lib/msun/src/e_jn.c

   1 /*
   2  * ====================================================
   3  * Copyright (C) 1993 by Sun Microsystems, Inc. All rights reserved.
   4  *
   5  * Developed at SunSoft, a Sun Microsystems, Inc. business.
   6  * Permission to use, copy, modify, and distribute this
   7  * software is freely granted, provided that this notice
   8  * is preserved.
   9  * ====================================================
  10  */
  11
  12 /*
  13  * jn(n, x), yn(n, x)
  14  * floating point Bessel's function of the 1st and 2nd kind
  15  * of order n
  16  *
  17  * Special cases:
  18  *      y0(0)=y1(0)=yn(n,0) = -inf with division by zero signal;
  19  *      y0(-ve)=y1(-ve)=yn(n,-ve) are NaN with invalid signal.
  20  * Note 2. About jn(n,x), yn(n,x)
  21  *      For n=0, j0(x) is called.
  22  *      For n=1, j1(x) is called.
  23  *      For n<x, forward recursion is used starting
  24  *      from values of j0(x) and j1(x).
  25  *      For n>x, a continued fraction approximation to
  26  *      j(n,x)/j(n-1,x) is evaluated and then backward
  27  *      recursion is used starting from a supposed value
  28  *      for j(n,x). The resulting values of j(0,x) or j(1,x) are
  29  *      compared with the actual values to correct the
  30  *      supposed value of j(n,x).
  31  *
  32  *      yn(n,x) is similar in all respects, except
  33  *      that forward recursion is used for all
  34  *      values of n>1.
  35  */
  36
  37 #include "math.h"
  38 #include "math_private.h"
  39
  40 static const volatile double vone = 1, vzero = 0;
  41
  42 static const double
  43 invsqrtpi=  5.64189583547756279280e-01, /* 0x3FE20DD7, 0x50429B6D */
  44 two   =  2.00000000000000000000e+00, /* 0x40000000, 0x00000000 */
  45 one   =  1.00000000000000000000e+00; /* 0x3FF00000, 0x00000000 */
  46
  47 static const double zero  =  0.00000000000000000000e+00;
  48
  49 double
  50 jn(int n, double x)
  51 {
  52         int32_t i,hx,ix,lx, sgn;
  53         double a, b, c, s, temp, di;
  54         double z, w;
  55
  56     /* J(-n,x) = (-1)^n * J(n, x), J(n, -x) = (-1)^n * J(n, x)
  57      * Thus, J(-n,x) = J(n,-x)
  58      */
  59         EXTRACT_WORDS(hx,lx,x);
  60         ix = 0x7fffffff&hx;
  61     /* if J(n,NaN) is NaN */
  62         if((ix|((u_int32_t)(lx|-lx))>>31)>0x7ff00000) return x+x;
  63         if(n<0){
  64                 n = -n;
  65                 x = -x;
  66                 hx ^= 0x80000000;
  67         }
  68         if(n==0) return(j0(x));
  69         if(n==1) return(j1(x));
  70         sgn = (n&1)&(hx>>31);   /* even n -- 0, odd n -- sign(x) */
  71         x = fabs(x);
  72         if((ix|lx)==0||ix>=0x7ff00000)  /* if x is 0 or inf */
  73             b = zero;
  74         else if((double)n<=x) {
  75                 /* Safe to use J(n+1,x)=2n/x *J(n,x)-J(n-1,x) */
  76             if(ix>=0x52D00000) { /* x > 2**302 */
  77     /* (x >> n**2)
  78      *      Jn(x) = cos(x-(2n+1)*pi/4)*sqrt(2/x*pi)
  79      *      Yn(x) = sin(x-(2n+1)*pi/4)*sqrt(2/x*pi)
  80      *      Let s=sin(x), c=cos(x),
  81      *          xn=x-(2n+1)*pi/4, sqt2 = sqrt(2), then
  82      *
  83      *             n    sin(xn)*sqt2    cos(xn)*sqt2
  84      *          ----------------------------------
  85      *             0     s-c             c+s
  86      *             1    -s-c            -c+s
  87      *             2    -s+c            -c-s
  88      *             3     s+c             c-s
  89      */
  90                 sincos(x, &s, &c);
  91                 switch(n&3) {
  92                     case 0: temp =  c+s; break;
  93                     case 1: temp = -c+s; break;
  94                     case 2: temp = -c-s; break;
  95                     case 3: temp =  c-s; break;
  96                 }
  97                 b = invsqrtpi*temp/sqrt(x);
  98             } else {
  99                 a = j0(x);
 100                 b = j1(x);
 101                 for(i=1;i<n;i++){
 102                     temp = b;
 103                     b = b*((double)(i+i)/x) - a; /* avoid underflow */
 104                     a = temp;
 105                 }
 106             }
 107         } else {
 108             if(ix<0x3e100000) { /* x < 2**-29 */
 109     /* x is tiny, return the first Taylor expansion of J(n,x)
 110      * J(n,x) = 1/n!*(x/2)^n  - ...
 111      */
 112                 if(n>33)        /* underflow */
 113                     b = zero;
 114                 else {
 115                     temp = x*0.5; b = temp;
 116                     for (a=one,i=2;i<=n;i++) {
 117                         a *= (double)i;         /* a = n! */
 118                         b *= temp;              /* b = (x/2)^n */
 119                     }
 120                     b = b/a;
 121                 }
 122             } else {
 123                 /* use backward recurrence */
 124                 /*                      x      x^2      x^2
 125                  *  J(n,x)/J(n-1,x) =  ----   ------   ------   .....
