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.     

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,     

The Borel—Bernstein Theorem for multidimensional continued fractions

A central result in the metric theory of continued fractions, the Borel—Bernstein Theorem gives statistical information on the rate of increase of the partial quotients. We introduce a geometrical interpretation of the continued fraction algorithm; then, using this set-up, we generalize it to higher dimensions. In this manner, we can define known multidimensional algorithms such as Jacobi—Perron, Poincaré, Brun, Rauzy induction process for interval exchange transformations, etc. For the standard continued fractions, partial quotients become return times in the geometrical approach. The same definition holds for the multidimensional case. We prove that the Borel—Bernstein Theorem holds for recurrent multidimensional continued fraction algorithms. Supported by a grant from the CNP _q -Brazil, 301456/80, and FINEP/CNP _q /MCT 41.96.0923.00 (PRONEX).     

On Schweiger’s problems on fully subtractive algorithms

In his monograph [6] on multi-dimensional continued fractions, Schweiger has presented two conjectures on fully subtractive algorithms. We affirm one and refute another.     

Exact arithmetic on the Stern–Brocot tree

In this paper we present the Stern–Brocot tree as a basis for performing exact arithmetic on rational numbers. There exists an elegant binary representation for positive rational numbers based on this tree [Graham et al., Concrete Mathematics, 1994]. We will study this representation by investigating various algorithms to perform exact rational arithmetic using an adaptation of the homographic and the quadratic algorithms that were first proposed by Gosper for computing with continued fractions. We will show generalisations of homographic and quadratic algorithms to multilinear forms in n variables. Finally, we show an application of the algorithms for evaluating polynomials.     

Computation of poles of two-point Padé approximants and their limits

Algorithms are developed to compute simultaneously the poles of functions represented by continued fractions whose approximants lie on the main diagonal of two-point Padé tables. The algorithms are based on equations recently developed by McCabe and Murphy for producing continued fraction expansions of a pair of power series. Sufficient conditions are given to ensure that the computations can be carried out and that the resulting approximations converge geometrically to the desired poles. As a by-product, an algorithm is obtained for computing zeros of polynomials. The theory and methods are illustrated by means of numerical examples.     

Discussion of algorithms for the computation of generalized continued fractions

Four algorithms for the computation of convergents of generalized continued fractions are defined and studied with respect to numerical effort, error propagation, and practical aspects. Some conclusions from numerical tests are deduced.     

Evaluation of Branched Continued Fractions using Block-Tridiagonal Linear Systems

The convergent of an ordinary continued fraction can be computedby solving a tridiagonal linear system for its first unknown.In this paper, this approach is generalized to branched continuedfractions, and it is shown how the convergent of a branchedcontinued fraction can be considered as the first unknown ofa block-tridiagonal linear system. Hence algorithms for thesolution of such systems of equations can be used for the computationof convergents of branched continued fractions, which have applicationsin approximation theory, systems theory, etc. In future research,special attention will be paid to the use of parallel algorithms.     

On the Khintchine Constant

We present rapidly converging series for the Khintchine constant and for general ``Khintchine means' of continued fractions. We show that each of these constants can be cast in terms of an efficient free-parameter series, each series involving values of the Riemann zeta function, rationals, and logarithms of rationals. We provide an alternative, polylogarithm series for the Khintchine constant and indicate means to accelerate such series. We discuss properties of some explicit continued fractions, constructing specific fractions that have limiting geometric mean equal to the Khintchine constant. We report numerical evaluations of such special numbers and of various Khintchine means. In particular, we used an optimized series and a collection of fast algorithms to evaluate the Khintchine constant to more than 7000 decimal places.

     

The origins of Euler's early work on continued fractions

In this paper, I examine Euler's early work on the elementary properties of continued fractions in the 1730s, and investigate its possible links to previous writings on continued fractions by authors such as William Brouncker. By analysing the content of Euler's first paper on continued fractions, ‘De fractionibus continuis dissertatio’ (1737, published 1744) I conclude that, contrary to what one might expect, Euler's work on continued fractions initially arose not from earlier writings on continued fractions, but from a wish to solve the Riccati differential equation.     

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.     

Continued fractions associated with the Newton-Padé table

Summary We discuss first the block structure of the Newton-Padé table (or, rational interpolation table) corresponding to the double sequence of rational interpolants for the data{(z _k, h(z_k)} _k =0. (The (m, n)-entry of this table is the rational function of type (m,n) solving the linearized rational interpolation problem on the firstm+n+1 data.) We then construct continued fractions that are associated with either a diagonal or two adjacent diagonals of this Newton-Padé table in such a way that the convergents of the continued fractions are equal to the distinct entries on this diagonal or this pair of diagonals, respectively. The resulting continued fractions are generalizations of Thiele fractions and of Magnus'sP-fractions. A discussion of an some new results on related algorithms of Werner and Graves-Morris and Hopkins are also given.Dedicated to the memory of Helmut Werner (1931–1985)     

Convergence of matrix continued fractions

The aim of this work is to give some criteria on the convergence of matrix continued fractions. We begin by presenting some new results which generalize the links between the convergent elements of real continued fractions. Secondly, we give necessary and sufficient conditions for the convergence of continued fractions of matrix arguments. This paper will be completed by illustrating the theoretical results with some examples.     

Two types of explicit continued fractions

Summary Two types of explicit continued fractions are presented. The continued fractions of the first type include those discovered by Shallit in 1979 and 1982, which were later generalized by Pethő. They are further extended here using Peth\H o's method. The continued fractions of the second type include those whose partial denominators form an arithmetic progression as expounded by Lehmer in 1973. We give here another derivation based on a modification of Komatsu's method and derive its generalization. Similar results are also established for continued fractions in the field of formal series over a finite base field.     

A note on extending Euler's connection between continued fractions and power series

Euler's Connection describes an exact equivalence between certain continued fractions and power series. If the partial numerators and denominators of the continued fractions are perturbed slightly, the continued fractions equal power series plus easily computed error terms. These continued fractions may be integrated by the series with another easily computed error term.     

Division algorithms for continued fractions and the Padé table

It is known [26] that the Viskovatoff algorithm can be generalized to cover the computation of continued fractions whose successive convergents form the Padé approximants of a descending staircase or diagonal, even in the case of a non-normal Padé table. It is the intention of the author to generalize this idea to other paths of the Padé table and in this way link together some algorithms scattered in literature.     

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.     

Solution formulas for linear difference equations with applications to continued fractions

For any system of linear difference equations of arbitrary order, a family of solution formulas is constructed explicitly by way of relating the given system to simpler neighboring systems. These formulas are then used to investigate the asymptotic behavior of the solutions. When applying this difference equation method to second-order equations that belong to neighboring continued fractions, new results concerning convergence of continued fractions as well as meromorphic extension of analytic continued fractions beyond their convergence region are provided. This is demonstrated for analytic continued fractions whose elements tend to infinity. Finally, a recent result on the existence of limits of solutions to real difference equations having infinite order is extended to complex equations.     

A class of algorithms for obtaining rational approximants to functions which are defined by power series

The problem of converting power series to different types of continued fractions is treated by demonstrating the generality of application of an often neglected class of algorithms. As an example, a new expansion is obtained for the gamma function.

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Zusammenfassung Die Arbeit behandelt die Umwandlung von Potenzreihen in entsprechende Kettenbrüche. Es wird die allgemeine Anwendbarkeit einer häufig unberücksichtigten Klasse von Algorithmen gezeigt. Als Anwendung wird eine neue Entwicklung der Gammafunktion hergeleitet.

     

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.