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The paper is devoted to the problem of smooth approximation of data. We are concerned with the exact interpolation of the data at nodes and, at the same time, with the smoothness of the interpolating curve and its derivatives. The same procedure is applied to the fitting of data (smoothing of data), too. The approximating curve is defined as the solution of a variational problem with constraints (interpolation conditions at nodes, data fitting property of the curve) the existence of whose unique solution we prove. Except for the constraints, the formulation of the problem of data approximation is not unique in general as our requirements on the behavior of the approximating curve between the nodes can be very subjective. The smooth approximation of this kind can lead, e.g., to the cubic spline interpolation.We discuss the choice of basis systems for this way of approximation and present results of several 1D numerical examples that show some advantages and drawbacks of smooth approximation. [ABSTRACT FROM AUTHOR]

This paper presents meshless method using RBF collocation scheme for the coupled Schrödinger-KdV equations. Instead of traditional mesh oriented methods such as finite element method (FEM) or finite difference method (FDM), this method requires only a scattered set of nodes in the domain. For this scheme, error estimates and stability analysis are studied. L and L error norms between the results and exact solution is used as a performance measure. Moreover the results of numerical experiments are presented, and are compared with the findings of Finite Element method, finite difference Crank-Nicolson (CN) scheme and analytical solution to confirm the good accuracy of the presented scheme. [ABSTRACT FROM AUTHOR]

In this work, we develop a simple yet practical algorithm for constructing derivative free iterative methods of higher convergence orders. The algorithm can be easily implemented in software packages for achieving desired convergence orders. Convergence analysis shows that the algorithm can develop methods of various convergence orders which is also supported through the numerical work. Computational results ascertain that the developed algorithm is efficient and demonstrate equal or better performance as compared with other well known methods. [ABSTRACT FROM AUTHOR]

In this paper, we present general formulae for the mask of (2 b + 4)-point n-ary approximating as well as interpolating subdivision schemes for any integers $${b\,\geqslant\,0}$$ and $${n\,\geqslant\,2}$$. These formulae corresponding to the mask not only generalize and unify several well-known schemes but also provide the mask of higher arity schemes. Moreover, the 4-point and 6-point a-ary schemes introduced by Lian [Appl Appl Math Int J 3(1):18–29, 2008] are special cases of our general formulae. [ABSTRACT FROM AUTHOR]

We develop a scheme for constructing G1 triangular spline surfaces of arbitrary topological type. To assure that the scheme is local and singularity-free, we analyze the selection of scalar weight functions and the construction of the boundary curve network in detail. With the further requirements of interpolating positions, normals, and surface curvatures, we show that the minimum degree of such a triangular spline surface is 6. And we present a method for constructing boundary curves network, which consists of cubic Bézier curves. To deal with certain singular cases, the base mesh must be locally subdivided and we proposed an adaptive subdivision strategy for it. An application of our G1 triangular spline surfaces to the approximation of implicit surfaces is described. The visual quality of this scheme is demonstrated by some examples. [ABSTRACT FROM AUTHOR]

We provide estimates for the maximum error of polynomial tensor product interpolation on regular grids in $${\mathbb{R}^d}$$. The set of partial derivatives required to form these bounds depends on the clustering of interpolation nodes. Also bounds on the partial derivatives of the error are derived. [ABSTRACT FROM AUTHOR]

In this paper we present a new nonstationary, interpolatory, curve subdivision scheme. We show that the scheme converges and the subdivision curve is continuous. Moreover, starting with the chord length parametrization of the initial polygon, we obtain a subdivision curve parametrized by a multiple of the arc-length. The proposed method focuses on convexity preservation, limiting the oscillations of the subdivision curve and avoiding artifacts. A bound for the Hausdorff distance between the limit curve and the initial polygon is also obtained. [ABSTRACT FROM AUTHOR]

A roundoff error analysis of formulae for evaluating polynomials is performed. The considered formulae are linear combinations of basis functions, which can be computed with high relative accuracy. We have taken into account that all steps but the last one can be computed to high relative accuracy. The exactness of the initial data is crucial for obtaining low error bounds. The Lagrange interpolation formula and related formulae are considered and numerical experiments are provided. [ABSTRACT FROM AUTHOR]

An interpolant defined via moments is investigated for triangles, quadrilaterals, tetrahedra, and hexahedra and arbitrarily high polynomial degree. The elements are allowed to have diameters with different asymptotic behavior in different space directions. Anisotropic interpolation error estimates are proved. [ABSTRACT FROM AUTHOR]

