START-INFO-DIR-ENTRY * gmp: (gmp). GNU Multiple Precision Arithmetic Library. END-INFO-DIR-ENTRY GNU MP ****** This manual describes how to install and use the GNU multiple precision arithmetic library, version 4.1. 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 *Note GNU Free Documentation License::. ...Table of Contents... GNU MP Copying Conditions ************************* This library is "free"; this means that everyone is free to use it and free to redistribute it on a free basis. The library is not in the public domain; it is copyrighted and there are restrictions on its distribution, but these restrictions are designed to permit everything that a good cooperating citizen would want to do. What is not allowed is to try to prevent others from further sharing any version of this library that they might get from you. Specifically, we want to make sure that you have the right to give away copies of the library, that you receive source code or else can get it if you want it, that you can change this library or use pieces of it in new free programs, and that you know you can do these things. To make sure that everyone has such rights, we have to forbid you to deprive anyone else of these rights. For example, if you distribute copies of the GNU MP library, you must give the recipients all the rights that you have. You must make sure that they, too, receive or can get the source code. And you must tell them their rights. Also, for our own protection, we must make certain that everyone finds out that there is no warranty for the GNU MP library. If it is modified by someone else and passed on, we want their recipients to know that what they have is not what we distributed, so that any problems introduced by others will not reflect on our reputation. 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'. Introduction to GNU MP ********************** GNU MP is a portable library written in C for arbitrary precision arithmetic on integers, rational numbers, and floating-point numbers. It aims to provide the fastest possible arithmetic for all applications that need higher precision than is directly supported by the basic C types. Many applications use just a few hundred bits of precision; but some applications may need thousands or even millions of bits. GMP is designed to give good performance for both, by choosing algorithms based on the sizes of the operands, and by carefully keeping the overhead at a minimum. The speed of GMP is achieved by using fullwords as the basic arithmetic type, by using sophisticated algorithms, by including carefully optimized assembly code for the most common inner loops for many different CPUs, and by a general emphasis on speed (as opposed to simplicity or elegance). There is 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 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. How to use this Manual ====================== Everyone should read *Note GMP Basics::. If you need to install the library yourself, then read *Note Installing GMP::. If you have a system with multiple ABIs, then read *Note ABI and ISA::, for the compiler options that must be used on applications. The rest of the manual can be used for later reference, although it is probably a good idea to glance through it. Installing GMP ************** GMP has an autoconf/automake/libtool based configuration system. On a Unix-like system a basic build can be done with ./configure make Some self-tests can be run with make check And you can install (under `/usr/local' by default) with make install If you experience problems, please report them to . See *Note Reporting Bugs::, for information on what to include in useful bug reports. Build Options ============= All the usual autoconf configure options are available, run `./configure --help' for a summary. The file `INSTALL.autoconf' has some generic installation information too. Non-Unix Systems `configure' requires various Unix-like tools. On an MS-DOS system DJGPP can be used, and on MS Windows Cygwin or MINGW can be used, `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, see `macos/README'. Note that MacOS X "Darwin" should use the normal Unix-style `./configure'. It might be possible to build without the help of `configure', certainly all the code is there, but unfortunately you'll be on your own. Build Directory To compile in a separate build directory, `cd' to that directory, and prefix the configure command with the path to the GMP source directory. For example cd /my/build/dir /my/sources/gmp-4.1/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. `--disable-shared', `--disable-static' By default both shared and static libraries are built (where possible), but one or other can be disabled. Shared libraries result in smaller executables and permit code sharing between separate running processes, but on some CPUs are slightly slower, having a small cost on each function call. Native Compilation, `--build=CPU-VENDOR-OS' For normal native compilation, the system can be specified with `--build'. By default `./configure' uses the output from running `./config.guess'. On some systems `./config.guess' can determine the exact CPU type, on others it will be necessary to give it explicitly. For example, ./configure --build=ultrasparc-sun-solaris2.7 In all cases the `OS' part is important, since it controls how libtool generates shared libraries. Running `./config.guess' is the simplest way to see what it should be, if you don't know already. Cross Compilation, `--host=CPU-VENDOR-OS' When cross-compiling, the system used for compiling is given by `--build' and the system where the library will run is given by `--host'. For example when using a FreeBSD Athlon system to build GNU/Linux m68k binaries, ./configure --build=athlon-pc-freebsd3.5 --host=m68k-mac-linux-gnu Compiler tools are sought first with the host system type as a prefix. For example `m68k-mac-linux-gnu-ranlib' is tried, then plain `ranlib'. This makes it possible for a set of cross-compiling tools to co-exist with native tools. The prefix is the argument to `--host', and this can be an alias, such as `m68k-linux'. But note that tools don't have to be setup this way, it's enough to just have a `PATH' with a suitable cross-compiling `cc' etc. Compiling for a different CPU in the same family as the build system is a form of cross-compilation, though very possibly this would merely be special options on a native compiler. In any case `./configure' avoids depending on being able to run code on the build system, which is important when creating binaries for a newer CPU since they very possibly won't run on the build system. In all cases the compiler must be able to produce an executable (of whatever format) from a standard C `main'. Although only object files will go to make up `libgmp', `./configure' uses linking tests for various purposes, such as determining what functions are available on the host system. Currently a warning is given unless an explicit `--build' is used when cross-compiling, because it may not be possible to correctly guess the build system type if the `PATH' has only a cross-compiling `cc'. Note that the `--target' option is not appropriate for GMP. It's for use when building compiler tools, with `--host' being where they will run, and `--target' what they'll produce code for. Ordinary programs or libraries like GMP are only interested in the `--host' part, being where they'll run. (Some past versions of GMP used `--target' incorrectly.) CPU types In general, if you want a library that runs as fast as possible, you should configure GMP for the exact CPU type your system uses. However, this may mean the binaries won't run on older members of the family, and might run slower on other members, older or newer. The best idea is always to build GMP for the exact machine type you intend to run it on. The following CPUs have specific support. See `configure.in' for details of what code and compiler options they select. * Alpha: alpha, alphaev5, alphaev56, alphapca56, alphapca57, alphaev6, alphaev67, alphaev68 * Cray: c90, j90, t90, sv1 * HPPA: hppa1.0, hppa1.1, hppa2.0, hppa2.0n, hppa2.0w * MIPS: mips, mips3, mips64 * Motorola: m68k, m68000, m68010, m68020, m68030, m68040, m68060, m68302, m68360, m88k, m88110 * POWER: power, power1, power2, power2sc * PowerPC: powerpc, powerpc64, powerpc401, powerpc403, powerpc405, powerpc505, powerpc601, powerpc602, powerpc603, powerpc603e, powerpc604, powerpc604e, powerpc620, powerpc630, powerpc740, powerpc7400, powerpc7450, powerpc750, powerpc801, powerpc821, powerpc823, powerpc860, * SPARC: sparc, sparcv8, microsparc, supersparc, sparcv9, ultrasparc, ultrasparc2, ultrasparc2i, ultrasparc3, sparc64 * 80x86 family: i386, i486, i586, pentium, pentiummmx, pentiumpro, pentium2, pentium3, pentium4, k6, k62, k63, athlon * Other: a29k, arm, clipper, i960, ns32k, pyramid, sh, sh2, vax, z8k CPUs not listed will use generic C code. Generic C Build If some of the assembly code causes problems, or if otherwise desired, the generic C code can be selected with CPU `none'. For example, ./configure --host=none-unknown-freebsd3.5 Note that this will run quite slowly, but it should be portable and should at least make it possible to get something running if all else fails. `ABI' On some systems GMP supports multiple ABIs (application binary interfaces), meaning data type sizes and calling conventions. By default GMP chooses the best ABI available, but a particular ABI can be selected. For example ./configure --host=mips64-sgi-irix6 ABI=n32 See *Note ABI and ISA::, for the available choices on relevant CPUs, and what applications need to do. `CC', `CFLAGS' By default the C compiler used is chosen from among some likely candidates, with `gcc' normally preferred if it's present. The usual `CC=whatever' can be passed to `./configure' to choose something different. For 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). `CPPFLAGS' Any flags like `-D' defines or `-I' includes required by the preprocessor should be set in `CPPFLAGS' rather than `CFLAGS'. Compiling is done with both `CPPFLAGS' and `CFLAGS', but preprocessing uses just `CPPFLAGS'. This distinction is because most preprocessors won't accept all the flags the compiler does. Preprocessing is done separately in some configure tests, and in the `ansi2knr' support for K&R compilers. C++ Support, `--enable-cxx' C++ support in GMP can be enabled with `--enable-cxx', in which case a C++ compiler will be required. As a convenience `--enable-cxx=detect' can be used to enable C++ support only if a compiler can be found. The C++ support consists of a library `libgmpxx.la' and header file `gmpxx.h'. A separate `libgmpxx.la' has been adopted rather than having C++ objects within `libgmp.la' in order to ensure dynamic linked C programs aren't bloated by a dependency on the C++ standard library, and to avoid any chance that the C++ compiler could be required when linking plain C programs. `libgmpxx.la' will use certain internals from `libgmp.la' and can only be expected to work with `libgmp.la' from the same GMP version. Future changes to the relevant internals will be accompanied by renaming, so a mismatch will cause unresolved symbols rather than perhaps mysterious misbehaviour. In general `libgmpxx.la' will be usable only with the C++ compiler that built it, since name mangling and runtime support are usually incompatible between different compilers. `CXX', `CXXFLAGS' When C++ support is enabled, the C++ compiler and its flags can be set with variables `CXX' and `CXXFLAGS' in the usual way. The default for `CXX' is the first compiler that works from a list of likely candidates, with `g++' normally preferred when available. The default for `CXXFLAGS' is to try `CFLAGS', `CFLAGS' without `-g', then for `g++' either `-g -O2' or `-O2', or for other compilers `-g' or nothing. Trying `CFLAGS' this way is convenient when using `gcc' and `g++' together, since the flags for `gcc' will usually suit `g++'. It's important that the C and C++ compilers match, meaning