# An Efficient Class of Traub–Steffensen-Type Methods for Computing Multiple Zeros

^{1}

^{2}

^{*}

## Abstract

**:**

## 1. Introduction

## 2. The Method

**Theorem**

**1.**

**Proof.**

#### Some Concrete Forms of H(u)

- (1)
- $H\left(u\right)=mu$,
- (2)
- $H\left(u\right)=\frac{mu}{1+u},$
- (3)
- $H\left(u\right)=\frac{mu}{1-u}$,
- (4)
- $H\left(u\right)=\frac{mu}{1+mu},$
- (5)
- $H\left(u\right)=mlog(u+1)$,
- (6)
- $H\left(u\right)=m({e}^{u}-1).$

- Method 1 (M1):$${x}_{n+1}={y}_{n}-mu\frac{f\left({x}_{n}\right)}{f[{x}_{n},{w}_{n}]}.$$
- Method 2 (M2):$${x}_{n+1}=\phantom{\rule{0.166667em}{0ex}}{y}_{n}-\frac{mu}{1+u}\frac{f\left({x}_{n}\right)}{f[{x}_{n},{w}_{n}]}.$$
- Method 3 (M3):$${x}_{n+1}=\phantom{\rule{0.166667em}{0ex}}{y}_{n}-\frac{mu}{1-u}\frac{f\left({x}_{n}\right)}{f[{x}_{n},{w}_{n}]}.$$
- Method 4 (M4):$${x}_{n+1}=\phantom{\rule{0.166667em}{0ex}}{y}_{n}-\frac{mu}{1+mu}\frac{f\left({x}_{n}\right)}{f[{x}_{n},{w}_{n}]}.$$
- Method 5 (M5):$${x}_{n+1}=\phantom{\rule{0.166667em}{0ex}}{y}_{n}-mlog(u+1)\frac{f\left({x}_{n}\right)}{f[{x}_{n},{w}_{n}]}.$$
- Method 6 (M6):$${x}_{n+1}=\phantom{\rule{0.166667em}{0ex}}{y}_{n}-m({e}^{u}-1)\frac{f\left({x}_{n}\right)}{f[{x}_{n},{w}_{n}]}.$$

