and Lucas numbers are related sequences, so it's never
necessary to call both mpz_fib2_ui and mpz_lucnum2_ui. The
formulas for going from Fibonacci to Lucas can be found in 16.7.4 Lucas Numbers, the reverse is straightforward too.
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Note that mpz_cmp_ui and mpz_cmp_si are macros and will evaluate
their arguments more than once.
Note that mpz_cmpabs_si is a macro and will evaluate its arguments more
than once.
This function is actually implemented as a macro. It evaluates its argument multiple times.
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These functions behave as if twos complement arithmetic were used (although sign-magnitude is the actual implementation). The least significant bit is number 0.
unsigned long.
unsigned long.
If the bit at starting_bit is already what's sought, then starting_bit is returned.
If there's no bit found, then MAX_ULONG is returned. This will happen
in mpz_scan0 past the end of a positive number, or mpz_scan1
past the end of a negative.
Functions that perform input from a stdio stream, and functions that output to
a stdio stream. Passing a NULL pointer for a stream argument to any of
these functions will make them read from stdin and write to
stdout, respectively.
When using any of these functions, it is a good idea to include `stdio.h' before `gmp.h', since that will allow `gmp.h' to define prototypes for these functions.
Return the number of bytes written, or if an error occurred, return 0.
Return the number of bytes read, or if an error occurred, return 0.
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.
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.
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 9. Random Number Functions for more information on how to use and not to use random number functions.
The variable state must be initialized by calling one of the
gmp_randinit functions (9.1 Random State Initialization) before
invoking this function.
The variable state must be initialized by calling one of the
gmp_randinit functions (9.1 Random State Initialization)
before invoking this function.
The variable state must be initialized by calling one of the
gmp_randinit functions (9.1 Random State Initialization)
before invoking this function.
This function is obsolete. Use mpz_urandomb or
mpz_urandomm instead.
This function is obsolete. Use mpz_rrandomb instead.
unsigned long int,
signed long int, unsigned int, signed int, unsigned
short int, or signed short int, respectively. Otherwise, return zero.
- Macro: int mpz_odd_p (mpz_t op)
-
- Macro: int mpz_even_p (mpz_t op)
- Determine whether op is odd or even, respectively. Return non-zero if
yes, zero if no. These macros evaluate their argument more than once.
- Function: size_t mpz_size (mpz_t op)
- Return the size of op measured in number of limbs. If op is zero,
the returned value will be zero.
- Function: size_t mpz_sizeinbase (mpz_t op, int base)
- Return the size of op measured in number of digits in base base.
The base may vary from 2 to 36. The sign of op is ignored, just the
absolute value is used. The returned value will be exact or 1 too big. If
base is a power of 2, the returned value will always be exact.
This function is useful in order to allocate the right amount of space before
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).
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.
6.1 Initialization and Assignment Functions 6.2 Conversion Functions 6.3 Arithmetic Functions 6.4 Comparison Functions 6.5 Applying Integer Functions to Rationals 6.6 Input and Output Functions
6.1 Initialization and Assignment Functions
- Function: void mpq_init (mpq_t dest_rational)
- Initialize dest_rational and set it to 0/1. Each variable should
normally only be initialized once, or at least cleared out (using the function
mpq_clear) between each initialization.
- Function: void mpq_clear (mpq_t rational_number)
- Free the space occupied by rational_number. Make sure to call this
function for all
mpq_t variables when you are done with them.
- Function: void mpq_set (mpq_t rop, mpq_t op)
-
- Function: void mpq_set_z (mpq_t rop, mpz_t op)
- Assign rop from op.
- Function: void mpq_set_ui (mpq_t rop, 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, 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 (see section 6. 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 (see section 5.2 Assignment Functions). White space is allowed in
the string, and is simply ignored. The base can vary from 2 to 36, or
if base is 0 then the leading characters are used: 0x for hex,
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.
6.2 Conversion Functions
- Function: double mpq_get_d (mpq_t op)
- Convert op to a
double.
- Function: void mpq_set_d (mpq_t rop, double op)
-
- Function: void mpq_set_f (mpq_t rop, mpf_t op)
- Set rop to the value of op, without rounding.
- Function: char * mpq_get_str (char *str, int base, mpq_t op)
- Convert op to a string of digits in base base. The base may vary
from 2 to 36. The string will be of the form `num/den', or if the
denominator is 1 then just `num'.
If str is NULL, the result string is allocated using the current
allocation function (see section 14. 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.
6.3 Arithmetic Functions
- Function: void mpq_add (mpq_t sum, mpq_t addend1, mpq_t addend2)
- Set sum to addend1 + addend2.
- Function: void mpq_sub (mpq_t difference, mpq_t minuend, mpq_t subtrahend)
- Set difference to minuend - subtrahend.
- Function: void mpq_mul (mpq_t product, mpq_t multiplier, mpq_t multiplicand)
- Set product to multiplier times.
- Function: void mpq_mul_2exp (mpq_t rop, mpq_t op1, unsigned long int op2)
Set rop to \N\ \times 2^{op2, op1 times 2 raised to
op2.
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To determine if two rationals are equal, mpq_equal is faster than
mpq_cmp.
num2 and den2 are allowed to have common factors.
These functions are implemented as a macros and evaluate their arguments multiple times.
This function is actually implemented as a macro. It evaluates its arguments multiple times.
mpq_cmp can be used for the same purpose, this
function is much faster.
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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
(see section 6. Rational Number Functions) then mpq_canonicalize must be
called before any other mpq functions are applied to that mpq_t.
mpz functions can be used on the result of these macros.
mpz_set with an appropriate mpq_numref or
mpq_denref. Direct use of mpq_numref or mpq_denref is
recommended instead of these functions.
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When using any of these functions, it's a good idea to include `stdio.h' before `gmp.h', since that will allow `gmp.h' to define prototypes for these functions.
Passing a NULL pointer for a stream argument to any of these
functions will make them read from stdin and write to stdout,
respectively.
Return the number of bytes written, or if an error occurred, return 0.
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 (see section 6. 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.
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GMP floating point numbers are stored in objects of type mpf_t and
functions operating on them have an mpf_ prefix.
The mantissa of each float has a user-selectable precision, limited only by available memory. Each variable has its own precision, and that can be increased or decreased at any time.
The exponent of each float is a fixed precision, one machine word on most
systems. In the current implementation the exponent is a count of limbs, so
for example on a 32-bit system this means a range of roughly
2^-68719476768 to 2^68719476736, or on a 64-bit system this
will be greater. Note however mpf_get_str can only return an exponent
which fits an mp_exp_t and currently mpf_set_str doesn't accept
exponents bigger than a long.
Each variable keeps a size for the mantissa data actually in use. This means that if a float is exactly represented in only a few bits then only those bits will be used in a calculation, even if the selected precision is high.
All calculations are performed to the precision of the destination variable. Each function is defined to calculate with "infinite precision" followed by a truncation to the destination precision, but of course the work done is only what's needed to determine a result under that definition.
The precision selected for a variable is a minimum value, GMP may increase it a little to facilitate efficient calculation. Currently this means rounding up to a whole limb, and then sometimes having a further partial limb, depending on the high limb of the mantissa. But applications shouldn't be concerned by such details.
mpf functions and variables have no special notion of infinity or
not-a-number, and applications must take care not to overflow the exponent or
results will be unpredictable. This might change in a future release.
Note that the mpf functions are not intended as a smooth
extension to IEEE P754 arithmetic. In particular results obtained on one
computer often differ from the results on a computer with a different word
size.
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mpf_init will use this precision, but previously
initialized variables are unaffected.
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.
mpf_clear, between initializations. The
precision of x is undefined unless a default precision has already been
established by a call to mpf_set_default_prec.
mpf_clear, between initializations.
mpf_t variables when you are done with them.
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.
This function requires a call to realloc, and so should not be used in
a tight loop.
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.
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These functions assign new values to already initialized floats (see section 7.1 Initialization Functions).
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.
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.
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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!
The precision of rop will be taken from the active default precision, as
set by mpf_set_default_prec.
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.
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double.
frexp.
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
(see section 7.8 Miscellaneous Functions).
If str is NULL, the result string is allocated using the current
allocation function (see section 14. 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.
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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.
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Caution: Currently only whole limbs are compared, and only in an exact fashion. In the future values like 1000 and 0111 may be considered the same to 3 bits (on the basis that their difference is that small).
This function is actually implemented as a macro. It evaluates its arguments multiple times.
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Functions that perform input from a stdio stream, and functions that output to
a stdio stream. Passing a NULL pointer for a stream argument to
any of these functions will make them read from stdin and write to
stdout, respectively.
When using any of these functions, it is a good idea to include `stdio.h' before `gmp.h', since that will allow `gmp.h' to define prototypes for these functions.
The mantissa is prefixed with an `0.' and is in the given base,
which may vary from 2 to 36. An exponent then printed, separated by an
`e', or if base is greater than 10 then by an `@'. The
exponent is always in decimal. The decimal point follows the current locale,
on systems providing localeconv.
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.
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.
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mpf_ceil rounds to the
next higher integer, mpf_floor to the next lower, and mpf_trunc
to the integer towards zero.
The variable state must be initialized by calling one of the
gmp_randinit functions (9.1 Random State Initialization) before
invoking this function.
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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) |
In the notation used here, a source operand is identified by the pointer to the least significant limb, and the limb count in braces. For example, {s1p, s1n}.
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.
This function requires that s1n is greater than or equal to s2n.
This is the lowest-level function for subtraction. It is the preferred function for subtraction, since it is written in assembly for most CPUs.
This function requires that s1n is greater than or equal to s2n.
The destination has to have space for 2*n limbs, even if the product's most significant limb is zero.
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.
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.
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.
The destination has to have space for s1n + s2n limbs, even if the result might be one limb smaller.
This function requires that s1n is greater than or equal to s2n. The destination must be distinct from both input operands.
No overlap is permitted between arguments. nn must be greater than or equal to dn. The most significant limb of dp must be non-zero. The qxn operand must be zero.
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.
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.
mpn_tdiv_qr instead for best
performance.]