their startup and runtime support routines are compatible and that they generate code in the same ABI (if there's a choice of ABIs on the system). `./configure' isn't currently able to check these things very well itself, so for that reason `--disable-cxx' is the default, to avoid a build failure due to a compiler mismatch. Perhaps this will change in the future. Incidentally, it's normally not good enough to set `CXX' to the same as `CC'. Although `gcc' for instance recognises `foo.cc' as C++ code, only `g++' will invoke the linker the right way when building an executable or shared library from object files. Temporary Memory, `--enable-alloca=' GMP allocates temporary workspace using one of the following three methods, which can be selected with for instance `--enable-alloca=malloc-reentrant'. * `alloca' - C library or compiler builtin. * `malloc-reentrant' - the heap, in a re-entrant fashion. * `malloc-notreentrant' - the heap, with global variables. For convenience, the following choices are also available. `--disable-alloca' is the same as `--enable-alloca=no'. * `yes' - a synonym for `alloca'. * `no' - a synonym for `malloc-reentrant'. * `reentrant' - `alloca' if available, otherwise `malloc-reentrant'. This is the default. * `notreentrant' - `alloca' if available, otherwise `malloc-notreentrant'. `alloca' is reentrant and fast, and is recommended, 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. *Note Custom Allocation::. An additional choice `--enable-alloca=debug' is available, to help when debugging memory related problems (*note Debugging::). FFT Multiplication, `--disable-fft' By default multiplications are done using Karatsuba, 3-way Toom-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. Berkeley MP, `--enable-mpbsd' The Berkeley MP compatibility library (`libmp') and header file (`mp.h') are built and installed only if `--enable-mpbsd' is used. *Note BSD Compatible Functions::. MPFR, `--enable-mpfr' 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 (*note Introduction to MPFR: (mpfr)Introduction to MPFR.). Assertion Checking, `--enable-assert' This option enables some consistency checking within the library. This can be of use while debugging, *note Debugging::. Execution Profiling, `--enable-profiling=prof/gprof' 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. *Note Profiling::. `MPN_PATH' Various assembler versions of each mpn subroutines are provided. For a given CPU, a search is made though a path to choose a version of each. For example `sparcv8' has MPN_PATH="sparc32/v8 sparc32 generic" which means look first for v8 code, then plain sparc32 (which is v7), and finally fall back on generic C. Knowledgeable users with special requirements can specify a different path. Normally this is completely unnecessary. Demonstration Programs The `demos' subdirectory has some sample programs using GMP. These aren't built or installed, but there's a `Makefile' with rules for them. For instance, make pexpr ./pexpr 68^975+10 Documentation 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 *Note Generating HTML: (texinfo)makeinfo html. Or alternately `texi2html', see *Note Texinfo to HTML: (texi2html)Top. PDF can be produced with `texi2dvi --pdf' (*note PDF: (texinfo)PDF Output.) or with `pdftex'. Some supplementary notes can be found in the `doc' subdirectory. ABI and ISA =========== ABI (Application Binary Interface) refers to the calling conventions between functions, meaning what registers are used and what sizes the various C data types are. ISA (Instruction Set Architecture) refers to the instructions and registers a CPU has available. Some 64-bit ISA CPUs have both a 64-bit ABI and a 32-bit ABI defined, the latter for compatibility with older CPUs in the family. GMP supports some CPUs like this in both ABIs. In fact within GMP `ABI' means a combination of chip ABI, plus how GMP chooses to use it. For example in some 32-bit ABIs, GMP may support a limb as either a 32-bit `long' or a 64-bit `long long'. By default GMP chooses the best ABI available for a given system, and this generally gives significantly greater speed. But an ABI can be chosen explicitly to make GMP compatible with other libraries, or particular application requirements. For example, ./configure ABI=32 In all cases it's vital that all object code used in a given program is compiled for the same ABI. Usually a limb is implemented as a `long'. When a `long long' limb is used this is encoded in the generated `gmp.h'. This is convenient for applications, but it does mean that `gmp.h' will vary, and can't be just copied around. `gmp.h' remains compiler independent though, since all compilers for a particular ABI will be expected to use the same limb type. Currently no attempt is made to follow whatever conventions a system has for installing library or header files built for a particular ABI. This will probably only matter when installing multiple builds of GMP, and it might be as simple as configuring with a special `libdir', or it might require more than that. Note that builds for different ABIs need to done separately, with a fresh `./configure' and `make' each. HPPA 2.0 (`hppa2.0*') `ABI=2.0w' 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 `ABI=2.0n' 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 `ABI=1.0' HPPA 2.0 CPUs can run all HPPA 1.0 and 1.1 code in the 32-bit HPPA 1.0 ABI. No special compiler options are needed for applications. All three ABIs are available for CPUs `hppa2.0w' and `hppa2.0', but for CPU `hppa2.0n' only 2.0n or 1.0 are allowed. MIPS under IRIX 6 (`mips*-*-irix[6789]') 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). `ABI=n32' The n32 ABI is 32-bit pointers and integers, but with a 64-bit limb using a `long long'. Applications must be compiled with gcc -mabi=n32 cc -n32 `ABI=64' The 64-bit ABI is 64-bit pointers and integers. Applications must be compiled with gcc -mabi=64 cc -64 Note that MIPS GNU/Linux, as of kernel version 2.2, doesn't have the necessary support for n32 or 64 and so only gets a 32-bit limb and the MIPS 2 code. PowerPC 64 (`powerpc64*') `ABI=aix64' 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 `ABI=32L' 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 `ABI=32' This is the basic 32-bit PowerPC ABI. No special compiler options are needed for applications. Sparc V9 (`sparcv9' and `ultrasparc*') `ABI=64' 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 `ABI=32' 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 Notes for Package Builds ======================== GMP should present no great difficulties for packaging in a binary distribution. Libtool is used to build the library and `-version-info' is set appropriately, having started from `3:0:0' in GMP 3.0. The GMP 4 series will be upwardly binary compatible in each release and will be upwardly binary compatible with all of the GMP 3 series. Additional function interfaces may be added in each release, so on systems where libtool versioning is not fully checked by the loader an auxiliary mechanism may be needed to express that a dynamic linked application depends on a new enough GMP. An auxiliary mechanism may also be needed to express that `libgmpxx.la' (from `--enable-cxx', *note Build Options::) requires `libgmp.la' from the same GMP version, since this is not done by the libtool versioning, nor otherwise. A mismatch will result in unresolved symbols from the linker, or perhaps the loader. Using `DESTDIR' or a `prefix' override with `make install' and a shared `libgmpxx' may run into a libtool relinking problem, see *Note Known Build Problems::. 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. Notes for Particular Systems ============================ AIX 3 and 4 On systems `*-*-aix[34]*' shared libraries are disabled by default, since some versions of the native `ar' fail on the convenience libraries used. A shared build can be attempted with ./configure --enable-shared --disable-static Note that the `--disable-static' is necessary because in a shared build libtool makes `libgmp.a' a symlink to `libgmp.so', apparently for the benefit of old versions of `ld' which only recognise `.a', but unfortunately this is done even if a fully functional `ld' is available. ARM On systems `arm*-*-*', versions of GCC up to and including 2.95.3 have a bug in unsigned division, giving wrong results for some operands. GMP `./configure' will demand GCC 2.95.4 or later. Microsoft Windows On systems `*-*-cygwin*', `*-*-mingw*' and `*-*-pw32*' by default GMP builds only a static library, but a DLL can be built instead using ./configure --disable-static --enable-shared Static and DLL libraries can't both be built, since certain export directives in `gmp.h' must be different. `--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. Motorola 68k CPU Types `m68k' is taken to mean 68000. `m68020' or higher will give a performance boost on applicable CPUs. `m68360' can be used for CPU32 series chips. `m68302' can be used for "Dragonball" series chips, though this is merely a synonym for `m68000'. OpenBSD 2.6 `m4' in this release of OpenBSD has a bug in `eval' that makes it unsuitable for `.asm' file processing. `./configure' will detect the problem and either abort or choose another m4 in the `PATH'. The bug is fixed in OpenBSD 2.7, so either upgrade or use GNU m4. Power CPU Types In GMP, CPU types `power*' and `powerpc*' will each use instructions not available on the other, so it's important to choose the right one for the CPU that will be used. Currently GMP has no 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'. Sparc CPU Types `sparcv8' or `supersparc' on relevant systems will give a significant performance increase over the V7 code. SunOS 4 `/usr/bin/m4' lacks various features needed to process `.asm' files, and instead `./configure' will automatically use `/usr/5bin/m4', which we believe is always available (if not then use GNU m4). x86 CPU Types `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 speedup over the generic `i386' code. x86 MMX and SSE2 Code If the CPU selected has MMX code but the assembler doesn't support it, a warning is given and non-MMX code is used instead. This will be an inferior build, since the MMX code that's present is there because it's faster than the corresponding plain integer code. The same applies to SSE2. Old versions of `gas' don't support MMX instructions, in particular version 1.92.3 that comes with FreeBSD 2.2.8 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. x86 GCC `-march=pentiumpro' 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. Known Build Problems ==================== You might find more up-to-date information at `http://swox.com/gmp/'. DJGPP 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. `DESTDIR' and shared `libgmpxx' `make install DESTDIR=/my/staging/area' or the same with a `prefix' override to install to a temporary directory is not fully supported by current versions of libtool when building a shared version of a library which depends on another being built at the same time, like `libgmpxx' and `libgmp'. The problem is that `libgmpxx' is relinked at the install stage to ensure that if the system puts a hard-coded path to `libgmp' within `libgmpxx' then that path will be correct. Naturally the linker is directed to look only at the final location, not the staging area, so if `libgmp' is not already in that final location then the link will fail. On systems which don't hard-code library paths, for instance SVR4 style systems such as GNU/Linux, a workaround is to insert a suitable `-L' in the `relink_command' of `libgmpxx.la' after building but before installing. This can be automated with something like sed '/^relink_command/s:libgmp.la:-L /my/staging/area libgmp.la:' \ libgmpxx.new mv libgmpxx.new libgmpxx.la GNU binutils `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. `make' syntax error On certain versions of SCO OpenServer 5 and IRIX 6.5 the native `make' is unable to handle the long dependencies list for `libgmp.la'. The symptom is a "syntax error" on the following line of the top-level `Makefile'. libgmp.la: $(libgmp_la_OBJECTS) $(libgmp_la_DEPENDENCIES) Either use GNU Make, or as a workaround remove `$(libgmp_la_DEPENDENCIES)' from that line (which will make the initial build work, but if any recompiling is done `libgmp.la' might not be rebuilt). NeXT prior to 3.3 The system compiler on old versions of NeXT was a massacred and old GCC, even if it called itself `cc'. This compiler cannot be used to build GMP, you need to get a real GCC, and install that. (NeXT may have fixed this in release 3.3 of their system.) POWER and PowerPC Bugs in GCC 2.7.2 (and 2.6.3) mean it can't be used to compile GMP on POWER or PowerPC. If you want to use GCC for these machines, get GCC 2.7.2.1 (or later). Sequent Symmetry Use the GNU assembler instead of the system assembler, since the latter has serious bugs. Solaris 2.6 The system `sed' prints an error "Output line too long" when libtool builds `libgmp.la'. This doesn't seem to cause any obvious ill effects, but GNU `sed' is recommended, to avoid any doubt. Sparc Solaris 2.7 with gcc 2.95.2 in ABI=32 A shared library build of GMP seems to fail in this combination, it builds but then fails the tests, apparently due to some incorrect data relocations within `gmp_randinit_lc_2exp_size'. The exact cause is unknown, `--disable-shared' is recommended. Windows DLL test programs 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'. GMP Basics ********** *Using functions, macros, data types, etc. not documented in this manual is strongly discouraged. If you do so your application is guaranteed to be incompatible with future versions of GMP.