# Parametric Family of Root-Finding Iterative Methods: Fractals of the Basins of Attraction

^{1}

^{2}

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Convergence Analysis

**Theorem**

**1.**

**Proof.**

## 3. Stability Analysis

- Attractive if $|{R}^{\prime}\left({z}^{F}\right)|1$.
- Parabolic or Neutral if $|{R}^{\prime}\left({z}^{F}\right)|=1$.
- Repulsive if $|{R}^{\prime}\left({z}^{F}\right)|1$.
- Superattractive if ${R}^{\prime}\left({z}^{F}\right)=0$.

- (a)
- If ${O}_{\alpha}\left({z}^{F}\right)={z}^{F}$, then ${O}_{\alpha}(1/{z}^{F})=1/{z}^{F}$.
- (b)
- Except for some specific values of the $\alpha $ simplifying the operator, $z=1$ is an strange fixed point of rational operator ${O}_{\alpha}\left(z\right)$.
- (c)
- The stability function of two conjugate fixed points coincide,$${O}_{\alpha}^{\prime}\left({z}^{F}\right)=1/{O}_{\alpha}^{\prime}\left({z}^{F}\right).$$

#### 3.1. Performance of the Strange Fixed Points

**Theorem**

**2.**

**Proof.**

**Proposition**

**1.**

- ${f}_{1}\left(\alpha \right)$and${f}_{2}\left(\alpha \right)$are conjugate and repulsive, with independence of the value of parameter α.
- ${f}_{3}\left(\alpha \right)$and${f}_{4}\left(\alpha \right)$are attractors for values of α in small regions of the complex plane, inside the complex area$[-0.26,-0.24]\times [-0.36,-0.34]$and$[-0.26,-0.24]\times [0.34,0.36]$. Moreover, both are superattracting for$\alpha \approx -0.250121\pm 0.348771i$.
- ${f}_{5}\left(\alpha \right)$and${f}_{6}\left(\alpha \right)$are conjugate and attractors for values of α inside the complex area$[10,40]\times [-15,15]$. Moreover, both are superattracting for$\alpha \approx 20.3811$.

#### 3.2. Critical Points and Parameter Planes

**Proposition**

**2.**

- One, if$\alpha =0$or$\alpha =-\frac{135}{8}$. In these cases, the reduced rational operator is:$$\begin{array}{ccc}\hfill {O}_{-\frac{135}{8}}\left(z\right)& =& \frac{{x}^{4}\left(27{x}^{4}+162{x}^{3}+378{x}^{2}+378x-945\right)}{-945{x}^{4}+378{x}^{3}+378{x}^{2}+162x+27},\hfill \\ \hfill {O}_{0}\left(z\right)& =& \frac{{x}^{4}\left(27{x}^{4}+162{x}^{3}+378{x}^{2}+378x+135\right)}{135{x}^{4}+378{x}^{3}+378{x}^{2}+162x+27},\hfill \end{array}$$
- Three, if$\alpha \ne 0$and$\alpha \ne -\frac{135}{8}$, as in this case, they are defined as:$$\begin{array}{ccc}\hfill c{r}_{1}\left(\alpha \right)& =& -1,\hfill \\ \hfill c{r}_{2}\left(\alpha \right)& =& {\displaystyle \frac{-135+48\alpha -4\sqrt{14}\sqrt{-135\alpha -8{\alpha}^{2}}}{135+64\alpha}},\hfill \\ \hfill c{r}_{3}\left(\alpha \right)& =& {\displaystyle \frac{-135+48\alpha +4\sqrt{14}\sqrt{-135\alpha -8{\alpha}^{2}}}{135+64\alpha}}.\hfill \end{array}$$

- (a)
- If $\alpha =0$, then $c{r}_{2}=c{r}_{3}=-1$, and it is a pre-image of the fixed point $z=1$: ${O}_{0}\left(-1\right)=1$. As $z=1$ is repulsive for $\alpha =0$, $z=-1\in \mathcal{J}\left({O}_{\alpha}\right)$. Thus, ${O}_{p}\left(z\right)$ has only two invariant Fatou components, $\mathcal{A}\left(0\right)$ and $\mathcal{A}\left(\infty \right)$.
- (b)
- If $\alpha =-{\displaystyle \frac{135}{8}}$, then $c{r}_{2}=c{r}_{3}=-1$, and ${O}_{-\frac{135}{8}}\left(-1\right)=1$. As $z=1$ is not a fixed point when $\alpha =-{\displaystyle \frac{135}{8}}$, then $z=-1\in \mathcal{J}\left({O}_{\alpha}\right)$ and its orbit will remain at Julia set until the rounding error makes it fall into the basin of attraction of $z=0$ or $z=\infty $.
- (c)
- For the rest of the values of $\alpha \in \mathbf{C}$, we gave three critical points.

