<s>
In	O
mathematics	O
,	O
particularly	O
matrix	B-Architecture
theory	I-Architecture
,	O
a	O
band	B-Algorithm
matrix	I-Algorithm
or	O
banded	B-Algorithm
matrix	I-Algorithm
is	O
a	O
sparse	B-Algorithm
matrix	I-Algorithm
whose	O
non-zero	O
entries	O
are	O
confined	O
to	O
a	O
diagonal	O
band	O
,	O
comprising	O
the	O
main	B-Algorithm
diagonal	I-Algorithm
and	O
zero	O
or	O
more	O
diagonals	O
on	O
either	O
side	O
.	O
</s>
<s>
For	O
k1	O
=	O
k2	O
=	O
2	O
one	O
has	O
a	O
pentadiagonal	B-Algorithm
matrix	I-Algorithm
and	O
so	O
on	O
.	O
</s>
<s>
similarly	O
,	O
for	O
k1	O
=	O
n1	O
,	O
k2	O
=	O
0	O
one	O
obtains	O
a	O
lower	B-Algorithm
triangular	I-Algorithm
matrix	I-Algorithm
.	O
</s>
<s>
Toeplitz	B-Algorithm
matrices	I-Algorithm
when	O
bandwidth	B-Algorithm
is	O
limited	O
.	O
</s>
<s>
The	O
inverses	O
of	O
Lehmer	B-Algorithm
matrices	I-Algorithm
are	O
constant	O
tridiagonal	B-Algorithm
matrices	O
,	O
and	O
are	O
thus	O
band	O
matrices	O
.	O
</s>
<s>
In	O
numerical	B-General_Concept
analysis	I-General_Concept
,	O
matrices	O
from	O
finite	B-Application
element	I-Application
or	O
finite	B-Algorithm
difference	I-Algorithm
problems	O
are	O
often	O
banded	O
.	O
</s>
<s>
For	O
instance	O
,	O
a	O
partial	O
differential	O
equation	O
on	O
a	O
square	O
domain	O
(	O
using	O
central	B-Algorithm
differences	I-Algorithm
)	O
will	O
yield	O
a	O
matrix	O
with	O
a	O
bandwidth	B-Algorithm
equal	O
to	O
the	O
square	O
root	O
of	O
the	O
matrix	O
dimension	O
,	O
but	O
inside	O
the	O
band	O
only	O
5	O
diagonals	O
are	O
nonzero	O
.	O
</s>
<s>
Unfortunately	O
,	O
applying	O
Gaussian	B-Algorithm
elimination	I-Algorithm
(	O
or	O
equivalently	O
an	O
LU	O
decomposition	O
)	O
to	O
such	O
a	O
matrix	O
results	O
in	O
the	O
band	O
being	O
filled	O
in	O
by	O
many	O
non-zero	O
elements	O
.	O
</s>
<s>
For	O
example	O
,	O
a	O
tridiagonal	B-Algorithm
matrix	I-Algorithm
has	O
bandwidth	B-Algorithm
1	O
.	O
</s>
<s>
A	O
further	O
saving	O
is	O
possible	O
when	O
the	O
matrix	O
is	O
symmetric	B-Algorithm
.	O
</s>
<s>
For	O
example	O
,	O
consider	O
a	O
symmetric	B-Algorithm
6-by-6	O
matrix	O
with	O
an	O
upper	O
bandwidth	B-Algorithm
of	O
2	O
:	O
</s>
<s>
From	O
a	O
computational	O
point	O
of	O
view	O
,	O
working	O
with	O
band	O
matrices	O
is	O
always	O
preferential	O
to	O
working	O
with	O
similarly	O
dimensioned	O
square	B-Algorithm
matrices	I-Algorithm
.	O
</s>
<s>
A	O
band	B-Algorithm
matrix	I-Algorithm
can	O
be	O
likened	O
in	O
complexity	O
to	O
a	O
rectangular	B-Architecture
matrix	I-Architecture
whose	O
row	O
dimension	O
is	O
equal	O
to	O
the	O
bandwidth	B-Algorithm
of	O
the	O
band	B-Algorithm
matrix	I-Algorithm
.	O
</s>
<s>
As	O
sparse	B-Algorithm
matrices	I-Algorithm
lend	O
themselves	O
to	O
more	O
efficient	O
computation	O
than	O
dense	O
matrices	O
,	O
as	O
well	O
as	O
in	O
more	O
efficient	O
utilization	O
of	O
computer	O
storage	O
,	O
there	O
has	O
been	O
much	O
research	O
focused	O
on	O
finding	O
ways	O
to	O
minimise	O
the	O
bandwidth	B-Algorithm
(	O
or	O
directly	O
minimise	O
the	O
fill-in	O
)	O
by	O
applying	O
permutations	O
to	O
the	O
matrix	O
,	O
or	O
other	O
such	O
equivalence	O
or	O
similarity	O
transformations	O
.	O
</s>
<s>
The	O
Cuthill	B-Algorithm
–	I-Algorithm
McKee	I-Algorithm
algorithm	I-Algorithm
can	O
be	O
used	O
to	O
reduce	O
the	O
bandwidth	B-Algorithm
of	O
a	O
sparse	O
symmetric	B-Algorithm
matrix	I-Algorithm
.	O
</s>
<s>
There	O
are	O
,	O
however	O
,	O
matrices	O
for	O
which	O
the	O
reverse	B-Algorithm
Cuthill	I-Algorithm
–	I-Algorithm
McKee	I-Algorithm
algorithm	I-Algorithm
performs	O
better	O
.	O
</s>
<s>
The	O
problem	O
of	O
finding	O
a	O
representation	O
of	O
a	O
matrix	O
with	O
minimal	O
bandwidth	B-Algorithm
by	O
means	O
of	O
permutations	O
of	O
rows	O
and	O
columns	O
is	O
NP-hard	O
.	O
</s>
