## 2 A Reduction of Spectral Radius  
Through the descent vector and the \(k\)th step residual vector  
\[\mathbf{u}=\mathbf{x}-\mathbf{x}_{k}, \tag{10}\] \[\mathbf{r}_{k}=\mathbf{b}-\mathbf{A}\mathbf{x}_{k}, \tag{11}\]  
Equation (1) can be written as  
\[\mathbf{A}\mathbf{u}=\mathbf{r}_{k}. \tag{12}\]  
Equation (12) shows that the best descent vector is \(\mathbf{u}=\mathbf{A}^{-1}\mathbf{r}_{k}\). However, if we can find \(\mathbf{A}^{-1}\), the solution of Equation (1) is already given by \(\mathbf{x}=\mathbf{A}^{-1}\mathbf{b}\) exactly. The main difficulty is that finding \(\mathbf{A}^{-1}\) is absolutely not an easy task. **Lemma 1**. : _The \((\mathbf{M},\mathbf{N})\) splitting iterative scheme (5) for solving Equation (1) can be reformulated as_  
\[\mathbf{x}_{k+1}=\mathbf{x}_{k}+\mathbf{u}_{k},\ \ \mathbf{M}\mathbf{u}_{k}= \mathbf{r}_{k}. \tag{13}\]  
Proof. : By means of Equation (5), we have  
\[\mathbf{M}\mathbf{x}_{k+1}=\mathbf{M}\mathbf{x}_{k}+(\mathbf{N}-\mathbf{M}) \mathbf{x}_{k}+\mathbf{b};\]  
it becomes, with the aids of Equations (4) and (11),  
\[\mathbf{M}(\mathbf{x}_{k+1}-\mathbf{x}_{k})=\mathbf{r}_{k}.\]  
We end the proof of Lemma 1 by setting \(\mathbf{u}_{k}=\mathbf{x}_{k+1}-\mathbf{x}_{k}\) in view of Equation (10). We modify Equation (13) and prove the following result. **Theorem 1**. : _Inserting an acceleration parameter \(\alpha\) preceding \(\mathbf{u}_{k}\) in Equation (13), a new splitting iterative scheme for Equation (1) reads as_  
\[\mathbf{x}_{k+1}=\mathbf{x}_{k}+\alpha\mathbf{u}_{k},\ \ \mathbf{M}\mathbf{u}_{k}= \mathbf{r}_{k}, \tag{14}\]  
Under the following condition:  
\[1<\alpha<\frac{1}{1-\rho(\mathbf{G})}, \tag{15}\]  
where \(\mathbf{G}=\mathbf{M}^{-1}\mathbf{N}\) is the iteration matrix for Equation (5), the spectral radius for Equation (14), denoted by \(\rho_{\alpha}\), satisfies. \[\rho_{\alpha}<\rho(\mathbf{G}). \tag{16}\]  
Proof. : When the first one in Equation (14) is multiplied by \(\mathbf{M}\), and Equations (4) and (11), and the second one in Equation (14) are taken into account, it generates  
\[\mathbf{M}\mathbf{x}_{k+1}=\mathbf{M}\mathbf{x}_{k}+\alpha\mathbf{ M}\mathbf{u}_{k}=\mathbf{M}\mathbf{x}_{k}+\alpha\mathbf{r}_{k}\] \[\mathbf{r}_{k}=\mathbf{b}-\mathbf{A}\mathbf{x}_{k}, \tag{11}\]0 \[\mathbf{r}_{k}=\mathbf{b}-\mathbf{A}\mathbf{x}_{k}, \tag{11}\]1  
The associated iteration matrix is  
\[\mathbf{r}_{k}=\mathbf{b}-\mathbf{A}\mathbf{x}_{k}, \tag{11}\]2  
It follows that  
\[\mathbf{r}_{k}=\mathbf{b}-\mathbf{A}\mathbf{x}_{k}, \tag{11}\]3  
If \(\alpha>1\) and \(0<\rho(\mathbf{G})<1\) owing to Equation (6), then  
\[\mathbf{r}_{k}=\mathbf{b}-\mathbf{A}\mathbf{x}_{k}, \tag{11}\]4  
implies  
\[\mathbf{r}_{k}=\mathbf{b}-\mathbf{A}\mathbf{x}_{k}, \tag{11}\]5  
To meet the requirement of  
\[\mathbf{r}_{k}=\mathbf{b}-\mathbf{A}\mathbf{x}_{k}, \tag{11}\]6  
\(\alpha<1/[1-\rho(\mathbf{G})]\) follows readily. Equation (14) can be also derived from Equation (5) after multiplying \(\alpha\) and using a substitution technique:  
\[\mathbf{r}_{k}=\mathbf{b}-\mathbf{A}\mathbf{x}_{k}, \tag{11}\]7 \[\mathbf{r}_{k}=\mathbf{b}-\mathbf{A}\mathbf{x}_{k}, \tag{11}\]8It results in  
\[\mathbf{r}_{k}=\mathbf{b}-\mathbf{A}\mathbf{x}_{k}, \tag{11}\]9 \[\mathbf{A}\mathbf{u}=\mathbf{r}_{k}. \tag{12}\]0 \[\mathbf{A}\mathbf{u}=\mathbf{r}_{k}. \tag{12}\]1 \[\mathbf{A}\mathbf{u}=\mathbf{r}_{k}. \tag{12}\]2 \[\mathbf{A}\mathbf{u}=\mathbf{r}_{k}. \tag{12}\]3  
This is the usual extrapolation method used in the iterative scheme with \(\alpha\) as an extrapolation parameter [33]. The extrapolation technique consists of a multiplication in Equation (19), and an extrapolation in Equation (20), which will be named an extrapolation technique for the \((\mathbf{M},\mathbf{N})\) splitting iterative scheme (5) with the parameter \(\alpha\), or simply, an \(\alpha\)-extrapolation.