#### 3.3. Dynamical Planes

## 4. Numerical Results

- ${f}_{1}\left(x\right)=si{n}^{2}x-{x}^{2}+1$, with two real roots at ${x}_{1}^{*}\approx $$-1.4044$ and ${x}_{2}^{*}\approx 1.4044$.
- ${f}_{2}\left(x\right)=cosx-x{e}^{x}$, with real roots at ${x}_{1}^{*}\approx 0.517$, ${x}_{2}^{*}\approx -14.137$ and ${x}_{3}^{*}\approx -17.278$, among others.
- Colebrook-White function [27] ${f}_{3}\left(x\right)=\sqrt{\frac{1}{f}}+0.86\mathrm{ln}\left(\frac{{10}^{-4}}{3.7}+\frac{2.51}{{10}^{5}\sqrt{\frac{1}{f}}}\right)$, with a real root at ${x}^{*}\approx 0.01885050$, among others.

## 5. Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## References

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$\mathit{\alpha}$ | $\left|\mathit{f}\right({\mathit{x}}_{\mathit{k}+1}\left)\right|$ | $|{\mathit{x}}_{\mathit{k}+1}-{\mathit{x}}_{\mathit{k}}|$ | Solution | Iterations | ACOC | Time (s) |
---|---|---|---|---|---|---|

−50 | 7.0883 $\times {10}^{-1007}$ | 8.4884 $\times {10}^{-253}$ | −1.4044 | 8 | 4.0 | 0.3223 ± 0.2001 |

−40 | 0 | 1.4357 $\times {10}^{-402}$ | −1.4044 | 29 | 4.0 | 0.9086 ± 0.5325 |

−20 + 45i | d | d | d | d | d | d |

−16 − 45i | 2.4842 $\times {10}^{-1432}$ | 5.1013 $\times {10}^{-425}$ | 1.4044 | 10 | 4.0 | 0.3597 ± 0.1498 |

1 | 0 | 1.8974 $\times {10}^{-331}$ | 1.4044 | 6 | 4.0 | 0.2188 0.2253 ± 0.0838 |

−20i | 7.0302 $\times {10}^{-1505}$ | 8.8753 $\times {10}^{-498}$ | 1.4044 | 7 | 4.0 | 0.2487 ± 0.0415 |

5 − 10i | 0.0 | 1.4936 $\times {10}^{-755}$ | 1.4044 | 7 | 4.0 | 0.2481 0.2419 ± 0.0176 |

−4.5 + 10i | 3.5457 $\times {10}^{-891}$ | 1.0501 $\times {10}^{-223}$ | 1.4044 | 6 | 4.0 | 0.2223 ± 0.0224 |

$Newton$ | 1.4479 $\times {10}^{-514}$ | 8.6274 $\times {10}^{-258}$ | 1.4044 | 10 | 2.0 | 0.2033 ± 0.0248 |

$Jarratt$ | 0.0 | 9.6997 $\times {10}^{-510}$ | 1.4044 | 6 | 4.0 | 0.1970 ± 0.0200 |

$CM1$ | 0.0 | 4.9393 $\times {10}^{-810}$ | 1.4044 | 5 | 6.0 | 0.2264 ± 0.0419 |

$CM2$ | 0.0 | 1.5533 $\times {10}^{-693}$ | 1.4044 | 5 | 6.0 | 0.2334 ± 0.0619 |

$KM$ | 0.0 | 9.821 $\times {10}^{-761}$ | 1.4044 | 5 | 6.0 | 0.2158 ± 0.0185 |

$\mathit{\alpha}$ | $\left|\mathit{f}\right({\mathit{x}}_{\mathit{k}+1}\left)\right|$ | $|{\mathit{x}}_{\mathit{k}+1}-{\mathit{x}}_{\mathit{k}}|$ | Solution | Iterations | ACOC | Time (s) |
---|---|---|---|---|---|---|

−50 | 0 | 1.5957 $\times {10}^{-569}$ | 1.4044 | 7 | 4.0 | 0.2347 ± 0.0293 |

−40 | 0 | 3.5896 $\times {10}^{-451}$ | 1.4044 | 7 | 4.0 | 0.2420 ± 0.0614 |

−20 + 45i | 1.1188 $\times {10}^{-1416}$ | 1.475 $\times {10}^{-409}$ | 1.4044 | 8 | 4.0 | 0.2589 ± 0.0262 |

