xref: /netbsd-src/external/lgpl3/mpc/dist/doc/mpc.info (revision 367b82799ab709709d3c3b541df56a2a14644d3e)

This is mpc.info, produced by makeinfo version 7.0 from mpc.texi.

This manual is for GNU MPC, a library for multiple precision complex
arithmetic, version 1.3.1 of December 2022.

   Copyright © 2002, 2003, 2004, 2005, 2006, 2007, 2008, 2009, 2010,
2011, 2012, 2013, 2016, 2018, 2020, 2022 INRIA

     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.  A copy of the license is
     included in the section entitled “GNU Free Documentation License.”
INFO-DIR-SECTION GNU Packages
START-INFO-DIR-ENTRY
* mpc: (mpc)Multiple Precision Complex Library.
END-INFO-DIR-ENTRY


File: mpc.info,  Node: Top,  Next: Copying,  Up: (dir)

GNU MPC
*******

This manual documents how to install and use the GNU Multiple Precision
Complex Library, version 1.3.1

* Menu:

* Copying::                     GNU MPC Copying Conditions (LGPL).
* Introduction to GNU MPC::     Brief introduction to GNU MPC.
* Installing GNU MPC::          How to configure and compile the GNU MPC library.
* Reporting Bugs::              How to usefully report bugs.
* GNU MPC Basics::              What every GNU MPC user should know.
* Complex Functions::           Functions for arithmetic on complex numbers.
* Ball Arithmetic::             Types and functions for complex balls.
* References::
* Concept Index::
* Function Index::
* Type Index::
* GNU Free Documentation License::


File: mpc.info,  Node: Copying,  Next: Introduction to GNU MPC,  Prev: Top,  Up: Top

GNU MPC Copying Conditions
**************************

GNU MPC is free software; you can redistribute it and/or modify it under
the terms of the GNU Lesser General Public License as published by the
Free Software Foundation; either version 3 of the License, or (at your
option) any later version.

   GNU MPC 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 Lesser
General Public License for more details.

   You should have received a copy of the GNU Lesser General Public
License along with this program.  If not, see
<http://www.gnu.org/licenses/>.


File: mpc.info,  Node: Introduction to GNU MPC,  Next: Installing GNU MPC,  Prev: Copying,  Up: Top

1 Introduction to GNU MPC
*************************

GNU MPC is a portable library written in C for arbitrary precision
arithmetic on complex numbers providing correct rounding.  It implements
a multiprecision equivalent of the C99 standard.  It builds upon the GNU
MP and the GNU MPFR libraries.

1.1 How to use this Manual
==========================

Everyone should read *note GNU MPC Basics::.  If you need to install the
library yourself, you need to read *note Installing GNU MPC::, too.

   The remainder of the manual can be used for later reference, although
it is probably a good idea to skim through it.


File: mpc.info,  Node: Installing GNU MPC,  Next: Reporting Bugs,  Prev: Introduction to GNU MPC,  Up: Top

2 Installing GNU MPC
********************

To build GNU MPC, you first have to install GNU MP (version 5.0.0 or
higher) and GNU MPFR (version 4.1.0 or higher) on your computer.  You
need a C compiler; GCC version 4.4 or higher is recommended, since GNU
MPC may trigger a bug in previous versions, see the thread at
<https://sympa.inria.fr/sympa/arc/mpc-discuss/2011-02/msg00024.html>.
And you need a standard Unix ‘make’ program, plus some other standard
Unix utility programs.

   Here are the steps needed to install the library on Unix systems:

  1. ‘tar xzf mpc-1.3.1.tar.gz’

  2. ‘cd mpc-1.3.1’

  3. ‘./configure’

     if GMP and GNU MPFR are installed into standard directories, that
     is, directories that are searched by default by the compiler and
     the linking tools.

     ‘./configure --with-gmp=<gmp_install_dir>’

     is used to indicate a different location where GMP is installed.
     Alternatively, you can specify directly GMP include and GMP lib
     directories with ‘./configure --with-gmp-lib=<gmp_lib_dir>
     --with-gmp-include=<gmp_include_dir>’.

     ‘./configure --with-mpfr=<mpfr_install_dir>’

     is used to indicate a different location where GNU MPFR is
     installed.  Alternatively, you can specify directly GNU MPFR
     include and GNU MPFR lib directories with ‘./configure
     --with-mpf-lib=<mpfr_lib_dir>
     --with-mpfr-include=<mpfr_include_dir>’.

     Another useful parameter is ‘--prefix’, which can be used to
     specify an alternative installation location instead of
     ‘/usr/local’; see ‘make install’ below.

     To enable checking for memory leaks using ‘valgrind’ during ‘make
     check’, add the parameter ‘--enable-valgrind-tests’.

     If for debugging purposes you wish to log calls to GNU MPC
     functions from within your code, add the parameter
     ‘--enable-logging’.  In your code, replace the inclusion of ‘mpc.h’
     by ‘mpc-log.h’ and link the executable dynamically.  Then all calls
     to functions with only complex arguments are printed to ‘stderr’ in
     the following form: First, the function name is given, followed by
     its type such as ‘c_cc’, meaning that the function has one complex
     result (one ‘c’ in front of the ‘_’), computed from two complex
     arguments (two ‘c’ after the ‘_’).  Then, the precisions of the
     real and the imaginary part of the first result is given, followed
     by the second one and so on.  Finally, for each argument, the
     precisions of its real and imaginary part are specified and the
     argument itself is printed in hexadecimal via the function
     ‘mpc_out_str’ (*note String and Stream Input and Output::).  The
     option requires a dynamic library, so it may not be combined with
     ‘--disable-shared’.

     Use ‘./configure --help’ for an exhaustive list of parameters.

  4. ‘make’

     This compiles GNU MPC in the working directory.

  5. ‘make check’

     This will make sure GNU MPC was built correctly.

     If you get error messages, please report them to
     ‘mpc-discuss@inria.fr’ (*Note Reporting Bugs::, for information on
     what to include in useful bug reports).

  6. ‘make install’

     This will copy the file ‘mpc.h’ to the directory
     ‘/usr/local/include’, the file ‘libmpc.a’ to the directory
     ‘/usr/local/lib’, and the file ‘mpc.info’ to the directory
     ‘/usr/local/share/info’ (or if you passed the ‘--prefix’ option to
     ‘configure’, using the prefix directory given as argument to
     ‘--prefix’ instead of ‘/usr/local’).  Note: you need write
     permissions on these directories.

2.1 Other ‘make’ Targets
========================

There are some other useful make targets:

   • ‘info’

     Create an info version of the manual, in ‘mpc.info’.

   • ‘pdf’

     Create a PDF version of the manual, in ‘doc/mpc.pdf’.

   • ‘dvi’

     Create a DVI version of the manual, in ‘doc/mpc.dvi’.

   • ‘ps’

     Create a Postscript version of the manual, in ‘doc/mpc.ps’.

   • ‘html’

     Create an HTML version of the manual, in several pages in the
     directory ‘doc/mpc.html’; if you want only one output HTML file,
     then type ‘makeinfo --html --no-split mpc.texi’ instead.

   • ‘clean’

     Delete all object files and archive files, but not the
     configuration files.

   • ‘distclean’

     Delete all files not included in the distribution.

   • ‘uninstall’

     Delete all files copied by ‘make install’.

2.2 Known Build Problems
========================

On AIX, if GMP was built with the 64-bit ABI, before building and
testing GNU MPC, it might be necessary to set the ‘OBJECT_MODE’
environment variable to 64 by, e.g.,

   ‘export OBJECT_MODE=64’

   This has been tested with the C compiler IBM XL C/C++ Enterprise
Edition V8.0 for AIX, version: 08.00.0000.0021, GMP 4.2.4 and GNU MPFR
2.4.1.

   Please report any other problems you encounter to
‘mpc-discuss@inria.fr’.  *Note Reporting Bugs::.


File: mpc.info,  Node: Reporting Bugs,  Next: GNU MPC Basics,  Prev: Installing GNU MPC,  Up: Top

3 Reporting Bugs
****************

If you think you have found a bug in the GNU MPC library, please
investigate and report it.  We have made this library available to you,
and it is not to ask too much from you, to ask you to report the bugs
that you find.

   There are a few things you should think about when you put your bug
report together.

   You have to send us a test case that makes it possible for us to
reproduce the bug.  Include instructions on how to run the test case.

   You also have to explain what is wrong; if you get a crash, or if the
results printed are incorrect and in that case, in what way.

   Please include compiler version information in your bug report.  This
can be extracted using ‘gcc -v’, or ‘cc -V’ on some machines.  Also,
include the output from ‘uname -a’.

   If your bug report is good, we will do our best to help you to get a
corrected version of the library; if the bug report is poor, we will not
do anything about it (aside of chiding you to send better bug reports).

   Send your bug report to: ‘mpc-discuss@inria.fr’.

   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.


File: mpc.info,  Node: GNU MPC Basics,  Next: Complex Functions,  Prev: Reporting Bugs,  Up: Top

4 GNU MPC Basics
****************

All declarations needed to use GNU MPC are collected in the include file
‘mpc.h’.  It is designed to work with both C and C++ compilers.  You
should include that file in any program using the GNU MPC library by
adding the line
        #include "mpc.h"

4.1 Nomenclature and Types
==========================

“Complex number” or “Complex” for short, is a pair of two arbitrary
precision floating-point numbers (for the real and imaginary parts).
The C data type for such objects is ‘mpc_t’.

The “Precision” is the number of bits used to represent the mantissa of
the real and imaginary parts; the corresponding C data type is
‘mpfr_prec_t’.  For more details on the allowed precision range, *note
(mpfr.info)Nomenclature and Types::.

The “rounding mode” specifies the way to round the result of a complex
operation, in case the exact result can not be represented exactly in
the destination mantissa; the corresponding C data type is ‘mpc_rnd_t’.
A complex rounding mode is a pair of two rounding modes: one for the
real part, one for the imaginary part.

4.2 Function Classes
====================

There is only one class of functions in the GNU MPC library, namely
functions for complex arithmetic.  The function names begin with ‘mpc_’.
The associated type is ‘mpc_t’.

4.3 GNU MPC Variable Conventions
================================

As a general rule, all GNU MPC functions expect output arguments before
input arguments.  This notation is based on an analogy with the
assignment operator.

   GNU MPC allows you to use the same variable for both input and output
in the same expression.  For example, the main function for
floating-point multiplication, ‘mpc_mul’, can be used like this:
‘mpc_mul (x, x, x, rnd_mode)’.  This computes the square of X with
rounding mode ‘rnd_mode’ and puts the result back in X.

   Before you can assign to an GNU MPC variable, you need to initialise
it by calling one of the special initialization functions.  When you are
done with a variable, you need to clear it out, using one of the
functions for that purpose.

   A variable should only be initialised once, or at least cleared out
between each initialization.  After a variable has been initialised, it
may be assigned to any number of times.

   For efficiency reasons, avoid to initialise and clear out a variable
in loops.  Instead, initialise it before entering the loop, and clear it
out after the loop has exited.

   You do not need to be concerned about allocating additional space for
GNU MPC variables, since each of its real and imaginary part has a
mantissa of fixed size.  Hence unless you change its precision, or clear
and reinitialise it, a complex variable will have the same allocated
space during all its life.

4.4 Rounding Modes
==================

A complex rounding mode is of the form ‘MPC_RNDxy’ where ‘x’ and ‘y’ are
one of ‘N’ (to nearest), ‘Z’ (towards zero), ‘U’ (towards plus
infinity), ‘D’ (towards minus infinity), ‘A’ (away from zero, that is,
towards plus or minus infinity depending on the sign of the number to be
rounded).  The first letter refers to the rounding mode for the real
part, and the second one for the imaginary part.  For example
‘MPC_RNDZU’ indicates to round the real part towards zero, and the
imaginary part towards plus infinity.

