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Low rank solution of Lyapunov equations. (English) Zbl 1016.65024

The Cholesky factor-alternating direction implicit algorithm is presented to compute a low rank approximation to the solution \(X\) of the Lyapunov equation \(AX+XA^T=-BB^T\) with large matrix \(A\) and right hand side of low rank. The algorithm requires only matrix-vector products and linear solvers.
The dominant invariant subspace of the solution is also approximated and a spanning set for the range of \(X\) is characterized. Numerical examples show that the rational Krylov subspace generated by the new 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 \(X\).

MSC:

65F30 Other matrix algorithms (MSC2010)
65F10 Iterative numerical methods for linear systems
93C05 Linear systems in control theory
15A24 Matrix equations and identities
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