1 This is mpc.info, produced by makeinfo version 4.13 from mpc.texi. 2 3 This manual is for GNU MPC, a library for multiple precision complex 4 arithmetic, version 1.0.1 of September 2012. 5 6 Copyright (C) 2002, 2003, 2004, 2005, 2006, 2007, 2008, 2009, 2010, 7 2011, 2012 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." 14 15 INFO-DIR-SECTION GNU Packages 16 START-INFO-DIR-ENTRY 17 * mpc: (mpc)Multiple Precision Complex Library. 18 END-INFO-DIR-ENTRY 19 20 21 File: mpc.info, Node: Top, Next: Copying, Up: (dir) 22 23 GNU MPC 24 ******* 25 26 This manual documents how to install and use the GNU Multiple Precision 27 Complex Library, version 1.0.1 28 29 * Menu: 30 31 * Copying:: GNU MPC Copying Conditions (LGPL). 32 * Introduction to GNU MPC:: Brief introduction to GNU MPC. 33 * Installing GNU MPC:: How to configure and compile the GNU MPC library. 34 * Reporting Bugs:: How to usefully report bugs. 35 * GNU MPC Basics:: What every GNU MPC user should know. 36 * Complex Functions:: Functions for arithmetic on complex numbers. 37 * References:: 38 * Concept Index:: 39 * Function Index:: 40 * GNU Free Documentation License:: 41 42 43 File: mpc.info, Node: Copying, Next: Introduction to GNU MPC, Prev: Top, Up: Top 44 45 GNU MPC Copying Conditions 46 ************************** 47 48 GNU MPC is free software; you can redistribute it and/or modify it under 49 the terms of the GNU Lesser General Public License as published by the 50 Free Software Foundation; either version 3 of the License, or (at your 51 option) any later version. 52 53 GNU MPC is distributed in the hope that it will be useful, but 54 WITHOUT ANY WARRANTY; without even the implied warranty of 55 MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the GNU Lesser 56 General Public License for more details. 57 58 You should have received a copy of the GNU Lesser General Public 59 License along with this program. If not, see 60 `http://www.gnu.org/licenses/'. 61 62 63 File: mpc.info, Node: Introduction to GNU MPC, Next: Installing GNU MPC, Prev: Copying, Up: Top 64 65 1 Introduction to GNU MPC 66 ************************* 67 68 GNU MPC is a portable library written in C for arbitrary precision 69 arithmetic on complex numbers providing correct rounding. It implements 70 a multiprecision equivalent of the C99 standard. It builds upon the 71 GNU MP and the GNU MPFR libraries. 72 73 1.1 How to use this Manual 74 ========================== 75 76 Everyone should read *note GNU MPC Basics::. If you need to install 77 the library yourself, you need to read *note Installing GNU MPC::, too. 78 79 The remainder of the manual can be used for later reference, 80 although it is probably a good idea to skim through it. 81 82 83 File: mpc.info, Node: Installing GNU MPC, Next: Reporting Bugs, Prev: Introduction to GNU MPC, Up: Top 84 85 2 Installing GNU MPC 86 ******************** 87 88 To build GNU MPC, you first have to install GNU MP (version 4.3.2 or 89 higher) and GNU MPFR (version 2.4.2 or higher) on your computer. You 90 need a C compiler; GCC version 4.4 or higher is recommended, since GNU 91 MPC may trigger a bug in previous versions, see the thread at 92 `http://lists.gforge.inria.fr/pipermail/mpc-discuss/2011-February/000823.html'. 93 And you need a standard Unix `make' program, plus some other standard 94 Unix utility programs. 95 96 Here are the steps needed to install the library on Unix systems: 97 98 1. `tar xzf mpc-1.0.1.tar.gz' 99 100 2. `cd mpc-1.0.1' 101 102 3. `./configure' 103 104 if GMP and GNU MPFR are installed into standard directories, that 105 is, directories that are searched by default by the compiler and 106 the linking tools. 107 108 `./configure --with-gmp=<gmp_install_dir>' 109 110 is used to indicate a different location where GMP is installed. 111 Alternatively, you can specify directly GMP include and GMP lib 112 directories with `./configure --with-gmp-lib=<gmp_lib_dir> 113 --with-gmp-include=<gmp_include_dir>'. 114 115 `./configure --with-mpfr=<mpfr_install_dir>' 116 117 is used to indicate a different location where GNU MPFR is 118 installed. Alternatively, you can specify directly GNU MPFR 119 include and GNU MPFR lib directories with `./configure 120 --with-mpf-lib=<mpfr_lib_dir> 121 --with-mpfr-include=<mpfr_include_dir>'. 122 123 Another useful parameter is `--prefix', which can be used to 124 specify an alternative installation location instead of 125 `/usr/local'; see `make install' below. 126 127 To enable checking for memory leaks using `valgrind' during `make 128 check', add the parameter `--enable-valgrind-tests'. 129 130 If for debugging purposes you wish to log calls to GNU MPC 131 functions from within your code, add the parameter 132 `--enable-logging'. In your code, replace the inclusion of 133 `mpc.h' by `mpc-log.h' and link the executable dynamically. Then 134 all calls to functions with only complex arguments are printed to 135 `stderr' in the following form: First, the function name is given, 136 followed by its type such as `c_cc', meaning that the function has 137 one complex result (one `c' in front of the `_'), computed from 138 two complex arguments (two `c' after the `_'). Then, the 139 precisions of the real and the imaginary part of the first result 140 is given, followed by the second one and so on. Finally, for each 141 argument, the precisions of its real and imaginary part are 142 specified and the argument itself is printed in hexadecimal via 143 the function `mpc_out_str' (*note String and Stream Input and 144 Output::). The option requires a dynamic library, so it may not 145 be combined with `--disable-shared'. 146 147 Use `./configure --help' for an exhaustive list of parameters. 148 149 4. `make' 150 151 This compiles GNU MPC in the working directory. 152 153 5. `make check' 154 155 This will make sure GNU MPC was built correctly. 156 157 If you get error messages, please report them to 158 `mpc-discuss (a] lists.gforge.inria.fr' (*Note Reporting Bugs::, for 159 information on what to include in useful bug reports). 160 161 6. `make install' 162 163 This will copy the file `mpc.h' to the directory 164 `/usr/local/include', the file `libmpc.a' to the directory 165 `/usr/local/lib', and the file `mpc.info' to the directory 166 `/usr/local/share/info' (or if you passed the `--prefix' option to 167 `configure', using the prefix directory given as argument to 168 `--prefix' instead of `/usr/local'). Note: you need write 169 permissions on these directories. 170 171 172 2.1 Other `make' Targets 173 ======================== 174 175 There are some other useful make targets: 176 177 * `info' 178 179 Create an info version of the manual, in `mpc.info'. 180 181 * `pdf' 182 183 Create a PDF version of the manual, in `doc/mpc.pdf'. 184 185 * `dvi' 186 187 Create a DVI version of the manual, in `doc/mpc.dvi'. 188 189 * `ps' 190 191 Create a Postscript version of the manual, in `doc/mpc.ps'. 192 193 * `html' 194 195 Create an HTML version of the manual, in several pages in the 196 directory `doc/mpc.html'; if you want only one output HTML file, 197 then type `makeinfo --html --no-split mpc.texi' instead. 198 199 * `clean' 200 201 Delete all object files and archive files, but not the 202 configuration files. 203 204 * `distclean' 205 206 Delete all files not included in the distribution. 207 208 * `uninstall' 209 210 Delete all files copied by `make install'. 211 212 2.2 Known Build Problems 213 ======================== 214 215 On AIX, if GMP was built with the 64-bit ABI, before building and 216 testing GNU MPC, it might be necessary to set the `OBJECT_MODE' 217 environment variable to 64 by, e.g., 218 219 `export OBJECT_MODE=64' 220 221 This has been tested with the C compiler IBM XL C/C++ Enterprise 222 Edition V8.0 for AIX, version: 08.00.0000.0021, GMP 4.2.4 and GNU MPFR 223 2.4.1. 224 225 Please report any other problems you encounter to 226 `mpc-discuss (a] lists.gforge.inria.fr'. *Note Reporting Bugs::. 227 228 229 File: mpc.info, Node: Reporting Bugs, Next: GNU MPC Basics, Prev: Installing GNU MPC, Up: Top 230 231 3 Reporting Bugs 232 **************** 233 234 If you think you have found a bug in the GNU MPC library, please 235 investigate and report it. We have made this library available to you, 236 and it is not to ask too much from you, to ask you to report the bugs 237 that you find. 238 239 There are a few things you should think about when you put your bug 240 report together. 241 242 You have to send us a test case that makes it possible for us to 243 reproduce the bug. Include instructions on how to run the test case. 244 245 You also have to explain what is wrong; if you get a crash, or if 246 the results printed are incorrect and in that case, in what way. 247 248 Please include compiler version information in your bug report. 249 This can be extracted using `gcc -v', or `cc -V' on some machines. 250 Also, include the output from `uname -a'. 251 252 If your bug report is good, we will do our best to help you to get a 253 corrected version of the library; if the bug report is poor, we will 254 not do anything about it (aside of chiding you to send better bug 255 reports). 256 257 Send your bug report to: `mpc-discuss (a] lists.gforge.inria.fr'. 258 259 If you think something in this manual is unclear, or downright 260 incorrect, or if the language needs to be improved, please send a note 261 to the same address. 262 263 264 File: mpc.info, Node: GNU MPC Basics, Next: Complex Functions, Prev: Reporting Bugs, Up: Top 265 266 4 GNU MPC Basics 267 **************** 268 269 All declarations needed to use GNU MPC are collected in the include file 270 `mpc.h'. It is designed to work with both C and C++ compilers. You 271 should include that file in any program using the GNU MPC library by 272 adding the line 273 #include "mpc.h" 274 275 4.1 Nomenclature and Types 276 ========================== 277 278 "Complex number" or "Complex" for short, is a pair of two arbitrary 279 precision floating-point numbers (for the real and imaginary parts). 280 The C data type for such objects is `mpc_t'. 