An optimal three-point eighth-order iterative method without memory for solving nonlinear equations with its dynamics
Gunar Matthies $^a$\thanks{gunar.matthies@tu-dresden.de
We present a three-point iterative method without memory for solving nonlinear equations in one variable. The proposed method provides convergence order eight with four function evaluations per iteration. Hence, it possesses a very high computational efficiency and supports Kung and Traub's conjecture. The construction, the convergence analysis, and the numerical implementation of the method will be presented. Using several test problems, the proposed method will be compared with existing methods of convergence order eight concerning accuracy and basin of attraction. Furthermore, some measures are used to judge methods with respect to their performance in finding the basin of attraction. \medskip \noindent Keywords: Optimal multi-point iterative methods; Simple root; Order of convergence; Kung and Traub's conjecture; Basins of attraction. \medskip \noindent Mathematics Subject Classification: 65H05, 37F10
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%--------------------------------------------------------------%
%\title{Title}

\title{An optimal three-point eighth-order iterative method without memory for
solving nonlinear equations with its dynamics}

\author{Gunar Matthies $^a$\thanks{gunar.matthies@tu-dresden.de}
\and Mehdi Salimi $^{b,c}$\thanks{mehdi.salimi@tu-dresden.de} \and
Somayeh Sharifi $^{d}$\thanks{somayeh.sharifi@medalics.org} \and
Juan Luis Varona $^{e}$\thanks{jvarona@unirioja.es}\\[9pt]
\small
$^{a}$Institut f{\"u}r Numerische Mathematik, Technische Universit{\"a}t Dresden, Germany\\[-3pt]
\small
$^{b}$Center for Dynamics, Department of Mathematics, Technische Universit{\"a}t Dresden, Germany\\[-3pt]
\small
$^{c}$Department of Mathematics, Universiti Putra Malaysia, 43400 UPM Serdang, Selangor, Malaysia\\[-3pt]
\small
$^{d}$MEDAlics, Research \small Center at Universit\`{a} per
Stranieri Dante Alighieri, Reggio Calabria, Italy\\[-3pt]
\small $^{e}$Departamento de Matem{\'a}ticas y Computaci{\'o}n,
Universidad de La Rioja, Logro{\~n}o, Spain }

\date{}
%--------------------------------------------------------------%

%--------------------------------------------------------------%
\begin{document}
%--------------------------------------------------------------%

%--------------------------------------------------------------%
\maketitle
%--------------------------------------------------------------%

\begin{abstract}
We present a three-point iterative method without memory for
solving nonlinear equations in one variable. The proposed method
provides convergence order eight with four function evaluations
per iteration. Hence, it possesses a very high computational
efficiency and supports Kung and Traub's conjecture. The
construction, the convergence analysis, and the numerical
implementation of the method will be presented. Using several test
problems, the proposed method will be compared with existing
methods of convergence order eight concerning accuracy and basin
of attraction. Furthermore, some measures are used to judge
methods with respect to their performance in finding the basin of
attraction.
\medskip

\noindent \textbf{Keywords}: Optimal multi-point iterative
methods; Simple root; Order of convergence; Kung and Traub's
conjecture; Basins of attraction.
\medskip

\noindent \textbf{Mathematics Subject Classification}: 65H05, 37F10
\end{abstract}

%--------------------------------------------------------------%
\section{Introduction}
%--------------------------------------------------------------%
Solving nonlinear equations is a basic and extremely valuable tool
in all fields in science and engineering. One can distinguish
between two general approaches for solving nonlinear equations
numerically, namely, one-point and multi-point methods. The basic
optimality theorem shows that an analytic one-point method based on $k$
evaluations is of order at most $k$, see~\cite[\S\,5.4]{Traub}
or~\cite{Kung} for an improved proof. The Newton--Raphson method
$x_{n+1}:=x_n - \frac{f(x_n)}{f'(x_n)}$ is probably the most widely used
algorithms for finding roots. It requires two evaluations per iteration
step, one for $f$ and one for $f'$, and results in second order
convergence which is optimal for this one-point method.

Some computational issues encountered by one-point methods are overcome by
multi-point methods since they allow to achieve greater accuracy with the
same number of function evaluations. Important aspects related to these
methods are convergence order and efficiency. It is favorable to attain
with a fixed number of function evaluations per iteration step a
convergence order which is as high as possible. A central role in this
context plays the unproved conjecture by Kung and Traub~\cite{Kung} which
states that an optimal multi-point method without memory provides a
convergence order of~$2^k$ while using $k+1$ evaluations in each iteration
step. The efficiency index for a method with $k$ evaluations and convergence
order $p$ and $k$ evaluations is given by $E(k,p)=\sqrt[k]{p}$,
see~\cite{Ostrowski}. Hence, the efficiency of a method supporting Kung and
Traub's conjecture is $\sqrt[k+1]{2^k}$. In particular, an optimal method
with convergence order eight has an efficiency index
$\sqrt[4]{8}\simeq 1.68179$.

A large number of multi-point methods for finding simple roots of a
nonlinear equation $f(x)=0$ with a scalar function
$f: D\subset\mathbb{R} \to \mathbb{R}$ which is defined on
an open interval~$D$ (or $f: D\subset\mathbb{C} \to \mathbb{C}$
defined on a region $D$ in the complex plane~$\mathbb{C}$) have been developed
and analyzed for improving the convergence order of classical methods like
the Newton--Raphson iteration.

Some well known two-point methods without memory are described
e.g.\ in Jarratt~\cite{Jarratt}, King~\cite{King}, and
Ostrowski~\cite{Ostrowski}. Using inverse interpolation, Kung and
Traub \cite{Kung} constructed two general optimal classes without
memory. Since then, there have been many attempts to construct
optimal multi-point methods, utilizing e.g.\ weight functions, see
in particular~\cite{Babajee, Bi, Chun1, Lotfi2, Petkovic,
Sharifi1, Sharifi2, Sharifi3, Sharma1, Thukral, Wang}.

We will construct a three-point method of convergence order eight
which is free from second order derivatives, uses $4$ evaluations, and
provides the efficiency index $\sqrt[4]{8} \simeq 1.68179$.

A wide used criterion to judge and rank different methods for
solving nonlinear equations is the basin of attraction. We will
use two measures to assess the performance in finding the basin of
attraction~\cite{Varona}.

The paper is organized as follows. Section~\ref{sec:description}
introduces the new methods based on a Newton step and Newton's
interpolation. Moreover, details of the new method and the proof
of its optimal convergence order eight are given. The numerical
performance of the proposed method compared to other methods are
illustrated in Section~\ref{sec:examples}. We approximate and
visualize the basins of attraction in Section~\ref{sec:dynamic}
for the proposed method and several existing methods, both
graphically and by mean of introduced numerical performance
measures~\cite{Varona}. Finally, we conclude in
Section~\ref{sec:conclusion}.

%--------------------------------------------------------------%
\section{Description of the method and convergence analysis}
\label{sec:description}
%--------------------------------------------------------------%

We construct in this section a new optimal three-point method for
solving nonlinear equations by using a Newton-step and Newton's
interpolation polynomial of degree three which was also applied
in~\cite{Sharifi2}.

\paragraph{Method 1:}
The new method is given by
\begin{equation}
\label{a1}
\left\{
\begin{aligned}
y_{n} & := x_{n} - u_n(x_n), \\[1.2ex]
z_{n} & := x_{n} -
u_n(x_n) \left(1+\dfrac{f(y_n)}{f(x_n)}
+ \left(1+\dfrac{1}{1+u_n(x_n)}\right) \left(\dfrac{f(y_n)}{f(x_n)}\right)^2\right),
\\[1.2ex]
x_{n+1} & := z_{n} -
\dfrac{f(z_n)}{f[z_n,y_n]+(z_n-y_n)f[z_n,y_n,x_n]+(z_n-y_n)(z_n-x_n)f[z_n,y_n,x_n,x_n]},
\end{aligned}
\right.
\end{equation}
where $u_n(x_n)=\frac{f(x_{n})}{f'(x_{n})}$.
The standard notation for divided differences
in Newton's interpolation
\[
  g[t_{\nu},t_{\nu+1}, \dots, t_{\nu+j}]
  = \frac{g[t_{\nu+1}, \dots, t_{\nu+j}] - g[t_\nu, \dots,
  t_{\nu+j-1}]}{t_{\nu+j}-t_{\nu}},
\]
with $g[t_\nu] = g(t_\nu)$ and $g[t_{\nu},t_{\nu}] = g'(t_{\nu})$ are used.

