THE HYPERBOLIC QUADRATIC EIGENVALUE PROBLEM

Xin Liang,Ren-Cang Li

doi:10.1017/fms.2015.14

Abstract

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.

Highlights

It was argued in [27] that the hyperbolic quadratic eigenvalue problem (HQEP) is the closest analog to the standard Hermitian eigenvalue problem among quadratic eigenvalue problems (QEPs)(λ2 A + λB + C)x = 0. (1.1)In many ways, both problems share common properties: the eigenvalues are all real and semisimple, and for the HQEP there is a version of the min–max principles [13, 1955] that are very much like the Courant–Fischer min–max principles.One source of QEPs (1.1) is dynamical systems with friction, where A and C are associated with the kinetic-energy and potential-energy quadratic forms, respectively, and B is associated with the Rayleigh dissipation function [17, 67]
We argue that for an HQEP it is in general unavoidable because the results in Theorem 6.1 are consistent with the asymptotic expression (6.6) after dropping terms of order 2 or higher in a, b, and c
Single-vector CG type methods heavily rely on the line-search problem (8.5)–(8.7) which was solved by projecting the original order-n HQEP for Q(λ) to an order-2 HQEP for Y H Q(λ)Y without computing the optimal parameter topt, and the approximation y as in (8.7) for the steepest descent/ascent method and Downloaded from https://www.cambridge.org/core

Summary

Introduction

It was argued in [27] that the hyperbolic quadratic eigenvalue problem (HQEP) is the closest analog to the standard Hermitian eigenvalue problem among quadratic eigenvalue problems (QEPs). An HQEP is slightly more general than an overdamped QEP in that B and C are no longer required to be positive definite or positive semidefinite, respectively. We undertake a systematic study of the HQEP both in theory and in numerical computation that further reinforces the belief that this class of QEP is the closest analog to the standard Hermitian eigenvalue problem. Propose extended steepest descent/ascent and conjugate gradient type methods for computing extreme eigenpairs;. The generic notation eig( · ) is the set of all eigenvalues, counting algebraic multiplicities, of a matrix or a matrix pencil, depending on its argument(s): eig(A) is for a square matrix A, and eig(A, B) is for a square matrix pencil A − λB

Hyperbolic quadratic matrix polynomial

The HQEP and linearization

Variational principles

Perturbation analysis

11. Conjugate gradient method

12. Numerical examples