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Mathematics > Numerical Analysis

arXiv:2206.00585 (math)
[Submitted on 1 Jun 2022]

Title:Convergence rates of individual Ritz values in block preconditioned gradient-type eigensolvers

Authors:Ming Zhou, Klaus Neymeyr
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Abstract:Many popular eigensolvers for large and sparse Hermitian matrices or matrix pairs can be interpreted as accelerated block preconditioned gradient (BPG) iterations in order to analyze their convergence behavior by composing known estimates. An important feature of BPG is the cluster robustness, i.e., reasonable performance for computing clustered eigenvalues is ensured by a sufficiently large block size. This feature can easily be explained for exact-inverse (exact shift-inverse) preconditioning by adapting classical estimates on nonpreconditioned eigensolvers, whereas the existing results for more general preconditioning are still improvable. We expect to extend certain sharp estimates for the corresponding vector iterations to BPG where proper bounds of convergence rates of individual Ritz values are to be derived. Such an extension has been achieved for BPG with fixed step sizes in [Math. Comp. 88 (2019), 2737--2765]. The present paper deals with the more practical case that the step sizes are implicitly optimized by the Rayleigh-Ritz method. Our new estimates improve some previous ones in view of concise and more flexible bounds.
Comments: 21 pages, 4 figures
Subjects: Numerical Analysis (math.NA)
MSC classes: 65F15, 65N12, 65N25
Cite as: arXiv:2206.00585 [math.NA]
  (or arXiv:2206.00585v1 [math.NA] for this version)
  https://doi.org/10.48550/arXiv.2206.00585

Submission history

From: Ming Zhou [view email]
[v1] Wed, 1 Jun 2022 15:56:05 UTC (263 KB)
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