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Mathematics of Computation

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Expansion and Estimation of the Range of Nonlinear Functions

Author: S. M. Rump
Journal: Math. Comp. 65 (1996), 1503-1512
MSC (1991): Primary 65G10; Secondary 65D15
MathSciNet review: 1361812
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Abstract: Many verification algorithms use an expansion $f(x) \in f(\tilde {x}) + S \cdot (x - \tilde {x})$, $f : \mathbb {R} ^n \rightarrow \mathbb {R} ^n$ for $x \in X$, where the set of matrices $S$ is usually computed as a gradient or by means of slopes. In the following, an expansion scheme is described which frequently yields sharper inclusions for $S$. This allows also to compute sharper inclusions for the range of $f$ over a domain. Roughly speaking, $f$ has to be given by means of a computer program. The process of expanding $f$ can then be fully automatized. The function $f$ need not be differentiable. For locally convex or concave functions special improvements are described. Moreover, in contrast to other methods, $\tilde {x} \ \cap \ X$ may be empty without implying large overestimations for $S$. This may be advantageous in practical applications.

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S. M. Rump
Affiliation: Arbeitsbereich Informatik III, Technische Universität Hamburg-Harburg, D-21071 Hamburg, Germany

Received by editor(s): January 11, 1995
Received by editor(s) in revised form: November 2, 1995
Article copyright: © Copyright 1996 American Mathematical Society

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