Srinivasarao Thota     (Department of Mathematics, Amrita School of Physical Sciences, Amrita Vishwa Vidyapeetham, Amaravati, Andhra Pradesh–522503, India)
Tekle Gemechu     (Department of Applied Mathematics, School of Applied Natural Sciences, Adama Science and Technology University, Post Box No. 1888, Adama, Ethiopia)
Abayomi Ayotunde Ayoade     (Department of Mathematics, University of Lagos, Lagos, Lagos State, Nigeria)


The objective of this paper is to propose two new hybrid root finding algorithms for solving transcendental equations. The proposed algorithms are based on the well-known root finding methods namely the Halley's method, regula-falsi method and exponential method. We show using numerical examples that the proposed algorithms converge faster than other related methods. The first hybrid algorithm consists of regula-falsi method and exponential method (RF-EXP). In the second hybrid algorithm, we use regula falsi method and Halley's method (RF-Halley). Several numerical examples are presented to illustrate the proposed algorithms, and comparison of these algorithms with other existing methods are presented to show the efficiency and accuracy. The implementation of the proposed algorithms is presented in Microsoft Excel (MS Excel) and the mathematical software tool Maple.


Hybrid method, Halley's method, Regula-falsi method, Transcendental equations, Root-finding algorithms

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  1. Badr E., Attiya H., El Ghamry A. Novel hybrid algorithms for root determining using advantages of open methods and bracketing methods. Alexandria Eng. J., 2022. Vol. 61, No. 12. P. 11579–11588. DOI: 10.1016/j.aej.2022.05.007
  2. Badr E., Almotairi S., El Ghamry A. A Comparative study among new hybrid root finding algorithms and traditional methods. Mathematics, 2021. Vol. 9, No. 11. Art. no. 1306. DOI: 10.3390/math9111306
  3. Badr E.M., ElGendy H.S. A hybrid water cycle-particle swarm optimization for solving the fuzzy underground water confined steady flow. Indones. J. Electr. Eng. Comput. Sci., 2020. Vol. 19, No. 1. P. 492–504. DOI: 10.11591/ijeecs.v19.i1.pp492-504
  4. Baskar S., Ganesh S.S. Introduction to Numerical Analysis. Powai, Mumbai, India: Depart. Math., Indian Inst. Tech. Bombay, 2016. 230 p.
  5. Brent R.P. Algorithms for Minimization without Derivatives. Englewood Cliffs, NJ: Prentice-Hall, 1973. 195 p.
  6. Burden R.L., J. Douglas Faires. Numerical Analysis, 3rd ed. Baston, USA: PWS Publishing, 1985. 676 p.
  7. Chapra S.C., Canale R.P. Numerical Methods for Engineers, 7th ed. Boston, MA, USA: McGraw-Hill, 2015. 970 p.
  8. Dekker T.J. Finding a zero by means of successive linear interpolation. In: Constructive Aspects of the Fundamental Theorem of Algebra, B. Dejon, P. Henrici (eds.). London: Wiley-Interscience, 1969. P. 37–48.
  9. Fink K.D., Mathews J.H. Numerical Methods Using Matlab, 4th ed. Upper Saddle River, NJ, USA: Prentice-Hall Inc., 2004. 696 p.
  10. Gemechu T., Thota S. On new root finding algorithms for solving nonlinear transcendental equations. Int. J. Chem., Math. Phys., 2020. Vol. 4, No. 2. P. 18–24. DOI: 10.22161/ijcmp.4.2.1
  11. Hasan A. Numerical study of some iterative methods for solving nonlinear equations. Int. J. Eng. Sci. Invent., 2016. Vol. 5, No. 2. P. 1–10.
  12. Hoffman J.D. Numerical Methods for Engineers and Scientists, 2nd ed. NY, Basel: Marcel Dekker Inc., 2001. 823 p.
  13. Kiusalaas J. Numerical Methods in Engineering with Python, 2nd ed. Cambridge: Cambridge University Press, 2010. 432 p.
  14. Novak E., Ritter K., Woźniakowski H. Average-case optimality of a hybrid secant-bisection method. Math. Comput., 1995. Vol. 64, No. 212. P. 1517–1539. DOI: 10.2307/2153369
  15. Parveen T., Singh S., Thota S., Srivastav V.K. A new hybrid root-finding algorithm for transcendental equations using bisection, regula-falsi and Newton-Raphson methods. In: National Conf. Sustainable & Recent Innovation in Science and Engineering (SUNRISE-19), 2019.
  16. Ridders C. A new algorithm for computing a single root of a real continuous function. IEEE Trans. Circuits Syst., 1979. Vol. 26, No. 11. P. 979–980. DOI: 10.1109/TCS.1979.1084580
  17. Sabharwal C.L. Blended root finding algorithm outperforms bisection and regula falsi algorithms. Mathematics, 2019. Vol. 7, No. 11. Art. no. 1118. DOI: 10.3390/math7111118
  18. Srivastav V.K., Thota S., Kumar M. A new trigonometrical algorithm for computing real root of non-linear transcendental equations. Int. J. Appl. Comput. Math., 2019. Vol. 5. Art. no. 44. DOI: 10.1007/s40819-019-0600-8
  19. Thota S., Srivastav V.K. An algorithm to compute real root of transcendental equations using hyperbolic tangent function. Int. J. Open Problems Compt. Math., 2021. Vol. 14, No. 2. P. 1–14.
  20. Thota S. A numerical algorithm to find a root of non-linear equations using householder’s method. Int. J. Adv. Appl. Sci., 2021. Vol. 10, No. 2. P. 141–148. DOI: 10.11591/ijaas.v10.i2.pp141-148
  21. Thota S., Gemechu T., Shanmugasundaram P. New algorithms for computing a root of non-linear equations using exponential series. Palestine J. Math., 2021. Vol. 10, No. 1. P. 128–134.
  22. Thota S., Gemechu T. A new algorithm for computing a root of transcendental equations using series expansion. Southeast Asian J. Sci., 2019. Vol. 7, No. 2. P. 106–114.
  23. Thota S. A new root-finding algorithm using exponential series. Ural Math. J., 2019. Vol. 5, No. 1. P. 83–90. DOI: 10.15826/umj.2019.1.008
  24. Thota S., Srivastav V.K. Quadratically convergent algorithm for computing real root of non-linear transcendental equations. BMC Res. Notes, 2018. Vol. 11. Art. no. 909. DOI: 10.1186/s13104-018-4008-z
  25. Thota S., Srivastav V.K. Interpolation based hybrid algorithm for computing real root of non-linear transcendental functions. Int. J. Math. Comp. Res., 2014. Vol. 2, No. 11. P. 729–735.
  26. Thota S. A new hybrid halley-false position type root finding algorithm to solve transcendental equations. In: Istanbul International Modern Scientific Research Congress-III, 06–08 May 2022. Istanbul, Turkey: Istanbul Gedik University, 2022. P. 1–2.

DOI: http://dx.doi.org/10.15826/umj.2023.1.016

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