Eulalia Martínez

A modified Newton-Jarratt’s composition

A reduced composition technique has been used on Newton and Jarratt’s methods in order to obtain an optimal relation between convergence order, functional evaluations and number of operations. Following this aim, a family of methods is obtained whose efficiency indices are proved to be better for systems of nonlinear equations.

New modifications of Potra-Pták's method with optimal fourth and eighth orders of convergence

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-Ptaks 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 Traubs conjecture Kung and Traub (1974) [2], that establishes for an iterative method based on n evaluations an optimal order p=2^n^-^1 is fulfilled, getting the highest efficiency indices for orders p=4 and p=8, 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-Ptaks method from which they have been derived, and with other recently published eighth-order methods.

Steffensen type methods for solving nonlinear equations

In the present paper, by approximating the derivatives in the well known fourth-order Ostrowskis method and in a sixth-order improved Ostrowskis method by central-difference quotients, we obtain new modifications of these methods free from derivatives. We prove the important fact that the methods obtained preserve their convergence orders 4 and 6, respectively, without calculating any derivatives. Finally, numerical tests confirm the theoretical results and allow us to compare these variants with the corresponding methods that make use of derivatives and with the classical Newtons method.

Increasing the convergence order of an iterative method for nonlinear systems

Abstract In this work we introduce a technique for solving nonlinear systems that improves the order of convergence of any given iterative method which uses the Newton iteration as a predictor. The main idea is to compose a given iterative method of order p with a modification of the Newton method that introduces just one evaluation of the function, obtaining a new method of order p + 2 . By applying this procedure to known methods of order three and four, we obtain new methods of order five and six, respectively. The efficiency index and the computational effort of the new methods are checked. We also perform different numerical tests that confirm the theoretical results and allow us to compare these methods with the ones from which have been derived and with the classical Newton method.

A way to optimally solve a time-dependent Vehicle Routing Problem with Time Windows

In this paper we deal with a generalization of the Vehicle Routing Problem with Time Windows that considers time-dependent travel times and costs. Through several steps we transform this extension into an Asymmetric Capacitated Vehicle Routing Problem, so it can be solved both optimally and heuristically with known codes.

Iterative methods of order four and five for systems of nonlinear equations

The Adomian decomposition is used in order to obtain a family of methods to solve systems of nonlinear equations. The order of convergence of these methods is proved to be p>=2, under the same conditions as the classical Newton method. Also, numerical examples will confirm the theoretical results.

The Rural Postman Problem on mixed graphs with turn penalties

In this paper we deal with a problem which generalizes the Rural Postman Problem defined on a mixed graph (MRPP). The generalization consists of associating a non-negative penalty to every turn as well as considering the existence of forbidden turns. This new problem fits real-world situations more closely than other simpler problems. A solution tour must traverse all the requiring service arcs and edges of the graph while not making forbidden turns. Its total cost will be the sum of the costs of the traversed arcs and edges together with the penalties associated with the turns done. The Mixed Rural Postman Problem with Turn Penalties (MRPPTP) consists of finding such a tour with a total minimum cost. We show that the new problem is NP-hard, even in some particular cases. In order to solve the MRPPTP, a polynomial transformation of the problem into the Asymmetric Traveling Salesman Problem (ATSP) can be done and then apply heuristics and exact methods for the ATSP. This transformation is summarized here and a specific heuristic algorithm, based on two recent heuristics for the MRPP, is also presented. Extensive computational experiments with more than 250 instances establish the effectiveness of our procedures.

A family of iterative methods with sixth and seventh order convergence for nonlinear equations

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 Newtons 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.

A transformation for the mixed general routing problem with turn penalties

In this paper, we study a generalization of the Mixed General Routing Problem (MGRP) with turn penalties and forbidden turns. Thus, we present a unified model of this kind of extended versions for both node- and arc-routing problems with a single vehicle. We provide a polynomial transformation of this generalization into an asymmetric travelling salesman problem, which can be considered a particular case of the MGRP. We show computational results on the exact resolution on a set of 128 instances of the new problem using a recently developed code for the MGRP.

Convergence, efficiency and dynamics of new fourth and sixth order families of iterative methods for nonlinear systems

In this work we present a new family of iterative methods for solving nonlinear systems that are optimal in the sense of Kung and Traubs 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-Sanchez et al. A comparison with already known methods is done by studying the dynamics of these methods in an example system.