2636 lines
130 KiB
Plaintext
2636 lines
130 KiB
Plaintext
This is /home/tege/prec/gmp42/doc/gmp.info, produced by makeinfo
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version 4.6 from /home/tege/prec/gmp42/doc/gmp.texi.
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This manual describes how to install and use the GNU multiple precision
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arithmetic library, version 4.2.1.
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Copyright 1991, 1993, 1994, 1995, 1996, 1997, 1998, 1999, 2000,
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2001, 2002, 2003, 2004, 2005, 2006 Free Software Foundation, Inc.
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Permission is granted to copy, distribute and/or modify this
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document under the terms of the GNU Free Documentation License, Version
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1.2 or any later version published by the Free Software Foundation;
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with no Invariant Sections, with the Front-Cover Texts being "A GNU
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Manual", and with the Back-Cover Texts being "You have freedom to copy
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and modify this GNU Manual, like GNU software". A copy of the license
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is included in *Note GNU Free Documentation License::.
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INFO-DIR-SECTION GNU libraries
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START-INFO-DIR-ENTRY
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* gmp: (gmp). GNU Multiple Precision Arithmetic Library.
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END-INFO-DIR-ENTRY
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File: gmp.info, Node: Radix to Binary, Prev: Binary to Radix, Up: Radix Conversion Algorithms
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Radix to Binary
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---------------
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Conversions from a power-of-2 radix into binary use a simple and fast
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O(N) bitwise concatenation algorithm.
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Conversions from other radices use one of two algorithms. Sizes
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below `SET_STR_THRESHOLD' use a basic O(N^2) method. Groups of n
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digits are converted to limbs, where n is the biggest power of the base
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b which will fit in a limb, then those groups are accumulated into the
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result by multiplying by b^n and adding. This saves multi-precision
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operations, as per Knuth section 4.4 part E (*note References::). Some
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special case code is provided for decimal, giving the compiler a chance
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to optimize multiplications by 10.
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Above `SET_STR_THRESHOLD' a sub-quadratic algorithm is used. First
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groups of n digits are converted into limbs. Then adjacent limbs are
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combined into limb pairs with x*b^n+y, where x and y are the limbs.
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Adjacent limb pairs are combined into quads similarly with x*b^(2n)+y.
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This continues until a single block remains, that being the result.
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The advantage of this method is that the multiplications for each x
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are big blocks, allowing Karatsuba and higher algorithms to be used.
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But the cost of calculating the powers b^(n*2^i) must be overcome.
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`SET_STR_THRESHOLD' usually ends up quite big, around 5000 digits, and
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on some processors much bigger still.
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`SET_STR_THRESHOLD' is based on the input digits (and tuned for
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decimal), though it might be better based on a limb count, so as to be
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independent of the base. But that sort of count isn't used by the base
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case and so would need some sort of initial calculation or estimate.
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The main reason `SET_STR_THRESHOLD' is so much bigger than the
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corresponding `GET_STR_PRECOMPUTE_THRESHOLD' is that `mpn_mul_1' is
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much faster than `mpn_divrem_1' (often by a factor of 10, or more).
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File: gmp.info, Node: Other Algorithms, Next: Assembler Coding, Prev: Radix Conversion Algorithms, Up: Algorithms
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Other Algorithms
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================
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* Menu:
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* Prime Testing Algorithm::
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* Factorial Algorithm::
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* Binomial Coefficients Algorithm::
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* Fibonacci Numbers Algorithm::
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* Lucas Numbers Algorithm::
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* Random Number Algorithms::
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File: gmp.info, Node: Prime Testing Algorithm, Next: Factorial Algorithm, Prev: Other Algorithms, Up: Other Algorithms
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Prime Testing
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-------------
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The primality testing in `mpz_probab_prime_p' (*note Number Theoretic
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Functions::) first does some trial division by small factors and then
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uses the Miller-Rabin probabilistic primality testing algorithm, as
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described in Knuth section 4.5.4 algorithm P (*note References::).
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For an odd input n, and with n = q*2^k+1 where q is odd, this
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algorithm selects a random base x and tests whether x^q mod n is 1 or
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-1, or an x^(q*2^j) mod n is 1, for 1<=j<=k. If so then n is probably
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prime, if not then n is definitely composite.
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Any prime n will pass the test, but some composites do too. Such
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composites are known as strong pseudoprimes to base x. No n is a
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strong pseudoprime to more than 1/4 of all bases (see Knuth exercise
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22), hence with x chosen at random there's no more than a 1/4 chance a
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"probable prime" will in fact be composite.
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In fact strong pseudoprimes are quite rare, making the test much more
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powerful than this analysis would suggest, but 1/4 is all that's proven
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for an arbitrary n.
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File: gmp.info, Node: Factorial Algorithm, Next: Binomial Coefficients Algorithm, Prev: Prime Testing Algorithm, Up: Other Algorithms
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Factorial
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---------
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Factorials are calculated by a combination of removal of twos,
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powering, and binary splitting. The procedure can be best illustrated
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with an example,
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23! = 1.2.3.4.5.6.7.8.9.10.11.12.13.14.15.16.17.18.19.20.21.22.23
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has factors of two removed,
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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
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and the resulting terms collected up according to their multiplicity,
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23! = 2^19.(3.5)^3.(7.9.11)^2.(13.15.17.19.21.23)
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Each sequence such as 13.15.17.19.21.23 is evaluated by splitting
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into every second term, as for instance (13.17.21).(15.19.23), and the
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same recursively on each half. This is implemented iteratively using
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some bit twiddling.
