Annotation of OpenXM_contrib/gmp/mpfr/const_euler.c, Revision 1.1
1.1 ! ohara 1: /* mpfr_const_euler -- Euler's constant
! 2:
! 3: Copyright 2001 Free Software Foundation.
! 4:
! 5: This file is part of the MPFR Library.
! 6:
! 7: The MPFR Library is free software; you can redistribute it and/or modify
! 8: it under the terms of the GNU Lesser General Public License as published by
! 9: the Free Software Foundation; either version 2.1 of the License, or (at your
! 10: option) any later version.
! 11:
! 12: The MPFR Library is distributed in the hope that it will be useful, but
! 13: WITHOUT ANY WARRANTY; without even the implied warranty of MERCHANTABILITY
! 14: or FITNESS FOR A PARTICULAR PURPOSE. See the GNU Lesser General Public
! 15: License for more details.
! 16:
! 17: You should have received a copy of the GNU Lesser General Public License
! 18: along with the MPFR Library; see the file COPYING.LIB. If not, write to
! 19: the Free Software Foundation, Inc., 59 Temple Place - Suite 330, Boston,
! 20: MA 02111-1307, USA. */
! 21:
! 22: #include <stdio.h>
! 23: #include <stdlib.h>
! 24: #include "gmp.h"
! 25: #include "gmp-impl.h"
! 26: #include "longlong.h"
! 27: #include "mpfr.h"
! 28: #include "mpfr-impl.h"
! 29:
! 30: static void mpfr_const_euler_S _PROTO ((mpfr_ptr, unsigned long));
! 31: static void mpfr_const_euler_R _PROTO ((mpfr_ptr, unsigned long));
! 32:
! 33: int
! 34: mpfr_const_euler (mpfr_t x, mp_rnd_t rnd)
! 35: {
! 36: mp_prec_t prec = MPFR_PREC(x), m, log2m;
! 37: mpfr_t y, z;
! 38: unsigned long n;
! 39:
! 40: log2m = _mpfr_ceil_log2 ((double) prec);
! 41: m = prec + log2m;
! 42:
! 43: mpfr_init (y);
! 44: mpfr_init (z);
! 45:
! 46: do
! 47: {
! 48: m += BITS_PER_MP_LIMB;
! 49: n = 1 + (unsigned long)((double) m * LOG2 / 2.0);
! 50: if (n < 9)
! 51: n = 9;
! 52: MPFR_ASSERTD (n >= 9);
! 53: mpfr_set_prec (y, m + log2m);
! 54: mpfr_set_prec (z, m + log2m);
! 55: mpfr_const_euler_S (y, n);
! 56: mpfr_set_ui (z, n, GMP_RNDN);
! 57: mpfr_log (z, z, GMP_RNDD);
! 58: mpfr_sub (y, y, z, GMP_RNDN); /* S'(n) - log(n) */
! 59: mpfr_set_prec (z, m);
! 60: mpfr_const_euler_R (z, n);
! 61: mpfr_sub (y, y, z, GMP_RNDN);
! 62: }
! 63: while (!mpfr_can_round (y, m - 3, GMP_RNDN, rnd, prec));
! 64:
! 65: mpfr_set (x, y, rnd);
! 66:
! 67: mpfr_clear (y);
! 68: mpfr_clear (z);
! 69:
! 70: return 1; /* always inexact */
! 71: }
! 72:
! 73: /* computes S(n) = sum(n^k*(-1)^(k-1)/k!/k, k=1..ceil(4.319136566 * n))
! 74: with an error of at most ulp(x).
