Basic Integration in F# QuickStart Sample

Illustrates the basic numerical integration classes in F#.

This sample is also available in: C#, Visual Basic, IronPython.

Overview

This QuickStart sample demonstrates how to use the basic numerical integration classes in Numerics.NET.

The sample shows how to use three different non-adaptive numerical integration algorithms:

Simpson’s Rule - A simple and widely used integration method
Gauss-Kronrod Integration - An optimized fixed-point scheme using carefully chosen evaluation points
Romberg Integration - A method that combines Simpson’s Rule with convergence acceleration

For each integration method, the sample demonstrates:

How to configure the integrator (tolerances, convergence criteria)
How to perform the actual integration
How to access results and diagnostic information (error estimates, iteration counts)
The strengths and limitations of each method

The sample concludes by showing how simpler integration methods can break down when dealing with difficult integrands that have singularities or discontinuities, setting up the motivation for using adaptive integration methods covered in other samples.

The code

//=====================================================================
//
//  File: basic-integration.fs
//
//---------------------------------------------------------------------
//
//  This file is part of the Numerics.NET Code Samples.
//
//  Copyright (c) 2004-2025 ExoAnalytics Inc. All rights reserved.
//
//=====================================================================

module BasicIntegration

#light

open System

// The numerical integration classes reside in the
// Numerics.NET.Calculus namespace.
open Numerics.NET.Calculus
// Function delegates reside in the Numerics.NET
// namespace.
open Numerics.NET

// The license is verified at runtime. We're using a 30 day trial key here.
// For more information, see:
//     https://numerics.net/trial-key
let licensed = Numerics.NET.License.Verify("your-trial-key-here")

// Illustrates the basic use of the numerical integration
// classes in the Numerics.NET.Calculus namespace of Numerics.NET.

// Numerical integration algorithms fall into two
// main categories: adaptive and non-adaptive.
// 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.

printfn "sin(x) on [0,2]"

//
// SimpsonIntegrator
//

// The simplest numerical integration algorithm
// is Simpson's rule.
printfn "Simpson integrator:"

let simpson = SimpsonIntegrator()
// You can set the relative or absolute tolerance
// to which to evaluate the integral.
simpson.RelativeTolerance <- 1e-5
// You can select the type of tolerance using the
// ConvergenceCriterion property:
simpson.ConvergenceCriterion <- ConvergenceCriterion.WithinRelativeTolerance
// The Integrate method performs the actual integration:
let result = simpson.Integrate(Func<_,_> Math.Sin, 0.0, 2.0)
// The result is also available in the Result property:
printfn "  Value: %A" simpson.Result
// To see whether the algorithm ended normally,
// inspect the Status property:
printfn "  Status: %A" simpson.Status
// You can find out the estimated error of the result
// through the EstimatedError property:
printfn "  Estimated error: %A" simpson.EstimatedError
// The number of iterations to achieve the result
// is available through the IterationsNeeded property.
printfn "  Iterations: %d" simpson.IterationsNeeded
// The number of function evaluations is available
// through the EvaluationsNeeded property.
printfn "  Function evaluations: %d" simpson.EvaluationsNeeded

//
// Gauss-Kronrod Integration
//

printfn "Non-adaptive Gauss-Kronrod rule:"

// 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.
let nagk = NonAdaptiveGaussKronrodIntegrator()
let result2 = nagk.Integrate(Func<_,_> Math.Sin, 0.0, 2.0)
printfn "  Value: %A" nagk.Result
printfn "  Status: %A" nagk.Status
printfn "  Estimated error: %A" nagk.EstimatedError
printfn "  Iterations: %d" nagk.IterationsNeeded
printfn "  Function evaluations: %d" nagk.EvaluationsNeeded

//
// Romberg Integration
//

printfn "Romberg integration:"

// Romberg integration combines Simpson's Rule
// with a scheme to accelerate convergence.
// This algorithm is useful for smooth integrands.
let romberg = RombergIntegrator()
let result3 = romberg.Integrate(Func<_,_> Math.Sin, 0.0, 2.0)
printfn "  Value: %A" romberg.Result
printfn "  Status: %A" romberg.Status
printfn "  Estimated error: %A" romberg.EstimatedError
printfn "  Iterations: %d" romberg.IterationsNeeded
printfn "  Function evaluations: %d" romberg.EvaluationsNeeded

// However, it breaks down if the integration
// algorithm contains singularities or
// discontinuities.

// The AdaptiveIntegrator can handle this type
// of integrand, and many other difficult cases.
// See the AdvancedIntegration QuickStart sample
// for details.
printfn "Romberg on hard integrand:"

let result4 = romberg.Integrate((fun x -> if x <= 0.0 then 0.0 else x**(-0.9) * log(1.0/x)), 0.0, 1.0)
printfn "  Value: %A" romberg.Result
printfn "  Actual value: 100"
printfn "  Status: %A" romberg.Status
printfn "  Estimated error: %A" romberg.EstimatedError
printfn "  Iterations: %d" romberg.IterationsNeeded
printfn "  Function evaluations: %d" romberg.EvaluationsNeeded

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