Efficient methods for solving a nonsymmetric algebraic Riccati equation arising in stochastic fluid models

We consider the nonsymmetric algebraic Riccati equation XM12X+XM11+M22X+M21=0XM12X+XM11+M22X+M21=0, where M11,M12,M21,M22M11,M12,M21,M22 are real matrices of sizes n×n,n×m,m×n,m×mn×n,n×m,m×n,m×m, respectively, and M=[Mij]i,j=12 is an irreducible singular M  -matrix with zero row sums. The equation plays an important role in the study of stochastic fluid models, where the matrix -M-M is the generator of a Markov chain. The solution of practical interest is the minimal nonnegative solution. This solution may be found by basic fixed-point iterations, Newton's method and the Schur method. However, these methods run into difficulties in certain situations. In this paper we provide two efficient methods that are able to find the solution with high accuracy even for these difficult situations.