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| 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.) |
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| * SAGE has fairly complete symbolic manipulation capabilities, including symbolic and numerical integration, differentiation, limits, etc. == Coding theory == * A wide range of basic functionality. |
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| * Computing a Groebner basis is fast because of the SINGULAR computer algebra system. | * Fast computation of Groebner basis. |
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| == Crypto == | == Cryptography == |
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| * 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. |
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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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| * 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. |
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| * Permutations groups * Abelian groups * Matrix groups (in particular, classical groups over finite fields) |
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| * 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. |
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| * 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. |
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| * 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. |
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| * Fast arithmetic and special functions with double precision real and complex numbers. * Matrix and vector arithmetic, QR decomposition, system solving. |
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| * 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).