Gillespie, Mary Irene2024-08-132024-08-1319921992https://hdl.handle.net/1828/17895The QR factorization of a matrix A is commonly used to solve least squares problems. One way to compute such a factorization is by using Givens transformations. When A is sparse, the precise order in which the transformations are applied can affect the amount of storage required. In this work we present an ordering for the Givens transformations that is opti­mal with regard to storage (a so-called "tight" ordering) and that preserves sparsity by restricting fill to those locations in R that are necessarily nonzero when A has the Hall property. This ordering is of particular interest when A does not have the strong Hall property and is not permuted into block upper trapezoidal form. We describe a bipartite graph model of sparse matrix structures, and summarize the characterization of the structures of the factors Q and R. We define the product of structures of matrices, determine the product of the structures of a sequence of Givens transformations, and define a tight ordering for the Givens transformations We then present a family of tight orderings for a given matrix structure.75 pagesAvailable to the World Wide WebUN SDG 11: Sustainable Cities and CommunitiesThesis