C# Integration Example

using System;

using CenterSpace.NMath.Core;

namespace CenterSpace.NMath.Examples.CSharp
{
  /// <summary>
  /// A .NET example in C# showing how to create integrator objects for greater control
  /// over how numerical integration is performed.
  /// </summary>
  class IntegrationExample
  {

    static double MyFunction( double x )
    {
      return Math.Sqrt( x );
    }

    static void Main( string[] args )
    {
      Console.WriteLine();

      Console.WriteLine( "Known exact integral of f from 1 to 4 = 4.6666666...\n" );

      // As shown in the OneVariableFunctionExample, the Integrate() method
      // on OneVariableFunction computes the integral of a function over a
      // given interval.
      var d = new Func<double, double>( MyFunction );
      OneVariableFunction f = d;
      Console.WriteLine( "Estimate using default Romberg integrator = {0}",
                         f.Integrate( 1, 4 ) );

      // To perform integration, every OneVariableFunction has an IIntegrator
      // object associated with it. NMath Core integration classes such as
      // RombergIntegrator and GaussKronrodIntegrator implement the IIntegrator
      // interface. The default integrator for a OneVariableFunction is an instance
      // of RombergIntegrator.

      // Instances of class RombergIntegrator compute successive Romberg
      // approximations of increasing order until the estimated error in the
      // approximation is less than a specified error tolerance, or until the
      // maximum order is reached. To achieve greater control over how integration
      // is performed, you can instantiate your own RombergIntegrator.
      var romberg = new RombergIntegrator();

      // The Tolerance property gets and sets the error tolerance used in computing
      // integrals. MaximumOrder gets and sets the maximum order.
      romberg.Tolerance = 1e-10;
      romberg.MaximumOrder = 20;

      // The Integrate() method on RombergIntegrator accepts a
      // OneVariableFunction and an interval over which to integrate.
      Console.WriteLine( "Estimate using customized Romberg integrator = {0}",
                         romberg.Integrate( f, 1, 4 ) );

      // After computing an estimate, a RombergIntegrator holds a record of
      // the iteration process. Read-only properties are provided for accessing
      // this information.
      Console.WriteLine( "Order of estimate = {0}", romberg.Order );
      Console.WriteLine( "Error estimate = {0}", romberg.RombergErrorEstimate );

      // ToleranceMet gets a boolean value indicating whether or not the error
      // estimate for the integral approximation is less than the tolerance.
      if ( romberg.ToleranceMet )
      {
        Console.WriteLine( "The estimate is within the error tolerance.\n" );
      }
      else
      {
        Console.WriteLine( "The estimate is NOT within the error tolerance.\n" );
      }

      // Tableau gets the entire matrix of successive approximations
      // computed while computing a Romberg estimate. The rows are the order
      // of approximation. The columns are the level of approximation.
      // The first column contains the trapezoidal approximations,
      // the second column the Simpson’s rule approximations, the third
      // column the Boole’s rule approximations, and so on, up to the Order
      // of the approximation just computed.
      Console.WriteLine( "Romberg Tableau:" );
      Console.WriteLine( romberg.Tableau.ToTabDelimited( "F5" ) );

      // The automatic GaussKronrodIntegrator class uses Gauss-Kronrod rules with
      // increasing number of points. Approximation ends when the estimated error
      // is less than a specified error tolerance, or when the maximum number of
      // points is reached. The Gauss-Kronrod method is especially suited for
      // non-singular oscillating integrands.
      var gk = new GaussKronrodIntegrator();
      Console.WriteLine( "\nEstimate using Gauss-Kronrod automatic integrator = {0}",
                        gk.Integrate( f, 1, 4 ) );

      // NMath Core also includes Gauss-Kronrod classes for different numbers of
      // Kronrod points (2n+1, beginning with a Gauss 10-point rule). For instance,
      // GaussKronrod21Integrator approximates integrals using the Gauss 10-point
      // and the Kronrod 21-point rule.
      var gk21 = new GaussKronrod21Integrator();
      Console.WriteLine( "Estimate using Gauss-Kronrod 21 integrator = {0}",
                        gk21.Integrate( f, 1, 4 ) );

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

    }

  }// class

}// namespace