nsz Git - libm/blob - src/math/jn.c

   1 /* origin: FreeBSD /usr/src/lib/msun/src/e_jn.c */
   2 /*
   3  * ====================================================
   4  * Copyright (C) 1993 by Sun Microsystems, Inc. All rights reserved.
   5  *
   6  * Developed at SunSoft, 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  * jn(n, x), 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 "libm.h"
  39
  40 static const double
  41 invsqrtpi = 5.64189583547756279280e-01, /* 0x3FE20DD7, 0x50429B6D */
  42 two       = 2.00000000000000000000e+00, /* 0x40000000, 0x00000000 */
  43 one       = 1.00000000000000000000e+00; /* 0x3FF00000, 0x00000000 */
  44
  45 static const double zero = 0.00000000000000000000e+00;
  46
  47 double jn(int n, double x)
  48 {
  49         int32_t i,hx,ix,lx,sgn;
  50         double a, b, temp, di;
  51         double z, w;
  52
  53         /* J(-n,x) = (-1)^n * J(n, x), J(n, -x) = (-1)^n * J(n, x)
  54          * Thus, J(-n,x) = J(n,-x)
  55          */
  56         EXTRACT_WORDS(hx, lx, x);
  57         ix = 0x7fffffff & hx;
  58         /* if J(n,NaN) is NaN */
  59         if ((ix|((uint32_t)(lx|-lx))>>31) > 0x7ff00000)
  60                 return x+x;
  61         if (n < 0) {
  62                 n = -n;
  63                 x = -x;
  64                 hx ^= 0x80000000;
  65         }
  66         if (n == 0) return j0(x);
  67         if (n == 1) return j1(x);
  68
  69         sgn = (n&1)&(hx>>31);  /* even n -- 0, odd n -- sign(x) */
  70         x = fabs(x);
  71         if ((ix|lx) == 0 || ix >= 0x7ff00000)  /* if x is 0 or inf */
  72                 b = zero;
  73         else if ((double)n <= x) {
  74                 /* Safe to use J(n+1,x)=2n/x *J(n,x)-J(n-1,x) */
  75                 if (ix >= 0x52D00000) { /* x > 2**302 */
  76                         /* (x >> n**2)
  77                          *      Jn(x) = cos(x-(2n+1)*pi/4)*sqrt(2/x*pi)
  78                          *      Yn(x) = sin(x-(2n+1)*pi/4)*sqrt(2/x*pi)
  79                          *      Let s=sin(x), c=cos(x),
  80                          *          xn=x-(2n+1)*pi/4, sqt2 = sqrt(2),then
  81                          *
  82                          *             n    sin(xn)*sqt2    cos(xn)*sqt2
  83                          *          ----------------------------------
  84                          *             0     s-c             c+s
  85                          *             1    -s-c            -c+s
  86                          *             2    -s+c            -c-s
  87                          *             3     s+c             c-s
  88                          */
  89                         switch(n&3) {
  90                         case 0: temp =  cos(x)+sin(x); break;
  91                         case 1: temp = -cos(x)+sin(x); break;
  92                         case 2: temp = -cos(x)-sin(x); break;
  93                         case 3: temp =  cos(x)-sin(x); break;
  94                         }
  95                         b = invsqrtpi*temp/sqrt(x);
  96                 } else {
  97                         a = j0(x);
  98                         b = j1(x);
  99                         for (i=1; i<n; i++){
 100                                 temp = b;
 101                                 b = b*((double)(i+i)/x) - a; /* avoid underflow */
 102                                 a = temp;
 103                         }
 104                 }
 105         } else {
 106                 if (ix < 0x3e100000) { /* x < 2**-29 */
 107                         /* x is tiny, return the first Taylor expansion of J(n,x)
 108                          * J(n,x) = 1/n!*(x/2)^n  - ...
 109                          */
 110                         if (n > 33)  /* underflow */
 111                                 b = zero;
 112                         else {
 113                                 temp = x*0.5;
 114                                 b = temp;
 115                                 for (a=one,i=2; i<=n; i++) {
 116                                         a *= (double)i; /* a = n! */
 117                                         b *= temp;      /* b = (x/2)^n */
 118                                 }
 119                                 b = b/a;
 120                         }
 121                 } else {
 122                         /* use backward recurrence */
 123                         /*                      x      x^2      x^2
 124                          *  J(n,x)/J(n-1,x) =  ----   ------   ------   .....
