Stair Matrices and Their Generalizations with Applications to Iterative Methods I: A Generalization of the Successive Overrelaxation Method

Stair matrices and their generalizations are introduced. Some properties of the matrices are presented. Like triangular matrices this class of matrices provides bases of matrix splittings for iterative methods. A remarkable feature of iterative methods based on the new class of matrices is that the methods are easily implemented for parallel computation. In particular, a generalization of the successive overrelaxation (SOR) method is introduced. The SOR theory on determination of the optimal parameter is extended to the generalized method to include a wide class of matrices. The asymptotic rate of convergence of the new method is derived for Hermitian positive definite matrices using bounds of the eigenvalues of Jacobi matrices and numerical radius. Finally, numerical tests are presented to corroborate the obtained results.