Algorithms for computing shape preserving spline interpolations to data

Abstract

Abstract.Algorithms are presented for computing a smooth piecewise polynomial interpolation which preserves the monotonicity and/or convexity of the data.

On Monotone and Convex Spline Interpolation

“…Some Remarks. As previously mentioned in the introduction, the well-known results of [9], [7] can be deduced from those presented here. In this paper all the lemmas and theorems are stated for k fixed.…”

“…Such splines are obtained from Bernstein polynomials of monotone and/or convex interpolating linear splines with two knots in (x,, x, + 1), i = 0,..., N -1. The well-known theorems of [9] and [7] (see also [8] for related results), where linear splines with one knot in (x,, x, + 1), i = 0,...,N -1, are used, can be deduced from those presented here. This paper is divided into three parts.…”

“…However, they are still valid if we introduce, together with the polynomial degrees «,, /' = 0,..., N -1, the sequence k,, i = 0,..., N -1. This means (see (2.4)) that t2l + x = x, + k,h¡/n¡, t2i+2 = x, + (n,-k,)h,/n,, i = 0,...,JV -1, and that n = max{«" i = 0,. to Alpha-Algorithm and Theorem 3.2 of [7].…”

Interpolation by Convex Quadratic Splines

“…Furthermore, if such ^-admissible points exist, then such an / can be constructed as follows: The following then is a special case of a theorem in [5]. Theorem 2.2.…”

“…Interpolation to convex data by a convex polynomial spline has been investigated by Passow and Roulier in [6]. McAllister, Passow, and Roulier [5] present an efficient computational algorithm for such interpolation. In both of these papers, the knots of the spline are the interpolation points.…”