Comparison of Halley-Chebyshev Method with Several Nonlinear equation Solving Methods Methods

In this paper, we present one of the most important numerical analysis problems that we find in the roots of the nonlinear equation. In numerical analysis and numerical computing, there are many methods that we can approximate the roots of this equation. We present here several different methods, such as Halley's method, Chebyshev's method, Newton's method, and other new methods presented in papers and journals, and compare them. In the end, we get a good and attractive result

Several New Families of Jarratt’s Method for Solving Systems of Nonlinear Equations

In this study, we suggest and analyze a new and wide general class of Jarratt’s method for solving systems of nonlinear equations. These methods have fourth-order convergence and do not require the evaluation of any second or higher-order Fréchet derivatives. In terms of computational cost, all these methods require evaluations of one function and two first-order Fréchet derivatives. The performance of proposed methods is compared with their closest competitors in a series of numerical experiments. It is worth mentioning that all the methods considered here are found to be effective and comparable to the robust methods available in the literature

ROOT FINDING FOR NONLINEAR EQUATIONS

Nonlinear equations /systems appear in most science and engineering models. For example, when solving eigen value problems, optimization problems, differential equations, in circuit analysis, analysis of state equations for a real gas, in mechanical motions /oscillations, weather forecasting, integral equations, image processing and many other fields of engineering designing processes. Nonlinear systems /problems are difficult to solve manually but they occur naturally in fluid motions, heat transfer, wave motions, chemical reactions, etc. This study deals with construction of iterative methods for nonlinear root finding, applying Taylor’s series approximation of a nonlinear function f(x) combined with a new correction term in a quadratic or cubic model. Competent iterative algorithms of higher order were investigated. For test of convergence and efficiency, we applied basic theorems and solved some equations in C++. Keywords – nonlinear equations, Taylor’s approximation, iterative algorithms for roots, error correctio

Iterative Algorithms for Nonlinear Equations and Dynamical Behaviors: Applications

Numerical iteration methods for solving the roots of nonlinear transcendental or algebraic model equations (in 1D, 2D or 3D) are useful in most applied sciences (Biology, physics, mathematics, Chemistry…) and in engineering, for example, problems of beam deflections. This article presents new iterative algorithms for finding roots of nonlinear equations applying some fixed point transformation and interpolation. A method for solving nonlinear systems (in higher dimensions, for multi-variables) is also considered. Our main focus is on methods not involving the equation f(x) in problem and or its derivatives. These new algorithm can be considered as the acceleration convergence of several existing methods. For convergence and efficiency proofs and applications, we solve deflection of a beam differential equation and some test experiments in in Matlab.  Different (real & complex) dynamical (convergence plane) analyzes are also shown graphically. Keywords: nonlinear equations, deflection of beam, iterations, dynamical analysis, applications, 2

Some Root Finding With Extensions to Higher Dimensions

Root finding is an issue in scientific computing. Because most nonlinear problems in science and engineering can be considered as the root finding problems, directly or indirectly. The research in numerical modeling for root finding is still going on. In this study, fixed point iterative methods for solving simple real roots of nonlinear equations, which improve convergence of some existing methods, are thorough. Derivative estimations up to the third order (in root finding, some recent ideas) are applied in Taylor’s approximation of a nonlinear equation by a cubic model to achieve efficient iterative methods. We may also discuss possible extensions to two dimensions and consider Newton’s method and Halley’s method in 1D and 2D problem solving. Several examples for test of efficiency and convergence analyses using C++ are offered. And some engineering applications of root finding are conferred. Graphical demonstrations are supported with matlab basic tools. Keywords: engineering applications, derivative estimations, iterative methods, simple roots, Taylor’s approximation

