Shape - preserving Knot

Starting with a shape-preserving C 1 quadratic spline, we show how knots can be removed to produce a new spline which is within a speciied tolerance of the original one, and which has the same shape properties. We give speciic algorithms and some numerical examples, and also show how the method can be used to compute approximate best free-knot splines. Finally, we discuss how to handle noisy data, and develop an analogous knot removal algorithm for a monotonicity preserving surface method. The idea of removing knots from a spline function in order to produce a good approximation with fewer parameters has been discussed in a number of recent papers Lyche & MMrken '87a, '87b, '88, Arge et al '90, Lyche '92]. See also the book Goldman & Lyche '93]. Starting with a given B-spline expansion f, these authors construct another B-spline expansion g with fewer knots which diiers from f by less than some given tolerance. The method in Lyche & MMrken '87a, '87b, '88, Lyche '92] uses a discrete least squares approximation process to construct g, and to decide which knots should be removed. In many practical applications, it is important to construct a spline whose shape accurately models the shape of the input data. In Arge et al '90], the authors extended the knot removal method to enforce constraints such as positivity, monotonicity, and convexity. Their method requires repeated solutions of quadratic minimization problems with linear constraints. In the rst part of this paper we consider knot removal based on the shape-preserving C 1 quadratic interpolating splines discussed in McAllister & Roulier '81, Schumaker '83, DeVore & Yan '86]. We believe this approach has several advantages: 1) shape-preservation is built into the algorithm, 2) in contrast to B-spline based methods, our algorithm is completely local,