On numerical differentiation
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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.
- Ström, T. (1973) Monotonicity and error bounds in schemes of Romberg Type for the computation of f(n)(0) by central differences and extrapolation. Num. Math. 21: pp. 14-21 CrossRef
- Curry, H. B., Schoenberg, I. J. (1966) On Polya frequency functions IV: The fundamental spline functions and their limits. J. Anal. Math. 17: pp. 71-107
- Lyness, J. N., Moler, C. B. (1966) Van der Monde systems and numerical differentiation. Num. Math. 8: pp. 458-464
- Oliver, J., Ruffhead, A. (1975) The selection of interpolation points in numerical differentiation. BIT 15: pp. 283-295
- On numerical differentiation
BIT Numerical Mathematics
Volume 15, Issue 3 , pp 314-322
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