Generalized preconditioning strategies

Over the past decade Professor David J. Evans [1968] has suggested the use of ‘Preconditioning’ in iterative methods for solving large, sparse systems of linear equations, which arise from the finite difference approximations to the partial differential equations. Since then, certain aspects on preconditioning have appeared in the literature and a whole new theory constructed. The versatility of the preconditioning concept is shown by the stimulating exploration of new numerical algorithms and methods of their realization. The aim of this thesis is to emphasise in the theory we use and develop together with the practice we state. This study led to a new form of preconditioning, which has not yet appeared in the literature. Specifically, we consider the conditioning matrix factorized into two rectangular matrices, so as to develop a new preconditioned iterative method and its related properties as well. It requires the selection of two parameters to be applied, a preconditioning parameter at its optimal value and an acceleration parameter in such a fashion that a simultaneous displacement method is applicable. [Continues.]