# Some Real-Life Applications of a Newly Constructed Derivative Free Iterative Scheme

^{1}

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## Abstract

**:**

## 1. Introduction

## 2. Construction of the Proposed Scheme

## 3. Convergence Analysis

**Theorem**

**1.**

**Proof.**

**Remark**

**1.**

## 4. Numerical Examples

- We choose an optimal derivative free fourth-order method (6) suggested by Cordero and Torregrosa [3]. Then, we have$$\left(\right),$$

**Example**

**1.**

**Example**

**2.**

**Example**

**3.**

**Example**

**4.**

**Example**

**5.**

## 5. Graphical Comparison by Means of Attraction Basins

## 6. Conclusions

## Author Contributions

## Funding

## Conflicts of Interest

## References

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**Figure 1.**The dynamical behavior of our methods namely, $PM{1}_{8}$, $PM{2}_{8}$, $PM{3}_{8}$ and $PM{4}_{8}$, respectively, from left to right for test problem ${p}_{1}\left(z\right)$.

**Figure 2.**The dynamical behavior of methods $S{M}_{8}$, $K{T}_{8}$ and $K{M}_{8}$, respectively, from left to right for test problem ${p}_{1}\left(z\right)$.

**Figure 3.**The dynamical behavior of our methods namely, $PM{1}_{8}$, $PM{2}_{8}$, $PM{3}_{8}$ and $PM{4}_{8}$, respectively, from left to right for test problem ${p}_{2}\left(z\right)$.

**Figure 4.**The dynamical behavior of methods $S{M}_{8}$, $K{T}_{8}$ and $K{M}_{8}$, respectively from left to right for test problem ${p}_{2}\left(z\right)$.

**Table 1.**Convergence performance of distinct 8-order optimal derivative free methods for ${f}_{1}\left(x\right)$.

Cases | j | ${\mathit{x}}_{\mathit{j}}$ | $\left|\mathit{f}\right({\mathit{x}}_{\mathit{j}}\left)\right|$ | $|{\mathit{x}}_{\mathit{j}+1}-{\mathit{x}}_{\mathit{j}}|$ | $\mathit{\rho}$ | $\left(\right)open="|"\; close="|">\frac{{\mathit{x}}_{\mathit{j}+1}-{\mathit{x}}_{\mathit{j}}}{{({\mathit{x}}_{\mathit{j}}-{\mathit{x}}_{\mathit{n}-1})}^{8}}$ | $\mathit{\eta}$ |
---|---|---|---|---|---|---|---|

1 | $0.75742117642117592668$ | $2.0(-3)$ | $2.5(-5)$ | ||||

$K{T}_{8}$ | 2 | $0.75739624625375387946$ | $1.0(-19)$ | $1.3(-21)$ | $8.387722076(+15)$ | $8.409575862(+15)$ | |

3 | $0.75739624625375387946$ | $4.0(-150)$ | $5.1(-152)$ | $7.9999$ | $8.409575862(+15)$ | ||

1 | $0.75739472392262620965$ | $1.2(-4)$ | $1.5(-6)$ | ||||

$K{M}_{8}$ | 2 | $0.75739624625375387946$ | $1.2(-34)$ | $1.5(-36)$ | $5.252005934(+10)$ | $2.765111335(+10)$ | |

3 | $0.75739624625375387946$ | $6.1(-275)$ | $7.7(-277)$ | $8.0093$ | $2.765111335(+10)$ | ||

1 | $0.75726839017571335554$ | $1.0(-2)$ | $1.3(-4)$ | ||||

$S{V}_{8}$ | 2 | $0.75739624625375406009$ | $1.4(-14)$ | $1.8(-16)$ | $2.529459671(+15)$ | $1.540728199(+14)$ | |

3 | $0.75739624625375387946$ | $1.4(-110)$ | $1.7(-112)$ | $8.1026$ | $1.540728199(+14)$ | ||

1 | $0.75739624679631343572$ | $4.3(-8)$ | $5.4(-10)$ | ||||

$PM{1}_{8}$ | 2 | $0.75739624625375387946$ | $7.9(-60)$ | $9.9(-62)$ | $1.318011692(+13)$ | $1.318013290(+13)$ | |

3 | $0.75739624625375387946$ | $9.7(-474)$ | $1.2(-475)$ | $8.0000$ | $1.318013290(+13)$ | ||