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Such splitting is more efficient than repeated Nx1 multiplies since
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it forms big multiplies, allowing Karatsuba and higher algorithms to be
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used. And even below the Karatsuba threshold a big block of work can
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be more efficient for the basecase algorithm.
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Splitting into subsequences of every second term keeps the resulting
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products more nearly equal in size than would the simpler approach of
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say taking the first half and second half of the sequence. Nearly
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equal products are more efficient for the current multiply
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implementation.
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File: gmp.info, Node: Binomial Coefficients Algorithm, Next: Fibonacci Numbers Algorithm, Prev: Factorial Algorithm, Up: Other Algorithms
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Binomial Coefficients
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---------------------
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Binomial coefficients C(n,k) are calculated by first arranging k <= n/2
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using C(n,k) = C(n,n-k) if necessary, and then evaluating the following
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product simply from i=2 to i=k.
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k (n-k+i)
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C(n,k) = (n-k+1) * prod -------
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i=2 i
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It's easy to show that each denominator i will divide the product so
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far, so the exact division algorithm is used (*note Exact Division::).
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The numerators n-k+i and denominators i are first accumulated into
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as many fit a limb, to save multi-precision operations, though for
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`mpz_bin_ui' this applies only to the divisors, since n is an `mpz_t'
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and n-k+i in general won't fit in a limb at all.
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File: gmp.info, Node: Fibonacci Numbers Algorithm, Next: Lucas Numbers Algorithm, Prev: Binomial Coefficients Algorithm, Up: Other Algorithms
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Fibonacci Numbers
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-----------------
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The Fibonacci functions `mpz_fib_ui' and `mpz_fib2_ui' are designed for
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calculating isolated F[n] or F[n],F[n-1] values efficiently.
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For small n, a table of single limb values in `__gmp_fib_table' is
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used. On a 32-bit limb this goes up to F[47], or on a 64-bit limb up
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to F[93]. For convenience the table starts at F[-1].
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Beyond the table, values are generated with a binary powering
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algorithm, calculating a pair F[n] and F[n-1] working from high to low
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across the bits of n. The formulas used are
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F[2k+1] = 4*F[k]^2 - F[k-1]^2 + 2*(-1)^k
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F[2k-1] = F[k]^2 + F[k-1]^2
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F[2k] = F[2k+1] - F[2k-1]
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At each step, k is the high b bits of n. If the next bit of n is 0
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then F[2k],F[2k-1] is used, or if it's a 1 then F[2k+1],F[2k] is used,
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and the process repeated until all bits of n are incorporated. Notice
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these formulas require just two squares per bit of n.
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It'd be possible to handle the first few n above the single limb
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table with simple additions, using the defining Fibonacci recurrence
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F[k+1]=F[k]+F[k-1], but this is not done since it usually turns out to
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be faster for only about 10 or 20 values of n, and including a block of
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code for just those doesn't seem worthwhile. If they really mattered
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it'd be better to extend the data table.
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Using a table avoids lots of calculations on small numbers, and
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makes small n go fast. A bigger table would make more small n go fast,
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it's just a question of balancing size against desired speed. For GMP
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the code is kept compact, with the emphasis primarily on a good
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powering algorithm.
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`mpz_fib2_ui' returns both F[n] and F[n-1], but `mpz_fib_ui' is only
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interested in F[n]. In this case the last step of the algorithm can
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become one multiply instead of two squares. One of the following two
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formulas is used, according as n is odd or even.
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F[2k] = F[k]*(F[k]+2F[k-1])
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F[2k+1] = (2F[k]+F[k-1])*(2F[k]-F[k-1]) + 2*(-1)^k
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F[2k+1] here is the same as above, just rearranged to be a multiply.
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For interest, the 2*(-1)^k term both here and above can be applied
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just to the low limb of the calculation, without a carry or borrow into
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further limbs, which saves some code size. See comments with
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`mpz_fib_ui' and the internal `mpn_fib2_ui' for how this is done.
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File: gmp.info, Node: Lucas Numbers Algorithm, Next: Random Number Algorithms, Prev: Fibonacci Numbers Algorithm, Up: Other Algorithms
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Lucas Numbers
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-------------
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`mpz_lucnum2_ui' derives a pair of Lucas numbers from a pair of
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Fibonacci numbers with the following simple formulas.
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L[k] = F[k] + 2*F[k-1]
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L[k-1] = 2*F[k] - F[k-1]
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`mpz_lucnum_ui' is only interested in L[n], and some work can be
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saved. Trailing zero bits on n can be handled with a single square
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each.
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L[2k] = L[k]^2 - 2*(-1)^k
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And the lowest 1 bit can be handled with one multiply of a pair of
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Fibonacci numbers, similar to what `mpz_fib_ui' does.
