GNU MPRIA
MPRIA Copying Conditions
1 Introduction to MPRIA
1.1 Description
1.2 Up-to-date Material
1.3 Mailing Lists
1.4 How to use this Manual
2 Installing MPRIA
2.1 How to Install
2.2 Other 'make' Targets
2.3 Known Build Problems
2.4 Getting the Latest Version
3 Reporting Bugs
4 MPRIA Basics
4.1 Headers and Libraries
4.2 Nomenclature and Types
4.3 Function Classes
4.4 Variable Conventions
4.5 Precision Handling and Surrounding Modes
4.6 Assignment Modes
4.7 Memory Management
4.8 Autoconf
5 Rational Interval Functions
5.1 Initialisation Functions
5.2 Assignment Functions
5.3 Interval Conversion Functions
5.4 Interval Comparison Functions
5.5 Interval Basic Functions
5.6 Interval Arithmetic Functions
5.7 Interval Approximation of Elementary Functions
6 Low-Level Rational Interval Functions
6.1 Low-Level Interval Elementary Functions
6.2 Hard-Coded Numbers
7 Extra Number Functions
7.1 Extra Rational Number Functions
7.2 Extra Signed Integer Functions
8 General Library Functions
8.1 Library Version Handling
8.2 Miscellaneous Utilities
Appendix A References
Appendix B GNU General Public License
Appendix C GNU Free Documentation License
Appendix D Indices
D.1 Concept Index
D.2 Type Index
D.3 Variable Index
D.4 Function Index
GNU MPRIA
*********
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, .
Copyright (C) 2009-2016 Jerome 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 *note
GNU Free Documentation License::.
MPRIA Copying Conditions
************************
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
*note GNU General Public License::.
1 Introduction to MPRIA
***********************
1.1 Description
===============
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
.
1.2 Up-to-date Material
=======================
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
.
1.3 Mailing Lists
=================
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'. *Note
Reporting Bugs::, for information about reporting bugs.
1.4 How to use this Manual
==========================
Everyone should read *note MPRIA Basics::. If you need to install the
library yourself, then read *note Installing MPRIA::. To use the
library you will need to refer to *note Rational Interval Functions::;
for more advanced usage you want to peruse *note 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.
2 Installing MPRIA
******************
2.1 How to Install
==================
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.
1. './configure'
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.
2. 'make'
This will compile MPRIA and create library files with respect to
your platform and environment.
3. 'make check'
This will make sure MPRIA was built correctly. If you get error
messages, please send a bug report to 'bug-mpria@gnu.org'. *Note
Reporting Bugs::, for information about reporting bugs.
4. 'make install'
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').
2.2 Other 'make' Targets
========================
There are some other useful 'make' targets:
* 'mpria.info' or 'info'
Create or update an info version of the manual, in 'mpria.info';
this file is already provided in the MPRIA source tarball.
* 'mpria.pdf' or 'pdf'
Create a PDF version of the manual, in 'mpria.pdf'; this file is
already provided in the MPRIA source tarball.
* 'mpria.dvi' or 'dvi'
Create a DVI version of the manual, in 'mpria.dvi'.
* 'mpria.ps' or 'ps'
Create a PostScript version of the manual, in 'mpria.ps'.
* 'mpria.html' or 'html'
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.
* 'clean'
Delete all object files and archive files, but not the
configuration files.
* 'distclean'
Delete all generated files not included in the distribution.
* 'uninstall'
Delete all files copied by 'make install'.
2.3 Known Build Problems
========================
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'. *Note Reporting
Bugs::, for information about reporting bugs.
2.4 Getting the Latest Version
==============================
The latest stable version of MPRIA is available from
.
3 Reporting Bugs
****************
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:
* The MPRIA library version, along with the involved GMP library
version.
* A test case that makes it possible to reproduce the bug; do not
forget to include instructions on how to run the test case.
* A description of what goes wrong; please clearly explain what is
incorrect and in what way, whether or not you get a crash.
* Options given to 'configure' other than specifying installation
directories.
* The output from running './configure', as printed to 'stdout', with
any options used.
* The name of the involved compiler and its version; for 'gcc', get
the version with 'gcc -v', otherwise perhaps 'what `which cc`', or
similar.
* Hardware and operating system names, versions and details; the
output from 'uname -a' along with the output from running
'./build-aux/config.guess' should be sufficient.
* If the bug is related to 'configure', then attach the compressed
contents of 'config.log'.
* Anything else that you think would be helpful; when in doubt
whether something is needed or not, include it since it is better
to include too much than to leave out something important.
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.
4 MPRIA Basics
**************
As MPRIA is built upon GMP, it is very advisable to read the GMP Manual
first.
4.1 Headers and Libraries
=========================
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
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 (*note (libtool)Integrating
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.
4.2 Nomenclature and Types
==========================
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 follows(1):
* A "valid rational interval" can have finite or infinite endpoints,
but its left endpoint is not larger than its right endpoint and
cannot be +infinity (+1/0) while the right endpoint cannot be
-infinity (-1/0). Whenever the left and right endpoints are equal
to a same rational Q, the valid rational interval reduces then to
the singleton interval [Q,Q] which represents exactly the rational
Q; conversely, any rational Q is perfectly represented by the
singleton interval [Q,Q].
MPRIA functions may return intervals that are not valid as input value;
their semantic is defined as follows(2):
* Whenever the left endpoint or the right endpoint is NaN (0/0), it
indicates that an "invalid operation" has been performed and that
the resulting rational interval has no mathematical meaning.
* Whenever the left endpoint is strictly greater than the right
endpoint, it means that the resulting rational interval is the
"empty interval".
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_' (*note (gmp)Rational Number Functions::).
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.
---------- Footnotes ----------
(1) The definition of a valid rational interval might be refined in
future releases of MPRIA.
(2) 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' (*note References::).
4.3 Function Classes
====================
There are four classes of functions in the MPRIA library:
1. Functions for intervals computation based on rational numbers:
their names begin with '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.
(*Note Rational Interval Functions::.)
2. Low-level functions for rational interval approximations of
analytic mathematical functions: their names are both prepended by
'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. (*Note Low-Level Rational Interval Functions::.)
3. Fast and convenient low-level functions that operate on signed
integers and rational numbers: their names begin with '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. (*Note Extra Number Functions::.)
4. Miscellaneous functions. As memory management is inherited from
the GNU MP library by design, this miscellanea essentially concerns
functions for handling up different versions of the library. Two
kinds of version handling function are distinguished: the functions
that treat the version data of the library against which the
application is effectively compiled, as such they act at compile
time; the functions that deal with the version data of the library
against which the application is dynamically linked, therefore they
rather serve at run time. The formers are C preprocessor macros
with names beginning with 'MPRIA_VERSION_', the latters are C plain
functions with names beginning with 'mpria_libversion_'. (*Note
General Library Functions::.)
4.4 Variable Conventions
========================
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.
4.5 Precision Handling and Surrounding Modes
============================================
The following six PRECision parameters are predefined with respect to
the 'IEEE-754' standard (*note 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.
4.6 Assignment Modes
====================
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.
4.7 Memory Management
=====================
Basically MPRIA mimics and relays to the GNU MP memory management,
except notably for temporary use (*note (gmp)Memory Management::).
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' (*note
(gmp)Custom Allocation::).
While GMP uses temporary memory on the stack (via 'alloca'), MPRIA
creates, passes along and intensively reuses "workspace"s for internal
computation; the various created workspaces are freed before exiting
with the help of the standard C 'atexit' function (*note (libc)Cleanups
on Exit::), therefore no memory leaks should be reported by tools like
'valgrind' ().
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.
4.8 Autoconf
============
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:
* '--with-mpria-prefix=PREFIX' assumes that MPRIA is installed in the
'PREFIX' directory, the default assumption being '/usr/local';
* '--with-mpria-include=PATH' specifies that 'PATH' is the MPRIA
'include' directory, the default being 'PREFIX/include';
* '--with-mpria-lib=PATH' specifies that 'PATH' is the MPRIA 'lib'
directory, the default being 'PREFIX/lib'.
