On the convergence of high-order Ehrlich-type iterative methods for approximating all zeros of a polynomial simultaneously

We study a family of high order Ehrlich-type methods for approximating all zeros of a polynomial simultaneously. Let us denote by $T^{(1)}$ the famous Ehrlich method (1967). Starting from $T^{(1)}$, Kjurkchiev and Andreev (1987) have introduced recursively a sequence ${(T^{(N)})_{N=1}^\infty}$ of iterative methods for simultaneous finding polynomial zeros. For given $N \ge 1$, the Ehrlich-type method $T^{(N)}$ has the order of convergence ${2 N + 1}$. In this paper, we establish two new local convergence theorems as well as a semilocal convergence theorem (under computationally verifiable initial conditions and with a posteriori error estimate) for the Ehrlich-type methods $T^{(N)}$. Our first local convergence theorem generalizes a result of Proinov (2015) and improves the result of Kjurkchiev and Andreev (1987). The second local convergence theorem generalizes another recent result of Proinov (2015), but only in the case of maximum-norm. Our semilocal convergence theorem is the first result in this direction.


Introduction
Throughout this paper (K, | · |) denotes an algebraically closed field and K[z] denotes the ring of polynomials (in one variable) over K. For a given vector x in K n , x i always denotes the ith coordinate of x. In particular, if F is a map with values in K n , then F i (x) denotes the ith coordinate of the vector F (x). We endow the vector space K n with a norm x p defined as usual: and with coordinate-wise ordering defined by x y if and only if x i ≤ y i for each i = 1, . . . , n (1.1) Then (K n , · , ) is a cone normed space over R n (see, e.g., Proinov [1]). Let f ∈ K[z] be a polynomial of degree n ≥ 2. A vector ξ ∈ K n is said to be a root-vector of f if f (z) = a 0 n i=1 (z − ξ i ) for all z ∈ K, where a 0 ∈ K. We denote with sep(f ) the separation number of f which is defined to be the minimum distance between two distinct zeros of f .

The Weierstrass method and Weierstrass correction
In the literature, there are a lot of iterative methods for finding all zeros of f simultaneously (see, e.g., the monographs of Sendov, Andreev and Kjurkchiev [2], Kjurkchiev [3], McNamee [4] and Petković [5] and references given therein). In 1891, Weierstrass [6] published his famous iterative method for simultaneous computation of all zeros of f . The Weierstrass method is defined by the following iteration where the operator W f : D ⊂ K n → K n is defined by where a 0 ∈ K is the leading coefficient of f and the domain D of W is the set of all vectors in K n with distinct components. The Weierstrass method (1.2) has secondorder of convergence provided that all zeros of f are simple. The operator W f is called Weierstrass correction. We should note that W f plays an important role in many semilocal convergence theorems for simultaneous methods.

