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
Read more about this topic: Magma Computer Algebra System
Famous quotes containing the words mathematical, areas, covered and/or system:
“The most distinct and beautiful statement of any truth must take at last the mathematical form.”
—Henry David Thoreau (1817–1862)
“Helping children at a level of genuine intellectual inquiry takes imagination on the part of the adult. Even more, it takes the courage to become a resource in unfamiliar areas of knowledge and in ones for which one has no taste. But parents, no less than teachers, must respect a child’s mind and not exploit it for their own vanity or ambition, or to soothe their own anxiety.”
—Dorothy H. Cohen (20th century)
“At Eutaw Springs the valiant died;
Their limbs with dust are covered o’er—
Weep on, ye springs, your tearful tide;
How many heroes are no more!”
—Philip Freneau (1752–1832)
“An avant-garde man is like an enemy inside a city he is bent on destroying, against which he rebels; for like any system of government, an established form of expression is also a form of oppression. The avant-garde man is the opponent of an existing system.”
—Eugène Ionesco (b. 1912)