Efficient cyclic reduction for Quasi-Birth-Death problems with rank structured blocks

We provide effective algorithms for solving block tridiagonal block Toeplitz systems with m×m quasiseparable blocks, as well as quadratic matrix equations with m×m quasiseparable coefficients, based on cyclic reduction and on the technology of rank-structured matrices. The algorithms rely on the exponential decay of the singular values of the off-diagonal submatrices generated by cyclic reduction. We provide a formal proof of this decay in the Markovian framework. The results of the numerical experiments that we report confirm a significant speed up over the general algorithms, already starting with the moderately small size m≈102.