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##

Inverse Iteration

The drawbacks of the power method can be partially overcome
by applying the power method to
instead of , where is called a *shift*. This
will let the method converge to the eigenvalue closest to
, rather than just
. This method is
called *inverse iteration*, or the *inverse power
method*.

Note that there is no need to compute the
inverse
explicitly at step (4). We only have to solve a system like
for efficiently, which formally gives
.

As in the power method, we should expect the sequence of
vectors generated by the inverse power method to become
increasingly parallel to the eigenvector corresponding to
the largest eigenvalue of
in
magnitude. More specifically, assume that and
are an eigenvalue and eigenvector pair of so that
is the largest eigenvalue of
in magnitude. The inverse power method
converges if is not perpendicular to . The
convergence rate is
, where is an eigenvalue of
such that
is the second
largest eigenvalue of
in magnitude.

One advantage of inverse iteration over the power method is
the ability to converge to any desired eigenvalue (the one
nearest ). By choosing very close to a
desired eigenvalue, inverse iteration can converge very
quickly. This method is particularly effective when we have
a good approximation to an eigenvalue and only want to
compute this eigenvalue and its corresponding
eigenvector. However, inverse iteration does require a
factorization of the matrix , making it less
attractive when this factorization is expensive.

** Next:** Rayleigh Quotient Iteration
** Up:** Single- and Multiple-Vector Iterations
** Previous:** Power Method
** Contents**
** Index**
Susan Blackford
2000-11-20