求解大规模矩阵问题的Krylov子空间方法

摘要

Solving large scale matrix problems including large linear systems, eigenvalue problems, etc. , is a vital subject in computational mathematics and scientific engineering computing. There have been important advances in the subject in recent years. A comprehensive survey of some developments including the authors＇ works regarding Krylov subspace methods for solving large linear systems and eigenvalue problems is given. Specific topics include: conjugate gradient algorithm, SYMMLQ algorithm, MINRES algorithm, GMRES algorithm , Lanczos biorthogonalization algorithm, QMR algornrithm and their block versions for solving large linear systems, Lanczos algorithm and its block version for solving large symmetric eigenvalue problems, Lanczos algorithm, Arnoldi algorithm and their block versions for solving large unsymmetric eigenvalue problems. The acceleration techniques and preconditioning techniques for large scale matrix problems are discussed. Some problems that need to be further studied are presented.%求解大规模矩阵问题包括线性方程组和特征值问题等是计算数学和科学工程计算中的重大课题。最近几年，其研究工作取得了许多重大进展。文中给出大型线性方程组和特征值问题Krylov子空间方法若干进展的一个概述，其中包括作者对这些问题的研究成果。涉及的专题包括求解大型线性方程组的共轭梯度法、SYMMLQrn算法、MINRES算法、GMRES算法、Lanczos双正交化算法、QMR算法以及这些算法的块格式；求解大型对称特征值问题的Lanczos算法和块Lanczos算法；求解大型非对称特征值问题的Lanczos算法、Atnoldi算法以及这些算法的块推广。讨论求解大规模矩阵问题的加速技术和预处理技术。提出了一些有待进一步研究的问题。