The extended Numerov scheme of Chawla, adopted for nonuniform grids, is a useful compact finite-difference discretisation, suitable for the numerical solution of boundary value problems in singularly perturbed second order non-linear ordinary differential equations. A new set of three-point compact approximations to first and second derivatives, related to the Chawla scheme and valid for nonuniform grids, is developed in the present work. The approximations economically re-use intermediate quantities occurring in the Chawla scheme. The theoretical orders of accuracy are equal four for the central and one-sided first derivative approximations obtained, whereas the central second derivative formula is either fourth, third, or second order accurate, depending on the grid ratio. The approximations can be used for accurate a posteriori derivative evaluations. A Hermitian interpolation polynomial, consistent with the derivative approximations, is also derived. The values of the polynomial can be used, among other things, for guiding adaptive grid refinement. Accuracy orders of the new derivative approximations, and of the interpolating polynomial, are verified by computational experiments. [ABSTRACT FROM AUTHOR]

In this paper, we propose an approach to the computation of more accurate divided differences for the interpolation in the Newton form of the matrix exponential propagator φ( hA) v, φ ( z) = ( e z − 1)/ z. In this way, it is possible to approximate φ ( hA) v with larger time step size h than with traditionally computed divided differences, as confirmed by numerical examples. The technique can be also extended to “higher” order φ k functions, k≥0. [ABSTRACT FROM AUTHOR]

We propose a local method of constructing piecewise G 1 Bézier patches to span a Bézier curve network with odd- and 4-valent node points. We analyze all possible singular cases of the G 1 condition that is to be met by the curve network interpolation and propose a new G 1 continuity condition using linear and quartic scalar weight functions. Using this condition, a curve network can be interpolated without modification at 4-valent nodes with two collinear tangent vectors, even in the presence of singularities. We demonstrate our approach by generating G 1 surfaces over the curve network which includes singularities at its node vertices and edges. [ABSTRACT FROM AUTHOR]

Curves are commonly drawn by piecewise linear interpolation, but to worry about the error is rather seldom. In the present paper we give a strong mathematical error analysis for curve segments with bounded curvature and length. Though the result seems very clear, the proof turned out to be unexpectedly hard, comparable to that of the famous four vertex theorem. [ABSTRACT FROM AUTHOR]

We present refinement properties of cubic parametric B-splines: a family of functions introduced some years ago in the context of shape-preserving approximation of curves. The refinement equations determine a related subdivision scheme which turns out to be irregular, nonstationary and nonuniform. [ABSTRACT FROM AUTHOR]

In this paper, we present an iterative algorithm to generate a nice triangular curvilinear mesh on a regular parametric surface. The main idea is to obtain a reparametrization of the surface that behaves approximately like a conformal map in a finite number of points. These points are the vertices of a planar triangulation T op in the parameter space, which we would like to lift on the surface. The image of T op by means of the reparametrization provides us a triangular curvilinear mesh on the surface which reflects the properties of T op . [ABSTRACT FROM AUTHOR]

We present a multiresolution morphing algorithm using ``as-rigid-as-possible'' shape interpolation combined with an angle-length based multiresolution decomposition of simple 2D piecewise curves. This novel multiresolution representation is defined intrinsically and has the advantage that the details' orientation follows any deformation naturally. The multiresolution morphing algorithm consists of transforming separately the coarse and detail coefficients of the multiresolution decomposition. Thus all LoD (level of detail) applications like LoD display, compression, LoD editing etc. can be applied directly to all morphs without any extra computation. Furthermore, the algorithm can robustly morph between very large size polygons with many local details as illustrated in numerous figures. The intermediate morphs behave natural and least-distorting due to the particular intrinsic multiresolution representation. [ABSTRACT FROM AUTHOR]

A new backward stable algorithm (Algorithm 2) for polynomial interpolation based on the Lagrange and the Newton interpolation forms is proposed. It is shown that the Aitken algorithm and the scheme of the divided differences can be significantly less accurate than the proposed unconditionally stable Algorithm 2. Numerical examples that illustrate the advantage of a new algorithm are also given. [ABSTRACT FROM AUTHOR]

A simple algorithm for finding the implicit equation of a parametric plane curve given by its parametric equations is presented. The algorithm is based on an efficient computation of the Bézout resultant and Lagrange interpolation. One of main features of our approach is the fact that it considerably reduces the problem of computing intermediate expressions. [ABSTRACT FROM AUTHOR]

In his paper ``Lagrange interpolation on Chebyshev points of two variables'' (J. Approx. Theor. 87, 220–238, 1996), Y. Xu proposed a set of Chebyshev like points for polynomial interpolation in the square [−1,1]2, and derived a compact form of the corresponding Lagrange interpolation formula. We investigate computational aspects of the Xu polynomial interpolation formula like numerical stability and efficiency, the behavior of the Lebesgue constant, and its application to the reconstruction of various test functions. [ABSTRACT FROM AUTHOR]

