For eigenvalue computations, one is usually free to choose
the right and left starting vectors for the band Lanczos method.
However, there are other important applications where the
starting vectors are given as part of the problem.
One such application is reduced-order modeling of
time-invariant linear dynamical systems with multiple inputs
and multiple outputs; see, e.g., [176] for a recent survey.
Such systems are characterized by matrix-valued transfer functions
of the form
For reduced-order modeling, the entries and
with negative indices
are also used.
More precisely, in this case, step (16) in Algorithm 7.16
is augmented by the following six lines:
ifHere again we use the convention that entries,
setfor all
with
,
and set![]()
if,
setfor all
with
,
and set![]()
The matrix is upper triangular and of size
.
Here,
is defined as the value of
at
iteration of Algorithm 7.16 for which
is
reached.
It turns out that
is just the value of
minus
the number of right initial vectors
that have
been deflated.
In particular,
, and
if and only if none of the
right starting vectors has been deflated.
The matrix
is upper triangular and of size
.
Here,
is defined as the value of
at the iteration
of Algorithm 7.16 for which
is reached.
The number
is the value of
minus
the number of left initial vectors
that have
been deflated.
In particular,
, and
if and only if none of
the left starting vectors has been deflated.
The entries of
are the
coefficients used to turn the right starting vectors into
the first
right Lanczos vectors, and
the entries of
are the
coefficients used to turn the left starting vectors into
the first
left Lanczos vectors.
Now let
, and let
,
,
and
be the matrices generated after
iterations
of Algorithm 7.16.
These three matrices then define a
th-order reduced model
of the original transfer function (7.83) as follows: