New iterative schemes for nonlinear fixed point problems, with applications to problems with bifurcations and incomplete-data problems

Authors:
Ch. Roland;R. Varadhan
Affiliations:
Laboratoire Paul Painlevé, UFR de Mathématiques Pures et Appliquées -- M3, Université des Sciences et Technologies de Lille, 59655 Villeneuve d'Ascq cedex, France;The Center on Aging and Health, Johns Hopkins University, 2024E. Monument Street, Suite 2-700, Baltimore, Maryland 21205-2223, USA
Venue:
Applied Numerical Mathematics
Year:
2005

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Abstract

In this paper, we propose a new method which could be considered as a modification of the @D^k-method introduced for solving nonlinear fixed point problems. At each iteration of the new scheme, we evaluate the @D^k steplength once and we use it twice. Various numerical results illustrate the efficiency of the new scheme. They concern the solution of a reaction-diffusion problem which exhibits a bifurcation. An additional example, involving a mixture of Poisson distributions, will be given and suggest that the new scheme could be adapted with success for an important statistical problem called the expectation-maximization problem.