<s>
In	O
numerical	O
linear	O
algebra	O
,	O
the	O
method	O
of	O
successive	B-Algorithm
over-relaxation	I-Algorithm
(	O
SOR	O
)	O
is	O
a	O
variant	O
of	O
the	O
Gauss	B-Algorithm
–	I-Algorithm
Seidel	I-Algorithm
method	I-Algorithm
for	O
solving	O
a	O
linear	O
system	O
of	O
equations	O
,	O
resulting	O
in	O
faster	O
convergence	O
.	O
</s>
<s>
A	O
similar	O
method	O
can	O
be	O
used	O
for	O
any	O
slowly	O
converging	O
iterative	B-Algorithm
process	I-Algorithm
.	O
</s>
<s>
Then	O
A	O
can	O
be	O
decomposed	O
into	O
a	O
diagonal	B-Algorithm
component	O
D	O
,	O
and	O
strictly	O
lower	O
and	O
upper	O
triangular	O
components	O
L	O
and	O
U	O
:	O
</s>
<s>
The	O
method	O
of	O
successive	B-Algorithm
over-relaxation	I-Algorithm
is	O
an	O
iterative	B-Algorithm
technique	I-Algorithm
that	O
solves	O
the	O
left	O
hand	O
side	O
of	O
this	O
expression	O
for	O
x	O
,	O
using	O
the	O
previous	O
value	O
for	O
x	O
on	O
the	O
right	O
hand	O
side	O
.	O
</s>
<s>
In	O
1947	O
,	O
Ostrowski	O
proved	O
that	O
if	O
is	O
symmetric	B-Algorithm
and	O
positive-definite	B-Algorithm
then	O
for	O
.	O
</s>
<s>
The	O
convergence	O
rate	O
for	O
the	O
SOR	B-Algorithm
method	I-Algorithm
can	O
be	O
analytically	O
derived	O
.	O
</s>
<s>
Jacobi	B-Algorithm
's	I-Algorithm
method	I-Algorithm
is	O
convergent	O
:	O
</s>
<s>
In	O
particular	O
,	O
for	O
(	O
Gauss-Seidel	B-Algorithm
)	O
it	O
holds	O
that	O
.	O
</s>
<s>
The	O
last	O
assumption	O
is	O
satisfied	O
for	O
tridiagonal	B-Algorithm
matrices	I-Algorithm
since	O
for	O
diagonal	B-Algorithm
with	O
entries	O
and	O
.	O
</s>
<s>
According	O
to	O
the	O
successive	B-Algorithm
over-relaxation	I-Algorithm
algorithm	O
,	O
the	O
following	O
table	O
is	O
obtained	O
,	O
representing	O
an	O
exemplary	O
iteration	O
with	O
approximations	O
,	O
which	O
ideally	O
,	O
but	O
not	O
necessarily	O
,	O
finds	O
the	O
exact	O
solution	O
,	O
,	O
in	O
38	O
steps	O
.	O
</s>
<s>
A	O
similar	O
technique	O
can	O
be	O
used	O
for	O
any	O
iterative	B-Algorithm
method	I-Algorithm
.	O
</s>
<s>
Values	O
of	O
are	O
used	O
to	O
speed	O
up	O
convergence	O
of	O
a	O
slow-converging	O
process	O
,	O
while	O
values	O
of	O
are	O
often	O
used	O
to	O
help	O
establish	O
convergence	O
of	O
a	O
diverging	O
iterative	B-Algorithm
process	I-Algorithm
or	O
speed	O
up	O
the	O
convergence	O
of	O
an	O
overshooting	O
process	O
.	O
</s>
