The NumericalIntegrator Class
The NumericalIntegrator class is the abstract
base class for all numerical integration classes.
It inherits from IterativeAlgorithm.
properties set the desired precision as specified by
the ConvergenceCriterion property. The default
value for both tolerances is 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
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.
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 FuncT, TResult delegate that specifies the target function. The remaining
two parameters are Double values that specify the lower and upper bounds of the
Code examples are given in later sections.
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.
The algorithm has not been executed.
The algorithm ended normally. The desired accuracy has been achieved.
The number of iterations needed to achieve the desired accuracy is greater than MaxIterations.
The number of function evaluations needed to achieve the desired accuracy is greater than MaxEvaluations.
Round-off prevented the algorithm from achieving the desired accuracy.
Bad behavior of the target function prevented the algorithm from achieving the desired accuracy.
The integral appears to diverge.