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1.
If the system of linear equations defining a multivariate rational interpolant is singular, then the table of multivariate rational interpolants displays a structure where the basic building block is a hexagon. Remember that for univariate rational interpolation the structure is built by joining squares. In this paper we associate with every entry of the table of rational interpolants a well-defined determinant representation, also when this entry has a nonunique solution. These determinant formulas are crucial if one wants to develop a recursive computation scheme.In section 2 we repeat the determinant representation for nondegenerate solutions (nonsingular systems of interpolation conditions). In theorem 1 this is generalized to an isolated hexagon in the table. In theorem 2 the existence of such a determinant formula is proven for each entry in the table. We conclude with an example in section 5. 相似文献
2.
Annie Cuyt 《BIT Numerical Mathematics》1988,28(1):98-112
The problem of constructing a univariate rational interpolant or Padé approximant for given data can be solved in various equivalent ways: one can compute the explicit solution of the system of interpolation or approximation conditions, or one can start a recursive algorithm, or one can obtain the rational function as the convergent of an interpolating or corresponding continued fraction.In case of multivariate functions general order systems of interpolation conditions for a multivariate rational interpolant and general order systems of approximation conditions for a multivariate Padé approximant were respectively solved in [6] and [9]. Equivalent recursive computation schemes were given in [3] for the rational interpolation case and in [5] for the Padé approximation case. At that moment we stated that the next step was to write the general order rational interpolants and Padé approximants as the convergent of a multivariate continued fraction so that the univariate equivalence of the three main defining techniques was also established for the multivariate case: algebraic relations, recurrence relations, continued fractions. In this paper a multivariate qd-like algorithm is developed that serves this purpose. 相似文献
3.
Jesus M. Carnicer 《Numerical Algorithms》2010,55(2-3):223-232
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. 相似文献
4.
We improve upon the method of Zhu and Zhu [A method for directly finding the denominator values of rational interpolants, J. Comput. Appl. Math. 148 (2002) 341–348] for finding the denominator values of rational interpolants, reducing considerably the number of arithmetical operations required for their computation. In a second stage, we determine the points (if existent) which can be discarded from the rational interpolation problem. Furthermore, when the interpolant has a linear denominator, we obtain a formula for the barycentric weights which is simpler than the one found by Berrut and Mittelmann [Matrices for the direct determination of the barycentric weights of rational interpolation, J. Comput. Appl. Math. 78 (1997) 355–370]. Subsequently, we give a necessary and sufficient condition for the rational interpolant to have a pole. 相似文献
5.
Using the multivariateqdg-algorithm developed in [5], it is possible to compute the partial numerators and denominators of a continued fraction representation associated with a descending staircase in a table of multivariate rational interpolants, more precisely, multivariate Newton-Padé approximants. The algorithm is only applicable if every three successive elements on the staircase are different. If a singularity occurs in the defining system of equations for the multivariate rational interpolant then singular rules must be developed. For the univariate Newton-Padé approximant this was done in [3] by Claessens and Wuytack. The idea to perturb the initial staircase and walk around the block structure in the table in order to avoid the singularity, is explored now in a multivariate setting. Another approach would be to use block bordering methods in combination with reverse bordering [4] in order to solve the rank deficient linear system of interpolation conditions (Newton-Padé approximation conditions) recursively. Since this last technique can also be used for scattered multivariate data exhibiting near-singularity, we describe the second approach in a separate paper [7]. Here we deal only with partially grid-structured data (satisfying the so-called rectangle rule or inclusion property). 相似文献
6.
The aim of this paper is to construct rational approximants for multivariate functions given by their expansion in an orthogonal polynomial system. This will be done by generalizing the concept of multivariate Padé approximation. After defining the multivariate Frobenius–Padé approximants, we will be interested in the two following problems: the first one is to develop recursive algorithms for the computation of the value of a sequence of approximants at a given point. The second one is to compute the coefficients of the numerator and denominator of the approximants by solving a linear system. For some particular cases we will obtain a displacement rank structure for the matrix of the system we have to solve. The case of a Tchebyshev expansion is considered in more detail. 相似文献
7.
In this paper, we describe a recursive method for computing interpolants defined in a space spanned by a finite number of continuous functions in Rd. We apply this method to construct several interpolants such as spline interpolants, tensor product interpolants and multivariate polynomial interpolants. We also give a simple algorithm for solving a multivariate polynomial interpolation problem and constructing the minimal interpolation space for a given finite set of interpolation points. 相似文献
8.
