This is /home/tege/prec/gmp42/doc/gmp.info, produced by makeinfo version 4.6 from /home/tege/prec/gmp42/doc/gmp.texi. This manual describes how to install and use the GNU multiple precision arithmetic library, version 4.2.1. Copyright 1991, 1993, 1994, 1995, 1996, 1997, 1998, 1999, 2000, 2001, 2002, 2003, 2004, 2005, 2006 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 1.2 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: Radix to Binary, Prev: Binary to Radix, Up: Radix Conversion Algorithms Radix to Binary --------------- 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_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_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_THRESHOLD' usually ends up quite big, around 5000 digits, and on some processors much bigger still. `SET_STR_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_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 10, or more).  File: gmp.info, Node: Other Algorithms, Next: Assembler Coding, Prev: Radix Conversion Algorithms, Up: Algorithms 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 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 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 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 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 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 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 transfered 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: Assembler SIMD Instructions, Next: Assembler Software Pipelining, Prev: Assembler Floating Point, Up: Assembler Coding 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: Assembler Software Pipelining, Next: Assembler Loop Unrolling, Prev: Assembler SIMD Instructions, Up: Assembler Coding 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 Assembler Loop Unrolling::).  File: gmp.info, Node: Assembler Loop Unrolling, Next: Assembler Writing Guide, Prev: Assembler Software Pipelining, Up: Assembler Coding 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: Assembler Writing Guide, Prev: Assembler Loop Unrolling, Up: Assembler Coding Writing Guide ------------- This is a guide to writing software pipelined loops for processing limb vectors in assembler. 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 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 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 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 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. `_mp_size' and `_mp_prec' 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. 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 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 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, unsigned long int 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, unsigned long int 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 Contributors ************ Torbjorn Granlund wrote the original GMP library and is still developing and maintaining it. Several other individuals and organizations have contributed to GMP in various ways. Here is a list in chronological order: Gunnar Sjoedin and Hans Riesel helped with mathematical problems in early versions of the library. Richard Stallman contributed to 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 of Inria sparked the development of GMP 2, with his comparisons between bignum packages. Ken Weber (Kent State University, Universidade Federal do Rio Grande do Sul) contributed `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). 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). GNU MP 2 was finished and released by SWOX AB, SWEDEN, in cooperation with the IDA Center for Computing Sciences, USA. Robert Harley of Inria, France and David Seal of ARM, England, suggested clever improvements for population count. Robert Harley also wrote highly optimized Karatsuba and 3-way Toom multiplication functions for GMP 3. He also 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. Paul Zimmermann wrote the Divide and Conquer division code, the REDC code, the REDC-based mpz_powm code, the FFT multiply code, and the Karatsuba square root code. He also rewrote the Toom3 code for GMP 4.2. The ECMNET project Paul is organizing was a driving force behind many of the optimizations in GMP 3. Linus Nordberg wrote the new configure system based on autoconf and implemented the new random functions. Kent Boortz made the Mac OS 9 port. Kevin Ryde worked on a 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. 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. GNU MP 4 was finished and released by Torbjorn Granlund and Kevin Ryde. Torbjorn's work was partially funded by the IDA Center for Computing Sciences, USA. Jason Moxham rewrote `mpz_fac_ui'. Pedro Gimeno implemented the Mersenne Twister and made other random number improvements. (This list is chronological, not ordered after significance. If you have contributed to GMP but are not listed above, please tell about the omission!) 