public void SmallSVD()
{
SquareMatrix A0 = new SquareMatrix(1);
A0[0, 0] = 0.0;
SingularValueDecomposition SVD0 = A0.SingularValueDecomposition();
Assert.IsTrue(SVD0.Contributors.Count == 1);
Assert.IsTrue(SVD0.Contributors[0].SingularValue == 0.0);
SquareMatrix A1 = new SquareMatrix(1);
A1[0, 0] = 1.0;
SingularValueDecomposition SVD1 = A1.SingularValueDecomposition();
Assert.IsTrue(SVD1.Contributors.Count == 1);
Assert.IsTrue(SVD1.Contributors[0].SingularValue == 1.0);
SquareMatrix A2 = new SquareMatrix(2);
A2[0, 0] = 0.0; A2[0, 1] = 1.0;
A2[1, 0] = 0.0; A2[1, 1] = 1.0;
// Singular values should be Sqrt(2), 0
double[] s = new double[] { Math.Sqrt(2.0), 0.0 };
SingularValueDecomposition SVD2 = A2.SingularValueDecomposition();
SquareMatrix S2 = SVD2.LeftTransformMatrix.Transpose * A2 * SVD2.RightTransformMatrix;
Assert.IsTrue(SVD2.Contributors.Count == 2);
for (int i = 0; i < SVD2.Contributors.Count; i++)
{
double w = SVD2.Contributors[i].SingularValue;
Assert.IsTrue(TestUtilities.IsNearlyEqual(w, s[i]));
Assert.IsTrue(TestUtilities.IsNearlyEqual(S2[i, i], w));
}
}
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 SquareMatrixNotInvertable()
{
SquareMatrix A = new SquareMatrix(2);
A[1, 1] = 1.0;
// Inverting should throw
try {
A.Inverse();
Assert.IsTrue(false);
} catch (DivideByZeroException) {
}
// LU Decomposing should throw
try {
A.LUDecomposition();
Assert.IsTrue(false);
} catch (DivideByZeroException) {
}
// SVD should succeed, and give infinite condition number
SingularValueDecomposition SVD = A.SingularValueDecomposition();
Assert.IsTrue(Double.IsInfinity(SVD.ConditionNumber));
}
public void SquareMatrixSVD()
{
for (int d = 4; d < 50; d += 7)
{
SquareMatrix A = CreateSquareRandomMatrix(d, d);
SingularValueDecomposition SVD = A.SingularValueDecomposition();
Assert.IsTrue(SVD.Dimension == A.Dimension);
ColumnVector x = new ColumnVector(d);
for (int i = 0; i < d; i++)
{
x[i] = i;
}
ColumnVector b = A * x;
ColumnVector b1 = SVD.LeftTransformMatrix().Transpose() * b;
for (int i = 0; i < SVD.Rank; i++)
{
b1[i] = b1[i] / SVD.SingularValue(i);
}
ColumnVector b2 = SVD.RightTransformMatrix() * b1;
ColumnVector y = SVD.Solve(b);
Assert.IsTrue(TestUtilities.IsNearlyEqual(x, y));
}
}
public void HilbertMatrixSVD()
{
int n = 4;
SquareMatrix H = new SquareMatrix(n);
for (int r = 0; r < n; r++)
{
for (int c = 0; c < n; c++)
{
H[r, c] = 1.0 / (r + c + 1);
}
}
SingularValueDecomposition SVD = H.SingularValueDecomposition();
for (int i = 0; i < n; i++)
{
Console.WriteLine(SVD.SingularValue(i));
}
SquareMatrix S = SVD.LeftTransformMatrix().Transpose() * H * SVD.RightTransformMatrix();
for (int i = 0; i < SVD.Dimension; i++)
{
Console.WriteLine(S[i, i]);
Assert.IsTrue(TestUtilities.IsNearlyEqual(S[i, i], SVD.SingularValue(i)));
}
}
public void SmallSVD()
{
SquareMatrix A0 = new SquareMatrix(1);
A0[0, 0] = 0.0;
SingularValueDecomposition SVD0 = A0.SingularValueDecomposition();
Console.WriteLine(SVD0.SingularValue(0));
Assert.IsTrue(SVD0.SingularValue(0) == 0.0);
SquareMatrix A1 = new SquareMatrix(1);
A1[0, 0] = 1.0;
SingularValueDecomposition SVD1 = A1.SingularValueDecomposition();
Console.WriteLine(SVD1.SingularValue(0));
//Assert.IsTrue(SVD1.SingularValue(0) == 1.0);
SquareMatrix A2 = new SquareMatrix(2);
A2[0, 0] = 0.0; A2[0, 1] = 1.0;
A2[1, 0] = 0.0; A2[1, 1] = 1.0;
// Singular values Sqrt(2), 0
SingularValueDecomposition SVD2 = A2.SingularValueDecomposition();
SquareMatrix S2 = SVD2.LeftTransformMatrix().Transpose() * A2 * SVD2.RightTransformMatrix();
for (int i = 0; i < SVD2.Dimension; i++)
{
Console.WriteLine("{0} {1}", S2[i, i], SVD2.SingularValue(i));
Assert.IsTrue(TestUtilities.IsNearlyEqual(S2[i, i], SVD2.SingularValue(i)));
}
}
public void Bug7685()
{
// This SVD failed with a NonconvergenceException.
