Inner iterations in the shift-invert residual Arnoldi method and the Jacobi-Davidson method  

作　　者：JIA ZhongXiao LI Cen 

机构地区：[1]Department of Mathematical Sciences, Tsinghua University

出　　处：《Science China Mathematics》2014年第8期1733-1752,共20页中国科学：数学（英文版）

基　　金：supported by National Basic Research Program of China(Grant No.2011CB302400);National Natural Science Foundation of China(Grant No.11071140)

摘　　要：We establish a general convergence theory of the Shift-Invert Residual Arnoldi(SIRA)method for computing a simple eigenvalue nearest to a given targetσand the associated eigenvector.In SIRA,a subspace expansion vector at each step is obtained by solving a certain inner linear system.We prove that the inexact SIRA method mimics the exact SIRA well,i.e.,the former uses almost the same outer iterations to achieve the convergence as the latter does if all the inner linear systems are iteratively solved with low or modest accuracy during outer iterations.Based on the theory,we design practical stopping criteria for inner solves.Our analysis is on one step expansion of subspace and the approach applies to the Jacobi-Davidson(JD)method with the fixed targetσas well,and a similar general convergence theory is obtained for it.Numerical experiments confirm our theory and demonstrate that the inexact SIRA and JD are similarly effective and are considerably superior to the inexact SIA.We establish a general convergence theory of the Shift-Invert Residual Arnoldi（SIRA） method for computing a simple eigenvalue nearest to a given target σ and the associated eigenvector. In SIRA, a subspace expansion vector at each step is obtained by solving a certain inner linear system. We prove that the inexact SIRA method mimics the exact SIRA well, i.e., the former uses almost the same outer iterations to achieve the convergence as the latter does if all the inner linear systems are iteratively solved with low or modest accuracy during outer iterations. Based on the theory, we design practical stopping criteria for inner solves. Our analysis is on one step expansion of subspace and the approach applies to the Jacobi-Davidson（JD） method with the fixed target σ as well, and a similar general convergence theory is obtained for it. Numerical experiments confirm our theory and demonstrate that the inexact SIRA and JD are similarly effective and are considerably superior to the inexact SIA.

关 键 词：subspace expansion expansion vector inexact low or modest accuracy the SIRA method the JD method inner iteration outer iteration 

分 类 号：O241.6[理学—计算数学] U270.331[理学—数学]