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