The paper presents a modified Newton iterative method (MNI) for solving non-symmetric algebraic Riccati equations arising in transport theory:
\[ XCX-XE-AX+B=0 \tag{1} \] where \(A,B,C,E\in{\mathbb R}^{n\times n}\) are given by
\[ A=\Delta-eq^T,\quad B=ee^T,\quad C=qq^T,\quad E=D-qe^T \]
Here, \(e=(1,1,\dots,1)^T\) and \(q=(q_1,q_2,\dots,q_n)^T\) with \(q_i=\frac{c_i}{2\omega_i},\)
\[ \begin{aligned} \Delta&= \text{diag}(\delta_1,\delta_2,\dots,\delta_n)\quad \text{with}\quad \delta_i=\frac{1}{c\omega_i(1+\alpha)}\\ D&= \text{diag}(d_1,d_2,\dots,d_n) \quad\text{with}\quad d_i=\frac{1}{c\omega_i(1-\alpha)}. \end{aligned} \]
and
\[ 0<c\leq 1,\;0\leq\alpha<1,\;0<\omega_n<\dots<\omega_2<\omega_1<1,\;\sum_{i=1}^nc_i=1,\;c_i>0,\;i=1,2,\dots,n. \]
(1) has a positive solution (in the component-wise sense). Since only the minimal positive solution is physically meaningful, one uses the modified Newton method for computing this one.
The first section is an introduction in nature. The authors present the Hadamard product of two matrices, the \(Z\)-matrix and the nonsingular \(M\)-matrix.
The second section presents the modified Newton iterative procedure and the corresponding algorithm for practical implementation. Thus for the vector equation
\[ f(u,v)=0 \tag{2} \]
the Newton method is given by
\[ f'(\tilde u_k,\tilde v_k) \left[\left( \begin{matrix} \tilde u_{k+1} \\ \tilde v_{k+1}\\ \end{matrix} \right)-\left( \begin{matrix} \tilde u_{k} \\ \tilde v_{k}\\ \end{matrix} \right)\right]=-f(\tilde u_k,\tilde v_k) \tag{3} \]
where for any \(u,v\in{\mathbb R}^n,\) \(f'(\tilde u_k,\tilde v_k)\) represents the Jacobian matrix, while the modified Newton iterative scheme is the following
\[ \begin{aligned} f'(u_k,v_k) \left[\left( \begin{matrix} \tilde u_{k} \\ \tilde v_{k}\\ \end{matrix} \right)-\left( \begin{matrix} u_{k} \\ v_{k}\\ \end{matrix} \right)\right]&= f(u_k,v_k)\\ f'(u_k,v_k) \left[\left( \begin{matrix} u_{k+1} \\ v_{k+1}\\ \end{matrix} \right)-\left( \begin{matrix} \tilde u_{k} \\ \tilde v_{k}\\ \end{matrix} \right)\right]&= -f(\tilde u_k,\tilde v_k),\quad k=0,1,2\dots\end{aligned}\tag{4} \]
In the third section the authors give some convergence results. The authors show that the sequence of vectors generated by the MNI is monotonically increasing and converges to the minimal positive solution of the vector equation (2).
The numerical test presented in the fourth section illustrates the performance of the modified Newton iterative method in comparison with the Newton iterative method.
The last section contains the conclusions: the modified Newton iterative method is feasible and effective, and outperforms the Newton iterative method.