Inertia‐based spectrum slicing for symmetric quadratic eigenvalue problems

Summary In the quadratic eigenvalue problem (QEP) with all coefficient matrices symmetric, there can be complex eigenvalues. However, some applications need to compute real eigenvalues only. We propose a Lanczos‐based method for computing all real eigenvalues contained in a given interval of large‐s...

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Bibliographic Details
Published inNumerical linear algebra with applications Vol. 27; no. 4
Main Authors Campos, Carmen, Roman, Jose E.
Format Journal Article
LanguageEnglish
Published Oxford Wiley Subscription Services, Inc 01.08.2020
Subjects
Computation
Eigenvalues
inertia
Mathematical analysis
Matrix methods
parallel computing
Polynomials
pseudo‐Lanczos
quadratic eigenvalue problem
SLEPc
Slicing
symmetric linearization
Tensors
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Summary:Summary In the quadratic eigenvalue problem (QEP) with all coefficient matrices symmetric, there can be complex eigenvalues. However, some applications need to compute real eigenvalues only. We propose a Lanczos‐based method for computing all real eigenvalues contained in a given interval of large‐scale symmetric QEPs. The method uses matrix inertias of the quadratic polynomial evaluated at different shift values. In this way, for hyperbolic problems, it is possible to make sure that all eigenvalues in the interval have been computed. We also discuss the general nonhyperbolic case. Our implementation is memory‐efficient by representing the computed pseudo‐Lanczos basis in a compact tensor product representation. We show results of computational experiments with a parallel implementation in the SLEPc library.
Bibliography:Funding information
Agencia Estatal de Investigación, TIN2016‐75985‐P
ISSN:1070-5325
1099-1506
DOI:10.1002/nla.2293