lib/msun/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  * $FreeBSD: src/lib/msun/src/e_jn.c,v 1.6 1999/08/28 00:06:33 peter Exp $
  13  * $DragonFly: src/lib/msun/src/Attic/e_jn.c,v 1.2 2003/06/17 04:26:52 dillon Exp $
  14  */
  15
  16 /*
  17  * __ieee754_jn(n, x), __ieee754_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 #ifdef __STDC__
  46 static const double
  47 #else
  48 static double
  49 #endif
  50 invsqrtpi=  5.64189583547756279280e-01, /* 0x3FE20DD7, 0x50429B6D */
  51 two   =  2.00000000000000000000e+00, /* 0x40000000, 0x00000000 */
  52 one   =  1.00000000000000000000e+00; /* 0x3FF00000, 0x00000000 */
  53
  54 #ifdef __STDC__
  55 static const double zero  =  0.00000000000000000000e+00;
  56 #else
  57 static double zero  =  0.00000000000000000000e+00;
  58 #endif
  59
  60 #ifdef __STDC__
  61         double __ieee754_jn(int n, double x)
  62 #else
  63         double __ieee754_jn(n,x)
  64         int n; double x;
  65 #endif
  66 {
  67         int32_t i,hx,ix,lx, sgn;
  68         double a, b, temp, di;
  69         double z, w;
  70
  71     /* J(-n,x) = (-1)^n * J(n, x), J(n, -x) = (-1)^n * J(n, x)
  72      * Thus, J(-n,x) = J(n,-x)
  73      */
  74         EXTRACT_WORDS(hx,lx,x);
  75         ix = 0x7fffffff&hx;
  76     /* if J(n,NaN) is NaN */
  77         if((ix|((u_int32_t)(lx|-lx))>>31)>0x7ff00000) return x+x;
  78         if(n<0){
  79                 n = -n;
  80                 x = -x;
  81                 hx ^= 0x80000000;
  82         }
  83         if(n==0) return(__ieee754_j0(x));
  84         if(n==1) return(__ieee754_j1(x));
  85         sgn = (n&1)&(hx>>31);   /* even n -- 0, odd n -- sign(x) */
  86         x = fabs(x);
  87         if((ix|lx)==0||ix>=0x7ff00000)  /* if x is 0 or inf */
  88             b = zero;
  89         else if((double)n<=x) {
  90                 /* Safe to use J(n+1,x)=2n/x *J(n,x)-J(n-1,x) */
  91             if(ix>=0x52D00000) { /* x > 2**302 */
  92     /* (x >> n**2)
  93      *      Jn(x) = cos(x-(2n+1)*pi/4)*sqrt(2/x*pi)
  94      *      Yn(x) = sin(x-(2n+1)*pi/4)*sqrt(2/x*pi)
  95      *      Let s=sin(x), c=cos(x),
  96      *          xn=x-(2n+1)*pi/4, sqt2 = sqrt(2),then
  97      *
  98      *             n    sin(xn)*sqt2    cos(xn)*sqt2
  99      *          ----------------------------------
 100      *             0     s-c             c+s
 101      *             1    -s-c            -c+s
 102      *             2    -s+c            -c-s
 103      *             3     s+c             c-s
 104      */
 105                 switch(n&3) {
 106                     case 0: temp =  cos(x)+sin(x); break;
 107                     case 1: temp = -cos(x)+sin(x); break;
 108                     case 2: temp = -cos(x)-sin(x); break;
 109                     case 3: temp =  cos(x)-sin(x); break;
 110                 }
 111                 b = invsqrtpi*temp/sqrt(x);
 112             } else {
 113                 a = __ieee754_j0(x);
 114                 b = __ieee754_j1(x);
 115                 for(i=1;i<n;i++){
 116                     temp = b;
 117                     b = b*((double)(i+i)/x) - a; /* avoid underflow */
 118                     a = temp;
 119                 }
 120             }
 121         } else {
 122             if(ix<0x3e100000) { /* x < 2**-29 */
 123     /* x is tiny, return the first Taylor expansion of J(n,x)
 124      * J(n,x) = 1/n!*(x/2)^n  - ...
 125      */
 126                 if(n>33)        /* underflow */
 127                     b = zero;
 128                 else {
 129                     temp = x*0.5; b = temp;
 130                     for (a=one,i=2;i<=n;i++) {
 131                         a *= (double)i;         /* a = n! */
 132                         b *= temp;              /* b = (x/2)^n */
 133                     }
 134                     b = b/a;
 135                 }
 136             } else {
 137                 /* use backward recurrence */
 138                 /*                      x      x^2      x^2
 139                  *  J(n,x)/J(n-1,x) =  ----   ------   ------   .....
 140                  *                      2n  - 2(n+1) - 2(n+2)
 141                  *
 142                  *                      1      1        1
 143                  *  (for large x)   =  ----  ------   ------   .....
 144                  *                      2n   2(n+1)   2(n+2)
 145                  *                      -- - ------ - ------ -
 146                  *                       x     x         x
 147                  *
 148                  * Let w = 2n/x and h=2/x, then the above quotient
 149                  * is equal to the continued fraction:
 150                  *                  1
 151                  *      = -----------------------
 152                  *                     1
 153                  *         w - -----------------
 154                  *                        1
 155                  *              w+h - ---------
 156                  *                     w+2h - ...
