(py)oscode: Oscillatory ordinary differential equation solver

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oscode is a C++ tool with a Python interface that solves oscillatory ordinary differential equations efficiently. It is designed to deal with equations of the form

\[\ddot{x}(t) + 2\gamma(t)\dot{x}(t) + \omega^2(t)x(t) = 0,\]

where \(\gamma(t)\) and \(\omega(t)\) can be given as arrays.

oscode makes use of an analytic approximation of \(x(t)\) embedded in a stepping procedure to skip over long regions of oscillations, giving a reduction in computing time. The approximation is valid when the frequency \(\omega(t)\) changes slowly relative to the timescales of integration, it is therefore worth applying when this condition holds for at least some part of the integration range.

For the details of the numerical method used by oscode, see the Citations section.



Basic requirements for using the C++ interface:

  • C++11 or later

  • Eigen (a header-only library included in this source)

The strictly necessary Python dependencies are automatically installed when you use pip or the setup.py. They are:

The optional dependencies are:


pyoscode can be installed via pip

pip install pyoscode

or via the setup.py

git clone https://github.com/fruzsinaagocs/oscode
cd oscode
python setup.py install --user

You can then import pyoscode from anywhere. Omit the --user option if you wish to install globally or in a virtual environment. If you have any difficulties, check out the FAQs section below.

You can check that things are working by running tests/ (also ran by Travis continuous integration):

pytest tests/


oscode is a header-only C++ package, it requires no installation.

git clone https://github.com/fruzsinaagocs/oscode

and then include the relevant header files in your C++ code:

#include "solver.hpp"
#include "system.hpp"

Quick start

Try the following quick examples. They are available in the examples.


Introduction to pyoscode


Cosmology examples




Introduction to oscode


To plot results from burst.cpp


To compile and run:

g++ -g -Wall -std=c++11 -c -o burst.o burst.cpp
g++ -g -Wall -std=c++11 -o burst burst.o


Documentation is hosted at readthedocs.

To build your own local copy of the documentation you can run:

cd pyoscode/docs
make html


If you use oscode to solve equations for a publication, please cite:


Any comments and improvements to this project are welcome. You can contribute by:

  • Opening and issue to report bugs and propose new features.

  • Making a pull request.

Further help

You can get help by submitting an issue or posting a message on Gitter.



  1. Eigen import errors:
    pyoscode/_pyoscode.hpp:6:10: fatal error: Eigen/Dense: No such file or directory
     #include <Eigen/Dense>

    Try explicitly including the location of your Eigen library via the CPLUS_INCLUDE_PATH environment variable, for example:

    CPLUS_INCLUDE_PATH=/usr/include/eigen3 python setup.py install --user
    # or
    CPLUS_INCLUDE_PATH=/usr/include/eigen3 pip install pyoscode

    where /usr/include/eigen3 should be replaced with your system-specific eigen location.


Many thanks to Will Handley, Lukas Hergt, Anthony Lasenby, and Mike Hobson for their support and advice regarding the algorithm behind oscode. There are many packages without which some part of oscode (e.g. testing and examples) wouldn’t run as nicely and smoothly, thank you all developers for making and maintaining these open-source projects. A special thanks goes to the devs of exhale for making the beautiful C++ documentation possible.


  • 1.0.0: current version
    • Dense output

    • Arrays for frequency and damping term need not be evenly spaced

    • Automatic C++ documentation on readthedocs

    • Eigen included in source for pip installability

    • First pip release :)

  • 0.1.2:
    • Bug that occurred when beginning and end of integration coincided corrected

  • 0.1.1:
    • Automatic detection of direction of integration

  • 0.1.0:
    • Memory leaks at python interface fixed

    • C++ documentation added