# On Derivative Free Multiple-Root Finders with Optimal Fourth Order Convergence

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## Abstract

**:**

## 1. Introduction

## 2. Development of a Novel Scheme

**Theorem**

**1.**

**Proof.**

**Theorem**

**2.**

**Proof.**

**Remark**

**1.**

## 3. Main Result

**Theorem**

**3.**

**Proof.**

**Remark**

**2.**

**Remark**

**3.**

#### Some Special Cases

- Method 1 (M1) :$${u}_{k+1}={z}_{k}-\frac{\mu \phantom{\rule{0.166667em}{0ex}}h(1+3\phantom{\rule{0.166667em}{0ex}}h)}{2}\left(1+\frac{1}{{y}_{k}}\right)\frac{\psi \left({u}_{k}\right)}{\psi [{v}_{k},{u}_{k}]}.$$
- Method 2 (M2) :$${u}_{k+1}={z}_{k}-\frac{\mu \phantom{\rule{0.166667em}{0ex}}h}{2-6h}\left(1+\frac{1}{{y}_{k}}\right)\frac{\psi \left({u}_{k}\right)}{\psi [{v}_{k},{u}_{k}]}.$$
- Method 3 (M3) :$${u}_{k+1}={z}_{k}-\frac{\mu \phantom{\rule{0.166667em}{0ex}}h(\mu -2h)}{2(\mu -(2+3\mu )h+2\mu {h}^{2})}\left(1+\frac{1}{{y}_{k}}\right)\frac{\psi \left({u}_{k}\right)}{\psi [{v}_{k},{u}_{k}]}.$$
- Method 4 (M4) :$${u}_{k+1}={z}_{k}-\frac{\mu \phantom{\rule{0.166667em}{0ex}}h(3-h)}{6-20h}\left(1+\frac{1}{{y}_{k}}\right)\frac{\psi \left({u}_{k}\right)}{\psi [{v}_{k},{u}_{k}]}.$$

## 4. Basins of Attraction

**Test problem 1**. Consider the polynomial ${\psi}_{1}\left(z\right)={({z}^{2}+z+1)}^{2}$ having two zeros $\{-0.5-0.866025i,-0.5+0.866025i\}$ with multiplicity $\mu =2$. The attraction basins for this polynomial are shown in Figure 1, Figure 2 and Figure 3 corresponding to the choices $0.01,\phantom{\rule{0.166667em}{0ex}}{10}^{-4},\phantom{\rule{0.166667em}{0ex}}{10}^{-6}$ of parameter $\beta $. A color is assigned to each basin of attraction of a zero. In particular, red and green colors have been allocated to the basins of attraction of the zeros $-0.5-0.866025i$ and $-0.5+0.866025i$, respectively.

**Test problem 2**. Consider the polynomial ${\psi}_{2}\left(z\right)={\left({z}^{3}+\frac{1}{4}z\right)}^{3}$ which has three zeros $\{-\frac{i}{2},\frac{i}{2},0\}$ with multiplicities $\mu =3$. Basins of attractors assessed by methods for this polynomial are drawn in Figure 4, Figure 5 and Figure 6 corresponding to choices $\beta =0.01,\phantom{\rule{0.166667em}{0ex}}{10}^{-4},\phantom{\rule{0.166667em}{0ex}}{10}^{-6}.$ The corresponding basin of a zero is identified by a color assigned to it. For example, green, red, and blue colors have been assigned corresponding to $-\frac{i}{2}$, $\frac{i}{2}$, and 0.

**Test problem 3**. Next, let us consider the polynomial ${\psi}_{3}\left(z\right)={\left({z}^{3}+\frac{1}{z}\right)}^{4}$ that has four zeros $\{-0.707107+0.707107i,-0.707107-0.707107i,0.707107+0.707107i,0.707107-0.707107i\}$ with multiplicity $\mu =4$. The basins of attractors of zeros are shown in Figure 7, Figure 8 and Figure 9, for choices of the parameter $\beta =0.01,\phantom{\rule{0.166667em}{0ex}}{10}^{-4},\phantom{\rule{0.166667em}{0ex}}{10}^{-6}.$ A color is assigned to each basin of attraction of a zero. In particular, we assign yellow, blue, red, and green colors to $-0.707107+0.707107i$, $-0.707107-0.707107i$, $0.707107+0.707107i$ and $0.707107-0.707107i$, respectively.

## 5. Numerical Results

## 6. Conclusions

## Author Contributions

## Funding

## Conflicts of Interest

## References

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**Figure 1.**Basins of attraction by M-1–M-4 $(\beta =0.01)$ for polynomial ${\psi}_{1}\left(z\right)$.

