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In this paper, we are interested to justified two typical hypotheses that appear in the convergence analysis, $$|\lambda |\le 2$$ and $$z_0$$ sufficient close to $$z^*$$ . In order to proof these ideas, the dynamics of a damped two-step Newton-type method for solving nonlinear equations and systems is presented. We present the parameter space for values of the damping factor in the complex plane, focusing our attention in such values for which the fixed points related to the roots are attracting. Moreover, we study the stability of the strange fixed points, showing that there exists attracting cycles and chaotical behavior for some choices of the damping factor.

One of the most studied problems in numerical analysis is the approximation of nonlinear equations using iterative methods. In the last years, attention has been paid in studying Newton's method on manifolds. In this paper, we generalize this study considering some Newton-type iterative methods. A characterization of the convergence under Kantorovich type conditions and optimal estimates of the error are found. Using normal coordinates the order of convergence is derived. The sufficient semilocal convergence criteria are weaker and the majorizing sequences are tighter for the special cases of simplified Newton and Newton methods than in earlier studies such as Argyros (2004, 2007, 2008) [6,8,12] and Kantorovich and Akilov (1964) [32].

The aim of this paper is the approximation of nonlinear equations using iterative methods. We present a unified convergence analysis for some two-point type methods. This way we compare specializations of our method using not necessarily the same convergence criteria. We consider both semilocal and local analysis. In the first one, the hypotheses are imposed on the initial guess and in the second on the solution. The results can be applied for smooth and nonsmooth operators.

A generalized k-step iterative application of Newton's method with frozen derivative is studied and used to solve a system of nonlinear equations. The maximum computational efficiency is computed. A sequence that approximates the order of convergence is generated for the examples, and it numerically confirms the calculation of the order of the method and computational efficiency. This type of method appears in many applications where the authors have heuristically chosen a given number of steps with frozen derivatives. An example is shown in which the total variation (TV) minimization model is approximated using the schemes described in this paper.

In this paper a nonlinear wavelet approximation to design an adaptive two-step Newton-like scheme is applied. Local and semilocal convergence results are presented.

The geometrical interpretation of a family of higher order iterative methods for solving nonlinear scalar equations was presented in [S. Amat, S. Busquier, J.M. Gutirrez, Geometric constructions of iterative functions to solve nonlinear equations. J. Comput. Appl. Math. 157(1) (2003) 197-205]. This family includes, as particular cases, some of the most famous third-order iterative methods: Chebyshev methods, Halley methods, super-Halley methods, C-methods and Newton-type two-step methods. The aim of the present paper is to analyze the convergence of this family for equations defined between two Banach spaces by using a technique developed in [J.A. Ezquerro, M.A. Hernndez, Halley's method for operators with unbounded second derivative. Appl. Numer. Math. 57(3) (2007) 354-360]. This technique allows us to obtain a general semilocal convergence result for these methods, where the usual conditions on the second derivative are relaxed. On the other hand, the main practical difficulty related to the classical third-order iterative methods is the evaluation of bilinear operators, typically second-order Frchet derivatives. However, in some cases, the second derivative is easy to evaluate. A clear example is provided by the approximation of Hammerstein equations, where it is diagonal by blocks. We finish the paper by applying our methods to some nonlinear integral equations of this type. © 2010 Elsevier B.V. All rights reserved.

In this paper, a modified Steffensen's type iterative scheme for the numerical solution of a system of nonlinear equations is studied. Two convergence theorems are presented. The numerical solution of boundary-value problems by the multiple shooting method using the proposed iterative scheme is analyzed.

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.

We use a classical third order root-finding iterative method for approximating roots of nonlinear equations. We present a procedure for constructing polynomials so that super-attracting periodic orbits of any prescribed period occur when this method is applied. This note can be considered as the second part of our previous study [S. Amat, S. Busquier, S. Plaza, A construction of attracting periodic orbits for some classical third order iterative methods, J. Comput. Appl. Math. 189(1–2) (2006) 22–33].