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
\N\}, b=2^mp_bits_per_limb}. 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
is given by 3-c, because b == 1 mod (when
mp_bits_per_limb is even, which is always so currently).
mp\_bits\_per\_limb limbs of q =
{s1p, s1n}/{s2p, s2n} mod 2^d at
rp, and returns the high d mod mp_bits_per_limb bits of
q.
{s1p, s1n} - q * {s2p, s2n} mod \N\{2
\GMPraise{s1n*mp\_bits\_per\_limb,
2^(s1n*mp\_bits\_per\_limb) is placed at s1p. Since the
low floor(d)/mp\_bits\_per\_limb limbs of this
difference are zero, it is possible to overwrite the low limbs at s1p
with this difference, provided rp <=.
This function requires that s1n * mp\_bits\_per\_limb
>=, and that {s2p, s2n} is odd.
This interface is preliminary. It might change incompatibly in future revisions.
count must be in the range 1 to mp_bits_per_limb-1. The
regions {sp, n} and {rp, n} may overlap, provided
rp >=.
This function is written in assembly for most CPUs.
count must be in the range 1 to mp_bits_per_limb-1. The
regions {sp, n} and {rp, n} may overlap, provided
rp <=.
This function is written in assembly for most CPUs.
{s1p, s1n} must have at least as many bits as {s2p, s2n}. {s2p, s2n} must be odd. Both operands must have non-zero most significant limbs.
{s1p, s1n} >=, s2n} is required, and both must be non-zero. The regions {s1p, s1n+1} and {s2p, s2n+1} are destroyed (i.e. the operands plus an extra limb past the end of each).
The cofactor r1 will satisfy r2*s1 + k*s2 = r1. The second cofactor k is not calculated but can easily be obtained from (r1 - r2*s1) / s2.
The most significant limb of {sp, n} must be non-zero. The areas {r1p, ceil(n)/2} and {sp, n} must be completely separate. The areas {r2p, n} and {sp, n} must be either identical or completely separate.
If the remainder is not wanted then r2p can be NULL, and in this
case the return value is zero or non-zero according to whether the remainder
would have been zero or non-zero.
A return value of zero indicates a perfect square. See also
mpz_perfect_square_p.
The most significant limb of the input {s1p, s1n} must be non-zero. The area {s1p, s1n+1} is clobbered.
The area at str has to have space for the largest possible number represented by a s1n long limb array, plus one extra character.
Return the number of limbs stored in r1p.
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.
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.
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.
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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 5.13 Random Number Functions, and 7.8 Miscellaneous 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.
9.1 Random State Initialization 9.2 Random State Seeding
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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.
gmp_randinit_lc_2exp. a, c and m2exp are selected
from a table, chosen so that size bits (or more) of each X will be
used, ie. m2exp >=/2.
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.
Initialize state with an algorithm selected by alg. The only
choice is GMP_RAND_ALG_LC, which is gmp_randinit_lc_2exp_size.
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 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.
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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.
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10.1 Format Strings 10.2 Functions 10.3 C++ Formatted Output
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gmp_printf and friends accept format strings similar to the standard C
printf (see section `Formatted Output' in The GNU C Library Reference Manual). 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. `Z' and `Q' 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);
|
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 `'' (see section `Locales and Internationalization' in The GNU C Library Reference Manual) is only for the standard C types (not the GMP types), and only if the C library supports it.
0pad 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 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.
hshorthhcharjintmax_toruintmax_tllongorwchar_tllsame as LLlong longorlong doubleqquad_toru_quad_ttptrdiff_tzsize_t
The GMP types are
Fmpf_t, float conversionsQmpq_t, integer conversionsZmpz_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.
aAhex floats, GLIBC style ccharacter ddecimal integer eEscientific format float ffixed point float isame as dgGfixed or scientific float mstrerrorstring, GLIBC stylencharacters written so far ooctal integer ppointer sstring uunsigned integer xXhex integer
`o', `x' and `X' are unsigned for the standard C types, but for
`Z' and `Q' a sign is included. `u' is not meaningful for
Z and Q.
`n' can be used with any of the types, 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 it's usual meaning for integer `Z' and float `F' types, but is currently undefined for `Q' and should not be used with that.
mpf_t conversions only ever generate as many digits as can be
accurately represented by the operand, the same as mpf_get_str does.
Zeros will be used if necessary to pad to the requested precision. This
happens even for an `f' conversion of an mpf_t which is an
integer, for instance 2^1024 in an mpf_t of 128 bits precision
will only produce about 20 digits, then pad with zeros to the decimal point.
An empty precision field like `%.Fe' or `%.Ff' can be used to
specifically request all significant digits.
The decimal point character (or string) is taken from the current locale
settings on systems which provide localeconv (see section `Locales and Internationalization' in The GNU C Library Reference Manual). The C
library will normally do the same for standard float output.
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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 section `Variadic Functions' in The GNU C Library Reference Manual, 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. All the functions can return -1 if
the C library printf variant in use returns -1, but this shouldn't
normally occur.
stdout. Return the number of characters
written, or -1 if an error occurred.
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.
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.
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.
obstack_printf. Return the number of characters written. A
null-terminator is not written.
fmt cannot be within the current obstack object, since the 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 section `Obstacks' in The GNU C Library Reference Manual.
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The following functions are provided in `libgmpxx', which is built if C++
support is enabled (see section 2.1 Build Options). Prototypes are available from
<gmp.h>.
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.
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).
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 current
locale, on systems providing localeconv.
Hex and octal are supported, unlike the standard operator<< routines on
double etc. 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.
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, see section 11.3 C++ Formatted Input) is the only overloading available and using for instance
+ with an mpz_t will have unpredictable results.
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11.1 Formatted Input Strings 11.2 Formatted Input Functions 11.3 C++ Formatted Input
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gmp_scanf and friends accept format strings similar to the standard C
scanf (see section `Formatted Input' in The GNU C Library Reference Manual). 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 aallocate a buffer (string conversions) 'group 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.
hshorthhcharjintmax_toruintmax_tllongorwchar_tllsame as LLlong longorlong doubleqquad_toru_quad_ttptrdiff_tzsize_t
The GMP types are
Fmpf_t, float conversionsQmpq_t, integer conversionsZmpz_t, integer conversions
The conversions accepted are as follows. `p' and `[' will depend on support from the C library, the rest are standard.
ccharacter or characters ddecimal integer eEfgGfloat iinteger with base indicator ncharacters written so far ooctal integer ppointer sstring of non-whitespace characters udecimal integer xXhex 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 `e' or `E' for the exponent in scientific format.
`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 (see strtoul, section `Parsing of Integers' in The GNU C Library Reference Manual). For GMP types there are no overflows, and
`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 (see section 6. 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 the field would then 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 (see section `Locales and Internationalization' in The GNU C Library Reference Manual). The C library will normally do the same for
standard float input.
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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 section `Variadic Functions' in The GNU C Library Reference Manual, 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.
stdin.
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 file or file error, or end of string, is reached when a match is required, and when no previous non-suppressed fields have matched, then the return value is EOF instead of 0. A match is required for a literal character in the format string or a field other than `%n'. Whitespace in the format string is only an optional match and won't induce an EOF in this fashion. Leading whitespace read and discarded for a field doesn't count as a match.
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The following functions are provided in `libgmpxx', which is built only
if C++ support is enabled (see section 2.1 Build Options). Prototypes are available
from <gmp.h>.
ios formatting settings.
ios formatting settings.
An integer like `123' will be read, or a fraction like `5/9'. If
the fraction is not in canonical form then mpq_canonicalize must be
called (see section 6. Rational Number Functions).
ios formatting settings.
Hex or octal floats are not supported, but might be 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, see section 10.3 C++ Formatted Output) is the only overloading available and using for instance
+ with an mpz_t will have unpredictable results.
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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. For GCC this means version 2.91 or later.
Everything described in this chapter is to be considered preliminary and might be subject to incompatible changes if some unforeseen difficulty reveals itself.
12.1 C++ Interface General 12.2 C++ Interface Integers 12.3 C++ Interface Rationals 12.4 C++ Interface Floats 12.5 C++ Interface MPFR 12.6 C++ Interface Random Numbers 12.7 C++ Interface Limitations
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All the C++ classes and functions are available with
#include <gmpxx.h> |
The classes defined are
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 C++ types.
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,
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.
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mpz_class. All the standard C++ types may be used, except
long long and long double, and all the GMP C++ classes can be
used. Any necessary conversion follows the corresponding C function, for
example double follows mpz_set_d (see section 5.2 Assignment Functions).
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.
mpz_class converted from a string using
mpz_set_str, (see section 5.2 Assignment Functions). If the base is not
given then 0 is used.
mpz_class round towards zero, as per the
mpz_tdiv_q and mpz_tdiv_r functions (see section 5.6 Division Functions).
This corresponds to the rounding used for plain int calculations on
most machines.
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()); |
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.
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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.
mpq_class. The initial value can be a single value of any
type, or a pair of integers (mpz_class or standard C++ integer types)
representing a fraction, except that long long and long double
are not supported. For example,
mpq_class q (99); mpq_class q (1.75); mpq_class q (1, 3); |
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.
mpq_class converted from a string using
mpq_set_str, (see section 6.1 Initialization and Assignment Functions). If the base is
not given then 0 is used.
mpq_class into canonical form, as per 6. Rational Number Functions. All arithmetic operators require their operands in canonical
form, and will return results in canonical form.
cmp can be used with any of the classes or the standard C++ types,
except long long and long double.
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.
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.
ios formatting settings,
the same as mpq_t operator>> (see section 11.3 C++ Formatted Input).
If the rop read might not be in canonical form then
mpq_class::canonicalize must be called.
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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.
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, string, or C++
builtin type will give the default mpf precision (see section 7.1 Initialization Functions). 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 |
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.
mpf_class.
The restrictions described for mpf_set_prec_raw (see section 7.1 Initialization Functions) 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.
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The C++ class interface to MPFR is provided if MPFR is enabled (see section 2.1 Build Options). This interface must be regarded as preliminary and possibly subject to incompatible changes in the future, since MPFR itself is preliminary. All definitions can be obtained with
#include <mpfrxx.h> |
This defines
which behaves similarly to mpf_class (see section 12.4 C++ Interface Floats).
