Модифицированный метод парабол поиска корня

Модифицированный метод парабол поиска корня

Богданов В. В., Волков Ю. С.
Сибирский журнал индустриальной математики, 26, 3(95), 5-13 (2023)
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УДК 519.65 
DOI: 10.33048/SIBJIM.2023.26.301

Аннотация:

Предлагается модификация метода парабол для поиска корня непрерывной функции, состоящая в том, что одновременно строятся две параболы, интерполирующие значения заданной функции. Показано, что если третья производная исходной функции знакопостоянна на рассматриваемом промежутке локализации искомого корня, то корень лежит между корнями парабол. Это позволяет существенно сузить интервал локализации корня и уменьшить количество шагов для вычисления корня с заданной точностью. Предложенная модификация метода парабол используется в задаче расчёта изолиний при моделировании универсальной характеристики гидротурбин.

Литература:
  1. Ostrowski A. M. Solution of equations and systems of equations. N. Y.: Academ. Press, 1960.
     
  2. Bakhvalov N. S., Zhidkov N. P., Kobelkov G. M. Chislennye metody [Numerical methods]. Moscow: Binom, 2012 (in Russian).
     
  3. Berezin I. S., Zhidkov N. P. Computing Methods. Vol. 1. Oxford: Pergamon Press, 1965.
     
  4. Voevodin V. V. Chislennye metody algebry (teoria i algorifmy) [Numerical methods of algebra (theory and algorithms)]. Moscow: Nauka, 1966 (in Russian).
     
  5. Kalitkin N. N. Chislennye metody [Numerical methods]. Moscow: Nauka, 1978 (in Russian).
     
  6. Costabile F., Gualtieri M. I., Luceri R. A modification of Muller’s method. Calcolo, 2006, Vol. 43, No. 1, pp. 39–50. 
     
  7. Gemechu T., Thota S. On new root finding algorithms for solving nonlinear transcendental equations. Internat. J. Chem., Math. Phys., 2020, Vol. 4, No. 2, pp. 18–24.
     
  8. Cordero A., Garrido N., Torregrosa J. R., Triguero—Navarro P. Iterative schemes for finding all roots simultaneously of nonlinear equations. Appl. Math. Lett., 2022, Vol. 134, article 108325.
     
  9. Kalitkin N. N., Kuz’mina L. V. Calculation of roots and there multiplicity for nonlinear equation. Math. Models Comput. Simul., 2011, Vol. 3, No. 1, pp. 65–80.
     
  10. Intep S. A review of bracketing methods for finding zeros of nonlinear functions. Appl. Math. Sci., 2018, Vol. 12, No. 3, pp. 137–146.
     
  11. Kovalev N. N. Gidroturbiny. Konstrukcii i voprosy proektirovanija [Hydraulic turbines. Constructions and design issues]. Leningrad: Mashinostroenie, 1971 (in Russian).
     
  12. Barlit V. V. Gidravlicheskie turbiny [Hydraulic turbines]. Kiev: Vishcha shkola, 1977 (in Russian).
     
  13. Krivchenko G. I. Gidravlicheskie mashiny: turbiny i nasosy [Hydraulic machines: turbines and pumps]. Moscow: Jenergoatomizdat, 1983 (in Russian).
     
  14. Bronshtejn L. Ya., German A. N., Goldin V. E. and others. Spravochnik konstruktora gidroturbin [Handbook of the designer of hydraulic turbines]. Leningrad: Mashinostroenie, 1971 (in Russian).
     
  15. Volkov Yu. S., Miroshnichenko V. L. Postroenie matematicheskoj modeli universal’noj harakteristiki radial’no-osevoj gidroturbiny [Development of a mathematical model of an universal characteristic a francis turbine] // Sib. Zhurn. Indust. Mat., 1988, Vol. 1, No. 1, pp. 77–88 (in Russian).
     
  16. Volkov Yu. S., Miroshnichenko V. L., Salienko A. E. Matematicheskoe modelirovanie universal’noj harakteristiki povorotno-lopastnoj gidroturbiny [Mathematical modeling of hill diagram for Kaplan turbine]. Machine Learning and Data Analysis, 2014, Vol. 1, No. 10, pp. 1439–1450.
     
  17. Bogdanov V. V., Karsten W. V., Miroshnichenko V. L., Volkov Yu. S. Application of splines for determining the velocity characteristic of a medium from a vertical seismic survey. Central European J. Math., 2013, Vol. 11, No. 4, pp. 779–786.
     
  18. Anikonov Yu. E., Bogdanov V. V., Volkov Yu. S., Derevtsov E. Yu. On the Determination of the Velocity and Elastic Parameters of a Medium in the Focal Zone from Earthquake Hodographs. J. Appl. Indust. Math., 2021, Vol. 24, No. 4, pp. 569–585.
     
  19. Wendland H. Scattered Data Approximation. Cambridge: Cambridge Univ. Press, 2005.
     
  20. Ignatov M. I., Pevnyi A. B. Natural’nye splajny mnogih peremennyh [Natural splines of many variables]. Leningrad: Nauka, 1991 (in Russian).
     
  21. Schaback R. Native Hilbert Spaces for Radial Basis Functions. I. New Developments in Approximation Theory. Basel: Birkhauser, 1999, pp. 255–282.

