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:
“An accurate charting of the American woman’s progress through history might look more like a corkscrew tilted slightly to one side, its loops inching closer to the line of freedom with the passage of time—but like a mathematical curve approaching infinity, never touching its goal. . . . Each time, the spiral turns her back just short of the finish line.”
—Susan Faludi (20th century)
“Adults understandably assume that the level of verbal proficiency a five-year-old displays represents his level of proficiency in all areas of functioning—if he talks like an adult, he must think and feel like one. However, five-year-olds,... belie the promise of adult-like behavior with their child-like, impulsive actions.”
—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)
“Our system of government, in spite of Vietnam, Cambodia, CIA, Watergate, is still the best system of government on earth. And the greatest resource of all are the 215 million Americans who still have within us the strength, the character, the intelligence, the experience, the patriotism, the idealism, the compassion, the sense of brotherhood on which we can rely in the future to restore the greatness to our country.”
—Jimmy Carter (James Earl Carter, Jr.)