**Remark**

**1.**

## 3. Numerical Tests

#### 3.1. Basins of Attraction

**Problem**

**1.**

**Problem**

**2.**

**Problem**

**3.**

#### 3.2. Applications

- Dong’s method (DM):$$\left\{\begin{array}{c}{y}_{n}={x}_{n}-\sqrt{m}\frac{f\left({x}_{n}\right)}{{f}^{\prime}\left({x}_{n}\right)},\hfill \\ {x}_{n+1}={y}_{n}-m{\left(1-\frac{1}{\sqrt{m}}\right)}^{1-m}\frac{f\left({y}_{n}\right)}{{f}^{\prime}\left({x}_{n}\right)}.\hfill \end{array}\right.$$
- Halley’s method (HM):$${x}_{n+1}={x}_{n}-\frac{f\left({x}_{n}\right)}{\frac{m+1}{2m}{f}^{\prime}\left({x}_{n}\right)-\frac{f\left({x}_{n}\right){f}^{\u2033}\left({x}_{n}\right)}{2{f}^{\prime}\left({x}_{n}\right)}}.$$
- Chebyshev’s method (CM):$${x}_{n+1}={x}_{n}-\frac{m(3-m)}{2}\frac{f\left({x}_{n}\right)}{{f}^{\prime}\left({x}_{n}\right)}-\frac{{m}^{2}}{2}\frac{{f}^{2}\left({x}_{n}\right){f}^{\u2033}\left({x}_{n}\right)}{{f}^{\prime 3}\left({x}_{n}\right)}.$$
- Osada’s method (OM):$${x}_{n+1}={x}_{n}-\frac{1}{2}m(m+1)\frac{f\left({x}_{n}\right)}{{f}^{\prime}({x}_{n}}+\frac{1}{2}{(m-1)}^{2}\frac{{f}^{\prime}\left({x}_{n}\right)}{{f}^{\u2033}\left({x}_{n}\right)}.$$
- Victory-Neta method (VNM):$$\left\{\begin{array}{c}{y}_{n}=\phantom{\rule{0.166667em}{0ex}}{x}_{n}-\frac{f\left({x}_{n}\right)}{{f}^{\prime}\left({x}_{n}\right)},\hfill \\ {x}_{n+1}=\phantom{\rule{0.166667em}{0ex}}{y}_{n}-\frac{f\left({y}_{n}\right)}{{f}^{\prime}\left({x}_{n}\right)}\frac{f\left({x}_{n}\right)+Af\left({y}_{n}\right)}{f\left({x}_{n}\right)+Bf\left({y}_{n}\right)},\hfill \end{array}\right.$$$$A={\mu}^{2m}-{\mu}^{m+1},\phantom{\rule{4pt}{0ex}}\phantom{\rule{4pt}{0ex}}\phantom{\rule{4pt}{0ex}}\phantom{\rule{4pt}{0ex}}\phantom{\rule{4pt}{0ex}}B=-\frac{{\mu}^{m}(m-2)(m-1)+1}{{(m-1)}^{2}}\phantom{\rule{4pt}{0ex}}\phantom{\rule{4pt}{0ex}}\mathrm{and}\phantom{\rule{4pt}{0ex}}\phantom{\rule{4pt}{0ex}}\phantom{\rule{4pt}{0ex}}\mu =\frac{m}{m-1}.$$

**Example**

**1**(

**Eigenvalue**

**problem**)

**.**

**Example**

**2**(

**Manning**

**equation**)

**.**

**Example**

**3.**

**Example**

**4.**

## 4. 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] - Behl, R.; Cordero, A.; Motsa, S.S.; Torregrosa, J.R. On developing fourth-order optimal families of methods for multiple roots and their dynamics. Appl. Math. Comput.
**2015**, 265, 520–532. [Google Scholar] [CrossRef] [Green Version] - Behl, R.; Cordero, A.; Motsa, S.S.; Torregrosa, J.R.; Kanwar, V. An optimal fourth-order family of methods for multiple roots and its dynamics. Numer. Algor.
**2016**, 71, 775–796. [Google Scholar] [CrossRef] - Behl, R.; Zafar, F.; Alshormani, A.S.; Junjua, M.U.D.; Yasmin, N. An optimal eighth-order scheme for multiple zeros of unvariate functions. Int. J. Comput. Meth.
**2018**, 15. [Google Scholar] [CrossRef] - Dong, C. A basic theorem of constructing an iterative formula of the higher order for computing multiple roots of an equation. Math. Numer. Sin.
**1982**, 11, 445–450. [Google Scholar] - 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] - Hansen, E.; Patrick, M. A family of root finding methods. Numer. Math.
**1977**, 27, 257–269. [Google Scholar] [CrossRef] - 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] - 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] - 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] - Neta, B. New third order nonlinear solvers for multiple roots. Appl. Math. Comput.
**2008**, 202, 162–170. [Google Scholar] [CrossRef] [Green Version] - Osada, N. An optimal multiple root-finding method of order three. J. Comput. Appl. Math.
**1994**, 51, 131–133. [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] - 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] - Zhou, X.; Chen, X.; Song, Y. Constructing higher-order methods for obtaining the muliplte roots of nonlinear equations. J. Comput. Appl. Math.
**2011**, 235, 4199–4206. [Google Scholar] [CrossRef] - Traub, J.F. Iterative Methods for the Solution of Equations; Chelsea Publishing Company: New York, NY, USA, 1982. [Google Scholar]
- Ostrowski, A.M. Solution of Equations and Systems of Equations; Academic Press: New York, NY, USA, 1966. [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] - Scott, M.; Neta, B.; Chun, C. Basin attractors for various methods. Appl. Math. Comput.
**2011**, 218, 2584–2599. [Google Scholar] [CrossRef] - Lotfi, T.; Sharifi, S.; Salimi, M.; Siegmund, S. A new class of three-point methods with optimal convergence order eight and its dynamics. Numer. Algor.
**2015**, 68, 261–288. [Google Scholar] [CrossRef] - 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] - Hoffman, J.D. Numerical Methods for Engineers and Scientists; McGraw-Hill Book Company: New York, NY, USA, 1992. [Google Scholar]

