Topics in matrix analysis
Iterative solution of the Lyapunov matrix equation
Applied Mathematics Letters
Alternating direction implicit iteration for systems with complex spectra
SIAM Journal on Numerical Analysis
Krylov subspace methods for solving large Lyapunov equations
SIAM Journal on Numerical Analysis
Oblique Projection Methods for Large Scale Model Reduction
SIAM Journal on Matrix Analysis and Applications
Extended Krylov Subspaces: Approximation of the Matrix Square Root and Related Functions
SIAM Journal on Matrix Analysis and Applications
New ADI model problem applications
ACM '86 Proceedings of 1986 ACM Fall joint computer conference
Global FOM and GMRES algorithms for matrix equations
Applied Numerical Mathematics
The Faber operator and its boundedness
Journal of Approximation Theory
A Cyclic Low-Rank Smith Method for Large Sparse Lyapunov Equations
SIAM Journal on Scientific Computing
Low Rank Solution of Lyapunov Equations
SIAM Journal on Matrix Analysis and Applications
Approximation of Large-Scale Dynamical Systems (Advances in Design and Control) (Advances in Design and Control)
A New Iterative Method for Solving Large-Scale Lyapunov Matrix Equations
SIAM Journal on Scientific Computing
Algorithm 882: Near-Best Fixed Pole Rational Interpolation with Applications in Spectral Methods
ACM Transactions on Mathematical Software (TOMS)
$\mathcal{H}_2$ 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
Convergence Analysis of Projection Methods for the Numerical Solution of Large Lyapunov Equations
SIAM Journal on Numerical Analysis
SIAM Journal on Numerical Analysis
Solution of Large Scale Evolutionary Problems Using Rational Krylov Subspaces with Optimized Shifts
SIAM Journal on Scientific Computing
Error Estimates and Evaluation of Matrix Functions via the Faber Transform
SIAM Journal on Numerical Analysis
On Adaptive Choice of Shifts in Rational Krylov Subspace Reduction of Evolutionary Problems
SIAM Journal on Scientific Computing
Convergence analysis of the extended Krylov subspace method for the Lyapunov equation
Numerische Mathematik
An Error Analysis for Rational Galerkin Projection Applied to the Sylvester Equation
SIAM Journal on Numerical Analysis
On the ADI method for the Sylvester equation and the optimal-H2 points
Applied Numerical Mathematics
Computers & Mathematics with Applications
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For large scale problems, an effective approach for solving the algebraic Lyapunov equation consists of projecting the problem onto a significantly smaller space and then solving the reduced order matrix equation. Although Krylov subspaces have been used for a long time, only more recent developments have shown that rational Krylov subspaces can be a competitive alternative to the classical and very popular alternating direction implicit (ADI) recurrence. In this paper we develop a convergence analysis of the rational Krylov subspace method (RKSM) based on the Kronecker product formulation and on potential theory. Moreover, we propose new enlightening relations between this approach and the ADI method. Our results provide solid theoretical ground for recent numerical evidence of the superiority of RKSM over ADI when the involved parameters cannot be computed optimally, as is the case in many practical application problems.