Quadratic/linear rational spline interpolation

Abstract

We describe the construction of an interpolating quadratic/linear rational spline S of smoothness class C ² for a strictly convex (or strictly concave) function y on [a, b]. On uniform mesh x _i = a + ih, i = 0,..., n, in the case of sufficiently smooth function y the expansions of S and its derivatives are obtained. They give the superconvergence of order h ⁴ for the first derivative, of order h ³ for the second derivative and of order h ²for the third derivative of S in certain points. Corresponding numerical examples are given.