A Condition Analysis of the Weighted Linear Least Squares Problem Using Dual Norms
For researchers and practitioners solving weighted least squares problems, this work offers efficient and accurate condition number estimation, though it is an incremental refinement of existing perturbation theory.
This paper defines condition numbers for linear solution functions of weighted linear least squares problems using dual norms and adjoint operators, deriving explicit expressions and efficient estimators. Numerical experiments show the estimators provide accurate perturbation bounds and reveal componentwise conditioning.
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.