# Development of Optimal Eighth Order Derivative-Free Methods for Multiple Roots of Nonlinear Equations

^{1}

^{2}

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Development of Method

**Theorem**

**1.**

**Proof.**

**Remark**

**1.**

**Remark**

**2.**

**Remark**

**3.**

#### Some Particular Forms of Proposed Family

- (1)
- Let us consider the following function $G(h,t)$ which satisfies the conditions of Theorem 1$$G(h,t)=1+2h+t-2{h}^{2}+4ht-12{h}^{3}.$$Then, the corresponding eighth-order iterative scheme is given by$$\begin{array}{cc}\hfill {x}_{k+1}=& \phantom{\rule{4pt}{0ex}}{z}_{k}-mut\left[1+2h+t-2{h}^{2}+4ht-12{h}^{3}\right]\frac{f\left({x}_{k}\right)}{f[{w}_{k},{x}_{k}]}.\hfill \end{array}$$
- (2)
- Next, consider the rational function$$G(h,t)=\frac{1+2h+2t-2{h}^{2}+6ht-12{h}^{3}}{1+t}$$Satisfying the conditions of Theorem 1. Then, corresponding eighth-order iterative scheme is given by$$\begin{array}{cc}\hfill {x}_{k+1}=& \phantom{\rule{4pt}{0ex}}{z}_{k}-mut\left[\frac{1+2h+2t-2{h}^{2}+6ht-12{h}^{3}}{1+t}\right]\frac{f\left({x}_{k}\right)}{f[{w}_{k},{x}_{k}]}.\hfill \end{array}$$
- (3)
- Consider another rational function satisfying the conditions of Theorem 1, which is given by$$G(h,t)=\frac{1+3h+t+5ht-14{h}^{3}-12{h}^{4}}{1+h}.$$Then, corresponding eighth-order iterative scheme is given by$$\begin{array}{cc}\hfill {x}_{k+1}=& \phantom{\rule{4pt}{0ex}}{z}_{k}-mut\left[\frac{1+3h+t+5ht-14{h}^{3}-12{h}^{4}}{1+h}\right]\frac{f\left({x}_{k}\right)}{f[{w}_{k},{x}_{k}]}.\hfill \end{array}$$
- (4)
- Next, we suggest another rational function satisfying the conditions of Theorem 1, which is given by$$G(h,t)=\frac{1+3h+2t+8ht-14{h}^{3}}{(1+h)(1+t)}.$$Then, corresponding eighth-order iterative scheme is given by$$\begin{array}{cc}\hfill {x}_{k+1}=& \phantom{\rule{4pt}{0ex}}{z}_{k}-mut\left[\frac{1+3h+2t+8ht-14{h}^{3}}{(1+h)(1+t)}\right]\frac{f\left({x}_{k}\right)}{f[{w}_{k},{x}_{k}]}.\hfill \end{array}$$
- (5)
- Lastly, we consider yet another function satisfying the conditions of Theorem 1$$G(h,t)=\frac{1+t-2h(2+t)-2{h}^{2}(6+11t)+{h}^{3}(4+8t)}{2{h}^{2}-6h+1}.$$Then, the corresponding eighth-order method is given as$$\begin{array}{cc}\hfill {x}_{k+1}=& \phantom{\rule{4pt}{0ex}}{z}_{k}-mut\left[\frac{1+t-2h(2+t)-2{h}^{2}(6+11t)+{h}^{3}(4+8t)}{2{h}^{2}-6h+1}\right]\frac{f\left({x}_{k}\right)}{f[{w}_{k},{x}_{k}]}.\hfill \end{array}$$

## 3. Complex Dynamics of Methods

**Test problem 1**. Consider the polynomial ${P}_{1}\left(z\right)={({z}^{2}-1)}^{2}$ having two zeros $\{-1,1\}$ with multiplicities $m=2$. The basin of attractors for this polynomial are shown in Figure 1, Figure 2 and Figure 3, for different choices of $\beta =0.01,{10}^{-6},{10}^{-10}.$ A color is assigned to each basin of attraction of a zero. In particular, to obtain the basin of attraction, the red and green colors have been assigned for the zeros $-1$ and 1, respectively. Looking at the behavior of the methods, we see that the method M-2 and M-4 possess less number of divergent points and therefore have better convergence than rest of the methods. Observe that there is a small difference among the basins for the remaining methods with the same value of $\beta $. Note also that the basins are becoming larger as the parameter $\beta $ assumes smaller values.

