Convergence Of Continued Fractions Research Articles

On the convergence of periodic integral continued fractions with variable limits of integration

We establish estimates for the “tails” of periodic integral continued fractions with variable upper limits of integration. We prove a theorem on the uniform and absolute convergence of such fractions, and we obtain an estimate of their rate of convergence.

Convergence of two-dimensional continued fractions with complex elements

We prove an analog of Vorpits'kii's theorem for two-dimensional continued fractions, applying the formulas for computing their absolute error.

Une application du théorème ergodique sous-additif à la théorie métrique des fractions continues

For any irrational x ∈[0, 1] we denote by p n ( x )/ q n ( x ), n =1, 2, … the sequence of its continued fraction convergents and define θ n ( x ) ≔ q n | q n x − p n |. Also let T : [0, 1]→[0, 1] be defined by T (0)=0 and T ( x )=1/ x −[1/ x ] if x ≠0. For some random variables X 1 , X 2 , …, which are connected with the regular continued fraction expansion, the subadditive ergodic theorem yields to the existence of a function ω satisfying: for all z ∈ R , lim n→+∞ 1 n #{1⩽i⩽n/X i (z)⩽z}=ω(z) for almost everyx. In particular, for X n = θ n , using this study and a result of Knuth, we give another proof of the following conjecture of Lenstra (the first proof of this conjecture has been given by Bosma, Jager, and Wiedijk): for all z ∈[0, 1],[formula]for almost every x . Furthermore, for X n = θ n ∘ T n and X n =( q n −1 / q n )∘ T n , the functions ω are explicitly determined. The above results show that the subadditive ergodic theorem can be useful in the metric theory of continued fraction.

Shadowing by computable orbits of continued fraction convergents for algebraic numbers. II

We slate and prove a theorem on a recurrence relation satisfied by a sequence of functions (f λ(z))λ⩽2 such that f λ converges to a quadratic irrational with an order of convergence equal to k. We define recursively a sequence of functions (F λ(z))λ⩽1 that shadows orbits generated by successive approximations and that are identical to (f k (z))k ⩽1 quadratic irrationals. By defining F 2 as Newton's Map we show that F 3, is Halley's Map. We prove that members of the sequence (F λ(z))λ⩽2 have shadowing properties with order of convergence equal to k

Representations of Algebras Associated with a Möbius Transformation

Abstract The Hilbert space representations of a class of commutation relations associated with a Mobius transformation is studied using results on convergence of continued fractions.

Some remarks on a probability limit theorem for continued fractions

It is shown that if a certain condition on the variances of the partial sums is satisfied then a theorem of Philipp and Stout, which implies the asymptotic fluctuation results known for independent random variables, can be applied to some quantities related to continued fractions. Previous results on the behavior of the approximation by the continued fraction convergents to a random real number are improved.

Short Periods of Continued Fraction Convergents Modulo M: A Generalization of the Fibonacci Case

Short Periods of Continued Fraction Convergents Modulo M: A Generalization of the Fibonacci Case

Computation of limit periodic continued fractions. A survey

Over the last 20 years a large number of algorithms has been published to improve the speed and domain of convergence of continued fractions. In this survey we show that these algorithms are strongly related. Actually, they essentially boil down to two main principles.

Shadowing by computable orbits of continued fraction convergents for algebraic numbers

We state and prove several results on the shadowing of a subsequence of a computable orbit of a discrete dynamical system by computable orbits of other discrete dynamical systems. We define ∊-shadowing and asymptotic ∊-shadowing. By shadowing part of an orbit of a dynamical system by a second orbit of a different dynamical system we have removed the usual emphasis on phase. Our results compare sub-sequences of computable orbits that converge linearly to an algebraic number to computable orbits of Newton's and other methods which converge to the same algebraic number with orders of convergence greater than one. We study algebraic numbers and show relations between global complex analysis and number theory.

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.

On Symmetrical and Asymmetric Diophantine Approximation by Continued Fractions

Let (Pn/Qn)n ≥ 0 be the sequence of regular continued fraction convergents of the real irrational number x. Define the approximation constantsdn, n ≥ 0 by dn = dn(x) ≔ Qn + 1|Qnx − Pn|, n ≥ 0. In this paper the distribution for almost all x of the sequences (dn − 1, dn)n ≥ 1 and (dn − 1, dn, dn + 1)n ≥ 1 is discussed. Furthermore, two recent theorems by Jingcheng Tong concerning arithmetical properties of the dn′s are generalized.

A note on separate convergence for continued fractions

Let A n and B n denote the canonical numerators and denominators of a continued fraction K( a n / b n ). Let { t̃ n } be a tail sequence for an auxiliary continued fraction K( ã n / b̃ n ) close to K( a n / b n ). We give sufficient conditions for the convergence of the four sequences A n /Π n k=1 (− t̃ k ) and B n /Π n k=1 (− t̃ k ), ( A n + A n−1 t̃ n )/Π n k=1 ( b̃ k + t̃ k and ( B n + B n−1 t̃ n )/Π n k=1 ( b̃ k + t̃ k ).

An O(log n) algorithm for computing periodic continued fractions and its applications

An O(log n) algorithm for computing the nth convergents of periodic continued fractions is presented. It can be applied to solve the second-order linear recurrences with constant coeffients very efficiently. We also use it to approximate the quadratic numbers in O(log m) time for a result with m-digit precision. Our method can be thought as a generalization of the matrix-vector approach for computing the Fibonacci numbers. It is easy to implement because there are only some matrix multiplications and a division operation involved.

On the convergence of limit periodic continued fractions K(a n/1), wherea n→−1/4. Part III

A previous divergence criterion by A. Magnus and the author is generalized. We show that K(an/1) diverges (by oscillation) if an=−1/4−(c+en)/n(n+1), where c>1/16 and en e ℝ, |e|=O(1/logαn), α>2.

Convergence Acceleration for the Numerical Solution of Second-Order Linear Recurrence Relations

The concept of modification used for accelerating the convergence of ordinary continued fractions is adapted to the case of the Gautschi–Aggarwal–Burgmeier algorithm for the computation of nondominant solutions of nonhomogeneous second-order linear recurrence relations.

Some probabilistic results on the convergents of continued fractions

Some probabilistic results on the convergents of continued fractions

Approximation by mediants

The distribution is determined of some sequences that measure how well a number is approximated by its mediants (or intermediate continued fraction convergents). The connection with a theorem of Fatou, as well as a new proof of this, is given.

Approximation by Mediants

The distribution is determined of some sequences that measure how well a number is approximated by its mediants (or intermediate continued fraction convergents). The connection with a theorem of Fatou, as well as a new proof of this, is given.

Pringsheim's theorem revisited

In this paper we prove a generalization to higher-order linear recurrence relations of Pringsheim's theorem on the convergence of ordinary continued fractions.

Convergence acceleration of continued fractions of poincaré's type

This paper investigates the convergence properties of continued fractions of Poincaré's type with the aid of Pincherle's theorem, extends the investigation to the case where a tail is added to the approximants, and distinguishes between accelarative and decelarative tails. It shows that classical sequence transforms like θ or Levin/ u algorithms accelerate the sequence of the approximants modified by the optimal accelerative tail. A two-step procedure is deduced which performs the acceleration of the convergence of continued fractions of Poincaré's type.