Connolly, Michael P. and Higham, Nicholas J. (2022) Probabilistic Rounding Error Analysis of Householder QR Factorization. [MIMS Preprint] (Submitted)
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Abstract
The standard worstcase 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 twosided transformations by Householder matrices and hence to standard QRtype algorithms for computing eigenvalues and singular values. It also applies to Givens QR factorization. These results complement recent probabilistic rounding error analysis results for innerproduct 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 worstcase bounds.
Item Type:  MIMS Preprint 

Uncontrolled Keywords:  floatingpoint 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 
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Probabilistic Rounding Error Analysis of Householder QR Factorization. (deposited 15 Feb 2022 14:20)

Probabilistic Rounding Error Analysis of Householder QR Factorization. (deposited 08 Aug 2022 18:59)
 Probabilistic Rounding Error Analysis of Householder QR Factorization. (deposited 13 Apr 2023 08:35) [Currently Displayed]

Probabilistic Rounding Error Analysis of Householder QR Factorization. (deposited 08 Aug 2022 18:59)
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