Mixed and componentwise condition numbers for a linear function of the solution of the linear least squares problem with equality constrains
Provides more accurate perturbation analysis for a specific constrained least squares problem, which is incremental for numerical linear algebra researchers.
The paper derives explicit expressions for mixed and componentwise condition numbers for a linear function of the solution to the linear least squares problem with equality constraints, and obtains sharp upper bounds that can be efficiently estimated. Numerical examples show these condition numbers provide sharp perturbation bounds, while normwise condition numbers overestimate relative errors.
In this paper, we consider the mixed and componentwise condition numbers for a linear function of the solution to the linear least squares problem with equality constrains (LSE). We derive the explicit expressions of the mixed and componentwise condition numbers through the dual techniques. The sharp upper bounds for the derived mixed and componentwise condition numbers are obtained, which can be estimated efficiently by means of the classical Hager-Higham algorithm for estimating matrix one-norm during using the generalized QR factorization method for solving LSE. The numerical examples show that the derived condition numbers can give sharp perturbation bounds, on the other hand normwise condition numbers can severely overestimate the relative errors because normwise condition numbers ignore the data sparsity and scaling.