This file documents GNU MP, a library for arbitrary-precision
arithmetic.
Copyright (C) 1991, 1993, 1994, 1995, 1996, 1997, 1998, 1999, 2000
Free Software Foundation, Inc.
Permission is granted to make and distribute verbatim copies of this
manual provided the copyright notice and this permission notice are
preserved on all copies.
Permission is granted to copy and distribute modified versions of
this manual under the conditions for verbatim copying, provided that
the entire resulting derived work is distributed under the terms of a
permission notice identical to this one.
Permission is granted to copy and distribute translations of this
manual into another language, under the above conditions for modified
versions, except that this permission notice may be stated in a
translation approved by the Foundation.
GNU MP
******
This manual documents how to install and use the GNU multiple
precision arithmetic library, version 3.1.1.
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 that accompany the source
code.
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 and Athlon, Hitachi
SuperH and SH-2, HPPA 1.0, 1.1 and 2.0, Intel Pentium, Pentium
Pro/Pentium II, generic x86, Intel 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
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 Home Pages at
`http://www.swox.com/gmp/'.
How to use this Manual
======================
Everyone should read *Note GMP Basics::. If you need to install the
library yourself, you need to read *Note Installing GMP::, too.
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 .
(*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.
Non-Unix Systems
`configure' needs various Unix-like tools installed. On an MS-DOS
system cygwin or djgpp should work. 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.
Object Directory
To compile in a separate object directory, `cd' to that directory,
and prefix the configure command with the path to the GMP source
directory. For example `../src/gmp/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 from a separate object 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 are
very slightly slower, having a small cost on each function call,
but result in smaller executables and permit code sharing between
separate running processes.
`--target=CPU-VENDOR-OS'
The build target can be specified in the usual way, for either
native or cross compilation.
If `--target' isn't given, `./configure' builds for the host
system as determined by `./config.guess'. On some systems this
can't distinguish between different CPUs in a family, and you
should check the guess. Running `./config.guess' on the target
system will also show the relevant `VENDOR-OS', if you don't
already know what it should be.
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 CPU targets have specific assembly code support. See
`configure.in' for which `mpn' subdirectories get used by each.
* Alpha: `alpha', `alphaev5', `alphaev6'
* Hitachi: `sh', `sh2'
* HPPA: `hppa1.0', `hppa1.1', `hppa2.0', `hppa2.0w'
* MIPS: `mips', `mips3',
* Motorola: `m68000', `m68k', `m88k', `m88110'
* POWER: `power1', `power2', `power2sc', `powerpc', `powerpc64'
* SPARC: `sparc', `sparcv8', `microsparc', `supersparc',
`sparcv9', `ultrasparc', `sparc64'
* 80x86 family: `i386', `i486', `i586', `pentium', `pentiummmx',
`pentiumpro', `pentium2', `pentium3', `k6', `k62', `k63',
`athlon'
* Other: `a29k', `arm', `clipper', `i960', `ns32k', `pyramid',
`vax', `z8k'
CPUs not listed use generic C code. If some of the assembly code
causes problems, the generic C code can be selected with CPU
`none'.
`CC', `CFLAGS'
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 configurations specific compiler flags are set based on
the target CPU and compiler, see `CFLAGS' in the generated
`Makefile's. 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.
Note that if `CC' is set then `CFLAGS' must also be set. This
applies even if `CC' is merely one of the choices GMP would make
itself. This may change in a future release.
`--disable-alloca'
By default, GMP allocates temporary workspace using `alloca' if
that function is available, or `malloc' if not. If you're working
with large numbers and `alloca' overflows the available stack
space, you can build with `--disable-alloca' to use `malloc'
instead. `malloc' will probably be slightly slower than `alloca'.
When not using `alloca', it's actually the allocation function
selected with `mp_set_memory_functions' that's used, this being
`malloc' by default. *Note Custom Allocation::.
Depending on your system, the only indication of stack overflow
might be a segmentation violation. It might be possible to
increase available stack space with `limit', `ulimit' or
`setrlimit', or under DJGPP with `stubedit' or `_stklen'.
`--enable-fft'
By default multiplications are done using Karatsuba and 3-way
Toom-Cook algorithms, but a Fermat FFT can be enabled, for use on
large to very large operands. Currently the FFT is recommended
only for knowledgeable users who check the algorithm thresholds
for their CPU.
`--enable-mpbsd'
The Berkeley MP compatibility library (`libmp.a') and header file
(`mp.h') are built and installed only if `--enable-mpbsd' is used.
*Note BSD Compatible Functions::.
`MPN_PATH'
Various assembler versions of mpn subroutines are provided, and,
for a given CPU target, a search is made though a path to choose a
version of each. For example `sparcv8' has path `"sparc32/v8
sparc32 generic"', which means it looks first for v8 code, falls
back on plain sparc32, and finally falls back on generic C.
Knowledgeable users with special requirements can specify a path
with `MPN_PATH="dir list"'. This will normally be unnecessary
because all sensible paths should be available under one or other
CPU target.
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' and then `./pexpr 68^975+10'.
Documentation
The document you're now reading is `gmp.texi'. The usual automake
targets are available to make `gmp.ps' and/or `gmp.dvi'. 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 chooses the best ABI available for a given target system, and this
generally gives significantly greater speed.
The burden is on application programs and cooperating libraries to
ensure they match the ABI chosen by GMP. Fortunately this presents a
difficulty only on a few systems, and if you have one of them then the
performance gains are enough to make it worth the trouble.
Some of what's described in this section may change in future
releases of GMP.
HPPA 2.0
CPU target `hppa2.0' uses the hppa2.0n 32-bit ABI, but either a
32-bit or 64-bit limb.
A 64-bit limb is available on HP-UX 10 or up when using `c89'. No
`gcc' support is planned for 64-bit operations in this ABI.
Applications must be compiled with the same options as GMP, which
means
c89 +DA2.0 +e -D_LONG_LONG_LIMB
A 32-bit limb is used in other cases, and no special compiler
options are needed.
CPU target `hppa2.0w' uses the hppa2.0w 64-bit ABI, which is
available on HP-UX 11 or up when using `c89'. `gcc' support for
this is in progress. Applications must be compiled for the same
ABI, which means
c89 +DD64
MIPS 3 and 4 under IRIX 6
Targets `mips*-*-irix6*' use the n32 ABI and a 64-bit limb.
Applications must be compiled for the same ABI, which means either
gcc -mabi=n32
cc -n32
PowerPC 64
CPU target `powerpc64' uses either the 32-bit ABI or the AIX
64-bit ABI. The latter is used on targets `powerpc64-*-aix*' and
applications must be compiled using either
gcc -maix64
xlc -q64
On other systems the 32-bit ABI is used, but with 64-bit limbs
provided by `long long' in `gcc'. Applications must be compiled
using
gcc -D_LONG_LONG_LIMB
Sparc V9
On a sparc v9 CPU, either the v8plus 32-bit ABI or v9 64-bit ABI
is used. Targets `ultrasparc*-*-solaris2.[7-9]',
`sparcv9-*-solaris2.[7-9]' and `sparc64-*-linux*' use the v9 ABI,
if the compiler supports it. Other targets use the v8plus ABI
(but with as much of the v9 ISA as possible in the circumstances).
Note that Solaris prior to 2.7 doesn't save all registers
properly, and hence uses the v8plus ABI.
For the v8plus ABI, applications can be compiled with either
gcc -mv8plus
cc -xarch=v8plus
For the v9 ABI, applications must be compiled with either
gcc -m64 -mptr64 -Wa,-xarch=v9 -mcpu=v9
cc -xarch=v9
Don't be confused by the names of these options, they're called
`arch' but they effectively control the ABI.
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 3 series
will be upwardly binary compatible in each release, but may be adding
additional function interfaces. 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
minor version of GMP.
When building a package for a CPU family, care should be taken to use
`--target' 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 `--target' in a build so
`./config.guess' will detect the CPU. But a way to manually specify a
`--target' will be wanted for systems where `./config.guess' is inexact.
Notes for Particular Systems
============================
AIX 4.3
Targets `*-*-aix4.[3-9]*' have shared libraries disabled since
they seem to fail on AIX 4.3.
OpenBSD 2.6
`m4' in this release of OpenBSD has a bug in `eval' that makes it
unsuitable for `.asm' file processing. `./configure' will detect
the problem and either abort or choose another m4 in the `PATH'.
The bug is fixed in OpenBSD 2.7, so either upgrade or use GNU m4.
Sparc V8
Using CPU target `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 Pentium and PentiumPro
The Intel Pentium P5 code is good for its intended P5, but quite
slow when run on Intel P6 class chips (PPro, P-II, P-III). `i386'
is a better choice if you're making binaries that must run on both.
x86 MMX and old GAS
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).
If the target CPU 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.
x86 GCC 2.95.2 `-march=pentiumpro'
GCC 2.95.2 miscompiles `mpz/powm.c' when `-march=pentiumpro' is
used, so that option is omitted from the `CFLAGS' chosen for
relevant CPUs. The problem is believed to be fixed in GCC 2.96.
Known Build Problems
====================
You might find more up-to-date information at
`http://www.swox.com/gmp/'.
Generic C on a 64-bit system
When making a generic C build using `--target=none' on a 64-bit
system (meaning where `unsigned long' is 64 bits),
`BITS_PER_MP_LIMB', `BITS_PER_LONGINT' and `BYTES_PER_MP_LIMB' in
`mpn/generic/gmp-mparam.h' need to be changed to 64 and 8. This
will hopefully be automated in a future version of GMP.
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
before you compile GMP. (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.
Stripped Libraries
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' extracts archives into a
single directory, 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.
SunOS 4 Native Tools
The setting for `GSYM_PREFIX' in `config.m4' may be incorrectly
determined when using the native `grep', leading at link-time to
undefined symbols like `___gmpn_add_n'. To fix this, after running
`./configure', change the relevant line in `config.m4' to
`define(, <_>)'.
The `ranlib' command will need to be run manually when building a
static library with the native `ar'. After `make', run `ranlib
.libs/libgmp.a', and when using `--enable-mpbsd' run `ranlib
.libs/libmp.a' too.
VAX running Ultrix
You need to build and install the GNU assembler before you compile
GMP. The VAX assembly in GMP uses an instruction (`jsobgtr') that
cannot be assembled by the Ultrix assembler.
GMP Basics
**********
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.
*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.*
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 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 contains 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 100
functions in this class.
2. Functions for rational number arithmetic, with names beginning with
`mpq_'. The associated type is `mpq_t'. There are about 20
functions in this class, but the functions in the previous class
can be used for performing 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 50
functions is this class.
4. Functions compatible with Berkeley GMP, 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_'. There are about 30
(hard-to-use) functions in this class.
The associated type is array of `mp_limb_t'.
6. Miscellaneous functions. Functions for setting up custom
allocation and functions for generating random numbers.
GMP Variable Conventions
========================
As a general rule, all GMP functions expect output arguments before
input arguments. This notation is based on an analogy with the
assignment operator. (The BSD MP compatibility functions disobey this
rule, having the output argument(s) 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 out
between each initialization. After a variable has been initialized, it
may be assigned to any number of times.
For efficiency reasons, avoid initializing and clearing out a GMP
variable in a loop. Instead, initialize it before entering the loop,
and clear it out after the loop has exited.
GMP variables are small, containing only a couple of sizes, and
pointers to allocated data. Once you have initialized a GMP variable,
you don't need to worry about space allocation. All functions in GMP
automatically allocate additional space when a variable does not
already have enough. They do not, however, reduce the space when a
smaller value is stored. Most of the time this policy is best, since
it avoids frequent re-allocation.
When a variable of type `mpz_t' is used as a function parameter, it's
effectively a call-by-reference, meaning anything the function does to
it will be be done to the original in the caller. When a function is
going to return an `mpz_t' result, it should provide a separate
parameter or parameters that it sets, like the GMP library functions
do. A `return' of an `mpz_t' doesn't return the object, only a pointer
to it, and this is almost certainly not what you want. All this
applies to `mpq_t' and `mpf_t' too.
Here's an example function accepting an `mpz_t' parameter, doing a
certain calculation, and returning a result.
void
myfunction (mpz_t result, mpz_t param, unsigned long n)
{
unsigned long i;
mpz_mul_ui (result, param, n);
for (i = 1; i < n; i++)
mpz_add_ui (result, result, i*7);
}
int
main (void)
{
mpz_t r, n;
mpz_init (r);
mpz_init_set_str (n, "123456", 0);
myfunction (r, n, 20L);
mpz_out_str (stdout, 10, r); printf ("\n");
return 0;
}
This example will work if `result' and `param' are the same
variable, just like the library functions. But sometimes this is
tricky to arrange, and an application might not want to bother for its
own subroutines.
`mpz_t' is actually implemented as a one-element array of a certain
structure type. This is why using it to declare a variable gives an
object with the fields GMP needs, but then using it as a parameter
passes a pointer to the object. Note that the actual contents of an
`mpz_t' are for internal use only and you should not access them
directly if you want your code to be compatible with future GMP
releases.
GMP and Reentrancy
==================
The GMP code is reentrant and thread-safe, with some exceptions:
* The function `mpf_set_default_prec' saves the selected precision in
a global variable.
* The function `mp_set_memory_functions' uses several global
variables for storing 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, GMP will not be reentrant either.
* The old random number functions (`mpz_random', etc) use a random
number generator from the C library, usually `mrand48' or
`random'. These routines are not reentrant, since they rely on
global state. (However the newer random number functions that
accept a `gmp_randstate_t' parameter are reentrant.)
* If `alloca' is not available, or GMP is configured with
`--disable-alloca', the library is not reentrant, due to the
current implementation of `stack-alloc.c'. In the generated
`config.h', `USE_STACK_ALLOC' set to 1 will mean not reentrant.
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.
Compatibility with older versions
=================================
This version of GMP is upwardly binary compatible with versions 3.0
and 3.0.1, and upwardly compatible at the source level with versions
2.0, 2.0.1, and 2.0.2, 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 has reverted to the 2.0.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
3. Please see the GMP 2 manual for details.
Getting the Latest Version of GMP
=================================
The latest version of the GMP library is available at
`ftp://ftp.gnu.org/pub/gnu/gmp'. Many sites around the world mirror
`ftp.gnu.org'; please use a mirror site near you, see
`http://www.gnu.org/order/ftp.html'.
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, you may want to check
`http://www.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'.
* 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'.
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'.
- Function: void mpz_init (mpz_t INTEGER)
Initialize INTEGER with limb space and set the initial numeric
value to 0. Each variable should normally only be initialized
once, or at least cleared out (using `mpz_clear') between each
initialization.
Here is an example of using `mpz_init':
{
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_clear (mpz_t INTEGER)
Free the limb space occupied by INTEGER. Make sure to call this
function for all `mpz_t' variables when you are done with them.
- Function: void * _mpz_realloc (mpz_t INTEGER, mp_size_t NEW_ALLOC)
Change the limb space allocation to NEW_ALLOC limbs. This
function is not normally called from user code, but it can be used
to give memory back to the heap, or to increase the space of a
variable to avoid repeated automatic re-allocation.
- Function: void mpz_array_init (mpz_t INTEGER_ARRAY[], size_t
ARRAY_SIZE, mp_size_t FIXED_NUM_BITS)
Allocate *fixed* limb space for all ARRAY_SIZE integers in
INTEGER_ARRAY. The fixed allocation for each integer in the array
is enough to store FIXED_NUM_BITS. If the fixed space will be
insufficient for storing the result of a subsequent calculation,
the result is unpredictable.
This function is useful for decreasing the working set for some
algorithms that use large integer arrays.
There is no way to de-allocate the storage allocated by this
function. Don't call `mpz_clear'!
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.
- Function: int mpz_set_str (mpz_t ROP, char *STR, int BASE)
Set the value of ROP from STR, a '\0'-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 up to the '\0' 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. Please tell us your opinion about this change. 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: mp_limb_t mpz_getlimbn (mpz_t OP, mp_size_t N)
Return limb #N from OP. This function allows for very efficient
decomposition of a number in its limbs.
The function `mpz_size' can be used to determine the useful range
for N.
- Function: unsigned long int mpz_get_ui (mpz_t OP)
Return the least significant part from OP. This function combined
with
`mpz_tdiv_q_2exp(..., OP, CHAR_BIT*sizeof(unsigned long int))' can
be used to decompose an integer into unsigned longs.
- 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 large to fit in a `signed long int', the returned
result is probably not very useful. To find out if the value will
fit, use the function `mpz_fits_slong_p'.
- Function: double mpz_get_d (mpz_t OP)
Convert OP to a double.
- 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', space for the result string is allocated using
the default allocation function.
If STR is not `NULL', it should point to a block of storage enough
large for the result. To find out the right amount of space to
provide for STR, use `mpz_sizeinbase (OP, BASE) + 2'. The two
extra bytes are for a possible minus sign, and for the terminating
null character.
A pointer to the result string is returned. This pointer will
will either equal STR, or if that is `NULL', will point to the
allocated storage.
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)
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_ui (mpz_t ROP, mpz_t OP1, unsigned long
int OP2)
Add OP1 times OP2 to ROP.
- 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, OP2 steps.
- 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, and passing a zero
divisor to the divide or modulo functions, as well passing a zero mod
argument to the `mpz_powm' and `mpz_powm_ui' 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.
There are three main groups of division functions:
* Functions that truncate the quotient towards 0. The names of
these functions start with `mpz_tdiv'. The `t' in the name is
short for `truncate'.
* Functions that round the quotient towards -infinity). The names
of these routines start with `mpz_fdiv'. The `f' in the name is
short for `floor'.
* Functions that round the quotient towards +infinity. The names of
these routines start with `mpz_cdiv'. The `c' in the name is
short for `ceil'.
For each rounding mode, there are a couple of variants. Here `q'
means that the quotient is computed, while `r' means that the remainder
is computed. Functions that compute both the quotient and remainder
have `qr' in the name.
- Function: void mpz_tdiv_q (mpz_t Q, mpz_t N, mpz_t D)
- Function: unsigned long int mpz_tdiv_q_ui (mpz_t Q, mpz_t N,
unsigned long int D)
Set Q to [N/D], truncated towards 0.
The function `mpz_tdiv_q_ui' returns the absolute value of the true
remainder.
- Function: void mpz_tdiv_r (mpz_t R, mpz_t N, mpz_t D)
- Function: unsigned long int mpz_tdiv_r_ui (mpz_t R, mpz_t N,
unsigned long int D)
Set R to (N - [N/D] * D), where the quotient is truncated towards
0. Unless R becomes zero, it will get the same sign as N.
The function `mpz_tdiv_r_ui' returns the absolute value of the
remainder.
- Function: void mpz_tdiv_qr (mpz_t Q, mpz_t R, mpz_t N, mpz_t D)
- Function: unsigned long int mpz_tdiv_qr_ui (mpz_t Q, mpz_t R, mpz_t
N, unsigned long int D)
Set Q to [N/D], truncated towards 0. Set R to (N - [N/D] * D).
Unless R becomes zero, it will get the same sign as N. If Q and R
are the same variable, the results are undefined.
The function `mpz_tdiv_qr_ui' returns the absolute value of the
remainder.
- Function: unsigned long int mpz_tdiv_ui (mpz_t N, unsigned long int
D)
Like `mpz_tdiv_r_ui', but the remainder is not stored anywhere; its
absolute value is just returned.
- Function: void mpz_fdiv_q (mpz_t Q, mpz_t N, mpz_t D)
- Function: unsigned long int mpz_fdiv_q_ui (mpz_t Q, mpz_t N,
unsigned long int D)
Set Q to N/D, rounded towards -infinity.
The function `mpz_fdiv_q_ui' returns the remainder.
- Function: void mpz_fdiv_r (mpz_t R, mpz_t N, mpz_t D)
- Function: unsigned long int mpz_fdiv_r_ui (mpz_t R, mpz_t N,
unsigned long int D)
Set R to (N - N/D * D), where the quotient is rounded towards
-infinity. Unless R becomes zero, it will get the same sign as D.
The function `mpz_fdiv_r_ui' returns the remainder.
- Function: void mpz_fdiv_qr (mpz_t Q, mpz_t R, mpz_t N, mpz_t D)
- Function: unsigned long int mpz_fdiv_qr_ui (mpz_t Q, mpz_t R, mpz_t
N, unsigned long int D)
Set Q to N/D, rounded towards -infinity. Set R to (N - N/D * D).
Unless R becomes zero, it will get the same sign as D. If Q and R
are the same variable, the results are undefined.
The function `mpz_fdiv_qr_ui' returns the remainder.
- Function: unsigned long int mpz_fdiv_ui (mpz_t N, unsigned long int
D)
Like `mpz_fdiv_r_ui', but the remainder is not stored anywhere; it
is just returned.
- Function: void mpz_cdiv_q (mpz_t Q, mpz_t N, mpz_t D)
- Function: unsigned long int mpz_cdiv_q_ui (mpz_t Q, mpz_t N,
unsigned long int D)
Set Q to N/D, rounded towards +infinity.
The function `mpz_cdiv_q_ui' returns the negated remainder.
- Function: void mpz_cdiv_r (mpz_t R, mpz_t N, mpz_t D)
- Function: unsigned long int mpz_cdiv_r_ui (mpz_t R, mpz_t N,
unsigned long int D)
Set R to (N - N/D * D), where the quotient is rounded towards
+infinity. Unless R becomes zero, it will get the opposite sign
as D.
The function `mpz_cdiv_r_ui' returns the negated remainder.
- Function: void mpz_cdiv_qr (mpz_t Q, mpz_t R, mpz_t N, mpz_t D)
- Function: unsigned long int mpz_cdiv_qr_ui (mpz_t Q, mpz_t R, mpz_t
N, unsigned long int D)
Set Q to N/D, rounded towards +infinity. Set R to (N - N/D * D).
Unless R becomes zero, it will get the opposite sign as D. If Q
and R are the same variable, the results are undefined.
The function `mpz_cdiv_qr_ui' returns the negated remainder.
- Function: unsigned long int mpz_cdiv_ui (mpz_t N, unsigned long int
D)
Like `mpz_tdiv_r_ui', but the remainder is not stored anywhere; its
negated value is just returned.
- Function: void mpz_mod (mpz_t R, mpz_t N, mpz_t D)
- Function: unsigned long int mpz_mod_ui (mpz_t R, mpz_t N, unsigned
long int D)
Set R to N `mod' D. The sign of the divisor is ignored; the
result is always non-negative.
The function `mpz_mod_ui' returns the remainder.
- Function: void mpz_divexact (mpz_t Q, mpz_t N, mpz_t D)
Set Q to N/D. This function produces correct results only when it
is known in advance that D divides N.
Since mpz_divexact is much faster than any of the other routines
that produce the quotient (*note References:: Jebelean), it is the
best choice for instances in which exact division is known to
occur, such as reducing a rational to lowest terms.
- Function: void mpz_tdiv_q_2exp (mpz_t Q, mpz_t N, unsigned long int
D)
Set Q to N divided by 2 raised to D. The quotient is truncated
towards 0.
- Function: void mpz_tdiv_r_2exp (mpz_t R, mpz_t N, unsigned long int
D)
Divide N by (2 raised to D), rounding the quotient towards 0, and
put the remainder in R. Unless it is zero, R will have the same
sign as N.
- Function: void mpz_fdiv_q_2exp (mpz_t Q, mpz_t N, unsigned long int
D)
Set Q to N divided by 2 raised to D, rounded towards -infinity.
This operation can also be defined as arithmetic right shift D bit
positions.
- Function: void mpz_fdiv_r_2exp (mpz_t R, mpz_t N, unsigned long int
D)
Divide N by (2 raised to D), rounding the quotient towards
-infinity, and put the remainder in R. The sign of R will always
be positive. This operation can also be defined as masking of the
D least significant bits.
Exponentiation Functions
========================
- Function: void mpz_powm (mpz_t ROP, mpz_t BASE, mpz_t EXP, mpz_t MOD)
- Function: void mpz_powm_ui (mpz_t ROP, mpz_t BASE, unsigned long int
EXP, mpz_t MOD)
Set ROP to (BASE raised to EXP) `mod' MOD. If EXP is negative,
the result is undefined.
- Function: void mpz_pow_ui (mpz_t ROP, mpz_t BASE, unsigned long int
EXP)
- Function: void mpz_ui_pow_ui (mpz_t ROP, unsigned long int BASE,
unsigned long int EXP)
Set ROP to BASE raised to EXP. The case of 0^0 yields 1.
