Author Archives: Christopher Rackauckas

A Comparison Between Differential Equation Solver Suites In MATLAB, R, Julia, Python, C, Mathematica, Maple, and Fortran

By: Christopher Rackauckas

Re-posted from: http://www.stochasticlifestyle.com/comparison-differential-equation-solver-suites-matlab-r-julia-python-c-fortran/

Many times a scientist is choosing a programming language or a software for a specific purpose. For the field of scientific computing, the methods for solving differential equations are what’s important. What I would like to do is take the time to compare and contrast between the most popular offerings.
This is a good way to reflect upon what’s available and find out where there is room for improvement. I hope that by giving you the details for how each suite was put together (and the “why”, as gathered from software publications) you can come to your own conclusion as to which suites are right for you.

(Full disclosure, I am the lead developer of DifferentialEquations.jl. You will see at the end that DifferentialEquations.jl does offer pretty much everything from the other suite combined, but that’s no accident: our software organization came last and we used these suites as a guiding hand for how to design ours.)

Quick Summary Table

If you just want a quick summary, I created a table which has all of this information. You can find it here (click for PDF):

Comparison Of Differential Equation Solver Software

MATLAB’s Built-In Methods

Due to its popularity, let’s start with MATLAB’s built in differential equation solvers. MATLAB’s differential equation solver suite was described in a research paper by its creator Lawerance Shampine, and this paper is one of the most highly cited SIAM Scientific Computing publications. Shampine also had a few other papers at this time developing the idea of a “methods for a problem solving environment” or a PSE. The idea is pretty simple: users of a problem solving environment (the examples from his papers are MATLAB and Maple) do not have the same requirements as more general users of scientific computing. Instead of focusing on efficiency, they key for this group is to have a clear and neatly defined (universal) interface which has a lot of flexibility.

The MATLAB ODE Suite does extremely well at hitting these goals. MATLAB documents its ODE solvers very well, there’s a similar interface for using each of the different methods, and it tells you in a table in which cases you should use the different methods.

But the modifications to the methods goes even further. Let’s take for example the classic ode45. That method just works and creates good plots, right? Well, Shampine added a little trick to it. When you solve an equation using ode45, the Runge-Kutta method uses a “free” interpolation to fill in some extra points. So between any two steps that the solver takes, it automatically adds in 4 extra points using a 4th order interpolation. This is because high order ODE solvers are good enough at achieving “standard user error tolerances” that they actually achieve quite large timesteps, and in doing so step too infrequently to make a good plot. Shampine’s scheme is a good quick fix to this problem which most people probably never knew was occurring under the hood!

There’s quite a bit of flexibility here. The methods allow you to use complex numbers. You’re given access to the “dense output function” (this is the function which computes the interpolations). There’s a few options you can tweak. Every one of these methods is setup with event handling, and there are methods which can handle differential-algebraic equations. There are also dde23 and ddesd for delay differential equations, and in the financial toolbox there’s an Euler-Maruyama method for SDEs.

While MATLAB does an excellent job at giving a large amount of easily available functionality, where it lacks is performance. There’s a few reasons for this. For one thing, these modifications like adding extra points to the solution array can really increase the amount of memory consumed if the ODE system is large! This actually has an impact in other ways. There’s a very good example of this in ode45. ode45 is based on the Dormand-Prince 5(4) pair. However, in 1999, the same year the MATLAB ODE Suite was published, Shampine released a paper with a new 5(4) pair which was more efficient than the Dormand-Prince method. So that begs the question, why wasn’t this used in the MATLAB ODE Suite (it’s clear Shampine new about it!)? (I actually asked him in an email) The reason is because its interpolation requires calculating some extra steps, so it’s less efficient if you are ALWAYS interpolating. But since ode45 is always interpolating in order to make the plots look nicer, this would get in the way. Essentially, it can be more efficient, but MATLAB sets things up for nice plotting and not pure efficiency.

But there are other areas where more efficient methods were passed up during the development phase of the ODE suite. For example, Hairer’s benchmarks in his book Solving Ordinary Differential Equations I and II (the second is for stiff problems), along with the benchmarks from the Julia DifferentialEquations.jl suite, consistently show that high order Runge-Kutta methods are usually the most efficient methods for high accuracy solving of nonstiff ODEs. These benchmarks both consistently show that, for the same error, high order Runge-Kutta methods (like order >6) can solve the equation much faster than methods like Adams methods. But MATLAB does not offer high order Runge-Kutta methods and only offers ode113 (an Adams method) for high-accuracy solving.

Some of this is due to a limitation within MATLAB itself. MATLAB’s ODE solver requires taking in a user-defined function, and since this function is defined in MATLAB its function calls are very inefficient and expensive. Thus MATLAB’s ODE solver suite can become more efficient by using methods which reduce the number of function calls (which multistep methods do). But this isn’t the only case where efficient methods are missing. Both Hairer and the JuliaDiffEq benchmarks show that high-order Rosenbrock methods are the most efficient for low-medium accuracy stiff ODEs, but MATLAB doesn’t offer these methods. It does offer ode23s, a low-order Rosenbrock method, and ode15s which is a multistep method. ode15s is supposed to be the “meat and potatoes of the MATLAB Suite”, but it cannot handle all equations of since its higher order methods (it’s adaptive order) are not L-stable (and not even A-stable). For this reason there’s a few other low order SDIRK methods (ode23tb, an ESDIRK method for highly stiff problems) which are recommended to fill in the gaps, but none of the higher order variants which are known to be more efficient for many equations.

