Algorithms for projection methods for solving linear systems of equations

AbstractThe solution of linear systems of equations using various projection algorithms is considered. Since nonsingularity of the coefficient matrix is the only requirement for convergence, techniques for increasing the rate of convergence are presented. Various criteria for the selection of two and three-dimensional dimensional subspaces to project the residual vector at any step are reviewed in the literature. These algorithms are called quasi-optimal since the subspaces formed by the column vectors of the coefficient matrix are done a priori. They are shown to significantly reduce the number of cycles required for convergence and compare favorably with standard methods. A new class of projection algorithms is presented which is proven to be superior to conventional projection algorithms. It is shown that the new algorithms are equivalent to the conventional projection algorithms, but require less than half the number of arithmetic computations per iterative step and that the number of computations required per iterative step per component is independent of the dimension of the algorithm used