# Optimal Derivative-Free One-Point Algorithms for Computing Multiple Zeros of Nonlinear Equations

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## Abstract

**:**

## 1. Introduction

## 2. Development of Novel Scheme

**Theorem**

**1.**

**Proof.**

**Theorem**

**2.**

**Proof.**

**Theorem**

**3.**

**Proof.**

**Remark**

**1.**

**Remark**

**2.**

## 3. Numerical Results

## 4. Conclusions

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

## References

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Problems | Root | Multiplicity | Initial Guess |
---|---|---|---|

Kepler’s problem [37] | |||

${\Phi}_{1}\left(y\right)=y-\frac{1}{4}\mathrm{sin}\left(y\right)-\frac{\pi}{5}$ | 0.8093… | 1 | 0.6 & 1 |

Van der Waals problem [38] | |||

${\Phi}_{2}\left(y\right)={y}^{3}-5.22{y}^{2}+9.0825y-5.2675$ | 1.75 | 2 | 2.2 & 2.5 |

Continuous-Stirred Tank Reactor [38,39] | |||

${\Phi}_{3}\left(y\right)={y}^{4}+11.50{y}^{3}+47.49{y}^{2}+83.06325y+51.23266875$ | −2.85 | 2 | −3.5 &−3.8 |

Academic problem [29] | |||

${\Phi}_{4}\left(y\right)=-\frac{{y}^{4}}{12}+\frac{{y}^{2}}{2}+y+{e}^{y}(y-3)+siny+3$ | 0 | 3 | −0.2 & 0.6 |

Complex root problem [28] | |||

${\Phi}_{5}\left(y\right)=y({y}^{2}+1)(2{e}^{{y}^{2}+1}+{y}^{2}-1){\left(\right)}^{cosh}3$ | i | 5 | 0.9i & 1.1i |

Root-Clustering problem [40] | |||

${\Phi}_{6}\left(y\right)={(y-1)}^{20}{(y-2)}^{15}{(y-3)}^{10}{(y-4)}^{5}$ | 3 | 10 | 2.9 |

1 | 20 | 0.7 |

Methods | q | $|{\mathit{y}}_{\mathit{q}-3}-{\mathit{y}}_{\mathit{q}-4}|$ | $|{\mathit{y}}_{\mathit{q}-2}-{\mathit{y}}_{\mathit{q}-3}|$ | $|{\mathit{y}}_{\mathit{q}-1}-{\mathit{y}}_{\mathit{q}-2}|$ | $|{\mathit{y}}_{\mathit{q}}-{\mathit{y}}_{\mathit{q}-1}|$ | COC |
---|---|---|---|---|---|---|

${y}_{0}=0.6$ | ||||||

TM | 6 | $1.96\times {10}^{-6}$ | $4.15\times {10}^{-13}$ | $1.87\times {10}^{-26}$ | $3.78\times {10}^{-53}$ | 2 |

KM1 | 35 | $8.28\times {10}^{-91}$ | $9.79\times {10}^{-94}$ | $1.16\times {10}^{-96}$ | $1.37\times {10}^{-99}$ | 1 |

KM2 | 39 | $4.03\times {10}^{-90}$ | $1.11\times {10}^{-92}$ | $3.07\times {10}^{-95}$ | $8.46\times {10}^{-98}$ | 1 |

KM3 | 38 | $1.20\times {10}^{-91}$ | $2.48\times {10}^{-94}$ | $5.14\times {10}^{-97}$ | $1.06\times {10}^{-99}$ | 1 |

KM4 | 35 | $8.02\times {10}^{-89}$ | $1.11\times {10}^{-91}$ | $1.52\times {10}^{-94}$ | $2.10\times {10}^{-97}$ | 1 |

NM1 | 6 | $1.38\times {10}^{-8}$ | $3.60\times {10}^{-18}$ | $2.44\times {10}^{-37}$ | $1.13\times {10}^{-75}$ | 2 |

NM2 | 6 | $4.38\times {10}^{-7}$ | $1.23\times {10}^{-14}$ | $9.74\times {10}^{-30}$ | $6.08\times {10}^{-60}$ | 2 |

