Extreme Optimization > QuickStart Samples > Basic Integration QuickStart Sample (F#)

Extreme Optimization QuickStart Samples

Basic Integration QuickStart Sample (F#)

Illustrates the basic numerical integration (quadrature) classes (Extreme.Mathematics.Calculus namespace) in F#.

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#light
 
open System
// The numerical integration classes reside in the
// Extreme.Mathematics.Calculus namespace.
open Extreme.Mathematics.Calculus
// Function delegates reside in the Extreme.Mathematics
// namespace.
open Extreme.Mathematics
 
// Illustrates the basic use of the numerical integration
// classes in the Extreme.Mathematics.Calculus namespace of the Extreme
// Optimization Mathematics Library for .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.
 
//
// The integrand
//
 
// The function we are integrating must be
// provided as a RealFunction. For more
// information about this delegate, see the
// FunctionDelegates QuickStart sample.
let f = new RealFunction(Math.Sin)
 
// Function that will cause difficulties to the
// simplistic integration algorithms.
let HardIntegrand x = if (x <= 0.0) then 0.0 else Math.Pow(x,-0.9) * Math.Log(1.0/x)
 
//
// SimpsonIntegrator
// 
 
// The simplest numerical integration algorithm
// is Simpson's rule. 
let simpson = new 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 = Extreme.Numerics.ConvergenceCriterion.WithinRelativeTolerance
// The Integrate method performs the actual 
// integration:
let result = simpson.Integrate(f, 0.0, 2.0)
Console.WriteLine("sin(x) on [0,2]")
Console.WriteLine("Simpson integrator:")
// The result is also available in the Result 
// property:
Console.WriteLine("  Value: {0}", simpson.Result)
// To see whether the algorithm ended normally,
// inspect the Status property:
Console.WriteLine("  Status: {0}", simpson.Status)
// You can find out the estimated error of the result
// through the EstimatedError property:
Console.WriteLine("  Estimated error: {0}", simpson.EstimatedError)
// The number of iterations to achieve the result
// is available through the IterationsNeeded property.
Console.WriteLine("  Iterations: {0}", simpson.IterationsNeeded)
// The number of function evaluations is available 
// through the FunctionEvaluationsNeeded property.
Console.WriteLine("  Function evaluations: {0}", simpson.FunctionEvaluationsNeeded)
 
//
// 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 Extreme.Mathematics.Calculus namespace contains a series
// of Gauss-Kronrod integrators that are mainly used
// as an integration rule for the adaptive 
// integrator. No iteration is used.
// Here's the 21-point rule:
let gk21 = new GaussKronrodIntegrator21()
gk21.Integrate(f, 0.0, 2.0)
Console.WriteLine("21 point Gauss-Kronrod rule:")
Console.WriteLine("  Value: {0}", gk21.Result)
Console.WriteLine("  Status: {0}", gk21.Status)
Console.WriteLine("  Estimated error: {0}", gk21.EstimatedError)
Console.WriteLine("  Iterations: {0}", gk21.IterationsNeeded)
Console.WriteLine("  Function evaluations: {0}", gk21.FunctionEvaluationsNeeded)
 
// The NonAdaptiveGaussKronrodIntegrator uses a
// succession of 10, 21, 43, and 87 point rules
// to approximate the integral.
let nagk = new NonAdaptiveGaussKronrodIntegrator()
nagk.Integrate(f, 0.0, 2.0)
Console.WriteLine("Non-adaptive Gauss-Kronrod rule:")
Console.WriteLine("  Value: {0}", nagk.Result)
Console.WriteLine("  Status: {0}", nagk.Status)
Console.WriteLine("  Estimated error: {0}", nagk.EstimatedError)
Console.WriteLine("  Iterations: {0}", nagk.IterationsNeeded)
Console.WriteLine("  Function evaluations: {0}", nagk.FunctionEvaluationsNeeded)
 
//
// Romberg Integration
//
// Romberg integration combines Simpson's Rule
// with a scheme to accelerate convergence.
// This algorithm is useful for smooth integrands.
let romberg = new RombergIntegrator()
result = romberg.Integrate(f, 0.0, 2.0)
Console.WriteLine("Romberg integration:")
Console.WriteLine("  Value: {0}", romberg.Result)
Console.WriteLine("  Status: {0}", romberg.Status)
Console.WriteLine("  Estimated error: {0}", romberg.EstimatedError)
Console.WriteLine("  Iterations: {0}", romberg.IterationsNeeded)
Console.WriteLine("  Function evaluations: {0}", romberg.FunctionEvaluationsNeeded)
 
// However, it breaks down if the integration
// algorithm contains singularities or 
// discontinuities.
f = new RealFunction(HardIntegrand)
result = romberg.Integrate(f, 0.0, 1.0)
Console.WriteLine("Romberg on hard integrand:")
Console.WriteLine("  Value: {0}", romberg.Result)
Console.WriteLine("  Actual value: 100")
Console.WriteLine("  Status: {0}", romberg.Status)
Console.WriteLine("  Estimated error: {0}", romberg.EstimatedError)
Console.WriteLine("  Iterations: {0}", romberg.IterationsNeeded)
Console.WriteLine("  Function evaluations: {0}", romberg.FunctionEvaluationsNeeded)
 
Console.Write("Press Enter key to exit...")
Console.ReadLine()

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