Block methods are used for two major reasons. The first one is to aid in
reliably determining multiple and/or clustered eigenvalues.
Although [26]
indicates that an unblocked Arnoldi
method coupled with an appropriate deflation strategy may be used to compute
multiple and/or clustered eigenvalues, a relatively small convergence tolerance
is required to reliably compute clustered eigenvalues. Many problems do not require
this much accuracy, and such a criterion can result in unnecessary computation.
The second reason for using a block formulation is related to computational
efficiency.
Often, when a matrix-vector product with is very costly, it is possible to
compute the action of
on several vectors at once with roughly the same cost
as computing a single matrix-vector product. This can happen, for example, when
the matrix is so large that it must be retrieved from disk each time a
matrix-vector product
is performed. In this situation, a block method may have considerable advantages.
The performance tradeoffs of block methods and potential improvements to deflation techniques are under investigation.