1This is ../../gmp/doc/gmp.info, produced by makeinfo version 4.13 from 2../../gmp/doc/gmp.texi. 3 4This manual describes how to install and use the GNU multiple precision 5arithmetic library, version 5.1.3. 6 7 Copyright 1991, 1993, 1994, 1995, 1996, 1997, 1998, 1999, 2000, 82001, 2002, 2003, 2004, 2005, 2006, 2007, 2008, 2009, 2010, 2011, 2012, 92013 Free Software Foundation, Inc. 10 11 Permission is granted to copy, distribute and/or modify this 12document under the terms of the GNU Free Documentation License, Version 131.3 or any later version published by the Free Software Foundation; 14with no Invariant Sections, with the Front-Cover Texts being "A GNU 15Manual", and with the Back-Cover Texts being "You have freedom to copy 16and modify this GNU Manual, like GNU software". A copy of the license 17is included in *note GNU Free Documentation License::. 18 19INFO-DIR-SECTION GNU libraries 20START-INFO-DIR-ENTRY 21* gmp: (gmp). GNU Multiple Precision Arithmetic Library. 22END-INFO-DIR-ENTRY 23 24 25File: gmp.info, Node: Extended GCD, Next: Jacobi Symbol, Prev: Subquadratic GCD, Up: Greatest Common Divisor Algorithms 26 2715.3.4 Extended GCD 28------------------- 29 30The extended GCD function, or GCDEXT, calculates gcd(a,b) and also 31cofactors x and y satisfying a*x+b*y=gcd(a,b). All the algorithms used 32for plain GCD are extended to handle this case. The binary algorithm is 33used only for single-limb GCDEXT. Lehmer's algorithm is used for sizes 34up to `GCDEXT_DC_THRESHOLD'. Above this threshold, GCDEXT is 35implemented as a loop around HGCD, but with more book-keeping to keep 36track of the cofactors. This gives the same asymptotic running time as 37for GCD and HGCD, O(M(N)*log(N)) 38 39 One difference to plain GCD is that while the inputs a and b are 40reduced as the algorithm proceeds, the cofactors x and y grow in size. 41This makes the tuning of the chopping-point more difficult. The current 42code chops off the most significant half of the inputs for the call to 43HGCD in the first iteration, and the most significant two thirds for 44the remaining calls. This strategy could surely be improved. Also the 45stop condition for the loop, where Lehmer's algorithm is invoked once 46the inputs are reduced below `GCDEXT_DC_THRESHOLD', could maybe be 47improved by taking into account the current size of the cofactors. 48 49 50File: gmp.info, Node: Jacobi Symbol, Prev: Extended GCD, Up: Greatest Common Divisor Algorithms 51 5215.3.5 Jacobi Symbol 53-------------------- 54 55[This section is obsolete. The current Jacobi code actually uses a very 56efficient algorithm.] 57 58 `mpz_jacobi' and `mpz_kronecker' are currently implemented with a 59simple binary algorithm similar to that described for the GCDs (*note 60Binary GCD::). They're not very fast when both inputs are large. 61Lehmer's multi-step improvement or a binary based multi-step algorithm 62is likely to be better. 63 64 When one operand fits a single limb, and that includes 65`mpz_kronecker_ui' and friends, an initial reduction is done with 66either `mpn_mod_1' or `mpn_modexact_1_odd', followed by the binary 67algorithm on a single limb. The binary algorithm is well suited to a 68single limb, and the whole calculation in this case is quite efficient. 69 70 In all the routines sign changes for the result are accumulated 71using some bit twiddling, avoiding table lookups or conditional jumps. 72 73 74File: gmp.info, Node: Powering Algorithms, Next: Root Extraction Algorithms, Prev: Greatest Common Divisor Algorithms, Up: Algorithms 75 7615.4 Powering Algorithms 77======================== 78 79* Menu: 80 81* Normal Powering Algorithm:: 82* Modular Powering Algorithm:: 83 84 85File: gmp.info, Node: Normal Powering Algorithm, Next: Modular Powering Algorithm, Prev: Powering Algorithms, Up: Powering Algorithms 86 8715.4.1 Normal Powering 88---------------------- 89 90Normal `mpz' or `mpf' powering uses a simple binary algorithm, 91successively squaring and then multiplying by the base when a 1 bit is 92seen in the exponent, as per Knuth section 4.6.3. The "left to right" 93variant described there is used rather than algorithm A, since it's 94just as easy and can be done with somewhat less temporary memory. 95 96 97File: gmp.info, Node: Modular Powering Algorithm, Prev: Normal Powering Algorithm, Up: Powering Algorithms 98 9915.4.2 Modular Powering 100----------------------- 101 102Modular powering is implemented using a 2^k-ary sliding window 103algorithm, as per "Handbook of Applied Cryptography" algorithm 14.85 104(*note References::). k is chosen according to the size of the 105exponent. Larger exponents use larger values of k, the choice being 106made to minimize the average number of multiplications that must 107supplement the squaring. 108 109 The modular multiplies and squarings use either a simple division or 110the REDC method by Montgomery (*note References::). REDC is a little 111faster, essentially saving N single limb divisions in a fashion similar 112to an exact remainder (*note Exact Remainder::). 113 114 115File: gmp.info, Node: Root Extraction Algorithms, Next: Radix Conversion Algorithms, Prev: Powering Algorithms, Up: Algorithms 116 11715.5 Root Extraction Algorithms 118=============================== 119 120* Menu: 121 122* Square Root Algorithm:: 123* Nth Root Algorithm:: 124* Perfect Square Algorithm:: 125* Perfect Power Algorithm:: 126 127 128File: gmp.info, Node: Square Root Algorithm, Next: Nth Root Algorithm, Prev: Root Extraction Algorithms, Up: Root Extraction Algorithms 129 13015.5.1 Square Root 131------------------ 132 133Square roots are taken using the "Karatsuba Square Root" algorithm by 134Paul Zimmermann (*note References::). 135 136 An input n is split into four parts of k bits each, so with b=2^k we 137have n = a3*b^3 + a2*b^2 + a1*b + a0. Part a3 must be "normalized" so 138that either the high or second highest bit is set. In GMP, k is kept 139on a limb boundary and the input is left shifted (by an even number of 140bits) to normalize. 141 142 The square root of the high two parts is taken, by recursive 143application of the algorithm (bottoming out in a one-limb Newton's 144method), 145 146 s1,r1 = sqrtrem (a3*b + a2) 147 148 This is an approximation to the desired root and is extended by a 149division to give s,r, 150 151 q,u = divrem (r1*b + a1, 2*s1) 152 s = s1*b + q 153 r = u*b + a0 - q^2 154 155 The normalization requirement on a3 means at this point s is either 156correct or 1 too big. r is negative in the latter case, so 157 158 if r < 0 then 159 r = r + 2*s - 1 160 s = s - 1 161 162 The algorithm is expressed in a divide and conquer form, but as 163noted in the paper it can also be viewed as a discrete variant of 164Newton's method, or as a variation on the schoolboy method (no longer 165taught) for square roots two digits at a time. 166 167 If the remainder r is not required then usually only a few high limbs 168of r and u need to be calculated to determine whether an adjustment to 169s is required. This optimization is not currently implemented. 170 171 In the Karatsuba multiplication range this algorithm is 172O(1.5*M(N/2)), where M(n) is the time to multiply two numbers of n 173limbs. In the FFT multiplication range this grows to a bound of 174O(6*M(N/2)). In practice a factor of about 1.5 to 1.8 is found in the 175Karatsuba and Toom-3 ranges, growing to 2 or 3 in the FFT range. 176 177 The algorithm does all its calculations in integers and the resulting 178`mpn_sqrtrem' is used for both `mpz_sqrt' and `mpf_sqrt'. The extended 179precision given by `mpf_sqrt_ui' is obtained by padding with zero limbs. 180 181 182File: gmp.info, Node: Nth Root Algorithm, Next: Perfect Square Algorithm, Prev: Square Root Algorithm, Up: Root Extraction Algorithms 183 18415.5.2 Nth Root 185--------------- 186 187Integer Nth roots are taken using Newton's method with the following 188iteration, where A is the input and n is the root to be taken. 189 190 1 A 191 a[i+1] = - * ( --------- + (n-1)*a[i] ) 192 n a[i]^(n-1) 193 194 The initial approximation a[1] is generated bitwise by successively 195powering a trial root with or without new 1 bits, aiming to be just 196above the true root. The iteration converges quadratically when 197started from a good approximation. When n is large more initial bits 198are needed to get good convergence. The current implementation is not 199particularly well optimized. 200 201 202File: gmp.info, Node: Perfect Square Algorithm, Next: Perfect Power Algorithm, Prev: Nth Root Algorithm, Up: Root Extraction Algorithms 203 20415.5.3 Perfect Square 205--------------------- 206 207A significant fraction of non-squares can be quickly identified by 208checking whether the input is a quadratic residue modulo small integers. 209 210 `mpz_perfect_square_p' first tests the input mod 256, which means 211just examining the low byte. Only 44 different values occur for 212squares mod 256, so 82.8% of inputs can be immediately identified as 213non-squares. 214 215 On a 32-bit system similar tests are done mod 9, 5, 7, 13 and 17, 216for a total 99.25% of inputs identified as non-squares. On a 64-bit 217system 97 is tested too, for a total 99.62%. 218 219 These moduli are chosen because they're factors of 2^24-1 (or 2^48-1 220for 64-bits), and such a remainder can be quickly taken just using 221additions (see `mpn_mod_34lsub1'). 222 223 When nails are in use moduli are instead selected by the `gen-psqr.c' 224program and applied with an `mpn_mod_1'. The same 2^24-1 or 2^48-1 225could be done with nails using some extra bit shifts, but this is not 226currently implemented. 227 228 In any case each modulus is applied to the `mpn_mod_34lsub1' or 229`mpn_mod_1' remainder and a table lookup identifies non-squares. By 230using a "modexact" style calculation, and suitably permuted tables, 231just one multiply each is required, see the code for details. Moduli 232are also combined to save operations, so long as the lookup tables 233don't become too big. `gen-psqr.c' does all the pre-calculations. 234 235 A square root must still be taken for any value that passes these 236tests, to verify it's really a square and not one of the small fraction 237of non-squares that get through (i.e. a pseudo-square to all the tested 238bases). 239 240 Clearly more residue tests could be done, `mpz_perfect_square_p' only 241uses a compact and efficient set. Big inputs would probably benefit 242from more residue testing, small inputs might be better off with less. 243The assumed distribution of squares versus non-squares in the input 244would affect such considerations. 245 246 247File: gmp.info, Node: Perfect Power Algorithm, Prev: Perfect Square Algorithm, Up: Root Extraction Algorithms 248 24915.5.4 Perfect Power 250-------------------- 251 252Detecting perfect powers is required by some factorization algorithms. 253Currently `mpz_perfect_power_p' is implemented using repeated Nth root 254extractions, though naturally only prime roots need to be considered. 255(*Note Nth Root Algorithm::.) 256 257 If a prime divisor p with multiplicity e can be found, then only 258roots which are divisors of e need to be considered, much reducing the 259work necessary. To this end divisibility by a set of small primes is 260checked. 261 262 263File: gmp.info, Node: Radix Conversion Algorithms, Next: Other Algorithms, Prev: Root Extraction Algorithms, Up: Algorithms 264 26515.6 Radix Conversion 266===================== 267 268Radix conversions are less important than other algorithms. A program 269dominated by conversions should probably use a different data 270representation. 271 272* Menu: 273 274* Binary to Radix:: 275* Radix to Binary:: 276 277 278File: gmp.info, Node: Binary to Radix, Next: Radix to Binary, Prev: Radix Conversion Algorithms, Up: Radix Conversion Algorithms 279 28015.6.1 Binary to Radix 281---------------------- 282 283Conversions from binary to a power-of-2 radix use a simple and fast 284O(N) bit extraction algorithm. 285 286 Conversions from binary to other radices use one of two algorithms. 287Sizes below `GET_STR_PRECOMPUTE_THRESHOLD' use a basic O(N^2) method. 288Repeated divisions by b^n are made, where b is the radix and n is the 289biggest power that fits in a limb. But instead of simply using the 290remainder r from such divisions, an extra divide step is done to give a 291fractional limb representing r/b^n. The digits of r can then be 292extracted using multiplications by b rather than divisions. Special 293case code is provided for decimal, allowing multiplications by 10 to 294optimize to shifts and adds. 295 296 Above `GET_STR_PRECOMPUTE_THRESHOLD' a sub-quadratic algorithm is 297used. For an input t, powers b^(n*2^i) of the radix are calculated, 298until a power between t and sqrt(t) is reached. t is then divided by 299that largest power, giving a quotient which is the digits above that 300power, and a remainder which is those below. These two parts are in 301turn divided by the second highest power, and so on recursively. When 302a piece has been divided down to less than `GET_STR_DC_THRESHOLD' 303limbs, the basecase algorithm described above is used. 304 305 The advantage of this algorithm is that big divisions can make use 306of the sub-quadratic divide and conquer division (*note Divide and 307Conquer Division::), and big divisions tend to have less overheads than 308lots of separate single limb divisions anyway. But in any case the 309cost of calculating the powers b^(n*2^i) must first be overcome. 310 311 `GET_STR_PRECOMPUTE_THRESHOLD' and `GET_STR_DC_THRESHOLD' represent 312the same basic thing, the point where it becomes worth doing a big 313division to cut the input in half. `GET_STR_PRECOMPUTE_THRESHOLD' 314includes the cost of calculating the radix power required, whereas 315`GET_STR_DC_THRESHOLD' assumes that's already available, which is the 316case when recursing. 317 318 Since the base case produces digits from least to most significant 319but they want to be stored from most to least, it's necessary to 320calculate in advance how many digits there will be, or at least be sure 321not to underestimate that. For GMP the number of input bits is 322multiplied by `chars_per_bit_exactly' from `mp_bases', rounding up. 323The result is either correct or one too big. 324 325 Examining some of the high bits of the input could increase the 326chance of getting the exact number of digits, but an exact result every 327time would not be practical, since in general the difference between 328numbers 100... and 99... is only in the last few bits and the work to 329identify 99... might well be almost as much as a full conversion. 330 331 `mpf_get_str' doesn't currently use the algorithm described here, it 332multiplies or divides by a power of b to move the radix point to the 333just above the highest non-zero digit (or at worst one above that 334location), then multiplies by b^n to bring out digits. This is O(N^2) 335and is certainly not optimal. 336 337 The r/b^n scheme described above for using multiplications to bring 338out digits might be useful for more than a single limb. Some brief 339experiments with it on the base case when recursing didn't give a 340noticeable improvement, but perhaps that was only due to the 341implementation. Something similar would work for the sub-quadratic 342divisions too, though there would be the cost of calculating a bigger 343radix power. 344 345 Another possible improvement for the sub-quadratic part would be to 346arrange for radix powers that balanced the sizes of quotient and 347remainder produced, i.e. the highest power would be an b^(n*k) 348approximately equal to sqrt(t), not restricted to a 2^i factor. That 349ought to smooth out a graph of times against sizes, but may or may not 350be a net speedup. 351 352 353File: gmp.info, Node: Radix to Binary, Prev: Binary to Radix, Up: Radix Conversion Algorithms 354 35515.6.2 Radix to Binary 356---------------------- 357 358*This section needs to be rewritten, it currently describes the 359algorithms used before GMP 4.3.* 360 361 Conversions from a power-of-2 radix into binary use a simple and fast 362O(N) bitwise concatenation algorithm. 363 364 Conversions from other radices use one of two algorithms. Sizes 365below `SET_STR_PRECOMPUTE_THRESHOLD' use a basic O(N^2) method. Groups 366of n digits are converted to limbs, where n is the biggest power of the 367base b which will fit in a limb, then those groups are accumulated into 368the result by multiplying by b^n and adding. This saves 369multi-precision operations, as per Knuth section 4.4 part E (*note 370References::). Some special case code is provided for decimal, giving 371the compiler a chance to optimize multiplications by 10. 372 373 Above `SET_STR_PRECOMPUTE_THRESHOLD' a sub-quadratic algorithm is 374used. First groups of n digits are converted into limbs. Then adjacent 375limbs are combined into limb pairs with x*b^n+y, where x and y are the 376limbs. Adjacent limb pairs are combined into quads similarly with 377x*b^(2n)+y. This continues until a single block remains, that being 378the result. 379 380 The advantage of this method is that the multiplications for each x 381are big blocks, allowing Karatsuba and higher algorithms to be used. 382But the cost of calculating the powers b^(n*2^i) must be overcome. 383`SET_STR_PRECOMPUTE_THRESHOLD' usually ends up quite big, around 5000 384digits, and on some processors much bigger still. 385 386 `SET_STR_PRECOMPUTE_THRESHOLD' is based on the input digits (and 387tuned for decimal), though it might be better based on a limb count, so 388as to be independent of the base. But that sort of count isn't used by 389the base case and so would need some sort of initial calculation or 390estimate. 391 392 The main reason `SET_STR_PRECOMPUTE_THRESHOLD' is so much bigger 393than the corresponding `GET_STR_PRECOMPUTE_THRESHOLD' is that 394`mpn_mul_1' is much faster than `mpn_divrem_1' (often by a factor of 5, 395or more). 