sysdeps/ieee754/dbl-64/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 /*
  14  * __ieee754_jn(n, x), __ieee754_yn(n, x)
  15  * floating point Bessel's function of the 1st and 2nd kind
  16  * of order n
  17  *
  18  * Special cases:
  19  *      y0(0)=y1(0)=yn(n,0) = -inf with overflow signal;
  20  *      y0(-ve)=y1(-ve)=yn(n,-ve) are NaN with invalid signal.
  21  * Note 2. About jn(n,x), yn(n,x)
  22  *      For n=0, j0(x) is called,
  23  *      for n=1, j1(x) is called,
  24  *      for n<x, forward recursion us used starting
  25  *      from values of j0(x) and j1(x).
  26  *      for n>x, a continued fraction approximation to
  27  *      j(n,x)/j(n-1,x) is evaluated and then backward
  28  *      recursion is used starting from a supposed value
  29  *      for j(n,x). The resulting value of j(0,x) is
  30  *      compared with the actual value to correct the
  31  *      supposed value of j(n,x).
  32  *
  33  *      yn(n,x) is similar in all respects, except
  34  *      that forward recursion is used for all
  35  *      values of n>1.
  36  *
  37  */
  38
  39 #include <errno.h>
  40 #include <math.h>
  41 #include <math_private.h>
  42
  43 static const double
  44   invsqrtpi = 5.64189583547756279280e-01, /* 0x3FE20DD7, 0x50429B6D */
  45   two = 2.00000000000000000000e+00,  /* 0x40000000, 0x00000000 */
  46   one = 1.00000000000000000000e+00;  /* 0x3FF00000, 0x00000000 */
  47
  48 static const double zero = 0.00000000000000000000e+00;
  49
  50 double
  51 __ieee754_jn (int n, double x)
  52 {
  53   int32_t i, hx, ix, lx, sgn;
  54   double a, b, temp, di;
  55   double z, w;
  56
  57   /* J(-n,x) = (-1)^n * J(n, x), J(n, -x) = (-1)^n * J(n, x)
  58    * Thus, J(-n,x) = J(n,-x)
  59    */
  60   EXTRACT_WORDS (hx, lx, x);
  61   ix = 0x7fffffff & hx;
  62   /* if J(n,NaN) is NaN */
  63   if (__builtin_expect ((ix | ((u_int32_t) (lx | -lx)) >> 31) > 0x7ff00000, 0))
  64     return x + x;
  65   if (n < 0)
  66     {
  67       n = -n;
  68       x = -x;
  69       hx ^= 0x80000000;
  70     }
  71   if (n == 0)
  72     return (__ieee754_j0 (x));
  73   if (n == 1)
  74     return (__ieee754_j1 (x));
  75   sgn = (n & 1) & (hx >> 31);   /* even n -- 0, odd n -- sign(x) */
  76   x = fabs (x);
  77   if (__builtin_expect ((ix | lx) == 0 || ix >= 0x7ff00000, 0))
  78     /* if x is 0 or inf */
  79     b = zero;
  80   else if ((double) n <= x)
  81     {
  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           double s;
  98           double c;
  99           __sincos (x, &s, &c);
 100           switch (n & 3)
 101             {
 102             case 0: temp = c + s; break;
 103             case 1: temp = -c + s; break;
 104             case 2: temp = -c - s; break;
 105             case 3: temp = c - s; break;
 106             }
 107           b = invsqrtpi * temp / __ieee754_sqrt (x);
 108         }
 109       else
 110         {
 111           a = __ieee754_j0 (x);
 112           b = __ieee754_j1 (x);
 113           for (i = 1; i < n; i++)
 114             {
 115               temp = b;
 116               b = b * ((double) (i + i) / x) - a; /* avoid underflow */
 117               a = temp;
 118             }
 119         }
 120     }
 121   else
 122     {
 123       if (ix < 0x3e100000)      /* x < 2**-29 */
 124         { /* x is tiny, return the first Taylor expansion of J(n,x)
 125                          * J(n,x) = 1/n!*(x/2)^n  - ...
 126                          */
 127           if (n > 33)           /* underflow */
 128             b = zero;
 129           else
 130             {
 131               temp = x * 0.5; b = temp;
 132               for (a = one, i = 2; i <= n; i++)
 133                 {
 134                   a *= (double) i;              /* a = n! */
 135                   b *= temp;                    /* b = (x/2)^n */
 136                 }
 137               b = b / a;
 138             }
 139         }
 140       else
 141         {
 142           /* use backward recurrence */
 143           /*                    x      x^2      x^2
 144            *  J(n,x)/J(n-1,x) =  ----   ------   ------   .....
 145            *                    2n  - 2(n+1) - 2(n+2)
 146            *
 147            *                    1      1        1
 148            *  (for large x)   =  ----  ------   ------   .....
 149            *                    2n   2(n+1)   2(n+2)
 150            *                    -- - ------ - ------ -
 151            *                     x     x         x
 152            *
 153            * Let w = 2n/x and h=2/x, then the above quotient
 154            * is equal to the continued fraction:
 155            *                1
 156            *    = -----------------------
 157            *                   1
 158            *       w - -----------------
 159            *                      1
 160            *            w+h - ---------
 161            *                   w+2h - ...
