Authors:
Sanaullah Jamali,Zubair Ahmed Kalhoro,Abdul Wasim Shaikh,Muhammad Saleem Chandio,DOI NO:
https://doi.org/10.26782/jmcms.2021.04.00006Keywords:
Nonlinear equation,Interpolation,convergence,number of iteration,Accuracy ,Abstract
Its most important task in numerical analysis to find roots of nonlinear equations, several methods already exist in literature to find roots but in this paper, we introduce a unique idea by using the interpolation technique. The proposed method derived from the newton backward interpolation technique and the convergence of the proposed method is quadratic, all types of problems (taken from literature) have been solved by this method and compared their results with another existing method (bisection method (BM), regula falsi method (RFM), secant method (SM) and newton raphson method (NRM)) it's observed that the proposed method have fast convergence. MATLAB/C++ software is used to solve problems by different methods.Refference:
I. Ababneh, O. Y. (2016). New Fourth Order Iterative Methods Second Derivative Free. Journal of Applied Mathematics and Physics, 04(03), 519–523. https://doi.org/10.4236/jamp.2016.43058
II. Ali, A., Saeed Ahmed, Muhmmmad Tanveer, M., Mehmood, Q., & Wazeer, W. (2015). MODIFIED TWO-STEP FIXED POINT ITERATIVE METHOD FOR SOLVING NONLINEAR FUNCTIONAL EQUATIONS. Sci.Int.(Lahore), 27(3), 1737–1739.
III. Allame, M., & Azad, N. (2012). On Modified Newton Method for Solving a Nonlinear Algebraic Equations by Mid-Point. World Applied Sciences Journal, 17(12), 1546–1548.
IV. Hafiz, M. A., & Bahgat, M. S. M. (2013). Solving Nonlinear Equations Using Two-Step Optimal Methods. In Annual Review of Chaos Theory, Bifurcations and Dynamical Systems (Vol. 3). www.arctbds.com.
V. Homeier, H. H. H. (2005). On Newton-type methods with cubic convergence. Journal of Computational and Applied Mathematics, 176(2), 425–432. https://doi.org/10.1016/j.cam.2004.07.027
VI. Intep, S. (2018). A review of bracketing methods for finding zeros of nonlinear functions. Applied Mathematical Sciences, 12(3), 137–146. https://doi.org/10.12988/ams.2018.811
VII. Li, Y.-T., & Jiao, A.-Q. (2009). Some Variants of Newton’s Method with Fifth-Order and Fourth-Order Convergence for Solving Nonlinear Equations. In Darbose International Journal of Applied Mathematics and Computation (Vol. 1, Issue 1). http//:ijamc.darbose.com
VIII. Liu, X., & Wang, X. (2013). A Family of Methods for Solving Nonlinear Equations with Twelfth-Order Convergence. Applied Mathematics, 04(02), 326–329. https://doi.org/10.4236/am.2013.42049
IX. Mehtre, V. V. (2019). Root Finding Methods: Newton Raphson Method. International Journal for Research in Applied Science and Engineering Technology, 7(11), 411–414. https://doi.org/10.22214/ijraset.2019.11065
X. Ozbzn, A. Y (2004). Some New Variants of Newton’s Method. Applied Mathematics Letters, 17, 677–682. https://doi.org/10.1016/j
XI. Omran, H. H. (2013). Modified Third Order Iterative Method for Solving Nonlinear Equations. In Journal of Al-Nahrain University (Vol. 16, Issue 3).
XII. Qureshi ++, U. K., Shaikh, A. A., & Solangi, M. A. (2017). Modified Free Derivative Open Method for Solving Non-Linear Equations. SINDH UNIVERSITY RESEARCH JOURNAL (SCIENCESERIES), 49(004), 821–824. https://doi.org/10.26692/sujo/2017.12.0065
XIII. Qureshi, U. K, Ansari, ++ M Y, & Syed, M. R. (2018). Super Linear Iterated Method for Solving Non-Linear Equations. SINDH UNIVERSITY RESEARCH JOURNAL (SCIENCESERIES) Super, 50(1), 137–140. https://doi.org/10.26692/sujo/2018.1.0024
XIV. Qureshi, U. K, Bhatti, A. A., Kalhoro, Z. A., & Ali, Z. (2019). On the Development of Numerical Iterated Method of Newton Raphson method for Estimating Nonlinear Equations. University of Sindh Journal of Information and Communication Technology (USJICT), 3(2). http://sujo2.usindh.edu.pk/index.php/USJICT/
XV. Qureshi, U. K, Jamali, S., & Kalhoro, Z. A. (2021). MODIFIED QUADRATURE ITERATED METHODS OF BOOLE RULE AND WEDDLE RULE FOR SOLVING NON-LINEAR EQUATIONS. J. Mech. Cont. & Math. Sci., Vol.-16, No.-2, February (2021) pp 87-101. DOI : 10.26782/jmcms.2021.02.00008
XVI. Qureshi, U. K, Kalhoro, Z. A., Shaikh, A. A., & Jamali, S. (2020). 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.
XVII. Qureshi, U. K., Solanki, N., & Ansari, M. Y. (2019). Algorithm of Difference Operator for Computing a Single Root of Nonlinear Equations. In Punjab University Journal of Mathematics (Vol. 51, Issue 4).
XVIII. Sangah, A. A., Shaikh, A. A., & Shah, S. F. (2016). Comparative Study of Existing Bracketing Methods with Modified Bracketing Algorithm for Solving Nonlinear Equations in Single Variable. SINDH UNIVERSITY RESEARCH JOURNAL (SCIENCE SERIES), 48(1), 171–174.
XIX. Shah, F. A., Noor, M. A., & Batool, M. (2014). Derivative-free iterative methods for solving nonlinear equations. Applied Mathematics and Information Sciences, 8(5), 2189–2193. https://doi.org/10.12785/amis/080512
XX. Subash, R., & Sathya, S. (2011). NUMERICAL SOLUTION OF FUZZY MODIFIED NEWTON-RAPHSON METHOD FOR SOLVING NON-LINEAR EQUATIONS. International Journal of Current Research, 3(11), 390–392. http://www.journalcra.com
XXI. Suhadolnik, A. (2013). Superlinear bracketing method for solving nonlinear equations. Applied Mathematics and Computation, 219(14), 7369–7376. https://doi.org/10.1016/j.amc.2012.12.064
XXII. Umar Sehrish, Muhammad Mujtaba Shaikh, Abdul Wasim Shaikh. : A NEW QUADRATURE-BASED ITERATIVE METHOD FOR SCALAR NONLINEAR EQUATIONS. J. Mech. Cont.& Math. Sci., Vol.-15, No.-10, October (2020) pp 79-93. DOI : 10.26782/jmcms.2020.10.00006
XXIII. Yasmin, N., & Junjua, M. (2012). SOME DERIVATIVE FREE ITERATIVE METHODS FOR SOLVING NONLINEAR EQUATIONS. Academic Research International, 2 (1), 75–82. www.savap.org.pkwww.journals.savap.org.pk