On the Q-Convergence and Dynamics of a Modified Weierstrass Method for the Simultaneous Extraction of Polynomial Zeros

Abstract

In the present paper, we prove a new local convergence theorem with initial conditions and error estimates that ensure the Q-quadratic convergence of a modification of the famous Weierstrass method. Afterward, we prove a semilocal convergence theorem that is of great practical importance owing to its computable initial condition. The obtained theorems improve and complement all existing such kind of convergence results about this method. At the end of the paper, we provide three numerical examples to show the applicability of our semilocal theorem to some physics problems. Within the examples, we propose a new algorithm for the experimental study of the dynamics of the simultaneous methods and compare the convergence and dynamical behaviors of the modified and the classical Weierstrass methods.

1. Introduction

In the present paper, $\mathbb{C}\left[z\right]$ shall denote the ring of univariate polynomials over $\mathbb{C}$. Furthermore, the vector space ${\mathbb{C}}^n$ is supplied with p-norm ${\parallel ·\parallel }_p$ for $1\le p\le \infty$ and with vector norm $\parallel ·\parallel$ defined by

$${\parallel x\parallel }_p={\left(\sum _{i=1}^n{\left|x_i\right|}^p\right)}^{1/p}\mathrm{and}\parallel x\parallel =\left(\right|x_1|,\dots ,|x_n\left|\right).$$

Also, $\xi \in {\mathbb{C}}^n$ shall be called a root vector of f if

$$f\left(z\right)=a_0\prod _{i=1}^n(z-{\xi }_i)\mathrm{for}\mathrm{all}z\in \mathbb{C},$$

where $a_0$ is the leading coefficient of f.

In 1891, Weierstrass [1] derived his famous iterative method for approximating the root vector of f. It can be defined in ${\mathbb{C}}^n$ by the following iteration:

$$x^{(k+1)}=x^{\left(k\right)}-W_f\left(x^{\left(k\right)}\right),k=0,1,2,\dots ,$$

(1)

where the Weierstrass’ correction $W_f:\mathcal{D}\subset {\mathbb{C}}^n\to {\mathbb{C}}^n$ is defined by

$$W_f\left(x\right)=(W_1\left(x\right),\dots ,W_n\left(x\right))\mathrm{with}{W}_i\left(x\right)={\displaystyle \frac{f\left(x_i\right)}{a_0{\prod }_{j\ne i}(x_i-x_j)}}(i,j\in I_n).$$

(2)

Here, and in the whole paper, $I_n=\{1,2,\dots ,n\}$ and $\mathcal{D}$ denotes the set of all vectors in ${\mathbb{C}}^n$ with pairwise distinct components, that is, $\mathcal{D}=\left\{x\in {\mathbb{C}}^n:x_i\ne x_j\mathrm{whenever}i\ne j\right\}.$

In 1962, Dochev [2] obtained a local convergence theorem about Weierstrass’ method (1) with initial condition of the type $\parallel x^{\left(0\right)}{-\xi \parallel }_{\infty }<C(n,\xi ),$ where $C(n,\xi )$ is a real constant dependent on the degree n and the root vector $\xi$ of the polynomial f. In 1991, Wang and Zhao ([3], Theorem 2.5) managed to prove a local convergence theorem under a weaker initial condition of the type $\parallel x^{\left(0\right)}{-\xi \parallel }_{\infty }<C(n,x^{\left(0\right)}),$ i.e., with computable right hand side. In light of ([4], Definition 2.1), these two types of initial conditions will be further called initial condition of the first kind and initial condition of the second kind, respectively. We have to note that, in 2016, all convergence results about Weierstrass’ method (1) were improved by Proinov [5].

Since Duran [6] and Dochev [2], Weierstrass’ method (1) draws a great interest among the mathematical community as a very efficient second-order simultaneous method. In particular, it has been used for the construction of many higher-order simultaneous methods (see Ivanov [7] and references therein) but so far there exist only a few second-order such methods in the literature and one of them is the following modification of Weierstrass’ method (Nedzhibov [8]):

$$x^{(k+1)}=T\left(x^{\left(k\right)}\right),k=0,1,2,\dots ,$$

(3)

where the iteration function $T:D\subset {\mathbb{C}}^n\to {\mathbb{C}}^n$ is defined by

$${\displaystyle T\left(x\right)=(T_1\left(x\right),\dots ,T_n\left(x\right))\mathrm{and}{T}_i\left(x\right)=\frac{x_i^2}{x_i+W_i\left(x\right)}(i\in I_n),}$$

(4)

with domain:

$$D=\left\{x\in \mathcal{D}:x_i+W_i\left(x\right)\ne 0\mathrm{for}\mathrm{all}i\in I_n\right\}.$$

(5)

Local convergence theorems with initial conditions of the first and the second kind for the method (3) have been obtained by Nedzhibov in ([8], Theorem 3.1) and [9] (see Theorem 5) while a local convergence theorem with initial condition of the first kind that provides the Q-convergence of the method (3) has been established in [10]. In [9], using the methods of Proinov [4], a semilocal convergence theorem (Theorem 6) about the iteration (3) has been proved as well. Very recently, all convergence results about the method (3) including Theorems 5 and 6 have been improved in [11]. For a more detailed overview of Weierstrass’ method and its modifications, we refer the reader to Sendov, Andreev and Kyurkchiev ([12], Ch. IV), Wang and Zhao [13], Kyurkchiev ([14], Ch. 1), Petković [15], Hristov, Kyurkchiev and Iliev [16], Proinov [5], and references therein.

The aim of the present paper is two fold:

• Theoretical: To improve and complement all existing local and semilocal convergence theorems about the modified Weierstrass method (3);
• Practical: (1) To apply our semilocal convergence result for the numerical proof of the Q-quadratic convergence of the modified Weierstrass method (3) and for the numerical guarantee of the accuracy of approximations of the roots of three polynomials two of which arise from some very important physics problems. (2) To provide a new algorithm for the experimental study of the dynamics of the simultaneous methods, which can be used for influencing the convergence of a simultaneous method, and to use it for the comparison of the convergence and dynamics of the modified Weierstrass method (3) and the famous Weierstrass’ method (1).

