1/* mpfr_jn_asympt, mpfr_yn_asympt -- shared code for mpfr_jn and mpfr_yn
2
3Copyright 2007, 2008, 2009, 2010, 2011, 2012, 2013 Free Software Foundation, Inc.
4Contributed by the AriC and Caramel 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*/
40static int
41FUNCTION (mpfr_ptr res, long n, mpfr_srcptr z, mpfr_rnd_t r)
42{
43  mpfr_t s, c, P, Q, t, iz, err_t, err_s, err_u;
44  mpfr_prec_t w;
45  long k;
46  int inex, stop, diverge = 0;
47  mpfr_exp_t err2, err;
48  MPFR_ZIV_DECL (loop);
49
50  mpfr_init (c);
51
52  w = MPFR_PREC(res) + MPFR_INT_CEIL_LOG2(MPFR_PREC(res)) + 4;
53
54  MPFR_ZIV_INIT (loop, w);
55  for (;;)
56    {
57      mpfr_set_prec (c, w);
58      mpfr_init2 (s, w);
59      mpfr_init2 (P, w);
60      mpfr_init2 (Q, w);
61      mpfr_init2 (t, w);
62      mpfr_init2 (iz, w);
63      mpfr_init2 (err_t, 31);
64      mpfr_init2 (err_s, 31);
65      mpfr_init2 (err_u, 31);
66
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);
71      if (MPFR_IS_NEG(z))
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);
76      mpfr_swap (s, t);
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). */
79
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);
83
84      /* compute P and Q */
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++)
91        {
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 */
106          if (stop >= 2)
107            {
108              /* take into account the neglected terms: t * 2^w */
109              mpfr_div_2ui (err_s, err_s, w, MPFR_RNDU);
110              if (MPFR_IS_POS(t))
111                mpfr_add (err_s, err_s, t, MPFR_RNDU);
112              else
113                mpfr_sub (err_s, err_s, t, MPFR_RNDU);
114              mpfr_mul_2ui (err_s, err_s, w, MPFR_RNDU);
115              stop ++;
116            }
117          /* if k is odd, add to Q, otherwise to P */
118          else if (k & 1)
119            {
120              /* if k = 1 mod 4, add, otherwise subtract */
121              if ((k & 2) == 0)
122                mpfr_add (Q, Q, t, MPFR_RNDN);
123              else
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))
129                stop ++;
130              else
131                stop = 0;
132            }
133          else
134            {
135              /* if k = 0 mod 4, add, otherwise subtract */
136              if ((k & 2) == 0)
137                mpfr_add (P, P, t, MPFR_RNDN);
138              else
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))
142                stop ++;
143              else
144                stop = 0;
145            }
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
148             err_s * 2^(-w) */
149
150          /* stop when start to diverge */
151          if (stop < 2 &&
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)))
154            {
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 */
159              diverge = 1;
160              stop = 2; /* force stop */
161            }
162        }
163      /* the sum of the total errors on P and Q is bounded by err_s * 2^(-w) */
164
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 */
169        {
170#ifdef MPFR_JN
171          mpfr_mul (c, c, Q, MPFR_RNDN); /* Q * (sin - cos) */
172          mpfr_mul (s, s, P, MPFR_RNDN); /* P * (sin + cos) */
173#else
174          mpfr_mul (c, c, P, MPFR_RNDN); /* P * (sin - cos) */
175          mpfr_mul (s, s, Q, MPFR_RNDN); /* Q * (sin + cos) */
176#endif
177          err = MPFR_EXP(c);
178          if (MPFR_EXP(s) > err)
179            err = MPFR_EXP(s);
180#ifdef MPFR_JN
181          mpfr_sub (s, s, c, MPFR_RNDN);
182#else
183          mpfr_add (s, s, c, MPFR_RNDN);
184#endif
185        }
186      else /* n odd: P * (sin - cos) + Q (cos + sin) for jn,
187                     Q * (sin - cos) - P (cos + sin) for yn */
188        {
189#ifdef MPFR_JN
190          mpfr_mul (c, c, P, MPFR_RNDN); /* P * (sin - cos) */
191          mpfr_mul (s, s, Q, MPFR_RNDN); /* Q * (sin + cos) */
192#else
193          mpfr_mul (c, c, Q, MPFR_RNDN); /* Q * (sin - cos) */
194          mpfr_mul (s, s, P, MPFR_RNDN); /* P * (sin + cos) */
195#endif
196          err = MPFR_EXP(c);
197          if (MPFR_EXP(s) > err)
198            err = MPFR_EXP(s);
199#ifdef MPFR_JN
200          mpfr_add (s, s, c, MPFR_RNDN);
201#else
202          mpfr_sub (s, c, s, MPFR_RNDN);
203#endif
204        }
205      if ((n & 2) != 0)
206        mpfr_neg (s, s, MPFR_RNDN);
207      if (MPFR_EXP(s) > err)
208        err = MPFR_EXP(s);
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) */
219
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) */
230      err2 += MPFR_EXP(c);
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) */
241
242      mpfr_clear (s);
243      mpfr_clear (P);
244      mpfr_clear (Q);
245      mpfr_clear (t);
246      mpfr_clear (iz);
247      mpfr_clear (err_t);
248      mpfr_clear (err_s);
249      mpfr_clear (err_u);
250
251      err -= MPFR_EXP(c);
252      if (MPFR_LIKELY (MPFR_CAN_ROUND (c, w - err, MPFR_PREC(res), r)))
253        break;
254      if (diverge != 0)
255        {
256          mpfr_set (c, z, r); /* will force inex=0 below, which means the
257                               asymptotic expansion failed */
258          break;
259        }
260      MPFR_ZIV_NEXT (loop, w);
261    }
262  MPFR_ZIV_FREE (loop);
263
264  inex = (MPFR_IS_POS(z) || ((n & 1) == 0)) ? mpfr_set (res, c, r)
265    : mpfr_neg (res, c, r);
266  mpfr_clear (c);
267
268  return inex;
269}
270