<s>
Magma	B-General_Concept
is	O
a	O
computer	B-General_Concept
algebra	I-General_Concept
system	I-General_Concept
designed	O
to	O
solve	O
problems	O
in	O
algebra	O
,	O
number	O
theory	O
,	O
geometry	O
and	O
combinatorics	O
.	O
</s>
<s>
It	O
is	O
named	O
after	O
the	O
algebraic	O
structure	O
magma	B-General_Concept
.	O
</s>
<s>
It	O
runs	O
on	O
Unix-like	B-Operating_System
operating	I-Operating_System
systems	I-Operating_System
,	O
as	O
well	O
as	O
Windows	B-Application
.	O
</s>
<s>
Magma	B-General_Concept
is	O
produced	O
and	O
distributed	O
by	O
the	O
within	O
the	O
School	O
of	O
Mathematics	O
and	O
Statistics	O
at	O
the	O
University	O
of	O
Sydney	O
.	O
</s>
<s>
The	O
Magma	B-General_Concept
system	O
is	O
used	O
extensively	O
within	O
pure	O
mathematics	O
.	O
</s>
<s>
The	O
Computational	O
Algebra	O
Group	O
maintain	O
a	O
list	O
of	O
publications	O
that	O
cite	O
Magma	B-General_Concept
,	O
and	O
as	O
of	O
2010	O
there	O
are	O
about	O
2600	O
citations	O
,	O
mostly	O
in	O
pure	O
mathematics	O
,	O
but	O
also	O
including	O
papers	O
from	O
areas	O
as	O
diverse	O
as	O
economics	O
and	O
geophysics	O
.	O
</s>
<s>
The	O
predecessor	O
of	O
the	O
Magma	B-General_Concept
system	O
was	O
named	O
Cayley	O
(	O
1982	O
–	O
1993	O
)	O
,	O
after	O
Arthur	O
Cayley	O
.	O
</s>
<s>
Magma	B-General_Concept
was	O
officially	O
released	O
in	O
August	O
1993	O
(	O
version	O
1.0	O
)	O
.	O
</s>
<s>
Version	O
2.0	O
of	O
Magma	B-General_Concept
was	O
released	O
in	O
June	O
1996	O
and	O
subsequent	O
versions	O
of	O
2.X	O
have	O
been	O
released	O
approximately	O
once	O
per	O
year	O
.	O
</s>
<s>
In	O
2013	O
,	O
the	O
Computational	O
Algebra	O
Group	O
finalized	O
an	O
agreement	O
with	O
the	O
Simons	O
Foundation	O
,	O
whereby	O
the	O
Simons	O
Foundation	O
will	O
underwrite	O
all	O
costs	O
of	O
providing	O
Magma	B-General_Concept
to	O
all	O
U.S.	O
nonprofit	O
,	O
non-governmental	O
scientific	O
research	O
or	O
educational	O
institutions	O
.	O
</s>
<s>
All	O
students	O
,	O
researchers	O
and	O
faculty	O
associated	O
with	O
a	O
participating	O
institution	O
will	O
be	O
able	O
to	O
access	O
Magma	B-General_Concept
for	O
free	O
,	O
through	O
that	O
institution	O
.	O
</s>
<s>
Magma	B-General_Concept
includes	O
permutation	B-Algorithm
,	O
matrix	B-Architecture
,	O
finitely	O
presented	O
,	O
soluble	O
,	O
abelian	O
(	O
finite	O
or	O
infinite	O
)	O
,	O
polycyclic	O
,	O
braid	B-Application
and	O
straight-line	O
program	O
groups	O
.	O
</s>
<s>
Magma	B-General_Concept
contains	O
asymptotically	O
fast	O
algorithms	O
for	O
all	O
fundamental	O
integer	O
and	O
polynomial	O
operations	O
,	O
such	O
as	O
the	O
Schönhage	B-Algorithm
–	I-Algorithm
Strassen	I-Algorithm
algorithm	I-Algorithm
for	O
fast	O
multiplication	O
of	O
integers	O
and	O
polynomials	O
.	O
</s>
<s>
Integer	O
factorization	O
algorithms	O
include	O
the	O
Elliptic	B-Algorithm
Curve	I-Algorithm
Method	I-Algorithm
,	O
the	O
Quadratic	B-Algorithm
sieve	I-Algorithm
and	O
the	O
Number	B-Algorithm
field	I-Algorithm
sieve	I-Algorithm
.	O
</s>
<s>
Magma	B-General_Concept
includes	O
the	O
KANT	B-General_Concept
computer	B-General_Concept
algebra	I-General_Concept
system	I-General_Concept
for	O
comprehensive	O
computations	O
in	O
algebraic	O
number	O
fields	O
.	O
</s>
<s>
Magma	B-General_Concept
contains	O
asymptotically	O
fast	O
algorithms	O
for	O
all	O
fundamental	O
dense	B-Algorithm
matrix	I-Algorithm
operations	O
,	O
such	O
as	O
Strassen	B-Algorithm
multiplication	I-Algorithm
.	O
</s>
<s>
Magma	B-General_Concept
contains	O
the	O
structured	O
Gaussian	B-Algorithm
elimination	I-Algorithm
and	O
Lanczos	O
algorithms	O
for	O
reducing	O
sparse	B-Algorithm
systems	I-Algorithm
which	O
arise	O
in	O
index	O
calculus	O
methods	O
,	O
while	O
Magma	B-General_Concept
uses	O
Markowitz	O
pivoting	B-Algorithm
for	O
several	O
other	O
sparse	O
linear	B-Language
algebra	I-Language
problems	O
.	O
</s>
<s>
Magma	B-General_Concept
has	O
a	O
provable	O
implementation	O
of	O
fpLLL	O
,	O
which	O
is	O
an	O
LLL	O
algorithm	O
for	O
integer	O
matrices	O
which	O
uses	O
floating	O
point	O
numbers	O
for	O
the	O
Gram	B-Algorithm
–	I-Algorithm
Schmidt	I-Algorithm
coefficients	I-Algorithm
,	O
but	O
such	O
that	O
the	O
result	O
is	O
rigorously	O
proven	O
to	O
be	O
LLL-reduced	O
.	O
</s>
<s>
Magma	B-General_Concept
has	O
an	O
efficient	O
implementation	O
of	O
the	O
Faugère	B-Algorithm
F4	I-Algorithm
algorithm	I-Algorithm
for	O
computing	O
Gröbner	O
bases	O
.	O
</s>
<s>
Magma	B-General_Concept
has	O
extensive	O
tools	O
for	O
computing	O
in	O
representation	O
theory	O
,	O
including	O
the	O
computation	O
of	O
character	O
tables	O
of	O
finite	O
groups	O
and	O
the	O
Meataxe	O
algorithm	O
.	O
</s>
<s>
Magma	B-General_Concept
has	O
a	O
type	O
for	O
invariant	O
rings	O
of	O
finite	O
groups	O
,	O
for	O
which	O
one	O
can	O
primary	O
,	O
secondary	O
and	O
fundamental	O
invariants	O
,	O
and	O
compute	O
with	O
the	O
module	O
structure	O
.	O
</s>
