public void KnownEigenvalues()
{
// This example is presented in http://people.inf.ethz.ch/arbenz/ewp/Lnotes/chapter3.pdf
// It has eigenvalues 1 \pm 2i, 3, 4, 5 \pm 6i
double[,] souce = new double[, ] {
{ 7, 3, 4, -11, -9, -2 },
{ -6, 4, -5, 7, 1, 12 },
{ -1, -9, 2, 2, 9, 1 },
{ -8, 0, -1, 5, 0, 8 },
{ -4, 3, -5, 7, 2, 10 },
{ 6, 1, 4, -11, -7, -1 }
};
SquareMatrix A = new SquareMatrix(6);
for (int r = 0; r < souce.GetLength(0); r++)
{
for (int c = 0; c < souce.GetLength(1); c++)
{
A[r, c] = souce[r, c];
}
}
Complex[] eigenvalues = A.Eigenvalues();
foreach (Complex eigenvalue in eigenvalues)
{
Console.WriteLine(eigenvalue);
}
}
public void CirculantEigenvalues()
{
int n = 50;
double[] x = new double[n];
for (int i = 0; i < x.Length; i++)
{
x[i] = 1.0 / (i + 1);
}
SquareMatrix A = CreateCirculantMatrix(x);
Complex[] u = RootsOfUnity(n);
Complex[] v = new Complex[n];
for (int i = 0; i < v.Length; i++)
{
for (int j = 0; j < n; j++)
{
v[i] += x[j] * u[(i * (n - j)) % n];
}
}
Complex[][] w = new Complex[n][];
for (int i = 0; i < n; i++)
{
w[i] = new Complex[n];
for (int j = 0; j < n; j++)
{
w[i][j] = u[i * j % n];
}
}
Complex[] e = A.Eigenvalues();
}
public void CirculantEigenvalues()
{
// See https://en.wikipedia.org/wiki/Circulant_matrix
// Definition is C_{ij} = c_{|i - j|} for any n-vector c.
// jth eigenvector known to be (1, \omega_j, \omega_j^2, \cdots, \omega_j^{n-1}) where \omega_j = \exp(2 \pi i j / n) are nth roots of unity
// jth eigenvalue known to be c_0 + c_{n-1} \omega_j + c_{n-2} \omega_j^2 + \cdots + c_1 \omega_j^{n-1}
int n = 12;
double[] x = new double[n];
for (int i = 0; i < x.Length; i++)
{
x[i] = 1.0 / (i + 1);
}
SquareMatrix A = CreateCirculantMatrix(x);
Complex[] u = RootsOfUnity(n);
Complex[] v = new Complex[n];
for (int i = 0; i < v.Length; i++)
{
for (int j = 0; j < n; j++)
{
v[i] += x[j] * u[(i * (n - j)) % n];
}
}
Complex[][] w = new Complex[n][];
for (int i = 0; i < n; i++)
{
w[i] = new Complex[n];
for (int j = 0; j < n; j++)
{
w[i][j] = u[i * j % n];
}
}
Complex[] eValues = A.Eigenvalues();
Complex eProduct = 1.0;
foreach (Complex eValue in eValues)
{
eProduct *= eValue;
}
// v and eValues should be equal. By inspection they are,
// but how to verify this given floating point jitter?
// Verify that eigenvalue product equals determinant.
SquareQRDecomposition QR = A.QRDecomposition();
double det = QR.Determinant();
Assert.IsTrue(TestUtilities.IsNearlyEqual(eProduct, det));
}
public void CirculantEigenvalues()
{
int n = 12;
double[] x = new double[n];
for (int i = 0; i < x.Length; i++)
{
x[i] = 1.0 / (i + 1);
}
SquareMatrix A = CreateCirculantMatrix(x);
Complex[] u = RootsOfUnity(n);
Complex[] v = new Complex[n];
for (int i = 0; i < v.Length; i++)
{
for (int j = 0; j < n; j++)
{
v[i] += x[j] * u[(i * (n - j)) % n];
}
}
Complex[][] w = new Complex[n][];
for (int i = 0; i < n; i++)
{
w[i] = new Complex[n];
for (int j = 0; j < n; j++)
{
w[i][j] = u[i * j % n];
}
}
Complex[] eValues = A.Eigenvalues();
Complex eProduct = 1.0;
foreach (Complex eValue in eValues)
{
eProduct *= eValue;
}
// v and eValues should be equal. By inspection they are,
// but how to verify this given floating point jitter?
