Parallel Strategies for Rank-k Updating of the QR Decomposition

Parallel strategies based on Givens rotations are proposed for updating the QR decomposition of an n × n matrix after a rank-k change (k < n). The complexity analyses of the Givens algorithms are based on the total number of Givens rotations applied to a 2-element vector. The algorithms, which are extensions of the rank-1 updating method, achieve the updating using approximately 2(k + n) compound disjoint Givens rotations (CDGRs) with elements annihilated by rotations in adjacent planes. Block generalization of the serial rank-1 algorithms are also presented. The algorithms are rich in level 3 BLAS operations, making them suitable for implementation on large scale parallel systems. The performance of some of the algorithms on a 2-D SIMD (single instruction stream–multiple instruction stream) array processor is discussed.