SageMath (previously Sage or SAGE, "System for Algebra and Geometry Experimentation") is a computer algebra system with features covering many aspects of mathematics, including algebra, combinatorics, graph theory, numerical analysis, number theory, calculus and statistics.
|Initial release||24 February 2005|
8.9 / 29 September 2019
9.0.beta9 / 8 December 2019
|Written in||Python, Cython|
|Operating system||Linux, macOS, Microsoft Windows, Solaris, Android, iOS|
|Size||Approx. 112–3319 MB|
|Type||Computer algebra system|
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.
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.
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.
|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|
|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|
|8.0||July 2017||First version with fully working Windows support|
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.
Licensing and availability
- 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.
- 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. As of SageMath 8.0 (July 2017), with development funded by the OpenDreamKit project, it successfully builds on Cygwin, and a binary installer for 64-bit versions of Windows is available.
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 "sage-on-gentoo" overlay. The package used by NixOS is available for use on other distributions, due the distribution-agnostic 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 open-source libraries wherever they exist. Therefore, it uses many libraries from other projects.
|Algebra||GAP, Singular, FLINT|
|Arbitrary precision arithmetic||MPIR, MPFR, MPFI, NTL, mpmath, Arb|
|Arithmetic geometry||PARI/GP, NTL, mwrank, ECM|
|Calculus||Maxima, SymPy, GiNaC, Giac, FriCAS|
|Linear algebra||ATLAS, BLAS, LAPACK, NumPy, LinBox, IML, GSL|
|Numerical computation||GSL, SciPy, NumPy, ATLAS|
|Number theory||PARI/GP, FLINT, NTL|
|Statistical computing||R, SciPy|
|Graphical interface||SageMath Notebook, jsMath|
|Graphics||matplotlib, Tachyon, GD, Jmol|
|Interactive programming language||Python|
and Tensor Calculus
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]
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)
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] # The Moore-Penrose pseudo-inverse sage: C = matrix([[1 , 1], [2 , 2]]) sage: C.pseudoinverse() [1/10 1/5] [1/10 1/5] # Alternatively, call NumPy for the pseudo-inverse # (only numerical) import numpy C = matrix([[1 , 1], [2 , 2]]) matrix(numpy.linalg.pinv(C)) # 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)
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
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]
- "COPYING.txt – sage.git". The Sage Repository. Retrieved 4 April 2017.
- "Sagemath.org Site Info". Alexa Internet. Retrieved 2018-02-13.
- 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.
- "sage-5.0.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 / Code-in 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 Sage-Combinat". 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.
- "sage-on-gentoo Wiki". Retrieved 6 January 2017.
- "Standard Packages". doc.sagemath.org. Retrieved 6 January 2017.
|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
- Sage-Related 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 F-Droid repository