Skip to main content
SpringerLink
Log in

A Modified Quadratic Interpolation Method for Root Finding

  • Published:
Journal of Applied and Industrial Mathematics Aims and scope Submit manuscript

Abstract

A modification of the quadratic interpolation method for finding the root of a continuous function is proposed. Two quadratic interpolation polynomials are simultaneously constructed. It is shown that if the third derivative of the original function does not change sign on the considered interval of localization of the required root, then the root lies between the roots of the quadratic functions. This allows one to substantially narrow the localization interval and reduce the number of steps to calculate the root with a given accuracy. The proposed modification of the quadratic interpolation method is used in the problem of calculating isolines when modeling the hill diagram of hydraulic turbines.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Fig. 1.
Fig. 2.

REFERENCES

  1. A. M. Ostrowski, Solution of Equations and Systems of Equations (Academic Press, New York, 1960).

    MATH  Google Scholar 

  2. N. S. Bakhvalov, N. P. Zhidkov, and G. M. Kobel’kov, Numerical Methods (Binom, Moscow, 2012) [in Russian].

    MATH  Google Scholar 

  3. I. S. Berezin and N. P. Zhidkov, Computing Methods. Vol. 1 (Pergamon Press, Oxford, 1965).

    MATH  Google Scholar 

  4. V. V. Voevodin, Numerical Methods (Theory and Algorithms) (Nauka, Moscow, 1966) [in Russian].

    Google Scholar 

  5. N. N. Kalitkin, Numerical Methods (Nauka, Moscow, 1978) [in Russian].

    Google Scholar 

  6. F. Costabile, M. I. Gualtieri, and R. Luceri, “A modification of Muller’s method,” Calcolo 43 (1), 39–50 (2006).

    Article  MathSciNet  MATH  Google Scholar 

  7. T. Gemechu and S. Thota, “On new root finding algorithms for solving nonlinear transcendental equations,” Int. J. Chem. Math. Phys. 4 (2), 18–24 (2020).

    Article  Google Scholar 

  8. A. Cordero, N. Garrido, J. R. Torregrosa, and P. Triguero-Navarro, “Iterative schemes for finding all roots simultaneously of nonlinear equations,” Appl. Math. Lett. 134, 108325 (2022).

    Article  MathSciNet  MATH  Google Scholar 

  9. N. N. Kalitkin and L. V. Kuz’mina, “Computation of roots of an equation and determination of their multiplicity,” Mat. Model. 22 (7), 33–52 (2010) [Math. Models Comput. Simul. 3 (1), 65–80 (2011)].

    Article  MathSciNet  Google Scholar 

  10. S. Intep, “A review of bracketing methods for finding zeros of nonlinear functions,” Appl. Math. Sci. 12 (3), 137–146 (2018).

    Google Scholar 

  11. N. N. Kovalev, Hydroturbines. Constructions and Issues of Design (Mashinostroenie, Leningrad, 1971) [in Russian].

    Google Scholar 

  12. V. V. Barlit, Hydraulic Turbines (Vishcha Shk., Kiev, 1977) [in Russian].

    Google Scholar 

  13. G. I. Krivchenko, Hydraulic Machines: Turbines and Pumps (Energoatomizdat, Moscow, 1983) [in Russian].

    Google Scholar 

  14. L. Ya. Bronstein, A. N. German, V. E. Gol’din, et al., Handbook of the Designer of Hydraulic Turbines (Mashinostroenie, Leningrad, 1971) [in Russian].

    Google Scholar 

  15. Yu. S. Volkov and V. L. Miroshnichenko, “Development of a mathematical model of the hill diagram of a Francis turbine,” Sib. Zh. Ind. Mat. 1 (1), 77–88 (1998) [in Russian].

    Google Scholar 

  16. Yu. S. Volkov, V. L. Miroshnichenko, and A. E. Salienko, “Mathematical modeling of the hill diagram of a Kaplan hydroturbine,” Mash. Obuchenie Anal. Dannykh 1 (10), 1439–1450 (2014) [in Russian].

    Google Scholar 

  17. V. V. Bogdanov, W. V. Karsten, V. L. Miroshnichenko, and Yu. S. Volkov, “Application of splines for determining the velocity characteristic of a medium from a vertical seismic survey,” Central Eur. J. Math. 11 (4), 779–786 (2013).

    MathSciNet  MATH  Google Scholar 

  18. Yu. E. Anikonov, V. V. Bogdanov, Yu. S. Volkov, and E. Yu. Derevtsov, “On the determination of the velocity and elastic parameters of the medium of the focal zone from earthquake hodographs,” Sib. Zh. Ind. Mat. 24 (4), 1–18 (2021) [in Russian].

    Article  MATH  Google Scholar 

  19. H. Wendland, Scattered Data Approximation (Cambridge Univ. Press, Cambridge, 2005).

    MATH  Google Scholar 

  20. M. I. Ignatov and A. B. Pevnyi, Natural Splines of Many Variables (Nauka, Leningrad, 1991) [in Russian].

    Google Scholar 

  21. R. Schaback, Native Hilbert Spaces for Radial Basis Functions. I. New Developments in Approximation Theory (Birkhäuser, Basel, 1999), 255—282.

Download references

Funding

This work was carried out within the framework of the state task for Sobolev Institute of Mathematics of the Siberian Branch of the Russian Academy of Sciences, project no. FWNF-2022-0015.

Author information

Authors and Affiliations

  1. Sobolev Institute of Mathematics, Siberian Branch, Russian Academy of Sciences, Novosibirsk, 630090, Russia

    V. V. Bogdanov & Yu. S. Volkov

Authors
  1. V. V. Bogdanov
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Yu. S. Volkov
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding authors

Correspondence to V. V. Bogdanov or Yu. S. Volkov.

Additional information

Translated by V. Potapchouck

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Bogdanov, V.V., Volkov, Y.S. A Modified Quadratic Interpolation Method for Root Finding. J. Appl. Ind. Math. 17, 491–497 (2023). https://doi.org/10.1134/S1990478923030031

Download citation

  • Received:

  • Revised:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1134/S1990478923030031

Keywords

Search

Navigation