Generic numerical linear algebra for matrices with elements in one of eight number systems – real numbers, complkex numbers, quaternions, reduced biquaternions and dual numbers of those four types of numbers:
accurate and fast matrix algorithms
solving eigenvalue and singular value problems
QR factorization
algorithms for structured matrices
perturbation theory fot the above problems
Applications of generic algorithms to:
solving least squares problems
image analysis and classification
computation of generalized inverses
computing zeros of polynomials
Operator theory
Fast optimal damping of oscillating systems
Discrete evolutionary models for periodical organisms
Applications of quaternions and reduced biquaternions on image processing
Analysis of symmetric polynomials and their combinatorial interpretations and applications
Description of laboratory and equipment
Four computers and two workstations.
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 1-year term
An improved algorithm for deflating singular values of semi-arrow matrices will be developed. In addition to the standard approach, the algorithm is based on a detailed analysis of 2×2 submatrices.
Generic algorithms will be developed for computing eigenvalues and vectors, singular values and vectors, and QR decompositions for matrices whose elements are in one of eight number systems – real numbers, complex numbers, quaternions, reduced biquaternions, and dual numbers of the previously mentioned numbers. The only parts where the algorithms may differ are the computation of the Householder reflector and Givens rotation.
The algorithms from the previous point will be applied to solving the least squares problem, image processing and classification, computing generalized inverses, and computing zeros of polynomials.
In a similar way, a generic algorithm will be developed for the Falk-Langemeyer method for computing eigenvalues of vectors of definite pairs of matrices.
All algorithms will be implemented in the Julia programming language.
The possible application of algorithms for arrow, DPR1 and DPRk matrices, and other matrix algorithms to the analysis of ultrasound images will be examined.
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