This manual describes how to install and use the GNU Multi-Precision Rational Interval Arithmetic Library, release 0.7.3. Please report any errors in this manual to ‘bug-mpria@gnu.org’.
More information about the GNU MPRIA Library can be found at the project homepage, http://www.gnu.org/software/mpria/.
Copyright © 2009-2016 Jérôme Benoit
Permission is granted to copy, distribute and/or modify this document under the terms of the GNU Free Documentation License, Version 1.3 or any later version published by the Free Software Foundation; with no Invariant Sections, no Front-Cover Texts, and no Back-Cover Texts. A copy of the license is included in GNU Free Documentation License.
| • Copying: | MPRIA Copying Conditions (GPL). | |
| • Introduction to MPRIA: | Brief introduction to GNU MPRIA. | |
| • Installing MPRIA: | How to configure and compile the MPRIA library. | |
| • Reporting Bugs: | How to usefully report bugs. | |
| • MPRIA Basics: | What every MPRIA user should now. | |
| • Rational Interval Functions: | Functions for computation on rational intervals. | |
| • Low-Level Rational Interval Functions: | Low-level functions for computation on rational intervals. | |
| • Extra Number Functions: | Extra functions for rational numbers and signed integers. | |
| • General Library Functions: | General-purpose library functions. | |
| • References: | ||
| • GNU General Public License: | ||
| • GNU Free Documentation License: | ||
| • Indices: |
Next: Introduction to MPRIA, Previous: Top, Up: Top [Index]
The GNU MPRIA Library (or MPRIA for short) is free software: 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 MPRIA 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 MPRIA 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 MPRIA Library are found in the General Public License version 3 that accompanies the source code, see COPYING. A copy of the license is also included in GNU General Public License.
Next: Installing MPRIA, Previous: Copying, Up: Top [Index]
| • Description: | ||
| • Up-to-date Material: | ||
| • Mailing Lists: | ||
| • How to use this Manual: |
Next: Up-to-date Material, Up: Introduction to MPRIA [Index]
GNU MPRIA is intended to be a portable mathematical library written in C for rational interval arithmetic computations with arbitrary precision.
The basic principle of rational interval arithmetic consists in enclosing every number by a rational interval containing it: each number is stored as its lower and upper endpoints and these bounds are rational numbers; their absolute difference measures the precision. The purpose is on the right hand to obtain guaranteed results, thanks to interval computation, and on the left hand to compute accurate results, thanks to arbitrary precision arithmetic.
The arithmetic operations are extended for interval operands in such a way that the exact result of the operation belongs to the computed rational interval.
The GNU MPRIA library is built upon the GNU MP library for operating on rational numbers; see
Next: Mailing Lists, Previous: Description, Up: Introduction to MPRIA [Index]
The latest information about the library can be found at the project homepage
while the primary distribution point for stable releases is at
Many sites around the world mirror ‘ftp.gnu.org’, please use a mirror near you; for a full list, see
Next: How to use this Manual, Previous: Up-to-date Material, Up: Introduction to MPRIA [Index]
There are three public mailing lists of interest: one for release announcements, one for general questions and discussions about usage of the GNU MPRIA Library and one for bug reports. For more information, visit
The proper place for bug reports is ‘bug-mpria@gnu.org’. See Reporting Bugs, for information about reporting bugs.
Previous: Mailing Lists, Up: Introduction to MPRIA [Index]
Everyone should read MPRIA Basics. If you need to install the library yourself, then read Installing MPRIA. To use the library you will need to refer to Rational Interval Functions; for more advanced usage you want to peruse Low-Level Rational Interval Functions.
The rest of the manual can be used for later reference, although it is probably a good idea to glance through it.
Next: Reporting Bugs, Previous: Introduction to MPRIA, Up: Top [Index]
| • How to Install: | ||
| • Other make Targets: | ||
| • Known Build Problems: | ||
| • Getting the Latest Version: |
Next: Other make Targets, Up: Installing MPRIA [Index]
For a generic installation of the MPRIA library,
you have first to install a recent version of the GNU MP on your computer.
You need a C compiler, preferably gcc, but any reasonable C compiler should work.
And you need the standard Unix make command,
plus some other standard Unix utility commands.
Then, in the MPRIA build directory, type the following commands.
This will prepare the build and setup the options according to your system. You can give options to specify the install directories (instead of the default /usr/local), threading support, and so on. See the INSTALL file or the output of ‘./configure --help’ for detailed information, in particular if you get error messages.
This will compile MPRIA and create library files with respect to your platform and environment.
This will make sure MPRIA was built correctly. If you get error messages, please send a bug report to ‘bug-mpria@gnu.org’. See Reporting Bugs, for information about reporting bugs.
