lib/msun/src/e_jn.c

   1
   2 /* @(#)e_jn.c 1.4 95/01/18 */
   3 /*
   4  * ====================================================
   5  * Copyright (C) 1993 by Sun Microsystems, Inc. All rights reserved.
   6  *
   7  * Developed at SunSoft, a Sun Microsystems, Inc. business.
   8  * Permission to use, copy, modify, and distribute this
   9  * software is freely granted, provided that this notice
  10  * is preserved.
  11  * ====================================================
  12  */
  13
  14 #include <sys/cdefs.h>
  15 __FBSDID("$FreeBSD$");
  16
  17 /*
  18  * __ieee754_jn(n, x), __ieee754_yn(n, x)
  19  * floating point Bessel's function of the 1st and 2nd kind
  20  * of order n
  21  *
  22  * Special cases:
  23  *      y0(0)=y1(0)=yn(n,0) = -inf with division by zero signal;
  24  *      y0(-ve)=y1(-ve)=yn(n,-ve) are NaN with invalid signal.
  25  * Note 2. About jn(n,x), yn(n,x)
  26  *      For n=0, j0(x) is called,
  27  *      for n=1, j1(x) is called,
  28  *      for n<x, forward recursion us used starting
  29  *      from values of j0(x) and j1(x).
  30  *      for n>x, a continued fraction approximation to
  31  *      j(n,x)/j(n-1,x) is evaluated and then backward
  32  *      recursion is used starting from a supposed value
  33  *      for j(n,x). The resulting value of j(0,x) is
  34  *      compared with the actual value to correct the
  35  *      supposed value of j(n,x).
  36  *
  37  *      yn(n,x) is similar in all respects, except
  38  *      that forward recursion is used for all
  39  *      values of n>1.
  40  *
  41  */
  42
  43 #include "math.h"
  44 #include "math_private.h"
  45
  46 static const volatile double vone = 1, vzero = 0;
  47
  48 static const double
  49 invsqrtpi=  5.64189583547756279280e-01, /* 0x3FE20DD7, 0x50429B6D */
  50 two   =  2.00000000000000000000e+00, /* 0x40000000, 0x00000000 */
  51 one   =  1.00000000000000000000e+00; /* 0x3FF00000, 0x00000000 */
  52
  53 static const double zero  =  0.00000000000000000000e+00;
  54
  55 double
  56 __ieee754_jn(int n, double x)
  57 {
  58         int32_t i,hx,ix,lx, sgn;
  59         double a, b, temp, di;
  60         double z, w;
  61
  62     /* J(-n,x) = (-1)^n * J(n, x), J(n, -x) = (-1)^n * J(n, x)
  63      * Thus, J(-n,x) = J(n,-x)
  64      */
  65         EXTRACT_WORDS(hx,lx,x);
  66         ix = 0x7fffffff&hx;
  67     /* if J(n,NaN) is NaN */
  68         if((ix|((u_int32_t)(lx|-lx))>>31)>0x7ff00000) return x+x;
  69         if(n<0){
  70                 n = -n;
  71                 x = -x;
  72                 hx ^= 0x80000000;
  73         }
  74         if(n==0) return(__ieee754_j0(x));
  75         if(n==1) return(__ieee754_j1(x));
  76         sgn = (n&1)&(hx>>31);   /* even n -- 0, odd n -- sign(x) */
  77         x = fabs(x);
  78         if((ix|lx)==0||ix>=0x7ff00000)  /* if x is 0 or inf */
  79             b = zero;
  80         else if((double)n<=x) {
  81                 /* Safe to use J(n+1,x)=2n/x *J(n,x)-J(n-1,x) */
  82             if(ix>=0x52D00000) { /* x > 2**302 */
  83     /* (x >> n**2)
  84      *      Jn(x) = cos(x-(2n+1)*pi/4)*sqrt(2/x*pi)
  85      *      Yn(x) = sin(x-(2n+1)*pi/4)*sqrt(2/x*pi)
  86      *      Let s=sin(x), c=cos(x),
  87      *          xn=x-(2n+1)*pi/4, sqt2 = sqrt(2),then
  88      *
  89      *             n    sin(xn)*sqt2    cos(xn)*sqt2
  90      *          ----------------------------------
  91      *             0     s-c             c+s
  92      *             1    -s-c            -c+s
  93      *             2    -s+c            -c-s
  94      *             3     s+c             c-s
  95      */