**Test problem 2.**Let ${P}_{2}\left(z\right)={({z}^{3}+z)}^{2}$ having three zeros $\{-i,0,i\}$ with multiplicities $m=2$. The basin of attractors for this polynomial are shown in Figure 4, Figure 5 and Figure 6, for different choices of $\beta =0.01,{10}^{-6},{10}^{-10}.$ A color is allocated to each basin of attraction of a zero. For example, we have assigned the colors: green, red and blue corresponding to the basins of the zeros $-i$, i and 0, From graphics, we see that the methods M-2 and M-4 have better convergence due to a lesser number of divergent points. Also observe that in each case, the basins are getting broader with the smaller values of $\beta $. The basins in methods M-1, M-3 are almost the same and method M-5 has more divergent points.

**Test problem 3.**Let ${P}_{3}\left(z\right)=({z}^{2}-\frac{1}{4})({z}^{2}+\frac{9}{4})$ having four simple zeros $\{-\frac{1}{2},\frac{1}{2},-\frac{3}{2}i,\frac{3}{2}i,\}$. To see the dynamical view, we allocate the colors green, red, blue and yellow corresponding to basins of the zeros $-\frac{1}{2}$, $\frac{1}{2}$, $-\frac{3}{2}i$ and $\frac{3}{2}i$. The basin of attractors for this polynomial are shown in Figure 7, Figure 8 and Figure 9, for different choices of $\beta =0.01,{10}^{-6},{10}^{-10}.$ Looking at the graphics, we conclude that the methods M-2 and M-4 have better convergence behavior since they have lesser number of divergent points. The remaining methods have almost similar basins with the same value of $\beta $. Notice also that the basins are becoming larger with the smaller values of $\beta $.

## 4. Numerical Results

`f[x_ ]=x9-29x8+349x7-2261x6+8455x5-17663x4+15927x3+6993x2-24732x+12960;`

`a=2; b=3.5; k=1; x0=0.5*(a+b+Sign[f[a]]*NIntegrate[Tanh[k *f[x]],{x,a,b}])`

**Example**

**1.**

**Example**

**2.**

**Example**

**3.**

**Example**

**4.**

**Example**

**5.**

**Example**

**6.**

## 5. Conclusions

## Author Contributions

## Funding

## Conflicts of Interest

## References

- Traub, J.F. Iterative Methods for the Solution of Equations; Chelsea Publishing Company: New York, NY, USA, 1982. [Google Scholar]
- Hoffman, J.D. Numerical Methods for Engineers and Scientists; McGraw-Hill Book Company: New York, NY, USA, 1992. [Google Scholar]
- Argyros, I.K. Convergence and Applications of Newton-Type Iterations; Springer-Verlag: New York, NY, USA, 2008. [Google Scholar]
- Argyros, I.K.; Magreñán, Á.A. Iterative Methods and Their Dynamics with Applications; CRC Press: New York, NY, USA, 2017. [Google Scholar]
- Schröder, E. Über unendlich viele Algorithmen zur Auflösung der Gleichungen. Math. Ann.
**1870**, 2, 317–365. [Google Scholar] [CrossRef] - 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. Appl. 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. Math. Appl.
**2011**, 235, 4199–4206. [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] - Geum, Y.H.; Kim, Y.I.; Neta, B. A class of two-point sixth-order multiplezero finders of modified double-Newton type and their dynamics. Appl. Math. Comput.
**2015**, 270, 387–400. [Google Scholar] - 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] - 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.
**2017**. [Google Scholar] [CrossRef] - Zafar, F.; Cordero, A.; Quratulain, R.; Torregrosa, J.R. Optimal iterative methods for finding multiple roots of nonlinear equations using free parameters. J. Math. Chem.
**2017**. [Google Scholar] [CrossRef] - Zafar, F.; Cordero, A.; Sultana, S.; Torregrosa, J.R. Optimal iterative methods for finding multiple roots of nonlinear equations using weight functions and dynamics. J. Comp. Appl. Math.
**2018**, 342, 352–374. [Google Scholar] [CrossRef] - Zafar, F.; Cordero, A.; Torregrosa, J.R. An efficient family of optimal eighth-order multiple root finders. Mathematics
**2018**, 6, 310. [Google Scholar] [CrossRef] - Behl, R.; Cordero, A.; Motsa, S.S.; Torregrosa, J.R. An eighth-order family of optimal multiple root finders and its dynamics. Numer. Algor.
**2018**, 77, 1249–1272. [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**. [Google Scholar] [CrossRef] - Behl, R.; Alshomrani, A.S.; Motsa, S.S. An optimal scheme for multiple roots of nonlinear equations with eighth-order convergence. J. Math. Chem.
**2018**. [Google Scholar] [CrossRef] - Sharma, J.R.; Kumar, D.; Argyros, I.K. An efficient class of Traub-Steffensen-like seventh order multiple-root solvers with applications. Symmetry
**2019**, 11, 518. [Google Scholar] [CrossRef] - 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] - 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. Algorithms
**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] - Wolfram, S. The Mathematica Book, 5th ed.; Wolfram Media: Champaign, IL, USA, 2003. [Google Scholar]
- Yun, B.I. A non-iterative method for solving non-linear equations. Appl. Math. Comput.
**2008**, 198, 691–699. [Google Scholar] [CrossRef] - Bradie, B. A Friendly Introduction to Numerical Analysis; Pearson Education Inc.: New Delhi, India, 2006. [Google Scholar]

