Document - Nonsingular systems of generalized Sylvester equations: An algorithmic approach

2019

Journal article Open Access

Nonsingular systems of generalized Sylvester equations: An algorithmic approach

De Teran F., Iannazzo B., Poloni F., Robol L.

Formal matrix product QZ algorithm Matemáticas Sylvester and -Sylvester equations FOS: Mathematics Periodic Schur decomposition Mathematics - Numerical Analysis 15A24 Periodic QR Periodic QR/QZ algorithm Algebra and Number Theory 15A22 65F15 Systems of linear matrix equations Sylvester and ⋆‐Sylvester equations Applied Mathematics Numerical Analysis (math.NA) Matrix pencils

We consider the uniqueness of solution (i.e., nonsingularity) of systems of r generalized Sylvester and -Sylvester equations with nxn 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(n(3)r) algorithm for computing the (unique) solution.

Source: Numerical linear algebra with applications 26 (2019). doi:10.1002/nla.2261

Publisher: Wiley., Chichester, West Sussex, Regno Unito

Citations

[1] P. Anderson, R. Granat, I. Jonsson, and B. K˚agstro¨m. Parallel algorithms for triangular periodic Sylvester-type matrix equations. In Lecture Notes in Computer Science, pages 169-174. Euro-Par 2008-Parallel Processing, Springer, 2007.
[2] P. Benner, V. Mehrmann, V. Sima, S. Van Huffel, and A. Varga. SLICOT - a subroutine library in systems and control theory. In Biswa Nath Datta, editor, Applied and Computational Control, Signals, and Circuits (1997), chapter 10, pages 499-539. Birkh¨auser Boston, Boston, MA, 1997.
[3] A. W Bojanczyk, G. H. Golub, and P. Van Dooren. Periodic Schur decomposition: algorithms and applications. In Proc. SPIE Conference, pages 31-42. International Society for Optics and Photonics, 1992.

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@article{oai:it.cnr:prodotti:424803,
	title = {Nonsingular systems of generalized Sylvester equations: An algorithmic approach},
	author = {De Teran F. and Iannazzo B. and Poloni F. and Robol L.},
	publisher = {Wiley., Chichester, West Sussex, Regno Unito},
	doi = {10.1002/nla.2261 and 10.48550/arxiv.1709.03783},
	journal = {Numerical linear algebra with applications},
	volume = {26},
	year = {2019}
}