A de Montessus type convergence study for a vector-valued rational interpolation procedure

In a recent paper of the author [8], three new interpolation procedures for vector-valued functions F(z), where F: ℂ → ℂ^N, were proposed, and some of their algebraic properties were studied. In the present work, we concentrate on one of these procedures, denoted IMMPE, and study its convergence properties when it is applied to meromorphic functions. We prove de Montessus and Koenig type theorems in the presence of simple poles when the points of interpolation are chosen appropriately. We also provide simple closed-form expressions for the error in case the function F(z) in question is itself a vector-valued rational function whose denominator polynomial has degree greater than that of the interpolant.     

Hybrid polynomial approximation to higher derivatives of rational curves

In this paper, we extend the results published in JCAM volume 214 pp. 163-174 in 2008. Based on the bound estimates of higher derivatives of both Bernstein basis functions and rational Bézier curves, we prove that for any given rational Bézier curve, if the convergence condition of the corresponding hybrid polynomial approximation is satisfied, then not only the l-th (l=1,2,3) derivatives of its hybrid polynomial approximation curve uniformly converge to the corresponding derivatives of the rational Bézier curve, but also this conclusion is tenable in the case of any order derivative. This result can expand the area of applications of hybrid polynomial approximation to rational curves in geometric design and geometric computation.     

Convergence Conditions for Interpolation Fractions at the Nodes Distinct from the Singular Points of a Function

We consider an interpolation process for the class of functions with finitely many singular points by means of the rational functions whose poles coincide with the singular points of the function under interpolation. The interpolation nodes constitute a triangular matrix and are distinct from the singular points of the function. We find a necessary and sufficient condition for uniform convergence of sequences of interpolation fractions to the function under interpolation on every compact set disjoint from the singular points of the function and other conditions for convergence.Original Russian Text Copyright © 2005 Lipchinskii A. G.__________Translated from Sibirskii Matematicheskii Zhurnal, Vol. 46, No. 4, pp. 822–833, July–August, 2005.     

Technical Note: Reduction of rational integrals related to linear and convex quadrilateral finite elements

We reduce a rational function of bivariate nth degree polynomial numerator with a linear denominator to a simple bivariate polynomial of degree (n ? 1) and a rational function of a single variate nth degree polynomial numerator with the same bivariate linear denominator. This has very greatly contributed to the evaluation of (n + 1)(n + 2)/2 rational integrals in bivariates to mere (n + 1) rational integral of a single variate and an integration of simple polynomial in bivariates. Thus the effort of integration is reduced several times and leads to simple analytical expressions in terms of the nodal coordinates. In order to illustrate the numerical process two examples are considered. © 2002 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 18: 759–770, 2002; Published online in Wiley InterScience (www.interscience.wiley.com); DOI 10.1002/num.10026.     

Vector-valued,rational interpolants III

In 1963, Wynn proposed a method for rational interpolation of vector-valued quantities given on a set of distinct interpolation points. He used continued fractions, and generalized inverses for the reciprocals of vector-valued quantities. In this paper, we present an axiomatic approach to vector-valued rational interpolation. Uniquely defined interpolants are constructed for vector-valued data so that the components of the resulting vector-valued rational interpolant share a common denominator polynomial. An explicit determinantal formula is given for the denominator polynomial for the cases of (i) vector-valued rational interpolation on distinct real or complex points and (ii) vector-valued Padé approximation. We derive the connection with theε-algorithm of Wynn and Claessens, and we establish a five-term recurrence relation for the denominator polynomials.     

Weighted interpolation for equidistant nodes

Weighted Lagrange interpolation is proposed for solving Lagrange interpolation problems on equidistant or almost equidistant data. Good condition numbers are found in the case of rational interpolants whose denominator has degree about twice the number of data to be interpolated. Since the degree of the denominator is higher than that of the numerator, simple functions like constants and linear polynomials will not be reproduced. Furthermore, the interpolant cannot be expressed by a barycentric formula. As a counterpart, the interpolation algorithm is simple and leads to small Lebesgue constants.     

