# VB Quasi Random Example

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```ï»¿Imports System
Imports System.Collections

Imports CenterSpace.NMath.Core

Namespace CenterSpace.NMath.Core.Examples.VisualBasic

' <summary>
' A .NET example in Visual Basic showing how to generate a sequence of quasirandom points.
' </summary>
Module QuasiRandomExample

Sub Main()

' A quasirandom sequence is a sequence of n-tuples that fills n-space
' more uniformly than uncorrelated random points. NMath provides both
' Sobol and Niederreiter quasirandom number generators.

' Create a Niederreiter quasirandom number generator object with dimension 1
' to generate quasirandom numbers (not vectors).
Dim Dimensions As Integer = 1
Dim nqrg As New NiederreiterQuasiRandomGenerator(Dimensions)

' You can fill an existing matrix or array. (The points are the columns of the matrix,
' so the number of rows in the given matrix must be equal to the Dimension of the
' quasirandom number generator.)

' Here we create and fill a 1 x numPts matrix with quasirandom numbers from a uniform
' (0,1) distribution. The resulting numbers will occupy the first row of A.
Dim NumPts As Integer = 100
Dim A As New DoubleMatrix(nqrg.Dimension, NumPts)
nqrg.Fill(A)

' Create a histogram to verify that the quasirandom numbers are uniformly
' distributed in the interval (0,1). We create 10 bins expecting that each
' bin will contain approximately numPts/10 points.
Dim H As New Histogram(10, 0, 1)

Console.WriteLine()

Console.WriteLine("{0} uniform (0,1) quasirandom points:", NumPts)
Console.WriteLine(H.StemLeaf())

' Compare the above results with those from a pseudo-random generator. One expects
' a not-so uniform distribution in space.
Dim Seed As Integer = &H345
Dim Stream As New RandomNumberStream(Seed, RandomNumberStream.BasicRandGenType.MersenneTwister)
Dim UnifDist As New DoubleRandomUniformDistribution()
Dim pseudoRandom As New DoubleVector(NumPts, Stream, UnifDist)
H.Reset()
Console.WriteLine()
Console.WriteLine("{0} uniform (0,1) pseudo-random points:", NumPts)
Console.WriteLine(H.StemLeaf())

' Quasi-random numbers are useful for evaluating the integral of a function in the
' unit n-dimensional cube - essentially calculating the average of the function
' at a set of randomly selected points.

' In the following example we use a Sobol generator to approximate the integral
' of a function F over the 6-dimensional unit cube. The function is defined by
' F(x) = 1*cos(1*x1) * 2*cos(2*x2) * 3*cos(3*x3) *...* 6*cos(6*x6)
' where x = (x1, x2,..., x6) is a point in 6-dimensional space.
Dimensions = 6

' In this example we provide our own primitive polynomials for initializing
' the generator. Primitive polynomials have coefficients in {0, 1} and are
' specified by BitArray's containing the polynomial coefficients with the
' leading coefficient at index 0.

' We can supply either dim polynomials, or dim - 1 polynomials. If we specify
' dim - 1 polynomials the primitive polynomial for the first dimension will
' be initialized with a default value.
Dim primitivePolys(Dimensions - 1) As BitArray
primitivePolys(0) = New BitArray(New Boolean() {True, True}) ' x + 1
primitivePolys(1) = New BitArray(New Boolean() {True, True, True}) ' x^2 + x + 1
primitivePolys(2) = New BitArray(New Boolean() {True, False, True, True}) ' x^3 + x + 1
primitivePolys(3) = New BitArray(New Boolean() {True, True, False, True}) ' x^3 + x^2 + 1
primitivePolys(4) = New BitArray(New Boolean() {True, False, False, True, True}) ' x^4 + x + 1
primitivePolys(5) = New BitArray(New Boolean() {True, True, False, False, True}) ' x^4 + x^3 + 1

Dim Sobol As New SobolQuasiRandomGenerator(Dimensions, primitivePolys)
Dim NumPoints As Integer = 180000
Dim Points As New DoubleMatrix(Dimensions, NumPoints)
Sobol.Fill(Points)
Dim Sum As Double = 0

For I = 0 To NumPoints - 1
Sum += F(Points.Col(I))
Next

Sum /= NumPoints
Dim actualIntegralValue As Double = -0.022
Console.WriteLine("Actual integral value = " & actualIntegralValue)
Console.WriteLine()
Console.WriteLine("Monte-Carlo approximated integral value = " & Sum)

Console.WriteLine()
Console.WriteLine("Press Enter Key")

End Sub

' <summary>
' F(x) = cos(x(0))*2cos(2x(1))*...*6cos(6*x(5))
' </summary>
' <param name="x"></param>
' <returns></returns>
Function F(ByVal X As DoubleVector) As Double

If (X.Length <> 6) Then
Throw New InvalidArgumentException("x must have length 6!")
End If

Dim Y As Double = 1

For i = 1 To X.Length
Y *= i * Math.Cos(i * X(i - 1))
Next
Return Y
End Function

End Module
End Namespace
```
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