mathnet-numerics/src/UnitTests/LinearAlgebraTests/Complex/Factorization/UserEvdTests.cs

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

namespace MathNet.Numerics.UnitTests.LinearAlgebraTests.Complex.Factorization
{
#if NOSYSNUMERICS
    using Complex = Numerics.Complex;
#else
    using Complex = System.Numerics.Complex;
#endif

    /// <summary>
    /// Eigenvalues factorization tests for an user matrix.
    /// </summary>
    [TestFixture, Category("LAFactorization")]
    public class UserEvdTests
    {
        /// <summary>
        /// Can factorize identity matrix.
        /// </summary>
        /// <param name="order">Matrix order.</param>
        [TestCase(1)]
        [TestCase(10)]
        [TestCase(100)]
        public void CanFactorizeIdentity(int order)
        {
            var matrixI = UserDefinedMatrix.Identity(order);
            var factorEvd = matrixI.Evd();
            var eigenValues = factorEvd.EigenValues;
            var eigenVectors = factorEvd.EigenVectors;
            var d = factorEvd.D;

            Assert.AreEqual(matrixI.RowCount, eigenVectors.RowCount);
            Assert.AreEqual(matrixI.RowCount, eigenVectors.ColumnCount);

            Assert.AreEqual(matrixI.ColumnCount, d.RowCount);
            Assert.AreEqual(matrixI.ColumnCount, d.ColumnCount);

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

        /// <summary>
        /// Can factorize a random square matrix.
        /// </summary>
        /// <param name="order">Matrix order.</param>
        [TestCase(1)]
        [TestCase(2)]
        [TestCase(5)]
        [TestCase(10)]
        [TestCase(50)]
        [TestCase(100)]
        public void CanFactorizeRandomMatrix(int order)
        {
            var matrixA = new UserDefinedMatrix(Matrix<Complex>.Build.Random(order, order, 1).ToArray());
            var factorEvd = matrixA.Evd();
            var eigenVectors = factorEvd.EigenVectors;
            var d = factorEvd.D;

            Assert.AreEqual(order, eigenVectors.RowCount);
            Assert.AreEqual(order, eigenVectors.ColumnCount);

            Assert.AreEqual(order, d.RowCount);
            Assert.AreEqual(order, d.ColumnCount);

            // Make sure the A*V = λ*V 
            var matrixAv = matrixA * eigenVectors;
            var matrixLv = eigenVectors * d;

            for (var i = 0; i < matrixAv.RowCount; i++)
            {
                for (var j = 0; j < matrixAv.ColumnCount; j++)
                {
                    AssertHelpers.AlmostEqualRelative(matrixAv[i, j], matrixLv[i, j], 7);
                }
            }
        }

        /// <summary>
        /// Can factorize a symmetric random square matrix.
        /// </summary>
        /// <param name="order">Matrix order.</param>
        [TestCase(1)]
        [TestCase(2)]
        [TestCase(5)]
        [TestCase(10)]
        [TestCase(50)]
        [TestCase(100)]
        public void CanFactorizeRandomSymmetricMatrix(int order)
        {
            var matrixA = new UserDefinedMatrix(Matrix<Complex>.Build.RandomPositiveDefinite(order, 1).ToArray());
            var factorEvd = matrixA.Evd();
            var eigenVectors = factorEvd.EigenVectors;
            var d = factorEvd.D;

            Assert.AreEqual(order, eigenVectors.RowCount);
            Assert.AreEqual(order, eigenVectors.ColumnCount);

            Assert.AreEqual(order, d.RowCount);
            Assert.AreEqual(order, d.ColumnCount);

            // Make sure the A = V*λ*VT 
            var matrix = eigenVectors * d * eigenVectors.ConjugateTranspose();

            for (var i = 0; i < matrix.RowCount; i++)
            {
                for (var j = 0; j < matrix.ColumnCount; j++)
                {
                    AssertHelpers.AlmostEqualRelative(matrix[i, j], matrixA[i, j], 7);
                }
            }
        }

