166 lines
4.7 KiB
C
166 lines
4.7 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 "arith.h"
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static void
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__ramanujan_even_common_denom(fmpz * num, fmpz * den, slong start, slong n)
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{
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fmpz_t t, c, d, cden;
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slong j, k, m, mcase;
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int prodsize;
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if (start >= n)
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return;
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fmpz_init(t);
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fmpz_init(c);
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fmpz_init(d);
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fmpz_init(cden);
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/* Common denominator */
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arith_primorial(cden, n + 1);
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start += start % 2;
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/* Convert initial values to common denominator */
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for (k = 0; k < start; k += 2)
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{
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fmpz_divexact(t, cden, den + k);
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fmpz_mul(num + k, num + k, t);
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}
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/* Ramanujan's recursive formula */
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for (m = start; m < n; m += 2)
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{
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mcase = m % 6;
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fmpz_mul_ui(num + m, cden, m + UWORD(3));
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fmpz_divexact_ui(num + m, num + m, UWORD(3));
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if (mcase == 4)
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{
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fmpz_neg(num + m, num + m);
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fmpz_divexact_ui(num + m, num + m, UWORD(2));
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}
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/* All factors are strictly smaller than m + 4; choose prodsize such
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that (m + 4)^prodsize fits in an slong. */
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{
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#if FLINT64
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if (m < WORD(1444)) prodsize = 6;
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else if (m < WORD(2097148)) prodsize = 3;
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else if (m < WORD(3037000495)) prodsize = 2; /* not very likely... */
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else abort();
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#else
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if (m < WORD(32)) prodsize = 6;
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else if (m < WORD(1286)) prodsize = 3;
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else if (m < WORD(46336)) prodsize = 2;
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else abort();
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#endif
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}
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/* c = t = binomial(m+3, m) */
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fmpz_set_ui(t, m + UWORD(1));
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fmpz_mul_ui(t, t, m + UWORD(2));
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fmpz_mul_ui(t, t, m + UWORD(3));
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fmpz_divexact_ui(t, t, UWORD(6));
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fmpz_set(c, t);
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for (j = 6; j <= m; j += 6)
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{
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slong r = m - j;
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/* c = binomial(m+3, m-j); */
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switch (prodsize)
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{
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case 2:
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fmpz_mul_ui(c, c, (r+6)*(r+5));
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fmpz_mul_ui(c, c, (r+4)*(r+3));
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fmpz_mul_ui(c, c, (r+2)*(r+1));
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fmpz_set_ui(d, (j+0)*(j+3));
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fmpz_mul_ui(d, d, (j-2)*(j+2));
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fmpz_mul_ui(d, d, (j-1)*(j+1));
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fmpz_divexact(c, c, d);
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break;
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case 3:
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fmpz_mul_ui(c, c, (r+6)*(r+5)*(r+4));
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fmpz_mul_ui(c, c, (r+3)*(r+2)*(r+1));
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fmpz_set_ui(d, (j+0)*(j+3)*(j-2));
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fmpz_mul_ui(d, d, (j+2)*(j-1)*(j+1));
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fmpz_divexact(c, c, d);
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break;
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case 6:
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fmpz_mul_ui(c, c, (r+6)*(r+5)*(r+4)*(r+3)*(r+2)*(r+1));
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fmpz_divexact_ui(c, c, (j+0)*(j+3)*(j-2)*(j+2)*(j-1)*(j+1));
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break;
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}
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fmpz_submul(num + m, c, num + (m - j));
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}
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fmpz_divexact(num + m, num + m, t);
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}
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/* Convert to separate denominators */
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for (k = 0; k < n; k += 2)
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{
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arith_bernoulli_number_denom(den + k, k);
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fmpz_divexact(t, cden, den + k);
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fmpz_divexact(num + k, num + k, t);
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}
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fmpz_clear(t);
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fmpz_clear(c);
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fmpz_clear(d);
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fmpz_clear(cden);
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}
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void _arith_bernoulli_number_vec_recursive(fmpz * num, fmpz * den, slong n)
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{
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slong i, start;
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fmpz_t t;
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fmpz_t d;
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fmpz_init(t);
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fmpz_init(d);
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start = FLINT_MIN(BERNOULLI_SMALL_NUMER_LIMIT, n);
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/* Initial values */
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for (i = 0; i < start; i += 2)
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_arith_bernoulli_number(num + i, den + i, i);
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__ramanujan_even_common_denom(num, den, start, n);
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/* Odd values */
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for (i = 1; i < n; i += 2)
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_arith_bernoulli_number(num + i, den + i, i);
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fmpz_clear(d);
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fmpz_clear(t);
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}
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