# Stability Analysis of Simple Root Seeker for Nonlinear Equation

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Stability of Iterative Method under Fractal Study

#### 2.1. Rational Operator

**Proposition**

**1.**

**Proof.**

#### 2.2. Fixed Points and Critical Points

**Proposition**

**2.**

- When $\lambda =0$, ${\rho}_{\lambda}\left(x\right)$ and ${\theta}_{\lambda}\left(x\right)$ have common factors ${(1+x)}^{2}{(1+{x}^{2})}^{2}$.
- When $\lambda =-6848$, ${\rho}_{\lambda}\left(x\right)$ has a factor ${(x-1)}^{2}$.
- When $\lambda =1344$, ${\theta}_{\lambda}\left(x\right)$ has a factor $(x-1)$.

**Proof.**

**Proposition**

**3.**

- $e{x}_{1}=1$ (when $\lambda \ne 1344$) and $e{x}_{i}\left(\lambda \right)$, which correspond to the 18 roots of polynomial $1+7x+27{x}^{2}+77{x}^{3}+176{x}^{4}+336{x}^{5}+552{x}^{6}+779{x}^{7}+956{x}^{8}+(1026+\lambda ){x}^{9}+956{x}^{10}+779{x}^{11}+552{x}^{12}+336{x}^{13}+176{x}^{14}+77{x}^{15}+27{x}^{16}+7{x}^{17}+{x}^{18},$ where $i=2,3,\cdots ,19$.

- ${R}_{\lambda}\left(x\right)$ has 21 fixed points when $\lambda \in \mathbb{C}$ and $\lambda \notin \{0,1344\}$.
- ${R}_{\lambda}\left(x\right)$ has 20 fixed points excluding $e{x}_{1}=1$ when $\lambda =1344$.
- ${R}_{\lambda}\left(x\right)$ has 15 fixed points when $\lambda =0$.
- ${R}_{\lambda}\left(x\right)$ has 21 fixed points, and $e{x}_{1}=1$ is a triple root when $\lambda =-6848$.
- The strange fixed points of ${R}_{\lambda}\left(x\right)$ satisfy $e{x}_{m}=\frac{1}{e{x}_{n}}$ for $m\ne n$; each pair is conjugate to each other, being $e{x}_{2}$ and $e{x}_{3}$, $e{x}_{4}$ and $e{x}_{5}$, $e{x}_{6}$ and $e{x}_{7}$, $e{x}_{8}$ and $e{x}_{9}$, $e{x}_{10}$ and $e{x}_{11}$, $e{x}_{12}$ and $e{x}_{13}$, $e{x}_{14}$ and $e{x}_{15}$, $e{x}_{16}$ and $e{x}_{17}$, and $e{x}_{18}$ and $e{x}_{19}$.

**Proof.**

**Proposition**

**4.**

- (1)
- $e{x}_{1}$ is a repulsive point when $|-\frac{8192}{-1344+\lambda}|>1$, that is $|\lambda -1344|>8192$;
- (2)
- $e{x}_{1}$ is an attractive point when $|-\frac{8192}{-1344+\lambda}|<1$, that is $|\lambda -1344|<8192$;
- (3)
- $e{x}_{1}$ is a parabolic point when $|-\frac{8192}{-1344+\lambda}|=1$, that is $|\lambda -1344|=8192$;
- (4)
- $e{x}_{1}$ is never a superattracting point because $|-\frac{8192}{-1344+\lambda}|\ne 0$.

