Topic
Monotone cubic interpolation
About: Monotone cubic interpolation is a research topic. Over the lifetime, 1740 publications have been published within this topic receiving 38111 citations.
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01 Jan 2001
14 citations
01 Feb 1975
TL;DR: In this article, it was shown that there exists an algebraic polynomial p(x) with properties (i) and (ii) that interpolate piecewise monotonely; in case y. > y., for all j (or if y • < y ■_ j for all /), p{x) is simply said to interpolate monotonally.
Abstract: Let 0 = xQ < x.< "• < x, = 1 and let y Q, y., • • •, y, be real numbers such that y. , * y., j = 1, 2, • • •, k. Estimates are obtained on the degree of an algebraic polynomial p{x) that interpolates the given data piecewise monotonely; i.e., such that (i) p{x .) = y ., / = 0, 1, • • •, k, and such that (ii) p(x) is increasing on I. =(x . ,x ■) ií y . < y . , and decreasing on /. if y . < y ._., j = 1, 2, • • •, k. The problem is seen to be related to the problem of monotone approximation. Let 0 = x 0< x < • • • < x, = 1 and let y 0, y,, • • •, y. be real numbers such that y ._, 7= y , / = 1, 2, • • • , fe. It is a result of Wolibner [7], Kammerer [2], and Young [8] that there exists an algebraic polynomial p(x) such that: (i) Pixj) = y;.» 7 = 0| 1» • * " » *, and (ii) p{x) is increasing on /. = (x ._,, x .) if y . > y ._ , and decreasing on 7 if yj< y,_i» /'" ii2.-"» *'• A polynomial p(x) with properties (i) and (ii) is said to interpolate piecewise monotonely; in case y . > y . , for all j (or if y • < y ■_ j for all /), p{x) is simply said to interpolate monotonely. The smallest degree of a polynomial that interpolates the values Y = jy0, y,,'" , y A at the points X = jx0> x j, • • •, xA (piecewise) monotonely is called the degree of (piecewise) monotone interpolation of Y with respect to X, and is denoted by N = N{X; Y). Rubinstein has obtained estimates for the degree of monotone interpolation for the special case k = 2 [6]. We seek general estimates on N{X; Y). Let y. — y . . A = MY) = min |y. -y. |, and M = M(X; Y) = max !<7~L \
14 citations
01 Jan 1984
TL;DR: In this article, a rational cubic function is presented which has shape preserving interpolation properties and can be used to construct C 2 rational spline interpolants to monotonic or convex sets of data which are defined on a partition x 1 < x2 < … < xn of the real interval [x 1, xn].
Abstract: A rational cubic function is presented which has shape preserving interpolation properties. It is shown that the rational cubic can be used to construct C2 rational spline interpolants to monotonic or convex sets of data which are defined on a partition x1 < x2 < … < xn of the real interval [x1, xn].
14 citations
TL;DR: This work rigorously proves that the PH interpolant it selects doesn’t depend on the unit pure vector chosen for representing its hodograph in quaternion form, and evaluates the corresponding interpolation scheme from a theoretical point of view, proving with the help of symbolic computation that it has fourth approximation order.
14 citations
TL;DR: A trivariate Lagrange interpolation method based on C 1 cubic splines is described, which is local and stable, provides optimal order approximation, and has linear complexity.
Abstract: A trivariate Lagrange interpolation method based on C 1 cubic splines is described. The splines are defined over a special refinement of the Freudenthal partition of a cube partition. The interpolating splines are uniquely determined by data values, but no derivatives are needed. The interpolation method is local and stable, provides optimal order approximation, and has linear complexity.
14 citations