More precisely, in this paper we prove a new local convergence theorem (Theorem 1), under initial condition of the second kind, which ensures the Q-convergence of the modified Weierstrass iteration (3) and refines the local convergence result of [11]. Afterward, using the methods of [4] we transform Theorem 1 into a new semilocal convergence theorem (Theorem 2), which is of significant practical importance because of its computable initial condition. In Section 4, we show that Theorem 1 improves Theorems 5 and 6 and we give some examples to show that Theorem 2 improves the semilocal result of [11] (Theorem 7). In Section 5, we conduct three numerical examples to apply our semilocal theorem for the numerical proof of the convergence and the guaranteed accuracy of approximations at each iteration for the modified Weierstrass method (3) when applied to some physics problems. In any example, by implementing a new algorithm for experimental study of the dynamics of the simultaneous methods, we depict the dynamical planes of the two methods to compare their dynamical behavior. It is worth noting that plenty of papers have been devoted to the study of the dynamics of iterative methods for the individual approximation of polynomial and non-polynomial zeros (see, e.g., [17,18,19,20] and references therein) but only a few about the dynamics of simultaneous methods are available [21,22,23,24] and only for low degree polynomials.

2. Main Results

From now on, we endow ${\mathbb{R}}^n$ with a partial ordering ⪯ defined by $x⪯y$ if and only if $x_i\le y_i$ for each $i\in I_n$. For a given p ($1\le p\le \infty$), we define q by

$$1\le q\le \infty \mathrm{with}1/p+1/q=1$$

and for a given $n\ge 2$, we use the denotations

$$a={(n-1)}^{1/q},b=2^{1/q}.$$

(6)

Furthermore, we define the function $\Delta :{\mathbb{C}}^n\to {\mathbb{R}}^n$ by

$$\Delta \left(x\right)=({\Delta }_1\left(x\right),\dots ,{\Delta }_n\left(x\right))\mathrm{with}{\Delta }_i\left(x\right)=min\left\{\right|x_i|,d_i\left(x\right)\}(i\in I_n),$$

(7)

where the function $d:{\mathbb{C}}^n\to {\mathbb{R}}^n$ is defined by

$$d\left(x\right)=(d_1\left(x\right),\dots ,d_n\left(x\right))\mathrm{with}{d}_i\left(x\right)=\underset{j\ne i}{min}|x_i-x_j|(i,j\in I_n).$$

(8)

Also, we define the functions $\delta ,\tilde{\Delta }:{\mathbb{C}}^n\to {\mathbb{R}}_{+}$ by

$$\delta \left(x\right)=\underset{i\ne j}{min}|x_i-x_j|\mathrm{and}\tilde{\Delta }\left(x\right)=min{\Delta }_i\left(x\right)(i\in I_n).$$

(9)

Finally, for a vector $x\in {\mathbb{C}}^n$ and a vector $y\in {\mathbb{R}}^n$ with nonzero components, we use the denotation $x/y$ for the vector in ${\mathbb{R}}^n$ defined by

$${\displaystyle \frac{x}{y}}=\left({\displaystyle \frac{|x_1|}{y_1}},\cdots ,{\displaystyle \frac{|x_n|}{y_n}}\right).$$

Let $f\in \mathbb{C}\left[z\right]$ be a polynomial of degree $n\ge 2$ with only simple roots in $\mathbb{C}$, $\xi$ be a root vector of f and $x\in \mathcal{D}$ be a vector with nonzero components. In this paper, we study the convergence of the modified Weierstrass method (3) under the following two functions:

$$E\left(x\right)={∥{\displaystyle \frac{x-\xi }{\Delta \left(x\right)}}∥}_p\mathrm{and}{E}_f\left(x\right)={∥{\displaystyle \frac{W_f\left(x\right)}{\Delta \left(x\right)}}∥}_p(1\le p\le \infty ),$$

(10)

where $W_f$ is Weierstrass’ correction defined by (2).

For the proposes of the main results, we define the functions $\gamma :[0,\infty )\to {\mathbb{R}}_{+}$ and $\beta :[0,\eta )\to {\mathbb{R}}_{+}$ by

$$\gamma \left(t\right)={\left(1+{\displaystyle \frac{at}{n-1}}\right)}^{n-1}\mathrm{and}\beta \left(t\right)={\displaystyle \frac{\gamma \left(t\right)-1+t\gamma \left(t\right)}{1-t\gamma \left(t\right)}},$$

(11)

where $\eta$ is the unique solution of the equation $t\gamma \left(t\right)=1$ in $[0,\infty )$.

The following local convergence theorem is our first main result.

Theorem 1. 

Let $f\in \mathbb{C}\left[z\right]$ be a polynomial of degree $n\ge 2$ with only simple roots and ξ be a root vector of f. Suppose $x^{\left(0\right)}\in \mathcal{D}$ is an initial approximation with nonzero components satisfying

$$\Phi \left(E\left(x^{\left(0\right)}\right)\right)\le 2,$$

(12)

where E is defined by (10) and the function Φ is defined by

$$\Phi \left(t\right)=(1+(2+b\left)t\right)\gamma \left(t\right)$$

(13)

with γ defined by (11). Then, the modified Weierstrass method (3) is well defined and converges Q-quadratically to ξ with error estimates for all $k\ge 0$

$$\parallel x^{(k+1)}-\xi \parallel ⪯\theta {\lambda }^{2^k}\parallel x^{\left(k\right)}-\xi \parallel \mathit{and}\parallel x^{\left(k\right)}-\xi \parallel ⪯{\theta }^k{\lambda }^{2^k-1}\parallel x^{\left(0\right)}-\xi \parallel ,$$

(14)

where $\lambda =\varphi \left(E\left(x^{\left(0\right)}\right)\right)$, $\theta =\psi \left(E\left(x^{\left(0\right)}\right)\right)$, $\varphi =\beta /\psi$ and $\psi \left(t\right)=1-bt(1+\beta (t\left)\right)$ with β defined by (11). Also, we have the following a posteriori error estimate:

$$\parallel x^{(k+1)}{-\xi \parallel }_p\le c_k{\parallel x^{\left(k\right)}-\xi \parallel }_p^2\mathit{for}\mathit{all}k\ge 0,$$

where the sequence ${\left(c_k\right)}_{k=0}^{\infty }$ is defined by $c_k=\beta \left(E\left(x^{\left(k\right)}\right)\right)/\left(E\left(x^{\left(k\right)}\right)\delta \left(x^{\left(k\right)}\right)\right)$.

Before stating our next main result, let us define the number $\tau$ by

$$\tau ={\displaystyle \frac{1}{{(1+\sqrt{a})}^2}},$$

(15)

and the functions $\alpha ,h,\Omega :[0,\tau ]\to {\mathbb{R}}_{+}$ by

$$\alpha \left(t\right)={\displaystyle \frac{2}{1-(a-1)t+\sqrt{{(1-(a-1)t)}^2-4t}}},h\left(t\right)=t\alpha \left(t\right)\mathrm{and}\Omega \left(t\right)=\Phi \left(h\left(t\right)\right),$$

(16)

where $\Phi$ is defined by (13).

