SageMath

SageMath

The SageMath notebook document interface in a web browser
Initial release	24 February 2005 (2005-02-24)
Stable release	8.3 / 3 August 2018 (2018-08-03)
Preview release	8.4.beta7 / 30 September 2018 (2018-09-30)
Repository	https://git.sagemath.org/sage.git/
Written in	Python, Cython
Operating system	Linux, macOS, Microsoft Windows, Solaris, Android, iOS
Platform	Personal computers and web platform IA-32, x86-64, ARM, Itanium, SPARC
Size	Approx. 112–3319 MB
Type	Computer algebra system
License	GPLv3[1]
Alexa rank	95,815 (October 2018)[2]
Website	www.sagemath.org

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.

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".^[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 object-oriented constructs.

Features

Equation solving and typesetting using the SageMath notebook web interface

Features of SageMath include:^[6]

• 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^[7].
• Graph theory visualization and analysis tools.
• Libraries of number theory functions.
• Support for complex numbers, arbitrary-precision arithmetic, and symbolic computation.
• Technical word processing including formula editing and embedding SageMath within LaTeX documents^[8].
• 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^[9].
• SageMath can pull up Mathematica within a program.^[10] Interfacing this way is documented officially to Sage.^[11]

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.^[12]

SageMath uses both students and professionals for development. The development of SageMath is supported by both volunteer work and grants.^[13] However, it was not until 2016 that the first full-time Sage developer was hired (funded by an EU grant).^[14] 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.^[14]

Achievements

• 2007: first prize in the scientific software division of Les Trophées du Libre, an international competition for free software.^[18]
• 2012: one of the projects selected for the Google Summer of Code.^[19]
• 2013: ACM/SIGSAM Jenks Prize.^[20]
• SageMath has been cited in a variety of publications.^[21][22]

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.^[23]

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.
• Users could have used 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.^[24]
• 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.^[25] As of SageMath 8.0 (July 2017), with development funded by the OpenDreamKit project^[14], it successfully builds on Cygwin, and a binary installer for 64-bit versions of Windows is available.^[26]

Linux distributions in which SageMath is available as a package are Mandriva, Fedora, Arch Linux, Debian and NixOS. It is also available as a dedicated Ubuntu PPA.^[27] In Gentoo, it's available via layman in the "sage-on-gentoo"^[28] 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.

Usage examples

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]

See also

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

References

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
• SageMath Cloud – Use SageMath online
• AMS Notices Opinion – Open Source Mathematical Software
• 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