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This work is concerned with the interpolation of a function f when using a low number of interpolation points, as required by the finite element method for solving PDEs numerically. The function f is assumed to have a jump or a steep derivative, and our goal is to minimize the oscillations produced by the Gibbs phenomenon while preserving the approximation properties for smoother functions. This is achieved by interpolating the transform f ^ = g ∘ f using Lagrange polynomials, where g is a rational transformation chosen by minimizing a suitable functional depending on the values of f . The mapping g is monotonic and constructed to possess boundary layers that remove the Gibbs phenomenon. No previous knowledge of the location of the jump is required. The extension to functions of several variables is straightforward, of which we provide several examples. Finally, we show how the interpolation fits the finite element method and compare it with known strategies. [ABSTRACT FROM AUTHOR]

We construct some smooth functions defined over a sphere that interpolate large sets of scattered data, using some modified Shepard methods, the least squares thin-plate spline and the inverse multiquadric functions. We illustrate the benefits of our methods in numerical examples for some test functions and some real data applications. [ABSTRACT FROM AUTHOR]

In this work, a general class of interpolation and smoothing natural exponential splines with respect to fourth order differential operators with two real parameters is considered. Some sufficient conditions for the associated matrix R to be a diagonally dominant matrix are given. Based on these, fast algorithms for computing the coefficients of this general class of exponential splines are developed. The obtained splines have C 2 continuity and are the minimum solution of the combination of interpolation and smoothing energy integral. The performances of the resulting splines in financial data from the S &P500 index are given. Numerical experiments show that the resulting splines have more freedom to adjust the shape and control the energy of the curves. Cross-validation and generalized cross-validation for determining an appropriate smoothing parameter are also given. [ABSTRACT FROM AUTHOR]

The best polynomial approximation and the Chebyshev approximation are both important in numerical analysis. In tradition, the best approximation is regarded as better than the Chebyshev approximation, because it is usually considered in the uniform norm. However, it is not always superior to the latter, as noticed by Trefethen (Math. Today 47:184–188, 2011) for the function f (x) = | x - 0.25 | . Recently, Wang (arxiv, 2106.03456, 2021) observed a similar phenomenon for a function with an algebraic singularity and proved it in theory. In this paper, we find that for functions with logarithmic endpoint singularities, the pointwise errors of the Chebyshev approximation are smaller than those of the best approximations of the same degree, except only at the narrow boundary layer. The pointwise error for the Chebyshev series truncated at the degree n is O (n - κ) ( κ = min { 2 γ + 1 , 2 δ + 1 } ), but increases as O (n - (κ - 1)) at the endpoint singularities. Theorems are given to explain this effect. [ABSTRACT FROM AUTHOR]

In this work, our motivation is to design an impressive new numerical approximation on non-uniform grid points for the Caputo fractional derivative in time 0 C D t α with the order α ∈ (1,2). An adaptive high-order stable implicit difference scheme is developed for the time-fractional diffusion wave equations (TFDWEs) by using estimation of order O (N t α - 5) for the Caputo derivative in the time domain on non-uniform mesh and well-known second-order central difference approximation for estimating the spatial derivative on a uniform mesh. The designed algorithm allows one to build adaptive nature where the scheme is adjusted according to the behaviour of α in order to keep the numerical errors very small and converge to the solution very fast as compared to the previously investigated scheme. We rigorously analyze the local truncation errors, unconditional stability of the proposed method, and its convergence of (5 − α)-th order in time and second-order in space for all values of α ∈ (1,2). A reduced order technique is implemented by using moving mesh refinement and assemble with the derived scheme in order to improve the temporal accuracy at several starting time levels. Furthermore, the numerical stability of the derived adaptive scheme is verified by imposing random external noises. Some numerical tests are given to show that the numerical results are consistent with the theoretical results. [ABSTRACT FROM AUTHOR]

