Numerical linear algebra (accurate and fast matrix algorithms, solving eigenvalue and singular value problems, perturbation theory, eigenvalue and singular value problems for matrices with elements from noncommutative fields and other numbers types)
Operator theory
Applications: inverse problem for ultrasound image acquisition and analysis, factorizations of matrices with low displacement rank and high sparsity, zeros of polynomials
Fast optimal damping of oscillating systems
Discrete evolutionary models for periodical organisms
Applications of quaternions and reduced biquaternions on image processing
Generalized inverses of matrices
Analysis of symmetric polynomials and their combinatorial interpretations and applications
Description of laboratory and equipment
Five offices with 4 computers and two workstations, one laboratory (lecture room).
Contacts with academic and other institutions
Indian Institute of Technology Indore (IIT), India
University of Cádiz, Spain
Berlin University of Technology, Germany
The Pennsylvania State University, USA
École Polytechnique Fédérale de Lausanne (EPFL), Switzerland
Josip Juraj Strossmayer University of Osijek, School of Applied Mathematics and Informatics (MATHOS), Croatia
Massachusetts Institute of Technology (MIT), USA
Utah State University, USA
Adolfo Ibáñez University, Santiago, Chile
University of Zagreb, Faculty of Science (PMF), Croatia
project title
Matrix Algorithms and Applications (MATAL)
Description of research in a 5-year term
The ideas used during the development of the forward stable algorithm for computing eigenvalues and eigenvectors of arrowhead matrices and rank-one modifications of diagonal (DPR1) matrices – (published papers by N. Jakovčević Stor, I. Slapničar and J. Barlow) will be applied to new classes of matrices: DPRk matrices, block matrices, complex matrices, matrices of quaternions, …, and new problems: updating SVD decomposition; nonnegative factorizations; inverses, eigen- and singular values and vectors in noncommutative fields, Takagi factorization, etc. Also, for the stated problems we will develop deflation theory for more efficient computing. The methods will also be applied to various structured matrices like Toeplitz, Hankel, and Cauchy matrices.
Computing zeros of polynomials over different fields (real numbers, complex numbers, quaternions, reduced biquaternions and dual numbers).
Solving inverse problems for matrices that arise in the analysis of ultrasound images. This is joint research with The Pennsylvania State University.
Development of fast algorithms for optimization of vibrating systems which depend on parameters. This is joint research with Josip Juraj Strossmayer University in Osijek. We will develop an algorithm for computing eigenvalue decomposition of complex and quaternionic DPRk matrices which uses fast multiplication of Cauchy-like matrices.
Generalizing the finished work on a discrete universal evolutionary model for pair of periodical organisms, including analysis of this model with respect to convergence, local minima, and eventual loops, to the case of three periodical organisms
Developing matrix algorithms in noncommutative associative algebras (HRZZ research project MANAA), in particular in algebras of quaternions and tensors, but also in other number systems. Standard and Jacobi-type methods for eigenvalue and singular value problems of one matrix or a pair of matrices in such algebras will be developed and analyzed.
Analysis of symmetric polynomials and their combinatorial interpretations and applications. In particular, the analysis of applications to graphs associated to various number sequences and connections to set partitions. Proving properties of induced combinatorial structures.
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