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Projection methods for large-scale T-Sylvester equations

Authors: Froilán M. Dopico, Javier González, Daniel Kressner and Valeria Simoncini
Journal: Math. Comp. 85 (2016), 2427-2455
MSC (2010): Primary 65F10, 65F30, 15A06
Published electronically: January 12, 2016
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Abstract: The matrix Sylvester equation for congruence, or T-Sylvester
equation, has recently attracted considerable attention as a consequence of its close relation to palindromic eigenvalue problems. The theory concerning T-Sylvester equations is rather well understood, and there are stable and efficient numerical algorithms which solve these equations for small- to medium-sized matrices. However, developing numerical algorithms for solving large-scale T-Sylvester equations still remains an open problem. In this paper, we present several projection algorithms based on different Krylov spaces for solving this problem when the right-hand side of the T-Sylvester equation is a low-rank matrix. The new algorithms have been extensively tested, and the reported numerical results show that they work very well in practice, offering clear guidance on which algorithm is the most convenient in each situation.

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Additional Information

Froilán M. Dopico
Affiliation: Departamento de Matemáticas, Universidad Carlos III de Madrid, Avda. Universidad 30, 28911 Leganés, Spain

Javier González
Affiliation: Departamento de Matemáticas, Universidad Carlos III de Madrid, Avda. Universidad 30, 28911 Leganés, Spain

Daniel Kressner
Affiliation: ANCHP, MATHICSE, EPF Lausanne, Station 8, 1015 Lausanne, Switzerland

Valeria Simoncini
Affiliation: Dipartimento di Matematica, Università di Bologna, Piazza di Porta San Donato 5, I-40127 Bologna, Italy

Keywords: Matrix equations, Krylov subspace, iterative methods, large-scale equations, Sylvester equation, Sylvester equation for congruence.
Received by editor(s): April 10, 2014
Received by editor(s) in revised form: February 20, 2015
Published electronically: January 12, 2016
Additional Notes: The first and second authors were partially supported by Ministerio de Economía y Competitividad of Spain through grant MTM2012-32542.
Article copyright: © Copyright 2016 American Mathematical Society