Local Convergence Analysis Research Articles

Unified local convergence for a certain family of methods in Banach space

We provide a local convergence analysis for a unified family of methods in order to approximate a locally unique solution of a nonlinear equation in a Banach space setting. These methods have fourth-order convergence, when specialized in the finite dimensional Euclidean space. In contrast to the earlier studies using hypotheses up to the fifth Frechet-derivative, we only use hypotheses on the first Frechet-derivative and Lipschitz constants. The applicability of these methods is expanded this way. Numerical examples are presented to show that we can solve equations in cases not possible with earlier approaches.

Local convergence and a chemical application of derivative free root finding methods with one parameter based on interpolation

We present a local convergence analysis of a derivative free fourth order method with one parameter based on rational interpolation in order to approximate a locally unique root of a function. The method is optimal in the sense of Traub. In earlier studies such as Steffensen (Scand Actuar J 16(1):64–72, 1933) and Zafer et al. (Sci World J, 2015. doi:10.1155/2015/934260) the convergence was based on hypotheses on the third derivative or even higher. We extend the applicability of theses methods using only the first derivative. Moreover, we provide computable radii and error bounds based on Lipschitz constants. Furthermore, the dynamics of this method are studied in order to find the best choice of the parameter in terms of convergence. An application is also presented in this study.

Local analysis of a spectral correction for the Gauss-Newton model applied to quadratic residual problems

A simple spectral correction for the Gauss-Newton model applied to nonlinear least squares problems is presented. Such a correction consists in adding a sign-free multiple of the identity to the Hessian of the Gauss-Newton model, being the multiple based on spectral approximations for the Hessians of the residual functions. A detailed local convergence analysis is provided for the resulting method applied to the class of quadratic residual problems. Under mild assumptions, the proposed method is proved to be convergent for problems for which the convergence of the Gauss-Newton method might not be ensured. Moreover, the rate of linear convergence is proved to be better than the Gauss-Newton's one for a class of non-zero residue problems. These theoretical results are illustrated by numerical examples with quadratic and non-quadratic residual problems.

Local convergence for a derivative free method of order three under weak conditions

A local convergence analysis for a family of a third-order method in order to approximate a solution of a nonlinear equation is presented in this paper. We use hypotheses only on the first derivative in contrast to earlier studies such as Parhi and Gupta (2007, 2010) and Zhu and Wu (2003) using hypotheses only on the first derivative. This way the applicability of these methods is extended under weaker hypotheses. Moreover the radius of convergence and computable error bounds on the distances involved are also given in this study. Numerical examples are also presented in this study.

Enlarging the convergence ball of Newton's method on Lie groups

We present a local convergence analysis of Newton's method for approximating a zero of a mapping from a Lie group into its Lie algebra. Using more precise estimates than before [55, 56] and under the same computational cost, we obtain a larger convergence ball and more precise error bounds on the distances involved. Some examples are presented to further validate the theoretical results.

The Local Convergence Analysis of Inexact Quasi-Gauss–Newton Method Under the Hölder Condition

In this article, we introduce the local convergence of the inexact quasi-Gauss---Newton method when the first Frechet derivative of operator involved is Holder condition. Furthermore, we give some results on the existence and uniqueness of the solution for a nonlinear function. Based on this study, the R-order of convergence is proved.

Ball convergence for Traub-Steffensen like methods in Banach space

Abstract We present a local convergence analysis for two Traub-Steffensen-like methods in order to approximate a locally unique solution of an equation in a Banach space setting. In earlier studies such as [16, 23] Taylor expansions and hypotheses up to the third Fréchet-derivative are used. We expand the applicability of these methods using only hypotheses on the first Fréchet derivative. Moreover, we obtain a radius of convergence and computable error bounds using Lipschitz constants not given before. Numerical examples are also presented in this study.

Improved local convergence for Euler–Halley-like methods with a parameter

We present a local convergence analysis for Euler–Halley-like methods with a parameter in order to approximate a locally unique solution of an equation in a Banach space setting. Using more flexible Lipschitz-type hypotheses than in earlier studies such as Huang and Ma (Numer Algorith 52:419–433, 2009), we obtain a larger radius of convergence as well as more precise error estimates on the distances involved. Numerical examples justify our theoretical results.

