A new three step derivative free method using weight function for numerical solution of non-linear equations arises in application problems
AbstractAbstract In this paper a three-step numerical method, using weight function, has been derived for ﬁnding the root of non-linear equations. The proposed method possesses the accuracy of order eight with four functional evaluations.
The eﬃciency index of the derived scheme is 1.682. Numerical examples, application problems are used to demonstrate the performance of the presented schemes and compare them to other available methods in the literature of the same order. Matlab, Mathematica 2021 & Maple 2021 software were used for numerical results.
D. Kincaid and W. Cheney. Numerical Analysis. Brooks/Cole Publishing Company, 2009.
A. Cordero, J. L. Hueso, E. Martínez, and J. R. Torregrosa. Generating optimal derivative free iterative methods for nonlinear equations by using polynomial interpolation. Math. Comput. Model, 57(7):1950–1956, 2013. doi: https://doi.org/10.1016/j.mcm.2012.01.012.
S. Jamali, Z. A. Kalhoro, A. W. Shaikh, and M. S. Chandio. A New Second Order Derivative Free Method for Numerical Solution of Non-Linear Algebraic and Transcendental Equations using Interpolation Technique. Journal of Mechanics of Continua Mathematical Sciences, 16(4):75–84, apr 2021. ISSN 09738975.
S. Jamali, Z. A. Kalhoro, A. W. Shaikh, and M. S. Chandio. An Iterative, Bracketing & Derivative-Free Method for Numerical Solution of Non-Linear Equations using Stirling Interpolation Technique. Journal of Mechanics of Continua Mathematical Sciences, 16(6):13–27, jun 2021. ISSN 0973-8975. doi: https://doi.org/10.26782/jmcms.2021.06.00002.
A. Cordero and J. R. Torregrosa. Low-complexity root-ﬁnding iteration functions with no derivatives of any order of convergence. J. Comput. Appl. Math, 275:502–515, February 2015. doi:
B. Neta. A new derivative-free method to solve nonlinear equations. Mathematics, 9(6):1–5, 2021. doi: https://doi.org/10.3390/math9060583.
I. K. Argyros, M. Kansal, V. Kanwar, and S. Bajaj. Higher-order derivative-free families of chebyshev– halley type methods with or without memory for solving nonlinear equations. Appl. Math, 315:224–245, December 2017. doi: https://doi.org/10.1016/j.amc.2017.07.051.
A. Suhadolnik. Combined bracketing methods for solving nonlinear equations. Appl. Math, 25(11): 1755–1760, 2012. doi: https://doi.org/10.1016/j.aml.2012.02.006.
U. K. Qureshi, Z. A. Kalhoro, A. A. Shaikh, and S. Jamali. Sixth Order Numerical Iterated Method of Open Methods for Solving Nonlinear Applications Problems. Proceedings of the Pakistan Academy of Sciences: A. Physical and Computational Sciences, 57(November):35–40, 2020.
M. K. Mahmood and F. Ahmad. Recursive elucidation of polynomial congruences using root-ﬁnding numerical techniques. Abstract and Applied Analysis, 2014:9, 2014. doi:
S. Jamali, Z. A. Kalhoro, A. W. Shaikh, and M. S. Chandio. An Iterative, Bracketing & Derivative-Free
Method for Numerical Solution of Non-Linear Equations using Stirling Interpolation Technique. Journal
of Mechanics of Continua Mathematical Sciences, 16(6):13–27, jun 2021. ISSN 0973-8975. doi: https://doi.org/10.26782/jmcms.2021.06.00002.
Farooq Ahmed Shah and Ehsan Ul Haq. Some new multi-step derivative-free iterative methods for solving nonlinear equations. App. and Eng. Math., 10(4):951–963, 2020.
U. K. Qureshi, S. Jamali, Z. A. Kalhoro, and A. G. Shaikh. Modiﬁed Quadrature Iterated Methods of Boole Rule and Weddle Rule for Solving non-Linear Equations. Journal of Mechanics of continua and Mathematical sciences, 16(2):87–101, feb 2021. ISSN 0973-8975. doi: https:doi.org/10.26782/jmcms.2021.02.00008.
A. Suhadolnik. Superlinear bracketing method for solving nonlinear equations. Appl. Math, 219(14): 7369–7376, 2013. doi: https://doi.org/10.1016/j.amc.2012.12.064.
F. Soleymani, D. K. R. Babajee, and M. Shariﬁ. Modiﬁed jarratt method without memory with twelfthorder convergence. Annals of the University of Craiova, Mathematics and Computer Science Series, 39(1): 21–34, 2012. ISSN 1223-6934.
S. Jamali, Z. A. Kalhoro, A. W. Shaikh, M. S. Chandio, and S. Dehraj. A novel two point optimal derivative free method for numerical solution of nonlinear algebraic, transcendental equations and application problems using weight function. VFAST Transactions on Mathematics, 10(2):137–146, 2022. ISSN 23090022.
M. S. Petković, S. Ilic, and J. Dzunic. Derivative free two-point methods with and without memory for solving nonlinear equations. Appl. Math, 217:1887–1895, 2010. doi: https: doi.org/10.1016/j.amc.2010.06.043.
A. Cordero, J. L. Hueso, E. Martínez, and J. R. Torregrosa. Steffensen type methods for solving nonlinear equations . J. Comput. Appl. Math, 236(12):3058–3064, 2012. doi: https: doi.org/10.1016/j.cam.2010.08.043.
B. Kong-ied. Two new eighth and twelfth order iterative methods for solving nonlinear equations. Int. J. Math, 16(1):333–344, 2021.
D. Jain. Families of newton-like methods with fourth-order convergence. Int. J. Comput. Math, 90(5): 1072–1082, 2013. doi: https://doi.org/10.1080/00207160.2012.746677.
P. Sivakumar and J. Jayaraman. Some new higher order weighted newton methods for solving nonlinear equation with applications. Math. Comput. Appl., 24(2):1–16, 2019. doi:
F. Soleymani. Eﬃcient optimal eighth-order derivative-free methods for nonlinear equations. Jpn. J. Ind. Appl. Math, 30(2):287–306, June 2013. doi: https://doi.org/10.100/s13160-013-0103-7.
N. Raﬁq, M. Shams, N. A. Mir, and Y. U. Gaba. A highly eﬃcient computer method for solving polynomial equations appearing in engineering problems. Math. Probl. Eng, 2021, 2021. doi:
F. Zafar, A. Cordero, and J. R. Torregrosa. An eﬃcient family of optimal eighth-order multiple root ﬁnders. Mathematics, 6(12):1–16, 2018. doi: https://doi.org/10.3390/math6120310.
M. Shams, N. Ra, N. Kausar, N. A. Mir, and A. Alalyani. Computer oriented numerical scheme for solving engineering problems. Comput. Syst. Sci, 42(2):689–701, 2022. doi:
Authors who publish with this journal agree to the following terms:
- 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.
- 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.
- 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