Initial Approximation and Root Finding Methods (Mathematical Research S.) - Tapa blanda

De la contraportada

One of the crucial problems in solving polynomial equations is the construction of such initial conditions which provide a safe convergence of the considered numerical method. In general the problem of the choice of initial approximations is a very difficult one and the convergence of any iterative algorithm for finding all roots of a given polynomial is strongly connected with the distribution of its zeros. The author?s investigations concern the opposite problem, i. e. the divergent feature: Can we state such initial conditions under which the considered numerical method fails? Situations are considered in which the critical initial approximations fail, when the iterative methods of Weierstrass, Euler-Chebyshev, Alefeld-Herzberger, Nourein, Dvorcuk, Kyurkchiev-Andreev and Wang-Zheng are applied. Investigations of divergent starting point for every numerical method show that for any monic polynomial f of degree n there exists a set G_f ??? Cⁿ such that these methods, starting from z_o = z &Egr; G_f, do not converge to the roots of f. This NS-set yields divergent starting points and is obtained as the set of solutions of a non-linear system of n equations. The study of the distribution and measure of NS-sets is very complicated and does not appear in the literature. Non-attractive starting points for iterative algorithms are presented which facilitate the choice of the initial approximations for the user. This gives an improvement of all results available until now.