1 /* mpfr_const_catalan -- compute Catalan's constant. 2 3 Copyright 2005-2016 Free Software Foundation, Inc. 4 Contributed by the AriC and Caramba 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 #ifdef MPFR_WIN_THREAD_SAFE_DLL 30 mpfr_cache_t * 31 __gmpfr_cache_const_catalan_f() 32 { 33 return &__gmpfr_cache_const_catalan; 34 } 35 #endif 36 37 /* Set User Interface */ 38 #undef mpfr_const_catalan 39 int 40 mpfr_const_catalan (mpfr_ptr x, mpfr_rnd_t rnd_mode) { 41 return mpfr_cache (x, __gmpfr_cache_const_catalan, rnd_mode); 42 } 43 44 /* return T, Q such that T/Q = sum(k!^2/(2k)!/(2k+1)^2, k=n1..n2-1) */ 45 static void 46 S (mpz_t T, mpz_t P, mpz_t Q, unsigned long n1, unsigned long n2) 47 { 48 if (n2 == n1 + 1) 49 { 50 if (n1 == 0) 51 { 52 mpz_set_ui (P, 1); 53 mpz_set_ui (Q, 1); 54 } 55 else 56 { 57 mpz_set_ui (P, 2 * n1 - 1); 58 mpz_mul_ui (P, P, n1); 59 mpz_ui_pow_ui (Q, 2 * n1 + 1, 2); 60 mpz_mul_2exp (Q, Q, 1); 61 } 62 mpz_set (T, P); 63 } 64 else 65 { 66 unsigned long m = (n1 + n2) / 2; 67 mpz_t T2, P2, Q2; 68 S (T, P, Q, n1, m); 69 mpz_init (T2); 70 mpz_init (P2); 71 mpz_init (Q2); 72 S (T2, P2, Q2, m, n2); 73 mpz_mul (T, T, Q2); 74 mpz_mul (T2, T2, P); 75 mpz_add (T, T, T2); 76 mpz_mul (P, P, P2); 77 mpz_mul (Q, Q, Q2); 78 mpz_clear (T2); 79 mpz_clear (P2); 80 mpz_clear (Q2); 81 } 82 } 83 84 /* Don't need to save/restore exponent range: the cache does it. 85 Catalan's constant is G = sum((-1)^k/(2*k+1)^2, k=0..infinity). 86 We compute it using formula (31) of Victor Adamchik's page 87 "33 representations for Catalan's constant" 88 http://www-2.cs.cmu.edu/~adamchik/articles/catalan/catalan.htm 89 90 G = Pi/8*log(2+sqrt(3)) + 3/8*sum(k!^2/(2k)!/(2k+1)^2,k=0..infinity) 91 */ 92 int 93 mpfr_const_catalan_internal (mpfr_ptr g, mpfr_rnd_t rnd_mode) 94 { 95 mpfr_t x, y, z; 96 mpz_t T, P, Q; 97 mpfr_prec_t pg, p; 98 int inex; 99 MPFR_ZIV_DECL (loop); 100 MPFR_GROUP_DECL (group); 101 102 MPFR_LOG_FUNC (("rnd_mode=%d", rnd_mode), 103 ("g[%Pu]=%.*Rg inex=%d", mpfr_get_prec (g), mpfr_log_prec, g, inex)); 104 105 /* Here are the WC (max prec = 100.000.000) 106 Once we have found a chain of 11, we only look for bigger chain. 107 Found 3 '1' at 0 108 Found 5 '1' at 9 109 Found 6 '0' at 34 110 Found 9 '1' at 176 111 Found 11 '1' at 705 112 Found 12 '0' at 913 113 Found 14 '1' at 12762 114 Found 15 '1' at 152561 115 Found 16 '0' at 171725 116 Found 18 '0' at 525355 117 Found 20 '0' at 529245 118 Found 21 '1' at 6390133 119 Found 22 '0' at 7806417 120 Found 25 '1' at 11936239 121 Found 27 '1' at 51752950 122 */ 123 pg = MPFR_PREC (g); 124 p = pg + MPFR_INT_CEIL_LOG2 (pg) + 7; 125 126 MPFR_GROUP_INIT_3 (group, p, x, y, z); 127 mpz_init (T); 128 mpz_init (P); 129 mpz_init (Q); 130 131 MPFR_ZIV_INIT (loop, p); 132 for (;;) { 133 mpfr_sqrt_ui (x, 3, MPFR_RNDU); 134 mpfr_add_ui (x, x, 2, MPFR_RNDU); 135 mpfr_log (x, x, MPFR_RNDU); 136 mpfr_const_pi (y, MPFR_RNDU); 137 mpfr_mul (x, x, y, MPFR_RNDN); 138 S (T, P, Q, 0, (p - 1) / 2); 139 mpz_mul_ui (T, T, 3); 140 mpfr_set_z (y, T, MPFR_RNDU); 141 mpfr_set_z (z, Q, MPFR_RNDD); 142 mpfr_div (y, y, z, MPFR_RNDN); 143 mpfr_add (x, x, y, MPFR_RNDN); 144 mpfr_div_2ui (x, x, 3, MPFR_RNDN); 145 146 if (MPFR_LIKELY (MPFR_CAN_ROUND (x, p - 5, pg, rnd_mode))) 147 break; 148 149 MPFR_ZIV_NEXT (loop, p); 150 MPFR_GROUP_REPREC_3 (group, p, x, y, z); 151 } 152 MPFR_ZIV_FREE (loop); 153 inex = mpfr_set (g, x, rnd_mode); 154 155 MPFR_GROUP_CLEAR (group); 156 mpz_clear (T); 157 mpz_clear (P); 158 mpz_clear (Q); 159 160 return inex; 161 } 162