// Probably this was due to a zero appearing in the super-diagonal band in an intermediate position, since the code change to handle that eliminated the problem.
SquareMatrix A = new SquareMatrix(new double[, ] {
{ 1.900019E+01, 0.000000E+00, 0.000000E+00, -1.385471E+02, 1.027977E+05, 1.520100E+04 },
{ 0.000000E+00, 1.900019E+01, 0.000000E+00, -1.026921E+05, -1.499209E+02, 1.410499E+05 },
{ 0.000000E+00, 0.000000E+00, 1.900019E+01, -1.499527E+04, -1.410719E+05, 0.000000E+00 },
{ -1.385471E+02, -1.026921E+05, -1.499527E+04, 5.669219E+08, 1.101144E+08, -7.618976E+08 },
{ 1.027977E+05, -1.499209E+02, -1.410719E+05, 1.101144E+08, 1.670648E+09, 8.113594E+07 },
{ 1.520100E+04, 1.410499E+05, 0.000000E+00, -7.618976E+08, 8.113594E+07, 1.126334E+09 }
});
SingularValueDecomposition SVD = A.SingularValueDecomposition();
foreach (SingularValueContributor contributor in SVD.Contributors)
{
Console.WriteLine(contributor.SingularValue);
}
}
public void SquareMatrixSVD()
{
for (int d = 4; d < 64; d += 7)
{
SquareMatrix A = CreateSquareRandomMatrix(d, d);
SingularValueDecomposition SVD = A.SingularValueDecomposition();
Assert.IsTrue(SVD.Dimension == A.Dimension);
// U has right dimensions and is orthogonal
SquareMatrix U = SVD.LeftTransformMatrix;
Assert.IsTrue(U.Dimension == A.Dimension);
Assert.IsTrue(TestUtilities.IsNearlyEqual(U.MultiplyTransposeBySelf(), UnitMatrix.OfDimension(U.Dimension)));
// V has right dimensions and is orthogonal
SquareMatrix V = SVD.RightTransformMatrix;
Assert.IsTrue(V.Dimension == A.Dimension);
Assert.IsTrue(TestUtilities.IsNearlyEqual(V.MultiplyTransposeBySelf(), UnitMatrix.OfDimension(V.Dimension)));
Assert.IsTrue(SVD.Dimension == A.Dimension);
// The transforms decompose the matrix with the claimed singular values
SquareMatrix S = U.Transpose * A * V;
for (int i = 0; i < SVD.Contributors.Count; i++)
{
SingularValueContributor t = SVD.Contributors[i];
Assert.IsTrue(t.SingularValue >= 0.0);
Assert.IsTrue(TestUtilities.IsNearlyEqual(S[i, i], t.SingularValue));
}
// We can solve a rhs using the SVD
ColumnVector x = new ColumnVector(d);
for (int i = 0; i < d; i++)
{
x[i] = i;
}
ColumnVector b = A * x;
ColumnVector y = SVD.Solve(b);
Assert.IsTrue(TestUtilities.IsNearlyEqual(x, y));
}
}
public void SquareMatrixLowRankSVD()
{
// Form a rank-2 dimension-3 square matrix
ColumnVector c1 = new ColumnVector(1.0, 1.0, 0.0);
ColumnVector c2 = new ColumnVector(1.0, -1.0, 0.0);
SquareMatrix A = (SquareMatrix)(1.0E6 * c1 * c1.Transpose + 1.03 * c2 * c2.Transpose);
SingularValueDecomposition SVD = A.SingularValueDecomposition();
// We should see the rank
Assert.IsTrue(SVD.Rank == 2);
// Solve a solve-able problem even though matrix is singular
ColumnVector b = new ColumnVector(2.0, 1.0, 0.0);
ColumnVector x = SVD.Solve(b);
Assert.IsTrue(TestUtilities.IsNearlyEqual(A * x, b, 1.0E-6));
// Increasing the tolerance should decrease the rank
SVD.Tolerance = 1.0E-4;
Assert.IsTrue(SVD.Tolerance == 1.0E-4);
Assert.IsTrue(SVD.Rank == 1);
}