**Proof.**

**Proposition**

**5.**

- $c{r}_{1}=-1$;
- $c{r}_{2}=i$;
- $c{r}_{3}=-i$;
- $c{r}_{4}=\frac{1}{2}(-\frac{21}{26}+M\left(\lambda \right)+\frac{1}{2}\sqrt{P\left(\lambda \right)-N\left(\lambda \right)}-\sqrt{{\displaystyle -4+{(\frac{21}{26}-M\left(\lambda \right)-\frac{1}{2}\sqrt{P\left(\lambda \right)-N\left(\lambda \right)})}^{2}}})$;
- $c{r}_{5}=\frac{1}{2}(-\frac{21}{26}+M\left(\lambda \right)+\frac{1}{2}\sqrt{P\left(\lambda \right)-N\left(\lambda \right)}+\sqrt{{\displaystyle -4+{(\frac{21}{26}-M\left(\lambda \right)-\frac{1}{2}\sqrt{P\left(\lambda \right)-N\left(\lambda \right)})}^{2}}})$;
- $c{r}_{6}=\frac{1}{2}(-\frac{21}{26}+M\left(\lambda \right)-\frac{1}{2}\sqrt{P\left(\lambda \right)-N\left(\lambda \right)}-\sqrt{{\displaystyle -4+{(\frac{21}{26}-M\left(\lambda \right)+\frac{1}{2}\sqrt{P\left(\lambda \right)-N\left(\lambda \right)})}^{2}}})$;
- $c{r}_{7}=\frac{1}{2}(-\frac{21}{26}+M\left(\lambda \right)-\frac{1}{2}\sqrt{P\left(\lambda \right)-N\left(\lambda \right)}+\sqrt{{\displaystyle -4+{(\frac{21}{26}-M\left(\lambda \right)+\frac{1}{2}\sqrt{P\left(\lambda \right)-N\left(\lambda \right)})}^{2}}})$;
- $c{r}_{8}=\frac{1}{2}(-\frac{21}{26}-M\left(\lambda \right)+\frac{1}{2}\sqrt{P\left(\lambda \right)+N\left(\lambda \right)}-\sqrt{{\displaystyle -4+{(\frac{21}{26}+M\left(\lambda \right)-\frac{1}{2}\sqrt{P\left(\lambda \right)+N\left(\lambda \right)})}^{2}}})$;
- $c{r}_{9}=\frac{1}{2}(-\frac{21}{26}-M\left(\lambda \right)+\frac{1}{2}\sqrt{P\left(\lambda \right)+N\left(\lambda \right)}+\sqrt{{\displaystyle -4+{(\frac{21}{26}+M\left(\lambda \right)-\frac{1}{2}\sqrt{P\left(\lambda \right)+N\left(\lambda \right)})}^{2}}})$;
- $c{r}_{10}=\frac{1}{2}(-\frac{21}{26}-M\left(\lambda \right)-\frac{1}{2}\sqrt{P\left(\lambda \right)+N\left(\lambda \right)}-\sqrt{{\displaystyle -4+{(\frac{21}{26}+M\left(\lambda \right)+\frac{1}{2}\sqrt{P\left(\lambda \right)+N\left(\lambda \right)})}^{2}}})$;
- $c{r}_{11}=\frac{1}{2}(-\frac{21}{26}-M\left(\lambda \right)-\frac{1}{2}\sqrt{P\left(\lambda \right)+N\left(\lambda \right)}+\sqrt{{\displaystyle -4+{(\frac{21}{26}+M\left(\lambda \right)+\frac{1}{2}\sqrt{P\left(\lambda \right)+N\left(\lambda \right)})}^{2}}})$;

- $M\left(\lambda \right)=\frac{1}{2}\sqrt{a\left(\lambda \right)+b\left(\lambda \right)+c\left(\lambda \right)}$,
- $P\left(\lambda \right)=\frac{882}{169}+\frac{1}{39}(-128+5\lambda )-b\left(\lambda \right)-c\left(\lambda \right)-N\left(\lambda \right)$;
- $N\left(\lambda \right)=\frac{g\left(\lambda \right)}{h\left(\lambda \right)}$;
- $a\left(\lambda \right)=\frac{441}{169}+\frac{1}{156}(128-5\lambda )+\frac{1}{52}(-128+5\lambda )$;
- $b\left(\lambda \right)=\frac{d\left(\lambda \right)}{156{(e\left(\lambda \right)+18\sqrt{f\left(\lambda \right)})}^{1/3}}$;
- $c\left(\lambda \right)=\frac{1}{156}{(e\left(\lambda \right)+18\sqrt{f\left(\lambda \right)})}^{1/3}$;
- $d\left(\lambda \right)=16384+4696\lambda +25{\lambda}^{2}$;
- $e\left(\lambda \right)=2097152+1160832\lambda +139578{\lambda}^{2}-125{\lambda}^{3}$;
- $f\left(\lambda \right)=3355443200\lambda +2558365696{\lambda}^{2}+643302144{\lambda}^{3}+54034441{\lambda}^{4}-134875{\lambda}^{5}$;
- $g\left(\lambda \right)=\frac{74088}{2197}-\frac{42}{169}(128-5\lambda )+\frac{6\lambda}{13}$;
- $h\left(\lambda \right)=4\sqrt{{\displaystyle \frac{441}{169}+\frac{1}{156}(128-5\lambda )+\frac{1}{52}(-128+5\lambda )+b\left(\lambda \right)+c\left(\lambda \right)}}$.

