Numerical solution to low rank perturbed Lyapunov equations by the sign function method

Abstract

AbstractWe investigate the numerical solution to a low rank perturbed Lyapunov equation ATX + 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 + UCVT, where B is a blockdiagonal matrix and UCVT 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)