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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]

This article explores various techniques for generating fractal-like Bézier curves in both 2D and 3D environments. It delves into methods such as subdivision schemes, Iterated Function System (IFS) theory, perturbation of Bézier curves, and perturbation of Bézier basis functions. The article outlines conditions on subdivision matrices necessary for convergence and demonstrates their use in creating an IFS with an attractor aligned to the convergent point of the subdivision scheme based on specified initial data. Additionally, it discusses conditions for obtaining a one-sided approximation of a given Bézier curve through perturbation. The article also addresses considerations for perturbed Bézier basis functions to construct fractal-like Bézier curves that remain within the convex hull polygon/polyhedron defined by control points. These methods find applications in various fields, including computer graphics, art, and design. [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]

In this paper, we consider the structure-preserving model order reduction problem for multi-input/multi-output bilinear control systems by tangential interpolation. We propose a new type of tangential interpolation problem for structured bilinear systems, for which we develop a new structure-preserving interpolation framework. This new framework extends and generalizes different formulations of tangential interpolation for bilinear systems from the literature and also provides a unifying framework. We then derive explicit conditions on the projection spaces to enforce tangential interpolation in different settings, including conditions for tangential Hermite interpolation. The analysis is illustrated by means of three numerical examples. [ABSTRACT FROM AUTHOR]

A novel approach to zipper fractal interpolation theory for functions of several variables is presented. Multivariate zipper fractal functions are constructed and then perturbed through free choices of base functions, scaling functions, and a binary matrix called signature to obtain their zipper α-fractal versions. In particular, we propose a multivariate Bernstein zipper fractal function and study its coordinate-wise monotonicity which depends on the values of signature. We derive bounds for the graph of a multivariate zipper fractal function by imposing conditions on the scaling factors and the Hölder exponent of the associated germ function and base function. The box dimension result for multivariate Bernstein zipper fractal function is derived. Finally, we study some constrained approximation properties for multivariate zipper Bernstein fractal functions. [ABSTRACT FROM AUTHOR]

We introduce interpolation operators with approximation and stability properties suited for parabolic problems in primal and mixed formulations. We derive localized error estimates for tensor product meshes (occurring in classical time-marching schemes) as well as locally in space-time refined meshes. [ABSTRACT FROM AUTHOR]

Hermite interpolation is classically used to reconstruct smooth data when the function and its first order derivatives are available at certain nodes. If first order derivatives are not available, it is easy to set a system of equations imposing some regularity conditions at the data nodes in order to obtain them. This process leads to the construction of a Hermite spline. The problem of the described Hermite splines is that the accuracy is lost if the data contains singularities. The consequence is the appearance of oscillations, if there is a jump discontinuity in the function, that globally affects the accuracy of the spline, or the smearing of singularities, if the discontinuities are in the derivatives of the function. This paper is devoted to the construction and analysis of a new technique that allows for the computation of accurate first order derivatives of a function close to singularities using a Hermite spline. The idea is to correct the system of equations of the spline in order to attain the desired accuracy even close to the singularities. Once we have computed the first order derivatives with enough accuracy, a correction term is added to the Hermite spline in the intervals that contain a singularity. The aim is to reconstruct piecewise smooth functions with O (h 4) accuracy even close to the singularities. The process of adaption will require some knowledge about the position of the singularity and the jumps of the function and some of its derivatives at the singularity. The whole process can be used as a post-processing, where a correction term is added to the classical cubic Hermite spline. Proofs for the accuracy and regularity of the corrected spline and its derivatives are given. We also analyse the mechanism that eliminates the Gibbs phenomenon close to jump discontinuities in the function. The numerical experiments presented confirm the theoretical results obtained. [ABSTRACT FROM AUTHOR]

