On the convergence of highorder Ehrlichtype iterative methods for approximating all zeros of a polynomial simultaneously
 Petko D Proinov^{1}Email author and
 Maria T Vasileva^{1}
https://doi.org/10.1186/s1366001508555
© Proinov and Vasileva 2015
Received: 14 August 2015
Accepted: 5 October 2015
Published: 19 October 2015
Abstract
We study a family of highorder Ehrlichtype 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 \(N \ge1\), the Ehrlichtype 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 an a posteriori error estimate) for the Ehrlichtype 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 the maximum norm. Our semilocal convergence theorem is the first result in this direction.
Keywords
MSC
1 Introduction
1.1 The Weierstrass method and Weierstrass correction
1.2 The Ehrlich 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.
Theorem 1.1
(Proinov [11])
Theorem 1.2
(Proinov [11])
1.3 A family of highorder Ehrlichtype methods
Definition 1.3
It is easy to see that in the case \({N = 1}\) the Ehrlichtype method (1.18) coincides with the classical Ehrlich method (1.7). The order of convergence of the Ehrlichtype method (1.18) is \({2 N + 1}\).
Kjurkchiev and Andreev [16] established the following convergence result for the Ehrlichtype methods (1.18). This result and its proof can also be found in the monographs of Sendov, Andreev and Kjurkchiev [2], Section 19 and Kyurkchiev [3], Chapter 9.2).
Theorem 1.4
(Kjurkchiev and Andreev [16])
1.4 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 an a posteriori error estimate) for Ehrlichtype methods (1.18). 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 \({p = \infty}\). Furthermore, several numerical examples are provided to show some practical applications of our semilocal convergence result.
2 A general convergence theorem
Definition 2.1
([18])
If m functions \({\varphi_{1},\ldots,\varphi_{m}}\) are quasihomogeneous on J of degree \({r_{1},\ldots,r_{m}}\), then their product \({\varphi_{1} \cdots\varphi_{m}}\) is a quasihomogeneous function of degree \({r_{1} + \cdots+ r_{m}}\) on J. Note also that a function φ is quasihomogeneous of degree 0 on J if and only it is nondecreasing on J.
Definition 2.2
([17])
 (i)
φ is quasihomogeneous of degree r on J;
 (ii)
\({\varphi(t) \le t}\) for all \({t \in J}\).
The following is a sufficient condition for a gauge function of order r.
Lemma 2.3
([18])
If \({\varphi\colon J \to \mathbb{R}_{+}}\) is a quasihomogeneous function of degree \({r \ge1}\) on an interval J and \({R > 0}\) is a fixed point of φ in J, then φ is a gauge function of order r on \({[0, R]}\). Moreover, if \({r > 1}\), then function φ is a strict gauge of order r on \({J = [0, R)}\).
Definition 2.4
([17])
Definition 2.5
([17])
Let \({T \colon D \subset X \to X}\) be a map on an arbitrary set X, and \(E \colon D \to \mathbb{R}_{+}\) be a function of the initial conditions of T with a gauge function on J. Then a point \({x \in D}\) is said to be an initial point of T (with respect to E) if \({E(x) \in J}\) and all of the iterates \({T^{k}x}\) (\(k = 0,1,2,\ldots \)) are well defined and belong to D.
The following is a simple sufficient condition for initial points.
Theorem 2.6
([18])
Let \({T \colon D \subset X \to X}\) be a map on an arbitrary set X and \({E \colon D \to \mathbb{R}_{+}}\) be a function of the initial conditions of T with a gauge function φ on J. Suppose that \({x \in D}\) with \({E(x) \in J}\) implies \({Tx \in D}\). Then every point \({x_{0} \in D}\) such that \({E(x_{0}) \in J}\) is an initial points of T.
Definition 2.7
([19])
The following fixed point theorem plays an important role in our paper.
Theorem 2.8
(Proinov [19])
 (i)
The point ξ is a unique fixed point of T in the set \({U = \{ x \in D : E(x) \in J \}}\).
 (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 estimatesfor all \({k \ge0}\), where \({\lambda= \phi(E(x^{(0)}))}\) and \({\theta = \psi(E(x^{(0)}))}\).$$ \bigl\Vert x^{(k+1)}  \xi\bigr\Vert \preceq\theta \lambda^{r^{k}} \bigl\Vert x^{(k)}  \xi\bigr\Vert \quad \textit{and}\quad \bigl\Vert x^{(k)}  \xi\bigr\Vert \preceq \theta^{k} \lambda^{S_{k}(r)} \bigl\Vert x^{(0)}  \xi \bigr\Vert $$(2.8)
In the case \({\beta\equiv\phi}\), Theorem 2.8 reduces to the following result.