The iteration method~\eqref{a1} and all forthcoming methods are applied for
$n = 0, 1, \dots$ where $x_0$ denotes an initial approximation of the simple
root~$x^{\ast}$ of the function $f$. The method~\eqref{a1} uses four
evaluations per iteration step, three for $f$ and one for $f'$. Note
that~\eqref{a1} works for real and complex functions.

The convergence order of method~\eqref{a1} is given in the following theorem.

\begin{thm}
\label{thm:main}
Let $f:D\subset \mathbb{R} \to \mathbb{R}$ be an
eight times continuously differentiable function with a simple
zero $x^{\ast}\in D$. If the initial point $x_{0}$ is sufficiently
close to $x^{\ast}$ then the method defined by \eqref{a1} converges
to $x^{\ast}$ with order eight.
\end{thm}

\begin{proof}
Let $e_{n} := x_{n}-x^{\ast}$, $e_{n,y}:=y_{n}-x^{*}$,
$e_{n,z} := z_{n}-x^{\ast}$ and
$c_{n} := \frac{f^{(n)}(x^{\ast})}{n!\,f'(x^{*})}$ for $n\in \mathbb{N}$.
Using the fact that $f(x^{\ast})=0$, the Taylor expansion of $f$ at
$x^{\ast}$ yields
\begin{equation}
\label{a10}
f(x_{n}) = f'(x^{\ast}) \left(e_{n}
+ c_{2}e_{n}^{2}+c_{3}e_{n}^{3} + \cdots
+ c_{8}e_{n}^{8}\right) + O(e_n^{9})
\end{equation}
and
\begin{equation}
\label{a11}
f'(x_{n}) = f'(x^{\ast}) \left(1 +
2c_{2}e_{n}+3c_{3}e_{n}^{2}+4c_{4}e_{n}^{3} + \cdots
+ 9c_9e_n^8\right) + O(e_n^{9}).
\end{equation}
Therefore, we have
\begin{equation*}
\begin{split}
\frac{f(x_{n})}{f'(x_{n})}
&= e_{n}-c_{2}e_{n}^{2} + \left(2c_{2}^{2}-2c_{3}\right) e_{n}^{3}
+ \left(-4c_2^3+7c_2c_3-3c_4\right)e_n^4 \\
& \qquad + \left(8c_2^4-20c_2^2c_3+6c_3^2+10c_2c_4-4c_5\right)e_n^5 \\
& \qquad + \left(-16c_2^5+52c_2^3c_3-28c_2^2c_4
+ 17c_3c_4-c_2(33c_3^2-13c_5)\right)e_n^6 + O(e_n^{7}),
\end{split}
\end{equation*}
and
\begin{equation*}
\begin{split}
e_{n,y} = y_n-x^{\ast} & = c_{2}e_{n}^{2} + \left(-2c_2^2+2c_3\right)e_n^3
+ \left(4c_2^3-7c_2c_3+3c_4\right)e_n^4\\
& \qquad + \left(-8c_2^4+20c_2^2c_3-6c_3^2-10c_2c_4+4c_5\right)e_n^5\\
& \qquad + \left(16c_2^5-52c_2^3c_3+28c_2^2c_4-17c_3c_4
+ c_2(33c_3^2-13c_5)\right)e_n^6 + O(e_n^{7}).
\end{split}
\end{equation*}
We have
\begin{equation}
\label{a12a}
f(y_{n}) = f'(x^{\ast}) \left(e_{n,y} + c_{2}e_{n,y}^{2} + c_{3}e_{n,y}^{3}
+ \cdots + c_{8}e_{n,y}^{8}\right) + O(e_{n,y}^{9})
\end{equation}
by a Taylor expansion of $f$ at $x^{\ast}$.
By substituting \eqref{a10}--\eqref{a12a} into \eqref{a1}, we get
\begin{equation*}
\begin{split}
\label{a13a}
e_{n,z} = z_n-x^{\ast}
&= c_2 \left(c_2+5c_2^2-c_3\right)e_n^4 \\
& \qquad + \left(-8c_2^3-36c_2^4-2c_3^2+c_2^2(-1+32c_3) + c_2(4c_3-2c_4)\right)e_n^5 + O(e_n^6).
\end{split}
\end{equation*}
We obtain
\begin{equation}
\label{a12b}
f(z_{n}) = f'(x^{\ast}) \left(e_{n,z} +
c_{2}e_{n,z}^{2}+c_{3}e_{n,z}^{3}+\cdots+c_{8}e_{n,z}^{8}\right) + O(e_{n,z}^{9})
\end{equation}
by using again a Taylor expansion of $f$ at $x^{\ast}$.
Substituting \eqref{a10}--\eqref{a12b} into \eqref{a1}, we get
\begin{equation}
\label{erreq}
e_{n+1} = x_{n+1}-x^{\ast} =
c_2^2 \left(c_2+5c_2^2-c_3\right) \left(c_2^2-5c_2^3-c_2c_3+c_4\right)e_n^8 + O(e_n^9),
\end{equation}
which finishes the proof of the theorem.
\end{proof}

We will compare the new method~\eqref{a1} with some existing
optimal three-point methods of order eight having the same optimal
computational efficiency index equal to $\sqrt[4]{8} \simeq
1.68179$, see~\cite{Ostrowski, Traub}.

The existing methods that we are going to use to compare are the following:

\paragraph{Method 2:}
The method by Chun and Lee \cite{Chun1} is given by
\begin{equation}
\label{o1}
%\begin{cases}
%y_{n} = x_{n} - \dfrac{f(x_{n})}{f'(x_{n})}, \\[0.8ex]
%z_{n} = y_{n} - \dfrac{f(y_n)}{f'(x_n)}
%\cdot \dfrac{1}{\left(1-\frac{f(y_n)}{f(x_n)}\right)^2}, \\[0.8ex]
%x_{n+1} = z_{n} - \dfrac{f(z_n)}{f'(x_n)}
%\cdot \dfrac{1}{\left(1-H(t_n)-J(s_n)-P(u_n)\right)^2}
%\end{cases}
\left\{
\begin{aligned}
y_{n} & := x_{n} - \dfrac{f(x_{n})}{f'(x_{n})}, \\[0.8ex]
z_{n} & := y_{n} - \dfrac{f(y_n)}{f'(x_n)}
\cdot \dfrac{1}{\left(1-\frac{f(y_n)}{f(x_n)}\right)^2}, \\[0.8ex]
x_{n+1} & := z_{n} - \dfrac{f(z_n)}{f'(x_n)}
\cdot \dfrac{1}{\left(1-H(t_n)-J(s_n)-P(u_n)\right)^2}
\end{aligned}
\right.
\end{equation}
with weight functions
\begin{equation*}
%\label{t1}
H(t_n) = -\beta-\gamma+t_n+\frac{t_n^2}{2}-\frac{t_n^3}{2}, \quad
J(s_n) = \beta+\frac{s_n}{2}, \quad P(u_n) = \gamma+\frac{u_n}{2},
\end{equation*}
where $t_n = \frac{f(y_n)}{f(x_n)}$, $s_n = \frac{f(z_n)}{f(x_n)}$,
$u_n = \frac{f(z_n)}{f(y_n)}$, and $\beta, \gamma \in \mathbb{R}$.
Note that the parameters $\beta$ and $\gamma$ cancel when used
in~\eqref{o1}. Hence, their choice has no contribution to the method.

