Successive Overrelaxation Iterative Methods

The point successice overrelaxation iterative method of Chap.3 was simultaneously introduced by Frankel (1950) and Young (1950). Whereas Frankel considered the special case of the numerical solutation of the Dirichlet problem for a rectangle and showed for this case that the point successive overrelaxation iterative method, with suitable chosen relaxation factor, gave substantially larger (by an order of magnitude) asymptotic rates of convergence than those for the point Jacobi and point Gauss-Seidel iterative methods, Young (1950)and Yound (1954a) showed that these conclusions held more generally for matrices satisfying his definition of propertly A, and that these results could be rigorously applied to the iterative solution of matrix equations arising from discrete approximations to a large class of elliptic partial differential equations for general regions. Then, Arms, Gates, and Zondek (1956) with their definition of property AΠ generalized Young's results. In so doing, they enlarged the class of matrix equations to which the basic results of Young and Frankel, on the successive overrelaxation iterative method, could be rigorously applied.