public static void QRDecomposition()
{
SquareMatrix A = new SquareMatrix(new double[, ] {
{ 1, -2, 3 },
{ 2, -5, 12 },
{ 0, 2, -10 }
});
ColumnVector b = new ColumnVector(2, 8, -4);
SquareQRDecomposition qrd = A.QRDecomposition();
ColumnVector x = qrd.Solve(b);
PrintMatrix("x", x);
SquareMatrix Q = qrd.QMatrix;
SquareMatrix R = qrd.RMatrix;
PrintMatrix("QR", Q * R);
PrintMatrix("Q Q^T", Q.MultiplySelfByTranspose());
SquareMatrix AI = qrd.Inverse();
PrintMatrix("A^{-1}", AI);
PrintMatrix("A^{-1} A", AI * A);
}
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));
}
public void doWork()
{
matrix.Fill((j, i) => array[i, j]);
SquareQRDecomposition qr = matrix.QRDecomposition();
SquareLUDecomposition lu = matrix.LUDecomposition();
ColumnVector qrresult = qr.Solve(vector);
ColumnVector luresult = lu.Solve(vector);
ColumnVector c6 = matrix.Column(6);
RowVector r2 = matrix.Row(2);
SquareMatrix inv = matrix.Inverse();
ColumnVector result = new ColumnVector(inv.Dimension);
for (int i = 0; i < matrix.Dimension; ++i)
{
for (int j = 0; j < matrix.Dimension; ++j)
{
result[i] += matrix[i, j] * qrresult[j];
}
}
}
public static QuadraticInterpolationModel Construct(double[][] points, double[] values)
{
int m = points.Length;
int d = points[0].Length;
// find the minimum point, use it as the origin
int iMin = 0; double fMin = values[0];
for (int i = 1; i < values.Length; i++)
{
if (values[i] < fMin)
{
iMin = i; fMin = values[i];
}
}
SquareMatrix A = new SquareMatrix(m);
int c = 0;
for (int r = 0; r < m; r++)
{
A[r, 0] = 1.0;
}
for (int i = 0; i < d; i++)
{
c++;
for (int r = 0; r < m; r++)
{
A[r, c] = points[r][i] - points[iMin][i];
}
}
for (int i = 0; i < d; i++)
{
for (int j = 0; j <= i; j++)
{
c++;
for (int r = 0; r < m; r++)
{
A[r, c] = (points[r][i] - points[iMin][i]) * (points[r][j] - points[iMin][j]);
}
}
}
ColumnVector b = new ColumnVector(values);
SquareQRDecomposition QR = A.QRDecomposition();
ColumnVector a = QR.Solve(b);
QuadraticInterpolationModel model = new QuadraticInterpolationModel();
model.d = d;
model.origin = points[iMin];
model.f0 = a[0];
model.g = new double[d];
c = 0;
for (int i = 0; i < d; i++)
{
c++;
model.g[i] = a[c];
}
model.h = new double[d][];
for (int i = 0; i < d; i++)
{
model.h[i] = new double[d];
}
for (int i = 0; i < d; i++)
{
for (int j = 0; j <= i; j++)
{
c++;
if (i == j)
{
model.h[i][j] = 2.0 * a[c];
}
else
{
model.h[i][j] = a[c];
model.h[j][i] = a[c];
}
}
}
return(model);
}
