/// <summary>
/// Reduces a complex hermitian matrix to a real symmetric tridiagonal matrix using unitary similarity transformations.
/// </summary>
/// <param name="matrixA">Source matrix to reduce</param>
/// <param name="d">Output: Arrays for internal storage of real parts of eigenvalues</param>
/// <param name="e">Output: Arrays for internal storage of imaginary parts of eigenvalues</param>
/// <param name="tau">Output: Arrays that contains further information about the transformations.</param>
/// <param name="order">Order of initial matrix</param>
/// <remarks>This is derived from the Algol procedures HTRIDI by
/// Smith, Boyle, Dongarra, Garbow, Ikebe, Klema, Moler, and Wilkinson, Handbook for
/// Auto. Comp., Vol.ii-Linear Algebra, and the corresponding
/// Fortran subroutine in EISPACK.</remarks>
private static void SymmetricTridiagonalize(Complex[,] matrixA, double[] d, double[] e, Complex[] tau, int order)
{
double hh;
tau[order - 1] = Complex.One;
for (var i = 0; i < order; i++)
{
d[i] = matrixA[i, i].Real;
}
// Householder reduction to tridiagonal form.
for (var i = order - 1; i > 0; i--)
{
// Scale to avoid under/overflow.
var scale = 0.0;
var h = 0.0;
for (var k = 0; k < i; k++)
{
scale = scale + Math.Abs(matrixA[i, k].Real) + Math.Abs(matrixA[i, k].Imaginary);
}
if (scale == 0.0)
{
tau[i - 1] = Complex.One;
e[i] = 0.0;
}
else
{
for (var k = 0; k < i; k++)
{
matrixA[i, k] /= scale;
h += matrixA[i, k].MagnitudeSquared();
}
Complex g = Math.Sqrt(h);
e[i] = scale * g.Real;
Complex temp;
var f = matrixA[i, i - 1];
if (f.Magnitude != 0)
{
temp = -(matrixA[i, i - 1].Conjugate() * tau[i].Conjugate()) / f.Magnitude;
h += f.Magnitude * g.Real;
g = 1.0 + (g / f.Magnitude);
matrixA[i, i - 1] *= g;
}
else
{
temp = -tau[i].Conjugate();
matrixA[i, i - 1] = g;
}
if ((f.Magnitude == 0) || (i != 1))
{
f = Complex.Zero;
for (var j = 0; j < i; j++)
{
var tmp = Complex.Zero;
// Form element of A*U.
for (var k = 0; k <= j; k++)
{
tmp += matrixA[j, k] * matrixA[i, k].Conjugate();
}
for (var k = j + 1; k <= i - 1; k++)
{
tmp += matrixA[k, j].Conjugate() * matrixA[i, k].Conjugate();
}
// Form element of P
tau[j] = tmp / h;
f += (tmp / h) * matrixA[i, j];
}
hh = f.Real / (h + h);
// Form the reduced A.
for (var j = 0; j < i; j++)
{
f = matrixA[i, j].Conjugate();
g = tau[j] - (hh * f);
tau[j] = g.Conjugate();
for (var k = 0; k <= j; k++)
{
matrixA[j, k] -= (f * tau[k]) + (g * matrixA[i, k]);
}
}
}
for (var k = 0; k < i; k++)
{
matrixA[i, k] *= scale;
}
tau[i - 1] = temp.Conjugate();
}
hh = d[i];
d[i] = matrixA[i, i].Real;
matrixA[i, i] = new Complex(hh, scale * Math.Sqrt(h));
}
hh = d[0];
d[0] = matrixA[0, 0].Real;
matrixA[0, 0] = hh;
e[0] = 0.0;
}
