Magma is a

computer algebra system A computer algebra system (CAS) or symbolic algebra system (SAS) is any mathematical software with the ability to manipulate mathematical expressions in a way similar to the traditional manual computations of mathematicians and scientists. The d ...

designed to solve problems in

algebra Algebra () is one of the broad areas of mathematics. Roughly speaking, algebra is the study of mathematical symbols and the rules for manipulating these symbols in formulas; it is a unifying thread of almost all of mathematics. Elementary a ...

number theory Number theory (or arithmetic or higher arithmetic in older usage) is a branch of pure mathematics devoted primarily to the study of the integers and integer-valued functions. German mathematician Carl Friedrich Gauss (1777–1855) said, "Mathe ...

geometry Geometry (; ) is, with arithmetic, one of the oldest branches of mathematics. It is concerned with properties of space such as the distance, shape, size, and relative position of figures. A mathematician who works in the field of geometry is ...

and combinatorics. It is named after the

algebraic structure In mathematics, an algebraic structure consists of a nonempty set ''A'' (called the underlying set, carrier set or domain), a collection of operations on ''A'' (typically binary operations such as addition and multiplication), and a finite set ...

magma Magma () is the molten or semi-molten natural material from which all igneous rocks are formed. Magma is found beneath the surface of the Earth, and evidence of magmatism has also been discovered on other terrestrial planets and some natural s ...

. It runs on Unix-like

operating system An operating system (OS) is system software that manages computer hardware, software resources, and provides common daemon (computing), services for computer programs. Time-sharing operating systems scheduler (computing), schedule tasks for ef ...

s, as well as

Windows Windows is a group of several Proprietary software, proprietary graphical user interface, graphical operating system families developed and marketed by Microsoft. Each family caters to a certain sector of the computing industry. For example, W ...

Introduction

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

University of Sydney The University of Sydney (USYD), also known as Sydney University, or informally Sydney Uni, is a public research university located in Sydney, Australia. Founded in 1850, it is the oldest university in Australia and is one of the country's six ...

. In late 2006, the boo
Discovering Mathematics with Magma
was published by

Springer Springer or springers may refer to: Publishers * Springer Science+Business Media, aka Springer International Publishing, a worldwide publishing group founded in 1842 in Germany formerly known as Springer-Verlag. ** Springer Nature, a multinationa ...

as volume 19 of the Algorithms and Computations in Mathematics series. The Magma system is used extensively within pure mathematics. The Computational Algebra Group maintain a list of publications that cite Magma, and as of 2010 there are about 2600 citations, mostly in pure mathematics, but also including papers from areas as diverse as economics and geophysics.

History

The predecessor of the Magma system was named Cayley (1982–1993), after

Arthur Cayley Arthur Cayley (; 16 August 1821 – 26 January 1895) was a prolific British mathematician who worked mostly on algebra. He helped found the modern British school of pure mathematics. As a child, Cayley enjoyed solving complex maths problems ...

. Magma was officially released in August 1993 (version 1.0). Version 2.0 of Magma was released in June 1996 and subsequent versions of 2.X have been released approximately once per year. In 2013, the Computational Algebra Group finalized an agreement with the

Simons Foundation The Simons Foundation is a private foundation established in 1994 by Marilyn and Jim Simons with offices in New York City. As one of the largest charitable organizations in the US with assets of over $5 billion in 2022, the foundation's mission ...

, whereby the Simons Foundation will underwrite all costs of providing Magma to all U.S.

nonprofit A nonprofit organization (NPO) or non-profit organisation, also known as a non-business entity, not-for-profit organization, or nonprofit institution, is a legal entity organized and operated for a collective, public or social benefit, in co ...

, non-governmental scientific research or educational institutions. All students, researchers and faculty associated with a participating institution will be able to access Magma for free, through that institution.

Mathematical areas covered by the system

Group theory In abstract algebra, group theory studies the algebraic structures known as groups. The concept of a group is central to abstract algebra: other well-known algebraic structures, such as rings, fields, and vector spaces, can all be seen as ...

