1 /* mpfr_jn_asympt, mpfr_yn_asympt -- shared code for mpfr_jn and mpfr_yn
3 Copyright 2007, 2008, 2009, 2010, 2011 Free Software Foundation, Inc.
4 Contributed by the Arenaire and Caramel projects, INRIA.
6 This file is part of the GNU MPFR Library.
8 The GNU MPFR Library is free software; you can redistribute it and/or modify
9 it under the terms of the GNU Lesser General Public License as published by
10 the Free Software Foundation; either version 3 of the License, or (at your
11 option) any later version.
13 The GNU MPFR Library is distributed in the hope that it will be useful, but
14 WITHOUT ANY WARRANTY; without even the implied warranty of MERCHANTABILITY
15 or FITNESS FOR A PARTICULAR PURPOSE. See the GNU Lesser General Public
16 License for more details.
18 You should have received a copy of the GNU Lesser General Public License
19 along with the GNU MPFR Library; see the file COPYING.LESSER. If not, see
20 http://www.gnu.org/licenses/ or write to the Free Software Foundation, Inc.,
21 51 Franklin St, Fifth Floor, Boston, MA 02110-1301, USA. */
24 # define FUNCTION mpfr_jn_asympt
27 # define FUNCTION mpfr_yn_asympt
29 # error "neither MPFR_JN nor MPFR_YN is defined"
33 /* Implements asymptotic expansion for jn or yn (formulae 9.2.5 and 9.2.6
34 from Abramowitz & Stegun).
35 Assumes |z| > p log(2)/2, where p is the target precision
36 (z can be negative only for jn).
37 Return 0 if the expansion does not converge enough (the value 0 as inexact
38 flag should not happen for normal input).
41 FUNCTION (mpfr_ptr res, long n, mpfr_srcptr z, mpfr_rnd_t r)
43 mpfr_t s, c, P, Q, t, iz, err_t, err_s, err_u;
46 int inex, stop, diverge = 0;
52 w = MPFR_PREC(res) + MPFR_INT_CEIL_LOG2(MPFR_PREC(res)) + 4;
54 MPFR_ZIV_INIT (loop, w);
63 mpfr_init2 (err_t, 31);
64 mpfr_init2 (err_s, 31);
65 mpfr_init2 (err_u, 31);
67 /* Approximate sin(z) and cos(z). In the following, err <= k means that
68 the approximate value y and the true value x are related by
69 y = x * (1 + u)^k with |u| <= 2^(-w), following Higham's method. */
70 mpfr_sin_cos (s, c, z, MPFR_RNDN);
72 mpfr_neg (s, s, MPFR_RNDN); /* compute jn/yn(|z|), fix sign later */
73 /* The absolute error on s/c is bounded by 1/2 ulp(1/2) <= 2^(-w-1). */
74 mpfr_add (t, s, c, MPFR_RNDN);
75 mpfr_sub (c, s, c, MPFR_RNDN);
77 /* now s approximates sin(z)+cos(z), and c approximates sin(z)-cos(z),
78 with total absolute error bounded by 2^(1-w). */
80 /* precompute 1/(8|z|) */
81 mpfr_si_div (iz, MPFR_IS_POS(z) ? 1 : -1, z, MPFR_RNDN); /* err <= 1 */
82 mpfr_div_2ui (iz, iz, 3, MPFR_RNDN);
85 mpfr_set_ui (P, 1, MPFR_RNDN);
86 mpfr_set_ui (Q, 0, MPFR_RNDN);
87 mpfr_set_ui (t, 1, MPFR_RNDN); /* current term */
88 mpfr_set_ui (err_t, 0, MPFR_RNDN); /* error on t */
