# A Family of Higher Order Scheme for Multiple Roots

^{1}

^{2}

^{3}

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Construction of the Fifth-Order Family

**Theorem**

**1.**

**Proof of Theorem**

**1.**

## 3. Special Cases

## 4. Basins of Attraction

- The first rule is to select a rectangular area and make sure that every root of the considered polynomial lies inside this region.
- Each initial guess of a root is given a color in the basins of attraction. Similarly, another color is given to each initial guess of another root in the basins of attraction. For each of the complex polynomial’s roots, this operation is repeated.
- If the iterative formula which started with the initial approximation ${w}_{0}$ converges to a root, then the basins of attraction display the color which is allotted to the initial guess of that root at point ${w}_{0}.$ Otherwise, the initial guess ${w}_{0}$ is painted with black color.

`TM1`,

`TM2`,

`TM3`and

`TM4`, with some existing schemes of the same nature. For comparison, we consider the three subcases of the scheme by Sharma et al. [40] and three subcases of the family presented by Chanu et al. [41]. The first method developed by Sharma et al. [40], which we denoted by

`SM1`, is given as:

`SM2`, which is given below:

`SM1`and

`SM2`.

`SM3`, which is given below:

`NPM1`. The scheme for

`NPM1`is:

`NPM2`. This scheme is given by:

`NPM3`. The method is presented by:

`NPM1`,

`NPM2`and

`NPM3`.

`TM1`,

`TM2`,

`TM3`and

`TM4`with

`SM1`,

`SM2`,

`SM3`,

`NPM1`,

`NPM2`and

`NPM3`by using the following four problems through basins of attraction. To view dynamical vision, we assume a rectangle $D=[-3,3]\times [-3,3]\in \mathbb{C}$ with $500\times 500$ grid points.

**Problem**

**1.**

`TM1`,

`TM2`,

`TM3`,

`TM4`,

`SM1`,

`SM2`,

`SM3`,

`NPM1`,

`NPM2`and

`NPM3`. From Figure 1, one can easily observe that the methods

`TM1`,

`TM2`,

`TM3`and

`TM4`divide the complex plane into two equal halves without any disturbance in the regions, but in case of the methods

`SM1`,

`SM2`,

`SM3`,

`NPM1`,

`NPM2`and

`NPM3`, the complex plane is divided into two equal halves, which have disturbances. Therefore, the basins of attraction for the proposed methods,

`TM1`,

`TM2`,

`TM3`and

`TM4`, are more stable than those for

`SM1`,

`SM2`,

`SM3`,

`NPM1`,

`NPM2`and

`NPM3`. Among the newly proposed methods, all performed equally well.

**Problem**

**2.**

`TM1`,

`TM2`,

`TM3`,

`TM4`,

`SM1`,

`SM2`,

`SM3`,

`NPM1`,

`NPM2`and

`NPM3`. Black spots in Figure 2 show the points which are not convergent to any of the roots. It is clear from Figure 2 that the convergence area is more for methods

`TM1`,

`TM2`,

`TM3`,

`TM4`and

`SM3`as compared to the methods

`SM1`,

`SM2`,

`NPM1`,

`NPM2`and

`NPM3`. Furthermore, it can be seen that the new proposed methods,

`SM2`and

`SM3`, do not contain any black spots which indicate the divergent points.

**Problem**

**3.**

`TM1`,

`TM2`,

`TM3`and

`TM4`, as well as the method

`SM3`, are adequate to converge to all the roots, and the divergent points occur in case of the methods

`SM1`,

`SM2`,

`NPM1`,

`NPM2`and

`NPM3`.

**Problem**

**4.**

`TM1`,

`TM2`,

`TM3`,

`TM4`and

`SM3`do not contain any black spots, but the methods

`SM1`,

`SM2`,

`NPM1`,

`NPM2`and

`NPM3`do contain them, which means that the methods

`TM1`,

`TM2`,

`TM3`,

`TM4`and

`SM3`have performed better than the methods

`SM1`,

`SM2`,

`NPM1`,

`NPM2`and

`NPM3`.

