Abstract
In this paper, we present a method for finding safe starting regions for a given system of non-linear equations. The method is an improvement of the usual method which is based on the fixed point theorem. The improvement is obtained by enclosing the components of the equation system by univariante interval polynomials whose zero sets are found. This operation is called “tightening”. Preliminary experiments show that the tightening operation usually reduces the number of bisections, and thus the computing time. The reduction seems to become more dramatic when the number of variables increases.
Zusammenfassung
In dieser Arbeit wird eine Methode zur Bestimmung von Startintervallen mit garantierter Konvergenz für ein gegebenes nichtlineares Gleichungssystem vorgestellt. Die Methode ist eine verbesserung der gebräuchlichen, auf dem Fixpunkt Theorem basierenden Methode. Die Verbesserung wird durch Einschließen der Komponenten des Gleichungssystems durch univariate Intervallpolynome, deren Lösungsmengen berechnet werden, erzielt. Diese Operation wird “Einengung” genannt. Erste experimentelle Untersuchungen zeigen, daß Einengung im allgemeinen die Anzahl der Intervallhalbierungen und somit die Rechenzeit reduziert. Die Reduktion scheint umso signifikanter, je höher die Anzahl der Variablen ist.
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The research was done within the framework of the ACCLAIM project sponsored by European Community Basic Research Action (ESPRIT 7195) and Austrian Science Foundation (P9374-PHY).
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Hong, H., Stahl, V. Safe starting regions by fixed points and tightening. Computing 53, 323–335 (1994). https://doi.org/10.1007/BF02307383
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DOI: https://doi.org/10.1007/BF02307383