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A new version of the alternating directions implicit (ADI) iteration for thesolution of large-scale Lyapunov equations is introduced. It generalizes thehitherto existing iteration, by incorporating tangential directions in the waythey are already available for rational Krylov subspaces. Additionally, firststrategies to adaptively select shifts and tangential directions in eachiteration are presented. Numerical examples emphasize the potential of the newresults.

ADI iteration for Lyapunov equations: a tangential approach and adaptive shift selection