1 /* mpfr_jn_asympt, mpfr_yn_asympt -- shared code for mpfr_jn and mpfr_yn
3 Copyright 2007, 2008, 2009, 2010, 2011, 2012, 2013 Free Software Foundation, Inc.
4 Contributed by the AriC 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
);