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MiS Preprint

A Multigrid Method to Solve Large Scale Sylvester Equations

Lars Grasedyck and Wolfgang Hackbusch


We consider the Sylvester equation $AX-XB+C=0$, where the $n\times m$ matrix is of low rank and the spectra of the $n\times n$ and $m\times m$ matrices $A$,$B$ are separated by a line. The solution $X$ can be approximated in a data-sparse format and we develop a multigrid algorithm that computes the solution in this format. For the multigrid method to work we need a hierarchy of discretisations, i.e., the matrices $A$ and $B$ each stem from the discretisation of some partial differential operator of elliptic type. The algorithm is of complexity $\mathcal{O}(n+m)$, or, more precisely, if the solution can be represented with $(n+m)\log(n+m)$ data then the complexity of the algorithm is $\mathcal{O}((n+m)\log(n+m)^{2})$.

MSC Codes:
65F05, 65F30, 65F50
sylvester equation, low rank approximation, multigrid method

Related publications

2007 Repository Open Access
Lars Grasedyck and Wolfgang Hackbusch

A multigrid method to solve large scale Sylvester equations

In: SIAM journal on matrix analysis and applications, 29 (2007) 3, pp. 870-894