−16 − 45i | 2.9743 $\times {10}^{-1380}$ | 8.4532 $\times {10}^{-372}$ | 1.4044 | 8 | 4.0 | 0.2603 ± 0.0305 |

1 | 0 | 7.1697 $\times {10}^{-507}$ | 1.4044 | 7 | 4.0 | 0.2263 ± 0.0155 |

−20i | 1.1363 $\times {10}^{-995}$ | 6.6845 $\times {10}^{-250}$ | 1.4044 | 7 | 4.0 | 0.2347 ± 0.0266 |

5 − 10i | 3.0563 $\times {10}^{-1262}$ | 2.6576 $\times {10}^{-316}$ | 1.4044 | 7 | 4.0 | 0.2314 ± 0.0152 |

−4.5 + 10i | 0 | 3.7277 $\times {10}^{-384}$ | 1.4044 | 7 | 4.0 | 0.2302 ± 0.0129 |

$Newton$ | 1.6796 $\times {10}^{-543}$ | 2.9384 $\times {10}^{-272}$ | 1.4044 | 11 | 2.0 | 0.2013 ± 0.0128 |

$Jarratt$ | 0.0 | 1.7167 $\times {10}^{-754}$ | 1.4044 | 7 | 4.0 | 0.1970 ± 0.0200 |

$CM1$ | 0.0 | 7.3559 $\times {10}^{-390}$ | 1.4044 | 5 | 6.0 | 0.2177 ± 0.0203 |

$CM2$ | 0.0 | 2.2786 $\times {10}^{-230}$ | 1.4044 | 5 | 6.0 | 0.2100 ± 0.0144 |

$KM$ | 0.0 | 9.2909 $\times {10}^{-265}$ | 1.4044 | 5 | 6.0 | 0.2083 ± 0.0436 |

$\mathit{\alpha}$ | $\left|\mathit{f}\right({\mathit{x}}_{\mathit{k}+1}\left)\right|$ | $|{\mathit{x}}_{\mathit{k}+1}-{\mathit{x}}_{\mathit{k}}|$ | Solution | Iterations | ACOC | Time (s) |
---|---|---|---|---|---|---|

−50 | 1.0082 $\times {10}^{-1007}$ | 4.6325 $\times {10}^{-399}$ | −14.137 | 8 | 4.0 | 0.2728 ± 0.0657 |

−40 | d | d | d | d | d | d |

−20 + 45i | 2.9787 $\times {10}^{-1009}$ | 2.2945 $\times {10}^{-324}$ | −17.278 | 12 | 4.0 | 0.3858 ± 0.0756 |

−16 − 45i | d | d | d | d | d | d |

1 | 0 | 5.7578 $\times {10}^{-315}$ | 0.517 | 6 | 4.0 | 0.2247 ± 0.0474 |

−20i | 1.9236 $\times {10}^{-1366}$ | 6.3944 $\times {10}^{-359}$ | 0.517 | 7 | 4.0 | 0.2566 ± 0.0551 |

5 − 10i | 0.0 | 6.7181 $\times {10}^{-632}$ | 0.517 | 7 | 4.0 | 0.2797 ± 0.0715 |

−4.5 + 10i | 3.0281 $\times {10}^{-1747}$ | 1.7704 $\times {10}^{-740}$ | 0.517 | 7 | 4.0 | 0.3034 ± 0.0963 |

$Newton$ | 1.4521 $\times {10}^{-498}$ | 7.5503 $\times {10}^{-250}$ | 0.517 | 10 | 2.0 | 0.2075 ± 0.0499 |

$Jarratt$ | 0.0 | 3.9685 $\times {10}^{-570}$ | 0.517 | 6 | 4.0 | 0.2198 ± 0.0529 |

$CM1$ | 0.0 | 4.9393 $\times {10}^{-810}$ | 1.4044 | 5 | 6.0 | 0.1923 ± 0.0152 |

$CM2$ | 0.0 | 1.5533 $\times {10}^{-693}$ | 1.4044 | 5 | 6.0 | 0.1922 ± 0.0168 |

$KM$ | 0.0 | 9.821 $\times {10}^{-761}$ | 1.4044 | 5 | 6.0 | 0.2158 ± 0.0185 |

$\mathit{\alpha}$ | $\left|\mathit{f}\right({\mathit{x}}_{\mathit{k}+1}\left)\right|$ | $|{\mathit{x}}_{\mathit{k}+1}-{\mathit{x}}_{\mathit{k}}|$ | Solution | Iterations | ACOC | Time (s) |
---|---|---|---|---|---|---|

