static ScalarValue LogGamma_Stirling(ScalarValue z)
{
if (z.Im < 0.0)
{
return(LogGamma_Stirling(z.Conjugate()).Conjugate());
}
var f = (z - 0.5) * z.Ln() - z + Math.Log(2.0 * Math.PI) / 2.0;
var reduce = f.Im / (2.0 * Math.PI);
reduce = f.Im - (int)(reduce) * 2.0 * Math.PI;
f = new ScalarValue(f.Re, reduce);
var zsqu = z * z;
var zp = z.Clone();
for (var i = 1; i < 10; i++)
{
var f_old = f.Clone();
f += Helpers.BernoulliNumbers[i] / (2 * i) / (2 * i - 1) / zp;
if (f == f_old)
{
return(f);
}
zp = zp * zsqu;
}
throw new YAMPNotConvergedException("gamma");
}
/// <summary>
/// The Faddeeva function or Kramp function is a scaled complex complementary error function.
/// </summary>
/// <param name="z">The argument z.</param>
/// <returns>The evaluated value.</returns>
public static ScalarValue Faddeeva(ScalarValue z)
{
if (z.Im < 0.0)
{
return(2.0 * (-z * z).Exp() - Faddeeva(-z));
}
if (z.Re < 0.0)
{
return(Faddeeva(-z.Conjugate()).Conjugate());
}
var r = z.Abs();
if (r < 2.0)
{
return((-z * z).Exp() * (1.0 - Erf_Series(-ScalarValue.I * z)));
}
else if ((z.Im < 0.1) && (z.Re < 30.0))
{
return(Taylor(new ScalarValue(z.Re), Math.Exp(-z.Re * z.Re) + 2.0 * Dawson.DawsonIntegral(z.Re) / Helpers.SqrtPI * ScalarValue.I, new ScalarValue(0.0, z.Im)));
}
else if (r > 7.0)
{
return(ContinuedFraction(z));
}
return(Weideman(z));
}
/// <summary>
/// Cholesky algorithm for symmetric and positive definite matrix.
/// </summary>
/// <param name="Arg">Square, symmetric matrix.</param>
/// <returns>Structure to access L and isspd flag.</returns>
public CholeskyDecomposition(MatrixValue Arg)
{
// Initialize.
var A = Arg.GetComplexMatrix();
n = Arg.DimensionY;
L = new ScalarValue[n][];
for (int i = 0; i < n; i++)
{
L[i] = new ScalarValue[n];
}
isspd = Arg.DimensionX == n;
// Main loop.
for (int i = 0; i < n; i++)
{
var Lrowi = L[i];
var d = ScalarValue.Zero;
for (int j = 0; j < i; j++)
{
var Lrowj = L[j];
var s = new ScalarValue();
for (int k = 0; k < j; k++)
{
s += Lrowi[k] * Lrowj[k].Conjugate();
}
s = (A[i][j] - s) / L[j][j];
Lrowi[j] = s;
d += s * s.Conjugate();
isspd = isspd && (A[j][i] == A[i][j]);
}
d = A[i][i] - d;
isspd = isspd & (d.Abs() > 0.0);
L[i][i] = d.Sqrt();
for (int k = i + 1; k < n; k++)
{
L[i][k] = ScalarValue.Zero;
}
}
}
protected override ScalarValue GetValue(ScalarValue value)
{
return(value.Conjugate());
}
protected override ScalarValue GetValue(ScalarValue value)
{
return value.Conjugate();
}
static ScalarValue LogGamma_Stirling(ScalarValue z)
{
if (z.Im < 0.0)
{
return LogGamma_Stirling(z.Conjugate()).Conjugate();
}
var f = (z - 0.5) * z.Ln() - z + Math.Log(2.0 * Math.PI) / 2.0;
var reduce = f.Im / (2.0 * Math.PI);
reduce = f.Im - (int)(reduce) * 2.0 * Math.PI;
f = new ScalarValue(f.Re, reduce);
var zsqu = z * z;
var zp = z.Clone();
for (var i = 1; i < 10; i++)
{
var f_old = f.Clone();
f += Helpers.BernoulliNumbers[i] / (2 * i) / (2 * i - 1) / zp;
if (f == f_old)
{
return (f);
}
zp = zp * zsqu;
}
throw new YAMPNotConvergedException("gamma");
}
/// <summary>
/// The Faddeeva function or Kramp function is a scaled complex complementary error function.
/// </summary>
/// <param name="z">The argument z.</param>
/// <returns>The evaluated value.</returns>
public static ScalarValue Faddeeva(ScalarValue z)
{
if (z.Im < 0.0)
return 2.0 * (-z * z).Exp() - Faddeeva(-z);
if (z.Re < 0.0)
return Faddeeva(-z.Conjugate()).Conjugate();
var r = z.Abs();
if (r < 2.0)
return (-z * z).Exp() * (1.0 - Erf_Series(-ScalarValue.I * z));
else if ((z.Im < 0.1) && (z.Re < 30.0))
return Taylor(new ScalarValue(z.Re), Math.Exp(-z.Re * z.Re) + 2.0 * Dawson.DawsonIntegral(z.Re) / Helpers.SqrtPI * ScalarValue.I, new ScalarValue(0.0, z.Im));
else if (r > 7.0)
return ContinuedFraction(z);
return Weideman(z);
}