1 | $0.75739624527627277118$ | $7.8(-8)$ | $9.8(-10)$ | ||||

$PM{2}_{8}$ | 2 | $0.75739624625375387946$ | $5.3(-58)$ | $6.7(-60)$ | $8.002563231(+12)$ | $8.002546457(+12)$ | |

3 | $0.75739624625375387946$ | $2.5(-459)$ | $3.1(-461)$ | $8.0000$ | $8.002546457(+12)$ | ||

1 | $0.75739624669712714014$ | $3.5(-8)$ | $4.4(-10)$ | ||||

$PM{3}_{8}$ | 2 | $0.75739624625375387946$ | $1.6(-60)$ | $2.0(-62)$ | $1.316590806(+13)$ | $1.316592111(+13)$ | |

3 | $0.75739624625375387946$ | $2.3(-479)$ | $2.9(-481)$ | $8.0000$ | $1.316592111(+13)$ | ||

1 | $0.75739625664695918279$ | $8.3(-7)$ | $1.0(-8)$ | ||||

$PM{4}_{8}$ | 2 | $0.75739624625375387946$ | $1.7(-49)$ | $2.1(-51)$ | $1.522844707(+13)$ | $1.522886893(+13)$ | |

3 | $0.75739624625375387946$ | $4.1(-391)$ | $5.2(-393)$ | $8.0000$ | $1.522886893(+13)$ |

**Table 2.**Convergence performance of distinct 8-order optimal derivative free methods for ${f}_{2}\left(x\right)$.

Cases | j | ${\mathit{x}}_{\mathit{j}}$ | $\left|\mathit{f}\right({\mathit{x}}_{\mathit{j}}\left)\right|$ | $|{\mathit{e}}_{\mathit{j}}|$ | $\mathit{\rho}$ | $\left(\right)open="|"\; close="|">\frac{{\mathit{e}}_{\mathit{j}+1}}{{\mathit{e}}_{\mathit{j}}^{8}}$ | $\mathit{\eta}$ |
---|---|---|---|---|---|---|---|

1 | $1.9299358075659180242$ | $7.7(-6)$ | $9.0(-5)$ | ||||

$K{T}_{8}$ | 2 | $1.9298462428478622185$ | $9.2(-28)$ | $1.1(-26)$ | $2.570367432(+6)$ | $2.580781373(+6)$ | |

3 | $1.9298462428478622185$ | $3.7(-203)$ | $4.3(-202)$ | $7.9999$ | $2.580781373(+6)$ | ||

1 | $1.9300063313329939091$ | $1.4(-5)$ | $1.6(-4)$ | ||||

$K{M}_{8}$ | 2 | $1.9298462428478622185$ | $7.0(-26)$ | $8.1(-25)$ | $1.872886840(+6)$ | $1.859196359(+6)$ | |

3 | $1.9298462428478622185$ | $2.9(-188)$ | $3.4(-187)$ | $8.0002$ | $1.859196359(+6)$ | ||

1 | $1.9299298655571245217$ | $7.2(-6)$ | $8.4(-5)$ | ||||

$S{V}_{8}$ | 2 | $1.9298462428478622185$ | $2.6(-30)$ | $3.0(-29)$ | $1.272677056(+4)$ | $5.345691399(+3)$ | |

3 | $1.9298462428478622185$ | $3.4(-226)$ | $3.9(-225)$ | $8.0154$ | $5.345691399(+3)$ | ||

1 | $1.9298703396056890283$ | $2.1(-6)$ | $2.4(-5)$ | ||||

$PM{1}_{8}$ | 2 | $1.9298462428478622185$ | $3.2(-33)$ | $3.7(-32)$ | $3.292189981(+5)$ | $3.294743419(+5)$ | |

3 | $1.9298462428478622185$ | $1.1(-247)$ | $1.3(-246)$ | $8.0000$ | $3.294743419(+5)$ | ||

1 | $1.9299039277100182896$ | $5.0(-6)$ | $5.8(-5)$ | ||||

$PM{2}_{8}$ | 2 | $1.9298462428478622185$ | $1.5(-29)$ | $1.7(-28)$ | $1.415845181(+6)$ | $1.419322205(+6)$ | |

3 | $1.9298462428478622185$ | $1.0(-217)$ | $1.2(-216)$ | $8.0000$ | $1.419322205(+6)$ | ||

1 | $1.9298835516272248348$ | $3.2(-6)$ | $3.7(-5)$ | ||||

$PM{3}_{8}$ | 2 | $1.9298462428478622185$ | $2.0(-31)$ | $2.3(-30)$ | $6.132728979(+5)$ | $6.140666943(+5)$ | |

3 | $1.9298462428478622185$ | $4.2(-233)$ | $4.8(-232)$ | $8.0000$ | $6.140666943(+5)$ | ||

