An Improved Arc Algorithm for Detecting Definite Hermitian Pairs

Guo, Chun-Hua and Higham, Nicholas J. and Tisseur, Françoise (2008) An Improved Arc Algorithm for Detecting Definite Hermitian Pairs. [MIMS Preprint]

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Abstract

A 25-year old and somewhat neglected algorithm of Crawford and Moon attempts to determine whether a given Hermitian matrix pair $(A,B)$ is definite by exploring the range of the function $f(x) = x^*(A+iB)x / | x^*(A+iB)x |$, which is a subset of the unit circle. We revisit the algorithm and show that with suitable modifications and careful attention to implementation details it provides a reliable and efficient means of testing definiteness. A clearer derivation of the basic algorithm is given that emphasizes an arc expansion viewpoint and makes no assumptions about the definiteness of the pair. Convergence of the algorithm is proved for all $(A,B$), definite or not. It is shown that proper handling of three details of the algorithm is crucial to the efficiency and reliability: how the midpoint of an arc is computed, whether shrinkage of an arc is permitted, and how directions of negative curvature are computed. For the latter, several variants of Cholesky factorization with complete pivoting are explored and the benefits of pivoting demonstrated. The overall cost of our improved algorithm is typically just a few Cholesky factorizations. Applications of the algorithm are described to testing the hyperbolicity of a Hermitian quadratic matrix polynomial, constructing conjugate gradient methods for sparse linear systems in saddle point form, and computing the Crawford number of the pair $(A,B)$ via a quasiconvex univariate minimization problem.

Item Type: MIMS Preprint
Additional Information: To appear in the SIAM Journal on Matrix Analysis and Applications
Uncontrolled Keywords: definite pair, pencil, Hermitian generalized eigenvalue problem, direction of negative curvature, Crawford number, hyperbolic quadratic eigenvalue problem, saddle point linear system
Subjects: MSC 2010, the AMS's Mathematics Subject Classification > 15 Linear and multilinear algebra; matrix theory
MSC 2010, the AMS's Mathematics Subject Classification > 65 Numerical analysis
Depositing User: Nick Higham
Date Deposited: 30 Jun 2009
Last Modified: 08 Nov 2017 18:18
URI: https://eprints.maths.manchester.ac.uk/id/eprint/1282

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