共查询到20条相似文献,搜索用时 15 毫秒
1.
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. 相似文献
2.
3.
王家正 《应用数学与计算数学学报》2006,20(2):77-82
Stieltjes型分叉连分式在有理插值问题中有着重要的地位,它通过定义反差商和混合反差商构造给定结点上的二元有理函数,我们将Stieltjes型分叉连分式与二元多项式结合起来,构造Stieltje- Newton型有理插值函数,通过定义差商和混合反差商,建立递推算法,构造的Stieltjes-Newton型有理插值函数满足有理插值问题中所给的插值条件,并给出了插值的特征定理及其证明,最后给出的数值例子,验证了所给算法的有效性. 相似文献
4.
《Journal of Computational and Applied Mathematics》2004,163(1):219-232
As we know, the classical Neville's algorithm is an effective method used to solve the interpolation problem by polynomials. In this paper, we adopt the idea of the Neville's algorithm to construct a kind of blending rational interpolants via continued fractions. For a given set of support points, there are many ways to build up the interpolation schemes, by which we mean that there are many choices to make to determine the initial interpolants on subsets of support points and then update them step by step to form a solution to the full interpolation problem. Numerical examples are given to show the advantage of our method and a multivariate analogy is also discussed. 相似文献
5.
Lagrange基函数的复矩阵有理插值及连分式插值 总被引:1,自引:0,他引:1
顾传青 《高等学校计算数学学报》1998,20(4):306-314
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 相似文献
6.
Qianjin Zhao Jieqing Tan 《高等学校计算数学学报(英文版)》2007,16(1):63-73
This paper constructs a new kind of block based bivariate blending rational interpolation via symmetric branched continued fractions. The construction process may be outlined as follows. The first step is to divide the original set of support points into some subsets (blocks). Then construct each block by using symmetric branched continued fraction. Finally assemble these blocks by Newton’s method to shape the whole interpolation scheme. Our new method offers many flexible bivariate blending rational interpolation schemes which include the classical bivariate Newton’s polynomial interpolation and symmetric branched continued fraction interpolation as its special cases. The block based bivariate blending rational interpolation is in fact a kind of tradeoff between the purely linear interpolation and the purely nonlinear interpolation. Finally, numerical examples are given to show the effectiveness of the proposed method. 相似文献
7.
It is well-known that osculatory rational interpolation sometimes gives better approximation than Hermite interpolation, especially for large sequences of points. However, it is difficult to solve the problem of convergence and control the occurrence of poles. In this paper, we propose and study a family of barycentric osculatory rational interpolation function, the proposed function and its derivative function both have no real poles and arbitrarily high approximation orders on any real interval. 相似文献
8.
It is well known that rational interpolation sometimes gives better approximations than polynomial interpolation, especially
for large sequences of points, but it is difficult to control the occurrence of poles. In this paper we propose and study
a family of barycentric rational interpolants that have no real poles and arbitrarily high approximation orders on any real
interval, regardless of the distribution of the points. These interpolants depend linearly on the data and include a construction
of Berrut as a special case. 相似文献
9.
A matrix rational Lanczos method for model reduction in large‐scale first‐ and second‐order dynamical systems 下载免费PDF全文
In the present paper, we describe an adaptive modified rational global Lanczos algorithm for model‐order reduction problems using multipoint moment matching‐based methods. The major problem of these methods is the selection of some interpolation points. We first propose a modified rational global Lanczos process and then we derive Lanczos‐like equations for the global case. Next, we propose adaptive techniques for choosing the interpolation points. Second‐order dynamical systems are also considered in this paper, and the adaptive modified rational global Lanczos algorithm is applied to an equivalent state space model. Finally, some numerical examples will be given. 相似文献
10.
We consider a Nevanlinna-Pick type interpolation problem for Carathéodory functions, where the values of the function and its derivatives up to certain orders are given at finitely many points of the unit disk. The set of all solutions of this problem is described by means of the orthogonal rational functions which play here a similar role as the orthogonal polynomials on the unit circle in the classical case of the trigonometric moment problem. In particular, we use a connection between Szegö and Schur parameters which in the classical situation was discovered by Ja.L. Geronimus. 相似文献
11.
