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 __jnl = jnl 31 #pragma weak __ynl = ynl 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 "longdouble.h" 60 #include <float.h> /* LDBL_MAX */ 61 62 #define GENERIC long double 63 64 static const GENERIC 65 invsqrtpi = 5.641895835477562869480794515607725858441e-0001L, 66 two = 2.0L, 67 zero = 0.0L, 68 one = 1.0L; 69 70 GENERIC 71 jnl(int n, GENERIC x) 72 { 73 int i, sgn; 74 GENERIC a, b, temp, z, w; 75 76 /* 77 * J(-n,x) = (-1)^n * J(n, x), J(n, -x) = (-1)^n * J(n, x) 78 * Thus, J(-n,x) = J(n,-x) 79 */ 80 if (n < 0) { 81 n = -n; 82 x = -x; 83 } 84 if (n == 0) 85 return (j0l(x)); 86 if (n == 1) 87 return (j1l(x)); 88 if (x != x) 89 return (x+x); 90 if ((n&1) == 0) 91 sgn = 0; /* even n */ 92 else 93 sgn = signbitl(x); /* old n */ 94 x = fabsl(x); 95 if (x == zero || !finitel(x)) b = zero; 96 else if ((GENERIC)n <= x) { 97 /* 98 * Safe to use 99 * J(n+1,x)=2n/x *J(n,x)-J(n-1,x) 100 */ 101 if (x > 1.0e91L) { 102 /* 103 * x >> n**2 104 * Jn(x) = cos(x-(2n+1)*pi/4)*sqrt(2/x*pi) 105 * Yn(x) = sin(x-(2n+1)*pi/4)*sqrt(2/x*pi) 106 * Let s=sin(x), c=cos(x), 107 * xn=x-(2n+1)*pi/4, sqt2 = sqrt(2),then 108 * 109 * n sin(xn)*sqt2 cos(xn)*sqt2 110 * ---------------------------------- 111 * 0 s-c c+s 112 * 1 -s-c -c+s 113 * 2 -s+c -c-s 114 * 3 s+c c-s 115 */ 116 switch (n&3) { 117 case 0: 118 temp = cosl(x)+sinl(x); 119 break; 120 case 1: 121 temp = -cosl(x)+sinl(x); 122 break; 123 case 2: 124 temp = -cosl(x)-sinl(x); 125 break; 126 case 3: 127 temp = cosl(x)-sinl(x); 128 break; 129 } 130 b = invsqrtpi*temp/sqrtl(x); 131 } else { 132 a = j0l(x); 133 b = j1l(x); 134 for (i = 1; i < n; i++) { 135 temp = b; 136 /* avoid underflow */ 137 b = b*((GENERIC)(i+i)/x) - a; 138 a = temp; 139 } 140 } 141 } else { 142 if (x < 1e-17L) { /* use J(n,x) = 1/n!*(x/2)^n */ 143 b = powl(0.5L*x, (GENERIC)n); 144 if (b != zero) { 145 for (a = one, i = 1; i <= n; i++) 146 a *= (GENERIC)i; 147 b = b/a; 148 } 149 } else { 150 /* use backward recurrence */ 151 /* BEGIN CSTYLED */ 152 /* 153 * x x^2 x^2 154 * J(n,x)/J(n-1,x) = ---- ------ ------ ..... 155 * 2n - 2(n+1) - 2(n+2) 156 * 157 * 1 1 1 158 * (for large x) = ---- ------ ------ ..... 159 * 2n 2(n+1) 2(n+2) 160 * -- - ------ - ------ - 161 * x x x 162 * 163 * Let w = 2n/x and h=2/x, then the above quotient 164 * is equal to the continued fraction: 165 * 1 166 * = ----------------------- 167 * 1 168 * w - ----------------- 169 * 1 170 * w+h - --------- 171 * w+2h - ... 