Document Type

Conference Proceeding

Publication Date

2010

Keywords

Separable parameterized equations, rank deficiency, Golub-Pereyra variable projection method, bordered matrix, singular value decomposition, Newton's method

Abstract

Many applications give rise to separable parameterized equations of the form A(y,µ)z + b(y, µ) = 0, where y ∈ Rn, z ∈ RN and the parameter µ ∈ R; here A(y,µ) is an (N + n) × N matrix and b(y, µ) ∈ RN +n. Under the assumption that A(y, µ) has full rank we showed in [21] that bifurcation points can be located by solving a reduced equation of the form f (y, µ) = 0. In this paper we extend that method to the case that A(y, µ) has rank deficiency one at the bifurcation point. At such a point the solution curve (y, µ, z) branches into infinitely many additional solutions,which form a straight line. A numerical method for reducing the problem to a smaller space and locating such a bifurcation point is given. Applications to equilibrium solutions of nonlinear ordinary equations and solutions of discretized partial differential equations are provided.

Publication Title

Electronic Journal of Differential Equations

Volume

19

First Page

254

Last Page

255

Required Publisher's Statement

Published by the Department of Mathematics, Texas State University

Comments

Eighth Mississippi State - UAB Conference on Differential Equations and Computational Simulations

This is an open access journal.

Subjects - Topical (LCSH)

Seperable algebras; Bifurcation theory

Genre/Form

conference proceedings

Type

Text

Rights

Copying of this document in whole or in part is allowable only for scholarly purposes. It is understood, however, that any copying or publication of this document for commercial purposes, or for financial gain, shall not be allowed without the author’s written permission.

Language

English

Format

application/pdf

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Included in

Mathematics Commons

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