jyn_asympt.c revision 1.1.1.5
1/* mpfr_jn_asympt, mpfr_yn_asympt -- shared code for mpfr_jn and mpfr_yn 2 3Copyright 2007-2023 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 int ok = 0; 73 74 mpfr_set_prec (c, w); 75 mpfr_init2 (s, w); 76 mpfr_init2 (P, w); 77 mpfr_init2 (Q, w); 78 mpfr_init2 (t, w); 79 mpfr_init2 (iz, w); 80 mpfr_init2 (err_t, 31); 81 mpfr_init2 (err_s, 31); 82 mpfr_init2 (err_u, 31); 83 84 /* Approximate sin(z) and cos(z). In the following, err <= k means that 85 the approximate value y and the true value x are related by 86 y = x * (1 + u)^k with |u| <= 2^(-w), following Higham's method. */ 87 mpfr_sin_cos (s, c, z, MPFR_RNDN); 88 if (MPFR_IS_NEG(z)) 89 mpfr_neg (s, s, MPFR_RNDN); /* compute jn/yn(|z|), fix sign later */ 90 /* The absolute error on s/c is bounded by 1/2 ulp(1/2) <= 2^(-w-1). */ 91 mpfr_add (t, s, c, MPFR_RNDN); 92 mpfr_sub (c, s, c, MPFR_RNDN); 93 mpfr_swap (s, t); 94 /* now s approximates sin(z)+cos(z), and c approximates sin(z)-cos(z), 95 with total absolute error bounded by 2^(1-w). */ 96 97 /* if s or c is zero, MPFR_GET_EXP will fail below */ 98 if (MPFR_IS_ZERO(s) || MPFR_IS_ZERO(c)) 99 goto clear; /* with ok=0 */ 100 101 /* precompute 1/(8|z|) */ 102 mpfr_si_div (iz, MPFR_IS_POS(z) ? 1 : -1, z, MPFR_RNDN); /* err <= 1 */ 103 mpfr_div_2ui (iz, iz, 3, MPFR_RNDN); 104 105 /* compute P and Q */ 106 mpfr_set_ui (P, 1, MPFR_RNDN); 107 mpfr_set_ui (Q, 0, MPFR_RNDN); 108 mpfr_set_ui (t, 1, MPFR_RNDN); /* current term */ 109 mpfr_set_ui (err_t, 0, MPFR_RNDN); /* error on t */ 110 mpfr_set_ui (err_s, 0, MPFR_RNDN); /* error on P and Q (sum of errors) */ 111 for (k = 1, stop = 0; stop < 4; k++) 112 { 113 /* compute next term: t(k)/t(k-1) = (2n+2k-1)(2n-2k+1)/(8kz) */ 114 MPFR_LOG_MSG (("loop (k,stop) = (%ld,%d)\n", k, stop)); 115 mpfr_mul_si (t, t, 2 * (n + k) - 1, MPFR_RNDN); /* err <= err_k + 1 */ 116 mpfr_mul_si (t, t, 2 * (n - k) + 1, MPFR_RNDN); /* err <= err_k + 2 */ 117 mpfr_div_ui (t, t, k, MPFR_RNDN); /* err <= err_k + 3 */ 118 mpfr_mul (t, t, iz, MPFR_RNDN); /* err <= err_k + 5 */ 119 /* the relative error on t is bounded by (1+u)^(5k)-1, which is 120 bounded by 6ku for 6ku <= 0.02: first |5 log(1+u)| <= |5.5u| 121 for |u| <= 0.15, then |exp(5.5u)-1| <= 6u for |u| <= 0.02. */ 122 mpfr_mul_ui (err_t, t, 6 * k, MPFR_IS_POS(t) ? MPFR_RNDU : MPFR_RNDD); 123 mpfr_abs (err_t, err_t, MPFR_RNDN); /* exact */ 124 /* the absolute error on t is bounded by err_t * 2^(-w) */ 125 mpfr_abs (err_u, t, MPFR_RNDU); 126 mpfr_mul_2ui (err_u, err_u, w, MPFR_RNDU); /* t * 2^w */ 127 mpfr_add (err_u, err_u, err_t, MPFR_RNDU); /* max|t| * 2^w */ 128 if (stop >= 2) 129 { 130 /* take into account the neglected terms: t * 2^w */ 131 mpfr_div_2ui (err_s, err_s, w, MPFR_RNDU); 132 if (MPFR_IS_POS(t)) 133 mpfr_add (err_s, err_s, t, MPFR_RNDU); 134 else 135 mpfr_sub (err_s, err_s, t, MPFR_RNDU); 