A General Optimal Iterative Scheme with Arbitrary Order of Convergence

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

doi:10.3390/sym13050884

Cited by 10 publications

(10 citation statements)

References 22 publications

(31 reference statements)

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“…In [11] Cordero et al, in 2021, proposed a method that consists of a four-step iterative method (O82) with the eighth-order of convergence. The efficiency index of this scheme is about 1.5157, which is better than NRM and some other existing methods.…”

Section: Methodsmentioning

confidence: 99%

“…The use and recommendation of higher-order nonlinear methods for scalar equations are also encouraged in previous and current articles. For example, the eighth-order method in [9][10][11], and fifteenth-order method in [14], have been devised. While solving nonlinear models, we have employed three methods known as the proposed method given in (12), the well-known Newton's method given in Eq.…”

Section: Mathematical Analysesmentioning

confidence: 99%

“…For example, Jaiswal and Choubey [9] presented a three-step method of eighth-order with five function evaluations. Similarly, Liu et al in [10] developed a three-step iterative scheme of eighth-order of convergence with five function evaluations, and Cordero et al in [11] also showed a four-step technique of order eight with five function evaluations.…”

Section: Introductionmentioning

confidence: 99%

See 2 more Smart Citations

A novel hybrid iterative method for applied mathematical models with time-efficiency

Jamali¹,

Solangi²,

Shaikh³

2022

J. Appl. Math. Comput. Mech.

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Non-linear phenomena appear in many fields of engineering and science. Research on numerical methods is continually extending with the improvement of the latest computing tools. In today's computational field, one requires maximum achievement in a minimum amount of time. Therefore, there is a need to modify the Newton-type method to achieve higher-order convergence to solve non-linear equations. While the modified methods are expected to be higher-order convergent, the minor computational information and the maximum time efficiency are also important factors. We propose a three-step hybrid iterative method having a non-linear nature. Per iteration, the method requires three function evaluations and three first-order derivatives. The method is theoretically proven to be tenth-order convergent. The mathematical results of the proposed strategy to solve models from fluid dynamics, electric field, and real gases demonstrated better performance. In light of error analysis, computational productivity, and CPU times, the proposed method's performance is compared to the famous Newton and a recently proposed tenth-order method.

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Section: Methodsmentioning

confidence: 99%

Section: Mathematical Analysesmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

See 1 more Smart Citation

A novel hybrid iterative method for applied mathematical models with time-efficiency

Jamali¹,

Solangi²,

Shaikh³

2022

J. Appl. Math. Comput. Mech.

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show abstract

“…In this section, we perform different numerical experiments in order to observe the behavior of the proposed methods. In this case, we modify, as discussed in the previous section, Newton's method, Steffensen's method [8], the N 4 and N 8 methods designed in [9], and the M 4 and M 6 schemes constructed in [10]. We denote these methods in the same way as in the previous section, that is, if the method is denoted by ϕ, then its variant with the added step is denoted by ϕ s .…”

Section: Numerical Experimentsmentioning

confidence: 99%

Iterative schemes for finding all roots simultaneously of nonlinear equations

Cordero

Garrido

Torregrosa

et al. 2022

Applied Mathematics Letters

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“…The following Theorem is devoted to studying the order of convergence of MIP, for which the Taylor expansion technique has been used (e.g. see [52] for further details). Theorem 3.…”

Section: A Order Of Convergence Of Mipmentioning

confidence: 99%

Mann-Iteration Process for Power Flow Calculation of Large-Scale Ill-Conditioned Systems: Theoretical Analysis and Numerical Results

et al. 2021

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Power Flow solution of realistic ill-conditioned systems has recently attracted huge attention. Nevertheless, there are still some gaps in this field. For example, most of available references do not provide exhaustive theoretical analysis about convergence properties of proposed approaches. In addition, efficient solution of large-scale ill-conditioned systems is still an open topic. This paper tackles these issues by comprehensively studying the suitability of the Mann Iteration Process for the solution of ill-conditioned systems. A comprehensive theoretical analysis is provided, from which is demonstrated that the Mann Iteration Process has with asymptotic stability, may achieve a high convergence rate and constitutes a robust methodology, improving the contractive properties of the Newton-Raphson method. Moreover, some interesting links with other Power-Flow approaches are obtained as by-product. Several numerical experiments serve to confirm the theoretical findings and to compare the performance of the Mann Iteration Process with other well-known PF solvers. In all cases, the results obtained with the Mann Iteration Process are superior to that obtained using other methodologies, being able to efficiently solve various large-scale ill-conditioned systems.INDEX TERMS power-flow analysis, ill-conditioned systems, large-scale systems, Mann iteration process

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A General Optimal Iterative Scheme with Arbitrary Order of Convergence

Cited by 10 publications

References 22 publications

A novel hybrid iterative method for applied mathematical models with time-efficiency

A novel hybrid iterative method for applied mathematical models with time-efficiency

Iterative schemes for finding all roots simultaneously of nonlinear equations

Mann-Iteration Process for Power Flow Calculation of Large-Scale Ill-Conditioned Systems: Theoretical Analysis and Numerical Results

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