# On Constructing a Family of Sixth-Order Methods for Multiple Roots

## Abstract

**:**

## 1. Introduction

**Definition**

**1.**

**Definition**

**2.**

**Definition**

**3.**

## 2. Convergence Analysis

**Theorem**

**1.**

## 3. Numerical Experiments

**CPU**,

**tcon**,

**avg**, and

**tdiv**denote the value of the CPU time for sixth-order convergence, the total number of convergent points, the value of the average iterative number for sixth-order convergence, and the number of divergent points. Table 3 shows the statistical data for the basins of attraction.

**Example**

**1.**

**CPU**, and M4 is better in terms of

**avg**for ${p}_{1}\left(z\right)$. There are similar lobes along the boundaries of the basins in (a), (b), and (c). Picture (d) has colored (yellow, magenta, green, blue, and so on) carrot shapes around the boundary. Methods (X1) and (X2) do not have the value of the total convergent point, and pictures (e) and (f) show the black region.

**Example**

**2.**

**avg**. Basins for $\alpha =-1$ are painted red and basins for $\alpha =1$ are blue. There are similar lobes along the boundaries of the basins in Figure 2a–d has colored (blue, green, magenta) carrot shapes around the boundary. Methods (X1) and (X2) have considerable black points and circular patterns made up of blue diamond shapes in Figure 2e,f.

**Example**

**3.**

**CPU**and

**avg**. The basins of method (X1) are represented by black regions and the basins of method (X2) are represented by circular light blue patterns in Figure 3.

## 4. Conclusions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## References

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**Figure 1.**Basins of attraction for ${P}_{1}\left(z\right)={(2{z}^{2}+1)}^{2}$. (

**a**) $M1$; (

**b**) $M2$; (

**c**) $M3$; (

**d**) $M4$; (

**e**) $X1$; (

**f**) $X2$.

**Figure 2.**Basins of attraction for ${P}_{2}\left(z\right)={({z}^{2}-1)}^{3}$. (

**a**) $M1$; (

**b**) $M2$; (

**c**) $M3$; (

**d**) $M4$; (

**e**) $X1$; (

**f**) $X2$.

**Figure 3.**Basins of attraction for ${P}_{3}\left(z\right)={({z}^{3}-1)}^{2}$. (

**a**) $M1$; (

**b**) $M2$; (

**c**) $M3$; (

**d**) $M4$; (

**e**) $X1$; (

**f**) $X2$.

Method | Function | n | ${\mathit{x}}_{\mathit{n}}$ | $|{\mathit{x}}_{\mathit{n}}-\mathit{\alpha}|$ | $|{\mathit{e}}_{\mathit{n}}/{{\mathit{e}}_{\mathit{n}-1}}^{6}|$ | ${\mathit{p}}_{\mathit{n}}$ |
---|---|---|---|---|---|---|

$M1$ | ${h}_{1}$ | 0 | $-2.1$ | 0.0652751 | ||

1 | $-2.0372492017726$ | $4.913\times {10}^{-8}$ | 0.3089431095 | 6.07933 | ||

2 | $-2.0372476627913$ | $5.378\times {10}^{-45}$ | 0.428220700 | 6.00000 | ||

3 | $-2.0372476627913$ | $0.0\times {10}^{-98}$ | ||||

$M2$ | ${h}_{2}$ | 0 | $2.0$ | 0.0350903 | ||

1 | $2.0350902813353$ | $2.389\times {10}^{-7}$ | 26.31721953 | 6.10370 | ||

2 | $2.0350903306632$ | $7.978\times {10}^{-45}$ | 38.017167 | 6.00000 | ||

3 | $2.0350903306632$ | $0.0\times {10}^{-99}$ | ||||

$M3$ | ${h}_{3}$ | 0 | $1.084$ | 0.0425181 | ||

1 | $1.04148199694193$ | $1.263\times {10}^{-7}$ | 21.38733354 | 6.06612 | ||

2 | $1.041481808433$ | $1.076\times {10}^{-40}$ | 26.44205449 | 6.00000 | ||

3 | $1.041481808433$ | $0.0\times {10}^{-99}$ | ||||

$M4$ | ${h}_{4}$ | 0 | $1.35$ | 0.0582667 | ||

1 | $1.29173359504767$ | $3.07\times {10}^{-7}$ | 7.7330687 | 6.23175 | ||

2 | $1.29173329265770$ | $9.550\times {10}^{-40}$ | 12.73465793 | 6.00000 | ||

3 | $1.29173329265770$ | $0.0\times {10}^{-99}$ |

**Table 2.**Comparison of $|{x}_{n}-\alpha |$ for ${f}_{1}\left(x\right),{f}_{2}\left(x\right)$ and ${f}_{3}\left(x\right)$.