Numerical analysts and scientists working in applications often observe that once they improve their techniques to get a better accuracy, some instability of the evaluation creeps in through the back door. This paper shows for a large class of numerical methods that such a Trade-off Principle between error and evaluation stability is unavoidable. It is an instance of a no free lunch theorem. Here, evaluation is the mathematical map that takes input data to output data. This is independent from the numerical routine that calculates the output. Therefore, evaluation stability is different from computational stability. The setting is confined to recovery of functions from data, but it includes solving differential equations by writing such methods as a recovery of functions under constraints imposed by differential operators and boundary values. The trade-off principle bounds the product of two terms from below. The first is related to errors, and the second turns out to be related to evaluation instability. Under certain conditions satisfied for splines and kernel-based interpolation, both can be minimized. Then the lower bound is attained, and the error term is the inverse of the instability term. As a byproduct, it is shown that Kansa’s Unsymmetric Collocation Method sacrifices accuracy for improved evaluation stability, when compared to symmetric collocation. [ABSTRACT FROM AUTHOR]

When one solves differential equations by a spectral method, it is often convenient to shift from Chebyshev polynomials Tn(x) with coefficients an to modified basis functions that incorporate the boundary conditions. For homogeneous Dirichlet boundary conditions, u(±1) = 0, popular choices include the "Chebyshev difference basis" ςn(x) ≡ Tn+2(x)−Tn(x) with coefficients here denoted by bn and the "quadratic factor basis" ϱn(x) ≡ (1 − x2)Tn(x) with coefficients cn. If u(x) is weakly singular at the boundary, then the coefficients an decrease proportionally to O (A (n) / n κ) for some positive constant κ, where A(n) is a logarithm or a constant. We prove that the Chebyshev difference coefficients bn decrease more slowly by a factor of 1/n while the quadratic factor coefficients cn decrease more slowly still as O (A (n) / n κ − 2 ) . The error for the unconstrained Chebyshev series, truncated at degree n = N, is O (| A (N) | / N κ) in the interior, but is worse by one power of N in narrow boundary layers near each of the endpoints. Despite having nearly identical error norms in interpolation, the error in the Chebyshev basis is concentrated in boundary layers near both endpoints, whereas the error in the quadratic factor and difference basis sets is nearly uniformly oscillating over the entire interval in x. Meanwhile, for Chebyshev polynomials, the values of their derivatives at the endpoints are O (N 2) , but only O (N) for the difference basis. Furthermore, we give the asymptotic coefficients and rigorous error estimates of the approximations in these three bases, solved by the least squares method. We also find an interesting fact that on the face of it, the aliasing error is regarded as a bad thing; actually, the error norm associated with the downward curving spectral coefficients decreases even faster than the error norm of infinite truncation. But the premise is under the same basis, and when involving different bases, it may not be established yet. [ABSTRACT FROM AUTHOR]

We present a precise anisotropic interpolation error estimate for the Morley finite element method (FEM) and apply it to fourth-order elliptic equations. We do not impose the shape-regularity mesh condition in the analysis. Anisotropic meshes can be used for this purpose. The main contributions of this study include providing a new proof of the term consistency. This enables us to obtain an anisotropic consistency error estimate. The core idea of the proof involves using the relationship between the Raviart-Thomas and Morley finite-element spaces. Our results indicate optimal convergence rates and imply that the modified Morley FEM may be effective for errors. [ABSTRACT FROM AUTHOR]

In neuroscience, the distribution of a decision time is modelled by means of a one-dimensional Fokker–Planck equation with time-dependent boundaries and space-time-dependent drift. Efficient approximation of the solution to this equation is required, e.g., for model evaluation and parameter fitting. However, the prescribed boundary conditions lead to a strong singularity and thus to slow convergence of numerical approximations. In this article we demonstrate that the solution can be related to the solution of a parabolic PDE on a rectangular space-time domain with homogeneous initial and boundary conditions by transformation and subtraction of a known function. We verify that the solution of the new PDE is indeed more regular than the solution of the original PDE and proceed to discretize the new PDE using a space-time minimal residual method. We also demonstrate that the solution depends analytically on the parameters determining the boundaries as well as the drift. This justifies the use of a sparse tensor product interpolation method to approximate the PDE solution for various parameter ranges. The predicted convergence rates of the minimal residual method and that of the interpolation method are supported by numerical simulations. [ABSTRACT FROM AUTHOR]

