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Diffstat (limited to 'src/jn.c')
| -rw-r--r-- | src/jn.c | 329 |
1 files changed, 329 insertions, 0 deletions
diff --git a/src/jn.c b/src/jn.c new file mode 100644 index 0000000..f857207 --- /dev/null +++ b/src/jn.c @@ -0,0 +1,329 @@ +/* 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, 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" + +/* 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, mpfr_rnd_t); + +int +mpfr_j0 (mpfr_ptr res, mpfr_srcptr z, mpfr_rnd_t r) +{ + return mpfr_jn (res, 0, z, r); +} + +int +mpfr_j1 (mpfr_ptr res, mpfr_srcptr z, mpfr_rnd_t r) +{ + return mpfr_jn (res, 1, z, r); +} + +/* Estimate k1 such that z^2/4 = k1 * (k1 + n) + i.e., k1 = (sqrt(n^2+z^2)-n)/2 = n/2 * (sqrt(1+(z/n)^2) - 1) if n != 0. + Return k0 = min(2*k1/log(2), ULONG_MAX). +*/ +static unsigned long +mpfr_jn_k0 (unsigned long n, mpfr_srcptr z) +{ + mpfr_t t, u; + unsigned long k0; + + mpfr_init2 (t, 32); + mpfr_init2 (u, 32); + if (n == 0) + { + mpfr_abs (t, z, MPFR_RNDN); /* t = 2*k1 */ + } + else + { + mpfr_div_ui (t, z, n, MPFR_RNDN); + mpfr_sqr (t, t, MPFR_RNDN); + mpfr_add_ui (t, t, 1, MPFR_RNDN); + mpfr_sqrt (t, t, MPFR_RNDN); + mpfr_sub_ui (t, t, 1, MPFR_RNDN); + mpfr_mul_ui (t, t, n, MPFR_RNDN); /* t = 2*k1 */ + } + /* the following is a 32-bit approximation to nearest to 1/log(2) */ + mpfr_set_str_binary (u, "1.0111000101010100011101100101001"); + mpfr_mul (t, t, u, MPFR_RNDN); + if (mpfr_fits_ulong_p (t, MPFR_RNDN)) + k0 = mpfr_get_ui (t, MPFR_RNDN); + else + k0 = ULONG_MAX; + mpfr_clear (t); + mpfr_clear (u); + return k0; +} + +int +mpfr_jn (mpfr_ptr res, long n, mpfr_srcptr z, mpfr_rnd_t r) +{ + int inex; + int exception = 0; + unsigned long absn; + mpfr_prec_t prec, pbound, err; + mpfr_uprec_t uprec; + mpfr_exp_t exps, expT, diffexp; + mpfr_t y, s, t, absz; + unsigned long k, zz, k0; + MPFR_GROUP_DECL(g); + MPFR_SAVE_EXPO_DECL (expo); + MPFR_ZIV_DECL (loop); + + MPFR_LOG_FUNC + (("n=%d x[%Pu]=%.*Rg rnd=%d", n, mpfr_get_prec (z), mpfr_log_prec, z, r), + ("res[%Pu]=%.*Rg inexact=%d", + mpfr_get_prec (res), mpfr_log_prec, res, inex)); + + 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); + } + } + + MPFR_SAVE_EXPO_MARK (expo); + + /* 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, inex = _inexact; goto end); + + /* 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. */ + /* TODO: add a test to trigger an error when + inex = _inexact; goto end + is forgotten in MPFR_FAST_COMPUTE_IF_SMALL_INPUT below. */ + 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. But we + must also handle the underflow case (an overflow is not possible + for small inputs). If an underflow occurred in mpfr_round_near_x, + the rounding was to zero or equivalent, and the result is 0, so + that the division by 2 will give the wanted result. Otherwise... + The rounded result in unbounded exponent range is res/2. If the + division by 2 doesn't underflow, it is exact, and we can return + this result. And an underflow in the division is a real underflow. + In case of directed rounding mode, the result is correct. But in + case of rounding to nearest, there is a double rounding problem, + and the result is 0 iff the result before the division is the + minimum positive number and _inexact has the same sign as z; + but in rounding to nearest, res/2 will yield 0 iff |res| is the + minimum positive number, so that we just need to test the result + of the division and the sign of _inexact. */ + mpfr_clear_flags (); + MPFR_FAST_COMPUTE_IF_SMALL_INPUT + (res, z, -2 * MPFR_GET_EXP (z), 3, 0, r, { + int inex2 = mpfr_div_2ui (res, res, 1, r); + if (MPFR_UNLIKELY (r == MPFR_RNDN && MPFR_IS_ZERO (res)) && + (MPFR_ASSERTN (inex2 != 0), SIGN (_inexact) != MPFR_SIGN (z))) + { + mpfr_nexttoinf (res); + inex = - inex2; + } + else + inex = inex2 != 0 ? inex2 : _inexact; + MPFR_SAVE_EXPO_UPDATE_FLAGS (expo, __gmpfr_flags); + goto end; + }); + } + + /* 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) + goto end; + } + + MPFR_GROUP_INIT_3 (g, 32, y, s, t); + + /* check underflow case: |j(n,z)| <= 1/sqrt(2 Pi n) (ze/2n)^n + (see algorithms.tex) */ + /* FIXME: the code below doesn't detect all the underflow cases. Either + this should be done, or the generic code