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'.
5 Rational Interval Functions
*****************************
5.1 Initialisation Functions
============================
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.
-- Inline Function: void mpri_init (mpri_t X)
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.
-- Inline Function: void mpri_clear (mpri_t X)
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.
5.2 Assignment Functions
========================
These functions and macros assign new values to already initialised
rational intervals.
-- Inline Function: void mpri_set (mpri_t ROP, const mpri_t OP)
Assign ROP from OP.
-- Macro: void MPRI_SET_ZERO (mpri_t OP)
-- Macro: void MPRI_SET_NAN (mpri_t OP)
Set the value of OP to the singleton intervals [0/1,0/1] (zero) and
[0/0,0/0] (NaN), respectively.
-- Macro: void MPRI_SET_Q (mpri_t ROP, const mpq_t OP)
Set the value of ROP to the singleton interval [OP,OP].
-- Macro: void mpri_set_qi_z (mpri_t ROP, const mpz_t OP1, const mpz_t
OP2, const mpz_t OP3, mpri_prec_t PREC, mpri_srnd_t SRND)
-- Inline Function: void mpri_set_qi_q (mpri_t ROP, const mpq_t OP1,
const mpq_t OP2, const mpq_t OP3, mpri_prec_t PREC,
mpri_srnd_t SRND)
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'.
-- Inline Function: void mpri_set_q (mpri_t ROP, const mpq_t OP,
mpri_prec_t PREC, mpri_srnd_t SRND)
-- Inline Function: void mpri_set_d (mpri_t ROP, double OP, mpri_prec_t
PREC, mpri_srnd_t SRND)
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.
-- Inline Function: void mpri_set_sqrt_q (mpri_t ROP, const mpq_t OP,
mpri_prec_t PREC, mpri_srnd_t SRND)
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'.
-- Inline Function: void mpri_set_rsqrt_q (mpri_t ROP, const mpq_t OP,
mpri_prec_t PREC, mpri_srnd_t SRND)
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'.
-- Inline Function: void mpri_swap (mpri_t ROP1, mpri_t ROP2)
Swap the values ROP1 and ROP2 efficiently.
5.3 Interval Conversion Functions
=================================
-- Inline Function: void mpri_get_q (mpq_t ROP, const mpri_t OP)
Convert OP to a rational number, which is its centre.(1)
-- Function: double mpri_get_d (const mpri_t OP)
Convert OP to a 'double', this conversion is the composition of
'mpri_get_q' and 'mpq_get_d'.
---------- Footnotes ----------
(1) 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.
5.4 Interval Comparison Functions
=================================
-- Inline Function: int mpri_equal (const mpri_t OP1, const mpri_t OP2)
Return either '1' (read true) if the rational intervals OP1 and OP2
are equal or '0' (read false) if they are non-equal.
-- Inline Function: int mpri_is_zero (const mpri_t OP)
Return '1' (read true) if the rational interval OP is the singleton
interval [0/1,0/1] (zero), '0' (read false) otherwise.
-- Inline Function: int mpri_is_nonzero (const mpri_t OP)
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.
-- Inline Function: int mpri_has_zero (const mpri_t OP)
Return '1' (read true) if zero belongs to the rational interval OP,
'0' (read false) otherwise.
-- Inline Function: int mpri_hasnot_zero (const mpri_t OP)
Return either '-1' if the rational interval OP is strictly
negative, or '+1' if it is strictly positive, or '0' if it contains
zero
5.5 Interval Basic Functions
============================
Some MPRIA functions on rational intervals return rational results, such
as the diameter or the centre of a rational interval.
-- Inline Function: void mpri_diam_abs (mpq_t ROP, const mpri_t OP)
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.
-- Function: void mpri_diam_rel (mpq_t ROP, const mpri_t OP)
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.
-- Inline Function: void mpri_diam (mpq_t ROP, const mpri_t OP)
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.
-- Inline Function: void mpri_mig (mpq_t ROP, const mpri_t OP)
-- Inline Function: void mpri_mag (mpq_t ROP, const mpri_t OP)
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.
-- Inline Function: void mpri_mid (mpq_t ROP, const mpri_t OP)
Set the value of ROP to the value of the middle of the rational
interval OP, namely, to the half sum of its endpoints.
-- Macro: mpq_t mpri_lepref (const mpri_t OP)
-- Macro: mpq_t mpri_repref (const mpri_t OP)
Return a reference to the left and right endpoint of the rational
interval OP, respectively.
-- Inline Function: void mpri_get_left (mpq_t ROP, const mpri_t OP)
-- Inline Function: void mpri_get_right (mpq_t ROP, const mpri_t OP)
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.
-- Function: void mpri_urandomm (mpq_t ROP, const mpri_t OP,
gmp_randstate_t STATE)
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 (*note (gmp)Random State
Initialization::) before invoking this functions.
5.6 Interval Arithmetic Functions
=================================
-- Inline Function: void mpri_add (mpri_t ROP, const mpri_t OP1, const
mpri_t OP2)
-- Inline Function: void mpri_add_q (mpri_t ROP, const mpri_t OP1,
const mpq_t OP2)
Set ROP to OP1 + OP2.
-- Inline Function: void mpri_sub (mpri_t ROP, const mpri_t OP1, const
mpri_t OP2)
-- Inline Function: void mpri_sub_q (mpri_t ROP, const mpri_t OP1,
const mpq_t OP2)
-- Inline Function: void mpri_q_sub (mpri_t ROP, const mpq_t OP1, const
mpri_t OP2)
Set ROP to OP1 - OP2.
-- Function: void mpri_mul (mpri_t ROP, const mpri_t OP1, const mpri_t
OP2)
-- Inline Function: void mpri_mul_q (mpri_t ROP, const mpri_t OP1,
const mpq_t 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].
-- Function: void mpri_div (mpri_t ROP, const mpri_t OP1, const mpri_t
OP2)
-- Inline Function: void mpri_div_q (mpri_t ROP, const mpri_t OP1,
const mpq_t OP2)
-- Inline Function: void mpri_q_div (mpri_t ROP, const mpq_t OP1, const
mpri_t OP2)
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.
-- Inline Function: void mpri_neg (mpri_t ROP, const mpri_t OP)
Set ROP to -OP.
-- Inline Function: void mpri_abs (mpri_t ROP, const mpri_t OP)
Set ROP to abs(OP), the absolute value of OP.
-- Inline Function: void mpri_inv (mpri_t ROP, const mpri_t OP)
Set ROP to 1/OP when the rational interval OP does not contains
zero, to [0/0,0/0] (NaN) otherwise.
-- Inline Function: void mpri_sqr (mpri_t ROP, const mpri_t OP)
Set ROP to the square of OP.
-- Inline Function: void mpri_sqrt (mpri_t ROP, const mpri_t OP,
mpri_prec_t PREC)
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'.
-- Inline Function: void mpri_rsqrt (mpri_t ROP, const mpri_t OP,
mpri_prec_t PREC)
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'.
-- Inline Function: void mpri_mul_2exp (mpri_t ROP, const mpri_t OP,
unsigned long int EXPONENT)
Set ROP to OP times 2 raised to EXPONENT.
-- Inline Function: void mpri_div_2exp (mpri_t ROP, const mpri_t OP,
unsigned long int EXPONENT)
Set ROP to OP divided by 2 raised to EXPONENT.
5.7 Interval Approximation of Elementary Functions
==================================================
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.
-- Inline Function: void mpri_atan (mpri_t ROP, const mpri_t OP,
mpri_prec_t PREC)
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'.
-- Function: void mpri_2exp_atan (mpri_t ROP, unsigned long int
EXPONENT, const mpri_t OP, mpri_prec_t PREC)
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.