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||
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L[2k+1] = 5*F[k-1]*(2*F[k]+F[k-1]) - 4*(-1)^k
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File: gmp.info, Node: Random Number Algorithms, Prev: Lucas Numbers Algorithm, Up: Other Algorithms
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Random Numbers
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--------------
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For the `urandomb' functions, random numbers are generated simply by
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concatenating bits produced by the generator. As long as the generator
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has good randomness properties this will produce well-distributed N bit
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numbers.
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For the `urandomm' functions, random numbers in a range 0<=R<N are
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generated by taking values R of ceil(log2(N)) bits each until one
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satisfies R<N. This will normally require only one or two attempts,
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but the attempts are limited in case the generator is somehow
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degenerate and produces only 1 bits or similar.
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The Mersenne Twister generator is by Matsumoto and Nishimura (*note
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References::). It has a non-repeating period of 2^19937-1, which is a
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Mersenne prime, hence the name of the generator. The state is 624
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words of 32-bits each, which is iterated with one XOR and shift for each
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32-bit word generated, making the algorithm very fast. Randomness
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properties are also very good and this is the default algorithm used by
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GMP.
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Linear congruential generators are described in many text books, for
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instance Knuth volume 2 (*note References::). With a modulus M and
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parameters A and C, a integer state S is iterated by the formula S <-
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A*S+C mod M. At each step the new state is a linear function of the
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previous, mod M, hence the name of the generator.
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In GMP only moduli of the form 2^N are supported, and the current
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implementation is not as well optimized as it could be. Overheads are
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significant when N is small, and when N is large clearly the multiply
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at each step will become slow. This is not a big concern, since the
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Mersenne Twister generator is better in every respect and is therefore
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recommended for all normal applications.
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For both generators the current state can be deduced by observing
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enough output and applying some linear algebra (over GF(2) in the case
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of the Mersenne Twister). This generally means raw output is
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unsuitable for cryptographic applications without further hashing or
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the like.
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File: gmp.info, Node: Assembler Coding, Prev: Other Algorithms, Up: Algorithms
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Assembler Coding
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================
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The assembler subroutines in GMP are the most significant source of
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speed at small to moderate sizes. At larger sizes algorithm selection
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becomes more important, but of course speedups in low level routines
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will still speed up everything proportionally.
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Carry handling and widening multiplies that are important for GMP
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can't be easily expressed in C. GCC `asm' blocks help a lot and are
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provided in `longlong.h', but hand coding low level routines invariably
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offers a speedup over generic C by a factor of anything from 2 to 10.
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* Menu:
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* Assembler Code Organisation::
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* Assembler Basics::
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* Assembler Carry Propagation::
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* Assembler Cache Handling::
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* Assembler Functional Units::
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* Assembler Floating Point::
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* Assembler SIMD Instructions::
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* Assembler Software Pipelining::
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* Assembler Loop Unrolling::
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* Assembler Writing Guide::
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File: gmp.info, Node: Assembler Code Organisation, Next: Assembler Basics, Prev: Assembler Coding, Up: Assembler Coding
|
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Code Organisation
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-----------------
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The various `mpn' subdirectories contain machine-dependent code, written
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in C or assembler. The `mpn/generic' subdirectory contains default
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code, used when there's no machine-specific version of a particular
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||
file.
|
||
|
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Each `mpn' subdirectory is for an ISA family. Generally 32-bit and
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64-bit variants in a family cannot share code and have separate
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directories. Within a family further subdirectories may exist for CPU
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variants.
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||
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In each directory a `nails' subdirectory may exist, holding code with
|
||
nails support for that CPU variant. A `NAILS_SUPPORT' directive in each
|
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file indicates the nails values the code handles. Nails code only
|
||
exists where it's faster, or promises to be faster, than plain code.
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There's no effort put into nails if they're not going to enhance a
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given CPU.
|
||
|
||
|
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File: gmp.info, Node: Assembler Basics, Next: Assembler Carry Propagation, Prev: Assembler Code Organisation, Up: Assembler Coding
|
||
|
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Assembler Basics
|
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----------------
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||
|
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`mpn_addmul_1' and `mpn_submul_1' are the most important routines for
|
||
overall GMP performance. All multiplications and divisions come down to
|
||
repeated calls to these. `mpn_add_n', `mpn_sub_n', `mpn_lshift' and
|
||
`mpn_rshift' are next most important.
|
||
|
||
On some CPUs assembler versions of the internal functions
|
||
`mpn_mul_basecase' and `mpn_sqr_basecase' give significant speedups,
|
||
mainly through avoiding function call overheads. They can also
|
||
potentially make better use of a wide superscalar processor, as can
|
||
bigger primitives like `mpn_addmul_2' or `mpn_addmul_4'.
|
||
|
||
The restrictions on overlaps between sources and destinations (*note
|
||
Low-level Functions::) are designed to facilitate a variety of
|
||
implementations. For example, knowing `mpn_add_n' won't have partly
|
||
overlapping sources and destination means reading can be done far ahead
|
||
of writing on superscalar processors, and loops can be vectorized on a
|
||
vector processor, depending on the carry handling.
