Consider well-known three-step Newton methods [26] of convergence order eight as follows:
$$ v^{(t)}=u^{(t)}-\frac{f(u^{(t)})}{f^{\prime }(u^{(t)})}, $$
(9)
where \(u^{(t)}=s^{(t)}-\frac{f(s^{(t)})}{f^{\prime }(s^{(t)})}\) and \(s^{(t)}=r^{(t)}-\frac{f(r^{(t)})}{f^{\prime }(r^{(t)})}\). Taking Weierstrass correction [24]
$$ \frac{f(r_{i}^{(t)})}{f^{\prime }(r_{i}^{(t)})}=w\bigl(r_{i}^{(t)}\bigr)= \frac{f(r_{i}^{(t)})}{\prod_{\substack{j=1\\ j\neq i}}^{n} (r_{i}^{(t)}-r_{j}^{(t)})}, $$
(10)
and replacing \(r_{j}^{(t)}=\overset{\ast }{s}{}_{j}^{(t)}\) in (10), we have
$$ \frac{f(r_{i}^{(t)})}{f^{\prime }(r_{i}^{(t)})}= \frac{f(r_{i}^{(t)})}{\prod_{\substack{j=1\\ j\neq i}}^{n}(r_{i}^{(t)}-\overset{\ast }{s}{}_{j}^{(t)})}, $$
(11)
where \(\overset{\ast }{s}{}_{j}^{(t)}=r_{j}^{(t)}- \frac{\alpha ( f(r_{j}^{(t)}) ) ^{2}}{f(r_{j}^{(t)}+\alpha f(r_{j}^{(t)}))-f(r_{j}^{(t)})}\). Using \(\frac{f(r_{i}^{(t)})}{f^{\prime }(r_{i}^{(t)})}= \frac{f(r_{i}^{(t)})}{\prod_{\substack{j=1\\ j\neq i}}^{n}(r_{i}^{(t)}-\overset{\ast }{s}{}_{j}^{(t)})}\), \(\frac{f(s_{i}^{(t)})}{f^{\prime }(s_{i}^{(t)})}= \frac{f(s_{i}^{(t)})}{\prod_{\substack{j=1\\ j\neq i}}^{n}(s_{i}^{(t)}-s_{j}^{(t)})}\), and \(\frac{f(u_{i}^{(t)})}{f^{\prime }(u_{i}^{(t)})}= \frac{f(u_{i}^{(t)})}{\prod_{\substack{j=1\\ j\neq i}}^{n}(u_{i}^{(t)}-u_{j}^{(t)})}\) in (9), we have
$$ v_{i}^{(t)}=u_{i}^{(t)}- \frac{f(u_{i}^{(t)})}{\prod_{\substack{j=1\\ j\neq i}}^{n}(u_{i}^{(t)}-u_{j}^{(t)})}, $$
(12)
where \(u_{i}^{(t)}=s_{i}^{(t)}- \frac{f(s_{i}^{(t)})}{\prod_{\substack{j=1\\ j\neq i}}^{n}(s_{i}^{(t)}-s_{j}^{(t)})}\) and \(s_{i}^{(t)}=r_{i}^{(t)}- \frac{f(r_{i}^{(t)})}{\prod_{\substack{j=1\\ j\neq i}}^{n}(r_{i}^{(t)}-\overset{\ast }{s}{}_{j}^{(t)})}\).
Thus, we have constructed a new simultaneous iterative method (12), which is abbreviated as NIM12.
2.1 Convergence aspect
In this section, we prove that method NIM12 has local convergence order 12.
Theorem 1
Let \(\zeta _{{1}}, \ldots, \zeta _{n}\) be the n simple roots of (1). If \(r_{1}^{(0)}, \ldots, r_{n}^{(0)}\) are the initial estimates of the roots respectively and sufficiently close to actual roots, then NIM12 has a convergence order 12.