\paragraph{Method 3:}
The method by B. Neta~\cite{Neta0}, see also~\cite[formula~(9)]{Neta1}, is
given by
\begin{equation}
\label{o2}
%\begin{cases}
%y_{n} = x_{n} - \dfrac{f(x_n)}{f'(x_n)}, \\[0.8ex]
%z_{n} = y_{n} - \dfrac{f(x_n)+Af(y_n)}{f(x_n)+(A-2)f(y_n)}\cdot
%\dfrac{f(y_n)}{f'(x_n)},
%\quad A\in \mathbb{R}, \\[0.8ex]
%x_{n+1} = y_n + \delta_1f^2(x_n)+\delta_2f^3(x_n),
%\end{cases}
\left\{
\begin{aligned}
y_{n} & := x_{n} - \dfrac{f(x_n)}{f'(x_n)}, \\[0.8ex]
z_{n} & := y_{n} - \dfrac{f(x_n)+Af(y_n)}{f(x_n)+(A-2)f(y_n)} \cdot
\dfrac{f(y_n)}{f'(x_n)},
\quad A\in \mathbb{R}, \\[0.8ex]
x_{n+1} & := y_n + \delta_1f^2(x_n)+\delta_2f^3(x_n),
\end{aligned}
\right.
\end{equation}
where
%\begin{equation*}
%F_y = f(y_n)-f(x_n), \quad F_z = f(z_n)-f(x_n),
%\end{equation*}
%\begin{equation*}
%\zeta_y = \dfrac{1}{F_y}
%\left(\dfrac{y_n-x_n}{F_y}-\dfrac{1}{f'(x_n)}\right),
%\quad
%\zeta_z = \dfrac{1}{F_z}
%\left(\dfrac{z_n-x_n}{F_z}-\dfrac{1}{f'(x_n)}\right),
%\end{equation*}
%\begin{equation*}
%\delta_2 = -\dfrac{\zeta_y-\zeta_z}{F_y-F_z},
%\quad \quad
%\delta_1 = \zeta_y+\delta_2F_y.
%\end{equation*}
\begin{alignat*}{2}
F_y & = f(y_n)-f(x_n), &\qquad F_z &= f(z_n)-f(x_n),\\
\zeta_y & = \dfrac{1}{F_y}
\left(\dfrac{y_n-x_n}{F_y}-\dfrac{1}{f'(x_n)}\right),
&\qquad
\zeta_z & = \dfrac{1}{F_z}
\left(\dfrac{z_n-x_n}{F_z}-\dfrac{1}{f'(x_n)}\right),\\
\delta_2 & = -\dfrac{\zeta_y-\zeta_z}{F_y-F_z},
&\qquad
\delta_1 & = \zeta_y+\delta_2F_y.
\end{alignat*}
We will use $A = 0$ in the numerical experiments of this paper.

\paragraph{Method 4:}
The Sharma and Sharma method \cite{Sharma1} is given by
\begin{equation}
\label{o3}
%\begin{cases}
%y_{n} = x_{n} - \dfrac{f(x_n)}{f'(x_n)}, \\[0.8ex]
%z_{n} = y_{n} - \dfrac{f(y_n)}{f'(x_n)} \cdot \dfrac{f(x_n)}{f(x_n)-2f(y_n)}, \\[0.8ex]
%x_{n+1} = z_{n} - \dfrac{f[x_n,y_n]f(z_n)}{f[x_n,z_n]f[y_n,z_n]}\,W(t_n),
%\end{cases}
\left\{
\begin{aligned}
y_{n} & := x_{n} - \dfrac{f(x_n)}{f'(x_n)}, \\[0.8ex]
z_{n} & := y_{n} - \dfrac{f(y_n)}{f'(x_n)} \cdot \dfrac{f(x_n)}{f(x_n)-2f(y_n)}, \\[0.8ex]
x_{n+1} & := z_{n} - \dfrac{f[x_n,y_n]f(z_n)}{f[x_n,z_n]f[y_n,z_n]}\,W(t_n),
\end{aligned}
\right.
\end{equation}
with the weight function
\begin{equation*}
%\label{d6}
W(t_n) = 1+\frac{t_n}{1+\alpha t_n},
\quad \alpha\in \mathbb{R},
\end{equation*}
and $t_n = \frac{f(z_n)}{f(x_n)}$.
We will use $\alpha=1$ in the numerical experiments of this paper.

\paragraph{Method 5:}
The method from Babajee, Cordero, Soleymani and Torregrosa \cite{Babajee}
is given by
\begin{equation}
\label{o4}
%\begin{cases}
%y_{n} = x_{n} - \dfrac{f(x_n)}{f'(x_n)}
%\left( 1 + \left(\dfrac{f(x_n)}{f'(x_n)}\right)^5 \right), \\[0.8ex]
%z_{n} = y_{n} - \dfrac{f(y_n)}{f'(x_n)}
%\left( 1 - \dfrac{f(y_n)}{f(x_n)}\right)^{-2}, \\[0.8ex]
%x_{n+1} = z_n - \dfrac{f(z_n)}{f'(x_n)}
%\cdot \dfrac{1+\left(\dfrac{f(y_n)}{f(x_n)}\right)^2 + 5\left(\dfrac{f(y_n)}{f(x_n)}\right)^4
%+ \dfrac{f(z_n)}{f(y_n)}}
%{\left( 1 - \dfrac{f(y_n)}{f(x_n)} - \dfrac{f(z_n)}{f(x_n)} \right)^2} .
%\end{cases}
\left\{
\begin{aligned}
y_{n} & := x_{n} - \dfrac{f(x_n)}{f'(x_n)}
\left( 1 + \left(\dfrac{f(x_n)}{f'(x_n)}\right)^5 \right), \\[0.8ex]
z_{n} & := y_{n} - \dfrac{f(y_n)}{f'(x_n)}
\left( 1 - \dfrac{f(y_n)}{f(x_n)}\right)^{-2}, \\[0.8ex]
x_{n+1} & := z_n - \dfrac{f(z_n)}{f'(x_n)}
\cdot \dfrac{1+\left(\dfrac{f(y_n)}{f(x_n)}\right)^2 + 5\left(\dfrac{f(y_n)}{f(x_n)}\right)^4
+ \dfrac{f(z_n)}{f(y_n)}}
{\left( 1 - \dfrac{f(y_n)}{f(x_n)} - \dfrac{f(z_n)}{f(x_n)} \right)^2} .
\end{aligned}
\right.
\end{equation}

\paragraph{Method 6:}
The method from Thukral and Petkovi\'{c} \cite{Thukral} is given by
\begin{equation}
\label{o5}
\left\{
\begin{aligned}
y_{n} & := x_{n} - \dfrac{f(x_n)}{f'(x_n)}, \\[0.8ex]
z_{n} & := y_{n} - \dfrac{f(y_n)}{f'(x_n)}\cdot
\dfrac{f(x_n)+\beta f(y_n)}{f(x_n)+(\beta-2)f(y_n)}, \qquad \beta\in \mathbb{R}, \\[0.8ex]
x_{n+1} & := z_n - \dfrac{f(z_n)}{f'(x_n)} \cdot
\left(\varphi(t_n)+\psi(s_n)+\omega(u_n)\right),
\end{aligned}
\right.
\end{equation}
where weight functions are
\begin{equation*}
\varphi(t_n) = \left(1+\dfrac{t_n}{1-2t_n}\right)^2,
\qquad \psi(s_n) = \dfrac{s_n}{1-\alpha s_n},
\quad \alpha \in\mathbb{R},\qquad \omega(u_n)=4u_n,
\end{equation*}
and $t_n=\frac{f(y_n)}{f(x_n)}$, $s_n=\frac{f(z_n)}{f(y_n)}$
and $u_n=\frac{f(z_n)}{f(x_n)}$. We will use $\beta=0$ and
$\alpha=1$ in the numerical experiments of this paper.

%--------------------------------------------------------------%
\section{Numerical examples}
\label{sec:examples}
%--------------------------------------------------------------%

The new three-point method~\eqref{a1} is tested on several
nonlinear equations. To obtain high accuracy and avoid the loss of
significant digits, we employed multi-precision arithmetic with
20\,000 significant decimal digits in the programming package
Mathematica.
% (we have used version~8).
% \cite{Hazrat}.

We are going to perform numerical experiments with the four test functions
$f_j$, $j=1,\dots,4$, which appear in Table~\ref{table1}. We are going to
reach the given root $x^{\ast}$ starting with the mentioned $x_0$ for the
four functions and the six methods of convergence order eight.

\begin{table}[htb!]
\begin{center}
\begin{tabular}{lcc}
   \toprule
    test function $f_j$ & root $x^{\ast}$ & initial guess $x_0$
  \\ \midrule
    $f_1(x) = \ln (1+x^2)+e^{x^2-3x}\sin x$ & $0$ & $0.35$
  \\[0.5ex]
  $f_2(x) = 1+e^{2+x-x^2}+x^3-\cos(1+x)$ & $-1$ & $-0.3$
  \\[0.5ex]
   $f_3(x) = (1+x^2)\cos\frac{\pi x}{2}+\frac{\ln(x^2+2x+2)}{1+x^2}$ & $-1$ & $-1.1$
  \\[0.5ex]
   $f_4(x) = x^4+\sin\frac{\pi}{x^2}-5$ & $\sqrt{2}$ & $1.5$
  \\[0.5ex]
  \bottomrule
\end{tabular}
\caption{Test functions $f_1, \dots, f_4$, root
$x^{\ast}$, and initial guess $x_0$.}
\label{table1}
\end{center}
\end{table}

In order to test our proposed method~\eqref{a1} and
compare it with the methods \eqref{o1}--\eqref{o5},
we compute the error, the computational
order of convergence (COC) by the approximate formula~\cite{coc}
\begin{equation}
\label{coc}
\textup{COC} \approx \frac{\ln|(x_{n+1}-x^{\ast})/(x_{n}-x^{*})|}{\ln|(x_{n}-x^{*})/(x_{n-1}-x^{*})|},
\end{equation}
and the approximated computational order of convergence (ACOC) by
the formula~\cite{acoc}
\begin{equation}
\label{acoc}
\textup{ACOC} \approx \frac{\ln|(x_{n+1}-x_{n})/(x_{n}-x_{n-1})|}{\ln|(x_{n}-x_{n-1})/(x_{n-1}-x_{n-2})|}.
\end{equation}

It is worth noting that COC has been used in the recent years.
Nevertheless, ACOC is more practical because it does not require to know the
root~$x^{\ast}$. See~\cite{acoc-et-al} for a comparison among several
convergence orders. Note that these formulas may result for particular
examples in convergence orders which are higher than expected. The reason
is that the error equation~\eqref{erreq} contains problem-dependent
coefficients which may vanish for some nonlinear functions $f$. However,
the formulas~\eqref{coc} and~\eqref{acoc} will provide for a ``random''
example good approximations for the convergence order of the method.

