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master's thesis
Parallel computation of the hyperbolic QR factorization
Zagreb: University of Zagreb, Faculty of Science, 2018. urn:nbn:hr:217:024075
Čaklović, Gayatri
University of Zagreb
Faculty of Science
Department of Mathematics

Institutional repository: Repository of the Faculty of Science

Cite this document

Čaklović, G. (2018). Parallel computation of the hyperbolic QR factorization (Master's thesis). Zagreb: University of Zagreb, Faculty of Science. Retrieved from https://urn.nsk.hr/urn:nbn:hr:217:024075

Čaklović, Gayatri. "Parallel computation of the hyperbolic QR factorization." Master's thesis, University of Zagreb, Faculty of Science, 2018. https://urn.nsk.hr/urn:nbn:hr:217:024075

Čaklović, Gayatri. "Parallel computation of the hyperbolic QR factorization." Master's thesis, University of Zagreb, Faculty of Science, 2018. https://urn.nsk.hr/urn:nbn:hr:217:024075

Čaklović, G. (2018). 'Parallel computation of the hyperbolic QR factorization', Master's thesis, University of Zagreb, Faculty of Science, accessed 23 December 2024, https://urn.nsk.hr/urn:nbn:hr:217:024075

Čaklović G. Parallel computation of the hyperbolic QR factorization [Master's thesis]. Zagreb: University of Zagreb, Faculty of Science; 2018 [cited 2024 December 23] Available at: https://urn.nsk.hr/urn:nbn:hr:217:024075

G. Čaklović, "Parallel computation of the hyperbolic QR factorization", Master's thesis, University of Zagreb, Faculty of Science, Zagreb, 2018. Available at: https://urn.nsk.hr/urn:nbn:hr:217:024075

Metadata
TitleParallel computation of the hyperbolic QR factorization
AuthorGayatri Čaklović
MentorSanja Singer (mentor)
Committee memberSanja Singer (predsjednik povjerenstva)
Committee memberEduard Marušić-Paloka (član povjerenstva)
Committee memberBoris Muha (član povjerenstva)
Committee memberTomislav Pejković (član povjerenstva)
GranterUniversity of Zagreb
Faculty of Science
(Department of Mathematics)
Zagreb
Defense date and country2018-07-18-07-18, Croatia
Scientific / art field,
discipline and subdiscipline		NATURAL SCIENCES
Mathematics
Abstract
U ovom radu prezentirali smo kako računati hiperboličku QR \(J\)-faktorizaciju. Prvo je postavljena teorija koja nam daje dva načina redukcije matrice \(G \in \mathbb{C}^{m \times n}, m \geq n,\) na blok gornjetrokutastu formu. Jedan način je redukcija jednog stupca pomoću \(J\)-Householderovog reflektora. Razjašnjeni su nužni i dovoljni uvjeti postojanja takvih operatora. Drugi način je redukcija dva stupca koristeći Givensove rotacije. U tom poglavlju je obrađeno što sve zovemo pravilnom (’proper’) formom, kako svesti matrice na pravilnu formu, te kako tu pravilnu formu do kraja reducirati \(J\)-unitarnim matricama manjih dimenzija. Nadalje, indefinitni QR povezali smo sa još jednom faktorizacijom, hermitskom indefinitnom faktorizacijom. Pokazali smo kako su te dvije faktorizacije povezane i koja je optimalna strategija pivotiranja u hermitskoj indefinitnoj faktorizaciji (ona koja ima najmanji pivotni rast u svakom slučaju izbora strategije, nebitno biraju li se dva ili jedan stupac za redukciju). Ista pivotna strategija primijenjena je i na QR faktorizaciju. Naposljetku, prezentiran je sekvencijalni algoritam redukcije matrice \(G\) na gorenje blok trokutastu formu, kao i njegovi dijelovi koji su paralelizirani. Pri optimizaciji koda, u obzir je uzeta i arhitektura memorije računala, te način funkcioniranja biblioteka OpenMP i MKL koje smo koristili za paralelizaciju. Testiranja na umjetno generiranim matricama izvedena su na Intelovom Xeon Phi 7210 računalu, gdje je također u obzir uzeta posebna memorijska arhitektura računala.
Abstract (english)
In this thesis, a way of computing a \(J\)-unitary QR factorization was presented. The theoretical part was set first, in order to explain two possible ways of transforming a matrix \(G \in \mathbb{C}^{m \times n}, m \geq n,\) into a block upper triangular matrix. One way to do this is with \(J\)-unitary Householder like reflectors. The necessary and sufficient conditions for their existence were formed. The other way to do this is using Givens rotations. In that chapter, a term proper form was defined, how to transform matrices into proper forms and, in the end, how are those proper forms fully reduced with \(J\)-unitary matrices of smaller dimensions. Furthermore, we showed how indefinite QR is connected to the Hermitian indefinite factorization. An optimal pivoting strategy for the Hermitian indefinite factorization was presented, based on minimizing the pivot growth (regardless of the fact that one or two columns were chosen as pivotal). The same strategy was then used in the QR factorization. At last, a sequential version of the algorithm for reducing the matrix \(G\) to a block upper triangular form was presented, as well as with the parallelised segments of it. The memory architecture was took into account while optimizing the code as well as the optimal usage of OpenMP and MKL libraries. The tests on randomly generated matrices were performed on Intel’s Xeon Phi 7210. The special architecture of Xeon Phi was also taken into account.
Keywords
Keywords (english)
Languagecroatian
URN:NBNurn:nbn:hr:217:024075
Study programmeTitle: Applied Mathematics
Study programme type: university
Study level: graduate
Academic / professional title: magistar/magistra matematike (magistar/magistra matematike)
Type of resourceText
File originBorn digital
Access conditionsOpen access
Terms of useLicence In copyright icon
Created on2018-11-22 12:58:01