Solving the nonnegative solution for a (shifted) nonsymmetric algebraic Riccati equation in the critical case

We are interested in computing the nonnegative solution of a nonsymmetric algebraic Riccati equation arising in transport theory. The coefficient matrices of this equation have two parameters c and α. There have been some iterative methods presented by Lu in [13] and Bai et al. in [2] to solve the minimal positive solution for c≠1 or α≠0. While the equation has a unique nonnegative solution when c=1 and α=0, all the methods presented by Lu and Bai cannot be used to find the nonnegative solution. To cope with this problem, a shifted technique is used in this paper to transform the original Riccati equation into a new one so that all the methods can be effectively employed to solve the nonnegative solution. Numerical experiments are given to illustrate the results.