We develop optimal quadratic and cubic spline collocation methods for solving linear second-order two-point boundary value problems on non-uniform partitions. To develop optimal nonuniform partition methods, we use a mapping function from uniform to nonuniform partitions and develop expansions of the error at the nonuniform collocation points of some appropriately defined spline interpolants. The existence and uniqueness of the spline collocation approximations are shown, under some conditions. Optimal global and local orders of convergence of the spline collocation approximations and derivatives are derived, similar to those of the respective methods for uniform partitions. Numerical results on a variety of problems, including a boundary-layer problem, and a nonlinear problem, verify the optimal convergence of the methods, even under more relaxed conditions than those assumed by theory. [ABSTRACT FROM AUTHOR]

We study cubature formulas on relatively small scattered samples in the unit square, obtained by integrating radial basis function (RBF) interpolants. Numerical tests show that, due to the small size of the corresponding weights (which are not all positive in general), thin-plate splines and Wendland's compactly supported radial functions give the most reliable RBF cubature methods. [ABSTRACT FROM AUTHOR]

We integrate optimal quadratic and cubic spline collocation methods for second-order two-point boundary value problems with adaptive grid techniques, and grid size and error estimators. Some adaptive grid techniques are based on the construction of a mapping function that maps uniform to non-uniform points, placed appropriately to minimize a certain norm of the error. One adaptive grid technique for cubic spline collocation is mapping-free and resembles the technique used in COLSYS (COLNEW) [2], [4]. Numerical results on a variety of problems, including problems with boundary or interior layers, and singular perturbation problems indicate that, for most problems, the cubic spline collocation method requires less computational effort for the same error tolerance, and has equally reliable error estimators, when compared to Hermite piecewise cubic collocation. Comparison results with quadratic spline collocation are also presented. [ABSTRACT FROM AUTHOR]

Anisotropic local interpolation error estimates are derived for quadrilateral and hexahedral Lagrangian finite elements with straight edges. These elements are allowed to have diameters with different asymptotic behaviour in different space directions. The case of affine elements (parallel-epipeds) with arbitrarily high degree of the shape functions is considered first. Then, a careful examination of the multi-linear map leads to estimates for certain classes of more general, isoparametric elements. As an application, the Galerkin finite element method for a reaction diffusion problem in a polygonal domain is considered. The boundary layers are resolved using anisotropic trapezoidal elements. [ABSTRACT FROM AUTHOR]

The application of Powell-Sabin’s or Clough-Tocher’s schemes to scattered data problems, as known requires the knowledge of the partial derivatives of first order at the vertices of an underlying triangulation. We study a local method for generating partial derivatives based on the minimization of the energy functional on the star of triangles sharing a node that we called a cell. The functional is associated to some piecewise polynomial function interpolating the points. The proposed method combines the global Method II by Renka and Cline (cf. [16, pp. 230–231]) with the variational approach suggested by Alfeld (cf. [2]) with care to efficiency in the computations. The locality together with some implementation strategies produces a method well suited for the treatment of a big amount of data. An improvement of the estimates is also proposed. [ABSTRACT FROM AUTHOR]

Many applications in inventory control, reliability engineering and preventive maintenance involve frequent calculations of probabilities and partial expectations. In the design of high-volume computer-based applications recourse to internal tables may therefore be preferable to an import of statistical packages. While interpolation in tabulated cdf’s will often prove sufficiently accurate from the point of view of statistical representation of the underlying problem, tables of compatible (partial) expectations need to be constructed with regard to the method of interpolation employed. The mathematics for establishing such tables differ from standard textbook procedures. This paper develops appropriate expressions in general terms and gives explicit results for the Gamma family of distributions, which is of particular interest in the areas of application mentioned. [ABSTRACT FROM AUTHOR]

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Copyright of Computing is the property of Springer Nature and its content may not be copied or emailed to multiple sites or posted to a listserv without the copyright holder's express written permission. However, users may print, download, or email articles for individual use. This abstract may be abridged. No warranty is given about the accuracy of the copy. Users should refer to the original published version of the material for the full abstract. (Copyright applies to all Abstracts.)

Copyright of Computing is the property of Springer Nature and its content may not be copied or emailed to multiple sites or posted to a listserv without the copyright holder's express written permission. However, users may print, download, or email articles for individual use. This abstract may be abridged. No warranty is given about the accuracy of the copy. Users should refer to the original published version of the material for the full abstract. (Copyright applies to all Abstracts.)

Copyright of Computing is the property of Springer Nature and its content may not be copied or emailed to multiple sites or posted to a listserv without the copyright holder's express written permission. However, users may print, download, or email articles for individual use. This abstract may be abridged. No warranty is given about the accuracy of the copy. Users should refer to the original published version of the material for the full abstract. (Copyright applies to all Abstracts.)

