Data Analysis Mathematics Linear Algebra Statistics

Try it now

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.

View this sample in: C# F# IronPython

``````Option Infer On

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

' 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()
' The license is verified at runtime. We're using
' https://www.extremeoptimization.com/trialkey

' 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.
'
' 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.
'
' 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 Factor 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())
' 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...")