: Magma includes

permutation In mathematics, a permutation of a set is, loosely speaking, an arrangement of its members into a sequence or linear order, or if the set is already ordered, a rearrangement of its elements. The word "permutation" also refers to the act or pr ...

matrix Matrix most commonly refers to: * ''The Matrix'' (franchise), an American media franchise ** ''The Matrix'', a 1999 science-fiction action film ** "The Matrix", a fictional setting, a virtual reality environment, within ''The Matrix'' (franchis ...

, finitely presented,

soluble In chemistry, solubility is the ability of a substance, the solute, to form a solution with another substance, the solvent. Insolubility is the opposite property, the inability of the solute to form such a solution. The extent of the solubi ...

, abelian (finite or infinite), polycyclic,

braid A braid (also referred to as a plait) is a complex structure or pattern formed by interlacing two or more strands of flexible material such as textile yarns, wire, or hair. The simplest and most common version is a flat, solid, three-strande ...

and

straight-line program In mathematics, more specifically in computational algebra, a straight-line program (SLP) for a finite group ''G'' = ⟨''S''⟩ is a finite sequence ''L'' of elements of ''G'' such that every element of ''L'' either belongs to ''S'', ...

groups. Several databases of groups are also included. *

Number theory Number theory (or arithmetic or higher arithmetic in older usage) is a branch of pure mathematics devoted primarily to the study of the integers and integer-valued functions. German mathematician Carl Friedrich Gauss (1777–1855) said, "Mathe ...

: Magma contains asymptotically fast algorithms for all fundamental integer and polynomial operations, such as the

Schönhage–Strassen algorithm The Schönhage–Strassen algorithm is an asymptotically fast multiplication algorithm for large integers. It was developed by Arnold Schönhage and Volker Strassen in 1971.A. Schönhage and V. Strassen,Schnelle Multiplikation großer Zahlen, ''C ...

for fast multiplication of integers and polynomials.

Integer factorization In number theory, integer factorization is the decomposition of a composite number into a product of smaller integers. If these factors are further restricted to prime numbers, the process is called prime factorization. When the numbers are suf ...

algorithms include the

Elliptic Curve Method The Lenstra elliptic-curve factorization or the elliptic-curve factorization method (ECM) is a fast, sub- exponential running time, algorithm for integer factorization, which employs elliptic curves. For general-purpose factoring, ECM is the th ...

, the

Quadratic sieve The quadratic sieve algorithm (QS) is an integer factorization algorithm and, in practice, the second fastest method known (after the general number field sieve). It is still the fastest for integers under 100 decimal digits or so, and is considera ...

and the

Number field sieve In number theory, the general number field sieve (GNFS) is the most efficient classical algorithm known for factoring integers larger than . Heuristically, its complexity for factoring an integer (consisting of bits) is of the form :\exp\left( ...

. *

Algebraic number theory Algebraic number theory is a branch of number theory that uses the techniques of abstract algebra to study the integers, rational numbers, and their generalizations. Number-theoretic questions are expressed in terms of properties of algebraic ob ...

: Magma includes the

KANT Immanuel Kant (, , ; 22 April 1724 – 12 February 1804) was a German philosopher and one of the central Enlightenment thinkers. Born in Königsberg, Kant's comprehensive and systematic works in epistemology, metaphysics, ethics, and ae ...

computer algebra system for comprehensive computations in algebraic number fields. A special type also allows one to compute in the

algebraic closure In mathematics, particularly abstract algebra, an algebraic closure of a field ''K'' is an algebraic extension of ''K'' that is algebraically closed. It is one of many closures in mathematics. Using Zorn's lemmaMcCarthy (1991) p.21Kaplansky ( ...

of a field. *

Module theory In mathematics, a module is a generalization of the notion of vector space in which the field of scalars is replaced by a ring. The concept of ''module'' generalizes also the notion of abelian group, since the abelian groups are exactly the mo ...

and

linear algebra Linear algebra is the branch of mathematics concerning linear equations such as: :a_1x_1+\cdots +a_nx_n=b, linear maps such as: :(x_1, \ldots, x_n) \mapsto a_1x_1+\cdots +a_nx_n, and their representations in vector spaces and through matrices. ...

