jyn_asympt.c revision 1.1.1.4
1/* mpfr_jn_asympt, mpfr_yn_asympt -- shared code for mpfr_jn and mpfr_yn 2 3Copyright 2007-2020 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 20https://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_init2 (c, 64); 53 54 /* The terms of the asymptotic expansion grow like mu^(2k)/(8z)^(2k), where 55 mu = 4n^2, thus we need mu < 8|z| so that it converges, 56 i.e., n^2/2 < |z| */ 57 MPFR_ASSERTD (n >= 0); 58 mpfr_set_ui (c, n, MPFR_RNDU); 59 mpfr_mul_ui (c, c, n, MPFR_RNDU); 60 mpfr_div_2ui (c, c, 1, MPFR_RNDU); 61 if (mpfr_cmpabs (c, z) >= 0) 62 { 63 mpfr_clear (c); 64 return 0; /* asymptotic expansion failed */ 65 } 66 67 w = MPFR_PREC(res) + MPFR_INT_CEIL_LOG2(MPFR_PREC(res)) + 4; 68 69 MPFR_ZIV_INIT (loop, w); 70 for (;;) 71 { 72 mpfr_set_prec (c, w); 73 mpfr_init2 (s, w); 74 mpfr_init2 (P, w); 75 mpfr_init2 (Q, w); 76 mpfr_init2 (t, w); 77 mpfr_init2 (iz, w); 78 mpfr_init2 (err_t, 31); 79 mpfr_init2 (err_s, 31); 80 mpfr_init2 (err_u, 31); 81 82 /* Approximate sin(z) and cos(z). In the following, err <= k means that 83 the approximate value y and the true value x are related by 84 y = x * (1 + u)^k with |u| <= 2^(-w), following Higham's method. */ 85 mpfr_sin_cos (s, c, z, MPFR_RNDN); 86 if (MPFR_IS_NEG(z)) 87 mpfr_neg (s, s, MPFR_RNDN); /* compute jn/yn(|z|), fix sign later */ 88 /* The absolute error on s/c is bounded by 1/2 ulp(1/2) <= 2^(-w-1). */ 89 mpfr_add (t, s, c, MPFR_RNDN); 90 mpfr_sub (c, s, c, MPFR_RNDN); 91 mpfr_swap (s, t); 92 /* now s approximates sin(z)+cos(z), and c approximates sin(z)-cos(z), 93 with total absolute error bounded by 2^(1-w). */ 94 95 /* precompute 1/(8|z|) */ 96 mpfr_si_div (iz, MPFR_IS_POS(z) ? 1 : -1, z, MPFR_RNDN); /* err <= 1 */ 97 mpfr_div_2ui (iz, iz, 3, MPFR_RNDN); 98 99 /* compute P and Q */ 100 mpfr_set_ui (P, 1, MPFR_RNDN); 101 mpfr_set_ui (Q, 0, MPFR_RNDN); 102 mpfr_set_ui (t, 1, MPFR_RNDN); /* current term */ 103 mpfr_set_ui (err_t, 0, MPFR_RNDN); /* error on t */ 104 mpfr_set_ui (err_s, 0, MPFR_RNDN); /* error on P and Q (sum of errors) */ 105 for (k = 1, stop = 0; stop < 4; k++) 106 { 107 /* compute next term: t(k)/t(k-1) = (2n+2k-1)(2n-2k+1)/(8kz) */ 108 MPFR_LOG_MSG (("loop (k,stop) = (%ld,%d)\n", k, stop)); 109 mpfr_mul_si (t, t, 2 * (n + k) - 1, MPFR_RNDN); /* err <= err_k + 1 */ 110 mpfr_mul_si (t, t, 2 * (n - k) + 1, MPFR_RNDN); /* err <= err_k + 2 */ 111 mpfr_div_ui (t, t, k, MPFR_RNDN); /* err <= err_k + 3 */ 112 mpfr_mul (t, t, iz, MPFR_RNDN); /* err <= err_k + 5 */ 113 /* the relative error on t is bounded by (1+u)^(5k)-1, which is 114 bounded by 6ku for 6ku <= 0.02: first |5 log(1+u)| <= |5.5u| 115 for |u| <= 0.15, then |exp(5.5u)-1| <= 6u for |u| <= 0.02. */ 116 mpfr_mul_ui (err_t, t, 6 * k, MPFR_IS_POS(t) ? MPFR_RNDU : MPFR_RNDD); 117 mpfr_abs (err_t, err_t, MPFR_RNDN); /* exact */ 118 /* the absolute error on t is bounded by err_t * 2^(-w) */ 119 mpfr_abs (err_u, t, MPFR_RNDU); 120 mpfr_mul_2ui (err_u, err_u, w, MPFR_RNDU); /* t * 2^w */ 121 mpfr_add (err_u, err_u, err_t, MPFR_RNDU); /* max|t| * 2^w */ 122 if (stop >= 2) 123 { 124 /* take into account the neglected terms: t * 2^w */ 125 mpfr_div_2ui (err_s, err_s, w, MPFR_RNDU); 126 if (MPFR_IS_POS(t)) 127 mpfr_add (err_s, err_s, t, MPFR_RNDU); 128 else 129 mpfr_sub (err_s, err_s, t, MPFR_RNDU); 130 mpfr_mul_2ui (err_s, err_s, w, MPFR_RNDU); 131 stop ++; 132 } 133 /* if k is odd, add to Q, otherwise to P */ 134 else if (k & 1) 135 { 136 /* if k = 1 mod 4, add, otherwise subtract */ 137 if ((k & 2) == 0) 138 mpfr_add (Q, Q, t, MPFR_RNDN); 139 else 140 mpfr_sub (Q, Q, t, MPFR_RNDN); 141 /* check if the next term is smaller than ulp(Q): if EXP(err_u) 142 <= EXP(Q), since the current term is bounded by 143 err_u * 2^(-w), it is bounded by ulp(Q) */ 144 if (MPFR_GET_EXP (err_u) <= MPFR_GET_EXP (Q)) 145 stop ++; 146 else 147 stop = 0; 148 } 149 else 150 { 151 /* if k = 0 mod 4, add, otherwise subtract */ 152 if ((k & 2) == 0) 153 mpfr_add (P, P, t, MPFR_RNDN); 154 else 155 mpfr_sub (P, P, t, MPFR_RNDN); 156 /* check if the next term is smaller than ulp(P) */ 157 if (MPFR_GET_EXP (err_u) <= MPFR_GET_EXP (P)) 158 stop ++; 159 else 160 stop = 0; 161 } 162 mpfr_add (err_s, err_s, err_t, MPFR_RNDU); 163 /* the sum of the rounding errors on P and Q is bounded by 164 err_s * 2^(-w) */ 165 166 /* stop when start to diverge */ 167 if (stop < 2 && 168 ((MPFR_IS_POS(z) && mpfr_cmp_ui (z, (k + 1) / 2) < 0) || 169 (MPFR_IS_NEG(z) && mpfr_cmp_si (z, - ((k + 1) / 2)) > 0))) 170 { 171 /* if we have to stop the series because it diverges, then 172 increasing the precision will most probably fail, since 173 we will stop to the same point, and thus compute a very 174 similar approximation */ 175 diverge = 1; 176 stop = 2; /* force stop */ 177 } 178 } 179 /* the sum of the total errors on P and Q is bounded by err_s * 2^(-w) */ 180 181 /* Now combine: the sum of the rounding errors on P and Q is bounded by 182 err_s * 2^(-w), and the absolute error on s/c is bounded by 2^(1-w) */ 183 if ((n & 1) == 0) /* n even: P * (sin + cos) + Q (cos - sin) for jn 184 Q * (sin + cos) + P (sin - cos) for yn */ 185 { 186#ifdef MPFR_JN 187 mpfr_mul (c, c, Q, MPFR_RNDN); /* Q * (sin - cos) */ 188 mpfr_mul (s, s, P, MPFR_RNDN); /* P * (sin + cos) */ 189#else 190 mpfr_mul (c, c, P, MPFR_RNDN); /* P * (sin - cos) */ 191 mpfr_mul (s, s, Q, MPFR_RNDN); /* Q * (sin + cos) */ 192#endif 193 err = MPFR_GET_EXP (c); 194 if (MPFR_GET_EXP (s) > err) 195 err = MPFR_EXP (s); 196#ifdef MPFR_JN 197 mpfr_sub (s, s, c, MPFR_RNDN); 198#else 199 mpfr_add (s, s, c, MPFR_RNDN); 200#endif 201 } 202 else /* n odd: P * (sin - cos) + Q (cos + sin) for jn, 203 Q * (sin - cos) - P (cos + sin) for yn */ 204 { 205#ifdef MPFR_JN 206 mpfr_mul (c, c, P, MPFR_RNDN); /* P * (sin - cos) */ 207 mpfr_mul (s, s, Q, MPFR_RNDN); /* Q * (sin + cos) */ 208#else 209 mpfr_mul (c, c, Q, MPFR_RNDN); /* Q * (sin - cos) */ 210 mpfr_mul (s, s, P, MPFR_RNDN); /* P * (sin + cos) */ 