# GSoC 2011 Ideas

goodok edited this page Nov 20, 2011 · 45 revisions

# GSoC 2011 Project Ideas

This is the list of ideas for students wishing to apply for Google Summer of Code 2011.

If you are a student interested in applying, please get in touch with us on our mailing list, so that we can help you with the application.

## Ideas

Please add new ideas here. This is for inspiration, you can apply with something completely different if you like. The best project for you is one you are interested in, and are knowledgeable about. That way, you will be the most successful in your project and have the most fun doing it, while we will be the most confident in your commitment and your ability to complete it.

The ideas are in no particular order. Ideas in bold are ones that we would really like to have implemented. Some of these things are already partially implemented. Checkout the current source or ask on the mailing list to see where things presently stand.

The second section contains more detailed projects that can be done.

• anything from our roadmap
• optimize the core using Cython (see this proof-of-concept)
• asymptotic series
• port to Python 3.0 (issue 1262)
• Linear algebra
• Rewrite the Matrices module to be more like the polys module, i.e., allow Matrix to use the polys ground types, and separate the internal data (sparse vs. dense) from the Matrix interface. The goal is to make the matrices in SymPy much faster and more modular than they are now.
• Implement a sparse matrix representation for Matrix, so we can efficiently manipulate large sparse matrices. This is also related to the previous bullet point.
• improve the integration algorithm, so that SymPy can integrate anything that can be integrated.
• integration of functions on domains of maximum extent, etc.
• definite integration & integration on complex plane using residues
• Groebner bases and their applications in geometry, simplification and integration
• improve Buchberger's algorithm and implement Faugere F4 (compare their speed)
• improve polynomial algorithms (gcd, factorization) by allowing coefficients in algebraic extensions of the ground domain
• implement efficient multivariate polynomials (arithmetics, gcd, factorization)
• choose a polynomial representation (e.g. recursive dense) or use task dependent representations
• implement efficient arithmetics (e.g. using geobuckets (Yan) or heaps (Monagan & Pearce))
• implement factorization algorithm (Musser's or Wang's EEZ (better)) and gcd (e.g. EEZ-GCD)
• provide high-level OO abstraction over polynomial tools you've developed
• improve SymPy's pattern matching abilities (efficiency and generality)
• implement similarity measure between expression trees
• expression complexity measures (e.g. Kolmogorov's complexity)
• implement expressions signatures and heuristic equivalence testing
• implement semantic matching (e.g. expression: cos(x), pattern: sin(a*x) + b)
• e.g by using power series for this purpose (improve series speed)
• improve simplification and term rewriting algorithms
• add (improve) verbatim and semi-verbatim modes (more control on expression rewriting)
• fix annoying minus sign rewriting problem (separate internal representation and printing)
• extend simplify() function to support something more than rational functions (see trim())
• implement more expression rewrite functions (to an exact form that user specifies)
• maybe put transformation rules in an external database (e.g. prolog), what about speed?
• improve context (e.g. input) depended simplification steps in different algorithms
• e.g. the integrator needs different sets of rules to return "better" output for different input
• but there are more: recurrences, summations, solvers, polynomials with arbitrary coefficients
• what about information carried by expressions?
• what is simpler: chebyshevt(1, x) or x ?
• what is simpler: chebyshevt(1000, x) or (...) ?
• improve trigonometric simplification
• implement symbolic (formal) logics and set theory
• implement predicate (e.g. first-order), modal, temporal, description logics
• implement multivalued logics; fuzzy and uncertain logics and variables
• implement rewriting, minimization, normalization (e.g. Skolem) of expressions
• implement set theory, cardinal numbers, relations etc.
• implement symbolic global optimization (value, argument) with/without constraints, use assumptions
• improve series expansions (relevant issues)
• formal power series
• improve limits - make sure all basic limits work
• continue work on objects with indices (tensors)
• include the index simplification algorithms used in xAct and cadabra.
• improve the plotting module:
• Make the Plot() command use more backends: matplotlib, google chart API link, pyglet, asciart...
• Implement high level features, so that it works like in Mathematica (http://reference.wolfram.com/mathematica/ref/Plot.html)
• Make sure that all SymPy functions/expressions can be nicely plotted
• Fix related things/bugs in SymPy
• generalized functions -- Dirac delta, P(1/x), etc... Convolution, Fourier and Laplace transforms
• vector calculus, differential fields, maybe Lie algebras & groups
• parametric integrals asymptotic expansion (integral series)
• ordinary differential equations. Currently, SymPy only supports many basic types of differential equations, but there are plenty of methods that are not implemented. Maybe support for using Lie groups to help solve ODEs. See the ODE docs and the current source for information on what methods are currently implemented. Also, there is no support currently for solving systems of ODEs. You also might want to look at Manuel Bronstein's sumit.
• partial differential equations. Currently, SymPy can solve only very simple case of separable PDEs.
• increase image processing of PIL+SymPy functionality to match that of octave or matlab
• improve SymPy's interoperatibility with other CAS software
• implement general code parsing (Mathematica, Maxima, Axiom etc.) using e.g. pyparsing
• implement Python + SymPy (structure + semantics) translation to your favorite CAS (other than SymPy :)
• Symbolic quantum mechanics in SymPy. See below for details on projects related to this.