%On the other hand, it is nice to note that, given an iterative method,
%computing COC or ACOC on several examples is a good experiment to check
%theoretical errors in the deduction of the method and to check practical
%errors in the implementation of the method in a computer.
%For a general problem it will be difficult that COC or ACOC approach
%the theoretical order of convergence by chance.

We have used both COC and ACOC to check the accuracy of the considered
methods. Note that both COC and ACOC give already for small values of $n$
good experimental approximations to convergence order.

%In Table~\ref{table2} our new three-point methods \eqref{e1}, \eqref{e2} and
%\eqref{e3} are tested on the four nonlinear equations $f_j(x)=0$, $j=1,2,3,4$.
%Notice that, to estimate the COC and the ACOC, it has been enough to use
%$n=3$ or $4$ in \eqref{coc} and \eqref{acoc} to get excellent approximations
%of the order of convergence.

The comparison of our method~\eqref{a1} with the methods
\eqref{o1}--\eqref{o5} applied to the four nonlinear equations $f_j(x)=0$,
$j=1,\dots,4,$ are presented in in Table~\ref{table2-3}. We
abbreviate~\eqref{a1} by M1 and \eqref{o1}--\eqref{o5} as M2--M6,
respectively. The computational convergence order COC and ACOC are given
$n=3$. Note that they are for all problems and methods in excellent with
the theoretical order of convergence.

%%% We can write both tables together using \small
\begin{table}[htb!]\small
\begin{center}
\begin{tabular}{cllllll}
 \toprule
  & \multicolumn{1}{c}{M1} & \multicolumn{1}{c}{M2}
  & \multicolumn{1}{c}{M3} & \multicolumn{1}{c}{M4} & \multicolumn{1}{c}{M5}& \multicolumn{1}{c}{M6} \\
\midrule
$f_1$, $x_0=0.35$\\
$|x_{1}-x^{\ast}|$ & $0.610\mathrm{e}{-}6$ & $0.721\mathrm{e}{-}4$
& $0.893\mathrm{e}{-}4$ & $0.753\mathrm{e}{-}4$ & $0.347\mathrm{e}{-}3$& $0.328\mathrm{e}{-}3$ \\
$|x_{2}-x^{\ast}|$ & $0.319\mathrm{e}{-}46$& $0.230\mathrm{e}{-}30$
& $0.126\mathrm{e}{-}30$ & $0.619\mathrm{e}{-}31$ & $0.471\mathrm{e}{-}25$ & $0.256\mathrm{e}{-}25$\\
$|x_{3}-x^{\ast}|$ & $0.179\mathrm{e}{-}368$ &
  $0.252\mathrm{e}{-}242$
& $0.200\mathrm{e}{-}245$ & $0.128\mathrm{e}{-}247$ & $0.546\mathrm{e}{-}200$& $0.345\mathrm{e}{-}202$ \\
COC &  $8.0000$ & $8.0000$ & $8.0000$ & $8.0000$ & $8.0000$ & $8.0000$\\
ACOC & $8.0000$ & $7.9999$ & $7.9999$ & $7.9999$ & $7.9999$ & $7.9999$\\%[0.5ex]
\midrule
%
$f_2$, $x_0=-0.3$\\
$|x_{1}-x^{\ast}|$ & $0.248\mathrm{e}{-}3$ & $0.157\mathrm{e}{-}3$
& $0.763\mathrm{e}{-}4$ & $0.871\mathrm{e}{-}4$ & $0.411\mathrm{e}{-}3$ & $0.273\mathrm{e}{-}4$\\
$|x_{2}-x^{\ast}|$ & $0.582\mathrm{e}{-}32$& $0.119\mathrm{e}{-}33$
& $0.540\mathrm{e}{-}35$ & $0.134\mathrm{e}{-}34$ & $0.377\mathrm{e}{-}29$ & $0.321\mathrm{e}{-}38$\\
$|x_{3}-x^{\ast}|$ & $0.532\mathrm{e}{-}261$ &
 $0.138\mathrm{e}{-}274$
& $0.342\mathrm{e}{-}284$ & $0.438\mathrm{e}{-}281$ & $0.189\mathrm{e}{-}237$& $0.117\mathrm{e}{-}309$ \\
COC &  $8.0000$ &  $8.0000$ & $8.0000$ & $8.0000$ & $8.0000$& $8.0000$ \\
ACOC & $8.0000$ & $7.9998$ & $7.9999$ & $7.9999$ & $7.9999$& $7.9999$ \\%[0.5ex]
\midrule
%
$f_3$, $x_0=-1.1$\\
$|x_{1}-x^{\ast}|$ & $0.106\mathrm{e}{-}7$ & $0.614\mathrm{e}{-}8$
& $0.388\mathrm{e}{-}8$ & $0.175\mathrm{e}{-}8$ & $0.554\mathrm{e}{-}8$ & $0.100\mathrm{e}{-}7$  \\
$|x_{2}-x^{\ast}|$ & $0.482\mathrm{e}{-}63$ & $0.328\mathrm{e}{-}65$
& $0.254\mathrm{e}{-}67$ & $0.154\mathrm{e}{-}70$ & $0.426\mathrm{e}{-}66$& $0.136\mathrm{e}{-}63$ \\
$|x_{3}-x^{\ast}|$ & $0.833\mathrm{e}{-}506$ &
 $0.217\mathrm{e}{-}523$
& $0.877\mathrm{e}{-}541$ & $0.582\mathrm{e}{-}567$ & $0.528\mathrm{e}{-}531$ & $0.154\mathrm{e}{-}510$\\
COC &  $8.0000$  & $8.0000$ & $8.0000$ & $8.0000$ & $8.0000$& $8.0000$ \\
ACOC & $7.9999$  & $8.0000$ & $7.9999$ & $8.0000$ & $8.0000$& $7.9999$ \\%[0.5ex]
\midrule
%
$f_4$, $x_0=1.5$\\
$|x_{1}-x^{\ast}|$ & $0.148\mathrm{e}{-}7$ & $0.433\mathrm{e}{-}8$
& $0.327\mathrm{e}{-}10$ & $0.642\mathrm{e}{-}10$ & $0.281\mathrm{e}{-}8$ & $0.727\mathrm{e}{-}10$ \\
$|x_{2}-x^{\ast}|$ & $0.138\mathrm{e}{-}61$& $0.134\mathrm{e}{-}66$
& $0.369\mathrm{e}{-}84$ & $0.101\mathrm{e}{-}81$ & $0.341\mathrm{e}{-}68$ & $0.543\mathrm{e}{-}81$ \\
$|x_{3}-x^{\ast}|$ & $0.769\mathrm{e}{-}494$ &
 $0.116\mathrm{e}{-}534$
& $0.967\mathrm{e}{-}676$ & $0.389\mathrm{e}{-}656$ & $0.161\mathrm{e}{-}547$& $0.530\mathrm{e}{-}650$ \\
COC &  $8.0000$  & $8.0000$ & $8.0000$ & $8.0000$ & $8.0000$ & $8.0000$ \\
ACOC & $8.0000$ & $7.9999$ & $7.9999$ & $7.9999$ & $8.0000$ & $7.9999$\\
\bottomrule
\end{tabular}
\caption{Errors, COC, and ACOC for the iterative methods
\eqref{a1} and \eqref{o1}--\eqref{o5} (abbreviated as M1--M6)
applied to the find the root of test functions $f_1,\dots,f_4$
given in Table~\ref{table1}.} \label{table2-3}
\end{center}
\end{table}

%\newpage

%--------------------------------------------------------------%
\section{Dynamic behavior}
\label{sec:dynamic}
%--------------------------------------------------------------%

We have already observed that all methods converge if the initial
guess is chosen suitably. We now investigate the regions where the
initial point has to chosen in order to achieve the root. In other
words, we will numerically approximate the domain of attraction of
the zeros as a qualitative measure of how the method depends on
the choice of the initial approximation of the root. To answer
this important question on the dynamical behavior of the
algorithms, we will investigate the dynamics of the new
method~\eqref{a1} and compare it with the methods
\eqref{o1}--\eqref{o5}.

