<s>
In	O
the	O
mathematical	O
discipline	O
of	O
numerical	O
linear	O
algebra	O
,	O
a	O
matrix	B-Algorithm
splitting	I-Algorithm
is	O
an	O
expression	O
which	O
represents	O
a	O
given	O
matrix	B-Architecture
as	O
a	O
sum	O
or	O
difference	O
of	O
matrices	O
.	O
</s>
<s>
Many	O
iterative	B-Algorithm
methods	I-Algorithm
(	O
for	O
example	O
,	O
for	O
systems	O
of	O
differential	O
equations	O
)	O
depend	O
upon	O
the	O
direct	O
solution	O
of	O
matrix	B-Architecture
equations	I-Architecture
involving	O
matrices	O
more	O
general	O
than	O
tridiagonal	B-Algorithm
matrices	I-Algorithm
.	O
</s>
<s>
These	O
matrix	B-Architecture
equations	I-Architecture
can	O
often	O
be	O
solved	O
directly	O
and	O
efficiently	O
when	O
written	O
as	O
a	O
matrix	B-Algorithm
splitting	I-Algorithm
.	O
</s>
<s>
where	O
A	O
is	O
a	O
given	O
n	O
×	O
n	O
non-singular	O
matrix	B-Architecture
,	O
and	O
k	O
is	O
a	O
given	O
column	O
vector	O
with	O
n	O
components	O
.	O
</s>
<s>
If	O
,	O
for	O
an	O
arbitrary	O
n	O
×	O
n	O
matrix	B-Architecture
M	O
,	O
M	O
has	O
nonnegative	O
entries	O
,	O
we	O
write	O
M	O
0	O
.	O
</s>
<s>
Similarly	O
,	O
if	O
the	O
matrix	B-Architecture
M1	O
M2	O
has	O
nonnegative	O
entries	O
,	O
we	O
write	O
M1	O
M2	O
.	O
</s>
<s>
The	O
matrix	B-Architecture
D	O
=	O
B−1C	O
has	O
nonnegative	O
entries	O
if	O
(	O
)	O
represents	O
a	O
regular	O
splitting	O
of	O
A	O
.	O
</s>
<s>
It	O
can	O
be	O
shown	O
that	O
if	O
A−1	O
0	O
,	O
then	O
<	O
1	O
,	O
where	O
represents	O
the	O
spectral	O
radius	O
of	O
D	O
,	O
and	O
thus	O
D	O
is	O
a	O
convergent	B-Algorithm
matrix	I-Algorithm
.	O
</s>
<s>
As	O
a	O
consequence	O
,	O
the	O
iterative	B-Algorithm
method	I-Algorithm
(	O
)	O
is	O
necessarily	O
convergent	O
.	O
</s>
<s>
If	O
,	O
in	O
addition	O
,	O
the	O
splitting	O
(	O
)	O
is	O
chosen	O
so	O
that	O
the	O
matrix	B-Architecture
B	O
is	O
a	O
diagonal	B-Algorithm
matrix	I-Algorithm
(	O
with	O
the	O
diagonal	O
entries	O
all	O
non-zero	O
,	O
since	O
B	O
must	O
be	O
invertible	O
)	O
,	O
then	O
B	O
can	O
be	O
inverted	O
in	O
linear	O
time	O
(	O
see	O
Time	O
complexity	O
)	O
.	O
</s>
<s>
Many	O
iterative	B-Algorithm
methods	I-Algorithm
can	O
be	O
described	O
as	O
a	O
matrix	B-Algorithm
splitting	I-Algorithm
.	O
</s>
<s>
where	O
D	O
is	O
the	O
diagonal	O
part	O
of	O
A	O
,	O
and	O
U	O
and	O
L	O
are	O
respectively	O
strictly	O
upper	O
and	O
lower	B-Algorithm
triangular	I-Algorithm
n	O
×	O
n	O
matrices	O
,	O
then	O
we	O
have	O
the	O
following	O
.	O
</s>
<s>
Let	O
us	O
apply	O
the	O
splitting	O
(	O
)	O
which	O
is	O
used	O
in	O
the	O
Jacobi	B-Algorithm
method	I-Algorithm
:	O
we	O
split	O
A	O
in	O
such	O
a	O
way	O
that	O
B	O
consists	O
of	O
all	O
of	O
the	O
diagonal	O
elements	O
of	O
A	O
,	O
and	O
C	O
consists	O
of	O
all	O
of	O
the	O
off-diagonal	O
elements	O
of	O
A	O
,	O
negated	O
.	O
</s>
<s>
(	O
Of	O
course	O
this	O
is	O
not	O
the	O
only	O
useful	O
way	O
to	O
split	O
a	O
matrix	B-Architecture
into	O
two	O
matrices	O
.	O
)	O
</s>
<s>
(	O
The	O
approximate	O
eigenvalues	O
of	O
D	O
are	O
)	O
Hence	O
,	O
the	O
matrix	B-Architecture
D	O
is	O
convergent	O
and	O
the	O
method	O
(	O
)	O
necessarily	O
converges	O
for	O
the	O
problem	O
(	O
)	O
.	O
</s>
<s>
Note	O
that	O
the	O
diagonal	O
elements	O
of	O
A	O
are	O
all	O
greater	O
than	O
zero	O
,	O
the	O
off-diagonal	O
elements	O
of	O
A	O
are	O
all	O
less	O
than	O
zero	O
and	O
A	O
is	O
strictly	B-Algorithm
diagonally	I-Algorithm
dominant	I-Algorithm
.	O
</s>
<s>
As	O
stated	O
above	O
,	O
the	O
Jacobi	B-Algorithm
method	I-Algorithm
(	O
)	O
is	O
the	O
same	O
as	O
the	O
specific	O
regular	O
splitting	O
(	O
)	O
demonstrated	O
above	O
.	O
</s>
<s>
From	O
the	O
table	O
one	O
can	O
see	O
that	O
the	O
method	O
is	O
evidently	O
converging	O
to	O
the	O
solution	O
(	O
)	O
,	O
somewhat	O
faster	O
than	O
the	O
Jacobi	B-Algorithm
method	I-Algorithm
described	O
above	O
.	O
</s>
<s>
From	O
the	O
table	O
one	O
can	O
see	O
that	O
the	O
method	O
is	O
evidently	O
converging	O
to	O
the	O
solution	O
(	O
)	O
,	O
slightly	O
faster	O
than	O
the	O
Gauss	B-Algorithm
–	I-Algorithm
Seidel	I-Algorithm
method	I-Algorithm
described	O
above	O
.	O
</s>
