Introducing memory to a family of multi-step multidimensional iterative methods with weight function

Cordero, Alicia; Villalba, Eva G.; Torregrosa, Juan R.; Triguero‐Navarro, Paula

doi:10.1016/j.exmath.2023.04.004

Cited by 6 publications

(13 citation statements)

References 11 publications

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“…To avoid the generally expensive computation of the Fréchet derivative P ′ (x k ) of the operator P and increase the convergence to be higher than one other ITs have been developed [1,[3][4][5][6][7] using divided differences of order one [1,2]. The methods of chords or Regula falsi or the Secant method are some of the most used ITs for solving Equation (1).…”

Section: Introductionmentioning

confidence: 99%

“…which is also of convergence order two [4,5]. But the divided difference [•, •; P] is used instead of the derivative P ′ .…”

Section: Introductionmentioning

confidence: 99%

“…The weight B(s) is any real matrix function that satisfies B(0) = 1, B(1) = −1 and ∥B ′′ (0)∥ < ∞. The convergence order 2 m is proven in [5] using the Taylor series for E = R n . A detailed favorable comparison with other competing methods using similar information has been carried out in [5,10,11].…”

Section: Introductionmentioning

confidence: 99%

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Symmetric-Type Multi-Step Difference Methods for Solving Nonlinear Equations

Argyros,

Shakhno,

Regmi

et al. 2024

Symmetry

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Symmetric-type methods (STM) without derivatives have been used extensively to solve nonlinear equations in various spaces. In particular, multi-step STMs of a higher order of convergence are very useful. By freezing the divided differences in the methods and using a weight operator a method is generated using m steps (m a natural number) of convergence order 2 m. This method avoids a large increase in the number of operator evaluations. However, there are several problems with the conditions used to show the convergence: the existence of high order derivatives is assumed, which are not in the method; there are no a priori results for the error distances or information on the uniqueness of the solutions. Therefore, the earlier studies cannot guarantee the convergence of the method to solve nondifferentiable equations. However, the method may converge to the solution. Thus, the convergence conditions can be weakened. These problems arise since the convergence order is determined using the Taylor series which requires the existence of high-order derivatives which are not present in the method, and they may not even exist. These concerns are our motivation for authoring this article. Moreover, the novelty of this article is that all the aforementioned problems are addressed positively, and by using conditions only related to the divided differences in the method. Furthermore, a more challenging and important semi-local analysis of convergence is presented utilizing majorizing sequences in combination with the concept of the generalized continuity of the divided difference involved. The convergence is also extended from the Euclidean to the Banach space. We have chosen to demonstrate our technique in the present method. But it can be used in other studies using the Taylor series to show the convergence of the method. The applicability of other single- or multi-step methods using the inverses of linear operators with or without derivatives can also be extended with the same methodology along the same lines. Several examples are provided to test the theoretical results and validate the performance of the method.

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Section: Introductionmentioning

confidence: 99%

“…which is also of convergence order two [4,5]. But the divided difference [•, •; P] is used instead of the derivative P ′ .…”

Section: Introductionmentioning

confidence: 99%

See 1 more Smart Citation

Symmetric-Type Multi-Step Difference Methods for Solving Nonlinear Equations

Argyros,

Shakhno,

Regmi

et al. 2024

Symmetry

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“…Cordero et al constructed a derivative-free multi-step iterative scheme based on Steffensen's method. In fact, to avoid excessively increasing the number of functional evaluations and, at the same time, to increase the order of convergence, the divided differences and a weight function on already evaluated functions were used [23].…”

Section: Introductionmentioning

confidence: 99%

A family of iterative methods to solve nonlinear problems with applications in fractional differential equations

Erfanifar,

Hajarian,

Sayevand

2023

Math Methods in App Sciences

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In this work, first, a family of fourth‐order methods is proposed to solve nonlinear equations. The methods satisfy the Kung‐Traub optimality conjecture. By developing the methods into memory methods, their efficiency indices are increased. Then, the methods are extended to the multi‐step methods for finding the solutions to systems of problems. The formula for the order of convergence of the multi‐step iterative methods is , where is the step number of the methods. It is clear that computing the Jacobian matrix derivative evaluation and its inversion are expensive; therefore, we compute them only once in every cycle of the methods. The important feature of these multi‐step methods is their high‐efficiency index. Numerical examples that confirm the theoretical results are performed. In applications, some nonlinear problems related to the numerical approximation of fractional differential equations (FDEs) are constructed and solved by the proposed methods.

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“…These new schemes can be with or without memory, increasing in the first case the convergence order without adding computational cost. Some references in this area are [10][11][12][13] and the references therein.…”

Section: Introductionmentioning

confidence: 99%

Increasing in three units the order of convergence of iterative methods for solving nonlinear systems

Cordero,

Leonardo-Sepúlveda,

Torregrosa

et al. 2023

Preprint

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This manuscript presents an innovative technique for high-precision iterative methods in nonlinear systems. The goal is to enhance the convergence order by three units with just one additional functional evaluation per iteration, while keeping the computational cost low. This new approach is incorporated into existing methods of order p, allowing for more accurate and efficient results. Numerical tests support the theoretical validity of the technique, highlighting its superiority over conventional p-order methods and its potential for solving complex problems. In summary, the manuscript introduces a promising technique that combines high precision and efficiency in approximating solutions for nonlinear systems, increasing the convergence order of iterative methods from p to p+3.

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Introducing memory to a family of multi-step multidimensional iterative methods with weight function

Cited by 6 publications

References 11 publications

Symmetric-Type Multi-Step Difference Methods for Solving Nonlinear Equations

Symmetric-Type Multi-Step Difference Methods for Solving Nonlinear Equations

A family of iterative methods to solve nonlinear problems with applications in fractional differential equations

Increasing in three units the order of convergence of iterative methods for solving nonlinear systems

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