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locks; strict; comment @# @; 1.1 date 2011.06.20.05.54.51; author mrg; state Exp; branches 1.1.1.1; next ; 1.1.1.1 date 2011.06.20.05.54.51; author mrg; state Exp; branches 1.1.1.1.2.1 1.1.1.1.8.1; next 1.1.1.2; 1.1.1.2 date 2013.11.29.07.49.48; author mrg; state Exp; branches; next 1.1.1.3; commitid L2Av4PuGmdoL39fx; 1.1.1.3 date 2017.08.22.09.40.49; author mrg; state Exp; branches; next 1.1.1.4; commitid W5kmAIk8hwVpSb4A; 1.1.1.4 date 2020.09.27.00.27.06; author mrg; state Exp; branches; next 1.1.1.5; commitid BWuUFuEU17KgrCpC; 1.1.1.5 date 2021.07.11.21.14.45; author mrg; state Exp; branches; next ; commitid saHZLNSlLcjF8C0D; 1.1.1.1.2.1 date 2014.05.22.14.08.59; author yamt; state Exp; branches; next ; commitid nx2BSsHy0NPeAxBx; 1.1.1.1.8.1 date 2014.08.19.23.59.48; author tls; state Exp; branches; next ; commitid jTnpym9Qu0o4R1Nx; desc @@ 1.1 log @Initial revision @ text @This is ../../gmp/doc/gmp.info, produced by makeinfo version 4.8 from ../../gmp/doc/gmp.texi. This manual describes how to install and use the GNU multiple precision arithmetic library, version 5.0.2. Copyright 1991, 1993, 1994, 1995, 1996, 1997, 1998, 1999, 2000, 2001, 2002, 2003, 2004, 2005, 2006, 2007, 2008, 2009, 2010, 2011 Free Software Foundation, Inc. Permission is granted to copy, distribute and/or modify this document under the terms of the GNU Free Documentation License, Version 1.3 or any later version published by the Free Software Foundation; with no Invariant Sections, with the Front-Cover Texts being "A GNU Manual", and with the Back-Cover Texts being "You have freedom to copy and modify this GNU Manual, like GNU software". A copy of the license is included in *Note GNU Free Documentation License::. INFO-DIR-SECTION GNU libraries START-INFO-DIR-ENTRY * gmp: (gmp). GNU Multiple Precision Arithmetic Library. END-INFO-DIR-ENTRY  File: gmp.info, Node: Jacobi Symbol, Prev: Extended GCD, Up: Greatest Common Divisor Algorithms 16.3.5 Jacobi Symbol -------------------- `mpz_jacobi' and `mpz_kronecker' are currently implemented with a simple binary algorithm similar to that described for the GCDs (*note Binary GCD::). They're not very fast when both inputs are large. Lehmer's multi-step improvement or a binary based multi-step algorithm is likely to be better. When one operand fits a single limb, and that includes `mpz_kronecker_ui' and friends, an initial reduction is done with either `mpn_mod_1' or `mpn_modexact_1_odd', followed by the binary algorithm on a single limb. The binary algorithm is well suited to a single limb, and the whole calculation in this case is quite efficient. In all the routines sign changes for the result are accumulated using some bit twiddling, avoiding table lookups or conditional jumps.  File: gmp.info, Node: Powering Algorithms, Next: Root Extraction Algorithms, Prev: Greatest Common Divisor Algorithms, Up: Algorithms 16.4 Powering Algorithms ======================== * Menu: * Normal Powering Algorithm:: * Modular Powering Algorithm::  File: gmp.info, Node: Normal Powering Algorithm, Next: Modular Powering Algorithm, Prev: Powering Algorithms, Up: Powering Algorithms 16.4.1 Normal Powering ---------------------- Normal `mpz' or `mpf' powering uses a simple binary algorithm, successively squaring and then multiplying by the base when a 1 bit is seen in the exponent, as per Knuth section 4.6.3. The "left to right" variant described there is used rather than algorithm A, since it's just as easy and can be done with somewhat less temporary memory.  File: gmp.info, Node: Modular Powering Algorithm, Prev: Normal Powering Algorithm, Up: Powering Algorithms 16.4.2 Modular Powering ----------------------- Modular powering is implemented using a 2^k-ary sliding window algorithm, as per "Handbook of Applied Cryptography" algorithm 14.85 (*note References::). k is chosen according to the size of the exponent. Larger exponents use larger values of k, the choice being made to minimize the average number of multiplications that must supplement the squaring. The modular multiplies and squares use either a simple division or the REDC method by Montgomery (*note References::). REDC is a little faster, essentially saving N single limb divisions in a fashion similar to an exact remainder (*note Exact Remainder::).  File: gmp.info, Node: Root Extraction Algorithms, Next: Radix Conversion Algorithms, Prev: Powering Algorithms, Up: Algorithms 16.5 Root Extraction Algorithms =============================== * Menu: * Square Root Algorithm:: * Nth Root Algorithm:: * Perfect Square Algorithm:: * Perfect Power Algorithm::  File: gmp.info, Node: Square Root Algorithm, Next: Nth Root Algorithm, Prev: Root Extraction Algorithms, Up: Root Extraction Algorithms 16.5.1 Square Root ------------------ Square roots are taken using the "Karatsuba Square Root" algorithm by Paul Zimmermann (*note References::). An input n is split into four parts of k bits each, so with b=2^k we have n = a3*b^3 + a2*b^2 + a1*b + a0. Part a3 must be "normalized" so that either the high or second highest bit is set. In GMP, k is kept on a limb boundary and the input is left shifted (by an even number of bits) to normalize. The square root of the high two parts is taken, by recursive application of the algorithm (bottoming out in a one-limb Newton's method), s1,r1 = sqrtrem (a3*b + a2) This is an approximation to the desired root and is extended by a division to give s,r, q,u = divrem (r1*b + a1, 2*s1) s = s1*b + q r = u*b + a0 - q^2 The normalization requirement on a3 means at this point s is either correct or 1 too big. r is negative in the latter case, so if r < 0 then r = r + 2*s - 1 s = s - 1 The algorithm is expressed in a divide and conquer form, but as noted in the paper it can also be viewed as a discrete variant of Newton's method, or as a variation on the schoolboy method (no longer taught) for square roots two digits at a time. If the remainder r is not required then usually only a few high limbs of r and u need to be calculated to determine whether an adjustment to s is required. This optimization is not currently implemented. In the Karatsuba multiplication range this algorithm is O(1.5*M(N/2)), where M(n) is the time to multiply two numbers of n limbs. In the FFT multiplication range this grows to a bound of O(6*M(N/2)). In practice a factor of about 1.5 to 1.8 is found in the Karatsuba and Toom-3 ranges, growing to 2 or 3 in the FFT range. The algorithm does all its calculations in integers and the resulting `mpn_sqrtrem' is used for both `mpz_sqrt' and `mpf_sqrt'. The extended precision given by `mpf_sqrt_ui' is obtained by padding with zero limbs.  File: gmp.info, Node: Nth Root Algorithm, Next: Perfect Square Algorithm, Prev: Square Root Algorithm, Up: Root Extraction Algorithms 16.5.2 Nth Root --------------- Integer Nth roots are taken using Newton's method with the following iteration, where A is the input and n is the root to be taken. 1 A a[i+1] = - * ( --------- + (n-1)*a[i] ) n a[i]^(n-1) The initial approximation a[1] is generated bitwise by successively powering a trial root with or without new 1 bits, aiming to be just above the true root. The iteration converges quadratically when started from a good approximation. When n is large more initial bits are needed to get good convergence. The current implementation is not particularly well optimized.  File: gmp.info, Node: Perfect Square Algorithm, Next: Perfect Power Algorithm, Prev: Nth Root Algorithm, Up: Root Extraction Algorithms 16.5.3 Perfect Square --------------------- A significant fraction of non-squares can be quickly identified by checking whether the input is a quadratic residue modulo small integers. `mpz_perfect_square_p' first tests the input mod 256, which means just examining the low byte. Only 44 different values occur for squares mod 256, so 82.8% of inputs can be immediately identified as non-squares. On a 32-bit system similar tests are done mod 9, 5, 7, 13 and 17, for a total 99.25% of inputs identified as non-squares. On a 64-bit system 97 is tested too, for a total 99.62%. These moduli are chosen because they're factors of 2^24-1 (or 2^48-1 for 64-bits), and such a remainder can be quickly taken just using additions (see `mpn_mod_34lsub1'). When nails are in use moduli are instead selected by the `gen-psqr.c' program and applied with an `mpn_mod_1'. The same 2^24-1 or 2^48-1 could be done with nails using some extra bit shifts, but this is not currently implemented. In any case each modulus is applied to the `mpn_mod_34lsub1' or `mpn_mod_1' remainder and a table lookup identifies non-squares. By using a "modexact" style calculation, and suitably permuted tables, just one multiply each is required, see the code for details. Moduli are also combined to save operations, so long as the lookup tables don't become too big. `gen-psqr.c' does all the pre-calculations. A square root must still be taken for any value that passes these tests, to verify it's really a square and not one of the small fraction of non-squares that get through (ie. a pseudo-square to all the tested bases). Clearly more residue tests could be done, `mpz_perfect_square_p' only uses a compact and efficient set. Big inputs would probably benefit from more residue testing, small inputs might be better off with less. The assumed distribution of squares versus non-squares in the input would affect such considerations.  File: gmp.info, Node: Perfect Power Algorithm, Prev: Perfect Square Algorithm, Up: Root Extraction Algorithms 16.5.4 Perfect Power -------------------- Detecting perfect powers is required by some factorization algorithms. Currently `mpz_perfect_power_p' is implemented using repeated Nth root extractions, though naturally only prime roots need to be considered. (*Note Nth Root Algorithm::.) If a prime divisor p with multiplicity e can be found, then only roots which are divisors of e need to be considered, much reducing the work necessary. To this end divisibility by a set of small primes is checked.  File: gmp.info, Node: Radix Conversion Algorithms, Next: Other Algorithms, Prev: Root Extraction Algorithms, Up: Algorithms 16.6 Radix Conversion ===================== Radix conversions are less important than other algorithms. A program dominated by conversions should probably use a different data representation. * Menu: * Binary to Radix:: * Radix to Binary::  File: gmp.info, Node: Binary to Radix, Next: Radix to Binary, Prev: Radix Conversion Algorithms, Up: Radix Conversion Algorithms 16.6.1 Binary to Radix ---------------------- Conversions from binary to a power-of-2 radix use a simple and fast O(N) bit extraction algorithm. Conversions from binary to other radices use one of two algorithms. Sizes below `GET_STR_PRECOMPUTE_THRESHOLD' use a basic O(N^2) method. Repeated divisions by b^n are made, where b is the radix and n is the biggest power that fits in a limb. But instead of simply using the remainder r from such divisions, an extra divide step is done to give a fractional limb representing r/b^n. The digits of r can then be extracted using multiplications by b rather than divisions. Special case code is provided for decimal, allowing multiplications by 10 to optimize to shifts and adds. Above `GET_STR_PRECOMPUTE_THRESHOLD' a sub-quadratic algorithm is used. For an input t, powers b^(n*2^i) of the radix are calculated, until a power between t and sqrt(t) is reached. t is then divided by that largest power, giving a quotient which is the digits above that power, and a remainder which is those below. These two parts are in turn divided by the second highest power, and so on recursively. When a piece has been divided down to less than `GET_STR_DC_THRESHOLD' limbs, the basecase algorithm described above is used. The advantage of this algorithm is that big divisions can make use of the sub-quadratic divide and conquer division (*note Divide and Conquer Division::), and big divisions tend to have less overheads than lots of separate single limb divisions anyway. But in any case the cost of calculating the powers b^(n*2^i) must first be overcome. `GET_STR_PRECOMPUTE_THRESHOLD' and `GET_STR_DC_THRESHOLD' represent the same basic thing, the point where it becomes worth doing a big division to cut the input in half. `GET_STR_PRECOMPUTE_THRESHOLD' includes the cost of calculating the radix power required, whereas `GET_STR_DC_THRESHOLD' assumes that's already available, which is the case when recursing. Since the base case produces digits from least to most significant but they want to be stored from most to least, it's necessary to calculate in advance how many digits there will be, or at least be sure not to underestimate that. For GMP the number of input bits is multiplied by `chars_per_bit_exactly' from `mp_bases', rounding up. The result is either correct or one too big. Examining some of the high bits of the input could increase the chance of getting the exact number of digits, but an exact result every time would not be practical, since in general the difference between numbers 100... and 99... is only in the last few bits and the work to identify 99... might well be almost as much as a full conversion. `mpf_get_str' doesn't currently use the algorithm described here, it multiplies or divides by a power of b to move the radix point to the just above the highest non-zero digit (or at worst one above that location), then multiplies by b^n to bring out digits. This is O(N^2) and is certainly not optimal. The r/b^n scheme described above for using multiplications to bring out digits might be useful for more than a single limb. Some brief experiments with it on the base case when recursing didn't give a noticeable improvement, but perhaps that was only due to the implementation. Something similar would work for the sub-quadratic divisions too, though there would be the cost of calculating a bigger radix power. Another possible improvement for the sub-quadratic part would be to arrange for radix powers that balanced the sizes of quotient and remainder produced, ie. the highest power would be an b^(n*k) approximately equal to sqrt(t), not restricted to a 2^i factor. That ought to smooth out a graph of times against sizes, but may or may not be a net speedup.  File: gmp.info, Node: Radix to Binary, Prev: Binary to Radix, Up: Radix Conversion Algorithms 16.6.2 Radix to Binary ---------------------- *This section needs to be rewritten, it currently describes the algorithms used before GMP 4.3.* Conversions from a power-of-2 radix into binary use a simple and fast O(N) bitwise concatenation algorithm. Conversions from other radices use one of two algorithms. Sizes below `SET_STR_PRECOMPUTE_THRESHOLD' use a basic O(N^2) method. Groups of n digits are converted to limbs, where n is the biggest power of the base b which will fit in a limb, then those groups are accumulated into the result by multiplying by b^n and adding. This saves multi-precision operations, as per Knuth section 4.4 part E (*note References::). Some special case code is provided for decimal, giving the compiler a chance to optimize multiplications by 10. Above `SET_STR_PRECOMPUTE_THRESHOLD' a sub-quadratic algorithm is used. First groups of n digits are converted into limbs. Then adjacent limbs are combined into limb pairs with x*b^n+y, where x and y are the limbs. Adjacent limb pairs are combined into quads similarly with x*b^(2n)+y. This continues until a single block remains, that being the result. The advantage of this method is that the multiplications for each x are big blocks, allowing Karatsuba and higher algorithms to be used. But the cost of calculating the powers b^(n*2^i) must be overcome. `SET_STR_PRECOMPUTE_THRESHOLD' usually ends up quite big, around 5000 digits, and on some processors much bigger still. `SET_STR_PRECOMPUTE_THRESHOLD' is based on the input digits (and tuned for decimal), though it might be better based on a limb count, so as to be independent of the base. But that sort of count isn't used by the base case and so would need some sort of initial calculation or estimate. The main reason `SET_STR_PRECOMPUTE_THRESHOLD' is so much bigger than the corresponding `GET_STR_PRECOMPUTE_THRESHOLD' is that `mpn_mul_1' is much faster than `mpn_divrem_1' (often by a factor of 5, or more).  File: gmp.info, Node: Other Algorithms, Next: Assembly Coding, Prev: Radix Conversion Algorithms, Up: Algorithms 16.7 Other Algorithms ===================== * Menu: * Prime Testing Algorithm:: * Factorial Algorithm:: * Binomial Coefficients Algorithm:: * Fibonacci Numbers Algorithm:: * Lucas Numbers Algorithm:: * Random Number Algorithms::  File: gmp.info, Node: Prime Testing Algorithm, Next: Factorial Algorithm, Prev: Other Algorithms, Up: Other Algorithms 16.7.1 Prime Testing -------------------- The primality testing in `mpz_probab_prime_p' (*note Number Theoretic Functions::) first does some trial division by small factors and then uses the Miller-Rabin probabilistic primality testing algorithm, as described in Knuth section 4.5.4 algorithm P (*note References::). For an odd input n, and with n = q*2^k+1 where q is odd, this algorithm selects a random base x and tests whether x^q mod n is 1 or -1, or an x^(q*2^j) mod n is 1, for 1<=j<=k. If so then n is probably prime, if not then n is definitely composite. Any prime n will pass the test, but some composites do too. Such composites are known as strong pseudoprimes to base x. No n is a strong pseudoprime to more than 1/4 of all bases (see Knuth exercise 22), hence with x chosen at random there's no more than a 1/4 chance a "probable prime" will in fact be composite. In fact strong pseudoprimes are quite rare, making the test much more powerful than this analysis would suggest, but 1/4 is all that's proven for an arbitrary n.  File: gmp.info, Node: Factorial Algorithm, Next: Binomial Coefficients Algorithm, Prev: Prime Testing Algorithm, Up: Other Algorithms 16.7.2 Factorial ---------------- Factorials are calculated by a combination of removal of twos, powering, and binary splitting. The procedure can be best illustrated with an example, 23! = 1.2.3.4.5.6.7.8.9.10.11.12.13.14.15.16.17.18.19.20.21.22.23 has factors of two removed, 23! = 2^19.1.1.3.1.5.3.7.1.9.5.11.3.13.7.15.1.17.9.19.5.21.11.23 and the resulting terms collected up according to their multiplicity, 23! = 2^19.(3.5)^3.(7.9.11)^2.(13.15.17.19.21.23) Each sequence such as 13.15.17.19.21.23 is evaluated by splitting into every second term, as for instance (13.17.21).(15.19.23), and the same recursively on each half. This is implemented iteratively using some bit twiddling. Such splitting is more efficient than repeated Nx1 multiplies since it forms big multiplies, allowing Karatsuba and higher algorithms to be used. And even below the Karatsuba threshold a big block of work can be more efficient for the basecase algorithm. Splitting into subsequences of every second term keeps the resulting products more nearly equal in size than would the simpler approach of say taking the first half and second half of the sequence. Nearly equal products are more efficient for the current multiply implementation.  File: gmp.info, Node: Binomial Coefficients Algorithm, Next: Fibonacci Numbers Algorithm, Prev: Factorial Algorithm, Up: Other Algorithms 16.7.3 Binomial Coefficients ---------------------------- Binomial coefficients C(n,k) are calculated by first arranging k <= n/2 using C(n,k) = C(n,n-k) if necessary, and then evaluating the following product simply from i=2 to i=k. k (n-k+i) C(n,k) = (n-k+1) * prod ------- i=2 i It's easy to show that each denominator i will divide the product so far, so the exact division algorithm is used (*note Exact Division::). The numerators n-k+i and denominators i are first accumulated into as many fit a limb, to save multi-precision operations, though for `mpz_bin_ui' this applies only to the divisors, since n is an `mpz_t' and n-k+i in general won't fit in a limb at all.  File: gmp.info, Node: Fibonacci Numbers Algorithm, Next: Lucas Numbers Algorithm, Prev: Binomial Coefficients Algorithm, Up: Other Algorithms 16.7.4 Fibonacci Numbers ------------------------ The Fibonacci functions `mpz_fib_ui' and `mpz_fib2_ui' are designed for calculating isolated F[n] or F[n],F[n-1] values efficiently. For small n, a table of single limb values in `__gmp_fib_table' is used. On a 32-bit limb this goes up to F[47], or on a 64-bit limb up to F[93]. For convenience the table starts at F[-1]. Beyond the table, values are generated with a binary powering algorithm, calculating a pair F[n] and F[n-1] working from high to low across the bits of n. The formulas used are F[2k+1] = 4*F[k]^2 - F[k-1]^2 + 2*(-1)^k F[2k-1] = F[k]^2 + F[k-1]^2 F[2k] = F[2k+1] - F[2k-1] At each step, k is the high b bits of n. If the next bit of n is 0 then F[2k],F[2k-1] is used, or if it's a 1 then F[2k+1],F[2k] is used, and the process repeated until all bits of n are incorporated. Notice these formulas require just two squares per bit of n. It'd be possible to handle the first few n above the single limb table with simple additions, using the defining Fibonacci recurrence F[k+1]=F[k]+F[k-1], but this is not done since it usually turns out to be faster for only about 10 or 20 values of n, and including a block of code for just those doesn't seem worthwhile. If they really mattered it'd be better to extend the data table. Using a table avoids lots of calculations on small numbers, and makes small n go fast. A bigger table would make more small n go fast, it's just a question of balancing size against desired speed. For GMP the code is kept compact, with the emphasis primarily on a good powering algorithm. `mpz_fib2_ui' returns both F[n] and F[n-1], but `mpz_fib_ui' is only interested in F[n]. In this case the last step of the algorithm can become one multiply instead of two squares. One of the following two formulas is used, according as n is odd or even. F[2k] = F[k]*(F[k]+2F[k-1]) F[2k+1] = (2F[k]+F[k-1])*(2F[k]-F[k-1]) + 2*(-1)^k F[2k+1] here is the same as above, just rearranged to be a multiply. For interest, the 2*(-1)^k term both here and above can be applied just to the low limb of the calculation, without a carry or borrow into further limbs, which saves some code size. See comments with `mpz_fib_ui' and the internal `mpn_fib2_ui' for how this is done.  File: gmp.info, Node: Lucas Numbers Algorithm, Next: Random Number Algorithms, Prev: Fibonacci Numbers Algorithm, Up: Other Algorithms 16.7.5 Lucas Numbers -------------------- `mpz_lucnum2_ui' derives a pair of Lucas numbers from a pair of Fibonacci numbers with the following simple formulas. L[k] = F[k] + 2*F[k-1] L[k-1] = 2*F[k] - F[k-1] `mpz_lucnum_ui' is only interested in L[n], and some work can be saved. Trailing zero bits on n can be handled with a single square each. L[2k] = L[k]^2 - 2*(-1)^k And the lowest 1 bit can be handled with one multiply of a pair of Fibonacci numbers, similar to what `mpz_fib_ui' does. L[2k+1] = 5*F[k-1]*(2*F[k]+F[k-1]) - 4*(-1)^k  File: gmp.info, Node: Random Number Algorithms, Prev: Lucas Numbers Algorithm, Up: Other Algorithms 16.7.6 Random Numbers --------------------- For the `urandomb' functions, random numbers are generated simply by concatenating bits produced by the generator. As long as the generator has good randomness properties this will produce well-distributed N bit numbers. For the `urandomm' functions, random numbers in a range 0<=R48 bit pieces is convenient. With some care though six 21x32->53 bit products can be used, if one of the lower two 21-bit pieces also uses the sign bit. For the `mpn_mul_1' family of functions on a 64-bit machine, the invariant single limb is split at the start, into 3 or 4 pieces. Inside the loop, the bignum operand is split into 32-bit pieces. Fast conversion of these unsigned 32-bit pieces to floating point is highly machine-dependent. In some cases, reading the data into the integer unit, zero-extending to 64-bits, then transferring to the floating point unit back via memory is the only option. Converting partial products back to 64-bit limbs is usually best done as a signed conversion. Since all values are smaller than 2^53, signed and unsigned are the same, but most processors lack unsigned conversions. Here is a diagram showing 16x32 bit products for an `mpn_mul_1' or `mpn_addmul_1' with a 64-bit limb. The single limb operand V is split into four 16-bit parts. The multi-limb operand U is split in the loop into two 32-bit parts. +---+---+---+---+ |v48|v32|v16|v00| V operand +---+---+---+---+ +-------+---+---+ x | u32 | u00 | U operand (one limb) +---------------+ --------------------------------- +-----------+ | u00 x v00 | p00 48-bit products +-----------+ +-----------+ | u00 x v16 | p16 +-----------+ +-----------+ | u00 x v32 | p32 +-----------+ +-----------+ | u00 x v48 | p48 +-----------+ +-----------+ | u32 x v00 | r32 +-----------+ +-----------+ | u32 x v16 | r48 +-----------+ +-----------+ | u32 x v32 | r64 +-----------+ +-----------+ | u32 x v48 | r80 +-----------+ p32 and r32 can be summed using floating-point addition, and likewise p48 and r48. p00 and p16 can be summed with r64 and r80 from the previous iteration. For each loop then, four 49-bit quantities are transferred to the integer unit, aligned as follows, |-----64bits----|-----64bits----| +------------+ | p00 + r64' | i00 +------------+ +------------+ | p16 + r80' | i16 +------------+ +------------+ | p32 + r32 | i32 +------------+ +------------+ | p48 + r48 | i48 +------------+ The challenge then is to sum these efficiently and add in a carry limb, generating a low 64-bit result limb and a high 33-bit carry limb (i48 extends 33 bits into the high half).  File: gmp.info, Node: Assembly SIMD Instructions, Next: Assembly Software Pipelining, Prev: Assembly Floating Point, Up: Assembly Coding 16.8.7 SIMD Instructions ------------------------ The single-instruction multiple-data support in current microprocessors is aimed at signal processing algorithms where each data point can be treated more or less independently. There's generally not much support for propagating the sort of carries that arise in GMP. SIMD multiplications of say four 16x16 bit multiplies only do as much work as one 32x32 from GMP's point of view, and need some shifts and adds besides. But of course if say the SIMD form is fully pipelined and uses less instruction decoding then it may still be worthwhile. On the x86 chips, MMX has so far found a use in `mpn_rshift' and `mpn_lshift', and is used in a special case for 16-bit multipliers in the P55 `mpn_mul_1'. SSE2 is used for Pentium 4 `mpn_mul_1', `mpn_addmul_1', and `mpn_submul_1'.  File: gmp.info, Node: Assembly Software Pipelining, Next: Assembly Loop Unrolling, Prev: Assembly SIMD Instructions, Up: Assembly Coding 16.8.8 Software Pipelining -------------------------- Software pipelining consists of scheduling instructions around the branch point in a loop. For example a loop might issue a load not for use in the present iteration but the next, thereby allowing extra cycles for the data to arrive from memory. Naturally this is wanted only when doing things like loads or multiplies that take several cycles to complete, and only where a CPU has multiple functional units so that other work can be done in the meantime. A pipeline with several stages will have a data value in progress at each stage and each loop iteration moves them along one stage. This is like juggling. If the latency of some instruction is greater than the loop time then it will be necessary to unroll, so one register has a result ready to use while another (or multiple others) are still in progress. (*note Assembly Loop Unrolling::).  File: gmp.info, Node: Assembly Loop Unrolling, Next: Assembly Writing Guide, Prev: Assembly Software Pipelining, Up: Assembly Coding 16.8.9 Loop Unrolling --------------------- Loop unrolling consists of replicating code so that several limbs are processed in each loop. At a minimum this reduces loop overheads by a corresponding factor, but it can also allow better register usage, for example alternately using one register combination and then another. Judicious use of `m4' macros can help avoid lots of duplication in the source code. Any amount of unrolling can be handled with a loop counter that's decremented by N each time, stopping when the remaining count is less than the further N the loop will process. Or by subtracting N at the start, the termination condition becomes when the counter C is less than 0 (and the count of remaining limbs is C+N). Alternately for a power of 2 unroll the loop count and remainder can be established with a shift and mask. This is convenient if also making a computed jump into the middle of a large loop. The limbs not a multiple of the unrolling can be handled in various ways, for example * A simple loop at the end (or the start) to process the excess. Care will be wanted that it isn't too much slower than the unrolled part. * A set of binary tests, for example after an 8-limb unrolling, test for 4 more limbs to process, then a further 2 more or not, and finally 1 more or not. This will probably take more code space than a simple loop. * A `switch' statement, providing separate code for each possible excess, for example an 8-limb unrolling would have separate code for 0 remaining, 1 remaining, etc, up to 7 remaining. This might take a lot of code, but may be the best way to optimize all cases in combination with a deep pipelined loop. * A computed jump into the middle of the loop, thus making the first iteration handle the excess. This should make times smoothly increase with size, which is attractive, but setups for the jump and adjustments for pointers can be tricky and could become quite difficult in combination with deep pipelining.  