125 lines
3.1 KiB
C
125 lines
3.1 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 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 slong
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_bell_series_cutoff(slong n)
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{
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double N, log_N, log_pow, log_fac;
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N = n;
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log_N = (N==0 ? 0 : log(N));
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log_pow = n * log_N;
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log_fac = N*log_N - N;
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while (log_pow - log_fac >= -2)
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{
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N++;
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log_N = log(N);
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log_pow = n * log_N;
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log_fac += log_N;
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}
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return N;
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}
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static void
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_mpz_bell_bsplit(mpz_t P, mpz_t Q, slong a, slong b, slong n, slong bmax)
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{
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if (b - a < 20)
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{
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mpz_t u;
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slong k;
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mpz_init(u);
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flint_mpz_set_ui(P, UWORD(0));
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flint_mpz_set_ui(Q, UWORD(0));
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flint_mpz_set_ui(Q, (b - 1 == bmax) ? UWORD(1) : b);
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for (k = b - 1; k >= a; k--)
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{
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flint_mpz_set_ui(u, k);
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flint_mpz_pow_ui(u, u, n);
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mpz_addmul(P, Q, u);
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if (k != a)
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flint_mpz_mul_ui(Q, Q, k);
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}
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mpz_clear(u);
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}
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else
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{
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slong m;
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mpz_t P1, Q2;
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m = (a + b) / 2;
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mpz_init(P1);
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mpz_init(Q2);
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_mpz_bell_bsplit(P1, Q, a, m, n, bmax);
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_mpz_bell_bsplit(P, Q2, m, b, n, bmax);
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mpz_mul(Q, Q, Q2);
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mpz_addmul(P, P1, Q2);
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mpz_clear(P1);
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mpz_clear(Q2);
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}
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}
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void
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arith_bell_number_bsplit(fmpz_t b, ulong n)
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{
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slong N;
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mp_bitcnt_t prec;
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mpz_t P, Q;
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mpfr_t Pf, Qf, E, one;
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N = _bell_series_cutoff(n);
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mpz_init(P);
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mpz_init(Q);
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_mpz_bell_bsplit(P, Q, 1, N + 1, n, N);
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prec = mpz_sizeinbase(P, 2) - mpz_sizeinbase(Q, 2) + 10;
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mpfr_init2(Pf, prec);
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mpfr_init2(Qf, prec);
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mpfr_init2(E, prec);
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mpfr_init2(one, 2);
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mpfr_set_z(Pf, P, GMP_RNDN);
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mpfr_set_z(Qf, Q, GMP_RNDN);
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mpfr_set_ui(one, 1, GMP_RNDN);
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mpfr_exp(E, one, GMP_RNDN);
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mpfr_mul(Qf, Qf, E, GMP_RNDN);
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mpfr_div(Pf, Pf, Qf, GMP_RNDN);
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mpfr_get_z(P, Pf, GMP_RNDN);
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fmpz_set_mpz(b, P);
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mpfr_clear(one);
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mpfr_clear(Pf);
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mpfr_clear(Qf);
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mpfr_clear(E);
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mpz_clear(P);
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mpz_clear(Q);
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}
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