Effective condition number for finite difference method

For solving the linear algebraic equations Ax=bAx=b with the symmetric and positive definite matrix AA, from elliptic equations, the traditional condition number in the 2-norm is defined by Cond.=λ1/λnCond.=λ1/λn, where λ1λ1 and λnλn are the maximal and minimal eigenvalues of the matrix AA, respectively. The condition number is used to provide the bounds of the relative errors from the perturbation of both AA and bb. Such a Cond. can only be reached by the worst situation of all rounding errors and all bb. For the given bb, the true relative errors may be smaller, or even much smaller than the Cond., which is called the effective condition number in Chan and Foulser [Effectively well-conditioned linear systems, SIAM J. Sci. Statist. Comput. 9 (1988) 963–969] and Christiansen and Hansen [The effective condition number applied to error analysis of certain boundary collocation methods, J. Comput. Appl. Math. 54(1) (1994) 15–36]. In this paper, we propose the new computational formulas for effective condition number Cond_eff, and define the new simplified effective condition number Cond_E. For the latter, we only need the eigenvector corresponding to the minimal eigenvalue of AA, which can be easily obtained by the inverse power method. In this paper, we also apply the effective condition number for the finite difference method for Poisson's equation. The difference grids are not supposed to be quasiuniform. Under a non-orthogonality assumption, the effective condition number is proven to be O(1)O(1) for the homogeneous boundary conditions. Such a result is extraordinary, compared with the traditional Cond.=O(hmin-2), where hminhmin is the minimal meshspacing of the difference grids used. For the non-homogeneous Neumann and Dirichlet boundary conditions, the effective condition number is proven to be O(h-1/2)O(h-1/2) and O(h-1/2hmin-1), respectively, where h is the maximal meshspacing of the difference grids. Numerical experiments are carried out to verify the analysis made.