## 1. Introduction

**Figure 1.**Basins of attraction of the relaxed Newton’s method given in Equation (1) applied to the polynomial $p\left(z\right)={z}^{3}-1$ for $h=1$, $h=2/3$ and $h=1/3$, respectively.

_{0}∈ $\mathbb{C}$, where p is a non-constant complex polynomial is known as continuous Newton’s method. We refer to [1] for the theoretical basis of continuous Newton’s method. In particular, it is shown that solutions $z\left(t\right)$ of Equation (2) (or Equation (3)) flow to a zero of p while keeping the argument of $p\left(z\right(t\left)\right)$ constant at $arg\left(p\left({z}_{0}\right)\right)$.

## 2. Numerical Algorithms Applied to Continuous Newton’s Method

- Euler’s method:$${y}_{n+1}={y}_{n}+hf({t}_{n},{y}_{n})$$
- Refined Euler’s method:$$\left\{\begin{array}{ccc}\hfill {y}^{*}& =& {y}_{n}+h/2f({t}_{n},{y}_{n})\hfill \\ \hfill {y}_{n+1}& =& {y}_{n}+hf({t}_{n}+h/2,{y}^{*})\hfill \end{array}\right.$$
- Heun’s method:$$\left\{\begin{array}{ccc}\hfill {y}^{*}& =& {y}_{n}+hf({t}_{n},{y}_{n})\hfill \\ \hfill {y}_{n+1}& =& {y}_{n}+h/2\left(f({t}_{n},{y}_{n})+f({t}_{n}+h,{y}^{*})\right)\hfill \end{array}\right.$$
- Runge-Kutta method of order 2:$$\left\{\begin{array}{ccc}\hfill {y}^{*}& =& {y}_{n}+2h/3f({t}_{n},{y}_{n})\hfill \\ \hfill {y}_{n+1}& =& {y}_{n}+h/4\left(f({t}_{n},{y}_{n})+3f({t}_{n}+2h/3,{y}^{*})\right)\hfill \end{array}\right.$$
- Runge-Kutta method of order 4:$$\left\{\begin{array}{ccc}\hfill {k}_{1}& =& hf({t}_{n},{y}_{n})\hfill \\ \hfill {k}_{2}& =& hf({t}_{n}+h/2,{y}_{n}+{k}_{1}/2)\hfill \\ \hfill {k}_{3}& =& hf({t}_{n}+h/2,{y}_{n}+{k}_{2}/2)\hfill \\ \hfill {k}_{4}& =& hf({t}_{n}+h,{y}_{n}+{k}_{3})\hfill \\ \hfill {y}_{n+1}& =& {y}_{n}+1/6\left({k}_{1}+2{k}_{2}+2{k}_{3}+{k}_{4}\right)\hfill \end{array}\right.$$
- Adams-Bashforth method of order 2:$$\left\{\begin{array}{ccc}\hfill {y}^{*}& =& {y}_{0}+2h/3f({t}_{n},{y}_{0})\hfill \\ \hfill {y}_{1}& =& {y}_{0}+h/4\left(f({t}_{n},{y}_{0})+3f({t}_{n}+2h/3,{y}^{*})\right)\hfill \\ \hfill {y}_{n+1}& =& {y}_{n}+{y}_{n}+h\left(3/2f({t}_{n},{y}_{n})-1/2f({t}_{n-1},{y}_{n-1})\right)\hfill \end{array}\right.$$

**Table 1.**Some numerical properties of the root-finding methods obtained after applying methods Equations (5)–(9) to continuous Newton’s method Equation (2).

Method | A.E.C. | I.C. | ${\mathit{h}}^{\mathbf{*}}$ |
---|---|---|---|

Euler Equation (5) | $1-h$ | $(0,2)$ | 1 |

Refined Euler Equation (6) | $(1+{(1-h)}^{2})/2$ | $(0,2)$ | 1 |

Heun Equation (7) | $(1+{(1-h)}^{2})/2$ | $(0,2)$ | 1 |

Runge-Kutta 2 Equation (8) | $(1+{(1-h)}^{2})/2$ | $(0,2)$ | 1 |

Runge-Kutta 4 Equation (9) | $({h}^{4}-4{h}^{3}+12{h}^{2}-24h+24)/24$ | $(0,2.7853)$ | $1.5961$ |

## 3. Numerical Algorithms with Non-Constant Step Size

**Figure 2.**Basins of attraction of the Chebyshev, Halley and super-Halley methods ($\alpha =0$, $\alpha =1/2$ and $\alpha =1$, respectively, in Equation (19)) applied to the polynomial $p\left(z\right)={z}^{3}-1$.

## 4. Conclusions

## Acknowledgments

## Conflicts of Interest

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