## 3 A Steffensen-Type Family of Super Convergence with Memory  
The added high-order terms in the denominators in Equations (4) and (5) at least have no bad effect by now. Furthermore, by adjusting these coefficients of the high-order terms, _i.e._, only using several arithmetic operations of old evaluations of \(f\) to express the parameters, the asymptotic convergence constants of the optimal second-order and fourth-order methods can tend to zero, respectively, and the obtained methods of super-convergence can exceed SASM and RWBM, respectively. For example:  
The super second-order method: Iterate Equation (4) with  
\[\mu_{n}=\frac{1+\gamma_{n}f[x_{n},z_{n}]}{\gamma_{n}f[x_{n},z_{n}]}f[z_{n-1}, x_{n},z_{n}] \tag{10}\]  
The super fourth-order method: Iterate Equation (5) with  
\[\alpha_{n}=f[x_{n-1},x_{n},z_{n},y_{n}]-\frac{f[x_{n},z_{n},y_{n}]^{2}}{f[x_{n},y_{n}]} \tag{11}\]  
**Theorem 3**. : _Let \(f:D\rightarrow\Re\) be a sufficiently differentiable function with a simple root \(a\in D\), \(D\subset\Re\) be an open set, \(x_{0}\) be close enough to \(a\), then the methods Equations (10) and (11) satisfy the following error equations:_  
\[e_{n+1}=-(1+\gamma_{n}f^{\prime}(a))[c_{3}e_{n-1}e_{n}^{2}+O(e_{n-1}^{2}e_{n}^{ 2})] \tag{12}\]\[e_{n+1}=(1+\gamma_{n}f^{\prime}(a))^{2}[c_{2}c_{4}e_{n-1}e_{n}^{4}+O(e_{n-1}^{2}e_{ n}^{4})] \tag{13}\]  
_where \(c_{k}=\frac{f^{(k)}(a)}{k!f^{\prime}(a)}\), \(e_{n}=x_{n}-a,n=0,1,2,\ldots\), and achieve convergence of order at least \(1+\sqrt{2}\) and \(2+\sqrt{5}\) respectively._  
**Proof. ** By the definition of divided difference and Taylor formula, we also have  
\[f[z_{n-1},x_{n},z_{n}] = \frac{f^{\prime\prime}(a)}{2}+\frac{f^{\prime\prime\prime}(a)}{6} e_{n-1}+O(e_{n-1}^{2}),\] \[f[x_{n-1},x_{n},z_{n},y_{n}] = \frac{f^{\prime\prime\prime}(a)}{6}+\frac{f^{(4)}(a)}{4!}e_{n-1}+ O((e_{n-1})^{2})\]  
Equation (12) follows from Equation (6) by  
\[\mu_{n} = \frac{1+\gamma_{n}f^{\prime}(a)}{\gamma_{n}f^{\prime}(a)}(\frac{ f^{\prime\prime}(a)}{2!}+\frac{f^{\prime\prime\prime}(a)}{3!}e_{n-1}+O(e_{n-1}^{2}))\]  
The order \(1+\sqrt{2}\approx 2.4142\) is obtained as the positive root by solving \(s^{2}-2s-1=0\). Equation (13) follows from Equation (7) by  
\[\alpha_{n} = f^{\prime}(a)[c_{3}-c_{2}^{2}+c_{4}e_{n-1}+O(e_{n-1}^{2})]\]  
The order \(2+\sqrt{5}\approx 4.2361\) is obtained as the positive root by solving \(s^{2}-4s-1=0\). Generally, we have the super \(2^{m}\)th-order Steffensen-type family: Iterate Equation (8) with  
\[\
u_{n}=f[y_{m-2},y_{m-1}](\tilde{c}_{m+1}+(-1)^{m-1}\frac{\bar{c}_{2}\bar{D} _{m-1}}{\bar{D}_{-1}\cdots\bar{D}_{m-2}}) \tag{14}\]  
where \(\bar{D}_{-1}=1+\gamma_{n}f[y_{m-2},y_{m-1}],\bar{D}_{0}=1\), \(\bar{D}_{1}=(1+\gamma_{n}f[y_{m-2},y_{m-1}])\bar{c}_{2}\), \(\cdots\), \(\bar{D}_{m-1}=\bar{D}_{m-2}[\bar{c}_{2}\bar{D}_{m-2}+(-1)^{m-2}\ \bar{c}_{m}\bar{D}_{m-3}\cdots\bar{D}_{-1}]\), \(\bar{c}_{m}=\frac{f[y_{m-1},\cdots,y_{m-1}]}{f[y_{m-2},y_{m-1}]}\) and \(\tilde{c}_{m+1}=\frac{f[x_{n-1},y_{-1},\cdots,y_{m-1}]}{f[y_{m-2},y_{m-1}]}\). **Theorem 4**. : _Let \(f:D\to\Re\) be a sufficiently differentiable function with a simple root \(a\in D\), \(D\subset\Re\) be an open set, \(x_{0}\) be close enough to \(a\), then the family Equation (14) is super \(2^{m}\)th-order convergent, and satisfies the following error equation:_  
\[e_{n+1}=(-1)^{m}D_{m-1}\cdots D_{-1}c_{m+2}e_{n-1}e_{n}^{2^{m}}+O(e_{n-1}^{2}e _{n}^{2^{m}}),m>1 \tag{15}\]  
_where \(D_{-1}=1+\gamma_{n}f^{\prime}(a),D_{0}=1\), \(D_{1}=(1+\gamma_{n}f^{\prime}(a))c_{2}\), \(\cdots\), \(D_{m-1}=D_{m-2}[c_{2}D_{m-2}+(-1)^{m-2}\)\(c_{m}D_{m-3}\cdots D_{-1}]\), \(c_{m}=\frac{f^{(m)}(a)}{m!f^{\prime}(a)}\), and \(e_{n}=x_{n}-a\) for \(n=0,1,\cdots\)._  
**Proof.