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glibc  2.9
Functions | Variables
e_jnl.c File Reference
#include "math.h"
#include "math_private.h"

Go to the source code of this file.

Functions

long double __ieee754_jnl (int n, long double x)
long double __ieee754_ynl (int n, long double x)

Variables

static long double invsqrtpi = 5.6418958354775628694807945156077258584405E-1L
static long double two = 2.0e0L
static long double one = 1.0e0L
static long double zero = 0.0L

Function Documentation

long double __ieee754_jnl ( int  n,
long double  x 
)

Definition at line 78 of file e_jnl.c.

{
  u_int32_t se;
  int32_t i, ix, sgn;
  long double a, b, temp, di;
  long double z, w;
  ieee854_long_double_shape_type u;


  /* J(-n,x) = (-1)^n * J(n, x), J(n, -x) = (-1)^n * J(n, x)
   * Thus, J(-n,x) = J(n,-x)
   */

  u.value = x;
  se = u.parts32.w0;
  ix = se & 0x7fffffff;

  /* if J(n,NaN) is NaN */
  if (ix >= 0x7fff0000)
    {
      if ((u.parts32.w0 & 0xffff) | u.parts32.w1 | u.parts32.w2 | u.parts32.w3)
       return x + x;
    }

  if (n < 0)
    {
      n = -n;
      x = -x;
      se ^= 0x80000000;
    }
  if (n == 0)
    return (__ieee754_j0l (x));
  if (n == 1)
    return (__ieee754_j1l (x));
  sgn = (n & 1) & (se >> 31);      /* even n -- 0, odd n -- sign(x) */
  x = fabsl (x);

  if (x == 0.0L || ix >= 0x7fff0000)      /* if x is 0 or inf */
    b = zero;
  else if ((long double) n <= x)
    {
      /* Safe to use J(n+1,x)=2n/x *J(n,x)-J(n-1,x) */
      if (ix >= 0x412D0000)
       {                    /* x > 2**302 */

         /* ??? Could use an expansion for large x here.  */

         /* (x >> n**2)
          *      Jn(x) = cos(x-(2n+1)*pi/4)*sqrt(2/x*pi)
          *      Yn(x) = sin(x-(2n+1)*pi/4)*sqrt(2/x*pi)
          *      Let s=sin(x), c=cos(x),
          *          xn=x-(2n+1)*pi/4, sqt2 = sqrt(2),then
          *
          *             n    sin(xn)*sqt2    cos(xn)*sqt2
          *          ----------------------------------
          *             0     s-c             c+s
          *             1    -s-c            -c+s
          *             2    -s+c            -c-s
          *             3     s+c             c-s
          */
         long double s;
         long double c;
         __sincosl (x, &s, &c);
         switch (n & 3)
           {
           case 0:
             temp = c + s;
             break;
           case 1:
             temp = -c + s;
             break;
           case 2:
             temp = -c - s;
             break;
           case 3:
             temp = c - s;
             break;
           }
         b = invsqrtpi * temp / __ieee754_sqrtl (x);
       }
      else
       {
         a = __ieee754_j0l (x);
         b = __ieee754_j1l (x);
         for (i = 1; i < n; i++)
           {
             temp = b;
             b = b * ((long double) (i + i) / x) - a;   /* avoid underflow */
             a = temp;
           }
       }
    }
  else
    {
      if (ix < 0x3fc60000)
       {                    /* x < 2**-57 */
         /* x is tiny, return the first Taylor expansion of J(n,x)
          * J(n,x) = 1/n!*(x/2)^n  - ...
          */
         if (n >= 400)             /* underflow, result < 10^-4952 */
           b = zero;
         else
           {
             temp = x * 0.5;
             b = temp;
             for (a = one, i = 2; i <= n; i++)
              {
                a *= (long double) i;     /* a = n! */
                b *= temp;  /* b = (x/2)^n */
              }
             b = b / a;
           }
       }
      else
       {
         /* use backward recurrence */
         /*                      x      x^2      x^2
          *  J(n,x)/J(n-1,x) =  ----   ------   ------   .....
          *                      2n  - 2(n+1) - 2(n+2)
          *
          *                      1      1        1
          *  (for large x)   =  ----  ------   ------   .....
          *                      2n   2(n+1)   2(n+2)
          *                      -- - ------ - ------ -
          *                       x     x         x
          *
          * Let w = 2n/x and h=2/x, then the above quotient
          * is equal to the continued fraction:
          *                  1
          *      = -----------------------
          *                     1
          *         w - -----------------
          *                        1
          *              w+h - ---------
          *                     w+2h - ...
          *
          * To determine how many terms needed, let
          * Q(0) = w, Q(1) = w(w+h) - 1,
          * Q(k) = (w+k*h)*Q(k-1) - Q(k-2),
          * When Q(k) > 1e4      good for single
          * When Q(k) > 1e9      good for double
          * When Q(k) > 1e17     good for quadruple
          */
         /* determine k */
         long double t, v;
         long double q0, q1, h, tmp;
         int32_t k, m;
         w = (n + n) / (long double) x;
         h = 2.0L / (long double) x;
         q0 = w;
         z = w + h;
         q1 = w * z - 1.0L;
         k = 1;
         while (q1 < 1.0e17L)
           {
             k += 1;
             z += h;
             tmp = z * q1 - q0;
             q0 = q1;
             q1 = tmp;
           }
         m = n + n;
         for (t = zero, i = 2 * (n + k); i >= m; i -= 2)
           t = one / (i / x - t);
         a = t;
         b = one;
         /*  estimate log((2/x)^n*n!) = n*log(2/x)+n*ln(n)
          *  Hence, if n*(log(2n/x)) > ...
          *  single 8.8722839355e+01
          *  double 7.09782712893383973096e+02
          *  long double 1.1356523406294143949491931077970765006170e+04
          *  then recurrent value may overflow and the result is
          *  likely underflow to zero
          */
         tmp = n;
         v = two / x;
         tmp = tmp * __ieee754_logl (fabsl (v * tmp));

