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Algorithm for summation of divergent continued fractions and some applications

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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 nth-degree polynomials is constructed. The r/φ-algorithm proposed is also used to solve infinite systems of linear algebraic equations.

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Authors and Affiliations

  1. Southern Federal University, per. Nekrasovskii 44, Taganrog, 347928, Russia

    G. A. Kirichenko

  2. Southern Scientific Center, Russian Academy of Sciences, pr. Chekhova 41, Rostov-on-Don, 344006, Russia

    V. I. Shmoylov

Authors
  1. G. A. Kirichenko
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  2. V. I. Shmoylov
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Correspondence to G. A. Kirichenko.

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Original Russian Text © G.A. Kirichenko, V.I. Shmoylov, 2015, published in Zhurnal Vychislitel’noi Matematiki i Matematicheskoi Fiziki, 2015, Vol. 55, No. 4, pp. 558–573.

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Kirichenko, G.A., Shmoylov, V.I. Algorithm for summation of divergent continued fractions and some applications. Comput. Math. and Math. Phys. 55, 549–563 (2015). https://doi.org/10.1134/S0965542515040132

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  • DOI: https://doi.org/10.1134/S0965542515040132

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