# American Institute of Mathematical Sciences

2011, 1(2): 301-316. doi: 10.3934/naco.2011.1.301

## Multiplicative perturbation analysis for QR factorizations

 1 School of Computer Science, McGill University, Montreal, Quebec, Canada H3A 2A7, Canada 2 Department of Mathematics, University of Texas at Arlington, Arlington, TX 76019-0408, United States

Received  December 2010 Revised  May 2011 Published  June 2011

This paper is concerned with how the QR factors change when a real matrix $A$ suffers from a left or right multiplicative perturbation, where $A$ is assumed to have full column rank. It is proved that for a left multiplicative perturbation the relative changes in the QR factors in norm are no bigger than a small constant multiple of the norm of the difference between the perturbation and the identity matrix. One of common cases for a left multiplicative perturbation case naturally arises from the computation of the QR factorization. The newly established bounds can be used to explain the accuracy in the computed QR factors. For a right multiplicative perturbation, the bounds on the relative changes in the QR factors are still dependent upon the condition number of the scaled $R$-factor, however. Some optimized'' bounds are also obtained by taking into account certain invariant properties in the factors.
Citation: Xiao-Wen Chang, Ren-Cang Li. Multiplicative perturbation analysis for QR factorizations. Numerical Algebra, Control & Optimization, 2011, 1 (2) : 301-316. doi: 10.3934/naco.2011.1.301
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