* Headers and Libraries ===================== All declarations needed to use GMP are collected in the include file `gmp.h'. It is designed to work with both C and C++ compilers. #include Note however that prototypes for GMP functions with `FILE *' parameters are only provided if `' is included too. #include #include Likewise `' (or `') is required for prototypes with `va_list' parameters, such as `gmp_vprintf'. And `' for prototypes with `struct obstack' parameters, such as `gmp_obstack_printf', when available. All programs using GMP must link against the `libgmp' library. On a typical Unix-like system this can be done with `-lgmp', for example gcc myprogram.c -lgmp GMP C++ functions are in a separate `libgmpxx' library. This is built and installed if C++ support has been enabled (*note Build Options::). For example, g++ mycxxprog.cc -lgmpxx -lgmp GMP is built using Libtool and an application can use that to link if desired, *note Shared library support for GNU: (libtool)Top. If GMP has been installed to a non-standard location then it may be necessary to use `-I' and `-L' compiler options to point to the right directories, and some sort of run-time path for a shared library. Consult your compiler documentation, for instance *Note Introduction: (gcc)Top. Nomenclature and Types ====================== In this manual, "integer" usually means a multiple precision integer, as defined by the GMP library. The C data type for such integers is `mpz_t'. Here are some examples of how to declare such integers: mpz_t sum; struct foo { mpz_t x, y; }; mpz_t vec[20]; "Rational number" means a multiple precision fraction. The C data type for these fractions is `mpq_t'. For example: mpq_t quotient; "Floating point number" or "Float" for short, is an arbitrary precision mantissa with a limited precision exponent. The C data type for such objects is `mpf_t'. 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'. Function Classes ================ There are six classes of functions in the GMP library: 1. Functions for signed integer arithmetic, with names beginning with `mpz_'. The associated type is `mpz_t'. There are about 150 functions in this class. 2. Functions for rational number arithmetic, with names beginning with `mpq_'. The associated type is `mpq_t'. There are about 40 functions in this class, but the integer functions can be used for arithmetic on the numerator and denominator separately. 3. Functions for floating-point arithmetic, with names beginning with `mpf_'. The associated type is `mpf_t'. There are about 60 functions is this class. 4. Functions compatible with Berkeley MP, such as `itom', `madd', and `mult'. The associated type is `MINT'. 5. Fast low-level functions that operate on natural numbers. These are used by the functions in the preceding groups, and you can also call them directly from very time-critical user programs. These functions' names begin with `mpn_'. The associated type is array of `mp_limb_t'. There are about 30 (hard-to-use) functions in this class. 6. Miscellaneous functions. Functions for setting up custom allocation and functions for generating random numbers. Variable Conventions ==================== GMP functions generally have output arguments before input arguments. This notation is by analogy with the assignment operator. The BSD MP compatibility functions are exceptions, having the output arguments last. GMP lets you use the same variable for both input and output in one call. For example, the main function for integer multiplication, `mpz_mul', can be used to square `x' and put the result back in `x' with mpz_mul (x, x, x); Before you can assign to a GMP variable, you need to initialize it by calling one of the special initialization functions. When you're done with a variable, you need to clear it out, using one of the functions for that purpose. Which function to use depends on the type of variable. See the chapters on integer functions, rational number functions, and floating-point functions for details. A variable should only be initialized once, or at least cleared between each initialization. After a variable has been initialized, it may be assigned to any number of times. For efficiency reasons, avoid excessive initializing and clearing. In general, initialize near the start of a function and clear near the end. For example, void foo (void) { mpz_t n; int i; mpz_init (n); for (i = 1; i < 100; i++) { mpz_mul (n, ...); mpz_fdiv_q (n, ...); ... } mpz_clear (n); } Parameter Conventions ===================== When a GMP variable is used as a function parameter, it's effectively a call-by-reference, meaning if the function stores a value there it will change the original in the caller. Parameters which are input-only can be designated `const' to provoke a compiler error or warning on attempting to modify them. When a function is going to return a GMP result, it should designate a parameter that it sets, like the library functions do. More than one value can be returned by having more than one output parameter, again like the library functions. A `return' of an `mpz_t' etc doesn't return the object, only a pointer, and this is almost certainly not what's wanted. Here's an example accepting an `mpz_t' parameter, doing a calculation, and storing the result to the indicated parameter. void foo (mpz_t result, const mpz_t param, unsigned long n) { unsigned long i; mpz_mul_ui (result, param, n); for (i = 1; i < n; i++) mpz_add_ui (result, result, i*7); } int main (void) { mpz_t r, n; mpz_init (r); mpz_init_set_str (n, "123456", 0); foo (r, n, 20L); gmp_printf ("%Zd\n", r); return 0; } `foo' works even if the mainline passes the same variable for `param' and `result', just like the library functions. But sometimes it's tricky to make that work, and an application might not want to bother supporting that sort of thing. For interest, the GMP types `mpz_t' etc are implemented as one-element arrays of certain structures. This is why declaring a variable creates an object with the fields GMP needs, but then using it as a parameter passes a pointer to the object. Note that the actual fields in each `mpz_t' etc are for internal use only and should not be accessed directly by code that expects to be compatible with future GMP releases. Memory Management ================= The GMP types like `mpz_t' are small, containing only a couple of sizes, and pointers to allocated data. Once a variable is initialized, GMP takes care of all space allocation. Additional space is allocated whenever a variable doesn't have enough. `mpz_t' and `mpq_t' variables never reduce their allocated space. Normally this is the best policy, since it avoids frequent reallocation. Applications that need to return memory to the heap at some particular point can use `mpz_realloc2', or clear variables no longer needed. `mpf_t' variables, in the current implementation, use a fixed amount of space, determined by the chosen precision and allocated at initialization, so their size doesn't change. All memory is allocated using `malloc' and friends by default, but this can be changed, see *Note Custom Allocation::. Temporary memory on the stack is also used (via `alloca'), but this can be changed at build-time if desired, see *Note Build Options::. Reentrancy ========== GMP is reentrant and thread-safe, with some exceptions: * If configured with `--enable-alloca=malloc-notreentrant' (or with `--enable-alloca=notreentrant' when `alloca' is not available), then naturally GMP is not reentrant. * `mpf_set_default_prec' and `mpf_init' use a global variable for the selected precision. `mpf_init2' can be used instead. * `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' uses global variables to store the selected memory allocation functions. * If the memory allocation functions set by a call to `mp_set_memory_functions' (or `malloc' and friends by default) are not reentrant, then GMP will not be reentrant either. * If the standard I/O functions such as `fwrite' are not reentrant then the GMP I/O functions using them will not be reentrant either. * It's safe for two threads to read from the same GMP variable simultaneously, but it's not safe for one to read while the another might be writing, nor for two threads to write simultaneously. It's not safe for two threads to generate a random number from the same `gmp_randstate_t' simultaneously, since this involves an update of that variable. * On SCO systems the default `' 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. Useful Macros and Constants =========================== - Global Constant: const int mp_bits_per_limb The number of bits per limb. - Macro: __GNU_MP_VERSION - Macro: __GNU_MP_VERSION_MINOR - Macro: __GNU_MP_VERSION_PATCHLEVEL The major and minor GMP version, and patch level, respectively, as integers. For GMP i.j, these numbers will be i, j, and 0, respectively. For GMP i.j.k, these numbers will be i, j, and k, respectively. - Global Constant: const char * const gmp_version The GMP version number, as a null-terminated string, in the form "i.j" or "i.j.k". This release is "4.1". Compatibility with older versions ================================= This version of GMP is upwardly binary compatible with all 4.x and 3.x versions, and upwardly compatible at the source level with all 2.x 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 (*note BSD Compatible Functions::) is source and binary compatible with the standard `libmp'. Efficiency ========== Small operands On small operands, the time for function call overheads and memory allocation can be significant in comparison to actual calculation. This is unavoidable in a general purpose variable precision library, although GMP attempts to be as efficient as it can on both large and small operands. Static Linking On some CPUs, in particular the x86s, the static `libgmp.a' should be used for maximum speed, since the PIC code in the shared `libgmp.so' will have a small overhead on each function call and global data address. For many programs this will be insignificant, but for long calculations there's a gain to be had. Initializing and clearing Avoid excessive initializing and clearing of variables, since this can be quite time consuming, especially in comparison to otherwise fast operations like addition. A language interpreter might want to keep a free list or stack of initialized variables ready for use. It