  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.
