\documentclass[12pt,twoside]{article} \usepackage{amssymb,amsfonts,amsmath} %packages for mathematics formulas, equations and fonts \usepackage{fancyhdr,setspace} % packages for fancy header and spaces \usepackage{graphicx,graphics, float} % packages for graphics \usepackage{tabularx,multicol,multirow,adjustbox} %packages for tables \usepackage{booktabs,bigstrut} %packages for tables \usepackage{hyperref} % package for hyperlink \usepackage{array} % array package \usepackage{vector} % vector package \newtheorem{theorem}{Theorem}[section] \newtheorem{lemma}[theorem]{Lemma} \newtheorem{proposition}[theorem]{Proposition} \newtheorem{corollary}[theorem]{Corollary} \newtheorem{definition}[theorem]{Definition} \newtheorem{remark}[theorem]{Remark} \newtheorem{example}[theorem]{Example} \newtheorem{algorithm}[theorem]{Algorithm} \setcounter{page}{1} \setlength{\textheight}{21.6cm} \setlength{\textwidth}{14cm} \setlength{\oddsidemargin}{1cm} \setlength{\evensidemargin}{1cm} \pagestyle{myheadings} \thispagestyle{empty} \markboth{\small{J. Sabi'u}}{\small{J. Sabi'u}} \date{} \begin{document} \centerline{\Large{\bf Enhanced derivative-free conjugate gradient method }} \centerline{} \centerline{\Large{\bf for solving symmetric nonlinear equations}} \centerline{} \centerline{\bf {Jamilu Sabi'u}} \centerline{} \centerline{Department of Mathematics} \centerline{ Faculty of Science, Northwest University Kano} \centerline{Kano, Nigeria,} \centerline{sabiujamilu@gmail.com} \newtheorem{Theorem}{\quad Theorem}[section] \newtheorem{Definition}[Theorem]{\quad Definition} \newtheorem{Corollary}[Theorem]{\quad Corollary} \newtheorem{Lemma}[Theorem]{\quad Lemma} \newtheorem{Example}[Theorem]{\quad Example} \centerline{} \begin{abstract} In this article, an enhanced conjugate gradient approach for solving symmetric nonlinear equations is propose without computing the Jacobian matrix. This approach is completely derivative and matrix free. Using classical assumptions the proposed method has global convergence with nonmonotone line search. Some reported numerical results shows the approach is promising. \end{abstract} {\bf Keywords:} Backtracking line search, hyperplane, symmetric nonlinear equations and conjugate gradient method. \section{Introduction} Let us consider the systems of nonlinear equations \begin{equation}\label{EQ12} F({x})=0, \end{equation} where $F:R^n \rightarrow R^n$ is a nonlinear mapping. Often, the mapping, $F$ is assumed to satisfying the following assumptions:\\ A1. There exists an ${x} ^* \in R^n$ s.t $F({x}^*)=0$\\ A2. $F$ is a continuously differentiable mapping in a neighborhood of ${x}^*$\\ A3. $F'({x}^*)$ is invertible\\ A4. The Jacobian $F^{'}(x)$ is symmetric.\\ Monotonicity means \begin{equation} (F(x)-F(y))^T(x-y)\geq 0,\quad \forall x,y\in R^n \end{equation}. The prominent method for finding the solution of \eqref{EQ12}, is the classical Newton's method which generates a sequence of iterates $\{x_k\}$ from a given initial point $x_0$ via \begin{equation}\label{EQ11a} x_{k+1}=x_k - (F^\prime(x_k))^{-1}F(x_k), \end{equation}where $k=0,1,2\ldots$. The attractive features of this method are; rapid convergence and easy to implement. Nevertheless, Newton's method requires the computation of the Jacobian matrix, which require the first-order derivative of the systems. In practice, computations of some functions derivatives are quite costly