We construct new families of direct serendipity and direct mixed finite elements on general planar, strictly convex polygons that are H1 and H(div) conforming, respectively, and possess optimal order of accuracy for any order. They have a minimal number of degrees of freedom subject to the conformity and accuracy constraints. The name arises because the shape functions are defined directly on the physical elements, i.e., without using a mapping from a reference element. The finite element shape functions are defined to be the full spaces of scalar or vector polynomials plus a space of supplemental functions. The direct serendipity elements are the precursors of the direct mixed elements in a de Rham complex. The convergence properties of the finite elements are shown under a regularity assumption on the shapes of the polygons in the mesh, as well as some mild restrictions on the choices one can make in the construction of the supplemental functions. Numerical experiments on various meshes exhibit the performance of these new families of finite elements. [ABSTRACT FROM AUTHOR]

Under appropriate technical assumptions, the simple-loop theory allows to derive various types of asymptotic expansions for the eigenvalues of Toeplitz matrices generated by a function f. Independently and under the milder hypothesis that f is even and monotone over [0,π], matrix-less algorithms have been developed for the fast eigenvalue computation of large Toeplitz matrices, within a linear complexity in the matrix order: behind the high efficiency of such algorithms there are the expansions predicted by the simple-loop theory, combined with the extrapolation idea. Here we focus our attention on a change of variable, followed by the asymptotic expansion of the new variable, and we adapt the matrix-less algorithm to the considered new setting. Numerical experiments show a higher precision (till machine precision) and the same linear computation cost, when compared with the matrix-less procedures already presented in the relevant literature. Among the advantages, we concisely mention the following: (a) when the coefficients of the simple-loop function are analytically known, the algorithm computes them perfectly; (b) while the proposed algorithm is better or at worst comparable to the previous ones for computing the inner eigenvalues, it is vastly better for the computation of the extreme eigenvalues; a mild deterioration in the quality of the numerical experiments is observed when dense Toeplitz matrices are considered, having generating function of low smoothness and not satisfying the simple-loop assumptions. [ABSTRACT FROM AUTHOR]

In this paper, the solution to a bivariate Appell interpolation problem proposed in a previous work is given. Bounds of the truncation error are considered. Ten new interpolants for real, regular, bivariate functions are constructed. Numerical examples and comparisons with bivariate Bernstein polynomials are considered. [ABSTRACT FROM AUTHOR]

An inverse central ordering of the nodes is proposed for the Newton interpolation formula. This ordering may improve the stability for certain distributions of nodes. For equidistant nodes, an upper bound of the conditioning is provided. This bound is close to the bound of the conditioning in the Lagrange interpolation formula, whose conditioning is the lowest. This ordering is related to a pivoting strategy of a matrix elimination procedure called Neville elimination. The results are illustrated with examples. [ABSTRACT FROM AUTHOR]

We propose a fast solver for the variable-order (VO) time-fractional diffusion equation. Due to the impact of the time-dependent VO function, the resulting coefficient matrix of the large linear system assembling discrete equations of all time levels is a block lower triangular matrix without the block Toeplitz structure. Here, we approximate the off-diagonal blocks by low-rank matrices based on the polynomial interpolation, which can be constructed in O (M log 2 M) operations with the same number storage requirement, where M is the number of time steps. Furthermore, a divide-and-conquer method is developed to fast solve the approximated linear system. The proposed solver can be implemented in O (N M log 2 M) complexity with N being the degree of freedom in space. The accuracy of approximation is theoretically studied, and the stability and convergence of the proposed fast method are also investigated. Numerical experiments are carried out to exemplify the accuracy and efficiency of the proposed method. [ABSTRACT FROM AUTHOR]

We present an asymptotic analysis of adaptive methods for Lp approximation of functions f ∈ Cr([a, b]), where 1 ≤ p ≤ + ∞ . The methods rely on piecewise polynomial interpolation of degree r − 1 with adaptive strategy of selecting m subintervals. The optimal speed of convergence is in this case of order m−r and it is already achieved by the uniform (nonadaptive) subdivision of the initial interval; however, the asymptotic constant crucially depends on the chosen strategy. We derive asymptotically best adaptive strategies and show their applicability to automatic Lp approximation with a given accuracy ε. [ABSTRACT FROM AUTHOR]

