Convergence rates of derivatives of a family of barycentric rational interpolants

In polynomial and spline interpolation the k-th derivative of the interpolant, as a function of the mesh size h, typically converges at the rate of O(hd+1−k) as h→0, where d is the degree of the polynomial or spline. In this paper we establish, in the important cases k=1,2, the same convergence rate for a recently proposed family of barycentric rational interpolants based on blending polynomial interpolants of degree d.     

切触有理插值新方法

针对传统连分式插值,计算复杂度高,计算过程中分母为零的不可预知性及插值函数不满足某些给定条件,应用不方便等问题,利用已知节点、函数值、导数值,构造两个多项式,分别作为有理插值函数的分子和分母,得出各阶导数条件下切触有理插值的新公式,并给出特殊情形的表达式.若添加适当的参数,可任意降低插值函数次数.该方法计算简洁,应用方便,插值函数的分母在节点处不为零且满足全部插值条件.数值例子验证了新方法的可行性、有效性和实用性.     

Block based Newton-like blending osculatory rational interpolation

With Newton’s interpolating formula, we construct a kind of block based Newton-like blending osculatory interpolation.The interpolation provides us many flexible interpolation schemes for choices which include the expansive Newton’s polynomial interpolation as its special case. A bivariate analogy is also discussed and numerical examples are given to show the effectiveness of the interpolation.     

A new algorithm for osculatory rational interpolation

Summary A new algorithm is derived for computing continued fractions whose convergents form the elements of the osculatory rational interpolation table.     

Padé–type rational and barycentric interpolation

In this paper, we consider the particular case of the general rational Hermite interpolation problem where only the value of the function is interpolated at some points, and where the function and its first derivatives agree at the origin. Thus, the interpolants constructed in this way possess a Padé–type property at 0. Numerical examples show the interest of the procedure. The interpolation procedure can be easily modified to introduce a partial knowledge on the poles and the zeros of the function to approximated. A strategy for removing the spurious poles is explained. A formula for the error is proved in the real case. Applications are given.     

The barycentric weights of rational interpolation with prescribed poles

We introduce a method for calculating rational interpolants when some (but not necessarily all) of their poles are prescribed. The algorithm determines the weights in the barycentric representation of the rationals; it simply consists in multiplying each interpolated value by a certain number, computing the weights of a rational interpolant without poles, and finally multiplying the weights by those same numbers. The supplementary cost in comparison with interpolation without poles is about (v + 2)N, where v is the number of poles and N the number of interpolation points. We also give a condition under which the computed rational interpolation really shows the desired poles.     

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.     

Convergence rates for exponential interpolation series

An analysis of the rate of convergence is made for the interpolation series based on the biorthogonal system (_n^Δ)ƒ(0) and e_n(z) = Δⁿx^z ¦_x−1, which was recently shown to be convergent for certain entire functions of exponential type. An error bound is obtained which is shown to vary as a negative power of the number of terms in the partial sum. Comparison is made with numerical calculations in a few simple cases and certain practical applications are mentioned.     

Linear barycentric rational quadrature

Linear interpolation schemes very naturally lead to quadrature rules. Introduced in the eighties, linear barycentric rational interpolation has recently experienced a boost with the presentation of new weights by Floater and Hormann. The corresponding interpolants converge in principle with arbitrary high order of precision. In the present paper we employ them to construct two linear rational quadrature rules. The weights of the first are obtained through the direct numerical integration of the Lagrange fundamental rational functions; the other rule, based on the solution of a simple boundary value problem, yields an approximation of an antiderivative of the integrand. The convergence order in the first case is shown to be one unit larger than that of the interpolation, under some restrictions. We demonstrate the efficiency of both approaches with numerical tests.     

Convergence of osculatory quadrature formulae

The regions of analyticity of functions to be integrated using equally spaced osculatory quadrature formulae are obtained. As a by-product it is noted that the asymptotic forms used are applicable to estimating or placing bounds on errors.     

Convergence rates for lacunary trigonometric interpolation

Conditions for the solvability of lacunary trigonometric interpolation have been given by Cavaretta, Sharma, and Varga. When these conditions are satisfied, linear operators are defined on the space of continuous periodic functions. In this paper, the saturation rate and class of many of these operators is determined.     

Error estimates for generalized barycentric interpolation

We prove the optimal convergence estimate for first-order interpolants used in finite element methods based on three major approaches for generalizing barycentric interpolation functions to convex planar polygonal domains. The Wachspress approach explicitly constructs rational functions, the Sibson approach uses Voronoi diagrams on the vertices of the polygon to define the functions, and the Harmonic approach defines the functions as the solution of a PDE. We show that given certain conditions on the geometry of the polygon, each of these constructions can obtain the optimal convergence estimate. In particular, we show that the well-known maximum interior angle condition required for interpolants over triangles is still required for Wachspress functions but not for Sibson functions.     

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.     

Barycentric rational interpolation with no poles and high rates of approximation

It is well known that rational interpolation sometimes gives better approximations than polynomial interpolation, especially for large sequences of points, but it is difficult to control the occurrence of poles. In this paper we propose and study a family of barycentric rational interpolants that have no real poles and arbitrarily high approximation orders on any real interval, regardless of the distribution of the points. These interpolants depend linearly on the data and include a construction of Berrut as a special case.     

Chebyshev rational interpolation

The Lanczos method and its variants can be used to solve efficiently the rational interpolation problem. In this paper we present a suitable fast modification of a general look-ahead version of the Lanczos process in order to deal with polynomials expressed in the Chebyshev orthogonal basis. The proposed approach is particularly suited for approximating analytic functions by means of rational interpolation at certain nodes located on the boundary of an elliptical region of the complex plane. In fact, in this case it overcomes some of the numerical difficulties which limited the applicability of the look-ahead Lanczos process for determining the coefficients both of the numerators and of the denominators with respect to the standard power basis. This revised version was published online in August 2006 with corrections to the Cover Date.     

Fourier and barycentric formulae for equidistant Hermite trigonometric interpolation

We consider the Hermite trigonometric interpolation problem of order 1 for equidistant nodes, i.e., the problem of finding a trigonometric polynomial t that interpolates the values of a function and of its derivative at equidistant points. We give a formula for the Fourier coefficients of t in terms of those of the two classical trigonometric polynomials interpolating the values and those of the derivative separately. This formula yields the coefficients with a single FFT. It also gives an aliasing formula for the error in the coefficients which, on its turn, yields error bounds and convergence results for differentiable as well as analytic functions. We then consider the Lagrangian formula and eliminate the unstable factor by switching to the barycentric formula. We also give simplified formulae for even and odd functions, as well as consequent formulae for Hermite interpolation between Chebyshev points.     

On the denominator values and barycentric weights of rational interpolants

We improve upon the method of Zhu and Zhu [A method for directly finding the denominator values of rational interpolants, J. Comput. Appl. Math. 148 (2002) 341–348] for finding the denominator values of rational interpolants, reducing considerably the number of arithmetical operations required for their computation. In a second stage, we determine the points (if existent) which can be discarded from the rational interpolation problem. Furthermore, when the interpolant has a linear denominator, we obtain a formula for the barycentric weights which is simpler than the one found by Berrut and Mittelmann [Matrices for the direct determination of the barycentric weights of rational interpolation, J. Comput. Appl. Math. 78 (1997) 355–370]. Subsequently, we give a necessary and sufficient condition for the rational interpolant to have a pole.