 
Summary: Computer Physics Communications 161 (2004) 109118
www.elsevier.com/locate/cpc
On the use of higherorder formula for numerical derivatives
in scientific computing
N. Mohankumar a,c,
, Scott M. Auerbach a,b
a Department of Chemistry, University of Massachusetts, Amherst, MA 01003, USA
b Department of Chemical Engineering, University of Massachusetts, Amherst, MA 01003, USA
c Radiological Safety Division, Indira Gandhi Centre for Atomic Research, Kalpakkam 603 102, India
Received 17 November 2003; accepted 15 February 2004
Available online 2 July 2004
Abstract
In many situations, the numerical derivative of a function at a point x must be calculated since the function is not defined by a
closedform 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 2 or 3point formula to approximate the
derivative. On the other hand, the use of a higherorder formula, such as a 7 or even a 10point 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 higherorder formulas for the first derivative in two important problems: the
calculation of quantum mechanical reaction rates using the MillerSchwartzTromp correlation function, and the calculation of
the radioactivity migration in a porous medium.
