scholarly journals Convergence and Dynamics of a Higher-Order Method

Symmetry ◽  
2020 ◽  
Vol 12 (3) ◽  
pp. 420 ◽  
Author(s):  
Alejandro Moysi ◽  
Ioannis K. Argyros ◽  
Samundra Regmi ◽  
Daniel González ◽  
Á. Alberto Magreñán ◽  
...  
Keyword(s):  
Nonlinear Equation ◽  
Iterative Method ◽  
Higher Order ◽  
High Order ◽  
Local Form ◽  
Order Method ◽  
First Derivative ◽  
The Family ◽  
Higher Order Method

Solving problems in various disciplines such as biology, chemistry, economics, medicine, physics, and engineering, to mention a few, reduces to solving an equation. Its solution is one of the greatest challenges. It involves some iterative method generating a sequence approximating the solution. That is why, in this work, we analyze the convergence in a local form for an iterative method with a high order to find the solution of a nonlinear equation. We extend the applicability of previous results using only the first derivative that actually appears in the method. This is in contrast to either works using a derivative higher than one, or ones not in this method. Moreover, we consider the dynamics of some members of the family in order to see the existing differences between them.

scholarly journals Ball Convergence of an Efficient Eighth Order Iterative Method Under Weak Conditions

Mathematics ◽  
2018 ◽  
Vol 6 (11) ◽  
pp. 260 ◽  
Author(s):  
Janak Sharma ◽  
Ioannis Argyros ◽  
Sunil Kumar
Keyword(s):  
Iterative Method ◽  
Iterative Methods ◽  
Higher Order ◽  
High Order ◽  
Convergence Order ◽  
Eighth Order ◽  
First Derivative ◽  
Ball Convergence ◽  
Derivatives Of

The convergence order of numerous iterative methods is obtained using derivatives of a higher order, although these derivatives are not involved in the methods. Therefore, these methods cannot be used to solve equations with functions that do not have such high-order derivatives, since their convergence is not guaranteed. The convergence in this paper is shown, relying only on the first derivative. That is how we expand the applicability of some popular methods.

scholarly journals After notes on Chebyshev’s iterative method

2017 ◽  
Vol 2 (1) ◽  
pp. 1-12 ◽  
Author(s):  
S. Amat ◽  
S. Busquier
Keyword(s):  
Global Convergence ◽  
Iterative Method ◽  
Newton’S Method ◽  
Newton's Method ◽  
Geometric Interpretation ◽  
Higher Order ◽  
Order Method ◽  
Chebyshev’S Method ◽  
Higher Order Method ◽  
Dynamical Properties

AbstractThis paper is a small review of Chebyshev’s method. The geometric interpretation as a generalization of Newton’s method is derived. Using this interpretation its global convergence is proved. Some dynamical properties are studied. As a higher order method, they are more complicated than in Newton’s method. Finally, some applications are revisited pointing out the advantages of Chebyshev’s method with respect Newton’s method.

Sign in / Sign up

Export Citation Format

Share Document