scholarly journals The Difference Between the Hurwitz Continued Fraction Expansions of a Complex Number and its Rational Approximations

Fractals ◽  
2021 ◽  
Author(s):  
Yubin He ◽  
Ying Xiong
Keyword(s):  
Complex Number ◽  
Continued Fraction ◽  
Rational Approximations ◽  
The Difference ◽  
Continued Fraction Expansions

Reversibility, Continued Fractions, and Infinite Meander Permutations of Planar Homoclinic Orbits in Linear Hyperbolic Anosov Maps

2014 ◽  
Vol 24 (08) ◽  
pp. 1440008
Author(s):  
Bernold Fiedler
Keyword(s):  
Continued Fractions ◽  
Continued Fraction ◽  
Singularity Theory ◽  
Global Attractors ◽  
Homoclinic Orbits ◽  
Stable And Unstable Manifolds ◽  
Unstable Manifolds ◽  
The Difference ◽  
Continued Fraction Expansions

Meander permutations have been encountered in the context of Gauss words, singularity theory, Sturm global attractors, plane Cartesian billiards, and Temperley–Lieb algebras, among others. In this spirit, we attempt to investigate the difference of orderings of homoclinic orbits on the stable and unstable manifolds of a planar saddle. As an example, we consider reversible linear Anosov maps on the 2-torus, and their relation to continued fraction expansions.

scholarly journals Rational Approximations for Irrational Functions using Collocating Continued Fraction Expansions

2021 ◽  
Author(s):  
philip olivier
Keyword(s):  
Complex Variable ◽  
Continued Fraction ◽  
Rational Approximations ◽  
Mathematical Tool ◽  
Continued Fraction Expansion ◽  
Continued Fraction Expansions ◽  
Irrational Function

<div> <div> <div> <p>This paper is motivated by the need in certain engineering contexts to construct approximations for irrational functions in the complex variable z. The main mathematical tool that will be used is a special continued fraction expansion that cause the rational approximant to collocate the irrational function at specific important values of z. This paper introduces two theorems that facilitate the construction of the rational approxima- tion. </p> </div> </div> </div>

scholarly journals Rational Approximations for Irrational Functions using Collocating Continued Fraction Expansions

2021 ◽  
Author(s):  
philip olivier
Keyword(s):  
Complex Variable ◽  
Continued Fraction ◽  
Rational Approximations ◽  
Mathematical Tool ◽  
Continued Fraction Expansion ◽  
Continued Fraction Expansions ◽  
Irrational Function

<div> <div> <div> <p>This paper is motivated by the need in certain engineering contexts to construct approximations for irrational functions in the complex variable z. The main mathematical tool that will be used is a special continued fraction expansion that cause the rational approximant to collocate the irrational function at specific important values of z. This paper introduces two theorems that facilitate the construction of the rational approxima- tion. </p> </div> </div> </div>

On Newton's Method and Rational Approximations to Quadratic Irrationals

2004 ◽  
Vol 47 (1) ◽  
pp. 12-16
Author(s):  
Edward B. Burger
Keyword(s):  
Differentiable Function ◽  
Newton’S Method ◽  
Newton's Method ◽  
Continued Fraction ◽  
Analogous Result ◽  
Golden Ratio ◽  
Fibonacci Number ◽  
Rational Approximations ◽  
Initial Value ◽  
Continued Fraction Expansions

AbstractIn 1988 Rieger exhibited a differentiable function having a zero at the golden ratio (−1 + )/2 for which when Newton's method for approximating roots is applied with an initial value x0 = 0, all approximates are so-called “best rational approximates”—in this case, of the form F2n/F2n+1, where Fn denotes the n-th Fibonacci number. Recently this observation was extended by Komatsu to the class of all quadratic irrationals whose continued fraction expansions have period length 2. Here we generalize these observations by producing an analogous result for all quadratic irrationals and thus provide an explanation for these phenomena.

scholarly journals A Lochs-Type Approach via Entropy in Comparing the Efficiency of Different Continued Fraction Algorithms

Mathematics ◽  
2021 ◽  
Vol 9 (3) ◽  
pp. 255
Author(s):  
Dan Lascu ◽  
Gabriela Ileana Sebe
Keyword(s):  
Dynamical Systems ◽  
Continued Fractions ◽  
Continued Fraction ◽  
Unit Interval ◽  
Probability Measures ◽  
Absolutely Continuous ◽  
Continued Fraction Expansions ◽  
Absolutely Continuous Invariant

We investigate the efficiency of several types of continued fraction expansions of a number in the unit interval using a generalization of Lochs theorem from 1964. Thus, we aim to compare the efficiency by describing the rate at which the digits of one number-theoretic expansion determine those of another. We study Chan’s continued fractions, θ-expansions, N-continued fractions, and Rényi-type continued fractions. A central role in fulfilling our goal is played by the entropy of the absolutely continuous invariant probability measures of the associated dynamical systems.

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