A Modified Hybrid Method For Solving Non-Linear Equations With Computational Efficiency

Authors

  • Shehzad Ali Soomro Department of Basic Sciences and Related Studies, Mehran University of Engineering & Technology, Jamshoro, Sindh, Pakistan https://orcid.org/0009-0007-1241-8083
  • Asif Ali Shaikh Department of Basic Sciences and Related Studies, Mehran University of Engineering & Technology, Jamshoro, Sindh, Pakistan https://orcid.org/0000-0002-7225-2309
  • Sania Qureshi 3 Department of Computer Science and Mathematics, Lebanese American University, Beirut, Lebanon
  • Bahadur Ali 2People’s University of Medical and Health Sciences for Women Shaheed Benazirabad, Sindh, Pakistan https://orcid.org/0000-0001-9806-8211

DOI:

https://doi.org/10.21015/vtm.v11i2.1620

Abstract

This paper proposes a modified hybrid method for solving non-linear equations that improves computational efficiency while maintaining accuracy. The proposed method combines the advantages of the traditional Halley’s and mean-based methods, resulting in a more efficient algorithm. The modified hybrid method starts with Halley’s method and then switches to the mean-based method for rapid convergence. To further improve the efficiency of the algorithm, the proposed method incorporates a dynamic selection criterion to choose the appropriate method at each iteration. Numerical experiments are performed to evaluate the performance of the proposed method in comparison to other existing methods. The results show that the modified hybrid method is computationally efficient and can achieve high accuracy in a shorter time than other commonly used methods having similar features. The proposed method is applicable to a wide range of non-linear equations and can be used in various fields of science and engineering where non-linear equations arise. The modified hybrid method provides an effective tool for solving non-linear equations, offering significant improvements in computational efficiency over existing methods.

References

Ababneh, O. Y. [2012], ‘New newton’s method with third-order convergence for solving nonlinear equations’, World Academy of Science, Engineering and Technology 61(198), 1071–1073.

Awadalla, M., Qureshi, S., Soomro, A. and Abuasbeh, K. [2023], ‘A novel three-step numerical solver for physical models under fractal behavior’, Symmetry 15(2), 330.

Behl, R., Bhalla, S., Magreñán, Á. A. and Kumar, S. [2022], ‘An efficient high order iterative scheme for large nonlinear systems with dynamics’, Journal of Computational and Applied Mathematics 404, 113249.

Chanu, W. H., Panday, S. and Thangkhenpau, G. [2022], ‘Development of optimal iterative methods with their applications and basins of attraction’, Symmetry 14(10), 2020.

Chapra, S. [2011], EBOOK: Applied Numerical Methods with MATLAB for Engineers and Scientists, McGraw Hill.

Cordero, A., Torregrosa, J. R. and Triguero-Navarro, P. [2021], ‘A general optimal iterative scheme with arbitrary order of convergence’, Symmetry 13(5), 884.

Jaiswal, J. and Choubey, N. [2013], ‘A new efficient optimal eighth-order iterative method for solving nonlinear equations’, arXiv preprint arXiv:1304.4702 .

Jamali, K., Solangi, M. A. and Qureshi, S. [2022], ‘A novel hybrid iterative method for applied mathematical models with time-efficiency’, Journal of Applied Mathematics and Computational Mechanics 21(3).

Liu, L. and Wang, X. [2010], ‘Eighth-order methods with high efficiency index for solving nonlinear equations’, Applied Mathematics and Computation 215(9), 3449–3454.

Lukić, T. and Ralević, N. M. [2008], ‘Geometric mean newton’s method for simple and multiple roots’, Applied Mathematics Letters 21(1), 30–36.

Mastoi, V., Ali, A., Shaikh, M. M. and Shaikh, A. W. [2020], ‘A new third-order derivative-based iterative method for nonlinear equations’, J. Mech. Cont. & Math. Sci 15(10), 110–123.

Naseem, A., Rehman, M. and Abdeljawad, T. [2020], ‘Higher-order root-finding algorithms and their basins of attraction’, Journal of Mathematics 2020, 1–11.

Naseem, A., Rehman, M. and Abdeljawad, T. [2022], ‘A novel root-finding algorithm with engineering applications and its dynamics via computer technology’, IEEE Access 10, 19677–19684.

Nguyen, Q. and Yang, C.-y. [2016], ‘A modified newton–raphson method to estimate the temperature dependent absorption coefficient in laser welding process’, International Journal of Heat and Mass

Transfer 102, 1222–1229.

Özban, A. Y. [2004], ‘Some new variants of newton’s method’, Applied Mathematics Letters 17(6), 677– 682.

Panday, S., Sharma, A. and Thangkhenpau, G. [2023], ‘Optimal fourth and eighth-order iterative methods for non-linear equations’, Journal of Applied Mathematics and Computing 69(1), 953–971.

Planck, M. [1914], The theory of heat radiation, Blakiston.

Qureshi, S., Argyros, I. K., Soomro, A., Gdawiec, K., Shaikh, A. A. and Hincal, E. [2023], ‘A new optimal root-finding iterative algorithm: local and semilocal analysis with polynomiography’, Numerical Algorithms pp. 1–31.

Qureshi, S., Ramos, H. and Soomro, A. K. [2021], ‘A new nonlinear ninth-order root-finding method with error analysis and basins of attraction’, Mathematics 9(16), 1996.

Singh, M. K. and Singh, A. K. [2017], ‘A new-mean type variant of newton’s method for simple and multiple roots’, Int. J. Math. Trends Technol 49, 174–177.

Tassaddiq, A., Qureshi, S., Soomro, A., Hincal, E., Baleanu, D. and Shaikh, A. A. [2021], ‘A new three step root-finding numerical method and its fractal global behavior’, Fractal and Fractional 5(4), 204.

Weerakoon, S. and Fernando, T. [2000], ‘A variant of newton’s method with accelerated third-order

convergence’, Applied mathematics letters 13(8), 87–93.

Yaseen, S. and Zafar, F. [2022], ‘A new sixth-order jarratt-type iterative method for systems of nonlinear equations’, Arabian Journal of Mathematics 11(3), 585–599.

Zhou, X. [2007], ‘A class of newton’s methods with third-order convergence’, Applied Mathematics Letters 20(9), 1026–1030.

Downloads

Published

2023-11-25

How to Cite

Shehzad Ali Soomro, Asif Ali Shaikh, Qureshi, S., & Bahadur Ali. (2023). A Modified Hybrid Method For Solving Non-Linear Equations With Computational Efficiency. VFAST Transactions on Mathematics, 11(2), 126–137. https://doi.org/10.21015/vtm.v11i2.1620

Issue

Vol. 11 No. 2 (2023): July-December

Section

Articles

License

Authors who publish with this journal agree to the following terms:

  1. Authors retain copyright and grant the journal right of first publication with the work simultaneously licensed under a Creative Commons Attribution License (CC-By) that allows others to share the work with an acknowledgment of the work's authorship and initial publication in this journal.
  2. Authors are able to enter into separate, additional contractual arrangements for the non-exclusive distribution of the journal's published version of the work (e.g., post it to an institutional repository or publish it in a book), with an acknowledgement of its initial publication in this journal.
  3. Authors are permitted and encouraged to post their work online (e.g., in institutional repositories or on their website) prior to and during the submission process, as it can lead to productive exchanges, as well as earlier and greater citation of published work (See The Effect of Open Access).

                                      

This work is licensed under a Creative Commons Attribution  License CC BY