Expansion and estimation of the range of nonlinear functions
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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.References
- Götz Alefeld, Intervallanalytische Methoden bei nichtlinearen Gleichungen, Jahrbuch Überblicke Mathematik, 1979, Bibliographisches Inst., Mannheim, 1979, pp. 63–78 (German). MR 554359
- Götz Alefeld and Jürgen Herzberger, Introduction to interval computations, Computer Science and Applied Mathematics, Academic Press, Inc. [Harcourt Brace Jovanovich, Publishers], New York, 1983. Translated from the German by Jon Rokne. MR 733988
- C. G. Broyden, A new method of solving nonlinear simultaneous equations, Comput. J. 12 (1969/70), 94–99. MR 245197, DOI 10.1093/comjnl/12.1.94
- Andreas Griewank, On automatic differentiation, Mathematical programming (Tokyo, 1988) Math. Appl. (Japanese Ser.), vol. 6, SCIPRESS, Tokyo, 1989, pp. 83–107. MR 1114312
- E. Hansen, A generalized interval arithmetic, Interval Mathematics (K. Nickel, editor), vol. 29, Springer, Berlin, 1975, pp. 7–18.
- Eldon R. Hansen, On solving systems of equations using interval arithmetic, Math. Comp. 22 (1968), 374–384. MR 229411, DOI 10.1090/S0025-5718-1968-0229411-4
- G. Heindl, Zur numerischen Einschließung der Werte von Peanofunktionalen, Z. Angew. Math. Mech. 75 (1995), S637–S638.
- D. Husung, Precompiler for scientific computation (TPX), Technical Report 91.1, Inst. for Computer Science III, TU Hamburg-Harburg, 1989.
- R. Krawczyk and A. Neumaier, Interval slopes for rational functions and associated centered forms, SIAM J. Numer. Anal. 22 (1985), no. 3, 604–616. MR 787580, DOI 10.1137/0722037
- Ramon E. Moore, Interval analysis, Prentice-Hall, Inc., Englewood Cliffs, N.J., 1966. MR 0231516
- Arnold Neumaier, Interval methods for systems of equations, Encyclopedia of Mathematics and its Applications, vol. 37, Cambridge University Press, Cambridge, 1990. MR 1100928
- L.B. Rall, Automatic differentiation: techniques and applications, Lecture Notes in Comput. Sci., vol. 120, Springer Verlag, Berlin–Heidelberg–New York, 1981.
- J.W. Schmidt, Die Regula Falsi für Operatioren in Banachräumen, Z. Angew. Math. Mech. 41 (1961), T61–T63.
Additional Information
- S. M. Rump
- Affiliation: Arbeitsbereich Informatik III, Technische Universität Hamburg-Harburg, D-21071 Hamburg, Germany
- MR Author ID: 151815
- Email: rump@tu-harburg.d400.de
- Received by editor(s): January 11, 1995
- Received by editor(s) in revised form: November 2, 1995
- © Copyright 1996 American Mathematical Society
- Journal: Math. Comp. 65 (1996), 1503-1512
- MSC (1991): Primary 65G10; Secondary 65D15
- DOI: https://doi.org/10.1090/S0025-5718-96-00773-9
- MathSciNet review: 1361812