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Diffstat (limited to 'Build/source/libs/mpfr/mpfr-3.1.2/src/bernoulli.c')
-rw-r--r-- | Build/source/libs/mpfr/mpfr-3.1.2/src/bernoulli.c | 80 |
1 files changed, 80 insertions, 0 deletions
diff --git a/Build/source/libs/mpfr/mpfr-3.1.2/src/bernoulli.c b/Build/source/libs/mpfr/mpfr-3.1.2/src/bernoulli.c new file mode 100644 index 00000000000..85381550120 --- /dev/null +++ b/Build/source/libs/mpfr/mpfr-3.1.2/src/bernoulli.c @@ -0,0 +1,80 @@ +/* bernoulli -- internal function to compute Bernoulli numbers. + +Copyright 2005, 2006, 2007, 2008, 2009, 2010, 2011, 2012, 2013 Free Software Foundation, Inc. +Contributed by the AriC and Caramel projects, INRIA. + +This file is part of the GNU MPFR Library. + +The GNU MPFR Library is free software; you can redistribute it and/or modify +it under the terms of the GNU Lesser General Public License as published by +the Free Software Foundation; either version 3 of the License, or (at your +option) any later version. + +The GNU MPFR Library 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 Lesser General Public +License for more details. + +You should have received a copy of the GNU Lesser General Public License +along with the GNU MPFR Library; see the file COPYING.LESSER. If not, see +http://www.gnu.org/licenses/ or write to the Free Software Foundation, Inc., +51 Franklin St, Fifth Floor, Boston, MA 02110-1301, USA. */ + +#include "mpfr-impl.h" + +/* assuming b[0]...b[2(n-1)] are computed, computes and stores B[2n]*(2n+1)! + + t/(exp(t)-1) = sum(B[j]*t^j/j!, j=0..infinity) + thus t = (exp(t)-1) * sum(B[j]*t^j/j!, n=0..infinity). + Taking the coefficient of degree n+1 > 1, we get: + 0 = sum(1/(n+1-k)!*B[k]/k!, k=0..n) + which gives: + B[n] = -sum(binomial(n+1,k)*B[k], k=0..n-1)/(n+1). + + Let C[n] = B[n]*(n+1)!. + Then C[n] = -sum(binomial(n+1,k)*C[k]*n!/(k+1)!, k=0..n-1), + which proves that the C[n] are integers. +*/ +mpz_t* +mpfr_bernoulli_internal (mpz_t *b, unsigned long n) +{ + if (n == 0) + { + b = (mpz_t *) (*__gmp_allocate_func) (sizeof (mpz_t)); + mpz_init_set_ui (b[0], 1); + } + else + { + mpz_t t; + unsigned long k; + + b = (mpz_t *) (*__gmp_reallocate_func) + (b, n * sizeof (mpz_t), (n + 1) * sizeof (mpz_t)); + mpz_init (b[n]); + /* b[n] = -sum(binomial(2n+1,2k)*C[k]*(2n)!/(2k+1)!, k=0..n-1) */ + mpz_init_set_ui (t, 2 * n + 1); + mpz_mul_ui (t, t, 2 * n - 1); + mpz_mul_ui (t, t, 2 * n); + mpz_mul_ui (t, t, n); + mpz_fdiv_q_ui (t, t, 3); /* exact: t=binomial(2*n+1,2*k)*(2*n)!/(2*k+1)! + for k=n-1 */ + mpz_mul (b[n], t, b[n-1]); + for (k = n - 1; k-- > 0;) + { + mpz_mul_ui (t, t, 2 * k + 1); + mpz_mul_ui (t, t, 2 * k + 2); + mpz_mul_ui (t, t, 2 * k + 2); + mpz_mul_ui (t, t, 2 * k + 3); + mpz_fdiv_q_ui (t, t, 2 * (n - k) + 1); + mpz_fdiv_q_ui (t, t, 2 * (n - k)); + mpz_addmul (b[n], t, b[k]); + } + /* take into account C[1] */ + mpz_mul_ui (t, t, 2 * n + 1); + mpz_fdiv_q_2exp (t, t, 1); + mpz_sub (b[n], b[n], t); + mpz_neg (b[n], b[n]); + mpz_clear (t); + } + return b; +} |