1. Rate of convergence of a

Journal of Computational and Applied Mathematics, 1987

It is well known that limit periodic continued fractions can be accelerated by modifications using converging factors. In this paper the repeated use of modifications is studied in the case of a constant converging factor and in the case of Aitken's A* process. Based on these modifications, a method for controlling the error is proposed.

1999

In this paper we give the relationship between the regular continued fraction and the Lehner fractions, using a procedure known as insertion. Starting from the regular continued fraction expansion of any real irrational x, when the maximal number of insertions is applied one obtains the Lehner fraction of x. Insertions (and singularizations) show h o w these (and other) continued fractions expansions are related. We will also investigate the relation between the Lehner fractions and the Farey expansion, and obtain the ergodic system underlying the Farey expansion. 2 1991 Mathematics Subject Classi cation. 28D05 (11K55).

In this article, the theory of continued fractions is presented. There aretwo types of continued fraction, one is the finite continued fraction and the other is the infinite continued fraction. A rational number can be expressed as a finite continued fraction. The value of an infinite continued fraction is an irrational number. The ratio of two successive Fibonacci numbers, which is a rational number, can be written as a simple finite continued fraction. The golden ratio can be expressed as an infinite continued fraction. The concept of golden ratio finds application in architecture. Using the convergents of finite continued fraction, the relation between Fibonacci numbers can be calculated and Linear Diophantine equations will be solved.

Journal of Computational and Applied Mathematics, 1988

In this note we compare two recent methods of convergence acceleration the first one introduced by Thron and Waadeland [13], and further developed by Jacobsen and Waadeland [4,5].

2000

The notion of 'bifurcating continued fractions' is introduced. Two coupled sequences of non-negative integers are obtained from an ordered pair of positive real numbers in a manner that generalizes the notion of continued fractions. These sequences enable simple representations of roots of cubic equations. In particular, remarkably simple and elegant 'bifurcating continued fraction' representations of Tribonacci and Moore numbers, the

2013

Abstract. If the equation of the title has an integer solution with k ≥ 2, then m> 109.3·106. This was the current best result and proved using a method due to L. Moser (1953). This approach cannot be improved to reach the benchmark m> 10107. Here we achieve m> 10109 by showing that 2k/(2m−3) is a convergent of log 2 and making an extensive continued fraction digits calculation of (log 2)/N, with N an appropriate integer. This method is very different from that of Moser. Indeed, our result seems to give one of very few instances where a large scale computation of a numerical constant has an application. 1.

Transactions of the American Mathematical Society, 1988

The main result in this paper is the proof of convergence acceleration for a suitable modification (as defined by de Bruin and Jacobsen) in the case of an n-fraction for which the underlying recurrence relation is of Perron-Kreuser type. It is assumed that the characteristic equations for this recurrence relation have only simple roots with differing absolute values.

2000

The notion of 'bifurcating continued fractions' is introduced. Two coupled sequences of non-negative integers are obtained from an ordered pair of positive real numbers in a manner that generalizes the notion of continued fractions. These sequences enable simple representations of roots of cubic equations. In particular, remarkably simple and elegant 'bifurcating continued fraction' representations of Tribonacci and Moore numbers, the cubic variations of the 'golden mean', are obtained. This is further generalized to associate m non-negative integer sequences with a set of m given real numbers so as to provide simple 'bifurcating continued fraction' representation of roots of polynomial equations of degree m+1.

On numerical stability of continued fractions, 2024

The paper considers the numerical stability of the backward recurrence algorithm (BR-algorithm) for computing approximants of the continued fraction with complex elements. The new method establishes sufficient conditions for the numerical stability of this algorithm and the error bounds of the calculation of the nth approximant of the continued fraction with complex elements. It follows from the obtained conditions that the numerical stability of the algorithm depends not only on the rounding errors of the elements and errors of machine operations but also on the value sets and the element sets of the continued fraction. The obtained results were used to study the numerical stability of the BR-algorithm for computing the approximants of the continued fraction expansion of the ratio of Horn’s confluent functions H7. Bidisc and bicardioid regions are established, which guarantee the numerical stability of the BR-algorithm. The obtained result is applied to the study of the numerical stability of computing approximants of the continued fraction expansion of the ratio of Horn’s confluent function H7 with complex parameters. In addition, the analysis of the relative errors arising from the computation of approximants using the backward recurrence algorithm, the forward recurrence algorithm, and Lenz’s algorithm is given. The method for studying the numerical stability of the BR-algorithm proposed in the paper can be used to study the numerical stability of the branched continued fraction expansions and numerical branched continued fractions with elements in angular and parabolic domains. © (2024), (VNTL Publishers). All rights reserved.

Atlantis Studies in Mathematics for Engineering and Science, 2008

Aims and scope of the series The series 'Atlantis Studies in Mathematics for Engineering and Science'(AMES) publishes high quality monographs in applied mathematics, computational mathematics, and statistics that have the potential to make a significant impact on the advancement of engineering and science on the one hand, and economics and commerce on the other. We welcome submission of book proposals and manuscripts from mathematical scientists worldwide who share our vision of mathematics as the engine of progress in the disciplines mentioned above. All books in this series are co-published with World Scientific.