Rational interpolation through the optimal attachment of poles to the interpolating polynomial

After recalling some pitfalls of polynomial interpolation (in particular, slopes limited by Markov's inequality) and rational interpolation (e.g., unattainable points, poles in the interpolation interval, erratic behavior of the error for small numbers of nodes), we suggest an alternative for the case when the function to be interpolated is known everywhere, not just at the nodes. The method consists in replacing the interpolating polynomial with a rational interpolant whose poles are all prescribed, written in its barycentric form as in [4], and optimizing the placement of the poles in such a way as to minimize a chosen norm of the error. This revised version was published online in June 2006 with corrections to the Cover Date.     

Thiele重心型矩阵值混合有理插值算法

利用Samelson型矩阵广义逆,构造了一种基于Thiele型连分式插值与重心有理插值的相结合的二元矩阵值混合有理插值格式,这种新的混合矩阵值有理插值函数继承了连分式插值和重心插值的优点,它的表达式简单,计算方便,数值稳定性好.该算法满足有理插值问题所给的插值条件,同时给出了误差估计分析.最后用数值算例验证了插值算法的有效性.     

Radial basis functions under tension

We discuss multivariate interpolation with some radial basis function, called radial basis function under tension (RBFT). The RBFT depends on a positive parameter which provides a convenient way of controlling the behavior of the interpolating surface. We show that our RBFT is conditionally positive definite of order at least one and give a construction of the native space, namely a semi-Hilbert space with a semi-norm, minimized by such an interpolant. Error estimates are given in terms of this semi-norm and numerical examples illustrate the behavior of interpolating surfaces.     

The numerical stability of barycentric Lagrange interpolation

The Lagrange representation of the interpolating polynomialcan be rewritten in two more computationally attractive forms:a modified Lagrange form and a barycentric form. We give anerror analysis of the evaluation of the interpolating polynomialusing these two forms. The modified Lagrange formula is shownto be backward stable. The barycentric formula has a less favourableerror analysis, but is forward stable for any set of interpolatingpoints with a small Lebesgue constant. Therefore the barycentricformula can be significantly less accurate than the modifiedLagrange formula only for a poor choice of interpolating points.This analysis provides further weight to the argument of Berrutand Trefethen that barycentric Lagrange interpolation shouldbe the polynomial interpolation method of choice.     

Transfinite mean value interpolation in general dimension

Mean value interpolation is a simple, fast, linearly precise method of smoothly interpolating a function given on the boundary of a domain. For planar domains, several properties of the interpolant were established in a recent paper by Dyken and the second author, including: sufficient conditions on the boundary to guarantee interpolation for continuous data; a formula for the normal derivative at the boundary; and the construction of a Hermite interpolant when normal derivative data is also available. In this paper we generalize these results to domains in arbitrary dimension.     

Weighted interpolation for equidistant nodes

Weighted Lagrange interpolation is proposed for solving Lagrange interpolation problems on equidistant or almost equidistant data. Good condition numbers are found in the case of rational interpolants whose denominator has degree about twice the number of data to be interpolated. Since the degree of the denominator is higher than that of the numerator, simple functions like constants and linear polynomials will not be reproduced. Furthermore, the interpolant cannot be expressed by a barycentric formula. As a counterpart, the interpolation algorithm is simple and leads to small Lebesgue constants.     

Barycentric coordinates for convex sets

In this paper we provide an extension of barycentric coordinates from simplices to arbitrary convex sets. Barycentric coordinates over convex 2D polygons have found numerous applications in various fields as they allow smooth interpolation of data located on vertices. However, no explicit formulation valid for arbitrary convex polytopes has been proposed to extend this interpolation in higher dimensions. Moreover, there has been no attempt to extend these functions into the continuous domain, where barycentric coordinates are related to Green’s functions and construct functions that satisfy a boundary value problem. First, we review the properties and construction of barycentric coordinates in the discrete domain for convex polytopes. Next, we show how these concepts extend into the continuous domain to yield barycentric coordinates for continuous functions. We then provide a proof that our functions satisfy all the desirable properties of barycentric coordinates in arbitrary dimensions. Finally, we provide an example of constructing such barycentric functions over regions bounded by parametric curves and show how they can be used to perform freeform deformations.      

