# One-Point Optimal Family of Multiple Root Solvers of Second-Order

^{1}

^{2}

^{*}

## Abstract

**:**

## 1. Introduction

## 2. The Method

**Theorem**

**1.**

**Proof.**

#### Some Particular Forms of $G\left({\nu}_{n}\right)$

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

- Method 1 (M1):
- $${x}_{n+1}=\phantom{\rule{0.166667em}{0ex}}{x}_{n}-m{\nu}_{n}(1+{a}_{1}{\nu}_{n}).$$
- Method 2 (M2):
- $${x}_{n+1}=\phantom{\rule{0.166667em}{0ex}}{x}_{n}-\frac{m{\nu}_{n}}{1+{a}_{2}{\nu}_{n}}.$$
- Method 3 (M3):
- $${x}_{n+1}=\phantom{\rule{0.166667em}{0ex}}{x}_{n}-\frac{m{\nu}_{n}}{1+{a}_{3}m{\nu}_{n}}.$$
- Method 4 (M4):
- $${x}_{n+1}=\phantom{\rule{0.166667em}{0ex}}{x}_{n}-m({e}^{{\nu}_{n}}-1).$$
- Method 5 (M5):
- $${x}_{n+1}=\phantom{\rule{0.166667em}{0ex}}{x}_{n}-mlog({\nu}_{n}+1).$$
- Method 6 (M6):
- $${x}_{n+1}=\phantom{\rule{0.166667em}{0ex}}{x}_{n}-msin{\nu}_{n}.$$
- Method 7 (M7):
- $${x}_{n+1}=\phantom{\rule{0.166667em}{0ex}}{x}_{n}-\frac{{\nu}_{n}}{{(\frac{1}{\sqrt{m}}+{a}_{4}{\nu}_{n})}^{2}}.$$
- Method 8 (M8):
- $${x}_{n+1}=\phantom{\rule{0.166667em}{0ex}}{x}_{n}-\frac{{\nu}_{n}^{2}+{\nu}_{n}}{\frac{1}{m}+{a}_{5}{\nu}_{n}}.$$

**Remark**

**1.**

**Remark**

**2.**

## 3. Complex Dynamics of Methods

**Problem**

**1.**

**Problem**

**2.**

**Problem**

**3.**

**Problem**

**4.**

## 4. Numerical Results

**Example**

**1**(Eigenvalue problem)

**.**

**Example**

**2**(Beam Designing Model)

**.**

**Example**

**3.**

**Example**

**4.**

## 5. Conclusions

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

## References

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

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 | $2.79\times {10}^{-21}$ | $1.90\times {10}^{-42}$ | $8.77\times {10}^{-85}$ | $9.90\times {10}^{-674}$ | 2.000 |

M2 | 7 | $2.99\times {10}^{-24}$ | $1.56\times {10}^{-48}$ | $4.27\times {10}^{-97}$ | $8.37\times {10}^{-773}$ | 2.000 |

M3 | 6 | $2.17\times {10}^{-13}$ | $6.50\times {10}^{-27}$ | $5.80\times {10}^{-54}$ | $3.68\times {10}^{-428}$ | 2.000 |

M4 | 7 | $3.39\times {10}^{-18}$ | $4.18\times {10}^{-36}$ | $6.32\times {10}^{-72}$ | $3.53\times {10}^{-570}$ | 2.000 |

M5 | 6 | $5.79\times {10}^{-15}$ | $3.77\times {10}^{-30}$ | $1.60\times {10}^{-60}$ | $5.51\times {10}^{-481}$ | 2.000 |

M6 | 7 | $1.93\times {10}^{-21}$ | $8.86\times {10}^{-43}$ | $1.86\times {10}^{-85}$ | $3.69\times {10}^{-679}$ | 2.000 |

M7 | 6 | $2.28\times {10}^{-13}$ | $7.15\times {10}^{-27}$ | $7.03\times {10}^{-54}$ | $1.71\times {10}^{-427}$ | 2.000 |

M8 | 7 | $6.63\times {10}^{-20}$ | $1.26\times {10}^{-39}$ | $4.60\times {10}^{-79}$ | $1.09\times {10}^{-627}$ | 2.000 |

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

MNM | 7 | $1.61\times {10}^{-20}$ | $6.50\times {10}^{-41}$ | $1.06\times {10}^{-81}$ | $1.86\times {10}^{-324}$ | 2.000 |

M1 | 7 | $3.37\times {10}^{-21}$ | $2.55\times {10}^{-42}$ | $1.47\times {10}^{-84}$ | $5.64\times {10}^{-336}$ | 2.000 |

M2 | 7 | $7.19\times {10}^{-18}$ | $1.94\times {10}^{-35}$ | $1.41\times {10}^{-70}$ | $1.34\times {10}^{-279}$ | 2.000 |

M3 | 7 | $2.52\times {10}^{-18}$ | $2.22\times {10}^{-36}$ | $1.73\times {10}^{-72}$ | $2.63\times {10}^{-287}$ | 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 | $6.46\times {10}^{-16}$ | $2.09\times {10}^{-31}$ | $2.17\times {10}^{-62}$ | $1.34\times {10}^{-246}$ | 2.000 |