The barycentric forms of polynomial and rational interpolation have recently gained popularity, because they can be computed with simple, efficient, and numerically stable algorithms. In this paper, we show more generally that the evaluation of any function that can be expressed as r (x) = ∑ i = 0 n a i (x) f i / ∑ j = 0 m b j (x) in terms of data values f i and some functions a i and b j for i = 0 , ... , n and j = 0 , ⋯ , m with a simple algorithm that first sums up the terms in the numerator and the denominator, followed by a final division, is forward and backward stable under certain assumptions. This result includes the two barycentric forms of rational interpolation as special cases. Our analysis further reveals that the stability of the second barycentric form depends on the Lebesgue constant associated with the interpolation nodes, which typically grows with n, whereas the stability of the first barycentric form depends on a similar, but different quantity, that can be bounded in terms of the mesh ratio, regardless of n. We support our theoretical results with numerical experiments. [ABSTRACT FROM AUTHOR]

We propose to extend results on the interpolation theory for scalar functions to the case of differential k-forms. More precisely, we consider the interpolation of fields in P r - Λ k (T) , the finite element spaces of trimmed polynomial k-forms of arbitrary degree r ≥ 1 , from their weights, namely their integrals on k-chains. These integrals have a clear physical interpretation, such as circulations along curves, fluxes across surfaces, densities in volumes, depending on the value of k. In this work, for k = 1 , we rely on the flexibility of the weights with respect to their geometrical support, to study different sets of 1-chains in T for a high order interpolation of differential 1-forms, constructed starting from "good" sets of nodes for a high order multi-variate polynomial representation of scalar fields, namely 0-forms. We analyse the growth of the generalized Lebesgue constant with the degree r and preliminary numerical results for edge elements support the nonuniform choice, in agreement with the well-known nodal case. [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]

We present a number of new piecewise-polynomial kernels for image interpolation. The kernels are constructed by optimizing a measure of interpolation quality based on the magnitude of anisotropic artifacts. The kernel design process is performed symbolically using the Mathematica computer algebra system. An experimental evaluation involving 14 image quality assessment methods demonstrates that our results compare favorably with the existing linear interpolators. [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]

The algebraic characterization of dual univariate interpolating subdivision schemes is investigated. Specifically, we provide a constructive approach for finding dual univariate interpolating subdivision schemes based on the solutions of certain associated polynomial equations. The proposed approach also makes it possible to identify conditions for the existence of the sought schemes. [ABSTRACT FROM AUTHOR]

In this paper, we construct a piecewise interpolation method of approximate solution of the transport equation based on the Newton interpolation polynomial of two variables. We transform the polynomial to the algebraic form with numerical coefficients; this leads us to a sequence of iterations, which improves the accuracy of the approximation. The method is implemented in software and numerical experiments are performed. The possibility of generalizations to systems of partial differential equations and integro-differential equations is discussed. [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 this paper, we consider general cases of linear Volterra integral equations under minimal assumptions on their weakly singular kernels and introduce a new product integration method in which we involve the linear interpolation to get a better approximate solution, figure out its effect and also we provide a convergence proof. Furthermore, we apply our method to a numerical example and conclude this paper by adding a conclusion [ABSTRACT FROM AUTHOR]

Fractal approximation is a well studied concept, but the convergence of all the existing fractal approximants towards the original function follows usually if the magnitude of the corresponding scaling factors approaches zero. In this article, for a given function f ∈ C (I) , by exploiting fractal approximation theory and considering the classical q-Bernstein polynomials as base functions, we construct a sequence { f n (q , α) (x) } n = 1 ∞ of (q , α) -fractal functions that converges uniformly to f even if the norm/magnitude of the scaling functions/scaling factors does not tend to zero. The convergence of the sequence { f n (q , α) (x) } n = 1 ∞ of (q , α) -fractal functions towards f follows from the convergence of the sequence of q-Bernstein polynomials of f towards f. If we consider a sequence { f m (x) } m = 1 ∞ of positive functions on a compact real interval that converges uniformly to a function f, we develop a double sequence { { f m , n (q , α) (x) } n = 1 ∞ } m = 1 ∞ of (q , α) -fractal functions that converges uniformly to f. [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]