should be possible to integrate something like that with a garbage collector too. Reallocations An `mpz_t' or `mpq_t' variable used to hold successively increasing values will have its memory repeatedly `realloc'ed, which could be quite slow or could fragment memory, depending on the C library. If an application can estimate the final size then `mpz_init2' or `mpz_realloc2' can be called to allocate the necessary space from the beginning (*note Initializing Integers::). It doesn't matter if a size set with `mpz_init2' or `mpz_realloc2' is too small, since all functions will do a further reallocation if necessary. Badly overestimating memory required will waste space though. `2exp' functions It's up to an application to call functions like `mpz_mul_2exp' when appropriate. General purpose functions like `mpz_mul' make no attempt to identify powers of two or other special forms, because such inputs will usually be very rare and testing every time would be wasteful. `ui' and `si' functions The `ui' functions and the small number of `si' functions exist for convenience and should be used where applicable. But if for example an `mpz_t' contains a value that fits in an `unsigned long' there's no need extract it and call a `ui' function, just use the regular `mpz' function. In-Place Operations `mpz_abs', `mpq_abs', `mpf_abs', `mpz_neg', `mpq_neg' and `mpf_neg' are fast when used for in-place operations like `mpz_abs(x,x)', since in the current implementation only a single field of `x' needs changing. On suitable compilers (GCC for instance) this is inlined too. `mpz_add_ui', `mpz_sub_ui', `mpf_add_ui' and `mpf_sub_ui' benefit from an in-place operation like `mpz_add_ui(x,x,y)', since usually only one or two limbs of `x' will need to be changed. The same applies to the full precision `mpz_add' etc if `y' is small. If `y' is big then cache locality may be helped, but that's all. `mpz_mul' is currently the opposite, a separate destination is slightly better. A call like `mpz_mul(x,x,y)' will, unless `y' is only one limb, make a temporary copy of `x' before forming the result. Normally that copying will only be a tiny fraction of the time for the multiply, so this is not a particularly important consideration. `mpz_set', `mpq_set', `mpq_set_num', `mpf_set', etc, make no attempt to recognise a copy of something to itself, so a call like `mpz_set(x,x)' will be wasteful. Naturally that would never be written deliberately, but if it might arise from two pointers to the same object then a test to avoid it might be desirable. if (x != y) mpz_set (x, y); Note that it's never worth introducing extra `mpz_set' calls just to get in-place operations. If a result should go to a particular variable then just direct it there and let GMP take care of data movement. Divisibility Testing (Small Integers) `mpz_divisible_ui_p' and `mpz_congruent_ui_p' are the best functions for testing whether an `mpz_t' is divisible by an individual small integer. They use an algorithm which is faster than `mpz_tdiv_ui', but which gives no useful information about the actual remainder, only whether it's zero (or a particular value). However when testing divisibility by several small integers, it's best to take a remainder modulo their product, to save multi-precision operations. For instance to test whether a number is divisible by any of 23, 29 or 31 take a remainder modulo 23*29*31 = 20677 and then test that. The division functions like `mpz_tdiv_q_ui' which give a quotient as well as a remainder are generally a little slower than the remainder-only functions like `mpz_tdiv_ui'. If the quotient is only rarely wanted then it's probably best to just take a remainder and then go back and calculate the quotient if and when it's wanted (`mpz_divexact_ui' can be used if the remainder is zero). Rational Arithmetic The `mpq' functions operate on `mpq_t' values with no common factors in the numerator and denominator. Common factors are checked-for and cast out as necessary. In general, cancelling factors every time is the best approach since it minimizes the sizes for subsequent operations. However, applications that know something about the factorization of the values they're working with might be able to avoid some of the GCDs used for canonicalization, or swap them for divisions. For example when multiplying by a prime it's enough to check for factors of it in the denominator instead of doing a full GCD. Or when forming a big product it might be known that very little cancellation will be possible, and so canonicalization can be left to the end. The `mpq_numref' and `mpq_denref' macros give access to the numerator and denominator to do things outside the scope of the supplied `mpq' functions. *Note Applying Integer Functions::. The canonical form for rationals allows mixed-type `mpq_t' and integer additions or subtractions to be done directly with multiples of the denominator. This will be somewhat faster than `mpq_add'. For example, /* mpq increment */ mpz_add (mpq_numref(q), mpq_numref(q), mpq_denref(q)); /* mpq += unsigned long */ mpz_addmul_ui (mpq_numref(q), mpq_denref(q), 123UL); /* mpq -= mpz */ mpz_submul (mpq_numref(q), mpq_denref(q), z); Number Sequences Functions like `mpz_fac_ui', `mpz_fib_ui' and `mpz_bin_uiui' are designed for calculating isolated values. If a range of values is wanted it's probably best to call to get a starting point and iterate from there. Text Input/Output Hexadecimal or octal are suggested for input or output in text form. Power-of-2 bases like these can be converted much more efficiently than other bases, like decimal. For big numbers there's usually nothing of particular interest to be seen in the digits, so the base doesn't matter much. Maybe we can hope octal will one day become the normal base for everyday use, as proposed by King Charles XII of Sweden and later reformers. Debugging ========= Stack Overflow Depending on the system, a segmentation violation or bus error might be the only indication of stack overflow. See `--enable-alloca' choices in *Note Build Options::, for how to address this. Heap Problems The most likely cause of application problems with GMP is heap corruption. Failing to `init' GMP variables will have unpredictable effects, and corruption arising elsewhere in a program may well affect GMP. Initializing GMP variables more than once or failing to clear them will cause memory leaks. In all such cases a malloc debugger is recommended. On a GNU or BSD system the standard C library `malloc' has some diagnostic facilities, see *Note Allocation Debugging: (libc)Allocation Debugging, or `man 3 malloc'. Other possibilities, in no particular order, include `http://www.inf.ethz.ch/personal/biere/projects/ccmalloc' `http://quorum.tamu.edu/jon/gnu' (debauch) `http://dmalloc.com' `http://www.perens.com/FreeSoftware' (electric fence) `http://packages.debian.org/fda' `http://www.gnupdate.org/components/leakbug' `http://people.redhat.com/~otaylor/memprof' `http://www.cbmamiga.demon.co.uk/mpatrol' The GMP default allocation routines in `memory.c' also have a simple sentinel scheme which can be enabled with `#define DEBUG' in that file. This is mainly designed for detecting buffer overruns during GMP development, but might find other uses. Stack Backtraces On some systems the compiler options GMP uses by default can interfere with debugging. In particular on x86 and 68k systems `-fomit-frame-pointer' is used and this generally inhibits stack backtracing. Recompiling without such options may help while debugging, though the usual caveats about it potentially moving a memory problem or hiding a compiler bug will apply. GNU Debugger A sample `.gdbinit' is included in the distribution, showing how to call some undocumented dump functions to print GMP variables from within GDB. Note that these functions shouldn't be used in final application code since they're undocumented and may be subject to incompatible changes in future versions of GMP. Source File Paths GMP has multiple source files with the same name, in different directories. For example `mpz', `mpq', `mpf' and `mpfr' each have an `init.c'. If the debugger can't already determine the right one it may help to build with absolute paths on each C file. One way to do that is to use a separate object directory with an absolute path to the source directory. cd /my/build/dir /my/source/dir/gmp-4.1/configure This works via `VPATH', and might require GNU `make'. Alternately it might be possible to change the `.c.lo' rules appropriately. Assertion Checking The build option `--enable-assert' is available to add some consistency checks to the library (see *Note Build Options::). These are likely to be of limited value to most applications. Assertion failures are just as likely to indicate memory corruption as a library or compiler bug. Applications using the low-level `mpn' functions, however, will benefit from `--enable-assert' since it adds checks on the parameters of most such functions, many of which have subtle restrictions on their usage. Note however that only the generic C code has checks, not the assembler code, so CPU `none' should be used for maximum checking. Temporary Memory Checking The build option `--enable-alloca=debug' arranges that each block of temporary memory in GMP is allocated with a separate call to `malloc' (or the allocation function set with `mp_set_memory_functions'). This can help a malloc debugger detect accesses outside the intended bounds, or detect memory not released. In a normal build, on the other hand, temporary memory is allocated in blocks which GMP divides up for its own use, or may be allocated with a compiler builtin `alloca' which will go nowhere near any malloc debugger hooks. Checker The checker program (`http://savannah.gnu.org/projects/checker') can be used with GMP. It contains a stub library which means GMP applications compiled with checker can use a normal GMP build. A build of GMP with checking within GMP itself can be made. This will run very very slowly. Configure with ./configure --host=none-pc-linux-gnu CC=checkergcc `--host=none' must be used, since the GMP assembler code doesn't support the checking scheme. The GMP C++ features cannot be used, since current versions of checker (0.9.9.1) don't yet support the standard C++ library. Valgrind The valgrind program (`http://devel-home.kde.org/~sewardj') is a memory checker for x86s. It translates and emulates machine instructions to do strong checks for uninitialized data (at the level of individual bits), memory accesses through bad pointers, and memory leaks. Current versions (20020226 snapshot) don't support MMX or SSE, so GMP must be configured for an x86 without those (eg. plain `i386'), or with a special `MPN_PATH' that excludes those subdirectories (*note Build Options::). Other Problems Any suspected bug in GMP itself should be isolated to make sure it's not an application problem, see *Note Reporting Bugs::. Profiling ========= Running a program under a profiler is a good way to find where it's spending most time and where improvements can be best sought. 