**Figure 2.**Basins of attraction by M-1–M-4 $(\beta ={10}^{-4})$ for polynomial ${\psi}_{1}\left(z\right)$.

**Figure 3.**Basins of attraction by M-1–M-4 $(\beta ={10}^{-6})$ for polynomial ${\psi}_{1}\left(z\right)$.

**Figure 4.**Basins of attraction by M-1–M-4 $(\beta =0.01)$ for polynomial ${\psi}_{2}\left(z\right)$.

**Figure 5.**Basins of attraction by M-1–M-4 $(\beta ={10}^{-4})$ for polynomial ${\psi}_{2}\left(z\right)$.

**Figure 6.**Basins of attraction by methods M-1–M-4 $(\beta ={10}^{-6})$ for polynomial ${\psi}_{2}\left(z\right)$.

**Figure 7.**Basins of attraction by M-1–M-4 $(\beta =0.01)$ for polynomial ${\psi}_{3}\left(z\right)$.

**Figure 8.**Basins of attraction by M-1–M-4 $(\beta ={10}^{-4})$ for polynomial ${\psi}_{3}\left(z\right)$.

**Figure 9.**Basins of attraction by M-1–M-4 $(\beta ={10}^{-6})$ for polynomial ${\psi}_{3}\left(z\right)$.

Methods | k | $|{\mathit{u}}_{2}-{\mathit{u}}_{1}|$ | $|{\mathit{u}}_{3}-{\mathit{u}}_{2}|$ | $|{\mathit{u}}_{4}-{\mathit{u}}_{3}|$ | CCO | CPU-Time |
---|---|---|---|---|---|---|

${\psi}_{1}\left(u\right)$ | ||||||

LLCM | 6 | $7.84\times {10}^{-2}$ | $6.31\times {10}^{-3}$ | $1.06\times {10}^{-5}$ | 4.000 | 0.0784 |

LCNM | 6 | $7.84\times {10}^{-2}$ | $6.31\times {10}^{-3}$ | $1.006\times {10}^{-5}$ | 4.000 | 0.0822 |

SSM | 6 | $7.99\times {10}^{-2}$ | $6.78\times {10}^{-3}$ | $1.44\times {10}^{-5}$ | 4.000 | 0.0943 |

ZCSM | 6 | $8.31\times {10}^{-2}$ | $7.83\times {10}^{-3}$ | $2.76\times {10}^{-5}$ | 4.000 | 0.0956 |

SBLM | 6 | $7.84\times {10}^{-2}$ | $6.31\times {10}^{-3}$ | $1.06\times {10}^{-5}$ | 4.000 | 0.0874 |

KKBM | 6 | $7.74\times {10}^{-2}$ | $5.97\times {10}^{-3}$ | $7.31\times {10}^{-6}$ | 4.000 | 0.0945 |

M1 | 6 | $9.20\times {10}^{-2}$ | $1.16\times {10}^{-2}$ | $1.16\times {10}^{-4}$ | 4.000 | 0.0774 |

M2 | 6 | $6.90\times {10}^{-2}$ | $3.84\times {10}^{-3}$ | $1.03\times {10}^{-6}$ | 4.000 | 0.0794 |

M3 | 6 | $6.21\times {10}^{-2}$ | $2.39\times {10}^{-3}$ | $7.06\times {10}^{-8}$ | 4.000 | 0.0626 |

M4 | 6 | $6.29\times {10}^{-2}$ | $2.54\times {10}^{-3}$ | $9.28\times {10}^{-8}$ | 4.000 | 0.0785 |

${\psi}_{2}\left(u\right)$ | ||||||

LLCM | 4 | $2.02\times {10}^{-4}$ | $2.11\times {10}^{-17}$ | $2.51\times {10}^{-69}$ | 4.000 | 0.7334 |

LCNM | 4 | $2.02\times {10}^{-4}$ | $2.12\times {10}^{-17}$ | $2.54\times {10}^{-69}$ | 4.000 | 1.0774 |

SSM | 4 | $2.02\times {10}^{-4}$ | $2.12\times {10}^{-17}$ | $2.60\times {10}^{-69}$ | 4.000 | 1.0765 |

ZCSM | 4 | $2.02\times {10}^{-4}$ | $2.15\times {10}^{-17}$ | $2.75\times {10}^{-69}$ | 4.000 | 1.1082 |

SBLM | 4 | $2.02\times {10}^{-4}$ | $2.13\times {10}^{-17}$ | $2.62\times {10}^{-69}$ | 4.000 | 1.2950 |

