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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

 (189)

Here, is a square, in general non-Hermitian matrix, and
 (190)

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:


if ,

set  for              all  with ,

and set

if ,

set  for       all  with ,

and set

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

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:

 (191)

Here,

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.

Next: Variants Up: Band Lanczos Method   Previous: Algorithm   Contents   Index
Susan Blackford 2000-11-20