  96                 switch(n&3) {
  97                     case 0: temp =  cos(x)+sin(x); break;
  98                     case 1: temp = -cos(x)+sin(x); break;
  99                     case 2: temp = -cos(x)-sin(x); break;
 100                     case 3: temp =  cos(x)-sin(x); break;
 101                 }
 102                 b = invsqrtpi*temp/sqrt(x);
 103             } else {
 104                 a = __ieee754_j0(x);
 105                 b = __ieee754_j1(x);
 106                 for(i=1;i<n;i++){
 107                     temp = b;
 108                     b = b*((double)(i+i)/x) - a; /* avoid underflow */
 109                     a = temp;
 110                 }
 111             }
 112         } else {
 113             if(ix<0x3e100000) { /* x < 2**-29 */
 114     /* x is tiny, return the first Taylor expansion of J(n,x)
 115      * J(n,x) = 1/n!*(x/2)^n  - ...
 116      */
 117                 if(n>33)        /* underflow */
 118                     b = zero;
 119                 else {
 120                     temp = x*0.5; b = temp;
 121                     for (a=one,i=2;i<=n;i++) {
 122                         a *= (double)i;         /* a = n! */
 123                         b *= temp;              /* b = (x/2)^n */
 124                     }
 125                     b = b/a;
 126                 }
 127             } else {
 128                 /* use backward recurrence */
 129                 /*                      x      x^2      x^2
 130                  *  J(n,x)/J(n-1,x) =  ----   ------   ------   .....
 131                  *                      2n  - 2(n+1) - 2(n+2)
 132                  *
 133                  *                      1      1        1
 134                  *  (for large x)   =  ----  ------   ------   .....
 135                  *                      2n   2(n+1)   2(n+2)
 136                  *                      -- - ------ - ------ -
 137                  *                       x     x         x
 138                  *
 139                  * Let w = 2n/x and h=2/x, then the above quotient
 140                  * is equal to the continued fraction:
 141                  *                  1
 142                  *      = -----------------------
 143                  *                     1
 144                  *         w - -----------------
 145                  *                        1
 146                  *              w+h - ---------
 147                  *                     w+2h - ...
 148                  *
 149                  * To determine how many terms needed, let
 150                  * Q(0) = w, Q(1) = w(w+h) - 1,
 151                  * Q(k) = (w+k*h)*Q(k-1) - Q(k-2),
 152                  * When Q(k) > 1e4      good for single
 153                  * When Q(k) > 1e9      good for double
 154                  * When Q(k) > 1e17     good for quadruple
 155                  */
 156             /* determine k */
 157                 double t,v;
 158                 double q0,q1,h,tmp; int32_t k,m;
 159                 w  = (n+n)/(double)x; h = 2.0/(double)x;
 160                 q0 = w;  z = w+h; q1 = w*z - 1.0; k=1;
 161                 while(q1<1.0e9) {
 162                         k += 1; z += h;
 163                         tmp = z*q1 - q0;
 164                         q0 = q1;
 165                         q1 = tmp;
 166                 }
 167                 m = n+n;
 168                 for(t=zero, i = 2*(n+k); i>=m; i -= 2) t = one/(i/x-t);
 169                 a = t;
 170                 b = one;
 171                 /*  estimate log((2/x)^n*n!) = n*log(2/x)+n*ln(n)
 172                  *  Hence, if n*(log(2n/x)) > ...
