The algorithms of constructing the continued fractions for any rations of the hypergeometric Gaussian functions

Manzii, O. S.; Gladun, Valeri V.; Вентик, Л.; Manziy, Oleksandra; Hladun, Volodymyr; Ventyk, L.

doi:10.23939/mmc2017.01.048

Cited by 8 publications

(7 citation statements)

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“…. Taking into account the conditions (24) and the inequalities ( 14), ( 15), (19), we obtain the estimate (25) the inequalities |ε (s) | < ε, s = 0, 1, 2, . .…”

Section: Examplementioning

confidence: 99%

Convergence sets and relative stability to perturbations of a branched continued fraction with positive elements

Hladun,

Bodnar,

Rusyn

2024

Carpathian Math. Publ.

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In the paper, the problems of convergence and relative stability to perturbations of a branched continued fraction with positive elements and a fixed number of branching branches are investigated. The conditions under which the sets of elements \[\Omega_0 = ( {0,\mu _0^{(2)}} ] \times [ {\nu _0^{(1)}, + \infty } ),\quad \Omega _{i(k)}=[ {\mu _k^{(1)},\mu _k^{(2)}} ] \times [ {\nu _k^{(1)},\nu _k^{(2)}} ],\]\[i(k) \in {I_k}, \quad k = 1,2,\ldots,\] where $\nu _0^{(1)}>0,$ $0 < \mu _k^{(1)} < \mu _k^{(2)},$ $0 < \nu _k^{(1)} < \nu _k^{(2)},$ $k = 1,2,\ldots,$ are a sequence of sets of convergence and relative stability to perturbations of the branched continued fraction \[\frac{a_0}{b_0}{\atop+}\sum_{i_1=1}^N\frac{a_{i(1)}}{b_{i(1)}}{\atop+}\sum_{i_2=1}^N\frac{a_{i(2)}}{b_{i(2)}}{\atop+}\ldots{\atop+} \sum_{i_k=1}^N\frac{a_{i(k)}}{b_{i(k)}}{\atop+}\ldots\] have been established. The obtained conditions require the boundedness or convergence of the sequences whose members depend on the values $\mu _k^{(j)},$ $\nu _k^{(j)},$ $j=1,2.$ If the sets of elements of the branched continued fraction are sets ${\Omega _{i(k)}} = ( {0,{\mu _k}} ] \times [ {{\nu _k}, + \infty } )$, $i(k) \in {I_k}$, $k = 0,1,\ldots,$ where ${\mu _k} > 0$, ${\nu _k} > 0$, $k = 0,1,\ldots,$ then the conditions of convergence and stability to perturbations are formulated through the convergence of series whose terms depend on the values $\mu _k,$ $\nu _k.$ The conditions of relative resistance to perturbations of the branched continued fraction are also established if the partial numerators on the even floors of the fraction are perturbed by a shortage and on the odd ones by an excess, i.e. under the condition that the relative errors of the partial numerators alternate in sign. In all cases, we obtained estimates of the relative errors of the approximants that arise as a result of perturbation of the elements of the branched continued fraction.

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“…. Taking into account the conditions (24) and the inequalities ( 14), ( 15), (19), we obtain the estimate (25) the inequalities |ε (s) | < ε, s = 0, 1, 2, . .…”

Section: Examplementioning

confidence: 99%

Convergence sets and relative stability to perturbations of a branched continued fraction with positive elements

Hladun,

Bodnar,

Rusyn

2024

Carpathian Math. Publ.

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“…Numerical aspects related to the backward recurrence algorithm for computing the approximants of continued fractions were considered in [7,9,27,28,30]. Some analogous results concerning branched continued fractions can be found in [19,21,22,25,31,32].…”

Section: Introductionmentioning

confidence: 99%

Numerical stability of the branched continued fraction expansion of Horn's hypergeometric function $H_4$

Dmytryshyn,

Cesarano,

Lutsiv

et al. 2024

Mat. Stud.

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In this paper, we consider some numerical aspects of branched continued fractions as special families of functions to represent and expand analytical functions of several complex variables, including generalizations of hypergeometric functions. The backward recurrence algorithm is one of the basic tools of computation approximants of branched continued fractions. Like most recursive processes, it is susceptible to error growth. Each cycle of the recursive process not only generates its own rounding errors but also inherits the rounding errors committed in all the previous cycles. On the other hand, in general, branched continued fractions are a non-linear object of study (the sum of two fractional-linear mappings is not always a fractional-linear mapping). In this work, we are dealing with a confluent branched continued fraction, which is a continued fraction in its form. The essential difference here is that the approximants of the continued fraction are the so-called figure approximants of the branched continued fraction. An estimate of the relative rounding error, produced by the backward recurrence algorithm in calculating an nth approximant of the branched continued fraction expansion of Horn’s hypergeometric function H4, is established. The derivation uses the methods of the theory of branched continued fractions, which are essential in developing convergence criteria. The numerical examples illustrate the numerical stability of the backward recurrence algorithm.

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“…In the 1976, D. Bodnar introduced and applied the functional BCF to approximate the functions of several variables [13,14]. The advantage of using BCF is a small accumulation of computational errors [15] and a wider convergence domain compared to the convergence domain of hypergeometric series [16]. Therefore, when we research the approximation of the multiple hypergeometric functions by the branched continued fraction, we should construct the expansion of the multiple hypergeometric function into BCF; investigate the convergence of this expansion; prove that the BCF converges to the function, which is an analytic continuation of a multiple hypergeometric function in some domain.…”

Section: Introductionmentioning

confidence: 99%

“…There are some new papers dealt with the problem of constructions and investigation of BCF expansions of special functions of several variables [15][16][17][18].…”

Section: Introductionmentioning

confidence: 99%

On convergence of function F4(1,2;2,2;z1,z2) expansion into a branched continued fraction

Hladun¹,

Hoyenko²,

Manziy³

et al. 2022

Math. Model. Comput.

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In the paper, the possibility of the Appell hypergeometric function F4(1,2;2,2;z1,z2) approximation by a branched continued fraction of a special form is analysed. The correspondence of the constructed branched continued fraction to the Appell hypergeometric function F4 is proved. The convergence of the obtained branched continued fraction in some polycircular domain of two-dimensional complex space is established, and numerical experiments are carried out. The results of the calculations confirmed the efficiency of approximating the Appell hypergeometric function F4(1,2;2,2;z1,z2) by a branched continued fraction of special form and illustrated the hypothesis of the existence of a wider domain of convergence of the obtained expansion.

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The algorithms of constructing the continued fractions for any rations of the hypergeometric Gaussian functions

Cited by 8 publications

References 0 publications

Convergence sets and relative stability to perturbations of a branched continued fraction with positive elements

Convergence sets and relative stability to perturbations of a branched continued fraction with positive elements

Numerical stability of the branched continued fraction expansion of Horn's hypergeometric function $H_4$

On convergence of function F4(1,2;2,2;z1,z2) expansion into a branched continued fraction

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