281 282 The "Precision" is the number of bits used to represent the mantissa of 283 the real and imaginary parts; the corresponding C data type is 284 `mpfr_prec_t'. For more details on the allowed precision range, *note 285 Nomenclature and Types: (mpfr.info)Nomenclature and Types. 286 287 The "rounding mode" specifies the way to round the result of a complex 288 operation, in case the exact result can not be represented exactly in 289 the destination mantissa; the corresponding C data type is `mpc_rnd_t'. 290 A complex rounding mode is a pair of two rounding modes: one for the 291 real part, one for the imaginary part. 292 293 4.2 Function Classes 294 ==================== 295 296 There is only one class of functions in the GNU MPC library, namely 297 functions for complex arithmetic. The function names begin with `mpc_'. 298 The associated type is `mpc_t'. 299 300 4.3 GNU MPC Variable Conventions 301 ================================ 302 303 As a general rule, all GNU MPC functions expect output arguments before 304 input arguments. This notation is based on an analogy with the 305 assignment operator. 306 307 GNU MPC allows you to use the same variable for both input and 308 output in the same expression. For example, the main function for 309 floating-point multiplication, `mpc_mul', can be used like this: 310 `mpc_mul (x, x, x, rnd_mode)'. This computes the square of X with 311 rounding mode `rnd_mode' and puts the result back in X. 312 313 Before you can assign to an GNU MPC variable, you need to initialize 314 it by calling one of the special initialization functions. When you 315 are done with a variable, you need to clear it out, using one of the 316 functions for that purpose. 317 318 A variable should only be initialized once, or at least cleared out 319 between each initialization. After a variable has been initialized, it 320 may be assigned to any number of times. 321 322 For efficiency reasons, avoid to initialize and clear out a variable 323 in loops. Instead, initialize it before entering the loop, and clear 324 it out after the loop has exited. 325 326 You do not need to be concerned about allocating additional space 327 for GNU MPC variables, since each of its real and imaginary part has a 328 mantissa of fixed size. Hence unless you change its precision, or 329 clear and reinitialize it, a complex variable will have the same 330 allocated space during all its life. 331 332 4.4 Rounding Modes 333 ================== 334 335 A complex rounding mode is of the form `MPC_RNDxy' where `x' and `y' 336 are one of `N' (to nearest), `Z' (towards zero), `U' (towards plus 337 infinity), `D' (towards minus infinity). The first letter refers to 338 the rounding mode for the real part, and the second one for the 339 imaginary part. For example `MPC_RNDZU' indicates to round the real 340 part towards zero, and the imaginary part towards plus infinity. 341 342 The `round to nearest' mode works as in the IEEE P754 standard: in 343 case the number to be rounded lies exactly in the middle of two 344 representable numbers, it is rounded to the one with the least 345 significant bit set to zero. For example, the number 5, which is 346 represented by (101) in binary, is rounded to (100)=4 with a precision 347 of two bits, and not to (110)=6. 348 349 4.5 Return Value 350 ================ 351 352 Most GNU MPC functions have a return value of type `int', which is used 353 to indicate the position of the rounded real and imaginary parts with 354 respect to the exact (infinite precision) values. If this integer is 355 `i', the macros `MPC_INEX_RE(i)' and `MPC_INEX_IM(i)' give 0 if the 356 corresponding rounded value is exact, a negative value if the rounded 357 value is less than the exact one, and a positive value if it is greater 358 than the exact one. Similarly, functions computing a result of type 359 `mpfr_t' return an integer that is 0, positive or negative depending on 360 whether the rounded value is the same, larger or smaller then the exact 361 result. 362 363 Some functions, such as `mpc_sin_cos', compute two complex results; 364 the macros `MPC_INEX1(i)' and `MPC_INEX2(i)', applied to the return 365 value `i' of such a function, yield the exactness value corresponding 366 to the first or the second computed value, respectively. 367 368 4.6 Branch Cuts And Special Values 369 ================================== 370 371 Some complex functions have branch cuts, across which the function is 372 discontinous. In GNU MPC, the branch cuts chosen are the same as those 373 specified for the corresponding functions in the ISO C99 standard. 374 375 Likewise, when evaluated at a point whose real or imaginary part is 376 either infinite or a NaN or a signed zero, a function returns the same 377 value as those specified for the corresponding function in the ISO C99 378 standard. 379 380 381 File: mpc.info, Node: Complex Functions, Next: References, Prev: GNU MPC Basics, Up: Top 382 383 5 Complex Functions 384 ******************* 385 386 The complex functions expect arguments of type `mpc_t'. 387 388 The GNU MPC floating-point functions have an interface that is 389 similar to the GNU MP integer functions. The function prefix for 390 operations on complex numbers is `mpc_'. 391 392 The precision of a computation is defined as follows: Compute the 393 requested operation exactly (with "infinite precision"), and round the 394 result to the destination variable precision with the given rounding 395 mode. 396 397 The GNU MPC complex functions are intended to be a smooth extension 398 of the IEEE P754 arithmetic. The results obtained on one computer 399 should not differ from the results obtained on a computer with a 400 different word size. 401 402 * Menu: 403 404 * Initializing Complex Numbers:: 405 * Assigning Complex Numbers:: 406 * Converting Complex Numbers:: 407 * String and Stream Input and Output:: 408 * Complex Comparison:: 409 * Projection & Decomposing:: 410 * Basic Arithmetic:: 411 * Power Functions and Logarithm:: 412 * Trigonometric Functions:: 413 * Miscellaneous Complex Functions:: 414 * Advanced Functions:: 415 * Internals:: 416 417 418 File: mpc.info, Node: Initializing Complex Numbers, Next: Assigning Complex Numbers, Up: Complex Functions 419 420 5.1 Initialization Functions 421 ============================ 422 423 An `mpc_t' object must be initialized before storing the first value in 424 it. The functions `mpc_init2' and `mpc_init3' are used for that 425 purpose. 426 427 -- Function: void mpc_init2 (mpc_t Z, mpfr_prec_t PREC) 428 Initialize Z to precision PREC bits and set its real and imaginary 429 parts to NaN. Normally, a variable should be initialized once only 430 or at least be cleared, using `mpc_clear', between initializations. 431 432 -- Function: void mpc_init3 (mpc_t Z, mpfr_prec_t PREC_R, mpfr_prec_t 433 PREC_I) 434 Initialize Z with the precision of its real part being PREC_R bits 435 and the precision of its imaginary part being PREC_I bits, and set 436 the real and imaginary parts to NaN. 437 438 -- Function: void mpc_clear (mpc_t Z) 439 Free the space occupied by Z. Make sure to call this function for 440 all `mpc_t' variables when you are done with them. 441 442 Here is an example on how to initialize complex variables: 443 { 444 mpc_t x, y; 445 mpc_init2 (x, 256); /* precision _exactly_ 256 bits */ 446 mpc_init3 (y, 100, 50); /* 100/50 bits for the real/imaginary part */ 447 ... 448 mpc_clear (x); 449 mpc_clear (y); 450 } 451 452 The following function is useful for changing the precision during a 453 calculation. A typical use would be for adjusting the precision 454 gradually in iterative algorithms like Newton-Raphson, making the 455 computation precision closely match the actual accurate part of the 456 numbers. 457 458 -- Function: void mpc_set_prec (mpc_t X, mpfr_prec_t PREC) 459 Reset the precision of X to be *exactly* PREC bits, and set its 460 real/imaginary parts to NaN. The previous value stored in X is 461 lost. It is equivalent to a call to `mpc_clear(x)' followed by a 462 call to `mpc_init2(x, prec)', but more efficient as no allocation 463 is done in case the current allocated space for the mantissa of X 464 is sufficient. 465 466 -- Function: mpfr_prec_t mpc_get_prec (mpc_t X) 467 If the real and imaginary part of X have the same precision, it is 468 returned, otherwise, 0 is returned. 469 470 -- Function: void mpc_get_prec2 (mpfr_prec_t* PR, mpfr_prec_t* PI, 471 mpc_t X) 472 Returns the precision of the real part of X via PR and of its 473 imaginary part via PI. 474 475 476 File: mpc.info, Node: Assigning Complex Numbers, Next: Converting Complex Numbers, Prev: Initializing Complex Numbers, Up: Complex Functions 477 478 5.2 Assignment Functions 479 ======================== 480 481 These functions assign new values to already initialized complex numbers 482 (*note Initializing Complex Numbers::). When using any functions with 483 `intmax_t' or `uintmax_t' parameters, you must include `<stdint.h>' or 484 `<inttypes.h>' _before_ `mpc.h', to allow `mpc.h' to define prototypes 485 for these functions. Similarly, functions with parameters of type 486 `complex' or `long complex' are defined only if `<complex.h>' is 487 included _before_ `mpc.h'. If you need assignment functions that are 488 not in the current API, you can define them using the `MPC_SET_X_Y' 489 macro (*note Advanced Functions::). 490 491 -- Function: int mpc_set (mpc_t ROP, mpc_t OP, mpc_rnd_t RND) 492 Set the value of ROP from OP, rounded to the precision of ROP with 493 the given rounding mode RND. 494 495 -- Function: int mpc_set_ui (mpc_t ROP, unsigned long int OP, 496 mpc_rnd_t RND) 497 -- Function: int mpc_set_si (mpc_t ROP, long int OP, mpc_rnd_t RND) 498 -- Function: int mpc_set_uj (mpc_t ROP, uintmax_t OP, mpc_rnd_t RND) 499 -- Function: int mpc_set_sj (mpc_t ROP, intmax_t OP, mpc_rnd_t RND) 500 -- Function: int mpc_set_d (mpc_t ROP, double OP, mpc_rnd_t RND) 501 -- Function: int mpc_set_ld (mpc_t ROP, long double OP, mpc_rnd_t RND) 502 -- Function: int mpc_set_dc (mpc_t ROP, double _Complex OP, mpc_rnd_t 503 RND) 504 -- Function: int mpc_set_ldc (mpc_t ROP, long double _Complex OP, 505 mpc_rnd_t RND) 506 -- Function: int mpc_set_z (mpc_t ROP, mpz_t OP mpc_rnd_t RND) 507 -- Function: int mpc_set_q (mpc_t ROP, mpq_t OP mpc_rnd_t RND) 508 -- Function: int mpc_set_f (mpc_t ROP, mpf_t OP mpc_rnd_t RND) 509 -- Function: int mpc_set_fr (mpc_t ROP, mpfr_t OP, mpc_rnd_t RND) 510 Set the value of ROP from OP, rounded to the precision of ROP with 511 the given rounding mode RND. The argument OP is interpreted as 512 real, so the imaginary part of ROP is set to zero with a positive 513 sign. Please note that even a `long int' may have to be rounded, 514 if the destination precision is less than the machine word width. 