File: gmp.info, Node: Assembly Writing Guide, Prev: Assembly Loop Unrolling, Up: Assembly Coding 16.8.10 Writing Guide --------------------- This is a guide to writing software pipelined loops for processing limb vectors in assembly. First determine the algorithm and which instructions are needed. Code it without unrolling or scheduling, to make sure it works. On a 3-operand CPU try to write each new value to a new register, this will greatly simplify later steps. Then note for each instruction the functional unit and/or issue port requirements. If an instruction can use either of two units, like U0 or U1 then make a category "U0/U1". Count the total using each unit (or combined unit), and count all instructions. Figure out from those counts the best possible loop time. The goal will be to find a perfect schedule where instruction latencies are completely hidden. The total instruction count might be the limiting factor, or perhaps a particular functional unit. It might be possible to tweak the instructions to help the limiting factor. Suppose the loop time is N, then make N issue buckets, with the final loop branch at the end of the last. Now fill the buckets with dummy instructions using the functional units desired. Run this to make sure the intended speed is reached. Now replace the dummy instructions with the real instructions from the slow but correct loop you started with. The first will typically be a load instruction. Then the instruction using that value is placed in a bucket an appropriate distance down. Run the loop again, to check it still runs at target speed. Keep placing instructions, frequently measuring the loop. After a few you will need to wrap around from the last bucket back to the top of the loop. If you used the new-register for new-value strategy above then there will be no register conflicts. If not then take care not to clobber something already in use. Changing registers at this time is very error prone. The loop will overlap two or more of the original loop iterations, and the computation of one vector element result will be started in one iteration of the new loop, and completed one or several iterations later. The final step is to create feed-in and wind-down code for the loop. A good way to do this is to make a copy (or copies) of the loop at the start and delete those instructions which don't have valid antecedents, and at the end replicate and delete those whose results are unwanted (including any further loads). The loop will have a minimum number of limbs loaded and processed, so the feed-in code must test if the request size is smaller and skip either to a suitable part of the wind-down or to special code for small sizes.  File: gmp.info, Node: Internals, Next: Contributors, Prev: Algorithms, Up: Top 17 Internals ************ *This chapter is provided only for informational purposes and the various internals described here may change in future GMP releases. Applications expecting to be compatible with future releases should use only the documented interfaces described in previous chapters.* * Menu: * Integer Internals:: * Rational Internals:: * Float Internals:: * Raw Output Internals:: * C++ Interface Internals::  File: gmp.info, Node: Integer Internals, Next: Rational Internals, Prev: Internals, Up: Internals 17.1 Integer Internals ====================== `mpz_t' variables represent integers using sign and magnitude, in space dynamically allocated and reallocated. The fields are as follows. `_mp_size' The number of limbs, or the negative of that when representing a negative integer. Zero is represented by `_mp_size' set to zero, in which case the `_mp_d' data is unused. `_mp_d' A pointer to an array of limbs which is the magnitude. These are stored "little endian" as per the `mpn' functions, so `_mp_d[0]' is the least significant limb and `_mp_d[ABS(_mp_size)-1]' is the most significant. Whenever `_mp_size' is non-zero, the most significant limb is non-zero. Currently there's always at least one limb allocated, so for instance `mpz_set_ui' never needs to reallocate, and `mpz_get_ui' can fetch `_mp_d[0]' unconditionally (though its value is then only wanted if `_mp_size' is non-zero). `_mp_alloc' `_mp_alloc' is the number of limbs currently allocated at `_mp_d', and naturally `_mp_alloc >= ABS(_mp_size)'. When an `mpz' routine is about to (or might be about to) increase `_mp_size', it checks `_mp_alloc' to see whether there's enough space, and reallocates if not. `MPZ_REALLOC' is generally used for this. The various bitwise logical functions like `mpz_and' behave as if negative values were twos complement. But sign and magnitude is always used internally, and necessary adjustments are made during the calculations. Sometimes this isn't pretty, but sign and magnitude are best for other routines. Some internal temporary variables are setup with `MPZ_TMP_INIT' and these have `_mp_d' space obtained from `TMP_ALLOC' rather than the memory allocation functions. Care is taken to ensure that these are big enough that no reallocation is necessary (since it would have unpredictable consequences). `_mp_size' and `_mp_alloc' are `int', although `mp_size_t' is usually a `long'. This is done to make the fields just 32 bits on some 64 bits systems, thereby saving a few bytes of data space but still providing plenty of range.  File: gmp.info, Node: Rational Internals, Next: Float Internals, Prev: Integer Internals, Up: Internals 17.2 Rational Internals ======================= `mpq_t' variables represent rationals using an `mpz_t' numerator and denominator (*note Integer Internals::). The canonical form adopted is denominator positive (and non-zero), no common factors between numerator and denominator, and zero uniquely represented as 0/1. It's believed that casting out common factors at each stage of a calculation is best in general. A GCD is an O(N^2) operation so it's better to do a few small ones immediately than to delay and have to do a big one later. Knowing the numerator and denominator have no common factors can be used for example in `mpq_mul' to make only two cross GCDs necessary, not four. This general approach to common factors is badly sub-optimal in the presence of simple factorizations or little prospect for cancellation, but GMP has no way to know when this will occur. As per *Note Efficiency::, that's left to applications. The `mpq_t' framework might still suit, with `mpq_numref' and `mpq_denref' for direct access to the numerator and denominator, or of course `mpz_t' variables can be used directly.  File: gmp.info, Node: Float Internals, Next: Raw Output Internals, Prev: Rational Internals, Up: Internals 17.3 Float Internals ==================== Efficient calculation is the primary aim of GMP floats and the use of whole limbs and simple rounding facilitates this. `mpf_t' floats have a variable precision mantissa and a single machine word signed exponent. The mantissa is represented using sign and magnitude. most least significant significant limb limb _mp_d |---- _mp_exp ---> | _____ _____ _____ _____ _____ |_____|_____|_____|_____|_____| . <------------ radix point <-------- _mp_size ---------> The fields are as follows. `_mp_size' The number of limbs currently in use, or the negative of that when representing a negative value. Zero is represented by `_mp_size' and `_mp_exp' both set to zero, and in that case the `_mp_d' data is unused. (In the future `_mp_exp' might be undefined when representing zero.) `_mp_prec' The precision of the mantissa, in limbs. In any calculation the aim is to produce `_mp_prec' limbs of result (the most significant being non-zero). `_mp_d' A pointer to the array of limbs which is the absolute value of the mantissa. These are stored "little endian" as per the `mpn' functions, so `_mp_d[0]' is the least significant limb and `_mp_d[ABS(_mp_size)-1]' the most significant. The most significant limb is always non-zero, but there are no other restrictions on its value, in particular the highest 1 bit can be anywhere within the limb. `_mp_prec+1' limbs are allocated to `_mp_d', the extra limb being for convenience (see below). There are no reallocations during a calculation, only in a change of precision with `mpf_set_prec'. `_mp_exp' The exponent, in limbs, determining the location of the implied radix point. Zero means the radix point is just above the most significant limb. Positive values mean a radix point offset towards the lower limbs and hence a value >= 1, as for example in the diagram above. Negative exponents mean a radix point further above the highest limb. Naturally the exponent can be any value, it doesn't have to fall within the limbs as the diagram shows, it can be a long way above or a long way below. Limbs other than those included in the `{_mp_d,_mp_size}' data are treated as zero. The `_mp_size' and `_mp_prec' fields are `int', although the `mp_size_t' type is usually a `long'. The `_mp_exp' field is usually `long'. This is done to make some fields just 32 bits on some 64 bits systems, thereby saving a few bytes of data space but still providing plenty of precision and a very large range. The following various points should be noted. Low Zeros The least significant limbs `_mp_d[0]' etc can be zero, though such low zeros can always be ignored. Routines likely to produce low zeros check and avoid them to save time in subsequent calculations, but for most routines they're quite unlikely and aren't checked. Mantissa Size Range The `_mp_size' count of limbs in use can be less than `_mp_prec' if the value can be represented in less. This means low precision values or small integers stored in a high precision `mpf_t' can still be operated on efficiently. `_mp_size' can also be greater than `_mp_prec'. Firstly a value is allowed to use all of the `_mp_prec+1' limbs available at `_mp_d', and secondly when `mpf_set_prec_raw' lowers `_mp_prec' it leaves `_mp_size' unchanged and so the size can be arbitrarily bigger than `_mp_prec'. Rounding All rounding is done on limb boundaries. Calculating `_mp_prec' limbs with the high non-zero will ensure the application requested minimum precision is obtained. The use of simple "trunc" rounding towards zero is efficient, since there's no need to examine extra limbs and increment or decrement. Bit Shifts Since the exponent is in limbs, there are no bit shifts in basic operations like `mpf_add' and `mpf_mul'. When differing exponents are encountered all that's needed is to adjust pointers to line up the relevant limbs. Of course `mpf_mul_2exp' and `mpf_div_2exp' will require bit shifts, but the choice is between an exponent in limbs which requires shifts there, or one in bits which requires them almost everywhere else. Use of `_mp_prec+1' Limbs The extra limb on `_mp_d' (`_mp_prec+1' rather than just `_mp_prec') helps when an `mpf' routine might get a carry from its operation. `mpf_add' for instance will do an `mpn_add' of `_mp_prec' limbs. If there's no carry then that's the result, but if there is a carry then it's stored in the extra limb of space and `_mp_size' becomes `_mp_prec+1'. Whenever `_mp_prec+1' limbs are held in a variable, the low limb is not needed for the intended precision, only the `_mp_prec' high limbs. But zeroing it out or moving the rest down is unnecessary. Subsequent routines reading the value will simply take the high limbs they need, and this will be `_mp_prec' if their target has that same precision. This is no more than a pointer adjustment, and must be checked anyway since the destination precision can be different from the sources. Copy functions like `mpf_set' will retain a full `_mp_prec+1' limbs if available. This ensures that a variable which has `_mp_size' equal to `_mp_prec+1' will get its full exact value copied. Strictly speaking this is unnecessary since only `_mp_prec' limbs are needed for the application's requested precision, but it's considered that an `mpf_set' from one variable into another of the same precision ought to produce an exact copy. Application Precisions `__GMPF_BITS_TO_PREC' converts an application requested precision to an `_mp_prec'. The value in bits is rounded up to a whole limb then an extra limb is added since the most significant limb of `_mp_d' is only non-zero and therefore might contain only one bit. `__GMPF_PREC_TO_BITS' does the reverse conversion, and removes the extra limb from `_mp_prec' before converting to bits. The net effect of reading back with `mpf_get_prec' is simply the precision rounded up to a multiple of `mp_bits_per_limb'. Note that the extra limb added here for the high only being non-zero is in addition to the extra limb allocated to `_mp_d'. For example with a 32-bit limb, an application request for 250 bits will be rounded up to 8 limbs, then an extra added for the high being only non-zero, giving an `_mp_prec' of 9. `_mp_d' then gets 10 limbs allocated. Reading back with `mpf_get_prec' will take `_mp_prec' subtract 1 limb and multiply by 32, giving 256 bits. Strictly speaking, the fact the high limb has at least one bit means that a float with, say, 3 limbs of 32-bits each will be holding at least 65 bits, but for the purposes of `mpf_t' it's considered simply to be 64 bits, a nice multiple of the limb size.  File: gmp.info, Node: Raw Output Internals, Next: C++ Interface Internals, Prev: Float Internals, Up: Internals 17.4 Raw Output Internals ========================= `mpz_out_raw' uses the following format. +------+------------------------+ | size | data bytes | +------+------------------------+ The size is 4 bytes written most significant byte first, being the number of subsequent data bytes, or the twos complement negative of that when a negative integer is represented. The data bytes are the absolute value of the integer, written most significant byte first. The most significant data byte is always non-zero, so the output is the same on all systems, irrespective of limb size. In GMP 1, leading zero bytes were written to pad the data bytes to a multiple of the limb size. `mpz_inp_raw' will still accept this, for compatibility. The use of "big endian" for both the size and data fields is deliberate, it makes the data easy to read in a hex dump of a file. Unfortunately it also means that the limb data must be reversed when reading or writing, so neither a big endian nor little endian system can just read and write `_mp_d'.  File: gmp.info, Node: C++ Interface Internals, Prev: Raw Output Internals, Up: Internals 17.5 C++ Interface Internals ============================ A system of expression templates is used to ensure something like `a=b+c' turns into a simple call to `mpz_add' etc. For `mpf_class' the scheme also ensures the precision of the final destination is used for any temporaries within a statement like `f=w*x+y*z'. These are important features which a naive implementation cannot provide. A simplified description of the scheme follows. The true scheme is complicated by the fact that expressions have different return types. For detailed information, refer to the source code. To perform an operation, say, addition, we first define a "function object" evaluating it, struct __gmp_binary_plus { static void eval(mpf_t f, mpf_t g, mpf_t h) { mpf_add(f, g, h); } }; And an "additive expression" object, __gmp_expr<__gmp_binary_expr > operator+(const mpf_class &f, const mpf_class &g) { return __gmp_expr <__gmp_binary_expr >(f, g); } The seemingly redundant `__gmp_expr<__gmp_binary_expr<...>>' is used to encapsulate any possible kind of expression into a single template type. In fact even `mpf_class' etc are `typedef' specializations of `__gmp_expr'. Next we define assignment of `__gmp_expr' to `mpf_class'. template mpf_class & mpf_class::operator=(const __gmp_expr &expr) { expr.eval(this->get_mpf_t(), this->precision()); return *this; } template void __gmp_expr<__gmp_binary_expr >::eval (mpf_t f, mp_bitcnt_t precision) { Op::eval(f, expr.val1.get_mpf_t(), expr.val2.get_mpf_t()); } where `expr.val1' and `expr.val2' are references to the expression's operands (here `expr' is the `__gmp_binary_expr' stored within the `__gmp_expr'). This way, the expression is actually evaluated only at the time of assignment, when the required precision (that of `f') is known. Furthermore the target `mpf_t' is now available, thus we can call `mpf_add' directly with `f' as the output argument. Compound expressions are handled by defining operators taking subexpressions as their arguments, like this: template __gmp_expr <__gmp_binary_expr<__gmp_expr, __gmp_expr, __gmp_binary_plus> > operator+(const __gmp_expr &expr1, const __gmp_expr &expr2) { return __gmp_expr <__gmp_binary_expr<__gmp_expr, __gmp_expr, __gmp_binary_plus> > (expr1, expr2); } And the corresponding specializations of `__gmp_expr::eval': template void __gmp_expr <__gmp_binary_expr<__gmp_expr, __gmp_expr, Op> >::eval (mpf_t f, mp_bitcnt_t precision) { // declare two temporaries mpf_class temp1(expr.val1, precision), temp2(expr.val2, precision); Op::eval(f, temp1.get_mpf_t(), temp2.get_mpf_t()); } The expression is thus recursively evaluated to any level of complexity and all subexpressions are evaluated to the precision of `f'.  File: gmp.info, Node: Contributors, Next: References, Prev: Internals, Up: Top Appendix A Contributors *********************** Torbjo"rn Granlund wrote the original GMP library and is still the main developer. Code not explicitly attributed to others, was contributed by Torbjo"rn. Several other individuals and organizations have contributed GMP. Here is a list in chronological order on first contribution: Gunnar Sjo"din and Hans Riesel helped with mathematical problems in early versions of the library. Richard Stallman helped with the interface design and revised the first version of this manual. Brian Beuning and Doug Lea helped with testing of early versions of the library and made creative suggestions. John Amanatides of York University in Canada contributed the function `mpz_probab_prime_p'. Paul Zimmermann wrote the REDC-based mpz_powm code, the Scho"nhage-Strassen FFT multiply code, and the Karatsuba square root code. He also improved the Toom3 code for GMP 4.2. Paul sparked the development of GMP 2, with his comparisons between bignum packages. The ECMNET project Paul is organizing was a driving force behind many of the optimizations in GMP 3. Paul also wrote the new GMP 4.3 nth root code (with Torbjo"rn). Ken Weber (Kent State University, Universidade Federal do Rio Grande do Sul) contributed now defunct versions of `mpz_gcd', `mpz_divexact', `mpn_gcd', and `mpn_bdivmod', partially supported by CNPq (Brazil) grant 301314194-2. Per Bothner of Cygnus Support helped to set up GMP to use Cygnus' configure. He has also made valuable suggestions and tested numerous intermediary releases. Joachim Hollman was involved in the design of the `mpf' interface, and in the `mpz' design revisions for version 2. Bennet Yee contributed the initial versions of `mpz_jacobi' and `mpz_legendre'. Andreas Schwab contributed the files `mpn/m68k/lshift.S' and `mpn/m68k/rshift.S' (now in `.asm' form). Robert Harley of Inria, France and David Seal of ARM, England, suggested clever improvements for population count. Robert also wrote highly optimized Karatsuba and 3-way Toom multiplication functions for GMP 3, and contributed the ARM assembly code. Torsten Ekedahl of the Mathematical department of Stockholm University provided significant inspiration during several phases of the GMP development. His mathematical expertise helped improve several algorithms. Linus Nordberg wrote the new configure system based on autoconf and implemented the new random functions. Kevin Ryde worked on a large number of things: optimized x86 code, m4 asm macros, parameter tuning, speed measuring, the configure system, function inlining, divisibility tests, bit scanning, Jacobi symbols, Fibonacci and Lucas number functions, printf and scanf functions, perl interface, demo expression parser, the algorithms chapter in the manual, `gmpasm-mode.el', and various miscellaneous improvements elsewhere. Kent Boortz made the Mac OS 9 port. Steve Root helped write the optimized alpha 21264 assembly code. Gerardo Ballabio wrote the `gmpxx.h' C++ class interface and the C++ `istream' input routines. Jason Moxham rewrote `mpz_fac_ui'. Pedro Gimeno implemented the Mersenne Twister and made other random number improvements. Niels Mo"ller wrote the sub-quadratic GCD and extended GCD code, the quadratic Hensel division code, and (with Torbjo"rn) the new divide and conquer division code for GMP 4.3. Niels also helped implement the new Toom multiply code for GMP 4.3 and implemented helper functions to simplify Toom evaluations for GMP 5.0. He wrote the original version of mpn_mulmod_bnm1. Alberto Zanoni and Marco Bodrato suggested the unbalanced multiply strategy, and found the optimal strategies for evaluation and interpolation in Toom multiplication. Marco Bodrato helped implement the new Toom multiply code for GMP 4.3 and implemented most of the new Toom multiply and squaring code for 5.0. He is the main author of the current mpn_mulmod_bnm1 and mpn_mullo_n. Marco also wrote the functions mpn_invert and mpn_invertappr. David Harvey suggested the internal function `mpn_bdiv_dbm1', implementing division relevant to Toom multiplication. He also worked on fast assembly sequences, in particular on a fast AMD64 `mpn_mul_basecase'. Martin Boij wrote `mpn_perfect_power_p'. (This list is chronological, not ordered after significance. If you have contributed to GMP but are not listed above, please tell about the omission!) The development of floating point functions of GNU MP 2, were supported in part by the ESPRIT-BRA (Basic Research Activities) 6846 project POSSO (POlynomial System SOlving). The development of GMP 2, 3, and 4 was supported in part by the IDA Center for Computing Sciences. Thanks go to Hans Thorsen for donating an SGI system for the GMP test system environment.  File: gmp.info, Node: References, Next: GNU Free Documentation License, Prev: Contributors, Up: Top Appendix B References ********************* B.1 Books ========= * Jonathan M. Borwein and Peter B. Borwein, "Pi and the AGM: A Study in Analytic Number Theory and Computational Complexity", Wiley, 1998. * Richard Crandall and Carl Pomerance, "Prime Numbers: A Computational Perspective", 2nd edition, Springer-Verlag, 2005. `http://math.dartmouth.edu/~carlp/' * Henri Cohen, "A Course in Computational Algebraic Number Theory", Graduate Texts in Mathematics number 138, Springer-Verlag, 1993. `http://www.math.u-bordeaux.fr/~cohen/' * Donald E. Knuth, "The Art of Computer Programming", volume 2, "Seminumerical Algorithms", 3rd edition, Addison-Wesley, 1998. `http://www-cs-faculty.stanford.edu/~knuth/taocp.html' * John D. Lipson, "Elements of Algebra and Algebraic Computing", The Benjamin Cummings Publishing Company Inc, 1981. * Alfred J. Menezes, Paul C. van Oorschot and Scott A. Vanstone, "Handbook of Applied Cryptography", `http://www.cacr.math.uwaterloo.ca/hac/' * Richard M. Stallman and the GCC Developer Community, "Using the GNU Compiler Collection", Free Software Foundation, 2008, available online `http://gcc.gnu.org/onlinedocs/', and in the GCC package `ftp://ftp.gnu.org/gnu/gcc/' B.2 Papers ========== * Yves Bertot, Nicolas Magaud and Paul Zimmermann, "A Proof of GMP Square Root", Journal of Automated Reasoning, volume 29, 2002, pp. 225-252. Also available online as INRIA Research Report 4475, June 2001, `http://www.inria.fr/rrrt/rr-4475.html' * Christoph Burnikel and Joachim Ziegler, "Fast Recursive Division", Max-Planck-Institut fuer Informatik Research Report MPI-I-98-1-022, `http://data.mpi-sb.mpg.de/internet/reports.nsf/NumberView/1998-1-022' * Torbjo"rn Granlund and Peter L. Montgomery, "Division by Invariant Integers using Multiplication", in Proceedings of the SIGPLAN PLDI'94 Conference, June 1994. Also available `ftp://ftp.cwi.nl/pub/pmontgom/divcnst.psa4.gz' (and .psl.gz). * Niels Mo"ller and Torbjo"rn Granlund, "Improved division by invariant integers", IEEE Transactions on Computers, 11 June 2010. `http://gmplib.org/~tege/division-paper.pdf' * Torbjo"rn Granlund and Niels Mo"ller, "Division of integers large and small", to appear. * Tudor Jebelean, "An algorithm for exact division", Journal of Symbolic Computation, volume 15, 1993, pp. 169-180. Research report version available `ftp://ftp.risc.uni-linz.ac.at/pub/techreports/1992/92-35.ps.gz' * Tudor Jebelean, "Exact Division with Karatsuba Complexity - Extended Abstract", RISC-Linz technical report 96-31, `ftp://ftp.risc.uni-linz.ac.at/pub/techreports/1996/96-31.ps.gz' * Tudor Jebelean, "Practical Integer Division with Karatsuba Complexity", ISSAC 97, pp. 339-341. Technical report available `ftp://ftp.risc.uni-linz.ac.at/pub/techreports/1996/96-29.ps.gz' * Tudor Jebelean, "A Generalization of the Binary GCD Algorithm", ISSAC 93, pp. 111-116. Technical report version available `ftp://ftp.risc.uni-linz.ac.at/pub/techreports/1993/93-01.ps.gz' * Tudor Jebelean, "A Double-Digit Lehmer-Euclid Algorithm for Finding the GCD of Long Integers", Journal of Symbolic Computation, volume 19, 1995, pp. 145-157. Technical report version also available `ftp://ftp.risc.uni-linz.ac.at/pub/techreports/1992/92-69.ps.gz' * Werner Krandick and Tudor Jebelean, "Bidirectional Exact Integer Division", Journal of Symbolic Computation, volume 21, 1996, pp. 441-455. Early technical report version also available `ftp://ftp.risc.uni-linz.ac.at/pub/techreports/1994/94-50.ps.gz' * Makoto Matsumoto and Takuji Nishimura, "Mersenne Twister: A 623-dimensionally equidistributed uniform pseudorandom number generator", ACM Transactions on Modelling and Computer Simulation, volume 8, January 1998, pp. 3-30. Available online `http://www.math.sci.hiroshima-u.ac.jp/~m-mat/MT/ARTICLES/mt.ps.gz' (or .pdf) * R. Moenck and A. Borodin, "Fast Modular Transforms via Division", Proceedings of the 13th Annual IEEE Symposium on Switching and Automata Theory, October 1972, pp. 90-96. Reprinted as "Fast Modular Transforms", Journal of Computer and System Sciences, volume 8, number 3, June 1974, pp. 366-386. * Niels Mo"ller, "On Scho"nhage's algorithm and subquadratic integer GCD computation", in Mathematics of Computation, volume 77, January 2008, pp. 589-607. * Peter L. Montgomery, "Modular Multiplication Without Trial Division", in Mathematics of Computation, volume 44, number 170, April 1985. * Arnold Scho"nhage and Volker Strassen, "Schnelle Multiplikation grosser Zahlen", Computing 7, 1971, pp. 281-292. * Kenneth Weber, "The accelerated integer GCD algorithm", ACM Transactions on Mathematical Software, volume 21, number 1, March 1995, pp. 111-122. * Paul Zimmermann, "Karatsuba Square Root", INRIA Research Report 3805, November 1999, `http://www.inria.fr/rrrt/rr-3805.html' * Paul Zimmermann, "A Proof of GMP Fast Division and Square Root Implementations", `http://www.loria.fr/~zimmerma/papers/proof-div-sqrt.ps.gz' * Dan Zuras, "On Squaring and Multiplying Large Integers", ARITH-11: IEEE Symposium on Computer Arithmetic, 1993, pp. 260 to 271. Reprinted as "More on Multiplying and Squaring Large Integers", IEEE Transactions on Computers, volume 43, number 8, August 1994, pp. 899-908.  File: gmp.info, Node: GNU Free Documentation License, Next: Concept Index, Prev: References, Up: Top Appendix C GNU Free Documentation License ***************************************** Version 1.3, 3 November 2008 Copyright (C) 2000, 2001, 2002, 2007, 2008 Free Software Foundation, Inc. `http://fsf.org/' Everyone is permitted to copy and distribute verbatim copies of this license document, but changing it is not allowed. 0. PREAMBLE The purpose of this License is to make a manual, textbook, or other functional and useful document "free" in the sense of freedom: to assure everyone the effective freedom to copy and redistribute it, with or without modifying it, either commercially or noncommercially. Secondarily, this License preserves for the author and publisher a way to get credit for their work, while not being considered responsible for modifications made by others. 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If you have Invariant Sections, Front-Cover Texts and Back-Cover Texts, replace the "with...Texts." line with this: with the Invariant Sections being LIST THEIR TITLES, with the Front-Cover Texts being LIST, and with the Back-Cover Texts being LIST. If you have Invariant Sections without Cover Texts, or some other combination of the three, merge those two alternatives to suit the situation. If your document contains nontrivial examples of program code, we recommend releasing these examples in parallel under your choice of free software license, such as the GNU General Public License, to permit their use in free software.  