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Numerical solution to low rank perturbed Lyapunov equations by the sign function method
Author(s) -
Benner Peter,
Denißen Jonas
Publication year - 2016
Publication title -
pamm
Language(s) - English
Resource type - Journals
ISSN - 1617-7061
DOI - 10.1002/pamm.201610350
Subject(s) - mathematics , lyapunov function , perturbation (astronomy) , sign function , rank (graph theory) , sign (mathematics) , lyapunov equation , numerical analysis , mathematical analysis , combinatorics , physics , nonlinear system , quantum mechanics
Abstract We investigate the numerical solution to a low rank perturbed Lyapunov equation A T X + XA = W via the sign function method (SFM). The sign function method has been proposed to solve Lyapunov equations, see e.g. [1], but here we focus on a framework where the matrix A has a special structure, i. e. A = B + UCV T , where B is a blockdiagonal matrix and UCV T is a low rank perturbation. We show that this structure can be kept throughout the sign function iteration but the rank of the perturbation doubles per iteration. Therefore, we apply a low rank approximation to the perturbation in order to keep its numerical rank small. We compare the standard SFM with its structure preserving variant presented in this paper by means of numerical examples from viscously damped mechanical systems. (© 2016 Wiley‐VCH Verlag GmbH & Co. KGaA, Weinheim)