<s>
In	O
numerical	B-General_Concept
mathematics	I-General_Concept
,	O
relaxation	B-Algorithm
methods	I-Algorithm
are	O
iterative	B-Algorithm
methods	I-Algorithm
for	O
solving	O
systems	O
of	O
equations	O
,	O
including	O
nonlinear	O
systems	O
.	O
</s>
<s>
Relaxation	B-Algorithm
methods	I-Algorithm
were	O
developed	O
for	O
solving	O
large	O
sparse	B-Algorithm
linear	O
systems	O
,	O
which	O
arose	O
as	O
finite-difference	B-Algorithm
discretizations	B-Algorithm
of	O
differential	O
equations	O
.	O
</s>
<s>
They	O
are	O
also	O
used	O
for	O
the	O
solution	O
of	O
linear	O
equations	O
for	O
linear	B-Algorithm
least-squares	I-Algorithm
problems	O
and	O
also	O
for	O
systems	O
of	O
linear	O
inequalities	O
,	O
such	O
as	O
those	O
arising	O
in	O
linear	B-Algorithm
programming	I-Algorithm
.	O
</s>
<s>
Relaxation	B-Algorithm
methods	I-Algorithm
are	O
important	O
especially	O
in	O
the	O
solution	O
of	O
linear	O
systems	O
used	O
to	O
model	O
elliptic	O
partial	O
differential	O
equations	O
,	O
such	O
as	O
Laplace	O
's	O
equation	O
and	O
its	O
generalization	O
,	O
Poisson	O
's	O
equation	O
.	O
</s>
<s>
Relaxation	B-Algorithm
methods	I-Algorithm
are	O
used	O
to	O
solve	O
the	O
linear	O
equations	O
resulting	O
from	O
a	O
discretization	B-Algorithm
of	O
the	O
differential	O
equation	O
,	O
for	O
example	O
by	O
finite	B-Algorithm
differences	I-Algorithm
.	O
</s>
<s>
Iterative	O
relaxation	B-Algorithm
of	O
solutions	O
is	O
commonly	O
dubbed	O
smoothing	B-Application
because	O
with	O
certain	O
equations	O
,	O
such	O
as	O
Laplace	O
's	O
equation	O
,	O
it	O
resembles	O
repeated	O
application	O
of	O
a	O
local	O
smoothing	B-Application
filter	O
to	O
the	O
solution	O
vector	O
.	O
</s>
<s>
These	O
are	O
not	O
to	O
be	O
confused	O
with	O
relaxation	B-Algorithm
methods	I-Algorithm
in	O
mathematical	O
optimization	O
,	O
which	O
approximate	B-Algorithm
a	O
difficult	O
problem	O
by	O
a	O
simpler	O
problem	O
whose	O
"	O
relaxed	O
"	O
solution	O
provides	O
information	O
about	O
the	O
solution	O
of	O
the	O
original	O
problem	O
.	O
</s>
<s>
To	O
approximate	B-Algorithm
the	O
solution	O
of	O
the	O
Poisson	O
equation	O
:	O
</s>
<s>
Nonetheless	O
,	O
the	O
study	O
of	O
relaxation	B-Algorithm
methods	I-Algorithm
remains	O
a	O
core	O
part	O
of	O
linear	O
algebra	O
,	O
because	O
the	O
transformations	O
of	O
relaxation	B-Algorithm
theory	O
provide	O
excellent	O
preconditioners	O
for	O
new	O
methods	O
.	O
</s>
<s>
Indeed	O
,	O
the	O
choice	O
of	O
preconditioner	O
is	O
often	O
more	O
important	O
than	O
the	O
choice	O
of	O
iterative	B-Algorithm
method	I-Algorithm
.	O
</s>
<s>
One	O
can	O
first	O
compute	O
an	O
approximation	O
on	O
a	O
coarser	O
grid	O
–	O
usually	O
the	O
double	O
spacing	O
2h	O
–	O
and	O
use	O
that	O
solution	O
with	O
interpolated	B-Algorithm
values	O
for	O
the	O
other	O
grid	O
points	O
as	O
the	O
initial	O
assignment	O
.	O
</s>