This will copy the C header file mpria.h
to the ‘include’ directory /usr/local/include,
the library files
(as the share object file libmpria.so on GNU/Linux computers)
to the ‘lib’ directory /usr/local/lib,
possibly the file mpria.info to the ‘info’ directory /usr/local/share/info,
and some other documentation files into the document folder /usr/local/share/doc/mpria
(or, if you passed the --prefix option to configure,
using the prefix directory given as argument to --prefix instead of /usr/local).
Next: Known Build Problems, Previous: How to Install, Up: Installing MPRIA [Index]
There are some other useful ‘make’ targets:
Create or update an info version of the manual, in mpria.info; this file is already provided in the MPRIA source tarball.
Create a PDF version of the manual, in mpria.pdf; this file is already provided in the MPRIA source tarball.
Create a DVI version of the manual, in mpria.dvi.
Create a PostScript version of the manual, in mpria.ps.
Create a HTML version of the manual, in several pages in the folder doc/mpria.html; to obtain one single page HTML document, type ‘makeinfo --html --no-split mpria.texi’ from the ‘doc’ directory instead.
Delete all object files and archive files, but not the configuration files.
Delete all generated files not included in the distribution.
Delete all files copied by ‘make install’.
Next: Getting the Latest Version, Previous: Other make Targets, Up: Installing MPRIA [Index]
The installation procedure and the GNU MPRIA library itself have been only tested in some Unix-like environments. Because it has not been yet intensively tested, you may discover that the GNU MPRIA library suffers from all bugs of the underlying GNU MP library, plus many many more.
Please report any problem to ‘bug-mpria@gnu.org’. See Reporting Bugs, for information about reporting bugs.
Previous: Known Build Problems, Up: Installing MPRIA [Index]
The latest stable version of MPRIA is available from
Next: MPRIA Basics, Previous: Installing MPRIA, Up: Top [Index]
If you think you have found a bug in the MPRIA library, please investigate it and report it. Likewise, if you think you have figure out a valuable enhancement for the MPRIA library, please mature it and suggest it. This library has been made available to you: it is expected you will report the bugs that you find or you will suggest the enhancements that you wish.
For bug reports, please include enough information to reproduce the problem. Generally speaking, that means:
configure
other than specifying installation directories.
stdout,
with any options used.
gcc, get the version with ‘gcc -v’,
otherwise perhaps ‘what `which cc`’, or similar.
configure,
then attach the compressed contents of config.log.
If your bug report is good, I will do my best to help you to get a corrected version of the library; if the bug report is poor, I will not do anything about it (aside of chiding you to send better bug reports).
Patches are welcome; if possible, please make them with ‘diff -u’ and include ChangeLog entries. Please follow the existing coding style (even if you do not like it).
Please send your bug reports, your suggestions, your patches or your comments to:
‘bug-mpria@gnu.org’.
If you think something in this manual is unclear, or downright incorrect, or if the language needs to be improved, please send a note to the same address.
Next: Rational Interval Functions, Previous: Reporting Bugs, Up: Top [Index]
As MPRIA is built upon GMP, it is very advisable to read the GMP Manual first.
| • Headers and Libraries: | ||
| • Nomenclature and Types: | ||
| • Function Classes: | ||
| • Variable Conventions: | ||
| • Precision Handling and Surrounding Modes: | ||
| • Assignment Modes: | ||
| • Memory Management: | ||
| • Autoconf: |
Next: Nomenclature and Types, Up: MPRIA Basics [Index]
All declarations needed to use MPRIA are collected in the C header file mpria.h; it is designed to work with both C and C++ compilers. You should include this file in any program using MPRIA:
#include <mpria.h>
All programs using MPRIA must link against both libmpria and libgmp libraries. On typical Unix-like systems this can be done with ‘-lmpria -lgmp’ (in that order), for example:
gcc -o myprogram myprogram.c -lmpria -lgmp
GMP and MPRIA libraries are both built using Libtool, thus an application can use that to link if desired (see Integrating libtool in GNU Libtool).
If GMP or MPRIA have been installed to non-standard locations then it may be necessary to use ‘-I’ and ‘-L’ compiler options to point to the right directories, and some sort of run-time path for shared libraries.
Next: Function Classes, Previous: Headers and Libraries, Up: MPRIA Basics [Index]
A rational interval is a closed connected set of rational numbers,
it is represented in MPRIA by its endpoints which are GMP rational numbers.
The C data type for these objects is mpri_t.
MPRIA functions operate on valid rational intervals, while their behaviour remains undefined with non-valid rational intervals; a valid rational interval is defined as follows1:
MPRIA functions may return intervals that are not valid as input value; their semantic is defined as follows2:
Some functions on rational intervals return a rational number.