## 5. Numerical Results

`NPM1`,

`NPM2`and

`NPM3`have the best residuals, but the major drawback of these methods is that they do not retain their order of convergence. Therefore, our proposed methods have performed better and possess lower residual error as well as low CPU-time for functions ${f}_{1}\left(\mathtt{x}\right)$ and ${f}_{2}\left(\mathtt{x}\right)$. They also preserve their fifth order of convergence. The numerical data for the function ${f}_{3}\left(\mathtt{x}\right)$ shown in Table 5 demonstrate that

`TM4`performs better than the existing methods, with low error and precise result estimations. Further, the elapsed CPU time is minimal in the case of

`TM3`. The numerical results obtained for function ${f}_{4}\left(\mathtt{x}\right)$ are shown in Table 6. One can observe that each scheme has extremely high precision, with the lowest residual calculated by

`TM4`. Table 7 shows the numerical results of the methods for function ${f}_{5}\left(\mathtt{x}\right)$. The best performer for this function is

`TM4`. Furthermore, it is worth mentioning here that the methods NPM1–NPM3 do not converge to the required root for the problems ${f}_{3}\left(\mathtt{x}\right)$ and ${f}_{5}\left(\mathtt{x}\right)$, or we can say that they are divergent for these problems.

## 6. Concluding Remarks

## Author Contributions

## Funding

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## References

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**Figure 1.**Basins of attraction of

`TM1`–

`TM4`,

`SM1`–

`SM3`and

`NPM1`–

`NPM3`for polynomial ${g}_{1}\left(w\right)$.

**Figure 2.**Basins of attraction of

`TM1`–

`TM4`,

`SM1`–

`SM3`and

`NPM1`–

`NPM3`for polynomial ${g}_{2}\left(w\right)$.

**Figure 3.**Basins of attraction of

`TM1`–

`TM4`,

`SM1`–

`SM3`and

`NPM1`–

`NPM3`for polynomial ${g}_{3}\left(w\right)$.

**Figure 4.**Basins of attraction of

`TM1`–

`TM4`,

`SM1`–

`SM3`and

`NPM1`–

`NPM3`for polynomial ${g}_{4}\left(w\right)$.

**Table 1.**Some special cases of Mechanism (5).

Cases (Naming) | Weight Functions | Equivalent Iterative Method |
---|---|---|

Case-1 (TM1) | $R\left({\upsilon}_{t}\right)=1+{\upsilon}_{t}^{2}$ | ${\mathtt{y}}_{t}={\mathtt{x}}_{t}-\tilde{m}\frac{g\left({\mathtt{x}}_{t}\right)}{{g}^{\prime}\left({\mathtt{x}}_{t}\right)},$ |

${\mathtt{x}}_{t+1}={\mathtt{y}}_{t}-\tilde{m}(1+{\upsilon}_{t}^{2})\frac{g\left({\mathtt{y}}_{t}\right)}{{g}^{\prime}\left({\mathtt{y}}_{t}\right)}$. | ||

Case-2 (TM2) | $R\left({\upsilon}_{t}\right)={e}^{-{\upsilon}_{t}}+{\upsilon}_{t}+\frac{{\upsilon}_{t}^{2}}{2}$ | ${\mathtt{y}}_{t}={\mathtt{x}}_{t}-\tilde{m}\frac{g\left({\mathtt{x}}_{t}\right)}{{g}^{\prime}\left({\mathtt{x}}_{t}\right)},$ |

${\mathtt{x}}_{t+1}={\mathtt{y}}_{t}-\tilde{m}({e}^{-{\upsilon}_{t}}+{\upsilon}_{t}+\frac{{\upsilon}_{t}^{2}}{2})\frac{g\left({\mathtt{y}}_{t}\right)}{{g}^{\prime}\left({\mathtt{y}}_{t}\right)}$. | ||

Case-3 (TM3) | $R\left({\upsilon}_{t}\right)=(1-{\upsilon}_{t}){e}^{{\upsilon}_{t}}+\frac{3}{2}{\upsilon}_{t}^{2}$ | ${\mathtt{y}}_{t}={\mathtt{x}}_{t}-\tilde{m}\frac{g\left({\mathtt{x}}_{t}\right)}{{g}^{\prime}\left({\mathtt{x}}_{t}\right)},$ |

