{"id":18332,"date":"2024-01-30T21:06:57","date_gmt":"2024-01-30T21:06:57","guid":{"rendered":"https:\/\/science-hub.click\/%E3%82%B3%E3%83%AC%E3%82%B9%E3%82%AD%E3%83%BC%E5%9B%A0%E6%95%B0%E5%88%86%E8%A7%A3-%E5%AE%9A%E7%BE%A9\/"},"modified":"2024-01-30T21:06:57","modified_gmt":"2024-01-30T21:06:57","slug":"%E3%82%B3%E3%83%AC%E3%82%B9%E3%82%AD%E3%83%BC%E5%9B%A0%E6%95%B0%E5%88%86%E8%A7%A3-%E5%AE%9A%E7%BE%A9","status":"publish","type":"post","link":"https:\/\/science-hub.click\/?p=18332","title":{"rendered":"\u30b3\u30ec\u30b9\u30ad\u30fc\u56e0\u6570\u5206\u89e3 – \u5b9a\u7fa9"},"content":{"rendered":"

\u30a2\u30f3\u30c9\u30ec\u30fb\u30eb\u30a4\u30fb\u30b3\u30ec\u30b9\u30ad\u30fc\u306b\u3061\u306a\u3093\u3067\u540d\u4ed8\u3051\u3089\u308c\u305f\u30b3\u30ec\u30b9\u30ad\u30fc\u5206\u89e3\u306f<\/strong>\u3001\u6b63\u5b9a\u5bfe\u79f0\u884c\u5217 A \u306b\u5bfe\u3057\u3066\u3001A=LL T<\/sup>\u306e\u3088\u3046\u306a\u4e0b\u4e09\u89d2\u884c\u5217 L \u3092\u6c7a\u5b9a\u3059\u308b\u3053\u3068\u304b\u3089\u69cb\u6210\u3055\u308c\u307e\u3059\u3002<\/p>

\u884c\u5217 L \u306f\u3001\u3042\u308b\u610f\u5473 A \u306e\u300c\u5e73\u65b9\u6839\u300d\u3067\u3059\u3002\u3053\u306e\u5206\u89e3<\/a><\/span>\u306b\u3088\u308a\u3001\u7279\u306b\u9006<\/a><\/span>\u884c\u5217 A -1<\/sup>\u3092\u8a08\u7b97\u3057\u3001A \u306e\u884c\u5217\u5f0f (\u6b21\u306e\u5bfe\u89d2\u8981\u7d20\u306e\u7a4d\u306e2 \u4e57<\/a><\/span>\u306b\u7b49\u3057\u3044) \u3092\u8a08\u7b97\u3059\u308b\u3053\u3068\u304c\u53ef\u80fd\u306b\u306a\u308a\u307e\u3059\u3002 L) \u307e\u305f\u306f\u591a\u6b63\u898f\u5247\u3092\u30b7\u30df\u30e5\u30ec\u30fc\u30c8\u3059\u308b\u3053\u3068\u3055\u3048\u3067\u304d\u307e\u3059\u3002<\/p>

\u4f8b<\/span><\/h2>

\u5bfe\u79f0\u884c\u5217<\/a><\/span>A:<\/p>

\u306f\u4e09\u89d2\u884c\u5217<\/a><\/span>L \u306e\u53f3\u5074\u306e\u7a4d\u306b\u7b49\u3057\u3044\uff1a<\/p>

\u3068\u305d\u306e\u8ee2\u7f6e L T\u3002<\/sup><\/p>

\n\"\u30b3\u30ec\u30b9\u30ad\u30fc\u56e0\u6570\u5206\u89e3<\/figure>

\u5b9a\u7406<\/span><\/span><\/h2>

\u884c\u5217\u306e\u30b3\u30ec\u30b9\u30ad\u30fc\u5206\u89e3<\/a><\/span>\uff1a<\/p>

