The convergence of the generalised Selmer algorithm

The obvious way to compute the continued fraction of a real number α > 1 is to compute a very accurate numerical approximation of α, and then to iterate the well-known truncate-and-invert step which computes the next partial quotient a = bαc and the next complete quotient α′ = 1/(α − a). This method is called the basic method. In the course of this process precision is lost, and one has to take precautions to stop before the partial quotients become incorrect. Lehmer [6] gives a safe stopping-criterion, and a trick to reduce the amount of multi-length arithmetic. Schonhage [12] describes an algorithm for computing the greatest common divisor of u and v and the related continued fraction expansion of u/v in O(n log n log log n) steps if both u and v do not exceed 2. A disadvantage of this approach is that if we wish to extend the list of partial quotients computed from an initial approximation of α, we have to compute a more accurate initial approximation of α, compute the new complete quotient using this new approximation and the partial quotients already computed from the old approximation, and then extend the list of partial quotients using that new complete quotient (we notice that Shiu in [13, p.1312] incorrectly states that all the previous calculations have to be repeated, but the partial quotients computed so far don’t have to be recomputed). Bombieri and Van der Poorten [1], and Shiu [13] have recently proposed a remedy for this problem. They give a formula for computing a rational approximation of the next complete convergent from the first n partial quotients. From that complete convergent about n new partial quotients can be computed. So each step gives an approximate doubling of the number of partial quotients. To start the method, a few partial quotients have to be computed with the basic or indirect method. In [1] this method is proposed for algebraic numbers (which are zeros of polynomials) of degree ≥ 3, whereas Shiu also applies it to more general numbers, namely to transcendental numbers that can be defined as the zero of a function for which the logarithmic derivative at a rational point can be computed with arbitrary precision. This includes numbers like π, log π, and log 2. For each of thirteen different numbers, Shiu computes 10000 partial quotients. Their frequency distributions are compared with the one which almost all numbers should obey, according to the Khintchine– Levy theory [3, 7]. No significant deviations from this theory are reported. Shiu calls his method the direct method. Curiously, Shiu does not refer to what we would call the polynomial method for algebraic numbers [2, 5, 11] of degree ≥ 3, which computes the partial quotients of α using only the coefficients of its defining polynomial. Moreover, Shiu gives neither implementational

It has been a long-standing problem to find good symbolic codings for translations on the d -dimensional torus that enjoy the beautiful properties of Sturmian sequences like low factor complexity and good local discrepancy properties. Inspired by Rauzy’s approach we construct such codings in terms of multidimensional continued fraction algorithms that are realized by sequences of substitutions. In particular, given any exponentially convergent continued fraction algorithm, these sequences lead to renormalization schemes which produce symbolic codings of toral translations and bounded remainder sets at all scales in a natural way. The exponential convergence properties of a continued fraction algorithm can be viewed in terms of a Pisot type condition imposed on an attached symbolic dynamical system. Using this fact, our approach provides a systematic way to confirm purely discrete spectrum results for wide classes of symbolic dynamical systems. Indeed, as our examples illustrate, we are able to confirm the Pisot conjecture for many well-known families of sequences of substitutions. These examples include classical algorithms like the Jacobi–Perron, Brun, Cassaigne–Selmer, and Arnoux–Rauzy algorithms. As a consequence, we gain symbolic codings of almost all translations of the 2-dimensional torus having factor complexity 2n + 1 that are balanced for words, which leads to multiscale bounded remainder sets. Using the Brun algorithm, we also give symbolic codings of almost all 3-dimensional toral translations having multiscale bounded remainder sets.