In order to solve Prandtl-type equations we propose a collocation-quadrature method based on de la Vallée Poussin (briefly VP) filtered interpolation at Chebyshev nodes. Uniform convergence and stability are proved in a couple of Hölder-Zygmund spaces of locally continuous functions. With respect to classical methods based on Lagrange interpolation at the same collocation nodes, we succeed in reproducing the optimal convergence rates of the L2 case and cut off the typical log factor which seemed inevitable dealing with uniform norms. Such an improvement does not require a greater computational effort. In particular, we propose a fast algorithm based on the solution of a simple 2-bandwidth linear system and prove that, as its dimension tends to infinity, the sequence of the condition numbers (in any natural matrix norm) tends to a finite limit. [ABSTRACT FROM AUTHOR]

We offer the algorithm for choosing tension parameters of the generalized splines for convexity preserving interpolation. The resulting spline minimally differs from the classical cubic spline and coincides with it if sufficient convexity conditions are satisfied for the last one. We consider specific algorithms for different generalized cubic splines such as rational, exponential, variable power, hyperbolic splines, and splines with additional knots. [ABSTRACT FROM AUTHOR]

Scattered data interpolation can be used to approximate a multivariate function by a linear combination of positive definite radial basis functions (RBFs). In practice, the approximation error stagnates (due to numerical instability) even if the function is smooth and the number of data centers is increased. A smaller approximation error can be obtained using multilevel interpolation on a sequence of nested subsets of the initial set of centers. For the construction of these nested subsets, we compare two thinning algorithms from the literature, a greedy algorithm based on nearest neighbor computations and a Poisson point process. The main novelty of our approach lies in the use of H -matrices both for the solution of linear systems and for the evaluation of residual errors at each level. For the solution of linear systems, we use GMRes combined with a domain decomposition preconditioner. Using H -matrices allows us to solve larger problems more efficiently compared with multilevel interpolation based on dense matrices. Numerical experiments with up to 50,000 scattered centers in two and three spatial dimensions demonstrate that the computational time required for the construction of the multilevel interpolant using H -matrices is of almost linear complexity with respect to the number of centers. [ABSTRACT FROM AUTHOR]

The Floater–Hormann family of the barycentric rational interpolants has recently gained popularity because of its excellent stability properties and highly order of convergence. The purpose of this paper is to design highly accurate and stable schemes based on this family of interpolants for the numerical solution of stiff Volterra integral equations of the second kind. [ABSTRACT FROM AUTHOR]

The paper provides a general and simple approach for explicitly constructing multiscale radial kernels with high-order generalized Strang-Fix conditions from a given univariate generator. The resulting kernels are constructed by taking a linear functional to the scaled f -form of the generator with respect to the scale variable. Equivalent divided difference forms of the kernels are also derived; based on which, a pyramid-like algorithm for fast and stable computation of multiscale radial kernels is proposed. In addition, characterizations of the kernels in both the spatial and frequency domains are given, which show that the generalized Strang-Fix condition, the moment condition, and the condition of polynomial reproduction in the convolution sense are equivalent to each other. Hence, as a byproduct, the paper provides a unified view of these three classical concepts. These kernels can be used to construct quasi-interpolation with high approximation accuracy and construct convolution operators with high approximation orders, to name a few. As an example, we construct a quasi-interpolation scheme for irregularly spaced data and derived its error estimates and choices of scale parameters of multiscale radial kernels. Numerical results of approximating a bivariate Franke function using our quasi-interpolation are presented at the end of the paper. Both theoretical and numerical results show that quasi-interpolation with multiscale radial kernels satisfying high-order generalized Strang-Fix conditions usually provides high approximation orders. [ABSTRACT FROM AUTHOR]

