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Diffstat (limited to 'src/agm.c')
| -rw-r--r-- | src/agm.c | 319 |
1 files changed, 319 insertions, 0 deletions
diff --git a/src/agm.c b/src/agm.c new file mode 100644 index 0000000..567bc46 --- /dev/null +++ b/src/agm.c @@ -0,0 +1,319 @@ +/* mpfr_agm -- arithmetic-geometric mean of two floating-point numbers + +Copyright 1999, 2000, 2001, 2002, 2003, 2004, 2005, 2006, 2007, 2008, 2009, 2010, 2011, 2012, 2013 Free Software Foundation, Inc. +Contributed by the AriC and Caramel 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 3 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.LESSER. If not, see +http://www.gnu.org/licenses/ or 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, mpfr_rnd_t rnd_mode) +{ + int compare, inexact; + mp_size_t s; + mpfr_prec_t p, q; + mp_limb_t *up, *vp, *ufp, *vfp; + mpfr_t u, v, uf, vf, sc1, sc2; + mpfr_exp_t scaleop = 0, scaleit; + unsigned long n; /* number of iterations */ + MPFR_ZIV_DECL (loop); + MPFR_TMP_DECL(marker); + MPFR_SAVE_EXPO_DECL (expo); + + MPFR_LOG_FUNC + (("op2[%Pu]=%.*Rg op1[%Pu]=%.*Rg rnd=%d", + mpfr_get_prec (op2), mpfr_log_prec, op2, + mpfr_get_prec (op1), mpfr_log_prec, op1, rnd_mode), + ("r[%Pu]=%.*Rg inexact=%d", + mpfr_get_prec (r), mpfr_log_prec, 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 */ + } + } + + /* 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 = MPFR_PREC2LIMBS (p); + + /* 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_SAVE_EXPO_MARK (expo); + + MPFR_TMP_MARK(marker); + + /* Main loop */ + MPFR_ZIV_INIT (loop, p); + for (;;) + { + mpfr_prec_t eq; + unsigned long err = 0; /* must be set to 0 at each Ziv iteration */ + MPFR_BLOCK_DECL (flags); + + /* Init temporary vars */ + MPFR_TMP_INIT (up, u, p, s); + MPFR_TMP_INIT (vp, v, p, s); + MPFR_TMP_INIT (ufp, uf, p, s); + MPFR_TMP_INIT (vfp, vf, p, s); + + /* Calculus of un and vn */ + retry: + MPFR_BLOCK (flags, + mpfr_mul (u, op1, op2, MPFR_RNDN); + /* mpfr_mul(...): faster since PREC(op) < PREC(u) */ + mpfr_add (v, op1, op2, MPFR_RNDN); + /* mpfr_add with !=prec is still good */); + if (MPFR_UNLIKELY (MPFR_OVERFLOW (flags) || MPFR_UNDERFLOW (flags))) + { + mpfr_exp_t e1 , e2; + + MPFR_ASSERTN (scaleop == 0); + e1 = MPFR_GET_EXP (op1); + e2 = MPFR_GET_EXP (op2); + + /* Let's determine scaleop to avoid an overflow/underflow. */ + if (MPFR_OVERFLOW (flags)) + { + /* Let's recall that emin <= e1 <= e2 <= emax. + There has been an overflow. Thus e2 >= emax/2. + If the mpfr_mul overflowed, then e1 + e2 > emax. + If the mpfr_add overflowed, then e2 = emax. + We want: (e1 + scale) + (e2 + scale) <= emax, + i.e. scale <= (emax - e1 - e2) / 2. Let's take + scale = min(floor((emax - e1 - e2) / 2), -1). + This is OK, as: + 1. emin <= scale <= -1. + 2. e1 + scale >= emin. Indeed: + * If e1 + e2 > emax, then + e1 + scale >= e1 + (emax - e1 - e2) / 2 - 1 + >= (emax + e1 - emax) / 2 - 1 + >= e1 / 2 - 1 >= emin. + * Otherwise, mpfr_mul didn't overflow, therefore + mpfr_add overflowed and e2 = emax, so that + e1 > emin (see restriction below). + e1 + scale > emin - 1, thus e1 + scale >= emin. + 3. e2 + scale <= emax, since scale < 0. */ + if (e1 + e2 > MPFR_EXT_EMAX) + { + scaleop = - (((e1 + e2) - MPFR_EXT_EMAX + 1) / 2); + MPFR_ASSERTN (scaleop < 0); + } + else + { + /* The addition necessarily overflowed. */ + MPFR_ASSERTN (e2 == MPFR_EXT_EMAX); + /* The case where e1 = emin and e2 = emax is not supported + here. This would mean that the precision of e2 would be + huge (and possibly not supported in practice anyway). */ + MPFR_ASSERTN (e1 > MPFR_EXT_EMIN); + scaleop = -1; + } + + } + else /* underflow only (in the multiplication) */ + { + /* We have e1 + e2 <= emin (so, e1 <= e2 <= 0). + We want: (e1 + scale) + (e2 + scale) >= emin + 1, + i.e. scale >= (emin + 1 - e1 - e2) / 2. let's take + scale = ceil((emin + 1 - e1 - e2) / 2). This is OK, as: + 1. 