Optimal Eighth-Order Solver for Nonlinear Equations with Applications in Chemical Engineering

Solaiman, Obadah Said; Hashim, Ishak

doi:10.32604/iasc.2021.015285

Cited by 17 publications

(12 citation statements)

References 25 publications

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“…In the same manner, we can find the next approximation 2 by finding the -intercept of the tangent line to the graph of ( 1 , ( 1 )) and so on, as illustrated in Figure 1 [57]. Newton's method has been widely studied and modified to increase the accuracy of the approximate solutions of the nonlinear problems and increase the order of convergence that reflects consecutively on the speed of obtaining the desired solution [40]- [42].…”

Section: A Newton's Methodsmentioning

confidence: 99%

“…The most widespread iterative method is the one proposed by Isaak Newton, known as Newton's method or Newton-Raphson method. Newton's method has been widely studied and modified to increase both the accuracy of the approximate solutions and the convergence rate to the desired solution [40]- [42] Most of the proposed optimization algorithms in the literature deal with the systems of nonlinear problems using a single optimization algorithm, or by using a hybridized algorithm of two different algorithms. In the proposed work, we the systems of nonlinear problems using a combination of the well-known HHO optimizer and Newton's iterative technique to benefit from the fast convergence of Newton's method towards the desired solution.…”

Section: Introductionmentioning

confidence: 99%

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A Hybrid Approach for Solving Systems of Nonlinear Equations Using Harris Hawks Optimization and Newton’s Method

et al. 2021

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Systems of nonlinear equations are known as the basis for many models of engineering and data science, and their accurate solutions are very critical in achieving progress in these fields. However, solving a system with multiple nonlinear equations, usually, is not an easy task. Consequently, finding a robust and accurate solution can be a very challenging problem in complex systems. In this work, a novel hybrid method namely Newton-Harris hawks optimization (NHHO) for solving systems of nonlinear equations is proposed. The proposed NHHO combines Newton's method, with a second-order convergence where the correct digits roughly double in every step, and the Harris hawks optimization (HHO) to enhance the search mechanism, avoid local optima, improve convergence speed, and find more accurate solutions. We tested a group of six well-known benchmark systems of nonlinear equations to evaluate the efficiency of NHHO. Further, comparisons between NHHO and other optimization algorithms, including the original HHO algorithm, Particle Swarm Optimization (PSO), Ant Lion Optimizer (ALO), Butterfly Optimization Algorithm (BOA), and Equilibrium Optimization (EO) were performed. The norm of the equation system was calculated as a fitness function to measure the optimization algorithms' performance. A solution with less fitness value is considered a better solution. Furthermore, the experimental results confirmed the superiority of NHHO over the other optimization algorithms, in the comparisons, in different aspects, including best solution, average fitness value, and convergence speed. Accordingly, the proposed NHHO is powerful and more effective in all benchmark problems in solving systems of nonlinear equations compared to the other optimization algorithms. Finally, NHHO overcomes the limitations of Newton's method, including selecting the initial point and divergence problems.INDEX TERMS Hybridization; Harris hawks optimization; Newton's method; optimization; systems of nonlinear equations.

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Section: A Newton's Methodsmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

A Hybrid Approach for Solving Systems of Nonlinear Equations Using Harris Hawks Optimization and Newton’s Method

et al. 2021

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“…The basins of attraction [19][20][21][22] is a graphical representation of how root-finding algorithms respond to different initial estimate points. It is more than a graphical illustration of how a root-finding approach works; it also enables the comparsion of qualitative issues.…”

Section: Dynamical Planesmentioning

confidence: 99%

Efficient Numerical Scheme for Solving Large System of Nonlinear Equations

Shams¹,

Kausar²,

Ahmed³

et al. 2023

Computers, Materials &Amp; Continua

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A fifth-order family of an iterative method for solving systems of nonlinear equations and highly nonlinear boundary value problems has been developed in this paper. Convergence analysis demonstrates that the local order of convergence of the numerical method is five. The computer algebra system CAS-Maple, Mathematica, or MATLAB was the primary tool for dealing with difficult problems since it allows for the handling and manipulation of complex mathematical equations and other mathematical objects.Several numerical examples are provided to demonstrate the properties of the proposed rapidly convergent algorithms. A dynamic evaluation of the presented methods is also presented utilizing basins of attraction to analyze their convergence behavior. Aside from visualizing iterative processes, this methodology provides useful information on iterations, such as the number of diverging-converging points and the average number of iterations as a function of initial points. Solving numerous highly nonlinear boundary value problems and large nonlinear systems of equations of higher dimensions demonstrate the performance, efficiency, precision, and applicability of a newly presented technique.

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“…Recently, Vahidi et al [9] and Sharma et al [10] proposed new effective solvers for solution of nonlinear systems. Massive amount of existing literature [11][12][13][14][15][16][17][18] is dedicated to improve the convergence order of the classical deterministic recursive or iterative methods. The most recent trend for solving system of nonlinear equations is to transform it into an equivalent global optimization problem of the following form.…”

Section: Introductionmentioning

confidence: 99%

A Multi-Simplex Imperialist Competitive Paradigm for Solving Nonlinear Physical Systems

Ali¹,

Iqbal²,

Alkhalaf³

2022

Intelligent Automation &Amp; Soft Computing

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This paper proposes a novel gradient free multi-simplex topology fabric aided imperialist competitive algorithm (ICA) for solving nonlinear systems of algebraic equations by transforming them to equivalent global optimization problems. The high dependence of traditional gradient based solvers of such systems on initial guesses and the Jacobeans resulting in false convergence is the main motivation behind the present work. The present work provides a mechanism for enhancing exploitation powers of imperialist search phase of the algorithm and hence improves the convergence speed. The variants emerging from the proposed approach are applied to diverse nonlinear systems arising in different scientific areas and the results so obtained are analyzed in details. Based on the analysis of empirical results on complex benchmark models, it is observed that the proposed enhancement of ICA has not only boosted the problem solving power of the imperialist system when applied multidimensional nonlinear physical systems but also emerged as an efficient gradient/initial guess free alternative solver with high accuracy for nonlinear systems.

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Optimal Eighth-Order Solver for Nonlinear Equations with Applications in Chemical Engineering

Cited by 17 publications

References 25 publications

A Hybrid Approach for Solving Systems of Nonlinear Equations Using Harris Hawks Optimization and Newton’s Method

A Hybrid Approach for Solving Systems of Nonlinear Equations Using Harris Hawks Optimization and Newton’s Method

Efficient Numerical Scheme for Solving Large System of Nonlinear Equations

A Multi-Simplex Imperialist Competitive Paradigm for Solving Nonlinear Physical Systems

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