and sometime they are not available or could not be done precisely. In this case Newton's method cannot be applied directly. In this work, we are interested in handling large-scale problems for which the Jacobian is either not available or requires a low amount storage, the best method is CG approach. It is vital to mention that, the conjugate gradient methods are among the popular used methods for unconstrained optimization problems. They are particularly efficient for handling large-scale problems due to their convergence properties, simply to implement and low storage \cite {jam13}. Not withstanding, the study of conjugate gradient methods for large-scale symmetric nonlinear systems of equations is scanty, this is what motivated us to have this paper. In general, CG methods for solving nonlinear systems of equations generates an iterative points $\{x_k\}$ from initial given point $x_0$ using \begin{equation}\label{EQ11wa} x_{k+1}=x_k +\alpha_k d_k, \end{equation} where $\alpha_k>0$ ia attained via line search, and direction $d_k$ are obtained using \begin{equation}\label{EQ11da} d_k= \begin{cases} -F(x_k) \quad \quad \quad \quad \quad if \quad k=0\\ -F(x_k)+\beta_kd_{k} \quad if \quad k\geq1 \end{cases} \end{equation} $\beta_k$ is termed as conjugate gradient parameter. This problems under study, may arise from an unconstrained optimization problem, a saddle point problem, Karush-Kuhn-Tucker (KKT) of equality constrained optimization problem, the discritized two-point boundary value problem, the discritized elliptic boundary value problem, and etc. Equation (1) is the first-order necessary condition for the unconstrained optimization problem when F is the gradient mapping of some function $f:R ^n\longrightarrow R$, \begin{equation} minf(x), \quad x\epsilon R^n. \end{equation} For the equality constrained problem \begin{equation} minf(x), \end{equation} \begin{equation*} s.t \quad h(z)=0, \end{equation*}where $h$ is a vector-valued function, the KKT conditions can be represented as the system (1) with $x=(z,v),$ and \begin{equation} F(z,v)=(\nabla F(z)+\nabla h(z)v,h(z)), \end{equation}where v is the vector of Lagrange multipliers. Notice that the Jacobian $ \nabla F(z,v)$ is symmetric for all $(z,v)$ (see, e.g., \cite{jam8}). \\ Problem (1) can be converted to the following global optimization problem(5) with our function $f$ defined by \begin{equation}\label{EQ111wa} f(x)=\frac{1}{2} ||F(x)||^2. \end{equation} A large number of efficient solvers for large-scale symmetric nonlinear equations have been proposed, analyzed, and tested in the last decade. Among them are \cite{jam5, jam3, jam12}. Still the matrix storage and solving of n-linear system are required in the BFGS type methods presented in the literature. The recent designed nonmonotone spectral gradient algorithm \cite{jam2} falls within the frame work of matrix-free. \\ The conjugate gradient methods for symmetric nonlinear equations has received a good attension and take an appropriate progress. However, Li and Wang \cite{jam6} proposed a modified Flectcher-Reeves conjugate gradient method which is based on the work of Zhang et al. \cite{jam4}, and the results illustrate that their proposed conjugate gradient method is promising. In line with this development, further studies on conjugate gradient are \cite{jam9, jam13,jam11,jam17}. Extensive numerical experiments showed that