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.
"On a Newton-Type Method for Differential-Algebraic Equations." J. Appl. Math. 2012 (SI06) 1 - 15, 2012. https://doi.org/10.1155/2012/718608