Root Extraction Functions
=========================
- Function: int mpz_root (mpz_t ROP, mpz_t OP, unsigned long int N)
Set ROP to the truncated integer part of the Nth root of OP.
Return non-zero if the computation was exact, i.e., if OP is ROP
to the Nth power.
- Function: void mpz_sqrt (mpz_t ROP, mpz_t OP)
Set ROP to the truncated integer part of the square root of OP.
- Function: void mpz_sqrtrem (mpz_t ROP1, mpz_t ROP2, mpz_t OP)
Set ROP1 to the truncated integer part of the square root of OP,
like `mpz_sqrt'. Set ROP2 to OP-ROP1*ROP1, (i.e., zero if OP is a
perfect square).
If ROP1 and ROP2 are the same variable, the results are undefined.
- Function: int mpz_perfect_power_p (mpz_t OP)
Return non-zero if OP is a perfect power, i.e., if there exist
integers A and B, with B > 1, such that OP equals a raised to b.
Return zero otherwise.
- 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. Return zero otherwise.
Number Theoretic Functions
==========================
- Function: int mpz_probab_prime_p (mpz_t N, int REPS)
If this function returns 0, N is definitely not prime. If it
returns 1, then N is `probably' prime. If it returns 2, then N is
surely prime. Reasonable values of reps vary from 5 to 10; a
higher value lowers the probability for a non-prime to pass as a
`probable' prime.
The function uses Miller-Rabin's probabilistic test.
- Function: int 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,
but for for practical purposes it's adequate, since 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 either of 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)
Compute G, S, and T, such that AS + BT = G = `gcd'(A, B). If T is
`NULL', that argument is not computed.
- Function: void mpz_lcm (mpz_t ROP, mpz_t OP1, mpz_t OP2)
Set ROP to the least common multiple of OP1 and OP2.
- 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.
Return non-zero if an inverse exists, zero otherwise. When the
function returns zero, ROP is undefined.
- Function: int mpz_jacobi (mpz_t OP1, mpz_t OP2)
- Function: int mpz_legendre (mpz_t OP1, mpz_t OP2)
Compute the Jacobi and Legendre symbols, respectively. OP2 should
be odd and must be positive.
- Function: int mpz_si_kronecker (long A, mpz_t B);
- Function: int mpz_ui_kronecker (unsigned long A, mpz_t B);
- Function: int mpz_kronecker_si (mpz_t A, long B);
- Function: int mpz_kronecker_ui (mpz_t A, unsigned long B);
Calculate the value of the Kronecker/Jacobi symbol (A/B), with the
Kronecker extension (a/2)=(2/a) when a odd, or (a/2)=0 when a even.
All values of A and B give a well-defined result. See Henri
Cohen, section 1.4.2, for more information (*note References::).
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 ROP, unsigned long int N)
Compute the Nth Fibonacci number and store the result in ROP.
Comparison Functions
====================
- Function: int mpz_cmp (mpz_t OP1, mpz_t OP2)
Compare OP1 and OP2. Return a positive value if OP1 > OP2, zero
if OP1 = OP2, and a negative value if OP1 < OP2.
- Macro: int mpz_cmp_ui (mpz_t OP1, unsigned long int OP2)
- Macro: int mpz_cmp_si (mpz_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.
These functions are actually implemented as macros. They evaluate
their arguments multiple times.
- Function: int mpz_cmpabs (mpz_t OP1, mpz_t 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 OP1 > OP2, zero if OP1 = OP2, and a negative value if OP1
< OP2.
- 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
arguments multiple times.
Logical and Bit Manipulation Functions
======================================
These functions behave as if two's complement arithmetic were used
(although sign-magnitude is used by the actual implementation).
- 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)
For non-negative numbers, return the population count of OP. For
negative numbers, return the largest possible value (MAX_ULONG).
- Function: unsigned long int mpz_hamdist (mpz_t OP1, mpz_t OP2)
If OP1 and OP2 are both non-negative, return the hamming distance
between the two operands. Otherwise, return the largest possible
value (MAX_ULONG).
It is possible to extend this function to return a useful value
when the operands are both negative, but the current
implementation returns MAX_ULONG in this case. *Do not depend on
this behavior, since it will change in a future release.*
- Function: unsigned long int mpz_scan0 (mpz_t OP, unsigned long int
STARTING_BIT)
Scan OP, starting with bit STARTING_BIT, towards more significant
bits, until the first clear bit is found. Return the index of the
found bit.
- Function: unsigned long int mpz_scan1 (mpz_t OP, unsigned long int
STARTING_BIT)
Scan OP, starting with bit STARTING_BIT, towards more significant
bits, until the first set bit is found. Return the index of the
found bit.
- 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)
Check 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.
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
arguments 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 returned value will be exact
or 1 too big. If BASE is a power of 2, the returned value will
always be exact.
This function is useful in order to allocate the right amount of
space before converting OP to a string. The right amount of
allocation is normally two more than the value returned by
`mpz_sizeinbase' (one extra for a minus sign and one for the
terminating '\0').
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. *Note that this is an incompatible change from version 1 of
the library.*
- 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: void mpq_swap (mpq_t ROP1, mpq_t ROP2)
Swap the values ROP1 and ROP2 efficiently.
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_div (mpq_t QUOTIENT, mpq_t DIVIDEND, mpq_t
DIVISOR)
Set QUOTIENT to DIVIDEND/DIVISOR.
- Function: void mpq_neg (mpq_t NEGATED_OPERAND, mpq_t OPERAND)
Set NEGATED_OPERAND to -OPERAND.
- 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)
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.
This routine allows that NUM2 and DEN2 have common factors.
This function is actually implemented as a macro. It evaluates its
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. But there are two macros
that allow us to apply any `mpz' function on the numerator or
denominator of a rational number. If these macros are used to assign
to the rational number, `mpq_canonicalize' normally need to be called
afterwards.
- 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.
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 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.
Miscellaneous Functions
=======================
- Function: double mpq_get_d (mpq_t OP)
Convert OP to a double.
- Function: void mpq_set_d (mpq_t ROP, double D)
Set ROP to the value of d, without rounding.
These functions assign between either the numerator or denominator
of a rational, and an integer. Instead of using these functions, it is
preferable to use the more general mechanisms `mpq_numref' and
`mpq_denref', together with `mpz_set'.
- Function: void mpq_set_num (mpq_t RATIONAL, mpz_t NUMERATOR)
Copy NUMERATOR to the numerator of RATIONAL. When this risks to
make the numerator and denominator of RATIONAL have common
factors, you have to pass RATIONAL to `mpq_canonicalize' before
any operations are performed on RATIONAL.
This function is equivalent to `mpz_set (mpq_numref (RATIONAL),
NUMERATOR)'.
- Function: void mpq_set_den (mpq_t RATIONAL, mpz_t DENOMINATOR)
Copy DENOMINATOR to the denominator of RATIONAL. When this risks
to make the numerator and denominator of RATIONAL have common
factors, or if the denominator might be negative, you have to pass
RATIONAL to `mpq_canonicalize' before any operations are performed
on RATIONAL.
*In version 1 of the library, negative denominators were handled by
copying the sign to the numerator. That is no longer done.*
This function is equivalent to `mpz_set (mpq_denref (RATIONAL),
DENOMINATORS)'.
- Function: void mpq_get_num (mpz_t NUMERATOR, mpq_t RATIONAL)
Copy the numerator of RATIONAL to the integer NUMERATOR, to
prepare for integer operations on the numerator.
This function is equivalent to `mpz_set (NUMERATOR, mpq_numref
(RATIONAL))'.
- Function: void mpq_get_den (mpz_t DENOMINATOR, mpq_t RATIONAL)
Copy the denominator of RATIONAL to the integer DENOMINATOR, to
prepare for integer operations on the denominator.
This function is equivalent to `mpz_set (DENOMINATOR, mpq_denref
(RATIONAL))'.
Floating-point Functions
************************
This chapter describes the GMP functions for performing floating
point arithmetic. These functions start with the prefix `mpf_'.
GMP floating point numbers are stored in objects of type `mpf_t'.
The GMP floating-point functions have an interface that is similar
to the GMP integer functions. The function prefix for floating-point
operations is `mpf_'.
There is one significant characteristic of floating-point numbers
that has motivated a difference between this function class and other
GMP function classes: the inherent inexactness of floating point
arithmetic. The user has to specify the precision of each variable. A
computation that assigns a variable will take place with the precision
of the assigned variable; the precision of variables used as input is
ignored.
The precision of a calculation is defined as follows: Compute the
requested operation exactly (with "infinite precision"), and truncate
the result to the destination variable precision. Even if the user has
asked for a very high precision, GMP will not calculate with
superfluous digits. For example, if two low-precision numbers of
nearly equal magnitude are added, the precision of the result will be
limited to what is required to represent the result accurately.
The GMP floating-point functions are _not_ intended as a smooth
extension to the IEEE P754 arithmetic. Specifically, the results
obtained on one computer often differs from the results obtained 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.
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: void mpf_set_prec (mpf_t ROP, unsigned long int PREC)
Set the precision of ROP to be *at least* PREC bits. Since
changing the precision involves calls to `realloc', this routine
should not be called in a tight loop.
- Function: unsigned long int mpf_get_prec (mpf_t OP)
Return the precision actually used for assignments of OP.
- Function: void mpf_set_prec_raw (mpf_t ROP, unsigned long int PREC)
Set the precision of ROP to be *at least* PREC bits. This is a
low-level function that does not change the allocation. The PREC
argument must not be larger that the precision previously returned
by `mpf_get_prec'. It is crucial that the precision of ROP is
ultimately reset to exactly the value returned by `mpf_get_prec'
before the first call to `mpf_set_prec_raw'.
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 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 up to the '\0' is a
valid number in base BASE. Otherwise it returns -1.
- Function: void mpf_swap (mpf_t ROP1, mpf_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 `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: 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. The base may vary
from 2 to 36. Generate at most N_DIGITS significant digits, or if
N_DIGITS is 0, the maximum number of digits accurately
representable by OP.
If STR is `NULL', space for the mantissa is allocated using the
default allocation function.
If STR is not `NULL', it should point to a block of storage enough
large for the mantissa, i.e., N_DIGITS + 2. The two extra bytes
are for a possible minus sign, and for the terminating null
character.
The exponent is written through the pointer EXPPTR.
If N_DIGITS is 0, the maximum number of digits meaningfully
achievable from the precision of OP will be generated. Note that
the space requirements for STR in this case will be impossible for
the user to predetermine. Therefore, you need to pass `NULL' for
the string argument whenever N_DIGITS is 0.
The generated string is a fraction, with an implicit radix point
immediately to the left of the first digit. For example, the
number 3.1416 would be returned as "31416" in the string and 1
written at EXPPTR.
A pointer to the result string is returned. This pointer will
will either equal STR, or if that is `NULL', will point to the
allocated storage.
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_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.
- 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.
- 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 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 mpf_out_str (FILE *STREAM, int BASE, size_t
N_DIGITS, mpf_t OP)
Output OP on stdio stream STREAM, as a string of digits in base
BASE. The base may vary from 2 to 36. Print at most N_DIGITS
significant digits, or if N_DIGITS is 0, the maximum number of
digits accurately representable by OP.
In addition to the significant digits, a leading `0.' and a
trailing exponent, in the form `eNNN', are printed. If BASE is
greater than 10, `@' will be used instead of `e' as exponent
delimiter.
Return the number of bytes written, or if an error occurred,
return 0.
- Function: size_t mpf_inp_str (mpf_t ROP, FILE *STREAM, int BASE)
Input a string in base BASE from stdio stream STREAM, and put the
read float in ROP. The string is of the form `M@N' or, if the
base is 10 or less, alternatively `MeN'. `M' is the mantissa and
`N' is the exponent. The mantissa is always in the specified
base. The exponent is either in the specified base or, if BASE is
negative, in decimal.
The argument BASE may be in the ranges 2 to 36, or -36 to -2.
Negative values are used to specify that the exponent is in
decimal.
Unlike the corresponding `mpz' function, the base will not be
determined from the leading characters of the string if BASE is 0.
This is so that numbers like `0.23' are not interpreted as octal.
Return the number of bytes read, or if an error occurred, return 0.