Methods | $|{\mathit{x}}_{3}-{\mathit{x}}_{2}|$ | $|{\mathit{x}}_{4}-{\mathit{x}}_{3}|$ | $|{\mathit{x}}_{5}-{\mathit{x}}_{4}|$ | n | COC | CPU-Time |
---|---|---|---|---|---|---|

$\mathrm{DM}$ | $9.90\phantom{\rule{3.33333pt}{0ex}}\times \phantom{\rule{3.33333pt}{0ex}}{10}^{-11}$ | $1.52\phantom{\rule{3.33333pt}{0ex}}\times \phantom{\rule{3.33333pt}{0ex}}{10}^{-31}$ | $5.49\phantom{\rule{3.33333pt}{0ex}}\times \phantom{\rule{3.33333pt}{0ex}}{10}^{-94}$ | 5 | 3.0000 | 0.01942 |

$\mathrm{HM}$ | $5.84\phantom{\rule{3.33333pt}{0ex}}\times \phantom{\rule{3.33333pt}{0ex}}{10}^{-10}$ | $4.61\phantom{\rule{3.33333pt}{0ex}}\times \phantom{\rule{3.33333pt}{0ex}}{10}^{-29}$ | $2.24\phantom{\rule{3.33333pt}{0ex}}\times \phantom{\rule{3.33333pt}{0ex}}{10}^{-86}$ | 5 | 3.0000 | 0.01950 |

$\mathrm{CM}$ | $9.54\phantom{\rule{3.33333pt}{0ex}}\times \phantom{\rule{3.33333pt}{0ex}}{10}^{-10}$ | $2.47\phantom{\rule{3.33333pt}{0ex}}\times \phantom{\rule{3.33333pt}{0ex}}{10}^{-28}$ | $4.30\phantom{\rule{3.33333pt}{0ex}}\times \phantom{\rule{3.33333pt}{0ex}}{10}^{-84}$ | 5 | 3.0000 | 0.01885 |

$\mathrm{OM}$ | $1.26\phantom{\rule{3.33333pt}{0ex}}\times \phantom{\rule{3.33333pt}{0ex}}{10}^{-9}$ | $6.52\phantom{\rule{3.33333pt}{0ex}}\times \phantom{\rule{3.33333pt}{0ex}}{10}^{-28}$ | $8.94\phantom{\rule{3.33333pt}{0ex}}\times \phantom{\rule{3.33333pt}{0ex}}{10}^{-83}$ | 5 | 3.0000 | 0.01955 |

$\mathrm{VNM}$ | $2.50\phantom{\rule{3.33333pt}{0ex}}\times \phantom{\rule{3.33333pt}{0ex}}{10}^{-10}$ | $2.92\phantom{\rule{3.33333pt}{0ex}}\times \phantom{\rule{3.33333pt}{0ex}}{10}^{-30}$ | $4.68\phantom{\rule{3.33333pt}{0ex}}\times \phantom{\rule{3.33333pt}{0ex}}{10}^{-90}$ | 5 | 3.0000 | 0.02250 |

$\mathrm{M}1$ | $1.51\phantom{\rule{3.33333pt}{0ex}}\times \phantom{\rule{3.33333pt}{0ex}}{10}^{-12}$ | $3.91\phantom{\rule{3.33333pt}{0ex}}\times \phantom{\rule{3.33333pt}{0ex}}{10}^{-37}$ | 0 | 4 | 3.0000 | 0.00775 |