The following semilocal convergence theorem is our second main result. This theorem is of great practical importance owing to its computationally verifiable initial condition.

Theorem 2. 

Let $f\in \mathbb{C}\left[z\right]$ be a polynomial of degree $n\ge 2$ and let $x^{\left(0\right)}\in \mathcal{D}$ be an initial approximation with nonzero components satisfying

$$E_f\left(x^{\left(0\right)}\right)<\tau \mathit{and}\Omega \left(E_f\left(x^{\left(0\right)}\right)\right)\le 2,$$

(17)

where τ is the number given in (15), $E_f$ is defined by (10) and Ω is defined by (16). Then, f has only simple zeros in $\mathbb{C}$ and the modified Weierstrass method (3) is well defined and converges Q-quadratically to a root vector of f.

3. Proof of the Main Results

Recently, Proinov [25] has provided general local convergence theorems about the Picard iteration

$$x^{(k+1)}=T\left(x^{\left(k\right)}\right),k=0,1,2,\dots ,$$

(18)

where T is an iteration function. We start this section by recalling some crucial definitions and results from [5,25] that are needed to prove of our main results.

Definition 1 

([25]). A function $f:J\to {\mathbb{R}}_{+}$ is said to be quasi-homogeneous of exact degree $m\ge 0$ on an interval $J\subset {\mathbb{R}}_{+}$ containing 0 if it satisfies the following two conditions:

(i)

${\displaystyle f\left(\lambda t\right)\le {\lambda }^{m}f\left(t\right)\mathit{for}\mathit{all}\lambda \in [0,1]\mathit{and}t\in J;}$

(ii)

${\displaystyle \underset{t\to 0^{+}}{lim}f\left(t\right)/t^m\ne 0.}$

Definition 2 

([25], Definition 10). A function $F:D\subset {\mathbb{C}}^n\to {\mathbb{C}}^n$ is said to be an iteration function of second kind at a point $\xi \in {\mathbb{C}}^n$ if there is a nonzero quasi-homogeneous function $\beta :J\to {\mathbb{R}}_{+}$ such that for each vector $x\in \mathcal{D}$ with $E\left(x\right)\in J$, the following conditions hold:

$$x\in D\mathit{and}\parallel F\left(x\right)-\xi \parallel ⪯\beta \left(E\right(x\left)\right)\parallel x-\xi \parallel ,$$

(19)

where the function $E:\mathcal{D}\to {\mathbb{R}}_{+}$ is defined by (10). The function β is said to be a control function of F.

Lemma 1 

([25], Example 1). Let $n\in \mathbb{N}$ and f be a quasi-homogeneous function of exact degree $r>0$ on an interval J; then, the function

$$g\left(t\right)={(1+f\left(t\right))}^n-1$$

is also a quasi-homogeneous function of exact degree r on J.

Lemma 2 

([5], Proposition 5.5). If $u\in {\mathbb{C}}^n$ and $1\le p\le \infty$, then

$$\left|\prod _{j=1}^n(1+u_j)\right|\le {\left(1+{\displaystyle \frac{{\parallel u\parallel }_p}{n^{1/p}}}\right)}^{n}and\left|\prod _{j=1}^n(1+u_j)-1\right|\le {\left(1+{\displaystyle \frac{{\parallel u\parallel }_p}{n^{1/p}}}\right)}^n-1.$$

To prove Theorem 1, we shall apply the following general local convergence theorem of Proinov [25].

Theorem 3 

([25], Theorem 4). Let $T:D\subset {\mathbb{C}}^n\to {\mathbb{C}}^n$ be an iteration function of second kind at a point $\xi \in {\mathbb{C}}^n$ with a nonzero control function $\beta :J\to {\mathbb{R}}_{+}$ of exact degree $m\ge 0$ and let $x^{\left(0\right)}\in {\mathbb{C}}^n$ be an initial approximation with distinct components such that

$$E\left(x^{\left(0\right)}\right)\in Jand\Psi \left(E\left(x^{\left(0\right)}\right)\right)\ge 0,$$

(20)

where $\Psi :J\to \mathbb{R}$ is defined by

$$\Psi \left(t\right)=1-bt-\beta \left(t\right)(1+bt),$$

(21)

with $b=2^{1/q}$. Then, ξ is a fixed point of T with pairwise distinct components and Picard iteration (18) is well defined and converges to ξ with Q-order $r=m+1$ and with error estimates for all $k\ge 0$:

$$\parallel x^{\left(k\right)}-\xi \parallel ⪯{\theta }^k{\lambda }^{S_k\left(r\right)}\parallel x^{\left(0\right)}-\xi \parallel and\parallel x^{(k+1)}-\xi \parallel ⪯\theta {\lambda }^{r^k}\parallel x^{\left(k\right)}-\xi \parallel$$

(22)

where $\lambda =\varphi \left(E\left(x^{\left(0\right)}\right)\right)$, $\theta =\psi \left(E\left(x^{\left(0\right)}\right)\right)$ and the functions ψ and ϕ are defined by

$$\psi \left(t\right)=1-bt(1+\beta (t\left)\right)and\varphi \left(t\right)=\beta \left(t\right)/\psi \left(t\right).$$

Also, the following a posteriori error estimate holds for all $k\ge 0$:

$$\parallel x^{(k+1)}{-\xi \parallel }_p\le c_k{\parallel x^{\left(k\right)}-\xi \parallel }_p^r,$$

where the sequence ${\left(c_k\right)}_{k=0}^{\infty }$ is defined by $c_k=\sigma \left(E\left(x^{\left(k\right)}\right)\right)/\delta {\left(x^{\left(k\right)}\right)}^m$ and $\sigma :J\to {\mathbb{R}}_{+}$ is defined by $\sigma \left(0\right)=0$ and $\sigma \left(t\right)=\beta \left(t\right)/t^m$ for $t>0$.

The aim of the next lemma is to provide a sufficient condition for the modified Weierstrass iteration function (4) to be an iteration function of second kind at $\xi \in {\mathbb{C}}^n$ with control function of first degree.

Lemma 3. 

Let $f\in \mathbb{C}\left[z\right]$ be a polynomial of degree $n\ge 2$ with only simple roots and $\xi \in {\mathbb{C}}^n$ be a root vector of f. Then, T defined by (4) is an iteration function of second kind at a point $\xi \in {\mathbb{C}}^n$ with control function β of exact degree $m=1$ defined by (11).

Proof. 

