A barycentric trigonometric Hermite interpolant via an iterative approach

In this paper we extend and generalise an interative approach for constructing the Hermite interpolant introduced by Cirillo and Hormann (2018) for the Floater-Hormann family of interpolants. In particular, we apply that scheme to produce an effective barycentric rational trigonometric Hermite interpolant using the basis function of the interpolant introduced by Berrut (1988). In order to give an easy construction of such an interpolant we compute the differentation matrix analytically and we conclude with various examples and a numerical study of the rate of convergence at equidistant nodes

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

On the Lebesgue constant of barycentric rational Hermite interpolants at equidistant nodes

Barycentric rational Floater–Hormann interpolants compare favourably to classical polynomial interpolants in the case of equidistant nodes, because the Lebesgue constant associated with these interpolants grows logarithmically in this setting, in contrast to the exponential growth experienced by polynomials. In the Hermite setting, in which also the first derivatives of the interpolant are prescribed at the nodes, the same exponential growth has been proven for polynomial interpolants, and the main goal of this paper is to show that much better results can be obtained with a recent generalization of Floater–Hormann interpolants. After summarizing the construction of these barycentric rational Hermite interpolants, we study the behaviour of the corresponding Lebesgue constant and prove that it is bounded from above by a constant. Several numerical examples confirm this result

High-Order, Stable, And Efficient Pseudospectral Method Using Barycentric Gegenbauer Quadratures

The work reported in this article presents a high-order, stable, and efficient Gegenbauer pseudospectral method to solve numerically a wide variety of mathematical models. The proposed numerical scheme exploits the stability and the well-conditioning of the numerical integration operators to produce well-conditioned systems of algebraic equations, which can be solved easily using standard algebraic system solvers. The core of the work lies in the derivation of novel and stable Gegenbauer quadratures based on the stable barycentric representation of Lagrange interpolating polynomials and the explicit barycentric weights for the Gegenbauer-Gauss (GG) points. A rigorous error and convergence analysis of the proposed quadratures is presented along with a detailed set of pseudocodes for the established computational algorithms. The proposed numerical scheme leads to a reduction in the computational cost and time complexity required for computing the numerical quadrature while sharing the same exponential order of accuracy achieved by Elgindy and Smith-Miles (2013). The bulk of the work includes three numerical test examples to assess the efficiency and accuracy of the numerical scheme. The present method provides a strong addition to the arsenal of numerical pseudospectral methods, and can be extended to solve a wide range of problems arising in numerous applications.Comment: 30 pages, 10 figures, 1 tabl

Eigenvalue Methods for Interpolation Bases

This thesis investigates eigenvalue techniques for the location of roots of polynomials expressed in the Lagrange basis. Polynomial approximations to functions arise in almost all areas of computational mathematics, since polynomial expressions can be manipulated in ways that the original function cannot. Polynomials are most often expressed in the monomial basis; however, in many applications polynomials are constructed by interpolating data at a series ofpoints. The roots of such polynomial interpolants can be found by computing the eigenvalues of a generalized companion matrix pair constructed directly from the values of the interpolant. This affords the opportunity to work with polynomials expressed directly in the interpolation basis in which they were posed, avoiding the often ill-conditioned transformation between bases. Working within this framework, this thesis demonstrates that computing the roots of polynomials via these companion matrices is numerically stable, and the matrices involved can be reduced in such a way as to significantly lower the number of operations required to obtain the roots. Through examination of these various techniques, this thesis offers insight into the speed, stability, and accuracy of rootfinding algorithms for polynomials expressed in alternative bases

Linear barycentric rational interpolation method for solving Kuramoto-Sivashinsky equation

The Kuramoto-Sivashinsky (KS) equation being solved by the linear barycentric rational interpolation method (LBRIM) is presented. Three kinds of linearization schemes, direct linearization, partial linearization and Newton linearization, are presented to get the linear equation of the Kuramoto-Sivashinsky equation. Matrix equations of the discrete Kuramoto-Sivashinsky equation are also given. The convergence rate of LBRIM for solving the KS equation is also proved. At last, two examples are given to prove the theoretical analysis