This pattern goes beyond the ODE solvers. The DDE solvers are all low order, and in the case of ddesd, it’s a low accuracy method which is fast for getting plots correct but not something which converges to many decimal places all too well since it doesn’t explicitly track discontinuities. This is even seen in the paper on the method which shows the convergence only matches dde23 to graphical accuracy on a constant-delay problem. Again, this fits in with the mantra of the suite, but may not hit all demographics. Shampine specifically made a separate version of ddesd for Fortran for people who are interested in efficiency, which is another way of noting that the key of ddesd is features and automatic usage, and not hardcore scientific computing efficiency. The mentioned SDE solver from the financial toolbox is only order 0.5 and thus requires quite a small dt to be accurate.

And I can keep going, but I think you get the moral of the story. This suite was created with one purpose in mind: to make it very easy to solve a wide array of differential equations and get a nice plot out. It does a very good job at doing so. But it wasn’t made with efficiency in mind, and so it’s missing a lot of methods that may be useful if you want high efficiency or high accuracy. Adding these wouldn’t make sense to the MATLAB suite anyways since it would clutter the offering. MATLAB is about ease of use, and not efficiency, and it does extremely well at what it set out to do. For software that was first made before the Y2K crisis with just a few methods added later, it still holds up very well.

Hairer’s Solvers (Fortran)

Next I want to bring up some Fortran solvers because they will come up later. Hairer’s Fortran solvers are a set methods with similar interfaces that were designed with efficiency in mind. Many of these methods are classics: dopri5, dop853, radau, and rodas will specifically show up in many of the suites which are discussed later. These methods are not too flexible: they don’t allow event handling (though with enough gusto you can use the dense output to write your own), or numbers that aren’t double-precision floating point numbers (it’s Fortran). They have a good set of options for tweaking parameters to make the adaptive timestepping more efficient, though you may have to read a few textbooks to know exactly what they do. And that’s the key to them: they will solve an ODE, stiff or non-stiff, and they will do so pretty efficiently, but nothing more. But even then, they show some age which don’t make them “perfectly efficient”. These solvers include their own linear algebra routines which don’t multithread like standard BLAS and LINPACK implementations, meaning that they won’t make full use of modern CPU architectures. The computations don’t necessarily SIMD or use FMA. But most of all, to use it directly you need to use Fortran which would be turn off for many people.

There is some flexibility in the offering. There are symplectic solvers for second order ODEs, the stiff solvers allow for solving DAEs in mass matrix form, there’s a constant-lag nonstiff delay differential equation solver (RETARD), there is a fantastic generalization of radau to stiff state-dependent delay differential equations (RADAR5), and there’s some solvers specifically for some “mechanical ODEs” commonly found in physical problems. Of course, to get this all working you’ll need to be pretty familiar with Fortran, but this is a good suite to look at if you are.

ODEPACK and Netlib ODE Solvers (Fortran)

ODEPACK is an old set of ODE solvers which accumulated over time. You can learn a bit about its history by reading this interview with Alan Hindenmarsh. I also bundle the Netlib suite together with it. There’s a wide variety of methods in there. There’s old Runge-Kutta methods due to Shampine, some of Shampine’s old multistep methods ddebdf and ddeabm, etc. The reason why this pack is really important though is because of the Lawarance Livermore set of methods, specifically LSODE and its related methods (LSODA, LSODR, VODE, etc.). It includes methods for implicit ODEs (DAEs) as well. These are a group of multistep methods which are descendent from GEAR, the original BDF code. They are pretty low level and thus allow you to control the solver step by step, and some of them have “rootfinding capabilities”. This means that you can use these to add event handling, though an event handling interface will take some coding. There’s lots of options and these are generally pretty performant for a large array of problems, but they do show their age in the same way that the Hairer codes do. Their linear algebra setups do not make use of modern BLAS and LINPACK, which in practical terms means they don’t make full use of modern computer architectures to speed things up. They don’t always make use of SIMD and other modern CPU acceleration features. They are the vanilla ODE solvers which have existed through time. In particular, LSODA is popular because this is the most widely distributed method which automatically detects stiffness and swtiches between integration methods, though it should be pointed out that there is a performance penalty from this feature.

Sundials and ARKCODE (C++ and Fortran)

Sundials‘ CVODE is a re-write of VODE to C which is a descendent of LSODE which is a descendent itself of the original GEAR multistep code. Yes, this has a long history. But you should think of it as “LSODE upgraded”: it makes use of modern BLAS/LINPACK, and also a bunch of other efficient C/Fortran linear solvers, to give a very efficient Adams and BDF method. Its solver IDA is like CVODE but handles implicit ODEs (DAEs). The interface for these is very similar to the ODEPACK interface, which means you can control it step by step and use the rootfinding capabilities to write your own event handling interface. Since the Adams methods handle nonstiff ODEs and the BDF methods handle stiff ODEs, this performance plus flexibility makes it the “one-stop-shop” for ODE solving. Many different scientific computing software internally are making use of Sundials because it can handle just about anything. Well, it does have many limitations. For one, this only works with standard C++ numbers and arrays. There’s no stochasticity or delays allowed. But more importantly, Hairer and DifferentialEquations.jl’s show that these multistep methods are usually not the most efficient methods since high order Rosenbrock, (E)SDIRK, and fully implicit RK methods are usually faster for the same accuracy for stiff problems, and high order Runge-Kutta are faster than the Adams methods for the same accuracy. Multistep methods are also not very stable at their higher orders, and so at higher tolerances (lower accuracy) these methods may fail to have their steps converge on standard test problems (see this note in the ROBER tests), meaning that you may have to increase the accuracy (and thus computational cost) due to stiffness issues. But, since multistep methods only require a single function evaluation per step (or are implicit in only single steps), they are the most efficient for asymptotically hard problems (i.e. when the derivative calculation is very expensive or the ODE is a large 10,000+ system). For this reason, these methods excel at solving large discretizations of PDEs. To top it off, there are parallel (MPI) versions for using CVODE/IDA for HPC applications.