Ball convergence of a sixth order iterative method with one parameter for solving equations under weak conditions

We present a local convergence analysis of a sixth order iterative method for approximate a locally unique solution of an equation defined on the real line. Earlier studies such as Sharma et al. (Appl Math Comput 190:111---115, 2007) have shown convergence of these methods under hypotheses up to the fifth derivative of the function although only the first derivative appears in the method. In this study we expand the applicability of these methods using only hypotheses up to the first derivative of the function. Numerical examples are also presented in this study.

Local Convergence of an Efficient High Convergence Order Method Using Hypothesis Only on the First Derivative

We present a local convergence analysis of an eighth order three step methodin order to approximate a locally unique solution of nonlinear equation in a Banach spacesetting. In an earlier study by Sharma and Arora (2015), the order of convergence wasshown using Taylor series expansions and hypotheses up to the fourth order derivative oreven higher of the function involved which restrict the applicability of the proposed scheme. However, only first order derivative appears in the proposed scheme. In order to overcomethis problem, we proposed the hypotheses up to only the first order derivative. In this way,we not only expand the applicability of the methods but also propose convergence domain. Finally, where earlier studies cannot be applied, a variety of concrete numerical examplesare proposed to obtain the solutions of nonlinear equations. Our study does not exhibit thistype of problem/restriction.

On a sixth-order Jarratt-type method in Banach spaces

We present a local convergence analysis of a sixth-order Jarratt-type method in order to approximate a solution of a nonlinear equation in a Banach space. Our sufficient convergence conditions involve only hypotheses on the first Fréchet-derivative of the operator involved. Earlier studies such as [X. Wang, J. Kou and C. Gu, Semilocal convergence of a sixth-order Jarratt method in Banach spaces, Numer. Algorithms 57 (2011) 441–456.] require hypotheses up to the third Fréchet-derivative. Numerical examples are also provided in this study.

Ball Convergence for an Inverse Free Jarratt-Type Method Under Hölder Conditions

We present a local convergence analysis of an inverse free Jarratt-type method in order to approximate a locally unique solution of an equation in a Banach space setting. Earlier studies have used hypotheses up to the third Frechet-derivative of the operator involved to show convergence although only the first derivative is used in the method. We show convergence using only the first Frechet derivative under Holder conditions. This way we expand the applicability of the method. Numerical examples are also provided in this study.

New modifications of Hansen–Patrick’s family with optimal fourth and eighth orders of convergence

In this paper, we present two new second-derivative free classes of iterative methods based on Hansen-Patrick’s family (Hansen and Patrick, 1977 [4]) for solving nonlinear equations numerically. In terms of computational cost, both families require only three and four functional evaluations to achieve optimal fourth and eighth orders of convergence, respectively. Moreover, the local convergence analysis of the proposed methods is also given using hypotheses only on the first derivative and Lipschitz constants. Furthermore, the proposed schemes can also determine the complex zeros without having to start from a complex initial guess as would be necessary with other methods. Numerical examples and comparisons with some existing methods are included to confirm the theoretical results and high computational efficiency.

On the local convergence of Ehrlich method for numerical computation of polynomial zeros

In this paper, we provide a detailed local convergence analysis of the Ehrlich method for numerical computation of all zeros of a polynomial simultaneously. Our results generalize and improve the results of Kyurkchiev and Tashev (C. R. Acad. Bulg. Sci. 34:1053---1055, 1981), Wang and Zhao (Computing 43:187---197, 1989) and Tilli (Calcolo 35:3---15, 1998).

Ball convergence theorems for Maheshwari-type eighth-order methods under weak conditions

We present a local convergence analysis for a family of Maheshwari-type eighth-order methods in order to approximate a solution of a nonlinear equation. We use hypotheses up to the first derivative in contrast to earlier studies such as Cordero et al. (J Comput Appl Math 291(1):348–357, 2016), Maheshwari (Appl Math Comput 211:283–391, 2009), Petkovic et al. (Multipoint methods for solving nonlinear equations. Elsevier, Amsterdam, 2013) using hypotheses up to the seventh derivative. This way the applicability of these methods is extended under weaker hypotheses. Moreover the radius of convergence and computable error bounds on the distances involved are also given in this study. Numerical examples are also presented in this study.

