<s>
Levinson	B-Algorithm
recursion	I-Algorithm
or	O
Levinson	O
–	O
Durbin	O
recursion	O
is	O
a	O
procedure	O
in	O
linear	B-Language
algebra	I-Language
to	O
recursively	O
calculate	O
the	O
solution	O
to	O
an	O
equation	O
involving	O
a	O
Toeplitz	B-Algorithm
matrix	I-Algorithm
.	O
</s>
<s>
The	O
algorithm	O
runs	O
in	O
time	O
,	O
which	O
is	O
a	O
strong	O
improvement	O
over	O
Gauss	B-Algorithm
–	I-Algorithm
Jordan	I-Algorithm
elimination	I-Algorithm
,	O
which	O
runs	O
in	O
Θ(n3 )	O
.	O
</s>
<s>
In	O
comparison	O
to	O
these	O
,	O
Levinson	B-Algorithm
recursion	I-Algorithm
(	O
particularly	O
split	O
Levinson	B-Algorithm
recursion	I-Algorithm
)	O
tends	O
to	O
be	O
faster	O
computationally	O
,	O
but	O
more	O
sensitive	O
to	O
computational	O
inaccuracies	O
like	O
round-off	B-Algorithm
errors	I-Algorithm
.	O
</s>
<s>
The	O
Bareiss	B-Algorithm
algorithm	I-Algorithm
for	O
Toeplitz	B-Algorithm
matrices	I-Algorithm
(	O
not	O
to	O
be	O
confused	O
with	O
the	O
general	O
Bareiss	B-Algorithm
algorithm	I-Algorithm
)	O
runs	O
about	O
as	O
fast	O
as	O
Levinson	B-Algorithm
recursion	I-Algorithm
,	O
but	O
it	O
uses	O
space	O
,	O
whereas	O
Levinson	B-Algorithm
recursion	I-Algorithm
uses	O
only	O
O(n )	O
space	O
.	O
</s>
<s>
The	O
Bareiss	B-Algorithm
algorithm	I-Algorithm
,	O
though	O
,	O
is	O
numerically	B-Algorithm
stable	I-Algorithm
,	O
whereas	O
Levinson	B-Algorithm
recursion	I-Algorithm
is	O
at	O
best	O
only	O
weakly	O
stable	O
(	O
i.e.	O
</s>
<s>
it	O
exhibits	O
numerical	B-Algorithm
stability	I-Algorithm
for	O
well-conditioned	B-Algorithm
linear	O
systems	O
)	O
.	O
</s>
<s>
Levinson	B-Algorithm
recursion	I-Algorithm
remains	O
popular	O
for	O
several	O
reasons	O
;	O
for	O
one	O
,	O
it	O
is	O
relatively	O
easy	O
to	O
understand	O
in	O
comparison	O
;	O
for	O
another	O
,	O
it	O
can	O
be	O
faster	O
than	O
a	O
superfast	O
algorithm	O
for	O
small	O
n	O
(	O
usually	O
n	O
<	O
256	O
)	O
.	O
</s>
<s>
The	O
Levinson	O
–	O
Durbin	O
algorithm	O
may	O
be	O
used	O
for	O
any	O
such	O
equation	O
,	O
as	O
long	O
as	O
M	O
is	O
a	O
known	O
Toeplitz	B-Algorithm
matrix	I-Algorithm
with	O
a	O
nonzero	O
main	O
diagonal	O
.	O
</s>
<s>
For	O
example	O
(	O
and	O
definition	O
)	O
,	O
in	O
this	O
article	O
,	O
the	O
matrix	O
Tn	O
is	O
an	O
n×n	O
matrix	O
that	O
copies	O
the	O
upper	O
left	O
n×n	O
block	B-Algorithm
from	O
M	O
–	O
that	O
is	O
,	O
Tnij	O
=	O
Mij	O
.	O
</s>
<s>
An	O
important	O
simplification	O
can	O
occur	O
when	O
M	O
is	O
a	O
symmetric	B-Algorithm
matrix	I-Algorithm
;	O
then	O
the	O
two	O
vectors	O
are	O
related	O
by	O
bni	O
=	O
fnn+1−i	O
—	O
that	O
is	O
,	O
they	O
are	O
row-reversals	O
of	O
each	O
other	O
.	O
</s>
<s>
Even	O
if	O
the	O
matrix	O
is	O
not	O
symmetric	B-Algorithm
,	O
then	O
the	O
nth	O
forward	O
and	O
backward	O
vector	O
may	O
be	O
found	O
from	O
the	O
vectors	O
of	O
length	O
n−1	O
as	O
follows	O
.	O
</s>
<s>
If	O
M	O
is	O
not	O
strictly	O
Toeplitz	O
,	O
but	O
block	B-Algorithm
Toeplitz	O
,	O
the	O
Levinson	B-Algorithm
recursion	I-Algorithm
can	O
be	O
derived	O
in	O
much	O
the	O
same	O
way	O
by	O
regarding	O
the	O
block	B-Algorithm
Toeplitz	B-Algorithm
matrix	I-Algorithm
as	O
a	O
Toeplitz	B-Algorithm
matrix	I-Algorithm
with	O
matrix	O
elements	O
(	O
Musicus	O
1988	O
)	O
.	O
</s>
<s>
Block	B-Algorithm
Toeplitz	B-Algorithm
matrices	I-Algorithm
arise	O
naturally	O
in	O
signal	O
processing	O
algorithms	O
when	O
dealing	O
with	O
multiple	O
signal	O
streams	O
(	O
e.g.	O
,	O
in	O
MIMO	O
systems	O
)	O
or	O
cyclo-stationary	O
signals	O
.	O
</s>
