On the convergence of Halley’s method for simultaneous computation of polynomial zeros

Abstract

In this paper we study the convergence of Halley’s method as a method for finding all zeros of a polynomial simultaneously. We present two types of local convergence theorems as well as a semilocal convergence theorem for Halley’s method for simultaneous computation of polynomial zeros.

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].…”

“…where σ i , A i , B i , and C i are defined by (19) (20), (21), and (22). It follows from (28) that Z i (x) = 0 is equivalent to 2(1 (20), the triangle inequality, Lemma 1, and Hölder's inequality, we obtain:…”

“…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:…”

“…The proof is similar to the proof of Lemma 5. We again define the quantities σ i , A i , B i , and C i by (19), (20), (21), and (22). Using Lemma 2, we obtain the following estimates:…”

Relationships between different types of initial conditions for simultaneous root finding methods

“…Corollary 6.2 was stated without proof in [9], where it was used for obtaining a semilocal convergence result for the two-step Weierstrass method. Another application of Corollary 6.2 can be found in [10].…”