scholarly journals A New Optimal Family of Schröder’s Method for Multiple Zeros

Mathematics ◽  
2019 ◽  
Vol 7 (11) ◽  
pp. 1076 ◽  
Author(s):  
Ramandeep Behl ◽  
Arwa Jeza Alsolami ◽  
Bruno Antonio Pansera ◽  
Waleed M. Al-Hamdan ◽  
Mehdi Salimi ◽  
...  
Keyword(s):  
Fourth Order ◽  
High Order ◽  
Numerical Results ◽  
Convergence Order ◽  
Optimal Variant ◽  
Multiple Zeros ◽  
Optimal Convergence ◽  
New Family ◽  
Schröder’S Method

Here, we suggest a high-order optimal variant/modification of Schröder’s method for obtaining the multiple zeros of nonlinear uni-variate functions. Based on quadratically convergent Schröder’s method, we derive the new family of fourth -order multi-point methods having optimal convergence order. Additionally, we discuss the theoretical convergence order and the properties of the new scheme. The main finding of the present work is that one can develop several new and some classical existing methods by adjusting one of the parameters. Numerical results are given to illustrate the execution of our multi-point methods. We observed that our schemes are equally competent to other existing methods.

scholarly journals A New Family of Fourth-Order Optimal Iterative Schemes and Remark on Kung and Traub’s Conjecture

2021 ◽  
Vol 2021 ◽  
pp. 1-9
Author(s):  
Chein-Shan Liu ◽  
Tsung-Lin Lee
Keyword(s):  
Weight Function ◽  
Fourth Order ◽  
Iterative Scheme ◽  
Convergence Order ◽  
Function Technique ◽  
Iterative Schemes ◽  
Optimal Convergence ◽  
New Family ◽  
Kung And Traub’S Conjecture

Kung and Traub conjectured that a multipoint iterative scheme without memory based on m evaluations of functions has an optimal convergence order p = 2 m − 1 . In the paper, we first prove that the two-step fourth-order optimal iterative schemes of the same class have a common feature including a same term in the error equations, resorting on the conjecture of Kung and Traub. Based on the error equations, we derive a constantly weighting algorithm obtained from the combination of two iterative schemes, which converges faster than the departed ones. Then, a new family of fourth-order optimal iterative schemes is developed by using a new weight function technique, which needs three evaluations of functions and whose convergence order is proved to be p = 2 3 − 1 = 4 .

scholarly journals A New Family of High-Order Ehrlich-Type Iterative Methods

Mathematics ◽  
2021 ◽  
Vol 9 (16) ◽  
pp. 1855 ◽  
Author(s):  
Petko D. Proinov ◽  
Maria T. Vasileva
Keyword(s):  
Iterative Methods ◽  
Local Convergence ◽  
Semilocal Convergence ◽  
High Order ◽  
Convergence Theorems ◽  
New Family ◽  
Special Cases ◽  
Iteration Functions ◽  
Iteration Function ◽  
Local Convergence Analysis

One of the famous third-order iterative methods for finding simultaneously all the zeros of a polynomial was introduced by Ehrlich in 1967. In this paper, we construct a new family of high-order iterative methods as a combination of Ehrlich’s iteration function and an arbitrary iteration function. We call these methods Ehrlich’s methods with correction. The paper provides a detailed local convergence analysis of presented iterative methods for a large class of iteration functions. As a consequence, we obtain two types of local convergence theorems as well as semilocal convergence theorems (with computer verifiable initial condition). As special cases of the main results, we study the convergence of several particular iterative methods. The paper ends with some experiments that show the applicability of our semilocal convergence theorems.

Using the generalized Adams-Bashforth-Moulton method for obtaining the numerical solution of some variable-order fractional dynamical models

2020 ◽  
Vol 0 (0) ◽  
Author(s):  
Mohamed M. Khader
Keyword(s):  
Differential Equations ◽  
Fractional Derivative ◽  
Numerical Solution ◽  
Numerical Results ◽  
Convergence Order ◽  
Variable Order ◽  
Numerical Treatment ◽  
Dynamical Models ◽  
Fractional Modeling ◽  
Dynamics Problems

AbstractThis paper is devoted to introduce a numerical treatment using the generalized Adams-Bashforth-Moulton method for some of the variable-order fractional modeling dynamics problems, such as Riccati and Logistic differential equations. The fractional derivative is described in Caputo variable-order fractional sense. The obtained numerical results of the proposed models show the simplicity and efficiency of the proposed method. Moreover, the convergence order of the method is also estimated numerically.

A new family of high-order compact upwind difference schemes with good spectral resolution

2007 ◽  
Vol 227 (2) ◽  
pp. 1306-1339 ◽  
Author(s):  
Qiang Zhou ◽  
Zhaohui Yao ◽  
Feng He ◽  
M.Y. Shen
Keyword(s):  
Spectral Resolution ◽  
High Order ◽  
Difference Schemes ◽  
New Family

A new class of three-point methods with optimal convergence order eight and its dynamics

2014 ◽  
Vol 68 (2) ◽  
pp. 261-288 ◽  
Author(s):  
Taher Lotfi ◽  
Somayeh Sharifi ◽  
Mehdi Salimi ◽  
Stefan Siegmund
Keyword(s):  
Convergence Order ◽  
Optimal Convergence ◽  
New Class

A new family of high-order directions for unconstrained optimization inspired by Chebyshev and Shamanskii methods

2012 ◽  
Vol 29 (1) ◽  
pp. 68-87 ◽  
Author(s):  
Bilel Kchouk ◽  
Jean-Pierre Dussault
Keyword(s):  
Unconstrained Optimization ◽  
High Order ◽  
New Family

Two general higher-order derivative free iterative techniques having optimal convergence order

2019 ◽  
Vol 57 (3) ◽  
pp. 918-938
Author(s):  
Ramandeep Behl ◽  
Ali Saleh Alshomrani ◽  
Á. A. Magreñán
Keyword(s):  
Higher Order ◽  
Convergence Order ◽  
Order Derivative ◽  
Iterative Techniques ◽  
Optimal Convergence ◽  
Derivative Free ◽  
Higher Order Derivative

Reachability of Optimal Convergence Rate Estimates for High-Order Numerical Convex Optimization Methods

2019 ◽  
Vol 99 (1) ◽  
pp. 91-94
Author(s):  
A. V. Gasnikov ◽  
E. A. Gorbunov ◽  
D. A. Kovalev ◽  
A. A. M. Mokhammed ◽  
E. O. Chernousova
Keyword(s):  
Convex Optimization ◽  
Convergence Rate ◽  
Optimization Methods ◽  
High Order ◽  
Optimal Convergence Rate ◽  
Optimal Convergence
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