SAGE 4.0.1

Sage is free open source math software that supports research and teaching in algebra, geometry, number theory, cryptography, and related areas. Both the Sage development model and the technology in Sage itself are distinguished by an extremely strong emphasis on openness, community, cooperation, and collaboration: we are building the car, not reinventing the wheel.
General and Advanced Pure and Applied Mathematics
Use SAGE for studying a huge range of mathematics, including algebra, calculus, elementary to very advanced number theory, cryptography, numerical computation, commutative algebra, group theory, combinatorics, graph theory, and exact linear algebra.
Use an Open Source Alternative
By using SAGE you help to support a viable open source alternative to Magma, Maple, Mathematica, and MATLAB. SAGE includes many high-quality open source math packages.
Use Most Mathematics Software from Within SAGE
SAGE makes it easy for you to use most mathematics software together. SAGE includes interfaces to Magma, Maple, Mathematica, MATLAB, and MuPAD, and the free programs Axiom, GAP, GP/PARI, Macaulay2, Maxima, Octave, and Singular.
Use a Standard Programming Language
You work with SAGE using the highly regarded scripting language Python instead of an obscure language designed for a particular mathematics program. You can write programs that combine serious mathematics with anything else.
Create Notebooks with your Web Browser
Use SAGE from your web browser, which connects either to a program running on your computer, or a program running elsewhere. With the SAGE notebook you can create embedded graphics, beautifully typeset mathematical expressions, add and delete input, and start up and interrupt multiple calculations.
Be Curious
SAGE provides you with easy access to documentation and source code. Type plot? for help on the plot command and plot?? to see the source code. If X is anything, type X.[tab key] to see all commands that apply to X.

Enhancements
Algebra:
- Division over integers (Robert Bradshaw) -- A much simpler and faster algorithm for the divisors function over integers. The new optimized code is faster than a similar integer divisor function in the version of PARI/GP thats bundled with Sage 3.2.1, as well as outperforming a similar integer divisor function found in the version of Magma that Sage 3.2.1 interfaces with.
- Finite field operations (John Palmieri) -- A few methods for finite field elements including additive order, p-th power, and p-th root where p is the characteristic of the field.
Basic arithmetic:
- Polynomials over a field (Burcin Erocal) -- Improving the user interface of polynomial classes.
- Polynomial square roots (John Palmieri, Carl Witty) -- A method to test whether a polynomial is square over the field it is defined. If the polynomial is square, then the method has the option of returning a square root.
Build:
- Improve sage -upgrade (William Stein, Michael Abshoff) -- The Sage upgrade command can now take an optional URL from which it will pull all spkgs, and this URL can be a Sage install. The upgrade command lists packages that will be upgraded before upgrading them, and autodownloads a new version of any spkg that hasnt successfully been installed before upgrading it.
- Problematic CPU flags (William Stein, Michael Abshoff) -- Binary distributions of Sage for Linux (e.g. Ubuntu) may not work properly once installed. The following CPU flags are known to prevent Sage from running properly: sse, 3d, mmx, pni, and cmov.
Calculus:
- Gamma and factorial functions (Mike Hansen, Burcin Erocal, Wilfried Huss) -- Symbolic gamma and factorial functions.
- Update to sympy-0.6.3 (Ondrej Certik) -- Update to the latest upstream of SymPy (sympy-0.6.3), which is a Python library for symbolic mathematics. For more information about SymPy, please visit http://code.google.com/p/sympy/.
- Numerical trigonometry (Robert Bradshaw) -- Optimized floating point evaluation of trigonometric functions such as sine and cosine. For example, numerical calculation of sine via _fast_float_ is now twice as fast as math.sin.
- Floating point calculation (Robert Bradshaw) -- Changing the parsing code for numerical computation to use RDF, which is a better reflection of the underlying precision. For calculus expressions involving real numbers, redundant trailing zeros are removed.
Coercion:
- Coercion API (Robert Bradshaw) -- Some simplification of the coercion interface.
Combinatorics:
- Coding theory (David Joyner) -- Several changes in linear_codes.py which should speed up (and in some cases do:-) some coding theory computations considerably. It adds interfaces to Cython and C functions of Robert Miller, CJ Tjhal, and Jeffery Leon. Speed up of minimum_distance (for codes over GF(2) and GF(3)), the spectrum (=weight_distribution), and permutation_automorphism_group are expected and in most cases achieved. (Also a new function is_permutation_equivalent was added, which interfaces with Robert Millers double coset partition refinement code.)
- Incidence structures and block designs (David Joyner) -- Beginning of an incidence structure class and an implementation of some basic block design algorithms. A few functions require GAPs Design package (which is included in gap_packages-4.4.10_6.spkg) but calling GAP or GAPs Design was only done when the corresponding Sage functionality was missing. Robert Millers recent code on computing the automorphism group of a non-linear binary code was used to implement the automorphism group of a block design.
Testing:
- Added only_optional doctest option (William Stein) -- Added a new option sage -t -only_optional=component that allows one to run only the optional doctests that depend on a given component. Thus instead of much of the optional functionality of Sage being broken, it will now be much easier to automatically test it.