  We investigate the SPAI and PSAI preconditioning procedures and shed light on
two important features of them: (i) For the large linear system $Ax=b$ with $A$
irregular sparse, i.e., with $A$ having $s$ relatively dense columns, SPAI may
be very costly to implement, and the resulting sparse approximate inverses may
be ineffective for preconditioning. PSAI can be effective for preconditioning
but may require excessive storage and be unacceptably time consuming; (ii) the
situation is improved drastically when $A$ is regular sparse, that is, all of
its columns are sparse. In this case, both SPAI and PSAI are efficient.
Moreover, SPAI and, especially, PSAI are more likely to construct effective
preconditioners. Motivated by these features, we propose an approach to making
SPAI and PSAI more practical for $Ax=b$ with $A$ irregular sparse. We first
split $A$ into a regular sparse $\tilde A$ and a matrix of low rank $s$. Then
exploiting the Sherman--Morrison--Woodbury formula, we transform $Ax=b$ into
$s+1$ new linear systems with the same coefficient matrix $\tilde A$, use SPAI
and PSAI to compute sparse approximate inverses of $\tilde A$ efficiently and
apply Krylov iterative methods to solve the preconditioned linear systems.
Theoretically, we consider the non-singularity and conditioning of $\tilde A$
obtained from some important classes of matrices. We show how to recover an
approximate solution of $Ax=b$ from those of the $s+1$ new systems and how to
design reliable stopping criteria for the $s+1$ systems to guarantee that the
approximate solution of $Ax=b$ satisfies a desired accuracy. Given the fact
that irregular sparse linear systems are common in applications, this approach
widely extends the practicability of SPAI and PSAI. Numerical results
demonstrate the considerable superiority of our approach to the direct
application of SPAI and PSAI to $Ax=b$.
