contrib/mpfr/const_catalan.c

   1 /* mpfr_const_catalan -- compute Catalan's constant.
   2
   3 Copyright 2005, 2006, 2007, 2008, 2009 Free Software Foundation, Inc.
   4 Contributed by the Arenaire and Cacao 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 2.1 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.LIB.  If not, write to
  20 the Free Software Foundation, Inc., 51 Franklin St, Fifth Floor, Boston,
  21 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, mp_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, mp_rnd_t rnd_mode)
  86 {
  87   mpfr_t x, y, z;
  88   mpz_t T, P, Q;
  89   mp_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), ("g[%#R]=%R inex=%d", g, g, inex));
  95
  96   /* Here are the WC (max prec = 100.000.000)
  97      Once we have found a chain of 11, we only look for bigger chain.
  98      Found 3 '1' at 0
  99      Found 5 '1' at 9
 100      Found 6 '0' at 34
 101      Found 9 '1' at 176
 102      Found 11 '1' at 705
 103      Found 12 '0' at 913
 104      Found 14 '1' at 12762
 105      Found 15 '1' at 152561
 106      Found 16 '0' at 171725
 107      Found 18 '0' at 525355
 108      Found 20 '0' at 529245
 109      Found 21 '1' at 6390133
 110      Found 22 '0' at 7806417
 111      Found 25 '1' at 11936239
 112      Found 27 '1' at 51752950
 113   */
 114   pg = MPFR_PREC (g);
 115   p = pg + 9;
 116   p += MPFR_INT_CEIL_LOG2 (p);
 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, GMP_RNDU);
 126     mpfr_add_ui (x, x, 2, GMP_RNDU);
 127     mpfr_log (x, x, GMP_RNDU);
 128     mpfr_const_pi (y, GMP_RNDU);
 129     mpfr_mul (x, x, y, GMP_RNDN);
 130     S (T, P, Q, 0, (p - 1) / 2);
 131     mpz_mul_ui (T, T, 3);
 132     mpfr_set_z (y, T, GMP_RNDU);
 133     mpfr_set_z (z, Q, GMP_RNDD);
 134     mpfr_div (y, y, z, GMP_RNDN);
 135     mpfr_add (x, x, y, GMP_RNDN);
 136     mpfr_div_2ui (x, x, 3, GMP_RNDN);
 137
 138     if (MPFR_LIKELY (MPFR_CAN_ROUND (x, p - 5, pg, rnd_mode)))
 139       break;
 140     /* Fixme: Is it possible? */
 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 }