Among such functions, there are mpri_get_left and mpri_get_right
that respectively return the left and right endpoints of a rational interval,
and there is mpri_diam_abs that computes the width of a rational interval.
Rational numbers (or rationals for short) and rational arithmetic functions
are brought as is from the GMP library.
The C data type for rationals is mpq_t,
while their related functions start with the prefix mpq_
(see Rational Number Functions in The GNU MP Manual).
For rational intervals,
because their endpoints are numbers exactly representable
that are meant to enclose a result not exactly representable,
the notion of precision is essentially related to their width
which is meant to be arbitrarily small.
The precision of a rational interval designs
the integer binary logarithm of the reciprocal of its width;
as such, it expresses in bits.
The corresponding C data type is mpri_prec_t.
When a MPRIA function implements some sort of convergent algorithm to return rational intervals,
besides passing a precision parameter in bits to terminate the computation,
a surrounding mode parameter specifies whether to place the best convert
either at the left endpoint, at the right endpoint or arbitrarily.
The C data type for these modes is mpri_srnd_t.
Typically it concerns implementations based on the Euclidean algorithm (which are omnipresent).
Some MPRIA functions that involve heavy computations admit as last parameter
an assignment mode which specifies whether to assign
either only the left endpoint, only the right endpoint, or the two endpoints.
The C data type for these modes is mpri_asgmt_t.
Those functions are considered as low-level and are both
appended with the capitalised suffix _ASGMT
and wrapped by a macro that assigns the two endpoints.
Next: Variable Conventions, Previous: Nomenclature and Types, Up: MPRIA Basics [Index]
There are four classes of functions in the MPRIA library:
mpri_ and their associated type is mpri_t.
This class gathers
the standard computing assignment methods and concomitants,
computing subroutines for rational interval approximations of quadratic irrational numbers,
the four basic binary arithmetic operations and the classic unary operators built around them,
and
computing subroutines for rational interval approximations of elementary analytic mathematical functions.
(See Rational Interval Functions.)
mpri_ and appended by _ASGMT,
their associated type is mpri_t
while their last parameter is an assignment mode of type mpri_asgmt_t.
These low-level functions are not meant to be called directly
but rather efficiently enwrapped within inline or macro functions.
(See Low-Level Rational Interval Functions.)
mpria_mpq_ and mpria_mpz_, respectively;
their associated type are mpz_t and mpq_t, respectively.
Implemented with great efficiency and handiness in mind,
these functions are mainly inline and macro functions
that are intensively used by the functions in the precedent categories;
you are highly encouraged to employ them directly
within time-critical or intricate subroutines.
They intently complete rather than substitute
their already furnished alikes in the GNU MP library,
the prefix mpria_ preventing from possible naming conflicts.
(See Extra Number Functions.)
MPRIA_VERSION_,
the latters are C plain functions with names beginning with mpria_libversion_.
(See General Library Functions.)
Next: Precision Handling and Surrounding Modes, Previous: Function Classes, Up: MPRIA Basics [Index]
MPRIA functions expect output arguments before input arguments. This general rule, which is inherited from the GNU MP library, is based on an analogy with the assignment operator.
As a matter of fact,
the analogy has been pushed further by allowing
to use the same variable for both input and output in the same expression;
this extension of the general rule is also inherited from the GNU MP library.
For example,
the square function, mpri_sqr, can be used as follows:
mpri_sqr (x, x);
what computes the set of squares of every rational number belonging to x
and puts the results back in x.
As for MP variables, MPRIA variables must be initialised once before any assignment and may be cleared out after use. A (MP or) MPRIA variable should be initialised only once, or at least be cleared out between each initialisation. After such a variable has been initialised, it can be assigned numerous times; it will have the same allocated space during all its lifetime.
For efficiency reasons, avoid excessive initialising and clearing out: as a rule of thumb, initialise near the beginning of an application and clear out near its ending; better still, implement workspaces or garbage collections to pass and reuse these variables all along the computing process.
Next: Assignment Modes, Previous: Variable Conventions, Up: MPRIA Basics [Index]
The following six PRECision parameters are predefined with respect to the IEEE-754 standard (see References), except notably for the meaningless precision:
MPRI_PREC_BITS_NIL: meaningless precision,
MPRI_PREC_BITS_HALF: half precision (binary16) or 11 bits,
MPRI_PREC_BITS_SINGLE: single precision (binary32) or 24 bits,
MPRI_PREC_BITS_DOUBLE: double precision (binary64) or 53 bits,
MPRI_PREC_BITS_QUADRUPLE: quadruple precision (binary128) or 113 bits,
MPRI_PREC_BITS_OCTUPLE: octuple precision or 237 bits.
