# A New Third-Order Family of Multiple Root-Findings Based on Exponential Fitted Curve

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## Abstract

**:**

## 1. Introduction

## 2. Construction of Higher-Order Scheme

- 1.
- For ${q}_{\sigma}=0$, we have$${x}_{\sigma +1}={x}_{\sigma}-\left[1+\frac{1}{2}{L}_{f}\right]\frac{mf\left({x}_{\sigma}\right)}{{f}^{\prime}\left({x}_{\sigma}\right)-m\alpha f\left({x}_{\sigma}\right)}.$$
- 2.
- For ${q}_{\sigma}=-\frac{mf\left({x}_{\sigma}\right)\left({f}^{\u2033}\left({x}_{\sigma}\right)+m{\alpha}^{2}f\left({x}_{\sigma}\right)\right)-{f}^{\prime}{\left({x}_{\sigma}\right)}^{2}(m-1)-2m\alpha f\left({x}_{\sigma}\right){f}^{\prime}\left({x}_{\sigma}\right)}{2mf\left({x}_{\sigma}\right)({f}^{\prime}\left({x}_{\sigma}\right)-m\alpha f\left({x}_{\sigma}\right))}$, we obtain$${x}_{\sigma +1}={x}_{\sigma}-\left[\frac{2}{2-{L}_{f}}\right]\frac{mf\left({x}_{\sigma}\right)}{{f}^{\prime}\left({x}_{\sigma}\right)-m\alpha f\left({x}_{\sigma}\right)}.$$
- 3.
- For ${q}_{\sigma}=-\frac{mf\left({x}_{\sigma}\right)\left({f}^{\u2033}\left({x}_{\sigma}\right)+m{\alpha}^{2}f\left({x}_{\sigma}\right)\right)-{f}^{\prime}{\left({x}_{\sigma}\right)}^{2}(m-1)-2m\alpha f\left({x}_{\sigma}\right){f}^{\prime}\left({x}_{\sigma}\right)}{mf\left({x}_{\sigma}\right)({f}^{\prime}\left({x}_{\sigma}\right)-m\alpha f\left({x}_{\sigma}\right))}$, it yields$${x}_{\sigma +1}={x}_{\sigma}-\left[1+\frac{1}{2}\frac{{L}_{f}}{1-{L}_{f}}\right]\frac{mf\left({x}_{\sigma}\right)}{{f}^{\prime}\left({x}_{\sigma}\right)-m\alpha f\left({x}_{\sigma}\right)}.$$
- 4.
- For ${q}_{\sigma}=-\beta \frac{mf\left({x}_{\sigma}\right)\left({f}^{\u2033}\left({x}_{\sigma}\right)+m{\alpha}^{2}f\left({x}_{\sigma}\right)\right)-{f}^{\prime}{\left({x}_{\sigma}\right)}^{2}(m-1)-2m\alpha f\left({x}_{\sigma}\right){f}^{\prime}\left({x}_{\sigma}\right)}{mf\left({x}_{\sigma}\right)({f}^{\prime}\left({x}_{\sigma}\right)-m\alpha f\left({x}_{\sigma}\right))}$, we get$${x}_{\sigma +1}={x}_{\sigma}-\left[1+\frac{1}{2}\frac{{L}_{f}}{1-\beta {L}_{f}}\right]\frac{mf\left({x}_{\sigma}\right)}{{f}^{\prime}\left({x}_{\sigma}\right)-m\alpha f\left({x}_{\sigma}\right)}.$$
- 5.