- ${R}_{\lambda}\left(x\right)$ has 13 critical points, when $\lambda \notin \{0,1344\}$ and $\lambda \in \mathbb{C}$;
- ${R}_{\lambda}\left(x\right)$ has 7 critical points, when $\lambda =0$;
- ${R}_{\lambda}\left(x\right)$ has 11 critical points, when $\lambda =1344$;
- The free critical points of ${R}_{\lambda}\left(x\right)$ satisfy $c{r}_{m}=\frac{1}{c{r}_{n}}$ for $m\ne n$; each pair is conjugate to each other, being $c{r}_{2}$ and $c{r}_{3}$, $c{r}_{4}$ and $c{r}_{5}$, $c{r}_{6}$ and $c{r}_{7}$, $c{r}_{8}$ and $c{r}_{9}$, and $c{r}_{10}$ and $c{r}_{11}$.

**Proof.**

#### 2.3. Parameter Spaces and Dynamical Planes

## 3. Numerical Experiment and Practical Application

- When $\lambda =0$, ${X}_{n+1}=128{X}_{n}^{5}{(I+{X}_{n}^{2})}^{4}{[I-9{X}_{n}^{2}+57{X}_{n}^{4}+247{X}_{n}^{6}+763{X}_{n}^{8}+637{X}_{n}^{10}+331{X}_{n}^{12}+21{X}_{n}^{14}]}^{-1}$;
- When $\lambda =-50$, ${X}_{n+1}=8192{X}_{n}^{7}{(I+{X}_{n}^{2})}^{6}{[50{(-I+{X}_{n}^{2})}^{10}+64{({X}_{n}+{X}_{n}^{3})}^{2}(I-9{X}_{n}^{2}+57{X}_{n}^{4}+247{X}_{n}^{6}+763{X}_{n}^{8}+637{X}_{n}^{10}+331{X}_{n}^{12}+21{X}_{n}^{14})]}^{-1}$;
- When $\lambda =1300$, ${X}_{n+1}=8192{X}_{n}^{7}{(I+{X}_{n}^{2})}^{6}{[-1300{(-I+{X}_{n}^{2})}^{10}+64{({X}_{n}+{X}_{n}^{3})}^{2}(I-9{X}_{n}^{2}+57{X}_{n}^{4}+247{X}_{n}^{6}+763{X}_{n}^{8}+637{X}_{n}^{10}+331{X}_{n}^{12}+21{X}_{n}^{14})]}^{-1}$;
- When $\lambda =1340$, ${X}_{n+1}=8192{X}_{n}^{7}{(I+{X}_{n}^{2})}^{6}{[-1340{(-I+{X}_{n}^{2})}^{10}+64{({X}_{n}+{X}_{n}^{3})}^{2}(I-9{X}_{n}^{2}+57{X}_{n}^{4}+247{X}_{n}^{6}+763{X}_{n}^{8}+637{X}_{n}^{10}+331{X}_{n}^{12}+21{X}_{n}^{14})]}^{-1}$;
- When $\lambda =-1500$, ${X}_{n+1}=8192{X}_{n}^{7}{(I+{X}_{n}^{2})}^{6}{[1500{(-I+{X}_{n}^{2})}^{10}+64{({X}_{n}+{X}_{n}^{3})}^{2}(I-9{X}_{n}^{2}+57{X}_{n}^{4}+247{X}_{n}^{6}+763{X}_{n}^{8}+637{X}_{n}^{10}+331{X}_{n}^{12}+21{X}_{n}^{14})]}^{-1}$.

