Fractal Newton Methods

Akgül, Ali; Grow, David E.

doi:10.3390/math11102277

Cited by 2 publications

(2 citation statements)

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“…In recent papers, some fractional Newton-type iterative procedures with Caputo and Riemann-Liouville derivatives (see [4][5][6][7][8][9]), and fractal Newton-type iterative methods (see [10]), have been designed in order to find the solution x ∈ R of a nonlinear function f (x), where f :I ⊆ R → R is a continuous function in I ⊆ R, but the theoretical order of convergence is neither preserved nor held in practice. Additionally, optimal conformable Newton-type schemes were proposed in [11,12] (in scalar and vectorial versions, respectively) by using the conformable derivative/Jacobian, and the order of convergence was obtained in theory and in practice too.…”

Section: Introductionmentioning

confidence: 99%

Solving Nonlinear Transcendental Equations by Iterative Methods with Conformable Derivatives: A General Approach

et al. 2023

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In recent years, some Newton-type schemes with noninteger derivatives have been proposed for solving nonlinear transcendental equations by using fractional derivatives (Caputo and Riemann–Liouville) and conformable derivatives. It has also been shown that the methods with conformable derivatives improve the performance of classical schemes. In this manuscript, we design point-to-point higher-order conformable Newton-type and multipoint procedures for solving nonlinear equations and propose a general technique to deduce the conformable version of any classical iterative method with integer derivatives. A convergence analysis is given and the expected orders of convergence are obtained. As far as we know, these are the first optimal conformable schemes, beyond the conformable Newton procedure, that have been developed. The numerical results support the theory and show that the new schemes improve the performance of the original methods in some aspects. Additionally, the dependence on initial guesses is analyzed, and these schemes show good stability properties.

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Section: Introductionmentioning

confidence: 99%

Solving Nonlinear Transcendental Equations by Iterative Methods with Conformable Derivatives: A General Approach

et al. 2023

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“…In Ref. [5], the first Newton's methods with fractal derivatives are presented, whose order of convergence is quadratic. With regard to iterative schemes with fractional derivatives, the authors in Ref.…”

Section: Introductionmentioning

confidence: 99%

Derivative-Free Conformable Iterative Methods for Solving Nonlinear Equations

et al. 2023

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In this manuscript, we use approximations of conformable derivatives for designing iterative methods to solve nonlinear algebraic or trascendental equations. We adapt the approximation of conformable derivatives in order to design conformable derivative-free iterative schemes to solve nonlinear equations: Steffensen and Secant-type methods. To our knowledge, these are the first conformable derivative-free schemes in the literature, where the Steffensen conformable method is also optimal; moreover, the Secant conformable scheme is also a procedure with memory. A convergence analysis is made, preserving the order of classical cases, and the numerical performance is studied in order to confirm the theoretical results. It is shown that these methods can present some numerical advantages versus their classical partners, with wide sets of converging initial estimations.

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Fractal Newton Methods

Abstract: We introduce fractal Newton methods for solving f(x)=0 that generalize and improve the classical Newton method. We compare the theoretical efficacy of the classical and fractal Newton methods and illustrate the theory with examples.

Cited by 2 publications

References 12 publications

Solving Nonlinear Transcendental Equations by Iterative Methods with Conformable Derivatives: A General Approach

Solving Nonlinear Transcendental Equations by Iterative Methods with Conformable Derivatives: A General Approach

Derivative-Free Conformable Iterative Methods for Solving Nonlinear Equations

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