SageMath
SageMath (previously Sage or SAGE, "System for Algebra and Geometry Experimentation"[3]) is a computer algebra system with features covering many aspects of mathematics, including algebra, combinatorics, graph theory, numerical analysis, number theory, calculus and statistics.
Sagemath document (Jupyter Notebook) inside a web browser  
Initial release  24 February 2005 

Stable release  8.9
/ 29 September 2019 
Preview release  9.0.beta9
/ 8 December 2019 
Repository  
Written in  Python, Cython 
Operating system  Linux, macOS, Microsoft Windows, Solaris, Android, iOS 
Platform 

Size  Approx. 112–3319 MB 
Type  Computer algebra system 
License  GPLv3[1] 
Alexa rank  
Website  www 
The first version of SageMath was released on 24 February 2005 as free and opensource 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".[4] The originator and leader of the SageMath project, William Stein, is a mathematician at the University of Washington.
SageMath uses a syntax resembling Python's,[5] supporting procedural, functional and objectoriented constructs.
Development
William Stein realized when designing Sage that there were many opensource 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.[6]
SageMath uses both students and professionals for development. The development of SageMath is supported by both volunteer work and grants.[7] However, it was not until 2016 that the first fulltime Sage developer was hired (funded by an EU grant).[8] 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 fulltime on the project in a newly founded company, SageMath, Inc.[8]
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.[9]
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 OS X support 
5.0  May 2012[10]  OS X Lion support 
6.0  December 2013  SageMath Development moved to Git[11] 
7.0  January 2016  Massive GUI improvement 
8.0  July 2017  First version with fully working Windows support 
Achievements
 2007: first prize in the scientific software division of Les Trophées du Libre, an international competition for free software.[12]
 2012: one of the projects selected for the Google Summer of Code.[13]
 2013: ACM/SIGSAM Jenks Prize.[14]
 SageMath has been cited in a variety of publications.[15][16]
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.[17]
Licensing and availability
SageMath is free software, distributed under the terms of the GNU General Public License version 3.[1] 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. Many Linux distributions also include SageMath in their repositories [see below].
 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.
 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.[18]
 A new online SageMath notebook is available at cocalc.com.
Full documentation for installation is provided at doc.sagemath.org/html/en/installation/.
Although Microsoft was sponsoring a native version of SageMath for the Windows operating system, prior to 2016 there were no plans for a native port, and users of Windows had to use virtualization technology such as VirtualBox to run SageMath.[19] As of SageMath 8.0 (July 2017), with development funded by the OpenDreamKit project[8], it successfully builds on Cygwin, and a binary installer for 64bit versions of Windows is available.[20]
Linux distributions in which SageMath is available as a package are Fedora, Arch Linux, Debian, Ubuntu and NixOS. In Gentoo, it is available via layman in the "sageongentoo"[21] overlay. The package used by NixOS is available for use on other distributions, due the distributionagnostic nature of its package manager, Nix.
Gentoo prefix also provides Sage on other operating systems.
Software packages contained in SageMath
The philosophy of SageMath is to use existing opensource libraries wherever they exist. Therefore, it uses many libraries from other projects.
Mathematics packages contained in SageMath[22] 
Algebra  GAP, Singular, FLINT 

Algebraic geometry  Singular  
Arbitrary precision arithmetic  MPIR, MPFR, MPFI, NTL, mpmath, Arb  
Arithmetic geometry  PARI/GP, NTL, mwrank, ECM  
Calculus  Maxima, SymPy, GiNaC, Giac, FriCAS  
Combinatorics  Symmetrica, SageCombinat  
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 
Commandline 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()
[5.55111512313e17 0.5 0.5]
[ 0.25 0.25 1.0]
[ 0.5 0.0 0.5]
# The MoorePenrose pseudoinverse
sage: C = matrix([[1 , 1], [2 , 2]])
sage: C.pseudoinverse()
[1/10 1/5]
[1/10 1/5]
# Alternatively, call NumPy for the pseudoinverse
# (only numerical)
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
Commutative algebra
sage: P.<x, y, z> = PolynomialRing(QQ) # polynomial ring in x, y, z over the rationals
sage: (x**3 + y**3 + z**3  3*x*y*z).factor()
(x + y + z) * (x^2  x*y + y^2  x*z  y*z + z^2)
sage: I = P.ideal(x**2, x*y  y**2) # ideals defined by generators
sage: I.groebner_basis()
[y^3, x^2, x*y  y^2]
See also
References
 "COPYING.txt – sage.git". The Sage Repository. Retrieved 4 April 2017.
 "Sagemath.org Site Info". Alexa Internet. Retrieved 20180213.
 Stein, William. "SAGE: A Computer System for Algebra and Geometry Experimentation". Retrieved 30 March 2012.
 Stein, William (12 June 2007). "Sage Days 4" (PDF). Archived from the original (PDF) on 27 June 2007. Retrieved 2 August 2007.
 Anastassiou, George A.; Mezei, Razvan A. (2015). Numerical Analysis Using Sage. New York: Springer. pp. x1 and 1. ISBN 9783319167381.
 "Sage Days 7: Combinatorics". SageWiki. 14 November 2008. Retrieved 9 December 2013.
 "Sage – Acknowledgement to Supporters". Retrieved 6 January 2017.
 William Stein: The origins of SageMath – creating a viable open source alternative to Magma, Maple, Mathematica, and Matlab (presentation, 11 June 2016)
 "Source Code for Old Versions". 18 December 2015. Retrieved 6 January 2017.
 "sage5.0.txt". Retrieved 6 January 2017.
 "Installing and using SageMath just got even easier". 18 December 2013. Retrieved 6 January 2017.
 "Free Software Brings Affordability, Transparency To Mathematics". Science Daily. 7 December 2007. Retrieved 6 January 2017.
 "Sage Mathematical Software System". Google Summer of Code / Codein Archive. Retrieved 6 January 2017.
 "Richard Dimick Jenks Memorial Prize 2013 Award". Association for Computing Machinery – SIGSAM. Retrieved 6 January 2017.
 "Publications Citing Sage". Retrieved 6 January 2017.
 "Publications Citing SageCombinat". Retrieved 6 January 2017.
 Stein, William (3 November 2010). "Cython, Sage, and the Need for Speed". Retrieved 6 January 2017.
 "About SageMathCell". sagecell.sagemath.org. Retrieved 6 January 2017.
 Stein, William (16 March 2012). "Re: Question about Sage". Retrieved 6 January 2017.
 Lelievre, Samuel (18 August 2017). "SageMath 8.0 installer for Windows". Retrieved 28 August 2017.
 "sageongentoo Wiki". Retrieved 6 January 2017.
 "Standard Packages". doc.sagemath.org. Retrieved 6 January 2017.
External links
Wikibooks has a book on the topic of: Sage 
Wikimedia Commons has media related to Sage (mathematics software). 
 Official website
 Official SageMath documentation, reference, and tutorials
 SageMath introduction videos
 Free license book: "Computational Mathematics with SageMath"
 AMS Notices Opinion – Open Source Mathematical Software
 SageRelated Stuff and resources – Gregory V. Bard
 W. Stein's blog post on history of Sage
 Sage on GitHub (main)
 Sage Math on Google Play
 Sage Android package at the FDroid repository
Related projects
 GNU Octave A numerical computation software platform which is also a GNUlicensed alternative to MATLAB.
 CoCalc Hosted version of SageMath online
 LMFDB database of Lfunctions, modular forms, and related objects
 FindStat database of combinatorial statistics