PROBABILISTIC ROUNDING ERROR ANALYSIS OF HOUSEHOLDER QR FACTORIZATION

Abstract

The standard worst-case normwise backward error bound for Householder QR factorization of an m×n matrix is proportional to mnu, where u is the unit roundoff. We prove that the bound can be replaced by one proportional to √mnu that holds with high probability if the rounding errors are mean independent and of mean zero and if the normwise backward errors in applying a sequence of m × m Householder matrices to a vector satisfy bounds proportional to √mu with probability 1. The proof makes use of a matrix concentration inequality. The same square rooting of the error constant applies to two-sided transformations by Householder matrices and hence to standard QR-type algorithms for computing eigenvalues and singular values. It also applies to Givens QR factorization. These results complement recent probabilistic rounding error analysis results for inner product-based algorithms and show that the square rooting effect is widespread in numerical linear algebra. Our numerical experiments, which make use of a new backward error formula for QR factorization, show that the probabilistic bounds give a much better indicator of the actual backward errors and their rate of growth than the worst-case bounds.