This manual documents how to install and use the Multiple Precision Complex Library, version 0.8.2
The MPC Library 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 2.1 of the License, or (at your option) any later version, see the file COPYING.LIB.
The MPC Library 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.
MPC is a portable library written in C for arbitrary precision arithmetic on complex numbers providing correct rounding. Ultimately, it should implement a multiprecision equivalent of the C99 standard. It builds upon the GNU MP and the GNU MPFR libraries.
Everyone should read MPC Basics. If you need to install the library yourself, you need to read Installing MPC, too.
The remainder of the manual can be used for later reference, although it is probably a good idea to skim through it.
To build MPC, you first have to install GNU MP (version 4.2 or higher) and GNU MPFR (version 2.3.1 or higher) on your computer. You need a C compiler, preferably GCC, but any reasonable compiler should work. 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:
if GMP and 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 MPFR is installed. Alternatively, you can specify directly MPFR include and 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.
Use ‘./configure --help’ for an exhaustive list of parameters.
This compiles MPC in the working directory.
This will make sure MPC was built correctly.
If you get error messages, please report them to ‘mpc-discuss@lists.gforge.inria.fr’ (See Reporting Bugs, for information on what to include in useful bug reports).
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.
There are some other useful make targets:
Create an info version of the manual, in mpc.info.
Create a PDF version of the manual, in doc/mpc.pdf.
Create a DVI version of the manual, in doc/mpc.dvi.
Create a Postscript version of the manual, in doc/mpc.ps.
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.
Delete all object files and archive files, but not the configuration files.
Delete all files not included in the distribution.
Delete all files copied by ‘make install’.
On AIX, if GMP was built with the 64-bit ABI, before building and testing 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 MPFR 2.4.1.
Please report any other problems you encounter to ‘mpc-discuss@lists.gforge.inria.fr’. See Reporting Bugs.
If you think you have found a bug in the 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@lists.gforge.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.
All declarations needed to use 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 MPC library by adding the line
#include "mpc.h"
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 mp_prec_t.
For more details on the allowed precision range,
see Section “Nomenclature and Types” in MPFR.
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.
There is only one class of functions in the MPC library, namely functions for
complex arithmetic. The function names begin with mpc_. The
associated type is mpc_t.
As a general rule, all MPC functions expect output arguments before input arguments. This notation is based on an analogy with the assignment operator.
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 MPC variable, you need to initialize 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 initialized once, or at least cleared out between each initialization. After a variable has been initialized, it may be assigned to any number of times.
For efficiency reasons, avoid to initialize and clear out a variable in loops. Instead, initialize 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 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 reinitialize it, a complex variable will have the same allocated space during all its life.
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).
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.
Most 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 complex functions have branch cuts, across which the function is discontinous. In 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.
The complex functions expect arguments of type mpc_t.
The 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 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.
An mpc_t object must be initialized before storing the first value in
it. The functions mpc_init2 and mpc_init3
are used for that purpose.
Initialize z to precision prec bits and set its real and imaginary parts to NaN. Normally, a variable should be initialized once only or at least be cleared, using
mpc_clear, between initializations.
Initialize 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.
Free the space occupied by z. Make sure to call this function for all
mpc_tvariables when you are done with them.
Here is an example on how to initialize 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.
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 tompc_init2(x, prec), but more efficient as no allocation is done in case the current allocated space for the mantissa of x is sufficient.
If the real and imaginary part of x have the same precision, it is returned, otherwise, 0 is returned.
Returns the precision of the real part of x via pr and of its imaginary part via pi.
These functions assign new values to already initialized complex numbers
(see Initializing Complex Numbers).
When using any functions with intmax_t parameter, you must include
<stdint.h> or <inttypes.h> before mpc.h, to allow
mpc.h to define prototypes for these functions.
If you need assignment functions that are not in the current API, you can
define them using the MPC_SET_X_Y macro (see Advanced Functions).