The following three SuRrouNDing modes are supported:
MPRI_SRND_BCAL: Best Convert At Left endpoint,
MPRI_SRND_BCAA: Best Convert At Any endpoint,
MPRI_SRND_BCAR: Best Convert At Right endpoint.
Next: Memory Management, Previous: Precision Handling and Surrounding Modes, Up: MPRIA Basics [Index]
The following three ASsiGnMenT modes are supported:
MPRI_ASGMT_OL: assign Only Left endpoint,
MPRI_ASGMT_LR: assign Left and Right endpoints,
MPRI_ASGMT_OR: assign Only Right endpoint.
Next: Autoconf, Previous: Assignment Modes, Up: MPRIA Basics [Index]
Basically MPRIA mimics and relays to the GNU MP memory management, except notably for temporary use (see Memory Management in The GNU MP Manual).
The mpq_t type is for the implementation of the mpri_t type what
the mpz_t type is for the implementation of the mpq_t type itself:
mpri_t variables never reduce their allocated space, as mpq_t variables.
All memory is allocated, reallocated and freed by passing on
to the GNU MP memory functions as grabbed from mp_get_memory_functions
(see Custom Allocation in The GNU MP Manual).
While GMP uses temporary memory on the stack (via alloca),
MPRIA creates, passes along and intensively reuses workspaces for internal computation;
the various created workspaces are freed before exiting
with the help of the standard C atexit function
(see Cleanups on Exit in The GNU C Library Reference Manual),
therefore no memory leaks should be reported by tools like valgrind
(http://valgrind.org/).
Teething Note: At the time of writing, this internal workspace machinery is robust but global, read not yet thread safe, and no high-level function is yet implemented to free the created workspaces, or part of them, from time to time.
Previous: Memory Management, Up: MPRIA Basics [Index]
For applications using autoconf and its friends,
the macro mpria_AM_PATH_MPRIA
available in the file mpria.m4
can be employed to link with the MPRIA automatically
from the configure script.
As preliminary work,
this macro checks whether MPRIA is properly installed
and performs compatibility test against
either a specified version of the library
or a default workable version of a recent major release of the library.
To use this macro simply add the following line
to the configure.ac autoconf input file:
mpria_AM_PATH_MPRIA([MPRIA_VERSION], [action-if-found], [action-if-not-found])
where the arguments are optional.
The first argument MPRIA_VERSION should be
either the one digit version number MAJOR,
the two digit dotted version number MAJOR.MINOR
or the three digit dotted version number MAJOR.MINOR.MICRO
of the required release of the GNU MPRIA library.
While action-if-found might be worthily empty or :,
a suitable choice for action-if-not-found is
AC_MSG_ERROR([no suitable GNU MPRIA library found])
Then the variables MPRIA_CPPFLAGS, MPRIA_CFLAGS,
MPRIA_LDFLAGS and MPRIA_LIBS
can be added to the Makefile.am automake input files
to obtain the correct preprocessor, compiler and linker flags.
For example:
libfoo_la_CPPFLAGS = $(MPRIA_CPPFLAGS) $(GMP_CPPFLAGS) libfoo_la_CFLAGS = $(MPRIA_CFLAGS) $(GMP_CFLAGS) libfoo_la_SOURCES = foo-dim.c foo-dam.c foo-dom.c libfoo_la_LDFLAGS = $(MPRIA_LDFLAGS) $(GMP_LDFLAGS) libfoo_la_LIBADD = $(MPRIA_LIBS) $(GMP_LIBS) $(LIBM)
Note that the macro mpria_AM_PATH_MPRIA requires
the macro mpria_AM_PATH_GMP which is provided
in the file mpria_ax_prog_path_gmp_cc.m4;
as you have already guessed,
the macro mpria_AM_PATH_GMP is for the GNU MP library
what the macro mpria_AM_PATH_MPRIA is for the GNU MPRIA library.
So, in the configure.ac file, the macro mpria_AM_PATH_GMP
must precede the macro mpria_AM_PATH_MPRIA.
In the previous example,
the variables GMP_CPPFLAGS, GMP_CFLAGS,
GMP_LDFLAGS and GMP_LIBS
are furnished by the macro mpria_AM_PATH_GMP;
the variable LIBM being set up by
the Libtool macro LT_LIB_M.
For building more closely to the GNU MP library built,
further tweaks are required.
The main difficulty is to grab and use at proper time
the compiler information stored at GNU MP build-time
in the two macros __GMP_CC and __GMP_CFLAGS,
which are defined in the header file gmp.h.