515 For `mpc_set_d', be careful that the input number OP may not be 516 exactly representable as a double-precision number (this happens 517 for 0.1 for instance), in which case it is first rounded by the C 518 compiler to a double-precision number, and then only to a complex 519 number. 520 521 -- Function: int mpc_set_ui_ui (mpc_t ROP, unsigned long int OP1, 522 unsigned long int OP2, mpc_rnd_t RND) 523 -- Function: int mpc_set_si_si (mpc_t ROP, long int OP1, long int OP2, 524 mpc_rnd_t RND) 525 -- Function: int mpc_set_uj_uj (mpc_t ROP, uintmax_t OP1, uintmax_t 526 OP2, mpc_rnd_t RND) 527 -- Function: int mpc_set_sj_sj (mpc_t ROP, intmax_t OP1, intmax_t OP2, 528 mpc_rnd_t RND) 529 -- Function: int mpc_set_d_d (mpc_t ROP, double OP1, double OP2, 530 mpc_rnd_t RND) 531 -- Function: int mpc_set_ld_ld (mpc_t ROP, long double OP1, long 532 double OP2, mpc_rnd_t RND) 533 -- Function: int mpc_set_z_z (mpc_t ROP, mpz_t OP1, mpz_t OP2, 534 mpc_rnd_t RND) 535 -- Function: int mpc_set_q_q (mpc_t ROP, mpq_t OP1, mpq_t OP2, 536 mpc_rnd_t RND) 537 -- Function: int mpc_set_f_f (mpc_t ROP, mpf_t OP1, mpf_t OP2, 538 mpc_rnd_t RND) 539 -- Function: int mpc_set_fr_fr (mpc_t ROP, mpfr_t OP1, mpfr_t OP2, 540 mpc_rnd_t RND) 541 Set the real part of ROP from OP1, and its imaginary part from 542 OP2, according to the rounding mode RND. 543 544 Beware that the behaviour of `mpc_set_fr_fr' is undefined if OP1 545 or OP2 is a pointer to the real or imaginary part of ROP. To 546 exchange the real and the imaginary part of a complex number, 547 either use `mpfr_swap (mpc_realref (rop), mpc_imagref (rop))', 548 which also exchanges the precisions of the two parts; or use a 549 temporary variable. 550 551 For functions assigning complex variables from strings or input 552 streams, *note String and Stream Input and Output::. 553 554 -- Function: void mpc_set_nan (mpc_t ROP) 555 Set ROP to Nan+i*NaN. 556 557 -- Function: void mpc_swap (mpc_t OP1, mpc_t OP2) 558 Swap the values of OP1 and OP2 efficiently. Warning: The 559 precisions are exchanged, too; in case these are different, 560 `mpc_swap' is thus not equivalent to three `mpc_set' calls using a 561 third auxiliary variable. 562 563 564 File: mpc.info, Node: Converting Complex Numbers, Next: String and Stream Input and Output, Prev: Assigning Complex Numbers, Up: Complex Functions 565 566 5.3 Conversion Functions 567 ======================== 568 569 The following functions are available only if `<complex.h>' is included 570 _before_ `mpc.h'. 571 572 -- Function: double _Complex mpc_get_dc (mpc_t OP, mpc_rnd_t RND) 573 -- Function: long double _Complex mpc_get_ldc (mpc_t OP, mpc_rnd_t RND) 574 Convert OP to a C complex number, using the rounding mode RND. 575 576 For functions converting complex variables to strings or stream 577 output, *note String and Stream Input and Output::. 578 579 580 File: mpc.info, Node: String and Stream Input and Output, Next: Complex Comparison, Prev: Converting Complex Numbers, Up: Complex Functions 581 582 5.4 String and Stream Input and Output 583 ====================================== 584 585 -- Function: int mpc_strtoc (mpc_t ROP, const char *NPTR, char 586 **ENDPTR, int BASE, mpc_rnd_t RND) 587 Read a complex number from a string NPTR in base BASE, rounded to 588 the precision of ROP with the given rounding mode RND. The BASE 589 must be either 0 or a number from 2 to 36 (otherwise the behaviour 590 is undefined). If NPTR starts with valid data, the result is 591 stored in ROP, the usual inexact value is returned (*note Return 592 Value: return-value.) and, if ENDPTR is not the null pointer, 593 *ENDPTR points to the character just after the valid data. 594 Otherwise, ROP is set to `NaN + i * NaN', -1 is returned and, if 595 ENDPTR is not the null pointer, the value of NPTR is stored in the 596 location referenced by ENDPTR. 597 598 The expected form of a complex number string is either a real 599 number (an optional leading whitespace, an optional sign followed 600 by a floating-point number), or a pair of real numbers in 601 parentheses separated by whitespace. If a real number is read, the 602 missing imaginary part is set to +0. The form of a floating-point 603 number depends on the base and is described in the documentation 604 of `mpfr_strtofr' (*note Assignment Functions: 605 (mpfr.info)Assignment Functions.). For instance, `"3.1415926"', 606 `"(1.25e+7 +.17)"', `"(@nan@ 2)"' and `"(-0 -7)"' are valid 607 strings for BASE = 10. If BASE = 0, then a prefix may be used to 608 indicate the base in which the floating-point number is written. 609 Use prefix '0b' for binary numbers, prefix '0x' for hexadecimal 610 numbers, and no prefix for decimal numbers. The real and 611 imaginary part may then be written in different bases. For 612 instance, `"(1.024e+3 +2.05e+3)"' and `"(0b1p+10 +0x802)"' are 613 valid strings for `base'=0 and represent the same value. 614 615 -- Function: int mpc_set_str (mpc_t ROP, const char *S, int BASE, 616 mpc_rnd_t rnd) 617 Set ROP to the value of the string S in base BASE, rounded to the 618 precision of ROP with the given rounding mode RND. See the 619 documentation of `mpc_strtoc' for a detailed description of the 620 valid string formats. Contrarily to `mpc_strtoc', `mpc_set_str' 621 requires the _whole_ string to represent a valid complex number 622 (potentially followed by additional white space). This function 623 returns the usual inexact value (*note Return Value: 624 return-value.) if the entire string up to the final null character 625 is a valid number in base BASE; otherwise it returns -1, and ROP 626 is set to NaN+i*NaN. 627 628 -- Function: char * mpc_get_str (int B, size_t N, mpc_t OP, mpc_rnd_t 629 RND) 630 Convert OP to a string containing its real and imaginary parts, 631 separated by a space and enclosed in a pair of parentheses. The 632 numbers are written in base B (which may vary from 2 to 36) and 633 rounded according to RND. The number of significant digits, at 634 least 2, is given by N. It is also possible to let N be zero, in 635 which case the number of digits is chosen large enough so that 636 re-reading the printed value with the same precision, assuming 637 both output and input use rounding to nearest, will recover the 638 original value of OP. Note that `mpc_get_str' uses the decimal 639 point of the current locale if available, and `.' otherwise. 640 641 The string is generated using the current memory allocation 642 function (`malloc' by default, unless it has been modified using 643 the custom memory allocation interface of `gmp'); once it is not 644 needed any more, it should be freed by calling `mpc_free_str'. 645 646 -- Function: void mpc_free_str (char *STR) 647 Free the string STR, which needs to have been allocated by a call 648 to `mpc_get_str'. 649 650 The following two functions read numbers from input streams and write 651 them to output streams. When using any of these functions, you need to 652 include `stdio.h' _before_ `mpc.h'. 653 654 -- Function: int mpc_inp_str (mpc_t ROP, FILE *STREAM, size_t *READ, 655 int BASE, mpc_rnd_t RND) 656 Input a string in base BASE in the same format as for `mpc_strtoc' 657 from stdio stream STREAM, rounded according to RND, and put the 658 read complex number into ROP. If STREAM is the null pointer, ROP 659 is read from `stdin'. Return the usual inexact value; if an error 660 occurs, set ROP to `NaN + i * NaN' and return -1. If READ is not 661 the null pointer, it is set to the number of read characters. 662 663 Unlike `mpc_strtoc', the function `mpc_inp_str' does not possess 664 perfect knowledge of the string to transform and has to read it 665 character by character, so it behaves slightly differently: It 666 tries to read a string describing a complex number and processes 667 this string through a call to `mpc_set_str'. Precisely, after 668 skipping optional whitespace, a minimal string is read according 669 to the regular expression `mpfr | '(' \s* mpfr \s+ mpfr \s* ')'', 670 where `\s' denotes a whitespace, and `mpfr' is either a string 671 containing neither whitespaces nor parentheses, or 672 `nan(n-char-sequence)' or `@nan@(n-char-sequence)' (regardless of 673 capitalisation) with `n-char-sequence' a string of ascii letters, 674 digits or `'_''. 675 676 For instance, upon input of `"nan(13 1)"', the function 677 `mpc_inp_str' starts to recognise a value of NaN followed by an 678 n-char-sequence indicated by the opening parenthesis; as soon as 679 the space is reached, it becocmes clear that the expression in 680 parentheses is not an n-char-sequence, and the error flag -1 is 681 returned after 6 characters have been consumed from the stream 682 (the whitespace itself remaining in the stream). The function 683 `mpc_strtoc', on the other hand, may track back when reaching the 684 whitespace; it treats the string as the two successive complex 685 numbers `NaN + i * 0' and `13 + i'. It is thus recommended to 686 have a whitespace follow each floating point number to avoid this 687 problem. 688 689 -- Function: size_t mpc_out_str (FILE *STREAM, int BASE, size_t 690 N_DIGITS, mpc_t OP, mpc_rnd_t RND) 691 Output OP on stdio stream STREAM in base BASE, rounded according 692 to RND, in the same format as for `mpc_strtoc' If STREAM is the 693 null pointer, ROP is written to `stdout'. 694 695 Return the number of characters written. 