{
    "info": {
        "author": "Nico Schl\u00f6mer",
        "author_email": "nico.schloemer@gmail.com",
        "bugtrack_url": null,
        "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": "<p align=\"center\">\n  <a href=\"https://github.com/nschloe/orthopy\"><img alt=\"orthopy\" src=\"https://nschloe.github.io/orthopy/orthopy-logo-with-text.png\" width=\"30%\"></a>\n  <p align=\"center\">All about orthogonal polynomials.</p>\n</p>\n\n[![CircleCI](https://img.shields.io/circleci/project/github/nschloe/orthopy/master.svg?style=flat-square)](https://circleci.com/gh/nschloe/orthopy/tree/master)\n[![codecov](https://img.shields.io/codecov/c/github/nschloe/orthopy.svg?style=flat-square)](https://codecov.io/gh/nschloe/orthopy)\n[![Code style: black](https://img.shields.io/badge/code%20style-black-000000.svg?style=flat-square)](https://github.com/ambv/black)\n[![orthogonal](https://img.shields.io/badge/orthogonal-definitely-ff69b4.svg?style=flat-square)](https://github.com/nschloe/orthopy)\n[![PyPi Version](https://img.shields.io/pypi/v/orthopy.svg?style=flat-square)](https://pypi.org/project/orthopy)\n[![DOI](https://zenodo.org/badge/DOI/10.5281/zenodo.1173151.svg?style=flat-square)](https://doi.org/10.5281/zenodo.1173151)\n[![GitHub stars](https://img.shields.io/github/stars/nschloe/orthopy.svg?style=flat-square&logo=github&label=Stars&logoColor=white)](https://github.com/nschloe/orthopy)\n\northopy provides various orthogonal polynomial classes for\n[lines](#line-segment--1-1-with-weight-function-1-x%CE%B1-1-x%CE%B2),\n[triangles](#triangle),\n[quadrilaterals](#quadrilateral),\n[disks](#disk),\n[spheres](#sphere),\n[hexahedra](#hexahedron), and\n[n-cubes](#n-cube).\nAll computations are done using numerically stable recurrence schemes.\nFurthermore, all functions are fully vectorized and can return results in [_exact\narithmetic_](#symbolic-and-numerical-computation).\n\n_Note:_ In previous versions, orthopy contained tools for working with Gauss quadrature\nrules as well. Those have moved over to [quadpy](https://github.com/nschloe/quadpy).\n\n### Line segment [-1, +1] with weight function (1-x)<sup>\u03b1</sup> (1-x)<sup>\u03b2</sup>\n\n<img src=\"https://nschloe.github.io/orthopy/line-segment-legendre.png\" width=\"25%\">\n\nJacobi, Gegenbauer (\u03b1=\u03b2), Chebyshev 1 (\u03b1=\u03b2=-1/2), Chebyshev 2 (\u03b1=\u03b2=1/2), Legendre\n(\u03b1=\u03b2=0) polynomials.\n\n```python\nvals = orthopy.line_segment.tree_jacobi(x, 4, alpha, beta, 'normal', symbolic=False)\n```\n\nRecurrence coefficients can be explicitly retrieved by\n```python\np0, a, b, c = orthopy.line_segment.recurrence_coefficients.jacobi(n, a, b, 'monic')\n```\nPossible choices for the standardization are `'monic'`, `'p(1)=(n+alpha over\nn)'`, and `'normal'`.\n\n\n#### Associated Legendre polynomials\n\n<img src=\"https://nschloe.github.io/orthopy/line-segment-alp.png\" width=\"25%\">\n\n```python\nvals = orthopy.line_segment.tree_alp(\n    x, 4, phi=None, standardization='natural', with_condon_shortley_phase=True,\n    symbolic=False\n    )\n```\n\n### 1D half-space with weight function x<sup>\u03b1</sup> exp(-r)\n<img src=\"https://nschloe.github.io/orthopy/e1r.png\" width=\"25%\">\n\n(Generalized) Laguerre polynomials.\n```python\nvals = orthopy.e1r.tree(x, 4, alpha=0, standardization='normal', symbolic=False)\n```\n\n\n### 1D space with weight function exp(-r<sup>2</sup>)\n<img src=\"https://nschloe.github.io/orthopy/e1r2.png\" width=\"25%\">\n\nHermite polynomials.\n```python\nvals = orthopy.e1r2.tree(x, 4, 'normal', symbolic=False)\n```\nAll polynomials are normalized over the measure.\n\n\n### Triangle\n\n<img src=\"https://nschloe.github.io/orthopy/triangle-1-0.png\" width=\"70%\"> |\n<img src=\"https://nschloe.github.io/orthopy/triangle-2-1.png\" width=\"70%\"> |\n<img src=\"https://nschloe.github.io/orthopy/triangle-3-1.png\" width=\"70%\">\n:-------------------:|:------------------:|:----------:|\nn=1, k=0             |  n=2, k=1          |  n=3, k=1  |\n\n```python\nvals = orthopy.triangle.tree(x, 4, 'normal', symbolic=False)\n```\nAvailable standardizations are\n  * `'normal'` (normalized polynomials, i.e., the integral of the squared function equals 1) and\n  * `'1'` where the polynomial is `1` in at least one corner of the triangle.