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Mathematics of Computation

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Adaptive numerical differentiation

Authors: R. S. Stepleman and N. D. Winarsky
Journal: Math. Comp. 33 (1979), 1257-1264
MSC: Primary 65D25
MathSciNet review: 537969
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Abstract: It is well known that the calculation of an accurate approximate derivative $ f\prime(x)$ of a nontabular function $ f(x)$ on a finite-precision computer by the formula $ d(h) = (f(x + h) - f(x - h))/2h$ is a delicate task. If h is too large, truncation errors cause poor answers, while if h is too small, cancellation and other "rounding" errors cause poor answers. We will show that by using simple results on the nature of the asymptotic convergence of $ d(h)$ to $ f\prime$, a reliable numerical method can be obtained which can yield efficiently the theoretical maximum number of accurate digits for the given machine precision.

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Article copyright: © Copyright 1979 American Mathematical Society