R. Dmytryshyn, C. Cesarano, I.-A.V. Lutsiv, M. Dmytryshyn
{"title":"Numerical stability of the branched continued fraction expansion of Horn's hypergeometric function $H_4$","authors":"R. Dmytryshyn, C. Cesarano, I.-A.V. Lutsiv, M. Dmytryshyn","doi":"10.30970/ms.61.1.51-60","DOIUrl":null,"url":null,"abstract":"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.","PeriodicalId":37555,"journal":{"name":"Matematychni Studii","volume":null,"pages":null},"PeriodicalIF":0.0000,"publicationDate":"2024-03-19","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":null,"platform":"Semanticscholar","paperid":null,"PeriodicalName":"Matematychni Studii","FirstCategoryId":"1085","ListUrlMain":"https://doi.org/10.30970/ms.61.1.51-60","RegionNum":0,"RegionCategory":null,"ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q3","JCRName":"Mathematics","Score":null,"Total":0}
引用次数: 0
Abstract
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.
在本文中,我们考虑了支链续分数的一些数值方面的问题,支链续分数是表示和扩展多个复变函数的解析函数的特殊函数族,包括超几何函数的广义。后向递推算法是计算支化连续分数近似值的基本工具之一。与大多数递推过程一样,它容易受到误差增长的影响。递归过程的每个循环不仅会产生自己的舍入误差,还会继承之前所有循环的舍入误差。另一方面,一般来说,分支续分数是一种非线性研究对象(两个分数线性映射之和并不总是分数线性映射)。在这项工作中,我们处理的是汇合支链续分数,它在形式上是一种续分数。这里的本质区别在于,续分数的近似值就是所谓的支链续分数的图近似值。在计算霍恩超几何函数 H4 的支链续分数展开的 n 次近似值时,建立了对后向递推算法所产生的相对舍入误差的估计。推导过程使用了支链续分数理论的方法,这些方法对于制定收敛标准至关重要。数值示例说明了后向递推算法的数值稳定性。