396 397 398File: gmp.info, Node: Other Algorithms, Next: Assembly Coding, Prev: Radix Conversion Algorithms, Up: Algorithms 399 40015.7 Other Algorithms 401===================== 402 403* Menu: 404 405* Prime Testing Algorithm:: 406* Factorial Algorithm:: 407* Binomial Coefficients Algorithm:: 408* Fibonacci Numbers Algorithm:: 409* Lucas Numbers Algorithm:: 410* Random Number Algorithms:: 411 412 413File: gmp.info, Node: Prime Testing Algorithm, Next: Factorial Algorithm, Prev: Other Algorithms, Up: Other Algorithms 414 41515.7.1 Prime Testing 416-------------------- 417 418The primality testing in `mpz_probab_prime_p' (*note Number Theoretic 419Functions::) first does some trial division by small factors and then 420uses the Miller-Rabin probabilistic primality testing algorithm, as 421described in Knuth section 4.5.4 algorithm P (*note References::). 422 423 For an odd input n, and with n = q*2^k+1 where q is odd, this 424algorithm selects a random base x and tests whether x^q mod n is 1 or 425-1, or an x^(q*2^j) mod n is 1, for 1<=j<=k. If so then n is probably 426prime, if not then n is definitely composite. 427 428 Any prime n will pass the test, but some composites do too. Such 429composites are known as strong pseudoprimes to base x. No n is a 430strong pseudoprime to more than 1/4 of all bases (see Knuth exercise 43122), hence with x chosen at random there's no more than a 1/4 chance a 432"probable prime" will in fact be composite. 433 434 In fact strong pseudoprimes are quite rare, making the test much more 435powerful than this analysis would suggest, but 1/4 is all that's proven 436for an arbitrary n. 437 438 439File: gmp.info, Node: Factorial Algorithm, Next: Binomial Coefficients Algorithm, Prev: Prime Testing Algorithm, Up: Other Algorithms 440 44115.7.2 Factorial 442---------------- 443 444Factorials are calculated by a combination of two algorithms. An idea is 445shared among them: to compute the odd part of the factorial; a final 446step takes account of the power of 2 term, by shifting. 447 448 For small n, the odd factor of n! is computed with the simple 449observation that it is equal to the product of all positive odd numbers 450smaller than n times the odd factor of [n/2]!, where [x] is the integer 451part of x, and so on recursively. The procedure can be best illustrated 452with an example, 453 454 23! = (23.21.19.17.15.13.11.9.7.5.3)(11.9.7.5.3)(5.3)2^19 455 456 Current code collects all the factors in a single list, with a loop 457and no recursion, and compute the product, with no special care for 458repeated chunks. 459 460 When n is larger, computation pass trough prime sieving. An helper 461function is used, as suggested by Peter Luschny: 462 463 n 464 ----- 465 n! | | L(p,n) 466 msf(n) = -------------- = | | p 467 [n/2]!^2.2^k p=3 468 469 Where p ranges on odd prime numbers. The exponent k is chosen to 470obtain an odd integer number: k is the number of 1 bits in the binary 471representation of [n/2]. The function L(p,n) can be defined as zero 472when p is composite, and, for any prime p, it is computed with: 473 474 --- 475 \ n 476 L(p,n) = / [---] mod 2 <= log (n) . 477 --- p^i p 478 i>0 479 480 With this helper function, we are able to compute the odd part of n! 481using the recursion implied by n!=[n/2]!^2*msf(n)*2^k. The recursion 482stops using the small-n algorithm on some [n/2^i]. 483 484 Both the above algorithms use binary splitting to compute the 485product of many small factors. At first as many products as possible 486are accumulated in a single register, generating a list of factors that 487fit in a machine word. This list is then split into halves, and the 488product is computed recursively. 489 490 Such splitting is more efficient than repeated Nx1 multiplies since 491it forms big multiplies, allowing Karatsuba and higher algorithms to be 492used. And even below the Karatsuba threshold a big block of work can 493be more efficient for the basecase algorithm. 494 495 496File: gmp.info, Node: Binomial Coefficients Algorithm, Next: Fibonacci Numbers Algorithm, Prev: Factorial Algorithm, Up: Other Algorithms 497 49815.7.3 Binomial Coefficients 499---------------------------- 500 501Binomial coefficients C(n,k) are calculated by first arranging k <= n/2 502using C(n,k) = C(n,n-k) if necessary, and then evaluating the following 503product simply from i=2 to i=k. 504 505 k (n-k+i) 506 C(n,k) = (n-k+1) * prod ------- 507 i=2 i 508 509 It's easy to show that each denominator i will divide the product so 510far, so the exact division algorithm is used (*note Exact Division::). 511 512 The numerators n-k+i and denominators i are first accumulated into 513as many fit a limb, to save multi-precision operations, though for 514`mpz_bin_ui' this applies only to the divisors, since n is an `mpz_t' 515and n-k+i in general won't fit in a limb at all. 516 517 518File: gmp.info, Node: Fibonacci Numbers Algorithm, Next: Lucas Numbers Algorithm, Prev: Binomial Coefficients Algorithm, Up: Other Algorithms 519 52015.7.4 Fibonacci Numbers 521------------------------ 522 523The Fibonacci functions `mpz_fib_ui' and `mpz_fib2_ui' are designed for 524calculating isolated F[n] or F[n],F[n-1] values efficiently. 525 526 For small n, a table of single limb values in `__gmp_fib_table' is 527used. On a 32-bit limb this goes up to F[47], or on a 64-bit limb up 528to F[93]. For convenience the table starts at F[-1]. 529 530 Beyond the table, values are generated with a binary powering 531algorithm, calculating a pair F[n] and F[n-1] working from high to low 532across the bits of n. The formulas used are 533 534 F[2k+1] = 4*F[k]^2 - F[k-1]^2 + 2*(-1)^k 535 F[2k-1] = F[k]^2 + F[k-1]^2 536 537 F[2k] = F[2k+1] - F[2k-1] 538 539 At each step, k is the high b bits of n. If the next bit of n is 0 540then F[2k],F[2k-1] is used, or if it's a 1 then F[2k+1],F[2k] is used, 541and the process repeated until all bits of n are incorporated. Notice 542these formulas require just two squares per bit of n. 543 544 It'd be possible to handle the first few n above the single limb 545table with simple additions, using the defining Fibonacci recurrence 546F[k+1]=F[k]+F[k-1], but this is not done since it usually turns out to 547be faster for only about 10 or 20 values of n, and including a block of 548code for just those doesn't seem worthwhile. If they really mattered 549it'd be better to extend the data table. 550 551 Using a table avoids lots of calculations on small numbers, and 552makes small n go fast. A bigger table would make more small n go fast, 553it's just a question of balancing size against desired speed. For GMP 554the code is kept compact, with the emphasis primarily on a good 555powering algorithm. 556 557 `mpz_fib2_ui' returns both F[n] and F[n-1], but `mpz_fib_ui' is only 558interested in F[n]. In this case the last step of the algorithm can 559become one multiply instead of two squares. One of the following two 560formulas is used, according as n is odd or even. 561 562 F[2k] = F[k]*(F[k]+2F[k-1]) 563 564 F[2k+1] = (2F[k]+F[k-1])*(2F[k]-F[k-1]) + 2*(-1)^k 565 566 F[2k+1] here is the same as above, just rearranged to be a multiply. 567For interest, the 2*(-1)^k term both here and above can be applied just 568to the low limb of the calculation, without a carry or borrow into 569further limbs, which saves some code size. See comments with 570`mpz_fib_ui' and the internal `mpn_fib2_ui' for how this is done. 571 572 573File: gmp.info, Node: Lucas Numbers Algorithm, Next: Random Number Algorithms, Prev: Fibonacci Numbers Algorithm, Up: Other Algorithms 574 57515.7.5 Lucas Numbers 576-------------------- 577 578`mpz_lucnum2_ui' derives a pair of Lucas numbers from a pair of 579Fibonacci numbers with the following simple formulas. 580 581 L[k] = F[k] + 2*F[k-1] 582 L[k-1] = 2*F[k] - F[k-1] 583 584 `mpz_lucnum_ui' is only interested in L[n], and some work can be 585saved. Trailing zero bits on n can be handled with a single square 586each. 587 588 L[2k] = L[k]^2 - 2*(-1)^k 589 590 And the lowest 1 bit can be handled with one multiply of a pair of 591Fibonacci numbers, similar to what `mpz_fib_ui' does. 592 593 L[2k+1] = 5*F[k-1]*(2*F[k]+F[k-1]) - 4*(-1)^k 594 595 596File: gmp.info, Node: Random Number Algorithms, Prev: Lucas Numbers Algorithm, Up: Other Algorithms 597 59815.7.6 Random Numbers 599--------------------- 600 601For the `urandomb' functions, random numbers are generated simply by 602concatenating bits produced by the generator. As long as the generator 603has good randomness properties this will produce well-distributed N bit 604numbers. 605 606 For the `urandomm' functions, random numbers in a range 0<=R<N are 607generated by taking values R of ceil(log2(N)) bits each until one 608satisfies R<N. This will normally require only one or two attempts, 609but the attempts are limited in case the generator is somehow 610degenerate and produces only 1 bits or similar. 611 612 The Mersenne Twister generator is by Matsumoto and Nishimura (*note 613References::). It has a non-repeating period of 2^19937-1, which is a 614Mersenne prime, hence the name of the generator. The state is 624 615words of 32-bits each, which is iterated with one XOR and shift for each 61632-bit word generated, making the algorithm very fast. Randomness 617properties are also very good and this is the default algorithm used by 618GMP. 619 620 Linear congruential generators are described in many text books, for 621instance Knuth volume 2 (*note References::). With a modulus M and 622parameters A and C, an integer state S is iterated by the formula S <- 623A*S+C mod M. At each step the new state is a linear function of the 624previous, mod M, hence the name of the generator. 625 626 In GMP only moduli of the form 2^N are supported, and the current 627implementation is not as well optimized as it could be. Overheads are 628significant when N is small, and when N is large clearly the multiply 629at each step will become slow. This is not a big concern, since the 630Mersenne Twister generator is better in every respect and is therefore 631recommended for all normal applications. 632 633 For both generators the current state can be deduced by observing 634enough output and applying some linear algebra (over GF(2) in the case 635of the Mersenne Twister). This generally means raw output is 636unsuitable for cryptographic applications without further hashing or 637the like. 638 639 640File: gmp.info, Node: Assembly Coding, Prev: Other Algorithms, Up: Algorithms 641 64215.8 Assembly Coding 643==================== 644 645The assembly subroutines in GMP are the most significant source of 646speed at small to moderate sizes. At larger sizes algorithm selection 647becomes more important, but of course speedups in low level routines 648will still speed up everything proportionally. 649 650 Carry handling and widening multiplies that are important for GMP 651can't be easily expressed in C. GCC `asm' blocks help a lot and are 652provided in `longlong.h', but hand coding low level routines invariably 653offers a speedup over generic C by a factor of anything from 2 to 10. 654 655* Menu: 656 657* Assembly Code Organisation:: 658* Assembly Basics:: 659* Assembly Carry Propagation:: 660* Assembly Cache Handling:: 661* Assembly Functional Units:: 662* Assembly Floating Point:: 663* Assembly SIMD Instructions:: 664* Assembly Software Pipelining:: 665* Assembly Loop Unrolling:: 666* Assembly Writing Guide:: 667 668 669File: gmp.info, Node: Assembly Code Organisation, Next: Assembly Basics, Prev: Assembly Coding, Up: Assembly Coding 670 67115.8.1 Code Organisation 672------------------------ 673 674The various `mpn' subdirectories contain machine-dependent code, written 675in C or assembly. The `mpn/generic' subdirectory contains default code, 676used when there's no machine-specific version of a particular file. 677 678 Each `mpn' subdirectory is for an ISA family. Generally 32-bit and 67964-bit variants in a family cannot share code and have separate 680directories. Within a family further subdirectories may exist for CPU 681variants. 682 683 In each directory a `nails' subdirectory may exist, holding code with 684nails support for that CPU variant. A `NAILS_SUPPORT' directive in each 685file indicates the nails values the code handles. Nails code only 686exists where it's faster, or promises to be faster, than plain code. 687There's no effort put into nails if they're not going to enhance a 688given CPU. 689 690 691File: gmp.info, Node: Assembly Basics, Next: Assembly Carry Propagation, Prev: Assembly Code Organisation, Up: Assembly Coding 692 69315.8.2 Assembly Basics 694---------------------- 695 696`mpn_addmul_1' and `mpn_submul_1' are the most important routines for 697overall GMP performance. All multiplications and divisions come down to 698repeated calls to these. `mpn_add_n', `mpn_sub_n', `mpn_lshift' and 699`mpn_rshift' are next most important. 700 701 On some CPUs assembly versions of the internal functions 702`mpn_mul_basecase' and `mpn_sqr_basecase' give significant speedups, 703mainly through avoiding function call overheads. They can also 704potentially make better use of a wide superscalar processor, as can 705bigger primitives like `mpn_addmul_2' or `mpn_addmul_4'. 706 707 The restrictions on overlaps between sources and destinations (*note 708Low-level Functions::) are designed to facilitate a variety of 709implementations. For example, knowing `mpn_add_n' won't have partly 710overlapping sources and destination means reading can be done far ahead 711of writing on superscalar processors, and loops can be vectorized on a 712vector processor, depending on the carry handling. 713 714 715File: gmp.info, Node: Assembly Carry Propagation, Next: Assembly Cache Handling, Prev: Assembly Basics, Up: Assembly Coding 716 71715.8.3 Carry Propagation 718------------------------ 719 720The problem that presents most challenges in GMP is propagating carries 721from one limb to the next. In functions like `mpn_addmul_1' and 722`mpn_add_n', carries are the only dependencies between limb operations. 723 724 On processors with carry flags, a straightforward CISC style `adc' is 725generally best. AMD K6 `mpn_addmul_1' however is an example of an 726unusual set of circumstances where a branch works out better. 727 728 On RISC processors generally an add and compare for overflow is 729used. This sort of thing can be seen in `mpn/generic/aors_n.c'. Some 730carry propagation schemes require 4 instructions, meaning at least 4 731cycles per limb, but other schemes may use just 1 or 2. On wide 732superscalar processors performance may be completely determined by the 733number of dependent instructions between carry-in and carry-out for 734each limb. 735 736 On vector processors good use can be made of the fact that a carry 737bit only very rarely propagates more than one limb. When adding a 738single bit to a limb, there's only a carry out if that limb was 739`0xFF...FF' which on random data will be only 1 in 2^mp_bits_per_limb. 740`mpn/cray/add_n.c' is an example of this, it adds all limbs in 741parallel, adds one set of carry bits in parallel and then only rarely 742needs to fall through to a loop propagating further carries. 743 744 On the x86s, GCC (as of version 2.95.2) doesn't generate 745particularly good code for the RISC style idioms that are necessary to 746handle carry bits in C. Often conditional jumps are generated where 747`adc' or `sbb' forms would be better. And so unfortunately almost any 748loop involving carry bits needs to be coded in assembly for best 749results. 750 751 752File: gmp.info, Node: Assembly Cache Handling, Next: Assembly Functional Units, Prev: Assembly Carry Propagation, Up: Assembly Coding 753 75415.8.4 Cache Handling 755--------------------- 756 757GMP aims to perform well both on operands that fit entirely in L1 cache 758and those which don't. 759 760 Basic routines like `mpn_add_n' or `mpn_lshift' are often used on 761large operands, so L2 and main memory performance is important for them. 762`mpn_mul_1' and `mpn_addmul_1' are mostly used for multiply and square 763basecases, so L1 performance matters most for them, unless assembly 764versions of `mpn_mul_basecase' and `mpn_sqr_basecase' exist, in which 765case the remaining uses are mostly for larger operands. 766 767 For L2 or main memory operands, memory access times will almost 768certainly be more than the calculation time. The aim therefore is to 769maximize memory throughput, by starting a load of the next cache line 770while processing the contents of the previous one. Clearly this is 771only possible if the chip has a lock-up free cache or some sort of 772prefetch instruction. Most current chips have both these features. 773 774 Prefetching sources combines well with loop unrolling, since a 775prefetch can be initiated once per unrolled loop (or more than once if 776the loop covers more than one cache line). 777 778 On CPUs without write-allocate caches, prefetching destinations will 779ensure individual stores don't go further down the cache hierarchy, 780limiting bandwidth. Of course for calculations which are slow anyway, 781like `mpn_divrem_1', write-throughs might be fine. 782 783 The distance ahead to prefetch will be determined by memory latency 784versus throughput. The aim of course is to have data arriving 785continuously, at peak throughput. Some CPUs have limits on the number 786of fetches or prefetches in progress. 787 788 If a special prefetch instruction doesn't exist then a plain load 789can be used, but in that case care must be taken not to attempt to read 790past the end of an operand, since that might produce a segmentation 791violation. 