${\mathtt{x}}_{t+1}={\mathtt{y}}_{t}-\tilde{m}((1-{\upsilon}_{t}){e}^{{\upsilon}_{t}}+\frac{3}{2}{\upsilon}_{t}^{2})\frac{g\left({\mathtt{y}}_{t}\right)}{{g}^{\prime}\left({\mathtt{y}}_{t}\right)}$. | ||

Case-4 (TM4) | $R\left({\upsilon}_{t}\right)=2-{\upsilon}_{t}-\frac{1}{1+{\upsilon}_{t}}+2{\upsilon}_{t}^{2}$ | |

${\mathtt{x}}_{t+1}={\mathtt{y}}_{t}-\tilde{m}(2-{\upsilon}_{t}-\frac{1}{1+{\upsilon}_{t}}+2{\upsilon}_{t}^{2})\frac{g\left({\mathtt{y}}_{t}\right)}{{g}^{\prime}\left({\mathtt{y}}_{t}\right)}$. |

Test Functions | Root | p | ${\mathtt{x}}_{0}$ |
---|---|---|---|

${f}_{1}\left(\mathtt{x}\right)={\left(\right)}^{{\mathtt{sin}}^{2}}2$ | 1.4044916 | 2 | 2 |

${f}_{2}\left(\mathtt{x}\right)={\mathtt{x}}^{3}-5.22{\mathtt{x}}^{2}+9.0825\mathtt{x}-5.2675$ | 1.7500000 | 2 | 2.2 |

${f}_{3}\left(\mathtt{x}\right)={(\mathtt{x}-2)}^{15}{(\mathtt{x}-4)}^{5}{(\mathtt{x}-3)}^{10}{(\mathtt{x}-1)}^{20}$ | 1.0000000 | 20 | 0.7 |

${f}_{4}\left(\mathtt{x}\right)={\mathtt{x}}^{9}-29{\mathtt{x}}^{8}+349{\mathtt{x}}^{7}-2261{\mathtt{x}}^{6}+8455{\mathtt{x}}^{5}-17663{\mathtt{x}}^{4}+15927{\mathtt{x}}^{3}+6993{\mathtt{x}}^{2}-24732\mathtt{x}+12960$ | 3.0000000 | 4 | 2.5 |

${f}_{5}\left(\mathtt{x}\right)={[{(\mathtt{x}-1)}^{3}-1]}^{50}$ | 2.0000000 | 50 | 2.1 |

Methods | t | $\left|{\mathit{e}}_{\mathit{t}-2}\right|$ | $\left|{\mathit{e}}_{\mathit{t}-1}\right|$ | $\left|{\mathit{e}}_{\mathit{t}}\right|$ | $\left|\mathit{f}\left({\mathtt{x}}_{\mathit{t}+1}\right)\right|$ | COC | CPU Time |
---|---|---|---|---|---|---|---|

TM1 | 5 | $9.65\times {10}^{-12}$ | $5.42\times {10}^{-56}$ | $3.03\times {10}^{-277}$ | $1.66\times {10}^{-2765}$ | 5.00000 | 0.00186139 |

TM2 | 5 | $1.21\times {10}^{-11}$ | $1.87\times {10}^{-55}$ | $1.64\times {10}^{-274}$ | $4.44\times {10}^{-2738}$ | 5.00000 | 0.00216832 |

TM3 | 5 | $1.57\times {10}^{-11}$ | $7.36\times {10}^{-55}$ | $1.67\times {10}^{-271}$ | $6.13\times {10}^{-2708}$ | 5.00000 | 0.00231683 |

TM4 | 5 | $1.87\times {10}^{-12}$ | $6.12\times {10}^{-60}$ | $2.32\times {10}^{-297}$ | $2.00\times {10}^{-2967}$ | 5.00000 | 0.00247525 |