   The ‘round to nearest’ mode works as in the IEEE P754 standard: in
case the number to be rounded lies exactly in the middle of two
representable numbers, it is rounded to the one with the least
significant bit set to zero.  For example, the number 5, which is
represented by (101) in binary, is rounded to (100)=4 with a precision
of two bits, and not to (110)=6.

4.5 Return Value
================

Most GNU MPC functions have a return value of type ‘int’, which is used
to indicate the position of the rounded real and imaginary parts with
respect to the exact (infinite precision) values.  If this integer is
‘i’, the macros ‘MPC_INEX_RE(i)’ and ‘MPC_INEX_IM(i)’ give 0 if the
corresponding rounded value is exact, a negative value if the rounded
value is less than the exact one, and a positive value if it is greater
than the exact one.  Similarly, functions computing a result of type
‘mpfr_t’ return an integer that is 0, positive or negative depending on
whether the rounded value is the same, larger or smaller then the exact
result.

   Some functions, such as ‘mpc_sin_cos’, compute two complex results;
the macros ‘MPC_INEX1(i)’ and ‘MPC_INEX2(i)’, applied to the return
value ‘i’ of such a function, yield the exactness value corresponding to
the first or the second computed value, respectively.

4.6 Branch Cuts And Special Values
==================================

Some complex functions have branch cuts, across which the function is
discontinous.  In GNU MPC, the branch cuts chosen are the same as those
specified for the corresponding functions in the ISO C99 standard.

   Likewise, when evaluated at a point whose real or imaginary part is
either infinite or a NaN or a signed zero, a function returns the same
value as those specified for the corresponding function in the ISO C99
standard.


File: mpc.info,  Node: Complex Functions,  Next: Ball Arithmetic,  Prev: GNU MPC Basics,  Up: Top

5 Complex Functions
*******************

The complex functions expect arguments of type ‘mpc_t’.

   The GNU MPC floating-point functions have an interface that is
similar to the GNU MP integer functions.  The function prefix for
operations on complex numbers is ‘mpc_’.

   The precision of a computation is defined as follows: Compute the
requested operation exactly (with “infinite precision”), and round the
result to the destination variable precision with the given rounding
mode.

   The GNU MPC complex functions are intended to be a smooth extension
of the IEEE P754 arithmetic.  The results obtained on one computer
should not differ from the results obtained on a computer with a
different word size.

* Menu:

* Initializing Complex Numbers::
* Assigning Complex Numbers::
* Converting Complex Numbers::
* String and Stream Input and Output::
* Complex Comparison::
* Projection & Decomposing::
* Basic Arithmetic::
* Power Functions and Logarithm::
* Trigonometric Functions::
* Modular Functions::
* Miscellaneous Complex Functions::
* Advanced Functions::
* Internals::


File: mpc.info,  Node: Initializing Complex Numbers,  Next: Assigning Complex Numbers,  Up: Complex Functions

5.1 Initialization Functions
============================

An ‘mpc_t’ object must be initialised before storing the first value in
it.  The functions ‘mpc_init2’ and ‘mpc_init3’ are used for that
purpose.

 -- Function: void mpc_init2 (mpc_t Z, mpfr_prec_t PREC)
     Initialise Z to precision PREC bits and set its real and imaginary
     parts to NaN. Normally, a variable should be initialised once only
     or at least be cleared, using ‘mpc_clear’, between initializations.

 -- Function: void mpc_init3 (mpc_t Z, mpfr_prec_t PREC_R, mpfr_prec_t
          PREC_I)
     Initialise Z with the precision of its real part being PREC_R bits
     and the precision of its imaginary part being PREC_I bits, and set
     the real and imaginary parts to NaN.

 -- Function: void mpc_clear (mpc_t Z)
     Free the space occupied by Z.  Make sure to call this function for
     all ‘mpc_t’ variables when you are done with them.

   Here is an example on how to initialise complex variables:
     {
       mpc_t x, y;
       mpc_init2 (x, 256);		/* precision _exactly_ 256 bits */
       mpc_init3 (y, 100, 50);	/* 100/50 bits for the real/imaginary part */
       ...
       mpc_clear (x);
       mpc_clear (y);
     }

   The following function is useful for changing the precision during a
calculation.  A typical use would be for adjusting the precision
gradually in iterative algorithms like Newton-Raphson, making the
computation precision closely match the actual accurate part of the
numbers.

 -- Function: void mpc_set_prec (mpc_t X, mpfr_prec_t PREC)
     Reset the precision of X to be *exactly* PREC bits, and set its
     real/imaginary parts to NaN. The previous value stored in X is
     lost.  It is equivalent to a call to ‘mpc_clear(x)’ followed by a
     call to ‘mpc_init2(x, prec)’, but more efficient as no allocation
     is done in case the current allocated space for the mantissa of X
     is sufficient.

 -- Function: mpfr_prec_t mpc_get_prec (const mpc_t X)
     If the real and imaginary part of X have the same precision, it is
     returned, otherwise, 0 is returned.

 -- Function: void mpc_get_prec2 (mpfr_prec_t* PR, mpfr_prec_t* PI,
          const mpc_t X)
     Returns the precision of the real part of X via PR and of its
     imaginary part via PI.


File: mpc.info,  Node: Assigning Complex Numbers,  Next: Converting Complex Numbers,  Prev: Initializing Complex Numbers,  Up: Complex Functions

5.2 Assignment Functions
========================

These functions assign new values to already initialised complex numbers
(*note Initializing Complex Numbers::).  When using any functions with
‘intmax_t’ or ‘uintmax_t’ parameters, you must include ‘<stdint.h>’ or
‘<inttypes.h>’ _before_ ‘mpc.h’, to allow ‘mpc.h’ to define prototypes
for these functions.  Similarly, functions with parameters of type
‘complex’ or ‘long complex’ are defined only if ‘<complex.h>’ is
included _before_ ‘mpc.h’.  If you need assignment functions that are
not in the current API, you can define them using the ‘MPC_SET_X_Y’
macro (*note Advanced Functions::).

 -- Function: int mpc_set (mpc_t ROP, const mpc_t OP, mpc_rnd_t RND)
     Set the value of ROP from OP, rounded to the precision of ROP with
     the given rounding mode RND.

 -- Function: int mpc_set_ui (mpc_t ROP, unsigned long int OP, mpc_rnd_t
          RND)
 -- Function: int mpc_set_si (mpc_t ROP, long int OP, mpc_rnd_t RND)
 -- Function: int mpc_set_uj (mpc_t ROP, uintmax_t OP, mpc_rnd_t RND)
 -- Function: int mpc_set_sj (mpc_t ROP, intmax_t OP, mpc_rnd_t RND)
 -- Function: int mpc_set_d (mpc_t ROP, double OP, mpc_rnd_t RND)
 -- Function: int mpc_set_ld (mpc_t ROP, long double OP, mpc_rnd_t RND)
 -- Function: int mpc_set_dc (mpc_t ROP, double _Complex OP, mpc_rnd_t
          RND)
 -- Function: int mpc_set_ldc (mpc_t ROP, long double _Complex OP,
          mpc_rnd_t RND)
 -- Function: int mpc_set_z (mpc_t ROP, const mpz_t OP mpc_rnd_t RND)
 -- Function: int mpc_set_q (mpc_t ROP, const mpq_t OP mpc_rnd_t RND)
 -- Function: int mpc_set_f (mpc_t ROP, const mpf_t OP mpc_rnd_t RND)
 -- Function: int mpc_set_fr (mpc_t ROP, const mpfr_t OP, mpc_rnd_t RND)
     Set the value of ROP from OP, rounded to the precision of ROP with
     the given rounding mode RND.  The argument OP is interpreted as
     real, so the imaginary part of ROP is set to zero with a positive
     sign.  Please note that even a ‘long int’ may have to be rounded,
     if the destination precision is less than the machine word width.
     For ‘mpc_set_d’, be careful that the input number OP may not be
     exactly representable as a double-precision number (this happens
     for 0.1 for instance), in which case it is first rounded by the C
     compiler to a double-precision number, and then only to a complex
     number.

 -- Function: int mpc_set_ui_ui (mpc_t ROP, unsigned long int OP1,
          unsigned long int OP2, mpc_rnd_t RND)
 -- Function: int mpc_set_si_si (mpc_t ROP, long int OP1, long int OP2,
          mpc_rnd_t RND)
 -- Function: int mpc_set_uj_uj (mpc_t ROP, uintmax_t OP1, uintmax_t
          OP2, mpc_rnd_t RND)
 -- Function: int mpc_set_sj_sj (mpc_t ROP, intmax_t OP1, intmax_t OP2,
          mpc_rnd_t RND)
 -- Function: int mpc_set_d_d (mpc_t ROP, double OP1, double OP2,
          mpc_rnd_t RND)
 -- Function: int mpc_set_ld_ld (mpc_t ROP, long double OP1, long double
          OP2, mpc_rnd_t RND)
 -- Function: int mpc_set_z_z (mpc_t ROP, const mpz_t OP1, const mpz_t
          OP2, mpc_rnd_t RND)
 -- Function: int mpc_set_q_q (mpc_t ROP, const mpq_t OP1, const mpq_t
          OP2, mpc_rnd_t RND)
 -- Function: int mpc_set_f_f (mpc_t ROP, const mpf_t OP1, const mpf_t
          OP2, mpc_rnd_t RND)
 -- Function: int mpc_set_fr_fr (mpc_t ROP, const mpfr_t OP1, const
          mpfr_t OP2, mpc_rnd_t RND)
     Set the real part of ROP from OP1, and its imaginary part from OP2,
     according to the rounding mode RND.

     Beware that the behaviour of ‘mpc_set_fr_fr’ is undefined if OP1 or
     OP2 is a pointer to the real or imaginary part of ROP.  To exchange
     the real and the imaginary part of a complex number, either use
     ‘mpfr_swap (mpc_realref (rop), mpc_imagref (rop))’, which also
     exchanges the precisions of the two parts; or use a temporary
     variable.

   For functions assigning complex variables from strings or input
streams, *note String and Stream Input and Output::.

 -- Function: void mpc_set_nan (mpc_t ROP)
     Set ROP to Nan+i*NaN.

 -- Function: void mpc_swap (mpc_t OP1, mpc_t OP2)
     Swap the values of OP1 and OP2 efficiently.  Warning: The
     precisions are exchanged, too; in case these are different,
     ‘mpc_swap’ is thus not equivalent to three ‘mpc_set’ calls using a
     third auxiliary variable.


File: mpc.info,  Node: Converting Complex Numbers,  Next: String and Stream Input and Output,  Prev: Assigning Complex Numbers,  Up: Complex Functions

5.3 Conversion Functions
========================

The following functions are available only if ‘<complex.h>’ is included
_before_ ‘mpc.h’.

 -- Function: double _Complex mpc_get_dc (const mpc_t OP, mpc_rnd_t RND)
 -- Function: long double _Complex mpc_get_ldc (mpc_t OP, mpc_rnd_t RND)
     Convert OP to a C complex number, using the rounding mode RND.

   For functions converting complex variables to strings or stream
output, *note String and Stream Input and Output::.


File: mpc.info,  Node: String and Stream Input and Output,  Next: Complex Comparison,  Prev: Converting Complex Numbers,  Up: Complex Functions

5.4 String and Stream Input and Output
======================================

 -- Function: int mpc_strtoc (mpc_t ROP, const char *NPTR, char
          **ENDPTR, int BASE, mpc_rnd_t RND)
     Read a complex number from a string NPTR in base BASE, rounded to
     the precision of ROP with the given rounding mode RND.  The BASE
     must be either 0 or a number from 2 to 36 (otherwise the behaviour
     is undefined).  If NPTR starts with valid data, the result is
     stored in ROP, the usual inexact value is returned (*note Return
     Value: return-value.) and, if ENDPTR is not the null pointer,
     *ENDPTR points to the character just after the valid data.
     Otherwise, ROP is set to ‘NaN + i * NaN’, -1 is returned and, if
     ENDPTR is not the null pointer, the value of NPTR is stored in the
     location referenced by ENDPTR.