Miscellaneous Functions
=======================
- Function: void mpf_ceil (mpf_t ROP, mpf_t OP)
- Function: void mpf_floor (mpf_t ROP, mpf_t OP)
- Function: void mpf_trunc (mpf_t ROP, mpf_t OP)
Set ROP to OP rounded to an integer. `mpf_ceil' rounds to the
next higher integer, `mpf_floor' to the next lower, and
`mpf_trunc' to the integer towards zero.
- Function: void mpf_urandomb (mpf_t ROP, gmp_randstate_t STATE,
unsigned long int NBITS)
Generate a uniformly distributed random float in ROP, such that 0
<= ROP < 1, with NBITS significant bits in the mantissa.
The variable STATE must be initialized by calling one of the
`gmp_randinit' functions (*Note Random State Initialization::)
before invoking this function.
- Function: void mpf_random2 (mpf_t ROP, mp_size_t MAX_SIZE, mp_exp_t
MAX_EXP)
Generate a random float of at most MAX_SIZE limbs, with long
strings of zeros and ones in the binary representation. The
exponent of the number is in the interval -EXP to EXP. This
function is 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.
Low-level Functions
*******************
This chapter describes low-level GMP functions, used to implement
the high-level GMP functions, but also intended for time-critical user
code.
These functions start with the prefix `mpn_'.
The `mpn' functions are designed to be as fast as possible, *not* to
provide a coherent calling interface. The different functions have
somewhat similar interfaces, but there are variations that make them
hard to use. These functions do as little as possible apart from the
real multiple precision computation, so that no time is spent on things
that not all callers need.
A source operand is specified by a pointer to the least significant
limb and a limb count. A destination operand is specified by just a
pointer. It is the responsibility of the caller to ensure that the
destination has enough space for storing the result.
With this way of specifying operands, it is possible to perform
computations on subranges of an argument, and store the result into a
subrange of a destination.
A common requirement for all functions is that each source area
needs at least one limb. No size argument may be zero. Unless
otherwise stated, in-place operations are allowed where source and
destination are the same, but not where they only partly overlap.
The `mpn' functions are the base for the implementation of the
`mpz_', `mpf_', and `mpq_' functions.
This example adds the number beginning at S1P and the number
beginning at S2P and writes the sum at DESTP. All areas have SIZE
limbs.
cy = mpn_add_n (destp, s1p, s2p, size)
In the notation used here, a source operand is identified by the
pointer to the least significant limb, and the limb count in braces.
For example, {s1p, s1size}.
- Function: mp_limb_t mpn_add_n (mp_limb_t *RP, const mp_limb_t *S1P,
const mp_limb_t *S2P, mp_size_t SIZE)
Add {S1P, SIZE} and {S2P, SIZE}, and write the SIZE least
significant limbs of the result to RP. Return carry, either 0 or
1.
This is the lowest-level function for addition. It is the
preferred function for addition, since it is written in assembly
for most targets. For addition of a variable to itself (i.e., S1P
equals S2P, use `mpn_lshift' with a count of 1 for optimal speed.
- Function: mp_limb_t mpn_add_1 (mp_limb_t *RP, const mp_limb_t *S1P,
mp_size_t SIZE, mp_limb_t S2LIMB)
Add {S1P, SIZE} and S2LIMB, and write the SIZE least significant
limbs of the result to RP. Return carry, either 0 or 1.
- Function: mp_limb_t mpn_add (mp_limb_t *RP, const mp_limb_t *S1P,
mp_size_t S1SIZE, const mp_limb_t *S2P, mp_size_t S2SIZE)
Add {S1P, S1SIZE} and {S2P, S2SIZE}, and write the S1SIZE least
significant limbs of the result to RP. Return carry, either 0 or
1.
This function requires that S1SIZE is greater than or equal to
S2SIZE.
- Function: mp_limb_t mpn_sub_n (mp_limb_t *RP, const mp_limb_t *S1P,
const mp_limb_t *S2P, mp_size_t SIZE)
Subtract {S2P, S2SIZE} from {S1P, SIZE}, and write the SIZE least
significant limbs of the result to RP. Return borrow, either 0 or
1.
This is the lowest-level function for subtraction. It is the
preferred function for subtraction, since it is written in
assembly for most targets.
- Function: mp_limb_t mpn_sub_1 (mp_limb_t *RP, const mp_limb_t *S1P,
mp_size_t SIZE, mp_limb_t S2LIMB)
Subtract S2LIMB from {S1P, SIZE}, and write the SIZE least
significant limbs of the result to RP. Return borrow, either 0 or
1.
- Function: mp_limb_t mpn_sub (mp_limb_t *RP, const mp_limb_t *S1P,
mp_size_t S1SIZE, const mp_limb_t *S2P, mp_size_t S2SIZE)
Subtract {S2P, S2SIZE} from {S1P, S1SIZE}, and write the S1SIZE
least significant limbs of the result to RP. Return borrow,
either 0 or 1.
This function requires that S1SIZE is greater than or equal to
S2SIZE.
- Function: void mpn_mul_n (mp_limb_t *RP, const mp_limb_t *S1P, const
mp_limb_t *S2P, mp_size_t SIZE)
Multiply {S1P, SIZE} and {S2P, SIZE}, and write the *entire*
result to RP.
The destination has to have space for 2*SIZE limbs, even if the
significant result might be one limb smaller.
- Function: mp_limb_t mpn_mul_1 (mp_limb_t *RP, const mp_limb_t *S1P,
mp_size_t SIZE, mp_limb_t S2LIMB)
Multiply {S1P, SIZE} and S2LIMB, and write the SIZE least
significant limbs of the product to RP. Return the most
significant limb of the product.
This is a low-level function that is a building block for general
multiplication as well as other operations in GMP. It is written
in assembly for most targets.
Don't call this function if S2LIMB is a power of 2; use
`mpn_lshift' with a count equal to the logarithm of S2LIMB
instead, for optimal speed.
- Function: mp_limb_t mpn_addmul_1 (mp_limb_t *RP, const mp_limb_t
*S1P, mp_size_t SIZE, mp_limb_t S2LIMB)
Multiply {S1P, SIZE} and S2LIMB, and add the SIZE least
significant limbs of the product to {RP, SIZE} and write the
result to RP. Return the most significant limb of the product,
plus carry-out from the addition.
This is a low-level function that is a building block for general
multiplication as well as other operations in GMP. It is written
in assembly for most targets.
- Function: mp_limb_t mpn_submul_1 (mp_limb_t *RP, const mp_limb_t
*S1P, mp_size_t SIZE, mp_limb_t S2LIMB)
Multiply {S1P, SIZE} and S2LIMB, and subtract the SIZE least
significant limbs of the product from {RP, SIZE} and write the
result to RP. Return the most significant limb of the product,
minus borrow-out from the subtraction.
This is a low-level function that is a building block for general
multiplication and division as well as other operations in GMP.
It is written in assembly for most targets.
- Function: mp_limb_t mpn_mul (mp_limb_t *RP, const mp_limb_t *S1P,
mp_size_t S1SIZE, const mp_limb_t *S2P, mp_size_t S2SIZE)
Multiply {S1P, S1SIZE} and {S2P, S2SIZE}, and write the result to
RP. Return the most significant limb of the result.
The destination has to have space for S1SIZE + S2SIZE limbs, even
if the result might be one limb smaller.
This function requires that S1SIZE is greater than or equal to
S2SIZE. The destination must be distinct from either input
operands.
- Function: void mpn_tdiv_qr (mp_limb_t *QP, mp_limb_t *RP, mp_size_t
QXN, const mp_limb_t *NP, mp_size_t NN, const mp_limb_t *DP,
mp_size_t DN)
Divide {NP, NN} by {DP, DN}. Write the quotient at QP and the
remainder at RP.
The quotient written at QP will be NN - DN + 1 limbs. The
remainder written at RP will be DN limbs.
It is required that NN is greater than or equal to DN. The QXN
operand must be zero.
The quotient is rounded towards 0.
No overlap between arguments is permitted.
- Function: mp_limb_t mpn_divrem (mp_limb_t *R1P, mp_size_t XSIZE,
mp_limb_t *RS2P, mp_size_t RS2SIZE, const mp_limb_t *S3P,
mp_size_t S3SIZE)
[This function is obsolete. Please call `mpn_tdiv_qr' instead for
best performance.]
Divide {RS2P, RS2SIZE} by {S3P, S3SIZE}, and write the quotient at
R1P, with the exception of the most significant limb, which is
returned. The remainder replaces the dividend at RS2P; it will be
S3SIZE limbs long (i.e., as many limbs as the divisor).
In addition to an integer quotient, XSIZE fraction limbs are
developed, and stored after the integral limbs. For most usages,
XSIZE will be zero.
It is required that RS2SIZE is greater than or equal to S3SIZE.
It is required that the most significant bit of the divisor is set.
If the quotient is not needed, pass RS2P + S3SIZE as R1P. Aside
from that special case, no overlap between arguments is permitted.
Return the most significant limb of the quotient, either 0 or 1.
The area at R1P needs to be RS2SIZE - S3SIZE + XSIZE limbs large.
- Function: mp_limb_t mpn_divrem_1 (mp_limb_t *R1P, mp_size_t XSIZE,
mp_limb_t *S2P, mp_size_t S2SIZE, mp_limb_t S3LIMB)
- Macro: mp_limb_t mpn_divmod_1 (mp_limb_t *R1P, mp_limb_t *S2P,
mp_size_t S2SIZE, mp_limb_t S3LIMB)
Divide {S2P, S2SIZE} by S3LIMB, and write the quotient at R1P.
Return the remainder.
The integer quotient is written to {R1P+XSIZE, S2SIZE} and in
addition XSIZE fraction limbs are developed and written to {R1P,
XSIZE}. Either or both S2SIZE and XSIZE can be zero. For most
usages, XSIZE will be zero.
`mpn_divmod_1' exists for upward source compatibility and is
simply a macro calling `mpn_divrem_1' with an XSIZE of 0.
The areas at R1P and S2P have to be identical or completely
separate, not partially overlapping.
- Function: mp_limb_t mpn_divmod (mp_limb_t *R1P, mp_limb_t *RS2P,
mp_size_t RS2SIZE, const mp_limb_t *S3P, mp_size_t S3SIZE)
*This interface is obsolete. It will disappear from future
releases. Use `mpn_divrem' in its stead.*
- Macro: mp_limb_t mpn_divexact_by3 (mp_limb_t *RP, mp_limb_t *SP,
mp_size_t SIZE)
- Function: mp_limb_t mpn_divexact_by3c (mp_limb_t *RP, mp_limb_t *SP,
mp_size_t SIZE, mp_limb_t CARRY)
Divide {SP, SIZE} by 3, expecting it to divide exactly, and
writing the result to {RP, SIZE}. If 3 divides exactly, the
return value is zero and the result is the quotient. If not, the
return value is non-zero and the result won't be anything useful.
`mpn_divexact_by3c' takes an initial carry parameter, which can be
the return value from a previous call, so a large calculation can
be done piece by piece. `mpn_divexact_by3' is simply a macro
calling `mpn_divexact_by3c' with a 0 carry parameter.
These routines use a multiply-by-inverse and will be faster than
`mpn_divrem_1' on CPUs with fast multiplication but slow division.
The source a, result q, size n, initial carry i, and return value
c satisfy c*b^n + a-i = 3*q, where b is the size of a limb (2^32
or 2^64). c is always 0, 1 or 2, and the initial carry must also
be 0, 1 or 2 (these are both borrows really). When c=0, clearly
q=(a-i)/3. When c!=0, the remainder (a-i) mod 3 is given by 3-c,
because b == 1 mod 3.
- Function: mp_limb_t mpn_mod_1 (mp_limb_t *S1P, mp_size_t S1SIZE,
mp_limb_t S2LIMB)
Divide {S1P, S1SIZE} by S2LIMB, and return the remainder. S1SIZE
can be zero.
- Function: mp_limb_t mpn_preinv_mod_1 (mp_limb_t *S1P, mp_size_t
S1SIZE, mp_limb_t S2LIMB, mp_limb_t S3LIMB)
*This interface is obsolete. It will disappear from future
releases. Use `mpn_mod_1' in its stead.*
- Function: mp_limb_t mpn_bdivmod (mp_limb_t *RP, mp_limb_t *S1P,
mp_size_t S1SIZE, const mp_limb_t *S2P, mp_size_t S2SIZE,
unsigned long int D)
The function puts the low [D/BITS_PER_MP_LIMB] limbs of Q = {S1P,
S1SIZE}/{S2P, S2SIZE} mod 2^D at RP, and returns the high D mod
BITS_PER_MP_LIMB bits of Q.
