<s>
In	O
numerical	O
linear	O
algebra	O
,	O
the	O
Gauss	B-Algorithm
–	I-Algorithm
Seidel	I-Algorithm
method	I-Algorithm
,	O
also	O
known	O
as	O
the	O
Liebmann	O
method	O
or	O
the	O
method	O
of	O
successive	O
displacement	O
,	O
is	O
an	O
iterative	B-Algorithm
method	I-Algorithm
used	O
to	O
solve	O
a	O
system	O
of	O
linear	O
equations	O
.	O
</s>
<s>
It	O
is	O
named	O
after	O
the	O
German	O
mathematicians	O
Carl	O
Friedrich	O
Gauss	O
and	O
Philipp	O
Ludwig	O
von	O
Seidel	O
,	O
and	O
is	O
similar	O
to	O
the	O
Jacobi	B-Algorithm
method	I-Algorithm
.	O
</s>
<s>
Though	O
it	O
can	O
be	O
applied	O
to	O
any	O
matrix	O
with	O
non-zero	O
elements	O
on	O
the	O
diagonals	O
,	O
convergence	O
is	O
only	O
guaranteed	O
if	O
the	O
matrix	O
is	O
either	O
strictly	B-Algorithm
diagonally	I-Algorithm
dominant	I-Algorithm
,	O
or	O
symmetric	B-Algorithm
and	O
positive	B-Algorithm
definite	I-Algorithm
.	O
</s>
<s>
The	O
Gauss	B-Algorithm
–	I-Algorithm
Seidel	I-Algorithm
method	I-Algorithm
is	O
an	O
iterative	B-Algorithm
technique	I-Algorithm
for	O
solving	O
a	O
square	O
system	O
of	O
linear	O
equations	O
.	O
</s>
<s>
When	O
and	O
are	O
known	O
,	O
and	O
is	O
unknown	O
,	O
we	O
can	O
use	O
the	O
Gauss	B-Algorithm
–	I-Algorithm
Seidel	I-Algorithm
method	I-Algorithm
to	O
approximate	O
.	O
</s>
<s>
where	O
the	O
matrix	O
is	O
decomposed	O
into	O
a	O
lower	B-Algorithm
triangular	I-Algorithm
component	O
,	O
and	O
a	O
strictly	O
upper	B-Algorithm
triangular	I-Algorithm
component	O
such	O
that	O
.	O
</s>
<s>
The	O
Gauss	B-Algorithm
–	I-Algorithm
Seidel	I-Algorithm
method	I-Algorithm
now	O
solves	O
the	O
left	O
hand	O
side	O
of	O
this	O
expression	O
for	O
,	O
using	O
previous	O
value	O
for	O
on	O
the	O
right	O
hand	O
side	O
.	O
</s>
<s>
However	O
,	O
by	O
taking	O
advantage	O
of	O
the	O
triangular	B-Algorithm
form	I-Algorithm
of	O
,	O
the	O
elements	O
of	O
can	O
be	O
computed	O
sequentially	O
for	O
each	O
row	O
using	O
forward	O
substitution	O
:	O
</s>
<s>
The	O
procedure	O
is	O
generally	O
continued	O
until	O
the	O
changes	O
made	O
by	O
an	O
iteration	O
are	O
below	O
some	O
tolerance	O
,	O
such	O
as	O
a	O
sufficiently	O
small	O
residual	B-Algorithm
.	O
</s>
<s>
The	O
element-wise	O
formula	O
for	O
the	O
Gauss	B-Algorithm
–	I-Algorithm
Seidel	I-Algorithm
method	I-Algorithm
is	O
similar	O
to	O
that	O
of	O
the	O
Jacobi	B-Algorithm
method	I-Algorithm
.	O
</s>
<s>
This	O
means	O
that	O
,	O
unlike	O
the	O
Jacobi	B-Algorithm
method	I-Algorithm
,	O
only	O
one	O
storage	O
vector	O
is	O
required	O
as	O
elements	O
can	O
be	O
overwritten	O
as	O
they	O
are	O
computed	O
,	O
which	O
can	O
be	O
advantageous	O
for	O
very	O
large	O
problems	O
.	O
</s>
<s>
However	O
,	O
unlike	O
the	O
Jacobi	B-Algorithm
method	I-Algorithm
,	O
the	O
computations	O
for	O
each	O
element	O
are	O
generally	O
much	O
harder	O
to	O
implement	O
in	O
parallel	B-Operating_System
,	O
since	O
they	O
can	O
have	O
a	O
very	O
long	O
critical	B-Operating_System
path	I-Operating_System
,	O
and	O
are	O
thus	O
most	O
feasible	O
for	O
sparse	B-Algorithm
matrices	I-Algorithm
.	O
</s>
<s>
Gauss-Seidel	B-Algorithm
is	O
the	O
same	O
as	O
SOR	O
(	O
successive	B-Algorithm
over-relaxation	I-Algorithm
)	O
with	O
.	O
</s>
<s>
The	O
convergence	O
properties	O
of	O
the	O
Gauss	B-Algorithm
–	I-Algorithm
Seidel	I-Algorithm
method	I-Algorithm
are	O
dependent	O
on	O
the	O
matrix	O
A	O
.	O
Namely	O
,	O
the	O
procedure	O
is	O
known	O
to	O
converge	O
if	O
either	O
:	O
</s>
<s>
is	O
strictly	O
or	O
irreducibly	O
diagonally	B-Algorithm
dominant	I-Algorithm
.	O
</s>
<s>
The	O
Gauss	B-Algorithm
–	I-Algorithm
Seidel	I-Algorithm
method	I-Algorithm
sometimes	O
converges	O
even	O
if	O
these	O
conditions	O
are	O
not	O
satisfied	O
.	O
</s>
<s>
We	O
must	O
decompose	O
into	O
the	O
sum	O
of	O
a	O
lower	B-Algorithm
triangular	I-Algorithm
component	O
and	O
a	O
strict	O
upper	B-Algorithm
triangular	I-Algorithm
component	O
:	O
</s>
<s>
In	O
fact	O
,	O
the	O
matrix	O
is	O
strictly	B-Algorithm
diagonally	I-Algorithm
dominant	I-Algorithm
(	O
but	O
not	O
positive	B-Algorithm
definite	I-Algorithm
)	O
.	O
</s>
<s>
We	O
must	O
decompose	O
into	O
the	O
sum	O
of	O
a	O
lower	B-Algorithm
triangular	I-Algorithm
component	O
and	O
a	O
strict	O
upper	B-Algorithm
triangular	I-Algorithm
component	O
:	O
</s>
<s>
In	O
fact	O
,	O
the	O
matrix	O
A	O
is	O
neither	O
diagonally	B-Algorithm
dominant	I-Algorithm
nor	O
positive	B-Algorithm
definite	I-Algorithm
.	O
</s>
<s>
Using	O
the	O
approximations	O
obtained	O
,	O
the	O
iterative	B-Algorithm
procedure	I-Algorithm
is	O
repeated	O
until	O
the	O
desired	O
accuracy	O
has	O
been	O
reached	O
.	O
</s>