## Some Detailed Ideas (Projects)

### Symbolic quantum mechanics (sympy.physics.quantum)

Please contact the sympy list (or Brian E. Granger, Ondrej Certik) for questions about the physics related topics.

#### Abstract Dirac notation

• Status: Last summer we had two successful GSoC projects working on this topic. The code has been merged into sympy master (in sympy.physics.quantum), but there is a ton left to do.
• Idea: Continue to improve this code by adding any number of features. The sky is the limit on this one. You could basically pick anything from quantum mechanics and implement it (a physical system, scattering theory, perturbation theory, etc.). A student interested in working on this should minimally have already taken an upper division (undergrad) quantum course and have a solid understanding of quantum mechanics.
• Rating: 3 (moderate)

#### Implement All Known Analytical Solutions to Quantum Mechanical Systems

• Status: Currently, we have Hydrogen Schroedinger/Dirac energies and nonrelativistic wavefunctions implemented in sympy.physics.hydrogen.
• Idea: Implement all known analytical formulas for energies and wavefunctions for all QM systems that can be solved analytically (there are not that many). In particular, 1d: finite/infinite well, oscillator, Coulombic field, 1d radial equation: Coulombic field, oscillator, finite/infinite well, Dirac wavefunctions, 2d: Coulombic field, oscillator, finite/infinite well, 3D: finite/infinite well. Then there are systems in cylindrical coordinates and other things. There will be a submodule in the quantum, that would contain as much analytical knowledge about these systems as possible, and one would retrieve the knowledge using physical parameters (e.g. (Z, n) for Hydrogen states, (n1, n2, n3) for 3D oscillator, and so on). When this is done, more things can be added --- there are many systems, whose energies can be obtained with a simple numerical procedure (like zeros of bessel functions), and systems whose solution is given as a series. The devil is in little details, and there is high value in having all these systems implemented and tested, that they are correct. Usage of this is for testing numerical codes, that they can reproduce analytical results, as well as when playing with and learning about QM. This project should be quite easy in terms of Python/SymPy programming, but it requires knowledge about QM.
• Rating: 1 (easy)

#### Spin states and operators for arbitrary spin

• Status: Spin is a critical part of quantum mechanics. The spin algebra in quantum mechanics is rich and complex, even for spin 1/2. We would like to have a complete implementation of various spin states and operators for arbitrary spin s. We currently have a solid draft of this in sympy.physics.quantum.spin.
• Idea: Continue work on sympy.physics.quantum.spin. This would include rotation operators, testing, symbolic Clebsch-Gordon coefficients and Wiger 3j/6j symbols (we already have numerical versions), Wigner-Eckart theorem, tensors, etc. While SymPy already has spherical harmonics, it would be nice to integrate those into the treatment of orbital angular momentum as well. Basically pick up Zare or Varshalovich and start implementing things. These ideas could also be expanded to SU(N) for N>2 as well (very interesting and non-trivial).
• Rating: 3-4 (moderate-hard)