File: gmp.info, Node: Concept Index, Next: Function Index, Prev: GNU Free Documentation License, Up: Top Concept Index ************* [index] * Menu: * #include: Headers and Libraries. (line 6) * --build: Build Options. (line 52) * --disable-fft: Build Options. (line 317) * --disable-shared: Build Options. (line 45) * --disable-static: Build Options. (line 45) * --enable-alloca: Build Options. (line 278) * --enable-assert: Build Options. (line 327) * --enable-cxx: Build Options. (line 230) * --enable-fat: Build Options. (line 164) * --enable-mpbsd: Build Options. (line 322) * --enable-profiling <1>: Profiling. (line 6) * --enable-profiling: Build Options. (line 331) * --exec-prefix: Build Options. (line 32) * --host: Build Options. (line 66) * --prefix: Build Options. (line 32) * -finstrument-functions: Profiling. (line 66) * 2exp functions: Efficiency. (line 43) * 68000: Notes for Particular Systems. (line 80) * 80x86: Notes for Particular Systems. (line 126) * ABI <1>: Build Options. (line 171) * ABI: ABI and ISA. (line 6) * About this manual: Introduction to GMP. (line 58) * AC_CHECK_LIB: Autoconf. (line 11) * AIX <1>: ABI and ISA. (line 184) * AIX <2>: Notes for Particular Systems. (line 7) * AIX: ABI and ISA. (line 169) * Algorithms: Algorithms. (line 6) * alloca: Build Options. (line 278) * Allocation of memory: Custom Allocation. (line 6) * AMD64: ABI and ISA. (line 44) * Anonymous FTP of latest version: Introduction to GMP. (line 38) * Application Binary Interface: ABI and ISA. (line 6) * Arithmetic functions <1>: Float Arithmetic. (line 6) * Arithmetic functions <2>: Integer Arithmetic. (line 6) * Arithmetic functions: Rational Arithmetic. (line 6) * ARM: Notes for Particular Systems. (line 20) * Assembly cache handling: Assembly Cache Handling. (line 6) * Assembly carry propagation: Assembly Carry Propagation. (line 6) * Assembly code organisation: Assembly Code Organisation. (line 6) * Assembly coding: Assembly Coding. (line 6) * Assembly floating Point: Assembly Floating Point. (line 6) * Assembly loop unrolling: Assembly Loop Unrolling. (line 6) * Assembly SIMD: Assembly SIMD Instructions. (line 6) * Assembly software pipelining: Assembly Software Pipelining. (line 6) * Assembly writing guide: Assembly Writing Guide. (line 6) * Assertion checking <1>: Debugging. (line 79) * Assertion checking: Build Options. (line 327) * Assignment functions <1>: Assigning Floats. (line 6) * Assignment functions <2>: Initializing Rationals. (line 6) * Assignment functions <3>: Simultaneous Integer Init & Assign. (line 6) * Assignment functions <4>: Simultaneous Float Init & Assign. (line 6) * Assignment functions: Assigning Integers. (line 6) * Autoconf: Autoconf. (line 6) * Basics: GMP Basics. (line 6) * Berkeley MP compatible functions <1>: Build Options. (line 322) * Berkeley MP compatible functions: BSD Compatible Functions. (line 6) * Binomial coefficient algorithm: Binomial Coefficients Algorithm. (line 6) * Binomial coefficient functions: Number Theoretic Functions. (line 100) * Binutils strip: Known Build Problems. (line 28) * Bit manipulation functions: Integer Logic and Bit Fiddling. (line 6) * Bit scanning functions: Integer Logic and Bit Fiddling. (line 38) * Bit shift left: Integer Arithmetic. (line 35) * Bit shift right: Integer Division. (line 53) * Bits per limb: Useful Macros and Constants. (line 7) * BSD MP compatible functions <1>: Build Options. (line 322) * BSD MP compatible functions: BSD Compatible Functions. (line 6) * Bug reporting: Reporting Bugs. (line 6) * Build directory: Build Options. (line 19) * Build notes for binary packaging: Notes for Package Builds. (line 6) * Build notes for particular systems: Notes for Particular Systems. (line 6) * Build options: Build Options. (line 6) * Build problems known: Known Build Problems. (line 6) * Build system: Build Options. (line 52) * Building GMP: Installing GMP. (line 6) * Bus error: Debugging. (line 7) * C compiler: Build Options. (line 182) * C++ compiler: Build Options. (line 254) * C++ interface: C++ Class Interface. (line 6) * C++ interface internals: C++ Interface Internals. (line 6) * C++ istream input: C++ Formatted Input. (line 6) * C++ ostream output: C++ Formatted Output. (line 6) * C++ support: Build Options. (line 230) * CC: Build Options. (line 182) * CC_FOR_BUILD: Build Options. (line 217) * CFLAGS: Build Options. (line 182) * Checker: Debugging. (line 115) * checkergcc: Debugging. (line 122) * Code organisation: Assembly Code Organisation. (line 6) * Compaq C++: Notes for Particular Systems. (line 25) * Comparison functions <1>: Integer Comparisons. (line 6) * Comparison functions <2>: Comparing Rationals. (line 6) * Comparison functions: Float Comparison. (line 6) * Compatibility with older versions: Compatibility with older versions. (line 6) * Conditions for copying GNU MP: Copying. (line 6) * Configuring GMP: Installing GMP. (line 6) * Congruence algorithm: Exact Remainder. (line 29) * Congruence functions: Integer Division. (line 124) * Constants: Useful Macros and Constants. (line 6) * Contributors: Contributors. (line 6) * Conventions for parameters: Parameter Conventions. (line 6) * Conventions for variables: Variable Conventions. (line 6) * Conversion functions <1>: Converting Integers. (line 6) * Conversion functions <2>: Converting Floats. (line 6) * Conversion functions: Rational Conversions. (line 6) * Copying conditions: Copying. (line 6) * CPPFLAGS: Build Options. (line 208) * CPU types <1>: Introduction to GMP. (line 24) * CPU types: Build Options. (line 108) * Cross compiling: Build Options. (line 66) * Custom allocation: Custom Allocation. (line 6) * CXX: Build Options. (line 254) * CXXFLAGS: Build Options. (line 254) * Cygwin: Notes for Particular Systems. (line 43) * Darwin: Known Build Problems. (line 51) * Debugging: Debugging. (line 6) * Demonstration programs: Demonstration Programs. (line 6) * Digits in an integer: Miscellaneous Integer Functions. (line 23) * Divisibility algorithm: Exact Remainder. (line 29) * Divisibility functions: Integer Division. (line 124) * Divisibility testing: Efficiency. (line 91) * Division algorithms: Division Algorithms. (line 6) * Division functions <1>: Rational Arithmetic. (line 22) * Division functions <2>: Integer Division. (line 6) * Division functions: Float Arithmetic. (line 33) * DJGPP <1>: Notes for Particular Systems. (line 43) * DJGPP: Known Build Problems. (line 18) * DLLs: Notes for Particular Systems. (line 56) * DocBook: Build Options. (line 354) * Documentation formats: Build Options. (line 347) * Documentation license: GNU Free Documentation License. (line 6) * DVI: Build Options. (line 350) * Efficiency: Efficiency. (line 6) * Emacs: Emacs. (line 6) * Exact division functions: Integer Division. (line 102) * Exact remainder: Exact Remainder. (line 6) * Example programs: Demonstration Programs. (line 6) * Exec prefix: Build Options. (line 32) * Execution profiling <1>: Profiling. (line 6) * Execution profiling: Build Options. (line 331) * Exponentiation functions <1>: Integer Exponentiation. (line 6) * Exponentiation functions: Float Arithmetic. (line 41) * Export: Integer Import and Export. (line 45) * Expression parsing demo: Demonstration Programs. (line 18) * Extended GCD: Number Theoretic Functions. (line 45) * Factor removal functions: Number Theoretic Functions. (line 90) * Factorial algorithm: Factorial Algorithm. (line 6) * Factorial functions: Number Theoretic Functions. (line 95) * Factorization demo: Demonstration Programs. (line 25) * Fast Fourier Transform: FFT Multiplication. (line 6) * Fat binary: Build Options. (line 164) * FFT multiplication <1>: FFT Multiplication. (line 6) * FFT multiplication: Build Options. (line 317) * Fibonacci number algorithm: Fibonacci Numbers Algorithm. (line 6) * Fibonacci sequence functions: Number Theoretic Functions. (line 108) * Float arithmetic functions: Float Arithmetic. (line 6) * Float assignment functions <1>: Simultaneous Float Init & Assign. (line 6) * Float assignment functions: Assigning Floats. (line 6) * Float comparison functions: Float Comparison. (line 6) * Float conversion functions: Converting Floats. (line 6) * Float functions: Floating-point Functions. (line 6) * Float initialization functions <1>: Simultaneous Float Init & Assign. (line 6) * Float initialization functions: Initializing Floats. (line 6) * Float input and output functions: I/O of Floats. (line 6) * Float internals: Float Internals. (line 6) * Float miscellaneous functions: Miscellaneous Float Functions. (line 6) * Float random number functions: Miscellaneous Float Functions. (line 27) * Float rounding functions: Miscellaneous Float Functions. (line 9) * Float sign tests: Float Comparison. (line 33) * Floating point mode: Notes for Particular Systems. (line 34) * Floating-point functions: Floating-point Functions. (line 6) * Floating-point number: Nomenclature and Types. (line 21) * fnccheck: Profiling. (line 77) * Formatted input: Formatted Input. (line 6) * Formatted output: Formatted Output. (line 6) * Free Documentation License: GNU Free Documentation License. (line 6) * frexp <1>: Converting Floats. (line 23) * frexp: Converting Integers. (line 42) * FTP of latest version: Introduction to GMP. (line 38) * Function classes: Function Classes. (line 6) * FunctionCheck: Profiling. (line 77) * GCC Checker: Debugging. (line 115) * GCD algorithms: Greatest Common Divisor Algorithms. (line 6) * GCD extended: Number Theoretic Functions. (line 45) * GCD functions: Number Theoretic Functions. (line 30) * GDB: Debugging. (line 58) * Generic C: Build Options. (line 153) * GMP Perl module: Demonstration Programs. (line 35) * GMP version number: Useful Macros and Constants. (line 12) * gmp.h: Headers and Libraries. (line 6) * gmpxx.h: C++ Interface General. (line 8) * GNU Debugger: Debugging. (line 58) * GNU Free Documentation License: GNU Free Documentation License. (line 6) * GNU strip: Known Build Problems. (line 28) * gprof: Profiling. (line 41) * Greatest common divisor algorithms: Greatest Common Divisor Algorithms. (line 6) * Greatest common divisor functions: Number Theoretic Functions. (line 30) * Hardware floating point mode: Notes for Particular Systems. (line 34) * Headers: Headers and Libraries. (line 6) * Heap problems: Debugging. (line 24) * Home page: Introduction to GMP. (line 34) * Host system: Build Options. (line 66) * HP-UX: ABI and ISA. (line 107) * HPPA: ABI and ISA. (line 68) * I/O functions <1>: I/O of Integers. (line 6) * I/O functions <2>: I/O of Rationals. (line 6) * I/O functions: I/O of Floats. (line 6) * i386: Notes for Particular Systems. (line 126) * IA-64: ABI and ISA. (line 107) * Import: Integer Import and Export. (line 11) * In-place operations: Efficiency. (line 57) * Include files: Headers and Libraries. (line 6) * info-lookup-symbol: Emacs. (line 6) * Initialization functions <1>: Initializing Integers. (line 6) * Initialization functions <2>: Initializing Rationals. (line 6) * Initialization functions <3>: Random State Initialization. (line 6) * Initialization functions <4>: Simultaneous Float Init & Assign. (line 6) * Initialization functions <5>: Simultaneous Integer Init & Assign. (line 6) * Initialization functions: Initializing Floats. (line 6) * Initializing and clearing: Efficiency. (line 21) * Input functions <1>: I/O of Integers. (line 6) * Input functions <2>: I/O of Rationals. (line 6) * Input functions <3>: I/O of Floats. (line 6) * Input functions: Formatted Input Functions. (line 6) * Install prefix: Build Options. (line 32) * Installing GMP: Installing GMP. (line 6) * Instruction Set Architecture: ABI and ISA. (line 6) * instrument-functions: Profiling. (line 66) * Integer: Nomenclature and Types. (line 6) * Integer arithmetic functions: Integer Arithmetic. (line 6) * Integer assignment functions <1>: Simultaneous Integer Init & Assign. (line 6) * Integer assignment functions: Assigning Integers. (line 6) * Integer bit manipulation functions: Integer Logic and Bit Fiddling. (line 6) * Integer comparison functions: Integer Comparisons. (line 6) * Integer conversion functions: Converting Integers. (line 6) * Integer division functions: Integer Division. (line 6) * Integer exponentiation functions: Integer Exponentiation. (line 6) * Integer export: Integer Import and Export. (line 45) * Integer functions: Integer Functions. (line 6) * Integer import: Integer Import and Export. (line 11) * Integer initialization functions <1>: Simultaneous Integer Init & Assign. (line 6) * Integer initialization functions: Initializing Integers. (line 6) * Integer input and output functions: I/O of Integers. (line 6) * Integer internals: Integer Internals. (line 6) * Integer logical functions: Integer Logic and Bit Fiddling. (line 6) * Integer miscellaneous functions: Miscellaneous Integer Functions. (line 6) * Integer random number functions: Integer Random Numbers. (line 6) * Integer root functions: Integer Roots. (line 6) * Integer sign tests: Integer Comparisons. (line 28) * Integer special functions: Integer Special Functions. (line 6) * Interix: Notes for Particular Systems. (line 51) * Internals: Internals. (line 6) * Introduction: Introduction to GMP. (line 6) * Inverse modulo functions: Number Theoretic Functions. (line 60) * IRIX <1>: Known Build Problems. (line 38) * IRIX: ABI and ISA. (line 132) * ISA: ABI and ISA. (line 6) * istream input: C++ Formatted Input. (line 6) * Jacobi symbol algorithm: Jacobi Symbol. (line 6) * Jacobi symbol functions: Number Theoretic Functions. (line 66) * Karatsuba multiplication: Karatsuba Multiplication. (line 6) * Karatsuba square root algorithm: Square Root Algorithm. (line 6) * Kronecker symbol functions: Number Theoretic Functions. (line 78) * Language bindings: Language Bindings. (line 6) * Latest version of GMP: Introduction to GMP. (line 38) * LCM functions: Number Theoretic Functions. (line 55) * Least common multiple functions: Number Theoretic Functions. (line 55) * Legendre symbol functions: Number Theoretic Functions. (line 69) * libgmp: Headers and Libraries. (line 22) * libgmpxx: Headers and Libraries. (line 27) * Libraries: Headers and Libraries. (line 22) * Libtool: Headers and Libraries. (line 33) * Libtool versioning: Notes for Package Builds. (line 9) * License conditions: Copying. (line 6) * Limb: Nomenclature and Types. (line 31) * Limb size: Useful Macros and Constants. (line 7) * Linear congruential algorithm: Random Number Algorithms. (line 25) * Linear congruential random numbers: Random State Initialization. (line 32) * Linking: Headers and Libraries. (line 22) * Logical functions: Integer Logic and Bit Fiddling. (line 6) * Low-level functions: Low-level Functions. (line 6) * Lucas number algorithm: Lucas Numbers Algorithm. (line 6) * Lucas number functions: Number Theoretic Functions. (line 119) * MacOS X: Known Build Problems. (line 51) * Mailing lists: Introduction to GMP. (line 45) * Malloc debugger: Debugging. (line 30) * Malloc problems: Debugging. (line 24) * Memory allocation: Custom Allocation. (line 6) * Memory management: Memory Management. (line 6) * Mersenne twister algorithm: Random Number Algorithms. (line 17) * Mersenne twister random numbers: Random State Initialization. (line 13) * MINGW: Notes for Particular Systems. (line 43) * MIPS: ABI and ISA. (line 132) * Miscellaneous float functions: Miscellaneous Float Functions. (line 6) * Miscellaneous integer functions: Miscellaneous Integer Functions. (line 6) * MMX: Notes for Particular Systems. (line 132) * Modular inverse functions: Number Theoretic Functions. (line 60) * Most significant bit: Miscellaneous Integer Functions. (line 34) * mp.h: BSD Compatible Functions. (line 21) * MPN_PATH: Build Options. (line 335) * MS Windows: Notes for Particular Systems. (line 56) * MS-DOS: Notes for Particular Systems. (line 43) * Multi-threading: Reentrancy. (line 6) * Multiplication algorithms: Multiplication Algorithms. (line 6) * Nails: Low-level Functions. (line 479) * Native compilation: Build Options. (line 52) * NeXT: Known Build Problems. (line 57) * Next prime function: Number Theoretic Functions. (line 23) * Nomenclature: Nomenclature and Types. (line 6) * Non-Unix systems: Build Options. (line 11) * Nth root algorithm: Nth Root Algorithm. (line 6) * Number sequences: Efficiency. (line 147) * Number theoretic functions: Number Theoretic Functions. (line 6) * Numerator and denominator: Applying Integer Functions. (line 6) * obstack output: Formatted Output Functions. (line 81) * OpenBSD: Notes for Particular Systems. (line 86) * Optimizing performance: Performance optimization. (line 6) * ostream output: C++ Formatted Output. (line 6) * Other languages: Language Bindings. (line 6) * Output functions <1>: I/O of Floats. (line 6) * Output functions <2>: I/O of Rationals. (line 6) * Output functions <3>: Formatted Output Functions. (line 6) * Output functions: I/O of Integers. (line 6) * Packaged builds: Notes for Package Builds. (line 6) * Parameter conventions: Parameter Conventions. (line 6) * Parsing expressions demo: Demonstration Programs. (line 21) * Particular systems: Notes for Particular Systems. (line 6) * Past GMP versions: Compatibility with older versions. (line 6) * PDF: Build Options. (line 350) * Perfect power algorithm: Perfect Power Algorithm. (line 6) * Perfect power functions: Integer Roots. (line 27) * Perfect square algorithm: Perfect Square Algorithm. (line 6) * Perfect square functions: Integer Roots. (line 36) * perl: Demonstration Programs. (line 35) * Perl module: Demonstration Programs. (line 35) * Postscript: Build Options. (line 350) * Power/PowerPC <1>: Known Build Problems. (line 63) * Power/PowerPC: Notes for Particular Systems. (line 92) * Powering algorithms: Powering Algorithms. (line 6) * Powering functions <1>: Float Arithmetic. (line 41) * Powering functions: Integer Exponentiation. (line 6) * PowerPC: ABI and ISA. (line 167) * Precision of floats: Floating-point Functions. (line 6) * Precision of hardware floating point: Notes for Particular Systems. (line 34) * Prefix: Build Options. (line 32) * Prime testing algorithms: Prime Testing Algorithm. (line 6) * Prime testing functions: Number Theoretic Functions. (line 7) * printf formatted output: Formatted Output. (line 6) * Probable prime testing functions: Number Theoretic Functions. (line 7) * prof: Profiling. (line 24) * Profiling: Profiling. (line 6) * Radix conversion algorithms: Radix Conversion Algorithms. (line 6) * Random number algorithms: Random Number Algorithms. (line 6) * Random number functions <1>: Integer Random Numbers. (line 6) * Random number functions <2>: Miscellaneous Float Functions. (line 27) * Random number functions: Random Number Functions. (line 6) * Random number seeding: Random State Seeding. (line 6) * Random number state: Random State Initialization. (line 6) * Random state: Nomenclature and Types. (line 46) * Rational arithmetic: Efficiency. (line 113) * Rational arithmetic functions: Rational Arithmetic. (line 6) * Rational assignment functions: Initializing Rationals. (line 6) * Rational comparison functions: Comparing Rationals. (line 6) * Rational conversion functions: Rational Conversions. (line 6) * Rational initialization functions: Initializing Rationals. (line 6) * Rational input and output functions: I/O of Rationals. (line 6) * Rational internals: Rational Internals. (line 6) * Rational number: Nomenclature and Types. (line 16) * Rational number functions: Rational Number Functions. (line 6) * Rational numerator and denominator: Applying Integer Functions. (line 6) * Rational sign tests: Comparing Rationals. (line 27) * Raw output internals: Raw Output Internals. (line 6) * Reallocations: Efficiency. (line 30) * Reentrancy: Reentrancy. (line 6) * References: References. (line 6) * Remove factor functions: Number Theoretic Functions. (line 90) * Reporting bugs: Reporting Bugs. (line 6) * Root extraction algorithm: Nth Root Algorithm. (line 6) * Root extraction algorithms: Root Extraction Algorithms. (line 6) * Root extraction functions <1>: Float Arithmetic. (line 37) * Root extraction functions: Integer Roots. (line 6) * Root testing functions: Integer Roots. (line 36) * Rounding functions: Miscellaneous Float Functions. (line 9) * Sample programs: Demonstration Programs. (line 6) * Scan bit functions: Integer Logic and Bit Fiddling. (line 38) * scanf formatted input: Formatted Input. (line 6) * SCO: Known Build Problems. (line 38) * Seeding random numbers: Random State Seeding. (line 6) * Segmentation violation: Debugging. (line 7) * Sequent Symmetry: Known Build Problems. (line 68) * Services for Unix: Notes for Particular Systems. (line 51) * Shared library versioning: Notes for Package Builds. (line 9) * Sign tests <1>: Float Comparison. (line 33) * Sign tests <2>: Integer Comparisons. (line 28) * Sign tests: Comparing Rationals. (line 27) * Size in digits: Miscellaneous Integer Functions. (line 23) * Small operands: Efficiency. (line 7) * Solaris <1>: ABI and ISA. (line 201) * Solaris: Known Build Problems. (line 78) * Sparc: Notes for Particular Systems. (line 108) * Sparc V9: ABI and ISA. (line 201) * Special integer functions: Integer Special Functions. (line 6) * Square root algorithm: Square Root Algorithm. (line 6) * SSE2: Notes for Particular Systems. (line 132) * Stack backtrace: Debugging. (line 50) * Stack overflow <1>: Debugging. (line 7) * Stack overflow: Build Options. (line 278) * Static linking: Efficiency. (line 14) * stdarg.h: Headers and Libraries. (line 17) * stdio.h: Headers and Libraries. (line 11) * Stripped libraries: Known Build Problems. (line 28) * Sun: ABI and ISA. (line 201) * SunOS: Notes for Particular Systems. (line 120) * Systems: Notes for Particular Systems. (line 6) * Temporary memory: Build Options. (line 278) * Texinfo: Build Options. (line 347) * Text input/output: Efficiency. (line 153) * Thread safety: Reentrancy. (line 6) * Toom multiplication <1>: Other Multiplication. (line 6) * Toom multiplication <2>: Toom 4-Way Multiplication. (line 6) * Toom multiplication: Toom 3-Way Multiplication. (line 6) * Types: Nomenclature and Types. (line 6) * ui and si functions: Efficiency. (line 50) * Unbalanced multiplication: Unbalanced Multiplication. (line 6) * Upward compatibility: Compatibility with older versions. (line 6) * Useful macros and constants: Useful Macros and Constants. (line 6) * User-defined precision: Floating-point Functions. (line 6) * Valgrind: Debugging. (line 130) * Variable conventions: Variable Conventions. (line 6) * Version number: Useful Macros and Constants. (line 12) * Web page: Introduction to GMP. (line 34) * Windows: Notes for Particular Systems. (line 56) * x86: Notes for Particular Systems. (line 126) * x87: Notes for Particular Systems. (line 34) * XML: Build Options. (line 354)  File: gmp.info, Node: Function Index, Prev: Concept Index, Up: Top Function and Type Index *********************** [index] * Menu: * __GMP_CC: Useful Macros and Constants. (line 23) * __GMP_CFLAGS: Useful Macros and Constants. (line 24) * __GNU_MP_VERSION: Useful Macros and Constants. (line 10) * __GNU_MP_VERSION_MINOR: Useful Macros and Constants. (line 11) * __GNU_MP_VERSION_PATCHLEVEL: Useful Macros and Constants. (line 12) * _mpz_realloc: Integer Special Functions. (line 51) * abs <1>: C++ Interface Floats. (line 79) * abs <2>: C++ Interface Rationals. (line 43) * abs: C++ Interface Integers. (line 42) * ceil: C++ Interface Floats. (line 80) * cmp <1>: C++ Interface Floats. (line 81) * cmp <2>: C++ Interface Integers. (line 43) * cmp <3>: C++ Interface Floats. (line 82) * cmp <4>: C++ Interface Rationals. (line 45) * cmp: C++ Interface Integers. (line 44) * floor: C++ Interface Floats. (line 89) * gcd: BSD Compatible Functions. (line 82) * gmp_asprintf: Formatted Output Functions. (line 65) * gmp_errno: Random State Initialization. (line 55) * GMP_ERROR_INVALID_ARGUMENT: Random State Initialization. (line 55) * GMP_ERROR_UNSUPPORTED_ARGUMENT: Random State Initialization. (line 55) * gmp_fprintf: Formatted Output Functions. (line 29) * gmp_fscanf: Formatted Input Functions. (line 25) * GMP_LIMB_BITS: Low-level Functions. (line 509) * GMP_NAIL_BITS: Low-level Functions. (line 507) * GMP_NAIL_MASK: Low-level Functions. (line 517) * GMP_NUMB_BITS: Low-level Functions. (line 508) * GMP_NUMB_MASK: Low-level Functions. (line 518) * GMP_NUMB_MAX: Low-level Functions. (line 526) * gmp_obstack_printf: Formatted Output Functions. (line 79) * gmp_obstack_vprintf: Formatted Output Functions. (line 81) * gmp_printf: Formatted Output Functions. (line 24) * GMP_RAND_ALG_DEFAULT: Random State Initialization. (line 49) * GMP_RAND_ALG_LC: Random State Initialization. (line 49) * gmp_randclass: C++ Interface Random Numbers. (line 7) * gmp_randclass::get_f: C++ Interface Random Numbers. (line 45) * gmp_randclass::get_z_bits: C++ Interface Random Numbers. (line 38) * gmp_randclass::get_z_range: C++ Interface Random Numbers. (line 42) * gmp_randclass::gmp_randclass: C++ Interface Random Numbers. (line 27) * gmp_randclass::seed: C++ Interface Random Numbers. (line 34) * gmp_randclear: Random State Initialization. (line 62) * gmp_randinit: Random State Initialization. (line 47) * gmp_randinit_default: Random State Initialization. (line 7) * gmp_randinit_lc_2exp: Random State Initialization. (line 18) * gmp_randinit_lc_2exp_size: Random State Initialization. (line 32) * gmp_randinit_mt: Random State Initialization. (line 13) * gmp_randinit_set: Random State Initialization. (line 43) * gmp_randseed: Random State Seeding. (line 7) * gmp_randseed_ui: Random State Seeding. (line 9) * gmp_randstate_t: Nomenclature and Types. (line 46) * gmp_scanf: Formatted Input Functions. (line 21) * gmp_snprintf: Formatted Output Functions. (line 46) * gmp_sprintf: Formatted Output Functions. (line 34) * gmp_sscanf: Formatted Input Functions. (line 29) * gmp_urandomb_ui: Random State Miscellaneous. (line 8) * gmp_urandomm_ui: Random State Miscellaneous. (line 14) * gmp_vasprintf: Formatted Output Functions. (line 66) * gmp_version: Useful Macros and Constants. (line 18) * gmp_vfprintf: Formatted Output Functions. (line 30) * gmp_vfscanf: Formatted Input Functions. (line 26) * gmp_vprintf: Formatted Output Functions. (line 25) * gmp_vscanf: Formatted Input Functions. (line 22) * gmp_vsnprintf: Formatted Output Functions. (line 48) * gmp_vsprintf: Formatted Output Functions. (line 35) * gmp_vsscanf: Formatted Input Functions. (line 31) * hypot: C++ Interface Floats. (line 90) * itom: BSD Compatible Functions. (line 29) * madd: BSD Compatible Functions. (line 43) * mcmp: BSD Compatible Functions. (line 85) * mdiv: BSD Compatible Functions. (line 53) * mfree: BSD Compatible Functions. (line 105) * min: BSD Compatible Functions. (line 89) * MINT: BSD Compatible Functions. (line 21) * mout: BSD Compatible Functions. (line 94) * move: BSD Compatible Functions. (line 39) * mp_bitcnt_t: Nomenclature and Types. (line 42) * mp_bits_per_limb: Useful Macros and Constants. (line 7) * mp_exp_t: Nomenclature and Types. (line 27) * mp_get_memory_functions: Custom Allocation. (line 93) * mp_limb_t: Nomenclature and Types. (line 31) * mp_set_memory_functions: Custom Allocation. (line 21) * mp_size_t: Nomenclature and Types. (line 37) * mpf_abs: Float Arithmetic. (line 47) * mpf_add: Float Arithmetic. (line 7) * mpf_add_ui: Float Arithmetic. (line 9) * mpf_ceil: Miscellaneous Float Functions. (line 7) * mpf_class: C++ Interface General. (line 20) * mpf_class::fits_sint_p: C++ Interface Floats. (line 83) * mpf_class::fits_slong_p: C++ Interface Floats. (line 84) * mpf_class::fits_sshort_p: C++ Interface Floats. (line 85) * mpf_class::fits_uint_p: C++ Interface Floats. (line 86) * mpf_class::fits_ulong_p: C++ Interface Floats. (line 87) * mpf_class::fits_ushort_p: C++ Interface Floats. (line 88) * mpf_class::get_d: C++ Interface Floats. (line 91) * mpf_class::get_mpf_t: C++ Interface General. (line 66) * mpf_class::get_prec: C++ Interface Floats. (line 109) * mpf_class::get_si: C++ Interface Floats. (line 92) * mpf_class::get_str: C++ Interface Floats. (line 94) * mpf_class::get_ui: C++ Interface Floats. (line 95) * mpf_class::mpf_class: C++ Interface Floats. (line 12) * mpf_class::operator=: C++ Interface Floats. (line 56) * mpf_class::set_prec: C++ Interface Floats. (line 110) * mpf_class::set_prec_raw: C++ Interface Floats. (line 111) * mpf_class::set_str: C++ Interface Floats. (line 97) * mpf_clear: Initializing Floats. (line 37) * mpf_clears: Initializing Floats. (line 41) * mpf_cmp: Float Comparison. (line 7) * mpf_cmp_d: Float Comparison. (line 8) * mpf_cmp_si: Float Comparison. (line 10) * mpf_cmp_ui: Float Comparison. (line 9) * mpf_div: Float Arithmetic. (line 29) * mpf_div_2exp: Float Arithmetic. (line 53) * mpf_div_ui: Float Arithmetic. (line 33) * mpf_eq: Float Comparison. (line 17) * mpf_fits_sint_p: Miscellaneous Float Functions. (line 20) * mpf_fits_slong_p: Miscellaneous Float Functions. (line 18) * mpf_fits_sshort_p: Miscellaneous Float Functions. (line 22) * mpf_fits_uint_p: Miscellaneous Float Functions. (line 19) * mpf_fits_ulong_p: Miscellaneous Float Functions. (line 17) * mpf_fits_ushort_p: Miscellaneous Float Functions. (line 21) * mpf_floor: Miscellaneous Float Functions. (line 8) * mpf_get_d: Converting Floats. (line 7) * mpf_get_d_2exp: Converting Floats. (line 16) * mpf_get_default_prec: Initializing Floats. (line 12) * mpf_get_prec: Initializing Floats. (line 62) * mpf_get_si: Converting Floats. (line 27) * mpf_get_str: Converting Floats. (line 37) * mpf_get_ui: Converting Floats. (line 28) * mpf_init: Initializing Floats. (line 19) * mpf_init2: Initializing Floats. (line 26) * mpf_init_set: Simultaneous Float Init & Assign. (line 16) * mpf_init_set_d: Simultaneous Float Init & Assign. (line 19) * mpf_init_set_si: Simultaneous Float Init & Assign. (line 18) * mpf_init_set_str: Simultaneous Float Init & Assign. (line 25) * mpf_init_set_ui: Simultaneous Float Init & Assign. (line 17) * mpf_inits: Initializing Floats. (line 31) * mpf_inp_str: I/O of Floats. (line 39) * mpf_integer_p: Miscellaneous Float Functions. (line 14) * mpf_mul: Float Arithmetic. (line 19) * mpf_mul_2exp: Float Arithmetic. (line 50) * mpf_mul_ui: Float Arithmetic. (line 21) * mpf_neg: Float Arithmetic. (line 44) * mpf_out_str: I/O of Floats. (line 19) * mpf_pow_ui: Float Arithmetic. (line 41) * mpf_random2: Miscellaneous Float Functions. (line 37) * mpf_reldiff: Float Comparison. (line 29) * mpf_set: Assigning Floats. (line 10) * mpf_set_d: Assigning Floats. (line 13) * mpf_set_default_prec: Initializing Floats. (line 7) * mpf_set_prec: Initializing Floats. (line 65) * mpf_set_prec_raw: Initializing Floats. (line 72) * mpf_set_q: Assigning Floats. (line 15) * mpf_set_si: Assigning Floats. (line 12) * mpf_set_str: Assigning Floats. (line 18) * mpf_set_ui: Assigning Floats. (line 11) * mpf_set_z: Assigning Floats. (line 14) * mpf_sgn: Float Comparison. (line 33) * mpf_sqrt: Float Arithmetic. (line 36) * mpf_sqrt_ui: Float Arithmetic. (line 37) * mpf_sub: Float Arithmetic. (line 12) * mpf_sub_ui: Float Arithmetic. (line 16) * mpf_swap: Assigning Floats. (line 52) * mpf_t: Nomenclature and Types. (line 21) * mpf_trunc: Miscellaneous Float Functions. (line 9) * mpf_ui_div: Float Arithmetic. (line 31) * mpf_ui_sub: Float Arithmetic. (line 14) * mpf_urandomb: Miscellaneous Float Functions. (line 27) * mpn_add: Low-level Functions. (line 69) * mpn_add_1: Low-level Functions. (line 64) * mpn_add_n: Low-level Functions. (line 54) * mpn_addmul_1: Low-level Functions. (line 148) * mpn_and_n: Low-level Functions. (line 421) * mpn_andn_n: Low-level Functions. (line 436) * mpn_cmp: Low-level Functions. (line 284) * mpn_com: Low-level Functions. (line 461) * mpn_copyd: Low-level Functions. (line 470) * mpn_copyi: Low-level Functions. (line 466) * mpn_divexact_by3: Low-level Functions. (line 229) * mpn_divexact_by3c: Low-level Functions. (line 231) * mpn_divmod: Low-level Functions. (line 224) * mpn_divmod_1: Low-level Functions. (line 208) * mpn_divrem: Low-level Functions. (line 182) * mpn_divrem_1: Low-level Functions. (line 206) * mpn_gcd: Low-level Functions. (line 289) * mpn_gcd_1: Low-level Functions. (line 299) * mpn_gcdext: Low-level Functions. (line 305) * mpn_get_str: Low-level Functions. (line 346) * mpn_hamdist: Low-level Functions. (line 410) * mpn_ior_n: Low-level Functions. (line 426) * mpn_iorn_n: Low-level Functions. (line 441) * mpn_lshift: Low-level Functions. (line 260) * mpn_mod_1: Low-level Functions. (line 255) * mpn_mul: Low-level Functions. (line 114) * mpn_mul_1: Low-level Functions. (line 133) * mpn_mul_n: Low-level Functions. (line 103) * mpn_nand_n: Low-level Functions. (line 446) * mpn_neg: Low-level Functions. (line 98) * mpn_nior_n: Low-level Functions. (line 451) * mpn_perfect_square_p: Low-level Functions. (line 416) * mpn_popcount: Low-level Functions. (line 406) * mpn_random: Low-level Functions. (line 395) * mpn_random2: Low-level Functions. (line 396) * mpn_rshift: Low-level Functions. (line 272) * mpn_scan0: Low-level Functions. (line 380) * mpn_scan1: Low-level Functions. (line 388) * mpn_set_str: Low-level Functions. (line 361) * mpn_sqr: Low-level Functions. (line 125) * mpn_sqrtrem: Low-level Functions. (line 328) * mpn_sub: Low-level Functions. (line 90) * mpn_sub_1: Low-level Functions. (line 85) * mpn_sub_n: Low-level Functions. (line 76) * mpn_submul_1: Low-level Functions. (line 159) * mpn_tdiv_qr: Low-level Functions. (line 171) * mpn_xnor_n: Low-level Functions. (line 456) * mpn_xor_n: Low-level Functions. (line 