jyn_asympt.c revision 1.1.1.3 
1/* mpfr_jn_asympt, mpfr_yn_asympt -- shared code for mpfr_jn and mpfr_yn
2
3Copyright 2007-2018 Free Software Foundation, Inc.
4Contributed by the AriC and Caramba projects, INRIA.
5
6This file is part of the GNU MPFR Library.
7
8The GNU MPFR Library is free software; you can redistribute it and/or modify
9it under the terms of the GNU Lesser General Public License as published by
10the Free Software Foundation; either version 3 of the License, or (at your
11option) any later version.
12
13The GNU MPFR Library is distributed in the hope that it will be useful, but
14WITHOUT ANY WARRANTY; without even the implied warranty of MERCHANTABILITY
15or FITNESS FOR A PARTICULAR PURPOSE.  See the GNU Lesser General Public
16License for more details.
17
18You should have received a copy of the GNU Lesser General Public License
19along with the GNU MPFR Library; see the file COPYING.LESSER.  If not, see
20http://www.gnu.org/licenses/ or write to the Free Software Foundation, Inc.,
2151 Franklin St, Fifth Floor, Boston, MA 02110-1301, USA. */
22
23#ifdef MPFR_JN
24# define FUNCTION mpfr_jn_asympt
25#else
26# ifdef MPFR_YN
27#  define FUNCTION mpfr_yn_asympt
28# else
29#  error "neither MPFR_JN nor MPFR_YN is defined"
30# endif
31#endif
32
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).
39   Note: for MPFR_RNDF, it returns 0 if the expansion failed, and a non-zero
40   value otherwise (with no other meaning).
41*/
42static int
43FUNCTION (mpfr_ptr res, long n, mpfr_srcptr z, mpfr_rnd_t r)
44{
45  mpfr_t s, c, P, Q, t, iz, err_t, err_s, err_u;
46  mpfr_prec_t w;
47  long k;
48  int inex, stop, diverge = 0;
49  mpfr_exp_t err2, err;
50  MPFR_ZIV_DECL (loop);
51
52  mpfr_init (c);
53
54  w = MPFR_PREC(res) + MPFR_INT_CEIL_LOG2(MPFR_PREC(res)) + 4;
55
56  MPFR_ZIV_INIT (loop, w);
57  for (;;)
58    {
59      mpfr_set_prec (c, w);
60      mpfr_init2 (s, w);
61      mpfr_init2 (P, w);
62      mpfr_init2 (Q, w);
63      mpfr_init2 (t, w);
64      mpfr_init2 (iz, w);
65      mpfr_init2 (err_t, 31);
66      mpfr_init2 (err_s, 31);
67      mpfr_init2 (err_u, 31);
68
69      /* Approximate sin(z) and cos(z). In the following, err <= k means that
70         the approximate value y and the true value x are related by
71         y = x * (1 + u)^k with |u| <= 2^(-w), following Higham's method. */
72      mpfr_sin_cos (s, c, z, MPFR_RNDN);
73      if (MPFR_IS_NEG(z))
74        mpfr_neg (s, s, MPFR_RNDN); /* compute jn/yn(|z|), fix sign later */
75      /* The absolute error on s/c is bounded by 1/2 ulp(1/2) <= 2^(-w-1). */
76      mpfr_add (t, s, c, MPFR_RNDN);
77      mpfr_sub (c, s, c, MPFR_RNDN);
78      mpfr_swap (s, t);
79      /* now s approximates sin(z)+cos(z), and c approximates sin(z)-cos(z),
80         with total absolute error bounded by 2^(1-w). */
81
82      /* precompute 1/(8|z|) */
83      mpfr_si_div (iz, MPFR_IS_POS(z) ? 1 : -1, z, MPFR_RNDN);   /* err <= 1 */
84      mpfr_div_2ui (iz, iz, 3, MPFR_RNDN);
85
86      /* compute P and Q */
87      mpfr_set_ui (P, 1, MPFR_RNDN);
88      mpfr_set_ui (Q, 0, MPFR_RNDN);
89      mpfr_set_ui (t, 1, MPFR_RNDN); /* current term */
90      mpfr_set_ui (err_t, 0, MPFR_RNDN); /* error on t */
91      mpfr_set_ui (err_s, 0, MPFR_RNDN); /* error on P and Q (sum of errors) */
92      for (k = 1, stop = 0; stop < 4; k++)
93        {
94          /* compute next term: t(k)/t(k-1) = (2n+2k-1)(2n-2k+1)/(8kz) */
95          mpfr_mul_si (t, t, 2 * (n + k) - 1, MPFR_RNDN); /* err <= err_k + 1 */
96          mpfr_mul_si (t, t, 2 * (n - k) + 1, MPFR_RNDN); /* err <= err_k + 2 */
97          mpfr_div_ui (t, t, k, MPFR_RNDN);               /* err <= err_k + 3 */
98          mpfr_mul (t, t, iz, MPFR_RNDN);                 /* err <= err_k + 5 */
99          /* the relative error on t is bounded by (1+u)^(5k)-1, which is