136 mpfr_mul_2ui (err_s, err_s, w, MPFR_RNDU); 137 stop ++; 138 } 139 /* if k is odd, add to Q, otherwise to P */ 140 else if (k & 1) 141 { 142 /* if k = 1 mod 4, add, otherwise subtract */ 143 if ((k & 2) == 0) 144 mpfr_add (Q, Q, t, MPFR_RNDN); 145 else 146 mpfr_sub (Q, Q, t, MPFR_RNDN); 147 /* check if the next term is smaller than ulp(Q): if EXP(err_u) 148 <= EXP(Q), since the current term is bounded by 149 err_u * 2^(-w), it is bounded by ulp(Q) */ 150 if (MPFR_GET_EXP (err_u) <= MPFR_GET_EXP (Q)) 151 stop ++; 152 else 153 stop = 0; 154 } 155 else 156 { 157 /* if k = 0 mod 4, add, otherwise subtract */ 158 if ((k & 2) == 0) 159 mpfr_add (P, P, t, MPFR_RNDN); 160 else 161 mpfr_sub (P, P, t, MPFR_RNDN); 162 /* check if the next term is smaller than ulp(P) */ 163 if (MPFR_GET_EXP (err_u) <= MPFR_GET_EXP (P)) 164 stop ++; 165 else 166 stop = 0; 167 } 168 mpfr_add (err_s, err_s, err_t, MPFR_RNDU); 169 /* the sum of the rounding errors on P and Q is bounded by 170 err_s * 2^(-w) */ 171 172 /* stop when start to diverge */ 173 if (stop < 2 && 174 ((MPFR_IS_POS(z) && mpfr_cmp_ui (z, (k + 1) / 2) < 0) || 175 (MPFR_IS_NEG(z) && mpfr_cmp_si (z, - ((k + 1) / 2)) > 0))) 176 { 177 /* if we have to stop the series because it diverges, then 178 increasing the precision will most probably fail, since 179 we will stop to the same point, and thus compute a very 180 similar approximation */ 181 diverge = 1; 182 stop = 2; /* force stop */ 183 } 184 } 185 /* the sum of the total errors on P and Q is bounded by err_s * 2^(-w) */ 186 187 /* Now combine: the sum of the rounding errors on P and Q is bounded by 188 err_s * 2^(-w), and the absolute error on s/c is bounded by 2^(1-w) */ 189 if ((n & 1) == 0) /* n even: P * (sin + cos) + Q (cos - sin) for jn 190 Q * (sin + cos) + P (sin - cos) for yn */ 191 { 192#ifdef MPFR_JN 193 mpfr_mul (c, c, Q, MPFR_RNDN); /* Q * (sin - cos) */ 194 mpfr_mul (s, s, P, MPFR_RNDN); /* P * (sin + cos) */ 195#else 196 mpfr_mul (c, c, P, MPFR_RNDN); /* P * (sin - cos) */ 197 mpfr_mul (s, s, Q, MPFR_RNDN); /* Q * (sin + cos) */ 198#endif 199 err = MPFR_GET_EXP (c); 200 if (MPFR_GET_EXP (s) > err) 201 err = MPFR_EXP (s); 202#ifdef MPFR_JN 203 mpfr_sub (s, s, c, MPFR_RNDN); 204#else 205 mpfr_add (s, s, c, MPFR_RNDN); 206#endif 207 } 208 else /* n odd: P * (sin - cos) + Q (cos + sin) for jn, 209 Q * (sin - cos) - P (cos + sin) for yn */ 210 { 211#ifdef MPFR_JN 212 mpfr_mul (c, c, P, MPFR_RNDN); /* P * (sin - cos) */ 213 mpfr_mul (s, s, Q, MPFR_RNDN); /* Q * (sin + cos) */ 214#else 215 mpfr_mul (c, c, Q, MPFR_RNDN); /* Q * (sin - cos) */ 216 mpfr_mul (s, s, P, MPFR_RNDN); /* P * (sin + cos) */ 217#endif 218 err = MPFR_GET_EXP (c); 219 if (MPFR_GET_EXP (s) > err) 220 err = MPFR_EXP (s); 221#ifdef MPFR_JN 222 mpfr_add (s, s, c, MPFR_RNDN); 223#else 224 mpfr_sub (s, c, s, MPFR_RNDN); 225#endif 226 } 227 