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 1.14 2010/11/29 15:10:06 drochner Exp $
  13  */
  14
  15 /*
  16  * jn(n, x), yn(n, x)
  17  * floating point Bessel's function of the 1st and 2nd kind
  18  * of order n
  19  *
  20  * Special cases:
  21  *      y0(0)=y1(0)=yn(n,0) = -inf with division by zero signal;
  22  *      y0(-ve)=y1(-ve)=yn(n,-ve) are NaN with invalid signal.
  23  * Note 2. About jn(n,x), yn(n,x)
  24  *      For n=0, j0(x) is called,
  25  *      for n=1, j1(x) is called,
  26  *      for n<x, forward recursion us used starting
  27  *      from values of j0(x) and j1(x).
  28  *      for n>x, a continued fraction approximation to
  29  *      j(n,x)/j(n-1,x) is evaluated and then backward
  30  *      recursion is used starting from a supposed value
  31  *      for j(n,x). The resulting value of j(0,x) is
  32  *      compared with the actual value to correct the
  33  *      supposed value of j(n,x).
  34  *
  35  *      yn(n,x) is similar in all respects, except
  36  *      that forward recursion is used for all
  37  *      values of n>1.
  38  *
  39  */
  40
  41 #include <math.h>
  42 #include "math_private.h"
  43
  44 static const double
  45 invsqrtpi=  5.64189583547756279280e-01, /* 0x3FE20DD7, 0x50429B6D */
  46 two   =  2.00000000000000000000e+00, /* 0x40000000, 0x00000000 */
  47 one   =  1.00000000000000000000e+00; /* 0x3FF00000, 0x00000000 */
  48
  49 static const double zero  =  0.00000000000000000000e+00;
  50
  51 double
  52 jn(int n, double x)
  53 {
  54         int32_t i,hx,ix,lx, sgn;
  55         double a, b, temp, di;
  56         double z, w;
  57
  58         temp = 0;
  59     /* J(-n,x) = (-1)^n * J(n, x), J(n, -x) = (-1)^n * J(n, x)
  60      * Thus, J(-n,x) = J(n,-x)
  61      */
  62         EXTRACT_WORDS(hx,lx,x);
  63         ix = 0x7fffffff&hx;
  64     /* if J(n,NaN) is NaN */
  65         if((ix|((u_int32_t)(lx|-lx))>>31)>0x7ff00000) return x+x;
  66         if(n<0){
  67                 n = -n;
  68                 x = -x;
  69                 hx ^= 0x80000000;
  70         }
  71         if(n==0) return(j0(x));
  72         if(n==1) return(j1(x));
  73         sgn = (n&1)&(hx>>31);   /* even n -- 0, odd n -- sign(x) */
  74         x = fabs(x);
  75         if((ix|lx)==0||ix>=0x7ff00000)  /* if x is 0 or inf */
  76             b = zero;
  77         else if((double)n<=x) {
  78                 /* Safe to use J(n+1,x)=2n/x *J(n,x)-J(n-1,x) */
  79             if(ix>=0x52D00000) { /* x > 2**302 */
  80     /* (x >> n**2)
  81      *      Jn(x) = cos(x-(2n+1)*pi/4)*sqrt(2/x*pi)
  82      *      Yn(x) = sin(x-(2n+1)*pi/4)*sqrt(2/x*pi)
  83      *      Let s=sin(x), c=cos(x),
  84      *          xn=x-(2n+1)*pi/4, sqt2 = sqrt(2),then
  85      *
  86      *             n    sin(xn)*sqt2    cos(xn)*sqt2
  87      *          ----------------------------------
  88      *             0     s-c             c+s
  89      *             1    -s-c            -c+s
  90      *             2    -s+c            -c-s
  91      *             3     s+c             c-s
  92      */
  93                 switch(n&3) {
  94                     case 0: temp =  cos(x)+sin(x); break;
  95                     case 1: temp = -cos(x)+sin(x); break;
  96                     case 2: temp = -cos(x)-sin(x); break;
  97                     case 3: temp =  cos(x)-sin(x); break;
  98                 }
  99                 b = invsqrtpi*temp/sqrt(x);
 100             } else {
 101                 a = j0(x);
 102                 b = j1(x);
 103                 for(i=1;i<n;i++){
 104                     temp = b;
 105                     b = b*((double)(i+i)/x) - a; /* avoid underflow */
 106                     a = temp;
 107                 }
 108             }
 109         } else {
 110             if(ix<0x3e100000) { /* x < 2**-29 */
 111     /* x is tiny, return the first Taylor expansion of J(n,x)
 112      * J(n,x) = 1/n!*(x/2)^n  - ...
 113      */
 114                 if(n>33)        /* underflow */
 115                     b = zero;
 116                 else {
 117                     temp = x*0.5; b = temp;
 118                     for (a=one,i=2;i<=n;i++) {
 119                         a *= (double)i;         /* a = n! */
 120                         b *= temp;              /* b = (x/2)^n */
 121                     }
 122                     b = b/a;
 123                 }
 124             } else {
 125                 /* use backward recurrence */
 126                 /*                      x      x^2      x^2
 127                  *  J(n,x)/J(n-1,x) =  ----   ------   ------   .....
 128                  *                      2n  - 2(n+1) - 2(n+2)
 129                  *
 130                  *                      1      1        1
 131                  *  (for large x)   =  ----  ------   ------   .....