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gmp_randclass to hold an algorithm selection and current state, as per
gmp_randstate_t.
gmp_randclass, using a call to the given randinit
function (see section 9.1 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_randinit_lc_2exp_size can fail if the size requested is too big,
the behaviour of gmp_randclass::gmp_randclass is undefined in this case
(perhaps this will change in the future).
gmp_randclass using the same parameters as
gmp_randinit (see section 9.1 Random State Initialization). This function is
obsolete and the above randinit style should be preferred.
gmp_randclass r; ... mpf_class f (0, 512); // 512 bits precision f = r.get_f(); // random number, 512 bits |
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mpq_class and Templated Reading
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 bug-gmp@gnu.org.
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.
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
}
|
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These functions are intended to be fully compatible with the Berkeley MP library which is available on many BSD derived U*ix systems. The `--enable-mpbsd' option must be used when building GNU MP to make these available (see section 2. Installing GMP).
The original Berkeley MP library has a usage restriction: you cannot use the same variable as both source and destination in a single function call. The compatible functions in GNU MP do not share this restriction--inputs and outputs may overlap.
It is not recommended that new programs are written using these functions.
Apart from the incomplete set of functions, the interface for initializing
MINT objects is more error prone, and the pow function collides
with pow in `libm.a'.
Include the header `mp.h' to get the definition of the necessary types and functions. If you are on a BSD derived system, make sure to include GNU `mp.h' if you are going to link the GNU `libmp.a' to your program. This means that you probably need to give the `-I<dir>' option to the compiler, where `<dir>' is the directory where you have GNU `mp.h'.
MINT object and dynamic limb space.
Initialize the integer to initial_value. Return a pointer to the
MINT object.
MINT object and dynamic limb space.
Initialize the integer from initial_value, a hexadecimal,
null-terminated C string. Return a pointer to the MINT object.
Some implementations of these functions work differently--or not at all--for negative arguments.
mpz_sqrt. Set remainder to
\N\ - root^2), op-root*root}, i.e.
zero if op is a perfect square.
If root and remainder are the same variable, the results are undefined.
stdin, and put the read integer in
dest. SPC and TAB are allowed in the number string, and are ignored.
stdout, as a decimal string. Also output a newline.
malloc by default.
itom or xtom.
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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.
This feature is available in the Berkeley compatibility library (see section 13. Berkeley MP Compatible Functions) as well as the main GMP library.
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 (see section 2.1 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:
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.
ptr is never NULL, it's always a previously allocated block of
size bytes.
A byte here means the unit used by the sizeof operator.
The old_size parameters to reallocate_function and
deallocate_function are passed for convenience, but of course can be
ignored if not needed. The default functions using malloc and friends
for instance don't use them.
No error return is allowed from any of these functions, if they return then
they must have performed the specified operation. In particular note that
allocate_function or reallocate_function mustn't return
NULL.
Getting a different fatal error action is a good use for custom allocation
functions, for example giving a graphical dialog rather than the default print
to stderr. How much is possible when genuinely out of memory is
another question though.
There's currently no defined way for the allocation functions to recover from
an error such as out of memory, they must terminate program execution. A
longjmp or throwing a C++ exception will have undefined results. This
may change in the future.
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.
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The following packages and projects offer access to GMP from languages other than C, though perhaps with varying levels of functionality and efficiency.
mpeval.
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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.
16.1 Multiplication 16.2 Division Algorithms 16.3 Greatest Common Divisor 16.4 Powering Algorithms 16.5 Root Extraction Algorithms 16.6 Radix Conversion 16.7 Other Algorithms 16.8 Assembler Coding
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Nx{}N limb multiplications and squares are done using one of four algorithms, as the size N increases.
Algorithm Threshold Basecase (none) Karatsuba KARATSUBA_MUL_THRESHOLDToom-3 TOOM3_MUL_THRESHOLDFFT FFT_MUL_THRESHOLD
Similarly for squaring, with the SQR thresholds. Note though that the
FFT is only used if GMP is configured with `--enable-fft', see section 2.1 Build Options.
Nx{}M multiplications of operands with different sizes above
KARATSUBA_MUL_THRESHOLD are currently done by splitting into Mx{}M
pieces. The Karatsuba and Toom-3 routines then operate only on equal size
operands. This is not very efficient, and is slated for improvement in the
future.
16.1.1 Basecase Multiplication 16.1.2 Karatsuba Multiplication 16.1.3 Toom-Cook 3-Way Multiplication 16.1.4 FFT Multiplication 16.1.5 Other Multiplication
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Basecase Nx{}M multiplication is a straightforward rectangular set of cross-products, the same as long multiplication done by hand and for that reason sometimes known as the schoolbook or grammar school method. This is an O(N*M) algorithm. See Knuth section 4.3.1 algorithm M (see section B. References), and the `mpn/generic/mul_basecase.c' code.
Assembler implementations of mpn_mul_basecase are essentially the same
as the generic C code, but have all the usual assembler tricks and
obscurities introduced for speed.
A square can be done in roughly half the time of a multiply, by using the fact that the cross products above and below the diagonal are the same. A triangle of products below the diagonal is formed, doubled (left shift by one bit), and then the products on the diagonal added. This can be seen in `mpn/generic/sqr_basecase.c'. Again the assembler implementations take essentially the same approach.
u0 u1 u2 u3 u4 +---+---+---+---+---+ u0 | d | | | | | +---+---+---+---+---+ u1 | | d | | | | +---+---+---+---+---+ u2 | | | d | | | +---+---+---+---+---+ u3 | | | | d | | +---+---+---+---+---+ u4 | | | | | d | +---+---+---+---+---+ |
In practice squaring isn't a full 2x{} faster than multiplying, it's
usually around 1.5x 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. BASECASE_SQR_THRESHOLD is the size at which
to use mpn_sqr_basecase, this will be zero if that routine should be
used always.
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The Karatsuba multiplication algorithm is described in Knuth section 4.3.3 part A, and various other textbooks. A brief description is given here.
The inputs x and y are treated as each split into two parts of equal length (or the most significant part one limb shorter if N is odd).
high low +----------+----------+ | x1 | x0 | +----------+----------+ +----------+----------+ | y1 | y0 | +----------+----------+ |
Let b be the power of 2 where the split occurs, ie. if x0 is k limbs (y0 the same) then \N\}, b=2^(k*mp_bits_per_limb)}. With that x=x1*b+x0 and y=y1*b+y0, and the following holds,
\N\{xy = (b^2+b)x_1y_1 - b(x_1-x_0)(y_1-y_0) + (b+1)x_0y_0,
x*y = (b^2+b)*x1*y1 - b*(x1-x0)*(y1-y0) + (b+1)*x0*y0}
|
This formula means doing only three multiplies of (N/2)x{}(N/2) limbs, whereas a basecase multiply of Nx{}N limbs is equivalent to four multiplies of (N/2)x 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 \N\(x_0y_0)+\mathop{\rm low}(x_1y_1), 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,
\N\{x^2 = (b^2+b)x_1^2 - b(x_1-x_0)^2 + (b+1)x_0^2,
x^2 = (b^2+b)*x1^2 - b*(x1-x0)^2 + (b+1)*x0^2}
|
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.
KARATSUBA_MUL_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 \N\{K(N) = 3M(N/2) + dN + e, K(N) = 3*M(N/2) + d*N +
e}. Clearly per-crossproduct speedups in the basecase code reduce a and
decrease the threshold, but linear style speedups reducing b will
actually increase the threshold. 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.
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The Karatsuba formula is the simplest case of a general approach to splitting inputs that leads to both Toom-Cook and FFT algorithms. A description of Toom-Cook 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 others).
high low +----------+----------+----------+ | x2 | x1 | x0 | +----------+----------+----------+ +----------+----------+----------+ | y2 | y1 | y0 | +----------+----------+----------+ |
These parts are treated as the coefficients of two polynomials
\N\{X(t) = x_2t^2 + x_1t + x_0,
X(t) = x2*t^2 + x1*t + x0}
\N\{Y(t) = y_2t^2 + y_1t + y_0,
Y(t) = y2*t^2 + y1*t + y0}
|
Again let b equal the power of 2 which is the size of the x0, x1, y0 and y1 pieces, ie. if they're k limbs each then \N\}, 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
\N\{W(t) = w_4t^4 + w_3t^3 + w_2t^2 + w_1t + w_0,
W(t) = w4*t^4 + w3*t^3 + w2*t^2 + w1*t + w0}
|
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. The points used can be chosen in various ways, but in GMP the following are used
Point Value t=0 x0*y0, which gives w0 immediately t=2 (4*x2+2*x1+x0)*(4*y2+2*y1+y0) t=1 (x2+x1+x0)*(y2+y1+y0) \N\,t=1/2} (x2+2*x1+4*x0)*(y2+2*y1+4*y0) t=inf x2*y2, which gives w4 immediately
At \N\,t=1/2} the value calculated is actually \N\)Y({1\over2}), 16*X(1/2)*Y(1/2)}, giving a value for \N\),16*W(1/2)}, and this is always an integer. At t=inf the value is actually \N\ {X(t)Y(t)\over t^4}, X(t)*Y(t)/t^4 in the limit as t approaches infinity}, but it's much easier to think of as simply x2*y2 giving w4 immediately (much like x0*y0 at t=0 gives w0 immediately).
Now 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 16*W(1/2) = w4 + 2*w3 + 4*w2 + 8*w1 + 16*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], by subtracting multiples of one equation from another.
In the code the set of five values W(0),...,W(inf) will
represent those certain linear combinations. By adding or subtracting one
from another as necessary, values which are each w[i] alone are
arrived at. This involves only a few subtractions of small multiples (some of
which are powers of 2), and so is fast. A couple of divisions remain by
powers of 2 and one division by 3 (or by 6 rather), and that last uses the
special mpn_divexact_by3 (see section 16.2.4 Exact Division).
