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

$$H(s) = C^T \left(I - s A\right)^{-1} B, \quad s\in {\cal C}.$$ (189)

Here, $A$ is a square, in general non-Hermitian matrix, and

$$B = \left[ \begin{array}{cccc} b_1 & b_2 & \cdots & b_m \e... ...in{array}{cccc} c_1 & c_2 & \cdots & c_p \end{array} \right]$$ (190)

are rectangular matrices with $m$ and $p$ columns, respectively. Moreover, $m$ is the number of inputs, $p$ is the number of outputs, and $m$ and $p$ are different in general. The band Lanczos method (applied to $A$ and with the columns of the matrices (7.84), $B$ and $C$, 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 $t_{jk}$ and $\tilde{t}_{jk}$ with negative indices $k\leq 0$ are also used. More precisely, in this case, step (16) in Algorithm 7.16 is augmented by the following six lines:

 		 		 		 if $j\leq m_c$,


set $\rho_{jk}=t_{j,k-m}$ for              all $k$ with $j-m_c+m \leq k \leq m$,


and set $\rho = \left[\, \rho_{ik} \, \right]_{i=1,2,\ldots,j,\, k=1,2,\ldots,m}$


if $j\leq p_c$, 


set $\eta_{jk}=\tilde{t}_{j,k-p}$ for       all $k$ with $j-p_c+p \leq k \leq p$,


and set $\eta = \left[\, \eta_{ik} \, \right]_{i=1,2,\ldots,j,\, k=1,2,\ldots,p}$

Here again we use the convention that entries $\rho_{ik}$ and $\eta_{ik}$ that are not explicitly defined in Algorithm 7.16 are set to be zero.

The matrix $\rho$ is upper triangular and of size $m_1 \times m$. Here, $m_1$ is defined as the value of $m_c=m_c(j)$ at
iteration $j$ of Algorithm 7.16 for which $j=m_c$ is reached. It turns out that $m_1$ is just the value of $m$ minus the number of right initial vectors $b_1,b_2,\ldots,b_m$ that have been deflated. In particular, $m_1\leq m$, and $m_1 = m$ if and only if none of the right starting vectors has been deflated. The matrix $\eta$ is upper triangular and of size $p_1 \times p$. Here, $p_1$ is defined as the value of $p_c=p_c(j)$ at the iteration $j$ of Algorithm 7.16 for which $j=p_c$ is reached. The number $p_1$ is the value of $p$ minus the number of left initial vectors $c_1,c_2,\ldots,c_p$ that have been deflated. In particular, $p_1\leq p$, and $p_1 = p$ if and only if none of the left starting vectors has been deflated. The entries of $\rho$ are the coefficients used to turn the right starting vectors into the first $m_1$ right Lanczos vectors, and the entries of $\eta$ are the coefficients used to turn the left starting vectors into the first $p_1$ left Lanczos vectors.

Now let $j\geq \max\{m_1,p_1\}$, and let $T_j^{\rm (pr)}$, $\rho$, and $\eta$ be the matrices generated after $j$ iterations of Algorithm 7.16. These three matrices then define a $j$th-order reduced model of the original transfer function (7.83) as follows:

$$H_j(s) = \eta_j^T D_j \left(I_j - s T_j^{\rm (pr)}\right)^{-1} \rho_j, \quad s\in {\cal C}.$$ (191)

Here,

$$\rho_j = \left[ \begin{array}{c} \rho\\ 0 \end{array} \... ...{c} \eta\\ 0 \end{array} \right] \in {\cal C}^{j\times p}$$

are the matrices $\rho$ and $\eta$ with $j-m_1$, respectively $j-p_1$, 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.