211#endif 212 err = MPFR_GET_EXP (c); 213 if (MPFR_GET_EXP (s) > err) 214 err = MPFR_EXP (s); 215#ifdef MPFR_JN 216 mpfr_add (s, s, c, MPFR_RNDN); 217#else 218 mpfr_sub (s, c, s, MPFR_RNDN); 219#endif 220 } 221 if ((n & 2) != 0) 222 mpfr_neg (s, s, MPFR_RNDN); 223 if (MPFR_GET_EXP (s) > err) 224 err = MPFR_EXP (s); 225 /* the absolute error on s is bounded by P*err(s/c) + Q*err(s/c) 226 + err(P)*(s/c) + err(Q)*(s/c) + 3 * 2^(err - w - 1) 227 <= (|P|+|Q|) * 2^(1-w) + err_s * 2^(1-w) + 2^err * 2^(1-w), 228 since |c|, |old_s| <= 2. */ 229 err2 = (MPFR_GET_EXP (P) >= MPFR_GET_EXP (Q)) 230 ? MPFR_EXP (P) + 2 : MPFR_EXP (Q) + 2; 231 /* (|P| + |Q|) * 2^(1 - w) <= 2^(err2 - w) */ 232 err = MPFR_GET_EXP (err_s) >= err ? MPFR_EXP (err_s) + 2 : err + 2; 233 /* err_s * 2^(1-w) + 2^old_err * 2^(1-w) <= 2^err * 2^(-w) */ 234 err2 = (err >= err2) ? err + 1 : err2 + 1; 235 /* now the absolute error on s is bounded by 2^(err2 - w) */ 236 237 /* multiply by sqrt(1/(Pi*z)) */ 238 mpfr_const_pi (c, MPFR_RNDN); /* Pi, err <= 1 */ 239 mpfr_mul (c, c, z, MPFR_RNDN); /* err <= 2 */ 240 mpfr_si_div (c, MPFR_IS_POS(z) ? 1 : -1, c, MPFR_RNDN); /* err <= 3 */ 241 mpfr_sqrt (c, c, MPFR_RNDN); /* err<=5/2, thus the absolute error is 242 bounded by 3*u*|c| for |u| <= 0.25 */ 243 mpfr_mul (err_t, c, s, MPFR_SIGN(c)==MPFR_SIGN(s) ? MPFR_RNDU : MPFR_RNDD); 244 mpfr_abs (err_t, err_t, MPFR_RNDU); 245 mpfr_mul_ui (err_t, err_t, 3, MPFR_RNDU); 246 /* 3*2^(-w)*|old_c|*|s| [see below] is bounded by err_t * 2^(-w) */ 247 err2 += MPFR_GET_EXP (c); 248 /* |old_c| * 2^(err2 - w) [see below] is bounded by 2^(err2-w) */ 249 mpfr_mul (c, c, s, MPFR_RNDN); /* the absolute error on c is bounded by 250 1/2 ulp(c) + 3*2^(-w)*|old_c|*|s| 251 + |old_c| * 2^(err2 - w) */ 252 /* compute err_t * 2^(-w) + 1/2 ulp(c) = (err_t + 2^EXP(c)) * 2^(-w) */ 253 err = (MPFR_GET_EXP (err_t) > MPFR_GET_EXP (c)) ? 254 MPFR_EXP (err_t) + 1 : MPFR_EXP (c) + 1; 255 /* err_t * 2^(-w) + 1/2 ulp(c) <= 2^(err - w) */ 256 /* now err_t * 2^(-w) bounds 1/2 ulp(c) + 3*2^(-w)*|old_c|*|s| */ 257 err = (err >= err2) ? err + 1 : err2 + 1; 258 /* the absolute error on c is bounded by 2^(err - w) */ 259 260 mpfr_clear (s); 261 mpfr_clear (P); 262 mpfr_clear (Q); 263 mpfr_clear (t); 264 mpfr_clear (iz); 265 mpfr_clear (err_t); 266 mpfr_clear (err_s); 267 mpfr_clear (err_u); 268 269 err -= MPFR_GET_EXP (c); 270 if (MPFR_LIKELY (MPFR_CAN_ROUND (c, w - err, MPFR_PREC(res), r))) 271 break; 272 if (diverge != 0) 273 { 274 MPFR_ZIV_FREE (loop); 275 mpfr_clear (c); 276 return 0; /* means that the asymptotic expansion failed */ 277 } 278 MPFR_ZIV_NEXT (loop, w); 279 } 280 MPFR_ZIV_FREE (loop); 281 282 inex = (MPFR_IS_POS(z) || ((n & 1) == 0)) ? mpfr_set (res, c, r) 283 : mpfr_neg (res, c, r); 284 mpfr_clear (c); 285 286 /* for RNDF, mpfr_set or mpfr_neg may return 0, but if we return 0, it 287 would mean the asymptotic expansion failed, thus we return 1 instead */ 288 return (r != MPFR_RNDF) ? inex : 1; 289} 290