11297Salm/* origin: FreeBSD /usr/src/lib/msun/src/e_jn.c */ 21297Salm/* 31297Salm * ==================================================== 41297Salm * Copyright (C) 1993 by Sun Microsystems, Inc. All rights reserved. 51297Salm * 61297Salm * Developed at SunSoft, a Sun Microsystems, Inc. business. 71297Salm * Permission to use, copy, modify, and distribute this 81297Salm * software is freely granted, provided that this notice 91297Salm * is preserved. 101297Salm * ==================================================== 111297Salm */ 121297Salm/* 131297Salm * jn(n, x), yn(n, x) 141297Salm * floating point Bessel's function of the 1st and 2nd kind 151297Salm * of order n 161297Salm * 171297Salm * Special cases: 181297Salm * y0(0)=y1(0)=yn(n,0) = -inf with division by zero signal; 191297Salm * y0(-ve)=y1(-ve)=yn(n,-ve) are NaN with invalid signal. 201297Salm * Note 2. About jn(n,x), yn(n,x) 211297Salm * For n=0, j0(x) is called, 221297Salm * for n=1, j1(x) is called, 231297Salm * for n<=x, forward recursion is used starting 241297Salm * from values of j0(x) and j1(x). 251297Salm * for n>x, a continued fraction approximation to 261297Salm * j(n,x)/j(n-1,x) is evaluated and then backward 271297Salm * recursion is used starting from a supposed value 281297Salm * for j(n,x). The resulting value of j(0,x) is 291297Salm * compared with the actual value to correct the 3081220Smike * supposed value of j(n,x). 3150471Speter * 321297Salm * yn(n,x) is similar in all respects, except 331297Salm * that forward recursion is used for all 3467183Sbrian * values of n>1. 3567183Sbrian */ 361297Salm 371297Salm#include "libm.h" 381297Salm 391297Salmstatic const double invsqrtpi = 5.64189583547756279280e-01; /* 0x3FE20DD7, 0x50429B6D */ 401297Salm 411297Salmdouble jn(int n, double x) 421297Salm{ 431297Salm uint32_t ix, lx; 441297Salm int nm1, i, sign; 451297Salm double a, b, temp; 461297Salm 471297Salm EXTRACT_WORDS(ix, lx, x); 481297Salm sign = ix>>31; 491297Salm ix &= 0x7fffffff; 501297Salm 511297Salm if ((ix | (lx|-lx)>>31) > 0x7ff00000) /* nan */ 521297Salm return x; 531297Salm 5481220Smike /* J(-n,x) = (-1)^n * J(n, x), J(n, -x) = (-1)^n * J(n, x) 551297Salm * Thus, J(-n,x) = J(n,-x) 561297Salm */ 571297Salm /* nm1 = |n|-1 is used instead of |n| to handle n==INT_MIN */ 581297Salm if (n == 0) 591297Salm return j0(x); 601297Salm if (n < 0) { 611297Salm nm1 = -(n+1); 621297Salm x = -x; 631297Salm sign ^= 1; 641297Salm } else 651297Salm nm1 = n-1; 661297Salm if (nm1 == 0) 671297Salm return j1(x); 681297Salm 691297Salm sign &= n; /* even n: 0, odd n: signbit(x) */ 701297Salm x = fabs(x); 711297Salm if ((ix|lx) == 0 || ix == 0x7ff00000) /* if x is 0 or inf */ 721297Salm b = 0.0; 731297Salm else if (nm1 < x) { 741297Salm /* Safe to use J(n+1,x)=2n/x *J(n,x)-J(n-1,x) */ 751297Salm if (ix >= 0x52d00000) { /* x > 2**302 */ 761297Salm /* (x >> n**2) 771297Salm * Jn(x) = cos(x-(2n+1)*pi/4)*sqrt(2/x*pi) 781297Salm * Yn(x) = sin(x-(2n+1)*pi/4)*sqrt(2/x*pi) 791297Salm * Let s=sin(x), c=cos(x), 801297Salm * xn=x-(2n+1)*pi/4, sqt2 = sqrt(2),then 811297Salm * 821297Salm * n sin(xn)*sqt2 cos(xn)*sqt2 831297Salm * ---------------------------------- 841297Salm * 0 s-c c+s 851297Salm * 1 -s-c -c+s 861297Salm * 2 -s+c -c-s 871297Salm * 3 s+c c-s 881297Salm */ 891297Salm switch(nm1&3) { 901297Salm case 0: temp = -cos(x)+sin(x); break; 911297Salm case 1: temp = -cos(x)-sin(x); break; 921297Salm case 2: temp = cos(x)-sin(x); break; 931297Salm default: 941297Salm case 3: temp = cos(x)+sin(x); break; 951297Salm } 961297Salm b = invsqrtpi*temp/sqrt(x); 971297Salm } else { 981297Salm a = j0(x); 991297Salm b = j1(x); 1001297Salm for (i=0; i<nm1; ) { 1011297Salm i++; 1021297Salm temp = b; 1031297Salm b = b*(2.0*i/x) - a; /* avoid underflow */ 1041297Salm a = temp; 1051297Salm } 1061297Salm } 1071297Salm } else { 1081297Salm if (ix < 0x3e100000) { /* x < 2**-29 */ 1091297Salm /* x is tiny, return the first Taylor expansion of J(n,x) 1101297Salm * J(n,x) = 1/n!