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Other extensions of multigrid include techniques where no PDE and no geometrical problem background is used to construct the multilevel hierarchy. Such '''algebraic multigrid methods''' (AMG) construct their hierarchy of operators directly from the system matrix and thus become true black-box solvers for [[sparse matrices]].
== Algorithm ==
There are many variations of multigrid algorithms, but the common features are that a hierarchy of discretisations (grids) is considered. The important steps are:
* '''Smoothing''' – reducing high frequency errors, for example using a few iterations of the [[Gauss-Seidel method]].
* '''Restriction''' – downsampling the [[residual (mathematics)|residual]] error to a coarser grid.
* '''Prolongation''' – interpolating a correction computed on a coarser grid onto a finer grid.
== Convergence rate ==
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