lib/libm/src/e_jn.c

   1
   2 /* @(#)e_jn.c 1.4 95/01/18 */
   3 /* $FreeBSD: head/lib/msun/src/e_jn.c 215237 2010-11-13 10:54:10Z uqs $ */
   4 /*
   5  * ====================================================
   6  * Copyright (C) 1993 by Sun Microsystems, Inc. All rights reserved.
   7  *
   8  * Developed at SunSoft, a Sun Microsystems, Inc. business.
   9  * Permission to use, copy, modify, and distribute this
  10  * software is freely granted, provided that this notice
  11  * is preserved.
  12  * ====================================================
  13  */
  14
  15 /*
  16  * __ieee754_jn(n, x), __ieee754_yn(n, x)
  17  * floating point Bessel's function of the 1st and 2nd kind
  18  * of order n
  19  *
  20  * Special cases:
  21  *      y0(0)=y1(0)=yn(n,0) = -inf with division by zero signal;
  22  *      y0(-ve)=y1(-ve)=yn(n,-ve) are NaN with invalid signal.
  23  * Note 2. About jn(n,x), yn(n,x)
  24  *      For n=0, j0(x) is called,
  25  *      for n=1, j1(x) is called,
  26  *      for n<x, forward recursion us used starting
  27  *      from values of j0(x) and j1(x).
  28  *      for n>x, a continued fraction approximation to
  29  *      j(n,x)/j(n-1,x) is evaluated and then backward
  30  *      recursion is used starting from a supposed value
  31  *      for j(n,x). The resulting value of j(0,x) is
  32  *      compared with the actual value to correct the
  33  *      supposed value of j(n,x).
  34  *
  35  *      yn(n,x) is similar in all respects, except
  36  *      that forward recursion is used for all
  37  *      values of n>1.
  38  *
  39  */
  40
  41 #include "math.h"
  42 #include "math_private.h"
  43
  44 static const double
  45 invsqrtpi=  5.64189583547756279280e-01, /* 0x3FE20DD7, 0x50429B6D */
  46 two   =  2.00000000000000000000e+00, /* 0x40000000, 0x00000000 */
  47 one   =  1.00000000000000000000e+00; /* 0x3FF00000, 0x00000000 */
  48
  49 static const double zero  =  0.00000000000000000000e+00;
  50
  51 double
  52 __ieee754_jn(int n, double x)
  53 {
  54         int32_t i,hx,ix,lx, sgn;
  55         double a, b, temp, di;
  56         double z, w;
  57
  58     /* J(-n,x) = (-1)^n * J(n, x), J(n, -x) = (-1)^n * J(n, x)
  59      * Thus, J(-n,x) = J(n,-x)
  60      */
  61         EXTRACT_WORDS(hx,lx,x);
  62         ix = 0x7fffffff&hx;
  63     /* if J(n,NaN) is NaN */
  64         if((ix|((u_int32_t)(lx|-lx))>>31)>0x7ff00000) return x+x;
  65         if(n<0){
  66                 n = -n;
  67                 x = -x;
  68                 hx ^= 0x80000000;
  69         }
  70         if(n==0) return(__ieee754_j0(x));
  71         if(n==1) return(__ieee754_j1(x));
  72         sgn = (n&1)&(hx>>31);   /* even n -- 0, odd n -- sign(x) */
  73         x = fabs(x);
  74         if((ix|lx)==0||ix>=0x7ff00000)  /* if x is 0 or inf */
  75             b = zero;
  76         else if((double)n<=x) {
  77                 /* Safe to use J(n+1,x)=2n/x *J(n,x)-J(n-1,x) */
  78             if(ix>=0x52D00000) { /* x > 2**302 */
  79     /* (x >> n**2)
  80      *      Jn(x) = cos(x-(2n+1)*pi/4)*sqrt(2/x*pi)
  81      *      Yn(x) = sin(x-(2n+1)*pi/4)*sqrt(2/x*pi)
  82      *      Let s=sin(x), c=cos(x),
  83      *          xn=x-(2n+1)*pi/4, sqt2 = sqrt(2),then
  84      *
  85      *             n    sin(xn)*sqt2    cos(xn)*sqt2
  86      *          ----------------------------------
  87      *             0     s-c             c+s
  88      *             1    -s-c            -c+s
  89      *             2    -s+c            -c-s
  90      *             3     s+c             c-s
  91      */
  92                 switch(n&3) {
