A coarse-grained parallel QR-factorization algorithm for sparse least squares problems

Abstract

A sparse QR-factorization algorithm SPARQR for coarse-grained parallel computations is described. The coefficient matrix, which is assumed to be general sparse, is reordered in an attempt to bring as many zero elements in the lower left corner as possible. The reordered matrix is then partitioned into block rows, and Givens plane rotations are applied in each block-row. These are independent tasks and can be done in parallel. Row and column permutations are carried out within the diagonal blocks in an attempt to preserve better the sparsity of the matrix. The algorithm can be used for solving least squares problems either directly or combined with an iterative method (preconditioned conjugate gradients are used). Small non-zero elements can optionally be dropped in the latter case. This leads to a better preservation of the sparsity and, therefore, to a faster factorization. The price which has to be paid is some loss of accuracy. The iterative method is used to regain the accuracy lost during the factorization. Numerical results from several experiments with matrices from the well-known Harwell–Boeing collection as well as with some larger sparse matrices are presented in this work. An SGI Power Challenge computer with 16 processors has been used in the experiments.