mpc.info revision 1.1.1.5
1This is mpc.info, produced by makeinfo version 6.7 from mpc.texi. 2 3This manual is for GNU MPC, a library for multiple precision complex 4arithmetic, version 1.2.1 of October 2020. 5 6 Copyright (C) 2002, 2003, 2004, 2005, 2006, 2007, 2008, 2009, 2010, 72011, 2012, 2013, 2016, 2018, 2020 INRIA 8 9 Permission is granted to copy, distribute and/or modify this 10 document under the terms of the GNU Free Documentation License, 11 Version 1.3 or any later version published by the Free Software 12 Foundation; with no Invariant Sections. A copy of the license is 13 included in the section entitled "GNU Free Documentation License." 14INFO-DIR-SECTION GNU Packages 15START-INFO-DIR-ENTRY 16* mpc: (mpc)Multiple Precision Complex Library. 17END-INFO-DIR-ENTRY 18 19 20File: mpc.info, Node: Top, Next: Copying, Up: (dir) 21 22GNU MPC 23******* 24 25This manual documents how to install and use the GNU Multiple Precision 26Complex Library, version 1.2.1 27 28* Menu: 29 30* Copying:: GNU MPC Copying Conditions (LGPL). 31* Introduction to GNU MPC:: Brief introduction to GNU MPC. 32* Installing GNU MPC:: How to configure and compile the GNU MPC library. 33* Reporting Bugs:: How to usefully report bugs. 34* GNU MPC Basics:: What every GNU MPC user should know. 35* Complex Functions:: Functions for arithmetic on complex numbers. 36* References:: 37* Concept Index:: 38* Function Index:: 39* GNU Free Documentation License:: 40 41 42File: mpc.info, Node: Copying, Next: Introduction to GNU MPC, Prev: Top, Up: Top 43 44GNU MPC Copying Conditions 45************************** 46 47GNU MPC is free software; you can redistribute it and/or modify it under 48the terms of the GNU Lesser General Public License as published by the 49Free Software Foundation; either version 3 of the License, or (at your 50option) any later version. 51 52 GNU MPC is distributed in the hope that it will be useful, but 53WITHOUT ANY WARRANTY; without even the implied warranty of 54MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the GNU Lesser 55General Public License for more details. 56 57 You should have received a copy of the GNU Lesser General Public 58License along with this program. If not, see 59<http://www.gnu.org/licenses/>. 60 61 62File: mpc.info, Node: Introduction to GNU MPC, Next: Installing GNU MPC, Prev: Copying, Up: Top 63 641 Introduction to GNU MPC 65************************* 66 67GNU MPC is a portable library written in C for arbitrary precision 68arithmetic on complex numbers providing correct rounding. It implements 69a multiprecision equivalent of the C99 standard. It builds upon the GNU 70MP and the GNU MPFR libraries. 71 721.1 How to use this Manual 73========================== 74 75Everyone should read *note GNU MPC Basics::. If you need to install the 76library yourself, you need to read *note Installing GNU MPC::, too. 77 78 The remainder of the manual can be used for later reference, although 79it is probably a good idea to skim through it. 80 81 82File: mpc.info, Node: Installing GNU MPC, Next: Reporting Bugs, Prev: Introduction to GNU MPC, Up: Top 83 842 Installing GNU MPC 85******************** 86 87To build GNU MPC, you first have to install GNU MP (version 5.0.0 or 88higher) and GNU MPFR (version 4.1.0 or higher) on your computer. You 89need a C compiler; GCC version 4.4 or higher is recommended, since GNU 90MPC may trigger a bug in previous versions, see the thread at 91<http://lists.gforge.inria.fr/pipermail/mpc-discuss/2011-February/000823.html>. 92And you need a standard Unix 'make' program, plus some other standard 93Unix utility programs. 94 95 Here are the steps needed to install the library on Unix systems: 96 97 1. 'tar xzf mpc-1.2.1.tar.gz' 98 99 2. 'cd mpc-1.2.1' 100 101 3. './configure' 102 103 if GMP and GNU MPFR are installed into standard directories, that 104 is, directories that are searched by default by the compiler and 105 the linking tools. 106 107 './configure --with-gmp=<gmp_install_dir>' 108 109 is used to indicate a different location where GMP is installed. 110 Alternatively, you can specify directly GMP include and GMP lib 111 directories with './configure --with-gmp-lib=<gmp_lib_dir> 112 --with-gmp-include=<gmp_include_dir>'. 113 114 './configure --with-mpfr=<mpfr_install_dir>' 115 116 is used to indicate a different location where GNU MPFR is 117 installed. Alternatively, you can specify directly GNU MPFR 118 include and GNU MPFR lib directories with './configure 119 --with-mpf-lib=<mpfr_lib_dir> 120 --with-mpfr-include=<mpfr_include_dir>'. 121 122 Another useful parameter is '--prefix', which can be used to 123 specify an alternative installation location instead of 124 '/usr/local'; see 'make install' below. 125 126 To enable checking for memory leaks using 'valgrind' during 'make 127 check', add the parameter '--enable-valgrind-tests'. 128 129 If for debugging purposes you wish to log calls to GNU MPC 130 functions from within your code, add the parameter 131 '--enable-logging'. In your code, replace the inclusion of 'mpc.h' 132 by 'mpc-log.h' and link the executable dynamically. Then all calls 133 to functions with only complex arguments are printed to 'stderr' in 134 the following form: First, the function name is given, followed by 135 its type such as 'c_cc', meaning that the function has one complex 136 result (one 'c' in front of the '_'), computed from two complex 137 arguments (two 'c' after the '_'). Then, the precisions of the 138 real and the imaginary part of the first result is given, followed 139 by the second one and so on. Finally, for each argument, the 140 precisions of its real and imaginary part are specified and the 141 argument itself is printed in hexadecimal via the function 142 'mpc_out_str' (*note String and Stream Input and Output::). The 143 option requires a dynamic library, so it may not be combined with 144 '--disable-shared'. 145 146 Use './configure --help' for an exhaustive list of parameters. 147 148 4. 'make' 149 150 This compiles GNU MPC in the working directory. 151 152 5. 'make check' 153 154 This will make sure GNU MPC was built correctly. 155 156 If you get error messages, please report them to 157 'mpc-discuss@lists.gforge.inria.fr' (*Note Reporting Bugs::, for 158 information on what to include in useful bug reports). 159 160 6. 'make install' 161 162 This will copy the file 'mpc.h' to the directory 163 '/usr/local/include', the file 'libmpc.a' to the directory 164 '/usr/local/lib', and the file 'mpc.info' to the directory 165 '/usr/local/share/info' (or if you passed the '--prefix' option to 166 'configure', using the prefix directory given as argument to 167 '--prefix' instead of '/usr/local'). Note: you need write 168 permissions on these directories. 169 1702.1 Other 'make' Targets 171======================== 172 173There are some other useful make targets: 174 175 * 'info' 176 177 Create an info version of the manual, in 'mpc.info'. 178 179 * 'pdf' 180 181 Create a PDF version of the manual, in 'doc/mpc.pdf'. 182 183 * 'dvi' 184 185 Create a DVI version of the manual, in 'doc/mpc.dvi'. 186 187 * 'ps' 188 189 Create a Postscript version of the manual, in 'doc/mpc.ps'. 190 191 * 'html' 192 193 Create an HTML version of the manual, in several pages in the 194 directory 'doc/mpc.html'; if you want only one output HTML file, 195 then type 'makeinfo --html --no-split mpc.texi' instead. 196 197 * 'clean' 198 199 Delete all object files and archive files, but not the 200 configuration files. 