Ordering Givens Rotations for Sparse $QR$ Factorization

Abstract

The $QR$ factorization of a large, sparse matrix A is frequently computed using Givens rotations. The precise order in which the rotations are applied can affect the amount of storage required. We present an ordering for the Givens rotations that, when A has the Hall property, is optimal with regard to storage for Q (a so-called “tight” ordering) and that preserves sparsity by restricting fill to those locations in R that are necessarily nonzero. 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 rotations, and specify a tight ordering for these transformations.