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 References ********** Books ===== * Jonathan M. Borwein and Peter B. Borwein, "Pi and the AGM: A Study in Analytic Number Theory and Computational Complexity", Wiley, 1998. * 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, "Using and Porting GCC", Free Software Foundation, 1999, available online `http://gcc.gnu.org/onlinedocs/', and in the GCC package `ftp://ftp.gnu.org/gnu/gcc/' 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' * Torbjorn 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). * 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.keio.ac.jp/~nisimura/random/doc/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. * 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 GNU Free Documentation License ****************************** Version 1.2, November 2002 Copyright (C) 2000,2001,2002 Free Software Foundation, Inc. 51 Franklin Street, Fifth Floor, Boston, MA 02110-1301, USA 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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File: gmp.info, Node: Concept Index, Next: Function Index, Prev: GNU Free Documentation License, Up: Top Concept Index ************* * Menu: * #include: Headers and Libraries. * --build: Build Options. * --disable-fft: Build Options. * --disable-shared: Build Options. * --disable-static: Build Options. * --enable-alloca: Build Options. * --enable-assert: Build Options. * --enable-cxx: Build Options. * --enable-fat: Build Options. * --enable-mpbsd: Build Options. * --enable-profiling <1>: Build Options. * --enable-profiling: Profiling. * --exec-prefix: Build Options. * --host: Build Options. * --prefix: Build Options. * -finstrument-functions: Profiling. * 2exp functions: Efficiency. * 68000: Notes for Particular Systems. * 80x86: Notes for Particular Systems. * ABI <1>: Build Options. * ABI: ABI and ISA. * About this manual: Introduction to GMP. * AC_CHECK_LIB: Autoconf. * AIX <1>: Notes for Particular Systems. * AIX: ABI and ISA. * Algorithms: Algorithms. * alloca: Build Options. * Allocation of memory: Custom Allocation. * AMD64: ABI and ISA. * Anonymous FTP of latest version: Introduction to GMP. * Application Binary Interface: ABI and ISA. * Arithmetic functions <1>: Float Arithmetic. * Arithmetic functions <2>: Integer Arithmetic. * Arithmetic functions: Rational Arithmetic. * ARM: Notes for Particular Systems. * Assembler cache handling: Assembler Cache Handling. * Assembler carry propagation: Assembler Carry Propagation. * Assembler code organisation: Assembler Code Organisation. * Assembler coding: Assembler Coding. * Assembler floating Point: Assembler Floating Point. * Assembler loop unrolling: Assembler Loop Unrolling. * Assembler SIMD: Assembler SIMD Instructions. * Assembler software pipelining: Assembler Software Pipelining. * Assembler writing guide: Assembler Writing Guide. * Assertion checking <1>: Build Options. * Assertion checking: Debugging. * Assignment functions <1>: Assigning Floats. * Assignment functions <2>: Simultaneous Integer Init & Assign. * Assignment functions <3>: Initializing Rationals. * Assignment functions <4>: Simultaneous Float Init & Assign. * Assignment functions: Assigning Integers. * Autoconf: Autoconf. * Basics: GMP Basics. * Berkeley MP compatible functions <1>: BSD Compatible Functions. * Berkeley MP compatible functions: Build Options. * Binomial coefficient algorithm: Binomial Coefficients Algorithm. * Binomial coefficient functions: Number Theoretic Functions. * Binutils strip: Known Build Problems. * Bit manipulation functions: Integer Logic and Bit Fiddling. * Bit scanning functions: Integer Logic and Bit Fiddling. * Bit shift left: Integer Arithmetic. * Bit shift right: Integer Division. * Bits per limb: Useful Macros and Constants. * BSD MP compatible functions <1>: Build Options. * BSD MP compatible functions: BSD Compatible Functions. * Bug reporting: Reporting Bugs. * Build directory: Build Options. * Build notes for binary packaging: Notes for Package Builds. * Build notes for particular systems: Notes for Particular Systems. * Build options: Build Options. * Build problems known: Known Build Problems. * Build system: Build Options. * Building GMP: Installing GMP. * Bus error: Debugging. * C compiler: Build Options. * C++ compiler: Build Options. * C++ interface: C++ Class Interface. * C++ interface internals: C++ Interface Internals. * C++ istream input: C++ Formatted Input. * C++ ostream output: C++ Formatted Output. * C++ support: Build Options. * CC: Build Options. * CC_FOR_BUILD: Build Options. * CFLAGS: Build Options. * Checker: Debugging. * checkergcc: Debugging. * Code organisation: Assembler Code Organisation. * Compaq C++: Notes for Particular Systems. * Comparison functions <1>: Float Comparison. * Comparison functions <2>: Comparing Rationals. * Comparison functions: Integer Comparisons. * Compatibility with older versions: Compatibility with older versions. * Conditions for copying GNU MP: Copying. * Configuring GMP: Installing GMP. * Congruence algorithm: Exact Remainder. * Congruence functions: Integer Division. * Constants: Useful Macros and Constants. * Contributors: Contributors. * Conventions for parameters: Parameter Conventions. * Conventions for variables: Variable Conventions. * Conversion functions <1>: Converting Floats. * Conversion functions <2>: Rational