## 4 Comparison with Previous Methods  
In this section, we use the proposed modification of [PERSON]'s method with three-step (14) (MNM) (the Formula (14) in our paper) to solve some nonlinear equations which the multiplicity of the roots is a single multiplicity, and compare them with [PERSON]'s method [17] (KM, (4), (13)) (\(G=\frac{f(x)}{\frac{f(x-f(x))-f(x)}{(x-f(x))-x}}=\frac{-f^{2}(x)}{f(x-f(x))- f(x)}\) (4), \(x_{2}=x_{1}-(x_{0}-x_{1})\frac{G_{1}}{G_{0}-G_{1}}\) (13)), the high-order convergence iteration methods without employing derivatives given in [18] (WFM, (6)) (\(x_{n+1}=x_{n}-\frac{F^{2}(x_{n})}{pF^{2}(x_{n})+F(x_{n})-F(x_{n}-F(x_{n}))}\) (6)), the improved method for finding multiple roots and itars multiplicity of nonlinear equations given in [22] (PGM, ((6), (11))) (\(G(x)=\left\{ \begin{array}{l}\frac{f^{2}(x)}{\delta+f(x+f(x))-f(x)},\,if\,f(x )\
eq 0\\ 0,\,\,if\,f(x)=0\end{array} \right.\), where \(\delta=sign(f(x+f(x))-f(x))f^{2}(x)\) (6), \(x_{n+1}=x_{n}-\frac{G^{2}(x_{n})}{pG^{2}(x_{n})+G(x_{n})-G(x_{n}-G(x_{n}))}\) (11)), the derivative free iterative method for finding multiple roots of nonlinear equations given in [23] (YM, ((7), (10))) (\(H_{\varepsilon}(x)=\frac{\varepsilon f(x)^{2}}{f(x+\varepsilon f(x))-f(x)}\) (7), \(x_{k+1}=x_{k}-\frac{2(x_{k}-x_{k-1})H_{\varepsilon}(x_{k})}{H_{\varepsilon}(2 x_{k}-x_{k-1})-H_{\varepsilon}(x_{k-1})}\) (10)), transformation methods for finding multiple roots of nonlinear equations [24] (YM, ((10), (12))) (\(K(x)=\left\{ \begin{array}{l}\frac{\varepsilon f(x)^{2}}{f_{\varepsilon}(x)}, \,if\,f(x)\
eq 0\\ 0,\,if\,f(x)=0.\end{array} \right.\), where \(f_{\varepsilon}(x)\approx\varepsilon f(x)f^{\prime}(x)\) (10), \(p_{k+1}=p_{k}-\frac{\mu K(p_{k})^{2}}{K(p_{k}+\mu K(p_{k}))-K(p_{k})}\) (12)), as well as [PERSON]'s first order method(NM) \(x_{k+1}=x_{k}-\frac{f(x_{k})}{f^{\prime}(x_{k})}\). Displayed in Table 1 are the number of iterations satisfied such that\(|f(x_{n})|<1.E-17\), \(|x_{n}-x^{*}|<1.E-17\). All the computations were done by using Visual C++ 6.0. Since King's method and Yun'smethod require two starting values, we have used \(x_{1}=x_{0}-0.1\). We used the following test functions and obtained the approximate zeros \(x^{*}\) round up to the 17 th decimal place:  
\[f_{1}(x)=\frac{(x-\sqrt{5})^{4}}{(x-1)^{2}+1},\quad m=4,\quad x^{*}=2.236067977499 790\]  
\[f_{2}(x)=(\sin^{2}(x)-2x+1)^{5},\quad m=5,\quad x^{*}=0.71483582544138924\]  
\[f_{3}(x)=(8 xe^{-x^{2}}-2x-3)^{8},\quad m=8,\quad x^{*}=-1.7903531791589544\]  
\[f_{4}(x)=\frac{(2x\cos(x)+x^{2}-3)^{10}}{(x^{2}+1)},\quad m=10,\quad x^{*}=2.98 06452794385368\]  
\[f_{5}(x)=(e^{-x^{2}+x+3}-x+2)^{9},\quad m=9,\quad x^{*}=2.4905398276083051\]  
\[f_{6}(x)=(e^{-x}+2\sin(x))^{4},\quad m=4,\quad x^{*}=3.1627488709263654\]  
\[f_{7}(x)=(\ln(x^{2}+3x+5)-2x+7)^{8},\quad m=8,\quad x^{*}=5.4690123359101421\]  
\[f_{8}(x)=(\sqrt{x^{2}+2x+5}-2\sin(x)-x^{2}+3)^{5},\quad m=5,\quad x^{*}=2.33 19676558839640\]  
\[f_{9}(x)=(x-2)^{4}/((x-1)^{2}+1),\quad m=4,\quad x^{*}=2.00000000000000000\]  
\[f_{10}(x)=(x-2.5)^{\frac{15}{4}}e^{x},\quad m=\frac{15}{4},\quad x^{*}=2.500 00000000000000\]  
\[f_{2}(x)=(\sin^{2}(x)-2x+1)^{5},\quad m=5,\quad x^{*}=0.71483582544138924\]0  
\[f_{2}(x)=(\sin^{2}(x)-2x+1)^{5},\quad m=5,\quad x^{*}=0.71483582544138924\]1  
\[f_{2}(x)=(\sin^{2}(x)-2x+1)^{5},\quad m=5,\quad x^{*}=0.71483582544138924\]2  
\[f_{2}(x)=(\sin^{2}(x)-2x+1)^{5},\quad m=5,\quad x^{*}=0.71483582544138924\]3  
\[f_{2}(x)=(\sin^{2}(x)-2x+1)^{5},\quad m=5,\quad x^{*}=0.71483582544138924\]4  
Note that we used NC in Table 1 to mean that the method does not converge to the root. And these methods can converge to root by using closer initial values. The computational results in Table 1 demonstrate that our proposed iterative method (MNM) requires less number of iterations than those of KM, WFM, PGM, YM, and NM. Therefore, it is significant and applicable and can compete with other existing methods.