Doctoral thesis

Advances in barycentric rational interpolation of a function and its derivatives


138 p

Thèse de doctorat: Università della Svizzera italiana, 2019

English 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.
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