A \u304c\u6b63\u5b9a\u5bfe\u79f0\u884c\u5217\u306e\u5834\u5408\u3001\u6b21\u306e\u3088\u3046\u306a\u4e0b\u4e09\u89d2\u5b9f\u884c\u5217 L \u304c\u5c11\u306a\u304f\u3068\u3082 1 \u3064\u5b58\u5728\u3057\u307e\u3059\u3002<\/p>

A= LLT<\/sup><\/dd><\/dl>

\u307e\u305f\u3001\u884c\u5217 L \u306e\u5bfe\u89d2\u8981\u7d20\u304c\u3059\u3079\u3066\u6b63\u3067\u3042\u308a\u3001\u5bfe\u5fdc\u3059\u308b\u56e0\u6570\u5206\u89e3\u304c\u4e00\u610f\u3067\u3042\u308b\u3068\u4eee\u5b9a\u3059\u308b\u3053\u3068\u3082\u3067\u304d\u307e\u3059\u3002<\/p>

\n\"\u30b3\u30ec\u30b9\u30ad\u30fc\u56e0\u6570\u5206\u89e3<\/figure>

\u30a2\u30eb\u30b4\u30ea\u30ba\u30e0<\/span><\/h2>

\u30de\u30c8\u30ea\u30c3\u30af\u30b9\u3092\u63a2\u3057\u307e\u3059\u3002 <\/p>

$$ {L=\\begin{bmatrix} l_{11}\\\\ l_{21} & l_{22}\\\\ \\vdots & \\vdots & \\ddots\\\\ l_{n1} & l_{n2} & \\cdots & l_{nn} \\end{bmatrix}} $$<\/div><\/dd><\/dl>

A=LL T \u306e<\/sup>\u7b49\u5f0f\u304b\u3089\u3001\u6b21\u306e\u3053\u3068\u304c\u5c0e\u304d\u51fa\u3055\u308c\u307e\u3059\u3002 <\/p>

$$ {a_{ij}=\\left(LL^{T}\\right)_{ij}={\\sum_{k=1}^{n}l_{ik}l_{jk}}={\\sum_{k=1}^{\\min\\left\\{ i,j\\right\\} }l_{ik}l_{jk}},\\;1\\leq i,j\\leq n} $$<\/div><\/dd><\/dl>

1\u2264p \u306e\u5834\u5408 l pq<\/sub> =0 \u306a\u306e\u3067

\u884c\u5217 A \u306f\u5bfe\u79f0\u3067\u3042\u308b\u305f\u3081\u3001\u4e0a\u8a18\u306e\u95a2\u4fc2\u304c i\u2264j \u306b\u3064\u3044\u3066\u691c\u8a3c\u3055\u308c\u308b\u3060\u3051\u3067\u5341\u5206\u3067\u3059\u3002\u3064\u307e\u308a\u3001\u884c\u5217 L \u306e\u8981\u7d20 l ij \u304c<\/sub>\u6b21\u3092\u6e80\u305f\u3055\u306a\u3051\u308c\u3070\u306a\u308a\u307e\u305b\u3093\u3002 <\/p>

$$ {a_{ij}={\\sum_{k=1}^{i}l_{ik}l_{jk}},\\;1\\leq i,j\\leq n} $$<\/div><\/dd><\/dl>

i=1 \u306e\u5834\u5408\u3001L \u306e\u6700\u521d\u306e\u5217\u3092\u6c7a\u5b9a\u3057\u307e\u3059\u3002<\/p>

(j=1) a<\/i> 11<\/sub> = l<\/i> 11<\/sub> l<\/i> 11<\/sub><\/span>\u3053\u3053\u304b\u3089
$$ {l_{11}=\\sqrt{a_{11}}} $$<\/div><\/dd>
(j=2) a<\/i> 12<\/sub> = l<\/i> 11<\/sub> l<\/i> 21<\/sub><\/span>\u3057\u305f\u304c\u3063\u3066
$$ {l_{21}=\\frac{a_{12}}{l_{11}}} $$<\/div><\/dd>
…<\/dd>
(j=n) a<\/i> 1 n<\/i><\/sub> = l<\/i> 11<\/sub> l<\/i> n<\/i> 1<\/sub><\/span>\u3057\u305f\u304c\u3063\u3066
$$ {l_{n1}=\\frac{a_{1n}}{l_{11}}} $$<\/div><\/dd><\/dl>