696 697 698 File: mpc.info, Node: Complex Comparison, Next: Projection & Decomposing, Prev: String and Stream Input and Output, Up: Complex Functions 699 700 5.5 Comparison Functions 701 ======================== 702 703 -- Function: int mpc_cmp (mpc_t OP1, mpc_t OP2) 704 -- Function: int mpc_cmp_si_si (mpc_t OP1, long int OP2R, long int 705 OP2I) 706 -- Macro: int mpc_cmp_si (mpc_t OP1, long int OP2) 707 Compare OP1 and OP2, where in the case of `mpc_cmp_si_si', OP2 is 708 taken to be OP2R + i OP2I. The return value C can be decomposed 709 into `x = MPC_INEX_RE(c)' and `y = MPC_INEX_IM(c)', such that X is 710 positive if the real part of OP1 is greater than that of OP2, zero 711 if both real parts are equal, and negative if the real part of OP1 712 is less than that of OP2, and likewise for Y. Both OP1 and OP2 713 are considered to their full own precision, which may differ. It 714 is not allowed that one of the operands has a NaN (Not-a-Number) 715 part. 716 717 The storage of the return value is such that equality can be 718 simply checked with `mpc_cmp (op1, op2) == 0'. 719 720 721 File: mpc.info, Node: Projection & Decomposing, Next: Basic Arithmetic, Prev: Complex Comparison, Up: Complex Functions 722 723 5.6 Projection and Decomposing Functions 724 ======================================== 725 726 -- Function: int mpc_real (mpfr_t ROP, mpc_t OP, mpfr_rnd_t RND) 727 Set ROP to the value of the real part of OP rounded in the 728 direction RND. 729 730 -- Function: int mpc_imag (mpfr_t ROP, mpc_t OP, mpfr_rnd_t RND) 731 Set ROP to the value of the imaginary part of OP rounded in the 732 direction RND. 733 734 -- Macro: mpfr_t mpc_realref (mpc_t OP) 735 -- Macro: mpfr_t mpc_imagref (mpc_t OP) 736 Return a reference to the real part and imaginary part of OP, 737 respectively. The `mpfr' functions can be used on the result of 738 these macros (note that the `mpfr_t' type is itself a pointer). 739 740 -- Function: int mpc_arg (mpfr_t ROP, mpc_t OP, mpfr_rnd_t RND) 741 Set ROP to the argument of OP, with a branch cut along the 742 negative real axis. 743 744 -- Function: int mpc_proj (mpc_t ROP, mpc_t OP, mpc_rnd_t RND) 745 Compute a projection of OP onto the Riemann sphere. Set ROP to OP 746 rounded in the direction RND, except when at least one part of OP 747 is infinite (even if the other part is a NaN) in which case the 748 real part of ROP is set to plus infinity and its imaginary part to 749 a signed zero with the same sign as the imaginary part of OP. 750 751 752 File: mpc.info, Node: Basic Arithmetic, Next: Power Functions and Logarithm, Prev: Projection & Decomposing, Up: Complex Functions 753 754 5.7 Basic Arithmetic Functions 755 ============================== 756 757 All the following functions are designed in such a way that, when 758 working with real numbers instead of complex numbers, their complexity 759 should essentially be the same as with the GNU MPFR library, with only 760 a marginal overhead due to the GNU MPC layer. 761 762 -- Function: int mpc_add (mpc_t ROP, mpc_t OP1, mpc_t OP2, mpc_rnd_t 763 RND) 764 -- Function: int mpc_add_ui (mpc_t ROP, mpc_t OP1, unsigned long int 765 OP2, mpc_rnd_t RND) 766 -- Function: int mpc_add_fr (mpc_t ROP, mpc_t OP1, mpfr_t OP2, 767 mpc_rnd_t RND) 768 Set ROP to OP1 + OP2 rounded according to RND. 769 770 -- Function: int mpc_sub (mpc_t ROP, mpc_t OP1, mpc_t OP2, mpc_rnd_t 771 RND) 772 -- Function: int mpc_sub_fr (mpc_t ROP, mpc_t OP1, mpfr_t OP2, 773 mpc_rnd_t RND) 774 -- Function: int mpc_fr_sub (mpc_t ROP, mpfr_t OP1, mpc_t OP2, 775 mpc_rnd_t RND) 776 -- Function: int mpc_sub_ui (mpc_t ROP, mpc_t OP1, unsigned long int 777 OP2, mpc_rnd_t RND) 778 -- Macro: int mpc_ui_sub (mpc_t ROP, unsigned long int OP1, mpc_t OP2, 779 mpc_rnd_t RND) 780 -- Function: int mpc_ui_ui_sub (mpc_t ROP, unsigned long int RE1, 781 unsigned long int IM1, mpc_t OP2, mpc_rnd_t RND) 782 Set ROP to OP1 - OP2 rounded according to RND. For 783 `mpc_ui_ui_sub', OP1 is RE1 + IM1. 784 785 -- Function: int mpc_neg (mpc_t ROP, mpc_t OP, mpc_rnd_t RND) 786 Set ROP to -OP rounded according to RND. Just changes the sign if 787 ROP and OP are the same variable. 788 789 -- Function: int mpc_mul (mpc_t ROP, mpc_t OP1, mpc_t OP2, mpc_rnd_t 790 RND) 791 -- Function: int mpc_mul_ui (mpc_t ROP, mpc_t OP1, unsigned long int 792 OP2, mpc_rnd_t RND) 793 -- Function: int mpc_mul_si (mpc_t ROP, mpc_t OP1, long int OP2, 794 mpc_rnd_t RND) 795 -- Function: int mpc_mul_fr (mpc_t ROP, mpc_t OP1, mpfr_t OP2, 796 mpc_rnd_t RND) 797 Set ROP to OP1 times OP2 rounded according to RND. Note: for 798 `mpc_mul', in case OP1 and OP2 have the same value, use `mpc_sqr' 799 for better efficiency. 800 801 -- Function: int mpc_mul_i (mpc_t ROP, mpc_t OP, int SGN, mpc_rnd_t 802 RND) 803 Set ROP to OP times the imaginary unit i if SGN is non-negative, 804 set ROP to OP times -i otherwise, in both cases rounded according 805 to RND. 806 807 -- Function: int mpc_sqr (mpc_t ROP, mpc_t OP, mpc_rnd_t RND) 808 Set ROP to the square of OP rounded according to RND. 809 810 -- Function: int mpc_fma (mpc_t ROP, mpc_t OP1, mpc_t OP2, mpc_t OP3, 811 mpc_rnd_t RND) 812 Set ROP to OP1*OP2+OP3, rounded according to RND, with only one 813 final rounding. 814 815 -- Function: int mpc_div (mpc_t ROP, mpc_t OP1, mpc_t OP2, mpc_rnd_t 816 RND) 817 -- Function: int mpc_div_ui (mpc_t ROP, mpc_t OP1, unsigned long int 818 OP2, mpc_rnd_t RND) 819 -- Function: int mpc_div_fr (mpc_t ROP, mpc_t OP1, mpfr_t OP2, 820 mpc_rnd_t RND) 821 -- Function: int mpc_ui_div (mpc_t ROP, unsigned long int OP1, mpc_t 822 OP2, mpc_rnd_t RND) 823 -- Function: int mpc_fr_div (mpc_t ROP, mpfr_t OP1, mpc_t OP2, 824 mpc_rnd_t RND) 825 Set ROP to OP1/OP2 rounded according to RND. 826 827 -- Function: int mpc_conj (mpc_t ROP, mpc_t OP, mpc_rnd_t RND) 828 Set ROP to the conjugate of OP rounded according to RND. Just 829 changes the sign of the imaginary part if ROP and OP are the same 830 variable. 831 832 -- Function: int mpc_abs (mpfr_t ROP, mpc_t OP, mpfr_rnd_t RND) 833 Set the floating-point number ROP to the absolute value of OP, 834 rounded in the direction RND. 835 836 -- Function: int mpc_norm (mpfr_t ROP, mpc_t OP, mpfr_rnd_t RND) 837 Set the floating-point number ROP to the norm of OP (i.e., the 838 square of its absolute value), rounded in the direction RND. 839 840 -- Function: int mpc_mul_2ui (mpc_t ROP, mpc_t OP1, unsigned long int 841 OP2, mpc_rnd_t RND) 842 -- Function: int mpc_mul_2si (mpc_t ROP, mpc_t OP1, long int OP2, 843 mpc_rnd_t RND) 844 Set ROP to OP1 times 2 raised to OP2 rounded according to RND. 845 Just modifies the exponents of the real and imaginary parts by OP2 846 when ROP and OP1 are identical. 847 848 -- Function: int mpc_div_2ui (mpc_t ROP, mpc_t OP1, unsigned long int 849 OP2, mpc_rnd_t RND) 850 -- Function: int mpc_div_2si (mpc_t ROP, mpc_t OP1, long int OP2, 851 mpc_rnd_t RND) 852 Set ROP to OP1 divided by 2 raised to OP2 rounded according to 853 RND. Just modifies the exponents of the real and imaginary parts 854 by OP2 when ROP and OP1 are identical. 855 856 857 File: mpc.info, Node: Power Functions and Logarithm, Next: Trigonometric Functions, Prev: Basic Arithmetic, Up: Complex Functions 858 859 5.8 Power Functions and Logarithm 860 ================================= 861 862 -- Function: int mpc_sqrt (mpc_t ROP, mpc_t OP, mpc_rnd_t RND) 863 Set ROP to the square root of OP rounded according to RND. The 864 returned value ROP has a non-negative real part, and if its real 865 part is zero, a non-negative imaginary part. 866 867 -- Function: int mpc_pow (mpc_t ROP, mpc_t OP1, mpc_t OP2, mpc_rnd_t 868 RND) 869 -- Function: int mpc_pow_d (mpc_t ROP, mpc_t OP1, double OP2, 870 mpc_rnd_t RND) 871 -- Function: int mpc_pow_ld (mpc_t ROP, mpc_t OP1, long double OP2, 872 mpc_rnd_t RND) 873 -- Function: int mpc_pow_si (mpc_t ROP, mpc_t OP1, long OP2, mpc_rnd_t 874 RND) 875 -- Function: int mpc_pow_ui (mpc_t ROP, mpc_t OP1, unsigned long OP2, 876 mpc_rnd_t RND) 877 -- Function: int mpc_pow_z (mpc_t ROP, mpc_t OP1, mpz_t OP2, mpc_rnd_t 878 RND) 879 -- Function: int mpc_pow_fr (mpc_t ROP, mpc_t OP1, mpfr_t OP2, 880 mpc_rnd_t RND) 881 Set ROP to OP1 raised to the power OP2, rounded according to RND. 882 For `mpc_pow_d', `mpc_pow_ld', `mpc_pow_si', `mpc_pow_ui', 883 `mpc_pow_z' and `mpc_pow_fr', the imaginary part of OP2 is 884 considered as +0. When both OP1 and OP2 are zero, the result has 885 real part 1, and imaginary part 0, with sign being the opposite of 886 that of OP2. 887 888 -- Function: int mpc_exp (mpc_t ROP, mpc_t OP, mpc_rnd_t RND) 889 Set ROP to the exponential of OP, rounded according to RND with 890 the precision of ROP. 891 892 -- Function: int mpc_log (mpc_t ROP, mpc_t OP, mpc_rnd_t RND) 893 -- Function: int mpc_log10 (mpc_t ROP, mpc_t OP, mpc_rnd_t RND) 894 Set ROP to the natural and base-10 logarithm of OP respectively, 895 rounded according to RND with the precision of ROP. The principal 896 branch is chosen, with the branch cut on the negative real axis, 897 so that the imaginary part of the result lies in ]-\pi , \pi] and 898 ]-\pi/log(10) , \pi/log(10)] respectively. 