431) * mpn_zero: Low-level Functions. (line 473) * mpq_abs: Rational Arithmetic. (line 31) * mpq_add: Rational Arithmetic. (line 7) * mpq_canonicalize: Rational Number Functions. (line 22) * mpq_class: C++ Interface General. (line 19) * mpq_class::canonicalize: C++ Interface Rationals. (line 37) * mpq_class::get_d: C++ Interface Rationals. (line 46) * mpq_class::get_den: C++ Interface Rationals. (line 58) * mpq_class::get_den_mpz_t: C++ Interface Rationals. (line 68) * mpq_class::get_mpq_t: C++ Interface General. (line 65) * mpq_class::get_num: C++ Interface Rationals. (line 57) * mpq_class::get_num_mpz_t: C++ Interface Rationals. (line 67) * mpq_class::get_str: C++ Interface Rationals. (line 47) * mpq_class::mpq_class: C++ Interface Rationals. (line 30) * mpq_class::set_str: C++ Interface Rationals. (line 48) * mpq_clear: Initializing Rationals. (line 16) * mpq_clears: Initializing Rationals. (line 20) * mpq_cmp: Comparing Rationals. (line 7) * mpq_cmp_si: Comparing Rationals. (line 17) * mpq_cmp_ui: Comparing Rationals. (line 15) * mpq_denref: Applying Integer Functions. (line 18) * mpq_div: Rational Arithmetic. (line 22) * mpq_div_2exp: Rational Arithmetic. (line 25) * mpq_equal: Comparing Rationals. (line 33) * mpq_get_d: Rational Conversions. (line 7) * mpq_get_den: Applying Integer Functions. (line 24) * mpq_get_num: Applying Integer Functions. (line 23) * mpq_get_str: Rational Conversions. (line 22) * mpq_init: Initializing Rationals. (line 7) * mpq_inits: Initializing Rationals. (line 12) * mpq_inp_str: I/O of Rationals. (line 26) * mpq_inv: Rational Arithmetic. (line 34) * mpq_mul: Rational Arithmetic. (line 15) * mpq_mul_2exp: Rational Arithmetic. (line 18) * mpq_neg: Rational Arithmetic. (line 28) * mpq_numref: Applying Integer Functions. (line 17) * mpq_out_str: I/O of Rationals. (line 18) * mpq_set: Initializing Rationals. (line 24) * mpq_set_d: Rational Conversions. (line 17) * mpq_set_den: Applying Integer Functions. (line 26) * mpq_set_f: Rational Conversions. (line 18) * mpq_set_num: Applying Integer Functions. (line 25) * mpq_set_si: Initializing Rationals. (line 31) * mpq_set_str: Initializing Rationals. (line 36) * mpq_set_ui: Initializing Rationals. (line 29) * mpq_set_z: Initializing Rationals. (line 25) * mpq_sgn: Comparing Rationals. (line 27) * mpq_sub: Rational Arithmetic. (line 11) * mpq_swap: Initializing Rationals. (line 56) * mpq_t: Nomenclature and Types. (line 16) * mpz_abs: Integer Arithmetic. (line 42) * mpz_add: Integer Arithmetic. (line 7) * mpz_add_ui: Integer Arithmetic. (line 9) * mpz_addmul: Integer Arithmetic. (line 25) * mpz_addmul_ui: Integer Arithmetic. (line 27) * mpz_and: Integer Logic and Bit Fiddling. (line 11) * mpz_array_init: Integer Special Functions. (line 11) * mpz_bin_ui: Number Theoretic Functions. (line 98) * mpz_bin_uiui: Number Theoretic Functions. (line 100) * mpz_cdiv_q: Integer Division. (line 13) * mpz_cdiv_q_2exp: Integer Division. (line 24) * mpz_cdiv_q_ui: Integer Division. (line 17) * mpz_cdiv_qr: Integer Division. (line 15) * mpz_cdiv_qr_ui: Integer Division. (line 21) * mpz_cdiv_r: Integer Division. (line 14) * mpz_cdiv_r_2exp: Integer Division. (line 25) * mpz_cdiv_r_ui: Integer Division. (line 19) * mpz_cdiv_ui: Integer Division. (line 23) * mpz_class: C++ Interface General. (line 18) * mpz_class::fits_sint_p: C++ Interface Integers. (line 45) * mpz_class::fits_slong_p: C++ Interface Integers. (line 46) * mpz_class::fits_sshort_p: C++ Interface Integers. (line 47) * mpz_class::fits_uint_p: C++ Interface Integers. (line 48) * mpz_class::fits_ulong_p: C++ Interface Integers. (line 49) * mpz_class::fits_ushort_p: C++ Interface Integers. (line 50) * mpz_class::get_d: C++ Interface Integers. (line 51) * mpz_class::get_mpz_t: C++ Interface General. (line 64) * mpz_class::get_si: C++ Interface Integers. (line 52) * mpz_class::get_str: C++ Interface Integers. (line 53) * mpz_class::get_ui: C++ Interface Integers. (line 54) * mpz_class::mpz_class: C++ Interface Integers. (line 20) * mpz_class::set_str: C++ Interface Integers. (line 55) * mpz_clear: Initializing Integers. (line 44) * mpz_clears: Initializing Integers. (line 48) * mpz_clrbit: Integer Logic and Bit Fiddling. (line 54) * mpz_cmp: Integer Comparisons. (line 7) * mpz_cmp_d: Integer Comparisons. (line 8) * mpz_cmp_si: Integer Comparisons. (line 9) * mpz_cmp_ui: Integer Comparisons. (line 10) * mpz_cmpabs: Integer Comparisons. (line 18) * mpz_cmpabs_d: Integer Comparisons. (line 19) * mpz_cmpabs_ui: Integer Comparisons. (line 20) * mpz_com: Integer Logic and Bit Fiddling. (line 20) * mpz_combit: Integer Logic and Bit Fiddling. (line 57) * mpz_congruent_2exp_p: Integer Division. (line 124) * mpz_congruent_p: Integer Division. (line 121) * mpz_congruent_ui_p: Integer Division. (line 123) * mpz_divexact: Integer Division. (line 101) * mpz_divexact_ui: Integer Division. (line 102) * mpz_divisible_2exp_p: Integer Division. (line 112) * mpz_divisible_p: Integer Division. (line 110) * mpz_divisible_ui_p: Integer Division. (line 111) * mpz_even_p: Miscellaneous Integer Functions. (line 18) * mpz_export: Integer Import and Export. (line 45) * mpz_fac_ui: Number Theoretic Functions. (line 95) * mpz_fdiv_q: Integer Division. (line 27) * mpz_fdiv_q_2exp: Integer Division. (line 38) * mpz_fdiv_q_ui: Integer Division. (line 31) * mpz_fdiv_qr: Integer Division. (line 29) * mpz_fdiv_qr_ui: Integer Division. (line 35) * mpz_fdiv_r: Integer Division. (line 28) * mpz_fdiv_r_2exp: Integer Division. (line 39) * mpz_fdiv_r_ui: Integer Division. (line 33) * mpz_fdiv_ui: Integer Division. (line 37) * mpz_fib2_ui: Number Theoretic Functions. (line 108) * mpz_fib_ui: Number Theoretic Functions. (line 106) * mpz_fits_sint_p: Miscellaneous Integer Functions. (line 10) * mpz_fits_slong_p: Miscellaneous Integer Functions. (line 8) * mpz_fits_sshort_p: Miscellaneous Integer Functions. (line 12) * mpz_fits_uint_p: Miscellaneous Integer Functions. (line 9) * mpz_fits_ulong_p: Miscellaneous Integer Functions. (line 7) * mpz_fits_ushort_p: Miscellaneous Integer Functions. (line 11) * mpz_gcd: Number Theoretic Functions. (line 30) * mpz_gcd_ui: Number Theoretic Functions. (line 35) * mpz_gcdext: Number Theoretic Functions. (line 45) * mpz_get_d: Converting Integers. (line 27) * mpz_get_d_2exp: Converting Integers. (line 35) * mpz_get_si: Converting Integers. (line 18) * mpz_get_str: Converting Integers. (line 46) * mpz_get_ui: Converting Integers. (line 11) * mpz_getlimbn: Integer Special Functions. (line 60) * mpz_hamdist: Integer Logic and Bit Fiddling. (line 29) * mpz_import: Integer Import and Export. (line 11) * mpz_init: Initializing Integers. (line 26) * mpz_init2: Initializing Integers. (line 33) * mpz_init_set: Simultaneous Integer Init & Assign. (line 27) * mpz_init_set_d: Simultaneous Integer Init & Assign. (line 30) * mpz_init_set_si: Simultaneous Integer Init & Assign. (line 29) * mpz_init_set_str: Simultaneous Integer Init & Assign. (line 34) * mpz_init_set_ui: Simultaneous Integer Init & Assign. (line 28) * mpz_inits: Initializing Integers. (line 29) * mpz_inp_raw: I/O of Integers. (line 61) * mpz_inp_str: I/O of Integers. (line 30) * mpz_invert: Number Theoretic Functions. (line 60) * mpz_ior: Integer Logic and Bit Fiddling. (line 14) * mpz_jacobi: Number Theoretic Functions. (line 66) * mpz_kronecker: Number Theoretic Functions. (line 74) * mpz_kronecker_si: Number Theoretic Functions. (line 75) * mpz_kronecker_ui: Number Theoretic Functions. (line 76) * mpz_lcm: Number Theoretic Functions. (line 54) * mpz_lcm_ui: Number Theoretic Functions. (line 55) * mpz_legendre: Number Theoretic Functions. (line 69) * mpz_lucnum2_ui: Number Theoretic Functions. (line 119) * mpz_lucnum_ui: Number Theoretic Functions. (line 117) * mpz_mod: Integer Division. (line 91) * mpz_mod_ui: Integer Division. (line 93) * mpz_mul: Integer Arithmetic. (line 19) * mpz_mul_2exp: Integer Arithmetic. (line 35) * mpz_mul_si: Integer Arithmetic. (line 20) * mpz_mul_ui: Integer Arithmetic. (line 22) * mpz_neg: Integer Arithmetic. (line 39) * mpz_nextprime: Number Theoretic Functions. (line 23) * mpz_odd_p: Miscellaneous Integer Functions. (line 17) * mpz_out_raw: I/O of Integers. (line 45) * mpz_out_str: I/O of Integers. (line 18) * mpz_perfect_power_p: Integer Roots. (line 27) * mpz_perfect_square_p: Integer Roots. (line 36) * mpz_popcount: Integer Logic and Bit Fiddling. (line 23) * mpz_pow_ui: Integer Exponentiation. (line 31) * mpz_powm: Integer Exponentiation. (line 8) * mpz_powm_sec: Integer Exponentiation. (line 18) * mpz_powm_ui: Integer Exponentiation. (line 10) * mpz_probab_prime_p: Number Theoretic Functions. (line 7) * mpz_random: Integer Random Numbers. (line 42) * mpz_random2: Integer Random Numbers. (line 51) * mpz_realloc2: Initializing Integers. (line 52) * mpz_remove: Number Theoretic Functions. (line 90) * mpz_root: Integer Roots. (line 7) * mpz_rootrem: Integer Roots. (line 13) * mpz_rrandomb: Integer Random Numbers. (line 31) * mpz_scan0: Integer Logic and Bit Fiddling. (line 37) * mpz_scan1: Integer Logic and Bit Fiddling. (line 38) * mpz_set: Assigning Integers. (line 10) * mpz_set_d: Assigning Integers. (line 13) * mpz_set_f: Assigning Integers. (line 15) * mpz_set_q: Assigning Integers. (line 14) * mpz_set_si: Assigning Integers. (line 12) * mpz_set_str: Assigning Integers. (line 21) * mpz_set_ui: Assigning Integers. (line 11) * mpz_setbit: Integer Logic and Bit Fiddling. (line 51) * mpz_sgn: Integer Comparisons. (line 28) * mpz_si_kronecker: Number Theoretic Functions. (line 77) * mpz_size: Integer Special Functions. (line 68) * mpz_sizeinbase: Miscellaneous Integer Functions. (line 23) * mpz_sqrt: Integer Roots. (line 17) * mpz_sqrtrem: Integer Roots. (line 20) * mpz_sub: Integer Arithmetic. (line 12) * mpz_sub_ui: Integer Arithmetic. (line 14) * mpz_submul: Integer Arithmetic. (line 30) * mpz_submul_ui: Integer Arithmetic. (line 32) * mpz_swap: Assigning Integers. (line 37) * mpz_t: Nomenclature and Types. (line 6) * mpz_tdiv_q: Integer Division. (line 41) * mpz_tdiv_q_2exp: Integer Division. (line 52) * mpz_tdiv_q_ui: Integer Division. (line 45) * mpz_tdiv_qr: Integer Division. (line 43) * mpz_tdiv_qr_ui: Integer Division. (line 49) * mpz_tdiv_r: Integer Division. (line 42) * mpz_tdiv_r_2exp: Integer Division. (line 53) * mpz_tdiv_r_ui: Integer Division. (line 47) * mpz_tdiv_ui: Integer Division. (line 51) * mpz_tstbit: Integer Logic and Bit Fiddling. (line 60) * mpz_ui_kronecker: Number Theoretic Functions. (line 78) * mpz_ui_pow_ui: Integer Exponentiation. (line 33) * mpz_ui_sub: Integer Arithmetic. (line 16) * mpz_urandomb: Integer Random Numbers. (line 14) * mpz_urandomm: Integer Random Numbers. (line 23) * mpz_xor: Integer Logic and Bit Fiddling. (line 17) * msqrt: BSD Compatible Functions. (line 63) * msub: BSD Compatible Functions. (line 46) * mtox: BSD Compatible Functions. (line 98) * mult: BSD Compatible Functions. (line 49) * operator%: C++ Interface Integers. (line 30) * operator/: C++ Interface Integers. (line 29) * operator<<: C++ Formatted Output. (line 11) * operator>> <1>: C++ Formatted Input. (line 11) * operator>> <2>: C++ Interface Rationals. (line 77) * operator>>: C++ Formatted Input. (line 14) * pow: BSD Compatible Functions. (line 71) * rpow: BSD Compatible Functions. (line 79) * sdiv: BSD Compatible Functions. (line 55) * sgn <1>: C++ Interface Rationals. (line 50) * sgn <2>: C++ Interface Integers. (line 57) * sgn: C++ Interface Floats. (line 98) * sqrt <1>: C++ Interface Floats. (line 99) * sqrt: C++ Interface Integers. (line 58) * trunc: C++ Interface Floats. (line 100) * xtom: BSD Compatible Functions. (line 34) @ 1.1.1.1 log @initial import of GMP 5.0.2. GNU MP is a library for arbitrary precision arithmetic, operating on signed integers, rational numbers, and floating point numbers. It has a rich set of functions, and the functions have a regular interface. GMP is necessary for GCC >= 4.2. @ text @@ 1.1.1.1.8.1 log @Rebase to HEAD as of a few days ago. @ text @d1 1 a1 1 This is ../../gmp/doc/gmp.info, produced by makeinfo version 4.13 from d4 2 a5 2 This manual describes how to install and use the GNU multiple precision arithmetic library, version 5.1.3. d8 2 a9 2 2001, 2002, 2003, 2004, 2005, 2006, 2007, 2008, 2009, 2010, 2011, 2012, 2013 Free Software Foundation, Inc. d17 1 a17 1 is included in *note GNU Free Documentation License::. a24 25 File: gmp.info, Node: Extended GCD, Next: Jacobi Symbol, Prev: Subquadratic GCD, Up: Greatest Common Divisor Algorithms 15.3.4 Extended GCD ------------------- The extended GCD function, or GCDEXT, calculates gcd(a,b) and also cofactors x and y satisfying a*x+b*y=gcd(a,b). All the algorithms used for plain GCD are extended to handle this case. The binary algorithm is used only for single-limb GCDEXT. Lehmer's algorithm is used for sizes up to `GCDEXT_DC_THRESHOLD'. Above this threshold, GCDEXT is implemented as a loop around HGCD, but with more book-keeping to keep track of the cofactors. This gives the same asymptotic running time as for GCD and HGCD, O(M(N)*log(N)) One difference to plain GCD is that while the inputs a and b are reduced as the algorithm proceeds, the cofactors x and y grow in size. This makes the tuning of the chopping-point more difficult. The current code chops off the most significant half of the inputs for the call to HGCD in the first iteration, and the most significant two thirds for the remaining calls. This strategy could surely be improved. Also the stop condition for the loop, where Lehmer's algorithm is invoked once the inputs are reduced below `GCDEXT_DC_THRESHOLD', could maybe be improved by taking into account the current size of the cofactors.  d27 1 a27 1 15.3.5 Jacobi Symbol d30 1 a30 4 [This section is obsolete. The current Jacobi code actually uses a very efficient algorithm.] `mpz_jacobi' and `mpz_kronecker' are currently implemented with a d48 1 a48 1 15.4 Powering Algorithms d59 1 a59 1 15.4.1 Normal Powering d71 1 a71 1 15.4.2 Modular Powering d81 1 a81 1 The modular multiplies and squarings use either a simple division or d89 1 a89 1 15.5 Root Extraction Algorithms d102 1 a102 1 15.5.1 Square Root d156 1 a156 1 15.5.2 Nth Root d176 1 a176 1 15.5.3 Perfect Square d209 1 a209 1 of non-squares that get through (i.e. a pseudo-square to all the tested d221 1 a221 1 15.5.4 Perfect Power d237 1 a237 1 15.6 Radix Conversion d252 1 a252 1 15.6.1 Binary to Radix d319 1 a319 1 remainder produced, i.e. the highest power would be an b^(n*k) d327 1 a327 1 15.6.2 Radix to Binary d372 1 a372 1 15.7 Other Algorithms d387 1 a387 1 15.7.1 Prime Testing d413 1 a413 1 15.7.2 Factorial d416 2 a417 8 Factorials are calculated by a combination of two algorithms. An idea is shared among them: to compute the odd part of the factorial; a final step takes account of the power of 2 term, by shifting. For small n, the odd factor of n! is computed with the simple observation that it is equal to the product of all positive odd numbers smaller than n times the odd factor of [n/2]!, where [x] is the integer part of x, and so on recursively. The procedure can be best illustrated d420 7 a426 1 23! = (23.21.19.17.15.13.11.9.7.5.3)(11.9.7.5.3)(5.3)2^19 d428 6 a433 33 Current code collects all the factors in a single list, with a loop and no recursion, and compute the product, with no special care for repeated chunks. When n is larger, computation pass trough prime sieving. An helper function is used, as suggested by Peter Luschny: n ----- n! | | L(p,n) msf(n) = -------------- = | | p [n/2]!^2.2^k p=3 Where p ranges on odd prime numbers. The exponent k is chosen to obtain an odd integer number: k is the number of 1 bits in the binary representation of [n/2]. The function L(p,n) can be defined as zero when p is composite, and, for any prime p, it is computed with: --- \ n L(p,n) = / [---] mod 2 <= log (n) . --- p^i p i>0 With this helper function, we are able to compute the odd part of n! using the recursion implied by n!=[n/2]!^2*msf(n)*2^k. The recursion stops using the small-n algorithm on some [n/2^i]. Both the above algorithms use binary splitting to compute the product of many small factors. At first as many products as possible are accumulated in a single register, generating a list of factors that fit in a machine word. This list is then split into halves, and the product is computed recursively. d440 6 d449 1 a449 1 15.7.3 Binomial Coefficients d471 1 a471 1 15.7.4 Fibonacci Numbers d518 2 a519 2 For interest, the 2*(-1)^k term both here and above can be applied just to the low limb of the calculation, without a carry or borrow into d526 1 a526 1 15.7.5 Lucas Numbers d549 1 a549 1 15.7.6 Random Numbers d573 1 a573 1 parameters A and C, an integer state S is iterated by the formula S <- d593 1 a593 1 15.8 Assembly Coding d622 1 a622 1 15.8.1 Code Organisation d644 1 a644 1 15.8.2 Assembly Basics d668 1 a668 1 15.8.3 Carry Propagation d705 1 a705 1 15.8.4 Cache Handling d751 1 a751 1 15.8.5 Functional Units d787 1 a787 1 15.8.6 Floating Point d884 1 a884 1 15.8.7 SIMD Instructions d905 1 a905 1 15.8.8 Software Pipelining d930 1 a930 1 15.8.9 Loop Unrolling d977 1 a977 1 15.8.10 Writing Guide d1036 1 a1036 1 16 Internals d1055 1 a1055 1 16.1 Integer Internals d1105 1 a1105 1 16.2 Rational Internals d1124 1 a1124 1 but GMP has no way to know when this will occur. As per *note d1133 1 a1133 1 16.3 Float Internals d1297 1 a1297 1 16.4 Raw Output Internals d1327 1 a1327 1 16.5 C++ Interface Internals d1421 1 a1421 1 Torbjörn Granlund wrote the original GMP library and is still the main d1423 1 a1423 1 Torbjörn. Several other individuals and organizations have contributed d1426 1 a1426 1 Gunnar Sjödin and Hans Riesel helped with mathematical problems in d1439 1 a1439 1 Schönhage-Strassen FFT multiply code, and the Karatsuba square root d1444 1 a1444 1 root code (with Torbjörn). d1497 6 a1502 7 Niels Möller wrote the sub-quadratic GCD, extended GCD and jacobi code, the quadratic Hensel division code, and (with Torbjörn) the new divide and conquer division code for GMP 4.3. Niels also helped implement the new Toom multiply code for GMP 4.3 and implemented helper functions to simplify Toom evaluations for GMP 5.0. He wrote the original version of mpn_mulmod_bnm1, and he is the main author of the mini-gmp package used for gmp bootstrapping. d1512 1 a1512 2 mpn_invertappr. He is the author of the current combinatorial functions: binomial, factorial, multifactorial, primorial. d1517 1 a1517 3 `mpn_mul_basecase'. He wrote the internal middle product functions `mpn_mulmid_basecase', `mpn_toom42_mulmid', `mpn_mulmid_n' and related helper routines. a1520 8 Marc Glisse improved `gmpxx.h': use fewer temporaries (faster), specializations of `numeric_limits' and `common_type', C++11 features (move constructors, explicit bool conversion, UDL), make the conversion from `mpq_class' to `mpz_class' explicit, optimize operations where one argument is a small compile-time constant, replace some heap allocations by stack allocations. He also fixed the eofbit handling of C++ streams, and removed one division from `mpq/aors.c'. d1550 1 a1550 1 `http://www.math.dartmouth.edu/~carlp/' d1578 1 a1578 1 June 2002, `http://hal.inria.fr/docs/00/07/21/13/PDF/RR-4475.pdf' d1584 1 a1584 1 * Torbjörn Granlund and Peter L. Montgomery, "Division by Invariant d1587 1 a1587 1 `http://gmplib.org/~tege/divcnst-pldi94.pdf'. d1589 2 a1590 2 * Niels Möller and Torbjörn Granlund, "Improved division by invariant integers", IEEE Transactions on Computers, 11 June 2010. d1593 2 a1594 2 * Torbjörn Granlund and Niels Möller, "Division of integers large and small", to appear. d1637 1 a1637 1 * Niels Möller, "On Schönhage's algorithm and subquadratic integer d1645 1 a1645 1 * Arnold Schönhage and Volker Strassen, "Schnelle Multiplikation d1653 1 a1653 2 3805, November 1999, `http://hal.inria.fr/inria-00072854/PDF/RR-3805.pdf' d2163 1 a2163 1 * --disable-fft: Build Options. (line 314) d2166 7 a2172 6 * --enable-alloca: Build Options. (line 275) * --enable-assert: Build Options. (line 320) * --enable-cxx: Build Options. (line 227) * --enable-fat: Build Options. (line 162) * --enable-profiling <1>: Build Options. (line 324) * --enable-profiling: Profiling. (line 6) d2181 2 a2182 2 (line 127) * ABI <1>: Build Options. (line 169) d2186 2 a2187 1 * AIX <1>: Notes for Particular Systems. d2191 1 a2191 1 * alloca: Build Options. (line 275) d2196 3 a2198 3 * Arithmetic functions <1>: Rational Arithmetic. (line 6) * Arithmetic functions <2>: Float Arithmetic. (line 6) * Arithmetic functions: Integer Arithmetic. (line 6) d2218 3 a2220 3 * Assertion checking <1>: Build Options. (line 320) * Assertion checking: Debugging. (line 79) * Assignment functions <1>: Assigning Integers. (line 6) d2223 2 a2224 1 * Assignment functions <3>: Assigning Floats. (line 6) d2227 1 a2227 2 * Assignment functions: Simultaneous Integer Init & Assign. (line 6) d2230 3 d2236 1 a2236 1 (line 128) d2242 3 a2244 3 (line 40) * Bit shift left: Integer Arithmetic. (line 38) * Bit shift right: Integer Division. (line 62) d2247 3 d2262 2 a2263 2 * C compiler: Build Options. (line 180) * C++ compiler: Build Options. (line 251) d2270 4 a2273 4 * C++ support: Build Options. (line 227) * CC: Build Options. (line 180) * CC_FOR_BUILD: Build Options. (line 214) * CFLAGS: Build Options. (line 180) d2280 3 a2282 3 * Comparison functions <1>: Comparing Rationals. (line 6) * Comparison functions <2>: Float Comparison. (line 6) * Comparison functions: Integer Comparisons. (line 6) d2287 2 a2288 2 * Congruence algorithm: Exact Remainder. (line 30) * Congruence functions: Integer Division. (line 137) d2301 1 a2301 1 * CPPFLAGS: Build Options. (line 206) d2306 2 a2307 2 * CXX: Build Options. (line 251) * CXXFLAGS: Build Options. (line 251) d2317 2 a2318 2 * Divisibility algorithm: Exact Remainder. (line 30) * Divisibility functions: Integer Division. (line 137) d2321 3 a2323 3 * Division functions <1>: Rational Arithmetic. (line 24) * Division functions <2>: Float Arithmetic. (line 33) * Division functions: Integer Division. (line 6) d2330 2 a2331 2 * DocBook: Build Options. (line 347) * Documentation formats: Build Options. (line 340) d2334 1 a2334 1 * DVI: Build Options. (line 343) d2337 1 a2337 1 * Exact division functions: Integer Division. (line 112) d2343 2 a2344 3 * Execution profiling: Build Options. (line 324) * Exponentiation functions <1>: Float Arithmetic. (line 41) * Exponentiation functions: Integer Exponentiation. d2346 1 d2352 1 a2352 1 (line 49) d2354 1 a2354 1 (line 108) d2357 1 a2357 1 (line 116) d2361 3 a2363 3 * Fat binary: Build Options. (line 162) * FFT multiplication <1>: Build Options. (line 314) * FFT multiplication: FFT Multiplication. (line 6) d2367 1 a2367 1 (line 136) d2387 1 a2387 1 * Float sign tests: Float Comparison. (line 35) d2399 2 a2400 2 * frexp <1>: Converting Integers. (line 43) * frexp: Converting Floats. (line 24) d2408 1 a2408 1 (line 49) d2410 1 a2410 1 (line 32) d2430 1 a2430 1 (line 32) d2440 1 a2440 1 * I/O functions <1>: I/O of Floats. (line 6) d2442 1 a2442 1 * I/O functions: I/O of Integers. (line 6) d2444 1 a2444 1 (line 127) d2452 1 a2452 1 * Initialization functions <1>: Simultaneous Float Init & Assign. d2454 1 a2454 1 * Initialization functions <2>: Random State Initialization. d2456 1 a2456 2 * Initialization functions <3>: Initializing Floats. (line 6) * Initialization functions <4>: Simultaneous Integer Init & Assign. d2458 1 a2458 1 * Initialization functions <5>: Initializing Rationals. d2460 1 a2460 1 * Initialization functions: Initializing Integers. d2462 1 d2466 2 a2467 1 * Input functions <3>: Formatted Input Functions. a2468 1 * Input functions: I/O of Floats. (line 6) d2476 1 a2476 2 * Integer assignment functions <1>: Assigning Integers. (line 6) * Integer assignment functions: Simultaneous Integer Init & Assign. d2478 1 d2512 1 a2512 1 (line 76) d2520 1 a2520 1 (line 83) d2526 1 a2526 1 (line 95) d2530 1 a2530 1 (line 70) d2532 1 a2532 1 (line 70) d2534 1 a2534 1 (line 86) d2562 1 a2562 1 (line 147) d2582 1 a2582 1 (line 133) d2584 1 a2584 1 (line 76) d2587 3 a2589 1 * MPN_PATH: Build Options. (line 328) d2591 1 a2591 1 (line 43) d2597 1 a2597 1 * Nails: Low-level Functions. (line 485) d2602 1 a2602 1 (line 25) d2621 3 a2623 1 * Output functions <1>: Formatted Output Functions. a2624 2 * Output functions <2>: I/O of Rationals. (line 6) * Output functions <3>: I/O of Floats. (line 6) d2631 1 a2631 1 (line 15) d2636 1 a2636 1 * PDF: Build Options. (line 343) d2639 1 a2639 1 * Perfect power functions: Integer Roots. (line 28) d2642 1 a2642 1 * Perfect square functions: Integer Roots. (line 37) d2647 4 a2650 2 * Postscript: Build Options. (line 343) * Power/PowerPC <1>: Notes for Particular Systems. a2651 2 * Power/PowerPC: Known Build Problems. (line 63) d2653 2 a2654 1 * Powering functions <1>: Integer Exponentiation. a2655 1 * Powering functions: Float Arithmetic. (line 41) a2665 2 * Primorial functions: Number Theoretic Functions. (line 121) d2675 3 a2677 1 * Random number functions <1>: Miscellaneous Float Functions. d2679 1 a2679 3 * Random number functions <2>: Random Number Functions. (line 6) * Random number functions: Integer Random Numbers. d2711 1 a2711 1 (line 108) d2716 3 a2718 3 * Root extraction functions <1>: Integer Roots. (line 6) * Root extraction functions: Float Arithmetic. (line 37) * Root testing functions: Integer Roots. (line 37) d2724 1 a2724 1 (line 40) d2737 3 a2739 3 * Sign tests <1>: Integer Comparisons. (line 28) * Sign tests <2>: Comparing Rationals. (line 27) * Sign tests: Float Comparison. (line 35) d2743 1 a2743 3 * Solaris <1>: Known Build Problems. (line 72) * Solaris <2>: ABI and ISA. (line 199) d2747 2 a2748 2 (line 109) * Sparc V9: ABI and ISA. (line 199) d2754 1 a2754 1 (line 133) d2756 2 a2757 2 * Stack overflow <1>: Build Options. (line 275) * Stack overflow: Debugging. (line 7) d2765 1 a2765 1 * Sun: ABI and ISA. (line 199) d2767 1 a2767 1 (line 121) d2770 2 a2771 2 * Temporary memory: Build Options. (line 275) * Texinfo: Build Options. (line 340) d2776 1 a2776 3 * Toom multiplication <2>: Toom 3-Way Multiplication. (line 6) * Toom multiplication <3>: Higher degree Toom'n'half. d2778 1 a2778 1 * Toom multiplication: Toom 4-Way Multiplication. d2800 1 a2800 1 (line 127) d2803 1 a2803 1 * XML: Build Options. (line 347) d2826 4 a2829 4 * abs <1>: C++ Interface Rationals. (line 49) * abs <2>: C++ Interface Floats. (line 83) d2831 1 a2831 1 (line 47) d2833 3 a2835 3 (line 84) * cmp <1>: C++ Interface Rationals. (line 51) d2837 7 a2843 5 (line 49) * cmp <3>: C++ Interface Rationals. (line 50) * cmp: C++ Interface Floats. (line 86) d2845 3 a2847 1 (line 93) d2860 6 a2865 6 * GMP_LIMB_BITS: Low-level Functions. (line 515) * GMP_NAIL_BITS: Low-level Functions. (line 513) * GMP_NAIL_MASK: Low-level Functions. (line 523) * GMP_NUMB_BITS: Low-level Functions. (line 514) * GMP_NUMB_MASK: Low-level Functions. (line 524) * GMP_NUMB_MAX: Low-level Functions. (line 532) d2879 1 a2879 1 (line 46) d2885 1 a2885 1 (line 13) d2887 1 a2887 1 (line 33) d2903 1 a2903 1 (line 8) d2905 1 a2905 1 (line 10) d2939 16 d2956 2 d2964 1 a2964 1 * mp_get_memory_functions: Custom Allocation. (line 90) d2967 1 a2967 1 * mp_set_memory_functions: Custom Allocation. (line 18) d2978 1 a2978 1 (line 87) d2980 1 a2980 1 (line 88) d2982 1 a2982 1 (line 89) d2984 1 a2984 1 (line 90) d2986 1 a2986 1 (line 91) d2988 1 a2988 1 (line 92) d2990 1 a2990 1 (line 95) d2994 1 a2994 1 (line 115) d2996 1 a2996 1 (line 96) d2998 1 a2998 1 (line 98) d3000 1 a3000 1 (line 99) d3004 1 a3004 1 (line 60) d3006 1 a3006 1 (line 116) d3008 1 a3008 1 (line 117) d3010 1 a3010 3 (line 101) * mpf_class::swap: C++ Interface Floats. (line 104) d3018 1 a3018 1 * mpf_div_2exp: Float Arithmetic. (line 55) d3020 1 a3020 1 * mpf_eq: Float Comparison. (line 18) d3036 1 a3036 1 * mpf_get_d_2exp: Converting Floats. (line 17) d3039 3 a3041 3 * mpf_get_si: Converting Floats. (line 28) * mpf_get_str: Converting Floats. (line 38) * mpf_get_ui: Converting Floats. (line 29) d3051 1 a3051 1 (line 26) d3059 1 a3059 1 * mpf_mul_2exp: Float Arithmetic. (line 51) d3066 1 a3066 1 * mpf_reldiff: Float Comparison. (line 31) d3077 1 a3077 1 * mpf_sgn: Float Comparison. (line 35) d3095 2 a3096 2 * mpn_and_n: Low-level Functions. (line 427) * mpn_andn_n: Low-level Functions. (line 442) d3098 3 a3100 3 * mpn_com: Low-level Functions. (line 467) * mpn_copyd: Low-level Functions. (line 476) * mpn_copyi: Low-level Functions. (line 472) d3110 4 a3113 4 * mpn_get_str: Low-level Functions. (line 352) * mpn_hamdist: Low-level Functions. (line 416) * mpn_ior_n: Low-level Functions. (line 432) * mpn_iorn_n: Low-level Functions. (line 447) d3119 1 a3119 1 * mpn_nand_n: Low-level Functions. (line 452) d3121 5 a3125 5 * mpn_nior_n: Low-level Functions. (line 457) * mpn_perfect_square_p: Low-level Functions. (line 422) * mpn_popcount: Low-level Functions. (line 412) * mpn_random: Low-level Functions. (line 401) * mpn_random2: Low-level Functions. (line 402) d3127 3 a3129 3 * mpn_scan0: Low-level Functions. (line 386) * mpn_scan1: Low-level Functions. (line 394) * mpn_set_str: Low-level Functions. (line 367) d3131 1 a3131 1 * mpn_sqrtrem: Low-level Functions. (line 334) d3137 5 a3141 5 * mpn_xnor_n: Low-level Functions. (line 462) * mpn_xor_n: Low-level Functions. (line 437) * mpn_zero: Low-level Functions. (line 479) * mpq_abs: Rational Arithmetic. (line 34) * mpq_add: Rational Arithmetic. (line 8) d3147 1 a3147 1 (line 43) d3149 1 a3149 1 (line 52) d3151 1 a3151 1 (line 66) d3153 1 a3153 1 (line 76) d3157 1 a3157 1 (line 65) d3159 1 a3159 1 (line 75) d3161 1 a3161 1 (line 53) d3163 1 a3163 1 (line 23) d3165 1 a3165 3 (line 54) * mpq_class::swap: C++ Interface Rationals. (line 57) d3175 2 a3176 2 * mpq_div: Rational Arithmetic. (line 24) * mpq_div_2exp: Rational Arithmetic. (line 28) d3190 5 a3194 5 * mpq_inp_str: I/O of Rationals. (line 27) * mpq_inv: Rational Arithmetic. (line 37) * mpq_mul: Rational Arithmetic. (line 16) * mpq_mul_2exp: Rational Arithmetic. (line 20) * mpq_neg: Rational Arithmetic. (line 31) d3197 1 a3197 1 * mpq_out_str: I/O of Rationals. (line 19) d3217 1 a3217 1 * mpq_sub: Rational Arithmetic. (line 12) d3222 1 a3222 3 * mpz_2fac_ui: Number Theoretic Functions. (line 114) * mpz_abs: Integer Arithmetic. (line 45) d3225 2 a3226 2 * mpz_addmul: Integer Arithmetic. (line 26) * mpz_addmul_ui: Integer Arithmetic. (line 28) d3232 1 a3232 1 (line 126) d3234 1 a3234 1 (line 128) d3236 4 a3239 4 * mpz_cdiv_q_2exp: Integer Division. (line 26) * mpz_cdiv_q_ui: Integer Division. (line 18) * mpz_cdiv_qr: Integer Division. (line 16) * mpz_cdiv_qr_ui: Integer Division. (line 22) d3241 3 a3243 3 * mpz_cdiv_r_2exp: Integer Division. (line 