Ideally this information should be first obtained
with the help of a C PreProcessor (CPP)
in such a way that the C Compiler (CC)
could be then set up accordingly.
Unfortunately, at the time of writing,
the only ready-to-use autoconf macro
meant to set up the C preprocessor to be employed,
that is to say AC_PROG_CPP,
depends to do so on the autoconf macro AC_PROG_CC,
which determines with no easy comeback the C compiler to be employed:
in short, the difficulty is harder than expected.
As a matter of fact, the file mpria_ax_prog_path_gmp_cc.m4
contains a bunch of macros that allows to overcome the issue
in a transparent way for the final developer:
the macro mpria_AC_PROG_GMP_CC have to be used
instead of the macro AC_PROG_CC.
Typically the configure.ac file
may so contain something similar to the following scrap of code:
dnl Setup CC and CFLAGS wrt GMP: mpria_AC_PROG_GMP_CC dnl Checks for libraries: dnl the math library: LT_LIB_M dnl the GMP libray: mpria_AM_PATH_GMP([6.1.0]) dnl the GNU MPRIA library: mpria_AM_PATH_MPRIA([0.7.3])
Besides, the usage of mpria_AC_PROG_GMP_CC reinforces
the checks done by mpria_AM_PATH_GMP.
To allow code readability improvement,
the two latter macros have been combined
into the single macro mpria_AC_PROG_PATH_GMP_CC.
The above scrap of code can thus be rewritten as follows:
dnl Setup CC and CFLAGS wrt GMP: mpria_AC_PROG_PATH_GMP_CC([6.1.0]) dnl Checks for libraries: dnl the math library: LT_LIB_M dnl the GNU MPRIA library: mpria_AM_PATH_MPRIA([0.7.3])
Last but not least,
non-standard installation locations of the MPRIA and GMP libraries
are handled with respect to customary use;
in particular, command line options are implemented in the configure script
to specify these locations.
The macro mpria_AM_PATH_MPRIA affords
the following command line options which accept an absolute path as compulsory argument:
The macros mpria_AC_PROG_GMP_CC,
mpria_AM_PATH_GMP and mpria_AC_PROG_PATH_GMP_CC
implement command line options that have exactly the same usage but for the GMP library instead:
--with-gmp-prefix, --with-gmp-include and --with-gmp-lib,
respectively.
In addition,
these macros declare the environment variable GMP_GPP as precious:
this advanced feature enables to specify a Generic PreProcessor command
for early processing of the header file gmp.h.
Next: Low-Level Rational Interval Functions, Previous: MPRIA Basics, Up: Top [Index]
Next: Assignment Functions, Up: Rational Interval Functions [Index]
An mpri_t object must be initialised before storing the first value in it:
the function mpri_init is used for that purpose,
the function mpri_clear clears it out.
Initialise x and set it to the singleton interval [0/1,0/1].
Normally, a variable should be initialised once only
or at least be cleared out (using mpri_clear) between consecutive initialisation.
Free the space occupied by the endpoints of x.
Make sure to call this function for all mpri_t variables when you are done with them.
Next: Interval Conversion Functions, Previous: Initialisation Functions, Up: Rational Interval Functions [Index]
These functions and macros assign new values to already initialised rational intervals.
Assign rop from op.
Set the value of op to the singleton intervals [0/1,0/1] (zero) and [0/0,0/0] (NaN), respectively.
Set the value of rop to the singleton interval [op,op].
Set the value of rop to the best rational interval approximation
of the quadratic irrational number (op1+sqrt(op2))/op3
with a guaranteed precision of at least prec bits
and with respect to the surrounding srnd.
The result remains undefined
if the radicand op2 is negative
or if the divisor op3 is zero.
While the macro mpri_set_qi_z is its natural high-level wrapper,
the inline function mpri_set_qi_q belongs to one of the efficient wrappers
implemented around the low-level function mpri_set_qi_z_ASGMT.
Set the value of rop to the best rational interval approximation
of the number op
(respectively, a rational number and a double)
with a guaranteed precision of at least prec bits
and with respect to the surrounding srnd.
Both are inline wrappers efficiently built
around the low-level function mpri_set_qi_z_ASGMT;
a rational being a degenerate quadratic irrational,
a double an approximative rational representation of a real number.
Set the value of rop to the best rational interval approximation
of the square root of op,
sqrt(op),
with a guaranteed precision of at least prec bits
and with respect to the surrounding srnd.
The result is undefined if the radicand op is negative.
It is an inline function that efficiently wraps
around the low-level function mpri_set_qi_z_ASGMT.