Proof
Let \(\epsilon _{i}=r_{i}^{(t)}-\zeta _{i}\), \(\epsilon _{i}^{\prime }=s_{i}^{(t)}- \zeta _{i}\), \(\epsilon _{i}^{{\prime \prime }}=u_{i}^{(t)}-\zeta _{i}\), and \(\epsilon _{i}^{{\prime \prime \prime }}=v_{i}^{(t)}-\zeta _{i}\) be the errors in \(r_{i}\), \(s_{i}\), \(u_{i}\), and \(v_{i}\), respectively. From (12), the first step of NIM12, we have
$$\begin{aligned}& s_{i}^{(t)}-\zeta _{i} = r_{i}^{(t)}- \zeta _{i}-w_{i}\bigl(r_{i}^{(t)} \bigr), \\& \epsilon _{i}^{\prime } = \epsilon _{i}- \epsilon _{i} \frac{w_{i}(r_{i}^{(t)})}{\epsilon _{i}}, \\& \epsilon _{i}^{\prime } = \epsilon _{i}(1-E_{i}), \end{aligned}$$
(13)
where
$$\begin{aligned}& E_{i}=\frac{w_{i}(r_{i}^{(t)})}{\epsilon _{i}}=\prod_{\substack{j\neq i\\j=1}}^{n} \frac{(r_{i}^{(t)}-\zeta _{j})}{(r_{i}^{(t)}-\overset{\ast }{s}{}_{j}^{(t)})}, \\& \frac{r_{i}^{(t)}-\zeta _{j}}{r_{i}^{(t)}-\overset{\ast }{s}{}_{j}^{(t)}}=1+\frac{\overset{\ast }{s}{}_{j}^{(t)}-\zeta _{j}}{r_{i}^{(t)}-\overset{\ast }{s}{}_{j}^{(t)}}=1+O\bigl(\epsilon ^{2}\bigr), \end{aligned}$$
(14)
and \(\overset{\ast }{s}{}_{j}^{(t)}-\zeta _{j}=O(\epsilon ^{2})\) see [2]. For a simple root ζ and small enough ϵ, \(\vert r_{i}^{(t)}-\overset{\ast }{s}{}_{j}^{(t)} \vert \) is bounded away from zero, and so
$$\begin{aligned}& \prod_{\substack{j\neq i\\j=1}}^{n} \frac{(r_{i}^{(t)}-\zeta _{j})}{(r_{i}^{(t)}-\overset{\ast }{s}{}_{j}^{(t)}))}= \bigl(1+O\bigl(\epsilon ^{2}\bigr)\bigr)^{ n-1}=1+(n-1)O \bigl(\epsilon ^{2}\bigr)=1+O\bigl(\epsilon ^{2}\bigr), \\& E_{i} =1+O\bigl(\epsilon ^{2}\bigr), \\& E_{i}-1 =O\bigl(\epsilon ^{2}\bigr). \end{aligned}$$
Thus, (13) gives
$$ \epsilon _{i}^{{\prime }}=O(\epsilon )^{3}. $$
(15)
From the second step of NIM12, we have
$$\begin{aligned}& u_{i}^{(t)}-\zeta _{i} = s_{i}^{(t)}- \zeta _{i}-w_{i}\bigl(s_{i}^{(t)} \bigr), \\& \epsilon _{i}^{\prime \prime } = \epsilon _{i}^{\prime }- \epsilon _{i}^{ \prime }\frac{w_{i}(s_{i}^{(t)})}{\epsilon _{i}^{\prime }}, \\& \epsilon _{i}^{\prime \prime } = \epsilon _{i}^{\prime }(1-U_{i}), \end{aligned}$$
(16)
where
$$\begin{aligned}& U_{i}=\frac{w_{i}(s_{i}^{(t)})}{\epsilon _{i}^{\prime }}=\prod_{\substack{j\neq i\\j=1}}^{n} \frac{(s_{i}^{(t)}-\zeta _{j})}{(s_{i}^{(t)}-s_{j}^{(t)})}, \\& \frac{s_{i}^{(t)}-\zeta _{j}}{s_{i}^{(t)}-s_{j}^{(t)}}=1+ \frac{s_{i}^{(t)}-\zeta _{j}}{s_{i}^{(t)}-s_{j}^{(t)}}=1+O\bigl(\epsilon _{i}^{ \prime }\bigr). \end{aligned}$$
