<s>
In	O
linear	B-Language
algebra	I-Language
,	O
the	O
Strassen	B-Algorithm
algorithm	I-Algorithm
,	O
named	O
after	O
Volker	O
Strassen	O
,	O
is	O
an	O
algorithm	O
for	O
matrix	O
multiplication	O
.	O
</s>
<s>
The	O
Strassen	B-Algorithm
algorithm	I-Algorithm
is	O
slower	O
than	O
the	O
fastest	O
known	O
algorithms	O
for	O
extremely	O
large	O
matrices	O
,	O
but	O
such	O
galactic	O
algorithms	O
are	O
not	O
useful	O
in	O
practice	O
,	O
as	O
they	O
are	O
much	O
slower	O
for	O
matrices	O
of	O
practical	O
size	O
.	O
</s>
<s>
Strassen	B-Algorithm
's	I-Algorithm
algorithm	I-Algorithm
works	O
for	O
any	O
ring	O
,	O
such	O
as	O
plus/multiply	O
,	O
but	O
not	O
all	O
semirings	O
,	O
such	O
as	O
min-plus	O
or	O
boolean	O
algebra	O
,	O
where	O
the	O
naive	O
algorithm	O
still	O
works	O
,	O
and	O
so	O
called	O
combinatorial	O
matrix	O
multiplication	O
.	O
</s>
<s>
The	O
Strassen	B-Algorithm
algorithm	I-Algorithm
's	O
publication	O
resulted	O
in	O
more	O
research	O
about	O
matrix	O
multiplication	O
that	O
led	O
to	O
both	O
asymptotically	O
lower	O
bounds	O
and	O
improved	O
computational	O
upper	O
bounds	O
.	O
</s>
<s>
Let	O
,	O
be	O
two	O
square	B-Algorithm
matrices	I-Algorithm
over	O
a	O
ring	O
,	O
for	O
example	O
matrices	O
whose	O
entries	O
are	O
integers	O
or	O
the	O
real	O
numbers	O
.	O
</s>
<s>
This	O
construction	O
does	O
not	O
reduce	O
the	O
number	O
of	O
multiplications	O
:	O
8	O
multiplications	O
of	O
matrix	B-Algorithm
blocks	I-Algorithm
are	O
still	O
needed	O
to	O
calculate	O
the	O
matrices	O
,	O
the	O
same	O
number	O
of	O
multiplications	O
needed	O
when	O
using	O
standard	O
matrix	O
multiplication	O
.	O
</s>
<s>
The	O
Strassen	B-Algorithm
algorithm	I-Algorithm
defines	O
instead	O
new	O
matrices	O
:	O
</s>
<s>
Practical	O
implementations	O
of	O
Strassen	B-Algorithm
's	I-Algorithm
algorithm	I-Algorithm
switch	O
to	O
standard	O
methods	O
of	O
matrix	O
multiplication	O
for	O
small	O
enough	O
submatrices	O
,	O
for	O
which	O
those	O
algorithms	O
are	O
more	O
efficient	O
.	O
</s>
<s>
The	O
particular	O
crossover	O
point	O
for	O
which	O
Strassen	B-Algorithm
's	I-Algorithm
algorithm	I-Algorithm
is	O
more	O
efficient	O
depends	O
on	O
the	O
specific	O
implementation	O
and	O
hardware	O
.	O
</s>
<s>
Earlier	O
authors	O
had	O
estimated	O
that	O
Strassen	B-Algorithm
's	I-Algorithm
algorithm	I-Algorithm
is	O
faster	O
for	O
matrices	O
with	O
widths	O
from	O
32	O
to	O
128	O
for	O
optimized	O
implementations	O
.	O
</s>
<s>
However	O
,	O
it	O
has	O
been	O
observed	O
that	O
this	O
crossover	O
point	O
has	O
been	O
increasing	O
in	O
recent	O
years	O
,	O
and	O
a	O
2010	O
study	O
found	O
that	O
even	O
a	O
single	O
step	O
of	O
Strassen	B-Algorithm
's	I-Algorithm
algorithm	I-Algorithm
is	O
often	O
not	O
beneficial	O
on	O
current	O
architectures	O
,	O
compared	O
to	O
a	O
highly	O
optimized	O
traditional	O
multiplication	O
,	O
until	O
matrix	O
sizes	O
exceed	O
1000	O
or	O
more	O
,	O
and	O
even	O
for	O
matrix	O
sizes	O
of	O
several	O
thousand	O
the	O
benefit	O
is	O
typically	O
marginal	O
at	O
best	O
(	O
around	O
10%	O
or	O
less	O
)	O
.	O
</s>
<s>
The	O
question	O
then	O
is	O
how	O
many	O
operations	O
exactly	O
one	O
needs	O
for	O
Strassen	B-Algorithm
's	I-Algorithm
algorithms	I-Algorithm
,	O
and	O
how	O
this	O
compares	O
with	O
the	O
standard	O
matrix	O
multiplication	O
that	O
takes	O
approximately	O
(	O
where	O
)	O
arithmetic	O
operations	O
,	O
i.e.	O
</s>
<s>
The	O
number	O
of	O
additions	O
and	O
multiplications	O