The existence of fundamental cardinal exponential B-splines of positive real order σ is established subject to two conditions on σ and their construction is implemented. A sampling result for these fundamental cardinal exponential B-splines is also presented. [ABSTRACT FROM AUTHOR]

In this paper, we concern with the data interpolation by using extended cubic uniform B-splines with shape parameters. Two iterative formats, namely the progressive iterative approximation (PIA) and the weighted progressive iterative approximation (WPIA), are proposed to interpolate given data points. We study the optimal shape parameter and the optimal weight for the proposed methods by solving the eigenvalues of the collocation matrix. The optimal shape parameter can make the iterative methods not only have the fastest convergence speed but also have smallest initial interpolation error. Numerical experiments are given to illustrate the effectiveness of the proposed methods. [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]

In this paper a special class of one-dimensional L-splines of order 4 is studied, which naturally appear in the computation of interpolation and smoothing with multivariate polysplines. Fast algorithms are provided for interpolation and smoothing with this class of L-splines, as well as a generalization of the Reinsch algorithm to this setting. The explicit description of all mathematical expressions permits a simple and direct numerical implementation. Applications are provided to financial data of the index S&P500, for the fast calculation of statistically interesting quantities, as cross validation (scores), generalized cross validation (scores) for finding the best smoothing parameter α , and the residual sum of squares. [ABSTRACT FROM AUTHOR]

Polynomial interpolation or regression models are an important tool in Derivative-free Optimization, acting as surrogates of the real function. In this work, we propose the use of these models in the multiobjective framework of directional direct search, namely the one of Direct Multisearch. Previously evaluated points are used to build quadratic polynomial models, which are minimized in an attempt of generating nondominated points of the true function, defining a search step for the algorithm. Numerical results state the competitiveness of the proposed approach. [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, a kind of univariate multiquadric quasi-interpolants with the derivatives of approximated function is proposed by combining a univariate multiquadric quasi-interpolant with Lidstone interpolation polynomials proposed in Lidstone (Proc Edinb Math Soc 2:16–19, 1929), Costabile and Dell' Accio (App Numer Math 52:339–361, 2005) and Catinas (J Appl Funct Anal 4:425–439, 2006). For practical purposes, another kind of approximation operators without any derivative of the approximated function is given using divided differences to approximate the derivatives. Some error bounds and the convergence rates of new operators are derived, which demonstrates that our operators could provide the desired precision by choosing a suitable shape-preserving parameter c and a non-negative integer n. Finally, we make extensive comparison with the other existing methods and give some numerical examples. Moreover, the associated algorithm is easily implemented. [ABSTRACT FROM AUTHOR]

As an extension of Lagrange interpolation, we introduce a class of interpolation formulas and study their existence and uniqueness. In the sequel, we consider some particular cases and construct the corresponding weighted quadrature rules. Numerical examples are finally given and compared. [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]

We discuss the problem of fitting a smooth regular curve γ : [ 0 , T ] → E n based on reduced data Q m = { q i } i = 0 m in arbitrary Euclidean space E n . The respective interpolation knots T = { t i } i = 0 m satisfying q i = γ (t i) are assumed to be unknown. In our setting the substitutes T λ = { t ^ i } i = 0 m of T are selected according to the so-called exponential parameterization governed by Q m and λ ∈ [ 0 , 1 ] . A modified Hermite interpolant γ ^ H introduced in Kozera and Noakes (Fundam Inf 61(3–4):285–301, 2004) is used here to fit (T ^ λ , Q m) . The case of λ = 1 (i.e. for cumulative chords) for general class of admissible samplings yields a sharp quartic convergence order in estimating γ ∈ C 4 by γ ^ H [see Kozera (Stud Inf 25(4B–61):1–140, 2004) and Kozera and Noakes (Fundam Inf 61(3–4):285–301, 2004)]. As recently shown in Kozera and Wilkołazka (Math Comput Sci, 2018. 10.1007/s11786-018-0362-4) the remaining λ ∈ [ 0 , 1) render a linear convergence order in γ ^ H ≈ γ for any Q m sampled more-or-less uniformly. The related analysis relies on comparing the difference γ - γ ^ H ∘ ϕ H in which ϕ H forms a special mapping between [0, T] and [ 0 , T ^ ] with T ^ = t ^ m . In this paper: (a) several sufficient conditions enforcing ϕ H to yield a genuine reparameterization are first formulated and then analytically and symbolically simplified. The latter covers also the asymptotic case expressed in a simple form. Ultimately, the reformulated conditions can be algebraically verified and/or geometrically visualized, (b) additionally in Sect. 3, the sharpness of the asymptotics of γ - γ ^ H ∘ ϕ H [from Kozera and Wilkołazka (Math Comput Sci, 2018. 10.1007/s11786-018-0362-4)] is proved upon applying symbolic and analytic calculations in Mathematica. [ABSTRACT FROM AUTHOR]

