SVIBOR - Project code: 1-01-252

MINISTRY OF SCIENCE AND TECHNOLOGY

Strossmayerov trg 4, HR - 10000 ZAGREB
tel.: +385 1 459 44 44, fax: +385 1 459 44 69
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Project code: 1-01-252


Numerical methods of linear algebra


Main researcher: HARI, VJERAN (15335)


Assistants
Type of research: basic
Duration from: 01/01/91. to 12/31/93.
Papers on project (total): 28
Papers on project quoted in Current Contents: 11
Institution name: Prirodoslovno-matematički fakultet, Matematički odjel, Zagreb (37)
Department/Institute: Department of Mathematics, University of Zagreb
Address: Bijenička cesta 30
City: 10000 - Zagreb, Croatia
Communication
Phone: (385 1) 4555720/117
E-mail: hari@math.hr
Fax: (385 1) 432484
Summary: The work is devoted to matrix algorithms. The main field of research is diagonalization methods for eigenvalue and singular value problems. Special attention is paid to derivation, construction and investigation of algorithms which can compute eigendecompositions and singular value decompositions with high relative accuracy. It is known that only a few of the standard methods and under special constraints have that property. The task was to find new (primarily diagonalization) methods that will have this property under more general constraints. From this area, as part of the project, two Ph. D. theses (I. Slapničar, Z. Drmač), two M.S. theses (from project only Sanja Singer) and a series of notable articles (some of them already published) have been written. The second field of research is connected with asymptotical behavior of diagonalization methods. Rate of convergence described in terms of linear, quadratic and cubic convergence essentially determines numerical efficiency of algorithms. Very sharp quadratic and cubic convergence bounds for the symmetric and skew-symmetric Jacobi methods have been proved (Hari). Sharp quadratic convergence bounds for J-symmetric method (Drmač, Hari) and for method for the simultaneous diagonalization of a pair of symmetric matrices (Slapničar, Hari) have been proved. Special investigations have been devoted to (refers to published articles only): 1. Studying structure of a pair of almost diagonal Hermitian matrices with multiple eigenvalues (Hari), 2. Construction of a modified Jacobi method for Hermitian matrices which requires 25% less flops per cycle than the existing Jacobi methods (Hari), 3. Construction of a parallel SVD algorithm (Manger, Slapničar), 4. Implementation of parallel "for" loops on multiprocessors (Manger), 5. Relative perturbations of Hermitian matrices and perturbations of spectral projectors (Slapničar).

Keywords: Eigenvalues and eigenvectors, SVD, diagonalization methods, global and asymptotic convergence, relative accuracy.

Research goals: As was said in the proposal of the project five years ago, the aim of the project was to form a group of researchers recognized in the European (or maybe in the world) community of numerical linear algebraists. This demanding task presumed presence of several devoted young researchers and a good leadership in scientific research. To this end a good collaboration was established with the known expert Professor K. Veselic from Hagen, Germany. The idea behind was that young researchers finish grade level study (which ends with an M.S. thesis) in Zagreb and then continue their research in Hagen by dr Veselic. Having more free time (no lectures to attend, less assistant duties), better computing facilities and an excellent supervisor, they could finish their Ph. D. theses in Hagen. After comeback they continue their work, publish articles and with time show that there is a recognized group of numerical linear algebraists in Croatia. As such, the members of the group will be visiting foreign scientific centers and come back with even better research experience and knowledge. Since all the researchers included in the project are working at Croatian universities this will also revive or/and enhance the level of scientific work at domestic universities.

OTHER ACHIEVEMENTS


  1. Name: I. Slapničar je za svoj doktorat 1992. dobio nagradu "Preis zur Forderung junger Wissenschaftler der Fernuniversitat Hagen".
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