Let's recall some basic concepts such as basin of attraction. For
more details and many other examples of the study of the dynamic
behavior of iterative methods, one can consult~\cite{Amat5,
Babajee, Chicharro, Ezquerro2, Ezquerro1, Fer, GutiMaVar, Paricio,
Stewart, Varona}.

Let $Q:\mathbb{C} \to \mathbb{C}$ be a rational map on the
complex plane. For $z\in \mathbb{C} $, we define its orbit as the
set $\operatorname{orb}(z) = \{z,\,Q(z),\,Q^2(z),\dots\}$. The convergence
$\operatorname{orb}(z)\to z^{\ast}$ is understood in the sense
$\lim\limits_{k\to\infty} Q^k(z) = z^{\ast}$. A point $z_0 \in
\mathbb{C}$ is called periodic point with minimal period $m$ if
$Q^m(z_0) = z_0$ where $m$ is the smallest positive integer with this
property (and thus $\{z_0,Q(z_0),\dots, Q^{m-1}(z_0)\}$ is a cycle).
A periodic point with minimal period $1$ is called fixed
point.
%Moreover, a fixed point $z_0$ is called attracting if
%$|Q'(z_0)|<1$, repelling if $|Q'(z_0)|>1$, and neutral
%otherwise. The Julia set of a nonlinear map $Q(z)$, denoted by
%$J(Q)$, is the closure of the set of its repelling periodic
%points. The complement of $J(Q)$ is the Fatou set $F(Q)$.
Moreover, a periodic point $z_0$ with period $m$ is called attracting if
$|(Q^m)'(z_0)|<1$, repelling if $|(Q^m)'(z_0)|>1$, and neutral
otherwise. The Julia set of a nonlinear map $Q(z)$, denoted by
$J(Q)$, is the closure of the set of its repelling periodic
points. The complement of $J(Q)$ is the Fatou set $F(Q)$.

The six methods \eqref{a1} and \eqref{o1}--\eqref{o5}
provide iterative rational maps $Q$ when they are applied to
find roots of complex polynomials $p$. In particular, we
are interested in the basins of attraction of the roots of the
polynomials where the basin of attraction of a root $z^{\ast}$ is the
complex set
$\{ z_0 \in \mathbb{C} : \operatorname{orb}(z_0)  \to z^{\ast} \}$.
It is well known that the basins of attraction of the
different roots lie in the Fatou set $F(Q)$. The Julia set $J(Q)$
is, in general, a fractal and the rational map $Q$ is unstable there.

For the dynamical and graphical point of view, we take a
$600 \times 600$ grid of the square $[-3,3]\times[-3,3] \subset \mathbb{C}$
and assign a color to each point $z_0\in D$ according to the simple
root to which the corresponding orbit of the iterative method
starting from $z_0$ converges, and we mark the point as black if
the orbit does not converge to a root in the sense that after at
most $15$ iterations it has a distance to any of the roots which
is larger than $10^{-3}$. We have used only $15$ iterations because
we are using methods of convergence order eight which, if they
converge, do this very fast. The basins of attraction are distinguished
by their color.

\begin{table}[htb!]
\centering
\begin{tabular}{l @{\qquad} c}
   \toprule
    Test polynomials & Roots
   \\
   \midrule
   $p_1(z) = z^2-1$ & $1,\quad -1$
  \\[0.5ex]
   $p_2(z) = z^3-z $ & $0,\quad 1,\quad -1$
  \\[0.5ex]
   $p_3(z) = z(z^2+1)(z^2+4) $ & $0, \quad 2i, \quad -2i,\quad i, \quad -i$
  \\[0.5ex]
   $p_4(z) = (z^4-1)(z^2+2i)$ & $1, \quad i, \quad -1,\quad -i, \quad -1+i, \quad 1-i$
  \\[0.5ex]
   $p_5(z) = z^7-1$ & $e^{2k\pi i/7}, \qquad k=0,\dots,6$
  \\[0.5ex]
   $p_6(z) = (10z^5-1)(z^5+10)$ & ${\big(\tfrac{1}{10}\big)}^{1/5} e^{2k\pi i/5},
         \quad (-10)^{1/5} e^{2k\pi i/5}, \qquad k=0,\dots,4$
  \\[0.5ex]
  \bottomrule
\end{tabular}
\caption{Test polynomials $p_1(z),\dots, p_6(z)$ and their roots.}
\label{table4}
\end{table}

Different colors are used for different roots. In the basins of attraction,
the number of iterations needed to achieve the root is shown by the
brightness. Brighter color means less iteration steps. Note that black
color denotes lack of convergence to any of the roots. This happens, in
particular, when the method converges to a fixed point that is not a root or
if it ends in a periodic cycle or at infinity. Actually and although we have
not done it in this paper, infinity can be considered an ordinary point if
we consider the Riemann sphere instead of the complex plane. In this case,
we can assign a new ``ordinary color'' for the basin of attraction of
infinity. Details for this idea can be found in~\cite{Paricio}.

\begin{figure}
\centering
\includegraphics[width=0.9\textwidth]{p1.pdf}
\caption{Comparison of basins of attraction of methods \eqref{a1}
and \eqref{o1}--\eqref{o5} for the test problem $p_1(z)=
z^2-1=0$.} \label{fig:figure1}
\end{figure}

\begin{figure}
\centering
\includegraphics[width=0.9\textwidth]{p2.pdf}
\caption{Comparison of basins of attraction of methods \eqref{a1}
and \eqref{o1}--\eqref{o5} for the test problem $p_2(z)=
z^3-z=0$.} \label{fig:figure2}
\end{figure}


Basins of attraction for the six methods \eqref{a1} and
\eqref{o1}--\eqref{o5} for the six test problems $p_i(z)=0$,
$i=1,\dots,6$, are illustrated in
Figures~\ref{fig:figure1}--\ref{fig:figure6} from left to right
and from top to bottom.
%$p_2(z)=0$ in Figure~\ref{fig:figure2}, for $p_3(z)=0$ in
%Figure~\ref{fig:figure3}, for $p_4(z)=0$ in
%Figure~\ref{fig:figure4}, for $p_5(z)=0$ in
%Figure~\ref{fig:figure5} and for $p_6(z)=0$ in
%Figure~\ref{fig:figure6}.

\begin{figure}
\centering
\includegraphics[width=0.9\textwidth]{p3.pdf}
\caption{Comparison of basins of attraction of methods \eqref{a1}
and \eqref{o1}--\eqref{o5} for the test problem $p_3(z)=
z(z^2+1)(z^2+4)=0$.} \label{fig:figure3}
\end{figure}

\begin{figure}
\centering
\includegraphics[width=0.9\textwidth]{p4.pdf}
\caption{Comparison of basins of attraction of methods \eqref{a1}
and \eqref{o1}--\eqref{o5} for the test problem $p_4(z)=
(z^4-1)(z^2+2i)=0$.} \label{fig:figure4}
\end{figure}

From the pictures, we can easily judge the behavior and suitability
of any method depending on the circumstances.
% This is good entertainment.
If we choose an initial point $z_0$
in a zone where different basins of attraction touch each other, it is
impossible to predict which root is going to be reached by the
iterative method that starts in~$z_0$. Hence, $z_0$ is
not a good choice. Both the black zones and the zones with a lot
of colors are not suitable for choosing the initial guess $z_0$ if
precise root should be reached. Although the most attractive pictures appear
when we have very intricate frontiers between basins of attraction, they
correspond to the cases where the dynamic behavior of the method is more
unpredictable and the method is more demanding with respect to choice of
the initial point.

\begin{figure}
\centering
\includegraphics[width=0.9\textwidth]{p5.pdf}
\caption{Comparison of basins of attraction of methods \eqref{a1}
and \eqref{o1}--\eqref{o5} for the test problem $p_5(z)=
z^7-1=0$.} \label{fig:figure5}
\end{figure}

\begin{figure}
\centering
\includegraphics[width=0.9\textwidth]{p6.pdf}
\caption{Comparison of basins of attraction of methods \eqref{a1}
and \eqref{o1}--\eqref{o5} for the test problem $p_6(z)=
(10z^5-1)(z^5+10)=0$.} \label{fig:figure6}
\end{figure}

