19.1 Introduction

In numerical simulations one often encounters functions that are only given at discrete points, while values and derivatives are required for other values of the argument. Examples are:

1. the reconstruction of an interaction curve based on discrete points, for example, obtained from extensive quantum calculations;

2. recovery of function values and derivatives for arbitrary arguments from tabulated values, for example, for potentials and forces in MD simulations;

3. the estimation of a definite or indefinite integral of a function based on a discrete set of derivatives, for example, the free energy in thermodynamic integration methods;

4. the construction of a density distribution from a number of discrete events, for example, a radial distribution function.

In all these cases one looks for an interpolation scheme to construct a complete curve from discrete points or nodes. Considerations that influence the construction process are (i) the smoothness of the curve, (ii) the accuracy of the data points, and (iii) the complexity of the construction process.

Smoothness is not a well-defined property, but it has to do with two aspects: the number of continuous derivatives (continuous at the nodes), and the integrated curvature C, which can be defined as the integral over the square of the second derivative over the relevant interval:

or, in the multidimensional case:

The term curvature is used very loosely here. ^[1] If data points are not infinitely accurate, there is no reason why the curve should go exactly...