89 mpfr_set_ui (err_s, 0, MPFR_RNDN); /* error on P and Q (sum of errors) */
90 for (k = 1, stop = 0; stop < 4; k++)
92 /* compute next term: t(k)/t(k-1) = (2n+2k-1)(2n-2k+1)/(8kz) */
93 mpfr_mul_si (t, t, 2 * (n + k) - 1, MPFR_RNDN); /* err <= err_k + 1 */
94 mpfr_mul_si (t, t, 2 * (n - k) + 1, MPFR_RNDN); /* err <= err_k + 2 */
95 mpfr_div_ui (t, t, k, MPFR_RNDN); /* err <= err_k + 3 */
96 mpfr_mul (t, t, iz, MPFR_RNDN); /* err <= err_k + 5 */
97 /* the relative error on t is bounded by (1+u)^(5k)-1, which is
98 bounded by 6ku for 6ku <= 0.02: first |5 log(1+u)| <= |5.5u|
99 for |u| <= 0.15, then |exp(5.5u)-1| <= 6u for |u| <= 0.02. */
100 mpfr_mul_ui (err_t, t, 6 * k, MPFR_IS_POS(t) ? MPFR_RNDU : MPFR_RNDD);
101 mpfr_abs (err_t, err_t, MPFR_RNDN); /* exact */
102 /* the absolute error on t is bounded by err_t * 2^(-w) */
103 mpfr_abs (err_u, t, MPFR_RNDU);
104 mpfr_mul_2ui (err_u, err_u, w, MPFR_RNDU); /* t * 2^w */
105 mpfr_add (err_u, err_u, err_t, MPFR_RNDU); /* max|t| * 2^w */
108 /* take into account the neglected terms: t * 2^w */
109 mpfr_div_2ui (err_s, err_s, w, MPFR_RNDU);
111 mpfr_add (err_s, err_s, t, MPFR_RNDU);
113 mpfr_sub (err_s, err_s, t, MPFR_RNDU);
114 mpfr_mul_2ui (err_s, err_s, w, MPFR_RNDU);
117 /* if k is odd, add to Q, otherwise to P */
120 /* if k = 1 mod 4, add, otherwise subtract */
122 mpfr_add (Q, Q, t, MPFR_RNDN);
124 mpfr_sub (Q, Q, t, MPFR_RNDN);
125 /* check if the next term is smaller than ulp(Q): if EXP(err_u)
126 <= EXP(Q), since the current term is bounded by
127 err_u * 2^(-w), it is bounded by ulp(Q) */
128 if (MPFR_EXP(err_u) <= MPFR_EXP(Q))
135 /* if k = 0 mod 4, add, otherwise subtract */
137 mpfr_add (P, P, t, MPFR_RNDN);
139 mpfr_sub (P, P, t, MPFR_RNDN);
140 /* check if the next term is smaller than ulp(P) */
141 if (MPFR_EXP(err_u) <= MPFR_EXP(P))
146 mpfr_add (err_s, err_s, err_t, MPFR_RNDU);
147 /* the sum of the rounding errors on P and Q is bounded by
150 /* stop when start to diverge */
152 ((MPFR_IS_POS(z) && mpfr_cmp_ui (z, (k + 1) / 2) < 0) ||
153 (MPFR_IS_NEG(z) && mpfr_cmp_si (z, - ((k + 1) / 2)) > 0)))
155 /* if we have to stop the series because it diverges, then
156 increasing the precision will most probably fail, since
157 we will stop to the same point, and thus compute a very
158 similar approximation */
160 stop = 2; /* force stop */
163 /* the sum of the total errors on P and Q is bounded by err_s * 2^(-w) */
165 /* Now combine: the sum of the rounding errors on P and Q is bounded by
166 err_s * 2^(-w), and the absolute error on s/c is bounded by 2^(1-w) */
167 if ((n & 1) == 0) /* n even: P * (sin + cos) + Q (cos - sin) for jn
168 Q * (sin + cos) + P (sin - cos) for yn */
171 mpfr_mul (c, c, Q, MPFR_RNDN); /* Q * (sin - cos) */
172 mpfr_mul (s, s, P, MPFR_RNDN); /* P * (sin + cos) */