each over mentioned method performs quite well. \\ We organized the paper as follows: In the next section, we present the details of the proposed method. Convergence results are presented in Section 3. Some numerical results are reported in Section 4. Finally, conclusions are made in Section 5. \section{Enhanced derivative-free (EDF)} Given an initial point $x_0$, an iterative scheme for problem \eqref{EQ12} generally generates a sequence of iterates \begin{equation}\label{j2} x_{k}=x_{k-1}+\alpha_kd_{k-1}, \quad k=1,2\dots \end{equation} which employs a line search procedure along the direction $d_k$ to compute the stepsize $\alpha_k$. Typical line searches include Backtracking, Armijo or Wolfe line searches. Let $z_k=x_k+\alpha_kd_{k-1}$, the hyperplane \begin{equation}\label{j3} H_k =\left\{ x \in R^n | (x-z_k)^TF(z_k)=0\right\} \end{equation} strictly separates $x_k$ from the solution set of \eqref{EQ12}. Therefore, from this facts Solodov and Svaiter\cite{jam7} advised to let the next iterate $x_{k+1}$ be the projection of $x_k$ onto this hyperplane $H_k$. That is, $x_{k+1}$ is defined by \begin{equation}\label{j4} x_{k+1}=x_k-\frac{F(z_k)^T(x_k-z_k)}{||F(z_k)||^2}F(z_k). \end{equation} In this paper, the direction $d_k$ is base on \cite{jam17}, specifically \begin{equation} \label{j5} d_k= \begin{cases} - F_k \quad \quad \quad \quad \quad if \quad k=0\\ - \theta_kF_k+\beta_kd_{k} \quad if \quad k\geq1 \end{cases} \end{equation} where $F_k$ means $F(x_k)$ and $\beta_k$ defined as \begin{equation} \label{EQa11i} \beta_k= \frac{( \theta_ky_k-s_k)^T} { y_k^T d_k}F_{k} \end{equation} \begin{equation}\theta_k=\frac{s^Ts_k}{s_k^Ty_k}\end{equation}. Throughout this paper, $|| . ||$ is the Euclidean norm, $s_k =x_{k}-x_{k-1}$ and $y_k= F_k- F_{k-1}$. However, to compute the stepsize $\alpha_k$, nonmonotone line search proposed by \cite{jam5} is an interesting idea that avoids the necessity of descent directions to guarantee that each iteration is well defined. Let $\omega_1>0$, $\omega_2>0$, $r\in (0,1)$ be constants and $\left \{ \eta_k \right \} $ be a given positive sequence such that \begin{equation}\label{j13} \sum_{k=0}^{\infty} \eta_k<\infty. \end{equation} Let $\alpha_k=max\left \{ 1,r^k\right \}$ that satisfy \begin{equation}\label{j14} ||F(x_k+\alpha_kd_k)||^2-||F(x_k)||^2 \leq -\omega_1 ||\alpha_k F(x_k)||^2-\omega_2||\alpha_kd_k||^2+\eta_k||F(x_k)||^2. \end{equation} \textbf{EDF Algorithm }\\ \textbf{Step $1$} : Given $x_0$ ,$\alpha >0$ , $\omega \in(0,1)$, $r\in (0,1)$ and a positive sequence $\eta_k$ satisfying \eqref{j13}, and set $k=0$ .\\ \textbf{Step $2$ :} Test a stopping criterion. If yes, then stop; otherwise continue with Step 3. \\ \textbf{Step $3$} : Compute $d_k$ by \eqref{j5}.\\ \textbf{Step $4$} : Compute $\alpha_k$ by the line search \eqref{j14}.\\ \textbf{Step $5$} : Compute $ x_{k+1}=x_k-\frac{F(z_k)^T(x_k-z_k)}{||F(z_k)||^2}F(z_k)$.\\ \textbf{Step $6$} : Consider $k = k + 1$ and go to step 2.\\ \section{Global convergence} This section presents global convergence results of the Enhanced derivative-free conjugate gradient method. In order to analyze the convergence of our method, we will make the following assumptions on nonlinear systems $F$.