899 900 901 File: mpc.info, Node: Trigonometric Functions, Next: Miscellaneous Complex Functions, Prev: Power Functions and Logarithm, Up: Complex Functions 902 903 5.9 Trigonometric Functions 904 =========================== 905 906 -- Function: int mpc_sin (mpc_t ROP, mpc_t OP, mpc_rnd_t RND) 907 Set ROP to the sine of OP, rounded according to RND with the 908 precision of ROP. 909 910 -- Function: int mpc_cos (mpc_t ROP, mpc_t OP, mpc_rnd_t RND) 911 Set ROP to the cosine of OP, rounded according to RND with the 912 precision of ROP. 913 914 -- Function: int mpc_sin_cos (mpc_t ROP_SIN, mpc_t ROP_COS, mpc_t OP, 915 mpc_rnd_t RND_SIN, mpc_rnd_t RND_COS) 916 Set ROP_SIN to the sine of OP, rounded according to RND_SIN with 917 the precision of ROP_SIN, and ROP_COS to the cosine of OP, rounded 918 according to RND_COS with the precision of ROP_COS. 919 920 -- Function: int mpc_tan (mpc_t ROP, mpc_t OP, mpc_rnd_t RND) 921 Set ROP to the tangent of OP, rounded according to RND with the 922 precision of ROP. 923 924 -- Function: int mpc_sinh (mpc_t ROP, mpc_t OP, mpc_rnd_t RND) 925 Set ROP to the hyperbolic sine of OP, rounded according to RND 926 with the precision of ROP. 927 928 -- Function: int mpc_cosh (mpc_t ROP, mpc_t OP, mpc_rnd_t RND) 929 Set ROP to the hyperbolic cosine of OP, rounded according to RND 930 with the precision of ROP. 931 932 -- Function: int mpc_tanh (mpc_t ROP, mpc_t OP, mpc_rnd_t RND) 933 Set ROP to the hyperbolic tangent of OP, rounded according to RND 934 with the precision of ROP. 935 936 -- Function: int mpc_asin (mpc_t ROP, mpc_t OP, mpc_rnd_t RND) 937 -- Function: int mpc_acos (mpc_t ROP, mpc_t OP, mpc_rnd_t RND) 938 -- Function: int mpc_atan (mpc_t ROP, mpc_t OP, mpc_rnd_t RND) 939 Set ROP to the inverse sine, inverse cosine, inverse tangent of OP, 940 rounded according to RND with the precision of ROP. 941 942 -- Function: int mpc_asinh (mpc_t ROP, mpc_t OP, mpc_rnd_t RND) 943 -- Function: int mpc_acosh (mpc_t ROP, mpc_t OP, mpc_rnd_t RND) 944 -- Function: int mpc_atanh (mpc_t ROP, mpc_t OP, mpc_rnd_t RND) 945 Set ROP to the inverse hyperbolic sine, inverse hyperbolic cosine, 946 inverse hyperbolic tangent of OP, rounded according to RND with 947 the precision of ROP. The branch cut of MPC_ACOSH is (-\infty, 1). 948 949 950 File: mpc.info, Node: Miscellaneous Complex Functions, Next: Advanced Functions, Prev: Trigonometric Functions, Up: Complex Functions 951 952 5.10 Miscellaneous Functions 953 ============================ 954 955 -- Function: int mpc_urandom (mpc_t ROP, gmp_randstate_t STATE) 956 Generate a uniformly distributed random complex in the unit square 957 [0, 1] x [0, 1]. Return 0, unless an exponent in the real or 958 imaginary part is not in the current exponent range, in which case 959 that part is set to NaN and a zero value is returned. The second 960 argument is a `gmp_randstate_t' structure which should be created 961 using the GMP `rand_init' function, see the GMP manual. 962 963 -- Function: const char * mpc_get_version (void) 964 Return the GNU MPC version, as a null-terminated string. 965 966 -- Macro: MPC_VERSION 967 -- Macro: MPC_VERSION_MAJOR 968 -- Macro: MPC_VERSION_MINOR 969 -- Macro: MPC_VERSION_PATCHLEVEL 970 -- Macro: MPC_VERSION_STRING 971 `MPC_VERSION' is the version of GNU MPC as a preprocessing 972 constant. `MPC_VERSION_MAJOR', `MPC_VERSION_MINOR' and 973 `MPC_VERSION_PATCHLEVEL' are respectively the major, minor and 974 patch level of GNU MPC version, as preprocessing constants. 975 `MPC_VERSION_STRING' is the version as a string constant, which 976 can be compared to the result of `mpc_get_version' to check at run 977 time the header file and library used match: 978 if (strcmp (mpc_get_version (), MPC_VERSION_STRING)) 979 fprintf (stderr, "Warning: header and library do not match\n"); 980 Note: Obtaining different strings is not necessarily an error, as 981 in general, a program compiled with some old GNU MPC version can be 982 dynamically linked with a newer GNU MPC library version (if 983 allowed by the library versioning system). 984 985 -- Macro: long MPC_VERSION_NUM (MAJOR, MINOR, PATCHLEVEL) 986 Create an integer in the same format as used by `MPC_VERSION' from 987 the given MAJOR, MINOR and PATCHLEVEL. Here is an example of how 988 to check the GNU MPC version at compile time: 989 #if (!defined(MPC_VERSION) || (MPC_VERSION<MPC_VERSION_NUM(2,1,0))) 990 # error "Wrong GNU MPC version." 991 #endif 992 993 994 File: mpc.info, Node: Advanced Functions, Next: Internals, Prev: Miscellaneous Complex Functions, Up: Complex Functions 995 996 5.11 Advanced Functions 997 ======================= 998 999 -- Macro: MPC_SET_X_Y (REAL_SUFFIX, IMAG_SUFFIX, ROP, REAL, IMAG, RND) 1000 The macro MPC_SET_X_Y is designed to serve as the body of an 1001 assignment function and cannot be used by itself. The REAL_SUFFIX 1002 and IMAG_SUFFIX parameters are the types of the real and imaginary 1003 part, that is, the `x' in the `mpfr_set_x' function one would use 1004 to set the part; for the mpfr type, use `fr'. REAL (respectively 1005 IMAG) is the value you want to assign to the real (resp. 1006 imaginary) part, its type must conform to REAL_SUFFIX (resp. 1007 IMAG_SUFFIX). RND is the `mpc_rnd_t' rounding mode. The return 1008 value is the usual inexact value (*note Return Value: 1009 return-value.). 1010 1011 For instance, you can define mpc_set_ui_fr as follows: 1012 int mpc_set_ui_fr (mpc_t rop, long int re, double im, mpc_rnd_t rnd) 1013 MPC_SET_X_Y (ui, fr, rop, re, im, rnd); 1014 1015 1016 File: mpc.info, Node: Internals, Prev: Advanced Functions, Up: Complex Functions 1017 1018 5.12 Internals 1019 ============== 1020 1021 These macros and functions are mainly designed for the implementation 1022 of GNU MPC, but may be useful for users too. However, no upward 1023 compatibility is guaranteed. You need to include `mpc-impl.h' to use 1024 them. 1025 1026 The macro `MPC_MAX_PREC(z)' gives the maximum of the precisions of 1027 the real and imaginary parts of a complex number. 1028 1029 1030 File: mpc.info, Node: References, Next: Concept Index, Prev: Complex Functions, Up: Top 1031 1032 References 1033 ********** 1034 1035 * Torbjo"rn Granlund et al. `gmp' - GNU multiprecision library. 1036 Version 4.2.4, `http://gmplib.org/'. 1037 1038 * Guillaume Hanrot, Vincent Lefe`vre, Patrick Pe'lissier, Paul 1039 Zimmermann et al. `mpfr' - A library for multiple-precision 1040 floating-point computations with exact rounding. Version 2.4.1, 1041 `http://www.mpfr.org'. 1042 1043 * IEEE standard for binary floating-point arithmetic, Technical 1044 Report ANSI-IEEE Standard 754-1985, New York, 1985. Approved 1045 March 21, 1985: IEEE Standards Board; approved July 26, 1985: 1046 American National Standards Institute, 18 pages. 1047 1048 * Donald E. Knuth, "The Art of Computer Programming", vol 2, 1049 "Seminumerical Algorithms", 2nd edition, Addison-Wesley, 1981. 1050 1051 * ISO/IEC 9899:1999, Programming languages C. 1052 1053 1054 1055 File: mpc.info, Node: Concept Index, Next: Function Index, Prev: References, Up: Top 1056 1057 Concept Index 1058 ************* 1059 1060 [index] 1061 * Menu: 1062 1063 * Arithmetic functions: Basic Arithmetic. (line 6) 1064 * Comparison functions: Complex Comparison. (line 6) 1065 * Complex arithmetic functions: Basic Arithmetic. (line 6) 1066 * Complex assignment functions: Assigning Complex Numbers. 1067 (line 6) 1068 * Complex comparisons functions: Complex Comparison. (line 6) 1069 * Complex functions: Complex Functions. (line 6) 1070 * Complex number: GNU MPC Basics. (line 15) 1071 * Conditions for copying GNU MPC: Copying. (line 6) 1072 * Conversion functions: Converting Complex Numbers. 1073 (line 6) 1074 * Copying conditions: Copying. (line 6) 1075 * Installation: Installing GNU MPC. (line 6) 1076 * Logarithm: Power Functions and Logarithm. 1077 (line 6) 1078 * Miscellaneous complex functions: Miscellaneous Complex Functions. 1079 (line 6) 1080 * mpc.h: GNU MPC Basics. (line 6) 1081 * Power functions: Power Functions and Logarithm. 1082 (line 6) 1083 * Precision: GNU MPC Basics. (line 19) 1084 * Projection and Decomposing Functions: Projection & Decomposing. 1085 (line 6) 1086 * Reporting bugs: Reporting Bugs. (line 6) 1087 * Rounding Mode: GNU MPC Basics. (line 24) 1088 * String and stream input and output: String and Stream Input and Output. 1089 (line 6) 1090 * Trigonometric functions: Trigonometric Functions. 1091 (line 6) 1092 * User-defined precision: Complex Functions. (line 12) 1093 1094 1095 File: mpc.info, Node: Function Index, Next: GNU Free Documentation License, Prev: Concept Index, Up: Top 1096 1097 Function Index 1098 ************** 1099 1100 [index] 1101 * Menu: 1102 1103 * _Complex: Converting Complex Numbers. 1104 (line 10) 1105 * mpc_abs: Basic Arithmetic. (line 82) 1106 * mpc_acos: Trigonometric Functions. 1107 (line 38) 1108 * mpc_acosh: Trigonometric Functions. 1109 (line 44) 1110 * mpc_add: Basic Arithmetic. (line 13) 1111 * mpc_add_fr: Basic Arithmetic. (line 17) 1112 * mpc_add_ui: Basic Arithmetic. (line 15) 1113 * mpc_arg: Projection & Decomposing. 1114 (line 21) 1115 * mpc_asin: Trigonometric Functions. 1116 (line 37) 1117 * mpc_asinh: Trigonometric Functions. 1118 (line 43) 1119 * mpc_atan: Trigonometric Functions. 1120 (line 39) 1121 * mpc_atanh: Trigonometric Functions. 1122 (line 45) 1123 * mpc_clear: Initializing Complex Numbers. 1124 (line 22) 1125 * mpc_cmp: Complex Comparison. (line 7) 1126 * mpc_cmp_si: Complex Comparison. (line 10) 1127 * mpc_cmp_si_si: Complex Comparison. (line 9) 1128 * mpc_conj: Basic Arithmetic. (line 77) 1129 * mpc_cos: Trigonometric Functions. 1130 (line 11) 1131 * mpc_cosh: Trigonometric Functions. 1132 (line 29) 1133 * mpc_div: Basic Arithmetic. (line 66) 1134 * mpc_div_2si: Basic Arithmetic. (line 101) 1135 * mpc_div_2ui: Basic Arithmetic. (line 99) 1136 * mpc_div_fr: Basic Arithmetic. (line 70) 1137 * mpc_div_ui: Basic Arithmetic. (line 68) 1138 * mpc_exp: Power Functions and Logarithm. 1139 (line 33) 1140 * mpc_fma: Basic Arithmetic. (line 61) 1141 * mpc_fr_div: Basic Arithmetic. (line 74) 1142 * mpc_fr_sub: Basic Arithmetic. (line 25) 1143 * mpc_free_str: String and Stream Input and Output. 1144 (line 68) 1145 * mpc_get_ldc: Converting Complex Numbers. 1146 (line 11) 1147 * mpc_get_prec: Initializing Complex Numbers. 1148 (line 50) 1149 * mpc_get_prec2: Initializing Complex Numbers. 1150 (line 55) 1151 * mpc_get_str: String and Stream Input and Output. 1152 (line 51) 1153 * mpc_get_version: Miscellaneous Complex Functions. 1154 (line 15) 1155 * mpc_imag: Projection & Decomposing. 1156 (line 11) 1157 * mpc_imagref: Projection & Decomposing. 1158 (line 16) 1159 * mpc_init2: Initializing Complex Numbers. 1160 (line 11) 1161 * mpc_init3: Initializing Complex Numbers. 1162 (line 17) 1163 * mpc_inp_str: String and Stream Input and Output. 1164 (line 77) 1165 * mpc_log: Power Functions and Logarithm. 1166 (line 37) 1167 * mpc_log10: Power Functions and Logarithm. 