         if (tmp < 1.1356523406294143949491931077970765006170e+04L)
           {
             for (i = n - 1, di = (long double) (i + i); i > 0; i--)
              {
                temp = b;
                b *= di;
                b = b / x - a;
                a = temp;
                di -= two;
              }
           }
         else
           {
             for (i = n - 1, di = (long double) (i + i); i > 0; i--)
              {
                temp = b;
                b *= di;
                b = b / x - a;
                a = temp;
                di -= two;
                /* scale b to avoid spurious overflow */
                if (b > 1e100L)
                  {
                    a /= b;
                    t /= b;
                    b = one;
                  }
              }
           }
         b = (t * __ieee754_j0l (x) / b);
       }
    }
  if (sgn == 1)
    return -b;
  else
    return b;
}

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long double __ieee754_ynl ( int  n,
long double  x 
)

Definition at line 302 of file e_jnl.c.

{
  u_int32_t se;
  int32_t i, ix;
  int32_t sign;
  long double a, b, temp;
  ieee854_long_double_shape_type u;

  u.value = x;
  se = u.parts32.w0;
  ix = se & 0x7fffffff;

  /* if Y(n,NaN) is NaN */
  if (ix >= 0x7fff0000)
    {
      if ((u.parts32.w0 & 0xffff) | u.parts32.w1 | u.parts32.w2 | u.parts32.w3)
       return x + x;
    }
  if (x <= 0.0L)
    {
      if (x == 0.0L)
       return -HUGE_VALL + x;
      if (se & 0x80000000)
       return zero / (zero * x);
    }
  sign = 1;
  if (n < 0)
    {
      n = -n;
      sign = 1 - ((n & 1) << 1);
    }
  if (n == 0)
    return (__ieee754_y0l (x));
  if (n == 1)
    return (sign * __ieee754_y1l (x));
  if (ix >= 0x7fff0000)
    return zero;
  if (ix >= 0x412D0000)
    {                       /* x > 2**302 */

      /* ??? See comment above on the possible futility of this.  */

      /* (x >> n**2)
       *      Jn(x) = cos(x-(2n+1)*pi/4)*sqrt(2/x*pi)
       *      Yn(x) = sin(x-(2n+1)*pi/4)*sqrt(2/x*pi)
       *      Let s=sin(x), c=cos(x),
       *          xn=x-(2n+1)*pi/4, sqt2 = sqrt(2),then
       *
       *             n    sin(xn)*sqt2    cos(xn)*sqt2
       *          ----------------------------------
       *             0     s-c             c+s
       *             1    -s-c            -c+s
       *             2    -s+c            -c-s
       *             3     s+c             c-s
       */
      long double s;
      long double c;
      __sincosl (x, &s, &c);
      switch (n & 3)
       {
       case 0:
         temp = s - c;
         break;
       case 1:
         temp = -s - c;
         break;
       case 2:
         temp = -s + c;
         break;
       case 3:
         temp = s + c;
         break;
       }
      b = invsqrtpi * temp / __ieee754_sqrtl (x);
    }
  else
    {
      a = __ieee754_y0l (x);
      b = __ieee754_y1l (x);
      /* quit if b is -inf */
      u.value = b;
      se = u.parts32.w0 & 0xffff0000;
      for (i = 1; i < n && se != 0xffff0000; i++)
       {
         temp = b;
         b = ((long double) (i + i) / x) * b - a;
         u.value = b;
         se = u.parts32.w0 & 0xffff0000;
         a = temp;
       }
    }
  if (sign > 0)
    return b;
  else
    return -b;
}

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Variable Documentation

long double invsqrtpi = 5.6418958354775628694807945156077258584405E-1L [static]

Definition at line 67 of file e_jnl.c.

long double one = 1.0e0L [static]

Definition at line 69 of file e_jnl.c.

long double two = 2.0e0L [static]

Definition at line 68 of file e_jnl.c.

long double zero = 0.0L [static]

Definition at line 70 of file e_jnl.c.