should detect underflows. */ + if (absn > 0) + { + /* the following is an upper 32-bit approximation to exp(1)/2 */ + mpfr_set_str_binary (y, "1.0101101111110000101010001011001"); + if (MPFR_SIGN(z) > 0) + mpfr_mul (y, y, z, MPFR_RNDU); + else + { + mpfr_mul (y, y, z, MPFR_RNDD); + mpfr_neg (y, y, MPFR_RNDU); + } + mpfr_div_ui (y, y, absn, MPFR_RNDU); + /* now y is an upper approximation to |ze/2n|: y < 2^EXP(y), + thus |j(n,z)| < 1/2*y^n < 2^(n*EXP(y)-1). + If n*EXP(y) < emin then we have an underflow. + Note that if emin = MPFR_EMIN_MIN and j = 1, this inequality + will never be satisfied. + Warning: absn is an unsigned long. */ + if ((MPFR_GET_EXP (y) < 0 && absn > - expo.saved_emin) + || (absn <= - MPFR_EMIN_MIN && + MPFR_GET_EXP (y) < expo.saved_emin / (mpfr_exp_t) absn)) + { + MPFR_GROUP_CLEAR (g); + MPFR_SAVE_EXPO_FREE (expo); + return mpfr_underflow (res, (r == MPFR_RNDN) ? MPFR_RNDZ : r, + (n % 2) ? ((n > 0) ? MPFR_SIGN(z) : -MPFR_SIGN(z)) + : MPFR_SIGN_POS); + } + } + + /* 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 */ + /* The following operations avoid integer overflow and ensure that + prec <= MPFR_PREC_MAX (prec = MPFR_PREC_MAX won't prevent an abort, + but the failure should be handled cleanly). */ + k0 = mpfr_jn_k0 (absn, z); + MPFR_LOG_MSG (("k0 = %lu\n", k0)); + uprec = MPFR_PREC_MAX - 2 * MPFR_INT_CEIL_LOG2 (MPFR_PREC_MAX) - 3; + if (k0 < uprec) + uprec = k0; + uprec += MPFR_PREC (res) + 2 * MPFR_INT_CEIL_LOG2 (MPFR_PREC (res)) + 3; + prec = uprec < MPFR_PREC_MAX ? (mpfr_prec_t) uprec : MPFR_PREC_MAX; + + MPFR_ZIV_INIT (loop, prec); + for (;;) + { + MPFR_BLOCK_DECL (flags); + + MPFR_GROUP_REPREC_3 (g, prec, y, s, t); + MPFR_BLOCK (flags, { + mpfr_pow_ui (t, z, absn, MPFR_RNDN); /* z^|n| */ + mpfr_mul (y, z, z, MPFR_RNDN); /* z^2 */ + mpfr_clear_erangeflag (); + zz = mpfr_get_ui (y, MPFR_RNDU); + /* FIXME: The error analysis is incorrect in case of range error. */ + MPFR_ASSERTN (! mpfr_erangeflag_p ()); /* since mpfr_clear_erangeflag */ + mpfr_div_2ui (y, y, 2, MPFR_RNDN); /* z^2/4 */ + mpfr_fac_ui (s, absn, MPFR_RNDN); /* |n|! */ + mpfr_div (t, t, s, MPFR_RNDN); + if (absn > 0) + mpfr_div_2ui (t, t, absn, MPFR_RNDN); + mpfr_set (s, t, MPFR_RNDN); + /* note: we assume here that the maximal error bound is proportional to + 2^exps, which is true also in the case where s=0 */ + exps = MPFR_IS_ZERO (s) ? MPFR_EMIN_MIN : MPFR_GET_EXP (s); + expT = exps; + for (k = 1; ; k++) + { + MPFR_LOG_MSG (("loop on k, k = %lu\n", k)); + mpfr_mul (t, t, y, MPFR_RNDN); + mpfr_neg (t, t, MPFR_RNDN); + /* Mathematically: absn <= LONG_MAX + 1 <= (ULONG_MAX + 1) / 2, + and in practice, k is not very large, so that one should have + k + absn <= ULONG_MAX. */ + MPFR_ASSERTN (absn <= ULONG_MAX - k); + if (k + absn <= ULONG_MAX / k) + mpfr_div_ui (t, t, k * (k + absn), MPFR_RNDN); + else + { + mpfr_div_ui (t, t, k, MPFR_RNDN); + mpfr_div_ui (t, t, k + absn, MPFR_RNDN); + } + /* see above note */ + exps = MPFR_IS_ZERO (s) ? MPFR_EMIN_MIN : MPFR_GET_EXP (t); + if (exps > expT) + expT = exps; + mpfr_add (s, s, t, MPFR_RNDN); + exps = MPFR_IS_ZERO (s) ? MPFR_EMIN_MIN : MPFR_GET_EXP (s); + if (exps > expT) + expT = exps; + /* Above it has been checked that k + absn <= ULONG_MAX. */ + if (MPFR_GET_EXP (t) + (mpfr_exp_t) prec <= exps && + zz / (2 * k) < k + absn) + 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) */ + diffexp = expT - exps; + err = 2 * MPFR_INT_CEIL_LOG2(k + 2) + 2; + /* FIXME: Can an overflow occur in the following sum? */ + MPFR_ASSERTN (diffexp >= 0 && err >= 0 && + diffexp <= MPFR_PREC_MAX - err); + err += diffexp; + if (MPFR_LIKELY (MPFR_CAN_ROUND (s, prec - err, MPFR_PREC(res), r))) + { + if (MPFR_LIKELY (! (MPFR_UNDERFLOW (flags) || + MPFR_OVERFLOW (flags)))) + break; + /* The error analysis is incorrect in case of exception. + If an underflow or overflow occurred, try once more in + a larger precision, and if this happens a second time, + then abort to avoid a probable infinite loop. This is + a problem that must be fixed! */ + MPFR_ASSERTN (! exception); + exception = 1; + } + 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_GROUP_CLEAR (g); + + end: + MPFR_SAVE_EXPO_FREE (expo); + return mpfr_check_range (res, inex, r); +} + +#define MPFR_JN +#include "jyn_asympt.c" |