/// <summary>
/// Computes the logarithm of the Gamma function with double double precision.
/// </summary>
/// <param name="x">The argument, which must be non-negative.</param>
/// <returns>The value of ln(Γ(x)).</returns>
/// <exception cref="ArgumentOutOfRangeException"><paramref name="x"/> is less than zero.</exception>
/// <seealso cref="Meta.Numerics.Functions.AdvancedMath.LogGamma(double)"/>
public static DoubleDouble LogGamma(DoubleDouble x)
{
if (x < DoubleDouble.Zero)
{
throw new ArgumentOutOfRangeException(nameof(x));
}
else if (x < 0.5)
{
// Use x G(x) = G(1+x) to get G(x) from G(1+x)
return(LogGammaOnePlus(x) - DoubleDouble.Log(x));
}
else if (x < 1.5)
{
DoubleDouble y = x - 1.0;
Debug.Assert(DoubleDouble.Abs(y) <= 0.5);
return(LogGammaOnePlus(y));
}
else if (x < 16.0)
{
return(LogGamma_ShiftToZetaSeries(x));
}
else
{
return(LogGamma_ShiftToAsymptotic(x));
}
}
private static DoubleDouble LogGamma_ShiftToAsymptotic(DoubleDouble x)
{
Debug.Assert(x > 0.0);
DoubleDouble s = DoubleDouble.Zero;
while (x < 34.0)
{
s += DoubleDouble.Log(x);
x += DoubleDouble.One;
}
return(LogGamma_Asymptotic(x) - s);
}
private static DoubleDouble LogGamma_ShiftToZetaSeries(DoubleDouble x)
{
Debug.Assert(x >= 1.5);
DoubleDouble s = DoubleDouble.Zero;
while (x > 2.5)
{
x -= DoubleDouble.One;
s += DoubleDouble.Log(x);
}
DoubleDouble y = x - 2.0;
Debug.Assert(DoubleDouble.Abs(y) <= 0.5);
return(LogGammaTwoPlus(y) + s);
}
private static DoubleDouble LogGamma_Asymptotic(DoubleDouble x)
{
// Sum from smallest to largest terms to minimize error.
return(BernoulliSum(x) + halfLogTwoPi - x + (x - 0.5) * DoubleDouble.Log(x));
}