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 *Note 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. Autoconf ======== Autoconf based applications can easily check whether GMP is installed. The only thing to be noted is that GMP library symbols from version 3 onwards have prefixes like `__gmpz'. The following therefore would be a simple test, AC_CHECK_LIB(gmp, __gmpz_init) This just uses the default `AC_CHECK_LIB' actions for found or not found, but an application that must have GMP would want to generate an error if not found. For example, AC_CHECK_LIB(gmp, __gmpz_init, , [AC_MSG_ERROR( [GNU MP not found, see 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 ]) 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]) Reporting Bugs ************** If you think you have found a bug in the GMP library, please investigate it and report it. We have made this library available to you, and it is not too much to ask you to report the bugs you find. Before you report a bug, check it's not already addressed in *Note Known Build Problems::, or perhaps *Note Notes for Particular Systems::. You may also want to check `http://swox.com/gmp/' for patches for this release. Please include the following in any report, * The GMP version number, and if pre-packaged or patched then say so. * A test program that makes it possible for us to reproduce the bug. Include instructions on how to run the program. * A description of what is wrong. If the results are incorrect, in what way. If you get a crash, say so. * If you get a crash, include a stack backtrace from the debugger if it's informative (`where' in `gdb', or `$C' in `adb'). * Please do not send core dumps, executables or `strace's. * The configuration options you used when building GMP, if any. * The name of the compiler and its version. For `gcc', get the version with `gcc -v', otherwise perhaps `what `which cc`', or similar. * The output from running `uname -a'. * The output from running `./config.guess', and from running `./configfsf.guess' (might be the same). * If the bug is related to `configure', then the contents of `config.log'. * If the bug is related to an `asm' file not assembling, then the contents of `config.m4' and the offending line or lines from the temporary `mpn/tmp-.s'. Please make an effort to produce a self-contained report, with something definite that can be tested or debugged. Vague queries or piecemeal messages are difficult to act on and don't help the development effort. It is not uncommon that an observed problem is actually due to a bug in the compiler; the GMP code tends to explore interesting corners in compilers. If your bug report is good, we will do our best to help you get a corrected version of the library; if the bug report is poor, we won't do anything about it (except maybe ask you to send a better report). Send your report to: . 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. Integer Functions ***************** This chapter describes the GMP functions for performing integer arithmetic. These functions start with the prefix `mpz_'. GMP integers are stored in objects of type `mpz_t'. Initialization Functions ======================== The functions for integer arithmetic assume that all integer objects are initialized. You do that by calling the function `mpz_init'. For example, { mpz_t integ; mpz_init (integ); ... mpz_add (integ, ...); ... mpz_sub (integ, ...); /* Unless the program is about to exit, do ... */ mpz_clear (integ); } As you can see, you can store new values any number of times, once an object is initialized. - Function: void mpz_init (mpz_t INTEGER) Initialize INTEGER, and set its value to 0. - Function: void mpz_init2 (mpz_t INTEGER, unsigned long N) Initialize INTEGER, with space for N bits, and set its value to 0. N is only the initial space, INTEGER will grow automatically in the normal way, if necessary, for subsequent values stored. `mpz_init2' makes it possible to avoid such reallocations if a maximum size is known in advance. - Function: void mpz_clear (mpz_t INTEGER) Free the space occupied by INTEGER. Call this function for all `mpz_t' variables when you are done with them. - Function: void mpz_realloc2 (mpz_t INTEGER, unsigned long N) Change the space allocated for INTEGER to N bits. The value in INTEGER is preserved if it fits, or is set to 0 if not. This function can be used to increase the space for a variable in order to avoid repeated automatic reallocations, or to decrease it to give memory back to the heap. - Function: void mpz_array_init (mpz_t INTEGER_ARRAY[], size_t ARRAY_SIZE, mp_size_t FIXED_NUM_BITS) This is a special type of initialization. *Fixed* space of FIXED_NUM_BITS bits is allocated to each of the ARRAY_SIZE integers in INTEGER_ARRAY. 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'! - Function: void * _mpz_realloc (mpz_t INTEGER, mp_size_t NEW_ALLOC) Change the space for INTEGER to NEW_ALLOC limbs. The value in INTEGER is preserved if it fits, or is set to 0 if not. The return value is not useful to applications and should be ignored. `mpz_realloc2' is the preferred way to accomplish allocation changes like this. `mpz_realloc2' and `_mpz_realloc' are the same except that `_mpz_realloc' takes the new size in limbs. Assignment Functions ==================== These functions assign new values to already initialized integers (*note Initializing Integers::). - Function: void mpz_set (mpz_t ROP, mpz_t OP) - Function: void mpz_set_ui (mpz_t ROP, unsigned long int OP) - Function: void mpz_set_si (mpz_t ROP, signed long int OP) - Function: void mpz_set_d (mpz_t ROP, double OP) - Function: void mpz_set_q (mpz_t ROP, mpq_t OP) - Function: void mpz_set_f (mpz_t ROP, mpf_t OP) Set the value of ROP from OP. `mpz_set_d', `mpz_set_q' and `mpz_set_f' truncate OP to make it an integer. - Function: int mpz_set_str (mpz_t ROP, char *STR, int BASE) Set the value of ROP from STR, a null-terminated C string in base BASE. White space is allowed in the string, and is simply ignored. The base may vary from 2 to 36. If BASE is 0, the actual base is determined from the leading characters: if the first two characters are "0x" or "0X", hexadecimal is assumed, otherwise if the first character is "0", octal is assumed, otherwise decimal is assumed. 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 . Do you really want it to accept "3 14" as meaning 314 as it does now?] - Function: void mpz_swap (mpz_t ROP1, mpz_t ROP2) Swap the values ROP1 and ROP2 efficiently. Combined Initialization and Assignment Functions ================================================ For convenience, GMP provides a parallel series of initialize-and-set functions which initialize the output and then store the value there. These functions' names have the form `mpz_init_set...' Here is an example of using one: { mpz_t pie; mpz_init_set_str (pie, "3141592653589793238462643383279502884", 10); ... mpz_sub (pie, ...); ... mpz_clear (pie); } Once the integer has been initialized by any of the `mpz_init_set...' functions, it can be used as the source or destination operand for the ordinary integer functions. Don't use an initialize-and-set function on a variable already initialized! - Function: void mpz_init_set (mpz_t ROP, mpz_t OP) - Function: void mpz_init_set_ui (mpz_t ROP, unsigned long int OP) - Function: void mpz_init_set_si (mpz_t ROP, signed long int OP) - Function: void mpz_init_set_d (mpz_t ROP, double OP) Initialize ROP with limb space and set the initial numeric value from OP. - Function: int mpz_init_set_str (mpz_t ROP, char *STR, int BASE) Initialize ROP and set its value like `mpz_set_str' (see its 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.) Conversion Functions ==================== This section describes functions for converting GMP integers to standard C types. Functions for converting _to_ GMP integers are described in *Note Assigning Integers:: and *Note I/O of Integers::. - Function: unsigned long int mpz_get_ui (mpz_t OP) Return the value of OP as an `unsigned long'. If OP is too big to fit an `unsigned long' then just the least significant bits that do fit are returned. The sign of OP is ignored, only the absolute value is used. - Function: signed long int mpz_get_si (mpz_t OP) If OP fits into a `signed long int' return the value of OP. Otherwise return the least significant part of OP, with the same sign as OP. If OP is too big to fit in a `signed long int', the returned result is probably not very useful. To find out if the value will fit, use the function `mpz_fits_slong_p'. - Function: double mpz_get_d (mpz_t OP) Convert OP to a `double'. - Function: double mpz_get_d_2exp (signed long int *EXP, mpz_t OP) Find D and EXP such that D times 2 raised to EXP, with 0.5<=abs(D)<1, is a good approximation to OP. - Function: char * mpz_get_str (char *STR, int BASE, mpz_t OP) Convert OP to a string of digits in base BASE. The base may vary from 2 to 36. If STR is `NULL', the result string is allocated using the current allocation function (*note Custom Allocation::). The block will be `strlen(str)+1' bytes, that being exactly enough for the string and null-terminator. If STR is not `NULL', it should point to a block of storage large enough for the result, that being `mpz_sizeinbase (OP, BASE) + 2'. The two extra bytes are for a possible minus sign, and the null-terminator. A pointer to the result string is returned, being either the allocated block, or the given STR. - Function: mp_limb_t mpz_getlimbn (mpz_t OP, mp_size_t N) Return limb number N from OP. The sign of OP is ignored, just the absolute value is used. The least significant limb is number 0. `mpz_size' can be used to find how many limbs make up OP. `mpz_getlimbn' returns zero if N is outside the range 0 to `mpz_size(OP)-1'. Arithmetic Functions ==================== - Function: void mpz_add (mpz_t ROP, mpz_t OP1, mpz_t OP2) - Function: void mpz_add_ui (mpz_t ROP, mpz_t OP1, unsigned long int OP2) Set ROP to OP1 + OP2. - Function: void mpz_sub (mpz_t ROP, mpz_t OP1, mpz_t OP2) - Function: void mpz_sub_ui (mpz_t ROP, mpz_t OP1, unsigned long int OP2) - Function: void mpz_ui_sub (mpz_t ROP, unsigned long int OP1, mpz_t OP2) Set ROP to OP1 - OP2. - Function: void mpz_mul (mpz_t ROP, mpz_t OP1, mpz_t OP2) - Function: void mpz_mul_si (mpz_t ROP, mpz_t OP1, long int OP2) - Function: void mpz_mul_ui (mpz_t ROP, mpz_t OP1, unsigned long int OP2) Set ROP to OP1 times OP2. - Function: void mpz_addmul (mpz_t ROP, mpz_t OP1, mpz_t OP2) - Function: void mpz_addmul_ui (mpz_t ROP, mpz_t OP1, unsigned long int OP2) Set ROP to ROP + OP1 times OP2. - Function: void mpz_submul (mpz_t ROP, mpz_t OP1, mpz_t OP2) - Function: void mpz_submul_ui (mpz_t ROP, mpz_t OP1, unsigned long int OP2) Set ROP to ROP - OP1 times OP2. - Function: void mpz_mul_2exp (mpz_t ROP, mpz_t OP1, unsigned long int OP2) Set ROP to OP1 times 2 raised to OP2. This operation can also be defined as a left shift by OP2 bits. - Function: void mpz_neg (mpz_t ROP, mpz_t OP) Set ROP to -OP. - Function: void mpz_abs (mpz_t ROP, mpz_t OP) Set ROP to the absolute value of OP. Division Functions ================== Division is undefined if the divisor is zero. Passing a zero divisor to the division or modulo functions (including the modular powering functions `mpz_powm' and `mpz_powm_ui'), will cause an intentional division by zero. This lets a program handle arithmetic exceptions in these functions the same way as for normal C `int' arithmetic. - Function: void mpz_cdiv_q (mpz_t Q, mpz_t N, mpz_t D) - Function: void mpz_cdiv_r (mpz_t R, mpz_t N, mpz_t D) - Function: void mpz_cdiv_qr (mpz_t Q, mpz_t R, mpz_t N, mpz_t D) - Function: unsigned long int mpz_cdiv_q_ui (mpz_t Q, mpz_t N, unsigned long int D) - Function: unsigned long int mpz_cdiv_r_ui (mpz_t R, mpz_t N, unsigned long int D) - Function: unsigned long int mpz_cdiv_qr_ui (mpz_t Q, mpz_t R, mpz_t N, unsigned long int D) - Function: unsigned long int mpz_cdiv_ui (mpz_t N, unsigned long int D) - Function: void mpz_cdiv_q_2exp (mpz_t Q, mpz_t N, unsigned long int B) - Function: void mpz_cdiv_r_2exp (mpz_t R, mpz_t N, unsigned long int B) - Function: void mpz_fdiv_q (mpz_t Q, mpz_t N, mpz_t D) - Function: void mpz_fdiv_r (mpz_t R, mpz_t N, mpz_t D) - Function: void mpz_fdiv_qr (mpz_t Q, mpz_t R, mpz_t N, mpz_t D) - Function: unsigned long int mpz_fdiv_q_ui (mpz_t Q, mpz_t N, unsigned long int D) - Function: unsigned long int mpz_fdiv_r_ui (mpz_t R, mpz_t N, unsigned long int D) - Function: unsigned long int mpz_fdiv_qr_ui (mpz_t Q, mpz_t R, mpz_t N, unsigned long int D) - Function: unsigned long int mpz_fdiv_ui (mpz_t N, unsigned long int D) - Function: void mpz_fdiv_q_2exp (mpz_t Q, mpz_t N, unsigned long int B) - Function: void mpz_fdiv_r_2exp (mpz_t R, mpz_t N, unsigned long int B) - Function: void mpz_tdiv_q (mpz_t Q, mpz_t N, mpz_t D) - Function: void mpz_tdiv_r (mpz_t R, mpz_t N, mpz_t D) - Function: void mpz_tdiv_qr (mpz_t Q, mpz_t R, mpz_t N, mpz_t D) - Function: unsigned long int mpz_tdiv_q_ui (mpz_t