SymPy is a computer algebra system (CAS) written in pure Python. The core allows basic manipulation of expressions (like differentiation or expansion) and it contains many modules for common tasks (limits, integrals, differential equations, series, matrices, quantum physics, geometry, plotting, and code generation).

SymPy has participated in the Google Summer of Code program in previous years under the umbrellas of Python Software Foundation, Portland State University, and the Space Telescope Science Institute, where we were very successful. In fact, several of our core developers, including four of the mentors from this year, started working with SymPy as Google Summer of Code students. This was our first year participating as a standalone organization, and we would like to share our experience.

As part of the application process we required each student to submit a patch (as a GitHub pull request) that had to be reviewed and accepted. This allowed us to see that each applicant knew how to use git as well as communicate effectively during the review process.This also encouraged only serious applicants to apply. We had over 10 mentors available and we ended up with 9 students, all of whom were successful at final evaluations.

Tom implemented an algorithm for computing symbolic definite integrals that uses so-called Meijer G-functions. This is the state-of-the-art algorithm for computing definite integrals, and indeed the results of his project are very impressive. This project has pushed SymPy forward a long way to becoming the strongest open source computer algebra system with respect to symbolic definite integration.

Vladimir Peric - Porting to Python 3, mentored by Ronan Lamy
Vladimir ported SymPy to work on Python 3 and ported all testing infrastructure so that SymPy gets regularly tested in Python 2.x, 3.2 and PyPy. Thanks to Vladimir’s work, the next version of SymPy, 0.7.2, which will hopefully be released later this year, will work in both Python 2 and Python 3, and it may support PyPy as well.

Gilbert Gede - PyDy, mentored by Luke Peterson
Gilbert implemented a physics module to assist in generating symbolic equations of motion for complex multibody systems using Kane's Method. He expanded on the code written by his mentor, Luke, in 2009, and the module can now generate equations of motion for a bicycle. Gilbert also wrote very thorough documentation both for the Kane’s Method and the module in SymPy.

Tomo has greatly improved the quantum mechanics module by implementing position/momentum representations for operators and eigenstates in various coordinate systems (including cartesian, cylindrical, and spherical) that allows you to easily represent many of the "textbook" quantum mechanics systems, including particle in a box, simple harmonic oscillator, hydrogen atom, etc.

Saptarshi Mandal - Combinatorics package for Sympy, mentored by Christian Muise
Saptarshi’s project was to mimic the various capabilities of Combinatorica, a Mathematica package for combinatorics. Most of the functionality involving elementary combinatorial objects such as Permutations, Partitions, Subsets, Gray codes and Prufer codes are complete.

Sherjil Ozair - Symbolic Linear Algebra, mentored by Vinzent Steinberg
Sherjil improved the speed of the linear algebra module by using efficient coefficient types for values of entries of matrices. Previously, SymPy used generic expressions in this place, which slowed down computations considerably and caused trouble with solving of the zero equivalence problem. He also implemented sparse matrix representation and unified the API with dense matrices. In addition, Sherjil also added a few linear algebra related algorithms (e.g. Cholesky decomposition).

Matthew improved the statistics module to use symbolics and introduced a Random Variable type, with support for finite, continuous, and multivariable normal random variables. With these you can symbolically compute things like probabilities of a given condition, conditional spaces, and expectation values. As a side consequence of this project, he also improved some of our Sets classes and implemented a MatrixExpr class, which allows you to compute with matrices symbolically, including computing with block matrices.

Sean was working on the quantum mechanics module and has implemented symbolic Clebsch-Gordan coefficients, Wigner D function, and related mathematical concepts that are used very often in quantum physics when dealing with angular momentum and then the necessary classes to support coupled spin algebra.

Jeremias Yehdegho - Implementing F5, mentored by Mateusz Paprocki
Jeremias worked on implementing algorithms related to Groebner bases. Groebner bases are a useful tool in many areas of computer algebra. He implemented the F5B algorithm, which is an improved version of the classical Buchberger’s algorithm that was previously implemented in SymPy, and an algorithm for converting Groebner bases between different orders of monomials and worked on applications of Groebner bases. This allowed for handling problems of much larger size in SymPy.

The full report can be found here, where each student wrote a wiki page about their experience during the summer and you can also find their blogs and links to applications. Each student was required to blog about their progress each week and all blogs were synchronized at planet.sympy.org.

In previous years, there was usually one student from each summer who became a regular contributor and also a mentor for the next year. It has been a rewarding experience for the whole SymPy community.

By Ondřej Čertík, Aaron Meurer and Mateusz Paprocki, SymPy Google Summer of Code Mentors