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A quadratically convergent method for computing simple singular roots and its application to determining simple bifurcation points

Ein quadratisch konvergentes Verfahren zur Berechnung einfacher singulärer Nullstellen und dessen Anwendung zur Bestimmung einfacher Bifurkationspunkte

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Abstract

We present an efficient method for computing roots of mappings on ℝn in the case where the Jacobian has the rankn−1 at the root. For the accurate determination of such a rootx*∈ℝn an auxiliary system ofn equations inn+1 variables is constructed which possesses (x *, 1) as a turning point. This turning point can be computed by direct methods. We use an adapted method which requires only the solution of (n+1)-dimensional systems of linear equations and the evaluation of one Jacobian and 5 function values per step. This techniques is successfully applied to compute simple bifurcation points by means of a suitable system of nonlinear equations which has the properties mentioned above.

Zusammenfassung

Es wird ein effektives Verfahren zur Lösung nichtlinearer Gleichungssysteme hergeleitet, wobei die Jacobi-Matrix im Lösungspunktx*∈ℝn einen Rangabfall von 1 besitzt. Dem gegebenen Gleichungssystem der Dimensionn wird ein Hilfsproblem inn+1 Variablen zugeordnet, das den Rückkehrpunkt (x *, 1) besitzt. Die Bestimmung von (x *, 1) erfolgt mit einem angepaßten Verfahren, das ohne die Berechnung zweier Ableitungen auskommt und nur die Lösung linearer Gleichungssysteme der Dimensionn+1 erfordert. Das angegebene Prinzip wird erfolgreich zur Bestimmung einfacher Bifurkationspunkte eingesetzt, die sich als Lösungen nichtlinearer Gleichungssysteme mit singulären Jacobi-Matrizen charakterisieren lassen.

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Authors and Affiliations

  1. Sektion Mathematik, Technische Universität Dresden, Mommesenstrasse 13, DDR-8027, Dresden, German Democratic Republic

    R. Menzel &  G. Pönisch

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  1. R. Menzel
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  2. G. Pönisch
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Menzel, R., Pönisch, G. A quadratically convergent method for computing simple singular roots and its application to determining simple bifurcation points. Computing 32, 127–138 (1984). https://doi.org/10.1007/BF02253687

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