MATHEMATICS OF COMPUTATION, VOLUME 31, NUMBER 139 JULY 1977, PAGES 717-725 Algorithms for Computing Shape Preserving Spline Interpolations to Data By David F. McAllister*, Eli Passow** and John A. Roulier* Abstract. Algorithms are presented for computing a smooth piecewise polynomial interpolation which preserves the monotonicity and/or convexity of the data. 1. Introduction. Shape preserving polynomial or spline interpolation to mono- tonic and/or convex data has been investigated by several authors in recent years (see [1]—[5] and [7]—[16]). It is the purpose of this paper to develop and test algo- rithms for computing smooth monotonicity and/or convexity preserving spline inter- polation. The algorithms presented herein are based on the theory developed by Passow and Roulier [11]. The algorithms are tested on several examples and are compared with ordinary Lagrange and cubic spline interpolation of increasing convex data. It should be noted that algorithms for convexity preserving interpolation by exponential splines are given in [14]. The authors thank the referee of the original version of this article for pointing out numerous related articles on computer aided geometric design. See [1], [3], and [4]. In particular, see the articles in [1] by Gordon and Riesenfeld p. 95, Bezier p. 127, Forrest p. 17, Wielinga p. 153, and Nielson p. 209. Many of these articles refer to Bezier curves which are Bernstein polynomials of parametrized polygonal segments. While the ideas are similar to those presented here and make use of the convexity pre- serving properties of Bernstein polynomials, they deal more with construction of some suitable shape rather than interpolation of given data with shape preservation. Here we assume that the data is given and that no additional data points can be easily ob- tained. Moreover, the alpha algorithm and its use in Theorem 3.2 allows one to mini- mize the degree of the piecewise polynomials without losing the properties of inter- polation or convexity. This algorithm is automatic and requires no interaction on the part of the user as do many of the techniques in computer-aided geometric design. 2. Notation and Background. Let A = {xQ < xl < • • • < xN} be fixed real numbers and for / < n let S'n = S^A) be the set of splines of degree n and deficiency n - j on A. Thus, / £ 5¿(A) if and only if / € C1[x0, xN] and / is an algebraic poly- nomial of degree n or less on [xi_l, x¡] for /' = 1, 2, . . . , TV. Given corresponding real numbers y0, . . . , yN, define S¡ — (yi - y¡-i)l{x¡ - *,_,) for i = 1, 2, . . . , N. We say that the data are nondecreas- ing if y0 < y1 < • • • < yN, and the data are nonconcave if Sj < S2 < • • • < SN. If Received June 6, 1976; revised November 21, 1976. AMS (MOS) subject classifications (1970). Primary 41A0S, 41A1S. •Supported in part by NSF Grant # MCS 76-04033. "Supported in part by a Temple University Grant-in-Aid of Research. Copyright © 1977, American Mathematical Society 717 License or copyright restrictions may apply to redistribution; see https://www.ams.org/journal-terms-of-use