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Sage is a software application with features covering many aspects of mathematics, including algebra, combinatorics, numerical mathematics, and calculus.

The first version of Sage 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."[1] The lead developer of Sage, William Stein, is a mathematician at the University of Washington.

The Sage notebook document interface works with Firefox (and Mozilla), Opera, Konqueror, and Safari.
Equation solving and typesetting using the Sage notebook web interface.

Some of the many features of Sage include[2].

* A notebook document interface, for review and re-use of previous inputs and outputs, including graphics and text annotations usable from most web browsers including Firefox, Opera, Konqueror, and Safari. A secure connection via HTTPS to the notebook is supported when security or confidentiality are important, and allows Sage to be used both locally and remotely.
* A text-based command line interface using IPython
* The Python programming language supporting procedural, functional and object oriented constructs.
* Support for parallel processing using both multi-core processors found in many modern computers, multiple processors, in addition to 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 both functions and data.
* Matrix and data manipulation tools including support for sparse arrays
* Multivariate statistics libraries, using the functionality of R and SciPy.
* A toolkit for adding user interfaces to calculations and applications.[3]
* Tools for image processing[citation needed] using Pylab as well as the Python programming language.
* Graph theory visualization and analysis tools
* Libraries of number theory functions
* Import and export filters for data, images, video, sound, CAD, GIS, document and biomedical formats
* Support for complex number, arbitrary precision and symbolic computation for functions where this is appropriate.
* Technical word processing including formula editing and the ability to embed Sage inside LaTeX documents [4]
* Network tools for connecting to SQL, Java, .NET, C++, FORTRAN provide by Twisted, This supports a large number of protocols including HTTP, NNTP, IMAP, SSH, IRC, FTP and others.
* Interfaces to some third-party software like Mathematica, Magma, and Maple, which allows users to combine software and compare output and performance. It is thus also a "front-end" to other mathematical tools similar to GNU TeXmacs.
* MoinMoin as a Wiki system for knowledge management.
* Documentation using Sphinx.
* An automated test-suite, which allows for testing on an end-user's computer.

Although not provided by Sage directly, Sage can be called from within Mathematica.[5] A Mathematica notebook is available for this.[6]

Design Philosophy of Sage

William Stein realized several important facts when designing Sage.

* To create viable alternatives to Magma, Maple, Mathematica, and MATLAB, would take hundreds, or thousands of man-years if one started from the beginning.
* There was a large collection of open-source mathematics software already written, but which was written in different languages (C, C++, Fortran and Python being the most common).

So rather than start from the beginning, Sage (which is written in Python and Cython) integrates all their specialized mathematics software into a common interface. A user needs to know only Python.

Where no open-source option exists for a particular class of problem, then this would be written in Sage. But Sage does not reinvent the wheel. The same design philosophy is used in commercial mathematics programs such as Mathematica, but Sage can use a much wider range of open source software solutions than nonfree software, since some open source licensing imposes restrictions on proprietary use of code.

Sage development uses both students and professionals for development. The development of Sage is supported by both volunteer work and grants.[7]


Both binaries and source code are available for sage from the download page. If Sage 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

Sage is free software, distributed under the terms of the GNU General Public License version 2+. Sage is available in many ways:

* The source code can be downloaded from the downloads page. Although not recommended for end users, development releases of Sage are also available.
* Binaries can be downloaded for Linux, OS X and Solaris (both x86 and SPARC), though the latest version of Sage does not build on Solaris x86, although it does on SPARC.
* A live CD containing a bootable Linux operating system is also available. This allows Sage to be tried, without being installed.
* Users can use an online version of Sage at sagenb.org or http://t2nb.math.washington.edu:8000/, but with a limit to the amount of memory a user can use, as well as security restrictions.

Although Microsoft was sponsoring a native version of Sage for the Windows operating system [8], users of Windows will currently have to use virtualization technology and run Sage under one of the aforementioned operating systems. The preferred system is VirtualBox.

