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Perturbation Analyses for the QR Factorization (1997)
| Venue: | SIAM J. Matrix Anal. Appl |
| Citations: | 20 - 11 self |
BibTeX
@ARTICLE{Chang97perturbationanalyses,
author = {Xiao-wen Chang and Christopher. C. Paige and G. W. Stewart},
title = {Perturbation Analyses for the QR Factorization},
journal = {SIAM J. Matrix Anal. Appl},
year = {1997},
volume = {18},
pages = {775--791}
}
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Abstract
This paper gives perturbation analyses for Q 1 and R in the QR factorization A = Q 1 R, Q T 1 Q 1 = I, for a given real m \Theta n matrix A of rank n. The analyses more accurately reflect the sensitivity of the problem than previous normwise results. The condition numbers here are altered by any column pivoting used in AP = Q1R, and the condition numbers for R are bounded for a fixed n when the standard column pivoting strategy is used. This strategy tends to improve the condition of Q 1 , so the computed Q 1 and R will probably both have greatest accuracy when we use the standard column pivoting strategy. First order normwise perturbation analyses are given for both Q 1 and R. It is seen that the analysis for R may be approached in two ways --- a detailed "matrix--vector equation" analysis which provides tight bounds and resulting true condition numbers, which unfortunately are costly to compute and not very intuitive, and a perhaps simpler "matrix equation" analysis which provides results that are usually weaker but easier to interpret, and which allow efficient computation of a satisfactory estimate for the true condition number. Key Words. QR factorization, perturbation analysis, condition estimation, matrix equations, pivoting AMS Subject Classifications: 15A23, 65F35 1.
Keyphrases
perturbation analysis qr factorization standard column true condition number condition number column pivoting satisfactory estimate simpler matrix equation analysis detailed matrix vector equation analysis ap q1r previous normwise result tight bound matrix equation am subject classification condition estimation first order normwise perturbation analysis efficient computation key word
