1 /* origin: FreeBSD /usr/src/lib/msun/src/e_jn.c */ 2 /* 3 * ==================================================== 4 * Copyright (C) 1993 by Sun Microsystems, Inc. All rights reserved. 5 * 6 * Developed at SunSoft, a Sun Microsystems, Inc. business. 7 * Permission to use, copy, modify, and distribute this 8 * software is freely granted, provided that this notice 9 * is preserved. 10 * ==================================================== 11 */ 12 /* 13 * jn(n, x), yn(n, x) 14 * floating point Bessel's function of the 1st and 2nd kind 15 * of order n 16 * 17 * Special cases: 18 * y0(0)=y1(0)=yn(n,0) = -inf with division by zero signal; 19 * y0(-ve)=y1(-ve)=yn(n,-ve) are NaN with invalid signal. 20 * Note 2. About jn(n,x), yn(n,x) 21 * For n=0, j0(x) is called, 22 * for n=1, j1(x) is called, 23 * for n<=x, forward recursion is used starting 24 * from values of j0(x) and j1(x). 25 * for n>x, a continued fraction approximation to 26 * j(n,x)/j(n-1,x) is evaluated and then backward 27 * recursion is used starting from a supposed value 28 * for j(n,x). The resulting value of j(0,x) is 29 * compared with the actual value to correct the 30 * supposed value of j(n,x). 31 * 32 * yn(n,x) is similar in all respects, except 33 * that forward recursion is used for all 34 * values of n>1. 35 */ 36 37 #include "libm.h" 38 39 static const double invsqrtpi = 5.64189583547756279280e-01; /* 0x3FE20DD7, 0x50429B6D */ 40 41 double jn(int n, double x) 42 { 43 uint32_t ix, lx; 44 int nm1, i, sign; 45 double a, b, temp; 46 47 EXTRACT_WORDS(ix, lx, x); 48 sign = ix>>31; 49 ix &= 0x7fffffff; 50 51 if ((ix | (lx|-lx)>>31) > 0x7ff00000) /* nan */ 52 return x; 53 54 /* J(-n,x) = (-1)^n * J(n, x), J(n, -x) = (-1)^n * J(n, x) 55 * Thus, J(-n,x) = J(n,-x) 56 */ 57 /* nm1 = |n|-1 is used instead of |n| to handle n==INT_MIN */ 58 if (n == 0) 59 return j0(x); 60 if (n < 0) { 61 nm1 = -(n+1); 62 x = -x; 63 sign ^= 1; 64 } else 65 nm1 = n-1; 66 if (nm1 == 0) 67 return j1(x); 68 69 sign &= n; /* even n: 0, odd n: signbit(x) */ 70 x = fabs(x); 71 if ((ix|lx) == 0 || ix == 0x7ff00000) /* if x is 0 or inf */ 72 b = 0.0; 73 else if (nm1 < x) { 74 /* Safe to use J(n+1,x)=2n/x *J(n,x)-J(n-1,x) */ 75 if (ix >= 0x52d00000) { /* x > 2**302 */ 76 /* (x >> n**2) 77 * Jn(x) = cos(x-(2n+1)*pi/4)*sqrt(2/x*pi) 78 * Yn(x) = sin(x-(2n+1)*pi/4)*sqrt(2/x*pi) 79 * Let s=sin(x), c=cos(x), 80 * xn=x-(2n+1)*pi/4, sqt2 = sqrt(2),then 81 * 82 * n sin(xn)*sqt2 cos(xn)*sqt2 83 * ---------------------------------- 84 * 0 s-c c+s 85 * 1 -s-c -c+s 86 * 2 -s+c -c-s 87 * 3 s+c c-s 88 */ 89 switch(nm1&3) { 90 case 0: temp = -cos(x)+sin(x); break; 91 case 1: temp = -cos(x)-sin(x); break; 92 case 2: temp = cos(x)-sin(x); break; 93 default: 94 case 3: temp = cos(x)+sin(x); break; 95 } 96 b = invsqrtpi*temp/sqrt(x); 97 } else { 98 a = j0(x); 99 b = j1(x); 100 for (i=0; i<nm1; ) { 101 i++; 102 temp = b; 103 b = b*(2.0*i/x) - a; /* avoid underflow */ 104 a = temp; 105 } 106 } 107 } else { 108 if (ix < 0x3e100000) { /* x < 2**-29 */ 109 /* x is tiny, return the first Taylor expansion of J(n,x) 110 * J(n,x) = 1/n!*(x/2)^n - ... 111 */ 112 if (nm1 > 32) /* underflow */ 113 b = 0.0; 114 else { 115 temp = x*0.5; 116 b = temp; 117 a = 1.0; 118 for (i=2; i<=nm1+1; i++) { 119 a *= (double)i; /* a = n! */ 120 b *= temp; /* b = (x/2)^n */ 121 } 122 b = b/a; 123 } 124 } else { 125 /* use backward recurrence */ 126 /* x x^2 x^2 127 * J(n,x)/J(n-1,x) = ---- ------ ------ ..... 