#### Position and momentum basis functions

• Status: The position and momentum basis functions in quantum mechanics are somewhat pathological because the basis functions are not square integrable. In spite of this, these representations are immensely useful and are introduced to students early on. We would like to be able to handle position and momentum wavefunctions on arbitrary domains in 1D, 2D and 3D. We currently have very little on this in sympy.physics.quantum so there is a lot to do.
• Idea: Implement full machinery for position and momentum wavefunctions, including modified Hilbert spaces, various coordinate systems, Dirac delta functions, basis transformations in 1D, 2D and 3D. This should use the base layer of quantum states and operators in sympy.physics.quantum. The test suite/code should include classic examples from quantum mechanics, such as particle in box, H atom, simple harmonic oscillator, scattering, etc. There are some real subtle issues in this work in how operators, states and general quantum expressions are represented in the position and momentum bases.
• Rating: 3-4 (moderate-hard)

#### Symbolic quantum computing

• Status: Quantum computing refers to the usage of quantum states, operators, time-evolution and measurement to perform non-trivial computations. Sympy already has a quite sophisticated symbolic quantum computing library and work continues on this.
• Idea: Quantum error correction, density matrices, Solovay-Kitaev algorithm, gate+circuit simplification using genetic algorithms, etc.
• Rating: 3-5 (hard)

#### Second quantization capabilities

• Status: SymPy currently a solid module for second quantization implemented, but it needs to be updated to take advantage of the new stuff in sympy.physics.quantum. Interesting, but challenging work.
• Idea: Work on sympy.physics.secondquant in the following areas:
1. Refactor to use the new assumptions system.
2. Refactor to use the stuff in sympy.physics.quantum.
3. Implement Wick's theorem for Bosons, including the macroscopic population of the ground states.
4. Density matrices.
5. Time evolution.
6. Various approximations, such as perturbation theory, Hartree-Fock, time-dependent Hartree-Fock, Hartree-Fock-Bogoliubov, etc.
7. Implement new functions for simplifying products of second quantized operators in different ways (such as moving an annihilation operator R).
• Rating: 5 (hard)

#### Quantum mechanics on graphs

• Status: Using ideas from finite differences, the Laplacian can be generalize to general graphs. This allows a straightforward generalization of the Schrodinger equation to graphs. We have started to develop approaches for exploring this physics (please email Brian Granger for details).
• Idea: Write a general library for working with general symbolic Hamiltonians (including time dependent ones) on graphs.
• Rating: 3 (moderate)

### Polynomials module

#### Efficient Groebner bases and their applications

• Status: Groebner bases computation is one of the most important tools in computer algebra, which can be used for computing multivariate polynomial LCM and GCD, solving systems of polynomial equations, symbolic integration, simplification of rational expressions, etc. Currently there is an efficient version of Buchberger algorithm implemented, along with naive multivariate polynomial arithmetics in monomial form.
• Idea: Improve efficiency of Groebner basis algorithm by using better selection strategy (e.g. sugar method) and implement Faugere F4 or F5 algorithm and analyze which approach is better in what contexts. Apply Groebner bases in integration of rational and transcendental functions and simplification of rational expressions modulo a polynomial ideal (e.g. trigonometric functions).
• Rating: 5 (very hard)

#### Multivariate polynomials and factorization

• Status: Factorization of multivariate polynomials is an important tool in algebra systems, very useful by its own, also used in symbolic integration algorithms, simplification of expressions, partial fractions, etc. Currently multivariate factorization algorithm is based on Kronecker's method, which is impractical for real life problems. Undergo there is implementation of Wang's algorithm, the most widely used method for the task.
• Idea: Start with implementing efficient multivariate polynomial arithmetics and GCD algorithm. You do this by improving existing code, which is based on recursive dense representation or implement new methods based on your research in the field. There are many interesting methods, like Yan's geobuckets or heap based algorithms (Monagan & Pearce). Having this, implement efficient GCD algorithm over integers, which is not a heuristic, e.g. Zippel's SPMOD, Musser's EZ-GCD, Wang's EEZ-GCD. Help with implementing Wang's EEZ factorization algorithm or implement your favourite method, e.g. Gao's partial differential equations approach. You can go further and extend all this to polynomials with coefficients in algebraic domains or implement efficient multivariate factorization over finite fields.
• Rating: 4-5 (quite hard)

