--- /dev/null
+/* 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"
projects / msp430-gcc.git / blobdiff