256 Appendix E: A Detailed Look at f

If the interval of integration were (0, 10) so that the algorithm needed to sample the function only at values where it was interesting but relatively smooth, the sample points after the first few iterations would contribute no new information about the behavior of the function. Therefore, only a few iterations would be necessary before the disparity between successive approximations became sufficiently small that the algorithm could terminate with an approximation of a given accuracy.

On the other hand, if the interval of integration were more like the one shown in the graph on page 252, most of the sample points would capture the function in the region where its slope is not varying much. The few sample points at small values of x would find that values of the function changed appreciably from one iteration to the next. Consequently the function would have to be evaluated at additional sample points before the disparity between successive approximations would become sufficiently small.

In order for the integral to be approximated with the same accuracy over the larger interval as over the smaller interval, the density of the sample points must be the same in the region where the function is interesting. To achieve the same density of sample points, the total number of sample points required over the larger interval is much greater than the number required over the smaller interval. Consequently, several more iterations are required over the larger interval to achieve an approximation with the same accuracy, and therefore calculating the integral requires considerably more time.

Because the calculation time depends on how soon a certain density of sample points is achieved in the region where the function is interesting, the calculation of the integral of any function will be prolonged if the interval of integration includes mostly regions where the function is not interesting. Fortunately, if you must calculate such an integral, you can modify the problem so that the calculation time is considerably reduced. Two such techniques are subdividing the interval of integration and transformation of variables. These methods enable you to change the function or the limits of integration so that the integrand is better behaved over the interval(s) of integration. (These techniques are described in the HP-15C Advanced Functions Handbook.)