# Least Squares in Visual Basic QuickStart Sample

Illustrates how to solve least squares problems using classes in the Extreme.Mathematics.LinearAlgebra namespace in Visual Basic.

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```
Option Infer On
' The DenseMatrix and DoubleVector classes resides in the
' Extreme.Mathematics.LinearAlgebra namespace.
Imports Extreme.Mathematics
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 Extreme Numerics.NET.
Module LeastSquares
Sub Main()
' The license is verified at runtime. We're using
' a demo license here. For more information, see
' https://numerics.net/trial-key
Extreme.License.Verify("Demo license")
' A least squares problem consists in finding
' the solution to an overdetermined system of
' simultaneous linear equations so that the
' sum of the squares of the error is minimal.
'
' A common application is fitting data to a
' curve. See the CurveFitting sample application
' for a complete example.
' Let's start with a general matrix. This will be
' the matrix a in the left hand side ax=b:
Dim a = Matrix.Create(6, 4, New Double() _
{
1, 1, 1, 1, 1, 1,
1, 2, 3, 4, 5, 6,
1, 4, 9, 16, 25, 36,
1, 2, 1, 2, 1, 2
}, MatrixElementOrder.ColumnMajor)
' Here is the right hand side:
Dim b = Vector.Create(New Double() {1, 3, 6, 11, 15, 21})
Dim b2 = Matrix.Create(6, 2, New Double() _
{
1, 3, 6, 11, 15, 21,
1, 2, 3, 4, 5, 7
}, MatrixElementOrder.ColumnMajor)
Console.WriteLine("a = {0:F0}", a)
Console.WriteLine("b = {0:F0}", b)
'
' The LeastSquaresSolver class
'
' The following creates an instance of the
' LeastSquaresSolver class for our problem:
Dim solver = New LeastSquaresSolver(Of Double)(a, b)
' We can specify the solution method: normal
' equations or QR decomposition. In most cases,
' a QR decomposition is the most desirable:
solver.SolutionMethod = LeastSquaresSolutionMethod.QRDecomposition
' The Solve method calculates the solution:
Dim x = solver.Solve()
Console.WriteLine("x = {0:F4}", x)
' The Solution property also returns the solution:
Console.WriteLine("x = {0:F4}", solver.Solution)
' More detailed information is available from
' additional methods.
' The values of the right hand side predicted
' by the solution:
Console.WriteLine("Predictions = {0:F4}", solver.GetPredictions())
' The residuals (errors) of the solution:
Console.WriteLine("Residuals = {0:F4}", solver.GetResiduals())
' The total sum of squares of the residues:
Console.WriteLine("Residual square error = {0:F4}",
solver.GetResidualSumOfSquares())
'
' Direct normal equations
'
' Alternatively, you can create a least squares
' solution by providing the normal equations
' directly. This may be useful when it is easy
' to calculate the normal equations directly.
'
' Here, we'll just calculate the normal equation:
Dim aTa = SymmetricMatrix(Of Double).FromOuterProduct(a)
Dim aTb = Matrix.Multiply(a, TransposeOperation.Transpose, b)
' We find the solution by solving the normal equations
' directly:
x = aTa.Solve(aTb)
Console.WriteLine("x = {0:F4}", x)
' However, properties of the least squares solution, such as
' error estimates and residuals are not available.
Console.Write("Press Enter key to exit...")
Console.ReadLine()
End Sub
End Module
```