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Direct methods and ADI-preconditioned Krylov subspace methods for generalized Lyapunov equations. (English) Zbl 1212.65175
Summary: 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.

MSC:
65F30 Other matrix algorithms (MSC2010)
15A24 Matrix equations and identities
93E25 Computational methods in stochastic control (MSC2010)
65F10 Iterative numerical methods for linear systems
65F08 Preconditioners for iterative methods
65F05 Direct numerical methods for linear systems and matrix inversion
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