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Numerical Algorithms

, Volume 59, Issue 4, pp 623–638

# Recurrence relations for semilocal convergence of a fifth-order method in Banach spaces

• Lin Zheng
• Chuanqing Gu
Original Paper

## Abstract

In this paper, we study the semilocal convergence for a fifth-order method for solving nonlinear equations in Banach spaces. The semilocal convergence of this method is established by using recurrence relations. We prove an existence-uniqueness theorem and give a priori error bounds which demonstrates the R-order of the method. As compared with the Jarratt method in Hernández and Salanova (Southwest J Pure Appl Math 1:29–40, 1999) and the Multi-super-Halley method in Wang et al. (Numer Algorithms 56:497–516, 2011), the differentiability conditions of the convergence of the method in this paper are mild and the R-order is improved. Finally, we give some numerical applications to demonstrate our approach.

## Keywords

Nonlinear equations in Banach spaces Recurrence relations Semilocal convergence Newton-super-Halley method

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## Copyright information

© Springer Science+Business Media, LLC. 2011

## Authors and Affiliations

1. 1.Department of MathematicsShanghai UniversityShanghaiChina