1168 (line 38) 1169 * mpc_mul: Basic Arithmetic. (line 40) 1170 * mpc_mul_2si: Basic Arithmetic. (line 93) 1171 * mpc_mul_2ui: Basic Arithmetic. (line 91) 1172 * mpc_mul_fr: Basic Arithmetic. (line 46) 1173 * mpc_mul_i: Basic Arithmetic. (line 52) 1174 * mpc_mul_si: Basic Arithmetic. (line 44) 1175 * mpc_mul_ui: Basic Arithmetic. (line 42) 1176 * mpc_neg: Basic Arithmetic. (line 35) 1177 * mpc_norm: Basic Arithmetic. (line 86) 1178 * mpc_out_str: String and Stream Input and Output. 1179 (line 112) 1180 * mpc_pow: Power Functions and Logarithm. 1181 (line 13) 1182 * mpc_pow_d: Power Functions and Logarithm. 1183 (line 15) 1184 * mpc_pow_fr: Power Functions and Logarithm. 1185 (line 25) 1186 * mpc_pow_ld: Power Functions and Logarithm. 1187 (line 17) 1188 * mpc_pow_si: Power Functions and Logarithm. 1189 (line 19) 1190 * mpc_pow_ui: Power Functions and Logarithm. 1191 (line 21) 1192 * mpc_pow_z: Power Functions and Logarithm. 1193 (line 23) 1194 * mpc_proj: Projection & Decomposing. 1195 (line 25) 1196 * mpc_real: Projection & Decomposing. 1197 (line 7) 1198 * mpc_realref: Projection & Decomposing. 1199 (line 15) 1200 * mpc_rnd_t: GNU MPC Basics. (line 24) 1201 * mpc_set: Assigning Complex Numbers. 1202 (line 17) 1203 * mpc_set_d: Assigning Complex Numbers. 1204 (line 26) 1205 * mpc_set_d_d: Assigning Complex Numbers. 1206 (line 56) 1207 * mpc_set_dc: Assigning Complex Numbers. 1208 (line 29) 1209 * mpc_set_f: Assigning Complex Numbers. 1210 (line 34) 1211 * mpc_set_f_f: Assigning Complex Numbers. 1212 (line 64) 1213 * mpc_set_fr: Assigning Complex Numbers. 1214 (line 35) 1215 * mpc_set_fr_fr: Assigning Complex Numbers. 1216 (line 66) 1217 * mpc_set_ld: Assigning Complex Numbers. 1218 (line 27) 1219 * mpc_set_ld_ld: Assigning Complex Numbers. 1220 (line 58) 1221 * mpc_set_ldc: Assigning Complex Numbers. 1222 (line 31) 1223 * mpc_set_nan: Assigning Complex Numbers. 1224 (line 80) 1225 * mpc_set_prec: Initializing Complex Numbers. 1226 (line 42) 1227 * mpc_set_q: Assigning Complex Numbers. 1228 (line 33) 1229 * mpc_set_q_q: Assigning Complex Numbers. 1230 (line 62) 1231 * mpc_set_si: Assigning Complex Numbers. 1232 (line 23) 1233 * mpc_set_si_si: Assigning Complex Numbers. 1234 (line 50) 1235 * mpc_set_sj: Assigning Complex Numbers. 1236 (line 25) 1237 * mpc_set_sj_sj: Assigning Complex Numbers. 1238 (line 54) 1239 * mpc_set_str: String and Stream Input and Output. 1240 (line 38) 1241 * mpc_set_ui: Assigning Complex Numbers. 1242 (line 22) 1243 * mpc_set_ui_ui: Assigning Complex Numbers. 1244 (line 48) 1245 * mpc_set_uj: Assigning Complex Numbers. 1246 (line 24) 1247 * mpc_set_uj_uj: Assigning Complex Numbers. 1248 (line 52) 1249 * MPC_SET_X_Y: Advanced Functions. (line 7) 1250 * mpc_set_z: Assigning Complex Numbers. 1251 (line 32) 1252 * mpc_set_z_z: Assigning Complex Numbers. 1253 (line 60) 1254 * mpc_sin: Trigonometric Functions. 1255 (line 7) 1256 * mpc_sin_cos: Trigonometric Functions. 1257 (line 16) 1258 * mpc_sinh: Trigonometric Functions. 1259 (line 25) 1260 * mpc_sqr: Basic Arithmetic. (line 57) 1261 * mpc_sqrt: Power Functions and Logarithm. 1262 (line 7) 1263 * mpc_strtoc: String and Stream Input and Output. 1264 (line 8) 1265 * mpc_sub: Basic Arithmetic. (line 21) 1266 * mpc_sub_fr: Basic Arithmetic. (line 23) 1267 * mpc_sub_ui: Basic Arithmetic. (line 27) 1268 * mpc_swap: Assigning Complex Numbers. 1269 (line 83) 1270 * mpc_t: GNU MPC Basics. (line 15) 1271 * mpc_tan: Trigonometric Functions. 1272 (line 21) 1273 * mpc_tanh: Trigonometric Functions. 1274 (line 33) 1275 * mpc_ui_div: Basic Arithmetic. (line 72) 1276 * mpc_ui_sub: Basic Arithmetic. (line 29) 1277 * mpc_ui_ui_sub: Basic Arithmetic. (line 31) 1278 * mpc_urandom: Miscellaneous Complex Functions. 1279 (line 7) 1280 * MPC_VERSION: Miscellaneous Complex Functions. 1281 (line 18) 1282 * MPC_VERSION_MAJOR: Miscellaneous Complex Functions. 1283 (line 19) 1284 * MPC_VERSION_MINOR: Miscellaneous Complex Functions. 1285 (line 20) 1286 * MPC_VERSION_NUM: Miscellaneous Complex Functions. 1287 (line 37) 1288 * MPC_VERSION_PATCHLEVEL: Miscellaneous Complex Functions. 1289 (line 21) 1290 * MPC_VERSION_STRING: Miscellaneous Complex Functions. 1291 (line 22) 1292 * mpfr_prec_t: GNU MPC Basics. (line 19) 1293 1294 1295 File: mpc.info, Node: GNU Free Documentation License, Prev: Function Index, Up: Top 1296 1297 Anhang A GNU Free Documentation License 1298 *************************************** 1299 1300 Version 1.3, 3 November 2008 1301 1302 Copyright (C) 2000, 2001, 2002, 2007, 2008 Free Software Foundation, Inc. 1303 `http://fsf.org/' 1304 1305 Everyone is permitted to copy and distribute verbatim copies 1306 of this license document, but changing it is not allowed. 1307 1308 0. PREAMBLE 1309 1310 The purpose of this License is to make a manual, textbook, or other 1311 functional and useful document "free" in the sense of freedom: to 1312 assure everyone the effective freedom to copy and redistribute it, 1313 with or without modifying it, either commercially or 1314 noncommercially. Secondarily, this License preserves for the 1315 author and publisher a way to get credit for their work, while not 1316 being considered responsible for modifications made by others. 1317 1318 This License is a kind of "copyleft", which means that derivative 1319 works of the document must themselves be free in the same sense. 1320 It complements the GNU General Public License, which is a copyleft 1321 license designed for free software. 1322 1323 We have designed this License in order to use it for manuals for 1324 free software, because free software needs free documentation: a 1325 free program should come with manuals providing the same freedoms 1326 that the software does. But this License is not limited to 1327 software manuals; it can be used for any textual work, regardless 1328 of subject matter or whether it is published as a printed book. 1329 We recommend this License principally for works whose purpose is 1330 instruction or reference. 1331 1332 1. APPLICABILITY AND DEFINITIONS 1333 1334 This License applies to any manual or other work, in any medium, 1335 that contains a notice placed by the copyright holder saying it 1336 can be distributed under the terms of this License. Such a notice 1337 grants a world-wide, royalty-free license, unlimited in duration, 1338 to use that work under the conditions stated herein. The 1339 "Document", below, refers to any such manual or work. Any member 1340 of the public is a licensee, and is addressed as "you". You 1341 accept the license if you copy, modify or distribute the work in a 1342 way requiring permission under copyright law. 1343 1344 A "Modified Version" of the Document means any work containing the 1345 Document or a portion of it, either copied verbatim, or with 1346 modifications and/or translated into another language. 1347 1348 A "Secondary Section" is a named appendix or a front-matter section 1349 of the Document that deals exclusively with the relationship of the 1350 publishers or authors of the Document to the Document's overall 1351 subject (or to related matters) and contains nothing that could 1352 fall directly within that overall subject. (Thus, if the Document 1353 is in part a textbook of mathematics, a Secondary Section may not 1354 explain any mathematics.) The relationship could be a matter of 1355 historical connection with the subject or with related matters, or 1356 of legal, commercial, philosophical, ethical or political position 1357 regarding them. 1358 1359 The "Invariant Sections" are certain Secondary Sections whose 1360 titles are designated, as being those of Invariant Sections, in 1361 the notice that says that the Document is released under this 1362 License. If a section does not fit the above definition of 1363 Secondary then it is not allowed to be designated as Invariant. 1364 The Document may contain zero Invariant Sections. If the Document 1365 does not identify any Invariant Sections then there are none. 1366 1367 The "Cover Texts" are certain short passages of text that are 1368 listed, as Front-Cover Texts or Back-Cover Texts, in the notice 1369 that says that the Document is released under this License. A 1370 Front-Cover Text may be at most 5 words, and a Back-Cover Text may 1371 be at most 25 words. 1372 1373 A "Transparent" copy of the Document means a machine-readable copy, 1374 represented in a format whose specification is available to the 1375 general public, that is suitable for revising the document 1376 straightforwardly with generic text editors or (for images 1377 composed of pixels) generic paint programs or (for drawings) some 1378 widely available drawing editor, and that is suitable for input to 1379 text formatters or for automatic translation to a variety of 1380 formats suitable for input to text formatters. A copy made in an 1381 otherwise Transparent file format whose markup, or absence of 1382 markup, has been arranged to thwart or discourage subsequent 1383 modification by readers is not Transparent. An image format is 1384 not Transparent if used for any substantial amount of text. A 1385 copy that is not "Transparent" is called "Opaque". 1386 1387 Examples of suitable formats for Transparent copies include plain 1388 ASCII without markup, Texinfo input format, LaTeX input format, 1389 SGML or XML using a publicly available DTD, and 1390 standard-conforming simple HTML, PostScript or PDF designed for 1391 human modification. Examples of transparent image formats include 1392 PNG, XCF and JPG. Opaque formats include proprietary formats that 1393 can be read and edited only by proprietary word processors, SGML or 1394 XML for which the DTD and/or processing tools are not generally 1395 available, and the machine-generated HTML, PostScript or PDF 1396 produced by some word processors for output purposes only. 1397 1398 The "Title Page" means, for a printed book, the title page itself, 1399 plus such following pages as are needed to hold, legibly, the 1400 material this License requires to appear in the title page. For 1401 works in formats which do not have any title page as such, "Title 1402 Page" means the text near the most prominent appearance of the 1403 work's title, preceding the beginning of the body of the text. 1404 1405 The "publisher" means any person or entity that distributes copies 1406 of the Document to the public. 