A convergence analysis of a fourth-order method for computing all zeros of a polynomial simultaneously

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).…”

“…It easy to see that the function Λ is decreasing on [0, 1/n) and the function β is quasi-homogeneous of degree m = 3 on [0, ν). In the next lemma, we prove that the function β satisfies the assumptions (16) of Theorem 4 for T : D ⊂ K n → K n defined by (2).…”

“…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:…”

“…Proof. We will apply Proinov's Theorem 3 to the function T : D ⊂ K n → K n defined by (2). First, we prove that the initial Condition (39) implies condition:…”

“…Proof. Let T : D ⊂ K n → K n be defined by (2), and let the real function β be defined by (45). It follows from the first two inequalities of (50) that E(x (0) ) < ν, where ν is the unique zero of the function Λ in (0, 1/n).…”