On a global superconvergence of the gradient of linear triangular elements
Journal of Computational and Applied Mathematics
Effectively well-conditioned linear systems
SIAM Journal on Scientific and Statistical Computing
The algebraic eigenvalue problem
The algebraic eigenvalue problem
The effective condition number applied to error analysis of certain boundary collocation methods
Journal of Computational and Applied Mathematics
The conditioning of some numerical methods for first kind boundary integral equations
Journal of Computational and Applied Mathematics
Matrix computations (3rd ed.)
The symmetric eigenvalue problem
The symmetric eigenvalue problem
Journal of Computational and Applied Mathematics
Effective condition number of the Hermite finite element methods for biharmonic equations
Applied Numerical Mathematics
Effective condition number for the finite element method using local mesh refinements
Applied Numerical Mathematics
The Trefftz method using fundamental solutions for biharmonic equations
Journal of Computational and Applied Mathematics
Error analysis of the method of fundamental solutions for linear elastostatics
Journal of Computational and Applied Mathematics
Hi-index | 7.29 |
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. = λl/λn, where λl and λn 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(hmin-2), where hmin 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-1/2hmin-1), respectively, where h is the maximal meshspacing of the difference grids. Numerical experiments are carried out to verify the analysis made.