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

^{1}

^{2}

^{3}

^{*}

## 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) :$${u}_{k+1}={z}_{k}-\frac{\mu \phantom{\rule{0.166667em}{0ex}}h(1+3\phantom{\rule{0.166667em}{0ex}}h)}{2}\left(\right)open="("\; close=")">1+\frac{1}{{y}_{k}}$$
- Method 2 (M2) :$${u}_{k+1}={z}_{k}-\frac{\mu \phantom{\rule{0.166667em}{0ex}}h}{2-6h}\left(\right)open="("\; close=")">1+\frac{1}{{y}_{k}}$$
- Method 3 (M3) :$${u}_{k+1}={z}_{k}-\frac{\mu \phantom{\rule{0.166667em}{0ex}}h(\mu -2h)}{2(\mu -(2+3\mu )h+2\mu {h}^{2})}\left(\right)open="("\; close=")">1+\frac{1}{{y}_{k}}$$
- Method 4 (M4) :$${u}_{k+1}={z}_{k}-\frac{\mu \phantom{\rule{0.166667em}{0ex}}h(3-h)}{6-20h}\left(\right)open="("\; close=")">1+\frac{1}{{y}_{k}}$$

## 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 ${\psi}_{2}\left(z\right)={\left(\right)}^{{z}^{3}}3$ 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 ${\psi}_{3}\left(z\right)={\left(\right)}^{{z}^{3}}4$ 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

- Schröder, E. Über unendlich viele Algorithmen zur Auflösung der Gleichungen. Math. Ann.
**1870**, 2, 317–365. [Google Scholar] [CrossRef] [Green Version] - Hansen, E.; Patrick, M. A family of root finding methods. Numer. Math.
**1977**, 27, 257–269. [Google Scholar] [CrossRef] - Victory, H.D.; Neta, B. A higher order method for multiple zeros of nonlinear functions. Int. J. Comput. Math.
**1983**, 12, 329–335. [Google Scholar] [CrossRef] - Dong, C. A family of multipoint iterative functions for finding multiple roots of equations. Int. J. Comput. Math.
**1987**, 21, 363–367. [Google Scholar] [CrossRef] - Osada, N. An optimal multiple root-finding method of order three. J. Comput. Appl. Math.
**1994**, 51, 131–133. [Google Scholar] [CrossRef] [Green Version] - Neta, B. New third order nonlinear solvers for multiple roots. App. Math. Comput.
**2008**, 202, 162–170. [Google Scholar] [CrossRef] [Green Version] - Li, S.; Liao, X.; Cheng, L. A new fourth-order iterative method for finding multiple roots of nonlinear equations. Appl. Math. Comput.
**2009**, 215, 1288–1292. [Google Scholar] - Li, S.G.; Cheng, L.Z.; Neta, B. Some fourth-order nonlinear solvers with closed formulae for multiple roots. Comput Math. Appl.
**2010**, 59, 126–135. [Google Scholar] [CrossRef] [Green Version] - Sharma, J.R.; Sharma, R. Modified Jarratt method for computing multiple roots. Appl. Math. Comput.
**2010**, 217, 878–881. [Google Scholar] [CrossRef] - Zhou, X.; Chen, X.; Song, Y. Constructing higher-order methods for obtaining the multiple roots of nonlinear equations. J. Comput. Appl. Math.
**2011**, 235, 4199–4206. [Google Scholar] [CrossRef] [Green Version] - Sharifi, M.; Babajee, D.K.R.; Soleymani, F. Finding the solution of nonlinear equations by a class of optimal methods. Comput. Math. Appl.
**2012**, 63, 764–774. [Google Scholar] [CrossRef] [Green Version] - Soleymani, F.; Babajee, D.K.R.; Lotfi, T. On a numerical technique for finding multiple zeros and its dynamics. J. Egypt. Math. Soc.
**2013**, 21, 346–353. [Google Scholar] [CrossRef] - Geum, Y.H.; Kim, Y.I.; Neta, B. A class of two-point sixth-order multiple-zero finders of modified double-Newton type and their dynamics. Appl. Math. Comput.
**2015**, 270, 387–400. [Google Scholar] [CrossRef] [Green Version] - Kansal, M.; Kanwar, V.; Bhatia, S. On some optimal multiple root-finding methods and their dynamics. Appl. Appl. Math.
**2015**, 10, 349–367. [Google Scholar] - Traub, J.F. Iterative Methods for the Solution of Equations; Chelsea Publishing Company: New York, NY, USA, 1982. [Google Scholar]
- Sharma, J.R.; Kumar, S.; Jäntschi, L. On a class of optimal fourth order multiple root solvers without using derivatives. Symmetry
**2019**, 11, 766. [Google Scholar] [CrossRef] [Green Version] - Kung, H.T.; Traub, J.F. Optimal order of one-point and multipoint iteration. J. Assoc. Comput. Mach.
**1974**, 21, 643–651. [Google Scholar] [CrossRef] - Geum, Y.H.; Kim, Y.I.; Neta, B. Constructing a family of optimal eighth-order modified Newton-type multiple-zero finders along with the dynamics behind their purely imaginary extraneous fixed points. J. Comp. Appl. Math.
**2018**, 333, 131–156. [Google Scholar] [CrossRef] - Benbernou, S.; Gala, S.; Ragusa, M.A. On the regularity criteria for the 3D magnetohydrodynamic equations via two components in terms of BMO space. Math. Meth. Appl. Sci.
**2016**, 37, 2320–2325. [Google Scholar] [CrossRef] - Wolfram, S. The Mathematica Book, 5th ed.; Wolfram Media: Champaign, IL, USA, 2003. [Google Scholar]
- Vrscay, E.R.; Gilbert, W.J. Extraneous fixed points, basin boundaries and chaotic dynamics for Schröder and König rational iteration functions. Numer. Math.
**1988**, 52, 1–16. [Google Scholar] [CrossRef] - Varona, J.L. Graphic and numerical comparison between iterative methods. Math. Intell.
**2002**, 24, 37–46. [Google Scholar] [CrossRef] - Argyros, I.K.; Magreñán, Á.A. Iterative Methods and Their Dynamics with Applications; CRC Press: New York, NY, USA, 2017. [Google Scholar]
- Weerakoon, S.; Fernando, T.G.I. A variant of Newton’s method with accelerated third-order convergence. Appl. Math. Lett.
**2000**, 13, 87–93. [Google Scholar] [CrossRef]

**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 | |||

${\psi}_{3}\left(u\right)={\left(\right)}^{{e}^{-u}}4$ | 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 | |||

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

$\phantom{\rule{4pt}{0ex}}\phantom{\rule{4pt}{0ex}}\phantom{\rule{4pt}{0ex}}\phantom{\rule{4pt}{0ex}}\phantom{\rule{4pt}{0ex}}\phantom{\rule{4pt}{0ex}}\phantom{\rule{4pt}{0ex}}\phantom{\rule{4pt}{0ex}}\phantom{\rule{4pt}{0ex}}\phantom{\rule{4pt}{0ex}}-{tan}^{-1}\left(\right)open="("\; close=")">\frac{1}{2}\sqrt{\frac{5}{6}}$ | 1.8411294068… | 7 | 1.6 |

© 2020 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (http://creativecommons.org/licenses/by/4.0/).

## Share and Cite

**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