Sage (mathematics software)

Initial release	24 February 2005 (2005-02-24)
Stable release	5.9 (30 April 2013 (2013-04-30)) [±]
Written in	Python, Cython
Operating system	Cross-platform
Platform	Python
Size	411 MB download (Ubuntu 64-bit)
Type	Computer algebra system
License	GNU General Public License
Website	www.sagemath.org

Sage (previously SAGE, System for Algebra and Geometry Experimentation) is a 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". The originator 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.

Features

The Sage notebook document interface works with most web browsers.

Equation solving and typesetting using the Sage notebook web interface

Features of Sage include:

• A browser-based notebook for review and re-use of previous inputs and outputs, including graphics and text annotations. Compatible with Firefox, Opera, Konqueror, Google Chrome and Safari. Notebooks can be accessed locally or remotely and the connection can be secured with HTTPS.
• A text-based command-line interface using IPython
• Support for parallel processing using multi-core processors, multiple processors, or distributed computing
• Calculus using Maxima and SymPy
• Numerical linear algebra using the GSL, SciPy and NumPy
• Libraries of elementary and special mathematical functions
• 2D and 3D graphs of symbolic functions and numerical data
• Matrix manipulation, including sparse arrays
• Multivariate statistics libraries, using R and SciPy
• A toolkit for adding user interfaces to calculations and applications
• Graph theory visualization and analysis tools
• Libraries of number theory functions
• Support for complex numbers, arbitrary precision and symbolic computation
• Technical word processing including formula editing and embedding Sage within LaTeX documents
• The Python standard library, including tools for connecting to SQL, HTTP, HTTPS, NNTP, IMAP, SSH, IRC, FTP and others
• Interfaces to some third-party applications like Mathematica, Magma, R, and Maple
• MoinMoin as a Wiki system for knowledge management
• Documentation using Sphinx
• An automated test-suite
• Execution of Fortran, C, C++, and Cython code

Although not provided by Sage directly, Sage can be called from within Mathematica. as is done in this Mathematica notebook example

Development

William A. Stein

William Stein realized when designing Sage that there were many open-source mathematics software already written in different languages, namely C, C++, Common Lisp, 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.

Sage development uses both students and professionals for development. The development of Sage is supported by both volunteer work and grants.

Release history

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.

Sage versions

Version	Release Date	Description
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	Interacts, R interface
4.0	May 2009	Solaris 10 support, 64bit OSX support
5.0	May 2012	OSX Lion support

Achievements

In 2007, Sage won first prize in the scientific software division of Les Trophées du Libre, an international competition for free software. In 2012, it was one of the projects selected for the Google Summer of Code.

Sage has been cited in a variety of publications.

Performance

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.

Cython can increase the speed of Sage programs, as the Python code is converted into C.

Licensing and availability

Sage is free software, distributed under the terms of the GNU General Public License version 2+. Sage is available in many ways:

• The source code can be downloaded from the downloads page. Although not recommended for end users, development releases of Sage are also available.
• Binaries can be downloaded for Linux, OS X and Solaris (both x86 and SPARC).
• A live CD containing a bootable Linux operating system is also available. This allows usage of Sage without Linux installation.
• Users can use an online version of Sage at sagenb.org, but with a limit to the amount of memory a user can use.

Although Microsoft was sponsoring a native version of Sage for the Windows operating system, as of 2012 there were no plans for a native port, and users of Windows currently have to use virtualization technology such as VirtualBox to run Sage.

Linux distributions in which Sage is available as a package are Mandriva and Arch Linux. It is also available as a dedicated Ubuntu PPA. In Gentoo, it's available via layman in the "sage-on-gentoo" overlay. However, Sage can be installed to any Linux distribution.

Gentoo prefix also provides Sage on other operating systems.

Software packages contained in Sage

The philosophy of Sage is to use existing open-source libraries wherever they exist. Therefore it uses many libraries from other projects.

Mathematics packages contained in Sage

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

Other packages contained in Sage

Command-line shell	IPython
Database	ZODB, SQLite
Graphical interface	Sage Notebook, jsMath
Graphics	matplotlib, Tachyon3d, GD, Jmol
Interactive programming language	Python
Networking	Twisted

Usage examples

Algebra and calculus

x, a, b, c = var('x, a, b, c')
# Note that IPython also supports a faster way to do this, by calling 
# this equivalent expression starting with a comma:
# ,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]

Differential equations

t = var('t') # define a variable t
x = function('x', t) # define x to be a function of that variable
de = (diff(x, t) + x == 1)
desolve(de, [x, t]) # returns (c + e^t)*e^(-t)

Linear algebra

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)) # returns
'''[0.1 0.2]
   [0.1 0.2]'''

Number theory

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)

See also

• Comparison of computer algebra systems
• Comparison of statistical packages
• Comparison of numerical analysis software

External links

• Project home page
• Official Sage documentation, reference, and tutorials
• Sage introduction videos
• Use Sage online in your web browser
• Free software brings affordability, transparency to mathematics
• AMS Notices Opinion – Open Source Mathematical Software
• W. Stein's blog post on history of Sage