128 * 2n - 2(n+1) - 2(n+2) 129 * 130 * 1 1 1 131 * (for large x) = ---- ------ ------ ..... 132 * 2n 2(n+1) 2(n+2) 133 * -- - ------ - ------ - 134 * x x x 135 * 136 * Let w = 2n/x and h=2/x, then the above quotient 137 * is equal to the continued fraction: 138 * 1 139 * = ----------------------- 140 * 1 141 * w - ----------------- 142 * 1 143 * w+h - --------- 144 * w+2h - ... 145 * 146 * To determine how many terms needed, let 147 * Q(0) = w, Q(1) = w(w+h) - 1, 148 * Q(k) = (w+k*h)*Q(k-1) - Q(k-2), 149 * When Q(k) > 1e4 good for single 150 * When Q(k) > 1e9 good for double 151 * When Q(k) > 1e17 good for quadruple 152 */ 153 /* determine k */ 154 double t,q0,q1,w,h,z,tmp,nf; 155 int k; 156 157 nf = nm1 + 1.0; 158 w = 2*nf/x; 159 h = 2/x; 160 z = w+h; 161 q0 = w; 162 q1 = w*z - 1.0; 163 k = 1; 164 while (q1 < 1.0e9) { 165 k += 1; 166 z += h; 167 tmp = z*q1 - q0; 168 q0 = q1; 169 q1 = tmp; 170 } 171 for (t=0.0, i=k; i>=0; i--) 172 t = 1/(2*(i+nf)/x - t); 173 a = t; 174 b = 1.0; 175 /* estimate log((2/x)^n*n!) = n*log(2/x)+n*ln(n) 176 * Hence, if n*(log(2n/x)) > ... 177 * single 8.8722839355e+01 178 * double 7.09782712893383973096e+02 179 * long double 1.1356523406294143949491931077970765006170e+04 180 * then recurrent value may overflow and the result is 181 * likely underflow to zero 182 */ 183 tmp = nf*log(fabs(w)); 184 if (tmp < 7.09782712893383973096e+02) { 185 for (i=nm1; i>0; i--) { 186 temp = b; 187 b = b*(2.0*i)/x - a; 188 a = temp; 189 } 190 } else { 191 for (i=nm1; i>0; i--) { 192 temp = b; 193 b = b*(2.0*i)/x - a; 194 a = temp; 195 /* scale b to avoid spurious overflow */ 196 if (b > 0x1p500) { 197 a /= b; 198 t /= b; 199 b = 1.0; 200 } 201 } 202 } 203 z = j0(x); 204 w = j1(x); 205 if (fabs(z) >= fabs(w)) 206 b = t*z/b; 207 else 208 b = t*w/a; 209 } 210 } 211 return sign ? -b : b; 212 } 213 214 215 double yn(int n, double x) 216 { 217 uint32_t ix, lx, ib; 218 int nm1, sign, i; 219 double a, b, temp; 220 221 EXTRACT_WORDS(ix, lx, x); 222 sign = ix>>31; 223 ix &= 0x7fffffff; 224 225 if ((ix | (lx|-lx)>>31) > 0x7ff00000) /* nan */ 226 return x; 227 if (sign && (ix|lx)!=0) /* x < 0 */ 228 return 0/0.0; 229 if (ix == 0x7ff00000) 230 return 0.0; 231 232 if (n == 0) 233 return y0(x); 234 if (n < 0) { 235 nm1 = -(n+1); 236 sign = n&1; 237 } else { 238 nm1 = n-1; 239 sign = 0; 240 } 241 if (nm1 == 0) 242 return sign ? -y1(x) : y1(x); 243 244 if (ix >= 0x52d00000) { /* x > 2**302 */ 245 /* (x >> n**2) 246 * Jn(x) = cos(x-(2n+1)*pi/4)*sqrt(2/x*pi) 247 * Yn(x) = sin(x-(2n+1)*pi/4)*sqrt(2/x*pi) 248 * Let s=sin(x), c=cos(x), 249 * xn=x-(2n+1)*pi/4, sqt2 = sqrt(2),then 250 * 251 * n sin(xn)*sqt2 cos(xn)*sqt2 252 * ---------------------------------- 253 * 0 s-c c+s 254 * 1 -s-c -c+s 255 * 2 -s+c -c-s 256 * 3 s+c c-s 257 */ 258 switch(nm1&3) { 259 case 0: temp = -sin(x)-cos(x); break; 260 case 1: temp = -sin(x)+cos(x); break; 261 case 2: temp = sin(x)+cos(x); break; 262 default: 263 case 3: temp = sin(x)-cos(x); break; 264 } 265 b = invsqrtpi*temp/sqrt(x); 266 } else { 267 a = y0(x); 268 b = y1(x); 269 /* quit if b is -inf */ 270 GET_HIGH_WORD(ib, b); 271 for (i=0; i<nm1 && ib!=0xfff00000; ){ 272 i++; 273 temp = b; 274 b = (2.0*i/x)*b - a; 275 GET_HIGH_WORD(ib, b); 276 a = temp; 277 } 278 } 279 return sign ? -b : b; 280 } 281