Magma computer algebra system

Magma

Magma
Developer(s)	Computational Algebra Group, School of Mathematics and Statistics, University of Sydney
Stable release	2.15 / December 2008
Operating system	Cross-platform
Type	Computer algebra system
License	Cost recovery (non-commercial proprietary)
Website	http://magma.maths.usyd.edu.au/

Magma is a computer algebra system designed to solve problems in algebra, number theory, geometry and combinatorics. It is named after the algebraic structure magma. It runs on Unix-like and Linux based operating systems, as well as Windows.

Introduction

Magma is produced and distributed by the Computational Algebra Group within the School of Mathematics and Statistics at the University of Sydney.

In late 2006, the book Discovering Mathematics with Magma was published by Springer as volume 19 of the Algorithms and Computations in Mathematics series.

History

The predecessor of the Magma system was called Cayley (1982-1993).

Magma was officially released in August 1993 (version 1.0). Version 2.0 of Magma was released in June 1996 and subsequent versions of the form 2.X have been released approximately once per year.

Mathematical Areas Covered by the System

• Group theory

Magma includes permutation, matrix, finitely-presented, soluble, abelian (finite or infinite), polycyclic, braid and straight-line program groups. Several databases of groups are also included.

• Number theory

Magma contains 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 Elliptic Curve Method, the Quadratic sieve and the Number field sieve.

• Algebraic number theory

Magma includes the 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 asymptotically-fast algorithms for all fundamental dense matrix operations, such as 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 Markowitz pivoting for several other sparse linear algebra problems.

• Lattices and the LLL algorithm

Magma has a provable implementation of fpLLL , which is an LLL algorithm for integer matrices which uses floating point numbers for the 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 structures
• Cryptography
• Coding theory
• Optimization

See also

• Comparison of computer algebra systems
• Sage, an open source computer algebra system

External links

• Official website
• Magma Free Online Calculator
• Magma's High Performance for computing Groebner Bases
• Magma's High Performance for computing Hermite Normal Forms of integer matrices
• Magma V2.12 is apparently "Overall Best in the World at Polynomial GCD" :-)
• Magma example code