So how do we now keep a component in the direction of x0 from creeping in?  Well
let's take this method we just created.
Let's save it as -- let's tag a reorth on
there, to say that we're re-orthogonalizing at every step.  Let's rename this
reorth.  And we probably should move this down here.
We should probably recomment, but let's
not bother with that.  And the idea now is
that the same orthogonalization that we
did right here before we get started, we
really need to do every time through
this loop.  So every time we compute v1, we
subtract out the component in the
direction of x0.  And then we normalize
it to have length 1.  And then we just
proceed as we did before. Now again to
change our test routine, what we now want to do is say, "Okay, let's not bother with
executing the previous method we just
created. And let's instead execute the
one that we just modified."  And we save
that.  And then we can go to our command
window and we can execute it.  And notice that now we start homing in on the
eigenvalue equal to four.  That means that
our eigenvector is starting to point in
that direction or the fact that that's
being created by this power method. And
now, we just keep going.  And we keep
getting closer and closer to a value of
4.  Since we put a maximum number of
iterations at 30 we are now done.  And if
we now look at the graph that comes from this, it, you know, very nicely goes and
homes in on the eigenvalue equal to 4.
Now this is labelled lambda_0
because the way we report this, that's the
only eigenvalue that's being found.  So
this really should be lambda_1 being
reported right here.
