Effective condition number and its applications

Consider the over-determined system Fx = b where F ∈ Rm × n, m ≥ n and rank (F) = r ≤ n, the effective condition number is defined by Cond_eff = ||b||/σ1||x||, where the singular values of F are given as σmax = σ1 ≥ σ1 ≥...≥ σr > 0 and σr+1 = ... = σn = 0. For the general perturbed system (A + ΔA)(x+ Δx) = b+ Δb involving both ΔA and Δb, the new error bounds pertinent to Cond_eff are derived. Next, we apply the effective condition number to the solutions of Motz's problem by the collocation Trefftz methods (CTM). Motz's problem is the benchmark of singularity problems. We choose the general particular solutions vL = Σk=0L dk(r/Rp)k+1/2 cos(k + 1/2)θ with a radius parameter Rp. The CTM is used to seek the coefficients di by satisfying the boundary conditions only. Based on the new effective condition number, the optimal parameter Rp = 1 is found. which is completely in accordance with the numerical results. However, if based on the traditional condition number Cond, the optimal choice of Rp is misleading. Under the optimal choice Rp = 1, the Cond grows exponentially as L increases, but Cond_eff is only linear. The smaller effective condition number explains well the very accurate solutions obtained. The error analysis in [14, 15] and the stability analysis in this paper grant the CTM to become the most efficient and competent boundary method.