source: trunk/sys/libm/e_jn.c @ 80

/* @(#)e_jn.c 5.1 93/09/24 */
/*
 * ====================================================
 * Copyright (C) 1993 by Sun Microsystems, Inc. All rights reserved.
 *
 * Developed at SunPro, 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)=y1(0)=yn(n,0) = -inf with division by zero signal;
 *      y0(-ve)=y1(-ve)=yn(n,-ve) are NaN with invalid signal.
 * Note 2. About jn(n,x), yn(n,x)
 *      For n=0, j0(x) is called,
 *      for n=1, j1(x) is called,
 *      for n<x, forward recursion us used starting
 *      from values of j0(x) and j1(x).
 *      for n>x, 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 <libm/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,n0,hx,ix,lx, sgn;
        double a, b, temp=0, 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)
     */
        n0 = 1^((*(int*)&one)>>29);
        hx = *(n0+(int*)&x);
        ix = 0x7fffffff&hx;
        lx = *(1-n0+(int*)&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 = 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) = cos(x-(2n+1)*pi/4)*sqrt(2/x*pi)
     *      Yn(x) = sin(x-(2n+1)*pi/4)*sqrt(2/x*pi)
     *      Let s=sin(x), c=cos(x), 
     *          xn=x-(2n+1)*pi/4, sqt2 = 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 =  cos(x)+sin(x); break;
                    case 1: temp = -cos(x)+sin(x); break;
                    case 2: temp = -cos(x)-sin(x); break;
                    case 3: temp =  cos(x)-sin(x); break;
                }
                b = invsqrtpi*temp/sqrt(x);
            } else {    
                a = __ieee754_j0(x);
                b = __ieee754_j1(x);
                for(i=1;i<n;i++){
                    temp = b;
                    b = b*((double)(i+i)/x) - a; /* avoid underflow */
                    a = temp;
                }
            }
        } else {
            if(ix<0x3e100000) { /* x < 2**-29 */
    /* x is tiny, return the first Taylor expansion of J(n,x) 
     * J(n,x) = 1/n!*(x/2)^n  - ...
     */
                if(n>33)        /* 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 log((2/x)^n*n!) = n*log(2/x)+n*ln(n)
                 *  Hence, if n*(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(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,n0,hx,ix,lx;
        int sign;
        double a, b, temp=0;

        n0 = 1^((*(int*)&one)>>29);
        hx = *(n0+(int*)&x);
        ix = 0x7fffffff&hx;
        lx = *(1-n0+(int*)&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)<<2);
        }
        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) = cos(x-(2n+1)*pi/4)*sqrt(2/x*pi)
     *      Yn(x) = sin(x-(2n+1)*pi/4)*sqrt(2/x*pi)
     *      Let s=sin(x), c=cos(x), 
     *          xn=x-(2n+1)*pi/4, sqt2 = 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 =  sin(x)-cos(x); break;
                    case 1: temp = -sin(x)-cos(x); break;
                    case 2: temp = -sin(x)+cos(x); break;
                    case 3: temp =  sin(x)+cos(x); break;
                }
                b = invsqrtpi*temp/sqrt(x);
        } else {
            a = __ieee754_y0(x);
            b = __ieee754_y1(x);
        /* quit if b is -inf */
            for(i=1;i<n && ((unsigned int)(*(n0+(int*)&b))!= 0xfff00000);i++){ 
                temp = b;
                b = ((double)(i+i)/x)*b - a;
                a = temp;
            }
        }
        if(sign>0) return b; else return -b;
}