Data Analysis Mathematics Linear Algebra Statistics
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QuickStart Samples

# Linear Equations QuickStart Sample (Visual Basic)

Illustrates how to solve systems of simultaneous linear equations in Visual Basic.

```Option Infer On

' The DenseMatrix and LUDecomposition classes reside in the
' Extreme.Mathematics.LinearAlgebra namespace.
Imports Extreme.Mathematics
Imports Extreme.Mathematics.LinearAlgebra

Namespace Extreme.Numerics.QuickStart.VB

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

Sub Main()
' A system of simultaneous linear equations is
' defined by a square matrix A and a right-hand
' side B, which can be a vector or a matrix.
'
' You can use any matrix type for the matrix A.
' The optimal algorithm is automatically selected.

Dim m = Matrix.Create(4, 4, New Double() _
{1, 1, 1, 1, _
1, 2, 3, 4, _
1, 4, 9, 16, _
1, 2, 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("m = {0:F4}", m)

'
' 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 = m.Solve(b1, False)
Console.WriteLine("x1 = {0:F4}", x1)
' If the overwrite parameter is omitted, the
' right-hand-side is overwritten with the solution:
m.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 = m.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(m) = {0:F4}", _
m.IsSingular())
' The inverse matrix is returned by the GetInverse
' method:
Console.WriteLine("GetInverse(m) = {0:F4}", m.GetInverse())
' The determinant is also available:
Console.WriteLine("Det(m) = {0:F4}", m.GetDeterminant())
' The condition number is an estimate for the
' loss of precision in solving the equations
Console.WriteLine("Cond(m) = {0:F4}", m.EstimateConditionNumber())
Console.WriteLine()

'
' The LUDecomposition class
'

' If multiple operations need to be performed
' on the same matrix, it is more efficient to use
' the LUDecomposition class. This class does the
' bulk of the calculations only once.
Console.WriteLine("Using LU Decomposition:")
' The constructor takes an optional second argument
' indicating whether to overwrite the original
' matrix with its decomposition:
Dim lu = m.GetLUDecomposition(False)
' All methods mentioned earlier are still available:
x2 = lu.Solve(b2, False)
Console.WriteLine("x2 = {0:F4}", x2)
Console.WriteLine("IsSingular(m) = {0:F4}", _
lu.IsSingular())
Console.WriteLine("GetInverse(m) = {0:F4}", lu.GetInverse())
Console.WriteLine("Det(m) = {0:F4}", lu.GetDeterminant())
Console.WriteLine("Cond(m) = {0:F4}", lu.EstimateConditionNumber())