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Short introduction to SAGE: Software for Algebra and Geometry Experimentation

SAGE is a framework for number theory, algebra, and geometry computation. The implementation is due to William Stein and over 30 other contributors from around the world. It is open source and freely available under the terms of the GNU General Public License (GPL). SAGE is

• A unified distribution of free math software that mere mortals can build from source
• A new Python library for mathematical computation with graphical and command-line interfaces
• An interface to existing mathematical software.

It is written in Python, C++, and C (via Pyrex). Python is an open source object-oriented interpreted language, with a large number of libraries, e.g., for numerical analysis, which are available to users of SAGE. Python can also be accessed in library mode from C/C++ programs.

SAGE provides a unified interface to several important open source libraries, including SINGULAR (for commutative algebra) and GAP (for group theory), John Cremona's MWRANK library (for elliptic curves), the PARI library for number theory, and Shoup's number theory library NTL, and Maxima (for symbolic manipulation). There are also interfaces to many commercial or closed source mathematics software packages include Maple, MATLAB, Mathematica, KASH/KANT, MAGMA, Axiom, and many others. For graphics, SAGE includes Matplotlib (for 2-d and some 3-d graphics) and tachyon (a 3-d ray tracer).

The main longterm goals and guiding principles for SAGE are:

• Free and open source: The source code must be freely available and readable, so users can understand what the system is really doing and more easily extend it. Just as mathematicians gain a deeper understanding of a theorem by carefully reading or at least skimming the proof, people who do computations should be able to understand how the calculations work by reading documented source code. All software included in the SAGE core distribution must be free and open source, and arbitrary modifications and redistribution of every single line must be allowed.
• Good programming environment: It is hoped that SAGE will form a stable environment to begin programming new mathematical software projects. SAGE provides a model for the mathematical community of software development with a strong emphasis on openness, community, cooperation, and collaboration.
• Extensible: Be able to define new data types or derive from built-in types, and make code written in a favorite language (including C/C++) part of the system.
• User friendly: The hope is to eventually attain a high level of user support.
• Easy to compile: SAGE should be relatively easy to compile from source for Linux, OS X, and MS Windows users. This provides more flexibility in modifying the system.
• Cross-platform: SAGE runs under Linux, OS X, Windows.
• Comprehensive: Implement enough algorithms to be really useful. Unify free open source mathematics software. Be a comprehensive mainstream high quality open source free mathematics software system.
• Efficient: Be very fast---comparable to or faster than anything else available. This is very difficult, since many systems are closed source, algorithms are sometimes not published, and finding fast algorithms is often extremely difficult (years of work, Ph.D. theses, luck, etc.).
• Tools: Provide robust interfaces to some of the functionality of PARI, GAP, GMP, Maxima, SINGULAR, MWRANK, and NTL. These are all are GPL'd and SAGE provides (or will provide) a unified interface for using them.
• Well documented: Reference manual, Tutorial, API reference with examples for every function, and an extensive "How can ... be constructed in SAGE?" document.

Download SAGE, an installation guide and a tutorial, from its webpage

http://sage.scipy.org/.

Although SAGE uses Python and PARI and other packages, keep in mind that it is not necessary to have this software preinstalled on the computer. The installation of SAGE is really designed to be relatively painless, but if there are any problems, please ask (the web page has links to the email lists for SAGE support). There are installation instructions at http://sage.scipy.org/sage/doc/html/inst/index.html. Moreover, if SAGE is installed once, upgrading to the newest version is especially easy with the "upgrade" command option (this is described on the website given above and assumes wget is installed and an internet connection).

Once SAGE is installed into a directory such as sage-x.y.z. In Linux, cd to this directory and type the command "./sage" to start SAGE and display the SAGE prompt "sage:". This is one way to use SAGE. Another way is to next type into SAGE "notebook(open_viewer=True)" at the SAGE prompt, which (a) starts the SAGE server running (loading the previously used worksheet, if any) and (b) opens the firefox browser (or starts a new tab if it is already open) and displays the SAGE notebook webpage.

At the SAGE prompt, type "factor(100)" (hit return) and then enter Ctrl-d (hold the Ctrl key and d at the same time) or "quit" (return) SAGE will first factor 100 and then exit. To do this in the notebook, enter "factor(100)" into a "cell" (an empty white window in the www browser) and then enter shift-enter. SAGE will display the factorization in the space just below the cell. To exit, the notebook version of SAGE, first go back to the command line, and enter Ctrl-c. Next, exit the browser. This quits SAGE's notebook.

There are more examples, in the tutorial, constructions, and reference manuals, available on the SAGE webpage. Please try them out!

SAGE is in an early stage of development, but is actively growing, and is already usable. There is a SAGE discussion board, bug-tracker, and wish list. Please visit

http://sage.scipy.org/

or

http://sage.math.washington.edu/sage/

or

http://echidna.maths.usyd.edu.au/sage/

or

email William Stein at [email protected]

for more information. Above all, have fun with SAGE!

David Joyner [email protected]

William Stein [email protected]

Last Updated 11-19-2006.