Comonotone and coconvex rational interpolation and approximation

Comonotonicity and coconvexity are well-understood in uniform polynomial approximation and in piecewise interpolation. The covariance of a global (Hermite) rational interpolant under certain transformations, such as taking the reciprocal, is well-known, but its comonotonicity and its coconvexity are much less studied. In this paper we show how the barycentric weights in global rational (interval) interpolation can be chosen so as to guarantee the absence of unwanted poles and at the same time deliver comonotone and/or coconvex interpolants. In addition the rational (interval) interpolant is well-suited to reflect asymptotic behaviour or the like.     

On cubic Hermite coalescence hidden variable fractal interpolation functions

Hermite interpolation is a very important tool in approximation theory and numerical analysis, and provides a popular method for modeling in the area of computer aided geometric design. However, the classical Hermite interpolant is unique for a prescribed data set,and hence lacks freedom for the choice of an interpolating curve, which is a crucial requirement in design environment. Even though there is a rather well developed fractal theory for Hermite interpolation that offers a large flexibility in the choice of interpolants, it also has the shortcoming that the functions that can be well approximated are highly restricted to the class of self-affine functions. The primary objective of this paper is to suggest a C1-cubic Hermite interpolation scheme using a fractal methodology, namely, the coalescence hidden variable fractal interpolation, which works equally well for the approximation of a self-affine and non-self-affine data generating functions. The uniform error bound for the proposed fractal interpolant is established to demonstrate that the convergence properties are similar to that of the classical Hermite interpolant. For the Hermite interpolation problem, if the derivative values are not actually prescribed at the knots, then we assign these values so that the interpolant gains global C2-continuity. Consequently, the procedure culminates with the construction of cubic spline coalescence hidden variable fractal interpolants. Thus, the present article also provides an alternative to the construction of cubic spline coalescence hidden variable fractal interpolation functions through moments proposed by Chand and Kapoor [Fractals, 15(1)(2007), pp. 41-53].     

一类有理插值曲面模型及其可视化约束控制

本文构造一类新的基于函数值和偏导数值的双变量加权混合有理插值样条.与已有的有理插值样条相比,这类新的有理插值样条具有以下四方面的特性,其一,插值函数可以由简单的对称基函数来表示;其二,对任何正参数,插值函数满足C1连续,而且,在不限制参数取值的条件之下,插值曲面保持光滑;其三,插值函数不但含有参数,而且带有加权系数,增加了插值函数的自由度;其四,插值曲面的形状随着参数与加权系数的变化而变化.同时,本文讨论此类插值曲面的性质,包括基函数的性质、积分加权系数的性质和插值函数的边界性质.此类插值函数的优势在于,不改变给定插值数据的前提下,通过选择合适的参数和不同的加权系数,对插值区域内的任意点的函数值进行修改.因此可将其应用于曲面设计,根据实际设计需要,自由地修改曲面形状.数值实验表明,此类新的有理样条插值具有良好的约束控制性质.     

First applications of a formula for the error of finite sinc interpolation

In former articles we have given a formula for the error committed when interpolating a several times differentiable function by the sinc interpolant on a fixed finite interval. In the present work we demonstrate the relevance of the formula through several applications: correction of the interpolant through the insertion of derivatives to increase its order of convergence, improvement of the barycentric formula, rational sinc interpolants (with and without replacement of the (usually unknown) derivatives with finite differences), convergence acceleration through extrapolation and improvement of one-sided interpolants. Work partly supported by the Swiss National Science Foundation under grant Nr 200021-116122.     

有理插值问题存在性的一个判别准则

１引言我们知道,多项式Ｌａｇｒａｎｇｅ插值是适定的［１,２］,但有理插值函数却未必存在［８,３］．并且到目前为止,也没有类似于多项式Ｌａｇｒａｎｇｅ插值的能够揭示插值结构的显式插值公式．不过有理插值已有许多算法,比如Ｓｔｏｅｒ算法,Ｔｈｉｅｌｅ倒差商算法,Ｓａｌｚｅｒ算法以及Ｗｕｙｔａｃｋ算法等等,见［８,４,５,６］．本文为寻求尽可能接近显式的插值公式,进而揭示有理插值问题的内在结构,得到了有理插值函数存在的一个充要条件,同时也给出了有理插值函数的一种表现形式,参见［１１］．本文约定,所有矩阵…     

A matrix for determining lower complexity barycentric representations of rational interpolants

Among the representations of rational interpolants, the barycentric form has several advantages, for example, with respect to stability of interpolation, location of unattainable points and poles, and differentiation. But it also has some drawbacks, in particular the more costly evaluation than the canonical representation. In the present work we address this difficulty by diminishing the number of interpolation nodes embedded in the barycentric form. This leads to a structured matrix, made of two (modified) Vandermonde and one Löwner, whose kernel is the set of weights of the interpolant (if the latter exists). We accordingly modify the algorithm presented in former work for computing the barycentric weights and discuss its efficiency with several examples.     

