Naslov
Krylov type methods for large scale eigenvalue computations

Autori
Bujanović, Zvonimir

Vrsta, podvrsta i kategorija rada
Ocjenski radovi, doktorska disertacija

Fakultet
Prirodoslovno matematički fakultet - Matematički odsjek

Mjesto
Zagreb

Datum
01.04

Godina
2011

Stranica
158

Mentor
Drmač, Zlatko

Ključne riječi
matrix eigenvalue problem; Arnoldi method; Ritz values; Hessenberg reduction

Sažetak
This thesis is devoted to the large scale eigenvalue problem, in particular to the Arnoldi algorithm for computing a few eigenvalues of large matrices. It is shown how the Krylov-Schur restarting method can be used with any choice of shifts. On the other hand, the connection with the pole placement problem is made, and this problem is known to generally be ill conditioned. Next, geometry of the Ritz values for normal matrices is studied ; this is an important problem in convergence theory of the Arnoldi algorithm restarted using the exact shifts. The necessary and the sufficient condition for a given set of k complex numbers to appear as a set of Ritz values from a Krylov subspace is shown to be the existence of a positive solution to a linear system with a Cauchy matrix. This fact is used for derivation of simple proofs for some known facts for the Ritz values. An example of a normal matrix for which the restarted Arnoldi algorithm fails to compute the second largest eigenvalue is constructed. Also, a variant of the Cauchy interlacing lemma is shown to hold in the setting of normal matrices. Finally, a new blocked algorithm for the reduction of a matrix to the m-Hessenberg form is presented. This algorithm is superior to the existing implementation in the SLICOT software library. A variant of the algorithm that uses hybrid CPU+GPU computing is derived, exhibiting even higher performance.

Izvorni jezik
Engleski

Znanstvena područja
Matematika

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