## 1 Introduction
[PERSON] [12] developed a two term PRP-based descent method satisfying (7) by use of the projection technique for unconstrained optimization problem. [PERSON] et al. [13] proposed a modified form of \(\beta_{k}^{PRP}\) as follows:  
\[\beta_{k}^{DRP}=\beta_{k}^{PRP}-\mu\frac{\|y_{k}\|^{2}}{\|g_{k}\|^{4}}g_{k+1}^ {T}d_{k}.\]  
It is important that if \(\mu>\frac{1}{4}\), then the condition (7) is achieved with \(c=1-\frac{1}{4\mu}\). [PERSON] [14] present a new PRP formula defined by  
\[\beta_{k}^{MPRP}=\beta_{k}^{PRP}-min\{\beta_{k}^{PRP},\mu\frac{\|y_{k}\|^{2}}{ \|g_{k}\|^{4}}g_{k+1}^{T}d_{k}\},\]  
where \(\mu>\frac{1}{4}\) guaranteeing the descent property (7) and \(\beta_{k}^{MPRP}\geq 0\). [PERSON] and [PERSON] [15] proposed a new class of spectral conjugate gradient methods which ensures sufficient descent independent of the accuracy of the line search. [PERSON] et al. [16] gave a variant of the PRP method called the VPRP method. The parameter \(\beta_{k}\) in the VPRP method is given by  
\[\beta_{k}^{VPRP}=\frac{\|g_{k}\|^{2}-\frac{\|g_{k}\|}{\|g_{k-1}\|}\beta_{k}^{ T}g_{k-1}}{\|g_{k-1}\|^{2}}.\]Based on the VPRP method, [PERSON] [17] made a little modification and obtained the NPRP method as follows,  
\[\beta_{k}^{NPRP}=\frac{\|g_{k}\|^{2}-\frac{\|g_{k}\|}{\|g_{k-1}\|}|g_{k}^{T}g_{k -1}|}{\|g_{k-1}\|^{2}},\]  
and established the sufficient descent property (7) of the NPRP method. Recently, [PERSON] [18] proposed a three-term conjugate gradient method called MPRP method in which the direction \(d_{k}\) takes the following form:  
\[d_{k}=-g_{k}+\beta_{k}^{PRP}d_{k-1}-\frac{g_{k}^{T}d_{k-1}}{\|g_{k-1}\|^{2}}y_{ k-1}, \tag{8}\]  
leading to the MPRP method with the sufficient descent property. This property always holds independent of any line search and the convexity of the objective function. Under the following line search  
\[f(x_{k}+\alpha_{k}d_{k})\leq f(x_{k})-\delta\alpha_{k}^{2}\|d_{k}\|^{2}, \tag{9}\]  
where \(\alpha_{k}=\max\{\rho^{i},i=1,2,...\}\) with \(0<\rho,\delta<1\), the global convergence of the MPRP method is established. Note that the MPRP method in [18] will reduce to the standard PRP method if exact line search is used and converges globally under the line search (9). However, it fails to converge under the weak Wolfe line search (6). The main reason lies in the trust region property (Lemma 1 in Section 2) that is not satisfied by the MPRP method. Based on the method in [12; 18], [PERSON] et al. [19] propose a three-term PRP-type conjugate gradient method which always satisfies the sufficient descent condition independently of line searches employed. Motivated by the above observations, we propose a modified three-term PRP formula based on (8), which possesses not only the sufficient descent property but also the trust region feature. In the following, we first reformulate the search direction (8) into a new form, which can be written as follows:  
\[d_{k}=-g_{k}+\beta_{k}^{PRP}d_{k-1}-\beta_{k}^{PRP}\frac{g_{k}^{T}d_{k-1}}{ \frac{g_{k}^{T}}{g_{k}^{T}}y_{k-1}}y_{k-1}. \tag{10}\]  
Then, we can consider the following general iteration form:  
\[\beta_{k}^{HZ}=\beta_{k}^{HS}-2\frac{\|y_{k}\|^{2}g_{k}^{T}d_{k-1}}{(d_{k-1}^{ T}y_{k-1})^{2}}.\]0  
with any \(\beta_{k}\), it is not difficult to deduce that the direction defined by (11) satisfies  
\[\beta_{k}^{HZ}=\beta_{k}^{HS}-2\frac{\|y_{k}\|^{2}g_{k}^{T}d_{k-1}}{(d_{k-1}^{ T}y_{k-1})^{2}}.\]1  
which is independent of any line search and the convexity of the objective function. In this paper, we further study the PRP method and suggest a new three term PRP method to improve the numerical performance and obtain better property of the PRP method. The remaining of this paper is organized as follows. In Section 2, we present a modified PRP method by using a new technique and establish its global convergence. In Section 3, the new technique is extended to the HS and LS method. In the last section, some numerical results are reported to show the modified methods are efficient.