On Correction-Based Iterative Methods for Eigenvalue Problems

Takafumi MIYATA  

Publication
IEICE TRANSACTIONS on Fundamentals of Electronics, Communications and Computer Sciences   Vol.E101-A   No.10   pp.1668-1675
Publication Date: 2018/10/01
Online ISSN: 1745-1337
DOI: 10.1587/transfun.E101.A.1668
Type of Manuscript: PAPER
Category: Numerical Analysis and Optimization
Keyword: 
numerical analysis,  eigenvalue and eigenvector,  iterative method,  correction equation,  

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Summary: 
The Jacobi-Davidson method and the Riccati method for eigenvalue problems are studied. In the methods, one has to solve a nonlinear equation called the correction equation per iteration, and the difference between the methods comes from how to solve the equation. In the Jacobi-Davidson/Riccati method the correction equation is solved with/without linearization. In the literature, avoiding the linearization is known as an improvement to get a better solution of the equation and bring the faster convergence. In fact, the Riccati method showed superior convergence behavior for some problems. Nevertheless the advantage of the Riccati method is still unclear, because the correction equation is solved not exactly but with low accuracy. In this paper, we analyzed the approximate solution of the correction equation and clarified the point that the Riccati method is specialized for computing particular solutions of eigenvalue problems. The result suggests that the two methods should be selectively used depending on target solutions. Our analysis was verified by numerical experiments.