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| This is supposed to be a high-level overview and list of functionality that is easily available from the standard SAGE interface. In particular, don't list functionality that is only available, e.g., by directly calling GAP or other systems. Also, this is aimed at people who have never heard of programs like GAP, Singular, etc., and don't care how SAGE does things -- they just want to know exactly ''what'' SAGE can do. |
SAGE does a wider range of mathematics than every other open source mathematics software, partly by virtue of incorporating several of the other large mature systems. 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.). This should be aimed at SAGE-2.5, and will be released with it. |
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| * SAGE has fairly complete symbolic manipulation capabilities, including symbolic and numerical integration, differentiation, limits, etc. | |
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| * Many commands from GAP and GUAVA are wrapped and there are some native (Python/SAGE) commands. | * A wide range of basic functionality. |
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| * Basic arithmetic over finite extension fields is fast because of the Givaro library. | * Very fast arithmetic over finite fields and extensions of finite fields (especially up to cardinality 2^16). |
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Most of these are computed by wrapping GAP commands but some commands are native (Python/SAGE). |
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| * Optimized implementation of the SEA point counting algorithm for counting points modulo p when p is large. | * Optimized implementation of the Schoof-Elkies-Atkin point counting algorithm for counting points modulo p when p is large. |
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| * Extensive support for arithmetic with a range of different models of p-adic arithmetic. | |
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| * Exact convex hulls in any dimension can be quickly computed using polymake and the cddlib library. | * 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). |
What SAGE Can Do
SAGE does a wider range of mathematics than every other open source mathematics software, partly by virtue of incorporating several of the other large mature systems.
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.). This should be aimed at SAGE-2.5, and will be released with it.
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).