Local convergence of a parameter based iteration with Holder continuous derivative in Banach spaces

[EN] The local convergence analysis of a parameter based iteration with Hölder continuous first derivative is studied for finding solutions of nonlinear equations in Banach spaces. It generalizes the local convergence analysis under Lipschitz continuous first derivative. The main contribution is to show the applicability to those problems for which Lipschitz condition fails without using higher order derivatives. An existence-uniqueness theorem along with the derivation of error bounds for the solution is established. Different numerical examples including nonlinear Hammerstein equation are solved. The radii of balls of convergence for them are obtained. Substantial improvements of these radii are found in comparison to some other existing methods under similar conditions for all examples considered.The authors thank the referees for their valuable comments which have improved the presentation of the paper. The authors thankfully acknowledge the financial assistance provided by Council of Scientific and Industrial Research (CSIR), New Delhi, India.Singh, S.; Gupta, DK.; Badoni, RP.; Martínez Molada, E.; Hueso Pagoaga, JL. (2017). Local convergence of a parameter based iteration with Holder continuous derivative in Banach spaces. CALCOLO. 54(2):527-539. doi:10.1007/s10092-016-0197-9S527539542Argyros, I.K., Hilout, S.: Numerical methods in nonlinear analysis. World Scientific Publ. Comp, New Jersey (2013)Argyros, I.K., Hilout, S., Tabatabai, M.A.: Mathematical modelling with applications in biosciences and engineering. Nova Publishers, New York (2011)Singh, S., Gupta, D.K., Martínez, E., Hueso, J.L.: Semilocal and local convergence of a fifth order iteration with Fréchet derivative satisfying Hölder condition. Appl. Math. Comput. 276, 266–277 (2016)Traub, J.F.: Iterative methods for the solution of equations. Prentice-Hall, Englewood Cliffs (1964)Rall, L.B.: Computational solution of nonlinear operator equations, reprint edn. R. E. Krieger, New York (2007)Cordero, A., Ezquerro, J.A., Hernández-Verón, M.A., Torregrosa, J.R.: On the local convergence of a fifth-order iterative method in Banach spaces. Appl. Math. Comput. 251, 396–403 (2015)Argyros, I.K., Hilout, A.S.: On the local convergence of fast two-step Newton-like methods for solving nonlinear equations. J. Comput. Appl. Math. 245, 1–9 (2013)Argyros, I.K., Behl, R., Motsa, S.S.: Local convergence of an efficient high convergence order method using hypothesis only on the first derivative. Algorithms 8, 1076–1087 (2015)Kantorovich, L.V., Akilov, G.P.: Functional analysis. Pergamon Press, Oxford (1982)Argyros, I.K., Magreñán, A.A.: A study on the local convergence and dynamics of Chebyshev-Halley-type methods free from second derivative. Numer. Algorithms 71, 1–23 (2016)Li, D., Liu, P., Kou, J.: An improvement ofthe Chebyshev-Halley methods free from second derivative. Appl. Math. Comput. 235, 221–225 (2014)Argyros, I.K., George, S.: Local convergence of deformed Halley method in Banach space under Holder continuity conditions. J. Nonlinear Sci. Appl. 8, 246–254 (2015)Argyros, I.K., Khattri, S.K.: Local convergence for a family of third order methods in Banach spaces. J. Math. 46, 53–62 (2014)Argyros, I.K., George, S., Magreñán, A.A.: Local convergence for multi-point-parametric Chebyshev-Halley-type methods of higher convergence order. J. Comput. Appl. Math. 282, 215–224 (2015)Argyros, I.K., George, S.: Local convergence of modified Halley-like methods with less computation of inversion. Novi. Sad. J. Math. 45, 47–58 (2015)Xiao, X.Y., Yin, H.W.: Increasing the order of convergence for iterative methods to solve nonlinear systems. Calcolo (2015). doi: 10.1007/s10092-015-0149-9Martínez, E., Singh, S., Hueso, J.L., Gupta, D.K.: Enlarging the convergence domain in local convergence studies for iterative methods in Banach spaces. Appl. Math. Comput. 281, 252–265 (2016

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

Basins of attraction for several methods to find simple roots of nonlinear equations

The article of record as published may be located at http://dx.doi.org/10.1016/j.amc.2010.04.017There are many methods for solving a nonlinear algebraic equation. The methods are clas- sified by the order, informational efficiency and efficiency index. Here we consider other criteria, namely the basin of attraction of the method and its dependence on the order. We discuss several third and fourth order methods to find simple zeros. The relationship between the basins of attraction and the corresponding conjugacy maps will be discussed in numerical experiments. The effect of the extraneous roots on the basins is also discussed

A Note on the “Constructing” of Nonstationary Methods for Solving Nonlinear Equations with Raised Speed of Convergence

This paper is partially supported by project ISM-4 of Department for Scientific Research, “Paisii Hilendarski” University of Plovdiv.In this paper we give methodological survey of “contemporary methods” for solving the nonlinear equation f(x) = 0. The reason for this review is that many authors in present days rediscovered such classical methods. Here we develop one methodological schema for constructing nonstationary methods with a preliminary chosen speed of convergence

