{
  "__type": "IngestedDoc",
  "__tag": 4010,
  "_content": {
    "Notes": {
      "__type": "Section",
      "__tag": 4015,
      "children": [],
      "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": "J",
              "annotation": "csc_matrix, shape (n * m + k, n * m + k)",
              "desc": [
                {
                  "__type": "Paragraph",
                  "__tag": 4045,
                  "children": [
                    {
                      "__type": "Text",
                      "__tag": 4046,
                      "value": "Jacobian of the collocation system in a sparse form."
                    }
                  ]
                }
              ]
            }
          ]
        }
      ],
      "title": [],
      "level": 0,
      "target": null
    },
    "Summary": {
      "__type": "Section",
      "__tag": 4015,
      "children": [
        {
          "__type": "Paragraph",
          "__tag": 4045,
          "children": [
            {
              "__type": "Text",
              "__tag": 4046,
              "value": "Construct the Jacobian of the collocation system."
            }
          ]
        }
      ],
      "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": "Number of equations in the ODE system."
                    }
                  ]
                }
              ]
            },
            {
              "__type": "DocParam",
              "__tag": 4016,
              "name": "m",
              "annotation": "int",
              "desc": [
                {
                  "__type": "Paragraph",
                  "__tag": 4045,
                  "children": [
                    {
                      "__type": "Text",
                      "__tag": 4046,
                      "value": "Number of nodes in the mesh."
                    }
                  ]
                }
              ]
            },
            {
              "__type": "DocParam",
              "__tag": 4016,
              "name": "k",
              "annotation": "int",
              "desc": [
                {
                  "__type": "Paragraph",
                  "__tag": 4045,
                  "children": [
                    {
                      "__type": "Text",
                      "__tag": 4046,
                      "value": "Number of the unknown parameters."
                    }
                  ]
                }
              ]
            },
            {
              "__type": "DocParam",
              "__tag": 4016,
              "name": "i_jac, j_jac",
              "annotation": "ndarray",
              "desc": [
                {
                  "__type": "Paragraph",
                  "__tag": 4045,
                  "children": [
                    {
                      "__type": "Text",
                      "__tag": 4046,
                      "value": "Row and column indices returned by "
                    },
                    {
                      "__type": "CrossRef",
                      "__tag": 4002,
                      "value": "compute_jac_indices",
                      "reference": {
                        "__type": "LocalRef",
                        "__tag": 4022,
                        "kind": "module",
                        "path": "scipy.integrate._bvp:compute_jac_indices"
                      },
                      "kind": "module"
                    },
                    {
                      "__type": "Text",
                      "__tag": 4046,
                      "value": ". They represent different blocks in the Jacobian matrix in the following order (see the scheme above):"
                    }
                  ]
                },
                {
                  "__type": "Blockquote",
                  "__tag": 4059,
                  "children": [
                    {
                      "__type": "BulletList",
                      "__tag": 4053,
                      "ordered": false,
                      "start": 1,
                      "children": [
                        {
                          "__type": "ListItem",
                          "__tag": 4054,
                          "children": [
                            {
                              "__type": "Paragraph",
                              "__tag": 4045,
                              "children": [
                                {
                                  "__type": "Text",
                                  "__tag": 4046,
                                  "value": "1: m - 1 diagonal n x n blocks for the collocation residuals."
                                }
                              ]
                            }
                          ]
                        },
                        {
                          "__type": "ListItem",
                          "__tag": 4054,
                          "children": [
                            {
                              "__type": "Paragraph",
                              "__tag": 4045,
                              "children": [
                                {
                                  "__type": "Text",
                                  "__tag": 4046,
                                  "value": "2: m - 1 off-diagonal n x n blocks for the collocation residuals."
                                }
                              ]
                            }
                          ]
                        },
                        {
                          "__type": "ListItem",
                          "__tag": 4054,
                          "children": [
                            {
                              "__type": "Paragraph",
                              "__tag": 4045,
                              "children": [
                                {
                                  "__type": "Text",
                                  "__tag": 4046,
                                  "value": "3(n + k) x n block for the dependency of the boundary       conditions on ya."
