Effective condition number for numerical partial differential equations

Abstract

Abstract In this paper, the new computational formulas are derived for the effective condition number Cond_eff, and the new error bounds involved in both Cond and Cond_eff are developed. A theoretical analysis is provided to support some conclusions in Banoczi et al . ( SIAM J. Sci. Comput . 1998; 20 :203–227). For the linear algebraic equations solved by the Gaussian elimination or the QR factorization (QR), the direction of the right‐hand vector is insignificant for the solution errors, but such a conclusion is invalid for the finite difference method for Poisson's equation. The effective condition number is important to the numerical partial differential equations, because the discretization errors are dominant. Copyright © 2008 John Wiley & Sons, Ltd.