A condition analysis of the weighted linear least squares problem using dual norms

Abstract

In this paper, based on the theory of adjoint operators and dual norms, we define condition numbers for a linear solution function of the weighted linear least squares problem. The explicit expressions of the normwise and componentwise condition numbers derived in this paper can be computed at low cost when the dimension of the linear function is low due to dual operator theory. Moreover, we use the augmented system to perform a componentwise perturbation analysis of the solution and residual of the weighted linear least squares problems. We also propose two efficient condition number estimators. Our numerical experiments demonstrate that our condition numbers give accurate perturbation bounds and can reveal the conditioning of individual components of the solution. Our condition number estimators are accurate as well as efficient.

Keywords:

• Weighted least squares
• condition number
• dual norm
• adjoint operator
• componentwise perturbation

AMS Subject Classifications:

• 65F20
• 65F35

Acknowledgements

The authors would like to thank the referee for his/her constructive comments, which significantly improve an earlier version of this paper.

Notes

No potential conflict of interest was reported by the authors.

Additional information

Funding

The third author was partially supported by Natural Science and Engineering Research Council (NSERC) of Canada [grant number RGPIN-2014-04252]; International Cooperation Project of Shanghai Municipal Science and Technology Commission [grant number 16510711200].