jyn_asympt.c revision 1.1.1.1
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