In the code the values w4, w2 and w0 are formed in the
destination with pointers E, C and A, and w3 and
w1 in temporary space D and B are added to them. There
are extra limbs tD, tC and tB at the high end of
w3, w2 and w1 which are handled separately. The final
addition then is as follows.
high low
+-------+-------+-------+-------+-------+-------+
| E | C | A |
+-------+-------+-------+-------+-------+-------+
+------+-------++------+-------+
| D || B |
+------+-------++------+-------+
-- -- --
|tD| |tC| |tB|
-- -- --
|
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 can be chosen to make the linear equations come out with a convenient set of steps for isolating the w[i].
In `mpn/generic/mul_n.c' the interpolate3 routine performs the
interpolation. The open-coded one-pass version may be a bit hard to
understand, the steps performed can be better seen in the USE_MORE_MPN
version.
Squaring follows the same procedure as multiplication, but there's only one
X(t) and it's evaluated at 5 points, and those values squared to give
values of W(t). The interpolation is then identical, and in fact the
same interpolate3 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. This is an improvement over Karatsuba at O(N^1.585), though Toom-Cook 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.
TOOM3_MUL_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 above for the Karatsuba algorithm 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 (see section B. 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.
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At large to very large sizes a Fermat style FFT multiplication is used, following Schönhage and Strassen (see section B. 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\{N \ge \mathop{\rm bits}(x)+\mathop{\rm bits}(y), 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
\N\, (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 \N\+1, 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 \N\, g=2^(2N'/2^k)}. g is a 2^k'th root of unity mod \N\+1,2^N'+1}, which produces necessary cancellations at the interpolation stage, and it's also a power of 2 so the fast fourier transforms used for the evaluation and interpolation do only shifts, adds and negations.
The pointwise multiplications are done modulo \N\+1, 2^N'+1} and either recurse into a further FFT or use a plain multiplication (Toom-3, Karatsuba or basecase), whichever is optimal at the size N'. The interpolation is an inverse fast fourier transform. The resulting set of sums of \N\{x_iy_j, 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 \N\),
O(N^(k/(k-1)))} algorithm, the exponent representing 2^k recursed modular
multiplies each \N\,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, FFT_MUL_TABLE and
FFT_SQR_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
Nx2N 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 FFT_MODF_MUL_THRESHOLD and
FFT_MODF_SQR_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 \N\,1/2^(k-2)} the size of the inputs,
making it \N\), 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
FFT_MUL_THRESHOLD and FFT_SQR_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>= the effective N is a multiple
of \N\,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 \N\,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 FFT_MUL_TABLE and FFT_SQR_TABLE will be
needed.
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The 3-way Toom-Cook algorithm described above (see section 16.1.3 Toom-Cook 3-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, though it's possible a Toom-4 might fit in between Toom-3 and the FFTs. 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 \N\, 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 \N\, 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)).
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16.2.1 Single Limb Division 16.2.2 Basecase Division 16.2.3 Divide and Conquer Division 16.2.4 Exact Division 16.2.5 Exact Remainder 16.2.6 Small Quotient Division
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Nx1 divisions from high to low, either with a hardware divide instruction or a multiplication by inverse, whichever is best on a given CPU.
The multiply by inverse follows section 8 of "Division by Invariant Integers
using Multiplication" by Granlund and Montgomery (see section B. References) and is
implemented as udiv_qrnnd_preinv in `gmp-impl.h'. The idea is to
have a fixed-point approximation to 1/d (see invert_limb) and then
multiply by the high limb (plus one bit) of the dividend to get a quotient
q. With d normalized (high bit set), q is no more than 1 too
small. Subtracting q*d from the dividend gives a remainder, and
reveals whether q or q-1 is correct.
The result is a division done with two multiplications and four or five arithmetic operations. On CPUs with low latency multipliers this can be much faster than a hardware divide, though the cost of calculating the inverse at the start may mean it's only better on inputs bigger than say 4 or 5 limbs.
When a divisor must be normalized, either for the generic C
__udiv_qrnnd_c or the multiply by inverse, the division performed is
actually a*2^k by d*2^k where a is the dividend and
k is the power necessary to have the high bit of d*2^k set. The
bit shifts for the dividend are usually accomplished "on the fly" meaning by
extracting the appropriate bits at each step. Done this way the quotient
limbs come out aligned ready to store. When only the remainder is wanted, an
alternative is to take the dividend limbs unshifted and calculate \N\{r = a
\bmod d2^k, r = a mod d*2^k} followed by an extra final step \N\{r2^k \bmod
d2^k, r*2^k mod d*2^k}. This can help on CPUs with poor bit shifts or few
registers.
The multiply by inverse can be done two limbs at a time. The calculation is basically the same, but the inverse is two limbs and the divisor treated as if padded with a low zero limb. This means more work, since the inverse will need a 2x1s to do that are independent and can therefore be done partly or wholly in parallel. Likewise for a 2x. The net effect is to process two limbs with roughly the same two multiplies worth of latency that one limb at a time gives. This extends to 3 or 4 limbs at a time, though the extra work to apply the inverse will almost certainly soon reach the limits of multiplier throughput.
A similar approach in reverse can be taken to process just half a limb at a time if the divisor is only a half limb. In this case the 1x{}1 multiply for the inverse effectively becomes two \N\1, (1/2)x1} for each limb, which can be a saving on CPUs with a fast half limb multiply, or in fact if the only multiply is a half limb, and especially if it's not pipelined.
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Basecase Nx{}M division is like long division done by hand, but in base \N\}, 2^mp_bits_per_limb}. See Knuth section 4.3.1 algorithm D, and `mpn/generic/sb_divrem_mn.c'.
Briefly stated, while the dividend remains larger than the divisor, a high quotient limb is formed and the Nx subtracted at the top end of the dividend. With a normalized divisor (most significant bit set), each quotient limb can be formed with a 2x{}1 division and a 1x1 division is by the high limb of the divisor and is done either with a hardware divide or a multiply by inverse (the same as in 16.2.1 Single Limb Division) whichever is faster. Such a quotient is sometimes one too big, requiring an addback of the divisor, but that happens rarely.
With Q=N-M being the number of quotient limbs, this is an O(Q*M) algorithm and will run at a speed similar to a basecase Qx{}M multiplication, differing in fact only in the extra multiply and divide for each of the Q quotient limbs.
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For divisors larger than DC_THRESHOLD, division is done by dividing.
Or to be precise by a recursive divide and conquer algorithm based on work by
Moenck and Borodin, Jebelean, and Burnikel and Ziegler (see section B. References).
The algorithm consists essentially of recognising that a 2Nx{}N division can be done with the basecase division algorithm (see section 16.2.2 Basecase Division), but using N/2 limbs as a base, not just a single limb. This way the multiplications that arise are (N/2)x{}(N/2) and can take advantage of Karatsuba and higher multiplication algorithms (see section 16.1 Multiplication). The "digits" of the quotient are formed by recursive Nx{}(N/2) divisions.
If the (N/2)x{}(N/2) multiplies are done with a basecase multiplication
then the work is about the same as a basecase division, but with more function
call overheads and with some subtractions separated from the multiplies.
These overheads mean that it's only when N/2 is above
KARATSUBA_MUL_THRESHOLD that divide and conquer is of use.
DC_THRESHOLD is based on the divisor size N, so it will be somewhere
above twice KARATSUBA_MUL_THRESHOLD, but how much above depends on the
CPU. An optimized mpn_mul_basecase can lower DC_THRESHOLD a
little by offering a ready-made advantage over repeated mpn_submul_1
calls.
Divide and conquer is asymptotically O(M(N)*log(N)) where M(N) is the time for an Nx{}N multiplication done with FFTs. The actual time is a sum over multiplications of the recursed sizes, as can be seen near the end of section 2.2 of Burnikel and Ziegler. For example, within the Toom-3 range, divide and conquer is 2.63*M(N). With higher algorithms the M(N) term improves and the multiplier tends to \N\{\log N, log(N)}. In practice, at moderate to large sizes, a 2Nx{}N division is about 2 to 4 times slower than an Nx{}N multiplication.
Newton's method used for division is asymptotically O(M(N)) and should therefore be superior to divide and conquer, but it's believed this would only be for large to very large N.
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A so-called exact division is when the dividend is known to be an exact multiple of the divisor. Jebelean's exact division algorithm uses this knowledge to make some significant optimizations (see section B. References).
The idea can be illustrated in decimal for example with 368154 divided by 543. Because the low digit of the dividend is 4, the low digit of the quotient must be 8. This is arrived at from \N\ 7 \bmod 10, 4*7 mod 10}, using the fact 7 is the modular inverse of 3 (the low digit of the divisor), since \N\ 7 \mathop{\equiv} 1 \bmod 10, 3*7 == 1 mod 10}. So \N\543 = 4344,8*543=4344} can be subtracted from the dividend leaving 363810. Notice the low digit has become zero.
The procedure is repeated at the second digit, with the next quotient digit 7 (\N\ 7 \bmod 10, 7 == 1*7 mod 10}), subtracting \N\543 = 3801,7*543=3801}, leaving 325800. And finally at the third digit with quotient digit 6 (\N\ 7 \bmod 10, 8*7 mod 10}), subtracting \N\543 = 3258,6*543=3258} leaving 0. So the quotient is 678.
Notice however that the multiplies and subtractions don't need to extend past the low three digits of the dividend, since that's enough to determine the three quotient digits. For the last quotient digit no subtraction is needed at all. On a 2Nx{}N division like this one, only about half the work of a normal basecase division is necessary.
For an NxM quotient limbs, the saving over a normal basecase division is in two parts. Firstly, each of the Q quotient limbs needs only one multiply, not a 2x{}1 divide and multiply. Secondly, the crossproducts are reduced when Q>M to Q*M-M*(M+1)/2, or when Q<= to \N\{Q(Q-1)/2, Q*(Q-1)/2. Notice the savings are complementary. If Q is big then many divisions are saved, or if Q is small then the crossproducts reduce to a small number.
The modular inverse used is calculated efficiently by modlimb_invert in
`gmp-impl.h'. This does four multiplies for a 32-bit limb, or six for a
64-bit limb. `tune/modlinv.c' has some alternate implementations that
might suit processors better at bit twiddling than multiplying.