KKBM | 4 | $2.02\times {10}^{-4}$ | $2.08\times {10}^{-17}$ | $2.31\times {10}^{-69}$ | 4.000 | 1.1548 |

M1 | 4 | $1.01\times {10}^{-4}$ | $1.08\times {10}^{-18}$ | $1.43\times {10}^{-74}$ | 4.000 | 0.5612 |

M2 | 4 | $9.85\times {10}^{-5}$ | $4.94\times {10}^{-19}$ | $3.13\times {10}^{-76}$ | 4.000 | 0.5154 |

M3 | 4 | $9.85\times {10}^{-5}$ | $4.94\times {10}^{-19}$ | $3.13\times {10}^{-76}$ | 4.000 | 0.5311 |

M4 | 4 | $9.82\times {10}^{-5}$ | $4.35\times {10}^{-19}$ | $1.67\times {10}^{-76}$ | 4.000 | 0.5003 |

${\psi}_{3}\left(u\right)$ | ||||||

LLCM | 4 | $4.91\times {10}^{-5}$ | $5.70\times {10}^{-21}$ | $1.03\times {10}^{-84}$ | 4.000 | 0.6704 |

LCNM | 4 | $4.91\times {10}^{-5}$ | $5.70\times {10}^{-21}$ | $1.03\times {10}^{-84}$ | 4.000 | 0.9832 |

SSM | 4 | $4.92\times {10}^{-5}$ | $5.71\times {10}^{-21}$ | $1.04\times {10}^{-84}$ | 4.000 | 1.0303 |

ZCSM | 4 | $4.92\times {10}^{-5}$ | $5.72\times {10}^{-21}$ | $1.05\times {10}^{-84}$ | 4.000 | 1.0617 |

SBLM | 4 | $4.92\times {10}^{-5}$ | $5.73\times {10}^{-21}$ | $1.06\times {10}^{-84}$ | 4.000 | 1.2644 |

KKBM | 4 | $4.91\times {10}^{-5}$ | $5.66\times {10}^{-21}$ | $1.00\times {10}^{-84}$ | 4.000 | 1.0768 |

M1 | 3 | $6.35\times {10}^{-6}$ | $2.73\times {10}^{-25}$ | 0 | 4.000 | 0.3433 |

M2 | 3 | $4.94\times {10}^{-6}$ | $6.81\times {10}^{-26}$ | 0 | 4.000 | 0.2965 |

M3 | 3 | $5.02\times {10}^{-6}$ | $7.46\times {10}^{-26}$ | 0 | 4.000 | 0.3598 |

M4 | 3 | $4.77\times {10}^{-6}$ | $5.66\times {10}^{-26}$ | 0 | 4.000 | 0.3446 |

${\psi}_{4}\left(u\right)$ | ||||||

LLCM | 4 | $1.15\times {10}^{-4}$ | $5.69\times {10}^{-17}$ | $3.39\times {10}^{-66}$ | 4.000 | 1.4824 |

LCNM | 4 | $1.15\times {10}^{-4}$ | $5.70\times {10}^{-17}$ | $3.40\times {10}^{-66}$ | 4.000 | 2.5745 |

SSM | 4 | $1.15\times {10}^{-4}$ | $5.71\times {10}^{-17}$ | $3.44\times {10}^{-66}$ | 4.000 | 2.5126 |

ZCSM | 4 | $1.15\times {10}^{-4}$ | $5.72\times {10}^{-17}$ | $3.47\times {10}^{-66}$ | 4.000 | 2.5587 |

SBLM | 4 | $1.15\times {10}^{-4}$ | $5.83\times {10}^{-17}$ | $3.79\times {10}^{-66}$ | 4.000 | 3.1824 |

KKBM | 4 | $1.15\times {10}^{-4}$ | $5.63\times {10}^{-17}$ | $3.21\times {10}^{-66}$ | 4.000 | 2.4965 |

M1 | 4 | $4.18\times {10}^{-4}$ | $6.03\times {10}^{-19}$ | $2.60\times {10}^{-74}$ | 4.000 | 0.4993 |

M2 | 4 | $3.88\times {10}^{-5}$ | $2.24\times {10}^{-19}$ | $2.45\times {10}^{-76}$ | 4.000 | 0.5151 |

M3 | 4 | $3.92\times {10}^{-5}$ | $2.57\times {10}^{-19}$ | $4.80\times {10}^{-76}$ | 4.000 | 0.4996 |

M4 | 4 | $3.85\times {10}^{-5}$ | $1.92\times {10}^{-19}$ | $1.18\times {10}^{-76}$ | 4.000 | 0.4686 |