1407 1408 A section "Entitled XYZ" means a named subunit of the Document 1409 whose title either is precisely XYZ or contains XYZ in parentheses 1410 following text that translates XYZ in another language. (Here XYZ 1411 stands for a specific section name mentioned below, such as 1412 "Acknowledgements", "Dedications", "Endorsements", or "History".) 1413 To "Preserve the Title" of such a section when you modify the 1414 Document means that it remains a section "Entitled XYZ" according 1415 to this definition. 1416 1417 The Document may include Warranty Disclaimers next to the notice 1418 which states that this License applies to the Document. These 1419 Warranty Disclaimers are considered to be included by reference in 1420 this License, but only as regards disclaiming warranties: any other 1421 implication that these Warranty Disclaimers may have is void and 1422 has no effect on the meaning of this License. 1423 1424 2. VERBATIM COPYING 1425 1426 You may copy and distribute the Document in any medium, either 1427 commercially or noncommercially, provided that this License, the 1428 copyright notices, and the license notice saying this License 1429 applies to the Document are reproduced in all copies, and that you 1430 add no other conditions whatsoever to those of this License. You 1431 may not use technical measures to obstruct or control the reading 1432 or further copying of the copies you make or distribute. However, 1433 you may accept compensation in exchange for copies. If you 1434 distribute a large enough number of copies you must also follow 1435 the conditions in section 3. 1436 1437 You may also lend copies, under the same conditions stated above, 1438 and you may publicly display copies. 1439 1440 3. COPYING IN QUANTITY 1441 1442 If you publish printed copies (or copies in media that commonly 1443 have printed covers) of the Document, numbering more than 100, and 1444 the Document's license notice requires Cover Texts, you must 1445 enclose the copies in covers that carry, clearly and legibly, all 1446 these Cover Texts: Front-Cover Texts on the front cover, and 1447 Back-Cover Texts on the back cover. Both covers must also clearly 1448 and legibly identify you as the publisher of these copies. The 1449 front cover must present the full title with all words of the 1450 title equally prominent and visible. You may add other material 1451 on the covers in addition. Copying with changes limited to the 1452 covers, as long as they preserve the title of the Document and 1453 satisfy these conditions, can be treated as verbatim copying in 1454 other respects. 1455 1456 If the required texts for either cover are too voluminous to fit 1457 legibly, you should put the first ones listed (as many as fit 1458 reasonably) on the actual cover, and continue the rest onto 1459 adjacent pages. 1460 1461 If you publish or distribute Opaque copies of the Document 1462 numbering more than 100, you must either include a 1463 machine-readable Transparent copy along with each Opaque copy, or 1464 state in or with each Opaque copy a computer-network location from 1465 which the general network-using public has access to download 1466 using public-standard network protocols a complete Transparent 1467 copy of the Document, free of added material. If you use the 1468 latter option, you must take reasonably prudent steps, when you 1469 begin distribution of Opaque copies in quantity, to ensure that 1470 this Transparent copy will remain thus accessible at the stated 1471 location until at least one year after the last time you 1472 distribute an Opaque copy (directly or through your agents or 1473 retailers) of that edition to the public. 1474 1475 It is requested, but not required, that you contact the authors of 1476 the Document well before redistributing any large number of 1477 copies, to give them a chance to provide you with an updated 1478 version of the Document. 1479 1480 4. MODIFICATIONS 1481 1482 You may copy and distribute a Modified Version of the Document 1483 under the conditions of sections 2 and 3 above, provided that you 1484 release the Modified Version under precisely this License, with 1485 the Modified Version filling the role of the Document, thus 1486 licensing distribution and modification of the Modified Version to 1487 whoever possesses a copy of it. In addition, you must do these 1488 things in the Modified Version: 1489 1490 A. Use in the Title Page (and on the covers, if any) a title 1491 distinct from that of the Document, and from those of 1492 previous versions (which should, if there were any, be listed 1493 in the History section of the Document). You may use the 1494 same title as a previous version if the original publisher of 1495 that version gives permission. 1496 1497 B. List on the Title Page, as authors, one or more persons or 1498 entities responsible for authorship of the modifications in 1499 the Modified Version, together with at least five of the 1500 principal authors of the Document (all of its principal 1501 authors, if it has fewer than five), unless they release you 1502 from this requirement. 1503 1504 C. State on the Title page the name of the publisher of the 1505 Modified Version, as the publisher. 1506 1507 D. Preserve all the copyright notices of the Document. 1508 1509 E. Add an appropriate copyright notice for your modifications 1510 adjacent to the other copyright notices. 1511 1512 F. Include, immediately after the copyright notices, a license 1513 notice giving the public permission to use the Modified 1514 Version under the terms of this License, in the form shown in 1515 the Addendum below. 1516 1517 G. Preserve in that license notice the full lists of Invariant 1518 Sections and required Cover Texts given in the Document's 1519 license notice. 1520 1521 H. Include an unaltered copy of this License. 1522 1523 I. Preserve the section Entitled "History", Preserve its Title, 1524 and add to it an item stating at least the title, year, new 1525 authors, and publisher of the Modified Version as given on 1526 the Title Page. If there is no section Entitled "History" in 1527 the Document, create one stating the title, year, authors, 1528 and publisher of the Document as given on its Title Page, 1529 then add an item describing the Modified Version as stated in 1530 the previous sentence. 1531 1532 J. Preserve the network location, if any, given in the Document 1533 for public access to a Transparent copy of the Document, and 1534 likewise the network locations given in the Document for 1535 previous versions it was based on. These may be placed in 1536 the "History" section. You may omit a network location for a 1537 work that was published at least four years before the 1538 Document itself, or if the original publisher of the version 1539 it refers to gives permission. 1540 1541 K. For any section Entitled "Acknowledgements" or "Dedications", 1542 Preserve the Title of the section, and preserve in the 1543 section all the substance and tone of each of the contributor 1544 acknowledgements and/or dedications given therein. 1545 1546 L. Preserve all the Invariant Sections of the Document, 1547 unaltered in their text and in their titles. Section numbers 1548 or the equivalent are not considered part of the section 1549 titles. 1550 1551 M. Delete any section Entitled "Endorsements". Such a section 1552 may not be included in the Modified Version. 1553 1554 N. Do not retitle any existing section to be Entitled 1555 "Endorsements" or to conflict in title with any Invariant 1556 Section. 1557 1558 O. Preserve any Warranty Disclaimers. 1559 1560 If the Modified Version includes new front-matter sections or 1561 appendices that qualify as Secondary Sections and contain no 1562 material copied from the Document, you may at your option 1563 designate some or all of these sections as invariant. To do this, 1564 add their titles to the list of Invariant Sections in the Modified 1565 Version's license notice. These titles must be distinct from any 1566 other section titles. 1567 1568 You may add a section Entitled "Endorsements", provided it contains 1569 nothing but endorsements of your Modified Version by various 1570 parties--for example, statements of peer review or that the text 1571 has been approved by an organization as the authoritative 1572 definition of a standard. 1573 1574 You may add a passage of up to five words as a Front-Cover Text, 1575 and a passage of up to 25 words as a Back-Cover Text, to the end 1576 of the list of Cover Texts in the Modified Version. Only one 1577 passage of Front-Cover Text and one of Back-Cover Text may be 1578 added by (or through arrangements made by) any one entity. If the 1579 Document already includes a cover text for the same cover, 1580 previously added by you or by arrangement made by the same entity 1581 you are acting on behalf of, you may not add another; but you may 1582 replace the old one, on explicit permission from the previous 1583 publisher that added the old one. 1584 1585 The author(s) and publisher(s) of the Document do not by this 1586 License give permission to use their names for publicity for or to 1587 assert or imply endorsement of any Modified Version. 1588 1589 5. COMBINING DOCUMENTS 1590 1591 You may combine the Document with other documents released under 1592 this License, under the terms defined in section 4 above for 1593 modified versions, provided that you include in the combination 1594 all of the Invariant Sections of all of the original documents, 1595 unmodified, and list them all as Invariant Sections of your 1596 combined work in its license notice, and that you preserve all 1597 their Warranty Disclaimers. 