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