In this work, we use a kind of C 1 rational interpolation splines in one and two dimensions to generate curves and surfaces with region control. Simple data-dependent sufficient constraints are derived on the local control parameters to generate C 1 interpolation curves lying strictly between two given piecewise linear curves and C 1 interpolation surfaces lying strictly between two given piecewise bi-cubic blending linear interpolation surfaces. [ABSTRACT FROM AUTHOR]

A geometric construction of a Bézier curve is presented by a unifiable way from the mentioned literature with some modification. A closed-form solution to the inverse problem in cubic Bézier-spline interpolation will be obtained. Calculations in the given examples are performed by a Maple procedure using this solution. [ABSTRACT FROM AUTHOR]

We give compact formulae of the Lagrange interpolation polynomials and cubature formulae based on the common zeros of product Chebyshev polynomials of the second kind. Further, for 0 < p ≤ 2 , we study the mean convergence of the Lagrange interpolation polynomials. [ABSTRACT FROM AUTHOR]

It is known that discrete sets of uniformly distributed points on the hypersphere S d ⊂ ℝ d + 1 can be obtained from minimizing the energy functional corresponding to Riesz s-kernels k s (x , y) = ∥ x − y ∥ − s (s > 0) or the logarithmic kernel k log (x , y) = − log ∥ x − y ∥ + log 2 . We prove the same for the kernel k log (x , y) = ∥ x − y ∥ (log ∥ x − y ∥ 2 − 1) + 2 which is a front-extension of the sequence of derivatives k log , k 1 , k 2 , k 3 , ... , up to sign and constants. The boundedness of the kernel simplifies the classical potential-theoretical proof of the asymptotic uniformity of the point distributions. Still, the property of a singular derivative for x → y is preserved, with the physical interpretation of infinite repulsive forces for touching particles. The quality of the resulting point distributions is exemplary compared with that of Riesz- and classical logarithmic point sets, and found to be competitive. Originally motivated by problems of high-dimensional data, the applicability of log -optimal point sets with a novel concentric interpolation and differentiation scheme is demonstrated. The method is significantly optimized by the introduction of symmetrized kernels for both the generation of the minimum energy points and the spherical basis functions. Both the point generation and the Concentric Interpolation software are available as Open Source software and selected point sets are provided. [ABSTRACT FROM AUTHOR]

The stability properties of the Newton interpolation formula depend on the order of the nodes and can be measured through a condition number. Increasing and Leja orderings have been previously considered (Carnicer et al. in J Approx Theory, 2017. https://doi.org/10.1016/j.jat.2017.07.005; Reichel in BIT 30:332–346, 1990). We analyze central orderings for equidistant nodes on a bounded real interval. A bound for conditioning is given. We demonstrate in particular that this ordering provides a more stable Newton formula than the natural increasing order. We also analyze of a central ordering with respect to the evaluation point, which provides low bounds for the conditioning. Numerical examples are included. [ABSTRACT FROM AUTHOR]