6 Low-Level Rational Interval Functions
***************************************
6.1 Low-Level Interval Elementary Functions
===========================================
-- Function: void mpri_set_qi_z_ASGMT (mpri_t ROP, const mpz_t OP1,
const mpz_t OP2, const mpz_t OP3, mpri_prec_t PREC,
mpri_srnd_t SRND,
mpri_asgmt_t ASGMT)
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.
6.2 Hard-Coded Numbers
======================
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'.
-- Constant: const mpz_t __mpria_z_zero
-- Constant: const mpz_t __mpria_z_pos_one
-- Constant: const mpz_t __mpria_z_neg_one
-- Constant: const mpz_t __mpria_z_pos_two
-- Constant: const mpz_t __mpria_z_neg_two
Collection of 'mpz_t' signed integers with self-explanatory names.
-- Constant: const mpq_t __mpria_q_zero
-- Constant: const mpq_t __mpria_q_pos_one
-- Constant: const mpq_t __mpria_q_neg_one
-- Constant: const mpq_t __mpria_q_pos_two
-- Constant: const mpq_t __mpria_q_neg_two
Collection of 'mpq_t' rational numbers with self-explanatory names.
-- Constant: const mpri_t __mpria_ri_zero
-- Constant: const mpri_t __mpria_ri_pos_one
-- Constant: const mpri_t __mpria_ri_neg_one
Collection of 'mpri_t' rational singleton intervals with
self-explanatory names.
7 Extra Number Functions
************************
7.1 Extra Rational Number Functions
===================================
-- Macro: MPRIA_MPQ_SET_ZERO (Q)
-- Macro: MPRIA_MPQ_SET_POS_ONE (Q)
-- Macro: MPRIA_MPQ_SET_NEG_ONE (Q)
-- Macro: MPRIA_MPQ_SET_NAN (Q)
-- Macro: MPRIA_MPQ_SET_POS_INF (Q)
-- Macro: MPRIA_MPQ_SET_NEG_INF (Q)
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).
-- Macro: MPRIA_MPQ_IS_ZERO (Q)
-- Macro: MPRIA_MPQ_IS_NONZERO (Q)
-- Macro: MPRIA_MPQ_IS_POSITIVE (Q)
-- Macro: MPRIA_MPQ_IS_NEGATIVE (Q)
-- Macro: MPRIA_MPQ_IS_STRICTLY_POSITIVE (Q)
-- Macro: MPRIA_MPQ_IS_STRICTLY_NEGATIVE (Q)
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).
-- Inline Function: int mpria_mpq_is_nan (const mpq_t OP)
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.(1)
-- Inline Function: int mpria_mpq_is_infinite (const mpq_t OP)
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;(2) they are commonly written +infinity and
-infinity, respectively.
-- Inline Function: int mpria_mpq_is_finite (const mpq_t OP)
Return '1' (read true) if the rational number OP is finite, '0'
(read false) if it is either infinite or Not-a-Number.
-- Inline Function: int mpria_mpq_sgn (const mpq_t OP)
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.
-- Function: int mpria_mpq_cmpabs (const mpq_t OP1, const mpq_t OP2)
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).
-- Inline Function: void mpria_mpq_min3 (mpq_t ROP, const mpq_t OP1,
const mpq_t OP2, const mpq_t OP3)
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).
---------- Footnotes ----------
(1) At the time of writing, GMP does not support NaN for 'mpq_t'
numbers.
(2) At the time of writing, GMP does not support infinities for
'mpq_t' numbers.
7.2 Extra Signed Integer Functions
==================================
-- Macro: MPRIA_MPZ_SET_ZERO (Z)
-- Macro: MPRIA_MPZ_SET_POS_ONE (Z)
-- Macro: MPRIA_MPZ_SET_NEG_ONE (Z)
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).
-- Macro: MPRIA_MPZ_IS_ZERO (Z)
-- Macro: MPRIA_MPZ_IS_NONZERO (Z)
-- Macro: MPRIA_MPZ_IS_POSITIVE (Z)
-- Macro: MPRIA_MPZ_IS_NEGATIVE (Z)
-- Macro: MPRIA_MPZ_IS_STRICTLY_POSITIVE (Z)
-- Macro: MPRIA_MPZ_IS_STRICTLY_NEGATIVE (Z)
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).
-- Inline Function: int mpria_mpz_sgn (const mpz_t OP)
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.
-- Inline Function: void mpria_mpz_minabs3 (mpz_t ROP, const mpz_t OP1,
const mpz_t OP2, const mpz_t OP3)
Set the value of ROP to the minimum of the triplet
{abs(OP1),abs(OP2),abs(OP3)}.
8 General Library Functions
***************************
8.1 Library Version Handling
============================
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.
-- Macro: MPRIA_VERSION_MAJOR
-- Macro: MPRIA_VERSION_MINOR
-- Macro: MPRIA_VERSION_MICRO
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).
-- Function: void mpria_libversion_get_numbers ( int * MAJOR, int *
MINOR, int * MICRO)
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.
-- Function: int mpria_libversion_check_numbers (int MAJOR, int MINOR,
int MICRO)
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:
* 0 if the two version triplets are not compatible
(incompatibility);
* 1 if they are compatible and exactly the same (strict or
strong compatibility);
* 2 if they are compatible but not exactly the same (weak
compatibility).
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).
-- Macro: int mpria_libversion_check (void)
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 ();
}
-- Macro: MPRIA_VERSION_EXTRA
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 "".
-- Macro: MPRIA_VERSION_NUMBER_PACK (MAJOR, MINOR, MICRO)
-- Macro: MPRIA_VERSION_STRING_PACK (MAJOR, MINOR, MICRO, STREXTRA)
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.
-- Macro: MPRIA_VERSION_NUMBER
-- Macro: MPRIA_VERSION_STRING
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.
-- Function: int mpria_libversion_get_number (void)
-- Function: const char * mpria_libversion_get_string (void)
Retrieve the non-authoritative version number and string,
respectively, of the MPRIA library against which the application is
currently linked.
-- Macro: const char * mpria_libversion
-- Macro: const char * mpria_version
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.
8.2 Miscellaneous Utilities
===========================
-- Macro: MPRIA_STRINGIFY (TOKEN)
Stringify TOKEN.
Appendix A References
*********************
Teething Note: This is clearly a *non-exhaustive* list (in
progress) of references.
* IEEE-1788, Interval Standard Working Group:
.
* IEEE-754, Standard for Binary Floating-Point Arithmetic:
.
Appendix B GNU General Public License
*************************************
Version 3, 29 June 2007
Copyright (C) 2007 Free Software Foundation, Inc.
Everyone is permitted to copy and distribute verbatim copies of this
license document, but changing it is not allowed.
Preamble
========
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The "System Libraries" of an executable work include anything,
other than the work as a whole, that (a) is included in the normal
form of packaging a Major Component, but which is not part of that
Major Component, and (b) serves only to enable use of the work with
that Major Component, or to implement a Standard Interface for
which an implementation is available to the public in source code
form. A "Major Component", in this context, means a major
essential component (kernel, window system, and so on) of the
specific operating system (if any) on which the executable work
runs, or a compiler used to produce the work, or an object code
interpreter used to run it.
The "Corresponding Source" for a work in object code form means all
the source code needed to generate, install, and (for an executable
work) run the object code and to modify the work, including scripts
to control those activities. However, it does not include the
work's System Libraries, or general-purpose tools or generally
available free programs which are used unmodified in performing
those activities but which are not part of the work. For example,
Corresponding Source includes interface definition files associated
with source files for the work, and the source code for shared
libraries and dynamically linked subprograms that the work is
specifically designed to require, such as by intimate data
communication or control flow between those subprograms and other
parts of the work.
The Corresponding Source need not include anything that users can
regenerate automatically from other parts of the Corresponding
Source.
The Corresponding Source for a work in source code form is that
same work.
2. Basic Permissions.
All rights granted under this License are granted for the term of
copyright on the Program, and are irrevocable provided the stated
conditions are met. This License explicitly affirms your unlimited
permission to run the unmodified Program. The output from running
a covered work is covered by this License only if the output, given
its content, constitutes a covered work. This License acknowledges
your rights of fair use or other equivalent, as provided by
copyright law.
