EvdTests.cs

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using MathNet.Numerics.LinearAlgebra;
using MathNet.Numerics.LinearAlgebra.Complex32;
using NUnit.Framework;

namespace MathNet.Numerics.Tests.LinearAlgebraTests.Complex32.Factorization
{
    using Numerics;
    using Complex = System.Numerics.Complex;

    /// <summary>
    /// Eigenvalues factorization tests for a dense matrix.
    /// </summary>
    [TestFixture, Category("LAFactorization")]
    public class EvdTests
    {
        [Test]
        public void CanFactorizeIdentityMatrix([Values(1, 10, 100)] int order)
        {
            var matrix = Matrix<Complex32>.Build.DenseIdentity(order);
            var factorEvd = matrix.Evd();
            var eigenValues = factorEvd.EigenValues;
            var eigenVectors = factorEvd.EigenVectors;
            var d = factorEvd.D;

            Assert.AreEqual(matrix.RowCount, eigenVectors.RowCount);
            Assert.AreEqual(matrix.RowCount, eigenVectors.ColumnCount);
            Assert.AreEqual(matrix.ColumnCount, d.RowCount);
            Assert.AreEqual(matrix.ColumnCount, d.ColumnCount);

            for (var i = 0; i < eigenValues.Count; i++)
            {
                Assert.AreEqual(Complex.One, eigenValues[i]);
            }
        }

        [Test]
        public void CanFactorizeRandomSquareMatrix([Values(1, 2, 5, 10, 50, 100)] int order)
        {
            var A = Matrix<Complex32>.Build.Random(order, order, 1);
            var factorEvd = A.Evd();
            var V = factorEvd.EigenVectors;
            var λ = factorEvd.D;

            Assert.AreEqual(order, V.RowCount);
            Assert.AreEqual(order, V.ColumnCount);
            Assert.AreEqual(order, λ.RowCount);
            Assert.AreEqual(order, λ.ColumnCount);

            // Verify A*V = λ*V
            var Av = A * V;
            var Lv = V * λ;
            AssertHelpers.AlmostEqual(Av, Lv, 4);
        }

        [Test]
        public void CanFactorizeRandomSymmetricMatrix([Values(1, 2, 5, 10, 50, 100)] int order)
        {
            var A = Matrix<Complex32>.Build.RandomPositiveDefinite(order, 1);
            MatrixHelpers.ForceHermitian(A);
            var factorEvd = A.Evd(Symmetricity.Hermitian);
            var V = factorEvd.EigenVectors;
            var λ = factorEvd.D;

            Assert.AreEqual(order, V.RowCount);
            Assert.AreEqual(order, V.ColumnCount);
            Assert.AreEqual(order, λ.RowCount);
            Assert.AreEqual(order, λ.ColumnCount);

            // Verify A = V*λ*VT
            var matrix = V*λ*V.ConjugateTranspose();
            AssertHelpers.AlmostEqual(matrix, A, 3);
            AssertHelpers.AlmostEqualRelative(matrix, A, 1);
        }

        [Test]
        public void CanCheckRankSquare([Values(10, 50, 100)] int order)
        {
            var A = Matrix<Complex32>.Build.Random(order, order, 1);
            Assert.AreEqual(A.Evd().Rank, order);
        }

        [Test]
        public void CanCheckRankOfSquareSingular([Values(10, 50, 100)] int order)
        {
            var A = new DenseMatrix(order, order);
            A[0, 0] = 1;
            A[order - 1, order - 1] = 1;
            for (var i = 1; i < order - 1; i++)
            {
                A[i, i - 1] = 1;
                A[i, i + 1] = 1;
                A[i - 1, i] = 1;
                A[i + 1, i] = 1;
            }

            var factorEvd = A.Evd();

            Assert.AreEqual(factorEvd.Determinant, Complex32.Zero);
            Assert.AreEqual(factorEvd.Rank, order - 1);
        }

        [Test]
        public void IdentityDeterminantIsOne([Values(1, 10, 100)] int order)
        {
            var matrixI = DenseMatrix.CreateIdentity(order);
            var factorEvd = matrixI.Evd();
            Assert.AreEqual(Complex32.One, factorEvd.Determinant);
        }

        /// <summary>
        /// Can solve a system of linear equations for a random vector and symmetric matrix (Ax=b).
        /// </summary>
        /// <param name="order">Matrix order.</param>
        [Test]
        public void CanSolveForRandomVectorAndSymmetricMatrix([Values(1, 2, 5, 10, 50, 100)] int order)
        {
            var A = Matrix<Complex32>.Build.RandomPositiveDefinite(order, 1);
            MatrixHelpers.ForceHermitian(A);
            var ACopy = A.Clone();
            var evd = A.Evd(Symmetricity.Hermitian);

            var b = Vector<Complex32>.Build.Random(order, 2);
            var bCopy = b.Clone();

            var x = evd.Solve(b);

            var bReconstruct = A * x;

            // Check the reconstruction.
            AssertHelpers.AlmostEqual(b, bReconstruct, 2);

            // Make sure A/B didn't change.
            AssertHelpers.AlmostEqual(ACopy, A, 14);
            AssertHelpers.AlmostEqual(bCopy, b, 14);
        }