This work discusses the problem of fitting a regular curve γ based on reduced data points Q m = (q 0 , q 1 , ⋯ , q m) in arbitrary Euclidean space. The corresponding interpolation knots T = (t 0 , t 1 , ⋯ , t m) are assumed to be unknown. In this paper the missing knots are estimated by T λ = (t ^ 0 , t ^ 1 , ⋯ , t ^ m) in accordance with the so-called exponential parameterization (see Kvasov in Methods of shape-preserving spline approximation, World Scientific Publishing Company, Singapore, 2000) controlled by a single parameter λ ∈ [ 0 , 1 ] . In order to fit (T ^ λ , Q m) , a modified Hermite interpolant γ ^ H (a C 1 piecewise-cubic) introduced in Kozera and Noakes (Fundam Inf 61(3–4):285–301, 2004) is used. The sharp quartic convergence order for estimating γ ∈ C 4 by γ ^ H is proved in Kozera (Stud Inf 25(4B-61):1–140, 2004) and Kozera and Noakes (2004) only for λ = 1 and within the general class of admissible samplings. The main result of this paper extends the latter to the remaining cases of exponential parameterization covering all λ ∈ [ 0 , 1) . A slower linear convergence order in trajectory estimation is established for any λ ∈ [ 0 , 1) and arbitrary more-or-less uniform sampling. The numerical tests conducted in Mathematica indicate the sharpness of the latter and confirm the necessity of more-or-less uniformity. Other interpolation schemes used to fit reduced data Q m and based on T λ together with some relevant applications are also briefly recalled in this paper. [ABSTRACT FROM AUTHOR]

This paper examines the problem of extrapolation of an analytic function for x > 1 given N + 1 perturbed samples from an equally spaced grid on [ - 1 , 1 ] . For a function f on [ - 1 , 1 ] that is analytic in a Bernstein ellipse with parameter ρ > 1 , and for a uniform perturbation level ε on the function samples, we construct an asymptotically best extrapolant e(x) as a least squares polynomial approximant of degree M ∗ determined explicitly. We show that the extrapolant e(x) converges to f(x) pointwise in the interval I ρ ∈ [ 1 , (ρ + ρ - 1) / 2) as ε → 0 , at a rate given by a x-dependent fractional power of ε . More precisely, for each x ∈ I ρ we have | f (x) - e (x) | = O ε - log r (x) / log ρ , r (x) = x + x 2 - 1 ρ , up to log factors, provided that an oversampling conditioning is satisfied, viz. M ∗ ≤ 1 2 N , which is known to be needed from approximation theory. In short, extrapolation enjoys a weak form of stability, up to a fraction of the characteristic smoothness length. The number of function samples does not bear on the size of the extrapolation error provided that it obeys the oversampling condition. We also show that one cannot construct an asymptotically more accurate extrapolant from equally spaced samples than e(x), using any other linear or nonlinear procedure. The proofs involve original statements on the stability of polynomial approximation in the Chebyshev basis from equally spaced samples and these are expected to be of independent interest. [ABSTRACT FROM AUTHOR]

In this paper, a novel hybrid trust-region algorithm using radial basis function (RBF) interpolations is proposed. The new algorithm is an improved version of ORBIT algorithm based on two novel ideas. Because the accuracy and stability of RBF interpolation depends on a shape parameter, so it is more appropriate to select this parameter according to the optimization problem. In the new algorithm, the appropriate shape parameter value is determined according to the optimization problem based on an effective statistical approach, while the ORBIT algorithm in all problems uses a fixed shape parameter value. In addition, the new algorithm is equipped with a new intelligent nonmonotone strategy which improves the speed of convergence, while the monotonicity of the sequence of objective function values in the ORBIT may decrease the rate of convergence, especially when an iteration is trapped near a narrow curved valley. The global convergence of the new hybrid algorithm is analyzed under some mild assumptions. The numerical results significantly indicate the superiority of the new algorithm compared with the original version. [ABSTRACT FROM AUTHOR]

The aim of this piece of work is to study some topics related to an Hermite interpolation problem on the unit circle. We consider as nodal points the zeros of the para-orthogonal polynomials with respect to a measure in the Baxter class and such that the sequence of the first derivative of the reciprocal of the orthogonal polynomials is uniformly bounded on the unit circle. We study the convergence of the Hermite–Fejér interpolants related to piecewise continuous functions and we describe the sets in which the interpolants uniformly converge to the piecewise continuous function as well as the oscillatory behavior of the interpolants near the discontinuities, where a Gibbs–Wilbraham phenomenon appears. Finally we present some numerical experiments applying the main results and by considering nodal systems of interest in the theory of orthogonal polynomials. [ABSTRACT FROM AUTHOR]

