Interdisciplinary Applied Mathematics

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Next, we need to define a set of basis functions such that they span the desired Krylov subspaces. Let V = [v1 ,v2,v3,…,vn] and W = [w1,w2,w3,


…, wn ] be basis vectors such that


Kn(A, r) = span[v1,v2,v3,… ,Vn],    Kn(AT, l) = span[wbw2, w3,.. .,wri].


The Lanczos process is an elegant way of generating such basis vectors. The Lanczos vectors V and W are constructed to be biorthogonal, i.e.,


wT Vk =0    V j = k.


The Lanczos algorithm to generate V and W from A, r, and l can be found in (Freund, 1999).


The reduced system, using Krylov subspaces, can be generated using the following steps (see also Figure 17.10):


1.    Taking    so    as    the    expansion    point    of    equation    (17.9), A and r are


defined as


A = -(G + s0C)-1C    and r = (G + s0C)^1b.


2.    Using the Lanczos method, V and W are computed as


span[V] = Kn(A, r),    span[W] = Kn(AT, l).


3. The reduced matrices Cn, Gn, bn, ln are computed as follows: For double-sided projection


Cn = VTCW,    Gn = VTGW,    bn = WTb,    ln = VTl,


and for single-sided projection,


Cn = VTCV,    Gn = VTGV,    bn = VTb,    ln = VTl.


The double-sided projection formula does not always guarantee a stable reduced-order model except for certain trivial cases (e.g., RC networks), whereas single-sided projection onto V guarantees an unconditionally stable reduced-order system. However, double-sided projection generally gives more accurate results than single-sided projection. The order of the subspace is chosen according to the frequency range where matching is required. For matching of q resonant peaks, n has to be at least 2q.

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