−50 | 1.4177 $\times {10}^{-1008}$ | 5.6457 $\times {10}^{-294}$ | −1.8639 | 6 | 4.0 | 0.1848 ± 0.0133 |

−40 | 1.4177 $\times {10}^{-1008}$ | 5.1924 $\times {10}^{-678}$ | −1.8639 | 7 | 4.0 | 0.2048 ± 0.0155 |

−20 + 45i | d | d | d | d | d | d |

−16 − 45i | d | d | d | d | d | d |

1 | 0 | 1.6078 $\times {10}^{-702}$ | 0.5177 | 8 | 4.0 | 0.2273 ± 0.0189 |

−20i | 1.0214 $\times {10}^{-961}$ | 1.0043 $\times {10}^{-240}$ | −1.8639 | 9 | 4.0 | 0.2553 ± 0.0230 |

5 − 10i | 0.0 | 4.4053 $\times {10}^{-627}$ | 0.5177 | 22 | 4.0 | 0.5289 ± 0.0233 |

−4.5 + 10i | 1.4234 $\times {10}^{-979}$ | 2.2838 $\times {10}^{-242}$ | −29.8451 | 9 | 4.0 | 0.2653 ± 0.0487 |

$Newton$ | 2.4269 $\times {10}^{-437}$ | 3.0867 $\times {10}^{-219}$ | 0.5177 | 12 | 2.0 | 0.1956 ± 0.0260 |

$Jarratt$ | 0.0 | 1.0119 $\times {10}^{-612}$ | 0.5177 | 7 | 4.0 | 0.1909 ± 0.0243 |

$CM1$ | 0.0 | 4.8037 $\times {10}^{-559}$ | 0.5177 | 6 | 6.0 | 0.2230 ± 0.0273 |

$CM2$ | 0.0 | 1.5709 $\times {10}^{-609}$ | 0.5177 | 6 | 6.0 | 0.2208 ± 0.0261 |

$KM$ | 0.0 | 9.7594 $\times {10}^{-581}$ | 0.5177 | 5 | 6.0 | 0.2106 ± 0.0218 |

**Table 5.**${f}_{3}\left(x\right)=\sqrt{\frac{1}{f}}+0.86\mathrm{ln}\left(\frac{{10}^{-4}}{3.7}+\frac{2.51}{{10}^{5}\sqrt{\frac{1}{f}}}\right)$, ${x}_{0}=0.02$.

$\mathit{\alpha}$ | $\left|\mathit{f}\right({\mathit{x}}_{\mathit{k}}\left)\right|$ | $|{\mathit{x}}_{\mathit{k}+1}-{\mathit{x}}_{\mathit{k}}|$ | Solution | Iterations | ACOC | Time (s) |
---|---|---|---|---|---|---|

−50 | 2.2683 $\times {10}^{-1007}$ | 5.9163 $\times {10}^{-668}$ | 0.01885050 | 6 | 4.0 | 0.4098 ± 0.1367 |

−40 | 1.1341 $\times {10}^{-1007}$ | 3.1316 $\times {10}^{-702}$ | 0.01885050 | 6 | 4.0 | 0.3978 ± 0.0532 |

−20 + 45i | 2.4166 $\times {10}^{-667}$ | 0.0 | 0.01885050 | 6 | 4.0 | 0.4517 ± 0.1145 |

−16 − 45i | 1.5247 $\times {10}^{-1677}$ | 1.4073 $\times {10}^{-671}$ | 0.01885050 | 6 | 4.0 | 0.4555 ± 0.0999 |

1 | 1.1341 $\times {10}^{-1007}$ | 2.2713 $\times {10}^{-277}$ | 0.01885050 | 5 | 4.0 | 0.3361 ± 0.0419 |

−20i | 4.0218 $\times {10}^{-796}$ | 9.0943 $\times {10}^{-202}$ | 0.01885050 | 5 | 4.0 | 0.3386 ± 0.0255 |

5 − 10i | 1.5202 $\times {10}^{-872}$ | 8.1035 $\times {10}^{-221}$ | 0.01885050 | 5 | 4.0 | 0.3744 ± 0.0241 |

−4.5 + 10i | 3.2812 $\times {10}^{-893}$ | 5.7307 $\times {10}^{-226}$ | 0.01885050 | 5 | 4.0 | 0.3706 ± 0.0178 |

