# One Parameter Optimal Derivative-Free Family to Find the Multiple Roots of Algebraic Nonlinear Equations

^{1}

^{2}

^{3}

^{4}

^{5}

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Construction of the Second-Order Scheme

#### Convergence Analysis

**Lemma**

**1.**

**Proof.**

**Lemma**

**2.**

**Proof.**

**Theorem 1.**

**Proof.**

## 3. Numerical Illustration

**Example**

**1.**

**Example**

**2.**

**Example**

**3.**

**Example**

**4.**

**Example**

**5.**

**Example**

**6.**

## 4. Basin of Attraction

- a superattractor if $|{R}^{\prime}\left({z}_{0}\right)|=0$
- an attractor if $|{R}^{\prime}\left({z}_{0}\right)|<1$
- a repulsor if $|{R}^{\prime}\left({z}_{0}\right)|>1$
- and parabolic if $|{R}^{\prime}\left({z}_{0}\right)|=1$.

**Problem**

**1.**

**Problem**

**2.**

**Problem**

**3.**

## 5. Conclusions

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

## References

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Method Representation | $\mathit{G}\left(\mathit{\theta}\right)$ |
---|---|

$D{M}_{1}$ | $m\theta (1+\frac{1}{10}\theta )$ |

$D{M}_{2}$ | $\frac{m\theta}{1+\frac{1}{4}\theta}$ |

$D{M}_{3}$ | $\frac{m\theta}{1+m\frac{1}{10}\theta}$ |

$D{M}_{4}$ | $m{e}^{\theta}-1$ |

$D{M}_{5}$ | $mlog(\theta +1)$ |

$D{M}_{6}$ | $\frac{\theta}{{\left(\right)}^{\sqrt{(}}}$ |

$D{M}_{7}$ | $\frac{{\theta}^{2}+\theta}{\frac{1}{m}+\frac{1}{5}\theta}$ |

Schemes | $|{\mathit{x}}_{5}-{\mathit{x}}_{4}|$ | $|{\mathit{x}}_{6}-{\mathit{x}}_{5}|$ | $|{\mathit{x}}_{7}-{\mathit{x}}_{6}|$ | $\left|\mathit{g}\right({\mathit{x}}_{7}\left)\right|$ | $\mathit{\rho}$ |
---|---|---|---|---|---|

$TM$ | $1.9(-8)$ | $6.2(-15)$ | $6.3(-28)$ | $1.4(-108)$ | $2.000$ |

$D{M}_{1}$ | $1.8(-8)$ | $5.4(-15)$ | $4.9(-28)$ | $4.8(-109)$ | $2.000$ |

$D{M}_{2}$ | $2.2(-8)$ | $8.4(-15)$ | $1.2(-27)$ | $1.7(-107)$ | $2.000$ |

$D{M}_{3}$ | $2.2(-8)$ | $7.9(-15)$ | $1.1(-27)$ | $1.0(-107)$ | $2.000$ |

$D{M}_{4}$ | $1.4(-8)$ | $3.2(-15)$ | $1.7(-28)$ | $7.3(-111)$ | $2.000$ |

$D{M}_{5}$ | $2.6(-8)$ | $1.1(-14)$ | $2.2(-27)$ | $2.1(-106)$ | $2.000$ |

$D{M}_{6}$ | $2.3(-8)$ | $8.8(-15)$ | $1.3(-27)$ | $2.4(-107)$ | $2.000$ |

$D{M}_{7}$ | $1.3(-8)$ | $2.9(-15)$ | $1.3(-28)$ | $2.7(-111)$ | $2.000$ |

$P{M}_{1}$ | $1.9(-8)$ | $6.2(-15)$ | $6.3(-28)$ | $1.4(-108)$ | $2.000$ |

$P{M}_{2}$ | $1.9(-8)$ | $6.2(-15)$ | $6.3(-28)$ | $1.4(-108)$ | $2.000$ |

$P{M}_{3}$ | $1.9(-8)$ | $6.2(-15)$ | $6.3(-28)$ | $1.4(-108)$ | $2.000$ |

$P{M}_{4}$ | $1.9(-8)$ | $6.2(-15)$ | $6.3(-28)$ | $1.4(-108)$ | $2.000$ |

Schemes | $|{\mathit{x}}_{5}-{\mathit{x}}_{4}|$ | $|{\mathit{x}}_{6}-{\mathit{x}}_{5}|$ | $|{\mathit{x}}_{7}-{\mathit{x}}_{6}|$ | $\left|\mathit{g}\right({\mathit{x}}_{7}\left)\right|$ | $\mathit{\rho}$ |
---|---|---|---|---|---|

