Computation of a simple bifurcation point using one singular value decomposition nearby
type: Item
instance of: scholarly article
title: Computation of a simple bifurcation point using one singular value decomposition nearby
zbMATH Open document ID: 0880.65024
DOI: 10.1007/BF02684346
author: O. Diekmann
published in: Computing
publication date: 1998-02-02T00:00:00Z
review text: A direct method to locate simple bifurcation points of a system of nonlinear parameter dependent equations \(G(x,t)=0\), \(G:D_G\subset \mathbb{R}^n\times \mathbb{R}\to\mathbb{R}^n\) is developed. This method requires a good choice of an initial point \((x^{(0)},t^{(0)})\) only. Auxiliary vectors are defined then automatically on the base of the usage of a singular value decomposition of the Jacobian matrix \(G'(x^{(0)}, t^{(0)})\) near the simple bifurcation point. The suggested approach preserves the advantages of the method of \textit{V. Janovský} [Computing 43, No. 1, 27-36 (1989; Zbl 0695.65033)] by using one LU factorization in each iteration, but also improves it by providing a set of auxiliary vectors which always produces a well-conditioned bordered matrix. The quadratic convergence for the extended system proposed by \textit{G. Pönisch} [Computing 35, 277-294 (1985; Zbl 0569.65041)] is guaranteed.
Mathematics Subject Classification ID: 65H17
zbMATH DE Number: 1045486
zbMATH Keywords: direct method
zbMATH Keywords: simple bifurcation points
zbMATH Keywords: singular value decomposition
zbMATH Keywords: Jacobian matrix
zbMATH Keywords: LU factorization
zbMATH Keywords: bordered matrix
zbMATH Keywords: quadratic convergence
reviewed by: Boris V. Loginov
MaRDI profile type: Publication
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full work available at URL: https://doi.org/10.1007/bf02684346
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description: scientific article; zbMATH DE number 1045486