In his paper "Multidimensional continued fractions" ( Ann. Univ. Sci. Budapest. Eötvös Sect. Math. , v. 13, 1970, pp. 113-140), G. Szekeres introduced a new higher dimensional analogue of the ordinary continued fraction expansion of a single real number. The Szekeres algorithm associates with each k -tuple ( α 1 , … , α k ) ({\alpha _1}, \ldots ,{\alpha _k}) of real numbers (satisfying 0 > α i > 1 0 > {\alpha _i} > 1 ) a sequence b 1 , b 2 , … {b_1}, {b_2}, \ldots of positive integers; this sequence is called a continued k -fraction, and for k = 1 it is just the sequence of partial quotients of the ordinary continued fraction for α 1 {\alpha _1} . A simple recursive procedure applied to b 1 , b 2 , … {b_1}, {b_2}, \ldots produces a sequence a ( n ) = ( A n ( 1 ) / B n , … , A n ( k ) / B n ) ( n = 1 , 2 , … ; A n ( i ) ⩾ 0 a(n) = (A_n^{(1)}/{B_n}, \ldots ,A_n^{(k)}/{B_n})\;(n = 1,2, \ldots ;A_n^{(i)} \geqslant 0 and B n > 0 {B_n} > 0 are integers) of simultaneous rational approximations to ( α 1 , … , α k ) ({\alpha _1}, \ldots ,{\alpha _k}) and a sequence c ( n ) = ( c n 0 , c n 1 , … , c n k ) ( n = 1 , 2 , … ) c(n) = ({c_{n0}},{c_{n1}}, \ldots ,{c_{nk}})\;(n = 1,2, \ldots ) of integer ( k + 1 ) (k + 1) -tuples such that the linear combination c n 0 + c n 1 α 1 + ⋯ + c n k α k {c_{n0}} + {c_{n1}}{\alpha _1} + \cdots + {c_{nk}}{\alpha _k} approximates zero. Szekeres conjectured, on the basis of extensive computations, that the sequence a ( 1 ) , a ( 2 ) , … a(1),a(2), \ldots contains all of the "best" simultaneous rational approximations to ( α 1 , … , α k ) ({\alpha _1}, \ldots ,{\alpha _k}) and that the sequence c ( 1 ) , c ( 2 ) , … c(1),c(2), \ldots contains all of the "best" approximations to zero by the linear form x 0 + x 1 α 1 + ⋯ + x n α n {x_0} + {x_1}{\alpha _1} + \cdots + {x_n}{\alpha _n} . For the special case k = 2 and α 1 = θ 2 − 1 , α 2 = θ − 1 {\alpha _1} = {\theta ^2} - 1,{\alpha _2} = \theta - 1 (where θ = 2 cos ⁡ ( 2 π / 7 ) \theta = 2\cos (2\pi /7) is the positive root of x 3 + x 2 − 2 x − 1 = 0 {x^3} + {x^2} - 2x - 1 = 0 , Szekeres further conjectured that the 2-fraction b 1 , b 2 , … {b_1},{b_2}, \ldots is "almost periodic" in a precisely defined sense. In this paper the Szekeres conjectures concerning best approximations to zero by the linear form x 0 + x 1 ( θ 2 − 1 ) + x 2 ( θ − 1 ) {x_0} + {x_1}({\theta ^2} - 1) + {x_2}(\theta - 1) and concerning almost periodicity for the 2-fraction of ( θ 2 − 1 , θ − 1 ) ({\theta ^2} - 1,\theta - 1) are proved. The method used can be applied to other pairs of cubic irrationals α 1 , α 2 {\alpha _1},{\alpha _2} .

Let q, fj, t2, be a sequence of elements of a field F. We give a 2 3 continued fraction algorithm for t^x + txx + t2x + .If our sequence satisfies a linear recurrence, then the continued fraction algorithm is finite and produces this recurrence.More generally the algorithm produces a nontrivial solution of the system s ',+/*/.0</<*-l, /=0 for every positive integer s. 1.Let t0, tx, t2, be a sequence of elements of a field F. Set T-\ttft./=0 Let d be a nonnegative integer.We say that T* is an approximation of T of degree d if there exist polynomials V and W such that T* = V/W, deg V < d, deg W < d, x\w, and x2d\WT -V.We shall give a continued fraction expansion for xT.This yields polynomials V-, W, and integers d-, 0 = dx < d2 < d3 < , such that (V, W) = 1 and V,/W-is an approximation of T of degree d.Suppose T* is any approximation of T of some degree d.Then T* = V/W for that value of i such that d < d<di+x.If the sequence of the t satisfies a linear recurrence of degree d, but not one of smaller degree, then there is an m such that dm = d and the linear recurrence is given by the polynomial Wm.In this case, WmT = Vm, the continued fraction expansion, terminates at i = m, and we can determine Wm from the first 2d of the t:, i.e., from those t such that 0 </ < 2d.Our algorithm is closely related to Zierler's version of Berlekamp's algorithm [1].2. We consider continued fraction expansions of the form a = A, +-l A, + 2 A3+-where A,, A2, A3, are elements from some field E. We can write