Finally, we have included in Table~\ref{table5} the results of
some numerical experiments to measure the behavior of the five
iterative methods \eqref{a1} and \eqref{o1}--\eqref{o5} in finding
the roots of the test polynomials $p_j$, $j=1,\dots,6$. To
compute the data of this table, we have applied the six methods to
the six polynomials, starting at an initial points $z_0$ on a $600
\times 600$ grid in the rectangle $[-3,3] \times [-3,3]$ of the
complex plane. The same way was used in
Figures~\ref{fig:figure1}--\ref{fig:figure6} to show the basins of
attraction of the roots. In particular, we decide again that an
initial point $z_0$ has reached a root $z^{\ast}$ when its distance to
$z^{\ast}$ is less than $10^{-3}$ (in this case $z_0$ is in the basin
of attraction of~$z^{\ast}$) and we decide that the method starting in
$z_0$ diverges when no root is found in a maximum of $15$
iterations of the method. We say in this case that $z_0$ is a
``nonconvergent point''. In Table~\ref{table5}, we have
abbreviated the methods \eqref{a1} and \eqref{o1}--\eqref{o5} as
M1--M6, respectively. The column I/P shows the mean of iterations
per point until the algorithm decides that a root has been reached
or the point is declared nonconvergent. The column NC shows the
percentage of nonconvergent points, indicated as black zones in
the pictures of Figures~\ref{fig:figure1}--\ref{fig:figure6}.
It is clear that the nonconvergent points have a
great influence on the values of I/P since these points contribute
always with the maximum number of $15$ allowed iterations. In
contrast, ``convergent points'' are reached usually very fast due
to the fact that we are dealing with methods of order eight. To
reduce the effect of nonconvergent points, we have included the
column I$_\textrm{C}$/C which shows the mean number of iterations
per convergent point. If we use either the columns I/P or the
column I$_\textrm{C}$/C to compare the performance of the
iterative methods, we clearly obtain different conclusions.

%\newpage

\begin{table}[htbp!]
\begin{center}
\begin{tabular}{cclll}
\toprule
Polynomial & Method & I/P & NC (\%) & I$_\textrm{C}$/C \\
\midrule
$p_1(z)$ & M1 & 2.21 & 0.00111 & 2.21 \\
         & M2 & 2.19 & 0. & 2.19 \\
         & M3 & 2.16 & 0. & 2.16 \\
         & M4 & 2.11 & 0. & 2.11 \\
         & M5 & 6.01 & 71.0 & 2.09 \\
         & M6 & 2.30 & 0.0256 & 2.30 \\
\midrule
$p_2(z)$ & M1 & 2.90 & 0.125 & 2.89 \\
         & M2 & 2.88 & 0.00111 & 2.88 \\
         & M3 & 2.82 & 0.00444 & 2.82 \\
         & M4 & 2.73 & 0. & 2.73 \\
         & M5 & 4.32 & 27.5 & 2.81 \\
         & M6 & 3.21 & 0.216 & 3.18 \\
\midrule
$p_3(z)$ & M1 & 3.22 & 0.802 & 3.13 \\
         & M2 & 2.99 & 0.0178 & 2.99 \\
         & M3 & 2.94 & 0.0367 & 2.94 \\
         & M4 & 2.82 & 0. & 2.82 \\
         & M5 & 3.28 & 5.47 & 2.99 \\
         & M6 & 3.42 & 1.08 & 3.30 \\
%\midrule
%$p_4(z)$ & M1 & 6.00 & 17.7 & 4.06 \\
%         & M2 & 4.06 & 0.819 & 3.97 \\
%         & M3 & 4.21 & 1.82 & 4.01 \\
%         & M4 & 3.95 & 4.40 & 3.44 \\
%         & M5 & 4.44 & 20.0 & 3.57 \\
%         & M6 & 5.17 & 9.35 & 4.15 \\
%\midrule
%$p_5(z)$ & M1 & 6.89 & 24.4 & 4.27 \\
%         & M2 & 4.81 & 3.33 & 4.46 \\
%         & M3 & 5.07 & 5.70 & 4.46 \\
%         & M4 & 4.59 & 7.05 & 3.80 \\
%         & M5 & 5.02 & 21.4 & 4.02 \\
%         & M6 & 5.78 & 13.3 & 4.36 \\
%\midrule
%$p_6(z)$ & M1 & 6.72 & 18.2 & 4.88 \\
%         & M2 & 4.68 & 2.29 & 4.44 \\
%         & M3 & 4.89 & 4.04 & 4.46 \\
%         & M4 & 4.44 & 3.96 & 4.00 \\
%         & M5 & 5.26 & 11.8 & 4.71 \\
%         & M6 & 5.45 & 8.49 & 4.56 \\
\bottomrule
\end{tabular}%
\quad
\begin{tabular}{cclll}
\toprule
Polynomial & Method & I/P & NC (\%) & I$_\textrm{C}$/C \\
%\midrule
%$p_1(z)$ & M1 & 2.21 & 0.00111 & 2.21 \\
%         & M2 & 2.19 & 0. & 2.19 \\
%         & M3 & 2.16 & 0. & 2.16 \\
%         & M4 & 2.11 & 0. & 2.11 \\
%         & M5 & 6.01 & 71.0 & 2.09 \\
%         & M6 & 2.30 & 0.0256 & 2.30 \\
%\midrule
%$p_2(z)$ & M1 & 2.90 & 0.125 & 2.89 \\
%         & M2 & 2.88 & 0.00111 & 2.88 \\
%         & M3 & 2.82 & 0.00444 & 2.82 \\
%         & M4 & 2.73 & 0. & 2.73 \\
%         & M5 & 4.32 & 27.5 & 2.81 \\
%         & M6 & 3.21 & 0.216 & 3.18 \\
%\midrule
%$p_3(z)$ & M1 & 3.22 & 0.802 & 3.13 \\
%         & M2 & 2.99 & 0.0178 & 2.99 \\
%         & M3 & 2.94 & 0.0367 & 2.94 \\
%         & M4 & 2.82 & 0. & 2.82 \\
%         & M5 & 3.28 & 5.47 & 2.99 \\
%         & M6 & 3.42 & 1.08 & 3.30 \\
\midrule
$p_4(z)$ & M1 & 6.00 & 17.7 & 4.06 \\
         & M2 & 4.06 & 0.819 & 3.97 \\
         & M3 & 4.21 & 1.82 & 4.01 \\
         & M4 & 3.95 & 4.40 & 3.44 \\
         & M5 & 4.44 & 20.0 & 3.57 \\
         & M6 & 5.17 & 9.35 & 4.15 \\
\midrule
$p_5(z)$ & M1 & 6.89 & 24.4 & 4.27 \\
         & M2 & 4.81 & 3.33 & 4.46 \\
         & M3 & 5.07 & 5.70 & 4.46 \\
         & M4 & 4.59 & 7.05 & 3.80 \\
         & M5 & 5.02 & 21.4 & 4.02 \\
         & M6 & 5.78 & 13.3 & 4.36 \\
\midrule
$p_6(z)$ & M1 & 6.72 & 18.2 & 4.88 \\
         & M2 & 4.68 & 2.29 & 4.44 \\
         & M3 & 4.89 & 4.04 & 4.46 \\
         & M4 & 4.44 & 3.96 & 4.00 \\
         & M5 & 5.26 & 11.8 & 4.71 \\
         & M6 & 5.45 & 8.49 & 4.56 \\
\bottomrule
\end{tabular}
\caption{Measures of convergence of the iterative methods
\eqref{a1} and \eqref{o1}--\eqref{o5} (abbreviated as M1--M6)
applied to find the roots of the polynomials $p_j(z)$,
$j=1,\dots,6$.} \label{table5}
\end{center}
\end{table}

%--------------------------------------------------------------%
\section{Conclusion}
\label{sec:conclusion}
%--------------------------------------------------------------%

We have introduced a new optimal three-point method without memory
for approximating a simple root of a given nonlinear equation
which use only four function evaluations each iteration and result
in a method of convergence order eight. Therefore, Kung and
Traub's conjecture is supported. Numerical examples and
comparisons with some existing eighth-order methods are included
and confirm the theoretical results. The numerical experience
suggests that the new method is a valuable alternative for solving
these problems and finding simple roots. We used the basins of
attraction for comparing the iterative algorithms and we have
included some tables with comparative results.


