<s>
In	O
numerical	B-General_Concept
analysis	I-General_Concept
,	O
the	O
Crank	B-Algorithm
–	I-Algorithm
Nicolson	I-Algorithm
method	I-Algorithm
is	O
a	O
finite	B-Algorithm
difference	I-Algorithm
method	I-Algorithm
used	O
for	O
numerically	B-General_Concept
solving	O
the	O
heat	O
equation	O
and	O
similar	O
partial	O
differential	O
equations	O
.	O
</s>
<s>
It	O
is	O
implicit	B-Algorithm
in	O
time	O
,	O
can	O
be	O
written	O
as	O
an	O
implicit	B-Algorithm
Runge	O
–	O
Kutta	O
method	O
,	O
and	O
it	O
is	O
numerically	B-Algorithm
stable	I-Algorithm
.	O
</s>
<s>
For	O
diffusion	O
equations	O
(	O
and	O
many	O
other	O
equations	O
)	O
,	O
it	O
can	O
be	O
shown	O
the	O
Crank	B-Algorithm
–	I-Algorithm
Nicolson	I-Algorithm
method	I-Algorithm
is	O
unconditionally	O
stable	B-Algorithm
.	O
</s>
<s>
However	O
,	O
the	O
approximate	O
solutions	O
can	O
still	O
contain	O
(	O
decaying	O
)	O
spurious	O
oscillations	O
if	O
the	O
ratio	O
of	O
time	O
step	O
times	O
the	O
thermal	O
diffusivity	O
to	O
the	O
square	O
of	O
space	O
step	O
,	O
,	O
is	O
large	O
(	O
typically	O
,	O
larger	O
than	O
1/2	O
per	O
Von	B-Algorithm
Neumann	I-Algorithm
stability	I-Algorithm
analysis	I-Algorithm
)	O
.	O
</s>
<s>
For	O
this	O
reason	O
,	O
whenever	O
large	O
time	O
steps	O
or	O
high	O
spatial	O
resolution	O
is	O
necessary	O
,	O
the	O
less	O
accurate	O
backward	B-Algorithm
Euler	I-Algorithm
method	I-Algorithm
is	O
often	O
used	O
,	O
which	O
is	O
both	O
stable	B-Algorithm
and	O
immune	O
to	O
oscillations	O
.	O
</s>
<s>
The	O
Crank	B-Algorithm
–	I-Algorithm
Nicolson	I-Algorithm
method	I-Algorithm
is	O
based	O
on	O
the	O
trapezoidal	B-Algorithm
rule	I-Algorithm
,	O
giving	O
second-order	O
convergence	O
in	O
time	O
.	O
</s>
<s>
For	O
linear	O
equations	O
,	O
the	O
trapezoidal	B-Algorithm
rule	I-Algorithm
is	O
equivalent	O
to	O
the	O
implicit	B-Algorithm
midpoint	I-Algorithm
method	I-Algorithm
the	O
simplest	O
example	O
of	O
a	O
Gauss	B-Algorithm
–	I-Algorithm
Legendre	I-Algorithm
implicit	B-Algorithm
Runge	O
–	O
Kutta	O
method	O
which	O
also	O
has	O
the	O
property	O
of	O
being	O
a	O
geometric	B-Algorithm
integrator	I-Algorithm
.	O
</s>
<s>
Letting	O
and	O
evaluated	O
for	O
and	O
,	O
the	O
equation	O
for	O
Crank	B-Algorithm
–	I-Algorithm
Nicolson	I-Algorithm
method	I-Algorithm
is	O
a	O
combination	O
of	O
the	O
forward	B-Algorithm
Euler	I-Algorithm
method	I-Algorithm
at	O
and	O
the	O
backward	B-Algorithm
Euler	I-Algorithm
method	I-Algorithm
at	O
n+1	O
(	O
note	O
,	O
however	O
,	O
that	O
the	O
method	O
itself	O
is	O
not	O
simply	O
the	O
average	O
of	O
those	O
two	O
methods	O
,	O
as	O
the	O
backward	O
Euler	O
equation	O
has	O
an	O
implicit	B-Algorithm
dependence	O
on	O
the	O
solution	O
)	O
:	O
</s>
<s>
Note	O
that	O
this	O
is	O
an	O
implicit	B-Algorithm
method	I-Algorithm
:	O
to	O
get	O
the	O
"	O
next	O
"	O
value	O
of	O
u	O
in	O
time	O
,	O
a	O
system	O
of	O
algebraic	O
equations	O
must	O
be	O
solved	O
.	O
</s>
<s>
In	O
many	O
problems	O
,	O
especially	O
linear	O
diffusion	O
,	O
the	O
algebraic	O
problem	O
is	O
tridiagonal	B-Algorithm
and	O
may	O
be	O
efficiently	O
solved	O
with	O
the	O
tridiagonal	B-Language
matrix	I-Language
algorithm	I-Language
,	O
which	O
gives	O
a	O
fast	O
direct	O
solution	O
,	O
as	O
opposed	O
to	O
the	O
usual	O
for	O
a	O
full	O
matrix	O
,	O
in	O
which	O
indicates	O
the	O
matrix	O
size	O
.	O
</s>
<s>
The	O
Crank	B-Algorithm
–	I-Algorithm
Nicolson	I-Algorithm
method	I-Algorithm
is	O
often	O
applied	O
to	O
diffusion	O
problems	O
.	O
</s>
<s>
Given	O
that	O
the	O
terms	O
on	O
the	O
right-hand	O
side	O
of	O
the	O
equation	O
are	O
known	O
,	O
this	O
is	O
a	O
tridiagonal	B-Algorithm
problem	O
,	O
so	O
that	O
may	O
be	O
efficiently	O
solved	O
by	O
using	O
the	O
tridiagonal	B-Language
