base/math/e_jn.c

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/* @(#)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.
 * ====================================================
 */

#if defined(LIBM_SCCS) && !defined(lint)
static char rcsid[] = "$NetBSD: e_jn.c,v 1.9 1995/05/10 20:45:34 jtc Exp $";
#endif

/*
 * __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 "math.h"
#include "mathP.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 */

#ifdef __STDC__
static const double zero  =  0.00000000000000000000e+00;
#else
static double zero  =  0.00000000000000000000e+00;
#endif

#ifdef __STDC__
    double __ieee754_jn(int n, double x)
#else
    double __ieee754_jn(n,x)
    int n; double x;
#endif
{
    int32_t 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)
     */
    EXTRACT_WORDS(hx,lx,x);
    ix = 0x7fffffff&hx;
    /* if J(n,NaN) is NaN */
    if((ix|((u_int32_t)(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;
            default: temp = 0.0;
        }
        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; int32_t 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
{
    int32_t i,hx,ix,lx;
    int32_t sign;
    double a, b, temp;

    EXTRACT_WORDS(hx,lx,x);
    ix = 0x7fffffff&hx;
    /* if Y(n,NaN) is NaN */
    if((ix|((u_int32_t)(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) = 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;
            default: temp = 0.0;
        }
        b = invsqrtpi*temp/sqrt(x);
    } else {
        u_int32_t high;
        a = __ieee754_y0(x);
        b = __ieee754_y1(x);
    /* quit if b is -inf */
        GET_HIGH_WORD(high,b);
        for(i=1;i<n&&high!=0xfff00000;i++){ 
        temp = b;
        b = ((double)(i+i)/x)*b - a;
        GET_HIGH_WORD(high,b);
        a = temp;
        }
    }
    if(sign>0) return b; else return -b;
}

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