The compass identity (Wynn's five point star identity) for Padé approximants connects neighbouring elements called N, S, E, W and C in the Padé table. Its form has been extended to the cases of rational interpolation of ordinary (scalar) data and interpolation
of vector-valued data. In this paper, full specifications of the associated five point identity for the scalar denominator
polynomials and the vector numerator polynomials of the vector-valued rational interpolants on real data points are given,
as well as the related generalisations of Frobenius' identities. Unique minimal forms of the polynomials constituting the
interpolants and results about unattainable points correspond closely to their counterparts in the scalar case.
This revised version was published online in June 2006 with corrections to the Cover Date. 相似文献
9.
Jean-Paul Berrut 《Numerical Algorithms》2000,24(1-2):17-29
Among the representations of rational interpolants, the barycentric form has several advantages, for example, with respect to stability of interpolation, location of unattainable points and poles, and differentiation. But it also has some drawbacks, in particular the more costly evaluation than the canonical representation. In the present work we address this difficulty by diminishing the number of interpolation nodes embedded in the barycentric form. This leads to a structured matrix, made of two (modified) Vandermonde and one Löwner, whose kernel is the set of weights of the interpolant (if the latter exists). We accordingly modify the algorithm presented in former work for computing the barycentric weights and discuss its efficiency with several examples. 相似文献
10.
Celina Pestano-Gabino Concepción González-Concepción María Candelaria Gil-Fariña 《Journal of Computational and Applied Mathematics》2012
In this paper we define a type of matrix Padé approximant inspired by the identification stage of multivariate time series models considering scalar component models. Of course, the formalization of certain properties in the matrix Padé approximation framework can be applied to time series models and in other fields. Specifically, we want to study matrix Padé approximants as follows: to find rational representations (or rational approximations) of a matrix formal power series, with both matrix polynomials, numerator and denominator, satisfying three conditions: (a) minimum row degrees for the numerator and denominator, (b) an invertible denominator at the origin, and (c) canonical representation (without free parameters). 相似文献
11.
The Fitzpatrick algorithm, which seeks a Gr?bner basis for the solution of a system of polynomial congruences, can be applied to compute a rational interpolant. Based on the Fitzpatrick algorithm and the properties of an Hermite interpolation basis, we present a Neville-like algorithm for multivariate osculatory rational interpolation. It may be used to compute the values of osculatory rational interpolants at some points directly without computing the rational interpolation function explicitly. 相似文献
12.
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
13.
14.
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
15.
《Journal of Computational and Applied Mathematics》2002,148(2):341-348
In this paper, we give an algorithm for directly finding the denominator values of rational interpolants at the nodes, and present an expression for the corresponding rational interpolant when the latter exists. With these denominator values, our method also provides information concerning the existence of the interpolant and the presence of unattainable points and poles. 相似文献
16.
一种求二元有理插值函数的方法 总被引:11,自引:3,他引:8
给出一种方法可直接计算基于矩形节点的二元有理插值函数的分母在节点处的值 ,进而判断相应的二元有理插值函数是否存在 .此方法运用灵活 ,适用范围广 ,在相应的有理插值函数存在时 ,能给出它的具体表达式 .此外 ,我们还针对文中两个主要逆矩阵 ,给出了相应的递推公式 ,避免了求逆计算 . 相似文献
17.
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
18.
Note on Rational Interpolants 总被引:1,自引:0,他引:1
Tan Jieqing 《大学数学》1993,(3)
<正> In this note we present a constructive proof of symmetrical determinantal formulas forthe numerator and denominator of an ordinary rational interpolant,consider the confluencecase and give new determinantal formulas of the rational interpolant by means of Lagrange'sbasis functions. 相似文献
19.
Annie Cuyt 《BIT Numerical Mathematics》1986,26(1):71-79
Several definitions of multivariate Padé approximants have been introduced during the last decade. We will here consider all types of definitions based on the choice that the coefficients in numerator and denominator of the multivariate Padé approximant are defined by means of a linear system of equations. In this case a determinant representation for the multivariate Padé approximant exists. We will show that a general recursive algorithm can be formulated to compute a multivariate Padé approximant given by any definition of this type. Here intermediate results in the recursive computation scheme will also be multivariate Padé approximants. Up to now such a recursive computation of multivariate Padé approximants only seemed possible in some special cases. 相似文献
20.
ZhibingChen 《计算数学(英文版)》2003,21(2):157-166
A new kind of matrix-valued rational interpolants is recursively established by means of generalized Samelson iverse for matrices,with scalar numerator and matrix-valued denominatror.In this respect,it is essentially different form that of the previous works [7,9],where the matrix-valued rational interpolants is in Thiele-type continued fraction form with matrix-valued numerator and scalar denominator.For both univariate and bivariate cases,sufficient conditions for existence,characterisation and univquenese in some sense are proved respectively,and an error formula for the univariate interpolating function is also given.The results obtained in this paper are illustrated with some numerical examples. 相似文献