Effective condition number for finite difference method

Abstract

For solving the linear algebraic equations Ax = b with the symmetric and positive definite matrix A, from elliptic equations, the traditional condition number in the 2-norm is defined by Cond. = lambda(1)/lambda(n) where lambda(1) and lambda(n) are the maximal and minimal eigenvalues of the matrix A, respectively. The condition number is used to provide the bounds of the relative errors from the perturbation of both A and b. Such a Cond. can only be reached by the worst situation of all rounding errors and all b. For the given b, 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 A, 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) for the homogeneous boundary conditions. Such a result is extraordinary, compared with the traditional Cond. = O(h(min)(-2)), where h(min) 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)) and O(h(-1/2)h(min)(-1)), respectively, where It is the maximal meshspacing of the difference grids. Numerical experiments are carried out to verify the analysis made. (c) 2006 Elsevier B.V. All rights reserved

Effective condition number and its applications

“…The Cond is often too large, to mislead the true stability of the numerical solutions obtained. Hence, we propose the following effective condition number for better stability analysis in [11,12],…”

“…Recently, we develop the effective condition number in [11,12], and apply it to the symmetric and positive definite matrix F ∈ R n×n from the finite difference method. In this paper, we will apply the effective condition number for over-determined systems from the spectral and Trefftz methods.…”

“…On the other hand, when the matrix F is positive definite and symmetric, the effective condition numbers of this paper are all valid if letting σ i = λ i , where λ i are the eigenvalues of F, see [11,12].…”

“…The main concern for choosing the radius parameter R p is that whether or not the instability may damage the accuracy of the Motz solution under a certain working digits. From our recent study [11,12], the stability analysis should be made, based on Cond eff, but not on Cond.…”

“…In this paper, we will apply the effective condition number for over-determined systems from the spectral and Trefftz methods. Let the rank (F) = r ≤ n. for (1) and the perturbed system F(x + ∆x) = b + ∆b, there exists the bound in [11,12],…”