Note on Interlacing for Hyperbolic Quadratic Pencils

Veselić, Krešimir

doi:10.1007/978-3-0346-0180-1_17

Cited by 3 publications

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“…In particular, setting i j = j in (5.18a) and setting i j = j + n − k in (5.18b) gives [66]). Suppose that X ∈ C n×k with rank(X ) = k. Denote the eigenvalues of X H Q(λ)X by…”

Section: By Theorem 52 and Noticing Thatmentioning

confidence: 99%

The Hyperbolic Quadratic Eigenvalue Problem

Liang

2015

Forum math. Sigma

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The hyperbolic quadratic eigenvalue problem (HQEP) was shown to admit Courant-Fischer type min-max principles in 1955 by Duffin and Cauchy type interlacing inequalities in 2010 by Veselić. It can be regarded as the closest analog (among all kinds of quadratic eigenvalue problem) to the standard Hermitian eigenvalue problem (among all kinds of standard eigenvalue problem). In this paper, we conduct a systematic study on the HQEP both theoretically and numerically. On the theoretical front, we generalize Wielandt-Lidskii type min-max principles and, as a special case, Fan type trace min/max principles and establish Weyl type and Wielandt-Lidskii-Mirsky type perturbation results when an HQEP is perturbed to another HQEP. On the numerical front, we justify the natural generalization of the Rayleigh-Ritz procedure with existing principles and our new optimization principles, and, as consequences of these principles, we extend various current optimization approaches-steepest descent/ascent and nonlinear conjugate gradient type methods for the Hermitian eigenvalue problem-to calculate a few extreme eigenvalues (of both positive and negative type). A detailed convergence analysis is given for the steepest descent/ascent methods. The analysis reveals the intrinsic quantities that control convergence rates and consequently yields ways of constructing effective preconditioners. Numerical examples are presented to demonstrate the proposed theory and algorithms.2010 Mathematics Subject Classification: 15A42, 65F15 (primary); 15A18, 65F08, 65F50, 65G99 (secondary)

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“…In particular, setting i j = j in (5.18a) and setting i j = j + n − k in (5.18b) gives [66]). Suppose that X ∈ C n×k with rank(X ) = k. Denote the eigenvalues of X H Q(λ)X by…”

Section: By Theorem 52 and Noticing Thatmentioning

confidence: 99%

The Hyperbolic Quadratic Eigenvalue Problem

Liang

2015

Forum math. Sigma

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“…Courant-Fischer type min-max principles were known to Duffin [10,1955] and the Cauchy type interlacing inequalities to Veselić [52,2010]. Other min-max principles (Wielandt-Lidskii type, Ky Fan trace min/max type) are being discovered [28].…”

Section: Introductionmentioning

confidence: 99%

Rayleigh Quotient Based Optimization Methods for Eigenvalue Problems

2015

Series in Contemporary Applied Mathematics

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Four classes of eigenvalue problems that admit similar min-max principles and the Cauchy interlacing inequalities as the symmetric eigenvalue problem famously does are investigated. These min-max principles pave ways for efficient numerical solutions for extreme eigenpairs by optimizing the so-called Rayleigh quotient functions. In fact, scientists and engineers have already been doing that for computing the eigenvalues and eigenvectors of Hermitian matrix pencils A − λB with B positive definite, the first class of our eigenvalue problems. But little attention has gone to the other three classes: positive semidefinite pencils, linear response eigenvalue problems, and hyperbolic eigenvalue problems, in part because most min-max principles for the latter were discovered only very recently and some more are being discovered. It is expected that they will drive the effort to design better optimization based numerical methods for years to come.

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Detecting a hyperbolic quadratic eigenvalue problem by using a subspace algorithm

Pandur

2019

Numer Algor

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Note on Interlacing for Hyperbolic Quadratic Pencils

Cited by 3 publications

References 2 publications

The Hyperbolic Quadratic Eigenvalue Problem

The Hyperbolic Quadratic Eigenvalue Problem

Rayleigh Quotient Based Optimization Methods for Eigenvalue Problems

Detecting a hyperbolic quadratic eigenvalue problem by using a subspace algorithm

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