Appendix E: A Detailed Look at f 253
Note that the rapidity of variation in the function (or its low-order
derivatives) must be determined with respect to the width of the interval of
integration. With a given number of sample points, a function f(x) that has
three fluctuations can be better characterized by its samples when these
variations are spread out over most of the interval of integration than if they
are confined to only a small fraction of the interval. (These two situations
are shown in the next two illustrations.) Considering the variations or
fluctuations as a type of oscillation in the function, the criterion of interest
is the ratio of the period of the oscillations to the width of the interval of
integration: the larger this ratio, the more quickly the algorithm will
terminate, and the more reliable will be the resulting approximation.
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