  93                     case 0: temp =  cos(x)+sin(x); break;
  94                     case 1: temp = -cos(x)+sin(x); break;
  95                     case 2: temp = -cos(x)-sin(x); break;
  96                     case 3: temp =  cos(x)-sin(x); break;
  97                 }
  98                 b = invsqrtpi*temp/sqrt(x);
  99             } else {
 100                 a = __ieee754_j0(x);
 101                 b = __ieee754_j1(x);
 102                 for(i=1;i<n;i++){
 103                     temp = b;
 104                     b = b*((double)(i+i)/x) - a; /* avoid underflow */
 105                     a = temp;
 106                 }
 107             }
 108         } else {
 109             if(ix<0x3e100000) { /* x < 2**-29 */
 110     /* x is tiny, return the first Taylor expansion of J(n,x)
 111      * J(n,x) = 1/n!*(x/2)^n  - ...
 112      */
 113                 if(n>33)        /* underflow */
 114                     b = zero;
 115                 else {
 116                     temp = x*0.5; b = temp;
 117                     for (a=one,i=2;i<=n;i++) {
 118                         a *= (double)i;         /* a = n! */
 119                         b *= temp;              /* b = (x/2)^n */
 120                     }
 121                     b = b/a;
 122                 }
 123             } else {
 124                 /* use backward recurrence */
 125                 /*                      x      x^2      x^2
 126                  *  J(n,x)/J(n-1,x) =  ----   ------   ------   .....
 127                  *                      2n  - 2(n+1) - 2(n+2)
 128                  *
 129                  *                      1      1        1
 130                  *  (for large x)   =  ----  ------   ------   .....
 131                  *                      2n   2(n+1)   2(n+2)
 132                  *                      -- - ------ - ------ -
 133                  *                       x     x         x
 134                  *
 135                  * Let w = 2n/x and h=2/x, then the above quotient
 136                  * is equal to the continued fraction:
 137                  *                  1
 138                  *      = -----------------------
 139                  *                     1
 140                  *         w - -----------------
 141                  *                        1
 142                  *              w+h - ---------
 143                  *                     w+2h - ...
 144                  *
 145                  * To determine how many terms needed, let
 146                  * Q(0) = w, Q(1) = w(w+h) - 1,
 147                  * Q(k) = (w+k*h)*Q(k-1) - Q(k-2),
 148                  * When Q(k) > 1e4      good for single
 149                  * When Q(k) > 1e9      good for double
 150                  * When Q(k) > 1e17     good for quadruple
 151                  */
 152             /* determine k */
 153                 double t,v;
 154                 double q0,q1,h,tmp; int32_t k,m;
 155                 w  = (n+n)/(double)x; h = 2.0/(double)x;
 156                 q0 = w;  z = w+h; q1 = w*z - 1.0; k=1;
 157                 while(q1<1.0e9) {
 158                         k += 1; z += h;
 159                         tmp = z*q1 - q0;
 160                         q0 = q1;
 161                         q1 = tmp;
 162                 }
 163                 m = n+n;
 164                 for(t=zero, i = 2*(n+k); i>=m; i -= 2) t = one/(i/x-t);
 165                 a = t;
 166                 b = one;
 167                 /*  estimate log((2/x)^n*n!) = n*log(2/x)+n*ln(n)
 168                  *  Hence, if n*(log(2n/x)) > ...