Corollary 2.9
([19])
 (i)
The point ξ is a unique fixed point of T in the set \({U = \{ x \in D : E(x) \in J \}}\).
 (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 estimatesfor all \({k \ge0}\), where \({\lambda= \phi(E(x^{(0)}))}\).$$ \bigl\Vert x^{(k+1)}  \xi\bigr\Vert \preceq \lambda^{r^{k}} \bigl\Vert x^{(k)}  \xi\bigr\Vert \quad \textit{and}\quad \bigl\Vert x^{(k)}  \xi\bigr\Vert \preceq \lambda^{S_{k}(r)} \bigl\Vert x^{(0)}  \xi\bigr\Vert $$(2.9)
3 Some inequalities in \(\mathbb{K}^{n}\)
In this section, we present some useful inequalities in \(\mathbb{K}^{n}\) which play an important role in the paper.
Lemma 3.1
([20])
Lemma 3.2
([19])
Lemma 3.3
([21])
Lemma 3.4
Proof
Lemma 3.5
4 Local convergence theorem of the first type
Let \({a > 0}\) and \({b \ge1}\). Throughout this section, we define the function ϕ and the real number R by (1.10) and (1.11), respectively. It is easy to show that R is the smallest positive solution of the equation \({\phi(t) = 1}\). Note that ϕ is an increasing function which maps \([0,R]\) onto \([0,1]\). Besides, ϕ is quasihomogeneous of degree 2 on \({[0,R]}\). In the next definition, we introduce a sequence of such functions.
Definition 4.1
Proof of the correctness of Definition 4.1
Definition 4.2
In the next lemma, we present some properties of the functions \(\phi_{N}\) and \(\varphi_{N}\).
Lemma 4.3
 (i)
\(\phi_{N}\) is a quasihomogeneous function of degree 2N on \({[0,R]}\);
 (ii)
\({\phi_{N+1}(t) \le\phi(t) \phi_{N}(t)}\) for every \({t \in[0,R]}\);
 (iii)
\({\phi_{N+1}(t) \le\phi_{N}(t)}\) for every \({t \in[0,R]}\);
 (iv)
\({\phi_{N}(t) \le\phi(t)^{N}}\) for every \({t \in[0,R]}\);
 (v)
\(\varphi_{N}\) is a gauge function of order \({2 N + 1}\) on \({[0,R]}\).
Proof
Lemma 4.4
 (i)If \({x \mathbin{\#} T^{(N)}(x)}\), thenwhere \({\sigma_{i} \in \mathbb{K}}\) is defined by$$ \frac{f'(x_{i})}{f(x_{i})}  \sum_{j \ne i} \frac{1}{x_{i}  T_{j}^{(N)}(x)} = \frac{1  \sigma_{i}}{x_{i} \xi_{i}}, $$(4.5)$$ \sigma_{i} = (x_{i}  \xi_{i}) \sum_{j \ne i} \frac{ T_{j} ^{(N)}(x)  \xi_{j}}{(x_{i}  \xi_{j})(x_{i}  T_{j} ^{(N)}(x))} . $$(4.6)
 (ii)If \({x \in D_{N + 1}}\), then$$ T^{(N + 1)}_{i}(x)  \xi_{i} =  \frac{ \sigma_{i}}{1  \sigma_{i}} (x_{i}  \xi_{i}). $$(4.7)
Proof
Lemma 4.5
Proof
We shall prove statements by induction on N. If \({N = 0}\), then (4.10) holds trivially. Assume that (4.10) holds for some \({N \ge0}\).
Now we are ready to state the main result of this section. In the case \({N = 1}\) this result coincides with Theorem 1.1.