Motivated by problems in random continued fraction expansions, this research describes the design and implementation on θ-expansions of a real number, x in (0, θ) where 0 < θ < 1. Initially, for such a number θ, the convergent of θ-expansions is computed with comparison to the regular continued fraction (RCF)-convergent based on the samples of the approximated value, x. θ-expansions algorithm and continued fraction algorithm are applied in a Maple software to compute the θ-convergent and RCF-convergent. In this respect, the purpose of this research is to examine the growth rate of convergent for both expansions and provide the theoretical behaviours of θ-convergent in θ-expansions. Numerical results based on these samples show 3 conditions of convergence growth rate, which are C2 and C4 give the best decimal approximations in θ-expansions, whereas, Cn be the best approximations in RCF expansions. Therefore, this research reveals that various value of θ affect the performance on their growth rate of convergent for these two expansions. As θ approaching to 1, it reduces their convergence errors. The analysis of the experimental results reveals the RCF-convergent give a better performance compared to θ-convergent as its yield to less convergence errors. In addition, the value of convergent do converge to the approximated value, x and that is how the name of convergent comes.

The expected cell count for a 2 x 2 contingency table, governed by the noncentral (extended) hypergeometric distribution, is expressed as a terminating continued fraction. The coefficients in the continued fraction are better behaved than the multinomial coefficients required for the usual moment calculation. The expected cell count must be calculated repeatedly in a conditional maximum likelihood analysis of K2 x 2 contingency tables. Since the continued fraction can be easily evaluated, a rapid and numerically stable computational algorithm results. Once this first moment is known, higher moments can be obtained as shown by Harkness (1965, Annals of Mathematical Statistics 36, 938-945). A BASIC program to implement the continued fraction algorithm is given in an appendix.

Optimal continued fractions

Based on work done by Morse and Hedlund (1940) it was observed by Arnoux and Rauzy (1991) that the classical continued fraction algorithm provides a surprising link between arithmetic and diophantine properties of an irrational number $\alpha$, the rotation by $\alpha$ on the torus $\mathbb{T} = \mathbb{R}/\mathbb{Z}$, and combinatorial properties of the well known Sturmian sequences, a class of sequences on two letters with low subword complexity. It has been conjectured since the early 1990ies that this correspondence carries over to generalized continued fraction algorithms, rotations on higher dimensional tori, and so-called $S$-adic sequences generated by substitutions. The idea of working towards this generalization is known as Rauzy’s program. Although, starting with Rauzy (1982) a number of examples for such a generalization was devised, Cassaigne, Ferenczi, and Zamboni (2000) came up with a counterexample that showed the limitations of such a generalization. Nevertheless, recently Berthe, Steiner, and Thuswaldner (2016) made some further progress on Rauzy’s program and were able to set up a generalization of the above correspondences. They proved that the above conjecture is true under certain natural conditions. A prominent role in this generalization is played by tilings induced by generalizations of the classical Rauzy fractal introduced by Rauzy (1982). Another idea which is related to the above results goes back to Artin (1924), who observed that the classical continued fraction algorithm and its natural extension can be viewed as a Poincare section of the geodesic flow on the space $SL_2(\mathbb{Z}) \ SL_2(\mathbb{R})$. Arnoux and Fisher (2001) revisited Artin’s idea and showed that the above mentioned correspondence between continued fractions, rotations, and Sturmian sequences can be interpreted in a very nice way in terms of an extension of this geodesic flow which they called the scenery flow. Currently, Arnoux et al. are setting up elements of a generalization of this connection as well. It is the aim of my series of lectures to review the above results.

This is a bibliographical review that provides a brief glimpse into the beauty of continued fractions and how they can be very useful, both in obtaining good rational approximations of a given real number and in their representation as a continued fraction. Furthermore, it presents important properties, such as the relationship between quadratic irrationals and periodic continued fractions. On the other hand, with a view to a potential introduction of this topic in elementary education, a method for obtaining square root approximations through continued fractions is presented. Finally, using more advanced tools, we present an infinite continued fraction representation of the Euler number, which consequently implies the irrationality of e.