Conversions. * Conversion functions: Converting Integers. * Copying conditions: Copying. * CPPFLAGS: Build Options. * CPU types <1>: Introduction to GMP. * CPU types: Build Options. * Cross compiling: Build Options. * Custom allocation: Custom Allocation. * CXX: Build Options. * CXXFLAGS: Build Options. * Cygwin: Notes for Particular Systems. * Darwin: Known Build Problems. * Debugging: Debugging. * Demonstration programs: Demonstration Programs. * Digits in an integer: Miscellaneous Integer Functions. * Divisibility algorithm: Exact Remainder. * Divisibility functions: Integer Division. * Divisibility testing: Efficiency. * Division algorithms: Division Algorithms. * Division functions <1>: Float Arithmetic. * Division functions <2>: Rational Arithmetic. * Division functions: Integer Division. * DJGPP <1>: Notes for Particular Systems. * DJGPP: Known Build Problems. * DLLs: Notes for Particular Systems. * DocBook: Build Options. * Documentation formats: Build Options. * Documentation license: GNU Free Documentation License. * DVI: Build Options. * Efficiency: Efficiency. * Emacs: Emacs. * Exact division functions: Integer Division. * Exact remainder: Exact Remainder. * Example programs: Demonstration Programs. * Exec prefix: Build Options. * Execution profiling <1>: Profiling. * Execution profiling: Build Options. * Exponentiation functions <1>: Float Arithmetic. * Exponentiation functions: Integer Exponentiation. * Export: Integer Import and Export. * Expression parsing demo: Demonstration Programs. * Extended GCD: Number Theoretic Functions. * Factor removal functions: Number Theoretic Functions. * Factorial algorithm: Factorial Algorithm. * Factorial functions: Number Theoretic Functions. * Factorization demo: Demonstration Programs. * Fast Fourier Transform: FFT Multiplication. * Fat binary: Build Options. * FDL, GNU Free Documentation License: GNU Free Documentation License. * FFT multiplication <1>: FFT Multiplication. * FFT multiplication: Build Options. * Fibonacci number algorithm: Fibonacci Numbers Algorithm. * Fibonacci sequence functions: Number Theoretic Functions. * Float arithmetic functions: Float Arithmetic. * Float assignment functions <1>: Assigning Floats. * Float assignment functions: Simultaneous Float Init & Assign. * Float comparison functions: Float Comparison. * Float conversion functions: Converting Floats. * Float functions: Floating-point Functions. * Float initialization functions <1>: Initializing Floats. * Float initialization functions: Simultaneous Float Init & Assign. * Float input and output functions: I/O of Floats. * Float internals: Float Internals. * Float miscellaneous functions: Miscellaneous Float Functions. * Float random number functions: Miscellaneous Float Functions. * Float rounding functions: Miscellaneous Float Functions. * Float sign tests: Float Comparison. * Floating point mode: Notes for Particular Systems. * Floating-point functions: Floating-point Functions. * Floating-point number: Nomenclature and Types. * fnccheck: Profiling. * Formatted input: Formatted Input. * Formatted output: Formatted Output. * Free Documentation License: GNU Free Documentation License. * frexp <1>: Converting Floats. * frexp: Converting Integers. * FTP of latest version: Introduction to GMP. * Function classes: Function Classes. * FunctionCheck: Profiling. * GCC Checker: Debugging. * GCD algorithms: Greatest Common Divisor Algorithms. * GCD extended: Number Theoretic Functions. * GCD functions: Number Theoretic Functions. * GDB: Debugging. * Generic C: Build Options. * GMP Perl module: Demonstration Programs. * GMP version number: Useful Macros and Constants. * gmp.h: Headers and Libraries. * gmpxx.h: C++ Interface General. * GNU Debugger: Debugging. * GNU Free Documentation License: GNU Free Documentation License. * GNU strip: Known Build Problems. * gprof: Profiling. * Greatest common divisor algorithms: Greatest Common Divisor Algorithms. * Greatest common divisor functions: Number Theoretic Functions. * Hardware floating point mode: Notes for Particular Systems. * Headers: Headers and Libraries. * Heap problems: Debugging. * Home page: Introduction to GMP. * Host system: Build Options. * HP-UX: ABI and ISA. * HPPA: ABI and ISA. * I/O functions <1>: I/O of Floats. * I/O functions <2>: I/O of Integers. * I/O functions: I/O of Rationals. * i386: Notes for Particular Systems. * IA-64: ABI and ISA. * Import: Integer Import and Export. * In-place operations: Efficiency. * Include files: Headers and Libraries. * info-lookup-symbol: Emacs. * Initialization functions <1>: Simultaneous Integer Init & Assign. * Initialization functions <2>: Initializing Integers. * Initialization functions <3>: Simultaneous Float Init & Assign. * Initialization functions <4>: Initializing Floats. * Initialization functions <5>: Initializing Rationals. * Initialization functions: Random State Initialization. * Initializing and clearing: Efficiency. * Input functions <1>: I/O of Integers. * Input functions <2>: Formatted Input Functions. * Input functions <3>: I/O of Floats. * Input functions: I/O of Rationals. * Install prefix: Build Options. * Installing GMP: Installing GMP. * Instruction Set Architecture: ABI and