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252
Appendix E: A Detailed Look at
f
The graph is a spike very close to the origin. (Actually, to illustrate f(x) the
width of the spike has been considerably exaggerated. Shown in actual scale
over the interval of integration, the spike would be indistinguishable from
the vertical axis of the graph.) Because no sample point happened to
discover the spike, the algorithm assumed that f(x) was identically equal to
zero throughout the interval of integration. Even if you increased the
number of sample points by calculating the integral in i 9, none of the
additional sample points would discover the spike when this particular
function is integrated over this particular interval. (Better approaches to
problems such as this are mentioned at the end of the next topic, Conditions
That Prolong Calculation Time.)
You've seen how the f algorithm can give you an incorrect answer when
f(x) has a fluctuation somewhere that is very uncharacteristic of the
behavior of the function elsewhere. Fortunately, functions exhibiting such
aberrations are unusual enough that you are unlikely to have to integrate
one unknowingly.
Functions that could lead to incorrect results can be identified in simple
terms by how rapidly it and its low-order derivatives vary across the
interval of integration. Basically, the more rapid the variation in the
function or its derivatives, and the lower the order of such rapidly varying
derivatives, the less quickly will the f algorithm terminate, and the less
reliable will the resulting approximation be.
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