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.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Alefeld, G., Herzberger, J.: Introduction to interval computations. New York: Academic Press 1983.
Cleary, J. G.: Logical arithmetic. Future Comput. Syst. 125–149 (1987).
Hansen, E., Sengupta, S.: Bounding solutions of systems of equations using interval analysis. BIT21, 203–211 (1981).
Krawczyk, R.: Newton-Algorithmen zur Bestimmung von Nullstellen mit Fehlerschranken. Computing4, 187–201 (1969).
Mackworth, A. K.: Consistency in networks of relations. Art. Intell.8, 99–118 (1977).
Moore, R. E.: Interval analysis. Englewood Cliffs: Prentice-Hall 1966.
Moore, R. E.: A test for existence of solution to nonlinear systems. ISAM J. Numer. Anal.14, 611–615 (1977).
Moore, R. E.: A computational test for convergence of iterative methods for nonlinear systems. SIAM J. Numer. Anal.15, 1194–1196 (1978).
Moore, R. E., Jones, S. T.: Safe starting regions for iterative methods. SIAM J. Numer. Anal.14, 1051–1065 (1977).
Moore, R. E., Qi, L.: A successive interval test for nonlinear systems. SIAM J. Numer. Anal.19, 845–850 (1982).
Morgan, A.: Solving polynomial systems using continuation for engineering and scientific problems. Englewood Cliffs: Prentice-Hall 1987.
Neumaier, A.: Interval methods for systems of equations. Cambridge: Cambridge University Press 1990.
Older, W., Vellino, A.: Extending Prolog with constraint arithmetic on real intervals. In: Proceedings of the Eight Biennial Conference of the Canadian Society for Computational Studies of Intelligence, 1990.
Qi, L.: A note on the Moore test for nonlinear system. SIAM J. Numer. Anal.19, 851–857 (1982).
Rump, S. M.: Solving nonlinear systems with least significant bit accuracy. Computing29, 183–200 (1982).
Rump, S. M.: Solution of linear and nonlinear algebraic problems with sharp, guaranteed bounds. Computing [Suppl.]5, 147–168 (1984).
Shearer, J. M., Wolfe, M. A.: Some computable existence, uniquness, and convergence tests for nonlinear systems. SIAM J. Numer. Anal.22, 1200–1207 (1985).
Wolfe, M. A.: Interval methods for algebraic equations. In: Reliability in computing, pp. 229–248. London: Academic Press 1988.
Author information
Authors and Affiliations
Additional information
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).
Rights and permissions
About this article
Cite this article
Hong, H., Stahl, V. Safe starting regions by fixed points and tightening. Computing 53, 323–335 (1994). https://doi.org/10.1007/BF02307383
Received:
Revised:
Issue Date:
DOI: https://doi.org/10.1007/BF02307383