Copyright of Computing is the property of Springer Nature and its content may not be copied or emailed to multiple sites or posted to a listserv without the copyright holder's express written permission. However, users may print, download, or email articles for individual use. This abstract may be abridged. No warranty is given about the accuracy of the copy. Users should refer to the original published version of the material for the full abstract. (Copyright applies to all Abstracts.)

Copyright of Computing is the property of Springer Nature and its content may not be copied or emailed to multiple sites or posted to a listserv without the copyright holder's express written permission. However, users may print, download, or email articles for individual use. This abstract may be abridged. No warranty is given about the accuracy of the copy. Users should refer to the original published version of the material for the full abstract. (Copyright applies to all Abstracts.)

Copyright of Computing is the property of Springer Nature and its content may not be copied or emailed to multiple sites or posted to a listserv without the copyright holder's express written permission. However, users may print, download, or email articles for individual use. This abstract may be abridged. No warranty is given about the accuracy of the copy. Users should refer to the original published version of the material for the full abstract. (Copyright applies to all Abstracts.)

Copyright of Computing is the property of Springer Nature and its content may not be copied or emailed to multiple sites or posted to a listserv without the copyright holder's express written permission. However, users may print, download, or email articles for individual use. This abstract may be abridged. No warranty is given about the accuracy of the copy. Users should refer to the original published version of the material for the full abstract. (Copyright applies to all Abstracts.)

Copyright of Computing is the property of Springer Nature and its content may not be copied or emailed to multiple sites or posted to a listserv without the copyright holder's express written permission. However, users may print, download, or email articles for individual use. This abstract may be abridged. No warranty is given about the accuracy of the copy. Users should refer to the original published version of the material for the full abstract. (Copyright applies to all Abstracts.)

Copyright of Computing is the property of Springer Nature and its content may not be copied or emailed to multiple sites or posted to a listserv without the copyright holder's express written permission. However, users may print, download, or email articles for individual use. This abstract may be abridged. No warranty is given about the accuracy of the copy. Users should refer to the original published version of the material for the full abstract. (Copyright applies to all Abstracts.)

Copyright of Computing is the property of Springer Nature and its content may not be copied or emailed to multiple sites or posted to a listserv without the copyright holder's express written permission. However, users may print, download, or email articles for individual use. This abstract may be abridged. No warranty is given about the accuracy of the copy. Users should refer to the original published version of the material for the full abstract. (Copyright applies to all Abstracts.)

Copyright of Computing is the property of Springer Nature and its content may not be copied or emailed to multiple sites or posted to a listserv without the copyright holder's express written permission. However, users may print, download, or email articles for individual use. This abstract may be abridged. No warranty is given about the accuracy of the copy. Users should refer to the original published version of the material for the full abstract. (Copyright applies to all Abstracts.)

Copyright of Computing is the property of Springer Nature and its content may not be copied or emailed to multiple sites or posted to a listserv without the copyright holder's express written permission. However, users may print, download, or email articles for individual use. This abstract may be abridged. No warranty is given about the accuracy of the copy. Users should refer to the original published version of the material for the full abstract. (Copyright applies to all Abstracts.)

Copyright of Computing is the property of Springer Nature and its content may not be copied or emailed to multiple sites or posted to a listserv without the copyright holder's express written permission. However, users may print, download, or email articles for individual use. This abstract may be abridged. No warranty is given about the accuracy of the copy. Users should refer to the original published version of the material for the full abstract. (Copyright applies to all Abstracts.)

Copyright of Computing is the property of Springer Nature and its content may not be copied or emailed to multiple sites or posted to a listserv without the copyright holder's express written permission. However, users may print, download, or email articles for individual use. This abstract may be abridged. No warranty is given about the accuracy of the copy. Users should refer to the original published version of the material for the full abstract. (Copyright applies to all Abstracts.)

Copyright of Computing is the property of Springer Nature and its content may not be copied or emailed to multiple sites or posted to a listserv without the copyright holder's express written permission. However, users may print, download, or email articles for individual use. This abstract may be abridged. No warranty is given about the accuracy of the copy. Users should refer to the original published version of the material for the full abstract. (Copyright applies to all Abstracts.)

Copyright of Computing is the property of Springer Nature and its content may not be copied or emailed to multiple sites or posted to a listserv without the copyright holder's express written permission. However, users may print, download, or email articles for individual use. This abstract may be abridged. No warranty is given about the accuracy of the copy. Users should refer to the original published version of the material for the full abstract. (Copyright applies to all Abstracts.)

Copyright of Computing is the property of Springer Nature and its content may not be copied or emailed to multiple sites or posted to a listserv without the copyright holder's express written permission. However, users may print, download, or email articles for individual use. This abstract may be abridged. No warranty is given about the accuracy of the copy. Users should refer to the original published version of the material for the full abstract. (Copyright applies to all Abstracts.)