M6 | 7 | $1.23\times {10}^{-20}$ | $3.75\times {10}^{-41}$ | $3.52\times {10}^{-82}$ | $2.31\times {10}^{-326}$ | 2.000 |

M7 | 7 | $1.35\times {10}^{-17}$ | $7.17\times {10}^{-35}$ | $2.01\times {10}^{-69}$ | $6.02\times {10}^{-275}$ | 2.000 |

M8 | 6 | $1.34\times {10}^{-17}$ | $9.03\times {10}^{-36}$ | $4.08\times {10}^{-72}$ | $1.66\times {10}^{-287}$ | 2.000 |

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

MNM | 10 | $2.22\times {10}^{-22}$ | $8.24\times {10}^{-43}$ | $1.13\times {10}^{-83}$ | $1.36\times {10}^{-331}$ | 2.000 |

M1 | 10 | $1.49\times {10}^{-22}$ | $3.70\times {10}^{-43}$ | $2.27\times {10}^{-84}$ | $2.22\times {10}^{-334}$ | 2.000 |

M2 | 10 | $1.52\times {10}^{-21}$ | $3.88\times {10}^{-41}$ | $2.52\times {10}^{-80}$ | $3.43\times {10}^{-318}$ | 2.000 |

M3 | 10 | $1.05\times {10}^{-21}$ | $1.83\times {10}^{-41}$ | $5.64\times {10}^{-81}$ | $8.54\times {10}^{-321}$ | 2.000 |

M4 | 10 | $3.11\times {10}^{-24}$ | $1.59\times {10}^{-46}$ | $4.15\times {10}^{-91}$ | $2.39\times {10}^{-361}$ | 2.000 |

M5 | 10 | $8.82\times {10}^{-21}$ | $1.32\times {10}^{-39}$ | $2.93\times {10}^{-77}$ | $6.32\times {10}^{-306}$ | 2.000 |

M6 | 10 | $2.42\times {10}^{-22}$ | $9.73\times {10}^{-43}$ | $1.58\times {10}^{-83}$ | $5.16\times {10}^{-331}$ | 2.000 |

M7 | 10 | $1.95\times {10}^{-21}$ | $6.42\times {10}^{-41}$ | $6.92\times {10}^{-80}$ | $1.95\times {10}^{-316}$ | 2.000 |

M8 | 9 | $9.79\times {10}^{-17}$ | $1.57\times {10}^{-31}$ | $4.02\times {10}^{-61}$ | $2.10\times {10}^{-241}$ | 2.000 |

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

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 | $2.88\times {10}^{-15}$ | $2.76\times {10}^{-30}$ | $2.53\times {10}^{-60}$ | $6.21\times {10}^{-478}$ | 2.000 |

M2 | 7 | $3.41\times {10}^{-15}$ | $3.95\times {10}^{-30}$ | $5.30\times {10}^{-60}$ | $2.45\times {10}^{-475}$ | 2.000 |

M3 | 7 | $4.43\times {10}^{-15}$ | $6.84\times {10}^{-30}$ | $1.63\times {10}^{-59}$ | $2.14\times {10}^{-471}$ | 2.000 |

M4 | 7 | $5.70\times {10}^{-15}$ | $1.16\times {10}^{-29}$ | $4.75\times {10}^{-59}$ | $1.24\times {10}^{-467}$ | 2.000 |

M5 | 7 | $5.48\times {10}^{-15}$ | $1.07\times {10}^{-29}$ | $4.06\times {10}^{-59}$ | $3.53\times {10}^{-468}$ | 2.000 |

M6 | 7 | $2.99\times {10}^{-15}$ | $2.98\times {10}^{-30}$ | $2.95\times {10}^{-60}$ | $2.10\times {10}^{-477}$ | 2.000 |

M7 | 7 | $4.48\times {10}^{-15}$ | $6.97\times {10}^{-30}$ | $1.69\times {10}^{-59}$ | $2.90\times {10}^{-471}$ | 2.000 |

M8 | 7 | $3.37\times {10}^{-15}$ | $3.82\times {10}^{-30}$ | $4.91\times {10}^{-60}$ | $1.29\times {10}^{-475}$ | 2.000 |

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**MDPI and ACS Style**

Kumar, D.; Sharma, J.R.; Cesarano, C.
One-Point Optimal Family of Multiple Root Solvers of Second-Order. *Mathematics* **2019**, *7*, 655.
https://doi.org/10.3390/math7070655

**AMA Style**

Kumar D, Sharma JR, Cesarano C.
One-Point Optimal Family of Multiple Root Solvers of Second-Order. *Mathematics*. 2019; 7(7):655.
https://doi.org/10.3390/math7070655

**Chicago/Turabian Style**

Kumar, Deepak, Janak Raj Sharma, and Clemente Cesarano.
2019. "One-Point Optimal Family of Multiple Root Solvers of Second-Order" *Mathematics* 7, no. 7: 655.
https://doi.org/10.3390/math7070655