NM3 | 6 | $8.01\times {10}^{-7}$ | $5.09\times {10}^{-14}$ | $2.05\times {10}^{-28}$ | $3.32\times {10}^{-57}$ | 2 |

${y}_{0}=1$ | ||||||

TM | 6 | $1.91\times {10}^{-6}$ | $3.96\times {10}^{-13}$ | $1.70\times {10}^{-26}$ | $3.13\times {10}^{-53}$ | 2 |

KM1 | 35 | $9.42\times {10}^{-91}$ | $1.11\times {10}^{-93}$ | $1.32\times {10}^{-96}$ | $1.56\times {10}^{-99}$ | 1 |

KM2 | 39 | $5.37\times {10}^{-90}$ | $1.48\times {10}^{-92}$ | $4.09\times {10}^{-95}$ | $1.13\times {10}^{-97}$ | 1 |

KM3 | 38 | $1.49\times {10}^{-91}$ | $3.09\times {10}^{-94}$ | $6.39\times {10}^{-97}$ | $1.32\times {10}^{-99}$ | 1 |

KM4 | 35 | $9.30\times {10}^{-89}$ | $1.28\times {10}^{-91}$ | $1.77\times {10}^{-94}$ | $2.44\times {10}^{-97}$ | 1 |

NM1 | 6 | $7.68\times {10}^{-9}$ | $1.11\times {10}^{-18}$ | $2.33\times {10}^{-38}$ | $1.02\times {10}^{-77}$ | 2 |

NM2 | 6 | $3.74\times {10}^{-7}$ | $8.99\times {10}^{-15}$ | $5.18\times {10}^{-30}$ | $1.72\times {10}^{-60}$ | 2 |

NM3 | 6 | $7.22\times {10}^{-7}$ | $4.12\times {10}^{-14}$ | $1.35\times {10}^{-28}$ | $1.43\times {10}^{-57}$ | 2 |

Methods | q | $|{\mathit{y}}_{\mathit{q}-3}-{\mathit{y}}_{\mathit{q}-4}|$ | $|{\mathit{y}}_{\mathit{q}-2}-{\mathit{y}}_{\mathit{q}-3}|$ | $|{\mathit{y}}_{\mathit{q}-1}-{\mathit{y}}_{\mathit{q}-2}|$ | $|{\mathit{y}}_{\mathit{q}}-{\mathit{y}}_{\mathit{q}-1}|$ | COC |
---|---|---|---|---|---|---|

${y}_{0}=2.2$ | ||||||

TM | 10 | $4.28\times {10}^{-9}$ | $3.05\times {10}^{-16}$ | $1.55\times {10}^{-30}$ | $4.03\times {10}^{-59}$ | 2 |

KM1 | 10 | $4.37\times {10}^{-9}$ | $3.19\times {10}^{-16}$ | $1.69\times {10}^{-30}$ | $4.77\times {10}^{-59}$ | 2 |

KM2 | 10 | $4.50\times {10}^{-9}$ | $3.37\times {10}^{-16}$ | $1.90\times {10}^{-30}$ | $5.99\times {10}^{-59}$ | 2 |

KM3 | 10 | $4.44\times {10}^{-9}$ | $3.29\times {10}^{-16}$ | $1.80\times {10}^{-30}$ | $5.42\times {10}^{-59}$ | 2 |

KM4 | 10 | $4.39\times {10}^{-9}$ | $3.21\times {10}^{-16}$ | $1.72\times {10}^{-30}$ | $4.91\times {10}^{-59}$ | 2 |

NM1 | 10 | $5.05\times {10}^{-10}$ | $4.25\times {10}^{-18}$ | $3.01\times {10}^{-34}$ | $1.51\times {10}^{-66}$ | 2 |

NM2 | 10 | $1.99\times {10}^{-9}$ | $6.60\times {10}^{-17}$ | $7.26\times {10}^{-32}$ | $8.79\times {10}^{-62}$ | 2 |