792 793 Some CPUs or systems have hardware that detects sequential memory 794accesses and initiates suitable cache movements automatically, making 795life easy. 796 797 798File: gmp.info, Node: Assembly Functional Units, Next: Assembly Floating Point, Prev: Assembly Cache Handling, Up: Assembly Coding 799 80015.8.5 Functional Units 801----------------------- 802 803When choosing an approach for an assembly loop, consideration is given 804to what operations can execute simultaneously and what throughput can 805thereby be achieved. In some cases an algorithm can be tweaked to 806accommodate available resources. 807 808 Loop control will generally require a counter and pointer updates, 809costing as much as 5 instructions, plus any delays a branch introduces. 810CPU addressing modes might reduce pointer updates, perhaps by allowing 811just one updating pointer and others expressed as offsets from it, or 812on CISC chips with all addressing done with the loop counter as a 813scaled index. 814 815 The final loop control cost can be amortised by processing several 816limbs in each iteration (*note Assembly Loop Unrolling::). This at 817least ensures loop control isn't a big fraction the work done. 818 819 Memory throughput is always a limit. If perhaps only one load or 820one store can be done per cycle then 3 cycles/limb will the top speed 821for "binary" operations like `mpn_add_n', and any code achieving that 822is optimal. 823 824 Integer resources can be freed up by having the loop counter in a 825float register, or by pressing the float units into use for some 826multiplying, perhaps doing every second limb on the float side (*note 827Assembly Floating Point::). 828 829 Float resources can be freed up by doing carry propagation on the 830integer side, or even by doing integer to float conversions in integers 831using bit twiddling. 832 833 834File: gmp.info, Node: Assembly Floating Point, Next: Assembly SIMD Instructions, Prev: Assembly Functional Units, Up: Assembly Coding 835 83615.8.6 Floating Point 837--------------------- 838 839Floating point arithmetic is used in GMP for multiplications on CPUs 840with poor integer multipliers. It's mostly useful for `mpn_mul_1', 841`mpn_addmul_1' and `mpn_submul_1' on 64-bit machines, and 842`mpn_mul_basecase' on both 32-bit and 64-bit machines. 843 844 With IEEE 53-bit double precision floats, integer multiplications 845producing up to 53 bits will give exact results. Breaking a 64x64 846multiplication into eight 16x32->48 bit pieces is convenient. With 847some care though six 21x32->53 bit products can be used, if one of the 848lower two 21-bit pieces also uses the sign bit. 849 850 For the `mpn_mul_1' family of functions on a 64-bit machine, the 851invariant single limb is split at the start, into 3 or 4 pieces. 852Inside the loop, the bignum operand is split into 32-bit pieces. Fast 853conversion of these unsigned 32-bit pieces to floating point is highly 854machine-dependent. In some cases, reading the data into the integer 855unit, zero-extending to 64-bits, then transferring to the floating 856point unit back via memory is the only option. 857 858 Converting partial products back to 64-bit limbs is usually best 859done as a signed conversion. Since all values are smaller than 2^53, 860signed and unsigned are the same, but most processors lack unsigned 861conversions. 862 863 864 865 Here is a diagram showing 16x32 bit products for an `mpn_mul_1' or 866`mpn_addmul_1' with a 64-bit limb. The single limb operand V is split 867into four 16-bit parts. The multi-limb operand U is split in the loop 868into two 32-bit parts. 869 870 +---+---+---+---+ 871 |v48|v32|v16|v00| V operand 872 +---+---+---+---+ 873 874 +-------+---+---+ 875 x | u32 | u00 | U operand (one limb) 876 +---------------+ 877 878 --------------------------------- 879 880 +-----------+ 881 | u00 x v00 | p00 48-bit products 882 +-----------+ 883 +-----------+ 884 | u00 x v16 | p16 885 +-----------+ 886 +-----------+ 887 | u00 x v32 | p32 888 +-----------+ 889 +-----------+ 890 | u00 x v48 | p48 891 +-----------+ 892 +-----------+ 893 | u32 x v00 | r32 894 +-----------+ 895 +-----------+ 896 | u32 x v16 | r48 897 +-----------+ 898 +-----------+ 899 | u32 x v32 | r64 900 +-----------+ 901 +-----------+ 902 | u32 x v48 | r80 903 +-----------+ 904 905 p32 and r32 can be summed using floating-point addition, and 906likewise p48 and r48. p00 and p16 can be summed with r64 and r80 from 907the previous iteration. 908 909 For each loop then, four 49-bit quantities are transferred to the 910integer unit, aligned as follows, 911 912 |-----64bits----|-----64bits----| 913 +------------+ 914 | p00 + r64' | i00 915 +------------+ 916 +------------+ 917 | p16 + r80' | i16 918 +------------+ 919 +------------+ 920 | p32 + r32 | i32 921 +------------+ 922 +------------+ 923 | p48 + r48 | i48 924 +------------+ 925 926 The challenge then is to sum these efficiently and add in a carry 927limb, generating a low 64-bit result limb and a high 33-bit carry limb 928(i48 extends 33 bits into the high half). 929 930 931File: gmp.info, Node: Assembly SIMD Instructions, Next: Assembly Software Pipelining, Prev: Assembly Floating Point, Up: Assembly Coding 932 93315.8.7 SIMD Instructions 934------------------------ 935 936The single-instruction multiple-data support in current microprocessors 937is aimed at signal processing algorithms where each data point can be 938treated more or less independently. There's generally not much support 939for propagating the sort of carries that arise in GMP. 940 941 SIMD multiplications of say four 16x16 bit multiplies only do as much 942work as one 32x32 from GMP's point of view, and need some shifts and 943adds besides. But of course if say the SIMD form is fully pipelined 944and uses less instruction decoding then it may still be worthwhile. 945 946 On the x86 chips, MMX has so far found a use in `mpn_rshift' and 947`mpn_lshift', and is used in a special case for 16-bit multipliers in 948the P55 `mpn_mul_1'. SSE2 is used for Pentium 4 `mpn_mul_1', 949`mpn_addmul_1', and `mpn_submul_1'. 950 951 952File: gmp.info, Node: Assembly Software Pipelining, Next: Assembly Loop Unrolling, Prev: Assembly SIMD Instructions, Up: Assembly Coding 953 95415.8.8 Software Pipelining 955-------------------------- 956 957Software pipelining consists of scheduling instructions around the 958branch point in a loop. For example a loop might issue a load not for 959use in the present iteration but the next, thereby allowing extra 960cycles for the data to arrive from memory. 961 962 Naturally this is wanted only when doing things like loads or 963multiplies that take several cycles to complete, and only where a CPU 964has multiple functional units so that other work can be done in the 965meantime. 966 967 A pipeline with several stages will have a data value in progress at 968each stage and each loop iteration moves them along one stage. This is 969like juggling. 970 971 If the latency of some instruction is greater than the loop time 972then it will be necessary to unroll, so one register has a result ready 973to use while another (or multiple others) are still in progress. 974(*note Assembly Loop Unrolling::). 975 976 977File: gmp.info, Node: Assembly Loop Unrolling, Next: Assembly Writing Guide, Prev: Assembly Software Pipelining, Up: Assembly Coding 978 97915.8.9 Loop Unrolling 980--------------------- 981 982Loop unrolling consists of replicating code so that several limbs are 983processed in each loop. At a minimum this reduces loop overheads by a 984corresponding factor, but it can also allow better register usage, for 985example alternately using one register combination and then another. 986Judicious use of `m4' macros can help avoid lots of duplication in the 987source code. 988 989 Any amount of unrolling can be handled with a loop counter that's 990decremented by N each time, stopping when the remaining count is less 991than the further N the loop will process. Or by subtracting N at the 992start, the termination condition becomes when the counter C is less 993than 0 (and the count of remaining limbs is C+N). 994 995 Alternately for a power of 2 unroll the loop count and remainder can 996be established with a shift and mask. This is convenient if also 997making a computed jump into the middle of a large loop. 998 999 The limbs not a multiple of the unrolling can be handled in various 1000ways, for example 1001 1002 * A simple loop at the end (or the start) to process the excess. 1003 Care will be wanted that it isn't too much slower than the 1004 unrolled part. 1005 1006 * A set of binary tests, for example after an 8-limb unrolling, test 1007 for 4 more limbs to process, then a further 2 more or not, and 1008 finally 1 more or not. This will probably take more code space 1009 than a simple loop. 1010 1011 * A `switch' statement, providing separate code for each possible 1012 excess, for example an 8-limb unrolling would have separate code 1013 for 0 remaining, 1 remaining, etc, up to 7 remaining. This might 1014 take a lot of code, but may be the best way to optimize all cases 1015 in combination with a deep pipelined loop. 1016 1017 * A computed jump into the middle of the loop, thus making the first 1018 iteration handle the excess. This should make times smoothly 1019 increase with size, which is attractive, but setups for the jump 1020 and adjustments for pointers can be tricky and could become quite 1021 difficult in combination with deep pipelining. 1022 1023 1024File: gmp.info, Node: Assembly Writing Guide, Prev: Assembly Loop Unrolling, Up: Assembly Coding 1025 102615.8.10 Writing Guide 1027--------------------- 1028 1029This is a guide to writing software pipelined loops for processing limb 1030vectors in assembly. 1031 1032 First determine the algorithm and which instructions are needed. 1033Code it without unrolling or scheduling, to make sure it works. On a 10343-operand CPU try to write each new value to a new register, this will 1035greatly simplify later steps. 1036 1037 Then note for each instruction the functional unit and/or issue port 1038requirements. If an instruction can use either of two units, like U0 1039or U1 then make a category "U0/U1". Count the total using each unit 1040(or combined unit), and count all instructions. 1041 1042 Figure out from those counts the best possible loop time. The goal 1043will be to find a perfect schedule where instruction latencies are 1044completely hidden. The total instruction count might be the limiting 1045factor, or perhaps a particular functional unit. It might be possible 1046to tweak the instructions to help the limiting factor. 1047 1048 Suppose the loop time is N, then make N issue buckets, with the 1049final loop branch at the end of the last. Now fill the buckets with 1050dummy instructions using the functional units desired. Run this to 1051make sure the intended speed is reached. 1052 1053 Now replace the dummy instructions with the real instructions from 1054the slow but correct loop you started with. The first will typically 1055be a load instruction. Then the instruction using that value is placed 1056in a bucket an appropriate distance down. Run the loop again, to check 1057it still runs at target speed. 1058 1059 Keep placing instructions, frequently measuring the loop. After a 1060few you will need to wrap around from the last bucket back to the top 1061of the loop. If you used the new-register for new-value strategy above 1062then there will be no register conflicts. If not then take care not to 1063clobber something already in use. Changing registers at this time is 1064very error prone. 1065 1066 The loop will overlap two or more of the original loop iterations, 1067and the computation of one vector element result will be started in one 1068iteration of the new loop, and completed one or several iterations 1069later. 1070 1071 The final step is to create feed-in and wind-down code for the loop. 1072A good way to do this is to make a copy (or copies) of the loop at the 1073start and delete those instructions which don't have valid antecedents, 1074and at the end replicate and delete those whose results are unwanted 1075(including any further loads). 1076 1077 The loop will have a minimum number of limbs loaded and processed, 1078so the feed-in code must test if the request size is smaller and skip 1079either to a suitable part of the wind-down or to special code for small 1080sizes. 1081 1082 1083File: gmp.info, Node: Internals, Next: Contributors, Prev: Algorithms, Up: Top 1084 108516 Internals 1086************ 1087 1088*This chapter is provided only for informational purposes and the 1089various internals described here may change in future GMP releases. 1090Applications expecting to be compatible with future releases should use 1091only the documented interfaces described in previous chapters.* 1092 1093* Menu: 1094 1095* Integer Internals:: 1096* Rational Internals:: 1097* Float Internals:: 1098* Raw Output Internals:: 1099* C++ Interface Internals:: 1100 1101 1102File: gmp.info, Node: Integer Internals, Next: Rational Internals, Prev: Internals, Up: Internals 1103 110416.1 Integer Internals 1105====================== 1106 1107`mpz_t' variables represent integers using sign and magnitude, in space 1108dynamically allocated and reallocated. The fields are as follows. 1109 1110`_mp_size' 1111 The number of limbs, or the negative of that when representing a 1112 negative integer. Zero is represented by `_mp_size' set to zero, 1113 in which case the `_mp_d' data is unused. 1114 1115`_mp_d' 1116 A pointer to an array of limbs which is the magnitude. These are 1117 stored "little endian" as per the `mpn' functions, so `_mp_d[0]' 1118 is the least significant limb and `_mp_d[ABS(_mp_size)-1]' is the 1119 most significant. Whenever `_mp_size' is non-zero, the most 1120 significant limb is non-zero. 1121 1122 Currently there's always at least one limb allocated, so for 1123 instance `mpz_set_ui' never needs to reallocate, and `mpz_get_ui' 1124 can fetch `_mp_d[0]' unconditionally (though its value is then 1125 only wanted if `_mp_size' is non-zero). 1126 1127`_mp_alloc' 1128 `_mp_alloc' is the number of limbs currently allocated at `_mp_d', 1129 and naturally `_mp_alloc >= ABS(_mp_size)'. When an `mpz' routine 1130 is about to (or might be about to) increase `_mp_size', it checks 1131 `_mp_alloc' to see whether there's enough space, and reallocates 1132 if not. `MPZ_REALLOC' is generally used for this. 1133 1134 The various bitwise logical functions like `mpz_and' behave as if 1135negative values were twos complement. But sign and magnitude is always 1136used internally, and necessary adjustments are made during the 1137calculations. Sometimes this isn't pretty, but sign and magnitude are 1138best for other routines. 1139 1140 Some internal temporary variables are setup with `MPZ_TMP_INIT' and 1141these have `_mp_d' space obtained from `TMP_ALLOC' rather than the 1142memory allocation functions. Care is taken to ensure that these are 1143big enough that no reallocation is necessary (since it would have 1144unpredictable consequences). 1145 1146 `_mp_size' and `_mp_alloc' are `int', although `mp_size_t' is 1147usually a `long'. This is done to make the fields just 32 bits on some 114864 bits systems, thereby saving a few bytes of data space but still 1149providing plenty of range. 1150 1151 1152File: gmp.info, Node: Rational Internals, Next: Float Internals, Prev: Integer Internals, Up: Internals 1153 115416.2 Rational Internals 1155======================= 1156 1157`mpq_t' variables represent rationals using an `mpz_t' numerator and 1158denominator (*note Integer Internals::). 1159 1160 The canonical form adopted is denominator positive (and non-zero), 1161no common factors between numerator and denominator, and zero uniquely 1162represented as 0/1. 1163 1164 It's believed that casting out common factors at each stage of a 1165calculation is best in general. A GCD is an O(N^2) operation so it's 1166better to do a few small ones immediately than to delay and have to do 1167a big one later. Knowing the numerator and denominator have no common 1168factors can be used for example in `mpq_mul' to make only two cross 1169GCDs necessary, not four. 1170 1171 This general approach to common factors is badly sub-optimal in the 1172presence of simple factorizations or little prospect for cancellation, 1173but GMP has no way to know when this will occur. As per *note 1174Efficiency::, that's left to applications. The `mpq_t' framework might 1175still suit, with `mpq_numref' and `mpq_denref' for direct access to the 1176numerator and denominator, or of course `mpz_t' variables can be used 1177directly. 1178 1179 1180File: gmp.info, Node: Float Internals, Next: Raw Output Internals, Prev: Rational Internals, Up: Internals 1181 118216.3 Float Internals 1183==================== 1184 1185Efficient calculation is the primary aim of GMP floats and the use of 1186whole limbs and simple rounding facilitates this. 1187 1188 `mpf_t' floats have a variable precision mantissa and a single 1189machine word signed exponent. The mantissa is represented using sign 1190and magnitude. 1191 1192 most least 1193 significant significant 1194 limb limb 1195 1196 _mp_d 1197 |---- _mp_exp ---> | 1198 _____ _____ _____ _____ _____ 1199 |_____|_____|_____|_____|_____| 1200 . <------------ radix point 1201 1202 <-------- _mp_size ---------> 1203 1204The fields are as follows. 1205 1206`_mp_size' 1207 The number of limbs currently in use, or the negative of that when 1208 representing a negative value. Zero is represented by `_mp_size' 1209 and `_mp_exp' both set to zero, and in that case the `_mp_d' data 1210 is unused. (In the future `_mp_exp' might be undefined when 1211 representing zero.) 1212 1213`_mp_prec' 1214 The precision of the mantissa, in limbs. In any calculation the 1215 aim is to produce `_mp_prec' limbs of result (the most significant 1216 being non-zero). 