解大规模非对称线性方程组的Lanczos方法和精化Lanczos方法  

A LANCZOS METHOD AND A REFINED LANCZOS METHOD FOR LARGE UNSYMMETRIC LINEAR SYSTEMS

作　　者：贾仲孝[1] 李焱淼[2] 

机构地区：[1]清华大学数学科学系,100084 [2]大连理工大学应用数学系,116024

出　　处：《数值计算与计算机应用》2004年第1期48-59,共12页Journal on Numerical Methods and Computer Applications

基　　金：国家重点基础研究专项基金(G1999032805)

摘　　要：A large unsymmetric linear system problem is transformed into the problem of computing the eigenvector of a large symmetric nonnegative definite matrix associated with the eigenvalue zero, i.e., the computation of the elgenvector of the cross-product matrix of an augmented matrix associated with the eigenvalue zero. The standard Lanczos method and an improved refined Lanczos method are proposed that compute approximate eigenvectors and return approximate solutions of the linear system. An implicitly restarted Lanczos algorithm and its refined version are developed. Theoretical analysis and numerical experiments show the refined method is better than the standard one. If the large matrix has small eigenvalues, the two new algorithms are much faster than the unpreconditioned restarted GMRES.A large unsymmetric linear system problem is transformed into the problem of computing the eigenvector of a large symmetric nonnegative definite matrix associated with the eigenvalue zero, i.e., the computation of the eigenvector of the cross-product matrix of an augmented matrix associated with the eigenvalue zero. The standard Lanczos method and an improved refined Lanczos method are proposed that compute approximate eigenvectors and return approximate solutions of the linear system. An implicitly restarted Lanczos algorithm and its refined version are developed. Theoretical analysis and numerical experiments show the refined method is better than the standard one. If the large matrix has small eigenvalues, the two new algorithms are much faster than the unpreconditioned restarted GMRES.

关 键 词：非对称线性方程组 Lanczos法 增广矩阵 奇异向量 特征值 数值计算 

分 类 号：O241[理学—计算数学]