You may make, run and propagate covered works that you do not
convey, without conditions so long as your license otherwise
remains in force. You may convey covered works to others for the
sole purpose of having them make modifications exclusively for you,
or provide you with facilities for running those works, provided
that you comply with the terms of this License in conveying all
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or running the covered works for you must do so exclusively on your
behalf, under your direction and control, on terms that prohibit
them from making any copies of your copyrighted material outside
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Conveying under any other circumstances is permitted solely under
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10 makes it unnecessary.
3. Protecting Users' Legal Rights From Anti-Circumvention Law.
No covered work shall be deemed part of an effective technological
measure under any applicable law fulfilling obligations under
article 11 of the WIPO copyright treaty adopted on 20 December
1996, or similar laws prohibiting or restricting circumvention of
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When you convey a covered work, you waive any legal power to forbid
circumvention of technological measures to the extent such
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with respect to the covered work, and you disclaim any intention to
limit operation or modification of the work as a means of
enforcing, against the work's users, your or third parties' legal
rights to forbid circumvention of technological measures.
4. Conveying Verbatim Copies.
You may convey verbatim copies of the Program's source code as you
receive it, in any medium, provided that you conspicuously and
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keep intact all notices stating that this License and any
non-permissive terms added in accord with section 7 apply to the
code; keep intact all notices of the absence of any warranty; and
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You may charge any price or no price for each copy that you convey,
and you may offer support or warranty protection for a fee.
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You may convey a work based on the Program, or the modifications to
produce it from the Program, in the form of source code under the
terms of section 4, provided that you also meet all of these
conditions:
a. The work must carry prominent notices stating that you
modified it, and giving a relevant date.
b. The work must carry prominent notices stating that it is
released under this License and any conditions added under
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section 4 to "keep intact all notices".
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License to anyone who comes into possession of a copy. This
License will therefore apply, along with any applicable
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its parts, regardless of how they are packaged. This License
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it does not invalidate such permission if you have separately
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d. If the work has interactive user interfaces, each must display
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A compilation of a covered work with other separate and independent
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6. Conveying Non-Source Forms.
You may convey a covered work in object code form under the terms
of sections 4 and 5, provided that you also convey the
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a. Convey the object code in, or embodied in, a physical product
(including a physical distribution medium), accompanied by the
Corresponding Source fixed on a durable physical medium
customarily used for software interchange.
b. Convey the object code in, or embodied in, a physical product
(including a physical distribution medium), accompanied by a
written offer, valid for at least three years and valid for as
long as you offer spare parts or customer support for that
product model, to give anyone who possesses the object code
either (1) a copy of the Corresponding Source for all the
software in the product that is covered by this License, on a
durable physical medium customarily used for software
interchange, for a price no more than your reasonable cost of
physically performing this conveying of source, or (2) access
to copy the Corresponding Source from a network server at no
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c. Convey individual copies of the object code with a copy of the
written offer to provide the Corresponding Source. This
alternative is allowed only occasionally and noncommercially,
and only if you received the object code with such an offer,
in accord with subsection 6b.
d. Convey the object code by offering access from a designated
place (gratis or for a charge), and offer equivalent access to
the Corresponding Source in the same way through the same
place at no further charge. You need not require recipients
to copy the Corresponding Source along with the object code.
If the place to copy the object code is a network server, the
Corresponding Source may be on a different server (operated by
you or a third party) that supports equivalent copying
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object code saying where to find the Corresponding Source.
Regardless of what server hosts the Corresponding Source, you
remain obligated to ensure that it is available for as long as
needed to satisfy these requirements.
e. Convey the object code using peer-to-peer transmission,
provided you inform other peers where the object code and
Corresponding Source of the work are being offered to the
general public at no charge under subsection 6d.
A separable portion of the object code, whose source code is
excluded from the Corresponding Source as a System Library, need
not be included in conveying the object code work.
A "User Product" is either (1) a "consumer product", which means
any tangible personal property which is normally used for personal,
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incorporation into a dwelling. In determining whether a product is
a consumer product, doubtful cases shall be resolved in favor of
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"normally used" refers to a typical or common use of that class of
product, regardless of the status of the particular user or of the
way in which the particular user actually uses, or expects or is
expected to use, the product. A product is a consumer product
regardless of whether the product has substantial commercial,
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"Installation Information" for a User Product means any methods,
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User Product from a modified version of its Corresponding Source.
The information must suffice to ensure that the continued
functioning of the modified object code is in no case prevented or
interfered with solely because modification has been made.
If you convey an object code work under this section in, or with,
or specifically for use in, a User Product, and the conveying
occurs as part of a transaction in which the right of possession
and use of the User Product is transferred to the recipient in
perpetuity or for a fixed term (regardless of how the transaction
is characterized), the Corresponding Source conveyed under this
section must be accompanied by the Installation Information. But
this requirement does not apply if neither you nor any third party
retains the ability to install modified object code on the User
Product (for example, the work has been installed in ROM).
The requirement to provide Installation Information does not
include a requirement to continue to provide support service,
warranty, or updates for a work that has been modified or installed
by the recipient, or for the User Product in which it has been
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modification itself materially and adversely affects the operation
of the network or violates the rules and protocols for
communication across the network.
Corresponding Source conveyed, and Installation Information
provided, in accord with this section must be in a format that is
publicly documented (and with an implementation available to the
public in source code form), and must require no special password
or key for unpacking, reading or copying.
7. Additional Terms.
"Additional permissions" are terms that supplement the terms of
this License by making exceptions from one or more of its
conditions. Additional permissions that are applicable to the
entire Program shall be treated as though they were included in
this License, to the extent that they are valid under applicable
law. If additional permissions apply only to part of the Program,
that part may be used separately under those permissions, but the
entire Program remains governed by this License without regard to
the additional permissions.
When you convey a copy of a covered work, you may at your option
remove any additional permissions from that copy, or from any part
of it. (Additional permissions may be written to require their own
removal in certain cases when you modify the work.) You may place
additional permissions on material, added by you to a covered work,
for which you have or can give appropriate copyright permission.
Notwithstanding any other provision of this License, for material
you add to a covered work, you may (if authorized by the copyright
holders of that material) supplement the terms of this License with
terms:
a. Disclaiming warranty or limiting liability differently from
the terms of sections 15 and 16 of this License; or
b. Requiring preservation of specified reasonable legal notices
or author attributions in that material or in the Appropriate
Legal Notices displayed by works containing it; or
c. Prohibiting misrepresentation of the origin of that material,
or requiring that modified versions of such material be marked
in reasonable ways as different from the original version; or
d. Limiting the use for publicity purposes of names of licensors
or authors of the material; or
e. Declining to grant rights under trademark law for use of some
trade names, trademarks, or service marks; or
f. Requiring indemnification of licensors and authors of that
material by anyone who conveys the material (or modified
versions of it) with contractual assumptions of liability to
the recipient, for any liability that these contractual
assumptions directly impose on those licensors and authors.
All other non-permissive additional terms are considered "further
restrictions" within the meaning of section 10. If the Program as
you received it, or any part of it, contains a notice stating that
it is governed by this License along with a term that is a further
restriction, you may remove that term. If a license document
contains a further restriction but permits relicensing or conveying
under this License, you may add to a covered work material governed
by the terms of that license document, provided that the further
restriction does not survive such relicensing or conveying.
If you add terms to a covered work in accord with this section, you
must place, in the relevant source files, a statement of the
additional terms that apply to those files, or a notice indicating
where to find the applicable terms.
Additional terms, permissive or non-permissive, may be stated in
the form of a separately written license, or stated as exceptions;
the above requirements apply either way.
8. Termination.
You may not propagate or modify a covered work except as expressly
provided under this License. Any attempt otherwise to propagate or
modify it is void, and will automatically terminate your rights
under this License (including any patent licenses granted under the
third paragraph of section 11).
