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
  13 #include <sys/cdefs.h>
  14 #if defined(LIBM_SCCS) && !defined(lint)
  15 __RCSID("$NetBSD: e_jn.c,v 1.14 2010/11/29 15:10:06 drochner Exp $");
  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 "namespace.h"
  45 #include "math.h"
  46 #include "math_private.h"
  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         temp = 0;
  63     /* J(-n,x) = (-1)^n * J(n, x), J(n, -x) = (-1)^n * J(n, x)
  64      * Thus, J(-n,x) = J(n,-x)
  65      */
  66         EXTRACT_WORDS(hx,lx,x);
  67         ix = 0x7fffffff&hx;
  68     /* if J(n,NaN) is NaN */
  69         if((ix|((u_int32_t)(lx|-lx))>>31)>0x7ff00000) return x+x;
  70         if(n<0){
  71                 n = -n;
  72                 x = -x;
  73                 hx ^= 0x80000000;
  74         }
  75         if(n==0) return(__ieee754_j0(x));
  76         if(n==1) return(__ieee754_j1(x));
  77         sgn = (n&1)&(hx>>31);   /* even n -- 0, odd n -- sign(x) */
  78         x = fabs(x);
  79         if((ix|lx)==0||ix>=0x7ff00000)  /* if x is 0 or inf */
  80             b = zero;
  81         else if((double)n<=x) {
  82                 /* Safe to use J(n+1,x)=2n/x *J(n,x)-J(n-1,x) */
  83             if(ix>=0x52D00000) { /* x > 2**302 */
  84     /* (x >> n**2)
  85      *      Jn(x) = cos(x-(2n+1)*pi/4)*sqrt(2/x*pi)
  86      *      Yn(x) = sin(x-(2n+1)*pi/4)*sqrt(2/x*pi)
  87      *      Let s=sin(x), c=cos(x),
  88      *          xn=x-(2n+1)*pi/4, sqt2 = sqrt(2),then
  89      *
  90      *             n    sin(xn)*sqt2    cos(xn)*sqt2
  91      *          ----------------------------------
  92      *             0     s-c             c+s
  93      *             1    -s-c            -c+s
  94      *             2    -s+c            -c-s
  95      *             3     s+c             c-s
  96      */
  97                 switch(n&3) {
  98                     case 0: temp =  cos(x)+sin(x); break;
  99                     case 1: temp = -cos(x)+sin(x); break;
 100                     case 2: temp = -cos(x)-sin(x); break;
 101                     case 3: temp =  cos(x)-sin(x); break;
 102                 }
 103                 b = invsqrtpi*temp/sqrt(x);
 104             } else {
 105                 a = __ieee754_j0(x);
 106                 b = __ieee754_j1(x);
 107                 for(i=1;i<n;i++){
 108                     temp = b;
 109                     b = b*((double)(i+i)/x) - a; /* avoid underflow */
 110                     a = temp;
 111                 }
 112             }
 113         } else {
 114             if(ix<0x3e100000) { /* x < 2**-29 */
 115     /* x is tiny, return the first Taylor expansion of J(n,x)
 116      * J(n,x) = 1/n!*(x/2)^n  - ...
 117      */
 118                 if(n>33)        /* underflow */
 119                     b = zero;
 120                 else {
 121                     temp = x*0.5; b = temp;
 122                     for (a=one,i=2;i<=n;i++) {
 123                         a *= (double)i;         /* a = n! */
 124                         b *= temp;              /* b = (x/2)^n */
 125                     }
 126                     b = b/a;
 127                 }
 128             } else {
 129                 /* use backward recurrence */
 130                 /*                      x      x^2      x^2
 131                  *  J(n,x)/J(n-1,x) =  ----   ------   ------   .....
 132                  *                      2n  - 2(n+1) - 2(n+2)
 133                  *
 134                  *                      1      1        1
 135                  *  (for large x)   =  ----  ------   ------   .....
 136                  *                      2n   2(n+1)   2(n+2)
 137                  *                      -- - ------ - ------ -
 138                  *                       x     x         x
 139                  *
 140                  * Let w = 2n/x and h=2/x, then the above quotient
 141                  * is equal to the continued fraction:
 142                  *                  1
 143                  *      = -----------------------
 144                  *                     1
 145                  *         w - -----------------
 146                  *                        1
 147                  *              w+h - ---------
 148                  *                     w+2h - ...
 149                  *
 150                  * To determine how many terms needed, let
 151                  * Q(0) = w, Q(1) = w(w+h) - 1,
 152                  * Q(k) = (w+k*h)*Q(k-1) - Q(k-2),
 153                  * When Q(k) > 1e4      good for single
 154                  * When Q(k) > 1e9      good for double
 155                  * When Q(k) > 1e17     good for quadruple
 156                  */
 157             /* determine k */
 158                 double t,v;
 159                 double q0,q1,h,tmp; int32_t k,m;
 160                 w  = (n+n)/(double)x; h = 2.0/(double)x;
 161                 q0 = w;  z = w+h; q1 = w*z - 1.0; k=1;
 162                 while(q1<1.0e9) {
 163                         k += 1; z += h;
 164                         tmp = z*q1 - q0;
 165                         q0 = q1;
 166                         q1 = tmp;
 167                 }
 168                 m = n+n;
 169                 for(t=zero, i = 2*(n+k); i>=m; i -= 2) t = one/(i/x-t);
 170                 a = t;
 171                 b = one;
 172                 /*  estimate log((2/x)^n*n!) = n*log(2/x)+n*ln(n)
 173                  *  Hence, if n*(log(2n/x)) > ...
 174                  *  single 8.8722839355e+01
 175                  *  double 7.09782712893383973096e+02
 176                  *  long double 1.1356523406294143949491931077970765006170e+04
 177                  *  then recurrent value may overflow and the result is
 178                  *  likely underflow to zero
 179                  */
 180                 tmp = n;
 181                 v = two/x;
 182                 tmp = tmp*__ieee754_log(fabs(v*tmp));
 183                 if(tmp<7.09782712893383973096e+02) {
 184                     for(i=n-1,di=(double)(i+i);i>0;i--){
 185                         temp = b;
 186                         b *= di;
 187                         b  = b/x - a;
 188                         a = temp;
 189                         di -= two;
 190                     }
 191                 } else {
 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                     /* scale b to avoid spurious overflow */
 199                         if(b>1e100) {
 200                             a /= b;
 201                             t /= b;
 202                             b  = one;
 203                         }
 204                     }
 205                 }
 206                 z = __ieee754_j0(x);
 207                 w = __ieee754_j1(x);
 208                 if (fabs(z) >= fabs(w))
 209                         b = (t*z/b);
 210                 else
 211                         b = (t*w/a);
 212             }
 213         }
 214         if(sgn==1) return -b; else return b;
 215 }
 216
 217 double
 218 __ieee754_yn(int n, double x)
 219 {
 220         int32_t i,hx,ix,lx;
 221         int32_t sign;
 222         double a, b, temp;
 223
 224         temp = 0;
 225         EXTRACT_WORDS(hx,lx,x);
 226         ix = 0x7fffffff&hx;
 227     /* if Y(n,NaN) is NaN */
 228         if((ix|((u_int32_t)(lx|-lx))>>31)>0x7ff00000) return x+x;
 229         if((ix|lx)==0) return -one/zero;
 230         if(hx<0) return zero/zero;
 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 }