Numerical analysis of a quadratic matrix equation

Higham, Nicholas J and Kim, Hyun-Min (2000) Numerical analysis of a quadratic matrix equation. IMA Journal of Numerical Analysis, 20. pp. 499-519. ISSN 1464-3642

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Abstract

The quadratic matrix equation AX2+ BX + C = 0in n x nmatrices arises in applications and is of intrinsic interest as one of the simplest nonlinear matrix equations. We give a complete characterization of solutions in terms of the generalized Schur decomposition and describe and compare various numerical solution techniques. In particular, we give a thorough treatment of functional iteration methods based on Bernoulli’s method. Other methods considered include Newton’s method with exact line searches, symbolic solution and continued fractions. We show that functional iteration applied to the quadratic matrix equation can provide an efficient way to solve the associated quadratic eigenvalue problem ({lambda}2A + {lambda}B + C)x = 0.

Item Type: Article
Uncontrolled Keywords: quadratic matrix equation; solvent; generalized Schur decomposition; scaling; functional iteration; Bernoulli’s method; Newton’s method; exact line searches; continued fractions; quadratic eigenvalue problem
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: Ms Lucy van Russelt
Date Deposited: 27 Jun 2006
Last Modified: 20 Oct 2017 14:12
URI: http://eprints.maths.manchester.ac.uk/id/eprint/316

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