I also want to note that recently Sundials added ARKCODE, a set of Runge-Kutta methods. These include explicit Runge-Kutta methods and SDIRK methods, including additive Runge-Kutta methods for IMEX methods (i.e. you can split out a portion to solve explicitly so that the implicit portion is cheaper when you know your problem is only partly or semi stiff). This means that it covers the methods which I mentioned earlier are more efficient in many of the cases (though it is a bit lacking on the explicit tableaus and thus could be more efficient, but that’s just details).

If you are using C++ or Fortran and want to write to only one interface, the Sundials suite is a great Swiss Army knife. And if you have an asymtopically large problem or very expensive function evaluations, this will be the most efficient as well. Plus the parallel versions are cool! You do have to live with the limitations of low-level software forcing specific number and array types, along with the fact that you need to write your own event handling, but if you’re “hardcore” and writing in a compiled language this suite is a good bet.

SciPy’s Solvers (Python)

Now we come to SciPy’s suite. If you look at what it offers, the names should now start to sound familiar: LSODA, VODE, dopri5, dop853. That is write: SciPy is simply a wrapper over Hairer’s and ODEPACK’s methods. Because it writes a generic interface though, it doesn’t have the granularity that is offered by ODEPACK, meaning that you don’t have step-by-step control and no event handling. The tweaking options are very basic as well. Basically, they let you define a function in Python, say what timeframe to solve it on, give a few tolerances, and it calls Fortran code and spits back a solution. It only has ODE solvers, no differential-algebraic, delay, or stochastic solvers. It only allows the basic number types and does no event handling. Super vanilla, but gets the job done? As with the methods analysis, it has the high order Runge-Kutta methods for efficient solving of nonstiff equations, but it’s missing Rosenbrock and SDIRK methods entirely, opting to only provide the multistep methods.

I should note here that it has the same limitation as MATLAB though, namely that the user’s function is Python code. Since the derivative function is where the ODE solver spends most of its time (for sufficiently difficult problems), this means that even though you are calling Fortran code, you will lose out on a lot of efficiency. Still, if efficiency isn’t a big deal and you don’t need bells and whistles, this suite will do the basics.

deSolve Package (R)

There’s not much to say other than that deSolve is very much like SciPy’s suite. It wraps almost the same solvers, has pretty much the same limitations, and has the same efficiency problem since in this case it’s calling the user-provided R function most of the time. One advantage is that it does have event handling. Less vanilla with a little bit more features, but generally the same as SciPy.

PyDSTool (Python)

PyDSTool is an odd little beast. Part of the software is for analytic continuation (i.e. bifurcation plotting). But another part of it is for ODE solvers. It contains one ODE solver which is written in Python itself and it recommends against actually using this for efficiency reasons. Instead, it wraps some of the Hairer methods, specifically dopri5 and radau, and recommends these. But it’s different than SciPy in that it takes in the specification of the ODE as a string, and compiles it to a C function, and uses this inside the ODE solver. By doing so, it’s much more efficient. We still note that its array of available methods is small and it offers radau which is great for high accuracy ODEs and DAEs, but is not the most efficient at lower accuracy so it would’ve been nice to see Rosenbrock and ESDIRK methods. It has some basic event handling and methods for DDEs (again as wrappers to a Hairer method). This is a pretty good suite if you can get it working, though I do caution that getting the extra (non-Python) methods setup and compiled is nontrivial. One big point to note is that I find the documentation spectacularly difficult to parse. Together, it’s pretty efficient and has a good set of methods which will do basic event handling and solve problems at double precision.

JiTCODE and JiTCSDE (Python)

JiTCODE is another Python library which makes things efficient by compiling the function that the user provides. It uses the SciPy integrators and does something similar to PyDSTool in order to get efficiency. I haven’t tried it out myself but I’ll assume this will get you as efficient as though you used it from Fortran. However, it is lacking in the features department, not offering advanced things like arbitrary number types, event handling, etc. But if you have a vanilla ODE to solve and you want to easily do it efficiently in Python, this is a good option to look at.

Additionally, JiTCDDE is a version for constant-lag DDEs similar to dde23. JiTCSDE is a version for stochastic differential equations. It uses the high order (strong order 1.5) adaptive Runge-Kutta method for diagonal noise SDEs developed by Rackauckas (that’s me) and Nie which has been demonstrated as much more efficient than the low order and fixed timestep methods found in the other offerings. It employs the same compilation setup as JitCODE so it should create efficient code as well. I haven’t used this myself but it would probably be a very efficient ODE/DDE/SDE solver if you want to use Python and don’t need events and other sugar.

Boost ODEINT Solver Library (C++)

The Boost ODEINT solver library has some efficient implementations of some basic explicit Runge-Kutta methods (including high order RK methods) and some basic Rosenbrock methods (including a high order one). Thus it can be pretty efficient for solving the most standard stiff and nonstiff ODEs. However, its implementations do not make use of advanced timestepping techniques (PI-controllers and Gustofsson accleration) which makes it require more steps to achieve the same accuracy as some of the more advanced software, making it not really any more efficient in practice. It doesn’t have event handling, but it is flexible with the number and array types you can put in there via C++ templates. It and DifferentialEquations.jl are the only two suites that are mentioned that allow for solving the differential equations on the GPU. Thus if you are familiar with templates and really want to make use of them, this might be the library to look at, otherwise you’re probably better off looking elsewhere like Sundials.