The Ehrlich method
Another famous iterative method for finding simultaneously all zeros of a polynomial f was introduced by Ehrlich [7] in 1967. The Ehrlich method is defined by the following fixed point iteration: x (k+1) = T (x (k) ), k = 0, 1, 2, . . . , (1.4) where the operator T : D ⊂ K n → K n is defined by T (x) = (T 1 (x), . . . , T n (x)) with (i = 1, . . . , n) (1.5) and the domain of T is the set (1. 6) Here and throughout the paper, we denote by I n the set of indices 1, . . . , n, that is I n = {1, . . . , n}. The Ehrlich method has third-order of convergence if all zeros of f are simple. The Ehrlich method was rediscovered by Abert [8] in 1973. In 1970, Börsch-Supan [9] introduced another third-order method for numerical computation of all zeros of a polynomial simultaneously. In 1982, Werner [10] has proved that the both methods are identical. The Ehrlich method (1.4) is known also as "Ehrlich-Abert method", "Börsch-Supan method" and "Abert method". Recently, Proinov [11] obtained two local convergence theorems for Ehrlich method under different types of initial conditions. The first one generalizes and improves the results of Kyurkchiev and Tashev [12,13] and Wang and Zhao [14,Theorem 2.1]. The second one generalizes and improves the results of Wang and Zhao [14,Theorem 2.2] and Tilli [15,Theorem 3.3].
Before we state the two results of [11], we need some notations which will be used throughout the paper. For given vectors x ∈ K n and y ∈ R n , we define in R n the vector provided that y has no zero components. Given p such that 1 ≤ p ≤ ∞, we always denote by q the conjugate exponent of p, i.e. q is defined by means of 1 ≤ q ≤ ∞ and 1/p + 1/q = 1.
In the sequel, we use the function d : Let a > 0 and b ≥ 1. We define the real function φ by and the real number R as follows be a polynomial of degree n ≥ 2 which has only simple zeros, ξ ∈ K n be a root-vector of f and 1 ≤ p ≤ ∞. Suppose x (0) ∈ K n is an initial guess satisfying where a = (n − 1) 1/q and b = 2 1/q . Then Ehrlich iteration (1.4) is well-defined and converges cubically to ξ with error estimates ) and the function φ is defined by (1.7).
Theorem 1.2 (Proinov [11]) Let f ∈ K[z] be a polynomial of degree n ≥ 2, ξ ∈ K n be a root-vector of f and 1 ≤ p ≤ ∞. Suppose x (0) ∈ K n is a vector with distinct components satisfying where a = (n − 1) 1/q and b = 2 1/q . Then f has only simple zeros in K and Ehrlich iteration (1.4) is well-defined and converges to ξ with error estimates ) and the function φ is defined by (1.7) and the function ψ by Moreover, the method converges cubically to ξ provided that E(x (0) ) < R.

A family of high-order Ehrlich-type methods
In the following definition, we define a sequence (T (N ) ) ∞ N =0 of iteration functions in the vector space K n . In what follows, we define the binary relation # on K n by x # y ⇔ x i = y j for all i, j ∈ I n with i = j. (1.12) where the sequence of domains (D N ) ∞ N =0 is also defined recursively by setting D 0 = K n and (1.14) Given N ∈ N, the N th method of Kjurkchiev-Andreev's family can be defined by the following fixed-point iteration:  [16]) Let f ∈ C[z] be a polynomial of degree n ≥ 2 which has only simple zeros, ξ ∈ C n be a root-vector of f and N ≥ 1. Let 0 < h < 1 and c > 0 be such that where δ = sep(f ). Suppose x (0) ∈ C n is an initial guess satisfying the condition (1.17) Then the Ehrlich-type method (1.15) converges to ξ with error estimate The purpose of the paper In this paper, we present two new local convergence theorems as well as a semilocal convergence theorem (under computationally verifiable initial conditions and with a posteriori error estimate) for Ehrlich-type methods (1.15). Our first local convergence result (Theorem 4.6) generalizes Theorem 1.1 (Proinov [11]) and improves Theorem 1.4 (Kjurkchiev and Andreev [16]). Our second local convergence result (Theorem 5.4) generalizes Theorem 1.2 (Proinov [11]), but only in the case p = ∞. Furthermore, several numerical examples are provided to show some practical applications of our semilocal convergence result.

