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Keywords = Potra–Pták method

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21 pages, 1037 KiB  
Article
Design and Complex Dynamics of Potra–Pták-Type Optimal Methods for Solving Nonlinear Equations and Its Applications
by Prem B. Chand, Francisco I. Chicharro, Neus Garrido and Pankaj Jain
Mathematics 2019, 7(10), 942; https://doi.org/10.3390/math7100942 - 11 Oct 2019
Cited by 9 | Viewed by 3055
Abstract
In this paper, using the idea of weight functions on the Potra–Pták method, an optimal fourth order method, a non optimal sixth order method, and a family of optimal eighth order methods are proposed. These methods are tested on some numerical examples, and [...] Read more.
In this paper, using the idea of weight functions on the Potra–Pták method, an optimal fourth order method, a non optimal sixth order method, and a family of optimal eighth order methods are proposed. These methods are tested on some numerical examples, and the results are compared with some known methods of the corresponding order. It is proved that the results obtained from the proposed methods are compatible with other methods. The proposed methods are tested on some problems related to engineering and science. Furthermore, applying these methods on quadratic and cubic polynomials, their stability is analyzed by means of their basins of attraction. Full article
(This article belongs to the Special Issue Iterative Methods for Solving Nonlinear Equations and Systems)
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15 pages, 979 KiB  
Article
Modified Potra–Pták Multi-step Schemes with Accelerated Order of Convergence for Solving Systems of Nonlinear Equations
by Himani Arora, Juan R. Torregrosa and Alicia Cordero
Math. Comput. Appl. 2019, 24(1), 3; https://doi.org/10.3390/mca24010003 - 27 Dec 2018
Cited by 1 | Viewed by 2788
Abstract
In this study, an iterative scheme of sixth order of convergence for solving systems of nonlinear equations is presented. The scheme is composed of three steps, of which the first two steps are that of third order Potra–Pták method and last is weighted-Newton [...] Read more.
In this study, an iterative scheme of sixth order of convergence for solving systems of nonlinear equations is presented. The scheme is composed of three steps, of which the first two steps are that of third order Potra–Pták method and last is weighted-Newton step. Furthermore, we generalize our work to derive a family of multi-step iterative methods with order of convergence 3r+6,r=0,1,2,…. The sixth order method is the special case of this multi-step scheme for r=0. The family gives a four-step ninth order method for r=1. As much higher order methods are not used in practice, so we study sixth and ninth order methods in detail. Numerical examples are included to confirm theoretical results and to compare the methods with some existing ones. Different numerical tests, containing academical functions and systems resulting from the discretization of boundary problems, are introduced to show the efficiency and reliability of the proposed methods. Full article
(This article belongs to the Special Issue Mathematical Modelling in Engineering & Human Behaviour 2018)
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295 KiB  
Article
An Optimal Eighth-Order Derivative-Free Family of Potra-Pták’s Method
by Munish Kansal, Vinay Kanwar and Saurabh Bhatia
Algorithms 2015, 8(2), 309-320; https://doi.org/10.3390/a8020309 - 15 Jun 2015
Cited by 8 | Viewed by 5262
Abstract
In this paper, we present a new three-step derivative-free family based on Potra-Pták’s method for solving nonlinear equations numerically. In terms of computational cost, each member of the proposed family requires only four functional evaluations per full iteration to achieve optimal eighth-order convergence. [...] Read more.
In this paper, we present a new three-step derivative-free family based on Potra-Pták’s method for solving nonlinear equations numerically. In terms of computational cost, each member of the proposed family requires only four functional evaluations per full iteration to achieve optimal eighth-order convergence. Further, computational results demonstrate that the proposed methods are highly efficient as compared with many well-known methods. Full article
(This article belongs to the Special Issue Numerical Algorithms for Solving Nonlinear Equations and Systems)
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