        /// <summary>
        /// Can check rank of square matrix.
        /// </summary>
        /// <param name="order">Matrix order.</param>
        [TestCase(10)]
        [TestCase(50)]
        [TestCase(100)]
        public void CanCheckRankSquare(int order)
        {
            var matrixA = new UserDefinedMatrix(Matrix<Complex>.Build.Random(order, order, 1).ToArray());
            var factorEvd = matrixA.Evd();

            Assert.AreEqual(factorEvd.Rank, order);
        }

        /// <summary>
        /// Can check rank of square singular matrix.
        /// </summary>
        /// <param name="order">Matrix order.</param>
        [TestCase(10)]
        [TestCase(50)]
        [TestCase(100)]
        public void CanCheckRankOfSquareSingular(int order)
        {
            var matrixA = new UserDefinedMatrix(order, order);
            matrixA[0, 0] = 1;
            matrixA[order - 1, order - 1] = 1;
            for (var i = 1; i < order - 1; i++)
            {
                matrixA[i, i - 1] = 1;
                matrixA[i, i + 1] = 1;
                matrixA[i - 1, i] = 1;
                matrixA[i + 1, i] = 1;
            }

            var factorEvd = matrixA.Evd();

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

        /// <summary>
        /// Identity determinant is one.
        /// </summary>
        /// <param name="order">Matrix order.</param>
        [TestCase(1)]
        [TestCase(10)]
        [TestCase(100)]
        public void IdentityDeterminantIsOne(int order)
        {
            var matrixI = UserDefinedMatrix.Identity(order);
            var factorEvd = matrixI.Evd();
            Assert.AreEqual(Complex.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 = new UserDefinedMatrix(Matrix<Complex>.Build.RandomPositiveDefinite(order, 1).ToArray());
            MatrixHelpers.ForceHermitian(A);
            var ACopy = A.Clone();
            var evd = A.Evd();

            var b = new UserDefinedVector(Vector<Complex>.Build.Random(order, 1).ToArray());
            var bCopy = b.Clone();

            var x = evd.Solve(b);

            var bReconstruct = A * x;

            // Check the reconstruction.
            AssertHelpers.AlmostEqual(b, bReconstruct, 9);

            // 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 = new UserDefinedMatrix(Matrix<Complex>.Build.RandomPositiveDefinite(order, 1).ToArray());
            MatrixHelpers.ForceHermitian(A);
            var ACopy = A.Clone();
            var evd = A.Evd();

            var B = new UserDefinedMatrix(Matrix<Complex>.Build.Random(order, order, 1).ToArray());
            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, 9);

            // 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 matrix.
        /// </summary>
        /// <param name="order">Matrix order.</param>
        [Test]
        public void CanSolveForRandomVectorAndSymmetricMatrixWhenResultVectorGiven([Values(1, 2, 5, 10, 50, 100)] int order)
        {
            var A = new UserDefinedMatrix(Matrix<Complex>.Build.RandomPositiveDefinite(order, 1).ToArray());
            MatrixHelpers.ForceHermitian(A);
            var ACopy = A.Clone();
            var evd = A.Evd();

            var b = new UserDefinedVector(Vector<Complex>.Build.Random(order, 1).ToArray());
            var bCopy = b.Clone();

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

            var bReconstruct = A * x;

            // Check the reconstruction.
            AssertHelpers.AlmostEqual(b, bReconstruct, 9);

            // 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 = new UserDefinedMatrix(Matrix<Complex>.Build.RandomPositiveDefinite(order, 1).ToArray());
            MatrixHelpers.ForceHermitian(A);
            var ACopy = A.Clone();
            var evd = A.Evd();

            var B = new UserDefinedMatrix(Matrix<Complex>.Build.Random(order, order, 1).ToArray());
            var BCopy = B.Clone();

            var X = new UserDefinedMatrix(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, 9);

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