The article mainly introduces preprocessing algorithms for solving linear equation systems. This algorithm uses three algorithms as inner iterations, namely RPCG algorithm, ADI algorithm, and Kaczmarz algorithm. Then, it uses BA-GMRES as an outer iteration to solve the linear equation system. These three algorithms can indirectly generate preprocessing matrices, which are used for solving equation systems. In addition, we provide corresponding convergence analysis and numerical examples. Through numerical examples, we demonstrate the effectiveness and feasibility of these preprocessing methods. Furthermore, in the Kaczmarz algorithm, we introduce both constant step size and adaptive step size, and extend the parameter range of the Kaczmarz algorithm to $\alpha\in(0,\infty)$. We also study the solution rate of linear equation systems using different step sizes. Numerical examples show that both constant step size and adaptive step size have higher solution efficiency than the solving algorithm without preprocessing.

翻译：本文主要介绍求解线性方程组的预处理算法。该算法采用RPCG算法、ADI算法和Kaczmarz算法三种算法作为内迭代，然后使用BA-GMRES作为外迭代求解线性方程组。这三种算法可间接生成用于方程组求解的预处理矩阵。此外，我们提供了相应的收敛性分析与数值算例。通过数值算例验证了这些预处理方法的有效性与可行性。同时在Kaczmarz算法中引入了固定步长与自适应步长两种模式，并将Kaczmarz算法的参数范围扩展至$\alpha\in(0,\infty)$。我们还研究了不同步长下线性方程组的求解速率。数值算例表明，无论是固定步长还是自适应步长，其求解效率均高于无预处理的求解算法。