Differences between revisions 1 and 22 (spanning 21 versions)
 ⇤ ← Revision 1 as of 2007-04-16 17:59:23 → Size: 265 Editor: anonymous Comment: ← Revision 22 as of 2007-04-23 16:35:09 → ⇥ Size: 3155 Editor: wstein Comment: Deletions are marked like this. Additions are marked like this. Line 3: Line 3: 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.) Line 4: Line 6: * SAGE has fairly complete symbolic manipulation capabilities, including symbolic and numerical integration, differentiation, limits, etc. Line 5: Line 8: == Commutative Algebra == == 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. Line 9: Line 26: * Very fast arithmetic over finite fields and extensions of finite fields (especially up to cardinality 2^16). Line 10: Line 29: * 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. Line 13: Line 34: * 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. Line 15: Line 45: == Number Theory == * 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. Line 18: Line 59: * Fast arithmetic and special functions with double precision real and complex numbers. * Matrix and vector arithmetic, QR decomposition, system solving. Line 20: Line 63: * Extensive support for arithmetic with a range of different models of p-adic arithmetic. Line 22: Line 66: * 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).

# 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.

• 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).

cando (last edited 2008-11-14 13:42:15 by anonymous)