$\mathrm{M}2$ | $5.15\phantom{\rule{3.33333pt}{0ex}}\times \phantom{\rule{3.33333pt}{0ex}}{10}^{-12}$ | $2.30\phantom{\rule{3.33333pt}{0ex}}\times \phantom{\rule{3.33333pt}{0ex}}{10}^{-35}$ | 0 | 4 | 3.0000 | 0.01165 |

$\mathrm{M}3$ | $2.32\phantom{\rule{3.33333pt}{0ex}}\times \phantom{\rule{3.33333pt}{0ex}}{10}^{-13}$ | $7.01\phantom{\rule{3.33333pt}{0ex}}\times \phantom{\rule{3.33333pt}{0ex}}{10}^{-40}$ | 0 | 4 | 3.0000 | 0.01175 |

$\mathrm{M}4$ | $4.73\phantom{\rule{3.33333pt}{0ex}}\times \phantom{\rule{3.33333pt}{0ex}}{10}^{-11}$ | $3.59\phantom{\rule{3.33333pt}{0ex}}\times \phantom{\rule{3.33333pt}{0ex}}{10}^{-32}$ | $1.57\phantom{\rule{3.33333pt}{0ex}}\times \phantom{\rule{3.33333pt}{0ex}}{10}^{-95}$ | 5 | 3.0000 | 0.01285 |

$\mathrm{M}5$ | $2.94\phantom{\rule{3.33333pt}{0ex}}\times \phantom{\rule{3.33333pt}{0ex}}{10}^{-12}$ | $3.57\phantom{\rule{3.33333pt}{0ex}}\times \phantom{\rule{3.33333pt}{0ex}}{10}^{-36}$ | 0 | 4 | 3.0000 | 0.01775 |

$\mathrm{M}6$ | $6.71\phantom{\rule{3.33333pt}{0ex}}\times \phantom{\rule{3.33333pt}{0ex}}{10}^{-13}$ | $2.55\phantom{\rule{3.33333pt}{0ex}}\times \phantom{\rule{3.33333pt}{0ex}}{10}^{-38}$ | 0 | 4 | 3.0000 | 0.01115 |

Methods | $|{\mathit{x}}_{3}-{\mathit{x}}_{2}|$ | $|{\mathit{x}}_{4}-{\mathit{x}}_{3}|$ | $|{\mathit{x}}_{5}-{\mathit{x}}_{4}|$ | n | COC | CPU-Time |
---|---|---|---|---|---|---|

$\mathrm{DM}$ | $6.14\phantom{\rule{3.33333pt}{0ex}}\times \phantom{\rule{3.33333pt}{0ex}}{10}^{-9}$ | $1.48\phantom{\rule{3.33333pt}{0ex}}\times \phantom{\rule{3.33333pt}{0ex}}{10}^{-26}$ | $2.06\phantom{\rule{3.33333pt}{0ex}}\times \phantom{\rule{3.33333pt}{0ex}}{10}^{-79}$ | 5 | 3.0000 | 0.17575 |

$\mathrm{HM}$ | $5.10\phantom{\rule{3.33333pt}{0ex}}\times \phantom{\rule{3.33333pt}{0ex}}{10}^{-8}$ | $1.20\phantom{\rule{3.33333pt}{0ex}}\times \phantom{\rule{3.33333pt}{0ex}}{10}^{-23}$ | $1.54\phantom{\rule{3.33333pt}{0ex}}\times \phantom{\rule{3.33333pt}{0ex}}{10}^{-70}$ | 5 | 3.0000 | 0.20354 |

$\mathrm{CM}$ | $5.98\phantom{\rule{3.33333pt}{0ex}}\times \phantom{\rule{3.33333pt}{0ex}}{10}^{-8}$ | $2.17\phantom{\rule{3.33333pt}{0ex}}\times \phantom{\rule{3.33333pt}{0ex}}{10}^{-23}$ | $1.04\phantom{\rule{3.33333pt}{0ex}}\times \phantom{\rule{3.33333pt}{0ex}}{10}^{-69}$ | 5 | 3.0000 | 0.23560 |