1 <= scale <= emax. + 2. e1 + scale >= emin + 1 >= emin. + 3. e2 + scale <= scale <= emax. */ + MPFR_ASSERTN (e1 <= e2 && e2 <= 0); + scaleop = (MPFR_EXT_EMIN + 2 - e1 - e2) / 2; + MPFR_ASSERTN (scaleop > 0); + } + + MPFR_ALIAS (sc1, op1, MPFR_SIGN (op1), e1 + scaleop); + MPFR_ALIAS (sc2, op2, MPFR_SIGN (op2), e2 + scaleop); + op1 = sc1; + op2 = sc2; + MPFR_LOG_MSG (("Exception in pre-iteration, scale = %" + MPFR_EXP_FSPEC "d\n", scaleop)); + goto retry; + } + + mpfr_clear_flags (); + mpfr_sqrt (u, u, MPFR_RNDN); + mpfr_div_2ui (v, v, 1, MPFR_RNDN); + + scaleit = 0; + n = 1; + while (mpfr_cmp2 (u, v, &eq) != 0 && eq <= p - 2) + { + MPFR_BLOCK_DECL (flags2); + + MPFR_LOG_MSG (("Iteration n = %lu\n", n)); + + retry2: + mpfr_add (vf, u, v, MPFR_RNDN); /* No overflow? */ + mpfr_div_2ui (vf, vf, 1, MPFR_RNDN); + /* See proof in algorithms.tex */ + if (4*eq > p) + { + mpfr_t w; + MPFR_BLOCK_DECL (flags3); + + MPFR_LOG_MSG (("4*eq > p\n", 0)); + + /* vf = V(k) */ + mpfr_init2 (w, (p + 1) / 2); + MPFR_BLOCK + (flags3, + mpfr_sub (w, v, u, MPFR_RNDN); /* e = V(k-1)-U(k-1) */ + mpfr_sqr (w, w, MPFR_RNDN); /* e = e^2 */ + mpfr_div_2ui (w, w, 4, MPFR_RNDN); /* e*= (1/2)^2*1/4 */ + mpfr_div (w, w, vf, MPFR_RNDN); /* 1/4*e^2/V(k) */ + ); + if (MPFR_LIKELY (! MPFR_UNDERFLOW (flags3))) + { + mpfr_sub (v, vf, w, MPFR_RNDN); + err = MPFR_GET_EXP (vf) - MPFR_GET_EXP (v); /* 0 or 1 */ + mpfr_clear (w); + break; + } + /* There has been an underflow because of the cancellation + between V(k-1) and U(k-1). Let's use the conventional + method. */ + MPFR_LOG_MSG (("4*eq > p -> underflow\n", 0)); + mpfr_clear (w); + mpfr_clear_underflow (); + } + /* U(k) increases, so that U.V can overflow (but not underflow). */ + MPFR_BLOCK (flags2, mpfr_mul (uf, u, v, MPFR_RNDN);); + if (MPFR_UNLIKELY (MPFR_OVERFLOW (flags2))) + { + mpfr_exp_t scale2; + + scale2 = - (((MPFR_GET_EXP (u) + MPFR_GET_EXP (v)) + - MPFR_EXT_EMAX + 1) / 2); + MPFR_EXP (u) += scale2; + MPFR_EXP (v) += scale2; + scaleit += scale2; + MPFR_LOG_MSG (("Overflow in iteration n = %lu, scaleit = %" + MPFR_EXP_FSPEC "d (%" MPFR_EXP_FSPEC "d)\n", + n, scaleit, scale2)); + mpfr_clear_overflow (); + goto retry2; + } + mpfr_sqrt (u, uf, MPFR_RNDN); + mpfr_swap (v, vf); + n ++; + } + + MPFR_LOG_MSG (("End of iterations (n = %lu)\n", 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 = MPFR_PREC2LIMBS (p); + } + MPFR_ZIV_FREE (loop); + + if (MPFR_UNLIKELY ((__gmpfr_flags & (MPFR_FLAGS_ALL ^ MPFR_FLAGS_INEXACT)) + != 0)) + { + MPFR_ASSERTN (! mpfr_overflow_p ()); /* since mpfr_clear_flags */ + MPFR_ASSERTN (! mpfr_underflow_p ()); /* since mpfr_clear_flags */ + MPFR_ASSERTN (! mpfr_divby0_p ()); /* since mpfr_clear_flags */ + MPFR_ASSERTN (! mpfr_nanflag_p ()); /* since mpfr_clear_flags */ + } + + /* Setting of the result */ + inexact = mpfr_set (r, v, rnd_mode); + MPFR_EXP (r) -= scaleop + scaleit; + + /* Let's clean */ + MPFR_TMP_FREE(marker); + + MPFR_SAVE_EXPO_FREE (expo); + /* From the definition of the AGM, underflow and overflow + are not possible. */ + return mpfr_check_range (r, inexact, rnd_mode); + /* 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. */ +} |