# Generating Optimal Eighth Order Methods for Computing Multiple Roots

^{1}

^{2}

^{3}

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Construction of the Method

**Theorem**

**1.**

**Proof.**

## 3. Some Special Cases of Weight Functions of G(u) and H(v)

**Case**

**1.**

**Case**

**2.**

**Case**

**3.**

## 4. Numerical Results

## 5. Basins of Attraction

## 6. Conclusions

## Author Contributions

## Funding

## Conflicts of Interest

## References

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**Figure 1.**Basins of attraction of $BM1$, $BM2$, $BM3$, $FM1$, $FM2$, $NM1$, $NM2$, $NM3$ for polynomial ${P}_{1}\left(z\right)$.

**Figure 2.**Basins of attraction of $BM1$, $BM2$, $BM3$, $FM1$, $FM2$, $NM1$, $NM2$, $NM3$ for polynomial ${P}_{2}\left(z\right)$.

**Figure 3.**Basins of attraction of $BM1$, $BM2$, $BM3$, $FM1$, $FM2$, $NM1$, $NM2$, $NM3$ for polynomial ${P}_{3}\left(z\right)$.

Example | Root | Multiplicity | Initial Guess | |||
---|---|---|---|---|---|---|

$\left(\alpha \right)$ | $\left(\mathit{m}\right)$ | $\left({\mathit{x}}_{\mathbf{0}}\right)$ | ||||

Example 1: Standard nonlinear function [26]: | ||||||

${f}_{1}\left(x\right)={\left(xlogx-\sqrt{x}+{x}^{3}\right)}^{3}$ | 1.000 | 3 | 1.50 | |||

Example 2: Standard nonlinear function [17]: | ||||||

${f}_{2}\left(x\right)={\left(x{e}^{{x}^{2}}-{sin}^{2}x+3cosx+5\right)}^{4}$ | $-1.21596937\dots $ | 4 | $-1.50$ | |||

Example 3: Standard nonlinear function [26]: | ||||||

${f}_{3}\left(x\right)=\left(9-2x-2{x}^{4}+cos2x\right)\left(5-x-{x}^{4}-{sin}^{2}x\right)$ | 1.29179850… | 2 | 1.50 | |||

Example 4: Eigen value problem [18]: | ||||||

${f}_{4}\left(x\right)={x}^{7}-17{x}^{6}+116{x}^{5}-410{x}^{4}+809{x}^{3}-893{x}^{2}+514x-120$ | 1 | 2 | 0.50 |

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

BM1 | 4 | $6.33\times {10}^{-03}$ | $7.63\times {10}^{-16}$ | $3.69\times {10}^{-119}$ | $5.90\times {10}^{-2834}$ | 7.9957 | 0.078 |

BM2 | 4 | $1.40\times {10}^{-03}$ | $1.20\times {10}^{-21}$ | $1.10\times {10}^{-062}$ | $2.79\times {10}^{-0554}$ | 2.2710 | 0.140 |

BM3 | 4 | $2.05\times {10}^{-03}$ | $4.23\times {10}^{-21}$ | $4.89\times {10}^{-061}$ | $1.81\times {10}^{-0539}$ | 2.2582 | 0.094 |

FM1 | 4 | $4.34\times {10}^{-03}$ | $4.36\times {10}^{-17}$ | $4.93\times {10}^{-129}$ | $9.54\times {10}^{-3071}$ | 7.9964 | 0.078 |

FM2 | 4 | $4.24\times {10}^{-03}$ | $3.64\times {10}^{-17}$ | $1.15\times {10}^{-129}$ | $6.83\times {10}^{-3086}$ | 7.9965 | 0.078 |

NM1 | 4 | $4.51\times {10}^{-03}$ | $3.05\times {10}^{-17}$ | $1.40\times {10}^{-130}$ | $9.34\times {10}^{-3109}$ | 7.9969 | 0.093 |

NM2 | 4 | $3.15\times {10}^{-03}$ | $1.74\times {10}^{-18}$ | $1.60\times {10}^{-140}$ | $2.35\times {10}^{-3347}$ | 7.9980 | 0.094 |

NM2 | 4 | $3.91\times {10}^{-03}$ | $9.80\times {10}^{-18}$ | $1.62\times {10}^{-134}$ | $2.96\times {10}^{-3203}$ | 7.9974 | 0.094 |

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

BM1 | 4 | $8.32\times {10}^{-03}$ | $5.37\times {10}^{-14}$ | $1.68\times {10}^{-103}$ | $1.03\times {10}^{-3270}$ | 7.9947 | 0.360 |

BM2 | 4 | $4.34\times {10}^{-04}$ | $1.47\times {10}^{-09}$ | $5.79\times {10}^{-026}$ | $2.57\times {10}^{-0293}$ | 3.0000 | 0.344 |

BM3 | 4 | $2.14\times {10}^{-03}$ | $2.15\times {10}^{-19}$ | $1.80\times {10}^{-055}$ | $2.12\times {10}^{-0647}$ | 2.2550 | 0.344 |

FM1 | 4 | $5.65\times {10}^{-03}$ | $3.48\times {10}^{-15}$ | $7.80\times {10}^{-113}$ | $1.04\times {10}^{-3568}$ | 7.9947 | 0.360 |

FM2 | 4 | $5.53\times {10}^{-03}$ | $2.93\times {10}^{-15}$ | $1.98\times {10}^{-113}$ | $9.76\times {10}^{-3588}$ | 7.9948 | 0.344 |

NM1 | 4 | $5.51\times {10}^{-03}$ | $8.72\times {10}^{-16}$ | $3.53\times {10}^{-118}$ | $6.94\times {10}^{-3742}$ | 7.9968 | 0.343 |

