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QuickStart Samples

Quasi-Random Sequences QuickStart Sample (F#)

Illustrates how to generate quasi-random sequences like Fauré and Sobol sequences using classes in the Extreme.Statistics.Random namespace in F#.

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/// Illustrates the use of quasi-random sequences by computing
/// a multi-dimensional integral.

#light

open System

open Extreme.Mathematics
open Extreme.Mathematics.Random

// This QuickStart Sample demonstrates the use of
// quasi-random sequences by computing
// a multi-dimensional integral.

// We will use one million points.
let length = 10000
// The number of dimensions:
let dimension = 5

// We will evaluate the function
//
//    Product(i = 1 -> # dimensions) |4 x.[i] - 2|
//
// over the hypercube 0 <= x.[i] <= 1. The value of this integral
// is exactly 1.

// Compute the integral by summing over all points:
let f (point : Vector<float>) =
    let mutable functionValue = 1.0
    for j in 0..point.Length-1 do
        functionValue <- functionValue * abs(4.0*point.[j]-2.0)
    functionValue

let integrate f sequence = 
    printfn "# iter.  Estimate"
    let mutable sum = 0.0
    let mutable i = 0
    for point in sequence do
        if (i % 100000 = 0) then
            printfn "%6d  %8.4f" i (sum / float i)
        sum <- sum + f point
        i <- i + 1
    sum / (float i)
    
// Create the sequence:
let halton = QuasiRandom.HaltonSequence(dimension, length)
let haltonResult = integrate f halton

printfn "Final estimate (Halton): %8.4f" haltonResult
printfn "Exact value: 1.0000"

// Sobol sequences require more data and more initialization.
// Fortunately, different sequences of the same dimension
// can share much of the work and storage. The 
// SobolSequenceGenerator class should be used in this case:

let skip = 1000;
let sobol = new SobolSequenceGenerator(dimension, length + skip);
// Sobol sequences are more flexible: they let you skip
// a number of points at the start of the sequence.
// The cost of skipping points is O(1).
let sobolResult = integrate f (sobol.Generate(length, skip))

// Print the final result.
printfn "Final estimate (Sobol): %8.4f" sobolResult
printfn "Exact value: 1.0000"

printf "Press any key to exit."
Console.ReadLine() |> ignore