This paper extends, in a multi-dimensional framework, pattern recognition technics for generation or recognition of digital lines. More precisely, we show how the connection between chain codes of digital lines and continued fractions can be generalized by a connection between tilings and multi-dimensional continued fractions. This leads to a new approach for generating and recognizing digital hyperplanes.

Continued fraction techniques are equivalent to Berlekamp's algorithm. The sequence D(k), k \geq 0 , in Berlekamp's algorithm provides the information about when Berlekamp's algorithm completes one iterative step of the continued fraction. In fact, this happens when D(K) ; and when D(k) \neq D(k + 1) , it implies that Berlekamp's algorithm begins the next iterative step of the continued fraction.

Continued fractions (CFs) enable straightforward representation of elementary functions and rational approximations. We improve the positional algebraic algorithm, which computes homographic functions such as y equals ax+b/cx+d, given redundant continued fractions x,y, and integers a,b,c,d. The improved algorithm for the linear fractional transformation produces exact results, given regular continued fraction input. In case the input is in redundant continued fraction form, our improved linear algorithm increases the percentage of exact results with 12-bit state registers from 78% to 98%. The maximal error of non-exact results is improved from approximately 1 to 2<SUP>-8</SUP>. Indeed, by detecting a small number of cases, we can add a final correction step to improve the guaranteed accuracy of non-exact results. We refer to the fact that a few results may not be exact as 'Semi- Exact' arithmetic. We detail the adjustments to the positional algebraic algorithm concerning register overflow, the virtual singularities that occur during the computation, and the errors due to non-regular, redundant CF inputs.

This paper illustrates a TI N-Spire .tns file created by the author for generating continued fraction representations of real numbers and doing arithmetic with them. The continued fraction representation provides an alternative to the decimal representation. The .tns file can be used as tool for studying continued fractions and their properties as well as introducing the ideas of a best rational approximation to a real number and exploring some elementary ideas in error analysis. While the paper is a review of some of the classical works on continued factions, it does not present the material in a theorem/proof style. It motivates the results by citing examples and noting some interesting properties that will need to be verified. Notations are given to the references where exact and complete proofs of these facts can be found. The type of investigations done in this paper are motivational and the style of presentation is designed to pique students and teachers curiosity about the subject. Another purpose of this paper is to show that by looking at alternative representations of something that seems to be well known, such as the real numbers, can lead to exciting new and useful outcomes and applications.

The fractional order system (FOS) comprises fractional order operator. In order to obtain the discretized version of the fractional order system, the first step is to discretize the fractional order operator, commonly expressed as s?m, 0 &lt; m &lt; 1. The fractional order operator can be used as fractional order differentiator or integrator, depending upon the values of . In general, there are two approaches for discretization of fractional order operator, one is indirect method of discretization and another is direct method of discretization. The direct discretization method capitalizes the method of formation of generating function where fractional order operator s?m is expressed as a function of Z in the shift operator parameterization and continued fraction expansion (CFE) method is then utilized to get the corresponding discrete domain rational transfer function. There is an inherent problem with this discretization method using shift operator parameterization (discrete Z-domain). At fast sampling time, the discretized version of the continuous time operator or system should resemble that of the continuous time counterpart if the sampling theorem is satisfied. At very high sampling rate, the shift operator parameterized system fails to provide meaningful information due to its numerical ill conditioning. To overcome this problem, Delta operator parameterization for discretization is considered in this paper, where at fast sampling rate, the continuous time results can be obtained from the discrete time experiments and therefore a unified framework can be developed to get the discrete time results and continuous time results hand to hand. In this paper a new generating function is proposed to discretize the fractional order operator using the Gauss-Legendre 2-point quadrature rule. Additionally, the function has been expanded using the CFE in order to obtain rational approximation of the fractional order operator. The detailed mathematical formulations along with the simulation results in MATLAB, with different fractional order systems are considered, in order to prove the newness of this formulation for discretization of the FOS in complex Delta domain.

A simplex–karyon algorithm for expanding real numbers α = (α1,..., α d ) in multidimensional continued fractions is considered. The algorithm is based on a (d + 1)-dimensional superspace S with embedded hyperplanes: a karyon hyperplane K and a Farey hyperplane F. The approximation of numbers α by continued fractions is performed on the hyperplane F, and the degree of approximation is controlled on the hyperplane K. A local ℘(r)-strategy for constructing convergents is chosen, with a free objective function ℘(r) on the hyperplane K.