Effective condition number for weighted linear least squares problems and applications to the Trefftz method

Abstract

In [27], the effective condition number Cond_eff is developed for the linear least squares problem. In this paper, we extend the effective condition number for weighted linear least squares problem with both full rank and rank-deficient cases. We apply the effective condition number to the collocation Trefftz method (CTM) [29] for Laplace's equation with a crack singularity, to prove that Cond_eff = O ( L ) and Cond = O ( L 1 / 2 ( 2 ) L ) , where L is the number of singular particular solutions used. The Cond grows exponentially as L increases, but Cond_eff is only O ( L ) . The small effective condition number explains well the high accuracy of the TM solution, but the huge Cond cannot.