Theorem 4.6
Proof
We apply Corollary 2.9 to the iteration function \({T^{(N)} \colon D_{N} \subset \mathbb{K}^{n} \to \mathbb{K}^{n}}\) defined by Definition 1.3 and to the function \({E \colon \mathbb{K}^{n} \to \mathbb{R}_{+}}\) defined by (4.1). Let \({J = [0, R)}\). It follows from Lemma 4.5, Lemma 4.3(v), and Lemma 2.3 that E is a function of the initial conditions of \(T^{(N)}\) with a strict gauge function \(\varphi_{N}\) of order \({r = 2 N + 1}\) on J. Since ξ is a root vector of f, then \({E(\xi) = 0 \in J}\). It follows from Lemma 4.5 that \({T^{(N)}}\) is an iterated contraction at a point ξ with respect to E and with control function \(\phi_{N}\). The fact that \({x^{(0)}}\) is an initial point of \({T^{(N)}}\) follows from Lemma 4.5 and Theorem 2.6. Hence, all the assumptions of Corollary 2.9 are satisfied, and the statement of Theorem 4.6 follows from it. □
Corollary 4.7
Let \({0 < h < 1}\) be a given number. Solving the equation \({\phi(t) = h^{2}}\) in the interval \({(0,R)}\), we can reformulate Corollary 4.7 in the following equivalent form.
Corollary 4.8
Remark 4.9
5 Local convergence theorem of the second type
Definition 5.1
Proof of the correctness of Definition 5.1
Lemma 5.2
 (i)
\(\beta_{N}\) is a quasihomogeneous of degree 2N on \({[0,R]}\);
 (ii)
\({\beta_{N}(t) = \phi_{N}(t) \psi_{N}(t)}\) for every \({t \in[0,R]}\);
 (iii)
\({\beta_{N+1}(t) \le\beta_{N}(t)}\) for every \({t \in[0,R]}\);
 (iv)
\({\psi_{N + 1}(t) \ge\psi_{N}(t)}\) for every \({t \in[0,R]}\).
Proof
Lemma 5.3
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 \(\mathbb{K}\). We divide the proof into two steps.
Step 1. In this step, we prove \({x \in D_{N}}\) 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 \({x \in D_{N}}\) and the first inequality in (5.8) hold for some \({N \ge1}\).
Now we are able to state the main result of this section. In the case when \({N = 1}\) and \({p = \infty}\) this result reduces to Theorem 1.2.
Theorem 5.4
Proof
We apply Theorem 2.8 to the iteration function \({T^{(N)} \colon D_{N} \subset \mathbb{K}^{n} \to \mathbb{K}^{n}}\) together with the function \({E \colon D_{N} \to \mathbb{R}_{+}}\) defined by (5.1).
It follows from Lemma 5.3 and Lemma 4.3(v) that E is a function of the initial conditions of \({T^{(N)}}\) with gauge function \(\varphi_{N}\) of order \({r = 2 N + 1}\) on the interval \({J = [0, R]}\).
From Lemma 5.3, we see that \({T^{(N)}}\) is an iterated contraction at ξ with respect to E and with control function \({\beta_{N}}\). Also, it is easy to see that the functions \(\beta_{N}\), \(\phi_{N}\), \(\psi _{N}\), and \(\varphi_{N}\) have the properties (2.5), (2.6) and (2.7).
Now the statement of Theorem 5.4 follows from Theorem 2.8. □
6 Semilocal convergence theorem
Recently Proinov [22] has shown that there is a relationship between local and semilocal theorems for simultaneous rootfinding 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 5.4 we can obtain a semilocal convergence theorem for Ehrlichtype methods (1.18) under computationally verifiable initial conditions. For this purpose we need the following result.
Theorem 6.1
(Proinov [22])
Now, we are ready to state and prove the main result of this paper.
Theorem 6.2
Proof
Setting \({p = \infty}\) in Theorem 6.2, we obtain the following result.
Corollary 6.3
Setting \({p = 1}\) in Theorem 6.2 we obtain the following result.
Corollary 6.4
7 Numerical examples

to prove numerically that f has only simple zeros;

to prove numerically that Nth Ehrlichtype iteration (1.18) 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 the Nth Ehrlichtype method.
For given N we calculate the smallest \(m \ge0\) which satisfies the convergence condition (7.3), the smallest \(k \ge m\) for which the stopping criterion (7.6) is satisfied, as well as the value of \(\varepsilon_{k}\) for the last k.
In Table 2 the values of iterations are given to 15 decimal places. The values of other quantities (\(\mathscr{R}\), \(E_{f}(x^{(m)})\), etc.) are given to six decimal places.
Numerical calculations are made using the software package Mathematica [23].
