On the convergence of limit periodic continued fractions

On the convergence of limit periodic continued fractions by Lisa Jacobsen Download PDF EPUB FB2

Parusnikov V.I. () On the convergence of the multidimensional limit-periodic continued fractions. In: Gilewicz J., Pindor M., Siemaszko W. (eds) Rational Approximation and its Applications in Mathematics and Physics. Lecture Notes in Mathematics, vol Springer, Berlin, Heidelberg. First Online 10 September Jacobsen, L.

and Magnus, A., On the Convergence of Limit Periodic Continued Fractions K(a n /1), where a n → −1/4, Lecture Notes in Math. (), – Jacobsen L., Magnus A. () On the convergence of limit periodic continued fractions K(a n /1), where a 1 → −1/4.

In: Graves-Morris P.R., Saff E.B., Varga R.S. (eds) Rational Approximation and Interpolation. Lecture Notes in Mathematics, vol Springer, Berlin, Heidelberg. First Online 09 September On the Convergence of Limit-Periodic Continued Fractions.

We investigate the possibility of a new proof of the Lorentzen bestness theorem and gives a related convergence theorem together with a conjecture. We explore some connections between the limit-periodic continued fractions and other parts of mathematics and we give a few suggestions   Previous results on the convergence and divergence of K(a n/1).a n→−1/4, are generalized by constructing a sequence of reference continued fractions having explicit tails and associated chain sequences and then applying Pincherle's theorem together with a perturbation theory for solutions to the associated difference :// Journal of Computational and Applied Mathematics 19 () 75 North-Holland The linear convergence of limit periodic continued fractions C.

BREZINSKI and A. LEMBARKI UER IEEA, Laboratoire d Analyse Numique et d'Optimisation, Universitde Lille 1, Villeneuve d Ascq-Cedex, France Received 2 December Abstract: The only linearly convergent continued fractions are the limit periodic Key words: T+ transformation; Acceleration convergence; Limit periodic continued fraction 1.

The main result The T+ transformation is a sequence-to-sequence transformation method. InLembarki [3] obtained some good results for acceleration convergence of limit periodic continued fractions   The!inear convergence of limit periodic continued fractions C. BREZINSKI and A. LEMBARKI UER IEEA, Laboratoire d'Analyse Num&ique et d'Optimisation, Universit~ de Lille I, Villeneuve d'Ascq-Cedex, France Received 2 December Jacobsen, L.

and Magnus, A., On the convergence of limit periodic continued fractions K(a n /1), where a n →-1/4, Proceedings of a conference in TampaLecture Notes in MathematicsSpringer-Verlag (), – Google Scholar   The idea to write a Handbook of Continued fractions for Special functions originated more than 15 years ago, but the project only got started end of when a pair of Belgian and a pair of Norwegian authors agreed to join forces with the initiator W.B.

Jones. The book splits naturally Thus limit periodic continued fractions converge super linearly if and only if lim, ~ a, = 0.

In that case it is less crucial to be able to accelerate the convergence. (ii) If r = 1 then a = - This is the worth case since the convergence, when it occurs, is very slow (logarithmic convergence).

On the Convergence of Limit-Periodic Continued Fractions. By Nils Gaute Voll. Download PDF ( KB) Abstract We give a brief account of the general analytic theory of continued fractions and state and prove the Lorentzen bestness theorem.

We explore some connections between the limit-periodic continued fractions and other parts of In this paper, the composite sequence transformation is introduced for the approximant sequence of limit k periodic continued fractions.

Its convergence accle ration in the case of generalized Aitken's Δ 2—process is studied. Some results concerning   convergence of limit periodic fractions C. BREZINSKI and A. LEMBARKI UER IEEA, Luboratoire d’Anatjue NumPrique et d’Optimisation, Universitk de Lille I, Villeneuve d’Ascq-Cedex, France Received 2 December Abstract: The only linearly convergent continued fractions are the limit periodic ones.

Keywords: Continued :// On the Convergence of Limit-Periodic Continued Fractions. By Nils Gaute Voll. Abstract. We give a brief account of the general analytic theory of continued fractions and state and prove the Lorentzen bestness theorem.

We investigate the possibility of a new proof of the Lorentzen bestness theorem and gives a related convergence theorem Tang S,Zhu G Q On the acceleration convergence of limit periodic continued fractions by the T+m transformation, (荷兰)(SCI) ,No, 2.

朱功勤，唐烁 关于两种连分式加速收敛方法等价性的一般猜想的证明，数学研究与评论，，No.2,— Accelerating convergence of limit periodic continued fractionsK(an/1) It is shown that the convergence of limit periodic continued fractionsK(an/1) with liman=a can be substantially accelerated by replacing the sequence of approximations {Sn(0)} by the sequence {Sn(x1)},   Here one sees how continued fractions can be used to give better and better rational approximations to irrational numbers.

These and later results are closely connected with and supplement similar ideas developed in Niven's book, Numbers: Rational and Irrational. The periodic properties of continued fractions are discussed in Chapter ~sohum/ma/files/Continued   Infinite Continued Fractions. If is an infinite continued fraction, I want to define its value to be the limit of the convergents.

For this to make sense, I need to show that this limit exists. In what follows, take as given an infinite continued   2 Properties of Continued Fractions Finite Continued Fractions Rational Numbers Theorem Every rational number has a simple continued fraction expansion which is nite and every nite simple continued fraction expansion is a rational number.

Proof. Suppose we start with a rational number, then Euclid’s algorithm terminates in ~gautam/. Continued Fractions consists of two volumes — Volume 1: Convergence Theory; and Volume 2: Representation of Functions (tentative title), which is expected in Volume 1 is dedicated to the convergence and computation of continued fractions, while Volume 2 will treat representations of meromorphic functions by continued  › Mathematics › Algebra.

PERIODIC CONTINUED FRACTIONS JORDAN SCHETTLER Abstract. The goals of this project are to have the reader explore some of the basic properties of continued fractions and prove that 2R is a quadratic irrational i is equal to a periodic continued fraction.

1. Finite Continued Fractions Fix s= (a 0;(a 1;;a n)) 2Z Nn. The nite (simple) continued ~jcs/Journal of Computational and Applied Mathematics 19 () 67 North-Holland Successive modifications of limit periodic continued fractions C. BREZINSKI UER IEEA, Laboratoire d Analyse Numique et d'Optimisation, Universitde Lille 1, Villeneuve d Ascq-Cedex, France Received 2 December Abstract: It is well known that limit periodic continued fractions can be accelerated by