On a newton-type method under weak conditions with dynamics

2020 ◽  
pp. 2150145
Author(s):  
Manoj Kumar Singh ◽  
Arvind K. Singh
Keyword(s):  
Iterative Methods ◽  
Newton’S Method ◽  
Newton's Method ◽  
Second Order ◽  
Numerical Results ◽  
Order Derivative ◽  
Algebraic Equations ◽  
Type Method ◽  
Nonlinear Algebraic Equations ◽  
Fractal Patterns

In this paper, we present new cubically convergent Newton-type iterative methods with dynamics for solving nonlinear algebraic equations under weak conditions. The proposed methods are free from second-order derivative and work well when [Formula: see text]. Numerical results show that the proposed method performs better when Newton’s method fails or diverges and competes well with same order existing method. Fractal patterns of different methods also support the numerical results and explain the compactness regarding the convergence, divergence, and stability of the methods to different roots.

Application of the Monomial Method to Shape Optimization

1994 ◽  
Vol 116 (4) ◽  
pp. 1013-1018 ◽  
Author(s):  
S. A. Burns
Keyword(s):  
Shape Optimization ◽  
Newton’S Method ◽  
Structural Design ◽  
Newton's Method ◽  
Computational Effort ◽  
Algebraic Equations ◽  
Structural Shape ◽  
Continuum Structures ◽  
Structural Shape Optimization ◽  
Nonlinear Algebraic Equations

The monomial method is an alternative to Newton’s method for solving systems of nonlinear algebraic equations. It possesses several properties not shared by Newton’s method that enhance performance, yet does not require substantial computational effort beyond that required for Newton’s method. Previous work has demonstrated that the monomial method treats problems in structural design very effectively. This paper combines the monomial method with the method of generalized geometric programming to treat the problem of structural shape optimization of continuum structures modeled by finite elements.

Application of the Monomial Method to Shape Optimization

1993 ◽  
Author(s):  
Scott A. Burns
Keyword(s):  
Shape Optimization ◽  
Newton’S Method ◽  
Structural Design ◽  
Newton's Method ◽  
Computational Effort ◽  
Algebraic Equations ◽  
Structural Shape ◽  
Continuum Structures ◽  
Structural Shape Optimization ◽  
Nonlinear Algebraic Equations

Abstract The monomial method is an alternative to Newton’s method for solving systems of nonlinear algebraic equations. It possesses several properties not shared by Newton’s method that enhance performance, yet does not require substantial computational effort beyond that required for Newton’s method. Previous work has demonstrated that the monomial method treats problems in structural design very effectively. This paper combines the monomial method with the method of generalized geometric programming to treat the problem of structural shape optimization of continuum structures modeled by finite elements.

A Modification of Newton-Type Method with Sixth-Order Convergence for Solving Nonlinear Equations

2012 ◽  
Vol 490-495 ◽  
pp. 1839-1843
Author(s):  
Rui Chen ◽  
Liang Fang
Keyword(s):  
Nonlinear Equations ◽  
Newton’S Method ◽  
Newton's Method ◽  
Efficiency Index ◽  
Numerical Results ◽  
Sixth Order ◽  
Order Convergence ◽  
Type Method ◽  
Second Derivatives ◽  
Better Than

In this paper, we present and analyze a modified Newton-type method with oder of convergence six for solving nonlinear equations. The method is free from second derivatives. It requires three evaluations of the functions and two evaluations of derivatives in each step. Therefore the efficiency index of the presented method is 1.431 which is better than that of classical Newton’s method 1.414. Some numerical results illustrate that the proposed method is more efficient and performs better than classical Newton's method

scholarly journals Reducing Chaos and Bifurcations in Newton-Type Methods

2013 ◽  
Vol 2013 ◽  
pp. 1-10 ◽  
Author(s):  
S. Amat ◽  
S. Busquier ◽  
Á. A. Magreñán
Keyword(s):  
Iterative Methods ◽  
Newton’S Method ◽  
Newton's Method ◽  
High Order ◽  
Damping Factor ◽  
Type Method ◽  
Iterative Schemes

