{
  "__type": "IngestedDoc",
  "__tag": 4010,
  "_content": {
    "Notes": {
      "__type": "Section",
      "__tag": 4015,
      "children": [
        {
          "__type": "Paragraph",
          "__tag": 4045,
          "children": [
            {
              "__type": "Text",
              "__tag": 4046,
              "value": "Implemented by translating the Matlab code of "
            },
            {
              "__type": "FootnoteReference",
              "__tag": 4066,
              "label": "2"
            },
            {
              "__type": "Text",
              "__tag": 4046,
              "value": " to Python."
            }
          ]
        },
        {
          "__type": "Paragraph",
          "__tag": 4045,
          "children": [
            {
              "__type": "Text",
              "__tag": 4046,
              "value": "The available orders (argument "
            },
            {
              "__type": "ParamRef",
              "__tag": 4071,
              "name": "n"
            },
            {
              "__type": "Text",
              "__tag": 4046,
              "value": ") are      "
            }
          ]
        },
        {
          "__type": "Code",
          "__tag": 4050,
          "value": "3, 5, 7, 9, 11, 13, 15, 17,\n19, 21, 23, 25, 27, 29, 31, 35,\n41, 47, 53, 59, 65, 71, 77, 83,\n89, 95, 101, 107, 113, 119, 125, 131",
          "execution_status": null
        },
        {
          "__type": "Paragraph",
          "__tag": 4045,
          "children": [
            {
              "__type": "Text",
              "__tag": 4046,
              "value": "The corresponding degrees "
            },
            {
              "__type": "InlineCode",
              "__tag": 4051,
              "value": "m"
            },
            {
              "__type": "Text",
              "__tag": 4046,
              "value": " are      "
            }
          ]
        },
        {
          "__type": "Code",
          "__tag": 4050,
          "value": "6, 14, 26, 38, 50, 74, 86, 110,\n146, 170, 194, 230, 266, 302, 350, 434,\n590, 770, 974, 1202, 1454, 1730, 2030, 2354,\n2702, 3074, 3470, 3890, 4334, 4802, 5294, 5810",
          "execution_status": null
        },
        {
          "__type": "Paragraph",
          "__tag": 4045,
          "children": [
            {
              "__type": "Strong",
              "__tag": 4048,
              "children": [
                {
                  "__type": "Text",
                  "__tag": 4046,
                  "value": "Array API Standard Support"
                }
              ]
            }
          ]
        },
        {
          "__type": "Paragraph",
          "__tag": 4045,
          "children": [
            {
              "__type": "CrossRef",
              "__tag": 4002,
              "value": "lebedev_rule",
              "reference": {
                "__type": "LocalRef",
                "__tag": 4022,
                "kind": "module",
                "path": "scipy.integrate._lebedev:lebedev_rule"
              },
              "kind": "module"
            },
            {
              "__type": "Text",
              "__tag": 4046,
              "value": " has experimental support for Python Array API Standard compatible backends in addition to NumPy. Please consider testing these features by setting an environment variable "
            },
            {
              "__type": "InlineCode",
              "__tag": 4051,
              "value": "SCIPY_ARRAY_API=1"
            },
            {
              "__type": "Text",
              "__tag": 4046,
              "value": " and providing CuPy, PyTorch, JAX, or Dask arrays as array arguments. The following combinations of backend and device (or other capability) are supported."
            }
          ]
        },
        {
          "__type": "Code",
          "__tag": 4050,
          "value": "====================  ====================  ====================\nLibrary               CPU                   GPU\n====================  ====================  ====================\nNumPy                 ✅                     n/a                 \nCuPy                  n/a                   ⛔                   \nPyTorch               ⛔                     ⛔                   \nJAX                   ⛔                     ⛔                   \nDask                  ⛔                     n/a                 \n====================  ====================  ====================",
          "execution_status": null
        },
        {
          "__type": "Blockquote",
          "__tag": 4059,
          "children": [
            {
              "__type": "Paragraph",
              "__tag": 4045,
              "children": [
                {
                  "__type": "Text",
                  "__tag": 4046,
                  "value": "See "
                },
                {
                  "__type": "InlineRole",
                  "__tag": 4003,
                  "value": "dev-arrayapi",
                  "domain": null,
                  "role": "ref",
                  "inventory": null
                },
                {
                  "__type": "Text",
                  "__tag": 4046,
                  "value": " for more information."