ISA. * instrument-functions: Profiling. * Integer: Nomenclature and Types. * Integer arithmetic functions: Integer Arithmetic. * Integer assignment functions <1>: Assigning Integers. * Integer assignment functions: Simultaneous Integer Init & Assign. * Integer bit manipulation functions: Integer Logic and Bit Fiddling. * Integer comparison functions: Integer Comparisons. * Integer conversion functions: Converting Integers. * Integer division functions: Integer Division. * Integer exponentiation functions: Integer Exponentiation. * Integer export: Integer Import and Export. * Integer functions: Integer Functions. * Integer import: Integer Import and Export. * Integer initialization functions <1>: Simultaneous Integer Init & Assign. * Integer initialization functions: Initializing Integers. * Integer input and output functions: I/O of Integers. * Integer internals: Integer Internals. * Integer logical functions: Integer Logic and Bit Fiddling. * Integer miscellaneous functions: Miscellaneous Integer Functions. * Integer random number functions: Integer Random Numbers. * Integer root functions: Integer Roots. * Integer sign tests: Integer Comparisons. * Integer special functions: Integer Special Functions. * Interix: Notes for Particular Systems. * Internals: Internals. * Introduction: Introduction to GMP. * Inverse modulo functions: Number Theoretic Functions. * IRIX <1>: Known Build Problems. * IRIX: ABI and ISA. * ISA: ABI and ISA. * istream input: C++ Formatted Input. * Jacobi symbol algorithm: Jacobi Symbol. * Jacobi symbol functions: Number Theoretic Functions. * Karatsuba multiplication: Karatsuba Multiplication. * Karatsuba square root algorithm: Square Root Algorithm. * Kronecker symbol functions: Number Theoretic Functions. * Language bindings: Language Bindings. * Latest version of GMP: Introduction to GMP. * LCM functions: Number Theoretic Functions. * Least common multiple functions: Number Theoretic Functions. * Legendre symbol functions: Number Theoretic Functions. * libgmp: Headers and Libraries. * libgmpxx: Headers and Libraries. * Libraries: Headers and Libraries. * Libtool: Headers and Libraries. * Libtool versioning: Notes for Package Builds. * License conditions: Copying. * Limb: Nomenclature and Types. * Limb size: Useful Macros and Constants. * Linear congruential algorithm: Random Number Algorithms. * Linear congruential random numbers: Random State Initialization. * Linking: Headers and Libraries. * Logical functions: Integer Logic and Bit Fiddling. * Low-level functions: Low-level Functions. * Lucas number algorithm: Lucas Numbers Algorithm. * Lucas number functions: Number Theoretic Functions. * MacOS 9: Notes for Particular Systems. * MacOS X: Known Build Problems. * Mailing lists: Introduction to GMP. * Malloc debugger: Debugging. * Malloc problems: Debugging. * Memory allocation: Custom Allocation. * Memory management: Memory Management. * Mersenne twister algorithm: Random Number Algorithms. * Mersenne twister random numbers: Random State Initialization. * MINGW: Notes for Particular Systems. * MIPS: ABI and ISA. * Miscellaneous float functions: Miscellaneous Float Functions. * Miscellaneous integer functions: Miscellaneous Integer Functions. * MMX: Notes for Particular Systems. * Modular inverse functions: Number Theoretic Functions. * Most significant bit: Miscellaneous Integer Functions. * mp.h: BSD Compatible Functions. * MPN_PATH: Build Options. * MS Windows: Notes for Particular Systems. * MS-DOS: Notes for Particular Systems. * Multi-threading: Reentrancy. * Multiplication algorithms: Multiplication Algorithms. * Nails: Low-level Functions. * Native compilation: Build Options. * NeXT: Known Build Problems. * Next prime function: Number Theoretic Functions. * Nomenclature: Nomenclature and Types. * Non-Unix systems: Build Options. * Nth root algorithm: Nth Root Algorithm. * Number sequences: Efficiency. * Number theoretic functions: Number Theoretic Functions. * Numerator and denominator: Applying Integer Functions. * obstack output: Formatted Output Functions. * OpenBSD: Notes for Particular Systems. * Optimizing performance: Performance optimization. * ostream output: C++ Formatted Output. * Other languages: Language Bindings. * Output functions <1>: I/O of Floats. * Output functions <2>: I/O of Integers. * Output functions <3>: Formatted Output Functions. * Output functions: I/O of Rationals. * Packaged builds: Notes for Package Builds. * Parameter conventions: Parameter Conventions. * Parsing expressions demo: Demonstration Programs. * Particular systems: Notes for Particular Systems. * Past GMP versions: Compatibility with older versions. * PDF: Build Options. * Perfect power algorithm: Perfect Power Algorithm. * Perfect power functions: Integer Roots. * Perfect square algorithm: Perfect Square Algorithm. * Perfect square functions: Integer Roots. * perl: Demonstration Programs. * Perl module: Demonstration Programs. * Postscript: Build Options. * Power/PowerPC <1>: Known Build Problems. * Power/PowerPC: Notes for Particular Systems. * Powering algorithms: Powering Algorithms. * Powering functions <1>: Integer Exponentiation. * Powering functions: Float Arithmetic. * PowerPC: ABI and ISA. * Precision of