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"description": "\nDESolver\n========\n\n\n.. image:: https://travis-ci.com/Microno95/desolver.svg?branch=master\n :target: https://travis-ci.com/Microno95/desolver\n :alt: Build Status\n\n.. image:: https://readthedocs.org/projects/desolver/badge/?version=latest\n :target: https://desolver.readthedocs.io/en/latest/?badge=latest\n :alt: Documentation Status\n\n.. image:: https://codecov.io/gh/Microno95/desolver/branch/master/graph/badge.svg\n :target: https://codecov.io/gh/Microno95/desolver\n :alt: codecov\n\n.. image:: https://bettercodehub.com/edge/badge/Microno95/desolver?branch=master\n :target: https://bettercodehub.com/\n :alt: BCH compliance\n\n\nThis is a python package for solving Initial Value Problems using various numerical integrators.\nMany integration routines are included ranging from fixed step to symplectic to adaptive integrators.\n\nDocumentation\n=============\n\nDocumentation is now available at `desolver docs `_! This will be updated with new examples as they are written, currently the examples show the use of ``pyaudi``.\n\nLatest Release\n==============\n\n**4.2.0** - Improved performance of implicit methods, added embedded implicit methods following Kroul\u00edkov\u00e1 (2017) for fully implicit adaptive integration.\n\n**4.1.0** - Initial release of implicit integration schemes that use a basic newton-raphson algorithm to solve for the intermediate states.\n\n**3.0.0** - PyAudi support has been finalised. It is now possible to do numerical integrations using ``gdual`` variables such as ``gdual_double``\\ , ``gdual_vdouble`` and ``gdual_real128`` (only on select platforms, refer to `pyaudi docs `_ for more information). Install desolver with pyaudi support using ``pip install desolver[pyaudi]``. Documentation has also been added and is available at `desolver docs `_.\n\n**2.5.0** - Event detection has been added to the module. It is now possible to do numerical integration with terminal and non-terminal events.\n\n**2.2.0** - PyTorch backend is now implemented. It is now possible to numerically integrate a system of equations that use pytorch tensors and then compute gradients from these.\n\nUse of PyTorch backend requires installation of PyTorch from `here `_.\n\nTo Install:\n===========\n\nJust type\n\n``pip install desolver``\n\nImplemented Integration Methods\n-------------------------------\n\nExplicit Methods\n~~~~~~~~~~~~~~~~\n\nAdaptive Methods\n^^^^^^^^^^^^^^^^\n\n#. Runge-Kutta 14(12) (Feagin, 2009)\n#. Runge-Kutta 10(8) (Feagin, 2009)\n#. Runge-Kutta 8(7) (Dormand & Prince, 1980)\n#. Runge-Kutta 4(5) with Cash-Karp Coefficients\n#. Adaptive Heun-Euler Method\n\nFixed Step Methods\n^^^^^^^^^^^^^^^^^^\n\n#. Symplectic BABs9o7H Method (Mads & Nielsen, 2015, BAB's9o7H)\n#. Symplectic ABAs5o6HA Method (Mads & Nielsen, 2015, ABAs5o6H)\n#. Runge-Kutta 5 - The 5th order integrator from RK45 with Cash-Karp Coefficients.\n#. Runge-Kutta 4 - The classic RK4 integrator\n#. Midpoint Method\n#. Heun's Method\n#. Euler's Method\n#. Euler-Trapezoidal Method\n\nImplicit Methods [\\ **NEW**\\ ]\n~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~\n\nAdaptive Methods\n^^^^^^^^^^^^^^^^\n\n#. Lobatto IIIC 4(2) (Kroul\u00edkov\u00e1, 2017)\n#. Radau IIA 5(2) (Kroul\u00edkov\u00e1, 2017)\n\n\nFixed Step Methods\n^^^^^^^^^^^^^^^^^^\n\n#. Backward Euler\n#. Implicit Midpoint\n#. Crank-Nicolson\n#. Lobatto IIIA 2\n#. Lobatto IIIB 2\n#. Lobatto IIIC 2\n#. Radau IA 3\n#. Radau IIA 3\n#. Lobatto IIIA 4\n#. Lobatto IIIB 4\n#. Gauss-Legendre 4\n#. Radau IA 5\n#. Radau IIA 6\n\nMinimal Working Example\n=======================\n\nThis example shows the integration of a harmonic oscillator using DESolver.\n\n.. code-block:: python\n\n import desolver as de\n import desolver.backend as D\n\n def rhs(t, state, k, m, **kwargs):\n return D.array([[0.0, 1.0], [-k/m, 0.0]])@state\n\n y_init = D.array([1., 0.])\n\n a = de.OdeSystem(rhs, y0=y_init, dense_output=True, t=(0, 2*D.pi), dt=0.01, rtol=1e-9, atol=1e-9, constants=dict(k=1.0, m=1.0))\n\n print(a)\n\n a.integrate()\n\n print(a)\n\n print(\"If the integration was successful and correct, a[0].y and a[-1].y should be near identical.\")\n print(\"a[0].y = {}\".format(a[0].y))\n print(\"a[-1].y = {}\".format(a[-1].y))\n\n print(\"Maximum difference from initial state after one oscillation cycle: {}\".format(D.max(D.abs(a[0].y-a[-1].y))))\n\n\nReferences\n==========\n\nFeagin, T. (2009). High-Order Explicit Runge-Kutta Methods. Retrieved from `https://sce.uhcl.edu/rungekutta/ `_\n\nDormand, J. R. and Prince, P. J. (1980) A family of embedded Runge-Kutta formulae. *Journal of Computational and Applied Mathematics*, 6(1), 19-26. `https://doi.org/10.1016/0771-050X(80)90013-3 `_\n\nMads, K. and Nielsen, E. (2015). *Efficient fourth order symplectic integrators for near-harmonic separable Hamiltonian systems*. Retrieved from `https://arxiv.org/abs/1501.04345 `_\n\nKroul\u00edkov\u00e1, T. (2017). RUNGE-KUTTA METHODS (Master's thesis, BRNO UNIVERSITY OF TECHNOLOGY, Brno, Czechia). Retrieved from `https://www.vutbr.cz/www_base/zav_prace_soubor_verejne.php?file_id=174714 `_\n\n",
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