## 4. Conclusions

## Author Contributions

## Funding

## Data Availability Statement

## Conflicts of Interest

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$\mathsf{\lambda}$ | ${\mathit{ex}}_{\mathit{i}}$ | No. of ${\mathit{ex}}_{\mathit{i}}$ | ||
---|---|---|---|---|

$|{\mathit{R}}_{\mathsf{\lambda}}^{\prime}\left({\mathit{ex}}_{\mathit{i}}\right)|:\mathit{t}$${}^{\mathbf{1}}$ | ||||

0 | $-1.62291\pm 0.455126i$ | $-0.611529\pm 1.28544i$ | $0.234438\pm 1.48434i$ | 6 |

$6.31492:r$ | $4.91253:r$ | $7.6094:r$ | ||

1344 | $-2.57424$ | $-0.085032$ | $0.123846$ | 11 |

$4.9157:r$ | $1.17175\times {10}^{-6}:a$ | $6.42351\times {10}^{-6}:a$ | ||

$-2.16337\pm 1.21147i$ | $-1.09844\pm 2.04782i$ | $0.10715\pm 2.1662i$ | ||

$4.67559:r$ | $4.68214:r$ | 5.53101:r | ||

$0.922374\pm 1.48996i$ | ||||

$5.75625:r$ | ||||

$-6848$ | $-2.81972\pm 0.753267i$ | $-1.9133\pm 1.99919i$ | $-0.512724\pm 2.5536i$ | 12 |

$4.07847:r$ | $3.98574:r$ | $4.3251:r$ | ||

$-0.00328839\pm 0.0431351i$ | $0.793402\pm 2.17265i$ | $1.45563\pm 0.903056i$ | ||

$7.52394\times {10}^{-9}:a$ | $4.60277:r$ | $3.6229:r$ |

^{1}$|{R}_{\lambda}^{\prime}\left({\mathit{ex}}_{i}\right)|:t$ implies that ex

_{i}is attractive, parabolic, and repulsive, if $t=a\left(\right|{R}^{\prime}|<1)$, $t=a\left(\right|{R}^{\prime}|=1)$, and $t=a\left(\right|{R}^{\prime}|>1)$, respectively.

**Table 2.**Free critical points $c{r}_{i}$ from ${R}_{\lambda}^{\prime}\left(c{r}_{i}\right)$ for special $\lambda $-values.

$\mathsf{\lambda}$ | ${\mathit{cr}}_{\mathit{i}}$ | No. of ${\mathit{cr}}_{\mathit{i}}$ | ||
---|---|---|---|---|

0 | $-1$ | $\pm i$ | $0.615385\pm 0.788227i$ | 5 |

1344 | $-1$ | $\pm i$ | $-13.0923$ | 9 |

$-0.0763809$ | $0.109819$ | $9.10591$ | ||

$-0.638919\pm 0.769274i$ | ||||

$-6848$ | $-1$ | $\pm i$ | $-1.90256\pm 25.7656i$ | 11 |

$-0.639\pm 0.769207i$ | $-0.00285034\pm 0.038601i$ | $0.929025\pm 0.370017i$ |

$\mathsf{\lambda}=0$ | $\mathsf{\lambda}=-50$ | $\mathsf{\lambda}=1300$ | $\mathsf{\lambda}=1340$ | $\mathsf{\lambda}=-1500$ | ||||||
---|---|---|---|---|---|---|---|---|---|---|

Matrices | n${}^{1}$ | Time | n | Time | n | Time | n | Time | n | Time |

${R}_{4\times 4}$ | 3 | $0.006994$ | 5 | $0.079566$ | 6 | $0.074327$ | 6 | $0.103540$ | 5 | $0.108107$ |

${R}_{5\times 5}$ | 3 | $0.016522$ | 13 | $0.066437$ | 13 | $0.094517$ | 9 | $0.122004$ | 16 | $0.129702$ |

