<s>
In	O
mathematics	O
,	O
power	B-Language
iteration	I-Language
(	O
also	O
known	O
as	O
the	O
power	B-Language
method	I-Language
)	O
is	O
an	O
eigenvalue	O
algorithm	O
:	O
given	O
a	O
diagonalizable	B-Algorithm
matrix	I-Algorithm
,	O
the	O
algorithm	O
will	O
produce	O
a	O
number	O
,	O
which	O
is	O
the	O
greatest	O
(	O
in	O
absolute	O
value	O
)	O
eigenvalue	O
of	O
,	O
and	O
a	O
nonzero	O
vector	O
,	O
which	O
is	O
a	O
corresponding	O
eigenvector	O
of	O
,	O
that	O
is	O
,	O
.	O
</s>
<s>
The	O
algorithm	O
is	O
also	O
known	O
as	O
the	O
Von	B-Language
Mises	I-Language
iteration	I-Language
.	O
</s>
<s>
Power	B-Language
iteration	I-Language
is	O
a	O
very	O
simple	O
algorithm	O
,	O
but	O
it	O
may	O
converge	O
slowly	O
.	O
</s>
<s>
The	O
most	O
time-consuming	O
operation	O
of	O
the	O
algorithm	O
is	O
the	O
multiplication	O
of	O
matrix	B-Architecture
by	O
a	O
vector	O
,	O
so	O
it	O
is	O
effective	O
for	O
a	O
very	O
large	O
sparse	B-Algorithm
matrix	I-Algorithm
with	O
appropriate	O
implementation	O
.	O
</s>
<s>
The	O
power	B-Language
iteration	I-Language
algorithm	O
starts	O
with	O
a	O
vector	O
,	O
which	O
may	O
be	O
an	O
approximation	O
to	O
the	O
dominant	O
eigenvector	O
or	O
a	O
random	O
vector	O
.	O
</s>
<s>
So	O
,	O
at	O
every	O
iteration	O
,	O
the	O
vector	O
is	O
multiplied	O
by	O
the	O
matrix	B-Architecture
and	O
normalized	O
.	O
</s>
<s>
This	O
algorithm	O
is	O
used	O
to	O
calculate	O
the	B-Application
Google	I-Application
PageRank	B-Algorithm
.	O
</s>
<s>
Since	O
the	O
dominant	O
eigenvalue	O
of	O
is	O
unique	O
,	O
the	O
first	O
Jordan	O
block	O
of	O
is	O
the	O
matrix	B-Architecture
where	O
is	O
the	O
largest	O
eigenvalue	O
of	O
A	O
in	O
magnitude	O
.	O
</s>
<s>
Although	O
the	O
power	B-Language
iteration	I-Language
method	O
approximates	O
only	O
one	O
eigenvalue	O
of	O
a	O
matrix	B-Architecture
,	O
it	O
remains	O
useful	O
for	O
certain	O
computational	O
problems	O
.	O
</s>
<s>
For	O
instance	O
,	O
Google	B-Application
uses	O
it	O
to	O
calculate	O
the	O
PageRank	B-Algorithm
of	O
documents	O
in	O
their	O
search	O
engine	O
,	O
and	O
Twitter	B-Application
uses	O
it	O
to	O
show	O
users	O
recommendations	O
of	O
whom	O
to	O
follow	O
.	O
</s>
<s>
The	O
power	B-Language
iteration	I-Language
method	O
is	O
especially	O
suitable	O
for	O
sparse	B-Algorithm
matrices	I-Algorithm
,	O
such	O
as	O
the	O
web	O
matrix	B-Architecture
,	O
or	O
as	O
the	O
matrix-free	O
method	O
that	O
does	O
not	O
require	O
storing	O
the	O
coefficient	O
matrix	B-Architecture
explicitly	O
,	O
but	O
can	O
instead	O
access	O
a	O
function	O
evaluating	O
matrix-vector	O
products	O
.	O
</s>
<s>
For	O
non-symmetric	O
matrices	O
that	O
are	O
well-conditioned	O
the	O
power	B-Language
iteration	I-Language
method	O
can	O
outperform	O
more	O
complex	O
Arnoldi	O
iteration	O
.	O
</s>
<s>
For	O
symmetric	O
matrices	O
,	O
the	O
power	B-Language
iteration	I-Language
method	O
is	O
rarely	O
used	O
,	O
since	O
its	O
convergence	O
speed	O
can	O
be	O
easily	O
increased	O
without	O
sacrificing	O
the	O
small	O
cost	O
per	O
iteration	O
;	O
see	O
,	O
e.g.	O
,	O
Lanczos	O
iteration	O
and	O
LOBPCG	B-Application
.	O
</s>
<s>
Some	O
of	O
the	O
more	O
advanced	O
eigenvalue	O
algorithms	O
can	O
be	O
understood	O
as	O
variations	O
of	O
the	O
power	B-Language
iteration	I-Language
.	O
</s>
<s>
For	O
instance	O
,	O
the	O
inverse	O
iteration	O
method	O
applies	O
power	B-Language
iteration	I-Language
to	O
the	O
matrix	B-Architecture
.	O
</s>
