# On a Class of Optimal Fourth Order Multiple Root Solvers without Using Derivatives

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## Abstract

**:**

## 1. Introduction

## 2. Formulation of Method

**Theorem**

**1.**

**Proof.**

**Theorem**

**2.**

**Proof.**

**Theorem**

**3.**

**Theorem**

**4.**

**Theorem**

**5.**

**Remark**

**1.**

## 3. Main Result

**Theorem**

**6.**

**Proof.**

**Remark**

**2.**

**Remark**

**3.**

#### Some Special Cases

- $(1)$
- Let us choose the function$$H({x}_{k},{y}_{k})={x}_{k}+m\phantom{\rule{0.166667em}{0ex}}{x}_{k}^{2}+(m-1){y}_{k}+m\phantom{\rule{0.166667em}{0ex}}{x}_{k}\phantom{\rule{0.166667em}{0ex}}{y}_{k},$$$${t}_{k+1}={z}_{k}-\left({x}_{k}+m\phantom{\rule{0.166667em}{0ex}}{x}_{k}^{2}+(m-1){y}_{k}+m\phantom{\rule{0.166667em}{0ex}}{x}_{k}\phantom{\rule{0.166667em}{0ex}}{y}_{k}\right)\frac{f({t}_{k})}{f[{s}_{k},{t}_{k}]}.$$
- $(2)$
- Next, consider the rational function$$H({x}_{k},{y}_{k})=-\frac{{x}_{k}+m\phantom{\rule{0.166667em}{0ex}}{x}_{k}^{2}-(m-1){y}_{k}(m\phantom{\rule{0.166667em}{0ex}}{y}_{k}-1)}{m\phantom{\rule{0.166667em}{0ex}}{y}_{k}-1},$$$${t}_{k+1}={z}_{k}+\frac{{x}_{k}+m\phantom{\rule{0.166667em}{0ex}}{x}_{k}^{2}-(m-1){y}_{k}(m\phantom{\rule{0.166667em}{0ex}}{y}_{k}-1)}{m\phantom{\rule{0.166667em}{0ex}}{y}_{k}-1}\frac{f({t}_{k})}{f[{s}_{k},{t}_{k}]}.$$
- $(3)$
- Consider another rational function satisfying the conditions of Theorems 1, 2 and 6, which is given by$$\begin{array}{cc}\hfill H({x}_{k},{y}_{k})=& \phantom{\rule{4pt}{0ex}}\frac{{x}_{k}-{y}_{k}+m\phantom{\rule{0.166667em}{0ex}}{y}_{k}+2m\phantom{\rule{0.166667em}{0ex}}{x}_{k}\phantom{\rule{0.166667em}{0ex}}{y}_{k}-{m}^{2}\phantom{\rule{0.166667em}{0ex}}{x}_{k}\phantom{\rule{0.166667em}{0ex}}{y}_{k}}{1-m\phantom{\rule{0.166667em}{0ex}}{x}_{k}+{x}_{k}^{2}}.\hfill \end{array}$$$${t}_{k+1}={z}_{k}-\frac{{x}_{k}-{y}_{k}+m\phantom{\rule{0.166667em}{0ex}}{y}_{k}+2m\phantom{\rule{0.166667em}{0ex}}{x}_{k}\phantom{\rule{0.166667em}{0ex}}{y}_{k}-{m}^{2}\phantom{\rule{0.166667em}{0ex}}{x}_{k}\phantom{\rule{0.166667em}{0ex}}{y}_{k}}{1-m\phantom{\rule{0.166667em}{0ex}}{x}_{k}+{x}_{k}^{2}}\frac{f({t}_{k})}{f[{s}_{k},{t}_{k}]}.$$

## 4. Numerical Results

**Example**

**1.**

**Example**

**2.**

**Example**

**3.**

**Example**

**4.**

## 5. Conclusions

## Author Contributions

## Funding

## Conflicts of Interest

## References

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

LLC | 4 | $1.51\times {10}^{-5}$ | $1.47\times {10}^{-23}$ | $1.30\times {10}^{-95}$ | 4.000 | 0.4993 | ||||||

LCN | 4 | $1.55\times {10}^{-5}$ | $1.73\times {10}^{-23}$ | $2.65\times {10}^{-95}$ | 4.000 | 0.5302 | ||||||

SS | 4 | $1.52\times {10}^{-5}$ | $1.51\times {10}^{-23}$ | $1.47\times {10}^{-95}$ | 4.000 | 0.6390 | ||||||

ZCS | 4 | $1.57\times {10}^{-5}$ | $1.87\times {10}^{-23}$ | $3.75\times {10}^{-95}$ | 4.000 | 0.6404 | ||||||

SBL | 4 | $1.50\times {10}^{-5}$ | $1.43\times {10}^{-23}$ | $1.19\times {10}^{-95}$ | 4.000 | 0.8112 | ||||||

KKB | fails | - | - | - | - | - | ||||||

NM1 | 3 | $5.59\times {10}^{-6}$ | $1.35\times {10}^{-25}$ | 0 | 4.000 | 0.3344 | ||||||

NM2 | 3 | $5.27\times {10}^{-6}$ | $9.80\times {10}^{-26}$ | 0 | 4.000 | 0.3726 | ||||||

NM3 | 3 | $5.43\times {10}^{-6}$ | $1.16\times {10}^{-25}$ | 0 | 4.000 | 0.3475 |

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

LLC | 6 | $9.09\times {10}^{-2}$ | $8.03\times {10}^{-3}$ | $2.33\times {10}^{-5}$ | 4.000 | 0.0780 | ||||||

LCN | 6 | $9.09\times {10}^{-2}$ | $8.03\times {10}^{-3}$ | $2.33\times {10}^{-5}$ | 4.000 | 0.0784 | ||||||

