My Rhapsody

gsoc maths programming python

GSoC Week 3

So far

My PR 9262 finally got merged. However, we have not yet decided which algorithm to use for the trigonometric and hyperbolic functions. In most cases there are multiple options

  • Expansion using closed form formula
  • Expansion of tan and tanh using a form of Newton's method
  • Expansion of sin, cos, sinh and cosh from tan and tanh
Newton's Method

Newton's method is rather interesting. It lets you calculate the series expansion of a function, given the series expansion of its inverse. Now, the formula for the expansion of tan is much more complicated than that of atan. So using atan's series expansion for tan makes sense. The basic idea is as follows:

Say I have a continuous and differentiable function f(x), whose inverse is g(x), i.e, f(g(x)) = x. Assume that we have an efficient implementation for g(x) and we want to use it for f(x). Let h(x) = y - g(x). On solving this equation, we'll get a root c, such that: h(c) = 0 or y = g(c) or c = f(y)

Now, using Newton's method we have the following iteration: x_j+1 = x_j + (y - g(x_j)) / g'(x_j)

The more iterations we do, higher the precision of our series expansion. If you are interested in the mathematics behind it, you can refer to this

As of now, ring_series has the code for plain vanilla formula based expansion too. I need to properly benchmark the different methods to conclude anything about their relative performance.

Puiseux Series

This week I am working on PR 9495, which will let us manipulate Puiseux series in ring_series. A Puiseux is a series which can have fractional exponents. By definition, a polynomial should have only positive integer exponents. So, I was hoping that using a Rational doesn't break anything in the polys module. Unfortunately, my PR is causing some tests to fail. I hope I don't need to make major changes because of it.

Once Puiseux series gets up and running, I will add the remaining functions dependent on it.

Next Week

A Symbolic Ring

Just as the type of series decides how the exponents of the series will be, the coefficients are determined by the ring over which the polynomial is defined. To be able to expand functions with arguments that have a constant term with respect to the expansion variable, the series should be allowed to have symbolic expressions as coefficients. For example, a 2nd order expansion about 0 of sin(x + y), wrt x, is sin(y) + x*cos(y). To be able to do this with ring_series, the polynomial needs to be defined over the EX or expression ring, which is Sympy's implementation of a symbolic ring. Currently ring_series works with EX ring but it can not handle a series with constant terms.

So, my major tasks for next week are:

  • Get ring_series working with symbolic ring
  • Implement series expansion of polylog and series reversion
  • Discuss the structure of Series class and send a PR for it.

Cheers!

gsoc maths programming python

GSoC Week 2

The 2nd week is now coming to an end, and by now I have a pretty good idea about how things often don't work the way we want them to.

So far

I trimmed down PR 9262 to remove all the parts not yet ready. It is undergoing review and should get merged soon.

Last week, I had planned to complete work on Laurent series by now, but it is still a work in progress. Fortunately, while discussing it with my mentors, we realized that handling Puiseux series is not as difficult as I had thought initially.

Internally, all polynomials are stored in the form of a dictionary, with a tuple of exponents being the key. So x + x*y + x**2*y**3 is stored as {(1, 0): 1, (1, 1): 1, (2, 3): 1}. For Puiseux series I need to be able to have rational exponents as keys in the dict. This isn't an issue in Python 3 as it evaluates 1/4 to 0.25 and the uses the decimal value as key. It doesn't work in Python 2 as it evaluates 1/4 to 0 and hence all fractions less than 1 become 0. The solution to this is to use Sympy's Rational data type, which lets us use the exact fraction as a key. This means that, hopefully, I will not need to make any complex changes in the code of ring.

I still have a few days to go in this week, during which I will further explore how to make the required changes.

There hasn't been much progress on the hashtable as both me and Sumith have been busy with our PRs. Hopefully, we will look into it during the weekend.

Next Week

  • Make ring_series work with Puiseux series
  • Write an interface to better handle the series, especially so that it works with the rest of Sympy.

Cheers!

gsoc maths programming python

GSoC Week 1

The first week of the Coding period is now almost over. Though this post is quite late, I'll try to post updates at the start of each week from now.

What makes my project so interesting (and challenging!) is that I need to work on two separate code bases- SymPy and Symengine. After discussing with Ondrej, I have started the part of my project related to SymPy.

Following are the goals of Phase I:

  • Complete ring_series : There is already a partial implementation of ring_series in SymPy. I will add more elementary and user functions, so that it has similar capabilities as the series function.
  • Write a class structure on top of ring_series so that the user need not bother about calling separate functions for each function.

So far

My main task, this week, was to port PR 609 into ring_series . I have, hence been working on PR 9262. I am done porting most elementary functions and writing their tests, with the exception of rs_nth_root and rs_cot.

A major issue that cropped up while writing tests is concerning Laurent series. As of now, polys do not accept negative exponents. I will either need to rewrite poly rings to accept negative exponents or modify operations so that they can use the present poly structure to handle negative exponents.

Next Week

My targets for the next week are:

  • Make ring_series handle Laurent series.
  • Polish PR 9262 and get it merged. It has already grown big. I will probably need to send another PR.
  • Discuss a class structure for series expansion and send a PR for it.
  • Start working on the hashtable for Symengine with Sumith

Looking forward to another exciting week.

Cheers!

gsoc maths programming python

Beginning of GSoC 2015

gsoc

In an earlier post I had talked about my application for Google Summer of Code, 2015. I am extremely glad to inform that my project for CSymPy (now called SymEngine) and SymPy has been selected under Python Software Foundation. Ondrej Certik and Thilina Rathnayake will be mentoring me during the course of my project. Ondrej started the SymPy library and is currently leading the development of SymEngine, while Thilina is a two time GSoC'er with SymPy and SymEngine and has research interests in Symbolic computation. Needless to say, I am extremely lucky to work under such talented people.

My project involves writing polynomial based series expansion modules for SymEngine and SymPy. I have already had very productive discussions with my mentors and hope to do some good work.

Looking forward to a great summer!

Cheers!

gsoc maths programming python

Hey there!

I am applying for Google Summer of Code, 2015. It is an annual program sponsored by Google, that pays students to contribute to open source organizations of their choice. My application involves two symbolic computation libraries - Sympy and CSymPy, the former written in Python and the latter in C++. CSymPy is meant to be a fast cousin of Sympy and can be used with optional wrappers.

I am proposing to implement fast series expansion in CSymPy and Sympy. Sympy already has series expansion, but suffers from speed issues and lack of a class structure. You can look at my full proposal here

I will use this blog to write about my GSoC experience, if I am selected.

I welcome your suggestions.

Cheers!