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The main goal of this paper is to approximate the principal pth root of a matrix by using a family of high-order iterative methods. We analyse the semi-local convergence and the speed of convergence of these methods. Concerning stability, it is well known that even the simplified Newton method is unstable. Despite it, we present stable versions of our family of algorithms. We test numerically the methods: we check the numerical robustness and stability by considering matrices that are close to be singular and badly conditioned. We find algorithms of the family with better numerical behavior than the Newton and the Halley methods. These two algorithms are basically the iterative methods proposed in the literature to solve this problem. Copyright © 2015 John Wiley & Sons, Ltd. [ABSTRACT FROM AUTHOR]

In this paper, a semilocal convergence result in Banach spaces of an efficient fifth-order method is analyzed. Recurrence relations are used in order to prove this convergence, and some a priori error bounds are found. This scheme is finally used to estimate the solution of an integral equation and so, the theoretical results are numerically checked. We use this example to show the better efficiency of the current method compared with other existing ones, including Newton's scheme. [ABSTRACT FROM AUTHOR]

SUMMARY The main goal of this paper is to approximate inverse operators by high-order Newton-type methods with the important feature of not using inverse operators. We analyse the semilocal convergence, the speed of convergence, and the efficiency of these methods. We determine that Chebyshev's method is the most efficient method and test it on two problems: one associated to the heat equation and the other one to a boundary value problem. We consider examples with matrices that are close to be singular and/or are badly conditioned. We check the robustness and the stability of the methods by considering situations with many steps and noised data. Copyright © 2013 John Wiley & Sons, Ltd. [ABSTRACT FROM AUTHOR]

We analyze the semilocal convergence of Steffensen's method, using a novel technique, which is based on recurrence relations, for solving systems of nonlinear equations. This technique allows analyzing the convergence of Steffensen's method to solutions of equations, where the function involved can be both differentiable and nondifferentiable. Moreover, this technique also allows enlarging the domain of starting points for Steffensen's method from certain predictions with the simplified Steffensen method. [ABSTRACT FROM AUTHOR]

This paper is devoted to the approximation of solutions for nonlinear equations by using iterative methods. We present a unified convergence analysis for some Newton-type methods. We consider both semilocal and local analysis. In the first one, the hypotheses are imposed on the initial guess and in the second on the solution. The results can be applied for smooth and non-smooth operators. In the numerical section we study two applications, first one, it is devoted to a nonlinear integral equation of Hammerstein type and in second one, we approximate the solution of a nonlinear PDE related to image denoising. [ABSTRACT FROM AUTHOR]