\\ {\bf Assumption} \\(i) The level set $\Omega=\left \{ x|F(x) \leq e^nF(x_0) \right \}$ is bounded \\(ii) In some neighborhood $N$ of $\Omega$, $F(x)$ is Lipschitz continous i.e there exists a positive constant $L>0$ such that \begin{equation}\label{EQ01} \|F(x)-F(y)\|\leq L \|x-y\|, \end{equation} $\forall x ,y \in N$. Properties (i) and (ii) implies that there exists positive constants $M_1$, $M_2$ such that \begin{equation}\label{EQ112wa} ||F(x)||\leq M_1,\quad ||F(z)||\leq M_2,\quad \forall x\epsilon N, \end{equation} \begin{lemma}\cite{jam5}Let the sequence $\left \{ x_k\right \}$ be generated by the algorithms above. Then the sequence $\left \{ ||F_k|| \right \} $ converges and $x_k\in N$ for all $k\geq 0$. \end{lemma} \begin{lemma}Let the properties of \eqref{EQ12} above hold. Then we have \begin{equation}\label{EQ114wa} \lim_ {k \, \rightarrow\, \infty} ||\alpha_kd_k||=\lim_ {k \, \rightarrow\, \infty}||s_k||=0, \end{equation} \begin{equation}\label{EQ115wa} \lim_ {k \, \rightarrow\, \infty}||\alpha_kF_k||=0 \end{equation} \end{lemma} {\bf Proof}.by \eqref{j13} and \eqref{j14} we have for all $k>0$, \begin{equation}\label{EQ116wa} \omega_2 ||\alpha_k d_k||^2 \leq \omega_1||\alpha_kF(x_k)||^2+\omega_2 ||\alpha_kd_k||^2 \leq||F_k||^2-||F_{k+1}||^2+\eta_k||F_k||^2 \end{equation} by summing the above $k$ inequality, then we obtain: \begin{equation}\label{EQ117wa} \omega_2\sum_{i=0}^{k} ||\alpha_k d_k||^2 \leq ||F_k||^2\left \{ \sum_{i=0}^{k} (1-\eta_i)\right \}-||F_{k+1}||^2 \end{equation} so, from \eqref{EQ112wa} and the fact that $\left \{ \eta_k \right \}$ satisfies \eqref{j13} the series $\sum_{i=0}^{k} ||\alpha_k d_k||^2$ is convergent. This implies \eqref{EQ114wa}. By a similar way, we can prove that\eqref{EQ115wa} holds. \begin{Lemma} Suppose that $\left\{x_k\right\}$ is generated by EDF algorithm. Let $s_k =x_{k}-x_{k-1}.$ Then, we have \begin{equation}\label{EQ114wa} \lim_ {k \, \rightarrow\, \infty} \|\alpha_kd_k\|=\lim_ {k \, \rightarrow\, \infty}\|s_k\|=0, \end{equation} \end{Lemma} {\bf Proof}. \begin{equation} \|x_k-x_{k-1}\|=\frac{|F(z_k)^T(x_k-z_k)|}{\|F(z_k)\|}\leq \frac{\|F(z_k)\| \|x_k-z_k\|}{\|F(z_k)\|} \end{equation} \begin{equation} = \|x_k-z_k\|=\alpha_k \|d_k\|. \end{equation} The following theorem establishes the global convergence of the EDF algorithm. \begin{theorem} Let $\left\{x_k\right\}$ be generated by EDF algorithm. Then, we have \begin{equation}\label{mry} \liminf_ {k \, \rightarrow\, \infty} \| F(x_k)\|=0. \end{equation} \end{theorem} {\bf Proof}. We prove this theorem by contradiction. Suppose that \eqref{mry} does not hold, then there exists a positive constant $\tau$ such that \begin{equation}\label{EQ119wa} \|F(x_k)\|\geq \tau, \quad \forall k\geq 0. \end{equation} Clearly, $\|F_k\| \leq \| d_k\|$, which implies \begin{equation}\label{EQ120wa} \|d_k\|\geq \tau, \quad \forall k\geq 0. \end{equation} Observe that, \begin{equation} \|y_k\|=\|F(x_k)-F(x_{k-1})\| \leq L\|s_k\| \end{equation} \begin{equation}\label{EQ126wa} |\theta_k| \leq \frac {||s_k^T|| ||s_k||} {||s_k^T|| ||y_k||}\longrightarrow 0 \end{equation}meaning there exists a constant $\lambda \in (0,1)$ such that for sufficiently large $k$ \begin{equation}\label{EQ127wa} |\theta_k| \leq \lambda. \end{equation} Again from the definition of our $\beta_k^{*}$ we obtain \begin{equation}\label{EQ128wa} |\beta^{*}_k|\leq \frac {||\theta_ky_k-s_k|| ||F_k||} {||y^T_k||||s_k||} \leq M_1\frac {||\theta_ky_k-s_k|| } {||y^T_k||||s_k||} \longrightarrow 0 \end{equation}which implies there exists a constant $\rho \in (0,1)$ such that for sufficiently large $k$ \begin{equation}\label{EQ129wa} |\beta_k^{*}| \leq \rho. \end{equation}Without lost of generality, \begin{equation}\label{EQ130wa} ||d_{k}||\leq ||\theta_kg_{k+1}||+|\beta_k| || d_k||\leq \lambda M_1+\rho ||d_{k-1}|| \end{equation}which shows that the sequence $\left \{ d_{k} \right \}$ is bounded. This together with \eqref{EQ119wa} and \eqref{EQ120wa} yields a contradiction. Hence the proof is complete. \section{Numerical experiment} In this section, we compare the performance of our method for solving nonlinear equation (1) with A derivative-free conjugate gradient method and its global convergence for solving symmetric nonlinear equations \cite{jam17}. The two algorithms were implemented with the following parameters; $\omega_1=\omega_2=10^{-4}$, $\alpha_{k-1}=0.01$, $r=0.2$ and $\eta_k=\frac {1} {(k+1)^2}.$ The codes for both methods were written in Matlab 7.4 R2010a and run on a personal computer 1.8 GHz CPU processor and 4 GB RAM memory. We stopped the iteration if the toatal number of iterations exceeds 2000 or $||F_k|| \leq 10^{-4}$. "-" represent failure due one of the following: \\(i) insufficient memory \\(ii) Number of iteration exceed 2000 \\(ii) If $||F_k||$ is not a number(NaN) Problems 1-5 are from \cite{jam16} and the rest are artificial problems. \\Problem 1. \emph {The strictly convex function:} \\$F_i(x)=e^{x_i}-1\quad ;1,2,\dots ,n$ \\\\ Problem 2: .\emph{(n is multiple of 3)} for $ i =1,2,···,n/3,$ \\$ F_{3i-2}(x) = x_{3i-2}x_{3i-1} -x^2_{3i} -1,$ \\$ F_{3i-1}(x) = x_{3i-2}x_{3i-1}x_{3i} -x^2_ {3i-2} +x^2_{ 3i-1} -2,$ \\$F_{3i} (x) = e^{-x_{3i-2}} -e^{-x_{3i-1}}.$ \\\\ Problem 3. \emph {The Exponential function:} \\$F_i(x)=\frac {i} {10}(1-x_i^2-e^{-x_i^2})\quad ;1,2,\dots ,n-1$ \\$F_n(x)=\frac {n} {10}(1-e^{-x_n^2}).$ \\\\ Problem 4.\emph {Trigonometric Function:} \\$F_i(x)=2(n+i(1-cosx_i)-sinx_i-\sum_{j=1}^{n}cosx_j)(2sinx_i-cosx_i)$ for $ i =1,2,···,n$ \\\\ Problem 5.\emph { The discretized Chandrasehar's H-equation:} \\$F_i(x)=x_i-(1-\frac {c} {2n} \sum_{j=1}^{n} \frac {\mu_ix_j} {\mu_i+\mu_j})^{-1},\quad for i=1,2,\dots ,n,$ \\wth $c\in [0,1)$ and $\mu =\frac { i-0.5} {n},$ for $1\leq i \leq n.$ (In our experiment we take $c=0.9$). \\\\ Problem 6. \emph {Artificial Problem:} \\$F_{2i-1}(x)=x_{2i-1}+x_{2i}(-x_{2i}+5)(x_{2i}-2)-13\quad ;1,2,\dots ,\frac{n}{2}$ \\$F_{2i}(x)=x_{2i-1}+x_{2i}(x_{2i+1}x_{2i}-14)-29.$ \\\\ Problem 7.\emph {Artificial Problem: } \\$F_i(x)=3n-\sum_{i=1}^{n}cos(x_i-2)-\sum_{i=1}^{n}x_i-sin(x_i-2),\quad$ for $ i=1,2,\dots ,n,$ \\\\ Problem 8. \emph {Artificial Problem:} \\$F_i(x)=x_i-0.1x_{i+1}^2\quad ;1,2,\dots ,n-1$ \\$F_n(x)=x_n-0.1x_1^2.$ \\\\ Problem 9.\emph {Artificial Problem: } \\$F_i(x)=\sum_{i=1}^{n}x_i^2sinx_i-x_i^4+sinx_i^2,\quad$ for $ i=1,2,\dots ,n,$ \\\\ \begin{table}[htbp] \centering \caption{Test results for the two methods, where e =ones(n,1)} \begin{tabular}{rrrrrrrrr} \hline & & & \multicolumn{3}{c}{EDF} & \multicolumn{3}{c}{DFCG} \\ \hline \textbf{Problem (P)} & \textbf{$x_0$} & \textbf{n} & \textbf{Iter} & \textbf{Time} & \textbf{$||F_k||$} & \textbf{Iter} & \textbf{Time} & \textbf{$||F_k||$} \\ \hline P1 & e & 50 & 7 & 0.013401 & 1.01E-07 & - & - &- \\ & & 100 & 7 & 0.007188 & 1.43E-07 & - & - & - \\ & & 500 & 7 & 0.010701 & 3.19E-07 & - & - &- \\ & & 1000 & 7 & 0.014594 & 4.51E-07 & - & - & - \\ & & 10000 & 7 & 0.15437 & 1.43E-06 & - & - & - \\ & & 100000 & 7 & 0.505017 & 4.51E-06 & - & - & - \\ & & 500000 & 7 & 2.964011 & 1.01E-05 & - & - &- \\ & & 1000000 & 7 & 6.490167 & 1.43E-05 & - & - & - \\ & & & & & & & & \\ & 0.1e & 50 & 3 & 0.003589 & 4.66E-06 & 4 & 0.0056 & 2.71E-06 \\ & & 500 & 3 & 0.00514 & 1.47E-05 & 4 & 0.010405 & 0.010405 \\ & & 5000 & 3 & 0.022715 & 4.66E-05 & 4 & 0.072571 & 2.71E-05 \\ & & 50000 & 4 & 0.159338 & 1.87E-08 & 4 & 0.61336 & 8.58E-05 \\ & & 500000 & 4 & 1.769383 & 5.92E-08 & 5 & 7.388756 & 4.32E-08 \\ & & & & & & & & \\ P2 & e & 100 & 5 & 0.008126 & 1.21E-06 & 5 & 0.009352 & 8.32E-08 \\ & & 1000 & 5 & 0.016368 & 3.85E-06 & 5 & 0.023964 & 2.64E-07 \\ & & 10000 & 5 & 0.093909 & 1.22E-05 & 5 & 0.187341 & 8.36E-07 \\ & & 100000 & 5 & 0.606995 & 3.85E-05 & 5 & 1.640271 & 2.64E-06 \\ & & & & & & & & \\ & 0.1e & 500 & 3 & 0.00578 & 1.47E-05 & 4 & 0.012256 & 8.58E-06 \\ & & 5000 & 3 & 0.02127 & 4.66E-05 & 4 & 0.069252 & 2.71E-05 \\ & & 50000 & 4 & 0.161407 & 1.87E-08 & 4 & 0.600705 & 8.58E-05 \\ & & 500000 & 4 & 1.790072 & 5.92E-08 & 5 & 7.61203 & 4.32E-08 \\ & & & & & & & & \\ P3 & e & 500 & 13 & 0.011359 & 6.01E-05 & 11 & 0.054786 & 6.72E-05 \\ & & 1000 & 14 & 0.035401 & 5.18E-05 & 19 & 0.063617 & 5.81E-05 \\ & & 10000 & 10 & 0.179693 & 4.39E-05 & - & - & -\\ & & 50000 & 11 & 0.520165 & 8.38E-05 & - & - & - \\ & & & & & & & & \\ & 0.1e & 500 & 1 & 0.003457 & 1.24E-05 & 7 & 0.020203 & 8.87E-05 \\ & & 1000 & 12 & 0.029657 & 4.85E-05 & - & - & - \\ & & 5000 & 13 & 0.11261 & 9.26E-05 & - & - & - \\ & & 50000 & 12 & 0.597629 & 7.90E-05 & 95 & 6.054463 & 6.51E-05 \\ & & 100000 & 15 & 1.652158 & 9.51E-05 & - & - & - \\ & & 500000 & 17 & 8.94895 & 6.94E-05 & 23 & 14.79145 & 9.00E-05 \\ \hline \end{tabular}% \label{tab:addlabel}% \end{table}% \clearpage % Table generated by Excel2LaTeX from sheet 'Sheet4' \begin{table}[htbp] \centering \caption{Test results for the two methods, where e =ones(n,1)} \begin{tabular}{rrrrrrrrr} \hline & & & \multicolumn{3}{c}{EDF} & \multicolumn{3}{c}{DFCG} \\ \hline \textbf{Problem (P)} & \textbf{$x_0$} & \textbf{n} & \textbf{Iter} & \textbf{Time} & \textbf{$||F_||$} & \textbf{Iter} & \textbf{Time} & \textbf{$||F_k||$} \\ \hline P4 & e & 1000 & 26 & 0.149828 & 6.48E-05 & - & - & - \\ & & 10000 & 31 & 5.597554 & 4.53E-05 & - & - & -\\ & & 50000 & 37 & 9.69744 & 9.73E-07 & - & - &- \\ & & & & & & & & \\ & 0.1e & 1000 & 23 & 0.105184 & 3.13E-05 & - & - &- \\ & & 100000 & 34 & 5.610234 & 5.63E-06 & - & - & -\\ & & & & & & & & \\ P5 & e & 1000 & 24 & 0.067117 & 3.47E-05 & 51 & 0.223826 & 5.77E-05 \\ & & 5000 & 28 & 0.40068 & 5.09E-05 & 60 & 1.123916 & 8.65E-05 \\ & & 10000 & 29 & 0.644661 & 8.39E-05 & 61 & 1.916932 & 8.35E-05 \\ & & 50000 & 29 & 2.38082 & 9.88E-05 & - & - &- \\ & & & & & & & & \\ & 0.1e & 1000 & 19 & 0.064227 & 5.53E-05 & 215 & 1.107148 & 5.42E-05 \\ & & 5000 & 23 & 0.367682 & 6.63E-05 & 48 & 0.600497 & 9.04E-05 \\ & & 10000 & 25 & 0.577474 & 4.15E-05 & 51 & 1.578848 & 7.46E-05 \\ & & 50000 & 27 & 2.1800 & 7.02E-05 & 56 & 5.790128 & 8.08E-05 \\ & & & & & & & & \\ P6 & e & 500 & 546 & 0.639782 & 9.63E-05 & - & - & - \\ & & 1000 & 558 & 2.025724 & 9.86E-05 & - & - &- \\ & & 10000 & 605 & 9.510888 & 9.87E-05 & - & - & - \\ & & 50000 & 637 & 42.46983 & 9.69E-05 & - & - & - \\ & & & & & & & & \\ P7 & e & 500 & 8 & 0.02121 & 5.14E-09 & 9 & 0.034247 & 1.34E-05 \\ & & 5000 & 8 & 0.128485 & 2.69E-07 & 10 & 0.226067 & 2.24E-07 \\ & & 50000 & 8 & 0.658527 & 7.36E-05 & 9 & 1.520098 & 5.07E-05 \\ & & 100000 & 8 & 1.252776 & 5.68E-05 & 10 & 3.480638 & 1.50E-05 \\ & & & & & & & & \\ & 0.1e & 500 & 7 & 0.018371 & 1.61E-07 & 10 & 0.035181 & 1.08E-08 \\ & & 5000 & 7 & 0.154703 & 2.29E-07 & 10 & 0.226576 & 6.92E-07 \\ & & 50000 & 8 & 0.73483 & 1.07E-05 & 13 & 2.043117 & 7.45E-05 \\ & & 100000 & 10 & 1.8931 & 6.70E-05 & 10 & 3.425698 & 7.75E-05 \\ \hline \end{tabular}% \label{tab:addlabel}% \end{table}% \clearpage \begin{table}[htbp] \centering \caption{Test results for the two methods, where e =ones(n,1)} \begin{tabular}{rrrrrrrrr}\hline & & & \multicolumn{3}{c}{EDF} & \multicolumn{3}{c}{DFCG} \\ \hline \textbf{Problem (P)} & \textbf{$x_0$} & \textbf{n} & \textbf{Iter} & \textbf{Time} & \textbf{$||F_k||$} & \textbf{Iter} & \textbf{Time} & \textbf{$||F_k||$} \\ \hline P8 & e & 500 & 4 & 0.019037 & 2.85E-06 & 5 & 0.013187 & 5.32E-05 \\ & & 5000 & 4 & 0.031794 & 9.00E-06 & 6 & 0.139321 & 3.28E-08 \\ & & 50000 & 4 & 0.206865 & 2.85E-05 & 6 & 0.803616 & 1.04E-07 \\ & & 500000 & 4 & 2.693915 & 9.00E-05 & 6 & 7.289005 & 3.28E-07 \\ & & 1000000 & 5 & 5.726167 & 1.44E-09 & 6 & 14.47488 & 4.64E-07 \\ & & & & & & & & \\ & 0.1e & 1000 & 3 & 0.007335 & 3.19E-08 & 3 & 0.017376 & 3.45E-06 \\ & & 100000 & 3 & 0.423481 & 3.19E-07 & 3 & 1.642134 & 3.45E-05 \\ & & 1000000 & 3 & 3.699026 & 1.01E-06 & 4 & 12.16758 & 3.99E-09 \\ & & & & & & & & \\ P9 & e & 1000 & 23 & 0.137972 & 9.38E-05 & - & - &- \\ & & 10000 & 29 & 1.073695 & 9.28E-05 & - & - & - \\ & & 50000 & 32 & 3.617281 & 8.27E-05 & - & - &- \\ & & 100000 & 28 & 6.610935 & 8.65E-05 & - & - & -\\ & & 500000 & 30 & 24.53221 & 9.35E-05 & - & - & - \\ \hline \end{tabular}% \label{tab:addlabel}% \end{table}% \begin{figure}[htp] \begin{center} \includegraphics[angle=0,width=10cm]{pro2.eps} \end{center} \vspace{-1cm}{\textbf{\caption{Performance profile of EDF and DFCG conjugate gradient methods as the dimension increases(in term of CPU time)}\label{ds}}} \end{figure} \newpage \begin{figure}[htp1] \begin{center} \includegraphics[angle=0,width=10cm]{pro1.eps} \end{center} \vspace{-1cm}{\textbf{\caption{Performance profile of EDF and DFCG conjugate gradient methods as the dimension increases(in term of number of iterrations)}\label{ds}}} \end{figure} In the table above are numerical results, where "Iter" and "Time" stand for the total number of all iterations and the CPU time in seconds, respectively;$||F_k||$ is the norm of the residual at the stopping point. It is obvious that EDF algorithms has many advantages over DFCG irrespect of CPU time, number of iterations and norm of residual at the stopping. This is because, the Hyperplane $H_k$ strictly separates $x_k$ from the solution set of \eqref{EQ12} in EDF algorithm. However from the above figures, the performance profile of Dolan and Mor$\acute{e}$ \cite{jam15} shows that our claim is justified the superiority of the proposed algorithm. \section{Conclusion} In this paper, we developed an enhanced derivative-free conjugate gradient method with less number of iterations and CPU time compared to the existing methods. It is a fully derivative-free iterative method which possesses global convergence under some reasonable conditions. Numerical comparisons using a set of large-scale test problems show that the proposed method is promising. However to extend the method to general smooth and nonsmooth nonlinear equations will be our further research. \subsection{Conflict of Interests} There is no conflict of interest regarding the publication of this paper. %\section{Main Results} These are the main results of the paper. %{\bf Acknowledgements.