# SageMath

Initial release | 24 February 2005 |
---|---|

Stable release | 6.10 / 18 December 2015 |

Preview release | 7.0-beta1 / 25 December 2015 |

Written in | Python, Cython |

Operating system | Cross-platform |

Platform | Python |

Size | 411 MB download (Ubuntu 64-bit)^{[1]} |

Type | Computer algebra system |

License | GNU General Public License |

Website | www |

**SageMath** (previously **Sage** or **SAGE**, System for Algebra and Geometry Experimentation^{[2]}) 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".^{[3]} The originator and leader of the SageMath project, William Stein, is a mathematician at the University of Washington.

SageMath "uses a Python-like syntax,"^{[4]} supporting procedural, functional and object-oriented constructs.

## Contents

## Features

Features of SageMath include:^{[5]}

- 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
^{[6]} - 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
^{[7]} - 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
^{[8]} - Although not provided by SageMath directly, SageMath can be called from within Mathematica;
^{[9]}as is done in this Mathematica notebook example

## Development

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.^{[10]}

SageMath uses both students and professionals for development. The development of SageMath is supported by both volunteer work and grants.^{[11]}

### 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.^{[12]}

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 OSX support |

5.0 | May 2012^{[13]} |
OSX Lion support |

6.0 | December 2013 | SageMath Development moved to Git^{[14]} |

7.0 | 2016 (planned) |

## Achievements

- 2007: first prize in the scientific software division of Les Trophées du Libre, an international competition for free software.
^{[15]} - 2012: one of the projects selected for the Google Summer of Code.
^{[16]} - 2013: ACM/SIGSAM Jenks Prize.
^{[17]} - SageMath has been cited in a variety of publications.
^{[18]}^{[19]}

## 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.^{[20]}

## 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, OS X and Solaris (both x86 and SPARC).
- A live CD containing a bootable Linux operating system is also available. This allows usage of Sage 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 sage 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

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

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 |

Command-line shell | IPython |

Database | ZODB, SQLite |

Graphical interface | SageMath Notebook, jsMath |

Graphics | matplotlib, Tachyon3d, GD, Jmol |

Interactive programming language | Python |

Networking | Twisted |

Differential Geometry and Tensor Calculus | Sage Manifolds^{[28]} |

## 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
```

## 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

## See also

- Comparison of computer algebra systems
- Comparison of statistical packages
- Comparison of numerical analysis software

## References

- ↑ "SageMath Download".
- ↑ Stein, William. "SAGE: A Computer System for Algebra and Geometry Experimentation". Retrieved 30 March 2012.
- ↑ Stein, William (2007-06-12). "Sage Days 4" (PDF). Archived from the original (PDF) on 2007-06-27. Retrieved 2007-08-02.
- ↑ Anastassiou, George A. and Razvan A. Mezei (2015).
*Numerical Analysis Using Sage*. New York: Springer. pp. x1 and 1. ISBN 9783319167381. - ↑ "Sage documentation".
- ↑ "SageMath Interact functionality". Retrieved 2008-04-11.
- ↑ "Using SageTeX".
- ↑ "Using Compiled Code Interactively".
*SageMath Documentation*. Retrieved 14 July 2011. - ↑ "Calling SageMath from Mathematica".
- ↑ "days7 - Sage Wiki". Wiki.sagemath.org. 2008-11-14. Retrieved 2013-12-09.
- ↑ "Sage – Acknowledgement". Retrieved 2010-07-13.
- ↑ "SageMath Download - src-old". Retrieved 17 July 2011.
- ↑ "sage-5.0.txt". Retrieved 17 May 2012.
- ↑ "Installing and using SageMath just got even easier". Retrieved 12 July 2014.
- ↑ "Free Software Brings Affordability, Transparency To Mathematics". Science Daily. December 7, 2007. Retrieved 2008-07-20.
- ↑ "Sage Mathematical Software System". Retrieved 9 June 2012.
- ↑ "SIGSAM: Awards and prizes". Retrieved 2 Aug 2013.
- ↑ "Publications Citing Sage". Retrieved 14 July 2011.
- ↑ "Publications Citing Sage-Combinat". Retrieved 14 July 2011.
- ↑ "Cython, Sage, and the Need for Speed".
- ↑ "About the SageMath Cell Server". Sagecell.sagemath.org. Retrieved 2013-12-09.
- ↑ "Sage – Acknowledgment".
- ↑ Stein, William (16 March 2012). "Re: Question about Sage". Newsgroup: sage-devel Check
`|newsgroup=`

value (help). Retrieved March 18, 2012. - ↑ "Information for the port to Cygwin". Retrieved 12 June 2013.
- ↑ "Sage : "AIMS" team".
- ↑ "sage-on-gentoo".
- ↑ "SageMath Standard Packages". Retrieved 10 June 2012.
- ↑ http://sagemanifolds.obspm.fr/

## External links

Wikibooks has a book on the topic of: Sage |

Wikimedia Commons has media related to .Sage (mathematics software) |

- Project home page
- 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

- Pages using deprecated image syntax
- Computer algebra system software for Linux
- Computer algebra system software for OS X
- Computer algebra system software for Windows
- Free and open-source Android software
- Free computer algebra systems
- Free educational software
- Free mathematics software
- Free software programmed in Python
- Mathematical software
- Python scientific libraries
- Statistical software