Numerical methods for an algebraic Riccati equation arising in transport theory in the critical case

In this paper, we are interested in computing the minimal positive solution of an algebraic Riccati equation arising in transport theory. The coefficient matrices of this equation have two parameters a and c. When 0 < a < 1, 0 < c < 1, the existing numerical algorithms can solve its minimal positive solution very quickly and efficiently. However, these algorithms do not converge or converge slowly to the minimal positive solution for a = 0 and c = 1. In this case, we propose two methods to improve the effectiveness of these algorithms. The first method is a modified double shift technique to transform the original algebraic Riccati equation into a new one, and the two equations have the same minimal positive solution. The second method is a deflating technique to transform the original equation into a new low-order algebraic Riccati equation, and we give the relationship between the minimal positive solutions of these two equations. The minimal positive solutions of two new equations both can be effectively solved by the existing algorithms. We show that two new equations both maintain the symmetry. Moreover, two new equations are M-matrix algebraic Riccati equations. Numerical experiments are given to illustrate the results.