1217 1218`_mp_d' 1219 A pointer to the array of limbs which is the absolute value of the 1220 mantissa. These are stored "little endian" as per the `mpn' 1221 functions, so `_mp_d[0]' is the least significant limb and 1222 `_mp_d[ABS(_mp_size)-1]' the most significant. 1223 1224 The most significant limb is always non-zero, but there are no 1225 other restrictions on its value, in particular the highest 1 bit 1226 can be anywhere within the limb. 1227 1228 `_mp_prec+1' limbs are allocated to `_mp_d', the extra limb being 1229 for convenience (see below). There are no reallocations during a 1230 calculation, only in a change of precision with `mpf_set_prec'. 1231 1232`_mp_exp' 1233 The exponent, in limbs, determining the location of the implied 1234 radix point. Zero means the radix point is just above the most 1235 significant limb. Positive values mean a radix point offset 1236 towards the lower limbs and hence a value >= 1, as for example in 1237 the diagram above. Negative exponents mean a radix point further 1238 above the highest limb. 1239 1240 Naturally the exponent can be any value, it doesn't have to fall 1241 within the limbs as the diagram shows, it can be a long way above 1242 or a long way below. Limbs other than those included in the 1243 `{_mp_d,_mp_size}' data are treated as zero. 1244 1245 The `_mp_size' and `_mp_prec' fields are `int', although the 1246`mp_size_t' type is usually a `long'. The `_mp_exp' field is usually 1247`long'. This is done to make some fields just 32 bits on some 64 bits 1248systems, thereby saving a few bytes of data space but still providing 1249plenty of precision and a very large range. 1250 1251 1252The following various points should be noted. 1253 1254Low Zeros 1255 The least significant limbs `_mp_d[0]' etc can be zero, though 1256 such low zeros can always be ignored. Routines likely to produce 1257 low zeros check and avoid them to save time in subsequent 1258 calculations, but for most routines they're quite unlikely and 1259 aren't checked. 1260 1261Mantissa Size Range 1262 The `_mp_size' count of limbs in use can be less than `_mp_prec' if 1263 the value can be represented in less. This means low precision 1264 values or small integers stored in a high precision `mpf_t' can 1265 still be operated on efficiently. 1266 1267 `_mp_size' can also be greater than `_mp_prec'. Firstly a value is 1268 allowed to use all of the `_mp_prec+1' limbs available at `_mp_d', 1269 and secondly when `mpf_set_prec_raw' lowers `_mp_prec' it leaves 1270 `_mp_size' unchanged and so the size can be arbitrarily bigger than 1271 `_mp_prec'. 1272 1273Rounding 1274 All rounding is done on limb boundaries. Calculating `_mp_prec' 1275 limbs with the high non-zero will ensure the application requested 1276 minimum precision is obtained. 1277 1278 The use of simple "trunc" rounding towards zero is efficient, 1279 since there's no need to examine extra limbs and increment or 1280 decrement. 1281 1282Bit Shifts 1283 Since the exponent is in limbs, there are no bit shifts in basic 1284 operations like `mpf_add' and `mpf_mul'. When differing exponents 1285 are encountered all that's needed is to adjust pointers to line up 1286 the relevant limbs. 1287 1288 Of course `mpf_mul_2exp' and `mpf_div_2exp' will require bit 1289 shifts, but the choice is between an exponent in limbs which 1290 requires shifts there, or one in bits which requires them almost 1291 everywhere else. 1292 1293Use of `_mp_prec+1' Limbs 1294 The extra limb on `_mp_d' (`_mp_prec+1' rather than just 1295 `_mp_prec') helps when an `mpf' routine might get a carry from its 1296 operation. `mpf_add' for instance will do an `mpn_add' of 1297 `_mp_prec' limbs. If there's no carry then that's the result, but 1298 if there is a carry then it's stored in the extra limb of space and 1299 `_mp_size' becomes `_mp_prec+1'. 1300 1301 Whenever `_mp_prec+1' limbs are held in a variable, the low limb 1302 is not needed for the intended precision, only the `_mp_prec' high 1303 limbs. But zeroing it out or moving the rest down is unnecessary. 1304 Subsequent routines reading the value will simply take the high 1305 limbs they need, and this will be `_mp_prec' if their target has 1306 that same precision. This is no more than a pointer adjustment, 1307 and must be checked anyway since the destination precision can be 1308 different from the sources. 1309 1310 Copy functions like `mpf_set' will retain a full `_mp_prec+1' limbs 1311 if available. This ensures that a variable which has `_mp_size' 1312 equal to `_mp_prec+1' will get its full exact value copied. 1313 Strictly speaking this is unnecessary since only `_mp_prec' limbs 1314 are needed for the application's requested precision, but it's 1315 considered that an `mpf_set' from one variable into another of the 1316 same precision ought to produce an exact copy. 1317 1318Application Precisions 1319 `__GMPF_BITS_TO_PREC' converts an application requested precision 1320 to an `_mp_prec'. The value in bits is rounded up to a whole limb 1321 then an extra limb is added since the most significant limb of 1322 `_mp_d' is only non-zero and therefore might contain only one bit. 1323 1324 `__GMPF_PREC_TO_BITS' does the reverse conversion, and removes the 1325 extra limb from `_mp_prec' before converting to bits. The net 1326 effect of reading back with `mpf_get_prec' is simply the precision 1327 rounded up to a multiple of `mp_bits_per_limb'. 1328 1329 Note that the extra limb added here for the high only being 1330 non-zero is in addition to the extra limb allocated to `_mp_d'. 1331 For example with a 32-bit limb, an application request for 250 1332 bits will be rounded up to 8 limbs, then an extra added for the 1333 high being only non-zero, giving an `_mp_prec' of 9. `_mp_d' then 1334 gets 10 limbs allocated. Reading back with `mpf_get_prec' will 1335 take `_mp_prec' subtract 1 limb and multiply by 32, giving 256 1336 bits. 1337 1338 Strictly speaking, the fact the high limb has at least one bit 1339 means that a float with, say, 3 limbs of 32-bits each will be 1340 holding at least 65 bits, but for the purposes of `mpf_t' it's 1341 considered simply to be 64 bits, a nice multiple of the limb size. 1342 1343 1344File: gmp.info, Node: Raw Output Internals, Next: C++ Interface Internals, Prev: Float Internals, Up: Internals 1345 134616.4 Raw Output Internals 1347========================= 1348 1349`mpz_out_raw' uses the following format. 1350 1351 +------+------------------------+ 1352 | size | data bytes | 1353 +------+------------------------+ 1354 1355 The size is 4 bytes written most significant byte first, being the 1356number of subsequent data bytes, or the twos complement negative of 1357that when a negative integer is represented. The data bytes are the 1358absolute value of the integer, written most significant byte first. 1359 1360 The most significant data byte is always non-zero, so the output is 1361the same on all systems, irrespective of limb size. 1362 1363 In GMP 1, leading zero bytes were written to pad the data bytes to a 1364multiple of the limb size. `mpz_inp_raw' will still accept this, for 1365compatibility. 1366 1367 The use of "big endian" for both the size and data fields is 1368deliberate, it makes the data easy to read in a hex dump of a file. 1369Unfortunately it also means that the limb data must be reversed when 1370reading or writing, so neither a big endian nor little endian system 1371can just read and write `_mp_d'. 1372 1373 1374File: gmp.info, Node: C++ Interface Internals, Prev: Raw Output Internals, Up: Internals 1375 137616.5 C++ Interface Internals 1377============================ 1378 1379A system of expression templates is used to ensure something like 1380`a=b+c' turns into a simple call to `mpz_add' etc. For `mpf_class' the 1381scheme also ensures the precision of the final destination is used for 1382any temporaries within a statement like `f=w*x+y*z'. These are 1383important features which a naive implementation cannot provide. 1384 1385 A simplified description of the scheme follows. The true scheme is 1386complicated by the fact that expressions have different return types. 1387For detailed information, refer to the source code. 1388 1389 To perform an operation, say, addition, we first define a "function 1390object" evaluating it, 1391 1392 struct __gmp_binary_plus 1393 { 1394 static void eval(mpf_t f, mpf_t g, mpf_t h) { mpf_add(f, g, h); } 1395 }; 1396 1397And an "additive expression" object, 1398 1399 __gmp_expr<__gmp_binary_expr<mpf_class, mpf_class, __gmp_binary_plus> > 1400 operator+(const mpf_class &f, const mpf_class &g) 1401 { 1402 return __gmp_expr 1403 <__gmp_binary_expr<mpf_class, mpf_class, __gmp_binary_plus> >(f, g); 1404 } 1405 1406 The seemingly redundant `__gmp_expr<__gmp_binary_expr<...>>' is used 1407to encapsulate any possible kind of expression into a single template 1408type. In fact even `mpf_class' etc are `typedef' specializations of 1409`__gmp_expr'. 1410 1411 Next we define assignment of `__gmp_expr' to `mpf_class'. 1412 1413 template <class T> 1414 mpf_class & mpf_class::operator=(const __gmp_expr<T> &expr) 1415 { 1416 expr.eval(this->get_mpf_t(), this->precision()); 1417 return *this; 1418 } 1419 1420 template <class Op> 1421 void __gmp_expr<__gmp_binary_expr<mpf_class, mpf_class, Op> >::eval 1422 (mpf_t f, mp_bitcnt_t precision) 1423 { 1424 Op::eval(f, expr.val1.get_mpf_t(), expr.val2.get_mpf_t()); 1425 } 1426 1427 where `expr.val1' and `expr.val2' are references to the expression's 1428operands (here `expr' is the `__gmp_binary_expr' stored within the 1429`__gmp_expr'). 1430 1431 This way, the expression is actually evaluated only at the time of 1432assignment, when the required precision (that of `f') is known. 1433Furthermore the target `mpf_t' is now available, thus we can call 1434`mpf_add' directly with `f' as the output argument. 1435 1436 Compound expressions are handled by defining operators taking 1437subexpressions as their arguments, like this: 1438 1439 template <class T, class U> 1440 __gmp_expr 1441 <__gmp_binary_expr<__gmp_expr<T>, __gmp_expr<U>, __gmp_binary_plus> > 1442 operator+(const __gmp_expr<T> &expr1, const __gmp_expr<U> &expr2) 1443 { 1444 return __gmp_expr 1445 <__gmp_binary_expr<__gmp_expr<T>, __gmp_expr<U>, __gmp_binary_plus> > 1446 (expr1, expr2); 1447 } 1448 1449 And the corresponding specializations of `__gmp_expr::eval': 1450 1451 template <class T, class U, class Op> 1452 void __gmp_expr 1453 <__gmp_binary_expr<__gmp_expr<T>, __gmp_expr<U>, Op> >::eval 1454 (mpf_t f, mp_bitcnt_t precision) 1455 { 1456 // declare two temporaries 1457 mpf_class temp1(expr.val1, precision), temp2(expr.val2, precision); 1458 Op::eval(f, temp1.get_mpf_t(), temp2.get_mpf_t()); 1459 } 1460 1461 The expression is thus recursively evaluated to any level of 1462complexity and all subexpressions are evaluated to the precision of `f'. 1463 1464 1465File: gmp.info, Node: Contributors, Next: References, Prev: Internals, Up: Top 1466 1467Appendix A Contributors 1468*********************** 1469 1470Torbj�rn Granlund wrote the original GMP library and is still the main 1471developer. Code not explicitly attributed to others, was contributed by 1472Torbj�rn. Several other individuals and organizations have contributed 1473GMP. Here is a list in chronological order on first contribution: 1474 1475 Gunnar Sj�din and Hans Riesel helped with mathematical problems in 1476early versions of the library. 1477 1478 Richard Stallman helped with the interface design and revised the 1479first version of this manual. 1480 1481 Brian Beuning and Doug Lea helped with testing of early versions of 1482the library and made creative suggestions. 1483 1484 John Amanatides of York University in Canada contributed the function 1485`mpz_probab_prime_p'. 1486 1487 Paul Zimmermann wrote the REDC-based mpz_powm code, the 1488Sch�nhage-Strassen FFT multiply code, and the Karatsuba square root 1489code. He also improved the Toom3 code for GMP 4.2. Paul sparked the 1490development of GMP 2, with his comparisons between bignum packages. 1491The ECMNET project Paul is organizing was a driving force behind many 1492of the optimizations in GMP 3. Paul also wrote the new GMP 4.3 nth 1493root code (with Torbj�rn). 1494 1495 Ken Weber (Kent State University, Universidade Federal do Rio Grande 1496do Sul) contributed now defunct versions of `mpz_gcd', `mpz_divexact', 1497`mpn_gcd', and `mpn_bdivmod', partially supported by CNPq (Brazil) 1498grant 301314194-2. 1499 1500 Per Bothner of Cygnus Support helped to set up GMP to use Cygnus' 1501configure. He has also made valuable suggestions and tested numerous 1502intermediary releases. 1503 1504 Joachim Hollman was involved in the design of the `mpf' interface, 1505and in the `mpz' design revisions for version 2. 1506 1507 Bennet Yee contributed the initial versions of `mpz_jacobi' and 1508`mpz_legendre'. 1509 1510 Andreas Schwab contributed the files `mpn/m68k/lshift.S' and 1511`mpn/m68k/rshift.S' (now in `.asm' form). 1512 1513 Robert Harley of Inria, France and David Seal of ARM, England, 1514suggested clever improvements for population count. Robert also wrote 1515highly optimized Karatsuba and 3-way Toom multiplication functions for 1516GMP 3, and contributed the ARM assembly code. 1517 1518 Torsten Ekedahl of the Mathematical department of Stockholm 1519University provided significant inspiration during several phases of 1520the GMP development. His mathematical expertise helped improve several 1521algorithms. 1522 1523 Linus Nordberg wrote the new configure system based on autoconf and 1524implemented the new random functions. 1525 1526 Kevin Ryde worked on a large number of things: optimized x86 code, 1527m4 asm macros, parameter tuning, speed measuring, the configure system, 1528function inlining, divisibility tests, bit scanning, Jacobi symbols, 1529Fibonacci and Lucas number functions, printf and scanf functions, perl 1530interface, demo expression parser, the algorithms chapter in the 1531manual, `gmpasm-mode.el', and various miscellaneous improvements 1532elsewhere. 1533 1534 Kent Boortz made the Mac OS 9 port. 1535 1536 Steve Root helped write the optimized alpha 21264 assembly code. 1537 1538 Gerardo Ballabio wrote the `gmpxx.h' C++ class interface and the C++ 1539`istream' input routines. 1540 1541 Jason Moxham rewrote `mpz_fac_ui'. 1542 1543 Pedro Gimeno implemented the Mersenne Twister and made other random 1544number improvements. 1545 1546 Niels M�ller wrote the sub-quadratic GCD, extended GCD and jacobi 1547code, the quadratic Hensel division code, and (with Torbj�rn) the new 1548divide and conquer division code for GMP 4.3. Niels also helped 1549implement the new Toom multiply code for GMP 4.3 and implemented helper 1550functions to simplify Toom evaluations for GMP 5.0. He wrote the 1551original version of mpn_mulmod_bnm1, and he is the main author of the 1552mini-gmp package used for gmp bootstrapping. 1553 1554 Alberto Zanoni and Marco Bodrato suggested the unbalanced multiply 1555strategy, and found the optimal strategies for evaluation and 1556interpolation in Toom multiplication. 1557 1558 Marco Bodrato helped implement the new Toom multiply code for GMP 15594.3 and implemented most of the new Toom multiply and squaring code for 15605.0. He is the main author of the current mpn_mulmod_bnm1 and 1561mpn_mullo_n. Marco also wrote the functions mpn_invert and 1562mpn_invertappr. He is the author of the current combinatorial 1563functions: binomial, factorial, multifactorial, primorial. 1564 1565 David Harvey suggested the internal function `mpn_bdiv_dbm1', 1566implementing division relevant to Toom multiplication. He also worked 1567on fast assembly sequences, in particular on a fast AMD64 1568`mpn_mul_basecase'. He wrote the internal middle product functions 1569`mpn_mulmid_basecase', `mpn_toom42_mulmid', `mpn_mulmid_n' and related 1570helper routines. 1571 1572 Martin Boij wrote `mpn_perfect_power_p'. 1573 1574 Marc Glisse improved `gmpxx.h': use fewer temporaries (faster), 1575specializations of `numeric_limits' and `common_type', C++11 features 1576(move constructors, explicit bool conversion, UDL), make the conversion 1577from `mpq_class' to `mpz_class' explicit, optimize operations where one 1578argument is a small compile-time constant, replace some heap 1579allocations by stack allocations. He also fixed the eofbit handling of 1580C++ streams, and removed one division from `mpq/aors.c'. 1581 1582 (This list is chronological, not ordered after significance. If you 1583have contributed to GMP but are not listed above, please tell 1584<gmp-devel@gmplib.org> about the omission!) 1585 1586 The development of floating point functions of GNU MP 2, were 1587supported in part by the ESPRIT-BRA (Basic Research Activities) 6846 1588project POSSO (POlynomial System SOlving). 1589 1590 The development of GMP 2, 3, and 4 was supported in part by the IDA 1591Center for Computing Sciences. 1592 1593 Thanks go to Hans Thorsen for donating an SGI system for the GMP 1594test system environment. 1595 1596 1597File: gmp.info, Node: References, Next: GNU Free Documentation License, Prev: Contributors, Up: Top 1598 1599Appendix B References 1600********************* 1601 1602B.1 Books 1603========= 1604 1605 * Jonathan M. Borwein and Peter B. Borwein, "Pi and the AGM: A Study 1606 in Analytic Number Theory and Computational Complexity", Wiley, 1607 1998. 1608 1609 * Richard Crandall and Carl Pomerance, "Prime Numbers: A 1610 Computational Perspective", 2nd edition, Springer-Verlag, 2005. 1611 `http://www.math.dartmouth.edu/~carlp/' 1612 1613 * Henri Cohen, "A Course in Computational Algebraic Number Theory", 1614 Graduate Texts in Mathematics number 138, Springer-Verlag, 1993. 