Geodesic
On the Guaranteed Convergence of the Japanese Zero-finding Method
Publications de l'Institut Mathématique, _N_S_73 (2003) no. 87, p. 59 .

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One of the most important problems in solving nonlinear equations is the construction of such initial conditions which provide both the guaranteed and fast convergence of the considered numerical method. Smale's approach from 1981, known as ``point estimation theory", treats convergence conditions and the domain of convergence in solving an equation $f(z)=0$ using only the information of $f$ at the initial point $\boldsymbol{z}^{(0)}$. A procedure of this type is applied in this paper to the fourth order iterative method for the simultaneous approximation of simple zeros of polynomials, proposed by Sakurai, Torii and Sugiura in 1991. We have stated initial conditions which ensure the guaranteed convergence of this method. These conditions are of significant practical importance since they are computationally verifiable; they depend only on the coefficients of a given polynomial, its degree $n$ and initial approximations to polynomial zeros.
DOI : 10.2298/PIM0373059P
Classification : 65H05 
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     title = {On the {Guaranteed} {Convergence} of the {Japanese} {Zero-finding} {Method}},
     journal = {Publications de l'Institut Math\'ematique},
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Miodrag S. Petković; Lidija Rančić; Dušan Milošević. On the Guaranteed Convergence of the Japanese Zero-finding Method. Publications de l'Institut Mathématique, _N_S_73 (2003) no. 87, p. 59 . doi : 10.2298/PIM0373059P. http://geodesic.mathdoc.fr/articles/10.2298/PIM0373059P/

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