nsz Git - musl/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 invsqrtpi = 5.64189583547756279280e-01; /* 0x3FE20DD7, 0x50429B6D */
  41
  42 double jn(int n, double x)
  43 {
  44         int32_t i,hx,ix,lx,sgn;
  45         double a, b, temp, di;
  46         double z, w;
  47
  48         /* J(-n,x) = (-1)^n * J(n, x), J(n, -x) = (-1)^n * J(n, x)
  49          * Thus, J(-n,x) = J(n,-x)
  50          */
  51         EXTRACT_WORDS(hx, lx, x);
  52         ix = 0x7fffffff & hx;
  53         /* if J(n,NaN) is NaN */
  54         if ((ix|((uint32_t)(lx|-lx))>>31) > 0x7ff00000)
  55                 return x+x;
  56         if (n < 0) {
  57                 n = -n;
  58                 x = -x;
  59                 hx ^= 0x80000000;
  60         }
  61         if (n == 0) return j0(x);
  62         if (n == 1) return j1(x);
  63
  64         sgn = (n&1)&(hx>>31);  /* even n -- 0, odd n -- sign(x) */
  65         x = fabs(x);
  66         if ((ix|lx) == 0 || ix >= 0x7ff00000)  /* if x is 0 or inf */
  67                 b = 0.0;
  68         else if ((double)n <= x) {
  69                 /* Safe to use J(n+1,x)=2n/x *J(n,x)-J(n-1,x) */
  70                 if (ix >= 0x52D00000) { /* x > 2**302 */
  71                         /* (x >> n**2)
  72                          *      Jn(x) = cos(x-(2n+1)*pi/4)*sqrt(2/x*pi)
  73                          *      Yn(x) = sin(x-(2n+1)*pi/4)*sqrt(2/x*pi)
  74                          *      Let s=sin(x), c=cos(x),
  75                          *          xn=x-(2n+1)*pi/4, sqt2 = sqrt(2),then
  76                          *
  77                          *             n    sin(xn)*sqt2    cos(xn)*sqt2
  78                          *          ----------------------------------
  79                          *             0     s-c             c+s
  80                          *             1    -s-c            -c+s
  81                          *             2    -s+c            -c-s
  82                          *             3     s+c             c-s
  83                          */
  84                         switch(n&3) {
  85                         case 0: temp =  cos(x)+sin(x); break;
  86                         case 1: temp = -cos(x)+sin(x); break;
  87                         case 2: temp = -cos(x)-sin(x); break;
  88                         case 3: temp =  cos(x)-sin(x); break;
  89                         }
  90                         b = invsqrtpi*temp/sqrt(x);
  91                 } else {
  92                         a = j0(x);
  93                         b = j1(x);
  94                         for (i=1; i<n; i++){
  95                                 temp = b;
  96                                 b = b*((double)(i+i)/x) - a; /* avoid underflow */
  97                                 a = temp;
  98                         }
  99                 }
 100         } else {
 101                 if (ix < 0x3e100000) { /* x < 2**-29 */
 102                         /* x is tiny, return the first Taylor expansion of J(n,x)
 103                          * J(n,x) = 1/n!*(x/2)^n  - ...
 104                          */
 105                         if (n > 33)  /* underflow */
 106                                 b = 0.0;
 107                         else {
 108                                 temp = x*0.5;
 109                                 b = temp;
 110                                 for (a=1.0,i=2; i<=n; i++) {
 111                                         a *= (double)i; /* a = n! */
 112                                         b *= temp;      /* b = (x/2)^n */
 113                                 }
 114                                 b = b/a;
 115                         }
 116                 } else {
 117                         /* use backward recurrence */
 118                         /*                      x      x^2      x^2
 119                          *  J(n,x)/J(n-1,x) =  ----   ------   ------   .....
 120                          *                      2n  - 2(n+1) - 2(n+2)
 121                          *
 122                          *                      1      1        1
 123                          *  (for large x)   =  ----  ------   ------   .....
 124                          *                      2n   2(n+1)   2(n+2)
 125                          *                      -- - ------ - ------ -
 126                          *                       x     x         x
 127                          *
 128                          * Let w = 2n/x and h=2/x, then the above quotient
 129                          * is equal to the continued fraction:
 130                          *                  1
 131                          *      = -----------------------
 132                          *                     1
 133                          *         w - -----------------
 134                          *                        1
 135                          *              w+h - ---------
 136                          *                     w+2h - ...