Local convergence of a family of iterative methods for Hammerstein equations

[EN] In this paper we give a local convergence result for a uniparametric family of iterative methods for nonlinear equations in Banach spaces. We assume boundedness conditions involving only the first Fr,chet derivative, instead of using boundedness conditions for high order derivatives as it is usual in studies of semilocal convergence, which is a drawback for solving some practical problems. The existence and uniqueness theorem that establishes the convergence balls of these methods is obtained. We apply this theory to different examples, including a nonlinear Hammerstein equation that have many applications in chemistry and appears in problems of electro-magnetic fluid dynamics or in the kinetic theory of gases. With these examples we illustrate the advantages of these results. The global convergence of the method is addressed by analysing the behaviour of the methods on complex polynomials of second degree.This research was supported by Ministerio de Ciencia y Tecnologia MTM2014-52016-C2-02.This research was supported by Ministerio de Ciencia y Tecnología MTM2014-52016-C2-02.Martínez Molada, E.; Singh, S.; Hueso Pagoaga, JL.; Gupta, D. (2016). Local convergence of a family of iterative methods for Hammerstein equations. Journal of Mathematical Chemistry. 54(7):1370-1386. https://doi.org/10.1007/s10910-016-0602-2S13701386547I.K. Argyros, S. Hilout, M.A. Tabatabai, Mathematical Modelling with Applications in Biosciences and Engineering (Nova Publishers, New York, 2011)J.F. Traub, Iterative Methods for the Solution of Equations (Prentice-Hall, Englewood Cliffs, New Jersey, 1964)A.M. Ostrowski, Solutions of Equations in Euclidean and Banach Spaces (Academic Press, New York, 1973)I.K. Argyros, J.A. Ezquerro, J.M. Gutiárrez, M.A. Hernández, S. Hilout, On the semilocal convergence of efficient ChebyshevSecant-type methods. J. Comput. Appl. Math. 235, 3195–3206 (2011)José L. Hueso, E. Martínez, Semilocal convergence of a family of iterative methods in Banach spaces. Numer. Algorithms 67, 365–384 (2014)X. Wang, C. Gu, J. Kou, Semilocal convergence of a multipoint fourth-order super-Halley method in Banach spaces. Numer. Algorithms 54, 497–516 (2011)J. Kou, Y. Li, X. Wang, A variant of super Halley method with accelerated fourth-order convergence. Appl. Math. Comput. 186, 535–539 (2007)L. Zheng, C. Gu, Recurrence relations for semilocal convergence of a fifth-order method in Banach spaces. Numer. Algorithms 59, 623–638 (2012)S. Amat, M.A. Hernández, N. Romero, A modified Chebyshevs iterative method with at least sixth order of convergence. Appl. Math. Comput. 206, 164–174 (2008)X. Wang, J. Kou, C. Gu, Semilocal convergence of a sixth-order Jarratt method in Banach spaces. Numer. Algorithms 57, 441–456 (2011)A. Cordero, J.A. Ezquerro, M.A. Hernández-Verón, J.R. Torregrosa, On the local convergence of a fifth-order iterative method in Banach spaces. Appl. Math. Comput. 251, 396–403 (2015)I.K. Argyros, S. Hilout, On the local convergence of fast two-step Newton-like methods for solving nonlinear equations. J. Comput. Appl. Math. 245, 1–9 (2013)S. Weerakoon, T.G.I. Fernando, A variant of Newton’s method with accelerated third-order convergence. Appl. Math. Lett. 13(8), 87–93 (2000)X. Feng, Y. He, High order oterative methods without derivatives for solving nonlinear equations. Appl. Math. Comput. 186, 1617–1623 (2007)X. Wang, J. Kou, Y. Li, Modified Jarratt method with sixth-order convergence. Appl. Math. Lett. 22, 1798–1802 (2009)A.D. Polyanin, A.V. Manzhirov, Handbook of Integral Equations (CRC Press, Boca Raton, 1998)S. Plaza, N. Romero, Attracting cycles for the relaxed Newton’s method. J. Comput. Appl. Math. 235(10), 3238–3244 (2011)A. Cordero, J.R. Torregrosa, P. Vindel, Study of the dynamics of third-order iterative methods on quadratic polynomials. Int. J. Comput. Math. 89(13–14), 1826–1836 (2012)Gerardo Honorato, Sergio Plaza, Natalia Romero, Dynamics of a higher-order family of iterative methods. J. Complex. 27(2), 221–229 (2011)J.M. Gutirrez, M.A. Hernández, N. Romero, Dynamics of a new family of iterative processes for quadratic polynomials. J. Comput. Appl. Math. 233(10), 2688–2695 (2010)I.K. Argyros, A.A. Magreñan, A study on the local convergence and dynamics of Chebyshev-Halley-type methods free from second derivative. Numer. Algorithms. doi: 10.1007/s11075-015-9981-xI.K. Argyros, S. George, Local convergence of modified Halley-like methods with less computation of inversion (Novi Sad J. Math, Draft version, 2015

A Unified Convergence Analysis for Some Two-Point Type Methods for Nonsmooth Operators

The aim of this paper is the approximation of nonlinear equations using iterative methods. We present a unified convergence analysis for some two-point type methods. This way we compare specializations of our method using not necessarily the same convergence criteria. We consider both semilocal and local analysis. In the first one, the hypotheses are imposed on the initial guess and in the second on the solution. The results can be applied for smooth and nonsmooth operators.Research of the first and third authors supported in part by Programa de Apoyo a la investigación de la fundación Séneca-Agencia de Ciencia y Tecnología de la Región de Murcia 20928/PI/18 and by MTM2015-64382-P. Research of the fourth and fifth authors supported by Ministerio de Economía y Competitividad under grant MTM2014-52016-C2-1P. This research received no external funding

Basins of attraction for various Steffensen-Type methods

The dynamical behavior of different Steffensen-type methods is analyzed. We check the chaotic behaviors alongside the convergence radii (understood as the wideness of the basin of attraction) needed by Steffensen-type methods, that is, derivative-free iteration functions, to converge to a root and compare the results using different numerical tests. We will conclude that the convergence radii (and the stability) of Steffensen-type methods are improved by increasing the convergence order. The computer programming package MATHEMATICA provides a powerful but easy environment for all aspects of numerics. This paper puts on show one of the application of this computer algebra system in finding fixed points of iteration functions.The authors are indebted to the referees for some interesting comments and suggestions. This research was supported by Ministerio de Ciencia y Tecnologia MTM2011-28636-C02-02.Cordero Barbero, A.; Soleymani, F.; Torregrosa Sánchez, JR.; Shateyi, S. (2014). Basins of attraction for various Steffensen-Type methods. Journal of Applied Mathematics. 2014. https://doi.org/10.1155/2014/539707S2014Soleymani, F. (2011). Optimal fourth-order iterative method free from derivative. Miskolc Mathematical Notes, 12(2), 255. doi:10.18514/mmn.2011.303Zheng, Q., Zhao, P., Zhang, L., & Ma, W. (2010). Variants of Steffensen-secant method and applications. Applied Mathematics and Computation, 216(12), 3486-3496. doi:10.1016/j.amc.2010.04.058Neta, B., Scott, M., & Chun, C. (2012). Basins of attraction for several methods to find simple roots of nonlinear equations. Applied Mathematics and Computation, 218(21), 10548-10556. doi:10.1016/j.amc.2012.04.017Neta, B., & Scott, M. (2013). On a family of Halley-like methods to find simple roots of nonlinear equations. Applied Mathematics and Computation, 219(15), 7940-7944. doi:10.1016/j.amc.2013.02.035Neta, B., & Chun, C. (2013). On a family of Laguerre methods to find multiple roots of nonlinear equations. 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Convergence Theorem for a Family of New Modified Halley’s Method in Banach Space

We establish convergence theorems of Newton-Kantorovich type for a family of new modified Halley’s method in Banach space to solve nonlinear operator equations. We present the corresponding error estimate. To show the application of our theorems, two numerical examples are given