Modified Cubic Convergence Iterative Method for Estimating a Single Root of Nonlinear Equations

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1 J. Basic. Appl. Sci. Res., 8()9-1, 18 18, TextRoad Publication ISSN 9-44 Journal of Basic and Applied Scientific Research Modified Cubic Convergence Iterative Method for Estimating a Single Root of Nonlinear Equations Umair Khalid Qureshi *, Asif Ali Shaikh Department of Basic Science & Related Studies, Mehran University of Engineering and technology, Pakistan ABSTRACT Received: December, 17 Accepted: March 1, 18 This study has been developed a Modified Iterative Method for estimating a single root of nonlinear equations and analyzed. The proposed Modified Technique is a Modification of Regula-Falsi Method and Newton Raphson Method, and it is cubic order of convergence. The Modified Cubic Convergence Method faster than cubic Methods such as variant of Newton Raphson Method and. The comparison in table-1 for different test functions demonstrate the faster convergence of proposed method. EXCEL and C++ are implemented for the results and graphical representations. KEYWORD: Non-linear equations, cubic methods, order of convergence, Absolute percentage error, accuracy 1. INTRODUCTION Finding roots of nonlinear equations efficiently has widespread applications in numerous branches of pure science and applied science, it is deliberated in general framework of the nonlinear equations [], such as nonlinear equations For estimating a root of non-linear equations, utmost researchers and scientists took interest and lots of variants of accelerated methods had been introduced. Such as most commonly used bracketing techniques includes bisection method and regula-falsi method [1]. These methods are useful bracketing techniques which require two initial guesses. Both techniques are linear convergence order, while in some cases regula-falsi technique struggles due to sluggish convergence. Besides, Newton Raphson Method are fast converging numerical techniques but are not reliable because keeping a kind of pitfall []. Where n=,1,, However, it is most valuable and modest numerical techniques. In recent years, in literature several modifications had been done by using these techniques for solving nonlinear equations [4-6].Furthermore, modification in Newton Raphson Method to increasing order of convergence and computational efficacy a Variant of Newton Raphson Technique had been proposed by using Quadrature rule [9], such as Similar investigation, combined the Bisection, Regula-Falsi and Newton Raphson techniques are given some techniques for solving non-linear equations with better accuracy sight as well as iteration perspective [7-8]. Correspondingly, in this paper a Modified cubic iterated method has been suggested. The proposed method is assortment of regula-falsi method and Newton Raphson Method. The Modified cubic method has been compared Corresponding Author: Umair Khalid Qureshi, Department of Basic Science & Related Studies, Mehran University of Engineering and technology, Pakistan. khalidumair1@gmail.com 9

2 Qureshi and Shaikh, 18 with cubic method in reference [9, ]. The Modified Method is fast converging and more efficient to approaching the root.. MODIFIEDITERATEDMETHOD The new developed iterative method is based on Regula-Falsi Method and Newton Raphson Method, such as Or1 Where, By using 1, we get or Finally, we obtain Hence this is a proposed method.. CONVERGENCE ANALYSIS This segment has shown that the developed Method is keeping cubic convergence by using Taylor series expansion. Proof: Using the relation a in Taylor series, therefore from Taylor series we estimate, " with using this condition c and ignoring higher order term for easy to " solve, such as # 1# By using, we get # 1# $ 1# 1# $ $ 1# 1# $ 1# Thus, '( c c # % & c 1# ) *c 1#

3 J. Basic. Appl. Sci. Res., 8()9-1, 18 By using,, we get # # # 1# 1# c 1# [ 1# 1# $ ] 1# c 1# 1# [ 1# 1# ]1# c # % % 1# [ 1# # ]1# # % % 1 [ 1# # # # # % % ]1# % % $ 1# # % % 1# % % 1# # % % # % % #.. # / / # % # % & #. / # / # % # % & #. / # / Hence this has been proven that the established iterative method is cubic order of convergence. 4. NUMERICAL RESULTS This segment the established method is practical on few examples of nonlinear equations and tested developed method with the variant of Newton Raphson Method [11] and [1] From numerical result in table-1, it has been detected that the cubic iterative method is reducing the number of iterations which is less than the number of iteration of cubic methods and likewise accuracy side. Mathematical package such as C++ and EXCEL have used to justify the proposed Iterative Method. From the results and comparison of proposed cubic iterative method with the cubic methods that the proposed cubic iterative method is well execution than prevailing cubic methods. Table-1 FUNCTIONS METHODS ITERATIONS X A E% Sinx-x+1 X= e e -7.16e - x-lnx-7 X=4 x -9x+1 X= Cosx-x X=1 xe x X= e e -.86e -6 1.e e e e e e e e -8.98e -8 11

4 Qureshi and Shaikh, Figure-1: Comparative Illustration A E % of sinx-x+1 1 Figure-: Comparative Illustration A E% of x-lnx CONCLUSION This paper a Modified Iterated Method has been designed to find the root of nonlinear equations. The Modified Iterated Method has a cubic order of convergence, and it is derived from Regula-Falsi Method and Newton Raphson Method. Throughout the study, it can be concluded that the Modified Cubic Method is good execution and fast converging technique to approaching the root for the assessment of variant of Newton Raphson Method and. Henceforth the proposed method is superior and performing well for estimating a root of non-linear equations. REFERENCES [1] Solanki C., P. Thapliyal and K. Tomar, Role of Bisection Method, International Journal of Computer Applications. Technology and Research Volume Issue 8, -, 14. [] Iwetan, C. N., I. A. Fuwape, M. S. Olajide, and R. A. Adenodi, (1), Comparative Study of the Bisection and Newton Methods in solving for Zero and Extremes of a Single-Variable Function. J. of NAMP Vol [] Akram, S. and Q. U. Ann., 1, Newton Raphson Method, International Journal of Scientific &Engineering Research, Volume 6. [4] Siyal, A. A. R. A. Memon, N. M. Katbar, and F. Ahmad, 17, Modified Algorithm for Solving Nonlinear Equations in Single Variable, J. Appl. Environ. Biol. Sci., 7() , 17. [] Somroo, E., 16. On the Development of New Multi-Step Derivative Free Method to Accelerate the 1

5 J. Basic. Appl. Sci. Res., 8()9-1, 18 Convergence of Bracketing Methods for Solving, Sindh University Research Journal (Sci. Ser.) Vol. 48() [6] Sangah, A. A., 16. Comparative study of Existing bracketing methods with modified Bracketing algorithm for Solving Non-Linear Equation in single variable, Sindh University. Research Journal (Sci.Ser.) Vol. 48 (1) (16). [7] Siyal, A. A., 16. Hybrid Closed Algorithm for Solving Nonlinear Equations in one Variable, Sindh University Research Journal (Sci. Ser.) Vol. 48 (4) [8] Allame M., and N. Azad, 1.On Modified Newton Method for Solving a Nonlinear Algebraic Equations by Mid-Point, World Applied Sciences Journal 17 (1): , ISSN IDOSI Publications. [9] Weerakoon, S. And T. G. I. Fernando, A Variant of Newton s Method with Accelerated Third-Order Convergence, Applied Mathematics Letters 1, [] E. Halley, A new exact and easy method for finding the roots of equations generally and without any previous reduction, Phil. Roy. Soc. London 8,