Some Classes of third and Fourth-order iterative methods for solving nonlinear equations

The object of the present work is to present the new classes of third-order and fourth-order iterative methods for solving nonlinear equations. Our third-order method includes methods of Weerakoon \cite{Weerakoon}, Homeier \cite{Homeier2}, Chun \cite{Chun} e.t.c. as particular cases. After that we make this third-order method to fourth-order (optimal) by using a single weight function rather than using two different weight functions in \cite{Soleymani}. Finally some examples are given to illustrate the performance of the our method by comparing with new existing third and fourth-order methods.Comment: arXiv admin note: substantial text overlap with arXiv:1307.733

Evaluation of iterative methods for solving nonlinear scalar equations

This study is aimed at performing a comprehensive numerical evalua-tion of the iterative solution techniques without memory for solving non-linear scalar equations with simple real roots, in order to specify the most efficient and applicable methods for practical purposes. In this regard, the capabilities of the methods for applicable purposes are be evaluated, in which the ability of the methods to solve different types of nonlinear equations is be studied. First, 26 different iterative methods with the best performance are reviewed. These methods are selected based on performing more than 46000 analyses on 166 different available nonlinear solvers. For the easier application of the techniques, consistent mathematical notation is employed to present reviewed approaches. After presenting the diverse methodologies suggested for solving nonlinear equations, the performances of the reviewed methods are evaluated by solving 28 different nonlinear equations. The utilized test functions, which are selected from the re-viewed research works, are solved by all schemes and by assuming different initial guesses. To select the initial guesses, endpoints of five neighboring intervals with different sizes around the root of test functions are used. Therefore, each problem is solved by ten different starting points. In order to calculate novel computational efficiency indices and rank them accu-rately, the results of the obtained solutions are used. These data include the number of iterations, number of function evaluations, and convergence times. In addition, the successful runs for each process are used to rank the evaluated schemes. Although, in general, the choice of the method de-pends on the problem in practice, but in practical applications, especially in engineering, changing the solution method for different problems is not feasible all the time, and accordingly, the findings of the present study can be used as a guide to specify the fastest and most appropriate solution technique for solving nonlinear problems

Three New Optimal Fourth-Order Iterative Methods to Solve Nonlinear Equations

We present new modifications to Newton's method for solving nonlinear equations. The analysis of convergence shows that these methods have fourth-order convergence. Each of the three methods uses three functional evaluations. Thus, according to Kung-Traub's conjecture, these are optimal methods. With the previous ideas, we extend the analysis to functions with multiple roots. Several numerical examples are given to illustrate that the presented methods have better performance compared with Newton's classical method and other methods of fourth-order convergence recently published

On generalization based on Bi et al. Iterative methods with eighth-order convergence for solving nonlinear equations

The primary goal of this work is to provide a general optimal three-step class of iterative methods based on the schemes designed by Bi et al. (2009). Accordingly, it requires four functional evaluations per iteration with eighth-order convergence. Consequently, it satisfies Kung and Traub's conjecture relevant to construction optimal methods without memory. Moreover, some concrete methods of this class are shown and implemented numerically, showing their applicability and efficiency.The authors thank the anonymous referees for their valuable comments and for the suggestions to improve the readability of the paper. This research was supported by Islamic Azad University, Hamedan Branch, and Ministerio de Ciencia y Tecnologia MTM2011-28636-C02-02.Lotfi, T.; Cordero Barbero, A.; Torregrosa Sánchez, JR.; Abadi, MA.; Zadeh, MM. (2014). On generalization based on Bi et al. Iterative methods with eighth-order convergence for solving nonlinear equations. 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Eighth-order methods with high efficiency index for solving nonlinear equations. Applied Mathematics and Computation, 215(9), 3449-3454. doi:10.1016/j.amc.2009.10.040Wang, X., & Liu, L. (2010). New eighth-order iterative methods for solving nonlinear equations. Journal of Computational and Applied Mathematics, 234(5), 1611-1620. doi:10.1016/j.cam.2010.03.002Wang, X., & Liu, L. (2010). Modified Ostrowski’s method with eighth-order convergence and high efficiency index. Applied Mathematics Letters, 23(5), 549-554. doi:10.1016/j.aml.2010.01.009Sharma, J. R., & Sharma, R. (2009). A new family of modified Ostrowski’s methods with accelerated eighth order convergence. Numerical Algorithms, 54(4), 445-458. doi:10.1007/s11075-009-9345-5Soleymani, F. (2011). Novel Computational Iterative Methods with Optimal Order for Nonlinear Equations. Advances in Numerical Analysis, 2011, 1-10. doi:10.1155/2011/270903Soleymani, F., Sharifi, M., & Somayeh Mousavi, B. (2011). An Improvement of Ostrowski’s and King’s Techniques with Optimal Convergence Order Eight. Journal of Optimization Theory and Applications, 153(1), 225-236. doi:10.1007/s10957-011-9929-9Soleymani, F., Karimi Vanani, S., & Afghani, A. (2011). A General Three-Step Class of Optimal Iterations for Nonlinear Equations. Mathematical Problems in Engineering, 2011, 1-10. doi:10.1155/2011/469512Soleymani, F., Vanani, S. K., Khan, M., & Sharifi, M. (2012). Some modifications of King’s family with optimal eighth order of convergence. Mathematical and Computer Modelling, 55(3-4), 1373-1380. doi:10.1016/j.mcm.2011.10.016Soleymani, F., Karimi Vanani, S., & Jamali Paghaleh, M. (2012). A Class of Three-Step Derivative-Free Root Solvers with Optimal Convergence Order. Journal of Applied Mathematics, 2012, 1-15. doi:10.1155/2012/568740Thukral, R. (2010). A new eighth-order iterative method for solving nonlinear equations. 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