Citation:
De Terán, F, Iannazzo, B, Poloni, F, Robol, L. Nonsingular systems of generalized Sylvester equations: An algorithmic approach. Numer Linear Algebra Appl. 2019; 26:e2261
xmlui.dri2xhtml.METS-1.0.item-contributor-funder:
Ministerio de Economía y Competitividad (España)
Sponsor:
Ministerio de Economía y Competitividad of Spain. Grant Numbers: MTM2015-68805-REDT, MTM2015- 65798-P; Istituto Nazionale di Alta Matematica “Francesco Severi”. Grant Number: GNCS Project 2016; Research project of the Università di Perugia Soluzione numerica di problemi di algebra lineare strutturata
Project:
Gobierno de España. MTM2015-68805-REDT Gobierno de España. MTM2015- 65798-P
Keywords:
Formal matrix product
,
Matrix pencils
,
Periodic QR/QZ algorithm
,
Periodic Schur decomposition
,
Sylvester and ⋆‐Sylvester equations
,
Systems of linear matrix equations
We consider the uniqueness of solution (i.e., nonsingularity) of systems of r generalized Sylvester and ⋆‐Sylvester equations with n×n coefficients. After several reductions, we show that it is sufficient to analyze periodic systems having, at most, one generaWe consider the uniqueness of solution (i.e., nonsingularity) of systems of r generalized Sylvester and ⋆‐Sylvester equations with n×n coefficients. 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 a backward stable O(n3r) algorithm for computing the (unique) solution.[+][-]