The sub-quadratic exact division described by Jebelean in "Exact Division with Karatsuba Complexity" is not currently implemented. It uses a rearrangement similar to the divide and conquer for normal division (see section 16.2.3 Divide and Conquer Division), but operating from low to high. A further possibility not currently implemented is "Bidirectional Exact Integer Division" by Krandick and Jebelean which forms quotient limbs from both the high and low ends of the dividend, and can halve once more the number of crossproducts needed in a 2Nx{}N division.
A special case exact division by 3 exists in mpn_divexact_by3,
supporting Toom-3 multiplication and mpq canonicalizations. It forms
quotient digits with a multiply by the modular inverse of 3 (which is
0xAA..AAB) and uses two comparisons to determine a borrow for the next
limb. The multiplications don't need to be on the dependent chain, as long as
the effect of the borrows is applied. Only a few optimized assembler
implementations currently exist.
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If the exact division algorithm is done with a full subtraction at each stage and the dividend isn't a multiple of the divisor, then low zero limbs are produced but with a remainder in the high limbs. For dividend a, divisor d, quotient q, and \N\}, b = 2^mp_bits_per_limb}, then this remainder r is of the form
a = q*d + r*b^n |
n represents the number of zero limbs produced by the subtractions, that being the number of limbs produced for q. r will be in the range 0<= and can be viewed as a remainder, but one shifted up by a factor of b^n.
Carrying out full subtractions at each stage means the same number of cross products must be done as a normal division, but there's still some single limb divisions saved. When d is a single limb some simplifications arise, providing good speedups on a number of processors.
mpn_bdivmod, mpn_divexact_by3, mpn_modexact_1_odd and the
redc function in mpz_powm differ subtly in how they return
r, leading to some negations in the above formula, but all are
essentially the same.
Clearly r is zero when a is a multiple of d, and this leads to divisibility or congruence tests which are potentially more efficient than a normal division.
The factor of b^n on r can be ignored in a GCD when d is odd,
hence the use of mpn_bdivmod in mpn_gcd, and the use of
mpn_modexact_1_odd by mpn_gcd_1 and mpz_kronecker_ui etc
(see section 16.3 Greatest Common Divisor).
Montgomery's REDC method for modular multiplications uses operands of the form of \N\, x*b^-n and \N\, y*b^-n} and on calculating \N\) (yb^{-n}), (x*b^-n)*(y*b^-n)} uses the factor of b^n in the exact remainder to reach a product in the same form \N\, (x*y)*b^-n} (see section 16.4.2 Modular Powering).
Notice that r generally gives no useful information about the ordinary
remainder a mod d since b^n mod d could be anything. If however
b^n == 1 mod d, then r is the negative of the ordinary
remainder. This occurs whenever d is a factor of b^n-1, as for
example with 3 in mpn_divexact_by3. Other such factors include 5, 17
and 257, but no particular use has been found for this.
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An NxM is small can be optimized somewhat.
An ordinary basecase division normalizes the divisor by shifting it to make the high bit set, shifting the dividend accordingly, and shifting the remainder back down at the end of the calculation. This is wasteful if only a few quotient limbs are to be formed. Instead a division of just the top 2*Q limbs of the dividend by the top Q limbs of the divisor can be used to form a trial quotient. This requires only those limbs normalized, not the whole of the divisor and dividend.
A multiply and subtract then applies the trial quotient to the M-Q unused limbs of the divisor and N-Q dividend limbs (which includes Q limbs remaining from the trial quotient division). The starting trial quotient can be 1 or 2 too big, but all cases of 2 too big and most cases of 1 too big are detected by first comparing the most significant limbs that will arise from the subtraction. An addback is done if the quotient still turns out to be 1 too big.
This whole procedure is essentially the same as one step of the basecase algorithm done in a Q limb base, though with the trial quotient test done only with the high limbs, not an entire Q limb "digit" product. The correctness of this weaker test can be established by following the argument of Knuth section 4.3.1 exercise 20 but with the \N\{v_2 \GMPhat q > b \GMPhat r + u_2, v2*q>b*r+u2} condition appropriately relaxed.
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16.3.1 Binary GCD 16.3.2 Accelerated GCD 16.3.3 Extended GCD 16.3.4 Jacobi Symbol
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At small sizes GMP uses an O(N^2) binary style GCD. This is described in many textbooks, for example Knuth section 4.5.2 algorithm B. It simply consists of successively reducing operands a and b using gcd(min(a-b)), and also that if a and b are first made odd then abs is even and factors of two can be discarded.
Variants like letting a-b become negative and doing a different next step are of interest only as far as they suit particular CPUs, since on small operands it's machine dependent factors that determine performance.
The Euclidean GCD algorithm, as per Knuth algorithms E and A, reduces using a mod b but this has so far been found to be slower everywhere. One reason the binary method does well is that the implied quotient at each step is usually small, so often only one or two subtractions are needed to get the same effect as a division. Quotients 1, 2 and 3 for example occur 67.7% of the time, see Knuth section 4.5.3 Theorem E.
When the implied quotient is large, meaning b is much smaller than
a, then a division is worthwhile. This is the basis for the initial
a mod b reductions in mpn_gcd and mpn_gcd_1 (the latter
for both Nx1 cases). But after that initial reduction,
big quotients occur too rarely to make it worth checking for them.
For sizes above GCD_ACCEL_THRESHOLD, GMP uses the Accelerated GCD
algorithm described independently by Weber and Jebelean (the latter as the
"Generalized Binary" algorithm), see section B. References. This algorithm is
still O(N^2), but is much faster than the binary algorithm since it does
fewer multi-precision operations. It consists of alternating the k-ary
reduction by Sorenson, and a "dmod" exact remainder reduction.
For operands u and v the k-ary reduction replaces u with
n*v-d*u where n and d are single limb values chosen to
give two trailing zero limbs on that value, which can be stripped. n and
d are calculated using an algorithm similar to half of a two limb GCD
(see find_a in `mpn/generic/gcd.c').
When u and v differ in size by more than a certain number of bits, a dmod is performed to zero out bits at the low end of the larger. It consists of an exact remainder style division applied to an appropriate number of bits (see section 16.2.4 Exact Division, and see section 16.2.5 Exact Remainder). This is faster than a k-ary reduction but useful only when the operands differ in size. There's a dmod after each k-ary reduction, and if the dmod leaves the operands still differing in size then it's repeated.
The k-ary reduction step can introduce spurious factors into the GCD calculated, and these are eliminated at the end by taking GCDs with the original inputs gcd(v,g)) using the binary algorithm. Since g is almost always small this takes very little time.
At small sizes the algorithm needs a good implementation of find_a. At
larger sizes it's dominated by mpn_addmul_1 applying n and d.
The extended GCD calculates gcd and also cofactors x and y satisfying \N\b), a*x+b*y=gcd(a,. Lehmer's multi-step improvement of the extended Euclidean algorithm is used. See Knuth section 4.5.2 algorithm L, and `mpn/generic/gcdext.c'. This is an O(N^2) algorithm.
The multipliers at each step are found using single limb calculations for
sizes up to GCDEXT_THRESHOLD, or double limb calculations above that.
The single limb code is faster but doesn't produce full-limb multipliers,
hence not making full use of the mpn_addmul_1 calls.
When a CPU has a data-dependent multiplier, meaning one which is faster on
operands with fewer bits, the extra work in the double-limb calculation might
only save some looping overheads, leading to a large GCDEXT_THRESHOLD.
Currently the single limb calculation doesn't optimize for the small quotients
that often occur, and this can lead to unusually low values of
GCDEXT_THRESHOLD, depending on the CPU.
An analysis of double-limb calculations can be found in "A Double-Digit Lehmer-Euclid Algorithm" by Jebelean (see section B. References). The code in GMP was developed independently.
It should be noted that when a double limb calculation is used, it's used for the whole of that GCD, it doesn't fall back to single limb part way through. This is because as the algorithm proceeds, the inputs a and b are reduced, but the cofactors x and y grow, so the multipliers at each step are applied to a roughly constant total number of limbs.
mpz_jacobi and mpz_kronecker are currently implemented with a
simple binary algorithm similar to that described for the GCDs (see section 16.3.1 Binary GCD). They're not very fast when both inputs are large. Lehmer's multi-step
improvement or a binary based multi-step algorithm is likely to be better.
When one operand fits a single limb, and that includes mpz_kronecker_ui
and friends, an initial reduction is done with either mpn_mod_1 or
mpn_modexact_1_odd, followed by the binary algorithm on a single limb.
The binary algorithm is well suited to a single limb, and the whole
calculation in this case is quite efficient.
In all the routines sign changes for the result are accumulated using some bit twiddling, avoiding table lookups or conditional jumps.
16.4.1 Normal Powering 16.4.2 Modular Powering
Normal mpz or mpf powering uses a simple binary algorithm,
successively squaring and then multiplying by the base when a 1 bit is seen in
the exponent, as per Knuth section 4.6.3. The "left to right"
variant described there is used rather than algorithm A, since it's just as
easy and can be done with somewhat less temporary memory.
Modular powering is implemented using a 2^k-ary sliding window algorithm, as per "Handbook of Applied Cryptography" algorithm 14.85 (see section B. References). k is chosen according to the size of the exponent. Larger exponents use larger values of k, the choice being made to minimize the average number of multiplications that must supplement the squaring.
The modular multiplies and squares use either a simple division or the REDC
method by Montgomery (see section B. References). REDC is a little faster,
essentially saving N single limb divisions in a fashion similar to an exact
remainder (see section 16.2.5 Exact Remainder). The current REDC has some limitations.
It's only O(N^2) so above POWM_THRESHOLD division becomes faster
and is used. It doesn't attempt to detect small bases, but rather always uses
a REDC form, which is usually a full size operand. And lastly it's only
applied to odd moduli.
16.5.1 Square Root 16.5.2 Nth Root 16.5.3 Perfect Square 16.5.4 Perfect Power
Square roots are taken using the "Karatsuba Square Root" algorithm by Paul Zimmermann (see section B. References). This is expressed in a divide and conquer form, but as noted in the paper it can also be viewed as a discrete variant of Newton's method.