1615 `http://www.math.u-bordeaux.fr/~cohen/' 1616 1617 * Donald E. Knuth, "The Art of Computer Programming", volume 2, 1618 "Seminumerical Algorithms", 3rd edition, Addison-Wesley, 1998. 1619 `http://www-cs-faculty.stanford.edu/~knuth/taocp.html' 1620 1621 * John D. Lipson, "Elements of Algebra and Algebraic Computing", The 1622 Benjamin Cummings Publishing Company Inc, 1981. 1623 1624 * Alfred J. Menezes, Paul C. van Oorschot and Scott A. Vanstone, 1625 "Handbook of Applied Cryptography", 1626 `http://www.cacr.math.uwaterloo.ca/hac/' 1627 1628 * Richard M. Stallman and the GCC Developer Community, "Using the 1629 GNU Compiler Collection", Free Software Foundation, 2008, 1630 available online `http://gcc.gnu.org/onlinedocs/', and in the GCC 1631 package `ftp://ftp.gnu.org/gnu/gcc/' 1632 1633B.2 Papers 1634========== 1635 1636 * Yves Bertot, Nicolas Magaud and Paul Zimmermann, "A Proof of GMP 1637 Square Root", Journal of Automated Reasoning, volume 29, 2002, pp. 1638 225-252. Also available online as INRIA Research Report 4475, 1639 June 2002, `http://hal.inria.fr/docs/00/07/21/13/PDF/RR-4475.pdf' 1640 1641 * Christoph Burnikel and Joachim Ziegler, "Fast Recursive Division", 1642 Max-Planck-Institut fuer Informatik Research Report MPI-I-98-1-022, 1643 `http://data.mpi-sb.mpg.de/internet/reports.nsf/NumberView/1998-1-022' 1644 1645 * Torbj�rn Granlund and Peter L. Montgomery, "Division by Invariant 1646 Integers using Multiplication", in Proceedings of the SIGPLAN 1647 PLDI'94 Conference, June 1994. Also available 1648 `http://gmplib.org/~tege/divcnst-pldi94.pdf'. 1649 1650 * Niels M�ller and Torbj�rn Granlund, "Improved division by invariant 1651 integers", IEEE Transactions on Computers, 11 June 2010. 1652 `http://gmplib.org/~tege/division-paper.pdf' 1653 1654 * Torbj�rn Granlund and Niels M�ller, "Division of integers large and 1655 small", to appear. 1656 1657 * Tudor Jebelean, "An algorithm for exact division", Journal of 1658 Symbolic Computation, volume 15, 1993, pp. 169-180. Research 1659 report version available 1660 `ftp://ftp.risc.uni-linz.ac.at/pub/techreports/1992/92-35.ps.gz' 1661 1662 * Tudor Jebelean, "Exact Division with Karatsuba Complexity - 1663 Extended Abstract", RISC-Linz technical report 96-31, 1664 `ftp://ftp.risc.uni-linz.ac.at/pub/techreports/1996/96-31.ps.gz' 1665 1666 * Tudor Jebelean, "Practical Integer Division with Karatsuba 1667 Complexity", ISSAC 97, pp. 339-341. Technical report available 1668 `ftp://ftp.risc.uni-linz.ac.at/pub/techreports/1996/96-29.ps.gz' 1669 1670 * Tudor Jebelean, "A Generalization of the Binary GCD Algorithm", 1671 ISSAC 93, pp. 111-116. Technical report version available 1672 `ftp://ftp.risc.uni-linz.ac.at/pub/techreports/1993/93-01.ps.gz' 1673 1674 * Tudor Jebelean, "A Double-Digit Lehmer-Euclid Algorithm for 1675 Finding the GCD of Long Integers", Journal of Symbolic 1676 Computation, volume 19, 1995, pp. 145-157. Technical report 1677 version also available 1678 `ftp://ftp.risc.uni-linz.ac.at/pub/techreports/1992/92-69.ps.gz' 1679 1680 * Werner Krandick and Tudor Jebelean, "Bidirectional Exact Integer 1681 Division", Journal of Symbolic Computation, volume 21, 1996, pp. 1682 441-455. Early technical report version also available 1683 `ftp://ftp.risc.uni-linz.ac.at/pub/techreports/1994/94-50.ps.gz' 1684 1685 * Makoto Matsumoto and Takuji Nishimura, "Mersenne Twister: A 1686 623-dimensionally equidistributed uniform pseudorandom number 1687 generator", ACM Transactions on Modelling and Computer Simulation, 1688 volume 8, January 1998, pp. 3-30. Available online 1689 `http://www.math.sci.hiroshima-u.ac.jp/~m-mat/MT/ARTICLES/mt.ps.gz' 1690 (or .pdf) 1691 1692 * R. Moenck and A. Borodin, "Fast Modular Transforms via Division", 1693 Proceedings of the 13th Annual IEEE Symposium on Switching and 1694 Automata Theory, October 1972, pp. 90-96. Reprinted as "Fast 1695 Modular Transforms", Journal of Computer and System Sciences, 1696 volume 8, number 3, June 1974, pp. 366-386. 1697 1698 * Niels M�ller, "On Sch�nhage's algorithm and subquadratic integer 1699 GCD computation", in Mathematics of Computation, volume 77, 1700 January 2008, pp. 589-607. 1701 1702 * Peter L. Montgomery, "Modular Multiplication Without Trial 1703 Division", in Mathematics of Computation, volume 44, number 170, 1704 April 1985. 1705 1706 * Arnold Sch�nhage and Volker Strassen, "Schnelle Multiplikation 1707 grosser Zahlen", Computing 7, 1971, pp. 281-292. 1708 1709 * Kenneth Weber, "The accelerated integer GCD algorithm", ACM 1710 Transactions on Mathematical Software, volume 21, number 1, March 1711 1995, pp. 111-122. 1712 1713 * Paul Zimmermann, "Karatsuba Square Root", INRIA Research Report 1714 3805, November 1999, 1715 `http://hal.inria.fr/inria-00072854/PDF/RR-3805.pdf' 1716 1717 * Paul Zimmermann, "A Proof of GMP Fast Division and Square Root 1718 Implementations", 1719 `http://www.loria.fr/~zimmerma/papers/proof-div-sqrt.ps.gz' 1720 1721 * Dan Zuras, "On Squaring and Multiplying Large Integers", ARITH-11: 1722 IEEE Symposium on Computer Arithmetic, 1993, pp. 260 to 271. 1723 Reprinted as "More on Multiplying and Squaring Large Integers", 1724 IEEE Transactions on Computers, volume 43, number 8, August 1994, 1725 pp. 899-908. 1726 1727 1728File: gmp.info, Node: GNU Free Documentation License, Next: Concept Index, Prev: References, Up: Top 1729 1730Appendix C GNU Free Documentation License 1731***************************************** 1732 1733 Version 1.3, 3 November 2008 1734 1735 Copyright (C) 2000, 2001, 2002, 2007, 2008 Free Software Foundation, Inc. 1736 `http://fsf.org/' 1737 1738 Everyone is permitted to copy and distribute verbatim copies 1739 of this license document, but changing it is not allowed. 1740 1741 0. 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MODIFICATIONS 1914 1915 You may copy and distribute a Modified Version of the Document 1916 under the conditions of sections 2 and 3 above, provided that you 1917 release the Modified Version under precisely this License, with 1918 the Modified Version filling the role of the Document, thus 1919 licensing distribution and modification of the Modified Version to 1920 whoever possesses a copy of it. In addition, you must do these 1921 things in the Modified Version: 1922 1923 A. Use in the Title Page (and on the covers, if any) a title 1924 distinct from that of the Document, and from those of 1925 previous versions (which should, if there were any, be listed 1926 in the History section of the Document). You may use the 1927 same title as a previous version if the original publisher of 1928 that version gives permission. 1929 1930 B. List on the Title Page, as authors, one or more persons or 1931 entities responsible for authorship of the modifications in 1932 the Modified Version, together with at least five of the 1933 principal authors of the Document (all of its principal 1934 authors, if it has fewer than five), unless they release you 1935 from this requirement. 1936 1937 C. State on the Title page the name of the publisher of the 1938 Modified Version, as the publisher. 1939 1940 D. Preserve all the copyright notices of the Document. 1941 1942 E. Add an appropriate copyright notice for your modifications 1943 adjacent to the other copyright notices. 1944 1945 F. Include, immediately after the copyright notices, a license 1946 notice giving the public permission to use the Modified 1947 Version under the terms of this License, in the form shown in 1948 the Addendum below. 1949 1950 G. Preserve in that license notice the full lists of Invariant 1951 Sections and required Cover Texts given in the Document's 1952 license notice. 1953 1954 H. Include an unaltered copy of this License. 1955 1956 I. Preserve the section Entitled "History", Preserve its Title, 1957 and add to it an item stating at least the title, year, new 1958 authors, and publisher of the Modified Version as given on 1959 the Title Page. If there is no section Entitled "History" in 1960 the Document, create one stating the title, year, authors, 1961 and publisher of the Document as given on its Title Page, 1962 then add an item describing the Modified Version as stated in 1963 the previous sentence. 1964 1965 J. Preserve the network location, if any, given in the Document 1966 for public access to a Transparent copy of the Document, and 1967 likewise the network locations given in the Document for 1968 previous versions it was based on. These may be placed in 1969 the "History" section. You may omit a network location for a 1970 work that was published at least four years before the 1971 Document itself, or if the original publisher of the version 1972 it refers to gives permission. 1973 1974 K. For any section Entitled "Acknowledgements" or "Dedications", 1975 Preserve the Title of the section, and preserve in the 1976 section all the substance and tone of each of the contributor 1977 acknowledgements and/or dedications given therein. 1978 1979 L. Preserve all the Invariant Sections of the Document, 1980 unaltered in their text and in their titles. Section numbers 1981 or the equivalent are not considered part of the section 1982 titles. 1983 1984 M. Delete any section Entitled "Endorsements". Such a section 1985 may not be included in the Modified Version. 1986 1987 N. Do not retitle any existing section to be Entitled 1988 "Endorsements" or to conflict in title with any Invariant 1989 Section. 1990 1991 O. Preserve any Warranty Disclaimers. 1992 1993 If the Modified Version includes new front-matter sections or 1994 appendices that qualify as Secondary Sections and contain no 1995 material copied from the Document, you may at your option 1996 designate some or all of these sections as invariant. To do this, 1997 add their titles to the list of Invariant Sections in the Modified 1998 Version's license notice. These titles must be distinct from any 1999 other section titles. 2000 2001 You may add a section Entitled "Endorsements", provided it contains 2002 nothing but endorsements of your Modified Version by various 2003 parties--for example, statements of peer review or that the text 2004 has been approved by an organization as the authoritative 2005 definition of a standard. 2006 2007 You may add a passage of up to five words as a Front-Cover Text, 2008 and a passage of up to 25 words as a Back-Cover Text, to the end 2009 of the list of Cover Texts in the Modified Version. Only one 2010 passage of Front-Cover Text and one of Back-Cover Text may be 2011 added by (or through arrangements made by) any one entity. If the 2012 Document already includes a cover text for the same cover, 2013 previously added by you or by arrangement made by the same entity 2014 you are acting on behalf of, you may not add another; but you may 2015 replace the old one, on explicit permission from the previous 2016 publisher that added the old one. 2017 2018 The author(s) and publisher(s) of the Document do not by this 2019 License give permission to use their names for publicity for or to 2020 assert or imply endorsement of any Modified Version. 2021 2022 5. COMBINING DOCUMENTS 2023 2024 You may combine the Document with other documents released under 2025 this License, under the terms defined in section 4 above for 2026 modified versions, provided that you include in the combination 2027 all of the Invariant Sections of all of the original documents, 2028 unmodified, and list them all as Invariant Sections of your 2029 combined work in its license notice, and that you preserve all 2030 their Warranty Disclaimers. 2031 2032 The combined work need only contain one copy of this License, and 2033 multiple identical Invariant Sections may be replaced with a single 2034 copy. If there are multiple Invariant Sections with the same name 2035 but different contents, make the title of each such section unique 2036 by adding at the end of it, in parentheses, the name of the 2037 original author or publisher of that section if known, or else a 2038 unique number. Make the same adjustment to the section titles in 2039 the list of Invariant Sections in the license notice of the 2040 combined work. 2041 2042 In the combination, you must combine any sections Entitled 2043 "History" in the various original documents, forming one section 2044 Entitled "History"; likewise combine any sections Entitled 2045 "Acknowledgements", and any sections Entitled "Dedications". You 2046 must delete all sections Entitled "Endorsements." 2047 2048 6. COLLECTIONS OF DOCUMENTS 2049 2050 You may make a collection consisting of the Document and other 2051 documents released under this License, and replace the individual 2052 copies of this License in the various documents with a single copy 2053 that is included in the collection, provided that you follow the 2054 rules of this License for verbatim copying of each of the 2055 documents in all other respects. 2056 2057 You may extract a single document from such a collection, and 2058 distribute it individually under this License, provided you insert 2059 a copy of this License into the extracted document, and follow 2060 this License in all other respects regarding verbatim copying of 2061 that document. 2062 2063 7. AGGREGATION WITH INDEPENDENT WORKS 2064 2065 A compilation of the Document or its derivatives with other 2066 separate and independent documents or works, in or on a volume of 2067 a storage or distribution medium, is called an "aggregate" if the 2068 copyright resulting from the compilation is not used to limit the 2069 legal rights of the compilation's users beyond what the individual 2070 works permit. When the Document is included in an aggregate, this 2071 License does not apply to the other works in the aggregate which 2072 are not themselves derivative works of the Document. 2073 2074 If the Cover Text requirement of section 3 is applicable to these 2075 copies of the Document, then if the Document is less than one half 2076 of the entire aggregate, the Document's Cover Texts may be placed 2077 on covers that bracket the Document within the aggregate, or the 2078 electronic equivalent of covers if the Document is in electronic 2079 form. Otherwise they must appear on printed covers that bracket 2080 the whole aggregate. 2081 2082 8. TRANSLATION 2083 2084 Translation is considered a kind of modification, so you may 2085 distribute translations of the Document under the terms of section 2086 4. Replacing Invariant Sections with translations requires special 2087 permission from their copyright holders, but you may include 2088 translations of some or all Invariant Sections in addition to the 2089 original versions of these Invariant Sections. You may include a 2090 translation of this License, and all the license notices in the 2091 Document, and any Warranty Disclaimers, provided that you also 2092 include the original English version of this License and the 2093 original versions of those notices and disclaimers. In case of a 2094 disagreement between the translation and the original version of 2095 this License or a notice or disclaimer, the original version will 2096 prevail. 2097 2098 If a section in the Document is Entitled "Acknowledgements", 2099 "Dedications", or "History", the requirement (section 4) to 2100 Preserve its Title (section 1) will typically require changing the 2101 actual title. 2102 2103 9. TERMINATION 2104 2105 You may not copy, modify, sublicense, or distribute the Document 2106 except as expressly provided under this License. Any attempt 2107 otherwise to copy, modify, sublicense, or distribute it is void, 2108 and will automatically terminate your rights under this License. 2109 2110 However, if you cease all violation of this License, then your 2111 license from a particular copyright holder is reinstated (a) 2112 provisionally, unless and until the copyright holder explicitly 2113 and finally terminates your license, and (b) permanently, if the 2114 copyright holder fails to notify you of the violation by some 2115 reasonable means prior to 60 days after the cessation. 2116 2117 Moreover, your license from a particular copyright holder is 2118 reinstated permanently if the copyright holder notifies you of the 2119 violation by some reasonable means, this is the first time you have 2120 received notice of violation of this License (for any work) from 2121 that copyright holder, and you cure the violation prior to 30 days 2122 after your receipt of the notice. 2123 2124 Termination of your rights under this section does not terminate 2125 the licenses of parties who have received copies or rights from 2126 you under this License. If your rights have been terminated and 2127 not permanently reinstated, receipt of a copy of some or all of 2128 the same material does not give you any rights to use it. 2129 2130 10. FUTURE REVISIONS OF THIS LICENSE 2131 2132 The Free Software Foundation may publish new, revised versions of 2133 the GNU Free Documentation License from time to time. Such new 2134 versions will be similar in spirit to the present version, but may 2135 differ in detail to address new problems or concerns. See 2136 `http://www.gnu.org/copyleft/'. 2137 2138 Each version of the License is given a distinguishing version 2139 number. If the Document specifies that a particular numbered 2140 version of this License "or any later version" applies to it, you 2141 have the option of following the terms and conditions either of 2142 that specified version or of any later version that has been 2143 published (not as a draft) by the Free Software Foundation. If 2144 the Document does not specify a version number of this License, 2145 you may choose any version ever published (not as a draft) by the 2146 Free Software Foundation. If the Document specifies that a proxy 2147 can decide which future versions of this License can be used, that 2148 proxy's public statement of acceptance of a version permanently 2149 authorizes you to choose that version for the Document. 