 162            *
 163            * To determine how many terms needed, let
 164            * Q(0) = w, Q(1) = w(w+h) - 1,
 165            * Q(k) = (w+k*h)*Q(k-1) - Q(k-2),
 166            * When Q(k) > 1e4    good for single
 167            * When Q(k) > 1e9    good for double
 168            * When Q(k) > 1e17   good for quadruple
 169            */
 170           /* determine k */
 171           double t, v;
 172           double q0, q1, h, tmp; int32_t k, m;
 173           w = (n + n) / (double) x; h = 2.0 / (double) x;
 174           q0 = w;  z = w + h; q1 = w * z - 1.0; k = 1;
 175           while (q1 < 1.0e9)
 176             {
 177               k += 1; z += h;
 178               tmp = z * q1 - q0;
 179               q0 = q1;
 180               q1 = tmp;
 181             }
 182           m = n + n;
 183           for (t = zero, i = 2 * (n + k); i >= m; i -= 2)
 184             t = one / (i / x - t);
 185           a = t;
 186           b = one;
 187           /*  estimate log((2/x)^n*n!) = n*log(2/x)+n*ln(n)
 188            *  Hence, if n*(log(2n/x)) > ...
 189            *  single 8.8722839355e+01
 190            *  double 7.09782712893383973096e+02
 191            *  long double 1.1356523406294143949491931077970765006170e+04
 192            *  then recurrent value may overflow and the result is
 193            *  likely underflow to zero
 194            */
 195           tmp = n;
 196           v = two / x;
 197           tmp = tmp * __ieee754_log (fabs (v * tmp));
 198           if (tmp < 7.09782712893383973096e+02)
 199             {
 200               for (i = n - 1, di = (double) (i + i); i > 0; i--)
 201                 {
 202                   temp = b;
 203                   b *= di;
 204                   b = b / x - a;
 205                   a = temp;
 206                   di -= two;
 207                 }
 208             }
 209           else
 210             {
 211               for (i = n - 1, di = (double) (i + i); i > 0; i--)
 212                 {
 213                   temp = b;
 214                   b *= di;
 215                   b = b / x - a;
 216                   a = temp;
 217                   di -= two;
 218                   /* scale b to avoid spurious overflow */
 219                   if (b > 1e100)
 220                     {
 221                       a /= b;
 222                       t /= b;
 223                       b = one;
 224                     }
 225                 }
 226             }
 227           /* j0() and j1() suffer enormous loss of precision at and
 228            * near zero; however, we know that their zero points never
 229            * coincide, so just choose the one further away from zero.
 230            */
 231           z = __ieee754_j0 (x);
 232           w = __ieee754_j1 (x);
 233           if (fabs (z) >= fabs (w))
 234             b = (t * z / b);
 235           else
 236             b = (t * w / a);
 237         }
 238     }
 239   if (sgn == 1)
 240     return -b;
 241   else
 242     return b;
 243 }
 244 strong_alias (__ieee754_jn, __jn_finite)
 245
 246 double
 247 __ieee754_yn (int n, double x)
 248 {
 249   int32_t i, hx, ix, lx;
 250   int32_t sign;
 251   double a, b, temp;
 252
 253   EXTRACT_WORDS (hx, lx, x);
 254   ix = 0x7fffffff & hx;
 255   /* if Y(n,NaN) is NaN */
 256   if (__builtin_expect ((ix | ((u_int32_t) (lx | -lx)) >> 31) > 0x7ff00000, 0))
 257     return x + x;
 258   if (__builtin_expect ((ix | lx) == 0, 0))
 259     return -HUGE_VAL + x;
 260   /* -inf and overflow exception.  */;
 261   if (__builtin_expect (hx < 0, 0))
 262     return zero / (zero * x);
 263   sign = 1;
 264   if (n < 0)
 265     {
 266       n = -n;
 267       sign = 1 - ((n & 1) << 1);
 268     }
 269   if (n == 0)
 270     return (__ieee754_y0 (x));
 271   if (n == 1)
 272     return (sign * __ieee754_y1 (x));
 273   if (__builtin_expect (ix == 0x7ff00000, 0))
 274     return zero;
 275   if (ix >= 0x52D00000)      /* x > 2**302 */
 276     { /* (x >> n**2)
 277                  *          Jn(x) = cos(x-(2n+1)*pi/4)*sqrt(2/x*pi)
 278                  *          Yn(x) = sin(x-(2n+1)*pi/4)*sqrt(2/x*pi)
 279                  *          Let s=sin(x), c=cos(x),
 280                  *              xn=x-(2n+1)*pi/4, sqt2 = sqrt(2),then
 281                  *
 282                  *                 n    sin(xn)*sqt2    cos(xn)*sqt2
 283                  *              ----------------------------------
 284                  *                 0     s-c             c+s
 285                  *                 1    -s-c            -c+s
 286                  *                 2    -s+c            -c-s
 287                  *                 3     s+c             c-s
 288                  */
 289       double c;
 290       double s;
 291       __sincos (x, &s, &c);
 292       switch (n & 3)
 293         {
 294         case 0: temp = s - c; break;
 295         case 1: temp = -s - c; break;
 296         case 2: temp = -s + c; break;
 297         case 3: temp = s + c; break;
 298         }
 299       b = invsqrtpi * temp / __ieee754_sqrt (x);
 300     }
 301   else
 302     {
 303       u_int32_t high;
 304       a = __ieee754_y0 (x);
 305       b = __ieee754_y1 (x);
 306       /* quit if b is -inf */
 307       GET_HIGH_WORD (high, b);
 308       for (i = 1; i < n && high != 0xfff00000; i++)
 309         {
 310           temp = b;
 311           b = ((double) (i + i) / x) * b - a;
 312           GET_HIGH_WORD (high, b);
 313           a = temp;
 314         }
 315       /* If B is +-Inf, set up errno accordingly.  */
 316       if (!__finite (b))
 317         __set_errno (ERANGE);
 318     }
 319   if (sign > 0)
 320     return b;
 321   else
 322     return -b;
 323 }
 324 strong_alias (__ieee754_yn, __yn_finite)