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Scientific project: Accurate and fast matrix algorithms and applications

Project code  023-0372783-1289

Project name  Accurate and fast matrix algorithms and applications

Senior researcher  Ivan Slapničar

Institution of employment University of Split, Faculty of Electrical Engineering, Mechanical Engineering a...

Program name Matrix Algorithms and Applications

Scientific area  natural science

Peer review group Mathematics

Summary The research deals with seven algorithms of within numerical linear algebra. All assumptions are based on the existing results and the contributions of the collaborators. All algorithms have applications in one of the strategic research directions of the Republic of Croatia. A. We develop new implementation of the two-step method for solving the eigenvalue problem for symmetric indefinite matrix which will, as has already been done for the positive definite case, keep its property of attaining high relative acuracy, and at the same time be faster than the QR method. B. Clustering of the nodes of a graph is based on bipartitioning of the nodes according to the signs of the Fiedler vector of the Laplace matrix of the graph. Laplace matrix is a matrix of low displacement rank. This property is used to derive an algorithm for fast multiplication of the Laplace matrix and a vector, without forming the matrix itself, which will lead to a new fast algorithm for for graph bipartitioning. C. Computing eigenvalues and eigenvectors of arrowhead matrices is an important part of many algorithms. We shall develop new algorithm which will, unlike the present algorithms, compute all eigenvalues and all components of eigenvectors with high relative accuracy. D. We will develop improved algorithms for updating of matrix decompositions (SVD and ULV) in the cases when those decompositionsa are used for low rank approximations of given matrix. E. We will analyse posibilites of apllication of the Falk-Langemeyer metod for computing eigenvalues of definite matrix pairs to the problem of detecting definiteness of the pair, classification of quadratic eigenvalue problems and the problem of computing the distance to the nearest definite pair. F. We will develop a new algorithm for fast clustering of images and textual data based on the fast multipole method for solving the n body problem. G. Known matrix and operator models for spatio-temporal modelling of the invasion of fungal diseases in crops will be applied to modelling other types of infections. We will develop parallel versions of the algorithms in items A-F. The algorithms in item A. have applications in all natural and technical sciences. Algoritms in B., D. and F. will be applied to extraction of knowledge (data mining) in textual data bases, and especially for Croatian web space, for detection of forest fires, analysis of biological data and medical images.

Key words |eigenvalues and eigenvectors|Jacobi and Falk-Langemeyer method,graph nodes clustering,singular value decomposition,data mining,image processing

Contract from 2.1.2007

Croatian scientific bibliography http://bib.irb.hr/lista-radova?sif_proj=023-0372783-1289&print=true&lang=EN

Address for communication
 Name and surname Ivan Slapničar
 Street and no. Trondheimska 4c, 21000, Split
 E-mail ivan.slapnicar@fesb.hr
 Web address http://www.fesb.hr/~slap
Project assistants 
  Name and Surname  Status
Nevena Jakovčević Stor Scientific novice
Damir Krstinić Scientific novice
Ivančica Mirošević Scientific novice
Ivan Slapničar Project leader