In this paper, the properties of a new family of nonlinear dyadic subdivision schemes are presented and studied depending on the conditions imposed to the mean used to rewrite the linear scheme upon which the new scheme is based. The convergence, stability, and order of approximation of the schemes of the family are analyzed in general. Also, the elimination of the Gibbs oscillations close to discontinuities is proved in particular cases. It is proved that these schemes converge towards limit functions that are Hölder continuous with exponent larger than 1. The results are illustrated with several examples. [ABSTRACT FROM AUTHOR]

Given discrete function values sampled at uniform centers, the iterated quasi-interpolation approach for approximating the m th derivative consists of two steps. The first step adopts m successive applications of the operator DQ (the quasi-interpolation operator Q first, and then the differentiation operator D) to get approximated values of the m th derivative at uniform centers. Then, by one further application of the quasi-interpolation operator Q to corresponding approximated derivative values gives the final approximation of the m th derivative. The most salient feature of the approach is that it approximates all derivatives with the same convergence rate. In addition, it is valid for a general multivariate function, compared with the existing iterated interpolation approaches that are only valid for periodic functions, so far. Numerical examples of approximating high-order derivatives using both the iterated and direct approach based on B-spline quasi-interpolation and multiquadric quasi-interpolation are presented at the end of the paper, which demonstrate that the iterated quasi-interpolation approach provides higher approximation orders than the corresponding direct approach. [ABSTRACT FROM AUTHOR]

In this paper, we propose a variational method in order to solve Bratu's problem for two dimensions in an adequate space of biquadratic spline functions. The solution is obtained by resolving a sequence of boundary value problems. We study some characterizations of the functions of such sequence and we express them as some linear combination of biquadratic spline bases functions. We finish by showing some numerical and graphical examples in order to prove the validity and the effectiveness of our method. [ABSTRACT FROM AUTHOR]

We propose algorithms to take point sets for kernel-based interpolation of functions in reproducing kernel Hilbert spaces (RKHSs) by convex optimization. We consider the case of kernels with the Mercer expansion and propose an algorithm by deriving a second-order cone programming (SOCP) problem that yields n points at one sitting for a given integer n. In addition, by modifying the SOCP problem slightly, we propose another sequential algorithm that adds an arbitrary number of new points in each step. Numerical experiments show that in several cases the proposed algorithms compete with the P-greedy algorithm, which is known to provide nearly optimal points. [ABSTRACT FROM AUTHOR]

Interpolatory splines are usually useful to reconstruct data that present certain regularity. This paper is devoted to the construction and analysis of a new technique that allows to improve the accuracy of splines near corner singularities in the point values and jump discontinuities in the cell averages. The detection of discontinuities is easily done through the processing of the right hand side of the system of equations of the spline, that contains divided differences. The process of adaption will require some knowledge about the position of the discontinuity and the values of the function and its derivatives at the discontinuity. Using Harten's multiresolution we can adapt splines to the presence of corner singularities and jump discontinuities. Thanks to the adaption, the smearing of corner singularities does not appear in the reconstruction obtained in the point values and Gibbs phenomenon and diffusion of jump discontinuities is also eliminated. These numerical effects are a consequence of assuming that the data used in the reconstruction comes from the discretization of a continuous function with certain regularity and this might not be true. [ABSTRACT FROM AUTHOR]

In the past few years, Bogoya, Böttcher, Grudsky, and Maximenko obtained the precise asymptotic expansion for the eigenvalues of a Toeplitz matrix Tn(f), under suitable assumptions on the generating function f, as the matrix size n goes to infinity. On the basis of several numerical experiments, it was conjectured by Serra-Capizzano that a completely analogous expansion also holds for the eigenvalues of the preconditioned Toeplitz matrix Tn(u)− 1Tn(v), provided f = v/u is monotone and further conditions on u and v are satisfied. Based on this expansion, we here propose and analyze an interpolation-extrapolation algorithm for computing the eigenvalues of Tn(u)− 1Tn(v). The algorithm is suited for parallel implementation and it may be called "matrix-less" as it does not need to store the entries of the matrix. We illustrate the performance of the algorithm through numerical experiments and we also present its generalization to the case where f = v/u is non-monotone. [ABSTRACT FROM AUTHOR]

