Householder QR Factorization: Adding Randomization for Column Pivoting. FLAME Working Note #78

Per-Gunnar Martinsson, Gregorio Quintana-Orti, Nathan Heavner, Robert van de Geijn

Published 2015-12-08Version 1

A fundamental problem when adding column pivoting to the Householder QR factorization is that only about half of the computation can be cast in terms of high performing matrix-matrix multiplication, which greatly limits the benefits of so-called blocked algorithms. This paper describes a technique for selecting groups of pivot vectors by means of randomized projections. It is demonstrated that the asymptotic flop count for the proposed method is $2mn^2 - (2/3)n^3$ for an $m\times n$ matrix with $m \geq n$, identical to that of the best classical Householder QR factorization (with or without pivoting). Experiments demonstrate improvements in speed of substantial integer factors (between a factor of 3 and 5) relative to LAPACK's geqp3 implementation on a modern CPU.