In the Karatsuba multiplication range this is an \N\ M(N/2)),O(1.5*M(N/2)) algorithm, where M(n) is the time to multiply two numbers of n limbs. In the FFT multiplication range this grows to a bound of O(6*M(N/2)). In practice a factor of about 1.5 to 1.8 is found in the Karatsuba and Toom-3 ranges, growing to 2 or 3 in the FFT range.
The algorithm does all its calculations in integers and the resulting
mpn_sqrtrem is used for both mpz_sqrt and mpf_sqrt.
The extended precision given by mpf_sqrt_ui is obtained by
padding with zero limbs.
Integer Nth roots are taken using Newton's method with the following iteration, where A is the input and n is the root to be taken.
1 A
a[i+1] = - * ( --------- + (n-1)*a[i] )
n a[i]^(n-1)
|
The initial approximation a[1] is generated bitwise by successively powering a trial root with or without new 1 bits, aiming to be just above the true root. The iteration converges quadratically when started from a good approximation. When n is large more initial bits are needed to get good convergence. The current implementation is not particularly well optimized.
mpz_perfect_square_p is able to quickly exclude most non-squares by
checking whether the input is a quadratic residue modulo some small integers.
The first test is modulo 256 which means simply examining the least
significant byte. Only 44 different values occur as the low byte of a square,
so 82.8% of non-squares can be immediately excluded. Similar tests modulo
primes from 3 to 29 exclude 99.5% of those remaining, or if a limb is 64 bits
then primes up to 53 are used, excluding 99.99%. A single Nx{1
remainder using PP from `gmp-impl.h' quickly gives all these
remainders.
A square root must still be taken for any value that passes the residue tests, to verify it's really a square and not one of the 0.086% (or 0.000156% for 64 bits) non-squares that get through. See section 16.5.1 Square Root.
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Detecting perfect powers is required by some factorization algorithms.
Currently mpz_perfect_power_p is implemented using repeated Nth root
extractions, though naturally only prime roots need to be considered.
(See section 16.5.2 Nth Root.)
If a prime divisor p with multiplicity e can be found, then only roots which are divisors of e need to be considered, much reducing the work necessary. To this end divisibility by a set of small primes is checked.
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Radix conversions are less important than other algorithms. A program dominated by conversions should probably use a different data representation.
16.6.1 Binary to Radix 16.6.2 Radix to Binary
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Conversions from binary to a power-of-2 radix use a simple and fast O(N) bit extraction algorithm.
Conversions from binary to other radices use repeated divisions, first by the biggest power of the radix that fits in a single limb, then by the radix on the remainders. This is an O(N^2) algorithm and can be quite time-consuming on large inputs.
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Conversions from a power-of-2 radix into binary use a simple and fast O(N) bitwise concatenation algorithm.
Conversions from other radices use repeated multiplications, first accumulating as many digits as fit in a limb, then doing an Nx{}1 multi-precision multiplication. This is O(N^2) and is certainly sub-optimal on sizes above the Karatsuba multiply threshold.
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16.7.1 Factorial 16.7.2 Binomial Coefficients 16.7.3 Fibonacci Numbers 16.7.4 Lucas Numbers
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Factorials n! are calculated by a simple product from 1 to n, but arranged into certain sub-products.
First as many factors as fit in a limb are accumulated, then two of those multiplied to give a 2-limb product. When two 2-limb products are ready they're multiplied to a 4-limb product, and when two 4-limbs are ready they're multiplied to an 8-limb product, etc. A stack of outstanding products is built up, with two of the same size multiplied together when ready.
Arranging for multiplications to have operands the same (or nearly the same) size means the Karatsuba and higher multiplication algorithms can be used. And even on sizes below the Karatsuba threshold an Nx{}N multiply will give a basecase multiply more to work on.
An obvious improvement not currently implemented would be to strip factors of 2 from the products and apply them at the end with a bit shift. Another possibility would be to determine the prime factorization of the result (which can be done easily), and use a powering method, at each stage squaring then multiplying in those primes with a 1 in their exponent at that point. The advantage would be some multiplies turned into squares.
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Binomial coefficients \N\\atop{k}\right), C(n, are calculated by first arranging k <= using \N\\atop{k\right) = \left({n}\atop{n-k}\right), C(n,n-k)} if necessary, and then evaluating the following product simply from i=2 to i=k.
k (n-k+i)
C(n,k) = (n-k+1) * prod -------
i=2 i
|
It's easy to show that each denominator i will divide the product so far, so the exact division algorithm is used (see section 16.2.4 Exact Division).
The numerators n-k+i and denominators i are first accumulated into
as many fit a limb, to save multi-precision operations, though for
mpz_bin_ui this applies only to the divisors, since n is an
mpz_t and n-k+i in general won't fit in a limb at all.
An obvious improvement would be to strip factors of 2 from each multiplier and divisor and count them separately, to be applied with a bit shift at the end. Factors of 3 and perhaps 5 could even be handled similarly. Another possibility, if n is not too big, would be to determine the prime factorization of the result based on the factorials involved, and power up those primes appropriately. This would help most when k is near n/2.
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The Fibonacci functions mpz_fib_ui and mpz_fib2_ui are designed
for calculating isolated F[n] or F[n],\N\,F[n-1]}
values efficiently.
For small n, a table of single limb values in __gmp_fib_table is
used. On a 32-bit limb this goes up to \N\,F[47]}, or on a 64-bit limb
up to \N\,F[93]}. For convenience the table starts at \N\,F[-1]}.
Beyond the table, values are generated with a binary powering algorithm, calculating a pair F[n] and \N\,F[n-1]} working from high to low across the bits of n. The formulas used are
F[2k+1] = 4*F[k]^2 - F[k-1]^2 + 2*(-1)^k F[2k-1] = F[k]^2 + F[k-1]^2 F[2k] = F[2k+1] - F[2k-1] |
At each step, k is the high b bits of n. If the next bit of n is 0 then \N\,F[2k]},\N\,F[2k-1]} is used, or if it's a 1 then \N\,F[2k+1]},\N\,F[2k]} is used, and the process repeated until all bits of n are incorporated. Notice these formulas require just two squares per bit of n.
It'd be possible to handle the first few n above the single limb table with simple additions, using the defining Fibonacci recurrence \N\ = F_k + F_{k-1}, F[k+1]=F[k]+F[k-1]}, but this is not done since it usually turns out to be faster for only about 10 or 20 values of n, and including a block of code for just those doesn't seem worthwhile. If they really mattered it'd be better to extend the data table.
Using a table avoids lots of calculations on small numbers, and makes small n go fast. A bigger table would make more small n go fast, it's just a question of balancing size against desired speed. For GMP the code is kept compact, with the emphasis primarily on a good powering algorithm.
mpz_fib2_ui returns both F[n] and \N\,F[n-1]}, but
mpz_fib_ui is only interested in F[n]. In this case the last
step of the algorithm can become one multiply instead of two squares. One of
the following two formulas is used, according as n is odd or even.
F[2k] = F[k]*(F[k]+2F[k-1]) F[2k+1] = (2F[k]+F[k-1])*(2F[k]-F[k-1]) + 2*(-1)^k |
\N\,F[2k+1]} here is the same as above, just rearranged to be a
multiply. For interest, the 2*(-1)^k term both here and above
can be applied just to the low limb of the calculation, without a carry or
borrow into further limbs, which saves some code size. See comments with
mpz_fib_ui and the internal mpn_fib2_ui for how this is done.
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mpz_lucnum2_ui derives a pair of Lucas numbers from a pair of Fibonacci
numbers with the following simple formulas.
L[k] = F[k] + 2*F[k-1] L[k-1] = 2*F[k] - F[k-1] |
mpz_lucnum_ui is only interested in L[n], and some work can be
saved. Trailing zero bits on n can be handled with a single square each.
L[2k] = L[k]^2 - 2*(-1)^k |
And the lowest 1 bit can be handled with one multiply of a pair of Fibonacci
numbers, similar to what mpz_fib_ui does.
L[2k+1] = 5*F[k-1]*(2*F[k]+F[k-1]) - 4*(-1)^k |
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The assembler subroutines in GMP are the most significant source of speed at small to moderate sizes. At larger sizes algorithm selection becomes more important, but of course speedups in low level routines will still speed up everything proportionally.
Carry handling and widening multiplies that are important for GMP can't be
easily expressed in C. GCC asm blocks help a lot and are provided in
`longlong.h', but hand coding low level routines invariably offers a
speedup over generic C by a factor of anything from 2 to 10.
16.8.1 Code Organisation 16.8.2 Assembler Basics 16.8.3 Carry Propagation 16.8.4 Cache Handling 16.8.5 Floating Point 16.8.6 SIMD Instructions 16.8.7 Software Pipelining 16.8.8 Loop Unrolling
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The various `mpn' subdirectories contain machine-dependent code, written in C or assembler. The `mpn/generic' subdirectory contains default code, used when there's no machine-specific version of a particular file.
Each `mpn' subdirectory is for an ISA family. Generally 32-bit and 64-bit variants in a family cannot share code and will have separate directories. Within a family further subdirectories may exist for CPU variants.
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mpn_addmul_1 and mpn_submul_1 are the most important routines
for overall GMP performance. All multiplications and divisions come down to
repeated calls to these. mpn_add_n, mpn_sub_n,
mpn_lshift and mpn_rshift are next most important.
On some CPUs assembler versions of the internal functions
mpn_mul_basecase and mpn_sqr_basecase give significant speedups,
mainly through avoiding function call overheads. They can also potentially
make better use of a wide superscalar processor.
The restrictions on overlaps between sources and destinations
(see section 8. Low-level Functions) are designed to facilitate a variety of
implementations. For example, knowing mpn_add_n won't have partly
overlapping sources and destination means reading can be done far ahead of
writing on superscalar processors, and loops can be vectorized on a vector
processor, depending on the carry handling.
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The problem that presents most challenges in GMP is propagating carries from
one limb to the next. In functions like mpn_addmul_1 and
mpn_add_n, carries are the only dependencies between limb operations.
On processors with carry flags, a straightforward CISC style adc is
generally best. AMD K6 mpn_addmul_1 however is an example of an
unusual set of circumstances where a branch works out better.