A general convergence theorem
Recently, Proinov [17,18,19] has developed a general convergence theory for iterative processes of the type where T : D ⊂ X → X is an iteration function in a cone metric space X. In order to make this paper self-contained, we briefly review some basic definitions and results from this theory. Throughout this paper J denotes an interval on R + containing 0. For an integer k ≥ 1, we denote by S k (t) the following polynomial: S k (t) = 1 + t + . . . + t k−1 .
If k = 0 we assume that S k (t) ≡ 0. Throughout the paper we assume by definition that 0 0 = 1. If m functions ϕ 1 , . . . , ϕ m are quasi-homogeneous on J of degree r 1 , . . . , r m , then their product ϕ 1 . . . ϕ m is a quasi-homogeneous function of degree r 1 + . . . + r m on J. Note also that a function ϕ is quasi-homogeneous of degree 0 on J if and only it is nondecreasing on J. (i) ϕ is quasi-homogeneous of degree r on J; A gauge function ϕ of order r on J is said to be a strict gauge function if the inequality in (ii) holds strictly whenever t ∈ J\{0}.
The following is a sufficient condition for a gauge function of order r.  The following is a simple sufficient condition for initial points. Suppose that x ∈ D with E(x) ∈ J implies T x ∈ D. Then every point x 0 ∈ D such that E(x 0 ) ∈ J is an initial points of T .
The following fixed point theorem plays an important role in our paper.
Theorem 2.8 (Proinov [19]) Let T : D ⊂ X → X be an operator of a cone normed space (X, · ) over a solid vector space (Y, ), and let E : D → R + be a function of initial conditions of T with a gauge function ϕ of order r ≥ 1 on an interval J. Suppose T is an iterated contraction at a point ξ with respect to E with control function β such that t β(t) is a strict gauge function of order r on J (2.5) and there exist a function ψ : J → R + such that where φ : J → R + is a nondecreasing function satisfying Then the following statements hold true. (ii) Starting from each initial point x (0) of T , Picard iteration (2.1) remains in the set U and converges to ξ with error estimates In the case β ≡ φ, Theorem 2.8 reduces to the following result.

Corollary 2.9 ([19])
Let T : D ⊂ X → X be an operator in a cone normed space (X, · ) over a solid vector space (Y, ), and let E : D → R + be a function of initial conditions of T with a strict gauge function ϕ of order r ≥ 1 on an interval J. Suppose that T is an iterated contraction at a point ξ with respect to E and with control function φ satisfying (2.7). Then the following statements hold true.
(ii) Starting from each initial point x (0) of T , Picard iteration (2.1) remains in U and converges to ξ with order r and error estimates for all k ≥ 0, where λ = φ(E(x (0) )).

Some inequalities in K n
In this section, we present some useful inequalities in K n which play an important role in the paper.
then the vector u also has distinct components.
Then for all i, j ∈ I n , then for all i, j ∈ I n , Proof By the triangle inequality of cone norm in K n and (3.5), we obtain Taking the p-norm, we get From (3.1) and (3.7), we obtain (3.6) which completes the proof.
If v is a vector with distinct components such that (3.5) holds, then for all i, j ∈ I n , Proof From (3.2) and (3.7), we get (3.8) which completes the proof.

Local convergence theorem of the first type
Let f ∈ K[z] be a polynomial of degree n ≥ 2 which has only simple zeros in K, and let ξ ∈ K n be a root-vector of f . In this section we study the convergence of the Ehrlich-type methods (1.15) with respect to the function of initial conditions E : K n → R + defined as follows Let a > 0 and b ≥ 1. Throughout this section, we define the function φ and the real number R by (1.7) and (1.8), respectively. It is easy to show that R is the unique solution of the equation In the next definition, we introduce a sequence of such functions.
where a > 0 and b ≥ 1 are constants.
Proof of the correctness of Definition 4. 1 We prove the correctness of the definition by induction. For N = 0 it is obvious. Assume that for some N ≥ 0 the function φ N is well-defined and nondecreasing on [0, R] and φ N (R) = 1. We shall prove the same for φ N +1 . It follows from the induction hypothesis that which means that the function φ N +1 is well-defined on [0, R]. From (4.2) and the induction hypothesis, we deduce that φ N +1 is nondecreasing on [0,R]. From (4.2) and φ N (R) = 1, we obtain This completes the induction and the proof of the correctness of Definition 4.1.

