{
  "__type": "IngestedDoc",
  "__tag": 4010,
  "_content": {
    "Notes": {
      "__type": "Section",
      "__tag": 4015,
      "children": [
        {
          "__type": "Paragraph",
          "__tag": 4045,
          "children": [
            {
              "__type": "Text",
              "__tag": 4046,
              "value": "This file is a translation of the MATLAB implementation "
            },
            {
              "__type": "FootnoteReference",
              "__tag": 4066,
              "label": "2"
            },
            {
              "__type": "Text",
              "__tag": 4046,
              "value": "."
            }
          ]
        }
      ],
      "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": "The converged solution."
                    }
                  ]
                }
              ]
            },
            {
              "__type": "DocParam",
              "__tag": 4016,
              "name": "info",
              "annotation": "int",
              "desc": [
                {
                  "__type": "DefList",
                  "__tag": 4033,
                  "children": [
                    {
                      "__type": "DefListItem",
                      "__tag": 4037,
                      "dt": {
                        "__type": "Paragraph",
                        "__tag": 4045,
                        "children": [
                          {
                            "__type": "Text",
                            "__tag": 4046,
                            "value": "Provides convergence information:"
                          }
                        ]
                      },
                      "dd": [
                        {
                          "__type": "Paragraph",
                          "__tag": 4045,
                          "children": [
                            {
                              "__type": "Text",
                              "__tag": 4046,
                              "value": "0successful exit     >0 : convergence to tolerance not achieved, number of iterations     <0 : illegal input or breakdown"
                            }
                          ]
                        }
                      ]
                    }
                  ]
                }
              ]
            }
          ]
        }
      ],
      "title": [],
      "level": 0,
      "target": null
    },
    "Summary": {
      "__type": "Section",
      "__tag": 4015,
      "children": [
        {
          "__type": "Paragraph",
          "__tag": 4045,
          "children": [
            {
              "__type": "Text",
              "__tag": 4046,
              "value": "Solve "
            },
            {
              "__type": "InlineCode",
              "__tag": 4051,
              "value": "Ax = b"
            },
            {
              "__type": "Text",
              "__tag": 4046,
              "value": " with the MINimum RESidual method, for a real symmetric or complex hermitian matrix "
            },
            {
              "__type": "ParamRef",
              "__tag": 4071,
              "name": "A"
            },
            {
              "__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",
              "annotation": "{sparse array, ndarray, LinearOperator}",
              "desc": [
                {
                  "__type": "Paragraph",
                  "__tag": 4045,
                  "children": [
                    {
                      "__type": "Text",
                      "__tag": 4046,
                      "value": "The real symmetric or complex hermitian N-by-N matrix of the linear system. Alternatively, "
                    },
                    {
                      "__type": "ParamRef",
                      "__tag": 4071,
                      "name": "A"
                    },
                    {
                      "__type": "Text",
                      "__tag": 4046,
                      "value": " can be a linear operator which can produce "
                    },
                    {
                      "__type": "InlineCode",
                      "__tag": 4051,
                      "value": "Ax"
                    },
                    {
                      "__type": "Text",
                      "__tag": 4046,
                      "value": " using, e.g., "
                    },
                    {
                      "__type": "InlineCode",
                      "__tag": 4051,
                      "value": "scipy.sparse.linalg.LinearOperator"
                    },
                    {
                      "__type": "Text",
                      "__tag": 4046,
                      "value": "."
                    }
                  ]
                }
              ]
            },
            {
              "__type": "DocParam",
              "__tag": 4016,
              "name": "b",
              "annotation": "ndarray",
              "desc": [
                {
                  "__type": "Paragraph",
                  "__tag": 4045,
                  "children": [
                    {
                      "__type": "Text",
                      "__tag": 4046,
                      "value": "Right hand side of the linear system. Has shape (N,) or (N,1)."
