public void SquareRandomMatrixQRDecomposition()
{
for (int d = 1; d <= 100; d += 11)
{
Console.WriteLine("d={0}", d);
SquareMatrix M = CreateSquareRandomMatrix(d);
// QR decompose the matrix.
SquareQRDecomposition QRD = M.QRDecomposition();
// The dimension should be right.
Assert.IsTrue(QRD.Dimension == M.Dimension);
// Test that the decomposition works.
SquareMatrix Q = QRD.QMatrix;
SquareMatrix R = QRD.RMatrix;
Assert.IsTrue(TestUtilities.IsNearlyEqual(Q * R, M));
// Check that the inverse works.
SquareMatrix MI = QRD.Inverse();
Assert.IsTrue(TestUtilities.IsNearlyEqual(M * MI, UnitMatrix.OfDimension(d)));
// Test that a solution works.
ColumnVector t = new ColumnVector(d);
for (int i = 0; i < d; i++)
{
t[i] = i;
}
ColumnVector s = QRD.Solve(t);
Assert.IsTrue(TestUtilities.IsNearlyEqual(M * s, t));
}
}
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));
}