This paper is devoted to multiresolution schemes that use a stencil selection procedure in order to obtain adaptation to the presence of edges in the images. Since non adapted schemes, based on a centered stencil, are less affected by the presence of texture, we propose the introduction of some weight that leads to a more frequent use of the centered stencil in regions without edges. In these regions the different stencils have similar weights and therefore the selection becomes an ill-posed problem with high risk of instabilities. In particular, numerical artifacts appear in the decompressed images. Our attention is centered in ENO schemes, but similar ideas can be developed for other multiresolution schemes. A nonlinear multiresolution scheme corresponding to a nonlinear interpolatory technique is analyzed. It is based on a modification of classical ENO schemes. As the original ENO stencil selection, our algorithm chooses the stencil within a region of smoothness of the interpolated function if the jump discontinuity is sufficiently big. The scheme is tested, allowing to compare its performances with other linear and nonlinear schemes. The algorithm gives results that are at least competitive in all the analyzed cases. The problems of the original ENO interpolation with the texture of real images seem solved in our numerical experiments. Our modified ENO multiresolution will lead to a reconstructed image free of numerical artifacts or blurred regions, obtaining similar results than WENO schemes. Similar ideas can be used in multiresolution schemes based in other stencil selection algorithms.

Multiresolution transforms provide useful tools for image processing applications. For an optimal representation of the edges, it is crucial to develop nonlinear schemes which are not based on tensor product. This paper links the nonseparable quincunx pyramid and the nonlinear discrete Harten's multiresolution framework. In order to obtain the stability of these representations, an error-control multiresolution algorithm is introduced. A prescribed accuracy in various norms is ensured by these strategies. Explicit error bounds are presented. Finally, a nonlinear reconstruction is proposed and tested.

Multiresolution transforms are powerful tools in video processing applications because of its flexibility in representing nonstationary signals. For a proper adaptation to the singularities, it is crucial to develop nonlinear schemes. In these applications where some coefficients are modified or discarded we need to have some stability properties. In this paper, three-dimensional multiresolution processing algorithms that ensure this stability are introduced. A prescribed accuracy in various norms is ensured by these strategies. Explicit error bounds are presented. Finally, a numerical experiment using linear and non-linear schemes is performed. (© 2005 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)

In this paper we present the geometrical interpretation of several iterative methods to solve a nonlinear scalar equation. In addition, we also review the extension to general Banach spaces and some computational aspects of these methods. © 2003 Elsevier B.V. All rights reserved.

We consider a class of generalized Steffensen iterations procedure for solving nonlinear equations on Banach spaces without any derivative. We establish the convergence under the Kantarovich–Ostrowski's conditions. The majorizing sequence will be a Newton's type sequence, thus the convergence can have better properties. Finally, a numerical comparation with the classical methods is presented.

In this paper, a nonlinear multiresolution is constructed based on a modified stencil algorithm. The stability properties of the ENO (essential nonoscillatory) multiresolution are studied. A new data-compression strategy is presented in order to obtain explicit error bounds.

We present a semilocal convergence analysis of a bilinear operator free third order method on Riemannian manifolds. Using a combination of generalized Lipschitz and center-Lipschitz conditions, we provide a convergence analysis which expands the applicability of the method even in the setting of nonlinear equations considered in earlier studies such as (cf. [4,9,38]).

We study a general class of high order Newton type methods. The schemes consist of the application of several steps of Newton type methods with frozen derivatives. We are interested to improve the order of convergence in each sub-step. In particular, we should finish the computation after some stop criteria and before the full computation of the current approximation. We prove that only two sequences of parameters can be derived verifying these properties. One corresponds to a very well known family and the other is a little (but not natural) modification. Finally, we study some dynamical aspects of these families in order to find differences. Surprisingly, the less natural family seems to have a simpler dynamic.

This paper is devoted to the study of a class of high-order iterative methods for nonlinear equations on Banach spaces. An analysis of the convergence under Kantorovich-type conditions is proposed. Some numerical experiments, where the analyzed methods present better behavior than some classical schemes, are presented. These applications include the approximation of some quadratic and integral equations.

The dynamics of a family of Newton's type iterative methods for second-and third-degree complex polynomials is studied. The conjugacy classes of these methods are presented. Classical properties of rational maps are used.