**Theorem**

**1.**

**Proof.**

## 3. Stability Analysis

#### 3.1. Fixed Points and Stability

**Theorem**

**2.**

- (a)
- Coming from the roots of $p\left(z\right)$, $z=0$ is always superattracting and $z=\infty $ is a fixed point only if $\beta \ne -1$; moreover, $z=\infty $ is attracting only if $\Re \left(\beta \right)>-\frac{1}{2}$. It is parabolic if $\Re \left(\beta \right)=-\frac{1}{2}$ and repulsive in other cases.
- (b)
- $z=1$ is a parabolic strange fixed point, for any $\beta \in \mathbb{C}$.
- (c)
- There exist also four strange fixed points, denoted by ${s}_{i}\left(\beta \right)$, $i=1,2,3,4$, corresponding to the roots of the fourth-degree polynomial $q\left(t\right)=(2\beta +1){z}^{4}+(34-28\beta ){z}^{3}+(57-30\beta ){z}^{2}+4(6\beta -13)z+8$. These points can be attracting in small sets of the area $[-1,4]\times [-2.5,2.5]$ of the complex plane.

**Proof.**

#### 3.2. Dynamical Planes

## 4. Numerical Results

- Processor: Intel(R) Core(TM) i7-4790 CPU @ 3.60GHz
- RAM: 8:00GB
- System type: 64-bit-Operating System, x64-based processor.

**Example**

**1.**

**Van der Waals equation of state (Case of failure):**

**Example**

**2.**

**Planck’s radiation problem (Case of failure and divergence):**

**Example**

**3.**

**Jumping and Oscillating problem, when we have infinite numbers of roots:**

**Example**

**4.**

**Convergence to the undesired root problem:**

**Example**

**5.**

## 5. Concluding Remarks

## Author Contributions

## Funding

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## References

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**Table 1.**Comparison of distinct iterative functions based on absolute residual errors and absolute difference between two consecutive iterations. * stands for convergence to undesired root.

Methods | I.G. | $\left|\mathit{f}\right({\mathit{x}}_{\mathit{\sigma}}\left)\right|$ $\parallel {\mathit{x}}_{\mathbf{\sigma}+\mathbf{1}}-{\mathit{x}}_{\mathbf{\sigma}}\parallel $ | $\mathit{CS}$ | $\mathit{HS}$ | $\mathit{OS}$ | $\mathit{ONS}$ | $\mathit{CN}$ | $\mathit{MHS}1$ | $\mathit{MHS}2$ | $\mathit{MHS}3$ | $\mathit{MSHS}1$ | $\mathit{MSHS}2$ | $\mathit{MSHS}3$ |
---|---|---|---|---|---|---|---|---|---|---|---|---|---|

Ex. (1) | $1.73$ | $\left|f\right({x}_{6}\left)\right|$ | F | F | F | F | F | $1.3(-15)$ | $3.0(-10)$ | $2.0(-6)$ | $7.7(-102)$ | $2.0(-67)$ | $2.2(-45)$ |

$\parallel {x}_{7}-{x}_{6}\parallel $ | F | F | F | F | F | $2.1(-7)$ | $1.0(-4)$ | $8.9(-3)$ | $1.6(-50)$ | $2.6(-33)$ | $2.7(-22)$ | ||

Ex. (2) | $log5$ | $\left|f\right({x}_{6}\left)\right|$ | F | F | F | F | F | $3.2(-97)$ | $2.0(-228)$ | $3.8(-179)$ | $2.6(-122)$ | $2.7(-404)$ | $2.9(-924)$ |

$\parallel {x}_{7}-{x}_{6}\parallel $ | F | F | F | F | F | $3.5(-32)$ | $6.5(-76)$ | $1.7(-59)$ | $1.5(-40)$ | $1.5(-134)$ | $7.4(-308)$ | ||

$1.6$ | $\left|f\right({x}_{6}\left)\right|$ | $Div$ | $9.8(-2)$ | $7.2(-464)$ | $Div$ | $Div$ | $2.8(-97)$ | $1.4(-228)$ | $1.3(-179)$ | $2.1(-122)$ | $4.7(-404)$ | $1.0(-926)$ | |

$\parallel {x}_{7}-{x}_{6}\parallel $ | $Div$ | $9.3(-1)$ | $2.2(-154)$ | $Div$ | $Div$ | $3.4(-32)$ | $5.7(-76)$ | $1.2(-59)$ | $1.4(-40)$ | $1.9(-134)$ | $1.1(-308)$ | ||

