A note on solutions of nonlinear equations with singular Jacobian matrices
type: Item
instance of: scholarly article
title: A note on solutions of nonlinear equations with singular Jacobian matrices
zbMATH Open document ID: 0619.65041
DOI: 10.1016/0022-247X(87)90300-3
author: Norio Yamamoto
published in: Journal of Mathematical Analysis and Applications
publication date: 1987-01-01T00:00:00Z
review text: Let \(x^*\) be a solution of the nonlinear equation \(F(x)=0\) where the Jacobian of \(F: R^ n\to R^ n\) has rank n-2. The author considers an unfolded, augmented equation of the form \[ g(x)= \begin{pmatrix} c F(x)- \beta_ 1e_ i- \beta_ 2e_ j \\ F_ x(x)h \\ e^ T_ 1h-1 \\ e^ T_ 2h \end{pmatrix} \tag{1} \] and shows that, under appropriate assumptions and for suitable parameters, the Jacobian of G is nonsingular at \(x^*\) and hence (1) can be used to compute this solution. Such unfoldings and augmentations of singular problems have been studied extensively in connection with numerical bifurcation problems, but the author does not refer to that literature.
Mathematics Subject Classification ID: 65H10
zbMATH DE Number: 4005464
zbMATH Keywords: singular Jacobian matrices
zbMATH Keywords: unfoldings
zbMATH Keywords: augmentations
zbMATH Keywords: singular problems
zbMATH Keywords: numerical bifurcation problems
reviewed by: Werner C. Rheinboldt
MaRDI profile type: Publication
cites work: Newton’s Method for Singular Problems when the Dimension of the Null Space is $>1$
cites work: Q3219692
cites work: Q3678579
description: scientific article; zbMATH DE number 4005464