The two-level local projection stabilization as an enriched one-level approach

The two-level local projection stabilization is considered as a one-level approach in which the enrichments on each element are piecewise polynomial functions. The dimension of the enrichment space can be significantly reduced without losing the convergence order. On triangular meshes, for example, using continuous piecewise polynomials of degree r ≥ 1, only 2r − 1 functions per macro-cell are needed for the enrichment compared to r ² in the two-level approach. In case of the Stokes problem r − 1 functions per macro-cell are already sufficient to guarantee stability and to preserve convergence order. On quadrilateral meshes the corresponding reduction rates are even higher. We give examples of “reduced” two-level approaches and study how the constant in the local inf-sup condition for the one-level and different two-level approaches, respectively, depends on the polynomial degree r.     

Lagrange基函数的复矩阵有理插值及连分式插值

1引言 矩阵有理插值问题与系统线性理论中的模型简化问题和部分实现问题有着紧密的联系~[1][2]，在矩阵外推方法中也常常涉及线性或有理矩阵插值问题~[3]。按照文~[1]的阐述。目前已经研究的矩阵有理插值问题包括矩阵幂级数和Newton-Pade逼近。Hade逼近，联立Pade逼近，M-Pade逼近，多点Pade逼近等。显然，上述各种形式的矩阵Pade逼上梁山近是矩     

Optimal constrained polynomials approximation of hyperbolas based on Lupaş q‐Bézier curves

This paper presents an explicit optimal polynomial for approximating the quadratic Lupaş q‐Bézier curve. We first prove that the quadratic Lupaş q‐Bézier curve represents a hyperbola or a parabola. Then we research the approximation of quadratic Lupaş q‐Bézier curves by polynomials. Since the denominator of quadratic Lupaş q‐Bézier curves is a linear function, the explicit optimal constrained approximation is obtained. Finally, some numerical examples are presented to illustrate the effectiveness of the proposed method.     

Rational Approximation to the Exponential Function with Complex Conjugate Interpolation Points

In this paper, we study asymptotic properties of rational functions that interpolate the exponential function. The interpolation is performed with respect to a triangular scheme of complex conjugate points lying in bounded rectangular domains included in the horizontal strip |Im z|<2π. Moreover, the height of these domains cannot exceed some upper bound which depends on the type of rational functions. We obtain different convergence results and precise estimates for the error function in compact sets of that generalize the classical properties of Padé approximants to the exponential function. The proofs rely on, among others, Walsh's theorem on the location of the zeros of linear combinations of derivatives of a polynomial and on Rolle's theorem for real exponential polynomials in the complex domain.     

可调形三次三角Cardinal插值样条曲线

在三次Cardinal插值样条曲线的基础上,引入了三角函数多项式,得到一组带调形参数的三次三角Cardinal样条基函数,以此构造一种可调形的三次三角Cardinal插值样条曲线.该插值样条可以精确表示直线、圆弧、椭圆以及自由曲线,改变调形参数可以调控插值曲线的形状.该插值样条避免了使用有理形式,其表达式较为简洁,计算量也相对较少,从而为多种线段的构造与处理提供了一种通用与简便的方法.     

Application of degree reduction of polynomial Bézier curves to rational case

An algorithmic approach to degree reduction of rational Bézier curves is presented. The algorithms are based on the degree reduction of polynomial Bézier curves. The method is introduced with the following steps: (a) convert the rational Bézier curve to polynomial Bézier curve by using homogenous coordinates, (b) reduce the degree of polynomial Bézier curve, (c) determine weights of degree reduced curve, (d) convert the Bézier curve obtained through step (b) to rational Bézier curve with weights in step (c).     

In what sense is the rational interpolation problem well posed

It is well known that the nonlinear problem of interpolatingm+n+1 data by a rational function of type (m, n) may have no solution, but that the corresponding linearized problem (obtained by multiplying through by the denominator) always leads to a unique rational function, which is often still called the rational interpolant. For fixedm andn, and fixed (possibly multiple) interpolation points, the dependence of this interpolant on the prescribed function values is studied here. For ten notions of convergence in the space _{m, n} the question of the continuity of this interpolation operator is investigated.Communicated by William B. Gragg.AMS classification: 41A24, 30E05, 41A20, 65D05.     