However, if you cease all violation of this License, then your
license from a particular copyright holder is reinstated (a)
provisionally, unless and until the copyright holder explicitly and
finally terminates your license, and (b) permanently, if the
copyright holder fails to notify you of the violation by some
reasonable means prior to 60 days after the cessation.
Moreover, your license from a particular copyright holder is
reinstated permanently if the copyright holder notifies you of the
violation by some reasonable means, this is the first time you have
received notice of violation of this License (for any work) from
that copyright holder, and you cure the violation prior to 30 days
after your receipt of the notice.
Termination of your rights under this section does not terminate
the licenses of parties who have received copies or rights from you
under this License. If your rights have been terminated and not
permanently reinstated, you do not qualify to receive new licenses
for the same material under section 10.
9. Acceptance Not Required for Having Copies.
You are not required to accept this License in order to receive or
run a copy of the Program. Ancillary propagation of a covered work
occurring solely as a consequence of using peer-to-peer
transmission to receive a copy likewise does not require
acceptance. However, nothing other than this License grants you
permission to propagate or modify any covered work. These actions
infringe copyright if you do not accept this License. Therefore,
by modifying or propagating a covered work, you indicate your
acceptance of this License to do so.
10. Automatic Licensing of Downstream Recipients.
Each time you convey a covered work, the recipient automatically
receives a license from the original licensors, to run, modify and
propagate that work, subject to this License. You are not
responsible for enforcing compliance by third parties with this
License.
An "entity transaction" is a transaction transferring control of an
organization, or substantially all assets of one, or subdividing an
organization, or merging organizations. If propagation of a
covered work results from an entity transaction, each party to that
transaction who receives a copy of the work also receives whatever
licenses to the work the party's predecessor in interest had or
could give under the previous paragraph, plus a right to possession
of the Corresponding Source of the work from the predecessor in
interest, if the predecessor has it or can get it with reasonable
efforts.
You may not impose any further restrictions on the exercise of the
rights granted or affirmed under this License. For example, you
may not impose a license fee, royalty, or other charge for exercise
of rights granted under this License, and you may not initiate
litigation (including a cross-claim or counterclaim in a lawsuit)
alleging that any patent claim is infringed by making, using,
selling, offering for sale, or importing the Program or any portion
of it.
11. Patents.
A "contributor" is a copyright holder who authorizes use under this
License of the Program or a work on which the Program is based.
The work thus licensed is called the contributor's "contributor
version".
A contributor's "essential patent claims" are all patent claims
owned or controlled by the contributor, whether already acquired or
hereafter acquired, that would be infringed by some manner,
permitted by this License, of making, using, or selling its
contributor version, but do not include claims that would be
infringed only as a consequence of further modification of the
contributor version. For purposes of this definition, "control"
includes the right to grant patent sublicenses in a manner
consistent with the requirements of this License.
Each contributor grants you a non-exclusive, worldwide,
royalty-free patent license under the contributor's essential
patent claims, to make, use, sell, offer for sale, import and
otherwise run, modify and propagate the contents of its contributor
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In the following three paragraphs, a "patent license" is any
express agreement or commitment, however denominated, not to
enforce a patent (such as an express permission to practice a
patent or covenant not to sue for patent infringement). To "grant"
such a patent license to a party means to make such an agreement or
commitment not to enforce a patent against the party.
If you convey a covered work, knowingly relying on a patent
license, and the Corresponding Source of the work is not available
for anyone to copy, free of charge and under the terms of this
License, through a publicly available network server or other
readily accessible means, then you must either (1) cause the
Corresponding Source to be so available, or (2) arrange to deprive
yourself of the benefit of the patent license for this particular
work, or (3) arrange, in a manner consistent with the requirements
of this License, to extend the patent license to downstream
recipients. "Knowingly relying" means you have actual knowledge
that, but for the patent license, your conveying the covered work
in a country, or your recipient's use of the covered work in a
country, would infringe one or more identifiable patents in that
country that you have reason to believe are valid.
If, pursuant to or in connection with a single transaction or
arrangement, you convey, or propagate by procuring conveyance of, a
covered work, and grant a patent license to some of the parties
receiving the covered work authorizing them to use, propagate,
modify or convey a specific copy of the covered work, then the
patent license you grant is automatically extended to all
recipients of the covered work and works based on it.
A patent license is "discriminatory" if it does not include within
the scope of its coverage, prohibits the exercise of, or is
conditioned on the non-exercise of one or more of the rights that
are specifically granted under this License. You may not convey a
covered work if you are a party to an arrangement with a third
party that is in the business of distributing software, under which
you make payment to the third party based on the extent of your
activity of conveying the work, and under which the third party
grants, to any of the parties who would receive the covered work
from you, a discriminatory patent license (a) in connection with
copies of the covered work conveyed by you (or copies made from
those copies), or (b) primarily for and in connection with specific
products or compilations that contain the covered work, unless you
entered into that arrangement, or that patent license was granted,
prior to 28 March 2007.
Nothing in this License shall be construed as excluding or limiting
any implied license or other defenses to infringement that may
otherwise be available to you under applicable patent law.
12. No Surrender of Others' Freedom.
If conditions are imposed on you (whether by court order, agreement
or otherwise) that contradict the conditions of this License, they
do not excuse you from the conditions of this License. If you
cannot convey a covered work so as to satisfy simultaneously your
obligations under this License and any other pertinent obligations,
then as a consequence you may not convey it at all. For example,
if you agree to terms that obligate you to collect a royalty for
further conveying from those to whom you convey the Program, the
only way you could satisfy both those terms and this License would
be to refrain entirely from conveying the Program.
13. Use with the GNU Affero General Public License.
Notwithstanding any other provision of this License, you have
permission to link or combine any covered work with a work licensed
under version 3 of the GNU Affero General Public License into a
single combined work, and to convey the resulting work. The terms
of this License will continue to apply to the part which is the
covered work, but the special requirements of the GNU Affero
General Public License, section 13, concerning interaction through
a network will apply to the combination as such.
14. Revised Versions of this License.
The Free Software Foundation may publish revised and/or new
versions of the GNU General Public License from time to time. Such
new versions will be similar in spirit to the present version, but
may differ in detail to address new problems or concerns.
Each version is given a distinguishing version number. If the
Program specifies that a certain numbered version of the GNU
General Public License "or any later version" applies to it, you
have the option of following the terms and conditions either of
that numbered version or of any later version published by the Free
Software Foundation. If the Program does not specify a version
number of the GNU General Public License, you may choose any
version ever published by the Free Software Foundation.
If the Program specifies that a proxy can decide which future
versions of the GNU General Public License can be used, that
proxy's public statement of acceptance of a version permanently
authorizes you to choose that version for the Program.
Later license versions may give you additional or different
permissions. However, no additional obligations are imposed on any
author or copyright holder as a result of your choosing to follow a
later version.
15. Disclaimer of Warranty.
THERE IS NO WARRANTY FOR THE PROGRAM, TO THE EXTENT PERMITTED BY
APPLICABLE LAW. EXCEPT WHEN OTHERWISE STATED IN WRITING THE
COPYRIGHT HOLDERS AND/OR OTHER PARTIES PROVIDE THE PROGRAM "AS IS"
WITHOUT WARRANTY OF ANY KIND, EITHER EXPRESSED OR IMPLIED,
INCLUDING, BUT NOT LIMITED TO, THE IMPLIED WARRANTIES OF
MERCHANTABILITY AND FITNESS FOR A PARTICULAR PURPOSE. THE ENTIRE
RISK AS TO THE QUALITY AND PERFORMANCE OF THE PROGRAM IS WITH YOU.
SHOULD THE PROGRAM PROVE DEFECTIVE, YOU ASSUME THE COST OF ALL
NECESSARY SERVICING, REPAIR OR CORRECTION.