$\mathrm{OM}$ | $6.30\phantom{\rule{3.33333pt}{0ex}}\times \phantom{\rule{3.33333pt}{0ex}}{10}^{-8}$ | $2.63\phantom{\rule{3.33333pt}{0ex}}\times \phantom{\rule{3.33333pt}{0ex}}{10}^{-23}$ | $1.91\phantom{\rule{3.33333pt}{0ex}}\times \phantom{\rule{3.33333pt}{0ex}}{10}^{-69}$ | 5 | 3.0000 | 0.23225 |

$\mathrm{VNM}$ | $2.45\phantom{\rule{3.33333pt}{0ex}}\times \phantom{\rule{3.33333pt}{0ex}}{10}^{-8}$ | $1.17\phantom{\rule{3.33333pt}{0ex}}\times \phantom{\rule{3.33333pt}{0ex}}{10}^{-24}$ | $1.28\phantom{\rule{3.33333pt}{0ex}}\times \phantom{\rule{3.33333pt}{0ex}}{10}^{-73}$ | 5 | 3.0000 | 0.24875 |

$\mathrm{M}1$ | $1.28\phantom{\rule{3.33333pt}{0ex}}\times \phantom{\rule{3.33333pt}{0ex}}{10}^{-15}$ | $4.70\phantom{\rule{3.33333pt}{0ex}}\times \phantom{\rule{3.33333pt}{0ex}}{10}^{-47}$ | 0 | 4 | 3.0000 | 0.11121 |

$\mathrm{M}2$ | $2.95\phantom{\rule{3.33333pt}{0ex}}\times \phantom{\rule{3.33333pt}{0ex}}{10}^{-15}$ | $8.62\phantom{\rule{3.33333pt}{0ex}}\times \phantom{\rule{3.33333pt}{0ex}}{10}^{-46}$ | 0 | 4 | 3.0000 | 0.12100 |

$\mathrm{M}3$ | $3.86\phantom{\rule{3.33333pt}{0ex}}\times \phantom{\rule{3.33333pt}{0ex}}{10}^{-16}$ | $6.45\phantom{\rule{3.33333pt}{0ex}}\times \phantom{\rule{3.33333pt}{0ex}}{10}^{-49}$ | 0 | 4 | 3.0000 | 0.10925 |

$\mathrm{M}4$ | $4.96\phantom{\rule{3.33333pt}{0ex}}\times \phantom{\rule{3.33333pt}{0ex}}{10}^{-14}$ | $1.23\phantom{\rule{3.33333pt}{0ex}}\times \phantom{\rule{3.33333pt}{0ex}}{10}^{-41}$ | 0 | 4 | 3.0000 | 0.11725 |

$\mathrm{M}5$ | $2.00\phantom{\rule{3.33333pt}{0ex}}\times \phantom{\rule{3.33333pt}{0ex}}{10}^{-15}$ | $2.24\phantom{\rule{3.33333pt}{0ex}}\times \phantom{\rule{3.33333pt}{0ex}}{10}^{-46}$ | 0 | 4 | 3.0000 | 0.12125 |

$\mathrm{M}6$ | $7.54\phantom{\rule{3.33333pt}{0ex}}\times \phantom{\rule{3.33333pt}{0ex}}{10}^{-16}$ | $7.21\phantom{\rule{3.33333pt}{0ex}}\times \phantom{\rule{3.33333pt}{0ex}}{10}^{-48}$ | 0 | 4 | 3.0000 | 0.12525 |

Methods | $|{\mathit{x}}_{3}-{\mathit{x}}_{2}|$ | $|{\mathit{x}}_{4}-{\mathit{x}}_{3}|$ | $|{\mathit{x}}_{5}-{\mathit{x}}_{4}|$ | n | COC | CPU-Time |
---|---|---|---|---|---|---|

