A convergence analysis of the inexact Rayleigh quotient iteration and simplified Jacobi-Davidson method for the large Hermitian matrix eigenproblem  

作　　者：JIA ZhongXiao WANG Zhen 

机构地区：[1]Department of Mathematical Sciences,Tsinghua University,Beijing 100084,China

出　　处：《Science China Mathematics》2008年第12期2205-2216,共12页中国科学：数学（英文版）

基　　金：supported by the National Natural Science Foundation of China (Grant Nos. 10471074, 10771116);the Doctoral Program of the Ministry of Education of China (Grant No. 20060003003)

摘　　要：The inexact Rayleigh quotient iteration (RQI) is used for computing the smallest eigenpair of a large Hermitian matrix. Under certain condition, the method was proved to converge quadratically in literature. However, it is shown in this paper that under the original given condition the inexact RQI may not quadratically converge to the desired eigenpair and even may misconverge to some other undesired eigenpair. A new condition, called the uniform positiveness condition, is given that can fix misconvergence problem and ensure the quadratic convergence of the inexact RQI. An alternative to the inexact RQI is the Jacobi-Davidson (JD) method without subspace acceleration. A new proof of its linear convergence is presented and a sharper bound is established in the paper. All the results are verified and analyzed by numerical experiments.The inexact Rayleigh quotient iteration (RQI) is used for computing the smallest eigenpair of a large Hermitian matrix. Under certain condition, the method was proved to converge quadratically in literature. However, it is shown in this paper that under the original given condition the inexact RQI may not quadratically converge to the desired eigenpair and even may misconverge to some other undesired eigenpair. A new condition, called the uniform positiveness condition, is given that can fix misconvergence problem and ensure the quadratic convergence of the inexact RQI. An alternative to the inexact RQI is the Jacobi-Davidson (JD) method without subspace acceleration. A new proof of its linear convergence is presented and a sharper bound is established in the paper. All the results are verified and analyzed by numerical experiments.

关 键 词：eigenvalue EIGENVECTOR large Hermite matrix INEXACT RQI simplified JD convergence misconvergence the uniform positiveness condition 65F15 

分 类 号：O151.21[理学—数学]