Local convergence for composite Chebyshev-type methods

Yıl 2018, Cilt: 1 Sayı: 1, 84 - 90, 30.09.2018

İoannis K Argyros Santhosh George

https://doi.org/10.33434/cams.441220

Öz

We replace Chebyshev's method for solving equations requiring the second derivative by a Chebyshev-type second derivative free method. The local convergence analysis of the new method is provided using hypotheses only on the first derivative in contrast to the Chebyshev method using hypotheses on the second derivative. This way we extend the applicability of the method. Numerical examples are also used to test the convergence criteria and to obtain error bounds and also the radius of convergence.

Anahtar Kelimeler

Chebyshev method, Newton method, Fr\'echet derivative, Local convergence, Divided differences

Kaynakça

• [1] I.K. Argyros, Convergence and applications of Newton-type iteration, Springer, New York, 2008.
• [2] I.K Argyros,A unified local-semilocal convergence analysis and applications for two-point Newton-like methods in Banach spaces, J. Math. Appl., 288, (2004), 374-397.
• [3] I. K. Argyros,A. A. Magre˜na˜n, Iterative Methods and their dynamics with applications: A Contemporary Study, CRC Press, 2017.
• [4] I. K. Argyros, S. George, N. Thapa, Mathematical Modeling For The Solution Of Equations And Systems Of Equations With Applications, Volume-I, Nova Publishes, NY, 2018.
• [5] I. K. Argyros, S. George, N. Thapa, Mathematical Modeling For The Solution Of Equations And Systems Of Equations With Applications, Volume-II, Nova Publishes, NY, 2018.
• [6] M. Grau-Sanchez, A . Grau, M. Noguera, Ostrowski type methods for solving systems of nonlinear equations, Appl. Math. Comput., 218, (2011), 2377-2385.
• [7] J. Kou, Y. Li, X. Wang, Some variants of Ostrowski’s method with seventh-order convergence, J. Comput. Appl. Math., 209, (2007), 153-159.
• [8] H. T. Kung, J. F. Traub, Optimal order of one-point and multipoint iteration, J. ACM 21, (1974), 643-651.
• [9] A. A. Magrenan, Different anomalies in a Jarratt family of iterative root finding methods, Appl. Math. Comput. 233, (2014), 29-38.
• [10] A. A. Magren´an, A new tool to study real dynamics: The convergence plane, Appl. Math. Comput. 248, (2014), 29-38.
• [11] J. M. Ortega and R. C. Rheinboldt, Iterative solution of nonlinear equations in several variables, Academic Press, 1970, New York.
• [12] M. S. Petkovic, B. Neta, L. Petkovic, J. Dˇzuniˇc, Multipoint methods for solving nonlinear equations, Elsevier, 2013.
• [13] J.F.Traub, Iterative methods for the solution of equations, AMS Chelsea Publishing, 1982.
• [14] S. Weerkoon, T. G. I. Fernando, A variant of Newton’s method with accelerated third-order convergence, Appl. math. Lett., 13, (2000), 87-93.