X-Git-Url: https://oss.titaniummirror.com/gitweb?a=blobdiff_plain;f=mpfr%2Fjn.c;fp=mpfr%2Fjn.c;h=84153f9bf0179c2f3e6e4c3fe7a506e68fd580fa;hb=6fed43773c9b0ce596dca5686f37ac3fc0fa11c0;hp=0000000000000000000000000000000000000000;hpb=27b11d56b743098deb193d510b337ba22dc52e5c;p=msp430-gcc.git diff --git a/mpfr/jn.c b/mpfr/jn.c new file mode 100644 index 00000000..84153f9b --- /dev/null +++ b/mpfr/jn.c @@ -0,0 +1,243 @@ +/* mpfr_j0, mpfr_j1, mpfr_jn -- Bessel functions of 1st kind, integer order. + http://www.opengroup.org/onlinepubs/009695399/functions/j0.html + +Copyright 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" + +/* Relations: j(-n,z) = (-1)^n j(n,z) + j(n,-z) = (-1)^n j(n,z) +*/ + +static int mpfr_jn_asympt (mpfr_ptr, long, mpfr_srcptr, mp_rnd_t); + +int +mpfr_j0 (mpfr_ptr res, mpfr_srcptr z, mp_rnd_t r) +{ + return mpfr_jn (res, 0, z, r); +} + +int +mpfr_j1 (mpfr_ptr res, mpfr_srcptr z, mp_rnd_t r) +{ + return mpfr_jn (res, 1, z, r); +} + +/* Estimate k0 such that z^2/4 = k0 * (k0 + n) + i.e., (sqrt(n^2+z^2)-n)/2 = n/2 * (sqrt(1+(z/n)^2) - 1). + Return min(2*k0/log(2), ULONG_MAX). +*/ +static unsigned long +mpfr_jn_k0 (long n, mpfr_srcptr z) +{ + mpfr_t t, u; + unsigned long k0; + + mpfr_init2 (t, 32); + mpfr_init2 (u, 32); + mpfr_div_si (t, z, n, GMP_RNDN); + mpfr_sqr (t, t, GMP_RNDN); + mpfr_add_ui (t, t, 1, GMP_RNDN); + mpfr_sqrt (t, t, GMP_RNDN); + mpfr_sub_ui (t, t, 1, GMP_RNDN); + mpfr_mul_si (t, t, n, GMP_RNDN); + /* the following is a 32-bit approximation to nearest of log(2) */ + mpfr_set_str_binary (u, "0.10110001011100100001011111111"); + mpfr_div (t, t, u, GMP_RNDN); + if (mpfr_fits_ulong_p (t, GMP_RNDN)) + k0 = mpfr_get_ui (t, GMP_RNDN); + else + k0 = ULONG_MAX; + mpfr_clear (t); + mpfr_clear (u); + return k0; +} + +int +mpfr_jn (mpfr_ptr res, long n, mpfr_srcptr z, mp_rnd_t r) +{ + int inex; + unsigned long absn; + mp_prec_t prec, pbound, err; + mp_exp_t exps, expT; + mpfr_t y, s, t, absz; + unsigned long k, zz, k0; + MPFR_ZIV_DECL (loop); + + MPFR_LOG_FUNC (("x[%#R]=%R n=%d rnd=%d", z, z, n, r), + ("y[%#R]=%R", res, res)); + + absn = SAFE_ABS (unsigned long, n); + + if (MPFR_UNLIKELY (MPFR_IS_SINGULAR (z))) + { + if (MPFR_IS_NAN (z)) + { + MPFR_SET_NAN (res); + MPFR_RET_NAN; + } + /* j(n,z) tends to zero when z goes to +Inf or -Inf, oscillating around + 0. We choose to return +0 in that case. */ + else if (MPFR_IS_INF (z)) /* FIXME: according to j(-n,z) = (-1)^n j(n,z) + we might want to give a sign depending on + z and n */ + return mpfr_set_ui (res, 0, r); + else /* z=0: j(0,0)=1, j(n odd,+/-0) = +/-0 if n > 0, -/+0 if n < 0, + j(n even,+/-0) = +0 */ + { + if (n == 0) + return mpfr_set_ui (res, 1, r); + else if (absn & 1) /* n odd */ + return (n > 0) ? mpfr_set (res, z, r) : mpfr_neg (res, z, r); + else /* n even */ + return mpfr_set_ui (res, 0, r); + } + } + + /* check for tiny input for j0: j0(z) = 1 - z^2/4 + ..., more precisely + |j0(z) - 1| <= z^2/4 for -1 <= z <= 1. */ + if (n == 0) + MPFR_FAST_COMPUTE_IF_SMALL_INPUT (res, __gmpfr_one, -2 * MPFR_GET_EXP (z), + 2, 0, r, return _inexact); + + /* idem for j1: j1(z) = z/2 - z^3/16 + ..., more precisely + |j1(z) - z/2| <= |z^3|/16 for -1 <= z <= 1, with the sign of j1(z) - z/2 + being the opposite of that of z. */ + if (n == 1) + /* we first compute 2j1(z) = z - z^3/8 + ..., then divide by 2 using + the "extra" argument of