$\mathrm{DM}$ | $1.02\phantom{\rule{3.33333pt}{0ex}}\times \phantom{\rule{3.33333pt}{0ex}}{10}^{-9}$ | $3.43\phantom{\rule{3.33333pt}{0ex}}\times \phantom{\rule{3.33333pt}{0ex}}{10}^{-29}$ | $1.31\phantom{\rule{3.33333pt}{0ex}}\times \phantom{\rule{3.33333pt}{0ex}}{10}^{-87}$ | 5 | 3.0000 | 0.12925 |

$\mathrm{HM}$ | $2.58\phantom{\rule{3.33333pt}{0ex}}\times \phantom{\rule{3.33333pt}{0ex}}{10}^{-8}$ | $1.09\phantom{\rule{3.33333pt}{0ex}}\times \phantom{\rule{3.33333pt}{0ex}}{10}^{-24}$ | $8.36\phantom{\rule{3.33333pt}{0ex}}\times \phantom{\rule{3.33333pt}{0ex}}{10}^{-74}$ | 5 | 3.0000 | 0.14450 |

$\mathrm{CM}$ | $2.85\phantom{\rule{3.33333pt}{0ex}}\times \phantom{\rule{3.33333pt}{0ex}}{10}^{-8}$ | $1.65\phantom{\rule{3.33333pt}{0ex}}\times \phantom{\rule{3.33333pt}{0ex}}{10}^{-24}$ | $3.16\phantom{\rule{3.33333pt}{0ex}}\times \phantom{\rule{3.33333pt}{0ex}}{10}^{-73}$ | 5 | 3.0000 | 0.13275 |

$\mathrm{OM}$ | $3.13\phantom{\rule{3.33333pt}{0ex}}\times \phantom{\rule{3.33333pt}{0ex}}{10}^{-8}$ | $2.39\phantom{\rule{3.33333pt}{0ex}}\times \phantom{\rule{3.33333pt}{0ex}}{10}^{-24}$ | $1.06\phantom{\rule{3.33333pt}{0ex}}\times \phantom{\rule{3.33333pt}{0ex}}{10}^{-72}$ | 5 | 3.0000 | 0.12500 |

$\mathrm{VNM}$ | $5.37\phantom{\rule{3.33333pt}{0ex}}\times \phantom{\rule{3.33333pt}{0ex}}{10}^{-8}$ | $7.00\phantom{\rule{3.33333pt}{0ex}}\times \phantom{\rule{3.33333pt}{0ex}}{10}^{-27}$ | $1.56\phantom{\rule{3.33333pt}{0ex}}\times \phantom{\rule{3.33333pt}{0ex}}{10}^{-80}$ | 5 | 3.0000 | 0.21355 |

$\mathrm{M}1$ | $1.88\phantom{\rule{3.33333pt}{0ex}}\times \phantom{\rule{3.33333pt}{0ex}}{10}^{-13}$ | $9.27\phantom{\rule{3.33333pt}{0ex}}\times \phantom{\rule{3.33333pt}{0ex}}{10}^{-41}$ | 0 | 4 | 3.0000 | 0.05475 |

$\mathrm{M}2$ | $6.24\phantom{\rule{3.33333pt}{0ex}}\times \phantom{\rule{3.33333pt}{0ex}}{10}^{-13}$ | $5.05\phantom{\rule{3.33333pt}{0ex}}\times \phantom{\rule{3.33333pt}{0ex}}{10}^{-39}$ | 0 | 4 | 3.0000 | 0.05852 |

$\mathrm{M}3$ | $3.10\phantom{\rule{3.33333pt}{0ex}}\times \phantom{\rule{3.33333pt}{0ex}}{10}^{-14}$ | $2.06\phantom{\rule{3.33333pt}{0ex}}\times \phantom{\rule{3.33333pt}{0ex}}{10}^{-43}$ | 0 | 4 | 3.0000 | 0.05075 |