28) * mpz_cdiv_r_ui: Integer Division. (line 20) * mpz_cdiv_ui: Integer Division. (line 24) d3247 1 a3247 1 (line 50) d3249 1 a3249 1 (line 51) d3251 1 a3251 1 (line 52) d3253 1 a3253 1 (line 53) d3255 1 a3255 1 (line 54) d3257 1 a3257 1 (line 55) d3259 1 a3259 1 (line 56) d3263 1 a3263 1 (line 57) d3265 1 a3265 1 (line 58) d3267 1 a3267 1 (line 59) d3269 1 a3269 1 (line 7) d3271 1 a3271 3 (line 60) * mpz_class::swap: C++ Interface Integers. (line 64) d3273 1 a3273 1 (line 49) d3275 1 a3275 1 (line 53) d3277 1 a3277 1 (line 56) d3288 9 a3296 9 (line 59) * mpz_congruent_2exp_p: Integer Division. (line 137) * mpz_congruent_p: Integer Division. (line 133) * mpz_congruent_ui_p: Integer Division. (line 135) * mpz_divexact: Integer Division. (line 110) * mpz_divexact_ui: Integer Division. (line 112) * mpz_divisible_2exp_p: Integer Division. (line 123) * mpz_divisible_p: Integer Division. (line 120) * mpz_divisible_ui_p: Integer Division. (line 122) d3302 10 a3311 10 (line 113) * mpz_fdiv_q: Integer Division. (line 30) * mpz_fdiv_q_2exp: Integer Division. (line 43) * mpz_fdiv_q_ui: Integer Division. (line 35) * mpz_fdiv_qr: Integer Division. (line 33) * mpz_fdiv_qr_ui: Integer Division. (line 39) * mpz_fdiv_r: Integer Division. (line 31) * mpz_fdiv_r_2exp: Integer Division. (line 45) * mpz_fdiv_r_ui: Integer Division. (line 37) * mpz_fdiv_ui: Integer Division. (line 41) d3313 1 a3313 1 (line 136) d3315 1 a3315 1 (line 134) d3329 1 a3329 1 (line 32) d3331 1 a3331 1 (line 39) d3333 1 a3333 1 (line 49) d3335 1 a3335 1 * mpz_get_d_2exp: Converting Integers. (line 36) d3337 1 a3337 1 * mpz_get_str: Converting Integers. (line 47) d3356 1 a3356 1 (line 35) d3361 2 a3362 2 * mpz_inp_raw: I/O of Integers. (line 62) * mpz_inp_str: I/O of Integers. (line 31) d3364 1 a3364 1 (line 76) d3368 1 a3368 1 (line 83) d3370 1 a3370 1 (line 91) d3372 1 a3372 1 (line 92) d3374 1 a3374 1 (line 93) d3376 1 a3376 1 (line 68) d3378 1 a3378 1 (line 70) d3380 1 a3380 1 (line 86) d3382 1 a3382 1 (line 147) d3384 3 a3386 5 (line 145) * mpz_mfac_uiui: Number Theoretic Functions. (line 116) * mpz_mod: Integer Division. (line 100) * mpz_mod_ui: Integer Division. (line 102) d3388 1 a3388 1 * mpz_mul_2exp: Integer Arithmetic. (line 38) d3391 1 a3391 1 * mpz_neg: Integer Arithmetic. (line 42) d3393 1 a3393 1 (line 25) d3396 4 a3399 4 * mpz_out_raw: I/O of Integers. (line 46) * mpz_out_str: I/O of Integers. (line 19) * mpz_perfect_power_p: Integer Roots. (line 28) * mpz_perfect_square_p: Integer Roots. (line 37) a3409 2 * mpz_primorial_ui: Number Theoretic Functions. (line 121) d3417 1 a3417 1 (line 57) d3419 3 a3421 3 (line 108) * mpz_root: Integer Roots. (line 8) * mpz_rootrem: Integer Roots. (line 14) d3425 2 a3427 2 * mpz_scan1: Integer Logic and Bit Fiddling. (line 40) d3436 1 a3436 1 (line 53) d3439 1 a3439 1 (line 94) d3444 2 a3445 2 * mpz_sqrt: Integer Roots. (line 18) * mpz_sqrtrem: Integer Roots. (line 21) d3448 2 a3449 2 * mpz_submul: Integer Arithmetic. (line 32) * mpz_submul_ui: Integer Arithmetic. (line 34) d3453 9 a3461 9 * mpz_tdiv_q: Integer Division. (line 47) * mpz_tdiv_q_2exp: Integer Division. (line 60) * mpz_tdiv_q_ui: Integer Division. (line 52) * mpz_tdiv_qr: Integer Division. (line 50) * mpz_tdiv_qr_ui: Integer Division. (line 56) * mpz_tdiv_r: Integer Division. (line 48) * mpz_tdiv_r_2exp: Integer Division. (line 62) * mpz_tdiv_r_ui: Integer Division. (line 54) * mpz_tdiv_ui: Integer Division. (line 58) d3463 1 a3463 1 (line 62) d3465 1 a3465 1 (line 95) d3475 9 a3483 1 * operator"" <1>: C++ Interface Integers. a3484 6 * operator"" <2>: C++ Interface Floats. (line 56) * operator"": C++ Interface Rationals. (line 38) * operator%: C++ Interface Integers. (line 35) d3486 1 a3486 1 (line 34) d3488 11 a3498 4 (line 20) * operator>> <1>: C++ Interface Rationals. (line 85) * operator>>: C++ Formatted Input. (line 25) d3500 1 a3500 1 (line 56) d3502 1 a3502 1 (line 62) d3504 4 a3507 10 (line 102) * sqrt <1>: C++ Interface Integers. (line 63) * sqrt: C++ Interface Floats. (line 103) * swap <1>: C++ Interface Floats. (line 105) * swap <2>: C++ Interface Integers. (line 65) * swap: C++ Interface Rationals. d3510 3 a3512 2 (line 106) a3514 4  Local Variables: coding: iso-8859-1 End: @ 1.1.1.1.2.1 log @sync with head. for a reference, the tree before this commit was tagged as yamt-pagecache-tag8. this commit was splitted into small chunks to avoid a limitation of cvs. ("Protocol error: too many arguments") @ text @d1 1 a1 1 This is ../../gmp/doc/gmp.info, produced by makeinfo version 4.13 from d4 2 a5 2 This manual describes how to install and use the GNU multiple precision arithmetic library, version 5.1.3. d8 2 a9 2 2001, 2002, 2003, 2004, 2005, 2006, 2007, 2008, 2009, 2010, 2011, 2012, 2013 Free Software Foundation, Inc. d17 1 a17 1 is included in *note GNU Free Documentation License::. a24 25 File: gmp.info, Node: Extended GCD, Next: Jacobi Symbol, Prev: Subquadratic GCD, Up: Greatest Common Divisor Algorithms 15.3.4 Extended GCD ------------------- The extended GCD function, or GCDEXT, calculates gcd(a,b) and also cofactors x and y satisfying a*x+b*y=gcd(a,b). All the algorithms used for plain GCD are extended to handle this case. The binary algorithm is used only for single-limb GCDEXT. Lehmer's algorithm is used for sizes up to `GCDEXT_DC_THRESHOLD'. Above this threshold, GCDEXT is implemented as a loop around HGCD, but with more book-keeping to keep track of the cofactors. This gives the same asymptotic running time as for GCD and HGCD, O(M(N)*log(N)) One difference to plain GCD is that while the inputs a and b are reduced as the algorithm proceeds, the cofactors x and y grow in size. This makes the tuning of the chopping-point more difficult. The current code chops off the most significant half of the inputs for the call to HGCD in the first iteration, and the most significant two thirds for the remaining calls. This strategy could surely be improved. Also the stop condition for the loop, where Lehmer's algorithm is invoked once the inputs are reduced below `GCDEXT_DC_THRESHOLD', could maybe be improved by taking into account the current size of the cofactors.  d27 1 a27 1 15.3.5 Jacobi Symbol d30 1 a30 4 [This section is obsolete. The current Jacobi code actually uses a very efficient algorithm.] `mpz_jacobi' and `mpz_kronecker' are currently implemented with a d48 1 a48 1 15.4 Powering Algorithms d59 1 a59 1 15.4.1 Normal Powering d71 1 a71 1 15.4.2 Modular Powering d81 1 a81 1 The modular multiplies and squarings use either a simple division or d89 1 a89 1 15.5 Root Extraction Algorithms d102 1 a102 1 15.5.1 Square Root d156 1 a156 1 15.5.2 Nth Root d176 1 a176 1 15.5.3 Perfect Square d209 1 a209 1 of non-squares that get through (i.e. a pseudo-square to all the tested d221 1 a221 1 15.5.4 Perfect Power d237 1 a237 1 15.6 Radix Conversion d252 1 a252 1 15.6.1 Binary to Radix d319 1 a319 1 remainder produced, i.e. the highest power would be an b^(n*k) d327 1 a327 1 15.6.2 Radix to Binary d372 1 a372 1 15.7 Other Algorithms d387 1 a387 1 15.7.1 Prime Testing d413 1 a413 1 15.7.2 Factorial d416 2 a417 8 Factorials are calculated by a combination of two algorithms. An idea is shared among them: to compute the odd part of the factorial; a final step takes account of the power of 2 term, by shifting. For small n, the odd factor of n! is computed with the simple observation that it is equal to the product of all positive odd numbers smaller than n times the odd factor of [n/2]!, where [x] is the integer part of x, and so on recursively. The procedure can be best illustrated d420 7 a426 1 23! = (23.21.19.17.15.13.11.9.7.5.3)(11.9.7.5.3)(5.3)2^19 d428 6 a433 33 Current code collects all the factors in a single list, with a loop and no recursion, and compute the product, with no special care for repeated chunks. When n is larger, computation pass trough prime sieving. An helper function is used, as suggested by Peter Luschny: n ----- n! | | L(p,n) msf(n) = -------------- = | | p [n/2]!^2.2^k p=3 Where p ranges on odd prime numbers. The exponent k is chosen to obtain an odd integer number: k is the number of 1 bits in the binary representation of [n/2]. The function L(p,n) can be defined as zero when p is composite, and, for any prime p, it is computed with: --- \ n L(p,n) = / [---] mod 2 <= log (n) . --- p^i p i>0 With this helper function, we are able to compute the odd part of n! using the recursion implied by n!=[n/2]!^2*msf(n)*2^k. The recursion stops using the small-n algorithm on some [n/2^i]. Both the above algorithms use binary splitting to compute the product of many small factors. At first as many products as possible are accumulated in a single register, generating a list of factors that fit in a machine word. This list is then split into halves, and the product is computed recursively. d440 6 d449 1 a449 1 15.7.3 Binomial Coefficients d471 1 a471 1 15.7.4 Fibonacci Numbers d518 2 a519 2 For interest, the 2*(-1)^k term both here and above can be applied just to the low limb of the calculation, without a carry or borrow into d526 1 a526 1 15.7.5 Lucas Numbers d549 1 a549 1 15.7.6 Random Numbers d573 1 a573 1 parameters A and C, an integer state S is iterated by the formula S <- d593 1 a593 1 15.8 Assembly Coding d622 1 a622 1 15.8.1 Code Organisation d644 1 a644 1 15.8.2 Assembly Basics d668 1 a668 1 15.8.3 Carry Propagation d705 1 a705 1 15.8.4 Cache Handling d751 1 a751 1 15.8.5 Functional Units d787 1 a787 1 15.8.6 Floating Point d884 1 a884 1 15.8.7 SIMD Instructions d905 1 a905 1 15.8.8 Software Pipelining d930 1 a930 1 15.8.9 Loop Unrolling d977 1 a977 1 15.8.10 Writing Guide d1036 1 a1036 1 16 Internals d1055 1 a1055 1 16.1 Integer Internals d1105 1 a1105 1 16.2 Rational Internals d1124 1 a1124 1 but GMP has no way to know when this will occur. As per *note d1133 1 a1133 1 16.3 Float Internals d1297 1 a1297 1 16.4 Raw Output Internals d1327 1 a1327 1 16.5 C++ Interface Internals d1421 1 a1421 1 Torbjörn Granlund wrote the original GMP library and is still the main d1423 1 a1423 1 Torbjörn. Several other individuals and organizations have contributed d1426 1 a1426 1 Gunnar Sjödin and Hans Riesel helped with mathematical problems in d1439 1 a1439 1 Schönhage-Strassen FFT multiply code, and the Karatsuba square root d1444 1 a1444 1 root code (with Torbjörn). d1497 6 a1502 7 Niels Möller wrote the sub-quadratic GCD, extended GCD and jacobi code, the quadratic Hensel division code, and (with Torbjörn) the new divide and conquer division code for GMP 4.3. Niels also helped implement the new Toom multiply code for GMP 4.3 and implemented helper functions to simplify Toom evaluations for GMP 5.0. He wrote the original version of mpn_mulmod_bnm1, and he is the main author of the mini-gmp package used for gmp bootstrapping. d1512 1 a1512 2 mpn_invertappr. He is the author of the current combinatorial functions: binomial, factorial, multifactorial, primorial. d1517 1 a1517 3 `mpn_mul_basecase'. He wrote the internal middle product functions `mpn_mulmid_basecase', `mpn_toom42_mulmid', `mpn_mulmid_n' and related helper routines. a1520 8 Marc Glisse improved `gmpxx.h': use fewer temporaries (faster), specializations of `numeric_limits' and `common_type', C++11 features (move constructors, explicit bool conversion, UDL), make the conversion from `mpq_class' to `mpz_class' explicit, optimize operations where one argument is a small compile-time constant, replace some heap allocations by stack allocations. He also fixed the eofbit handling of C++ streams, and removed one division from `mpq/aors.c'. d1550 1 a1550 1 `http://www.math.dartmouth.edu/~carlp/' d1578 1 a1578 1 June 2002, `http://hal.inria.fr/docs/00/07/21/13/PDF/RR-4475.pdf' d1584 1 a1584 1 * Torbjörn Granlund and Peter L. Montgomery, "Division by Invariant d1587 1 a1587 1 `http://gmplib.org/~tege/divcnst-pldi94.pdf'. d1589 2 a1590 2 * Niels Möller and Torbjörn Granlund, "Improved division by invariant integers", IEEE Transactions on Computers, 11 June 2010. d1593 2 a1594 2 * Torbjörn Granlund and Niels Möller, "Division of integers large and small", to appear. d1637 1 a1637 1 * Niels Möller, "On Schönhage's algorithm and subquadratic integer d1645 1 a1645 1 * Arnold Schönhage and Volker Strassen, "Schnelle Multiplikation d1653 1 a1653 2 3805, November 1999, `http://hal.inria.fr/inria-00072854/PDF/RR-3805.pdf' d2163 1 a2163 1 * --disable-fft: Build Options. (line 314) d2166 7 a2172 6 * --enable-alloca: Build Options. (line 275) * --enable-assert: Build Options. (line 320) * --enable-cxx: Build Options. (line 227) * --enable-fat: Build Options. (line 162) * --enable-profiling <1>: Build Options. (line 324) * --enable-profiling: Profiling. (line 6) d2181 2 a2182 2 (line 127) * ABI <1>: Build Options. (line 169) d2186 2 a2187 1 * AIX <1>: Notes for Particular Systems. d2191 1 a2191 1 * alloca: Build Options. (line 275) d2196 3 a2198 3 * Arithmetic functions <1>: Rational Arithmetic. (line 6) * Arithmetic functions <2>: Float Arithmetic. (line 6) * Arithmetic functions: Integer Arithmetic. (line 6) d2218 3 a2220 3 * Assertion checking <1>: Build Options. (line 320) * Assertion checking: Debugging. (line 79) * Assignment functions <1>: Assigning Integers. (line 6) d2223 2 a2224 1 * Assignment functions <3>: Assigning Floats. (line 6) d2227 1 a2227 2 * Assignment functions: Simultaneous Integer Init & Assign. (line 6) d2230 3 d2236 1 a2236 1 (line 128) d2242 3 a2244 3 (line 40) * Bit shift left: Integer Arithmetic. (line 38) * Bit shift right: Integer Division. (line 62) d2247 3 d2262 2 a2263 2 * C compiler: Build Options. (line 180) * C++ compiler: Build Options. (line 251) d2270 4 a2273 4 * C++ support: Build Options. (line 227) * CC: Build Options. (line 180) * CC_FOR_BUILD: Build Options. (line 214) * CFLAGS: Build Options. (line 180) d2280 3 a2282 3 * Comparison functions <1>: Comparing Rationals. (line 6) * Comparison functions <2>: Float Comparison. (line 6) * Comparison functions: Integer Comparisons. (line 6) d2287 2 a2288 2 * Congruence algorithm: Exact Remainder. (line 30) * Congruence functions: Integer Division. (line 137) d2301 1 a2301 1 * CPPFLAGS: Build Options. (line 206) d2306 2 a2307 2 * CXX: Build Options. (line 251) * CXXFLAGS: Build Options. (line 251) d2317 2 a2318 2 * Divisibility algorithm: Exact Remainder. (line 30) * Divisibility functions: Integer Division. (line 137) d2321 3 a2323 3 * Division functions <1>: Rational Arithmetic. (line 24) * Division functions <2>: Float Arithmetic. (line 33) * Division functions: Integer Division. (line 6) d2330 2 a2331 2 * DocBook: Build Options. (line 347) * Documentation formats: Build Options. (line 340) d2334 1 a2334 1 * DVI: Build Options. (line 343) d2337 1 a2337 1 * Exact division functions: Integer Division. (line 112) d2343 2 a2344 3 * Execution profiling: Build Options. (line 324) * Exponentiation functions <1>: Float Arithmetic. (line 41) * Exponentiation functions: Integer Exponentiation. d2346 1 d2352 1 a2352 1 (line 49) d2354 1 a2354 1 (line 108) d2357 1 a2357 1 (line 116) d2361 3 a2363 3 * Fat binary: Build Options. (line 162) * FFT multiplication <1>: Build Options. (line 314) * FFT multiplication: FFT Multiplication. (line 6) d2367 1 a2367 1 (line 136) d2387 1 a2387 1 * Float sign tests: Float Comparison. (line 35) d2399 2 a2400 2 * frexp <1>: Converting Integers. (line 43) * frexp: Converting Floats. (line 24) d2408 1 a2408 1 (line 49) d2410 1 a2410 1 (line 32) d2430 1 a2430 1 (line 32) d2440 1 a2440 1 * I/O functions <1>: I/O of Floats. (line 6) d2442 1 a2442 1 * I/O functions: I/O of Integers. (line 6) d2444 1 a2444 1 (line 127) d2452 1 a2452 1 * Initialization functions <1>: Simultaneous Float Init & Assign. d2454 1 a2454 1 * Initialization functions <2>: Random State Initialization. d2456 1 a2456 2 * Initialization functions <3>: Initializing Floats. (line 6) * Initialization functions <4>: Simultaneous Integer Init & Assign. d2458 1 a2458 1 * Initialization functions <5>: Initializing Rationals. d2460 1 a2460 1 * Initialization functions: Initializing Integers. d2462 1 d2466 2 a2467 1 * Input functions <3>: Formatted Input Functions. a2468 1 * Input functions: I/O of Floats. (line 6) d2476 1 a2476 2 * Integer assignment functions <1>: Assigning Integers. (line 6) * Integer assignment functions: Simultaneous Integer Init & Assign. d2478 1 d2512 1 a2512 1 (line 76) d2520 1 a2520 1 (line 83) d2526 1 a2526 1 (line 95) d2530 1 a2530 1 (line 70) d2532 1 a2532 1 (line 70) d2534 1 a2534 1 (line 86) d2562 1 a2562 1 (line 147) d2582 1 a2582 1 (line 133) d2584 1 a2584 1 (line 76) d2587 3 a2589 1 * MPN_PATH: Build Options. (line 328) d2591 1 a2591 1 (line 43) d2597 1 a2597 1 * Nails: Low-level Functions. (line 485) d2602 1 a2602 1 (line 25) d2621 3 a2623 1 * Output functions <1>: Formatted Output Functions. a2624 2 * Output functions <2>: I/O of Rationals. (line 6) * Output functions <3>: I/O of Floats. (line 6) d2631 1 a2631 1 (line 15) d2636 1 a2636 1 * PDF: Build Options. (line 343) d2639 1 a2639 1 * Perfect power functions: Integer Roots. (line 28) d2642 1 a2642 1 * Perfect square functions: Integer Roots. (line 37) d2647 4 a2650 2 * Postscript: Build Options. (line 343) * Power/PowerPC <1>: Notes for Particular Systems. a2651 2 * Power/PowerPC: Known Build Problems. (line 63) d2653 2 a2654 1 * Powering functions <1>: Integer Exponentiation. a2655 1 * Powering functions: Float Arithmetic. (line 41) a2665 2 * Primorial functions: Number Theoretic Functions. (line 121) d2675 3 a2677 1 * Random number functions <1>: Miscellaneous Float Functions. d2679 1 a2679 3 * Random number functions <2>: Random Number Functions. (line 6) * Random number functions: Integer Random Numbers. d2711 1 a2711 1 (line 108) d2716 3 a2718 3 * Root extraction functions <1>: Integer Roots. (line 6) * Root extraction functions: Float Arithmetic. (line 37) * Root testing functions: Integer Roots. (line 37) d2724 1 a2724 1 (line 40) d2737 3 a2739 3 * Sign tests <1>: Integer Comparisons. (line 28) * Sign tests <2>: Comparing Rationals. (line 27) * Sign tests: Float Comparison. (line 35) d2743 1 a2743 3 * Solaris <1>: Known Build Problems. (line 72) * Solaris <2>: ABI and ISA. (line 199) d2747 2 a2748 2 (line 109) * Sparc V9: ABI and ISA. (line 199) d2754 1 a2754 1 (line 133) d2756 2 a2757 2 * Stack overflow <1>: Build Options. (line 275) * Stack overflow: Debugging. (line 7) d2765 1 a2765 1 * Sun: ABI and ISA. (line 199) d2767 1 a2767 1 (line 121) d2770 2 a2771 2 * Temporary memory: Build Options. (line 275) * Texinfo: Build Options. (line 340) d2776 1 a2776 3 * Toom multiplication <2>: Toom 3-Way Multiplication. (line 6) * Toom multiplication <3>: Higher degree Toom'n'half. d2778 1 a2778 1 * Toom multiplication: Toom 4-Way Multiplication. d2800 1 a2800 1 (line 127) d2803 1 a2803 1 * XML: Build Options. (line 347) d2826 4 a2829 4 * abs <1>: C++ Interface Rationals. (line 49) * abs <2>: C++ Interface Floats. (line 83) d2831 1 a2831 1 (line 47) d2833 3 a2835 3 (line 84) * cmp <1>: C++ Interface Rationals. (line 51) d2837 7 a2843 5 (line 49) * cmp <3>: C++ Interface Rationals. (line 50) * cmp: C++ Interface Floats. (line 86) d2845 3 a2847 1 (line 93) d2860 6 a2865 6 * GMP_LIMB_BITS: Low-level Functions. (line 515) * GMP_NAIL_BITS: Low-level Functions. (line 513) * GMP_NAIL_MASK: Low-level Functions. (line 523) * GMP_NUMB_BITS: Low-level Functions. (line 514) * GMP_NUMB_MASK: Low-level Functions. (line 524) * GMP_NUMB_MAX: Low-level Functions. (line 532) d2879 1 a2879 1 (line 46) d2885 1 a2885 1 (line 13) d2887 1 a2887 1 (line 33) d2903 1 a2903 1 (line 8) d2905 1 a2905 1 (line 10) d2939 16 d2956 2 d2964 1 a2964 1 * mp_get_memory_functions: Custom Allocation. (line 90) d2967 1 a2967 1 * mp_set_memory_functions: Custom Allocation. (line 18) d2978 1 a2978 1 (line 87) d2980 1 a2980 1 (line 88) d2982 1 a2982 1 (line 89) d2984 1 a2984 1 (line 90) d2986 1 a2986 1 (line 91) d2988 1 a2988 1 (line 92) d2990 1 a2990 1 (line 95) d2994 1 a2994 1 (line 115) d2996 1 a2996 1 (line 96) d2998 1 a2998 1 (line 98) d3000 1 a3000 1 (line 99) d3004 1 a3004 1 (line 60) d3006 1 a3006 1 (line 116) d3008 1 a3008 1 (line 117) d3010 1 a3010 3 (line 101) * mpf_class::swap: C++ Interface Floats. (line 104) d3018 1 a3018 1 * mpf_div_2exp: Float Arithmetic. (line 55) d3020 1 a3020 1 * mpf_eq: Float Comparison. (line 18) d3036 1 a3036 1 * mpf_get_d_2exp: Converting Floats. (line 17) d3039 3 a3041 3 * mpf_get_si: Converting Floats. (line 28) * mpf_get_str: Converting Floats. (line 38) * mpf_get_ui: Converting Floats. (line 29) d3051 1 a3051 1 (line 26) d3059 1 a3059 1 * mpf_mul_2exp: Float Arithmetic. (line 51) d3066 1 a3066 1 * mpf_reldiff: Float Comparison. (line 31) d3077 1 a3077 1 * mpf_sgn: Float Comparison. (line 35) d3095 2 a3096 2 * mpn_and_n: Low-level Functions. (line 427) * mpn_andn_n: Low-level Functions. (line 442) d3098 3 a3100 3 * mpn_com: Low-level Functions. (line 467) * mpn_copyd: Low-level Functions. (line 476) * mpn_copyi: Low-level Functions. (line 472) d3110 4 a3113 4 * mpn_get_str: Low-level Functions. (line 352) * mpn_hamdist: Low-level Functions. (line 416) * mpn_ior_n: Low-level Functions. (line 432) * mpn_iorn_n: Low-level Functions. (line 447) d3119 1 a3119 1 * mpn_nand_n: Low-level Functions. (line 452) d3121 5 a3125 5 * mpn_nior_n: Low-level Functions. (line 457) * mpn_perfect_square_p: Low-level Functions. (line 422) * mpn_popcount: Low-level Functions. (line 412) * mpn_random: Low-level Functions. (line 401) * mpn_random2: Low-level Functions. (line 402) d3127 3 a3129 3 * mpn_scan0: Low-level Functions. (line 386) * mpn_scan1: Low-level Functions. (line 394) * mpn_set_str: Low-level Functions. (line 367) d3131 1 a3131 1 * mpn_sqrtrem: Low-level Functions. (line 334) d3137 5 a3141 5 * mpn_xnor_n: Low-level Functions. (line 462) * mpn_xor_n: Low-level Functions. (line 437) * mpn_zero: Low-level Functions. (line 479) * mpq_abs: Rational Arithmetic. (line 34) * mpq_add: Rational Arithmetic. (line 8) d3147 1 a3147 1 (line 43) d3149 1 a3149 1 (line 52) d3151 1 a3151 1 (line 66) d3153 1 a3153 1 (line 76) d3157 1 a3157 1 (line 65) d3159 1 a3159 1 (line 75) d3161 1 a3161 1 (line 53) d3163 1 a3163 1 (line 23) d3165 1 a3165 3 (line 54) * mpq_class::swap: C++ Interface Rationals. (line 57) d3175 2 a3176 2 * mpq_div: Rational Arithmetic. (line 24) * mpq_div_2exp: Rational Arithmetic. (line 28) d3190 5 a3194 5 * mpq_inp_str: I/O of Rationals. (line 27) * mpq_inv: Rational Arithmetic. (line 37) * mpq_mul: Rational Arithmetic. (line 16) * mpq_mul_2exp: Rational Arithmetic. (line 20) * mpq_neg: Rational Arithmetic. (line 31) d3197 1 a3197 1 * mpq_out_str: I/O of Rationals. (line 19) d3217 1 a3217 1 * mpq_sub: Rational Arithmetic. (line 12) d3222 1 a3222 3 * mpz_2fac_ui: Number Theoretic Functions. (line 114) * mpz_abs: Integer Arithmetic. (line 45) d3225 2 a3226 2 * mpz_addmul: Integer Arithmetic. (line 26) * mpz_addmul_ui: Integer Arithmetic. (line 28) d3232 1 a3232 1 (line 126) d3234 1 a3234 1 (line 128) d3236 4 a3239 4 * mpz_cdiv_q_2exp: Integer Division. (line 26) * mpz_cdiv_q_ui: Integer Division. (line 18) * mpz_cdiv_qr: Integer Division. (line 16) * mpz_cdiv_qr_ui: Integer Division. (line 22) d3241 3 a3243 3 * mpz_cdiv_r_2exp: Integer Division. (line 28) * mpz_cdiv_r_ui: Integer Division. (line 20) * mpz_cdiv_ui: Integer Division. (line 24) d3247 1 a3247 1 (line 50) d3249 1 a3249 1 (line 51) d3251 1 a3251 1 (line 52) d3253 1 a3253 1 (line 53) d3255 1 a3255 1 (line 54) d3257 1 a3257 1 (line 55) d3259 1 a3259 1 (line 56) d3263 1 a3263 1 (line 57) d3265 1 a3265 1 (line 58) d3267 1 a3267 1 (line 59) d3269 1 a3269 1 (line 7) d3271 1 a3271 3 (line 60) * mpz_class::swap: C++ Interface Integers. (line 64) d3273 1 a3273 1 (line 49) d3275 1 a3275 1 (line 53) d3277 1 a3277 1 (line 56) d3288 9 a3296 9 (line 59) * mpz_congruent_2exp_p: Integer Division. (line 137) * mpz_congruent_p: Integer Division. (line 133) * mpz_congruent_ui_p: Integer Division. (line 135) * mpz_divexact: Integer Division. (line 110) * mpz_divexact_ui: Integer Division. (line 112) * mpz_divisible_2exp_p: Integer Division. (line 123) * mpz_divisible_p: Integer Division. (line 120) * mpz_divisible_ui_p: Integer Division. (line 122) d3302 10 a3311 10 (line 113) * mpz_fdiv_q: Integer Division. (line 30) * mpz_fdiv_q_2exp: Integer Division. (line 43) * mpz_fdiv_q_ui: Integer Division. (line 35) * mpz_fdiv_qr: Integer Division. (line 33) * mpz_fdiv_qr_ui: Integer Division. (line 39) * mpz_fdiv_r: Integer Division. (line 31) * mpz_fdiv_r_2exp: Integer Division. (line 45) * mpz_fdiv_r_ui: Integer Division. (line 37) * mpz_fdiv_ui: Integer Division. (line 41) d3313 1 a3313 1 (line 136) d3315 1 a3315 1 (line 134) d3329 1 a3329 1 (line 32) d3331 1 a3331 1 (line 39) d3333 1 a3333 1 (line 49) d3335 1 a3335 1 * mpz_get_d_2exp: Converting Integers. (line 36) d3337 1 a3337 1 * mpz_get_str: Converting Integers. (line 47) d3356 1 a3356 1 (line 35) d3361 2 a3362 2 * mpz_inp_raw: I/O of Integers. (line 62) * mpz_inp_str: I/O of Integers. (line 31) d3364 1 a3364 1 (line 76) d3368 1 a3368 1 (line 83) d3370 1 a3370 1 (line 91) d3372 1 a3372 1 (line 92) d3374 1 a3374 1 (line 93) d3376 1 a3376 1 (line 68) d3378 1 a3378 1 (line 70) d3380 1 a3380 1 (line 86) d3382 1 a3382 1 (line 147) d3384 3 a3386 5 (line 145) * mpz_mfac_uiui: Number Theoretic Functions. (line 116) * mpz_mod: Integer Division. (line 100) * mpz_mod_ui: Integer Division. (line 102) d3388 1 a3388 1 * mpz_mul_2exp: Integer Arithmetic. (line 38) d3391 1 a3391 1 * mpz_neg: Integer Arithmetic. (line 42) d3393 1 a3393 1 (line 25) d3396 4 a3399 4 * mpz_out_raw: I/O of Integers. (line 46) * mpz_out_str: I/O of Integers. (line 19) * mpz_perfect_power_p: Integer Roots. (line 28) * mpz_perfect_square_p: Integer Roots. (line 37) a3409 2 * mpz_primorial_ui: Number Theoretic Functions. (line 121) d3417 1 a3417 1 (line 57) d3419 3 a3421 3 (line 108) * mpz_root: Integer Roots. (line 8) * mpz_rootrem: Integer Roots. (line 14) d3425 2 a3427 2 * mpz_scan1: Integer Logic and Bit Fiddling. (line 40) d3436 1 a3436 1 (line 53) d3439 1 a3439 1 (line 94) d3444 2 a3445 2 * mpz_sqrt: Integer Roots. (line 18) * mpz_sqrtrem: Integer Roots. (line 21) d3448 2 a3449 2 * mpz_submul: Integer Arithmetic. (line 32) * mpz_submul_ui: Integer Arithmetic. (line 34) d3453 9 a3461 9 * mpz_tdiv_q: Integer Division. (line 47) * mpz_tdiv_q_2exp: Integer Division. (line 60) * mpz_tdiv_q_ui: Integer Division. (line 52) * mpz_tdiv_qr: Integer Division. (line 50) * mpz_tdiv_qr_ui: Integer Division. (line 56) * mpz_tdiv_r: Integer Division. (line 48) * mpz_tdiv_r_2exp: Integer Division. (line 62) * mpz_tdiv_r_ui: Integer Division. (line 54) * mpz_tdiv_ui: Integer Division. (line 58) d3463 1 a3463 1 (line 62) d3465 1 a3465 1 (line 95) d3475 9 a3483 1 * operator"" <1>: C++ Interface Integers. a3484 6 * operator"" <2>: C++ Interface Floats. (line 56) * operator"": C++ Interface Rationals. (line 38) * operator%: C++ Interface Integers. (line 35) d3486 1 a3486 1 (line 34) d3488 11 a3498 4 (line 20) * operator>> <1>: C++ Interface Rationals. (line 85) * operator>>: C++ Formatted Input. (line 25) d3500 1 a3500 1 (line 56) d3502 1 a3502 1 (line 62) d3504 4 a3507 10 (line 102) * sqrt <1>: C++ Interface Integers. (line 63) * sqrt: C++ Interface Floats. (line 103) * swap <1>: C++ Interface Floats. (line 105) * swap <2>: C++ Interface Integers. (line 65) * swap: C++ Interface Rationals. d3510 3 a3512 2 (line 106) a3514 4  Local Variables: coding: iso-8859-1 End: @ 1.1.1.2 log @initial import GMP 5.1.3 sources. changes include: fixes for: - mpn_sbpi1_div_qr_sec and mpn_sbpi1_div_r_sec - mpz_powm_ui - AMD family 11h - mpz_powm_sec and mpn_powm_sec - ASSERT() fixes - gcd, gcdext, and invert function fixes - some PPC division operations @ text @d1 1 a1 1 This is ../../gmp/doc/gmp.info, produced by makeinfo version 4.13 from d4 2 a5 2 This manual describes how to install and use the GNU multiple precision arithmetic library, version 5.1.3. d8 2 a9 2 2001, 2002, 2003, 2004, 2005, 2006, 2007, 2008, 2009, 2010, 2011, 2012, 2013 Free Software Foundation, Inc. d17 1 a17 1 is included in *note GNU Free Documentation License::. a24 25 File: gmp.info, Node: Extended GCD, Next: Jacobi Symbol, Prev: Subquadratic GCD, Up: Greatest Common Divisor Algorithms 15.3.4 Extended GCD ------------------- The extended GCD function, or GCDEXT, calculates gcd(a,b) and also cofactors x and y satisfying a*x+b*y=gcd(a,b). All the algorithms used for plain GCD are extended to handle this case. The binary algorithm is used only for single-limb GCDEXT. Lehmer's algorithm is used for sizes up to `GCDEXT_DC_THRESHOLD'. Above this threshold, GCDEXT is implemented as a loop around HGCD, but with more book-keeping to keep track of the cofactors. This gives the same asymptotic running time as for GCD and HGCD, O(M(N)*log(N)) One difference to plain GCD is that while the inputs a and b are reduced as the algorithm proceeds, the cofactors x and y grow in size. This makes the tuning of the chopping-point more difficult. The current code chops off the most significant half of the inputs for the call to HGCD in the first iteration, and the most significant two thirds for the remaining calls. This strategy could surely be improved. Also the stop condition for the loop, where Lehmer's algorithm is invoked once the inputs are reduced below `GCDEXT_DC_THRESHOLD', could maybe be improved by taking into account the current size of the cofactors.  d27 1 a27 1 15.3.5 Jacobi Symbol d30 1 a30 4 [This section is obsolete. The current Jacobi code actually uses a very efficient algorithm.] `mpz_jacobi' and `mpz_kronecker' are currently implemented with a d48 1 a48 1 15.4 Powering Algorithms d59 1 a59 1 15.4.1 Normal Powering d71 1 a71 1 15.4.2 Modular Powering d81 1 a81 1 The modular multiplies and squarings use either a simple division or d89 1 a89 1 15.5 Root Extraction Algorithms d102 1 a102 1 15.5.1 Square Root d156 1 a156 1 15.5.2 Nth Root d176 1 a176 1 15.5.3 Perfect Square d209 1 a209 1 of non-squares that get through (i.e. a pseudo-square to all the tested d221 1 a221 1 15.5.4 Perfect Power d237 1 a237 1 15.6 Radix Conversion d252 1 a252 1 15.6.1 Binary to Radix d319 1 a319 1 remainder produced, i.e. the highest power would be an b^(n*k) d327 1 a327 1 15.6.2 Radix to Binary d372 1 a372 1 15.7 Other Algorithms d387 1 a387 1 15.7.1 Prime Testing d413 1 a413 1 15.7.2 Factorial d416 2 a417 8 Factorials are calculated by a combination of two algorithms. An idea is shared among them: to compute the odd part of the factorial; a final step takes account of the power of 2 term, by shifting. For small n, the odd factor of n! is computed with the simple observation that it is equal to the product of all positive odd numbers smaller than n times the odd factor of [n/2]!, where [x] is the integer part of x, and so on recursively. The procedure can be best illustrated d420 7 a426 1 23! = (23.21.19.17.15.13.11.9.7.5.3)(11.9.7.5.3)(5.3)2^19 d428 6 a433 33 Current code collects all the factors in a single list, with a loop and no recursion, and compute the product, with no special care for repeated chunks. When n is larger, computation pass trough prime sieving. An helper function is used, as suggested by Peter Luschny: n ----- n! | | L(p,n) msf(n) = -------------- = | | p [n/2]!