A new fourth order Newton-type method for solution of system of nonlinear equations

Compactly supported orthogonal wavelet filters are extensively applied to the analysis and description of abrupt signals in fields such as multimedia. Based on the application of an elementary method for compactly supported orthogonal wavelet filters and the construction of a system of nonlinear equations for filter coefficients, we design compactly supported orthogonal wavelet filters, in which both the scaling and wavelet functions have many vanishing moments, by approximately solving the system of nonlinear equations. However, when solving such a system about filter coefficients of compactly supported wavelets, the most widely used method, the Newton Iteration method, cannot converge to the solution if the selected initial value is not near the exact solution. For such, we propose optimization algorithms for the Gauss-Newton type method that expand the selection range of initial values. The proposed method is optimal and promising when compared to other works, by analyzing the experimental results obtained in terms of accuracy, iteration times, solution speed, and complexity.

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On a Three-Step Efficient Fourth-Order Method for Computing the Numerical Solution of System of Nonlinear Equations and Its Applications

Proceedings of the National Academy of Sciences India Section A Physical Sciences ◽

10.1007/s40010-019-00617-4 ◽

2019 ◽

Vol 90 (4) ◽

pp. 709-716

Author(s):

Anuradha Singh

Keyword(s):

Numerical Solution ◽

Nonlinear Equations ◽

Fourth Order ◽

System Of Nonlinear Equations ◽

Order Method ◽

Fourth Order Method

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A New Jarratt-Type Fourth-Order Method for Solving System of Nonlinear Equations and Applications

Journal of Applied Mathematics ◽

10.1155/2015/805278 ◽

2015 ◽

Vol 2015 ◽

pp. 1-14 ◽

Cited By ~ 5

Author(s):

Moin-ud-Din Junjua ◽

Saima Akram ◽

Nusrat Yasmin ◽

Fiza Zafar

Keyword(s):

Nonlinear Equations ◽

Fourth Order ◽

Basins Of Attraction ◽

Systems Of Nonlinear Equations ◽

System Of Nonlinear Equations ◽

Numerical Tests ◽

New Methods ◽

New Family ◽

Engineering Problems ◽

Fourth Order Method

Solving systems of nonlinear equations plays a major role in engineering problems. We present a new family of optimal fourth-order Jarratt-type methods for solving nonlinear equations and extend these methods to solve system of nonlinear equations. Convergence analysis is given for both cases to show that the order of the new methods is four. Cost of computations, numerical tests, and basins of attraction are presented which illustrate the new methods as better alternates to previous methods. We also give an application of the proposed methods to well-known Burger's equation.

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An Iterative Solver in the Presence and Absence of Multiplicity for Nonlinear Equations

The Scientific World JOURNAL ◽

10.1155/2013/837243 ◽

2013 ◽

Vol 2013 ◽

pp. 1-9 ◽

Cited By ~ 3

Author(s):

Fazlollah Soleymani ◽

Stanford Shateyi ◽

Gülcan Özkum

Keyword(s):

Fixed Point ◽

Nonlinear Equations ◽

Fourth Order ◽

Chaotic Behavior ◽

Multiple Root ◽

Basins Of Attraction ◽

Iterative Solver ◽

Type Method ◽

New Class ◽

Point Type

We develop a high-order fixed point type method to approximate a multiple root. By using three functional evaluations per full cycle, a new class of fourth-order methods for this purpose is suggested and established. The methods from the class require the knowledge of the multiplicity. We also present a method in the absence of multiplicity for nonlinear equations. In order to attest the efficiency of the obtained methods, we employ numerical comparisons alongside obtaining basins of attraction to compare them in the complex plane according to their convergence speed and chaotic behavior.

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Solving System of Nonlinear Equations using Genetic Algorithm

Journal of Computer and Mathematical Sciences ◽

10.29055/jcms/1072 ◽

2019 ◽

Vol 10 (4) ◽

pp. 877-886 ◽

Cited By ~ 1

Author(s):

Chhavi Mangla ◽

Musheer Ahmad ◽

Moin Uddin

Keyword(s):

Genetic Algorithm ◽

Nonlinear Equations ◽

System Of Nonlinear Equations

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An Efficient Sixth-Order Convergent Newton-Type Iterative Method for Nonlinear Equations

Applied Mechanics and Materials ◽

10.4028/www.scientific.net/amm.220-223.2585 ◽

2012 ◽

Vol 220-223 ◽

pp. 2585-2588

Author(s):

Zhong Yong Hu ◽

Fang Liang ◽

Lian Zhong Li ◽

Rui Chen

Keyword(s):

Iterative Method ◽

Nonlinear Equations ◽

Newton’S Method ◽

Newton's Method ◽

Efficiency Index ◽

Sixth Order ◽

Type Method ◽

Second Derivatives ◽

Better Than

In this paper, we present a modified sixth order convergent Newton-type method for solving nonlinear equations. It is free from second derivatives, and requires three evaluations of the functions and two evaluations of derivatives per iteration. Hence the efficiency index of the presented method is 1.43097 which is better than that of classical Newton’s method 1.41421. Several results are given to illustrate the advantage and efficiency the algorithm.

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New fourth-order convergent algorithm for analysis of trusses with material and geometric nonlinearities

The Journal of Strain Analysis for Engineering Design ◽

10.1177/03093247211000528 ◽

2021 ◽

pp. 030932472110005

Author(s):

Luiz Antonio Farani de Souza ◽

Douglas Fernandes dos Santos ◽

Rodrigo Yukio Mizote Kawamoto ◽

Leandro Vanalli

Keyword(s):

Fourth Order ◽

Nonlinear Behavior ◽

Path Following ◽

System Of Nonlinear Equations ◽

The Finite Element Method ◽

Step Method ◽

Geometric Nonlinearities ◽

Standard Procedures ◽

Elastoplastic Constitutive Model ◽

Convergent Algorithm

This paper presents a new algorithm to solve the system of nonlinear equations that describes the static equilibrium of trusses with material and geometric nonlinearities, adapting a three-step method with fourth-order convergence found in the literature. The co-rotational formulation of the Finite Element Method is used in the discretization of structures. The nonlinear behavior of the material is characterized by an elastoplastic constitutive model. The equilibrium paths with limit points of load and displacement are obtained using the linearized Arc-Length path-following technique. The numerical results obtained with the free program Scilab show that the new algorithm converges faster than standard procedures and modified Newton-Raphson, since the approximate solution of the problem is obtained with a smaller number of accumulated iterations and less CPU time. The equilibrium paths show that the structures exhibit a completely different behavior when the material nonlinearity is considered in the analysis with large displacements.

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