/// <summary>
/// Upper incomplete gamma fraction
/// <para>Γ(a,z) = (z^a * e^-z) * UpperFraction(a,z)</para>
/// <para>Q(a,z) = (z^a * e^-z)/Γ(a) * UpperFraction(a,z)</para>
/// </summary>
/// <param name="a"></param>
/// <param name="z"></param>
/// <returns></returns>
public static double UpperFraction(double a, double z)
{
double z0 = z - a + 1;
int k = 0;
Func <(double, double)> f = () => {
k++;
return(k * (a - k), z0 + 2 * k);
};
return(1.0 / ContinuedFraction.Eval(z0, f));
}
/// <summary>
/// Return En(x) using continued fractions:
/// <para>En(x) = e^-x/(x + n + ContinuedFraction(-k * (k + n - 1), x + n + 2*k) k={1, ∞}</para>
/// </summary>
/// <param name="n"></param>
/// <param name="x">Requires x > 0</param>
/// <returns></returns>
/// <seealso href="http://functions.wolfram.com/GammaBetaErf/ExpIntegralE/10/0004/"/>
public static double En_Fraction(int n, double x)
{
double b = n + x;
double k = 0;
Func <(double, double)> fracF = () => {
++k;
return(-k * (k + n - 1), b + 2 * k);
};
double frac = ContinuedFraction.Eval(b, fracF);
return(Math.Exp(-x) / frac);
}
/// <summary>
/// Evaluate the incomplete beta via the continued fraction representation
/// </summary>
/// <param name="a"></param>
/// <param name="b"></param>
/// <param name="x"></param>
/// <param name="y"></param>
/// <param name="normalised"></param>
/// <returns></returns>
/// <remarks>
/// Continued fraction for Ibeta- See: http://dlmf.nist.gov/8.17.E22
/// Converges rapidly when x < (a+1)/(a+b+2); for x > (a+1)/(a+b+2) || 1-x < (b+1)/(a+b+2), use I{1-x}(a, b)
/// Continued fraction for the incomplete beta used when a > 1 and b > 1
/// <para>According to NR this is O(sqrt(max(a,b))</para>
/// </remarks>
public static double Fraction2(double a, double b, double x, double y, bool normalised)
{
Debug.Assert(x <= a / (a + b) || y <= b / (a + b));
double result = PowerTerms(a, b, x, y, normalised);
if (result == 0)
{
return(result);
}
// Define the continued fraction function
int m = 0;
Func <(double an, double bn)> fracF = () => {
double aN = (a + m - 1) * (a + b + m - 1) * m * (b - m) * x * x;
double denom = (a + 2 * m - 1);
aN /= denom * denom;
double bN = m;
bN += (m * (b - m) * x) / (a + 2 * m - 1);
bN += ((a + m) * (a - (a + b) * x + 1 + m * (2 - x))) / (a + 2 * m + 1);
++m;
return(aN, bN);
};
var(_, b0) = fracF(); // get b0 - skip a0;
double fract = ContinuedFraction.Eval(b0, fracF);
#if EXTRA_DEBUG
double cvg = (a + 1) / (a + b + 2);
Debug.WriteLine("CF B({0}, {1}, {2}): cvg = {3}; iterations = {4}", a, b, x, cvg, m);
#endif
return(result / fract);
}