floats: Floating-point Functions. * Precision of hardware floating point: Notes for Particular Systems. * Prefix: Build Options. * Prime testing algorithms: Prime Testing Algorithm. * Prime testing functions: Number Theoretic Functions. * printf formatted output: Formatted Output. * Probable prime testing functions: Number Theoretic Functions. * prof: Profiling. * Profiling: Profiling. * Radix conversion algorithms: Radix Conversion Algorithms. * Random number algorithms: Random Number Algorithms. * Random number functions <1>: Random Number Functions. * Random number functions <2>: Integer Random Numbers. * Random number functions: Miscellaneous Float Functions. * Random number seeding: Random State Seeding. * Random number state: Random State Initialization. * Random state: Nomenclature and Types. * Rational arithmetic: Efficiency. * Rational arithmetic functions: Rational Arithmetic. * Rational assignment functions: Initializing Rationals. * Rational comparison functions: Comparing Rationals. * Rational conversion functions: Rational Conversions. * Rational initialization functions: Initializing Rationals. * Rational input and output functions: I/O of Rationals. * Rational internals: Rational Internals. * Rational number: Nomenclature and Types. * Rational number functions: Rational Number Functions. * Rational numerator and denominator: Applying Integer Functions. * Rational sign tests: Comparing Rationals. * Raw output internals: Raw Output Internals. * Reallocations: Efficiency. * Reentrancy: Reentrancy. * References: References. * Remove factor functions: Number Theoretic Functions. * Reporting bugs: Reporting Bugs. * Root extraction algorithm: Nth Root Algorithm. * Root extraction algorithms: Root Extraction Algorithms. * Root extraction functions <1>: Integer Roots. * Root extraction functions: Float Arithmetic. * Root testing functions: Integer Roots. * Rounding functions: Miscellaneous Float Functions. * Sample programs: Demonstration Programs. * Scan bit functions: Integer Logic and Bit Fiddling. * scanf formatted input: Formatted Input. * SCO: Known Build Problems. * Seeding random numbers: Random State Seeding. * Segmentation violation: Debugging. * Sequent Symmetry: Known Build Problems. * Services for Unix: Notes for Particular Systems. * Shared library versioning: Notes for Package Builds. * Sign tests <1>: Integer Comparisons. * Sign tests <2>: Comparing Rationals. * Sign tests: Float Comparison. * Size in digits: Miscellaneous Integer Functions. * Small operands: Efficiency. * Solaris <1>: Known Build Problems. * Solaris <2>: ABI and ISA. * Solaris: Known Build Problems. * Sparc: Notes for Particular Systems. * Sparc V9: ABI and ISA. * Special integer functions: Integer Special Functions. * Square root algorithm: Square Root Algorithm. * SSE2: Notes for Particular Systems. * Stack backtrace: Debugging. * Stack overflow <1>: Build Options. * Stack overflow: Debugging. * Static linking: Efficiency. * stdarg.h: Headers and Libraries. * stdio.h: Headers and Libraries. * Stripped libraries: Known Build Problems. * Sun: ABI and ISA. * SunOS: Notes for Particular Systems. * Systems: Notes for Particular Systems. * Temporary memory: Build Options. * Texinfo: Build Options. * Text input/output: Efficiency. * Thread safety: Reentrancy. * Toom multiplication <1>: Toom 3-Way Multiplication. * Toom multiplication: Other Multiplication. * Types: Nomenclature and Types. * ui and si functions: Efficiency. * Upward compatibility: Compatibility with older versions. * Useful macros and constants: Useful Macros and Constants. * User-defined precision: Floating-point Functions. * Valgrind: Debugging. * Variable conventions: Variable Conventions. * Version number: Useful Macros and Constants. * Web page: Introduction to GMP. * Windows: Notes for Particular Systems. * x86: Notes for Particular Systems. * x87: Notes for Particular Systems. * XML: Build Options.  File: gmp.info, Node: Function Index, Prev: Concept Index, Up: Top Function and Type Index *********************** * Menu: * __GNU_MP_VERSION: Useful Macros and Constants. * __GNU_MP_VERSION_MINOR: Useful Macros and Constants. * __GNU_MP_VERSION_PATCHLEVEL: Useful Macros and Constants. * _mpz_realloc: Integer Special Functions. * abs <1>: C++ Interface Rationals. * abs <2>: C++ Interface Integers. * abs: C++ Interface Floats. * ceil: C++ Interface Floats. * cmp <1>: C++ Interface Integers. * cmp <2>: C++ Interface Rationals. * cmp <3>: C++ Interface Integers. * cmp: C++ Interface Floats. * floor: C++ Interface Floats. * gcd: BSD Compatible Functions. * gmp_asprintf: Formatted Output Functions. * gmp_errno: Random State Initialization. * GMP_ERROR_INVALID_ARGUMENT: Random State Initialization. * GMP_ERROR_UNSUPPORTED_ARGUMENT: Random State Initialization. * gmp_fprintf: Formatted Output Functions. * gmp_fscanf: Formatted Input Functions. * GMP_LIMB_BITS: Low-level Functions. * GMP_NAIL_BITS: Low-level Functions. * GMP_NAIL_MASK: Low-level Functions. * GMP_NUMB_BITS: Low-level Functions. * GMP_NUMB_MASK: Low-level Functions. * GMP_NUMB_MAX: Low-level Functions. * gmp_obstack_printf: Formatted Output Functions. * gmp_obstack_vprintf: Formatted Output Functions. * gmp_printf: Formatted