16. Limitation of Liability.
IN NO EVENT UNLESS REQUIRED BY APPLICABLE LAW OR AGREED TO IN
WRITING WILL ANY COPYRIGHT HOLDER, OR ANY OTHER PARTY WHO MODIFIES
AND/OR CONVEYS THE PROGRAM AS PERMITTED ABOVE, BE LIABLE TO YOU FOR
DAMAGES, INCLUDING ANY GENERAL, SPECIAL, INCIDENTAL OR
CONSEQUENTIAL DAMAGES ARISING OUT OF THE USE OR INABILITY TO USE
THE PROGRAM (INCLUDING BUT NOT LIMITED TO LOSS OF DATA OR DATA
BEING RENDERED INACCURATE OR LOSSES SUSTAINED BY YOU OR THIRD
PARTIES OR A FAILURE OF THE PROGRAM TO OPERATE WITH ANY OTHER
PROGRAMS), EVEN IF SUCH HOLDER OR OTHER PARTY HAS BEEN ADVISED OF
THE POSSIBILITY OF SUCH DAMAGES.
17. Interpretation of Sections 15 and 16.
If the disclaimer of warranty and limitation of liability provided
above cannot be given local legal effect according to their terms,
reviewing courts shall apply local law that most closely
approximates an absolute waiver of all civil liability in
connection with the Program, unless a warranty or assumption of
liability accompanies a copy of the Program in return for a fee.
END OF TERMS AND CONDITIONS
===========================
How to Apply These Terms to Your New Programs
=============================================
If you develop a new program, and you want it to be of the greatest
possible use to the public, the best way to achieve this is to make it
free software which everyone can redistribute and change under these
terms.
To do so, attach the following notices to the program. It is safest to
attach them to the start of each source file to most effectively state
the exclusion of warranty; and each file should have at least the
"copyright" line and a pointer to where the full notice is found.
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 .
Also add information on how to contact you by electronic and paper mail.
If the program does terminal interaction, make it output a short notice
like this when it starts in an interactive mode:
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 .
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 .
Appendix C GNU Free Documentation License
*****************************************
Version 1.3, 3 November 2008
Copyright (C) 2000, 2001, 2002, 2007, 2008 Free Software Foundation, Inc.
Everyone is permitted to copy and distribute verbatim copies
of this license document, but changing it is not allowed.
0. PREAMBLE
The purpose of this License is to make a manual, textbook, or other
functional and useful document "free" in the sense of freedom: to
assure everyone the effective freedom to copy and redistribute it,
with or without modifying it, either commercially or
noncommercially. Secondarily, this License preserves for the
author and publisher a way to get credit for their work, while not
being considered responsible for modifications made by others.
This License is a kind of "copyleft", which means that derivative
works of the document must themselves be free in the same sense.
It complements the GNU General Public License, which is a copyleft
license designed for free software.
We have designed this License in order to use it for manuals for
free software, because free software needs free documentation: a
free program should come with manuals providing the same freedoms
that the software does. But this License is not limited to
software manuals; it can be used for any textual work, regardless
of subject matter or whether it is published as a printed book. We
recommend this License principally for works whose purpose is
instruction or reference.
1. APPLICABILITY AND DEFINITIONS
This License applies to any manual or other work, in any medium,
that contains a notice placed by the copyright holder saying it can
be distributed under the terms of this License. Such a notice
grants a world-wide, royalty-free license, unlimited in duration,
to use that work under the conditions stated herein. The
"Document", below, refers to any such manual or work. Any member
of the public is a licensee, and is addressed as "you". You accept
the license if you copy, modify or distribute the work in a way
requiring permission under copyright law.
A "Modified Version" of the Document means any work containing the
Document or a portion of it, either copied verbatim, or with
modifications and/or translated into another language.
A "Secondary Section" is a named appendix or a front-matter section
of the Document that deals exclusively with the relationship of the
publishers or authors of the Document to the Document's overall
subject (or to related matters) and contains nothing that could
fall directly within that overall subject. (Thus, if the Document
is in part a textbook of mathematics, a Secondary Section may not
explain any mathematics.) The relationship could be a matter of
historical connection with the subject or with related matters, or
of legal, commercial, philosophical, ethical or political position
regarding them.
The "Invariant Sections" are certain Secondary Sections whose
titles are designated, as being those of Invariant Sections, in the
notice that says that the Document is released under this License.
If a section does not fit the above definition of Secondary then it
is not allowed to be designated as Invariant. The Document may
contain zero Invariant Sections. If the Document does not identify
any Invariant Sections then there are none.
The "Cover Texts" are certain short passages of text that are
listed, as Front-Cover Texts or Back-Cover Texts, in the notice
that says that the Document is released under this License. A
Front-Cover Text may be at most 5 words, and a Back-Cover Text may
be at most 25 words.
A "Transparent" copy of the Document means a machine-readable copy,
represented in a format whose specification is available to the
general public, that is suitable for revising the document
straightforwardly with generic text editors or (for images composed
of pixels) generic paint programs or (for drawings) some widely
available drawing editor, and that is suitable for input to text
formatters or for automatic translation to a variety of formats
suitable for input to text formatters. A copy made in an otherwise
Transparent file format whose markup, or absence of markup, has
been arranged to thwart or discourage subsequent modification by
readers is not Transparent. An image format is not Transparent if
used for any substantial amount of text. A copy that is not
"Transparent" is called "Opaque".
Examples of suitable formats for Transparent copies include plain
ASCII without markup, Texinfo input format, LaTeX input format,
SGML or XML using a publicly available DTD, and standard-conforming
simple HTML, PostScript or PDF designed for human modification.
Examples of transparent image formats include PNG, XCF and JPG.
Opaque formats include proprietary formats that can be read and
edited only by proprietary word processors, SGML or XML for which
the DTD and/or processing tools are not generally available, and
the machine-generated HTML, PostScript or PDF produced by some word
processors for output purposes only.
The "Title Page" means, for a printed book, the title page itself,
plus such following pages as are needed to hold, legibly, the
material this License requires to appear in the title page. For
works in formats which do not have any title page as such, "Title
Page" means the text near the most prominent appearance of the
work's title, preceding the beginning of the body of the text.
The "publisher" means any person or entity that distributes copies
of the Document to the public.
A section "Entitled XYZ" means a named subunit of the Document
whose title either is precisely XYZ or contains XYZ in parentheses
following text that translates XYZ in another language. (Here XYZ
stands for a specific section name mentioned below, such as
"Acknowledgements", "Dedications", "Endorsements", or "History".)
To "Preserve the Title" of such a section when you modify the
Document means that it remains a section "Entitled XYZ" according
to this definition.
The Document may include Warranty Disclaimers next to the notice
which states that this License applies to the Document. These
Warranty Disclaimers are considered to be included by reference in
this License, but only as regards disclaiming warranties: any other
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Appendix D Indices
******************
D.1 Concept Index
=================
* Menu:
* '#include': Headers and Libraries.
(line 325)
* +infinity: Extra Rational Number Functions.
(line 1030)
* -infinity: Extra Rational Number Functions.
(line 1030)
* '__GMP_CC': Autoconf. (line 596)
* '__GMP_CFLAGS': Autoconf. (line 596)
* About this manual: How to use this Manual.
(line 153)
* Anonymous FTP of latest version: Up-to-date Material.
(line 131)
* Arithmetic functions: Interval Arithmetic Functions.
(line 855)
* Assignment functions: Assignment Functions.
(line 688)
* Assignment mode: Nomenclature and Types.
(line 400)
* Assignments modes: Assignment Modes. (line 520)
* Autoconf: Autoconf. (line 556)
* Basic functions: Interval Basic Functions.
(line 793)
* Basics: MPRIA Basics. (line 319)
* Best convert: Nomenclature and Types.
(line 392)
* Bug reporting: Reporting Bugs. (line 269)
* Building MPRIA: Installing MPRIA. (line 165)
* Compacted version triplet: Library Version Handling.
(line 1181)
* Comparison functions: Interval Comparison Functions.
