An Overlapped Group Iterative Method for Solving Linear Systems

We propose a new iterative method for the numerical computation of the solution of linear systems of equations. The method is suited to problems exhibiting local dependencies. The technique can be envisaged as a generalization of the Gauss-Seidel point iterative method, being able to achieve greater convergence rates with, approximately, the same number of operations per iteration. On each iteration, a group of variables is treated as unknown while the others are assumed known; the equations associated to the mentioned group are then solved in order to the unknown variables. The following iterations do the same, choosing other groups of variables. The successive groups overlap each other, departing definitely from the group iterative perspective, since, in the latter case, groups are disjoint. The overlapped group (OG) method, herein introduced, is shown to converge for two classes of problems: (1) symmetric positive definite systems; (2) systems in which any principal submatrix is nonsingular and whose inverse matrix elements are null above (below) some upper (lower) diagonal. In class (2) the exact solution is reached in just one step. Comparisons of the convergence rate with those of other iterative and direct methods are provided illustrating the effectiveness of the OG scheme.