Guo, ChunHua and Higham, Nicholas J. and Tisseur, Françoise (2009) An Improved Arc Algorithm for Detecting Definite Hermitian Pairs. SIAM Journal On Matrix Analysis and Applications., 31 (3). 1131 1151. ISSN 10957162
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Abstract
A 25year 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:  Article 

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:  18 Sep 2009 
Last Modified:  20 Oct 2017 14:12 
URI:  https://eprints.maths.manchester.ac.uk/id/eprint/1307 
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An Improved Arc Algorithm for Detecting Definite Hermitian Pairs. (deposited 28 Nov 2008)

An Improved Arc Algorithm for Detecting Definite Hermitian Pairs. (deposited 30 Jun 2009)
 An Improved Arc Algorithm for Detecting Definite Hermitian Pairs. (deposited 18 Sep 2009) [Currently Displayed]

An Improved Arc Algorithm for Detecting Definite Hermitian Pairs. (deposited 30 Jun 2009)
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