if (MPFR_IS_ZERO(s)) 228 goto clear; /* with ok=0 */ 229 ok = 1; 230 if ((n & 2) != 0) 231 mpfr_neg (s, s, MPFR_RNDN); 232 if (MPFR_GET_EXP (s) > err) 233 err = MPFR_EXP (s); 234 /* the absolute error on s is bounded by P*err(s/c) + Q*err(s/c) 235 + err(P)*(s/c) + err(Q)*(s/c) + 3 * 2^(err - w - 1) 236 <= (|P|+|Q|) * 2^(1-w) + err_s * 2^(1-w) + 2^err * 2^(1-w), 237 since |c|, |old_s| <= 2. */ 238 err2 = (MPFR_GET_EXP (P) >= MPFR_GET_EXP (Q)) 239 ? MPFR_EXP (P) + 2 : MPFR_EXP (Q) + 2; 240 /* (|P| + |Q|) * 2^(1 - w) <= 2^(err2 - w) */ 241 err = MPFR_GET_EXP (err_s) >= err ? MPFR_EXP (err_s) + 2 : err + 2; 242 /* err_s * 2^(1-w) + 2^old_err * 2^(1-w) <= 2^err * 2^(-w) */ 243 err2 = (err >= err2) ? err + 1 : err2 + 1; 244 /* now the absolute error on s is bounded by 2^(err2 - w) */ 245 246 /* multiply by sqrt(1/(Pi*z)) */ 247 mpfr_const_pi (c, MPFR_RNDN); /* Pi, err <= 1 */ 248 mpfr_mul (c, c, z, MPFR_RNDN); /* err <= 2 */ 249 mpfr_si_div (c, MPFR_IS_POS(z) ? 1 : -1, c, MPFR_RNDN); /* err <= 3 */ 250 mpfr_sqrt (c, c, MPFR_RNDN); /* err<=5/2, thus the absolute error is 251 bounded by 3*u*|c| for |u| <= 0.25 */ 252 mpfr_mul (err_t, c, s, MPFR_SIGN(c)==MPFR_SIGN(s) ? MPFR_RNDU : MPFR_RNDD); 253 mpfr_abs (err_t, err_t, MPFR_RNDU); 254 mpfr_mul_ui (err_t, err_t, 3, MPFR_RNDU); 255 /* 3*2^(-w)*|old_c|*|s| [see below] is bounded by err_t * 2^(-w) */ 256 err2 += MPFR_GET_EXP (c); 257 /* |old_c| * 2^(err2 - w) [see below] is bounded by 2^(err2-w) */ 258 mpfr_mul (c, c, s, MPFR_RNDN); /* the absolute error on c is bounded by 259 1/2 ulp(c) + 3*2^(-w)*|old_c|*|s| 260 + |old_c| * 2^(err2 - w) */ 261 /* compute err_t * 2^(-w) + 1/2 ulp(c) = (err_t + 2^EXP(c)) * 2^(-w) */ 262 err = (MPFR_GET_EXP (err_t) > MPFR_GET_EXP (c)) ? 263 MPFR_EXP (err_t) + 1 : MPFR_EXP (c) + 1; 264 /* err_t * 2^(-w) + 1/2 ulp(c) <= 2^(err - w) */ 265 /* now err_t * 2^(-w) bounds 1/2 ulp(c) + 3*2^(-w)*|old_c|*|s| */ 266 err = (err >= err2) ? err + 1 : err2 + 1; 267 /* the absolute error on c is bounded by 2^(err - w) */ 268 269 err -= MPFR_GET_EXP (c); 270 271 clear: 272 mpfr_clear (s); 273 mpfr_clear (P); 274 mpfr_clear (Q); 275 mpfr_clear (t); 276 mpfr_clear (iz); 277 mpfr_clear (err_t); 278 mpfr_clear (err_s); 279 mpfr_clear (err_u); 280 281 if (ok && MPFR_LIKELY (MPFR_CAN_ROUND (c, w - err, MPFR_PREC(res), r))) 282 break; 283 if (diverge != 0) 284 { 285 MPFR_ZIV_FREE (loop); 286 mpfr_clear (c); 287 return 0; /* means that the asymptotic expansion failed */ 288 } 289 MPFR_ZIV_NEXT (loop, w); 290 } 291 MPFR_ZIV_FREE (loop); 292 293 inex = mpfr_set4 (res, c, r, MPFR_IS_POS (z) || (n & 1) == 0 ? 294 MPFR_SIGN (c) : - MPFR_SIGN (c)); 295 mpfr_clear (c); 296 297 /* for RNDF, mpfr_set or mpfr_neg may return 0, but if we return 0, it 298 would mean the asymptotic expansion failed, thus we return 1 instead */ 299 return (r != MPFR_RNDF) ? inex : 1; 300} 301