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Numerical analysis of a quadratic matrix equation
Authors:Higham  Nicholas J; Kim  Hyun-Min
Institution: 1 Department of Mathematics, University of Manchester, Manchester, M13 9PL, UK
Abstract:The quadratic matrix equation AX2+ BX + C = 0in n x nmatricesarises in applications and is of intrinsic interest as oneof the simplest nonlinear matrix equations. We give a completecharacterization of solutions in terms of the generalized Schurdecomposition and describe and compare various numerical solutiontechniques. In particular, we give a thorough treatment offunctional 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 quadraticmatrix equation can provide an efficient way to solve the associated quadratic eigenvalue problem ({lambda}2A + {lambda}B + C)x = 0.
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
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