Convergence Analysis of Iterative Solvers in Inexact Rayleigh Quotient Iteration
نویسندگان
چکیده
منابع مشابه
Convergence Analysis of Iterative Solvers in Inexact Rayleigh Quotient Iteration
Abstract. We present a detailed convergence analysis of preconditioned MINRES for approximately solving the linear systems that arise when Rayleigh Quotient Iteration is used to compute the lowest eigenpair of a symmetric positive definite matrix. We provide insight into the “slow start” of MINRES iteration in both a qualitative and quantitative way, and show that the convergence of MINRES main...
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We consider the computation of the smallest eigenvalue and associated eigenvector of a Hermitian positive definite pencil. Rayleigh quotient iteration (RQI) is known to converge cubically, and we first analyze how this convergence is affected when the arising linear systems are solved only approximately. We introduce a special measure of the relative error made in the solution of these systems ...
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For the Hermitian inexact Rayleigh quotient iteration (RQI), we present general convergence results, independent of iterative solvers for inner linear systems. We prove that the method converges quadratically at least under a new condition, called the uniform positiveness condition. This condition can be much weaker than the commonly used one that at outer iteration k, requires the relative res...
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We consider the computation of the smallest eigenvalue and associated eigenvector of a an Hermitian positive definite pencil. Rayleigh quotient iterations (RQI) are known to converge cubically and we first analyze how this convergence is affected when the arising linear systems are solved only approximately. We obtain an upper bound on the convergence factor which we show to be sharp for a wide...
متن کاملThe Rayleigh Quotient Iteration
The Rayleigh Quotient Iteration (RQI) was developed for real symmetric matrices. Its rapid local convergence is due to the stationarity of the Rayleigh Quotient at an eigenvector. Its excellent global properties are due to the monotonie decrease in the norms of the residuals. These facts are established for normal matrices. Both properties fail for nonnormal matrices and no generalization of th...
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ژورنال
عنوان ژورنال: SIAM Journal on Matrix Analysis and Applications
سال: 2010
ISSN: 0895-4798,1095-7162
DOI: 10.1137/080712908