GSL ODE Solvers (C)

To say it lightly, the GSL ODE solvers are kind of a tragedy. Actually, they are just kind of weird. When comparing their choices against what is known to be efficient according to the ODE research and benchmarks, the methods that they choose to implement are pretty bizarre like extrapolation methods which have repeatedly been shown to not be very efficient, while not included other more important methods. But they do have some of the basics like multistep Adams and BDF methods. This, like Boost, doesn’t do all of the fancy PI-controlled adaptivity and everything though, so YMMV. This benchmark, while not measuring runtime and only uses function evaluations (which can be very very different according to more
sophisticated benchmarks like the Hairer and DifferentialEquations.jl ones!), clearly shows that the GSL solvers can take way too many function evaluations because of this and thus, since it’s using methods similar to LSODA/LSODE/dopri5, probably have much higher runtimes than they should.

Mathematica’s ODE Solvers

Mathematica’s ODE solvers are very sophisticated. It has a lot of explicit Runge-Kutta methods, including newer high order efficient methods due to Verner and Shampine’s efficient method mentioned in the MATLAB section. These methods can be almost 5x faster than the older high order explicit RK methods which themselves are the most efficient class of methods for many nonstiff ODEs, and thus these do quite well. It
includes interpolations to make their solutions continuous functions that plot nicely. Its native methods are able to make full use of Mathematica and its arbitrary precision, but sadly most of the methods it uses are wrappers for the classic solvers. Its stiff solvers mainly call into Sundials or LSODA. By using LSODA, it tends to be able to do automatic stiffness detection by default. It also wraps Sundials’ IDA for DAEs. It uses a method of steps implementation over its explicit Runge-Kutta methods for solving nonstiff DDEs efficiently, and includes high order Runge-Kutta methods for stochastic differential equations (though it doesn’t do adaptive timestepping in this case). One nice feature is that all solutions come with an interpolation to make them continuous. Additionally, it can use the symbolic form of the user-provided equation in order to create a function for the Jacobian, and use that (instead of a numerical differentiation fallback like all of the previous methods) in the stiff solvers for more accuracy and efficiency. It also has symplectic methods for solving second order ODEs, and its event handling is very expressive. It’s very impressive, but since it’s using a lot of wrapped methods you cannot always make use of Mathematica’s arbitrary precision inside of these numerical methods. Additionally, its interface doesn’t give you control over all of the fine details of the solvers that it wraps.

Maple’s ODE Solvers

Maple’s ODE solvers are pretty basic. It defaults to a 6th order explicit RK method due to Verner (not the newer more efficient ones though), and for stiff problems it uses a high order Rosenbrock method. It also wraps LSODE like everything else. It has some basic event handling, but not much more. As another symbolic programming language, it computes Jacobians analytically to pass to the stiff solvers like Mathematica, helping it out in that respect, but its offerings pale in comparison to Mathematica.

FATODE (Fortran)

FATODE
is a set of methods written in Fortran. It includes explicit Runge-Kutta methods, SDIRK methods, Rosenbrock methods and fully implicit RK methods. Thus it has something that’s pretty efficient for pretty much every case. What makes it special is that it includes the ability to do sensitivity analysis calcuations. It can’t do anything but double-precision numbers and doesn’t have event handling, but the sensitivity calculations makes it quite special. If you’re a FORTRAN programmer, this is worth a look, especially if you want to do sensitivity analysis.

DifferentialEquations.jl (Julia)

Okay, time for DifferentialEquations.jl. I left it for last because it is by far the most complex of the solver suites, and pulls ideas from many of them. While most of the other suite offer no more than about 15 methods on the high end (with most offering about 8 or less), DifferentialEquations.jl offers 200+ methods and is continually growing. Like the standard Python and R suites, it offers wrappers to Sundials, ODEPACK, and Hairer methods. However, since Julia code is always JIT compiled, its wrappers are more akin to PyDSTool or JiTCODE in terms of efficiency. Thus all of the standard methods mentioned before are available in this suite.

But then there are the native Julia methods. For ODEs, these include explicit Runge-Kutta methods, (E)SDIRK methods, and Rosenbrock methods. In each of these categories it has a huge amount of variety, offering pretty much every method from the other suites along with some unique methods. Some unique methods to point out are that it has the only 5th order Rosenbrock method, it has the efficient Verner methods discussed in the Mathematica part, it has newer 5th order methods (i.e. it includes the Bogacki-Shampine method discussed as an alternative to ode45’s tableau, along with an even newer tableau due to Tsitorious which is even more efficient). It has methods specialized to reduce interpolation error (the OwrenZen methods), and methods which are strong-stability presurving (for hyperbolic PDEs). It by default the solutions as continuous functions via a high order interpolation (though this can be turned off to make the saving more efficient). Each of these implementations employ a lot of extra tricks for efficiency. For example, the interpolation is “lazy”, meaning that if the method requires extra function evaluations for the continuous form, it will only do those extra calculations when the continuous function is used (so when you directly ask for it or when you plot). This is just a peak at the special things the library does to gain an edge.

The native Julia methods benchmark very well as well, and all of the benchmarks are openly available. Essentially, these methods make use of the native multithreading of modern BLAS/LINPACK, FMA, SIMD, and all of the extra little compiler goodies that allows code to be efficient, along with newer solver methods which theoretically reduce the amount of work that’s required to get the same error. They even allow you to tweak a lot of the internals and swap out the linear algebra routines to use parallel solvers like PETSc. The result is that these methods usually outperform the classic C/Fortran methods which are wrapped. Additionally, it has ways to symbolically calculate Jacobians like Mathematica/Maple, and instead of defaulting to numerical differentiation the stiff solvers fall back to automatic differentiation which is more efficient and has much increased accuracy.

