1 /* 2 * CDDL HEADER START 3 * 4 * The contents of this file are subject to the terms of the 5 * Common Development and Distribution License (the "License"). 6 * You may not use this file except in compliance with the License. 7 * 8 * You can obtain a copy of the license at usr/src/OPENSOLARIS.LICENSE 9 * or http://www.opensolaris.org/os/licensing. 10 * See the License for the specific language governing permissions 11 * and limitations under the License. 12 * 13 * When distributing Covered Code, include this CDDL HEADER in each 14 * file and include the License file at usr/src/OPENSOLARIS.LICENSE. 15 * If applicable, add the following below this CDDL HEADER, with the 16 * fields enclosed by brackets "[]" replaced with your own identifying 17 * information: Portions Copyright [yyyy] [name of copyright owner] 18 * 19 * CDDL HEADER END 20 */ 21 22 /* 23 * Copyright 2011 Nexenta Systems, Inc. All rights reserved. 24 */ 25 /* 26 * Copyright 2006 Sun Microsystems, Inc. All rights reserved. 27 * Use is subject to license terms. 28 */ 29 30 #pragma weak __jn = jn 31 #pragma weak __yn = yn 32 33 /* 34 * floating point Bessel's function of the 1st and 2nd kind 35 * of order n: jn(n,x),yn(n,x); 36 * 37 * Special cases: 38 * y0(0)=y1(0)=yn(n,0) = -inf with division by zero signal; 39 * y0(-ve)=y1(-ve)=yn(n,-ve) are NaN with invalid signal. 40 * Note 2. About jn(n,x), yn(n,x) 41 * For n=0, j0(x) is called, 42 * for n=1, j1(x) is called, 43 * for n<x, forward recursion us used starting 44 * from values of j0(x) and j1(x). 45 * for n>x, a continued fraction approximation to 46 * j(n,x)/j(n-1,x) is evaluated and then backward 47 * recursion is used starting from a supposed value 48 * for j(n,x). The resulting value of j(0,x) is 49 * compared with the actual value to correct the 50 * supposed value of j(n,x). 51 * 52 * yn(n,x) is similar in all respects, except 53 * that forward recursion is used for all 54 * values of n>1. 55 * 56 */ 57 58 #include "libm.h" 59 #include <float.h> /* DBL_MIN */ 60 #include <values.h> /* X_TLOSS */ 61 #include "xpg6.h" /* __xpg6 */ 62 63 #define GENERIC double 64 65 static const GENERIC 66 invsqrtpi = 5.641895835477562869480794515607725858441e-0001, 67 two = 2.0, 68 zero = 0.0, 69 one = 1.0; 70 71 GENERIC 72 jn(int n, GENERIC x) 73 { 74 int i, sgn; 75 GENERIC a, b, temp = 0; 76 GENERIC z, w, ox, on; 77 78 /* 79 * J(-n,x) = (-1)^n * J(n, x), J(n, -x) = (-1)^n * J(n, x) 80 * Thus, J(-n,x) = J(n,-x) 81 */ 82 ox = x; 83 on = (GENERIC)n; 84 85 if (n < 0) { 86 n = -n; 87 x = -x; 88 } 89 if (isnan(x)) 90 return (x*x); /* + -> * for Cheetah */ 91 if (!((int)_lib_version == libm_ieee || 92 (__xpg6 & _C99SUSv3_math_errexcept) != 0)) { 93 if (fabs(x) > X_TLOSS) 94 return (_SVID_libm_err(on, ox, 38)); 95 } 96 if (n == 0) 97 return (j0(x)); 98 if (n == 1) 99 return (j1(x)); 100 if ((n&1) == 0) 101 sgn = 0; /* even n */ 102 else 103 sgn = signbit(x); /* old n */ 104 x = fabs(x); 105 if (x == zero||!finite(x)) b = zero; 106 else if ((GENERIC)n <= x) { 107 /* 108 * Safe to use 109 * J(n+1,x)=2n/x *J(n,x)-J(n-1,x) 110 */ 111 if (x > 1.0e91) { 112 /* 113 * x >> n**2 114 * Jn(x) = cos(x-(2n+1)*pi/4)*sqrt(2/x*pi) 115 * Yn(x) = sin(x-(2n+1)*pi/4)*sqrt(2/x*pi) 116 * Let s=sin(x), c=cos(x), 117 * xn=x-(2n+1)*pi/4, sqt2 = sqrt(2),then 118 * 119 * n sin(xn)*sqt2 cos(xn)*sqt2 120 * ---------------------------------- 121 * 0 s-c c+s 122 * 1 -s-c -c+s 123 * 2 -s+c -c-s 124 * 3 s+c c-s 125 */ 126 switch (n&3) { 127 case 0: 128 temp = cos(x)+sin(x); 129 break; 130 case 1: 131 temp = -cos(x)+sin(x); 132 break; 133 case 2: 134 temp = -cos(x)-sin(x); 135 break; 136 case 3: 137 temp = cos(x)-sin(x); 138 break; 139 } 140 b = invsqrtpi*temp/sqrt(x); 141 } else { 142 a = j0(x); 143 b = j1(x); 144 for (i = 1; i < n; i++) { 145 temp = b; 146 /* avoid underflow */ 147 b = b*((GENERIC)(i+i)/x) - a; 148 a = temp; 149 } 150 } 151 } else { 152 if (x < 1e-9) { /* use J(n,x) = 1/n!