NM3 | 10 | $2.74\times {10}^{-9}$ | $1.25\times {10}^{-16}$ | $2.62\times {10}^{-31}$ | $1.14\times {10}^{-60}$ | 2 |

${y}_{0}=2.5$ | ||||||

TM | 11 | $2.05\times {10}^{-12}$ | $7.01\times {10}^{-23}$ | $8.19\times {10}^{-44}$ | $1.12\times {10}^{-85}$ | 2 |

KM1 | 11 | $2.23\times {10}^{-12}$ | $8.31\times {10}^{-23}$ | $1.55\times {10}^{-43}$ | $2.21\times {10}^{-85}$ | 2 |

KM2 | 11 | $2.50\times {10}^{-12}$ | $1.04\times {10}^{-22}$ | $1.81\times {10}^{-43}$ | $5.43\times {10}^{-85}$ | 2 |

KM3 | 11 | $2.38\times {10}^{-12}$ | $9.43\times {10}^{-23}$ | $1.48\times {10}^{-43}$ | $3.66\times {10}^{-85}$ | 2 |

KM4 | 11 | $2.26\times {10}^{-12}$ | $8.55\times {10}^{-23}$ | $1.22\times {10}^{-43}$ | $2.47\times {10}^{-85}$ | 2 |

NM1 | 9 | $3.52\times {10}^{-12}$ | $2.06\times {10}^{-22}$ | $7.11\times {10}^{-43}$ | $8.42\times {10}^{-84}$ | 2 |

NM2 | 10 | $2.46\times {10}^{-8}$ | $1.01\times {10}^{-14}$ | $1.70\times {10}^{-27}$ | $4.80\times {10}^{-53}$ | 2 |

NM3 | 11 | $1.61\times {10}^{-13}$ | $4.29\times {10}^{-25}$ | $3.07\times {10}^{-48}$ | $1.58\times {10}^{-94}$ | 2 |

Methods | q | $|{\mathit{y}}_{\mathit{q}-3}-{\mathit{y}}_{\mathit{q}-4}|$ | $|{\mathit{y}}_{\mathit{q}-2}-{\mathit{y}}_{\mathit{q}-3}|$ | $|{\mathit{y}}_{\mathit{q}-1}-{\mathit{y}}_{\mathit{q}-2}|$ | $|{\mathit{y}}_{\mathit{q}}-{\mathit{y}}_{\mathit{q}-1}|$ | COC |
---|---|---|---|---|---|---|

${y}_{0}=-3.5$ | ||||||

TM | 7 | $1.65\times {10}^{-7}$ | $3.61\times {10}^{-16}$ | $1.73\times {10}^{-33}$ | $4.00\times {10}^{-68}$ | 2 |

KM1 | 7 | $2.60\times {10}^{-7}$ | $1.31\times {10}^{-15}$ | $3.29\times {10}^{-32}$ | $2.09\times {10}^{-65}$ | 2 |

KM2 | 7 | $3.96\times {10}^{-7}$ | $4.29\times {10}^{-15}$ | $5.03\times {10}^{-31}$ | $6.90\times {10}^{-63}$ | 2 |

KM3 | 7 | $3.36\times {10}^{-7}$ | $2.68\times {10}^{-15}$ | $1.71\times {10}^{-31}$ | $6.96\times {10}^{-64}$ | 2 |

KM4 | 7 | $2.77\times {10}^{-7}$ | $1.55\times {10}^{-15}$ | $4.90\times {10}^{-32}$ | $4.87\times {10}^{-65}$ | 2 |

NM1 | 9 | $8.03\times {10}^{-8}$ | $1.54\times {10}^{-16}$ | $5.63\times {10}^{-34}$ | $7.54\times {10}^{-69}$ | 2 |

NM2 | 7 | $4.91\times {10}^{-8}$ | $5.74\times {10}^{-17}$ | $7.83\times {10}^{-35}$ | $1.46\times {10}^{-70}$ | 2 |

NM3 | 7 | $2.27\times {10}^{-8}$ | $1.23\times {10}^{-17}$ | $3.61\times {10}^{-36}$ | $3.11\times {10}^{-73}$ | 2 |

