The Cubic X-Spline and its Application to Interpolation

The cubic X-spline is a generalization of the conventional cubic spline in which the second derivative is allowed to possess discontinuities at the internal knots. The magnitudes of these discontinuities are related to those of the discontinuities in the third derivative by means of a simple relationship which introduces one free parameter at each internal knot. In order to illustrate the usefulness of the cubic X-spline it is applied to the classical interpolation problem and compared with the conventional cubic spline. It is shown that the free parameters may be chosen so as to impart desirable properties to the interpolating X-spline in one of two alternative ways; either (i) the X-spline can be made more accurate than the conventional spline with no corresponding increase in computational labour, or (ii) with only a slight loss in accuracy, it can effect a considerable computational simplification. The second alternative may be of particular value in computer graph-plotting applications. A numerical example is given for each of the interpolating X-splines.