Example 7.1
Values of m , k , and \(\pmb{\varepsilon_{k}}\) for Example 7.1 ( \(\pmb{{\mathscr{R} = 0.125}}\) )
N  m  \(\boldsymbol{E_{f}(x^{(m)})}\)  \(\boldsymbol{\varepsilon_{m}}\)  k  \(\boldsymbol{\varepsilon_{k}}\)  \(\boldsymbol{\varepsilon_{k + 1}}\) 

1  2  0.010032  1.457548 × 10^{−2}  4  4.385760 × 10^{−21}  8.919073 × 10^{−63} 
2  1  0.067725  1.242914 × 10^{−1}  3  1.347060 × 10^{−38}  7.284576 × 10^{−193} 
3  1  0.015716  2.300541 × 10^{−2}  3  1.825502 × 10^{−106}  5.054741 × 10^{−744} 
4  1  0.002730  3.887455 × 10^{−3}  2  1.330837 × 10^{−25}  3.543773 × 10^{−230} 
5  1  0.001215  1.722883 × 10^{−3}  2  4.720064 × 10^{−37}  2.999643 × 10^{−407} 
6  1  0.000206  2.927439 × 10^{−4}  2  1.060096 × 10^{−50}  5.523501 × 10^{−657} 
7  1  0.000081  1.155284 × 10^{−4}  2  6.261239 × 10^{−67}  3.252761 × 10^{−1,002} 
8  1  0.000014  1.986052 × 10^{−5}  2  6.080606 × 10^{−85}  3.570038 × 10^{−1,439} 
9  1  0.000005  7.910775 × 10^{−6}  2  1.309022 × 10^{−105}  1.170454 × 10^{−2,002} 
10  1  0.000000  1.366899 × 10^{−6}  2  4.301615 × 10^{−128}  8.477451 × 10^{−2,683} 
100  1  0.000000  1.820743 × 10^{−57}  1  1.820743 × 10^{−57}  3.460397 × 10^{−11,451} 
Numerical results for Example 7.1 in the case \(\pmb{N = 10}\)
k  \(\boldsymbol{x_{1}^{(k)}}\)  \(\boldsymbol{x_{2}^{(k)}}\) 

0  0.5 + 0.5i  −1.36 + 0.42i 
1  1.000000380419496 + 0.000000816235730i  −1.000000220051461 − 0.000000495915480i 
2  1.000000000000000 + 0.000000000000000i  −1.000000000000000 + 0.000000000000000i 
k  \(\boldsymbol{x_{1}^{(k)}}\)  \(\boldsymbol{x_{2}^{(k)}}\) 
0  −0.25 + 1.28i  0.46 − 1.37i 
1  0.000000277962637 + 0.999999578393062i  −0.000000314533436 − 0.999998669784542i 
2  0.000000000000000 + 1.000000000000000i  0.000000000000000 − 1.000000000000000i 
Example 7.2
Values of m , k , and \(\pmb{\varepsilon_{k}}\) for Example 7.2 ( \(\pmb{{\mathscr{R} = 0.043061}}\) )
N  m  \(\boldsymbol{E_{f}(x^{(m)})}\)  \(\boldsymbol{\varepsilon_{m}}\)  k  \(\boldsymbol{\varepsilon_{k}}\)  \(\boldsymbol{\varepsilon_{k + 1}}\) 

1  6  0.036897  3.187918 × 10^{−2}  9  3.967908 × 10^{−36}  5.304009 × 10^{−106} 
2  5  0.000003  1.182714 × 10^{−6}  6  6.112531 × 10^{−28}  2.230412 × 10^{−134} 
3  4  0.000064  2.475020 × 10^{−5}  5  2.446120 × 10^{−29}  2.722168 × 10^{−197} 
4  4  0.000000  1.550670 × 10^{−11}  5  3.838741 × 10^{−93}  1.589981 × 10^{−827} 
5  3  0.005793  2.415745 × 10^{−3}  4  9.532339 × 10^{−24}  8.487351 × 10^{−248} 
6  3  0.000293  1.127450 × 10^{−4}  4  9.565008 × 10^{−45}  1.725858 × 10^{−565} 
7  3  0.000005  2.173198 × 10^{−6}  4  4.018844 × 10^{−77}  6.737932 × 10^{−1,138} 
8  3  0.000000  1.562375 × 10^{−8}  4  1.162424 × 10^{−123}  1.291370 × 10^{−2,080} 
9  3  0.000000  4.092421 × 10^{−11}  4  4.245137 × 10^{−187}  1.373908 × 10^{−3,530} 
