public void HilbertMatrixSVD()
{
int n = 4;
SquareMatrix H = TestUtilities.CreateSymmetricHilbertMatrix(n).ToSquareMatrix();
SingularValueDecomposition SVD = H.SingularValueDecomposition();
Assert.IsTrue(SVD.RowCount == n);
Assert.IsTrue(SVD.ColumnCount == n);
Assert.IsTrue(SVD.ConditionNumber > 1.0);
// Reconstruct matrix
SquareMatrix H2 = new SquareMatrix(n);
foreach (SingularValueContributor contributor in SVD.Contributors)
{
H2 += contributor.SingularValue * ((SquareMatrix)(contributor.LeftSingularVector * contributor.RightSingularVector.Transpose));
}
// Use SVD to solve
ColumnVector b = new ColumnVector(n);
for (int i = 0; i < b.Dimension; i++)
{
b[i] = i;
}
ColumnVector x = SVD.Solve(b);
Assert.IsTrue(TestUtilities.IsNearlyEqual(H * x, b));
}
public void SymmetricHilbertMatrixEigenvalues()
{
for (int d = 1; d <= 8; d++)
{
SymmetricMatrix H = TestUtilities.CreateSymmetricHilbertMatrix(d);
double tr = H.Trace();
double[] es = H.Eigenvalues();
Assert.IsTrue(TestUtilities.IsSumNearlyEqual(es, tr));
}
}
public void SymmetricHilbertMatrixInverse()
{
for (int d = 1; d < 4; d++)
{
SymmetricMatrix H = TestUtilities.CreateSymmetricHilbertMatrix(d);
SymmetricMatrix HI = H.Inverse();
Assert.IsTrue(TestUtilities.IsNearlyEqual(HI * H, UnitMatrix.OfDimension(d)));
}
// fails for d >= 4; look into this
}
public void SymmetricHilbertMatrixInverse()
{
for (int d = 1; d <= 4; d++)
{
Console.WriteLine("d={0}", d);
SquareMatrix I = TestUtilities.CreateSquareUnitMatrix(d);
SymmetricMatrix H = TestUtilities.CreateSymmetricHilbertMatrix(d);
SymmetricMatrix HI = H.Inverse();
Assert.IsTrue(TestUtilities.IsNearlyEqual(H * HI, I));
}
// fails for d > 4! look into this
}
public void SymmetricMatrixDecomposition()
{
for (int d = 1; d <= 4; d++)
{
SymmetricMatrix H = TestUtilities.CreateSymmetricHilbertMatrix(d);
CholeskyDecomposition CD = H.CholeskyDecomposition();
Assert.IsTrue(CD != null, String.Format("d={0} not positive definite", d));
Assert.IsTrue(CD.Dimension == d);
SymmetricMatrix HI = CD.Inverse();
SquareMatrix I = TestUtilities.CreateSquareUnitMatrix(d);
Assert.IsTrue(TestUtilities.IsNearlyEqual(H * HI, I));
}
}
public void HilbertMatrixCholeskyDecomposition()
{
for (int d = 1; d <= 4; d++)
{
SymmetricMatrix H = TestUtilities.CreateSymmetricHilbertMatrix(d);
// Decomposition succeeds
CholeskyDecomposition CD = H.CholeskyDecomposition();
Assert.IsTrue(CD != null);
Assert.IsTrue(CD.Dimension == d);
// Decomposition works
SquareMatrix S = CD.SquareRootMatrix();
Assert.IsTrue(TestUtilities.IsNearlyEqual(S * S.Transpose, H));
// Inverse works
SymmetricMatrix HI = CD.Inverse();
Assert.IsTrue(TestUtilities.IsNearlyEqual(H * HI, UnitMatrix.OfDimension(d)));
}
}