It follows from Lemma 1 and the properties of the quasi-homogeneous functions (see, e.g., ([25], Proposition 1)) that the function $\beta$ defined by (11) is quasi-homogeneous of exact degree $m=1$ in $[0,\eta )$. Now pick a vector $x\in \mathcal{D}$ such that

$$E\left(x\right)<\eta ,$$

(23)

where E is defined by (10). We need to prove that

$$x\in D\mathrm{and}|T_i\left(x\right)-{\xi }_i|\le \beta \left(E\left(x\right)\right)|x_i-{\xi }_i|\mathrm{for}\mathrm{each}i\in I_n.$$

(24)

In accordance with (5), to prove that x belongs to D we have to prove that

$$x_i+W_i\left(x\right)\ne 0\mathrm{for}\mathrm{each}i\in I_n.$$

(25)

Let $i\in I_n$ be fixed. From the assumptions of the lemma and the definition of $\Delta$, we have

$$|x_i|\ge {\Delta }_i\left(x\right)>0\mathrm{and}\left|{\displaystyle \frac{x_j-{\xi }_j}{x_i-x_j}}\right|\le {\displaystyle \frac{|x_j-{\xi }_j|}{{\Delta }_j\left(x\right)}}\le E\left(x\right)\mathrm{for}\mathrm{all}i\ne j.$$

(26)

So, from the definition of $W_i\left(x\right)$, the last inequality of (26) and using the first inequality of Lemma 2 with $u_j=(x_j-{\xi }_j)/(x_i-x_j)$, we obtain

$$\begin{array}{ccc}\hfill |W_i\left(x\right)|& =& |x_i-{\xi }_i|\left|\prod _{j\ne i}{\displaystyle \frac{x_i-{\xi }_j}{x_i-x_j}}\right|=|x_i-{\xi }_i|\left|\prod _{j\ne i}\left(1+{\displaystyle \frac{x_j-{\xi }_j}{x_i-x_j}}\right)\right|\hfill \\ & \le & |x_i-{\xi }_i|{\left(1+{\displaystyle \frac{E\left(x\right)}{{(n-1)}^{1/p}}}\right)}^{n-1}=|x_i-{\xi }_i|\gamma \left(E\left(x\right)\right).\hfill \end{array}$$

(27)

Now, from the triangle inequality, (26), (27) and (23), we obtain

$$\begin{array}{ccc}\hfill |x_i+W_i\left(x\right)|& \ge & |x_i|-|W_i\left(x\right)|\ge {\Delta }_i\left(x\right)-|x_i-{\xi }_i|\gamma \left(E\left(x\right)\right)\hfill \\ & \ge & (1-E\left(x\right)\gamma \left(E\left(x\right)\right)){\Delta }_i\left(x\right)>0\hfill \end{array}$$

which leads to (25).

It remains to prove the second part of (24) for each $i\in I_n$. If $x_i={\xi }_i$ for some i, then the inequality of (24) turns to equality. Suppose $x_i\ne {\xi }_i$. From (4), we obtain

$${\displaystyle T_i\left(x\right)-{\xi }_i=\frac{x_i^2}{x_i+W_i\left(x\right)}-{\xi }_i=x_i-{\xi }_i-\frac{W_i\left(x\right)}{{\displaystyle 1+\frac{W_i\left(x\right)}{x_i}}}={\sigma }_i(x_i-{\xi }_i),}$$

(28)

where ${\sigma }_i$ is defined by

$${\sigma }_i={\displaystyle \frac{{\displaystyle 1-\frac{W_i\left(x\right)}{x_i-{\xi }_i}+\frac{W_i\left(x\right)}{x_i}}}{{\displaystyle 1+\frac{W_i\left(x\right)}{x_i}}}}.$$

(29)

In order to estimate $|{\sigma }_i|$, we shall establish some auxiliary estimates. From (26) and (27), we obtain

$$\left|{\displaystyle \frac{W_i\left(x\right)}{x_i}}\right|\le {\displaystyle \frac{|x_i-{\xi }_i|\gamma \left(E\left(x\right)\right)}{{\Delta }_i\left(x\right)}}\le E\left(x\right)\gamma \left(E\left(x\right)\right).$$

(30)

From this and (23), we obtain

$$\left|1+{\displaystyle \frac{W_i\left(x\right)}{x_i}}\right|\ge 1-\left|{\displaystyle \frac{W_i\left(x\right)}{x_i}}\right|\ge 1-E\left(x\right)\gamma \left(E\left(x\right)\right)>0.$$

(31)

Now, using the second inequality of Lemma 2 with $u_j=(x_j-{\xi }_j)/(x_i-x_j)$ and the second estimate of (26), we reach the following estimate:

$$\left|1-{\displaystyle \frac{W_i\left(x\right)}{x_i-{\xi }_i}}\right|=\left|\prod _{j\ne i}{\displaystyle \frac{x_i-{\xi }_j}{x_i-x_j}}-1\right|\le {\left(1+{\displaystyle \frac{E\left(x\right)}{{(n-1)}^{1/p}}}\right)}^{n-1}-1=\gamma \left(E\left(x\right)\right)-1.$$

(32)

Hence, from (29) and the estimates (30), (31) and (32), we obtain

$$|{\sigma }_i|\le {\displaystyle \frac{{\displaystyle \left|{\displaystyle 1-\frac{W_i\left(x\right)}{x_i-{\xi }_i}}\right|+\left|\frac{W_i\left(x\right)}{x_i}\right|}}{\left|{\displaystyle 1+\frac{W_i\left(x\right)}{x_i}}\right|}}\le {\displaystyle \frac{\gamma \left(E\right(x\left)\right)-1+E\left(x\right)\gamma \left(E\right(x\left)\right)}{1-E\left(x\right)\gamma \left(E\right(x\left)\right)}}=\beta \left(E\left(x\right)\right)$$

(33)

which, together with (28), leads to the inequality of (24). □

Proof of Theorem 1. 

