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 int i, sgn; 74 GENERIC a, b, temp = 0; 75 GENERIC z, w, ox, on; 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 ox = x; on = (GENERIC)n; 82 if (n < 0) { 83 n = -n; 84 x = -x; 85 } 86 if (isnan(x)) 87 return (x*x); /* + -> * for Cheetah */ 88 if (!((int) _lib_version == libm_ieee || 89 (__xpg6 & _C99SUSv3_math_errexcept) != 0)) { 90 if (fabs(x) > X_TLOSS) 91 return (_SVID_libm_err(on, ox, 38)); 92 } 93 if (n == 0) 94 return (j0(x)); 95 if (n == 1) 96 return (j1(x)); 97 if ((n&1) == 0) 98 sgn = 0; /* even n */ 99 else 100 sgn = signbit(x); /* old n */ 101 x = fabs(x); 102 if (x == zero||!finite(x)) b = zero; 103 else if ((GENERIC)n <= x) { 104 /* 105 * Safe to use 106 * J(n+1,x)=2n/x *J(n,x)-J(n-1,x) 107 */ 108 if (x > 1.0e91) { 109 /* 110 * x >> n**2 111 * Jn(x) = cos(x-(2n+1)*pi/4)*sqrt(2/x*pi) 112 * Yn(x) = sin(x-(2n+1)*pi/4)*sqrt(2/x*pi) 113 * Let s=sin(x), c=cos(x), 114 * xn=x-(2n+1)*pi/4, sqt2 = sqrt(2),then 115 * 116 * n sin(xn)*sqt2 cos(xn)*sqt2 117 * ---------------------------------- 118 * 0 s-c c+s 119 * 1 -s-c -c+s 120 * 2 -s+c -c-s 121 * 3 s+c c-s 122 */ 123 switch (n&3) { 124 case 0: temp = cos(x)+sin(x); break; 125 case 1: temp = -cos(x)+sin(x); break; 126 case 2: temp = -cos(x)-sin(x); break; 127 case 3: temp = cos(x)-sin(x); break; 128 } 129 b = invsqrtpi*temp/sqrt(x); 130 } else { 131 a = j0(x); 132 b = j1(x); 133 for (i = 1; i < n; i++) { 134 temp = b; 135 b = b*((GENERIC)(i+i)/x) - a; /* avoid underflow */ 136 a = temp; 137 } 138 } 139 } else { 140 if (x < 1e-9) { /* use J(n,x) = 1/n!*(x/2)^n */ 141 b = pow(0.5*x, (GENERIC) n); 142 if (b != zero) { 143 for (a = one, i = 1; i <= n; i++) a *= (GENERIC)i; 144 b = b/a; 145 } 146 } else { 147 /* 148 * use backward recurrence 149 * x x^2 x^2 150 * J(n,x)/J(n-1,x) = ---- ------ ------ ..... 151 * 2n - 2(n+1) - 2(n+2) 152 * 153 * 1 1 1 154 * (for large x) = ---- ------ ------ ..... 155 * 2n 2(n+1) 2(n+2) 156 * -- - ------ - ------ - 157 * x x x 158 * 159 * Let w = 2n/x and h = 2/x, then the above quotient 160 * is equal to the continued fraction: 161 * 1 162 * = ----------------------- 163 * 1 164 * w - ----------------- 165 * 1 166 * w+h - --------- 167 * w+2h - ... 168 * 169 * To determine how many terms needed, let 170 * Q(0) = w, Q(1) = w(w+h) - 1, 171 * Q(k) = (w+k*h)*Q(k-1) - Q(k-2), 172 * When Q(k) > 1e4 good for single 173 * When Q(k) > 1e9 good for double 174 * When Q(k) > 1e17 good for quaduple 175 */ 176 /* determin k */ 177 GENERIC t, v; 178 double q0, q1, h, tmp; int k, m; 179 w = (n+n)/(double)x; h = 2.0/(double)x; 180 q0 = w; z = w + h; q1 = w*z - 1.0; k = 1; 181 while (q1 < 1.0e9) { 182 k += 1; z += h; 183 tmp = z*q1 - q0; 184 q0 = q1; 185 q1 = tmp; 186 } 187 m = n+n; 188 for (t = zero, i = 2*(n+k); i >= m; i -= 2) t = one/(i/x-t); 189 a = t; 190 b = one; 191 /* 192 * estimate log((2/x)^n*n!) = n*log(2/x)+n*ln(n) 193 * hence, if n*(log(2n/x)) > ... 