Effective condition number for numerical partial differential equations

For solving the linear algebraic equation Ax = b resulting from the numerical partial differential equations, the traditional conditions number is defined by Cond(A) = σ1 σn , where σ1 and σn are the maximal and minimal singular values of 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 worse situation of all rounding errors and all b. For given b the real relative errors may be smaller, or even much smaller than the Cond, which is called the effective condition number. In this report, we propose the new computational formulas for effective condition number Cond eff, and define the new simplified effective condition number Cond E and the simplest effective condition number Cond EE. We also applied the effective condition number to some numerical methods for Poisson’s equation, such as the finite element method, the finite difference method, etc. Numerical experiments are carried out to verify the analysis made.