## 1 Introduction  
Solving the nonlinear equation \(f(x)=0\) is a fundamental problem in scientific computation. Besides [PERSON]'s method (NM), [PERSON]'s method (SM):  
\[x_{n+1}=x_{n}-\frac{f^{2}(x_{n})}{f(x_{n}+f(x_{n}))-f(x_{n})},\ \ \ \ n=0,1,2,\ldots \tag{1}\]  
is also a famous method for dealing with such a problem, because it is derivative free and maintains quadratic convergence (see [1]). Since [PERSON] and [PERSON] conjectured in 1974 that a multipoint iteration based on \(m\) evaluations without memory has optimal order \(2^{m-1}\) of convergence (see [2]), NM and SM are methods of optimal order. The efficiency index of them is\(\sqrt{2}=1.4142\).  
In order to achieve higher order of convergence, the self-acceleration of SM (SASM) was introduced in [PERSON]'s book as follows (see [3]):  
\[x_{n+1}=x_{n}-\frac{\gamma_{n}f^{2}(x_{n})}{f(x_{n}+\gamma_{n}f(x_{n}))-f(x_{n})} \tag{2}\]  
where \(\gamma_{n}=-\frac{\gamma_{n-1}f(x_{n-1})}{f(x_{n-1}+\gamma_{n-1}f(x_{n-1}))-f( x_{n-1})}\), which was obtained recursively by using memory. SASM achieves super convergence of order \(1+\sqrt{2}=2.4142\). Its efficiency index is \(\sqrt{1+\sqrt{2}}=1.5538\). The other two choices were also introduced for Steffensen-type methods by [PERSON], _et al._, (see [4,5]): \(\gamma_{n}=-\frac{x_{n}-x_{n-1}}{f(x_{n})-f(x_{n-1})}\) and \(\gamma_{n}=\frac{x_{n}-x_{n-1}}{f(x_{n-1})}\). The latter is the same as the above expression of \(\gamma_{n}\) for SASM, but different from the above for the multi-step methods. These expressions of \(\gamma_{n}\) ensure the methods to achieve super convergence by using the same number of evaluations of \(f\) as before. Local and semilocal convergence of Steffensen-type methods and their applications in the solution of nonlinear systems and nonlinear differential equations were discussed in the literature (see [1,5,6]).  
Moreover, [PERSON], [PERSON] introduced generalized biparametric multipoint methods as follows (DPM, see [7]):  
\[\left\{\begin{array}{l}y_{k,1}=y_{k,0}+\gamma_{k}f(y_{k,0}),y_{k,0}=x_{k}, \\ y_{k,2}=y_{k,0}-\frac{f(y_{k,0})}{f[y_{k,0},y_{k,1}]+p_{k}f(y_{k,1})},\\ y_{k,j}=y_{k,j-1}-\frac{f(y_{k,j-1})}{N_{j-1}^{\prime}(y_{k,j-1}y_{k,0},y_{k,1 },\ldots,y_{k,j-1})},\quad j=3,\ldots,n,\\ x_{k+1}=y_{k,n}-\frac{f(y_{k,n})}{N_{h}(y_{k,n};y_{k,0},y_{k,1},\ldots,y_{k,n}) },\quad k=0,1,\ldots,\end{array}\right. \tag{3}\]  
where \(\gamma_{k}=-\frac{1}{N_{m}^{\prime}(y_{k,0})},p_{k}=-\frac{N_{m+1}^{\prime \prime}(y_{k,1})}{2N_{m+1}^{\prime}(y_{k,1})},(m=1,\ldots,n+1),\) and \(N_{j}(x;y_{k,0},y_{k,1},\ldots,y_{k,j})\)\((j=2,\ldots,n)\) was [PERSON]'s interpolating polynomial of degree \(j\).  
This paper is organized as the following. In Section 2, by using [PERSON]'s method for the direct Newtonian interpolation of the function, we construct an optimal Steffensen-type method of second-order which has one more parameter than that in SASM, establish an optimal Steffensen-type method of fourth-order which generalizes [PERSON]'s method (RWBM, see [8]), deduce their error equations and asymptotic convergence constants, and induce to a general optimal Steffensen-type family of \(2^{m-1}\)th-order without memory. Furthermore, in Section 3, we obtain the family of Steffensen-type methods by accelerating with memory, and Steffensen-type methods of super second-order and super fourth-order of convergence by doubly accelerating with memory. In Section 4, we compare the proposed families with NM, SM, SASM, RWBM and DPM by solving nonlinear equations in numerical examples. Finally we make conclusions in Section 5.