**Figure 1.**Basins of attraction for methods M-1 to M-5 $(\beta =0.01)$ in polynomial ${P}_{1}\left(z\right)$.

**Figure 2.**Basins of attraction for methods M-1 to M-5 $(\beta ={10}^{-6})$ in polynomial ${P}_{1}\left(z\right)$.

**Figure 3.**Basins of attraction for methods M-1 to M-5 $(\beta ={10}^{-10})$ in polynomial ${P}_{1}\left(z\right)$.

**Figure 4.**Basins of attraction for methods M-1 to M-5 $(\beta =0.01)$ in polynomial ${P}_{2}\left(z\right)$.

**Figure 5.**Basins of attraction for methods M-1 to M-5 $(\beta ={10}^{-6})$ in polynomial ${P}_{2}\left(z\right)$.

**Figure 6.**Basins of attraction for methods M-1 to M-5 $(\beta ={10}^{-10})$ in polynomial ${P}_{2}\left(z\right)$.

**Figure 7.**Basins of attraction for methods M-1 to M-5 $(\beta =0.01)$ in polynomial ${P}_{3}\left(z\right)$.

**Figure 8.**Basins of attraction for methods M-1 to M-5 $(\beta ={10}^{-6})$ in polynomial ${P}_{3}\left(z\right)$.

**Figure 9.**Basins of attraction for methods M-1 to M-5 $(\beta ={10}^{-10})$ in polynomial ${P}_{3}\left(z\right)$.

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

ZM−1 | $2.13$ | $4.82\times {10}^{-8}$ | $4.27\times {10}^{-67}$ | 4 | 8.000 | 0.608 | ||||||

ZM−2 | $6.43$ | $5.30\times {10}^{-7}$ | $6.10\times {10}^{-59}$ | 4 | 8.000 | 0.671 | ||||||

BM−1 | $1.03\times {10}^{-1}$ | $3.34\times {10}^{-6}$ | $9.73\times {10}^{-20}$ | 5 | 3.000 | 0.687 | ||||||

BM−2 | $1.03\times {10}^{-1}$ | $3.35\times {10}^{-6}$ | $9.82\times {10}^{-20}$ | 5 | 3.000 | 0.702 | ||||||

BM−3 | $1.85$ | $2.44\times {10}^{-8}$ | $1.15\times {10}^{-69}$ | 4 | 8.000 | 0.640 | ||||||

M−1 | $1.65$ | $1.86\times {10}^{-8}$ | $3.08\times {10}^{-70}$ | 4 | 8.000 | 0.452 | ||||||

M−2 | $9.64\times {10}^{-1}$ | $1.86\times {10}^{-9}$ | $5.08\times {10}^{-78}$ | 4 | 8.000 | 0.453 | ||||||

M−3 | $1.64$ | $1.81\times {10}^{-8}$ | $2.80\times {10}^{-70}$ | 4 | 8.000 | 0.468 | ||||||

M−4 | $9.55\times {10}^{-1}$ | $1.84\times {10}^{-9}$ | $5.09\times {10}^{-78}$ | 4 | 8.000 | 0.437 | ||||||

M−5 | $1.65$ | $1.86\times {10}^{-8}$ | $3.29\times {10}^{-70}$ | 4 | 8.000 | 0.421 |

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

ZM−1 | $2.24\times {10}^{-1}$ | $3.06\times {10}^{-8}$ | $3.36\times {10}^{-62}$ | 4 | 8.000 | 0.140 | ||||||

ZM−2 | $6.45\times {10}^{-1}$ | $1.99\times {10}^{-6}$ | $5.85\times {10}^{-48}$ | 4 | 8.000 | 0.187 | ||||||

BM−1 | $9.85\times {10}^{-3}$ | $4.51\times {10}^{-7}$ | $4.14\times {10}^{-20}$ | 5 | 3.000 | 0.140 | ||||||