|
||
|
||
|
||
File: gmp.info, Node: Assembler Carry Propagation, Next: Assembler Cache Handling, Prev: Assembler Basics, Up: Assembler Coding
|
||
|
||
Carry Propagation
|
||
-----------------
|
||
|
||
The problem that presents most challenges in GMP is propagating carries
|
||
from one limb to the next. In functions like `mpn_addmul_1' and
|
||
`mpn_add_n', carries are the only dependencies between limb operations.
|
||
|
||
On processors with carry flags, a straightforward CISC style `adc' is
|
||
generally best. AMD K6 `mpn_addmul_1' however is an example of an
|
||
unusual set of circumstances where a branch works out better.
|
||
|
||
On RISC processors generally an add and compare for overflow is
|
||
used. This sort of thing can be seen in `mpn/generic/aors_n.c'. Some
|
||
carry propagation schemes require 4 instructions, meaning at least 4
|
||
cycles per limb, but other schemes may use just 1 or 2. On wide
|
||
superscalar processors performance may be completely determined by the
|
||
number of dependent instructions between carry-in and carry-out for
|
||
each limb.
|
||
|
||
On vector processors good use can be made of the fact that a carry
|
||
bit only very rarely propagates more than one limb. When adding a
|
||
single bit to a limb, there's only a carry out if that limb was
|
||
`0xFF...FF' which on random data will be only 1 in 2^mp_bits_per_limb.
|
||
`mpn/cray/add_n.c' is an example of this, it adds all limbs in
|
||
parallel, adds one set of carry bits in parallel and then only rarely
|
||
needs to fall through to a loop propagating further carries.
|
||
|
||
On the x86s, GCC (as of version 2.95.2) doesn't generate
|
||
particularly good code for the RISC style idioms that are necessary to
|
||
handle carry bits in C. Often conditional jumps are generated where
|
||
`adc' or `sbb' forms would be better. And so unfortunately almost any
|
||
loop involving carry bits needs to be coded in assembler for best
|
||
results.
|
||
|
||
|
||
File: gmp.info, Node: Assembler Cache Handling, Next: Assembler Functional Units, Prev: Assembler Carry Propagation, Up: Assembler Coding
|
||
|
||
Cache Handling
|
||
--------------
|
||
|
||
GMP aims to perform well both on operands that fit entirely in L1 cache
|
||
and those which don't.
|
||
|
||
Basic routines like `mpn_add_n' or `mpn_lshift' are often used on
|
||
large operands, so L2 and main memory performance is important for them.
|
||
`mpn_mul_1' and `mpn_addmul_1' are mostly used for multiply and square
|
||
basecases, so L1 performance matters most for them, unless assembler
|
||
versions of `mpn_mul_basecase' and `mpn_sqr_basecase' exist, in which
|
||
case the remaining uses are mostly for larger operands.
|
||
|
||
For L2 or main memory operands, memory access times will almost
|
||
certainly be more than the calculation time. The aim therefore is to
|
||
maximize memory throughput, by starting a load of the next cache line
|
||
while processing the contents of the previous one. Clearly this is
|
||
only possible if the chip has a lock-up free cache or some sort of
|
||
prefetch instruction. Most current chips have both these features.
|
||
|
||
Prefetching sources combines well with loop unrolling, since a
|
||
prefetch can be initiated once per unrolled loop (or more than once if
|
||
the loop covers more than one cache line).
|
||
|
||
On CPUs without write-allocate caches, prefetching destinations will
|
||
ensure individual stores don't go further down the cache hierarchy,
|
||
limiting bandwidth. Of course for calculations which are slow anyway,
|
||
like `mpn_divrem_1', write-throughs might be fine.
|
||
|
||
The distance ahead to prefetch will be determined by memory latency
|
||
versus throughput. The aim of course is to have data arriving
|
||
continuously, at peak throughput. Some CPUs have limits on the number
|
||
of fetches or prefetches in progress.
|
||
|
||
If a special prefetch instruction doesn't exist then a plain load
|
||
can be used, but in that case care must be taken not to attempt to read
|
||
past the end of an operand, since that might produce a segmentation
|
||
violation.
|
||
|
||
Some CPUs or systems have hardware that detects sequential memory
|
||
accesses and initiates suitable cache movements automatically, making
|
||
life easy.
|
||
|
||
|
||
File: gmp.info, Node: Assembler Functional Units, Next: Assembler Floating Point, Prev: Assembler Cache Handling, Up: Assembler Coding
|
||
|
||
Functional Units
|
||
----------------
|
||
|
||
When choosing an approach for an assembler loop, consideration is given
|
||
to what operations can execute simultaneously and what throughput can
|
||
thereby be achieved. In some cases an algorithm can be tweaked to
|
||
accommodate available resources.
|
||
|
||
Loop control will generally require a counter and pointer updates,
|
||
costing as much as 5 instructions, plus any delays a branch introduces.
|
||
CPU addressing modes might reduce pointer updates, perhaps by allowing
|
||
just one updating pointer and others expressed as offsets from it, or
|
||
on CISC chips with all addressing done with the loop counter as a
|
||
scaled index.
|
||
|
||
The final loop control cost can be amortised by processing several
|
||
limbs in each iteration (*note Assembler Loop Unrolling::). This at
|
||
least ensures loop control isn't a big fraction the work done.