Q, mpz_t N, unsigned long int D) - Function: unsigned long int mpz_tdiv_r_ui (mpz_t R, mpz_t N, unsigned long int D) - Function: unsigned long int mpz_tdiv_qr_ui (mpz_t Q, mpz_t R, mpz_t N, unsigned long int D) - Function: unsigned long int mpz_tdiv_ui (mpz_t N, unsigned long int D) - Function: void mpz_tdiv_q_2exp (mpz_t Q, mpz_t N, unsigned long int B) - Function: void mpz_tdiv_r_2exp (mpz_t R, mpz_t N, unsigned long int B) Divide N by D, forming a quotient Q and/or remainder R. For the `2exp' functions, D=2^B. The rounding is in three styles, each suiting different applications. * `cdiv' rounds Q up towards +infinity, and R will have the opposite sign to D. The `c' stands for "ceil". * `fdiv' rounds Q down towards -infinity, and R will have the same sign as D. The `f' stands for "floor". * `tdiv' rounds Q towards zero, and R will have the same sign as N. The `t' stands for "truncate". In all cases Q and R will satisfy N=Q*D+R, and R will satisfy 0<=abs(R)1, such that OP equals A raised to the power B. Under this definition both 0 and 1 are considered to be perfect powers. Negative values of OP are accepted, but of course can only be odd perfect powers. - Function: int mpz_perfect_square_p (mpz_t OP) Return non-zero if OP is a perfect square, i.e., if the square root of OP is an integer. Under this definition both 0 and 1 are considered to be perfect squares. Number Theoretic Functions ========================== - Function: int mpz_probab_prime_p (mpz_t N, int REPS) Determine whether N is prime. Return 2 if N is definitely prime, return 1 if N is probably prime (without being certain), or return 0 if N is definitely composite. This function does some trial divisions, then some Miller-Rabin probabilistic primality tests. REPS controls how many such tests are done, 5 to 10 is a reasonable number, more will reduce the chances of a composite being returned as "probably prime". Miller-Rabin and similar tests can be more properly called compositeness tests. Numbers which fail are known to be composite but those which pass might be prime or might be composite. Only a few composites pass, hence those which pass are considered probably prime. - Function: void mpz_nextprime (mpz_t ROP, mpz_t OP) Set ROP to the next prime greater than OP. This function uses a probabilistic algorithm to identify primes. For practical purposes it's adequate, the chance of a composite passing will be extremely small. - Function: void mpz_gcd (mpz_t ROP, mpz_t OP1, mpz_t OP2) Set ROP to the greatest common divisor of OP1 and OP2. The result is always positive even if one or both input operands are negative. - Function: unsigned long int mpz_gcd_ui (mpz_t ROP, mpz_t OP1, unsigned long int OP2) Compute the greatest common divisor of OP1 and OP2. If ROP is not `NULL', store the result there. If the result is small enough to fit in an `unsigned long int', it is returned. If the result does not fit, 0 is returned, and the result is equal to the argument OP1. Note that the result will always fit if OP2 is non-zero. - Function: void mpz_gcdext (mpz_t G, mpz_t S, mpz_t T, mpz_t A, mpz_t B) Set G to the greatest common divisor of A and B, and in addition set S and T to coefficients satisfying A*S + B*T = G. G is always positive, even if one or both of A and B are negative. If T is `NULL' then that value is not computed. - Function: void mpz_lcm (mpz_t ROP, mpz_t OP1, mpz_t OP2) - Function: void mpz_lcm_ui (mpz_t ROP, mpz_t OP1, unsigned long OP2) Set ROP to the least common multiple of OP1 and OP2. ROP is always positive, irrespective of the signs of OP1 and OP2. ROP will be zero if either OP1 or OP2 is zero. - Function: int mpz_invert (mpz_t ROP, mpz_t OP1, mpz_t OP2) Compute the inverse of OP1 modulo OP2 and put the result in ROP. If the inverse exists, the return value is non-zero and ROP will satisfy 0 <= ROP < OP2. If an inverse doesn't exist the return value is zero and ROP is undefined. - Function: int mpz_jacobi (mpz_t A, mpz_t B) Calculate the Jacobi symbol (A/B). This is defined only for B odd. - Function: int mpz_legendre (mpz_t A, mpz_t P) Calculate the Legendre symbol (A/P). This is defined only for P an odd positive prime, and for such P it's identical to the Jacobi symbol. - Function: int mpz_kronecker (mpz_t A, mpz_t B) - Function: int mpz_kronecker_si (mpz_t A, long B) - Function: int mpz_kronecker_ui (mpz_t A, unsigned long B) - Function: int mpz_si_kronecker (long A, mpz_t B) - Function: int mpz_ui_kronecker (unsigned long A, mpz_t B) Calculate the Jacobi symbol (A/B) with the Kronecker extension (a/2)=(2/a) when a odd, or (a/2)=0 when a even. When B is odd the Jacobi symbol and Kronecker symbol are identical, so `mpz_kronecker_ui' etc can be used for mixed precision Jacobi symbols too. For more information see Henri Cohen section 1.4.2 (*note References::), or any number theory textbook. See also the example program `demos/qcn.c' which uses `mpz_kronecker_ui'. - Function: unsigned long int mpz_remove (mpz_t ROP, mpz_t OP, mpz_t F) Remove all occurrences of the factor F from OP and store the result in ROP. Return the multiplicity of F in OP. - Function: void mpz_fac_ui (mpz_t ROP, unsigned long int OP) Set ROP to OP!, the factorial of OP. - Function: void mpz_bin_ui (mpz_t ROP, mpz_t N, unsigned long int K) - Function: void mpz_bin_uiui (mpz_t ROP, unsigned long int N, unsigned long int K) Compute the binomial coefficient N over K and store the result in ROP. Negative values of N are supported by `mpz_bin_ui', using the identity bin(-n,k) = (-1)^k * bin(n+k-1,k), see Knuth volume 1 section 1.2.6 part G. - Function: void mpz_fib_ui (mpz_t FN, unsigned long int N) - Function: void mpz_fib2_ui (mpz_t FN, mpz_t FNSUB1, unsigned long int N) `mpz_fib_ui' sets FN to to F[n], the N'th Fibonacci number. `mpz_fib2_ui' sets FN to F[n], and FNSUB1 to F[n-1]. These functions are designed for calculating isolated Fibonacci numbers. When a sequence of values is wanted it's best to start with `mpz_fib2_ui' and iterate the defining F[n+1]=F[n]+F[n-1] or similar. - Function: void mpz_lucnum_ui (mpz_t LN, unsigned long int N) - Function: void mpz_lucnum2_ui (mpz_t LN, mpz_t LNSUB1, unsigned long int N) `mpz_lucnum_ui' sets LN to to L[n], the N'th Lucas number. `mpz_lucnum2_ui' sets LN to L[n], and LNSUB1 to L[n-1]. These functions are designed for calculating isolated Lucas numbers. When a sequence of values is wanted it's best to start with `mpz_lucnum2_ui' and iterate the defining L[n+1]=L[n]+L[n-1] or similar. The Fibonacci numbers and Lucas numbers are related sequences, so it's never necessary to call both `mpz_fib2_ui' and `mpz_lucnum2_ui'. The formulas for going from Fibonacci to Lucas can be found in *Note Lucas Numbers Algorithm::, the reverse is straightforward too. Comparison Functions ==================== - Function: int mpz_cmp (mpz_t OP1, mpz_t OP2) - Function: int mpz_cmp_d (mpz_t OP1, double OP2) - Macro: int mpz_cmp_si (mpz_t OP1, signed long int OP2) - Macro: int mpz_cmp_ui (mpz_t OP1, unsigned long int OP2) Compare OP1 and OP2. Return a positive value if OP1 > OP2, zero if OP1 = OP2, or a negative value if OP1 < OP2. Note that `mpz_cmp_ui' and `mpz_cmp_si' are macros and will evaluate their arguments more than once. - Function: int mpz_cmpabs (mpz_t OP1, mpz_t OP2) - Function: int mpz_cmpabs_d (mpz_t OP1, double OP2) - Function: int mpz_cmpabs_ui (mpz_t OP1, unsigned long int OP2) Compare the absolute values of OP1 and OP2. Return a positive value if abs(OP1) > abs(OP2), zero if abs(OP1) = abs(OP2), or a negative value if abs(OP1) < abs(OP2). Note that `mpz_cmpabs_si' is a macro and will evaluate its arguments more than once. - Macro: int mpz_sgn (mpz_t OP) Return +1 if OP > 0, 0 if OP = 0, and -1 if OP < 0. This function is actually implemented as a macro. It evaluates its argument multiple times. Logical and Bit Manipulation Functions ====================================== These functions behave as if twos complement arithmetic were used (although sign-magnitude is the actual implementation). The least significant bit is number 0. - Function: void mpz_and (mpz_t ROP, mpz_t OP1, mpz_t OP2) Set ROP to OP1 logical-and OP2. - Function: void mpz_ior (mpz_t ROP, mpz_t OP1, mpz_t OP2) Set ROP to OP1 inclusive-or OP2. - Function: void mpz_xor (mpz_t ROP, mpz_t OP1, mpz_t OP2) Set ROP to OP1 exclusive-or OP2. - Function: void mpz_com (mpz_t ROP, mpz_t OP) Set ROP to the one's complement of OP. - Function: unsigned long int mpz_popcount (mpz_t OP) If OP>=0, return the population count of OP, which is the number of 1 bits in the binary representation. If OP<0, the number of 1s is infinite, and the return value is MAX_ULONG, the largest possible `unsigned long'. - Function: unsigned long int mpz_hamdist (mpz_t OP1, mpz_t OP2) If OP1 and OP2 are both >=0 or both <0, return the hamming distance between the two operands, which is the number of bit positions where OP1 and OP2 have different bit values. If one operand is >=0 and the other <0 then the number of bits different is infinite, and the return value is MAX_ULONG, the largest possible `unsigned long'. - Function: unsigned long int mpz_scan0 (mpz_t OP, unsigned long int STARTING_BIT) - Function: unsigned long int mpz_scan1 (mpz_t OP, unsigned long int STARTING_BIT) Scan OP, starting from bit STARTING_BIT, towards more significant bits, until the first 0 or 1 bit (respectively) is found. Return the index of the found bit. If the bit at STARTING_BIT is already what's sought, then STARTING_BIT is returned. If there's no bit found, then 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. - Function: void mpz_setbit (mpz_t ROP, unsigned long int BIT_INDEX) Set bit BIT_INDEX in ROP. - Function: void mpz_clrbit (mpz_t ROP, unsigned long int BIT_INDEX) Clear bit BIT_INDEX in ROP. - Function: int mpz_tstbit (mpz_t OP, unsigned long int BIT_INDEX) Test bit BIT_INDEX in OP and return 0 or 1 accordingly. Input and Output Functions ========================== Functions that perform input from a stdio stream, and functions that output to a stdio stream. Passing a `NULL' pointer for a STREAM argument to any of these functions will make them read from `stdin' and write to `stdout', respectively. When using any of these functions, it is a good idea to include `stdio.h' before `gmp.h', since that will allow `gmp.h' to define prototypes for these functions. - Function: size_t mpz_out_str (FILE *STREAM, int BASE, mpz_t OP) Output OP on stdio stream STREAM, as a string of digits in base BASE. The base may vary from 2 to 36. Return the number of bytes written, or if an error occurred, return 0. - Function: size_t mpz_inp_str (mpz_t ROP, FILE *STREAM, int BASE) Input a possibly white-space preceded string in base BASE from stdio stream STREAM, and put the read integer in ROP. The base may vary from 2 to 36. If BASE is 0, the actual base is determined from the leading characters: if the first two characters are `0x' or `0X', hexadecimal is assumed, otherwise if the first character is `0', octal is assumed, otherwise decimal is assumed. Return the number of bytes read, or if an error occurred, return 0. - Function: size_t mpz_out_raw (FILE *STREAM, mpz_t OP) Output OP on stdio stream STREAM, in raw binary format. The integer is written in a portable format, with 4 bytes of size information, and that many bytes of limbs. Both the size and the limbs are written in decreasing significance order (i.e., in big-endian). The output can be read with `mpz_inp_raw'. Return the number of bytes written, or if an error occurred, return 0. The output of this can not be read by `mpz_inp_raw' from GMP 1, because of changes necessary for compatibility between 32-bit and 64-bit machines. - Function: size_t mpz_inp_raw (mpz_t ROP, FILE *STREAM) Input from stdio stream STREAM in the format written by `mpz_out_raw', and put the result in ROP. Return the number of bytes read, or if an error occurred, return 0. This routine can read the output from `mpz_out_raw' also from GMP 1, in spite of changes necessary for compatibility between 32-bit and 64-bit