New efficient methods for solving nonlinear systems of equations with arbitrary even order

Abbasbandy, Saeid; Bakhtiari, Parisa; Cordero, Alicia; Torregrosa, Juan R.; Lotfi, Taher

doi:10.1016/j.amc.2016.04.038

Cited by 17 publications

(15 citation statements)

References 12 publications

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“…In addition, we also compare them with an Ostrowski-type method proposed by Grau-Sánchez et al [22], where among them we choose the iterative scheme (7), denoted by (GR). Further, we contrast them with sixth-order iterative schemes presented by Sharma and Arora [23] (expression-18) and Abbasbandy et al [24] (expression-8), notated as (SA) and (AB), respectively. Furthermore, we contrast them with solution techniques of order six designed by Soleymani et al [25] (method-5) and Wang and Li [26] (method-6), known as (SO) and (WL), respectively.…”

Section: Numerical Experimentationmentioning

confidence: 99%

A New Higher-Order Iterative Scheme for the Solutions of Nonlinear Systems

Behl

Argyros

2020

Mathematics

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Many real-life problems can be reduced to scalar and vectorial nonlinear equations by using mathematical modeling. In this paper, we introduce a new iterative family of the sixth-order for a system of nonlinear equations. In addition, we present analyses of their convergences, as well as the computable radii for the guaranteed convergence of them for Banach space valued operators and error bounds based on the Lipschitz constants. Moreover, we show the applicability of them to some real-life problems, such as kinematic syntheses, Bratu's, Fisher's, boundary value, and Hammerstein integral problems. We finally wind up on the ground of achieved numerical experiments, where they perform better than other competing schemes.

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Section: Numerical Experimentationmentioning

confidence: 99%

A New Higher-Order Iterative Scheme for the Solutions of Nonlinear Systems

Behl

Argyros

2020

Mathematics

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“…We propose an algorithm, called Hit-Hit Moving Directives (HHMD-3), for solving unconstrained optimization problem (4). HHMD-3 is a probability-directed search technique that finds the global optimum (minimum or maximum)for unconstrained optimization problems such as the one in (4).…”

Section: The Hit-hit Moving Directives Algorithmmentioning

confidence: 99%

“…The comparison includes both the CPU time and the precision of the solution (in terms of aggregate violation). Table 4 compares our algorithm and the Modified Line Search (MLS) [4]. The entry "-" in some of the Figure 4 indicate that our algorithm outperformed MLS in both solution precisionand the CPU time.…”

Section: Benchmark I3mentioning

confidence: 99%

“…Researchers have proposed many methods for solving systems of equations [1] [7] [8] (see [5] for a good discussion of these methods). These methods either solve the equations directly by applying numerical methods [4] [9][10] [11] [17] [23] or solve them indirectly by transforming the system of equations into problem optimization [2] [3][4] [5] [12] [16] [19] [22]. This paper offers a technique for solving systems of linear or nonlinear equations.…”

Section: Introductionmentioning

confidence: 99%

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Probability-Directed Problem Optimization Technique for Solving Systems of Linear and Non-Linear Equations

Al-Muhammed¹

2019

Computer Science &Amp; Information Technology (CS &Amp; IT )

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Although many methods have been proposed for solving linear or nonlinear systems of equations, there is always a pressing need for more effective and efficient methods. Good methods should produce solutions with high precision and speed. This paper proposed an innovative method for solving systems of linear and nonlinear equations. This method transforms the problem into an optimization problem and uses a probability guided search technique for solving this optimization problem, which is the solution for the system of equations. The transformation results in an aggregate violation function and a criterion function. The aggregation violation function is composed of the constraints that represent the equations and whose satisfaction is a solution for the system of equations. The criterion function intelligently guides the search for the solution to the aggregate violation function by determining when the constraints must be checked; thereby avoiding unnecessary, timeintensive checks for the constraints. Experiments conducted with our prototype implementation showed that our method is effective in finding solutions with high precision and efficient in terms of CPU time.

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“…Sharma and Arora proposed a fourth order method in [19] requiring one F and two F in each iteration. Apart from these third and fourth-order methods, researchers have also proposed some higher order methods, see, for example [11,13,15,17,[20][21][22][23][24] and references therein.…”

Section: Introductionmentioning

confidence: 99%

On a Bi-Parametric Family of Fourth Order Composite Newton–Jarratt Methods for Nonlinear Systems

et al. 2019

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We present a new two-parameter family of fourth-order iterative methods for solving systems of nonlinear equations. The scheme is composed of two Newton–Jarratt steps and requires the evaluation of one function and two first derivatives in each iteration. Convergence including the order of convergence, the radius of convergence, and error bounds is presented. Theoretical results are verified through numerical experimentation. Stability of the proposed class is analyzed and presented by means of using new dynamics tool, namely, the convergence plane. Performance is exhibited by implementing the methods on nonlinear systems of equations, including those resulting from the discretization of the boundary value problem. In addition, numerical comparisons are made with the existing techniques of the same order. Results show the better performance of the proposed techniques than the existing ones.

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New efficient methods for solving nonlinear systems of equations with arbitrary even order

Cited by 17 publications

References 12 publications

A New Higher-Order Iterative Scheme for the Solutions of Nonlinear Systems

A New Higher-Order Iterative Scheme for the Solutions of Nonlinear Systems

Probability-Directed Problem Optimization Technique for Solving Systems of Linear and Non-Linear Equations

On a Bi-Parametric Family of Fourth Order Composite Newton–Jarratt Methods for Nonlinear Systems

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