Geodesic
Multi-step high convergence order methods for solving equations
Serdica Mathematical Journal, Tome 47 (2021) no. 1, pp. 1-12.

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The local convergence analysis of iterative methods is important since it indicates the degree of difficulty for choosing initial points. In the present study we introduce generalized multi-step high order methods for solving nonlinear equations. The local convergence analysis is given using hypotheses only on the first derivative which actually appears in the methods in contrast to earlier works using hypotheses on higher derivatives. This way we extend the applicability of these methods. The analysis includes computable radius of convergence as well as error bounds based on Lipschitz-type conditions not given in earlier studies. Numerical examples conclude this study.
Keywords: multi step method, local convergence, Fréchet derivative, system of equations, Banach space, 65H10, 47H17, 49M15, 65D10, 65G99 
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Argyros, Ioannis; George, Santhosh. Multi-step high convergence order  methods for solving equations. Serdica Mathematical Journal, Tome 47 (2021) no. 1, pp. 1-12. http://geodesic.mathdoc.fr/item/SMJ2_2021_47_1_a0/