Geodesic
Refinement of convergence conditions of the Chebyshev method
Sibirskij žurnal vyčislitelʹnoj matematiki, Tome 7 (2004) no. 3, pp. 249-260.

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The iterative Chebyshev method of an approximate solution of equations of the form $F(x)=0$ in Banach spaces is studied, assuming that $F''$ satisfies the Lipschitz condition. Accurate (attainable) estimates of the domains of existence and uniqueness of solution, non-refinable conditions of existence and convergence of the Chebyshev method as well as asymptotic estimates of the rate of convergence have been obtained. 
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M. I. Nechepurenko. Refinement of convergence conditions of the Chebyshev method. Sibirskij žurnal vyčislitelʹnoj matematiki, Tome 7 (2004) no. 3, pp. 249-260. http://geodesic.mathdoc.fr/item/SJVM_2004_7_3_a6/

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