Daniel González

Majorizing sequences for Newton's method from initial value problems

The most restrictive condition used by Kantorovich for proving the semilocal convergence of Newtons method in Banach spaces is relaxed in this paper, providing we can guarantee the semilocal convergence in situations that Kantorovich cannot. To achieve this, we use Kantorovichs technique based on majorizing sequences, but our majorizing sequences are obtained differently, by solving initial value problems.

A modification of the classic conditions of Newton–Kantorovich for Newton’s method☆

Abstract We study the semilocal convergence of Newton’s method in Banach spaces under a modification of the classic conditions of Kantorovich, which leads to a generalization of Kantorovich’s theory. We illustrate this study with two Hammerstein integral equations of the second kind, where the classic conditions of Kantorovich cannot be applied, but our modification of them can.

A Semilocal Convergence for a Uniparametric Family of Efficient Secant-Like Methods

We present a semilocal convergence analysis for a uniparametric family of efficient secant-like methods (including the secant and Kurchatov method as special cases) in a Banach space setting (Ezquerro et al., 2000–2012). Using our idea of recurrent functions and tighter majorizing sequences, we provide convergence results under the same or less computational cost than the ones of Ezquerro et al., (2013, 2010, and 2012) and Hernandez et al., (2000, 2005, and 2002) and with the following advantages: weaker sufficient convergence conditions, tighter error estimates on the distances involved, and at least as precise information on the location of the solution. Numerical examples validating our theoretical results are also provided in this study.

A semilocal convergence result for Newton’s method under generalized conditions of Kantorovich

Abstract From Kantorovich’s theory we establish a general semilocal convergence result for Newton’s method based fundamentally on a generalization required to the second derivative of the operator involved. As a consequence, we obtain a modification of the domain of starting points for Newton’s method and improve the a priori error estimates. Finally, we illustrate our study with an application to a special case of conservative problems.

A variant of the Newton–Kantorovich theorem for nonlinear integral equations of mixed Hammerstein type

Abstract We study nonlinear integral equations of mixed Hammerstein type using Newton’s method as follows. We investigate the theoretical significance of Newton’s method to draw conclusions about the existence and uniqueness of solutions of these equations. After that, we approximate the solutions of a particular nonlinear integral equation by Newton’s method. For this, we use the majorant principle, which is based on the concept of majorizing sequence given by Kantorovich, and milder convergence conditions than those of Kantorovich. Actually, we prove a semilocal convergence theorem which is applicable to situations where Kantorovich’s theorem is not.

Majorizing sequences for Newton's method under centred conditions for the derivative

We present semi-local and local convergence results for Newtons method in order to approximate a locally unique solution of a nonlinear equation in a Banach space setting. Our technique is more flexible than in earlier studies such that [J.A. Ezquerro, D. González, and M.A. Hernández, Majorizing sequences for Newtons method from initial value problems, J. Comput. Appl. Math. 236 (2012), pp. 2246–2258; J.A. Ezquerro, D. González, and M.A. Hernández, A general semi-local convergence result for Newtons method under centred conditions for the second derivative, ESAIM: Math. Model. Numer. Anal. 47 (2013), pp. 149–167]. The operator involved is twice Fréchet-differentiable. We also assume certain centred Lipschitz-type conditions for the derivative which are more precise than the Lipschitz conditions used in earlier works. Numerical examples are used to show that our results apply to solve equations but earlier ones do not in the semi-local case. In the local case we obtain a larger convergence ball. These advantages are obtained under the same computational cost as before [17,18].

On the local convergence of Newton’s method under generalized conditions of Kantorovich

Abstract Following an idea similar to that given by Dennis and Schnabel (1996) in [2] , we prove a local convergence result for Newton’s method under generalized conditions of Kantorovich type.

Unified majorizing sequences for Traub-type multipoint iterative procedures

We present a unified approach to generating majorizing sequences for multipoint iterative procedures in order to solve nonlinear equations in a Banach space setting. The semilocal convergence results have the following advantages over earlier work (under the same computational cost): weaker sufficient convergence conditions, more precise error bounds on the distances involved and more precise information on the location of the solution. Special cases and numerical examples are also provided in this study.

Local convergence of Cauchy-type methods under hypotheses on the first derivative

We present a local convergence analysis of Cauchy-type methods free of the second derivative using hypotheses only on the first derivative. In earlier studies such as Amat et al. (2003, 2008), Hernandez and Salanova (1999), Jarratt (1996), Kou (2007), Parhi and Gupta (2007), Rall (1979) and Ren et al. (2009) hypotheses up to the fourth derivative have been used to show convergence although the method requires evaluations of the function and its derivative. This way we extend the applicability of these methods. Numerical examples are provided in this study where earlier results cannot apply but the new results can apply to solve equations.

How to Apply Newton’s Method to Operators with Unbounded Second Derivative

The problem of finding the roots of a nonlinear equation has a long history. Although some nonlinear equations can be solved analytically, numerical approximations of the roots can be normally wanted. Moreover, since it is usually difficult or impossible to obtain an exact root of a nonlinear equation, we usually have to be satisfied with approximating the root numerically. To do this, we habitually approximate the root by iterative methods, which provide a sequence of approximations, from one or several initial approximations, that converges to the root of the nonlinear equation.