1 2 Se p 20 17 Nonsingular systems of generalized Sylvester equations : an algorithmic approach ∗

We consider the uniqueness of solution (nonsingularity) of systems of r generalized Sylvester and ⋆-Sylvester equations with n × n coefficient matrices. After several reductions, we show that it is sufficient to analyze periodic systems having, at most, one generalized ⋆-Sylvester equation. We provide characterizations for the nonsingularity in terms of spectral properties of either matrix pencils or formal matrix products, both constructed from the coefficients of the system. The proposed approach uses the periodic Schur decomposition, and leads to an O(n 3 r) algorithm for computing the (unique) solution. We prove that the proposed algorithm is backward stable. The asymptotic cost and the stability are then verified by some numerical experiments. ∗ 2010 Mathematics Subject Classification. Primary 15A22, 15A24, 65F15. This work was partially supported by the Ministerio de Economı́a y Competitividad of Spain through grants MTM2015-68805-REDT, and MTM2015-65798-P (F. De Terán), by an INdAM/GNCS Research Project 2016 (B. Iannazzo, F. Poloni, and L. Robol), and by the Region of Tuscany (PAR-FAS 2007 – 2013) and by MIUR, the Italian Ministry of Education, Universities and Research (FAR) within the Call FAR – FAS 2014 (MOSCARDO Project: ICT technologies for structural monitoring of age-old constructions based on wireless sensor networks and drones, 2016 – 2018) (L. Robol). Part of this work was done during a visit of the first author to the Università di Perugia as a Visiting Researcher.