Ex. (3) | $1.5$ | $\left|f\right({x}_{6}\left)\right|$ | $2.9(-274)$ * | $2.2(-129)$ | $7.6(-837)$ | $Div$ | $3.8(-582)$ * | $4.8(-593)$ | $2.2(-652)$ | $2.1(-294)$ | $4.7(-1197)$ | $8.4(-1217)$ | $1.4(-1304)$ |

$\parallel {x}_{7}-{x}_{6}\parallel $ | $2.0(-55)$ * | $1.9(-26)$ | $6.0(-168)$ | $Div$ | $5.2(-117)$ * | $3.4(-119)$ | $4.7(-131)$ | $1.8(-59)$ | $8.6(-240)$ | $6.1(-244)$ | $1.7(-261)$ | ||

Ex. (4) | $4.4$ | $\left|f\right({x}_{6}\left)\right|$ | $1.9(-142)$ * | $3.8(-375)$ | $2.2(-694)$ | $1.9(-142)$ * | $1.9(-142)$ * | $1.5(-497)$ | $1.5(-613)$ | $4.2(-462)$ | $5.9(-705)$ | $1.2(-828)$ | $1.3(-1005)$ |

$\parallel {x}_{7}-{x}_{6}\parallel $ | $5.8(-48)$ * | $1.5(-125)$ | $5.8(-232)$ | $5.8(-48)$ * | $5.8(-48)$ * | $2.4(-166)$ | $5.1(-205)$ | $1.6(-154)$ | $1.7(-235)$ | $1.0(-276)$ | $1.0(-335)$ | ||

$1.7$ | $\left|f\right({x}_{6}\left)\right|$ | $1.4(+7)$ | $5.9(-331)$ | $1.2(-646)$ | $1.4(+7)$ | $1.4(+7)$ | $4.8(-403)$ | $1.3(-566)$ | $8.7(-427)$ | $3.5(-669)$ | $1.7(-865)$ | $3.6(-885)$ | |

$\parallel {x}_{7}-{x}_{6}\parallel $ | $1.5$ | $8.1(-111)$ | $4.7(-216)$ | $1.5$ | $1.5$ | $7.5(-135)$ | $2.3(-189)$ | $9.2(-143)$ | $1.5(-223)$ | $5.3(-289)$ | $1.5(-295)$ | ||

Ex. (5) | $\left|f\right({x}_{6}\left)\right|$ | $5.7(-14,352)$ | $7.5(-21,328)$ | $4.7(-34,103)$ | $Div$ | $5.3(-21,087)$ | $3.1(-9344)$ | $1.1(-13,965)$ | $1.1(-19,577)$ | $3.2(-23,126)$ | $5.0(-40,450)$ | $2.9(-38,495)$ | |

$\parallel {x}_{7}-{x}_{6}\parallel $ | $1.0(-144)$ | $1.8(-214)$ | $3.2(-342)$ | $Div$ | $4.6(-212)$ | $1.2(-94)$ | $7.5(-141)$ | $5.7(-197)$ | $1.9(-232)$ | $1.1(-405)$ | $3.8(-386)$ |

**Table 2.**Contrast of distinct iterative methods based on number of iterations. * stands for convergence to undesired root.

Methods | I.G. | $\mathit{CS}$ | $\mathit{HS}$ | $\mathit{OS}$ | $\mathit{ONS}$ | $\mathit{CN}$ | $\mathit{MHS}1$ | $\mathit{MHS}2$ | $\mathit{MHS}3$ | $\mathit{MSHS}1$ | $\mathit{MSHS}2$ | $\mathit{MSHS}3$ |
---|---|---|---|---|---|---|---|---|---|---|---|---|

