# Low-Rank Solution of Lyapunov Equations

## Abstract

*X*of the Lyapunov equation

*AX*+

*XA*= -

^{T}*BB*. The coefficient matrix

^{T}*A*is assumed to be large, and the rank of the right-hand side -

*BB*is assumed to be much smaller than the size of

^{T}*A*. The CF--ADI algorithm requires only matrix-vector products and matrix-vector solves by shifts of

*A*. Hence, it enables one to take advantage of any sparsity or structure in

*A*.

*X*. We characterize a group of spanning sets for the range of

*X*. A connection is made between the approximation of the dominant invariant subspace of

*X*and the generation of various low-order Krylov and rational Krylov subspaces. It is shown by numerical examples that the rational Krylov subspace generated by the CF--ADI algorithm, where the shifts are obtained as the solution of a rational minimax problem, often gives the best approximation to the dominant invariant subspace of

*X*.

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**Published online**: 4 August 2006

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