                    }
                  ]
                }
              ]
            }
          ]
        }
      ],
      "title": [],
      "level": 0,
      "target": null
    },
    "Extended Summary": {
      "__type": "Section",
      "__tag": 4015,
      "children": [
        {
          "__type": "Paragraph",
          "__tag": 4045,
          "children": [
            {
              "__type": "Text",
              "__tag": 4046,
              "value": "MINRES minimizes "
            },
            {
              "__type": "InlineCode",
              "__tag": 4051,
              "value": "norm(Ax - b)"
            },
            {
              "__type": "Text",
              "__tag": 4046,
              "value": " for a real symmetric or complex hermitian matrix "
            },
            {
              "__type": "ParamRef",
              "__tag": 4071,
              "name": "A"
            },
            {
              "__type": "Text",
              "__tag": 4046,
              "value": ". Unlike the Conjugate Gradient method, "
            },
            {
              "__type": "ParamRef",
              "__tag": 4071,
              "name": "A"
            },
            {
              "__type": "Text",
              "__tag": 4046,
              "value": " can be indefinite or singular."
            }
          ]
        },
        {
          "__type": "Paragraph",
          "__tag": 4045,
          "children": [
            {
              "__type": "Text",
              "__tag": 4046,
              "value": "If "
            },
            {
              "__type": "InlineCode",
              "__tag": 4051,
              "value": "shift != 0"
            },
            {
              "__type": "Text",
              "__tag": 4046,
              "value": " then the method solves "
            },
            {
              "__type": "InlineCode",
              "__tag": 4051,
              "value": "(A - shift*I)x = b"
            },
            {
              "__type": "Text",
              "__tag": 4046,
              "value": "."
            }
          ]
        }
      ],
      "title": [],
      "level": 0,
      "target": null
    },
    "Other Parameters": {
      "__type": "Section",
      "__tag": 4015,
      "children": [
        {
          "__type": "Parameters",
          "__tag": 4026,
          "children": [
            {
              "__type": "DocParam",
              "__tag": 4016,
              "name": "x0",
              "annotation": "ndarray",
              "desc": [
                {
                  "__type": "Paragraph",
                  "__tag": 4045,
                  "children": [
                    {
                      "__type": "Text",
                      "__tag": 4046,
                      "value": "Starting guess for the solution."
                    }
                  ]
                }
              ]
            },
            {
              "__type": "DocParam",
              "__tag": 4016,
              "name": "rtol",
              "annotation": "float",
              "desc": [
                {
                  "__type": "Paragraph",
                  "__tag": 4045,
                  "children": [
                    {
                      "__type": "Text",
                      "__tag": 4046,
                      "value": "Tolerance to achieve. The algorithm terminates when the relative residual is below "
                    },
                    {
                      "__type": "InlineCode",
                      "__tag": 4051,
                      "value": "rtol"
                    },
                    {
                      "__type": "Text",
                      "__tag": 4046,
                      "value": "."
                    }
                  ]
                }
              ]
            },
            {
              "__type": "DocParam",
              "__tag": 4016,
              "name": "shift",
              "annotation": "float",
              "desc": [
                {
                  "__type": "Paragraph",
                  "__tag": 4045,
                  "children": [
                    {
                      "__type": "Text",
                      "__tag": 4046,
                      "value": "Value to apply to the system "
                    },
                    {
                      "__type": "InlineCode",
                      "__tag": 4051,
                      "value": "(A - shift * I)x = b"
                    },
                    {
                      "__type": "Text",
                      "__tag": 4046,
                      "value": ". Default is 0."
                    }
                  ]
                }
              ]
            },
            {
              "__type": "DocParam",
              "__tag": 4016,
              "name": "maxiter",
              "annotation": "int",
              "desc": [
                {
                  "__type": "Paragraph",
                  "__tag": 4045,
                  "children": [
                    {
                      "__type": "Text",
                      "__tag": 4046,
                      "value": "Maximum number of iterations.  Iteration will stop after maxiter steps even if the specified tolerance has not been achieved."
