Civil Engineering Reference
In-Depth Information
scaling in some circumstances‚ such an approach provides limited relief. The
use of Krylov subspace-based methods provides an effective way to overcome a
number of problems associated with numerical instability.
We begin by defining the span of a set of vectors‚ which in turn is used to
define a Krylov subspace
Definition 3.6.1
The span of a set of vectors is
the set of points that can be expressed as a linear combination of these vectors.
In other words‚
n-dimensional
implies that
for some
set of real values
Definition 3.6.2
A Krylov subspace of order associated with an
matrix
A and a vector
denoted by
is given by the span of the vectors
Interestingly‚ it can be proven that the sequence
converges to the eigen-
vector corresponding to the largest eigenvalue of
A
as
becomes large‚ regard-
less of the chosen (nonzero) value of
If the vector
is replaced by a
matrix
R
(where typically
)‚
then the subspace
is similarly defined as the span of the columns of
the matrices
and is referred to as a
block Krylov
subspace.
The relationship between the moment generation process and Krylov sub-
spaces can be seen from Equations (3.36) and (3.37). From the latter‚ it is easy
to see that the sequence of moments
can be rewritten as
These vectors together match the above definition of a Krylov subspace‚ where
and
and we will henceforth use
A
and
to denote these
terms.
Instead of working with directly‚ which is liable to introduce numerical
errors‚ we will work with an
orthonormal basis
matrix within the subspace: for
a Krylov subspace
an orthonormal basis
consists of a set
of vectors
such that
and the
vectors are all orthogonal‚ i.e.‚ and
Techniques such as Lanczos-based [FF95‚ FF96‚ FF98‚ Fre99] and Arnoldi-
based methods [SKEW96‚ EL97] overcome the numerical limitations associated
with practical implementations of AWE at the expense of slightly more complex
calculations. Other methods such as truncated balanced realizations have also
been studied recently [PDS03] to overcome some of the limitations in accuracy
of other methods‚ and to more easily widen their applicability to a larger class
of circuits.
