source/libs/mpfr/mpfr-3.1.2/src/bernoulli.c

   1 /* bernoulli -- internal function to compute Bernoulli numbers.
   2
   3 Copyright 2005, 2006, 2007, 2008, 2009, 2010, 2011, 2012, 2013 Free Software Foundation, Inc.
   4 Contributed by the AriC and Caramel projects, INRIA.
   5
   6 This file is part of the GNU MPFR Library.
   7
   8 The GNU MPFR Library is free software; you can redistribute it and/or modify
   9 it under the terms of the GNU Lesser General Public License as published by
  10 the Free Software Foundation; either version 3 of the License, or (at your
  11 option) any later version.
  12
  13 The GNU MPFR Library is distributed in the hope that it will be useful, but
  14 WITHOUT ANY WARRANTY; without even the implied warranty of MERCHANTABILITY
  15 or FITNESS FOR A PARTICULAR PURPOSE.  See the GNU Lesser General Public
  16 License for more details.
  17
  18 You should have received a copy of the GNU Lesser General Public License
  19 along with the GNU MPFR Library; see the file COPYING.LESSER.  If not, see
  20 http://www.gnu.org/licenses/ or write to the Free Software Foundation, Inc.,
  21 51 Franklin St, Fifth Floor, Boston, MA 02110-1301, USA. */
  22
  23 #include "mpfr-impl.h"
  24
  25 /* assuming b[0]...b[2(n-1)] are computed, computes and stores B[2n]*(2n+1)!
  26
  27    t/(exp(t)-1) = sum(B[j]*t^j/j!, j=0..infinity)
  28    thus t = (exp(t)-1) * sum(B[j]*t^j/j!, n=0..infinity).
  29    Taking the coefficient of degree n+1 > 1, we get:
  30    0 = sum(1/(n+1-k)!*B[k]/k!, k=0..n)
  31    which gives:
  32    B[n] = -sum(binomial(n+1,k)*B[k], k=0..n-1)/(n+1).
  33
  34    Let C[n] = B[n]*(n+1)!.
  35    Then C[n] = -sum(binomial(n+1,k)*C[k]*n!/(k+1)!,  k=0..n-1),
  36    which proves that the C[n] are integers.
  37 */
  38 mpz_t*
  39 mpfr_bernoulli_internal (mpz_t *b, unsigned long n)
  40 {
  41   if (n == 0)
  42     {
  43       b = (mpz_t *) (*__gmp_allocate_func) (sizeof (mpz_t));
  44       mpz_init_set_ui (b[0], 1);
  45     }
  46   else
  47     {
  48       mpz_t t;
  49       unsigned long k;
  50
  51       b = (mpz_t *) (*__gmp_reallocate_func)
  52         (b, n * sizeof (mpz_t), (n + 1) * sizeof (mpz_t));
  53       mpz_init (b[n]);
  54       /* b[n] = -sum(binomial(2n+1,2k)*C[k]*(2n)!/(2k+1)!,  k=0..n-1) */
  55       mpz_init_set_ui (t, 2 * n + 1);
  56       mpz_mul_ui (t, t, 2 * n - 1);
  57       mpz_mul_ui (t, t, 2 * n);
  58       mpz_mul_ui (t, t, n);
  59       mpz_fdiv_q_ui (t, t, 3); /* exact: t=binomial(2*n+1,2*k)*(2*n)!/(2*k+1)!
  60                                for k=n-1 */
  61       mpz_mul (b[n], t, b[n-1]);
  62       for (k = n - 1; k-- > 0;)
  63         {
  64           mpz_mul_ui (t, t, 2 * k + 1);
  65           mpz_mul_ui (t, t, 2 * k + 2);
  66           mpz_mul_ui (t, t, 2 * k + 2);
  67           mpz_mul_ui (t, t, 2 * k + 3);
  68           mpz_fdiv_q_ui (t, t, 2 * (n - k) + 1);
  69           mpz_fdiv_q_ui (t, t, 2 * (n - k));
  70           mpz_addmul (b[n], t, b[k]);
  71         }
  72       /* take into account C[1] */
  73       mpz_mul_ui (t, t, 2 * n + 1);
  74       mpz_fdiv_q_2exp (t, t, 1);
  75       mpz_sub (b[n], b[n], t);
  76       mpz_neg (b[n], b[n]);
  77       mpz_clear (t);
  78     }
  79   return b;
  80 }