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undergraduate thesis
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Methods for determing QR decomposition
Valentina Per (2018)
Josip Juraj Strossmayer University of Osijek
Department of Mathematics
Chair of Applied Mathematics
Applied Mathematics Research Group
Metadata
TitleMetode za određivanje QR dekompozicije
AuthorValentina Per
Mentor(s)Darija Marković (thesis advisor)
Abstract
U ovom radu najprije ćemo objasnit što je to QR dekompozicija matrice. Nakon toga ćemo navesti tri metode uz pomoću kojih možemo odrediti QR dekompoziciju dane matrice. Kao prvu od metoda za određivanje QR dekompozicije pojasnit ćemo Gram–Schmidtov postupak ortogonalizacije. Navest ćemo osnovne definicije i teoreme vezane za ortogonalnost te pojasniti sam Gram–Schmidtov postupak. Opisanu metodu ilustrirat ćemo primjerom. U drugom poglavlju pojasnit ćemo QR dekompoziciju pomoću Householderovih transformacija. Definirat ćemo Householderove matrice i Householderov vektor. Opisat ćemo kako odredujemo QR dekompoziciju dane matrice te metodu ilustrirati primjerom. U posljednjem poglavlju pojasnit ćemo kako možemo odrediti QR dekompoziciju pomoću Givensovih rotacija. Uvest ćemo pojam matrica rotacija te navesti neka njihova osnovna svojstva. Zatim ćemo pojasniti Givensove rotacije te kako uz pomoć njih odrđujemo QR dekompoziciju. Također ćemo kao i u prethodnim poglavljima opisanu metodu ilustrirati primjerom.
KeywordsQR decomposition Gram-Schmidt orthogonalization process Householder matrix Givens rotations
Parallel title (English)Methods for determing QR decomposition
GranterJosip Juraj Strossmayer University of Osijek
Department of Mathematics
Lower level organizational unitsChair of Applied Mathematics
Applied Mathematics Research Group
PlaceOsijek
StateCroatia
Scientific field, discipline, subdisciplineNATURAL SCIENCES
Mathematics
Algebra
Study programme typeuniversity
Study levelundergraduate
Study programmeUniversity undergraduate study programme in mathematics
Academic title abbreviationuniv.bacc.math.
Genreundergraduate thesis
Language Croatian
Defense date2018-09-20
Parallel abstract (English)
In this paper we will first explain what a QR decomposition of the matrix is. After that, we will name three methods which can help us determine the QR decomposition of any given matrix. First method to be explained will be the Gram-Schmidt orthogonalization process. We will list the main definitions and theorems significant for orthogonalization and the Gram-Schmidt process itself will be explained. Described method we will apply on a given example. In second chapter we will clarify QR decomposition using Householder transformations. Householder matrixes and Householder vector will be define. We will clarify how to determine QR decomposition of a given matrix and apply it on an example. In last chapter we will explain how to determine QR decomposition using Givens rotations. We will define the term of matrix rotation and name some basic properties. Then, we will clarify Givens rotations and how to use them to determine QR decomposition. Also, we will explain the procedure using an example.
Parallel keywords (Croatian)QR dekompozicija Gram-Schmidtov postupak ortogonalizacije Householderove matrice Givensove rotacije
Resource typetext
Access conditionOpen access
Terms of usehttp://rightsstatements.org/vocab/InC/1.0/
URN:NBNhttps://urn.nsk.hr/urn:nbn:hr:126:995552