*(x/2)^n - ... 1111297Salm */ 1121297Salm if (nm1 > 32) /* underflow */ 1131297Salm b = 0.0; 11481220Smike else { 1151297Salm temp = x*0.5; 1161297Salm b = temp; 1171297Salm a = 1.0; 1181297Salm for (i=2; i<=nm1+1; i++) { 1191297Salm a *= (double)i; /* a = n! */ 12081220Smike b *= temp; /* b = (x/2)^n */ 1211297Salm } 1221297Salm b = b/a; 1231297Salm } 1241297Salm } else { 1251297Salm /* use backward recurrence */ 1261297Salm /* x x^2 x^2 1271297Salm * J(n,x)/J(n-1,x) = ---- ------ ------ ..... 1281297Salm * 2n - 2(n+1) - 2(n+2) 1291297Salm * 1301297Salm * 1 1 1 1311297Salm * (for large x) = ---- ------ ------ ..... 1321297Salm * 2n 2(n+1) 2(n+2) 1331297Salm * -- - ------ - ------ - 1341297Salm * x x x 1351297Salm * 1367165Sjoerg * Let w = 2n/x and h=2/x, then the above quotient 1377165Sjoerg * is equal to the continued fraction: 1381297Salm * 1 1391297Salm * = ----------------------- 1401297Salm * 1 1411297Salm * w - ----------------- 1421297Salm * 1 1431297Salm * w+h - --------- 1441297Salm * w+2h - ... 1451297Salm * 1461297Salm * To determine how many terms needed, let 1471297Salm * Q(0) = w, Q(1) = w(w+h) - 1, 1481297Salm * Q(k) = (w+k*h)*Q(k-1) - Q(k-2), 1491297Salm * When Q(k) > 1e4 good for single 1501297Salm * When Q(k) > 1e9 good for double 1511297Salm * When Q(k) > 1e17 good for quadruple 1521297Salm */ 1531297Salm /* determine k */ 1541297Salm double t,q0,q1,w,h,z,tmp,nf; 1551297Salm int k; 1561297Salm 1571297Salm nf = nm1 + 1.0; 1581297Salm w = 2*nf/x; 1591297Salm h = 2/x; 1601297Salm z = w+h; 1611297Salm q0 = w; 1628855Srgrimes q1 = w*z - 1.0; 1631297Salm k = 1; 1641297Salm while (q1 < 1.0e9) { 16581220Smike k += 1; 1661297Salm z += h; 1671297Salm tmp = z*q1 - q0; 1681297Salm q0 = q1; 1691297Salm q1 = tmp; 1701297Salm } 1718855Srgrimes for (t=0.0, i=k; i>=0; i--) 1721297Salm t = 1/(2*(i+nf)/x - t); 1731297Salm a = t; 17481220Smike b = 1.0; 1751297Salm /* estimate log((2/x)^n*n!) = n*log(2/x)+n*ln(n) 1761297Salm * Hence, if n*(log(2n/x)) > ... 1771297Salm * single 8.8722839355e+01 1781297Salm * double 7.09782712893383973096e+02 1791297Salm * long double 1.1356523406294143949491931077970765006170e+04 1801297Salm * then recurrent value may overflow and the result is 1811297Salm * likely underflow to zero 1821297Salm */ 1831297Salm tmp = nf*log(fabs(w)); 1841297Salm if (tmp < 7.09782712893383973096e+02) { 1851297Salm for (i=nm1; i>0; i--) { 1861297Salm temp = b; 1871297Salm b = b*(2.0*i)/x - a; 1881297Salm a = temp; 1891297Salm } 1901297Salm } else { 1911297Salm for (i=nm1; i>0; i--) { 1921297Salm temp = b; 1931297Salm b = b*(2.0*i)/x - a; 1941297Salm a = temp; 1951297Salm /* scale b to avoid spurious overflow */ 1961297Salm if (b > 0x1p500) { 1971297Salm a /= b; 1981297Salm t /= b; 1991297Salm b = 1.0; 2001297Salm } 2011297Salm } 2021297Salm } 2031297Salm z = j0(x); 2041297Salm w = j1(x); 2051297Salm if (fabs(z) >= fabs(w)) 2061297Salm b = t*z/b; 2071297Salm else 2081297Salm b = t*w/a; 2091297Salm } 2101297Salm } 2111297Salm return sign ? -b : b; 2121297Salm} 2131297Salm 2141297Salm 2151297Salmdouble yn(int n, double x) 2161297Salm{ 2171297Salm uint32_t ix, lx, ib; 2181297Salm int nm1, sign, i; 2191297Salm double a, b, temp; 2201297Salm 2211297Salm EXTRACT_WORDS(ix, lx, x); 2221297Salm sign = ix>>31; 2231297Salm ix &= 0x7fffffff; 2241297Salm 2251297Salm if ((ix | (lx|-lx)>>31) > 0x7ff00000) /* nan */ 2261297Salm 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