Showing papers on "Monotone cubic interpolation published in 1992"
195 citations
01 Jan 1992
TL;DR: For Hermite-Birkhoff interpolation of scattered multidumensional data by radial basis function, existence and characterization theorems and a variational principle are proved.
Abstract: For Hermite-Birkhoff interpolation of scattered multidumensional data by radial basisfunction ,existence and characterization theorems and a variational principle are proved.Examples include (r)=r~b,Duchon’s thin-plate splines,Hardy’s multiquadrics,and inversemultiquadrics.
47 citations
TL;DR: This method provides not only a large variety of very interesting shape controls like biased, point, and interval tensions but, as a special case, also recovers the cubic B-spline curve and the rational cubic spline with tension of Gregory and Sarfraz.
24 citations
TL;DR: This method is tailor-made for reparameterizing B-spline curves in Geometric Modeling, and it is shown that it is able to interpolate a monotone data sequence (values and derivatives) with a C1monotone rational B- Spline of degree 1.
23 citations
TL;DR: In this article, a method for constructing sets of points which admit Lagrange and Hermite type interpolation by spaces of bivariate splines on rectangular and triangular partitions is presented.
Abstract: We develop methods for constructing sets of points which admit Lagrange and Hermite type interpolation by spaces of bivariate splines on rectangular and triangular partitions which are uniform, in general. These sets are generated by building up a net of lines and by placing points on these lines which satisfy interlacing properties for univariate spline spaces.
22 citations
TL;DR: Some shortcomings of a proposed choice of parameters in a shape-preserving quadratic spline interpolation are examined in this article, where the authors also propose a new shape-adaptive spline algorithm.
19 citations
TL;DR: In this article, the shape preserving C1-interpolation of data sets given on rectangular grids was studied and rational biquadratic splines were derived which are sufficient for the positivity, monotonicity, and S-convexity.
Abstract: This paper is concerned with shape preservingC1-interpolation of data sets given on rectangular grids. Using special rational biquadratic splines, criteria are derived which are sufficient for the positivity, monotonicity, andS-convexity and which, in addition, are satisfied for sufficiently large rationality parameters.
18 citations
TL;DR: A local interpolation method for curves in R2 or R3 offering G1 continuity is described, using an intuitive geometric, rule-based approach to find a ‘good’ default solution that produces pleasing-looking results even for highly irregular sets of data.
Abstract: A local interpolation method for curves in R2 or R3 offering G1 continuity is described. A curve is represented as a union of geometrically continuous cubic Bezier segments between each pair of adjacent vertices. At each interpolation point, the procedure determines a tangent direction and two derivative magnitudes on either side of the vertex. The method uses an intuitive geometric, rule-based approach to find a ‘good’ default solution that produces pleasing-looking results even for highly irregular sets of data Various spline properties and their relevance to the method are also discussed.
16 citations
TL;DR: In this paper, the interpolatory piecewise Hermite quartic polynomials induced by cubic spline were used to estimate the Cauchy principal value integrals and their derivatives.
11 citations
TL;DR: In this article, the authors provide a quantitative comparison of eight different cubic spline interpolation schemes for a variety of smoothness classes, using uniform and also several thousand random meshes, and the performance criteria used are the quantitative ones of exact operator and derived operator norms.
Abstract: The aim of this paper is to provide a quantitative comparison of eight different $C^1 $ and $C^2 $ cubic spline interpolation schemes. The $C^1 $ schemes discussed are local while the $C^2 $ ones are global.In practice cubic splines are often used when the smoothness of the function being interpolated/approximated is unknown. Also, it is often necessary, or advantageous, to use a nonuniform mesh. Therefore performance over a variety of smoothness classes, using uniform and also several thousand random meshes, is compared. The performance criteria used are the quantitative ones of exact operator and derived operator norms and best possible pointwise error estimates.
11 citations
TL;DR: In this paper, the main idea of the not-a-knot cubic spline was extended to all the interior points (knots) of the spline interval to obtain a piecewise interpolatory cubic polynomial.
TL;DR: A class of iterative formulae is derived to find numerically a factor of arbitrary degree of a polynomialf(x) based on the rational Hermite interpolation which has a high convergence order even for a factor which includes multiple zeros.
Abstract: We derive a class of iterative formulae to find numerically a factor of arbitrary degree of a polynomialf(x) based on the rational Hermite interpolation. The iterative formula generates the sequence of polynomials which converge to a factor off(x). It has a high convergence order even for a factor which includes multiple zeros. Some numerical examples are also included.
