<s>
In	O
numerical	B-General_Concept
analysis	I-General_Concept
,	O
nested	B-Algorithm
dissection	I-Algorithm
is	O
a	O
divide	B-Algorithm
and	I-Algorithm
conquer	I-Algorithm
heuristic	B-Algorithm
for	O
the	O
solution	O
of	O
sparse	B-Algorithm
symmetric	B-Algorithm
systems	O
of	O
linear	O
equations	O
based	O
on	O
graph	O
partitioning	O
.	O
</s>
<s>
Nested	B-Algorithm
dissection	I-Algorithm
was	O
introduced	O
by	O
;	O
the	O
name	O
was	O
suggested	O
by	O
Garrett	O
Birkhoff	O
.	O
</s>
<s>
Nested	B-Algorithm
dissection	I-Algorithm
consists	O
of	O
the	O
following	O
steps	O
:	O
</s>
<s>
Form	O
an	O
undirected	O
graph	O
in	O
which	O
the	O
vertices	O
represent	O
rows	O
and	O
columns	O
of	O
the	O
system	O
of	O
linear	O
equations	O
,	O
and	O
an	O
edge	O
represents	O
a	O
nonzero	O
entry	O
in	O
the	O
sparse	B-Algorithm
matrix	I-Algorithm
representing	O
the	O
system	O
.	O
</s>
<s>
Perform	O
Cholesky	O
decomposition	O
(	O
a	O
variant	O
of	O
Gaussian	B-Algorithm
elimination	I-Algorithm
for	O
symmetric	B-Algorithm
matrices	I-Algorithm
)	O
,	O
ordering	O
the	O
elimination	O
of	O
the	O
variables	O
by	O
the	O
recursive	O
structure	O
of	O
the	O
partition	O
:	O
each	O
of	O
the	O
two	O
subgraphs	O
formed	O
by	O
removing	O
the	O
separator	O
is	O
eliminated	O
first	O
,	O
and	O
then	O
the	O
separator	O
vertices	O
are	O
eliminated	O
.	O
</s>
<s>
In	O
particular	O
,	O
for	O
planar	O
graphs	O
(	O
frequently	O
arising	O
in	O
the	O
solution	O
of	O
sparse	B-Algorithm
linear	O
systems	O
derived	O
from	O
two-dimensional	O
finite	B-Application
element	I-Application
method	I-Application
meshes	O
)	O
the	O
resulting	O
matrix	O
has	O
O(nlogn )	O
nonzeros	O
,	O
due	O
to	O
the	O
planar	O
separator	O
theorem	O
guaranteeing	O
separators	O
of	O
size	O
O( )	O
.	O
</s>
<s>
For	O
arbitrary	O
graphs	O
there	O
is	O
a	O
nested	B-Algorithm
dissection	I-Algorithm
that	O
guarantees	O
fill-in	O
within	O
a	O
factor	O
of	O
optimal	O
,	O
where	O
d	O
is	O
the	O
maximum	O
degree	O
and	O
m	O
is	O
the	O
number	O
of	O
non-zeros	O
.	O
</s>
