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differential equation
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In § we pointed out that conjugate
gradient methods for non-selfadjoint systems require the storage of
previously calculated vectors. Therefore it is somewhat remarkable
that preconditioning by the symmetric part
of the coefficient matrix
leads to a method that does not need this extended storage.
Such a method was proposed by Concus and Golub [54]
and Widlund [211].
However, solving a system with the symmetric part of a matrix may be
no easier than solving a system with the full matrix. This problem may
be tackled by imposing a nested iterative method, where a
preconditioner based on the symmetric part is used.
Vassilevski [207] proved that the efficiency of this
preconditioner for the symmetric part carries over to the outer
method.