${y}_{0}=-3.8$ | ||||||

TM | D | D | D | D | D | D |

KM1 | D | D | D | D | D | D |

KM2 | D | D | D | D | D | D |

KM3 | D | D | D | D | D | D |

KM4 | D | D | D | D | D | D |

NM1 | D | D | D | D | D | D |

NM2 | 7 | $3.28\times {10}^{-6}$ | $2.56\times {10}^{-13}$ | $1.56\times {10}^{-27}$ | $5.81\times {10}^{-56}$ | 2 |

NM3 | 8 | $1.30\times {10}^{-10}$ | $4.01\times {10}^{-22}$ | $3.82\times {10}^{-45}$ | $3.48\times {10}^{-91}$ | 2 |

Methods | q | $|{\mathit{y}}_{\mathit{q}-3}-{\mathit{y}}_{\mathit{q}-4}|$ | $|{\mathit{y}}_{\mathit{q}-2}-{\mathit{y}}_{\mathit{q}-3}|$ | $|{\mathit{y}}_{\mathit{q}-1}-{\mathit{y}}_{\mathit{q}-2}|$ | $|{\mathit{y}}_{\mathit{q}}-{\mathit{y}}_{\mathit{q}-1}|$ | COC |
---|---|---|---|---|---|---|

${y}_{0}=-0.2$ | ||||||

TM | 6 | $1.62\times {10}^{-6}$ | $2.18\times {10}^{-13}$ | $3.96\times {10}^{-27}$ | $1.31\times {10}^{-54}$ | 2 |

KM1 | 6 | $1.62\times {10}^{-6}$ | $2.19\times {10}^{-13}$ | $4.00\times {10}^{-27}$ | $1.33\times {10}^{-54}$ | 2 |

KM2 | 6 | $1.63\times {10}^{-6}$ | $2.21\times {10}^{-13}$ | $4.05\times {10}^{-27}$ | $1.37\times {10}^{-54}$ | 2 |

KM3 | 6 | $1.62\times {10}^{-6}$ | $2.20\times {10}^{-13}$ | $4.03\times {10}^{-27}$ | $1.35\times {10}^{-54}$ | 2 |

KM4 | 6 | $1.62\times {10}^{-6}$ | $2.19\times {10}^{-13}$ | $4.01\times {10}^{-27}$ | $1.34\times {10}^{-54}$ | 2 |

NM1 | 6 | $1.65\times {10}^{-6}$ | $2.28\times {10}^{-13}$ | $4.33\times {10}^{-27}$ | $1.56\times {10}^{-54}$ | 2 |

NM2 | 6 | $1.64\times {10}^{-6}$ | $2.24\times {10}^{-13}$ | $4.18\times {10}^{-27}$ | $1.45\times {10}^{-54}$ | 2 |

NM3 | 6 | $1.63\times {10}^{-6}$ | $2.23\times {10}^{-13}$ | $4.13\times {10}^{-27}$ | $1.42\times {10}^{-54}$ | 2 |

${y}_{0}=0.6$ | ||||||

TM | 6 | $1.82\times {10}^{-6}$ | $2.76\times {10}^{-13}$ | $6.35\times {10}^{-27}$ | $3.36\times {10}^{-54}$ | 2 |

KM1 | 6 | $1.69\times {10}^{-6}$ | $2.39\times {10}^{-13}$ | $4.74\times {10}^{-27}$ | $1.87\times {10}^{-54}$ | 2 |

KM2 | 6 | $1.53\times {10}^{-6}$ | $1.95\times {10}^{-13}$ | $3.16\times {10}^{-27}$ | $8.33\times {10}^{-55}$ | 2 |

KM3 | 6 | $1.60\times {10}^{-6}$ | $2.13\times {10}^{-13}$ | $3.78\times {10}^{-27}$ | $1.19\times {10}^{-54}$ | 2 |

KM4 | 6 | $1.67\times {10}^{-6}$ | $2.33\times {10}^{-13}$ | $4.51\times {10}^{-27}$ | $1.70\times {10}^{-54}$ | 2 |