Output Functions. * GMP_RAND_ALG_DEFAULT: Random State Initialization. * GMP_RAND_ALG_LC: Random State Initialization. * gmp_randclass: C++ Interface Random Numbers. * gmp_randclass::get_f: C++ Interface Random Numbers. * gmp_randclass::get_z_bits: C++ Interface Random Numbers. * gmp_randclass::get_z_range: C++ Interface Random Numbers. * gmp_randclass::gmp_randclass: C++ Interface Random Numbers. * gmp_randclass::seed: C++ Interface Random Numbers. * gmp_randclear: Random State Initialization. * gmp_randinit: Random State Initialization. * gmp_randinit_default: Random State Initialization. * gmp_randinit_lc_2exp: Random State Initialization. * gmp_randinit_lc_2exp_size: Random State Initialization. * gmp_randinit_mt: Random State Initialization. * gmp_randinit_set: Random State Initialization. * gmp_randseed: Random State Seeding. * gmp_randseed_ui: Random State Seeding. * gmp_randstate_t: Nomenclature and Types. * gmp_scanf: Formatted Input Functions. * gmp_snprintf: Formatted Output Functions. * gmp_sprintf: Formatted Output Functions. * gmp_sscanf: Formatted Input Functions. * gmp_urandomb_ui: Random State Miscellaneous. * gmp_urandomm_ui: Random State Miscellaneous. * gmp_vasprintf: Formatted Output Functions. * gmp_version: Useful Macros and Constants. * gmp_vfprintf: Formatted Output Functions. * gmp_vfscanf: Formatted Input Functions. * gmp_vprintf: Formatted Output Functions. * gmp_vscanf: Formatted Input Functions. * gmp_vsnprintf: Formatted Output Functions. * gmp_vsprintf: Formatted Output Functions. * gmp_vsscanf: Formatted Input Functions. * hypot: C++ Interface Floats. * itom: BSD Compatible Functions. * madd: BSD Compatible Functions. * mcmp: BSD Compatible Functions. * mdiv: BSD Compatible Functions. * mfree: BSD Compatible Functions. * min: BSD Compatible Functions. * MINT: BSD Compatible Functions. * mout: BSD Compatible Functions. * move: BSD Compatible Functions. * mp_bits_per_limb: Useful Macros and Constants. * mp_exp_t: Nomenclature and Types. * mp_get_memory_functions: Custom Allocation. * mp_limb_t: Nomenclature and Types. * mp_set_memory_functions: Custom Allocation. * mp_size_t: Nomenclature and Types. * mpf_abs: Float Arithmetic. * mpf_add: Float Arithmetic. * mpf_add_ui: Float Arithmetic. * mpf_ceil: Miscellaneous Float Functions. * mpf_class: C++ Interface General. * mpf_class::fits_sint_p: C++ Interface Floats. * mpf_class::fits_slong_p: C++ Interface Floats. * mpf_class::fits_sshort_p: C++ Interface Floats. * mpf_class::fits_uint_p: C++ Interface Floats. * mpf_class::fits_ulong_p: C++ Interface Floats. * mpf_class::fits_ushort_p: C++ Interface Floats. * mpf_class::get_d: C++ Interface Floats. * mpf_class::get_mpf_t: C++ Interface General. * mpf_class::get_prec: C++ Interface Floats. * mpf_class::get_si: C++ Interface Floats. * mpf_class::get_str: C++ Interface Floats. * mpf_class::get_ui: C++ Interface Floats. * mpf_class::mpf_class: C++ Interface Floats. * mpf_class::operator=: C++ Interface Floats. * mpf_class::set_prec: C++ Interface Floats. * mpf_class::set_prec_raw: C++ Interface Floats. * mpf_class::set_str: C++ Interface Floats. * mpf_clear: Initializing Floats. * mpf_cmp: Float Comparison. * mpf_cmp_d: Float Comparison. * mpf_cmp_si: Float Comparison. * mpf_cmp_ui: Float Comparison. * mpf_div: Float Arithmetic. * mpf_div_2exp: Float Arithmetic. * mpf_div_ui: Float Arithmetic. * mpf_eq: Float Comparison. * mpf_fits_sint_p: Miscellaneous Float Functions. * mpf_fits_slong_p: Miscellaneous Float Functions. * mpf_fits_sshort_p: Miscellaneous Float Functions. * mpf_fits_uint_p: Miscellaneous Float Functions. * mpf_fits_ulong_p: Miscellaneous Float Functions. * mpf_fits_ushort_p: Miscellaneous Float Functions. * mpf_floor: Miscellaneous Float Functions. * mpf_get_d: Converting Floats. * mpf_get_d_2exp: Converting Floats. * mpf_get_default_prec: Initializing Floats. * mpf_get_prec: Initializing Floats. * mpf_get_si: Converting Floats. * mpf_get_str: Converting Floats. * mpf_get_ui: Converting Floats. * mpf_init: Initializing Floats. * mpf_init2: Initializing Floats. * mpf_init_set: Simultaneous Float Init & Assign. * mpf_init_set_d: Simultaneous Float Init & Assign. * mpf_init_set_si: Simultaneous Float Init & Assign. * mpf_init_set_str: Simultaneous Float Init & Assign. * mpf_init_set_ui: Simultaneous Float Init & Assign. * mpf_inp_str: I/O of Floats. * mpf_integer_p: Miscellaneous Float Functions. * mpf_mul: Float Arithmetic. * mpf_mul_2exp: Float Arithmetic. * mpf_mul_ui: Float Arithmetic. * mpf_neg: Float Arithmetic. * mpf_out_str: I/O of Floats. * mpf_pow_ui: Float Arithmetic. * mpf_random2: Miscellaneous Float Functions. * mpf_reldiff: Float Comparison. * mpf_set: Assigning Floats. * mpf_set_d: Assigning Floats. * mpf_set_default_prec: Initializing Floats. * mpf_set_prec: Initializing Floats. * mpf_set_prec_raw: Initializing Floats. * mpf_set_q: Assigning Floats. * mpf_set_si: Assigning Floats. * mpf_set_str: Assigning Floats. * mpf_set_ui: Assigning Floats. * mpf_set_z: Assigning Floats. * mpf_sgn: Float Comparison. * mpf_sqrt: Float Arithmetic. * mpf_sqrt_ui: Float Arithmetic. * mpf_sub: Float Arithmetic. * mpf_sub_ui: Float Arithmetic. * mpf_swap: Assigning Floats. * mpf_t: Nomenclature