<s>
In	O
mathematics	O
and	O
computer	B-General_Concept
science	I-General_Concept
,	O
computer	B-Algorithm
algebra	I-Algorithm
,	O
also	O
called	O
symbolic	B-Algorithm
computation	I-Algorithm
or	O
algebraic	B-Algorithm
computation	I-Algorithm
,	O
is	O
a	O
scientific	O
area	O
that	O
refers	O
to	O
the	O
study	O
and	O
development	O
of	O
algorithms	O
and	O
software	O
for	O
manipulating	O
mathematical	O
expressions	O
and	O
other	O
mathematical	O
objects	O
.	O
</s>
<s>
Although	O
computer	B-Algorithm
algebra	I-Algorithm
could	O
be	O
considered	O
a	O
subfield	O
of	O
scientific	O
computing	O
,	O
they	O
are	O
generally	O
considered	O
as	O
distinct	O
fields	O
because	O
scientific	O
computing	O
is	O
usually	O
based	O
on	O
numerical	B-General_Concept
computation	I-General_Concept
with	O
approximate	O
floating	B-Algorithm
point	I-Algorithm
numbers	I-Algorithm
,	O
while	O
symbolic	B-Algorithm
computation	I-Algorithm
emphasizes	O
exact	O
computation	O
with	O
expressions	O
containing	O
variables	O
that	O
have	O
no	O
given	O
value	O
and	O
are	O
manipulated	O
as	O
symbols	O
.	O
</s>
<s>
Software	O
applications	O
that	O
perform	O
symbolic	B-Algorithm
calculations	I-Algorithm
are	O
called	O
computer	B-General_Concept
algebra	I-General_Concept
systems	I-General_Concept
,	O
with	O
the	O
term	O
system	O
alluding	O
to	O
the	O
complexity	O
of	O
the	O
main	O
applications	O
that	O
include	O
,	O
at	O
least	O
,	O
a	O
method	O
to	O
represent	O
mathematical	O
data	O
in	O
a	O
computer	O
,	O
a	O
user	O
programming	O
language	O
(	O
usually	O
different	O
from	O
the	O
language	O
used	O
for	O
the	O
implementation	O
)	O
,	O
a	O
dedicated	O
memory	O
manager	O
,	O
a	O
user	B-Application
interface	I-Application
for	O
the	O
input/output	O
of	O
mathematical	O
expressions	O
,	O
a	O
large	O
set	O
of	O
routines	O
to	O
perform	O
usual	O
operations	O
,	O
like	O
simplification	O
of	O
expressions	O
,	O
differentiation	B-Algorithm
using	O
chain	O
rule	O
,	O
polynomial	O
factorization	O
,	O
indefinite	B-Algorithm
integration	I-Algorithm
,	O
etc	O
.	O
</s>
<s>
Computer	B-Algorithm
algebra	I-Algorithm
is	O
widely	O
used	O
to	O
experiment	O
in	O
mathematics	O
and	O
to	O
design	O
the	O
formulas	O
that	O
are	O
used	O
in	O
numerical	O
programs	O
.	O
</s>
<s>
It	O
is	O
also	O
used	O
for	O
complete	O
scientific	O
computations	O
,	O
when	O
purely	O
numerical	B-General_Concept
methods	I-General_Concept
fail	O
,	O
as	O
in	O
public	B-Application
key	I-Application
cryptography	I-Application
,	O
or	O
for	O
some	O
non-linear	O
problems	O
.	O
</s>
<s>
Some	O
authors	O
distinguish	O
computer	B-Algorithm
algebra	I-Algorithm
from	O
symbolic	B-Algorithm
computation	I-Algorithm
using	O
the	O
latter	O
name	O
to	O
refer	O
to	O
kinds	O
of	O
symbolic	B-Algorithm
computation	I-Algorithm
other	O
than	O
the	O
computation	O
with	O
mathematical	O
formulas	O
.	O
</s>
<s>
Some	O
authors	O
use	O
symbolic	B-Algorithm
computation	I-Algorithm
for	O
the	O
computer	B-General_Concept
science	I-General_Concept
aspect	O
of	O
the	O
subject	O
and	O
"	O
computer	B-Algorithm
algebra	I-Algorithm
"	O
for	O
the	O
mathematical	O
aspect	O
.	O
</s>
<s>
This	O