                                }
                              ]
                            }
                          ]
                        },
                        {
                          "__type": "ListItem",
                          "__tag": 4054,
                          "children": [
                            {
                              "__type": "Paragraph",
                              "__tag": 4045,
                              "children": [
                                {
                                  "__type": "Text",
                                  "__tag": 4046,
                                  "value": "4: (n + k) x n block for the dependency of the boundary       conditions on yb."
                                }
                              ]
                            }
                          ]
                        },
                        {
                          "__type": "ListItem",
                          "__tag": 4054,
                          "children": [
                            {
                              "__type": "Paragraph",
                              "__tag": 4045,
                              "children": [
                                {
                                  "__type": "Text",
                                  "__tag": 4046,
                                  "value": "5: (m - 1) * n x k block for the dependency of the collocation       residuals on p."
                                }
                              ]
                            }
                          ]
                        },
                        {
                          "__type": "ListItem",
                          "__tag": 4054,
                          "children": [
                            {
                              "__type": "Paragraph",
                              "__tag": 4045,
                              "children": [
                                {
                                  "__type": "Text",
                                  "__tag": 4046,
                                  "value": "6: (n + k) x k block for the dependency of the boundary       conditions on p."
                                }
                              ]
                            }
                          ]
                        }
                      ]
                    }
                  ]
                }
              ]
            },
            {
              "__type": "DocParam",
              "__tag": 4016,
              "name": "df_dy",
              "annotation": "ndarray, shape (n, n, m)",
              "desc": [
                {
                  "__type": "Paragraph",
                  "__tag": 4045,
                  "children": [
                    {
                      "__type": "Text",
                      "__tag": 4046,
                      "value": "Jacobian of f with respect to y computed at the mesh nodes."
                    }
                  ]
                }
              ]
            },
            {
              "__type": "DocParam",
              "__tag": 4016,
              "name": "df_dy_middle",
              "annotation": "ndarray, shape (n, n, m - 1)",
              "desc": [
                {
                  "__type": "Paragraph",
                  "__tag": 4045,
                  "children": [
                    {
                      "__type": "Text",
                      "__tag": 4046,
                      "value": "Jacobian of f with respect to y computed at the middle between the mesh nodes."
                    }
                  ]
                }
              ]
            },
            {
              "__type": "DocParam",
              "__tag": 4016,
              "name": "df_dp",
              "annotation": "ndarray with shape (n, k, m) or None",
              "desc": [
                {
                  "__type": "Paragraph",
                  "__tag": 4045,
                  "children": [
                    {
                      "__type": "Text",
                      "__tag": 4046,
                      "value": "Jacobian of f with respect to p computed at the mesh nodes."
                    }
                  ]
                }
              ]
            },
            {
              "__type": "DocParam",
              "__tag": 4016,
              "name": "df_dp_middle",
              "annotation": "ndarray with shape (n, k, m - 1) or None",
              "desc": [
                {
                  "__type": "Paragraph",
                  "__tag": 4045,
                  "children": [
                    {
                      "__type": "Text",
                      "__tag": 4046,
                      "value": "Jacobian of f with respect to p computed at the middle between the mesh nodes."
                    }
                  ]
                }
              ]
            },
            {
              "__type": "DocParam",
              "__tag": 4016,
              "name": "dbc_dya, dbc_dyb",
              "annotation": "ndarray, shape (n, n)",
              "desc": [
                {
                  "__type": "Paragraph",
                  "__tag": 4045,
                  "children": [
                    {
                      "__type": "Text",
                      "__tag": 4046,
                      "value": "Jacobian of bc with respect to ya and yb."
                    }
                  ]
                }
              ]
            },
            {
              "__type": "DocParam",
              "__tag": 4016,
              "name": "dbc_dp",
              "annotation": "ndarray with shape (n, k) or None",
              "desc": [
                {
                  "__type": "Paragraph",
                  "__tag": 4045,
                  "children": [
                    {
                      "__type": "Text",
                      "__tag": 4046,
                      "value": "Jacobian of bc with respect to p."
