예제 #1
0
        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));
            }
        }
예제 #2
0
        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));
        }
예제 #3
0
        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));
        }