Journal of Mathematical Sciences

, Volume 215, Issue 1, pp 11–25 | Cite as

Some Sets of Relative Stability Under Perturbations of Branched Continued Fractions with Complex Elements and a Variable Number of Branches

  • V. R. Hladun
Article
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The present paper deals with the investigation of the conditions under which infinite branched continued fractions are stable under perturbations of their elements. We establish the formulas for the relative errors of the approximants of branched continued fractions with complex partial denominators and numerators that are equal to one. By using the technique of the sets of elements and the corresponding sets of values of the tails of approximants, we construct the sets of relative stability under perturbations, namely, the angular sets and the sets representing the exterior domains of circles on the even floors of the fraction and half planes on its odd floors. We also establish estimates for the relative errors of approximants of these branched continued fractions.

Keywords

Relative Error Relative Stability Recurrence Relation Continue Fraction Exterior Domain 
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References

  1. 1.
    T. M. Antonova and V. R. Hladun, “Sufficient conditions of convergence and stability of branched continued fractions with alternating partial numerators,” Mat. Metody Fiz.-Mekh. Polya, 47, No. 4, 27–35 (2004).MathSciNetMATHGoogle Scholar
  2. 2.
    D. I. Bodnar, Branched Continued Fractions [in Russian], Naukova Dumka, Kiev (1986).Google Scholar
  3. 3.
    D. I. Bodnar, H. Waadeland, Kh. Yo. Kuchmins’ka, and O. M. Sus’, “On the stability of branched continued fractions,” Mat. Metody Fiz.-Mekh. Polya, Issue 37 (1994), pp. 3–7.Google Scholar
  4. 4.
    D. I. Bodnar and V. R. Hladun, “Some domains of stability of branched continued fractions with complex elements under perturbations,” in: Nauk. Visnyk Cherniv. Univ. [in Ukrainian], Ser. Matematyka, Issue 288 (2006), pp. 18–27.Google Scholar
  5. 5.
    D. I. Bodnar and V. R. Hladun, “On the stability of branched continued fractions with complex elements under perturbations,” Mat. Studii, 25, No. 2, 207–212 (2006).Google Scholar
  6. 6.
    V. R. Hladun, “Sets of absolute stability of branched continued fractions with real elements under perturbations,” in: Visnyk “L’vivs’ka Politekhnika” Nats. Univ. [in Ukrainian], Ser. Fiz.-Mat. Nauk, No. 768 (2013), pp. 63–70.Google Scholar
  7. 7.
    W. B. Jones and W. J. Thron, Continued Fractions: Analytic Theory and Applications, Addison-Wesley, Reading, MA (1980).MATHGoogle Scholar
  8. 8.
    Kh. Yo. Kuchmins’ka, Two-Dimensional Continued Fractions [in Ukrainian], Inst. Appl. Probl. Mech. Math., Lviv (2010).Google Scholar
  9. 9.
    V. Ya. Skorobogat’ko, Theory of Branched Continued Fractions and Its Application in Computational Mathematics [in Russian], Nauka, Moscow (1983).Google Scholar
  10. 10.
    F. Backeljauw, S. Becuwe, and A. Cuyt, “Validated evaluation of special mathematical functions,” Lect. Notes Comput. Sci., 5144, 206–216 (2008).CrossRefMATHGoogle Scholar
  11. 11.
    G. Blanch, “Numerical evaluation of continued fractions,” SIAM Rev., 6, 383–421 (1964).MathSciNetCrossRefMATHGoogle Scholar
  12. 12.
    C. M. Craviotto, W. B. Jones, and N. J. Wyshinski, “Computation of the Binet and Gamma functions by Stieltjes continued fractions,” in: W. B. Jones and A. Sri Ranga (editors), Orthogonal Functions, Moment Theory and Continued Fractions: Theory and Applications, Lect. Notes Pure Appl. Math., 199 (1998), pp. 151–179.Google Scholar
  13. 13.
    A. Cuyt and P. van der Cruyssen, “Rounding error analysis for forward continued fraction algorithms,” Comput. Math. Appl., 11, No. 6, 541–564 (1985).MathSciNetCrossRefMATHGoogle Scholar
  14. 14.
    A. Cuyt, V. B. Petersen, B. Verdonk, H. Waadeland, and W. B. Jones, Handbook of Continued Fractions for Special Functions, Springer, Berlin (2008).MATHGoogle Scholar
  15. 15.
    W. Gautschi, “Computational aspects of three-term recurrence relations,” SIAM Rev., 9, 24–82 (1967).MathSciNetCrossRefMATHGoogle Scholar
  16. 16.
    A. Gil, J. Segura, and N. M. Temme, Numerical Methods for Special Functions, SIAM, Philadelphia (2007).CrossRefMATHGoogle Scholar
  17. 17.
    N. J. Higham, Accuracy and Stability of Numerical Algorithms, SIAM, Philadelphia (2002).CrossRefMATHGoogle Scholar
  18. 18.
    N. Macon and M. Baskervill, “On the generation of errors in the digital evaluation of continued fractions,” J. Assoc. Comput. Math., 3, No. 3, 199–202 (1956).Google Scholar

Copyright information

© Springer Science+Business Media New York 2016

Authors and Affiliations

  • V. R. Hladun
    • 1
  1. 1.“L’vivs’ka Politekhnika” National UniversityLvivUkraine

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