NM1 | 6 | $3.26\times {10}^{-6}$ | $8.84\times {10}^{-15}$ | $6.51\times {10}^{-30}$ | $3.53\times {10}^{-60}$ | 2 |

NM2 | 6 | $1.05\times {10}^{-6}$ | $9.27\times {10}^{-14}$ | $7.16\times {10}^{-28}$ | $4.27\times {10}^{-56}$ | 2 |

NM3 | 6 | $1.27\times {10}^{-6}$ | $1.34\times {10}^{-13}$ | $1.49\times {10}^{-27}$ | $1.84\times {10}^{-55}$ | 2 |

Methods | q | $|{\mathit{y}}_{\mathit{q}-3}-{\mathit{y}}_{\mathit{q}-4}|$ | $|{\mathit{y}}_{\mathit{q}-2}-{\mathit{y}}_{\mathit{q}-3}|$ | $|{\mathit{y}}_{\mathit{q}-1}-{\mathit{y}}_{\mathit{q}-2}|$ | $|{\mathit{y}}_{\mathit{q}}-{\mathit{y}}_{\mathit{q}-1}|$ | COC |
---|---|---|---|---|---|---|

${y}_{0}=0.9i$ | ||||||

TM | 7 | $3.34\times {10}^{-12}$ | $2.98\times {10}^{-24}$ | $2.37\times {10}^{-48}$ | $1.50\times {10}^{-96}$ | 2 |

KM1 | 7 | $3.35\times {10}^{-12}$ | $3.00\times {10}^{-24}$ | $2.40\times {10}^{-48}$ | $1.54\times {10}^{-96}$ | 2 |

KM2 | 7 | $3.37\times {10}^{-12}$ | $3.03\times {10}^{-24}$ | $2.44\times {10}^{-48}$ | $1.59\times {10}^{-96}$ | 2 |

KM3 | 7 | $3.36\times {10}^{-12}$ | $3.01\times {10}^{-24}$ | $2.42\times {10}^{-48}$ | $1.57\times {10}^{-96}$ | 2 |

KM4 | 7 | $3.36\times {10}^{-12}$ | $3.00\times {10}^{-24}$ | $2.41\times {10}^{-48}$ | $1.54\times {10}^{-96}$ | 2 |

NM1 | 7 | $3.45\times {10}^{-12}$ | $3.17\times {10}^{-24}$ | $2.67\times {10}^{-48}$ | $1.91\times {10}^{-96}$ | 2 |

NM2 | 7 | $3.41\times {10}^{-12}$ | $3.09\times {10}^{-24}$ | $2.55\times {10}^{-48}$ | $1.74\times {10}^{-96}$ | 2 |

NM3 | 7 | $3.39\times {10}^{-12}$ | $3.07\times {10}^{-24}$ | $2.52\times {10}^{-48}$ | $1.69\times {10}^{-96}$ | 2 |

${y}_{0}=1.1i$ | ||||||

TM | 6 | $8.96\times {10}^{-7}$ | $2.14\times {10}^{-13}$ | $1.22\times {10}^{-26}$ | $4.00\times {10}^{-53}$ | 2 |

KM1 | 6 | $8.93\times {10}^{-7}$ | $2.13\times {10}^{-13}$ | $1.20\times {10}^{-26}$ | $3.87\times {10}^{-53}$ | 2 |

KM2 | 6 | $8.88\times {10}^{-7}$ | $2.10\times {10}^{-13}$ | $1.18\times {10}^{-26}$ | $3.71\times {10}^{-53}$ | 2 |

KM3 | 6 | $8.90\times {10}^{-7}$ | $2.11\times {10}^{-13}$ | $1.19\times {10}^{-26}$ | $3.78\times {10}^{-53}$ | 2 |

KM4 | 6 | $8.92\times {10}^{-7}$ | $2.12\times {10}^{-13}$ | $1.20\times {10}^{-26}$ | $3.85\times {10}^{-53}$ | 2 |

