Γραμμικοποιητές πολυωνυμικών πινάκων

Δημητριάδης, Γεώργιος-Μιχαήλ; Dimitriadis, Georgios-Michail

dc.contributor.author	Δημητριάδης, Γεώργιος-Μιχαήλ	el
dc.contributor.author	Dimitriadis, Georgios-Michail	en
dc.date.accessioned	2016-07-08T06:39:48Z
dc.date.available	2016-07-08T06:39:48Z
dc.date.issued	2016-07-08
dc.identifier.uri	https://dspace.lib.ntua.gr/xmlui/handle/123456789/43031
dc.identifier.uri	http://dx.doi.org/10.26240/heal.ntua.6738
dc.rights	Αναφορά Δημιουργού-Μη Εμπορική Χρήση-Όχι Παράγωγα Έργα 3.0 Ελλάδα	*
dc.rights.uri	http://creativecommons.org/licenses/by-nc-nd/3.0/gr/	*
dc.subject	Γραμμικοποιητές	el
dc.subject	Δέσμη	el
dc.subject	Πίνακες	el
dc.title	Γραμμικοποιητές πολυωνυμικών πινάκων	el
heal.type	bachelorThesis
heal.classification	Μαθηματικά	el
heal.language	el
heal.access	free
heal.recordProvider	ntua	el
heal.publicationDate	2016-03-03
heal.abstract	Γραμμικοποιητές πολυωνυμικών πινάκων, δέσμη πίνακα, συνοδεύουσες μορφές	el
heal.abstract	The classical approach to investigating polynomial eigenvalue problems is linearization, where the underlying matrix polynomial is converted into a larger matrix pencil with the same eigenvalues. For any polynomial there are infinitely many linearizations with widely varying properties, but in practice the companion forms are typically used. However, these companion forms are not always entirely satisfactory, and linearizations with special properties may sometimes be required. Given a matrix polynomial P, we develop a systematic approach to generating large classes of linearizations for P. We show how to simply construct two vector spaces of pencils that generalize the companion forms of P, and prove that almost all of these pencils are linearizations for P. Eigenvectors of these pencils are shown to be closely related to those of P. A distinguished subspace, denoted DL(P), is then isolated, and the special properties of these pencils are investigated. These spaces of pencils provide a convenient arena in which to look for structured linearizations of structured polynomials, as well as to try to optimize the conditioning of linearizations. Many applications give rise to nonlinear eigenvalue problems with an underlying structured matrix polynomial; perhaps the most well-known are symmetric and Hermitian polynomials. In this thesis we also identify several less well-known types of structured polynomial (e.g., palindromic, even, odd), explore the relationships between them, and illustrate their appearance in a variety of applications. Special classes of linearizations that reflect the structure of these polynomials, and therefore, preserve symmetries in their spectra, are introduced and investigated. We analyze the existence and uniqueness of such linearizations, and show how they may be systematically constructed. The infinitely many linearizations of any given polynomial P can have widely varying eigenvalue condition numbers. We investigate the conditioning of linearizations from DL(P), looking for the best conditioned linearization in that space andcomparing its conditioning with that of the original polynomial. We also analyze the eigenvalue conditioning of the widely used first and second companion linearizations, and find that they can potentially be much more ill conditioned than P. These results are phrased in terms of both the standard relative condition number and the condition number of Dedieu and Tisseur for the problem in homogeneous form, this latter condition number having the advantage of applying to zero and infinite eigenvalues.	en
heal.advisorName	Ψαρράκος, Παναγιώτης	el
heal.committeeMemberName	Αρβανιτάκης, Αλέξανδρος	el
heal.committeeMemberName	Κανελλόπουλος, Βασίλειος	el
heal.academicPublisher	Εθνικό Μετσόβιο Πολυτεχνείο. Σχολή Εφαρμοσμένων Μαθηματικών και Φυσικών Επιστημών. Τομέας Μαθηματικών	el
heal.academicPublisherID	ntua
heal.numberOfPages	75 σ.	el
heal.fullTextAvailability	true