A convergence theorem for Newton-like methods in Banach spaces
Numerische Mathematik
Secant-like methods for solving nonlinear integral equations of the Hammerstein type
Journal of Computational and Applied Mathematics - Proceedings of the 8th international congress on computational and applied mathematics
A class of quasi-Newton generalized Steffensen methods on Banach spaces
Journal of Computational and Applied Mathematics
Recurrence relations for a Newton-like method in Banach spaces
Journal of Computational and Applied Mathematics
Weaker conditions for the convergence of Newton's method
Journal of Complexity
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We present a new semilocal convergence analysis for the Secant method in order to approximate a locally unique solution of a nonlinear equation in a Banach space setting. Our analysis is based on the weaker center-Lipschitz concept instead of the stronger Lipschitz condition which has been ubiquitously employed in other studies such as Amat et al. (2004) [2], Bosarge and Falb (1969) [9], Dennis (1971) [10], Ezquerro et al. (2010) [11], Hernandez et al. (2005, 2000) [13,12], Kantorovich and Akilov (1982) [14], Laasonen (1969) [15], Ortega and Rheinboldt (1970) [16], Parida and Gupta (2007) [17], Potra (1982, 1984-1985, 1985) [18-20], Proinov (2009, 2010) [21,22], Schmidt (1978) [23], Wolfe (1978) [24] and Yamamoto (1987) [25] for computing the inverses of the linear operators. We also provide lower and upper bounds on the limit point of the majorizing sequences for the Secant method. Under the same computational cost, our error analysis is tighter than that proposed in earlier studies. Numerical examples illustrating the theoretical results are also given in this study.