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The convergence of harmonic Ritz values, harmonic Ritz vectors, and refined harmonic Ritz vectors

Author: Zhongxiao Jia
Journal: Math. Comp. 74 (2005), 1441-1456
MSC (2000): Primary 15A18, 65F15, 65F30
Published electronically: June 1, 2004
MathSciNet review: 2137011
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Abstract: This paper concerns a harmonic projection method for computing an approximation to an eigenpair $(\lambda, x)$ of a large matrix $A$. Given a target point $\tau$ and a subspace $\mathcal{W}$ that contains an approximation to $x$, the harmonic projection method returns an approximation $(\mu+\tau, \tilde x)$ to $(\lambda,x)$. Three convergence results are established as the deviation $\epsilon$ of $x$ from $\mathcal{W}$ approaches zero. First, the harmonic Ritz value $\mu+\tau$ converges to $\lambda$ if a certain Rayleigh quotient matrix is uniformly nonsingular. Second, the harmonic Ritz vector $\tilde x$ converges to $x$ if the Rayleigh quotient matrix is uniformly nonsingular and $\mu+\tau$ remains well separated from the other harmonic Ritz values. Third, better error bounds for the convergence of $\mu+\tau$ are derived when $\tilde x$ converges. However, we show that the harmonic projection method can fail to find the desired eigenvalue $\lambda$--in other words, the method can miss $\lambda$ if it is very close to $\tau$. To this end, we propose to compute the Rayleigh quotient $\rho$ of $A$ with respect to $\tilde x$ and take it as a new approximate eigenvalue. $\rho$ is shown to converge to $\lambda$ once $\tilde x$ tends to $x$, no matter how $\tau$ is close to $\lambda$. Finally, we show that if the Rayleigh quotient matrix is uniformly nonsingular, then the refined harmonic Ritz vector, or more generally the refined eigenvector approximation introduced by the author, converges. We construct examples to illustrate our theory.

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Additional Information

Zhongxiao Jia
Affiliation: Department of Mathematical Sciences, Tsinghua University, Beijing 100084, Peoples Republic of China

Keywords: Harmonic projection, refined harmonic projection, harmonic Ritz value, harmonic Ritz vector, refined harmonic Ritz vector, refined eigenvector approximation, convergence
Received by editor(s): June 7, 2002
Received by editor(s) in revised form: December 23, 2003
Published electronically: June 1, 2004
Additional Notes: Work supported by Special Funds for the State Major Basic Research Projects (G1999032805)
Article copyright: © Copyright 2004 American Mathematical Society