201 202 * 'distclean' 203 204 Delete all files not included in the distribution. 205 206 * 'uninstall' 207 208 Delete all files copied by 'make install'. 209 2102.2 Known Build Problems 211======================== 212 213On AIX, if GMP was built with the 64-bit ABI, before building and 214testing GNU MPC, it might be necessary to set the 'OBJECT_MODE' 215environment variable to 64 by, e.g., 216 217 'export OBJECT_MODE=64' 218 219 This has been tested with the C compiler IBM XL C/C++ Enterprise 220Edition V8.0 for AIX, version: 08.00.0000.0021, GMP 4.2.4 and GNU MPFR 2212.4.1. 222 223 Please report any other problems you encounter to 224'mpc-discuss@lists.gforge.inria.fr'. *Note Reporting Bugs::. 225 226 227File: mpc.info, Node: Reporting Bugs, Next: GNU MPC Basics, Prev: Installing GNU MPC, Up: Top 228 2293 Reporting Bugs 230**************** 231 232If you think you have found a bug in the GNU MPC library, please 233investigate and report it. We have made this library available to you, 234and it is not to ask too much from you, to ask you to report the bugs 235that you find. 236 237 There are a few things you should think about when you put your bug 238report together. 239 240 You have to send us a test case that makes it possible for us to 241reproduce the bug. Include instructions on how to run the test case. 242 243 You also have to explain what is wrong; if you get a crash, or if the 244results printed are incorrect and in that case, in what way. 245 246 Please include compiler version information in your bug report. This 247can be extracted using 'gcc -v', or 'cc -V' on some machines. Also, 248include the output from 'uname -a'. 249 250 If your bug report is good, we will do our best to help you to get a 251corrected version of the library; if the bug report is poor, we will not 252do anything about it (aside of chiding you to send better bug reports). 253 254 Send your bug report to: 'mpc-discuss@lists.gforge.inria.fr'. 255 256 If you think something in this manual is unclear, or downright 257incorrect, or if the language needs to be improved, please send a note 258to the same address. 259 260 261File: mpc.info, Node: GNU MPC Basics, Next: Complex Functions, Prev: Reporting Bugs, Up: Top 262 2634 GNU MPC Basics 264**************** 265 266All declarations needed to use GNU MPC are collected in the include file 267'mpc.h'. It is designed to work with both C and C++ compilers. You 268should include that file in any program using the GNU MPC library by 269adding the line 270 #include "mpc.h" 271 2724.1 Nomenclature and Types 273========================== 274 275"Complex number" or "Complex" for short, is a pair of two arbitrary 276precision floating-point numbers (for the real and imaginary parts). 277The C data type for such objects is 'mpc_t'. 278 279The "Precision" is the number of bits used to represent the mantissa of 280the real and imaginary parts; the corresponding C data type is 281'mpfr_prec_t'. For more details on the allowed precision range, *note 282(mpfr.info)Nomenclature and Types::. 283 284The "rounding mode" specifies the way to round the result of a complex 285operation, in case the exact result can not be represented exactly in 286the destination mantissa; the corresponding C data type is 'mpc_rnd_t'. 287A complex rounding mode is a pair of two rounding modes: one for the 288real part, one for the imaginary part. 289 2904.2 Function Classes 291==================== 292 293There is only one class of functions in the GNU MPC library, namely 294functions for complex arithmetic. The function names begin with 'mpc_'. 295The associated type is 'mpc_t'. 296 2974.3 GNU MPC Variable Conventions 298================================ 299 300As a general rule, all GNU MPC functions expect output arguments before 301input arguments. This notation is based on an analogy with the 302assignment operator. 303 304 GNU MPC allows you to use the same variable for both input and output 305in the same expression. For example, the main function for 306floating-point multiplication, 'mpc_mul', can be used like this: 307'mpc_mul (x, x, x, rnd_mode)'. This computes the square of X with 308rounding mode 'rnd_mode' and puts the result back in X. 309 310 Before you can assign to an GNU MPC variable, you need to initialise 311it by calling one of the special initialization functions. When you are 312done with a variable, you need to clear it out, using one of the 313functions for that purpose. 314 315 A variable should only be initialised once, or at least cleared out 316between each initialization. After a variable has been initialised, it 317may be assigned to any number of times. 318 319 For efficiency reasons, avoid to initialise and clear out a variable 320in loops. Instead, initialise it before entering the loop, and clear it 321out after the loop has exited. 322 323 You do not need to be concerned about allocating additional space for 324GNU MPC variables, since each of its real and imaginary part has a 325mantissa of fixed size. Hence unless you change its precision, or clear 326and reinitialise it, a complex variable will have the same allocated 327space during all its life. 328 3294.4 Rounding Modes 330================== 331 332A complex rounding mode is of the form 'MPC_RNDxy' where 'x' and 'y' are 333one of 'N' (to nearest), 'Z' (towards zero), 'U' (towards plus 334infinity), 'D' (towards minus infinity). The first letter refers to the 335rounding mode for the real part, and the second one for the imaginary 336part. For example 'MPC_RNDZU' indicates to round the real part towards 337zero, and the imaginary part towards plus infinity. 338 339 The 'round to nearest' mode works as in the IEEE P754 standard: in 340case the number to be rounded lies exactly in the middle of two 341representable numbers, it is rounded to the one with the least 342significant bit set to zero. For example, the number 5, which is 343represented by (101) in binary, is rounded to (100)=4 with a precision 344of two bits, and not to (110)=6. 345 3464.5 Return Value 347================ 348 349Most GNU MPC functions have a return value of type 'int', which is used 350to indicate the position of the rounded real and imaginary parts with 351respect to the exact (infinite precision) values. If this integer is 352'i', the macros 'MPC_INEX_RE(i)' and 'MPC_INEX_IM(i)' give 0 if the 353corresponding rounded value is exact, a negative value if the rounded 354value is less than the exact one, and a positive value if it is greater 355than the exact one. Similarly, functions computing a result of type 356'mpfr_t' return an integer that is 0, positive or negative depending on 357whether the rounded value is the same, larger or smaller then the exact 358result. 359 360 Some functions, such as 'mpc_sin_cos', compute two complex results; 361the macros 'MPC_INEX1(i)' and 'MPC_INEX2(i)', applied to the return 362value 'i' of such a function, yield the exactness value corresponding to 363the first or the second computed value, respectively. 364 3654.6 Branch Cuts And Special Values 366================================== 367 368Some complex functions have branch cuts, across which the function is 369discontinous. In GNU MPC, the branch cuts chosen are the same as those 370specified for the corresponding functions in the ISO C99 standard. 371 372 Likewise, when evaluated at a point whose real or imaginary part is 373either infinite or a NaN or a signed zero, a function returns the same 374value as those specified for the corresponding function in the ISO C99 375standard. 