Multiparameter iterative schemes for the solution of systems of linear and nonlinear equations

In this paper, we introduce multiparameter generalizations of the linear and nonlinear iterative Richardson methods for solving systems of linear and nonlinear equations. The new algorithms are based on using a (optimal) matricial relaxation instead of the (optimal) scalar relaxation of the steepest descent method. The optimal matrix, which is defined at each iteration by minimizing the current residual, is computed as the least squares solution of an associated problem whose dimension is generally much lower than that of the original problem. In particular, thanks to this approach, we construct multiparameter versions of the Delta(k) method introduced for solving nonlinear fixed point problems. Various numerical results illustrate the implementation of the new schemes. They concern the solution of a linear problem and of a nonlinear one which comes out from a reaction-diffusion problem which exhibits bifurcations. In both cases, the (optimal) multiparameter relaxation improves the convergence as compared to the (optimal) scalar one.