10  3  0.000000  3.904607 × 10^{−14}  4  4.643262 × 10^{−270}  2.543247 × 10^{−5,644} 
30  2  0.000055  2.129417 × 10^{−5}  3  5.721566 × 10^{−249}  2.377023 × 10^{−15,106} 
Example 7.3
Values of m , k , and \(\pmb{\varepsilon_{k}}\) for Example 7.3 ( \(\pmb{{\mathscr{R} = 0.033867}}\) )
N  m  \(\boldsymbol{E_{f}(x^{(m)})}\)  \(\boldsymbol{\varepsilon_{m}}\)  k  \(\boldsymbol{\varepsilon_{k}}\)  \(\boldsymbol{\varepsilon_{k + 1}}\) 

1  18  0.000060  6.095859 × 10^{−5}  20  1.620028 × 10^{−38}  4.276235 × 10^{−114} 
2  12  0.015335  2.153155 × 10^{−2}  14  1.095084 × 10^{−46}  1.779476 × 10^{−230} 
3  10  0.018005  2.769333 × 10^{−2}  12  8.917532 × 10^{−86}  4.482714 × 10^{−596} 
4  9  0.005514  6.130790 × 10^{−3}  10  4.221856 × 10^{−21}  7.250879 × 10^{−184} 
5  9  0.000000  1.159694 × 10^{−15}  10  5.021359 × 10^{−165}  5.118016 × 10^{−1,808} 
6  8  0.000237  2.386016 × 10^{−4}  9  8.455240 × 10^{−48}  1.280870 × 10^{−612} 
7  8  0.000000  2.723047 × 10^{−17}  8  2.723047 × 10^{−17}  8.926059 × 10^{−249} 
8  7  0.018995  2.934241 × 10^{−2}  8  2.885374 × 10^{−30}  4.152134 × 10^{−503} 
9  7  0.002180  2.274734 × 10^{−3}  8  3.792876 × 10^{−51}  1.140751 × 10^{−958} 
10  7  0.000000  5.185525 × 10^{−7}  8  1.620086 × 10^{−132}  2.936276 × 10^{−2,768} 
30  5  0.000181  1.821419 × 10^{−4}  6  1.395923 × 10^{−226}  1.902920 × 10^{−13,777} 
Example 7.4
Values of m , k , and \(\pmb{\varepsilon_{k}}\) for Example 7.4 ( \(\pmb{{\mathscr{R} = 0.018685}}\) )
N  m  \(\boldsymbol{E_{f}(x^{(m)})}\)  \(\boldsymbol{\varepsilon_{m}}\)  k  \(\boldsymbol{\varepsilon_{k}}\)  \(\boldsymbol{\varepsilon_{k + 1}}\) 

1  15  0.007235  1.588799 × 10^{−3}  17  1.057241 × 10^{−18}  1.574672 × 10^{−52} 
2  11  0.000001  1.731641 × 10^{−7}  12  2.763909 × 10^{−30}  2.863869 × 10^{−144} 
3  9  0.000026  4.171842 × 10^{−6}  10  5.167701 × 10^{−32}  2.328540 × 10^{−213} 
4  8  0.000032  5.141616 × 10^{−6}  9  7.830010 × 10^{−40}  3.487627 × 10^{−344} 
5  7  0.010766  2.954474 × 10^{−3}  8  1.468181 × 10^{−20}  2.870206 × 10^{−208} 
6  7  0.000002  4.201055 × 10^{−7}  8  7.096655 × 10^{−71}  6.481892 × 10^{−900} 
7  7  0.000000  9.445503 × 10^{−15}  8  3.169914 × 10^{−196}  2.445585 × 10^{−2,918} 
8  6  0.010675  2.911647 × 10^{−3}  7  8.218559 × 10^{−31}  3.538870 × 10^{−495} 
9  6  0.000281  4.462548 × 10^{−5}  7  2.324176 × 10^{−64}  1.205364 × 10^{−1,190} 
10  6  0.000000  1.231259 × 10^{−7}  7  1.392265 × 10^{−124}  1.840079 × 10^{−2,580} 
30  5  0.000000  2.416285 × 10^{−34}  5  2.416285 × 10^{−34}  1.294365 × 10^{−1,987} 
Declarations
Acknowledgements
This research is supported by the project NI15FMI004 of Plovdiv University.
Open Access This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.
Authors’ Affiliations
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