376 377 378File: mpc.info, Node: Complex Functions, Next: References, Prev: GNU MPC Basics, Up: Top 379 3805 Complex Functions 381******************* 382 383The complex functions expect arguments of type 'mpc_t'. 384 385 The GNU MPC floating-point functions have an interface that is 386similar to the GNU MP integer functions. The function prefix for 387operations on complex numbers is 'mpc_'. 388 389 The precision of a computation is defined as follows: Compute the 390requested operation exactly (with "infinite precision"), and round the 391result to the destination variable precision with the given rounding 392mode. 393 394 The GNU MPC complex functions are intended to be a smooth extension 395of the IEEE P754 arithmetic. The results obtained on one computer 396should not differ from the results obtained on a computer with a 397different word size. 398 399* Menu: 400 401* Initializing Complex Numbers:: 402* Assigning Complex Numbers:: 403* Converting Complex Numbers:: 404* String and Stream Input and Output:: 405* Complex Comparison:: 406* Projection & Decomposing:: 407* Basic Arithmetic:: 408* Power Functions and Logarithm:: 409* Trigonometric Functions:: 410* Miscellaneous Complex Functions:: 411* Advanced Functions:: 412* Internals:: 413 414 415File: mpc.info, Node: Initializing Complex Numbers, Next: Assigning Complex Numbers, Up: Complex Functions 416 4175.1 Initialization Functions 418============================ 419 420An 'mpc_t' object must be initialised before storing the first value in 421it. The functions 'mpc_init2' and 'mpc_init3' are used for that 422purpose. 423 424 -- Function: void mpc_init2 (mpc_t Z, mpfr_prec_t PREC) 425 Initialise Z to precision PREC bits and set its real and imaginary 426 parts to NaN. Normally, a variable should be initialised once only 427 or at least be cleared, using 'mpc_clear', between initializations. 428 429 -- Function: void mpc_init3 (mpc_t Z, mpfr_prec_t PREC_R, mpfr_prec_t 430 PREC_I) 431 Initialise Z with the precision of its real part being PREC_R bits 432 and the precision of its imaginary part being PREC_I bits, and set 433 the real and imaginary parts to NaN. 434 435 -- Function: void mpc_clear (mpc_t Z) 436 Free the space occupied by Z. Make sure to call this function for 437 all 'mpc_t' variables when you are done with them. 438 439 Here is an example on how to initialise complex variables: 440 { 441 mpc_t x, y; 442 mpc_init2 (x, 256); /* precision _exactly_ 256 bits */ 443 mpc_init3 (y, 100, 50); /* 100/50 bits for the real/imaginary part */ 444 ... 445 mpc_clear (x); 446 mpc_clear (y); 447 } 448 449 The following function is useful for changing the precision during a 450calculation. A typical use would be for adjusting the precision 451gradually in iterative algorithms like Newton-Raphson, making the 452computation precision closely match the actual accurate part of the 453numbers. 454 455 -- Function: void mpc_set_prec (mpc_t X, mpfr_prec_t PREC) 456 Reset the precision of X to be *exactly* PREC bits, and set its 457 real/imaginary parts to NaN. The previous value stored in X is 458 lost. It is equivalent to a call to 'mpc_clear(x)' followed by a 459 call to 'mpc_init2(x, prec)', but more efficient as no allocation 460 is done in case the current allocated space for the mantissa of X 461 is sufficient. 462 463 -- Function: mpfr_prec_t mpc_get_prec (const mpc_t X) 464 If the real and imaginary part of X have the same precision, it is 465 returned, otherwise, 0 is returned. 466 467 -- Function: void mpc_get_prec2 (mpfr_prec_t* PR, mpfr_prec_t* PI, 468 const mpc_t X) 469 Returns the precision of the real part of X via PR and of its 470 imaginary part via PI. 471 472 473File: mpc.info, Node: Assigning Complex Numbers, Next: Converting Complex Numbers, Prev: Initializing Complex Numbers, Up: Complex Functions 474 4755.2 Assignment Functions 476======================== 477 478These functions assign new values to already initialised complex numbers 479(*note Initializing Complex Numbers::). When using any functions with 480'intmax_t' or 'uintmax_t' parameters, you must include '<stdint.h>' or 481'<inttypes.h>' _before_ 'mpc.h', to allow 'mpc.h' to define prototypes 482for these functions. Similarly, functions with parameters of type 483'complex' or 'long complex' are defined only if '<complex.h>' is 484included _before_ 'mpc.h'. If you need assignment functions that are 485not in the current API, you can define them using the 'MPC_SET_X_Y' 486macro (*note Advanced Functions::). 487 488 -- Function: int mpc_set (mpc_t ROP, const mpc_t OP, mpc_rnd_t RND) 489 Set the value of ROP from OP, rounded to the precision of ROP with 490 the given rounding mode RND. 491 492 -- Function: int mpc_set_ui (mpc_t ROP, unsigned long int OP, mpc_rnd_t 493 RND) 494 -- Function: int mpc_set_si (mpc_t ROP, long int OP, mpc_rnd_t RND) 495 -- Function: int mpc_set_uj (mpc_t ROP, uintmax_t OP, mpc_rnd_t RND) 496 -- Function: int mpc_set_sj (mpc_t ROP, intmax_t OP, mpc_rnd_t RND) 497 -- Function: int mpc_set_d (mpc_t ROP, double OP, mpc_rnd_t RND) 498 -- Function: int mpc_set_ld (mpc_t ROP, long double OP, mpc_rnd_t RND) 499 -- Function: int mpc_set_dc (mpc_t ROP, double _Complex OP, mpc_rnd_t 500 RND) 501 -- Function: int mpc_set_ldc (mpc_t ROP, long double _Complex OP, 502 mpc_rnd_t RND) 503 -- Function: int mpc_set_z (mpc_t ROP, const mpz_t OP mpc_rnd_t RND) 504 -- Function: int mpc_set_q (mpc_t ROP, const mpq_t OP mpc_rnd_t RND) 505 -- Function: int mpc_set_f (mpc_t ROP, const mpf_t OP mpc_rnd_t RND) 506 -- Function: int mpc_set_fr (mpc_t ROP, const mpfr_t OP, mpc_rnd_t RND) 507 Set the value of ROP from OP, rounded to the precision of ROP with 508 the given rounding mode RND. The argument OP is interpreted as 509 real, so the imaginary part of ROP is set to zero with a positive 510 sign. Please note that even a 'long int' may have to be rounded, 511 if the destination precision is less than the machine word width. 512 For 'mpc_set_d', be careful that the input number OP may not be 513 exactly representable as a double-precision number (this happens 514 for 0.1 for instance), in which case it is first rounded by the C 515 compiler to a double-precision number, and then only to a complex 516 number. 517 518 -- Function: int mpc_set_ui_ui (mpc_t ROP, unsigned long int OP1, 519 unsigned long int OP2, mpc_rnd_t RND) 520 -- Function: int mpc_set_si_si (mpc_t ROP, long int OP1, long int OP2, 521 mpc_rnd_t RND) 522 -- Function: int mpc_set_uj_uj (mpc_t ROP, uintmax_t OP1, uintmax_t 523 OP2, mpc_rnd_t RND) 524 -- Function: int mpc_set_sj_sj (mpc_t ROP, intmax_t OP1, intmax_t OP2, 525 mpc_rnd_t RND) 526 -- Function: int mpc_set_d_d (mpc_t ROP, double OP1, double OP2, 527 mpc_rnd_t RND) 528 -- Function: int mpc_set_ld_ld (mpc_t ROP, long double OP1, long double 529 OP2, mpc_rnd_t RND) 530 -- Function: int mpc_set_z_z (mpc_t ROP, const mpz_t OP1, const mpz_t 531 OP2, mpc_rnd_t RND) 532 -- Function: int mpc_set_q_q (mpc_t ROP, const mpq_t OP1, const mpq_t 533 OP2, mpc_rnd_t RND) 534 -- Function: int mpc_set_f_f (mpc_t ROP, const mpf_t OP1, const mpf_t 535 OP2, mpc_rnd_t RND) 536 -- Function: int mpc_set_fr_fr (mpc_t ROP, const mpfr_t OP1, const 537 mpfr_t OP2, mpc_rnd_t RND) 538 Set the real part of ROP from OP1, and its imaginary part from OP2, 539 according to the rounding mode RND. 540 541 Beware that the behaviour of 'mpc_set_fr_fr' is undefined if OP1 or 542 OP2 is a pointer to the real or imaginary part of ROP. To exchange 543 the real and the imaginary part of a complex number, either use 544 'mpfr_swap (mpc_realref (rop), mpc_imagref (rop))', which also 545 exchanges the precisions of the two parts; or use a temporary 546 variable. 