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undergraduate thesis
Sylvesterova i Ljapunovljeva matrična jednadžba
2020. urn:nbn:hr:126:517550
Kovačević, Maja
Josip Juraj Strossmayer University of Osijek
Department of Mathematics
Chair of Applied Mathematics and Computer Science
Geometry and Mathematics Teaching Mathematics Research Group

Cite this document

Kovačević, M. (2020). Sylvesterova i Ljapunovljeva matrična jednadžba (Undergraduate thesis). Osijek: Josip Juraj Strossmayer University of Osijek, Department of Mathematics. Retrieved from https://urn.nsk.hr/urn:nbn:hr:126:517550

Kovačević, Maja. "Sylvesterova i Ljapunovljeva matrična jednadžba." Undergraduate thesis, Josip Juraj Strossmayer University of Osijek, Department of Mathematics, 2020. https://urn.nsk.hr/urn:nbn:hr:126:517550

Kovačević, Maja. "Sylvesterova i Ljapunovljeva matrična jednadžba." Undergraduate thesis, Josip Juraj Strossmayer University of Osijek, Department of Mathematics, 2020. https://urn.nsk.hr/urn:nbn:hr:126:517550

Kovačević, M. (2020). 'Sylvesterova i Ljapunovljeva matrična jednadžba', Undergraduate thesis, Josip Juraj Strossmayer University of Osijek, Department of Mathematics, accessed 16 November 2024, https://urn.nsk.hr/urn:nbn:hr:126:517550

Kovačević M. Sylvesterova i Ljapunovljeva matrična jednadžba [Undergraduate thesis]. Osijek: Josip Juraj Strossmayer University of Osijek, Department of Mathematics; 2020 [cited 2024 November 16] Available at: https://urn.nsk.hr/urn:nbn:hr:126:517550

M. Kovačević, "Sylvesterova i Ljapunovljeva matrična jednadžba", Undergraduate thesis, Josip Juraj Strossmayer University of Osijek, Department of Mathematics, Osijek, 2020. Available at: https://urn.nsk.hr/urn:nbn:hr:126:517550

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Metadata
TitleSylvesterova i Ljapunovljeva matrična jednadžba
Title (english)Sylvester and Lyapunov Matrix Equation
AuthorMaja Kovačević
MentorIvana Kuzmanović Ivičić (mentor)
GranterJosip Juraj Strossmayer University of Osijek
Department of Mathematics
(Chair of Applied Mathematics and Computer Science)
(Geometry and Mathematics Teaching Mathematics Research Group)
Osijek
Defense date and country2020-09-29, Croatia
Scientific / art field,
discipline and subdiscipline		NATURAL SCIENCES
Mathematics
Numerical Mathematics
Abstract
U ovom radu bavit ćemo se Sylvesterovom jednadžbom koja je dobila ime po engleskom matematičaru Jamesu Josephu Sylvesteru i njezinim specijalnim slučajem koji se naziva Ljapunovljeva matrična jednadžba po ruskom matematičaru Aleksandru Mikhailovichu Ljapunovu. U prvom poglavlju navest ćemo samu definiciju jednadžbi te definirati Kroneckerov produkt i operator vektorizacije koji će nam pomoći pri numeričkom rješavanju jednadžbi. Od velike važnosti su nam Schurova i Hessenbergova forma matrica koje ćemo korisiti u Bartels - Stewartovom algoritmu. U drugom poglavlju opisat ćemo postojanje i jedinstvenost rješenja jednadžbi te iskazati i dokazati pripadne tvrdnje. U trećem poglavlju predstavit ćemo najučinkovitiji Bartels - Stewartov algoritam za jednadžbe malih dimenzija pomoću kojeg rješavamo Sylvesterovu i Ljapunovljevu jednadžbu. Algoritam ćemo opisati u 4 koraka te ćemo ga provesti za Ljapunovljevu jednadžbu. Analogno tome provest ćemo Bartels - Stewartov algoritam za Sylvesterovu jednadžbu gdje će nam matrice A i B biti različitih dimenzija pa ćemo uvesti posebnu metodu koja se naziva Hessenberg - Schurova metoda koju ćemo koristiti u algoritmu za rješavanje jednadžbi tog tipa.
Abstract (english)
In this paper we will talk about Sylvester’s equation named after the English mathematician James Joseph Sylvester and its special case called Lyapunov’s matrix equation after the Russian mathematician Alexander Mikhailovich Lyapunov. In the first chapter, we will state the definition of equations and define the Kronecker product and vectorization operator that will help us in numerical solving of equations. Of great importance to us are the Schurov and Hessenberg form of matrix that we will use in the Bartels - Stewart algorithm. In the second chapter we will describe the existence and uniqueness of the solution of the equations and state and prove the corresponding theorems. In the third chapter we will present the most efficient Bartels - Stewart algorithm for small equations by which we solve the Sylvester and Lyapunov equation. We will describe the algorithm in 4 steps and implement it for Lyapunov’s equation. Analogously, we will implement the Bartels - Stewart algorithm for the Sylvester equation where the matrices A and B will be of different dimensions, so we will introduce a special method called Hessenberg - Schur method, which we will use in the algorithm to solve equations of this type.
Keywords
Keywords (english)
Languagecroatian
URN:NBNurn:nbn:hr:126:517550
Study programmeTitle: University undergraduate study programme in mathematics
Study programme type: university
Study level: undergraduate
Academic / professional title: sveučilišni/a prvostupnik/prvostupnica (baccalaureus/baccalaurea) matematike (sveučilišni/a prvostupnik/prvostupnica (baccalaureus/baccalaurea) matematike)
Type of resourceText
File originBorn digital
Access conditionsOpen access
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Created on2020-11-05 08:02:55