Extended convergence of two-step iterative methods for solving equations with applications

Ioannis K. Argyros; Santhosh George

doi:10.33993/jnaat532-1178

Authors

Ioannis K. Argyros Department of Mathematical Sciences, Cameron University, USA https://orcid.org/0000-0002-9189-9298
Santhosh George National Institute of Technology Karnataka, India https://orcid.org/0000-0002-3530-5539

DOI:

https://doi.org/10.33993/jnaat532-1178

Keywords:

Banach space, restricted convergence region, convergence of iterative method.

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Abstract

The convergence of two-step iterative methods of third and fourth order of convergence are studied under weaker hypotheses than in earlier works using our new idea of the restricted convergence region. This way, we obtain a finer semilocal and local convergence analysis, and under the same or weaker hypotheses. Hence, we extend the applicability of these methods in cases not covered before. Numerical examples are used to compare our results favorably to earlier ones.

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References

S. Amat, S. Busquier, J.M. Gutierrez, Geometric constructions of iterative functions to solve nonlinear equations, J. Comput. Appl. Math., 157 (2003), pp. 197-205. https://doi.org/10.1016/S0377-0427(03)00420-5 DOI: https://doi.org/10.1016/S0377-0427(03)00420-5

S. Amat, S. Busquier, Third-order iterative methods under Kantorovich conditions, J. Math. Anal. Appl., 336 (2007), pp. 243-261. https://doi.org/10.1016/j.jmaa.2007.02.052 DOI: https://doi.org/10.1016/j.jmaa.2007.02.052

S. Amat, S. Busquier, M. Negra, Adaptive approximation of nonlinear operators, Numer. Funct. Anal. Optim., 25 (2004), pp. 397-405. DOI: https://doi.org/10.1081/NFA-200042628

S. Amat, S. Busquier, A. Alberto Magrenan, Improving the dynamics of Steffensen-type methods, Appl. Math. Inf. Sci., 9 (2015) 5:2403. DOI: https://doi.org/10.1007/978-3-319-39228-8_2

I.K. Argyros, On the Newton-Kantorovich hypothesis for solving equations, J. Comput. Math., 169 (2004), pp. 315-332. https://doi.org/10.1016/j.cam.2004.01.029 DOI: https://doi.org/10.1016/j.cam.2004.01.029

I.K. Argyros, A semi-local convergence analysis for directional Newton methods, Math. Comput., 80 (2011), pp. 327-343. https://doi.org/10.1090/S0025-5718-2010-02398-1 DOI: https://doi.org/10.1090/S0025-5718-2010-02398-1

I.K. Argyros, D. Gonzalez, Extending the applicability of Newton’s method for k-Frechet differentiable operators in Banach spaces, Appl. Math. Comput., 234 (2014), pp. 167-178. https://doi.org/10.1016/j.amc.2014.02.046 DOI: https://doi.org/10.1016/j.amc.2014.02.046

I.K. Argyros, S. Hilout, Weaker conditions for the convergence of Newton’s method, J. Complexity, 28 (2012), pp. 364-387. https://doi.org/10.1016/j.jco.2011.12.003 DOI: https://doi.org/10.1016/j.jco.2011.12.003

I.K. Argyros, A.A. Magrenan, Iterative Methods and Their Dynamics with Applications: A Contemporary Study, CRC Press, Boca Raton, 2017. DOI: https://doi.org/10.1201/9781315153469

I.K. Argyros, S. Hilout, On an improved convergence analysis of Newton’s method, Appl. Math. Comput., 225 (2013), pp. 372-386. https://doi.org/10.1016/j.amc.2013.09.049 DOI: https://doi.org/10.1016/j.amc.2013.09.049

I.K. Argyros, R. Behl, S.S. Motsa, Unifying semilocal and local convergence of newton’s method on banach space with a convergence structure, Appl. Numer. Math., 115 (2017), pp. 225-234. https://doi.org/10.1016/j.apnum.2017.01.008 DOI: https://doi.org/10.1016/j.apnum.2017.01.008

I.K. Argyros, Y.J. Cho, S. Hilout, On the midpoint method for solving equations, Appl. Math. Comput., 216 (2010) no. 8, pp. 2321-2332. https://doi.org/10.1016/j.amc.2010.03.076 DOI: https://doi.org/10.1016/j.amc.2010.03.076

R. Behl, A. Cordero, S.S. Motsa, J.R. Torregrosa, Stable high-order iterative methods for solving nonlinear models, Appl. Math. Comput., 303 (2017), pp. 70-88. https://doi.org/10.1016/j.amc.2017.01.029 DOI: https://doi.org/10.1016/j.amc.2017.01.029

E. Catinas, A survey on the high convergence orders and computational convergence orders of sequences, Appl. Math. Comput., 343 (2019), pp. 1-20. https://doi.org/10.1016/j.amc.2018.08.006 DOI: https://doi.org/10.1016/j.amc.2018.08.006

J. Chen, I.K. Argyros, R.P. Agarwal, Majorizing functions and two-point newton-type methods, J. Comput. Appl. Math., 234 (2010) no. 5, pp. 1473-1484. https://doi.org/10.1016/j.cam.2010.02.024 DOI: https://doi.org/10.1016/j.cam.2010.02.024

J.A. Ezquerro, M.A. Hernandez, How to improve the domain of parameters for Newton’s method, Appl. Math. Lett., 48 (2015), pp. 91-101. https://doi.org/10.1016/j.aml.2015.03.018 DOI: https://doi.org/10.1016/j.aml.2015.03.018

L.V. Kantorovich, G.P. Akilov, Functional Analysis, Pergamon Press, Oxford, 1982.

A.A. Magrenan, I.K. Argyros, Two-step newton methods, J. Complexity, 30 (2014) no. 4, pp. 533–553. DOI: https://doi.org/10.1016/j.jco.2013.10.002

W.C. Rheinboldt, An adaptive continuation process for solving systems of nonlinear equations, Polish Academy of Science, Banach Ctr. Publ., 3 (1978) no. 1, pp. 129-142. DOI: https://doi.org/10.4064/-3-1-129-142

J.F. Traub, Iterative methods for the solution of equations, 312, Amer. Math. Soc., 1982.