// one-argument functions
/// <summary>
/// Computes the natural logarithm of the Gamma function.
/// </summary>
/// <param name="x">The argument, which must be positive.</param>
/// <returns>The log Gamma function ln(Γ(x)).</returns>
/// <remarks>
/// <para>Because Γ(x) grows rapidly for increasing positive x, it is often necessary to
/// work with its logarithm in order to avoid overflow. This function returns accurate
/// values of ln(Γ(x)) even for values of x which would cause Γ(x) to overflow.</para>
/// </remarks>
/// <exception cref="ArgumentOutOfRangeException"><paramref name="x"/> is negative.</exception>
/// <seealso cref="Gamma(double)" />
public static double LogGamma(double x)
{
if (x < 0.0)
{
throw new ArgumentOutOfRangeException(nameof(x));
}
else if (x < 16.0)
{
// For small arguments, use the Lanczos approximation.
return(Lanczos.LogGamma(x));
}
else if (x < Double.PositiveInfinity)
{
// For large arguments, the asymptotic series is even faster than the Lanczos approximation.
return(Stirling.LogGamma(x));
}
else if (x == Double.PositiveInfinity)
{
// Precisely at infinity x * Math.Log(x) - x => NaN, so special-case it.
return(Double.PositiveInfinity);
}
else
{
return(Double.NaN);
}
}
// one-argument functions
/// <summary>
/// Computes the natural logrithm of the Gamma function.
/// </summary>
/// <param name="x">The argument, which must be positive.</param>
/// <returns>The log Gamma function ln(Γ(x)).</returns>
/// <remarks>
/// <para>Because Γ(x) grows rapidly for increasing positive x, it is often necessary to
/// work with its logarithm in order to avoid overflow. This function returns accurate
/// values of ln(Γ(x)) even for values of x which would cause Γ(x) to overflow.</para>
/// </remarks>
/// <exception cref="ArgumentOutOfRangeException"><paramref name="x"/> is negative or zero.</exception>
/// <seealso cref="Gamma(double)" />
public static double LogGamma(double x)
{
if (x <= 0.0)
{
throw new ArgumentOutOfRangeException("x");
}
else if (x < 16.0)
{
// For small arguments, use the Lanczos approximation.
return(Lanczos.LogGamma(x));
}
else
{
// For large arguments, the asymptotic series is even faster than the Lanczos approximation.
return(Stirling.LogGamma(x));
//return (LogGamma_Stirling(x));
}
}
// one-argument functions
/// <summary>
/// Computes the natural logarithm of the Gamma function.
/// </summary>
/// <param name="x">The argument, which must be positive.</param>
/// <returns>The log Gamma function ln(Γ(x)).</returns>
/// <remarks>
/// <para>Because Γ(x) grows rapidly for increasing positive x, it is often necessary to
/// work with its logarithm in order to avoid overflow. This function returns accurate
/// values of ln(Γ(x)) even for values of x which would cause Γ(x) to overflow.</para>
/// </remarks>
/// <exception cref="ArgumentOutOfRangeException"><paramref name="x"/> is negative.</exception>
/// <seealso cref="Gamma(double)" />
public static double LogGamma(double x)
{
if (x < 0.0)
{
throw new ArgumentOutOfRangeException(nameof(x));
}
else if (x < 0.5)
{
// For small arguments, use the Lanczos approximation.
return(Lanczos.LogGamma(x));
}
else if (x < 1.5)
{
// Use the series expansion near 1.
return(GammaSeries.LogGammaOnePlus(x - 1.0));
}
else if (x < 2.5)
{
// The series expansion can be adapted near 2, too.
return(GammaSeries.LogGammaTwoPlus(x - 2.0));
}
else if (x < 16.0)
{
// In between, we still use Lanczos.
return(Lanczos.LogGamma(x));
}
else if (x < Double.PositiveInfinity)
{
// For large arguments, the asymptotic series is even faster than the Lanczos approximation.
return(Stirling.LogGamma(x));
}
else if (x == Double.PositiveInfinity)
{
// Precisely at infinity x * Math.Log(x) - x => NaN, so special-case it.
return(Double.PositiveInfinity);
}
else
{
return(Double.NaN);
}
}
/// <summary>
/// Compute the complex log Gamma function.
/// </summary>
/// <param name="z">The complex argument.</param>
/// <returns>The principal complex value y for which exp(y) = Γ(z).</returns>
/// <seealso cref="AdvancedMath.LogGamma" />
/// <seealso href="http://mathworld.wolfram.com/LogGammaFunction.html"/>
public static Complex LogGamma(Complex z)
{
if (z.Im == 0.0 && z.Re < 0.0)
{
// Handle the pure negative case explicitly.
double re = Math.Log(Math.PI / Math.Abs(MoreMath.SinPi(z.Re))) - AdvancedMath.LogGamma(1.0 - z.Re);
double im = Math.PI * Math.Floor(z.Re);
return(new Complex(re, im));
}
else if (z.Re > 16.0 || Math.Abs(z.Im) > 16.0)
{
// According to https://dlmf.nist.gov/5.11, the Stirling asymptoic series is valid everywhere
// except on the negative real axis. So at first I tried to use it for |z.Im| > 0, |z| > 16. But in practice,
// it exhibits false convergence close to the negative real axis, e.g. z = -16 + i. So I have
// moved to requiring |z| large and reasonably far from the negative real axis.
return(Stirling.LogGamma(z));
}
else if (z.Re >= 0.125)
{
return(Lanczos.LogGamma(z));
}
else
{
// For the remaining z < 0, we need to use the reflection formula.
// For large z.Im, SinPi(z) \propto e^{\pi |z.Im|} overflows even though its log does not.
// It's possible to do some algebra to get around that problem, but it's not necessary
// because for z.Im that big we would have used the Stirling series.
Complex f = ComplexMath.Log(Math.PI / ComplexMath.SinPi(z));
Complex g = Lanczos.LogGamma(1.0 - z);
// The reflection formula doesn't stay on the principal branch, so we need to add a multiple of 2 \pi i
// to fix it up. See Hare, "Computing the Principal Branch of Log Gamma" for how to do this.
// https://pdfs.semanticscholar.org/1c9d/8865836a312836500126cb47c3cbbed3043e.pdf
Complex h = new Complex(0.0, 2.0 * Math.PI * Math.Floor(0.5 * (z.Re + 0.5)));
if (z.Im < 0.0)
{
h = -h;
}
return(f - g + h);
}
}
