Differences between revisions 1 and 21 (spanning 20 versions)
Revision 1 as of 2007-04-16 17:59:23
Size: 265
Editor: mdhcp175
Comment:
Revision 21 as of 2007-04-23 16:34:28
Size: 3063
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 22: Line 65:
 * 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.

p-adic Numbers

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 localhost)