                    }
                  ]
                }
              ]
            }
          ]
        }
      ],
      "title": [],
      "level": 0,
      "target": null
    },
    "Extended Summary": {
      "__type": "Section",
      "__tag": 4015,
      "children": [
        {
          "__type": "Paragraph",
          "__tag": 4045,
          "children": [
            {
              "__type": "Text",
              "__tag": 4046,
              "value": "There are n * m + k functions: m - 1 collocations residuals, each containing n components, followed by n + k boundary condition residuals."
            }
          ]
        },
        {
          "__type": "Paragraph",
          "__tag": 4045,
          "children": [
            {
              "__type": "Text",
              "__tag": 4046,
              "value": "There are n * m + k variables: m vectors of y, each containing n components, followed by k values of vector p."
            }
          ]
        },
        {
          "__type": "Paragraph",
          "__tag": 4045,
          "children": [
            {
              "__type": "Text",
              "__tag": 4046,
              "value": "For example, let m = 4, n = 2 and k = 1, then the Jacobian will have the following sparsity structure:"
            }
          ]
        },
        {
          "__type": "Blockquote",
          "__tag": 4059,
          "children": [
            {
              "__type": "Paragraph",
              "__tag": 4045,
              "children": [
                {
                  "__type": "Text",
                  "__tag": 4046,
                  "value": "1 1 2 2 0 0 0 0  5     1 1 2 2 0 0 0 0  5     0 0 1 1 2 2 0 0  5     0 0 1 1 2 2 0 0  5     0 0 0 0 1 1 2 2  5     0 0 0 0 1 1 2 2  5"
                }
              ]
            },
            {
              "__type": "Paragraph",
              "__tag": 4045,
              "children": [
                {
                  "__type": "Text",
                  "__tag": 4046,
                  "value": "3 3 0 0 0 0 4 4  6     3 3 0 0 0 0 4 4  6     3 3 0 0 0 0 4 4  6"
                }
              ]
            }
          ]
        },
        {
          "__type": "Paragraph",
          "__tag": 4045,
          "children": [
            {
              "__type": "Text",
              "__tag": 4046,
              "value": "Zeros denote identically zero values, other values denote different kinds of blocks in the matrix (see below). The blank row indicates the separation of collocation residuals from boundary conditions. And the blank column indicates the separation of y values from p values."
            }
          ]
        },
        {
          "__type": "Paragraph",
          "__tag": 4045,
          "children": [
            {
              "__type": "Text",
              "__tag": 4046,
              "value": "Refer to "
            },
            {
              "__type": "FootnoteReference",
              "__tag": 4066,
              "label": "1"
            },
            {
              "__type": "Text",
              "__tag": 4046,
              "value": "  (p. 306) for the formula of n x n blocks for derivatives of collocation residuals with respect to y."