547 548 For functions assigning complex variables from strings or input 549streams, *note String and Stream Input and Output::. 550 551 -- Function: void mpc_set_nan (mpc_t ROP) 552 Set ROP to Nan+i*NaN. 553 554 -- Function: void mpc_swap (mpc_t OP1, mpc_t OP2) 555 Swap the values of OP1 and OP2 efficiently. Warning: The 556 precisions are exchanged, too; in case these are different, 557 'mpc_swap' is thus not equivalent to three 'mpc_set' calls using a 558 third auxiliary variable. 559 560 561File: mpc.info, Node: Converting Complex Numbers, Next: String and Stream Input and Output, Prev: Assigning Complex Numbers, Up: Complex Functions 562 5635.3 Conversion Functions 564======================== 565 566The following functions are available only if '<complex.h>' is included 567_before_ 'mpc.h'. 568 569 -- Function: double _Complex mpc_get_dc (const mpc_t OP, mpc_rnd_t RND) 570 -- Function: long double _Complex mpc_get_ldc (mpc_t OP, mpc_rnd_t RND) 571 Convert OP to a C complex number, using the rounding mode RND. 572 573 For functions converting complex variables to strings or stream 574output, *note String and Stream Input and Output::. 575 576 577File: mpc.info, Node: String and Stream Input and Output, Next: Complex Comparison, Prev: Converting Complex Numbers, Up: Complex Functions 578 5795.4 String and Stream Input and Output 580====================================== 581 582 -- Function: int mpc_strtoc (mpc_t ROP, const char *NPTR, char 583 **ENDPTR, int BASE, mpc_rnd_t RND) 584 Read a complex number from a string NPTR in base BASE, rounded to 585 the precision of ROP with the given rounding mode RND. The BASE 586 must be either 0 or a number from 2 to 36 (otherwise the behaviour 587 is undefined). If NPTR starts with valid data, the result is 588 stored in ROP, the usual inexact value is returned (*note Return 589 Value: return-value.) and, if ENDPTR is not the null pointer, 590 *ENDPTR points to the character just after the valid data. 591 Otherwise, ROP is set to 'NaN + i * NaN', -1 is returned and, if 592 ENDPTR is not the null pointer, the value of NPTR is stored in the 593 location referenced by ENDPTR. 594 595 The expected form of a complex number string is either a real 596 number (an optional leading whitespace, an optional sign followed 597 by a floating-point number), or a pair of real numbers in 598 parentheses separated by whitespace. If a real number is read, the 599 missing imaginary part is set to +0. The form of a floating-point 600 number depends on the base and is described in the documentation of 601 'mpfr_strtofr' (*note (mpfr.info)Assignment Functions::). For 602 instance, '"3.1415926"', '"(1.25e+7 +.17)"', '"(@nan@ 2)"' and 603 '"(-0 -7)"' are valid strings for BASE = 10. If BASE = 0, then a 604 prefix may be used to indicate the base in which the floating-point 605 number is written. Use prefix '0b' for binary numbers, prefix '0x' 606 for hexadecimal numbers, and no prefix for decimal numbers. The 607 real and imaginary part may then be written in different bases. 608 For instance, '"(1.024e+3 +2.05e+3)"' and '"(0b1p+10 +0x802)"' are 609 valid strings for 'base'=0 and represent the same value. 610 611 -- Function: int mpc_set_str (mpc_t ROP, const char *S, int BASE, 612 mpc_rnd_t rnd) 613 Set ROP to the value of the string S in base BASE, rounded to the 614 precision of ROP with the given rounding mode RND. See the 615 documentation of 'mpc_strtoc' for a detailed description of the 616 valid string formats. Contrarily to 'mpc_strtoc', 'mpc_set_str' 617 requires the _whole_ string to represent a valid complex number 618 (potentially followed by additional white space). This function 619 returns the usual inexact value (*note Return Value: return-value.) 620 if the entire string up to the final null character is a valid 621 number in base BASE; otherwise it returns -1, and ROP is set to 622 NaN+i*NaN. 623 624 -- Function: char * mpc_get_str (int B, size_t N, const mpc_t OP, 625 mpc_rnd_t RND) 626 Convert OP to a string containing its real and imaginary parts, 627 separated by a space and enclosed in a pair of parentheses. The 628 numbers are written in base B (which may vary from 2 to 36) and 629 rounded according to RND. The number of significant digits, at 630 least 2, is given by N. It is also possible to let N be zero, in 631 which case the number of digits is chosen large enough so that 632 re-reading the printed value with the same precision, assuming both 633 output and input use rounding to nearest, will recover the original 634 value of OP. Note that 'mpc_get_str' uses the decimal point of the 635 current locale if available, and '.' otherwise. 636 637 The string is generated using the current memory allocation 638 function ('malloc' by default, unless it has been modified using 639 the custom memory allocation interface of 'gmp'); once it is not 640 needed any more, it should be freed by calling 'mpc_free_str'. 641 642 -- Function: void mpc_free_str (char *STR) 643 Free the string STR, which needs to have been allocated by a call 644 to 'mpc_get_str'. 645 646 The following two functions read numbers from input streams and write 647them to output streams. When using any of these functions, you need to 648include 'stdio.h' _before_ 'mpc.h'. 649 650 -- Function: int mpc_inp_str (mpc_t ROP, FILE *STREAM, size_t *READ, 651 int BASE, mpc_rnd_t RND) 652 Input a string in base BASE in the same format as for 'mpc_strtoc' 653 from stdio stream STREAM, rounded according to RND, and put the 654 read complex number into ROP. If STREAM is the null pointer, ROP 655 is read from 'stdin'. Return the usual inexact value; if an error 656 occurs, set ROP to 'NaN + i * NaN' and return -1. If READ is not 657 the null pointer, it is set to the number of read characters. 658 659 Unlike 'mpc_strtoc', the function 'mpc_inp_str' does not possess 660 perfect knowledge of the string to transform and has to read it 661 character by character, so it behaves slightly differently: It 662 tries to read a string describing a complex number and processes 663 this string through a call to 'mpc_set_str'. Precisely, after 664 skipping optional whitespace, a minimal string is read according to 665 the regular expression 'mpfr | '(' \s* mpfr \s+ mpfr \s* ')'', 666 where '\s' denotes a whitespace, and 'mpfr' is either a string 667 containing neither whitespaces nor parentheses, or 668 'nan(n-char-sequence)' or '@nan@(n-char-sequence)' (regardless of 669 capitalisation) with 'n-char-sequence' a string of ascii letters, 670 digits or ''_''. 671 672 For instance, upon input of '"nan(13 1)"', the function 673 'mpc_inp_str' starts to recognise a value of NaN followed by an 674 n-char-sequence indicated by the opening parenthesis; as soon as 675 the space is reached, it becomes clear that the expression in 676 parentheses is not an n-char-sequence, and the error flag -1 is 677 returned after 6 characters have been consumed from the stream (the 678 whitespace itself remaining in the stream). The function 679 'mpc_strtoc', on the other hand, may track back when reaching the 680 whitespace; it treats the string as the two successive complex 681 numbers 'NaN + i * 0' and '13 + i'. It is thus recommended to have 682 a whitespace follow each floating point number to avoid this 683 problem. 