For a simple root ζ and small enough ϵ, \(\vert s_{i}^{(t)}-\overset{(t)}{s}{}_{j} \vert \) is bounded away from zero, so
$$\begin{aligned}& \prod_{\substack{j\neq i\\j=1}}^{n} \frac{(s_{i}^{(t)}-\zeta _{j})}{(s_{i}^{(t)}-s_{j}^{(t)})}= \bigl(1+O\bigl(\epsilon ^{\prime }\bigr)\bigr) ^{n-1}=1+(n-1)O \bigl(\epsilon ^{\prime }\bigr)=1+O\bigl(\epsilon ^{\prime }\bigr), \\& U_{i} = 1+O\bigl(\epsilon ^{\prime }\bigr), \\& 1-U_{i} = O\bigl(\epsilon ^{\prime }\bigr), \\& \epsilon _{i}^{{\prime \prime }}=O\bigl(\epsilon ^{\prime } \bigr)^{2}. \end{aligned}$$
Since from (15), \(\epsilon _{i}^{{\prime }}=O\) \((\epsilon )^{3}\). Thus,
$$\begin{aligned}& \epsilon _{i}^{{\prime \prime }} = O\bigl((\epsilon )^{3}\bigr) ^{2}, \\& \epsilon _{i}^{{\prime \prime }} = O(\epsilon )^{6}. \end{aligned}$$
(17)
From the third step of NIM12, we have
$$\begin{aligned}& v_{i}^{(t)}-\zeta _{i} = u_{i}^{(t)}- \zeta _{i}-w_{i}\bigl(u_{i}^{(t)} \bigr), \\& \epsilon _{i}^{{\prime \prime \prime }} = \epsilon _{i}^{{\prime \prime }}- \epsilon _{i}^{{\prime \prime }} \frac{w_{i}(u_{i}^{(t)})}{\epsilon _{i}^{{\prime \prime }}}, \\& \epsilon _{i}^{{\prime \prime \prime }} = \epsilon _{i}^{{\prime \prime }}(1-G_{i}), \end{aligned}$$
(18)
where
$$\begin{aligned}& G_{i}=\frac{w_{i}(u_{i})}{\epsilon _{i}^{{\prime \prime }}}=\prod_{\substack{j\neq i\\j=1}}^{n} \frac{(u_{i}^{(t)}-\zeta _{j})}{(u_{i}^{(t)}-u_{j}^{(t)})}, \\& \frac{u_{i}^{(t)}-\zeta _{j}}{u_{i}^{(t)}-u_{j}^{(t)}}=1+ \frac{u_{j}^{(t)}-\zeta _{j}}{u_{i}^{(t)}-u_{j}^{(t)}}=1+O\bigl(\epsilon _{i}^{{ \prime \prime }}\bigr). \end{aligned}$$
(19)
With the same argument used in (16), we have
$$ \prod_{\substack{j\neq i\\j=1}}^{n} \frac{(u_{i}^{(t)}-\zeta _{j})}{(u_{i}^{(t)}-u_{j}^{(t)})}= \bigl(1+O\bigl(\epsilon ^{{\prime \prime }}\bigr)\bigr)^{n-1}=1+(n-1)O \bigl( \epsilon ^{{\prime \prime }}\bigr)=1+O\bigl(\epsilon ^{{\prime \prime }}\bigr). $$
Therefore,
$$\begin{aligned}& G_{i} = 1+O\bigl(\epsilon ^{{\prime \prime }}\bigr), \\& 1-G_{i} = O\bigl(\epsilon ^{{\prime \prime }}\bigr), \\& \epsilon _{i}^{{\prime \prime \prime }}=O\bigl(\epsilon ^{{\prime \prime }} \bigr)^{2}. \end{aligned}$$
(20)
Since from (17) \(\epsilon _{i}^{{\prime \prime }}=\)O\(( \epsilon )^{6}\), we obtain
$$\begin{aligned}& \epsilon _{i}^{{\prime \prime \prime }} = O\bigl((\epsilon )^{6}\bigr)^{2}, \\& \epsilon _{i}^{{\prime \prime \prime }} = O(\epsilon )^{12}. \end{aligned}$$
(21)
Hence, (21) proves 12th order convergence. □