Sage (mathematics software)

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Sage

Sage animated .gif, y=x2 (red) vs. y=x3(blue)
Initial release	24 February 2005
Stable release	4.1 / 2009-07-09; 4 days ago
Written in	Python, Cython
Operating system	Cross-platform
Platform	Python
Type	Computer algebra system
License	GNU General Public License
Website	Sage: Open Source Mathematics Software

Sage is a software application which covers many aspects of mathematics, including algebra, combinatorics, numerical mathematics and calculus. It has a wide range area of application, including Engineering, Science etc.

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.

[edit] Features

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.
• Sliders and other interactive controls as input for calculations.^[3]
• 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.
• Tools for image processing using pylab as well as the python programming language.
• Tools for visualizing and analysing graphs
• 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.
• An automated test-suite, which allows for testing on an end-users computer.

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

[edit] 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 hundred, or thousands of man-years if one started from the beginning.
• There was a huge range of well-tested 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 the Python, which is a well known program, used in 1000s of different applications.

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 program such as Mathematica, but Sage can use a much wider range of software, since commercial software can only programs if their license allows the source code to be kept hidden. Since Sage is open-source, no such limitation applies, so a much wider range of software may be used.

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

In 2007, Sage won first prize in the scientific software division of Les Trophées du Libre, an international competition for free software.^[8]

[edit] Performance

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 processor, the size of their caches, whether there is hardware support for SSE instructions etc. Many of the developers are passionate about performance, and often push to produce the fastest implementation in the world for a given task.

[edit] 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 uses to use, development releases of Sage are also available.
• Binaries can be downloaded for Linux, OS X and Solaris (both x86 and SPARC). The Solaris binaries are currently considered experimental.
• 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 http://demo.sagenb.org, but for technical reasons is is necessary to limit the amount of memory a user can use, as well as impose security restrictions.

Although Microsoft are sponsoring a native version of Sage to the Windows operating system, users of Windows will currently have to install the free VMWare Player and install a VMWare image of the Sage, available from the Sage download page.

[edit] Software packages contained in Sage

As stated above, the philosophy of Sage is to use

[edit] Command interface examples

[edit] Calculus

x,a,b,c = var('x,a,b,c')
 
log(sqrt(a))                            # 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]

[edit] 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]

[edit] 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)

[edit] History

Only the major releases are listed below. Sage practices the "Release Early, Release Often" mantra, with releases every two to three weeks.

[edit] See also

• Mathematica
• MATLAB
• Macsyma
• Maple
• Comparison of computer algebra systems
• Comparison of statistical packages
• Comparison of numerical analysis software

[edit] References

[edit] External links

Free software portal

• Project home page
• Sage introduction videos
• Use Sage online in your web-browser
• SAGE Reference Manual
• Free software brings affordability, transparency to mathematics
• AMS Notices Opinion - Open Source Mathematical Software