An iterative scheme to obtain multiple solutions simultaneously

Cordero, Alicia; Garrido, Neus; Torregrosa, Juan R.; Triguero‐Navarro, Paula

doi:10.1016/j.aml.2023.108738

Cited by 4 publications

(2 citation statements)

References 7 publications

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“…In [5], the particular case in which the functions are polynomials is analysed, as well as a case in which the solutions of arbitrary nonlinear equations have different multiplicities.…”

Section: Introduction and Design Of The Iterative Stepmentioning

confidence: 99%

CMMSE: Jacobian-free vectorial iterative scheme to find several solutions simultaneously

Triguero-Navarro,

Cordero,

Torregrosa

2023

Preprint

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This manuscript is devoted to a derivative-free parametric iterative step to obtain roots simultaneously for both nonlinear systems and equations. We prove that when it is added to an arbitrary scheme, it doubles the convergence order of the original procedure and defines a new algorithm that obtains several solutions simultaneously. Different numerical experiments are carried out to check the behaviour of the iterative methods by changing the value of the parameter and the initial guesses. Also, it is perform a numerical example where the dynamical planes are carried out to see and compare the basins of attraction.

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“…In [5], the particular case in which the functions are polynomials is analysed, as well as a case in which the solutions of arbitrary nonlinear equations have different multiplicities.…”

Section: Introduction and Design Of The Iterative Stepmentioning

confidence: 99%

CMMSE: Jacobian-free vectorial iterative scheme to find several solutions simultaneously

Triguero-Navarro,

Cordero,

Torregrosa

2023

Preprint

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“…Method ( 2) is of optimal convergence order 2 with efficiency index 1.41. If we use Weierstrass' Correction [48][49][50]…”

mentioning

confidence: 99%

A New Approach to Multiroot Vectorial Problems: Highly Efficient Parallel Computing Schemes

Shams,

Rafiq,

Carpentieri

et al. 2024

Fractal Fract

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In this article, we construct an efficient family of simultaneous methods for finding all the distinct as well as multiple roots of polynomial equations. Convergence analysis proves that the order of convergence of newly constructed family of simultaneous methods is seventeen. Fractal-based simultaneous iterative algorithms are thoroughly examined. Using self-similar features, fractal-based simultaneous schemes can converge to solutions faster, saving computational time and resources necessary for solving nonlinear equations. Fractals analysis illustrates the newly developed method’s global convergence behavior when compared to single root-finding procedures for solving fractional order polynomials that arise in complex engineering applications. Some real problems from various branches of engineering along with some higher degree polynomials are considered as test examples to show the global convergence property of simultaneous methods, performance and efficiency of the proposed family of methods. Further computational efficiencies, CPU time and residual graphs are also drawn to validate the efficiency, robustness of the newly introduced family of methods as compared to the existing methods in the literature.

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Semi-Analytical Closed-Form Solutions of the Ball–Plate Problem

Ene,

Pop

2024

Processes

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Mathematical models and numerical simulations are necessary to understand the dynamical behaviors of complex systems. The aim of this work is to investigate closed-form solutions for the ball–plate problem considering a system derived from an optimal control problem for ball–plate dynamics. The nonlinear properties of ball and plate control system are presented in this work. To semi-analytically solve this system, we explored a second-order nonlinear differential equation. Consequently, we obtained the approximate closed-form solutions by the Optimal Parametric Iteration Method (OPIM) using only one iteration. A comparison between the analytical and corresponding numerical procedures reflects the advantages of the first one. The accordance between the obtained results and the numerical ones highlights that the procedure used is accurate, effective, and good to implement in applications such as sliding mode control to the ball-and-plate problem.

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An iterative scheme to obtain multiple solutions simultaneously

Cited by 4 publications

References 7 publications

CMMSE: Jacobian-free vectorial iterative scheme to find several solutions simultaneously

CMMSE: Jacobian-free vectorial iterative scheme to find several solutions simultaneously

A New Approach to Multiroot Vectorial Problems: Highly Efficient Parallel Computing Schemes

Semi-Analytical Closed-Form Solutions of the Ball–Plate Problem

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