        /// <summary>
        /// Can solve a system of linear equations for a random matrix and symmetric matrix (AX=B).
        /// </summary>
        /// <param name="order">Matrix order.</param>
        [Test]
        public void CanSolveForRandomMatrixAndSymmetricMatrix([Values(1, 2, 5, 10, 50, 100)] int order)
        {
            var A = Matrix<Complex32>.Build.RandomPositiveDefinite(order, 1);
            MatrixHelpers.ForceHermitian(A);
            var ACopy = A.Clone();
            var evd = A.Evd(Symmetricity.Hermitian);

            var B = Matrix<Complex32>.Build.Random(order, order, 2);
            var BCopy = B.Clone();

            var X = evd.Solve(B);

            // The solution X row dimension is equal to the column dimension of A
            Assert.AreEqual(A.ColumnCount, X.RowCount);

            // The solution X has the same number of columns as B
            Assert.AreEqual(B.ColumnCount, X.ColumnCount);

            var BReconstruct = A * X;

            // Check the reconstruction.
            AssertHelpers.AlmostEqual(B, BReconstruct, 1);

            // Make sure A/B didn't change.
            AssertHelpers.AlmostEqual(ACopy, A, 14);
            AssertHelpers.AlmostEqual(BCopy, B, 14);
        }

        /// <summary>
        /// Can solve a system of linear equations for a random vector and symmetric matrix (Ax=b) into a result vector.
        /// </summary>
        /// <param name="order">Matrix order.</param>
        [Test]
        public void CanSolveForRandomVectorAndSymmetricMatrixWhenResultVectorGiven([Values(1, 2, 5, 10, 50, 100)] int order)
        {
            var A = Matrix<Complex32>.Build.RandomPositiveDefinite(order, 1);
            MatrixHelpers.ForceHermitian(A);
            var ACopy = A.Clone();
            var evd = A.Evd(Symmetricity.Hermitian);

            var b = Vector<Complex32>.Build.Random(order, 2);
            var bCopy = b.Clone();

            var x = new DenseVector(order);
            evd.Solve(b, x);

            var bReconstruct = A * x;

            // Check the reconstruction.
            AssertHelpers.AlmostEqual(b, bReconstruct, 2);

            // Make sure A/B didn't change.
            AssertHelpers.AlmostEqual(ACopy, A, 14);
            AssertHelpers.AlmostEqual(bCopy, b, 14);
        }

        /// <summary>
        /// Can solve a system of linear equations for a random matrix and symmetric matrix (AX=B) into result matrix.
        /// </summary>
        /// <param name="order">Matrix order.</param>
        [Test]
        public void CanSolveForRandomMatrixAndSymmetricMatrixWhenResultMatrixGiven([Values(1, 2, 5, 10, 50, 100)] int order)
        {
            var A = Matrix<Complex32>.Build.RandomPositiveDefinite(order, 1);
            MatrixHelpers.ForceHermitian(A);
            var ACopy = A.Clone();
            var evd = A.Evd(Symmetricity.Hermitian);

            var B = Matrix<Complex32>.Build.Random(order, order, 2);
            var BCopy = B.Clone();

            var X = new DenseMatrix(order, order);
            evd.Solve(B, X);

            // The solution X row dimension is equal to the column dimension of A
            Assert.AreEqual(A.ColumnCount, X.RowCount);

            // The solution X has the same number of columns as B
            Assert.AreEqual(B.ColumnCount, X.ColumnCount);

            var BReconstruct = A * X;

            // Check the reconstruction.
            AssertHelpers.AlmostEqual(B, BReconstruct, 1);

            // Make sure A/B didn't change.
            AssertHelpers.AlmostEqual(ACopy, A, 14);
            AssertHelpers.AlmostEqual(BCopy, B, 14);
        }

        /// <summary>
        /// See: https://github.com/mathnet/mathnet-numerics/issues/595
        /// </summary>
        [Test]
        public void CanFactorizeMatrixWithZeroInternalNorm()
        {
            Complex32[,] data =
            {
                {new Complex32(0.0f, 0.0f), new Complex32(1.0f, 0.0f), new Complex32(0.0f, 0.0f), new Complex32(0.0f, 0.0f)},
                {new Complex32(2.25f, 0.0f), new Complex32(0.0f, 0.0f), new Complex32(0.0f, 0.0f), new Complex32(0.0f, 0.0f)},
                {new Complex32(0.0f, 0.0f), new Complex32(0.0f, 0.0f), new Complex32(0.0f, 0.0f), new Complex32(1.0f, 0.0f)},
                {new Complex32(0.0f, 0.0f), new Complex32(0.0f, 0.0f), new Complex32(2.25f, 0.0f), new Complex32(0.0f, 0.0f)}
            };

            var A = Matrix<Complex32>.Build.DenseOfArray(data);

            var factorEvd = A.Evd();
            var V = factorEvd.EigenVectors;
            var λ = factorEvd.D;

            // Verify A*V = λ*V
            var Av = A * V;
            var Lv = V * λ;
            AssertHelpers.AlmostEqual(Av, Lv, 4);
            AssertHelpers.AlmostEqualRelative(Av, Lv, 8);
        }
    }
}