| /* mpfr_agm -- arithmetic-geometric mean of two floating-point numbers |
| |
| Copyright 1999, 2000, 2001, 2002, 2003, 2004, 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" |
| |
| /* agm(x,y) is between x and y, so we don't need to save exponent range */ |
| int |
| mpfr_agm (mpfr_ptr r, mpfr_srcptr op2, mpfr_srcptr op1, mp_rnd_t rnd_mode) |
| { |
| int compare, inexact; |
| mp_size_t s; |
| mp_prec_t p, q; |
| mp_limb_t *up, *vp, *tmpp; |
| mpfr_t u, v, tmp; |
| unsigned long n; /* number of iterations */ |
| unsigned long err = 0; |
| MPFR_ZIV_DECL (loop); |
| MPFR_TMP_DECL(marker); |
| |
| MPFR_LOG_FUNC (("op2[%#R]=%R op1[%#R]=%R rnd=%d", op2,op2,op1,op1,rnd_mode), |
| ("r[%#R]=%R inexact=%d", r, r, inexact)); |
| |
| /* Deal with special values */ |
| if (MPFR_ARE_SINGULAR (op1, op2)) |
| { |
| /* If a or b is NaN, the result is NaN */ |
| if (MPFR_IS_NAN(op1) || MPFR_IS_NAN(op2)) |
| { |
| MPFR_SET_NAN(r); |
| MPFR_RET_NAN; |
| } |
| /* now one of a or b is Inf or 0 */ |
| /* If a and b is +Inf, the result is +Inf. |
| Otherwise if a or b is -Inf or 0, the result is NaN */ |
| else if (MPFR_IS_INF(op1) || MPFR_IS_INF(op2)) |
| { |
| if (MPFR_IS_STRICTPOS(op1) && MPFR_IS_STRICTPOS(op2)) |
| { |
| MPFR_SET_INF(r); |
| MPFR_SET_SAME_SIGN(r, op1); |
| MPFR_RET(0); /* exact */ |
| } |
| else |
| { |
| MPFR_SET_NAN(r); |
| MPFR_RET_NAN; |
| } |
| } |
| else /* a and b are neither NaN nor Inf, and one is zero */ |
| { /* If a or b is 0, the result is +0 since a sqrt is positive */ |
| MPFR_ASSERTD (MPFR_IS_ZERO (op1) || MPFR_IS_ZERO (op2)); |
| MPFR_SET_POS (r); |
| MPFR_SET_ZERO (r); |
| MPFR_RET (0); /* exact */ |
| } |
| } |
| MPFR_CLEAR_FLAGS (r); |
| |
| /* If a or b is negative (excluding -Infinity), the result is NaN */ |
| if (MPFR_UNLIKELY(MPFR_IS_NEG(op1) || MPFR_IS_NEG(op2))) |
| { |
| MPFR_SET_NAN(r); |
| MPFR_RET_NAN; |
| } |
| |
| /* Precision of the following calculus */ |
| q = MPFR_PREC(r); |
| p = q + MPFR_INT_CEIL_LOG2(q) + 15; |
| MPFR_ASSERTD (p >= 7); /* see algorithms.tex */ |
| s = (p - 1) / BITS_PER_MP_LIMB + 1; |
| |
| /* b (op2) and a (op1) are the 2 operands but we want b >= a */ |
| compare = mpfr_cmp (op1, op2); |
| if (MPFR_UNLIKELY( compare == 0 )) |
| { |
| mpfr_set (r, op1, rnd_mode); |
| MPFR_RET (0); /* exact */ |
| } |
| else if (compare > 0) |
| { |
| mpfr_srcptr t = op1; |
| op1 = op2; |
| op2 = t; |
| } |
| /* Now b(=op2) >= a (=op1) */ |
| |
| MPFR_TMP_MARK(marker); |
| |
| /* Main loop */ |
| MPFR_ZIV_INIT (loop, p); |
| for (;;) |
| { |
| mp_prec_t eq; |
| |
| /* Init temporary vars */ |
| MPFR_TMP_INIT (up, u, p, s); |
| MPFR_TMP_INIT (vp, v, p, s); |
| MPFR_TMP_INIT (tmpp, tmp, p, s); |
| |
| /* Calculus of un and vn */ |
| mpfr_mul (u, op1, op2, GMP_RNDN); /* Faster since PREC(op) < PREC(u) */ |
| mpfr_sqrt (u, u, GMP_RNDN); |
| mpfr_add (v, op1, op2, GMP_RNDN); /* add with !=prec is still good*/ |
| mpfr_div_2ui (v, v, 1, GMP_RNDN); |
| n = 1; |
| while (mpfr_cmp2 (u, v, &eq) != 0 && eq <= p - 2) |
| { |
| mpfr_add (tmp, u, v, GMP_RNDN); |
| mpfr_div_2ui (tmp, tmp, 1, GMP_RNDN); |
| /* See proof in algorithms.tex */ |
| if (4*eq > p) |
| { |
| mpfr_t w; |
| /* tmp = U(k) */ |
| mpfr_init2 (w, (p + 1) / 2); |
| mpfr_sub (w, v, u, GMP_RNDN); /* e = V(k-1)-U(k-1) */ |
| mpfr_sqr (w, w, GMP_RNDN); /* e = e^2 */ |
| mpfr_div_2ui (w, w, 4, GMP_RNDN); /* e*= (1/2)^2*1/4 */ |
| mpfr_div (w, w, tmp, GMP_RNDN); /* 1/4*e^2/U(k) */ |
| mpfr_sub (v, tmp, w, GMP_RNDN); |
| err = MPFR_GET_EXP (tmp) - MPFR_GET_EXP (v); /* 0 or 1 */ |
| mpfr_clear (w); |
| break; |
| } |
| mpfr_mul (u, u, v, GMP_RNDN); |
| mpfr_sqrt (u, u, GMP_RNDN); |
| mpfr_swap (v, tmp); |
| n ++; |
| } |
| /* the error on v is bounded by (18n+51) ulps, or twice if there |
| was an exponent loss in the final subtraction */ |
| err += MPFR_INT_CEIL_LOG2(18 * n + 51); /* 18n+51 should not overflow |
| since n is about log(p) */ |
| /* we should have n+2 <= 2^(p/4) [see algorithms.tex] */ |
| if (MPFR_LIKELY (MPFR_INT_CEIL_LOG2(n + 2) <= p / 4 && |
| MPFR_CAN_ROUND (v, p - err, q, rnd_mode))) |
| break; /* Stop the loop */ |
| |
| /* Next iteration */ |
| MPFR_ZIV_NEXT (loop, p); |
| s = (p - 1) / BITS_PER_MP_LIMB + 1; |
| } |
| MPFR_ZIV_FREE (loop); |
| |
| /* Setting of the result */ |
| inexact = mpfr_set (r, v, rnd_mode); |
| |
| /* Let's clean */ |
| MPFR_TMP_FREE(marker); |
| |
| return inexact; /* agm(u,v) can be exact for u, v rational only for u=v. |
| Proof (due to Nicolas Brisebarre): it suffices to consider |
| u=1 and v<1. Then 1/AGM(1,v) = 2F1(1/2,1/2,1;1-v^2), |
| and a theorem due to G.V. Chudnovsky states that for x a |
| non-zero algebraic number with |x|<1, then |
| 2F1(1/2,1/2,1;x) and 2F1(-1/2,1/2,1;x) are algebraically |
| independent over Q. */ |
| } |