Its event handling is the most advanced out of all of the offerings. You can change just about anything. You can make your ODE do things like change its size during the solving if you want, and you can make the event handling adjust and adapt internal solver parameters. It’s not a hyperbole to say that the user is given “full control” since the differential equation solvers themselves are written as essentially a method on the event handling interface, meaning that anything it can do internally you can do.

The variety of methods is also much more diverse than the other offerings. It has symplectic integrators like Harier’s suite, but has more high and low order methods. It has a range of Runge-Kutta Nystrom methods for efficiently solving second order ODEs. It has the same high order adaptive method for diagonal noise SDEs as JiTCSDE, but also includes high order adaptive methods specifically for additive noise SDEs. It also has methods for stiff SDEs in Ito and Stratanovich interpretations, and allows for event handling in the SDE case (with the full flexibility). It has DDE solvers for constant-lag and state-dependent delays, and it has stiff solvers for each of these cases. The stiff solvers also all allow for solving DAEs in mass matrix form (though fully implicit ODEs are possible, but can only be solved using a few methods like a wrapper to Sundials’ IDA and doesn’t include event handling here quite yet).

It allows arbitrary numbers and arrays like Boost. This means you can use arbitrary precision numbers in the native Julia methods, or you can use “array-like” objects to model multiscale biological organisms instead of always having to use simple contiguous arrays. It has addons for things like sensitivity analysis and parameter estimation. Also like Boost, it can solve equations on the
GPU by using a GPUArray.

It hits the strong points of each of the previously mentioned suites because it was designed from the get-go to do so. And it benchmarks extremely well. The only downside is that, because it is so feature and performance focused, its documentation is heavy. The beginning tutorial will give you the basics (making it hopefully as easy as MATLAB to pick up), but pretty much every control knob is available, making the extended portion of the documentation a long read.

Conclusion

Let’s take a step back and summarize this information. DifferentialEquations.jl objectively has the largest feature-set, swamping most of the others while wrapping all of the common solvers. Since it also features solvers for the non-ordinary differential equations and its unique Julia methods also benchmarks well, I think it’s clear that DifferentialEquations.jl is by far the best choice for “power-users” who are looking for a comprehensive suite.

As for the other choices from scripting languages, MATLAB wasn’t designed to have all of the most efficient methods, but it’ll handle basic equations with delays and events and output good plots. R’s deSolve is similar in most respects to MATLAB. SciPy’s offering is lacking in comparison to MATLAB and R’s due to the lack of event handling. But MATLAB/Python/R all have efficiency problems due to the fact that the user’s function is written in the scripting language. JiTCODE and PyDSTool are two Python offerings make the interface to the Fortran solvers more efficient than straight SciPy. Mathematica and Maple will do symbolic pre-calculations to speed things up and can JiT compile functions, along with offering pretty good event handling, and thus their wrappers are more like DifferentialEquations.jl in terms of flexibility and efficiency (and Mathematica had a few non-wrapper goodies mentioned as well). So in a pinch when not all of the bells and whistles are necessary, each of these scripting language suites will get you by. Behind DifferentialEquations.jl, I would definitely put Mathematica’s suite second for scripting languages with everything else much further behind.

If you’re already hardcore and writing in C++/Fortran, Sundials is a pretty good “one-stop-shop” to get everything you need, especially when you add in ARKCODE. Still, you’re going to have to write a lot of stuff yourself to make the rootfinding into an event handling interface, but if you put the work in
this will do you well. Hairer’s codes are a great set of classics which cover a wide variety of equations, and FATODE is the only non-DifferentialEquations.jl suite which offers a way to calculate sensitivity equations (and its sensitivity equations are more advanced). Any of these will do you well if you want to really get down-and-dirty in a compiled language and write a lot of the interfaces yourself, but they will be a sacrifice in productivity with no clear performance gain over the scripting language methods which also include some form of JIT compilation. With these in mind, I don’t really see a purpose for the GSL or Boost suites, and the ODEPACK methods are generally outdated.

I hope this review helps you make a choice which is right for you.

The post A Comparison Between Differential Equation Solver Suites In MATLAB, R, Julia, Python, C, Mathematica, Maple, and Fortran appeared first on Stochastic Lifestyle.

Video Introduction to DifferentialEquations.jl

By: Christopher Rackauckas

Re-posted from: http://www.stochasticlifestyle.com/video-introduction-differentialequations-jl/

Videos can be much easier to follow than text (though they usually have fewer details!). So, here’s a video introduction to DifferentialEquations.jl from JuliaCon. In this talk I walk through the major features of DifferentialEquations.jl by walking through the the tutorials in the documentation, highlighting usage details and explaining how to properly think about the code. I hope this helps make it easier to adopt DifferentialEquations.jl!

The post Video Introduction to DifferentialEquations.jl appeared first on Stochastic Lifestyle.

Type-Dispatch Design: Post Object-Oriented Programming for Julia

By: Christopher Rackauckas

Re-posted from: http://www.stochasticlifestyle.com/type-dispatch-design-post-object-oriented-programming-julia/

In this post I am going to try to explain in detail the type-dispatch design which is used in Julian software architectures. It’s modeled after the design of many different packages and Julia Base, and has been discussed in parts elsewhere. This is actually just a blog post translation from my “A Deep Introduction to Julia for Data Science and Scientific Computing” workshop notes. I think it’s an important enough topic to share more broadly.

The tl;dr: Julia is built around types. Software architectures in Julia are built around good use of the type system. This makes it easy to build generic code which works over a large range of types and gets good performance. The result is high-performance code that has many features. In fact, with generic typing, your code may have more features than you know of! The purpose of this tutorial is to introduce the multiple dispatch designs that allow this to happen.

Now let’s discuss the main components of this design!