$\mathrm{M}4$ | $3.15\phantom{\rule{3.33333pt}{0ex}}\times \phantom{\rule{3.33333pt}{0ex}}{10}^{-12}$ | $1.09\phantom{\rule{3.33333pt}{0ex}}\times \phantom{\rule{3.33333pt}{0ex}}{10}^{-36}$ | 0 | 4 | 3.0000 | 0.05475 |

$\mathrm{M}5$ | $3.60\phantom{\rule{3.33333pt}{0ex}}\times \phantom{\rule{3.33333pt}{0ex}}{10}^{-13}$ | $8.07\phantom{\rule{3.33333pt}{0ex}}\times \phantom{\rule{3.33333pt}{0ex}}{10}^{-40}$ | 0 | 4 | 3.0000 | 0.05475 |

$\mathrm{M}6$ | $8.56\phantom{\rule{3.33333pt}{0ex}}\times \phantom{\rule{3.33333pt}{0ex}}{10}^{-14}$ | $6.54\phantom{\rule{3.33333pt}{0ex}}\times \phantom{\rule{3.33333pt}{0ex}}{10}^{-42}$ | 0 | 4 | 3.0000 | 0.06253 |

Methods | $|{\mathit{x}}_{3}-{\mathit{x}}_{2}|$ | $|{\mathit{x}}_{4}-{\mathit{x}}_{3}|$ | $|{\mathit{x}}_{5}-{\mathit{x}}_{4}|$ | n | COC | CPU-Time |
---|---|---|---|---|---|---|

$\mathrm{DM}$ | $7.61\phantom{\rule{3.33333pt}{0ex}}\times \phantom{\rule{3.33333pt}{0ex}}{10}^{-9}$ | $1.42\phantom{\rule{3.33333pt}{0ex}}\times \phantom{\rule{3.33333pt}{0ex}}{10}^{-25}$ | $9.14\phantom{\rule{3.33333pt}{0ex}}\times \phantom{\rule{3.33333pt}{0ex}}{10}^{-76}$ | 5 | 3.0000 | 1.047 |

$\mathrm{HM}$ | $6.17\phantom{\rule{3.33333pt}{0ex}}\times \phantom{\rule{3.33333pt}{0ex}}{10}^{-8}$ | $1.12\phantom{\rule{3.33333pt}{0ex}}\times \phantom{\rule{3.33333pt}{0ex}}{10}^{-22}$ | $6.66\phantom{\rule{3.33333pt}{0ex}}\times \phantom{\rule{3.33333pt}{0ex}}{10}^{-67}$ | 5 | 3.0000 | 2.141 |

$\mathrm{CM}$ | $7.82\phantom{\rule{3.33333pt}{0ex}}\times \phantom{\rule{3.33333pt}{0ex}}{10}^{-8}$ | $2.81\phantom{\rule{3.33333pt}{0ex}}\times \phantom{\rule{3.33333pt}{0ex}}{10}^{-22}$ | $1.31\phantom{\rule{3.33333pt}{0ex}}\times \phantom{\rule{3.33333pt}{0ex}}{10}^{-65}$ | 5 | 3.0000 | 1.985 |

$\mathrm{OM}$ | $8.97\phantom{\rule{3.33333pt}{0ex}}\times \phantom{\rule{3.33333pt}{0ex}}{10}^{-8}$ | $4.78\phantom{\rule{3.33333pt}{0ex}}\times \phantom{\rule{3.33333pt}{0ex}}{10}^{-22}$ | $7.22\phantom{\rule{3.33333pt}{0ex}}\times \phantom{\rule{3.33333pt}{0ex}}{10}^{-65}$ | 5 | 3.0000 | 1.906 |