Optimal local spline interpolants

A constructive proof is given of the existence of a local spline interpolant which also approximates optimally in the sense that its associated operator reproduces polynomials of maximal order. First, it is shown that such an interpolant does not exist for orders higher than the linear case if the partition points of the appropriate spline space coincide with the given interpolation points. Next, in the main result, the desired existence of an optimal local spline interpolant for all orders is proved by increasing, in a specified manner, the set of partition points. Although our interpolant reproduces a more restricted function space than its quasi-interpolant counterpart constructed by De Boor and Fix [1], it has the advantage of interpolating every real function at a given set of points. Finally, we do some explicit calculations in the quadratic case.     

Construction of asymptotically best quadrature formulas

Summary. We show that, for integrals with arbitrary integrable weight functions, asymptotically best quadrature formulas with equidistant nodes can be obtained by applying a certain scheme of piecewise polynomial interpolation to the function to be integrated, and then integrating this interpolant. Received August 7, 1991     

ON TRIANGULAR C~1 SCHEMES: A NOVEL CONSTRUCTION

1. IntroductionThe smooth interpolation on a triangulation of a planar region is of great importancein most applied areas) such as computation of finite element method, computer aided(geometric) design and scattered data processing.Let A be a triangulation of a polygonal domain fi C RZ and Ac, al and aZ the setso f venices, edges and triangles in a respectively. Usually the triangulation in practiceis formed by a mass of scattered nodes that, covered by the region fi, are carryingsimilar typ…     

On the Lebesgue constant of barycentric rational interpolation at equidistant nodes

Recent results reveal that the family of barycentric rational interpolants introduced by Floater and Hormann is very well-suited for the approximation of functions as well as their derivatives, integrals and primitives. Especially in the case of equidistant interpolation nodes, these infinitely smooth interpolants offer a much better choice than their polynomial analogue. A natural and important question concerns the condition of this rational approximation method. In this paper we extend a recent study of the Lebesgue function and constant associated with Berrut’s rational interpolant at equidistant nodes to the family of Floater–Hormann interpolants, which includes the former as a special case.     

Spherical Scattered Data Quasi-interpolation by Gaussian Radial Basis Function

Since the spherical Gaussian radial function is strictly positive definite, the authors use the linear combinations of translations of the Gaussian kernel to interpolate the scattered data on spheres in this article. Seeing that target functions are usually outside the native spaces, and that one has to solve a large scaled system of linear equations to obtain combinatorial coefficients of interpolant functions, the authors first probe into some problems about interpolation with Gaussian radial functions. Then they construct quasiinterpolation operators by Gaussian radial function, and get the degrees of approximation. Moreover, they show the error relations between quasi-interpolation and interpolation when they have the same basis functions. Finally, the authors discuss the construction and approximation of the quasi-interpolant with a local support function.     

基于二元连分式的散乱数据递推插值格式

本文首先基于新的非张量积型偏逆差商递推算法,分别构造奇数与偶数个插值节点上的二元连分式散乱数据插值格式,进而得到被插函数与二元连分式间的恒等式.接着,利用连分式三项递推关系式,提出特征定理来研究插值连分式的分子分母次数.然后,数值算例表明新的递推格式可行有效,同时,通过比较二元Thiele型插值连分式的分子分母次数,发现新的二元插值连分式的分子分母次数较低,这主要归功于节省了冗余的插值节点. 最后,计算此有理函数插值所需要的四则运算次数少于计算径向基函数插值.     

曲边三角形及曲边四边形上的混合函数插值格式

一、引言 二元函数在标准三角形上的混合函数插值格式在许多文献,例如,Birkhofft,Barnhill,Gordon及Gregory等的文章中都有讨论。在三角形周边上对高阶偏导数进行插值,而且计算比较简单的是J.A.Gregory的文章中所给出的一种混合函数插值格式。这种格式是由简单函数的线性组合所构成的,而且格式是对称的,因此计算比较简便。但是J.A.Gregory只是对直边三角形给出了格式。本文企图推广Gregory的格式,给出曲边三角形上对高阶偏导数进行插值的插值格式。我们还进一步给出了曲边四边形上