Convergence of the Weierstrass Method for Simultaneous Approximation of Polynomial Zeros

Abstract

In 1891, Weierstrass presented his famous iterative method for finding all the zeros of a polynomial simultaneously. In this paper we establish three new local convergence theorems for the Weierstrass method with a posteriori and a priori error estimates. The main result of the paper generalizes, improves and complements some well known results of Dochev (1962), Kjurkchiev and Markov (1983) and Yakoubsohn (2002). The results are given for polynomials over an arbitrary normed field.

Local and Semilocal Convergence of Wang-Zheng’s Method for Simultaneous Finding Polynomial Zeros

“…Finally, we refer the reader to some recent papers [2,17,20,[22][23][24][25]28,29], which investigate initial conditions of the type (66).…”

“…On the basis of this theory lays the notion function of initial conditions of T since the convergence of any iterative method of the type (1) is studied with respect to some function of initial conditions (see [35,36]). Some applications of this theory can be found in [1,2,5,7,8,[14][15][16][17][18][19][20][21][22][23][24][25][26][27][28][29][35][36][37][38]. Let R n be equipped with coordinate-wise ordering defined by:…”

“…where E is defined by (12) and the positive number µ is the unique zero in (0, 1/n) of the function g defined in: (25). Then:…”

Weierstrass method for quaternionic polynomial root‐finding

“…If the roots ζ 1 ,…, ζ n are distinct and are sufficiently good initial approximations to these roots, then the method converges at a quadratic rate, as was firstly proven by Dochev . A complete history and an improvement of Dochev's theorem can be found in Proinov and Petkova (see also Proinov, section 6). For multiple roots, the method still converges (locally), but the quadratic convergence is lost; see, eg, Fraigniaud.…”

Convergence of the modified inverse Weierstrass method for simultaneous approximation of polynomial zeros

“…The iteration method (1.2)-(1.3) is introduced for the first time by Weierstrass in 1891 [1], rediscovered later by Durand [2], Dochev [3], Kerner [4], Prešić [5], and since then it has been investigated by many authors (see [6,7,8,9,10,11,12,13,14,15,16]). In the literature the method (1.2)-(1.3) is also called Weierstrass-Dochev, Durand-Kerner, or shorter the WDK-method.…”