%%%%%%%%%%%%%%%%%%%%%% ACKNOWLEDGMENTS %%%%%%%%%%%%%%%%%%%%%%%%%%
\paragraph*{Acknowledgments.}
The research of the fourth author is supported by grant
MTM2015-65888-C4-4 from DGI (Spanish Government).


%--------------------------------------------------------------%
\begin{thebibliography}{99}
%--------------------------------------------------------------%

\small

%\bibitem{Amat1}
%Amat, S., Busquier, S., Plaza, S.:
%Iterative root-finding methods,
%Unpublished report (2004).

%% The final publication of this "Unpublished report" is the following:

% MR2127479
%\bibitem{Amat1}
%Amat, S., Busquier, S., Plaza, S.: Review of some iterative
%root-finding methods from a dynamical point of view, Sci. Ser. A
%Math. Sci. (N.S.) 10, 3--35 (2004).
%% SCIENTIA, Series A: Math. Sciences {10}, 3--35 (2004).
% http://www.mat.utfsm.cl/scientia/
% http://www.mat.utfsm.cl/scientia/archivos/vol10/part_2.pdf

% MR2139283
%\bibitem{Amat4} Amat, S., Busquier, S., Plaza, S.: Dynamics of
%the King and Jarratt iterations, Aequationes Math. 69, 212--223
%(2005).

\bibitem{Amat5}
Amat, S., Busquier, S., Magre\~n\'an, \'A.A.: Reducing chaos and
bifurcations in Newton-type methods, Abstr. Appl. Anal. 2013,
Art.\ ID 726701, 10 pages (2013).

\bibitem{Babajee}
Babajee, D.K.R., Cordero, A., Soleymani, F., Torregrosa, J.R.: On
improved three-step schemes with high efficiency index and their
dynamics, Numer. Algorithms 65, 153--169 (2014).

% MR2490175
\bibitem{Bi}
Bi, W., Ren, H., Wu, Q.: Three-step iterative methods with
eighth-order convergence for solving nonlinear equations, J.
Comput. Appl. Math. 225, 105--112 (2009).

\bibitem{Chicharro}
Chicharro, F., Cordero, A., Guti\'errez, J.M., Torregrosa, J.R.:
Complex dynamics of derivative-free methods for nonlinear
equations, Appl. Math. Comput. 219, 7023--7035 (2013).

\bibitem{Chun1}
Chun, C., Lee, M.Y.: A new optimal eighth-order family of
iterative methods for the solution of nonlinear equations, Appl.
Math. Comput. 223, 506--519 (2013).


\bibitem{acoc}
Cordero, A., Torregrosa, J.R.: Variants of Newton�'s method using
fifth-order quadrature formulas, Appl. Math. Comput. 190, 686--698
(2007).

\bibitem{Ezquerro2}
Ezquerro, J.A., Hern\'andez, M.A.: An improvement of the region of
accessibility of Chebyshev's method from Newton's method, Math.
Comp. 78, 1613--1627 (2009).

\bibitem{Ezquerro1}
Ezquerro, J.A., Hern\'andez, M.A.: An optimization of Chebyshev's
method, J. Complexity 25, 343--361 (2009).

\bibitem{Fer}
Ferrara, M., Sharifi, S., Salimi, M.: Computing multiple zeros by
using a parameter in Newton--Secant method, SeMA Journal, accepted
(2016).

\bibitem{acoc-et-al}
Grau-S\'anchez, M., Noguera, M., Guti\'errez, J.M.: On some
computational orders of convergence, Appl. Math. Lett. 23(4),
472--478 (2010).

\bibitem{GutiMaVar}
Guti\'errez, J.M., Magre\~n\'an, \'A.A., Varona, J.L.: The
``Gauss-Seidelization'' of iterative methods for solving nonlinear
equations in the complex plane, Appl. Math. Comput. 218,
2467--2479 (2011).

%\bibitem{Hazrat}
%Hazrat, R.:
%Mathematica\textsuperscript{\textregistered}: A Problem-Centered Approach,
%Springer-Verlag (2010).

\bibitem{Paricio}
Hern\'andez-Paricio, L.J., Mara\~n\'on-Grandes, M.,
Rivas-Rodr\'iguez, M.T.: Plotting basins of end points of rational
maps with Sage, Tbil. Math. J. 5(2), 71--99 (2012).
% http://tcms.org.ge/Journals/TMJ/

% http://www.ams.org/journals/mcom/1966-20-095/S0025-5718-66-99924-8/
\bibitem{Jarratt}
Jarratt, P.: Some fourth order multipoint iterative methods for
solving equations, Math. Comp. 20, 434--437 (1966).

% MR0343585
\bibitem{King}
King, R.F.: A family of fourth order methods for nonlinear
equations, SIAM J. Numer. Anal. 10, 876--879 (1973).

% MR0353657
\bibitem{Kung}
Kung, H.T., Traub, J.F.: Optimal order of one-point and multipoint
iteration, J. Assoc. Comput. Mach. 21, 634--651 (1974).

% MR3304844
\bibitem{Lotfi2}
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 68, 261--288 (2015).

\bibitem{Neta0}
Neta, B.: On a family of multipoint methods for nonlinear
equations, Internat. J. Comput. Math. 9, 353--361 (1981).

% MR3146342
\bibitem{Neta1}
Neta, B., Chun, C., Scott, M.: Basins of attraction for optimal
eighth order methods to find simple roots of nonlinear equations,
Appl. Math. Comput. 227, 567--592 (2014).

% MR0216746
\bibitem{Ostrowski}
Ostrowski, A.M.:
Solution of Equations and Systems of Equations, 2nd ed.,
Academic Press, New York (1966).

\bibitem{Petkovic}
Petkovi\'c, M.S., Neta, B., Petkovi\'c, L.D., D\v{z}uni\'c, J.:
Multipoint Methods for Solving Nonlinear Equations,
Elsevier/Academic Press, Amsterdam (2013).

% MR2838167
%\bibitem{Scott}
%Scott, M., Neta, B., Chun, C.: Basin attractors for various
%methods, Appl. Math. Comput. 218, 2584--2599 (2011).

\bibitem{Sharifi1}
Sharifi, S., Ferrara, M., Salimi, M., Siegmund, S.: New
modification of Maheshwari method with optimal eighth order of
convergence for solving nonlinear equations, preprint (2015).
% Under revision (2015).

\bibitem{Sharifi2}
Sharifi, S., Siegmund, S., Salimi, M.: Solving nonlinear equations
by a derivative-free form of the King's family with memory,
Calcolo, doi: 10.1007/s10092-015-0144-1 (2015).

\bibitem{Sharifi3}
Sharifi, S., Salimi, M., Siegmund, S., Lotfi, T.: A new class of
optimal four-point methods with convergence order $16$ for solving
nonlinear equations, Math. Comput. Simulation 119, 69--90 (2016).

\bibitem{Sharma1}
Sharma, J.R., Sharma, R.: A new family of modified Ostrowski's
methods with accelerated eighth order convergence, Numer.
Algorithms 54, 445--458 (2010).

\bibitem{Stewart}
Stewart, B.D.:
Attractor Basins of Various Root-Finding Methods,
M.S. thesis, Naval Postgraduate School,
% Department of Applied Mathematics,
Monterey, CA (2001).
% http://calhoun.nps.edu/bitstream/handle/10945/10997/ADA397512.pdf
% http://hdl.handle.net/10945/10997

% MR2577765
\bibitem{Thukral}
Thukral, R., Petkovi\'c, M.S.: A family of three-point methods of
optimal order for solving nonlinear equations, J. Comput. Appl.
Math. 233, 2278--2284 (2010).

% MR0169356
\bibitem{Traub}
Traub, J.F.:
Iterative Methods for the Solution of Equations,
Prentice Hall, Englewood Cliffs, N.J. (1964).

\bibitem{Varona}
Varona, J.L.: Graphic and numerical comparison between iterative
methods, Math. Intelligencer 24(1), 37--46 (2002).

% MR0918313
%\bibitem{Vrscay}
%Vrscay, E.R., Gilbert, W.J.: Extraneous fixed points, basin
%boundaries and chaotic dynamics for Schr\"oder and K\"onig
%rational iteration functions, Numer. Math. 52, 1--16 (1988).

% MR2610375
\bibitem{Wang}
Wang, X., Liu, L.: New eighth-order iterative methods for solving
nonlinear equations, J. Comput. Appl. Math. 234, 1611--1620
(2010).

\bibitem{coc}
Weerakoon, S., Fernando, T.G.I.: A variant of Newton's method with
accelerated third-order convergence, Appl. Math. Lett. 13(8),
87--93 (2000).

