On a family of Weierstrass-type root-finding methods with accelerated convergence

Abstract

Kyurkchiev and Andreev (1985) constructed an infinite sequence of Weierstrass-type iterative methods for approximating all zeros of a polynomial simultaneously. The first member of this sequence of iterative methods is the famous method of Weierstrass (1891) and the second one is the method of Nourein (1977). For a given integer N ≥ 1, the Nth method of this family has the order of convergence N + 1. Currently in the literature, there are only local convergence results for these methods. The main purpose of this paper is to present semilocal convergence results for the Weierstrass-type methods under computationally verifiable initial conditions and with computationally verifiable a posteriori error estimates.

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

“…Our local convergence result of the second type is the first result of this type for Wang-Zheng's method (1). The convergence results of the second type for other simultaneous methods can be found in [17,[22][23][24]28,29].…”

“…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:…”

“…It follows from Condition (26) and the monotonicity of g that g(E(x)) > g(µ) = 0. Now, using the triangle inequality and the inequalities (27), (29), (31), and (33), we get:…”

Inverse Numerical Iterative Technique for Finding all Roots of Nonlinear Equations with Engineering Applications

“…ere is another class of derivative-free iterative methods which approximates all roots of (1) simultaneously. e simultaneous iterative methods for approximating all roots of (1) are very popular due to their global convergence and parallel implementation on computer (see, e.g., Weierstrass [3], Kanno [4], Proinov [5], Petkovi´c [6], Mir [7], Nourein [8], Aberth [9], and reference cited there in [10][11][12][13][14][15][16][17][18][19][20][21][22]).…”