A New Trigonometrical Algorithm for Computing Real Root of Non-linear Transcendental Equations

  • Vivek Kumar SrivastavEmail author
  • Srinivasarao Thota
  • Manoj Kumar
Original Paper


This paper presents a new algorithm to find a non-zero real root of the transcendental equations using trigonometrical formula. Indeed, the new proposed algorithm is based on the combination of inverse of sine series and Newton Raphson method, which produces better approximate root than Newton Raphson method. The implementation of the proposed algorithm in MATLAB is also discussed. Certain numerical examples are presented to show the efficiency of the proposed algorithm. This algorithm will help to implement in the commercial package for finding a real root of a given transcendental equation.


Algebraic equations Transcendental equations Newton Raphson method Sine inverse function 

Mathematics Subject Classification

65Hxx 65H04 



  1. 1.
    Chitode, J.S.: Numerical Techniques, 2nd edn. Technical Publications, Pune (2008)Google Scholar
  2. 2.
    Somesundaram, R.M., Chandrasekaran, R.M.: Numerical Methods with \(C++\) Programming. Prentice-Hall of India, Delhi (2005)Google Scholar
  3. 3.
    Novak, E., Ritter, K., Wozniakowski, H.: Average-case ompitmality of a hybrid secant-bisection method. Math. Comput. 64(212), 1517–1539 (1995)CrossRefGoogle Scholar
  4. 4.
    Eiger, A., Sikorski, K., Stenger, F.: A bisection method for systems of nonlinear equations. ACM Trans. Math. Softw. 10(4), 367–377 (1984)MathSciNetCrossRefGoogle Scholar
  5. 5.
    Vrahatis, M.N., Iordanidis, K.I.: A rapid generalized method of bisection for solving system of non-linear equation. Numer. Math. 49, 123–138 (1986)MathSciNetCrossRefGoogle Scholar
  6. 6.
    Bachrathy, D., Stépán, G.: Bisection method in higher dimensions and the efficiency number. Mech. Eng. 56(2), 81–86 (2012)Google Scholar
  7. 7.
    Wood, G.R.: The bisection method in higher dimensions. Math. Program. 55, 319–337 (1992)MathSciNetCrossRefGoogle Scholar
  8. 8.
    Wu, X.: Improved Muller method and bisection method with global and asymptotic superlinear convergence of both point and interval for solving nonlinear equations. Appl. Math. Comput. 166, 299–311 (2005)MathSciNetzbMATHGoogle Scholar
  9. 9.
    Yakoubsohn, J.-C.: Numerical analysis of a bisection-exclusion method to find zeros of univariate analytic functions. J. Complex. 21, 652–690 (2005)MathSciNetCrossRefGoogle Scholar
  10. 10.
    Gutierrez, C., Gutierrez, F., Rivara, M.-C.: Complexity of the bisection method. Theor. Comput. Sci. 382, 131–138 (2007)MathSciNetCrossRefGoogle Scholar
  11. 11.
    Wu, X., Kanwar, V., Xia, J.: An improved regula falsi method with quadratic convergence of both diameter and point for enclosing simple zeros of nonlinear equations. Appl. Math. Comput. 144, 381–388 (2003)MathSciNetzbMATHGoogle Scholar
  12. 12.
    Dowell-Jarratt, A.: modified Regula-Falsi method for computing the real root of an equation. BIT Numer. Math. 11, 168–174 (1971)CrossRefGoogle Scholar
  13. 13.
    Saied, A., Liao, S.: A new modification of False-Position method based on homotopy analysis method. Appl. Math. Mech. 29(2), 223–228 (2003)MathSciNetGoogle Scholar
  14. 14.
    Wu, X., Wu, H.: On a class of quadratic convergence iteration formulae without derivatives. Appl. Math. Comput. 107, 77–80 (2000)MathSciNetzbMATHGoogle Scholar
  15. 15.
    Zhu, Y.R., Wu, X.Y.: A free derivative iteration method of order three having convergence of both point and interval for non-linear equations. Appl. Math. Comput. 137, 49–55 (2003)MathSciNetzbMATHGoogle Scholar
  16. 16.
    Johan, V., Ronald, C.: The Newton–Raphson method. Int. J. Math. Educ. Sci. Technol. 26(2), 177–193 (1995)CrossRefGoogle Scholar
  17. 17.
    Mamta, V.K., Kukreja, V.K., Singh, S.: On a class of quadratically convergent iteration formulae. Appl. Math. Comput. 166, 633–637 (2005)MathSciNetzbMATHGoogle Scholar
  18. 18.
    Mamta, V.K., Kukreja, V.K., Singh, S.: On some third-order iterative methods for solving nonlinear equations. Appl. Math. Comput. 171, 272–280 (2005)MathSciNetzbMATHGoogle Scholar
  19. 19.
    Sharma, J.R., Goyal, R.K.: Fourth-order derivative-free methods for solving non-linear equations. Int. J. Comput. Math. 83(1), 101–106 (2006)MathSciNetCrossRefGoogle Scholar
  20. 20.
    Noor, M.A., Noor, K.I., Khan, W.A., Ahmad, F.: On iterative methods for nonlinear equations. Appl. Math. Comput. 183, 128–133 (2006)MathSciNetzbMATHGoogle Scholar
  21. 21.
    Noor, M.A., Ahmad, F.: Numerical comparison of iterative methods for solving nonlinear equations. Appl. Math. Comput. 180, 167–172 (2006)MathSciNetzbMATHGoogle Scholar
  22. 22.
    Noor, M.A., Noor, K.I.: Three-step iterative methods for nonlinear equations. Appl. Math. Comput. 183, 322–327 (2006)MathSciNetzbMATHGoogle Scholar
  23. 23.
    Chen, J., Li, W.: An exponential regula falsi method for solving nonlinear equations. Numer. Algorithms 41, 327–338 (2006)MathSciNetCrossRefGoogle Scholar
  24. 24.
    Chen, J., Li, W.: An improved exponential regula falsi methods with quadratic convergence of both diameter and point for solving nonlinear equations. Appl. Numer. Math. 57, 80–88 (2007)MathSciNetCrossRefGoogle Scholar
  25. 25.
    Chen, J.: New modified regula falsi method for nonlinear equations. Appl. Math. Comput. 184, 965–971 (2007)MathSciNetzbMATHGoogle Scholar
  26. 26.
    Sagraloff, M., Mehlhorn, K.: Computing real roots of real polynomials. pp. 1–44, (2015). arXiv:1308.4088v2 [cs.SC]
  27. 27.
    Abbott, J.: Quadratic interval refinement for real roots. In: ACM Communications in Computer Algebra, vol. 48, no. 1 (2014)Google Scholar

Copyright information

© Springer Nature India Private Limited 2019

Authors and Affiliations

  • Vivek Kumar Srivastav
    • 2
    Email author
  • Srinivasarao Thota
    • 1
  • Manoj Kumar
    • 3
  1. 1.Department of Applied Mathematics, School of Applied Natural SciencesAdama Science and Technology UniversityAdamaEthiopia
  2. 2.Department of Mathematics and ComputingMotihari College of EngineeringMotihariIndia
  3. 3.Department of MathematicsMotilal Nehru National Institute of TechnologyAllahabadIndia

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