An Improved Arc Algorithm for Detecting Definite Hermitian Pairs - Guo, Chun-Hua; Higham, Nicholas J.; Tisseur, Françoise (SIAM journal on matrix analysis and applications, 2009)
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.)