MAGMA COMPUTER ALGEBRA SYSTEM

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'Magma' is a high-performance computer algebra system designed to solve problems in algebra, number theory, geometry and combinatorics. It is named after the algebraic structure magma. The current version as of July 2006 is 2.13. It is non-commercial proprietary software, distributed under a cost recovery licence. It runs mainly on the Unix-like and Linux based operating systems, but also supports Windows.

Contents

Introduction

History

Mathematical Areas Covered by the System

References

External links

Introduction

Magma is produced and distributed by the Computational Algebra Group within the School of Mathematics and Statistics of the University of Sydney.
Magma contains many of the most advanced and efficient known algorithms for the areas which it covers.

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 the only provable implementation of ''fp''LLL ^[1], which is a 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

References

1. Magma 2.13 release notes Cannon J.

External links

★ Magma 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