(line 768)
* Conditions for copying MPRIA: Copying. (line 68)
* 'configure.ac': Autoconf. (line 556)
* Configuring MPRIA: Installing MPRIA. (line 165)
* Constant numbers: Hard-Coded Numbers. (line 970)
* Contributing: Reporting Bugs. (line 304)
* Conversion functions: Interval Conversion Functions.
(line 752)
* Copying conditions: Copying. (line 68)
* Elementary functions: Interval Approximation of Elementary Functions.
(line 931)
* Extra number functions: Extra Number Functions.
(line 998)
* Extra rational number functions: Extra Rational Number Functions.
(line 1001)
* Extra signed integer functions: Extra Signed Integer Functions.
(line 1074)
* Finite number: Extra Rational Number Functions.
(line 1038)
* FTP of latest version: Up-to-date Material.
(line 131)
* General library functions: General Library Functions.
(line 1107)
* GNU Free Documentation License: GNU Free Documentation License.
(line 1932)
* GNU General Public License: GNU General Public License.
(line 1220)
* Hard-coded numbers: Hard-Coded Numbers. (line 970)
* Headers: Headers and Libraries.
(line 325)
* Homepage for MPRIA: Up-to-date Material.
(line 127)
* Include files: Headers and Libraries.
(line 325)
* infinity: Extra Rational Number Functions.
(line 1030)
* Initialisation functions: Initialisation Functions.
(line 670)
* Interface: Rational Interval Functions.
(line 667)
* Interval approximation of elementary functions: Interval Approximation of Elementary Functions.
(line 931)
* Interval arithmetic functions: Interval Arithmetic Functions.
(line 855)
* Interval assignment functions: Assignment Functions.
(line 688)
* Interval basic functions: Interval Basic Functions.
(line 793)
* Interval comparisons functions: Interval Comparison Functions.
(line 768)
* Interval conversion functions: Interval Conversion Functions.
(line 752)
* Interval initialisation functions: Initialisation Functions.
(line 670)
* Latest information about MPRIA: Up-to-date Material.
(line 127)
* Latest version of MPRIA: Up-to-date Material.
(line 131)
* 'libgmp': Headers and Libraries.
(line 331)
* 'libmpria': Headers and Libraries.
(line 331)
* Libraries: Headers and Libraries.
(line 325)
* Libraries <1>: Headers and Libraries.
(line 331)
* Library version handling: Library Version Handling.
(line 1110)
* Libtool: Headers and Libraries.
(line 337)
* License conditions: Copying. (line 68)
* Linking: Headers and Libraries.
(line 331)
* Low-level elementary functions: Low-Level Interval Elementary Functions.
(line 956)
* Low-level interface: Low-Level Rational Interval Functions.
(line 953)
* Low-level interval elementary functions: Low-Level Interval Elementary Functions.
(line 956)
* Low-level rational interval elementary functions: Low-Level Interval Elementary Functions.
(line 956)
* Low-level rational number functions: Extra Rational Number Functions.
(line 1001)
* Low-level signed integer functions: Extra Signed Integer Functions.
(line 1074)
* Mailing lists: Mailing Lists. (line 141)
* Major patch level: Library Version Handling.
(line 1119)
* Major version number: Library Version Handling.
(line 1119)
* 'Makefile.am': Autoconf. (line 556)
* Memory management: Memory Management. (line 529)
* Micro version number: Library Version Handling.
(line 1119)
* Minor version number: Library Version Handling.
(line 1119)
* Miscellaneous utilities: Miscellaneous Utilities.
(line 1203)
* 'mpria.h': Headers and Libraries.
(line 325)
* 'mpria.m4': Autoconf. (line 556)
* 'mpria_ax_prog_path_gmp_cc.m4': Autoconf. (line 556)
* NaN: Extra Rational Number Functions.
(line 1024)
* negative infinity: Extra Rational Number Functions.
(line 1030)
* Nomenclature: Nomenclature and Types.
(line 348)
* Not-a-Number: Extra Rational Number Functions.
(line 1024)
* Patches: Reporting Bugs. (line 304)
* positive infinity: Extra Rational Number Functions.
(line 1030)
* Precision: Nomenclature and Types.
(line 384)
* Primary distribution point: Up-to-date Material.
(line 131)
* Problems: Reporting Bugs. (line 269)
* Rational Interval: Nomenclature and Types.
(line 348)
* Rational interval approximation of elementary functions: Interval Approximation of Elementary Functions.
(line 931)
* Rational interval arithmetic functions: Interval Arithmetic Functions.
(line 855)
* Rational interval assignment functions: Assignment Functions.
(line 688)
* Rational Interval basic functions: Interval Basic Functions.
(line 793)
* Rational interval comparisons functions: Interval Comparison Functions.
(line 768)
* Rational interval conversion functions: Interval Conversion Functions.
(line 752)
* Rational interval initialisation functions: Initialisation Functions.
(line 670)
* Rational number functions: Extra Rational Number Functions.
(line 1001)
* Rational numbers: Nomenclature and Types.
(line 379)
* References: References. (line 1209)
* Reporting bugs: Reporting Bugs. (line 269)
* Revision number: Library Version Handling.
(line 1119)
* Signed integer functions: Extra Signed Integer Functions.
(line 1074)
* Stringified version triplet: Library Version Handling.
(line 1181)
* Surrounding mode: Nomenclature and Types.
(line 392)
* Types: Nomenclature and Types.
(line 348)
* Version number: Library Version Handling.
(line 1181)
* Version numbers: Library Version Handling.
(line 1110)
* Version string: Library Version Handling.
(line 1181)
* Version triplet: Library Version Handling.
(line 1119)
D.2 Type Index
==============
* Menu:
* 'mpq_t': Nomenclature and Types.
(line 379)
* 'mpri_asgmt_t': Nomenclature and Types.
(line 400)
* 'mpri_prec_t': Nomenclature and Types.
(line 384)
* 'mpri_srnd_t': Nomenclature and Types.
(line 392)
* 'mpri_t': Nomenclature and Types.
(line 348)
D.3 Variable Index
==================
* Menu:
* __mpria_q_neg_one: Hard-Coded Numbers. (line 984)
* __mpria_q_neg_two: Hard-Coded Numbers. (line 986)
* __mpria_q_pos_one: Hard-Coded Numbers. (line 983)
* __mpria_q_pos_two: Hard-Coded Numbers. (line 985)
* __mpria_q_zero: Hard-Coded Numbers. (line 982)
* __mpria_ri_neg_one: Hard-Coded Numbers. (line 991)
* __mpria_ri_pos_one: Hard-Coded Numbers. (line 990)
* __mpria_ri_zero: Hard-Coded Numbers. (line 989)
* __mpria_z_neg_one: Hard-Coded Numbers. (line 977)
* __mpria_z_neg_two: Hard-Coded Numbers. (line 979)
* __mpria_z_pos_one: Hard-Coded Numbers. (line 976)
* __mpria_z_pos_two: Hard-Coded Numbers. (line 978)
* __mpria_z_zero: Hard-Coded Numbers. (line 975)
* mpria_libversion: Library Version Handling.
(line 1192)
* mpria_version: Library Version Handling.
(line 1193)
* MPRIA_VERSION_EXTRA: Library Version Handling.
(line 1167)
* MPRIA_VERSION_MAJOR: Library Version Handling.
(line 1116)
* MPRIA_VERSION_MICRO: Library Version Handling.
(line 1118)
* MPRIA_VERSION_MINOR: Library Version Handling.
(line 1117)
* MPRIA_VERSION_NUMBER: Library Version Handling.
(line 1179)
* MPRIA_VERSION_STRING: Library Version Handling.
(line 1180)
* 'MPRI_ASGMT_LR': Assignment Modes. (line 522)
* 'MPRI_ASGMT_OL': Assignment Modes. (line 522)
* 'MPRI_ASGMT_OR': Assignment Modes. (line 522)
* 'MPRI_PREC_BITS_DOUBLE': Precision Handling and Surrounding Modes.
(line 503)
* 'MPRI_PREC_BITS_HALF': Precision Handling and Surrounding Modes.