We study the dynamics of some Newton-type iterative methods when they are applied of polynomials degrees two and three. The methods are free of high-order derivatives which are the main limitation of the classical high-order iterative schemes. The iterative schemes consist of several steps of damped Newton's method with the same derivative. We introduce a damping factor in order to reduce thebadzones of convergence. The conclusion is that the damped schemes become real alternative to the classical Newton-type method since both chaos and bifurcations of the original schemes are reduced. Therefore, the new schemes can be utilized to obtain good starting points for the original schemes.

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

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.

Legendre Collocation Method to Solve the Riccati Equations with Functional Arguments

2020 ◽  
Vol 17 (10) ◽  
pp. 2050011
Author(s):  
Şuayip Yüzbaşı ◽  
Gamze Yıldırım
Keyword(s):  
Approximate Solutions ◽  
Coefficient Matrix ◽  
Numerical Results ◽  
Matrix Form ◽  
Algebraic Equations ◽  
Mixed Condition ◽  
Collocation Points ◽  
Nonlinear Algebraic Equations ◽  
The Matrix ◽  
Legendre Collocation Method

In this study, a method for numerically solving Riccatti type differential equations with functional arguments under the mixed condition is presented. For the method, Legendre polynomials, the solution forms and the required expressions are written in the matrix form and the collocation points are defined. Then, by using the obtained matrix relations and the collocation points, the Riccati problem is reduced to a system of nonlinear algebraic equations. The condition in the problem is written in the matrix form and a new system of the nonlinear algebraic equations is found with the aid of the obtained matrix relation. This system is solved and thus the coefficient matrix is detected. This coefficient matrix is written in the solution form and hence approximate solution is obtained. In addition, by defining the residual function, an error problem is established and approximate solutions which give better numerical results are obtained. To demonstrate that the method is trustworthy and convenient, the presented method and error estimation technique are explicated by numerical examples. Consequently, the numerical results are shown more clearly with the aid of the tables and graphs and also the results are compared with the results of other methods.

scholarly journals Three New Optimal Fourth-Order Iterative Methods to Solve Nonlinear Equations

2013 ◽  
Vol 2013 ◽  
pp. 1-8 ◽  
Author(s):  
Gustavo Fernández-Torres ◽  
Juan Vásquez-Aquino
Keyword(s):  
Iterative Methods ◽  
Nonlinear Equations ◽  
Fourth Order ◽  
Newton’S Method ◽  
Newton's Method ◽  
Classical Method ◽  
Multiple Roots ◽  
Order Convergence ◽  
Numerical Examples ◽  
Optimal Methods

We present new modifications to Newton's method for solving nonlinear equations. The analysis of convergence shows that these methods have fourth-order convergence. Each of the three methods uses three functional evaluations. Thus, according to Kung-Traub's conjecture, these are optimal methods. With the previous ideas, we extend the analysis to functions with multiple roots. Several numerical examples are given to illustrate that the presented methods have better performance compared with Newton's classical method and other methods of fourth-order convergence recently published.

Newton’s Method Applied to Problems in Nonlinear Mechanics

1965 ◽  
Vol 32 (2) ◽  
pp. 383-388 ◽  
Author(s):  
G. A. Thurston
Keyword(s):  
Boundary Value Problems ◽  
Newton’S Method ◽  
Newton's Method ◽  
Nonlinear Boundary ◽  
Boundary Value ◽  
Practical Method ◽  
Algebraic Equations ◽  
Nonlinear Mechanics ◽  
Nonlinear Boundary Value Problems ◽  
Nonlinear Boundary Value

Many problems in mechanics are formulated as nonlinear boundary-value problems. A practical method of solving such problems is to extend Newton’s method for calculating roots of algebraic equations. Three problems are treated in this paper to illustrate the use of this method and compare it with other methods.

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