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GSoC 2009 Report Aaron Meurer: ODEs Module
See my blog post for the full report.
See my blog for more details on my 2009 Google Summer of Code experiences. Also, the blog post has some advice for anyone who may want to apply for GSoC in the future.
My name is Aaron Meurer, and I am a second year math and computer science major at New Mexico Tech in Socorro, New Mexico, USA.
My project was to implement Ordinary Differential Equations solvers in sympy. The final module that I completed is able to do the following:
- Solve ODEs using the following methods (with
dsolve()):- 1st order separable differential equations
- 1st order differential equations whose coefficients or dx and dy are functions homogeneous of the same order.
- 1st order exact differential equations.
- 1st order linear differential equations
- 1st order Bernoulli differential equations.
- 2nd order Liouville differential equations.
- nth order linear homogeneous differential equation with constant coefficients.
- nth order linear inhomogeneous differential equation with constant coefficients using the method of undetermined coefficients.
- nth order linear inhomogeneous differential equation with constant coefficients using the method of variation of parameters.
- Classify an ODE that fits one of the above methods into that method using the
classify_ode()function. - Check a solution to an ODE using the
checkodesol()function. - Separate the variables multiplicatively in an expression using the
separatevars()function. - Determine the homogeneous order of an expression using the
homogeneous_order()function.
In addition, I also did work to do the following.
- Remove automatic combination of exponents
(exp(x)*exp(y) => exp(x + y)). - Expand hints for the expand() function.
- Refactor powsimp() to be able to simplify only the exponent or only the base.
Let me start from the beginning. Around late February to early March of this year, I discovered the existence of Google Summer of Code. I knew that I wanted to do some kind of work this summer, preferably an internship, so it piqued my interest. At that time, the mentoring organizations were still applying for GSoC 2009, so I could only look at the ones from 2008. Most of them were either Linux things or Web things, neither of which I had any experience in or am I much interested in. I took a free course in Python at my University the previous semester, and it was the programming language that I knew best at the time. I had learned some Java in my first semester CS class (did I mention that this was my first year at college?), and I hated it, and I was still learning C for my second semester CS class. So I looked at what the Python Foundation had to offer. I am a double major in math and computer science, so I looked under the math/science heading. That's when I saw SymPy.
I should not that I have been ahead in Math. It was my second semester, and I was taking Discrete Mathematics, Ordinary Differential Equations, Basic Concepts of Math, and Vector Analysis. So I looked for project ideas on the SymPy page that related to what I knew. The only one that I saw, other than core improvements, was to improve the ODE solving capabilities. I got into contact with the community and looked at the source, finding that it was only capable of solving 1st order linear equations and some special cases of 2nd order linear homogeneous equations with constant coefficients. I already at that point knew several methods from my ODE course, and I knew much of what I would learn.
The most difficult part of the Google Summer of Code, in my opinion, is the application period. For starters, you have to do it while you are still in classes, so you pretty much have to do it in your free time. Also, if you have never applied for Google Summer of Code before, you do not really know what a good application should look like. I have long had my application available on the SymPy Wiki, and I will reference it here a few times. First off, it was recommended to me by some of the SymPy developers that I put as many potential things that I could do in the summer in my application as I though I could do. I was still only about half way through my ODEs course when I wrote the application, but I had the syllabus so I knew the methods I would be learning at least by name. So that is exactly what I did: I packed my application with every possible thing that I knew we would be learning about in ODEs.
After I felt that I had a strong application, and Ondrej had proofread it for me, I submitted it. There were actually two identical applications, one for the Python Software Foundation, and one for Portland State University. This is because SymPy was not accepted as a mentoring organization directly, so they had to use those two foundations as proxies. A requirement of acceptance is to submit a patch that passes review. I decided to add a Bernoulli solver, because Bernoulli can be solved in the general case much like the 1st order linear ODE, which was already implemented.
After I applied, there was an acceptance period. I used that period to become aquatinted with the SymPy community and code base. A good way to do this is to try to fix EasyToFix issues. I found issue 694, which is to implement a bunch of tests from a paper by Michael Wester for testing computer algebra systems. The tests cover every possible thing that a full featured CAS could do, so it was a great way to learn SymPy. The issue is still unfinished, so working on it is still a good way to learn how to use SymPy.
Also, it was important to learn git, SymPy's version control system. The learning curve it pretty steep if you have never used version control system before, but once you can use it, it becomes a great tool at your disposal.
After being accepted, I toned down my work with SymPy to work on finishing up my classes. My classes finished a few weeks before the official start, so I used that period to get a jump start on my project.
For the start of the period, I followed my timeline. I implemented 1st order ODEs with homogeneous coefficients and 1st order exact ODEs. These were both pretty simple to do, as I expected.
