For an iterative method, as described in Chapter 7,
for the reduced standard eigenvalue problem (8.3)
or (8.4), it is not necessary to evaluate
the product or . This is a key observation
in the treatment of large sparse and .
One only needs to evaluate matrix-vector products, like
(a) form ,Similarly, the vector , if necessary, can be evaluated as
(b) solve for .
(a) solve for ,One can exploit sparsity in each of these steps. In general, the iterative methods, with the exception of the Jacobi-Davidson method, require accurate solution of the linear systems at step (b) (and (b)). If possible, a direct linear system solver, say, using LU factorization of , is preferable. See §10.3 for the discussions on dense or sparse LU factorization.
(b) form .
The error introduced by this transformation to standard form can be proportional to . If is ill-conditioned, then the approach is potentially suspect. In that situation, one may consider the SI for the transformation or the usage of the Jacobi-Davidson method discussed below.