2150 2151 11. RELICENSING 2152 2153 "Massive Multiauthor Collaboration Site" (or "MMC Site") means any 2154 World Wide Web server that publishes copyrightable works and also 2155 provides prominent facilities for anybody to edit those works. A 2156 public wiki that anybody can edit is an example of such a server. 2157 A "Massive Multiauthor Collaboration" (or "MMC") contained in the 2158 site means any set of copyrightable works thus published on the MMC 2159 site. 2160 2161 "CC-BY-SA" means the Creative Commons Attribution-Share Alike 3.0 2162 license published by Creative Commons Corporation, a not-for-profit 2163 corporation with a principal place of business in San Francisco, 2164 California, as well as future copyleft versions of that license 2165 published by that same organization. 2166 2167 "Incorporate" means to publish or republish a Document, in whole or 2168 in part, as part of another Document. 2169 2170 An MMC is "eligible for relicensing" if it is licensed under this 2171 License, and if all works that were first published under this 2172 License somewhere other than this MMC, and subsequently 2173 incorporated in whole or in part into the MMC, (1) had no cover 2174 texts or invariant sections, and (2) were thus incorporated prior 2175 to November 1, 2008. 2176 2177 The operator of an MMC Site may republish an MMC contained in the 2178 site under CC-BY-SA on the same site at any time before August 1, 2179 2009, provided the MMC is eligible for relicensing. 2180 2181 2182ADDENDUM: How to use this License for your documents 2183==================================================== 2184 2185To use this License in a document you have written, include a copy of 2186the License in the document and put the following copyright and license 2187notices just after the title page: 2188 2189 Copyright (C) YEAR YOUR NAME. 2190 Permission is granted to copy, distribute and/or modify this document 2191 under the terms of the GNU Free Documentation License, Version 1.3 2192 or any later version published by the Free Software Foundation; 2193 with no Invariant Sections, no Front-Cover Texts, and no Back-Cover 2194 Texts. A copy of the license is included in the section entitled ``GNU 2195 Free Documentation License''. 2196 2197 If you have Invariant Sections, Front-Cover Texts and Back-Cover 2198Texts, replace the "with...Texts." line with this: 2199 2200 with the Invariant Sections being LIST THEIR TITLES, with 2201 the Front-Cover Texts being LIST, and with the Back-Cover Texts 2202 being LIST. 2203 2204 If you have Invariant Sections without Cover Texts, or some other 2205combination of the three, merge those two alternatives to suit the 2206situation. 2207 2208 If your document contains nontrivial examples of program code, we 2209recommend releasing these examples in parallel under your choice of 2210free software license, such as the GNU General Public License, to 2211permit their use in free software. 2212 2213 2214File: gmp.info, Node: Concept Index, Next: Function Index, Prev: GNU Free Documentation License, Up: Top 2215 2216Concept Index 2217************* 2218 2219�[index�] 2220* Menu: 2221 2222* #include: Headers and Libraries. 2223 (line 6) 2224* --build: Build Options. (line 52) 2225* --disable-fft: Build Options. (line 314) 2226* --disable-shared: Build Options. (line 45) 2227* --disable-static: Build Options. (line 45) 2228* --enable-alloca: Build Options. (line 275) 2229* --enable-assert: Build Options. (line 320) 2230* --enable-cxx: Build Options. (line 227) 2231* --enable-fat: Build Options. (line 162) 2232* --enable-profiling <1>: Build Options. (line 324) 2233* --enable-profiling: Profiling. (line 6) 2234* --exec-prefix: Build Options. (line 32) 2235* --host: Build Options. (line 66) 2236* --prefix: Build Options. (line 32) 2237* -finstrument-functions: Profiling. (line 66) 2238* 2exp functions: Efficiency. (line 43) 2239* 68000: Notes for Particular Systems. 2240 (line 80) 2241* 80x86: Notes for Particular Systems. 2242 (line 127) 2243* ABI <1>: Build Options. (line 169) 2244* ABI: ABI and ISA. (line 6) 2245* About this manual: Introduction to GMP. (line 58) 2246* AC_CHECK_LIB: Autoconf. (line 11) 2247* AIX <1>: Notes for Particular Systems. 2248 (line 7) 2249* AIX: ABI and ISA. (line 169) 2250* Algorithms: Algorithms. (line 6) 2251* alloca: Build Options. (line 275) 2252* Allocation of memory: Custom Allocation. (line 6) 2253* AMD64: ABI and ISA. (line 44) 2254* Anonymous FTP of latest version: Introduction to GMP. (line 38) 2255* Application Binary Interface: ABI and ISA. (line 6) 2256* Arithmetic functions <1>: Rational Arithmetic. (line 6) 2257* Arithmetic functions <2>: Float Arithmetic. (line 6) 2258* Arithmetic functions: Integer Arithmetic. (line 6) 2259* ARM: Notes for Particular Systems. 2260 (line 20) 2261* Assembly cache handling: Assembly Cache Handling. 2262 (line 6) 2263* Assembly carry propagation: Assembly Carry Propagation. 2264 (line 6) 2265* Assembly code organisation: Assembly Code Organisation. 2266 (line 6) 2267* Assembly coding: Assembly Coding. (line 6) 2268* Assembly floating Point: Assembly Floating Point. 2269 (line 6) 2270* Assembly loop unrolling: Assembly Loop Unrolling. 2271 (line 6) 2272* Assembly SIMD: Assembly SIMD Instructions. 2273 (line 6) 2274* Assembly software pipelining: Assembly Software Pipelining. 2275 (line 6) 2276* Assembly writing guide: Assembly Writing Guide. 2277 (line 6) 2278* Assertion checking <1>: Build Options. (line 320) 2279* Assertion checking: Debugging. (line 79) 2280* Assignment functions <1>: Assigning Integers. (line 6) 2281* Assignment functions <2>: Initializing Rationals. 2282 (line 6) 2283* Assignment functions <3>: Assigning Floats. (line 6) 2284* Assignment functions <4>: Simultaneous Float Init & Assign. 2285 (line 6) 2286* Assignment functions: Simultaneous Integer Init & Assign. 2287 (line 6) 2288* Autoconf: Autoconf. (line 6) 2289* Basics: GMP Basics. (line 6) 2290* Binomial coefficient algorithm: Binomial Coefficients Algorithm. 2291 (line 6) 2292* Binomial coefficient functions: Number Theoretic Functions. 2293 (line 128) 2294* Binutils strip: Known Build Problems. 2295 (line 28) 2296* Bit manipulation functions: Integer Logic and Bit Fiddling. 2297 (line 6) 2298* Bit scanning functions: Integer Logic and Bit Fiddling. 2299 (line 40) 2300* Bit shift left: Integer Arithmetic. (line 38) 2301* Bit shift right: Integer Division. (line 62) 2302* Bits per limb: Useful Macros and Constants. 2303 (line 7) 2304* Bug reporting: Reporting Bugs. (line 6) 2305* Build directory: Build Options. (line 19) 2306* Build notes for binary packaging: Notes for Package Builds. 2307 (line 6) 2308* Build notes for particular systems: Notes for Particular Systems. 2309 (line 6) 2310* Build options: Build Options. (line 6) 2311* Build problems known: Known Build Problems. 2312 (line 6) 2313* Build system: Build Options. (line 52) 2314* Building GMP: Installing GMP. (line 6) 2315* Bus error: Debugging. (line 7) 2316* C compiler: Build Options. (line 180) 2317* C++ compiler: Build Options. (line 251) 2318* C++ interface: C++ Class Interface. (line 6) 2319* C++ interface internals: C++ Interface Internals. 2320 (line 6) 2321* C++ istream input: C++ Formatted Input. (line 6) 2322* C++ ostream output: C++ Formatted Output. 2323 (line 6) 2324* C++ support: Build Options. (line 227) 2325* CC: Build Options. (line 180) 2326* CC_FOR_BUILD: Build Options. (line 214) 2327* CFLAGS: Build Options. (line 180) 2328* Checker: Debugging. (line 115) 2329* checkergcc: Debugging. (line 122) 2330* Code organisation: Assembly Code Organisation. 2331 (line 6) 2332* Compaq C++: Notes for Particular Systems. 2333 (line 25) 2334* Comparison functions <1>: Comparing Rationals. (line 6) 2335* Comparison functions <2>: Float Comparison. (line 6) 2336* Comparison functions: Integer Comparisons. (line 6) 2337* Compatibility with older versions: Compatibility with older versions. 2338 (line 6) 2339* Conditions for copying GNU MP: Copying. (line 6) 2340* Configuring GMP: Installing GMP. (line 6) 2341* Congruence algorithm: Exact Remainder. (line 30) 2342* Congruence functions: Integer Division. (line 137) 2343* Constants: Useful Macros and Constants. 2344 (line 6) 2345* Contributors: Contributors. (line 6) 2346* Conventions for parameters: Parameter Conventions. 2347 (line 6) 2348* Conventions for variables: Variable Conventions. 2349 (line 6) 2350* Conversion functions <1>: Converting Integers. (line 6) 2351* Conversion functions <2>: Converting Floats. (line 6) 2352* Conversion functions: Rational Conversions. 2353 (line 6) 2354* Copying conditions: Copying. (line 6) 2355* CPPFLAGS: Build Options. (line 206) 2356* CPU types <1>: Introduction to GMP. (line 24) 2357* CPU types: Build Options. (line 108) 2358* Cross compiling: Build Options. (line 66) 2359* Custom allocation: Custom Allocation. (line 6) 2360* CXX: Build Options. (line 251) 2361* CXXFLAGS: Build Options. (line 251) 2362* Cygwin: Notes for Particular Systems. 2363 (line 43) 2364* Darwin: Known Build Problems. 2365 (line 51) 2366* Debugging: Debugging. (line 6) 2367* Demonstration programs: Demonstration Programs. 2368 (line 6) 2369* Digits in an integer: Miscellaneous Integer Functions. 2370 (line 23) 2371* Divisibility algorithm: Exact Remainder. (line 30) 2372* Divisibility functions: Integer Division. (line 137) 2373* Divisibility testing: Efficiency. (line 91) 2374* Division algorithms: Division Algorithms. (line 6) 2375* Division functions <1>: Rational Arithmetic. (line 24) 2376* Division functions <2>: Float Arithmetic. (line 33) 2377* Division functions: Integer Division. (line 6) 2378* DJGPP <1>: Notes for Particular Systems. 2379 (line 43) 2380* DJGPP: Known Build Problems. 2381 (line 18) 2382* DLLs: Notes for Particular Systems. 2383 (line 56) 2384* DocBook: Build Options. (line 347) 2385* Documentation formats: Build Options. (line 340) 2386* Documentation license: GNU Free Documentation License. 2387 (line 6) 2388* DVI: Build Options. (line 343) 2389* Efficiency: Efficiency. (line 6) 2390* Emacs: Emacs. (line 6) 2391* Exact division functions: Integer Division. (line 112) 2392* Exact remainder: Exact Remainder. (line 6) 2393* Example programs: Demonstration Programs. 2394 (line 6) 2395* Exec prefix: Build Options. (line 32) 2396* Execution profiling <1>: Profiling. (line 6) 2397* Execution profiling: Build Options. (line 324) 2398* Exponentiation functions <1>: Float Arithmetic. (line 41) 2399* Exponentiation functions: Integer Exponentiation. 2400 (line 6) 2401* Export: Integer Import and Export. 2402 (line 45) 2403* Expression parsing demo: Demonstration Programs. 2404 (line 18) 2405* Extended GCD: Number Theoretic Functions. 2406 (line 49) 2407* Factor removal functions: Number Theoretic Functions. 2408 (line 108) 2409* Factorial algorithm: Factorial Algorithm. (line 6) 2410* Factorial functions: Number Theoretic Functions. 2411 (line 116) 2412* Factorization demo: Demonstration Programs. 2413 (line 25) 2414* Fast Fourier Transform: FFT Multiplication. (line 6) 2415* Fat binary: Build Options. (line 162) 2416* FFT multiplication <1>: Build Options. (line 314) 2417* FFT multiplication: FFT Multiplication. (line 6) 2418* Fibonacci number algorithm: Fibonacci Numbers Algorithm. 2419 (line 6) 2420* Fibonacci sequence functions: Number Theoretic Functions. 2421 (line 136) 2422* Float arithmetic functions: Float Arithmetic. (line 6) 2423* Float assignment functions <1>: Simultaneous Float Init & Assign. 2424 (line 6) 2425* Float assignment functions: Assigning Floats. (line 6) 2426* Float comparison functions: Float Comparison. (line 6) 2427* Float conversion functions: Converting Floats. (line 6) 2428* Float functions: Floating-point Functions. 2429 (line 6) 2430* Float initialization functions <1>: Simultaneous Float Init & Assign. 2431 (line 6) 2432* Float initialization functions: Initializing Floats. (line 6) 2433* Float input and output functions: I/O of Floats. (line 6) 2434* Float internals: Float Internals. (line 6) 2435* Float miscellaneous functions: Miscellaneous Float Functions. 2436 (line 6) 2437* Float random number functions: Miscellaneous Float Functions. 2438 (line 27) 2439* Float rounding functions: Miscellaneous Float Functions. 2440 (line 9) 2441* Float sign tests: Float Comparison. (line 35) 2442* Floating point mode: Notes for Particular Systems. 2443 (line 34) 2444* Floating-point functions: Floating-point Functions. 2445 (line 6) 2446* Floating-point number: Nomenclature and Types. 2447 (line 21) 2448* fnccheck: Profiling. (line 77) 2449* Formatted input: Formatted Input. (line 6) 2450* Formatted output: Formatted Output. (line 6) 2451* Free Documentation License: GNU Free Documentation License. 2452 (line 6) 2453* frexp <1>: Converting Integers. (line 43) 2454* frexp: Converting Floats. (line 24) 2455* FTP of latest version: Introduction to GMP. (line 38) 2456* Function classes: Function Classes. (line 6) 2457* FunctionCheck: Profiling. (line 77) 2458* GCC Checker: Debugging. (line 115) 2459* GCD algorithms: Greatest Common Divisor Algorithms. 2460 (line 6) 2461* GCD extended: Number Theoretic Functions. 2462 (line 49) 2463* GCD functions: Number Theoretic Functions. 2464 (line 32) 2465* GDB: Debugging. (line 58) 2466* Generic C: Build Options. (line 153) 2467* GMP Perl module: Demonstration Programs. 2468 (line 35) 2469* GMP version number: Useful Macros and Constants. 2470 (line 12) 2471* gmp.h: Headers and Libraries. 2472 (line 6) 2473* gmpxx.h: C++ Interface General. 2474 (line 8) 2475* GNU Debugger: Debugging. (line 58) 2476* GNU Free Documentation License: GNU Free Documentation License. 2477 (line 6) 2478* GNU strip: Known Build Problems. 2479 (line 28) 2480* gprof: Profiling. (line 41) 2481* Greatest common divisor algorithms: Greatest Common Divisor Algorithms. 2482 (line 6) 2483* Greatest common divisor functions: Number Theoretic Functions. 2484 (line 32) 2485* Hardware floating point mode: Notes for Particular Systems. 2486 (line 34) 2487* Headers: Headers and Libraries. 2488 (line 6) 2489* Heap problems: Debugging. (line 24) 2490* Home page: Introduction to GMP. (line 34) 2491* Host system: Build Options. (line 66) 2492* HP-UX: ABI and ISA. (line 107) 2493* HPPA: ABI and ISA. (line 68) 2494* I/O functions <1>: I/O of Floats. (line 6) 2495* I/O functions <2>: I/O of Rationals. (line 6) 2496* I/O functions: I/O of Integers. (line 6) 2497* i386: Notes for Particular Systems. 2498 (line 127) 2499* IA-64: ABI and ISA. (line 107) 2500* Import: Integer Import and Export. 2501 (line 11) 2502* In-place operations: Efficiency. (line 57) 2503* Include files: Headers and Libraries. 2504 (line 6) 2505* info-lookup-symbol: Emacs. (line 6) 2506* Initialization functions <1>: Simultaneous Float Init & Assign. 2507 (line 6) 2508* Initialization functions <2>: Random State Initialization. 2509 (line 6) 2510* Initialization functions <3>: Initializing Floats. (line 6) 2511* Initialization functions <4>: Simultaneous Integer Init & Assign. 2512 (line 6) 2513* Initialization functions <5>: Initializing Rationals. 2514 (line 6) 2515* Initialization functions: Initializing Integers. 2516 (line 6) 2517* Initializing and clearing: Efficiency. (line 21) 2518* Input functions <1>: I/O of Integers. (line 6) 2519* Input functions <2>: I/O of Rationals. (line 6) 2520* Input functions <3>: Formatted Input Functions. 2521 (line 6) 2522* Input functions: I/O of Floats. (line 6) 2523* Install prefix: Build Options. (line 32) 2524* Installing GMP: Installing GMP. (line 6) 2525* Instruction Set Architecture: ABI and ISA. (line 6) 2526* instrument-functions: Profiling. (line 66) 2527* Integer: Nomenclature and Types. 2528 (line 6) 2529* Integer arithmetic functions: Integer Arithmetic. (line 6) 2530* Integer assignment functions <1>: Assigning Integers. (line 6) 2531* Integer assignment functions: Simultaneous Integer Init & Assign. 2532 (line 6) 2533* Integer bit manipulation functions: Integer Logic and Bit Fiddling. 2534 (line 6) 2535* Integer comparison functions: Integer Comparisons. (line 6) 2536* Integer conversion functions: Converting Integers. (line 6) 2537* Integer division functions: Integer Division. (line 6) 2538* Integer exponentiation functions: Integer Exponentiation. 2539 (line 6) 2540* Integer export: Integer Import and Export. 2541 (line 45) 2542* Integer functions: Integer Functions. (line 6) 2543* Integer import: Integer Import and Export. 2544 (line 11) 2545* Integer initialization functions <1>: Simultaneous Integer Init & Assign. 2546 (line 6) 2547* Integer initialization functions: Initializing Integers. 2548 (line 6) 2549* Integer input and output functions: I/O of Integers. (line 6) 2550* Integer internals: Integer Internals. (line 6) 2551* Integer logical functions: Integer Logic and Bit Fiddling. 2552 (line 6) 2553* Integer miscellaneous functions: Miscellaneous Integer Functions. 2554 (line 6) 2555* Integer random number functions: Integer Random Numbers. 2556 (line 6) 2557* Integer root functions: Integer Roots. (line 6) 2558* Integer sign tests: Integer Comparisons. (line 28) 2559* Integer special functions: Integer Special Functions. 2560 (line 6) 2561* Interix: Notes for Particular Systems. 2562 (line 51) 2563* Internals: Internals. (line 6) 2564* Introduction: Introduction to GMP. (line 6) 2565* Inverse modulo functions: Number Theoretic Functions. 2566 (line 76) 2567* IRIX <1>: Known Build Problems. 2568 (line 38) 2569* IRIX: ABI and ISA. (line 132) 2570* ISA: ABI and ISA. (line 6) 2571* istream input: C++ Formatted Input. (line 6) 2572* Jacobi symbol algorithm: Jacobi Symbol. (line 6) 2573* Jacobi symbol functions: Number Theoretic Functions. 2574 (line 83) 2575* Karatsuba multiplication: Karatsuba Multiplication. 2576 (line 6) 2577* Karatsuba square root algorithm: Square Root Algorithm. 