## 5 Reaccelerated over Relaxation Method  
[PERSON] et al. [30] proposed a reaccelerated over relaxation (ROR) method as follows:  
\[(\mathbf{D}-w\mathbf{L})\mathbf{x}_{k+1}=r\mathbf{b}+[(1-r)\mathbf{D}+r \mathbf{U}+(r-w)\mathbf{L}]\mathbf{x}_{k}+rw(\mathbf{Ax}_{k}-\mathbf{b}). \tag{41}\]  
**Theorem 7**. : _A new iterative form of the ROR method is_  
\[\mathbf{x}_{k+1}=\mathbf{x}_{k}+\mathbf{u}_{k},\ (\mathbf{D}-w\mathbf{L}) \mathbf{u}_{k}=r(1-w)\mathbf{r}_{k}. \tag{42}\]  
Proof. : With the aid of Equation (11), we rewrite Equation (41) to  
\[(\mathbf{D}-w\mathbf{L})\mathbf{x}_{k+1}=(\mathbf{D}-w\mathbf{L})\mathbf{x}_ {k}+r\mathbf{b}+[(1-r)\mathbf{D}+r\mathbf{U}+(r-w)\mathbf{L}]\mathbf{x}_{k}-( \mathbf{D}-w\mathbf{L})\mathbf{x}_{k}-rw\mathbf{r}_{k},\]  
and then to  
\[(\mathbf{D}-w\mathbf{L})\mathbf{x}_{k+1}=(\mathbf{D}-w\mathbf{L})\mathbf{x}_ {k}+r\mathbf{b}-r[\mathbf{D}-\mathbf{U}-\mathbf{L}]\mathbf{x}_{k}-rw\mathbf{r }_{k}.\]  
It follows from Equations (8) and (11) that  
\[(\mathbf{D}-w\mathbf{L})\mathbf{x}_{k+1}=(\mathbf{D}-w\mathbf{L}) \mathbf{x}_{k}+r\mathbf{b}-r\mathbf{Ax}_{k}-rw\mathbf{r}_{k}\] \[=(\mathbf{D}-w\mathbf{L})\mathbf{x}_{k}+r\mathbf{r}_{k}-rw\mathbf{ r}_{k},\]and that  
\[(\mathbf{D}-w\mathbf{L})(\mathbf{x}_{k+1}-\mathbf{x}_{k})=r\mathbf{r}_{k}-rw \mathbf{r}_{k};\]  
Upon using \(\mathbf{u}_{k}=\mathbf{x}_{k+1}-\mathbf{x}_{k}\), Equation (42) is proven. Theorem 4.1: _The AOR method can be recast to_  
\[\mathbf{x}_{k+1}=\mathbf{x}_{k}+\mathbf{u}_{k},\ (\mathbf{D}-w\mathbf{L}) \mathbf{u}_{k}=r\mathbf{r}_{k}. \tag{43}\]  
_The ROR method is obtained from the AOR method upon applying the extrapolation technique (19) and (20) by a parameter \(\beta=1-w\) on Equation (43); it generates the following iterative form:_  
\[\mathbf{x}_{k+1}=\mathbf{x}_{k}+(1-w)\mathbf{u}_{k},\ (\mathbf{D}-w\mathbf{L}) \mathbf{u}_{k}=r\mathbf{r}_{k}. \tag{44}\]  
Proof. : Equation (43) is obtained from Equation (42) by deleting \(rw\mathbf{r}_{k}\) in the right-hand side of the second equation. Applying the extrapolation technique (19) and (20) on Equation (43) leads to  
\[\beta\mathbf{x}_{k+1}=\beta\mathbf{x}_{k}+\beta\mathbf{u}_{k},\] \[\mathbf{x}_{k+1}=\mathbf{x}_{k}+\mathbf{u}_{k},\ (\mathbf{D}-w\mathbf{L}) \mathbf{u}_{k}=r(1-w)\mathbf{r}_{k}. \tag{42}\]0 \[\mathbf{x}_{k+1}=\mathbf{x}_{k}+\mathbf{u}_{k},\ (\mathbf{D}-w\mathbf{L}) \mathbf{u}_{k}=r(1-w)\mathbf{r}_{k}. \tag{42}\]1  
which is the first one in Equation (44), if we take \(\beta=1-w\). In Equation (42), let  
\[\mathbf{x}_{k+1}=\mathbf{x}_{k}+\mathbf{u}_{k},\ (\mathbf{D}-w\mathbf{L}) \mathbf{u}_{k}=r(1-w)\mathbf{r}_{k}. \tag{42}\]2  
we have  
\[\mathbf{x}_{k+1}=\mathbf{x}_{k}+\mathbf{u}_{k},\ (\mathbf{D}-w\mathbf{L}) \mathbf{u}_{k}=r(1-w)\mathbf{r}_{k}. \tag{42}\]3  
Comparing Equation (45), with \(\mathbf{w}_{k}\) replaced by \(\mathbf{u}_{k}\), to Equation (44), they are the same. Theorem 4.2: _Given an initial value \(\mathbf{x}_{0}\), \(\mathbf{r}_{0}=\mathbf{b}-\mathbf{A}\mathbf{x}_{0}\) is known. For the ROR method in Equation (42) if \(w\) is given, then_  
\[\mathbf{x}_{k+1}=\mathbf{x}_{k}+\mathbf{u}_{k},\ (\mathbf{D}-w\mathbf{L}) \mathbf{u}_{k}=r(1-w)\mathbf{r}_{k}. \tag{42}\]4  
_is the optimal value of \(r\), where \(\mathbf{v}_{0}\) is obtained from a forward solution of_  
\[\mathbf{x}_{k+1}=\mathbf{x}_{k}+\mathbf{u}_{k},\ (\mathbf{D}-w\mathbf{L}) \mathbf{u}_{k}=r(1-w)\mathbf{r}_{k}. \tag{42}\]5  
Proof. : Letting  
\[\mathbf{x}_{k+1}=\mathbf{x}_{k}+\mathbf{u}_{k},\ (\mathbf{D}-w\mathbf{L}) \mathbf{u}_{k}=r(1-w)\mathbf{r}_{k}. \tag{42}\]6  
Equation (47) is a direct result of the second one in Equation (42) with \(k=0\). The first one in Equation (42) with \(k=0\) is  
\[\mathbf{x}_{k+1}=\mathbf{x}_{k}+\mathbf{u}_{k},\ (\mathbf{D}-w\mathbf{L}) \mathbf{u}_{k}=r(1-w)\mathbf{r}_{k}. \tag{42}\]7  
Other processes are similar to that in Theorem 4.2. Equation (46) is a simple equation for \(r\), because \(\mathbf{v}_{0}=(1-w)(\mathbf{D}-w\mathbf{L})^{-1}\mathbf{r}_{0}\) is not a function of \(r\).