Improved conditioning of the Floater--Hormann interpolants

The Floater--Hormann family of rational interpolants do not have spurious poles or unattainable points, are efficient to calculate, and have arbitrarily high approximation orders. One concern when using them is that the amplification of rounding errors increases with approximation order, and can make balancing the interpolation error and rounding error difficult. This article proposes to modify the Floater--Hormann interpolants by including additional local polynomial interpolants at the ends of the interval. This appears to improve the conditioning of the interpolants and allow higher approximation orders to be used in practice.Comment: 13 pages, 4 figures, 1 tabl

Convergence of linear barycentric rational interpolation for analytic functions

Polynomial interpolation to analytic functions can be very accurate, depending on the distribution of the interpolation nodes. However, in equispaced nodes and the like, besides being badly conditioned, these interpolants fail to converge even in exact arithmetic in some cases. Linear barycentric rational interpolation with the weights presented by Floater and Hormann can be viewed as blended polynomial interpolation and often yields better approximation in such cases. This has been proven for differentiable functions and indicated in several experiments for analytic functions. So far, these rational interpolants have been used mainly with a constant parameter usually denoted by d, the degree of the blended polynomials, which leads to small condition numbers but to merely algebraic convergence. With the help of logarithmic potential theory we derive asymptotic convergence results for analytic functions when this parameter varies with the number of nodes. Moreover, we present suggestions on how to choose d in order to observe fast and stable convergence, even in equispaced nodes where stable geometric convergence is provably impossible. We demonstrate our results with several numerical examples

Treating the Gibbs phenomenon in barycentric rational interpolation and approximation via the S-Gibbs algorithm

In this work, we extend the so-called mapped bases or fake nodes approach to the barycentric rational interpolation of Floater-Hormann and to AAA approximants. More precisely, we focus on the reconstruction of discontinuous functions by the S-Gibbs algorithm introduced in [12]. Numerical tests show that it yields an accurate approximation of discontinuous functions

On the constrained mock-Chebyshev least-squares

The algebraic polynomial interpolation on uniformly distributed nodes is affected by the Runge phenomenon, also when the function to be interpolated is analytic. Among all techniques that have been proposed to defeat this phenomenon, there is the mock-Chebyshev interpolation which is an interpolation made on a subset of the given nodes whose elements mimic as well as possible the Chebyshev-Lobatto points. In this work we use the simultaneous approximation theory to combine the previous technique with a polynomial regression in order to increase the accuracy of the approximation of a given analytic function. We give indications on how to select the degree of the simultaneous regression in order to obtain polynomial approximant good in the uniform norm and provide a sufficient condition to improve, in that norm, the accuracy of the mock-Chebyshev interpolation with a simultaneous regression. Numerical results are provided.Comment: 17 pages, 9 figure

A generalization of Floater--Hormann interpolants

In this paper the interpolating rational functions introduced by Floater and Hormann are generalized leading to a whole new family of rational functions depending on $\gamma$, an additional positive integer parameter. For $\gamma = 1$, the original Floater--Hormann interpolants are obtained. When $\gamma>1$ we prove that the new rational functions share a lot of the nice properties of the original Floater--Hormann functions. Indeed, for any configuration of nodes, they have no real poles, interpolate the given data, preserve the polynomials up to a certain fixed degree, and have a barycentric-type representation. Moreover, we estimate the associated Lebesgue constants in terms of the minimum ($h^*$) and maximum ($h$) distance between two consecutive nodes. It turns out that, in contrast to the original Floater-Hormann interpolants, for all $\gamma > 1$ we get uniformly bounded Lebesgue constants in the case of equidistant and quasi-equidistant nodes configurations (i.e., when $h\sim h^*$). In such cases, we also estimate the uniform and the pointwise approximation errors for functions having different degree of smoothness. Numerical experiments illustrate the theoretical results and show a better error profile for less smooth functions compared to the original Floater-Hormann interpolants.Comment: 29 page

On the Lebesgue constant of the trigonometric Floater-Hormann rational interpolant at equally spaced nodes

It is well known that the classical polynomial interpolation gives bad approximation if the nodes are equispaced. A valid alternative is the family of barycentric rational interpolants introduced by Berrut in [4], analyzed in terms of stability by Berrut and Mittelmann in [5] and their extension done by Floater and Hormann in [8]. In this paper firstly we extend them to the trigonometric case, then as in the Floater-Hormann classical interpolant, we study the growth of the Lebesgue constant on equally spaced points. We show that the growth is logarithmic providing a stable interpolation operato

Advances in barycentric rational interpolation of a function and its derivatives

Linear barycentric rational interpolants are a particular kind of rational interpolants, defined by weights that are independent of the function f. Such interpolants have recently proved to be a viable alternative to more classical interpolation methods, such as global polynomial interpolants and splines, especially in the equispaced setting. Other kinds of interpolants might indeed suffer from the use of floating point arithmetic, while the particular form of barycentric rational interpolants guarantees that the interpolation of data is achieved even if rounding errors affect the computation of the weights, as long as they are non zero. This dissertation is mainly concerned with the analysis of the convergence of a particular family of barycentric rational interpolants, the so-called Floater-Hormann family. Such functions are based on the blend of local polynomial interpolants of fixed degree d with rational blending functions, and we investigate their behavior in the interpolation of the derivatives of a function f. In the first part we focus on the approximation of the k-th derivative of the function f with classical Floater-Hormann interpolants. We first introduce the Floater-Hormann interpolation scheme and present the main advantages and disadvantages of these functions compared to polynomial and classical rational interpolants. We then proceed by recalling some previous result regarding the convergence rate of the k-th derivatives of these interpolants and extend these results. In particular, we prove that the k-th derivative of the Floater-Hormann interpolant converges to f^(k) at the rate of O(h_j^(d+1-k), for any k >= 0 and any set of well-spaced nodes, where h_j is the local mesh size. In the second part we instead focus on the interpolation of the derivatives of a function up to some order m. We first present several theorems regarding this kind of interpolation, both for polynomials and barycentric rational functions, and then we introduce a new iterative approach that allows us to generalise the Floater-Hormann family to this new setting. The resulting rational Hermite interpolants have numerator and denominator of degree at most (m+1)(n+1)-1 and (m+1) (n-d), respectively, and converge to the function at the rate of O(h^((m+1)(d+1))) as the mesh size h converges to zero. Next, we focus on the conditioning of the interpolants, presenting some classical results regarding polynomials and showing the reasons that make these functions unsuited to fit any kind of equispaced data. We then compare these results with the ones regarding Floater-Hormann interpolants at equispaced nodes, showing again the advantages of this interpolation scheme in this setting. Finally, we extend these conclusions to the Hermite setting, first introducing the generalisation of the results presented for polynomial Lagrange interpolants and then bounding the condition number of our Hermite interpolant at equispaced nodes by a constant independent of n. The comparison between this result and the equivalent for polynomials shows that our barycentric rational interpolants should be in many cases preferred to polynomials

Linear rational finite differences from derivatives of barycentric rational interpolants

Derivatives of polynomial interpolants lead in a natural way to approximations of derivatives of the interpolated function, e.g., through finite differences. We extend a study of the approximation of derivatives of linear barycentric rational interpolants and present improved finite difference formulas arising from these interpolants. The formulas contain the classical finite differences as a special case and are more stable for calculating one-sided derivatives as well as derivatives close to boundaries