A New Version of the Accelerated Overrelaxation Iterative Method

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, and L-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.


Introduction
Consider the following linear system: where  ∈ R × ,  ∈ R  are given and  ∈ R  is unknown.System of form (1) appears in many applications such as linear elasticity, fluid dynamics, and constrained quadratic programming [1][2][3][4].When the coefficient matrix of the linear system (1) is large and sparse, iterative methods are recommended against direct methods.In order to solve (1) more effectively by using the iterative methods, usually, efficient splittings of the coefficient matrix  are required.For example, the classical Jacobi and Gauss-Seidel iterations are obtained by splitting the matrix  into its diagonal and offdiagonal parts.
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.In certain cases the AOR method has better convergence rate than Jacobi, JOR, Gauss-Seidel, or SOR method [5,6].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].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].Although Krylov subspace methods [4,22] are considered as one kind of the important and efficient iterative techniques for solving the large sparse linear systems because these methods are cheap to be implemented and are able to fully exploit the sparsity of the coefficient matrix, Krylov subspace methods are very slow or even fail to converge when the coefficient matrix of (1) is often extremely ill-conditioned and highly indefinite.
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, -matrices, symmetric positive definite matrices, and matrices.
The remainder of the paper is organized as follows.In Section 2 the QAOR method is derived.In Section 3, some convergence results are given for the QAOR method when the coefficient matrices are irreducible diagonal dominant, -matrices, symmetric positive definite matrices, and matrices.A relational graph for QAOR and AOR is presented

The QAOR Method
To introduce the QAOR method, firstly, a brief review of the classical AOR method is required.
For any splitting,  =  −  with det() ̸ = 0, the basic iterative method for solving (1) is Let where  is a nonsingular diagonal matrix and   and   are strictly lower and upper triangular matrices, respectively.Then the classical AOR method in [5] is defined: where  is an acceleration parameter and  is an overrelaxation parameter.Its iterative matrix is where  =  −1   and  =  −1   .Obviously, the iterative matrix of the Jacobi method is  0,1 , the iterative matrix of the Gauss-Seidel method is  1,1 , and the iterative matrix of the successive overrelaxation (SOR) method is  , .
In fact, if we introduce matrices then Therefore, one can readily verify that the AOR method can be induced by the matrix splitting  = (1/)( 1 −  1 ).
To establish the QAOR method, we consider the following matrix splitting of the coefficient matrix ; that is to say, Then Based on the above matrix splitting (8), the QAOR method is defined as follows: and its iterative matrix is Comparing the QAOR method with the AOR method, it is easy to see that the iteration matrix of the QAOR method is similar to that of the AOR method.Based on this fact, the QAOR method may conserve all the advantages of the AOR method.If  = , the QAOR reduces to the QSOR method.The QSOR method is called the KSOR method as well [23,24].
Next, we will discuss some sufficient conditions for the convergence of the QAOR method when the coefficient matrices are irreducible diagonal dominant, -matrices, symmetric positive definite matrices, and -matrices.

Main Results
When  is an irreducible matrix with weak diagonal dominance, obviously, both the coefficient matrix  and the corresponding diagonal matrix  are nonsingular.Based on this case, we have the following theorem for the QAOR method.
Theorem 1.If  is an irreducible matrix with weak diagonal dominance, then the QAOR method converges for all −1 ≤  ≤ 1 and  > 0.
Proof.We assume that for the eigenvalue  of  , we have || ≥ 1.For this eigenvalue the relationship below holds: By performing a simple series of transformations, we have where The coefficients of  and  in ( 14) are less than one in modulus.To prove this it is sufficient and necessary to prove that If  −1 =   where  and  are real with 0 <  ≤ 1, then the first inequality in ( 15) is equivalent to which holds for  = −1 (in this case, obviously, (1 −  2 )(2 + ) ≥ 0).Since (1 −  2 ) + (1 + ) ≥ 0, (16) holds for all real  if and only if it holds for cos  = 1.Thus, ( 16) is equivalent to which is true.The second inequality in ( 15) is equivalent to which, for the same reason, must be satisfied for cos  = 1.Thus, we have which is also true.That is, for all −1 ≤  ≤ 1 and  > 0,  is nonsingular which contradicts with det() = 0. Therefore, ( , ) < 1.

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When  =  −   −   is symmetric positive definite, obviously,  is nonsingular.It is easy to see that In this case, the QAOR method converges if  =  +   −  is positive definite [2].By the simple computations, we have That is to say, the QAOR method converges if then the QAOR method converges.Therefore, we have the following theorem.
Further, we have the following theorem.
Some remarks on (43) are given as follows.