Application to Reduced-Order Modeling

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

are rectangular matrices with and columns, respectively. Moreover, is the number of inputs, is the number of outputs, and and are different in general. The band Lanczos method (applied to and with the columns of the matrices (7.84), and , as right and left starting vectors) can be used to generate a reduced-order model of the linear dynamical system described by (7.83).

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:

Here again we use the convention that entries and that are not explicitly defined in Algorithm 7.16 are set to be zero.if ,

set for all with ,

and set

if ,

set for 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:

are the matrices and with , respectively , rows of zeros added. The reduced-order model (7.85) can be shown to be characterized as a certain matrix-Padé approximation of the original transfer function (7.83); see [176] and the references given there.