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More about Integration D–1
File name 32sii-Manual-E-0424
Printed Date : 2003/4/24 Size : 17.7 x 25.2 cm
D
More about Integration
This appendix provides information about integration beyond that given in
chapter 8.
How the Integral Is Evaluated
The algorithm used by the integration operation,
∫
Gº
, 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) — always provide
an exact answer.
Evaluating the function at an infinite number of sample points would take
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 integration 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
(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).