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

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(L1/2(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.

► For weighted linear least squares problems, effective condition numbers Cond_eff are explored.
► The extremely accurate leading coefficient of Motz's problem is explained by very small Cond_eff.
► The effective condition number may become a new trend of stability analysis of numerical PDE.