Στοχαστικοί εκτιμητές του ίχνους συναρτήσεων πινάκων - Φίκα, Παρασκευή (Εθνικό Με5τσόβιο Πολυτεχνείο. Σχολή Εφαρμοσμένων Μαθηματικών και Φυσικών Επιστημών, 2016)
In this postgraduate work, stochastic estimates for the trace of functions of matrices, denoted as Tr(f(A)), are developed, where f is a suitable function and A is a symmetric matrix. This aims to the avoidance of the explicit computation of the matrix f(A). The trace of a matrix function arises in many disciplines such as in physics, in statistics, in network analysis, etc. The computation of a function of a matrix A can have a high computational cost, especially for large matrices that are required in applications. Therefore, it is necessary to estimate the trace of f(A). Stochastic estimates have been proposed, which are based on the computation of the quantity \sum_1^N x_i^T f(A)x_i /N for different sample vectors x_i, where N is the sample size (the number of the sample vectors). It can be proved that the expected value of the quantity x^T f(A)x, for suitable vector x and symmetric matrix A, equals to the Tr(f(A)). Moreover, since the proposed estimates are of a statistical nature, it is natural to expect that the application of some statistical designs may improve their quality and their effectiveness in estimating the trace Tr(f(A)). Various statistical designs will be compared and numerical examples stemming from real applications will be discussed.