QuickStart Samples

# Advanced Integration QuickStart Sample (IronPython)

Illustrates more advanced numerical integration using the AdaptiveIntegrator class in IronPython.

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import numerics from math import * # The numerical integration classes reside in the # Extreme.Mathematics.Calculus namespace. from Extreme.Mathematics.Calculus import * # Function delegates reside in the Extreme.Mathematics # namespace. from Extreme.Mathematics import * #/ AdaptiveGaussKronrodIntegrator numerical integrator class #/ classes in the Extreme.Mathematics.Calculus namespace of the Extreme #/ Optimization Numerical Libraries for .NET. #/ </summary> # Numerical integration algorithms fall into two # main categories: adaptive and non-adaptive. # This QuickStart Sample illustrates the use of # the adaptive numerical integrator implemented by # the AdaptiveIntegrator class. This class is the # most advanced of the numerical integration # classes. # # 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 Func<double, double>. For more # information about this delegate, see the # FunctionDelegates QuickStart sample. # # Construct an instance of the integrator class: integrator = AdaptiveIntegrator() # # Adaptive integrator basics # # All the properties and methods defined by the # NumericalIntegrator base class are available. # See the BasicIntegration QuickStart Sample # for details. The AdaptiveIntegrator class defines # the following additional properties: # # The IntegrationRule property gets or sets the # integration rule that is to be used for # integrating subintervals. It can be any # object derived from IntegrationRule. # # For convenience, a series of Gauss-Kronrod # integration rules of order 15, 21, 31, 41, 51, # and 61 have been provided. integrator.IntegrationRule = IntegrationRule.CreateGaussKronrod15PointRule() # The UseAcceleration property specifies whether # precautions should be taken for singularities # in the integration interval. integrator.UseExtrapolation = False # Finally, the Singularities property allows you # to specify singularities or discontinuities # inside the integration interval. See the # sample below for details. # # Integration over infinite intervals # integrator.AbsoluteTolerance = 1e-8 integrator.ConvergenceCriterion = ConvergenceCriterion.WithinAbsoluteTolerance # The Integrate method performs the actual # integration. To integrate over an infinite # interval, simply use either or both of # float.PositiveInfinity and # float.NegativeInfinity as bounds: result = integrator.Integrate(lambda x: exp(-x - x*x), float.NegativeInfinity, float.PositiveInfinity) print "Exp(-x^2-x) on [-inf,inf]" print " Value: ", integrator.Result print " Exact value:", exp(0.25) * Constants.SqrtPi # To see whether the algorithm ended normally, # inspect the Status property: print " Status:", integrator.Status print " Estimated error:", integrator.EstimatedError print " Iterations:", integrator.IterationsNeeded print " Function evaluations:", integrator.EvaluationsNeeded # If you just want the result, you can also call the Integrate # extension method directly on the integrand: integrand = lambda(x): exp(-x - x ** 2) result = FunctionMath.Integrate(integrand, float.NegativeInfinity, float.PositiveInfinity) print " Value: ", result # # Functions with singularities at the end points # of the integration interval. # # Thanks to the adaptive nature of the algorithm, # special measures can be taken to accelerate # convergence near singularities. To enable this # acceleration, set the Singularities property # to True. integrator.UseExtrapolation = True # We'll use the function that gives the Romberg # integrator in the BasicIntegration QuickStart # sample trouble. result = integrator.Integrate(lambda x: x ** -0.9 * log(1/x), 0.0, 1.0) print "Singularities on boundary:" print " Value: ", integrator.Result print " Exact value: 100" print " Status:", integrator.Status print " Estimated error:", integrator.EstimatedError # Where Romberg integration failed after 1,000,000 # function evaluations, we find the correct answer # to within tolerance using only 135 function # evaluations! print " Iterations:", integrator.IterationsNeeded print " Function evaluations:", integrator.EvaluationsNeeded # # Functions with singularities or discontinuities # inside the interval. # integrator.UseExtrapolation = True # We will pass an array containing the interior # singularities to the integrator through the # Singularities property: integrator.SetSingularities(1, sqrt(2)) integrator.Integrate(lambda x: x**3 * log(abs((x**2-1) * (x**2 - 2))), 0.0, 3.0) print "Singularities inside the interval:" print " Value: ", integrator.Result print " Exact value: 52.740748383471444998" print " Status:", integrator.Status print " Estimated error:", integrator.EstimatedError print " Iterations:", integrator.IterationsNeeded print " Function evaluations:", integrator.EvaluationsNeeded

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