Ex. (1) | $1.73$ | F | F | F | F | F | 9 | 10 | 11 | 7 | 8 | 8 |

Ex. (2) | $log5$ | F | F | F | F | F | 8 | 7 | 7 | 7 | 6 | 5 |

$1.61$ | $Div$ | 12 | 6 | $Div$ | $Div$ | 8 | 7 | 7 | 7 | 6 | 5 | |

Ex. (3) | $1.5$ | 7 * | 8 | 6 | $Div$ | 6 * | 6 | 6 | 7 | 6 | 6 | 6 |

Ex. (4) | $4.4$ | 7 * | 6 | 6 | 7 * | 7 * | 6 | 6 | 6 | 6 | 6 | 5 |

$1.7$ | 1280 | 6 | 6 | 1280 | 1280 | 6 | 6 | 6 | 6 | 6 | 6 | |

Ex. (5) | $1.5$ | 6 | 6 | 5 | $Div$ | 6 | 7 | 6 | 6 | 6 | 5 | 5 |

**Table 3.**Contrast of distinct iterative methods based on Computational order of convergence. * stands for convergence to undesired root.

Methods | I.G. | $\mathit{CS}$ | $\mathit{HS}$ | $\mathit{OS}$ | $\mathit{ONS}$ | $\mathit{CN}$ | $\mathit{MHS}1$ | $\mathit{MHS}2$ | $\mathit{MHS}3$ | $\mathit{MSHS}1$ | $\mathit{MSHS}2$ | $\mathit{MSHS}3$ |
---|---|---|---|---|---|---|---|---|---|---|---|---|

Ex. (1) | $1.73$ | $dne$ | $dne$ | $dne$ | $dne$ | $dne$ | $3.000$ | $3.000$ | $3.000$ | $3.000$ | $3.000$ | $3.000$ |

Ex. (2) | $log5$ | $dne$ | $dne$ | $dne$ | $dne$ | $dne$ | $3.000$ | $3.000$ | $3.000$ | $2.999$ | $3.002$ | $2.993$ |

$1.61$ | $Div$ | $3.000$ | $3.000$ | $dne$ | $dne$ | $3.000$ | $3.000$ | $3.000$ | $2.999$ | $3.002$ | $3.002$ | |

Ex. (3) | $1.5$ | $3.000$ * | $3.000$ | $3.000$ | $dne$ | $3.000$ | $3.000$ | $3.000$ | $3.000$ | $3.000$ | $3.000$ | $3.000$ |

Ex. (4) | $4.4$ | $3.000$ * | $3.000$ | $3.000$ | $3.000$ * | $3.000$ * | $3.000$ | $3.000$ | $3.000$ | $3.002$ | $3.000$ | $3.000$ |

$1.7$ | $3.000$ | $3.000$ | $3.000$ | $3.000$ | $3.000$ | $3.000$ | $3.000$ | $3.000$ | $2.998$ | $3.000$ | $3.000$ | |

Ex. (5) | $3.000$ | $3.000$ | $3.000$ | $3.000$ | $dne$ | $3.000$ | $3.000$ | $3.000$ | $3.000$ | $3.000$ | $2.975$ |

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## Share and Cite

**MDPI and ACS Style**

Kanwar, V.; Cordero, A.; Torregrosa, J.R.; Rajput, M.; Behl, R.
A New Third-Order Family of Multiple Root-Findings Based on Exponential Fitted Curve. *Algorithms* **2023**, *16*, 156.
https://doi.org/10.3390/a16030156

**AMA Style**

Kanwar V, Cordero A, Torregrosa JR, Rajput M, Behl R.
A New Third-Order Family of Multiple Root-Findings Based on Exponential Fitted Curve. *Algorithms*. 2023; 16(3):156.
https://doi.org/10.3390/a16030156

**Chicago/Turabian Style**

Kanwar, Vinay, Alicia Cordero, Juan R. Torregrosa, Mithil Rajput, and Ramandeep Behl.
2023. "A New Third-Order Family of Multiple Root-Findings Based on Exponential Fitted Curve" *Algorithms* 16, no. 3: 156.
https://doi.org/10.3390/a16030156