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本文首先利用Vandermonde矩阵得到矩形网格上二元多项式插值公式,然后利用该公式建立一类二元有理插值问题的存在性判别准则及有理插值函数的表现公式,并给出数值例子 相似文献
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矩形网格上一类二元有理插值问题 总被引:7,自引:0,他引:7
本首先利用Vandermonde矩阵得到矩形网格上二元多项式插值公式.然后利用该公式建立一类二元有理插值问题的存在性判别准则及有理插值函数的表现公式,并给出数值例于。 相似文献
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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
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二元切触有理插值是有理插值的一个重要内容,而降低其函数的次数和解决其函数的存在性是有理插值的一个重要问题.二元切触有理插值算法的可行性大都是有条件的,且计算复杂度较大,有理函数的次数较高.利用二元Hermite(埃米特)插值基函数的方法和二元多项式插值误差性质,构造出了一种二元切触有理插值算法并将其推广到向量值情形.较之其它算法,有理插值函数的次数和计算量较低.最后通过数值实例说明该算法的可行性是无条件的,且计算量低. 相似文献
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经慧芹 《纯粹数学与应用数学》2018,(1):15-25
针对传统连分式插值,计算复杂度高,计算过程中分母为零的不可预知性及插值函数不满足某些给定条件,应用不方便等问题,利用已知节点、函数值、导数值,构造两个多项式,分别作为有理插值函数的分子和分母,得出各阶导数条件下切触有理插值的新公式,并给出特殊情形的表达式.若添加适当的参数,可任意降低插值函数次数.该方法计算简洁,应用方便,插值函数的分母在节点处不为零且满足全部插值条件.数值例子验证了新方法的可行性、有效性和实用性. 相似文献
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Matrix-valued spherical functions related to the quantum symmetric pair for the quantum analogue of \((\mathrm{SU}(2) \times \mathrm{SU}(2), \mathrm{diag})\) are introduced and studied in detail. The quantum symmetric pair is given in terms of a quantised universal enveloping algebra with a coideal subalgebra. The matrix-valued spherical functions give rise to matrix-valued orthogonal polynomials, which are matrix-valued analogues of a subfamily of Askey–Wilson polynomials. For these matrix-valued orthogonal polynomials, a number of properties are derived using this quantum group interpretation: the orthogonality relations from the Schur orthogonality relations, the three-term recurrence relation and the structure of the weight matrix in terms of Chebyshev polynomials from tensor product decompositions, and the matrix-valued Askey–Wilson type q-difference operators from the action of the Casimir elements. A more analytic study of the weight gives an explicit LDU-decomposition in terms of continuous q-ultraspherical polynomials. The LDU-decomposition gives the possibility to find explicit expressions of the matrix entries of the matrix-valued orthogonal polynomials in terms of continuous q-ultraspherical polynomials and q-Racah polynomials. 相似文献
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In many applications it is of major interest to decide whether a given formal power series with matrix-valued coefficients of arbitrary dimensions results from a matrix-valued rational function. As the main result of this paper we provide an answer to this question in terms of Matrix Padé Approximants of the given power series. Furthermore, given a matrix rational function, the smallest degrees of the matrix polynomials which represent it are not necessarily unique. Therefore we study a certain minimality-type, that is, minimum degrees. We aim to obtain all the minimum degrees for the polynomials which represent the function as equivalents. In addition, given that the rational representation of the function for the same pair of degrees need not be unique, we have obtained conditions to study the uniqueness of said representation. All the results obtained are presented graphically in tables setting out the above information. They lead to a number of properties concerning special structures, staired blocks, in the Padé Table. 相似文献
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The relation between the spectral decomposition of a self-adjoint operator which is realizable as a higher order recurrence operator and matrix-valued orthogonal polynomials is investigated. A general construction of such operators from scalar-valued orthogonal polynomials is presented. Two examples of matrix-valued orthogonal polynomials with explicit orthogonality relations and three-term recurrence relation are presented, which both can be considered as 2×2-matrix-valued analogues of subfamilies of Askey–Wilson polynomials. 相似文献
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We give an analog of exceptional polynomials in the matrix-valued setting by considering suitable factorizations of a given second-order differential operator and performing Darboux transformations. Orthogonality and density of the exceptional sequence are discussed in detail. We give an example of matrix-valued exceptional Laguerre polynomials of arbitrary size. 相似文献
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Admissible slopes for monotone and convex interpolation 总被引:1,自引:0,他引:1
Summary In many applications, interpolation of experimental data exhibiting some geometric property such as nonnegativity, monotonicity or convexity is unacceptable unless the interpolant reflects these characteristics. This paper identifies admissible slopes at data points of variousC
1 interpolants which ensure a desirable shape. We discuss this question, in turn for the following function classes commonly used for shape preserving interpolations: monotone polynomials,C
1 monotone piecewise polynomials, convex polynomials, parametric cubic curves and rational functions. 相似文献
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The spectral decomposition for an explicit second-order differential operator T is determined. The spectrum consists of a continuous part with multiplicity two, a continuous part with multiplicity one, and a finite discrete part with multiplicity one. The spectral analysis gives rise to a generalized Fourier transform with an explicit hypergeometric function as a kernel. Using Jacobi polynomials, the operator T can also be realized as a five-diagonal operator, leading to orthogonality relations for 2×2-matrix-valued polynomials. These matrix-valued polynomials can be considered as matrix-valued generalizations of Wilson polynomials. 相似文献
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Rodica D. Costin 《Journal of Approximation Theory》2009,161(2):787-801
A hierarchy of matrix-valued polynomials which generalize the Jacobi polynomials is found. Defined by a Rodrigues formula, they are also products of a sequence of differential operators. Each class of polynomials is complete, satisfies a three term recurrence relation, integral inter-relations, and weak orthogonality relations. 相似文献
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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
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Hardin Douglas P. Hogan Thomas A. Sun Qiyu 《Advances in Computational Mathematics》2004,20(4):367-384
We use the matrix-valued Fejér–Riesz lemma for Laurent polynomials to characterize when a univariate shift-invariant space has a local orthonormal shift-invariant basis, and we apply the above characterization to study local dual frame generators, local orthonormal bases of wavelet spaces, and MRA-based affine frames. Also we provide a proof of the matrix-valued Fejér–Riesz lemma for Laurent polynomials. 相似文献
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《Journal of Computational and Applied Mathematics》1997,80(1):71-82
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. 相似文献