$TM$ | $6.0(-10)$ | $3.3(-26)$ | $1.0(-40)$ | $5.5(-242)$ | $2.000$ |

$D{M}_{1}$ | $6.7(-10)$ | $5.7(-20)$ | $4.2(-40)$ | $5.9(-238)$ | $2.000$ |

$D{M}_{2}$ | $4.5(-11)$ | $2.0(-23)$ | $4.1(-48)$ | $2.8(-289)$ | $2.000$ |

$D{M}_{3}$ | $1.3(-10)$ | $1.1(-22)$ | $8.1(-47)$ | $4.3(-282)$ | $2.000$ |

$D{M}_{4}$ | $1.1(-13)$ | $2.9(-27)$ | $2.2(-54)$ | $1.1(-322)$ | $2.000$ |

$D{M}_{5}$ | $2.9(-8)$ | $6.1(-17)$ | $2.7(-34)$ | $8.5(-204)$ | $2.000$ |

$D{M}_{6}$ | $1.2(-9)$ | $3.0(-20)$ | $2.0(-41)$ | $4.1(-248)$ | $2.000$ |

$D{M}_{7}$ | $3.4(-11)$ | $2.6(-22)$ | $1.5(-44)$ | $7.4(-264)$ | $2.000$ |

$P{M}_{1}$ | $8.4(-17)$ | $6.7(-34)$ | $4.1(-68)$ | $2.2(-406)$ | $2.000$ |

$P{M}_{2}$ | $1.5(-21)$ | $2.0(-43)$ | $3.7(-87)$ | $1.2(-520)$ | $2.000$ |

$P{M}_{3}$ | $8.8(-16)$ | $7.2(-32)$ | $4.8(-64)$ | $5.5(-382)$ | $2.000$ |

$P{M}_{4}$ | $1.2(-16)$ | $1.2(-33)$ | $1.4(-67)$ | $3.9(-403)$ | $2.000$ |

Schemes | $|{\mathit{x}}_{5}-{\mathit{x}}_{4}|$ | $|{\mathit{x}}_{6}-{\mathit{x}}_{5}|$ | $|{\mathit{x}}_{7}-{\mathit{x}}_{6}|$ | $\left|\mathit{g}\right({\mathit{x}}_{7}\left)\right|$ | $\mathit{\rho}$ |
---|---|---|---|---|---|

