Convergence Analysis of the Modified Chebyshev’s Method for Finding Multiple Roots

2021 ◽  
Author(s):  
Rongfei Lin ◽  
Hongmin Ren ◽  
Qingbiao Wu ◽  
Yasir Khan ◽  
Juelian Hu
Keyword(s):  
Convergence Analysis ◽  
Multiple Roots ◽  
Chebyshev’S Method

A CONTINUATION METHOD AND ITS CONVERGENCE FOR SOLVING NONLINEAR EQUATIONS IN BANACH SPACES

2013 ◽  
Vol 10 (04) ◽  
pp. 1350021 ◽  
Author(s):  
M. PRASHANTH ◽  
D. K. GUPTA
Keyword(s):  
Convergence Analysis ◽  
Banach Spaces ◽  
Nonlinear Equations ◽  
Error Bounds ◽  
Newton’S Method ◽  
Newton's Method ◽  
Continuation Method ◽  
Continuity Condition ◽  
Majorizing Sequences ◽  
Chebyshev’S Method

A continuation method is a parameter based iterative method establishing a continuous connection between two given functions/operators and used for solving nonlinear equations in Banach spaces. The semilocal convergence of a continuation method combining Chebyshev's method and Convex acceleration of Newton's method for solving nonlinear equations in Banach spaces is established in [J. A. Ezquerro, J. M. Gutiérrez and M. A. Hernández [1997] J. Appl. Math. Comput.85: 181–199] using majorizing sequences under the assumption that the second Frechet derivative satisfies the Lipschitz continuity condition. The aim of this paper is to use recurrence relations instead of majorizing sequences to establish the convergence analysis of such a method. This leads to a simpler approach with improved results. An existence–uniqueness theorem is given. Also, a closed form of error bounds is derived in terms of a real parameter α ∈ [0, 1]. Four numerical examples are worked out to demonstrate the efficacy of our convergence analysis. On comparing the existence and uniqueness region and error bounds for the solution obtained by our analysis with those obtained by using majorizing sequences, it is found that our analysis gives better results in three examples, whereas in one example it gives the same results. Further, we have observed that for particular values of the α, our analysis reduces to those for Chebyshev's method (α = 0) and Convex acceleration of Newton's method (α = 1) respectively with improved results.

scholarly journals On an efficient method for the simultaneous approximation of polynomial multiple roots

2014 ◽  
Vol 8 (1) ◽  
pp. 73-94 ◽  
Author(s):  
Miodrag Petkovic ◽  
Ljiljana Petkovic ◽  
Jovana Dzunic
Keyword(s):  
Convergence Analysis ◽  
Computational Efficiency ◽  
Positive Root ◽  
Single Step ◽  
Multiple Roots ◽  
Step Method ◽  
Computational Aspects ◽  
Parallel Mode ◽  
Fourth Order Method ◽  
High Computational Efficiency

An iterative method in parallel mode for the simultaneous determination of multiple roots of algebraic polynomials is stated together with its single-step variant. These methods are more efficient compared to all simultaneous methods based on fixed point relations. To attain very high computational efficiency, a suitable correction resulting from Li-Liao-Cheng?s two-point fourth order method of low computational complexity and Gauss-Seidel?s approach are applied. Considerable increase of the convergence rate is obtained applying only n additional polynomial evaluations per iteration, where n is the number of distinct roots. A special emphasis is given to the convergence analysis and computational efficiency of the proposed methods. The presented convergence analysis shows that the R-order of convergence of the proposed single-step method is at least 2 + ?v; where ?v?2 (4,6) is the unique positive root of the polynomial gv(t) = tn-4n-1 t-22n-1: The convergence order of the corresponding total-step method is six. Computational aspects and some numerical examples are given to demonstrate high computational efficiency and very fast convergence of the proposed methods.

scholarly journals An Optimal Derivative Free Family of Chebyshev–Halley’s Method for Multiple Zeros

Mathematics ◽  
2021 ◽  
Vol 9 (5) ◽  
pp. 546
Author(s):  
Ramandeep Behl ◽  
Sonia Bhalla ◽  
Ángel Alberto Magreñán ◽  
Alejandro Moysi
Keyword(s):  
Nonlinear Equation ◽  
Weight Function ◽  
Convergence Analysis ◽  
Nonlinear Equations ◽  
Fourth Order ◽  
Order Of Convergence ◽  
Higher Order ◽  
Multiple Roots ◽  
Iterative Technique ◽  
Derivative Free

In this manuscript, we introduce the higher-order optimal derivative-free family of Chebyshev–Halley’s iterative technique to solve the nonlinear equation having the multiple roots. The designed scheme makes use of the weight function and one parameter α to achieve the fourth-order of convergence. Initially, the convergence analysis is performed for particular values of multiple roots. Afterward, it concludes in general. Moreover, the effectiveness of the presented methods are certified on some applications of nonlinear equations and compared with the earlier derivative and derivative-free schemes. The obtained results depict better performance than the existing methods.

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