Convergence of iterative methods for multiple zeros

2018 ◽  
pp. 367-380
Author(s):  
Á. Alberto Magreñán ◽  
Ioannis K. Argyros
Keyword(s):  
Iterative Methods ◽  
Multiple Zeros

scholarly journals A Two-Parameter Family of Fourth-Order Iterative Methods with Optimal Convergence for Multiple Zeros

2013 ◽  
Vol 2013 ◽  
pp. 1-7 ◽  
Author(s):  
Young Ik Kim ◽  
Young Hee Geum
Keyword(s):  
Iterative Methods ◽  
Nonlinear Equations ◽  
Fourth Order ◽  
Numerical Experiments ◽  
Harmonic Mean ◽  
Optimal Order ◽  
Multiple Roots ◽  
Multiple Zeros ◽  
Asymptotic Error ◽  
Two Parameter

We develop a family of fourth-order iterative methods using the weighted harmonic mean of two derivative functions to compute approximate multiple roots of nonlinear equations. They are proved to be optimally convergent in the sense of Kung-Traub’s optimal order. Numerical experiments for various test equations confirm well the validity of convergence and asymptotic error constants for the developed methods.

scholarly journals Review of some iterative methods for solving nonlinear equations with multiple zeros

2019 ◽  
Vol 30 (3-4) ◽  
pp. 355-369 ◽  
Author(s):  
N. A. A. Jamaludin ◽  
N. M. A. Nik Long ◽  
Mehdi Salimi ◽  
Somayeh Sharifi
Keyword(s):  
Iterative Methods ◽  
Nonlinear Equations ◽  
Multiple Zeros

scholarly journals Introduction to Higher-Order Iterative Methods for Finding Multiple Roots of Nonlinear Equations

2013 ◽  
Vol 2013 ◽  
pp. 1-3 ◽  
Author(s):  
Rajinder Thukral
Keyword(s):  
Iterative Methods ◽  
Nonlinear Equations ◽  
Higher Order ◽  
Multiple Roots ◽  
Multiple Zeros ◽  
New Methods

We introduce two higher-order iterative methods for finding multiple zeros of nonlinear equations. Per iteration the new methods require three evaluations of function and one of its first derivatives. It is proved that the two methods have a convergence of order five or six.

scholarly journals An Efficient Class of Traub–Steffensen-Type Methods for Computing Multiple Zeros

Axioms ◽  
2019 ◽  
Vol 8 (2) ◽  
pp. 65 ◽  
Author(s):  
Deepak Kumar ◽  
Janak Raj Sharma ◽  
Clemente Cesarano
Keyword(s):  
Iterative Methods ◽  
Basins Of Attraction ◽  
Higher Order ◽  
Order Derivative ◽  
Third Order ◽  
Higher Order Methods ◽  
Multiple Zeros ◽  
Derivative Free ◽  
Graphical Tool ◽  
Numerical Experimentation

Numerous higher-order methods with derivative evaluations are accessible in the literature for computing multiple zeros. However, higher-order methods without derivatives are very rare for multiple zeros. Encouraged by this fact, we present a family of third-order derivative-free iterative methods for multiple zeros that require only evaluations of three functions per iteration. Convergence of the proposed class is demonstrated by means of using a graphical tool, namely basins of attraction. Applicability of the methods is demonstrated through numerical experimentation on different functions that illustrates the efficient behavior. Comparison of numerical results shows that the presented iterative methods are good competitors to the existing techniques.

scholarly journals Inclusion Weierstrass-like root-finders with corrections

Filomat ◽  
2003 ◽  
pp. 143-154 ◽  
Author(s):  
Ljiljana Petkovic ◽  
Miodrag Petkovic ◽  
Dusan Milosevic
Keyword(s):  
Iterative Methods ◽  
Computational Efficiency ◽  
Order Of Convergence ◽  
Numerical Results ◽  
Interval Method ◽  
Multiple Zeros ◽  
Convergence Behavior

In this paper we present iterative methods of Weierstrass's type for the simultaneous inclusion of all multiple zeros of a polynomial. The order of convergence of the proposed interval method is 1 + ?2 ? 2.414 or 3, depending on the type of the applied disk inversion. The criterion for the choice of a proper circular root-set is given. This criterion uses the already calculated entries which increases the computational efficiency of the presented algorithms. Numerical results are given to demonstrate the convergence behavior.

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