# Local and Semilocal Convergence of Nourein’s Iterative Method for Finding All Zeros of a Polynomial Simultaneously

^{*}

## Abstract

**:**

## 1. Introduction

**Weierstrass’ method and elementary symmetric functions.**Historically, the first iterative method for simultaneous finding all zeros of a polynomial was constructed by Weierstrass [3] in 1891. Let

**Ehrlich’s method.**In 1967, Ehrlich [6] introduced a third-order simultaneous method defined by the following iteration:

**Local and semilocal convergence analysis.**Recently, a general convergence theory of iterative methods of the type ${x}^{(k+1)}=T\left({x}^{\left(k\right)}\right)$, where $T:D\subset X\to X$ is an iteration function of a metric space X, was developed in [12,13]. Central to this theory is the concept of the function of initial approximations (see ([13], Section 3)). Roughly speaking, this is a real-valued function $E:D\subset X\to {\mathbb{R}}_{+}$ that sets the initial conditions. The initial condition of any convergence theorem of an iterative method can be represented in the form

**Definition**

**1.**

**Nourein’s method for simple zeros.**There are different ways to increase the convergence order of an iterative method for simultaneous computation of polynomial zeros. In 1977, Nourein [1,15] constructed three simultaneous methods that increase the convergence order of Weierstrass’s, Ehrlich’s and Börsch-Supan’s methods. Each of these three methods was constructed as a combination of two already known iterative methods. In particular, combining Ehrlich’s method and the classical Newton’s method, Nourein [1] constructed in ${\mathbb{K}}^{n}$ the following fourth-order iterative method (for simple zeros):

**Theorem**

**1**

**Nourein’s method for multiple zeros.**Nourein’s method (7) has a well-known but not well-studied generalization for the simultaneous finding of multiple polynomial zeros. Let $f\in \mathbb{K}\left[z\right]$ be a polynomial of degree $n\ge 2$ which splits in $\mathbb{K}$, and let ${\xi}_{1},\dots ,{\xi}_{s}$ (s is a positive integer such that $1\le s\le n$) be all distinct zeros of f with multiplicity ${m}_{1},\dots ,{m}_{s}$$({m}_{1}+\dots +{m}_{s}=n)$, respectively.

**Contributions.**In this paper, we present a detailed local convergence analysis for generalized Nourein’s method (11) for multiple zeros. As a consequence of these results, we obtain two types of local convergence theorems as well as semilocal convergence theorems (with verifiable initial condition and a posteriori error bound) for the classical Nourein’s method (7). Each of the new semilocal convergence results improves Theorem 1 in several directions.

## 2. Notations

**Definition**

**2**

## 3. Local Convergence Theorem of the First Kind for Multiple Zeros

**Lemma**

**1**

**Lemma**

**2.**

**Proof.**

**Theorem**

**2**

**Lemma**

**3**

**Lemma**

**4.**

**Proof.**

**Lemma**

**5.**

**Proof.**

**Theorem**

**3.**

**Proof.**

## 4. Local Convergence Theorem of the Second Kind for Multiple Zeros

**Lemma**

**6**

**Lemma**

**7**

**Theorem**

**4**

**Lemma**

**8.**

**Proof.**

**Lemma**

**9.**

**Proof.**

**Theorem**

**5.**

**Proof.**

## 5. Local Convergence Theorem of the First Kind for Simple Zeros

**Theorem**

**6.**

**Corollary**

**1.**

**Proof.**

**Corollary**

**2.**

**Remark**

**1.**

## 6. Local Convergence Theorem of the Second Kind for Simple Zeros

**Theorem**

**7.**

**Theorem**

**8.**

**Proof.**

**Corollary**

**3.**

**Proof.**

**Corollary**

**4.**

**Remark**

**2.**

## 7. Semilocal Convergence Analysis for Simple Zeros

**Theorem**

**9**

**Theorem**

**10**

**Theorem**

**11.**

**Proof.**

**Theorem**

**12.**

**Proof.**

**Corollary**

**5.**

**Proof.**

**Remark**

**3.**

## 8. Numerical Examples

- Convergence criterion that determines whether Nourein’s method is convergent.
- Accuracy criterion that determines whether Nourein’s method has reached a preset accuracy $\epsilon >0$. It can be used as stopping criterion.

