## 4 Numerical Experimental Simulation and Analysis
### Algorithm Performance Analysis
#### 4.3.2 Comparative Analysis of Solving Time After Preconditioning Optimization  
Solution efficiency is a crucial metric for assessing the performance of a solver, with the duration of the solving time directly impacting the progress of simulation projects. The most straightforward way to compare solution efficiency is by directly comparing the solving times. For methods that use iterative solvers without preconditioning, the total time consists solely of the matrix iteration solving time. In contrast, for solvers that incorporate preconditioning, the total time includes two components--the preconditioning time and the subsequent matrix solving time. The effectiveness of the preconditioning step depends on whether the sum of the preconditioning time and the matrix solving time can outperform direct solving using an iterative method.  
To evaluate the effectiveness of the G-Pre in accelerating the solution of matrix problems, we compared the total solving time, preconditioning time, and solving time with those of two different cases--no preconditioning and ILU factorization. Solving experiments were performed on the four matrices described in Section 4.1, and the results are presented in Figure 5. The efficiency of matrix solving was significantly improved with preconditioning, and both the overall time and the solving time for G-Pre preconditioning and ILU factorization were considerably reduced compared to the case without preconditioning.  
In the four matrix-solving tests, the efficiency of G-Pre solving improved by 53.5%, 42.1%, 77.6%, and 88.8%, respectively, compared to solving directly with GMRES, and improved by 44.9%, 2.8%, 64.8%, and 79.9%, respectively, compared to solving with ILU factorization. Comparing G-Pre preconditioning and ILU factorization, G-Pre preconditioning exhibited greater speedup, more than doubling the performance for matrices 3 and 4, indicating higher solution efficiency in G-Pre.  
However, when comparing matrices 1 and 2, the solving times were similar for both methods when the matrix size was small. In addition, in the case of Matrix 2, the ILU factorization solving time was significantly shorter than that of G-Pre preconditioning. This discrepancy may be due to the influence of graph partitioning and subsequent parallelism on solving, leading to the lower-than-expected solution efficiency in small-scale matrix computations. Additionally, larger-scale matrix solving experiments demonstrated that  
\begin{table}
\begin{tabular}{c c c c} \hline \hline
**Matrix** & **Original** & **ILU** & **G-Pre** \\ \hline Matrix 1 & 1.2297 \(\times\) 10\({}^{8}\) & 7.8489 \(\times\) 10\({}^{5}\) & 6.5499 \(\times\) 10\({}^{2}\) \\ Matrix 2 & 2.5738 \(\times\) 10\({}^{29}\) & 1.3881 \(\times\) 10\({}^{21}\) & 9.8851 \(\times\) 10\({}^{4}\) \\ Matrix 3 & 4.6577 \(\times\) 10\({}^{43}\) & 3.5248 \(\times\) 10\({}^{32}\) & 1.3415 \(\times\) 10\({}^{15}\) \\ \hline \hline \end{tabular}
\end{table}
Table 2: Comparison of matrix condition numbers.  
G-Pre provides a higher optimization of solution efficiency. In large-scale matrix solving, graph partitioning and rearrangement make the non-zero elements of the matrix more concentrated, improving the cache hit rate and enhancing the parallel processing effect.