## 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 |

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 |