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{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,2,21]],"date-time":"2025-02-21T19:14:05Z","timestamp":1740165245835,"version":"3.37.3"},"reference-count":26,"publisher":"MDPI AG","issue":"4","license":[{"start":{"date-parts":[[2018,11,9]],"date-time":"2018-11-09T00:00:00Z","timestamp":1541721600000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Axioms"],"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.<\/jats:p>","DOI":"10.3390\/axioms7040081","type":"journal-article","created":{"date-parts":[[2018,11,13]],"date-time":"2018-11-13T08:27:31Z","timestamp":1542097651000},"page":"81","source":"Crossref","is-referenced-by-count":3,"title":["The Generalized Schur Algorithm and Some Applications"],"prefix":"10.3390","volume":"7","author":[{"ORCID":"https:\/\/orcid.org\/0000-0002-2046-0456","authenticated-orcid":false,"given":"Teresa","family":"Laudadio","sequence":"first","affiliation":[{"name":"Istituto per le Applicazioni del Calcolo \u201cM. Picone\u201d, CNR, Sede di Bari, via G. Amendola 122\/D, 70126 Bari, Italy"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-0045-2862","authenticated-orcid":false,"given":"Nicola","family":"Mastronardi","sequence":"additional","affiliation":[{"name":"Istituto per le Applicazioni del Calcolo \u201cM. Picone\u201d, CNR, Sede di Bari, via G. Amendola 122\/D, 70126 Bari, Italy"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-0115-9932","authenticated-orcid":false,"given":"Paul","family":"Van Dooren","sequence":"additional","affiliation":[{"name":"Catholic University of Louvain, Department of Mathematical Engineering, Avenue Georges Lemaitre 4, B-1348 Louvain-la-Neuve, Belgium"}]}],"member":"1968","published-online":{"date-parts":[[2018,11,9]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","unstructured":"Kailath, T., and Sayed, A.H. (1999). 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[4th ed.]."},{"key":"ref_16","doi-asserted-by":"crossref","unstructured":"Boito, P. (2011). Structured Matrix Based Methods for Approximate Polynomial GCD, Edizioni Della Normale.","DOI":"10.1007\/978-88-7642-381-9"},{"key":"ref_17","doi-asserted-by":"crossref","first-page":"155","DOI":"10.1007\/978-3-7643-8996-3_6","article-title":"A Fast Algorithm for Approximate Polynomial GCD Based on Structured Matrix Computations","volume":"Volume 199","author":"Bini","year":"2010","journal-title":"Numerical Methods for Structured Matrices and Applications"},{"key":"ref_18","doi-asserted-by":"crossref","first-page":"203","DOI":"10.1016\/j.laa.2004.07.006","article-title":"Implementation of the regularized structured total least squares algorithms for blind image deblurring","volume":"391","author":"Mastronardi","year":"2004","journal-title":"Linear Algebra Its Appl."},{"key":"ref_19","doi-asserted-by":"crossref","first-page":"287","DOI":"10.1016\/j.laa.2003.11.030","article-title":"A robust solution of the generalized polynomial Bezout identity","volume":"385","author":"Basilio","year":"2004","journal-title":"Linear Algebra Its Appl."},{"key":"ref_20","unstructured":"Kailath, T. 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