SM1 | 5 | $6.71\times {10}^{-11}$ | $1.90\times {10}^{-51}$ | $3.48\times {10}^{-254}$ | $3.11\times {10}^{-2534}$ | 5.00000 | 0.00279208 |

SM2 | 5 | $1.17\times {10}^{-10}$ | $3.82\times {10}^{-50}$ | $1.45\times {10}^{-247}$ | $8.09\times {10}^{-2468}$ | 5.00000 | 0.00278218 |

SM3 | 5 | $5.76\times {10}^{-11}$ | $8.90\times {10}^{-52}$ | $7.82\times {10}^{-256}$ | $1.03\times {10}^{-2550}$ | 5.00000 | 0.00263366 |

NPM1 | 5 | $2.83\times {10}^{-15}$ | $3.23\times {10}^{-88}$ | $7.21\times {10}^{-526}$ | $4.85\times {10}^{-6302}$ | 6.00000 | 0.00279208 |

NPM2 | 5 | $2.83\times {10}^{-15}$ | $3.23\times {10}^{-88}$ | $7.21\times {10}^{-526}$ | $4.85\times {10}^{-6302}$ | 6.00000 | 0.00279208 |

NPM3 | 5 | $2.83\times {10}^{-15}$ | $3.23\times {10}^{-88}$ | $7.21\times {10}^{-526}$ | $4.85\times {10}^{-6302}$ | 6.00000 | 0.00279208 |

Methods | t | $\left|{\mathit{e}}_{\mathit{t}-2}\right|$ | $\left|{\mathit{e}}_{\mathit{t}-1}\right|$ | $\left|{\mathit{e}}_{\mathit{t}}\right|$ | $\left|\mathit{f}\left({\mathtt{x}}_{\mathit{t}+1}\right)\right|$ | COC | CPU Time |
---|---|---|---|---|---|---|---|

TM1 | 6 | $4.66\times {10}^{-12}$ | $5.07\times {10}^{-52}$ | $7.73\times {10}^{-252}$ | $1.22\times {10}^{-2502}$ | 5.00000 | 0.00154455 |

TM2 | 6 | $5.77\times {10}^{-12}$ | $1.57\times {10}^{-51}$ | $2.31\times {10}^{-249}$ | $7.59\times {10}^{-2478}$ | 5.00000 | 0.00170297 |

TM3 | 6 | $7.45\times {10}^{-12}$ | $5.92\times {10}^{-51}$ | $1.87\times {10}^{-246}$ | $1.02\times {10}^{-2448}$ | 5.00000 | 0.00138614 |

TM4 | 6 | $1.29\times {10}^{-12}$ | $5.41\times {10}^{-55}$ | $7.17\times {10}^{-267}$ | $2.57\times {10}^{-2653}$ | 5.00000 | 0.00169307 |

SM1 | 6 | $3.21\times {10}^{-11}$ | $1.31\times {10}^{-47}$ | $1.46\times {10}^{-229}$ | $2.01\times {10}^{-2279}$ | 5.00000 | 0.00215842 |

SM2 | 6 | $5.73\times {10}^{-11}$ | $2.87\times {10}^{-46}$ | $8.98\times {10}^{-223}$ | $2.18\times {10}^{-2211}$ | 5.00000 | 0.00262376 |

SM3 | 6 | $2.65\times {10}^{-11}$ | $5.00\times {10}^{-48}$ | $1.21\times {10}^{-231}$ | $3.03\times {10}^{-2300}$ | 5.00000 | 0.00231683 |

NPM1 | 6 | $1.48\times {10}^{-18}$ | $1.03\times {10}^{-100}$ | $1.16\times {10}^{-593}$ | $1.72\times {10}^{-7103}$ | 6.00000 | 0.00232673 |

NPM2 | 6 | $1.48\times {10}^{-18}$ | $1.03\times {10}^{-100}$ | $1.16\times {10}^{-593}$ | $1.72\times {10}^{-7103}$ | 6.00000 | 0.00247525 |

NPM3 | 6 | $1.48\times {10}^{-18}$ | $1.03\times {10}^{-100}$ | $1.16\times {10}^{-593}$ | $1.72\times {10}^{-7103}$ | 6.00000 | 0.00200990 |

