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Zh. Vychisl. Mat. Mat. Fiz., 2015, Volume 55, Number 4, Pages 558–573 (Mi zvmmf10183)  

Algorithm for summation of divergent continued fractions and some applications

G. A. Kirichenkoa, V. I. Shmoylovb

a Southern Federal University, per. Nekrasovskii 44, Taganrog, 347928, Russia
b Southern Scientific Center, Russian Academy of Sciences, pr. Chekhova 41, Rostov-on-Don, 344006, Russia

Abstract: The convergence of continued fractions is defined in a manner other than the conventional definition. A new summation method is used to determine the values of continued fractions and series that diverge in the classical sense. The method is applicable not only to ordinary continued fractions, but also to ones of other classes, for example, to Hessenberg continued fractions. As a result, an original algorithm for finding zeros of $n$th-degree polynomials is constructed. The $r/\varphi$-algorithm proposed is also used to solve infinite systems of linear algebraic equations.

Key words: high-degree algebraic equations, divergent continued fractions, infinite systems of linear algebraic equations, summation algorithm for divergent continued fractions.

DOI: https://doi.org/10.7868/S0044466915040146

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English version:
Computational Mathematics and Mathematical Physics, 2015, 55:4, 549–563

Bibliographic databases:

UDC: 519.651
MSC: Primary 30B70; Secondary 11A55, 40A15, 65B99
Received: 23.04.2013

Citation: G. A. Kirichenko, V. I. Shmoylov, “Algorithm for summation of divergent continued fractions and some applications”, Zh. Vychisl. Mat. Mat. Fiz., 55:4 (2015), 558–573; Comput. Math. Math. Phys., 55:4 (2015), 549–563

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  • Журнал вычислительной математики и математической физики Computational Mathematics and Mathematical Physics
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