Definition 4.2 For any integer
where the function φ N is defined by Definition 4.1.
In the next lemma, we present some properties of the functions φ N and ϕ N .
Proof Claim (i) can easily be proved by induction. From (4.2) and (4.3), we get which proves (ii). Claim (iii) is a trivial consequence from (ii). Claim (iv) follows from (ii) by induction. Claim (v) follows from (i) and the definition of ϕ N .
where σ i ∈ K is defined by .
Proof (i) Taking into account that ξ is a root-vector of f , we get Then from (1.13) and (4.5), we obtain which completes the proof.
be a polynomial of degree n ≥ 2 which has only simple zeros in K, ξ ∈ K n be a root-vector of f , N ≥ 0 and 1 ≤ p ≤ ∞. Suppose x ∈ K n is a vector satisfying the following condition where the function E : K n → R + is defined by (4.1), a = (n − 1) 1/q and b = 2 1/q . Then Proof We shall prove statements by induction on N . If N = 0, then (4.10) holds trivially. Assume that (4.10) holds for some N ≥ 0. First, we show that x ∈ D N +1 , i.e. x # T (N ) (x) and (4.8) holds for every i ∈ I n . It follows from the first inequality in (4.10) that the inequality (3.3) is satisfied with u = x and v = T (N ) (x). Then by Lemma 3.3 and (4.9), we obtain for every j = i. Consequently, x # T (N ) (x). It remains to prove (4.8) for every i ∈ I n . Let i ∈ I n be fixed. We shall consider only the non-trivial case f (x i ) = 0. In this case (4.8) is equivalent to We define σ i by (4.6). It follows from Lemma 4.4(i) that (4.12) is equivalent to σ i = 1. By Lemma 3.1 with u = x and v = ξ and (4.9), we get for every j = i. From the triangle inequality in K, (4.11), (4.13), induction hypothesis and Hölder's inequality, we get . (4.14) From this, φ N (E(x)) ≤ 1 and (4.9), we obtain which yields σ i = 1 and so (4.8) holds. Hence, x ∈ D N +1 . Second, we show that the inequalities in (4.10) hold for N + 1. The first inequality for N + 1 is equivalent to (4.15) Let i ∈ I n be fixed. If x i = ξ i , then T (N +1) i (x) = ξ i and so (4.15) becomes an equality. Suppose x i = ξ i . By Lemma 4.4(ii), the triangle inequality in K and the estimate (4.14), we get (4.15). Dividing both sides of the inequality (4.15) by d i (ξ) and taking the p-norm, we obtain which proves that the second inequality in (4.10) holds for N + 1. This completes the induction and the proof of the lemma.
Now we are ready to state the main result of this section. In the case N = 1 this result coincides with Theorem 1.1.
be a polynomial of degree n ≥ 2 which has only simple zeros in K, ξ ∈ K n be a root-vector of f , N ≥ 1 and 1 ≤ p ≤ ∞. Suppose x (0) ∈ K n is an initial guess satisfying where the function E : K n → R + is defined by (4.1), a = (n − 1) 1/q and b = 2 1/q . Then the Ehrlich-type iteration (1.15) is well-defined and converges to ξ with error estimates Corollary 4.7 Let f ∈ K[z] be a polynomial of degree n ≥ 2 which has only simple zeros in K, ξ ∈ K n be a root-vector of f , N ≥ 1 and 1 ≤ p ≤ ∞. Suppose x (0) ∈ K n is an initial guess satisfying (4.16). Then the Ehrlich-type iteration (1.15) is welldefined and converges to ξ with error estimates for all k ≥ 0, where λ = φ(E(x (0) )) and φ is a real function defined by (1.7).
Proof It follows from Theorem 4.6 and Lemma 4.3(iv).
Let 0 < h < 1 be a given number. Solving the equation φ(t) = h 2 in the interval (0, R), we can reformulate Corollary 4.7 in the following equivalent form.
Corollary 4.8 Let f ∈ K[z] be a polynomial of degree n ≥ 2 which has n simple zeros in K, ξ ∈ K n be a root-vector of f , N ≥ 1, 1 ≤ p ≤ ∞ and 0 < h < 1. Suppose x (0) ∈ K n is an initial guess which satisfies where a = (n − 1) 1/q and b = 2 1/q . Then the Ehrlich-type method (1.15) is welldefined and converges to ξ with error estimates for all k ≥ 0.

