Circulant preconditioners for stochastic automata networks

Stochastic Automata Networks (SANs) are widely used in modeling communication systems , manufacturing systems and computer systems. The solution of the steady state distribution cannot be obtained eeciently by direct methods such as LU decomposition due to the huge size of the generator matrix. EEcient numerical algorithm should make use of the tensor structure of SANs' generator matrix. The generalized Conjugate Gradient (CG) methods are possible choices. However, their convergence rates are slow in general. To speed up the convergence rate, we consider preconditioned CG methods. In this paper, we construct circulant preconditioners for the SANs. The preconditioners considered here are easy to construct, can be inverted eeciently and give very fast convergence rate.