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https://hdl.handle.net/123456789/1568| Τύπος: | Άρθρο σε επιστημονικό περιοδικό |
| Τίτλος: | Eigenvalue estimates via pseudospectra |
| Εναλλακτικός τίτλος: | Προσέγγιση ιδιοτιμών μέσω ψευδοφάσματος |
| Συγγραφέας: | [EL] Κατσουλέας, Γεώργιος[EN] Katsouleas, Georgios [EL] Παναγάκου, Βασιλική[EN] Panagakou, Vasiliki [EL] Ψαρράκος, Παναγιώτης[EN] Psarrakos, Panayiotis |
| Ημερομηνία: | 22/07/2021 |
| Περίληψη: | In this note, given a matrix A (or a general matrix polynomial P(z)) and an arbitrary scalar μ, we show how to define a sequence of complex numbers which converges to some element of its spectrum. The scalar μ serves as initial term, while additional terms are constructed through a recursive procedure, exploiting the fact that each term mk of this sequence is in fact a point lying on the boundary curve of some pseudospectral set of A (or P(z)). Then, the next term in the sequence is detected in the direction which is normal to this curve at the point. Repeating the construction for additional initial points, it is possible to approximate peripheral eigenvalues, localize the spectrum and even obtain spectral enclosures. Hence, as a by-product of our method, a computationally cheap procedure for approximate pseudospectra computations emerges. An advantage of the proposed approach is that it does not make any assumptions on the location of the spectrum. The fact that all computations are performed on some dynamically chosen locations on the complex plane which converge to the eigenvalues, rather than on a large number of predefined points on a rigid grid, can be used to accelerate conventional grid algorithms. Parallel implementation of the method or use in conjunction with randomization techniques can lead to further computational savings when applied to large-scale matrices. |
| Γλώσσα: | Αγγλικά |
| Σελίδες: | 18 |
| DOI: | 10.3390/math9151729 |
| EISSN: | 2227-7390 |
| Θεματική κατηγορία: | [EL] Εφαρμοσμένα μαθηματικά[EN] Applied Mathematics |
| Λέξεις-κλειδιά: | pseudospectra; eigenvalues; matrix polynomial; perturbation; Perron root; large-scale matrices; approximation algorithm |
| Κάτοχος πνευματικών δικαιωμάτων: | © 2021 by the authors. Licensee MDPI, Basel, Switzerland. |
| Όροι και προϋποθέσεις δικαιωμάτων: | This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (https:// creativecommons.org/licenses/by/ 4.0/). |
| Ηλεκτρονική διεύθυνση του τεκμηρίου στον εκδότη: | https://www.mdpi.com/2227-7390/9/15/1729 |
| Ηλεκτρονική διεύθυνση περιοδικού: | https://www.mdpi.com/journal/mathematics |
| Τίτλος πηγής δημοσίευσης: | Mathematics |
| Τεύχος: | 15 |
| Τόμος: | 9 |
| Σελίδες τεκμηρίου (στην πηγή): | Article no 1729 |
| Σημειώσεις: | This research is carried out/funded in the context of the project “Approximation algorithms and randomized methods for large-scale problems in computational linear algebra” (MIS 5049095) under the call for proposals “Researchers’ support with an emphasis on young researchers–2nd Cycle’.” The project is co-financed by Greece and the European Union (European Social Fund—ESF) by the Operational Programme Human Resources Development, Education and Lifelong Learning 2014–2020. |
| Εμφανίζεται στις συλλογές: | Ερευνητικές ομάδες |
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