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We propose a new method for the approximate solution of the Lyapunov equationwith rank-$1$ right-hand side, which is based on extended rational Krylovsubspace approximation with adaptively computed shifts. The shift selection isobtained from the connection between the Lyapunov equation, solution of systemsof linear ODEs and alternating least squares method for low-rank approximation.The numerical experiments confirm the effectiveness of our approach.

From low-rank approximation to an efficient rational Krylov subspace method for the Lyapunov equation