/// <summary>
/// Natural logarithm of the gamma function.
/// </summary>
///
public static double Log(double x)
{
if (x == 0)
{
return(Double.PositiveInfinity);
}
double p, q, w, z;
if (x < -34.0)
{
q = -x;
w = Log(q);
p = Math.Floor(q);
if (p == q)
{
throw new OverflowException();
}
z = q - p;
if (z > 0.5)
{
p += 1.0;
z = p - q;
}
z = q * Math.Sin(System.Math.PI * z);
if (z == 0.0)
{
throw new OverflowException();
}
z = Constants.LogPI - Math.Log(z) - w;
return(z);
}
if (x < 13.0)
{
z = 1.0;
while (x >= 3.0)
{
x -= 1.0;
z *= x;
}
while (x < 2.0)
{
if (x == 0.0)
{
throw new OverflowException();
}
z /= x;
x += 1.0;
}
if (z < 0.0)
{
z = -z;
}
if (x == 2.0)
{
return(System.Math.Log(z));
}
x -= 2.0;
p = x * Polynomial.Evaluate(x, log_B, 5) / Polynomial.EvaluateSpecial(x, log_C, 6);
return(Math.Log(z) + p);
}
if (x > 2.556348e305)
{
throw new OverflowException();
}
q = (x - 0.5) * Math.Log(x) - x + 0.91893853320467274178;
if (x > 1.0e8)
{
return(q);
}
p = 1.0 / (x * x);
if (x >= 1000.0)
{
q += ((7.9365079365079365079365e-4 * p
- 2.7777777777777777777778e-3) * p
+ 0.0833333333333333333333) / x;
}
else
{
q += Polynomial.Evaluate(p, log_A, 4) / x;
}
return(q);
}
/// <summary>
/// Normal (Gaussian) inverse cumulative distribution function.
/// </summary>
///
/// <remarks>
/// <para>
/// For small arguments <c>0 < y < exp(-2)</c>, the program computes <c>z =
/// sqrt( -2.0 * log(y) )</c>; then the approximation is <c>x = z - log(z)/z -
/// (1/z) P(1/z) / Q(1/z)</c>.</para>
/// <para>
/// There are two rational functions P/Q, one for <c>0 < y < exp(-32)</c> and
/// the other for <c>y</c> up to <c>exp(-2)</c>. For larger arguments, <c>w = y - 0.5</c>,
/// and <c>x/sqrt(2pi) = w + w^3 * R(w^2)/S(w^2))</c>.</para>
/// </remarks>
///
/// <returns>
/// Returns the value, <c>x</c>, for which the area under the Normal (Gaussian)
/// probability density function (integrated from minus infinity to <c>x</c>) is
/// equal to the argument <c>y</c> (assumes mean is zero, variance is one).
/// </returns>
///
public static double Inverse(double y0)
{
if (y0 <= 0.0)
{
if (y0 == 0)
{
return(Double.NegativeInfinity);
}
throw new ArgumentOutOfRangeException("y0");
}
if (y0 >= 1.0)
{
if (y0 == 1)
{
return(Double.PositiveInfinity);
}
throw new ArgumentOutOfRangeException("y0");
}
double s2pi = Math.Sqrt(2.0 * Math.PI);
int code = 1;
double y = y0;
double x;
if (y > 0.8646647167633873)
{
y = 1.0 - y;
code = 0;
}
if (y > 0.1353352832366127)
{
y -= 0.5;
double y2 = y * y;
x = y + y * ((y2 * Polynomial.Evaluate(y2, inverse_P0, 4)) / Polynomial.EvaluateSpecial(y2, inverse_Q0, 8));
x *= s2pi;
return(x);
}
x = Math.Sqrt(-2.0 * Math.Log(y));
double x0 = x - Math.Log(x) / x;
double z = 1.0 / x;
double x1;
if (x < 8.0)
{
x1 = (z * Polynomial.Evaluate(z, inverse_P1, 8)) / Polynomial.EvaluateSpecial(z, inverse_Q1, 8);
}
else
{
x1 = (z * Polynomial.Evaluate(z, inverse_P2, 8)) / Polynomial.EvaluateSpecial(z, inverse_Q2, 8);
}
x = x0 - x1;
if (code != 0)
{
x = -x;
}
return(x);
}