Precise calculation of numerical derivatives using Lagrange and Hermite interpolation

Abstract

The derivatives of a function at equidistant nodes can be calculated numerically using formulae based on the classical Lagrange and Hermite interpolation. Such formulae are derived here using the REDUCE language for algebraic manipulation. They are available for 11 nodes as FORTRAN subroutines which can be used on a personal computer. As an example, the derivatives of a sinusoidal function are calculated. The results show an excellent precision, consistently better than the upper bounds of the remainder terms.