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Mathematics of Computation

ISSN 1088-6842(online) ISSN 0025-5718(print)



For numerical differentiation,
dimensionality can be a blessing!

Authors: Robert S. Anderssen and Markus Hegland
Journal: Math. Comp. 68 (1999), 1121-1141
MSC (1991): Primary 65D25
Published electronically: February 10, 1999
MathSciNet review: 1620207
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Abstract: Finite difference methods, such as the mid-point rule, have been applied successfully to the numerical solution of ordinary and partial differential equations. If such formulas are applied to observational data, in order to determine derivatives, the results can be disastrous. The reason for this is that measurement errors, and even rounding errors in computer approximations, are strongly amplified in the differentiation process, especially if small step-sizes are chosen and higher derivatives are required. A number of authors have examined the use of various forms of averaging which allows the stable computation of low order derivatives from observational data. The size of the averaging set acts like a regularization parameter and has to be chosen as a function of the grid size $h$. In this paper, it is initially shown how first (and higher) order single-variate numerical differentiation of higher dimensional observational data can be stabilized with a reduced loss of accuracy than occurs for the corresponding differentiation of one-dimensional data. The result is then extended to the multivariate differentiation of higher dimensional data. The nature of the trade-off between convergence and stability is explicitly characterized, and the complexity of various implementations is examined.

References [Enhancements On Off] (What's this?)

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Additional Information

Robert S. Anderssen
Affiliation: CSIRO Mathematical and Information Sciences, GPO Box 1965, Canberra ACT 2601, Australia

Markus Hegland
Affiliation: Computer Sciences Laboratory, Australian National University, Canberra ACT 0200, Australia

Keywords: Numerical differentiation
Received by editor(s): April 29, 1997
Received by editor(s) in revised form: October 9, 1997
Published electronically: February 10, 1999
Article copyright: © Copyright 1999 American Mathematical Society