On the convergence of the sixth order Homeier like method in Banach spaces

Abstract

A sixth order Homeier-like method is introduced for approximating a solution of the non-linear equation in Banach space. Assumptions only on first and second derivatives are used to obtain a sixth order convergence. Our proof does not depend on Taylor series expansions as in the earlier studies for the similar methods.

Keywords

• Frechet derivative
• Frechet derivative
• Homeier method
• Local convergence
• Iterative method
• Banach Space

References

1. 1. I.K. Argyros, S. Hailout, Computational methods in nonlinear analysis: efficient algorithms, fixed point theory and applications, World Scientific (2013).
2. 2. A. Cordero, A. Franques, J.R. Torregrosa, Chaos and convergence of a family generalizing homeier's method with damping parameters, Nonlinear Dynamics 85(3) (2016) 1939-1954.
3. 3. A. Cordero, M.A. Hernandez-Veron, N. Romero, J.R. Torregrosa, Semilocal Convergence by using recurrence relations for a fifth-order method in banach spaces, Journal of Computational and Applied Mathematics 273 (2015) 205-213.
4. 4. S. George, I.K. Argyros, K. Senapati, K. Kanagaraj, Local convergence analysis of two iterative methods, The Journal of Analysis (2022) 1-12.
5. 5. M. Grau-Sanchez, A. Grau, M. Noguera, On the computational efficiency index and some iterative methods for solving systems of nonlinear equations, Journal of Computational and Applied Mathematics 236(6) (2011) 1259-1266.
6. 6. H.H.H. Homeier, A modified newton method with cubic convergence: the multiverse case, Journal of Computational and Applied Mathematics 168(1) (2004) 161-169.
7. 7. P. Jarratt, Some fourth order multipoint iterative methods for solving equations, Mathematics of computation 20(95) (1966) 434-437.
8. 8. C.T. Kelley, Iterative methods for linear and nonlinear equations, SIAM (1995).

9. 9. J.M. Ortega, W.C. Rheinboldt, Iterative solution of nonlinear equations in several variables, SIAM (2000).
10. 10. A.M. Ostrowski, Solution of equations and system of equations, Pure and Applied Mathematics: A series of monographs and textbooks volume 9 (2016).
11. 11. L.I. Piscoran, D. Miclaus, A new steffensen homeier iterative method for solving nonlinear equations, Investigacion Operacional 40(1) (2019) 74-80.
12. 12. J.R. Sharma, P. Gupta, An efficient fifth order method for solving systems of nonlinear equations, Computers and Mathematics with Applications 67(3) (2014) 591-601.
13. 13. S. Singh, D.K. Gupta, E. Martinez, J.L Hueso, Semilocal convergence analysis of an iteration of order five using recurrence relations in banach spaces, Mediterranean Journal of Mathematics 13(6) (2016) 4219-4235.
14. 14. O.S. Solaiman, I. Hashim, An iterative scheme of arbitrary odd order and its basins of attraction for nonlinear systems, Computers, Materials and Continua Computers 66 (2021) 1427-1444.
15. 15. J.F. Traub, Iterative methods for the solution of equations, American Mathematical Society volume 312 (2013).
16. 16. S. Weerakoon, T.G.I. Fernando, A variant of newton's method with accelerated third-order convergence, Applied Mathematics Letters 13(8) (2000) 87-93.