Condition numbers for a linear function of the solution of the linear least squares problem with equality constraints

Abstract

In this paper, we consider the normwise, mixed and componentwise condition numbers for a linear function L x of the solution x to the linear least squares problem with equality constraints (LSE). The explicit expressions of the normwise, mixed and componentwise condition numbers are derived. Also, we revisit some previous results on the condition numbers of linear least squares problem (LS) and LSE. It is shown that some previous explicit condition number expressions on LS and LSE can be recovered from our new derived condition numbers’ formulas. The sharp upper bounds for the derived normwise, 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. Moreover, the proposed condition estimation methods can be incorporated into the generalized QR factorization method for solving LSE. The numerical examples show that when the coefficient matrices of LSE are sparse and badly-scaled, the mixed and componentwise condition numbers can give sharp perturbation bounds, on the other hand normwise condition numbers can severely overestimate the exact relative errors because normwise condition numbers ignore the data sparsity and scaling. However, from the numerical experiments for random LSE problems, if the data is not either sparse or badly scaled, it is more suitable to adopt the normwise condition number to measure the conditioning of LSE since the explicit formula of the normwise condition number is more compact.