  Recently a new algorithm for model reduction of second order linear dynamical
systems with proportional damping, the Adaptive Iterative Rational Global
Arnoldi (AIRGA) algorithm, has been proposed. The main computational cost of
the AIRGA algorithm is in solving a sequence of linear systems. These linear
systems do change only slightly from one iteration step to the next. Here we
focus on efficiently solving these systems by iterative methods and the choice
of an appropriate preconditioner. We propose the use of relevant iterative
algorithm and the Sparse Approximate Inverse (SPAI) preconditioner. A technique
to cheaply update the SPAI preconditioner in each iteration step of the model
order reduction process is given. Moreover, it is shown that under certain
conditions the AIRGA algorithm is stable with respect to the error introduced
by iterative methods. Our theory is illustrated by experiments. It is
demonstrated that SPAI preconditioned Conjugate Gradient (CG) works well for
model reduction of a one dimensional beam model with AIRGA algorithm. Moreover,
the computation time of preconditioner with update is on an average 2/3 rd of
the computation time of preconditioner without update. With average timings
running into hours for very large systems, such savings are substantial.
