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A multigrid method with an intentionally reduced tolerance can be used as an efficient [[preconditioning|preconditioner]] for an external iterative solver, e.g.,<ref>Andrew V Knyazev, Klaus Neymeyr. [http://etna.mcs.kent.edu/volumes/2001-2010/vol15/abstract.php?vol=15&pages=38-55 Efficient solution of symmetric eigenvalue problems using multigrid preconditioners in the locally optimal block conjugate gradient method]. Electronic Transactions on Numerical Analysis, 15, 38–55, 2003.</ref> The solution may still be obtained in <math>O(N)</math> time as well as in the case where the multigrid method is used as a solver. Multigrid preconditioning is used in practice even for linear systems, typically with one cycle per iteration, e.g., in [[Hypre]]. Its main advantage versus a purely multigrid solver is particularly clear for nonlinear problems, e.g., [[eigenvalue]] problems.
If the matrix of the original equation or an eigenvalue problem is symmetric positive definite (SPD), the preconditioner is commonly constructed to be SPD as well, so that the standard [[conjugate gradient]] (CG) [[iterative methods]] can still be used. Such imposed SPD constraints may complicate the construction of the preconditioner, e.g., requiring coordinated pre- and post-smoothing. However, [[preconditioning|preconditioned]] [[steepest descent]] and [[Conjugate gradient method#The flexible preconditioned conjugate gradient method|flexible CG methods]] for SPD linear systems and [[LOBPCG]] for symmetric eigenvalue problems are all shown<ref>
==Bramble–Pasciak–Xu preconditioner==
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