 169                  *  single 8.8722839355e+01
 170                  *  double 7.09782712893383973096e+02
 171                  *  long double 1.1356523406294143949491931077970765006170e+04
 172                  *  then recurrent value may overflow and the result is
 173                  *  likely underflow to zero
 174                  */
 175                 tmp = n;
 176                 v = two/x;
 177                 tmp = tmp*__ieee754_log(fabs(v*tmp));
 178                 if(tmp<7.09782712893383973096e+02) {
 179                     for(i=n-1,di=(double)(i+i);i>0;i--){
 180                         temp = b;
 181                         b *= di;
 182                         b  = b/x - a;
 183                         a = temp;
 184                         di -= two;
 185                     }
 186                 } else {
 187                     for(i=n-1,di=(double)(i+i);i>0;i--){
 188                         temp = b;
 189                         b *= di;
 190                         b  = b/x - a;
 191                         a = temp;
 192                         di -= two;
 193                     /* scale b to avoid spurious overflow */
 194                         if(b>1e100) {
 195                             a /= b;
 196                             t /= b;
 197                             b  = one;
 198                         }
 199                     }
 200                 }
 201                 z = __ieee754_j0(x);
 202                 w = __ieee754_j1(x);
 203                 if (fabs(z) >= fabs(w))
 204                     b = (t*z/b);
 205                 else
 206                     b = (t*w/a);
 207             }
 208         }
 209         if(sgn==1) return -b; else return b;
 210 }
 211
 212 double
 213 __ieee754_yn(int n, double x)
 214 {
 215         int32_t i,hx,ix,lx;
 216         int32_t sign;
 217         double a, b, temp;
 218
 219         EXTRACT_WORDS(hx,lx,x);
 220         ix = 0x7fffffff&hx;
 221     /* if Y(n,NaN) is NaN */
 222         if((ix|((u_int32_t)(lx|-lx))>>31)>0x7ff00000) return x+x;
 223         if((ix|lx)==0) return -one/zero;
 224         if(hx<0) return zero/zero;
 225         sign = 1;
 226         if(n<0){
 227                 n = -n;
 228                 sign = 1 - ((n&1)<<1);
 229         }
 230         if(n==0) return(__ieee754_y0(x));
 231         if(n==1) return(sign*__ieee754_y1(x));
 232         if(ix==0x7ff00000) return zero;
 233         if(ix>=0x52D00000) { /* x > 2**302 */
 234     /* (x >> n**2)
 235      *      Jn(x) = cos(x-(2n+1)*pi/4)*sqrt(2/x*pi)
 236      *      Yn(x) = sin(x-(2n+1)*pi/4)*sqrt(2/x*pi)
 237      *      Let s=sin(x), c=cos(x),
 238      *          xn=x-(2n+1)*pi/4, sqt2 = sqrt(2),then
 239      *
 240      *             n    sin(xn)*sqt2    cos(xn)*sqt2
 241      *          ----------------------------------
 242      *             0     s-c             c+s
 243      *             1    -s-c            -c+s
 244      *             2    -s+c            -c-s
 245      *             3     s+c             c-s
 246      */
 247                 switch(n&3) {
 248                     case 0: temp =  sin(x)-cos(x); break;
 249                     case 1: temp = -sin(x)-cos(x); break;
 250                     case 2: temp = -sin(x)+cos(x); break;
 251                     case 3: temp =  sin(x)+cos(x); break;
 252                 }
 253                 b = invsqrtpi*temp/sqrt(x);
 254         } else {
 255             u_int32_t high;
 256             a = __ieee754_y0(x);
 257             b = __ieee754_y1(x);
 258         /* quit if b is -inf */
 259             GET_HIGH_WORD(high,b);
 260             for(i=1;i<n&&high!=0xfff00000;i++){
 261                 temp = b;
 262                 b = ((double)(i+i)/x)*b - a;
 263                 GET_HIGH_WORD(high,b);
 264                 a = temp;
 265             }
 266         }
 267         if(sign>0) return b; else return -b;
 268 }