01 Jan 1992
TL;DR: In this article, the authors discuss the structure of the table of URI, a recursive computation scheme and a continued fraction representation both in the normal case and the non-normal case and a convergence theorem for rational Hermite interpolants of meromorphic functions.
Abstract: In the first 4 sections we discuss topics from univariate rational H ermite interpolation (URI). These topics include the structure of the table of URI, a recursive computation scheme and a continued fraction representation both in the normal case and the non-normal case and a convergence theorem for rational Hermite interpolants of meromorphic functions.
TL;DR: This paper applies a vectorial Bézier technique and later a periodic 5-spline method to new efficient real algorithms for periodic Hermite spline interpolation based on the de Casteljau algorithm and the de Boor algorithm, respectively.
Abstract: Periodic Hermite spline interpolants on an equidistant lattice are represented by the Bézier technique as well as by the fi-spline method. Circulant matrices are used to derive new explicit formulas for the periodic Hermite splines of degree m and defect r (1 < r < m). Applying the known de Casteljau algorithm and the de Boor algorithm, respectively, we obtain new efficient real algorithms for periodic Hermite spline interpolation. 0. INTRODUCTION This paper deals with periodic Hermite spline interpolation on the equidistant lattice Z. Other approaches to this problem use Euler-Frobenius polynomials and complex line integrals (see [4-6]) or Euler-Frobenius polynomials and circulant matrices (see [7, 8]). Similar to [7, 8], we prefer a real-algebraic method for periodic Hermite spline interpolation. Contrary to [4-8], we apply a vectorial Bézier technique and later a periodic 5-spline method in this note. This leads to new efficient real algorithms for periodic Hermite spline interpolation. These methods are based on the de Casteljau algorithm and the de Boor algorithm, respectively. Both procedures possess a low arithmetic complexity. Further, one can see that the generalized Euler-Frobenius polynomials are very important for periodic Hermite spline interpolation. Note that our methods can be extended to periodic Hermite spline interpolation with shifted nodes too. 1. Preliminaries In this paper we use standard notations. First we recall some facts concerning circulant matrices, which form the background of the considerations in §§3 and 4 (cf. [2]). Let N £ N (N > 1) be fixed. For a = (ax, ... , aN)T £ RN , let ax a2 ■ ■ ■ un ' aw ax ■■■ üM-i
TL;DR: The method offers a free choice of a tangent direction, two derivatives, and a curvature at each interpolation point, and can be achieved by constructing two cubic segments for each adjacent pair of vertices.
Abstract: A local G2 curve interpolation method in R 2 is described. The method offers a free choice of a tangent direction, two derivatives, and a curvature at each interpolation point. This flexibility can be achieved by constructing two cubic segments for each adjacent pair of vertices. The method is also compared with other local schemes for curve construction.
TL;DR: This note introduces a simple recursive algorithm for computation of the values of neighbours of the singular block along antidiagonals even in the presence of singular blocks and discusses a facility to monitor the stability.
Abstract: Claessens' cross rule [8] enables simple computation of the values of the rational interpolation table if the table is normal, i.e. if the denominators in the cross rule are non-zero. In the exceptional case of a vanishing denominator a singular block is detected having certain structural properties so that some values are known without further computations. Nevertheless there remain entries which cannot be determined using only the cross rule.
01 Jan 1992
TL;DR: In this article, the Hermite and Hermite-Fejer interpolation based on the zeros of Jacobi polynomials plus additional nodes is considered and it is shown that such procedures can always well approximate a function and its derivatives simultaneously.
Abstract: The authors consider two procedures of Hermite and Hermite-Fejer interpolation based on the zeros of Jacobi polynomials plus additional nodes and prove that such procedures can always well approximate a function and its derivatives simultaneously.
TL;DR: Several investigations on cubic polynomials are presented in this article, which deal with certain relationships which hold between the roots of the general cubic equation and other appropriate parameters derived from the same polynomial.
Abstract: Several investigations on cubic polynomials are presented. The studies reported here deal with certain relationships which hold between the roots of the general cubic equation and other appropriate parameters derived from the same polynomial. Each section is organized in a monotonic sequence; that is, statement, demonstration and verification.
TL;DR: In this paper, the authors studied cubic spline interpolation with less restrictive continuity requirements at the knots and established an interpolant which can interpolate at any point of the partition and also match the area with certain mean over a greater partition length.