} This is a text of acknowledgements. \begin{thebibliography}{99} \bibitem{jam2} Cheng, W., Chen, Z., \emph{ Nonmonotone Spectral method for large-Scale symmetric nonlinear equations. Numer. Algorithms}, 62(2013) 149-162. \bibitem{jam3} Gu, G., Z., Li, D,-H., Qi, L., Zhou, S.-Z.,� \emph{Descent direction of quasi-Newton methods for symmetric nonlinear equations}. SIAM J. Numer. Anal. 40(2002), 1763-1774. \bibitem{jam4}Zhang, L.,Zhou, W., Li, D.-H., \emph{ Global convergence of a modified Fletcher-Reeves conjugate gradient method with Armijo-type line search}. Numer. Math. 104(2006), 561-572. \bibitem{jam5} Li, D.H., Fukushima, \emph{A globally and superlinearly convergent Gauss-Newton-based BFGS methods for symmetric nonlinear equations}. \emph{SIAM J. Numer. Anal. 37(1999), 152-172} \bibitem{jam6} Li, D.-H., Wang, X.,\emph{ A modified Fletcher-Reeves-type derivative-free method for symmetric nonlinear equations} . Numer. Algebra Control Optim. 1(2011), 71-82. \bibitem{jam7} M.V. Solodov, B.F. Svaiter,\emph {A globally convergent inexact Newton method for systems of monotone equations, in: M. Fukushima, L. Qi (Eds.), Reformulation: Nonsmooth, Piecewise Smooth, Semismooth and Smoothing Methods}, Kluwe rAcademic Publishers,1998, pp.355–369.\bibitem{jam9}Zhou, W., Shen, D., \emph{ Convergence properties of an iterative method for solving symmetric nonlinear equations }, J. Optim. Theory Appl.(2014) doi: 10. 1007/s10957-014-0547-1. %\bibitem{jam10}Hager, W.W., Zhang, H., \emph{ A new conjugate gradient method with guaranteed descent and an efficient line search}, SIAM J. Optim. 16(2005), 170-192 \bibitem{jam11} Yunhai, X., Chunjie,W., Soon, Y.W.,\emph{ Norm descent conjugate gradient method for solving symmetric nonlinear equations}, J. Glo. Optim.(2014) DOI 10.1007/s10898-014-0218-7. \bibitem{jam12} Yuan, G., Lu, X., Wei, Z.,\emph{BFGS trust-region method for symmetric nonlinear equations}, J. Comput. Appl. Math. 230(2009), 44-58. \bibitem{jam13}Zhou, W., Shen, D.,\emph{ An inexact PRP conjugate gradient method for symmetric nonlinear equations} Numer. Functional Anal. Opt. 35(2014), 370-388. %\bibitem{jam14}]Zhang, L., Zhou, W., Li, D.-H.,\emph{ A descent modified polak-Rebiere-Polyak conjugate gradient method and its global convergence}, IMA J. Numer. Anal. 26(2006), 629-640. \bibitem{jam15} Dolan, E.D., More, J.J., \emph {Benchmarking Optimization software with Performance profiles}. Maths. program. 91(2002), 201-213. \bibitem{jam16} La Cruz, W., Martinez, J.M., Raydan, M.,\emph{spetral residual method without gradient information for solving large-scale nonlinear systems of equations: Theory and experiments},P. optimization76(2004),79.1008-00. \bibitem{jam17} M.Y. Waziri, J. Sabi'u,\emph{ A derivative-free conjugate gradient method and its global convergence for solving symmetric nonlinear equations}. International J. of mathematics and mathematical science vol.(2015), doi:10.1155/2015/961487. \bibitem{jam18}W.W. Hager,H.Zhang, A New conjugate gradient Method with Guaranteed Descent and an efficient line search.SIAM J. Optim. 16(2005), 170-192. \end{thebibliography} \end{document}