Two Extrapolation Techniques on Splitting Iterative Schemes to Accelerate the Convergence Speed for Solving Linear Systems  
[PERSON]  
1 Center of Excellence for Ocean Engineering, National Taiwan Ocean University, Keelung 202301, Taiwan; [EMAIL_ADDRESS]  
[PERSON]  
2 School of Mathematics and  
The best descent vector \(\mathbf{u}\in\mathcal{K}_{m}\) is sought to minimize the residual [3]:  
\[\min\|\mathbf{r}\|,\]  
where  
\[\mathbf{r}=\mathbf{b}-\mathbf{A}\mathbf{x}=\mathbf{r}_{0}-\mathbf{A}\mathbf{u}\]  
is the residual vector, whose Euclidean norm is denoted by \(\|\mathbf{r}\|\). Equation (3) is one of the Petrov-Galerkin method to search \(\mathbf{u}\in[PERSON]mathcal{K}_{m}\). The perpendicularity is a vital guidance for developing effective iterative algorithms, which was [PERSON] in [7, 8] to enhance the stability of GMRES and SOR. [PERSON] and [PERSON] [9] introduced the Krylov subspace method as the top ten algorithms. [PERSON] [10] demonstrated the motivations and realizations of Krylov subspace methods for large sparse linear systems. Recently, [PERSON] et al. [11] provided a unified approach to the Krylov subspace methods for solving linear systems. Suppose that \(\mathbf{A}\) is decomposed by  
\[\mathbf{A}=\mathbf{M}-\mathbf{N}, \tag{4}\]  
where \(\mathbf{M}\) is a nonsingular matrix. The \((\mathbf{M},\mathbf{N})\) splitting iterative scheme for Equation (1) is [12]  
[PERSON]mathbf{M}\mathbf{x}_{k+1}=\mathbf{N}\mathbf{x}_{k}+\mathbf{b}, \tag{5}\]  
where \(\mathbf{x}_{k}\) is the \(k\)th step value of \(\mathbf{x}\). The convergence of the iteration in Equation (5) is guaranteed if  
\[\rho=\rho(\mathbf{G})<1, \tag{6}\]  
where \(\mathbf{G}=\mathbf{M}^{-1}\mathbf{N}\) is the iteration matrix, and \(\rho\) is the spectral radius of \(\mathbf{G}\). The Gaussian elimination method is a classical exact method, but it is not suitable for use in either scientific or engineering practice. The Gauss-Seidel method is a semi-exact method by finding the exact values of variables at each iteration step by a forward-substitution technique. Without needing extra parameter is its main advantage; however, it is convergent slowly. Many methods are special cases of the \((\mathbf{M},\mathbf{N})\) splitting iterative scheme, namely the Jacobi method, the Gauss-Seidel method, the successive overrelaxation (SOR) method, and the accelerated overrelaxation (AOR) method. The SOR method was developed in [13]. For multi-parameter splitting methods, the two-parameter AOR method [14]:  
\[(\mathbf{D}-w\mathbf{L})\mathbf{x}_{k+1}=r\mathbf{b}+[(1-r)\mathbf{D}+r \mathbf{U}+(r-w)\mathbf{L}]\mathbf{x}_{k}, \tag{7}\]  
is a generalization of SOR, which is equipped with a relaxation parameter \(w\) and an acceleration parameter \(r\) for controlling the convergence behavior. The \((\mathbf{M},\mathbf{N})\) splitting for Equation (7) is  
\[\mathbf{A}=\mathbf{M}-\mathbf{N}=\mathbf{D}-\mathbf{U}-\mathbf{L}, \tag{8}\]  
where  
\[\mathbf{M}=\frac{1}{r}(\mathbf{D}-w\mathbf{L}),\ \mathbf{N}=\frac{1}{r}[(1-r) \mathbf{D}+r\mathbf{U}+(r-w)\mathbf{L}].\]  
If \(r=w\), the SOR iterative method  
\[(\mathbf{D}-w\mathbf{L})\mathbf{x}_{k+1}=w\mathbf{b}+(1-w)\mathbf{D}\mathbf{ x}_{k}+w\mathbf{U}\mathbf{x}_{k} \tag{9}\]  
is recovered from Equation (7). The generalizations of AOR method for different linear problems can refer to [15; 16; 17]; the convergence behaviors were analyzed in [18; 19]. To improve the convergence speed the preconditioned AOR methods were developed. Different kinds of preconditioners have been examined to improve the speed of convergence of the iterative methods. A growing interest is to study the preconditioners to ameliorate the speed of convergence of iterative methods in a lot of literature [20; 21; 22; 23; 24; 25; 26; 27]. [PERSON] and his co-workers [28; 29] reformulated SOR and AOR in terms of descent and residual vectors. The reaccelerated version of AOR was discussed in [30], and the generalization to the three-parameter method of AOR was carried out in [31].