/// <summary>
/// Transforms the matrix to Schur form and calculates the eigenvalues.
/// </summary>
/// <param name="matrix">Matrix to transform.</param>
/// <returns>the <see cref="SchurTransformer">Shur transform</see> for this matrix
/// </returns>
private SchurTransformer transformToSchur(RealMatrix matrix)
{
SchurTransformer schurTransform = new SchurTransformer(matrix);
double[][] matT = schurTransform.getT().getData();
realEigenvalues = new double[matT.Length];
imagEigenvalues = new double[matT.Length];
for (int i = 0; i < realEigenvalues.Length; i++)
{
if (i == (realEigenvalues.Length - 1) ||
Precision.equals(matT[i + 1][i], 0.0, EPSILON))
{
realEigenvalues[i] = matT[i][i];
}
else
{
double x = matT[i + 1][i + 1];
double p = 0.5 * (matT[i][i] - x);
double z = FastMath.sqrt(FastMath.abs(p * p + matT[i + 1][i] * matT[i][i + 1]));
realEigenvalues[i] = x + p;
imagEigenvalues[i] = z;
realEigenvalues[i + 1] = x + p;
imagEigenvalues[i + 1] = -z;
i++;
}
}
return(schurTransform);
}
/// <summary>
/// Find eigenvectors from a matrix transformed to Schur form.
/// </summary>
/// <param name="schur">the schur transformation of the matrix</param>
/// <exception cref="MathArithmeticException"> if the Schur form has a norm of zero
/// </exception>
private void findEigenVectorsFromSchur(SchurTransformer schur)
{
double[][] matrixT = schur.getT().getData();
double[][] matrixP = schur.getP().getData();
int n = matrixT.Length;
// compute matrix norm
double norm = 0.0;
for (int i = 0; i < n; i++)
{
for (int j = FastMath.max(i - 1, 0); j < n; j++)
{
norm += FastMath.abs(matrixT[i][j]);
}
}
// we can not handle a matrix with zero norm
if (Precision.equals(norm, 0.0, EPSILON))
{
throw new MathArithmeticException(new LocalizedFormats("ZERO_NORM"));
}
// Backsubstitute to find vectors of upper triangular form
double r = 0.0;
double s = 0.0;
double z = 0.0;
for (int idx = n - 1; idx >= 0; idx--)
{
double p = realEigenvalues[idx];
double q = imagEigenvalues[idx];
if (Precision.equals(q, 0.0))
{
// Real vector
int l = idx;
matrixT[idx][idx] = 1.0;
for (int i = idx - 1; i >= 0; i--)
{
double w = matrixT[i][i] - p;
r = 0.0;
for (int j = l; j <= idx; j++)
{
r += matrixT[i][j] * matrixT[j][idx];
}
if (Precision.compareTo(imagEigenvalues[i], 0.0, EPSILON) < 0)
{
z = w;
s = r;
}
else
{
l = i;
if (Precision.equals(imagEigenvalues[i], 0.0))
{
if (w != 0.0)
{
matrixT[i][idx] = -r / w;
}
else
{
matrixT[i][idx] = -r / (Precision.EPSILON * norm);
}
}
else
{
// Solve real equations
double x = matrixT[i][i + 1];
double y = matrixT[i + 1][i];
q = (realEigenvalues[i] - p) * (realEigenvalues[i] - p) +
imagEigenvalues[i] * imagEigenvalues[i];
double t = (x * s - z * r) / q;
matrixT[i][idx] = t;
if (FastMath.abs(x) > FastMath.abs(z))
{
matrixT[i + 1][idx] = (-r - w * t) / x;
}
else
{
matrixT[i + 1][idx] = (-s - y * t) / z;
}
}
// Overflow control