matrix	I-Language
algorithm	I-Language
in	O
favor	O
over	O
the	O
much	O
more	O
costly	O
matrix	O
inversion	O
.	O
</s>
<s>
Other	O
times	O
,	O
it	O
may	O
be	O
possible	O
to	O
estimate	O
using	O
an	O
explicit	B-Algorithm
method	I-Algorithm
and	O
maintain	O
stability	O
.	O
</s>
<s>
The	O
Crank	B-Algorithm
–	I-Algorithm
Nicolson	I-Algorithm
method	I-Algorithm
(	O
where	O
i	O
represents	O
position	O
,	O
and	O
j	O
time	O
)	O
transforms	O
each	O
component	O
of	O
the	O
PDE	O
into	O
the	O
following	O
:	O
</s>
<s>
When	O
extending	O
into	O
two	O
dimensions	O
on	O
a	O
uniform	O
Cartesian	O
grid	O
,	O
the	O
derivation	O
is	O
similar	O
and	O
the	O
results	O
may	O
lead	O
to	O
a	O
system	O
of	O
band-diagonal	B-Algorithm
equations	O
rather	O
than	O
tridiagonal	B-Algorithm
ones	O
.	O
</s>
<s>
For	O
the	O
Crank	B-Algorithm
–	I-Algorithm
Nicolson	I-Algorithm
numerical	O
scheme	O
,	O
a	O
low	O
CFL	B-Algorithm
number	I-Algorithm
is	O
not	O
required	O
for	O
stability	O
,	O
however	O
,	O
it	O
is	O
required	O
for	O
numerical	O
accuracy	O
.	O
</s>
<s>
Hence	O
an	O
alternating-direction	B-Algorithm
implicit	I-Algorithm
method	I-Algorithm
can	O
be	O
implemented	O
to	O
solve	O
the	O
numerical	O
PDE	O
,	O
whereby	O
one	O
dimension	O
is	O
treated	O
implicitly	O
,	O
and	O
other	O
dimension	O
explicitly	O
for	O
half	O
of	O
the	O
assigned	O
time	O
step	O
and	O
conversely	O
for	O
the	O
remainder	O
half	O
of	O
the	O
time	O
step	O
.	O
</s>
<s>
The	O
benefit	O
of	O
this	O
strategy	O
is	O
that	O
the	O
implicit	B-Algorithm
solver	I-Algorithm
only	O
requires	O
a	O
tridiagonal	B-Language
matrix	I-Language
algorithm	I-Language
to	O
be	O
solved	O
.	O
</s>
<s>
The	O
difference	O
between	O
the	O
true	O
Crank	B-Algorithm
–	I-Algorithm
Nicolson	I-Algorithm
solution	O
and	O
ADI	O
approximated	O
solution	O
has	O
an	O
order	O
of	O
accuracy	O
of	O
and	O
hence	O
can	O
be	O
ignored	O
with	O
a	O
sufficiently	O
small	O
time	O
step	O
.	O
</s>
<s>
Because	O
the	O
Crank-Nicolson	B-Algorithm
method	I-Algorithm
is	O
implicit	B-Algorithm
,	O
it	O
is	O
generally	O
impossible	O
to	O
solve	O
exactly	O
.	O
</s>
<s>
For	O
a	O
high-dimensional	O
system	O
like	O
those	O
in	O
computational	O
fluid	O
dynamics	O
or	O
numerical	B-Algorithm
relativity	I-Algorithm
,	O
it	O
may	O
be	O
infeasible	O
to	O
compute	O
this	O
Jacobian	O
.	O
</s>
<s>
If	O
is	O
the	O
velocity	O
of	O
the	O
system	O
,	O
then	O
the	O
Crank-Nicolson	B-Algorithm
prediction	O
will	O
be	O
a	O
fixed	O
point	O
of	O
the	O
map	O
.	O
</s>
<s>
Because	O
a	O
number	O
of	O
other	O
phenomena	O
can	O
be	O
modeled	O
with	O
the	O
heat	O
equation	O
(	O
often	O
called	O
the	O
diffusion	O
equation	O
in	O
financial	O
mathematics	O
)	O
,	O
the	O
Crank	B-Algorithm
–	I-Algorithm
Nicolson	I-Algorithm
method	I-Algorithm
has	O
been	O
applied	O
to	O
those	O
areas	O
as	O
well	O
.	O
</s>
<s>
Particularly	O
,	O
the	O
Black	O
–	O
Scholes	O
option	O
pricing	O
model	O
's	O
differential	O
equation	O
can	O
be	O
transformed	O
into	O
the	O
heat	O
equation	O
,	O
and	O
thus	O
numerical	B-General_Concept
solutions	I-General_Concept
for	O
option	O
pricing	O
can	O
be	O
obtained	O
with	O
the	O
Crank	B-Algorithm
–	I-Algorithm
Nicolson	I-Algorithm
method	I-Algorithm
.	O
</s>
<s>
Note	O
however	O
,	O
that	O
for	O
non-smooth	O
final	O
conditions	O
(	O
which	O
happen	O
for	O
most	O
financial	O
instruments	O
)	O
,	O
the	O
Crank	B-Algorithm
–	I-Algorithm
Nicolson	I-Algorithm
method	I-Algorithm
is	O
not	O
satisfactory	O
as	O
numerical	O
oscillations	O
are	O
not	O
damped	O
.	O
</s>
<s>
Therefore	O
,	O
special	O
damping	O
initialization	O
steps	O
are	O
necessary	O
(	O
e.g.	O
,	O
fully	O
implicit	B-Algorithm
finite	B-Algorithm
difference	I-Algorithm
method	I-Algorithm
)	O
.	O
</s>
