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The purpose of this article is to present results that amount to a description of the conjugacy classes of two fourth-order root-finding iterative methods, namely King's family of iterative methods and Jarratt's iterative method, for complex polynomials of degrees two, three and four. For degrees two and three, a full description of the conjugacy classes is accomplished, in each case, by a one-parameter family of polynomials. This is done in such a way that, when one applies one of these two root-finding iterative methods to the elements of these parametrized families, a family of iterative methods is obtained, and its dynamics represents, up to conjugacy, the dynamics of the corresponding iterative root-finding method applied to any complex polynomial having the same degree. For degree four, partial results analogous to the ones just described are presented.

The dynamics of a classical third-order Newton-type iterative method is studied when it is applied to degrees two and three polynomials. The method is free of second derivatives which is the main limitation of the classical third-order iterative schemes for systems. Moreover, each iteration consists only in two steps of Newton's method having the same derivative. With these two properties the scheme becomes a real alternative to the classical Newton method. Affine conjugacy class of the method when is applied to a differentiable function is given. Chaotic dynamics have been investigated in several examples. Applying the root-finding method to a family of degree three polynomials, we have find a bifurcation diagram as those that appear in the bifurcation of the logistic map in the interval.

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We study a class of third-order iterative methods for nonlinear equations on Banach spaces. A characterization of the convergence under Kantorovich type conditions and optimal estimates of the error are found. Though, in general, these methods are not very extended due to their computational costs, we will show some examples in which they are competitive and even cheaper than other simpler methods. We center our analysis in both, analytic and computational, aspects.

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A simplification of a third order iterative method is proposed. The main advantage of this method is that it does not need to evaluate neither any Fréchet derivative nor any bilinear operator. A semilocal convergence theorem in Banach spaces, under modified Kantorovich conditions, is analyzed. A local convergence analysis is also performed. Finally, some numerical results are presented.

We provide sufficient conditions for the semilocal convergence of a family of two-step Steffensen's iterative methods on a Banach space. The main advantage of this family is that it does not need to evaluate neither any Fr4chet derivative nor any bilinear operator, but having a high speed of convergence. Some numerical experiments are also presented. (~)