1598 1599 The combined work need only contain one copy of this License, and 1600 multiple identical Invariant Sections may be replaced with a single 1601 copy. If there are multiple Invariant Sections with the same name 1602 but different contents, make the title of each such section unique 1603 by adding at the end of it, in parentheses, the name of the 1604 original author or publisher of that section if known, or else a 1605 unique number. Make the same adjustment to the section titles in 1606 the list of Invariant Sections in the license notice of the 1607 combined work. 1608 1609 In the combination, you must combine any sections Entitled 1610 "History" in the various original documents, forming one section 1611 Entitled "History"; likewise combine any sections Entitled 1612 "Acknowledgements", and any sections Entitled "Dedications". You 1613 must delete all sections Entitled "Endorsements." 1614 1615 6. COLLECTIONS OF DOCUMENTS 1616 1617 You may make a collection consisting of the Document and other 1618 documents released under this License, and replace the individual 1619 copies of this License in the various documents with a single copy 1620 that is included in the collection, provided that you follow the 1621 rules of this License for verbatim copying of each of the 1622 documents in all other respects. 1623 1624 You may extract a single document from such a collection, and 1625 distribute it individually under this License, provided you insert 1626 a copy of this License into the extracted document, and follow 1627 this License in all other respects regarding verbatim copying of 1628 that document. 1629 1630 7. AGGREGATION WITH INDEPENDENT WORKS 1631 1632 A compilation of the Document or its derivatives with other 1633 separate and independent documents or works, in or on a volume of 1634 a storage or distribution medium, is called an "aggregate" if the 1635 copyright resulting from the compilation is not used to limit the 1636 legal rights of the compilation's users beyond what the individual 1637 works permit. When the Document is included in an aggregate, this 1638 License does not apply to the other works in the aggregate which 1639 are not themselves derivative works of the Document. 1640 1641 If the Cover Text requirement of section 3 is applicable to these 1642 copies of the Document, then if the Document is less than one half 1643 of the entire aggregate, the Document's Cover Texts may be placed 1644 on covers that bracket the Document within the aggregate, or the 1645 electronic equivalent of covers if the Document is in electronic 1646 form. Otherwise they must appear on printed covers that bracket 1647 the whole aggregate. 1648 1649 8. TRANSLATION 1650 1651 Translation is considered a kind of modification, so you may 1652 distribute translations of the Document under the terms of section 1653 4. Replacing Invariant Sections with translations requires special 1654 permission from their copyright holders, but you may include 1655 translations of some or all Invariant Sections in addition to the 1656 original versions of these Invariant Sections. You may include a 1657 translation of this License, and all the license notices in the 1658 Document, and any Warranty Disclaimers, provided that you also 1659 include the original English version of this License and the 1660 original versions of those notices and disclaimers. In case of a 1661 disagreement between the translation and the original version of 1662 this License or a notice or disclaimer, the original version will 1663 prevail. 1664 1665 If a section in the Document is Entitled "Acknowledgements", 1666 "Dedications", or "History", the requirement (section 4) to 1667 Preserve its Title (section 1) will typically require changing the 1668 actual title. 1669 1670 9. TERMINATION 1671 1672 You may not copy, modify, sublicense, or distribute the Document 1673 except as expressly provided under this License. Any attempt 1674 otherwise to copy, modify, sublicense, or distribute it is void, 1675 and will automatically terminate your rights under this License. 1676 1677 However, if you cease all violation of this License, then your 1678 license from a particular copyright holder is reinstated (a) 1679 provisionally, unless and until the copyright holder explicitly 1680 and finally terminates your license, and (b) permanently, if the 1681 copyright holder fails to notify you of the violation by some 1682 reasonable means prior to 60 days after the cessation. 1683 1684 Moreover, your license from a particular copyright holder is 1685 reinstated permanently if the copyright holder notifies you of the 1686 violation by some reasonable means, this is the first time you have 1687 received notice of violation of this License (for any work) from 1688 that copyright holder, and you cure the violation prior to 30 days 1689 after your receipt of the notice. 1690 1691 Termination of your rights under this section does not terminate 1692 the licenses of parties who have received copies or rights from 1693 you under this License. If your rights have been terminated and 1694 not permanently reinstated, receipt of a copy of some or all of 1695 the same material does not give you any rights to use it. 1696 1697 10. FUTURE REVISIONS OF THIS LICENSE 1698 1699 The Free Software Foundation may publish new, revised versions of 1700 the GNU Free Documentation License from time to time. Such new 1701 versions will be similar in spirit to the present version, but may 1702 differ in detail to address new problems or concerns. See 1703 `http://www.gnu.org/copyleft/'. 1704 1705 Each version of the License is given a distinguishing version 1706 number. If the Document specifies that a particular numbered 1707 version of this License "or any later version" applies to it, you 1708 have the option of following the terms and conditions either of 1709 that specified version or of any later version that has been 1710 published (not as a draft) by the Free Software Foundation. If 1711 the Document does not specify a version number of this License, 1712 you may choose any version ever published (not as a draft) by the 1713 Free Software Foundation. If the Document specifies that a proxy 1714 can decide which future versions of this License can be used, that 1715 proxy's public statement of acceptance of a version permanently 1716 authorizes you to choose that version for the Document. 1717 1718 11. RELICENSING 1719 1720 "Massive Multiauthor Collaboration Site" (or "MMC Site") means any 1721 World Wide Web server that publishes copyrightable works and also 1722 provides prominent facilities for anybody to edit those works. A 1723 public wiki that anybody can edit is an example of such a server. 1724 A "Massive Multiauthor Collaboration" (or "MMC") contained in the 1725 site means any set of copyrightable works thus published on the MMC 1726 site. 1727 1728 "CC-BY-SA" means the Creative Commons Attribution-Share Alike 3.0 1729 license published by Creative Commons Corporation, a not-for-profit 1730 corporation with a principal place of business in San Francisco, 1731 California, as well as future copyleft versions of that license 1732 published by that same organization. 1733 1734 "Incorporate" means to publish or republish a Document, in whole or 1735 in part, as part of another Document. 1736 1737 An MMC is "eligible for relicensing" if it is licensed under this 1738 License, and if all works that were first published under this 1739 License somewhere other than this MMC, and subsequently 1740 incorporated in whole or in part into the MMC, (1) had no cover 1741 texts or invariant sections, and (2) were thus incorporated prior 1742 to November 1, 2008. 1743 1744 The operator of an MMC Site may republish an MMC contained in the 1745 site under CC-BY-SA on the same site at any time before August 1, 1746 2009, provided the MMC is eligible for relicensing. 1747 1748 1749 ADDENDUM: How to use this License for your documents 1750 ==================================================== 1751 1752 To use this License in a document you have written, include a copy of 1753 the License in the document and put the following copyright and license 1754 notices just after the title page: 1755 1756 Copyright (C) YEAR YOUR NAME. 1757 Permission is granted to copy, distribute and/or modify this document 1758 under the terms of the GNU Free Documentation License, Version 1.3 1759 or any later version published by the Free Software Foundation; 1760 with no Invariant Sections, no Front-Cover Texts, and no Back-Cover 1761 Texts. A copy of the license is included in the section entitled ``GNU 1762 Free Documentation License''. 1763 1764 If you have Invariant Sections, Front-Cover Texts and Back-Cover 1765 Texts, replace the "with...Texts." line with this: 1766 1767 with the Invariant Sections being LIST THEIR TITLES, with 1768 the Front-Cover Texts being LIST, and with the Back-Cover Texts 1769 being LIST. 1770 1771 If you have Invariant Sections without Cover Texts, or some other 1772 combination of the three, merge those two alternatives to suit the 1773 situation. 1774 1775 If your document contains nontrivial examples of program code, we 1776 recommend releasing these examples in parallel under your choice of 1777 free software license, such as the GNU General Public License, to 1778 permit their use in free software. 1779 1780 1781 1782 Tag Table: 1783 Node: Top738 1784 Node: Copying1445 1785 Node: Introduction to GNU MPC2216 1786 Node: Installing GNU MPC2934 1787 Node: Reporting Bugs8016 1788 Node: GNU MPC Basics9357 1789 Ref: return-value13055 1790 Node: Complex Functions14505 1791 Node: Initializing Complex Numbers15664 1792 Node: Assigning Complex Numbers18052 1793 Node: Converting Complex Numbers22451 1794 Node: String and Stream Input and Output23076 1795 Node: Complex Comparison29655 1796 Node: Projection & Decomposing30734 1797 Node: Basic Arithmetic32109 1798 Node: Power Functions and Logarithm36682 1799 Node: Trigonometric Functions38749 1800 Node: Miscellaneous Complex Functions40974 1801 Node: Advanced Functions43148 1802 Node: Internals44221 1803 Node: References44672 1804 Node: Concept Index45578 1805 Node: Function Index47892 1806 Node: GNU Free Documentation License61676 1807 1808 End Tag Table 1809