BM−2 | $9.86\times {10}^{-3}$ | $4.52\times {10}^{-7}$ | $4.18\times {10}^{-20}$ | 5 | 3.000 | 0.140 | ||||||

BM−3 | $1.97\times {10}^{-1}$ | $5.21\times {10}^{-9}$ | $4.23\times {10}^{-69}$ | 4 | 8.000 | 0.125 | ||||||

M−1 | $2.07\times {10}^{-1}$ | $6.58\times {10}^{-8}$ | $5.78\times {10}^{-59}$ | 4 | 8.000 | 0.125 | ||||||

M−2 | $1.21\times {10}^{-1}$ | $2.12\times {10}^{-9}$ | $1.01\times {10}^{-70}$ | 4 | 8.000 | 0.110 | ||||||

M−3 | $2.05\times {10}^{-1}$ | $6.68\times {10}^{-8}$ | $7.64\times {10}^{-59}$ | 4 | 8.000 | 0.125 | ||||||

M−4 | $1.20\times {10}^{-1}$ | $2.24\times {10}^{-9}$ | $1.79\times {10}^{-70}$ | 4 | 8.000 | 0.109 | ||||||

M−5 | $2.07\times {10}^{-1}$ | $8.86\times {10}^{-8}$ | $7.65\times {10}^{-58}$ | 4 | 8.000 | 0.093 |

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

ZM−1 | $3.19\times {10}^{-2}$ | $2.77\times {10}^{-16}$ | 0 | 3 | 7.995 | 2.355 | ||||||

ZM−2 | $7.25\times {10}^{-2}$ | $5.76\times {10}^{-14}$ | 0 | 3 | 7.986 | 2.371 | ||||||

BM−1 | $5.84\times {10}^{-4}$ | $1.78\times {10}^{-11}$ | $5.08\times {10}^{-34}$ | 4 | 3.000 | 2.683 | ||||||

BM−2 | $5.84\times {10}^{-4}$ | $1.78\times {10}^{-11}$ | $5.09\times {10}^{-34}$ | 4 | 3.000 | 2.777 | ||||||

BM−3 | $3.07\times {10}^{-2}$ | $4.39\times {10}^{-17}$ | 0 | 3 | 8.002 | 2.324 | ||||||

M−1 | $3.05\times {10}^{-2}$ | $4.52\times {10}^{-16}$ | 0 | 3 | 7.993 | 1.966 | ||||||

M−2 | $1.96\times {10}^{-2}$ | $2.65\times {10}^{-17}$ | 0 | 3 | 7.996 | 1.982 | ||||||

M−3 | $3.04\times {10}^{-2}$ | $5.46\times {10}^{-16}$ | 0 | 3 | 7.993 | 1.965 | ||||||

M−4 | $1.96\times {10}^{-2}$ | $3.05\times {10}^{-17}$ | 0 | 3 | 7.996 | 1.981 | ||||||

M−5 | $3.05\times {10}^{-2}$ | $5.43\times {10}^{-16}$ | 0 | 3 | 7.992 | 1.903 |

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

ZM−1 | $2.21\times {10}^{-1}$ | $1.83\times {10}^{-1}$ | $7.19\times {10}^{-3}$ | 6 | 8.000 | 0.124 | ||||||

ZM−2 | Fails | – | – | – | – | – | ||||||

BM−1 | $1.15$ | $1.06$ | $5.83\times {10}^{-2}$ | 7 | 3.000 | 0.109 | ||||||

BM−2 | $2.44\times {10}^{-2}$ | $4.15\times {10}^{-3}$ | $5.41\times {10}^{-4}$ | 7 | 3.000 | 0.110 | ||||||

BM−3 | $2.67\times {10}^{-2}$ | $3.06\times {10}^{-3}$ | $9.21\times {10}^{-4}$ | 5 | 7.988 | 0.109 | ||||||

M−1 | $3.55\times {10}^{-2}$ | $2.32\times {10}^{-3}$ | $1.42\times {10}^{-10}$ | 5 | 8.000 | 0.084 | ||||||

M−2 | $3.05\times {10}^{-2}$ | $7.06\times {10}^{-3}$ | $2.94\times {10}^{-3}$ | 6 | 8.000 | 0.093 | ||||||

M−3 | $3.30\times {10}^{-2}$ | $5.82\times {10}^{-4}$ | $4.26\times {10}^{-5}$ | 5 | 8.000 | 0.095 | ||||||

M−4 | $2.95\times {10}^{-2}$ | $1.22\times {10}^{-2}$ | $6.70\times {10}^{-3}$ | 6 | 8.000 | 0.094 | ||||||

