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The simplest eigenvalue problem is to compute just the dominating
eigenvalue along with its eigenvector.
The power method presented in Algorithm 4.1
is the simplest iterative
method for this task. Under mild assumptions it finds the
eigenvalue of
which has the largest absolute value, and a
corresponding eigenvector.
Let
be the eigenvector corresponding to
. The angle
between
and
is defined by the relation
If the starting vector
and the eigenvector
are perpendicular to each other, then
. In this case the power method does not converge
in exact arithmetic.
On the other hand, if
, the power method
generates a sequence of vectors that become increasingly
parallel to
. This condition on
is true with
very high probability if
is chosen at random.
The convergence rate of the power method depends on
, where
is the second largest
eigenvalue of
in magnitude.
This ratio is generally
smaller than
, allowing adequate convergence. But there
are cases where this ratio can be very close to
, causing very
slow convergence. For detailed discussions on the power
method, see Demmel [114, Chap. 4], Golub and Van
Loan [198], and Parlett [353].
Next: Inverse Iteration
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Susan Blackford
2000-11-20