On algebraic Riccati equations associated with regular singular M-matrices

We further study algebraic Riccati equations associated with regular singular M-matrices. An M-matrix M is said to be regular if Mv≥0 for some v>0, so every irreducible singular M-matrix is a regular singular M-matrix. We prove a property about the product of the minimal nonnegative solution of such an algebraic Riccati equation and the minimal nonnegative solution of its dual equation. In the critical case, we show that the alternating-directional doubling algorithm (which includes the structure-preserving double algorithm as a special case) has linear convergence with rate 1/2. The results enhance our understanding of the behaviour of doubling algorithms for finding the minimal nonnegative solutions.