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Overview of Numerical Integration

The NumericalIntegrator Class

The NumericalIntegrator class is the abstract base class for all numerical integration classes. It inherits from IterativeAlgorithm. The AbsoluteTolerance and RelativeTolerance properties set the desired precision as specified by the ConvergenceCriterion property. The default value for both tolerances is MachineConstants.SqrtEpsilon (roughly 10^-8). MaxIterations sets the maximum number of iterations. The default value for this property depends on the algorithm used. IterationsNeeded returns the actual number of iterations performed after the algorithm has completed.

Specific to this class are the Order and FunctionEvaluationsNeeded properties, and the overloaded Integrate method.

The Order property gives the order of the integration algorithm. The order of an integration algorithm is the highest degree of a general polynomial whose integral is calculated exactly by the algorithm. A method of order three integrates cubic polynomials exactly. Many methods have a fixed order. For some algorithms, the order depends on the input values.

The FunctionEvaluationsNeeded property returns the total number of times the target function was evaluated while approximating the integral. This property is a more reliable indication of the efficiency of an algorithm than IterationsNeeded. For some algorithms, the number of function evaluations grows exponentially with each iteration, while for others it is a simple multiple. Even though higher order methods are slower, they usually require less subdivisions of the integration interval, which makes them more desirable for smooth target functions. For target functions with integrable singularities, a low order method is usually preferred.

The Integrate method does the actual work of numerically integrating a target function. It has three overloads. With no parameters, the method uses the values supplied through the Integrand, LowerBound, and UpperBound properties.

The remaining two overloads take two or three parameters. The first parameter, if present, is a RealFunction delegate that specifies the target function. The remaining two parameters are Double values that specify the lower and upper bounds of the integration interval.

Code examples are given in later sections.

Result codes

The Integrate method always returns the best estimate for the integral. Successive calls to the Result property will also return this value, until the next call to Integrate.

If the ThrowExceptionOnFailure property is set to true, an exception is thrown if the algorithm has failed to obtain the integral with the desired accuracy. If false, the Integrate method returns the best approximation to the integral, regardless of whether it is within the requested tolerance.

The Status property indicates how the algorithm terminated. Its possible values and their meaning are listed below.

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