|
||
|
||
Memory throughput is always a limit. If perhaps only one load or
|
||
one store can be done per cycle then 3 cycles/limb will the top speed
|
||
for "binary" operations like `mpn_add_n', and any code achieving that
|
||
is optimal.
|
||
|
||
Integer resources can be freed up by having the loop counter in a
|
||
float register, or by pressing the float units into use for some
|
||
multiplying, perhaps doing every second limb on the float side (*note
|
||
Assembler Floating Point::).
|
||
|
||
Float resources can be freed up by doing carry propagation on the
|
||
integer side, or even by doing integer to float conversions in integers
|
||
using bit twiddling.
|
||
|
||
|
||
File: gmp.info, Node: Assembler Floating Point, Next: Assembler SIMD Instructions, Prev: Assembler Functional Units, Up: Assembler Coding
|
||
|
||
Floating Point
|
||
--------------
|
||
|
||
Floating point arithmetic is used in GMP for multiplications on CPUs
|
||
with poor integer multipliers. It's mostly useful for `mpn_mul_1',
|
||
`mpn_addmul_1' and `mpn_submul_1' on 64-bit machines, and
|
||
`mpn_mul_basecase' on both 32-bit and 64-bit machines.
|
||
|
||
With IEEE 53-bit double precision floats, integer multiplications
|
||
producing up to 53 bits will give exact results. Breaking a 64x64
|
||
multiplication into eight 16x32->48 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<mpf_class, mpf_class, __gmp_binary_plus> >
|
||
operator+(const mpf_class &f, const mpf_class &g)
|
||
{
|
||
return __gmp_expr
|
||
<__gmp_binary_expr<mpf_class, mpf_class, __gmp_binary_plus> >(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 <class T>
|
||
mpf_class & mpf_class::operator=(const __gmp_expr<T> &expr)
|
||
{
|
||
expr.eval(this->get_mpf_t(), this->precision());
|
||
return *this;
|
||
}
|
||
|
||
template <class Op>
|
||
void __gmp_expr<__gmp_binary_expr<mpf_class, mpf_class, Op> >::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 <class T, class U>
|
||
__gmp_expr
|
||
<__gmp_binary_expr<__gmp_expr<T>, __gmp_expr<U>, __gmp_binary_plus> >
|
||
operator+(const __gmp_expr<T> &expr1, const __gmp_expr<U> &expr2)
|
||
{
|
||
return __gmp_expr
|
||
<__gmp_binary_expr<__gmp_expr<T>, __gmp_expr<U>, __gmp_binary_plus> >
|
||
(expr1, expr2);
|
||
}
|
||
|
||
And the corresponding specializations of `__gmp_expr::eval':
|
||
|
||
template <class T, class U, class Op>
|
||
void __gmp_expr
|
||
<__gmp_binary_expr<__gmp_expr<T>, __gmp_expr<U>, 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
|
||
<tege@swox.com> 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.
|
||
|
||
This License is a kind of "copyleft", which means that derivative
|
||
works of the document must themselves be free in the same sense.
|
||
It complements the GNU General Public License, which is a copyleft
|
||
license designed for free software.
|
||
|
||
We have designed this License in order to use it for manuals for
|
||
free software, because free software needs free documentation: a
|
||
free program should come with manuals providing the same freedoms
|
||
that the software does. But this License is not limited to
|
||
software manuals; it can be used for any textual work, regardless
|
||
of subject matter or whether it is published as a printed book.
|
||
We recommend this License principally for works whose purpose is
|
||
instruction or reference.
|
||
|
||
1. APPLICABILITY AND DEFINITIONS
|
||
|
||
This License applies to any manual or other work, in any medium,
|
||
that contains a notice placed by the copyright holder saying it
|
||
can be distributed under the terms of this License. Such a notice
|
||
grants a world-wide, royalty-free license, unlimited in duration,
|
||
to use that work under the conditions stated herein. The
|
||
"Document", below, refers to any such manual or work. Any member
|
||
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.
|
||
|
||
A "Modified Version" of the Document means any work containing the
|
||
Document or a portion of it, either copied verbatim, or with
|
||
modifications and/or translated into another language.
|
||
|
||
A "Secondary Section" is a named appendix or a front-matter section
|
||
of the Document that deals exclusively with the relationship of the
|
||
publishers or authors of the Document to the Document's overall
|
||
subject (or to related matters) and contains nothing that could
|
||
fall directly within that overall subject. (Thus, if the Document
|
||
is in part a textbook of mathematics, a Secondary Section may not
|
||
explain any mathematics.) The relationship could be a matter of
|
||
historical connection with the subject or with related matters, or
|
||
of legal, commercial, philosophical, ethical or political position
|
||
regarding them.
|
||
|
||
The "Invariant Sections" are certain Secondary Sections whose
|
||
titles are designated, as being those of Invariant Sections, in
|
||
the notice that says that the Document is released under this
|
||
License. If a section does not fit the above definition of
|
||
Secondary then it is not allowed to be designated as Invariant.
|
||
The Document may contain zero Invariant Sections. If the Document
|
||
does not identify any Invariant Sections then there are none.
|
||
|
||
The "Cover Texts" are certain short passages of text that are
|
||
listed, as Front-Cover Texts or Back-Cover Texts, in the notice
|
||
that says that the Document is released under this License. A
|
||
Front-Cover Text may be at most 5 words, and a Back-Cover Text may
|
||
be at most 25 words.
|
||
|
||
A "Transparent" copy of the Document means a machine-readable copy,
|
||
represented in a format whose specification is available to the
|
||
general public, that is suitable for revising the document
|
||
straightforwardly with generic text editors or (for images
|
||
composed of pixels) generic paint programs or (for drawings) some
|
||
widely available drawing editor, and that is suitable for input to
|
||
text formatters or for automatic translation to a variety of
|
||
formats suitable for input to text formatters. A copy made in an
|
||
otherwise Transparent file format whose markup, or absence of
|
||
markup, has been arranged to thwart or discourage subsequent
|
||
modification by readers is not Transparent. An image format is
|
||
not Transparent if used for any substantial amount of text. A
|
||
copy that is not "Transparent" is called "Opaque".
|
||
|
||
Examples of suitable formats for Transparent copies include plain
|
||
ASCII without markup, Texinfo input format, LaTeX input format,
|
||
SGML or XML using a publicly available DTD, and
|
||
standard-conforming simple HTML, PostScript or PDF designed for
|
||
human modification. Examples of transparent image formats include
|
||
PNG, XCF and JPG. Opaque formats include proprietary formats that
|
||
can be read and edited only by proprietary word processors, SGML or
|
||
XML for which the DTD and/or processing tools are not generally
|
||
available, and the machine-generated HTML, PostScript or PDF
|
||
produced by some word processors for output purposes only.
|
||
|
||
The "Title Page" means, for a printed book, the title page itself,
|
||
plus such following pages as are needed to hold, legibly, the
|
||
material this License requires to appear in the title page. For
|
||
works in formats which do not have any title page as such, "Title
|
||
Page" means the text near the most prominent appearance of the
|
||
work's title, preceding the beginning of the body of the text.
|
||
|
||
A section "Entitled XYZ" means a named subunit of the Document
|
||
whose title either is precisely XYZ or contains XYZ in parentheses
|
||
following text that translates XYZ in another language. (Here XYZ
|
||
stands for a specific section name mentioned below, such as
|
||
"Acknowledgements", "Dedications", "Endorsements", or "History".)
|
||
To "Preserve the Title" of such a section when you modify the
|
||
Document means that it remains a section "Entitled XYZ" according
|
||
to this definition.
|
||
|
||
The Document may include Warranty Disclaimers next to the notice
|
||
which states that this License applies to the Document. These
|
||
Warranty Disclaimers are considered to be included by reference in
|
||
this License, but only as regards disclaiming warranties: any other
|
||
implication that these Warranty Disclaimers may have is void and
|
||
has no effect on the meaning of this License.
|
||
|
||
2. VERBATIM COPYING
|
||
|
||
You may copy and distribute the Document in any medium, either
|
||
commercially or noncommercially, provided that this License, the
|
||
copyright notices, and the license notice saying this License
|
||
applies to the Document are reproduced in all copies, and that you
|
||
add no other conditions whatsoever to those of this License. You
|
||
may not use technical measures to obstruct or control the reading
|
||
or further copying of the copies you make or distribute. However,
|
||
you may accept compensation in exchange for copies. If you
|
||
distribute a large enough number of copies you must also follow
|
||
the conditions in section 3.
|
||
|
||
You may also lend copies, under the same conditions stated above,
|
||
and you may publicly display copies.
|
||
|
||
3. COPYING IN QUANTITY
|
||
|
||
If you publish printed copies (or copies in media that commonly
|
||
have printed covers) of the Document, numbering more than 100, and
|
||
the Document's license notice requires Cover Texts, you must
|
||
enclose the copies in covers that carry, clearly and legibly, all
|
||
these Cover Texts: Front-Cover Texts on the front cover, and
|
||
Back-Cover Texts on the back cover. Both covers must also clearly
|
||
and legibly identify you as the publisher of these copies. The
|
||
front cover must present the full title with all words of the
|
||
title equally prominent and visible. You may add other material
|
||
on the covers in addition. Copying with changes limited to the
|
||
covers, as long as they preserve the title of the Document and
|
||
satisfy these conditions, can be treated as verbatim copying in
|
||
other respects.
|
||
|
||
If the required texts for either cover are too voluminous to fit
|
||
legibly, you should put the first ones listed (as many as fit
|
||
reasonably) on the actual cover, and continue the rest onto
|
||
adjacent pages.
|
||
|
||
If you publish or distribute Opaque copies of the Document
|
||
numbering more than 100, you must either include a
|
||
machine-readable Transparent copy along with each Opaque copy, or
|
||
state in or with each Opaque copy a computer-network location from
|
||
which the general network-using public has access to download
|
||
using public-standard network protocols a complete Transparent
|
||
copy of the Document, free of added material. If you use the
|
||
latter option, you must take reasonably prudent steps, when you
|
||
begin distribution of Opaque copies in quantity, to ensure that
|
||
this Transparent copy will remain thus accessible at the stated
|
||
location until at least one year after the last time you
|
||
distribute an Opaque copy (directly or through your agents or
|
||
retailers) of that edition to the public.
