Sources of error in computation
20. R. P. Brent, Sources of error in computation,
in Error, Approximation and Accuracy
(edited by F. R. de Hoog and C. L. Jarvis),
University of Queensland Press, Brisbane, 1973, 122-128.
MR 52#2124, 54#9069.
Paper: pdf (201K).
Abstract
In the numerical approximation or prediction of a physical situtation, errors
may arise in the following ways:
- There may be errors or oversimplifications in the formulation of the
mathematical model.
- To make a computational solution possible the model may have to be
discretized. For example, integrals may be approximated by finite sums.
Thus truncation errors may be introduced.
- There may be errors in the data. Such errors are usually
unavoidable if the data are obtained from physical measurements or
preliminary computations.
- Rounding errors may occur during the computation.
Although model and truncation errors are extremely important, they are
not the subject of this paper, which considers data and rounding errors.
With numerically unstable methods rounding errors may be amplified by
"catastrophic cancellation". With numerically stable methods this can
not occur, but many small errors may accumulate. The amount of
accumulation depends on the number system used, and different number
systems are compared.
Comments
This is a brief, expository paper on the subject of computational errors,
with examples drawn from Brent
[17],
Forsythe ["Pitfalls in computation, or why a Math Book isn't enough",
American Math. Monthly 77 (1970), 931-956],
and Wilkinson [ Rounding errors in algebraic processes,
Prentice-Hall, 1963].
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