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

**Accuracy criterion (stopping criterion).**Let $\epsilon >0$. If there exists an integer $k\ge 0$, such that

**First type of initial approximations.**For a monic polynomial

**Second type of initial approximations.**We choose the coordinates ${x}_{1}^{\left(0\right)},\dots ,{x}_{n}^{\left(0\right)}$ of the initial vector ${x}^{\left(0\right)}\in {\mathbb{C}}^{n}$ randomly in the square

**Example**

**1.**

**Example**

**2.**

**Example**

**3.**

**Numerical results.**In Table 1 are presented the results for the considered examples, we exhibit the values of m, ${E}_{f}\left({x}^{\left(m\right)}\right)$, $\Omega \left({E}_{f}\left({x}^{\left(m\right)}\right)\right)$, k and ${\epsilon}_{k}$. We recall that:

- m is the smallest nonnegative integer that satisfies the convergence criterion (88);
- ${\epsilon}_{m}$ is defined in (89) and denotes the guaranteed accuracy (by Theorem 12) for the approximation ${x}_{m}$ of the zeros of f;
- k is the smallest nonnegative integer that satisfies convergence accuracy criterion (89) with the preset accuracy $\epsilon ={10}^{-15}$;
- ${\epsilon}_{k}$ is the guaranteed accuracy (Theorem 12) for the approximation ${x}_{k}$ of the zeros of f.

## 9. Conclusions

## Author Contributions

## Funding

## Conflicts of Interest

## References

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Example | m | ${\mathit{E}}_{\mathit{f}}\left({\mathit{x}}^{\left(\mathit{m}\right)}\right)$ | $\mathbf{\Omega}\left({\mathit{E}}_{\mathit{f}}\left({\mathit{x}}^{\left(\mathit{m}\right)}\right)\right)$ | ${\mathit{\epsilon}}_{\mathit{m}}$ | k | ${\mathit{\epsilon}}_{\mathit{k}}$ | ${\mathit{\epsilon}}_{\mathit{k}+1}$ |
---|---|---|---|---|---|---|---|

First type initial approximations | |||||||

Example 1 | 31 | $6.254\times {10}^{-3}$ | $0.848$ | $6.988\times {10}^{-3}$ | 33 | $1.042\times {10}^{-41}$ | $1.442\times {10}^{-167}$ |

Example 2 | 20 | $2.845\times {10}^{-5}$ | $0.999$ | $3.498\times {10}^{-6}$ | 22 | $5.275\times {10}^{-20}$ | $5.711\times {10}^{-76}$ |

Example 3 | 14 | $6.688\times {10}^{-5}$ | $0.998$ | $4.139\times {10}^{-5}$ | 15 | $2.719\times {10}^{-17}$ | $5.946\times {10}^{-66}$ |

Second type initial approximations | |||||||

Example 1 | 31 | $2.774\times {10}^{-5}$ | $0.999$ | $2.775\times {10}^{-5}$ | 32 | $5.366\times {10}^{-20}$ | $1.170\times {10}^{-78}$ |

Example 2 | 18 | $1.945\times {10}^{-3}$ | $0.960$ | $2.460\times {10}^{-4}$ | 20 | $2.684\times {10}^{-51}$ | $6.697\times {10}^{-202}$ |

Example 3 | 14 | $1.045\times {10}^{-6}$ | $0.999$ | $6.463\times {10}^{-7}$ | 15 | $1.622\times {10}^{-24}$ | $7.769\times {10}^{-95}$ |

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Proinov, P.D.; Vasileva, M.T.
Local and Semilocal Convergence of Nourein’s Iterative Method for Finding All Zeros of a Polynomial Simultaneously. *Symmetry* **2020**, *12*, 1801.
https://doi.org/10.3390/sym12111801

**AMA Style**

Proinov PD, Vasileva MT.
Local and Semilocal Convergence of Nourein’s Iterative Method for Finding All Zeros of a Polynomial Simultaneously. *Symmetry*. 2020; 12(11):1801.
https://doi.org/10.3390/sym12111801

**Chicago/Turabian Style**

Proinov, Petko D., and Maria T. Vasileva.
2020. "Local and Semilocal Convergence of Nourein’s Iterative Method for Finding All Zeros of a Polynomial Simultaneously" *Symmetry* 12, no. 11: 1801.
https://doi.org/10.3390/sym12111801