Editing Magma (computer algebra system) (section)

== Mathematical areas covered by the system ==
* [[Group theory]]
: Magma includes [[permutation group|permutation]], [[matrix (mathematics)|matrix]], [[finitely presented group|finitely presented]], [[solvable group|soluble]], [[abelian group|abelian]] (finite or infinite), [[polycyclic group|polycyclic]], [[braid group|braid]] and [[straight-line program]] [[group (mathematics)|groups]]. Several databases of groups are also included.
* [[Number theory]]
: Magma contains [[Big O notation|asymptotically fast]] algorithms for all fundamental integer and polynomial operations, such as the [[Schönhage–Strassen algorithm]] for fast multiplication of integers and polynomials. [[Integer factorization]] algorithms include the [[Lenstra elliptic curve factorization|Elliptic Curve Method]], the [[Quadratic sieve]] and the [[General number field sieve|Number field sieve]].
* [[Algebraic number theory]]
: Magma includes the [[KANT (software)|KANT]] computer algebra system for comprehensive computations in algebraic number fields. A special type also allows one to compute in the [[algebraic closure]] of a field.
* [[Module theory]] and [[linear algebra]]
: Magma contains [[Big O notation|asymptotically fast]] algorithms for all fundamental dense matrix operations, such as [[Strassen algorithm|Strassen multiplication]].
* [[Sparse matrices]]
: Magma contains the structured [[Gaussian elimination]] and [[Lanczos]] algorithms for reducing sparse systems which arise in [[index calculus]] methods, while Magma uses [[Harry Markowitz|Markowitz]] [[Pivot element|pivoting]] for several other sparse linear algebra problems.
* [[Lattice (group)|Lattices]] and the [[LLL algorithm]]
: Magma has a provable implementation of [[fpLLL|''fp''LLL]],<ref>{{cite web |author=John Cannon |url=http://magma.maths.usyd.edu.au/magma/releasenotes/2/13/#section_13 |title=Magma 2.13 release notes |date=July 2006 }}</ref> which is an LLL algorithm for integer matrices which uses floating point numbers for the [[Gram–Schmidt process|Gram–Schmidt coefficients]], but such that the result is rigorously proven to be LLL-reduced.
* [[Commutative algebra]] and [[Gröbner bases]]
: Magma has an efficient implementation of the [[Faugère F4 algorithm]] for computing [[Gröbner bases]].
* [[Representation theory]]
: Magma has extensive tools for computing in representation theory, including the computation of [[character tables]] of finite groups and the [[Meataxe]] algorithm.
* [[Invariant theory]]
: Magma has a type for invariant rings of finite groups, for which one can primary, secondary and fundamental invariants, and compute with the module structure.
* [[Lie theory]]
* [[Algebraic geometry]]
* [[Arithmetic geometry]]
* Finite [[incidence structure]]s
* [[Cryptography]]
* [[Coding theory]]
* [[Optimization (mathematics)|Optimization]]