 173                  *  single 8.8722839355e+01
 174                  *  double 7.09782712893383973096e+02
 175                  *  long double 1.1356523406294143949491931077970765006170e+04
 176                  *  then recurrent value may overflow and the result is
 177                  *  likely underflow to zero
 178                  */
 179                 tmp = n;
 180                 v = two/x;
 181                 tmp = tmp*__ieee754_log(fabs(v*tmp));
 182                 if(tmp<7.09782712893383973096e+02) {
 183                     for(i=n-1,di=(double)(i+i);i>0;i--){
 184                         temp = b;
 185                         b *= di;
 186                         b  = b/x - a;
 187                         a = temp;
 188                         di -= two;
 189                     }
 190                 } else {
 191                     for(i=n-1,di=(double)(i+i);i>0;i--){
 192                         temp = b;
 193                         b *= di;
 194                         b  = b/x - a;
 195                         a = temp;
 196                         di -= two;
 197                     /* scale b to avoid spurious overflow */
 198                         if(b>1e100) {
 199                             a /= b;
 200                             t /= b;
 201                             b  = one;
 202                         }
 203                     }
 204                 }
 205                 z = __ieee754_j0(x);
 206                 w = __ieee754_j1(x);
 207                 if (fabs(z) >= fabs(w))
 208                     b = (t*z/b);
 209                 else
 210                     b = (t*w/a);
 211             }
 212         }
 213         if(sgn==1) return -b; else return b;
 214 }
 215
 216 double
 217 __ieee754_yn(int n, double x)
 218 {
 219         int32_t i,hx,ix,lx;
 220         int32_t sign;
 221         double a, b, temp;
 222
 223         EXTRACT_WORDS(hx,lx,x);
 224         ix = 0x7fffffff&hx;
 225         /* yn(n,NaN) = NaN */
 226         if((ix|((u_int32_t)(lx|-lx))>>31)>0x7ff00000) return x+x;
 227         /* yn(n,+-0) = -inf and raise divide-by-zero exception. */
 228         if((ix|lx)==0) return -one/vzero;
 229         /* yn(n,x<0) = NaN and raise invalid exception. */
 230         if(hx<0) return vzero/vzero;
 231         sign = 1;
 232         if(n<0){
 233                 n = -n;
 234                 sign = 1 - ((n&1)<<1);
 235         }
 236         if(n==0) return(__ieee754_y0(x));
 237         if(n==1) return(sign*__ieee754_y1(x));
 238         if(ix==0x7ff00000) return zero;
 239         if(ix>=0x52D00000) { /* x > 2**302 */
 240     /* (x >> n**2)
 241      *      Jn(x) = cos(x-(2n+1)*pi/4)*sqrt(2/x*pi)
 242      *      Yn(x) = sin(x-(2n+1)*pi/4)*sqrt(2/x*pi)
 243      *      Let s=sin(x), c=cos(x),
 244      *          xn=x-(2n+1)*pi/4, sqt2 = sqrt(2),then
 245      *
 246      *             n    sin(xn)*sqt2    cos(xn)*sqt2
 247      *          ----------------------------------
 248      *             0     s-c             c+s
 249      *             1    -s-c            -c+s
 250      *             2    -s+c            -c-s
 251      *             3     s+c             c-s
 252      */
 253                 switch(n&3) {
 254                     case 0: temp =  sin(x)-cos(x); break;
 255                     case 1: temp = -sin(x)-cos(x); break;
 256                     case 2: temp = -sin(x)+cos(x); break;
 257                     case 3: temp =  sin(x)+cos(x); break;
 258                 }
 259                 b = invsqrtpi*temp/sqrt(x);
 260         } else {
 261             u_int32_t high;
 262             a = __ieee754_y0(x);
 263             b = __ieee754_y1(x);
 264         /* quit if b is -inf */
 265             GET_HIGH_WORD(high,b);
 266             for(i=1;i<n&&high!=0xfff00000;i++){
 267                 temp = b;
 268                 b = ((double)(i+i)/x)*b - a;
 269                 GET_HIGH_WORD(high,b);
 270                 a = temp;
 271             }
 272         }
 273         if(sign>0) return b; else return -b;
 274 }