Monotone and convex interpolation by weighted quadratic splines

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Monotone and convex interpolation by weighted quadratic splines Boris I. Kvasov

Received: 31 March 2012 / Accepted: 20 March 2013 / Published online: 19 April 2013 © Springer Science+Business Media New York 2013

Abstract In this paper we discuss the design of algorithms for interpolating discrete data by using weighted C 1 quadratic splines in such a way that the monotonicity and convexity of the data are preserved. The analysis culminates in two algorithms with automatic selection of the shape control parameters: one to preserve the data monotonicity and other to retain the data convexity. Weighted C 1 quadratic B-splines and control point approximation are also considered. Keywords Monotone and convex interpolation · Weighted C 1 quadratic splines · Adaptive choice of shape control parameters · Weighted B-splines · Control point approximation. Mathematics Subject Classifications 2010 41A50 · 65D07 · 65D17

1 Introduction We are interested in numerically fitting a curve through a given finite set of points Pi = (xi , fi ), i = 0, . . . , N + 1, in the plane, with a = x0 < x1 < · · · < xN+1 = b. These points can be thought of as coming from the graph of some function f defined on [a, b]. We are particularly interested in algorithms which preserve local monotonicity and convexity of the data (or function). Here, we shall focus only on those algorithms which use C 1 piecewise quadratic interpolants.

Communicated by: L. L. Schumaker B. I. Kvasov () Department of Mathematical Modeling, Institute of Computational Technologies, Russian Academy of Sciences, Lavrentiev Avenue 6, Novosibirsk 630090, Russia e-mail: [email protected]

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Monotone and convex local C 1 quadratic splines with perhaps one additional knot in each subinterval between data points were considered by L.L. Schumaker [26] (see also [2, 5],[17]–[21], and references therein). In these interactive algorithms the location of additional knots allows the user to adjust spline to the data and to take full advantage of the flexibility which quadratic splines permit. Some improvements of these algorithms were suggested in [6, 15]. Very similar algorithms were also obtained in [29]. The best known algorithm of this type for piecewise cubics was given in [3, 9, 16]. In contrast with the previously published algorithms for shape preserving quadratic splines which rely on local schemes, our algorithms are based on global weighted C 1 quadratic splines. Such splines generalize global quadratic splines introduced by Yu.N. Subbotin [28] and are similar to weighted C 1 cubic splines [8, 22, 24]. We let the additional knots be the midpoints in each subinterval to have actually their optimal location. While there are many methods available for the solution of the shape-preserving interpolation problem (see a very detailed literature review in [10]), the shape parameters are mainly viewed as an interactive design tool for manipulating shape of a spline curve. The main challenge of this paper is to present algorithms that select shape control parameters (w