Probabilistic Rounding Error Analysis of Householder QR Factorization

Abstract

The standard worst-case normwise backward error bound for Householder QR factorization of an $m\times n$ matrix is proportional to $mnu$, where $u$ is the unit roundoff. We prove that the bound can be replaced by one proportional to $\sqrt{mn}u$ 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 $m\times m$ Householder matrices to a vector satisfy bounds proportional to $\sqrt{m}u$ 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