Convergence of the GAOR Method for One Subclass ofH-Matrix

We discuss the convergence of the GAORmethod to solve linear systemwhich occurred in solving the weighted linear least squares problem. Moreover, we present one convergence theorem of the GAOR method when the coefficient matrix is a strictly doubly α diagonally dominant matrix which is a nonsingular H-matrix. Finally, we show that our results are better than previous ones by using four numerical examples.


Introduction
Consider the weighted linear least squares problem min ∈  ( − )   −1 ( − ) , where  is the variance-covariance matrix.The problem has many scientific applications.A typical source is parameter estimation in mathematical modeling.
In order to solve it, we have to solve the following linear system: where is invertible.For example, in the variance-covariance matrix [1], a generalized SOR (GSOR) method to solve linear system (2) was proposed in [2], afterwards, a generalized AOR (GAOR) method to solve linear system (2) was established in [3] as follows: where In [4][5][6], authors studied the convergence of the GAOR method for solving the linear system  = .In [4,5], authors studied the convergence of the GAOR method for diagonally dominant coefficient matrices and gave the regions of convergence.In [6], authors studied the convergence of the GAOR method for strictly doubly diagonally dominant coefficient matrices and gave the regions of convergence.In [7], authors studied the preconditioned generalized AOR method for solving linear systems.They proposed two kinds of preconditioning that each one contains three preconditioners.They showed that the convergence rate of the preconditioned generalized AOR methods is better than that of the original method, whenever the original method is convergent.In [8], authors presented three kinds of preconditioners for preconditioned modified AOR method to solve systems of linear equations.They showed that the convergence rate of the preconditioned modified AOR method is better than that of the original method, whenever the original method is convergent.
Sometimes, the coefficient matrices of linear systems  =  are not strictly diagonally dominant or strictly doubly diagonally dominant.In this paper, we will discuss the convergence of the GAOR method when the coefficient matrices are strictly doubly  diagonally dominant.
Throughout this paper, we denote the -row sums of the modulus of the entries of  and  by   and   , the -column sums of the modulus of the entries of  and  by    and    , respectively.And we denote the spectral radius of iterative matrix  , by ( , ): Definition 1 (see [9]).
Obviously, a strictly doubly diagonally dominant matrix is a strictly doubly  diagonally dominant matrix, but not vice versa.
For example,
In this paper, we study the convergence of the GAOR method for solving the linear system  =  for strictly doubly  diagonally dominant coefficient matrices.Firstly, we obtain upper bound for the spectral radius of the matrix  , which is the iterative matrix of the GAOR iterative method.Moreover, we present one convergence theorem of the GAOR method.Finally, we present four numerical examples.

Upper Bound of the Spectral Radius of 𝐿 𝜔,𝑟
In this section, we obtain upper bound of the spectral radius of the iterative matrix  , .Theorem 3. Let  ∈ (), then ( , ) satisfies the following inequality: Proof.Let  be an arbitrary eigenvalue of the iterative matrix  , , then that is, From Lemma 2, we know that then  is not an eigenvalue of the iterative matrix  , .
However,  is an eigenvalue of the iterative matrix  , , then there exists at least a couple of ,  ∈  ( ̸ = ), such that That is, It is easy to find that the discriminant of a curve of second order then the solution of (18) satisfies So, ( , ) satisfies the following inequality: (21)

Convergence of the GAOR Method
In this section, we investigate the convergence of the GAOR method to solve linear system (2).We assume that  is a strictly doubly  diagonally dominant coefficient matrix and obtain the regions of convergence of the GAOR method.
then the GAOR method converge if ,  satisfy either where That is, Firstly, when 0 <  ≤ 1, we have That is, Then, we have the following conditions.

Examples
In this section, we give four numerical examples to show that our results are better than previous ones. where It is easy to know that  ∈ (1/2) and  = ( ( ) , So By Theorem 4, we obtain the following regions of convergence: From Figure 1, we know that the regions of convergence got by Theorem 4 in this paper (bounded by blue lines) are larger than ones got by Theorem 2 of paper [6] (bounded by green lines).From Example 1 of paper [6], we know that the regions of convergence got by Theorem 2 of paper [6] are larger than ones of paper [4] and paper [5].So, the regions of convergence got by Theorem 4 in this paper are larger than ones of paper [4][5][6].