{
  "$schema": "../../../schemas/mko-v0.4.schema.json",
  "schema_version": "mko-v0.4",
  "id": "mko-newton-method",
  "semantic_family_id": "sf-newton-method",
  "type": "algorithm",
  "version": "0.10.0",
  "language": "zh-Hant",
  "titles": {
    "zh-Hant": "Newton根求解方法"
  },
  "summary": {
    "zh-Hant": "在導數非零的目前點，以切線零點更新x至x-f(x)/f'(x)；收斂是另需局部假設的責任。"
  },
  "statement": {
    "zh-Hant": "對可微實函數f，只有在目前iterate x_k位於domain且f'(x_k)不等於0時，Newton update x_{k+1}=x_k-f(x_k)/f'(x_k)才有定義。本輪exact slice使用低次整係數多項式與有理iterate，逐步精確重算f、f'及更新；若已到f(x_k)=0，先以root_found停止，不再對可能同時為0的導數做0/0。有限orbit、residual下降或某一步接近根都不證一般收斂、二次收斂或global root；residual |f(x)|也不是未給定真根時的forward error。"
  },
  "formula": {
    "tex": "N(f,x)=y",
    "mathml": "<math xmlns=\"http://www.w3.org/1998/Math/MathML\"><mrow><mrow><mi>N</mi><mo>(</mo><mi>f</mi><mo>,</mo><mi>x</mi><mo>)</mo></mrow><mo>=</mo><mi>y</mi></mrow></math>",
    "semantic_ast": {
      "type": "equation",
      "lhs": {
        "type": "function_call",
        "callee": {
          "type": "symbol",
          "name": "N",
          "glyph": "N"
        },
        "arguments": [
          {
            "type": "symbol",
            "name": "f",
            "glyph": "f"
          },
          {
            "type": "symbol",
            "name": "x",
            "glyph": "x"
          }
        ]
      },
      "rhs": {
        "type": "symbol",
        "name": "y",
        "glyph": "y"
      }
    },
    "compiler": {
      "id": "ocme-formula-core",
      "version": "0.4.0",
      "source_sha256": "6ab3f28090c308e2d4f029108ca531328b0ed4ff4931a52652c6379fdecc9773"
    }
  },
  "assumptions": [
    {
      "id": "a1",
      "zh-Hant": "多項式係數按低次至高次排列，至少含一個非零非定數係數；iterate為bounded exact rational。"
    },
    {
      "id": "a2",
      "zh-Hant": "每一步先核f(x)=0，再核f'(x)不等於0；步數只產生有限trace。"
    },
    {
      "id": "a3",
      "zh-Hant": "square-root special slice要求正radicand與正iterate，仍只輸出有限exact orbit。"
    }
  ],
  "symbols": [
    {
      "token": "N",
      "role": "newton_update_operator",
      "role_zh": "Newton更新算子",
      "scope": "representative_formula"
    },
    {
      "token": "f",
      "role": "differentiable_root_function",
      "role_zh": "待求根可微函數",
      "scope": "representative_formula"
    },
    {
      "token": "x",
      "role": "current_iterate",
      "role_zh": "目前iterate",
      "scope": "representative_formula"
    },
    {
      "token": "y",
      "role": "next_iterate_or_status",
      "role_zh": "下一iterate或停止狀態",
      "scope": "representative_formula"
    }
  ],
  "dependencies": [
    {
      "id": "mko-derivative",
      "title_zh": "實函數在一點的導數與差商",
      "reason_zh": "update需要點導數。"
    },
    {
      "id": "mko-numerical-approximation-error",
      "title_zh": "數值近似與誤差",
      "reason_zh": "residual與forward error分界。"
    },
    {
      "id": "mko-algorithm-correctness",
      "title_zh": "演算法正確性",
      "reason_zh": "step precondition與停止狀態。"
    },
    {
      "id": "mko-equation",
      "title_zh": "方程式",
      "reason_zh": "目標是f(x)=0。"
    },
    {
      "id": "mko-sequence",
      "title_zh": "數列",
      "reason_zh": "iterate形成有序有限prefix或無限序列。"
    },
    {
      "id": "mko-derivative-rules",
      "title_zh": "多項式的和、乘積與合成微分",
      "reason_zh": "題族精確重算多項式導數。"
    }
  ],
  "explanation": {
    "paragraphs_zh": [
      "每一步的domain、root stop與nonzero derivative責任先於除法。",
      "exact finite orbit是演算法trace，不是收斂證明。",
      "residual、forward error、局部收斂率與global basin分開。"
    ],
    "common_misconception_zh": "看到幾步快速下降便宣稱必然或二次收斂，或在f=f'=0時仍執行0/0更新。"
  },
  "proofs": [],
  "computational_companions": [],
  "verification": {
    "computational_status": "not_evaluated_no_computational_claim",
    "warning_zh": "本輪不宣稱一般Newton收斂定理、root唯一性或floating-point stability。",
    "evidence_refs": [],
    "producers": [
      {
        "id": "lean-mathlib",
        "adapter": "formal_system",
        "config_path": "formal/lean/lakefile.toml",
        "status": "configured"
      }
    ]
  },
  "formalization": {
    "status": "not_formalized",
    "target_systems": [
      "Lean 4",
      "Mathlib"
    ],
    "next_obligation_zh": "加入Kantorovich／simple-root局部二次收斂條件與roundoff分析。"
  },
  "provenance": {
    "origin": "ocme_local_ai_materialization_candidate",
    "note_zh": "依 research/atlas-newton-method.md 整理；2026-10-15 脈衝 R01（使用者手動接續 2026-11-19 工作序列；實際日期仍為 2026-10-15），英文及 Lean 依使用者指示延後。",
    "content_license": "CC-BY-SA-4.0",
    "code_license": "MIT",
    "created_at": "2026-10-15"
  }
}