Due to Lemma 3, the modified Weierstrass iteration function (4) is an iteration function of second kind at a point $\xi \in {\mathbb{C}}^n$ with control function $\beta$ defined by (11). So, it follows from Theorem 3 that the conclusions of Theorem 1 hold under the condition

$$E\left(x^{\left(0\right)}\right)<\eta \mathrm{and}\Psi \left(E\left(x^{\left(0\right)}\right)\right)\ge 0,$$

(34)

where E is defined by (10) and $\Psi$ is defined by (21). Thus, the proof ends since $\Phi \left(t\right)\le 2$ and $\Psi \left(t\right)\ge 0$ are equivalent in $[0,\eta )$. □

Recently, Proinov [4] showed that a local convergence theorem for a simultaneous method can be transformed into a semilocal one for the same method. For the proof of Theorem 2, we shall use the following transformation theorem:

Theorem 4 

([4], Theorem 5.1). Let $f\in \mathbb{C}\left[z\right]$ be a polynomial of degree $n\ge 2$ and let there exists a vector $x\in \mathcal{D}$ such that

$$E_f\left(x\right)\le \tau ,$$

(35)

where $E_f$ is defined by (10) and τ is defined by (15). Then f has only simple roots and there is a root vector $\xi \in {\mathbb{C}}^n$ of f such that

$$E\left(x\right)\le h\left(E_f\left(x\right)\right)\mathrm{and}\parallel x-\xi \parallel ⪯\alpha \left({\mathcal{E}}_f\left(x\right)\right)\parallel W_f\left(x\right)\parallel ,$$

(36)

where E is defined by (10), the functions α and h are defined by (16) and ${\mathcal{E}}_f\left(x\right)={\parallel W_f\left(x\right)/d\left(x\right)\parallel }_p$.

Proof of Theorem 2. 

From the first inequality of (17) and Theorem 4, we deduce that f has only simple roots and that there is a root vector $\xi \in {\mathbb{C}}^n$ of f such that

$$E\left(x^{\left(0\right)}\right)\le h\left(E_f\left(x^{\left(0\right)}\right)\right).$$

(37)

Define the function $\Phi$ by (13), then from (37), the increasing of $\Phi$ and the second inequality of (17), we obtain

$$\Phi \left(E\left(x^{\left(0\right)}\right)\right)\le \Phi \left(h\left(E_f\left(x^{\left(0\right)}\right)\right)\right)=\Omega \left(E_f\left(x^{\left(0\right)}\right)\right)\le 2.$$

Therefore, it follows from Theorem 1 that the modified Weierstrass iteration (3) is well defined and converges Q-quadratically to $\xi$, which completes the proof. □

4. Comparison with Previous Results

As it was mentioned, in [9] Nedzhibov proved the following local and semilocal convergence theorems about the modified Weierstrass method (3):

Theorem 5 

([9], Theorem 3). Let $f\in \mathbb{C}\left[z\right]$ be a monic polynomial of degree $n\ge 2$ with only simple roots and such that $f\left(0\right)\ne 0$. Let also $\xi \in {\mathbb{C}}^n$ be a root vector of f and $x^{\left(0\right)}\in {\mathbb{C}}^n$ be an initial guess with distinct nonzero components satisfying

$$\tilde{E}\left(x^{\left(0\right)}\right)={\displaystyle \frac{\parallel x^{\left(0\right)}{-\xi \parallel }_{\infty }}{\tilde{\Delta }\left(x^{\left(0\right)}\right)}}\le \mu ={\displaystyle \frac{\sqrt[n+1]{2}-1}{4\sqrt[n+1]{2}-3}},$$

(38)

where the function $\tilde{\Delta }$ is defined by (9). Then, the modified Weierstrass iteration (3) is well defined and converges quadratically to the root vector ξ of f with error estimates

$$\parallel x^{(k+1)}{-\xi \parallel }_{\infty }\le {\tilde{\lambda }}^{2^k}\parallel x^{\left(k\right)}{-\xi \parallel }_{\infty }and\parallel x^{\left(k\right)}{-\xi \parallel }_{\infty }\le {\tilde{\lambda }}^{2^k-1}{\parallel x^{\left(0\right)}-\xi \parallel }_{\infty },$$

(39)

where $\tilde{\lambda }=\tilde{E}\left(x^{\left(0\right)}\right)/\tilde{\mu }$ with $\tilde{\mu }=(\sqrt[n+1]{2}-1)/(2\sqrt[n+1]{2}-1)$.

Theorem 6 

([9], Theorem 4). Let $f\in \mathbb{C}\left[z\right]$ be a monic polynomial of degree $n\ge 2$ and let there exist a vector $x^{\left(0\right)}\in {\mathbb{C}}^n$ with distinct nonzero components such that

$${\displaystyle \frac{\parallel W_f\left(x^{\left(0\right)}\right){\parallel }_{\infty }}{\tilde{\Delta }\left(x^{\left(0\right)}\right)}}\le {\displaystyle \frac{\mu (1-\mu )}{1+(n-2)\mu }},$$

(40)

where $\tilde{\Delta }$ is defined by (9) and μ is defined in (38). Then f has only simple zeros and the iteration (3) is well defined and converges quadratically to the root vector ξ of f with error estimates (39).

Remark 1. 

Theorem 1 generalizes Theorem 5 since the initial condition (38) implies (12). Indeed, taking $p=\infty$, we have $a=n-1$ and $b=2$, and so

$$\Phi \left(\mu \right)=(1+4\mu ){(1+\mu )}^{n-1}=\left(1+4{\displaystyle \frac{\sqrt[n+1]{2}-1}{4\sqrt[n+1]{2}-3}}\right){\left(1+{\displaystyle \frac{\sqrt[n+1]{2}-1}{4\sqrt[n+1]{2}-3}}\right)}^{n-1}\le 2$$

which follows from the fact that the left hand side of the inequality increases on $n\ge 2$ and tends to 2 as $n\to \infty$.

Theorem 1 improves and complements Theorem 5 with the guaranteed Q-quadratic convergence of the method (3) and the improved error estimates.

Theorem 1 generalizes Theorem 6 because it is a direct consequence of Theorem 5.

Remark 2. 

Theorem 2 improves and complements Theorem 6 with the guaranteed Q-quadratic convergence of the method (3).

Very recently, the following semilocal convergence theorem about the modified Weierstrass method (3) has been proven in [11]:

Theorem 7 

([11], Theorem 9). Let $f\in \mathbb{C}\left[z\right]$ be a polynomial of degree $n\ge 2$ and $x^{\left(0\right)}\in {\mathbb{C}}^n$ be an initial guess with pairwise distinct nonzero components satisfying

$$E_f\left(x^{\left(0\right)}\right)={∥{\displaystyle \frac{W_f\left(x^{\left(0\right)}\right)}{\Delta \left(x^{\left(0\right)}\right)}}∥}_p<R={\displaystyle \frac{\nu (1+(b-1\left)\nu \right)}{(1+b\nu )(1+(a+b-1\left)\nu \right)}},$$

(41)

for some $1\le p\le \infty$, where $W_f$ is Weierstrass’ correction defined by (2) and ν is defined by

$$\nu ={\displaystyle \frac{\sqrt[n-1]{l}-1}{b(\sqrt[n-1]{l}-1)+a/(n-1)}},$$

(42)

where $a,b$ are defined by (6) and l is defined by

$$l={\displaystyle \frac{3b-a-1+\sqrt{{(3b-a-1)}^2+8(b+1)(a+1-b)}}{2(b+1)}}.$$

(43)

Then f has only simple roots and the iteration (3) is well defined and converges quadratically to a root vector ξ of f

Remark 3. 

