<s>
The	O
method	B-Algorithm
of	I-Algorithm
moments	I-Algorithm
(	O
MoM	O
)	O
,	O
also	O
known	O
as	O
the	O
moment	O
method	O
and	O
method	O
of	O
weighted	O
residuals	B-Algorithm
,	O
is	O
a	O
numerical	B-General_Concept
method	I-General_Concept
in	O
computational	O
electromagnetics	O
.	O
</s>
<s>
It	O
is	O
used	O
in	O
computer	O
programs	O
that	O
simulate	O
the	O
interaction	O
of	O
electromagnetic	O
fields	O
such	O
as	O
radio	O
waves	O
with	O
matter	O
,	O
for	O
example	O
antenna	O
simulation	O
programs	O
like	O
NEC	B-Protocol
that	O
calculate	O
the	O
radiation	B-Device
pattern	I-Device
of	O
an	O
antenna	O
.	O
</s>
<s>
Generally	O
being	O
a	O
frequency-domain	O
method	O
,	O
it	O
involves	O
the	O
projection	O
of	O
an	O
integral	B-Algorithm
equation	I-Algorithm
into	O
a	O
system	O
of	O
linear	O
equations	O
by	O
the	O
application	O
of	O
appropriate	O
boundary	O
conditions	O
.	O
</s>
<s>
This	O
is	O
done	O
by	O
using	O
discrete	O
meshes	O
as	O
in	O
finite	B-Algorithm
difference	I-Algorithm
and	O
finite	B-Application
element	I-Application
methods	I-Application
,	O
often	O
for	O
the	O
surface	O
.	O
</s>
<s>
Green	O
's	O
functions	O
and	O
Galerkin	B-Algorithm
method	I-Algorithm
play	O
a	O
central	O
role	O
in	O
the	O
method	B-Algorithm
of	I-Algorithm
moments	I-Algorithm
.	O
</s>
<s>
For	O
many	O
applications	O
,	O
the	O
method	B-Algorithm
of	I-Algorithm
moments	I-Algorithm
is	O
identical	O
to	O
the	O
boundary	B-Algorithm
element	I-Algorithm
method	I-Algorithm
.	O
</s>
<s>
It	O
is	O
one	O
of	O
the	O
most	O
common	O
methods	O
in	O
microwave	B-Algorithm
and	O
antenna	O
engineering	O
.	O
</s>
<s>
Development	O
of	O
boundary	B-Algorithm
element	I-Algorithm
method	I-Algorithm
and	O
other	O
similar	O
methods	O
for	O
different	O
engineering	O
applications	O
is	O
associated	O
with	O
the	O
advent	O
of	O
digital	O
computing	O
in	O
the	O
1960s	O
.	O
</s>
<s>
Prior	O
to	O
this	O
,	O
variational	B-Algorithm
methods	I-Algorithm
were	O
applied	O
to	O
engineering	O
problems	O
at	O
microwave	B-Algorithm
frequencies	O
by	O
the	O
time	O
of	O
World	O
War	O
II	O
.	O
</s>
<s>
The	O
concept	O
was	O
later	O
shown	O
to	O
be	O
equivalent	O
to	O
the	O
Galerkin	B-Algorithm
method	I-Algorithm
.	O
</s>
<s>
In	O
the	O
late	O
1950s	O
,	O
an	O
early	O
version	O
of	O
the	O
method	B-Algorithm
of	I-Algorithm
moments	I-Algorithm
was	O
introduced	O
by	O
Yuen	O
Tze	O
Lo	O
at	O
a	O
course	O
on	O
mathematical	O
methods	O
in	O
electromagnetic	O
theory	O
at	O
University	O
of	O
Illinois	O
.	O
</s>
<s>
In	O
the	O
same	O
decade	O
,	O
the	O
systematic	O
theory	O
for	O
the	O
method	B-Algorithm
of	I-Algorithm
moments	I-Algorithm
in	O
electromagnetics	O
was	O
largely	O
formalized	O
by	O
Roger	O
F	O
.	O
Harrington	O
.	O
</s>
<s>
While	O
the	O
term	O
"	O
the	O
method	B-Algorithm
of	I-Algorithm
moments	I-Algorithm
"	O
was	O
coined	O
earlier	O
by	O
Leonid	O
Kantorovich	O
and	O
Gleb	O
P	O
.	O
Akilov	O
for	O
analogous	O
numerical	O
applications	O
,	O
Harrington	O
has	O
adapted	O
the	O
term	O
for	O
the	O
electromagnetic	O
formulation	O
.	O
</s>
<s>
The	O
development	O
of	O
the	O
