HP 15c - Manual (Page 256)

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.)