${R}_{10\times 10}$ | 5 | $0.069684$ | 10 | $0.139051$ | 27 | $0.136659$ | 41 | $0.157447$ | 15 | $0.059954$ |

${R}_{15\times 15}$ | 6 | $0.070848$ | 21 | $0.160829$ | 22 | $0.187887$ | 13 | $0.158099$ | 86 | $0.279862$ |

${R}_{20\times 20}$ | 6 | $0.071416$ | 19 | $0.180107$ | 19 | $0.186666$ | 17 | $0.162618$ | 103 | $0.718463$ |

${R}_{25\times 25}$ | 9 | $0.111521$ | 14 | $0.143166$ | 20 | $0.209045$ | 13 | $0.158114$ | 97 | $0.380726$ |

${R}_{30\times 30}$ | 8 | $0.136161$ | 28 | $0.203730$ | 16 | $0.177654$ | 15 | $0.176462$ | 191 | $0.832457$ |

^{1}n represents the number of iterations.

n | $\mathit{LWM}$ | $\mathit{SM}$ | $\mathit{SSM}$ | |||
---|---|---|---|---|---|---|

$\mathbf{|}{\mathit{x}}_{\mathit{n}}\mathbf{-}{\mathit{x}}_{\mathit{n}\mathbf{-}\mathbf{1}}\mathbf{|}$ | $\mathbf{\left|}\mathit{f}\mathbf{\right(}{\mathit{x}}_{\mathit{n}}\mathbf{\left)}\mathbf{\right|}$ | $\mathbf{|}{\mathit{x}}_{\mathit{n}}\mathbf{-}{\mathit{x}}_{\mathit{n}\mathbf{-}\mathit{1}}\mathbf{|}$ | $\mathbf{\left|}\mathit{f}\mathbf{\right(}{\mathit{x}}_{\mathit{n}}\mathbf{\left)}\mathbf{\right|}$ | $\mathbf{|}{\mathit{x}}_{\mathit{n}}\mathbf{-}{\mathit{x}}_{\mathit{n}\mathbf{-}\mathbf{1}}\mathbf{|}$ | $\mathbf{\left|}\mathit{f}\mathbf{\right(}{\mathit{x}}_{\mathit{n}}\mathbf{\left)}\mathbf{\right|}$ | |

1 | $5.7639\times {10}^{-4}$ | $1.818\times {10}^{-21}$ | $1.1528\times {10}^{-3}$ | $1.593\times {10}^{-1}$ | $5.7639\times {10}^{-4}$ | $1.4902\times {10}^{-19}$ |

2 | $6.5748\times {10}^{-24}$ | $4.1048\times {10}^{-47}$ | $1.1528\times {10}^{-3}$ | $1.5945\times {10}^{-1}$ | $5.3893\times {10}^{-22}$ | $1.6823\times {10}^{-45}$ |

3 | $1.4846\times {10}^{-49}$ | $9.2685\times {10}^{-73}$ | $1.1528\times {10}^{-3}$ | $1.593\times {10}^{-1}$ | $6.0843\times {10}^{-48}$ | $1.8993\times {10}^{-71}$ |

4 | $3.352\times {10}^{-75}$ | $2.0928\times {10}^{-98}$ | $1.1528\times {10}^{-3}$ | $1.5945\times {10}^{-1}$ | $6.869\times {10}^{-74}$ | $2.1442\times {10}^{-97}$ |

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Wang, X.; Li, W.
Stability Analysis of Simple Root Seeker for Nonlinear Equation. *Axioms* **2023**, *12*, 215.
https://doi.org/10.3390/axioms12020215

**AMA Style**

Wang X, Li W.
Stability Analysis of Simple Root Seeker for Nonlinear Equation. *Axioms*. 2023; 12(2):215.
https://doi.org/10.3390/axioms12020215

**Chicago/Turabian Style**

Wang, Xiaofeng, and Wenshuo Li.
2023. "Stability Analysis of Simple Root Seeker for Nonlinear Equation" *Axioms* 12, no. 2: 215.
https://doi.org/10.3390/axioms12020215