From Davidenko Method to Zhang Dynamics for Nonlinear Equation Systems Solving

2017 ◽  
Vol 47 (11) ◽  
pp. 2817-2830 ◽  
Author(s):  
Yunong Zhang ◽  
Yinyan Zhang ◽  
Dechao Chen ◽  
Zhengli Xiao ◽  
Xiaogang Yan
Keyword(s):  
Nonlinear Equation ◽  
Nonlinear Equation Systems

scholarly journals Modified Flower Pollination Algorithm for Global Optimization

Mathematics ◽  
2021 ◽  
Vol 9 (14) ◽  
pp. 1661
Author(s):  
Mohamed Abdel-Basset ◽  
Reda Mohamed ◽  
Safaa Saber ◽  
S. S. Askar ◽  
Mohamed Abouhawwash
Keyword(s):  
Global Optimization ◽  
Nonlinear Equation ◽  
Differential Evolution ◽  
Flower Pollination Algorithm ◽  
Test Cases ◽  
Test Functions ◽  
Flower Pollination ◽  
New Variant ◽  
Experimental Findings ◽  
Nonlinear Equation Systems

In this paper, a modified flower pollination algorithm (MFPA) is proposed to improve the performance of the classical algorithm and to tackle the nonlinear equation systems widely used in engineering and science fields. In addition, the differential evolution (DE) is integrated with MFPA to strengthen its exploration operator in a new variant called HFPA. Those two algorithms were assessed using 23 well-known mathematical unimodal and multimodal test functions and 27 well-known nonlinear equation systems, and the obtained outcomes were extensively compared with those of eight well-known metaheuristic algorithms under various statistical analyses and the convergence curve. The experimental findings show that both MFPA and HFPA are competitive together and, compared to the others, they could be superior and competitive for most test cases.

scholarly journals Multiparameter extrapolation and deflation methods for solving equation systems

1984 ◽  
Vol 7 (4) ◽  
pp. 793-802 ◽  
Author(s):  
A. J. Hughes Hallett
Keyword(s):  
Applied Sciences ◽  
Nonlinear Equation ◽  
Newton Method ◽  
Single Parameter ◽  
Iterative Techniques ◽  
First Order ◽  
Convergence Results ◽  
Nonlinear Equation Systems

Most models in economics and the applied sciences are solved by first order iterative techniques, usually those based on the Gauss-Seidel algorithm. This paper examines the convergence of multiparameter extrapolations (accelerations) of first order iterations, as an improved approximation to the Newton method for solving arbitrary nonlinear equation systems. It generalises my earlier results on single parameter extrapolations. Richardson's generalised method and the deflation method for detecting successive solutions in nonlinear equation systems are also presented as multiparameter extrapolations of first order iterations. New convergence results are obtained for those methods.

scholarly journals An Integrated Genetic Algorithm and Homotopy Analysis Method to Solve Nonlinear Equation Systems

2021 ◽  
Vol 2021 ◽  
pp. 1-14
Author(s):  
Hala A. Omar
Keyword(s):  
Genetic Algorithm ◽  
Nonlinear Equation ◽  
Homotopy Analysis Method ◽  
Optimal Solution ◽  
Initial Guess ◽  
Analysis Method ◽  
Optimization Task ◽  
Homotopy Analysis ◽  
Newton Homotopy ◽  
Nonlinear Equation Systems

Solving nonlinear equation systems for engineering applications is one of the broadest and most essential numerical studies. Several methods and combinations were developed to solve such problems by either finding their roots mathematically or formalizing such problems as an optimization task to obtain the optimal solution using a predetermined objective function. This paper proposes a new algorithm for solving square and nonsquare nonlinear systems combining the genetic algorithm (GA) and the homotopy analysis method (HAM). First, the GA is applied to find out the solution. If it is realized, the algorithm is terminated at this stage as the target solution is determined. Otherwise, the HAM is initiated based on the GA stage’s computed initial guess and linear operator. Moreover, the GA is utilized to calculate the optimum value of the convergence control parameter (h) algebraically without plotting the h-curves or identifying the valid region. Four test functions are examined in this paper to verify the proposed algorithm’s accuracy and efficiency. The results are compared to the Newton HAM (NHAM) and Newton homotopy differential equation (NHDE). The results corroborated the superiority of the proposed algorithm in solving nonlinear equation systems efficiently.

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