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We study the local convergence of a Steffensen-King-type method to approximate a locally unique solution of a nonlinear equation. Earlier studies such as [14, 15, 171 show convergence under hypotheses on the third derivative or even higher. The convergence in this study is shown under hypotheses on the first derivative. Hence, the applicability of the method is expanded. Finally, numerical examples are also provided in this study. [ABSTRACT FROM AUTHOR]

We present a local convergence analysis of a sixth order iterative method using only the first derivative for solving equations in order to approximate a solution of a nonlinear equation. In earlier studies such as [28] hypotheses on the fifth derivatives are used. Thus by using only first crivative, we extended the applicability of these methods. Moreover the radius of convergence and computable error bounds on the distances involved are also given in this study. Numerical examples are also presented in this study. [ABSTRACT FROM AUTHOR]

We present a local convergence analysis for eighth-order variants of Newton's method in order to approximate a solution of a nonlinear equation. We use hypotheses up to the first derivative in contrast to earlier studies such as [7]-[11], [20] using hypotheses up to the seventh derivative. This way the applicability of these methods is extended under weaker hypotheses. Moreover the radius of convergence and computable error bounds on the distances involved are also given in this study. Numerical examples are also presented in this study. [ABSTRACT FROM AUTHOR]

We present a local convergence analysis of a class of multi-point Super-Halley methods in order to approximate a solution of nonlinear equations in a Banach space setting. The semilocal convergence analysis of this method was studied by Wang et. al. [24], This work provides computable convergence ball and computable error bounds under weaker conditions than in [24], Numerical examples are also provided in this study. [ABSTRACT FROM AUTHOR]

We present a unified local and semilocal convergence analysis for secant-type methods in order to approximate a locally unique solution of a nonlinear equation in a Banach space setting. Our analysis includes the computation of the bounds on the limit points of the majorizing sequences involved. Under the same computational cost, using both Lipschtiz and center Lipschitz conditions, our convergence criteria can be: weaker; the error bounds more precise and the convergence balls larger than in earlier studies. Special cases such us Newton's method or Secant method are also presented. Numerical examples, including a Chandrasekhar equation and a boundary value problem, are also presented to illustrate the theoretical results obtained in this study. [ABSTRACT FROM AUTHOR]