194 * single 8.8722839355e+01 195 * double 7.09782712893383973096e+02 196 * long double 1.1356523406294143949491931077970765006170e+04 197 * then recurrent value may overflow and the result is 198 * likely underflow to zero 199 */ 200 tmp = n; 201 v = two/x; 202 tmp = tmp*log(fabs(v*tmp)); 203 if (tmp < 7.09782712893383973096e+02) { 204 for (i = n-1; i > 0; i--) { 205 temp = b; 206 b = ((i+i)/x)*b - a; 207 a = temp; 208 } 209 } else { 210 for (i = n-1; i > 0; i--) { 211 temp = b; 212 b = ((i+i)/x)*b - a; 213 a = temp; 214 if (b > 1e100) { 215 a /= b; 216 t /= b; 217 b = 1.0; 218 } 219 } 220 } 221 b = (t*j0(x)/b); 222 } 223 } 224 if (sgn == 1) 225 return (-b); 226 else 227 return (b); 228 } 229 230 GENERIC 231 yn(int n, GENERIC x) { 232 int i; 233 int sign; 234 GENERIC a, b, temp = 0, ox, on; 235 236 ox = x; on = (GENERIC)n; 237 if (isnan(x)) 238 return (x*x); /* + -> * for Cheetah */ 239 if (x <= zero) { 240 if (x == zero) { 241 /* return -one/zero; */ 242 return (_SVID_libm_err((GENERIC)n, x, 12)); 243 } else { 244 /* return zero/zero; */ 245 return (_SVID_libm_err((GENERIC)n, x, 13)); 246 } 247 } 248 if (!((int) _lib_version == libm_ieee || 249 (__xpg6 & _C99SUSv3_math_errexcept) != 0)) { 250 if (x > X_TLOSS) 251 return (_SVID_libm_err(on, ox, 39)); 252 } 253 sign = 1; 254 if (n < 0) { 255 n = -n; 256 if ((n&1) == 1) sign = -1; 257 } 258 if (n == 0) 259 return (y0(x)); 260 if (n == 1) 261 return (sign*y1(x)); 262 if (!finite(x)) 263 return (zero); 264 265 if (x > 1.0e91) { 266 /* 267 * x >> n**2 268 * Jn(x) = cos(x-(2n+1)*pi/4)*sqrt(2/x*pi) 269 * Yn(x) = sin(x-(2n+1)*pi/4)*sqrt(2/x*pi) 270 * Let s = sin(x), c = cos(x), 271 * xn = x-(2n+1)*pi/4, sqt2 = sqrt(2), then 272 * 273 * n sin(xn)*sqt2 cos(xn)*sqt2 274 * ---------------------------------- 275 * 0 s-c c+s 276 * 1 -s-c -c+s 277 * 2 -s+c -c-s 278 * 3 s+c c-s 279 */ 280 switch (n&3) { 281 case 0: temp = sin(x)-cos(x); break; 282 case 1: temp = -sin(x)-cos(x); break; 283 case 2: temp = -sin(x)+cos(x); break; 284 case 3: temp = sin(x)+cos(x); break; 285 } 286 b = invsqrtpi*temp/sqrt(x); 287 } else { 288 a = y0(x); 289 b = y1(x); 290 /* 291 * fix 1262058 and take care of non-default rounding 292 */ 293 for (i = 1; i < n; i++) { 294 temp = b; 295 b *= (GENERIC) (i + i) / x; 296 if (b <= -DBL_MAX) 297 break; 298 b -= a; 299 a = temp; 300 } 301 } 302 if (sign > 0) 303 return (b); 304 else 305 return (-b); 306 } 307