The ideas used during 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 and new problems: complex matrices, Takagi factorization, block matrices, matrices of quaternions, 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).

Solving inverse problems for matrices which arrise in analysis of ultrasound images. The is joint research with The Pennsylvania State University.  Development of fast algorithms for optimization of vibrationg 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 DPR1 matrices which uses fast multiplication of Cauchy-like matrices. We plan to develop discrete evolutionary model for periodical organisms, and to analyse this model with respect to convergence, local minima and eventual loops.   Methods for tensor decompositions and their applications will also be developed. We will study recompression of thensors in TT format and algorith which uses randomised rank-one method. This is joint work with EPF Lausanne.  We will also study eigenvalues of infinite Hamiltonian, mathematical theory for particle interaction in infinite quantum systems, and devlopment of algorithm which uses infinite tensor rings. This is joint work with Lawrence Berkeley National Laboratory. For all methods we will develop Julia programs and make them publicly accessible on GutHub.