We investigate the following problem: For which open simply connected domains do there exist interpolation schemes (a set of interpolation points) such that for any analytic function defined in the domain the corresponding interpolating polynomials converge to the function when the degree of the polynomials tends to infinity? We also study similar problems for rational interpolants. These problems are connected to the balayage (sweeping out) problems of measures. 相似文献
12.
A. G. Lipchinski? 《Siberian Mathematical Journal》2012,53(5):821-838
We consider an interpolation process for the class of functions with finitely many singular points by means of rational functions whose poles coincide with the singular points of the function under interpolation. The interpolation nodes form a triangular matrix. We find necessary and sufficient conditions for the 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. We generalize and improve the familiar results on the interpolation of functions with finitely many singular points by rational fractions and of entire functions by polynomials. 相似文献
13.
Given a continuous function on a closed interval, a sequence of rational interpolation splines is constructed which converges uniformly on this closed interval to the given function for any sequence of grids with step width tending to zero. The derivatives possess this unconditional convergence property as well. Estimates of the rate of convergence are given.
相似文献14.
In this work we propose three different procedures for vector-valued rational interpolation of a function F(z), where , and develop algorithms for constructing the resulting rational functions. We show that these procedures also cover the general case in which some or all points of interpolation coalesce. In particular, we show that, when all the points of interpolation collapse to the same point, the procedures reduce to those presented and analyzed in an earlier paper (J. Approx. Theory 77 (1994) 89) by the author, for vector-valued rational approximations from Maclaurin series of F(z). Determinant representations for the relevant interpolants are also derived. 相似文献
15.
Dietrich Braess 《Numerische Mathematik》1974,22(3):219-232
This paper is concerned with interpolation by rational functions. Conditions are given which ensure that the solution has no poles between the points of interpolation. Comparison theorems on the error of the interpolation are also derived. 相似文献
16.
17.
Andreas Lasarow 《Mathematische Nachrichten》2008,281(1):75-108
We consider an interpolation problem of Nevanlinna–Pick type for matrix‐valued Carathéodory functions, where the values of the functions and its derivatives up to certain orders are given at finitely many points of the open unit disk. For the non‐degenerate case, i.e., in the particular situation that a specific block matrix (which is formed by the given data in the problem) is positive Hermitian, the solution set of this problem is described in terms of orthogonal rational matrix‐valued functions. These rational matrix functions play here a similar role as Szegő's orthogonal polynomials on the unit circle in the classical case of the trigonometric moment problem. In particular, we present and use a connection between Szegő and Schur parameters for orthogonal rational matrix‐valued functions which in the primary situation of orthogonal polynomials was found by Geronimus. (© 2007 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim) 相似文献
18.
A. G. Lipchinskii 《Siberian Mathematical Journal》2005,46(4):652-660
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. 相似文献
19.
Newton-Thiele's rational interpolants 总被引:13,自引:0,他引:13
It is well known that Newton's interpolation polynomial is based on divided differences which produce useful intermediate
results and allow one to compute the polynomial recursively. Thiele's interpolating continued fraction is aimed at building
a rational function which interpolates the given support points. It is interesting to notice that Newton's interpolation polynomials
and Thiele's interpolating continued fractions can be incorporated in tensor‐product‐like manner to yield four kinds of bivariate
interpolation schemes. Among them are classical bivariate Newton's interpolation polynomials which are purely linear interpolants,
branched continued fractions which are purely nonlinear interpolants and have been studied by Chaffy, Cuyt and Verdonk, Kuchminska,
Siemaszko and many other authors, and Thiele-Newton's bivariate interpolating continued fractions which are investigated in
another paper by one of the authors. In this paper, emphasis is put on the study of Newton-Thiele's bivariate rational interpolants.
By introducing so‐called blending differences which look partially like divided differences and partially like inverse differences,
we give a recursive algorithm accompanied with a numerical example. Moreover, we bring out the error estimation and discuss
the limiting case.
This revised version was published online in June 2006 with corrections to the Cover Date. 相似文献