{S1P, S1SIZE} - Q * {S2P, S2SIZE} mod 2^(S1SIZE*BITS_PER_MP_LIMB)
is placed at S1P. Since the low [D/BITS_PER_MP_LIMB] limbs of
this difference are zero, it is possible to overwrite the low
limbs at S1P with this difference, provided RP <= S1P.
This function requires that S1SIZE * BITS_PER_MP_LIMB >= D, and
that {S2P, S2SIZE} is odd.
*This interface is preliminary. It might change incompatibly in
future revisions.*
- Function: mp_limb_t mpn_lshift (mp_limb_t *RP, const mp_limb_t
*SRC_PTR, mp_size_t SRC_SIZE, unsigned long int COUNT)
Shift {SRC_PTR, SRC_SIZE} COUNT bits to the left, and write the
SRC_SIZE least significant limbs of the result to RP. COUNT might
be in the range 1 to n - 1, on an n-bit machine. The bits shifted
out to the left are returned.
Overlapping of the destination space and the source space is
allowed in this function, provided RP >= SRC_PTR.
This function is written in assembly for most targets.
- Function: mp_limp_t mpn_rshift (mp_limb_t *RP, const mp_limb_t
*SRC_PTR, mp_size_t SRC_SIZE, unsigned long int COUNT)
Shift {SRC_PTR, SRC_SIZE} COUNT bits to the right, and write the
SRC_SIZE most significant limbs of the result to RP. COUNT might
be in the range 1 to n - 1, on an n-bit machine. The bits shifted
out to the right are returned.
Overlapping of the destination space and the source space is
allowed in this function, provided RP <= SRC_PTR.
This function is written in assembly for most targets.
- Function: int mpn_cmp (const mp_limb_t *S1P, const mp_limb_t *S2P,
mp_size_t SIZE)
Compare {S1P, SIZE} and {S2P, SIZE} and return a positive value if
s1 > src2, 0 of they are equal, and a negative value if s1 < src2.
- Function: mp_size_t mpn_gcd (mp_limb_t *RP, mp_limb_t *S1P,
mp_size_t S1SIZE, mp_limb_t *S2P, mp_size_t S2SIZE)
Puts at RP the greatest common divisor of {S1P, S1SIZE} and {S2P,
S2SIZE}; both source operands are destroyed by the operation. The
size in limbs of the greatest common divisor is returned.
{S1P, S1SIZE} must have at least as many bits as {S2P, S2SIZE},
and {S2P, S2SIZE} must be odd.
- Function: mp_limb_t mpn_gcd_1 (const mp_limb_t *S1P, mp_size_t
S1SIZE, mp_limb_t S2LIMB)
Return the greatest common divisor of {S1P, S1SIZE} and S2LIMB,
where S2LIMB (as well as S1SIZE) must be different from 0.
- Function: mp_size_t mpn_gcdext (mp_limb_t *R1P, mp_limb_t *R2P,
mp_size_t *R2SIZE, mp_limb_t *S1P, mp_size_t S1SIZE,
mp_limb_t *S2P, mp_size_t S2SIZE)
Compute the greatest common divisor of {S1P, S1SIZE} and {S2P,
S2SIZE}. Store the gcd at R1P and return its size in limbs.
Write the first cofactor at R2P and store its size in *R2SIZE. If
the cofactor is negative, *R2SIZE is negative and R2P is the
absolute value of the cofactor.
{S1P, S1SIZE} must be greater than or equal to {S2P, S2SIZE}.
Neither operand may equal 0. Both source operands are destroyed,
plus one limb past the end of each, ie. {S1P, S1SIZE+1} and {S2P,
S2SIZE+1}.
- Function: mp_size_t mpn_sqrtrem (mp_limb_t *R1P, mp_limb_t *R2P,
const mp_limb_t *SP, mp_size_t SIZE)
Compute the square root of {SP, SIZE} and put the result at R1P.
Write the remainder at R2P, unless R2P is `NULL'.
Return the size of the remainder, whether R2P was `NULL' or
non-`NULL'. Iff the operand was a perfect square, the return
value will be 0.
The areas at R1P and SP have to be distinct. The areas at R2P and
SP have to be identical or completely separate, not partially
overlapping.
The area at R1P needs to have space for ceil(SIZE/2) limbs. The
area at R2P needs to be SIZE limbs large.
- Function: mp_size_t mpn_get_str (unsigned char *STR, int BASE,
mp_limb_t *S1P, mp_size_t S1SIZE)
Convert {S1P, S1SIZE} to a raw unsigned char array in base BASE.
The string is not in ASCII; to convert it to printable format, add
the ASCII codes for `0' or `A', depending on the base and range.
There may be leading zeros in the string.
The area at S1P is clobbered.
Return the number of characters in STR.
The area at STR has to have space for the largest possible number
represented by a S1SIZE long limb array, plus one extra character.
- Function: mp_size_t mpn_set_str (mp_limb_t *R1P, const char *STR,
size_t STRSIZE, int BASE)
Convert the raw unsigned char array at STR of length STRSIZE to a
limb array {S1P, S1SIZE}. The base of STR is BASE.
Return the number of limbs stored in R1P.
- Function: unsigned long int mpn_scan0 (const mp_limb_t *S1P,
unsigned long int BIT)
Scan S1P from bit position BIT for the next clear bit.
It is required that there be a clear bit within the area at S1P at
or beyond bit position BIT, so that the function has something to
return.
- Function: unsigned long int mpn_scan1 (const mp_limb_t *S1P,
unsigned long int BIT)
Scan S1P from bit position BIT for the next set bit.
It is required that there be a set bit within the area at S1P at or
beyond bit position BIT, so that the function has something to
return.
- Function: void mpn_random (mp_limb_t *R1P, mp_size_t R1SIZE)
- Function: void mpn_random2 (mp_limb_t *R1P, mp_size_t R1SIZE)
Generate a random number of length R1SIZE and store it at R1P.
The most significant limb is always non-zero. `mpn_random'
generates uniformly distributed limb data, `mpn_random2' generates
long strings of zeros and ones in the binary representation.
`mpn_random2' is intended for testing the correctness of the `mpn'
routines.
- Function: unsigned long int mpn_popcount (const mp_limb_t *S1P,
unsigned long int SIZE)
Count the number of set bits in {S1P, SIZE}.
- Function: unsigned long int mpn_hamdist (const mp_limb_t *S1P, const
mp_limb_t *S2P, unsigned long int SIZE)
Compute the hamming distance between {S1P, SIZE} and {S2P, SIZE}.
- Function: int mpn_perfect_square_p (const mp_limb_t *S1P, mp_size_t
SIZE)
Return non-zero iff {S1P, SIZE} is a perfect square.
Random Number Functions
***********************
There are two groups of random number functions in GNU MP; older
functions that call C library random number generators, rely on a global
state, and aren't very random; and newer functions that don't have these
problems. The newer functions are self-contained, they accept a random
state parameter that supplants global state, and generate good random
numbers.
The random state parameter is of the type `gmp_randstate_t'. It
must be initialized by a call to one of the `gmp_randinit' functions
(*Note Random State Initialization::). The initial seed is set using
one of the `gmp_randseed' functions (*Note Random State
Initialization::).
The size of the seed determines the number of different sequences of
random numbers that is possible to generate. The "quality" of the seed
is the randomness of a given seed compared to the previous seed used
and affects the randomness of separate number sequences.
The algorithm for assigning seed is critical if the generated random
numbers are to be used for important applications, such as generating
cryptographic keys.
The traditional method is to use the current system time for
seeding. One has to be careful when using the current time though. If
the application seeds the random functions very often, say several
times per second, and the resolution of the system clock is
comparatively low, like one second, the same sequence of numbers will
be generated until the system clock ticks. Furthermore, the current
system time is quite easy to guess, so a system depending on any
unpredictability of the random number sequence should absolutely not
use that as its only source for a seed value.
On some systems there is a special device, often called
`/dev/random', which provides a source of somewhat random numbers more
usable as seed.
The functions actually generating random functions are documented
under "Miscellaneous Functions" in their respective function class:
*Note Miscellaneous Integer Functions::, *Note Miscellaneous Float
Functions::.
Random State Initialization
===========================
See *Note Random Number Functions:: for a discussion on how to
choose the initial seed value passed to these functions.
- Function: void gmp_randinit (gmp_randstate_t STATE, gmp_randalg_t
ALG, ...)
Initialize random state variable STATE.
ALG denotes what algorithm to use for random number generation.
Use one of
- GMP_RAND_ALG_LC -- Linear congruential.
A fast generator defined by X = (aX + c) mod m.
A third argument SIZE of type unsigned long int is required.
SIZE is the size of the largest good quality random number to
be generated, expressed in number of bits. If the random
generation functions are asked for a bigger random number
than indicated by this parameter, two or more numbers of SIZE
bits will be generated and concatenated, resulting in a "bad"
random number. This can be used to generate big random
numbers relatively cheap if the quality of randomness isn't
of great importance.
a, c, and m are picked from a table where the modulus (m) is
a power of 2 and the multiplier is congruent to 5 (mod 8).
The choice is based on the SIZE parameter. The maximum SIZE
supported by this algorithm is 128. If you need bigger
random numbers, use your own scheme and call one of the other
`gmp_randinit' functions.
If ALG is 0 or GMP_RAND_ALG_DEFAULT, the default algorithm is
used. The default algorithm is typically a fast algorithm like
the linear congruential and requires a third SIZE argument (see
GMP_RAND_ALG_LC).
When you're done with a STATE variable, call `gmp_randclear' to
deallocate any memory allocated by this function.
`gmp_randinit' may set the following bits in GMP_ERRNO:
* GMP_ERROR_UNSUPPORTED_ARGUMENT -- ALG is unsupported
* GMP_ERROR_INVALID_ARGUMENT -- SIZE is too big
- Function: void gmp_randinit_lc_2exp (gmp_randstate_t STATE, mpz_t A,
unsigned long int C, unsigned long int M2EXP)
Initialize random state variable STATE with given linear
congruential scheme.
Parameters A, C, and M2EXP are the multiplier, adder, and modulus
for the linear congruential scheme to use, respectively. The
modulus is expressed as a power of 2, so that M = 2^M2EXP.
The least significant bits of a random number generated by the
linear congruential algorithm where the modulus is a power of two
are not very random. Therefore, the lower half of a random number
generated by an LC scheme initialized with this function is
discarded. Thus, the size of a random number is M2EXP / 2
(rounded upwards) bits when this function has been used for
initializing the random state.
When you're done with a STATE variable, call `gmp_randclear' to
deallocate any memory allocated by this function.
- Function: void gmp_randseed (gmp_randstate_t STATE, mpz_t SEED)
- Function: void gmp_randseed_ui (gmp_randstate_t STATE, unsigned long
int SEED)
Set the initial seed value.
Parameter SEED is the initial random seed. The function
`gmp_randseed_ui' takes the SEED as an unsigned long int rather
than as an mpz_t.
- Function: void gmp_randclear (gmp_randstate_t STATE)
Free all memory occupied by STATE. Make sure to call this
function for all `gmp_randstate_t' variables when you are done with
them.
Berkeley MP Compatible Functions
********************************
These functions are intended to be fully compatible with the
Berkeley MP library which is available on many BSD derived U*ix
systems. The `--enable-mpbsd' option must be used when building GNU MP
to make these available (*note Installing GMP::).
The original Berkeley MP library has a usage restriction: you cannot
use the same variable as both source and destination in a single
function call. The compatible functions in GNU MP do not share this
restriction--inputs and outputs may overlap.
It is not recommended that new programs are written using these
functions. Apart from the incomplete set of functions, the interface
for initializing `MINT' objects is more error prone, and the `pow'
function collides with `pow' in `libm.a'.
Include the header `mp.h' to get the definition of the necessary
types and functions. If you are on a BSD derived system, make sure to
include GNU `mp.h' if you are going to link the GNU `libmp.a' to your
program. This means that you probably need to give the -I option
to the compiler, where is the directory where you have GNU `mp.h'.