100             bounded by 6ku for 6ku <= 0.02: first |5 log(1+u)| <= |5.5u|
101             for |u| <= 0.15, then |exp(5.5u)-1| <= 6u for |u| <= 0.02. */
102          mpfr_mul_ui (err_t, t, 6 * k, MPFR_IS_POS(t) ? MPFR_RNDU : MPFR_RNDD);
103          mpfr_abs (err_t, err_t, MPFR_RNDN); /* exact */
104          /* the absolute error on t is bounded by err_t * 2^(-w) */
105          mpfr_abs (err_u, t, MPFR_RNDU);
106          mpfr_mul_2ui (err_u, err_u, w, MPFR_RNDU); /* t * 2^w */
107          mpfr_add (err_u, err_u, err_t, MPFR_RNDU); /* max|t| * 2^w */
108          if (stop >= 2)
109            {
110              /* take into account the neglected terms: t * 2^w */
111              mpfr_div_2ui (err_s, err_s, w, MPFR_RNDU);
112              if (MPFR_IS_POS(t))
113                mpfr_add (err_s, err_s, t, MPFR_RNDU);
114              else
115                mpfr_sub (err_s, err_s, t, MPFR_RNDU);
116              mpfr_mul_2ui (err_s, err_s, w, MPFR_RNDU);
117              stop ++;
118            }
119          /* if k is odd, add to Q, otherwise to P */
120          else if (k & 1)
121            {
122              /* if k = 1 mod 4, add, otherwise subtract */
123              if ((k & 2) == 0)
124                mpfr_add (Q, Q, t, MPFR_RNDN);
125              else
126                mpfr_sub (Q, Q, t, MPFR_RNDN);
127              /* check if the next term is smaller than ulp(Q): if EXP(err_u)
128                 <= EXP(Q), since the current term is bounded by
129                 err_u * 2^(-w), it is bounded by ulp(Q) */
130              if (MPFR_EXP(err_u) <= MPFR_EXP(Q))
131                stop ++;
132              else
133                stop = 0;
134            }
135          else
136            {
137              /* if k = 0 mod 4, add, otherwise subtract */
138              if ((k & 2) == 0)
139                mpfr_add (P, P, t, MPFR_RNDN);
140              else
141                mpfr_sub (P, P, t, MPFR_RNDN);
142              /* check if the next term is smaller than ulp(P) */
143              if (MPFR_EXP(err_u) <= MPFR_EXP(P))
144                stop ++;
145              else
146                stop = 0;
147            }
148          mpfr_add (err_s, err_s, err_t, MPFR_RNDU);
149          /* the sum of the rounding errors on P and Q is bounded by
150             err_s * 2^(-w) */
151
152          /* stop when start to diverge */
153          if (stop < 2 &&
154              ((MPFR_IS_POS(z) && mpfr_cmp_ui (z, (k + 1) / 2) < 0) ||
155               (MPFR_IS_NEG(z) && mpfr_cmp_si (z, - ((k + 1) / 2)) > 0)))
156            {
157              /* if we have to stop the series because it diverges, then
158                 increasing the precision will most probably fail, since
159                 we will stop to the same point, and thus compute a very
160                 similar approximation */
161              diverge = 1;
162              stop = 2; /* force stop */
163            }
164        }
165      /* the sum of the total errors on P and Q is bounded by err_s * 2^(-w) */
166
167      /* Now combine: the sum of the rounding errors on P and Q is bounded by
168         err_s * 2^(-w), and the absolute error on s/c is bounded by 2^(1-w) */
169      if ((n & 1) == 0) /* n even: P * (sin + cos) + Q (cos - sin) for jn
170                                   Q * (sin + cos) + P (sin - cos) for yn */
171        {
172#ifdef MPFR_JN
173          mpfr_mul (c, c, Q, MPFR_RNDN); /* Q * (sin - cos) */
174          mpfr_mul (s, s, P, MPFR_RNDN); /* P * (sin + cos) */
175#else
176          mpfr_mul (c, c, P, MPFR_RNDN); /* P * (sin - cos) */
177          mpfr_mul (s, s, Q, MPFR_RNDN); /* Q * (sin + cos) */
178#endif
179          err = MPFR_EXP(c);
180          if (MPFR_EXP(s) > err)
181            err = MPFR_EXP(s);
182#ifdef MPFR_JN
183          mpfr_sub (s, s, c, MPFR_RNDN);
184#else
185          mpfr_add (s, s, c, MPFR_RNDN);