                }
              ]
            }
          ]
        }
      ],
      "title": [],
      "level": 0,
      "target": null
    },
    "Warns": {
      "__type": "Section",
      "__tag": 4015,
      "children": [],
      "title": [],
      "level": 0,
      "target": null
    },
    "Raises": {
      "__type": "Section",
      "__tag": 4015,
      "children": [],
      "title": [],
      "level": 0,
      "target": null
    },
    "Yields": {
      "__type": "Section",
      "__tag": 4015,
      "children": [],
      "title": [],
      "level": 0,
      "target": null
    },
    "Methods": {
      "__type": "Section",
      "__tag": 4015,
      "children": [],
      "title": [],
      "level": 0,
      "target": null
    },
    "Returns": {
      "__type": "Section",
      "__tag": 4015,
      "children": [
        {
          "__type": "Parameters",
          "__tag": 4026,
          "children": [
            {
              "__type": "DocParam",
              "__tag": 4016,
              "name": "x",
              "annotation": "ndarray of shape ``(3, m)``",
              "desc": [
                {
                  "__type": "Paragraph",
                  "__tag": 4045,
                  "children": [
                    {
                      "__type": "Text",
                      "__tag": 4046,
                      "value": "Sample points on the unit sphere in Cartesian coordinates. "
                    },
                    {
                      "__type": "InlineCode",
                      "__tag": 4051,
                      "value": "m"
                    },
                    {
                      "__type": "Text",
                      "__tag": 4046,
                      "value": " is the \"degree\" corresponding with the specified order; see Notes."
                    }
                  ]
                }
              ]
            },
            {
              "__type": "DocParam",
              "__tag": 4016,
              "name": "w",
              "annotation": "ndarray of shape ``(m,)``",
              "desc": [
                {
                  "__type": "Paragraph",
                  "__tag": 4045,
                  "children": [
                    {
                      "__type": "Text",
                      "__tag": 4046,
                      "value": "Weights"
                    }
                  ]
                }
              ]
            }
          ]
        }
      ],
      "title": [],
      "level": 0,
      "target": null
    },
    "Summary": {
      "__type": "Section",
      "__tag": 4015,
      "children": [
        {
          "__type": "Paragraph",
          "__tag": 4045,
          "children": [
            {
              "__type": "Text",
              "__tag": 4046,
              "value": "Lebedev quadrature."
            }
          ]
        }
      ],
      "title": [],
      "level": 0,
      "target": null
    },
    "Receives": {
      "__type": "Section",
      "__tag": 4015,
      "children": [],
      "title": [],
      "level": 0,
      "target": null
    },
    "Warnings": {
      "__type": "Section",
      "__tag": 4015,
      "children": [],
      "title": [],
      "level": 0,
      "target": null
    },
    "Attributes": {
      "__type": "Section",
      "__tag": 4015,
      "children": [],
      "title": [],
      "level": 0,
      "target": null
    },
    "Parameters": {
      "__type": "Section",
      "__tag": 4015,
      "children": [
        {
          "__type": "Parameters",
          "__tag": 4026,
          "children": [
            {
              "__type": "DocParam",
              "__tag": 4016,
              "name": "n",
              "annotation": "int",
              "desc": [
                {
                  "__type": "Paragraph",
                  "__tag": 4045,
                  "children": [
                    {
                      "__type": "Text",
                      "__tag": 4046,
                      "value": "Quadrature order. See Notes for supported values."
                    }
                  ]
                }
              ]
            }
          ]
        }
      ],
      "title": [],
      "level": 0,
      "target": null
    },
    "Extended Summary": {
      "__type": "Section",
      "__tag": 4015,
      "children": [
        {
          "__type": "Paragraph",
          "__tag": 4045,
          "children": [
            {
              "__type": "Text",
              "__tag": 4046,
              "value": "Compute the sample points and weights for Lebedev quadrature "
            },
            {
              "__type": "FootnoteReference",
              "__tag": 4066,
              "label": "1"
            },
            {
              "__type": "Text",
              "__tag": 4046,
              "value": " for integration of a function over the surface of a unit sphere."
