The Neville-like form of the Fitzpatrick algorithm for rational interpolation

The Fitzpatrick algorithm, which seeks a Gr?bner basis for the solution of a system of polynomial congruences, can be applied to compute a rational interpolant. Based on the Fitzpatrick algorithm and the properties of an Hermite interpolation basis, we present a Neville-like algorithm for multivariate osculatory rational interpolation. It may be used to compute the values of osculatory rational interpolants at some points directly without computing the rational interpolation function explicitly.     

Barycentric rational interpolation with asymptotically monitored poles

We present a method for asymptotically monitoring poles to a rational interpolant written in barycentric form. Theoretical and numerical results are given to show the potential of the proposed interpolant.     

A method for directly finding the denominator values of rational interpolants

In this paper, we give an algorithm for directly finding the denominator values of rational interpolants at the nodes, and present an expression for the corresponding rational interpolant when the latter exists. With these denominator values, our method also provides information concerning the existence of the interpolant and the presence of unattainable points and poles.     

A multivariate QD-like algorithm

The problem of constructing a univariate rational interpolant or Padé approximant for given data can be solved in various equivalent ways: one can compute the explicit solution of the system of interpolation or approximation conditions, or one can start a recursive algorithm, or one can obtain the rational function as the convergent of an interpolating or corresponding continued fraction.In case of multivariate functions general order systems of interpolation conditions for a multivariate rational interpolant and general order systems of approximation conditions for a multivariate Padé approximant were respectively solved in [6] and [9]. Equivalent recursive computation schemes were given in [3] for the rational interpolation case and in [5] for the Padé approximation case. At that moment we stated that the next step was to write the general order rational interpolants and Padé approximants as the convergent of a multivariate continued fraction so that the univariate equivalence of the three main defining techniques was also established for the multivariate case: algebraic relations, recurrence relations, continued fractions. In this paper a multivariate qd-like algorithm is developed that serves this purpose.     

Bivariate composite vector valued rational interpolation

In this paper we point out that bivariate vector valued rational interpolants (BVRI) have much to do with the vector-grid to be interpolated. When a vector-grid is well-defined, one can directly design an algorithm to compute the BVRI. However, the algorithm no longer works if a vector-grid is ill-defined. Taking the policy of ``divide and conquer', we define a kind of bivariate composite vector valued rational interpolant and establish the corresponding algorithm. A numerical example shows our algorithm still works even if a vector-grid is ill-defined.

     

Shape Preserving Interpolation to Convex Data

1.IntroductionA fundamental problem in computer graphics is the drawing of a smooth curve through aset of data points(xi,fi) (i=0 ,1 ,… ,n) .In many applications,particularly in scientificvisualisation,the y- values are depenenton the x- values and it is…     

Practical, Reliable, Rational Interpolation

An algorithm incorporating features essential for practical,reliable, rational interpolation is explained. This algorithmgenerates a Thiele-Werner continued fraction representationof the interpolant. A backward error analysis is presented forthe algorithm, as well as for its special cases of Newton polynomialinterpolation and Thiele rational interpolation. This is madepossible by introducing into the Newton method, Thiele methodand Werner method a strategy for selecting the interpolationpoints in an optimal order.     

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.     

Shape-Preserving Local Interpolation for Plotting Solutions of ODEs

We investigate the use of piecewise rational interpolants ofDelbourgo and Gregory in an important and widely occurring application.We propose the following algorithm for visually pleasing plotsof the solution of an ordinary differential equation (ODE):use piecewise cubic Hermite interpolation where it can be shownto preserve shape (monotonicity and/or convexity) and also wherethere is no shape to preserve, otherwise use the appropriateconvex or monotone piecewise rational interpolant. Bounds arederived which enable efficient plotting of the rational interpolants.This scheme should be useful in any context where both solutionand derivative of a function are available as data.     

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…     

Note on Rational Interpolants

<正> In this note we present a constructive proof of symmetrical determinantal formulas forthe numerator and denominator of an ordinary rational interpolant,consider the confluencecase and give new determinantal formulas of the rational interpolant by means of Lagrange'sbasis functions.     

C1 monotone cubic Hermite interpolant

Constraining an interpolation to be shape preserving is a well established technique for modelling scientific data. Many techniques express the constraint variables in terms of abstract quantities that are difficult to relate to either physical values or the geometric properties of the interpolant. In this paper, we construct a piecewise monotonic interpolant where the degrees of freedom are expressed in terms of the weights of the rational Bézier cubic interpolant.     

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.     

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.     

一种求二元有理插值函数的方法

给出一种方法可直接计算基于矩形节点的二元有理插值函数的分母在节点处的值 ,进而判断相应的二元有理插值函数是否存在 .此方法运用灵活 ,适用范围广 ,在相应的有理插值函数存在时 ,能给出它的具体表达式 .此外 ,我们还针对文中两个主要逆矩阵 ,给出了相应的递推公式 ,避免了求逆计算 .     

C2-rational cubic spline involving tension parameters

In the present paper, C¹-piecewise rational cubic spline function involving tension parameters is considered which produces a monotonie interpolant to a given monotonie data set. It is observed that under certain conditions the interpolant preserves the convexity property of the data set. The existence and uniqueness of a C²-rational cubic spline interpolant are established. The error analysis of the spline interpolant is also given.     

Constrained interpolation using rational Cubic Spline with linear denominators

In this paper, a rational cubic interpolant spline with linear denominator has been constructed, and it is used to constrain interpolation curves to be bounded in the given region. Necessary and sufficient conditions for the interpolant to satisfy the constraint have been developed. The existence conditions are computationally efficient and easy to apply. Finally, the approximation properties have been studied.     

A rational Lanczos algorithm for model reduction

This paper presents a model reduction method for large-scale linear systems that is based on a Lanczos-type approach. A variant of the nonsymmetric Lanczos method, rational Lanczos, is shown to yield a rational interpolant (multi-point Padé approximant) for the large-scale system. An exact expression for the error in the interpolant is derived. Examples are utilized to demonstrate that the rational Lanczos method provides opportunities for significant improvements in the rate of convergence over single-point Lanczos approaches.     

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.     

On the structure of a table of multivariate rational interpolants

In the table of multivariate rational interpolants the entries are arranged such that the row index indicates the number of numerator coefficients and the column index the number of denominator coefficients. If the homogeneous system of linear equations defining the denominator coefficients has maximal rank, then the rational interpolant can be represented as a quotient of determinants. If this system has a rank deficiency, then we identify the rational interpolant with another element from the table using less interpolation conditions for its computation and we describe the effect this dependence of interpolation conditions has on the structure of the table of multivariate rational interpolants. In the univariate case the table of solutions to the rational interpolation problem is composed of triangles of so-called minimal solutions, having minimal degree in numerator and denominator and using a minimal number of interpolation conditions to determine the solution.Communicated by Dietrich Braess.