Local convergence theorem of the second type
Let f ∈ K[z] be a polynomial of degree n ≥ 2. We study the convergence of the Ehrlich-type method (1.15) with respect to the function of initial conditions E : D → R + defined by In the previous section, we introduce the functions φ N , ϕ N and the real number R with two parameters a > 0 and b ≥ 1. In this section, we consider a special case of φ N , ϕ N and R when b = 2. In other words, now we define R by Furthermore, we define the functions φ N and ϕ N by Definitions 4.1 and 4.2, respectively, but with instead of (4.2), where a > 0 is a constant.
Definition 5.1 For a given integer N ≥ 1, we define the increasing function and we define the decreasing function ψ N : [0, R] → (0, 1] as follows Proof of the correctness of Definition 5.1 The functions β N and ψ N are well-defined on [0, R] since The monotonicity of β N and ψ N is obvious. It remains to prove that β N (R) < 1 and ψ N (R) > 0. Since φ N (R) = 1, we obtain which completes the proof of the correctness of Definition 5.1 Proof The function β N can be presented in the form β Claim (iii) follows from Lemma 4.3(iii) and (5.4). Claim (iv) follows from (iii) and (5.5).
be a polynomial of degree n ≥ 2 which has only simple zeros in K, ξ ∈ K n a root-vector of f , N ≥ 1 and 1 ≤ p ≤ ∞. Suppose x ∈ K n is a vector with distinct components such that where the function E : D → R + is defined by (5.1) and a = (n − 1) 1/q . Then f has only simple zeros in K, Besides, the vector T (N ) (x) has pairwise distinct components.
Proof It follows from (5.7) and R < 1/2 that E(x) < 1/2. Then it follows from Lemma 3.2 that the vector ξ has distinct components, which means that f has only simple zeros in K. We divide the proof into two steps.
Step 1 In this step, we prove x ∈ D N and the first inequality in (5.8) by induction on N . If N = 1, the proof of the claims can be found in [11]. Assume that x ∈ D N and the first inequality in (5.8) hold for some N ≥ 1.
First we show that x ∈ D N +1 i.e. x # T (N ) (x) and (4.8) holds for every i ∈ I n . It follows from the first inequality in (5.8) that (3.5) holds with v = x, u = T (N ) (x) and α = 1. Therefore by Lemma 3.4, (5.7) and R < 1/2, we obtain for every j = i. Consequently, x # T (N ) (x). It remains to prove (4.8) for every i ∈ I n . Let i ∈ I n be fixed. We shall consider only the non-trivial case f (x i ) = 0. In this case (4.8) is equivalent to (4.12). On the other hand, it follows from Lemma 4.4(i) that (4.12) is equivalent to σ i = 1, where σ i is defined by (4.6). By Lemma 3.1 with u = ξ and v = x and (5.7), we get for every j = i. Hence, we obtain x # ξ. From induction hypothesis, we get Combining the triangle inequality in K, (5.10), (5.9) and (5.11), we obtain which, using Hölder's inequality, yields . (5.12) From this and (5.7), we deduce which yields σ i = 1, and so (4.12) holds. Thus we prove that x ∈ D N +1 . Now we have to prove that the first inequality in (5.8) is satisfied for N + 1, which is equivalent to (5.13) Let i ∈ I n be fixed. If x i = ξ i , then T (N +1) i (x) = ξ i and the inequality (5.13) becomes an equality. Suppose x i = ξ i . It follows from Lemma 4.4(ii), the triangle inequality in K and the estimate (5.12) that From this inequality, Lemma 5.2(ii), ψ N (t) ≤ 1, (5.3) and Lemma 5.2(iv), we obtain which proves (5.13). This completes the induction.
Step 2 In this step we prove the second inequality in (5.8) and that T (N ) (x) has distinct components. First inequality in (5.8) allow us to apply Lemma 3.5 with u = T (N ) (x), v = x and α = β N (E(x)). By Lemma 3.5 and (5.5), we deduce By taking the minimum over all j ∈ I n such that j = i, we obtain which implies that T (N ) (x) has distinct components. It follows from (5.11), (5.14) and Lemma 5.2(ii) that By taking the p-norm, we obtain which proves the second inequality in (5.8). This completes the proof. Now we are able to state the main result of this section. In the case when N = 1 and p = ∞ this result reduces to Theorem 1.2.
Theorem 5.4 Let f ∈ K[z] be a polynomial of degree n ≥ 2 which splits over K, ξ ∈ K n be a root-vector of f , N ≥ 1 and 1 ≤ p ≤ ∞. Suppose x (0) ∈ K n is an initial guess with distinct components such that where the function E is defined by (5.1) and a = (n − 1) 1/q . Then f has only simple zeros in K and the Ehrlich-type iteration (1.15) is well-defined and converges to ξ with error estimates for all k ≥ 0, where λ = φ N (E(x (0) )), θ = ψ N (E(x (0) )). Moreover, the method is convergent with order 2N + 1 provided that E(x (0) ) < R.
Proof We apply Theorem 2.8 to the iteration function T (N ) : D N ⊂ K n → K n together with the function E : D N → R + defined by (5.1). It follows from Lemma 5.3 and Lemma 4.3(v) that E is a function of initial conditions of T (N ) with gauge function ϕ N of order r = 2N + 1 on the interval J = [0, R].
From Lemma 5.3, we get that T (N ) is an iterated contraction at ξ with respect to E and with control function β N . Also, it is easy to see that the functions β N , φ N , ψ N and ϕ N have the properties (2.5), (2.6) and (2.7).
It follows from Lemma 5.3 that x (0) ∈ D N . According to Theorem 2.6 to prove that x (0) is an initial point of T (N ) it is sufficient to prove that

