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A parameter free ADI-like method for the numerical solution of large scale Lyapunov equations. (English) Zbl 1372.65131

Glowinski, Roland (ed.) et al., Splitting methods in communication and imaging, science, and engineering. Cham: Springer (ISBN 978-3-319-41587-1/hbk; 978-3-319-41589-5/ebook). Scientific Computation, 409-425 (2016).
Summary: This work presents an algorithm for constructing an approximate numerical solution to a large scale Lyapunov equation in low rank factored form. The algorithm is based upon a synthesis of an approximate power method and an alternating direction implicit (ADI) method. The former is parameter-free and tends to be efficient in practice, but there is little theoretical understanding of its convergence properties. The ADI method has a well-understood convergence theory, but the method relies upon selection of shift parameters, and a poor shift selection can lead to very slow convergence in practice. The algorithm presented here uses an approximate power method iteration to obtain a basis update and then constructs a re-weighting of this basis to provide a factorization update that satisfies ADI-like convergence properties.
For the entire collection see [Zbl 1362.65002].

MSC:

65F30 Other matrix algorithms (MSC2010)
65F10 Iterative numerical methods for linear systems
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