name	O
reflects	O
the	O
ties	O
this	O
field	O
has	O
with	O
formal	B-Architecture
methods	I-Architecture
.	O
</s>
<s>
Symbolic	B-Algorithm
computation	I-Algorithm
has	O
also	O
been	O
referred	O
to	O
,	O
in	O
the	O
past	O
,	O
as	O
symbolic	O
manipulation	O
,	O
algebraic	O
manipulation	O
,	O
symbolic	B-Algorithm
processing	I-Algorithm
,	O
symbolic	B-Algorithm
mathematics	I-Algorithm
,	O
or	O
symbolic	B-General_Concept
algebra	I-General_Concept
,	O
but	O
these	O
terms	O
,	O
which	O
also	O
refer	O
to	O
non-computational	O
manipulation	O
,	O
are	O
no	O
longer	O
used	O
in	O
reference	O
to	O
computer	B-Algorithm
algebra	I-Algorithm
.	O
</s>
<s>
There	O
are	O
several	O
annual	O
conferences	O
on	O
computer	B-Algorithm
algebra	I-Algorithm
,	O
the	O
premier	O
being	O
ISSAC	B-Algorithm
(	O
International	O
Symposium	O
on	O
Symbolic	B-Algorithm
and	I-Algorithm
Algebraic	I-Algorithm
Computation	I-Algorithm
)	O
,	O
which	O
is	O
regularly	O
sponsored	O
by	O
SIGSAM	B-General_Concept
.	O
</s>
<s>
There	O
are	O
several	O
journals	O
specializing	O
in	O
computer	B-Algorithm
algebra	I-Algorithm
,	O
the	O
top	O
one	O
being	O
Journal	O
of	O
Symbolic	B-Algorithm
Computation	I-Algorithm
founded	O
in	O
1985	O
by	O
Bruno	O
Buchberger	O
.	O
</s>
<s>
There	O
are	O
also	O
several	O
other	O
journals	O
that	O
regularly	O
publish	O
articles	O
in	O
computer	B-Algorithm
algebra	I-Algorithm
.	O
</s>
<s>
As	O
numerical	O
software	O
is	O
highly	O
efficient	O
for	O
approximate	O
numerical	B-General_Concept
computation	I-General_Concept
,	O
it	O
is	O
common	O
,	O
in	O
computer	B-Algorithm
algebra	I-Algorithm
,	O
to	O
emphasize	O
exact	O
computation	O
with	O
exactly	O
represented	O
data	O
.	O
</s>
<s>
The	O
usual	O
numbers	O
systems	O
used	O
in	O
numerical	B-General_Concept
computation	I-General_Concept
are	O
floating	B-Algorithm
point	I-Algorithm
numbers	I-Algorithm
and	O
integers	O
of	O
a	O
fixed	O
bounded	O
size	O
.	O
</s>
<s>
None	O
of	O
these	O
is	O
convenient	O
for	O
computer	B-Algorithm
algebra	I-Algorithm
,	O
due	O
to	O
expression	O
swell	O
.	O
</s>
<s>
Therefore	O
,	O
the	O
basic	O
numbers	O
used	O
in	O
computer	B-Algorithm
algebra	I-Algorithm
are	O
the	O
integers	O
of	O
the	O
mathematicians	O
,	O
commonly	O
represented	O
by	O
an	O
unbounded	O
signed	O
sequence	O
of	O
digits	O
in	O
some	O
base	O
of	O
numeration	O
,	O
usually	O
the	O
largest	O
base	O
allowed	O
by	O
the	O
machine	O
word	O
.	O
</s>
<s>
Therefore	O
,	O
most	O
free	O
computer	B-General_Concept
algebra	I-General_Concept
systems	I-General_Concept
and	O
some	O
commercial	O
ones	O
such	O
as	O
Mathematica	B-Language
and	O
Maple	B-Language
(	O
software	O
)	O
,	O
use	O
the	O
GMP	B-Application
library	I-Application
,	O
which	O
is	O
thus	O
a	O
de	O
facto	O
standard	O
.	O
</s>
<s>
In	O
computer	B-Algorithm
algebra	I-Algorithm
software	O
,	O
the	O
expressions	O
are	O
usually	O
represented	O
in	O
this	O
way	O
.	O
</s>
<s>
This	O
process	O
of	O
delayed	O
evaluation	O
is	O
fundamental	O
in	O
computer	B-Algorithm
algebra	I-Algorithm
.	O
</s>
<s>
For	O
example	O
,	O
the	O
operator	O
"	O
=	O
"	O
of	O
the	O
equations	O
is	O
also	O
,	O
in	O
most	O
