SageMath

The SageMath notebook document interface in a web browser
Initial release	24 February 2005 (2005-02-24)
Stable release	7.4 / 18 October 2016 (2016-10-18)
Preview release	7.5.beta5 / 1 December 2016 (2016-12-01)
Repository	github.com/sagemath/sage
Written in	Python, Cython
Operating system	Linux, macOS, Microsoft Windows, Solaris, iOS, Android
Platform	Personal computers and web platform IA-32, x86-64, Itanium, ARM, SPARC
Size	Approx. 112–3319 MB
Type	Computer algebra system
License	GPLv2
Website	www.sagemath.org

SageMath (previously Sage or SAGE, System for Algebra and Geometry Experimentation) is mathematical software with features covering many aspects of mathematics, including algebra, combinatorics, numerical mathematics, number theory, and calculus.

The first version of SageMath was released on 24 February 2005 as free and open source software under the terms of the GNU General Public License version 2, with the initial goals of creating an "open source alternative to Magma, Maple, Mathematica, and MATLAB". The originator and leader of the SageMath project, William Stein, is a mathematician at the University of Washington.

SageMath "uses a Python-like syntax," supporting procedural, functional and object-oriented constructs.

Features

Equation solving and typesetting using the SageMath notebook web interface

Features of SageMath 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 SageMath 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 SageMath directly, SageMath 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 packages 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 packages 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.

SageMath uses both students and professionals for development. The development of SageMath is supported by both volunteer work and grants. However, it was not until 2016 that the first full-time Sage developer was hired (funded by an EU grant). The same year, Stein described his disappointment with a lack of academic funding and credentials for software development, citing it as the reason for his decision to leave his tenured academic position to work full-time on the project in a newly founded company, SageMath, Inc.

Release history

Only the major releases are listed below. SageMath 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.

SageMath 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, 64-bit OS X support
5.0	May 2012	OS X Lion support
6.0	December 2013	SageMath Development moved to Git
7.0	January 2016	Massive GUI improvement

Achievements

• 2007: first prize in the scientific software division of Les Trophées du Libre, an international competition for free software.
• 2012: one of the projects selected for the Google Summer of Code.
• 2013: ACM/SIGSAM Jenks Prize.
• SageMath has been cited in a variety of publications.

Performance

Both binaries and source code are available for SageMath from the download page. If SageMath 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 SageMath programs, as the Python code is converted into C.

Licensing and availability

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

• The source code can be downloaded from the downloads page. Although not recommended for end users, development releases of SageMath are also available.
• Binaries can be downloaded for Linux, macOS and Solaris (both x86 and SPARC).
• A live CD containing a bootable Linux operating system is also available. This allows usage of SageMath without Linux installation.
• Users could use an online version of SageMath at sagenb.org, but it has been discontinued in April 2015.
• Users can use an online "single cell" version of SageMath at sagecell.sagemath.org or embed a single SageMath cell into any web page. Users can also create permalinks to SageMath computations using the cell server.
• A new online SageMath notebook is available at cloud.sagemath.com.

Although Microsoft was sponsoring a native version of SageMath 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 SageMath. As of SageMath 5.9, it mostly successfully builds on Cygwin.

Linux distributions in which SageMath is available as a package are Mandriva, Fedora, 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, SageMath can be installed to any Linux distribution.

Gentoo prefix also provides Sage on other operating systems.

Software packages contained in SageMath

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

Mathematics packages contained in SageMath	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 SageMath	Command-line shell	IPython
	Database	ZODB, SQLite
	Graphical interface	SageMath Notebook, jsMath
	Graphics	matplotlib, Tachyon, GD, Jmol
	Interactive programming language	Python
	Networking	Twisted
Other Mathematics package available for SageMath	Differential Geometry and Tensor Calculus	Sage Manifolds

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]

# same matrix, but over the ring of doubles (not rationals, as above)
sage: B = matrix(RDF, [[1, 2, 3], [3, 2, 1], [1, 2, 1]])
sage: B.inverse()

[-5.55111512313e-17                0.5               -0.5]
[             -0.25              -0.25                1.0]
[               0.5                0.0               -0.5]

# Call NumPy for the Moore-Penrose pseudo-inverse, 
# since SageMath 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)

sage: E2 = EllipticCurve(CC, [0,0,-2,1,1])
sage: E2
Elliptic Curve defined by y^2 + (-2.00000000000000)*y = 
         x^3 + 1.00000000000000*x + 1.00000000000000 over 
         Complex Field with 53 bits of precision
sage: E2.j_invariant()
61.7142857142857

See also

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

External links

• Official website
• Official SageMath documentation, reference, and tutorials
• SageMath introduction videos
• Use SageMath 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
• Sage on GitHub
• Sage Math on Google Play
• Sage Android package at the F-Droid repository

Related projects

• Sagemath Cloud computational mathematics in the cloud
• Sage Math for Android to access Sagemath Cloud from Android
• LMFDB database of L-functions, modular forms, and related objects
• FindStat database of combinatorial statistics