Abstract
Hadjidimos (1978) proposed a classical accelerated overrelaxation (AOR) iterative method to solve the system of linear equations, and discussed its convergence under the conditions that the coefficient matrices are irreducible diagonal dominant,L-matrices, and consistently orders matrices. In this paper, a new version of the AOR method is presented. Some convergence results are derived when the coefficient matrices are irreducible diagonal dominant,H-matrices, symmetric positive definite matrices, andL-matrices. A relational graph for the new AOR method and the original AOR method is presented. Finally, a numerical example is presented to illustrate the efficiency of the proposed method.
Highlights
Consider the following linear system: Ax = b, (1)where A ∈ Rn×n, b ∈ Rn are given and x ∈ Rn is unknown
For the numerical solution of (1), the accelerated overrelaxation (AOR) method was introduced by Hadjidimos in [5] and is a two-parameter generalization of the successive overrelaxation (SOR) method
To improve the convergence rate of the AOR method, the preconditioned AOR (PAOR) method has been considered by many authors including [15,16,17,18,19,20,21]
Summary
Where A ∈ Rn×n, b ∈ Rn are given and x ∈ Rn is unknown. System of form (1) appears in many applications such as linear elasticity, fluid dynamics, and constrained quadratic programming [1,2,3,4]. Sufficient conditions for the convergence of the AOR method have been considered by many authors including [6,7,8,9,10,11,12,13,14]. The purpose of this paper is to present a new version of the accelerated overrelaxation (AOR) method for the linear system (1), which is called the quasi accelerated overrelaxation (QAOR) method. We discuss some sufficient conditions for the convergence of the QAOR method when the coefficient matrices are irreducible diagonal dominant, H-matrices, symmetric positive definite matrices, and Lmatrices.
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