Probabilistic Rounding Error Analysis of Householder QR Factorization

Connolly, Michael P. and Higham, Nicholas J. (2022) Probabilistic Rounding Error Analysis of Householder QR Factorization. [MIMS Preprint] (Submitted)

This is the latest version of this item.

[thumbnail of paper.pdf] Text
paper.pdf
Download (430kB)

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.

Item Type: MIMS Preprint
Uncontrolled Keywords: floating-point arithmetic, backward error analysis, backward error, probabilistic rounding error analysis, Givens QR factorization, Householder QR factorization, matrix concentration inequality
Subjects: MSC 2010, the AMS's Mathematics Subject Classification > 15 Linear and multilinear algebra; matrix theory
MSC 2010, the AMS's Mathematics Subject Classification > 65 Numerical analysis
Depositing User: Nick Higham
Date Deposited: 13 Apr 2023 08:35
Last Modified: 13 Apr 2023 08:35
URI: https://eprints.maths.manchester.ac.uk/id/eprint/2885

Available Versions of this Item

Actions (login required)

View Item View Item