Convergence Criteria for Successive Overrelaxation

Abstract

Cyclic iterative methods for solving systems of linear equations are investigated with reference to necessary and sufficient conditions for convergence. For matrices with positive diagonal elements and nonpositive off-diagonal elements (so-called L-matrices), a “generalized” diagonal dominance is found to be necessary for convergence of the Gauss–Seidel and Jacobi methods and for convergence of a certain range of relaxation. It is shown that convergent matrices of this type can be characterized in terms of strict diagonal dominance under row or column scaling. In general, scaling the coefficient matrix by rows or columns has no effect on the asymptotic convergence properties. For symmetric matrices of the same special type, it is shown that positive definiteness can be characterized in terms of scaling and strict diagonal dominance.