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 /* @(#)e_jn.c 1.4 95/01/18 */ /* * ==================================================== * Copyright (C) 1993 by Sun Microsystems, Inc. All rights reserved. * * Developed at SunSoft, a Sun Microsystems, Inc. business. * Permission to use, copy, modify, and distribute this * software is freely granted, provided that this notice * is preserved. * ==================================================== */ /* * __ieee754_jn(n, x), __ieee754_yn(n, x) * floating point Bessel's function of the 1st and 2nd kind * of order n * * Special cases: * y0(0)=ieee_y1(0)=ieee_yn(n,0) = -inf with division by zero signal; * y0(-ve)=ieee_y1(-ve)=ieee_yn(n,-ve) are NaN with invalid signal. * Note 2. About ieee_jn(n,x), ieee_yn(n,x) * For n=0, ieee_j0(x) is called, * for n=1, ieee_j1(x) is called, * for nx, a continued fraction approximation to * j(n,x)/j(n-1,x) is evaluated and then backward * recursion is used starting from a supposed value * for j(n,x). The resulting value of j(0,x) is * compared with the actual value to correct the * supposed value of j(n,x). * * yn(n,x) is similar in all respects, except * that forward recursion is used for all * values of n>1. * */ #include "fdlibm.h" #ifdef __STDC__ static const double #else static double #endif invsqrtpi= 5.64189583547756279280e-01, /* 0x3FE20DD7, 0x50429B6D */ two = 2.00000000000000000000e+00, /* 0x40000000, 0x00000000 */ one = 1.00000000000000000000e+00; /* 0x3FF00000, 0x00000000 */ static double zero = 0.00000000000000000000e+00; #ifdef __STDC__ double __ieee754_jn(int n, double x) #else double __ieee754_jn(n,x) int n; double x; #endif { int i,hx,ix,lx, sgn; double a, b, temp, di; double z, w; /* J(-n,x) = (-1)^n * J(n, x), J(n, -x) = (-1)^n * J(n, x) * Thus, J(-n,x) = J(n,-x) */ hx = __HI(x); ix = 0x7fffffff&hx; lx = __LO(x); /* if J(n,NaN) is NaN */ if((ix|((unsigned)(lx|-lx))>>31)>0x7ff00000) return x+x; if(n<0){ n = -n; x = -x; hx ^= 0x80000000; } if(n==0) return(__ieee754_j0(x)); if(n==1) return(__ieee754_j1(x)); sgn = (n&1)&(hx>>31); /* even n -- 0, odd n -- sign(x) */ x = ieee_fabs(x); if((ix|lx)==0||ix>=0x7ff00000) /* if x is 0 or inf */ b = zero; else if((double)n<=x) { /* Safe to use J(n+1,x)=2n/x *J(n,x)-J(n-1,x) */ if(ix>=0x52D00000) { /* x > 2**302 */ /* (x >> n**2) * Jn(x) = ieee_cos(x-(2n+1)*pi/4)*ieee_sqrt(2/x*pi) * Yn(x) = ieee_sin(x-(2n+1)*pi/4)*ieee_sqrt(2/x*pi) * Let s=ieee_sin(x), c=ieee_cos(x), * xn=x-(2n+1)*pi/4, sqt2 = ieee_sqrt(2),then * * n sin(xn)*sqt2 cos(xn)*sqt2 * ---------------------------------- * 0 s-c c+s * 1 -s-c -c+s * 2 -s+c -c-s * 3 s+c c-s */ switch(n&3) { case 0: temp = ieee_cos(x)+ieee_sin(x); break; case 1: temp = -ieee_cos(x)+ieee_sin(x); break; case 2: temp = -ieee_cos(x)-ieee_sin(x); break; case 3: temp = ieee_cos(x)-ieee_sin(x); break; } b = invsqrtpi*temp/ieee_sqrt(x); } else { a = __ieee754_j0(x); b = __ieee754_j1(x); for(i=1;i33) /* underflow */ b = zero; else { temp = x*0.5; b = temp; for (a=one,i=2;i<=n;i++) { a *= (double)i; /* a = n! */ b *= temp; /* b = (x/2)^n */ } b = b/a; } } else { /* use backward recurrence */ /* x x^2 x^2 * J(n,x)/J(n-1,x) = ---- ------ ------ ..... * 2n - 2(n+1) - 2(n+2) * * 1 1 1 * (for large x) = ---- ------ ------ ..... * 2n 2(n+1) 2(n+2) * -- - ------ - ------ - * x x x * * Let w = 2n/x and h=2/x, then the above quotient * is equal to the continued fraction: * 1 * = ----------------------- * 1 * w - ----------------- * 1 * w+h - --------- * w+2h - ... * * To determine how many terms needed, let * Q(0) = w, Q(1) = w(w+h) - 1, * Q(k) = (w+k*h)*Q(k-1) - Q(k-2), * When Q(k) > 1e4 good for single * When Q(k) > 1e9 good for double * When Q(k) > 1e17 good for quadruple */ /* determine k */ double t,v; double q0,q1,h,tmp; int k,m; w = (n+n)/(double)x; h = 2.0/(double)x; q0 = w; z = w+h; q1 = w*z - 1.0; k=1; while(q1<1.0e9) { k += 1; z += h; tmp = z*q1 - q0; q0 = q1; q1 = tmp; } m = n+n; for(t=zero, i = 2*(n+k); i>=m; i -= 2) t = one/(i/x-t); a = t; b = one; /* estimate ieee_log((2/x)^n*n!) = n*ieee_log(2/x)+n*ln(n) * Hence, if n*(ieee_log(2n/x)) > ... * single 8.8722839355e+01 * double 7.09782712893383973096e+02 * long double 1.1356523406294143949491931077970765006170e+04 * then recurrent value may overflow and the result is * likely underflow to zero */ tmp = n; v = two/x; tmp = tmp*__ieee754_log(ieee_fabs(v*tmp)); if(tmp<7.09782712893383973096e+02) { for(i=n-1,di=(double)(i+i);i>0;i--){ temp = b; b *= di; b = b/x - a; a = temp; di -= two; } } else { for(i=n-1,di=(double)(i+i);i>0;i--){ temp = b; b *= di; b = b/x - a; a = temp; di -= two; /* scale b to avoid spurious overflow */ if(b>1e100) { a /= b; t /= b; b = one; } } } b = (t*__ieee754_j0(x)/b); } } if(sgn==1) return -b; else return b; } #ifdef __STDC__ double __ieee754_yn(int n, double x) #else double __ieee754_yn(n,x) int n; double x; #endif { int i,hx,ix,lx; int sign; double a, b, temp; hx = __HI(x); ix = 0x7fffffff&hx; lx = __LO(x); /* if Y(n,NaN) is NaN */ if((ix|((unsigned)(lx|-lx))>>31)>0x7ff00000) return x+x; if((ix|lx)==0) return -one/zero; if(hx<0) return zero/zero; sign = 1; if(n<0){ n = -n; sign = 1 - ((n&1)<<1); } if(n==0) return(__ieee754_y0(x)); if(n==1) return(sign*__ieee754_y1(x)); if(ix==0x7ff00000) return zero; if(ix>=0x52D00000) { /* x > 2**302 */ /* (x >> n**2) * Jn(x) = ieee_cos(x-(2n+1)*pi/4)*ieee_sqrt(2/x*pi) * Yn(x) = ieee_sin(x-(2n+1)*pi/4)*ieee_sqrt(2/x*pi) * Let s=ieee_sin(x), c=ieee_cos(x), * xn=x-(2n+1)*pi/4, sqt2 = ieee_sqrt(2),then * * n sin(xn)*sqt2 cos(xn)*sqt2 * ---------------------------------- * 0 s-c c+s * 1 -s-c -c+s * 2 -s+c -c-s * 3 s+c c-s */ switch(n&3) { case 0: temp = ieee_sin(x)-ieee_cos(x); break; case 1: temp = -ieee_sin(x)-ieee_cos(x); break; case 2: temp = -ieee_sin(x)+ieee_cos(x); break; case 3: temp = ieee_sin(x)+ieee_cos(x); break; } b = invsqrtpi*temp/ieee_sqrt(x); } else { a = __ieee754_y0(x); b = __ieee754_y1(x); /* quit if b is -inf */ for(i=1;i0) return b; else return -b; }