Design and analysis of a faster King-Werner-type derivative free method
DOI:
https://doi.org/10.5269/bspm.44132Abstract
We introduce a new faster King-Werner-type derivative-free method for solving nonlinear equations. The local as well as semi-local convergence analysis is presented under weak center Lipschitz and Lipschitz conditions. The convergence order as well as the convergence radii are also provided. The radii are compared to the corresponding ones from similar methods. Numerical examples further validate the theoretical results.
References
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25. Weerakoon, S., Fernando, T. G. I., A variant of Newton's method with accelerated third order convergence, Appl. Math. Lett. 13, 87-93, (2000). https://doi.org/10.1016/S0893-9659(00)00100-2
26. Cordero, A., Torregrosa, J. R., Variants of Newton's method using fifth-order quadrature formulas, Appl. Math. Comput. 199, 686-698, (2007). https://doi.org/10.1016/j.amc.2007.01.062
2. Argyros, I. K., Covergence and Applications of Newton-type Iterations, Springer-Verlag, New York, 2008.
3. King, R. F., Tangent methods for nonlinear equations, Numer. Math. 18, 298-304, (1972). https://doi.org/10.1007/BF01404680
4. Gutierrez, J. M., Hernandez, M. A., A family of Chebyshev-Halley type methods in Banach spaces, Bull. Aust. Math. Soc. 55, 113-130, (1997). https://doi.org/10.1017/S0004972700030586
5. Hernandez, M. A., Chebyshev's approximation algorithms and applications, Comput. Math. Appl. 41, 433-455, (2001). https://doi.org/10.1016/S0898-1221(00)00286-8
6. Babajee, D. K. R., Dauhoo, M. Z., Darvishi, M. T., A. Karami and 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, 2002-2012, (2010). https://doi.org/10.1016/j.cam.2009.09.035
7. Grau-Sanchez, M., Grau, A., Noguera, M., Ostrowski type methods for solving systems of nonlinear equations, Appl. Math. Comput. 218, 2377-2385, (2011). https://doi.org/10.1016/j.amc.2011.08.011
8. Darvishi, M. T., Some three-step iterative methods free from second order derivative for finding solutions of systems of nonlinear equations, Int. J. Pure Appl. Math. 57, 557-573, (2010).
9. Ezquerro, J. A., Grau-Sanchez, M., Hernandez, M. A., Solving non-differentiability equations by a new one point iterative method with memory, J. Complexity 28, 48-58, (2012). https://doi.org/10.1016/j.jco.2011.06.002
10. MagreËœn'an, A. A., A new tool to study real dynamics: The convergence plane, Appl. Math. Comput. 248, 215-224, (2014). https://doi.org/10.1016/j.amc.2014.09.061
11. Ezquerro, J. A., Hernandez, M. A., Enlarging the domain of starting points for Newton's method under center conditions on the first Fr'echet-derivative, J. Complexity 33, 89-106, (2016). https://doi.org/10.1016/j.jco.2015.09.007
12. Behl, R., Cordero, A., Motsa, S. S., Torregrosa, J. R., Stable high-order iterative methods for solving nonlinear models, Appl. Math. Comput. 303, 70-88, (2017). https://doi.org/10.1016/j.amc.2017.01.029
13. Cordero, A., Ezquerro, J. A., Hernandez-Veron, M. A., Torregrosa, J.R., On the local convergence of fifth-order iterative method in Banach spaces, Appl. Math. Comput. 251, 396-403, (2015). https://doi.org/10.1016/j.amc.2014.11.084
14. Jaiswal, J. P., Semilocal convergnece of an eighth-order method in Banach spaces and its computational efficiency, Numer. Algor. 71, 933-951, (2016). https://doi.org/10.1007/s11075-015-0031-5
15. Narang, M., Bhatia, S., Kanwar, V., New two-parameter Chebyshev-Halley-like family of fourth and sixth-order methods for systems of nonlinear equations, Appl. Math. Comput. 275, 394-403, (2016). https://doi.org/10.1016/j.amc.2015.11.063
16. Sharma, J. R., Arora, H., Improved Newton-like methods for solving systems of nonlinear equations, SeMA 74, 147-163 (2017). https://doi.org/10.1007/s40324-016-0085-x
17. Sharma, J. R., Sharma, R., Bahl, A., An improved Newton-Traub composition for solving systems of nonlinear equations, Appl. Math. Comput. 290, 98-110, (2016). https://doi.org/10.1016/j.amc.2016.05.051
18. Soleymani, F., Lotfi, T., Bakhtiari, P., A multi-step class of iterative methods for nonlinear systems, Optim. Lett. 8, 1001-1015, (2014). https://doi.org/10.1007/s11590-013-0617-6
19. Argyros, I. K., Ren, H., On the convergence of efficient King-Werner-type methods of order 1 + √ 2, J. Comput. Appl. Math. 285, 169-180, (2015). https://doi.org/10.1016/j.cam.2015.02.021
20. Werner, W., Some supplementary results on the 1 + √ 2 order method for the solution of nonlinear equations, Numer. Math. 38, 383-392, (1982). https://doi.org/10.1007/BF01396439
21. Ren, H., Argyors, I. K., On the convergence of King-Werner-type methods of order 1 + √ 2 free of derivative, Appl. Math. Comput. 256, 148-159, (2015). https://doi.org/10.1016/j.cam.2015.02.021
22. Werner, W., Uber ein Verfahren der Ordung 1 + √ 2 zur Nullstellenbestimmung, Numer. Math. 32, 333-342, (1979). https://doi.org/10.1007/BF01397005
23. McDougall, T. J., Wotherspoon, S.J., A simple modification of Newton's method to achieve convergence of order 1+√ 2, Appl. Math. Lett. 29, 20-25, (2014).
24. Argyros, I. K., Cho, Y. J., Hilout, S., Numerical Methods for Equations and Its Applications, CRC Press, New York, 2012. https://doi.org/10.1201/b12297
25. Weerakoon, S., Fernando, T. G. I., A variant of Newton's method with accelerated third order convergence, Appl. Math. Lett. 13, 87-93, (2000). https://doi.org/10.1016/S0893-9659(00)00100-2
26. Cordero, A., Torregrosa, J. R., Variants of Newton's method using fifth-order quadrature formulas, Appl. Math. Comput. 199, 686-698, (2007). https://doi.org/10.1016/j.amc.2007.01.062
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