Set the value of rop to the best rational interval approximation
of the reciprocal square root of op,
literally sqrt(op)/op,
with a guaranteed precision of at least prec bits
and with respect to the surrounding srnd.
The result stays undefined if the operand op is either negative or zero.
This inline function is an efficient wrapper built
around the low-level function mpri_set_qi_z_ASGMT.
Swap the values rop1 and rop2 efficiently.
Next: Interval Comparison Functions, Previous: Assignment Functions, Up: Rational Interval Functions [Index]
Convert op to a rational number, which is its centre.3
Convert op to a double,
this conversion is the composition of mpri_get_q and mpq_get_d.
Next: Interval Basic Functions, Previous: Interval Conversion Functions, Up: Rational Interval Functions [Index]
Return either 1
(read true)
if the rational intervals op1 and op2 are equal
or
0
(read false)
if they are non-equal.
Return 1
(read true)
if the rational interval op is the singleton interval [0/1,0/1] (zero),
0
(read false)
otherwise.
Return 1
(read true)
if the rational interval op does not reduce to the singleton interval [0/1,0/1] (zero),
0
(read false)
otherwise.
Return 1
(read true)
if zero belongs to the rational interval op,
0
(read false)
otherwise.
Return either -1
if the rational interval op is strictly negative,
or +1 if it is strictly positive,
or 0
if it contains zero
Next: Interval Arithmetic Functions, Previous: Interval Comparison Functions, Up: Rational Interval Functions [Index]
Some MPRIA functions on rational intervals return rational results, such as the diameter or the centre of a rational interval.
Set the value of rop to the absolute diameter of the rational interval op, that is to say, to the difference between its right endpoint and its left one.
Set the value of rop to the relative diameter of the rational interval op, in other words, either to the difference between its right endpoint and its left one divided by the absolute value of its centre when it is not symmetric or to NaN ([0/0,0/0]) when it is symmetric.
Set the value of rop to the relative diameter of the rational interval op if it does not contains zero and to its absolute diameter otherwise.
Set the value of rop to the mignitude and magnitude of the rational interval op, respectively, that is to say, to the smallest and largest absolute value of its elements, respectively.
Set the value of rop to the value of the middle of the rational interval op, namely, to the half sum of its endpoints.
Return a reference to the left and right endpoint of the rational interval op, respectively.
Set the value of rop to the left and right endpoint
of the rational interval op, respectively.
These functions are equivalent to calling mpq_set
with an appropriate mpri_lepref or mpri_repref.
Direct use of mpri_lepref or mpri_repref is recommended
instead of these functions.
Set the value of rop to a rational number picked up at random in the rational interval op according to a uniform distribution. If the rational interval op is not valid, the generator returns NaN, namely 0/0.
Teething Note: At the time of writing, it is not clear to the author which value the generator should return when the rational interval op is valid but infinite: as caveat, the actual infinite endpoint is returned.
The argument state must be initialized by calling one of the GMP random state initialization functions (see Random State Initialization in The GNU MP Manual) before invoking this functions.
Next: Interval Approximation of Elementary Functions, Previous: Interval Basic Functions, Up: Rational Interval Functions [Index]
Set rop to op1 + op2.
Set rop to op1 - op2.
Set rop to op1 * op2. Multiplication by zero, passed as singleton interval [0/1,0/1] or literally, gives the singleton interval [0/1,0/1].
Set rop to op1/op2. When the dividend op1 reduces to the singleton interval [0/1,0/1], viz. zero, the division returns the singleton interval [0/1,0/1] as result; when the divisor op2 contains zero, the division returns [0/0,0/0], namely NaN.
Set rop to -op.
Set rop to abs(op), the absolute value of op.
Set rop to 1/op when the rational interval op does not contains zero, to [0/0,0/0] (NaN) otherwise.
Set rop to the square of op.
Set rop to the best rational interval approximation
of the square root of op,
sqrt(op),
with a guaranteed precision of at least prec bits.
If the rational interval radicand op is not positive,
the return interval is [0/0,0/0], namely NaN.
This inline function implements an efficient wrapper
around the low-level function mpri_set_qi_z_ASGMT.
Set rop to the best rational interval approximation
of the reciprocal square root of op,
literally sqrt(op)/op,
with a guaranteed precision of at least prec bits.
If the rational interval operand op is not strictly positive,
the return interval is [0/0,0/0], to wit NaN.
This inline function efficiently implements a wrapper
around the low-level function mpri_set_qi_z_ASGMT.
Set rop to op times 2 raised to exponent.
Set rop to op divided by 2 raised to exponent.
