scholarly journals A continued fraction algorithm for the computation of higher transcendental functions in the complex plane

1967 ◽  
Vol 21 (97) ◽  
pp. 18-18 ◽  
Author(s):  
I. Gargantini ◽  
P. Henrici
Keyword(s):  
Complex Plane ◽  
Continued Fraction ◽  
Transcendental Functions ◽  
Fraction Algorithm

scholarly journals A continued fraction algorithm for real algebraic numbers

1972 ◽  
Vol 26 (119) ◽  
pp. 785-785 ◽  
Author(s):  
David G. Cantor ◽  
Paul H. Galyean ◽  
Horst G. Zimmer
Keyword(s):  
Continued Fraction ◽  
Algebraic Numbers ◽  
Fraction Algorithm

scholarly journals Truncation error bounds for branched continued fraction whose partial denominators are equal to unity

2020 ◽  
Vol 54 (1) ◽  
pp. 3-14
Author(s):  
R. I. Dmytryshyn ◽  
T. M. Antonova
Keyword(s):  
Complex Plane ◽  
Error Bounds ◽  
Continued Fraction ◽  
Truncation Error ◽  
Independent Variables ◽  
Continued Fraction Expansions ◽  
Branched Continued Fraction

The paper deals with the problem of obtaining error bounds for branched continued fraction of the form $\sum_{i_1=1}^N\frac{a_{i(1)}}{1}{\atop+}\sum_{i_2=1}^{i_1}\frac{a_{i(2)}}{1}{\atop+}\sum_{i_3=1}^{i_2}\frac{a_{i(3)}}{1}{\atop+}\ldots$. By means of fundamental inequalities method the truncation error bounds are obtained for the above mentioned branched continued fraction providing its elements belong to some rectangular sets ofa complex plane. Applications are considered for several classes of branched continued fraction expansions including the multidimensional \emph{S}-, \emph{A}-, \emph{J}-fractions with independent variables.

scholarly journals Complex Method on Octagonal Number

2020 ◽  
Vol 8 (4S5) ◽  
pp. 11-13
Keyword(s):  
Number Theory ◽  
Continued Fraction ◽  
Complex Method ◽  
Polygonal Numbers ◽  
Fraction Algorithm

In Number theory Study of polygonal numbers is rich in varity. In this paper we establish a Complex Octagonal Number using Continued Fraction algorithm.

A NEW SYMMETRIC TWO-DIMENSIONAL ALGORITHM

2006 ◽  
Vol 02 (04) ◽  
pp. 489-498
Author(s):  
PEDRO FORTUNY AYUSO ◽  
FRITZ SCHWEIGER
Keyword(s):  
Continued Fractions ◽  
Continued Fraction ◽  
Dual Graph ◽  
Plane Curve ◽  
Singularity Theory ◽  
Main Topic ◽  
Two Dimensional ◽  
Dimension 2 ◽  
Multidimensional Continued Fraction ◽  
Fraction Algorithm

Continued fractions are deeply related to Singularity Theory, as the computation of the Puiseux exponents of a plane curve from its dual graph clearly shows. Another closely related topic is Euclid's Algorithm for computing the gcd of two integers (see [2] for a detailed overview). In the first section, we describe a subtractive algorithm for computing the gcd of n integers, related to singularities of curves in affine n-space. This gives rise to a multidimensional continued fraction algorithm whose version in dimension 2 is the main topic of the paper.

scholarly journals The length of the continued fraction expansion for a class of rational functions in

1991 ◽  
Vol 34 (1) ◽  
pp. 7-17 ◽  
Author(s):  
Arnold Knopfmacher
Keyword(s):  
Finite Field ◽  
Continued Fraction ◽  
Rational Functions ◽  
Euclidean Algorithm ◽  
Continued Fraction Expansion ◽  
Partial Quotients ◽  
Fraction Algorithm

A study is made of the length L(h, k) of the continued fraction algorithm for h/k where h and k are co-prime polynomials in a finite field. In addition we investigate the sum of the degrees of the partial quotients in this expansion for h/k, h, k in . The above continued fraction is determined by means of the Euclidean algorithm for the polynomials h, k in .

scholarly journals Continued fraction algorithm for Sturmian colorings of trees

2017 ◽  
Vol 39 (9) ◽  
pp. 2541-2569
Author(s):  
DONG HAN KIM ◽  
SEONHEE LIM
Keyword(s):  
Continued Fraction ◽  
Vertex Coloring ◽  
Regular Tree ◽  
Sturmian Words ◽  
Number Of Classes ◽  
Fraction Algorithm

Factor complexity $b_{n}(\unicode[STIX]{x1D719})$ for a vertex coloring $\unicode[STIX]{x1D719}$ of a regular tree is the number of classes of $n$-balls up to color-preserving automorphisms. Sturmian colorings are colorings of minimal unbounded factor complexity $b_{n}(\unicode[STIX]{x1D719})=n+2$. In this article, we prove an induction algorithm for Sturmian colorings using colored balls in a way analogous to the continued fraction algorithm for Sturmian words. Furthermore, we characterize Sturmian colorings in terms of the data appearing in the induction algorithm.

scholarly journals Good’s theorem for Hurwitz continued fractions

2020 ◽  
Vol 16 (07) ◽  
pp. 1433-1447
Author(s):  
Gerardo Gonzalez Robert
Keyword(s):  
Hausdorff Dimension ◽  
Complex Plane ◽  
Continued Fractions ◽  
Continued Fraction ◽  
Analogous Result ◽  
Complex Numbers ◽  
Real Numbers ◽  
Dimension Formula

Good’s Theorem for regular continued fraction states that the set of real numbers [Formula: see text] such that [Formula: see text] has Hausdorff dimension [Formula: see text]. We show an analogous result for the complex plane and Hurwitz Continued Fractions: the set of complex numbers whose Hurwitz Continued fraction [Formula: see text] satisfies [Formula: see text] has Hausdorff dimension [Formula: see text], half of the ambient space’s dimension.

A continued fraction algorithm

1981 ◽  
Vol 37 (1) ◽  
pp. 149-156 ◽  
Author(s):  
P. Van der Cruyssen
Keyword(s):  
Continued Fraction ◽  
Fraction Algorithm
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