Iterative solution of the Lyapunov matrix equation
Applied Mathematics Letters
Optimal alternating direction implicit parameters for nonsymmetric systems of linear equations
SIAM Journal on Numerical Analysis
A new algorithm for L2 optimal model reduction
Automatica (Journal of IFAC)
Feje´r-Walsh points for rational functions and their use in the ADI iterative method
Journal of Computational and Applied Mathematics - Special issue on computational complex analysis
Krylov subspace methods for solving large Lyapunov equations
SIAM Journal on Numerical Analysis
A Cyclic Low-Rank Smith Method for Large Sparse Lyapunov Equations
SIAM Journal on Scientific Computing
Solution of the matrix equation AX + XB = C [F4]
Communications of the ACM
Low Rank Solution of Lyapunov Equations
SIAM Journal on Matrix Analysis and Applications
Low-Rank Solution of Lyapunov Equations
SIAM Review
Approximation of Large-Scale Dynamical Systems (Advances in Design and Control) (Advances in Design and Control)
Low rank approximate solutions to large Sylvester matrix equations
Applied Mathematics and Computation
A new projection method for solving large Sylvester equations
Applied Numerical Mathematics
A New Iterative Method for Solving Large-Scale Lyapunov Matrix Equations
SIAM Journal on Scientific Computing
SIAM Journal on Scientific Computing
Trail to a Lyapunov equation solver
Computers & Mathematics with Applications
$\mathcal{H}_2$ Model Reduction for Large-Scale Linear Dynamical Systems
SIAM Journal on Matrix Analysis and Applications
On the ADI method for Sylvester equations
Journal of Computational and Applied Mathematics
h2-norm optimal model reduction for large scale discrete dynamical MIMO systems
Journal of Computational and Applied Mathematics
On the Convergence of Rational Ritz Values
SIAM Journal on Matrix Analysis and Applications
Krylov subspace methods for projected Lyapunov equations
Applied Numerical Mathematics
Analysis of the Rational Krylov Subspace and ADI Methods for Solving the Lyapunov Equation
SIAM Journal on Numerical Analysis
Structure-preserving tangential interpolation for model reduction of port-Hamiltonian systems
Automatica (Journal of IFAC)
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The ADI iteration is closely related to the rational Krylov projection methods for constructing low rank approximations to the solution of Sylvester equations. In this paper we show that the ADI and rational Krylov approximations are in fact equivalent when a special choice of shifts are employed in both methods. We will call these shifts pseudo H"2-optimal shifts. These shifts are also optimal in the sense that for the Lyapunov equation, they yield a residual which is orthogonal to the rational Krylov projection subspace. Via several examples, we show that the pseudo H"2-optimal shifts consistently yield nearly optimal low rank approximations to the solutions of the Lyapunov equations.