Civil Engineering Reference
In-Depth Information
3.6.2
The block Arnoldi method
A block Arnoldi method for solving circuit equations is proposed in [SKEW96].
We present a description of the procedure based on the explanation in [OCP98].
Consider a circuit whose inputs and outputs are all considered to be the ports
of the system (we will show a specific example in Section 3.6.3). Its equations
are given by
where
G
and
C
are matrices‚
B
and
L
are matrices‚ and
X
‚
u
and
y
are column vectors of dimension and respectively. This is essentially
similar to the descriptions used earlier: the first of these equations is almost
identical to Equation (3.33). The second‚ the output equation‚ represents the
values at the ports (including the outputs of interest) contained in
which are represented as linear functions of the
X
variables: for
instance‚ if the output of interest is the
element of
X
‚ then
and
for all other
From Equation (3.53)‚ simple matrix algebraic techniques can be used to
obtain
where
as before‚ and
This may be expanded about
to obtain
or in other words‚ the
matrix representing the
moment
Note that the entry in the position
of matrix
corresponds to the
moment of transfer function that relates the
entry of
y
with the
entry
of
u.
Instead of using moments‚ for numerical stability‚ an Arnoldi-based method
generates an orthonormal basis
for the block Krylov subspace
The
following properties of
follow as a natural consequence:
where
I
is the identity matrix.
For a
order reduction‚ an Arnoldi-based method uses the
matrix
matrix to apply the variable transformation:
