263 lines
10 KiB
Plaintext
263 lines
10 KiB
Plaintext
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/*=============================================================================
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This file is part of FLINT.
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FLINT is free software; you can redistribute it and/or modify
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it under the terms of the GNU General Public License as published by
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the Free Software Foundation; either version 2 of the License, or
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(at your option) any later version.
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FLINT is distributed in the hope that it will be useful,
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but WITHOUT ANY WARRANTY; without even the implied warranty of
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MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the
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GNU General Public License for more details.
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You should have received a copy of the GNU General Public License
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along with FLINT; if not, write to the Free Software
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Foundation, Inc., 51 Franklin St, Fifth Floor, Boston, MA 02110-1301 USA
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=============================================================================*/
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/******************************************************************************
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Copyright (C) 2010 Fredrik Johansson
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******************************************************************************/
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*******************************************************************************
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Macros
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*******************************************************************************
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MACRO MPN_NORM(a, an)
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Normalise \code{(a, an)} so that either \code{an} is zero or
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\code{a[an - 1]} is nonzero.
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MACRO MPN_SWAP(a, an, b, bn)
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Swap \code{(a, an)} and \code{(b, bn)}, i.e. swap pointers and sizes.
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*******************************************************************************
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Utility functions
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*******************************************************************************
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void flint_mpn_debug(mp_srcptr x, mp_size_t xsize)
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Prints debug information about \code{(x, xsize)} to \code{stdout}.
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In particular, this will print binary representations of all the limbs.
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int flint_mpn_zero_p(mp_srcptr x, mp_size_t xsize)
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Returns $1$ if all limbs of \code{(x, xsize)} are zero, otherwise $0$.
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*******************************************************************************
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Divisibility
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*******************************************************************************
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int flint_mpn_divisible_1_p(x, xsize, d) (macro)
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Expression determining whether \code{(x, xsize)} is divisible by the
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\code{mp_limb_t d} which is assumed to be odd-valued and at least~$3$.
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This function is implemented as a macro.
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mp_size_t flint_mpn_divexact_1(mp_ptr x, mp_size_t xsize, mp_limb_t d)
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Divides $x$ once by a known single-limb divisor, returns the new size.
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mp_size_t flint_mpn_remove_2exp(mp_ptr x, mp_size_t xsize, mp_bitcnt_t *bits)
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Divides \code{(x, xsize)} by $2^n$ where $n$ is the number of trailing
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zero bits in $x$. The new size of $x$ is returned, and $n$ is stored in
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the bits argument. $x$ may not be zero.
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mp_size_t flint_mpn_remove_power_ascending(mp_ptr x, mp_size_t xsize,
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mp_ptr p, mp_size_t psize, ulong *exp)
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Divides \code{(x, xsize)} by the largest power $n$ of \code{(p, psize)}
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that is an exact divisor of $x$. The new size of $x$ is returned, and
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$n$ is stored in the \code{exp} argument. $x$ may not be zero, and $p$
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must be greater than $2$.
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This function works by testing divisibility by ascending squares
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$p, p^2, p^4, p^8, \dotsc$, making it efficient for removing potentially
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large powers. Because of its high overhead, it should not be used as
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the first stage of trial division.
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int
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flint_mpn_factor_trial(mp_srcptr x, mp_size_t xsize, slong start, slong stop)
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Searches for a factor of \code{(x, xsize)} among the primes in positions
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\code{start, ..., stop-1} of \code{flint_primes}. Returns $i$ if
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\code{flint_primes[i]} is a factor, otherwise returns $0$ if no factor
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is found. It is assumed that \code{start >= 1}.
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*******************************************************************************
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Division
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*******************************************************************************
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int flint_mpn_divides(mp_ptr q, mp_srcptr array1,
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mp_size_t limbs1, mp_srcptr arrayg, mp_size_t limbsg, mp_ptr temp)
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If \code{(arrayg, limbsg)} divides \code{(array1, limbs1)} then
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\code{(q, limbs1 - limbsg + 1)} is set to the quotient and 1 is
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returned, otherwise 0 is returned. The temporary space \code{temp}
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must have space for \code{limbsg} limbs.
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Assumes limbs1 \code{limbs1 >= limbsg > 0}.
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mp_limb_t flint_mpn_preinv1(mp_limb_t d, mp_limb_t d2)
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Computes a precomputed inverse from the leading two limbs of the
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divisor \code{b, n} to be used with the \code{preinv1} functions.
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We require the most significant bit of \code{b, n} to be 1.
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mp_limb_t flint_mpn_divrem_preinv1(mp_ptr q, mp_ptr a,
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mp_size_t m, mp_srcptr b, mp_size_t n, mp_limb_t dinv)
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Divide \code{a, m} by \code{b, n}, returning the high limb of the
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quotient (which will either be 0 or 1), storing the remainder in-place
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in \code{a, n} and the rest of the quotient in \code{q, m - n}.
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We require the most significant bit of \code{b, n} to be 1.
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dinv must be computed from \code{b[n - 1]}, \code{b[n - 2]} by
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\code{flint_mpn_preinv1}. We also require \code{m >= n >= 2}.
