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Axioms 2018, 7(4), 81; https://doi.org/10.3390/axioms7040081

The Generalized Schur Algorithm and Some Applications

1
Istituto per le Applicazioni del Calcolo “M. Picone”, CNR, Sede di Bari, via G. Amendola 122/D, 70126 Bari, Italy
2
Catholic University of Louvain, Department of Mathematical Engineering, Avenue Georges Lemaitre 4, B-1348 Louvain-la-Neuve, Belgium
*
Author to whom correspondence should be addressed.
Received: 2 October 2018 / Revised: 5 November 2018 / Accepted: 7 November 2018 / Published: 9 November 2018
(This article belongs to the Special Issue Advanced Numerical Methods in Applied Sciences)
Full-Text   |   PDF [361 KB, uploaded 9 November 2018]   |  

Abstract

The generalized Schur algorithm is a powerful tool allowing to compute classical decompositions of matrices, such as the Q R and L U factorizations. When applied to matrices with particular structures, the generalized Schur algorithm computes these factorizations with a complexity of one order of magnitude less than that of classical algorithms based on Householder or elementary transformations. In this manuscript, we describe the main features of the generalized Schur algorithm. We show that it helps to prove some theoretical properties of the R factor of the Q R factorization of some structured matrices, such as symmetric positive definite Toeplitz and Sylvester matrices, that can hardly be proven using classical linear algebra tools. Moreover, we propose a fast implementation of the generalized Schur algorithm for computing the rank of Sylvester matrices, arising in a number of applications. Finally, we propose a generalized Schur based algorithm for computing the null-space of polynomial matrices. View Full-Text
Keywords: generalized Schur algorithm; null-space; displacement rank; structured matrices generalized Schur algorithm; null-space; displacement rank; structured matrices
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Laudadio, T.; Mastronardi, N.; Van Dooren, P. The Generalized Schur Algorithm and Some Applications. Axioms 2018, 7, 81.

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