Numerical analysis of a quadratic matrix equation |
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Authors: | Higham, Nicholas J. Kim, Hyun-Min |
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Affiliation: | 1 Department of Mathematics, University of Manchester, Manchester, M13 9PL, UK |
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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 Bernoullis method.Other methods considered include Newtons 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 (2A + B + C)x = 0. |
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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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