Multiparameter extrapolation and deflation methods for solving 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.

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The Convergence Ball and Error Analysis of the Relaxed Secant Method

Advances in Mathematical Physics ◽

10.1155/2017/6976205 ◽

2017 ◽

Vol 2017 ◽

pp. 1-7

Author(s):

Rongfei Lin ◽

Qingbiao Wu ◽

Minhong Chen ◽

Lu Liu

Keyword(s):

Nonlinear Equation ◽

Error Analysis ◽

Error Estimate ◽

Secant Method ◽

Speed Of Convergence ◽

Convergence Ball ◽

Numerical Examples ◽

Lipschitz Continuous ◽

First Order ◽

Nonlinear Equation Systems

A relaxed secant method is proposed. Radius estimate of the convergence ball of the relaxed secant method is attained for the nonlinear equation systems with Lipschitz continuous divided differences of first order. The error estimate is also established with matched convergence order. From the radius and error estimate, the relation between the radius and the speed of convergence is discussed with parameter. At last, some numerical examples are given.

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Solve nonlinear equation systems using midpoint Newton method

10.1063/5.0017834 ◽

2020 ◽

Author(s):

Fitri Nur Hidayah ◽

Yundari ◽

Mariatul Kiftiah

Keyword(s):

Nonlinear Equation ◽

Newton Method ◽

Nonlinear Equation Systems

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Modified Flower Pollination Algorithm for Global Optimization

Mathematics ◽

10.3390/math9141661 ◽

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.

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