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1.
Jean-Paul Berrut 《Numerische Mathematik》1989,55(6):747-718
Summary We present a barycentric representation of cardinal interpolants, as well as a weighted barycentric formula for their efficient evaluation. We also propose a rational cardinal function which in some cases agrees with the corresponding cardinal interpolant and, in other cases, is even more accurate.In numerical examples, we compare the relative accuracy of those various interpolants with one another and with a rational interpolant proposed in former work.Dedicated to the memory of Peter HenriciThis work was done at the University of California at San Diego, La Jolla 相似文献
2.
André Pierro de Camargo 《Numerical Algorithms》2016,72(1):131-152
In this note we extend the definition of the first barycentric formula for Lagrange interpolation to Floater-Hormann interpolants and present an algorithm to evaluate it which is backward stable on the entire real line. We also discuss in detail the numerical stability of the second barycentric formula for Floater-Hormann interpolants. 相似文献
3.
Numerical Algorithms - For their several attractive features from the viewpoint of the numerical computations, linear barycentric rational interpolants have been recently used to construct various... 相似文献
4.
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. 相似文献
5.
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. 相似文献
6.
用广义交互确认方法选择良好参数进行多元多项式散乱数据自然样条光顺 总被引:2,自引:1,他引:2
1.简介多元样条函数在多元逼近中发挥很大作用,已有数量相当多的综合报告和研究论文正式发表,就在1996年6月在法国召开的第三届国际曲线与曲面会议上便有不少多元样条方面的报告,不过总的感觉是仍然缺乏对噪声数据特别是散乱数据的有效光顺方法.李岳生、崔锦泰、关履泰、胡日章等讨论广义调配样条与张量积函数,并用希氏空间样条方法处理多元散乱数据样条插值与光顺,提出多元多项式自然样条,推广了相应一元的结果.我们知道,在样条光顺中有一个如何选择参数的问题,用广义交互确认方法(generalizedcross-validation,以下简称GC… 相似文献
7.
Andrew Gillette Alexander Rand Chandrajit Bajaj 《Advances in Computational Mathematics》2012,37(3):417-439
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. 相似文献
8.
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. 相似文献
9.
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. 相似文献
10.
《Journal of Computational and Applied Mathematics》1997,86(1):45-52
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. 相似文献
11.
12.
We present a methodology for fitting time-varying paired comparisons models in which the parameters are allowed to vary deterministically, as opposed to stochastically, with time. Our dynamic paired comparisons model is based on a new closed-form for Stern’s continuum of paired comparisons models which include the Bradley–Terry model and the Thurstone–Mosteller model. The dynamic element of our model is facilitated by utilising barycentric rational interpolants BRIs. An incidental result of our work is to show that BRIs often provide a better fit to data than the obvious alternative of spline interpolation. We use our model to shed light on the debate of who is the greatest tennis player of the Open Era of men’s professional tennis since 1968. Constructing a single rankings list from our model is not trivial as there are many alternative metrics that could be used to identify which player was the best ever. We present three alternative rankings lists derived from our model. In general our rankings lists largely agree with the rankings list based on number of Grand Slam titles won, which, to some extent, validates our choice of metrics. So who is the greatest tennis player of the Open Era? Roger Federer seems like the most likely candidate, with Bjorn Borg and Jimmy Connors close behind. 相似文献
13.
Jean-Paul Berrut 《Numerical Algorithms》2000,24(1-2):17-29
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. 相似文献
14.
R.L Eubank 《Statistics & probability letters》1984,2(1):9-14
The matrix which transforms the data vector to the vector of fitted values for smoothing splines is termed the hat matrix. This matrix is shown to have many of the same properties, and is seen to play the same role in the variances and covariances of the residuals, as its regression analysis counterpart. This fact is utilized to propose several possible diagnostic measures for use with smoothing splines. The extension of these results to include multivariate Laplacian smoothing spline is also indicated. 相似文献
15.
An increasingly popular method for smoothing noisy data is penalized regression spline fitting. In this paper a new procedure
is proposed for fitting robust penalized regression splines. This procedure is computationally fast, straightforward to implement,
and can be paired with any smoothing parameter selection method. In addition, it can also be extended to other settings, such
as additive mixed modeling. Both simulated and real data examples are used to illustrate the effectiveness of the procedure. 相似文献
16.
Jean-Paul Berrut 《Numerische Mathematik》2009,112(3):341-361
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. 相似文献
17.
《Journal of Computational and Applied Mathematics》1999,103(2):221-237
The construction of range restricted univariate and bivariate interpolants to gridded data is considered. We apply Gregory's rational cubic C1 splines as well as related rational quintic C2 splines. Assume that the lower and upper obstacles are compatible with the data set. Then the tension parameters occurring in the mentioned spline classes can be always determined in such a way that range restricted interpolation is successful. 相似文献
18.
F. Utreras Diaz 《Numerische Mathematik》1980,34(1):15-28
Summary We consider the problem of approximating an unknown functionf, known with error atn equally spaced points of the real interval [a, b].To solve this problem, we use the natural polynomial smoothing splines. We show that the eigenvalues associated to these splines converge to the eigenvalues of a differential operator and we use this fact to obtain an algorithm, based on the Generalized Cross Validation method, to calculate the smoothing parameter.With this algorithm, we divide byn the time used by classical methods. 相似文献
19.
G. Klein 《Journal of Computational and Applied Mathematics》1979,5(4):259-264
Interpolating natural splines are used for the algebraization and smoothing regularization of linear Fredholm integral equations of the first kind. A simplified version of statistical regularization is presented and, in turn, applied to data graduation by smoothing natural splines. 相似文献
20.
In this paper, we study the local asymptotic behavior of the regression spline estimator in the framework of marginal semiparametric model. Similarly to Zhu, Fung and He (2008), we give explicit expression for the asymptotic bias of regression spline estimator for nonparametric function f. Our results also show that the asymptotic bias of the regression spline estimator does not depend on the working covariance matrix, which distinguishes the regression splines from the smoothing splines and the seemingly u... 相似文献