# A Class of Sixth-Order Iterative Methods for Solving Nonlinear Systems: The Convergence and Fractals of Attractive Basins

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Preparation for Convergence Analysis

- •
- On the interval ${I}_{1}=[0,\infty )$, the function ${\mathcal{D}}_{1}:{I}_{1}\to {I}_{1}$ exists and satisfies condition ${\mathcal{D}}_{1}(0)=0$.
- •
- Let ${s}_{min}$ exist, where ${s}_{min}$ is the least positive solution satisfying ${\mathcal{D}}_{1}(s)=1$.
- •
- On the interval ${I}_{2}=[0,{s}_{min})$, the function ${\mathcal{D}}_{2}:{I}_{2}\to {I}_{1}$ exists and satisfies condition ${\mathcal{D}}_{2}(0)=0$.
- •
- On the interval ${I}_{2}=[0,{s}_{min})$, the function ${\mathcal{D}}_{3}:{I}_{2}\to {I}_{1}$ exists and satisfies condition ${\mathcal{D}}_{3}(0)<1$.

## 3. Analysis of Convergence

**Theorem**

**1.**

**Proof.**

- •
- ${D}_{2}=-2{A}_{2};$
- •
- ${D}_{3}=4{A}_{2}^{2}-3{A}_{3};$
- •
- ${D}_{4}=-8{A}_{2}^{3}+12{A}_{2}{A}_{3}-4{A}_{4}$;
- •
- ${D}_{5}=16{A}_{2}^{4}+9{A}_{3}^{2}+16{A}_{2}{A}_{4}-36{A}_{2}^{2}{A}_{3}-5{A}_{5}.$

**Theorem**

**2.**

**Proof.**

## 4. Fractals of Attractive Basins

## 5. Numerical Experiments and Practical Applications

#### 5.1. Solving Nonlinear Systems

#### 5.2. Solving the Matrix Sign Function

- •
- ${X}_{n+1}=8192{X}_{n}^{19}{[-I+4{X}_{n}^{2}-27{X}_{n}^{4}+120{X}_{n}^{6}-306{X}_{n}^{8}-2174{X}_{n}^{12}+4104{X}_{n}^{14}-7421{X}_{n}^{16}\phantom{\rule{0ex}{0ex}}+\mathrm{11,068}{X}_{n}^{18}+1737{X}_{n}^{20}]}^{-1}$,
- •
- ${X}_{n+1}=1024{X}_{n}^{13}{[I-13{X}_{n}^{2}+85{X}_{n}^{4}-305{X}_{n}^{6}+659{X}_{n}^{8}-951{X}_{n}^{10}+1303{X}_{n}^{12}+245{X}_{n}^{14}]}^{-1}$,
- •
- ${X}_{n+1}=-{(I+{X}_{n}^{2})}^{3}{\left[2{X}_{n}^{3}(3+8X{n}^{2}+9{X}_{n}^{4})\right]}^{-1}$,
- •
- ${X}_{n+1}=-\mathrm{82,944}{X}_{n}^{14}(-I+4{X}_{n}^{4})[-160I+320{X}_{n}^{2}+3408{X}_{n}^{3}+3040{X}_{n}^{4}-6816{X}_{n}^{5}\phantom{\rule{0ex}{0ex}}-2512{X}_{n}^{6}-\mathrm{53,424}{X}_{n}^{7}-\mathrm{18,656}{X}_{n}^{8}-\mathrm{104,856}{X}_{n}^{9}+\mathrm{24,560}{X}_{n}^{10}+\mathrm{625,584}{X}_{n}^{11}-\mathrm{960,810}{X}_{n}^{12}\phantom{\rule{0ex}{0ex}}+\mathrm{1,057,896}{X}_{n}^{13}+\mathrm{1,020,244}{X}_{n}^{14}-\mathrm{4,451,451}{X}_{n}^{15}+\mathrm{3,955,646}{X}_{n}^{16}+\mathrm{55,830}{X}_{n}^{17}\phantom{\rule{0ex}{0ex}}-\mathrm{4,435,280}{X}_{n}^{18}+\mathrm{5,390,181}{X}_{n}^{19}+\mathrm{89,000}{X}_{n}^{20}-\mathrm{3,637,632}{X}_{n}^{21}+\mathrm{160,768}{X}_{n}^{22}+\mathrm{872,448}{X}_{n}^{23}\phantom{\rule{0ex}{0ex}}+\mathrm{163,840}{X}_{n}^{24}]\text{respectively.}$

#### 5.3. Practical Applications

^{2}and b = 0.0371 L/mol. Then, the volume of the container is found by considering the pressure of 945.36 kPa (9.33 atm) and the temperature of 300.2 K with 2 mol nitrogen. Finally, by substituting the data into (60), we obtain

## 6. Conclusions and Discussion

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## References

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**Figure 1.**The attractive basins of Iterative Methods $M1$–$M4$ under the nonlinear function $f(x)={x}^{2}-1$.