NM1 | 6 | $8.88\times {10}^{-7}$ | $2.10\times {10}^{-13}$ | $1.18\times {10}^{-26}$ | $3.71\times {10}^{-53}$ | 2 |

NM2 | 6 | $8.89\times {10}^{-7}$ | $2.11\times {10}^{-13}$ | $1.18\times {10}^{-26}$ | $3.72\times {10}^{-53}$ | 2 |

NM3 | 6 | $8.88\times {10}^{-7}$ | $2.10\times {10}^{-13}$ | $1.18\times {10}^{-26}$ | $3.71\times {10}^{-53}$ | 2 |

Methods | q | $|{\mathit{y}}_{\mathit{q}-3}-{\mathit{y}}_{\mathit{q}-4}|$ | $|{\mathit{y}}_{\mathit{q}-2}-{\mathit{y}}_{\mathit{q}-3}|$ | $|{\mathit{y}}_{\mathit{q}-1}-{\mathit{y}}_{\mathit{q}-2}|$ | $|{\mathit{y}}_{\mathit{q}}-{\mathit{y}}_{\mathit{q}-1}|$ | COC |
---|---|---|---|---|---|---|

${y}_{0}=2.9,\lambda =10$ | ||||||

TM | 8 | $4.74\times {10}^{-11}$ | $4.50\times {10}^{-21}$ | $4.05\times {10}^{-41}$ | $3.27\times {10}^{-81}$ | 2 |

KM1 | 8 | $4.74\times {10}^{-11}$ | $4.50\times {10}^{-21}$ | $4.05\times {10}^{-41}$ | $3.27\times {10}^{-81}$ | 2 |

KM2 | 8 | $4.74\times {10}^{-11}$ | $4.50\times {10}^{-21}$ | $4.04\times {10}^{-41}$ | $3.27\times {10}^{-81}$ | 2 |

KM3 | 8 | $4.74\times {10}^{-11}$ | $4.50\times {10}^{-21}$ | $4.04\times {10}^{-41}$ | $3.27\times {10}^{-81}$ | 2 |

KM4 | 8 | $4.74\times {10}^{-11}$ | $4.50\times {10}^{-21}$ | $4.05\times {10}^{-41}$ | $3.27\times {10}^{-81}$ | 2 |

NM1 | 8 | $4.74\times {10}^{-11}$ | $4.49\times {10}^{-21}$ | $4.03\times {10}^{-41}$ | $3.24\times {10}^{-81}$ | 2 |

NM2 | 8 | $4.74\times {10}^{-11}$ | $4.49\times {10}^{-21}$ | $4.03\times {10}^{-41}$ | $3.26\times {10}^{-81}$ | 2 |

NM3 | 8 | $4.74\times {10}^{-11}$ | $4.49\times {10}^{-21}$ | $4.04\times {10}^{-41}$ | $3.26\times {10}^{-81}$ | 2 |

${y}_{0}=0.7,\lambda =20$ | ||||||

TM | 8 | $2.52\times {10}^{-10}$ | $6.88\times {10}^{-20}$ | $5.13\times {10}^{-39}$ | $2.85\times {10}^{-77}$ | 2 |

KM1 | 8 | $2.50\times {10}^{-10}$ | $6.75\times {10}^{-20}$ | $4.93\times {10}^{-39}$ | $2.63\times {10}^{-77}$ | 2 |

KM2 | 8 | $2.46\times {10}^{-10}$ | $6.57\times {10}^{-20}$ | $4.67\times {10}^{-39}$ | $2.37\times {10}^{-77}$ | 2 |

KM3 | 8 | $2.48\times {10}^{-10}$ | $6.65\times {10}^{-20}$ | $4.78\times {10}^{-39}$ | $2.48\times {10}^{-77}$ | 2 |

KM4 | 8 | $2.49\times {10}^{-10}$ | $6.72\times {10}^{-20}$ | $4.90\times {10}^{-39}$ | $2.60\times {10}^{-77}$ | 2 |

NM1 | 8 | $2.24\times {10}^{-10}$ | $5.45\times {10}^{-20}$ | $3.22\times {10}^{-39}$ | $1.12\times {10}^{-77}$ | 2 |

