Endpoint formulas for interpolatory cubic splines

Abstract

In the absence of known endpoint derivatives, the usual procedure is to use a “natural” spline interpolant which Kershaw has shown to have O ( h 4 ) \mathcal {O}({h^4}) error except near the endpoints. This note observes that either the use of appropriate finite-difference approximations for the endpoint derivatives or a proposed modification of the interpolation algorithm leads to O ( h 4 ) \mathcal {O}({h^4}) error uniformly in the interval of approximation.