# Basic Integration in Visual Basic QuickStart Sample

Illustrates the basic numerical integration classes in Visual Basic.

View this sample in: C# F# IronPython

``````Option Infer On

' The numerical integration classes reside in the
' Numerics.NET.Calculus namespace.
Imports Numerics.NET.Calculus
' Function delegates reside in the Numerics.NET
' namespace.
Imports Numerics.NET

' Illustrates the use of the Newton-Raphson equation solver
' in the Numerics.NET.EquationSolvers namespace of Numerics.NET.
Module BasicIntegration

Sub Main()
' The license is verified at runtime. We're using
' a 30 day trial key here. For more information, see
'     https://numerics.net/trial-key

' Numerical integration algorithms fall into two
' This QuickStart Sample illustrates the use of
' the non-adaptive numerical integrators.
'
' All numerical integration classes derive from
' NumericalIntegrator. This abstract base class
' defines properties and methods that are shared
' by all numerical integration classes.

'
' The integrand
'
' The function we are integrating must be
' provided as a Func(Of Double, Double). For more
' Functions QuickStart sample.
Dim f As Func(Of Double, Double) = AddressOf Math.Sin
' Variable to hold the result:
Dim result As Double

'
' SimpsonIntegrator
'

' The simplest numerical integration algorithm
' is Simpson's rule.
Dim simpson As New SimpsonIntegrator()
' You can set the relative or absolute tolerance
' to which to evaluate the integral.
simpson.RelativeTolerance = 0.00001
' You can select the type of tolerance using the
' ConvergenceCriterion property:
simpson.ConvergenceCriterion = ConvergenceCriterion.WithinRelativeTolerance
' The Integrate method performs the actual
' integration:
result = simpson.Integrate(f, 0, 5)
Console.WriteLine("sin(x) on [0,2]")
Console.WriteLine("Simpson integrator:")
' The result is also available in the Result
' property:
Console.WriteLine(\$"  Value: {simpson.Result}")
' To see whether the algorithm ended normally,
' inspect the Status property:
Console.WriteLine(\$"  Status: {simpson.Status}")
' You can find out the estimated error of the result
' through the EstimatedError property:
Console.WriteLine(\$"  Estimated error: {simpson.EstimatedError}")
' The number of iterations to achieve the result
' is available through the IterationsNeeded property.
Console.WriteLine(\$"  Iterations: {simpson.IterationsNeeded}")
' The number of function evaluations is available
' through the EvaluationsNeeded property.
Console.WriteLine(\$"  Function evaluations: {simpson.EvaluationsNeeded}")

'
' Gauss-Kronrod Integration
'

' Gauss-Kronrod integrators also use a fixed point
' scheme, but with certain optimizations in the
' choice of points where the integrand is evaluated.

' The NonAdaptiveGaussKronrodIntegrator uses a
' succession of 10, 21, 43, and 87 point rules
' to approximate the integral.
Dim nagk As New NonAdaptiveGaussKronrodIntegrator()
nagk.Integrate(f, 0, 5)
Console.WriteLine(\$"  Value: {nagk.Result}")
Console.WriteLine(\$"  Status: {nagk.Status}")
Console.WriteLine(\$"  Estimated error: {nagk.EstimatedError}")
Console.WriteLine(\$"  Iterations: {nagk.IterationsNeeded}")
Console.WriteLine(\$"  Function evaluations: {nagk.EvaluationsNeeded}")

'
' Romberg Integration
'

' Romberg integration combines Simpson's Rule
' with a scheme to accelerate convergence.
' This algorithm is useful for smooth integrands.
Dim romberg As New RombergIntegrator()
result = romberg.Integrate(f, 0, 5)
Console.WriteLine("Romberg integration:")
Console.WriteLine(\$"  Value: {romberg.Result}")
Console.WriteLine(\$"  Status: {romberg.Status}")
Console.WriteLine(\$"  Estimated error: {romberg.EstimatedError}")
Console.WriteLine(\$"  Iterations: {romberg.IterationsNeeded}")
Console.WriteLine(\$"  Function evaluations: {romberg.EvaluationsNeeded}")

' However, it breaks down if the integration
' algorithm contains singularities or
' discontinuities.
f = AddressOf HardIntegrand
result = romberg.Integrate(f, 0, 1)
Console.WriteLine("Romberg on hard integrand:")
Console.WriteLine(\$"  Value: {romberg.Result}")
Console.WriteLine("  Actual value: 100")
Console.WriteLine(\$"  Status: {romberg.Status}")
Console.WriteLine(\$"  Estimated error: {romberg.EstimatedError}")
Console.WriteLine(\$"  Iterations: {romberg.IterationsNeeded}")
Console.WriteLine(\$"  Function evaluations: {romberg.EvaluationsNeeded}")

Console.Write("Press Enter key to exit...")
End Sub

' Function that will cause difficulties to the
' simplistic integration algorithms.
Private Function HardIntegrand(x As Double) As Double
' This is put in because some integration rules
' evaluate the function at x=0.
If (x <= 0) Then
Return 0
End If
Return Math.Pow(x, -0.9) * Math.Log(1 / x)
End Function

End Module``````