NM2 | 4 | $2.02\times {10}^{-03}$ | $2.92\times {10}^{-19}$ | $5.64\times {10}^{-146}$ | $2.28\times {10}^{-4631}$ | 7.9990 | 0.328 |

NM2 | 4 | $4.13\times {10}^{-03}$ | $8.67\times {10}^{-17}$ | $3.37\times {10}^{-126}$ | $1.64\times {10}^{-3998}$ | 7.9977 | 0.360 |

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

BM1 | 4 | $6.52\times {10}^{-05}$ | $2.95\times {10}^{-32}$ | $5.20\times {10}^{-251}$ | $4.85\times {10}^{-3999}$ | 7.9999 | 0.250 |

BM2 | 4 | $6.81\times {10}^{-09}$ | $2.06\times {10}^{-24}$ | $5.69\times {10}^{-71}$ | $2.96\times {10}^{-418}$ | 3.0000 | 0.219 |

BM3 | 4 | $7.63\times {10}^{-06}$ | $1.35\times {10}^{-40}$ | $1.61\times {10}^{-119}$ | $1.53\times {10}^{-709}$ | 2.2711 | 0.219 |

FM1 | 4 | $4.88\times {10}^{-05}$ | $3.85\times {10}^{-33}$ | $5.74\times {10}^{-258}$ | $1.10\times {10}^{-4085}$ | 7.9999 | 0.187 |

FM2 | 4 | $4.81\times {10}^{-05}$ | $3.40\times {10}^{-33}$ | $2.13\times {10}^{-258}$ | 0 | 7.9999 | 0.203 |

NM1 | 4 | $3.59\times {10}^{-05}$ | $1.29\times {10}^{-34}$ | $3.67\times {10}^{-270}$ | 0 | 7.9999 | 0.188 |

NM2 | 4 | $2.73\times {10}^{-05}$ | $1.44\times {10}^{-35}$ | $8.58\times {10}^{-278}$ | 0 | 7.9999 | 0.203 |

NM2 | 4 | $3.17\times {10}^{-05}$ | $4.79\times {10}^{-35}$ | $1.31\times {10}^{-273}$ | 0 | 7.9999 | 0.204 |

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

BM1 | 4 | $1.27\times {10}^{-03}$ | $1.47\times {10}^{-22}$ | $4.85\times {10}^{-147}$ | $7.90\times {10}^{-4155}$ | 7.9995 | 0.078 |

BM2 | 4 | $2.13\times {10}^{-04}$ | $1.91\times {10}^{-29}$ | $2.02\times {10}^{-086}$ | $3.25\times {10}^{-0769}$ | 2.2747 | 0.172 |

BM3 | 4 | $3.30\times {10}^{-04}$ | $5.27\times {10}^{-29}$ | $4.24\times {10}^{-085}$ | $2.55\times {10}^{-0757}$ | 2.2621 | 0.157 |

FM1 | 4 | $8.52\times {10}^{-04}$ | $6.89\times {10}^{-24}$ | $1.28\times {10}^{-184}$ | $1.48\times {10}^{-4408}$ | 7.9996 | 0.109 |

FM2 | 4 | $8.35\times {10}^{-04}$ | $5.89\times {10}^{-24}$ | $3.65\times {10}^{-185}$ | $1.23\times {10}^{-4421}$ | 7.9996 | 0.093 |

NM1 | 4 | $8.50\times {10}^{-04}$ | $3.78\times {10}^{-24}$ | $5.84\times {10}^{-187}$ | $1.59\times {10}^{-4465}$ | 7.9997 | 0.109 |

NM2 | 4 | $6.65\times {10}^{-04}$ | $5.32\times {10}^{-25}$ | $8.92\times {10}^{-194}$ | $4.20\times {10}^{-4629}$ | 7.9997 | 0.109 |

NM2 | 4 | $7.63\times {10}^{-04}$ | $1.60\times {10}^{-24}$ | $5.87\times {10}^{-190}$ | $1.81\times {10}^{-4537}$ | 7.9997 | 0.125 |

S. No. | Test Problems | m | Roots | Color of Fractal | Best Performer | Poor Performer |
---|---|---|---|---|---|---|

1 | ${P}_{1}\left(z\right)={({z}^{2}-1)}^{3}$ | 3 | $-1$ | green | BM1, NM2, NM3, | FM1, FM2 |

1 | red | BM2, BM3, NM1 | ||||

2 | ${P}_{2}\left(z\right)={({z}^{3}-z)}^{3}$ | 3 | $-1$ | red | BM3, BM2, NM2 | FM1, FM2 |

0 | green | NM1, BM1 | ||||

1 | blue | |||||

3 | ${P}_{3}\left(z\right)={z}^{4}-6{z}^{2}+8$ | 1 | $-2$ | red | BM3, NM2, BM2 | FM1, FM2 |

−1.414 | green | NM3, NM1, BM1 | ||||

1.414 | yellow | |||||

2 | blue |

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**MDPI and ACS Style**

Kumar, D.; Kumar, S.; Sharma, J.R.; d’Amore, M.
Generating Optimal Eighth Order Methods for Computing Multiple Roots. *Symmetry* **2020**, *12*, 1947.
https://doi.org/10.3390/sym12121947

**AMA Style**

Kumar D, Kumar S, Sharma JR, d’Amore M.
Generating Optimal Eighth Order Methods for Computing Multiple Roots. *Symmetry*. 2020; 12(12):1947.
https://doi.org/10.3390/sym12121947

**Chicago/Turabian Style**

Kumar, Deepak, Sunil Kumar, Janak Raj Sharma, and Matteo d’Amore.
2020. "Generating Optimal Eighth Order Methods for Computing Multiple Roots" *Symmetry* 12, no. 12: 1947.
https://doi.org/10.3390/sym12121947