Fourth-order convergence theorem by using majorizing functions for super-Halley method in Banach spaces

Abstract

In this paper, we study the semilocal convergence of a multipoint fourth-order super-Halley method for solving nonlinear equations in Banach spaces. We establish the Newton–Kantorovich-type convergence theorem for the method by using majorizing functions. We also get the error estimate. In comparison with the results obtained in Wang et al. [X.H. Wang, C.Q. Gu, and J.S. Kou, Semilocal convergence of a multipoint fourth-order super-Halley method in Banach spaces, Numer. Algorithms 56 (2011), pp. 497–516], we can provide a larger convergence radius. Finally, we report some numerical applications to demonstrate our approach.

Keywords:

• nonlinear equations in Banach spaces
• super-Halley method
• majorizing functions
• semilocal convergence
• Newton–Kantorovich theorem

2000 AMS Subject Classifications:

• 65J15
• 65H10
• 65G99
• 47J25
• 49M15

Acknowledgements

The authors thank the referees for their corrections and many valuable suggestions. This work was supported by Shanghai Natural Science Foundation (10ZR1410900), by Key Disciplines of Shanghai Municipality (S30104) and by Natural Science Foundation of Universities of Anhui Province (KJ2012Z347).