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

The Jacobi–Davidson iteration method is very efficient in solving Hermitian eigenvalue problems. If the correction equation involved in the Jacobi–Davidson iteration is solved accurately, the simplified Jacobi–Davidson iteration is equivalent to the Rayleigh quotient iteration which achieves cubic convergence rate locally. When the involved linear system is solved by an iteration method, these two methods are also equivalent. In this paper, we present the convergence analysis of the simplified Jacobi–Davidson method and present the estimate of iteration numbers of the inner correction equation. Furthermore, based on the convergence factor, we can see how the accuracy of the inner iteration controls the outer iteration.

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Convergence Analysis of Inexact Rayleigh Quotient Iteration

SIAM Journal on Matrix Analysis and Applications ◽

10.1137/s0895479801399596 ◽

2003 ◽

Vol 24 (3) ◽

pp. 627-644 ◽

Cited By ~ 31

Author(s):

Yvan Notay

Keyword(s):

Convergence Analysis ◽

Rayleigh Quotient ◽

Rayleigh Quotient Iteration

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Rayleigh quotient iteration and simplified Jacobi–Davidson method with preconditioned iterative solves

Linear Algebra and its Applications ◽

10.1016/j.laa.2007.11.013 ◽

2008 ◽

Vol 428 (8-9) ◽

pp. 2049-2060 ◽

Cited By ~ 22

Author(s):

M.A. Freitag ◽

A. Spence

Keyword(s):

Rayleigh Quotient ◽

Davidson Method ◽

Preconditioned Iterative ◽

Rayleigh Quotient Iteration

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Efficient Preconditioned Inner Solves For Inexact Rayleigh Quotient Iteration And Their Connections To The Single-Vector Jacobi–Davidson Method

SIAM Journal on Matrix Analysis and Applications ◽

10.1137/100807922 ◽

2011 ◽

Vol 32 (3) ◽

pp. 993-1018 ◽

Cited By ~ 14

Author(s):

Daniel B. Szyld ◽

Fei Xue

Keyword(s):

Rayleigh Quotient ◽

Davidson Method ◽

Rayleigh Quotient Iteration

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The Inexact Rayleigh Quotient Iteration for the Large Hermitian Matrix Eigenproblem

Advanced Materials Research ◽

10.4028/www.scientific.net/amr.403-408.5230 ◽

2011 ◽

Vol 403-408 ◽

pp. 5230-5234

Author(s):

Qing Guo Qu

Keyword(s):

Convergence Theorem ◽

Numerical Experiments ◽

Hermitian Matrix ◽

Rayleigh Quotient ◽

Rayleigh Quotient Iteration

The inexact Rayleigh quotient iteration (RQI) is used for computing the smallest eigenpair of a large Hermitian matrix. It’s shown in this paper that under the uniform positiveness condition a new convergence theorem of the inexact RQI is presented and proved by the nature of eigenvalues. All the results are verified and analyzed by numerical experiments.

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