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.

Piecewise monotone interpolation and approximation with Muntz polynomials

Abstract: The possibility (subject to certain restrictions) of solving the following approximation and interpolation problem with a given set of \"Muntz polynomials\" on a real interval is demonstrated: (i) approximation of a continuous function by a \"copositive\" Muntz polynomial; (ii) approximation of a continuous function by a \"comonotone\" Muntz polynomial; (iii) approximation of a continuous function with a monotone fcth difference by a Muntz polynomial with a monotone fcth derivative; (iv) interpolation by piecewise monotone Muntz polynomials—i. e., polynomials that are monotone on each of the intervals determined by the points of interpolation. The strong interrelationship of these problems is shown implicitly in the proofs. The following related questions have been settled: I iMonotone Approximation). Let fix) he a continuous function with the property that the /th difference u¿f> 0 on [0, 1] where / is some nonnegative integer. Must there be for a given e > 0 a corresponding polynomial p(x) with p0)(x) > 0 on [0, 1] such that ||/-p|| = sup \\f(x)-p(x)\\ 0 must there be a corresponding polynomial p(x) that has the same monotonicity as fix) on each of the intervals (*,_!, xj), i = 1, 2,.... k, and such that ||/-p|| < e? Received by the editors October 31, 1973. AMS (MOS) subject classifications (1970). Primary 41A05, 41A10, 41A30, 41A25.

Piecewise Cubic Interpolation

Abstract: This chapter discusses the basic ideas behind piecewise cubic interpolation. A large variety of interpolation methods exists that are designed to cope with special problems. The most popular class of methods is that of piecewise polynomial schemes. All these methods construct curves that consist of polynomial pieces of the same degree and that are of a prescribed overall smoothness. The given data are usually points and parameter values; sometimes, tangent information is added as well. In a surface generation environment, one is often given a set of points p i ∈ 3 and a surface normal vector n i at each data point. Thus, one only knows the tangent plane of the desired surface at each data point, not the actual endpoint derivatives of the patch boundary curves.

Nonnegativity Preserving Interpolation by 1 Bivariate Rational Spline Surface

Abstract: This paper is concerned with the nonnegativity preserving interpolation of data on rectangular grids. The function is a kind of bivariate rational interpolation spline with parameters, which is 𝐶1 in the whole interpolation region. Sufficient conditions are derived on coefficients in the rational spline to ensure that the surfaces are always nonnegative if the original data are nonnegative. The gradients at the data sites are modified if necessary to ensure that the nonnegativity conditions are fulfilled. Some numerical examples are illustrated in the end of this paper.