Semilocal convergence theorem
In this section we establish semilocal convergence theorems for Ehrlich-type methods (1.15) for finding all zeros of a polynomial simultaneously. We study the convergence of these methods with respect to the function of initial conditions E : D → R + defined by Recently Proinov [22] has shown that there is a relationship between local and semilocal theorems for simultaneous root-finding methods. It turns out that from any local convergence theorem for a simultaneous method one can obtain as a consequence a semilocal theorem for the same method. In particular, from Theorem 4.6 we can obtain a semilocal convergence theorem for Ehrlich-type methods (1.15) under computationally verifiable initial conditions. For this purpose we need the following result. Theorem 6.1 (Proinov [22]) Let f ∈ K[z] be a polynomial of degree n ≥ 2. Suppose x ∈ K n is an initial guess with distinct components such that for some 1 ≤ p ≤ ∞ and 0 < R ≤ 1/(1 + √ a), where a = (n − 1) 1/q . In the case n = 2 and p = ∞ we assume that inequality in (6.2) is strict. Then f has only simple zeros in K and there exists a root-vector ξ ∈ K n of f such that where the real function α is defined by If the inequality (6.2) is strict, then the second inequality in (6.3) is strict too.
Now, we are ready to state and prove the main result of this paper.
Suppose x (0) ∈ K n is an initial guess with distinct components such that where the function E f is defined by (6.1) and a = (n − 1) 1/q . Then f has only simple zeros in K and the Ehrlich-type iteration (1.15) is well-defined and converges to a root-vector ξ of f with order of convergence 2N + 1 and with a posteriori error estimate where the function α is defined by (6.4).
Proof Let us define R by (5.2). It is easy to calculate that R < 1/(1 + √ a) and Therefore, (6.5) can be written in the form Then it follows from Theorem 6.1 that f has only simple zeros in K and there exists a root-vector ξ ∈ K n of f such that Now Theorem 5.4 implies that the Ehrlich-type iteration (1.15) converges to ξ with order of convergence 2N + 1. It remains to prove the error estimate (6.6). Suppose that for some k ≥ 0, Then it follows from Theorem 6.1 that there exists a root-vector η ∈ K n of f such that ) and From the second inequality in (6.8) and Theorem 5.4, we conclude that the Ehrlichtype iteration (1.15) converges to η. By the uniqueness of the limit, we get η = ξ. Therefore, the error estimate (6.6) follows from the first inequality in (6.8). This completes the proof.
Setting p = ∞ in Theorem 6.2, we obtain the following result.
be a polynomial of degree n ≥ 2 and N ≥ 1. Suppose x (0) ∈ K is an initial guess with distinct components such that Then f has only simple zeros in K and the Ehrlich-type iteration (1.15) is welldefined and converges to a root-vector ξ of f with order of convergence 2N + 1 and with error estimate (6.6) for p = ∞.
Setting p = 1 in Theorem 6.2 we obtain the following result.
be a polynomial of degree n ≥ 2 and N ≥ 1. Suppose x (0) ∈ K n is an initial guess with distinct components such that Then f has only simple zeros in K and the Ehrlich-type iteration (1.15) is welldefined and converges with order 2N + 1 to a root-vector ξ of f with error estimate (6.6) for p = 1.