$Newton$ | 2.9268 $\times {10}^{-693}$ | 5.9571 $\times {10}^{-349}$ | 0.01885050 | 9 | 2.0 | 0.3630 ± 0.0332 |

$Jarratt$ | 1.1341 $\times {10}^{-1007}$ | 2.0266 $\times {10}^{-489}$ | 0.01885050 | 5 | 4.0 | 0.3319 ± 0.0655 |

$CM1$ | 1.1341 $\times {10}^{-1007}$ | 2.0266 $\times {10}^{-489}$ | 0.01885050 | 5 | 4.0 | 0.3447 ± 0.0301 |

$CM2$ | 1.1341 $\times {10}^{-1007}$ | 2.0266 $\times {10}^{-489}$ | 0.01885050 | 5 | 4.0 | 0.3494 ± 0.0252 |

$KM$ | 1.1341 $\times {10}^{-1007}$ | 2.0266 $\times {10}^{-489}$ | 0.01885050 | 5 | 4.0 | 0.3400 ± 0.0208 |

**Table 6.**${f}_{3}\left(x\right)=\sqrt{\frac{1}{f}}+0.86\mathrm{ln}\left(\frac{{10}^{-4}}{3.7}+\frac{2.51}{{10}^{5}\sqrt{\frac{1}{f}}}\right)$, ${x}_{0}=0.009$.

$\mathit{\alpha}$ | $\left|\mathit{f}\right({\mathit{x}}_{\mathit{k}}\left)\right|$ | $|{\mathit{x}}_{\mathit{k}+1}-{\mathit{x}}_{\mathit{k}}|$ | Solution | Iterations | ACOC | Time (s) |
---|---|---|---|---|---|---|

−50 | d | d | d | d | d | d |

−40 | d | d | d | d | d | d |

−20 + 45i | d | d | d | d | d | d |

−16 − 45i | d | d | d | d | d | d |

1 | 2.3085 $\times {10}^{-906}$ | 4.1347 $\times {10}^{-229}$ | 0.01885050 | 6 | 4.0 | 0.3880 ± 0.0206 |

−20i | 2.2683 $\times {10}^{-1007}$ | 1.9203 $\times {10}^{-445}$ | 0.01885050 | 9 | 4.0 | 0.6564 ± 0.0519 |

5 − 10i | 1.2533 $\times {10}^{-1118}$ | 2.48 $\times {10}^{-282}$ | 0.01885050 | 7 | 4.0 | 0.5059 ± 0.0328 |

−4.5 + 10i | 2.2683 $\times {10}^{-1007}$ | 2.5742 $\times {10}^{-397}$ | 0.01885050 | 7 | 4.0 | 0.5020 ± 0.0177 |

$Newton$ | 2.0584 $\times {10}^{-397}$ | 4.9958 $\times {10}^{-201}$ | 0.01885050 | 10 | 2.0 | 0.3942 ± 0.0227 |

$Jarratt$ | 1.1341 $\times {10}^{-1007}$ | 1.1036 $\times {10}^{-272}$ | 0.01885050 | 5 | 4.0 | 0.3200 ± 0.0233 |

$CM1$ | 0 | 1.8787 $\times {10}^{-675}$ | 0.01885050 | 5 | 6.0 | 0.4191 ± 0.0255 |

$CM2$ | 0 | 1.8799 $\times {10}^{-675}$ | 0.01885050 | 5 | 6.0 | 0.4194 ± 0.0354 |

$KM$ | 0 | 1.8793 $\times {10}^{-675}$ | 0.01885050 | 5 | 6.0 | 0.4064 ± 0.0168 |

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**MDPI and ACS Style**

Padilla, J.J.; Chicharro, F.I.; Cordero, A.; Torregrosa, J.R.
Parametric Family of Root-Finding Iterative Methods: Fractals of the Basins of Attraction. *Fractal Fract.* **2022**, *6*, 572.
https://doi.org/10.3390/fractalfract6100572

**AMA Style**

Padilla JJ, Chicharro FI, Cordero A, Torregrosa JR.
Parametric Family of Root-Finding Iterative Methods: Fractals of the Basins of Attraction. *Fractal and Fractional*. 2022; 6(10):572.
https://doi.org/10.3390/fractalfract6100572

**Chicago/Turabian Style**

Padilla, José J., Francisco I. Chicharro, Alicia Cordero, and Juan R. Torregrosa.
2022. "Parametric Family of Root-Finding Iterative Methods: Fractals of the Basins of Attraction" *Fractal and Fractional* 6, no. 10: 572.
https://doi.org/10.3390/fractalfract6100572