## 1 Introduction  
Solving nonlinear equations by iterative methods have been of great interest to numerical analysts. The most famous one-point iterative method is probably [PERSON]'s Equation [1]: \(x_{k+1}=x_{k}-f(x_{k})/f^{\prime}(x_{k}),\) which converges quadratically. However, the condition \(f^{\prime}(x)\
eq 0\) in a neighborhood of the required root is severe indeed for convergence of [PERSON] method, which restricts its applications in practical. For resolving this problem, [PERSON] in [2] proposed the following one-point iterative method  
\[x_{k+1}=x_{k}-\frac{f(x_{k})}{\lambda f(x_{k})+f^{\prime}(x_{k})} \tag{1}\]  
where \(\lambda\in R,\)\(0<|\lambda|<+\infty\) and \(\lambda\) is chosen such that the corresponding function values \(\lambda f(x_{k})\) and \(f^{\prime}(x_{k})\) have the same signs. This method converges quadratically under the condition \(\lambda f(x_{k})+f^{\prime}(x_{k})\
eq 0,\) while \(f^{\prime}(x_{k})=0\) in some points is permitted. [PERSON] and [PERSON] in [3] obtained the error equation of the Equation (1) as follows  
\[e_{k+1}=(c_{2}+\lambda)e_{k}^{2}+O(e_{k}^{3}) \tag{2}\]  
where \(e_{k}=x_{k}-a,\ c_{k}=(1/k!)f^{(k)}(a)/f^{\prime}(a),\ k=2,3,\cdots\) and \(a\) is the root of the nonlinear equation \(f(x)=0.\)  
The convergence order and computational efficiency of the one-point iterative methods are lower than multipoint iterative methods. Multipoint iterative methods can overcome theoretical limits of one-point methods concerning the convergence order and computational efficiency. In recent years, many multipoint iterative methods have been proposed for solving nonlinear equations, see [4; 5; 6; 7; 8; 9; 10; 11; 12; 13; 14; 15; 16; 17; 18]. [PERSON] and [PERSON] in [4] developed the following eighth-order iterative method without memory by Hermite interpolation methods  
\[\left\{\begin{aligned} y_{k}=& x_{k}-\frac{f(x_{k})}{f^{ \prime}(x_{n})}\\ z_{k}=& y_{k}-\frac{f(y_{k})}{2f[y_{k},x_{k}]-f^{ \prime}(x_{k})}\\ x_{k+1}=& z_{k}-\frac{f(z_{k})}{N(x_{k},y_{k},z_{k })}\end{aligned}\right. \tag{3}\]  
where \(N(x_{k},y_{k},z_{k})=f[z_{k},y_{k}]+2f[z_{k},x_{k}]-2f[y_{k},x_{k}]+f[y_{k},x_ {k},x_{k}](y_{k}-z_{k}).\) Using the same strategy, [PERSON] in [5] presented a family of eighth-order iterative method without memory. The Equation (3) is a special case of the [PERSON]'s method. [PERSON] in [6] claimed a general class of optimal \(n\)-point methods without memory by Hermite interpolation methods, which have the order of convergence \(2^{n}\) and require evaluations of \(n\) functions and one first-order derivative. The Equation (3) is a special case of the [PERSON]'s \(n\)-point Method for \(n=3.\) But, the [PERSON]'s \(n\)-point method gives no specific iterative scheme and error relation for \(n\geq 4.\) In this paper, we construct a class of \(n\)-point iterative methods withand without memory by Hermite interpolation methods and give the specific iterative scheme and error relation for all \(n\geq 2.\)  
This paper is organized as follows. In Section 2, based on [PERSON]'s Equation [2] and [PERSON]'s \(n\)-point Equation [6], we derive a family of \(n\)-point iterative methods without memory for solving nonlinear equations. We prove that the order of convergence of the \(n\)-point methods without memory is \(2^{n}\) requiring the evaluations of \(n\) functions and one first-order derivative in per full iteration. [PERSON] and [PERSON] in [7] conjectured that a multipoint iteration without memory based on \(n\) functional evaluations could achieve an optimal convergence of order \(2^{n-1}\). The new methods without memory agree with the conjecture. Further accelerations of convergence speed are attained in Section 3. A family of \(n\)-point iterative methods with memory is obtained by using a self-accelerating parameter in per full iteration. This self-accelerating parameter is calculated using information available from the current and previous iterations. Numerical examples are given in Section 4 to confirm theoretical results.