More Efficient Iterative Methods than Newton's Method for Solving Nonlinear Systems

2010 ◽  
Author(s):  
J.A. Ezquerro ◽  
M.A. Hernández
Keyword(s):  
Nonlinear Systems ◽  
Iterative Methods ◽  
Newton’S Method ◽  
Newton's Method

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.

CONSTRUCTION OF DERIVATIVE-FREE ITERATIVE METHODS FROM CHEBYSHEV'S METHOD

2013 ◽  
Vol 11 (03) ◽  
pp. 1350009 ◽  
Author(s):  
J. A. EZQUERRO ◽  
A. GRAU ◽  
M. GRAU-SÁNCHEZ ◽  
M. A. HERNÁNDEZ
Keyword(s):  
Iterative Methods ◽  
Newton’S Method ◽  
Newton's Method ◽  
Classical Method ◽  
Secant Method ◽  
Chebyshev’S Method ◽  
Derivative Free

From some modifications of Chebyshev's method, we consider a uniparametric family of iterative methods that are more efficient than Newton's method, and we then construct two iterative methods in a similar way to the Secant method from Newton's method. These iterative methods do not use derivatives in their algorithms and one of them is more efficient than the Secant method, which is the classical method with this feature.

Discrete Newton's method with local variations for solving large-scale nonlinear systems

Optimization ◽  
2003 ◽  
Vol 52 (4-5) ◽  
pp. 417-440 ◽  
Author(s):  
Maria A. Diniz-Ehrhardt ◽  
Márcia A. Gomes-Ruggiero† ◽  
Vera L. Rocha Lopes‡ ◽  
José Mario Martínez**
Keyword(s):  
Nonlinear Systems ◽  
Newton’S Method ◽  
Newton's Method ◽  
Large Scale

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.

Contributions to Synthesis of Plane Linkages by the Use of Zero Order Transmission Functions

1972 ◽  
Vol 94 (3) ◽  
pp. 849-852
Author(s):  
V. Handra-Luca
Keyword(s):  
Nonlinear Systems ◽  
Newton’S Method ◽  
Newton's Method ◽  
Zero Order ◽  
Complex Numbers ◽  
Systems Of Equations ◽  
Dimensional Synthesis ◽  
Nonlinear Systems Of Equations ◽  
Single Mechanism

A method for the dimensional synthesis of a plane mechanism with more than one degree of mobility is developed when the positions of the driven link are given in correlation with those of the driving links. The synthesis is developed both for associated relative positions and, also, for associated absolute positions. It is assumed that the number of positions given is the maximum for which, through synthesis, a single mechanism is obtained. For the solution of the problem complex numbers are used and Newton’s method is recommended for solving the nonlinear systems of equations.

scholarly journals A new family of fourth-order methods for multiple roots of nonlinear equations

2013 ◽  
Vol 18 (2) ◽  
pp. 143-152 ◽  
Author(s):  
Baoqing Liu ◽  
Xiaojian Zhou
Keyword(s):  
Iterative Methods ◽  
Nonlinear Equations ◽  
Fourth Order ◽  
Newton’S Method ◽  
Newton's Method ◽  
Optimal Order ◽  
Multiple Roots ◽  
First Derivative ◽  
New Family ◽  
Modified Newton’S Method

Recently, some optimal fourth-order iterative methods for multiple roots of nonlinear equations are presented when the multiplicity m of the root is known. Different from these optimal iterative methods known already, this paper presents a new family of iterative methods using the modified Newton’s method as its first step. The new family, requiring one evaluation of the function and two evaluations of its first derivative, is of optimal order. Numerical examples are given to suggest that the new family can be competitive with other fourth-order methods and the modified Newton’s method for multiple roots.

scholarly journals Elegant Iterative Methods for Solving a Nonlinear Matrix Equation X-A* eX A=I

2021 ◽  
Vol 47 (3) ◽  
pp. 1033-1040
Author(s):  
Chacha S Chacha
Keyword(s):  
Fixed Point ◽  
Iterative Methods ◽  
Newton’S Method ◽  
Matrix Equation ◽  
Newton's Method ◽  
Fixed Point Method ◽  
Point Method ◽  
Computational Time ◽  
Nonlinear Matrix Equation ◽  
Nonlinear Matrix

The nonlinear matrix equation   was solved by Gao (2016) via standard fixed point method. In this paper, three more elegant iterative methods are proposed to find the approximate solution of the nonlinear matrix equation  namely: Newton’s method; modified fixed point method and a combination of Newton’s method and fixed point method. The convergence of Newton’s method and modified fixed point method are derived. Comparative numerical experimental results indicate that the new developed algorithms have both less computational time and good convergence properties when compared to their respective standard algorithms. Keywords: Hermitian positive definite solution; nonlinear matrix equation; modified fixed point method; iterative method

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