347 lines
10 KiB
C
347 lines
10 KiB
C
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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) 2011 William Hart
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******************************************************************************/
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#include <gmp.h>
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#include "flint.h"
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#include "fmpz.h"
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#include "fmpz_vec.h"
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#include "fmpz_poly.h"
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#include "mpn_extras.h"
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void _fmpz_poly_gcd_modular(fmpz * res, const fmpz * poly1, slong len1,
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const fmpz * poly2, slong len2)
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{
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mp_bitcnt_t bits1, bits2, nb1, nb2, bits_small, pbits, curr_bits = 0, new_bits;
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fmpz_t ac, bc, hc, d, g, l, eval_A, eval_B, eval_GCD, modulus;
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fmpz * A, * B, * Q, * lead_A, * lead_B;
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mp_ptr a, b, h;
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mp_limb_t p, h_inv, g_mod;
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nmod_t mod;
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slong i, n, n0, unlucky, hlen, bound;
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int g_pm1;
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fmpz_init(ac);
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fmpz_init(bc);
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fmpz_init(d);
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/* compute gcd of content of poly1 and poly2 */
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_fmpz_vec_content(ac, poly1, len1);
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_fmpz_vec_content(bc, poly2, len2);
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fmpz_gcd(d, ac, bc);
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/* special case, one of the polys is a constant */
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if (len2 == 1) /* if len1 == 1 then so does len2 */
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{
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fmpz_set(res, d);
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fmpz_clear(ac);
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fmpz_clear(bc);
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fmpz_clear(d);
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return;
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}
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/* divide poly1 and poly2 by their content */
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A = _fmpz_vec_init(len1);
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B = _fmpz_vec_init(len2);
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_fmpz_vec_scalar_divexact_fmpz(A, poly1, len1, ac);
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_fmpz_vec_scalar_divexact_fmpz(B, poly2, len2, bc);
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fmpz_clear(ac);
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fmpz_clear(bc);
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/* get bound on size of gcd coefficients */
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lead_A = A + len1 - 1;
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lead_B = B + len2 - 1;
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bits1 = _fmpz_vec_max_bits(A, len1); bits1 = FLINT_ABS(bits1);
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bits2 = _fmpz_vec_max_bits(B, len2); bits2 = FLINT_ABS(bits2);
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fmpz_init(l);
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if (len1 < 64 && len2 < 64) /* compute the squares of the 2-norms */
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{
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fmpz_set_ui(l, 0);
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for (i = 0; i < len1; i++)
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fmpz_addmul(l, A + i, A + i);
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nb1 = fmpz_bits(l);
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fmpz_set_ui(l, 0);
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for (i = 0; i < len2; i++)
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fmpz_addmul(l, B + i, B + i);
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nb2 = fmpz_bits(l);
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} else /* approximate to save time */
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{
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nb1 = 2*bits1 + FLINT_BIT_COUNT(len1);
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nb2 = 2*bits2 + FLINT_BIT_COUNT(len2);
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}
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/* get gcd of leading coefficients */
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fmpz_init(g);
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fmpz_gcd(g, lead_A, lead_B);
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fmpz_mul(l, lead_A, lead_B);
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g_pm1 = fmpz_is_pm1(g);
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/* evaluate -A at -1 */
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fmpz_init(eval_A);
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for (i = 0; i < len1; i++)
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{
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if (i & 1) fmpz_add(eval_A, eval_A, A + i);
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else fmpz_sub(eval_A, eval_A, A + i);
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}
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/* evaluate -B at -1 */
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fmpz_init(eval_B);
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for (i = 0; i < len2; i++)
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{
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if (i & 1) fmpz_add(eval_B, eval_B, B + i);
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else fmpz_sub(eval_B, eval_B, B + i);
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}
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/* compute the gcd of eval(-A, -1) and eval(-B, -1) */
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fmpz_init(eval_GCD);
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fmpz_gcd(eval_GCD, eval_A, eval_B);
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/* compute a heuristic bound after which we should begin checking if we're done */
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bits_small = FLINT_MAX(fmpz_bits(eval_GCD), fmpz_bits(g));
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if (bits_small < WORD(2)) bits_small = 2;
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fmpz_clear(eval_GCD);
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fmpz_clear(eval_A);
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fmpz_clear(eval_B);
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/* set size of first prime */
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pbits = FLINT_BITS - 1;
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p = (UWORD(1)<<pbits);
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fmpz_init(modulus);
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fmpz_init(hc);
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Q = _fmpz_vec_init(len1);
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/* make space for polynomials mod p */
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a = _nmod_vec_init(len1);
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b = _nmod_vec_init(len2);
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h = _nmod_vec_init(len2);
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/* zero entire output */
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_fmpz_vec_zero(res, len2);
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n = len2;
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/* current bound on length of result
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the bound we use is from section 6 of
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http://cs.nyu.edu/~yap/book/alge/ftpSite/l4.ps.gz
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*/
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n0 = len1 - 1;
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bound = (n0 + 3)*FLINT_MAX(nb1, nb2) + (n0 + 1); /* initialise bound */
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unlucky = 0;
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for (;;)
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{
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/* get new prime and initialise modulus */
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p = n_nextprime(p, 0);
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if (fmpz_fdiv_ui(l, p) == 0)
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{
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unlucky += pbits;
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continue;
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}
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nmod_init(&mod, p);
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/* reduce polynomials modulo p */
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_fmpz_vec_get_nmod_vec(a, A, len1, mod);
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_fmpz_vec_get_nmod_vec(b, B, len2, mod);
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/* compute gcd over Z/pZ */
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hlen = _nmod_poly_gcd(h, a, len1, b, len2, mod);
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if (hlen == 1) /* gcd is 1 */
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{
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fmpz_one(res);
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_fmpz_vec_zero(res + 1, len2 - 1);
