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 #ifndef lint
  15 static char rcsid[] = "$FreeBSD$";
  16 #endif
  17
  18 /*
  19  * __ieee754_jn(n, x), __ieee754_yn(n, x)
  20  * floating point Bessel's function of the 1st and 2nd kind
  21  * of order n
  22  *
  23  * Special cases:
  24  *      y0(0)=y1(0)=yn(n,0) = -inf with division by zero signal;
  25  *      y0(-ve)=y1(-ve)=yn(n,-ve) are NaN with invalid signal.
  26  * Note 2. About jn(n,x), yn(n,x)
  27  *      For n=0, j0(x) is called,
  28  *      for n=1, j1(x) is called,
  29  *      for n<x, forward recursion us used starting
  30  *      from values of j0(x) and j1(x).
  31  *      for n>x, a continued fraction approximation to
  32  *      j(n,x)/j(n-1,x) is evaluated and then backward
  33  *      recursion is used starting from a supposed value
  34  *      for j(n,x). The resulting value of j(0,x) is
  35  *      compared with the actual value to correct the
  36  *      supposed value of j(n,x).
  37  *
  38  *      yn(n,x) is similar in all respects, except
  39  *      that forward recursion is used for all
  40  *      values of n>1.
  41  *
  42  */
  43
  44 #include "math.h"
  45 #include "math_private.h"
  46
  47 static const double
  48 invsqrtpi=  5.64189583547756279280e-01, /* 0x3FE20DD7, 0x50429B6D */
  49 two   =  2.00000000000000000000e+00, /* 0x40000000, 0x00000000 */
  50 one   =  1.00000000000000000000e+00; /* 0x3FF00000, 0x00000000 */
  51
  52 static const double zero  =  0.00000000000000000000e+00;
  53
  54 double
  55 __ieee754_jn(int n, double x)
  56 {
  57         int32_t i,hx,ix,lx, sgn;
  58         double a, b, temp, di;
  59         double z, w;
  60
  61     /* J(-n,x) = (-1)^n * J(n, x), J(n, -x) = (-1)^n * J(n, x)
  62      * Thus, J(-n,x) = J(n,-x)
  63      */
  64         EXTRACT_WORDS(hx,lx,x);
  65         ix = 0x7fffffff&hx;
  66     /* if J(n,NaN) is NaN */
  67         if((ix|((u_int32_t)(lx|-lx))>>31)>0x7ff00000) return x+x;
  68         if(n<0){
  69                 n = -n;
  70                 x = -x;
  71                 hx ^= 0x80000000;
  72         }
  73         if(n==0) return(__ieee754_j0(x));
  74         if(n==1) return(__ieee754_j1(x));
  75         sgn = (n&1)&(hx>>31);   /* even n -- 0, odd n -- sign(x) */
  76         x = fabs(x);
  77         if((ix|lx)==0||ix>=0x7ff00000)  /* if x is 0 or inf */
  78             b = zero;
  79         else if((double)n<=x) {
  80                 /* Safe to use J(n+1,x)=2n/x *J(n,x)-J(n-1,x) */
  81             if(ix>=0x52D00000) { /* x > 2**302 */
  82     /* (x >> n**2)
  83      *      Jn(x) = cos(x-(2n+1)*pi/4)*sqrt(2/x*pi)
  84      *      Yn(x) = sin(x-(2n+1)*pi/4)*sqrt(2/x*pi)
  85      *      Let s=sin(x), c=cos(x),
  86      *          xn=x-(2n+1)*pi/4, sqt2 = sqrt(2),then
  87      *
  88      *             n    sin(xn)*sqt2    cos(xn)*sqt2
  89      *          ----------------------------------
  90      *             0     s-c             c+s
  91      *             1    -s-c            -c+s
  92      *             2    -s+c            -c-s
  93      *             3     s+c             c-s
  94      */
  95                 switch(n&3) {
  96                     case 0: temp =  cos(x)+sin(x); break;
  97                     case 1: temp = -cos(x)+sin(x); break;
  98                     case 2: temp = -cos(x)-sin(x); break;
  99                     case 3: temp =  cos(x)-sin(x); break;
 100                 }
 101                 b = invsqrtpi*temp/sqrt(x);
 102             } else {
 103                 a = __ieee754_j0(x);
 104                 b = __ieee754_j1(x);
 105                 for(i=1;i<n;i++){
 106                     temp = b;
 107                     b = b*((double)(i+i)/x) - a; /* avoid underflow */
 108                     a = temp;
 109                 }
 110             }
 111         } else {
 112             if(ix<0x3e100000) { /* x < 2**-29 */
 113     /* x is tiny, return the first Taylor expansion of J(n,x)
 114      * J(n,x) = 1/n!*(x/2)^n  - ...