684 685 -- Function: size_t mpc_out_str (FILE *STREAM, int BASE, size_t 686 N_DIGITS, const mpc_t OP, mpc_rnd_t RND) 687 Output OP on stdio stream STREAM in base BASE, rounded according to 688 RND, in the same format as for 'mpc_strtoc' If STREAM is the null 689 pointer, ROP is written to 'stdout'. 690 691 Return the number of characters written. 692 693 694File: mpc.info, Node: Complex Comparison, Next: Projection & Decomposing, Prev: String and Stream Input and Output, Up: Complex Functions 695 6965.5 Comparison Functions 697======================== 698 699 -- Function: int mpc_cmp (const mpc_t OP1, const mpc_t OP2) 700 -- Function: int mpc_cmp_si_si (const mpc_t OP1, long int OP2R, long 701 int OP2I) 702 -- Macro: int mpc_cmp_si (mpc_t OP1, long int OP2) 703 704 Compare OP1 and OP2, where in the case of 'mpc_cmp_si_si', OP2 is 705 taken to be OP2R + i OP2I. The return value C can be decomposed 706 into 'x = MPC_INEX_RE(c)' and 'y = MPC_INEX_IM(c)', such that X is 707 positive if the real part of OP1 is greater than that of OP2, zero 708 if both real parts are equal, and negative if the real part of OP1 709 is less than that of OP2, and likewise for Y. Both OP1 and OP2 are 710 considered to their full own precision, which may differ. It is 711 not allowed that one of the operands has a NaN (Not-a-Number) part. 712 713 The storage of the return value is such that equality can be simply 714 checked with 'mpc_cmp (op1, op2) == 0'. 715 716 -- Function: int mpc_cmp_abs (const mpc_t OP1, const mpc_t OP2) 717 718 Compare the absolute values of OP1 and OP2. The return value is 0 719 if both are the same (including infinity), positive if the absolute 720 value of OP1 is greater than that of OP2, and negative if it is 721 smaller. If OP1 or OP2 has a real or imaginary part which is NaN, 722 the function behaves like 'mpfr_cmp' on two real numbers of which 723 at least one is NaN. 724 725 726File: mpc.info, Node: Projection & Decomposing, Next: Basic Arithmetic, Prev: Complex Comparison, Up: Complex Functions 727 7285.6 Projection and Decomposing Functions 729======================================== 730 731 -- Function: int mpc_real (mpfr_t ROP, const mpc_t OP, mpfr_rnd_t RND) 732 Set ROP to the value of the real part of OP rounded in the 733 direction RND. 734 735 -- Function: int mpc_imag (mpfr_t ROP, const mpc_t OP, mpfr_rnd_t RND) 736 Set ROP to the value of the imaginary part of OP rounded in the 737 direction RND. 738 739 -- Macro: mpfr_t mpc_realref (mpc_t OP) 740 -- Macro: mpfr_t mpc_imagref (mpc_t OP) 741 Return a reference to the real part and imaginary part of OP, 742 respectively. The 'mpfr' functions can be used on the result of 743 these macros (note that the 'mpfr_t' type is itself a pointer). 744 745 -- Function: int mpc_arg (mpfr_t ROP, const mpc_t OP, mpfr_rnd_t RND) 746 Set ROP to the argument of OP, with a branch cut along the negative 747 real axis. 748 749 -- Function: int mpc_proj (mpc_t ROP, const mpc_t OP, mpc_rnd_t RND) 750 Compute a projection of OP onto the Riemann sphere. Set ROP to OP 751 rounded in the direction RND, except when at least one part of OP 752 is infinite (even if the other part is a NaN) in which case the 753 real part of ROP is set to plus infinity and its imaginary part to 754 a signed zero with the same sign as the imaginary part of OP. 755 756 757File: mpc.info, Node: Basic Arithmetic, Next: Power Functions and Logarithm, Prev: Projection & Decomposing, Up: Complex Functions 758 7595.7 Basic Arithmetic Functions 760============================== 761 762All the following functions are designed in such a way that, when 763working with real numbers instead of complex numbers, their complexity 764should essentially be the same as with the GNU MPFR library, with only a 765marginal overhead due to the GNU MPC layer. 766 767 -- Function: int mpc_add (mpc_t ROP, const mpc_t OP1, const mpc_t OP2, 768 mpc_rnd_t RND) 769 -- Function: int mpc_add_ui (mpc_t ROP, const mpc_t OP1, unsigned long 770 int OP2, mpc_rnd_t RND) 771 -- Function: int mpc_add_fr (mpc_t ROP, const mpc_t OP1, const mpfr_t 772 OP2, mpc_rnd_t RND) 773 Set ROP to OP1 + OP2 rounded according to RND. 774 775 -- Function: int mpc_sub (mpc_t ROP, const mpc_t OP1, const mpc_t OP2, 776 mpc_rnd_t RND) 777 -- Function: int mpc_sub_fr (mpc_t ROP, const mpc_t OP1, const mpfr_t 778 OP2, mpc_rnd_t RND) 779 -- Function: int mpc_fr_sub (mpc_t ROP, const mpfr_t OP1, const mpc_t 780 OP2, mpc_rnd_t RND) 781 -- Function: int mpc_sub_ui (mpc_t ROP, const mpc_t OP1, unsigned long 782 int OP2, mpc_rnd_t RND) 783 -- Macro: int mpc_ui_sub (mpc_t ROP, unsigned long int OP1, const mpc_t 784 OP2, mpc_rnd_t RND) 785 -- Function: int mpc_ui_ui_sub (mpc_t ROP, unsigned long int RE1, 786 unsigned long int IM1, mpc_t OP2, mpc_rnd_t RND) 787 Set ROP to OP1 - OP2 rounded according to RND. For 788 'mpc_ui_ui_sub', OP1 is RE1 + IM1. 789 790 -- Function: int mpc_neg (mpc_t ROP, const mpc_t OP, mpc_rnd_t RND) 791 Set ROP to -OP rounded according to RND. Just changes the sign if 792 ROP and OP are the same variable. 793 794 -- Function: int mpc_sum (mpc_t ROP, const mpc_ptr* OP, unsigned long 795 N, mpc_rnd_t RND) 796 Set ROP to the sum of the elements in the array OP of length N, 797 rounded according to RND. 798 799 -- Function: int mpc_mul (mpc_t ROP, const mpc_t OP1, const mpc_t OP2, 800 mpc_rnd_t RND) 801 -- Function: int mpc_mul_ui (mpc_t ROP, const mpc_t OP1, unsigned long 802 int OP2, mpc_rnd_t RND) 803 -- Function: int mpc_mul_si (mpc_t ROP, const mpc_t OP1, long int OP2, 804 mpc_rnd_t RND) 805 -- Function: int mpc_mul_fr (mpc_t ROP, const mpc_t OP1, const mpfr_t 806 OP2, mpc_rnd_t RND) 807 Set ROP to OP1 times OP2 rounded according to RND. Note: for 808 'mpc_mul', in case OP1 and OP2 have the same value, use 'mpc_sqr' 809 for better efficiency. 810 811 -- Function: int mpc_mul_i (mpc_t ROP, const mpc_t OP, int SGN, 812 mpc_rnd_t RND) 813 Set ROP to OP times the imaginary unit i if SGN is non-negative, 814 set ROP to OP times -i otherwise, in both cases rounded according 815 to RND. 816 817 -- Function: int mpc_sqr (mpc_t ROP, const mpc_t OP, mpc_rnd_t RND) 818 Set ROP to the square of OP rounded according to RND. 819 820 -- Function: int mpc_fma (mpc_t ROP, const mpc_t OP1, const mpc_t OP2, 821 const mpc_t OP3, mpc_rnd_t RND) 822 Set ROP to OP1*OP2+OP3, rounded according to RND, with only one 823 final rounding. 824 825 -- Function: int mpc_dot (mpc_t ROP, const mpc_ptr* OP1, mpc_ptr* OP2, 826 unsigned long N, mpc_rnd_t RND) 827 Set ROP to the dot product of the elements in the arrays OP1 and 828 OP2, both of length N, rounded according to RND. 829 830 -- Function: int mpc_div (mpc_t ROP, const mpc_t OP1, const mpc_t OP2, 831 mpc_rnd_t RND) 832 -- Function: int mpc_div_ui (mpc_t ROP, const mpc_t OP1, unsigned long 833 int OP2, mpc_rnd_t RND) 834 -- Function: int mpc_div_fr (mpc_t ROP, const mpc_t OP1, const mpfr_t 835 OP2, mpc_rnd_t RND) 836 -- Function: int mpc_ui_div (mpc_t ROP, unsigned long int OP1, const 837 mpc_t OP2, mpc_rnd_t RND) 838 -- Function: int mpc_fr_div (mpc_t ROP, const mpfr_t OP1, const mpc_t 839 OP2, mpc_rnd_t RND) 840 Set ROP to OP1/OP2 rounded according to RND. 841 842 -- Function: int mpc_conj (mpc_t ROP, const mpc_t OP, mpc_rnd_t RND) 843 Set ROP to the conjugate of OP rounded according to RND. Just 844 changes the sign of the imaginary part if ROP and OP are the same 845 variable. 