Nonlinear equations in Banach spaces Newton's method Chebyshev's method Multipoint iteration Semilocal convergence R-order of convergence Region of accessibility Attraction basin a b s t r a c t From Chebyshev's method, new third-order multipoint iterations are constructed with their efficiency close to that of Newton's method and the same region of accessibility.

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From a one-point iterative method of R-order at least three, we construct new two-point iterations to solve nonlinear equations in Banach spaces such that the computational cost is reduced, whereas the R-order of convergence is increased to at least four.Keywords Nonlinear equations in Banach spaces · Semilocal convergence · Recurrence relations · A priori error estimates · R-order of convergence Mathematics Subject Classification (2000) 45G10 · 47H17 · 65J15

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An R-order bound for the Halley method is obtained in this work, where an analysis of the convergence of the method is also presented under mild differentiability conditions. To do this, a new technique is developed, where the involved operator must satisfy some recurrence relations.  2004 Elsevier Inc. All rights reserved.

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