Работа выполнена в рамках государственного задания Института математики СО РАН (проект FWNF–2022–0015).

В. В. Богданов
  1. Институт математики им. С. Л. Соболева СО РАН, 
    просп. Акад. Коптюга, 4, г. Новосибирск 630090, Россия

E-mail: bogdanov@math.nsc.ru

Ю. С. Волков
  1. Институт математики им. С. Л. Соболева СО РАН, 
    просп. Акад. Коптюга, 4, г. Новосибирск 630090, Россия

E-mail: volkov@math.nsc.ru

Статья поступила 27.04.2023 г. 
После доработки — 26.05.2023 г.
Принята к публикации 07.06.2023 г.

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 to significantly 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.

References:
  1. Ostrowski A. M. Solution of equations and systems of equations. N. Y.: Academ. Press, 1960.
     
  2. Bakhvalov N. S., Zhidkov N. P., Kobelkov G. M. Chislennye metody [Numerical methods]. Moscow: Binom, 2012 (in Russian).
     
  3. Berezin I. S., Zhidkov N. P. Computing Methods. Vol. 1. Oxford: Pergamon Press, 1965.
     
  4. Voevodin V. V. Chislennye metody algebry (teoria i algorifmy) [Numerical methods of algebra (theory and algorithms)]. Moscow: Nauka, 1966 (in Russian).
     
  5. Kalitkin N. N. Chislennye metody [Numerical methods]. Moscow: Nauka, 1978 (in Russian).
     
  6. Costabile F., Gualtieri M. I., Luceri R. A modification of Muller’s method. Calcolo, 2006, Vol. 43, No. 1, pp. 39–50. 
     
  7. Gemechu T., Thota S. On new root finding algorithms for solving nonlinear transcendental equations. Internat. J. Chem., Math. Phys., 2020, Vol. 4, No. 2, pp. 18–24.
     
  8. Cordero A., Garrido N., Torregrosa J. R., Triguero—Navarro P. Iterative schemes for finding all roots simultaneously of nonlinear equations. Appl. Math. Lett., 2022, Vol. 134, article 108325.
     
  9. Kalitkin N. N., Kuz’mina L. V. Calculation of roots and there multiplicity for nonlinear equation. Math. Models Comput. Simul., 2011, Vol. 3, No. 1, pp. 65–80.
     
  10. Intep S. A review of bracketing methods for finding zeros of nonlinear functions. Appl. Math. Sci., 2018, Vol. 12, No. 3, pp. 137–146.
     
  11. Kovalev N. N. Gidroturbiny. Konstrukcii i voprosy proektirovanija [Hydraulic turbines. Constructions and design issues]. Leningrad: Mashinostroenie, 1971 (in Russian).
     
  12. Barlit V. V. Gidravlicheskie turbiny [Hydraulic turbines]. Kiev: Vishcha shkola, 1977 (in Russian).
     
  13. Krivchenko G. I. Gidravlicheskie mashiny: turbiny i nasosy [Hydraulic machines: turbines and pumps]. Moscow: Jenergoatomizdat, 1983 (in Russian).
     
  14. Bronshtejn L. Ya., German A. N., Goldin V. E. and others. Spravochnik konstruktora gidroturbin [Handbook of the designer of hydraulic turbines]. Leningrad: Mashinostroenie, 1971 (in Russian).
     
  15. Volkov Yu. S., Miroshnichenko V. L. Postroenie matematicheskoj modeli universal’noj harakteristiki radial’no-osevoj gidroturbiny [Development of a mathematical model of an universal characteristic a francis turbine] // Sib. Zhurn. Indust. Mat., 1988, Vol. 1, No. 1, pp. 77–88 (in Russian).
     
  16. Volkov Yu. S., Miroshnichenko V. L., Salienko A. E. Matematicheskoe modelirovanie universal’noj harakteristiki povorotno-lopastnoj gidroturbiny [Mathematical modeling of hill diagram for Kaplan turbine]. Machine Learning and Data Analysis, 2014, Vol. 1, No. 10, pp. 1439–1450.
     
  17. Bogdanov V. V., Karsten W. V., Miroshnichenko V. L., Volkov Yu. S. Application of splines for determining the velocity characteristic of a medium from a vertical seismic survey. Central European J. Math., 2013, Vol. 11, No. 4, pp. 779–786.
     
  18. Anikonov Yu. E., Bogdanov V. V., Volkov Yu. S., Derevtsov E. Yu. On the Determination of the Velocity and Elastic Parameters of a Medium in the Focal Zone from Earthquake Hodographs. J. Appl. Indust. Math., 2021, Vol. 24, No. 4, pp. 569–585.
     
  19. Wendland H. Scattered Data Approximation. Cambridge: Cambridge Univ. Press, 2005.
     
  20. Ignatov M. I., Pevnyi A. B. Natural’nye splajny mnogih peremennyh [Natural splines of many variables]. Leningrad: Nauka, 1991 (in Russian).
     
  21. Schaback R. Native Hilbert Spaces for Radial Basis Functions. I. New Developments in Approximation Theory. Basel: Birkhauser, 1999, pp. 255–282.