and Types. * mpf_trunc: Miscellaneous Float Functions. * mpf_ui_div: Float Arithmetic. * mpf_ui_sub: Float Arithmetic. * mpf_urandomb: Miscellaneous Float Functions. * mpn_add: Low-level Functions. * mpn_add_1: Low-level Functions. * mpn_add_n: Low-level Functions. * mpn_addmul_1: Low-level Functions. * mpn_bdivmod: Low-level Functions. * mpn_cmp: Low-level Functions. * mpn_divexact_by3: Low-level Functions. * mpn_divexact_by3c: Low-level Functions. * mpn_divmod: Low-level Functions. * mpn_divmod_1: Low-level Functions. * mpn_divrem: Low-level Functions. * mpn_divrem_1: Low-level Functions. * mpn_gcd: Low-level Functions. * mpn_gcd_1: Low-level Functions. * mpn_gcdext: Low-level Functions. * mpn_get_str: Low-level Functions. * mpn_hamdist: Low-level Functions. * mpn_lshift: Low-level Functions. * mpn_mod_1: Low-level Functions. * mpn_mul: Low-level Functions. * mpn_mul_1: Low-level Functions. * mpn_mul_n: Low-level Functions. * mpn_perfect_square_p: Low-level Functions. * mpn_popcount: Low-level Functions. * mpn_random: Low-level Functions. * mpn_random2: Low-level Functions. * mpn_rshift: Low-level Functions. * mpn_scan0: Low-level Functions. * mpn_scan1: Low-level Functions. * mpn_set_str: Low-level Functions. * mpn_sqrtrem: Low-level Functions. * mpn_sub: Low-level Functions. * mpn_sub_1: Low-level Functions. * mpn_sub_n: Low-level Functions. * mpn_submul_1: Low-level Functions. * mpn_tdiv_qr: Low-level Functions. * mpq_abs: Rational Arithmetic. * mpq_add: Rational Arithmetic. * mpq_canonicalize: Rational Number Functions. * mpq_class: C++ Interface General. * mpq_class::canonicalize: C++ Interface Rationals. * mpq_class::get_d: C++ Interface Rationals. * mpq_class::get_den: C++ Interface Rationals. * mpq_class::get_den_mpz_t: C++ Interface Rationals. * mpq_class::get_mpq_t: C++ Interface General. * mpq_class::get_num: C++ Interface Rationals. * mpq_class::get_num_mpz_t: C++ Interface Rationals. * mpq_class::get_str: C++ Interface Rationals. * mpq_class::mpq_class: C++ Interface Rationals. * mpq_class::set_str: C++ Interface Rationals. * mpq_clear: Initializing Rationals. * mpq_cmp: Comparing Rationals. * mpq_cmp_si: Comparing Rationals. * mpq_cmp_ui: Comparing Rationals. * mpq_denref: Applying Integer Functions. * mpq_div: Rational Arithmetic. * mpq_div_2exp: Rational Arithmetic. * mpq_equal: Comparing Rationals. * mpq_get_d: Rational Conversions. * mpq_get_den: Applying Integer Functions. * mpq_get_num: Applying Integer Functions. * mpq_get_str: Rational Conversions. * mpq_init: Initializing Rationals. * mpq_inp_str: I/O of Rationals. * mpq_inv: Rational Arithmetic. * mpq_mul: Rational Arithmetic. * mpq_mul_2exp: Rational Arithmetic. * mpq_neg: Rational Arithmetic. * mpq_numref: Applying Integer Functions. * mpq_out_str: I/O of Rationals. * mpq_set: Initializing Rationals. * mpq_set_d: Rational Conversions. * mpq_set_den: Applying Integer Functions. * mpq_set_f: Rational Conversions. * mpq_set_num: Applying Integer Functions. * mpq_set_si: Initializing Rationals. * mpq_set_str: Initializing Rationals. * mpq_set_ui: Initializing Rationals. * mpq_set_z: Initializing Rationals. * mpq_sgn: Comparing Rationals. * mpq_sub: Rational Arithmetic. * mpq_swap: Initializing Rationals. * mpq_t: Nomenclature and Types. * mpz_abs: Integer Arithmetic. * mpz_add: Integer Arithmetic. * mpz_add_ui: Integer Arithmetic. * mpz_addmul: Integer Arithmetic. * mpz_addmul_ui: Integer Arithmetic. * mpz_and: Integer Logic and Bit Fiddling. * mpz_array_init: Integer Special Functions. * mpz_bin_ui: Number Theoretic Functions. * mpz_bin_uiui: Number Theoretic Functions. * mpz_cdiv_q: Integer Division. * mpz_cdiv_q_2exp: Integer Division. * mpz_cdiv_q_ui: Integer Division. * mpz_cdiv_qr: Integer Division. * mpz_cdiv_qr_ui: Integer Division. * mpz_cdiv_r: Integer Division. * mpz_cdiv_r_2exp: Integer Division. * mpz_cdiv_r_ui: Integer Division. * mpz_cdiv_ui: Integer Division. * mpz_class: C++ Interface General. * mpz_class::fits_sint_p: C++ Interface Integers. * mpz_class::fits_slong_p: C++ Interface Integers. * mpz_class::fits_sshort_p: C++ Interface Integers. * mpz_class::fits_uint_p: C++ Interface Integers. * mpz_class::fits_ulong_p: C++ Interface Integers. * mpz_class::fits_ushort_p: C++ Interface Integers. * mpz_class::get_d: C++ Interface Integers. * mpz_class::get_mpz_t: C++ Interface General. * mpz_class::get_si: C++ Interface Integers. * mpz_class::get_str: C++ Interface Integers. * mpz_class::get_ui: C++ Interface Integers. * mpz_class::mpz_class: C++ Interface Integers. * mpz_class::set_str: C++ Interface Integers. * mpz_clear: Initializing Integers. * mpz_clrbit: Integer Logic and Bit Fiddling. * mpz_cmp: Integer Comparisons. * mpz_cmp_d: Integer Comparisons. * mpz_cmp_si: Integer Comparisons. * mpz_cmp_ui: Integer Comparisons. * mpz_cmpabs: Integer Comparisons. * mpz_cmpabs_d: Integer Comparisons. * mpz_cmpabs_ui: Integer Comparisons. * mpz_com: Integer Logic and Bit Fiddling. * mpz_combit: Integer Logic and Bit Fiddling. * mpz_congruent_2exp_p: Integer Division. * mpz_congruent_p: Integer Division. * mpz_congruent_ui_p: Integer Division. * mpz_divexact: Integer Division. * mpz_divexact_ui: Integer Division. * mpz_divisible_2exp_p: Integer