method	O
and	O
its	O
indications	O
in	O
radar	B-Application
and	O
antenna	O
engineering	O
attracted	O
interest	O
;	O
MoM	O
research	O
was	O
subsequently	O
supported	O
United	O
States	O
government	O
.	O
</s>
<s>
The	O
method	O
was	O
further	O
popularized	O
by	O
the	O
introduction	O
of	O
generalized	O
antenna	O
modeling	O
codes	O
such	O
as	O
Numerical	B-Protocol
Electromagnetics	I-Protocol
Code	I-Protocol
,	O
which	O
was	O
released	O
into	O
public	O
domain	O
by	O
the	O
United	O
States	O
government	O
in	O
the	O
late	O
1980s	O
.	O
</s>
<s>
In	O
the	O
1990s	O
,	O
introduction	O
of	O
fast	B-Algorithm
multipole	I-Algorithm
and	O
multilevel	B-Algorithm
fast	I-Algorithm
multipole	I-Algorithm
methods	I-Algorithm
enabled	O
efficient	O
MoM	O
solutions	O
to	O
problems	O
with	O
millions	O
of	O
unknowns	O
.	O
</s>
<s>
Being	O
one	O
of	O
the	O
most	O
common	O
simulation	O
techniques	O
in	O
RF	B-Application
and	O
microwave	B-Algorithm
engineering	I-Algorithm
,	O
the	O
method	B-Algorithm
of	I-Algorithm
moments	I-Algorithm
forms	O
the	O
basis	O
of	O
many	O
commercial	O
design	O
software	O
such	O
as	O
FEKO	B-Language
.	O
</s>
<s>
In	O
addition	O
to	O
its	O
use	O
in	O
electrical	O
engineering	O
,	O
the	O
method	B-Algorithm
of	I-Algorithm
moments	I-Algorithm
has	O
been	O
applied	O
to	O
light	O
scattering	O
and	O
plasmonic	O
problems	O
.	O
</s>
<s>
An	O
inhomogeneous	O
integral	B-Algorithm
equation	I-Algorithm
can	O
be	O
expressed	O
as	O
:	O
</s>
<s>
where	O
L	O
denotes	O
a	O
linear	B-Architecture
operator	I-Architecture
,	O
g	O
denotes	O
the	O
known	O
forcing	O
function	O
and	O
f	O
denotes	O
the	O
unknown	O
function	O
.	O
</s>
<s>
We	O
can	O
also	O
define	O
a	O
residual	B-Algorithm
for	O
this	O
expression	O
,	O
which	O
denotes	O
the	O
difference	O
between	O
the	O
actual	O
and	O
the	O
approximate	O
solution	O
:	O
</s>
<s>
The	O
aim	O
of	O
the	O
method	B-Algorithm
of	I-Algorithm
moments	I-Algorithm
is	O
to	O
minimize	O
this	O
residual	B-Algorithm
,	O
which	O
can	O
be	O
done	O
by	O
using	O
appropriate	O
weighting	O
or	O
testing	O
functions	O
,	O
hence	O
the	O
name	O
method	O
of	O
weighted	O
residuals	B-Algorithm
.	O
</s>
<s>
For	O
large	O
matrices	O
with	O
a	O
large	O
number	O
of	O
unknowns	O
,	O
iterative	B-Algorithm
methods	I-Algorithm
such	O
as	O
conjugate	B-Algorithm
gradient	I-Algorithm
method	I-Algorithm
can	O
be	O
used	O
for	O
acceleration	B-Algorithm
.	O
</s>
<s>
Choice	O
of	O
Dirac	O
delta	O
function	O
as	O
basis	O
function	O
is	O
known	O
as	O
point-matching	O
or	O
collocation	B-Algorithm
.	O
</s>
<s>
The	O
testing	O
and	O
basis	O
functions	O
are	O
often	O
chosen	O
to	O
be	O
the	O
same	O
;	O
this	O
is	O
known	O
as	O
the	O
Galerkin	B-Algorithm
method	I-Algorithm
.	O
</s>
<s>
For	O
surface	O
problems	O
,	O
common	O
integral	B-Algorithm
equation	I-Algorithm
formulations	O
include	O
electric	O
field	O
integral	B-Algorithm
equation	I-Algorithm
(	O
EFIE	O
)	O
,	O
magnetic	O
field	O
integral	B-Algorithm
equation	I-Algorithm
(	O
MFIE	O
)	O
and	O
mixed-potential	O
integral	B-Algorithm
equation	I-Algorithm
(	O
MPIE	O
)	O
.	O
</s>
<s>
Hallén	O
