Differences between revisions 12 and 13
 ⇤ ← Revision 12 as of 2007-04-17 18:40:25 → Size: 1110 Editor: c-67-183-64-183 Comment: ← Revision 13 as of 2007-04-18 16:31:33 → ⇥ Size: 2483 Editor: mdhcp175 Comment: Deletions are marked like this. Additions are marked like this. Line 2: Line 2: This is supposed to be a high-level overview and list of functionality that is easily availablefrom 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 heardof programs like GAP, Singular, etc., and don't care how SAGE does things -- they just want to knowexactly *what* SAGE can do. Line 7: Line 13: * Computing a Groebner basis is fast because of the SINGULAR computer algebra system. * Fast computation of Groebner basis. Line 9: Line 15: == Crypto == == Cryptography == Line 15: Line 21: * The notebook is a useful tool for basic math education because of its flexible visualization/output capabilities. * The SAGE notebook (a graphical interface) is a useful tool for basic math education because of its flexible visualization/output capabilities. Line 27: Line 33: * SAGE provides interpreter interfaces to Axiom, CoCoA, GAP, KASH, Macaulay2, Magma, Maple, Mathematica, Matlab, Maxima, MuPAD, Octave, and Singular. * SAGE provides C/C++-library interfaces to NTL, PARI, Linbox, and mwrank. * 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 32: Line 38: * The reduced row echelon form of e.g. dense 20,000x20,000 matrices over GF(2) can be computed in seconds and 50MB of RAM. * Computation of reduced row echelon forms of sparse matrices is supported. * 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. Line 36: Line 43: * 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 SEA 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 38: Line 52: * Fast arithmetic and special functions * Matrix and vector arithmetic, QR decomposition, system solving Line 42: Line 58: * SAGE provides 2d plotting functionality similar to Mathematica's.  * SAGE provides limited 3d plotting via an included ray tracer.

# What SAGE Can Do

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.

## 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

• Basic arithmetic over finite extension fields is fast because of the Givaro library.

## 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 SEA 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
• Matrix and vector arithmetic, QR decomposition, system solving