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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 îš± 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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