scholarly journals On convergence $(2,1,\dots,1)$-periodic branched continued fraction of the special form

2015 ◽  
Vol 7 (2) ◽  
pp. 148-154
Author(s):  
D.I. Bodnar ◽  
M.M. Bubniak
Keyword(s):  
Error Bounds ◽  
Special Form ◽  
Continued Fraction ◽  
Truncation Error ◽  
Branched Continued Fraction

$(2,1,\dots,1)$-periodic branched continued fraction of the special form is defined. Conditions of convergence are established for 2-periodic continued fraction and $(2,1,\dots,1)$-periodic branched continued fraction of the special form. Truncation error bounds are estimated for these fractions under additional conditions.

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.

Error Bounds of Continued Fractions for Complex Transport Coefficients and Spectral Functions

1978 ◽  
Vol 33 (11) ◽  
pp. 1380-1382 ◽  
Author(s):  
P. Hänggi
Keyword(s):  
Error Bounds ◽  
Continued Fractions ◽  
Continued Fraction ◽  
Truncation Error ◽  
Transport Coefficients ◽  
Power Spectra ◽  
A Posteriori ◽  
Spectral Functions ◽  
Dynamical Equilibrium ◽  
Complex Continued Fractions

We study the calculation of complex transport coefficients x (ω) and power spectra in terms of complex continued fractions. In particular, we establish classes of dynamical equilibrium and non-equilibrium systems for which we can obtain a posteriori bounds for the truncation error | x (ω) - x(n)(ω)| ≦ c (ω) | x(n)(ω) - x(n-1)(ω)| when the transport coefficient is approximated by its n-th continued fraction approximant x(n)(ω).

scholarly journals A Worpitzky boundary theorem for branched continued fractions of the special form

2016 ◽  
Vol 8 (2) ◽  
pp. 272-278 ◽  
Author(s):  
Kh.Yo. Kuchminska
Keyword(s):  
Continued Fractions ◽  
Special Form ◽  
Continued Fraction ◽  
Value Set ◽  
Limit Value ◽  
Branched Continued Fraction ◽  
Element Set

For a branched continued fraction of a special form we propose the limit value set for the Worpitzky-like theorem when the element set of the branched continued fraction is replaced by its boundary.

scholarly journals Some properties of branched continued fractions of special form

2015 ◽  
Vol 7 (1) ◽  
pp. 72-77
Author(s):  
R.I. Dmytryshyn
Keyword(s):  
Uniform Convergence ◽  
Continued Fractions ◽  
Special Form ◽  
Continued Fraction ◽  
Positive Definite ◽  
Branched Continued Fraction

The fact that the values of the approximates of the positive definite branched continued fraction of special form are all in a certain circle is established for the certain conditions. The uniform convergence of branched continued fraction of special form, which is a particular case of the mentioned fraction, in the some limited parabolic region is investigated.

scholarly journals On the convergence of multidimensional regular C-fractions with independent variables

2018 ◽  
Vol 26 (1) ◽  
pp. 18 ◽  
Author(s):  
R.I. Dmytryshyn
Keyword(s):  
Power Series ◽  
Continued Fractions ◽  
Continued Fraction ◽  
Open Disk ◽  
Efficient Tool ◽  
Independent Variables ◽  
Multidimensional Generalization ◽  
Multiple Power ◽  
Multiple Power Series ◽  
Branched Continued Fraction

In this paper, we investigate the convergence of multidimensional regular С-fractions with independent variables, which are a multidimensional generalization of regular С-fractions. These branched continued fractions are an efficient tool for the approximation of multivariable functions, which are represented by formal multiple power series. We have shown that the intersection of the interior of the parabola and the open disk is the domain of convergence of a multidimensional regular С-fraction with independent variables. And, in addition, we have shown that the interior of the parabola is the domain of convergence of a branched continued fraction, which is reciprocal to the multidimensional regular С-fraction with independent variables.

scholarly journals DISCRETE LAPLACE–BELTRAMI OPERATOR ON SPHERE AND OPTIMAL SPHERICAL TRIANGULATIONS

2006 ◽  
Vol 16 (01) ◽  
pp. 75-93 ◽  
Author(s):  
GUOLIANG XU
Keyword(s):  
Error Bounds ◽  
Truncation Error ◽  
Beltrami Operator ◽  
Discrete Operator ◽  
Convergence Properties

In this paper we first modify a widely used discrete Laplace-Beltrami operator proposed by Meyer et al over triangular surfaces, and then we show that the modified discrete operator has some convergence properties over the triangulated spheres. A sequence of spherical triangulations which is optimal in certain sense and leads to smaller truncation error of the discrete Laplace-Beltrami operator is constructed. Optimal hierarchical spherical triangulations are also given. Truncation error bounds of the discrete Laplace-Beltrami operator over the constructed triangulations are provided.

scholarly journals On truncations for weakly ergodic inhomogeneous birth and death processes

2014 ◽  
Vol 24 (3) ◽  
pp. 503-518 ◽  
Author(s):  
Alexander Zeifman ◽  
Yacov Satin ◽  
Victor Korolev ◽  
Sergey Shorgin
Keyword(s):  
Error Bounds ◽  
Truncation Error ◽  
Time Error ◽  
Birth And Death Processes ◽  
Intensity Matrix ◽  
Intensity Functions ◽  
Arbitrary Intensity ◽  
Reduced Intensity

Abstract We investigate a class of exponentially weakly ergodic inhomogeneous birth and death processes. We consider special transformations of the reduced intensity matrix of the process and obtain uniform (in time) error bounds of truncations. Our approach also guarantees that we can find limiting characteristics approximately with an arbitrarily fixed error. As an example, we obtain the respective bounds of the truncation error for an Mt/Mt/S queue for any number of servers S. Arbitrary intensity functions instead of periodic ones can be considered in the same manner.

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