/// <summary>
/// Multinomial Beta function.
/// </summary>
///
/// <example>
/// Please see <see cref="Beta"/>
/// </example>
///
public static double Multinomial(params double[] x)
{
double sum = 0;
double prd = 1;
for (int i = 0; i < x.Length; i++)
{
sum += x[i];
prd *= Gamma.Function(x[i]);
}
return(prd / Gamma.Function(sum));
}
/// <summary>
/// Power series for incomplete beta integral. Use when b*x
/// is small and x not too close to 1.
/// </summary>
///
/// <example>
/// Please see <see cref="Beta"/>
/// </example>
///
public static double PowerSeries(double a, double b, double x)
{
double s, t, u, v, n, t1, z, ai;
ai = 1.0 / a;
u = (1.0 - b) * x;
v = u / (a + 1.0);
t1 = v;
t = u;
n = 2.0;
s = 0.0;
z = Constants.DoubleEpsilon * ai;
while (System.Math.Abs(v) > z)
{
u = (n - b) * x / n;
t *= u;
v = t / (a + n);
s += v;
n += 1.0;
}
s += t1;
s += ai;
u = a * System.Math.Log(x);
if ((a + b) < Gamma.GammaMax && System.Math.Abs(u) < Constants.LogMax)
{
t = Gamma.Function(a + b) / (Gamma.Function(a) * Gamma.Function(b));
s = s * t * System.Math.Pow(x, a);
}
else
{
t = Gamma.Log(a + b) - Gamma.Log(a) - Gamma.Log(b) + u + System.Math.Log(s);
if (t < Constants.LogMin)
{
s = 0.0;
}
else
{
s = System.Math.Exp(t);
}
}
return(s);
}
/// <summary>
/// Incomplete (regularized) Beta function Ix(a, b).
/// </summary>
///
/// <example>
/// Please see <see cref="Beta"/>
/// </example>
///
public static double Incomplete(double a, double b, double x)
{
double aa, bb, t, xx, xc, w, y;
bool flag;
if (a <= 0.0)
{
throw new ArgumentOutOfRangeException("a", "Lower limit must be greater than zero.");
}
if (b <= 0.0)
{
throw new ArgumentOutOfRangeException("b", "Upper limit must be greater than zero.");
}
if ((x <= 0.0) || (x >= 1.0))
{
if (x == 0.0)
{
return(0.0);
}
if (x == 1.0)
{
return(1.0);
}
throw new ArgumentOutOfRangeException("x", "Value must be between 0 and 1.");
}
flag = false;
if ((b * x) <= 1.0 && x <= 0.95)
{
t = PowerSeries(a, b, x);
return(t);
}
w = 1.0 - x;
if (x > (a / (a + b)))
{
flag = true;
aa = b;
bb = a;
xc = x;
xx = w;
}
else
{
aa = a;
bb = b;
xc = w;
xx = x;
}
if (flag && (bb * xx) <= 1.0 && xx <= 0.95)
{
t = PowerSeries(aa, bb, xx);
if (t <= Constants.DoubleEpsilon)
{
t = 1.0 - Constants.DoubleEpsilon;
}
else
{
t = 1.0 - t;
}
return(t);
}
y = xx * (aa + bb - 2.0) - (aa - 1.0);
if (y < 0.0)
{
w = Incbcf(aa, bb, xx);
}
else
{
w = Incbd(aa, bb, xx) / xc;
}
y = aa * System.Math.Log(xx);
t = bb * System.Math.Log(xc);
if ((aa + bb) < Gamma.GammaMax && System.Math.Abs(y) < Constants.LogMax && System.Math.Abs(t) < Constants.LogMax)
{
t = System.Math.Pow(xc, bb);
t *= System.Math.Pow(xx, aa);
t /= aa;
t *= w;
t *= Gamma.Function(aa + bb) / (Gamma.Function(aa) * Gamma.Function(bb));
if (flag)
{
if (t <= Constants.DoubleEpsilon)
{
t = 1.0 - Constants.DoubleEpsilon;
}
else
{
t = 1.0 - t;
}
}
return(t);
}
y += t + Gamma.Log(aa + bb) - Gamma.Log(aa) - Gamma.Log(bb);
y += System.Math.Log(w / aa);
if (y < Constants.LogMin)
{
t = 0.0;
}
else
{
t = System.Math.Exp(y);
}
if (flag)
{
if (t <= Constants.DoubleEpsilon)
{
t = 1.0 - Constants.DoubleEpsilon;
}
else
{
t = 1.0 - t;
}
}
return(t);
}
/// <summary>
/// Returns the extended factorial definition of a real number.
/// </summary>
///
public static double Factorial(double n)
{
return(Gamma.Function(n + 1.0));
}