required	O
in	O
the	O
Strassen	B-Algorithm
algorithm	I-Algorithm
can	O
be	O
calculated	O
as	O
follows	O
:	O
let	O
be	O
the	O
number	O
of	O
operations	O
for	O
a	O
matrix	O
.	O
</s>
<s>
Then	O
by	O
recursive	O
application	O
of	O
the	O
Strassen	B-Algorithm
algorithm	I-Algorithm
,	O
we	O
see	O
that	O
,	O
for	O
some	O
constant	O
that	O
depends	O
on	O
the	O
number	O
of	O
additions	O
performed	O
at	O
each	O
application	O
of	O
the	O
algorithm	O
.	O
</s>
<s>
Hence	O
,	O
i.e.	O
,	O
the	O
asymptotic	O
complexity	O
for	O
multiplying	O
matrices	O
of	O
size	O
using	O
the	O
Strassen	B-Algorithm
algorithm	I-Algorithm
is	O
.	O
</s>
<s>
The	O
reduction	O
in	O
the	O
number	O
of	O
arithmetic	O
operations	O
however	O
comes	O
at	O
the	O
price	O
of	O
a	O
somewhat	O
reduced	O
numerical	B-Algorithm
stability	I-Algorithm
,	O
and	O
the	O
algorithm	O
also	O
requires	O
significantly	O
more	O
memory	O
compared	O
to	O
the	O
naive	O
algorithm	O
.	O
</s>
<s>
Strassen	B-Algorithm
's	I-Algorithm
algorithm	I-Algorithm
needs	O
to	O
be	O
compared	O
to	O
the	O
"	O
naive	O
"	O
way	O
of	O
doing	O
the	O
matrix	O
multiplication	O
that	O
would	O
require	O
8	O
instead	O
of	O
7	O
multiplications	O
of	O
sub-blocks	O
.	O
</s>
<s>
The	O
comparison	O
of	O
these	O
two	O
algorithms	O
shows	O
that	O
asymptotically	O
,	O
Strassen	B-Algorithm
's	I-Algorithm
algorithm	I-Algorithm
is	O
faster	O
:	O
There	O
exists	O
a	O
size	O
so	O
that	O
matrices	O
that	O
are	O
larger	O
are	O
more	O
efficiently	O
multiplied	O
with	O
Strassen	B-Algorithm
's	I-Algorithm
algorithm	I-Algorithm
than	O
the	O
"	O
traditional	O
"	O
way	O
.	O
</s>
<s>
However	O
,	O
the	O
asymptotic	O
statement	O
does	O
not	O
imply	O
that	O
Strassen	B-Algorithm
's	I-Algorithm
algorithm	I-Algorithm
is	O
always	O
faster	O
even	O
for	O
small	O
matrices	O
,	O
and	O
in	O
practice	O
this	O
is	O
in	O
fact	O
not	O
the	O
case	O
:	O
For	O
small	O
matrices	O
,	O
the	O
cost	O
of	O
the	O
additional	O
additions	O
of	O
matrix	B-Algorithm
blocks	I-Algorithm
outweighs	O
the	O
savings	O
in	O
the	O
number	O
of	O
multiplications	O
.	O
</s>
<s>
As	O
a	O
consequence	O
of	O
these	O
sorts	O
of	O
considerations	O
,	O
Strassen	B-Algorithm
's	I-Algorithm
algorithm	I-Algorithm
is	O
typically	O
only	O
used	O
on	O
"	O
large	O
"	O
matrices	O
.	O
</s>
<s>
The	O
existence	O
of	O
Strassen	B-Algorithm
's	I-Algorithm
algorithm	I-Algorithm
shows	O
that	O
the	O
rank	O
of	O
matrix	O
multiplication	O
is	O
no	O
more	O
than	O
seven	O
.	O
</s>
<s>
Strassen	B-Algorithm
's	I-Algorithm
algorithm	I-Algorithm
is	O
cache	B-Application
oblivious	I-Application
.	O
</s>
<s>
cache	B-General_Concept
misses	O
during	O
its	O
execution	O
,	O
assuming	O
an	O
idealized	O
cache	B-General_Concept
of	O
size	O
(	O
i.e.	O
</s>
<s>
It	O
is	O
not	O
necessary	O
or	O
desirable	O
to	O
use	O
the	O
Strassen	B-Algorithm
algorithm	I-Algorithm
down	O
to	O
the	O
limit	O
of	O
scalars	O
.	O
</s>
<s>
The	O
method	O
can	O
indeed	O
be	O
applied	O
to	O
square	B-Algorithm
matrices	I-Algorithm
of	O
any	O
dimension	O
.	O
</s>
<s>
In	O
practice	O
,	O
Strassen	B-Algorithm
's	I-Algorithm
algorithm	I-Algorithm
can	O
be	O
implemented	O
to	O
attain	O
better	O
performance	O
than	O
conventional	O
multiplication	O
even	O
for	O
small	O
matrices	O
,	O
for	O
matrices	O
that	O
are	O
not	O
at	O
all	O
square	O
,	O
and	O
without	O
requiring	O
workspace	O
beyond	O
buffers	O
that	O
are	O
already	O
needed	O
for	O
a	O
high-performance	O
conventional	O
multiplication	O
.	O
</s>