174 mpfr_mul (c, c, P, MPFR_RNDN); /* P * (sin - cos) */
175 mpfr_mul (s, s, Q, MPFR_RNDN); /* Q * (sin + cos) */
178 if (MPFR_EXP(s) > err)
181 mpfr_sub (s, s, c, MPFR_RNDN);
183 mpfr_add (s, s, c, MPFR_RNDN);
186 else /* n odd: P * (sin - cos) + Q (cos + sin) for jn,
187 Q * (sin - cos) - P (cos + sin) for yn */
190 mpfr_mul (c, c, P, MPFR_RNDN); /* P * (sin - cos) */
191 mpfr_mul (s, s, Q, MPFR_RNDN); /* Q * (sin + cos) */
193 mpfr_mul (c, c, Q, MPFR_RNDN); /* Q * (sin - cos) */
194 mpfr_mul (s, s, P, MPFR_RNDN); /* P * (sin + cos) */
197 if (MPFR_EXP(s) > err)
200 mpfr_add (s, s, c, MPFR_RNDN);
202 mpfr_sub (s, c, s, MPFR_RNDN);
206 mpfr_neg (s, s, MPFR_RNDN);
207 if (MPFR_EXP(s) > err)
209 /* the absolute error on s is bounded by P*err(s/c) + Q*err(s/c)
210 + err(P)*(s/c) + err(Q)*(s/c) + 3 * 2^(err - w - 1)
211 <= (|P|+|Q|) * 2^(1-w) + err_s * 2^(1-w) + 2^err * 2^(1-w),
212 since |c|, |old_s| <= 2. */
213 err2 = (MPFR_EXP(P) >= MPFR_EXP(Q)) ? MPFR_EXP(P) + 2 : MPFR_EXP(Q) + 2;
214 /* (|P| + |Q|) * 2^(1 - w) <= 2^(err2 - w) */
215 err = MPFR_EXP(err_s) >= err ? MPFR_EXP(err_s) + 2 : err + 2;
216 /* err_s * 2^(1-w) + 2^old_err * 2^(1-w) <= 2^err * 2^(-w) */
217 err2 = (err >= err2) ? err + 1 : err2 + 1;
218 /* now the absolute error on s is bounded by 2^(err2 - w) */
220 /* multiply by sqrt(1/(Pi*z)) */
221 mpfr_const_pi (c, MPFR_RNDN); /* Pi, err <= 1 */
222 mpfr_mul (c, c, z, MPFR_RNDN); /* err <= 2 */
223 mpfr_si_div (c, MPFR_IS_POS(z) ? 1 : -1, c, MPFR_RNDN); /* err <= 3 */
224 mpfr_sqrt (c, c, MPFR_RNDN); /* err<=5/2, thus the absolute error is
225 bounded by 3*u*|c| for |u| <= 0.25 */
226 mpfr_mul (err_t, c, s, MPFR_SIGN(c)==MPFR_SIGN(s) ? MPFR_RNDU : MPFR_RNDD);
227 mpfr_abs (err_t, err_t, MPFR_RNDU);
228 mpfr_mul_ui (err_t, err_t, 3, MPFR_RNDU);
229 /* 3*2^(-w)*|old_c|*|s| [see below] is bounded by err_t * 2^(-w) */
231 /* |old_c| * 2^(err2 - w) [see below] is bounded by 2^(err2-w) */
232 mpfr_mul (c, c, s, MPFR_RNDN); /* the absolute error on c is bounded by
233 1/2 ulp(c) + 3*2^(-w)*|old_c|*|s|
234 + |old_c| * 2^(err2 - w) */
235 /* compute err_t * 2^(-w) + 1/2 ulp(c) = (err_t + 2^EXP(c)) * 2^(-w) */
236 err = (MPFR_EXP(err_t) > MPFR_EXP(c)) ? MPFR_EXP(err_t) + 1 : MPFR_EXP(c) + 1;
237 /* err_t * 2^(-w) + 1/2 ulp(c) <= 2^(err - w) */
238 /* now err_t * 2^(-w) bounds 1/2 ulp(c) + 3*2^(-w)*|old_c|*|s| */
239 err = (err >= err2) ? err + 1 : err2 + 1;
240 /* the absolute error on c is bounded by 2^(err - w) */
252 if (MPFR_LIKELY (MPFR_CAN_ROUND (c, w - err, MPFR_PREC(res), r)))
256 mpfr_set (c, z, r); /* will force inex=0 below, which means the
257 asymptotic expansion failed */
260 MPFR_ZIV_NEXT (loop, w);
262 MPFR_ZIV_FREE (loop);
264 inex = (MPFR_IS_POS(z) || ((n & 1) == 0)) ? mpfr_set (res, c, r)
265 : mpfr_neg (res, c, r);