Bogoya, Böttcher, Grudsky, and Maximenko have recently obtained the precise asymptotic expansion for the eigenvalues of a sequence of Toeplitz matrices {Tn(f)}, under suitable assumptions on the associated generating function f. In this paper, we provide numerical evidence that some of these assumptions can be relaxed and extended to the case of a sequence of preconditioned Toeplitz matrices {Tn−1(g)Tn(f)}, for f trigonometric polynomial, g nonnegative, not identically zero trigonometric polynomial, r = f/g, and where the ratio r plays the same role as f in the nonpreconditioned case. Moreover, based on the eigenvalue asymptotics, we devise an extrapolation algorithm for computing the eigenvalues of preconditioned banded symmetric Toeplitz matrices with a high level of accuracy, with a relatively low computational cost, and with potential application to the computation of the spectrum of differential operators. [ABSTRACT FROM AUTHOR]

Some families of Haar spaces in ℝd,d≥1, whose basis functions are d-variate piecewise polynomials, are highlighted. The starting point is a sequence of univariate piecewise polynomials, called Lobachevsky splines, arised in probability theory and asymptotically related to the normal density function. Then, it is shown that d-variate Lobachevsky splines can be expressed as products of Lobachevsky splines. All these splines have simple analytic expressions and subsets of them are suitable for scattered data interpolation, allowing efficient computation and plain error analysis. [ABSTRACT FROM AUTHOR]

We consider the wide class of all piecewise Chebyshevian splines with connection matrices at the knots. We prove that a spline space of this class is “good for interpolation” if and only if the spline space obtained by integration is “good for design”. As a consequence, this provides us with a simple practical description of all such spline spaces which can be used for solving Hermite interpolation problems. These results strongly rely on the properties of blossoms. [ABSTRACT FROM AUTHOR]

Both barycentric Lagrange interpolation and barycentric rational interpolation are thought to be stable and effective methods for approximating a given function on some special point sets. A direct evaluation of these interpolants due to N interpolation points at M sampling points requires $\mathcal {O}(NM)$ arithmetic operations. In this paper, we introduce two fast multipole methods to reduce the complexity to $\mathcal {O}(\max \left \{N,M\right \})$ . The convergence analysis is also presented in this paper. [ABSTRACT FROM AUTHOR]

The Piecewise Polynomial Harmonic Method (PPHM) is a local third order accurate shock capturing method for hyperbolic conservation laws. In this paper, theoretical stability properties are presented. Using these properties, the convergence of the scheme for scalar conservation laws is obtained. A direct adaptation of these results should be used to derive the convergence of the PHM (Marquina, SIAM J. Sci. Comput. 15(4), 892-915, 1994). Finally, the method is tested in several classical problems in order to explore its numerical behavior. [ABSTRACT FROM AUTHOR]

The Lagrange interpolation problem on spaces of symmetric bivariate polynomials is considered to reduce the interpolation problem to problems of approximately half dimension. The Berzolari-Radon construction is adapted to these kinds of problems by considering nodes placed on symmetric lines or symmetric pairs of lines. A Newton formula for the symmetric interpolant using the Berzolari-Radon construction is proposed. [ABSTRACT FROM AUTHOR]

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. [ABSTRACT FROM AUTHOR]

This paper is devoted to the compression of colour images using a new nonlinear cell-average multiresolution scheme. The aim is to obtain similar compression properties as linear multiresolution schemes but eliminating the classical Gibbs phenomenon of this type of reconstructions near the edges. The algorithm is based on a nonlinear reconstruction operator (using a nonlinear trigonometric mean). The new reconstruction is third-order accurate in smooth regions and adapted to the presence of discontinuities. The data used are always centred with optimal support. Some theoretical properties of this scheme are analysed (order of approximation, convergence, elimination of Gibbs effect and stability). [ABSTRACT FROM AUTHOR]

