Stability analysis of a parametric family of iterative methods for solving nonlinear models

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• 作者：Alicia Cordero^a ; ^{acordero@mat.upv.es" class="auth_mail" title="E-mail the corresponding author} ; José ; M. Gutié ; rrez^b ; ^{jmguti@unirioja.es" class="auth_mail" title="E-mail the corresponding author} ; Á ; . Alberto Magre&ntilde ; á ; n^c ; ^{alberto.magrenan@unir.net" class="auth_mail" title="E-mail the corresponding author} ; ^{alberto.magrenan@gmail.com" class="auth_mail" title="E-mail the corresponding author} ; Juan R. Torregrosa ; ^a ; ^{jrtorre@mat.upv.es" class="auth_mail" title="E-mail the corresponding author}
• 关键词：Stability ; Nonlinear problems ; Parameter space ; Iterative methods ; Basins of attraction ; Complex dynamics
• 刊名：Applied Mathematics and Computation
• 出版年：2016
• 出版时间：20 July 2016
• 年：2016
• 卷：285
• 期：Complete
• 页码：26-40
• 全文大小：6653 K

文摘

A one-parametric family of fourth-order iterative methods for solving nonlinear systems is presented, proving the fourth-order of convergence of all members in this family, except one of them whose order is five. The methods in our family are numerically compared with other known methods in terms of the classical efficiency index (order of convergence and number of functional evaluations) and in terms of the operational efficiency index, which also takes into account the total number of product-quotients per iteration. In order to analyze its stability and its dynamical properties, the parameter space for quadratic polynomials is shown. The stability of the strange fixed points is studied in this case. We note that even for this particular case, the family presents a very interesting dynamical behavior. The analysis of the parameter plane allows us to find values for the involved parameter with good stability properties as well as other values with bad numerical behavior. Finally, amongst the first ones, there is a special value of the parameter related to a fifth-order method in the family.