<s>
The	O
Neumann	B-Algorithm
series	I-Algorithm
is	O
used	O
in	O
functional	B-Application
analysis	I-Application
.	O
</s>
<s>
It	O
forms	O
the	O
basis	O
of	O
the	O
Liouville-Neumann	O
series	O
,	O
which	O
is	O
used	O
to	O
solve	O
Fredholm	B-Algorithm
integral	I-Algorithm
equations	I-Algorithm
.	O
</s>
<s>
If	O
the	O
Neumann	B-Algorithm
series	I-Algorithm
converges	B-Algorithm
in	O
the	O
operator	O
norm	O
,	O
then	O
is	O
invertible	O
and	O
its	O
inverse	O
is	O
the	O
series	O
:	O
</s>
<s>
One	O
case	O
in	O
which	O
convergence	B-Algorithm
is	O
guaranteed	O
is	O
when	O
is	O
a	O
Banach	O
space	O
and	O
in	O
the	O
operator	O
norm	O
or	O
is	O
convergent	O
.	O
</s>
<s>
However	O
,	O
there	O
are	O
also	O
results	O
which	O
give	O
weaker	O
conditions	O
under	O
which	O
the	O
series	O
converges	B-Algorithm
.	O
</s>
<s>
A	O
truncated	O
Neumann	B-Algorithm
series	I-Algorithm
can	O
be	O
used	O
for	O
approximate	O
matrix	O
inversion	O
.	O
</s>
<s>
Since	O
,	O
the	O
Neumann	B-Algorithm
series	I-Algorithm
is	O
convergent	O
.	O
</s>
<s>
The	O
Neumann	B-Algorithm
series	I-Algorithm
has	O
been	O
used	O
for	O
linear	O
data	O
detection	O
in	O
massive	O
multiuser	O
multiple-input	O
multiple-output	O
(	O
MIMO	O
)	O
wireless	O
systems	O
.	O
</s>
<s>
Using	O
a	O
truncated	O
Neumann	B-Algorithm
series	I-Algorithm
avoids	O
computation	O
of	O
an	O
explicit	O
matrix	O
inverse	O
,	O
which	O
reduces	O
the	O
complexity	O
of	O
linear	O
data	O
detection	O
from	O
cubic	O
to	O
square	O
.	O
</s>
<s>
Another	O
application	O
is	O
the	O
theory	O
of	O
Propagation	O
graphs	O
which	O
takes	O
advantage	O
of	O
Neumann	B-Algorithm
series	I-Algorithm
to	O
derive	O
closed	O
form	O
expression	O
for	O
the	O
transfer	O
function	O
.	O
</s>
