Iterative schemes for finding all roots simultaneously of nonlinear equations

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

doi:10.1016/j.aml.2022.108325

Cited by 10 publications

(13 citation statements)

References 12 publications

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“…It can be demonstrated in a simple way, as in [6], that if we combine any iterative method for systems with the iterative step (2), we will obtain a new method that obtains several solutions simultaneously with duplicated order regarding the original scheme for systems.…”

Section: Contains the Elements Of Thementioning

confidence: 99%

“…The results obtain for the Freudenstein-Roth function are shown in Table 2. The methods employed are P S , from [6], and J F S by changing the values of the parameter β , which in this case is equal for all the components of the initial guess. Here it is shown that depending on how we change the values of the parameter, the results obtained also vary.…”

Section: Freudenstein-roth Functionmentioning

confidence: 99%

“…For this reason, we highlight the relevance of the study we carried out in [6], where we study how to add the Ehrlich step to any other iterative scheme, thus obtaining a new method that obtains solutions simultaneously. In this paper, the order of convergence is analysed for arbitrary equations, not only polynomials.…”

Section: Introduction and Design Of The Iterative Stepmentioning

confidence: 99%

“…In addition, in [4], the iterative step studied in [6] is modified in order to obtain a compatible iterative process for systems, a subject that, to our knowledge, has not been done for systems of arbitrary nonlinear equations.…”

Section: Introduction and Design Of The Iterative Stepmentioning

confidence: 99%

See 3 more Smart Citations

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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Section: Contains the Elements Of Thementioning

confidence: 99%

Section: Freudenstein-roth Functionmentioning

confidence: 99%

Section: Introduction and Design Of The Iterative Stepmentioning

confidence: 99%

Section: Introduction and Design Of The Iterative Stepmentioning

confidence: 99%

See 2 more Smart Citations

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

Triguero-Navarro,

Cordero,

Torregrosa

2023

Preprint

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show abstract

“…Nonlinear equations are widely used in computational science and engineering modeling because of their ability to accurately represent the complexities of real-world phenomena, resulting in more precise predictions, optimizations, and insights into system behaviors in a wide range of scientific and engineering disciplines [1][2][3][4][5], including fluid dynamics [6], quantum mechanics [7], electromagnetism, and computational biology processes [8], to name only a few. They are especially important in chaos theory and complexity, where they can model systems with great sensitivity to initial conditions, e.g., in meteorology, population dynamics, and in the financial sector [9,10].…”

Section: Introductionmentioning

confidence: 99%

Q-Analogues of Parallel Numerical Scheme Based on Neural Networks and Their Engineering Applications

Shams,

Carpentieri

2024

Applied Sciences

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Quantum calculus can provide new insights into the nonlinear behaviour of functions and equations, addressing problems that may be difficult to tackle by classical calculus due to high nonlinearity. Iterative methods for solving nonlinear equations can benefit greatly from the mathematical theory and tools provided by quantum calculus, e.g., using the concept of q-derivatives, which extends beyond classical derivatives. In this paper, we develop parallel numerical root-finding algorithms that approximate all distinct roots of nonlinear equations by utilizing q-analogies of the function derivative. Furthermore, we utilize neural networks to accelerate the convergence rate by providing accurate initial guesses for our parallel schemes. The global convergence of the q-parallel numerical techniques is demonstrated using random initial approximations on selected biomedical applications, and the efficiency, stability, and consistency of the proposed hybrid numerical schemes are analyzed.

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A new multi-step method for solving nonlinear systems with high efficiency indices

Erfanifar,

Hajarian

2024

Numer Algor

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Iterative schemes for finding all roots simultaneously of nonlinear equations

Cited by 10 publications

References 12 publications

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

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

Q-Analogues of Parallel Numerical Scheme Based on Neural Networks and Their Engineering Applications

A new multi-step method for solving nonlinear systems with high efficiency indices

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