**Figure 2.**The attractive basins of Iterative Methods $M1$–$M4$ under the nonlinear function $f(x)={x}^{3}-1$.

**Table 1.**Comparison of the average number of iterations after five iterations of Iterative Methods $M1$–$M4$.

$\mathit{M}1$ | $\mathit{M}2$ | $\mathit{M}3$ | $\mathit{M}4$ | |
---|---|---|---|---|

$f(x)={x}^{2}-1$ | $5.4303$ | $3.9349$ | $13.243$ | $3.9349$ |

$f(x)={x}^{3}-1$ | $11.827$ | $7.3072$ | $10.359$ | $7.2266$ |

The Iterative Method | k | $\parallel {\mathit{x}}^{(\mathit{k})}-{\mathit{x}}^{(\mathit{k}-1)}\parallel $ | $\parallel \mathcal{F}({\mathit{x}}^{(\mathit{k})})\parallel $ |
---|---|---|---|

$M2$ | 4 | $9.193\times {10}^{-158}$ | $7.704\times {10}^{-946}$ |

$M4$ | 4 | $8.793\times {10}^{-240}$ | $9.423\times {10}^{-1439}$ |

$M5$ | 4 | $1.350\times {10}^{-260}$ | $1.409\times {10}^{-1239}$ |

$M6$ | 4 | $2.427\times {10}^{-232}$ | $6.265\times {10}^{-1394}$ |

$M7$ | 5 | $7.121\times {10}^{-144}$ | $9.423\times {10}^{-575}$ |

The Iterative Method | k | $\parallel {\mathit{x}}^{(\mathit{k})}-{\mathit{x}}^{(\mathit{k}-1)}\parallel $ | $\parallel \mathcal{F}({\mathit{x}}^{(\mathit{k})})\parallel $ |
---|---|---|---|

$M2$ | 5 | $3.985\times {10}^{-562}$ | $1.000\times {10}^{-2048}$ |

$M4$ | 5 | $1.018\times {10}^{-568}$ | $1.000\times {10}^{-2048}$ |

$M5$ | 4 | $5.015\times {10}^{-138}$ | $1.409\times {10}^{-823}$ |

$M6$ | 5 | $9.737\times {10}^{-569}$ | $1.000\times {10}^{-2048}$ |

$M7$ | 4 | $1.483\times {10}^{-107}$ | $2.358\times {10}^{-639}$ |

The Iterative Method | k | $\parallel {\mathit{x}}^{(\mathit{k})}-{\mathit{x}}^{(\mathit{k}-1)}\parallel $ | $\parallel \mathcal{F}({\mathit{x}}^{(\mathit{k})})\parallel $ |
---|---|---|---|

$M2$ | 5 | $2.417\times {10}^{-297}$ | $1.111\times {10}^{-1779}$ |

$M4$ | 5 | $6.170\times {10}^{-464}$ | $6.000\times {10}^{-2048}$ |

$M5$ | 5 | $4.248\times {10}^{-465}$ | $3.000\times {10}^{-2048}$ |

$M6$ | 6 | $9.128\times {10}^{-409}$ | $3.000\times {10}^{-2048}$ |

$M7$ | 5 | $8.615\times {10}^{-290}$ | $9.599\times {10}^{-1734}$ |

The Iterative Method | Matrices | n | t |
---|---|---|---|

$M2$ | 1 | $nc$ | $nc$ |

4 | $nc$ | $nc$ | |

10 | $nc$ | $nc$ | |

15 | $nc$ | $nc$ | |

$M4$ | 1 | 1 | $0.008228$ |

4 | 18 | $0.040754$ | |

10 | 22 | $0.051711$ | |

15 | 36 | $0.089662$ | |

$M5$ | 1 | $nc$ | $nc$ |

4 | $nc$ | $nc$ | |

10 | $nc$ | $nc$ | |

15 | $nc$ | $nc$ | |

$M6$ | 1 | 1 | $0.035088$ |

4 | $nc$ | $nc$ | |

10 | $nc$ | $nc$ | |

15 | $nc$ | $nc$ |

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**MDPI and ACS Style**

Wang, X.; Li, W.
A Class of Sixth-Order Iterative Methods for Solving Nonlinear Systems: The Convergence and Fractals of Attractive Basins. *Fractal Fract.* **2024**, *8*, 133.
https://doi.org/10.3390/fractalfract8030133

**AMA Style**

Wang X, Li W.
A Class of Sixth-Order Iterative Methods for Solving Nonlinear Systems: The Convergence and Fractals of Attractive Basins. *Fractal and Fractional*. 2024; 8(3):133.
https://doi.org/10.3390/fractalfract8030133

**Chicago/Turabian Style**

Wang, Xiaofeng, and Wenshuo Li.
2024. "A Class of Sixth-Order Iterative Methods for Solving Nonlinear Systems: The Convergence and Fractals of Attractive Basins" *Fractal and Fractional* 8, no. 3: 133.
https://doi.org/10.3390/fractalfract8030133