What SAGE Can Do
This is a high-level overview and list of functionality that is easily available from the standard SAGE interface. (The intended reader has never heard of Maxima, GAP, Singular, Givaro, etc.)
Calculus
- SAGE has fairly complete symbolic manipulation capabilities, including symbolic and numerical integration, differentiation, limits, etc.
Coding theory
- A wide range of basic functionality.
Commutative Algebra
- Fast computation of Groebner basis.
Cryptography
- Classical ciphers are well supported.
Elementary Education
- The SAGE notebook (a graphical interface) is a useful tool for basic math education because of its flexible visualization/output capabilities.
Finite Fields
- Very fast arithmetic over finite fields and extensions of finite fields (especially up to cardinality 2^16).
Graphical Interface
- A web-browser based graphical interface, which anybody can easily use or share. The GUI can also be used for any math software that SAGE interfaces with.
- A wiki with math typesetting preconfigured.
Group Theory
- Permutations groups
- Abelian groups
- Matrix groups (in particular, classical groups over finite fields)
Interfaces
- Interpreter interfaces to Axiom, CoCoA, GAP, KASH, Macaulay2, Magma, Maple, Mathematica, Matlab, Maxima, MuPAD, Octave, and Singular.
- C/C++-library interfaces to NTL, PARI, Linbox, and mwrank.
Linear Algebra
- Compute the reduced row echelon form of e.g. dense 20,000x20,000 matrices over GF(2) in seconds and 50MB of RAM.
- Computation of reduced row echelon forms of sparse matrices.
- Fast matrix multiplication, characteristic polynomial and echelon forms of dense matrices over QQ.
Number Theory
- Compute Mordell-Weil groups of (many) elliptic curves using both invariants and algebraic 2-descents.
- A wide range of number theoretic functions, e.g., euler_phi, primes enumeration, sigma, tau_qexp, etc.
- Compute the number of points on an elliptic curve modulo p for all primes p less than a million in seconds.
- Optimized implementation of the Schoof-Elkies-Atkin point counting algorithm for counting points modulo p when p is large.
- An optimized modern quadratic Sieve for factoring integers n = p*q.
- Modular symbols for general weight, character, Gamma1, and GammaH.
Modular forms for general weight >= 2, character, Gamma1, and GammaH.
Numerical Computation
- Fast arithmetic and special functions with double precision real and complex numbers.
- Matrix and vector arithmetic, QR decomposition, system solving.
p-adic Numbers
- Extensive support for arithmetic with a range of different models of p-adic arithmetic.
Plotting
- SAGE provides 2d plotting functionality similar to Mathematica's.
- SAGE provides limited 3d plotting via an included ray tracer.
Polytopes
- State of the art support for computing with lattice polytopes.
- Exact convex hulls in any dimension can be quickly computed (requires the optional polymake package).