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A novel formulation of approximate truncated balanced realization (TBR) is introduced to unify three approaches: two iterative methods for solving the underlying Lyapunov equations - the alternating directions implicit (ADI) iteration and the rational Krylov subspace method (RKSM) - and a two-step procedure that performs a Krylov-based projection and subsequently direct TBR. The framework allows to compare the three methods with respect to solvability, fidelity, numerical effort, stability preservation and global error bounds, which suggests to merge ADI with the two-step procedure

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Model Order Reduction by Approximate Balanced Truncation: A Unifying Framework / Modellreduktion durch approximatives Balanciertes Abschneiden: eine vereinigende Formulierung