2023
DOI: 10.3390/sym15051116
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Extension of King’s Iterative Scheme by Means of Memory for Nonlinear Equations

Abstract: We developed a new family of optimal eighth-order derivative-free iterative methods for finding simple roots of nonlinear equations based on King’s scheme and Lagrange interpolation. By incorporating four self-accelerating parameters and a weight function in a single variable, we extend the proposed family to an efficient iterative scheme with memory. Without performing additional functional evaluations, the order of convergence is boosted from 8 to 15.51560, and the efficiency index is raised from 1.6817 to 1… Show more

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Cited by 6 publications
(2 citation statements)
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“…Other algorithms (hidden to the user) used in this article are as follows: (i) the arbitrary values of the parameters of the model given by the costumer-the producing of the expanded oscillator; (ii) specific computational procedures for comprehensive Hamiltonian research of differential systems and picturing the "level curves" (supposing the realization of software facilities in a user-chosen CAS for reliable computations); (iii) an algorithm for picking out the starting approximations when searching for a solution to the differential systems, which states its intriguing peculiarity and demeanor of the solution in confidential time intervals; (iv) numerical algorithms for solving the nonlinear equation, M(t 0 ) = 0 [33][34][35]-in several cases (when chaos is produced in dynamical models), M(t) is a polynomial and this necessitates the use of specialized algorithms to simultaneously find of all zeros [36][37][38][39][40][41]; (v) algorithms for generating the Melnikov functions of a higher type for analyzing the homo/heteroclinic bifurcation in mechanical systems; (vi) algorithms for generating bifurcation diagrams; (vii) algorithms for generating chaos in non-self-governmental model. Several of these computational procedures have been build on the basis of famous traditional and newer investigations [42][43][44][45][46][47][48][49][50].…”
Section: Discussionmentioning
confidence: 99%
“…Other algorithms (hidden to the user) used in this article are as follows: (i) the arbitrary values of the parameters of the model given by the costumer-the producing of the expanded oscillator; (ii) specific computational procedures for comprehensive Hamiltonian research of differential systems and picturing the "level curves" (supposing the realization of software facilities in a user-chosen CAS for reliable computations); (iii) an algorithm for picking out the starting approximations when searching for a solution to the differential systems, which states its intriguing peculiarity and demeanor of the solution in confidential time intervals; (iv) numerical algorithms for solving the nonlinear equation, M(t 0 ) = 0 [33][34][35]-in several cases (when chaos is produced in dynamical models), M(t) is a polynomial and this necessitates the use of specialized algorithms to simultaneously find of all zeros [36][37][38][39][40][41]; (v) algorithms for generating the Melnikov functions of a higher type for analyzing the homo/heteroclinic bifurcation in mechanical systems; (vi) algorithms for generating bifurcation diagrams; (vii) algorithms for generating chaos in non-self-governmental model. Several of these computational procedures have been build on the basis of famous traditional and newer investigations [42][43][44][45][46][47][48][49][50].…”
Section: Discussionmentioning
confidence: 99%
“…The extension of without-memory methods into with-memory methods using accelerating parameters have gained much attention in recent years [10][11][12][13]. In multi-point with memory iterative methods, the order of convergence is significantly increased without any additional function evaluation by using information from the current as well as the previous iterations.…”
Section: Introductionmentioning
confidence: 99%