     The expected form of a complex number string is either a real
     number (an optional leading whitespace, an optional sign followed
     by a floating-point number), or a pair of real numbers in
     parentheses separated by whitespace.  If a real number is read, the
     missing imaginary part is set to +0.  The form of a floating-point
     number depends on the base and is described in the documentation of
     ‘mpfr_strtofr’ (*note (mpfr.info)Assignment Functions::).  For
     instance, ‘"3.1415926"’, ‘"(1.25e+7 +.17)"’, ‘"(@nan@ 2)"’ and
     ‘"(-0 -7)"’ are valid strings for BASE = 10.  If BASE = 0, then a
     prefix may be used to indicate the base in which the floating-point
     number is written.  Use prefix ’0b’ for binary numbers, prefix ’0x’
     for hexadecimal numbers, and no prefix for decimal numbers.  The
     real and imaginary part may then be written in different bases.
     For instance, ‘"(1.024e+3 +2.05e+3)"’ and ‘"(0b1p+10 +0x802)"’ are
     valid strings for ‘base’=0 and represent the same value.

 -- Function: int mpc_set_str (mpc_t ROP, const char *S, int BASE,
          mpc_rnd_t rnd)
     Set ROP to the value of the string S in base BASE, rounded to the
     precision of ROP with the given rounding mode RND.  See the
     documentation of ‘mpc_strtoc’ for a detailed description of the
     valid string formats.  Contrarily to ‘mpc_strtoc’, ‘mpc_set_str’
     requires the _whole_ string to represent a valid complex number
     (potentially followed by additional white space).  This function
     returns the usual inexact value (*note Return Value: return-value.)
     if the entire string up to the final null character is a valid
     number in base BASE; otherwise it returns −1, and ROP is set to
     NaN+i*NaN.

 -- Function: char * mpc_get_str (int B, size_t N, const mpc_t OP,
          mpc_rnd_t RND)
     Convert OP to a string containing its real and imaginary parts,
     separated by a space and enclosed in a pair of parentheses.  The
     numbers are written in base B (which may vary from 2 to 36) and
     rounded according to RND.  The number of significant digits, at
     least 2, is given by N.  It is also possible to let N be zero, in
     which case the number of digits is chosen large enough so that
     re-reading the printed value with the same precision, assuming both
     output and input use rounding to nearest, will recover the original
     value of OP.  Note that ‘mpc_get_str’ uses the decimal point of the
     current locale if available, and ‘.’ otherwise.

     The string is generated using the current memory allocation
     function (‘malloc’ by default, unless it has been modified using
     the custom memory allocation interface of ‘gmp’); once it is not
     needed any more, it should be freed by calling ‘mpc_free_str’.

 -- Function: void mpc_free_str (char *STR)
     Free the string STR, which needs to have been allocated by a call
     to ‘mpc_get_str’.

   The following two functions read numbers from input streams and write
them to output streams.  When using any of these functions, you need to
include ‘stdio.h’ _before_ ‘mpc.h’.

 -- Function: int mpc_inp_str (mpc_t ROP, FILE *STREAM, size_t *READ,
          int BASE, mpc_rnd_t RND)
     Input a string in base BASE in the same format as for ‘mpc_strtoc’
     from stdio stream STREAM, rounded according to RND, and put the
     read complex number into ROP.  If STREAM is the null pointer, ROP
     is read from ‘stdin’.  Return the usual inexact value; if an error
     occurs, set ROP to ‘NaN + i * NaN’ and return -1.  If READ is not
     the null pointer, it is set to the number of read characters.

     Unlike ‘mpc_strtoc’, the function ‘mpc_inp_str’ does not possess
     perfect knowledge of the string to transform and has to read it
     character by character, so it behaves slightly differently: It
     tries to read a string describing a complex number and processes
     this string through a call to ‘mpc_set_str’.  Precisely, after
     skipping optional whitespace, a minimal string is read according to
     the regular expression ‘mpfr | '(' \s* mpfr \s+ mpfr \s* ')'’,
     where ‘\s’ denotes a whitespace, and ‘mpfr’ is either a string
     containing neither whitespaces nor parentheses, or
     ‘nan(n-char-sequence)’ or ‘@nan@(n-char-sequence)’ (regardless of
     capitalisation) with ‘n-char-sequence’ a string of ascii letters,
     digits or ‘'_'’.

     For instance, upon input of ‘"nan(13 1)"’, the function
     ‘mpc_inp_str’ starts to recognise a value of NaN followed by an
     n-char-sequence indicated by the opening parenthesis; as soon as
     the space is reached, it becomes clear that the expression in
     parentheses is not an n-char-sequence, and the error flag -1 is
     returned after 6 characters have been consumed from the stream (the
     whitespace itself remaining in the stream).  The function
     ‘mpc_strtoc’, on the other hand, may track back when reaching the
     whitespace; it treats the string as the two successive complex
     numbers ‘NaN + i * 0’ and ‘13 + i’.  It is thus recommended to have
     a whitespace follow each floating point number to avoid this
     problem.

 -- Function: size_t mpc_out_str (FILE *STREAM, int BASE, size_t
          N_DIGITS, const mpc_t OP, mpc_rnd_t RND)
     Output OP on stdio stream STREAM in base BASE, rounded according to
     RND, in the same format as for ‘mpc_strtoc’ If STREAM is the null
     pointer, ROP is written to ‘stdout’.

     Return the number of characters written.


File: mpc.info,  Node: Complex Comparison,  Next: Projection & Decomposing,  Prev: String and Stream Input and Output,  Up: Complex Functions

5.5 Comparison Functions
========================

 -- Function: int mpc_cmp (const mpc_t OP1, const mpc_t OP2)
 -- Function: int mpc_cmp_si_si (const mpc_t OP1, long int OP2R, long
          int OP2I)
 -- Macro: int mpc_cmp_si (mpc_t OP1, long int OP2)

     Compare OP1 and OP2, where in the case of ‘mpc_cmp_si_si’, OP2 is
     taken to be OP2R + i OP2I.  The return value C can be decomposed
     into ‘x = MPC_INEX_RE(c)’ and ‘y = MPC_INEX_IM(c)’, such that X is
     positive if the real part of OP1 is greater than that of OP2, zero
     if both real parts are equal, and negative if the real part of OP1
     is less than that of OP2, and likewise for Y.  Both OP1 and OP2 are
     considered to their full own precision, which may differ.  It is
     not allowed that one of the operands has a NaN (Not-a-Number) part.

     The storage of the return value is such that equality can be simply
     checked with ‘mpc_cmp (op1, op2) == 0’.

 -- Function: int mpc_cmp_abs (const mpc_t OP1, const mpc_t OP2)

     Compare the absolute values of OP1 and OP2.  The return value is 0
     if both are the same (including infinity), positive if the absolute
     value of OP1 is greater than that of OP2, and negative if it is
     smaller.  If OP1 or OP2 has a real or imaginary part which is NaN,
     the function behaves like ‘mpfr_cmp’ on two real numbers of which
     at least one is NaN.


File: mpc.info,  Node: Projection & Decomposing,  Next: Basic Arithmetic,  Prev: Complex Comparison,  Up: Complex Functions

5.6 Projection and Decomposing Functions
========================================

 -- Function: int mpc_real (mpfr_t ROP, const mpc_t OP, mpfr_rnd_t RND)
     Set ROP to the value of the real part of OP rounded in the
     direction RND.

 -- Function: int mpc_imag (mpfr_t ROP, const mpc_t OP, mpfr_rnd_t RND)
     Set ROP to the value of the imaginary part of OP rounded in the
     direction RND.

 -- Macro: mpfr_t mpc_realref (mpc_t OP)
 -- Macro: mpfr_t mpc_imagref (mpc_t OP)
     Return a reference to the real part and imaginary part of OP,
     respectively.  The ‘mpfr’ functions can be used on the result of
     these macros (note that the ‘mpfr_t’ type is itself a pointer).

 -- Function: int mpc_arg (mpfr_t ROP, const mpc_t OP, mpfr_rnd_t RND)
     Set ROP to the argument of OP, with a branch cut along the negative
     real axis.

 -- Function: int mpc_proj (mpc_t ROP, const mpc_t OP, mpc_rnd_t RND)
     Compute a projection of OP onto the Riemann sphere.  Set ROP to OP
     rounded in the direction RND, except when at least one part of OP
     is infinite (even if the other part is a NaN) in which case the
     real part of ROP is set to plus infinity and its imaginary part to
     a signed zero with the same sign as the imaginary part of OP.


File: mpc.info,  Node: Basic Arithmetic,  Next: Power Functions and Logarithm,  Prev: Projection & Decomposing,  Up: Complex Functions

5.7 Basic Arithmetic Functions
==============================

All the following functions are designed in such a way that, when
working with real numbers instead of complex numbers, their complexity
should essentially be the same as with the GNU MPFR library, with only a
marginal overhead due to the GNU MPC layer.

   For functions taking as input an integer argument (for example
‘mpc_add_ui’), when this argument is zero, it is considered as an
unsigned (that is, exact in this context) zero, and we follow the MPFR
conventions: (0) + (+0) = +0, (0) - (+0) = -0, (0) - (+0) = -0, (0) -
(-0) = +0.  The same applies for functions taking an argument of type
‘mpfr_t’, such as ‘mpc_add_fr’, of which the imaginary part is
considered to be an exact, unsigned zero.

 -- Function: int mpc_add (mpc_t ROP, const mpc_t OP1, const mpc_t OP2,
          mpc_rnd_t RND)
 -- Function: int mpc_add_ui (mpc_t ROP, const mpc_t OP1, unsigned long
          int OP2, mpc_rnd_t RND)
 -- Function: int mpc_add_fr (mpc_t ROP, const mpc_t OP1, const mpfr_t
          OP2, mpc_rnd_t RND)
     Set ROP to OP1 + OP2 rounded according to RND.

 -- Function: int mpc_sub (mpc_t ROP, const mpc_t OP1, const mpc_t OP2,
          mpc_rnd_t RND)
 -- Function: int mpc_sub_fr (mpc_t ROP, const mpc_t OP1, const mpfr_t
          OP2, mpc_rnd_t RND)
 -- Function: int mpc_fr_sub (mpc_t ROP, const mpfr_t OP1, const mpc_t
          OP2, mpc_rnd_t RND)
 -- Function: int mpc_sub_ui (mpc_t ROP, const mpc_t OP1, unsigned long
          int OP2, mpc_rnd_t RND)
 -- Macro: int mpc_ui_sub (mpc_t ROP, unsigned long int OP1, const mpc_t
          OP2, mpc_rnd_t RND)
 -- Function: int mpc_ui_ui_sub (mpc_t ROP, unsigned long int RE1,
          unsigned long int IM1, mpc_t OP2, mpc_rnd_t RND)
     Set ROP to OP1 − OP2 rounded according to RND.  For
     ‘mpc_ui_ui_sub’, OP1 is RE1 + IM1.

 -- Function: int mpc_neg (mpc_t ROP, const mpc_t OP, mpc_rnd_t RND)
     Set ROP to −OP rounded according to RND.  Just changes the sign if
     ROP and OP are the same variable.

 -- Function: int mpc_sum (mpc_t ROP, const mpc_ptr* OP, unsigned long
          N, mpc_rnd_t RND)
     Set ROP to the sum of the elements in the array OP of length N,
     rounded according to RND.

