CyRK

Documentation | GitHub

Runge-Kutta ODE Integrator Implemented in Cython and Numba

CyRK provides fast integration tools to solve systems of ODEs using an adaptive time stepping scheme. CyRK can accept differential equations that are written in pure Python, njited numba, or cython-based cdef. Implementing these types of functions is generally easier than doing so in pure C. Calling CyRK's functions and utilizing its results is also much easier to integrate into existing Python or Cython software. Using CyRK can speed up development time while avoiding the slow performance that comes with using pure Python-based solvers like SciPy's solve_ivp.

The purpose of this package is to provide some functionality of scipy's solve_ivp with greatly improved performance.

Currently, CyRK's numba-based (njit-safe) implementation is 8--500x faster than scipy's solve_ivp function. The cython-based pysolve_ivp function that works with python (or njit'd) functions is 10-40x faster than scipy. The cython-based cysolver_ivp function that works with cython-based cdef functions is 100-500x faster than scipy.

Details about CyRK's performance including more benchmarks can be found here.

CyRK's solvers can also be parallelized for even larger gains.

An additional benefit of the two cython implementations is that they are pre-compiled. This avoids most of the start-up performance hit experienced by just-in-time compilers like numba.

Overview

Supported Features

CyRK's pysolve_ivp (which works with pure python functions) and cysolve_ivp (which works with cython compiled functions) shares many of the same features as SciPy's solve_ivp:

• Triggerable events to track certain events or cause a early termination.
• Adaptive step size solver that uses relative and absolute error of the ODE to increase or decrease the step size as needed.
• Dense output to create callable functions to interpolate an ODE's solution between solution steps. Allowing user to take advantage of the much more efficient adaptive step size solver.
• A user-provided time domain or t_eval can be given to the solver to find solutions at specific time steps.

Note: nbsolve_ivp has a much more limited feature set than cysolve_ivp and pysolve_ivp. The latter methods are recommended where possible.

Supported Integrators

Currently, CyRK supports the following integration methods:

• "RK23" - Explicit Runge-Kutta method of order 3(2).
• "RK45" - Explicit Runge-Kutta method of order 5(4)
• "DOP853" - Explicit Runge-Kutta method of order 8. Error is controlled using a combination of 5th and 3rd order interpolators.
• "Tsit5" - Explicit Runge-Kutta method of order 5(4) due to Tsitouras, with a free fourth-order interpolant.
• "Vern7" - Explicit Runge-Kutta method of order 7(6) due to Verner, with a seventh-order interpolant.
• "Vern8" - Explicit Runge-Kutta method of order 8(7) due to Verner, with an eighth-order interpolant.
• "BDF" - Implicit multi-step method based on backward differentiation formulas of order 1 to 5.
• "LSODA" - Adams / BDF method with automatic stiffness detection and switching.
• "Radau" - Implicit Runge-Kutta method of the Radau IIA family of order 5.

"BDF", "LSODA", and "Radau" are implicit ODE methods used for stiff problems. See the implicit methods documentation for guidance on choosing between them and for the options that only they support. The integration methods documentation compares all of the methods and suggests which to use for different kinds of problems.

More methods will be added as the need arises. We are always looking for contributors if you'd like to see your favorite method added to CyRK!

Additional Features

In additional to improved performance, CyRK offers a few additional features that SciPy does not:

• Ability to capture extra outputs during integration so a user can record other parameters instead of just the dependent variables without having to make repeat calls to the differential equations.
• Ability to reuse solvers when the previous result is no longer needed or has been recorded (improving performance, particularly for problems with small integration domain sizes).
• CyRK provides a robust integration diagnostic feedback system to identify and help remedy issues with an ODE solution.
• CyRK offers much more control over the memory management and other aspects of the solver.

Limitations

There are some features that SciPy has that CyRK currently does not. A non-exhaustive list is:

• An analytic Jacobian can only be provided to the implicit methods at the C++ / Cython level; pysolve_ivp always estimates it with finite differences.
• cysolve_ivp and pysolve_ivp can only work with ODEs of double-precision floating point numbers. So complex numbers are not directly supported but systems of ODEs of complex numbers can be converted to systems of doubles for use with CyRK.

Installation

CyRK has been tested on Python 3.9--3.14; Windows, Ubuntu, and MacOS.

Install via pip:

pip install CyRK

conda:

conda install -c conda-forge CyRK

mamba:

mamba install cyrk

If not installing from a wheel, CyRK will attempt to install Cython and Numpy in order to compile the source code. A "C++ 20" compatible compiler is required. Compiling CyRK has been tested on the latest versions of Windows, Ubuntu, and MacOS. Your milage may vary if you are using an older or different operating system. Most platform's default compiler will work to build CyRK: MSVC on Windows, GCC on Linux, and Apple's clang on MacOS (xcode-select --install). After everything has been compiled, cython will be uninstalled and CyRK's runtime dependencies (see the pyproject.toml file for the latest list) will be installed instead.

A new installation of CyRK can be tested quickly by running the following from a python console.

from CyRK import test_pysolver, test_cysolver, test_nbrk
test_pysolver()
# Should see "CyRK's PySolver was tested successfully."
test_cysolver()
# Should see "CyRK's CySolver was tested successfully."
test_nbrk()
# Should see "CyRK's nbsolve_ivp was tested successfully."

Installation Troubleshooting

Please report installation issues. We will work on a fix and/or add workaround information here.