846 847 -- Function: int mpc_abs (mpfr_t ROP, const mpc_t OP, mpfr_rnd_t RND) 848 Set the floating-point number ROP to the absolute value of OP, 849 rounded in the direction RND. 850 851 -- Function: int mpc_norm (mpfr_t ROP, const mpc_t OP, mpfr_rnd_t RND) 852 Set the floating-point number ROP to the norm of OP (i.e., the 853 square of its absolute value), rounded in the direction RND. 854 855 -- Function: int mpc_mul_2ui (mpc_t ROP, const mpc_t OP1, unsigned long 856 int OP2, mpc_rnd_t RND) 857 -- Function: int mpc_mul_2si (mpc_t ROP, const mpc_t OP1, long int OP2, 858 mpc_rnd_t RND) 859 Set ROP to OP1 times 2 raised to OP2 rounded according to RND. 860 Just modifies the exponents of the real and imaginary parts by OP2 861 when ROP and OP1 are identical. 862 863 -- Function: int mpc_div_2ui (mpc_t ROP, const mpc_t OP1, unsigned long 864 int OP2, mpc_rnd_t RND) 865 -- Function: int mpc_div_2si (mpc_t ROP, const mpc_t OP1, long int OP2, 866 mpc_rnd_t RND) 867 Set ROP to OP1 divided by 2 raised to OP2 rounded according to RND. 868 Just modifies the exponents of the real and imaginary parts by OP2 869 when ROP and OP1 are identical. 870 871 872File: mpc.info, Node: Power Functions and Logarithm, Next: Trigonometric Functions, Prev: Basic Arithmetic, Up: Complex Functions 873 8745.8 Power Functions and Logarithm 875================================= 876 877 -- Function: int mpc_sqrt (mpc_t ROP, const mpc_t OP, mpc_rnd_t RND) 878 Set ROP to the square root of OP rounded according to RND. The 879 returned value ROP has a non-negative real part, and if its real 880 part is zero, a non-negative imaginary part. 881 882 -- Function: int mpc_pow (mpc_t ROP, const mpc_t OP1, const mpc_t OP2, 883 mpc_rnd_t RND) 884 -- Function: int mpc_pow_d (mpc_t ROP, const mpc_t OP1, double OP2, 885 mpc_rnd_t RND) 886 -- Function: int mpc_pow_ld (mpc_t ROP, const mpc_t OP1, long double 887 OP2, mpc_rnd_t RND) 888 -- Function: int mpc_pow_si (mpc_t ROP, const mpc_t OP1, long OP2, 889 mpc_rnd_t RND) 890 -- Function: int mpc_pow_ui (mpc_t ROP, const mpc_t OP1, unsigned long 891 OP2, mpc_rnd_t RND) 892 -- Function: int mpc_pow_z (mpc_t ROP, const mpc_t OP1, const mpz_t 893 OP2, mpc_rnd_t RND) 894 -- Function: int mpc_pow_fr (mpc_t ROP, const mpc_t OP1, const mpfr_t 895 OP2, mpc_rnd_t RND) 896 Set ROP to OP1 raised to the power OP2, rounded according to RND. 897 For 'mpc_pow_d', 'mpc_pow_ld', 'mpc_pow_si', 'mpc_pow_ui', 898 'mpc_pow_z' and 'mpc_pow_fr', the imaginary part of OP2 is 899 considered as +0. When both OP1 and OP2 are zero, the result has 900 real part 1, and imaginary part 0, with sign being the opposite of 901 that of OP2. 902 903 -- Function: int mpc_exp (mpc_t ROP, const mpc_t OP, mpc_rnd_t RND) 904 Set ROP to the exponential of OP, rounded according to RND with the 905 precision of ROP. 906 907 -- Function: int mpc_log (mpc_t ROP, const mpc_t OP, mpc_rnd_t RND) 908 -- Function: int mpc_log10 (mpc_t ROP, const mpc_t OP, mpc_rnd_t RND) 909 Set ROP to the natural and base-10 logarithm of OP respectively, 910 rounded according to RND with the precision of ROP. The principal 911 branch is chosen, with the branch cut on the negative real axis, so 912 that the imaginary part of the result lies in ]-Pi , Pi] and 913 ]-Pi/log(10) , Pi/log(10)] respectively. 914 915 -- Function: int mpc_rootofunity (mpc_t ROP, unsigned long int N, 916 unsigned long int K, mpc_rnd_t RND) 917 Set ROP to the standard primitive N-th root of unity raised to the 918 power K, that is, exp (2 Pi i k / n), rounded according to RND with 919 the precision of ROP. 920 921 922File: mpc.info, Node: Trigonometric Functions, Next: Miscellaneous Complex Functions, Prev: Power Functions and Logarithm, Up: Complex Functions 923 9245.9 Trigonometric Functions 925=========================== 926 927 -- Function: int mpc_sin (mpc_t ROP, const mpc_t OP, mpc_rnd_t RND) 928 -- Function: int mpc_cos (mpc_t ROP, const mpc_t OP, mpc_rnd_t RND) 929 -- Function: int mpc_tan (mpc_t ROP, const mpc_t OP, mpc_rnd_t RND) 930 Set ROP to the sine, cosine, tangent of OP, rounded according to 931 RND with the precision of ROP. 932 933 -- Function: int mpc_sin_cos (mpc_t ROP_SIN, mpc_t ROP_COS, const mpc_t 934 OP, mpc_rnd_t RND_SIN, mpc_rnd_t RND_COS) 935 Set ROP_SIN to the sine of OP, rounded according to RND_SIN with 936 the precision of ROP_SIN, and ROP_COS to the cosine of OP, rounded 937 according to RND_COS with the precision of ROP_COS. 938 939 -- Function: int mpc_sinh (mpc_t ROP, const mpc_t OP, mpc_rnd_t RND) 940 -- Function: int mpc_cosh (mpc_t ROP, const mpc_t OP, mpc_rnd_t RND) 941 -- Function: int mpc_tanh (mpc_t ROP, const mpc_t OP, mpc_rnd_t RND) 942 Set ROP to the hyperbolic sine, hyperbolic cosine, hyperbolic 943 tangent of OP, rounded according to RND with the precision of ROP. 944 945 -- Function: int mpc_asin (mpc_t ROP, const mpc_t OP, mpc_rnd_t RND) 946 -- Function: int mpc_acos (mpc_t ROP, const mpc_t OP, mpc_rnd_t RND) 947 -- Function: int mpc_atan (mpc_t ROP, const mpc_t OP, mpc_rnd_t RND) 948 Set ROP to the inverse sine, inverse cosine, inverse tangent of OP, 949 rounded according to RND with the precision of ROP. 950 951 -- Function: int mpc_asinh (mpc_t ROP, const mpc_t OP, mpc_rnd_t RND) 952 -- Function: int mpc_acosh (mpc_t ROP, const mpc_t OP, mpc_rnd_t RND) 953 -- Function: int mpc_atanh (mpc_t ROP, const mpc_t OP, mpc_rnd_t RND) 954 Set ROP to the inverse hyperbolic sine, inverse hyperbolic cosine, 955 inverse hyperbolic tangent of OP, rounded according to RND with the 956 precision of ROP. The branch cut of 'mpc_acosh' is (-Inf, 1) 957 958 959File: mpc.info, Node: Miscellaneous Complex Functions, Next: Advanced Functions, Prev: Trigonometric Functions, Up: Complex Functions 960 9615.10 Miscellaneous Functions 962============================ 963 964 -- Function: int mpc_urandom (mpc_t ROP, gmp_randstate_t STATE) 965 Generate a uniformly distributed random complex in the unit square 966 [0, 1] x [0, 1]. Return 0, unless an exponent in the real or 967 imaginary part is not in the current exponent range, in which case 968 that part is set to NaN and a zero value is returned. The second 969 argument is a 'gmp_randstate_t' structure which should be created 970 using the GMP 'rand_init' function, see the GMP manual. 971 972 -- Function: const char * mpc_get_version (void) 973 Return the GNU MPC version, as a null-terminated string. 974 975 -- Macro: MPC_VERSION 976 -- Macro: MPC_VERSION_MAJOR 977 -- Macro: MPC_VERSION_MINOR 978 -- Macro: MPC_VERSION_PATCHLEVEL 979 -- Macro: MPC_VERSION_STRING 980 'MPC_VERSION' is the version of GNU MPC as a preprocessing 981 constant. 'MPC_VERSION_MAJOR', 'MPC_VERSION_MINOR' and 982 'MPC_VERSION_PATCHLEVEL' are respectively the major, minor and 983 patch level of GNU MPC version, as preprocessing constants. 984 'MPC_VERSION_STRING' is the version as a string constant, which can 985 be compared to the result of 'mpc_get_version' to check at run time 986 the header file and library used match: 987 if (strcmp (mpc_get_version (), MPC_VERSION_STRING)) 988 fprintf (stderr, "Warning: header and library do not match\n"); 989 Note: Obtaining different strings is not necessarily an error, as 990 in general, a program compiled with some old GNU MPC version can be 991 dynamically linked with a newer GNU MPC library version (if allowed 992 by the library versioning system). 