- Function: MINT * itom (signed short int INITIAL_VALUE)
Allocate an integer consisting of a `MINT' object and dynamic limb
space. Initialize the integer to INITIAL_VALUE. Return a pointer
to the `MINT' object.
- Function: MINT * xtom (char *INITIAL_VALUE)
Allocate an integer consisting of a `MINT' object and dynamic limb
space. Initialize the integer from INITIAL_VALUE, a hexadecimal,
'\0'-terminate C string. Return a pointer to the `MINT' object.
- Function: void move (MINT *SRC, MINT *DEST)
Set DEST to SRC by copying. Both variables must be previously
initialized.
- Function: void madd (MINT *SRC_1, MINT *SRC_2, MINT *DESTINATION)
Add SRC_1 and SRC_2 and put the sum in DESTINATION.
- Function: void msub (MINT *SRC_1, MINT *SRC_2, MINT *DESTINATION)
Subtract SRC_2 from SRC_1 and put the difference in DESTINATION.
- Function: void mult (MINT *SRC_1, MINT *SRC_2, MINT *DESTINATION)
Multiply SRC_1 and SRC_2 and put the product in DESTINATION.
- Function: void mdiv (MINT *DIVIDEND, MINT *DIVISOR, MINT *QUOTIENT,
MINT *REMAINDER)
- Function: void sdiv (MINT *DIVIDEND, signed short int DIVISOR, MINT
*QUOTIENT, signed short int *REMAINDER)
Set QUOTIENT to DIVIDEND/DIVISOR, and REMAINDER to DIVIDEND mod
DIVISOR. The quotient is rounded towards zero; the remainder has
the same sign as the dividend unless it is zero.
Some implementations of these functions work differently--or not
at all--for negative arguments.
- Function: void msqrt (MINT *OPERAND, MINT *ROOT, MINT *REMAINDER)
Set ROOT to the truncated integer part of the square root of
OPERAND. Set REMAINDER to OPERAND-ROOT*ROOT, (i.e., zero if
OPERAND is a perfect square).
If ROOT and REMAINDER are the same variable, the results are
undefined.
- Function: void pow (MINT *BASE, MINT *EXP, MINT *MOD, MINT *DEST)
Set DEST to (BASE raised to EXP) modulo MOD.
- Function: void rpow (MINT *BASE, signed short int EXP, MINT *DEST)
Set DEST to BASE raised to EXP.
- Function: void gcd (MINT *OPERAND1, MINT *OPERAND2, MINT *RES)
Set RES to the greatest common divisor of OPERAND1 and OPERAND2.
- Function: int mcmp (MINT *OPERAND1, MINT *OPERAND2)
Compare OPERAND1 and OPERAND2. Return a positive value if
OPERAND1 > OPERAND2, zero if OPERAND1 = OPERAND2, and a negative
value if OPERAND1 < OPERAND2.
- Function: void min (MINT *DEST)
Input a decimal string from `stdin', and put the read integer in
DEST. SPC and TAB are allowed in the number string, and are
ignored.
- Function: void mout (MINT *SRC)
Output SRC to `stdout', as a decimal string. Also output a
newline.
- Function: char * mtox (MINT *OPERAND)
Convert OPERAND to a hexadecimal string, and return a pointer to
the string. The returned string is allocated using the default
memory allocation function, `malloc' by default.
- Function: void mfree (MINT *OPERAND)
De-allocate, the space used by OPERAND. *This function should
only be passed a value returned by `itom' or `xtom'.*
Custom Allocation
*****************
By default, GMP uses `malloc', `realloc' and `free' for memory
allocation. If `malloc' or `realloc' fails, GMP prints a message to
the standard error output and terminates execution.
Some applications might want to allocate memory in other ways, or
might not want a fatal error when there is no more memory available.
To accomplish this, you can specify alternative memory allocation
functions.
This can be done in the Berkeley compatibility library as well as
the main GMP library.
- Function: void mp_set_memory_functions (
void *(*ALLOC_FUNC_PTR) (size_t),
void *(*REALLOC_FUNC_PTR) (void *, size_t, size_t),
void (*FREE_FUNC_PTR) (void *, size_t))
Replace the current allocation functions from the arguments. If
an argument is `NULL', the corresponding default function is
retained.
*Be sure to call this function only when there are no active GMP
objects allocated using the previous memory functions! Usually,
that means that you have to call this function before any other
GMP function.*
The functions you supply should fit the following declarations:
- Function: void * allocate_function (size_t ALLOC_SIZE)
This function should return a pointer to newly allocated space
with at least ALLOC_SIZE storage units.
- Function: void * reallocate_function (void *PTR, size_t OLD_SIZE,
size_t NEW_SIZE)
This function should return a pointer to newly allocated space of
at least NEW_SIZE storage units, after copying at least the first
OLD_SIZE storage units from PTR. It should also de-allocate the
space at PTR.
You can assume that the space at PTR was formerly returned from
`allocate_function' or `reallocate_function', for a request for
OLD_SIZE storage units.
- Function: void deallocate_function (void *PTR, size_t SIZE)
De-allocate the space pointed to by PTR.
You can assume that the space at PTR was formerly returned from
`allocate_function' or `reallocate_function', for a request for
SIZE storage units.
(A "storage unit" is the unit in which the `sizeof' operator returns
the size of an object, normally an 8 bit byte.)
Contributors
************
Torbjorn Granlund wrote the original GMP library and is still
developing and maintaining it. Several other individuals and
organizations have contributed to GMP in various ways. Here is a list
in chronological order:
Gunnar Sjoedin and Hans Riesel helped with mathematical problems in
early versions of the library.
Richard Stallman contributed to the interface design and revised the
first version of this manual.
Brian Beuning and Doug Lea helped with testing of early versions of
the library and made creative suggestions.
John Amanatides of York University in Canada contributed the function
`mpz_probab_prime_p'.
Paul Zimmermann of Inria sparked the development of GMP 2, with his
comparisons between bignum packages.
Ken Weber (Kent State University, Universidade Federal do Rio Grande
do Sul) contributed `mpz_gcd', `mpz_divexact', `mpn_gcd', and
`mpn_bdivmod', partially supported by CNPq (Brazil) grant 301314194-2.
Per Bothner of Cygnus Support helped to set up GMP to use Cygnus'
configure. He has also made valuable suggestions and tested numerous
intermediary releases.
Joachim Hollman was involved in the design of the `mpf' interface,
and in the `mpz' design revisions for version 2.
Bennet Yee contributed the functions `mpz_jacobi' and `mpz_legendre'.
Andreas Schwab contributed the files `mpn/m68k/lshift.S' and
`mpn/m68k/rshift.S'.
The development of floating point functions of GNU MP 2, were
supported in part by the ESPRIT-BRA (Basic Research Activities) 6846
project POSSO (POlynomial System SOlving).
GNU MP 2 was finished and released by SWOX AB (formerly known as TMG
Datakonsult), Swedenborgsgatan 23, SE-118 27 STOCKHOLM, SWEDEN, in
cooperation with the IDA Center for Computing Sciences, USA.
Robert Harley of Inria, France and David Seal of ARM, England,
suggested clever improvements for population count.
Robert Harley also wrote highly optimized Karatsuba and 3-way Toom
multiplication functions for GMP 3. He also contributed the ARM
assembly code.
Torsten Ekedahl of the Mathematical department of Stockholm
University provided significant inspiration during several phases of
the GMP development. His mathematical expertise helped improve several
algorithms.
Paul Zimmermann wrote the Burnikel-Ziegler division code, the REDC
code, the REDC-based mpz_powm code, and the FFT multiply code. The
ECMNET project Paul is organizing has been a driving force behind many
of the optimization of GMP 3.
Linus Nordberg wrote the new configure system based on autoconf and
implemented the new random functions.
Kent Boortz made the Macintosh port.
Kevin Ryde wrote a lot of very high quality x86 code, optimized for
most CPU variants. He also made countless other valuable contributions.
Steve Root helped write the optimized alpha 21264 assembly code.
GNU MP 3.1 was finished and released by Torbjorn Granlund and Kevin
Ryde. Torbjorn's work was partially funded by the IDA Center for
Computing Sciences, USA.
(This list is chronological, not ordered after significance. If you
have contributed to GMP but are not listed above, please tell
about the omission!)
References
**********
* Donald E. Knuth, "The Art of Computer Programming", vol 2,
"Seminumerical Algorithms", 3rd edition, Addison-Wesley, 1988.
* John D. Lipson, "Elements of Algebra and Algebraic Computing", The
Benjamin Cummings Publishing Company Inc, 1981.
* Richard M. Stallman, "Using and Porting GCC", Free Software
Foundation, 1999, available online
`http://www.gnu.org/software/gcc/onlinedocs/', and in the GCC
package `ftp://ftp.gnu.org/pub/gnu/gcc/'.
* Peter L. Montgomery, "Modular Multiplication Without Trial
Division", in Mathematics of Computation, volume 44, number 170,
April 1985.
* Torbjorn Granlund and Peter L. Montgomery, "Division by Invariant
Integers using Multiplication", in Proceedings of the SIGPLAN
PLDI'94 Conference, June 1994. Available online,
`ftp://ftp.cwi.nl/pub/pmontgom/divcnst.psa4.gz' (and .psl.gz too).
* Tudor Jebelean, "An algorithm for exact division", Journal of
Symbolic Computation, v. 15, 1993, pp. 169-180. Research report
version available online
`ftp://ftp.risc.uni-linz.ac.at/pub/techreports/1992/92-35.ps.gz'
* Kenneth Weber, "The accelerated integer GCD algorithm", ACM
Transactions on Mathematical Software, v. 21 (March), 1995, pp.
111-122.
* Christoph Burnikel and Joachim Ziegler, "Fast Recursive Division",
Max-Planck-Institut fuer Informatik Research Report
MPI-I-98-1-022,
`http://www.mpi-sb.mpg.de/~ziegler/TechRep.ps.gz'.
* Alfred J. Menezes, Paul C. van Oorschot and Scott A. Vanstone,
"Handbook of Applied Cryptography",
`http://cacr.math.uwaterloo.ca/hac/'.
* Henri Cohen, "A Course in Computational Algebraic Number Theory",
Graduate Texts in Mathematics number 138, Springer-Verlag, 1993.
Errata available online
`http://www.math.u-bordeaux.fr/~cohen'
Concept Index
*************
ABI:
See ``ABI and ISA''.
About this manual:
See ``Introduction to GNU MP''.
alloca:
See ``Build Options''.
Allocation of memory:
See ``Custom Allocation''.
Anonymous FTP of latest version:
See ``Getting the Latest Version of GMP''.
Arithmetic functions <1>:
See ``Arithmetic Functions''.
Arithmetic functions <2>:
See ``Arithmetic Functions''.
Arithmetic functions:
See ``Arithmetic Functions''.
Assignment functions <1>:
See ``Assignment Functions''.
Assignment functions:
See ``Assignment Functions''.
Basics:
See ``GMP Basics''.
Berkeley MP compatible functions:
See ``Berkeley MP Compatible Functions''.
Binomial coefficient functions:
See ``Number Theoretic Functions''.
Bit manipulation functions:
See ``Logical and Bit Manipulation Functions''.
Bit shift left:
See ``Arithmetic Functions''.
Bit shift right:
See ``Division Functions''.
Bits per limb:
See ``Useful Macros and Constants''.
BSD MP compatible functions:
See ``Berkeley MP Compatible Functions''.
Bug reporting:
See ``Reporting Bugs''.
Build notes for binary packaging:
See ``Notes for Package Builds''.
Build notes for particular systems:
See ``Notes for Particular Systems''.
Build options:
See ``Build Options''.
Build problems known:
See ``Known Build Problems''.
Comparison functions <1>:
See ``Comparison Functions''.
Comparison functions <2>:
See ``Comparison Functions''.
Comparison functions:
See ``Comparison Functions''.
Compatibility with older versions:
See ``Compatibility with older versions''.
Conditions for copying GNU MP:
See ``GNU MP Copying Conditions''.
Configuring GMP:
See ``Installing GMP''.
Constants:
See ``Useful Macros and Constants''.
Contributors:
See ``Contributors''.
Conventions for variables:
See ``GMP Variable Conventions''.
Conversion functions <1>:
See ``Conversion Functions''.
Conversion functions:
See ``Conversion Functions''.