We provide an algorithm for constructing a generalized Hermite interpolant. [ABSTRACT FROM AUTHOR]

Kernel-based approximation methods-often in the form of radial basis functions-have been used for many years now and usually involve setting up a kernel matrix which may be ill-conditioned when the shape parameter of the kernel takes on extreme values, i.e., makes the kernel 'flat'. In this paper we present an algorithm we refer to as the Hilbert-Schmidt SVD and use it to emphasize two important points which-while not entirely new-present a paradigm shift under way in the practical application of kernel-based approximation methods: (i) it is not necessary to form the kernel matrix (in fact, it might even be a bad idea to do so), and (ii) it is not necessary to know the kernel in closed form. While the Hilbert-Schmidt SVD and its two implications apply to general positive definite kernels, we introduce in this paper a class of so-called iterated Brownian bridge kernels which allow us to keep the discussion as simple and accessible as possible. [ABSTRACT FROM AUTHOR]

Bivariate polynomials defined on the unit disk are used to reconstruct a wavefront from a data sample. We analyze the interpolation problem arising in critical sampling, that is, using a minimal sample. The interpolant is expressed as a linear combination of Zernike polynomials, whose coefficients represent relevant optical features of the wavefront. We study the propagation of errors of the polynomial values and their coefficients, obtaining bounds for the Lebesgue constants and condition numbers. A node distribution leading to low Lebesgue constants and condition numbers for degrees up to 20 is proposed. [ABSTRACT FROM AUTHOR]

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. [ABSTRACT FROM AUTHOR]

Blossoming is a useful technique to study bases and curve representations in computer-aided geometric design. Recently Simeonov et al. (Comput Aided Geom Des 28:549-565, ) have used a blossom generalization, namely the h-blossom, to derive new results about the h-Bernstein basis and h-Bézier curves that have previously been studied in approximation theory and computer-aided geometric design. This paper introduces a basis related to the h-Bernstein basis. There is a close relationship between this new basis and the h-Bernstein basis, between the new basis and the h-blossom, and between the new basis and 'progressive' curves. This paper explores these relationships and uses them to derive properties both of the new basis itself, and of curves represented in terms of the new basis. [ABSTRACT FROM AUTHOR]

The reciprocal polynomial extrapolation was introduced in Amat et al. (J Comput Math 22(1):1-10, ), where its accuracy and stability were studied and a linear scalar test problem was analyzed numerically. In the present work, a new step in the implementation of the reciprocal polynomial extrapolation, ensuring at least the same behavior as the Richardson extrapolation, is proposed. Looking at the reciprocal extrapolation as a Richardson extrapolation where the original data is nonlinearly modified, the improvements that we will obtain should be justified. Several theoretical analysis of the new extrapolation, including local error estimates and stability properties, are presented. A comparison between the two extrapolation techniques is performed for solving some boundary problems with perturbation controlled by a small parameter ϵ. Using two specific boundary problems, the error and the robustness of the new technique using centered divided differences in a uniform mesh are investigated numerically. They turn out to be better than those presented by the Richardson extrapolation. Finally, investigations on the accuracy when using a special non-uniform discretization mesh are presented. A numerical comparison with the Richardson extrapolation for this particular case, where we present some improvements, is also performed. [ABSTRACT FROM AUTHOR]

