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Direct methods and ADI‐preconditioned Krylov subspace methods for generalized Lyapunov equations
Author(s) -
Damm T.
Publication year - 2008
Publication title -
numerical linear algebra with applications
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 1.02
H-Index - 53
eISSN - 1099-1506
pISSN - 1070-5325
DOI - 10.1002/nla.603
Subject(s) - mathematics , krylov subspace , lyapunov equation , lyapunov function , preconditioner , operator (biology) , iterative method , linear system , mathematical analysis , mathematical optimization , nonlinear system , biochemistry , chemistry , physics , repressor , quantum mechanics , transcription factor , gene
We consider linear matrix equations where the linear mapping is the sum of a standard Lyapunov operator and a positive operator. These equations play a role in the context of stochastic or bilinear control systems. To solve them efficiently one can fall back on known efficient methods developed for standard Lyapunov equations. In this paper, we describe a direct and an iterative method based on this idea. The direct method is applicable if the generalized Lyapunov operator is a low‐rank perturbation of a standard Lyapunov operator; it is related to the Sherman–Morrison–Woodbury formula. The iterative method requires a stability assumption; it uses convergent regular splittings, an alternate direction implicit iteration as preconditioner, and Krylov subspace methods. Copyright © 2008 John Wiley & Sons, Ltd.