Over the past decade Professor David J. Evans  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
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.]
A Master's Thesis. Submitted in partial fulfilment of the requirements for the award of Master of Philosophy at Loughborough University.