On RISC processors generally an add and compare for overflow is used. This sort of thing can be seen in `mpn/generic/aors_n.c'. Some carry propagation schemes require 4 instructions, meaning at least 4 cycles per limb, but other schemes may use just 1 or 2. On wide superscalar processors performance may be completely determined by the number of dependent instructions between carry-in and carry-out for each limb.
On vector processors good use can be made of the fact that a carry bit only
very rarely propagates more than one limb. When adding a single bit to a
limb, there's only a carry out if that limb was 0xFF...FF which on
random data will be only 1 in \N\},
2^mp_bits_per_limb}. `mpn/cray/add_n.c' is an example of this, it adds
all limbs in parallel, adds one set of carry bits in parallel and then only
rarely needs to fall through to a loop propagating further carries.
On the x86s, GCC (as of version 2.95.2) doesn't generate particularly good code
for the RISC style idioms that are necessary to handle carry bits in
C. Often conditional jumps are generated where adc or sbb forms
would be better. And so unfortunately almost any loop involving carry bits
needs to be coded in assembler for best results.
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GMP aims to perform well both on operands that fit entirely in L1 cache and those that don't. In the assembler subroutines this means prefetching, either always or when large enough operands are presented.
Pre-fetching sources combines well with loop unrolling, since a prefetch can be initiated once per unrolled loop (or more than once if the loop processes more than one cache line).
Pre-fetching destinations won't be necessary if the CPU has a big enough store queue. Older processors without a write-allocate L1 however will want destination prefetching, to avoid repeated write-throughs, unless they can keep up with the rate at which destination limbs are produced.
The distance ahead to prefetch will be determined by the rate data is processed versus the time it takes to bring a line up to L1. Naturally the net data rate from L2 or RAM will always limit the rate of data processing. Prefetch distance may also be limited by the number of prefetches the processor can have in progress at any one time.
If a special prefetch instruction doesn't exist then a plain load can be used, so long as the CPU supports out-of-order loads. But this may mean having a second copy of a loop so that the last few limbs can be processed without prefetching, since reading past the end of an operand must be avoided.
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Floating point arithmetic is used in GMP for multiplications on CPUs with poor integer multipliers. Floating point generally doesn't suit other operations like additions or shifts, due to difficulties implementing carry handling.
With IEEE 53-bit double precision floats, integer multiplications producing up to 53 bits will give exact results. Breaking a multiplication into 16x48} bit pieces is convenient. With some care though three 21x53} bit products can be used to do a 64x32 parts uses the sign bit.
Generally limbs want to be treated as unsigned, but on some CPUs floating point conversions only treat integers as signed. Copying through a zero extended memory region or testing and adjusting for a sign bit may be necessary.
Currently floating point FFTs aren't used for large multiplications. On some processors they probably have a good chance of being worthwhile, if great care is taken with precision control.
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The single-instruction multiple-data support in current microprocessors is aimed at signal processing algorithms where each data point can be treated more or less independently. There's generally not much support for propagating the sort of carries that arise in GMP.
SIMD multiplications of say four 16x{}16 bit multiplies only do as much work as one 32x{}32 from GMP's point of view, and need some shifts and adds besides. But of course if say the SIMD form is fully pipelined and uses less instruction decoding then it may still be worthwhile.
On the 80x86 chips, MMX has so far found a use in mpn_rshift and
mpn_lshift since it allows 64-bit operations, and is used in a special
case for 16-bit multipliers in the P55 mpn_mul_1. 3DNow and SSE
haven't found a use so far.
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Software pipelining consists of scheduling instructions around the branch point in a loop. For example a loop taking a checksum of an array of limbs might have a load and an add, but the load wouldn't be for that add, rather for the one next time around the loop. Each load then is effectively scheduled back in the previous iteration, allowing latency to be hidden.
Naturally this is wanted only when doing things like loads or multiplies that take a few cycles to complete, and only where a CPU has multiple functional units so that other work can be done while waiting.
A pipeline with several stages will have a data value in progress at each stage and each loop iteration moves them along one stage. This is like juggling.
Within the loop some moves between registers may be necessary to have the right values in the right places for each iteration. Loop unrolling can help this, with each unrolled block able to use different registers for different values, even if some shuffling is still needed just before going back to the top of the loop.
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Loop unrolling consists of replicating code so that several limbs are
processed in each loop. At a minimum this reduces loop overheads by a
corresponding factor, but it can also allow better register usage, for example
alternately using one register combination and then another. Judicious use of
m4 macros can help avoid lots of duplication in the source code.
Unrolling is commonly done to a power of 2 multiple so the number of unrolled loops and the number of remaining limbs can be calculated with a shift and mask. But other multiples can be used too, just by subtracting each n limbs processed from a counter and waiting for less than n remaining (or offsetting the counter by n so it goes negative when there's less than n remaining).
The limbs not a multiple of the unrolling can be handled in various ways, for example
switch statement, providing separate code for each possible excess,
for example an 8-limb unrolling would have separate code for 0 remaining, 1
remaining, etc, up to 7 remaining. This might take a lot of code, but may be
the best way to optimize all cases in combination with a deep pipelined loop.
One way to write the setups and finishups for a pipelined unrolled loop is simply to duplicate the loop at the start and the end, then delete instructions at the start which have no valid antecedents, and delete instructions at the end whose results are unwanted. Sizes not a multiple of the unrolling can then be handled as desired.
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This chapter is provided only for informational purposes and the various internals described here may change in future GMP releases. Applications expecting to be compatible with future releases should use only the documented interfaces described in previous chapters.
17.1 Integer Internals 17.2 Rational Internals 17.3 Float Internals 17.4 Raw Output Internals 17.5 C++ Interface Internals
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mpz_t variables represent integers using sign and magnitude, in space
dynamically allocated and reallocated. The fields are as follows.
_mp_size
_mp_size set to zero, in which case
the _mp_d data is unused.
_mp_d
mpn functions, so _mp_d[0] is the
least significant limb and _mp_d[ABS(_mp_size)-1] is the most
significant. Whenever _mp_size is non-zero, the most significant limb
is non-zero.
Currently there's always at least one limb allocated, so for instance
mpz_set_ui never needs to reallocate, and mpz_get_ui can fetch
_mp_d[0] unconditionally (though its value is then only wanted if
_mp_size is non-zero).
_mp_alloc
_mp_alloc is the number of limbs currently allocated at _mp_d,
and naturally _mp_alloc >= ABS(_mp_size). When an mpz routine
is about to (or might be about to) increase _mp_size, it checks
_mp_alloc to see whether there's enough space, and reallocates if not.
MPZ_REALLOC is generally used for this.
The various bitwise logical functions like mpz_and behave as if
negative values were twos complement. But sign and magnitude is always used
internally, and necessary adjustments are made during the calculations.
Sometimes this isn't pretty, but sign and magnitude are best for other
routines.
Some internal temporary variables are setup with MPZ_TMP_INIT and these
have _mp_d space obtained from TMP_ALLOC rather than the memory
allocation functions. Care is taken to ensure that these are big enough that
no reallocation is necessary (since it would have unpredictable consequences).
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mpq_t variables represent rationals using an mpz_t numerator and
denominator (see section 17.1 Integer Internals).
The canonical form adopted is denominator positive (and non-zero), no common factors between numerator and denominator, and zero uniquely represented as 0/1.
It's believed that casting out common factors at each stage of a calculation
is best in general. A GCD is an O(N^2) operation so it's better to do a
few small ones immediately than to delay and have to do a big one later.
Knowing the numerator and denominator have no common factors can be used for
example in mpq_mul to make only two cross GCDs necessary, not four.
This general approach to common factors is badly sub-optimal in the presence
of simple factorizations or little prospect for cancellation, but GMP has no
way to know when this will occur. As per 3.9 Efficiency, that's left to
applications. The mpq_t framework might still suit, with
mpq_numref and mpq_denref for direct access to the numerator and
denominator, or of course mpz_t variables can be used directly.
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Efficient calculation is the primary aim of GMP floats and the use of whole limbs and simple rounding facilitates this.
mpf_t floats have a variable precision mantissa and a single machine
word signed exponent. The mantissa is represented using sign and magnitude.
most least
significant significant
limb limb
_mp_d
|---- _mp_exp ---> |
_____ _____ _____ _____ _____
|_____|_____|_____|_____|_____|
. <------------ radix point
<-------- _mp_size --------->
|
The fields are as follows.
_mp_size
_mp_size and
_mp_exp both set to zero, and in that case the _mp_d data is
unused. (In the future _mp_exp might be undefined when representing
zero.)
_mp_prec
_mp_prec limbs of result (the most significant being non-zero).
_mp_d
mpn functions, so
_mp_d[0] is the least significant limb and
_mp_d[ABS(_mp_size)-1] the most significant.
The most significant limb is always non-zero, but there are no other restrictions on its value, in particular the highest 1 bit can be anywhere within the limb.
_mp_prec+1 limbs are allocated to _mp_d, the extra limb being
for convenience (see below). There are no reallocations during a calculation,
only in a change of precision with mpf_set_prec.
_mp_exp
Naturally the exponent can be any value, it doesn't have to fall within the
limbs as the diagram shows, it can be a long way above or a long way below.
Limbs other than those included in the {_mp_d,_mp_size} data
are treated as zero.
The following various points should be noted.
_mp_d[0] etc can be zero, though such low
zeros can always be ignored. Routines likely to produce low zeros check and
avoid them to save time in subsequent calculations, but for most routines
they're quite unlikely and aren't checked.
_mp_size count of limbs in use can be less than _mp_prec if
the value can be represented in less. This means low precision values or
small integers stored in a high precision mpf_t can still be operated
on efficiently.
_mp_size can also be greater than _mp_prec. Firstly a value is
allowed to use all of the _mp_prec+1 limbs available at _mp_d,
and secondly when mpf_set_prec_raw lowers _mp_prec it leaves
_mp_size unchanged and so the size can be arbitrarily bigger than
_mp_prec.
_mp_prec limbs
with the high non-zero will ensure the application requested minimum precision
is obtained.
