Efficient Steffensen-type algorithms for solving nonlinear equations

Abstract

We introduce a Steffensen-type method (STTM) for solving nonlinear equations in a Banach space setting. Then, we present a local convergence analysis for (STTM) using recurrence relations. Numerical examples validating our theoretical results are also provided in this study to show that (STTM) is faster than other methods [I.K. Argyros, J. Ezquerro, J.M. Gutiérrez, M. Hernández, and S. Hilout, On the semilocal convergence of efficient Chebyshev-Secant-type methods, J. Comput. Appl. Math. 235 (2011), pp. 3195–3206; J.A. Ezquerro and M.A. Hernández, An optimization of Chebyshev's method, J. Complexity 25 (2009), pp. 343–361] using similar convergence conditions.

Keywords:

• Banach space
• derivative-free method
• Newton's method
• Steffensen-type method
• divided difference
• recurrence relations

2000 Mathematics Subject Classifications:

• 65J15
• 65G99
• 47H99
• 49M15

Acknowledgements

The research of the second author has been supported in part by the National Natural Science Foundation of China (Grant No. 10871178), Natural Science Foundation of Zhejiang Province of China (Grant No. Y606154), and Scientific Research Fund of Zhejiang Provincial Education Department of China (Grant No. 20071362).