# Optimal One-Point Iterative Function Free from Derivatives for Multiple Roots

^{1}

^{2}

^{3}

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Formulation of Method

**Theorem**

**1.**

**Proof.**

**Theorem**

**2.**

**Proof.**

**Remark**

**1.**

**Theorem**

**3.**

**Proof.**

**Remark**

**2.**

- (1)
- $G\left({\mathsf{\Theta}}_{n}\right)=m{\mathsf{\Theta}}_{n}(1+{a}_{1}{\mathsf{\Theta}}_{n})$
- (2)
- $G\left({\mathsf{\Theta}}_{n}\right)=\frac{m{\mathsf{\Theta}}_{n}}{1+{a}_{2}{\mathsf{\Theta}}_{n}}$
- (3)
- $G\left({\mathsf{\Theta}}_{n}\right)=\frac{m{\mathsf{\Theta}}_{n}}{1+{a}_{3}m{\mathsf{\Theta}}_{n}}$,
- (4)
- $G\left({\mathsf{\Theta}}_{n}\right)=m({e}^{{\mathsf{\Theta}}_{n}}-1)$
- (5)
- $G\left({\mathsf{\Theta}}_{n}\right)=mlog({\mathsf{\Theta}}_{n}+1)$
- (6)
- $G\left({\mathsf{\Theta}}_{n}\right)=msin{\mathsf{\Theta}}_{n}$
- (7)
- $G\left({\mathsf{\Theta}}_{n}\right)=\frac{{\mathsf{\Theta}}_{n}}{{(\frac{1}{\sqrt{m}}+{a}_{4}{\mathsf{\Theta}}_{n})}^{2}}$
- (8)
- $G\left({\mathsf{\Theta}}_{n}\right)=\frac{{\mathsf{\Theta}}_{n}^{2}+{\mathsf{\Theta}}_{n}}{\frac{1}{m}+{a}_{5}{\mathsf{\Theta}}_{n}}$,

## 3. Basins of Attraction

**Problem**

**1.**

**Problem**

**2.**

## 4. Numerical Results

**Example**

**1**

**Example**

**2**

**Example**

**3.**

**Example**

**4.**

## 5. Conclusions

## Author Contributions

## Funding

## Conflicts of Interest

## References

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Methods | n | $|{\mathit{e}}_{\mathit{n}-2}|$ | $|{\mathit{e}}_{\mathit{n}-1}|$ | $|{\mathit{e}}_{\mathit{n}}|$ | $\mathit{f}\left({\mathit{x}}_{\mathit{n}+1}\right)$ | $\mathit{COC}$ |
---|---|---|---|---|---|---|

MNM | 7 | $1.70\times {10}^{-21}$ | $6.84\times {10}^{-43}$ | $1.11\times {10}^{-85}$ | $5.90\times {10}^{-681}$ | 2.000 |

M1 | 7 | $2.93\times {10}^{-21}$ | $2.09\times {10}^{-42}$ | $1.07\times {10}^{-84}$ | $4.70\times {10}^{-673}$ | 2.000 |

M2 | 7 | $3.27\times {10}^{-24}$ | $1.87\times {10}^{-48}$ | $6.10\times {10}^{-97}$ | $1.43\times {10}^{-771}$ | 2.000 |

M3 | 7 | $3.66\times {10}^{-23}$ | $1.84\times {10}^{-46}$ | $4.67\times {10}^{-93}$ | $6.47\times {10}^{-741}$ | 2.000 |

M4 | 7 | $3.49\times {10}^{-18}$ | $4.41\times {10}^{-36}$ | $7.04\times {10}^{-72}$ | $8.30\times {10}^{-570}$ | 2.000 |

M5 | 6 | $6.85\times {10}^{-15}$ | $5.28\times {10}^{-30}$ | $3.14\times {10}^{-60}$ | $1.21\times {10}^{-478}$ | 2.000 |

M6 | 7 | $2.03\times {10}^{-21}$ | $9.79\times {10}^{-43}$ | $2.27\times {10}^{-85}$ | $1.82\times {10}^{-678}$ | 2.000 |