On the convergence of certain sequences of rational approximants to meromorphic functions in several variables

In a previous paper, the author introduced a new class of multivariate rational interpolants, which are called Optimal Padé-type Approximants (OPTA). There, for this class of rational interpolants, which extends classical univariate Padé Approximants, a direct extension of the “de Montessus de Ballore's Theorem” for meromorphic functions in several variables is established. In the univariate case, this theorem ensures uniform convergence of a row of Pade Approximants when the denominator degree equals the number of poles (counting multiplicities) in a certain disc. When one overshoots the number of poles when fixing the denominator degree, convergence in measure or capacity has been proved and, under certain additional restrictions, the uniform convergence of a subsequence of the row. The author tackles the latter case and studies its generalization to functions in several variables by using OPTA.     

On G 2 Continuous Spline Interpolation of Curves in ℝ d

In this paper the problem of G ² continuous interpolation of curves in ^d by polynomial splines of degree n is studied. The interpolation of the data points and two tangent directions at the boundary is considered. The case n = r + 2 = d, where r is the number of interior points interpolated by each segment of the spline curve, is studied in detail. It is shown that the problem is uniquely solvable asymptotically, e., when the data points are sampled regularly and sufficiently dense, and lie on a regular, convex parametric curve in ^d . In this case the optimal approximation order is also determined.     

Convergence rates of derivatives of a family of barycentric rational interpolants

In polynomial and spline interpolation the k-th derivative of the interpolant, as a function of the mesh size h, typically converges at the rate of O(hd+1−k) as h→0, where d is the degree of the polynomial or spline. In this paper we establish, in the important cases k=1,2, the same convergence rate for a recently proposed family of barycentric rational interpolants based on blending polynomial interpolants of degree d.     

Bivariate Thiele-type matrix-valued rational interpolants

A new method for the construction of bivariate matrix-valued rational interpolants on a rectangular grid is introduced in this paper. The rational interpolants are of the continued fraction form, with scalar denominator. In this respect the approach is essentially different from that of Bose and Basu (1980) where a rational matrix-valued approximant with matrix-valued numerator and denominator is used for the approximation of a bivariate matrix power series. The matrix quotients are based on the generalized inverse for a matrix introduced by Gu Chuanqing and Chen Zhibing (1995) which is found to be effective in continued fraction interpolation. A sufficient condition of existence is obtained. Some important conclusions such as characterisation and uniqueness are proven respectfully. The inner connection between two type interpolating functions is investigated. Some examples are given so as to illustrate the results in the paper.     

Biorthogonal Polynomial Bases and Vandermonde-like Matrices

This article considers a family of Gram matrices of pairs of bases of a finite dimensional vector space of polynomials with respect to certain indefinite inner products. Such a family includes all the generalized confluent Vandermonde matrices relative to any polynomial basis, like the Chebyshev-Vandermonde matrices, for example. Using the biorthogonality of pairs of bases with respect to a divided difference functional, properties of matrices and functionals, as well as interpolation formulas are obtained. I show that the computation of the inverse of a Vandermonde-like matrix is essentially equivalent to the computation of the partial fractions decompositions of a set of rational functions with a common denominator. I also explain why the various Chebyshev-Vandermonde matrices are the simplest generalizations of the classic Vandermonde matrices and describe a simple algorithm for the computation of their inverses, which requires a number of multiplications of the order of 3N².     

有理曲线的多项式逼近

利用曲线摄动的思想给出了用多项式曲线逼近有理曲线的一种新方法．其基本步骤是对有理曲线的控制顶点进行摄动，使之产生一多项式曲线，并使摄动误差在某种范数意义之下达到最小．同时，通过适当控制摄动曲线的顶点，使逼近多项式曲线与有理曲线在两端点保持一定的连续性．这一结果可以与细分（ｓｕｂｄｉｖｉｓｉｏｎ）技术结合给出有理曲线的整体光滑的分片多项式逼近．实例表明，在某些情况下本文中的方法要优于传统的Ｈｅｒｍｉｔｅ插值方法及Ｔ．Ｗ．Ｓｅｄｅｒｂｅｒｇ和Ｍ．Ｋａｋｉｍｏｔｏ（１９９１）提出的杂交曲线逼近算法．