Duck Typing

If it quacks like a duck, it might as well be a duck. This is the idea of defining an object by the way that it acts. This idea is central to type-based designs: abstract types are defined by how they act. For example, a `Number` is some type that can do things like +,-,*, and /. In this category we have things like Float64 and Int32. An AbstractFloat is some floating point number, and so it should have a dispatch of eps(T) that gives its machine epsilon. An AbstractArray is a type that can be indexed like `A[i]`. An AbstractArray may be mutable, meaning it can be “set”: A[i]=v.

These abstract types then have actions which abstract from their underlying implmentation. A.*B does element-wise multiplication, and in many cases it does not matter what kind of array this is done on. The default is Array which is a contiguous array on the CPU, but this action is common amongst AbstractArray types. If a user has a DistributedArray (DArray), then A.*B will work on multiple nodes of a cluster. If the user uses a `GPUArray`, then A.*B will be performed on the GPU. Thus, if you don’t restrict the usage of your algorithm to Array, then your algorithm actually “just works” as many different algorithms.

This is all well and good, but this would not be worthwhile if it were not performant. Thankfully, Julia has an answer to this. Every function auto-specializes on the types which it is given. Thus if you look at something like:

my_square(x) = x^2

then we see that this function will be efficient for the types that we give it. Looking at the generated code:

@code_llvm my_square(1)
define i64 @julia_my_square_72669(i64) #0 {
top:
  %1 = mul i64 %0, %0
  ret i64 %1
}
@code_llvm my_square(1.0)
define double @julia_my_square_72684(double) #0 {
top:
  %1 = fmul double %0, %0
  ret double %1
}

See that the function which is generated by the compiler is different in each case. The first specifically is an integer multiplication x*x of the input x. The other is a floating point multiplication x*x of the input x. But this means that it does not matter what kind of Number we put in here: this function will work as long as * is defined, and it will be efficient by Julia’s multiple dispatch design.

Thus we don’t need to restrict the types we allow in functions in order to get performance. That means that

my_restricted_square(x::Int) = x^2

is no more efficient than the version above, and actually generates the same exact compiled code:

@code_llvm my_restricted_square(1)
 
define i64 @julia_my_restricted_square_72686(i64) #0 {
top:
  %1 = mul i64 %0, %0
  ret i64 %1
}

Thus we can write generic and efficient code by leaving our functions unrestricted. This is the practice of duck-typing functions. We just let them work on any input types. If the type has the correct actions, the function will “just work”. If it does not have the correct actions, for our example above say * is undefined, then a MethodError saying the action is not defined will be thrown.

We can be slightly more conservative by restricting to abstract types. For example:

my_number_restricted_square(x::Number) = x^2

will allow any Number. There are things which can square which aren’t Numbers for which this will now throw an error (a matrix is a simple example). But, this can let us clearly define the interface for our package/script/code. Using these assertions, we can then dispatch differently for different type classes. For example:

my_number_restricted_square(x::AbstractArray) = (println(x);x.^2)

Now, my_number_restricted_square calculates x^2 on a Number, and for an array it will print the array and calculate x^2 element-wise. Thus we are controlling behavior with broad strokes using classes of types and their associated actions.

Type Hierarchies

This idea of control leads to type hierarchies. In object-oriented programming languages, you sort objects by their implementation. Fields, the pieces of data that an object holds, are what is inherited.

There is an inherent limitation to that kind of thinking when looking to achieve good performance. In many cases, you don’t need as much data to do an action. A good example of this is the range type, for example 1:10.

a = 1:10

This type is an abstract array:

typeof(a) <: AbstractArray
#true

It has actions like an Array













fieldnames(a)
 
# Output
2-element Array{Symbol,1}:
 :start
 :stop

It is an immutable type which just holds the start and stop values. This means that its indexing, A[i], is just a function. What’s nice about this is that means that no array is ever created. Creating large arrays can be a costly action:

@time collect(1:10000000)
0.038615 seconds (308 allocations: 76.312 MB, 45.16% gc time)

But creating an immutable type of two numbers is essentially free, no matter what those two numbers are:

@time 1:10000000
0.000001 seconds (5 allocations: 192 bytes)

The array takes \mathcal{O}(n) memory to store its values while this type is \mathcal{O}(1), using a constant 192 bytes (if the start and stop are Int64). Yet, in cases where we just want to index values, they act exactly the same.

Another nice example is the UniformScaling operator, which acts like an identity matrix without forming an identity matrix.

println(I[10,10]) # prints 1
println(I[10,2]) # prints 0

This can calculate expressions like A-b*I without ever forming the matrix (eye(n)) which would take \mathcal{O}(n^2) memory.

This means that a lot of efficiency can be gained by generalizing our algorithms to allow for generic typing and organization around actions. This means that, while in an object-oriented programming language you group by implementation details, in typed-dispatch programming you group by actions. Number is an abstract type for “things which act like numbers, i.e. do things like *”, while AbstractArray is for “things which index and sometimes set”.