|
||
|
||
It is requested, but not required, that you contact the authors of
|
||
the Document well before redistributing any large number of
|
||
copies, to give them a chance to provide you with an updated
|
||
version of the Document.
|
||
|
||
4. MODIFICATIONS
|
||
|
||
You may copy and distribute a Modified Version of the Document
|
||
under the conditions of sections 2 and 3 above, provided that you
|
||
release the Modified Version under precisely this License, with
|
||
the Modified Version filling the role of the Document, thus
|
||
licensing distribution and modification of the Modified Version to
|
||
whoever possesses a copy of it. In addition, you must do these
|
||
things in the Modified Version:
|
||
|
||
A. Use in the Title Page (and on the covers, if any) a title
|
||
distinct from that of the Document, and from those of
|
||
previous versions (which should, if there were any, be listed
|
||
in the History section of the Document). You may use the
|
||
same title as a previous version if the original publisher of
|
||
that version gives permission.
|
||
|
||
B. List on the Title Page, as authors, one or more persons or
|
||
entities responsible for authorship of the modifications in
|
||
the Modified Version, together with at least five of the
|
||
principal authors of the Document (all of its principal
|
||
authors, if it has fewer than five), unless they release you
|
||
from this requirement.
|
||
|
||
C. State on the Title page the name of the publisher of the
|
||
Modified Version, as the publisher.
|
||
|
||
D. Preserve all the copyright notices of the Document.
|
||
|
||
E. Add an appropriate copyright notice for your modifications
|
||
adjacent to the other copyright notices.
|
||
|
||
F. Include, immediately after the copyright notices, a license
|
||
notice giving the public permission to use the Modified
|
||
Version under the terms of this License, in the form shown in
|
||
the Addendum below.
|
||
|
||
G. Preserve in that license notice the full lists of Invariant
|
||
Sections and required Cover Texts given in the Document's
|
||
license notice.
|
||
|
||
H. Include an unaltered copy of this License.
|
||
|
||
I. Preserve the section Entitled "History", Preserve its Title,
|
||
and add to it an item stating at least the title, year, new
|
||
authors, and publisher of the Modified Version as given on
|
||
the Title Page. If there is no section Entitled "History" in
|
||
the Document, create one stating the title, year, authors,
|
||
and publisher of the Document as given on its Title Page,
|
||
then add an item describing the Modified Version as stated in
|
||
the previous sentence.
|
||
|
||
J. Preserve the network location, if any, given in the Document
|
||
for public access to a Transparent copy of the Document, and
|
||
likewise the network locations given in the Document for
|
||
previous versions it was based on. These may be placed in
|
||
the "History" section. You may omit a network location for a
|
||
work that was published at least four years before the
|
||
Document itself, or if the original publisher of the version
|
||
it refers to gives permission.
|
||
|
||
K. For any section Entitled "Acknowledgements" or "Dedications",
|
||
Preserve the Title of the section, and preserve in the
|
||
section all the substance and tone of each of the contributor
|
||
acknowledgements and/or dedications given therein.
|
||
|
||
L. Preserve all the Invariant Sections of the Document,
|
||
unaltered in their text and in their titles. Section numbers
|
||
or the equivalent are not considered part of the section
|
||
titles.
|
||
|
||
M. Delete any section Entitled "Endorsements". Such a section
|
||
may not be included in the Modified Version.
|
||
|
||
N. Do not retitle any existing section to be Entitled
|
||
"Endorsements" or to conflict in title with any Invariant
|
||
Section.
|
||
|
||
O. Preserve any Warranty Disclaimers.
|
||
|
||
If the Modified Version includes new front-matter sections or
|
||
appendices that qualify as Secondary Sections and contain no
|
||
material copied from the Document, you may at your option
|
||
designate some or all of these sections as invariant. To do this,
|
||
add their titles to the list of Invariant Sections in the Modified
|
||
Version's license notice. These titles must be distinct from any
|
||
other section titles.
|
||
|
||
You may add a section Entitled "Endorsements", provided it contains
|
||
nothing but endorsements of your Modified Version by various
|
||
parties--for example, statements of peer review or that the text
|
||
has been approved by an organization as the authoritative
|
||
definition of a standard.
|
||
|
||
You may add a passage of up to five words as a Front-Cover Text,
|
||
and a passage of up to 25 words as a Back-Cover Text, to the end
|
||
of the list of Cover Texts in the Modified Version. Only one
|
||
passage of Front-Cover Text and one of Back-Cover Text may be
|
||
added by (or through arrangements made by) any one entity. If the
|
||
Document already includes a cover text for the same cover,
|
||
previously added by you or by arrangement made by the same entity
|
||
you are acting on behalf of, you may not add another; but you may
|
||
replace the old one, on explicit permission from the previous
|
||
publisher that added the old one.
|
||
|
||
The author(s) and publisher(s) of the Document do not by this
|
||
License give permission to use their names for publicity for or to
|
||
assert or imply endorsement of any Modified Version.
