## 3 Proposed Scheme
### Overview of Iteration Method
#### 3.1.3 Chebyshev Acceleration  
Chebyshev acceleration is also called the Chebyshev semi-iterative method, which can be regarded as an extension of the extrapolation method [53]. It uses a feasible iteration scheme to provide an initial state to the recurrence relation of Chebyshev polynomials in order to derive coefficients adapted to each iteration, which helps to achieve accelerated convergence. Here, we consider the following iterative equation: \(\mathbf{x}^{(k)}=\mathbf{G}\mathbf{x}^{(k-1)}+\mathbf{d}\), where \(\mathbf{G}\) is the iteration matrix. Assume that the equation converges and that each iteration result is \(\mathbf{x}^{(0)}\), \(\mathbf{x}^{(1)}\), \(\mathbf{x}^{(2)}\), \(\ldots\), \(\mathbf{x}^{(k)}\), \(\ldots\), \(\mathbf{x}^{*}\); in other words, \(\mathbf{x}^{(k)}\) is the result of the \(kth\) iteration and \(\mathbf{x}^{*}\) is the final converged product. Further, the error of the \(kth\) iteration can be defined as follows:  
\[\epsilon^{(k)}=\mathbf{x}^{(k)}-\mathbf{x}^{*}. \tag{24}\]  
In a more advanced form, the iteration relationship among \(\{\epsilon^{(k)}\}\) can be rewritten as follows:  
\[\epsilon^{(k)}=\mathbf{G}\epsilon^{(k-1)}=\mathbf{G}^{2}\epsilon^{(k-2)}= \cdots=\mathbf{G}^{k}\epsilon^{(0)} \tag{25}\]  
where \(\mathbf{G}\) is the iteration matrix. We can find a more accurate result \(\mathbf{\tilde{x}}^{(k)}\) based on the known iteration set \(\{\mathbf{x}^{(0)},\ \mathbf{x}^{(1)},\ \mathbf{x}^{(2)},\ \ldots,\ \mathbf{x}^{(k)}\}\). Then, \(\mathbf{\tilde{x}}^{(k)}\) can be expressed in linear combination form as follows:  
\[\mathbf{\tilde{x}}^{(k)}=\sum\limits_{i=0}^{k}a_{i}\cdot\mathbf{x}^{(i)} \tag{26}\]  
where \(\{a_{i}\}\) is an undetermined coefficient set that plays a weighting role and needs to satisfy the restriction of \(\sum\limits_{i=0}^{k}a_{i}=1\)[52]; put another way, \(\mathbf{\tilde{x}}^{(k)}\) is the weighted average of \(\mathbf{x}^{(i)}\). Next, we use the distance between \(\mathbf{\tilde{x}}^{(k)}\) and \(\mathbf{x}^{*}\) to define the estimated error \(\tilde{\epsilon}^{(k)}\), as follows:  
\[\tilde{\epsilon}^{(k)}=\mathbf{\tilde{x}}^{(k)}-\mathbf{x}^{*}=\sum\limits_{i =0}^{k}a_{i}(\mathbf{x}^{(i)}-\mathbf{x}^{*})=\sum\limits_{i=0}^{k}a_{i} \mathbf{G}^{i}\tilde{\epsilon}^{(0)}\triangleq p_{k}(\mathbf{G})\epsilon^{(0)} \tag{27}\]  
where \(p_{k}(\mathbf{G})=\sum\limits_{i=0}^{k}a_{i}\mathbf{G}^{i}\) is a polynomial with degree \(k\). From Equation (27), we can obtain the following inequality:\[\left\|\mathbf{\tilde{x}}^{(k)}-\mathbf{x}^{*}\right\|_{2}=\left\|p_{k}(\mathbf{G}) \varepsilon^{(0)}\right\|_{2}\leq\left\|p_{k}(\mathbf{G})\right\|_{2}\left\| \varepsilon^{(0)}\right\|_{2}. \tag{28}\]  
Therefore, we can find that the upper bound of the estimated error \(\tilde{\varepsilon}^{(k)}\) is \(\left\|p_{k}(\mathbf{G})\right\|_{2}\left\|\varepsilon^{(0)}\right\|_{2}\). In other words, the estimated error is eliminated when \(\left\|p_{k}(\mathbf{G})\right\|_{2}\left\|\varepsilon^{(0)}\right\|_{2}\) equals zero. To achieve fast convergence, \(\left\|p_{k}(\mathbf{G})\right\|_{2}\) should be as small as possible [52]. Thus, the following minimization problem is derived:  
\[\min_{ \begin{subarray}{c}p_{k}\in\,P_{k}\\ p_{k}(\mathbf{I})=\mathbf{I}\end{subarray} }\left\|p_{k}(\mathbf{G})\right\|_{2} \tag{29}\]  
where \(P_{k}\) represents the set of all \(k\)-degree polynomials. Moreover, we assume that the iteration matrix \(\mathbf{G}\) is symmetric and that Equation (29) is derived as  
\[\min_{ \begin{subarray}{c}p_{k}\in\,P_{k}\\ p_{k}(\mathbf{I})=\mathbf{I}\end{subarray} }\left\|p_{k}(\mathbf{G})\right\|_{2 }=\min_{ \begin{subarray}{c}p_{k}\in\,P_{k}\\ p_{k}(\mathbf{I})=\mathbf{I}\end{subarray} }\max_{\lambda\in\left[\lambda_{ \text{min}}\lambda_{\text{max}}\right]}|p_{k}(\lambda)|, \tag{30}\]  
in which \(\lambda_{\text{min}}\) and \(\lambda_{\text{max}}\) are the minimum and maximum of the eigenvalues, respectively. In addition, the definition of \(\sigma(\mathbf{G})\) is the same as in Equation (15).