An Improved Arc Algorithm for Detecting Definite Hermitian Pairs

• Published in: SIAM journal on matrix analysis and applications Vol. 31; no. 3; pp. 1131 - 1151
• Main Authors: Guo, Chun-Hua, Higham, Nicholas J., Tisseur, Françoise
• Format: Journal Article
• Language: English
• Published: Philadelphia, PA Society for Industrial and Applied Mathematics 01.01.2009
• Subjects: Algebra; Algorithms; Applications; Computational efficiency; Convergence; Curvature; Derivation; Exact sciences and technology; Linear and multilinear algebra, matrix theory; Mathematical analysis; Mathematics; Moon; Numerical analysis; Numerical analysis. Scientific computation; Numerical linear algebra; Probability and statistics; Probability theory and stochastic processes; Reliability, life testing, quality control; Sciences and techniques of general use; Special processes (renewal theory, markov renewal processes, semi-markov processes, statistical mechanics type models, applications); Statistics; Symbols; Saddle point method; Modification; Costs; Hermitian generalized eigenvalue problem; Linear form; Unit circle; Shrinkage estimator; Hermitian matrix; hyperbolic quadratic eigenvalue problem; Polynomial method; Computing; saddle point linear system; Partial differential equation; Implementation; Convergence; 15A18; Statistical test; Efficiency; Cholesky method; 65F30; Crawford number; 65F15; Eigenvalue problem; Mathematical expansion; Expansion; Conjugate gradient method; definite pair; direction of negative curvature; Polynomial matrix; pencil; Algorithm; Factorization; Saddle point; Numerical analysis; Linear system; Linear algebra; Hyperbolic equation; Matrix method; Matrix polynomial; Reliability; Curvature
• ISSN: 0895-4798; 1095-7162
• DOI: 10.1137/08074218X

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 ..., which is a subset of the unit circle. The researchers 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.(ProQuest: ... denotes formulae/symbols omitted.)