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Appendix E
A Detailed Look at f
Section 14, Numerical Integration, presented the basic information you
need to use f. This appendix discusses more intricate aspects of f
that are of interest if you use f often.
How f Works
The f algorithm calculates the integral of a function f(x) by computing
a weighted average of the function’s values at many values of x (known
as sample points) within the interval of integration. The accuracy of the
result of any such sampling process depends on the number of sample
points considered: generally, the more sample points, the greater the
accuracy. If f (x) could be evaluated at an infinite number of sample
points, the algorithm could—neglecting the limitation imposed by the
inaccuracy in the calculated function f(x)—provide an exact answer.
Evaluating the function at an infinite number of sample points would take
a very long time (namely, forever). However, this is not necessary, since
the maximum accuracy of the calculated integral is limited by the
accuracy of the calculated function values. Using only a finite number of
sample points, the algorithm can calculate an integral that is as accurate
as is justified considering the inherent uncertainty in f (x).
The f algorithm at first considers only a few sample points, yielding
relatively inaccurate approximations. If these approximations are not yet
as accurate as the accuracy of f (x) would permit, the algorithm is iterated
(that is, repeated) with a larger number of sample points. These iterations
continue, using about twice as many sample points each time, until the
resulting approximation is as accurate as is justified considering the
inherent uncertainty in f (x).
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