<s>
The	O
pivot	O
or	O
pivot	B-Algorithm
element	I-Algorithm
is	O
the	O
element	O
of	O
a	O
matrix	B-Architecture
,	O
or	O
an	O
array	B-Data_Structure
,	O
which	O
is	O
selected	O
first	O
by	O
an	O
algorithm	O
(	O
e.g.	O
</s>
<s>
Gaussian	B-Algorithm
elimination	I-Algorithm
,	O
simplex	B-Algorithm
algorithm	I-Algorithm
,	O
etc	O
.	O
</s>
<s>
In	O
the	O
case	O
of	O
matrix	B-Architecture
algorithms	O
,	O
a	O
pivot	O
entry	O
is	O
usually	O
required	O
to	O
be	O
at	O
least	O
distinct	O
from	O
zero	O
,	O
and	O
often	O
distant	O
from	O
it	O
;	O
in	O
this	O
case	O
finding	O
this	O
element	O
is	O
called	O
pivoting	O
.	O
</s>
<s>
Pivoting	O
might	O
be	O
thought	O
of	O
as	O
swapping	O
or	O
sorting	O
rows	O
or	O
columns	O
in	O
a	O
matrix	B-Architecture
,	O
and	O
thus	O
it	O
can	O
be	O
represented	O
as	O
multiplication	O
by	O
permutation	B-Algorithm
matrices	I-Algorithm
.	O
</s>
<s>
However	O
,	O
algorithms	O
rarely	O
move	O
the	O
matrix	B-Architecture
elements	O
because	O
this	O
would	O
cost	O
too	O
much	O
time	O
;	O
instead	O
,	O
they	O
just	O
keep	O
track	O
of	O
the	O
permutations	O
.	O
</s>
<s>
Other	O
times	O
these	O
additional	O
operations	O
are	O
worthwhile	O
because	O
they	O
add	O
numerical	B-Algorithm
stability	I-Algorithm
to	O
the	O
final	O
result	O
.	O
</s>
<s>
In	O
the	O
case	O
of	O
Gaussian	B-Algorithm
elimination	I-Algorithm
,	O
the	O
algorithm	O
requires	O
that	O
pivot	B-Algorithm
elements	I-Algorithm
not	O
be	O
zero	O
.	O
</s>
<s>
Interchanging	O
rows	O
or	O
columns	O
in	O
the	O
case	O
of	O
a	O
zero	O
pivot	B-Algorithm
element	I-Algorithm
is	O
necessary	O
.	O
</s>
<s>
Furthermore	O
,	O
in	O
Gaussian	B-Algorithm
elimination	I-Algorithm
it	O
is	O
generally	O
desirable	O
to	O
choose	O
a	O
pivot	B-Algorithm
element	I-Algorithm
with	O
large	O
absolute	O
value	O
.	O
</s>
<s>
This	O
improves	O
the	O
numerical	B-Algorithm
stability	I-Algorithm
.	O
</s>
<s>
The	O
following	O
system	O
is	O
dramatically	O
affected	O
by	O
round-off	O
error	O
when	O
Gaussian	B-Algorithm
elimination	I-Algorithm
and	O
backwards	O
substitution	O
are	O
performed	O
.	O
</s>
<s>
In	O
partial	B-Algorithm
pivoting	I-Algorithm
,	O
the	O
algorithm	O
selects	O
the	O
entry	O
with	O
largest	O
absolute	O
value	O
from	O
the	O
column	O
of	O
the	O
matrix	B-Architecture
that	O
is	O
currently	O
being	O
considered	O
as	O
the	O
pivot	B-Algorithm
element	I-Algorithm
.	O
</s>
<s>
More	O
specifically	O
,	O
when	O
reducing	O
a	O
matrix	B-Architecture
to	O
row	O
echelon	O
form	O
,	O
partial	B-Algorithm
pivoting	I-Algorithm
swaps	O
rows	O
before	O
the	O
column	O