: Magma contains asymptotically fast algorithms for all fundamental dense matrix operations, such as Strassen multiplication. *

Sparse matrices In numerical analysis and scientific computing, a sparse matrix or sparse array is a matrix in which most of the elements are zero. There is no strict definition regarding the proportion of zero-value elements for a matrix to qualify as sparse b ...

: Magma contains the structured

Gaussian elimination In mathematics, Gaussian elimination, also known as row reduction, is an algorithm for solving systems of linear equations. It consists of a sequence of operations performed on the corresponding matrix of coefficients. This method can also be used ...

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 ''fp''LLL, 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 Commutative algebra, first known as ideal theory, is the branch of algebra that studies commutative rings, their ideals, and modules over such rings. Both algebraic geometry and algebraic number theory build on commutative algebra. Prominent ...

and Gröbner bases : Magma has an efficient implementation of the Faugère F4 algorithm for computing Gröbner bases. *

Representation theory Representation theory is a branch of mathematics that studies abstract algebraic structures by ''representing'' their elements as linear transformations of vector spaces, and studies modules over these abstract algebraic structures. In essen ...

: Magma has extensive tools for computing in representation theory, including the computation of

character tables In group theory, a branch of abstract algebra, a character table is a two-dimensional table whose rows correspond to irreducible representations, and whose columns correspond to conjugacy classes of group elements. The entries consist of characters ...

of finite groups and the Meataxe algorithm. *

Invariant theory Invariant theory is a branch of abstract algebra dealing with actions of groups on algebraic varieties, such as vector spaces, from the point of view of their effect on functions. Classically, the theory dealt with the question of explicit descr ...

: 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 In mathematics, the mathematician Sophus Lie ( ) initiated lines of study involving integration of differential equations, transformation groups, and contact of spheres that have come to be called Lie theory. For instance, the latter subject is ...

Algebraic geometry Algebraic geometry is a branch of mathematics, classically studying zeros of multivariate polynomials. Modern algebraic geometry is based on the use of abstract algebraic techniques, mainly from commutative algebra, for solving geometrica ...

Arithmetic geometry In mathematics, arithmetic geometry is roughly the application of techniques from algebraic geometry to problems in number theory. Arithmetic geometry is centered around Diophantine geometry, the study of rational points of algebraic varieties. ...

* Finite

incidence structure In mathematics, an incidence structure is an abstract system consisting of two types of objects and a single relationship between these types of objects. Consider the points and lines of the Euclidean plane as the two types of objects and ignore a ...

s *

Cryptography Cryptography, or cryptology (from grc, , translit=kryptós "hidden, secret"; and ''graphein'', "to write", or ''-logia'', "study", respectively), is the practice and study of techniques for secure communication in the presence of adve ...

Coding theory Coding theory is the study of the properties of codes and their respective fitness for specific applications. Codes are used for data compression, cryptography, error detection and correction, data transmission and data storage. Codes are studied ...

Optimization Mathematical optimization (alternatively spelled ''optimisation'') or mathematical programming is the selection of a best element, with regard to some criterion, from some set of available alternatives. It is generally divided into two subfi ...

References

External links

*
Magma Free Online CalculatorMagma's High Performance for computing Gröbner Bases
(2004)

* ttp://magma.maths.usyd.edu.au/users/allan/gcdcomp.html Magma V2.12 is apparently "Overall Best in the World at Polynomial GCD" :-)br>Magma example codeListe
von Publikationen, die Magma zitieren {{DEFAULTSORT:Magma Computer Algebra System Computer algebra system software for Linux Computer algebra system software for macOS Computer algebra system software for Windows Cross-platform software Functional languages Numerical programming languages Proprietary commercial software for Linux

Introduction

History

Mathematical areas covered by the system

See also

References

External links