#### Univariate polynomials over algebraic domains

• Status: Currently SymPy features efficient univariate polynomial arithmetics, GCD and factorization over Galois fields and integers (rationals). This is, however, insufficient in solving real life problems, and isn't very useful symbolic integration and simplification algorithms.
• Idea: Choose a univariate polynomial representation in which elements of algebraic domains will be efficiently encoded. By algebraic domains we mean algebraic numbers and algebraic function fields. Having a good representation, implement efficient arithmetics and GCD algorithm. You should refer to work due to Monagan, Pearce, van Hoeij et. al. Having this, implement your favourite algorithm for factorization over discussed domains. This will require algorithms for computing minimal polynomials (this can be done by using LLL or Groebner bases). You can also go ahead and do all this in multivariate case.
• Rating: 4-5 (quite hard)

### Integration module

#### Implement symbolic integration via Marichev-Adamchik Mellin transform

• Status: SymPy's integrator supports several classes of special function. This is, however, insufficient in solving advanced physics and engineering problems.
• Idea: Extend the integrator by implementing methods for integration of elementary and special functions via Meijer G-functions (also known as Marichev-Adamchik Mellin transform). This approach is useful in both indefinite and definite integration. The problems are how to convert arbitrary input expression to a G-function and how to rewrite the resulting expression in terms of more familiar functions (elementary, special, hypergeometric).

Look at the paper by Kelly Roach (K. Roach. Meijer g function representations. In ISSAC ’97: Proceedings of the 1997 international symposium on Symbolic and algebraic computation, pages 205–211, New York, NY, USA, 1997. ACM.)

• Rating: 3-5 (quite hard)

#### Implement definite integration algorithm using residues

• Status: Currently the integrator handles definite integration problem by applying Newton-Liebnitz theorem. This is, however, only useful when there are no poles on the path of integration.
• Idea: Improve the integrator by expressing the initial integral as an integral on the complex plane and applying Cauchy integral theorem.
• Rating: 3-4 (hard)

### Concrete module

#### Implement Karr algorithm, a decision procedure for symbolic summation

• Status: SymPy currently features Gosper algorithm and some heuristics for computing sums of expressions. Special preference is for summations of hypergeometric type. It would be very convenient to support more classes of expressions, like (generalized) harmonic numbers etc.
• Idea: Algorithm due to Karr is the most powerful tool in the field of symbolic summation, which you will implement in SymPy. There are strong similarities between this method and Risch algorithm for the integration problem. You will start with implementing the indefinite case and later can extend it to support definite summation (see work due to Schneider). Possibly you will also need to work on solving difference equations.
• Rating: 3-5 (quite hard)

### Plotting module

#### Improve the plotting module

• Status: Currently the Plot() command can only call pyglet, which is shipped with SymPy.
• Plan: Make the Plot() command use more backends: matplotlib, google chart API link, pyglet, asciart (and more...) Implement high level features, so that it works like in Mathematica (http://reference.wolfram.com/mathematica/ref/Plot.html). Make sure that all SymPy functions/expressions can be nicely plotted. Fix related things/bugs in SymPy
• Rating: 2-3 (easy, medium)

## Potential Mentors

• Aaron Meurer (asmeurer)
• Ondrej Certik (certik)
• Brian E. Granger: I am willing to mentor any of the physics related projects. (ellisonbg)
• Andy R. Terrel (aterrel)
• Mateusz Paprocki (mattpap)
• David joyner (wdjoyner)
• Christian Muise (haz)
• Ronan Lamy (rlamy)
• Santosh Lakshman (luckymurari)
• Luis Lira (lglira) Physics