The standard non-Hermitian Lanczos algorithm as presented in
§7.8 uses the Krylov
subspaces induced by the matrix and a pair of single right and
left starting vectors
and
, to produce approximate solutions
of the NHEP,
There are situations where the use of blocks of right
and left starting vectors, instead of a pair of single starting
vectors, is preferable.
One such case is eigenvalue computations for matrices with multiple
or closely clustered eigenvalues.
Another important application is reduced-order modeling of
linear dynamical systems.
Here, the right and left starting blocks are given as part of the
problem, as is described in more detail in §7.10.4 below.
Finally, the use of blocks of starting vectors is also beneficial
whenever computing matrix-matrix products and
, where
and
are blocks of vectors, is cheaper than sequentially
computing matrix-vector products
and
for all the
columns of
and
.
Block Lanczos methods for blocks of equal size were
discussed in §7.9.
In this section, we describe the non-Hermitian band Lanczos
method, which extends the standard non-Hermitian Lanczos algorithm
for single starting vectors to blocks of right and
left
starting vectors,
The matrix and the starting vectors (7.59) induce
the right block Krylov sequence
The non-Hermitian band Lanczos method discussed in this
section can also be viewed as an extension of the Hermitian band
Lanczos method described in §4.6 to
general square non-Hermitian matrices.
For the special case of right and left starting blocks of the same
size, i.e., , the band Lanczos method is also related to
the block Lanczos method
described in §7.9.
However, the band Lanczos method is more general in that it
can handle the case of arbitrary block sizes
.
Even for the special case
, there are advantages of
the band method over the block Lanczos method; see
§7.10.5 below.