2578 (line 6) 2579* Kronecker symbol functions: Number Theoretic Functions. 2580 (line 95) 2581* Language bindings: Language Bindings. (line 6) 2582* Latest version of GMP: Introduction to GMP. (line 38) 2583* LCM functions: Number Theoretic Functions. 2584 (line 70) 2585* Least common multiple functions: Number Theoretic Functions. 2586 (line 70) 2587* Legendre symbol functions: Number Theoretic Functions. 2588 (line 86) 2589* libgmp: Headers and Libraries. 2590 (line 22) 2591* libgmpxx: Headers and Libraries. 2592 (line 27) 2593* Libraries: Headers and Libraries. 2594 (line 22) 2595* Libtool: Headers and Libraries. 2596 (line 33) 2597* Libtool versioning: Notes for Package Builds. 2598 (line 9) 2599* License conditions: Copying. (line 6) 2600* Limb: Nomenclature and Types. 2601 (line 31) 2602* Limb size: Useful Macros and Constants. 2603 (line 7) 2604* Linear congruential algorithm: Random Number Algorithms. 2605 (line 25) 2606* Linear congruential random numbers: Random State Initialization. 2607 (line 32) 2608* Linking: Headers and Libraries. 2609 (line 22) 2610* Logical functions: Integer Logic and Bit Fiddling. 2611 (line 6) 2612* Low-level functions: Low-level Functions. (line 6) 2613* Lucas number algorithm: Lucas Numbers Algorithm. 2614 (line 6) 2615* Lucas number functions: Number Theoretic Functions. 2616 (line 147) 2617* MacOS X: Known Build Problems. 2618 (line 51) 2619* Mailing lists: Introduction to GMP. (line 45) 2620* Malloc debugger: Debugging. (line 30) 2621* Malloc problems: Debugging. (line 24) 2622* Memory allocation: Custom Allocation. (line 6) 2623* Memory management: Memory Management. (line 6) 2624* Mersenne twister algorithm: Random Number Algorithms. 2625 (line 17) 2626* Mersenne twister random numbers: Random State Initialization. 2627 (line 13) 2628* MINGW: Notes for Particular Systems. 2629 (line 43) 2630* MIPS: ABI and ISA. (line 132) 2631* Miscellaneous float functions: Miscellaneous Float Functions. 2632 (line 6) 2633* Miscellaneous integer functions: Miscellaneous Integer Functions. 2634 (line 6) 2635* MMX: Notes for Particular Systems. 2636 (line 133) 2637* Modular inverse functions: Number Theoretic Functions. 2638 (line 76) 2639* Most significant bit: Miscellaneous Integer Functions. 2640 (line 34) 2641* MPN_PATH: Build Options. (line 328) 2642* MS Windows: Notes for Particular Systems. 2643 (line 43) 2644* MS-DOS: Notes for Particular Systems. 2645 (line 43) 2646* Multi-threading: Reentrancy. (line 6) 2647* Multiplication algorithms: Multiplication Algorithms. 2648 (line 6) 2649* Nails: Low-level Functions. (line 485) 2650* Native compilation: Build Options. (line 52) 2651* NeXT: Known Build Problems. 2652 (line 57) 2653* Next prime function: Number Theoretic Functions. 2654 (line 25) 2655* Nomenclature: Nomenclature and Types. 2656 (line 6) 2657* Non-Unix systems: Build Options. (line 11) 2658* Nth root algorithm: Nth Root Algorithm. (line 6) 2659* Number sequences: Efficiency. (line 147) 2660* Number theoretic functions: Number Theoretic Functions. 2661 (line 6) 2662* Numerator and denominator: Applying Integer Functions. 2663 (line 6) 2664* obstack output: Formatted Output Functions. 2665 (line 81) 2666* OpenBSD: Notes for Particular Systems. 2667 (line 86) 2668* Optimizing performance: Performance optimization. 2669 (line 6) 2670* ostream output: C++ Formatted Output. 2671 (line 6) 2672* Other languages: Language Bindings. (line 6) 2673* Output functions <1>: Formatted Output Functions. 2674 (line 6) 2675* Output functions <2>: I/O of Rationals. (line 6) 2676* Output functions <3>: I/O of Floats. (line 6) 2677* Output functions: I/O of Integers. (line 6) 2678* Packaged builds: Notes for Package Builds. 2679 (line 6) 2680* Parameter conventions: Parameter Conventions. 2681 (line 6) 2682* Parsing expressions demo: Demonstration Programs. 2683 (line 15) 2684* Particular systems: Notes for Particular Systems. 2685 (line 6) 2686* Past GMP versions: Compatibility with older versions. 2687 (line 6) 2688* PDF: Build Options. (line 343) 2689* Perfect power algorithm: Perfect Power Algorithm. 2690 (line 6) 2691* Perfect power functions: Integer Roots. (line 28) 2692* Perfect square algorithm: Perfect Square Algorithm. 2693 (line 6) 2694* Perfect square functions: Integer Roots. (line 37) 2695* perl: Demonstration Programs. 2696 (line 35) 2697* Perl module: Demonstration Programs. 2698 (line 35) 2699* Postscript: Build Options. (line 343) 2700* Power/PowerPC <1>: Notes for Particular Systems. 2701 (line 92) 2702* Power/PowerPC: Known Build Problems. 2703 (line 63) 2704* Powering algorithms: Powering Algorithms. (line 6) 2705* Powering functions <1>: Integer Exponentiation. 2706 (line 6) 2707* Powering functions: Float Arithmetic. (line 41) 2708* PowerPC: ABI and ISA. (line 167) 2709* Precision of floats: Floating-point Functions. 2710 (line 6) 2711* Precision of hardware floating point: Notes for Particular Systems. 2712 (line 34) 2713* Prefix: Build Options. (line 32) 2714* Prime testing algorithms: Prime Testing Algorithm. 2715 (line 6) 2716* Prime testing functions: Number Theoretic Functions. 2717 (line 7) 2718* Primorial functions: Number Theoretic Functions. 2719 (line 121) 2720* printf formatted output: Formatted Output. (line 6) 2721* Probable prime testing functions: Number Theoretic Functions. 2722 (line 7) 2723* prof: Profiling. (line 24) 2724* Profiling: Profiling. (line 6) 2725* Radix conversion algorithms: Radix Conversion Algorithms. 2726 (line 6) 2727* Random number algorithms: Random Number Algorithms. 2728 (line 6) 2729* Random number functions <1>: Miscellaneous Float Functions. 2730 (line 27) 2731* Random number functions <2>: Random Number Functions. 2732 (line 6) 2733* Random number functions: Integer Random Numbers. 2734 (line 6) 2735* Random number seeding: Random State Seeding. 2736 (line 6) 2737* Random number state: Random State Initialization. 2738 (line 6) 2739* Random state: Nomenclature and Types. 2740 (line 46) 2741* Rational arithmetic: Efficiency. (line 113) 2742* Rational arithmetic functions: Rational Arithmetic. (line 6) 2743* Rational assignment functions: Initializing Rationals. 2744 (line 6) 2745* Rational comparison functions: Comparing Rationals. (line 6) 2746* Rational conversion functions: Rational Conversions. 2747 (line 6) 2748* Rational initialization functions: Initializing Rationals. 2749 (line 6) 2750* Rational input and output functions: I/O of Rationals. (line 6) 2751* Rational internals: Rational Internals. (line 6) 2752* Rational number: Nomenclature and Types. 2753 (line 16) 2754* Rational number functions: Rational Number Functions. 2755 (line 6) 2756* Rational numerator and denominator: Applying Integer Functions. 2757 (line 6) 2758* Rational sign tests: Comparing Rationals. (line 27) 2759* Raw output internals: Raw Output Internals. 2760 (line 6) 2761* Reallocations: Efficiency. (line 30) 2762* Reentrancy: Reentrancy. (line 6) 2763* References: References. (line 6) 2764* Remove factor functions: Number Theoretic Functions. 2765 (line 108) 2766* Reporting bugs: Reporting Bugs. (line 6) 2767* Root extraction algorithm: Nth Root Algorithm. (line 6) 2768* Root extraction algorithms: Root Extraction Algorithms. 2769 (line 6) 2770* Root extraction functions <1>: Integer Roots. (line 6) 2771* Root extraction functions: Float Arithmetic. (line 37) 2772* Root testing functions: Integer Roots. (line 37) 2773* Rounding functions: Miscellaneous Float Functions. 2774 (line 9) 2775* Sample programs: Demonstration Programs. 2776 (line 6) 2777* Scan bit functions: Integer Logic and Bit Fiddling. 2778 (line 40) 2779* scanf formatted input: Formatted Input. (line 6) 2780* SCO: Known Build Problems. 2781 (line 38) 2782* Seeding random numbers: Random State Seeding. 2783 (line 6) 2784* Segmentation violation: Debugging. (line 7) 2785* Sequent Symmetry: Known Build Problems. 2786 (line 68) 2787* Services for Unix: Notes for Particular Systems. 2788 (line 51) 2789* Shared library versioning: Notes for Package Builds. 2790 (line 9) 2791* Sign tests <1>: Integer Comparisons. (line 28) 2792* Sign tests <2>: Comparing Rationals. (line 27) 2793* Sign tests: Float Comparison. (line 35) 2794* Size in digits: Miscellaneous Integer Functions. 2795 (line 23) 2796* Small operands: Efficiency. (line 7) 2797* Solaris <1>: Known Build Problems. 2798 (line 72) 2799* Solaris <2>: ABI and ISA. (line 199) 2800* Solaris: Known Build Problems. 2801 (line 78) 2802* Sparc: Notes for Particular Systems. 2803 (line 109) 2804* Sparc V9: ABI and ISA. (line 199) 2805* Special integer functions: Integer Special Functions. 2806 (line 6) 2807* Square root algorithm: Square Root Algorithm. 2808 (line 6) 2809* SSE2: Notes for Particular Systems. 2810 (line 133) 2811* Stack backtrace: Debugging. (line 50) 2812* Stack overflow <1>: Build Options. (line 275) 2813* Stack overflow: Debugging. (line 7) 2814* Static linking: Efficiency. (line 14) 2815* stdarg.h: Headers and Libraries. 2816 (line 17) 2817* stdio.h: Headers and Libraries. 2818 (line 11) 2819* Stripped libraries: Known Build Problems. 2820 (line 28) 2821* Sun: ABI and ISA. (line 199) 2822* SunOS: Notes for Particular Systems. 2823 (line 121) 2824* Systems: Notes for Particular Systems. 2825 (line 6) 2826* Temporary memory: Build Options. (line 275) 2827* Texinfo: Build Options. (line 340) 2828* Text input/output: Efficiency. (line 153) 2829* Thread safety: Reentrancy. (line 6) 2830* Toom multiplication <1>: Other Multiplication. 2831 (line 6) 2832* Toom multiplication <2>: Toom 3-Way Multiplication. 2833 (line 6) 2834* Toom multiplication <3>: Higher degree Toom'n'half. 2835 (line 6) 2836* Toom multiplication: Toom 4-Way Multiplication. 2837 (line 6) 2838* Types: Nomenclature and Types. 2839 (line 6) 2840* ui and si functions: Efficiency. (line 50) 2841* Unbalanced multiplication: Unbalanced Multiplication. 2842 (line 6) 2843* Upward compatibility: Compatibility with older versions. 2844 (line 6) 2845* Useful macros and constants: Useful Macros and Constants. 2846 (line 6) 2847* User-defined precision: Floating-point Functions. 2848 (line 6) 2849* Valgrind: Debugging. (line 130) 2850* Variable conventions: Variable Conventions. 2851 (line 6) 2852* Version number: Useful Macros and Constants. 2853 (line 12) 2854* Web page: Introduction to GMP. (line 34) 2855* Windows: Notes for Particular Systems. 2856 (line 56) 2857* x86: Notes for Particular Systems. 2858 (line 127) 2859* x87: Notes for Particular Systems. 2860 (line 34) 2861* XML: Build Options. (line 347) 2862 2863 2864File: gmp.info, Node: Function Index, Prev: Concept Index, Up: Top 2865 2866Function and Type Index 2867*********************** 2868 2869�[index�] 2870* Menu: 2871 2872* __GMP_CC: Useful Macros and Constants. 2873 (line 23) 2874* __GMP_CFLAGS: Useful Macros and Constants. 2875 (line 24) 2876* __GNU_MP_VERSION: Useful Macros and Constants. 2877 (line 10) 2878* __GNU_MP_VERSION_MINOR: Useful Macros and Constants. 2879 (line 11) 2880* __GNU_MP_VERSION_PATCHLEVEL: Useful Macros and Constants. 2881 (line 12) 2882* _mpz_realloc: Integer Special Functions. 2883 (line 51) 2884* abs <1>: C++ Interface Rationals. 2885 (line 49) 2886* abs <2>: C++ Interface Floats. 2887 (line 83) 2888* abs: C++ Interface Integers. 2889 (line 47) 2890* ceil: C++ Interface Floats. 2891 (line 84) 2892* cmp <1>: C++ Interface Rationals. 2893 (line 51) 2894* cmp <2>: C++ Interface Integers. 2895 (line 49) 2896* cmp <3>: C++ Interface Rationals. 2897 (line 50) 2898* cmp: C++ Interface Floats. 2899 (line 86) 2900* floor: C++ Interface Floats. 2901 (line 93) 2902* gmp_asprintf: Formatted Output Functions. 2903 (line 65) 2904* gmp_errno: Random State Initialization. 2905 (line 55) 2906* GMP_ERROR_INVALID_ARGUMENT: Random State Initialization. 2907 (line 55) 2908* GMP_ERROR_UNSUPPORTED_ARGUMENT: Random State Initialization. 2909 (line 55) 2910* gmp_fprintf: Formatted Output Functions. 2911 (line 29) 2912* gmp_fscanf: Formatted Input Functions. 2913 (line 25) 2914* GMP_LIMB_BITS: Low-level Functions. (line 515) 2915* GMP_NAIL_BITS: Low-level Functions. (line 513) 2916* GMP_NAIL_MASK: Low-level Functions. (line 523) 2917* GMP_NUMB_BITS: Low-level Functions. (line 514) 2918* GMP_NUMB_MASK: Low-level Functions. (line 524) 2919* GMP_NUMB_MAX: Low-level Functions. (line 532) 2920* gmp_obstack_printf: Formatted Output Functions. 2921 (line 79) 2922* gmp_obstack_vprintf: Formatted Output Functions. 2923 (line 81) 2924* gmp_printf: Formatted Output Functions. 2925 (line 24) 2926* GMP_RAND_ALG_DEFAULT: Random State Initialization. 2927 (line 49) 2928* GMP_RAND_ALG_LC: Random State Initialization. 2929 (line 49) 2930* gmp_randclass: C++ Interface Random Numbers. 2931 (line 7) 2932* gmp_randclass::get_f: C++ Interface Random Numbers. 2933 (line 46) 2934* gmp_randclass::get_z_bits: C++ Interface Random Numbers. 2935 (line 38) 2936* gmp_randclass::get_z_range: C++ Interface Random Numbers. 2937 (line 42) 2938* gmp_randclass::gmp_randclass: C++ Interface Random Numbers. 2939 (line 13) 2940* gmp_randclass::seed: C++ Interface Random Numbers. 2941 (line 33) 2942* gmp_randclear: Random State Initialization. 2943 (line 62) 2944* gmp_randinit: Random State Initialization. 2945 (line 47) 2946* gmp_randinit_default: Random State Initialization. 2947 (line 7) 2948* gmp_randinit_lc_2exp: Random State Initialization. 2949 (line 18) 2950* gmp_randinit_lc_2exp_size: Random State Initialization. 2951 (line 32) 2952* gmp_randinit_mt: Random State Initialization. 2953 (line 13) 2954* gmp_randinit_set: Random State Initialization. 2955 (line 43) 2956* gmp_randseed: Random State Seeding. 2957 (line 8) 2958* gmp_randseed_ui: Random State Seeding. 2959 (line 10) 2960* gmp_randstate_t: Nomenclature and Types. 2961 (line 46) 2962* gmp_scanf: Formatted Input Functions. 2963 (line 21) 2964* gmp_snprintf: Formatted Output Functions. 2965 (line 46) 2966* gmp_sprintf: Formatted Output Functions. 2967 (line 34) 2968* gmp_sscanf: Formatted Input Functions. 2969 (line 29) 2970* gmp_urandomb_ui: Random State Miscellaneous. 2971 (line 8) 2972* gmp_urandomm_ui: Random State Miscellaneous. 2973 (line 14) 2974* gmp_vasprintf: Formatted Output Functions. 2975 (line 66) 2976* gmp_version: Useful Macros and Constants. 2977 (line 18) 2978* gmp_vfprintf: Formatted Output Functions. 2979 (line 30) 2980* gmp_vfscanf: Formatted Input Functions. 2981 (line 26) 2982* gmp_vprintf: Formatted Output Functions. 2983 (line 25) 2984* gmp_vscanf: Formatted Input Functions. 2985 (line 22) 2986* gmp_vsnprintf: Formatted Output Functions. 2987 (line 48) 2988* gmp_vsprintf: Formatted Output Functions. 2989 (line 35) 2990* gmp_vsscanf: Formatted Input Functions. 2991 (line 31) 2992* hypot: C++ Interface Floats. 2993 (line 94) 2994* mp_bitcnt_t: Nomenclature and Types. 2995 (line 42) 2996* mp_bits_per_limb: Useful Macros and Constants. 2997 (line 7) 2998* mp_exp_t: Nomenclature and Types. 2999 (line 27) 3000* mp_get_memory_functions: Custom Allocation. (line 90) 3001* mp_limb_t: Nomenclature and Types. 3002 (line 31) 3003* mp_set_memory_functions: Custom Allocation. (line 18) 3004* mp_size_t: Nomenclature and Types. 3005 (line 37) 3006* mpf_abs: Float Arithmetic. (line 47) 3007* mpf_add: Float Arithmetic. (line 7) 3008* mpf_add_ui: Float Arithmetic. (line 9) 3009* mpf_ceil: Miscellaneous Float Functions. 3010 (line 7) 3011* mpf_class: C++ Interface General. 3012 (line 20) 3013* mpf_class::fits_sint_p: C++ Interface Floats. 3014 (line 87) 3015* mpf_class::fits_slong_p: C++ Interface Floats. 3016 (line 88) 3017* mpf_class::fits_sshort_p: C++ Interface Floats. 3018 (line 89) 3019* mpf_class::fits_uint_p: C++ Interface Floats. 3020 (line 90) 3021* mpf_class::fits_ulong_p: C++ Interface Floats. 3022 (line 91) 3023* mpf_class::fits_ushort_p: C++ Interface Floats. 3024 (line 92) 3025* mpf_class::get_d: C++ Interface Floats. 3026 (line 95) 3027* mpf_class::get_mpf_t: C++ Interface General. 3028 (line 66) 3029* mpf_class::get_prec: C++ Interface Floats. 3030 (line 115) 3031* mpf_class::get_si: C++ Interface Floats. 3032 (line 96) 3033* mpf_class::get_str: C++ Interface Floats. 3034 (line 98) 3035* mpf_class::get_ui: C++ Interface Floats. 3036 (line 99) 3037* mpf_class::mpf_class: C++ Interface Floats. 3038 (line 12) 3039* mpf_class::operator=: C++ Interface Floats. 3040 (line 60) 3041* mpf_class::set_prec: C++ Interface Floats. 3042 (line 116) 3043* mpf_class::set_prec_raw: C++ Interface Floats. 3044 (line 117) 3045* mpf_class::set_str: C++ Interface Floats. 3046 (line 101) 3047* mpf_class::swap: C++ Interface Floats. 3048 (line 104) 3049* mpf_clear: Initializing Floats. (line 37) 3050* mpf_clears: Initializing Floats. (line 41) 3051* mpf_cmp: Float Comparison. (line 7) 3052* mpf_cmp_d: Float Comparison. (line 8) 3053* mpf_cmp_si: Float Comparison. (line 10) 3054* mpf_cmp_ui: Float Comparison. (line 9) 3055* mpf_div: Float Arithmetic. (line 29) 3056* mpf_div_2exp: Float Arithmetic. (line 55) 3057* mpf_div_ui: Float Arithmetic. (line 33) 3058* mpf_eq: Float Comparison. (line 18) 3059* mpf_fits_sint_p: Miscellaneous Float Functions. 3060 (line 20) 3061* mpf_fits_slong_p: Miscellaneous Float Functions. 3062 (line 18) 3063* mpf_fits_sshort_p: Miscellaneous Float Functions. 3064 (line 22) 3065* mpf_fits_uint_p: Miscellaneous Float Functions. 3066 (line 19) 3067* mpf_fits_ulong_p: Miscellaneous Float Functions. 3068 (line 17) 3069* mpf_fits_ushort_p: Miscellaneous Float Functions. 3070 (line 21) 3071* mpf_floor: Miscellaneous Float Functions. 3072 (line 8) 3073* mpf_get_d: Converting Floats. (line 7) 3074* mpf_get_d_2exp: Converting Floats. (line 17) 3075* mpf_get_default_prec: Initializing Floats. (line 12) 3076* mpf_get_prec: Initializing Floats. (line 62) 3077* mpf_get_si: Converting Floats. (line 28) 3078* mpf_get_str: Converting Floats. (line 38) 3079* mpf_get_ui: Converting Floats. (line 29) 3080* mpf_init: Initializing Floats. (line 19) 3081* mpf_init2: Initializing Floats. (line 26) 3082* mpf_init_set: Simultaneous Float Init & Assign. 