Two improved classes of Broyden's methods for solving nonlinear systems of equations

Al-Towaiq, Mohammad H.; Hour, Yousef S. Abu

doi:10.22436/jmcs.017.01.02

Cited by 8 publications

(6 citation statements)

References 11 publications

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“…Solving systems of nonlinear equations is one of the most important problems in numerical computations, especially for a diverse range of engineering applications, including applications in many scientific fields [13]. Several real -life problems can be reduced to solving systems of nonlinear equations, which is one of the most basic problems in mathematics [1]. Great efforts have been made by a lot of people and many constructive theories and algorithms are proposed to solve systems of nonlinear equations [12].…”

Section: Introductionmentioning

confidence: 99%

“…The Broyden method, which is a quasi-Newton method, has seen significant modifications and improvements and these have motivated other researchers to develop new methods capable of solving efficiently nonlinear systems of equations [1]. Many authors continue to present different techniques which are Newton-like schemes [23,22,8,6,7], Mixed Free Secant methods, or Quadrature formulas.…”

Section: Introductionmentioning

confidence: 99%

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A New Trapezoidal-Simpson 3/8 Method for Solving Systems of Nonlinear Equations

Azure¹,

Stephen²,

Seidu³

2021

AJMCM

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Since its introduction, the Broyden method has been used as the foundation to develop several other Broyden-like methods (or hybrid Broyden methods) which in many cases have turned out to be improved forms of the original method. The modified classical Broyden methods developed by many authors to solve system of nonlinear equations have been effective in overcoming the deficiency of the classical Newton Raphson method, however there are new trends of methods proposed by authors, which have proven to be more efficient than some already existing ones. This work introduces two Broyden-like method developed from a weighted combination of quadrature rules, namely the Trapizoidal, Simpson 3/8 and Simpson 1/3 quadrature rules. Hence the new Broyden-like methods named by the authors as TS-3/8 and TS -1/3 methods have been developed from these rules. After subjecting the proposed methods together with some other existing Broyden-like methods to solve four bench-mark problems, the results of numerical test confirm that the TS-3/8 method is promising (in terms of speed and in most cases accuracy) when compared with other proposed Broyden-like methods. Results gathered after the comparison of TS -3/8 with the other methods revealed that TS -3/8 method performed better than all the methods in terms of speed and the number of iterations needed to reach a solution. On the other hand, TS -1/3 method yielded results for all the benchmark problems but with a relatively higher number of iterations compared with the other methods selected for comparison.

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Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

A New Trapezoidal-Simpson 3/8 Method for Solving Systems of Nonlinear Equations

Azure¹,

Stephen²,

Seidu³

2021

AJMCM

View full text Add to dashboard Cite

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“…There are various methods that can be used in solving a nonlinear equations system such as Newton method, Secant method and Bisection method. In this study, Newton method is used because Newton method is fast in solving the system [6], simple to used [7,8] and very widely used in solving nonlinear equations system [9,10,11].…”

Section: Introductionmentioning

confidence: 99%

Optimisation of Biochemical Systems Production using Hybrid of Newton Method, Differential Evolution Algorithm and Cooperative Coevolution Algorithm

Ismail

Mezhuyev

Moorthy

et al. 2017

IJEECS

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<p>This paper present a hybrid method of Newton method, Differential Evolution Algorithm (DE) and Cooperative Coevolution Algorithm (CCA). The proposed method is used to solve the optimisation problem in optimise the production of biochemical systems. The problems are maximising the biochemical systems production and simultaneously minimising the total amount of chemical reaction concentration involves. Besides that, the size of biochemical systems also contributed to the problem in optimising the biochemical systems production. In the proposed method, the Newton method is used in dealing biochemical system, DE for optimisation process while CCA is used to increase the performance of DE. In order to evaluate the performance of the proposed method, the proposed method is tested on two benchmark biochemical systems. Then, the result that obtained by the proposed method is compare with other works and the finding shows that the proposed method performs well compare to the other works.</p>

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“…It is a hard task because the equations are usually nondeterministic polynomial equations. In addition, nonlinear equations system has a complex structure due to the number equations and variables, which makes it difficult to solve [1], [2]. Nowadays, many applications use nonlinear equations system such as in the chemistry domain [3], [4].…”

Section: Introductionmentioning

confidence: 99%

“…Among these methods, it was found that Newton method is the suitable method to be used in solving a nonlinear equation system. This is due to advantages offered by Newton method which is the convergence speed of Newton method [11]- [13] and the simplicity in using Newton method [2], [11]. Due to that, this study use Newton method in solving nonlinear equations system.…”

Section: Introductionmentioning

confidence: 99%

Multi-objective Optimization of Biochemical System Production Using an Improve Newton Competitive Differential Evolution Method

Ismail¹,

Mezhuyev²,

Deris³

et al. 2017

International Journal on Advanced Science, Engineering and Information Technology

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In this paper, an improved method of multi-objective optimization for biochemical system production is presented and discussed in detail. The optimization process of biochemical system production become hard and difficult when involved a large biochemical system that contains many components. In addition, the multi-objective problem also needs to be considered. Due to that, this study proposed and improved a method that comprises with Newton method, differential evolution algorithm (DE) and competitive co-evolutionary algorithm(ComCA). The aim of the proposed method is to maximize the production and simultaneously minimize the total amount of chemical concentrations involves. The operation of the proposed method starts with Newton method by dealing with biochemical system production as a nonlinear equations system. Then DE and ComCA are used to represent the variables in nonlinear equation system and tune the variables in order to find the best solution. The used of DE is to maximize the production while ComCA is to minimize the total amount of chemical concentrations involves. The effectiveness of the proposed method is evaluated using two benchmark biochemical systems, and the experimental results show that the proposed method performs well compared to other works.

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Two improved classes of Broyden's methods for solving nonlinear systems of equations

Cited by 8 publications

References 11 publications

A New Trapezoidal-Simpson 3/8 Method for Solving Systems of Nonlinear Equations

A New Trapezoidal-Simpson 3/8 Method for Solving Systems of Nonlinear Equations

Optimisation of Biochemical Systems Production using Hybrid of Newton Method, Differential Evolution Algorithm and Cooperative Coevolution Algorithm

Multi-objective Optimization of Biochemical System Production Using an Improve Newton Competitive Differential Evolution Method

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