M−5 | $5.01\times {10}^{-2}$ | $1.20\times {10}^{-2}$ | $5.06\times {10}^{-6}$ | 5 | 8.000 | 0.089 |

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

ZM−1 | $1.02\times {10}^{-2}$ | $1.56\times {10}^{-14}$ | 0 | 3 | 7.983 | 0.702 | ||||||

ZM−2 | $2.40\times {10}^{-2}$ | $5.32\times {10}^{-14}$ | $7.45\times {10}^{-89}$ | 4 | 8.000 | 0.873 | ||||||

BM−1 | $2.55\times {10}^{-4}$ | $7.84\times {10}^{-11}$ | $2.26\times {10}^{-30}$ | 5 | 3.000 | 0.920 | ||||||

BM−2 | $2.55\times {10}^{-4}$ | $7.84\times {10}^{-11}$ | $2.26\times {10}^{-30}$ | 5 | 3.000 | 0.795 | ||||||

BM−3 | $9.57\times {10}^{-3}$ | $2.50\times {10}^{-15}$ | 0 | 3 | 7.989 | 0.671 | ||||||

M−1 | $9.44\times {10}^{-3}$ | $2.07\times {10}^{-14}$ | 0 | 3 | 7.982 | 0.593 | ||||||

M−2 | $5.96\times {10}^{-3}$ | $1.02\times {10}^{-15}$ | 0 | 3 | 7.990 | 0.608 | ||||||

M−3 | $9.42\times {10}^{-3}$ | $2.48\times {10}^{-14}$ | 0 | 3 | 7.982 | 0.562 | ||||||

M−4 | $5.95\times {10}^{-3}$ | $1.18\times {10}^{-15}$ | 0 | 3 | 7.989 | 0.530 | ||||||

M−5 | $9.44\times {10}^{-3}$ | $2.62\times {10}^{-14}$ | 0 | 3 | 7.982 | 0.499 |

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

ZM−1 | $1.38\times {10}^{-2}$ | $5.09\times {10}^{-4}$ | $2.24\times {10}^{-27}$ | 4 | 8.000 | 1.217 | ||||||

ZM−2 | $3.13\times {10}^{-2}$ | $4.80\times {10}^{-3}$ | $7.00\times {10}^{-20}$ | 4 | 7.998 | 1.357 | ||||||

BM−1 | $4.76\times {10}^{-5}$ | $1.26\times {10}^{-36}$ | 0 | 3 | 8.000 | 0.874 | ||||||

BM−2 | $4.76\times {10}^{-5}$ | $2.57\times {10}^{-36}$ | 0 | 3 | 8.000 | 0.889 | ||||||

BM−3 | $1.37\times {10}^{-2}$ | $4.98\times {10}^{-4}$ | $3.38\times {10}^{-28}$ | 4 | 8.000 | 1.201 | ||||||

M−1 | $7.34\times {10}^{-6}$ | $1.14\times {10}^{-41}$ | 0 | 3 | 8.000 | 0.448 | ||||||

M−2 | $8.25\times {10}^{-6}$ | $4.84\times {10}^{-41}$ | 0 | 3 | 8.000 | 0.452 | ||||||

M−3 | $7.71\times {10}^{-6}$ | $2.09\times {10}^{-41}$ | 0 | 3 | 8.000 | 0.460 | ||||||

M−4 | $8.68\times {10}^{-6}$ | $8.58\times {10}^{-41}$ | 0 | 3 | 8.000 | 0.468 | ||||||

M−5 | $8.32\times {10}^{-6}$ | $4.03\times {10}^{-41}$ | 0 | 3 | 8.000 | 0.436 |

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## Share and Cite

**MDPI and ACS Style**

Sharma, J.R.; Kumar, S.; Argyros, I.K.
Development of Optimal Eighth Order Derivative-Free Methods for Multiple Roots of Nonlinear Equations. *Symmetry* **2019**, *11*, 766.
https://doi.org/10.3390/sym11060766

**AMA Style**

Sharma JR, Kumar S, Argyros IK.
Development of Optimal Eighth Order Derivative-Free Methods for Multiple Roots of Nonlinear Equations. *Symmetry*. 2019; 11(6):766.
https://doi.org/10.3390/sym11060766

**Chicago/Turabian Style**

Sharma, Janak Raj, Sunil Kumar, and Ioannis K. Argyros.
2019. "Development of Optimal Eighth Order Derivative-Free Methods for Multiple Roots of Nonlinear Equations" *Symmetry* 11, no. 6: 766.
https://doi.org/10.3390/sym11060766