Function | $|{\mathit{x}}_{\mathit{n}}-\mathit{\alpha}|$ | $\mathit{M}1$ | $\mathit{M}2$ | $\mathit{M}3$ | $\mathit{M}4$ | $\mathit{X}1$ | $\mathit{X}2$ |
---|---|---|---|---|---|---|---|

${f}_{1}$ | $|{x}_{1}-\alpha |$ | $2.31\times {10}^{-11}$ | $2.61\times {10}^{-7}$ | $7.76\times {10}^{-10}$ | $1.21\times {10}^{-8}$ | $3.11\times {10}^{-7}$ | $2.63\times {10}^{-6}$ |

$|{x}_{2}-\alpha |$ | $\mathbf{3}.\mathbf{71}\times {\mathbf{10}}^{-\mathbf{62}}$ | $2.35\times {10}^{-43}$ | $2.42\times {10}^{-55}$ | $3.51\times {10}^{-47}$ | $8.52\times {10}^{-40}$ | $7.16\times {10}^{-41}$ | |

${f}_{2}$ | $|{x}_{1}-\alpha |$ | $5.16\times {10}^{-7}$ | $5.57\times {10}^{-8}$ | $3.12\times {10}^{-7}$ | $2.12\times {10}^{-7}$ | $5.32\times {10}^{-6}$ | $2.22\times {10}^{-5}$ |

$|{x}_{2}-\alpha |$ | $\mathbf{2}.\mathbf{31}\times {\mathbf{10}}^{-\mathbf{41}}$ | $1.66\times {10}^{-40}$ | $4.16\times {10}^{-39}$ | $3.51\times {10}^{-38}$ | $2.12\times {10}^{-37}$ | $5.31\times {10}^{-39}$ | |

${f}_{3}$ | $|{x}_{1}-\alpha |$ | $2.71\times {10}^{-6}$ | $3.46\times {10}^{-6}$ | $1.67\times {10}^{-5}$ | $1.88\times {10}^{-6}$ | $7.62\times {10}^{-5}$ | $4.42\times {10}^{-8}$ |

$|{x}_{2}-\alpha |$ | $\mathbf{3}.\mathbf{71}\times {\mathbf{10}}^{-\mathbf{42}}$ | $5.23\times {10}^{-42}$ | $5.11\times {10}^{-41}$ | $6.75\times {10}^{-40}$ | $6.51\times {10}^{-35}$ | $4.12\times {10}^{-41}$ |

${\mathit{p}}_{\mathit{m}}$ | Method | CPU | tcon | avg | tdiv |
---|---|---|---|---|---|

${p}_{1}\left(z\right)$ | M1 | 120.922 | 360,000 | 8.73961 | 0 |

M2 | 165.969 | 360,000 | 8.96783 | 0 | |

M3 | 177.515 | 360,000 | 8.96783 | 0 | |

M4 | 183.984 | 354,440 | 8.15107 | 5560 | |

X1 | 1234.78 | 0 | - | 360,000 | |

X2 | 333.578 | 0 | - | 360,000 | |

${p}_{2}\left(z\right)$ | M1 | 274.985 | 360,000 | 9.09136 | 0 |

M2 | 340.906 | 360,000 | 9.19766 | 0 | |

M3 | 266.187 | 360,000 | 9.07998 | 0 | |

M4 | 306.485 | 360,000 | 7.92103 | 0 | |

X1 | 10.822 | 20,880 | 142.582 | 339,120 | |

X2 | 1575.34 | 6118 | 37.9915 | 353,882 | |

${p}_{3}\left(z\right)$ | M1 | 3104.25 | 360,000 | 9.30217 | 107,540 |

M2 | 3584.16 | 249,022 | 8.94829 | 110,978 | |

M3 | 3660.17 | 249,022 | 8.94829 | 110,978 | |

M4 | 497.906 | 287,654 | 5.48361 | 72,346 | |

X1 | 519.563 | 0 | - | 360,000 | |

X2 | 3443.61 | 1054 | 26.759 | 358,946 |

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**MDPI and ACS Style**

Geum, Y.H.
On Constructing a Family of Sixth-Order Methods for Multiple Roots. *Fractal Fract.* **2023**, *7*, 878.
https://doi.org/10.3390/fractalfract7120878

**AMA Style**

Geum YH.
On Constructing a Family of Sixth-Order Methods for Multiple Roots. *Fractal and Fractional*. 2023; 7(12):878.
https://doi.org/10.3390/fractalfract7120878

**Chicago/Turabian Style**

Geum, Young Hee.
2023. "On Constructing a Family of Sixth-Order Methods for Multiple Roots" *Fractal and Fractional* 7, no. 12: 878.
https://doi.org/10.3390/fractalfract7120878