NM2 | 8 | $2.42\times {10}^{-10}$ | $6.36\times {10}^{-20}$ | $4.38\times {10}^{-39}$ | $2.08\times {10}^{-77}$ | 2 |

NM3 | 8 | $2.45\times {10}^{-10}$ | $6.48\times {10}^{-20}$ | $4.55\times {10}^{-39}$ | $2.25\times {10}^{-77}$ | 2 |

Problems | TM | KM1 | KM2 | KM3 | KM4 | NM1 | NM2 | NM3 |
---|---|---|---|---|---|---|---|---|

${\Phi}_{1}\left(y\right)$ | ||||||||

${y}_{0}=0.6$ | 0.6241 | 1.8892 | 2.1255 | 2.1376 | 2.1061 | 0.4217 | 0.4537 | 0.4641 |

${y}_{0}=1$ | 0.6417 | 2.0912 | 2.3059 | 2.2125 | 2.0249 | 0.4998 | 0.5014 | 0.5267 |

${\Phi}_{2}\left(y\right)$ | ||||||||

${y}_{0}=2.2$ | 0.0622 | 0.0631 | 0.0763 | 0.0872 | 0.0801 | 0.0591 | 0.0613 | 0.0624 |

${y}_{0}=2.5$ | 0.0590 | 0.0781 | 0.0782 | 0.0772 | 0.0792 | 0.0499 | 0.0581 | 0.0589 |

${\Phi}_{3}\left(y\right)$ | ||||||||

${y}_{0}=-3.5$ | 0.0779 | 0.0723 | 0.0781 | 0.0776 | 0.0765 | 0.0813 | 0.0697 | 0.0682 |

${y}_{0}=-3.8$ | - | - | - | - | - | - | 0.0741 | 0.0782 |

${\Phi}_{4}\left(y\right)$ | ||||||||

${y}_{0}=-0.2$ | 0.0498 | 0.0501 | 0.0598 | 0.0500 | 0.0499 | 0.0458 | 0.0478 | 0.0489 |

${y}_{0}=0.6$ | 0.0671 | 0.0690 | 0.0688 | 0.0683 | 0.0670 | 0.0655 | 0.0670 | 0.0669 |

${\Phi}_{5}\left(y\right)$ | ||||||||

${y}_{0}=0.9i$ | 0.5997 | 0.5675 | 0.5567 | 0.5431 | 0.5520 | 0.5317 | 0.5236 | 0.5224 |

${y}_{0}=1.1i$ | 0.5431 | 0.5155 | 0.4997 | 0.5107 | 0.4923 | 0.4991 | 0.4897 | 0.4872 |

${\Phi}_{6}\left(y\right)$ | ||||||||

${y}_{0}=2.9,\lambda =10$ | 0.1199 | 0.1123 | 0.1251 | 0.1370 | 0.1357 | 0.1080 | 0.1091 | 0.1110 |

${y}_{0}=0.7,\lambda =20$ | 0.1210 | 0.1321 | 0.1298 | 0.1234 | 0.1299 | 0.1199 | 0.1201 | 0.1190 |

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## Share and Cite

**MDPI and ACS Style**

Kumar, S.; Bhagwan, J.; Jäntschi, L.
Optimal Derivative-Free One-Point Algorithms for Computing Multiple Zeros of Nonlinear Equations. *Symmetry* **2022**, *14*, 1881.
https://doi.org/10.3390/sym14091881

**AMA Style**

Kumar S, Bhagwan J, Jäntschi L.
Optimal Derivative-Free One-Point Algorithms for Computing Multiple Zeros of Nonlinear Equations. *Symmetry*. 2022; 14(9):1881.
https://doi.org/10.3390/sym14091881

**Chicago/Turabian Style**

Kumar, Sunil, Jai Bhagwan, and Lorentz Jäntschi.
2022. "Optimal Derivative-Free One-Point Algorithms for Computing Multiple Zeros of Nonlinear Equations" *Symmetry* 14, no. 9: 1881.
https://doi.org/10.3390/sym14091881