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 [56] and Widlund [216].

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 [212] proved that the efficiency of this preconditioner for the symmetric part carries over to the outer method.

Mon Nov 20 08:52:54 EST 1995