(line 503)
* 'MPRI_PREC_BITS_NIL': Precision Handling and Surrounding Modes.
(line 503)
* 'MPRI_PREC_BITS_OCTUPLE': Precision Handling and Surrounding Modes.
(line 503)
* 'MPRI_PREC_BITS_QUADRUPLE': Precision Handling and Surrounding Modes.
(line 503)
* 'MPRI_PREC_BITS_SINGLE': Precision Handling and Surrounding Modes.
(line 503)
* 'MPRI_SRND_BCAA': Precision Handling and Surrounding Modes.
(line 513)
* 'MPRI_SRND_BCAL': Precision Handling and Surrounding Modes.
(line 513)
* 'MPRI_SRND_BCAR': Precision Handling and Surrounding Modes.
(line 513)
D.4 Function Index
==================
* Menu:
* mpria_libversion_check: Library Version Handling.
(line 1147)
* mpria_libversion_check_numbers: Library Version Handling.
(line 1130)
* mpria_libversion_get_number: Library Version Handling.
(line 1186)
* mpria_libversion_get_numbers: Library Version Handling.
(line 1124)
* mpria_libversion_get_string: Library Version Handling.
(line 1187)
* mpria_mpq_cmpabs: Extra Rational Number Functions.
(line 1049)
* mpria_mpq_is_finite: Extra Rational Number Functions.
(line 1037)
* mpria_mpq_is_infinite: Extra Rational Number Functions.
(line 1029)
* mpria_mpq_is_nan: Extra Rational Number Functions.
(line 1023)
* MPRIA_MPQ_IS_NEGATIVE: Extra Rational Number Functions.
(line 1015)
* MPRIA_MPQ_IS_NONZERO: Extra Rational Number Functions.
(line 1013)
* MPRIA_MPQ_IS_POSITIVE: Extra Rational Number Functions.
(line 1014)
* MPRIA_MPQ_IS_STRICTLY_NEGATIVE: Extra Rational Number Functions.
(line 1017)
* MPRIA_MPQ_IS_STRICTLY_POSITIVE: Extra Rational Number Functions.
(line 1016)
* MPRIA_MPQ_IS_ZERO: Extra Rational Number Functions.
(line 1012)
* mpria_mpq_min3: Extra Rational Number Functions.
(line 1057)
* MPRIA_MPQ_SET_NAN: Extra Rational Number Functions.
(line 1004)
* MPRIA_MPQ_SET_NEG_INF: Extra Rational Number Functions.
(line 1006)
* MPRIA_MPQ_SET_NEG_ONE: Extra Rational Number Functions.
(line 1003)
* MPRIA_MPQ_SET_POS_INF: Extra Rational Number Functions.
(line 1005)
* MPRIA_MPQ_SET_POS_ONE: Extra Rational Number Functions.
(line 1002)
* MPRIA_MPQ_SET_ZERO: Extra Rational Number Functions.
(line 1001)
* mpria_mpq_sgn: Extra Rational Number Functions.
(line 1041)
* MPRIA_MPZ_IS_NEGATIVE: Extra Signed Integer Functions.
(line 1084)
* MPRIA_MPZ_IS_NONZERO: Extra Signed Integer Functions.
(line 1082)
* MPRIA_MPZ_IS_POSITIVE: Extra Signed Integer Functions.
(line 1083)
* MPRIA_MPZ_IS_STRICTLY_NEGATIVE: Extra Signed Integer Functions.
(line 1086)
* MPRIA_MPZ_IS_STRICTLY_POSITIVE: Extra Signed Integer Functions.
(line 1085)
* MPRIA_MPZ_IS_ZERO: Extra Signed Integer Functions.
(line 1081)
* mpria_mpz_minabs3: Extra Signed Integer Functions.
(line 1099)
* MPRIA_MPZ_SET_NEG_ONE: Extra Signed Integer Functions.
(line 1076)
* MPRIA_MPZ_SET_POS_ONE: Extra Signed Integer Functions.
(line 1075)
* MPRIA_MPZ_SET_ZERO: Extra Signed Integer Functions.
(line 1074)
* mpria_mpz_sgn: Extra Signed Integer Functions.
(line 1092)
* MPRIA_STRINGIFY: Miscellaneous Utilities.
(line 1203)
* MPRIA_VERSION_NUMBER_PACK: Library Version Handling.
(line 1172)
* MPRIA_VERSION_STRING_PACK: Library Version Handling.
(line 1173)
* mpri_2exp_atan: Interval Approximation of Elementary Functions.
(line 944)
* mpri_abs: Interval Arithmetic Functions.
(line 891)
* mpri_add: Interval Arithmetic Functions.
(line 855)
* mpri_add_q: Interval Arithmetic Functions.
(line 857)
* mpri_atan: Interval Approximation of Elementary Functions.
(line 937)
* mpri_clear: Initialisation Functions.
(line 680)
* mpri_diam: Interval Basic Functions.
(line 808)
* mpri_diam_abs: Interval Basic Functions.
(line 796)
* mpri_diam_rel: Interval Basic Functions.
(line 801)
* mpri_div: Interval Arithmetic Functions.
(line 877)
* mpri_div_2exp: Interval Arithmetic Functions.
(line 924)
* mpri_div_q: Interval Arithmetic Functions.
(line 879)
* mpri_equal: Interval Comparison Functions.
(line 768)
* mpri_get_d: Interval Conversion Functions.
(line 755)
* mpri_get_left: Interval Basic Functions.
(line 828)
* mpri_get_q: Interval Conversion Functions.
(line 752)
* mpri_get_right: Interval Basic Functions.
(line 829)
* mpri_hasnot_zero: Interval Comparison Functions.
(line 785)
* mpri_has_zero: Interval Comparison Functions.
(line 781)
* mpri_init: Initialisation Functions.
(line 674)
* mpri_inv: Interval Arithmetic Functions.
(line 894)
* mpri_is_nonzero: Interval Comparison Functions.
(line 776)
* mpri_is_zero: Interval Comparison Functions.
(line 772)
* mpri_lepref: Interval Basic Functions.
(line 823)
* mpri_mag: Interval Basic Functions.
(line 814)
* mpri_mid: Interval Basic Functions.
(line 819)
* mpri_mig: Interval Basic Functions.
(line 813)
* mpri_mul: Interval Arithmetic Functions.
(line 869)
* mpri_mul_2exp: Interval Arithmetic Functions.
(line 920)
* mpri_mul_q: Interval Arithmetic Functions.
(line 871)
* mpri_neg: Interval Arithmetic Functions.
(line 888)
* mpri_q_div: Interval Arithmetic Functions.
(line 881)
* mpri_q_sub: Interval Arithmetic Functions.
(line 865)
* mpri_repref: Interval Basic Functions.
(line 824)
* mpri_rsqrt: Interval Arithmetic Functions.
(line 910)
* mpri_set: Assignment Functions.
(line 691)
* mpri_set_d: Assignment Functions.
(line 718)
* MPRI_SET_NAN: Assignment Functions.
(line 695)
* MPRI_SET_Q: Assignment Functions.
(line 699)
* mpri_set_q: Assignment Functions.
(line 716)
* mpri_set_qi_q: Assignment Functions.
(line 704)
* mpri_set_qi_z: Assignment Functions.
(line 702)
* mpri_set_qi_z_ASGMT: Low-Level Interval Elementary Functions.
(line 956)
* mpri_set_rsqrt_q: Assignment Functions.
(line 737)
* mpri_set_sqrt_q: Assignment Functions.
(line 728)
* MPRI_SET_ZERO: Assignment Functions.
(line 694)
* mpri_sqr: Interval Arithmetic Functions.
(line 898)
* mpri_sqrt: Interval Arithmetic Functions.
(line 901)
* mpri_sub: Interval Arithmetic Functions.
(line 861)
* mpri_sub_q: Interval Arithmetic Functions.
(line 863)
* mpri_swap: Assignment Functions.
(line 746)
* mpri_urandomm: Interval Basic Functions.
(line 836)