 -- Function: int mpc_mul (mpc_t ROP, const mpc_t OP1, const mpc_t OP2,
          mpc_rnd_t RND)
 -- Function: int mpc_mul_ui (mpc_t ROP, const mpc_t OP1, unsigned long
          int OP2, mpc_rnd_t RND)
 -- Function: int mpc_mul_si (mpc_t ROP, const mpc_t OP1, long int OP2,
          mpc_rnd_t RND)
 -- Function: int mpc_mul_fr (mpc_t ROP, const mpc_t OP1, const mpfr_t
          OP2, mpc_rnd_t RND)
     Set ROP to OP1 times OP2 rounded according to RND.  Note: for
     ‘mpc_mul’, in case OP1 and OP2 have the same value, use ‘mpc_sqr’
     for better efficiency.

 -- Function: int mpc_mul_i (mpc_t ROP, const mpc_t OP, int SGN,
          mpc_rnd_t RND)
     Set ROP to OP times the imaginary unit i if SGN is non-negative,
     set ROP to OP times -i otherwise, in both cases rounded according
     to RND.

 -- Function: int mpc_sqr (mpc_t ROP, const mpc_t OP, mpc_rnd_t RND)
     Set ROP to the square of OP rounded according to RND.

 -- Function: int mpc_fma (mpc_t ROP, const mpc_t OP1, const mpc_t OP2,
          const mpc_t OP3, mpc_rnd_t RND)
     Set ROP to OP1*OP2+OP3, rounded according to RND, with only one
     final rounding.

 -- Function: int mpc_dot (mpc_t ROP, const mpc_ptr* OP1, mpc_ptr* OP2,
          unsigned long N, mpc_rnd_t RND)
     Set ROP to the dot product of the elements in the arrays OP1 and
     OP2, both of length N, rounded according to RND.

 -- Function: int mpc_div (mpc_t ROP, const mpc_t OP1, const mpc_t OP2,
          mpc_rnd_t RND)
 -- Function: int mpc_div_ui (mpc_t ROP, const mpc_t OP1, unsigned long
          int OP2, mpc_rnd_t RND)
 -- Function: int mpc_div_fr (mpc_t ROP, const mpc_t OP1, const mpfr_t
          OP2, mpc_rnd_t RND)
 -- Function: int mpc_ui_div (mpc_t ROP, unsigned long int OP1, const
          mpc_t OP2, mpc_rnd_t RND)
 -- Function: int mpc_fr_div (mpc_t ROP, const mpfr_t OP1, const mpc_t
          OP2, mpc_rnd_t RND)
     Set ROP to OP1/OP2 rounded according to RND.

 -- Function: int mpc_conj (mpc_t ROP, const mpc_t OP, mpc_rnd_t RND)
     Set ROP to the conjugate of OP rounded according to RND.  Just
     changes the sign of the imaginary part if ROP and OP are the same
     variable.

 -- Function: int mpc_abs (mpfr_t ROP, const mpc_t OP, mpfr_rnd_t RND)
     Set the floating-point number ROP to the absolute value of OP,
     rounded in the direction RND.

 -- Function: int mpc_norm (mpfr_t ROP, const mpc_t OP, mpfr_rnd_t RND)
     Set the floating-point number ROP to the norm of OP (i.e., the
     square of its absolute value), rounded in the direction RND.

 -- Function: int mpc_mul_2ui (mpc_t ROP, const mpc_t OP1, unsigned long
          int OP2, mpc_rnd_t RND)
 -- Function: int mpc_mul_2si (mpc_t ROP, const mpc_t OP1, long int OP2,
          mpc_rnd_t RND)
     Set ROP to OP1 times 2 raised to OP2 rounded according to RND.
     Just modifies the exponents of the real and imaginary parts by OP2
     when ROP and OP1 are identical.

 -- Function: int mpc_div_2ui (mpc_t ROP, const mpc_t OP1, unsigned long
          int OP2, mpc_rnd_t RND)
 -- Function: int mpc_div_2si (mpc_t ROP, const mpc_t OP1, long int OP2,
          mpc_rnd_t RND)
     Set ROP to OP1 divided by 2 raised to OP2 rounded according to RND.
     Just modifies the exponents of the real and imaginary parts by OP2
     when ROP and OP1 are identical.


File: mpc.info,  Node: Power Functions and Logarithm,  Next: Trigonometric Functions,  Prev: Basic Arithmetic,  Up: Complex Functions

5.8 Power Functions and Logarithm
=================================

 -- Function: int mpc_sqrt (mpc_t ROP, const mpc_t OP, mpc_rnd_t RND)
     Set ROP to the square root of OP rounded according to RND.  The
     returned value ROP has a non-negative real part, and if its real
     part is zero, a non-negative imaginary part.

 -- Function: int mpc_pow (mpc_t ROP, const mpc_t OP1, const mpc_t OP2,
          mpc_rnd_t RND)
 -- Function: int mpc_pow_d (mpc_t ROP, const mpc_t OP1, double OP2,
          mpc_rnd_t RND)
 -- Function: int mpc_pow_ld (mpc_t ROP, const mpc_t OP1, long double
          OP2, mpc_rnd_t RND)
 -- Function: int mpc_pow_si (mpc_t ROP, const mpc_t OP1, long OP2,
          mpc_rnd_t RND)
 -- Function: int mpc_pow_ui (mpc_t ROP, const mpc_t OP1, unsigned long
          OP2, mpc_rnd_t RND)
 -- Function: int mpc_pow_z (mpc_t ROP, const mpc_t OP1, const mpz_t
          OP2, mpc_rnd_t RND)
 -- Function: int mpc_pow_fr (mpc_t ROP, const mpc_t OP1, const mpfr_t
          OP2, mpc_rnd_t RND)
     Set ROP to OP1 raised to the power OP2, rounded according to RND.
     For ‘mpc_pow_d’, ‘mpc_pow_ld’, ‘mpc_pow_si’, ‘mpc_pow_ui’,
     ‘mpc_pow_z’ and ‘mpc_pow_fr’, the imaginary part of OP2 is
     considered as +0.  When both OP1 and OP2 are zero, the result has
     real part 1, and imaginary part 0, with sign being the opposite of
     that of OP2.

 -- Function: int mpc_exp (mpc_t ROP, const mpc_t OP, mpc_rnd_t RND)
     Set ROP to the exponential of OP, rounded according to RND with the
     precision of ROP.

 -- Function: int mpc_log (mpc_t ROP, const mpc_t OP, mpc_rnd_t RND)
 -- Function: int mpc_log10 (mpc_t ROP, const mpc_t OP, mpc_rnd_t RND)
     Set ROP to the natural and base-10 logarithm of OP respectively,
     rounded according to RND with the precision of ROP.  The principal
     branch is chosen, with the branch cut on the negative real axis, so
     that the imaginary part of the result lies in ]-Pi , Pi] and
     ]-Pi/log(10) , Pi/log(10)] respectively.

 -- Function: int mpc_rootofunity (mpc_t ROP, unsigned long int N,
          unsigned long int K, mpc_rnd_t RND)
     Set ROP to the standard primitive N-th root of unity raised to the
     power K, that is, exp (2 Pi i k / n), rounded according to RND with
     the precision of ROP.

 -- Function: int mpc_agm (mpc_t ROP, const mpc_t A, const mpc_t B,
          mpc_rnd_t RND)
     Set ROP to the arithmetic-geometric mean (AGM) of A and B, rounded
     according to RND with the precision of ROP.  Concerning the branch
     cut, the function is computed by homogeneity either as A AGM(1,b0)
     with b0=B/A if |A|>=|B|, or as B AGM(1,b0) with b0=A/B otherwise;
     then when b0 is real and negative, AGM(1,b0) is chosen to have
     positive imaginary part.


File: mpc.info,  Node: Trigonometric Functions,  Next: Modular Functions,  Prev: Power Functions and Logarithm,  Up: Complex Functions

5.9 Trigonometric Functions
===========================

 -- Function: int mpc_sin (mpc_t ROP, const mpc_t OP, mpc_rnd_t RND)
 -- Function: int mpc_cos (mpc_t ROP, const mpc_t OP, mpc_rnd_t RND)
 -- Function: int mpc_tan (mpc_t ROP, const mpc_t OP, mpc_rnd_t RND)
     Set ROP to the sine, cosine, tangent of OP, rounded according to
     RND with the precision of ROP.

 -- Function: int mpc_sin_cos (mpc_t ROP_SIN, mpc_t ROP_COS, const mpc_t
          OP, mpc_rnd_t RND_SIN, mpc_rnd_t RND_COS)
     Set ROP_SIN to the sine of OP, rounded according to RND_SIN with
     the precision of ROP_SIN, and ROP_COS to the cosine of OP, rounded
     according to RND_COS with the precision of ROP_COS.

 -- Function: int mpc_sinh (mpc_t ROP, const mpc_t OP, mpc_rnd_t RND)
 -- Function: int mpc_cosh (mpc_t ROP, const mpc_t OP, mpc_rnd_t RND)
 -- Function: int mpc_tanh (mpc_t ROP, const mpc_t OP, mpc_rnd_t RND)
     Set ROP to the hyperbolic sine, hyperbolic cosine, hyperbolic
     tangent of OP, rounded according to RND with the precision of ROP.

 -- Function: int mpc_asin (mpc_t ROP, const mpc_t OP, mpc_rnd_t RND)
 -- Function: int mpc_acos (mpc_t ROP, const mpc_t OP, mpc_rnd_t RND)
 -- Function: int mpc_atan (mpc_t ROP, const mpc_t OP, mpc_rnd_t RND)
     Set ROP to the inverse sine, inverse cosine, inverse tangent of OP,
     rounded according to RND with the precision of ROP.

 -- Function: int mpc_asinh (mpc_t ROP, const mpc_t OP, mpc_rnd_t RND)
 -- Function: int mpc_acosh (mpc_t ROP, const mpc_t OP, mpc_rnd_t RND)
 -- Function: int mpc_atanh (mpc_t ROP, const mpc_t OP, mpc_rnd_t RND)
     Set ROP to the inverse hyperbolic sine, inverse hyperbolic cosine,
     inverse hyperbolic tangent of OP, rounded according to RND with the
     precision of ROP.  The branch cut of ‘mpc_acosh’ is (-Inf, 1)


File: mpc.info,  Node: Modular Functions,  Next: Miscellaneous Complex Functions,  Prev: Trigonometric Functions,  Up: Complex Functions

5.10 Modular Functions
======================

The following function is experimental, not least because it depends on
the equally experimental ball arithmetic, see *note Ball Arithmetic::.
So its prototype may change in future releases, and it may be removed
altogether.

 -- Function: int mpc_eta_fund (mpc_t ROP, const mpc_t OP, mpc_rnd_t
          RND)
     Assuming that the argument OP lies in the fundamental domain for
     Sl_2(Z), that is, it has real part not below -1/2 and not above
     +1/2 and absolute value at least 1, return the value of the
     Dedekind eta-function in ROP.  For arguments outside the
     fundamental domain the function is expected to loop indefinitely.


File: mpc.info,  Node: Miscellaneous Complex Functions,  Next: Advanced Functions,  Prev: Modular Functions,  Up: Complex Functions

5.11 Miscellaneous Functions
============================

 -- Function: int mpc_urandom (mpc_t ROP, gmp_randstate_t STATE)
     Generate a uniformly distributed random complex in the unit square
     [0, 1] x [0, 1].  Return 0, unless an exponent in the real or
     imaginary part is not in the current exponent range, in which case
     that part is set to NaN and a zero value is returned.  The second
     argument is a ‘gmp_randstate_t’ structure which should be created
     using the GMP ‘rand_init’ function, see the GMP manual.

 -- Function: const char * mpc_get_version (void)
     Return the GNU MPC version, as a null-terminated string.