1 | $1.9298454768935056951$ | $6.6(-8)$ | $7.7(-7)$ | ||||

$PM{4}_{8}$ | 2 | $1.9298462428478622185$ | $1.6(-46)$ | $1.9(-45)$ | $1.600600022(+4)$ | $1.600542542(+4)$ | |

3 | $1.9298462428478622185$ | $2.3(-355)$ | $2.7(-354)$ | $8.0000$ | $1.600542542(+4)$ |

**Table 3.**Convergence performance of distinct 8-order optimal derivative free methods for ${f}_{3}\left(x\right)$.

Cases | j | ${\mathit{x}}_{\mathit{j}}$ | $\left|\mathit{f}\right({\mathit{x}}_{\mathit{j}}\left)\right|$ | $|{\mathit{e}}_{\mathit{j}}|$ | $\mathit{\rho}$ | $\left(\right)open="|"\; close="|">\frac{{\mathit{e}}_{\mathit{j}+1}}{{\mathit{e}}_{\mathit{j}}^{8}}$ | $\mathit{\eta}$ |
---|---|---|---|---|---|---|---|

1 | $3.94856259325568+0.31584953607444i$ | $2.8(-3)$ | $2.7(-4)$ | ||||

$K{T}_{8}$ | 2 | $3.94854244556204+0.31612357089701i$ | $1.1(-21)$ | $1.1(-22)$ | $3.278944412(+6)$ | $3.291035449(+6)$ | |

3 | $3.94854244556204+0.31612357089701i$ | $5.5(-169)$ | $5.5(-170)$ | $7.999$ | $3.291035449(+6)$ | ||

1 | $3.94541341953964+0.28830540896626i$ | $2.7(-1)$ | $2.8(-2)$ | ||||

$K{M}_{8}$ | 2 | $3.94854253806613+0.31612376121596i$ | $2.1(-6)$ | $2.1(-7)$ | $5.611004628(+5)$ | $1.267588109(+4)$ | |

3 | $3.94854244556204+0.31612357089701i$ | $5.2(-49)$ | $5.1(-50)$ | $8.3214$ | $1.267588109(+4)$ | ||

1 | $3.94857741336794+0.31574108761478i$ | $3.9(-3)$ | $3.8(-4)$ | ||||

$S{V}_{8}$ | 2 | $3.94854244556204+0.31612357089701i$ | $9.1(-21)$ | $9.0(-22)$ | $1.895162520(+6)$ | $1.896706799(+6)$ | |

3 | $3.94854244556204+0.31612357089701i$ | $8.1(-162)$ | $8.0(-163)$ | $8.0000$ | $1.896706799(+6)$ | ||

1 | $3.94848048827814+0.31602117152370i$ | $1.2(-3)$ | $1.2(-4)$ | ||||

$PM{1}_{8}$ | 2 | $3.94854244556204+0.31612357089701i$ | $2.5(-25)$ | $2.5(-26)$ | $5.923125406(+5)$ | $5.903970786(+5)$ | |

3 | $3.94854244556204+0.31612357089701i$ | $8.9(-199)$ | $8.8(-200)$ | $8.0001$ | $5.903970786(+5)$ | ||

1 | $3.94846874984553+0.31601667713734i$ | $1.3(-3)$ | $1.3(-4)$ | ||||

$PM{2}_{8}$ | 2 | $3.94854244556204+0.31612357089701i$ | $5.1(-25)$ | $5.0(-26)$ | $6.241093912(+5)$ | $6.214835024(+5)$ | |

3 | $3.94854244556204+0.31612357089701i$ | $2.6(-196)$ | $2.6(-197)$ | $8.0001$ | $6.214835024(+5)$ | ||

1 | $3.94848290176499+0.31601668833975i$ | $1.2(-3)$ | $1.2(-4)$ | ||||

$PM{3}_{8}$ | 2 | $3.94854244556204+0.31612357089701i$ | $3.1(-25)$ | $3.1(-26)$ | $6.078017700(+5)$ | $6.059534898(+5)$ | |

3 | $3.94854244556204+0.31612357089701i$ | $4.6(-198)$ | $4.6(-199)$ | $8.0001$ | $6.059534898(+5)$ | ||

1 | $3.94849208916059+0.31602400692668i$ | $1.1(-3)$ | $1.1(-4)$ | ||||

$PM{4}_{8}$ | 2 | $3.94854244556204+0.31612357089701i$ | $1.4(-25)$ | $1.4(-26)$ | $5.704624073(+5)$ | $5.691514905(+5)$ | |

