On the use of higher-order formula for numerical derivatives inscientific computing

In many situations, the numerical derivative of a function at a point x must be calculated since the function is not defined by a closed-form expression, but rather by values of the function at grid points at and around x. This typically arises when enforcing the boundary conditions while solving a differential equation. Usually, one employs a 2or 3-point formula to approximate the derivative. On the other hand, the use of a higher-order formula, such as a 7or even a 10-point approximation, based on the method of undetermined coefficients, can sometimes lead to better accuracy and enhanced computational efficiency. We show that significant improvements arise from using higher-order formulas for the first derivative in two important problems: the calculation of quantum mechanical reaction rates using the Miller–Schwartz–Tromp correlation function, and the calculation of the radioactivity migration in a porous medium.  2004 Elsevier B.V. All rights reserved.