Journal of Applied Mathematics
Volume 2012 (2012), Article ID 718608, 15 pages
Research Article

On a Newton-Type Method for Differential-Algebraic Equations

1Departamento de Matemática Aplicada y Estadística, Universidad Politécnica de Cartagena, Paseo de Alfonso XIII 52, Murcia, 30203 Cartagena, Spain
2ETSI Industriales, Universidad de Castilla-La Mancha, 13071 Ciudad Real, Spain

Received 18 September 2012; Accepted 27 November 2012

Academic Editor: Alicia Cordero

Copyright © 2012 S. Amat et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.


This paper deals with the approximation of systems of differential-algebraic equations based on a certain error functional naturally associated with the system. In seeking to minimize the error, by using standard descent schemes, the procedure can never get stuck in local minima but will always and steadily decrease the error until getting to the solution sought. Starting with an initial approximation to the solution, we improve it by adding the solution of some associated linear problems, in such a way that the error is significantly decreased. Some numerical examples are presented to illustrate the main theoretical conclusions. We should mention that we have already explored, in some previous papers (Amat et al., in press, Amat and Pedregal, 2009, and Pedregal, 2010), this point of view for regular problems. However, the main hypotheses in these papers ask for some requirements that essentially rule out the application to singular problems. We are also preparing a much more ambitious perspective for the theoretical analysis of nonlinear DAEs based on this same approach.