Local convergence for some high convergence order Newton-like methods with frozen derivatives

We present a local convergence analysis of some families of Newton-like methods with frozen derivatives in order to approximate a locally unique solution of an equation in a Banach space setting. In earlier studies such as Amat et al. (Appl Math Lett. 25:2209–2217, 2012), Petkovic (Multipoint methods for solving nonlinear equations, Elsevier, Amsterdam, 2013), Traub (Iterative methods for the solution of equations, AMS Chelsea Publishing, Providence, 1982) and Xiao and Yin (Appl Math Comput, 2015) the local convergence was proved based on hypotheses on the derivative of order higher than two although only the first derivative appears in these methods. In this paper we expand the applicability of these methods using only hypotheses on the first derivative and Lipschitz constants. Numerical examples are also presented in this study.

Local Convergence for an Efficient Eighth Order Iterative Method with a Parameter for Solving Equations Under Weak Conditions

We present a local convergence analysis of an efficient eighth order iterative method with a parameter for approximate a locally unique solution of an equation defined on the real line. Earlier studies such as Bi et al. (J Comput Appl Math 225:105–112, 2009) have shown convergence of these methods under hypotheses up to the seventh derivative of the function although only the first derivative appears in the method. In this study we expand the applicability of these methods using only hypotheses up to the first derivative of the function. Numerical examples are also presented in this study.

A study of optimization for Steffensen-type methods with frozen divided differences

A local convergence analysis for a generalization of a family of Steffensen-type iterative methods with three frozen steps is presented for solving nonlinear equations. From the use of three classical divided difference operators, we study four families of iterative methods with optimal local order of convergence. Then, new variants of the family of iterative methods is constructed, where a study of the computational efficiency is carried out. Moreover, the semilocal convergence for these families is also studied. Finally, an application of nonlinear integral equations of mixed Hammerstein type is presented, where multiple precision and a stopping criterion are implemented without using any known root. In addition, a study, where we compare orders, efficiencies and elapsed times of the methods suggested, supports the theoretical results obtained.

Local convergence analysis of inexact Gauss–Newton method for singular systems of equations under majorant and center-majorant condition

We present a new semi-local convergence analysis of the Gauss–Newton method for solving convex composite optimization problems using the concept of quasi-regularity for an initial point. The convergence analysis is based on a combination of a center-majorant and a majorant function. The results extend the applicability of the Gauss–Newton method under the same computational cost as in earlier studies. In particular, the advantages are: the error estimates on the distances involved are more precise and the convergence ball is at least as large. Numerical examples are also provided in this study.

Existe realmente convergência de renda entre países?

<title>Resumo</title><p>A maioria dos trabalhos sobre convergência de renda pressupõe que os países são unidades geográficas isoladas, não considerando os efeitos espaciais (dependência e heterogeneidade) subjacentes. Este artigo investiga a análise de convergência de renda entre países, incorporando o efeito de vizinhança proporcionado pela interação espacial, e, portanto, controlando-se para os efeitos espaciais, dependência espacial e heterogeneidade espacial observável e não observável. Para se conseguir isso, foi construído um modelo de convergência espacial local para 148 países no período quinquenal entre 1985 e 2005, em primeiras diferenças, usando a técnica da Regressão Ponderada Geograficamente. Os principais resultados revelam a existência de múltiplos equilíbrios quando considerada a análise de convergência local. Um grupo de países se aproxima de um equilíbrio particular em virtude da existência de atributos correlatos ou porque se encontrava em uma localização inicial correspondente àquele equilíbrio, fato em consonância com a convergência em clubes ou múltiplos equilíbrios. A diferença, todavia, repousa no fato de que os intervalos de convergência aqui definidos são consequência de como o crescimento de um país se processa, em termos de seu próprio ritmo condicionado às suas características socioeconômicas iniciais, relativamente ao crescimento de outros países e não em relação a si mesmo. Além disso, em contraste com os clubes de convergência, esses intervalos de convergência são definidos ex-post, levando em consideração a dinâmica de crescimento específica a cada país.</p>