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.     

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.     

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.     

Second derivative backward differentiation formulae for ODEs based on barycentric rational interpolants

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...     

Stability of rootfinding for barycentric Lagrange interpolants

Computing the roots of a univariate polynomial can be reduced to computing the eigenvalues of an associated companion matrix. For the monomial basis, these computations have been shown to be numerically stable under certain conditions. However, for certain applications, polynomials are more naturally expressed in other bases, such as the Lagrange basis or orthogonal polynomial bases. For the Lagrange basis, the equivalent stability results have not been published. We show that computing the roots of a polynomial expressed in barycentric form via the eigenvalues of an associated companion matrix pair is numerically stable, and give a bound for the backward error. Numerical experiments show that the error bound is approximately an order of magnitude larger than the backward error. We also discuss the matter of scaling and balancing the companion matrix to bring it closer to a normal pair. With balancing, we are able to produce roots with small backward error.     

Algebraic properties of some new vector-valued rational interpolants

In a recent paper of the author [A. Sidi, A new approach to vector-valued rational interpolation, J. Approx. Theory, 130 (2004) 177–187], three new interpolation procedures for vector-valued functions F(z), where , were proposed, and some of their properties were studied. In this work, after modifying their definition slightly, we continue the study of these interpolation procedures. We show that the interpolants produced via these procedures are unique in some sense and that they are symmetric functions of the points of interpolation. We also show that, under the conditions that guarantee uniqueness, they also reproduce F(z) in case F(z) is a rational function.     

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.     

Convergence rates of a family of barycentric osculatory rational interpolation

It is well-known that osculatory rational interpolation sometimes gives better approximation than Hermite interpolation, especially for large sequences of points. However, it is difficult to solve the problem of convergence and control the occurrence of poles. In this paper, we propose and study a family of barycentric osculatory rational interpolation function, the proposed function and its derivative function both have no real poles and arbitrarily high approximation orders on any real interval.     

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.     

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.     

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.     

Helix splines as an example of affine Tchebycheffian splines

The present paper summarizes the theory of affine Tchebycheffian splines and presents an interesting affine Tchebycheffian free-form scheme, the “helix scheme”. The curve scheme provides exact representations of straight lines, circles and helix curves in an arc length parameterization. The corresponding tensor product surfaces contain helicoidal surfaces, surfaces of revolution and patches on all types of quadrics. We also show an application to the construction of planarC ² motions interpolating a given set of positions. Because the spline curve segments are calculated using a subdivision algorithm, many algorithms, which are of fundamental importance in the B-spline technique, can be applied to helix splines as well. This paper should demonstrate how to create an affine free-form scheme fitting to certain special applications.     

Wachspress type rational complex planar splines of degree (3,1)

We study the existence, uniqueness and approximation properties of rational complex planar spline interpolants of order (3, 1). We also find sufficient conditions for such interpolants to be quasiregular and quasiconformal. Examples are given. This work was carried out with the aid of MACSYMA, a large symbolic manipulation program developed at the MIT Laboratory for Computer Science and supported from 1975 to 1983 by the National Aeronautics and Space Administration under grant NSG 1323, by the Office of Naval Research under grant N00014-77C-0641, by the U.S. Department of Energy under grant ET-78-C-024687, and by the U.S. Air Force under grant F49620-79-C-020, and since 1982 by Symbolics, Inc. of Burlington, MA.     

An evolutionary alternative to rational expectations models of stock markets

Recent time series analyses of stock returns suggest that they contain persistent, mean-reverting stochastic trends, a results that is difficult to explain with traditional rational expectations models of stock markets. A simple model of asset pricing, which uses results on the dynamics of evolutionary games, provides a possible micro-economic foundation for this result.     

A shape preserving area true approximation of histogram by rational splines

For a given histogram, we consider an application of a simple rational spline to a shape preserving area true approximation of the histogram. An algorithm for determination of the spline is as easy as one with a quadratic polynomial spline, while the latter does not always preserve the shape of the histogram. Some numerical examples are given at the end of the paper.     

An application of Kronecker's theorem to rational functions

Dedicated to the memory of S.K. Pichorides     

Approximation of an interval by splines

For a function defined on an interval we indicate a method of constructing an interpolational spline, which does not require knowledge of the derivatives of the function. The interpolational process possesses good convergence properties.Translated from Matematicheskie Zametki, Vol. 16, No. 3, pp. 491–500, September, 1974.The author thanks Yu. N. Subbotin for his statement of the problem.