! 75: [S(n) >= 2 for n >= 5]
! 76: */
! 77: void
! 78: mpfr_const_euler_S (mpfr_t x, unsigned long n)
! 79: {
! 80: unsigned long N, k, m;
! 81: mpz_t a, s, t;
! 82:
! 83: N = (long) (ALPHA * (double) n + 1.0); /* ceil(alpha * n) */
! 84:
! 85: m = MPFR_PREC(x) + (unsigned long) ((double) n / LOG2)
! 86: + _mpfr_ceil_log2 ((double) N) + 1;
! 87:
! 88: mpz_init_set_ui (a, 1);
! 89: mpz_mul_2exp (a, a, m); /* a=-2^m where m is the precision of x */
! 90: mpz_init_set_ui (s, 0);
! 91: mpz_init (t);
! 92:
! 93: /* here, a and s are exact */
! 94: for (k = 1; k <= N; k++)
! 95: {
! 96: mpz_mul_ui (a, a, n);
! 97: mpz_div_ui (a, a, k);
! 98: mpz_div_ui (t, a, k);
! 99: if (k % 2)
! 100: mpz_add (s, s, t);
! 101: else
! 102: mpz_sub (s, s, t);
! 103: }
! 104:
! 105: /* the error on s is at most N (e^n + 1),
! 106: thus that the error on x is at most one ulp */
! 107:
! 108: mpfr_set_z (x, s, GMP_RNDD);
! 109: mpfr_div_2ui (x, x, m, GMP_RNDD);
! 110:
! 111: mpz_clear (a);
! 112: mpz_clear (s);
! 113: mpz_clear (t);
! 114: }
! 115:
! 116: /* computes R(n) = exp(-n)/n * sum(k!/(-n)^k, k=0..n-2)
! 117: with error at most 4*ulp(x). Assumes n>=2.
! 118: Since x <= exp(-n)/n <= 1/8, then 4*ulp(x) <= ulp(1).
! 119: */
! 120: void
! 121: mpfr_const_euler_R (mpfr_t x, unsigned long n)
! 122: {
! 123: unsigned long k, m;
! 124: mpz_t a, s;
! 125: mpfr_t y;
! 126:
! 127: MPFR_ASSERTN (n >= 2); /* ensures sum(k!/(-n)^k, k=0..n-2) >= 2/3 */
! 128:
! 129: /* as we multiply the sum by exp(-n), we need only PREC(x) - n/LOG2 bits */
! 130: m = MPFR_PREC(x) - (unsigned long) ((double) n / LOG2);
! 131:
! 132: mpz_init_set_ui (a, 1);
! 133: mpz_mul_2exp (a, a, m);
! 134: mpz_init_set (s, a);
! 135:
! 136: for (k = 1; k <= n; k++)
! 137: {
! 138: mpz_mul_ui (a, a, k);
! 139: mpz_div_ui (a, a, n);
! 140: /* the error e(k) on a is e(k) <= 1 + k/n*e(k-1) with e(0)=0,
! 141: i.e. e(k) <= k */
! 142: if (k % 2)
! 143: mpz_sub (s, s, a);
! 144: else
! 145: mpz_add (s, s, a);
! 146: }
! 147: /* the error on s is at most 1+2+...+n = n*(n+1)/2 */
! 148: mpz_div_ui (s, s, n); /* err <= 1 + (n+1)/2 */
! 149: if (MPFR_PREC(x) < mpz_sizeinbase(s, 2))
! 150: {
! 151: fprintf (stderr, "prec(x) is too small in mpfr_const_euler_R\n");
! 152: exit (1);
! 153: }
! 154: mpfr_set_z (x, s, GMP_RNDD); /* exact */
! 155: mpfr_div_2ui (x, x, m, GMP_RNDD);
! 156: /* now x = 1/n * sum(k!/(-n)^k, k=0..n-2) <= 1/n */
! 157: /* err(x) <= (n+1)/2^m <= (n+1)*exp(n)/2^PREC(x) */
! 158:
! 159: mpfr_init2 (y, m);
! 160: mpfr_set_si (y, -n, GMP_RNDD); /* assumed exact */
! 161: mpfr_exp (y, y, GMP_RNDD); /* err <= ulp(y) <= exp(-n)*2^(1-m) */
! 162: mpfr_mul (x, x, y, GMP_RNDD);
! 163: /* err <= ulp(x) + (n + 1 + 2/n) / 2^prec(x)
! 164: <= ulp(x) + (n + 1 + 2/n) ulp(x)/x since x*2^(-prec(x)) < ulp(x)
! 165: <= ulp(x) + (n + 1 + 2/n) 3/(2n) ulp(x) since x >= 2/3*n for n >= 2
! 166: <= 4 * ulp(x) for n >= 2 */
! 167: mpfr_clear (y);
! 168:
! 169: mpz_clear (a);
! 170: mpz_clear (s);
! 171: }
FreeBSD-CVSweb <freebsd-cvsweb@FreeBSD.org>
![[BACK]](/icons/cvsweb/back.gif){kind=icon}
Return to const_euler.c CVS log ![[TXT]](/icons/cvsweb/text.gif){kind=icon}
![[DIR]](/icons/cvsweb/dir.gif){kind=icon}
Up to [local] / OpenXM_contrib / gmp / mpfr