3 | $3.94854244556204+0.31612357089701i$ | $7.1(-201)$ | $7.1(-202)$ | $8.0000$ | $5.691514905(+5)$ |

**Table 4.**Convergence performance of distinct 8-order optimal derivative free methods for ${f}_{4}\left(x\right)$.

Cases | j | ${\mathit{x}}_{\mathit{j}}$ | $\left|\mathit{f}\right({\mathit{x}}_{\mathit{j}}\left)\right|$ | $|{\mathit{e}}_{\mathit{j}}|$ | $\mathit{\rho}$ | $\left(\right)open="|"\; close="|">\frac{{\mathit{e}}_{\mathit{j}+1}}{{\mathit{e}}_{\mathit{j}}^{8}}$ | $\mathit{\eta}$ |
---|---|---|---|---|---|---|---|

1 | $4305.3099136661255630$ | $3.3(-19)$ | $1.5(-20)$ | ||||

$K{T}_{8}$ | 2 | $4305.3099136661255630$ | $1.1(-179)$ | $4.8(-181)$ | $2.234387851(-22)$ | $2.234387851(-22)$ | |

3 | $4305.3099136661255630$ | $1.4(-1463)$ | $6.3(-1465)$ | $8.0000$ | $2.234387851(-22)$ | ||

1 | $4305.4966166546986926$ | $4.2$ | $1.9(-1)$ | ||||

$K{M}_{8}$ | 2 | $4305.3099136647999238$ | $3.0(-8)$ | $1.3(-9)$ | $8.978735581(-4)$ | $1.132645694(-16)$ | |

3 | $4305.3099136661255630$ | $2.4(-86)$ | $1.1(-87)$ | $9.5830$ | $1.132645694(-16)$ | ||

1 | $4305.3099136661255630$ | $1.5(-19)$ | $6.9(-21)$ | ||||

$S{V}_{8}$ | 2 | $4305.3099136661255630$ | $1.2(-182)$ | $5.4(-184)$ | $1.038308478(-22)$ | $1.038308478(-22)$ | |

3 | $4305.3099136661255630$ | $1.6(-1487)$ | $7.1(-1489)$ | $8.0000$ | $1.038308478(-22)$ | ||

1 | $4305.3099136661255630$ | $3.5(-20)$ | $1.6(-21)$ | ||||

$PM{1}_{8}$ | 2 | $4305.3099136661255630$ | $2.1(-188)$ | $9.3(-190)$ | $2.393094045(-23)$ | $2.393094045(-23)$ | |

3 | $4305.3099136661255630$ | $3.1(-1534)$ | $1.4(-1535)$ | $8.0000$ | $2.393094045(-23)$ | ||

1 | $4305.3099136661255630$ | $4.0(-20)$ | $1.8(-21)$ | ||||

$PM{2}_{8}$ | 2 | $4305.3099136661255630$ | $5.8(-188)$ | $2.6(-189)$ | $2.683028981(-23)$ | $2.683028981(-23)$ | |

3 | $4305.3099136661255630$ | $1.3(-1530)$ | $5.6(-1532)$ | $8.0000$ | $2.683028981(-23)$ | ||

1 | $4305.3099136690636946$ | $6.6(-8)$ | $2.8(-9)$ | ||||

$PM{3}_{8}$ | 2 | $4305.3099136661255630$ | $8.8(-77)$ | $3.9(-78)$ | $7.055841652(-10)$ | $7.055841652(-10)$ | |

3 | $4305.3099136661255630$ | $8.8(-628)$ | $3.9(-629)$ | $8.0000$ | $7.055841652(-10)$ | ||

1 | $4305.3099136661255630$ | $4.0(-20)$ | $1.8(-21)$ | ||||

$PM{4}_{8}$ | 2 | $4305.3099136661255630$ | $5.8(-188)$ | $2.6(-189)$ | $2.119306545(-23)$ | $2.119306545(-23)$ | |

3 | $4305.3099136661255630$ | $1.3(-1530)$ | $5.6(-1532)$ | $8.0000$ | $2.119306545(-23)$ |

**Table 5.**Convergence performance of distinct 8-order optimal derivative free methods for ${f}_{5}\left(x\right)$.

Cases | j | ${\mathit{x}}_{\mathit{j}}$ | $\left|\mathit{f}\right({\mathit{x}}_{\mathit{j}}\left)\right|$ | $|{\mathit{e}}_{\mathit{j}}|$ | $\mathit{\rho}$ | $\mathit{\eta}$ | |
---|---|---|---|---|---|---|---|