${\psi}_{5}\left(u\right)$ | ||||||

LLCM | 4 | $2.16\times {10}^{-4}$ | $3.17\times {10}^{-17}$ | $1.48\times {10}^{-68}$ | 4.000 | 1.9042 |

LCNM | 4 | $2.16\times {10}^{-4}$ | $3.17\times {10}^{-17}$ | $1.47\times {10}^{-68}$ | 4.000 | 2.0594 |

SSM | 4 | $2.16\times {10}^{-4}$ | $3.16\times {10}^{-17}$ | $1.45\times {10}^{-68}$ | 4.000 | 2.0125 |

ZCSM | 4 | $2.16\times {10}^{-4}$ | $3.15\times {10}^{-17}$ | $1.43\times {10}^{-68}$ | 4.000 | 2.1530 |

SBLM | 4 | $2.16\times {10}^{-4}$ | $3.01\times {10}^{-17}$ | $1.15\times {10}^{-68}$ | 4.000 | 2.4185 |

KKBM | 4 | $2.16\times {10}^{-4}$ | $3.24\times {10}^{-17}$ | $1.63\times {10}^{-68}$ | 4.000 | 2.2153 |

M1 | 4 | $2.48\times {10}^{-4}$ | $7.62\times {10}^{-21}$ | $6.81\times {10}^{-83}$ | 4.000 | 1.6697 |

M2 | 4 | $2.15\times {10}^{-5}$ | $2.03\times {10}^{-21}$ | $1.63\times {10}^{-85}$ | 4.000 | 1.7793 |

M3 | 4 | $2.19\times {10}^{-5}$ | $2.51\times {10}^{-21}$ | $4.35\times {10}^{-85}$ | 4.000 | 1.7942 |

M4 | 4 | $2.11\times {10}^{-5}$ | $1.66\times {10}^{-21}$ | $6.29\times {10}^{-86}$ | 4.000 | 1.6855 |

Functions | Root ($\mathit{\alpha}$) | Multiplicity | Initial Guess | |||
---|---|---|---|---|---|---|

${\psi}_{1}\left(u\right)={u}^{3}-5.22{u}^{2}+9.0825u-5.2675$ | 1.75 | 2 | 2.4 | |||

${\psi}_{2}\left(u\right)=-\frac{{u}^{4}}{12}+\frac{{u}^{2}}{2}+u+{e}^{u}(u-3)+sinu+3$ | 0 | 3 | 0.6 | |||

${\psi}_{3}\left(u\right)={\left({e}^{-u}-1+\frac{u}{5}\right)}^{4}$ | 4.9651142317… | 4 | 5.5 | |||

${\psi}_{4}\left(u\right)=u({u}^{2}+1)(2{e}^{{u}^{2}+1}+{u}^{2}-1){cosh}^{4}\left(\frac{\pi u}{2}\right)$ | i | 6 | 1.2 i | |||

${\psi}_{5}\left(u\right)=[{tan}^{-1}\left(\frac{\sqrt{5}}{2}\right)-{tan}^{-1}\left(\sqrt{{u}^{2}-1}\right)+\sqrt{6}({tan}^{-1}\left(\sqrt{\frac{{u}^{2}-1}{6}}\right)$ | ||||||

$\phantom{\rule{4pt}{0ex}}\phantom{\rule{4pt}{0ex}}\phantom{\rule{4pt}{0ex}}\phantom{\rule{4pt}{0ex}}\phantom{\rule{4pt}{0ex}}\phantom{\rule{4pt}{0ex}}\phantom{\rule{4pt}{0ex}}\phantom{\rule{4pt}{0ex}}\phantom{\rule{4pt}{0ex}}\phantom{\rule{4pt}{0ex}}-{tan}^{-1}\left(\frac{1}{2}\sqrt{\frac{5}{6}}\right))-\frac{11}{63}{]}^{7}$ | 1.8411294068… | 7 | 1.6 |

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**MDPI and ACS Style**

Sharma, J.R.; Kumar, S.; Jäntschi, L.
On Derivative Free Multiple-Root Finders with Optimal Fourth Order Convergence. *Mathematics* **2020**, *8*, 1091.
https://doi.org/10.3390/math8071091

**AMA Style**

Sharma JR, Kumar S, Jäntschi L.
On Derivative Free Multiple-Root Finders with Optimal Fourth Order Convergence. *Mathematics*. 2020; 8(7):1091.
https://doi.org/10.3390/math8071091

**Chicago/Turabian Style**

Sharma, Janak Raj, Sunil Kumar, and Lorentz Jäntschi.
2020. "On Derivative Free Multiple-Root Finders with Optimal Fourth Order Convergence" *Mathematics* 8, no. 7: 1091.
https://doi.org/10.3390/math8071091