                    }
                  ]
                }
              ]
            },
            {
              "__type": "DocParam",
              "__tag": 4016,
              "name": "M",
              "annotation": "{sparse array, ndarray, LinearOperator}",
              "desc": [
                {
                  "__type": "Paragraph",
                  "__tag": 4045,
                  "children": [
                    {
                      "__type": "Text",
                      "__tag": 4046,
                      "value": "Preconditioner for "
                    },
                    {
                      "__type": "ParamRef",
                      "__tag": 4071,
                      "name": "A"
                    },
                    {
                      "__type": "Text",
                      "__tag": 4046,
                      "value": ".  The preconditioner should approximate the inverse of "
                    },
                    {
                      "__type": "ParamRef",
                      "__tag": 4071,
                      "name": "A"
                    },
                    {
                      "__type": "Text",
                      "__tag": 4046,
                      "value": ".  Effective preconditioning dramatically improves the rate of convergence, which implies that fewer iterations are needed to reach a given error tolerance."
                    }
                  ]
                }
              ]
            },
            {
              "__type": "DocParam",
              "__tag": 4016,
              "name": "callback",
              "annotation": "function",
              "desc": [
                {
                  "__type": "Paragraph",
                  "__tag": 4045,
                  "children": [
                    {
                      "__type": "Text",
                      "__tag": 4046,
                      "value": "User-supplied function to call after each iteration.  It is called as "
                    },
                    {
                      "__type": "InlineCode",
                      "__tag": 4051,
                      "value": "callback(xk)"
                    },
                    {
                      "__type": "Text",
                      "__tag": 4046,
                      "value": ", where "
                    },
                    {
                      "__type": "InlineCode",
                      "__tag": 4051,
                      "value": "xk"
                    },
                    {
                      "__type": "Text",
                      "__tag": 4046,
                      "value": " is the current solution vector."
                    }
                  ]
                }
              ]
            },
            {
              "__type": "DocParam",
              "__tag": 4016,
              "name": "show",
              "annotation": "bool",
              "desc": [
                {
                  "__type": "Paragraph",
                  "__tag": 4045,
                  "children": [
                    {
                      "__type": "Text",
                      "__tag": 4046,
                      "value": "If "
                    },
                    {
                      "__type": "InlineCode",
                      "__tag": 4051,
                      "value": "True"
                    },
                    {
                      "__type": "Text",
                      "__tag": 4046,
                      "value": ", print out a summary and metrics related to the solution during iterations. Default is "
                    },
                    {
                      "__type": "InlineCode",
                      "__tag": 4051,
                      "value": "False"
                    },
                    {
                      "__type": "Text",
                      "__tag": 4046,
                      "value": "."
                    }
                  ]
                }
              ]
            },
            {
              "__type": "DocParam",
              "__tag": 4016,
              "name": "check",
              "annotation": "bool",
              "desc": [
                {
                  "__type": "Paragraph",
                  "__tag": 4045,
                  "children": [
                    {
                      "__type": "Text",
                      "__tag": 4046,
                      "value": "If "
                    },
                    {
                      "__type": "InlineCode",
                      "__tag": 4051,
                      "value": "True"
                    },
                    {
                      "__type": "Text",
                      "__tag": 4046,
                      "value": ", run additional input validation to check that "
                    },
                    {
                      "__type": "ParamRef",
                      "__tag": 4071,
                      "name": "A"
                    },
                    {
                      "__type": "Text",
                      "__tag": 4046,
                      "value": " and "
                    },
                    {
                      "__type": "ParamRef",
                      "__tag": 4071,
                      "name": "M"
                    },
                    {
                      "__type": "Text",
                      "__tag": 4046,
                      "value": " (if specified) are symmetric. Default is "
                    },
                    {
                      "__type": "InlineCode",
                      "__tag": 4051,
                      "value": "False"
                    },
                    {
                      "__type": "Text",
                      "__tag": 4046,
                      "value": "."