 126                  *                      2n  - 2(n+1) - 2(n+2)
 127                  *
 128                  *                      1      1        1
 129                  *  (for large x)   =  ----  ------   ------   .....
 130                  *                      2n   2(n+1)   2(n+2)
 131                  *                      -- - ------ - ------ -
 132                  *                       x     x         x
 133                  *
 134                  * Let w = 2n/x and h=2/x, then the above quotient
 135                  * is equal to the continued fraction:
 136                  *                  1
 137                  *      = -----------------------
 138                  *                     1
 139                  *         w - -----------------
 140                  *                        1
 141                  *              w+h - ---------
 142                  *                     w+2h - ...
 143                  *
 144                  * To determine how many terms needed, let
 145                  * Q(0) = w, Q(1) = w(w+h) - 1,
 146                  * Q(k) = (w+k*h)*Q(k-1) - Q(k-2),
 147                  * When Q(k) > 1e4      good for single
 148                  * When Q(k) > 1e9      good for double
 149                  * When Q(k) > 1e17     good for quadruple
 150                  */
 151             /* determine k */
 152                 double t,v;
 153                 double q0,q1,h,tmp; int32_t k,m;
 154                 w  = (n+n)/(double)x; h = 2.0/(double)x;
 155                 q0 = w;  z = w+h; q1 = w*z - 1.0; k=1;
 156                 while(q1<1.0e9) {
 157                         k += 1; z += h;
 158                         tmp = z*q1 - q0;
 159                         q0 = q1;
 160                         q1 = tmp;
 161                 }
 162                 m = n+n;
 163                 for(t=zero, i = 2*(n+k); i>=m; i -= 2) t = one/(i/x-t);
 164                 a = t;
 165                 b = one;
 166                 /*  estimate log((2/x)^n*n!) = n*log(2/x)+n*ln(n)
 167                  *  Hence, if n*(log(2n/x)) > ...
 168                  *  single 8.8722839355e+01
 169                  *  double 7.09782712893383973096e+02
 170                  *  long double 1.1356523406294143949491931077970765006170e+04
 171                  *  then recurrent value may overflow and the result is
 172                  *  likely underflow to zero
 173                  */
 174                 tmp = n;
 175                 v = two/x;
 176                 tmp = tmp*log(fabs(v*tmp));
 177                 if(tmp<7.09782712893383973096e+02) {
 178                     for(i=n-1,di=(double)(i+i);i>0;i--){
 179                         temp = b;
 180                         b *= di;
 181                         b  = b/x - a;
 182                         a = temp;
 183                         di -= two;
 184                     }
 185                 } else {
 186                     for(i=n-1,di=(double)(i+i);i>0;i--){
 187                         temp = b;
 188                         b *= di;
 189                         b  = b/x - a;
 190                         a = temp;
 191                         di -= two;
 192                     /* scale b to avoid spurious overflow */
 193                         if(b>1e100) {
 194                             a /= b;
 195                             t /= b;
 196                             b  = one;
 197                         }
 198                     }
 199                 }
 200                 z = j0(x);
 201                 w = j1(x);
 202                 if (fabs(z) >= fabs(w))
 203                     b = (t*z/b);
 204                 else
 205                     b = (t*w/a);
 206             }
 207         }
 208         if(sgn==1) return -b; else return b;
 209 }
 210
 211 double
 212 yn(int n, double x)
 213 {
 214         int32_t i,hx,ix,lx;
 215         int32_t sign;
 216         double a, b, c, s, temp;
 217
 218         EXTRACT_WORDS(hx,lx,x);
 219         ix = 0x7fffffff&hx;
 220         /* yn(n,NaN) = NaN */
 221         if((ix|((u_int32_t)(lx|-lx))>>31)>0x7ff00000) return x+x;
 222         /* yn(n,+-0) = -inf and raise divide-by-zero exception. */
 223         if((ix|lx)==0) return -one/vzero;
 224         /* yn(n,x<0) = NaN and raise invalid exception. */
 225         if(hx<0) return vzero/vzero;
 226         sign = 1;
 227         if(n<0){
 228                 n = -n;
 229                 sign = 1 - ((n&1)<<1);
 230         }
 231         if(n==0) return(y0(x));
 232         if(n==1) return(sign*y1(x));
 233         if(ix==0x7ff00000) return zero;
 234         if(ix>=0x52D00000) { /* x > 2**302 */
 235     /* (x >> n**2)
 236      *      Jn(x) = cos(x-(2n+1)*pi/4)*sqrt(2/x*pi)
 237      *      Yn(x) = sin(x-(2n+1)*pi/4)*sqrt(2/x*pi)
 238      *      Let s=sin(x), c=cos(x),
 239      *          xn=x-(2n+1)*pi/4, sqt2 = sqrt(2), then
 240      *
 241      *             n    sin(xn)*sqt2    cos(xn)*sqt2
 242      *          ----------------------------------
 243      *             0     s-c             c+s
 244      *             1    -s-c            -c+s
 245      *             2    -s+c            -c-s
 246      *             3     s+c             c-s
 247      */
 248                 sincos(x, &s, &c);
 249                 switch(n&3) {
 250                     case 0: temp =  s-c; break;
 251                     case 1: temp = -s-c; break;
 252                     case 2: temp = -s+c; break;
 253                     case 3: temp =  s+c; 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 }