Initial release | 24 February 2005 |
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Stable release | 4.7.2 (October 29, 2011 )[1] [±] |
Written in | Python, Cython |
Operating system | Cross-platform |
Platform | Python |
Size | 411 MB download (Ubuntu 64-bit)[2] |
Type | Computer algebra system |
License | GNU General Public License |
Website | www.sagemath.org |
Sage (previously SAGE) is mathematical software with features covering many aspects of mathematics, including algebra, combinatorics, numerical mathematics, number theory, and calculus. Sage is sometimes called sagemath to distinguish it from other uses of the word.
The first version of Sage was released on 24 February 2005 as free and open source software under the terms of the GNU General Public License, with the initial goals of creating an "open source alternative to Magma, Maple, Mathematica, and MATLAB."[3] The starter and leader of the Sage project, William Stein, is a mathematician at the University of Washington.
Sage uses the Python programming language, supporting procedural, functional and object-oriented constructs.
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Features of Sage include:[4]
Although not provided by Sage directly, Sage can be called from within Mathematica.[8] A Mathematica notebook is available for this purpose.[9]
William Stein realized when designing Sage that there were many open-source mathematics software already written in different languages, namely C, C++, Fortran and Python.
Rather than reinventing the wheel, Sage (which is written mostly in Python and Cython) integrates many specialized mathematics software into a common interface, for which a user needs to know only Python. However, Sage contains hundreds of thousands of unique lines of code adding new functions and creating the interface between its components.[10]
Sage development uses both students and professionals for development. The development of Sage is supported by both volunteer work and grants.[11]
Only the major releases are listed below. Sage practices the "release early, release often" concept, with releases every few weeks or months. In total, there have been over 300 releases, although their frequency has decreased.[12]
Version | Release Date | Description |
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0.1 | January, 2005 | Included PARI, but not GAP or Singular |
0.2 – 0.4 | March to July 2005 | Cremona's database, multivariate polynomials, large finite fields and more documentation |
0.5 – 0.7 | August to September 2005 | Vector spaces, rings, modular symbols, and windows usage |
0.8 | October 2005 | Full distribution of GAP, Singular |
0.9 | November, 2005 | Maxima and clisp added |
1.0 | February, 2006 | |
2.0 | January, 2007 | |
3.0 | April, 2008 | |
4.0 | May, 2009 | |
5.0 | future | 5.0 milestone |
In 2007, Sage won first prize in the scientific software division of Les Trophées du Libre, an international competition for free software.[13]
Sage has been cited in a variety of publications.[14][15]
Both binaries and source code are available for Sage from the download page. If Sage is built from source code, many of the included libraries such as ATLAS, FLINT, and NTL will be tuned and optimized for that computer, taking into account the number of processors, the size of their caches, whether there is hardware support for SSE instructions, etc.
Sage's speed is competitive with other mathematical software.[16][17]
Cython can increase the speed of Sage programs, as the Python code is converted into C.[18]
Sage is free software, distributed under the terms of the GNU General Public License version 2+. Sage is available in many ways:
Although Microsoft was sponsoring a native version of Sage for the Windows operating system,[19] users of Windows currently have to use virtualization technology such as VirtualBox to run Sage under one of the aforementioned operating systems. A Cygwin port is also being worked on.[20]
Linux distributions in which Sage is available as a package are Mandriva and Arch Linux. In Gentoo, it's available via layman in the "sage-on-gentoo"[21] overlay. However, Sage can be installed to any Linux distribution.
Gentoo prefix also provides Sage on other operating systems.
The philosophy of Sage is to use existing open-source libraries wherever they exist. Therefore it uses many libraries from other projects.
Algebra | GAP, Maxima, Singular |
Algebraic geometry | Singular |
Arbitrary precision arithmetic | MPIR, MPFR, MPFI, NTL, mpmath |
Arithmetic geometry | PARI/GP, NTL, mwrank, ecm |
Calculus | Maxima, SymPy, GiNaC |
Combinatorics | Symmetrica, Sage-Combinat |
Linear algebra | ATLAS, BLAS, LAPACK, NumPy, LinBox, IML, GSL |
Graph theory | NetworkX |
Group theory | GAP |
Numerical computation | GSL, SciPy, NumPy, ATLAS |
Number theory | PARI/GP, FLINT, NTL |
Statistical computing | R, SciPy |
Command-line shell | IPython |
Database | ZODB, Python pickles, SQLite |
Graphical interface | Sage Notebook, jsmath |
Graphics | Matplotlib, Tachyon3d, GD, Jmol |
Interactive programming language | Python |
Networking | Twisted |
x, a, b, c = var('x, a, b, c') log(sqrt(a)).simplify_log() # returns 1/2*log(a) log(a / b).expand_log() # returns log(a) - log(b) sin(a + b).simplify_trig() # returns sin(a)*cos(b) + sin(b)*cos(a) cos(a + b).simplify_trig() # returns -sin(a)*sin(b) + cos(a)*cos(b) (a + b)^5 # returns (a + b)^5 expand((a + b) ^ 5) # a^5 + 5*a^4*b + 10*a^3*b^2 + 10*a^2*b^3 + 5*a*b^4 + b^5 limit((x ^ 2 + 1) / (2 + x + 3 * x ^ 2), x=Infinity) # returns 1/3 limit(sin(x) / x, x=0) # returns 1 diff(acos(x), x) # returns -1/sqrt(-x^2 + 1) f = exp(x) * log(x) f.diff(x, 3) # returns e^x*log(x) + 3*e^x/x - 3*e^x/x^2 + 2*e^x/x^3 solve(a * x ^ 2 + b * x + c, x) # returns [x == -1/2*(b + sqrt(-4*a*c + b^2))/a, x == -1/2*(b - sqrt(-4*a*c + b^2))/a] f = x ^ 2 + 432 / x solve(f.diff(x) == 0, x) # returns [x == 3*I*sqrt(3) - 3, x == -3*I*sqrt(3) - 3, x == 6]
t = var('t') # define a variable t x = function('x', t) # define x to be a function of that variable DE = lambda y: diff(y, t) + y - 1 desolve(DE(x(x=t)), [x, t]) # returns (c + e^t)*e^(-t)
A = Matrix([[1, 2, 3], [3, 2, 1], [1, 1, 1]]) y = vector([0, -4, -1]) A.solve_right(y) # returns (-2, 1, 0) A.eigenvalues() # returns [5, 0, -1] B = Matrix([[1, 2, 3], [3, 2, 1], [1, 2, 1]]) B.inverse() # returns '''[ 0 1/2 -1/2] [-1/4 -1/4 1] [ 1/2 0 -1/2]''' # Call NumPy for the Moore-Penrose pseudo-inverse, since Sage does not support that yet. import numpy C = Matrix([[1 , 1], [2 , 2]]) matrix(numpy.linalg.pinv(C.numpy())) # returns '''[0.1 0.2] [0.1 0.2]'''
prime_pi(1000000) # returns 78498, the number of primes less than one million E = EllipticCurve('389a') # construct an elliptic curve from its Cremona label P, Q = E.gens() 7 * P + Q # returns (24187731458439253/244328192262001 : 3778434777075334029261244/3819094217575529893001 : 1)
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