                    }
                  ]
                }
              ]
            }
          ]
        }
      ],
      "title": [],
      "level": 0,
      "target": null
    }
  },
  "_ordered_sections": [
    "Summary",
    "Extended Summary",
    "Parameters",
    "Attributes",
    "Methods",
    "Returns",
    "Yields",
    "Receives",
    "Other Parameters",
    "Raises",
    "Warns",
    "Warnings",
    "Notes"
  ],
  "item_file": "build-install/usr/lib/python3.14/site-packages/scipy/sparse/linalg/_isolve/minres.py",
  "item_line": 12,
  "item_type": "function",
  "aliases": [
    "scipy.sparse.linalg.minres"
  ],
  "example_section_data": {
    "__type": "Section",
    "__tag": 4015,
    "children": [
      {
        "__type": "Code",
        "__tag": 4050,
        "value": "import numpy as np\nfrom scipy.sparse import csc_array\nfrom scipy.sparse.linalg import minres\nA = csc_array([[3, 2, 0], [1, -1, 0], [0, 5, 1]], dtype=float)\nA = A + A.T\nb = np.array([2, 4, -1], dtype=float)\nx, exitCode = minres(A, b)\nprint(exitCode)            # 0 indicates successful convergence\nnp.allclose(A.dot(x), b)\n",
        "execution_status": "success",
        "out": "True\n"
      }
    ],
    "title": [],
    "level": 0,
    "target": null
  },
  "see_also": [],
  "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": "b",
        "annotation": {
          "__type": "Empty",
          "__tag": 4031
        },
        "kind": "POSITIONAL_OR_KEYWORD",
        "default": {
          "__type": "Empty",
          "__tag": 4031
        }
      },
      {
        "__type": "SigParam",
        "__tag": 4030,
        "name": "x0",
        "annotation": {
          "__type": "Empty",
          "__tag": 4031
        },
        "kind": "POSITIONAL_OR_KEYWORD",
        "default": "None"
      },
      {
        "__type": "SigParam",
        "__tag": 4030,
        "name": "rtol",
        "annotation": {
          "__type": "Empty",
          "__tag": 4031
        },
        "kind": "KEYWORD_ONLY",
        "default": "1e-05"
      },
      {
        "__type": "SigParam",
        "__tag": 4030,
        "name": "shift",
        "annotation": {
          "__type": "Empty",
          "__tag": 4031
        },
        "kind": "KEYWORD_ONLY",
        "default": "0.0"
      },
      {
        "__type": "SigParam",
        "__tag": 4030,
        "name": "maxiter",
        "annotation": {
          "__type": "Empty",
          "__tag": 4031
        },
        "kind": "KEYWORD_ONLY",
        "default": "None"
      },
      {
        "__type": "SigParam",
        "__tag": 4030,
        "name": "M",
        "annotation": {
          "__type": "Empty",
          "__tag": 4031
        },
        "kind": "KEYWORD_ONLY",
        "default": "None"
      },
      {
        "__type": "SigParam",
        "__tag": 4030,
        "name": "callback",
        "annotation": {
          "__type": "Empty",
          "__tag": 4031
        },
        "kind": "KEYWORD_ONLY",
        "default": "None"
      },
      {
        "__type": "SigParam",
        "__tag": 4030,
        "name": "show",
        "annotation": {
          "__type": "Empty",
          "__tag": 4031
        },
        "kind": "KEYWORD_ONLY",
        "default": "False"
      },
      {
        "__type": "SigParam",
        "__tag": 4030,
        "name": "check",
        "annotation": {
          "__type": "Empty",
          "__tag": 4031
        },
        "kind": "KEYWORD_ONLY",
        "default": "False"
      }
    ],
    "return_annotation": {
      "__type": "Empty",
      "__tag": 4031
    },
    "target_name": "minres"
  },
  "references": [
    ".. [1] Solution of sparse indefinite systems of linear equations,",
    "      C. C. Paige and M. A. Saunders (1975),",
    "      SIAM J. Numer. Anal. 12(4), pp. 617-629.",
    "      https://web.stanford.edu/group/SOL/software/minres/",
    ".. [2] https://web.stanford.edu/group/SOL/software/minres/minres-matlab.zip"
  ],
  "qa": "scipy.sparse.linalg._isolve.minres:minres",
  "arbitrary": [],
  "local_refs": [
    "A",
    "M",
    "b",
    "callback",
    "check",
    "info",
    "maxiter",
    "rtol",
    "shift",
    "show",
    "x",
    "x0"
  ]
}