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In this paper, we consider the problem of solving initial value problems and boundary value problems through the point of view of its continuous form. It is well known that in most cases these types of problems are solved numerically by performing a discretization and applying the finite difference technique to approximate the derivatives, transforming the equation into a finite‐dimensional nonlinear system of equations. However, we would like to focus on the continuous problem, and therefore, we try to set the domain of existence and uniqueness for its analytic solution. For this purpose, we study the semilocal convergence of a Newton‐type method with frozen first derivative in Banach spaces. We impose only the assumption that the Fréchet derivative satisfies the Lipschitz continuity condition and that it is bounded in the whole domain in order to obtain appropriate recurrence relations so that we may determine the domains of convergence and uniqueness for the solution. Our final aim is to apply these theoretical results to solve applied problems that come from integral equations, ordinary differential equations, and boundary value problems. [ABSTRACT FROM AUTHOR]

In this work, we introduce a modification into the technique, presented in A. Cordero, J.L. Hueso, E. Martínez, and J.R. Torregrosa [Increasing the convergence order of an iterative method for nonlinear systems, Appl. Math. Lett. 25 (2012), pp. 2369–2374], that increases by two units the convergence order of an iterative method. The main idea is to compose a given iterative method of order p with a modification of Newton's method that introduces just one evaluation of the function, obtaining a new method of order p+2, avoiding the need to compute more than one derivative, so we improve the efficiency index in the scalar case. This procedure can be repeated n times, with the same approximation to the derivative, obtaining new iterative methods of order p+2n. We perform different numerical tests that confirm the theoretical results. By applying this procedure to Newton's method one obtains the well known fourth order Ostrowski's method. We finally analyse its dynamical behaviour on second and third degree real polynomials. [ABSTRACT FROM AUTHOR]

In this paper we revise the proofs of the results obtained in “Convergence radius of Osada’s method under Hölder continuous condition” [4], because the remainder of the Taylor’s expansion used for the obtainment of the local convergence radius is not correct. So we perform the complete study in order to modify the equation for getting the local convergence radius, the uniqueness radius and the error bounds. Moreover a dynamical study for the third order Osada’s method is also developed. [ABSTRACT FROM AUTHOR]

In this work we introduce a new operator of divided differences that preserves the convergence order when it is used for approximating the Jacobian matrix in iterative method for solving nonlinear systems. We obtain derivative free iterative methods with lower computational cost than the corresponding ones with different operators of divided differences. We also study the global convergence of these methods by analyzing their dynamical behavior. [ABSTRACT FROM AUTHOR]

In this work we present a new family of iterative methods for solving nonlinear systems that are optimal in the sense of Kung and Traub’s conjecture for the unidimensional case. We generalize this family by performing a new step in the iterative method, getting a new family with order of convergence six. We study the efficiency of these families for the multidimensional case by introducing a new term in the computational cost defined by Grau-Sánchez et al. A comparison with already known methods is done by studying the dynamics of these methods in an example system. [ABSTRACT FROM AUTHOR]

Abstract: In this paper, we study a new family of iterative methods for solving nonlinear equations with sixth and seventh order convergence. The new methods are obtained by composing known methods of third and fourth order with Newton’s method and using an adequate approximation for the last derivative, which provides high order of convergence and reduces the required number of functional evaluations per step. The new methods attain efficiency indices of 1.5651 and 1.6266, which makes them competitive. We introduce a new efficiency index involving the computational effort as well as the functional evaluations per iteration. We use this new index, in combination with the usual efficiency index, in order to compare the methods described in the paper with other known methods and present several numerical tests. [Copyright &y& Elsevier]

Abstract: In this paper, we present two new iterative methods for solving nonlinear equations by using suitable Taylor and divided difference approximations. Both methods are obtained by modifying Potra–Pták’s method trying to get optimal order. We prove that the new methods reach orders of convergence four and eight with three and four functional evaluations, respectively. So, Kung and Traub’s conjecture Kung and Traub (1974) , that establishes for an iterative method based on evaluations an optimal order is fulfilled, getting the highest efficiency indices for orders and , which are 1.587 and 1.682. We also perform different numerical tests that confirm the theoretical results and allow us to compare these methods with Potra-Pták’s method from which they have been derived, and with other recently published eighth-order methods. [Copyright &y& Elsevier]