$TM$ | $6.0(-13)$ | $8.5(-26)$ | $1.7(-51)$ | $1.8(-407)$ | $2.000$ |

$D{M}_{1}$ | $5.1(-13)$ | $6.9(-26)$ | $1.2(-51)$ | $2.3(-408)$ | $2.000$ |

$D{M}_{2}$ | $5.2(-13)$ | $4.7(-26)$ | $3.8(-52)$ | $3.5(-413)$ | $2.000$ |

$D{M}_{3}$ | $3.2(-13)$ | $1.4(-26)$ | $2.6(-53)$ | $6.8(-423)$ | $2.000$ |

$D{M}_{4}$ | $8.4(-14)$ | $2.5(-27)$ | $2.3(-54)$ | $1.2(-429)$ | $2.000$ |

$D{M}_{5}$ | $2.0(-13)$ | $4.6(-27)$ | $2.4(-54)$ | $1.6(-431)$ | $2.000$ |

$D{M}_{6}$ | $3.0(-13)$ | $1.3(-26)$ | $2.2(-53)$ | $1.5(-423)$ | $2.000$ |

$D{M}_{7}$ | $2.9(-13)$ | $2.5(-26)$ | $1.7(-52)$ | $4.8(-415)$ | $2.000$ |

$P{M}_{1}$ | $8.3(-15)$ | $1.6(-29)$ | $6.5(-59)$ | $7.7(-467)$ | $2.000$ |

$P{M}_{2}$ | $1.5(-22)$ | $5.2(-45)$ | $6.4(-90)$ | $7.1(-715)$ | $2.000$ |

$P{M}_{3}$ | $2.7(-17)$ | $1.8(-34)$ | $7.3(-69)$ | $2.0(-546)$ | $2.000$ |

$P{M}_{4}$ | $3.1(-15)$ | $2.3(-30)$ | $1.3(-60)$ | $1.8(-480)$ | $2.000$ |

Schemes | $|{\mathit{x}}_{5}-{\mathit{x}}_{4}|$ | $|{\mathit{x}}_{6}-{\mathit{x}}_{5}|$ | $|{\mathit{x}}_{7}-{\mathit{x}}_{6}|$ | $\left|\mathit{g}\right({\mathit{x}}_{7}\left)\right|$ | $\mathit{\rho}$ |
---|---|---|---|---|---|

$TM$ | $4.0(-15)$ | $5.2(-30)$ | $9.1(-60)$ | $1.8(-473)$ | $2.000$ |

$D{M}_{1}$ | $4.6(-15)$ | $7.1(-30)$ | $1.7(-59)$ | $2.3(-471)$ | $2.000$ |

$D{M}_{2}$ | $8.4(-15)$ | $2.4(-29)$ | $2.0(-58)$ | $8.7(-463)$ | $2.000$ |

$D{M}_{3}$ | $2.3(-14)$ | $1.8(-28)$ | $1.1(-56)$ | $9.1(-449)$ | $2.000$ |

$D{M}_{4}$ | $5.4(-14)$ | $1.1(-27)$ | $3.9(-55)$ | $2.7(-436)$ | $2.000$ |

$D{M}_{5}$ | $4.3(-14)$ | $6.7(-28)$ | $1.6(-55)$ | $2.0(-439)$ | $2.000$ |

$D{M}_{6}$ | $2.4(-14)$ | $2.0(-28)$ | $1.4(-56)$ | $6.4(-448)$ | $2.000$ |

$D{M}_{7}$ | $9.3(-15)$ | $2.9(-29)$ | $2.9(-58)$ | $1.9(-461)$ | $2.000$ |

$P{M}_{1}$ | $8.9(-17)$ | $2.7(-33)$ | $2.3(-66)$ | $3.3(-526)$ | $2.000$ |

$P{M}_{2}$ | $6.0(-22)$ | $1.2(-43)$ | $4.9(-87)$ | $1.2(-691)$ | $2.000$ |

$P{M}_{3}$ | $8.6(-19)$ | $2.4(-37)$ | $2.0(-74)$ | $9.3(-591)$ | $2.000$ |

$P{M}_{4}$ | $3.9(-17)$ | $5.0(-34)$ | $8.3(-68)$ | $8.6(-538)$ | $2.000$ |

Schemes | $|{\mathit{x}}_{5}-{\mathit{x}}_{4}|$ | $|{\mathit{x}}_{6}-{\mathit{x}}_{5}|$ | $|{\mathit{x}}_{7}-{\mathit{x}}_{6}|$ | $\left|\mathit{g}\right({\mathit{x}}_{7}\left)\right|$ | $\mathit{\rho}$ |
---|---|---|---|---|---|