 137                          *
 138                          * To determine how many terms needed, let
 139                          * Q(0) = w, Q(1) = w(w+h) - 1,
 140                          * Q(k) = (w+k*h)*Q(k-1) - Q(k-2),
 141                          * When Q(k) > 1e4      good for single
 142                          * When Q(k) > 1e9      good for double
 143                          * When Q(k) > 1e17     good for quadruple
 144                          */
 145                         /* determine k */
 146                         double t,v;
 147                         double q0,q1,h,tmp;
 148                         int32_t k,m;
 149
 150                         w  = (n+n)/(double)x; h = 2.0/(double)x;
 151                         q0 = w;
 152                         z = w+h;
 153                         q1 = w*z - 1.0;
 154                         k = 1;
 155                         while (q1 < 1.0e9) {
 156                                 k += 1;
 157                                 z += h;
 158                                 tmp = z*q1 - q0;
 159                                 q0 = q1;
 160                                 q1 = tmp;
 161                         }
 162                         m = n+n;
 163                         for (t=0.0, i = 2*(n+k); i>=m; i -= 2)
 164                                 t = 1.0/(i/x-t);
 165                         a = t;
 166                         b = 1.0;
 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 = 2.0/x;
 177                         tmp = tmp*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 -= 2.0;
 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 -= 2.0;
 193                                         /* scale b to avoid spurious overflow */
 194                                         if (b > 1e100) {
 195                                                 a /= b;
 196                                                 t /= b;
 197                                                 b  = 1.0;
 198                                         }
 199                                 }
 200                         }
 201                         z = j0(x);
 202                         w = 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;
 210         return b;
 211 }
 212
 213
 214
 215 double yn(int n, double x)
 216 {
 217         int32_t i,hx,ix,lx;
 218         int32_t sign;
 219         double a, b, temp;
 220
 221         EXTRACT_WORDS(hx, lx, x);
 222         ix = 0x7fffffff & hx;
 223         /* if Y(n,NaN) is NaN */
 224         if ((ix|((uint32_t)(lx|-lx))>>31) > 0x7ff00000)
 225                 return x+x;
 226         if ((ix|lx) == 0)
 227                 return -1.0/0.0;
 228         if (hx < 0)
 229                 return 0.0/0.0;
 230         sign = 1;
 231         if (n < 0) {
 232                 n = -n;
 233                 sign = 1 - ((n&1)<<1);
 234         }
 235         if (n == 0)
 236                 return y0(x);
 237         if (n == 1)
 238                 return sign*y1(x);
 239         if (ix == 0x7ff00000)
 240                 return 0.0;
 241         if (ix >= 0x52D00000) { /* x > 2**302 */
 242                 /* (x >> n**2)
 243                  *      Jn(x) = cos(x-(2n+1)*pi/4)*sqrt(2/x*pi)
 244                  *      Yn(x) = sin(x-(2n+1)*pi/4)*sqrt(2/x*pi)
 245                  *      Let s=sin(x), c=cos(x),
 246                  *          xn=x-(2n+1)*pi/4, sqt2 = sqrt(2),then
 247                  *
 248                  *             n    sin(xn)*sqt2    cos(xn)*sqt2
 249                  *          ----------------------------------
 250                  *             0     s-c             c+s
 251                  *             1    -s-c            -c+s
 252                  *             2    -s+c            -c-s
 253                  *             3     s+c             c-s
 254                  */
 255                 switch(n&3) {
 256                 case 0: temp =  sin(x)-cos(x); break;
 257                 case 1: temp = -sin(x)-cos(x); break;
 258                 case 2: temp = -sin(x)+cos(x); break;
 259                 case 3: temp =  sin(x)+cos(x); break;
 260                 }
 261                 b = invsqrtpi*temp/sqrt(x);
 262         } else {
 263                 uint32_t high;
 264                 a = y0(x);
 265                 b = y1(x);
 266                 /* quit if b is -inf */
 267                 GET_HIGH_WORD(high, b);
 268                 for (i=1; i<n && high!=0xfff00000; i++){
 269                         temp = b;
 270                         b = ((double)(i+i)/x)*b - a;
 271                         GET_HIGH_WORD(high, b);
 272                         a = temp;
 273                 }
 274         }
 275         if (sign > 0) return b;
 276         return -b;
 277 }