A modified low-rank Smith method for large-scale Lyapunov equations. (English) Zbl 1034.93020

The authors present a modified cyclic low-rank Smith method to compute low-rank approximations to solutions of Lyapunov equations arising from large-scale dynamical systems. Unlike the original cyclic low-rank Smith method introduced by T. Penzl [SIAM J. Sci. Comput. 21, 1401–1418 (2000; Zbl 0958.65052)], the number of columns required by the modified method in the approximate solution does not necessarily increase at each step and is usually much lower than in the original cyclic low-rank Smith method. The modified method never requires more columns than the original one. Upper bounds are established for the errors of the low-rank approximate solutions and also for the errors in the resulting approximate Hankel singular values. Numerical results are given to verify the efficiency and accuracy of the new algorithm.


93B40 Computational methods in systems theory (MSC2010)
93A15 Large-scale systems
15A24 Matrix equations and identities
93B11 System structure simplification


Zbl 0958.65052
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