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QuickStart Samples

Structured Linear Equations QuickStart Sample (Visual Basic)

Illustrates how to solve systems of simultaneous linear equations that have special structure in Visual Basic.

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Option Infer On

' The structured matrix classes reside in the 
' Extreme.Mathematics.LinearAlgebra namespace.
Imports Extreme.Mathematics
Imports Extreme.Mathematics.LinearAlgebra

Namespace Extreme.Numerics.QuickStart.VB

    ' Illustrates solving symmetrical and triangular systems 
    ' of simultaneous linear equations using classes 
    ' in the Extreme.Mathematics.LinearAlgebra namespace of the Extreme 
    ' Optimization Numerical Libraries for .NET.
    Module StructuredLinearEquations

        Sub Main()
            ' To learn more about solving general systems of
            ' simultaneous linear equations, see the
            ' LinearEquations QuickStart Sample.
            '
            ' The methods and classes available for solving
            ' structured systems of equations are similar
            ' to those for general equations.

            '
            ' Triangular systems and matrices
            '

            Console.WriteLine("Triangular matrices:")
            ' For the basics of working with triangular 
            ' matrices, see the TriangularMatrices QuickStart
            ' Sample.
            '
            ' Let's start with a triangular matrix. Remember
            ' that elements are stored in column-major order
            ' by default.
            Dim t = Matrix.CreateUpperTriangular(
                4, 4, New Double() _
                   {1, 0, 0, 0,
                    1, 2, 0, 0,
                    1, 4, 1, 0,
                    1, 3, 1, 2}, MatrixElementOrder.ColumnMajor)
            Dim b1 = Vector.Create(New Double() {1, 3, 6, 3})
            Dim b2 = Matrix.Create(4, 2, New Double() _
                {1, 3, 6, 3, _
                 2, 3, 5, 8}, MatrixElementOrder.ColumnMajor)
            Console.WriteLine("t = {0:F4}", t)

            '
            ' The Solve method
            '

            ' The following solves m x = b1. The second 
            ' parameter specifies whether to overwrite the
            ' right-hand side with the result.
            Dim x1 = t.Solve(b1, False)
            Console.WriteLine("x1 = {0:F4}", x1)
            ' If the overwrite parameter is omitted, the
            ' right-hand-side is overwritten with the solution:
            t.Solve(b1)
            Console.WriteLine("b1 = {0:F4}", b1)
            ' You can solve for multiple right hand side 
            ' vectors by passing them in a DenseMatrix:
            Dim x2 = t.Solve(b2, False)
            Console.WriteLine("x2 = {0:F4}", x2)

            '
            ' Related Methods
            '

            ' You can verify whether a matrix is singular
            ' using the IsSingular method:
            Console.WriteLine("IsSingular(t) = {0:F4}", _
                t.IsSingular())
            ' The inverse matrix is returned by the GetInverse
            ' method:
            Console.WriteLine("GetInverse(t) = {0:F4}", t.GetInverse())
            ' The determinant is also available:
            Console.WriteLine("Det(t) = {0:F4}", t.GetDeterminant())
            ' The condition number is an estimate for the
            ' loss of precision in solving the equations
            Console.WriteLine("Cond(t) = {0:F4}", t.EstimateConditionNumber())
            Console.WriteLine()

            '
            ' Symmetric systems and matrices
            '

            Console.WriteLine("Symmetric matrices:")
            ' For the basics of working with symmetric 
            ' matrices, see the SymmetricMatrices QuickStart
            ' Sample.
            '
            ' Let's start with a symmetric matrix. Remember
            ' that elements are stored in column-major order
            ' by default.
            Dim s = Matrix.CreateSymmetric(4, New Double() _
             {1, 0, 0, 0,
              1, 2, 0, 0,
              1, 1, 2, 0,
              1, 0, 1, 4}, MatrixTriangle.Upper, MatrixElementOrder.ColumnMajor)
            b1 = Vector.Create(New Double() {1, 3, 6, 3})
            Console.WriteLine("s = {0:F4}", s)

            '
            ' The Solve method
            '

            ' The following solves m x = b1. The second 
            ' parameter specifies whether to overwrite the
            ' right-hand side with the result.
            x1 = s.Solve(b1, False)
            Console.WriteLine("x1 = {0:F4}", x1)
            ' If the overwrite parameter is omitted, the
            ' right-hand-side is overwritten with the solution:
            s.Solve(b1)
            Console.WriteLine("b1 = {0:F4}", b1)
            ' You can solve for multiple right hand side 
            ' vectors by passing them in a DenseMatrix:
            x2 = s.Solve(b2, False)
            Console.WriteLine("x2 = {0:F4}", x2)

            '
            ' Related Methods
            '

            ' You can verify whether a matrix is singular
            ' using the IsSingular method:
            Console.WriteLine("IsSingular(s) = {0}", _
                s.IsSingular())
            ' The inverse matrix is returned by the GetInverse
            ' method:
            Console.WriteLine("GetInverse(s) = {0:F4}", s.GetInverse())
            ' The determinant is also available:
            Console.WriteLine("Det(s) = {0:F4}", s.GetDeterminant())
            ' The condition number is an estimate for the
            ' loss of precision in solving the equations
            Console.WriteLine("Cond(s) = {0:F4}", s.EstimateConditionNumber())
            Console.WriteLine()

            '
            ' The CholeskyDecomposition class
            '

            ' If the symmetric matrix is positive definite,
            ' you can use the CholeskyDecomposition class
            ' to optimize performance if multiple operations 
            ' need to be performed. This class does the
            ' bulk of the calculations only once. This
            ' decomposes the matrix into G x transpose(G)
            ' where G is a lower triangular matrix.
            '
            ' If the matrix is indefinite, you need to use
            ' the LUDecomposition class instead. See the
            ' LinearEquations QuickStart Sample for details.
            Console.WriteLine("Using Cholesky Decomposition:")
            ' The constructor takes an optional second argument
            ' indicating whether to overwrite the original
            ' matrix with its decomposition:
            Dim cf = s.GetCholeskyDecomposition(False)
            ' The Factorize method performs the actual
            ' factorization. It is called automatically
            ' if needed.
            cf.Decompose()
            ' All methods mentioned earlier are still available:
            x2 = cf.Solve(b2, False)
            Console.WriteLine("x2 = {0:F4}", x2)
            Console.WriteLine("IsSingular(m) = {0}", _
                cf.IsSingular())
            Console.WriteLine("Inverse(m) = {0:F4}", cf.GetInverse())
            Console.WriteLine("Det(m) = {0:F4}", cf.GetDeterminant())
            Console.WriteLine("Cond(m) = {0:F4}", cf.EstimateConditionNumber())
            ' In addition, you have access to the
            ' triangular matrix, G, of the composition.
            Console.WriteLine("  G = {0:F4}", cf.LowerTriangularFactor)

            ' Note that if the matrix is indefinite,
            ' the factorization will fail and throw a
            ' MatrixNotPositiveDefiniteException.
            s(0, 0) = -99
            cf = s.GetCholeskyDecomposition()
            Try
                cf.Decompose()
            Catch e As MatrixNotPositiveDefiniteException
                Console.WriteLine(e.Message)
            End Try
            ' It is better to use the TryDecompose method,
            ' which returns true if the decomposition succeeded:
            If cf.TryDecompose() Then
                Console.WriteLine("The decomposition succeeded!")
            Else
                Console.WriteLine("The decomposition failed!")
            End If

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

    End Module

End Namespace