lib/libm/src/e_jn.c

   1 /* @(#)e_jn.c 5.1 93/09/24 */
   2 /*
   3  * ====================================================
   4  * Copyright (C) 1993 by Sun Microsystems, Inc. All rights reserved.
   5  *
   6  * Developed at SunPro, 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  * $NetBSD: e_jn.c,v 1.12 2002/05/26 22:01:50 wiz Exp $
  13  * $DragonFly: src/lib/libm/src/e_jn.c,v 1.1 2005/07/26 21:15:20 joerg Exp $
  14  */
  15
  16 /*
  17  * jn(n, x), yn(n, x)
  18  * floating point Bessel's function of the 1st and 2nd kind
  19  * of order n
  20  *
  21  * Special cases:
  22  *      y0(0)=y1(0)=yn(n,0) = -inf with division by zero signal;
  23  *      y0(-ve)=y1(-ve)=yn(n,-ve) are NaN with invalid signal.
  24  * Note 2. About jn(n,x), yn(n,x)
  25  *      For n=0, j0(x) is called,
  26  *      for n=1, j1(x) is called,
  27  *      for n<x, forward recursion us used starting
  28  *      from values of j0(x) and j1(x).
  29  *      for n>x, a continued fraction approximation to
  30  *      j(n,x)/j(n-1,x) is evaluated and then backward
  31  *      recursion is used starting from a supposed value
  32  *      for j(n,x). The resulting value of j(0,x) is
  33  *      compared with the actual value to correct the
  34  *      supposed value of j(n,x).
  35  *
  36  *      yn(n,x) is similar in all respects, except
  37  *      that forward recursion is used for all
  38  *      values of n>1.
  39  *
  40  */
  41
  42 #include <math.h>
  43 #include "math_private.h"
  44
  45 static const double
  46 invsqrtpi=  5.64189583547756279280e-01, /* 0x3FE20DD7, 0x50429B6D */
  47 two   =  2.00000000000000000000e+00, /* 0x40000000, 0x00000000 */
  48 one   =  1.00000000000000000000e+00; /* 0x3FF00000, 0x00000000 */
  49
  50 static const double zero  =  0.00000000000000000000e+00;
  51
  52 double
  53 jn(int n, double x)
  54 {
  55         int32_t i,hx,ix,lx, sgn;
  56         double a, b, temp, di;
  57         double z, w;
  58
  59         temp = 0;
  60     /* J(-n,x) = (-1)^n * J(n, x), J(n, -x) = (-1)^n * J(n, x)
  61      * Thus, J(-n,x) = J(n,-x)
  62      */
  63         EXTRACT_WORDS(hx,lx,x);
  64         ix = 0x7fffffff&hx;
  65     /* if J(n,NaN) is NaN */
  66         if((ix|((u_int32_t)(lx|-lx))>>31)>0x7ff00000) return x+x;
  67         if(n<0){
  68                 n = -n;
  69                 x = -x;
  70                 hx ^= 0x80000000;
  71         }
  72         if(n==0) return(j0(x));
  73         if(n==1) return(j1(x));
  74         sgn = (n&1)&(hx>>31);   /* even n -- 0, odd n -- sign(x) */
  75         x = fabs(x);
  76         if((ix|lx)==0||ix>=0x7ff00000)  /* if x is 0 or inf */
  77             b = zero;
  78         else if((double)n<=x) {
  79                 /* Safe to use J(n+1,x)=2n/x *J(n,x)-J(n-1,x) */
  80             if(ix>=0x52D00000) { /* x > 2**302 */
  81     /* (x >> n**2)
  82      *      Jn(x) = cos(x-(2n+1)*pi/4)*sqrt(2/x*pi)
  83      *      Yn(x) = sin(x-(2n+1)*pi/4)*sqrt(2/x*pi)
  84      *      Let s=sin(x), c=cos(x),
  85      *          xn=x-(2n+1)*pi/4, sqt2 = sqrt(2),then
  86      *
  87      *             n    sin(xn)*sqt2    cos(xn)*sqt2
  88      *          ----------------------------------
  89      *             0     s-c             c+s
  90      *             1    -s-c            -c+s
  91      *             2    -s+c            -c-s
  92      *             3     s+c             c-s
  93      */
  94                 switch(n&3) {
  95                     case 0: temp =  cos(x)+sin(x); break;
  96                     case 1: temp = -cos(x)+sin(x); break;
  97                     case 2: temp = -cos(x)-sin(x); break;
  98                     case 3: temp =  cos(x)-sin(x); break;
  99                 }
 100                 b = invsqrtpi*temp/sqrt(x);
 101             } else {
 102                 a = j0(x);
 103                 b = j1(x);
 104                 for(i=1;i<n;i++){
 105                     temp = b;
 106                     b = b*((double)(i+i)/x) - a; /* avoid underflow */
 107                     a = temp;
 108                 }
 109             }
 110         } else {
 111             if(ix<0x3e100000) { /* x < 2**-29 */
 112     /* x is tiny, return the first Taylor expansion of J(n,x)
 113      * J(n,x) = 1/n!*(x/2)^n  - ...
 114      */
 115                 if(n>33)        /* underflow */
 116                     b = zero;
 117                 else {
 118                     temp = x*0.5; b = temp;
 119                     for (a=one,i=2;i<=n;i++) {
 120                         a *= (double)i;         /* a = n! */
 121                         b *= temp;              /* b = (x/2)^n */
 122                     }
 123                     b = b/a;
 124                 }
 125             } else {
 126                 /* use backward recurrence */
 127                 /*                      x      x^2      x^2
 128                  *  J(n,x)/J(n-1,x) =  ----   ------   ------   .....
