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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  and Widlund .
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  proved that the efficiency of this preconditioner for the symmetric part carries over to the outer method.