Ball analysis for an efficient sixth convergence order scheme under weaker conditions

Yıl 2021, Cilt: 5 Sayı: 3, 445 - 453, 30.09.2021

Ioannis K Argyros Santhosh George

https://doi.org/10.31197/atnaa.746959

Cited By: 1

Öz

In this study we consider an efficient sixth order scheme for solving Banach space-valued equations. The convergence criteria in earlier studies involve higher-order derivatives limiting the applicability of these methods. In this study, we use the first derivative only in our analysis to expand the usage of these schemes. The technique we use can be used on other schemes to obtain the same advantages. Numerical experiments compare favorably our results to earlier ones.

Anahtar Kelimeler

Semi-local convergence, Banach Space, High convergence order

Kaynakça

• [1] S. Amat, S. Busquier and M. Negra, Adaptive approximation of nonlinear operators, Numer. Funct. Anal. Optim. 25 (2004), 397--405.
• [2] S. Amat, I.K. Argyros, S. Busquier and A. A. Magrenan, Local convergence and the dynamics of a two-point four parameter Jarratt-like method under weak conditions, Numer. Algor., (2017), DOI: 10.1007/s11075-016-0152-5.
• [3] I.K. Argyros, Computational Theory of Iterative Methods, Series: Studies in Computational Mathematics, 15, Editors: Chui C.K. and Wuytack L. Elsevier Publ. Company, New York (2007).
• [4] I.K. Argyros, S. George, Mathematical modeling for the solution of equations and systems of equations with applications, Volume-III, Nova Publishes, NY, 2019.
• [5] I.K. Argyros, S. George, Mathematical modeling for the solution of equations and systems of equations with applications, Volume-IV, Nova Publishes, NY, 2019.
• [6] I.K. Argyros, S. George, Magrenan, A.A., Local convergence for multi-point- parametric Chebyshev-Halley-type methods of higher convergence order, J. Comput. Appl. Math. 282, (2015), 215--224.
• [7] I.K. Argyros, A.A. Magrenan, Iterative methods and their dynamics with applications, CRC Press, New York, USA, 2017.
• [8] I.K. Argyros, A.A. Magrenan, A study on the local convergence and the dynamics of Chebyshev-Halley-type methods free from second derivative, Numer. Algorithms 71, (2015), 1--23.
• [9] A.K.H. Alzahrani, R. Behl, A.S. Alshomrani, Some higher-order iteration functions for solving nonlinear models, Appl. Math. Comput. 334, (2018), 80--93.
• [10] D.K.R. Babajee, M.Z. Dauhoo, M.T. Darvishi, A. Karami, A. Barati, Analysis of two Chebyshev-like third order methods free from second derivatives for solving systems of nonlinear equations, J. Comput. Appl. Math. 233, (2010), 2002--2012.
• [11] R. Behl, A. Cordero, S.S. Motsa, J.R, Torregrosa, Stable high-order iterative methods for solving nonlinear models. Appl. Math. Comput. 303, (2017), 70--88.
• [12] N. Choubey, B. Panday, J.P. Jaiswal, Several two-point with memory iterative methods for solving nonlinear equations, Afrika Matematika 29, (2018),435--449.
• [13] A. Cordero, J.R. Torregrosa, Variants of Newton's method for functions of several variables, Appl. Math. Comput. 183, (2006), 199--208.
• [14] A. Cordero, J.L. Hueso, E. Martinez, J.R. Torregrosa, A modified Newton-Jarratt's composition, Numer. Algor. 55, (2010), 87--99.
• [15] A. Cordero, J.R. Torregrosa, Variants of Newton's method using fifth-order quadrature formulas. Appl. Math. Com- put. 190, (2007),686--698.
• [16] A. Cordero, J.L. Hueso, E Martinez, J.R. Torregrosa, Increasing the convergence order of an iterative method for nonlinear systems. Appl. Math. Lett. 25, (2012),2369-2374.
• [17] M.T. Darvishi, A. Barati, Super cubic iterative methods to solve systems of nonlinear equations. Appl. Math. Comput. 188, (2007), 1678--1685.
• [18] H. Esmaeili, M. Ahmadi, An efficient three-step method to solve system of non linear equations. Appl. Math. Comput. 266, (2015), 1093--1101.
• [19] X. Fang, Q. Ni, M. Zeng, A modified quasi-Newton method for nonlinear equations. J. Comput. Appl. Math. 328, (2018), 44--58.
• [20] L. Fousse, G. Hanrot, V. Lefvre, P. Plissier, P. Zimmermann, MPFR: a multiple-precision binary floating-point library with correct rounding. ACM Trans. Math. Softw. 33(2), 15 (2007).
• [21] H.H.H. Homeier, A modified Newton method with cubic convergence: the multivariate case, J. Comput. Appl. Math. 169, (2004), 161--169.
• [22] L.O. Jay, A note on Q-order of convergence, BIT 41, 422--429 (2001).
• [23] T. Lotfo, P. Bakhtiari, A. Cordero, K. Mahdiani, J.R. Torregrosa, Some new efficient multipoint iterative methods for solving nonlinear systems of equations, Int. J. Comput. Math. 92, (2015), 1921-1934.
• [24] A.A. Magrenan, Different anomalies in a Jarratt family of iterative root finding methods, Appl. Math. Comput. 233, (2014), 29--38.
• [25] A.A. Magrenan, A new tool to study real dynamics: The convergence plane, Appl. Math. Comput. 248, (2014), 29--38.
• [26] J.M. McNamee, Numerical Methods for Roots of Polynomials, Part I, Elsevier, Amsterdam (2007).
• [27] M.A. Noor, M. Waseem, Some iterative methods for solving a system of nonlinear equations, Comput. Math. Appl. 57, (2009)19, 101--106.
• [28] J.M. Ortega, W.C. Rheinboldt, Iterative Solutions of Nonlinear Equations in Several Variables, Academic Press, New York, USA (1970).
• [29] A.M. Ostrowski, Solution of Equation and Systems of Equations, Academic Press, New York (1960).
• [30] J.R. Sharma, R. Sharma, A. Bahl, An improved Newton-Traub composition for solving systems of nonlinear equations, Appl. Math. Comput. 290, (2016), 98--110.
• [31] J.R. Sharma, H. Arora, Improved Newton-like methods for solving systems of nonlinear equations, SeMA 74,(2017), 147--163.
• [32] J.R. Sharma, H. Arora, Efficient derivative-free numerical methods for solving systems of nonlinear equations, Comput.Appl. Math. 35, (2016), 269--284.