Using dual techniques to derive componentwise and mixed condition numbers for a linear function of a linear least squares solution

Abstract

We prove duality results for adjoint operators and product norms in the framework of Euclidean spaces. We show how these results can be used to derive condition numbers especially when perturbations on data are measured componentwise relatively to the original data. We apply this technique to obtain formulas for componentwise and mixed condition numbers for a linear function of a linear least squares solution. These expressions are closed when perturbations of the solution are measured using a componentwise norm or the infinity norm and we get an upper bound for the Euclidean norm.

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

“…Similarly, for linear operators from G to E, the norm induced from the dual norms · E * and · G * , is denoted by · G * ,E * . For the adjoint operators and dual norms, we have the following result [3]:…”

“…Let v * be the dual norm of v and satisfy the usual inner product on R p , we are interested in the dual norm · v * of the product norm · v which satisfies the scalar product in E. The following result can be found in [3].…”

“…In particular, for the full rank linear least squares problem, [14] and [2] present the condition number when the perturbations in both the data and solution are measured by norms. In [3], the perturbation in the data is componentwise, whereas the perturbation in the solution can be either componentwise or normwise, leading to componentwise or mixed condition numbers.…”

A note on stable perturbations of Moore–Penrose inverses

“…Let be a matrix perturbed from A . Pursuing an upper bound for plays a significant role in investigating the perturbation behavior of linear least squares problems .…”

“…Let N A D A C A 2 R m n be a matrix perturbed from A. Pursuing an upper bound for k N A k plays a significant role in investigating the perturbation behavior of linear least squares problems [1,2,[4][5][6][7][8][9][10].…”