\u6700\u521d\u306e (j-1) \u5217\u3092\u8a08\u7b97\u3057\u305f\u5f8c\u3001L \u306ej<\/sup>\u756a\u76ee\u306e\u5217\u3092\u6c7a\u5b9a\u3057\u307e\u3059\u3002<\/p>

(i=j)
$$ {a_{ii}=l_{i1}l_{i1}+\\ldots+l_{ii}l_{ii}} $$<\/div>\u3057\u305f\u304c\u3063\u3066
$$ {l_{ii}=\\sqrt{a_{ii}-{\\sum_{k=1}^{i-1}l_{ik}^{2}}}} $$<\/div><\/dd>
(i=j+1)
$$ {a_{i,i+1}=l_{i1}l_{i+1,1}+\\ldots+l_{ii}l_{i+1,i}} $$<\/div>\u3057\u305f\u304c\u3063\u3066
$$ {l_{i+1,i}=\\frac{a_{i,i+1}-{\\sum_{k=1}^{i-1}l_{ik}l_{i+1,k}}}{l_{ii}}} $$<\/div><\/dd>
…<\/dd>
(i=n)
$$ {a_{i,n}=l_{i1}l_{n1}+\\ldots+l_{ii}l_{ni}} $$<\/div>\u3057\u305f\u304c\u3063\u3066
$$ {l_{ni}=\\frac{a_{in}-{\\sum_{k=1}^{i-1}l_{ik}l_{nk}}}{l_{ii}}} $$<\/div><\/dd><\/dl>

\u524d\u306e\u5b9a\u7406\u304b\u3089\u3001\u3059\u3079\u3066\u306e\u91cf\u3092\u78ba\u5b9f\u306b\u3057\u306a\u304c\u3089\u3001\u3059\u3079\u3066\u306e\u8981\u7d20 l ii<\/sub> >0 \u3092\u9078\u629e\u3059\u308b\u3053\u3068\u304c\u53ef\u80fd\u3067\u3042\u308b\u3053\u3068\u304c\u308f\u304b\u308a\u307e\u3059\u3002 <\/p>

$$ {a_{11},\\ldots,a_{ii}-{\\sum_{k=1}^{i-1}l_{ik}^{2}},\\ldots} $$<\/div><\/dd><\/dl>

\u30dd\u30b8\u30c6\u30a3\u30d6\u3067\u3059\u3002 <\/p><\/q\u2264n.<><\/p><\/div>

\n\"\u30b3\u30ec\u30b9\u30ad\u30fc\u56e0\u6570\u5206\u89e3<\/figure>

\u53c2\u8003\u8cc7\u6599<\/h2>
  1. Factoritzaci\u00f3 de Cholesky \u2013 catalan<\/a><\/li>
  2. Cholesk\u00e9ho rozklad \u2013 tch\u00e8que<\/a><\/li>
  3. Cholesky-Zerlegung \u2013 allemand<\/a><\/li>
  4. Cholesky decomposition \u2013 anglais<\/a><\/li>
  5. Factorizaci\u00f3n de Cholesky \u2013 espagnol<\/a><\/li>
  6. \u062a\u062c\u0632\u06cc\u0647 \u0686\u0648\u0644\u06cc\u0633\u06a9\u0627\u06cc \u2013 persan<\/a><\/li><\/ol><\/div>\n
    \n

    \n \u30b3\u30ec\u30b9\u30ad\u30fc\u56e0\u6570\u5206\u89e3 – \u5b9a\u7fa9\u30fb\u95a2\u9023\u52d5\u753b\n <\/h2>\n
    \n \n
    \n
    \n