machines. Random Number Functions ======================= The random number functions of GMP come in two groups; older function that rely on a global state, and newer functions that accept a state parameter that is read and modified. Please see the *Note Random Number Functions:: for more information on how to use and not to use random number functions. - Function: void mpz_urandomb (mpz_t ROP, gmp_randstate_t STATE, unsigned long int N) Generate a uniformly distributed random integer in the range 0 to 2^N-1, inclusive. The variable STATE must be initialized by calling one of the `gmp_randinit' functions (*Note Random State Initialization::) before invoking this function. - Function: void mpz_urandomm (mpz_t ROP, gmp_randstate_t STATE, mpz_t N) Generate a uniform random integer in the range 0 to N-1, inclusive. The variable STATE must be initialized by calling one of the `gmp_randinit' functions (*Note Random State Initialization::) before invoking this function. - Function: void mpz_rrandomb (mpz_t ROP, gmp_randstate_t STATE, unsigned long int N) Generate a random integer with long strings of zeros and ones in the binary representation. Useful for testing functions and algorithms, since this kind of random numbers have proven to be more likely to trigger corner-case bugs. The random number will be in the range 0 to 2^N-1, inclusive. The variable STATE must be initialized by calling one of the `gmp_randinit' functions (*Note Random State Initialization::) before invoking this function. - Function: void mpz_random (mpz_t ROP, mp_size_t MAX_SIZE) Generate a random integer of at most MAX_SIZE limbs. The generated random number doesn't satisfy any particular requirements of randomness. Negative random numbers are generated when MAX_SIZE is negative. This function is obsolete. Use `mpz_urandomb' or `mpz_urandomm' instead. - Function: void mpz_random2 (mpz_t ROP, mp_size_t MAX_SIZE) Generate a random integer of at most MAX_SIZE limbs, with long strings of zeros and ones in the binary representation. Useful for testing functions and algorithms, since this kind of random numbers have proven to be more likely to trigger corner-case bugs. Negative random numbers are generated when MAX_SIZE is negative. This function is obsolete. Use `mpz_rrandomb' instead. Integer Import and Export ========================= `mpz_t' variables can be converted to and from arbitrary words of binary data with the following functions. - Function: void mpz_import (mpz_t ROP, size_t COUNT, int ORDER, int SIZE, int ENDIAN, size_t NAILS, const void *OP) Set ROP from an array of word data at OP. The parameters specify the format of the data. COUNT many words are read, each SIZE bytes. ORDER can be 1 for most significant word first or -1 for least significant first. Within each word ENDIAN can be 1 for most significant byte first, -1 for least significant first, or 0 for the native endianness of the host CPU. The most significant NAILS bits of each word are skipped, this can be 0 to use the full words. There are no data alignment restrictions on OP, any address is allowed. Here's an example converting an array of `unsigned long' data, most significant element first and host byte order within each value. unsigned long a[20]; mpz_t z; mpz_import (z, 20, 1, sizeof(a[0]), 0, 0, a); This example assumes the full `sizeof' bytes are used for data in the given type, which is usually true, and certainly true for `unsigned long' everywhere we know of. However on Cray vector systems it may be noted that `short' and `int' are always stored in 8 bytes (and with `sizeof' indicating that) but use only 32 or 46 bits. The NAILS feature can account for this, by passing for instance `8*sizeof(int)-INT_BIT'. - Function: void *mpz_export (void *ROP, size_t *COUNT, int ORDER, int SIZE, int ENDIAN, size_t NAILS, mpz_t OP) Fill ROP with word data from OP. The parameters specify the format of the data produced. Each word will be SIZE bytes and ORDER can be 1 for most significant word first or -1 for least significant first. Within each word ENDIAN can be 1 for most significant byte first, -1 for least significant first, or 0 for the native endianness of the host CPU. The most significant NAILS bits of each word are unused and set to zero, this can be 0 to produce full words. The number of words produced is written to `*COUNT'. ROP must have enough space for the data, or if ROP is `NULL' then a result array of the necessary size is allocated using the current GMP allocation function (*note Custom Allocation::). In either case the return value is the destination used, ROP or the allocated block. If OP is non-zero then the most significant word produced will be non-zero. If OP is zero then the count returned will be zero and nothing written to ROP. If ROP is `NULL' in this case, no block is allocated, just `NULL' is returned. There are no data alignment restrictions on ROP, any address is allowed. The sign of OP is ignored, just the absolute value is used. When an application is allocating space itself the required size can be determined with a calculation like the following. Since `mpz_sizeinbase' always returns at least 1, `count' here will be at least one, which avoids any portability problems with `malloc(0)', though if `z' is zero no space at all is actually needed. numb = 8*size - nail; count = (mpz_sizeinbase (z, 2) + numb-1) / numb; p = malloc (count * size); Miscellaneous Functions ======================= - Function: int mpz_fits_ulong_p (mpz_t OP) - Function: int mpz_fits_slong_p (mpz_t OP) - Function: int mpz_fits_uint_p (mpz_t OP) - Function: int mpz_fits_sint_p (mpz_t OP) - Function: int mpz_fits_ushort_p (mpz_t OP) - Function: int mpz_fits_sshort_p (mpz_t OP) Return non-zero iff the value of OP fits in an `unsigned long int', `signed long int', `unsigned int', `signed int', `unsigned short int', or `signed short int', respectively. Otherwise, return zero. - Macro: int mpz_odd_p (mpz_t OP) - Macro: int mpz_even_p (mpz_t OP) Determine whether OP is odd or even, respectively. Return non-zero if yes, zero if no. These macros evaluate their argument more than once. - Function: size_t mpz_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 result will be exact or 1 too big. If BASE is a power of 2, the result will always be exact. If OP is zero the return value is always 1. 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). 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. 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 (*note Rational Number Functions::), or if not then `mpq_canonicalize' must be called. The numerator and optional denominator are parsed the same as in `mpz_set_str' (*note Assigning Integers::). White space is allowed in the string, and is simply ignored. The BASE can vary from 2 to 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. 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 (*note Custom Allocation::). The block will be `strlen(str)+1' bytes, that being exactly enough for the string and null-terminator. If STR is not `NULL', it should point to a block of storage large enough for the result, that being mpz_sizeinbase (mpq_numref(OP), BASE) + mpz_sizeinbase (mpq_denref(OP), BASE) + 3 The three extra bytes are for a possible minus sign, possible slash, and the null-terminator. A pointer to the result string is returned, being either the allocated block, or the given STR. Arithmetic Functions ==================== - Function: void mpq_add (mpq_t SUM, mpq_t ADDEND1, mpq_t ADDEND2) Set SUM to ADDEND1 + ADDEND2. - Function: void mpq_sub (mpq_t DIFFERENCE, mpq_t MINUEND, mpq_t SUBTRAHEND) Set DIFFERENCE to MINUEND - SUBTRAHEND. - Function: void mpq_mul (mpq_t PRODUCT, mpq_t MULTIPLIER, mpq_t MULTIPLICAND) Set PRODUCT to MULTIPLIER times MULTIPLICAND. - Function: void mpq_mul_2exp (mpq_t ROP, mpq_t OP1, unsigned long int OP2) Set ROP to OP1 times 2 raised to OP2. - Function: void mpq_div (mpq_t QUOTIENT, mpq_t DIVIDEND, mpq_t DIVISOR) Set QUOTIENT to DIVIDEND/DIVISOR. - Function: void mpq_div_2exp (mpq_t ROP, mpq_t OP1, unsigned long int OP2) Set ROP to OP1 divided by 2 raised to OP2. - Function: void mpq_neg (mpq_t NEGATED_OPERAND, mpq_t OPERAND) Set NEGATED_OPERAND to -OPERAND. - Function: void mpq_abs (mpq_t ROP, mpq_t OP) Set ROP to the absolute value of OP. - Function: void mpq_inv (mpq_t INVERTED_NUMBER, mpq_t NUMBER) Set INVERTED_NUMBER to 1/NUMBER. If the new denominator is zero, this routine will divide by zero. Comparison Functions ==================== - Function: int mpq_cmp (mpq_t OP1, mpq_t OP2) Compare OP1 and OP2. Return a positive value if OP1 > OP2, zero if OP1 = OP2, and a negative value if OP1 < OP2. To determine if two rationals are equal, `mpq_equal' is faster than `mpq_cmp'. - Macro: int mpq_cmp_ui (mpq_t OP1, unsigned long int NUM2, unsigned long int DEN2) - Macro: int mpq_cmp_si (mpq_t OP1, long int NUM2, unsigned long int DEN2) Compare OP1 and NUM2/DEN2. Return a positive value if OP1 > NUM2/DEN2, zero if OP1 = NUM2/DEN2, and a negative value if OP1 < NUM2/DEN2. NUM2 and DEN2 are allowed to have common factors. These functions are implemented as a macros and evaluate their arguments multiple times. - Macro: int mpq_sgn (mpq_t OP) Return +1 if OP > 0, 0 if OP = 0, and -1 if OP < 0. This function is actually implemented as a macro. It evaluates its arguments multiple times. - Function: int mpq_equal (mpq_t OP1, mpq_t OP2) Return non-zero if OP1 and OP2 are equal, zero if they are non-equal. Although `mpq_cmp' can be used for the same purpose, this function is much faster. Applying Integer Functions to Rationals ======================================= The set of `mpq' functions is quite small. In particular, there are few functions for either input or output. The following functions give direct access to the numerator and denominator of an `mpq_t'. Note that if an assignment to the numerator and/or denominator could take an `mpq_t' out of the canonical form described at the start of this chapter (*note Rational Number Functions::) then `mpq_canonicalize' must be called before any other `mpq' functions are applied to that `mpq_t'. - Macro: mpz_t mpq_numref (mpq_t OP) - Macro: mpz_t mpq_denref (mpq_t OP) Return a reference to the numerator and denominator of OP, respectively. The `mpz' functions can be used on the result of these macros. - Function: void mpq_get_num (mpz_t NUMERATOR, mpq_t RATIONAL) - Function: void mpq_get_den (mpz_t DENOMINATOR, mpq_t RATIONAL) - Function: void mpq_set_num (mpq_t RATIONAL, mpz_t NUMERATOR) - Function: void mpq_set_den (mpq_t RATIONAL, mpz_t DENOMINATOR) Get or set the numerator or denominator of a rational. These functions are equivalent to calling `mpz_set' with an appropriate `mpq_numref' or `mpq_denref'. Direct use of `mpq_numref' or `mpq_denref' is recommended instead of these functions. Input and Output Functions ========================== When using any of these functions, it's a good idea to include `stdio.h' before `gmp.h', since that will allow `gmp.h' to define prototypes for these functions. Passing a `NULL' pointer for a STREAM argument to any of these functions will make them read from `stdin' and write to `stdout', respectively. - Function: size_t mpq_out_str (FILE *STREAM, int BASE, mpq_t OP) Output OP on stdio stream STREAM, as a string of digits in base BASE. The base may vary from 2 to 36. Output is in the form `num/den' or if the denominator is 1 then just `num'. Return the number of bytes written, or if an error occurred, return 0. - Function: size_t mpq_inp_str (mpq_t ROP, FILE *STREAM, int BASE) Read a string of digits from STREAM and convert them to a rational in ROP. Any initial white-space characters are read and discarded. Return the number of characters read (including white space), or 0 if a rational could not be read. The input can be a fraction like `17/63' or just an integer like `123'. Reading stops at the first character not in this form, and white space is not permitted within the string. If the input might not be in canonical form, then `mpq_canonicalize' must be called (*note Rational Number Functions::). The BASE can be between 2 and 36, or can be 0 in which case the leading characters of the string determine the base, `0x' or `0X' for hexadecimal, `0' for octal, or decimal otherwise. The leading characters are examined separately for the numerator and denominator of a fraction, so for instance `0x10/11' is 16/11, whereas `0x10/0x11' is 16/17. Floating-point Functions ************************ GMP floating point numbers are stored in objects of type `mpf_t' and functions operating on them have an `mpf_' prefix. The mantissa of each float has a user-selectable precision, 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. Initialization Functions ======================== - Function: void mpf_set_default_prec (unsigned long int PREC) Set the default precision to be *at least* PREC bits. All subsequent calls to `mpf_init' will use this precision, but previously initialized variables are unaffected. - Function: unsigned long int mpf_get_default_prec (void) Return the default default precision actually used. An `mpf_t' object must be initialized before storing the first value in it. The functions `mpf_init' and `mpf_init2' are used for that purpose. - Function: void mpf_init (mpf_t X) Initialize X to 0. Normally, a variable should be initialized once only or at least be cleared, using `mpf_clear', between initializations. The precision of X is undefined unless a default precision has already been established by a call to `mpf_set_default_prec'. - Function: void mpf_init2 (mpf_t X, unsigned long int PREC) Initialize X to 0 and set its precision to be *at least* PREC bits. Normally, a variable should be initialized once only or at least be cleared, using `mpf_clear', between initializations. - Function: void mpf_clear (mpf_t X) Free the space occupied by X. Make sure to call this function for all `mpf_t' variables when you are done with them. Here is an example on how to initialize floating-point variables: { mpf_t x, y; mpf_init (x); /* use default precision */ mpf_init2 (y, 256); /* precision _at least_ 256 bits */ ... /* Unless the program is about to exit, do ... */ mpf_clear (x); mpf_clear (y); } The following three functions are useful for changing the precision during a calculation. A typical use would be for adjusting the precision gradually in iterative algorithms like Newton-Raphson, making the computation precision closely match the actual accurate part of the numbers. - Function: unsigned long int mpf_get_prec (mpf_t OP) Return the current precision of OP, in bits. - Function: void mpf_set_prec (mpf_t ROP, unsigned long int PREC) Set the precision of ROP to be *at least* PREC bits. The value in ROP will be truncated to the new precision. This function requires a call to `realloc', and so should not be used in a tight loop. - Function: void mpf_set_prec_raw (mpf_t ROP, unsigned long int PREC) Set the precision of ROP to be *at least* PREC bits, without changing the memory allocated. PREC must be no more than the allocated precision for ROP, that being the precision when ROP was initialized, or in the most recent `mpf_set_prec'. The value in ROP is unchanged, and in particular if it had a higher precision than PREC it will retain that higher precision. New values written to ROP will use the new PREC. Before calling `mpf_clear' or the full `mpf_set_prec', another `mpf_set_prec_raw' call must be made to restore ROP to its original allocated precision. Failing to do so will have unpredictable results. `mpf_get_prec' can be used before `mpf_set_prec_raw' to get the original allocated precision. After `mpf_set_prec_raw' it reflects the PREC value set. `mpf_set_prec_raw' is an efficient way to use an `mpf_t' variable at different precisions during a calculation, perhaps to gradually increase precision in an iteration, or just to use various different precisions for different purposes during a calculation. Assignment Functions ==================== These functions assign new values to already initialized floats (*note Initializing Floats::). - Function: void mpf_set (mpf_t ROP, mpf_t OP) - Function: void mpf_set_ui (mpf_t ROP, unsigned long int OP) - Function: void mpf_set_si (mpf_t ROP, signed long int OP) - Function: void mpf_set_d (mpf_t ROP, double OP) - Function: void mpf_set_z (mpf_t ROP, mpz_t OP) - Function: void mpf_set_q (mpf_t ROP, mpq_t OP) Set the value of ROP from OP. - Function: int mpf_set_str (mpf_t ROP, char *STR, int BASE) Set the value of ROP from the string in STR. The string is of the form `M@N' or, if the base is 10 or less, alternatively `MeN'. `M' is the mantissa and `N' is the exponent. The mantissa is always in the specified base. The exponent is either in the specified base or, if BASE is negative, in decimal. The decimal point expected is taken from the current locale, on systems providing `localeconv'. The argument BASE may be in the ranges 2 to 36, or -36 to -2. Negative values are used to specify that the exponent is in decimal. Unlike the corresponding `mpz' function, the base will not be determined from the leading characters of the string if BASE is 0. This is so that numbers like `0.23' are not interpreted as octal. White space is allowed in the string, and is simply ignored. [This is not really true; white-space is ignored in the beginning of the string and within the mantissa, but not in other places, such as after a minus sign or in the exponent. We are considering changing the definition of this function, making it fail when there is any white-space in the input, since that makes a lot of sense. Please tell us your opinion about this change. Do you really want it to accept "3 14" as meaning 314 as it does now?] This function returns 0 if the entire string is a valid number in base BASE. Otherwise it returns -1. - Function: void mpf_swap (mpf_t ROP1, mpf_t ROP2) Swap ROP1 and ROP2 efficiently. Both the values and the precisions of the two variables are swapped. Combined Initialization and Assignment Functions ================================================ For convenience, GMP provides a parallel series of initialize-and-set functions which initialize the output and then store the value there. These functions' names have the form `mpf_init_set...' Once the float has been initialized by any of the `mpf_init_set...' functions, it can be used as the source or destination operand for the ordinary float functions. Don't use an initialize-and-set function on a variable already initialized! - Function: void mpf_init_set (mpf_t ROP, mpf_t OP) - Function: void mpf_init_set_ui (mpf_t ROP, unsigned long int OP) - Function: void mpf_init_set_si (mpf_t ROP, signed long int OP) - Function: void mpf_init_set_d (mpf_t ROP, double OP) Initialize ROP and set its value from OP. The precision of ROP will be taken from the active default precision, as set by `mpf_set_default_prec'. - Function: int mpf_init_set_str (mpf_t ROP, char *STR, int BASE) Initialize ROP and set its value from the string in STR. See `mpf_set_str' above for details on the assignment operation. Note that ROP is initialized even if an error occurs. (I.e., you have to call `mpf_clear' for it.) The precision of ROP will be taken from the active default precision, as set by `mpf_set_default_prec'. Conversion Functions ==================== - Function: double mpf_get_d (mpf_t OP) Convert OP to a `double'. - Function: double mpf_get_d_2exp (signed long int EXP, mpf_t OP) Find D and EXP such that D times 2 raised to EXP, with 0.5<=abs(D)<1, is a good approximation to OP. This is similar to the standard C function `frexp'. - Function: long mpf_get_si (mpf_t OP) - Function: unsigned long mpf_get_ui (mpf_t OP) Convert OP to a `long' or `unsigned long', truncating any fraction part. If OP is too big for the return type, the result is undefined. See also `mpf_fits_slong_p' and `mpf_fits_ulong_p' (*note Miscellaneous Float Functions::). - Function: char * mpf_get_str (char *STR, mp_exp_t *EXPPTR, int BASE, size_t N_DIGITS, mpf_t OP) Convert OP to a string of digits in base BASE. BASE can be 2 to 36. Up to N_DIGITS digits will be generated. Trailing zeros are not returned. No more digits than can be accurately represented by OP are ever generated. If N_DIGITS is 0 then that accurate maximum number of digits are generated. If STR is `NULL', the result string is allocated using the current allocation function (*note Custom Allocation::). The block will be `strlen(str)+1' bytes, that being exactly enough for the string and null-terminator. If STR is not `NULL', it should point to a block of N\_DIGITS + 2 bytes, that being enough for the mantissa, a possible minus sign, and a null-terminator. When N_DIGITS is 0 to get all significant digits, an application won't be able to know the space required, and STR should be `NULL' in that case. The generated string is a fraction, with an implicit radix point immediately to the left of the first digit. The applicable exponent is written through the EXPPTR pointer. For example, the number 3.1416 would be returned as string "31416" and exponent 1. When OP is zero, an empty string is produced and the exponent returned is 0. A pointer to the result string is returned, being either the allocated block or the given STR. Arithmetic Functions ==================== - Function: void mpf_add (mpf_t ROP, mpf_t OP1, mpf_t OP2) - Function: void mpf_add_ui (mpf_t ROP, mpf_t OP1, unsigned long int OP2) Set ROP to OP1 + OP2. - Function: void mpf_sub (mpf_t ROP, mpf_t OP1, mpf_t OP2) - Function: void mpf_ui_sub (mpf_t ROP, unsigned long int OP1, mpf_t OP2) - Function: void mpf_sub_ui (mpf_t ROP, mpf_t OP1, unsigned long int OP2) Set ROP to OP1 - OP2. - Function: void mpf_mul (mpf_t ROP, mpf_t OP1, mpf_t OP2) - Function: void mpf_mul_ui (mpf_t ROP, mpf_t OP1, unsigned long int OP2) Set ROP to OP1 times OP2. Division is undefined if the divisor is zero, and passing a zero divisor to the divide functions will make these functions intentionally divide by zero. This lets the user handle arithmetic exceptions in these functions in the same manner as other arithmetic exceptions. - Function: void mpf_div (mpf_t ROP, mpf_t OP1, mpf_t OP2) - Function: void mpf_ui_div (mpf_t ROP, unsigned long int OP1, mpf_t OP2) - Function: void mpf_div_ui (mpf_t ROP, mpf_t OP1, unsigned long int OP2) Set ROP to OP1/OP2. - Function: void mpf_sqrt (mpf_t ROP, mpf_t OP) - Function: void mpf_sqrt_ui (mpf_t ROP, unsigned long int OP) Set ROP to the square root of OP. - Function: void mpf_pow_ui (mpf_t ROP, mpf_t OP1, unsigned long int OP2) Set ROP to OP1 raised to the power OP2. - Function: void mpf_neg (mpf_t ROP, mpf_t OP) Set ROP to -OP. - Function: void mpf_abs (mpf_t ROP, mpf_t OP) Set ROP to the absolute value of OP. - Function: void mpf_mul_2exp (mpf_t ROP, mpf_t OP1, unsigned long int OP2) Set ROP to OP1 times 2 raised to OP2. - Function: void mpf_div_2exp (mpf_t ROP, mpf_t OP1, unsigned long int OP2) Set ROP to OP1 divided by 2 raised to OP2. Comparison Functions ==================== - Function: int mpf_cmp (mpf_t OP1, mpf_t OP2) - Function: int mpf_cmp_d (mpf_t OP1, double OP2) - Function: int mpf_cmp_ui (mpf_t OP1, unsigned long int OP2) - Function: int mpf_cmp_si (mpf_t OP1, signed long int OP2) Compare OP1 and OP2. Return a positive value if OP1 > OP2, zero if OP1 = OP2, and a negative value if OP1 < OP2. - Function: int mpf_eq (mpf_t OP1, mpf_t OP2, unsigned long int op3) Return non-zero if the first OP3 bits of OP1 and OP2 are equal, zero otherwise. I.e., test of OP1 and OP2 are approximately equal. 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). - Function: void mpf_reldiff (mpf_t ROP, mpf_t OP1, mpf_t OP2) Compute the relative difference between OP1 and OP2 and store the result in ROP. This is abs(OP1-OP2)/OP1. - Macro: int mpf_sgn (mpf_t OP) Return +1 if OP > 0, 0 if OP = 0, and -1 if OP < 0. This function is actually implemented as a macro. It evaluates its arguments multiple times. Input and Output Functions ========================== Functions that perform input from a stdio stream, and functions that output to