## 2 Methods
### Matrix Solving Algorithm  
Many solution methods have been developed for matrix solving problems. Among them, iterative methods are more suitable for large-scale matrix solving problems due to the advantages of low storage requirements and the easy implementation of algorithms. Commonly used iterative methods for solving symmetric matrices include the CG method [46], the Jacobi iteration method [30], the Gauss-Seidel iteration method [47], and the successive over-relaxation (SOR) iteration method [31]. The Jacobi iteration method has the advantage of simplicity and ease of implementation, but its convergence speed is slower, and it is only suitable for the case of the diagonal dominance of the matrices. The Gauss-Seidel iteration method usually converges faster than the Jacobi iteration method, but it requires more time and computation to implement the algorithm. The SOR method requires the selection of a suitable relaxation factor \(w\) in order to effectively improve the convergence speed. The conjugate gradient method is an efficient stepwise approximation method that only needs to utilize the first-order derivative information and does not need to compute and store the Hesse matrix and its inverse matrix, thus reducing the high computational cost of [PERSON]'s method. At the same time, it overcomes the disadvantage of the slow convergence of the most rapid descent method, which is a very effective approach to solve large sparse symmetric matrix solution problems [20]. In asymmetric matrix solving problems, the most commonly used methods include the QR decomposition method [48], GMRES [21,22], and the biconjugate gradient stabilized (BiCGStab) method [49]. Among these, the former has a high computational complexity, usually \(O\left(n^{3}\right)\), and a low computational efficiency, which is not applicable for solving large-scale matrices. Although the stabilized dual conjugate gradient method converges faster, it is prone to the phenomenon of iterative oscillation and abnormal residual values, relying less on the preconditioning technology, which cannot effectively reflect the practical effect of the preconditioning algorithm proposed in this paper. As a mature iterative method, the generalized minimum residual method has better applicability and convergence, and the iterative process is more stable. Therefore, in this test study, CG and GMRES were used in order to fully verify the effectiveness of the preconditioning algorithm G-Pre that is proposed in this study. GMRES is an iterative technique for solving large sparse asymmetric linear systems of the form \(Au=b\). The core idea is to find an approximate solution that minimizes the residual \(r=\|b-Au\|\) by constructing the Krylov subspace \(K_{m}(A,r_{0})=span\Big{\{}r_{0},Ar_{0},A^{2}r_{0},\ldots A^{m-1}r_{0}\Big{\}}\) within this subspace [21,22]. The flow of the GMRES algorithm is as follows. It starts with an initial estimate of the solution \(u_{0}\) and computes the following initial residual:  
\[r_{0}=b-Au_{0}. \tag{4}\]  
Let \(\beta=\left\lvert\left\lvert r_{0}\right\rvert\right\rvert\) and set the basis vector as follows:  
\[v_{1}=r_{0}/\beta. \tag{5}\]  
The Arnold process is performed to construct orthogonal bases. For \(j=1,2,\ldots,m\), the following is computed:  
\[w\;=\;Av_{j}. \tag{6}\]  
Orthogonalizing \(w\) with all previously generated basis vectors \(v_{1}\), \(\ldots\), \(v_{j}\) using the modified Gram-Schmidt process, we obtain the following:  
\[h_{i,j}=v_{i}^{j}w, \tag{7}\]  
\[w=w-h_{i,j}v_{i},\;(i=1,2,\ldots j). \tag{8}\]  
Computing \(h_{j+1,j}=\left\lVert w\right\rVert\) and normalizing, we obtain the following new basis vector:  
\[v_{j+1}=w/h_{j+1,j}. \tag{9}\]  
Then, we construct the upper Hessenberg matrix \(H_{m}\in R^{(m+1)\times m}\) (consisting of \(h_{ij}\)) and minimize the residuals by solving the following least squares problem:  
\[y_{m}=\underset{y}{\arg\min}\|\beta e_{1}-H_{m}y\|,\infty, \tag{10}\]  
where \(e_{1}=[1,0,\ldots,0]^{\mathrm{T}}\in\mathbb{R}^{m+1}\).