186#endif
187        }
188      else /* n odd: P * (sin - cos) + Q (cos + sin) for jn,
189                     Q * (sin - cos) - P (cos + sin) for yn */
190        {
191#ifdef MPFR_JN
192          mpfr_mul (c, c, P, MPFR_RNDN); /* P * (sin - cos) */
193          mpfr_mul (s, s, Q, MPFR_RNDN); /* Q * (sin + cos) */
194#else
195          mpfr_mul (c, c, Q, MPFR_RNDN); /* Q * (sin - cos) */
196          mpfr_mul (s, s, P, MPFR_RNDN); /* P * (sin + cos) */
197#endif
198          err = MPFR_EXP(c);
199          if (MPFR_EXP(s) > err)
200            err = MPFR_EXP(s);
201#ifdef MPFR_JN
202          mpfr_add (s, s, c, MPFR_RNDN);
203#else
204          mpfr_sub (s, c, s, MPFR_RNDN);
205#endif
206        }
207      if ((n & 2) != 0)
208        mpfr_neg (s, s, MPFR_RNDN);
209      if (MPFR_EXP(s) > err)
210        err = MPFR_EXP(s);
211      /* the absolute error on s is bounded by P*err(s/c) + Q*err(s/c)
212         + err(P)*(s/c) + err(Q)*(s/c) + 3 * 2^(err - w - 1)
213         <= (|P|+|Q|) * 2^(1-w) + err_s * 2^(1-w) + 2^err * 2^(1-w),
214         since |c|, |old_s| <= 2. */
215      err2 = (MPFR_EXP(P) >= MPFR_EXP(Q)) ? MPFR_EXP(P) + 2 : MPFR_EXP(Q) + 2;
216      /* (|P| + |Q|) * 2^(1 - w) <= 2^(err2 - w) */
217      err = MPFR_EXP(err_s) >= err ? MPFR_EXP(err_s) + 2 : err + 2;
218      /* err_s * 2^(1-w) + 2^old_err * 2^(1-w) <= 2^err * 2^(-w) */
219      err2 = (err >= err2) ? err + 1 : err2 + 1;
220      /* now the absolute error on s is bounded by 2^(err2 - w) */
221
222      /* multiply by sqrt(1/(Pi*z)) */
223      mpfr_const_pi (c, MPFR_RNDN);     /* Pi, err <= 1 */
224      mpfr_mul (c, c, z, MPFR_RNDN);    /* err <= 2 */
225      mpfr_si_div (c, MPFR_IS_POS(z) ? 1 : -1, c, MPFR_RNDN); /* err <= 3 */
226      mpfr_sqrt (c, c, MPFR_RNDN);      /* err<=5/2, thus the absolute error is
227                                          bounded by 3*u*|c| for |u| <= 0.25 */
228      mpfr_mul (err_t, c, s, MPFR_SIGN(c)==MPFR_SIGN(s) ? MPFR_RNDU : MPFR_RNDD);
229      mpfr_abs (err_t, err_t, MPFR_RNDU);
230      mpfr_mul_ui (err_t, err_t, 3, MPFR_RNDU);
231      /* 3*2^(-w)*|old_c|*|s| [see below] is bounded by err_t * 2^(-w) */
232      err2 += MPFR_EXP(c);
233      /* |old_c| * 2^(err2 - w) [see below] is bounded by 2^(err2-w) */
234      mpfr_mul (c, c, s, MPFR_RNDN);    /* the absolute error on c is bounded by
235                                          1/2 ulp(c) + 3*2^(-w)*|old_c|*|s|
236                                          + |old_c| * 2^(err2 - w) */
237      /* compute err_t * 2^(-w) + 1/2 ulp(c) = (err_t + 2^EXP(c)) * 2^(-w) */
238      err = (MPFR_EXP(err_t) > MPFR_EXP(c)) ? MPFR_EXP(err_t) + 1 : MPFR_EXP(c) + 1;
239      /* err_t * 2^(-w) + 1/2 ulp(c) <= 2^(err - w) */
240      /* now err_t * 2^(-w) bounds 1/2 ulp(c) + 3*2^(-w)*|old_c|*|s| */
241      err = (err >= err2) ? err + 1 : err2 + 1;
242      /* the absolute error on c is bounded by 2^(err - w) */
243
244      mpfr_clear (s);
245      mpfr_clear (P);
246      mpfr_clear (Q);
247      mpfr_clear (t);
248      mpfr_clear (iz);
249      mpfr_clear (err_t);
250      mpfr_clear (err_s);
251      mpfr_clear (err_u);
252
253      err -= MPFR_EXP(c);
254      if (MPFR_LIKELY (MPFR_CAN_ROUND (c, w - err, MPFR_PREC(res), r)))
255        break;
256      if (diverge != 0)
257        {
258          MPFR_ZIV_FREE (loop);
259          mpfr_clear (c);
260          return 0; /* means that the asymptotic expansion failed */
261        }
262      MPFR_ZIV_NEXT (loop, w);
263    }
264  MPFR_ZIV_FREE (loop);
265
266  inex = (MPFR_IS_POS(z) || ((n & 1) == 0)) ? mpfr_set (res, c, r)
267    : mpfr_neg (res, c, r);
268  mpfr_clear (c);
269
270  /* for RNDF, mpfr_set or mpfr_neg may return 0, but if we return 0, it
271     would mean the asymptotic expansion failed, thus we return 1 instead */
272  return (r != MPFR_RNDF) ? inex : 1;
273}
274