^2.2^k p=3 Where p ranges on odd prime numbers. The exponent k is chosen to obtain an odd integer number: k is the number of 1 bits in the binary representation of [n/2]. The function L(p,n) can be defined as zero when p is composite, and, for any prime p, it is computed with: --- \ n L(p,n) = / [---] mod 2 <= log (n) . --- p^i p i>0 With this helper function, we are able to compute the odd part of n! using the recursion implied by n!=[n/2]!^2*msf(n)*2^k. The recursion stops using the small-n algorithm on some [n/2^i]. Both the above algorithms use binary splitting to compute the product of many small factors. At first as many products as possible are accumulated in a single register, generating a list of factors that fit in a machine word. This list is then split into halves, and the product is computed recursively. d440 6 d449 1 a449 1 15.7.3 Binomial Coefficients d471 1 a471 1 15.7.4 Fibonacci Numbers d518 2 a519 2 For interest, the 2*(-1)^k term both here and above can be applied just to the low limb of the calculation, without a carry or borrow into d526 1 a526 1 15.7.5 Lucas Numbers d549 1 a549 1 15.7.6 Random Numbers d573 1 a573 1 parameters A and C, an integer state S is iterated by the formula S <- d593 1 a593 1 15.8 Assembly Coding d622 1 a622 1 15.8.1 Code Organisation d644 1 a644 1 15.8.2 Assembly Basics d668 1 a668 1 15.8.3 Carry Propagation d705 1 a705 1 15.8.4 Cache Handling d751 1 a751 1 15.8.5 Functional Units d787 1 a787 1 15.8.6 Floating Point d884 1 a884 1 15.8.7 SIMD Instructions d905 1 a905 1 15.8.8 Software Pipelining d930 1 a930 1 15.8.9 Loop Unrolling d977 1 a977 1 15.8.10 Writing Guide d1036 1 a1036 1 16 Internals d1055 1 a1055 1 16.1 Integer Internals d1105 1 a1105 1 16.2 Rational Internals d1124 1 a1124 1 but GMP has no way to know when this will occur. As per *note d1133 1 a1133 1 16.3 Float Internals d1297 1 a1297 1 16.4 Raw Output Internals d1327 1 a1327 1 16.5 C++ Interface Internals d1421 1 a1421 1 Torbjörn Granlund wrote the original GMP library and is still the main d1423 1 a1423 1 Torbjörn. Several other individuals and organizations have contributed d1426 1 a1426 1 Gunnar Sjödin and Hans Riesel helped with mathematical problems in d1439 1 a1439 1 Schönhage-Strassen FFT multiply code, and the Karatsuba square root d1444 1 a1444 1 root code (with Torbjörn). d1497 6 a1502 7 Niels Möller wrote the sub-quadratic GCD, extended GCD and jacobi code, the quadratic Hensel division code, and (with Torbjörn) the new divide and conquer division code for GMP 4.3. Niels also helped implement the new Toom multiply code for GMP 4.3 and implemented helper functions to simplify Toom evaluations for GMP 5.0. He wrote the original version of mpn_mulmod_bnm1, and he is the main author of the mini-gmp package used for gmp bootstrapping. d1512 1 a1512 2 mpn_invertappr. He is the author of the current combinatorial functions: binomial, factorial, multifactorial, primorial. d1517 1 a1517 3 `mpn_mul_basecase'. He wrote the internal middle product functions `mpn_mulmid_basecase', `mpn_toom42_mulmid', `mpn_mulmid_n' and related helper routines. a1520 8 Marc Glisse improved `gmpxx.h': use fewer temporaries (faster), specializations of `numeric_limits' and `common_type', C++11 features (move constructors, explicit bool conversion, UDL), make the conversion from `mpq_class' to `mpz_class' explicit, optimize operations where one argument is a small compile-time constant, replace some heap allocations by stack allocations. He also fixed the eofbit handling of C++ streams, and removed one division from `mpq/aors.c'. d1550 1 a1550 1 `http://www.math.dartmouth.edu/~carlp/' d1578 1 a1578 1 June 2002, `http://hal.inria.fr/docs/00/07/21/13/PDF/RR-4475.pdf' d1584 1 a1584 1 * Torbjörn Granlund and Peter L. Montgomery, "Division by Invariant d1587 1 a1587 1 `http://gmplib.org/~tege/divcnst-pldi94.pdf'. d1589 2 a1590 2 * Niels Möller and Torbjörn Granlund, "Improved division by invariant integers", IEEE Transactions on Computers, 11 June 2010. d1593 2 a1594 2 * Torbjörn Granlund and Niels Möller, "Division of integers large and small", to appear. d1637 1 a1637 1 * Niels Möller, "On Schönhage's algorithm and subquadratic integer d1645 1 a1645 1 * Arnold Schönhage and Volker Strassen, "Schnelle Multiplikation d1653 1 a1653 2 3805, November 1999, `http://hal.inria.fr/inria-00072854/PDF/RR-3805.pdf' d2163 1 a2163 1 * --disable-fft: Build Options. (line 314) d2166 7 a2172 6 * --enable-alloca: Build Options. (line 275) * --enable-assert: Build Options. (line 320) * --enable-cxx: Build Options. (line 227) * --enable-fat: Build Options. (line 162) * --enable-profiling <1>: Build Options. (line 324) * --enable-profiling: Profiling. (line 6) d2181 2 a2182 2 (line 127) * ABI <1>: Build Options. (line 169) d2186 2 a2187 1 * AIX <1>: Notes for Particular Systems. d2191 1 a2191 1 * alloca: Build Options. (line 275) d2196 3 a2198 3 * Arithmetic functions <1>: Rational Arithmetic. (line 6) * Arithmetic functions <2>: Float Arithmetic. (line 6) * Arithmetic functions: Integer Arithmetic. (line 6) d2218 3 a2220 3 * Assertion checking <1>: Build Options. (line 320) * Assertion checking: Debugging. (line 79) * Assignment functions <1>: Assigning Integers. (line 6) d2223 2 a2224 1 * Assignment functions <3>: Assigning Floats. (line 6) d2227 1 a2227 2 * Assignment functions: Simultaneous Integer Init & Assign. (line 6) d2230 3 d2236 1 a2236 1 (line 128) d2242 3 a2244 3 (line 40) * Bit shift left: Integer Arithmetic. (line 38) * Bit shift right: Integer Division. (line 62) d2247 3 d2262 2 a2263 2 * C compiler: Build Options. (line 180) * C++ compiler: Build Options. (line 251) d2270 4 a2273 4 * C++ support: Build Options. (line 227) * CC: Build Options. (line 180) * CC_FOR_BUILD: Build Options. (line 214) * CFLAGS: Build Options. (line 180) d2280 3 a2282 3 * Comparison functions <1>: Comparing Rationals. (line 6) * Comparison functions <2>: Float Comparison. (line 6) * Comparison functions: Integer Comparisons. (line 6) d2287 2 a2288 2 * Congruence algorithm: Exact Remainder. (line 30) * Congruence functions: Integer Division. (line 137) d2301 1 a2301 1 * CPPFLAGS: Build Options. (line 206) d2306 2 a2307 2 * CXX: Build Options. (line 251) * CXXFLAGS: Build Options. (line 251) d2317 2 a2318 2 * Divisibility algorithm: Exact Remainder. (line 30) * Divisibility functions: Integer Division. (line 137) d2321 3 a2323 3 * Division functions <1>: Rational Arithmetic. (line 24) * Division functions <2>: Float Arithmetic. (line 33) * Division functions: Integer Division. (line 6) d2330 2 a2331 2 * DocBook: Build Options. (line 347) * Documentation formats: Build Options. (line 340) d2334 1 a2334 1 * DVI: Build Options. (line 343) d2337 1 a2337 1 * Exact division functions: Integer Division. (line 112) d2343 2 a2344 3 * Execution profiling: Build Options. (line 324) * Exponentiation functions <1>: Float Arithmetic. (line 41) * Exponentiation functions: Integer Exponentiation. d2346 1 d2352 1 a2352 1 (line 49) d2354 1 a2354 1 (line 108) d2357 1 a2357 1 (line 116) d2361 3 a2363 3 * Fat binary: Build Options. (line 162) * FFT multiplication <1>: Build Options. (line 314) * FFT multiplication: FFT Multiplication. (line 6) d2367 1 a2367 1 (line 136) d2387 1 a2387 1 * Float sign tests: Float Comparison. (line 35) d2399 2 a2400 2 * frexp <1>: Converting Integers. (line 43) * frexp: Converting Floats. (line 24) d2408 1 a2408 1 (line 49) d2410 1 a2410 1 (line 32) d2430 1 a2430 1 (line 32) d2440 1 a2440 1 * I/O functions <1>: I/O of Floats. (line 6) d2442 1 a2442 1 * I/O functions: I/O of Integers. (line 6) d2444 1 a2444 1 (line 127) d2452 1 a2452 1 * Initialization functions <1>: Simultaneous Float Init & Assign. d2454 1 a2454 1 * Initialization functions <2>: Random State Initialization. d2456 1 a2456 2 * Initialization functions <3>: Initializing Floats. (line 6) * Initialization functions <4>: Simultaneous Integer Init & Assign. d2458 1 a2458 1 * Initialization functions <5>: Initializing Rationals. d2460 1 a2460 1 * Initialization functions: Initializing Integers. d2462 1 d2466 2 a2467 1 * Input functions <3>: Formatted Input Functions. a2468 1 * Input functions: I/O of Floats. (line 6) d2476 1 a2476 2 * Integer assignment functions <1>: Assigning Integers. (line 6) * Integer assignment functions: Simultaneous Integer Init & Assign. d2478 1 d2512 1 a2512 1 (line 76) d2520 1 a2520 1 (line 83) d2526 1 a2526 1 (line 95) d2530 1 a2530 1 (line 70) d2532 1 a2532 1 (line 70) d2534 1 a2534 1 (line 86) d2562 1 a2562 1 (line 147) d2582 1 a2582 1 (line 133) d2584 1 a2584 1 (line 76) d2587 3 a2589 1 * MPN_PATH: Build Options. (line 328) d2591 1 a2591 1 (line 43) d2597 1 a2597 1 * Nails: Low-level Functions. (line 485) d2602 1 a2602 1 (line 25) d2621 3 a2623 1 * Output functions <1>: Formatted Output Functions. a2624 2 * Output functions <2>: I/O of Rationals. (line 6) * Output functions <3>: I/O of Floats. (line 6) d2631 1 a2631 1 (line 15) d2636 1 a2636 1 * PDF: Build Options. (line 343) d2639 1 a2639 1 * Perfect power functions: Integer Roots. (line 28) d2642 1 a2642 1 * Perfect square functions: Integer Roots. (line 37) d2647 4 a2650 2 * Postscript: Build Options. (line 343) * Power/PowerPC <1>: Notes for Particular Systems. a2651 2 * Power/PowerPC: Known Build Problems. (line 63) d2653 2 a2654 1 * Powering functions <1>: Integer Exponentiation. a2655 1 * Powering functions: Float Arithmetic. (line 41) a2665 2 * Primorial functions: Number Theoretic Functions. (line 121) d2675 3 a2677 1 * Random number functions <1>: Miscellaneous Float Functions. d2679 1 a2679 3 * Random number functions <2>: Random Number Functions. (line 6) * Random number functions: Integer Random Numbers. d2711 1 a2711 1 (line 108) d2716 3 a2718 3 * Root extraction functions <1>: Integer Roots. (line 6) * Root extraction functions: Float Arithmetic. (line 37) * Root testing functions: Integer Roots. (line 37) d2724 1 a2724 1 (line 40) d2737 3 a2739 3 * Sign tests <1>: Integer Comparisons. (line 28) * Sign tests <2>: Comparing Rationals. (line 27) * Sign tests: Float Comparison. (line 35) d2743 1 a2743 3 * Solaris <1>: Known Build Problems. (line 72) * Solaris <2>: ABI and ISA. (line 199) d2747 2 a2748 2 (line 109) * Sparc V9: ABI and ISA. (line 199) d2754 1 a2754 1 (line 133) d2756 2 a2757 2 * Stack overflow <1>: Build Options. (line 275) * Stack overflow: Debugging. (line 7) d2765 1 a2765 1 * Sun: ABI and ISA. (line 199) d2767 1 a2767 1 (line 121) d2770 2 a2771 2 * Temporary memory: Build Options. (line 275) * Texinfo: Build Options. (line 340) d2776 1 a2776 3 * Toom multiplication <2>: Toom 3-Way Multiplication. (line 6) * Toom multiplication <3>: Higher degree Toom'n'half. d2778 1 a2778 1 * Toom multiplication: Toom 4-Way Multiplication. d2800 1 a2800 1 (line 127) d2803 1 a2803 1 * XML: Build Options. (line 347) d2826 4 a2829 4 * abs <1>: C++ Interface Rationals. (line 49) * abs <2>: C++ Interface Floats. (line 83) d2831 1 a2831 1 (line 47) d2833 3 a2835 3 (line 84) * cmp <1>: C++ Interface Rationals. (line 51) d2837 7 a2843 5 (line 49) * cmp <3>: C++ Interface Rationals. (line 50) * cmp: C++ Interface Floats. (line 86) d2845 3 a2847 1 (line 93) d2860 6 a2865 6 * GMP_LIMB_BITS: Low-level Functions. (line 515) * GMP_NAIL_BITS: Low-level Functions. (line 513) * GMP_NAIL_MASK: Low-level Functions. (line 523) * GMP_NUMB_BITS: Low-level Functions. (line 514) * GMP_NUMB_MASK: Low-level Functions. (line 524) * GMP_NUMB_MAX: Low-level Functions. (line 532) d2879 1 a2879 1 (line 46) d2885 1 a2885 1 (line 13) d2887 1 a2887 1 (line 33) d2903 1 a2903 1 (line 8) d2905 1 a2905 1 (line 10) d2939 16 d2956 2 d2964 1 a2964 1 * mp_get_memory_functions: Custom Allocation. (line 90) d2967 1 a2967 1 * mp_set_memory_functions: Custom Allocation. (line 18) d2978 1 a2978 1 (line 87) d2980 1 a2980 1 (line 88) d2982 1 a2982 1 (line 89) d2984 1 a2984 1 (line 90) d2986 1 a2986 1 (line 91) d2988 1 a2988 1 (line 92) d2990 1 a2990 1 (line 95) d2994 1 a2994 1 (line 115) d2996 1 a2996 1 (line 96) d2998 1 a2998 1 (line 98) d3000 1 a3000 1 (line 99) d3004 1 a3004 1 (line 60) d3006 1 a3006 1 (line 116) d3008 1 a3008 1 (line 117) d3010 1 a3010 3 (line 101) * mpf_class::swap: C++ Interface Floats. (line 104) d3018 1 a3018 1 * mpf_div_2exp: Float Arithmetic. (line 55) d3020 1 a3020 1 * mpf_eq: Float Comparison. (line 18) d3036 1 a3036 1 * mpf_get_d_2exp: Converting Floats. (line 17) d3039 3 a3041 3 * mpf_get_si: Converting Floats. (line 28) * mpf_get_str: Converting Floats. (line 38) * mpf_get_ui: Converting Floats. (line 29) d3051 1 a3051 1 (line 26) d3059 1 a3059 1 * mpf_mul_2exp: Float Arithmetic. (line 51) d3066 1 a3066 1 * mpf_reldiff: Float Comparison. (line 31) d3077 1 a3077 1 * mpf_sgn: Float Comparison. (line 35) d3095 2 a3096 2 * mpn_and_n: Low-level Functions. (line 427) * mpn_andn_n: Low-level Functions. (line 442) d3098 3 a3100 3 * mpn_com: Low-level Functions. (line 467) * mpn_copyd: Low-level Functions. (line 476) * mpn_copyi: Low-level Functions. (line 472) d3110 4 a3113 4 * mpn_get_str: Low-level Functions. (line 352) * mpn_hamdist: Low-level Functions. (line 416) * mpn_ior_n: Low-level Functions. (line 432) * mpn_iorn_n: Low-level Functions. (line 447) d3119 1 a3119 1 * mpn_nand_n: Low-level Functions. (line 452) d3121 5 a3125 5 * mpn_nior_n: Low-level Functions. (line 457) * mpn_perfect_square_p: Low-level Functions. (line 422) * mpn_popcount: Low-level Functions. (line 412) * mpn_random: Low-level Functions. (line 401) * mpn_random2: Low-level Functions. (line 402) d3127 3 a3129 3 * mpn_scan0: Low-level Functions. (line 386) * mpn_scan1: Low-level Functions. (line 394) * mpn_set_str: Low-level Functions. (line 367) d3131 1 a3131 1 * mpn_sqrtrem: Low-level Functions. (line 334) d3137 5 a3141 5 * mpn_xnor_n: Low-level Functions. (line 462) * mpn_xor_n: Low-level Functions. (line 437) * mpn_zero: Low-level Functions. (line 479) * mpq_abs: Rational Arithmetic. (line 34) * mpq_add: Rational Arithmetic. (line 8) d3147 1 a3147 1 (line 43) d3149 1 a3149 1 (line 52) d3151 1 a3151 1 (line 66) d3153 1 a3153 1 (line 76) d3157 1 a3157 1 (line 65) d3159 1 a3159 1 (line 75) d3161 1 a3161 1 (line 53) d3163 1 a3163 1 (line 23) d3165 1 a3165 3 (line 54) * mpq_class::swap: C++ Interface Rationals. (line 57) d3175 2 a3176 2 * mpq_div: Rational Arithmetic. (line 24) * mpq_div_2exp: Rational Arithmetic. (line 28) d3190 5 a3194 5 * mpq_inp_str: I/O of Rationals. (line 27) * mpq_inv: Rational Arithmetic. (line 37) * mpq_mul: Rational Arithmetic. (line 16) * mpq_mul_2exp: Rational Arithmetic. (line 20) * mpq_neg: Rational Arithmetic. (line 31) d3197 1 a3197 1 * mpq_out_str: I/O of Rationals. (line 19) d3217 1 a3217 1 * mpq_sub: Rational Arithmetic. (line 12) d3222 1 a3222 3 * mpz_2fac_ui: Number Theoretic Functions. (line 114) * mpz_abs: Integer Arithmetic. (line 45) d3225 2 a3226 2 * mpz_addmul: Integer Arithmetic. (line 26) * mpz_addmul_ui: Integer Arithmetic. (line 28) d3232 1 a3232 1 (line 126) d3234 1 a3234 1 (line 128) d3236 4 a3239 4 * mpz_cdiv_q_2exp: Integer Division. (line 26) * mpz_cdiv_q_ui: Integer Division. (line 18) * mpz_cdiv_qr: Integer Division. (line 16) * mpz_cdiv_qr_ui: Integer Division. (line 22) d3241 3 a3243 3 * mpz_cdiv_r_2exp: Integer Division. (line 28) * mpz_cdiv_r_ui: Integer Division. (line 20) * mpz_cdiv_ui: Integer Division. (line 24) d3247 1 a3247 1 (line 50) d3249 1 a3249 1 (line 51) d3251 1 a3251 1 (line 52) d3253 1 a3253 1 (line 53) d3255 1 a3255 1 (line 54) d3257 1 a3257 1 (line 55) d3259 1 a3259 1 (line 56) d3263 1 a3263 1 (line 57) d3265 1 a3265 1 (line 58) d3267 1 a3267 1 (line 59) d3269 1 a3269 1 (line 7) d3271 1 a3271 3 (line 60) * mpz_class::swap: C++ Interface Integers. (line 64) d3273 1 a3273 1 (line 49) d3275 1 a3275 1 (line 53) d3277 1 a3277 1 (line 56) d3288 9 a3296 9 (line 59) * mpz_congruent_2exp_p: Integer Division. (line 137) * mpz_congruent_p: Integer Division. (line 133) * mpz_congruent_ui_p: Integer Division. (line 135) * mpz_divexact: Integer Division. (line 110) * mpz_divexact_ui: Integer Division. (line 112) * mpz_divisible_2exp_p: Integer Division. (line 123) * mpz_divisible_p: Integer Division. (line 120) * mpz_divisible_ui_p: Integer Division. (line 122) d3302 10 a3311 10 (line 113) * mpz_fdiv_q: Integer Division. (line 30) * mpz_fdiv_q_2exp: Integer Division. (line 43) * mpz_fdiv_q_ui: Integer Division. (line 35) * mpz_fdiv_qr: Integer Division. (line 33) * mpz_fdiv_qr_ui: Integer Division. (line 39) * mpz_fdiv_r: Integer Division. (line 31) * mpz_fdiv_r_2exp: Integer Division. (line 45) * mpz_fdiv_r_ui: Integer Division. (line 37) * mpz_fdiv_ui: Integer Division. (line 41) d3313 1 a3313 1 (line 136) d3315 1 a3315 1 (line 134) d3329 1 a3329 1 (line 32) d3331 1 a3331 1 (line 39) d3333 1 a3333 1 (line 49) d3335 1 a3335 1 * mpz_get_d_2exp: Converting Integers. (line 36) d3337 1 a3337 1 * mpz_get_str: Converting Integers. (line 47) d3356 1 a3356 1 (line 35) d3361 2 a3362 2 * mpz_inp_raw: I/O of Integers. (line 62) * mpz_inp_str: I/O of Integers. (line 31) d3364 1 a3364 1 (line 76) d3368 1 a3368 1 (line 83) d3370 1 a3370 1 (line 91) d3372 1 a3372 1 (line 92) d3374 1 a3374 1 (line 93) d3376 1 a3376 1 (line 68) d3378 1 a3378 1 (line 70) d3380 1 a3380 1 (line 86) d3382 1 a3382 1 (line 147) d3384 3 a3386 5 (line 145) * mpz_mfac_uiui: Number Theoretic Functions. (line 116) * mpz_mod: Integer Division. (line 100) * mpz_mod_ui: Integer Division. (line 102) d3388 1 a3388 1 * mpz_mul_2exp: Integer Arithmetic. (line 38) d3391 1 a3391 1 * mpz_neg: Integer Arithmetic. (line 42) d3393 1 a3393 1 (line 25) d3396 4 a3399 4 * mpz_out_raw: I/O of Integers. (line 46) * mpz_out_str: I/O of Integers. (line 19) * mpz_perfect_power_p: Integer Roots. (line 28) * mpz_perfect_square_p: Integer Roots. (line 37) a3409 2 * mpz_primorial_ui: Number Theoretic Functions. (line 121) d3417 1 a3417 1 (line 57) d3419 3 a3421 3 (line 108) * mpz_root: Integer Roots. (line 8) * mpz_rootrem: Integer Roots. (line 14) d3425 2 a3427 2 * mpz_scan1: Integer Logic and Bit Fiddling. (line 40) d3436 1 a3436 1 (line 53) d3439 1 a3439 1 (line 94) d3444 2 a3445 2 * mpz_sqrt: Integer Roots. (line 18) * mpz_sqrtrem: Integer Roots. (line 21) d3448 2 a3449 2 * mpz_submul: Integer Arithmetic. (line 32) * mpz_submul_ui: Integer Arithmetic. (line 34) d3453 9 a3461 9 * mpz_tdiv_q: Integer Division. (line 47) * mpz_tdiv_q_2exp: Integer Division. (line 60) * mpz_tdiv_q_ui: Integer Division. (line 52) * mpz_tdiv_qr: Integer Division. (line 50) * mpz_tdiv_qr_ui: Integer Division. (line 56) * mpz_tdiv_r: Integer Division. (line 48) * mpz_tdiv_r_2exp: Integer Division. (line 62) * mpz_tdiv_r_ui: Integer Division. (line 54) * mpz_tdiv_ui: Integer Division. (line 58) d3463 1 a3463 1 (line 62) d3465 1 a3465 1 (line 95) d3475 9 a3483 1 * operator"" <1>: C++ Interface Integers. a3484 6 * operator"" <2>: C++ Interface Floats. (line 56) * operator"": C++ Interface Rationals. (line 38) * operator%: C++ Interface Integers. (line 35) d3486 1 a3486 1 (line 34) d3488 11 a3498 4 (line 20) * operator>> <1>: C++ Interface Rationals. (line 85) * operator>>: C++ Formatted Input. (line 25) d3500 1 a3500 1 (line 56) d3502 1 a3502 1 (line 62) d3504 4 a3507 10 (line 102) * sqrt <1>: C++ Interface Integers. (line 63) * sqrt: C++ Interface Floats. (line 103) * swap <1>: C++ Interface Floats. (line 105) * swap <2>: C++ Interface Integers. (line 65) * swap: C++ Interface Rationals. d3510 3 a3512 2 (line 106) a3514 4  Local Variables: coding: iso-8859-1 End: @ 1.1.1.3 log @initial import of GMP 6.1.2. main changes from 5.1.3 below. notes: - support for thumb-less ARM chips was in our port of 5.1.3, but a similar method has been provided upstream now - someone should look at the AVX failure reports, and fix them Changes between GMP version 6.1.0 and 6.1.1 FEATURES * Work around faulty cpuid on some recent Intel chips (this allows GMP to run on Skylake Pentiums). * Support thumb-less ARM chips. Changes between GMP version 6.0.* and 6.1.0 BUGS FIXED * The public function mpn_com is now correctly declared in gmp.h. * Healed possible failures of mpn_sec_sqr for non-cryptographic sizes for some obsolete CPUs. * Various problems related to precision for mpf have been fixed. * Fixed ABI incompatible stack alignment in calls from assembly code. * Fixed PIC bug in popcount affecting Intel processors using the 32-bit ABI. SPEEDUPS * Speedup for Intel Broadwell and Skylake through assembly code making use of new ADX instructions. * Square root is now faster when the remainder is not needed. Also the speed to compute the k-th root improved, for small sizes. FEATURES * New C++ functions gcd and lcm for mpz_class. * New public mpn functions mpn_divexact_1, mpn_zero_p, and mpn_cnd_swap. * New public mpq_cmp_z function, to efficiently compare rationals with integers. * Support for more 32-bit arm processors. * Support for AVX-less modern x86 CPUs. (Such support might be missing either because the CPU vendor chose to disable AVX, or because the running kernel lacks AVX context switch support.) * Support for NetBSD under Xen; we switch off AVX unconditionally under NetBSD since a bug in NetBSD makes AVX fail under Xen. MISC * Tuned values for FFT multiplications are provided for larger number on many platforms. Changes between GMP version 5.1.* and 6.0.0 BUGS FIXED * The function mpz_invert now considers any number invertible in Z/1Z. * The mpn multiply code now handles operands of more than 2^31 limbs correctly. (Note however that the mpz code is limited to 2^32 bits on 32-bit hosts and 2^37 bits on 64-bit hosts.) SPEEDUPS * Plain division of large operands is faster and more monotonous in operand size. * Major speedup for ARM, in particular ARM Cortex-A15, thanks to improved assembly. * Speedup for Intel Sandy Bridge, Ivy Bridge, Haswell, thanks to rewritten and vastly expanded assembly support. Speedup also for the older Core 2 and Nehalem. * Faster mixed arithmetic between mpq_class and double. FEATURES * Support for new Intel and AMD CPUs. * New public functions mpn_sec_mul and mpn_sec_sqr, implementing side-channel silent multiplication and squaring. * New public functions mpn_sec_div_qr and mpn_sec_div_r, implementing side-channel silent division. * New public functions mpn_cnd_add_n and mpn_cnd_sub_n. Side-channel silent conditional addition and subtraction. * New public function mpn_sec_powm, implementing side-channel silent modexp. * New public function mpn_sec_invert, implementing side-channel silent modular inversion. * Better support for applications which use the mpz_t type, but nevertheless need to call some of the lower-level mpn functions. See the documentation for mpz_limbs_read and related functions. @ text @d1 2 a2 1 This is gmp.info, produced by makeinfo version 6.1 from gmp.texi. d5 1 a5 1 arithmetic library, version 6.1.2. d7 11 a17 1 Copyright 1991, 1993-2016 Free Software Foundation, Inc. a18 7 Permission is granted to copy, distribute and/or modify this document under the terms of the GNU Free Documentation License, Version 1.3 or any later version published by the Free Software Foundation; with no Invariant Sections, with the Front-Cover Texts being "A GNU Manual", and with the Back-Cover Texts being "You have freedom to copy and modify this GNU Manual, like GNU software". A copy of the license is included in *note GNU Free Documentation License::. a24 254 File: gmp.info, Node: Exact Remainder, Next: Small Quotient Division, Prev: Exact Division, Up: Division Algorithms 15.2.6 Exact Remainder ---------------------- If the exact division algorithm is done with a full subtraction at each stage and the dividend isn't a multiple of the divisor, then low zero limbs are produced but with a remainder in the high limbs. For dividend a, divisor d, quotient q, and b = 2^mp_bits_per_limb, this remainder r is of the form a = q*d + r*b^n n represents the number of zero limbs produced by the subtractions, that being the number of limbs produced for q. r will be in the range 0<=rb*r+u2 condition appropriately relaxed.  File: gmp.info, Node: Greatest Common Divisor Algorithms, Next: Powering Algorithms, Prev: Division Algorithms, Up: Algorithms 15.3 Greatest Common Divisor ============================ * Menu: * Binary GCD:: * Lehmer's Algorithm:: * Subquadratic GCD:: * Extended GCD:: * Jacobi Symbol::  File: gmp.info, Node: Binary GCD, Next: Lehmer's Algorithm, Prev: Greatest Common Divisor Algorithms, Up: Greatest Common Divisor Algorithms 15.3.1 Binary GCD ----------------- At small sizes GMP uses an O(N^2) binary style GCD. This is described in many textbooks, for example Knuth section 4.5.2 algorithm B. It simply consists of successively reducing odd operands a and b using a,b = abs(a-b),min(a,b) strip factors of 2 from a The Euclidean GCD algorithm, as per Knuth algorithms E and A, repeatedly computes the quotient q = floor(a/b) and replaces a,b by v, u - q v. The binary algorithm has so far been found to be faster than the Euclidean algorithm everywhere. One reason the binary method does well is that the implied quotient at each step is usually small, so often only one or two subtractions are needed to get the same effect as a division. Quotients 1, 2 and 3 for example occur 67.7% of the time, see Knuth section 4.5.3 Theorem E. When the implied quotient is large, meaning b is much smaller than a, then a division is worthwhile. This is the basis for the initial a mod b reductions in 'mpn_gcd' and 'mpn_gcd_1' (the latter for both Nx1 and 1x1 cases). But after that initial reduction, big quotients occur too rarely to make it worth checking for them. The final 1x1 GCD in 'mpn_gcd_1' is done in the generic C code as described above. For two N-bit operands, the algorithm takes about 0.68 iterations per bit. For optimum performance some attention needs to be paid to the way the factors of 2 are stripped from a. Firstly it may be noted that in twos complement the number of low zero bits on a-b is the same as b-a, so counting or testing can begin on a-b without waiting for abs(a-b) to be determined. A loop stripping low zero bits tends not to branch predict well, since the condition is data dependent. But on average there's only a few low zeros, so an option is to strip one or two bits arithmetically then loop for more (as done for AMD K6). Or use a lookup table to get a count for several bits then loop for more (as done for AMD K7). An alternative approach is to keep just one of a or b odd and iterate a,b = abs(a-b), min(a,b) a = a/2 if even b = b/2 if even This requires about 1.25 iterations per bit, but stripping of a single bit at each step avoids any branching. Repeating the bit strip reduces to about 0.9 iterations per bit, which may be a worthwhile tradeoff. Generally with the above approaches a speed of perhaps 6 cycles per bit can be achieved, which is still not terribly fast with for instance a 64-bit GCD taking nearly 400 cycles. It's this sort of time which means it's not usually advantageous to combine a set of divisibility tests into a GCD. Currently, the binary algorithm is used for GCD only when N < 3.  File: gmp.info, Node: Lehmer's Algorithm, Next: Subquadratic GCD, Prev: Binary GCD, Up: Greatest Common Divisor Algorithms 15.3.2 Lehmer's algorithm ------------------------- Lehmer's improvement of the Euclidean algorithms is based on the observation that the initial part of the quotient sequence depends only on the most significant parts of the inputs. The variant of Lehmer's algorithm used in GMP splits off the most significant two limbs, as suggested, e.g., in "A Double-Digit Lehmer-Euclid Algorithm" by Jebelean (*note References::). The quotients of two double-limb inputs are collected as a 2 by 2 matrix with single-limb elements. This is done by the function 'mpn_hgcd2'. The resulting matrix is applied to the inputs using 'mpn_mul_1' and 'mpn_submul_1'. Each iteration usually reduces the inputs by almost one limb. In the rare case of a large quotient, no progress can be made by examining just the most significant two limbs, and the quotient is computed using plain division. The resulting algorithm is asymptotically O(N^2), just as the Euclidean algorithm and the binary algorithm. The quadratic part of the work are the calls to 'mpn_mul_1' and 'mpn_submul_1'. For small sizes, the linear work is also significant. There are roughly N calls to the 'mpn_hgcd2' function. This function uses a couple of important optimizations: * It uses the same relaxed notion of correctness as 'mpn_hgcd' (see next section). This means that when called with the most significant two limbs of two large numbers, the returned matrix does not always correspond exactly to the initial quotient sequence for the two large numbers; the final quotient may sometimes be one off. * It takes advantage of the fact the quotients are usually small. The division operator is not used, since the corresponding assembler instruction is very slow on most architectures. (This code could probably be improved further, it uses many branches that are unfriendly to prediction). * It switches from double-limb calculations to single-limb calculations half-way through, when the input numbers have been reduced in size from two limbs to one and a half.  