Theorem 2 enlarges the convergence domain of Theorem 7 since every vector $x^{\left(0\right)}$ that satisfies the initial condition (41) satisfies (17) but not vice versa. In

Table 1. Numerical comparison between Theorem 2 and Theorem 7.

	$\mathit{p}=1$		$\mathit{p}=2$		$\mathit{p}=\mathit{\infty }$
$\mathit{n}$	$\mathit{R}$	$\Omega \left(\mathit{R}\right)$	$\mathit{R}$	$\Omega \left(\mathit{R}\right)$	$\mathit{R}$	$\Omega \left(\mathit{R}\right)$
2	0.147476	1.816214	0.136294	1.809102	$0.122449$	1.795918
3	0.142763	1.790711	0.115166	1.798972	$0.090245$	1.801232
4	0.141210	1.782394	0.103661	1.798553	$0.072327$	1.813296
5	0.140437	1.778270	0.095868	1.800342	$0.060653$	1.825738
10	0.139155	1.771449	0.075611	1.813043	$0.034149$	1.872578
15	0.138790	1.769513	0.065704	1.823715	$0.023943$	1.900192
20	0.138617	1.768597	0.059349	1.832223	$0.018471$	1.918019
25	0.138517	1.768063	0.054767	1.839215	$0.015047$	1.930447
30	0.138451	1.767714	0.051233	1.845121	$0.012699$	1.939602
50	0.138322	1.767030	0.042279	1.862230	0.007827	1.960427
100	0.138227	1.766530	0.032167	1.885650	0.003999	1.978739
1000	0.138144	1.766088	0.011831	1.949246	0.000408	1.997720
$10,000$	0.138135	1.766044	0.003971	1.981548	0.000040	1.999770

, we give some numerical data to verify the inequality $\Omega \left(R\right)<2$ in the cases $p=1,2$ and ∞.

Theorem 2 complements Theorem 7 with the provided Q-quadratic convergence of the method (3).

To highlight the advantages of Theorem 2 over Theorem 7, in Table 1, we exhibit the calculated values of R and $\Omega \left(R\right)$ for many choices of n up to $n=10,000$. It is clearly seen from the table that in all considered cases, we have $\Omega \left(R\right)<2$, which means that the convergence domain of Theorem 2 is larger than the one of Theorem 7. In fact, we have tested many more values of n than shown in the table, and we have found no contradictions.

5. Numerical Examples

Here, we show that Theorem 2 can be applied to some very important practical problems. Namely, we use it for a numerical proof of the Q-quadratic convergence of the modified Weierstrass method (MWM) (3) and for a numerical guarantee of the accuracy of approximation at each iteration. Also, we suggest a method for the experimental study of the dynamics of simultaneous methods and apply it to the modified Weierstrass method and the famous Weierstrass’ method (WM) (1). For our proposes, we apply the above-mentioned iteration methods to two very known physics problems and to a polynomial for which Wang and Zhao [26] have failed to find a suitable initial vector to use WM. For the sake of brevity, we shall consider only the case $p=\infty$.

Let $f\left(z\right)=a_0{z}^n+a_1{z}^{n-1}+\cdots +a_n$ be a polynomial of degree $n\ge 2$ with coefficients in $\mathbb{C}$ and $x^{\left(0\right)}\in {\mathbb{C}}^n$ be an initial guess. We utilize $MWM$ and $WM$ for computing all the zeros of f simultaneously.

Furthermore, from Theorems 2 and 4, we obtain the following two criteria:

(i)

Convergence criterion. If there exists an integer $m\ge 0$ such that

$$E_f\left(x^{\left(m\right)}\right)<{\tau }_n={\displaystyle \frac{1}{{(1+\sqrt{n-1})}^2}}\mathrm{and}\Omega \left(E_f\left(x^{\left(m\right)}\right)\right)\le 2,$$

(44)

where $E_f$ is defined by (10) and the function $\Omega$ is defined by (16), then f has only simple zeros and the iteration (3) is well defined and Q-quadratically convergent to a root vector $\xi$ of f.

(ii)

Accuracy criterion. If, for a preset accuracy, $\epsilon >0$, there exists an integer $k\ge 0$ for which

$${\mathcal{E}}_f\left(x^{\left(k\right)}\right)={∥{\displaystyle \frac{W_f\left(x^{\left(k\right)}\right)}{d\left(x^{\left(k\right)}\right)}}∥}_{\infty }<{\tau }_n\mathrm{and}{\epsilon }_k<\epsilon ,$$

(45)

where ${\epsilon }_k=\alpha \left({\mathcal{E}}_f\left(x^{\left(k\right)}\right)\right){\parallel W_f\left(x^{\left(k\right)}\right)\parallel }_{\infty }$ with $\alpha$ defined by (16), then the root vector $\xi$ of f is calculated with accuracy $\epsilon$. At the kth iteration, the guaranteed accuracy is ${\epsilon }_k$.

In any example, we calculate the smallest integer $m\ge 0$ that satisfies the convergence criterion (44) and the smallest integer $k\ge 0$ that satisfies the accuracy criterion (45) with accuracy $\epsilon ={10}^{-6}$. In the tables below, we give the values of m, $\Omega \left(E_f\left(x^{\left(m\right)}\right)\right)$, k, ${\mathcal{E}}_f\left(x^{\left(k\right)}\right)$, ${\tau }_n$ and ${\epsilon }_k$ with at least six decimal places. Since the convergence criterion concerns only MWM, we use the abbreviation $N/A$ for WM.

Furthermore, we propose the following algorithm for the experimental study of the dynamics of the simultaneous methods:

Step 1

Generate Aberth’s initial approximation $x^{\left(0\right)}\in {\mathbb{C}}^n$ defined by [27]

$$x_j^{\left(0\right)}=-{\displaystyle \frac{a_1}{n}}+r{e}^{i{\theta }_j},{\theta }_j={\displaystyle \frac{\pi }{n}}\left(2j-{\displaystyle \frac{3}{2}}\right),j\in I_n,$$

(46)

where $a_1$ and n are the second coefficient and the degree of f while r is chosen in accordance with a well-known Cauchy bound theorem (see, e.g., ([12], Theorem 11.4)). Namely, it is defined by $r=1+{max}_{i\in I_n}|a_i/a_0|$.

