Relaxing convergence conditions for an inverse-free Jarratt type approximation

We consider an inverse-free Jarratt-type approximation of order four in a Banach space (Argyros et al., 1996). We establish a convergence theorem by using recurrence relations. The purpose of this note is to relax convergence conditions and give an example where our convergence theorem can be applied but not the other ones

Increasing the applicability of Steffensen's method

We present an original alternative to the majorant principle of Kantorovich to study the semilocal convergence of Steffensen's method when it is applied to solve nonlinear systems which are differentiable. This alternative allows choosing starting points from which the convergence of Steffensen's method is guaranteed, but it is not from the majorant principle. Moreover, this study extends the applicability of Steffensen's method to the solution of nonlinear systems which are nondifferentiable and improves a previous result given by the authors. © 2014 Elsevier Inc. All rights reserved

Majorizing sequences for Newton’s method from initial value problems

AbstractThe most restrictive condition used by Kantorovich for proving the semilocal convergence of Newton’s 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 Kantorovich’s technique based on majorizing sequences, but our majorizing sequences are obtained differently, by solving initial value problems

An extension of Gander’s result for quadratic equations

AbstractIn the study of iterative methods with high order of convergence, Gander provides a general expression for iterative methods with order of convergence at least three in the scalar case. Taking into account an extension of this result, we define a family of iterations in Banach spaces with R-order of convergence at least four for quadratic equations

A construction procedure of iterative methods with cubical convergence II: Another convergence approach

We extend the analysis of convergence of the iterations considered in Ezquerro et al. [Appl. Math. Comput. 85 (1997) 181] for solving nonlinear operator equations in Banach spaces. We establish a different Kantorovich-type convergence theorem for this family and give some error estimates in terms of a real parameter [-5, 1). © 1998 Elsevier Science Inc. All rights reserved

Chronostratigraphy and new vertebrate sites from the upper Maastrichtian of Huesca (Spain), and their relation with the K/Pg boundary

The transitional-continental facies of the Tremp Formation within the South-Pyrenean Central Unit (Spain) contain one of the best continental vertebrate records of the Upper Cretaceous in Europe. This Pyrenean area is therefore an exceptional place to study the extinction of continental vertebrates across the Cretaceous/Paleogene (K/Pg) boundary, being one of the few places in Europe that has a relatively continuous record ranging from the upper Campanian to lower Eocene. The Serraduy area, located on the northwest flank of the Tremp syncline, has seen the discovery of abundant vertebrate remains in recent years, highlights being the presence of hadrosaurid dinosaurs and eusuchian crocodylomorphs. Nevertheless, although these deposits have been provisionally assigned a Maastrichtian age, they have not previously been dated with absolute or relative methods. This paper presents a detailed stratigraphic, magnetostratigraphic and biostratigraphic study for the first time in this area, making it possible to assign most vertebrate sites from the Serraduy area a late Maastrichtian age, specifically within polarity chron C29r. These results confirm that the vertebrate sites from Serraduy are among the most modern of the Upper Cretaceous in Europe, being very close to the K/Pg boundary.Spanish Ministry of Economy and Competitiveness (grant numbers CGL2014-53548-P, CGL2015-64422-P and CGL2017-85038-P), cofinanced by the European Regional Development Fund; and by the Department of Education and Science of the Aragonese Government (grant numbers DGA groups H54 and E05), cofinanced by the European Social Fund (ESF). The paleomagnetic study was possible thanks to the complementary grants (beneficiaries of FPU, grant number CGL2010-16447/BTE: Brief Stays and Temporary Transfers, year 2015) supported by the Spanish Ministry of Culture, Education and Sports; and the Laboratory of paleomagnetism of the University of Burgos (Spain). Eduardo Puértolas Pascual is the recipient of a postdoctoral grant (SFRH/BPD/116759/2016) funded by the Fundação para a Ciência e Tecnologia (FCT-MCTES)

A modification of the Chebyshev method

In this paper we use a one-parametric family of second-order iterations to solve a nonlinear operator equation in a Banach space. Two different analyses of convergence are shown. First, under standard Newton-Kantorovich conditions, we establish a Kantorovich-type convergence theorem. Second, another Kantorovich-type convergence theorem is proved, when the first Fréchet-derivative of the operator satisfies a Lipschitz condition. We also give an explicit expression for the error bound of the family of methods in terms of a real parameter a 0

Construccion de procesos iterativos mediante aceleraciones del metodo de Newton

Centro de Informacion y Documentacion Cientifica (CINDOC). C/Joaquin Costa, 22. 28002 Madrid. SPAIN / CINDOC - Centro de Informaciòn y Documentaciòn CientìficaSIGLEESSpai

On the R-order of the Halley method

An R-order bound for the Halley method is obtained in this work, where an analysis of the convergence of the method is also presented under mild differentiability conditions. To do this, a new technique is developed, where the involved operator must satisfy some recurrence relations. © 2004 Elsevier Inc. All rights reserved