993 994 -- Macro: long MPC_VERSION_NUM (MAJOR, MINOR, PATCHLEVEL) 995 Create an integer in the same format as used by 'MPC_VERSION' from 996 the given MAJOR, MINOR and PATCHLEVEL. Here is an example of how 997 to check the GNU MPC version at compile time: 998 #if (!defined(MPC_VERSION) || (MPC_VERSION<MPC_VERSION_NUM(2,1,0))) 999 # error "Wrong GNU MPC version." 1000 #endif 1001 1002 1003File: mpc.info, Node: Advanced Functions, Next: Internals, Prev: Miscellaneous Complex Functions, Up: Complex Functions 1004 10055.11 Advanced Functions 1006======================= 1007 1008 -- Macro: MPC_SET_X_Y (REAL_SUFFIX, IMAG_SUFFIX, ROP, REAL, IMAG, RND) 1009 The macro MPC_SET_X_Y is designed to serve as the body of an 1010 assignment function and cannot be used by itself. The REAL_SUFFIX 1011 and IMAG_SUFFIX parameters are the types of the real and imaginary 1012 part, that is, the 'x' in the 'mpfr_set_x' function one would use 1013 to set the part; for the mpfr type, use 'fr'. REAL (respectively 1014 IMAG) is the value you want to assign to the real (resp. 1015 imaginary) part, its type must conform to REAL_SUFFIX (resp. 1016 IMAG_SUFFIX). RND is the 'mpc_rnd_t' rounding mode. The return 1017 value is the usual inexact value (*note Return Value: 1018 return-value.). 1019 1020 For instance, you can define mpc_set_ui_fr as follows: 1021 int mpc_set_ui_fr (mpc_t rop, unsigned long int re, mpfr_t im, mpc_rnd_t rnd) 1022 MPC_SET_X_Y (ui, fr, rop, re, im, rnd); 1023 1024 1025File: mpc.info, Node: Internals, Prev: Advanced Functions, Up: Complex Functions 1026 10275.12 Internals 1028============== 1029 1030These macros and functions are mainly designed for the implementation of 1031GNU MPC, but may be useful for users too. However, no upward 1032compatibility is guaranteed. You need to include 'mpc-impl.h' to use 1033them. 1034 1035 The macro 'MPC_MAX_PREC(z)' gives the maximum of the precisions of 1036the real and imaginary parts of a complex number. 1037 1038 1039File: mpc.info, Node: References, Next: Concept Index, Prev: Complex Functions, Up: Top 1040 1041References 1042********** 1043 1044 * Torbj��rn Granlund et al. 'GMP' - GNU multiprecision library. 1045 Version 6.2.0, <http://gmplib.org>. 1046 1047 * Guillaume Hanrot, Vincent Lef��vre, Patrick P��lissier, Paul 1048 Zimmermann et al. 'MPFR' - A library for multiple-precision 1049 floating-point computations with exact rounding. Version 4.1.0, 1050 <http://www.mpfr.org>. 1051 1052 * IEEE Standard for Floating-Point Arithmetic, IEEE Computer Society, 1053 IEEE Std 754-2019, Approved 13 June 2019, 84 pages. 1054 1055 * Donald E. Knuth, "The Art of Computer Programming", vol 2, 1056 "Seminumerical Algorithms", 2nd edition, Addison-Wesley, 1981. 1057 1058 * ISO/IEC 9899:1999, Programming languages ��� C. 1059 1060 1061File: mpc.info, Node: Concept Index, Next: Function Index, Prev: References, Up: Top 1062 1063Concept Index 1064************* 1065 1066[index] 1067* Menu: 1068 1069* Arithmetic functions: Basic Arithmetic. (line 6) 1070* Comparison functions: Complex Comparison. (line 6) 1071* Complex arithmetic functions: Basic Arithmetic. (line 6) 1072* Complex assignment functions: Assigning Complex Numbers. 1073 (line 6) 1074* Complex comparisons functions: Complex Comparison. (line 6) 1075* Complex functions: Complex Functions. (line 6) 1076* Complex number: GNU MPC Basics. (line 15) 1077* Conditions for copying GNU MPC: Copying. (line 6) 1078* Conversion functions: Converting Complex Numbers. 1079 (line 6) 1080* Copying conditions: Copying. (line 6) 1081* Installation: Installing GNU MPC. (line 6) 1082* Logarithm: Power Functions and Logarithm. 1083 (line 6) 1084* Miscellaneous complex functions: Miscellaneous Complex Functions. 1085 (line 6) 1086* mpc.h: GNU MPC Basics. (line 6) 1087* Power functions: Power Functions and Logarithm. 1088 (line 6) 1089* Precision: GNU MPC Basics. (line 19) 1090* Projection and Decomposing Functions: Projection & Decomposing. 1091 (line 6) 1092* Reporting bugs: Reporting Bugs. (line 6) 1093* Rounding Mode: GNU MPC Basics. (line 24) 1094* String and stream input and output: String and Stream Input and Output. 1095 (line 6) 1096* Trigonometric functions: Trigonometric Functions. 1097 (line 6) 1098* User-defined precision: Complex Functions. (line 12) 1099 1100 1101File: mpc.info, Node: Function Index, Next: GNU Free Documentation License, Prev: Concept Index, Up: Top 1102 1103Function Index 1104************** 1105 1106[index] 1107* Menu: 1108 1109* _Complex: Converting Complex Numbers. 1110 (line 9) 1111* mpc_abs: Basic Arithmetic. (line 91) 1112* mpc_acos: Trigonometric Functions. 1113 (line 25) 1114* mpc_acosh: Trigonometric Functions. 1115 (line 31) 1116* mpc_add: Basic Arithmetic. (line 11) 1117* mpc_add_fr: Basic Arithmetic. (line 15) 1118* mpc_add_ui: Basic Arithmetic. (line 13) 1119* mpc_arg: Projection & Decomposing. 1120 (line 20) 1121* mpc_asin: Trigonometric Functions. 1122 (line 24) 1123* mpc_asinh: Trigonometric Functions. 1124 (line 30) 1125* mpc_atan: Trigonometric Functions. 1126 (line 26) 1127* mpc_atanh: Trigonometric Functions. 1128 (line 32) 1129* mpc_clear: Initializing Complex Numbers. 1130 (line 21) 1131* mpc_cmp: Complex Comparison. (line 6) 1132* mpc_cmp_abs: Complex Comparison. (line 23) 1133* mpc_cmp_si: Complex Comparison. (line 9) 1134* mpc_cmp_si_si: Complex Comparison. (line 7) 1135* mpc_conj: Basic Arithmetic. (line 86) 1136* mpc_cos: Trigonometric Functions. 1137 (line 7) 1138* mpc_cosh: Trigonometric Functions. 1139 (line 19) 1140* mpc_div: Basic Arithmetic. (line 74) 1141* mpc_div_2si: Basic Arithmetic. (line 109) 1142* mpc_div_2ui: Basic Arithmetic. (line 107) 1143* mpc_div_fr: Basic Arithmetic. (line 78) 1144* mpc_div_ui: Basic Arithmetic. (line 76) 1145* mpc_dot: Basic Arithmetic. (line 69) 1146* mpc_exp: Power Functions and Logarithm. 1147 (line 32) 1148* mpc_fma: Basic Arithmetic. (line 64) 1149* mpc_free_str: String and Stream Input and Output. 1150 (line 66) 1151* mpc_fr_div: Basic Arithmetic. (line 82) 1152* mpc_fr_sub: Basic Arithmetic. (line 23) 1153* mpc_get_ldc: Converting Complex Numbers. 1154 (line 10) 1155* mpc_get_prec: Initializing Complex Numbers. 1156 (line 49) 1157* mpc_get_prec2: Initializing Complex Numbers. 1158 (line 53) 1159* mpc_get_str: String and Stream Input and Output. 1160 (line 48) 1161* mpc_get_version: Miscellaneous Complex Functions. 1162 (line 14) 1163* mpc_imag: Projection & Decomposing. 1164 (line 10) 1165* mpc_imagref: Projection & Decomposing. 1166 (line 15) 1167* mpc_init2: Initializing Complex Numbers. 1168 (line 10) 1169* mpc_init3: Initializing Complex Numbers. 1170 (line 15) 1171* mpc_inp_str: String and Stream Input and Output. 1172 (line 74) 1173* mpc_log: Power Functions and Logarithm. 1174 (line 36) 1175* mpc_log10: Power Functions and Logarithm. 1176 (line 37) 1177* mpc_mul: Basic Arithmetic. (line 43) 1178* mpc_mul_2si: Basic Arithmetic. (line 101) 1179* mpc_mul_2ui: Basic Arithmetic. (line 99) 1180* mpc_mul_fr: Basic Arithmetic. (line 49) 1181* mpc_mul_i: Basic Arithmetic. (line 55) 1182* mpc_mul_si: Basic Arithmetic. (line 47) 1183* mpc_mul_ui: Basic Arithmetic. (line 45) 1184* mpc_neg: Basic Arithmetic. (line 34) 1185* mpc_norm: Basic Arithmetic. (line 95) 1186* mpc_out_str: String and Stream Input and Output. 1187 (line 109) 1188* mpc_pow: Power Functions and Logarithm. 1189 (line 11) 1190* mpc_pow_d: Power Functions and Logarithm. 1191 (line 13) 1192* mpc_pow_fr: Power Functions and Logarithm. 1193 (line 23) 1194* mpc_pow_ld: Power Functions and Logarithm. 1195 (line 15) 1196* mpc_pow_si: Power Functions and Logarithm. 1197 (line 17) 1198* mpc_pow_ui: Power Functions and Logarithm. 