<s>
Divide-and-conquer	B-Algorithm
eigenvalue	I-Algorithm
algorithms	I-Algorithm
are	O
a	O
class	O
of	O
eigenvalue	O
algorithms	O
for	O
Hermitian	B-Algorithm
or	O
real	O
symmetric	B-Algorithm
matrices	I-Algorithm
that	O
have	O
recently	O
(	O
circa	O
1990s	O
)	O
become	O
competitive	O
in	O
terms	O
of	O
stability	B-Algorithm
and	O
efficiency	O
with	O
more	O
traditional	O
algorithms	O
such	O
as	O
the	O
QR	O
algorithm	O
.	O
</s>
<s>
The	O
basic	O
concept	O
behind	O
these	O
algorithms	O
is	O
the	O
divide-and-conquer	B-Algorithm
approach	O
from	O
computer	B-General_Concept
science	I-General_Concept
.	O
</s>
<s>
Here	O
we	O
present	O
the	O
simplest	O
version	O
of	O
a	O
divide-and-conquer	B-Algorithm
algorithm	I-Algorithm
,	O
similar	O
to	O
the	O
one	O
originally	O
proposed	O
by	O
Cuppen	O
in	O
1981	O
.	O
</s>
<s>
As	O
with	O
most	O
eigenvalue	O
algorithms	O
for	O
Hermitian	B-Algorithm
matrices	I-Algorithm
,	O
divide-and-conquer	B-Algorithm
begins	O
with	O
a	O
reduction	O
to	O
tridiagonal	B-Algorithm
form	O
.	O
</s>
<s>
For	O
an	O
matrix	O
,	O
the	O
standard	O
method	O
for	O
this	O
,	O
via	O
Householder	B-Algorithm
reflections	I-Algorithm
,	O
takes	O
floating	B-Algorithm
point	I-Algorithm
operations	O
,	O
or	O
if	O
eigenvectors	O
are	O
needed	O
as	O
well	O
.	O
</s>
<s>
This	O
technique	O
can	O
be	O
used	O
to	O
improve	O
the	O
efficiency	O
of	O
many	O
eigenvalue	O
algorithms	O
,	O
but	O
it	O
has	O
special	O
significance	O
to	O
divide-and-conquer	B-Algorithm
.	O
</s>
<s>
For	O
the	O
rest	O
of	O
this	O
article	O
,	O
we	O
will	O
assume	O
the	O
input	O
to	O
the	O
divide-and-conquer	B-Algorithm
algorithm	I-Algorithm
is	O
an	O
real	O
symmetric	B-Algorithm
tridiagonal	B-Algorithm
matrix	I-Algorithm
.	O
</s>
<s>
Although	O
the	O
algorithm	O
can	O
be	O
modified	O
for	O
Hermitian	B-Algorithm
matrices	I-Algorithm
,	O
we	O
do	O
not	O
give	O
the	O
details	O
here	O
.	O
</s>
<s>
The	O
divide	O
part	O
of	O
the	O
divide-and-conquer	B-Algorithm
algorithm	I-Algorithm
comes	O
from	O
the	O
realization	O
that	O
a	O
tridiagonal	B-Algorithm
matrix	I-Algorithm
is	O
"	O
almost	O
"	O
block	O
diagonal	O
.	O
</s>
<s>
This	O
can	O
be	O
accomplished	O
with	O
recursive	O
calls	O
to	O
the	O
divide-and-conquer	B-Algorithm
algorithm	I-Algorithm
,	O
although	O
practical	O
implementations	O
often	O
switch	O
to	O
the	O
QR	O
algorithm	O
for	O
small	O
enough	O
submatrices	O
.	O
</s>
<s>
Given	O
the	O
diagonalizations	O
of	O
the	O
submatrices	O
,	O
calculated	O
above	O
,	O
how	O
do	O
we	O
find	O
the	O
diagonalization	B-Algorithm
of	O
the	O
original	O
matrix	O
?	O
</s>
<s>
All	O
general	O
eigenvalue	O
algorithms	O
must	O
be	O
iterative	O
,	O
and	O
the	O
divide-and-conquer	B-Algorithm
algorithm	I-Algorithm
is	O
no	O
different	O
.	O
</s>
<s>
As	O
is	O
common	O
for	O
divide	B-Algorithm
and	I-Algorithm
conquer	I-Algorithm
algorithms	I-Algorithm
,	O
we	O
will	O
use	O
the	O
master	B-Algorithm
theorem	I-Algorithm
for	I-Algorithm
