Matrix Decompositions QuickStart Sample (Visual Basic)

Option Infer On

Imports Extreme.Mathematics
' The DenseMatrix and DoubleVector classes resides in the 
' Extreme.Mathematics.LinearAlgebra namespace.
Imports Extreme.Mathematics.LinearAlgebra

'/ Illustrates the use of matrix decompositions for solving systems of
'/ simultaneous linear equations and related operations using the 
'/ Decomposition class and its derived classes from the
'/ Extreme.Mathematics.LinearAlgebra namespace of the Extreme Optimization
'/ Numerical Libraries for .NET.
Module MatrixDecompositions

    Sub Main()
        ' The license is verified at runtime. We're using
        ' a demo license here. For more information, see
        ' https://www.extremeoptimization.com/trialkey
        Extreme.License.Verify("Demo license")

        ' For details on the basic workings of Vector 
        ' objects, including constructing, copying and
        ' cloning vectors, see the BasicVectors QuickStart
        ' Sample.
        '
        ' For details on the basic workings of Matrix
        ' objects, including constructing, copying and
        ' cloning vectors, see the BasicVectors QuickStart
        ' Sample.
        '

        '
        ' LU Decomposition
        '

        ' The LU decomposition of a matrix rewrites a matrix A in the
        ' form A = PLU with P a permutation matrix, L a unit-
        ' lower triangular matrix, and U an upper triangular matrix.

        Dim aLU = Matrix.Create(4, 4,
             New Double() _
            {
             1.8, 5.25, 1.58, -1.11,
             2.88, -2.95, -2.69, -0.66,
             2.05, -0.95, -2.9, -0.59,
             -0.89, -3.8, -1.04, 0.8
            }, MatrixElementOrder.ColumnMajor)

        Dim bLU = Matrix.Create(4, 2, New Double() _
            {
             9.52, 24.35, 0.77, -6.22,
             18.47, 2.25, -13.28, -6.21
            }, MatrixElementOrder.ColumnMajor)

        ' The constructor takes one or two parameters. The second
        ' parameter is a bool value that indicates whether the
        ' matrix should be overwritten with its decomposition.
        Dim lu As LUDecomposition(Of Double) = aLU.GetLUDecomposition(True)
        Console.WriteLine("A = {0:F4}", aLU)

        ' The Decompose method performs the decomposition. You don't need
        ' to call it explicitly, as it is called automatically as needed.

        ' The IsSingular method checks for singularity.
        Console.WriteLine("'A is singular' is {0}.", lu.IsSingular())
        ' The LowerTriangularFactor and UpperTriangularFactor properties
        ' return the two main components of the decomposition.
        Console.WriteLine("L = {0:F4}", lu.LowerTriangularFactor)
        Console.WriteLine("U = {0:F4}", lu.UpperTriangularFactor)

        ' GetInverse() gives the matrix inverse, Determinant() the determinant:
        Console.WriteLine("Inv A = {0:F4}", lu.GetInverse())
        Console.WriteLine("Det A = {0:F4}", lu.GetDeterminant())

        ' The Solve method solves a system of simultaneous linear equations, with
        ' one or more right-hand-sides:
        Dim xLU = lu.Solve(bLU)
        Console.WriteLine("x = {0:F4}", xLU)

        ' The permutation is available through the RowPermutation property:
        Console.WriteLine("P = {0}", lu.RowPermutation)
        Console.WriteLine("Px = {0:F6}", xLU.PermuteRowsInPlace(lu.RowPermutation))

        '
        ' QR Decomposition
        '

        ' The QR decomposition of a matrix A rewrites the matrix
        ' in the form A = QR, with Q a square, orthogonal matrix,
        ' and R an upper triangular matrix.

