  A sparse matrix is called double irregular sparse if it has at least one
relatively dense column and row, and it is double regular sparse if all the
columns and rows of it are sparse. The sparse approximate inverse
preconditioning procedures SPAI, PSAI($tol$) and RSAI($tol$) are costly and
even impractical to construct preconditioners for a large sparse nonsymmetric
linear system with the coefficient matrix being double irregular sparse, but
they are efficient for double regular sparse problems. Double irregular sparse
linear systems have a wide range of applications, and 4.4\% of the nonsymmetric
matrices in the Florida University collection are double irregular sparse. For
this class of problems, we propose a transformation approach, which consists of
four steps: (i) transform a given double irregular sparse problem into a small
number of double regular sparse ones with the same coefficient matrix
$\hat{A}$, (ii) use SPAI, PSAI($tol$) and RSAI($tol$) to construct sparse
approximate inverses $M$ of $\hat{A}$, (iii) solve the preconditioned double
regular sparse linear systems by Krylov solvers, and (iv) recover an
approximate solution of the original problem with a prescribed accuracy from
those of the double regular sparse ones. A number of theoretical and practical
issues are considered on the transformation approach. Numerical experiments on
a number of real-world problems confirm the very sharp superiority of the
transformation approach to the standard approach that preconditions the
original double irregular sparse problem by SPAI, PSAI($tol$) or RSAI($tol$)
and solves the resulting preconditioned system by Krylov solvers.