The next thing I wanted to do was separable. My goal at this point was to get every relevant exercise from my textbook to work with my solvers. One of the exercises from my book (Pg. 56, No. 21) was dy/dx=exp**(x + y). I soon discovered that it was impossible for SymPy to separate e**(x + y) into exp(x)*exp(y), because the second would be automatically combined in the core. I also discovered that expand(), which should have been the function to split that, expanded using all possible methods indiscriminately. Part of my separatevars() function that I was writing to separate variables in expressions would be to split things like x + y*x into x*(1 + y) and 2*x*y + x**2 + y**2 into (x + y)**2, but expand()
as it was currently written would expand those.
So I spent a few weeks hacking on the core to make it not auto-combine exponents. I came up with a rule that exponents would only auto-combine if they had the same term minus the coefficient, the same rule that Add uses to determine what terms should auto combined by addition. So it would combine exp(x)*exp(x) into e**(2x), but exp(x)*exp(y) would be left alone. It turns out that some of our algorithms, namely the Gruntz limit algorithm, relies on auto-combining. We already had a function that could combine exponents, powsimp(), but it also combined bases, as in x**z*y**z into (x*y)**z, so I had to split the behavior so that it could act only as auto-combining once did (by the way, use powsimp(expr, combine='exp', deep=True) to do this). Then, after some help from Ondrej on pinpointing the exact location of the bugs, I just applied the function there. The last thing I did here was to split the behavior of expand, so that you could doexpand(x*(y + 1), mul=False) and it would leave it alone, butexpand(exp(x + y), mul=False) would return exp(x)*exp(y). This split behavior turned out to be useful in more than one place in my code.
This was the first non bug fix patch of mine that was pushed in to SymPy, and at the time of this writing, it is the last major one in the latest stable version. It took some major rebasing to get my convoluted commit history ready for submission, and it was during this phase that I git finally clicked for me, especially the git rebase command. This work took a few weeks from my ODEs time, and it became clear that I would not be doing every possible thing from my application. The reason that I included so much in my application was that my project was non-atomic. I could implement a little or a lot and still have a working useful module.
If you look at my timeline on my application, you can see that the first half is symbolic methods, and the second half is other methods, things like series. It turns out that we didn't really learn much about systems of ODEs in my course and we learned very little about numerical methods (and it would take much more to know how to implement them). We did learn series methods, but they were so annoying to do that I came to hate them with a passion. So I decided to just focus on symbolic methods, which were my favorite anyway. My goal was to implement as many as I could.
After I finished up separable equations, I came up with an idea that I did not have during the application period. dsolve() was becoming cluttered fast with all of my solution methods. The way that it worked was that it took an ODE and it tried to match methods one by one until it found one that worked, which it then used. This had some drawbacks. First, as I mentioned, the code was very cluttered. Second, the ODEs methods would have to be applied in a predetermined order. There are several ODEs that match more than one method. For example, 2*x*y + (x**2 + y**2)*dy/dx=0 has coefficients that are both homogeneous of order 2, and is also exact, so it can be solved by either method. The two solvers return differently formatted solutions for each one. A simpler example is that 1st order ODEs with homogeneous coefficients can be solved in two different ways. My working solution was to try them both and then apply some heuristics to return the simplest one. But sometimes, one way would create an impossible integral that would hand the integration engine. And it made debugging the two solvers more difficult because I had to override my heuristic. This also touches on the third point. Sometimes the solution to an ODE can only be represented in the form of an unevaluatable integral. SymPy's integrate() function is supposed to return an unevaluated Integral class if it cannot do it, but all too often it will just hang forever.
The solution I came up with was to rewrite dsolve using a hints method. I would create a new function called classify_ode() that would do all of the ODE classification, removing it from the solving code. By default, dsolve would still use a predetermined order of matching methods. But you could override it by passing a "hint" to dsolve for any matching method, and it would apply that method. There would also be options to only return unevaluated integrals when applicable.
I ended up doing this and more (see the docstrings for classify_ode() and dsolve() in the current git master branch), but before I could I needed to clean up some things. I needed to rewrite all of dsolve() and related functions. Before I started the program, there were some special cases in dsolve for second order linear homogeneous ODEs with constant coefficients and one very special case ODE for the expanded form of d^2/dx^2 x*e**(-y) = 0.
So the first thing I did was implement a solver for the general homogeneous linear with constant coefficients case. These are rather simple to do: you just find the roots of the characteristic polynomial built off of the coefficients, and then put the real parts of the roots in front of the argument of an exponential and the imaginary parts in front of the arguments of a sine and cosine (for example, 3 +/- 2i would give C1*e**(3*x)*sin(2*x) + C2*e**(3*x)*cos(2*x). The thing was, that if the imaginary part is 0, then you only have 1 arbitrary constant on the exponential, but if it is non-zero, you get 2, one for each trig function. The rest falls out nicely if you plug 0 in for b into e**(ax)*(C1*sin(b*x) + C2*cos(b*x) because the sine goes to 0 and the cosine becomes 1. But you would end up withC1 + C2 instead of just C1 in that case. I had already planned on doing arbitrary constant simplification as part of my project, so I figured I would put this on hold and do that first. Then, once that was done, the homogeneous case would be reduced to 1 case instead of the usual 2 or 3.
