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In this paper, we present a family of optimal, in the sense of Kung-Traub's conjecture, iterative methods for solving nonlinear equations with eighth-order convergence. Our methods are based on Chun's fourth-order method. We use the Ostrowski's efficiency index and several numerical tests in order to compare the new methods with other known eighth-order ones. We also extend this comparison to the dynamical study of the different methods. [ABSTRACT FROM AUTHOR]

The main goal of this paper is to study the order of convergence and the efficiency of four families of iterative methods using frozen divided differences. The first two families correspond to a generalization of the secant method and the implementation made by Schmidt and Schwetlick. The other two frozen schemes consist of a generalization of Kurchatov method and an improvement of this method applying the technique used by Schmidt and Schwetlick previously. An approximation of the local convergence order is generated by the examples, and it numerically confirms that the order of the methods is well deduced. Moreover, the computational efficiency indexes of the four algorithms are presented and computed in order to compare their efficiency. [ABSTRACT FROM AUTHOR]

In this paper, we present some new third-order iterative methods for finding a simple root α of nonlinear scalar equation f( x)=0 in R. A geometric approach based on the circle of curvature is used to construct the new methods. Analysis of convergence shows that the new methods have third-order convergence, that is, the sequence { x n} generated by each of the presented methods converges to α with the order of convergence three. The efficiency of the methods are tested on several numerical examples. It is observed that our methods can compete with Newton’s method and the classical third-order methods. [ABSTRACT FROM AUTHOR]

Many optimization problems reduce to the solution of a system of linear inequalities (SLI). Some solution methods use relaxed, averaged projections. Others invoke surrogate constraints (typically stemming from aggregation). This paper proposes a blend of these two approaches. A novelty comes from introducing as surrogate constraint a halfspace defined by differences of algorithmic iterates. The first iteration is identical to surrogate constraints methods. In next iterations, for a given approximation x¯, besides the violated constraints in x¯, we also take into consideration the surrogate inequality, which we have obtained in the previous iteration. The motivation for this research comes from the recent work of Scolnik et al. (Appl. Numer. Math. 41, 499—513, 2002), who studied some projection methods for a system of linear equations. [ABSTRACT FROM AUTHOR]

Time Discontinuous Galerkin methods require the factorization of a matrix larger than that exploited in standard implicit schemes. Therefore, they lend themselves to implementations based on predictor-multicorrector solution algorithms. In this paper, various convergent and computationally efficient iterative methods implemented in the unknown displacements for determining the solution of non linear systems are proposed. The iterative solutions presented here differ from those implemented in the unknown velocities in that they are computationally superior. The results of numerical simulations relevant to Duffing oscillators and to a stiff spring pendulum discretized with finite elements which are designed to evaluate the efficacy of these iterative methods with non-linear systems, show a low-computational expense when compared to earlier iterative schemes. [ABSTRACT FROM AUTHOR]