        Dim aQR = Matrix.Create(5, 3,
             New Double() _
             {
              2.0, 2.0, 1.6, 2.0, 1.2,
              2.5, 2.5, -0.4, -0.5, -0.3,
              2.5, 2.5, 2.8, 0.5, -2.9
             }, MatrixElementOrder.ColumnMajor)
        Dim bQR As DenseVector(Of Double) = Vector.Create(1.1, 0.9, 0.6, 0.0, -0.8)

        ' The constructor takes one or two parameters. The second
        ' parameter is a bool value that indicates whether the
        ' matrix should be overwritten with its decomposition.
        Dim qr As QRDecomposition(Of Double) = aQR.GetQRDecomposition(True)
        Console.WriteLine("A = {0:F4}", aQR)

        ' The Decompose method performs the decomposition. You don't need
        ' to call it explicitly, as it is called automatically as needed.

        ' The IsSingular method checks for singularity.
        Console.WriteLine("'A is singular' is {0}.", qr.IsSingular())

        ' GetInverse() gives the matrix inverse, GetDeterminant() the determinant.
        ' However, these are only defined for square matrices.

        ' The Solve method solves a system of simultaneous linear equations, with
        ' one or more right-hand-sides. If the matrix is over-determined, you can
        ' use the LeastSquaresSolve method to find a least squares solution:
        Dim xQR As DenseVector(Of Double) = CType(qr.LeastSquaresSolve(bQR), DenseVector(Of Double))
        Console.WriteLine("x = {0:F4}", xQR)

        ' You can also 
        ' The OrthogonalFactor and UpperTriangularFactor properties
        ' return the two main components of the decomposition.
        Console.WriteLine("Q = {0:F4}", qr.OrthogonalFactor.ToDenseMatrix())
        Console.WriteLine("R = {0:F4}", qr.UpperTriangularFactor)

        ' You don't usually need to form Q explicitly. You can multiply
        ' a vector or matrix on either side by Q using the ApplyQ method:
        Console.WriteLine("Qx = {0:F4}", qr.OrthogonalFactor * bQR)
        Console.WriteLine("transpose(Q)x = {0:F4}",
                qr.OrthogonalFactor.Transpose() * bQR)

        '
        ' Singular Value Decomposition
        '

        ' The singular value decomposition of a matrix A rewrites the matrix
        ' in the form A = USVt, with U and V orthogonal matrices, 
        ' S a diagonal matrix. The diagonal elements of S are called 
        ' the singular values.

        Dim aSvd = Matrix.Create(3, 5,
             New Double() _
             {
              2.0, 2.0, 1.6, 2.0, 1.2,
              2.5, 2.5, -0.4, -0.5, -0.3,
              2.5, 2.5, 2.8, 0.5, -2.9
             }, MatrixElementOrder.RowMajor)
        Dim bSvd As DenseVector(Of Double) = Vector.Create(1.1, 0.9, 0.6)

        ' The constructor takes one or two parameters. The second
        ' parameter indicates which parts of the decomposition
        ' are to be calculated. The default is All.
        Dim svd As SingularValueDecomposition(Of Double) = aSvd.GetSingularValueDecomposition()
        Console.WriteLine("A = {0:F1}", aSvd)

        ' The Decompose method performs the decomposition. You don't need
        ' to call it explicitly, as it is called automatically as needed.

        ' The IsSingular method checks for singularity.
        Console.WriteLine("'A is singular' is {0:F6}.", svd.IsSingular())

        ' Several methods are specific to this class. The GetPseudoInverse
        ' method returns the Moore-Penrose pseudo-inverse, a generalization
        ' of the inverse of a matrix to rectangular and/or singular matrices:
        Dim aInv = svd.GetPseudoInverse()

        ' It can be used to solve over- or under-determined systems.
        Dim xSvd = Matrix.Multiply(aInv, bSvd)
        Console.WriteLine("x = {0:F6}", xSvd)

        ' The SingularValues property returns a vector that contains
        ' the singular values in descending order:
        Console.WriteLine("S = {0:F6}", svd.SingularValues)

        ' The LeftSingularVectors and RightSingularVectors properties
        ' return matrices that contain the U and V factors
        ' of the decomposition.
        Console.WriteLine("U = {0:F6}", svd.LeftSingularVectors)
        Console.WriteLine("V = {0:F6}", svd.RightSingularVectors)

        '
        ' Cholesky decomposition
        '

        ' The Cholesky decomposition of a symmetric matrix A
        ' rewrites the matrix in the form A = GGt with
        ' G a lower-triangular matrix.

