{
    "info": {
        "author": "Nico Schl\u00f6mer",
        "author_email": "nico.schloemer@gmail.com",
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        "classifiers": [
            "Development Status :: 4 - Beta",
            "License :: OSI Approved :: MIT License",
            "Operating System :: OS Independent",
            "Programming Language :: Python",
            "Programming Language :: Python :: 3",
            "Topic :: Scientific/Engineering",
            "Topic :: Scientific/Engineering :: Mathematics"
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        "description": "# pacopy\n\n[![CircleCI](https://img.shields.io/circleci/project/github/nschloe/pacopy/master.svg)](https://circleci.com/gh/nschloe/pacopy/tree/master)\n[![codecov](https://img.shields.io/codecov/c/github/nschloe/pacopy.svg)](https://codecov.io/gh/nschloe/pacopy)\n[![Code style: black](https://img.shields.io/badge/code%20style-black-000000.svg)](https://github.com/ambv/black)\n[![Documentation Status](https://readthedocs.org/projects/pacopy/badge/?version=latest)](https://pacopy.readthedocs.org/en/latest/?badge=latest)\n[![PyPi Version](https://img.shields.io/pypi/v/pacopy.svg)](https://pypi.org/project/pacopy)\n[![GitHub stars](https://img.shields.io/github/stars/nschloe/pacopy.svg?logo=github&label=Stars&logoColor=white)](https://github.com/nschloe/pacopy)\n\npacopy provides various algorithms of [numerical parameter\ncontinuation](https://en.wikipedia.org/wiki/Numerical_continuation) for PDEs in Python.\n\npacopy is backend-agnostic, so it doesn't matter if your problem is formulated with\n[SciPy](https://www.scipy.org/), [FEniCS](https://fenicsproject.org/),\n[pyfvm](https://github.com/nschloe/pyfvm), or any other Python package. The only thing\nthe user must provide is a class with some simple methods, e.g., a function evaluation\n`f(u, lmbda)`, a Jacobian a solver `jacobian_solver(u, lmbda, rhs)` etc.\n\nSome pacopy documentation is available [here](https://pacopy.readthedocs.org/en/latest/?badge=latest).\n\n\n### Examples\n\n#### Bratu\n\n<img src=\"https://nschloe.github.io/pacopy/bratu1d.png\" width=\"30%\">\n\nThe classical [Bratu\nproblem](https://en.wikipedia.org/wiki/Liouville%E2%80%93Bratu%E2%80%93Gelfand_equation)\nin 1D with Dirichlet boundary conditions. To reproduce the plot, you first have to\nspecify the problem; this is the classical finite-difference approximation:\n```python\nclass Bratu1d(object):\n    def __init__(self):\n        self.n = 51\n        h = 1.0 / (self.n - 1)\n\n        self.H = numpy.full(self.n, h)\n        self.H[0] = h / 2\n        self.H[-1] = h / 2\n\n        self.A = (\n            scipy.sparse.diags([-1.0, 2.0, -1.0], [-1, 0, 1], shape=(self.n, self.n))\n            / h ** 2\n        )\n        return\n\n    def inner(self, a, b):\n        \"\"\"The inner product of the problem. Can be numpy.dot(a, b), but factoring in\n        the mesh width stays true to the PDE.\n        \"\"\"\n        return numpy.dot(a, self.H * b)\n\n    def norm2_r(self, a):\n        \"\"\"The norm in the range space; used to determine if a solution has been found.\n        \"\"\"\n        return numpy.dot(a, a)\n\n    def f(self, u, lmbda):\n        \"\"\"The evaluation of the function to be solved\n        \"\"\"\n        out = self.A.dot(u) - lmbda * numpy.exp(u)\n        out[0] = u[0]\n        out[-1] = u[-1]\n        return out\n\n    def df_dlmbda(self, u, lmbda):\n        \"\"\"The function's derivative with respect to the parameter. Used in Euler-Newton\n        continuation.\n        \"\"\"\n        out = -numpy.exp(u)\n        out[0] = 0.0\n        out[-1] = 0.0\n        return out\n\n    def jacobian_solver(self, u, lmbda, rhs):\n        \"\"\"A solver for the Jacobian problem.\n        \"\"\"\n        M = self.A.copy()\n        d = M.diagonal().copy()\n        d -= lmbda * numpy.exp(u)\n        M.setdiag(d)\n        # Dirichlet conditions\n        M.data[0][self.n - 2] = 0.0\n        M.data[1][0] = 1.0\n        M.data[1][self.n - 1] = 1.0\n        M.data[2][1] = 0.0\n        return scipy.sparse.linalg.spsolve(M.tocsr(), rhs)\n```\nThen pass the object to any of pacopy's methods, e.g., the Euler-Newton (arclength)\ncontinuation:\n```python\nproblem = Bratu1d()\n# Initial guess\nu0 = numpy.zeros(problem.n)\n# Initial parameter value\nlmbda0 = 0.0\n\nlmbda_list = []\nvalues_list = []\ndef callback(k, lmbda, sol):\n    # Use the callback for plotting, writing data to files etc.\n    fig = plt.figure()\n    ax1 = fig.add_subplot(111)\n    ax1.set_xlabel(\"$\\\\lambda$\")\n    ax1.set_ylabel(\"$||u||_2$\")\n    ax1.grid()\n\n    lmbda_list.append(lmbda)\n    values_list.append(numpy.sqrt(problem.inner(sol, sol)))\n\n    ax1.plot(lmbda_list, values_list, \"-x\", color=\"#1f77f4\")\n    ax1.set_xlim(0.0, 4.0)\n    ax1.set_ylim(0.0, 6.0)\n    return\n\n# Natural parameter continuation\n# pacopy.natural(problem, u0, lmbda0, callback, max_steps=100)\n\npacopy.euler_newton(\n    problem, u0, lmbda0, callback, max_steps=500, newton_tol=1.0e-10\n)\n```\n\n\n#### Ginzburg\u2013Landau\n\n![ginzburg-landau](https://nschloe.github.io/pacopy/ginzburg-landau.gif)\n\nThe [Ginzburg-Landau\nequations](https://en.wikipedia.org/wiki/Ginzburg%E2%80%93Landau_theory) model the\nbehavior of extreme type-II superconductors under a magnetic field. The above example\n(to be found in full detail\n[here](https://github.com/nschloe/pacopy/blob/master/test/test_ginzburg_landau.py))\nshows parameter continuation in the strength of the magnetic field. The plot on the\nright-hand side shows the complex-valued solution using\n[cplot](https://github.com/nschloe/cplot).\n\n\n### Installation\n\npacopy is [available from the Python Package\nIndex](https://pypi.org/project/pacopy/), so simply type\n```\npip install -U pacopy\n```\nto install or upgrade.\n\n### Testing\n\nTo run the pacopy unit tests, check out this repository and type\n```\npytest\n```\n\n### Distribution\n\nTo create a new release\n\n1. bump the `__version__` number,\n\n2. publish to PyPi and GitHub:\n    ```\n    make publish\n    ```\n\n### License\n\npacopy is published under the [MIT license](https://en.wikipedia.org/wiki/MIT_License).\n\n\n",
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