Advertisement

Numerische Mathematik

, Volume 88, Issue 2, pp 319–345 | Cite as

Componentwise perturbation analyses for the QR factorization

  • Xiao-Wen Chang
  • Christopher C. Paige
Orignial article

Summary.

This paper gives componentwise perturbation analyses for Q and R in the QR factorization A=QR, \(Q^\mathrm{T}Q=I\), R upper triangular, for a given real $m\times n$ matrix A of rank n. Such specific analyses are important for example when the columns of A are badly scaled. First order perturbation bounds are given for both Q and R. The analyses more accurately reflect the sensitivity of the problem than previous such results. The condition number for R is bounded for a fixed n when the standard column pivoting strategy is used. This strategy also tends to improve the condition of Q, so usually the computed Q and R will both have higher accuracy when we use the standard column pivoting strategy. Practical condition estimators are derived. The assumptions on the form of the perturbation \(\Delta A\) are explained and extended. Weaker rigorous bounds are also given.

Mathematics Subject Classification (1991): 15A23, 65F35 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

Copyright information

© Springer-Verlag Berlin Heidelberg 2000

Authors and Affiliations

  • Xiao-Wen Chang
    • 1
  • Christopher C. Paige
    • 1
  1. 1.School of Computer Science, McGill University, Montreal, Quebec, Canada H3A 2A7; e-mail: chang@cs.mcgill.ca, paige@cs.mcgill.ca CA

Personalised recommendations