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.     

Newton-Thiele's rational interpolants

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.     

形式级数域中具有某种连分数展式集合的Hausdorff维数

本文研究了形式级数域中若干连分数例外集.利用质量分布原理和构造特殊覆盖,得到了当连分数展式部分商的度分别以多项式速度和指数速度趋向无穷大时,分别对应例外集的Hausdorff维数.     

Groups of <Emphasis Type="Italic">S</Emphasis>-Units and the Problem of Periodicity of Continued Fractions in Hyperelliptic Fields

We construct a theory of periodic and quasiperiodic functional continued fractions in the field k((h)) for a linear polynomial h and in hyperelliptic fields. In addition, we establish a relationship between continued fractions in hyperelliptic fields, torsion in the Jacobians of the corresponding hyperelliptic curves, and S-units for appropriate sets S. We prove the periodicity of quasiperiodic elements of the form \(\sqrt f /d{h^s}\), where s is an integer, the polynomial f defines a hyperelliptic field, and the polynomial d is a divisor of f; such elements are important from the viewpoint of the torsion and periodicity problems. In particular, we show that the quasiperiodic element \(\sqrt f \) is periodic. We also analyze the continued fraction expansion of the key element \(\sqrt f /{h^{g + 1}}\), which defines the set of quasiperiodic elements of a hyperelliptic field.     

A refined Gallai-Edmonds structure theorem for weighted matching polynomials

In this work, we prove a refinement of the Gallai-Edmonds structure theorem for weighted matching polynomials by Ku and Wong. Our proof uses a connection between matching polynomials and branched continued fractions. We also show how this is related to a modification by Sylvester of the classical Sturm's theorem on the number of zeros of a real polynomial in an interval. In addition, we obtain some other results about zeros of matching polynomials.     

Inversion polynomials for 321-avoiding permutations

We prove a generalization of a conjecture of Dokos, Dwyer, Johnson, Sagan, and Selsor giving a recursion for the inversion polynomial of 321-avoiding permutations. We also answer a question they posed about finding a recursive formula for the major index polynomial of 321-avoiding permutations. Other properties of these polynomials are investigated as well. Our tools include Dyck and 2-Motzkin paths, polyominoes, and continued fractions.     

一种新形式的二元混合有理插值(英文)

Newton＇s polynomial interpolation may be the favorite linear interpolation,associated continued fractions interpolation is a new type nonlinear interpolation.We use those two interpolation to construct a new kind of bivariate blending rational interpolants.Characteristic theorem is discussed.We give some new blending interpolation formulae.     

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.     

New convergence results for continued fractions generated by four-term recurrence relations

In the study of simultaneous rational approximation of functions using rational functions with a common denominator (which can be viewed as the ‘German polynomial’ problem in simultaneous Pade´approximation, cf. [1]) the quest for convergence results lead to the study of generalized continued fractions, a type of Jacobi—Perron algorithm [2,3]. It then becomes important to exploit the connection between the convergence of the generalized continued fraction and the solutions of the associated difference equation (cf. [4,5]).     

基于Samelson逆的向量值连分式的收敛准则

The aim of this work is to give some criteria on the convergence of vector valued continued fractions defined by Samelson inverse. We give a new approach to prove the convergence theory of continued fractions. First, by means of the modified classical backward recurrence relation, we obtain a formula between the m-th and n-th convergence of vector valued continued fractions. Second, using this formula, we give necessary and sufficient conditions for the convergence of vector valued continued fractions.     

有理反插值

在解决反插值问题时,本文首次利用Thiele型连分式有理插值,得到了两种十分有效的方法:函数插值的有理反插法和反函数的有理插值法,同多项式反插值相比有较好的效果.数值例子说明了在解代数方程时有理反插法优于多项式反插法.     

Continued fractions and the second Kepler law

In this paper we introduce a link between geometry of ordinary continued fractions and trajectories of points that moves according to the second Kepler law. We expand geometric interpretation of ordinary continued fractions to the case of continued fractions with arbitrary elements.     

二元矩阵有理插值函数的构造

一般构造矩阵值有理函数的方法是利用连分式给出的,其算法的可行性不易预知,且计算量大.本文对于二元矩阵值有理插值的计算,通过引入多个参数,定义一对二元多项式:代数多项式和矩阵多项式,利用两多项式相等的充分必要条件通过求解线性方程组确定参数,并由此给出了矩阵值有理插值公式.该公式简单,具有广阔的应用前景.     

Matrix-valued continued fractions

We discuss the properties of matrix-valued continued fractions based on Samelson inverse. We begin to establish a recurrence relation for the approximants of matrix-valued continued fractions. Using this recurrence relation, we obtain a formula for the difference between mth and nth approximants of matrix-valued continued fractions. Based on this formula, we give some necessary and sufficient conditions for the convergence of matrix-valued continued fractions, and at the same time, we give the estimate of the rate of convergence. This paper shows that some famous results in the scalar case can be generalized to the matrix case, even some of them are exact generalizations of the scalar results.     

Block Based Bivariate Blending Rational Interpolation via Symmetric Branched Continued Fractions

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.     

An algorithm of infinite sums representations and Tasoev continued fractions

For any given real number, its corresponding continued fraction is unique. However, given an arbitrary continued fraction, there has been no general way to identify its corresponding real number. In this paper we shall show a general algorithm from continued fractions to real numbers via infinite sums representations. Using this algorithm, we obtain some new Tasoev continued fractions.

     

Linear recurrence sequences and periodicity of multidimensional continued fractions

Multidimensional continued fractions generalize classical continued fractions with the aim of providing periodic representations of algebraic irrationalities by means of integer sequences. We provide a characterization for periodicity of Jacobi–Perron algorithm by means of linear recurrence sequences. In particular, we prove that partial quotients of a multidimensional continued fraction are periodic if and only if numerators and denominators of convergents are linear recurrence sequences, generalizing similar results that hold for classical continued fractions.     

连分式的不等式

本文研究二元连分式G(m,λ)的单调性质,得到一些新的连分式不等式．     

Snake graphs and continued fractions

This paper is a sequel to our previous work in which we found a combinatorial realization of continued fractions as quotients of the number of perfect matchings of snake graphs. We show how this realization reflects the convergents of the continued fractions as well as the Euclidean division algorithm. We apply our findings to establish results on sums of squares, palindromic continued fractions, Markov numbers and other statements in elementary number theory.     

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．