SageMath

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SageMath
Sage logo new.png
Sage - plot.png
The Sage notebook document interface in a web browser
Initial release 24 February 2005; 11 years ago (2005-02-24)
Stable release

7.3

/ 4 August 2016; 24 days ago (2016-08-04)
Preview release

7.4.beta2

/ 26 August 2016; 2 days ago (2016-08-26)
Written in Python, Cython
Operating system Linux, OS X, Microsoft Windows, Solaris, iOS, Android
Platform
Size Approx. 112–3319 MB
Type Computer algebra system
License GNU General Public License
Website www.sagemath.org

SageMath (previously Sage or SAGE, System for Algebra and Geometry Experimentation[1]) 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, with the initial goals of creating an "open source alternative to Magma, Maple, Mathematica, and MATLAB".[2] The originator and leader of the SageMath project, William Stein, is a mathematician at the University of Washington.

SageMath "uses a Python-like syntax,"[3] supporting procedural, functional and object-oriented constructs.

Features[edit]

Equation solving and typesetting using the SageMath notebook web interface

Features of SageMath include:[4]

Development[edit]

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.[9]

SageMath uses both students and professionals for development. The development of SageMath is supported by both volunteer work and grants.[10] However, it was not until 2016 that the first full-time Sage developer was hired (funded by an EU grant).[11] 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.[11]

Release history[edit]

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.[12]

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[13] OS X Lion support
6.0 December 2013 SageMath Development moved to Git[14]
7.0 January 2016 Massive GUI improvement

Achievements[edit]

Performance[edit]

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.[20]

Licensing and availability[edit]

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, OS X 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.[21]
  • A new online SageMath notebook is available at cloud.sagemath.com.

Although Microsoft was sponsoring a native version of Sage for the Windows operating system,[22] 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.[23] As of Sage 5.9, it mostly successfully builds on Cygwin.[24]

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.[25] In Gentoo, it's available via layman in the "sage-on-gentoo"[26] overlay. However, SageMath can be installed to any Linux distribution.

Gentoo prefix also provides Sage on other operating systems.

Software packages contained in SageMath[edit]

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[27]
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[28]

Usage examples[edit]

Algebra and calculus[edit]

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[edit]

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[edit]

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[edit]

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[edit]

References[edit]

  1. ^ Stein, William. "SAGE: A Computer System for Algebra and Geometry Experimentation". Retrieved 30 March 2012. 
  2. ^ Stein, William (2007-06-12). "Sage Days 4" (PDF). Archived from the original (PDF) on 2007-06-27. Retrieved 2007-08-02. 
  3. ^ Anastassiou, George A.; Mezei, Razvan A. (2015). Numerical Analysis Using Sage. New York: Springer. pp. x1 and 1. ISBN 9783319167381. 
  4. ^ "Sage documentation". 
  5. ^ "SageMath Interact functionality". Retrieved 2008-04-11. 
  6. ^ "Using SageTeX". 
  7. ^ "Using Compiled Code Interactively". SageMath Documentation. Retrieved 14 July 2011. 
  8. ^ "Calling SageMath from Mathematica". 
  9. ^ "days7 - Sage Wiki". Wiki.sagemath.org. 2008-11-14. Retrieved 2013-12-09. 
  10. ^ "Sage – Acknowledgement". Retrieved 2010-07-13. 
  11. ^ a b William Stein: The origins of SageMath – creating a viable open source alternative to Magma, Maple, Mathematica, and Matlab (presentation, June 11, 2016)
  12. ^ "SageMath Download - src-old". Retrieved 17 July 2011. 
  13. ^ "sage-5.0.txt". Retrieved 17 May 2012. 
  14. ^ "Installing and using SageMath just got even easier". Retrieved 12 July 2014. 
  15. ^ "Free Software Brings Affordability, Transparency To Mathematics". Science Daily. December 7, 2007. Retrieved 2008-07-20. 
  16. ^ "Sage Mathematical Software System". Retrieved 9 June 2012. 
  17. ^ "SIGSAM: Awards and prizes". Retrieved 2 Aug 2013. 
  18. ^ "Publications Citing Sage". Retrieved 14 July 2011. 
  19. ^ "Publications Citing Sage-Combinat". Retrieved 14 July 2011. 
  20. ^ "Cython, Sage, and the Need for Speed". 
  21. ^ "About the SageMath Cell Server". Sagecell.sagemath.org. Retrieved 2013-12-09. 
  22. ^ "Sage – Acknowledgment". 
  23. ^ Stein, William (16 March 2012). "Re: Question about Sage". Google Groups. sage-devel. Retrieved March 18, 2012. 
  24. ^ "Information for the port to Cygwin". Retrieved 12 June 2013. 
  25. ^ "Sage : "AIMS" team". 
  26. ^ "sage-on-gentoo". 
  27. ^ "SageMath Standard Packages". Retrieved 10 June 2012. 
  28. ^ "SageManifolds: home". 

External links[edit]

Related projects[edit]