            }
          ]
        }
      ],
      "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/_bvp.py",
  "item_line": 159,
  "item_type": "function",
  "aliases": [
    "scipy.integrate._bvp.construct_global_jac"
  ],
  "example_section_data": {
    "__type": "Section",
    "__tag": 4015,
    "children": [],
    "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
        }
      },
      {
        "__type": "SigParam",
        "__tag": 4030,
        "name": "m",
        "annotation": {
          "__type": "Empty",
          "__tag": 4031
        },
        "kind": "POSITIONAL_OR_KEYWORD",
        "default": {
          "__type": "Empty",
          "__tag": 4031
        }
      },
      {
        "__type": "SigParam",
        "__tag": 4030,
        "name": "k",
        "annotation": {
          "__type": "Empty",
          "__tag": 4031
        },
        "kind": "POSITIONAL_OR_KEYWORD",
        "default": {
          "__type": "Empty",
          "__tag": 4031
        }
      },
      {
        "__type": "SigParam",
        "__tag": 4030,
        "name": "i_jac",
        "annotation": {
          "__type": "Empty",
          "__tag": 4031
        },
        "kind": "POSITIONAL_OR_KEYWORD",
        "default": {
          "__type": "Empty",
          "__tag": 4031
        }
      },
      {
        "__type": "SigParam",
        "__tag": 4030,
        "name": "j_jac",
        "annotation": {
          "__type": "Empty",
          "__tag": 4031
        },
        "kind": "POSITIONAL_OR_KEYWORD",
        "default": {
          "__type": "Empty",
          "__tag": 4031
        }
      },
      {
        "__type": "SigParam",
        "__tag": 4030,
        "name": "h",
        "annotation": {
          "__type": "Empty",
          "__tag": 4031
        },
        "kind": "POSITIONAL_OR_KEYWORD",
        "default": {
          "__type": "Empty",
          "__tag": 4031
        }
      },
      {
        "__type": "SigParam",
        "__tag": 4030,
        "name": "df_dy",
        "annotation": {
          "__type": "Empty",
          "__tag": 4031
        },
        "kind": "POSITIONAL_OR_KEYWORD",
        "default": {
          "__type": "Empty",
          "__tag": 4031
        }
      },
      {
        "__type": "SigParam",
        "__tag": 4030,
        "name": "df_dy_middle",
        "annotation": {
          "__type": "Empty",
          "__tag": 4031
        },
        "kind": "POSITIONAL_OR_KEYWORD",
        "default": {
          "__type": "Empty",
          "__tag": 4031
        }
      },
      {
        "__type": "SigParam",
        "__tag": 4030,
        "name": "df_dp",
        "annotation": {
          "__type": "Empty",
          "__tag": 4031
        },
        "kind": "POSITIONAL_OR_KEYWORD",
        "default": {
          "__type": "Empty",
          "__tag": 4031
        }
      },
      {
        "__type": "SigParam",
        "__tag": 4030,
        "name": "df_dp_middle",
        "annotation": {
          "__type": "Empty",
          "__tag": 4031
        },
        "kind": "POSITIONAL_OR_KEYWORD",
        "default": {
          "__type": "Empty",
          "__tag": 4031
        }
      },
      {
        "__type": "SigParam",
        "__tag": 4030,
        "name": "dbc_dya",
        "annotation": {
          "__type": "Empty",
          "__tag": 4031
        },
        "kind": "POSITIONAL_OR_KEYWORD",
        "default": {
          "__type": "Empty",
          "__tag": 4031
        }
      },
      {
        "__type": "SigParam",
        "__tag": 4030,
        "name": "dbc_dyb",
        "annotation": {
          "__type": "Empty",
          "__tag": 4031
        },
        "kind": "POSITIONAL_OR_KEYWORD",
        "default": {
          "__type": "Empty",
          "__tag": 4031
        }
      },
      {
        "__type": "SigParam",
        "__tag": 4030,
        "name": "dbc_dp",
        "annotation": {
          "__type": "Empty",
          "__tag": 4031
        },
        "kind": "POSITIONAL_OR_KEYWORD",
        "default": {
          "__type": "Empty",
          "__tag": 4031
        }
      }
    ],
    "return_annotation": {
      "__type": "Empty",
      "__tag": 4031
    },
    "target_name": "construct_global_jac"
  },
  "references": [
    ".. [1] J. Kierzenka, L. F. Shampine, \"A BVP Solver Based on Residual",
    "   Control and the Maltab PSE\", ACM Trans. Math. Softw., Vol. 27,",
    "   Number 3, pp. 299-316, 2001."
  ],
  "qa": "scipy.integrate._bvp:construct_global_jac",
  "arbitrary": [],
  "local_refs": [
    "J",
    "dbc_dp",
    "dbc_dya",
    "dbc_dyb",
    "df_dp",
    "df_dp_middle",
    "df_dy",
    "df_dy_middle",
    "i_jac",
    "j_jac",
    "k",
    "m",
    "n"
  ]
}