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Efficient cyclic reduction for quasi-birth-death problems with rank structured blocks. (English) Zbl 1372.65122
Summary: We provide effective algorithms for solving block tridiagonal block Toeplitz systems with $$m \times m$$ quasiseparable blocks, as well as quadratic matrix equations with $$m \times m$$ 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 $$m \approx 10^2$$.

##### MSC:
 65F30 Other matrix algorithms (MSC2010) 65F50 Computational methods for sparse matrices 15B05 Toeplitz, Cauchy, and related matrices 65C40 Numerical analysis or methods applied to Markov chains 15A24 Matrix equations and identities
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