Methods | t | $\left|{\mathit{e}}_{\mathit{t}-2}\right|$ | $\left|{\mathit{e}}_{\mathit{t}-1}\right|$ | $\left|{\mathit{e}}_{\mathit{t}}\right|$ | $\left|\mathit{f}\left({\mathtt{x}}_{\mathit{t}+1}\right)\right|$ | COC | CPU Time |
---|---|---|---|---|---|---|---|

TM1 | 5 | $1.52\times {10}^{-14}$ | $2.00\times {10}^{-69}$ | $7.74\times {10}^{-344}$ | $1.03\times {10}^{-34298}$ | 5.00000 | 0.00139604 |

TM2 | 5 | $2.12\times {10}^{-14}$ | $1.15\times {10}^{-68}$ | $5.39\times {10}^{-340}$ | $1.13\times {10}^{-33913}$ | 5.00000 | 0.00139604 |

TM3 | 5 | $3.08\times {10}^{-14}$ | $8.08\times {10}^{-68}$ | $9.98\times {10}^{-336}$ | $3.42\times {10}^{-33486}$ | 5.00000 | 0.00108911 |

TM4 | 5 | $1.46\times {10}^{-15}$ | $7.05\times {10}^{-75}$ | $1.84\times {10}^{-371}$ | $2.49\times {10}^{-37068}$ | 5.00000 | 0.00154455 |

SM1 | 5 | $2.29\times {10}^{-13}$ | $3.26\times {10}^{-63}$ | $1.92\times {10}^{-312}$ | $9.26\times {10}^{-31153}$ | 5.00000 | 0.00262376 |

SM2 | 5 | $4.77\times {10}^{-13}$ | $1.62\times {10}^{-61}$ | $7.34\times {10}^{-304}$ | $2.06\times {10}^{-30292}$ | 5.00000 | 0.00247525 |

SM3 | 5 | $1.84\times {10}^{-13}$ | $1.08\times {10}^{-63}$ | $7.68\times {10}^{-315}$ | $1.82\times {10}^{-31392}$ | 5.00000 | 0.00171683 |

NPM1 | Div | − | − | − | − | − | − |

NPM2 | Div | − | − | − | − | − | − |

NPM3 | Div | − | − | − | − | − | − |

Methods | t | $\left|{\mathit{e}}_{\mathit{t}-2}\right|$ | $\left|{\mathit{e}}_{\mathit{t}-1}\right|$ | $\left|{\mathit{e}}_{\mathit{t}}\right|$ | $\left|\mathit{f}\left({\mathtt{x}}_{\mathit{t}+1}\right)\right|$ | COC | CPU Time |
---|---|---|---|---|---|---|---|

TM1 | 5 | $9.88\times {10}^{-24}$ | $2.43\times {10}^{-117}$ | $2.19\times {10}^{-585}$ | $2.27\times {10}^{-11698}$ | 5.00000 | 0.00216832 |

TM2 | 5 | $1.02\times {10}^{-23}$ | $2.91\times {10}^{-117}$ | $5.48\times {10}^{-585}$ | $2.26\times {10}^{-11690}$ | 5.00000 | 0.00200990 |

TM3 | 5 | $1.05\times {10}^{-23}$ | $3.48\times {10}^{-117}$ | $1.37\times {10}^{-584}$ | $2.22\times {10}^{-11682}$ | 5.00000 | 0.00247525 |

TM4 | 5 | $8.10\times {10}^{-24}$ | $7.87\times {10}^{-118}$ | $6.84\times {10}^{-588}$ | $1.05\times {10}^{-11748}$ | 5.00000 | 0.00263366 |

SM1 | 5 | $1.40\times {10}^{-23}$ | $1.75\times {10}^{-116}$ | $5.27\times {10}^{-581}$ | $2.31\times {10}^{-11610}$ | 5.00000 | 0.00278218 |

SM2 | 5 | $1.64\times {10}^{-23}$ | $4.13\times {10}^{-116}$ | $4.26\times {10}^{-579}$ | $4.79\times {10}^{-11572}$ | 5.00000 | 0.00294059 |