's	O
row	B-Algorithm
reduction	I-Algorithm
to	O
make	O
the	O
pivot	B-Algorithm
element	I-Algorithm
have	O
the	O
largest	O
absolute	O
value	O
compared	O
to	O
the	O
elements	O
below	O
in	O
the	O
same	O
column	O
.	O
</s>
<s>
Partial	B-Algorithm
pivoting	I-Algorithm
is	O
generally	O
sufficient	O
to	O
adequately	O
reduce	O
round-off	O
error	O
.	O
</s>
<s>
However	O
,	O
for	O
certain	O
systems	O
and	O
algorithms	O
,	O
complete	B-Algorithm
pivoting	I-Algorithm
(	O
or	O
maximal	O
pivoting	O
)	O
may	O
be	O
required	O
for	O
acceptable	O
accuracy	O
.	O
</s>
<s>
Complete	B-Algorithm
pivoting	I-Algorithm
interchanges	O
both	O
rows	O
and	O
columns	O
in	O
order	O
to	O
use	O
the	O
largest	O
(	O
by	O
absolute	O
value	O
)	O
element	O
in	O
the	O
matrix	B-Architecture
as	O
the	O
pivot	O
.	O
</s>
<s>
Complete	B-Algorithm
pivoting	I-Algorithm
is	O
usually	O
not	O
necessary	O
to	O
ensure	O
numerical	B-Algorithm
stability	I-Algorithm
and	O
,	O
due	O
to	O
the	O
additional	O
cost	O
of	O
searching	O
for	O
the	O
maximal	O
element	O
,	O
the	O
improvement	O
in	O
numerical	B-Algorithm
stability	I-Algorithm
that	O
it	O
provides	O
is	O
typically	O
outweighed	O
by	O
its	O
reduced	O
efficiency	O
for	O
all	O
but	O
the	O
smallest	O
matrices	O
.	O
</s>
<s>
When	O
implemented	O
on	O
serial	O
computers	O
this	O
strategy	O
has	O
expected	O
cost	O
of	O
only	O
about	O
three	O
times	O
that	O
of	O
partial	B-Algorithm
pivoting	I-Algorithm
and	O
is	O
therefore	O
cheaper	O
than	O
complete	B-Algorithm
pivoting	I-Algorithm
.	O
</s>
<s>
Rook	O
pivoting	O
has	O
been	O
proved	O
to	O
be	O
more	O
stable	O
than	O
partial	B-Algorithm
pivoting	I-Algorithm
both	O
theoretically	O
and	O
in	O
practice	O
.	O
</s>
<s>
A	O
variation	O
of	O
the	O
partial	B-Algorithm
pivoting	I-Algorithm
strategy	O
is	O
scaled	O
pivoting	O
.	O
</s>
<s>
In	O
this	O
approach	O
,	O
the	O
algorithm	O
selects	O
as	O
the	O
pivot	B-Algorithm
element	I-Algorithm
the	O
entry	O
that	O
is	O
largest	O
relative	O
to	O
the	O
entries	O
in	O
its	O
row	O
.	O
</s>
<s>
In	O
the	O
example	O
below	O
,	O
it	O
would	O
be	O
desirable	O
to	O
interchange	O
the	O
two	O
rows	O
because	O
the	O
current	O
pivot	B-Algorithm
element	I-Algorithm
30	O
is	O
larger	O
than	O
5.291	O
but	O
it	O
is	O
relatively	O
small	O
compared	O
with	O
the	O
other	O
entries	O
in	O
its	O
row	O
.	O
</s>
<s>
A	O
pivot	O
position	O
in	O
a	O
matrix	B-Architecture
,	O
A	O
,	O
is	O
a	O
position	O
in	O
the	O
matrix	B-Architecture
that	O
corresponds	O
to	O
a	O
row	O
–	O
leading	O
1	O
in	O
the	O
reduced	O
row	O
echelon	O
form	O
of	O
A	O
.	O
</s>