File: gmp.info, Node: Subquadratic GCD, Next: Extended GCD, Prev: Lehmer's Algorithm, Up: Greatest Common Divisor Algorithms 15.3.3 Subquadratic GCD ----------------------- For inputs larger than 'GCD_DC_THRESHOLD', GCD is computed via the HGCD (Half GCD) function, as a generalization to Lehmer's algorithm. Let the inputs a,b be of size N limbs each. Put S = floor(N/2) + 1. Then HGCD(a,b) returns a transformation matrix T with non-negative elements, and reduced numbers (c;d) = T^{-1} (a;b). The reduced numbers c,d must be larger than S limbs, while their difference abs(c-d) must fit in S limbs. The matrix elements will also be of size roughly N/2. The HGCD base case uses Lehmer's algorithm, but with the above stop condition that returns reduced numbers and the corresponding transformation matrix half-way through. For inputs larger than 'HGCD_THRESHOLD', HGCD is computed recursively, using the divide and conquer algorithm in "On Schönhage's algorithm and subquadratic integer GCD computation" by Möller (*note References::). The recursive algorithm consists of these main steps. * Call HGCD recursively, on the most significant N/2 limbs. Apply the resulting matrix T_1 to the full numbers, reducing them to a size just above 3N/2. * Perform a small number of division or subtraction steps to reduce the numbers to size below 3N/2. This is essential mainly for the unlikely case of large quotients. * Call HGCD recursively, on the most significant N/2 limbs of the reduced numbers. Apply the resulting matrix T_2 to the full numbers, reducing them to a size just above N/2. * Compute T = T_1 T_2. * Perform a small number of division and subtraction steps to satisfy the requirements, and return. GCD is then implemented as a loop around HGCD, similarly to Lehmer's algorithm. Where Lehmer repeatedly chops off the top two limbs, calls 'mpn_hgcd2', and applies the resulting matrix to the full numbers, the sub-quadratic GCD chops off the most significant third of the limbs (the proportion is a tuning parameter, and 1/3 seems to be more efficient than, e.g, 1/2), calls 'mpn_hgcd', and applies the resulting matrix. Once the input numbers are reduced to size below 'GCD_DC_THRESHOLD', Lehmer's algorithm is used for the rest of the work. The asymptotic running time of both HGCD and GCD is O(M(N)*log(N)), where M(N) is the time for multiplying two N-limb numbers.  d31 4 a34 4 cofactors x and y satisfying a*x+b*y=gcd(a,b). All the algorithms used for plain GCD are extended to handle this case. The binary algorithm is used only for single-limb GCDEXT. Lehmer's algorithm is used for sizes up to 'GCDEXT_DC_THRESHOLD'. Above this threshold, GCDEXT is d36 1 a36 1 track of the cofactors. This gives the same asymptotic running time as d41 1 a41 1 This makes the tuning of the chopping-point more difficult. The current d43 5 a47 5 HGCD in the first iteration, and the most significant two thirds for the remaining calls. This strategy could surely be improved. Also the stop condition for the loop, where Lehmer's algorithm is invoked once the inputs are reduced below 'GCDEXT_DC_THRESHOLD', could maybe be improved by taking into account the current size of the cofactors. d58 1 a58 1 'mpz_jacobi' and 'mpz_kronecker' are currently implemented with a d65 4 a68 4 'mpz_kronecker_ui' and friends, an initial reduction is done with either 'mpn_mod_1' or 'mpn_modexact_1_odd', followed by the binary algorithm on a single limb. The binary algorithm is well suited to a single limb, and the whole calculation in this case is quite efficient. d70 2 a71 2 In all the routines sign changes for the result are accumulated using some bit twiddling, avoiding table lookups or conditional jumps. d90 1 a90 1 Normal 'mpz' or 'mpf' powering uses a simple binary algorithm, d93 2 a94 2 variant described there is used rather than algorithm A, since it's just as easy and can be done with somewhat less temporary memory. d138 2 a139 2 that either the high or second highest bit is set. In GMP, k is kept on a limb boundary and the input is left shifted (by an even number of d162 4 a165 4 The algorithm is expressed in a divide and conquer form, but as noted in the paper it can also be viewed as a discrete variant of Newton's method, or as a variation on the schoolboy method (no longer taught) for square roots two digits at a time. d168 2 a169 2 of r and u need to be calculated to determine whether an adjustment to s is required. This optimization is not currently implemented. d178 2 a179 2 'mpn_sqrtrem' is used for both 'mpz_sqrt' and 'mpf_sqrt'. The extended precision given by 'mpf_sqrt_ui' is obtained by padding with zero limbs. d196 4 a199 4 above the true root. The iteration converges quadratically when started from a good approximation. When n is large more initial bits are needed to get good convergence. The current implementation is not particularly well optimized. d210 3 a212 3 'mpz_perfect_square_p' first tests the input mod 256, which means just examining the low byte. Only 44 different values occur for squares mod 256, so 82.8% of inputs can be immediately identified as d215 3 a217 3 On a 32-bit system similar tests are done mod 9, 5, 7, 13 and 17, for a total 99.25% of inputs identified as non-squares. On a 64-bit system 97 is tested too, for a total 99.62%. d221 1 a221 1 additions (see 'mpn_mod_34lsub1'). d223 2 a224 2 When nails are in use moduli are instead selected by the 'gen-psqr.c' program and applied with an 'mpn_mod_1'. The same 2^24-1 or 2^48-1 d228 6 a233 6 In any case each modulus is applied to the 'mpn_mod_34lsub1' or 'mpn_mod_1' remainder and a table lookup identifies non-squares. By using a "modexact" style calculation, and suitably permuted tables, just one multiply each is required, see the code for details. Moduli are also combined to save operations, so long as the lookup tables don't become too big. 'gen-psqr.c' does all the pre-calculations. d240 1 a240 1 Clearly more residue tests could be done, 'mpz_perfect_square_p' only d253 1 a253 1 Currently 'mpz_perfect_power_p' is implemented using repeated Nth root d283 2 a284 2 Conversions from binary to a power-of-2 radix use a simple and fast O(N) bit extraction algorithm. d287 1 a287 1 Sizes below 'GET_STR_PRECOMPUTE_THRESHOLD' use a basic O(N^2) method. d296 1 a296 1 Above 'GET_STR_PRECOMPUTE_THRESHOLD' a sub-quadratic algorithm is d301 9 a309 9 turn divided by the second highest power, and so on recursively. When a piece has been divided down to less than 'GET_STR_DC_THRESHOLD' limbs, the basecase algorithm described above is used. The advantage of this algorithm is that big divisions can make use of the sub-quadratic divide and conquer division (*note Divide and Conquer Division::), and big divisions tend to have less overheads than lots of separate single limb divisions anyway. But in any case the cost of calculating the powers b^(n*2^i) must first be overcome. d311 1 a311 1 'GET_STR_PRECOMPUTE_THRESHOLD' and 'GET_STR_DC_THRESHOLD' represent d313 1 a313 1 division to cut the input in half. 'GET_STR_PRECOMPUTE_THRESHOLD' d315 1 a315 1 'GET_STR_DC_THRESHOLD' assumes that's already available, which is the d322 2 a323 2 multiplied by 'chars_per_bit_exactly' from 'mp_bases', rounding up. The result is either correct or one too big. d329 7 a335 1 identify 99... might well be almost as much as a full conversion. d365 1 a365 1 below 'SET_STR_PRECOMPUTE_THRESHOLD' use a basic O(N^2) method. Groups d368 4 a371 4 the result by multiplying by b^n and adding. This saves multi-precision operations, as per Knuth section 4.4 part E (*note References::). Some special case code is provided for decimal, giving the compiler a chance to optimize multiplications by 10. d373 1 a373 1 Above 'SET_STR_PRECOMPUTE_THRESHOLD' a sub-quadratic algorithm is d377 2 a378 2 x*b^(2n)+y. This continues until a single block remains, that being the result. d383 1 a383 1 'SET_STR_PRECOMPUTE_THRESHOLD' usually ends up quite big, around 5000 d386 1 a386 1 'SET_STR_PRECOMPUTE_THRESHOLD' is based on the input digits (and d392 4 a395 3 The main reason 'SET_STR_PRECOMPUTE_THRESHOLD' is so much bigger than the corresponding 'GET_STR_PRECOMPUTE_THRESHOLD' is that 'mpn_mul_1' is much faster than 'mpn_divrem_1' (often by a factor of 5, or more). d418 1 a418 1 The primality testing in 'mpz_probab_prime_p' (*note Number Theoretic d429 4 a432 4 composites are known as strong pseudoprimes to base x. No n is a strong pseudoprime to more than 1/4 of all bases (see Knuth exercise 22), hence with x chosen at random there's no more than a 1/4 chance a "probable prime" will in fact be composite. d444 2 a445 2 Factorials are calculated by a combination of two algorithms. An idea is shared among them: to compute the odd part of the factorial; a final d451 1 a451 1 part of x, and so on recursively. The procedure can be best illustrated d454 1 a454 1 23! = (23.21.19.17.15.13.11.9.7.5.3)(11.9.7.5.3)(5.3)2^{19} d460 1 a460 1 When n is larger, computation pass trough prime sieving. An helper d469 1 a469 1 Where p ranges on odd prime numbers. The exponent k is chosen to d471 1 a471 1 representation of [n/2]. The function L(p,n) can be defined as zero d481 1 a481 1 using the recursion implied by n!=[n/2]!^2*msf(n)*2^k. The recursion d484 5 a488 5 Both the above algorithms use binary splitting to compute the product of many small factors. At first as many products as possible are accumulated in a single register, generating a list of factors that fit in a machine word. This list is then split into halves, and the product is computed recursively. d492 2 a493 2 used. And even below the Karatsuba threshold a big block of work can be more efficient for the basecase algorithm. d512 3 a514 3 The numerators n-k+i and denominators i are first accumulated into as many fit a limb, to save multi-precision operations, though for 'mpz_bin_ui' this applies only to the divisors, since n is an 'mpz_t' d523 1 a523 1 The Fibonacci functions 'mpz_fib_ui' and 'mpz_fib2_ui' are designed for d526 3 a528 3 For small n, a table of single limb values in '__gmp_fib_table' is used. On a 32-bit limb this goes up to F[47], or on a 64-bit limb up to F[93]. For convenience the table starts at F[-1]. d551 5 a555 5 Using a table avoids lots of calculations on small numbers, and makes small n go fast. A bigger table would make more small n go fast, it's just a question of balancing size against desired speed. For GMP the code is kept compact, with the emphasis primarily on a good powering algorithm. d557 1 a557 1 'mpz_fib2_ui' returns both F[n] and F[n-1], but 'mpz_fib_ui' is only d570 1 a570 1 'mpz_fib_ui' and the internal 'mpn_fib2_ui' for how this is done. d578 1 a578 1 'mpz_lucnum2_ui' derives a pair of Lucas numbers from a pair of d584 1 a584 1 'mpz_lucnum_ui' is only interested in L[n], and some work can be d591 1 a591 1 Fibonacci numbers, similar to what 'mpz_fib_ui' does. d601 1 a601 1 For the 'urandomb' functions, random numbers are generated simply by d606 1 a606 1 For the 'urandomm' functions, random numbers in a range 0<=R48 bit pieces is convenient. With some care though six 21x32->53 bit products can be used, if one of the lower two 21-bit pieces also uses the sign bit. For the 'mpn_mul_1' family of functions on a 64-bit machine, the invariant single limb is split at the start, into 3 or 4 pieces. Inside the loop, the bignum operand is split into 32-bit pieces. Fast d855 2 a856 2 unit, zero-extending to 64-bits, then transferring to the floating point unit back via memory is the only option. d858 3 a860 3 Converting partial products back to 64-bit limbs is usually best done as a signed conversion. Since all values are smaller than 2^53, signed and unsigned are the same, but most processors lack unsigned d865 2 a866 2 Here is a diagram showing 16x32 bit products for an 'mpn_mul_1' or 'mpn_addmul_1' with a 64-bit limb. The single limb operand V is split d905 3 a907 3 p32 and r32 can be summed using floating-point addition, and likewise p48 and r48. p00 and p16 can be summed with r64 and r80 from the previous iteration. d943 2 a944 2 adds besides. But of course if say the SIMD form is fully pipelined and uses less instruction decoding then it may still be worthwhile. d946 4 a949 4 On the x86 chips, MMX has so far found a use in 'mpn_rshift' and 'mpn_lshift', and is used in a special case for 16-bit multipliers in the P55 'mpn_mul_1'. SSE2 is used for Pentium 4 'mpn_mul_1', 'mpn_addmul_1', and 'mpn_submul_1'. d959 2 a960 2 use in the present iteration but the next, thereby allowing extra cycles for the data to arrive from memory. d971 4 a974 4 If the latency of some instruction is greater than the loop time then it will be necessary to unroll, so one register has a result ready to use while another (or multiple others) are still in progress. (*note Assembly Loop Unrolling::). d986 1 a986 1 Judicious use of 'm4' macros can help avoid lots of duplication in the d992 2 a993 2 start, the termination condition becomes when the counter C is less than 0 (and the count of remaining limbs is C+N). d996 2 a997 2 be established with a shift and mask. This is convenient if also making a computed jump into the middle of a large loop. d1003 2 a1004 2 Care will be wanted that it isn't too much slower than the unrolled part. d1011 1 a1011 1 * A 'switch' statement, providing separate code for each possible d1038 3 a1040 3 requirements. If an instruction can use either of two units, like U0 or U1 then make a category "U0/U1". Count the total using each unit (or combined unit), and count all instructions. d1048 4 a1051 4 Suppose the loop time is N, then make N issue buckets, with the final loop branch at the end of the last. Now fill the buckets with dummy instructions using the functional units desired. Run this to make sure the intended speed is reached. d1054 4 a1057 4 the slow but correct loop you started with. The first will typically be a load instruction. Then the instruction using that value is placed in a bucket an appropriate distance down. Run the loop again, to check it still runs at target speed. d1060 2 a1061 2 few you will need to wrap around from the last bucket back to the top of the loop. If you used the new-register for new-value strategy above d1077 2 a1078 2 The loop will have a minimum number of limbs loaded and processed, so the feed-in code must test if the request size is smaller and skip d1107 1 a1107 1 'mpz_t' variables represent integers using sign and magnitude, in space d1110 1 a1110 1 '_mp_size' d1112 2 a1113 2 negative integer. Zero is represented by '_mp_size' set to zero, in which case the '_mp_d' data is unused. d1115 1 a1115 1 '_mp_d' d1117 4 a1120 4 stored "little endian" as per the 'mpn' functions, so '_mp_d[0]' is the least significant limb and '_mp_d[ABS(_mp_size)-1]' is the most significant. Whenever '_mp_size' is non-zero, the most significant limb is non-zero. d1123 10 a1132 10 instance 'mpz_set_ui' never needs to reallocate, and 'mpz_get_ui' can fetch '_mp_d[0]' unconditionally (though its value is then only wanted if '_mp_size' is non-zero). '_mp_alloc' '_mp_alloc' is the number of limbs currently allocated at '_mp_d', and naturally '_mp_alloc >= ABS(_mp_size)'. When an 'mpz' routine is about to (or might be about to) increase '_mp_size', it checks '_mp_alloc' to see whether there's enough space, and reallocates if not. 'MPZ_REALLOC' is generally used for this. d1134 1 a1134 1 The various bitwise logical functions like 'mpz_and' behave as if d1140 4 a1143 4 Some internal temporary variables are setup with 'MPZ_TMP_INIT' and these have '_mp_d' space obtained from 'TMP_ALLOC' rather than the memory allocation functions. Care is taken to ensure that these are big enough that no reallocation is necessary (since it would have d1146 4 a1149 4 '_mp_size' and '_mp_alloc' are 'int', although 'mp_size_t' is usually a 'long'. This is done to make the fields just 32 bits on some 64 bits systems, thereby saving a few bytes of data space but still providing plenty of range. d1157 1 a1157 1 'mpq_t' variables represent rationals using an 'mpz_t' numerator and d1160 2 a1161 2 The canonical form adopted is denominator positive (and non-zero), no common factors between numerator and denominator, and zero uniquely d1166 4 a1169 4 better to do a few small ones immediately than to delay and have to do a big one later. Knowing the numerator and denominator have no common factors can be used for example in 'mpq_mul' to make only two cross GCDs necessary, not four. d1174 3 a1176 3 Efficiency::, that's left to applications. The 'mpq_t' framework might still suit, with 'mpq_numref' and 'mpq_denref' for direct access to the numerator and denominator, or of course 'mpz_t' variables can be used d1188 1 a1188 1 'mpf_t' floats have a variable precision mantissa and a single a1203 1 d1206 1 a1206 1 '_mp_size' d1208 3 a1210 3 representing a negative value. Zero is represented by '_mp_size' and '_mp_exp' both set to zero, and in that case the '_mp_d' data is unused. (In the future '_mp_exp' might be undefined when d1213 1 a1213 1 '_mp_prec' d1215 1 a1215 1 aim is to produce '_mp_prec' limbs of result (the most significant d1218 1 a1218 1 '_mp_d' d1220 3 a1222 3 mantissa. These are stored "little endian" as per the 'mpn' functions, so '_mp_d[0]' is the least significant limb and '_mp_d[ABS(_mp_size)-1]' the most significant. d1228 1 a1228 1 '_mp_prec+1' limbs are allocated to '_mp_d', the extra limb being d1230 1 a1230 1 calculation, only in a change of precision with 'mpf_set_prec'. d1232 1 a1232 1 '_mp_exp' d1243 1 a1243 1 '{_mp_d,_mp_size}' data are treated as zero. d1245 3 a1247 3 The '_mp_size' and '_mp_prec' fields are 'int', although the 'mp_size_t' type is usually a 'long'. The '_mp_exp' field is usually 'long'. This is done to make some fields just 32 bits on some 64 bits d1255 5 a1259 4 The least significant limbs '_mp_d[0]' etc can be zero, though such low zeros can always be ignored. Routines likely to produce low zeros check and avoid them to save time in subsequent calculations, but for most routines they're quite unlikely and aren't checked. d1262 1 a1262 1 The '_mp_size' count of limbs in use can be less than '_mp_prec' if d1264 1 a1264 1 values or small integers stored in a high precision 'mpf_t' can d1267 5 a1271 5 '_mp_size' can also be greater than '_mp_prec'. Firstly a value is allowed to use all of the '_mp_prec+1' limbs available at '_mp_d', and secondly when 'mpf_set_prec_raw' lowers '_mp_prec' it leaves '_mp_size' unchanged and so the size can be arbitrarily bigger than '_mp_prec'. d1274 1 a1274 1 All rounding is done on limb boundaries. Calculating '_mp_prec' d1278 3 a1280 2 The use of simple "trunc" rounding towards zero is efficient, since there's no need to examine extra limbs and increment or decrement. d1284 1 a1284 1 operations like 'mpf_add' and 'mpf_mul'. When differing exponents d1288 1 a1288 1 Of course 'mpf_mul_2exp' and 'mpf_div_2exp' will require bit d1293 5 a1297 5 Use of '_mp_prec+1' Limbs The extra limb on '_mp_d' ('_mp_prec+1' rather than just '_mp_prec') helps when an 'mpf' routine might get a carry from its operation. 'mpf_add' for instance will do an 'mpn_add' of '_mp_prec' limbs. If there's no carry then that's the result, but d1299 1 a1299 1 '_mp_size' becomes '_mp_prec+1'. d1301 2 a1302 2 Whenever '_mp_prec+1' limbs are held in a variable, the low limb is not needed for the intended precision, only the '_mp_prec' high d1305 1 a1305 1 limbs they need, and this will be '_mp_prec' if their target has d1310 4 a1313 4 Copy functions like 'mpf_set' will retain a full '_mp_prec+1' limbs if available. This ensures that a variable which has '_mp_size' equal to '_mp_prec+1' will get its full exact value copied. Strictly speaking this is unnecessary since only '_mp_prec' limbs d1315 1 a1315 1 considered that an 'mpf_set' from one variable into another of the d1319 2 a1320 2 '__GMPF_BITS_TO_PREC' converts an application requested precision to an '_mp_prec'. The value in bits is rounded up to a whole limb d1322 1 a1322 1 '_mp_d' is only non-zero and therefore might contain only one bit. d1324 4 a1327 4 '__GMPF_PREC_TO_BITS' does the reverse conversion, and removes the extra limb from '_mp_prec' before converting to bits. The net effect of reading back with 'mpf_get_prec' is simply the precision rounded up to a multiple of 'mp_bits_per_limb'. d1330 7 a1336 6 non-zero is in addition to the extra limb allocated to '_mp_d'. For example with a 32-bit limb, an application request for 250 bits will be rounded up to 8 limbs, then an extra added for the high being only non-zero, giving an '_mp_prec' of 9. '_mp_d' then gets 10 limbs allocated. Reading back with 'mpf_get_prec' will take '_mp_prec' subtract 1 limb and multiply by 32, giving 256 bits. d1340 1 a1340 1 holding at least 65 bits, but for the purposes of 'mpf_t' it's d1349 1 a1349 1 'mpz_out_raw' uses the following format. d1356 3 a1358 3 number of subsequent data bytes, or the twos complement negative of that when a negative integer is represented. The data bytes are the absolute value of the integer, written most significant byte first. d1364 1 a1364 1 multiple of the limb size. 'mpz_inp_raw' will still accept this, for d1370 2 a1371 2 reading or writing, so neither a big endian nor little endian system can just read and write '_mp_d'. d1380 1 a1380 1 'a=b+c' turns into a simple call to 'mpz_add' etc. For 'mpf_class' the d1382 1 a1382 1 any temporaries within a statement like 'f=w*x+y*z'. These are d1394 1 a1394 4 static void eval(mpf_t f, const mpf_t g, const mpf_t h) { mpf_add(f, g, h); } d1406 1 a1406 1 The seemingly redundant '__gmp_expr<__gmp_binary_expr<...>>' is used d1408 2 a1409 2 type. In fact even 'mpf_class' etc are 'typedef' specializations of '__gmp_expr'. d1411 1 a1411 1 Next we define assignment of '__gmp_expr' to 'mpf_class'. d1427 3 a1429 3 where 'expr.val1' and 'expr.val2' are references to the expression's operands (here 'expr' is the '__gmp_binary_expr' stored within the '__gmp_expr'). d1432 3 a1434 3 assignment, when the required precision (that of 'f') is known. Furthermore the target 'mpf_t' is now available, thus we can call 'mpf_add' directly with 'f' as the output argument. d1449 1 a1449 1 And the corresponding specializations of '__gmp_expr::eval': d1462 1 a1462 1 complexity and all subexpressions are evaluated to the precision of 'f'. d1473 1 a1473 1 GMP. Here is a list in chronological order on first contribution: d1485 1 a1485 1 'mpz_probab_prime_p'. d1490 4 a1493 4 development of GMP 2, with his comparisons between bignum packages. The ECMNET project Paul is organizing was a driving force behind many of the optimizations in GMP 3. Paul also wrote the new GMP 4.3 nth root code (with Torbjörn). d1496 3 a1498 3 do Sul) contributed now defunct versions of 'mpz_gcd', 'mpz_divexact', 'mpn_gcd', and 'mpn_bdivmod', partially supported by CNPq (Brazil) grant 301314194-2. d1504 2 a1505 2 Joachim Hollman was involved in the design of the 'mpf' interface, and in the 'mpz' design revisions for version 2. d1507 2 a1508 2 Bennet Yee contributed the initial versions of 'mpz_jacobi' and 'mpz_legendre'. d1510 2 a1511 2 Andreas Schwab contributed the files 'mpn/m68k/lshift.S' and 'mpn/m68k/rshift.S' (now in '.asm' form). d1519 2 a1520 2 University provided significant inspiration during several phases of the GMP development. His mathematical expertise helped improve several d1526 2 a1527 2 Kevin Ryde worked on a large number of things: optimized x86 code, m4 asm macros, parameter tuning, speed measuring, the configure system, d1530 3 a1532 2 interface, demo expression parser, the algorithms chapter in the manual, 'gmpasm-mode.el', and various miscellaneous improvements elsewhere. d1538 2 a1539 2 Gerardo Ballabio wrote the 'gmpxx.h' C++ class interface and the C++ 'istream' input routines. d1541 1 a1541 1 Jason Moxham rewrote 'mpz_fac_ui'. d1558 6 a1563 7 Marco Bodrato helped implement the new Toom multiply code for GMP 4.3 and implemented most of the new Toom multiply and squaring code for 5.0. He is the main author of the current mpn_mulmod_bnm1, mpn_mullo_n, and mpn_sqrlo. Marco also wrote the functions mpn_invert and mpn_invertappr, and improved the speed of integer root extraction. He is the author of the current combinatorial functions: binomial, factorial, multifactorial, primorial. d1565 1 a1565 1 David Harvey suggested the internal function 'mpn_bdiv_dbm1', d1568 2 a1569 2 'mpn_mul_basecase'. He wrote the internal middle product functions 'mpn_mulmid_basecase', 'mpn_toom42_mulmid', 'mpn_mulmid_n' and related d1572 1 a1572 1 Martin Boij wrote 'mpn_perfect_power_p'. d1574 2 a1575 2 Marc Glisse improved 'gmpxx.h': use fewer temporaries (faster), specializations of 'numeric_limits' and 'common_type', C++11 features d1577 4 a1580 11 from 'mpq_class' to 'mpz_class' explicit, optimize operations where one argument is a small compile-time constant, replace some heap allocations by stack allocations. He also fixed the eofbit handling of C++ streams, and removed one division from 'mpq/aors.c'. David S Miller wrote assembly code for SPARC T3 and T4. Mark Sofroniou cleaned up the types of mul_fft.c, letting it work for huge operands. Ulrich Weigand ported GMP to the powerpc64le ABI. d1590 1 a1590 1 The development of GMP 2, 3, and 4.0 was supported in part by the IDA d1593 2 a1594 5 The development of GMP 4.3, 5.0, and 5.1 was supported in part by the Swedish Foundation for Strategic Research. Thanks go to Hans Thorsen for donating an SGI system for the GMP test system environment. d1611 1 a1611 1 d1615 1 a1615 1 d1619 1 a1619 1 d1626 1 a1626 1 d1628 4 a1631 4 * Richard M. Stallman and the GCC Developer Community, "Using the GNU Compiler Collection", Free Software Foundation, 2008, available online , and in the GCC package d1638 2 a1639 2 225-252. Also available online as INRIA Research Report 4475, June 2002, d1643 1 a1643 1 d1648 1 a1648 1 . d1652 1 a1652 1 d1660 1 a1660 1 d1664 1 a1664 1 d1668 1 a1668 1 d1672 1 a1672 1 d1674 5 a1678 4 * Tudor Jebelean, "A Double-Digit Lehmer-Euclid Algorithm for Finding the GCD of Long Integers", Journal of Symbolic Computation, volume 19, 1995, pp. 145-157. Technical report version also available d1683 1 a1683 1 d1689 1 a1689 1 d1699 2 a1700 2 GCD computation", in Mathematics of Computation, volume 77, January 2008, pp. 589-607. d1715 1 a1715 1 d1719 1 a1719 1 d1735 2 a1736 2 Copyright © 2000-2002, 2007, 2008 Free Software Foundation, Inc. d1761 2 a1762 2 of subject matter or whether it is published as a printed book. We recommend this License principally for works whose purpose is d1768 2 a1769 2 that contains a notice placed by the copyright holder saying it can be distributed under the terms of this License. Such a notice d1773 3 a1775 3 of the public is a licensee, and is addressed as "you". You accept the license if you copy, modify or distribute the work in a way requiring permission under copyright law. d1793 6 a1798 6 titles are designated, as being those of Invariant Sections, in the notice that says that the Document is released under this License. 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If your rights have been terminated and not permanently reinstated, receipt of a copy of some or all of the same material does not give you any rights to use it. d2130 1 a2130 1 10. FUTURE REVISIONS OF THIS LICENSE d2136 1 a2136 1 . d2143 5 a2147 5 published (not as a draft) by the Free Software Foundation. If the Document does not specify a version number of this License, you may choose any version ever published (not as a draft) by the Free Software Foundation. If the Document specifies that a proxy can decide which future versions of this License can be used, that d2151 1 a2151 1 11. RELICENSING d2181 1 d2198 1 a2198 1 Texts, replace the "with...Texts." line with this: d2209 3 a2211 3 recommend releasing these examples in parallel under your choice of free software license, such as the GNU General Public License, to permit their use in free software. d2224 10 a2233 10 * --build: Build Options. (line 51) * --disable-fft: Build Options. (line 307) * --disable-shared: Build Options. (line 44) * --disable-static: Build Options. (line 44) * --enable-alloca: Build Options. (line 273) * --enable-assert: Build Options. (line 313) * --enable-cxx: Build Options. (line 225) * --enable-fat: Build Options. (line 160) * --enable-profiling: Build Options. (line 317) * --enable-profiling <1>: Profiling. (line 6) d2235 1 a2235 1 * --host: Build Options. (line 65) d2240 1 a2240 1 (line 94) d2242 4 a2245 4 (line 150) * ABI: Build Options. (line 167) * ABI <1>: ABI and ISA. (line 6) * About this manual: Introduction to GMP. (line 57) a2246 1 * AIX: ABI and ISA. (line 174) d2249 1 d2251 1 a2251 1 * alloca: Build Options. (line 273) d2254 1 a2254 1 * Anonymous FTP of latest version: Introduction to GMP. (line 37) a2255 1 * Arithmetic functions: Integer Arithmetic. (line 6) d2258 1 d2278 3 a2280 5 * Assertion checking: Build Options. (line 313) * Assertion checking <1>: Debugging. (line 78) * Assignment functions: Assigning Integers. (line 6) * Assignment functions <1>: Simultaneous Integer Init & Assign. (line 6) d2286 2 d2293 1 a2293 1 (line 124) d2299 1 a2299 1 (line 39) d2301 1 a2301 1 * Bit shift right: Integer Division. (line 74) d2313 1 a2313 1 * Build system: Build Options. (line 51) d2316 2 a2317 2 * C compiler: Build Options. (line 178) * C++ compiler: Build Options. (line 249) d2324 6 a2329 6 * C++ support: Build Options. (line 225) * CC: Build Options. (line 178) * CC_FOR_BUILD: Build Options. (line 212) * CFLAGS: Build Options. (line 178) * Checker: Debugging. (line 114) * checkergcc: Debugging. (line 121) a2333 1 * Comparison functions: Integer Comparisons. (line 6) d2336 1 d2342 1 a2342 1 * Congruence functions: Integer Division. (line 150) d2350 3 a2352 2 * Conversion functions: Converting Integers. (line 6) * Conversion functions <1>: Rational Conversions. a2353 1 * Conversion functions <2>: Converting Floats. (line 6) d2355 4 a2358 5 * CPPFLAGS: Build Options. (line 204) * CPU types: Introduction to GMP. (line 24) * CPU types <1>: Build Options. (line 107) * Cross compiling: Build Options. (line 65) * Cryptography functions, low-level: Low-level Functions. (line 507) d2360 2 a2361 2 * CXX: Build Options. (line 249) * CXXFLAGS: Build Options. (line 249) d2363 1 a2363 1 (line 57) d2372 1 a2372 2 * Divisibility functions: Integer Division. (line 136) * Divisibility functions <1>: Integer Division. (line 150) a2374 1 * Division functions: Integer Division. (line 6) d2377 4 a2380 3 * DJGPP: Notes for Particular Systems. (line 57) * DJGPP <1>: Known Build Problems. d2383 3 a2385 3 (line 70) * DocBook: Build Options. (line 340) * Documentation formats: Build Options. (line 333) d2388 1 a2388 1 * DVI: Build Options. (line 336) d2391 1 a2391 1 * Exact division functions: Integer Division. (line 125) a2395 1 * Execution profiling: Build Options. (line 317) d2397 2 a2400 1 * Exponentiation functions <1>: Float Arithmetic. (line 41) d2404 1 a2404 5 (line 15) * Expression parsing demo <1>: Demonstration Programs. (line 17) * Expression parsing demo <2>: Demonstration Programs. (line 19) d2406 1 a2406 1 (line 43) d2408 1 a2408 1 (line 104) d2411 1 a2411 1 (line 112) d2413 1 a2413 1 (line 22) d2415 3 a2417 3 * Fat binary: Build Options. (line 160) * FFT multiplication: Build Options. (line 307) * FFT multiplication <1>: FFT Multiplication. (line 6) d2421 1 a2421 1 (line 132) a2422 1 * Float assignment functions: Assigning Floats. (line 6) d2425 1 a2429 1 * Float initialization functions: Initializing Floats. (line 6) d2432 1 d2441 1 a2441 1 * Float sign tests: Float Comparison. (line 34) d2453 3 a2455 7 * FreeBSD: Notes for Particular Systems. (line 43) * FreeBSD <1>: Notes for Particular Systems. (line 52) * frexp: Converting