TL;DR: A succinct introduction to splines, explaining how and why B-splines are used as a basis and how cubic and quadratic splines may be constructed, is given in this article.
18 Jun 1992
TL;DR: It is shown that exploiting the smooth behavior of the standing-wave's envelope near maxima yields better results in impedance and return loss computations, and the cubic spline interpolation of maxima is more reliable than that of minima.
Abstract: It is shown that exploiting the smooth behavior of the standing-wave's envelope near maxima yields better results in impedance and return loss computations. The improvement is achieved using a very simple algorithm to locate extrema in an envelope approximated by a cubic spline and without increasing the order of the moment matrix. The method consists of interpolating the continuous standing wave envelope by a cubic spline. The fit of a spline depends on the amount of discrete data points per wavelength and on the true shape of the envelope. An increase in the number of data points, or knots, along the feedline improves the fit of a spline, but at the cost of an increase in the order of the moment matrix. The objective is to obtain a satisfactory representation of the envelope with as few discrete data points along the feedline as possible. Based on the results, it appears that, for arbitrary values on the standing wave ratio, the cubic spline interpolation of maxima is more reliable than that of minima. >
TL;DR: In this article, a new stability property of the classical Hermite interpolation scheme in one variable has been proved, which refines the usual convergence property of this interpolation method.
TL;DR: In this article, a new finite element with dimension 16 of C 1 cubic splines which has interpolation schemes on the Morgan-Scott construction △ 0 of a triangle was given.
01 Sep 1992
TL;DR: The scheme is local, stable and provides parameters for modifying the shape of the curve and an explicit representation of the C2 piecewise rational cubic Bezier function which is used to interpolate the given monotone data.
Abstract: A scheme for generating interpolating monotone curves is described. The scheme is local, stable and provides parameters for modifying the shape of the curve. A sufficient condition for the curve to be monotone is derived and an explicit representation of the C2 piecewise rational cubic Bezier function which is used to interpolate the given monotone data is described. A number of examples are given.
01 Sep 1992
TL;DR: In this article, it was shown that the derivatives of U exhibit superconvergence at the nodes, and that U agrees with u to fourth order uniformly in, but away from the nodal points.
Abstract: This note is inspired by the paper of Bialecki, Fairweather, and Bennett [1] which studies a method for the numerical solution of u 00 = f on an interval, and u = f on a rectangle, with zero boundary data; their scheme calculates that C 1 piecewise cubic U for which U 00 (or U) agrees with f at the two (or four) Gauss quadrature points of each subdivision of the original domain. They observe that not only does their U approximate u to order h 4 at mesh nodes { h is the linear dimension of the subdivisions { but also U 0 agrees with u 0 to order h 4 in one dimension, and U x , U y , and U xy agree to order h 4 with u x , u y , and u xy in two dimensions. Agreement o f U with u to fourth order is perhaps not so surprising { Gaussian quadrature of this order is, after all, fourth order accurate and one could well expect this order of accuracy to be re BLOCKINected in a Gauss-inspired dierential equation solver. But fourth order agreement also for the derivatives is a surprise, and is due to the particular nature of the collocation scheme. Indeed, U agrees with u to fourth order uniformly in , but away from the nodal points their derivatives need not be so close. It is in this sense that the derivatives of U exhibit superconvergence at the nodes. We refer to the survey article of Fairweather and Meade [3] for a history of the subject and an extensive bibliography. This note consists of three sections. The rst section treats u 00 = f on [0; 1] by elementary methods, and admits a non-uniform mesh. At the mesh nodes, we show that u U and u 0 U 0 are dominated by a quantity on the order of h 4 (M 3 + M 4) (h is the maximum mesh size, M p is a bound for jf p j); this is a very special case of the general results of deBoor and Swartz [2]. A simple example shows that no derivative of lesser order than 4 can suce to give a general result of fourth order accuracy. The second section redoes the problem u 00 = f on [0; 1], this time with only a uniform mesh, by F …
TL;DR: In this article, it was shown that the matching of the area for the cubic spline does not follow from the corresponding result proved in [2] and that there is a unique quadratic spline which bounds the same area as that of the function.
Abstract: From the result in [1] it follows that there is a unique quadratic spline which bounds the same area as that of the function. The matching of the area for the cubic spline does not follow from the corresponding result proved in [2]. We obtain cubic splines which preserve the area of the function.