{
  "__type": "IngestedDoc",
  "__tag": 4010,
  "_content": {
    "Notes": {
      "__type": "Section",
      "__tag": 4015,
      "children": [
        {
          "__type": "Paragraph",
          "__tag": 4045,
          "children": [
            {
              "__type": "Text",
              "__tag": 4046,
              "value": "This section describes the available solvers that can be selected by the 'method' parameter. The default method is "
            },
            {
              "__type": "Emphasis",
              "__tag": 4047,
              "children": [
                {
                  "__type": "Text",
                  "__tag": 4046,
                  "value": "direct"
                }
              ]
            },
            {
              "__type": "Text",
              "__tag": 4046,
              "value": " if "
            },
            {
              "__type": "InlineCode",
              "__tag": 4051,
              "value": "M"
            },
            {
              "__type": "Text",
              "__tag": 4046,
              "value": " is less than 10 and "
            },
            {
              "__type": "InlineCode",
              "__tag": 4051,
              "value": "bilinear"
            },
            {
              "__type": "Text",
              "__tag": 4046,
              "value": " otherwise."
            }
          ]
        },
        {
          "__type": "Paragraph",
          "__tag": 4045,
          "children": [
            {
              "__type": "Text",
              "__tag": 4046,
              "value": "Method "
            },
            {
              "__type": "Emphasis",
              "__tag": 4047,
              "children": [
                {
                  "__type": "Text",
                  "__tag": 4046,
                  "value": "direct"
                }
              ]
            },
            {
              "__type": "Text",
              "__tag": 4046,
              "value": " uses a direct analytical solution to the discrete Lyapunov equation. The algorithm is given in, for example, "
            },
            {
              "__type": "FootnoteReference",
              "__tag": 4066,
              "label": "1"
            },
            {
              "__type": "Text",
              "__tag": 4046,
              "value": ". However, it requires the linear solution of a system with dimension "
            },
            {
              "__type": "InlineMath",
              "__tag": 4057,
              "value": "M^2"
            },
            {
              "__type": "Text",
              "__tag": 4046,
              "value": " so that performance degrades rapidly for even moderately sized matrices."
            }
          ]
        },
        {
          "__type": "Paragraph",
          "__tag": 4045,
          "children": [
            {
              "__type": "Text",
              "__tag": 4046,
              "value": "Method "
            },
            {
              "__type": "Emphasis",
              "__tag": 4047,
              "children": [
                {
                  "__type": "Text",
                  "__tag": 4046,
                  "value": "bilinear"
                }
              ]
            },
            {
              "__type": "Text",
              "__tag": 4046,
              "value": " uses a bilinear transformation to convert the discrete Lyapunov equation to a continuous Lyapunov equation "
            },
            {
              "__type": "InlineMath",
              "__tag": 4057,
              "value": "(BX+XB'=-C)"
            },
            {
              "__type": "Text",
              "__tag": 4046,
              "value": " where "
            },
            {
              "__type": "InlineMath",
              "__tag": 4057,
              "value": "B=(A-I)(A+I)^{-1}"
            },
            {
              "__type": "Text",
              "__tag": 4046,
              "value": " and "
            },
            {
              "__type": "InlineMath",
              "__tag": 4057,
              "value": "C=2(A' + I)^{-1} Q (A + I)^{-1}"
            },
            {
              "__type": "Text",
              "__tag": 4046,
              "value": ". The continuous equation can be efficiently solved since it is a special case of a Sylvester equation. The transformation algorithm is from Popov (1964) as described in "
            },
            {
              "__type": "FootnoteReference",
              "__tag": 4066,
              "label": "2"
            },
            {
              "__type": "Text",
              "__tag": 4046,
              "value": "."
            }
          ]
        },
        {
          "__type": "Admonition",
          "__tag": 4056,
          "kind": "versionadded",
          "base_type": "neutral",
          "children": [
            {
              "__type": "AdmonitionTitle",
              "__tag": 4055,
              "children": [
                {
                  "__type": "Text",
                  "__tag": 4046,
                  "value": "versionadded 0.11.0"
                }
              ]
            }
          ]
        }
      ],
      "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",
              "desc": [
                {
                  "__type": "Paragraph",
                  "__tag": 4045,
                  "children": [
                    {
                      "__type": "Text",
                      "__tag": 4046,
                      "value": "Solution to the discrete Lyapunov equation"
                    }
                  ]
                }
              ]
            }
          ]
        }
      ],
      "title": [],
      "level": 0,
      "target": null
    },
    "Summary": {
      "__type": "Section",
      "__tag": 4015,
      "children": [
        {
          "__type": "Paragraph",
          "__tag": 4045,
          "children": [
            {
              "__type": "Text",
              "__tag": 4046,
              "value": "Solves the discrete Lyapunov equation "
            },
            {
              "__type": "InlineMath",
              "__tag": 4057,
              "value": "AXA^H - X + Q = 0"
            },
            {
              "__type": "Text",
              "__tag": 4046,
              "value": "."
            }
          ]
        }
      ],
      "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, q",
              "annotation": "(M, M) array_like",
              "desc": [
                {
                  "__type": "Paragraph",
                  "__tag": 4045,
                  "children": [
                    {
                      "__type": "Text",
                      "__tag": 4046,
                      "value": "Square matrices corresponding to A and Q in the equation above respectively. Must have the same shape."
