A parallel sparse QR-factorization algorithm

Abstract

A sparse QR-factorization algorithm for coarse-grain parallel computations is described. Initially the coefficient matrix, which is assumed to be general sparse, is reordered properly in an attempt to bring as many zero elements in the lower left corner as possible. Then the matrix is partitioned into large blocks of rows and Givens rotations are applied in each block. These are independent tasks and can be done in parallel. Row and column permutations are carried out within the blocks to exploit the sparsity of the matrix.

The algorithm can be used for solving least squares problems either directly or combined with an appropriate iterative method (for example, the preconditioned conjugate gradients). In the latter case, dropping of numerically small elements is performed during the factorization stage, which often leads to a better preservation of sparsity and a faster factorization, but this also leads to a loss of accuracy. The iterative method is used to regain the accuracy lost during the factorization.

An SGI Power Challenge computer with 16 processors has been used in the experiments. Results from experiments with matrices from the Harwell-Boeing collection as well as with automatically generated large sparse matrices are presented in this work.