Computing multiple turning points by using simple extended systems and computational differentiation

A point (x * ,λ * ) is called a turning point of multiplicity p ≥1 of the nonlinear system if and if the Ljapunov-Schmidt reduced function has the normal form . A minimally extended system F(x, λ)=0 F(x, λ)=0 is proposed for defining turning points of multiplicity p, where is a scalar function which...

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Bibliographic Details
Published in:Optimization methods & software 1999-01, Vol.10 (4), p.639-668
Main Authors: Pönisch, G., Schnabel, U., Schwetlick, H.
Format: Article
Language:English
Subjects:
computational differentiation
minimally extended systems
multiple turning points
Newton's method
Parameterized nonlinear equations
singular points
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Summary:A point (x * ,λ * ) is called a turning point of multiplicity p ≥1 of the nonlinear system if and if the Ljapunov-Schmidt reduced function has the normal form . A minimally extended system F(x, λ)=0 F(x, λ)=0 is proposed for defining turning points of multiplicity p, where is a scalar function which is related to the pth order partial derivatives of g with respect to ξ. When Fdepends on m≤p-1 additional parameters the system F(x, λ α)=0 can be inflated by m + 1 scalar equations f 1 (x, λ α)=0,...f m +1(x, λ α)=0 The functions depend on certain partial derivatives of gwith respect to ξ where f m+1 corresponds to f The regular solution (x * , λ * ,α * ) of the extended system of n+m+1 equations delivers the desired turning point (x * , λ * ). For numerically solving these systems, two-stage New tonype methods are proposed, where only one LU decomposition of an (n+1) ×(++1) matrix and some back substitutions have to be preformed per iteration step if Gaussian elimination is used for solving the linear systems. Moreover, the methods require the computation of certain higher order partial derivatives of f with respect to low dimensional subspaces as well as derivatives of implicitly defined related functions. Both tasks are realized via computational differentiation, often also called automatic differentiation. For doing this the special structure of the higher order derivatives is exploited, and the problem is reformulated so that the Taylor coefficients technique of computational differentiation can efficiently be integrated into the algorithm. Some numerical tests show the behavior of the algorithms in case of turning points of multiplicity p= 2,3.
ISSN:1055-6788
1029-4937
DOI:10.1080/10556789908805731