The use of simple "trunc" rounding towards zero is efficient, since there's no need to examine extra limbs and increment or decrement.
mpf_add and mpf_mul. When differing exponents are
encountered all that's needed is to adjust pointers to line up the relevant
limbs.
Of course mpf_mul_2exp and mpf_div_2exp will require bit shifts,
but the choice is between an exponent in limbs which requires shifts there, or
one in bits which requires them almost everywhere else.
_mp_prec+1 Limbs
_mp_d (_mp_prec+1 rather than just
_mp_prec) helps when an mpf routine might get a carry from its
operation. mpf_add for instance will do an mpn_add of
_mp_prec limbs. If there's no carry then that's the result, but if
there is a carry then it's stored in the extra limb of space and
_mp_size becomes _mp_prec+1.
Whenever _mp_prec+1 limbs are held in a variable, the low limb is not
needed for the intended precision, only the _mp_prec high limbs. But
zeroing it out or moving the rest down is unnecessary. Subsequent routines
reading the value will simply take the high limbs they need, and this will be
_mp_prec if their target has that same precision. This is no more than
a pointer adjustment, and must be checked anyway since the destination
precision can be different from the sources.
Copy functions like mpf_set will retain a full _mp_prec+1 limbs
if available. This ensures that a variable which has _mp_size equal to
_mp_prec+1 will get its full exact value copied. Strictly speaking
this is unnecessary since only _mp_prec limbs are needed for the
application's requested precision, but it's considered that an mpf_set
from one variable into another of the same precision ought to produce an exact
copy.
__GMPF_BITS_TO_PREC converts an application requested precision to an
_mp_prec. The value in bits is rounded up to a whole limb then an
extra limb is added since the most significant limb of _mp_d is only
non-zero and therefore might contain only one bit.
__GMPF_PREC_TO_BITS does the reverse conversion, and removes the extra
limb from _mp_prec before converting to bits. The net effect of
reading back with mpf_get_prec is simply the precision rounded up to a
multiple of mp_bits_per_limb.
Note that the extra limb added here for the high only being non-zero is in
addition to the extra limb allocated to _mp_d. For example with a
32-bit limb, an application request for 250 bits will be rounded up to 8
limbs, then an extra added for the high being only non-zero, giving an
_mp_prec of 9. _mp_d then gets 10 limbs allocated. Reading
back with mpf_get_prec will take _mp_prec subtract 1 limb and
multiply by 32, giving 256 bits.
Strictly speaking, the fact the high limb has at least one bit means that a
float with, say, 3 limbs of 32-bits each will be holding at least 65 bits, but
for the purposes of mpf_t it's considered simply to be 64 bits, a nice
multiple of the limb size.
mpz_out_raw uses the following format.
+------+------------------------+ | size | data bytes | +------+------------------------+ |
The size is 4 bytes written most significant byte first, being the number of subsequent data bytes, or the twos complement negative of that when a negative integer is represented. The data bytes are the absolute value of the integer, written most significant byte first.
The most significant data byte is always non-zero, so the output is the same on all systems, irrespective of limb size.
In GMP 1, leading zero bytes were written to pad the data bytes to a multiple
of the limb size. mpz_inp_raw will still accept this, for
compatibility.
The use of "big endian" for both the size and data fields is deliberate, it
makes the data easy to read in a hex dump of a file. Unfortunately it also
means that the limb data must be reversed when reading or writing, so neither
a big endian nor little endian system can just read and write _mp_d.
A system of expression templates is used to ensure something like a=b+c
turns into a simple call to mpz_add etc. For mpf_class and
mpfr_class the scheme also ensures the precision of the final
destination is used for any temporaries within a statement like
f=w*x+y*z. These are important features which a naive implementation
cannot provide.
A simplified description of the scheme follows. The true scheme is complicated by the fact that expressions have different return types. For detailed information, refer to the source code.
To perform an operation, say, addition, we first define a "function object" evaluating it,
struct __gmp_binary_plus
{
static void eval(mpf_t f, mpf_t g, mpf_t h) { mpf_add(f, g, h); }
};
|
And an "additive expression" object,
__gmp_expr<__gmp_binary_expr<mpf_class, mpf_class, __gmp_binary_plus> >
operator+(const mpf_class &f, const mpf_class &g)
{
return __gmp_expr
<__gmp_binary_expr<mpf_class, mpf_class, __gmp_binary_plus> >(f, g);
}
|
The seemingly redundant __gmp_expr<__gmp_binary_expr<...>> is used to
encapsulate any possible kind of expression into a single template type. In
fact even mpf_class etc are typedef specializations of
__gmp_expr.
Next we define assignment of __gmp_expr to mpf_class.
template <class T>
mpf_class & mpf_class::operator=(const __gmp_expr<T> &expr)
{
expr.eval(this->get_mpf_t(), this->precision());
return *this;
}
template <class Op>
void __gmp_expr<__gmp_binary_expr<mpf_class, mpf_class, Op> >::eval
(mpf_t f, unsigned long int precision)
{
Op::eval(f, expr.val1.get_mpf_t(), expr.val2.get_mpf_t());
}
|
where expr.val1 and expr.val2 are references to the expression's
operands (here expr is the __gmp_binary_expr stored within the
__gmp_expr).
This way, the expression is actually evaluated only at the time of assignment,
when the required precision (that of f) is known. Furthermore the
target mpf_t is now available, thus we can call mpf_add directly
with f as the output argument.
Compound expressions are handled by defining operators taking subexpressions as their arguments, like this:
template <class T, class U>
__gmp_expr
<__gmp_binary_expr<__gmp_expr<T>, __gmp_expr<U>, __gmp_binary_plus> >
operator+(const __gmp_expr<T> &expr1, const __gmp_expr<U> &expr2)
{
return __gmp_expr
<__gmp_binary_expr<__gmp_expr<T>, __gmp_expr<U>, __gmp_binary_plus> >
(expr1, expr2);
}
|
And the corresponding specializations of __gmp_expr::eval:
template <class T, class U, class Op>
void __gmp_expr
<__gmp_binary_expr<__gmp_expr<T>, __gmp_expr<U>, Op> >::eval
(mpf_t f, unsigned long int precision)
{
// declare two temporaries
mpf_class temp1(expr.val1, precision), temp2(expr.val2, precision);
Op::eval(f, temp1.get_mpf_t(), temp2.get_mpf_t());
}
|
The expression is thus recursively evaluated to any level of complexity and
all subexpressions are evaluated to the precision of f.
Torbjorn Granlund wrote the original GMP library and is still developing and maintaining it. Several other individuals and organizations have contributed to GMP in various ways. Here is a list in chronological order:
Gunnar Sjoedin and Hans Riesel helped with mathematical problems in early versions of the library.
Richard Stallman contributed to the interface design and revised the first version of this manual.
Brian Beuning and Doug Lea helped with testing of early versions of the library and made creative suggestions.
John Amanatides of York University in Canada contributed the function
mpz_probab_prime_p.
Paul Zimmermann of Inria sparked the development of GMP 2, with his comparisons between bignum packages.
Ken Weber (Kent State University, Universidade Federal do Rio Grande do Sul)
contributed mpz_gcd, mpz_divexact, mpn_gcd, and
mpn_bdivmod, partially supported by CNPq (Brazil) grant 301314194-2.
Per Bothner of Cygnus Support helped to set up GMP to use Cygnus' configure. He has also made valuable suggestions and tested numerous intermediary releases.
Joachim Hollman was involved in the design of the mpf interface, and in
the mpz design revisions for version 2.
Bennet Yee contributed the initial versions of mpz_jacobi and
mpz_legendre.
Andreas Schwab contributed the files `mpn/m68k/lshift.S' and `mpn/m68k/rshift.S' (now in `.asm' form).
The development of floating point functions of GNU MP 2, were supported in part by the ESPRIT-BRA (Basic Research Activities) 6846 project POSSO (POlynomial System SOlving).
GNU MP 2 was finished and released by SWOX AB, SWEDEN, in cooperation with the IDA Center for Computing Sciences, USA.
Robert Harley of Inria, France and David Seal of ARM, England, suggested clever improvements for population count.
Robert Harley also wrote highly optimized Karatsuba and 3-way Toom multiplication functions for GMP 3. He also contributed the ARM assembly code.
Torsten Ekedahl of the Mathematical department of Stockholm University provided significant inspiration during several phases of the GMP development. His mathematical expertise helped improve several algorithms.
Paul Zimmermann wrote the Divide and Conquer division code, the REDC code, the REDC-based mpz_powm code, the FFT multiply code, and the Karatsuba square root. The ECMNET project Paul is organizing was a driving force behind many of the optimizations in GMP 3.
Linus Nordberg wrote the new configure system based on autoconf and implemented the new random functions.
Kent Boortz made the Macintosh port.
Kevin Ryde worked on a number of things: optimized x86 code, m4 asm macros, parameter tuning, speed measuring, the configure system, function inlining, divisibility tests, bit scanning, Jacobi symbols, Fibonacci and Lucas number functions, printf and scanf functions, perl interface, demo expression parser, the algorithms chapter in the manual, gmpasm-mode.el, and various miscellaneous improvements elsewhere.
Steve Root helped write the optimized alpha 21264 assembly code.
Gerardo Ballabio wrote the `gmpxx.h' C++ class interface and the C++ istream input routines.
GNU MP 4.0.1 was finished and released by Torbjorn Granlund and Kevin Ryde. Torbjorn's work was partially funded by the IDA Center for Computing Sciences, USA.
(This list is chronological, not ordered after significance. If you have contributed to GMP but are not listed above, please tell tege@swox.com about the omission!)
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| [Top] | [Contents] | [Index] | [ ? ] |
GNU MP Copying Conditions
1. Introduction to GNU MP
2. Installing GMP
3. GMP Basics
4. Reporting Bugs
5. Integer Functions
6. Rational Number Functions
7. Floating-point Functions
8. Low-level Functions
9. Random Number Functions
10. Formatted Output
11. Formatted Input
12. C++ Class Interface
13. Berkeley MP Compatible Functions
14. Custom Allocation
15. Language Bindings
16. Algorithms
17. Internals
A. Contributors
B. References
C. GNU Free Documentation License
Concept Index
Function and Type Index
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