  A preconditioning theory is presented which establishes sufficient conditions
for multiplicative and additive Schwarz algorithms to yield self-adjoint
positive definite preconditioners. It allows for the analysis and use of
non-variational and non-convergent linear methods as preconditioners for
conjugate gradient methods, and it is applied to domain decomposition and
multigrid. It is illustrated why symmetrizing may be a bad idea for linear
methods. It is conjectured that enforcing minimal symmetry achieves the best
results when combined with conjugate gradient acceleration. Also, it is shown
that absence of symmetry in the linear preconditioner is advantageous when the
linear method is accelerated by using the Bi-CGstab method. Numerical examples
are presented for two test problems which illustrate the theory and
conjectures.
