libm/e_jn.c

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