M7 | 6 | $2.46\times {10}^{-13}$ | $8.35\times {10}^{-27}$ | $9.60\times {10}^{-54}$ | $2.06\times {10}^{-426}$ | 2.000 |

M8 | 7 | $6.88\times {10}^{-20}$ | $1.36\times {10}^{-39}$ | $5.33\times {10}^{-79}$ | $3.56\times {10}^{-627}$ | 2.000 |

Methods | n | $|{\mathit{e}}_{\mathit{n}-2}|$ | $|{\mathit{e}}_{\mathit{n}-1}|$ | $|{\mathit{e}}_{\mathit{n}}|$ | $\mathit{f}\left({\mathit{x}}_{\mathit{n}+1}\right)$ | $\mathit{COC}$ |
---|---|---|---|---|---|---|

MNM | 7 | $1.61\times {10}^{-20}$ | $6.51\times {10}^{-41}$ | $1.06\times {10}^{-81}$ | $1.86\times {10}^{-324}$ | 2.000 |

M1 | 7 | $3.60\times {10}^{-21}$ | $2.94\times {10}^{-41}$ | $1.95\times {10}^{-84}$ | $1.77\times {10}^{-335}$ | 2.000 |

M2 | 7 | $7.50\times {10}^{-18}$ | $2.11\times {10}^{-35}$ | $1.68\times {10}^{-70}$ | $2.71\times {10}^{-279}$ | 2.000 |

M3 | 7 | $2.63\times {10}^{-18}$ | $2.44\times {10}^{-36}$ | $2.08\times {10}^{-72}$ | $5.58\times {10}^{-287}$ | 2.000 |

M4 | 6 | $1.85\times {10}^{-22}$ | $4.10\times {10}^{-47}$ | $2.02\times {10}^{-96}$ | $5.76\times {10}^{-388}$ | 2.000 |

M5 | 7 | $6.66\times {10}^{-16}$ | $2.23\times {10}^{-31}$ | $2.48\times {10}^{-62}$ | $2.30\times {10}^{-246}$ | 2.000 |

M6 | 7 | $1.30\times {10}^{-20}$ | $4.26\times {10}^{-41}$ | $4.57\times {10}^{-82}$ | $6.60\times {10}^{-326}$ | 2.000 |

M7 | 7 | $1.41\times {10}^{-17}$ | $7.79\times {10}^{-35}$ | $2.38\times {10}^{-69}$ | $1.19\times {10}^{-274}$ | 2.000 |

M8 | 6 | $1.16\times {10}^{-17}$ | $6.61\times {10}^{-36}$ | $2.13\times {10}^{-72}$ | $1.18\times {10}^{-288}$ | 2.000 |

Methods | n | $|{\mathit{e}}_{\mathit{n}-2}|$ | $|{\mathit{e}}_{\mathit{n}-1}|$ | $|{\mathit{e}}_{\mathit{n}}|$ | $\mathit{f}\left({\mathit{x}}_{\mathit{n}+1}\right)$ | $\mathit{COC}$ |
---|---|---|---|---|---|---|

MNM | 9 | $3.65\times {10}^{-12}$ | $2.22\times {10}^{-22}$ | $8.24\times {10}^{-43}$ | $3.84\times {10}^{-168}$ | 2.000 |

M1 | 9 | $2.99\times {10}^{-12}$ | $1.49\times {10}^{-22}$ | $3.70\times {10}^{-43}$ | $1.55\times {10}^{-169}$ | 2.000 |

M2 | 9 | $9.51\times {10}^{-12}$ | $1.52\times {10}^{-21}$ | $3.88\times {10}^{-41}$ | $1.92\times {10}^{-161}$ | 2.000 |

M3 | 9 | $7.90\times {10}^{-12}$ | $1.05\times {10}^{-21}$ | $1.84\times {10}^{-41}$ | $9.58\times {10}^{-163}$ | 2.000 |