 115      */
 116                 if(n>33)        /* underflow */
 117                     b = zero;
 118                 else {
 119                     temp = x*0.5; b = temp;
 120                     for (a=one,i=2;i<=n;i++) {
 121                         a *= (double)i;         /* a = n! */
 122                         b *= temp;              /* b = (x/2)^n */
 123                     }
 124                     b = b/a;
 125                 }
 126             } else {
 127                 /* use backward recurrence */
 128                 /*                      x      x^2      x^2
 129                  *  J(n,x)/J(n-1,x) =  ----   ------   ------   .....
 130                  *                      2n  - 2(n+1) - 2(n+2)
 131                  *
 132                  *                      1      1        1
 133                  *  (for large x)   =  ----  ------   ------   .....
 134                  *                      2n   2(n+1)   2(n+2)
 135                  *                      -- - ------ - ------ -
 136                  *                       x     x         x
 137                  *
 138                  * Let w = 2n/x and h=2/x, then the above quotient
 139                  * is equal to the continued fraction:
 140                  *                  1
 141                  *      = -----------------------
 142                  *                     1
 143                  *         w - -----------------
 144                  *                        1
 145                  *              w+h - ---------
 146                  *                     w+2h - ...
 147                  *
 148                  * To determine how many terms needed, let
 149                  * Q(0) = w, Q(1) = w(w+h) - 1,
 150                  * Q(k) = (w+k*h)*Q(k-1) - Q(k-2),
 151                  * When Q(k) > 1e4      good for single
 152                  * When Q(k) > 1e9      good for double
 153                  * When Q(k) > 1e17     good for quadruple
 154                  */
 155             /* determine k */
 156                 double t,v;
 157                 double q0,q1,h,tmp; int32_t k,m;
 158                 w  = (n+n)/(double)x; h = 2.0/(double)x;
 159                 q0 = w;  z = w+h; q1 = w*z - 1.0; k=1;
 160                 while(q1<1.0e9) {
 161                         k += 1; z += h;
 162                         tmp = z*q1 - q0;
 163                         q0 = q1;
 164                         q1 = tmp;
 165                 }
 166                 m = n+n;
 167                 for(t=zero, i = 2*(n+k); i>=m; i -= 2) t = one/(i/x-t);
 168                 a = t;
 169                 b = one;
 170                 /*  estimate log((2/x)^n*n!) = n*log(2/x)+n*ln(n)
 171                  *  Hence, if n*(log(2n/x)) > ...
 172                  *  single 8.8722839355e+01
 173                  *  double 7.09782712893383973096e+02
 174                  *  long double 1.1356523406294143949491931077970765006170e+04
 175                  *  then recurrent value may overflow and the result is
 176                  *  likely underflow to zero
 177                  */
 178                 tmp = n;
 179                 v = two/x;
 180                 tmp = tmp*__ieee754_log(fabs(v*tmp));
 181                 if(tmp<7.09782712893383973096e+02) {
 182                     for(i=n-1,di=(double)(i+i);i>0;i--){
 183                         temp = b;
 184                         b *= di;
 185                         b  = b/x - a;
 186                         a = temp;
 187                         di -= two;
 188                     }
 189                 } else {
 190                     for(i=n-1,di=(double)(i+i);i>0;i--){
 191                         temp = b;
 192                         b *= di;
 193                         b  = b/x - a;
 194                         a = temp;
 195                         di -= two;
 196                     /* scale b to avoid spurious overflow */
 197                         if(b>1e100) {
 198                             a /= b;
 199                             t /= b;
 200                             b  = one;
 201                         }
 202                     }
 203                 }
 204                 b = (t*__ieee754_j0(x)/b);
 205             }
 206         }
 207         if(sgn==1) return -b; else return b;
 208 }
 209
 210 double
 211 __ieee754_yn(int n, double x)
 212 {
 213         int32_t i,hx,ix,lx;
 214         int32_t sign;
 215         double a, b, temp;
 216
 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(__ieee754_y0(x));
 229         if(n==1) return(sign*__ieee754_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 = __ieee754_y0(x);
 255             b = __ieee754_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 }