Division. * mpz_divisible_p: Integer Division. * mpz_divisible_ui_p: Integer Division. * mpz_even_p: Miscellaneous Integer Functions. * mpz_export: Integer Import and Export. * mpz_fac_ui: Number Theoretic Functions. * mpz_fdiv_q: Integer Division. * mpz_fdiv_q_2exp: Integer Division. * mpz_fdiv_q_ui: Integer Division. * mpz_fdiv_qr: Integer Division. * mpz_fdiv_qr_ui: Integer Division. * mpz_fdiv_r: Integer Division. * mpz_fdiv_r_2exp: Integer Division. * mpz_fdiv_r_ui: Integer Division. * mpz_fdiv_ui: Integer Division. * mpz_fib2_ui: Number Theoretic Functions. * mpz_fib_ui: Number Theoretic Functions. * mpz_fits_sint_p: Miscellaneous Integer Functions. * mpz_fits_slong_p: Miscellaneous Integer Functions. * mpz_fits_sshort_p: Miscellaneous Integer Functions. * mpz_fits_uint_p: Miscellaneous Integer Functions. * mpz_fits_ulong_p: Miscellaneous Integer Functions. * mpz_fits_ushort_p: Miscellaneous Integer Functions. * mpz_gcd: Number Theoretic Functions. * mpz_gcd_ui: Number Theoretic Functions. * mpz_gcdext: Number Theoretic Functions. * mpz_get_d: Converting Integers. * mpz_get_d_2exp: Converting Integers. * mpz_get_si: Converting Integers. * mpz_get_str: Converting Integers. * mpz_get_ui: Converting Integers. * mpz_getlimbn: Integer Special Functions. * mpz_hamdist: Integer Logic and Bit Fiddling. * mpz_import: Integer Import and Export. * mpz_init: Initializing Integers. * mpz_init2: Initializing Integers. * mpz_init_set: Simultaneous Integer Init & Assign. * mpz_init_set_d: Simultaneous Integer Init & Assign. * mpz_init_set_si: Simultaneous Integer Init & Assign. * mpz_init_set_str: Simultaneous Integer Init & Assign. * mpz_init_set_ui: Simultaneous Integer Init & Assign. * mpz_inp_raw: I/O of Integers. * mpz_inp_str: I/O of Integers. * mpz_invert: Number Theoretic Functions. * mpz_ior: Integer Logic and Bit Fiddling. * mpz_jacobi: Number Theoretic Functions. * mpz_kronecker: Number Theoretic Functions. * mpz_kronecker_si: Number Theoretic Functions. * mpz_kronecker_ui: Number Theoretic Functions. * mpz_lcm: Number Theoretic Functions. * mpz_lcm_ui: Number Theoretic Functions. * mpz_legendre: Number Theoretic Functions. * mpz_lucnum2_ui: Number Theoretic Functions. * mpz_lucnum_ui: Number Theoretic Functions. * mpz_mod: Integer Division. * mpz_mod_ui: Integer Division. * mpz_mul: Integer Arithmetic. * mpz_mul_2exp: Integer Arithmetic. * mpz_mul_si: Integer Arithmetic. * mpz_mul_ui: Integer Arithmetic. * mpz_neg: Integer Arithmetic. * mpz_nextprime: Number Theoretic Functions. * mpz_odd_p: Miscellaneous Integer Functions. * mpz_out_raw: I/O of Integers. * mpz_out_str: I/O of Integers. * mpz_perfect_power_p: Integer Roots. * mpz_perfect_square_p: Integer Roots. * mpz_popcount: Integer Logic and Bit Fiddling. * mpz_pow_ui: Integer Exponentiation. * mpz_powm: Integer Exponentiation. * mpz_powm_ui: Integer Exponentiation. * mpz_probab_prime_p: Number Theoretic Functions. * mpz_random: Integer Random Numbers. * mpz_random2: Integer Random Numbers. * mpz_realloc2: Initializing Integers. * mpz_remove: Number Theoretic Functions. * mpz_root: Integer Roots. * mpz_rootrem: Integer Roots. * mpz_rrandomb: Integer Random Numbers. * mpz_scan0: Integer Logic and Bit Fiddling. * mpz_scan1: Integer Logic and Bit Fiddling. * mpz_set: Assigning Integers. * mpz_set_d: Assigning Integers. * mpz_set_f: Assigning Integers. * mpz_set_q: Assigning Integers. * mpz_set_si: Assigning Integers. * mpz_set_str: Assigning Integers. * mpz_set_ui: Assigning Integers. * mpz_setbit: Integer Logic and Bit Fiddling. * mpz_sgn: Integer Comparisons. * mpz_si_kronecker: Number Theoretic Functions. * mpz_size: Integer Special Functions. * mpz_sizeinbase: Miscellaneous Integer Functions. * mpz_sqrt: Integer Roots. * mpz_sqrtrem: Integer Roots. * mpz_sub: Integer Arithmetic. * mpz_sub_ui: Integer Arithmetic. * mpz_submul: Integer Arithmetic. * mpz_submul_ui: Integer Arithmetic. * mpz_swap: Assigning Integers. * mpz_t: Nomenclature and Types. * mpz_tdiv_q: Integer Division. * mpz_tdiv_q_2exp: Integer Division. * mpz_tdiv_q_ui: Integer Division. * mpz_tdiv_qr: Integer Division. * mpz_tdiv_qr_ui: Integer Division. * mpz_tdiv_r: Integer Division. * mpz_tdiv_r_2exp: Integer Division. * mpz_tdiv_r_ui: Integer Division. * mpz_tdiv_ui: Integer Division. * mpz_tstbit: Integer Logic and Bit Fiddling. * mpz_ui_kronecker: Number Theoretic Functions. * mpz_ui_pow_ui: Integer Exponentiation. * mpz_ui_sub: Integer Arithmetic. * mpz_urandomb: Integer Random Numbers. * mpz_urandomm: Integer Random Numbers. * mpz_xor: Integer Logic and Bit Fiddling. * msqrt: BSD Compatible Functions. * msub: BSD Compatible Functions. * mtox: BSD Compatible Functions. * mult: BSD Compatible Functions. * operator%: C++ Interface Integers. * operator/: C++ Interface Integers. * operator<<: C++ Formatted Output. * operator>> <1>: C++ Formatted Input. * operator>>: C++ Interface Rationals. * pow: BSD Compatible Functions. * rpow: BSD Compatible Functions. * sdiv: BSD Compatible Functions. * sgn <1>: C++ Interface Rationals. * sgn <2>: C++ Interface Floats. * sgn: C++ Interface Integers. * sqrt <1>: C++ Interface Floats. * sqrt: C++ Interface Integers. * trunc: C++ Interface Floats. * xtom: BSD Compatible Functions.