172 * 173 * To determine how many terms needed, let 174 * Q(0) = w, Q(1) = w(w+h) - 1, 175 * Q(k) = (w+k*h)*Q(k-1) - Q(k-2), 176 * When Q(k) > 1e4 good for single 177 * When Q(k) > 1e9 good for double 178 * When Q(k) > 1e17 good for quaduple 179 */ 180 /* END CSTYLED */ 181 /* determine k */ 182 GENERIC t, v; 183 double q0, q1, h, tmp; 184 int k, m; 185 w = (n+n)/(double)x; 186 h = 2.0/(double)x; 187 q0 = w; 188 z = w+h; 189 q1 = w*z - 1.0; 190 k = 1; 191 while (q1 < 1.0e17) { 192 k += 1; 193 z += h; 194 tmp = z*q1 - q0; 195 q0 = q1; 196 q1 = tmp; 197 } 198 m = n+n; 199 for (t = zero, i = 2*(n+k); i >= m; i -= 2) 200 t = one/(i/x-t); 201 a = t; 202 b = one; 203 /* 204 * estimate log((2/x)^n*n!) = n*log(2/x)+n*ln(n) 205 * hence, if n*(log(2n/x)) > ... 206 * single: 207 * 8.8722839355e+01 208 * double: 209 * 7.09782712893383973096e+02 210 * long double: 211 * 1.1356523406294143949491931077970765006170e+04 212 * then recurrent value may overflow and the result is 213 * likely underflow to zero 214 */ 215 tmp = n; 216 v = two/x; 217 tmp = tmp*logl(fabsl(v*tmp)); 218 if (tmp < 1.1356523406294143949491931077970765e+04L) { 219 for (i = n-1; i > 0; i--) { 220 temp = b; 221 b = ((i+i)/x)*b - a; 222 a = temp; 223 } 224 } else { 225 for (i = n-1; i > 0; i--) { 226 temp = b; 227 b = ((i+i)/x)*b - a; 228 a = temp; 229 if (b > 1e1000L) { 230 a /= b; 231 t /= b; 232 b = 1.0; 233 } 234 } 235 } 236 b = (t*j0l(x)/b); 237 } 238 } 239 if (sgn != 0) 240 return (-b); 241 else 242 return (b); 243 } 244 245 GENERIC 246 ynl(int n, GENERIC x) 247 { 248 int i; 249 int sign; 250 GENERIC a, b, temp; 251 252 if (x != x) 253 return (x+x); 254 if (x <= zero) { 255 if (x == zero) 256 return (-one/zero); 257 else 258 return (zero/zero); 259 } 260 sign = 1; 261 if (n < 0) { 262 n = -n; 263 if ((n&1) == 1) sign = -1; 264 } 265 if (n == 0) 266 return (y0l(x)); 267 if (n == 1) 268 return (sign*y1l(x)); 269 if (!finitel(x)) 270 return (zero); 271 272 if (x > 1.0e91L) { 273 /* 274 * x >> n**2 275 * Jn(x) = cos(x-(2n+1)*pi/4)*sqrt(2/x*pi) 276 * Yn(x) = sin(x-(2n+1)*pi/4)*sqrt(2/x*pi) 277 * Let s = sin(x), c = cos(x), 278 * xn = x-(2n+1)*pi/4, sqt2 = sqrt(2), then 279 * 280 * n sin(xn)*sqt2 cos(xn)*sqt2 281 * ---------------------------------- 282 * 0 s-c c+s 283 * 1 -s-c -c+s 284 * 2 -s+c -c-s 285 * 3 s+c c-s 286 */ 287 switch (n&3) { 288 case 0: 289 temp = sinl(x)-cosl(x); 290 break; 291 case 1: 292 temp = -sinl(x)-cosl(x); 293 break; 294 case 2: 295 temp = -sinl(x)+cosl(x); 296 break; 297 case 3: 298 temp = sinl(x)+cosl(x); 299 break; 300 } 301 b = invsqrtpi*temp/sqrtl(x); 302 } else { 303 a = y0l(x); 304 b = y1l(x); 305 /* 306 * fix 1262058 and take care of non-default rounding 307 */ 308 for (i = 1; i < n; i++) { 309 temp = b; 310 b *= (GENERIC) (i + i) / x; 311 if (b <= -LDBL_MAX) 312 break; 313 b -= a; 314 a = temp; 315 } 316 } 317 if (sign > 0) 318 return (b); 319 else 320 return (-b); 321 } 322