Integers. (line 43) * frexp <1>: Converting Floats. (line 24) * FTP of latest version: Introduction to GMP. (line 37) d2458 1 a2458 1 * GCC Checker: Debugging. (line 114) d2462 1 a2462 1 (line 43) d2464 3 a2466 3 (line 26) * GDB: Debugging. (line 57) * Generic C: Build Options. (line 151) d2468 1 a2468 1 (line 28) d2475 1 a2475 1 * GNU Debugger: Debugging. (line 57) d2484 1 a2484 1 (line 26) d2489 7 a2495 6 * Heap problems: Debugging. (line 23) * Home page: Introduction to GMP. (line 33) * Host system: Build Options. (line 65) * HP-UX: ABI and ISA. (line 76) * HP-UX <1>: ABI and ISA. (line 114) * HPPA: ABI and ISA. (line 76) a2496 2 * I/O functions <1>: I/O of Rationals. (line 6) * I/O functions <2>: I/O of Floats. (line 6) d2498 2 a2499 2 (line 150) * IA-64: ABI and ISA. (line 114) d2506 1 a2506 1 * Initialization functions: Initializing Integers. d2508 1 a2508 1 * Initialization functions <1>: Simultaneous Integer Init & Assign. d2510 2 a2511 1 * Initialization functions <2>: Initializing Rationals. d2513 1 a2513 2 * Initialization functions <3>: Initializing Floats. (line 6) * Initialization functions <4>: Simultaneous Float Init & Assign. d2515 1 a2515 1 * Initialization functions <5>: Random State Initialization. d2518 2 a2519 3 * Input functions: I/O of Integers. (line 6) * Input functions <1>: I/O of Rationals. (line 6) * Input functions <2>: I/O of Floats. (line 6) d2522 1 d2530 2 a2531 2 * Integer assignment functions: Assigning Integers. (line 6) * Integer assignment functions <1>: Simultaneous Integer Init & Assign. d2545 2 a2548 2 * Integer initialization functions <1>: Simultaneous Integer Init & Assign. (line 6) d2562 1 a2562 1 (line 65) d2566 1 a2566 2 (line 70) * IRIX: ABI and ISA. (line 139) d2569 1 d2574 1 a2574 1 (line 79) d2580 1 a2580 1 (line 91) d2582 1 a2582 1 * Latest version of GMP: Introduction to GMP. (line 37) d2584 1 a2584 1 (line 64) d2586 1 a2586 1 (line 64) d2588 1 a2588 1 (line 82) a2606 2 (line 18) * Linear congruential random numbers <1>: Random State Initialization. a2612 1 * Low-level functions for cryptography: Low-level Functions. (line 507) d2616 1 a2616 1 (line 143) d2619 3 a2621 3 * Mailing lists: Introduction to GMP. (line 44) * Malloc debugger: Debugging. (line 29) * Malloc problems: Debugging. (line 23) d2629 2 a2630 2 (line 57) * MIPS: ABI and ISA. (line 139) d2636 1 a2636 1 (line 156) d2638 1 a2638 1 (line 70) d2641 1 a2641 1 * MPN_PATH: Build Options. (line 321) d2643 1 a2643 3 (line 57) * MS Windows <1>: Notes for Particular Systems. (line 70) d2645 1 a2645 1 (line 57) d2649 2 a2650 4 * Nails: Low-level Functions. (line 685) * Native compilation: Build Options. (line 51) * NetBSD: Notes for Particular Systems. (line 100) d2654 1 a2654 1 (line 19) d2659 1 a2659 1 * Number sequences: Efficiency. (line 145) d2665 1 a2665 1 (line 79) d2667 1 a2667 1 (line 109) d2673 4 a2677 4 * Output functions <1>: I/O of Rationals. (line 6) * Output functions <2>: I/O of Floats. (line 6) * Output functions <3>: Formatted Output Functions. (line 6) a2683 4 * Parsing expressions demo <1>: Demonstration Programs. (line 17) * Parsing expressions demo <2>: Demonstration Programs. (line 19) d2688 1 a2688 1 * PDF: Build Options. (line 336) d2696 1 a2696 1 (line 28) d2698 5 a2702 5 (line 28) * Postscript: Build Options. (line 336) * Power/PowerPC: Notes for Particular Systems. (line 115) * Power/PowerPC <1>: Known Build Problems. d2705 1 a2705 1 * Powering functions: Integer Exponentiation. d2707 2 a2708 2 * Powering functions <1>: Float Arithmetic. (line 41) * PowerPC: ABI and ISA. (line 173) d2719 1 a2719 1 (line 117) a2728 2 * Random number functions: Integer Random Numbers. (line 6) d2733 2 d2741 1 a2741 1 * Rational arithmetic: Efficiency. (line 111) d2758 1 a2758 1 * Rational sign tests: Comparing Rationals. (line 28) d2763 1 a2763 1 * References: References. (line 5) d2765 1 a2765 1 (line 104) d2770 3 a2772 4 * Root extraction functions: Integer Roots. (line 6) * Root extraction functions <1>: Float Arithmetic. (line 37) * Root testing functions: Integer Roots. (line 28) * Root testing functions <1>: Integer Roots. (line 37) d2778 1 a2778 1 (line 39) d2788 1 a2788 1 (line 65) d2791 3 a2793 3 * Sign tests: Integer Comparisons. (line 28) * Sign tests <1>: Comparing Rationals. (line 28) * Sign tests <2>: Float Comparison. (line 34) a2796 1 * Solaris: ABI and ISA. (line 204) d2799 3 a2801 2 * Solaris <2>: Known Build Problems. (line 77) d2803 2 a2804 4 (line 127) * Sparc <1>: Notes for Particular Systems. (line 132) * Sparc V9: ABI and ISA. (line 204) d2810 4 a2813 4 (line 156) * Stack backtrace: Debugging. (line 49) * Stack overflow: Build Options. (line 273) * Stack overflow <1>: Debugging. (line 7) d2821 1 a2821 1 * Sun: ABI and ISA. (line 204) d2823 1 a2823 1 (line 144) d2826 3 a2828 3 * Temporary memory: Build Options. (line 273) * Texinfo: Build Options. (line 333) * Text input/output: Efficiency. (line 151) d2830 1 a2830 1 * Toom multiplication: Toom 3-Way Multiplication. d2832 1 a2832 1 * Toom multiplication <1>: Toom 4-Way Multiplication. d2834 1 a2834 1 * Toom multiplication <2>: Higher degree Toom'n'half. d2836 1 a2836 1 * Toom multiplication <3>: Other Multiplication. d2849 1 a2849 1 * Valgrind: Debugging. (line 129) d2854 1 a2854 1 * Web page: Introduction to GMP. (line 33) d2856 1 a2856 3 (line 57) * Windows <1>: Notes for Particular Systems. (line 70) d2858 1 a2858 1 (line 150) d2861 1 a2861 1 * XML: Build Options. (line 340) a2871 2 * _mpz_realloc: Integer Special Functions. (line 13) d2873 1 a2873 1 (line 22) d2875 1 a2875 1 (line 23) d2877 1 a2877 1 (line 9) d2879 1 a2879 1 (line 10) d2881 3 a2883 3 (line 11) * abs: C++ Interface Integers. (line 46) d2885 1 a2885 1 (line 47) a2886 2 (line 82) * ceil: C++ Interface Floats. d2888 1 a2888 1 * cmp: C++ Interface Integers. d2890 6 a2895 4 * cmp <1>: C++ Interface Integers. (line 48) * cmp <2>: C++ Interface Rationals. (line 48) d2897 3 a2899 5 (line 49) * cmp <4>: C++ Interface Floats. (line 84) * cmp <5>: C++ Interface Floats. (line 85) d2901 1 a2901 3 (line 95) * gcd: C++ Interface Integers. (line 68) d2903 1 a2903 1 (line 63) d2905 1 a2905 1 (line 56) d2907 1 a2907 1 (line 56) d2909 1 a2909 1 (line 56) d2911 1 a2911 1 (line 28) d2913 7 a2919 7 (line 24) * GMP_LIMB_BITS: Low-level Functions. (line 713) * GMP_NAIL_BITS: Low-level Functions. (line 711) * GMP_NAIL_MASK: Low-level Functions. (line 721) * GMP_NUMB_BITS: Low-level Functions. (line 712) * GMP_NUMB_MASK: Low-level Functions. (line 722) * GMP_NUMB_MAX: Low-level Functions. (line 730) d2921 1 a2921 1 (line 75) d2923 1 a2923 1 (line 77) d2925 5 a2929 1 (line 23) d2931 1 a2931 1 (line 6) d2933 1 a2933 3 (line 44) * gmp_randclass::get_f <1>: C++ Interface Random Numbers. (line 45) a2934 2 (line 37) * gmp_randclass::get_z_bits <1>: C++ Interface Random Numbers. d2937 1 a2937 1 (line 41) d2939 1 a2939 3 (line 11) * gmp_randclass::gmp_randclass <1>: C++ Interface Random Numbers. (line 26) a2940 2 (line 32) * gmp_randclass::seed <1>: C++ Interface Random Numbers. d2945 1 a2945 1 (line 45) d2947 1 a2947 1 (line 6) d2949 1 a2949 1 (line 16) d2951 1 a2951 1 (line 30) d2953 1 a2953 1 (line 12) d2955 1 a2955 1 (line 41) d2957 1 a2957 1 (line 6) d2959 1 a2959 1 (line 8) a2961 4 * GMP_RAND_ALG_DEFAULT: Random State Initialization. (line 50) * GMP_RAND_ALG_LC: Random State Initialization. (line 50) d2963 1 a2963 1 (line 20) d2965 1 a2965 1 (line 44) d2967 1 a2967 1 (line 33) d2969 1 a2969 1 (line 28) d2971 1 a2971 1 (line 6) d2973 1 a2973 1 (line 12) d2975 1 a2975 1 (line 64) d2979 1 a2979 1 (line 29) d2981 2 a2983 2 * gmp_vprintf: Formatted Output Functions. (line 24) d2985 1 a2985 1 (line 21) d2987 1 a2987 1 (line 46) d2989 1 a2989 1 (line 34) d2991 1 a2991 1 (line 29) d2993 16 a3008 6 (line 96) * lcm: C++ Interface Integers. (line 69) * mpf_abs: Float Arithmetic. (line 46) * mpf_add: Float Arithmetic. (line 6) * mpf_add_ui: Float Arithmetic. (line 7) d3010 1 a3010 1 (line 6) d3012 1 a3012 1 (line 19) d3020 2 d3023 1 a3023 1 * mpf_class::fits_ulong_p: C++ Interface Floats. a3024 2 * mpf_class::fits_ushort_p: C++ Interface Floats. (line 93) d3026 1 a3026 1 (line 98) d3028 1 a3028 1 (line 65) d3030 1 a3030 1 (line 120) d3032 1 a3032 1 (line 99) d3034 1 a3034 1 (line 100) d3036 1 a3036 1 (line 102) a3037 2 (line 11) * mpf_class::mpf_class <1>: C++ Interface Floats. a3038 12 * mpf_class::mpf_class <2>: C++ Interface Floats. (line 32) * mpf_class::mpf_class <3>: C++ Interface Floats. (line 33) * mpf_class::mpf_class <4>: C++ Interface Floats. (line 41) * mpf_class::mpf_class <5>: C++ Interface Floats. (line 42) * mpf_class::mpf_class <6>: C++ Interface Floats. (line 44) * mpf_class::mpf_class <7>: C++ Interface Floats. (line 45) d3040 1 a3040 1 (line 59) d3042 1 a3042 1 (line 121) d3044 1 a3044 1 (line 122) d3046 2 d3049 3 a3051 7 * mpf_class::set_str <1>: C++ Interface Floats. (line 105) * mpf_class::swap: C++ Interface Floats. (line 109) * mpf_clear: Initializing Floats. (line 36) * mpf_clears: Initializing Floats. (line 40) * mpf_cmp: Float Comparison. (line 6) d3055 4 a3058 5 * mpf_cmp_z: Float Comparison. (line 7) * mpf_div: Float Arithmetic. (line 28) * mpf_div_2exp: Float Arithmetic. (line 53) * mpf_div_ui: Float Arithmetic. (line 31) * mpf_eq: Float Comparison. (line 17) d3060 1 a3060 1 (line 19) d3062 1 a3062 1 (line 17) d3064 1 a3064 1 (line 21) d3066 1 a3066 1 (line 18) d3068 1 a3068 1 (line 16) d3070 1 a3070 1 (line 20) d3072 10 a3081 11 (line 7) * mpf_get_d: Converting Floats. (line 6) * mpf_get_default_prec: Initializing Floats. (line 11) * mpf_get_d_2exp: Converting Floats. (line 15) * mpf_get_prec: Initializing Floats. (line 61) * mpf_get_si: Converting Floats. (line 27) * mpf_get_str: Converting Floats. (line 36) * mpf_get_ui: Converting Floats. (line 28) * mpf_init: Initializing Floats. (line 18) * mpf_init2: Initializing Floats. (line 25) * mpf_inits: Initializing Floats. (line 30) d3083 1 a3083 1 (line 15) d3085 2 a3087 2 * mpf_init_set_si: Simultaneous Float Init & Assign. (line 17) d3089 1 a3089 1 (line 24) d3091 3 a3093 2 (line 16) * mpf_inp_str: I/O of Floats. (line 38) d3095 7 a3101 7 (line 13) * mpf_mul: Float Arithmetic. (line 18) * mpf_mul_2exp: Float Arithmetic. (line 49) * mpf_mul_ui: Float Arithmetic. (line 19) * mpf_neg: Float Arithmetic. (line 43) * mpf_out_str: I/O of Floats. (line 17) * mpf_pow_ui: Float Arithmetic. (line 39) d3103 18 a3120 18 (line 35) * mpf_reldiff: Float Comparison. (line 28) * mpf_set: Assigning Floats. (line 9) * mpf_set_d: Assigning Floats. (line 12) * mpf_set_default_prec: Initializing Floats. (line 6) * mpf_set_prec: Initializing Floats. (line 64) * mpf_set_prec_raw: Initializing Floats. (line 71) * mpf_set_q: Assigning Floats. (line 14) * mpf_set_si: Assigning Floats. (line 11) * mpf_set_str: Assigning Floats. (line 17) * mpf_set_ui: Assigning Floats. (line 10) * mpf_set_z: Assigning Floats. (line 13) * mpf_sgn: Float Comparison. (line 33) * mpf_sqrt: Float Arithmetic. (line 35) * mpf_sqrt_ui: Float Arithmetic. (line 36) * mpf_sub: Float Arithmetic. (line 11) * mpf_sub_ui: Float Arithmetic. (line 14) * mpf_swap: Assigning Floats. (line 50) d3124 3 a3126 3 (line 8) * mpf_ui_div: Float Arithmetic. (line 29) * mpf_ui_sub: Float Arithmetic. (line 12) d3128 4 a3131 2 (line 25) * mpn_add: Low-level Functions. (line 67) d3133 21 a3153 27 * mpn_add_1: Low-level Functions. (line 62) * mpn_add_n: Low-level Functions. (line 52) * mpn_andn_n: Low-level Functions. (line 462) * mpn_and_n: Low-level Functions. (line 447) * mpn_cmp: Low-level Functions. (line 293) * mpn_cnd_add_n: Low-level Functions. (line 540) * mpn_cnd_sub_n: Low-level Functions. (line 542) * mpn_cnd_swap: Low-level Functions. (line 567) * mpn_com: Low-level Functions. (line 487) * mpn_copyd: Low-level Functions. (line 496) * mpn_copyi: Low-level Functions. (line 492) * mpn_divexact_1: Low-level Functions. (line 231) * mpn_divexact_by3: Low-level Functions. (line 238) * mpn_divexact_by3c: Low-level Functions. (line 240) * mpn_divmod: Low-level Functions. (line 226) * mpn_divmod_1: Low-level Functions. (line 210) * mpn_divrem: Low-level Functions. (line 183) * mpn_divrem_1: Low-level Functions. (line 208) * mpn_gcd: Low-level Functions. (line 301) * mpn_gcdext: Low-level Functions. (line 316) * mpn_gcd_1: Low-level Functions. (line 311) * mpn_get_str: Low-level Functions. (line 371) * mpn_hamdist: Low-level Functions. (line 436) * mpn_iorn_n: Low-level Functions. (line 467) * mpn_ior_n: Low-level Functions. (line 452) * mpn_lshift: Low-level Functions. (line 269) * mpn_mod_1: Low-level Functions. (line 264) d3157 11 a3167 27 * mpn_nand_n: Low-level Functions. (line 472) * mpn_neg: Low-level Functions. (line 96) * mpn_nior_n: Low-level Functions. (line 477) * mpn_perfect_square_p: Low-level Functions. (line 442) * mpn_popcount: Low-level Functions. (line 432) * mpn_random: Low-level Functions. (line 422) * mpn_random2: Low-level Functions. (line 423) * mpn_rshift: Low-level Functions. (line 281) * mpn_scan0: Low-level Functions. (line 406) * mpn_scan1: Low-level Functions. (line 414) * mpn_sec_add_1: Low-level Functions. (line 553) * mpn_sec_div_qr: Low-level Functions. (line 629) * mpn_sec_div_qr_itch: Low-level Functions. (line 632) * mpn_sec_div_r: Low-level Functions. (line 648) * mpn_sec_div_r_itch: Low-level Functions. (line 650) * mpn_sec_invert: Low-level Functions. (line 664) * mpn_sec_invert_itch: Low-level Functions. (line 666) * mpn_sec_mul: Low-level Functions. (line 574) * mpn_sec_mul_itch: Low-level Functions. (line 577) * mpn_sec_powm: Low-level Functions. (line 604) * mpn_sec_powm_itch: Low-level Functions. (line 607) * mpn_sec_sqr: Low-level Functions. (line 590) * mpn_sec_sqr_itch: Low-level Functions. (line 592) * mpn_sec_sub_1: Low-level Functions. (line 555) * mpn_sec_tabselect: Low-level Functions. (line 621) * mpn_set_str: Low-level Functions. (line 386) * mpn_sizeinbase: Low-level Functions. (line 364) d3169 11 a3179 12 * mpn_sqrtrem: Low-level Functions. (line 346) * mpn_sub: Low-level Functions. (line 88) * mpn_submul_1: Low-level Functions. (line 160) * mpn_sub_1: Low-level Functions. (line 83) * mpn_sub_n: Low-level Functions. (line 74) * mpn_tdiv_qr: Low-level Functions. (line 172) * mpn_xnor_n: Low-level Functions. (line 482) * mpn_xor_n: Low-level Functions. (line 457) * mpn_zero: Low-level Functions. (line 500) * mpn_zero_p: Low-level Functions. (line 298) * mpq_abs: Rational Arithmetic. (line 33) * mpq_add: Rational Arithmetic. (line 6) d3181 1 a3181 1 (line 21) d3183 1 a3183 1 (line 18) d3185 1 a3185 1 (line 41) d3187 1 a3187 1 (line 51) d3189 1 a3189 1 (line 67) d3191 1 a3191 1 (line 77) d3193 1 a3193 1 (line 64) d3195 1 a3195 1 (line 66) d3197 1 a3197 1 (line 76) d3199 1 a3199 1 (line 52) d3201 1 a3201 9 (line 9) * mpq_class::mpq_class <1>: C++ Interface Rationals. (line 10) * mpq_class::mpq_class <2>: C++ Interface Rationals. (line 21) * mpq_class::mpq_class <3>: C++ Interface Rationals. (line 26) * mpq_class::mpq_class <4>: C++ Interface Rationals. (line 28) a3203 2 * mpq_class::set_str <1>: C++ Interface Rationals. (line 55) d3205 1 a3205 1 (line 58) d3207 1 a3207 1 (line 15) d3209 4 a3212 5 (line 19) * mpq_cmp: Comparing Rationals. (line 6) * mpq_cmp_si: Comparing Rationals. (line 16) * mpq_cmp_ui: Comparing Rationals. (line 14) * mpq_cmp_z: Comparing Rationals. (line 7) d3214 3 a3216 3 (line 16) * mpq_div: Rational Arithmetic. (line 22) * mpq_div_2exp: Rational Arithmetic. (line 26) d3219 1 a3219 1 (line 6) d3221 1 a3221 1 (line 22) d3223 1 a3223 1 (line 21) d3225 1 a3225 1 (line 21) d3227 1 a3227 1 (line 6) d3229 6 a3234 6 (line 11) * mpq_inp_str: I/O of Rationals. (line 26) * mpq_inv: Rational Arithmetic. (line 36) * mpq_mul: Rational Arithmetic. (line 14) * mpq_mul_2exp: Rational Arithmetic. (line 18) * mpq_neg: Rational Arithmetic. (line 30) d3236 2 a3237 2 (line 15) * mpq_out_str: I/O of Rationals. (line 17) d3239 1 a3239 1 (line 23) d3241 1 a3241 1 (line 16) d3243 1 a3243 1 (line 24) d3245 1 a3245 1 (line 17) d3247 1 a3247 1 (line 23) d3249 1 a3249 1 (line 29) d3251 1 a3251 1 (line 35) d3253 1 a3253 1 (line 27) d3255 1 a3255 1 (line 24) d3257 1 a3257 1 * mpq_sub: Rational Arithmetic. (line 10) d3259 1 a3259 1 (line 54) d3263 6 a3268 6 (line 109) * mpz_abs: Integer Arithmetic. (line 44) * mpz_add: Integer Arithmetic. (line 6) * mpz_addmul: Integer Arithmetic. (line 24) * mpz_addmul_ui: Integer Arithmetic. (line 26) * mpz_add_ui: Integer Arithmetic. (line 7) d3270 1 a3270 1 (line 10) d3272 1 a3272 1 (line 9) d3274 1 a3274 1 (line 120) d3276 2 a3277 4 (line 122) * mpz_cdiv_q: Integer Division. (line 12) * mpz_cdiv_qr: Integer Division. (line 14) * mpz_cdiv_qr_ui: Integer Division. (line 21) d3279 7 a3285 5 * mpz_cdiv_q_ui: Integer Division. (line 17) * mpz_cdiv_r: Integer Division. (line 13) * mpz_cdiv_r_2exp: Integer Division. (line 29) * mpz_cdiv_r_ui: Integer Division. (line 19) * mpz_cdiv_ui: Integer Division. (line 23) d3287 1 a3287 1 (line 17) d3295 2 d3298 1 a3298 1 * mpz_class::fits_ulong_p: C++ Interface Integers. d3300 1 a3300 1 * mpz_class::fits_ushort_p: C++ Interface Integers. a3301 2 * mpz_class::get_d: C++ Interface Integers. (line 58) d3303 1 a3303 1 (line 63) d3305 1 a3305 1 (line 59) d3307 1 a3307 1 (line 60) d3309 1 a3309 1 (line 61) d3311 1 a3311 7 (line 6) * mpz_class::mpz_class <1>: C++ Interface Integers. (line 14) * mpz_class::mpz_class <2>: C++ Interface Integers. (line 19) * mpz_class::mpz_class <3>: C++ Interface Integers. (line 21) d3313 2 a3314 2 (line 63) * mpz_class::set_str <1>: C++ Interface Integers. a3315 2 * mpz_class::swap: C++ Interface Integers. (line 71) d3317 1 a3317 1 (line 48) d3319 1 a3319 1 (line 52) d3321 8 a3328 8 (line 54) * mpz_cmp: Integer Comparisons. (line 6) * mpz_cmpabs: Integer Comparisons. (line 17) * mpz_cmpabs_d: Integer Comparisons. (line 18) * mpz_cmpabs_ui: Integer Comparisons. (line 19) * mpz_cmp_d: Integer Comparisons. (line 7) * mpz_cmp_si: Integer Comparisons. (line 8) * mpz_cmp_ui: Integer Comparisons. (line 9) d3330 1 a3330 1 (line 19) d3332 9 a3340 9 (line 57) * mpz_congruent_2exp_p: Integer Division. (line 148) * mpz_congruent_p: Integer Division. (line 144) * mpz_congruent_ui_p: Integer Division. (line 146) * mpz_divexact: Integer Division. (line 122) * mpz_divexact_ui: Integer Division. (line 123) * mpz_divisible_2exp_p: Integer Division. (line 135) * mpz_divisible_p: Integer Division. (line 132) * mpz_divisible_ui_p: Integer Division. (line 133) d3342 1 a3342 1 (line 17) d3344 1 a3344 1 (line 43) d3346 10 a3355 10 (line 108) * mpz_fdiv_q: Integer Division. (line 33) * mpz_fdiv_qr: Integer Division. (line 35) * mpz_fdiv_qr_ui: Integer Division. (line 42) * mpz_fdiv_q_2exp: Integer Division. (line 47) * mpz_fdiv_q_ui: Integer Division. (line 38) * mpz_fdiv_r: Integer Division. (line 34) * mpz_fdiv_r_2exp: Integer Division. (line 50) * mpz_fdiv_r_ui: Integer Division. (line 40) * mpz_fdiv_ui: Integer Division. (line 44) d3357 1 a3357 1 (line 130) d3359 1 a3359 1 (line 129) d3361 1 a3361 1 (line 9) d3363 1 a3363 1 (line 7) d3365 1 a3365 1 (line 11) d3367 1 a3367 1 (line 8) d3369 1 a3369 1 (line 6) d3371 1 a3371 1 (line 10) d3373 3 a3375 1 (line 25) d3377 6 a3382 3 (line 41) * mpz_gcd_ui: Number Theoretic Functions. (line 31) d3384 1 a3384 6 (line 22) * mpz_get_d: Converting Integers. (line 26) * mpz_get_d_2exp: Converting Integers. (line 34) * mpz_get_si: Converting Integers. (line 17) * mpz_get_str: Converting Integers. (line 46) * mpz_get_ui: Converting Integers. (line 10) d3386 1 a3386 1 (line 28) d3388 1 a3388 1 (line 9) d3390 1 a3390 1 (line 25) d3392 1 a3392 3 (line 32) * mpz_inits: Initializing Integers. (line 28) d3394 1 a3394 1 (line 26) d3396 2 a3398 2 * mpz_init_set_si: Simultaneous Integer Init & Assign. (line 28) d3400 1 a3400 1 (line 33) d3402 5 a3406 3 (line 27) * mpz_inp_raw: I/O of Integers. (line 61) * mpz_inp_str: I/O of Integers. (line 30) d3408 1 a3408 1 (line 68) d3410 1 a3410 1 (line 13) d3412 1 a3412 1 (line 78) d3414 1 a3414 1 (line 86) d3416 1 a3416 1 (line 87) d3418 1 a3418 1 (line 88) d3420 1 a3420 1 (line 61) d3422 1 a3422 1 (line 62) d3424 1 a3424 9 (line 81) * mpz_limbs_finish: Integer Special Functions. (line 47) * mpz_limbs_modify: Integer Special Functions. (line 40) * mpz_limbs_read: Integer Special Functions. (line 34) * mpz_limbs_write: Integer Special Functions. (line 39) d3426 1 a3426 1 (line 141) d3428 1 a3428 1 (line 140) d3430 8 a3437 8 (line 110) * mpz_mod: Integer Division. (line 112) * mpz_mod_ui: Integer Division. (line 113) * mpz_mul: Integer Arithmetic. (line 18) * mpz_mul_2exp: Integer Arithmetic. (line 36) * mpz_mul_si: Integer Arithmetic. (line 19) * mpz_mul_ui: Integer Arithmetic. (line 20) * mpz_neg: Integer Arithmetic. (line 41) d3439 1 a3439 1 (line 18) d3441 5 a3445 5 (line 16) * mpz_out_raw: I/O of Integers. (line 45) * mpz_out_str: I/O of Integers. (line 17) * mpz_perfect_power_p: Integer Roots. (line 27) * mpz_perfect_square_p: Integer Roots. (line 36) d3447 3 a3449 1 (line 22) d3451 1 a3451 1 (line 6) d3453 1 a3453 1 (line 16) d3455 1 a3455 3 (line 8) * mpz_pow_ui: Integer Exponentiation. (line 29) d3457 1 a3457 1 (line 116) d3459 1 a3459 1 (line 6) d3461 1 a3461 1 (line 41) d3463 1 a3463 1 (line 50) d3465 1 a3465 1 (line 56) d3467 3 a3469 7 (line 102) * mpz_roinit_n: Integer Special Functions. (line 67) * MPZ_ROINIT_N: Integer Special Functions. (line 83) * mpz_root: Integer Roots. (line 6) * mpz_rootrem: Integer Roots. (line 12) d3471 1 a3471 1 (line 29) d3473 1 a3473 1 (line 35) d3475 8 a3482 2 (line 37) * mpz_set: Assigning Integers. (line 9) d3484 4 a3487 8 (line 51) * mpz_set_d: Assigning Integers. (line 12) * mpz_set_f: Assigning Integers. (line 14) * mpz_set_q: Assigning Integers. (line 13) * mpz_set_si: Assigning Integers. (line 11) * mpz_set_str: Assigning Integers. (line 20) * mpz_set_ui: Assigning Integers. (line 10) * mpz_sgn: Integer Comparisons. (line 27) d3489 1 a3489 1 (line 30) d3491 8 a3498 10 (line 22) * mpz_si_kronecker: Number Theoretic Functions. (line 89) * mpz_sqrt: Integer Roots. (line 17) * mpz_sqrtrem: Integer Roots. (line 20) * mpz_sub: Integer Arithmetic. (line 11) * mpz_submul: Integer Arithmetic. (line 30) * mpz_submul_ui: Integer Arithmetic. (line 32) * mpz_sub_ui: Integer Arithmetic. (line 12) * mpz_swap: Assigning Integers. (line 36) d3501 9 a3509 9 * mpz_tdiv_q: Integer Division. (line 54) * mpz_tdiv_qr: Integer Division. (line 56) * mpz_tdiv_qr_ui: Integer Division. (line 63) * mpz_tdiv_q_2exp: Integer Division. (line 68) * mpz_tdiv_q_ui: Integer Division. (line 59) * mpz_tdiv_r: Integer Division. (line 55) * mpz_tdiv_r_2exp: Integer Division. (line 71) * mpz_tdiv_r_ui: Integer Division. (line 61) * mpz_tdiv_ui: Integer Division. (line 65) d3511 1 a3511 1 (line 60) d3513 1 a3513 1 (line 90) d3515 2 a3516 2 (line 31) * mpz_ui_sub: Integer Arithmetic. (line 14) d3518 1 a3518 1 (line 12) d3520 1 a3520 1 (line 21) d3522 3 a3524 17 (line 16) * mp_bitcnt_t: Nomenclature and Types. (line 42) * mp_bits_per_limb: Useful Macros and Constants. (line 7) * mp_exp_t: Nomenclature and Types. (line 27) * mp_get_memory_functions: Custom Allocation. (line 86) * mp_limb_t: Nomenclature and Types. (line 31) * mp_set_memory_functions: Custom Allocation. (line 14) * mp_size_t: Nomenclature and Types. (line 37) * operator"": C++ Interface Integers. (line 29) * operator"" <1>: C++ Interface Rationals. (line 36) d3526 3 a3528 1 (line 55) d3530 2 a3532 2 * operator/: C++ Interface Integers. (line 33) d3534 4 a3537 12 (line 10) * operator<< <1>: C++ Formatted Output. (line 19) * operator<< <2>: C++ Formatted Output. (line 32) * operator>>: C++ Formatted Input. (line 10) * operator>> <1>: C++ Formatted Input. (line 13) * operator>> <2>: C++ Formatted Input. (line 24) * operator>> <3>: C++ Interface Rationals. (line 86) * sgn: C++ Interface Integers. (line 65) d3540 15 a3554 1 * sgn <2>: C++ Interface Floats. a3555 12 * sqrt: C++ Interface Integers. (line 66) * sqrt <1>: C++ Interface Floats. (line 107) * swap: C++ Interface Integers. (line 72) * swap <1>: C++ Interface Rationals. (line 59) * swap <2>: C++ Interface Floats. (line 110) * trunc: C++ Interface Floats. (line 111) d3557 6 @ 1.1.1.4 log @initial import of GMP 6.2.0. changes include: - Bug fixes to gmp_snprintf, conversion to double, mpz_powm, and mpf_set_str. - New functions for factorial, primorial, fibonacci, mpz_2fac_ui, and mpz_mfac_uiui. - MIPS r6 cores are now supported. - Various speeds ups. @ text @d1 1 a1 1 This is gmp.info, produced by makeinfo version 6.6 from gmp.texi. d4 1 a4 1 arithmetic library, version 6.2.0. d6 1 a6 1 Copyright 1991, 1993-2016, 2018 Free Software Foundation, Inc. d305 2 a306 1 Jacobi symbol (A/B) d308 11 a318 30 Initially if either operand fits in a single limb, a reduction is done with either 'mpn_mod_1' or 'mpn_modexact_1_odd', followed by the binary algorithm on a single limb. The binary algorithm is well suited to a single limb, and the whole calculation in this case is quite efficient. For inputs larger than 'GCD_DC_THRESHOLD', 'mpz_jacobi', 'mpz_legendre' and 'mpz_kronecker' are computed via the HGCD (Half GCD) function, as a generalization to Lehmer's algorithm. Most GCD algorithms reduce a and b by repeatatily computing the quotient q = floor(a/b) and iteratively replacing a, b = b, a - q * b Different algorithms use different methods for calculating q, but the core algorithm is the same if we use *note Lehmer's Algorithm:: or *note HGCD: Subquadratic GCD. At each step it is possible to compute if the reduction inverts the Jacobi symbol based on the two least significant bits of A and B. For more details see "Efficient computation of the Jacobi symbol" by Möller (*note References::). A small set of bits is thus used to track state * current sign of result (1 bit) * two least significant bits of A and B (4 bits) * a pointer to which input is currently the denominator (1 bit) d321 1 a321 11 fast bit twiddling which avoids conditional jumps. The final result is calculated after verifying the inputs are coprime (GCD = 1) by raising (-1)^e Much of the HGCD code is shared directly with the HGCD implementations, such as the 2x2 matrix calculation, *Note Lehmer's Algorithm:: basecase and 'GCD_DC_THRESHOLD'. The asymptotic running time is O(M(N)*log(N)), where M(N) is the time for multiplying two N-limb numbers. d1353 1 a1353 1 in which case the '_mp_d' data is undefined. d1362 4 a1365 3 Currently there's always at least one readable limb, so for instance 'mpz_get_ui' can fetch '_mp_d[0]' unconditionally (though its value is undefined if '_mp_size' is zero). d1369 1 a1369 1 and normally '_mp_alloc >= ABS(_mp_size)'. When an 'mpz' routine a1373 5 'mpz_t' variables initialised with the 'mpz_roinit_n' function or the 'MPZ_ROINIT_N' macro have '_mp_alloc = 0' but can have a non-zero '_mp_size'. They can only be used as read-only constants. See *note Integer Special Functions:: for details. d1803 2 a1804 3 is the author of mini-mpq, an additional layer to mini-gmp; of most of the combinatorial functions and the BPSW primality testing implementation, for both the main library and the mini-gmp package. d1862 1 a1862 1 d1866 1 a1866 1 d1870 1 a1870 1 d1890 1 a1890 1 2002, d1894 1 a1894 1 d1939 2 a1940 1 d1950 1 a1950 2 2008, pp. 589-607, d1965 1 a1965 1 d1969 1 a1969 1 a1976 3 * Niels Möller, "Efficient computation of the Jacobi symbol", d2525 1 a2525 1 * Assertion checking <1>: Debugging. (line 74) d2539 1 a2539 1 (line 128) d2574 2 a2575 2 * Checker: Debugging. (line 110) * checkergcc: Debugging. (line 117) d2658 1 a2658 1 (line 47) d2660 1 a2660 1 (line 108) d2663 1 a2663 1 (line 116) d2673 1 a2673 1 (line 136) d2714 1 a2714 1 * GCC Checker: Debugging. (line 110) d2718 1 a2718 1 (line 47) d2720 2 a2721 2 (line 30) * GDB: Debugging. (line 53) d2731 1 a2731 1 * GNU Debugger: Debugging. (line 53) d2740 1 a2740 1 (line 30) d2823 1 a2823 1 (line 74) d2831 1 a2831 1 (line 83) d2837 1 a2837 1 (line 95) d2841 1 a2841 1 (line 68) d2843 1 a2843 1 (line 68) d2845 1 a2845 1 (line 86) d2876 1 a2876 1 (line 147) d2898 1 a2898 1 (line 74) d2911 1 a2911 1 * Nails: Low-level Functions. (line 686) d2918 1 a2918 1 (line 23) d2987 1 a2987 1 (line 121) d3033 1 a3033 1 (line 108) d3082 1 a3082 1 * Stack backtrace: Debugging. (line 45) d3120 1 a3120 1 * Valgrind: Debugging. (line 125) a3176 4 * factorial: C++ Interface Integers. (line 71) * fibonacci: C++ Interface Integers. (line 75) d3193 6 a3198 6 * GMP_LIMB_BITS: Low-level Functions. (line 714) * GMP_NAIL_BITS: Low-level Functions. (line 712) * GMP_NAIL_MASK: Low-level Functions. (line 722) * GMP_NUMB_BITS: Low-level Functions. (line 713) * GMP_NUMB_MASK: Low-level Functions. (line 723) * GMP_NUMB_MAX: Low-level Functions. (line 731) d3466 6 a3471 6 * mpn_sec_div_qr: Low-level Functions. (line 630) * mpn_sec_div_qr_itch: Low-level Functions. (line 633) * mpn_sec_div_r: Low-level Functions. (line 649) * mpn_sec_div_r_itch: Low-level Functions. (line 651) * mpn_sec_invert: Low-level Functions. (line 665) * mpn_sec_invert_itch: Low-level Functions. (line 667) d3479 1 a3479 1 * mpn_sec_tabselect: Low-level Functions. (line 622) d3556 1 a3556 1 * mpq_inp_str: I/O of Rationals. (line 32) d3589 1 a3589 1 (line 113) d3600 1 a3600 1 (line 124) d3602 1 a3602 1 (line 126) a3613 4 * mpz_class::factorial: C++ Interface Integers. (line 70) * mpz_class::fibonacci: C++ Interface Integers. (line 74) a3643 2 * mpz_class::primorial: C++ Interface Integers. (line 72) d3649 1 a3649 1 (line 77) d3680 1 a3680 1 (line 112) d3691 1 a3691 1 (line 134) d3693 1 a3693 1 (line 133) d3707 1 a3707 1 (line 29) d3709 1 a3709 1 (line 45) d3711 1 a3711 1 (line 35) d3742 1 a3742 1 (line 72) d3746 1 a3746 1 (line 82) d3748 1 a3748 1 (line 90) d3750 1 a3750 1 (line 91) d3752 1 a3752 1 (line 92) d3754 1 a3754 1 (line 65) d3756 1 a3756 1 (line 66) d3758 1 a3758 1 (line 85) d3768 1 a3768 1 (line 145) d3770 1 a3770 1 (line 144) d3772 1 a3772 1 (line 114) d3781 1 a3781 1 (line 22) d3799 1 a3799 1 (line 120) d3809 1 a3809 1 (line 106) d3837 1 a3837 1 (line 93) d3859 1 a3859 1 (line 94) a3901 2 * primorial: C++ Interface Integers. (line 73) d3913 1 a3913 1 (line 78) @ 1.1.1.5 log @initial import of GMP 6.2.1. from their NEWS: Changes between GMP version 6.2.0 and 6.2.1 BUGS FIXED * A possible overflow of type int is avoided for mpz_cmp on huge operands. * Overflows are more carefully detected and reported for mpz_pow_ui. * A bug in longlong.h for aarch64 sub_ddmmss, not affecting GMP, was healed. FEATURES * C90 compliance. * Initial support for Darwin on arm64, and improved portability. * Support for more processors. @ text @d1 1 a1 1 This is gmp.info, produced by makeinfo version 6.7 from gmp.texi. d4 1 a4 1 arithmetic library, version 6.2.1. d6 1 a6 1 Copyright 1991, 1993-2016, 2018-2020 Free Software Foundation, Inc. @