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Chebyshev‐like root‐finding methods with accelerated convergence
Author(s) -
Petković M. S.,
Rančić L.,
Petković L. D.,
Ilić S.
Publication year - 2009
Publication title -
numerical linear algebra with applications
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 1.02
H-Index - 53
eISSN - 1099-1506
pISSN - 1070-5325
DOI - 10.1002/nla.661
Subject(s) - mathematics , convergence (economics) , polynomial , root finding algorithm , chebyshev filter , chebyshev nodes , chebyshev iteration , newton's method , chebyshev polynomials , iterative method , verifiable secret sharing , local convergence , mathematical optimization , mathematical analysis , computer science , nonlinear system , physics , quantum mechanics , economics , economic growth , set (abstract data type) , programming language
Iterative methods for the simultaneous determination of simple or multiple complex zeros of a polynomial, based on a cubically convergent Chebyshev method, are considered. Using Newton's and Halley's corrections the convergence of the basic method of the fourth order is increased to five and six, respectively. The improved convergence is achieved with negligible number of additional calculations, which significantly increases the computational efficiency of the accelerated methods. One of the most important problems in solving polynomial equations, the construction of initial conditions that enable both guaranteed and fast convergence, is also studied for the proposed methods. These conditions are computationally verifiable since they depend only on initial approximations, the polynomial coefficients and the polynomial degree, which is of practical importance. Finally, modified methods of Chebyshev's type for finding multiple zeros and single‐step methods based on the Gauss–Seidel approach are constructed. Copyright © 2009 John Wiley & Sons, Ltd.