Numer. Math. 46, 159-174 (1985) Numerische Mal ematik 9 Springer-Verlag 1985 Algorithms for Computing Shape Preserving Spline Approximations to Data S.L. Dodd* and D.F. McAllister** North Carolina State University, Raleigh, NC 27650, USA Summary. We treat the problem of approximating data that are sampled with error from a function known to be convex and increasing. The approx- imating function is a polynomial spline with knots at the data points. This paper presents results (analogous to those in [7] and [9]) that describe some approximation properties of polynomial splines and algorithms for determining the existence of a "shape-preserving" approximant for given data. Subject Classifications: AMS(MOS): 41A15, 41A29: CR: G 1.2. 1. Introduction In recent years many authors have investigated ways to approximate data when the approximating function is required to satisfy certain convexity and/or monotonicity conditions (see [1, 2, 3, 5], and [12]), or it is required to make some tradeoff between the number of local optima of the highest derivative of the approximation and its fidelity to the data (see [4, 10], and [11]). In this paper we treat the problem of approximating data that are sampled from a function which is known to be convex and increasing, but the data are measured with an error at each point. We will assume the errors are described by ran- dom variables that have density functions with finite support and the approx- imating function is to be a polynomial spline with knots at the data points. We will present a theorem that shows that the degree of the piecewise polynomials which make up the spline may be forced to be arbitrarily high by a suitable choice of data points. We will then present a procedure that will determine whether or not a set of data with known error densities can be approximated by a shape preserving spline with knots at the data points. The results are an extension of the ideas presented in [7, 8], and [9]. * Formerly of the Graduate Program in Operations Research, NC State University. Author now at AT&T Bell Laboratories, Holmdel, NJ, USA ** Department of Computer Science, NC State University. Research supported in part by NASA Grant NAGI-103