 132                  *                      2n   2(n+1)   2(n+2)
 133                  *                      -- - ------ - ------ -
 134                  *                       x     x         x
 135                  *
 136                  * Let w = 2n/x and h=2/x, then the above quotient
 137                  * is equal to the continued fraction:
 138                  *                  1
 139                  *      = -----------------------
 140                  *                     1
 141                  *         w - -----------------
 142                  *                        1
 143                  *              w+h - ---------
 144                  *                     w+2h - ...
 145                  *
 146                  * To determine how many terms needed, let
 147                  * Q(0) = w, Q(1) = w(w+h) - 1,
 148                  * Q(k) = (w+k*h)*Q(k-1) - Q(k-2),
 149                  * When Q(k) > 1e4      good for single
 150                  * When Q(k) > 1e9      good for double
 151                  * When Q(k) > 1e17     good for quadruple
 152                  */
 153             /* determine k */
 154                 double t,v;
 155                 double q0,q1,h,tmp; int32_t k,m;
 156                 w  = (n+n)/(double)x; h = 2.0/(double)x;
 157                 q0 = w;  z = w+h; q1 = w*z - 1.0; k=1;
 158                 while(q1<1.0e9) {
 159                         k += 1; z += h;
 160                         tmp = z*q1 - q0;
 161                         q0 = q1;
 162                         q1 = tmp;
 163                 }
 164                 m = n+n;
 165                 for(t=zero, i = 2*(n+k); i>=m; i -= 2) t = one/(i/x-t);
 166                 a = t;
 167                 b = one;
 168                 /*  estimate log((2/x)^n*n!) = n*log(2/x)+n*ln(n)
 169                  *  Hence, if n*(log(2n/x)) > ...
 170                  *  single 8.8722839355e+01
 171                  *  double 7.09782712893383973096e+02
 172                  *  long double 1.1356523406294143949491931077970765006170e+04
 173                  *  then recurrent value may overflow and the result is
 174                  *  likely underflow to zero
 175                  */
 176                 tmp = n;
 177                 v = two/x;
 178                 tmp = tmp*log(fabs(v*tmp));
 179                 if(tmp<7.09782712893383973096e+02) {
 180                     for(i=n-1,di=(double)(i+i);i>0;i--){
 181                         temp = b;
 182                         b *= di;
 183                         b  = b/x - a;
 184                         a = temp;
 185                         di -= two;
 186                     }
 187                 } else {
 188                     for(i=n-1,di=(double)(i+i);i>0;i--){
 189                         temp = b;
 190                         b *= di;
 191                         b  = b/x - a;
 192                         a = temp;
 193                         di -= two;
 194                     /* scale b to avoid spurious overflow */
 195                         if(b>1e100) {
 196                             a /= b;
 197                             t /= b;
 198                             b  = one;
 199                         }
 200                     }
 201                 }
 202                 z = j0(x);
 203                 w = j1(x);
 204                 if (fabs(z) >= fabs(w))
 205                         b = (t*z/b);
 206                 else
 207                         b = (t*w/a);
 208             }
 209         }
 210         if(sgn==1) return -b; else return b;
 211 }
 212
 213 double
 214 yn(int n, double x)
 215 {
 216         int32_t i,hx,ix,lx;
 217         int32_t sign;
 218         double a, b, temp;
 219
 220         temp = 0;
 221         EXTRACT_WORDS(hx,lx,x);
 222         ix = 0x7fffffff&hx;
 223     /* if Y(n,NaN) is NaN */
 224         if((ix|((u_int32_t)(lx|-lx))>>31)>0x7ff00000) return x+x;
 225         if((ix|lx)==0) return -one/zero;
 226         if(hx<0) return zero/zero;
 227         sign = 1;
 228         if(n<0){
 229                 n = -n;
 230                 sign = 1 - ((n&1)<<1);
 231         }
 232         if(n==0) return(y0(x));
 233         if(n==1) return(sign*y1(x));
 234         if(ix==0x7ff00000) return zero;
 235         if(ix>=0x52D00000) { /* x > 2**302 */
 236     /* (x >> n**2)
 237      *      Jn(x) = cos(x-(2n+1)*pi/4)*sqrt(2/x*pi)
 238      *      Yn(x) = sin(x-(2n+1)*pi/4)*sqrt(2/x*pi)
 239      *      Let s=sin(x), c=cos(x),
 240      *          xn=x-(2n+1)*pi/4, sqt2 = sqrt(2),then
 241      *
 242      *             n    sin(xn)*sqt2    cos(xn)*sqt2
 243      *          ----------------------------------
 244      *             0     s-c             c+s
 245      *             1    -s-c            -c+s
 246      *             2    -s+c            -c-s
 247      *             3     s+c             c-s
 248      */
 249                 switch(n&3) {
 250                     case 0: temp =  sin(x)-cos(x); break;
 251                     case 1: temp = -sin(x)-cos(x); break;
 252                     case 2: temp = -sin(x)+cos(x); break;
 253                     case 3: temp =  sin(x)+cos(x); break;
 254                 }
 255                 b = invsqrtpi*temp/sqrt(x);
 256         } else {
 257             u_int32_t high;
 258             a = y0(x);
 259             b = y1(x);
 260         /* quit if b is -inf */
 261             GET_HIGH_WORD(high,b);
 262             for(i=1;i<n&&high!=0xfff00000;i++){
 263                 temp = b;
 264                 b = ((double)(i+i)/x)*b - a;
 265                 GET_HIGH_WORD(high,b);
 266                 a = temp;
 267             }
 268         }
 269         if(sign>0) return b; else return -b;
 270 }