We prove that any fat, subanalytic compact subset of $\mathbb R^N$ possesses a nearly optimal (polynomial) admissible mesh. It is related to particular results that have recently appeared in the literature for very special (globally semianalytic) sets like N-dimensional polynomial or analytic graph domains or polynomial and analytic polyhedrons. (Here a good source of references is the recent paper (Piazzon and Vianello, East J Approx 16(4):389-398, ).) We also show that an infinitely differentiable map f from a compact set Q in $\mathbb R^N$ onto a Markov compact set K in $\mathbb C^l$ ( l ≤ N) transforms a (weakly) admissible mesh in Q onto a (weakly) admissible mesh in K, which extends a result of Piazzon and Vianello (East J Approx 16(4):389-398, ) for analytic maps in case Q is a subset of $\mathbb R^N$. Versions for $\mathcal C^k$ maps with sufficiently large k are also given. [ABSTRACT FROM AUTHOR]

Starting from the function values on the roots of Jacobi polynomials, we construct a class of discrete de la Vallée Poussin means, by approximating the Fourier coefficients with a Gauss-Jacobi quadrature rule. Unlike the Lagrange interpolation polynomials, the resulting algebraic polynomials are uniformly convergent in suitable spaces of continuous functions, the order of convergence being comparable with the best polynomial approximation. Moreover, in the four Chebyshev cases the discrete de la Vallée Poussin means share the Lagrange interpolation property, which allows us to reduce the computational cost. [ABSTRACT FROM AUTHOR]

In order to interpolate 2 n + 1 points on the unit hypersphere $ \mathcal{S}^{d-1}$ with a vector-valued rational function, we use the Generalised Inverse Rational Interpolants (GIRI) of Graves-Morris. The construction process of these Thiele type rational interpolants is based on the Samelson's inverse for vectors. We show that in general any GIRI of 2 n + 1 points of $ \mathcal{S}^{d-1}$ lies on $ \mathcal{S}^{d-1}$. We also show that the stereographic projection induces a one-to-one correspondence between the set of vector-valued rational functions lying on $ \mathcal{S}^{d-1}$ and the set of Generalised Inverse Rational Fractions in the equator plane. [ABSTRACT FROM AUTHOR]

We prove that a general class of splines with sections in different Extended Chebyshev spaces or in different quasi Extended Chebyshev spaces can be viewed as quasi Chebyshevian splines, that is, as splines with all sections in a single convenient quasi Extended Chebyshev space. As a result, we can affirm the presence of blossoms in the corresponding spline spaces, with all the important consequences inherent in blossoms, namely, the possibility of developing all design algorithms for splines, the existence of B-splines bases, along with their optimality. [ABSTRACT FROM AUTHOR]

We introduce a unifying formulation of a number of related problems which can all be solved using a contour integral formula. Each of these problems requires finding a non-trivial linear combination of possibly some of the values of a function f, and possibly some of its derivatives, at a number of data points. This linear combination is required to have zero value when f is a polynomial of up to a specific degree p. Examples of this type of problem include Lagrange, Hermite and Hermite-Birkhoff interpolation; fixed-denominator rational interpolation; and various numerical quadrature and differentiation formulae. Other applications include the estimation of missing data and root-finding. [ABSTRACT FROM AUTHOR]

Sinc interpolation is a very efficient infinitely differentiable approximation scheme from equidistant data on the infinite line. We give a formula for the error committed when the function neither decreases rapidly nor is periodic, so that the sinc series must be truncated for practical purposes. To do so, we first complete a previous result for an odd number of points, before deriving a formula for the more involved case of an even number of points. [ABSTRACT FROM AUTHOR]

class of high-order data-bounded polynomials on general meshes are derived and analyzed in the context of numerical solutions of hyperbolic equations. Such polynomials make it possible to circumvent the problem of Runge-type oscillations by adaptively varying the stencil and order used, but at the cost of only enforcing C solution continuity at data points. It is shown that the use of these polynomials, based on extending the work of Berzins (SIAM Rev 1(4):624–627, ) to nonuniform meshes, provides a way to develop positivity preserving polynomial approximations of potentially high order for hyperbolic equations. The central idea is to use ENO (Essentially Non Oscillatory) type approximations but to enforce additional restrictions on how the polynomial order is increased. The question of how high a polynomial order should be used will be considered, with respect to typical numerical examples. The results show that this approach is successful but that it is necessary to provide sufficient resolution inside a front if high-order methods of this type are to be used, thus emphasizing the need to consider nonuniform meshes. [ABSTRACT FROM AUTHOR]