                    }
                  ]
                }
              ]
            },
            {
              "__type": "DocParam",
              "__tag": 4016,
              "name": "method",
              "annotation": "{'direct', 'bilinear'}, optional",
              "desc": [
                {
                  "__type": "Paragraph",
                  "__tag": 4045,
                  "children": [
                    {
                      "__type": "Text",
                      "__tag": 4046,
                      "value": "Type of solver."
                    }
                  ]
                },
                {
                  "__type": "Paragraph",
                  "__tag": 4045,
                  "children": [
                    {
                      "__type": "Text",
                      "__tag": 4046,
                      "value": "If not given, chosen to be "
                    },
                    {
                      "__type": "InlineCode",
                      "__tag": 4051,
                      "value": "direct"
                    },
                    {
                      "__type": "Text",
                      "__tag": 4046,
                      "value": " if "
                    },
                    {
                      "__type": "InlineCode",
                      "__tag": 4051,
                      "value": "M"
                    },
                    {
                      "__type": "Text",
                      "__tag": 4046,
                      "value": " is less than 10 and "
                    },
                    {
                      "__type": "InlineCode",
                      "__tag": 4051,
                      "value": "bilinear"
                    },
                    {
                      "__type": "Text",
                      "__tag": 4046,
                      "value": " otherwise."
                    }
                  ]
                }
              ]
            }
          ]
        }
      ],
      "title": [],
      "level": 0,
      "target": null
    },
    "Extended Summary": {
      "__type": "Section",
      "__tag": 4015,
      "children": [
        {
          "__type": "Paragraph",
          "__tag": 4045,
          "children": [
            {
              "__type": "Text",
              "__tag": 4046,
              "value": "The documentation is written assuming array arguments are of specified \"core\" shapes. However, 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. Note that calls with zero-size batches are unsupported and will raise a "
            },
            {
              "__type": "InlineCode",
              "__tag": 4051,
              "value": "ValueError"
            },
            {
              "__type": "Text",
              "__tag": 4046,
              "value": "."
            }
          ]
        }
      ],
      "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/_solvers.py",
  "item_line": 250,
  "item_type": "function",
  "aliases": [
    "scipy.linalg.solve_discrete_lyapunov"
  ],
  "example_section_data": {
    "__type": "Section",
    "__tag": 4015,
    "children": [
      {
        "__type": "Text",
        "__tag": 4046,
        "value": "Given `a` and `q` solve for `x`:\n\n"
      },
      {
        "__type": "Code",
        "__tag": 4050,
        "value": "import numpy as np\nfrom scipy import linalg\na = np.array([[0.2, 0.5],[0.7, -0.9]])\nq = np.eye(2)\nx = linalg.solve_discrete_lyapunov(a, q)\n",
        "execution_status": "success"
      },
      {
        "__type": "Code",
        "__tag": 4050,
        "value": "x\n",
        "execution_status": "failure"
      },
      {
        "__type": "Code",
        "__tag": 4050,
        "value": "np.allclose(a.dot(x).dot(a.T)-x, -q)\n",
        "execution_status": "success"
      }
    ],
    "title": [],
    "level": 0,
    "target": null
  },
  "see_also": [
    {
      "__type": "SeeAlsoItem",
      "__tag": 4028,
      "name": {
        "__type": "CrossRef",
        "__tag": 4002,
        "value": "solve_continuous_lyapunov",
        "reference": {
          "__type": "LocalRef",
          "__tag": 4022,
          "kind": "module",
          "path": "scipy.linalg._solvers:solve_continuous_lyapunov"
        },
        "kind": "module"
      },
      "descriptions": [
        {
          "__type": "Paragraph",
          "__tag": 4045,
          "children": [
            {
              "__type": "Text",
              "__tag": 4046,
              "value": "computes the solution to the continuous-time Lyapunov equation"
            }
          ]
        }
      ],
      "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
        }
      },
      {
        "__type": "SigParam",
        "__tag": 4030,
        "name": "q",
        "annotation": {
          "__type": "Empty",
          "__tag": 4031
        },
        "kind": "POSITIONAL_OR_KEYWORD",
        "default": {
          "__type": "Empty",
          "__tag": 4031
        }
      },
      {
        "__type": "SigParam",
        "__tag": 4030,
        "name": "method",
        "annotation": {
          "__type": "Empty",
          "__tag": 4031
        },
        "kind": "POSITIONAL_OR_KEYWORD",
        "default": "None"
      }
    ],
    "return_annotation": {
      "__type": "Empty",
      "__tag": 4031
    },
    "target_name": "solve_discrete_lyapunov"
  },
  "references": [
    ".. [1] \"Lyapunov equation\", Wikipedia,",
    "   https://en.wikipedia.org/wiki/Lyapunov_equation#Discrete_time",
    ".. [2] Gajic, Z., and M.T.J. Qureshi. 2008.",
    "   Lyapunov Matrix Equation in System Stability and Control.",
    "   Dover Books on Engineering Series. Dover Publications."
  ],
  "qa": "scipy.linalg._solvers:solve_discrete_lyapunov",
  "arbitrary": [],
  "local_refs": [
    "a",
    "method",
    "q",
    "x"
  ]
}