efee5258bc
The MPFR library is a C library for multiple-precision floating-point computations with exact rounding (also called correct rounding). It is based on the GMP multiple-precision library and should replace the MPF class in further releases of GMP. GCC >= 4.2 requires MPFR.
81 lines
2.8 KiB
C
81 lines
2.8 KiB
C
/* bernoulli -- internal function to compute Bernoulli numbers.
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Copyright 2005, 2006, 2007, 2008, 2009, 2010, 2011 Free Software Foundation, Inc.
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Contributed by the Arenaire and Cacao projects, INRIA.
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This file is part of the GNU MPFR Library.
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The GNU MPFR Library is free software; you can redistribute it and/or modify
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it under the terms of the GNU Lesser General Public License as published by
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the Free Software Foundation; either version 3 of the License, or (at your
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option) any later version.
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The GNU MPFR Library is distributed in the hope that it will be useful, but
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WITHOUT ANY WARRANTY; without even the implied warranty of MERCHANTABILITY
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or FITNESS FOR A PARTICULAR PURPOSE. See the GNU Lesser General Public
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License for more details.
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You should have received a copy of the GNU Lesser General Public License
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along with the GNU MPFR Library; see the file COPYING.LESSER. If not, see
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http://www.gnu.org/licenses/ or write to the Free Software Foundation, Inc.,
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51 Franklin St, Fifth Floor, Boston, MA 02110-1301, USA. */
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#include "mpfr-impl.h"
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/* assuming b[0]...b[2(n-1)] are computed, computes and stores B[2n]*(2n+1)!
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t/(exp(t)-1) = sum(B[j]*t^j/j!, j=0..infinity)
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thus t = (exp(t)-1) * sum(B[j]*t^j/j!, n=0..infinity).
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Taking the coefficient of degree n+1 > 1, we get:
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0 = sum(1/(n+1-k)!*B[k]/k!, k=0..n)
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which gives:
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B[n] = -sum(binomial(n+1,k)*B[k], k=0..n-1)/(n+1).
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Let C[n] = B[n]*(n+1)!.
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Then C[n] = -sum(binomial(n+1,k)*C[k]*n!/(k+1)!, k=0..n-1),
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which proves that the C[n] are integers.
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*/
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mpz_t*
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mpfr_bernoulli_internal (mpz_t *b, unsigned long n)
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{
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if (n == 0)
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{
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b = (mpz_t *) (*__gmp_allocate_func) (sizeof (mpz_t));
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mpz_init_set_ui (b[0], 1);
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}
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else
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{
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mpz_t t;
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unsigned long k;
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b = (mpz_t *) (*__gmp_reallocate_func)
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(b, n * sizeof (mpz_t), (n + 1) * sizeof (mpz_t));
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mpz_init (b[n]);
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/* b[n] = -sum(binomial(2n+1,2k)*C[k]*(2n)!/(2k+1)!, k=0..n-1) */
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mpz_init_set_ui (t, 2 * n + 1);
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mpz_mul_ui (t, t, 2 * n - 1);
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mpz_mul_ui (t, t, 2 * n);
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mpz_mul_ui (t, t, n);
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mpz_fdiv_q_ui (t, t, 3); /* exact: t=binomial(2*n+1,2*k)*(2*n)!/(2*k+1)!
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for k=n-1 */
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mpz_mul (b[n], t, b[n-1]);
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for (k = n - 1; k-- > 0;)
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{
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mpz_mul_ui (t, t, 2 * k + 1);
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mpz_mul_ui (t, t, 2 * k + 2);
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mpz_mul_ui (t, t, 2 * k + 2);
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mpz_mul_ui (t, t, 2 * k + 3);
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mpz_fdiv_q_ui (t, t, 2 * (n - k) + 1);
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mpz_fdiv_q_ui (t, t, 2 * (n - k));
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mpz_addmul (b[n], t, b[k]);
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}
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/* take into account C[1] */
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mpz_mul_ui (t, t, 2 * n + 1);
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mpz_fdiv_q_2exp (t, t, 1);
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mpz_sub (b[n], b[n], t);
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mpz_neg (b[n], b[n]);
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mpz_clear (t);
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}
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return b;
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}
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