integral	B-Algorithm
equation	I-Algorithm
,	O
published	O
by	O
E	O
.	O
Hallén	O
in	O
1938	O
,	O
can	O
be	O
given	O
as	O
:	O
</s>
<s>
The	O
general	O
form	O
of	O
electric	O
field	O
integral	B-Algorithm
equation	I-Algorithm
(	O
EFIE	O
)	O
can	O
be	O
written	O
as	O
:	O
</s>
<s>
The	O
boundary	O
conditions	O
are	O
met	O
at	O
a	O
defined	O
PEC	B-Algorithm
surface	O
.	O
</s>
<s>
EFIE	O
is	O
a	O
Fredholm	B-Algorithm
integral	I-Algorithm
equation	I-Algorithm
of	I-Algorithm
the	I-Algorithm
first	I-Algorithm
kind	I-Algorithm
.	O
</s>
<s>
Another	O
commonly	O
used	O
integral	B-Algorithm
equation	I-Algorithm
in	O
MoM	O
is	O
the	O
magnetic	O
field	O
integral	B-Algorithm
equation	I-Algorithm
(	O
MFIE	O
)	O
,	O
which	O
can	O
be	O
written	O
as	O
:	O
</s>
<s>
MFIE	O
is	O
often	O
formulated	O
to	O
be	O
a	O
Fredholm	B-Algorithm
integral	I-Algorithm
equation	I-Algorithm
of	I-Algorithm
the	I-Algorithm
second	I-Algorithm
kind	I-Algorithm
and	O
is	O
generally	O
well-posed	B-Algorithm
.	O
</s>
<s>
In	O
many	O
cases	O
,	O
EFIEs	O
are	O
converted	O
to	O
mixed	O
potential	O
integral	B-Algorithm
equations	I-Algorithm
(	O
MFIE	O
)	O
through	O
the	O
use	O
of	O
Lorenz	O
gauge	O
condition	O
;	O
this	O
aims	O
to	O
reduce	O
the	O
orders	O
of	O
singularities	O
through	O
the	O
use	O
of	O
magnetic	O
vector	O
and	O
scalar	O
electric	O
potentials	O
.	O
</s>
<s>
In	O
order	O
to	O
bypass	O
the	O
internal	O
resonance	O
problem	O
in	O
dielectric	O
scattering	O
calculations	O
,	O
combined-field	O
integral	B-Algorithm
equation	I-Algorithm
(	O
CFIE	O
)	O
and	O
Poggio	O
—	O
Miller	O
—	O
Chang	O
—	O
Harrington	O
—	O
Wu	O
—	O
Tsai	O
(	O
PMCHWT	O
)	O
formulations	O
are	O
also	O
used	O
.	O
</s>
<s>
Another	O
approach	O
,	O
the	O
volumetric	O
integral	B-Algorithm
equation	I-Algorithm
,	O
necessitates	O
the	O
discretization	O
of	O
the	O
volume	O
elements	O
and	O
is	O
often	O
computationally	O
expensive	O
.	O
</s>
<s>
MoM	O
can	O
also	O
be	O
integrated	O
with	O
physical	O
optics	O
theory	O
and	O
finite	B-Application
element	I-Application
method	I-Application
.	O
</s>
<s>
Nevertheless	O
,	O
this	O
involves	O
the	O
inverse	O
Hankel	B-Algorithm
transform	I-Algorithm
of	O
the	O
spectral	O
Green	O
's	O
function	O
,	O
which	O
is	O
defined	O
on	O
the	O
Sommerfeld	O
integration	O
path	O
.	O
</s>
<s>
This	O
integral	O
cannot	O
be	O
evaluated	O
analytically	O
,	O
and	O
its	O
numerical	B-General_Concept
evaluation	I-General_Concept
is	O
often	O
computationally	O
expensive	O
due	O
to	O
the	O
oscillatory	O
kernels	B-Algorithm
and	O
slowly-converging	O
nature	O
of	O
the	O
integral	O
.	O
</s>
<s>
Following	O
the	O
extraction	O
of	O
quasi-static	O
and	O
surface	O
pole	O
components	O
,	O
these	O
integrals	O
can	O
be	O
approximated	O
as	O
closed-form	O
complex	O
exponentials	O
through	O
Prony	B-Algorithm
's	I-Algorithm
method	I-Algorithm
or	O
generalized	B-Algorithm
pencil-of-function	I-Algorithm
method	I-Algorithm
;	O
thus	O
,	O
the	O
spatial	O
Green	O
's	O
functions	O
can	O
be	O
derived	O
through	O
the	O
use	O
of	O
appropriate	O
identities	O
such	O
as	O
Sommerfeld	O
identity	O
.	O
</s>