Set the value of rop from op, rounded to the precision of rop with the given rounding mode 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 intmay have to be rounded, if the destination precision is less than the machine word width. Formpc_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.
Set the real part of rop from op1, and its imaginary part from op2, according to the rounding mode rnd.
For functions assigning complex variables from strings or input streams, see String and Stream Input and Output.
Swap the values of op1 and op2 efficiently. Warning: The precisions are exchanged, too; in case these are different,
mpc_swapis thus not equivalent to threempc_setcalls using a third auxiliary variable.
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 (see 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_strtofrin the MPFR manual. 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 forbase=0 and represent the same value.
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_strtocfor a detailed description of the valid string formats. Contrarily tompc_strtoc,mpc_set_strrequires the whole string to represent a valid complex number (potentially followed by additional white space). This function returns the usual inexact value (see 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.
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_struses the decimal point of the current locale if available, and ‘.’ otherwise.The string is generated using the current memory allocation function (
mallocby default, unless it has been modified using the custom memory allocation interface ofgmp); once it is not needed any more, it should be freed by callingmpc_free_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.
Input a string in base base in the same format as for
mpc_strtocfrom stdio stream stream, rounded according to rnd, and put the read complex number into rop. If stream is the null pointer, rop is read fromstdin. Return the usual inexact value; if an error occurs, set rop toNaN + i * NaNand return -1. If read is not the null pointer, it is set to the number of read characters.Unlike
mpc_strtoc, the functionmpc_inp_strdoes 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 tompc_set_str. Precisely, after skipping optional whitespace, a minimal string is read according to the regular expressionmpfr | '(' \s* mpfr \s+ mpfr \s* ')', where\sdenotes a whitespace, andmpfris either a string containing neither whitespaces nor parentheses, ornan(n-char-sequence)or@nan@(n-char-sequence)(regardless of capitalisation) withn-char-sequencea string of ascii letters, digits or'_'.For instance, upon input of
"nan(13 1)", the functionmpc_inp_strstarts 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 becocmes 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 functionmpc_strtoc, on the other hand, may track back when reaching the whitespace; it treats the string as the two successive complex numbersNaN + i * 0and13 + i. It is thus recommended to have a whitespace follow each floating point number to avoid this problem.
Output op on stdio stream stream in base base, rounded according to rnd, in the same format as for
mpc_strtocIf stream is the null pointer, rop is written tostdout.Return the number of characters written.
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 intox = MPC_INEX_RE(c)andy = 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.
Set rop to the value of the real part of op rounded in the direction rnd.
Set rop to the value of the imaginary part of op rounded in the direction rnd.
Return a reference to the real part and imaginary part of op, respectively. The
mpfrfunctions can be used on the result of these macros (note that thempfr_ttype is itself a pointer).
Set rop to the argument of op, with a branch cut along the negative real axis.
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.
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 MPFR library, with only a marginal overhead due to the MPC layer.
Set rop to op1 + op2 rounded according to rnd.
Set rop to op1 − op2 rounded according to rnd. For
mpc_ui_ui_sub, op1 is re1 + im1.
Set rop to op1 times op2 rounded according to 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.
Set rop to the square of op rounded according to rnd.
Set rop to op1/op2 rounded according to rnd. For
mpc_divandmpc_ui_div, the return value may fail to recognize some exact results. The sign of returned value is significant only formpc_div_ui.
Set rop to −op rounded according to rnd. Just changes the sign if rop and op are the same variable.
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.
Set the floating-point number rop to the absolute value of op, rounded in the direction rnd. The returned value is zero iff the result is exact.
Set the floating-point number rop to the norm of op (i.e., the square of its absolute value), rounded in the direction rnd. The returned value is zero iff the result is exact. Note that the destination is of type
mpfr_t, notmpc_t.
Set rop to op1 times 2 raised to op2 rounded according to rnd. Just increases the exponents of the real and imaginary parts by op2 when rop and op1 are identical.
