A differentiation formula, with application to the two-dimensional Schr\"odinger equation

A method for obtaining discretization formulas for the derivatives of a function is presented, which relies on a generalization of divided differences. These modified divided differences essentially correspond to a change of the dependent variable. This method is applied to the numerical solution of the eigenvalue problem for the two-dimensional Schr"odinger equation, where standard methods converge very slowly while the approach proposed here gives accurate results.