            }
          ]
        }
      ],
      "title": [],
      "level": 0,
      "target": null
    },
    "Other Parameters": {
      "__type": "Section",
      "__tag": 4015,
      "children": [],
      "title": [],
      "level": 0,
      "target": null
    }
  },
  "_ordered_sections": [
    "Summary",
    "Extended Summary",
    "Parameters",
    "Attributes",
    "Methods",
    "Returns",
    "Yields",
    "Receives",
    "Other Parameters",
    "Raises",
    "Warns",
    "Warnings",
    "Notes"
  ],
  "item_file": "/scipy/integrate/_lebedev.py",
  "item_line": 5360,
  "item_type": "function",
  "aliases": [
    "scipy.integrate.lebedev_rule"
  ],
  "example_section_data": {
    "__type": "Section",
    "__tag": 4015,
    "children": [
      {
        "__type": "Text",
        "__tag": 4046,
        "value": "An example given in [3]_ is integration of :math:`f(x, y, z) = \\exp(x)` over a\nsphere of radius :math:`1`; the reference there is ``14.7680137457653``.\nShow the convergence to the expected result as the order increases:\n\n"
      },
      {
        "__type": "Code",
        "__tag": 4050,
        "value": "import matplotlib.pyplot as plt\nimport numpy as np\nfrom scipy.integrate import lebedev_rule\ndef f(x):\n    return np.exp(x[0])\nres = []\norders = np.arange(3, 20, 2)\nfor n in orders:\n    x, w = lebedev_rule(n)\n    res.append(w @ f(x))\nref = np.full_like(res, 14.7680137457653)\nerr = abs(res - ref)/abs(ref)\n",
        "execution_status": "success"
      },
      {
        "__type": "Code",
        "__tag": 4050,
        "value": "plt.semilogy(orders, err)\nplt.xlabel('order $n$')\nplt.ylabel('relative error')\nplt.title(r'Convergence for $f(x, y, z) = \\exp(x)$')\n",
        "execution_status": "failure"
      },
      {
        "__type": "Code",
        "__tag": 4050,
        "value": "plt.show()\n",
        "execution_status": "success"
      },
      {
        "__type": "Figure",
        "__tag": 4024,
        "value": {
          "__type": "RefInfo",
          "__tag": 4000,
          "module": "scipy",
          "version": "1.17.1",
          "kind": "assets",
          "path": "fig-10b2da0a9984c8c0.png"
        }
      }
    ],
    "title": [],
    "level": 0,
    "target": null
  },
  "see_also": [],
  "signature": {
    "__type": "SignatureNode",
    "__tag": 4029,
    "kind": "function",
    "parameters": [
      {
        "__type": "SigParam",
        "__tag": 4030,
        "name": "n",
        "annotation": {
          "__type": "Empty",
          "__tag": 4031
        },
        "kind": "POSITIONAL_OR_KEYWORD",
        "default": {
          "__type": "Empty",
          "__tag": 4031
        }
      }
    ],
    "return_annotation": {
      "__type": "Empty",
      "__tag": 4031
    },
    "target_name": "lebedev_rule"
  },
  "references": [
    ".. [1] V.I. Lebedev, and D.N. Laikov. \"A quadrature formula for the sphere of",
    "       the 131st algebraic order of accuracy\". Doklady Mathematics, Vol. 59,",
    "       No. 3, 1999, pp. 477-481.",
    ".. [2] R. Parrish. ``getLebedevSphere``. Matlab Central File Exchange.",
    "       https://www.mathworks.com/matlabcentral/fileexchange/27097-getlebedevsphere.",
    ".. [3] Bellet, Jean-Baptiste, Matthieu Brachet, and Jean-Pierre Croisille.",
    "       \"Quadrature and symmetry on the Cubed Sphere.\" Journal of Computational and",
    "       Applied Mathematics 409 (2022): 114142. :doi:`10.1016/j.cam.2022.114142`."
  ],
  "qa": "scipy.integrate._lebedev:lebedev_rule",
  "arbitrary": [],
  "local_refs": [
    "n",
    "w",
    "x"
  ]
}