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 #if defined(ELFOBJ) 31 #pragma weak jnl = __jnl 32 #pragma weak ynl = __ynl 33 #endif 34 35 /* 36 * floating point Bessel's function of the 1st and 2nd kind 37 * of order n: jn(n,x),yn(n,x); 38 * 39 * Special cases: 40 * y0(0)=y1(0)=yn(n,0) = -inf with division by zero signal; 41 * y0(-ve)=y1(-ve)=yn(n,-ve) are NaN with invalid signal. 42 * Note 2. About jn(n,x), yn(n,x) 43 * For n=0, j0(x) is called, 44 * for n=1, j1(x) is called, 45 * for n<x, forward recursion us used starting 46 * from values of j0(x) and j1(x). 47 * for n>x, a continued fraction approximation to 48 * j(n,x)/j(n-1,x) is evaluated and then backward 49 * recursion is used starting from a supposed value 50 * for j(n,x). The resulting value of j(0,x) is 51 * compared with the actual value to correct the 52 * supposed value of j(n,x). 53 * 54 * yn(n,x) is similar in all respects, except 55 * that forward recursion is used for all 56 * values of n>1. 57 * 58 */ 59 60 #include "libm.h" 61 #include "longdouble.h" 62 #include <float.h> /* LDBL_MAX */ 63 64 #define GENERIC long double 65 66 static const GENERIC 67 invsqrtpi = 5.641895835477562869480794515607725858441e-0001L, 68 two = 2.0L, 69 zero = 0.0L, 70 one = 1.0L; 71 72 GENERIC 73 jnl(n, x) int n; GENERIC x; { 74 int i, sgn; 75 GENERIC a, b, temp = 0, z, w; 76 77 /* 78 * J(-n,x) = (-1)^n * J(n, x), J(n, -x) = (-1)^n * J(n, x) 79 * Thus, J(-n,x) = J(n,-x) 80 */ 81 if (n < 0) { 82 n = -n; 83 x = -x; 84 } 85 if (n == 0) return (j0l(x)); 86 if (n == 1) return (j1l(x)); 87 if (x != x) return x+x; 88 if ((n&1) == 0) 89 sgn = 0; /* even n */ 90 else 91 sgn = signbitl(x); /* old n */ 92 x = fabsl(x); 93 if (x == zero || !finitel(x)) b = zero; 94 else if ((GENERIC)n <= x) { 95 /* 96 * Safe to use 97 * J(n+1,x)=2n/x *J(n,x)-J(n-1,x) 98 */ 99 if (x > 1.0e91L) { 100 /* 101 * x >> n**2 102 * Jn(x) = cos(x-(2n+1)*pi/4)*sqrt(2/x*pi) 103 * Yn(x) = sin(x-(2n+1)*pi/4)*sqrt(2/x*pi) 104 * Let s=sin(x), c=cos(x), 105 * xn=x-(2n+1)*pi/4, sqt2 = sqrt(2),then 106 * 107 * n sin(xn)*sqt2 cos(xn)*sqt2 108 * ---------------------------------- 109 * 0 s-c c+s 110 * 1 -s-c -c+s 111 * 2 -s+c -c-s 112 * 3 s+c c-s 113 */ 114 switch (n&3) { 115 case 0: temp = cosl(x)+sinl(x); break; 116 case 1: temp = -cosl(x)+sinl(x); break; 117 case 2: temp = -cosl(x)-sinl(x); break; 118 case 3: temp = cosl(x)-sinl(x); break; 119 } 120 b = invsqrtpi*temp/sqrtl(x); 121 } else { 122 a = j0l(x); 123 b = j1l(x); 124 for (i = 1; i < n; i++) { 125 temp = b; 126 b = b*((GENERIC)(i+i)/x) - a; /* avoid underflow */ 127 a = temp; 128 } 129 } 130 } else { 131 if (x < 1e-17L) { /* use J(n,x) = 1/n!*(x/2)^n */ 132 b = powl(0.5L*x, (GENERIC) n); 133 if (b != zero) { 134 for (a = one, i = 1; i <= n; i++) a *= (GENERIC)i; 135 b = b/a; 136 } 137 } else { 138 /* 139 * use backward recurrence 140 * x x^2 x^2 141 * J(n,x)/J(n-1,x) = ---- ------ ------ ..... 142 * 2n - 2(n+1) - 2(n+2) 143 * 144 * 1 1 1 145 * (for large x) = ---- ------ ------ ..... 