double tt = FastMath.abs(matrixT[i][idx]);
if ((Precision.EPSILON * tt) * tt > 1)
{
for (int j = i; j <= idx; j++)
{
matrixT[j][idx] /= tt;
}
}
}
}
}
else if (q < 0.0)
{
// Complex vector
int l = idx - 1;
// Last vector component imaginary so matrix is triangular
if (FastMath.abs(matrixT[idx][idx - 1]) > FastMath.abs(matrixT[idx - 1][idx]))
{
matrixT[idx - 1][idx - 1] = q / matrixT[idx][idx - 1];
matrixT[idx - 1][idx] = -(matrixT[idx][idx] - p) / matrixT[idx][idx - 1];
}
else
{
Complex result = cdiv(0.0, -matrixT[idx - 1][idx],
matrixT[idx - 1][idx - 1] - p, q);
matrixT[idx - 1][idx - 1] = result.getReal();
matrixT[idx - 1][idx] = result.getImaginary();
}
matrixT[idx][idx - 1] = 0.0;
matrixT[idx][idx] = 1.0;
for (int i = idx - 2; i >= 0; i--)
{
double ra = 0.0;
double sa = 0.0;
for (int j = l; j <= idx; j++)
{
ra += matrixT[i][j] * matrixT[j][idx - 1];
sa += matrixT[i][j] * matrixT[j][idx];
}
double w = matrixT[i][i] - p;
if (Precision.compareTo(imagEigenvalues[i], 0.0, EPSILON) < 0)
{
z = w;
r = ra;
s = sa;
}
else
{
l = i;
if (Precision.equals(imagEigenvalues[i], 0.0))
{
Complex c = cdiv(-ra, -sa, w, q);
matrixT[i][idx - 1] = c.getReal();
matrixT[i][idx] = c.getImaginary();
}
else
{
// Solve complex equations
double x = matrixT[i][i + 1];
double y = matrixT[i + 1][i];
double vr = (realEigenvalues[i] - p) * (realEigenvalues[i] - p) +
imagEigenvalues[i] * imagEigenvalues[i] - q * q;
double vi = (realEigenvalues[i] - p) * 2.0 * q;
if (Precision.equals(vr, 0.0) && Precision.equals(vi, 0.0))
{
vr = Precision.EPSILON * norm *
(FastMath.abs(w) + FastMath.abs(q) + FastMath.abs(x) +
FastMath.abs(y) + FastMath.abs(z));
}
Complex c = cdiv(x * r - z * ra + q * sa,
x * s - z * sa - q * ra, vr, vi);
matrixT[i][idx - 1] = c.getReal();
matrixT[i][idx] = c.getImaginary();
if (FastMath.abs(x) > (FastMath.abs(z) + FastMath.abs(q)))
{
matrixT[i + 1][idx - 1] = (-ra - w * matrixT[i][idx - 1] +
q * matrixT[i][idx]) / x;
matrixT[i + 1][idx] = (-sa - w * matrixT[i][idx] -
q * matrixT[i][idx - 1]) / x;
}
else
{
Complex c2 = cdiv(-r - y * matrixT[i][idx - 1],
-s - y * matrixT[i][idx], z, q);
matrixT[i + 1][idx - 1] = c2.getReal();
matrixT[i + 1][idx] = c2.getImaginary();
}
}
// Overflow control
double t = FastMath.max(FastMath.abs(matrixT[i][idx - 1]),
FastMath.abs(matrixT[i][idx]));
if ((Precision.EPSILON * t) * t > 1)
{
for (int j = i; j <= idx; j++)
{
matrixT[j][idx - 1] /= t;
matrixT[j][idx] /= t;
}
}
}
}
}
}
// Back transformation to get eigenvectors of original matrix
for (int j = n - 1; j >= 0; j--)
{
for (int i = 0; i <= n - 1; i++)
{
z = 0.0;
for (int k = 0; k <= FastMath.min(j, n - 1); k++)
{
z += matrixP[i][k] * matrixT[k][j];
}
matrixP[i][j] = z;
}
}
eigenvectors = new ArrayRealVector[n];
double[] tmp = new double[n];
for (int i = 0; i < n; i++)
{
for (int j = 0; j < n; j++)
{
tmp[j] = matrixP[j][i];
}
eigenvectors[i] = new ArrayRealVector(tmp);
}
}