SS | 6 | $9.26\times {10}^{-2}$ | $8.58\times {10}^{-3}$ | $3.11\times {10}^{-5}$ | 4.000 | 0.0945 | ||||||

ZCS | 6 | $9.62\times {10}^{-2}$ | $9.84\times {10}^{-3}$ | $5.64\times {10}^{-5}$ | 4.000 | 0.0792 | ||||||

SBL | 6 | $9.09\times {10}^{-2}$ | $8.03\times {10}^{-3}$ | $2.33\times {10}^{-5}$ | 4.000 | 0.0797 | ||||||

KKB | 6 | $8.97\times {10}^{-2}$ | $7.62\times {10}^{-3}$ | $1.68\times {10}^{-5}$ | 4.000 | 0.0934 | ||||||

NM1 | 6 | $9.91\times {10}^{-2}$ | $1.08\times {10}^{-2}$ | $8.79\times {10}^{-5}$ | 4.000 | 0.0752 | ||||||

NM2 | 6 | $8.06\times {10}^{-2}$ | $5.08\times {10}^{-3}$ | $2.81\times {10}^{-5}$ | 4.000 | 0.0684 | ||||||

NM3 | 6 | $8.78\times {10}^{-2}$ | $7.02\times {10}^{-3}$ | $1.31\times {10}^{-5}$ | 4.000 | 0.0788 |

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

LLC | 4 | $1.11\times {10}^{-4}$ | $9.02\times {10}^{-19}$ | $3.91\times {10}^{-75}$ | 4.000 | 2.5743 | ||||||

LCN | 4 | $1.11\times {10}^{-4}$ | $8.93\times {10}^{-19}$ | $3.72\times {10}^{-75}$ | 4.000 | 2.6364 | ||||||

SS | 4 | $1.11\times {10}^{-4}$ | $8.71\times {10}^{-19}$ | $3.29\times {10}^{-75}$ | 4.000 | 2.8718 | ||||||

ZCS | 4 | $1.11\times {10}^{-4}$ | $8.16\times {10}^{-19}$ | $2.38\times {10}^{-75}$ | 4.000 | 2.8863 | ||||||

SBL | 4 | $1.11\times {10}^{-4}$ | $8.63\times {10}^{-19}$ | $3.15\times {10}^{-75}$ | 4.000 | 3.2605 | ||||||

KKB | 4 | $1.11\times {10}^{-4}$ | $9.80\times {10}^{-19}$ | $5.87\times {10}^{-75}$ | 4.000 | 2.9011 | ||||||

NM1 | 4 | $2.31\times {10}^{-5}$ | $4.04\times {10}^{-21}$ | $3.78\times {10}^{-84}$ | 4.000 | 2.2935 | ||||||

NM2 | 4 | $2.07\times {10}^{-5}$ | $1.32\times {10}^{-21}$ | $2.18\times {10}^{-86}$ | 4.000 | 2.5287 | ||||||

NM3 | 4 | $2.11\times {10}^{-5}$ | $1.66\times {10}^{-21}$ | $6.36\times {10}^{-86}$ | 4.000 | 2.4964 |

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

LLC | 4 | $2.64\times {10}^{-4}$ | $2.13\times {10}^{-15}$ | $9.11\times {10}^{-60}$ | 4.000 | 1.7382 | ||||||

LCN | 4 | $2.64\times {10}^{-4}$ | $2.14\times {10}^{-15}$ | $9.39\times {10}^{-60}$ | 4.000 | 2.4035 | ||||||

SS | 4 | $2.64\times {10}^{-4}$ | $2.18\times {10}^{-15}$ | $1.01\times {10}^{-59}$ | 4.000 | 2.5431 | ||||||

ZCS | 4 | $2.65\times {10}^{-4}$ | $2.24\times {10}^{-15}$ | $1.14\times {10}^{-59}$ | 4.000 | 2.6213 | ||||||

SBL | 4 | $2.66\times {10}^{-4}$ | $2.28\times {10}^{-15}$ | $1.23\times {10}^{-59}$ | 4.000 | 3.2610 | ||||||

KKB | 4 | $2.61\times {10}^{-4}$ | $2.00\times {10}^{-15}$ | $6.83\times {10}^{-60}$ | 4.000 | 2.6524 | ||||||

NM1 | 4 | $1.43\times {10}^{-4}$ | $1.29\times {10}^{-16}$ | $8.61\times {10}^{-65}$ | 4.000 | 0.5522 | ||||||

NM2 | 4 | $4.86\times {10}^{-5}$ | $5.98\times {10}^{-20}$ | $1.36\times {10}^{-79}$ | 4.000 | 0.6996 | ||||||

NM3 | 4 | $6.12\times {10}^{-5}$ | $6.69\times {10}^{-19}$ | $9.54\times {10}^{-75}$ | 4.000 | 0.6837 |

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**MDPI and ACS Style**

Sharma, J.R.; Kumar, S.; Jäntschi, L. On a Class of Optimal Fourth Order Multiple Root Solvers without Using Derivatives. *Symmetry* **2019**, *11*, 1452.
https://doi.org/10.3390/sym11121452

**AMA Style**

Sharma JR, Kumar S, Jäntschi L. On a Class of Optimal Fourth Order Multiple Root Solvers without Using Derivatives. *Symmetry*. 2019; 11(12):1452.
https://doi.org/10.3390/sym11121452

**Chicago/Turabian Style**

Sharma, Janak Raj, Sunil Kumar, and Lorentz Jäntschi. 2019. "On a Class of Optimal Fourth Order Multiple Root Solvers without Using Derivatives" *Symmetry* 11, no. 12: 1452.
https://doi.org/10.3390/sym11121452