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QuickStart Samples

Least Squares QuickStart Sample (Visual Basic)

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

Namespace Extreme.Numerics.QuickStart.VB
    ' 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 the Extreme Optimization
    ' Numerical Libraries for .NET.
    Module LeastSquares

        Sub Main()
            ' 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

End Namespace