| /* mpfr_const_catalan -- compute Catalan's constant. |
| |
| Copyright 2005, 2006, 2007, 2008, 2009 Free Software Foundation, Inc. |
| Contributed by the Arenaire and Cacao 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 2.1 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.LIB. If not, write to |
| the Free Software Foundation, Inc., 51 Franklin St, Fifth Floor, Boston, |
| MA 02110-1301, USA. */ |
| |
| #define MPFR_NEED_LONGLONG_H |
| #include "mpfr-impl.h" |
| |
| /* Declare the cache */ |
| MPFR_DECL_INIT_CACHE(__gmpfr_cache_const_catalan, mpfr_const_catalan_internal); |
| |
| /* Set User Interface */ |
| #undef mpfr_const_catalan |
| int |
| mpfr_const_catalan (mpfr_ptr x, mp_rnd_t rnd_mode) { |
| return mpfr_cache (x, __gmpfr_cache_const_catalan, rnd_mode); |
| } |
| |
| /* return T, Q such that T/Q = sum(k!^2/(2k)!/(2k+1)^2, k=n1..n2-1) */ |
| static void |
| S (mpz_t T, mpz_t P, mpz_t Q, unsigned long n1, unsigned long n2) |
| { |
| if (n2 == n1 + 1) |
| { |
| if (n1 == 0) |
| { |
| mpz_set_ui (P, 1); |
| mpz_set_ui (Q, 1); |
| } |
| else |
| { |
| mpz_set_ui (P, 2 * n1 - 1); |
| mpz_mul_ui (P, P, n1); |
| mpz_ui_pow_ui (Q, 2 * n1 + 1, 2); |
| mpz_mul_2exp (Q, Q, 1); |
| } |
| mpz_set (T, P); |
| } |
| else |
| { |
| unsigned long m = (n1 + n2) / 2; |
| mpz_t T2, P2, Q2; |
| S (T, P, Q, n1, m); |
| mpz_init (T2); |
| mpz_init (P2); |
| mpz_init (Q2); |
| S (T2, P2, Q2, m, n2); |
| mpz_mul (T, T, Q2); |
| mpz_mul (T2, T2, P); |
| mpz_add (T, T, T2); |
| mpz_mul (P, P, P2); |
| mpz_mul (Q, Q, Q2); |
| mpz_clear (T2); |
| mpz_clear (P2); |
| mpz_clear (Q2); |
| } |
| } |
| |
| /* Don't need to save/restore exponent range: the cache does it. |
| Catalan's constant is G = sum((-1)^k/(2*k+1)^2, k=0..infinity). |
| We compute it using formula (31) of Victor Adamchik's page |
| "33 representations for Catalan's constant" |
| http://www-2.cs.cmu.edu/~adamchik/articles/catalan/catalan.htm |
| |
| G = Pi/8*log(2+sqrt(3)) + 3/8*sum(k!^2/(2k)!/(2k+1)^2,k=0..infinity) |
| */ |
| int |
| mpfr_const_catalan_internal (mpfr_ptr g, mp_rnd_t rnd_mode) |
| { |
| mpfr_t x, y, z; |
| mpz_t T, P, Q; |
| mp_prec_t pg, p; |
| int inex; |
| MPFR_ZIV_DECL (loop); |
| MPFR_GROUP_DECL (group); |
| |
| MPFR_LOG_FUNC (("rnd_mode=%d", rnd_mode), ("g[%#R]=%R inex=%d", g, g, inex)); |
| |
| /* Here are the WC (max prec = 100.000.000) |
| Once we have found a chain of 11, we only look for bigger chain. |
| Found 3 '1' at 0 |
| Found 5 '1' at 9 |
| Found 6 '0' at 34 |
| Found 9 '1' at 176 |
| Found 11 '1' at 705 |
| Found 12 '0' at 913 |
| Found 14 '1' at 12762 |
| Found 15 '1' at 152561 |
| Found 16 '0' at 171725 |
| Found 18 '0' at 525355 |
| Found 20 '0' at 529245 |
| Found 21 '1' at 6390133 |
| Found 22 '0' at 7806417 |
| Found 25 '1' at 11936239 |
| Found 27 '1' at 51752950 |
| */ |
| pg = MPFR_PREC (g); |
| p = pg + 9; |
| p += MPFR_INT_CEIL_LOG2 (p); |
| |
| MPFR_GROUP_INIT_3 (group, p, x, y, z); |
| mpz_init (T); |
| mpz_init (P); |
| mpz_init (Q); |
| |
| MPFR_ZIV_INIT (loop, p); |
| for (;;) { |
| mpfr_sqrt_ui (x, 3, GMP_RNDU); |
| mpfr_add_ui (x, x, 2, GMP_RNDU); |
| mpfr_log (x, x, GMP_RNDU); |
| mpfr_const_pi (y, GMP_RNDU); |
| mpfr_mul (x, x, y, GMP_RNDN); |
| S (T, P, Q, 0, (p - 1) / 2); |
| mpz_mul_ui (T, T, 3); |
| mpfr_set_z (y, T, GMP_RNDU); |
| mpfr_set_z (z, Q, GMP_RNDD); |
| mpfr_div (y, y, z, GMP_RNDN); |
| mpfr_add (x, x, y, GMP_RNDN); |
| mpfr_div_2ui (x, x, 3, GMP_RNDN); |
| |
| if (MPFR_LIKELY (MPFR_CAN_ROUND (x, p - 5, pg, rnd_mode))) |
| break; |
| /* Fixme: Is it possible? */ |
| MPFR_ZIV_NEXT (loop, p); |
| MPFR_GROUP_REPREC_3 (group, p, x, y, z); |
| } |
| MPFR_ZIV_FREE (loop); |
| inex = mpfr_set (g, x, rnd_mode); |
| |
| MPFR_GROUP_CLEAR (group); |
| mpz_clear (T); |
| mpz_clear (P); |
| mpz_clear (Q); |
| |
| return inex; |
| } |