{
  "__type": "IngestedDoc",
  "__tag": 4010,
  "_content": {
    "Notes": {
      "__type": "Section",
      "__tag": 4015,
      "children": [
        {
          "__type": "Admonition",
          "__tag": 4056,
          "kind": "versionadded",
          "base_type": "neutral",
          "children": [
            {
              "__type": "AdmonitionTitle",
              "__tag": 4055,
              "children": [
                {
                  "__type": "Text",
                  "__tag": 4046,
                  "value": "versionadded 1.3.0"
                }
              ]
            }
          ]
        },
        {
          "__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": "fiedler",
              "reference": {
                "__type": "LocalRef",
                "__tag": 4022,
                "kind": "module",
                "path": "scipy.linalg._special_matrices:fiedler"
              },
              "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": "F",
              "annotation": "(..., n, n) ndarray",
              "desc": [
                {
                  "__type": "Paragraph",
                  "__tag": 4045,
                  "children": [
                    {
                      "__type": "Text",
                      "__tag": 4046,
                      "value": "Fiedler matrix. For batch input, each slice of shape "
                    },
                    {
                      "__type": "InlineCode",
                      "__tag": 4051,
                      "value": "(n, n)"
                    },
                    {
                      "__type": "Text",
                      "__tag": 4046,
                      "value": " along the last two dimensions of the output corresponds with a slice of shape "
                    },
                    {
                      "__type": "InlineCode",
                      "__tag": 4051,
                      "value": "(n,)"
                    },
                    {
                      "__type": "Text",
                      "__tag": 4046,
                      "value": " along the last dimension of the input."
                    }
                  ]
                }
              ]
            }
          ]
        }
      ],
      "title": [],
      "level": 0,
      "target": null
    },
    "Summary": {
      "__type": "Section",
      "__tag": 4015,
      "children": [
        {
          "__type": "Paragraph",
          "__tag": 4045,
          "children": [
            {
              "__type": "Text",
              "__tag": 4046,
              "value": "Returns a symmetric Fiedler matrix"
            }
          ]
        }
      ],
      "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": "a",
              "annotation": "(..., n,) array_like",
              "desc": [
                {
                  "__type": "Paragraph",
                  "__tag": 4045,
                  "children": [
                    {
                      "__type": "Text",
                      "__tag": 4046,
                      "value": "Coefficient array. N-dimensional arrays are treated as a batch: each slice along the last axis is a 1-D coefficient array."
                    }
                  ]
                }
              ]
            }
          ]
        }
      ],
      "title": [],
      "level": 0,
      "target": null
    },
    "Extended Summary": {
      "__type": "Section",
      "__tag": 4015,
      "children": [
        {
          "__type": "Paragraph",
          "__tag": 4045,
          "children": [
            {
              "__type": "Text",
              "__tag": 4046,
              "value": "Given an sequence of numbers "
            },
            {
              "__type": "ParamRef",
              "__tag": 4071,
              "name": "a"
            },
            {
              "__type": "Text",
              "__tag": 4046,
              "value": ", Fiedler matrices have the structure "
            },
            {
              "__type": "InlineCode",
              "__tag": 4051,
              "value": "F[i, j] = np.abs(a[i] - a[j])"
            },
            {
              "__type": "Text",
              "__tag": 4046,
              "value": ", and hence zero diagonals and nonnegative entries. A Fiedler matrix has a dominant positive eigenvalue and other eigenvalues are negative. Although not valid generally, for certain inputs, the inverse and the determinant can be derived explicitly as given in "
            },
            {
              "__type": "FootnoteReference",
              "__tag": 4066,
              "label": "1"
            },
            {
              "__type": "Text",
              "__tag": 4046,
              "value": "."
            }
          ]
        },
        {
          "__type": "Paragraph",
          "__tag": 4045,
          "children": [
            {
              "__type": "Text",
              "__tag": 4046,
              "value": "Array argument(s) of this function may have additional \"batch\" dimensions prepended to the core shape. In this case, the array is treated as a batch of lower-dimensional slices; see "
            },
            {
              "__type": "InlineRole",
              "__tag": 4003,
              "value": "linalg_batch",
              "domain": null,
              "role": "ref",
              "inventory": null
            },
            {
              "__type": "Text",
              "__tag": 4046,
              "value": " for details."
            }
          ]
        }
      ],
      "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/linalg/_special_matrices.py",
  "item_line": 950,
  "item_type": "function",
  "aliases": [
    "scipy.linalg.fiedler"
  ],
  "example_section_data": {
    "__type": "Section",
    "__tag": 4015,
    "children": [
      {
        "__type": "Code",
        "__tag": 4050,
        "value": "import numpy as np\nfrom scipy.linalg import det, inv, fiedler\na = [1, 4, 12, 45, 77]\nn = len(a)\nA = fiedler(a)\nA\n",
        "execution_status": "success"
      },
      {
        "__type": "Text",
        "__tag": 4046,
        "value": "\nThe explicit formulas for determinant and inverse seem to hold only for\nmonotonically increasing/decreasing arrays. Note the tridiagonal structure\nand the corners.\n\n"
      },
      {
        "__type": "Code",
        "__tag": 4050,
        "value": "Ai = inv(A)\nAi[np.abs(Ai) < 1e-12] = 0.  # cleanup the numerical noise for display\n",
        "execution_status": "success"
      },
      {
        "__type": "Code",
        "__tag": 4050,
        "value": "Ai\ndet(A)\n(-1)**(n-1) * 2**(n-2) * np.diff(a).prod() * (a[-1] - a[0])\n",
        "execution_status": "failure"
      }
    ],
    "title": [],
    "level": 0,
    "target": null
  },
  "see_also": [
    {
      "__type": "SeeAlsoItem",
      "__tag": 4028,
      "name": {
        "__type": "CrossRef",
        "__tag": 4002,
        "value": "circulant",
        "reference": {
          "__type": "LocalRef",
          "__tag": 4022,
          "kind": "module",
          "path": "scipy.linalg._special_matrices:circulant"
        },
        "kind": "module"
      },
      "descriptions": [],
      "type": "func"
    },
    {
      "__type": "SeeAlsoItem",
      "__tag": 4028,
      "name": {
        "__type": "CrossRef",
        "__tag": 4002,
        "value": "toeplitz",
        "reference": {
          "__type": "LocalRef",
          "__tag": 4022,
          "kind": "module",
          "path": "scipy.linalg._special_matrices:toeplitz"
        },
        "kind": "module"
      },
      "descriptions": [],
      "type": "func"
    }
  ],
  "signature": {
    "__type": "SignatureNode",
    "__tag": 4029,
    "kind": "function",
    "parameters": [
      {
        "__type": "SigParam",
        "__tag": 4030,
        "name": "a",
        "annotation": {
          "__type": "Empty",
          "__tag": 4031
        },
        "kind": "POSITIONAL_OR_KEYWORD",
        "default": {
          "__type": "Empty",
          "__tag": 4031
        }
      }
    ],
    "return_annotation": {
      "__type": "Empty",
      "__tag": 4031
    },
    "target_name": "fiedler"
  },
  "references": [
    ".. [1] J. Todd, \"Basic Numerical Mathematics: Vol.2 : Numerical Algebra\",",
    "    1977, Birkhauser, :doi:`10.1007/978-3-0348-7286-7`"
  ],
  "qa": "scipy.linalg._special_matrices:fiedler",
  "arbitrary": [],
  "local_refs": [
    "F",
    "a"
  ]
}