 125                          *                      2n  - 2(n+1) - 2(n+2)
 126                          *
 127                          *                      1      1        1
 128                          *  (for large x)   =  ----  ------   ------   .....
 129                          *                      2n   2(n+1)   2(n+2)
 130                          *                      -- - ------ - ------ -
 131                          *                       x     x         x
 132                          *
 133                          * Let w = 2n/x and h=2/x, then the above quotient
 134                          * is equal to the continued fraction:
 135                          *                  1
 136                          *      = -----------------------
 137                          *                     1
 138                          *         w - -----------------
 139                          *                        1
 140                          *              w+h - ---------
 141                          *                     w+2h - ...
 142                          *
 143                          * To determine how many terms needed, let
 144                          * Q(0) = w, Q(1) = w(w+h) - 1,
 145                          * Q(k) = (w+k*h)*Q(k-1) - Q(k-2),
 146                          * When Q(k) > 1e4      good for single
 147                          * When Q(k) > 1e9      good for double
 148                          * When Q(k) > 1e17     good for quadruple
 149                          */
 150                         /* determine k */
 151                         double t,v;
 152                         double q0,q1,h,tmp;
 153                         int32_t k,m;
 154
 155                         w  = (n+n)/(double)x; h = 2.0/(double)x;
 156                         q0 = w;
 157                         z = w+h;
 158                         q1 = w*z - 1.0;
 159                         k = 1;
 160                         while (q1 < 1.0e9) {
 161                                 k += 1;
 162                                 z += h;
 163                                 tmp = z*q1 - q0;
 164                                 q0 = q1;
 165                                 q1 = tmp;
 166                         }
 167                         m = n+n;
 168                         for (t=zero, i = 2*(n+k); i>=m; i -= 2)
 169                                 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*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 = j0(x);
 207                         w = 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;
 215         return b;
 216 }
 217
 218
 219
 220 double yn(int n, double x)
 221 {
 222         int32_t i,hx,ix,lx;
 223         int32_t sign;
 224         double a, b, temp;
 225
 226         EXTRACT_WORDS(hx, lx, x);
 227         ix = 0x7fffffff & hx;
 228         /* if Y(n,NaN) is NaN */
 229         if ((ix|((uint32_t)(lx|-lx))>>31) > 0x7ff00000)
 230                 return x+x;
 231         if ((ix|lx) == 0)
 232                 return -one/zero;
 233         if (hx < 0)
 234                 return zero/zero;
 235         sign = 1;
 236         if (n < 0) {
 237                 n = -n;
 238                 sign = 1 - ((n&1)<<1);
 239         }
 240         if (n == 0)
 241                 return y0(x);
 242         if (n == 1)
 243                 return sign*y1(x);
 244         if (ix == 0x7ff00000)
 245                 return zero;
 246         if (ix >= 0x52D00000) { /* x > 2**302 */
 247                 /* (x >> n**2)
 248                  *      Jn(x) = cos(x-(2n+1)*pi/4)*sqrt(2/x*pi)
 249                  *      Yn(x) = sin(x-(2n+1)*pi/4)*sqrt(2/x*pi)
 250                  *      Let s=sin(x), c=cos(x),
 251                  *          xn=x-(2n+1)*pi/4, sqt2 = sqrt(2),then
 252                  *
 253                  *             n    sin(xn)*sqt2    cos(xn)*sqt2
 254                  *          ----------------------------------
 255                  *             0     s-c             c+s
 256                  *             1    -s-c            -c+s
 257                  *             2    -s+c            -c-s
 258                  *             3     s+c             c-s
 259                  */
 260                 switch(n&3) {
 261                 case 0: temp =  sin(x)-cos(x); break;
 262                 case 1: temp = -sin(x)-cos(x); break;
 263                 case 2: temp = -sin(x)+cos(x); break;
 264                 case 3: temp =  sin(x)+cos(x); break;
 265                 }
 266                 b = invsqrtpi*temp/sqrt(x);
 267         } else {
 268                 uint32_t high;
 269                 a = y0(x);
 270                 b = y1(x);
 271                 /* quit if b is -inf */
 272                 GET_HIGH_WORD(high, b);
 273                 for (i=1; i<n && high!=0xfff00000; i++){
 274                         temp = b;
 275                         b = ((double)(i+i)/x)*b - a;
 276                         GET_HIGH_WORD(high, b);
 277                         a = temp;
 278                 }
 279         }
 280         if (sign > 0) return b;
 281         return -b;
 282 }