1 // Copyright 2010 The Go Authors. All rights reserved. 2 // Use of this source code is governed by a BSD-style 3 // license that can be found in the LICENSE file. 4 5 package math 6 7 /* 8 Bessel function of the first and second kinds of order n. 9 */ 10 11 // The original C code and the long comment below are 12 // from FreeBSD's /usr/src/lib/msun/src/e_jn.c and 13 // came with this notice. The go code is a simplified 14 // version of the original C. 15 // 16 // ==================================================== 17 // Copyright (C) 1993 by Sun Microsystems, Inc. All rights reserved. 18 // 19 // Developed at SunPro, a Sun Microsystems, Inc. business. 20 // Permission to use, copy, modify, and distribute this 21 // software is freely granted, provided that this notice 22 // is preserved. 23 // ==================================================== 24 // 25 // __ieee754_jn(n, x), __ieee754_yn(n, x) 26 // floating point Bessel's function of the 1st and 2nd kind 27 // of order n 28 // 29 // Special cases: 30 // y0(0)=y1(0)=yn(n,0) = -inf with division by zero signal; 31 // y0(-ve)=y1(-ve)=yn(n,-ve) are NaN with invalid signal. 32 // Note 2. About jn(n,x), yn(n,x) 33 // For n=0, j0(x) is called, 34 // for n=1, j1(x) is called, 35 // for n<x, forward recursion is used starting 36 // from values of j0(x) and j1(x). 37 // for n>x, a continued fraction approximation to 38 // j(n,x)/j(n-1,x) is evaluated and then backward 39 // recursion is used starting from a supposed value 40 // for j(n,x). The resulting value of j(0,x) is 41 // compared with the actual value to correct the 42 // supposed value of j(n,x). 43 // 44 // yn(n,x) is similar in all respects, except 45 // that forward recursion is used for all 46 // values of n>1. 47 48 // Jn returns the order-n Bessel function of the first kind. 49 // 50 // Special cases are: 51 // Jn(n, Inf) = 0 52 // Jn(n, NaN) = NaN 53 func Jn(n int, x float64) float64 { 54 const ( 55 TwoM29 = 1.0 / (1 << 29) // 2**-29 0x3e10000000000000 56 Two302 = 1 << 302 // 2**302 0x52D0000000000000 57 ) 58 // special cases 59 switch { 60 case IsNaN(x): 61 return x 62 case IsInf(x, 0): 63 return 0 64 } 65 // J(-n, x) = (-1)**n * J(n, x), J(n, -x) = (-1)**n * J(n, x) 66 // Thus, J(-n, x) = J(n, -x) 67 68 if n == 0 { 69 return J0(x) 70 } 71 if x == 0 { 72 return 0 73 } 74 if n < 0 { 75 n, x = -n, -x 76 } 77 if n == 1 { 78 return J1(x) 79 } 80 sign := false 81 if x < 0 { 82 x = -x 83 if n&1 == 1 { 84 sign = true // odd n and negative x 85 } 86 } 87 var b float64 88 if float64(n) <= x { 89 // Safe to use J(n+1,x)=2n/x *J(n,x)-J(n-1,x) 90 if x >= Two302 { // x > 2**302 91 92 // (x >> n**2) 93 // Jn(x) = cos(x-(2n+1)*pi/4)*sqrt(2/x*pi) 94 // Yn(x) = sin(x-(2n+1)*pi/4)*sqrt(2/x*pi) 95 // Let s=sin(x), c=cos(x), 96 // xn=x-(2n+1)*pi/4, sqt2 = sqrt(2),then 97 // 98 // n sin(xn)*sqt2 cos(xn)*sqt2 99 // ---------------------------------- 100 // 0 s-c c+s 101 // 1 -s-c -c+s 102 // 2 -s+c -c-s 103 // 3 s+c c-s 104 105 var temp float64 106 switch n & 3 { 107 case 0: 108 temp = Cos(x) + Sin(x) 109 case 1: 110 temp = -Cos(x) + Sin(x) 111 case 2: 112 temp = -Cos(x) - Sin(x) 113 case 3: 114 temp = Cos(x) - Sin(x) 115 } 116 b = (1 / SqrtPi) * temp / Sqrt(x) 117 } else { 118 b = J1(x) 119 for i, a := 1, J0(x); i < n; i++ { 120 a, b = b, b*(float64(i+i)/x)-a // avoid underflow 121 } 122 } 123 } else { 124 if x < TwoM29 { // x < 2**-29 125 // x is tiny, return the first Taylor expansion of J(n,x) 126 // J(n,x) = 1/n!*(x/2)**n - ... 