\end{thebibliography}

\end{document}

1602.07026.extracted_bib
{Amat1 %Amat, S., Busquier, S., Plaza, S.: %Iterative root-finding methods, %Unpublished report (2004). %% The final publication of this "Unpublished report" is the following: % MR2127479 %
1602.07026.extracted_bib
{Amat1 %Amat, S., Busquier, S., Plaza, S.: Review of some iterative %root-finding methods from a dynamical point of view, Sci. Ser. A %Math. Sci. (N.S.) 10, 3--35 (2004). %% SCIENTIA, Series A: Math. Sciences {10, 3--35 (2004). % http://www.mat.utfsm.cl/scientia/ % http://www.mat.utfsm.cl/scientia/archivos/vol10/part_2.pdf % MR2139283 %
1602.07026.extracted_bib
{Amat4 Amat, S., Busquier, S., Plaza, S.: Dynamics of %the King and Jarratt iterations, Aequationes Math. 69, 212--223 %(2005).
1602.07026.extracted_bib
{Amat5 Amat, S., Busquier, S., Magre\~n\'an, \'A.A.: Reducing chaos and bifurcations in Newton-type methods, Abstr. Appl. Anal. 2013, Art.\ ID 726701, 10 pages (2013).
1602.07026.extracted_bib
{Babajee Babajee, D.K.R., Cordero, A., Soleymani, F., Torregrosa, J.R.: On improved three-step schemes with high efficiency index and their dynamics, Numer. Algorithms 65, 153--169 (2014). % MR2490175
1602.07026.extracted_bib
{Bi Bi, W., Ren, H., Wu, Q.: Three-step iterative methods with eighth-order convergence for solving nonlinear equations, J. Comput. Appl. Math. 225, 105--112 (2009).
1602.07026.extracted_bib
{Chicharro Chicharro, F., Cordero, A., Guti\'errez, J.M., Torregrosa, J.R.: Complex dynamics of derivative-free methods for nonlinear equations, Appl. Math. Comput. 219, 7023--7035 (2013).
1602.07026.extracted_bib
{Chun1 Chun, C., Lee, M.Y.: A new optimal eighth-order family of iterative methods for the solution of nonlinear equations, Appl. Math. Comput. 223, 506--519 (2013).
1602.07026.extracted_bib
{acoc Cordero, A., Torregrosa, J.R.: Variants of Newton�'s method using fifth-order quadrature formulas, Appl. Math. Comput. 190, 686--698 (2007).
1602.07026.extracted_bib
{Ezquerro2 Ezquerro, J.A., Hern\'andez, M.A.: An improvement of the region of accessibility of Chebyshev's method from Newton's method, Math. Comp. 78, 1613--1627 (2009).
1602.07026.extracted_bib
{Ezquerro1 Ezquerro, J.A., Hern\'andez, M.A.: An optimization of Chebyshev's method, J. Complexity 25, 343--361 (2009).
1602.07026.extracted_bib
{Fer Ferrara, M., Sharifi, S., Salimi, M.: Computing multiple zeros by using a parameter in Newton--Secant method, SeMA Journal, accepted (2016).
1602.07026.extracted_bib
{acoc-et-al Grau-S\'anchez, M., Noguera, M., Guti\'errez, J.M.: On some computational orders of convergence, Appl. Math. Lett. 23(4), 472--478 (2010).
1602.07026.extracted_bib
{GutiMaVar Guti\'errez, J.M., Magre\~n\'an, \'A.A., Varona, J.L.: The ``Gauss-Seidelization'' of iterative methods for solving nonlinear equations in the complex plane, Appl. Math. Comput. 218, 2467--2479 (2011). %
1602.07026.extracted_bib
{Hazrat %Hazrat, R.: %Mathematica\textsuperscript{\textregistered: A Problem-Centered Approach, %Springer-Verlag (2010).
1602.07026.extracted_bib
{Paricio Hern\'andez-Paricio, L.J., Mara\~n\'on-Grandes, M., Rivas-Rodr\'iguez, M.T.: Plotting basins of end points of rational maps with Sage, Tbil. Math. J. 5(2), 71--99 (2012). % http://tcms.org.ge/Journals/TMJ/ % http://www.ams.org/journals/mcom/1966-20-095/S0025-5718-66-99924-8/
1602.07026.extracted_bib
{Jarratt Jarratt, P.: Some fourth order multipoint iterative methods for solving equations, Math. Comp. 20, 434--437 (1966). % MR0343585
1602.07026.extracted_bib
{King King, R.F.: A family of fourth order methods for nonlinear equations, SIAM J. Numer. Anal. 10, 876--879 (1973). % MR0353657
1602.07026.extracted_bib
{Kung Kung, H.T., Traub, J.F.: Optimal order of one-point and multipoint iteration, J. Assoc. Comput. Mach. 21, 634--651 (1974). % MR3304844
1602.07026.extracted_bib
{Lotfi2 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 68, 261--288 (2015).
1602.07026.extracted_bib
{Neta0 Neta, B.: On a family of multipoint methods for nonlinear equations, Internat. J. Comput. Math. 9, 353--361 (1981). % MR3146342
1602.07026.extracted_bib
{Neta1 Neta, B., Chun, C., Scott, M.: Basins of attraction for optimal eighth order methods to find simple roots of nonlinear equations, Appl. Math. Comput. 227, 567--592 (2014). % MR0216746
1602.07026.extracted_bib
{Ostrowski Ostrowski, A.M.: Solution of Equations and Systems of Equations, 2nd ed., Academic Press, New York (1966).
1602.07026.extracted_bib
{Petkovic Petkovi\'c, M.S., Neta, B., Petkovi\'c, L.D., D\v{zuni\'c, J.: Multipoint Methods for Solving Nonlinear Equations, Elsevier/Academic Press, Amsterdam (2013). % MR2838167 %
1602.07026.extracted_bib
{Scott %Scott, M., Neta, B., Chun, C.: Basin attractors for various %methods, Appl. Math. Comput. 218, 2584--2599 (2011).
1602.07026.extracted_bib
{Sharifi1 Sharifi, S., Ferrara, M., Salimi, M., Siegmund, S.: New modification of Maheshwari method with optimal eighth order of convergence for solving nonlinear equations, preprint (2015). % Under revision (2015).
1602.07026.extracted_bib
{Sharifi2 Sharifi, S., Siegmund, S., Salimi, M.: Solving nonlinear equations by a derivative-free form of the King's family with memory, Calcolo, doi: 10.1007/s10092-015-0144-1 (2015).
1602.07026.extracted_bib
{Sharifi3 Sharifi, S., Salimi, M., Siegmund, S., Lotfi, T.: A new class of optimal four-point methods with convergence order $16$ for solving nonlinear equations, Math. Comput. Simulation 119, 69--90 (2016).
1602.07026.extracted_bib
{Sharma1 Sharma, J.R., Sharma, R.: A new family of modified Ostrowski's methods with accelerated eighth order convergence, Numer. Algorithms 54, 445--458 (2010).
1602.07026.extracted_bib
{Stewart Stewart, B.D.: Attractor Basins of Various Root-Finding Methods, M.S. thesis, Naval Postgraduate School, % Department of Applied Mathematics, Monterey, CA (2001). % http://calhoun.nps.edu/bitstream/handle/10945/10997/ADA397512.pdf % http://hdl.handle.net/10945/10997 % MR2577765
1602.07026.extracted_bib
{Thukral Thukral, R., Petkovi\'c, M.S.: A family of three-point methods of optimal order for solving nonlinear equations, J. Comput. Appl. Math. 233, 2278--2284 (2010). % MR0169356
1602.07026.extracted_bib
{Traub Traub, J.F.: Iterative Methods for the Solution of Equations, Prentice Hall, Englewood Cliffs, N.J. (1964).
1602.07026.extracted_bib
{Varona Varona, J.L.: Graphic and numerical comparison between iterative methods, Math. Intelligencer 24(1), 37--46 (2002). % MR0918313 %
1602.07026.extracted_bib
{Vrscay %Vrscay, E.R., Gilbert, W.J.: Extraneous fixed points, basin %boundaries and chaotic dynamics for Schr\"oder and K\"onig %rational iteration functions, Numer. Math. 52, 1--16 (1988). % MR2610375
1602.07026.extracted_bib
{Wang Wang, X., Liu, L.: New eighth-order iterative methods for solving nonlinear equations, J. Comput. Appl. Math. 234, 1611--1620 (2010).
1602.07026.extracted_bib
{coc Weerakoon, S., Fernando, T.G.I.: A variant of Newton's method with accelerated third-order convergence, Appl. Math. Lett. 13(8), 87--93 (2000).