Greedy givens algorithms for computing the rank-k updating of the qr decomposition

Abstract

A Greedy Givens algorithm for computing the rank-1 updating of the QR decomposition is proposed. An exclusive-read exclusive-write parallel random access machine computational model is assumed. The complexity of the algorithms is calculated in two different ways. In the unlimited parallelism case a single time unit is required to apply a compound disjoint Givens rotation of any size. In the limited parallelism case all the disjoint Givens rotations can be applied simultaneously, but one time unit is required to apply a rotation to a two-element vector. The proposed Greedy algorithm requires approximately 5/8 the number of steps performed by the conventional sequential Givens rank-1 algorithm under unlimited parallelism. A parallel implementation of the sequential Givens algorithm outperforms the Greedy one under limited parallelism. An adaptation of the Greedy algorithm to compute the rank-k updating of the QR decomposition has been developed. This algorithm outperforms a recently reported parallel method for small k, but its efficiency decreases as k increases