M4 | 9 | $4.35\times {10}^{-13}$ | $3.11\times {10}^{-24}$ | $1.59\times {10}^{-46}$ | $5.18\times {10}^{-183}$ | 2.000 |

M5 | 9 | $2.28\times {10}^{-11}$ | $8.82\times {10}^{-21}$ | $1.32\times {10}^{-39}$ | $2.58\times {10}^{-155}$ | 2.000 |

M6 | 9 | $3.81\times {10}^{-12}$ | $2.42\times {10}^{-22}$ | $9.74\times {10}^{-43}$ | $7.49\times {10}^{-168}$ | 2.000 |

M7 | 9 | $1.08\times {10}^{-11}$ | $1.95\times {10}^{-21}$ | $6.42\times {10}^{-41}$ | $1.44\times {10}^{-160}$ | 2.000 |

M8 | 9 | $7.78\times {10}^{-16}$ | $1.01\times {10}^{-29}$ | $1.69\times {10}^{-57}$ | $6.83\times {10}^{-227}$ | 2.000 |

Methods | n | $|{\mathit{e}}_{\mathit{n}-2}|$ | $|{\mathit{e}}_{\mathit{n}-1}|$ | $|{\mathit{e}}_{\mathit{n}}|$ | $\mathit{f}\left({\mathit{x}}_{\mathit{n}+1}\right)$ | $\mathit{COC}$ |
---|---|---|---|---|---|---|

MNM | 7 | $2.87\times {10}^{-15}$ | $2.75\times {10}^{-30}$ | $2.51\times {10}^{-60}$ | $5.82\times {10}^{-478}$ | 2.000 |

M1 | 7 | $2.88\times {10}^{-15}$ | $2.77\times {10}^{-30}$ | $2.55\times {10}^{-60}$ | $6.58\times {10}^{-478}$ | 2.000 |

M2 | 7 | $3.42\times {10}^{-15}$ | $3.97\times {10}^{-30}$ | $5.34\times {10}^{-60}$ | $2.59\times {10}^{-475}$ | 2.000 |

M3 | 7 | $4.44\times {10}^{-15}$ | $6.86\times {10}^{-30}$ | $1.64\times {10}^{-59}$ | $2.27\times {10}^{-471}$ | 2.000 |

M4 | 7 | $5.71\times {10}^{-15}$ | $1.16\times {10}^{-29}$ | $4.78\times {10}^{-59}$ | $1.30\times {10}^{-467}$ | 2.000 |

M5 | 7 | $5.49\times {10}^{-15}$ | $1.07\times {10}^{-29}$ | $4.09\times {10}^{-59}$ | $3.73\times {10}^{-468}$ | 2.000 |

M6 | 7 | $2.99\times {10}^{-15}$ | $2.99\times {10}^{-30}$ | $2.97\times {10}^{-60}$ | $2.22\times {10}^{-477}$ | 2.000 |

M7 | 7 | $4.48\times {10}^{-15}$ | $6.99\times {10}^{-30}$ | $1.70\times {10}^{-59}$ | $3.07\times {10}^{-471}$ | 2.000 |

M8 | 7 | $3.37\times {10}^{-15}$ | $3.83\times {10}^{-30}$ | $4.94\times {10}^{-60}$ | $1.37\times {10}^{-475}$ | 2.000 |

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**MDPI and ACS Style**

Kumar, D.; Sharma, J.R.; Argyros, I.K.
Optimal One-Point Iterative Function Free from Derivatives for Multiple Roots. *Mathematics* **2020**, *8*, 709.
https://doi.org/10.3390/math8050709

**AMA Style**

Kumar D, Sharma JR, Argyros IK.
Optimal One-Point Iterative Function Free from Derivatives for Multiple Roots. *Mathematics*. 2020; 8(5):709.
https://doi.org/10.3390/math8050709

**Chicago/Turabian Style**

Kumar, Deepak, Janak Raj Sharma, and Ioannis K. Argyros.
2020. "Optimal One-Point Iterative Function Free from Derivatives for Multiple Roots" *Mathematics* 8, no. 5: 709.
https://doi.org/10.3390/math8050709