本文主要研究了代数多重网格方法在方程组求解中的应用，通过实验得到代数多重网格方法在方程组求解中的收敛速度明显优于一般迭代法的结论。第一章介绍了代数多重网格法的研究背景，阐述了AMG方法的理论研究和应用发展现状；第二章介绍了一般的矩阵预条件子，并引入了代数多重网格方法；第三章构建了AMG方法的基本组件，详细阐述了代数多重网格方法的基本原理和流程；第四章介绍了经典的代数多重网格法的光滑过程、网格粗化算法和插值算子的计算；第五章通过数值算例验证了代数多重网格方法在方程组求解中的快速收敛和较低的计算复杂度。

This paper primarily investigates the application of algebraic multigrid methods in the solution of systems of equations, concluding through experimentation that the convergence rate of algebraic multigrid methods is significantly superior to that of general iterative methods. Chapter 1 introduces the research background of algebraic multigrid methods and elaborates on the theoretical research and current state of applications of AMG methods. Chapter 2 discusses general matrix preconditioners and introduces algebraic multigrid methods. Chapter 3 constructs the basic components of the AMG method, detailing the fundamental principles and processes of algebraic multigrid methods. Chapter 4 presents the smoothing process, grid coarsening algorithms, and the computation of interpolation operators of classic algebraic multigrid methods. Chapter 5 verifies through numerical examples the rapid convergence and lower computational complexity of algebraic multigrid methods in solving systems of equations.