// Verify that eigenvalue product equals determinant.
SquareQRDecomposition QR = A.QRDecomposition();
double det = QR.Determinant();
Assert.IsTrue(TestUtilities.IsNearlyEqual(eProduct, det));
}
public void Bug5504()
{
// this eigenvalue non-convergence is solved by an ad hoc shift
// these "cyclic matrices" are good examples of situations that are difficult for the QR algorithm
for (int d = 2; d <= 8; d++)
{
Console.WriteLine(d);
SquareMatrix C = CyclicMatrix(d);
Complex[] lambdas = C.Eigenvalues();
foreach (Complex lambda in lambdas)
{
Console.WriteLine("{0} ({1} {2})", lambda, ComplexMath.Abs(lambda), ComplexMath.Arg(lambda));
Assert.IsTrue(TestUtilities.IsNearlyEqual(ComplexMath.Abs(lambda), 1.0));
}
}
}
public void DifficultEigenvalue()
{
// This is from a paper describing difficult eigenvalue problems.
// https://www.mathworks.com/company/newsletters/news_notes/pdf/sum95cleve.pdf
SquareMatrix A = new SquareMatrix(4);
A[0, 0] = 0.0; A[0, 1] = 2.0; A[0, 2] = 0.0; A[0, 3] = -1.0;
A[1, 0] = 1.0; A[1, 1] = 0.0; A[1, 2] = 0.0; A[1, 3] = 0.0;
A[2, 0] = 0.0; A[2, 1] = 1.0; A[2, 2] = 0.0; A[2, 3] = 0.0;
A[3, 0] = 0.0; A[3, 1] = 0.0; A[3, 2] = 1.0; A[3, 3] = 0.0;
Complex[] zs = A.Eigenvalues();
foreach (Complex z in zs)
{
Console.WriteLine("{0} ({1} {2})", z, ComplexMath.Abs(z), ComplexMath.Arg(z));
}
}
public void Bug8021()
{
// User presented a matrix with a large number (138) of zero eigenvalues.
// QR algorithm got tripped up on high degeneracy, but eigenvalues could
// be revealed before QR by simple index permutation. Added code to isolate
// these "cheap" eigenvalues before starting QR algorithm.
int n = 276;
SquareMatrix A = new SquareMatrix(n);
using (System.IO.StreamReader reader = System.IO.File.OpenText(@"Bug8021.csv")) {
int r = 0;
while (!reader.EndOfStream)
{
string line = reader.ReadLine();
string[] cells = line.Split(',');
for (int c = 0; c < cells.Length; c++)
{
string cell = cells[c];
double value = Double.Parse(cell);
A[r, c] = value;
}
r++;
}
}
ComplexEigendecomposition S = A.Eigendecomposition();
foreach (ComplexEigenpair pair in S.Eigenpairs)
{
TestUtilities.IsNearlyEigenpair(A, pair.Eigenvector, pair.Eigenvalue);
}
Complex[] eigenvalues = A.Eigenvalues();
double trace = A.Trace();
Complex sum = 0.0;
for (int i = 0; i < eigenvalues.Length; i++)
{
sum += eigenvalues[i];
}
TestUtilities.IsNearlyEqual(trace, sum);
}
public void TimedEigenvalues()
{
int n = 200;
Random rng = new Random(1);
SquareMatrix A = new SquareMatrix(n);
for (int r = 0; r < n; r++)
{
for (int c = 0; c < n; c++)
{
A[r, c] = rng.NextDouble();
}
}
Stopwatch s = Stopwatch.StartNew();
Complex[] eigenvalues = A.Eigenvalues();
s.Stop();
Console.WriteLine(s.ElapsedMilliseconds);
}