Step 2

Take a square containing all n zeros of f and, using a mesh of $m\times m$ points and up to K iterations, ‘cover’ the square by picking one by one random number from any cell and using it as a coordinate of the initial vector $x^{\left(0\right)}$ instead of one of the coordinates of Aberth’s initial vector. Thus, the basins of attraction and the convergence areas, i.e., for preset $ϵ$, the areas with number of iterations $k\le K$ needed to meet the criterion

$$\underset{i\in I_n}{max}\left|f\left(x_i^{\left(k\right)}\right)\right|<ϵ,$$

(47)

are obtained.

In the following examples, we use a mesh of $400\times 400$ points, $K=80$ and $ϵ={10}^{-6}$ in squares selected to contain all zeros of f and to ensure a good view on them and their basins. The areas with initial points that generate iterative sequences such that ${max}_{i\in I_n}\left|f\left(x_i^{\left(k\right)}\right)\right|\in [{10}^{-6},{10}^{12}]$ are white colored and shell be called bad areas. In other words, if the ‘crawling’ coordinate is in a bad area; then, the method behaves very badly or even diverges. The red stars depict the exact zeros of the corresponding polynomial.

In the following example, we consider a polynomial with very close complex zeros.

Example 1 

(Van der Waals Equation (1873)). A well-known equation that describes the behavior of real gases is the so-called Van der Waals equation. In the case of Oxygen, it reads:

$$100000x^3-5622.002x^2+1.382x-0.04403052=0.$$

(48)

This equation has the following simple zeros ${\xi }_{1,2}=0.000053\pm 0.002800i$ and ${\xi }_3=0.056113$. Aberth’s initial guess, with $r=1$, is

$$x^{\left(0\right)}=(1874.91\dots +0.52\dots i,1873.08\dots +0.52\dots i,1874.00\dots -1.05\dots i).$$

The basins of attraction and the convergence areas are obtained by $MWM$ and $WM$ in the square

$$\{x\in \mathbb{C}:|Re\left(x\right)|\le 0.15\mathrm{and}|Im\left(x\right)|\le 0.15\}$$

and plotted in Figure 1 and Figure 2. Together with $x^{\left(0\right)}$, in this example, we also use the following initial vectors:

$$\begin{array}{ccc}\hfill y^{\left(0\right)}& =& (\begin{array}{c}\hfill -0.06\dots -0.13\dots i\end{array},1873.08\dots +0.52\dots i,1874.00\dots -1.05\dots i);\hfill \\ \hfill z^{\left(0\right)}& =& (1874.91\dots +0.52\dots i, \begin{array}{c}\hfill 0.12\dots -0.08\dots i\end{array},1874.00\dots -1.05\dots i);\hfill \\ \hfill t^{\left(0\right)}& =& (1874.91\dots +0.52\dots i,1873.08\dots +0.52\dots i,\begin{array}{c}\hfill -0.03\dots +0.03\dots i\end{array})\hfill \end{array}$$

one coordinate of any of which is randomly taken from the ‘red’, ‘green’ and ‘blue’ areas in Figure 1 instead of the first, second and third coordinates of $x^{\left(0\right)}$, respectively.

Table 2. Numerical data for Example 1.

Method	Initial Guess	$\mathit{m}$	$\Omega \left({\mathit{E}}_{\mathit{f}}\left({\mathit{x}}^{\left(\mathit{m}\right)}\right)\right)$	$\mathit{k}$	${\mathcal{E}}_{\mathit{f}}\left({\mathit{x}}^{\left(\mathit{k}\right)}\right)$	${\mathit{\tau }}_{\mathit{n}}$	${\mathit{\epsilon }}_{\mathit{k}}$
$MWM$	$x^{\left(0\right)}$	The method diverges
	$y^{\left(0\right)}$	18	$1.901188$	21	$1.877\times {10}^{-5}$	$0.171573$	$1.051\times {10}^{-7}$
	$z^{\left(0\right)}$	20	$1.431314$	23	$1.904\times {10}^{-7}$	$0.171573$	$1.067\times {10}^{-9}$
	$t^{\left(0\right)}$	17	$1.787441$	20	$7.297\times {10}^{-6}$	$0.171573$	$4.087\times {10}^{-8}$
$WM$	$x^{\left(0\right)}$	$N/A$	$N/A$	68	$4.589\times {10}^{-5}$	$0.171573$	$2.571\times {10}^{-7}$
	$y^{\left(0\right)}$	−	−	33	$4.304\times {10}^{-6}$	$0.171573$	$2.411\times {10}^{-8}$
	$z^{\left(0\right)}$	−	−	34	$6.942\times {10}^{-5}$	$0.171573$	$3.889\times {10}^{-7}$
	$t^{\left(0\right)}$	−	−	34	$3.435\times {10}^{-8}$	$0.171573$	$1.924\times {10}^{-10}$

shows that starting from $x^{\left(0\right)}$$MWM$ diverges but in all other cases it converges no matter the rough choice of the initial guess. Namely, starting from $y^{\left(0\right)}$$MWM$ meets the convergence criterion (44) at the eighteenth iteration and the accuracy criterion (45) at the twenty-first one, where the zeros are found with accuracy at least ${10}^{-6}$. On the other hand, $WM$ converges in any case but behaves quite better when starts from $y^{\left(0\right)}$, $z^{\left(0\right)}$ and $t^{\left(0\right)}$. So, the view on the basins of attraction gives us an opportunity to correct or even to restore the convergence of any simultaneous method.

The basins of attraction and the convergence areas are plotted in Figure 1 and Figure 2, where the color scales show the number of iterations that are needed for any method to meet the criterion (47). It is easy to see that the basins of attraction depend on the replaced coordinate in the initial vector, which confirms that the coordinates of the approximations in simultaneous methods influence each other.

In the next example, we consider a famous problem from Quantum Mechanics that leads to polynomial equations.