Software packages contained in Sage

As stated above, the philosophy of Sage is to use existing open source libraries wherever they exist. Therefore it borrows from many projects in producing the final product.
Mathematics packages contained in Sage Algebra GAP, Maxima, Singular
Algebraic Geometry Singular
Arbitrary Precision Arithmetic GMP, MPFR, MPFI, NTL
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 Sage Command line IPython
Database ZODB, Python Pickles, SQLite
Graphical Interface Sage Notebook, jsmath
Graphics Matplotlib, Tachyon3d, GD, Jmol
Interactive programming language Python
Networking Twisted

Command interface examples


x,a,b,c = var('x,a,b,c')

log(sqrt(a)).simplify_log() # returns log(a)/2
log(a/b).simplify_log() # returns log(a) - log(b)
sin(a+b).simplify_trig() # returns cos(a)*sin(b) + sin(a)*cos(b)
cos(a+b).simplify_trig() # returns cos(a)*cos(b) - sin(a)*sin(b)
(a+b)ˆ5 # returns (b + a)ˆ5
expand((a+b)ˆ5) # returns bˆ5 + 5*a*bˆ4 + 10*aˆ2*bˆ3 +
# 10*aˆ3*bˆ2 + 5*aˆ4*b + aˆ5

limit((xˆ2+1)/(2+x+3*xˆ2), x=infinity) # returns 1/3
limit(sin(x)/x, x=0) # returns 1

diff(acos(x),x) # returns -1/sqrt(1 - xˆ2)
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 == (-sqrt(b^2 - 4*a*c) - b)/(2*a),
# x == (sqrt(b^2 - 4*a*c) - b)/(2*a)]

f = xˆ2 + 432/x
solve(f.diff(x)==0,x) # returns [x == 3*sqrt(3)*I - 3,
# x == -3*sqrt(3)*I - 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 = lambda y: diff(y,t) + y - 1
desolve(DE(x(t)), [x,t]) # returns '%e^-t*(%e^t+%c)'

[edit] 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]

# Call numpy for the Moore-Penrose pseudo-inverse,
# since Sage does not support that yet.

import numpy
C = Matrix([[1 , 1], [2 , 2]])
matrix(numpy.linalg.pinv(C.numpy())) # 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 (2869/676 : -171989/17576 : 1)


Only the major releases are listed below. Sage practices the "release early, release often" concept, with releases every two to three weeks.
Sage 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
4.0 May, 2009
5.0 future 5.0 Milestone

In 2007, Sage won first prize in the scientific software division of Les Trophées du Libre, an international competition for free software.[9]
[edit] See also

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


1. ^ Stein, William (2007-06-12). "SAGE Days 4". http://www.sagemath.org/why/stein-sd4.pdf. Retrieved 2007-08-02.
2. ^ Sage documentation
3. ^ "Sage Interact functionality". http://wiki.sagemath.org/interact. Retrieved 2008-04-11.
4. ^ The TeX Catalogue OnLine, Entry for sagetex, Ctan Edition
5. ^ http://facstaff.unca.edu/mcmcclur/Mathematica/Sage/ Calling Sage from Mathematica
6. ^ http://facstaff.unca.edu/mcmcclur/Mathematica/Sage/UsingSage.nb A Mathematica notebook to call Sage from Mathematica.
7. ^ "Sage - Acknowledgement". http://www.sagemath.org/development-ack.html. Retrieved 2010-07-13.
8. ^ Sage - Acknowledgment
9. ^ "Free Software Brings Affordability, Transparency To Mathematics". Science Daily. December 7, 2007. http://www.sciencedaily.com/releases/2007/12/071206145213.htm. Retrieved 2008-07-20.

[edit] External links

* Project home page
* Official Sage Documentation Manual, Reference, Tutorials, ...
* Sage introduction videos
* Use Sage 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

Mathematics Software

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