{
  "A": [
    [
      4,
      1
    ],
    [
      2,
      3
    ]
  ],
  "b": [
    6,
    8
  ],
  "direct_exact_solution": [
    "1",
    "2"
  ],
  "synchronous_eight_steps": [
    {
      "step": 1,
      "old": [
        "0",
        "0"
      ],
      "new": [
        "3/2",
        "8/3"
      ],
      "residual_b_minus_Ax": [
        "-8/3",
        "-3"
      ],
      "residual_infinity": "3"
    },
    {
      "step": 2,
      "old": [
        "3/2",
        "8/3"
      ],
      "new": [
        "5/6",
        "5/3"
      ],
      "residual_b_minus_Ax": [
        "1",
        "4/3"
      ],
      "residual_infinity": "4/3"
    },
    {
      "step": 3,
      "old": [
        "5/6",
        "5/3"
      ],
      "new": [
        "13/12",
        "19/9"
      ],
      "residual_b_minus_Ax": [
        "-4/9",
        "-1/2"
      ],
      "residual_infinity": "1/2"
    },
    {
      "step": 4,
      "old": [
        "13/12",
        "19/9"
      ],
      "new": [
        "35/36",
        "35/18"
      ],
      "residual_b_minus_Ax": [
        "1/6",
        "2/9"
      ],
      "residual_infinity": "2/9"
    },
    {
      "step": 5,
      "old": [
        "35/36",
        "35/18"
      ],
      "new": [
        "73/72",
        "109/54"
      ],
      "residual_b_minus_Ax": [
        "-2/27",
        "-1/12"
      ],
      "residual_infinity": "1/12"
    },
    {
      "step": 6,
      "old": [
        "73/72",
        "109/54"
      ],
      "new": [
        "215/216",
        "215/108"
      ],
      "residual_b_minus_Ax": [
        "1/36",
        "1/27"
      ],
      "residual_infinity": "1/27"
    },
    {
      "step": 7,
      "old": [
        "215/216",
        "215/108"
      ],
      "new": [
        "433/432",
        "649/324"
      ],
      "residual_b_minus_Ax": [
        "-1/81",
        "-1/72"
      ],
      "residual_infinity": "1/72"
    },
    {
      "step": 8,
      "old": [
        "433/432",
        "649/324"
      ],
      "new": [
        "1295/1296",
        "1295/648"
      ],
      "residual_b_minus_Ax": [
        "1/216",
        "1/162"
      ],
      "residual_infinity": "1/162"
    }
  ],
  "status": "budget_exhausted",
  "in_place_first_step_different": [
    "3/2",
    "5/3"
  ],
  "SPD_does_not_guarantee_Jacobi_convergence": {
    "A": [
      [
        "1",
        "9/10",
        "9/10"
      ],
      [
        "9/10",
        "1",
        "9/10"
      ],
      [
        "9/10",
        "9/10",
        "1"
      ]
    ],
    "b": [
      "14/5",
      "14/5",
      "14/5"
    ],
    "three_independent_eigenvectors": [
      {
        "vector": [
          1,
          1,
          1
        ],
        "eigenvalue": "14/5"
      },
      {
        "vector": [
          1,
          -1,
          0
        ],
        "eigenvalue": "1/10"
      },
      {
        "vector": [
          1,
          0,
          -1
        ],
        "eigenvalue": "1/10"
      }
    ],
    "iteration_matrix": [
      [
        "0",
        "-9/10",
        "-9/10"
      ],
      [
        "-9/10",
        "0",
        "-9/10"
      ],
      [
        "-9/10",
        "-9/10",
        "0"
      ]
    ],
    "independent_iteration_eigenvectors": [
      {
        "vector": [
          1,
          1,
          1
        ],
        "eigenvalue": "-9/5"
      },
      {
        "vector": [
          1,
          -1,
          0
        ],
        "eigenvalue": "9/10"
      },
      {
        "vector": [
          1,
          0,
          -1
        ],
        "eigenvalue": "9/10"
      }
    ],
    "iteration_spectral_radius": "9/5",
    "iteration_on_equal_components_factor": "-9/5",
    "steps": [
      {
        "step": 1,
        "old": [
          "0",
          "0",
          "0"
        ],
        "new": [
          "14/5",
          "14/5",
          "14/5"
        ],
        "residual_b_minus_Ax": [
          "-126/25",
          "-126/25",
          "-126/25"
        ],
        "residual_infinity": "126/25"
      },
      {
        "step": 2,
        "old": [
          "14/5",
          "14/5",
          "14/5"
        ],
        "new": [
          "-56/25",
          "-56/25",
          "-56/25"
        ],
        "residual_b_minus_Ax": [
          "1134/125",
          "1134/125",
          "1134/125"
        ],
        "residual_infinity": "1134/125"
      },
      {
        "step": 3,
        "old": [
          "-56/25",
          "-56/25",
          "-56/25"
        ],
        "new": [
          "854/125",
          "854/125",
          "854/125"
        ],
        "residual_b_minus_Ax": [
          "-10206/625",
          "-10206/625",
          "-10206/625"
        ],
        "residual_infinity": "10206/625"
      },
      {
        "step": 4,
        "old": [
          "854/125",
          "854/125",
          "854/125"
        ],
        "new": [
          "-5936/625",
          "-5936/625",
          "-5936/625"
        ],
        "residual_b_minus_Ax": [
          "91854/3125",
          "91854/3125",
          "91854/3125"
        ],
        "residual_infinity": "91854/3125"
      },
      {
        "step": 5,
        "old": [
          "-5936/625",
          "-5936/625",
          "-5936/625"
        ],
        "new": [
          "62174/3125",
          "62174/3125",
          "62174/3125"
        ],
        "residual_b_minus_Ax": [
          "-826686/15625",
          "-826686/15625",
          "-826686/15625"
        ],
        "residual_infinity": "826686/15625"
      },
      {
        "step": 6,
        "old": [
          "62174/3125",
          "62174/3125",
          "62174/3125"
        ],
        "new": [
          "-515816/15625",
          "-515816/15625",
          "-515816/15625"
        ],
        "residual_b_minus_Ax": [
          "7440174/78125",
          "7440174/78125",
          "7440174/78125"
        ],
        "residual_infinity": "7440174/78125"
      },
      {
        "step": 7,
        "old": [
          "-515816/15625",
          "-515816/15625",
          "-515816/15625"
        ],
        "new": [
          "4861094/78125",
          "4861094/78125",
          "4861094/78125"
        ],
        "residual_b_minus_Ax": [
          "-66961566/390625",
          "-66961566/390625",
          "-66961566/390625"
        ],
        "residual_infinity": "66961566/390625"
      },
      {
        "step": 8,
        "old": [
          "4861094/78125",
          "4861094/78125",
          "4861094/78125"
        ],
        "new": [
          "-42656096/390625",
          "-42656096/390625",
          "-42656096/390625"
        ],
        "residual_b_minus_Ax": [
          "602654094/1953125",
          "602654094/1953125",
          "602654094/1953125"
        ],
        "residual_infinity": "602654094/1953125"
      }
    ],
    "status": "budget_exhausted",
    "closed_form_checked_every_step": "s_k=1-(-9/5)^k"
  },
  "invalid_cases_rejected": [
    "zero_diagonal",
    "ragged",
    "nonfinite",
    "bool_cap",
    "negative_tol"
  ],
  "scope": "small exact synchronous old/new vector iteration, not CG/Gauss-Seidel or direct solver; SPD eigenvalues positive but iteration spectral radius9/5; no speed/performance or real model claim"
}