This is the key idea to keep in mind when building type hierarchies: things which subtype are inheriting behavior. You should setup your abstract types to mean the existence or non-existence of some behavior. For example:

abstract AbstractPerson
abstract AbstractStudent <: AbstractPerson
abstract AbstractTeacher <: AbstractPerson
 
type Person <: AbstractPerson
  name::String    
end
 
type Student <: AbstractStudent
  name::String  
  grade::Int
  hobby::String
end
 
type MusicStudent <: AbstractStudent
  grade::Int
end
 
type Teacher <: AbstractTeacher
  name::String
  grade::Int
end

This can be interpreted as follows. At the top we have AbstractPerson. Our interface here is “a Person is someone who has a name which can be gotten by get_name”.

get_name(x::AbstractPerson) = x.name

Thus codes which are written for an AbstractPerson can “know” (by our informal declaration of the interface) that get_name will “just work” for its subtypes. However, notice that MusicStudent doesn’t have a name field. This is because MusicStudents just want to be named whatever the trendiest band is, so we can just replace the usage of the field by the action:

get_name(x::MusicStudent) = "Justin Bieber"

In this way, we can use get_name to get the name, and how it was implemented (whether it’s pulling something that had to be stored from memory, or if it’s something magically known in advance) does not matter. We can keep refining this: an AbstractStudent has a get_hobby, but a MusicStudent’s hobby is always Music, so there’s not reason to store that data in the type and instead just have its actions implicitly “know” this. In non-trivial examples (like the range and UniformScaling above), this distinction by action and abstraction away from the actual implementation of the types allows for full optimization of generic codes.

Small Functions and Constant Propagation

The next question to ask is, does storing information in functions and actions affect performance? The answer is yes, and in favor of the function approach! To see this, let’s see what happens when we use these functions. To make it simpler, let’s use a boolean function. Teachers are old and don’t like music, while students do like music. But generally people like music. This means that:

likes_music(x::AbstractTeacher) = false
likes_music(x::AbstractStudent) = true
likes_music(x::AbstractPerson) = true

Now how many records would these people buy at a record store? If they don’t like music, they will buy zero records. If they like music, then they will pick up a random number between 1 and 10. If they are a student, they will then double that (impulsive Millennials!).

function number_of_records(x::AbstractPerson)
    if !likes_music(x) 
      return 0
    end
    num_records = rand(10)
    if typeof(x) <: AbstractStudent
      return 2num_records
    else 
      return num_records
    end
end

Let’s check the code that is created:

x = Teacher("Randy",11)
println(number_of_records(x))
@code_llvm number_of_records(x)

on v0.6, we get:

on v0.6, we get:
; Function Attrs: uwtable
define i64 @julia_number_of_records_63848(i8** dereferenceable(16)) #0 !dbg !5 {
top:
  ret i64 0
}

Notice that the entire function compiled away, and the resulting compiled code is “return 0”! Then for a music student:

x = MusicStudent(10)
@code_typed number_of_records(x)
 
# Output
CodeInfo(:(begin 
        NewvarNode(:(num_records))
        goto 4 # line 30:
        4:  # line 32:
        @@0@@(Expr(:foreigncall, :(:jl_alloc_array_1d), Array{Float64,1}, svec(Any, Int64), Array{Float64,1}, 0, 10, 0))
        # meta: pop location
        # meta: pop location
        # meta: pop location
        # meta: pop location
        @@1@@(Expr(:invoke, MethodInstance for rand!(::MersenneTwister, ::Array{Float64,1}, ::Int64, ::Type{Base.Random.CloseOpen}), :(Base.Random.rand!), :(Base.Random.GLOBAL_RNG), SSAValue(1), :((Base.arraylen)(SSAValue(1))::Int64), :(Base.Random.CloseOpen))) # line 33:
        (MusicStudent <: Main.AbstractStudent)::Bool # line 34:
        return $(Expr(:invoke, MethodInstance for *(::Int64, ::Array{Float64,1}), :(Main.*), 2, :(num_records))) # line 36:
    end))=>Array{Float64,1}

we get a multiplication by 2, while for a regular person,

x = Person("Miguel")
@code_typed number_of_records(x)
 
# Output
CodeInfo(:(begin 
        NewvarNode(:(num_records))
        goto 4 # line 30:
        4:  # line 32:
        @@2@@(Expr(:foreigncall, :(:jl_alloc_array_1d), Array{Float64,1}, svec(Any, Int64), Array{Float64,1}, 0, 10, 0))
        # meta: pop location
        # meta: pop location
        # meta: pop location
        # meta: pop location
        @@3@@(Expr(:invoke, MethodInstance for rand!(::MersenneTwister, ::Array{Float64,1}, ::Int64, ::Type{Base.Random.CloseOpen}), :(Base.Random.rand!), :(Base.Random.GLOBAL_RNG), SSAValue(1), :((Base.arraylen)(SSAValue(1))::Int64), :(Base.Random.CloseOpen))) # line 33:
        (Person <: Main.AbstractStudent)::Bool
        goto 22 # line 34:
        22:  # line 36:
        return num_records
    end))=>Array{Float64,1}

we do not get a multiplication by 2. This is all in the compiled-code, so this means that in one case the *2 simply doesn’t exist at runtime, not even a check for whether to do it.

The key thing to see from the typed code is that the “branches” (the if statements) all compiled away. Since types are known at compile time (remember, functions specialize on types), the dispatch of likes_music is known at compile-time. But this means, since the result is directly inferred from the dispatch, the boolean value true/false is known at compile time. This means that the compiler can directly infer the answer to all of these checks, and will use this information to skip them at runtime.

This is the distinction between compile-time information and runtime information. At compile-time, what is known is:

  1. The types of the inputs
  2. Any types which can be inferred from the input types (via type-stability)
  3. The function dispatches that will be internally called (from types which have been inferred)

Note that what cannot be inferred by the compiler is the information in fields. Information in fields is strictly runtime information. This is easy to see since there is no way for the compiler to know that person’s name was “Miguel”: that’s ephemeral and part of the type instance we just created.

Thus by putting our information into our functions and dispatches, we are actually giving the compiler more information to perform more optimizations. Therefore using this “action-based design”, we are actually giving the compiler leeway to perform many extra optimizations on our code as long as we define our interfaces by the actions that are used. Of course, at the “very bottom” our algorithms have to use the fields of the types, but the full interface can then be built up using a simple set of functions which in many cases with replace runtime data with constants.

