A variant of the AOR method for augmented systems

A variant of the AOR method for augmented system

Generalized extrapolation principle and convergence of some generalized iterative methods

AbstractTo solve the linear system Ax = b, this paper presents a generalized extrapolated method by replacing the extrapolation parameter ω with the diagonal matrix Ω, and systematically gives the basic results for its convergence. Based upon these results, the paper considers the convergence of the GJ and GAOR iterative methods and, using the set of the equimodularized diagonally similar matrices defined here, gives some new further convergence results for H-matrices and their subclasses, strictly or irreducibly diagonally dominant matrices, which unify, improve, and extend previously given various results. Finally, conditions equivalent to the statement that A is a nonsingular H-matrix or a strictly (or an irreducibly) diagonally dominant matrix are given in connection with the GJ and GAOR methods

Modified successive overrelaxation (SOR) type methods for M-matrices

The SOR is a basic iterative method for solution of the linear system =. Such systems can easily be solved using direct methods such as Gaussian elimination. However, when the coefficient matrix is large and sparse, iterative methods such as the SOR become indispensable. A new preconditioner for speeding up the convergence of the SOR iterative method for solving the linear system = is proposed. Arising from the preconditioner, two new preconditioned iterative techniques of the SOR method are developed. The preconditioned iterations are applied to the linear system whose coefficient matrix is an −matrix. Convergence of the preconditioned iterations is established through standard procedures. Numerical examples and results comparison are in conformity with the analytic results. More so, it is established that the spectral radii of the proposed preconditioned SOR 1 and 2 are less than that of the classical SOR, which implies faster convergence