• If you see a "Can not load module: CyRK.cy" or similar error then the cython extensions likely did not compile during installation. Try running pip install CyRK --no-binary="CyRK" to force python to recompile the cython extensions locally (rather than via a prebuilt wheel).

• On MacOS: CyRK v0.19.0 and later compile with Apple's default clang, so the Homebrew llvm and libomp steps that older versions needed are no longer required (remove any CC/CXX/LDFLAGS/CPPFLAGS exports left over from them if a different compiler is being picked up). If you are writing your own Cython module that uses prange over CyRK's solvers, see the Parallelization documentation for the OpenMP flags that module needs.

• CyRK has a number of runtime status codes which can be used to help determine what failed during integration. Learn more about these codes https://cyrk.readthedocs.io/en/latest/Status_and_Error_Codes.html.

Development and Testing Dependencies

If you intend to work on CyRK's code base you will want to install the following dependencies in order to run CyRK's test suite and experimental notebooks.

conda install pytest scipy matplotlib jupyter

conda install can be replaced with pip install if you prefer.

Using CyRK

To learn how to use CyRK, please reference our detailed documentation on Read the Docs!

Read the Docs: CyRK Documentation

Citing CyRK

It is great to see CyRK used in other software or in scientific studies. We ask that you reference CyRK's GitHub website so interested parties can learn about this package. And please cite the following paper in any formal publication of your work.

@software{renaud_2022_cyrk,
  author       = {Renaud, Joe P.},
  title        = {CyRK - ODE Integrator Implemented in Cython and Numba},
  year         = {2022},
  publisher    = {Zenodo},
  doi          = {10.5281/zenodo.7093266},
  url          = {[https://doi.org/10.5281/zenodo.7093266](https://doi.org/10.5281/zenodo.7093266)}
}

It would also be great to hear about the work being done with CyRK and add your project to the list below, so get in touch!

In addition to citing CyRK, please strongly consider citing SciPy and its references for the specific integration method that was used in your work. CyRK is largely an adaptation of SciPy's functionality. Find more details here.

If you used the "LSODA" method then please also cite ODEPACK, which CyRK bundles a modified copy of (see the Third-Party Code section of LICENSE.md):

@article{hindmarsh1983odepack,
  author  = {Hindmarsh, Alan C.},
  title   = {{ODEPACK, A Systematized Collection of ODE Solvers}},
  journal = {IMACS Transactions on Scientific Computation},
  year    = {1983},
  volume  = {1},
  pages   = {55--64}
}

@article{petzold1983automatic,
  author  = {Petzold, Linda},
  title   = {{Automatic Selection of Methods for Solving Stiff and Nonstiff Systems of Ordinary Differential Equations}},
  journal = {SIAM Journal on Scientific and Statistical Computing},
  year    = {1983},
  volume  = {4},
  number  = {1},
  pages   = {136--148}
}

@article{virtanen2020scipy,
  author  = {Virtanen, Pauli and Gommers, Ralf and Oliphant, Travis E. and Haberland, Matt and Reddy, Tyler and Cournapeau, David and Burovski, Evgeni and Peterson, Pearu and Weckesser, Warren and Bright, Jonathan and van der Walt, St{\'e}fan J. and Brett, Matthew and Wilson, Joshua and Millman, K. Jarrod and Mayorov, Nikolay and Nelson, Andrew R. J. and Jones, Eric and Kern, Robert and Larson, Eric and Carey, CJ and Polat, {\.I}lhan and Feng, Yu and Moore, Eric W. and VanderPlas, Jake and Laxalde, Denis and Perktold, Josef and Cimrman, Robert and Henriksen, Ian and Quintero, E.A. and Harris, Charles R. and Archibald, Anne M. and Ribeiro, Ant{\^o}nio H. and Pedregosa, Fabian and van Mulbregt, Paul and {SciPy 1.0 Contributors}},
  title   = {{SciPy 1.0: Fundamental Algorithms for Scientific Computing in Python}},
  journal = {Nature Methods},
  year    = {2020},
  volume  = {17},
  number  = {3},
  pages   = {261--272}
}

If you used the "Tsit5", "Vern7", or "Vern8" methods then please also cite the paper that introduced them:

@article{tsitouras2011runge,
  author  = {Tsitouras, Ch.},
  title   = {{Runge--Kutta pairs of order 5(4) satisfying only the first column simplifying assumption}},
  journal = {Computers \& Mathematics with Applications},
  year    = {2011},
  volume  = {62},
  number  = {2},
  pages   = {770--775}
}

@article{verner2010numerically,
  author  = {Verner, J. H.},
  title   = {{Numerically optimal Runge--Kutta pairs with interpolants}},
  journal = {Numerical Algorithms},
  year    = {2010},
  volume  = {53},
  number  = {2--3},
  pages   = {383--396}
}

Projects using CyRK

Don't see your project here? Create a GitHub issue or otherwise get in touch!

• TidalPy
• Rydiqule
• KG-simulation
• twpaSolver
• NOrbit
• APPS-ODE

Contribute to CyRK

Please look here for an up-to-date list of contributors to the CyRK package.

CyRK is open-source and is distributed under the Apache 2.0 license. You are welcome to fork this repository and make any edits with attribution back to this project (please see the Citing CyRK section).

• We encourage users to report bugs or feature requests using GitHub Issues.
• If you would like to contribute but don't know where to start, check out the good first issue tag on GitHub.
• Users are welcome to submit pull requests and should feel free to create them before the final code is completed so that feedback and suggestions can be given early on.