Two shape preserving lagrangeC 2-interpolants

Abstract

Summary. We present a Lagrange c~2-interpolant to scattered convex data which preserves convexity. We also present a Lagrange c~2-interpolant to uniformly spaced monotone data sites which preserves monotonicity. In both cases no further conditions are required on the data values. These interpolants are explicitely described and local. Error is O(h 3) when the function to be interpolated is (~p3.

An always successful method in univariate convex $C^2$ interpolation

“…When C 2 splines are considered, the algorithms for choosing free nonlinearity parameters are, in comparison, fairly unsatisfactory; see, e.g., [23] for rational splines and [24] for rational lacunary splines. Recently by Archer and Le Gruyer [1] a direct method for computing C 2 interpolants was given which guarantee convexity preservation on the subinterval [x 1 , x n−1 ] ⊂ I . For reviews on shape preserving interpolation we refer to [9,30].…”

“…In general, the split points ξ i 0 and ξ i 1 have to be placed on I i rather unsymmetrically. The method [1] turns out to be a special case of this class, and is now completed to the whole interval I .…”

“…Further, we engage in positive and in monotone interpolations which are much easier to handle than convex interpolation, because the additional knots can be fixed a priori. In contrast to [1], where also monotone interpolation is considered, it is not necessary to assume the data sites to be uniformly spaced. Finally, we present some results of computational experiments.…”