1 /* mpfr_const_catalan -- compute Catalan's constant. 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 #define MPFR_NEED_LONGLONG_H 24 #include "mpfr-impl.h" 25 26 /* Declare the cache */ 27 MPFR_DECL_INIT_CACHE(__gmpfr_cache_const_catalan, mpfr_const_catalan_internal); 28 29 /* Set User Interface */ 30 #undef mpfr_const_catalan 31 int 32 mpfr_const_catalan (mpfr_ptr x, mpfr_rnd_t rnd_mode) { 33 return mpfr_cache (x, __gmpfr_cache_const_catalan, rnd_mode); 34 } 35 36 /* return T, Q such that T/Q = sum(k!^2/(2k)!/(2k+1)^2, k=n1..n2-1) */ 37 static void 38 S (mpz_t T, mpz_t P, mpz_t Q, unsigned long n1, unsigned long n2) 39 { 40 if (n2 == n1 + 1) 41 { 42 if (n1 == 0) 43 { 44 mpz_set_ui (P, 1); 45 mpz_set_ui (Q, 1); 46 } 47 else 48 { 49 mpz_set_ui (P, 2 * n1 - 1); 50 mpz_mul_ui (P, P, n1); 51 mpz_ui_pow_ui (Q, 2 * n1 + 1, 2); 52 mpz_mul_2exp (Q, Q, 1); 53 } 54 mpz_set (T, P); 55 } 56 else 57 { 58 unsigned long m = (n1 + n2) / 2; 59 mpz_t T2, P2, Q2; 60 S (T, P, Q, n1, m); 61 mpz_init (T2); 62 mpz_init (P2); 63 mpz_init (Q2); 64 S (T2, P2, Q2, m, n2); 65 mpz_mul (T, T, Q2); 66 mpz_mul (T2, T2, P); 67 mpz_add (T, T, T2); 68 mpz_mul (P, P, P2); 69 mpz_mul (Q, Q, Q2); 70 mpz_clear (T2); 71 mpz_clear (P2); 72 mpz_clear (Q2); 73 } 74 } 75 76 /* Don't need to save/restore exponent range: the cache does it. 77 Catalan's constant is G = sum((-1)^k/(2*k+1)^2, k=0..infinity). 78 We compute it using formula (31) of Victor Adamchik's page 79 "33 representations for Catalan's constant" 80 http://www-2.cs.cmu.edu/~adamchik/articles/catalan/catalan.htm 81 82 G = Pi/8*log(2+sqrt(3)) + 3/8*sum(k!^2/(2k)!/(2k+1)^2,k=0..infinity) 83 */ 84 int 85 mpfr_const_catalan_internal (mpfr_ptr g, mpfr_rnd_t rnd_mode) 86 { 87 mpfr_t x, y, z; 88 mpz_t T, P, Q; 89 mpfr_prec_t pg, p; 90 int inex; 91 MPFR_ZIV_DECL (loop); 92 MPFR_GROUP_DECL (group); 93 94 MPFR_LOG_FUNC (("rnd_mode=%d", rnd_mode), 95 ("g[%Pu]=%.*Rg inex=%d", mpfr_get_prec (g), mpfr_log_prec, g, inex)); 96 97 /* Here are the WC (max prec = 100.000.000) 98 Once we have found a chain of 11, we only look for bigger chain. 99 Found 3 '1' at 0 100 Found 5 '1' at 9 101 Found 6 '0' at 34 102 Found 9 '1' at 176 103 Found 11 '1' at 705 104 Found 12 '0' at 913 105 Found 14 '1' at 12762 106 Found 15 '1' at 152561 107 Found 16 '0' at 171725 108 Found 18 '0' at 525355 109 Found 20 '0' at 529245 110 Found 21 '1' at 6390133 111 Found 22 '0' at 7806417 112 Found 25 '1' at 11936239 113 Found 27 '1' at 51752950 114 */ 115 pg = MPFR_PREC (g); 116 p = pg + MPFR_INT_CEIL_LOG2 (pg) + 7; 117 118 MPFR_GROUP_INIT_3 (group, p, x, y, z); 119 mpz_init (T); 120 mpz_init (P); 121 mpz_init (Q); 122 123 MPFR_ZIV_INIT (loop, p); 124 for (;;) { 125 mpfr_sqrt_ui (x, 3, MPFR_RNDU); 126 mpfr_add_ui (x, x, 2, MPFR_RNDU); 127 mpfr_log (x, x, MPFR_RNDU); 128 mpfr_const_pi (y, MPFR_RNDU); 129 mpfr_mul (x, x, y, MPFR_RNDN); 130 S (T, P, Q, 0, (p - 1) / 2); 131 mpz_mul_ui (T, T, 3); 132 mpfr_set_z (y, T, MPFR_RNDU); 133 mpfr_set_z (z, Q, MPFR_RNDD); 134 mpfr_div (y, y, z, MPFR_RNDN); 135 mpfr_add (x, x, y, MPFR_RNDN); 136 mpfr_div_2ui (x, x, 3, MPFR_RNDN); 137 138 if (MPFR_LIKELY (MPFR_CAN_ROUND (x, p - 5, pg, rnd_mode))) 139 break; 140 141 MPFR_ZIV_NEXT (loop, p); 142 MPFR_GROUP_REPREC_3 (group, p, x, y, z); 143 } 144 MPFR_ZIV_FREE (loop); 145 inex = mpfr_set (g, x, rnd_mode); 146 147 MPFR_GROUP_CLEAR (group); 148 mpz_clear (T); 149 mpz_clear (P); 150 mpz_clear (Q); 151 152 return inex; 153 } 154