On Monotone and Convex Spline Interpolation

Abstract

Abstract. This paper is concerned with the problem of existence of monotone and/or convex splines, having degree n and order of continuity k, which interpolate to a set of data at the knots. The interpolating splines are obtained by using Bernstein polynomials of suitable continuous piecewise linear functions; they satisfy the inequality k < n -k. The theorems presented here are useful in developing algorithms for the construction of shape-preserving splines interpolating arbitrary sets of data points. Earlier results of McAllister, Passow and Roulier can be deduced from those given in this paper.

Blossoming beyond Extended Chebyshev Spaces

“…As discussed in Section 1, these examples include those introduced in [3,10] for tension methods for shape-preserving interpolation.…”

“…A different type of generalisation of P n was considered first by Costantini in [3], then by Kaklis and Pandelis in [10]. Motivated by tension methods for shape-preserving interpolation, they considered the space spanned by the functions [1, x, x 2+m , (1&x) 2+m ] on [0, 1], where m is any positive integer.…”

Convex data modelling using rational bi-cubic spline function

“…In contrast, the schemes 5, 6 achieve the required shape of data by inserting extra knots in the interval where the functions lose convexity. (iv) The present scheme is tested through several numerical examples and it is found to be local in comparison with global schemes 9 , time saving and computational economical due to second degree of denominator in rational function in comparison the function 3 has cubic denominator and produces smooth and visually pleasing surfaces than existing schemes 4 .…”

“…In contrast, firstly, a piecewise cubic Bernstein-Bézier polynomial function 5 , quadratic spline interpolation 6 , a piecewise quadratic polynomial 7 , and cubic Hermite interpolation 8 have been used to solve the problem of interpolating monotone and convex data in the sense of monotonicity and convexity preserving schemes which were very economical but the methods generally inserted extra knots in the interval to visualize and conserve the shape of data. Secondly, Costantini 9 developed a global scheme for preservation of convex surfaces on rectangular grid. The scheme 10 has some research gaps like the degree of interpolant in some rectangular patches which was too large, and the resulting surfaces were not visually pleasing and smooth.…”

Random variate generation by numerical inversion when only the density is known

“…There exists a lot of literature on shape-preserving (mostly cubic) spline interpolation. However, methods for constructing monotone polynomials that interpolate monotone points are rather complicated (e.g., Costantini [1986]). Thus we have decided to use a rather simple technique and only check the monotonicity at the points F −1 a (t i ) that we already use in (7) for estimating the u-error.…”