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break;
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}
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if (hlen > n + 1) /* discard this prime */
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{
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unlucky += pbits;
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continue;
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}
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/* scale new polynomial mod p appropriately */
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if (g_pm1) _nmod_poly_make_monic(h, h, hlen, mod);
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else
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{
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h_inv = n_invmod(h[hlen - 1], mod.n);
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g_mod = fmpz_fdiv_ui(g, mod.n);
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h_inv = n_mulmod2_preinv(h_inv, g_mod, mod.n, mod.ninv);
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_nmod_vec_scalar_mul_nmod(h, h, hlen, h_inv, mod);
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}
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if (hlen <= n) /* we have a new bound on size of result */
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{
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unlucky += fmpz_bits(modulus);
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_fmpz_vec_set_nmod_vec(res, h, hlen, mod);
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_fmpz_vec_zero(res + hlen, len2 - hlen);
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if (g_pm1)
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{
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/* are we done? */
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if (_fmpz_poly_divides(Q, B, len2, res, hlen) &&
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_fmpz_poly_divides(Q, A, len1, res, hlen))
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break;
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}
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else
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{
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if (pbits + unlucky >= bound) /* if we reach the bound with one prime */
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{
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_fmpz_vec_content(hc, res, hlen);
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/* divide by content */
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_fmpz_vec_scalar_divexact_fmpz(res, res, hlen, hc);
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break;
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}
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if (pbits >= bits_small) /* if one prime is already big enough to check */
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{
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/* divide by content */
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_fmpz_vec_content(hc, res, hlen);
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/* correct sign of leading term */
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if (fmpz_sgn(res + hlen - 1) < 0)
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fmpz_neg(hc, hc);
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_fmpz_vec_scalar_divexact_fmpz(res, res, hlen, hc);
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/* are we done? */
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if (_fmpz_poly_divides(Q, B, len2, res, hlen) &&
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_fmpz_poly_divides(Q, A, len1, res, hlen))
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break;
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/* no, so multiply by content again */
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_fmpz_vec_scalar_mul_fmpz(res, res, hlen, hc);
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}
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}
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curr_bits = FLINT_ABS(_fmpz_vec_max_bits(res, hlen));
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fmpz_set_ui(modulus, p);
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n = hlen - 1; /* if we reach this we have a new bound on length of result */
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continue;
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}
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_fmpz_poly_CRT_ui(res, res, hlen, modulus, h, hlen, mod.n, mod.ninv, 1);
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fmpz_mul_ui(modulus, modulus, mod.n);
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new_bits = _fmpz_vec_max_bits(res, hlen);
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new_bits = FLINT_ABS(new_bits);
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if (new_bits == curr_bits || fmpz_bits(modulus) >= bits_small)
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{
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if (!g_pm1)
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{
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_fmpz_vec_content(hc, res, hlen);
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/* correct sign of leading term */
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if (fmpz_sgn(res + hlen - 1) < 0)
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fmpz_neg(hc, hc);
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/* divide by content */
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_fmpz_vec_scalar_divexact_fmpz(res, res, hlen, hc);
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}
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if (fmpz_bits(modulus) + unlucky >= bound)
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break;
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/* are we done? */
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if (_fmpz_poly_divides(Q, B, len2, res, hlen) &&
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_fmpz_poly_divides(Q, A, len1, res, hlen))
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break;
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if (!g_pm1)
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{
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/* no, so multiply by content again */
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_fmpz_vec_scalar_mul_fmpz(res, res, hlen, hc);
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}
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}
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curr_bits = new_bits;
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}
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fmpz_clear(modulus);
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fmpz_clear(g);
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fmpz_clear(l);
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fmpz_clear(hc);
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_nmod_vec_clear(a);
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_nmod_vec_clear(b);
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_nmod_vec_clear(h);
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/* finally multiply by content */
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_fmpz_vec_scalar_mul_fmpz(res, res, hlen, d);
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fmpz_clear(d);
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_fmpz_vec_clear(A, len1);
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_fmpz_vec_clear(B, len2);
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_fmpz_vec_clear(Q, len1);
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}
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void
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fmpz_poly_gcd_modular(fmpz_poly_t res, const fmpz_poly_t poly1,
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const fmpz_poly_t poly2)
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{
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if (poly1->length < poly2->length)
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{
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fmpz_poly_gcd_modular(res, poly2, poly1);
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}
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else /* len1 >= len2 >= 0 */
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{
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const slong len1 = poly1->length;
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const slong len2 = poly2->length;
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if (len1 == 0) /* len1 = len2 = 0 */
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{
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fmpz_poly_zero(res);
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}
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else if (len2 == 0) /* len1 > len2 = 0 */
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{
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if (fmpz_sgn(poly1->coeffs + (len1 - 1)) > 0)
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fmpz_poly_set(res, poly1);
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else
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fmpz_poly_neg(res, poly1);
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}
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else /* len1 >= len2 >= 1 */
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{
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/* underscore function automatically aliases */
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fmpz_poly_fit_length(res, len2);
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_fmpz_poly_gcd_modular(res->coeffs, poly1->coeffs, len1,
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poly2->coeffs, len2);
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_fmpz_poly_set_length(res, len2);
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_fmpz_poly_normalise(res);
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}
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}
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}
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