SM3 | 5 | $1.40\times {10}^{-23}$ | $1.71\times {10}^{-116}$ | $4.76\times {10}^{-581}$ | $3.02\times {10}^{-11611}$ | 5.00000 | 0.00340595 |

NPM1 | 5 | $2.50\times {10}^{-18}$ | $3.16\times {10}^{-90}$ | $1.02\times {10}^{-449}$ | $1.21\times {10}^{-8984}$ | 5.00000 | 0.00309901 |

NPM2 | 5 | $2.70\times {10}^{-18}$ | $4.94\times {10}^{-90}$ | $1.00\times {10}^{-448}$ | $1.18\times {10}^{-8964}$ | 5.00000 | 0.00324752 |

NPM3 | 5 | $2.90\times {10}^{-18}$ | $7.52\times {10}^{-90}$ | $8.78\times {10}^{-448}$ | $1.04\times {10}^{-8945}$ | 5.00000 | 0.00308911 |

Methods | t | $\left|{\mathit{e}}_{\mathit{t}-2}\right|$ | $\left|{\mathit{e}}_{\mathit{t}-1}\right|$ | $\left|{\mathit{e}}_{\mathit{t}}\right|$ | $\left|\mathit{f}\left({\mathtt{x}}_{\mathit{t}+1}\right)\right|$ | COC | CPU Time |
---|---|---|---|---|---|---|---|

TM1 | 5 | $1.11\times {10}^{-25}$ | $2.25\times {10}^{-125}$ | $7.74\times {10}^{-624}$ | $1.96\times {10}^{-155748}$ | 5.00000 | 0.00200990 |

TM2 | 5 | $2.01\times {10}^{-25}$ | $4.90\times {10}^{-124}$ | $4.26\times {10}^{-617}$ | $8.03\times {10}^{-154061}$ | 5.00000 | 0.00170301 |

TM3 | 5 | $3.61\times {10}^{-25}$ | $1.02\times {10}^{-122}$ | $1.82\times {10}^{-610}$ | $1.64\times {10}^{-152400}$ | 5.00000 | 0.00200990 |

TM4 | 5 | $4.50\times {10}^{-28}$ | $6.15\times {10}^{-138}$ | $2.92\times {10}^{-687}$ | $2.36\times {10}^{-171634}$ | 5.00000 | 0.00170297 |

SM1 | 5 | $1.08\times {10}^{-23}$ | $4.79\times {10}^{-115}$ | $8.42\times {10}^{-572}$ | $2.36\times {10}^{-142719}$ | 5.00000 | 0.00247525 |

SM2 | 5 | $3.65\times {10}^{-23}$ | $2.79\times {10}^{-112}$ | $7.39\times {10}^{-558}$ | $6.33\times {10}^{-139228}$ | 5.00000 | 0.00231683 |

SM3 | 5 | $8.89\times {10}^{-24}$ | $1.85\times {10}^{-115}$ | $7.26\times {10}^{-574}$ | $1.87\times {10}^{-143235}$ | 5.00000 | 0.00757426 |

NPM1 | Div | − | − | − | − | − | − |

NPM2 | Div | − | − | − | − | − | − |

NPM3 | Div | − | − | − | − | − | − |

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## Share and Cite

**MDPI and ACS Style**

Singh, T.; Arora, H.; Jäntschi, L.
A Family of Higher Order Scheme for Multiple Roots. *Symmetry* **2023**, *15*, 228.
https://doi.org/10.3390/sym15010228

**AMA Style**

Singh T, Arora H, Jäntschi L.
A Family of Higher Order Scheme for Multiple Roots. *Symmetry*. 2023; 15(1):228.
https://doi.org/10.3390/sym15010228

**Chicago/Turabian Style**

Singh, Tajinder, Himani Arora, and Lorentz Jäntschi.
2023. "A Family of Higher Order Scheme for Multiple Roots" *Symmetry* 15, no. 1: 228.
https://doi.org/10.3390/sym15010228