<s>
In	O
linear	B-Language
algebra	I-Language
,	O
a	O
convergent	B-Algorithm
matrix	I-Algorithm
is	O
a	O
matrix	B-Architecture
that	O
converges	O
to	O
the	O
zero	B-Algorithm
matrix	I-Algorithm
under	O
matrix	B-Architecture
exponentiation	O
.	O
</s>
<s>
When	O
successive	O
powers	O
of	O
a	O
matrix	B-Architecture
T	O
become	O
small	O
(	O
that	O
is	O
,	O
when	O
all	O
of	O
the	O
entries	O
of	O
T	O
approach	O
zero	O
,	O
upon	O
raising	O
T	O
to	O
successive	O
powers	O
)	O
,	O
the	O
matrix	B-Architecture
T	O
converges	O
to	O
the	O
zero	B-Algorithm
matrix	I-Algorithm
.	O
</s>
<s>
A	O
regular	B-Algorithm
splitting	I-Algorithm
of	O
a	O
non-singular	O
matrix	B-Architecture
A	O
results	O
in	O
a	O
convergent	B-Algorithm
matrix	I-Algorithm
T	O
.	O
A	O
semi-convergent	O
splitting	O
of	O
a	O
matrix	B-Architecture
A	O
results	O
in	O
a	O
semi-convergent	O
matrix	B-Architecture
T	O
.	O
A	O
general	O
iterative	B-Algorithm
method	I-Algorithm
converges	O
for	O
every	O
initial	O
vector	O
if	O
T	O
is	O
convergent	O
,	O
and	O
under	O
certain	O
conditions	O
if	O
T	O
is	O
semi-convergent	O
.	O
</s>
<s>
T	O
is	O
a	O
convergent	B-Algorithm
matrix	I-Algorithm
.	O
</s>
<s>
Let	O
T	O
be	O
an	O
n	O
n	O
matrix	B-Architecture
.	O
</s>
<s>
The	O
following	O
properties	O
are	O
equivalent	O
to	O
T	O
being	O
a	O
convergent	B-Algorithm
matrix	I-Algorithm
:	O
</s>
<s>
For	O
any	O
initial	O
vector	O
x(0 )	O
,	O
the	O
sequence	O
defined	O
by	O
(	O
)	O
,	O
for	O
each	O
k	O
0	O
and	O
c	O
0	O
,	O
converges	O
to	O
the	O
unique	O
solution	O
of	O
(	O
)	O
if	O
and	O
only	O
if	O
ρ(T )	O
<	O
1	O
,	O
that	O
is	O
,	O
T	O
is	O
a	O
convergent	B-Algorithm
matrix	I-Algorithm
.	O
</s>
<s>
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>
The	O
expression	O
(	O
)	O
is	O
a	O
regular	B-Algorithm
splitting	I-Algorithm
of	O
A	O
if	O
and	O
only	O
if	O
B−1	O
0	O
and	O
C	O
0	O
,	O
that	O
is	O
,	O
and	O
C	O
have	O
only	O
nonnegative	O
entries	O
.	O
</s>
<s>
If	O
the	O
splitting	O
(	O
)	O
is	O
a	O
regular	B-Algorithm
splitting	I-Algorithm
of	O
the	O
matrix	B-Architecture
A	O
and	O
A−1	O
0	O
,	O
then	O
ρ(T )	O
<	O
1	O
and	O
T	O
is	O
a	O
convergent	B-Algorithm
matrix	I-Algorithm
.	O
</s>