 157                  *
 158                  * To determine how many terms needed, let
 159                  * Q(0) = w, Q(1) = w(w+h) - 1,
 160                  * Q(k) = (w+k*h)*Q(k-1) - Q(k-2),
 161                  * When Q(k) > 1e4      good for single
 162                  * When Q(k) > 1e9      good for double
 163                  * When Q(k) > 1e17     good for quadruple
 164                  */
 165             /* determine k */
 166                 double t,v;
 167                 double q0,q1,h,tmp; int32_t k,m;
 168                 w  = (n+n)/(double)x; h = 2.0/(double)x;
 169                 q0 = w;  z = w+h; q1 = w*z - 1.0; k=1;
 170                 while(q1<1.0e9) {
 171                         k += 1; z += h;
 172                         tmp = z*q1 - q0;
 173                         q0 = q1;
 174                         q1 = tmp;
 175                 }
 176                 m = n+n;
 177                 for(t=zero, i = 2*(n+k); i>=m; i -= 2) t = one/(i/x-t);
 178                 a = t;
 179                 b = one;
 180                 /*  estimate log((2/x)^n*n!) = n*log(2/x)+n*ln(n)
 181                  *  Hence, if n*(log(2n/x)) > ...
 182                  *  single 8.8722839355e+01
 183                  *  double 7.09782712893383973096e+02
 184                  *  long double 1.1356523406294143949491931077970765006170e+04
 185                  *  then recurrent value may overflow and the result is
 186                  *  likely underflow to zero
 187                  */
 188                 tmp = n;
 189                 v = two/x;
 190                 tmp = tmp*__ieee754_log(fabs(v*tmp));
 191                 if(tmp<7.09782712893383973096e+02) {
 192                     for(i=n-1,di=(double)(i+i);i>0;i--){
 193                         temp = b;
 194                         b *= di;
 195                         b  = b/x - a;
 196                         a = temp;
 197                         di -= two;
 198                     }
 199                 } else {
 200                     for(i=n-1,di=(double)(i+i);i>0;i--){
 201                         temp = b;
 202                         b *= di;
 203                         b  = b/x - a;
 204                         a = temp;
 205                         di -= two;
 206                     /* scale b to avoid spurious overflow */
 207                         if(b>1e100) {
 208                             a /= b;
 209                             t /= b;
 210                             b  = one;
 211                         }
 212                     }
 213                 }
 214                 b = (t*__ieee754_j0(x)/b);
 215             }
 216         }
 217         if(sgn==1) return -b; else return b;
 218 }
 219
 220 #ifdef __STDC__
 221         double __ieee754_yn(int n, double x)
 222 #else
 223         double __ieee754_yn(n,x)
 224         int n; double x;
 225 #endif
 226 {
 227         int32_t i,hx,ix,lx;
 228         int32_t sign;
 229         double a, b, temp;
 230
 231         EXTRACT_WORDS(hx,lx,x);
 232         ix = 0x7fffffff&hx;
 233     /* if Y(n,NaN) is NaN */
 234         if((ix|((u_int32_t)(lx|-lx))>>31)>0x7ff00000) return x+x;
 235         if((ix|lx)==0) return -one/zero;
 236         if(hx<0) return zero/zero;
 237         sign = 1;
 238         if(n<0){
 239                 n = -n;
 240                 sign = 1 - ((n&1)<<1);
 241         }
 242         if(n==0) return(__ieee754_y0(x));
 243         if(n==1) return(sign*__ieee754_y1(x));
 244         if(ix==0x7ff00000) return zero;
 245         if(ix>=0x52D00000) { /* x > 2**302 */
 246     /* (x >> n**2)
 247      *      Jn(x) = cos(x-(2n+1)*pi/4)*sqrt(2/x*pi)
 248      *      Yn(x) = sin(x-(2n+1)*pi/4)*sqrt(2/x*pi)
 249      *      Let s=sin(x), c=cos(x),
 250      *          xn=x-(2n+1)*pi/4, sqt2 = sqrt(2),then
 251      *
 252      *             n    sin(xn)*sqt2    cos(xn)*sqt2
 253      *          ----------------------------------
 254      *             0     s-c             c+s
 255      *             1    -s-c            -c+s
 256      *             2    -s+c            -c-s
 257      *             3     s+c             c-s
 258      */
 259                 switch(n&3) {
 260                     case 0: temp =  sin(x)-cos(x); break;
 261                     case 1: temp = -sin(x)-cos(x); break;
 262                     case 2: temp = -sin(x)+cos(x); break;
 263                     case 3: temp =  sin(x)+cos(x); break;
 264                 }
 265                 b = invsqrtpi*temp/sqrt(x);
 266         } else {
 267             u_int32_t high;
 268             a = __ieee754_y0(x);
 269             b = __ieee754_y1(x);
 270         /* quit if b is -inf */
 271             GET_HIGH_WORD(high,b);
 272             for(i=1;i<n&&high!=0xfff00000;i++){
 273                 temp = b;
 274                 b = ((double)(i+i)/x)*b - a;
 275                 GET_HIGH_WORD(high,b);
 276                 a = temp;
 277             }
 278         }
 279         if(sign>0) return b; else return -b;
 280 }