 129                  *                      2n  - 2(n+1) - 2(n+2)
 130                  *
 131                  *                      1      1        1
 132                  *  (for large x)   =  ----  ------   ------   .....
 133                  *                      2n   2(n+1)   2(n+2)
 134                  *                      -- - ------ - ------ -
 135                  *                       x     x         x
 136                  *
 137                  * Let w = 2n/x and h=2/x, then the above quotient
 138                  * is equal to the continued fraction:
 139                  *                  1
 140                  *      = -----------------------
 141                  *                     1
 142                  *         w - -----------------
 143                  *                        1
 144                  *              w+h - ---------
 145                  *                     w+2h - ...
 146                  *
 147                  * To determine how many terms needed, let
 148                  * Q(0) = w, Q(1) = w(w+h) - 1,
 149                  * Q(k) = (w+k*h)*Q(k-1) - Q(k-2),
 150                  * When Q(k) > 1e4      good for single
 151                  * When Q(k) > 1e9      good for double
 152                  * When Q(k) > 1e17     good for quadruple
 153                  */
 154             /* determine k */
 155                 double t,v;
 156                 double q0,q1,h,tmp; int32_t k,m;
 157                 w  = (n+n)/(double)x; h = 2.0/(double)x;
 158                 q0 = w;  z = w+h; q1 = w*z - 1.0; k=1;
 159                 while(q1<1.0e9) {
 160                         k += 1; z += h;
 161                         tmp = z*q1 - q0;
 162                         q0 = q1;
 163                         q1 = tmp;
 164                 }
 165                 m = n+n;
 166                 for(t=zero, i = 2*(n+k); i>=m; i -= 2) t = one/(i/x-t);
 167                 a = t;
 168                 b = one;
 169                 /*  estimate log((2/x)^n*n!) = n*log(2/x)+n*ln(n)
 170                  *  Hence, if n*(log(2n/x)) > ...
 171                  *  single 8.8722839355e+01
 172                  *  double 7.09782712893383973096e+02
 173                  *  long double 1.1356523406294143949491931077970765006170e+04
 174                  *  then recurrent value may overflow and the result is
 175                  *  likely underflow to zero
 176                  */
 177                 tmp = n;
 178                 v = two/x;
 179                 tmp = tmp*log(fabs(v*tmp));
 180                 if(tmp<7.09782712893383973096e+02) {
 181                     for(i=n-1,di=(double)(i+i);i>0;i--){
 182                         temp = b;
 183                         b *= di;
 184                         b  = b/x - a;
 185                         a = temp;
 186                         di -= two;
 187                     }
 188                 } else {
 189                     for(i=n-1,di=(double)(i+i);i>0;i--){
 190                         temp = b;
 191                         b *= di;
 192                         b  = b/x - a;
 193                         a = temp;
 194                         di -= two;
 195                     /* scale b to avoid spurious overflow */
 196                         if(b>1e100) {
 197                             a /= b;
 198                             t /= b;
 199                             b  = one;
 200                         }
 201                     }
 202                 }
 203                 b = (t*j0(x)/b);
 204             }
 205         }
 206         if(sgn==1) return -b; else return b;
 207 }
 208
 209 double
 210 yn(int n, double x)
 211 {
 212         int32_t i,hx,ix,lx;
 213         int32_t sign;
 214         double a, b, temp;
 215
 216         temp = 0;
 217         EXTRACT_WORDS(hx,lx,x);
 218         ix = 0x7fffffff&hx;
 219     /* if Y(n,NaN) is NaN */
 220         if((ix|((u_int32_t)(lx|-lx))>>31)>0x7ff00000) return x+x;
 221         if((ix|lx)==0) return -one/zero;
 222         if(hx<0) return zero/zero;
 223         sign = 1;
 224         if(n<0){
 225                 n = -n;
 226                 sign = 1 - ((n&1)<<1);
 227         }
 228         if(n==0) return(y0(x));
 229         if(n==1) return(sign*y1(x));
 230         if(ix==0x7ff00000) return zero;
 231         if(ix>=0x52D00000) { /* x > 2**302 */
 232     /* (x >> n**2)
 233      *      Jn(x) = cos(x-(2n+1)*pi/4)*sqrt(2/x*pi)
 234      *      Yn(x) = sin(x-(2n+1)*pi/4)*sqrt(2/x*pi)
 235      *      Let s=sin(x), c=cos(x),
 236      *          xn=x-(2n+1)*pi/4, sqt2 = sqrt(2),then
 237      *
 238      *             n    sin(xn)*sqt2    cos(xn)*sqt2
 239      *          ----------------------------------
 240      *             0     s-c             c+s
 241      *             1    -s-c            -c+s
 242      *             2    -s+c            -c-s
 243      *             3     s+c             c-s
 244      */
 245                 switch(n&3) {
 246                     case 0: temp =  sin(x)-cos(x); break;
 247                     case 1: temp = -sin(x)-cos(x); break;
 248                     case 2: temp = -sin(x)+cos(x); break;
 249                     case 3: temp =  sin(x)+cos(x); break;
 250                 }
 251                 b = invsqrtpi*temp/sqrt(x);
 252         } else {
 253             u_int32_t high;
 254             a = y0(x);
 255             b = y1(x);
 256         /* quit if b is -inf */
 257             GET_HIGH_WORD(high,b);
 258             for(i=1;i<n&&high!=0xfff00000;i++){
 259                 temp = b;
 260                 b = ((double)(i+i)/x)*b - a;
 261                 GET_HIGH_WORD(high,b);
 262                 a = temp;
 263             }
 264         }
 265         if(sign>0) return b; else return -b;
 266 }