Copying conditions:
See ``GNU MP Copying Conditions''.
CPUs supported:
See ``Introduction to GNU MP''.
Custom allocation:
See ``Custom Allocation''.
Demonstration programs:
See ``Build Options''.
Division functions <1>:
See ``Arithmetic Functions''.
Division functions <2>:
See ``Arithmetic Functions''.
Division functions:
See ``Division Functions''.
Exact division functions:
See ``Division Functions''.
Example programs:
See ``Build Options''.
Exponentiation functions <1>:
See ``Arithmetic Functions''.
Exponentiation functions:
See ``Exponentiation Functions''.
Extended GCD:
See ``Number Theoretic Functions''.
Factorial functions:
See ``Number Theoretic Functions''.
Fibonacci sequence functions:
See ``Number Theoretic Functions''.
Float arithmetic functions:
See ``Arithmetic Functions''.
Float assignment functions:
See ``Assignment Functions''.
Float comparison functions:
See ``Comparison Functions''.
Float conversion functions:
See ``Conversion Functions''.
Float functions:
See ``Floating-point Functions''.
Float init and assign functions:
See ``Combined Initialization and Assignment Functions''.
Float initialization functions:
See ``Initialization Functions''.
Float input and output functions:
See ``Input and Output Functions''.
Float miscellaneous functions:
See ``Miscellaneous Functions''.
Floating-point functions:
See ``Floating-point Functions''.
Floating-point number:
See ``Nomenclature and Types''.
FTP of latest version:
See ``Getting the Latest Version of GMP''.
Function classes:
See ``Function Classes''.
GMP version number:
See ``Useful Macros and Constants''.
gmp.h:
See ``GMP Basics''.
Greatest common divisor functions:
See ``Number Theoretic Functions''.
Home page:
See ``Introduction to GNU MP''.
I/O functions <1>:
See ``Input and Output Functions''.
I/O functions <2>:
See ``Input and Output Functions''.
I/O functions:
See ``Input and Output Functions''.
Initialization and assignment functions <1>:
See ``Combined Initialization and Assignment Functions''.
Initialization and assignment functions <2>:
See ``Initialization and Assignment Functions''.
Initialization and assignment functions:
See ``Combined Initialization and Assignment Functions''.
Initialization functions <1>:
See ``Initialization Functions''.
Initialization functions:
See ``Initialization Functions''.
Input functions <1>:
See ``Input and Output Functions''.
Input functions <2>:
See ``Input and Output Functions''.
Input functions:
See ``Input and Output Functions''.
Installing GMP:
See ``Installing GMP''.
Integer:
See ``Nomenclature and Types''.
Integer arithmetic functions:
See ``Arithmetic Functions''.
Integer assignment functions:
See ``Assignment Functions''.
Integer bit manipulation functions:
See ``Logical and Bit Manipulation Functions''.
Integer comparison functions:
See ``Comparison Functions''.
Integer conversion functions:
See ``Conversion Functions''.
Integer division functions:
See ``Division Functions''.
Integer exponentiation functions:
See ``Exponentiation Functions''.
Integer functions:
See ``Integer Functions''.
Integer init and assign:
See ``Combined Initialization and Assignment Functions''.
Integer initialization functions:
See ``Initialization Functions''.
Integer input and output functions:
See ``Input and Output Functions''.
Integer miscellaneous functions:
See ``Miscellaneous Functions''.
Integer random number functions:
See ``Random Number Functions''.
Integer root functions:
See ``Root Extraction Functions''.
Introduction:
See ``Introduction to GNU MP''.
ISA:
See ``ABI and ISA''.
Jabobi symbol functions:
See ``Number Theoretic Functions''.
Kronecker symbol functions:
See ``Number Theoretic Functions''.
Latest version of GMP:
See ``Getting the Latest Version of GMP''.
Least common multiple functions:
See ``Number Theoretic Functions''.
Libtool versioning:
See ``Notes for Package Builds''.
Limb:
See ``Nomenclature and Types''.
Limb size:
See ``Useful Macros and Constants''.
Logical functions:
See ``Logical and Bit Manipulation Functions''.
Low-level functions:
See ``Low-level Functions''.
Mailing list:
See ``Introduction to GNU MP''.
Memory allocation:
See ``Custom Allocation''.
Miscellaneous float functions:
See ``Miscellaneous Functions''.
Miscellaneous integer functions:
See ``Miscellaneous Functions''.
Miscellaneous rational functions:
See ``Miscellaneous Functions''.
Modular inverse functions:
See ``Number Theoretic Functions''.
mp.h:
See ``Berkeley MP Compatible Functions''.
Multi-threading:
See ``GMP and Reentrancy''.
Nomenclature:
See ``Nomenclature and Types''.
Number theoretic functions:
See ``Number Theoretic Functions''.
Numerator and denominator:
See ``Applying Integer Functions to Rationals''.
Output functions <1>:
See ``Input and Output Functions''.
Output functions <2>:
See ``Input and Output Functions''.
Output functions:
See ``Input and Output Functions''.
Packaged builds:
See ``Notes for Package Builds''.
Parameter conventions:
See ``GMP Variable Conventions''.
Precision of floats:
See ``Floating-point Functions''.
Prime testing functions:
See ``Number Theoretic Functions''.
Random number functions <1>:
See ``Random Number Functions''.
Random number functions:
See ``Random Number Functions''.
Random number state:
See ``Random State Initialization''.
Rational arithmetic functions:
See ``Arithmetic Functions''.
Rational comparison functions:
See ``Comparison Functions''.
Rational init and assign:
See ``Initialization and Assignment Functions''.
Rational input and output functions:
See ``Input and Output Functions''.
Rational miscellaneous functions:
See ``Miscellaneous Functions''.
Rational number:
See ``Nomenclature and Types''.
Rational number functions:
See ``Rational Number Functions''.
Rational numerator and denominator:
See ``Applying Integer Functions to Rationals''.
Reentrancy:
See ``GMP and Reentrancy''.
References:
See ``References''.
Reporting bugs:
See ``Reporting Bugs''.
Root extraction functions <1>:
See ``Arithmetic Functions''.
Root extraction functions:
See ``Root Extraction Functions''.
Stack overflow segfaults:
See ``Build Options''.
Stripped libraries:
See ``Known Build Problems''.
Thread safety:
See ``GMP and Reentrancy''.
Types:
See ``Nomenclature and Types''.
Upward compatibility:
See ``Compatibility with older versions''.
Useful macros and constants:
See ``Useful Macros and Constants''.
User-defined precision:
See ``Floating-point Functions''.
Variable conventions:
See ``GMP Variable Conventions''.
Version number:
See ``Useful Macros and Constants''.
Web page:
See ``Introduction to GNU MP''.
Function and Type Index
***********************
__GNU_MP_VERSION:
See ``Useful Macros and Constants''.
__GNU_MP_VERSION_MINOR:
See ``Useful Macros and Constants''.
__GNU_MP_VERSION_PATCHLEVEL:
See ``Useful Macros and Constants''.
_mpz_realloc:
See ``Initialization Functions''.
allocate_function:
See ``Custom Allocation''.
deallocate_function:
See ``Custom Allocation''.
gcd:
See ``Berkeley MP Compatible Functions''.
gmp_randclear:
See ``Random State Initialization''.
gmp_randinit:
See ``Random State Initialization''.
gmp_randinit_lc_2exp:
See ``Random State Initialization''.
gmp_randseed:
See ``Random State Initialization''.
gmp_randseed_ui:
See ``Random State Initialization''.
itom:
See ``Berkeley MP Compatible Functions''.
madd:
See ``Berkeley MP Compatible Functions''.
mcmp:
See ``Berkeley MP Compatible Functions''.
mdiv:
See ``Berkeley MP Compatible Functions''.
mfree:
See ``Berkeley MP Compatible Functions''.
min:
See ``Berkeley MP Compatible Functions''.
mout:
See ``Berkeley MP Compatible Functions''.
move:
See ``Berkeley MP Compatible Functions''.
mp_limb_t:
See ``Nomenclature and Types''.
mp_set_memory_functions:
See ``Custom Allocation''.
mpf_abs:
See ``Arithmetic Functions''.
mpf_add:
See ``Arithmetic Functions''.
mpf_add_ui:
See ``Arithmetic Functions''.
mpf_ceil:
See ``Miscellaneous Functions''.
mpf_clear:
See ``Initialization Functions''.
mpf_cmp:
See ``Comparison Functions''.
mpf_cmp_si:
See ``Comparison Functions''.
mpf_cmp_ui:
See ``Comparison Functions''.
mpf_div:
See ``Arithmetic Functions''.
mpf_div_2exp:
See ``Arithmetic Functions''.
mpf_div_ui:
See ``Arithmetic Functions''.
mpf_eq:
See ``Comparison Functions''.
mpf_floor:
See ``Miscellaneous Functions''.
mpf_get_d:
See ``Conversion Functions''.
mpf_get_prec:
See ``Initialization Functions''.
mpf_get_str:
See ``Conversion Functions''.
mpf_init:
See ``Initialization Functions''.
mpf_init2:
See ``Initialization Functions''.
mpf_init_set:
See ``Combined Initialization and Assignment Functions''.
mpf_init_set_d:
See ``Combined Initialization and Assignment Functions''.
mpf_init_set_si:
See ``Combined Initialization and Assignment Functions''.
mpf_init_set_str:
See ``Combined Initialization and Assignment Functions''.
mpf_init_set_ui:
See ``Combined Initialization and Assignment Functions''.
mpf_inp_str:
See ``Input and Output Functions''.
mpf_mul:
See ``Arithmetic Functions''.
mpf_mul_2exp:
See ``Arithmetic Functions''.
mpf_mul_ui:
See ``Arithmetic Functions''.
mpf_neg:
See ``Arithmetic Functions''.
mpf_out_str:
See ``Input and Output Functions''.
mpf_pow_ui:
See ``Arithmetic Functions''.
mpf_random2:
See ``Miscellaneous Functions''.
mpf_reldiff:
See ``Comparison Functions''.
mpf_set:
See ``Assignment Functions''.
mpf_set_d:
See ``Assignment Functions''.
mpf_set_default_prec:
See ``Initialization Functions''.
mpf_set_prec:
See ``Initialization Functions''.
mpf_set_prec_raw:
See ``Initialization Functions''.
mpf_set_q:
See ``Assignment Functions''.
mpf_set_si:
See ``Assignment Functions''.
mpf_set_str:
See ``Assignment Functions''.
mpf_set_ui:
See ``Assignment Functions''.
mpf_set_z:
See ``Assignment Functions''.
mpf_sgn:
See ``Comparison Functions''.
mpf_sqrt:
See ``Arithmetic Functions''.
mpf_sqrt_ui:
See ``Arithmetic Functions''.
mpf_sub:
See ``Arithmetic Functions''.
mpf_sub_ui:
See ``Arithmetic Functions''.
mpf_swap:
See ``Assignment Functions''.
mpf_t:
See ``Nomenclature and Types''.
mpf_trunc:
See ``Miscellaneous Functions''.
mpf_ui_div:
See ``Arithmetic Functions''.
mpf_ui_sub:
See ``Arithmetic Functions''.
mpf_urandomb:
See ``Miscellaneous Functions''.
mpn_add:
See ``Low-level Functions''.
mpn_add_1:
See ``Low-level Functions''.
mpn_add_n:
See ``Low-level Functions''.
mpn_addmul_1:
See ``Low-level Functions''.
mpn_bdivmod:
See ``Low-level Functions''.
mpn_cmp:
See ``Low-level Functions''.
mpn_divexact_by3:
See ``Low-level Functions''.
mpn_divexact_by3c:
See ``Low-level Functions''.
mpn_divmod:
See ``Low-level Functions''.
mpn_divmod_1:
See ``Low-level Functions''.
mpn_divrem:
See ``Low-level Functions''.
mpn_divrem_1:
See ``Low-level Functions''.
mpn_gcd:
See ``Low-level Functions''.
mpn_gcd_1:
See ``Low-level Functions''.
mpn_gcdext:
See ``Low-level Functions''.
mpn_get_str:
See ``Low-level Functions''.
mpn_hamdist:
See ``Low-level Functions''.
mpn_lshift:
See ``Low-level Functions''.
mpn_mod_1:
See ``Low-level Functions''.
mpn_mul:
See ``Low-level Functions''.