$TM$ | $9.9(-23)$ | $1.3(-46)$ | $2.3(-94)$ | $9.8(-379)$ | $2.000$ |

$D{M}_{1}$ | $2.3(-20)$ | $3.3(-41)$ | $7.0(-83)$ | $2.0(-331)$ | $2.000$ |

$D{M}_{2}$ | $6.5(-17)$ | $4.7(-34)$ | $2.5(-18)$ | $9.8(-273)$ | $2.000$ |

$D{M}_{3}$ | $3.6(-18)$ | $1.1(-36)$ | $1.1(-73)$ | $2.7(-294)$ | $2.000$ |

$D{M}_{4}$ | $6.7(-21)$ | $1.2(-41)$ | $3.7(-83)$ | $2.7(-331)$ | $2.000$ |

$D{M}_{5}$ | $5.4(-13)$ | $6.8(-26)$ | $1.1(-51)$ | $1.8(-205)$ | $2.000$ |

$D{M}_{6}$ | $3.0(-16)$ | $1.2(-32)$ | $1.7(-65)$ | $3.0(-261)$ | $2.000$ |

$D{M}_{7}$ | $5.7(-16)$ | $1.0(-31)$ | $3.2(-63)$ | $2.2(-251)$ | $2.000$ |

$P{M}_{1}$ | $4.4(-22)$ | $3.7(-45)$ | $2.6(-91)$ | $3.7(-366)$ | $2.000$ |

$P{M}_{2}$ | $1.8(-21)$ | $8.9(-44)$ | $2.2(-88)$ | $3.5(-354)$ | $2.000$ |

$P{M}_{3}$ | $1.0(-21)$ | $2.5(-44)$ | $1.5(-89)$ | $6.2(-359)$ | $2.000$ |

$P{M}_{4}$ | $5.4(-22)$ | $5.8(-45)$ | $6.9(-91)$ | $2.0(-364)$ | $2.000$ |

Schemes | $|{\mathit{x}}_{5}-{\mathit{x}}_{4}|$ | $|{\mathit{x}}_{6}-{\mathit{x}}_{5}|$ | $|{\mathit{x}}_{7}-{\mathit{x}}_{6}|$ | $\left|\mathit{g}\right({\mathit{x}}_{7}\left)\right|$ | $\mathit{\rho}$ |
---|---|---|---|---|---|

$TM$ | $2.2(-10)$ | $1.4(-20)$ | $5.6(-41)$ | $2.5(-406)$ | $2.000$ |

$D{M}_{1}$ | $1.2(-9)$ | $3.8(-19)$ | $3.8(-38)$ | $3.4(-378)$ | $2.000$ |

$D{M}_{2}$ | $2.4(-14)$ | $1.9(-28)$ | $1.3(-56)$ | $1.7(-562)$ | $2.000$ |

$D{M}_{3}$ | $2.1(-13)$ | $1.6(-26)$ | $9.9(-53)$ | $3.2(-523)$ | $2.000$ |

$D{M}_{4}$ | $2.8(-8)$ | $1.4(-16)$ | $3.5(-33)$ | $2.2(-329)$ | $2.000$ |

$D{M}_{5}$ | $4.8(-14)$ | $8.9(-28)$ | $3.0(-55)$ | $2.2(-548)$ | $2.000$ |

$D{M}_{6}$ | $1.8(-14)$ | $1.2(-28)$ | $5.1(-57)$ | $3.4(-566)$ | $2.000$ |

$D{M}_{7}$ | $2.2(-10)$ | $1.4(-20)$ | $5.6(-41)$ | $2.5(-406)$ | $2.000$ |

$P{M}_{1}$ | $3.7(-15)$ | $3.9(-30)$ | $4.2(-60)$ | $1.4(-597)$ | $2.000$ |

$P{M}_{2}$ | $1.2(-11)$ | $4.2(-23)$ | $4.9(-46)$ | $6.0(-457)$ | $2.000$ |

$P{M}_{3}$ | $1.1(-14)$ | $3.3(-29)$ | $3.0(-58)$ | $4.1(-579)$ | $2.000$ |

$P{M}_{4}$ | $1.9(-17)$ | $1.0(-34)$ | $2.9(-69)$ | $3.7(-689)$ | $2.000$ |

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**MDPI and ACS Style**

Kansal, M.; Alshomrani, A.S.; Bhalla, S.; Behl, R.; Salimi, M.
One Parameter Optimal Derivative-Free Family to Find the Multiple Roots of Algebraic Nonlinear Equations. *Mathematics* **2020**, *8*, 2223.
https://doi.org/10.3390/math8122223

**AMA Style**

Kansal M, Alshomrani AS, Bhalla S, Behl R, Salimi M.
One Parameter Optimal Derivative-Free Family to Find the Multiple Roots of Algebraic Nonlinear Equations. *Mathematics*. 2020; 8(12):2223.
https://doi.org/10.3390/math8122223

**Chicago/Turabian Style**

Kansal, Munish, Ali Saleh Alshomrani, Sonia Bhalla, Ramandeep Behl, and Mehdi Salimi.
2020. "One Parameter Optimal Derivative-Free Family to Find the Multiple Roots of Algebraic Nonlinear Equations" *Mathematics* 8, no. 12: 2223.
https://doi.org/10.3390/math8122223