<s>
Pseudo-spectral	B-Algorithm
methods	I-Algorithm
,	O
also	O
known	O
as	O
discrete	O
variable	O
representation	O
(	O
DVR	O
)	O
methods	O
,	O
are	O
a	O
class	O
of	O
numerical	B-Algorithm
methods	I-Algorithm
used	O
in	O
applied	O
mathematics	O
and	O
scientific	O
computing	O
for	O
the	O
solution	O
of	O
partial	O
differential	O
equations	O
.	O
</s>
<s>
They	O
are	O
closely	O
related	O
to	O
spectral	B-Algorithm
methods	I-Algorithm
,	O
but	O
complement	O
the	O
basis	O
by	O
an	O
additional	O
pseudo-spectral	O
basis	O
,	O
which	O
allows	O
representation	O
of	O
functions	O
on	O
a	O
quadrature	O
grid	O
.	O
</s>
<s>
In	O
the	O
spectral	B-Algorithm
method	I-Algorithm
,	O
the	O
solution	O
is	O
expanded	O
in	O
a	O
suitable	O
set	O
of	O
basis	O
functions	O
,	O
for	O
example	O
plane	O
waves	O
,	O
</s>
<s>
In	O
general	O
,	O
this	O
is	O
done	O
by	O
numerical	B-Algorithm
methods	I-Algorithm
,	O
such	O
as	O
Runge	B-Algorithm
–	I-Algorithm
Kutta	I-Algorithm
methods	I-Algorithm
.	O
</s>
<s>
For	O
the	O
numerical	B-General_Concept
solutions	I-General_Concept
,	O
the	O
right-hand	O
side	O
of	O
the	O
ordinary	O
differential	O
equation	O
has	O
to	O
be	O
evaluated	O
repeatedly	O
at	O
different	O
time	O
steps	O
.	O
</s>
<s>
At	O
this	O
point	O
,	O
the	O
spectral	B-Algorithm
method	I-Algorithm
has	O
a	O
major	O
problem	O
with	O
the	O
potential	O
term	O
.	O
</s>
<s>
In	O
the	O
pseudo-spectral	B-Algorithm
method	I-Algorithm
,	O
this	O
term	O
is	O
evaluated	O
differently	O
.	O
</s>
<s>
Given	O
the	O
coefficients	O
,	O
an	O
inverse	B-Algorithm
discrete	I-Algorithm
Fourier	I-Algorithm
transform	I-Algorithm
yields	O
the	O
value	O
of	O
the	O
function	O
at	O
discrete	O
grid	O
points	O
.	O
</s>
<s>
However	O
,	O
the	O
pseudo-spectral	B-Algorithm
method	I-Algorithm
allows	O
the	O
use	O
of	O
a	O
fast	O
Fourier	O
transform	O
,	O
which	O
scales	O
as	O
,	O
and	O
is	O
therefore	O
significantly	O
more	O
efficient	O
than	O
the	O
matrix	O
multiplication	O
.	O
</s>
<s>
In	O
a	O
more	O
abstract	O
way	O
,	O
the	O
pseudo-spectral	B-Algorithm
method	I-Algorithm
deals	O
with	O
the	O
multiplication	O
of	O
two	O
functions	O
and	O
as	O
part	O
of	O
a	O
partial	O
differential	O
equation	O
.	O
</s>
<s>
are	O
expanded	O
in	O
a	O
finite	O
set	O
of	O
basis	O
functions	O
(	O
this	O
is	O
the	O
spectral	B-Algorithm
method	I-Algorithm
)	O
.	O
</s>
<s>
This	O
forms	O
the	O
basis	O
of	O
the	O
spectral	B-Algorithm
method	I-Algorithm
.	O
</s>
<s>
Special	O
examples	O
are	O
the	O
Gaussian	B-Algorithm
quadrature	I-Algorithm
for	O
polynomials	O
and	O
the	O
Discrete	B-Algorithm
Fourier	I-Algorithm
Transform	I-Algorithm
for	O
plane	O
waves	O
.	O
</s>
<s>
However	O
,	O
using	O
the	O
spectral	B-Algorithm
method	I-Algorithm
,	O
the	O
same	O
coefficient	O
would	O
be	O
.	O
</s>
<s>
Thus	O
,	O
qualitatively	O
,	O
if	O
the	O
functions	O
can	O
be	O
represented	O
sufficiently	O
accurately	O
with	O
basis	O
functions	O
,	O
the	O
pseudo-spectral	B-Algorithm
method	I-Algorithm
gives	O
accurate	O
results	O
if	O
basis	O
functions	O
are	O
used	O
.	O
</s>
<s>
As	O
a	O
consequence	O
,	O
plane	O
waves	O
are	O
one	O
of	O
the	O
most	O
common	O
expansion	O
that	O
is	O
encountered	O
with	O
pseudo-spectral	B-Algorithm
methods	I-Algorithm
.	O
</s>
<s>
This	O
basis	O
,	O
together	O
with	O
the	O
quadrature	O
points	O
can	O
then	O
be	O
used	O
for	O
the	O
pseudo-spectral	B-Algorithm
method	I-Algorithm
.	O
</s>
<s>
For	O
the	O
discussion	O
of	O
the	O
error	O
,	O
note	O
that	O
if	O
is	O
well	O
represented	O
by	O
basis	O
functions	O
and	O
is	O
well	O
represented	O
by	O
a	O
polynomial	O
of	O
degree	O
,	O
their	O
product	O
can	O
be	O
expanded	O
in	O
the	O
first	O
basis	O
functions	O
,	O
and	O
the	O
pseudo-spectral	B-Algorithm
method	I-Algorithm
will	O
give	O
accurate	O
results	O
for	O
that	O
many	O
basis	O
functions	O
.	O
</s>