In this paper, three-term recurrence relations for branched continued fractions are determined. Based on the algorithm of partial inverse differences in tensor-product-like manner, the finite branched continued fractions can be applied to rational interpolation over pyramid-typed grids in R3. By means of the three-term recurrence relations, a characterization theorem is valid. Then an error estimation is worked out. Based on the relationship between the partial inverse differences and partial reciprocal ones, and the partial reciprocal derivatives as well, the BCFs osculatory interpolation with its algorithm is stated which shows it feasibility of partial derivable functions in BCFs expansion at one point. [ABSTRACT FROM AUTHOR]

When a Nyström-Broyden method is used to solve numerically Urysohn integral equations, the main problem is to evaluate the action of the integral operator at a low cost. Here we suitably approximate the relevant discrete integral operator of dimensionn by itsm-degree truncated Chebyshev series expansion (withm≪n), reducing the complexity from the basic to . A technique to evaluate cheaply suchm is presented. Several linear and nonlinear examples are considered. [ABSTRACT FROM AUTHOR]

It is well known that Newton's interpolation polynomial is based on divided differences which produce useful intermediate results and allow one to compute the polynomial recursively. Thiele's interpolating continued fraction is aimed at building a rational function which interpolates the given support points. It is interesting to notice that Newton's interpolation polynomials and Thiele's interpolating continued fractions can be incorporated in tensor product like manner to yield four kinds of bivariate interpolation schemes. Among them are classical bivariate Newton's interpolation polynomials which are purely linear interpolants, branched continued fractions which are purely nonlinear interpolants and have been studied by Chaffy, Cuyt and Verdonk, Kuchminska, Siemaszko and many other authors, and Thiele-Newton's bivariate interpolating continued fractions which are investigated in another paper by one of the authors. In this paper, emphasis is put on the study of Newton-Thiele's bivariate rational interpolants. By introducing so called blending differences which look partially like divided differences and partially like inverse differences, we give a recursive algorithm accompanied with a numerical example. Moreover, we bring out the error estimation and discuss the limiting case. [ABSTRACT FROM AUTHOR]

Let n be a positive integer, let $$a_{ - n{\text{ + }}1} ,...,a_{ - 1} ,a_0 ,a_1 ,...,a_{n - 1} $$ be complex numbers and let $$T: = {\text{ [}}a_{k - 1} {\text{]}}_{k,l = 0}^{n - 1} $$ be a nonsingular n × n complex Toeplitz matrix. We present a superfast algorithm for computing the determinant of T. Superfast means that the arithmetic complexity of our algorithm is $${\text{O(}}N\log ^2 N{\text{)}}$$ , where N denotes the smallest power of 2 that is larger than or equal to n. We show that det T can be computed from the determinant of a certain coupled Vandermonde matrix. The latter matrix is related to a linearized rational interpolation problem at roots of unity and we show how its determinant can be calculated by multiplying the pivots that appear in the superfast interpolation algorithm that we presented in a previous publication. [ABSTRACT FROM AUTHOR]

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. [ABSTRACT FROM AUTHOR]

After recalling some pitfalls of polynomial interpolation (in particular, slopes limited by Markov's inequality) and rational interpolation (e.g., unattainable points, poles in the interpolation interval, erratic behavior of the error for small numbers of nodes), we suggest an alternative for the case when the function to be interpolated is known everywhere, not just at the nodes. The method consists in replacing the interpolating polynomial with a rational interpolant whose poles are all prescribed, written in its barycentric form as in [4], and optimizing the placement of the poles in such a way as to minimize a chosen norm of the error. [ABSTRACT FROM AUTHOR]