My original plan was to make an arbitrary constant type that automatically simplified itself. So, for example, if you entered C1 + 2 + x with C1 an arbitrary constant, it would reduce to just C1 + x. I worked with Ondrej, including visiting him in Los Alamos, and we build up a class that worked. The problem was that, in order to have auto-simplification, I had to write the simplification directly into the core. Neither of us liked this, so we worked a little bit on a basic core that would allow auto-simplification to be written directly in the classes instead of in the Mul.flatten()and Add.flatten() methods. It turns out that my constant class isn't the only thing that would benefit from this. Things like the order class (O(x)) and the infinity class (oo) are auto-simplified in the core, and things could be much cleaner if they happened in the classes themselves. Unfortunately, modifying the core like this is not something that can happen overnight or even in a few weeks. For one thing, it needed to wait until we had the new assumptions system, which was another Google Summer of Code project running parallel to my own. So we decided to shelf the idea.
I still wanted constant simplification, so I settled with writing a function that could do it instead. There were some downsides to this. Making the function as general as the classes might have been would have been far too much work, so I settled on making it an internal-only function that only worked on symbols named C1, C2, etc. Also, unlike writing the simplification straight into Mul.flatten() which was as simple as removing any terms that were not dependent on x, writing a function that parsed an expression and simplified it was considerably harder to write. I managed to churn out something that worked, and so I was ready to finish up the solver I had started a few paragraphs ago.
After I finished that, I still needed to maintain the ability to solve that special case ODE. Apparently, it is an ODE that you would get somewhere in deriving something about relativity, because it was in the relativity.py example file. I used Maple's excellent odeanalyser() function (this is where I go the idea for my classify_ode())to find a simple general case ODE that it fit (Liouville ODE). After I finished this, I was ready to start working on the hints engine.
It took me about a week to move all classification code into classify_ode(), move all solvers into individual functions, separate simplification code into yet other functions, and tie it all together in dsolve(). In the end, the model worked very well. The modularization allowed me to do some other things that I had not considered, such as creating a special "best" hint that used some heuristics that I originally developed for first order homogeneous which always has two possible solutions to try to give the best formatted solution for any ODE that has more than one possible solution method. It also made debugging individual methods much easier, because I could just use the built in hint calls in dsolve() instead of commenting out lines of code in the source.
This was good, because there was one more method that I wanted to implement. I wanted to be able to solve the inhomogeneous case of a nth order linear ode with constant coefficients. This can be done in the general case using the method of variation of parameters. It was quite simple to set up variation of parameters up in the code. You only have to set up a system of integrals using the Wronskian of the general solutions. It would usually be a very poor choice of a method if you were trying to solve an ODE by hand because taking the Wronskian and computing n integrals is a lot of work. But for a CAS, the work is already there. I just have to set up the integrals.
It soon became clear that even though, in theory, the method of variation of parameters can solve any ODE of this type, in practice, it does not always work so well in SymPy. This is because SymPy have very poor simplification, especially trigonometric simplification, so sometimes there would be a trigonometric Wronskian that would be identically equal to some constant, but it could only simplify it to some very large rational function of sines and cosines. When these were passed to the integral engine, it would cause it to fail, because it could not find the integral for such a seemingly complex expression.
In addition, taking Wronskians, simplifying them, and then taking n integrals is a lot of work as I said, and even when SymPy could do it, it took a long time. There is another method for solving these types of equations called undetermined coefficients that does not require integration. It only works on a class of ODEs where the right hand side of the ODE is a simple combination of sines, cosines, exponentials, and polynomials in x. It turns out that these kinds of functions are common anyway, so most ODEs of this type that you would encounter could be solved with this method. Unlike variation of parameters, undetermined coefficients requires considerable setup, including checking for different cases. This would be the method that you would want to use if you had to solve the ODE by hand because, even with all the setup, it only requires solving a system of linear equations vs. solving n integrals with variation of parameters, but for a CAS, it is the setup that matters, so this was a difficult prospect.
I spent the last couple of weeks writing up the necessary algorithms to setup the required system of linear equations and handling the different cases. After I finally worked out all of the bugs, I ran some profiling against my variation of parameters solver. It turned out that for ODEs that had trigonometric solutions (which take longer to simplify), my undetermined coefficients solver was an order of magnitude faster than the variation of parameters solver (and that is just for the ODEs that the variation of parameters engine could even solve at all). For ODEs that only had exponentials, it was still 2-4 times faster.
I finished off the summer by writing extensive documentation for all of my solvers and functions. Hopefully someone who uses SymPy to solve ODEs can learn something about ODE solving methods as well as how to use the function I wrote when they read my documentation.
I plan on continuing development with SymPy now that the Google Summer of Code period is over. SymPy is an amazing project, mixing Python and Computer Algebra, and I want to help it grow. I may even apply again in a future year to implement some other thing in SymPy, or maybe apply as a mentor for SymPy to help someone else improve it.
All of my commits on this branch fell under the 2009 Google Summer of Code period (19b33d943e06a3b4ac9cdebb41901af853dc2224).