146 * 2n 2(n+1) 2(n+2) 147 * -- - ------ - ------ - 148 * x x x 149 * 150 * Let w = 2n/x and h=2/x, then the above quotient 151 * is equal to the continued fraction: 152 * 1 153 * = ----------------------- 154 * 1 155 * w - ----------------- 156 * 1 157 * w+h - --------- 158 * w+2h - ... 159 * 160 * To determine how many terms needed, let 161 * Q(0) = w, Q(1) = w(w+h) - 1, 162 * Q(k) = (w+k*h)*Q(k-1) - Q(k-2), 163 * When Q(k) > 1e4 good for single 164 * When Q(k) > 1e9 good for double 165 * When Q(k) > 1e17 good for quaduple 166 */ 167 /* determin k */ 168 GENERIC t, v; 169 double q0, q1, h, tmp; int k, m; 170 w = (n+n)/(double)x; h = 2.0/(double)x; 171 q0 = w; z = w+h; q1 = w*z - 1.0; k = 1; 172 while (q1 < 1.0e17) { 173 k += 1; z += h; 174 tmp = z*q1 - q0; 175 q0 = q1; 176 q1 = tmp; 177 } 178 m = n+n; 179 for (t = zero, i = 2*(n+k); i >= m; i -= 2) t = one/(i/x-t); 180 a = t; 181 b = one; 182 /* 183 * Estimate log((2/x)^n*n!) = n*log(2/x)+n*ln(n) 184 * hence, if n*(log(2n/x)) > ... 185 * single 8.8722839355e+01 186 * double 7.09782712893383973096e+02 187 * long double 1.1356523406294143949491931077970765006170e+04 188 * then recurrent value may overflow and the result is 189 * likely underflow to zero. 190 */ 191 tmp = n; 192 v = two/x; 193 tmp = tmp*logl(fabsl(v*tmp)); 194 if (tmp < 1.1356523406294143949491931077970765e+04L) { 195 for (i = n-1; i > 0; i--) { 196 temp = b; 197 b = ((i+i)/x)*b - a; 198 a = temp; 199 } 200 } else { 201 for (i = n-1; i > 0; i--) { 202 temp = b; 203 b = ((i+i)/x)*b - a; 204 a = temp; 205 if (b > 1e1000L) { 206 a /= b; 207 t /= b; 208 b = 1.0; 209 } 210 } 211 } 212 b = (t*j0l(x)/b); 213 } 214 } 215 if (sgn == 1) 216 return -b; 217 else 218 return b; 219 } 220 221 GENERIC 222 ynl(n, x) int n; GENERIC x; { 223 int i; 224 int sign; 225 GENERIC a, b, temp = 0; 226 227 if (x != x) 228 return x+x; 229 if (x <= zero) { 230 if (x == zero) 231 return -one/zero; 232 else 233 return zero/zero; 234 } 235 sign = 1; 236 if (n < 0) { 237 n = -n; 238 if ((n&1) == 1) sign = -1; 239 } 240 if (n == 0) return (y0l(x)); 241 if (n == 1) return (sign*y1l(x)); 242 if (!finitel(x)) return zero; 243 244 if (x > 1.0e91L) { 245 /* 246 * x >> n**2 247 * Jn(x) = cos(x-(2n+1)*pi/4)*sqrt(2/x*pi) 248 * Yn(x) = sin(x-(2n+1)*pi/4)*sqrt(2/x*pi) 249 * Let s=sin(x), c=cos(x), 250 * xn=x-(2n+1)*pi/4, sqt2 = sqrt(2),then 251 * 252 * n sin(xn)*sqt2 cos(xn)*sqt2 253 * ---------------------------------- 254 * 0 s-c c+s 255 * 1 -s-c -c+s 256 * 2 -s+c -c-s 257 * 3 s+c c-s 258 */ 259 switch (n&3) { 260 case 0: temp = sinl(x)-cosl(x); break; 261 case 1: temp = -sinl(x)-cosl(x); break; 262 case 2: temp = -sinl(x)+cosl(x); break; 263 case 3: temp = sinl(x)+cosl(x); break; 264 } 265 b = invsqrtpi*temp/sqrtl(x); 266 } else { 267 a = y0l(x); 268 b = y1l(x); 269 /* 270 * fix 1262058 and take care of non-default rounding 271 */ 272 for (i = 1; i < n; i++) { 273 temp = b; 274 b *= (GENERIC) (i + i) / x; 275 if (b <= -LDBL_MAX) 276 break; 277 b -= a; 278 a = temp; 279 } 280 } 281 if (sign > 0) 282 return b; 283 else 284 return -b; 285 } 286