\n\n\n### Quadrilateral\n\n<img src=\"https://nschloe.github.io/orthopy/quad-1-0.png\" width=\"70%\"> |\n<img src=\"https://nschloe.github.io/orthopy/quad-2-1.png\" width=\"70%\"> |\n<img src=\"https://nschloe.github.io/orthopy/quad-3-1.png\" width=\"70%\">\n:-------------------:|:------------------:|:----------:|\nn=1, k=0             |  n=2, k=1          |  n=3, k=1  |\n\n```python\nvals = orthopy.quadrilateral.tree(x, 4, symbolic=False)\n```\nAll polynomials are normalized on the quadrilateral.\n\n\n### Disk\n\n<img src=\"https://nschloe.github.io/orthopy/disk-1-0.png\" width=\"70%\"> |\n<img src=\"https://nschloe.github.io/orthopy/disk-2-1.png\" width=\"70%\"> |\n<img src=\"https://nschloe.github.io/orthopy/disk-4-3.png\" width=\"70%\">\n:-------------------:|:------------------:|:----------:|\nn=1, k=0             |  n=2, k=1          |  n=4, k=3  |\n\n```python\nvals = orthopy.disk.tree(x, 4, symbolic=False)\n```\nAll polynomials are normalized on the unit disk.\n\n\n### 2D space with weight function exp(-r<sup>2</sup>)\n\n<img src=\"https://nschloe.github.io/orthopy/e2r2-1-0.png\" width=\"70%\"> |\n<img src=\"https://nschloe.github.io/orthopy/e2r2-2-1.png\" width=\"70%\"> |\n<img src=\"https://nschloe.github.io/orthopy/e2r2-3-1.png\" width=\"70%\">\n:-------------------:|:------------------:|:----------:|\nn=1, k=0             |  n=2, k=1          |  n=3, k=1  |\n\n```python\nvals = orthopy.e2r2.tree(x, 4, symbolic=False)\n```\nAll polynomials are normalized over the measure.\n\n\n### Sphere\n\n<img src=\"https://nschloe.github.io/orthopy/sphere-1-0.png\" width=\"70%\"> |\n<img src=\"https://nschloe.github.io/orthopy/sphere-2-1.png\" width=\"70%\"> |\n<img src=\"https://nschloe.github.io/orthopy/sphere-5-3.png\" width=\"70%\">\n:-------------------:|:------------------:|:----------:|\nn=1, k=0             |  n=2, k=1          |  n=5, k=3  |\n\nComplex-valued _spherical harmonics,_ plotted with\n[cplot](https://github.com/nschloe/cplot/) coloring.\n\n```python\nvals = orthopy.sphere.tree_sph(\n    polar, azimuthal, n, standardization='quantum mechanic', symbolic=False\n    )\n```\n\n\n### Hexahedron\n\n<img src=\"https://nschloe.github.io/orthopy/hexa-1-0.png\" width=\"70%\"> |\n<img src=\"https://nschloe.github.io/orthopy/hexa-2-1.png\" width=\"70%\"> |\n<img src=\"https://nschloe.github.io/orthopy/hexa-5-5.png\" width=\"70%\">\n:-------------------:|:------------------:|:----------:|\nn=1, k=0             |  n=2, k=1          |  n=5, k=5  |\n\n```python\nvals = orthopy.hexahedron.tree(x, 3, symbolic=False)\n```\nAll polynomials are normalized on the hexahedron.\n\n\n### n-Cube\n\n```python\nvals = orthopy.ncube.tree(x, 6, symbolic=False)\n```\nAll polynomials are normalized on the n-dimensional cube. The dimensionality is\ndetermined by `X.shape[0]`.\n\n### nD space with weight function exp(-r<sup>2</sup>)\n\n```python\nvals = orthopy.enr2.tree(x, 4, symbolic=False)\n```\nAll polynomials are normalized over the measure. The dimensionality is\ndetermined by `X.shape[0]`.\n\n\n### Other tools\n\n * [Clenshaw algorithm](https://en.wikipedia.org/wiki/Clenshaw_algorithm) for\n   computing the weighted sum of orthogonal polynomials:\n   ```python\n   vals = orthopy.line_segment.clenshaw(a, alpha, beta, t)\n   ```\n\n\n#### Symbolic and numerical computation\n\nBy default, all operations are performed numerically. However, if\n`symbolic=True` is specified, all computations are performed symbolically. This\ncan be used, for example, to get explicit representations of the polynomials:\n```python\nimport numpy\nimport orthopy\nimport sympy\n\nb0, b1, b2 = sympy.Symbol('b0'), sympy.Symbol('b1'), sympy.Symbol('b2')\n\ntree = orthopy.triangle.tree(numpy.array([b0, b1, b2]), 3, 'normal', symbolic=True)\n\nprint(sympy.expand(tree[3][1]))\n```\n```\n42*sqrt(6)*b0*b2**2 - 24*sqrt(6)*b0*b2 + 2*sqrt(6)*b0 - 42*sqrt(6)*b1*b2**2\n+ 24*sqrt(6)*b1*b2 - 2*sqrt(6)*b1\n```\n\n\n### Installation\n\northopy is [available from the Python Package Index](https://pypi.python.org/pypi/orthopy/), so with\n```\npip install -U orthopy\n```\nyou can install/upgrade.\n\n### Testing\n\nTo run the tests, simply check out this repository and run\n```\npytest\n```\n\n### License\northopy is published under the [MIT license](https://en.wikipedia.org/wiki/MIT_License).\n\n\n",
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