*(x/2)^n */ 153 b = pow(0.5*x, (GENERIC) n); 154 if (b != zero) { 155 for (a = one, i = 1; i <= n; i++) 156 a *= (GENERIC)i; 157 b = b/a; 158 } 159 } else { 160 /* 161 * use backward recurrence 162 * x x^2 x^2 163 * J(n,x)/J(n-1,x) = ---- ------ ------ ..... 164 * 2n - 2(n+1) - 2(n+2) 165 * 166 * 1 1 1 167 * (for large x) = ---- ------ ------ ..... 168 * 2n 2(n+1) 2(n+2) 169 * -- - ------ - ------ - 170 * x x x 171 * 172 * Let w = 2n/x and h = 2/x, then the above quotient 173 * is equal to the continued fraction: 174 * 1 175 * = ----------------------- 176 * 1 177 * w - ----------------- 178 * 1 179 * w+h - --------- 180 * w+2h - ... 181 * 182 * To determine how many terms needed, let 183 * Q(0) = w, Q(1) = w(w+h) - 1, 184 * Q(k) = (w+k*h)*Q(k-1) - Q(k-2), 185 * When Q(k) > 1e4 good for single 186 * When Q(k) > 1e9 good for double 187 * When Q(k) > 1e17 good for quaduple 188 */ 189 /* determine k */ 190 GENERIC t, v; 191 double q0, q1, h, tmp; 192 int k, m; 193 w = (n+n)/(double)x; 194 h = 2.0/(double)x; 195 q0 = w; 196 z = w + h; 197 q1 = w*z - 1.0; 198 k = 1; 199 200 while (q1 < 1.0e9) { 201 k += 1; 202 z += h; 203 tmp = z*q1 - q0; 204 q0 = q1; 205 q1 = tmp; 206 } 207 m = n+n; 208 for (t = zero, i = 2*(n+k); i >= m; i -= 2) 209 t = one/(i/x-t); 210 a = t; 211 b = one; 212 /* 213 * estimate log((2/x)^n*n!) = n*log(2/x)+n*ln(n) 214 * hence, if n*(log(2n/x)) > ... 215 * single: 216 * 8.8722839355e+01 217 * double: 218 * 7.09782712893383973096e+02 219 * long double: 220 * 1.1356523406294143949491931077970765006170e+04 221 * then recurrent value may overflow and the result is 222 * likely underflow to zero 223 */ 224 tmp = n; 225 v = two/x; 226 tmp = tmp*log(fabs(v*tmp)); 227 if (tmp < 7.09782712893383973096e+02) { 228 for (i = n-1; i > 0; i--) { 229 temp = b; 230 b = ((i+i)/x)*b - a; 231 a = temp; 232 } 233 } else { 234 for (i = n-1; i > 0; i--) { 235 temp = b; 236 b = ((i+i)/x)*b - a; 237 a = temp; 238 if (b > 1e100) { 239 a /= b; 240 t /= b; 241 b = 1.0; 242 } 243 } 244 } 245 b = (t*j0(x)/b); 246 } 247 } 248 if (sgn != 0) 249 return (-b); 250 else 251 return (b); 252 } 253 254 GENERIC 255 yn(int n, GENERIC x) 256 { 257 int i; 258 int sign; 259 GENERIC a, b, temp = 0, ox, on; 260 261 ox = x; 262 on = (GENERIC)n; 263 if (isnan(x)) 264 return (x*x); /* + -> * for Cheetah */ 265 if (x <= zero) { 266 if (x == zero) { 267 /* return -one/zero; */ 268 return (_SVID_libm_err((GENERIC)n, x, 12)); 269 } else { 270 /* return zero/zero; */ 271 return (_SVID_libm_err((GENERIC)n, x, 13)); 272 } 273 } 274 if (!((int)_lib_version == libm_ieee || 275 (__xpg6 & _C99SUSv3_math_errexcept) != 0)) { 276 if (x > X_TLOSS) 277 return (_SVID_libm_err(on, ox, 39)); 278 } 279 sign = 1; 280 if (n < 0) { 281 n = -n; 282 if ((n&1) == 1) sign = -1; 283 } 284 if (n == 0) 285 return (y0(x)); 286 if (n == 1) 287 return (sign*y1(x)); 288 if (!finite(x)) 289 return (zero); 290 291 if (x > 1.0e91) { 292 /* 293 * x >> n**2 294 * Jn(x) = cos(x-(2n+1)*pi/4)*sqrt(2/x*pi) 295 * Yn(x) = sin(x-(2n+1)*pi/4)*sqrt(2/x*pi) 296 * Let s = sin(x), c = cos(x), 297 * xn = x-(2n+1)*pi/4, sqt2 = sqrt(2), then 298 * 299 * n sin(xn)*sqt2 cos(xn)*sqt2 300 * ---------------------------------- 301 * 0 s-c c+s 302 * 1 -s-c -c+s 303 * 2 -s+c -c-s 304 * 3 s+c c-s 305 */ 306 switch (n&3) { 307 case 0: 308 temp = sin(x)-cos(x); 309 break; 310 case 1: 311 temp = -sin(x)-cos(x); 312 break; 313 case 2: 314 temp = -sin(x)+cos(x); 315 break; 316 case 3: 317 temp = sin(x)+cos(x); 318 break; 319 } 320 b = invsqrtpi*temp/sqrt(x); 321 } else { 322 a = y0(x); 323 b = y1(x); 324 /* 325 * fix 1262058 and take care of non-default rounding 326 */ 327 for (i = 1; i < n; i++) { 328 temp = b; 329 b *= (GENERIC) (i + i) / x; 330 if (b <= -DBL_MAX) 331 break; 332 b -= a; 333 a = temp; 334 } 335 } 336 if (sign > 0) 337 return (b); 338 else 339 return (-b); 340 } 341