 -- Macro: MPC_VERSION
 -- Macro: MPC_VERSION_MAJOR
 -- Macro: MPC_VERSION_MINOR
 -- Macro: MPC_VERSION_PATCHLEVEL
 -- Macro: MPC_VERSION_STRING
     ‘MPC_VERSION’ is the version of GNU MPC as a preprocessing
     constant.  ‘MPC_VERSION_MAJOR’, ‘MPC_VERSION_MINOR’ and
     ‘MPC_VERSION_PATCHLEVEL’ are respectively the major, minor and
     patch level of GNU MPC version, as preprocessing constants.
     ‘MPC_VERSION_STRING’ is the version as a string constant, which can
     be compared to the result of ‘mpc_get_version’ to check at run time
     the header file and library used match:
          if (strcmp (mpc_get_version (), MPC_VERSION_STRING))
            fprintf (stderr, "Warning: header and library do not match\n");
     Note: Obtaining different strings is not necessarily an error, as
     in general, a program compiled with some old GNU MPC version can be
     dynamically linked with a newer GNU MPC library version (if allowed
     by the library versioning system).

 -- Macro: long MPC_VERSION_NUM (MAJOR, MINOR, PATCHLEVEL)
     Create an integer in the same format as used by ‘MPC_VERSION’ from
     the given MAJOR, MINOR and PATCHLEVEL.  Here is an example of how
     to check the GNU MPC version at compile time:
          #if (!defined(MPC_VERSION) || (MPC_VERSION<MPC_VERSION_NUM(2,1,0)))
          # error "Wrong GNU MPC version."
          #endif


File: mpc.info,  Node: Advanced Functions,  Next: Internals,  Prev: Miscellaneous Complex Functions,  Up: Complex Functions

5.12 Advanced Functions
=======================

 -- Macro: MPC_SET_X_Y (REAL_SUFFIX, IMAG_SUFFIX, ROP, REAL, IMAG, RND)
     The macro MPC_SET_X_Y is designed to serve as the body of an
     assignment function and cannot be used by itself.  The REAL_SUFFIX
     and IMAG_SUFFIX parameters are the types of the real and imaginary
     part, that is, the ‘x’ in the ‘mpfr_set_x’ function one would use
     to set the part; for the mpfr type, use ‘fr’.  REAL (respectively
     IMAG) is the value you want to assign to the real (resp.
     imaginary) part, its type must conform to REAL_SUFFIX (resp.
     IMAG_SUFFIX).  RND is the ‘mpc_rnd_t’ rounding mode.  The return
     value is the usual inexact value (*note Return Value:
     return-value.).

     For instance, you can define mpc_set_ui_fr as follows:
          int mpc_set_ui_fr (mpc_t rop, unsigned long int re, mpfr_t im, mpc_rnd_t rnd)
              MPC_SET_X_Y (ui, fr, rop, re, im, rnd);


File: mpc.info,  Node: Internals,  Prev: Advanced Functions,  Up: Complex Functions

5.13 Internals
==============

These macros and functions are mainly designed for the implementation of
GNU MPC, but may be useful for users too.  However, no upward
compatibility is guaranteed.  You need to include ‘mpc-impl.h’ to use
them.

   The macro ‘MPC_MAX_PREC(z)’ gives the maximum of the precisions of
the real and imaginary parts of a complex number.


File: mpc.info,  Node: Ball Arithmetic,  Next: References,  Prev: Complex Functions,  Up: Top

6 Ball Arithmetic
*****************

Since release 1.3.0, GNU MPC contains a simple and very limited
implementation of complex balls (or rather, circles).  This part is
experimental, its interface may vary and it may be removed completely in
future releases.

   A complex ball of the new type ‘mpcb_t’ is defined by a non-zero
centre c of the type ‘mpc_t’ and a relative radius r of the new type
‘mpcr_t’, and it represents all complex numbers z = c (1 + ϑ) with |ϑ| ≤
r, or equivalently the closed circle with centre c and radius r |c|.
The approach of using a relative error (or radius) instead of an
absolute one simplifies error analyses for multiplicative operations
(multiplication, division, square roots, and the AGM), at the expense of
making them more complicated for additive operations.  It has the major
drawback of not being able to represent balls centred at 0; in floating
point arithmetic, however, 0 is never reached by rounding, but only
through operations with exact result, which could be handled at a
higher, application level.  For more discussion on these issues, see the
file ‘algorithms.tex’.

6.1 Radius type and functions
=============================

The radius type is defined by
struct {
   int64_t mant;
   int64_t exp;
}
   with the usual trick in the GNU multiprecision libraries of defining
the main type ‘mpcr_t’ as a 1-dimensional array of this struct, and
variable and constant pointers ‘mpcr_ptr’ and ‘mpcr_srcptr’.  It can
contain the special values infinity or zero, or floating point numbers
encoded as m⋅2^{e} for a positive mantissa m and an arbitrary (usually
negative) exponent e.  Normalised finite radii use 31 bits for the
mantissa, that is, 2^{30}≤m≤2^{31} - 1.  The special values infinity and
0 are encoded through the sign of m, but should be tested for and set
using dedicated functions.

   Unless indicated otherwise, the following functions assume radius
arguments to be normalised, they return normalised results, and they
round their results up, not necessarily to the smallest representable
number, although reasonable effort is made to get a tight upper bound:
They only guarantee that their outputs are an upper bound on the true
results.  (There may be a trade-off between tightness of the result and
speed of computation.  For instance, when a 32-bit mantissa is
normalised, an even mantissa should be divided by 2, an odd mantissa
should be divided by 2 and 1 should be added, and then in both cases the
exponent must be increased by 1.  It might be more efficient to add 1
all the time instead of testing the last bit of the mantissa.)

 -- Function: int mpcr_inf_p (mpcr_srcptr R)
 -- Function: int mpcr_zero_p (mpcr_srcptr R)
     Test whether R is infinity or zero, respectively, and return a
     boolean.

 -- Function: int mpcr_lt_half_p (mpcr_srcptr R)
     Return ‘true’ if R<1/2, and ‘false’ otherwise.  (Everywhere in this
     document, ‘true’ means any non-zero value, and ‘false’ means zero.)

 -- Function: int mpcr_cmp (mpcr_srcptr R, mpcr_srcptr S)
     Return +1, 0 or -1 depending on whether R is larger than, equal to
     or less than S, with the natural total order on the compactified
     non-negative real axis letting 0 be smaller and letting infinity be
     larger than any finite real number.

 -- Function: void mpcr_set_inf (mpcr_ptr R)
 -- Function: void mpcr_set_zero (mpcr_ptr R)
 -- Function: void mpcr_set_one (mpcr_ptr R)
 -- Function: void mpcr_set (mpcr_ptr R, mpcr_srcptr S)
 -- Function: void mpcr_set_ui64_2si64 (mpcr_ptr R, uint64_t MANT,
          int64_t EXP)
     Set R to infinity, zero, 1, S or MANT⋅2^{EXP}, respectively.

 -- Function: void mpcr_max (mpcr_ptr R, mpcr_srcptr S, mpcr_srcptr T)
     Set R to the maximum of S and T.

 -- Function: int64_t mpcr_get_exp (mpcr_srcptr R)
     Assuming that R is neither infinity nor 0, return its exponent e
     when writing r = m⋅2^{e} with 1/2 ≤ m < 1.  (Notice that this is
     _not_ the same as the field ‘exp’ in the struct representing a
     radius, but that instead it is independent of the implementation.)
     Otherwise the behaviour is undefined.

 -- Function: void mpcr_out_str (FILE *F, mpcr_srcptr R)
     Output R on F, which may be ‘stdout’.  Caveat: This function so far
     serves mainly for debugging purposes, its behaviour will probably
     change in the future.

 -- Function: void mpcr_add (mpcr_ptr R, mpcr_srcptr S, mpcr_srcptr T)
 -- Function: void mpcr_sub (mpcr_ptr R, mpcr_srcptr S, mpcr_srcptr T)
 -- Function: void mpcr_mul (mpcr_ptr R, mpcr_srcptr S, mpcr_srcptr T)
 -- Function: void mpcr_div (mpcr_ptr R, mpcr_srcptr S, mpcr_srcptr T)
 -- Function: void mpcr_mul_2ui (mpcr_ptr R, mpcr_srcptr S, unsigned
          long int T)
 -- Function: void mpcr_div_2ui (mpcr_ptr R, mpcr_srcptr S, unsigned
          long int T)
 -- Function: void mpcr_sqr (mpcr_ptr R, mpcr_srcptr S)
 -- Function: void mpcr_sqrt (mpcr_ptr R, mpcr_srcptr S)
     Set R to the sum, difference, product or quotient of S and T, or to
     the product of S by 2^{T} or to the quotient of S by 2^{T}, or to
     the square or the square root of S.  If any of the arguments is
     infinity, or if a difference is negative, the result is infinity.

 -- Function: void mpcr_sub_rnd (mpcr_ptr R, mpcr_srcptr S, mpcr_srcptr
          T, mpfr_rnd_t RND)
     Set R to the difference of S and T, rounded into direction RND,
     which can be one of ‘MPFR_RNDU’ or ‘MPFR_RNDD’.  If one of the
     arguments is infinity or the difference is negative, the result is
     infinity.  Calling the function with ‘MPFR_RNDU’ is equivalent to
     calling ‘mpcr_sub’.

     This is one out of several functions taking a rounding parameter.
     Rounding down may be useful to obtain an upper bound when dividing
     by the result.

 -- Function: void mpcr_c_abs_rnd (mpcr_ptr R, mpc_srcptr Z, mpfr_rnd_t
          RND)
     Set R to the absolute value of the complex number Z, rounded in
     direction RND, which may be one of ‘MPFR_RNDU’ or ‘MPFR_RNDD’.

 -- Function: void mpcr_add_rounding_error (mpcr_ptr R, mpfr_prec_t P,
          mpfr_rnd_t RND)
     Set R to r + (1 + r) 2^{-p} if RND equals ‘MPFR_RNDN’, and to r +
     (1 + r) 2^{1-p} otherwise.  The idea is that if a (potentially not
     representable) centre of an ideal complex ball of radius R is
     rounded to a representable complex number at precision P, this
     shifts the centre by up to 1/2 ulp (for rounding to nearest) or 1
     ulp (for directed rounding of at least one of the real or imaginary
     parts), which increases the radius accordingly.  So this function
     is typically called internally at the end of each operation with
     complex balls to account for the error made by rounding the centre.

6.2 Ball type and functions
===========================

The ball type is defined by
typedef struct {
  mpc_t  c;
  mpcr_t r;
}
   or, more precisely, ‘mpcb_t’ is again a 1-dimensional array of such a
struct, and variable and constant pointer types are defined as
‘mpcb_ptr’ and ‘mpcb_srcptr’, respectively.  As usual, the components
should only be accessed through corresponding functions.

   To understand functions on balls, one needs to consider the balls
passed as arguments as sets of complex values, to which a mathematical
function is applied; the C function “rounds up” in the sense that it
returns a ball containing all possible values of the function in all the
possible input values.  Reasonable effort is made to return small balls,
but again there is no guarantee that the result is the smallest possible
one.  In the current implementation, the centre of a ball returned as a
value is obtained by applying the function to the centres of the balls
passed as arguments, and rounding.  While this is a natural approach, it
is not the only possible one; however, it also simplifies the error
analysis as already carried out for functions with regular complex
arguments.  Whenever the centre of a complex ball has a non-finite real
or imaginary part (positive or negative infinity or NaN) the radius is
set to infinity; this can be interpreted as the “useless ball”,
representing the whole complex plane, whatever the value of the centre
is.

