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

The modified inverse Weierstrass methods is constructed by combining the companion matrix method and similarity transformations between some companion matrices. In this paper we establish a new local convergence theorem for this method for simultaneous computation of polynomial zeros. The main result is analogous to a known result of the local convergence of the Weierstrass method.


Introduction
Let P(z) be a monic polynomial P(z) = a 0 + a 1 z + . . .+ a n−1 z n−1 + z n , (1.1) of degree n ≥ 2, with simple real or complex zeros α 1 , α 2 , . . ., α n , and let z k i (for i = 1, 2, . . ., n) be distinct reasonable close approximations of these zeros.One of the most famous iterative methods for simultaneously finding all the zeros of polynomials is the Weierstrass method, defined by where the term is the so-called Weierstrass' correction.
for all k ≥ 0.

Preliminary results
Our previous papers [17,18] are devoted to the investigation of the companion matrix method and similarity transformations between some companion matrices.We proposed two new modifications of the Weierstrass iterative method.The first modification (so-called inverse WDK method) is defined by [18] It is also shown that this iteration is equivalent form of the Weierstrass iteration in the case of a 0 ̸ = 0, i.e. if the roots of P(z) are α i ̸ = 0 for i = 1, 2, . . ., n.
The second modification (so-called modified inverse WDK method) is defined by where W i (z k ) is the Weierstrass' correction (1.3).We can also use some of the following equivalent forms of (2.4) ) ) In this paper we investigate the local convergence properties of the method (2.4).

Statement of the main result
First, we prove some auxiliary results.Proof.(i) From the inequality c < d an+1 , it follows that and a is the unique root of the equation t = e 1/t .Suppose that where 0 < c < d an+1 .Then the following statements hold true Proof.
From (3.8) and the choice of q, we get which implies the first claim.
(ii) We shall prove the second claim by means of the identity (given by [8]) ) .
From the first claim of the Lemma 3.2, we get where Substituting u i by z m i ,v j by α j and q by h 2 m in (3.12) and (3.13), and taking into account that we get From the assumption c ≤ d/(an + 4), it follows that This completes the proof.Now we will consider the case, when a 0 = 0 in (1.1), i.e. the polynomial P(z) has a root equal to 0. Without loss of generality we can assume that α n = 0.
(i) The first statement follows in a similar way by the proof of Theroem 3.1 and we omit it here.
(ii) Let us consider the case of i = n.From (2.5) and taking into account that α n = 0, we get Then, it follows Substituting u i by z k n , u j by z k j , v j by α j and q by h in (3.13) and from (3.22), it is easy to prove that

Conclusion
In this paper we provide a local convergence theorem for a new modification of Weierstrass iterative method.The main results are motivated by the convergence results to WDK-method by Kjurkchiev and Markov in [8].We prove our convergence theorems under conditions and assumptions similar to those in [8].We consider both cases, namely all the zeros of P are different from 0 and the case of all the zeros are different from 0 except only one.