Mentors: Joseph Prein (Former UNT Graduate student) and Dr. Nam Trang

Title: Investigating Exact Cancellations in Cholesky Decompositions Via Schur's Complements and Graphs

Abstract: This research project investigates solving systems of equations of the form Ax = b and its applications, where A is a sparse symmetric positive definite matrix, by considering the Cholesky factorization of A via graph-theoretic techniques, such as constructing fill graphs. The primary goal is to show whether or not there always exists permutations of a symmetric positive definite matrix with no exact cancellations in the fill graph. It has been shown so far that such permutations exist for the 3x3, 4x4, & the 5x5 matrices by focusing on specific subcases and showing that each subcase can be permuted so that there are no exact cancellations that aren't later filled-in after taking all the Schur's complements. The general nxn case is currently being investigated.