   Unlike for variables of ‘mpc_t’ type, where the precision needs to be
set explicitly at initialisation, variables of type ‘mpcb_t’ handle
their precision dynamically.  Ball centres always have the same
precision for their real and their imaginary parts (again this is a
choice of the implementation; if they are of very different sizes, one
could theoretically reduce the precision of the part that is smaller in
absolute value, which is more strongly affected by the common error
coded in the radius).  When setting a complex ball from a value of a
different type, an additional precision parameter is passed, which
determines the precision of the centre.  Functions on complex balls set
the precision of their result depending on the input.  In the current
implementation, this is the minimum of the argument precisions, so if
all balls are initially set to the same precision, this is preserved
throughout the computations.  (Notice that the exponent of the radius
encodes roughly the number of correct binary digits of the ball centre;
so it would also make sense to reduce the precision if the radius
becomes larger.)

   The following functions on complex balls are currently available; the
eclectic collection is motivated by the desire to provide an
implementation of the arithmetic-geometric mean of complex numbers
through the use of ball arithmetic.  As for functions taking complex
arguments, there may be arbitrary overlaps between variables
representing arguments and results; for instance ‘mpcb_mul (z, z, z)’ is
an allowed way of replacing the ball Z by its square.

 -- Function: void mpcb_init (mpcb_ptr Z)
 -- Function: void mpcb_clear (mpcb_ptr Z)
     Initialise or free memory for Z; ‘mpcb_init’ must be called once
     before using a variable, and ‘mpcb_clear’ must be called once
     before stopping to use a variable.  Unlike its ‘mpc_t’ counterpart,
     ‘mpcb_init’ does not fix the precision of Z, but it sets its radius
     to infinity, so that Z represents the whole complex plane.

 -- Function: mpfr_prec_t mpcb_get_prec (mpcb_srcptr Z)
     Return the (common) precision of the real and the complex parts of
     the centre of Z.

 -- Function: void mpcb_set (mpcb_ptr Z, mpcb_srcptr Z1)
     Set Z to Z1, preserving the precision of the centre.

 -- Function: void mpcb_set_inf (mpcb_ptr Z)
     Set Z to the whole complex plane.  This is intended to be used much
     in the spirit of an assertion: When a precondition is not satisfied
     inside a function, it can set its result to this value, which will
     propagate through further computations.

 -- Function: void mpcb_set_c (mpcb_ptr Z, mpc_srcptr C, mpfr_prec_t
          PREC, unsigned long int ERR_RE, unsigned long int ERR_IM)
     Set Z to a ball with centre C at precision PREC.  If PREC is at
     least the maximum of the precisions of the real and the imaginary
     parts of C and ERR_RE and ERR_IM are 0, then the resulting ball is
     exact with radius zero.  Using a larger value for PREC makes sense
     if C is considered exact and a larger target precision for the
     result is desired, or some leeway for the working precision is to
     be taken into account.  If PREC is less than the precision of C,
     then usually some rounding error occurs when setting the centre,
     which is taken into account in the radius.

     If ERR_RE and ERR_IM are non-zero, the argument C is considered as
     an inexact complex number, with a bound on the absolute error of
     its real part given in ERR_RE as a multiple of 1/2 ulp of the real
     part of C, and a bound on the absolute error of its imaginary part
     given in ERR_IM as a multiple of 1/2 ulp of the imaginary part of
     C.  (Notice that if the parts of C have different precisions or
     exponents, the absolute values of their ulp differ.)  Then Z is
     created as a ball with centre C and a radius taking these errors on
     C as well as the potential additional rounding error for the centre
     into account.  If the real part of C is 0, then ERR_RE must be 0,
     since ulp of 0 makes no sense; otherwise the radius is set to
     infinity.  The same remark holds for the imaginary part.

     Using ERR_RE and ERR_IM different from 0 is particularly useful in
     two settings: If C is itself the result of a call to an ‘mpc_’
     function with exact input and rounding mode ‘MPC_RNDNN’ of both
     parts to nearest, then its parts are known with errors of at most
     1/2 ulp, and setting ERR_RE and ERR_IM to 1 yields a ball which is
     known to contain the exact result (this motivates the strange unit
     of 1/2 ulp); if directed rounding was used, ERR_RE and ERR_IM can
     be set to 2 instead.

     And if C is the result of a sequence of calls to ‘mpc_’ functions
     for which some error analysis has been carried out (as is
     frequently the case internally when implementing complex
     functions), again the resulting ball Z is known to contain the
     exact result when using appropriate values for ERR_RE and ERR_IM.

 -- Function: void mpcb_set_ui_ui (mpcb_ptr Z, unsigned long int RE,
          unsigned long int IM, mpfr_prec_t PREC)
     Set Z to a ball with centre RE+I*IM at precision PREC or the size
     of an ‘unsigned long int’, whatever is larger.

 -- Function: void mpcb_neg (mpcb_ptr Z, mpcb_srcptr Z1)
 -- Function: void mpcb_add (mpcb_ptr Z, mpcb_srcptr Z1, mpcb_srcptr Z2)
 -- Function: void mpcb_mul (mpcb_ptr Z, mpcb_srcptr Z1, mpcb_srcptr Z2)
 -- Function: void mpcb_sqr (mpcb_ptr Z, mpcb_srcptr Z1)
 -- Function: void mpcb_pow_ui (mpcb_ptr Z, mpcb_srcptr Z1, unsigned
          long int E)
 -- Function: void mpcb_sqrt (mpcb_ptr Z, mpcb_srcptr Z1)
 -- Function: void mpcb_div (mpcb_ptr Z, mpcb_srcptr Z1, mpcb_srcptr Z2)
 -- Function: void mpcb_div_2ui (mpcb_ptr Z, mpcb_srcptr Z1, unsigned
          long int E)
     These are the exact counterparts of the corresponding functions
     ‘mpc_neg’, ‘mpc_add’ and so on, but on complex balls instead of
     complex numbers.

 -- Function: int mpcb_can_round (mpcb_srcptr Z, mpfr_prec_t PREC_RE,
          mpfr_prec_t PREC_IM, mpc_rnd_t RND)
     If the function returns ‘true’ (a non-zero number), then rounding
     any of the complex numbers in the ball to a complex number with
     precision PREC_RE of its real and precision PREC_IM of its
     imaginary part and rounding mode RND yields the same result and
     rounding direction value, cf.  *note return-value::.  If the
     function returns ‘false’ (that is, 0), then it could not conclude,
     or there are two numbers in the ball which would be rounded to a
     different complex number or in a different direction.  Notice that
     the function works in a best effort mode and errs on the side of
     caution by potentially returning ‘false’ on a roundable ball; this
     is consistent with computational functions not necessarily
     returning the smallest enclosing ball.

     If Z contains the result of evaluating some mathematical function
     through a sequence of calls to ‘mpcb’ functions, starting with
     exact complex numbers, that is, balls of radius 0, then a return
     value of ‘true’ indicates that rounding any value in the ball (its
     centre is readily available) in direction RND yields the correct
     result of the function and the correct rounding direction value
     with the usual MPC semantics.

     Notice that when the precision of Z is larger than PREC_RE or
     PREC_IM, the centre need not be representable at the desired
     precision, and in fact the ball need not contain a representable
     number at all to be “roundable”.  Even worse, when RND is a
     directed rounding mode for the real or the imaginary part and the
     ball of non-zero radius contains a representable number, the return
     value is necessarily ‘false’.  Even worse, when the rounding mode
     for one part is to nearest, the corresponding part of the centre of
     the ball is representable and the ball has a non-zero radius, then
     the return value is also necessarily ‘false’, since even if
     rounding may be possible, the rounding direction value cannot be
     determined.

 -- Function: int mpcb_round (mpc_ptr C, mpcb_srcptr Z, mpc_rnd_t RND)
     Set C to the centre of Z, rounded in direction RND, and return the
     corresponding rounding direction value.  If ‘mpcb_can_round’,
     called with Z, the precisions of C and the rounding mode RND
     returns ‘true’, then this function does what is expected, it
     “correctly rounds the ball” and returns a rounding direction value
     that is valid for all of the ball.  As explained above, the result
     is then not necessarily (in the presence of directed rounding with
     radius different from 0, it is rather necessarily not) an element
     of the ball.


File: mpc.info,  Node: References,  Next: Concept Index,  Prev: Ball Arithmetic,  Up: Top

References
**********

   • Torbjörn Granlund et al.  ‘GMP’ – GNU multiprecision library.
     Version 6.2.0, <http://gmplib.org>.

   • Guillaume Hanrot, Vincent Lefèvre, Patrick Pélissier, Paul
     Zimmermann et al.  ‘MPFR’ – A library for multiple-precision
     floating-point computations with exact rounding.  Version 4.1.0,
     <http://www.mpfr.org>.

   • IEEE Standard for Floating-Point Arithmetic, IEEE Computer Society,
     IEEE Std 754-2019, Approved 13 June 2019, 84 pages.

   • Donald E. Knuth, "The Art of Computer Programming", vol 2,
     "Seminumerical Algorithms", 2nd edition, Addison-Wesley, 1981.

   • ISO/IEC 9899:1999, Programming languages — C.


File: mpc.info,  Node: Concept Index,  Next: Function Index,  Prev: References,  Up: Top

Concept Index
*************

�[index�]
* Menu:

* Arithmetic functions:                  Basic Arithmetic.     (line  6)
* Ball arithmetic:                       Ball Arithmetic.      (line  6)
* Comparison functions:                  Complex Comparison.   (line  6)
* Complex arithmetic functions:          Basic Arithmetic.     (line  6)
* Complex assignment functions:          Assigning Complex Numbers.
                                                               (line  6)
* Complex comparisons functions:         Complex Comparison.   (line  6)
* Complex functions:                     Complex Functions.    (line  6)
* Complex number:                        GNU MPC Basics.       (line 15)
* Conditions for copying GNU MPC:        Copying.              (line  6)
* Conversion functions:                  Converting Complex Numbers.
                                                               (line  6)
* Copying conditions:                    Copying.              (line  6)
* Installation:                          Installing GNU MPC.   (line  6)
* Logarithm:                             Power Functions and Logarithm.
                                                               (line  6)
* Miscellaneous complex functions:       Miscellaneous Complex Functions.
                                                               (line  6)
* Modular functions:                     Modular Functions.    (line  6)
* mpc.h:                                 GNU MPC Basics.       (line  6)
* Power functions:                       Power Functions and Logarithm.
                                                               (line  6)
* Precision:                             GNU MPC Basics.       (line 19)
* Projection and Decomposing Functions:  Projection & Decomposing.
                                                               (line  6)
* Reporting bugs:                        Reporting Bugs.       (line  6)
* Rounding Mode:                         GNU MPC Basics.       (line 24)
* String and stream input and output:    String and Stream Input and Output.
                                                               (line  6)
* Trigonometric functions:               Trigonometric Functions.
                                                               (line  6)
* User-defined precision:                Complex Functions.    (line 12)


File: mpc.info,  Node: Function Index,  Next: Type Index,  Prev: Concept Index,  Up: Top

Function Index
**************

�[index�]
* Menu:

* _Complex:                              Converting Complex Numbers.
                                                              (line   9)
* mpcb_add:                              Ball Arithmetic.     (line 260)
* mpcb_can_round:                        Ball Arithmetic.     (line 273)
* mpcb_clear:                            Ball Arithmetic.     (line 194)
* mpcb_div:                              Ball Arithmetic.     (line 266)
* mpcb_div_2ui:                          Ball Arithmetic.     (line 267)
* mpcb_get_prec:                         Ball Arithmetic.     (line 201)
* mpcb_init:                             Ball Arithmetic.     (line 193)
* mpcb_mul:                              Ball Arithmetic.     (line 261)
* mpcb_neg:                              Ball Arithmetic.     (line 259)
* mpcb_pow_ui:                           Ball Arithmetic.     (line 263)
* mpcb_round:                            Ball Arithmetic.     (line 309)
* mpcb_set:                              Ball Arithmetic.     (line 205)
* mpcb_set_c:                            Ball Arithmetic.     (line 214)
* mpcb_set_inf:                          Ball Arithmetic.     (line 208)
* mpcb_set_ui_ui:                        Ball Arithmetic.     (line 254)
* mpcb_sqr:                              Ball Arithmetic.     (line 262)
* mpcb_sqrt:                             Ball Arithmetic.     (line 265)
* mpcr_add:                              Ball Arithmetic.     (line  93)
* mpcr_add_rounding_error:               Ball Arithmetic.     (line 125)
* mpcr_cmp:                              Ball Arithmetic.     (line  64)
* mpcr_c_abs_rnd:                        Ball Arithmetic.     (line 120)
* mpcr_div:                              Ball Arithmetic.     (line  96)
* mpcr_div_2ui:                          Ball Arithmetic.     (line  99)
* mpcr_get_exp:                          Ball Arithmetic.     (line  81)
* mpcr_inf_p:                            Ball Arithmetic.     (line  55)
* mpcr_lt_half_p:                        Ball Arithmetic.     (line  60)
* mpcr_max:                              Ball Arithmetic.     (line  78)
* mpcr_mul:                              Ball Arithmetic.     (line  95)
* mpcr_mul_2ui:                          Ball Arithmetic.     (line  97)
* mpcr_out_str:                          Ball Arithmetic.     (line  88)
* mpcr_set:                              Ball Arithmetic.     (line  73)
* mpcr_set_inf:                          Ball Arithmetic.     (line  70)
* mpcr_set_one:                          Ball Arithmetic.     (line  72)
* mpcr_set_ui64_2si64:                   Ball Arithmetic.     (line  74)
* mpcr_set_zero:                         Ball Arithmetic.     (line  71)
* mpcr_sqr:                              Ball Arithmetic.     (line 101)
* mpcr_sqrt:                             Ball Arithmetic.     (line 102)
* mpcr_sub:                              Ball Arithmetic.     (line  94)
* mpcr_sub_rnd:                          Ball Arithmetic.     (line 108)
* mpcr_zero_p:                           Ball Arithmetic.     (line  56)
* mpc_abs:                               Basic Arithmetic.    (line  99)
* mpc_acos:                              Trigonometric Functions.
                                                              (line  25)
* mpc_acosh:                             Trigonometric Functions.
                                                              (line  31)
* mpc_add:                               Basic Arithmetic.    (line  19)
* mpc_add_fr:                            Basic Arithmetic.    (line  23)
* mpc_add_ui:                            Basic Arithmetic.    (line  21)
* mpc_agm:                               Power Functions and Logarithm.
                                                              (line  50)
* mpc_arg:                               Projection & Decomposing.
                                                              (line  20)
* mpc_asin:                              Trigonometric Functions.
                                                              (line  24)
* mpc_asinh:                             Trigonometric Functions.
                                                              (line  30)
* mpc_atan:                              Trigonometric Functions.
                                                              (line  26)
* mpc_atanh:                             Trigonometric Functions.
                                                              (line  32)
* mpc_clear:                             Initializing Complex Numbers.
                                                              (line  21)
* mpc_cmp:                               Complex Comparison.  (line   6)
* mpc_cmp_abs:                           Complex Comparison.  (line  23)
* mpc_cmp_si:                            Complex Comparison.  (line   9)
* mpc_cmp_si_si:                         Complex Comparison.  (line   7)
* mpc_conj:                              Basic Arithmetic.    (line  94)
* mpc_cos:                               Trigonometric Functions.
                                                              (line   7)
* mpc_cosh:                              Trigonometric Functions.
                                                              (line  19)
* mpc_div:                               Basic Arithmetic.    (line  82)
* mpc_div_2si:                           Basic Arithmetic.    (line 117)
* mpc_div_2ui:                           Basic Arithmetic.    (line 115)
* mpc_div_fr:                            Basic Arithmetic.    (line  86)
* mpc_div_ui:                            Basic Arithmetic.    (line  84)
* mpc_dot:                               Basic Arithmetic.    (line  77)
* mpc_eta_fund:                          Modular Functions.   (line  11)
* mpc_exp:                               Power Functions and Logarithm.
                                                              (line  32)
* mpc_fma:                               Basic Arithmetic.    (line  72)
* mpc_free_str:                          String and Stream Input and Output.
                                                              (line  66)
* mpc_fr_div:                            Basic Arithmetic.    (line  90)
* mpc_fr_sub:                            Basic Arithmetic.    (line  31)
* mpc_get_ldc:                           Converting Complex Numbers.
                                                              (line  10)
* mpc_get_prec:                          Initializing Complex Numbers.
                                                              (line  49)
* mpc_get_prec2:                         Initializing Complex Numbers.
                                                              (line  53)
* mpc_get_str:                           String and Stream Input and Output.
                                                              (line  48)
* mpc_get_version:                       Miscellaneous Complex Functions.
                                                              (line  14)
* mpc_imag:                              Projection & Decomposing.
                                                              (line  10)
* mpc_imagref:                           Projection & Decomposing.
                                                              (line  15)
* mpc_init2:                             Initializing Complex Numbers.
                                                              (line  10)
* mpc_init3:                             Initializing Complex Numbers.
                                                              (line  15)
* mpc_inp_str:                           String and Stream Input and Output.
                                                              (line  74)
* mpc_log:                               Power Functions and Logarithm.
                                                              (line  36)
* mpc_log10:                             Power Functions and Logarithm.
                                                              (line  37)
* mpc_mul:                               Basic Arithmetic.    (line  51)
* mpc_mul_2si:                           Basic Arithmetic.    (line 109)
* mpc_mul_2ui:                           Basic Arithmetic.    (line 107)
* mpc_mul_fr:                            Basic Arithmetic.    (line  57)
* mpc_mul_i:                             Basic Arithmetic.    (line  63)
* mpc_mul_si:                            Basic Arithmetic.    (line  55)
* mpc_mul_ui:                            Basic Arithmetic.    (line  53)
* mpc_neg:                               Basic Arithmetic.    (line  42)
* mpc_norm:                              Basic Arithmetic.    (line 103)
* mpc_out_str:                           String and Stream Input and Output.
                                                              (line 109)
* mpc_pow:                               Power Functions and Logarithm.
                                                              (line  11)
* mpc_pow_d:                             Power Functions and Logarithm.
                                                              (line  13)
* mpc_pow_fr:                            Power Functions and Logarithm.
                                                              (line  23)
* mpc_pow_ld:                            Power Functions and Logarithm.
                                                              (line  15)
* mpc_pow_si:                            Power Functions and Logarithm.
                                                              (line  17)
* mpc_pow_ui:                            Power Functions and Logarithm.
                                                              (line  19)
* mpc_pow_z:                             Power Functions and Logarithm.
                                                              (line  21)
* mpc_proj:                              Projection & Decomposing.
                                                              (line  24)
* mpc_real:                              Projection & Decomposing.
                                                              (line   6)
* mpc_realref:                           Projection & Decomposing.
                                                              (line  14)
* mpc_rootofunity:                       Power Functions and Logarithm.
                                                              (line  44)
* mpc_set:                               Assigning Complex Numbers.
                                                              (line  16)
* mpc_set_d:                             Assigning Complex Numbers.
                                                              (line  25)
* mpc_set_dc:                            Assigning Complex Numbers.
                                                              (line  27)
* mpc_set_d_d:                           Assigning Complex Numbers.
                                                              (line  54)
* mpc_set_f:                             Assigning Complex Numbers.
                                                              (line  33)
* mpc_set_fr:                            Assigning Complex Numbers.
                                                              (line  34)
* mpc_set_fr_fr:                         Assigning Complex Numbers.
                                                              (line  64)
* mpc_set_f_f:                           Assigning Complex Numbers.
                                                              (line  62)
* mpc_set_ld:                            Assigning Complex Numbers.
                                                              (line  26)
* mpc_set_ldc:                           Assigning Complex Numbers.
                                                              (line  29)
* mpc_set_ld_ld:                         Assigning Complex Numbers.
                                                              (line  56)
* mpc_set_nan:                           Assigning Complex Numbers.
                                                              (line  79)
* mpc_set_prec:                          Initializing Complex Numbers.
                                                              (line  41)
* mpc_set_q:                             Assigning Complex Numbers.
                                                              (line  32)
* mpc_set_q_q:                           Assigning Complex Numbers.
                                                              (line  60)
* mpc_set_si:                            Assigning Complex Numbers.
                                                              (line  22)
* mpc_set_si_si:                         Assigning Complex Numbers.
                                                              (line  48)
* mpc_set_sj:                            Assigning Complex Numbers.
                                                              (line  24)
* mpc_set_sj_sj:                         Assigning Complex Numbers.
                                                              (line  52)
* mpc_set_str:                           String and Stream Input and Output.
                                                              (line  35)
* mpc_set_ui:                            Assigning Complex Numbers.
                                                              (line  20)
* mpc_set_ui_ui:                         Assigning Complex Numbers.
                                                              (line  46)
* mpc_set_uj:                            Assigning Complex Numbers.
                                                              (line  23)
* mpc_set_uj_uj:                         Assigning Complex Numbers.
                                                              (line  50)
* MPC_SET_X_Y:                           Advanced Functions.  (line   6)
* mpc_set_z:                             Assigning Complex Numbers.
                                                              (line  31)
* mpc_set_z_z:                           Assigning Complex Numbers.
                                                              (line  58)
* mpc_sin:                               Trigonometric Functions.
                                                              (line   6)
* mpc_sinh:                              Trigonometric Functions.
                                                              (line  18)
* mpc_sin_cos:                           Trigonometric Functions.
                                                              (line  12)
* mpc_sqr:                               Basic Arithmetic.    (line  69)
* mpc_sqrt:                              Power Functions and Logarithm.
                                                              (line   6)
* mpc_strtoc:                            String and Stream Input and Output.
                                                              (line   6)
* mpc_sub:                               Basic Arithmetic.    (line  27)
* mpc_sub_fr:                            Basic Arithmetic.    (line  29)
* mpc_sub_ui:                            Basic Arithmetic.    (line  33)
* mpc_sum:                               Basic Arithmetic.    (line  46)
* mpc_swap:                              Assigning Complex Numbers.
                                                              (line  82)
* mpc_tan:                               Trigonometric Functions.
                                                              (line   8)
* mpc_tanh:                              Trigonometric Functions.
                                                              (line  20)
* mpc_ui_div:                            Basic Arithmetic.    (line  88)
* mpc_ui_sub:                            Basic Arithmetic.    (line  35)
* mpc_ui_ui_sub:                         Basic Arithmetic.    (line  37)
* mpc_urandom:                           Miscellaneous Complex Functions.
                                                              (line   6)
* MPC_VERSION:                           Miscellaneous Complex Functions.
                                                              (line  17)
* MPC_VERSION_MAJOR:                     Miscellaneous Complex Functions.
                                                              (line  18)
* MPC_VERSION_MINOR:                     Miscellaneous Complex Functions.
                                                              (line  19)
* MPC_VERSION_NUM:                       Miscellaneous Complex Functions.
                                                              (line  36)
* MPC_VERSION_PATCHLEVEL:                Miscellaneous Complex Functions.
                                                              (line  20)
* MPC_VERSION_STRING:                    Miscellaneous Complex Functions.
                                                              (line  21)


File: mpc.info,  Node: Type Index,  Next: GNU Free Documentation License,  Prev: Function Index,  Up: Top

Type Index
**********

�[index�]
* Menu:

* mpcb_ptr:                              Ball Arithmetic.     (line 140)
* mpcb_srcptr:                           Ball Arithmetic.     (line 140)
* mpcb_t:                                Ball Arithmetic.     (line  11)
* mpcb_t <1>:                            Ball Arithmetic.     (line 140)
* mpcr_ptr:                              Ball Arithmetic.     (line  28)
* mpcr_srcptr:                           Ball Arithmetic.     (line  28)
* mpcr_t:                                Ball Arithmetic.     (line  28)
* mpc_ptr:                               GNU MPC Basics.      (line  15)
* mpc_rnd_t:                             GNU MPC Basics.      (line  24)
* mpc_srcptr:                            GNU MPC Basics.      (line  15)
* mpc_t:                                 GNU MPC Basics.      (line  15)
* mpfr_prec_t:                           GNU MPC Basics.      (line  19)