1 | $1.4142135646255204265$ | $6.4(-9)$ | $2.3(-9)$ | ||||

$K{T}_{8}$ | 2 | $1.4142135623730950488$ | $2.8(-69)$ | $9.8(-70)$ | $1.483428355$ | $1.483428382$ | |

3 | $1.4142135623730950488$ | $3.7(-552)$ | $1.3(-552)$ | $8.0000$ | $1.483428382$ | ||

1 | $1.4141886104951680577$ | $7.1(-5)$ | $2.5(-5)$ | ||||

$K{M}_{8}$ | 2 | $1.4142135641342028617$ | $5.0(-9)$ | $1.8(-9)$ | $1.171425936(+28)$ | $0.1339769256$ | |

3 | $1.4142135623730950488$ | $3.5(-71)$ | $1.2(-71)$ | $14.972$ | $0.1339769256$ | ||

1 | $1.4142135639458229191$ | $4.4(-9)$ | $1.6(-9)$ | ||||

$S{V}_{8}$ | 2 | $1.4142135623730950488$ | $8.4(-71)$ | $3.0(-71)$ | $0.7923194647$ | $0.7923194693$ | |

3 | $1.4142135623730950488$ | $1.3(-564)$ | $4.7(-564)$ | $8.0000$ | $0.7923194693$ | ||

1 | $1.4142135629037874832$ | $1.5(-9)$ | $5.3(-10)$ | ||||

$PM{1}_{8}$ | 2 | $1.4142135623730950488$ | $5.3(-75)$ | $1.9(-75)$ | $0.2966856754$ | $0.2966856763$ | |

3 | $1.4142135623730950488$ | $1.2(-598)$ | $4.4(-599)$ | $8.0000$ | $0.2966856763$ | ||

1 | $1.4142135630941303743$ | $2.0(-9)$ | $7.2(-10)$ | ||||

$PM{2}_{8}$ | 2 | $1.4142135623730950488$ | $8.7(-74)$ | $3.1(-74)$ | $0.4230499025$ | $0.4230499045$ | |

3 | $1.4142135623730950488$ | $1.0(-588)$ | $3.5(-589)$ | $8.0000$ | $0.4230499045$ | ||

1 | $1.4142135672540404368$ | $1.4(-8)$ | $4.9(-9)$ | ||||

$PM{3}_{8}$ | 2 | $1.4142135623730950488$ | $2.5(-66)$ | $8.8(-67)$ | $2.742159025$ | $2.742159103$ | |

3 | $1.4142135623730950488$ | $2.9(-528)$ | $1.0(-528)$ | $8.0000$ | $2.742159103$ | ||

1 | $1.4142135627314914846$ | $1.0(-9)$ | $3.6(-10)$ | ||||

$PM{4}_{8}$ | 2 | $1.4142135623730950488$ | $1.5(-76)$ | $5.2(-77)$ | $0.1905635592$ | $0.1905635596$ | |

3 | $1.4142135623730950488$ | $2.8(-611)$ | $1.0(-611)$ | $8.0000$ | $0.1905635596$ |

Test Problem | Roots |
---|---|

${p}_{1}\left(z\right)={z}^{2}-1$ | $1,-1$ |

${p}_{2}\left(z\right)={z}^{2}-z-1/z$ | $1.46557,\phantom{\rule{3.33333pt}{0ex}}-0.232786\pm 0.792552i$ |

© 2019 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (http://creativecommons.org/licenses/by/4.0/).

## Share and Cite

**MDPI and ACS Style**

Behl, R.; Salimi, M.; Ferrara, M.; Sharifi, S.; Alharbi, S.K.
Some Real-Life Applications of a Newly Constructed Derivative Free Iterative Scheme. *Symmetry* **2019**, *11*, 239.
https://doi.org/10.3390/sym11020239

**AMA Style**

Behl R, Salimi M, Ferrara M, Sharifi S, Alharbi SK.
Some Real-Life Applications of a Newly Constructed Derivative Free Iterative Scheme. *Symmetry*. 2019; 11(2):239.
https://doi.org/10.3390/sym11020239

**Chicago/Turabian Style**

Behl, Ramandeep, M. Salimi, M. Ferrara, S. Sharifi, and Samaher Khalaf Alharbi.
2019. "Some Real-Life Applications of a Newly Constructed Derivative Free Iterative Scheme" *Symmetry* 11, no. 2: 239.
https://doi.org/10.3390/sym11020239