A family of multiopoint iterative functions for finding multiple roots of equations

There is no doubt that the fourth-order King’s family is one of the important ones among its counterparts. However, it has two major problems: the first one is the calculation of the first-order derivative; secondly, it has a linear order of convergence in the case of multiple roots. In order to improve these complications, we suggested a new King’s family of iterative methods. The main features of our scheme are the optimal convergence order, being free from derivatives, and working for multiple roots (m≥2). In addition, we proposed a main theorem that illustrated the fourth order of convergence. It also satisfied the optimal Kung–Traub conjecture of iterative methods without memory. We compared our scheme with the latest iterative methods of the same order of convergence on several real-life problems. In accordance with the computational results, we concluded that our method showed superior behavior compared to the existing methods.

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

Vietnam Journal of Mathematics ◽

10.1007/s10013-020-00470-8 ◽

2021 ◽

Author(s):

Rongfei Lin ◽

Hongmin Ren ◽

Qingbiao Wu ◽

Yasir Khan ◽

Juelian Hu

Keyword(s):

Convergence Analysis ◽

Multiple Roots ◽

Chebyshev’S Method

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An efficient class of fourth-order derivative-free method for multiple-roots

International Journal of Nonlinear Sciences and Numerical Simulation ◽

10.1515/ijnsns-2020-0161 ◽

2021 ◽

Vol 0 (0) ◽

Author(s):

Sunil Kumar ◽

Deepak Kumar ◽

Janak Raj Sharma ◽

Ioannis K. Argyros

Keyword(s):

Multiple Root ◽

Second Step ◽

Optimal Order ◽

Multiple Roots ◽

Nonlinear Functions ◽

New Methods ◽

Derivative Free ◽

Derivative Free Methods ◽

Derivative Free Method ◽

Good Number

Abstract Many optimal order multiple root techniques, which use derivatives in the algorithm, have been proposed in literature. Many researchers tried to construct an optimal family of derivative-free methods for multiple roots, but they did not get success in this direction. With this as a motivation factor, here, we present a new optimal class of derivative-free methods for obtaining multiple roots of nonlinear functions. This procedure involves Traub–Steffensen iteration in the first step and Traub–Steffensen-like iteration in the second step. Efficacy is checked on a good number of relevant numerical problems that verifies the efficient convergent nature of the new methods. Moreover, we find that the new derivative-free methods are just as competent as the other existing robust methods that use derivatives.

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XI. Analysis of the roots of equations

Philosophical Transactions of the Royal Society of London ◽

10.1098/rstl.1837.0013 ◽

1837 ◽

Vol 127 ◽

pp. 161-178

Keyword(s):

Roots Of Unity ◽

Constituent Part ◽

Roots Of Equations

1. The object of this memoir is to show how the constituent parts of the roots of algebraical equations may be determined, by considering the conditions under which they vanish, and conversely to show the signification of each such constituent part. 2. In equations of degrees higher than the second the same constituent part of the root is found in several places governed by the same radical sign, but affected with the different corresponding roots of unity as multipliers.

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