, Volume 15, Issue 3, pp 314-322

On numerical differentiation

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Abstract

In a recent paper Ström analyzed a simple extrapolation algorithm for numerical differentiation and derived certain properties about the kernel function of the integral representation of the remainder term. These properties are useful for placing bounds on the error in cases when specified higher order derivatives are known not to change sign. The algorithm involves a separate Romberg table for each derivative and is rather inconvenient from the point of view of economizing the number of function values required.

In this paper we generalize Ström's results in two stages. First we show that they are valid for a very wide choice of definitions of the initial column of each Romberg table. Then we show that one such choice, making full use of the computed function values, gives results identical to those that can be obtained using an algorithm suggested by Lyness and Moler with a particular choice of sequence of function evaluations.

There is no detailed discussion of the effect of round-off error.

This work was performed under the auspices of the U.S. Atomic Energy Commission.