Kvadratični problem svojstvenih vrijednosti

master's thesis

• Author: Maša Avakumović
• Mentor: Zlatko Drmač (mentor)
• Committee member: Zlatko Drmač (predsjednik povjerenstva)
• Committee member: Josip Tambača (član povjerenstva)
• Committee member: Boris Muha (član povjerenstva)
• Committee member: Franka Miriam Bruckler (član povjerenstva)
• Granter: University of Zagreb; Faculty of Science; (Department of Mathematics); Zagreb
• Defense date and country: 2016-09-26, Croatia
• Scientific / art field, discipline and subdiscipline: NATURAL SCIENCES; Mathematics

Abstract

U ovom radu smo se bavili kvadratičnim problemom svojstvenih vrijednosti čiji je zadatak naći skalare \( \lambda \in \mathbb{C}\) i nenul vektore \(x \in \mathbb{C}^n\) tako da vrijedi \((M \lambda^2 + C\lambda + K)x = 0\), gdje su \(M, C, K \in \mathbb{C}^{n \times n}\). Ovako definiran problem ima \(2n\) svojstvenih vrijednosti kao rješenje, te svojstvene vrijednosti mogu biti i konačne i beskonačne. Prezentirali smo dvije različite numeričke metode koje rješavaju spomenuti problem; metoda quadeig i SOAR (eng. Second Order ARnoldi method) metoda. Metoda quadeig pripada klasi potpunih (direktnih) metoda, sto znači da računa svih 2n svojstvenih vrijednosti, a pogodna je za korištenje kada je dimenzija početnog problema relativno mala. Ta metoda se detaljnije bavi skaliranjem zadanih matrica M,C i K početnog problema, kao i problemom beskonačnih svojstvenih vrijednosti. Numeričkim eksperimentima smo usporedili quadeig s metodom polyeig koja je implementirana u paketu MATLAB-a. S druge strane, prezentirali smo SOAR metodu koja pripada klasi iterativnih metoda koje se primjenjuju kada je potrebno izračunati samo dio spektra koji je od interesa. Iterativne metode su iznimno efikasne u slučaju velikih dimenzija početnog problema, kada je primjena potpunih metoda iznimno skupa (memorijski i vremenski) ili cak nemoguća. SOAR metoda, koja je u ovom radu predložena kao efikasno rješenje kvadratičnih problema, je poopćenje Arnoldijeve metode za standardni svojstveni problem, a osnovna ideja joj je pronalazak pogodno odabranog potprostora koji će dobro aproksimirati prostor koji razapinju traženi svojstveni vektori. Jedan takav pogodan potprostor na kojem je bazirana SOAR metoda je Krilovljev potprostor drugog reda. Glavni problem iterativnih metoda je pitanje konvergencije. Na ubrzavanju konvergencije se i dalje aktivno radi. Svi numerički eksperimenti u ovom radu su napravljeni u MATLABu.

Abstract (english)

In this thesis we study the quadratic eigenvalue problem (QEP), that is for given \(M, C, K \in \mathbb{C}^{n \times n}\), task is to compute an eigenvalue \( \lambda \in \mathbb{C}\) and an eigenvector \(x \in \mathbb{C}^n\), \(x \neq 0\) \((M \lambda^2 + C\lambda + K)x = 0\). This QEP has \(2n\) eigenvaluess, some of them can even be infinite. We present two different numerical methods for solving QEP; quadeig method and SOAR (Second Order ARnoldi) method. Quadeig method is a complete (direct) method, meaning that it computes all \(2n\) eigenvalues, and we recommend applying this method when the dimension of the initial QEP is not too big. One of the improvements that quadeig offers is scaling of the given matrices M,C and K. The way that quadeig method deals with infinite eigenvalues and zero eigenvalues improves the accuracy of the approximations for the finite eigenvalues. Our numerical experiments compare quadeig method with MATLAB’s built-in function polyeig that solves the same QEP problem. Also, we investigate the SOAR method which belongs to the class of iterative methods. Unlike the complete methods, iterative methods are applied when we are interested only in a subset of the spectrum (only few eigenvalues). Those iterative methods are effective way of dealing with the problem of large dimension. Applying a complete method to such big problems would cause great memory cost and immense time consumption. SOAR method, that we suggest in this thesis for effectively solving QEP, is a generalization of the Arnoldi method for the standard eigenvalue problem and the basic idea is to find suitable subspace that is very close to the subspace that is spanned by the wanted eigenvectors. One of the good subspaces which is also the basis for the SOAR method is a second order Krylov subspace. The main problem of iterative methods is the convergence question. Enhancing the convergence is still one of the main topics in many researches and improvements are being made. All the numerical experiments in this thesis are made in MATLAB.

• Keywords: kvadratični problem svojstvenih vrijednosti; QEP; quadeig; SOAR; Second Order ARnoldi method; polyeig; MATLAB
• Keywords (english): quadratic eigenvalue problem; QEP; quadeig; SOAR; Second Order ARnoldi method; polyeig; MATLAB
• Language: croatian
• URN:NBN: urn:nbn:hr:217:674611
• Study programme: Title: Applied Mathematics; Study programme type: university; Study level: graduate; Academic / professional title: magistar/magistra matematike (magistar/magistra matematike)
• Type of resource: Text
• File origin: Born digital
• Access conditions: Open access
• Created on: 2017-05-05 12:31:14