We propose a factorized sparse approximate inverse preconditioner for
large real symmetric and indefinite linear systems. Factorized
approximate inverse preconditioners have been developed and used
successfully for symmetric positive definite and many nonsymmetric linear
systems, but it remains challenging to construct effective
preconditioners of this type in the symmetric and indefinite setting. In
this study, we show that whether the coefficient matrix is close to
diagonally dominant tends to have a greater impact on the density of
factorized approximate inverses than the inertia itself. To enhance the
diagonal dominance, we use HSL's symmetric MC64 subroutine for
preprocessing, which scales and relocates large entries to super- and
sub-diagonals near the diagonal. Our main incomplete conjugation
algorithm uses the Bunch-Kauffman partial pivoting. A
sparsest-columns-first dynamic reordering is proposed and shown effective
for maintaining the sparsity of the factorized preconditioners. Numerical
results are provided to illustrate the effectiveness of the new
preconditioner and its potential to be a competitive alternative to
incomplete LDL
preconditioners, especially on parallel
computers.