## 1 Introduction
The preconditioning technique is crucial for the successful application of the iterative method in real-world problems and has become one of the research hotspots for the efficient solution of sparse linear systems of equations [27]. There are various methods for constructing preprocessors, including incomplete LU (ILU) factorization preconditioning [28], sparse approximate inverse (SPAI) [29], polynomial preconditioning, classical iteration-based Jacobi [30], and symmetric successive over-relaxation (SSOR) [31] preconditioning. Among these methods, ILU factorization preconditioning has the advantages of low construction costs and a wide range of applications, and it can be adjusted by filling the parameters; however, it also encounters problems, whereby some indeterminate matrices cannot be applied and the storage computation is difficult to predict. The advantages of approximate inverse preconditioning are its good convergence and parallelism, as well as its ease of applicability; however, the construction and storage costs are too high and are not suitable for large-scale matrices. The advantages of polynomial preconditioning are that it is simple and easy to implement, and no matrix factorization is needed; however, the preconditioning effect is general and the convergence speed is slow. The advantages of the Jacobi method are that it is simple and easy to implement, but its convergence speed is slow, its preconditioning effect is limited, and it is only applicable in the case of the diagonal dominance of matrices. The advantages of the SSOR method are that it accelerates the convergence through the introduction of the relaxation factor w and that the iteration process is more stable, but it introduces additional computational effort, which is not suitable in the case of large-scale matrices. However, it introduces additional computational effort, which may lead to longer computation times when encountering large-scale matrix computation and is not suitable for asymmetric matrices. In practice, ILU factorization has become one of the most widely used preconditioning methods due to its good balance and scalability, wide range of applications, and mature technology development [32]. ILU factorization is a commonly used preconditioning technique, developed on the basis of standard LU factorization. The method reduces the computation and storage space by retaining only some non-zero elements in the factorization process, before obtaining the product of the lower triangular matrix L and the upper triangular matrix U in order to approximate the original matrix A. ILU factorization has a good preconditioning effect for both symmetric and asymmetric matrices. In practice, by discarding some elements, ILU factorization effectively weakens the fill-in phenomenon in the factorization process, further reducing the use of storage space. Among them, the commonly used ILU factorization has various forms. One of the standard and widely adopted padding selection methods is ILU (0) [33], which fills in zeros during the factorization process. Krylov subspace iterative solvers preprocessed with ILU (0) are extensively used across various engineering disciplines. Other commonly utilized padding selection methods include ILUT and ILUC [34; 35]. Additionally, ILUB, which leverages single instruction-multiple data (SIMD) instructions for enhanced performance, has been proposed [36]. However, in practice, conventional ILU factorization is difficult to parallelize and does not meet the demands of modern large-scale computation. The ILU factorization method with increased padding levels may change in accordance with the padding level, which, in turn, leads to a large change in the amount of storage and operations, making it difficult to control the factorization effectively. If zero padding is used, the element rounding of the matrix will also be further increased, which, in turn, amplifies the error of the factorization matrix.