151 lines
4.6 KiB
C
151 lines
4.6 KiB
C
/*=============================================================================
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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 Fredrik Johansson
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******************************************************************************/
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#include <math.h>
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#include "arith.h"
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static void
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__bernoulli_number_vec_mod_p(mp_ptr res, mp_ptr tmp, const fmpz * den,
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slong m, nmod_t mod)
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{
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mp_limb_t fac, c, t;
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slong k;
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/* x^2/(cosh(x)-1) = \sum_{k=0}^{\infty} 2(1-2k)/(2k)! B_2k x^(2k) */
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/* Divide by factorials */
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fac = n_factorial_mod2_preinv(2*m, mod.n, mod.ninv);
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c = n_invmod(fac, mod.n);
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for (k = m - 1; k >= 0; k--)
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{
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tmp[k] = c;
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c = n_mulmod2_preinv(c, (2*k+1)*(2*k+2), mod.n, mod.ninv);
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}
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_nmod_poly_inv_series(res, tmp, m, mod);
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res[0] = UWORD(1);
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/* N_(2k) = -1 * D_(2k) * (2k)! / (2k-1) */
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c = n_negmod(UWORD(1), mod.n);
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for (k = 1; k < m; k++)
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{
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t = fmpz_fdiv_ui(den + 2*k, mod.n);
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t = n_mulmod2_preinv(c, t, mod.n, mod.ninv);
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res[k] = n_mulmod2_preinv(res[k], t, mod.n, mod.ninv);
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c = n_mulmod2_preinv(c, 2*(k+1)*(2*k-1), mod.n, mod.ninv);
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}
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}
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#define CRT_MAX_RESOLUTION 16
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void _arith_bernoulli_number_vec_multi_mod(fmpz * num, fmpz * den, slong n)
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{
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fmpz_comb_t comb[CRT_MAX_RESOLUTION];
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fmpz_comb_temp_t temp[CRT_MAX_RESOLUTION];
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mp_limb_t * primes;
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mp_limb_t * residues;
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mp_ptr * polys;
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mp_ptr temppoly;
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nmod_t mod;
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slong i, j, k, m, num_primes, num_primes_k, resolution;
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mp_bitcnt_t size, prime_bits;
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if (n < 1)
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return;
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for (i = 0; i < n; i++)
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arith_bernoulli_number_denom(den + i, i);
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/* Number of nonzero entries (apart from B_1) */
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m = (n + 1) / 2;
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resolution = FLINT_MAX(1, FLINT_MIN(CRT_MAX_RESOLUTION, m / 16));
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/* Note that the denominators must be accounted for */
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size = arith_bernoulli_number_size(n) + _fmpz_vec_max_bits(den, n) + 2;
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prime_bits = FLINT_BITS - 1;
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num_primes = (size + prime_bits - 1) / prime_bits;
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primes = flint_malloc(num_primes * sizeof(mp_limb_t));
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residues = flint_malloc(num_primes * sizeof(mp_limb_t));
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polys = flint_malloc(num_primes * sizeof(mp_ptr));
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/* Compute Bernoulli numbers mod p */
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primes[0] = n_nextprime(UWORD(1)<<prime_bits, 0);
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for (k = 1; k < num_primes; k++)
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primes[k] = n_nextprime(primes[k-1], 0);
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temppoly = _nmod_vec_init(m);
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for (k = 0; k < num_primes; k++)
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{
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polys[k] = _nmod_vec_init(m);
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nmod_init(&mod, primes[k]);
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__bernoulli_number_vec_mod_p(polys[k], temppoly, den, m, mod);
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}
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/* Init CRT comb */
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for (i = 0; i < resolution; i++)
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{
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fmpz_comb_init(comb[i], primes, num_primes * (i + 1) / resolution);
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fmpz_comb_temp_init(temp[i], comb[i]);
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}
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/* Trivial entries */
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if (n > 1)
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fmpz_set_si(num + 1, WORD(-1));
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for (k = 3; k < n; k += 2)
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fmpz_zero(num + k);
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/* Reconstruction */
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for (k = 0; k < n; k += 2)
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{
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size = arith_bernoulli_number_size(k) + fmpz_bits(den + k) + 2;
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/* Use only as large a comb as needed */
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num_primes_k = (size + prime_bits - 1) / prime_bits;
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for (i = 0; i < resolution; i++)
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{
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if (comb[i]->num_primes >= num_primes_k)
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break;
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}
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num_primes_k = comb[i]->num_primes;
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for (j = 0; j < num_primes_k; j++)
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residues[j] = polys[j][k / 2];
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fmpz_multi_CRT_ui(num + k, residues, comb[i], temp[i], 1);
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}
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/* Cleanup */
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for (k = 0; k < num_primes; k++)
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_nmod_vec_clear(polys[k]);
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_nmod_vec_clear(temppoly);
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for (i = 0; i < resolution; i++)
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{
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fmpz_comb_temp_clear(temp[i]);
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fmpz_comb_clear(comb[i]);
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}
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flint_free(primes);
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flint_free(residues);
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flint_free(polys);
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}
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