127 128 if n > 33 { // underflow 129 b = 0 130 } else { 131 temp := x * 0.5 132 b = temp 133 a := 1.0 134 for i := 2; i <= n; i++ { 135 a *= float64(i) // a = n! 136 b *= temp // b = (x/2)**n 137 } 138 b /= a 139 } 140 } else { 141 // use backward recurrence 142 // x x**2 x**2 143 // J(n,x)/J(n-1,x) = ---- ------ ------ ..... 144 // 2n - 2(n+1) - 2(n+2) 145 // 146 // 1 1 1 147 // (for large x) = ---- ------ ------ ..... 148 // 2n 2(n+1) 2(n+2) 149 // -- - ------ - ------ - 150 // x x x 151 // 152 // Let w = 2n/x and h=2/x, then the above quotient 153 // is equal to the continued fraction: 154 // 1 155 // = ----------------------- 156 // 1 157 // w - ----------------- 158 // 1 159 // w+h - --------- 160 // w+2h - ... 161 // 162 // To determine how many terms needed, let 163 // Q(0) = w, Q(1) = w(w+h) - 1, 164 // Q(k) = (w+k*h)*Q(k-1) - Q(k-2), 165 // When Q(k) > 1e4 good for single 166 // When Q(k) > 1e9 good for double 167 // When Q(k) > 1e17 good for quadruple 168 169 // determine k 170 w := float64(n+n) / x 171 h := 2 / x 172 q0 := w 173 z := w + h 174 q1 := w*z - 1 175 k := 1 176 for q1 < 1e9 { 177 k += 1 178 z += h 179 q0, q1 = q1, z*q1-q0 180 } 181 m := n + n 182 t := 0.0 183 for i := 2 * (n + k); i >= m; i -= 2 { 184 t = 1 / (float64(i)/x - t) 185 } 186 a := t 187 b = 1 188 // estimate log((2/x)**n*n!) = n*log(2/x)+n*ln(n) 189 // Hence, if n*(log(2n/x)) > ... 190 // single 8.8722839355e+01 191 // double 7.09782712893383973096e+02 192 // long double 1.1356523406294143949491931077970765006170e+04 193 // then recurrent value may overflow and the result is 194 // likely underflow to zero 195 196 tmp := float64(n) 197 v := 2 / x 198 tmp = tmp * Log(Abs(v*tmp)) 199 if tmp < 7.09782712893383973096e+02 { 200 for i := n - 1; i > 0; i-- { 201 di := float64(i + i) 202 a, b = b, b*di/x-a 203 di -= 2 204 } 205 } else { 206 for i := n - 1; i > 0; i-- { 207 di := float64(i + i) 208 a, b = b, b*di/x-a 209 di -= 2 210 // scale b to avoid spurious overflow 211 if b > 1e100 { 212 a /= b 213 t /= b 214 b = 1 215 } 216 } 217 } 218 b = t * J0(x) / b 219 } 220 } 221 if sign { 222 return -b 223 } 224 return b 225 } 226 227 // Yn returns the order-n Bessel function of the second kind. 228 // 229 // Special cases are: 230 // Yn(n, +Inf) = 0 231 // Yn(n > 0, 0) = -Inf 232 // Yn(n < 0, 0) = +Inf if n is odd, -Inf if n is even 233 // Y1(n, x < 0) = NaN 234 // Y1(n, NaN) = NaN 235 func Yn(n int, x float64) float64 { 236 const Two302 = 1 << 302 // 2**302 0x52D0000000000000 237 // special cases 238 switch { 239 case x < 0 || IsNaN(x): 240 return NaN() 241 case IsInf(x, 1): 242 return 0 243 } 244 245 if n == 0 { 246 return Y0(x) 247 } 248 if x == 0 { 249 if n < 0 && n&1 == 1 { 250 return Inf(1) 251 } 252 return Inf(-1) 253 } 254 sign := false 255 if n < 0 { 256 n = -n 257 if n&1 == 1 { 258 sign = true // sign true if n < 0 && |n| odd 259 } 260 } 261 if n == 1 { 262 if sign { 263 return -Y1(x) 264 } 265 return Y1(x) 266 } 267 var b float64 268 if x >= Two302 { // x > 2**302 269 // (x >> n**2) 270 // Jn(x) = cos(x-(2n+1)*pi/4)*sqrt(2/x*pi) 271 // Yn(x) = sin(x-(2n+1)*pi/4)*sqrt(2/x*pi) 272 // Let s=sin(x), c=cos(x), 273 // xn=x-(2n+1)*pi/4, sqt2 = sqrt(2),then 274 // 275 // n sin(xn)*sqt2 cos(xn)*sqt2 276 // ---------------------------------- 277 // 0 s-c c+s 278 // 1 -s-c -c+s 279 // 2 -s+c -c-s 280 // 3 s+c c-s 281 282 var temp float64 283 switch n & 3 { 284 case 0: 285 temp = Sin(x) - Cos(x) 286 case 1: 287 temp = -Sin(x) - Cos(x) 288 case 2: 289 temp = -Sin(x) + Cos(x) 290 case 3: 291 temp = Sin(x) + Cos(x) 292 } 293 b = (1 / SqrtPi) * temp / Sqrt(x) 294 } else { 295 a := Y0(x) 296 b = Y1(x) 297 // quit if b is -inf 298 for i := 1; i < n && !IsInf(b, -1); i++ { 299 a, b = b, (float64(i+i)/x)*b-a 300 } 301 } 302 if sign { 303 return -b 304 } 305 return b 306 } 307
