矩阵连分式序列的收敛性

本文研究了矩阵连分式的性质，获得了关于矩阵连分式序列收敛性的一些结果。     

一个二元矩阵插值连分式的展开式

本文借助于文[1]定义的一种实用的矩阵广义逆,构造了一个二元Stieltjes型矩阵值插值连分式的展开式,它的截断分式可以定义二元矩阵值插值函数.     

关于Clifford连分式的三项递推关系和 Pincherle''''s 定理

本文建立了Clifford连分式的三项递推关系和Pincherle's定理,并给出了它们的应用,也获得了关于Clifford连分式的矩阵递推关系的最小解的几个性质.     

基于广义逆的多元矩阵有理插值

本文借助于文[5]给出的一种矩阵广义逆，构造了二元Stieltjes型矩阵连分式的截断连分式，以此首次定义了平面上拟三角形网格上的二元矩阵有理插道值函数。文中给出了存在性的一个有用的判别条件。重要的特征定理和唯一性定理得到证明，并借助了实例说明了本文的结果。     

Thiele重心型矩阵值混合有理插值算法

利用Samelson型矩阵广义逆,构造了一种基于Thiele型连分式插值与重心有理插值的相结合的二元矩阵值混合有理插值格式,这种新的混合矩阵值有理插值函数继承了连分式插值和重心插值的优点,它的表达式简单,计算方便,数值稳定性好.该算法满足有理插值问题所给的插值条件,同时给出了误差估计分析.最后用数值算例验证了插值算法的有效性.     

多项式矩阵求逆的连分式插值方法

本文借助于基于广义逆矩阵Thiele-型连分式插值的计算公式，建立了多项式矩阵求逆的一个新方法。关于多项式矩阵求逆的一个实例给出以说明本文的结果。     

e~T的矩阵连分式

<正> 设矩阵幂级数e~T=I+T+(T~2/2!)+(T~3/3!)+…,其中T是任何方阵,现用下列方法确定e~T的矩阵连分式     

矩阵有理逼近

本文利用Ｓａｍｅｌｓｏｎ逆变换得到了方阵逆阵的一个新方法，以此建立了Ｔｈｉｅｌｅ型矩阵连分式的展开式，将Ｔｈｉｅｌｅ定理推广到矩阵上去．首次引入了矩阵有理逼近的概念，并讨论和证明了它的特征性和唯一性．最后，给出了逼近式的余项公式．     

Thiele型向量连分式的收敛性定理

Thiele型向量连分式,不仅可用来解决一元和多元向量有理插值问题[1-3],一元和多元向量切触有理插值问题[3],还可用来研究向量Pade逼近及向量连分式逼近[1,3]。本文给出了这种连分式的收敛性定理,并把著名的Pringsheim定理推广到向量连分式上去。     

基于二元连分式的散乱数据递推插值格式

本文首先基于新的非张量积型偏逆差商递推算法,分别构造奇数与偶数个插值节点上的二元连分式散乱数据插值格式,进而得到被插函数与二元连分式间的恒等式.接着,利用连分式三项递推关系式,提出特征定理来研究插值连分式的分子分母次数.然后,数值算例表明新的递推格式可行有效,同时,通过比较二元Thiele型插值连分式的分子分母次数,发现新的二元插值连分式的分子分母次数较低,这主要归功于节省了冗余的插值节点. 最后,计算此有理函数插值所需要的四则运算次数少于计算径向基函数插值.     

On Sufficient Conditions for Convergence of Two-Dimensional Continued Fractions

By the method of majorant fractions and equivalent transformations, the analogies of leszyski–Pringsheim criteria for two-dimensional continued fractions are obtained.     

关于两种连分式加速收敛方法等价性的一般猜想的证明

本文证明了［２］提出的两种连分式加速收敛方法等价性的一般猜想是正确的。     

BIVARIATE RATIONAL INTERPOLANTS WITHRECTANGLE-HOLE-STRUCTURE

1.IntroductionGivenasetofdistinctrealpoints{xi,i~0,1,2,',n:xiER}andasetofcomplexvectordata{d'),i=0,1,2,',n:n)ECd},Graves-Momsshowed[5]thatthevectorvaluedThieletypecontinuedfractioncanservetointerpolatethegivenvectors.TheconstructionprocessiscloselyralatedtotheadoptionoftheSamelsoninverseforvectorswhere7denotesthecomplexconjugateofvector6.ItwasprovedthatS(x)isavectorvaluedrationalfunctionwithnumeratorbeingad-dimensionalpolynomialofdegreenanddenominatorbeingapolynomialofdegree2[n/2],here…     

ALGORITHMS FOR LACUNARY VECTOR VALUED RATIONAL INTERPOLANTS

Efficient algorithms are established for the computation of bivariate lacunary vector valued rational interpolants based on the branched continued fractions and a numerical example is given to show how the algorithms are implemented,     

Singular values of the Rogers-Ramanujan continued fraction

Let z∊ C be imaginary quadratic in the upper half plane. Then the Rogers-Ramanujan continued fraction evaluated at q = e ^{2π iz} is contained in a class field of Q(z). Ramanujan showed that for certain values of z, one can write these continued fractions as nested radicals. We use the Shimura reciprocity law to obtain such nested radicals whenever z is imaginary quadratic. 2000 Mathematics Subject Classification Primary—11Y65; Secondary—11Y40     

COMPUTATION OF VECTOR VALUED BLENDING RATIONAL INTERPOLANTS

As we know, Newton's interpolation polynomial is based on divided differences which can be calculated recursively by the divided-difference scheme while Thiele 's interpolating continued fractions are geared towards determining a rational function which can also be calculated recursively by so-called inverse differences. In this paper, both Newton's interpolation polynomial and Thiele's interpolating continued fractions are incorporated to yield a kind of bivariate vector valued blending rational interpolants by means of the Samelson inverse. Blending differences are introduced to calculate the blending rational interpolants recursively, algorithm and matrix-valued case are discussed and a numerical example is given to illustrate the efficiency of the algorithm.     

VECTOR VALUED RATIONAL INTERPOLANTS BY TRIPLE BRANCHED CONTINUED FRACTIONS

Triple branched continued fractions (TBCFs) are constructed by means of well-define Thiele-type partial inverted differences. The characterizatioon theorem, uniqueness theorem andsome projection identity properties are obtained for vector valued rational interpolants hy TBCFs.