computer	B-General_Concept
algebra	I-General_Concept
systems	I-General_Concept
,	O
the	O
name	O
of	O
the	O
program	O
of	O
the	O
equality	O
test	O
:	O
normally	O
,	O
the	O
evaluation	O
of	O
an	O
equation	O
results	O
in	O
an	O
equation	O
,	O
but	O
,	O
when	O
an	O
equality	O
test	O
is	O
needed	O
,	O
either	O
explicitly	O
asked	O
by	O
the	O
user	O
through	O
an	O
"	O
evaluation	O
to	O
a	O
Boolean	O
"	O
command	O
,	O
or	O
automatically	O
started	O
by	O
the	O
system	O
in	O
the	O
case	O
of	O
a	O
test	O
inside	O
a	O
program	O
,	O
then	O
the	O
evaluation	O
to	O
a	O
boolean	O
result	O
is	O
executed	O
.	O
</s>
<s>
As	O
the	O
size	O
of	O
the	O
operands	O
of	O
an	O
expression	O
is	O
unpredictable	O
and	O
may	O
change	O
during	O
a	O
working	O
session	O
,	O
the	O
sequence	O
of	O
the	O
operands	O
is	O
usually	O
represented	O
as	O
a	O
sequence	O
of	O
either	O
pointers	O
(	O
like	O
in	O
Macsyma	B-Language
)	O
or	O
entries	O
in	O
a	O
hash	B-Algorithm
table	I-Algorithm
(	O
like	O
in	O
Maple	B-Language
)	O
.	O
</s>
<s>
They	O
are	O
systematically	O
applied	O
in	O
computer	B-General_Concept
algebra	I-General_Concept
systems	I-General_Concept
.	O
</s>
<s>
To	O
address	O
this	O
,	O
Macsyma	B-Language
sorts	O
the	O
operands	O
of	O
sums	O
and	O
products	O
into	O
an	O
order	O
that	O
places	O
like	O
terms	O
in	O
consecutive	O
places	O
,	O
allowing	O
easy	O
detection	O
.	O
</s>
<s>
In	O
Maple	B-Language
,	O
a	O
hash	B-Error_Name
function	I-Error_Name
is	O
designed	O
for	O
generating	O
collisions	O
when	O
like	O
terms	O
are	O
entered	O
,	O
allowing	O
them	O
to	O
be	O
combined	O
as	O
soon	O
as	O
they	O
are	O
introduced	O
.	O
</s>
<s>
The	O
reverse	O
rewriting	O
rule	O
,	O
called	O
"	O
factor	O
"	O
,	O
requires	O
a	O
non-trivial	O
algorithm	O
,	O
which	O
is	O
thus	O
a	O
key	O
function	O
in	O
computer	B-General_Concept
algebra	I-General_Concept
systems	I-General_Concept
(	O
see	O
Polynomial	O
factorization	O
)	O
.	O
</s>
<s>
In	O
computer	B-Algorithm
algebra	I-Algorithm
,	O
"	O
canonical	O
form	O
"	O
and	O
"	O
normal	O
form	O
"	O
are	O
not	O
synonymous	O
.	O
</s>
<s>
Normal	O
forms	O
are	O
usually	O
preferred	O
in	O
computer	B-Algorithm
algebra	I-Algorithm
for	O
several	O
reasons	O
.	O
</s>
<s>
At	O
the	O
beginning	O
of	O
computer	B-Algorithm
algebra	I-Algorithm
,	O
circa	O
1970	O
,	O
when	O
the	O
long-known	O
algorithms	O
were	O
first	O
put	O
on	O
computers	O
,	O
they	O
turned	O
out	O
to	O
be	O
highly	O
inefficient	O
.	O
</s>
<s>
Therefore	O
,	O
a	O
large	O
part	O
of	O
the	O
work	O
of	O
the	O
researchers	O
in	O
the	O
field	O
consisted	O
in	O
revisiting	O
classical	O
algebra	O
in	O
order	O
to	O
make	O
it	O
effective	O
and	O
to	O
discover	O
efficient	B-General_Concept
algorithms	I-General_Concept
to	O
implement	O
this	O
effectiveness	O
.	O
</s>
<s>
A	O
typical	O
example	O
of	O
this	O
kind	O
of	O
work	O
is	O
the	O
computation	O
of	O
polynomial	B-Algorithm
greatest	I-Algorithm
common	I-Algorithm
divisors	I-Algorithm
,	O
which	O
is	O
required	O
to	O
simplify	O
fractions	O
.	O
</s>
<s>
The	O
same	O
was	O
also	O
true	O
for	O
the	O
classical	O
algorithms	O
from	O
linear	B-Language
algebra	I-Language
.	O
</s>