mpn_mul_1:
See ``Low-level Functions''.
mpn_mul_n:
See ``Low-level Functions''.
mpn_perfect_square_p:
See ``Low-level Functions''.
mpn_popcount:
See ``Low-level Functions''.
mpn_preinv_mod_1:
See ``Low-level Functions''.
mpn_random:
See ``Low-level Functions''.
mpn_random2:
See ``Low-level Functions''.
mpn_rshift:
See ``Low-level Functions''.
mpn_scan0:
See ``Low-level Functions''.
mpn_scan1:
See ``Low-level Functions''.
mpn_set_str:
See ``Low-level Functions''.
mpn_sqrtrem:
See ``Low-level Functions''.
mpn_sub:
See ``Low-level Functions''.
mpn_sub_1:
See ``Low-level Functions''.
mpn_sub_n:
See ``Low-level Functions''.
mpn_submul_1:
See ``Low-level Functions''.
mpn_tdiv_qr:
See ``Low-level Functions''.
mpq_add:
See ``Arithmetic Functions''.
mpq_canonicalize:
See ``Rational Number Functions''.
mpq_clear:
See ``Initialization and Assignment Functions''.
mpq_cmp:
See ``Comparison Functions''.
mpq_cmp_ui:
See ``Comparison Functions''.
mpq_denref:
See ``Applying Integer Functions to Rationals''.
mpq_div:
See ``Arithmetic Functions''.
mpq_equal:
See ``Comparison Functions''.
mpq_get_d:
See ``Miscellaneous Functions''.
mpq_get_den:
See ``Miscellaneous Functions''.
mpq_get_num:
See ``Miscellaneous Functions''.
mpq_init:
See ``Initialization and Assignment Functions''.
mpq_inv:
See ``Arithmetic Functions''.
mpq_mul:
See ``Arithmetic Functions''.
mpq_neg:
See ``Arithmetic Functions''.
mpq_numref:
See ``Applying Integer Functions to Rationals''.
mpq_out_str:
See ``Input and Output Functions''.
mpq_set:
See ``Initialization and Assignment Functions''.
mpq_set_d:
See ``Miscellaneous Functions''.
mpq_set_den:
See ``Miscellaneous Functions''.
mpq_set_num:
See ``Miscellaneous Functions''.
mpq_set_si:
See ``Initialization and Assignment Functions''.
mpq_set_ui:
See ``Initialization and Assignment Functions''.
mpq_set_z:
See ``Initialization and Assignment Functions''.
mpq_sgn:
See ``Comparison Functions''.
mpq_sub:
See ``Arithmetic Functions''.
mpq_swap:
See ``Initialization and Assignment Functions''.
mpq_t:
See ``Nomenclature and Types''.
mpz_abs:
See ``Arithmetic Functions''.
mpz_add:
See ``Arithmetic Functions''.
mpz_add_ui:
See ``Arithmetic Functions''.
mpz_addmul_ui:
See ``Arithmetic Functions''.
mpz_and:
See ``Logical and Bit Manipulation Functions''.
mpz_array_init:
See ``Initialization Functions''.
mpz_bin_ui:
See ``Number Theoretic Functions''.
mpz_bin_uiui:
See ``Number Theoretic Functions''.
mpz_cdiv_q:
See ``Division Functions''.
mpz_cdiv_q_ui:
See ``Division Functions''.
mpz_cdiv_qr:
See ``Division Functions''.
mpz_cdiv_qr_ui:
See ``Division Functions''.
mpz_cdiv_r:
See ``Division Functions''.
mpz_cdiv_r_ui:
See ``Division Functions''.
mpz_cdiv_ui:
See ``Division Functions''.
mpz_clear:
See ``Initialization Functions''.
mpz_clrbit:
See ``Logical and Bit Manipulation Functions''.
mpz_cmp:
See ``Comparison Functions''.
mpz_cmp_si:
See ``Comparison Functions''.
mpz_cmp_ui:
See ``Comparison Functions''.
mpz_cmpabs:
See ``Comparison Functions''.
mpz_cmpabs_ui:
See ``Comparison Functions''.
mpz_com:
See ``Logical and Bit Manipulation Functions''.
mpz_divexact:
See ``Division Functions''.
mpz_even_p:
See ``Miscellaneous Functions''.
mpz_fac_ui:
See ``Number Theoretic Functions''.
mpz_fdiv_q:
See ``Division Functions''.
mpz_fdiv_q_2exp:
See ``Division Functions''.
mpz_fdiv_q_ui:
See ``Division Functions''.
mpz_fdiv_qr:
See ``Division Functions''.
mpz_fdiv_qr_ui:
See ``Division Functions''.
mpz_fdiv_r:
See ``Division Functions''.
mpz_fdiv_r_2exp:
See ``Division Functions''.
mpz_fdiv_r_ui:
See ``Division Functions''.
mpz_fdiv_ui:
See ``Division Functions''.
mpz_fib_ui:
See ``Number Theoretic Functions''.
mpz_fits_sint_p:
See ``Miscellaneous Functions''.
mpz_fits_slong_p:
See ``Miscellaneous Functions''.
mpz_fits_sshort_p:
See ``Miscellaneous Functions''.
mpz_fits_uint_p:
See ``Miscellaneous Functions''.
mpz_fits_ulong_p:
See ``Miscellaneous Functions''.
mpz_fits_ushort_p:
See ``Miscellaneous Functions''.
mpz_gcd:
See ``Number Theoretic Functions''.
mpz_gcd_ui:
See ``Number Theoretic Functions''.
mpz_gcdext:
See ``Number Theoretic Functions''.
mpz_get_d:
See ``Conversion Functions''.
mpz_get_si:
See ``Conversion Functions''.
mpz_get_str:
See ``Conversion Functions''.
mpz_get_ui:
See ``Conversion Functions''.
mpz_getlimbn:
See ``Conversion Functions''.
mpz_hamdist:
See ``Logical and Bit Manipulation Functions''.
mpz_init:
See ``Initialization Functions''.
mpz_init_set:
See ``Combined Initialization and Assignment Functions''.
mpz_init_set_d:
See ``Combined Initialization and Assignment Functions''.
mpz_init_set_si:
See ``Combined Initialization and Assignment Functions''.
mpz_init_set_str:
See ``Combined Initialization and Assignment Functions''.
mpz_init_set_ui:
See ``Combined Initialization and Assignment Functions''.
mpz_inp_raw:
See ``Input and Output Functions''.
mpz_inp_str:
See ``Input and Output Functions''.
mpz_invert:
See ``Number Theoretic Functions''.
mpz_ior:
See ``Logical and Bit Manipulation Functions''.
mpz_jacobi:
See ``Number Theoretic Functions''.
mpz_kronecker_si:
See ``Number Theoretic Functions''.
mpz_kronecker_ui:
See ``Number Theoretic Functions''.
mpz_lcm:
See ``Number Theoretic Functions''.
mpz_legendre:
See ``Number Theoretic Functions''.
mpz_mod:
See ``Division Functions''.
mpz_mod_ui:
See ``Division Functions''.
mpz_mul:
See ``Arithmetic Functions''.
mpz_mul_2exp:
See ``Arithmetic Functions''.
mpz_mul_si:
See ``Arithmetic Functions''.
mpz_mul_ui:
See ``Arithmetic Functions''.
mpz_neg:
See ``Arithmetic Functions''.
mpz_nextprime:
See ``Number Theoretic Functions''.
mpz_odd_p:
See ``Miscellaneous Functions''.
mpz_out_raw:
See ``Input and Output Functions''.
mpz_out_str:
See ``Input and Output Functions''.
mpz_perfect_power_p:
See ``Root Extraction Functions''.
mpz_perfect_square_p:
See ``Root Extraction Functions''.
mpz_popcount:
See ``Logical and Bit Manipulation Functions''.
mpz_pow_ui:
See ``Exponentiation Functions''.
mpz_powm:
See ``Exponentiation Functions''.
mpz_powm_ui:
See ``Exponentiation Functions''.
mpz_probab_prime_p:
See ``Number Theoretic Functions''.
mpz_random:
See ``Random Number Functions''.
mpz_random2:
See ``Random Number Functions''.
mpz_remove:
See ``Number Theoretic Functions''.
mpz_root:
See ``Root Extraction Functions''.
mpz_rrandomb:
See ``Random Number Functions''.
mpz_scan0:
See ``Logical and Bit Manipulation Functions''.
mpz_scan1:
See ``Logical and Bit Manipulation Functions''.
mpz_set:
See ``Assignment Functions''.
mpz_set_d:
See ``Assignment Functions''.
mpz_set_f:
See ``Assignment Functions''.
mpz_set_q:
See ``Assignment Functions''.
mpz_set_si:
See ``Assignment Functions''.
mpz_set_str:
See ``Assignment Functions''.
mpz_set_ui:
See ``Assignment Functions''.
mpz_setbit:
See ``Logical and Bit Manipulation Functions''.
mpz_sgn:
See ``Comparison Functions''.
mpz_si_kronecker:
See ``Number Theoretic Functions''.
mpz_size:
See ``Miscellaneous Functions''.
mpz_sizeinbase:
See ``Miscellaneous Functions''.
mpz_sqrt:
See ``Root Extraction Functions''.
mpz_sqrtrem:
See ``Root Extraction Functions''.
mpz_sub:
See ``Arithmetic Functions''.
mpz_sub_ui:
See ``Arithmetic Functions''.
mpz_swap:
See ``Assignment Functions''.
mpz_t:
See ``Nomenclature and Types''.
mpz_tdiv_q:
See ``Division Functions''.
mpz_tdiv_q_2exp:
See ``Division Functions''.
mpz_tdiv_q_ui:
See ``Division Functions''.
mpz_tdiv_qr:
See ``Division Functions''.
mpz_tdiv_qr_ui:
See ``Division Functions''.
mpz_tdiv_r:
See ``Division Functions''.
mpz_tdiv_r_2exp:
See ``Division Functions''.
mpz_tdiv_r_ui:
See ``Division Functions''.
mpz_tdiv_ui:
See ``Division Functions''.
mpz_tstbit:
See ``Logical and Bit Manipulation Functions''.
mpz_ui_kronecker:
See ``Number Theoretic Functions''.
mpz_ui_pow_ui:
See ``Exponentiation Functions''.
mpz_urandomb:
See ``Random Number Functions''.
mpz_urandomm:
See ``Random Number Functions''.
mpz_xor:
See ``Logical and Bit Manipulation Functions''.
msqrt:
See ``Berkeley MP Compatible Functions''.
msub:
See ``Berkeley MP Compatible Functions''.
mtox:
See ``Berkeley MP Compatible Functions''.
mult:
See ``Berkeley MP Compatible Functions''.
pow:
See ``Berkeley MP Compatible Functions''.
reallocate_function:
See ``Custom Allocation''.
rpow:
See ``Berkeley MP Compatible Functions''.
sdiv:
See ``Berkeley MP Compatible Functions''.
xtom:
See ``Berkeley MP Compatible Functions''.
Table of Contents
*****************
GNU MP
GNU MP Copying Conditions
Introduction to GNU MP
How to use this Manual
Installing GMP
Build Options
ABI and ISA
Notes for Package Builds
Notes for Particular Systems
Known Build Problems
GMP Basics
Nomenclature and Types
Function Classes
GMP Variable Conventions
GMP and Reentrancy
Useful Macros and Constants
Compatibility with older versions
Getting the Latest Version of GMP
Reporting Bugs
Integer Functions
Initialization Functions
Assignment Functions
Combined Initialization and Assignment Functions
Conversion Functions
Arithmetic Functions
Division Functions
Exponentiation Functions
Root Extraction Functions
Number Theoretic Functions
Comparison Functions
Logical and Bit Manipulation Functions
Input and Output Functions
Random Number Functions
Miscellaneous Functions
Rational Number Functions
Initialization and Assignment Functions
Arithmetic Functions
Comparison Functions
Applying Integer Functions to Rationals
Input and Output Functions
Miscellaneous Functions
Floating-point Functions
Initialization Functions
Assignment Functions
Combined Initialization and Assignment Functions
Conversion Functions
Arithmetic Functions
Comparison Functions
Input and Output Functions
Miscellaneous Functions
Low-level Functions
Random Number Functions
Random State Initialization
Berkeley MP Compatible Functions
Custom Allocation
Contributors
References
Concept Index
Function and Type Index