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bell_number_bsplit.c
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bell_number_bsplit.c
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/*=============================================================================
This file is part of FLINT.
FLINT is free software; you can redistribute it and/or modify
it under the terms of the GNU General Public License as published by
the Free Software Foundation; either version 2 of the License, or
(at your option) any later version.
FLINT is distributed in the hope that it will be useful,
but WITHOUT ANY WARRANTY; without even the implied warranty of
MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the
GNU General Public License for more details.
You should have received a copy of the GNU General Public License
along with FLINT; if not, write to the Free Software
Foundation, Inc., 51 Franklin St, Fifth Floor, Boston, MA 02110-1301 USA
=============================================================================*/
/******************************************************************************
Copyright (C) 2011 Fredrik Johansson
******************************************************************************/
#include <math.h>
#include "arith.h"
static slong
_bell_series_cutoff(slong n)
{
double N, log_N, log_pow, log_fac;
N = n;
log_N = (N==0 ? 0 : log(N));
log_pow = n * log_N;
log_fac = N*log_N - N;
while (log_pow - log_fac >= -2)
{
N++;
log_N = log(N);
log_pow = n * log_N;
log_fac += log_N;
}
return N;
}
static void
_mpz_bell_bsplit(mpz_t P, mpz_t Q, slong a, slong b, slong n, slong bmax)
{
if (b - a < 20)
{
mpz_t u;
slong k;
mpz_init(u);
mpz_set_ui(P, UWORD(0));
mpz_set_ui(Q, UWORD(0));
mpz_set_ui(Q, (b - 1 == bmax) ? UWORD(1) : b);
for (k = b - 1; k >= a; k--)
{
mpz_set_ui(u, k);
mpz_pow_ui(u, u, n);
mpz_addmul(P, Q, u);
if (k != a)
mpz_mul_ui(Q, Q, k);
}
mpz_clear(u);
}
else
{
slong m;
mpz_t P1, Q2;
m = (a + b) / 2;
mpz_init(P1);
mpz_init(Q2);
_mpz_bell_bsplit(P1, Q, a, m, n, bmax);
_mpz_bell_bsplit(P, Q2, m, b, n, bmax);
mpz_mul(Q, Q, Q2);
mpz_addmul(P, P1, Q2);
mpz_clear(P1);
mpz_clear(Q2);
}
}
void
arith_bell_number_bsplit(fmpz_t b, ulong n)
{
slong N;
mp_bitcnt_t prec;
mpz_t P, Q;
mpfr_t Pf, Qf, E, one;
N = _bell_series_cutoff(n);
mpz_init(P);
mpz_init(Q);
_mpz_bell_bsplit(P, Q, 1, N + 1, n, N);
prec = mpz_sizeinbase(P, 2) - mpz_sizeinbase(Q, 2) + 10;
mpfr_init2(Pf, prec);
mpfr_init2(Qf, prec);
mpfr_init2(E, prec);
mpfr_init2(one, 2);
mpfr_set_z(Pf, P, GMP_RNDN);
mpfr_set_z(Qf, Q, GMP_RNDN);
mpfr_set_ui(one, 1, GMP_RNDN);
mpfr_exp(E, one, GMP_RNDN);
mpfr_mul(Qf, Qf, E, GMP_RNDN);
mpfr_div(Pf, Pf, Qf, GMP_RNDN);
mpfr_get_z(P, Pf, GMP_RNDN);
fmpz_set_mpz(b, P);
mpfr_clear(one);
mpfr_clear(Pf);
mpfr_clear(Qf);
mpfr_clear(E);
mpz_clear(P);
mpz_clear(Q);
}