<s>
In	O
numerical	O
linear	O
algebra	O
,	O
the	O
alternating-direction	O
implicit	O
(	O
ADI	O
)	O
method	O
is	O
an	O
iterative	O
method	O
used	O
to	O
solve	O
Sylvester	B-Algorithm
matrix	I-Algorithm
equations	I-Algorithm
.	O
</s>
<s>
The	O
ADI	B-Algorithm
method	I-Algorithm
is	O
a	O
two	O
step	O
iteration	O
process	O
that	O
alternately	O
updates	O
the	O
column	O
and	O
row	O
spaces	O
of	O
an	O
approximate	O
solution	O
to	O
.	O
</s>
<s>
If	O
and	O
,	O
then	O
can	O
be	O
solved	O
directly	O
in	O
using	O
the	O
Bartels-Stewart	B-Algorithm
method	I-Algorithm
.	O
</s>
<s>
However	O
,	O
the	O
ADI	B-Algorithm
method	I-Algorithm
performs	O
especially	O
well	O
when	O
and	O
are	O
well-separated	O
,	O
and	O
and	O
are	O
normal	B-Algorithm
matrices	I-Algorithm
.	O
</s>
<s>
These	O
assumptions	O
are	O
met	O
,	O
for	O
example	O
,	O
by	O
the	O
Lyapunov	O
equation	O
when	O
is	O
positive	B-Algorithm
definite	I-Algorithm
.	O
</s>
<s>
The	O
ADI	B-Algorithm
method	I-Algorithm
can	O
still	O
be	O
applied	O
when	O
the	O
above	O
assumptions	O
are	O
not	O
met	O
.	O
</s>
<s>
The	O
Lyapunov	O
equation	O
,	O
for	O
example	O
,	O
satisfies	O
these	O
assumptions	O
when	O
is	O
positive	B-Algorithm
definite	I-Algorithm
.	O
</s>
<s>
Historically	O
,	O
the	O
ADI	B-Algorithm
method	I-Algorithm
was	O
developed	O
to	O
solve	O
the	O
2D	O
diffusion	O
equation	O
on	O
a	O
square	O
domain	O
using	O
finite	O
differences	O
.	O
</s>
<s>
The	O
traditional	O
method	O
for	O
solving	O
the	O
heat	O
conduction	O
equation	O
numerically	O
is	O
the	O
Crank	B-Algorithm
–	I-Algorithm
Nicolson	I-Algorithm
method	I-Algorithm
.	O
</s>
<s>
The	O
advantage	O
of	O
the	O
ADI	B-Algorithm
method	I-Algorithm
is	O
that	O
the	O
equations	O
that	O
have	O
to	O
be	O
solved	O
in	O
each	O
step	O
have	O
a	O
simpler	O
structure	O
and	O
can	O
be	O
solved	O
efficiently	O
with	O
the	O
tridiagonal	B-Language
matrix	I-Language
algorithm	I-Language
.	O
</s>
<s>
The	O
implicit	O
Crank	B-Algorithm
–	I-Algorithm
Nicolson	I-Algorithm
method	I-Algorithm
produces	O
the	O
following	O
finite	O
difference	O
equation	O
:	O
</s>
<s>
After	O
performing	O
a	O
stability	B-Algorithm
analysis	I-Algorithm
,	O
it	O
can	O
be	O
shown	O
that	O
this	O
method	O
will	O
be	O
stable	O
for	O
any	O
.	O
</s>
<s>
A	O
disadvantage	O
of	O
the	O
Crank	B-Algorithm
–	I-Algorithm
Nicolson	I-Algorithm
method	I-Algorithm
is	O
that	O
the	O
matrix	O
in	O
the	O
above	O
equation	O
is	O
banded	B-Algorithm
with	O
a	O
band	O
width	O
that	O
is	O
generally	O
quite	O
large	O
.	O
</s>
<s>
This	O
makes	O
direct	O
solution	O
of	O
the	O
system	O
of	O
linear	O
equations	O
quite	O
costly	O
(	O
although	O
efficient	O
approximate	O
solutions	O
exist	O
,	O
for	O
example	O
use	O
of	O
the	O
conjugate	B-Algorithm
gradient	I-Algorithm
method	I-Algorithm
preconditioned	O
with	O
incomplete	O
Cholesky	O
factorization	O
)	O
.	O
</s>
<s>
The	O
idea	O
behind	O
the	O
ADI	B-Algorithm
method	I-Algorithm
is	O
to	O
split	O
the	O
finite	O
difference	O
equations	O
into	O
two	O
,	O
one	O
with	O
the	O
x-derivative	O
taken	O
implicitly	O
and	O
the	O
next	O
with	O
the	O
y-derivative	O
taken	O
implicitly	O
,	O
</s>
<s>
The	O
system	O
of	O
equations	O
involved	O
is	O
symmetric	B-Algorithm
and	O
tridiagonal	O
(	O
banded	B-Algorithm
with	O
bandwidth	B-Algorithm
3	O
)	O
,	O
and	O
is	O
typically	O
solved	O
using	O
tridiagonal	B-Language
matrix	I-Language
algorithm	I-Language
.	O
</s>
<s>
There	O
are	O
more	O
refined	O
ADI	B-Algorithm
methods	I-Algorithm
such	O
as	O
the	O
methods	O
of	O
Douglas	O
,	O
or	O
the	O
f-factor	O
method	O
which	O
can	O
be	O
used	O
for	O
three	O
or	O
more	O
dimensions	O
.	O
</s>
<s>
The	O
usage	O
of	O
the	O
ADI	B-Algorithm
method	I-Algorithm
as	O
an	O
operator	O
splitting	O
scheme	O
can	O
be	O
generalized	O
.	O
</s>
<s>
It	O
is	O
possible	O
to	O
simplify	O
the	O
conventional	O
ADI	B-Algorithm
method	I-Algorithm
into	O
Fundamental	O
ADI	B-Algorithm
method	I-Algorithm
,	O
which	O
only	O
has	O
the	O
similar	O
operators	O
at	O
the	O
left-hand	O
sides	O
while	O
being	O
operator-free	O
at	O
the	O
right-hand	O
sides	O
.	O
</s>
<s>
This	O
may	O
be	O
regarded	O
as	O
the	O
fundamental	O
(	O
basic	O
)	O
scheme	O
of	O
ADI	B-Algorithm
method	I-Algorithm
,	O
with	O
no	O
more	O
operator	O
(	O
to	O
be	O
reduced	O
)	O
at	O
the	O
right-hand	O
sides	O
,	O
unlike	O
most	O
traditional	O
implicit	O
methods	O
that	O
usually	O
consist	O
of	O
operators	O
at	O
both	O
sides	O
of	O
equations	O
.	O
</s>
<s>
The	O
FADI	O
method	O
leads	O
to	O
simpler	O
,	O
more	O
concise	O
and	O
efficient	O
update	O
equations	O
without	O
degrading	O
the	O
accuracy	O
of	O
conventional	O
ADI	B-Algorithm
method	I-Algorithm
.	O
</s>
<s>
Many	O
classical	O
implicit	O
methods	O
by	O
Peaceman-Rachford	O
,	O
Douglas-Gunn	O
,	O
D'Yakonov	O
,	O
Beam-Warming	O
,	O
Crank-Nicolson	B-Algorithm
,	O
etc.	O
,	O
may	O
be	O
simplified	O
to	O
fundamental	O
implicit	O
schemes	O
with	O
operator-free	O
right-hand	O
sides	O
.	O
</s>