$\mathrm{VNM}$ | $2.42\phantom{\rule{3.33333pt}{0ex}}\times \phantom{\rule{3.33333pt}{0ex}}{10}^{-8}$ | $5.53\phantom{\rule{3.33333pt}{0ex}}\times \phantom{\rule{3.33333pt}{0ex}}{10}^{-24}$ | $6.59\phantom{\rule{3.33333pt}{0ex}}\times \phantom{\rule{3.33333pt}{0ex}}{10}^{-71}$ | 5 | 3.0000 | 1.219 |

$\mathrm{M}1$ | $7.10\phantom{\rule{3.33333pt}{0ex}}\times \phantom{\rule{3.33333pt}{0ex}}{10}^{-12}$ | $7.96\phantom{\rule{3.33333pt}{0ex}}\times \phantom{\rule{3.33333pt}{0ex}}{10}^{-35}$ | 0 | 4 | 3.0000 | 0.390 |

$\mathrm{M}2$ | $1.88\phantom{\rule{3.33333pt}{0ex}}\times \phantom{\rule{3.33333pt}{0ex}}{10}^{-11}$ | $2.20\phantom{\rule{3.33333pt}{0ex}}\times \phantom{\rule{3.33333pt}{0ex}}{10}^{-33}$ | $3.54\phantom{\rule{3.33333pt}{0ex}}\times \phantom{\rule{3.33333pt}{0ex}}{10}^{-99}$ | 5 | 3.0000 | 0.516 |

$\mathrm{M}3$ | $1.72\phantom{\rule{3.33333pt}{0ex}}\times \phantom{\rule{3.33333pt}{0ex}}{10}^{-12}$ | $5.66\phantom{\rule{3.33333pt}{0ex}}\times \phantom{\rule{3.33333pt}{0ex}}{10}^{-37}$ | 0 | 4 | 3.0000 | 0.500 |

$\mathrm{M}4$ | $1.22\phantom{\rule{3.33333pt}{0ex}}\times \phantom{\rule{3.33333pt}{0ex}}{10}^{-10}$ | $1.22\phantom{\rule{3.33333pt}{0ex}}\times \phantom{\rule{3.33333pt}{0ex}}{10}^{-30}$ | $1.21\phantom{\rule{3.33333pt}{0ex}}\times \phantom{\rule{3.33333pt}{0ex}}{10}^{-90}$ | 5 | 3.0000 | 0.578 |

$\mathrm{M}5$ | $1.20\phantom{\rule{3.33333pt}{0ex}}\times \phantom{\rule{3.33333pt}{0ex}}{10}^{-11}$ | $4.74\phantom{\rule{3.33333pt}{0ex}}\times \phantom{\rule{3.33333pt}{0ex}}{10}^{-34}$ | 0 | 4 | 3.0000 | 0.359 |

$\mathrm{M}6$ | $3.80\phantom{\rule{3.33333pt}{0ex}}\times \phantom{\rule{3.33333pt}{0ex}}{10}^{-12}$ | $9.18\phantom{\rule{3.33333pt}{0ex}}\times \phantom{\rule{3.33333pt}{0ex}}{10}^{-36}$ | 0 | 4 | 3.0000 | 0.313 |

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**MDPI and ACS Style**

Kumar, D.; Sharma, J.R.; Cesarano, C.
An Efficient Class of Traub–Steffensen-Type Methods for Computing Multiple Zeros. *Axioms* **2019**, *8*, 65.
https://doi.org/10.3390/axioms8020065

**AMA Style**

Kumar D, Sharma JR, Cesarano C.
An Efficient Class of Traub–Steffensen-Type Methods for Computing Multiple Zeros. *Axioms*. 2019; 8(2):65.
https://doi.org/10.3390/axioms8020065

**Chicago/Turabian Style**

Kumar, Deepak, Janak Raj Sharma, and Clemente Cesarano.
2019. "An Efficient Class of Traub–Steffensen-Type Methods for Computing Multiple Zeros" *Axioms* 8, no. 2: 65.
https://doi.org/10.3390/axioms8020065