Abstract
Interval iteration can be used, in conjunction with other techniques, for rigorously bounding all solutions to a nonlinear system of equations within a given region, or for verifying approximate solutions. However, because of overestimation which occurs when the interval Jacobian matrix is accumulated and applied, straightforward linearization of the original nonlinear system sometimes leads to nonconvergent iteration.
In this paper, we examine interval iterations based on an expanded system obtained from the intermediate quantities in the original system. In this system, there is no overestimation in entries of the interval Jacobi matrix, and nonlinearities can be taken into account to obtain sharp bounds. We present an example in detail, algorithms, and detailed experimental results obtained from applying our algorithms to the example.
Zusammenfassung
Intervalliterationen können in Verbindung mit anderen Verfahren verwendet werden, um alle Lösungen eines nichlinearen Gleichungsystems in einem gegebenen Gebiet mit Sicherheit abzuschätzen, und auch um Approximationen der Lösungen solcher Systeme zu verifizieren. Die Abschätzungen in den Verfahren sind jedoch manchmal nicht hinreichend genau, da Überschätzungen in der Berechnung und in dem Gebrauch der Invervall-Jacobi Matrix auftreten.
In der vorliegenden Arbeit werden Intervalliterationen auf einem erweiterten Gleichungssystem behandelt. In diesem System gibt es keine Überschätzungen der Einzelkomponenten der Intervall-Jacobi Matrix, und für die Nichtlinearitären können Abschätzungen angegeben werden. Anhand eines Beispiels wird die Wirkungsweise der behandelten Algorithmen demonstriert.
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Kearfott, R.B. Decomposition of arithmetic expressions to improve the behavior of interval iteration for nonlinear systems. Computing 47, 169–191 (1991). https://doi.org/10.1007/BF02253433
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DOI: https://doi.org/10.1007/BF02253433