Previous: Interval Arithmetic Functions, Up: Rational Interval Functions [Index]
Teething Note: At the time of writing, this part of the library is clearly at a very early stage as it basically contains only one function: more functions may be furnished in the coming minor releases, the all set of elementary functions in the next major release.
Set rop to the best rational interval approximation
of the arc-tangent of op,
arctan(op),
with a guaranteed precision of at least prec bits.
This inline function straightforwardly wraps the function mpri_2exp_atan.
Set rop to the best rational interval approximation of 2 raised to exponent times the arc-tangent of op, with a guaranteed precision of at least prec bits.
Next: Extra Number Functions, Previous: Rational Interval Functions, Up: Top [Index]
| • Low-Level Interval Elementary Functions: | ||
| • Hard-Coded Numbers: |
Next: Hard-Coded Numbers, Up: Low-Level Rational Interval Functions [Index]
Set the value of rop to the best rational interval approximation of the quadratic irrational number (op1+sqrt(op2))/op3 with a guaranteed precision of at least prec bits and with respect to both the surrounding srnd and the assignment mode asgmt. The result remains undefined if the radicand op2 is negative or if the divisor op3 is zero.
Previous: Low-Level Interval Elementary Functions, Up: Low-Level Rational Interval Functions [Index]
The following collections of hard-coded numbers are mainly meant to serve the previous low-level functions within enwrapping inline functions or plain functions. For illustrations on how to wrap with them, peruse the header file mpria.h.
Collection of mpz_t signed integers with self-explanatory names.
Collection of mpq_t rational numbers with self-explanatory names.
Collection of mpri_t rational singleton intervals
with self-explanatory names.
Next: General Library Functions, Previous: Low-Level Rational Interval Functions, Up: Top [Index]
| • Extra Rational Number Functions: | ||
| • Extra Signed Integer Functions: |
Next: Extra Signed Integer Functions, Up: Extra Number Functions [Index]
Set the value of the rational number Q to
0, +1, -1, 0/0 (NaN),
+1/0 (+infinity) and -1/0 (-infinity), respectively.
These utility functions are implemented as plain macros (with self-explanatory names).
Return 1
(read true)
if the rational number Q
is either zero, nonzero, positive, negative, strictly positive or strictly negative,
respectively,
0
(read false)
otherwise.
These test functions are plain macro functions (with self-explanatory names).
Return 1
(read true)
if the rational number op is Not-a-Number,
0
(read false)
otherwise.
NaN, the acronym for Not-a-Number, has the representation 0/0.4
Return +1 if the rational number op is positive infinity,
-1 if it is negative infinity,
0 otherwise.
Positive and negative infinities have the representation +1/0 and -1/0, respectively;5 they are commonly written +infinity and -infinity, respectively.
Return 1
(read true)
if the rational number op is finite,
0
(read false)
if it is either infinite or Not-a-Number.
Return +1 if the rational op is strictly positive,
0 if it is zero,
or -1 if it is strictly negative.
Its behaviour stays undefined if its argument is NaN (0/0).
While its counterpart mpq_sgn is implemented as a macro,
this function is implemented as an inline function:
it evaluates its argument only once.
Compare the absolute values of the rational numbers op1 and op2. Return either a positive value if abs(op1) is strictly greater than abs(op2), zero if abs(op1) is equal to abs(op2), or a negative value if abs(op1) is strictly smaller than abs(op2). Its behaviour remains undefined if at least one of its arguments is either -infinity (-1/0), +infinity (+1/0), or NaN (0/0).
Set the value of rop to the minimum of the triplet {op1,op2,op3}. Its behaviour is undefined if the triplet contains -infinity (-1/0), +infinity (+1/0), or NaN (0/0).
Previous: Extra Rational Number Functions, Up: Extra Number Functions [Index]
Set the value of the signed integer Z to 0, +1 and -1,
respectively.
These utility functions are implemented as plain macros (with self-explanatory names).
Return 1
(read true)
if the signed integer Z
is either zero, nonzero, positive, negative, strictly positive or strictly negative,
respectively,
and
0
(read false)
otherwise.
These test functions are plain macro functions (with self-explanatory names).
Return +1 if the signed integer op is strictly positive,
0 if it is zero,
or -1 if it is strictly negative.
While its counterpart mpz_sgn is implemented as a macro,
this function is implemented as an inline function:
it evaluates its argument only once.
Set the value of rop to the minimum of the triplet {abs(op1),abs(op2),abs(op3)}.
Next: References, Previous: Extra Number Functions, Up: Top [Index]
| • Library Version Handling: | ||
| • Miscellaneous Utilities: |
Next: Miscellaneous Utilities, Up: General Library Functions [Index]
Different releases of the GNU MPRIA library are distinguished by an authoritative version triplet of nonnegative integer constants defined as macro constants. Utilities are implemented to efficiently check against, to numerically pack or to stringify this triplet; packed variants of the triplet are also defined as macro constants.