Set rop to op1 divided by 2 raised to op2 rounded according to rnd. Just decreases the exponents of the real and imaginary parts by op2 when rop and op1 are identical.
Set rop to the square root of op rounded according to 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_zandmpc_pow_fr, the imaginary part of op2 is considered as +0.
Set rop to the exponential of op, rounded according to rnd with the precision of rop.
Set rop to the logarithm of op, 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].
Set rop to the sine of op, rounded according to rnd with the precision of rop.
Set rop to the cosine of op, rounded according to rnd with the precision of rop.
Set rop to the tangent of op, rounded according to rnd with the precision of rop.
Set rop to the hyperbolic sine of op, rounded according to rnd with the precision of rop.
Set rop to the hyperbolic cosine of op, rounded according to rnd with the precision of rop.
Set rop to the hyperbolic tangent of op, rounded according to rnd with the precision of rop.
Set rop to the inverse sine, inverse cosine, inverse tangent of op, rounded according to rnd with the precision of rop.
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 (-\infty, 1).
Generate a uniformly distributed random complex in the unit square [0, 1] * [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_tstructure which should be created using the GMPrand_initfunction, see the GMP manual.
Return the MPC version, as a null-terminated string.
MPC_VERSIONis the version of MPC as a preprocessing constant.MPC_VERSION_MAJOR,MPC_VERSION_MINORandMPC_VERSION_PATCHLEVELare respectively the major, minor and patch level of MPC version, as preprocessing constants.MPC_VERSION_STRINGis the version as a string constant, which can be compared to the result ofmpc_get_versionto 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 MPC version can be dynamically linked with a newer MPC library version (if allowed by the library versioning system).
Create an integer in the same format as used by
MPC_VERSIONfrom the given major, minor and patchlevel. Here is an example of how to check the MPC version at compile time:#if (!defined(MPC_VERSION) || (MPC_VERSION<MPC_VERSION_NUM(2,1,0))) # error "Wrong MPC version." #endif
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
xin thempfr_set_xfunction one would use to set the part; for the mpfr type, usefr. 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 thempc_rnd_trounding mode. The return value is the usual inexact value (see Return Value).For instance, you can define mpc_set_ui_fr as follows:
int mpc_set_ui_fr (mpc_t rop, long int re, double im, mpc_rnd_t rnd) MPC_SET_X_Y (ui, fr, rop, re, im, rnd);
These macros and
functions are mainly designed for the implementation of 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.
The main developers of the MPC library are Andreas Enge,
Philippe Théveny and Paul Zimmermann.
Patrick Pélissier has helped cleaning up the code.
Marc Helbling contributed the mpc_ui_sub and
mpc_ui_ui_sub functions.
gmp – GNU multiprecision library.
Version 4.2.4, http://gmplib.org/.
mpfr – A library for multiple-precision floating-point computations with exact rounding.
Version 2.4.1, http://www.mpfr.org.
