The standard Hermitian Lanczos algorithm uses the Krylov
subspaces induced by the matrix and a single starting
vector
to produce approximate solutions of the
Hermitian eigenproblem
.
However, there are situations where the use of a block of
starting vectors, instead of a single starting vector, is preferable.
One such case is that of matrices with multiple or closely clustered eigenvalues.
To obtain basis vectors for the eigenspace corresponding to
such a cluster of
eigenvalues, block Krylov subspaces
induced by
and a block of
starting vectors need to be used.
An important application, where multiple starting vectors are given
as part of the problem, is reduced-order modeling of
-input
-output linear dynamical systems;
see §7.10.4 below.
While this application, in general, involves a non-Hermitian
square matrix
, a rectangular ``input'' matrix
with
columns, and a rectangular ``output'' matrix
with
columns, the special case where
is Hermitian and the input and output matrices
and
are identical is of particular importance.
For example, this special case arises in the context of
interconnect modeling of VLSI circuits; see, e.g., [176].
For this special case, an extension of the Hermitian Lanczos algorithm
to multiple starting vectors, namely, the
columns of
,
is needed.
Finally, employing block Krylov subspaces is also beneficial whenever
computing matrix-matrix products , where
is a matrix
with
columns, is cheaper than sequentially computing
matrix-vector products
for
vectors.
Lanczos methods based on block Krylov subspaces allow
computation of all necessary multiplications with
as
matrix-matrix products
with blocks
of size
, whereas
in the standard Hermitian Lanczos algorithm multiplications with
have to be computed as a sequence of matrix-vector products
.
In this section, we describe a band Lanczos method for the
Hermitian eigenvalue problem,