The authoritative version triplet, respectively, as nonnegative integer constants: the major version number, the minor version number (or revision number), the micro version number (or major patch level).
Retrieve the major, minor and micro version numbers
of the MPRIA library against which the application is currently linked.
The NULL pointer is accepted as argument.
Check the compatibility of the arbitrary major, minor and micro version numbers with their counterpart from the MPRIA library against which the application is currently linked. The returned response is as follows:
This function performs no action apart from checking and responding,
in particular it does not cause the application to abort
or to show up any kind of messages
(it may be enwrapped within a if else statement to do so).
Check the compatibility of the version triplet of the MPRIA library
with which an application was compiled with the version triplet of the MPRIA library
against which the application is currently linked.
This is a convenient wrapping macro that passes the authoritative macro version numbers
to the function mpria_libversion_check_numbers,
as such it acts similarly.
The most common cause for an incompatibility or a weak compatibility is that
an application was compiled against one version of the MPRIA library
while it is dynamically linked against a different one,
what might be due to a misconfiguration, a downgrading or an upgrading.
A typical usage may look like:
/* Check version of libmpria */
if (!(mpria_libversion_check ()))
{
fprintf (stderr,"version miss-compatibility\n");
fflush (stderr);
abort ();
}
The extra version string suffix, only meant for development purposes. For production releases, alpha and stable ones, it must be reset to the empty string "".
Compact, respectively stringify, the arbitrary version triplet [Major, Minor, Micro] into a single number, resp. into a null-terminated string to which is appended the arbitrary extra version string suffix StrExtra.
The non-authoritative version number, respectively string,
obtained by passing the authoritative version triplet
to MPRIA_VERSION_NUMBER_PACK,
resp. to MPRIA_VERSION_STRING_PACK
with MPRIA_VERSION_EXTRA as fourth argument.
Retrieve the non-authoritative version number and string, respectively, of the MPRIA library against which the application is currently linked.
The version string of the MPRIA library against which the application is currently linked.
While mpria_libversion is a convenient macro that wraps mpria_libversion_get_string,
mpria_version is defined as synonymous of mpria_libversion
with respect to the GNU MP naming scheme.
Previous: Library Version Handling, Up: General Library Functions [Index]
Stringify Token.
Next: GNU General Public License, Previous: General Library Functions, Up: Top [Index]
Teething Note: This is clearly a non-exhaustive list (in progress) of references.
Next: GNU Free Documentation License, Previous: References, Up: Top [Index]
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one line to give the program's name and a brief idea of what it does. Copyright (C) year name of author This program is free software: you can redistribute it and/or modify it under the terms of the GNU General Public License as published by the Free Software Foundation, either version 3 of the License, or (at your option) any later version. This program is distributed in the hope that it will be useful, but WITHOUT ANY WARRANTY; without even the implied warranty of MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the GNU General Public License for more details. You should have received a copy of the GNU General Public License along with this program. If not, see http://www.gnu.org/licenses/.
Also add information on how to contact you by electronic and paper mail.
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program Copyright (C) year name of author This program comes with ABSOLUTELY NO WARRANTY; for details type ‘show w’. This is free software, and you are welcome to redistribute it under certain conditions; type ‘show c’ for details.
The hypothetical commands ‘show w’ and ‘show c’ should show the appropriate parts of the General Public License. Of course, your program’s commands might be different; for a GUI interface, you would use an “about box”.
You should also get your employer (if you work as a programmer) or school, if any, to sign a “copyright disclaimer” for the program, if necessary. For more information on this, and how to apply and follow the GNU GPL, see http://www.gnu.org/licenses/.
The GNU General Public License does not permit incorporating your program into proprietary programs. If your program is a subroutine library, you may consider it more useful to permit linking proprietary applications with the library. If this is what you want to do, use the GNU Lesser General Public License instead of this License. But first, please read http://www.gnu.org/philosophy/why-not-lgpl.html.
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Previous: GNU Free Documentation License, Up: Top [Index]
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The definition of a valid rational interval might be refined in future releases of MPRIA.
The meaning of an invalid operation, the representation of the empty interval and their handling may evolve in future releases of MPRIA, according to the standardisation of interval arithmetic in IEEE-1788 (see References).
An other conversion choice might be made in future releases of MPRIA;
to explicitly obtain the centre of a rational interval,
use mpri_mid instead.
At the time of writing, GMP does not support NaN for mpq_t numbers.
At the time of writing, GMP does not support infinities for mpq_t numbers.