File: mpc.info,  Node: GNU Free Documentation License,  Prev: Type Index,  Up: Top

Appendix A GNU Free Documentation License
*****************************************

                     Version 1.3, 3 November 2008

     Copyright © 2000, 2001, 2002, 2007, 2008 Free Software Foundation, Inc.
     <http://fsf.org/>

     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
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     A “Secondary Section” is a named appendix or a front-matter section
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     The “Invariant Sections” are certain Secondary Sections whose
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     The “Cover Texts” are certain short passages of text that are
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     A “Transparent” copy of the Document means a machine-readable copy,
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     The “publisher” means any person or entity that distributes copies
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     A section “Entitled XYZ” means a named subunit of the Document
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     The Document may include Warranty Disclaimers next to the notice
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  2. VERBATIM COPYING

     You may copy and distribute the Document in any medium, either
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  3. COPYING IN QUANTITY

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     If the required texts for either cover are too voluminous to fit
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     If you publish or distribute Opaque copies of the Document
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     It is requested, but not required, that you contact the authors of
     the Document well before redistributing any large number of copies,
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  4. MODIFICATIONS

     You may copy and distribute a Modified Version of the Document
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     release the Modified Version under precisely this License, with the
     Modified Version filling the role of the Document, thus licensing
     distribution and modification of the Modified Version to whoever
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     the Modified Version:

       A. Use in the Title Page (and on the covers, if any) a title
          distinct from that of the Document, and from those of previous
          versions (which should, if there were any, be listed in the
          History section of the Document).  You may use the same title
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       B. List on the Title Page, as authors, one or more persons or
          entities responsible for authorship of the modifications in
          the Modified Version, together with at least five of the
          principal authors of the Document (all of its principal
          authors, if it has fewer than five), unless they release you
          from this requirement.

       C. State on the Title page the name of the publisher of the
          Modified Version, as the publisher.

       D. Preserve all the copyright notices of the Document.

       E. Add an appropriate copyright notice for your modifications
          adjacent to the other copyright notices.

       F. Include, immediately after the copyright notices, a license
          notice giving the public permission to use the Modified
          Version under the terms of this License, in the form shown in
          the Addendum below.

       G. Preserve in that license notice the full lists of Invariant
          Sections and required Cover Texts given in the Document’s
          license notice.

       H. Include an unaltered copy of this License.

       I. Preserve the section Entitled “History”, Preserve its Title,
          and add to it an item stating at least the title, year, new
          authors, and publisher of the Modified Version as given on the
          Title Page.  If there is no section Entitled “History” in the
          Document, create one stating the title, year, authors, and
          publisher of the Document as given on its Title Page, then add
          an item describing the Modified Version as stated in the
          previous sentence.

       J. Preserve the network location, if any, given in the Document
          for public access to a Transparent copy of the Document, and
          likewise the network locations given in the Document for
          previous versions it was based on.  These may be placed in the
          “History” section.  You may omit a network location for a work
          that was published at least four years before the Document
          itself, or if the original publisher of the version it refers
          to gives permission.

       K. For any section Entitled “Acknowledgements” or “Dedications”,
          Preserve the Title of the section, and preserve in the section
          all the substance and tone of each of the contributor
          acknowledgements and/or dedications given therein.

       L. Preserve all the Invariant Sections of the Document, unaltered
          in their text and in their titles.  Section numbers or the
          equivalent are not considered part of the section titles.

       M. Delete any section Entitled “Endorsements”.  Such a section
          may not be included in the Modified Version.

       N. Do not retitle any existing section to be Entitled
          “Endorsements” or to conflict in title with any Invariant
          Section.

       O. Preserve any Warranty Disclaimers.

     If the Modified Version includes new front-matter sections or
     appendices that qualify as Secondary Sections and contain no
     material copied from the Document, you may at your option designate
     some or all of these sections as invariant.  To do this, add their
     titles to the list of Invariant Sections in the Modified Version’s
     license notice.  These titles must be distinct from any other
     section titles.

     You may add a section Entitled “Endorsements”, provided it contains
     nothing but endorsements of your Modified Version by various
     parties—for example, statements of peer review or that the text has
     been approved by an organization as the authoritative definition of
     a standard.

     You may add a passage of up to five words as a Front-Cover Text,
     and a passage of up to 25 words as a Back-Cover Text, to the end of
     the list of Cover Texts in the Modified Version.  Only one passage
     of Front-Cover Text and one of Back-Cover Text may be added by (or
     through arrangements made by) any one entity.  If the Document
     already includes a cover text for the same cover, previously added
     by you or by arrangement made by the same entity you are acting on
     behalf of, you may not add another; but you may replace the old
     one, on explicit permission from the previous publisher that added
     the old one.

     The author(s) and publisher(s) of the Document do not by this
     License give permission to use their names for publicity for or to
     assert or imply endorsement of any Modified Version.

  5. COMBINING DOCUMENTS

     You may combine the Document with other documents released under
     this License, under the terms defined in section 4 above for
     modified versions, provided that you include in the combination all
     of the Invariant Sections of all of the original documents,
     unmodified, and list them all as Invariant Sections of your
     combined work in its license notice, and that you preserve all
     their Warranty Disclaimers.

     The combined work need only contain one copy of this License, and
     multiple identical Invariant Sections may be replaced with a single
     copy.  If there are multiple Invariant Sections with the same name
     but different contents, make the title of each such section unique
     by adding at the end of it, in parentheses, the name of the
     original author or publisher of that section if known, or else a
     unique number.  Make the same adjustment to the section titles in
     the list of Invariant Sections in the license notice of the
     combined work.

     In the combination, you must combine any sections Entitled
     “History” in the various original documents, forming one section
     Entitled “History”; likewise combine any sections Entitled
     “Acknowledgements”, and any sections Entitled “Dedications”.  You
     must delete all sections Entitled “Endorsements.”

  6. COLLECTIONS OF DOCUMENTS

     You may make a collection consisting of the Document and other
     documents released under this License, and replace the individual
     copies of this License in the various documents with a single copy
     that is included in the collection, provided that you follow the
     rules of this License for verbatim copying of each of the documents
     in all other respects.

     You may extract a single document from such a collection, and
     distribute it individually under this License, provided you insert
     a copy of this License into the extracted document, and follow this
     License in all other respects regarding verbatim copying of that
     document.

  7. AGGREGATION WITH INDEPENDENT WORKS

     A compilation of the Document or its derivatives with other
     separate and independent documents or works, in or on a volume of a
     storage or distribution medium, is called an “aggregate” if the
     copyright resulting from the compilation is not used to limit the
     legal rights of the compilation’s users beyond what the individual
     works permit.  When the Document is included in an aggregate, this
     License does not apply to the other works in the aggregate which
     are not themselves derivative works of the Document.

     If the Cover Text requirement of section 3 is applicable to these
     copies of the Document, then if the Document is less than one half
     of the entire aggregate, the Document’s Cover Texts may be placed
     on covers that bracket the Document within the aggregate, or the
     electronic equivalent of covers if the Document is in electronic
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  8. TRANSLATION

     Translation is considered a kind of modification, so you may
     distribute translations of the Document under the terms of section
     4.  Replacing Invariant Sections with translations requires special
     permission from their copyright holders, but you may include
     translations of some or all Invariant Sections in addition to the
     original versions of these Invariant Sections.  You may include a
     translation of this License, and all the license notices in the
     Document, and any Warranty Disclaimers, provided that you also
     include the original English version of this License and the
     original versions of those notices and disclaimers.  In case of a
     disagreement between the translation and the original version of
     this License or a notice or disclaimer, the original version will
     prevail.

     If a section in the Document is Entitled “Acknowledgements”,
     “Dedications”, or “History”, the requirement (section 4) to
     Preserve its Title (section 1) will typically require changing the
     actual title.

  9. TERMINATION

     You may not copy, modify, sublicense, or distribute the Document
     except as expressly provided under this License.  Any attempt
     otherwise to copy, modify, sublicense, or distribute it is void,
     and will automatically terminate your rights under this License.

     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
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     same material does not give you any rights to use it.

  10. FUTURE REVISIONS OF THIS LICENSE

     The Free Software Foundation may publish new, revised versions of
     the GNU Free Documentation 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.  See
     <http://www.gnu.org/copyleft/>.

     Each version of the License is given a distinguishing version
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     have the option of following the terms and conditions either of
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     Document does not specify a version number of this License, you may
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     Software Foundation.  If the Document specifies that a proxy can
     decide which future versions of this License can be used, that
     proxy’s public statement of acceptance of a version permanently
     authorizes you to choose that version for the Document.

  11. RELICENSING

     “Massive Multiauthor Collaboration Site” (or “MMC Site”) means any
     World Wide Web server that publishes copyrightable works and also
     provides prominent facilities for anybody to edit those works.  A
     public wiki that anybody can edit is an example of such a server.
     A “Massive Multiauthor Collaboration” (or “MMC”) contained in the
     site means any set of copyrightable works thus published on the MMC
     site.

     “CC-BY-SA” means the Creative Commons Attribution-Share Alike 3.0
     license published by Creative Commons Corporation, a not-for-profit
     corporation with a principal place of business in San Francisco,
     California, as well as future copyleft versions of that license
     published by that same organization.

     “Incorporate” means to publish or republish a Document, in whole or
     in part, as part of another Document.

     An MMC is “eligible for relicensing” if it is licensed under this
     License, and if all works that were first published under this
     License somewhere other than this MMC, and subsequently
     incorporated in whole or in part into the MMC, (1) had no cover
     texts or invariant sections, and (2) were thus incorporated prior
     to November 1, 2008.

     The operator of an MMC Site may republish an MMC contained in the
     site under CC-BY-SA on the same site at any time before August 1,
     2009, provided the MMC is eligible for relicensing.

ADDENDUM: How to use this License for your documents
====================================================

To use this License in a document you have written, include a copy of
the License in the document and put the following copyright and license
notices just after the title page:

       Copyright (C)  YEAR  YOUR NAME.
       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 the section entitled ``GNU
       Free Documentation License''.

   If you have Invariant Sections, Front-Cover Texts and Back-Cover
Texts, replace the “with...Texts.” line with this:

         with the Invariant Sections being LIST THEIR TITLES, with
         the Front-Cover Texts being LIST, and with the Back-Cover Texts
         being LIST.

   If you have Invariant Sections without Cover Texts, or some other
combination of the three, merge those two alternatives to suit the
situation.

   If your document contains nontrivial examples of program code, we
recommend releasing these examples in parallel under your choice of free
software license, such as the GNU General Public License, to permit
their use in free software.



Tag Table:
Node: Top769
Node: Copying1550
Node: Introduction to GNU MPC2322
Node: Installing GNU MPC3041
Node: Reporting Bugs8342
Node: GNU MPC Basics9689
Ref: return-value13571
Node: Complex Functions15058
Node: Initializing Complex Numbers16257
Node: Assigning Complex Numbers18684
Node: Converting Complex Numbers23230
Node: String and Stream Input and Output23869
Node: Complex Comparison30607
Node: Projection & Decomposing32170
Node: Basic Arithmetic33579
Node: Power Functions and Logarithm39308
Node: Trigonometric Functions42244
Node: Modular Functions44199
Node: Miscellaneous Complex Functions45034
Node: Advanced Functions47240
Node: Internals48338
Node: Ball Arithmetic48797
Node: References66134
Node: Concept Index66931
Node: Function Index69391
Node: Type Index86438
Node: GNU Free Documentation License87468

End Tag Table


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