We describe how to perform the backward error analysis for the approximation of exp(A)v by p(s-1A)sv, for any given polynomial p(x). The result of this analysis is an optimal choice of the scaling parameter s which assures a bound on the backward error, i.e. the equivalence of the approximation with the exponential of a slightly perturbed matrix. Thanks to the SageMath package expbea we have developed, one can optimize the performance of the given polynomial approximation. On the other hand, we employ the package for the analysis of polynomials interpolating the exponential function at so called Leja-Hermite points. The resulting method for the action of the matrix exponential can be considered an extension of both Taylor series approximation and Leja point interpolation. We illustrate the behavior of the new approximation with several numerical examples. [ABSTRACT FROM AUTHOR]

In this paper we study an iterative approach to the Hermite interpolation problem, which first constructs an interpolant of the function values at n+1 nodes and then successively adds m correction terms to fit the data up to the mth derivatives. In the case of polynomial interpolation, this simply reproduces the classical Hermite interpolant, but the approach is general enough to be used in other settings. In particular, we focus on the family of rational Floater-Hormann interpolants, which are based on blending local polynomial interpolants of degree d with rational blending functions. For this family, the proposed method results in rational Hermite interpolants, which depend linearly on the data, with numerator and denominator of degree at most (m+1)(n+1)-1 and (m+1)(n-d), respectively. They converge at the rate of O(h(m+1)(d+1)) as the mesh size h converges to 0. After deriving the barycentric form of these interpolants, we prove the convergence rate for m=1 and m=2, and show that the approximation results compare favourably with other constructions. [ABSTRACT FROM AUTHOR]

From spline theory it is well-known that univariate cubic spline interpolation, if carried out in its natural Hilbert space W 2 2 [ a , b ] and on point sets with fill distance h , converges only like O ( h 2 ) in L 2 [ a , b ] if no additional assumptions are made. But superconvergence up to order h 4 occurs if more smoothness is assumed and if certain additional boundary conditions are satisfied. This phenomenon was generalized in 1999 to multivariate interpolation in Reproducing Kernel Hilbert Spaces on domains Ω ⊂ R d for continuous positive definite Fourier-transformable shift-invariant kernels on R d . But the sufficient condition for superconvergence given in 1999 still needs further analysis, because the interplay between smoothness and boundary conditions is not clear at all. Furthermore, if only additional smoothness is assumed, superconvergence is numerically observed in the interior of the domain, but a theoretical foundation still is a challenging open problem. This paper first generalizes the “improved error bounds” of 1999 by an abstract theory that includes the Aubin–Nitsche trick and the known superconvergence results for univariate polynomial splines. Then the paper analyzes what is behind the sufficient conditions for superconvergence. They split into conditions on smoothness and localization , and these are investigated independently. If sufficient smoothness is present, but no additional localization conditions are assumed, it is numerically observed that superconvergence always occurs in the interior of the domain, and some supporting arguments are provided. If smoothness and localization interact in the kernel-based case on R d , weak and strong boundary conditions in terms of pseudodifferential operators occur. A special section on Mercer expansions is added, because Mercer eigenfunctions always satisfy the sufficient conditions for superconvergence. Numerical examples illustrate the theoretical findings. [ABSTRACT FROM AUTHOR]

Given m + 1 strictly decreasing numbers h 0 , h 1 , … , h m , we give an algorithm to construct a corresponding finite sequence of orthogonal polynomials p 0 , p 1 , … , p m such that p 0 = 1 , p j has degree j and p m − j ( h n ) = ( − 1 ) n p j ( h n ) for all j , n = 0 , 1 , … , m . Using these polynomials, we construct bivariate Lagrange polynomials and cubature formulas for nodes that are points in R 2 where the coordinates are taken from given finite decreasing sequences of the same length and where the indices have the same (or opposite) parity. [ABSTRACT FROM AUTHOR]