On an iterative method without inverses of derivatives for solving equations

Year 2020, Volume: 4 Issue: 2, 67 - 76, 30.06.2020

Santhosh George , İ. K Argyros

https://doi.org/10.31197/atnaa.624777

Cited By: 1

Abstract

We present the semi-local convergence analysis of a  Potra-type method to solve equations involving Banach space valued operators. The analysis is based on our ideas of  recurrent functions and restricted convergence region. The study is completed using numerical examples.

Keywords

multi step method, local convergence, system of equations, Frechet derivative, Banach space

References

• I.K. Argyros, Computational theory of iterative methods. Series: Studies in Computational Mathematics, 15, Editors: C.K.Chui and L. Wuytack, Elsevier Publ. Co. New York, U.S.A, 2007.
• I.K. Argyros, Convergence and applications of Newton-type iterations, Springer Verlag, New York, 2008.
• I. K. Argyros,A. A. Magre\~na\~n, Iterative Methods and their dynamics with applications: A Contemporary Study, CRC Press, 2017.
• R Behl, A. Cordero, S. S. Motsa, J. R. Torregrosa, Stable high order iterative methods for solving nonlinear models, Applied Mathematics and Computation, 303(15), (2017), 70--88.
• A. Cordero, J. R. Torregrosa, M. P. Vassileva, Pseudocomposition: a technique to design predictor-corrector methods for systems of nonlinear equations, Appl. Math. Comput., 218(2012), 11496--11504.
• K. Madru, J. Jayaraman, Some higher order Newton-like methods for solving system of nonlinear equations and its applications, Int. J. Appl. Comput. Math.,(2017) 3:2213--2230.
• J. A. Ezquerro, M.A. Hern\'{a}ndez, Recurrence relations for Chebyshev-type methods, Appl. Math. Optim. 41(2), (2000), 227-236 .
• J. A. Ezquerro, J.M. Guti\'errez, M.A. Hern\'andez, Avoiding the computation of the second-Fr\'echet derivative in the convex acceleration of Newton's method, J. Comput. Appl. Math., 96, (1998), 1-12.
• J. A. Ezquerro, M.A. Hern\'{a}ndez, Multipoint super-Halley type approximation algorithms in Banach spaces, Numer. Funct. Anal. Optimiz., 21 (7\&8), (2000), 845--858.
• J. A. Ezquerro, M.A. Hern\'{a}ndez, A modification of the super-Halley method under mild differentiability condition, J. Comput. Appl. Math., 114, (2000), 405--409.
• M. Grau-Sanchez, A. Grau, M. Noguera, Ostrowski type methods for solving systems of nonlinear equations, Appl. Math. Comput., 218 (2011), 2377--2385.
• J. M. Guti\'{e}rrez, A.A. Magre\={n}\'{a}n and N. Romero, On the semi-local convergence of Newton-Kantorovich method under center-Lipschitz conditions, Applied Mathematics and Computation, 221 (2013), 79-88.
• L.V. Kantorovich, G.P. Akilov, Functional Analysis, Pergamon Press, Oxford, 1982.
• A. A. Magre\'n\~an, Different anomalies in a Jarratt family of iterative root finding methods, Appl. Math. Comput. 233, (2014), 29-38. A. A. Magre\'n\~an, A new tool to study real dynamics: The convergence plane, Appl. Math. Comput. 248, (2014), 29-38.
• M. S. Petkovic, B. Neta, L. Petkovic, J. D\v{z}uni\v{c}, Multipoint methods for solving nonlinear equations, Elsevier, 2013.
• F. A. Potra and V. Pt\'{a}k, Nondiscrete Induction and Iterative Processes, in: Research Notes in Mathematics, Vol. 103, Pitman, Boston, 1984.
• F. A. Potra, On an iterative algorithm of order 1.839... for solving nonlinear operator equations, Numer. Funct. Anal. and Optimiz. 7, 1, (1984-85), 75--106.
• S. M. Shakhno, On a Kurchatov's method of linear interpolation for solving nonlinear equations, PAMM-Proc. Appl. Math. Mech.4, (2004), 650-651.
• S. M. Shakhno, On the difference method with quadratic convergence for solving nonlinear equations, Matem. Stud, 26, (2006), 105-110 (In Ukrainian).
• J.R. Sharma, H. Arora, On efficient weighted-Newton methods for solving system of nonlinear equations, Appl. Math. Comput., 222(2013), 497--506.
• J.R. Sharma, H. Arora, Efficient Jarratt-like methods for solving systems of nonlinear equations, Calcolo, 51(2014), 193--210.
• J.R. Sharma,P. Gaupta, An efficient fifth order method for solving systems of nonlinear equations, Comput. Math. Appl.,67(2014), 591-601.
• J.R. Sharma, R. K. Guha, R. Sharma, An efficient fourth-order weighted Newton method for systems of nonlinear equations, Numer. Algorithms, 62(2013), 307-323.
• J.F.Traub, Iterative methods for the solution of equations, AMS Chelsea Publishing, 1982.
• S. Weerakoon, T.G.I. Fernando, A variant of Newton's method with accelerated third-order convergence, Appl. Math. Lett. 13, (2000), 87-93.