Numerical examples
In this section, we present several numerical examples to show some applications of Theorem 6.2. Let f ∈ C[z] be a polynomial of degree n ≥ 2 and let x (0) ∈ C n be an initial guess. We show that Theorem 6.2 can be used: • to prove numerically that f has only simple zeros; • to prove numerically that N th Ehrlich-type iteration (1.15) starting from x (0) is well-defined and converges with order 2N + 1 to a root-vector of f ; • to guarantee the desired accuracy when calculating the roots of f via N th Ehrlich-type method.
In the examples below, we use the function of initial conditions E f : D → R + defined by where W f is the Weierstrass correction defined by (1.3). We consider only the case p = ∞ since the other cases are similar.
Also, we use the real function α defined by It follows from Theorem 6.2 that if there exists an integer m ≥ 0 such that 3) then f has only simple zeros and the Ehrlich-type iteration (1.15) is well-defined and converges to a root-vector ξ of f with order of convergence 2N + 1. Besides, for all k ≥ m such that the following a posteriori error estimate holds: In the examples, we apply the Ehrlich-type methods (1.15) for some N ≥ 1 using the following stopping criterion: ε k < 10 −15 and E f (x (k) ) < R (k ≥ m). (7.6) For given N we calculate the smallest m ≥ 0 which satisfies the convergence condition (7.3), the smallest k ≥ m for which the stopping criterion (7.6) is satisfied, as well as the value of ε k for the last k.
In Table 2 the values of iterations are given to 15 decimal places. The values of other quantities (R, E f (x (m) ), etc.) are given to 6 decimal places. which are taken from Zhang et al. [23]. We have R = 0.125 and E(x (0) ) = 0.506619. The results for this example are presented in Table 1. For example, we can see that for N = 10 at the first iteration we have proved that the Ehrlich-type method converges with order of convergence 21 and that at the second iteration we have calculated the zeros f with accuracy less than 10 −127 . Moreover, at the next iteration we obtain the zeros of f with accuracy less than 10 −2682 . Also, we can see that for N = 100 at the second iteration we have obtained the zeros of f with accuracy less than 10 −11450 . In Table 2, we present numerical results for Example 7.1 in the case N = 10. and Aberth's initial approximation x (0) ∈ C n given by (see Aberth [8] and Petković et al. [24]): x (0) ν = − a 1 n + r 0 exp (iθ ν ), θ ν = π n 2ν − 3 2 , ν = 1, . . . , n, (7.7) where a 1 = 1, r 0 = 2 and n = 15. We have R = 0.043061 and E(x (0) ) = 0.179999. The results for this example are presented in Table 3. For example, we can see that for N = 30 at the third iteration we have obtained the zeros of f with accuracy less than 10 −248 . Moreover, at the next iteration we get the zeros of f with accuracy less than 10 −15105 . and Abert's initial approximation (7.7) with a 1 = −120, r 0 = 20 and n = 20. We have R = 0.033867 and E(x (0) ) = 0.344409. The results foe Example 7.3 are shown in Table 4. For example, we for N = 100 at the seventh iteration we get the zeros of f with accuracy less than 10 −13776 . In the Figure 1, we present the trajectories of approximations generated by the method (1.15) for N = 30 after 6 iterations. In this example we use Abert's initial approximation (7.7) with a 1 = 0, r 0 = 2 and n = 40. We have R = 0.018685, E(x (0) ) = 0.159318. The results for Example 7.4 can be seen in Table 5.