        ' Remember the column-major storage mode: each line of
        ' components contains one COLUMN of the matrix.
        Dim aC As SymmetricMatrix(Of Double) = Matrix.CreateSymmetric(4,
             New Double() _
            {
             4.16, -3.12, 0.56, -0.1,
             0, 5.03, -0.83, 1.18,
             0, 0, 0.76, 0.34,
             0, 0, 0, 1.18
            }, MatrixTriangle.Lower, MatrixElementOrder.ColumnMajor)
        Dim bC = Matrix.Create(4, 2,
                New Double() {8.7, -13.35, 1.89, -4.14, 8.3, 2.13, 1.61, 5.0},
                MatrixElementOrder.ColumnMajor)

        ' The constructor takes one or two parameters. The second
        ' parameter is a bool value that indicates whether the
        ' matrix should be overwritten with its decomposition.
        Dim c As CholeskyDecomposition(Of Double) = aC.GetCholeskyDecomposition(True)
        Console.WriteLine("A = {0:F4}", aC)

        ' The Decompose method performs the decomposition. You don't need
        ' to call it explicitly, as it is called automatically as needed.

        ' The IsSingular method checks for singularity.
        Console.WriteLine("'A is singular' is {0}.", c.IsSingular())
        ' The LowerTriangularFactor returns the component of the decomposition.
        Console.WriteLine("L = {0:F4}", c.LowerTriangularFactor)

        ' GetInverse() gives the matrix inverse, Determinant() the determinant:
        Console.WriteLine("Inv A = {0:F4}", c.GetInverse())
        Console.WriteLine("Det A = {0:F4}", c.GetDeterminant())

        ' The Solve method solves a system of simultaneous linear equations, with
        ' one or more right-hand-sides:
        Dim xC = c.Solve(bC)
        Console.WriteLine("x = {0:F4}", xC)

        '
        ' Symmetric eigenvalue decomposition
        '

        ' The eigenvalue decomposition of a symmetric matrix A
        ' rewrites the matrix in the form A = XLXt with
        ' X an orthogonal matrix and L a diagonal matrix.
        ' The diagonal elements of L are the eigenvalues.
        ' The columns of X are the eigenvectors.

        ' Remember the column-major storage mode: each line of
        ' components contains one COLUMN of the matrix.
        Dim aEig As SymmetricMatrix(Of Double) = Matrix.CreateSymmetric(4,
             New Double() _
            {
               0.5, 0.0, 2.3, -2.6,
               0.0, 0.5, -1.4, -0.7,
               2.3, -1.4, 0.5, 0.0,
              -2.6, -0.7, 0.0, 0.5
            }, MatrixTriangle.Lower, MatrixElementOrder.ColumnMajor)

        ' The constructor takes one or two parameters. The second
        ' parameter is a bool value that indicates whether the
        ' matrix should be overwritten with its decomposition.
        Dim eig As EigenvalueDecomposition(Of Double) = aEig.GetEigenvalueDecomposition()
        Console.WriteLine("A = {0:F2}", aEig)

        ' The Decompose method performs the decomposition. You don't need
        ' to call it explicitly, as it is called automatically as needed.

        ' The IsSingular method checks for singularity.
        Console.WriteLine("'A is singular' is {0:F6}.", eig.IsSingular())
        ' The Eigenvalues property returns a vector containing the eigenvalues:
        Console.WriteLine("L = {0:F6}", eig.Eigenvalues)
        ' The Eigenvectors property returns a vector containing the eigenvectors:
        Console.WriteLine("X = {0:F6}", eig.Eigenvectors)

        ' GetInverse() gives the matrix inverse, Determinant() the determinant:
        Console.WriteLine("Inv A = {0:F6}", eig.GetInverse())
        Console.WriteLine("Det A = {0:F6}", eig.GetDeterminant())

        Console.Write("Press Enter key to exit...")
        Console.ReadLine()
    End Sub

End Module