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Structured Linear Equations QuickStart Sample (VB.NET)
Extreme Optimization QuickStart Samples
Structured Linear Equations QuickStart Sample (VB.NET)
Illustrates how to solve symmetrical and triangular systems of simultaneous
linear equations using classes from the Extreme.Mathematics.LinearAlgebra
namespace in Visual Basic .NET.
C# code
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' The structured matrix classes reside in the
' Extreme.Mathematics.LinearAlgebra namespace.
Imports Extreme.Mathematics.LinearAlgebra
Namespace Extreme.Mathematics.QuickStart.VB
' Illustrates solving symmetrical and triangular systems
' of simultaneous linear equations using classes
' in the Extreme.Mathematics.LinearAlgebra namespace of the Extreme
' Optimization Mathematics Library 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 As TriangularMatrix = New TriangularMatrix( _
MatrixTriangleMode.Upper, 4, New Double() _
{1, 0, 0, 0, _
1, 2, 0, 0, _
1, 4, 1, 0, _
1, 3, 1, 2})
Dim b1 As Vector = New GeneralVector(1, 3, 6, 3)
Dim b2 As GeneralMatrix = New GeneralMatrix(4, 2, New Double() _
{1, 3, 6, 3, _
2, 3, 5, 8})
Console.WriteLine("t = {0}", 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 As Vector = t.Solve(b1, False)
Console.WriteLine("x1 = {0}", x1)
' If the overwrite parameter is omitted, the
' right-hand-side is overwritten with the solution:
t.Solve(b1)
Console.WriteLine("b1 = {0}", b1)
' You can solve for multiple right hand side
' vectors by passing them in a GeneralMatrix:
Dim x2 As GeneralMatrix = t.Solve(b2, False)
Console.WriteLine("x2 = {0}", x2)
'
' Related Methods
'
' You can verify whether a matrix is singular
' using the IsSingular method:
Console.WriteLine("IsSingular(t) = {0}", _
t.IsSingular())
' The inverse matrix is returned by the GetInverse
' method:
Console.WriteLine("GetInverse(t) = {0}", t.GetInverse())
' The determinant is also available:
Console.WriteLine("Det(t) = {0}", t.GetDeterminant())
' The condition number is an estimate for the
' loss of precision in solving the equations
Console.WriteLine("Cond(t) = {0}", 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 As SymmetricMatrix = New SymmetricMatrix( _
MatrixTriangleMode.Upper, New Double() _
{1, 0, 0, 0, _
1, 2, 0, 0, _
1, 1, 2, 0, _
1, 0, 1, 4}, 4)
b1 = New GeneralVector(1, 3, 6, 3)
Console.WriteLine("s = {0}", 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}", x1)
' If the overwrite parameter is omitted, the
' right-hand-side is overwritten with the solution:
s.Solve(b1)
Console.WriteLine("b1 = {0}", b1)
' You can solve for multiple right hand side
' vectors by passing them in a GeneralMatrix:
x2 = s.Solve(b2, False)
Console.WriteLine("x2 = {0}", 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}", s.GetInverse())
' The determinant is also available:
Console.WriteLine("Det(s) = {0}", s.GetDeterminant())
' The condition number is an estimate for the
' loss of precision in solving the equations
Console.WriteLine("Cond(s) = {0}", 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 As CholeskyDecomposition = _
New CholeskyDecomposition(s, 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}", x2)
Console.WriteLine("IsSingular(m) = {0}", _
cf.IsSingular())
Console.WriteLine("GetInverse(m) = {0}", cf.GetInverse())
Console.WriteLine("Det(m) = {0}", cf.GetDeterminant())
Console.WriteLine("Cond(m) = {0}", cf.EstimateConditionNumber())
' In addition, you have access to the
' triangular matrix, G, of the composition.
Console.WriteLine(" G = {0}", cf.LowerTriangularFactor)
' Note that if the matrix is indefinite,
' the factorization will fail and throw a
' MatrixNotPositiveDefiniteException.
s(0, 0) = -99
cf = New CholeskyDecomposition(s)
Try
cf.Decompose()
Catch e As MatrixNotPositiveDefiniteException
Console.WriteLine(e.Message)
End Try
' The SymmetricMatrix class take care of this
' possibility automatically, but is much less
' efficient:
Console.WriteLine("Cond s = {0}", s.EstimateConditionNumber())
Console.Write("Press Enter key to exit...")
Console.ReadLine()
End Sub
End Module
End Namespace
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