1199 (line 19) 1200* mpc_pow_z: Power Functions and Logarithm. 1201 (line 21) 1202* mpc_proj: Projection & Decomposing. 1203 (line 24) 1204* mpc_real: Projection & Decomposing. 1205 (line 6) 1206* mpc_realref: Projection & Decomposing. 1207 (line 14) 1208* mpc_rnd_t: GNU MPC Basics. (line 24) 1209* mpc_rootofunity: Power Functions and Logarithm. 1210 (line 44) 1211* mpc_set: Assigning Complex Numbers. 1212 (line 16) 1213* mpc_set_d: Assigning Complex Numbers. 1214 (line 25) 1215* mpc_set_dc: Assigning Complex Numbers. 1216 (line 27) 1217* mpc_set_d_d: Assigning Complex Numbers. 1218 (line 54) 1219* mpc_set_f: Assigning Complex Numbers. 1220 (line 33) 1221* mpc_set_fr: Assigning Complex Numbers. 1222 (line 34) 1223* mpc_set_fr_fr: Assigning Complex Numbers. 1224 (line 64) 1225* mpc_set_f_f: Assigning Complex Numbers. 1226 (line 62) 1227* mpc_set_ld: Assigning Complex Numbers. 1228 (line 26) 1229* mpc_set_ldc: Assigning Complex Numbers. 1230 (line 29) 1231* mpc_set_ld_ld: Assigning Complex Numbers. 1232 (line 56) 1233* mpc_set_nan: Assigning Complex Numbers. 1234 (line 79) 1235* mpc_set_prec: Initializing Complex Numbers. 1236 (line 41) 1237* mpc_set_q: Assigning Complex Numbers. 1238 (line 32) 1239* mpc_set_q_q: Assigning Complex Numbers. 1240 (line 60) 1241* mpc_set_si: Assigning Complex Numbers. 1242 (line 22) 1243* mpc_set_si_si: Assigning Complex Numbers. 1244 (line 48) 1245* mpc_set_sj: Assigning Complex Numbers. 1246 (line 24) 1247* mpc_set_sj_sj: Assigning Complex Numbers. 1248 (line 52) 1249* mpc_set_str: String and Stream Input and Output. 1250 (line 35) 1251* mpc_set_ui: Assigning Complex Numbers. 1252 (line 20) 1253* mpc_set_ui_ui: Assigning Complex Numbers. 1254 (line 46) 1255* mpc_set_uj: Assigning Complex Numbers. 1256 (line 23) 1257* mpc_set_uj_uj: Assigning Complex Numbers. 1258 (line 50) 1259* MPC_SET_X_Y: Advanced Functions. (line 6) 1260* mpc_set_z: Assigning Complex Numbers. 1261 (line 31) 1262* mpc_set_z_z: Assigning Complex Numbers. 1263 (line 58) 1264* mpc_sin: Trigonometric Functions. 1265 (line 6) 1266* mpc_sinh: Trigonometric Functions. 1267 (line 18) 1268* mpc_sin_cos: Trigonometric Functions. 1269 (line 12) 1270* mpc_sqr: Basic Arithmetic. (line 61) 1271* mpc_sqrt: Power Functions and Logarithm. 1272 (line 6) 1273* mpc_strtoc: String and Stream Input and Output. 1274 (line 6) 1275* mpc_sub: Basic Arithmetic. (line 19) 1276* mpc_sub_fr: Basic Arithmetic. (line 21) 1277* mpc_sub_ui: Basic Arithmetic. (line 25) 1278* mpc_sum: Basic Arithmetic. (line 38) 1279* mpc_swap: Assigning Complex Numbers. 1280 (line 82) 1281* mpc_t: GNU MPC Basics. (line 15) 1282* mpc_tan: Trigonometric Functions. 1283 (line 8) 1284* mpc_tanh: Trigonometric Functions. 1285 (line 20) 1286* mpc_ui_div: Basic Arithmetic. (line 80) 1287* mpc_ui_sub: Basic Arithmetic. (line 27) 1288* mpc_ui_ui_sub: Basic Arithmetic. (line 29) 1289* mpc_urandom: Miscellaneous Complex Functions. 1290 (line 6) 1291* MPC_VERSION: Miscellaneous Complex Functions. 1292 (line 17) 1293* MPC_VERSION_MAJOR: Miscellaneous Complex Functions. 1294 (line 18) 1295* MPC_VERSION_MINOR: Miscellaneous Complex Functions. 1296 (line 19) 1297* MPC_VERSION_NUM: Miscellaneous Complex Functions. 1298 (line 36) 1299* MPC_VERSION_PATCHLEVEL: Miscellaneous Complex Functions. 1300 (line 20) 1301* MPC_VERSION_STRING: Miscellaneous Complex Functions. 1302 (line 21) 1303* mpfr_prec_t: GNU MPC Basics. (line 19) 1304 1305 1306File: mpc.info, Node: GNU Free Documentation License, Prev: Function Index, Up: Top 1307 1308Appendix A GNU Free Documentation License 1309***************************************** 1310 1311 Version 1.3, 3 November 2008 1312 1313 Copyright (C) 2000, 2001, 2002, 2007, 2008 Free Software Foundation, Inc. 1314 <http://fsf.org/> 1315 1316 Everyone is permitted to copy and distribute verbatim copies 1317 of this license document, but changing it is not allowed. 1318 1319 0. PREAMBLE 1320 1321 The purpose of this License is to make a manual, textbook, or other 1322 functional and useful document "free" in the sense of freedom: to 1323 assure everyone the effective freedom to copy and redistribute it, 1324 with or without modifying it, either commercially or 1325 noncommercially. Secondarily, this License preserves for the 1326 author and publisher a way to get credit for their work, while not 1327 being considered responsible for modifications made by others. 1328 1329 This License is a kind of "copyleft", which means that derivative 1330 works of the document must themselves be free in the same sense. 1331 It complements the GNU General Public License, which is a copyleft 1332 license designed for free software. 1333 1334 We have designed this License in order to use it for manuals for 1335 free software, because free software needs free documentation: a 1336 free program should come with manuals providing the same freedoms 1337 that the software does. But this License is not limited to 1338 software manuals; it can be used for any textual work, regardless 1339 of subject matter or whether it is published as a printed book. We 1340 recommend this License principally for works whose purpose is 1341 instruction or reference. 1342 1343 1. APPLICABILITY AND DEFINITIONS 1344 1345 This License applies to any manual or other work, in any medium, 1346 that contains a notice placed by the copyright holder saying it can 1347 be distributed under the terms of this License. Such a notice 1348 grants a world-wide, royalty-free license, unlimited in duration, 1349 to use that work under the conditions stated herein. The 1350 "Document", below, refers to any such manual or work. Any member 1351 of the public is a licensee, and is addressed as "you". 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A copy of the license is included in the section entitled ``GNU 1769 Free Documentation License''. 1770 1771 If you have Invariant Sections, Front-Cover Texts and Back-Cover 1772Texts, replace the "with...Texts." line with this: 1773 1774 with the Invariant Sections being LIST THEIR TITLES, with 1775 the Front-Cover Texts being LIST, and with the Back-Cover Texts 1776 being LIST. 1777 1778 If you have Invariant Sections without Cover Texts, or some other 1779combination of the three, merge those two alternatives to suit the 1780situation. 1781 1782 If your document contains nontrivial examples of program code, we 1783recommend releasing these examples in parallel under your choice of free 1784software license, such as the GNU General Public License, to permit 1785their use in free software. 1786 1787 1788 1789Tag Table: 1790Node: Top759 1791Node: Copying1466 1792Node: Introduction to GNU MPC2238 1793Node: Installing GNU MPC2957 1794Node: Reporting Bugs8042 1795Node: GNU MPC Basics9386 1796Ref: return-value13063 1797Node: Complex Functions14514 1798Node: Initializing Complex Numbers15674 1799Node: Assigning Complex Numbers18073 1800Node: Converting Complex Numbers22551 1801Node: String and Stream Input and Output23182 1802Node: Complex Comparison29750 1803Node: Projection & Decomposing31293 1804Node: Basic Arithmetic32694 1805Node: Power Functions and Logarithm37946 1806Node: Trigonometric Functions40379 1807Node: Miscellaneous Complex Functions42344 1808Node: Advanced Functions44520 1809Node: Internals45602 1810Node: References46053 1811Node: Concept Index46830 1812Node: Function Index49144 1813Node: GNU Free Documentation License63292 1814 1815End Tag Table 1816 1817 1818Local Variables: 1819coding: utf-8 1820End: 1821