MPFR_FAST_COMPUTE_IF_SMALL_INPUT. */ + MPFR_FAST_COMPUTE_IF_SMALL_INPUT (res, z, -2 * MPFR_GET_EXP (z), 3, + 0, r, mpfr_div_2ui (res, res, 1, r)); + + /* we can use the asymptotic expansion as soon as |z| > p log(2)/2, + but to get some margin we use it for |z| > p/2 */ + pbound = MPFR_PREC (res) / 2 + 3; + MPFR_ASSERTN (pbound <= ULONG_MAX); + MPFR_ALIAS (absz, z, 1, MPFR_EXP (z)); + if (mpfr_cmp_ui (absz, pbound) > 0) + { + inex = mpfr_jn_asympt (res, n, z, r); + if (inex != 0) + return inex; + } + + mpfr_init2 (y, 32); + + /* check underflow case: |j(n,z)| <= 1/sqrt(2 Pi n) (ze/2n)^n + (see algorithms.tex) */ + if (absn > 0) + { + /* the following is an upper 32-bit approximation of exp(1)/2 */ + mpfr_set_str_binary (y, "1.0101101111110000101010001011001"); + if (MPFR_SIGN(z) > 0) + mpfr_mul (y, y, z, GMP_RNDU); + else + { + mpfr_mul (y, y, z, GMP_RNDD); + mpfr_neg (y, y, GMP_RNDU); + } + mpfr_div_ui (y, y, absn, GMP_RNDU); + /* now y is an upper approximation of |ze/2n|: y < 2^EXP(y), + thus |j(n,z)| < 1/2*y^n < 2^(n*EXP(y)-1). + If n*EXP(y) < __gmpfr_emin then we have an underflow. + Warning: absn is an unsigned long. */ + if ((MPFR_EXP(y) < 0 && absn > (unsigned long) (-__gmpfr_emin)) + || (absn <= (unsigned long) (-MPFR_EMIN_MIN) && + MPFR_EXP(y) < __gmpfr_emin / (mp_exp_t) absn)) + { + mpfr_clear (y); + return mpfr_underflow (res, (r == GMP_RNDN) ? GMP_RNDZ : r, + (n % 2) ? ((n > 0) ? MPFR_SIGN(z) : -MPFR_SIGN(z)) + : MPFR_SIGN_POS); + } + } + + mpfr_init (s); + mpfr_init (t); + + /* the logarithm of the ratio between the largest term in the series + and the first one is roughly bounded by k0, which we add to the + working precision to take into account this cancellation */ + k0 = mpfr_jn_k0 (absn, z); + prec = MPFR_PREC (res) + k0 + 2 * MPFR_INT_CEIL_LOG2 (MPFR_PREC (res)) + 3; + + MPFR_ZIV_INIT (loop, prec); + for (;;) + { + mpfr_set_prec (y, prec); + mpfr_set_prec (s, prec); + mpfr_set_prec (t, prec); + mpfr_pow_ui (t, z, absn, GMP_RNDN); /* z^|n| */ + mpfr_mul (y, z, z, GMP_RNDN); /* z^2 */ + zz = mpfr_get_ui (y, GMP_RNDU); + MPFR_ASSERTN (zz < ULONG_MAX); + mpfr_div_2ui (y, y, 2, GMP_RNDN); /* z^2/4 */ + mpfr_fac_ui (s, absn, GMP_RNDN); /* |n|! */ + mpfr_div (t, t, s, GMP_RNDN); + if (absn > 0) + mpfr_div_2ui (t, t, absn, GMP_RNDN); + mpfr_set (s, t, GMP_RNDN); + exps = MPFR_EXP (s); + expT = exps; + for (k = 1; ; k++) + { + mpfr_mul (t, t, y, GMP_RNDN); + mpfr_neg (t, t, GMP_RNDN); + if (k + absn <= ULONG_MAX / k) + mpfr_div_ui (t, t, k * (k + absn), GMP_RNDN); + else + { + mpfr_div_ui (t, t, k, GMP_RNDN); + mpfr_div_ui (t, t, k + absn, GMP_RNDN); + } + exps = MPFR_EXP (t); + if (exps > expT) + expT = exps; + mpfr_add (s, s, t, GMP_RNDN); + exps = MPFR_EXP (s); + if (exps > expT) + expT = exps; + if (MPFR_EXP (t) + (mp_exp_t) prec <= MPFR_EXP (s) && + zz / (2 * k) < k + n) + break; + } + /* the error is bounded by (4k^2+21/2k+7) ulp(s)*2^(expT-exps) + <= (k+2)^2 ulp(s)*2^(2+expT-exps) */ + err = 2 * MPFR_INT_CEIL_LOG2(k + 2) + 2 + expT - MPFR_EXP (s); + if (MPFR_LIKELY (MPFR_CAN_ROUND (s, prec - err, MPFR_PREC(res), r))) + break; + MPFR_ZIV_NEXT (loop, prec); + } + MPFR_ZIV_FREE (loop); + + inex = ((n >= 0) || ((n & 1) == 0)) ? mpfr_set (res, s, r) + : mpfr_neg (res, s, r); + + mpfr_clear (y); + mpfr_clear (s); + mpfr_clear (t); + + return inex; +} + +#define MPFR_JN +#include "jyn_asympt.c"