3083 (line 16) 3084* mpf_init_set_d: Simultaneous Float Init & Assign. 3085 (line 19) 3086* mpf_init_set_si: Simultaneous Float Init & Assign. 3087 (line 18) 3088* mpf_init_set_str: Simultaneous Float Init & Assign. 3089 (line 26) 3090* mpf_init_set_ui: Simultaneous Float Init & Assign. 3091 (line 17) 3092* mpf_inits: Initializing Floats. (line 31) 3093* mpf_inp_str: I/O of Floats. (line 39) 3094* mpf_integer_p: Miscellaneous Float Functions. 3095 (line 14) 3096* mpf_mul: Float Arithmetic. (line 19) 3097* mpf_mul_2exp: Float Arithmetic. (line 51) 3098* mpf_mul_ui: Float Arithmetic. (line 21) 3099* mpf_neg: Float Arithmetic. (line 44) 3100* mpf_out_str: I/O of Floats. (line 19) 3101* mpf_pow_ui: Float Arithmetic. (line 41) 3102* mpf_random2: Miscellaneous Float Functions. 3103 (line 37) 3104* mpf_reldiff: Float Comparison. (line 31) 3105* mpf_set: Assigning Floats. (line 10) 3106* mpf_set_d: Assigning Floats. (line 13) 3107* mpf_set_default_prec: Initializing Floats. (line 7) 3108* mpf_set_prec: Initializing Floats. (line 65) 3109* mpf_set_prec_raw: Initializing Floats. (line 72) 3110* mpf_set_q: Assigning Floats. (line 15) 3111* mpf_set_si: Assigning Floats. (line 12) 3112* mpf_set_str: Assigning Floats. (line 18) 3113* mpf_set_ui: Assigning Floats. (line 11) 3114* mpf_set_z: Assigning Floats. (line 14) 3115* mpf_sgn: Float Comparison. (line 35) 3116* mpf_sqrt: Float Arithmetic. (line 36) 3117* mpf_sqrt_ui: Float Arithmetic. (line 37) 3118* mpf_sub: Float Arithmetic. (line 12) 3119* mpf_sub_ui: Float Arithmetic. (line 16) 3120* mpf_swap: Assigning Floats. (line 52) 3121* mpf_t: Nomenclature and Types. 3122 (line 21) 3123* mpf_trunc: Miscellaneous Float Functions. 3124 (line 9) 3125* mpf_ui_div: Float Arithmetic. (line 31) 3126* mpf_ui_sub: Float Arithmetic. (line 14) 3127* mpf_urandomb: Miscellaneous Float Functions. 3128 (line 27) 3129* mpn_add: Low-level Functions. (line 69) 3130* mpn_add_1: Low-level Functions. (line 64) 3131* mpn_add_n: Low-level Functions. (line 54) 3132* mpn_addmul_1: Low-level Functions. (line 148) 3133* mpn_and_n: Low-level Functions. (line 427) 3134* mpn_andn_n: Low-level Functions. (line 442) 3135* mpn_cmp: Low-level Functions. (line 284) 3136* mpn_com: Low-level Functions. (line 467) 3137* mpn_copyd: Low-level Functions. (line 476) 3138* mpn_copyi: Low-level Functions. (line 472) 3139* mpn_divexact_by3: Low-level Functions. (line 229) 3140* mpn_divexact_by3c: Low-level Functions. (line 231) 3141* mpn_divmod: Low-level Functions. (line 224) 3142* mpn_divmod_1: Low-level Functions. (line 208) 3143* mpn_divrem: Low-level Functions. (line 182) 3144* mpn_divrem_1: Low-level Functions. (line 206) 3145* mpn_gcd: Low-level Functions. (line 289) 3146* mpn_gcd_1: Low-level Functions. (line 299) 3147* mpn_gcdext: Low-level Functions. (line 305) 3148* mpn_get_str: Low-level Functions. (line 352) 3149* mpn_hamdist: Low-level Functions. (line 416) 3150* mpn_ior_n: Low-level Functions. (line 432) 3151* mpn_iorn_n: Low-level Functions. (line 447) 3152* mpn_lshift: Low-level Functions. (line 260) 3153* mpn_mod_1: Low-level Functions. (line 255) 3154* mpn_mul: Low-level Functions. (line 114) 3155* mpn_mul_1: Low-level Functions. (line 133) 3156* mpn_mul_n: Low-level Functions. (line 103) 3157* mpn_nand_n: Low-level Functions. (line 452) 3158* mpn_neg: Low-level Functions. (line 98) 3159* mpn_nior_n: Low-level Functions. (line 457) 3160* mpn_perfect_square_p: Low-level Functions. (line 422) 3161* mpn_popcount: Low-level Functions. (line 412) 3162* mpn_random: Low-level Functions. (line 401) 3163* mpn_random2: Low-level Functions. (line 402) 3164* mpn_rshift: Low-level Functions. (line 272) 3165* mpn_scan0: Low-level Functions. (line 386) 3166* mpn_scan1: Low-level Functions. (line 394) 3167* mpn_set_str: Low-level Functions. (line 367) 3168* mpn_sqr: Low-level Functions. (line 125) 3169* mpn_sqrtrem: Low-level Functions. (line 334) 3170* mpn_sub: Low-level Functions. (line 90) 3171* mpn_sub_1: Low-level Functions. (line 85) 3172* mpn_sub_n: Low-level Functions. (line 76) 3173* mpn_submul_1: Low-level Functions. (line 159) 3174* mpn_tdiv_qr: Low-level Functions. (line 171) 3175* mpn_xnor_n: Low-level Functions. (line 462) 3176* mpn_xor_n: Low-level Functions. (line 437) 3177* mpn_zero: Low-level Functions. (line 479) 3178* mpq_abs: Rational Arithmetic. (line 34) 3179* mpq_add: Rational Arithmetic. (line 8) 3180* mpq_canonicalize: Rational Number Functions. 3181 (line 22) 3182* mpq_class: C++ Interface General. 3183 (line 19) 3184* mpq_class::canonicalize: C++ Interface Rationals. 3185 (line 43) 3186* mpq_class::get_d: C++ Interface Rationals. 3187 (line 52) 3188* mpq_class::get_den: C++ Interface Rationals. 3189 (line 66) 3190* mpq_class::get_den_mpz_t: C++ Interface Rationals. 3191 (line 76) 3192* mpq_class::get_mpq_t: C++ Interface General. 3193 (line 65) 3194* mpq_class::get_num: C++ Interface Rationals. 3195 (line 65) 3196* mpq_class::get_num_mpz_t: C++ Interface Rationals. 3197 (line 75) 3198* mpq_class::get_str: C++ Interface Rationals. 3199 (line 53) 3200* mpq_class::mpq_class: C++ Interface Rationals. 3201 (line 23) 3202* mpq_class::set_str: C++ Interface Rationals. 3203 (line 54) 3204* mpq_class::swap: C++ Interface Rationals. 3205 (line 57) 3206* mpq_clear: Initializing Rationals. 3207 (line 16) 3208* mpq_clears: Initializing Rationals. 3209 (line 20) 3210* mpq_cmp: Comparing Rationals. (line 7) 3211* mpq_cmp_si: Comparing Rationals. (line 17) 3212* mpq_cmp_ui: Comparing Rationals. (line 15) 3213* mpq_denref: Applying Integer Functions. 3214 (line 18) 3215* mpq_div: Rational Arithmetic. (line 24) 3216* mpq_div_2exp: Rational Arithmetic. (line 28) 3217* mpq_equal: Comparing Rationals. (line 33) 3218* mpq_get_d: Rational Conversions. 3219 (line 7) 3220* mpq_get_den: Applying Integer Functions. 3221 (line 24) 3222* mpq_get_num: Applying Integer Functions. 3223 (line 23) 3224* mpq_get_str: Rational Conversions. 3225 (line 22) 3226* mpq_init: Initializing Rationals. 3227 (line 7) 3228* mpq_inits: Initializing Rationals. 3229 (line 12) 3230* mpq_inp_str: I/O of Rationals. (line 27) 3231* mpq_inv: Rational Arithmetic. (line 37) 3232* mpq_mul: Rational Arithmetic. (line 16) 3233* mpq_mul_2exp: Rational Arithmetic. (line 20) 3234* mpq_neg: Rational Arithmetic. (line 31) 3235* mpq_numref: Applying Integer Functions. 3236 (line 17) 3237* mpq_out_str: I/O of Rationals. (line 19) 3238* mpq_set: Initializing Rationals. 3239 (line 24) 3240* mpq_set_d: Rational Conversions. 3241 (line 17) 3242* mpq_set_den: Applying Integer Functions. 3243 (line 26) 3244* mpq_set_f: Rational Conversions. 3245 (line 18) 3246* mpq_set_num: Applying Integer Functions. 3247 (line 25) 3248* mpq_set_si: Initializing Rationals. 3249 (line 31) 3250* mpq_set_str: Initializing Rationals. 3251 (line 36) 3252* mpq_set_ui: Initializing Rationals. 3253 (line 29) 3254* mpq_set_z: Initializing Rationals. 3255 (line 25) 3256* mpq_sgn: Comparing Rationals. (line 27) 3257* mpq_sub: Rational Arithmetic. (line 12) 3258* mpq_swap: Initializing Rationals. 3259 (line 56) 3260* mpq_t: Nomenclature and Types. 3261 (line 16) 3262* mpz_2fac_ui: Number Theoretic Functions. 3263 (line 114) 3264* mpz_abs: Integer Arithmetic. (line 45) 3265* mpz_add: Integer Arithmetic. (line 7) 3266* mpz_add_ui: Integer Arithmetic. (line 9) 3267* mpz_addmul: Integer Arithmetic. (line 26) 3268* mpz_addmul_ui: Integer Arithmetic. (line 28) 3269* mpz_and: Integer Logic and Bit Fiddling. 3270 (line 11) 3271* mpz_array_init: Integer Special Functions. 3272 (line 11) 3273* mpz_bin_ui: Number Theoretic Functions. 3274 (line 126) 3275* mpz_bin_uiui: Number Theoretic Functions. 3276 (line 128) 3277* mpz_cdiv_q: Integer Division. (line 13) 3278* mpz_cdiv_q_2exp: Integer Division. (line 26) 3279* mpz_cdiv_q_ui: Integer Division. (line 18) 3280* mpz_cdiv_qr: Integer Division. (line 16) 3281* mpz_cdiv_qr_ui: Integer Division. (line 22) 3282* mpz_cdiv_r: Integer Division. (line 14) 3283* mpz_cdiv_r_2exp: Integer Division. (line 28) 3284* mpz_cdiv_r_ui: Integer Division. (line 20) 3285* mpz_cdiv_ui: Integer Division. (line 24) 3286* mpz_class: C++ Interface General. 3287 (line 18) 3288* mpz_class::fits_sint_p: C++ Interface Integers. 3289 (line 50) 3290* mpz_class::fits_slong_p: C++ Interface Integers. 3291 (line 51) 3292* mpz_class::fits_sshort_p: C++ Interface Integers. 3293 (line 52) 3294* mpz_class::fits_uint_p: C++ Interface Integers. 3295 (line 53) 3296* mpz_class::fits_ulong_p: C++ Interface Integers. 3297 (line 54) 3298* mpz_class::fits_ushort_p: C++ Interface Integers. 3299 (line 55) 3300* mpz_class::get_d: C++ Interface Integers. 3301 (line 56) 3302* mpz_class::get_mpz_t: C++ Interface General. 3303 (line 64) 3304* mpz_class::get_si: C++ Interface Integers. 3305 (line 57) 3306* mpz_class::get_str: C++ Interface Integers. 3307 (line 58) 3308* mpz_class::get_ui: C++ Interface Integers. 3309 (line 59) 3310* mpz_class::mpz_class: C++ Interface Integers. 3311 (line 7) 3312* mpz_class::set_str: C++ Interface Integers. 3313 (line 60) 3314* mpz_class::swap: C++ Interface Integers. 3315 (line 64) 3316* mpz_clear: Initializing Integers. 3317 (line 49) 3318* mpz_clears: Initializing Integers. 3319 (line 53) 3320* mpz_clrbit: Integer Logic and Bit Fiddling. 3321 (line 56) 3322* mpz_cmp: Integer Comparisons. (line 7) 3323* mpz_cmp_d: Integer Comparisons. (line 8) 3324* mpz_cmp_si: Integer Comparisons. (line 9) 3325* mpz_cmp_ui: Integer Comparisons. (line 10) 3326* mpz_cmpabs: Integer Comparisons. (line 18) 3327* mpz_cmpabs_d: Integer Comparisons. (line 19) 3328* mpz_cmpabs_ui: Integer Comparisons. (line 20) 3329* mpz_com: Integer Logic and Bit Fiddling. 3330 (line 20) 3331* mpz_combit: Integer Logic and Bit Fiddling. 3332 (line 59) 3333* mpz_congruent_2exp_p: Integer Division. (line 137) 3334* mpz_congruent_p: Integer Division. (line 133) 3335* mpz_congruent_ui_p: Integer Division. (line 135) 3336* mpz_divexact: Integer Division. (line 110) 3337* mpz_divexact_ui: Integer Division. (line 112) 3338* mpz_divisible_2exp_p: Integer Division. (line 123) 3339* mpz_divisible_p: Integer Division. (line 120) 3340* mpz_divisible_ui_p: Integer Division. (line 122) 3341* mpz_even_p: Miscellaneous Integer Functions. 3342 (line 18) 3343* mpz_export: Integer Import and Export. 3344 (line 45) 3345* mpz_fac_ui: Number Theoretic Functions. 3346 (line 113) 3347* mpz_fdiv_q: Integer Division. (line 30) 3348* mpz_fdiv_q_2exp: Integer Division. (line 43) 3349* mpz_fdiv_q_ui: Integer Division. (line 35) 3350* mpz_fdiv_qr: Integer Division. (line 33) 3351* mpz_fdiv_qr_ui: Integer Division. (line 39) 3352* mpz_fdiv_r: Integer Division. (line 31) 3353* mpz_fdiv_r_2exp: Integer Division. (line 45) 3354* mpz_fdiv_r_ui: Integer Division. (line 37) 3355* mpz_fdiv_ui: Integer Division. (line 41) 3356* mpz_fib2_ui: Number Theoretic Functions. 3357 (line 136) 3358* mpz_fib_ui: Number Theoretic Functions. 3359 (line 134) 3360* mpz_fits_sint_p: Miscellaneous Integer Functions. 3361 (line 10) 3362* mpz_fits_slong_p: Miscellaneous Integer Functions. 3363 (line 8) 3364* mpz_fits_sshort_p: Miscellaneous Integer Functions. 3365 (line 12) 3366* mpz_fits_uint_p: Miscellaneous Integer Functions. 3367 (line 9) 3368* mpz_fits_ulong_p: Miscellaneous Integer Functions. 3369 (line 7) 3370* mpz_fits_ushort_p: Miscellaneous Integer Functions. 3371 (line 11) 3372* mpz_gcd: Number Theoretic Functions. 3373 (line 32) 3374* mpz_gcd_ui: Number Theoretic Functions. 3375 (line 39) 3376* mpz_gcdext: Number Theoretic Functions. 3377 (line 49) 3378* mpz_get_d: Converting Integers. (line 27) 3379* mpz_get_d_2exp: Converting Integers. (line 36) 3380* mpz_get_si: Converting Integers. (line 18) 3381* mpz_get_str: Converting Integers. (line 47) 3382* mpz_get_ui: Converting Integers. (line 11) 3383* mpz_getlimbn: Integer Special Functions. 3384 (line 60) 3385* mpz_hamdist: Integer Logic and Bit Fiddling. 3386 (line 29) 3387* mpz_import: Integer Import and Export. 3388 (line 11) 3389* mpz_init: Initializing Integers. 3390 (line 26) 3391* mpz_init2: Initializing Integers. 3392 (line 33) 3393* mpz_init_set: Simultaneous Integer Init & Assign. 3394 (line 27) 3395* mpz_init_set_d: Simultaneous Integer Init & Assign. 3396 (line 30) 3397* mpz_init_set_si: Simultaneous Integer Init & Assign. 3398 (line 29) 3399* mpz_init_set_str: Simultaneous Integer Init & Assign. 3400 (line 35) 3401* mpz_init_set_ui: Simultaneous Integer Init & Assign. 3402 (line 28) 3403* mpz_inits: Initializing Integers. 3404 (line 29) 3405* mpz_inp_raw: I/O of Integers. (line 62) 3406* mpz_inp_str: I/O of Integers. (line 31) 3407* mpz_invert: Number Theoretic Functions. 3408 (line 76) 3409* mpz_ior: Integer Logic and Bit Fiddling. 3410 (line 14) 3411* mpz_jacobi: Number Theoretic Functions. 3412 (line 83) 3413* mpz_kronecker: Number Theoretic Functions. 3414 (line 91) 3415* mpz_kronecker_si: Number Theoretic Functions. 3416 (line 92) 3417* mpz_kronecker_ui: Number Theoretic Functions. 3418 (line 93) 3419* mpz_lcm: Number Theoretic Functions. 3420 (line 68) 3421* mpz_lcm_ui: Number Theoretic Functions. 3422 (line 70) 3423* mpz_legendre: Number Theoretic Functions. 3424 (line 86) 3425* mpz_lucnum2_ui: Number Theoretic Functions. 3426 (line 147) 3427* mpz_lucnum_ui: Number Theoretic Functions. 3428 (line 145) 3429* mpz_mfac_uiui: Number Theoretic Functions. 3430 (line 116) 3431* mpz_mod: Integer Division. (line 100) 3432* mpz_mod_ui: Integer Division. (line 102) 3433* mpz_mul: Integer Arithmetic. (line 19) 3434* mpz_mul_2exp: Integer Arithmetic. (line 38) 3435* mpz_mul_si: Integer Arithmetic. (line 20) 3436* mpz_mul_ui: Integer Arithmetic. (line 22) 3437* mpz_neg: Integer Arithmetic. (line 42) 3438* mpz_nextprime: Number Theoretic Functions. 3439 (line 25) 3440* mpz_odd_p: Miscellaneous Integer Functions. 3441 (line 17) 3442* mpz_out_raw: I/O of Integers. (line 46) 3443* mpz_out_str: I/O of Integers. (line 19) 3444* mpz_perfect_power_p: Integer Roots. (line 28) 3445* mpz_perfect_square_p: Integer Roots. (line 37) 3446* mpz_popcount: Integer Logic and Bit Fiddling. 3447 (line 23) 3448* mpz_pow_ui: Integer Exponentiation. 3449 (line 31) 3450* mpz_powm: Integer Exponentiation. 3451 (line 8) 3452* mpz_powm_sec: Integer Exponentiation. 3453 (line 18) 3454* mpz_powm_ui: Integer Exponentiation. 3455 (line 10) 3456* mpz_primorial_ui: Number Theoretic Functions. 3457 (line 121) 3458* mpz_probab_prime_p: Number Theoretic Functions. 3459 (line 7) 3460* mpz_random: Integer Random Numbers. 3461 (line 42) 3462* mpz_random2: Integer Random Numbers. 3463 (line 51) 3464* mpz_realloc2: Initializing Integers. 3465 (line 57) 3466* mpz_remove: Number Theoretic Functions. 3467 (line 108) 3468* mpz_root: Integer Roots. (line 8) 3469* mpz_rootrem: Integer Roots. (line 14) 3470* mpz_rrandomb: Integer Random Numbers. 3471 (line 31) 3472* mpz_scan0: Integer Logic and Bit Fiddling. 3473 (line 38) 3474* mpz_scan1: Integer Logic and Bit Fiddling. 3475 (line 40) 3476* mpz_set: Assigning Integers. (line 10) 3477* mpz_set_d: Assigning Integers. (line 13) 3478* mpz_set_f: Assigning Integers. (line 15) 3479* mpz_set_q: Assigning Integers. (line 14) 3480* mpz_set_si: Assigning Integers. (line 12) 3481* mpz_set_str: Assigning Integers. (line 21) 3482* mpz_set_ui: Assigning Integers. (line 11) 3483* mpz_setbit: Integer Logic and Bit Fiddling. 3484 (line 53) 3485* mpz_sgn: Integer Comparisons. (line 28) 3486* mpz_si_kronecker: Number Theoretic Functions. 3487 (line 94) 3488* mpz_size: Integer Special Functions. 3489 (line 68) 3490* mpz_sizeinbase: Miscellaneous Integer Functions. 3491 (line 23) 3492* mpz_sqrt: Integer Roots. (line 18) 3493* mpz_sqrtrem: Integer Roots. (line 21) 3494* mpz_sub: Integer Arithmetic. (line 12) 3495* mpz_sub_ui: Integer Arithmetic. (line 14) 3496* mpz_submul: Integer Arithmetic. (line 32) 3497* mpz_submul_ui: Integer Arithmetic. (line 34) 3498* mpz_swap: Assigning Integers. (line 37) 3499* mpz_t: Nomenclature and Types. 3500 (line 6) 3501* mpz_tdiv_q: Integer Division. (line 47) 3502* mpz_tdiv_q_2exp: Integer Division. (line 60) 3503* mpz_tdiv_q_ui: Integer Division. (line 52) 3504* mpz_tdiv_qr: Integer Division. (line 50) 3505* mpz_tdiv_qr_ui: Integer Division. (line 56) 3506* mpz_tdiv_r: Integer Division. (line 48) 3507* mpz_tdiv_r_2exp: Integer Division. (line 62) 3508* mpz_tdiv_r_ui: Integer Division. (line 54) 3509* mpz_tdiv_ui: Integer Division. (line 58) 3510* mpz_tstbit: Integer Logic and Bit Fiddling. 3511 (line 62) 3512* mpz_ui_kronecker: Number Theoretic Functions. 3513 (line 95) 3514* mpz_ui_pow_ui: Integer Exponentiation. 3515 (line 33) 3516* mpz_ui_sub: Integer Arithmetic. (line 16) 3517* mpz_urandomb: Integer Random Numbers. 3518 (line 14) 3519* mpz_urandomm: Integer Random Numbers. 3520 (line 23) 3521* mpz_xor: Integer Logic and Bit Fiddling. 3522 (line 17) 3523* operator"" <1>: C++ Interface Integers. 3524 (line 30) 3525* operator"" <2>: C++ Interface Floats. 3526 (line 56) 3527* operator"": C++ Interface Rationals. 3528 (line 38) 3529* operator%: C++ Interface Integers. 3530 (line 35) 3531* operator/: C++ Interface Integers. 3532 (line 34) 3533* operator<<: C++ Formatted Output. 3534 (line 20) 3535* operator>> <1>: C++ Interface Rationals. 3536 (line 85) 3537* operator>>: C++ Formatted Input. (line 25) 3538* sgn <1>: C++ Interface Rationals. 3539 (line 56) 3540* sgn <2>: C++ Interface Integers. 3541 (line 62) 3542* sgn: C++ Interface Floats. 3543 (line 102) 3544* sqrt <1>: C++ Interface Integers. 3545 (line 63) 3546* sqrt: C++ Interface Floats. 3547 (line 103) 3548* swap <1>: C++ Interface Floats. 3549 (line 105) 3550* swap <2>: C++ Interface Integers. 3551 (line 65) 3552* swap: C++ Interface Rationals. 3553 (line 58) 3554* trunc: C++ Interface Floats. 3555 (line 106) 3556 3557 3558 3559 3560Local Variables: 3561coding: iso-8859-1 3562End: 3563