Example 2 

(Quantum harmonic oscillator (1925)). The one-dimensional quantum harmonic oscillator is described by the following time-independent Schrödinger equation:

$$\left(-{\displaystyle \frac{{\hslash }^2}{2\mu }}{\displaystyle \frac{{\partial }^2}{\partial z^2}}+{\displaystyle \frac{\mu {\omega }^2{z}^2}{2}}\right)\Psi \left(z\right)=E\Psi \left(z\right),$$

(49)

where μ is the mass of the particle, ω is the angular frequency of the oscillator and E specifies the time-independent energy level. It is well known that the solutions of (49) involve the so-called physicist’s Hermite polynomials defined by

$$H_n\left(x\right)={(-1)}^n{e}^{x^2}{\displaystyle \frac{d^n}{d{x}^n}}\left(e^{-x^2}\right)n=0,1,\dots$$

which zeros represent the nodes of the wave function Ψ, i.e., the points where the particle has zero probability of being located at (see, e.g., ([28], Focus 7)). Here, we consider the eighth-degree physicist’s Hermite polynomial, which reads

$$f\left(z\right)=256x^8-3584x^6+13440x^4-13440x^2+1680,$$

(50)

and has the following zeros ${\xi }_{1,2}=\pm 2.93064$, ${\xi }_{3,4}=\pm 1.98166$, ${\xi }_{5,6}=\pm 1.15719$ and ${\xi }_{7,8}=\pm 0.381187$. The basins of attraction of the roots of (50) and the convergence areas are obtained by $MWM$ and $WM$ in the square

$$\{x\in \mathbb{C}:|Re\left(x\right)|\le 5\mathit{and}|Im\left(x\right)|\le 5\}$$

by replacing the first coordinate of Aberth’s initial approximation (46). For the numerical test, we pick Aberth’s initial guess $x^{\left(0\right)}$ and the initial vectors $y^{\left(0\right)}$ and $z^{\left(0\right)}$ which are obtained from $x^{\left(0\right)}$ replacing its first coordinate by the random numbers $-2.54\dots +1.67\dots i$ and $-2.70\dots -2.47\dots i$ taken from the colored area and the bad basins of $MWM$ (Figure 3), respectively.

One can see from

Table 3. Numerical data for Example 2.

Method	Initial Guess	m	$\mathbf{\Omega }\left({\mathit{E}}_{\mathit{f}}\left({\mathit{x}}^{\left(\mathit{m}\right)}\right)\right)$	k	${\mathcal{E}}_{\mathit{f}}\left({\mathit{x}}^{\left(\mathit{k}\right)}\right)$	${\mathit{\tau }}_{\mathit{n}}$	${\mathit{\epsilon }}_{\mathit{k}}$
$MWM$	$x^{\left(0\right)}$	37	$1.286425$	40	$4.938\times {10}^{-11}$	$0.075236$	$3.764\times {10}^{-11}$
	$y^{\left(0\right)}$	35	$1.303345$	37	$9.212\times {10}^{-8}$	$0.075236$	$7.148\times {10}^{-8}$
	$z^{\left(0\right)}$	The method diverges
$WM$	$x^{\left(0\right)}$	$N/A$	$N/A$	31	$4.716\times {10}^{-7}$	$0.075236$	$3.595\times {10}^{-7}$
	$y^{\left(0\right)}$	−	−	29	$1.159\times {10}^{-9}$	$0.075236$	$8.838\times {10}^{-10}$
	$z^{\left(0\right)}$	−	−	29	$3.487\times {10}^{-7}$	$0.075236$	$2.658\times {10}^{-7}$

that starting from Aberth’s initial guess $x^{\left(0\right)}$, the two methods converge but for $z^{\left(0\right)}$, i.e., when there is a coordinate from the bad basin, $MWM$ diverges, while $WM$ converges in any case.

The basins of attraction and the convergence areas for this example are plotted in Figure 3.

Example 3 

([26], Example 2, [29], Example 2). In this example, we consider the following polynomial:

$$f\left(z\right)=z^{20}-1$$

(51)

for which Wang and Zhao [26] have failed to find a suitable initial vector to use WM. The basins of attraction of the roots of (51) and the convergence areas are obtained by $MWM$ and $WM$ in the square

$$\{x\in \mathbb{C}:|Re\left(x\right)|\le 3\mathit{and}|Im\left(x\right)|\le 3\}$$

replacing the fifth coordinate (randomly chosen) of Aberth’s initial approximation (46). For the numerical test, we take Aberth’s initial guess $x^{\left(0\right)}$ and the initial vectors $y^{\left(0\right)}$ and $z^{\left(0\right)}$ which are obtained from $x^{\left(0\right)}$ replacing its fifth coordinate by the randomly generated numbers $0.89\dots +1.86\dots i$ and $0.43\dots -0.16\dots i$ from the colored area and the bad basins of $MWM$ (Figure 4), respectively.

As can be seen from

Table 4. Numerical data for Example 3.

Method	Initial Guess	m	$\mathbf{\Omega }\left({\mathit{E}}_{\mathit{f}}\left({\mathit{x}}^{\left(\mathit{m}\right)}\right)\right)$	k	${\mathcal{E}}_{\mathit{f}}\left({\mathit{x}}^{\left(\mathit{k}\right)}\right)$	${\mathit{\tau }}_{\mathit{n}}$	${\mathit{\epsilon }}_{\mathit{k}}$
$MWM$	$x^{\left(0\right)}$	17	$1.100417$	19	$7.706\times {10}^{-9}$	$0.034821$	$2.411\times {10}^{-9}$
	$y^{\left(0\right)}$	17	$1.051683$	19	$5.797\times {10}^{-10}$	$0.034821$	$1.813\times {10}^{-10}$
	$z^{\left(0\right)}$	20	$1.040132$	21	$2.554\times {10}^{-6}$	$0.034821$	$7.991\times {10}^{-7}$
$WM$	$x^{\left(0\right)}$	$N/A$	$N/A$	18	$2.376\times {10}^{-8}$	$0.034821$	$7.435\times {10}^{-9}$
	$y^{\left(0\right)}$	−	−	18	$1.159\times {10}^{-9}$	$0.034821$	$9.041\times {10}^{-10}$
	$z^{\left(0\right)}$	−	−	19	$3.487\times {10}^{-7}$	$0.034821$	$4.548\times {10}^{-7}$

the two methods converge in all considered cases but behave slightly worse when start from $z^{\left(0\right)}$.

6. Conclusions

In this paper, we have obtained a new kind of local convergence theorem (Theorem 1) that generalizes, improves and complements all existing such kind of results about the modified Weierstrass method (3). Thereafter, we have transformed our local convergence theorem into a semilocal convergence one (Theorem 2) which is of significant practical importance owing to its computationally verifiable initial condition. We have shown that this theorem improves all existing results about the studied method. At the end of the study, we have conducted three numerical examples to show the applicability of our semilocal convergence theorem. Within the examples, we have proposed an algorithm for the experimental study of the dynamics of the simultaneous methods and have applied it to the modified Weierstrass method and the famous Weierstrass’ one when solving some very well-known physics problems.