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void flint_mpn_mulmod_preinv1(mp_ptr r,
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mp_srcptr a, mp_srcptr b, mp_size_t n,
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mp_srcptr d, mp_limb_t dinv, ulong norm)
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Given a normalised integer $d$ with precomputed inverse \code{dinv}
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provided by \code{flint_mpn_preinv1}, computes $ab \pmod{d}$ and
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stores the result in $r$. Each of $a$, $b$ and $r$ is expected to
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have $n$ limbs of space, with zero padding if necessary.
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The value \code{norm} is provided for convenience. If $a$, $b$ and
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$d$ have been shifted left by \code{norm} bits so that $d$ is
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normalised, then $r$ will be shifted right by \code{norm} bits
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so that it has the same shift as all the inputs.
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We require $a$ and $b$ to be reduced modulo $n$ before calling the
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function.
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void flint_mpn_preinvn(mp_ptr dinv, mp_srcptr d, mp_size_t n)
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Compute an $n$ limb precomputed inverse \code{dinv} of the $n$ limb
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integer $d$.
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We require that $d$ is normalised, i.e. with the most significant
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bit of the most significant limb set.
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void flint_mpn_mod_preinvn(mp_ptr r, mp_srcptr a, mp_size_t m,
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mp_srcptr d, mp_size_t n, mp_srcptr dinv)
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Given a normalised integer $d$ of $n$ limbs, with precomputed inverse
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\code{dinv} provided by \code{flint_mpn_preinvn} and integer $a$ of $m$
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limbs, computes $a \pmod{d}$ and stores the result in-place in the lower
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$n$ limbs of $a$. The remaining limbs of $a$ are destroyed.
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We require $m \geq n$. No aliasing of $a$ with any of the other operands
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is permitted.
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Note that this function is not always as fast as ordinary division.
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mp_limb_t flint_mpn_divrem_preinvn(mp_ptr q, mp_ptr r, mp_srcptr a,
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mp_size_t m, mp_srcptr d, mp_size_t n, mp_srcptr dinv)
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Given a normalised integer $d$ with precomputed inverse \code{dinv}
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provided by \code{flint_mpn_preinvn}, computes the quotient of $a$ by $d$
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and stores the result in $q$ and the remainder in the lower $n$ limbs of
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$a$. The remaining limbs of $a$ are destroyed.
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The value $q$ is expected to have space for $m - n$ limbs and we require
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$m >= n$. No aliasing is permitted between $q$ and $a$ or between these
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and any of the other operands.
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Note that this function is not always as fast as ordinary division.
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void flint_mpn_mulmod_preinvn(mp_ptr r,
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mp_srcptr a, mp_srcptr b, mp_size_t n,
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mp_srcptr d, mp_srcptr dinv, ulong norm)
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Given a normalised integer $d$ with precomputed inverse \code{dinv}
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provided by \code{flint_mpn_preinvn}, computes $ab \pmod{d}$ and
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stores the result in $r$. Each of $a$, $b$ and $r$ is expected to
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have $n$ limbs of space, with zero padding if necessary.
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The value \code{norm} is provided for convenience. If $a$, $b$ and
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$d$ have been shifted left by \code{norm} bits so that $d$ is
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normalised, then $r$ will be shifted right by \code{norm} bits
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so that it has the same shift as all the inputs.
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We require $a$ and $b$ to be reduced modulo $n$ before calling the
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function.
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Note that this function is not always as fast as ordinary division.
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*******************************************************************************
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GCD
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*******************************************************************************
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mp_size_t flint_mpn_gcd_full(mp_ptr arrayg,
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mp_ptr array1, mp_size_t limbs1, mp_ptr array2, mp_size_t limbs2)
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Sets \code{(arrayg, retvalue)} to the gcd of \code{(array1, limbs1)} and
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\code{(array2, limbs2)}.
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The only assumption is that neither \code{limbs1} or \code{limbs2} is
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zero.
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*******************************************************************************
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Special numbers
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*******************************************************************************
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void flint_mpn_harmonic_odd_balanced(mp_ptr t, mp_size_t * tsize,
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mp_ptr v, mp_size_t * vsize,
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slong a, slong b, slong n, int d)
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Computes \code{(t,tsize)} and \code{(v,vsize)} such that
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$t/v = H_n = 1 + 1/2 + \dotsb + 1/n$.
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The computation is performed using recursive balanced summation
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over the odd terms. The resulting fraction will not generally be
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normalized. At the top level, this function should be called with
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$n > 0$, $a = 1$, $b = n$, and $d = 1$.
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Enough space should be allocated for $t$ and $v$ to fit the entire sum
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$1 + 1/2 + \dotsb + 1/n$ computed without normalization; i.e.\ $t$ and $v$
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should have room to fit $n!$ plus one extra limb.
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*******************************************************************************
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Random Number Generation
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*******************************************************************************
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void flint_mpn_rrandom(mp_limb_t *rp,
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gmp_randstate_t state, mp_size_t n)
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Generates a random number with \code{n} limbs and stores
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it on \code{rp}. The number it generates will tend to have
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long strings of zeros and ones in the binary representation.
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Useful for testing functions and algorithms, since this kind of random
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numbers have proven to be more likely to trigger corner-case bugs.
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void flint_mpn_urandomb(mp_limb_t *rp,
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gmp_randstate_t state, mp_bitcnt_t n)
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Generates a uniform random number \code{n} bits and stores
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it on \code{rp}.
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