Volume 43, pp. 45-59, 2014-2015.

A minimal residual norm method for large-scale Sylvester matrix equations

Said Agoujil, Abdeslem H. Bentbib, Khalide Jbilou, and El Mostafa Sadek


In this paper, we present a new method for solving large-scale Sylvester matrix equations with a low-rank right-hand side. The proposed method is an iterative method based on a projection onto an extended block Krylov subspace by minimization of the norm of the residual. The obtained reduced-order problem is solved via different direct or iterative solvers that exploit the structure of the linear operator associated with the obtained matrix equation. In particular, we use the global LSQR algorithm as iterative method for the derived low-order problem. Then, when convergence is achieved, a low-rank approximate solution is computed given as a product of two low-rank matrices, and a stopping procedure based on an economical computation of the norm of the residual is proposed. Different numerical examples are presented, and the proposed minimal residual approach is compared with the corresponding Galerkin-type approach.

Full Text (PDF) [178 KB]

Key words

extended block Krylov subspaces, low-rank approximation, Sylvester equation, minimal residual methods

AMS subject classifications

65F10, 65F30

Links to the cited ETNA articles

[9]Vol. 33 (2008-2009), pp. 53-62 M. Heyouni and K. Jbilou: An extended block Arnoldi algorithm for large-scale solutions of the continuous-time algebraic Riccati equation

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