divide-and-conquer	I-Algorithm
recurrences	I-Algorithm
to	O
analyze	O
the	O
running	O
time	O
.	O
</s>
<s>
In	O
the	O
notation	O
of	O
the	O
Master	B-Algorithm
theorem	I-Algorithm
,	O
and	O
thus	O
.	O
</s>
<s>
Remember	O
that	O
above	O
we	O
pointed	O
out	O
that	O
reducing	O
a	O
Hermitian	B-Algorithm
matrix	I-Algorithm
to	O
tridiagonal	B-Algorithm
form	O
takes	O
flops	O
.	O
</s>
<s>
This	O
dwarfs	O
the	O
running	O
time	O
of	O
the	O
divide-and-conquer	B-Algorithm
part	O
,	O
and	O
at	O
this	O
point	O
it	O
is	O
not	O
clear	O
what	O
advantage	O
the	O
divide-and-conquer	B-Algorithm
algorithm	I-Algorithm
offers	O
over	O
the	O
QR	O
algorithm	O
(	O
which	O
also	O
takes	O
flops	O
for	O
tridiagonal	B-Algorithm
matrices	O
)	O
.	O
</s>
<s>
The	O
advantage	O
of	O
divide-and-conquer	B-Algorithm
comes	O
when	O
eigenvectors	O
are	O
needed	O
as	O
well	O
.	O
</s>
<s>
If	O
this	O
is	O
the	O
case	O
,	O
reduction	O
to	O
tridiagonal	B-Algorithm
form	O
takes	O
,	O
but	O
the	O
second	O
part	O
of	O
the	O
algorithm	O
takes	O
as	O
well	O
.	O
</s>
<s>
For	O
the	O
QR	O
algorithm	O
with	O
a	O
reasonable	O
target	O
precision	O
,	O
this	O
is	O
,	O
whereas	O
for	O
divide-and-conquer	B-Algorithm
it	O
is	O
.	O
</s>
<s>
The	O
reason	O
for	O
this	O
improvement	O
is	O
that	O
in	O
divide-and-conquer	B-Algorithm
,	O
the	O
part	O
of	O
the	O
algorithm	O
(	O
multiplying	O
matrices	O
)	O
is	O
separate	O
from	O
the	O
iteration	O
,	O
whereas	O
in	O
QR	O
,	O
this	O
must	O
occur	O
in	O
every	O
iterative	O
step	O
.	O
</s>
<s>
Practical	O
use	O
of	O
the	O
divide-and-conquer	B-Algorithm
algorithm	I-Algorithm
has	O
shown	O
that	O
in	O
most	O
realistic	O
eigenvalue	O
problems	O
,	O
the	O
algorithm	O
actually	O
does	O
better	O
than	O
this	O
.	O
</s>
<s>
The	O
reason	O
is	O
that	O
very	O
often	O
the	O
matrices	O
and	O
the	O
vectors	O
tend	O
to	O
be	O
numerically	O
sparse	O
,	O
meaning	O
that	O
they	O
have	O
many	O
entries	O
with	O
values	O
smaller	O
than	O
the	O
floating	B-Algorithm
point	I-Algorithm
precision	O
,	O
allowing	O
for	O
numerical	O
deflation	O
,	O
i.e.	O
</s>
<s>
In	O
many	O
practical	O
implementations	O
,	O
more	O
complicated	O
rank-1	O
corrections	O
are	O
used	O
to	O
guarantee	O
stability	B-Algorithm
;	O
some	O
variants	O
even	O
use	O
rank-2	O
corrections	O
.	O
</s>
<s>
There	O
exist	O
specialized	O
root-finding	O
techniques	O
for	O
rational	O
functions	O
that	O
may	O
do	O
better	O
than	O
the	O
Newton-Raphson	O
method	O
in	O
terms	O
of	O
both	O
performance	O
and	O
stability	B-Algorithm
.	O
</s>
<s>
These	O
can	O
be	O
used	O
to	O
improve	O
the	O
iterative	O
part	O
of	O
the	O
divide-and-conquer	B-Algorithm
algorithm	I-Algorithm
.	O
</s>
<s>
The	O
divide-and-conquer	B-Algorithm
algorithm	I-Algorithm
is	O
readily	O
parallelized	B-Operating_System
,	O
and	O
linear	B-Language
algebra	I-Language
computing	O
packages	O
such	O
as	O
LAPACK	B-Application
contain	O
high-quality	O
parallel	O
implementations	O
.	O
</s>