Traits and THTT

What we just saw is a “trait”. Traits are compile-time designations about types which are distinct from their abstract hierarchy. likes_music is a trait which designates which people like music, and it could in cases not be defined using the abstract types. For example, we can, using dispatch, create a WeirdStudent which does not like music, and that will still be compile-time information which is fully optimized. This means that these small functions which have constant return values allow for compile-time inheritance of behavior, and these traits don’t have to be tied to abstract types (all of our examples were on AbstractPerson, but we could’ve said a GPUArray likes music if we felt like it!). Traits are multiple-inheritance for type systems.

Traits can be more refined than just true/false. This can be done by having the return be a type itself. For example, we can create music genre types:

abstract MusicGenres
abstract RockGenre <: MusicGenres
immutable ClassicRock <: RockGenre end
immutable AltRock <: RockGenre end
immutable Classical <: MusicGenres end

These “simple types” are known as singleton types. This means that we can have traits like:

favorite_genre(x::AbstractPerson) = ClassicRock()
favorite_genre(x::MusicStudent) = Classical()
favorite_genre(x::AbstractTeacher) = AltRock()

This gives us all of the tools we need to compile the most efficient code, and structure our code around types/actions/dispatch to get high performance out. The last thing we need is syntactic sugar. Since traits are compile-time information, the compiler could in theory dispatch on them. While this is currently not part of Julia, it’s scheduled to be part of a future version of Julia (2.0?). The design for this (since Julia is written in Julia!) is known as the Tim Holy Trait Trick (THTT), named after its inventor. It’s described in detail on this page. But in the end, macros can make this easier. A package which implements trait-dispatch is SimpleTraits.jl, which allows you to dispatch on a trait IsNice like:

@traitfn ft(x::::IsNice) = "Very nice!"
@traitfn ft(x::::(!IsNice)) = "Not so nice!"

Composition vs Inheritance

The last remark that is needed is a discussion of composition vs inheritance. While the previous discussions have all explained why “information not in fields” makes structural relations compile-time information and increases the efficiency. However, there are cases where we want to share runtime structure. Thus the great debate of composition vs inheritance, comes up.

Composition vs inheritance isn’t a Julia issue, it’s a long debate in object-oriented programming. The idea is that, inheritance is inherently (pun-inteded) inflexible. It forces an “is a” relation: A inherits from B means A is a B, and adds a few things. It copies behavior from something defined elsewere. This is a recipe for havoc. Here’s a few links which discuss this in more detail:

https://softwareengineering.stackexchange.com/questions/134097/why-should-i-prefer-composition-over-inheritance

https://en.wikipedia.org/wiki/Composition_over_inheritance

https://www.thoughtworks.com/insights/blog/composition-vs-inheritance-how-choose

So if possible, give composition a try. Say you have MyType, and it has some function f defined on it. This means that you can extend MyType by making it a field in another type:

type MyType2
    mt::MyType
    ... # Other stuff
end 
 
f(mt2::MyType2) = f(mt2.mt)

The pro here is that it’s explicit: you’ve made the choice for each extension. The con is that this can require some extra code, though this can be automated by metaprogramming.

What if you really really really want inheritance of fields? There are solutions via metaprogramming. One simple solution is the @def macro.

  macro def(name, definition)
      return quote
          macro $name()
              esc($(Expr(:quote, definition)))
          end
      end
  end

This macro is very simple. What it does is compile-time copy/paste. For example:

@def give_it_a_name begin
  a = 2
  println(a)
end

defines a macro @give_it_a_name that will paste in those two lines of code wherever it is used. As another example, the reused fields of Optim.jl’s solvers could be put into an @def:

@def add_generic_fields begin
        method_string::String
        n::Int64
        x::Array{T}
        f_x::T
        f_calls::Int64
        g_calls::Int64
        h_calls::Int64
end

and those fields can be copied around with

type LBFGSState{T}
    @add_generic_fields
    x_previous::Array{T}
    g::Array{T}
    g_previous::Array{T}
    rho::Array{T}
    # ... more fields ... 
end

Because @def works at compile-time, there is no cost associated with this. Similar metaprogramming can be used to build an “inheritance feature” for Julia. One package which does this is ConcreteAbstractions.jl which allows you to add fields to abstract types and make the child types inherit the fields:

# The abstract type
@base type AbstractFoo{T}
    a
    b::Int
    c::T
    d::Vector{T}
end
 
# Inheritance
@extend type Foo <: AbstractFoo
    e::T
end

where the @extend macro generates the type-definition:

type Foo{T} <: AbstractFoo
    a
    b::Int
    c::T
    d::Vector{T}
    e::T
end

But it’s just a package? Well, that’s the beauty of Julia. Most of Julia is written in Julia, and Julia code is first class and performant (here, this is all at compile-time, so again runtime is not affected at all). Honestly, if something ever gets added to Julia’s Base library for this, it will likely look very similar, and the only real difference to the user will be that the compiler will directly recognize the keywords, meaning you would use base and extend instead of @base and @extend. So if you have something that really really really needs inheritance, go for it: there’s no downsides to using a package + macro for this. But you should really try other means to reduce the runtime information and build a more performant and more Julian architecture first.

Conclusion

Programming for type systems has a different architecture than object-oriented systems. Instead of being oriented around the objects and their fields, type-dispatch systems are oriented around the actions of types. Using traits, multiple inheritance behavior can be given. Using this structure, the compiler can have maximal information, and use this to optimize the code. But also, this directly generalizes the vast majority of the code to not be “implementation-dependent”, allowing for duck-typed code to be fully performant, with all of the details handled by dispatch/traits/abstract types. The end result is flexible, generic, and high performance code.

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