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In some applications,
is Hermitian positive
definite and well-conditioned.
In this case, it is recommended to compute first a
sparse Cholesky decomposition
with
a lower triangular matrix; see §10.3.
The equivalent standard eigenvalue problem is
![\begin{displaymath}
(L^{-1} AL^{-\ast}) \tilde{x} = \lambda \tilde{x}
\quad \m...
...tilde{y}^{\ast} (L^{-1}AL^{-\ast}) = \lambda \tilde{y}^{\ast},
\end{displaymath}](img2649.png) |
(218) |
where
and
.
As in Invert B, matrices like
should never be
evaluated explicitly, more in particular because they will in general not
be sparse.
For the application of an iterative method for the standard eigenvalue problem,
we only need to provide the efficient evaluation of
matrix-vector products, like
or
where
and
are given vectors.
In practice, the vector
can be formed in three steps:
(a) solve
for
,
(b) form
,
(c) solve
for
.
The vector
, when necessary,
can be formed as follows:
(a)
solve
for
,
(b)
form
,
(c)
solve
for
.
Since
is a triangular matrix, the solutions of linear systems
with
or
can be obtained
by forward and backward substitutions.
Sparsity can be exploited in a straightforward manner in all these steps.
Next: Shift-and-Invert.
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Susan Blackford
2000-11-20