A study of the local convergence of a fifth order iterative method

[EN] We present a local convergence study of a fifth order iterative method to approximate a locally unique root of nonlinear equations. The analysis is discussed under the assumption that first order Frechet derivative satisfies the Lipschitz continuity condition. Moreover, we consider the derivative free method that obtained through approximating the derivative with divided difference along with the local convergence study. Finally, we provide computable radii and error bounds based on the Lipschitz constant for both cases. Some of the numerical examples are worked out and compared the results with existing methods.This research was partially supported by Ministerio de Economia y Competitividad under grant PGC2018-095896-B-C21-C22.Singh, S.; Martínez Molada, E.; Maroju, P.; Behl, R. (2020). A study of the local convergence of a fifth order iterative method. Indian Journal of Pure and Applied Mathematics. 51(2):439-455. https://doi.org/10.1007/s13226-020-0409-5S439455512A. Constantinides and N. Mostoufi, Numerical Methods for Chemical Engineers with MATLAB Applications, Prentice Hall PTR, New Jersey, (1999).J. M. Douglas, Process Dynamics and Control, Prentice Hall, Englewood Cliffs, (1972).M. Shacham, An improved memory method for the solution of a nonlinear equation, Chem. Eng. Sci., 44 (1989), 1495–1501.J. M. Ortega and W. C. Rheinboldt, Iterative solution of nonlinear equations in several variables, Academic Press, New-York, (1970).J. R. Sharma and H. Arora, A novel derivative free algorithm with seventh order convergence for solving systems of nonlinear equations, Numer. Algorithms, 67 (2014), 917–933.I. K. Argyros, A. A. Magreńan, and L. Orcos, Local convergence and a chemical application of derivative free root finding methods with one parameter based on interpolation, J. Math. Chem., 54 (2016), 1404–1416.E. L. Allgower and K. Georg, Lectures in Applied Mathematics, American Mathematical Society (Providence, RI) 26, 723–762.A. V. Rangan, D. Cai, and L. Tao, Numerical methods for solving moment equations in kinetic theory of neuronal network dynamics, J. Comput. Phys., 221 (2007), 781–798.A. Nejat and C. Ollivier-Gooch, Effect of discretization order on preconditioning and convergence of a high-order unstructured Newton-GMRES solver for the Euler equations, J. Comput. Phys., 227 (2008), 2366–2386.C. Grosan and A. Abraham, A new approach for solving nonlinear equations systems, IEEE Trans. Syst. Man Cybernet Part A: System Humans, 38 (2008), 698–714.F. Awawdeh, On new iterative method for solving systems of nonlinear equations, Numer. Algorithms, 54 (2010), 395–409.I. G. Tsoulos and A. Stavrakoudis, On locating all roots of systems of nonlinear equations inside bounded domain using global optimization methods, Nonlinear Anal. Real World Appl., 11 (2010), 2465–2471.E. Martínez, S. Singh, J. L. Hueso, and D. K. Gupta, Enlarging the convergence domain in local convergence studies for iterative methods in Banach spaces, Appl. Math. Comput., 281 (2016), 252–265.S. Singh, D. K. Gupta, E. Martínez, and J. L. Hueso, Semi local and local convergence of a fifth order iteration with Fréchet derivative satisfying Hölder condition, Appl. Math. Comput., 276 (2016), 266–277.I. K. Argyros and S. George, Local convergence of modified Halley-like methods with less computation of inversion, Novi. Sad.J. Math., 45 (2015), 47–58.I. K. Argyros, R. Behl, and S. S. Motsa, Local Convergence of an Efficient High Convergence Order Method Using Hypothesis Only on the First Derivative Algorithms 2015, 8, 1076–1087; doi:https://doi.org/10.3390/a8041076.A. Cordero, J. L. Hueso, E. Martínez, and J. R. Torregrosa, Increasing the convergence order of an iterative method for nonlinear systems, Appl. Math. Lett., 25 (2012), 2369–2374.I. K. Argyros and A. A. Magreñán, A study on the local convergence and dynamics of Chebyshev- Halley-type methods free from second derivative, Numer. Algorithms71 (2016), 1–23.M. Grau-Sánchez, Á Grau, asnd M. Noguera, Frozen divided difference scheme for solving systems of nonlinear equations, J. Comput. Appl. Math., 235 (2011), 1739–1743.M. Grau-Sánchez, M. Noguera, and S. Amat, On the approximation of derivatives using divided difference operators preserving the local convergence order of iterative methods, J. Comput. Appl. Math., 237 (2013), 363–372

New Iterative Method for Solving Nonlinear Equations

The aim of this paper is to propose an efficient three steps iterative method for finding the zeros of the nonlinear equation f(x)=0 . Starting with a suitably chosen , the method generates a sequence of iterates converging to the root. The convergence analysis is proved to establish its five order of convergence. Several examples are given to illustrate the efficiency of the proposed new method and its comparison with other methods