|
||
|
||
5. COMBINING DOCUMENTS
|
||
|
||
You may combine the Document with other documents released under
|
||
this License, under the terms defined in section 4 above for
|
||
modified versions, provided that you include in the combination
|
||
all of the Invariant Sections of all of the original documents,
|
||
unmodified, and list them all as Invariant Sections of your
|
||
combined work in its license notice, and that you preserve all
|
||
their Warranty Disclaimers.
|
||
|
||
The combined work need only contain one copy of this License, and
|
||
multiple identical Invariant Sections may be replaced with a single
|
||
copy. If there are multiple Invariant Sections with the same name
|
||
but different contents, make the title of each such section unique
|
||
by adding at the end of it, in parentheses, the name of the
|
||
original author or publisher of that section if known, or else a
|
||
unique number. Make the same adjustment to the section titles in
|
||
the list of Invariant Sections in the license notice of the
|
||
combined work.
|
||
|
||
In the combination, you must combine any sections Entitled
|
||
"History" in the various original documents, forming one section
|
||
Entitled "History"; likewise combine any sections Entitled
|
||
"Acknowledgements", and any sections Entitled "Dedications". You
|
||
must delete all sections Entitled "Endorsements."
|
||
|
||
6. COLLECTIONS OF DOCUMENTS
|
||
|
||
You may make a collection consisting of the Document and other
|
||
documents released under this License, and replace the individual
|
||
copies of this License in the various documents with a single copy
|
||
that is included in the collection, provided that you follow the
|
||
rules of this License for verbatim copying of each of the
|
||
documents in all other respects.
|
||
|
||
You may extract a single document from such a collection, and
|
||
distribute it individually under this License, provided you insert
|
||
a copy of this License into the extracted document, and follow
|
||
this License in all other respects regarding verbatim copying of
|
||
that document.
|
||
|
||
7. AGGREGATION WITH INDEPENDENT WORKS
|
||
|
||
A compilation of the Document or its derivatives with other
|
||
separate and independent documents or works, in or on a volume of
|
||
a storage or distribution medium, is called an "aggregate" if the
|
||
copyright resulting from the compilation is not used to limit the
|
||
legal rights of the compilation's users beyond what the individual
|
||
works permit. When the Document is included in an aggregate, this
|
||
License does not apply to the other works in the aggregate which
|
||
are not themselves derivative works of the Document.
|
||
|
||
If the Cover Text requirement of section 3 is applicable to these
|
||
copies of the Document, then if the Document is less than one half
|
||
of the entire aggregate, the Document's Cover Texts may be placed
|
||
on covers that bracket the Document within the aggregate, or the
|
||
electronic equivalent of covers if the Document is in electronic
|
||
form. Otherwise they must appear on printed covers that bracket
|
||
the whole aggregate.
|
||
|
||
8. TRANSLATION
|
||
|
||
Translation is considered a kind of modification, so you may
|
||
distribute translations of the Document under the terms of section
|
||
4. Replacing Invariant Sections with translations requires special
|
||
permission from their copyright holders, but you may include
|
||
translations of some or all Invariant Sections in addition to the
|
||
original versions of these Invariant Sections. You may include a
|
||
translation of this License, and all the license notices in the
|
||
Document, and any Warranty Disclaimers, provided that you also
|
||
include the original English version of this License and the
|
||
original versions of those notices and disclaimers. In case of a
|
||
disagreement between the translation and the original version of
|
||
this License or a notice or disclaimer, the original version will
|
||
prevail.
|
||
|
||
If a section in the Document is Entitled "Acknowledgements",
|
||
"Dedications", or "History", the requirement (section 4) to
|
||
Preserve its Title (section 1) will typically require changing the
|
||
actual title.
|
||
|
||
9. TERMINATION
|
||
|
||
You may not copy, modify, sublicense, or distribute the Document
|
||
except as expressly provided for under this License. Any other
|
||
attempt to copy, modify, sublicense or distribute the Document is
|
||
void, and will automatically terminate your rights under this
|
||
License. However, parties who have received copies, or rights,
|
||
from you under this License will not have their licenses
|
||
terminated so long as such parties remain in full compliance.
|
||
|
||
10. FUTURE REVISIONS OF THIS LICENSE
|
||
|
||
The Free Software Foundation may publish new, revised versions of
|
||
the GNU Free Documentation License from time to time. Such new
|
||
versions will be similar in spirit to the present version, but may
|
||
differ in detail to address new problems or concerns. See
|
||
`http://www.gnu.org/copyleft/'.
|
||
|
||
Each version of the License is given a distinguishing version
|
||
number. If the Document specifies that a particular numbered
|
||
version of this License "or any later version" applies to it, you
|
||
have the option of following the terms and conditions either of
|
||
that specified version or of any later version that has been
|
||
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.
|
||
|
||
ADDENDUM: How to use this License for your documents
|
||
====================================================
|
||
|
||
To use this License in a document you have written, include a copy of
|
||
the License in the document and put the following copyright and license
|
||
notices just after the title page:
|
||
|
||
Copyright (C) YEAR YOUR NAME.
|
||
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, no Front-Cover Texts, and no Back-Cover
|
||
Texts. A copy of the license is included in the section entitled ``GNU
|
||
Free Documentation License''.
|
||
|
||
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
|
||
*************
|
||
|
||
* 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.
|
||
|
||
|