mp_prec_t: MPC Basicsmpc_abs: Basic Arithmeticmpc_acos: Trigonometric Functionsmpc_acosh: Trigonometric Functionsmpc_add: Basic Arithmeticmpc_add_fr: Basic Arithmeticmpc_add_ui: Basic Arithmeticmpc_arg: Projection & Decomposingmpc_asin: Trigonometric Functionsmpc_asinh: Trigonometric Functionsmpc_atan: Trigonometric Functionsmpc_atanh: Trigonometric Functionsmpc_clear: Initializing Complex Numbersmpc_cmp: Complex Comparisonmpc_cmp_si: Complex Comparisonmpc_cmp_si_si: Complex Comparisonmpc_conj: Basic Arithmeticmpc_cos: Trigonometric Functionsmpc_cosh: Trigonometric Functionsmpc_div: Basic Arithmeticmpc_div_2exp: Basic Arithmeticmpc_div_fr: Basic Arithmeticmpc_div_ui: Basic Arithmeticmpc_exp: Power Functions and Logarithmmpc_fr_div: Basic Arithmeticmpc_fr_sub: Basic Arithmeticmpc_free_str: String and Stream Input and Outputmpc_get_prec: Initializing Complex Numbersmpc_get_prec2: Initializing Complex Numbersmpc_get_str: String and Stream Input and Outputmpc_get_version: Miscellaneous Complex Functionsmpc_imag: Projection & Decomposingmpc_imagref: Projection & Decomposingmpc_init2: Initializing Complex Numbersmpc_init3: Initializing Complex Numbersmpc_inp_str: String and Stream Input and Outputmpc_log: Power Functions and Logarithmmpc_mul: Basic Arithmeticmpc_mul_2exp: Basic Arithmeticmpc_mul_fr: Basic Arithmeticmpc_mul_i: Basic Arithmeticmpc_mul_si: Basic Arithmeticmpc_mul_ui: Basic Arithmeticmpc_neg: Basic Arithmeticmpc_norm: Basic Arithmeticmpc_out_str: String and Stream Input and Outputmpc_pow: Power Functions and Logarithmmpc_pow_d: Power Functions and Logarithmmpc_pow_fr: Power Functions and Logarithmmpc_pow_ld: Power Functions and Logarithmmpc_pow_si: Power Functions and Logarithmmpc_pow_ui: Power Functions and Logarithmmpc_pow_z: Power Functions and Logarithmmpc_proj: Projection & Decomposingmpc_real: Projection & Decomposingmpc_realref: Projection & Decomposingmpc_rnd_t: MPC Basicsmpc_set: Assigning Complex Numbersmpc_set_d: Assigning Complex Numbersmpc_set_d_d: Assigning Complex Numbersmpc_set_f: Assigning Complex Numbersmpc_set_f_f: Assigning Complex Numbersmpc_set_fr: Assigning Complex Numbersmpc_set_fr_fr: Assigning Complex Numbersmpc_set_ld: Assigning Complex Numbersmpc_set_ld_ld: Assigning Complex Numbersmpc_set_nan: Assigning Complex Numbersmpc_set_prec: Initializing Complex Numbersmpc_set_q: Assigning Complex Numbersmpc_set_q_q: Assigning Complex Numbersmpc_set_si: Assigning Complex Numbersmpc_set_si_si: Assigning Complex Numbersmpc_set_sj: Assigning Complex Numbersmpc_set_sj_sj: Assigning Complex Numbersmpc_set_str: String and Stream Input and Outputmpc_set_ui: Assigning Complex Numbersmpc_set_ui_ui: Assigning Complex Numbersmpc_set_uj: Assigning Complex Numbersmpc_set_uj_uj: Assigning Complex NumbersMPC_SET_X_Y: Advanced Functionsmpc_set_z: Assigning Complex Numbersmpc_set_z_z: Assigning Complex Numbersmpc_sin: Trigonometric Functionsmpc_sinh: Trigonometric Functionsmpc_sqr: Basic Arithmeticmpc_sqrt: Power Functions and Logarithmmpc_strtoc: String and Stream Input and Outputmpc_sub: Basic Arithmeticmpc_sub_fr: Basic Arithmeticmpc_sub_ui: Basic Arithmeticmpc_swap: Assigning Complex Numbersmpc_t: MPC Basicsmpc_tan: Trigonometric Functionsmpc_tanh: Trigonometric Functionsmpc_ui_div: Basic Arithmeticmpc_ui_sub: Basic Arithmeticmpc_ui_ui_sub: Basic Arithmeticmpc_urandom: Miscellaneous Complex FunctionsMPC_VERSION: Miscellaneous Complex FunctionsMPC_VERSION_MAJOR: Miscellaneous Complex FunctionsMPC_VERSION_MINOR: Miscellaneous Complex FunctionsMPC_VERSION_NUM: Miscellaneous Complex FunctionsMPC_VERSION_PATCHLEVEL: Miscellaneous Complex FunctionsMPC_VERSION_STRING: Miscellaneous Complex Functions