{"id":1001,"date":"2015-10-11T23:08:29","date_gmt":"2015-10-11T14:08:29","guid":{"rendered":"https:\/\/www.ktech.biz\/jp\/?p=1001"},"modified":"2020-07-28T22:33:41","modified_gmt":"2020-07-28T13:33:41","slug":"10-ode-1","status":"publish","type":"post","link":"https:\/\/www.ktech.biz\/jp\/num\/10-ode-1\/","title":{"rendered":"10. \u5e38\u5fae\u5206\u65b9\u7a0b\u5f0f (1) \u6982\u8981, \uff11\u6bb5\u6cd5, \u591a\u6bb5\u6cd5, \u4f7f\u308f\u308c\u306a\u304f\u306a\u3063\u305f\u89e3\u6cd5"},"content":{"rendered":"

\u76ee\u6b21<\/h3>\n

\u5e38\u5fae\u5206\u65b9\u7a0b\u5f0f (1): \u6982\u8981, \uff11\u6bb5\u6cd5, \u591a\u6bb5\u6cd5, \u4f7f\u308f\u308c\u306a\u304f\u306a\u3063\u305f\u89e3\u6cd5
\n\u5e38\u5fae\u5206\u65b9\u7a0b\u5f0f (2)<\/a>: \u8aa4\u5dee\u306e\u63a8\u5b9a, \u88dc\u5916\u6cd5, \u30b9\u30c6\u30c3\u30d7\u5e45\u306e\u81ea\u52d5\u8abf\u7bc0, \u5bc6\u51fa\u529b, 2\u968e\u5e38\u5fae\u5206\u65b9\u7a0b\u5f0f, \u9045\u5ef6\u5fae\u5206\u65b9\u7a0b\u5f0f
\n\u5e38\u5fae\u5206\u65b9\u7a0b\u5f0f (3)<\/a>: \u89e3\u6cd5\u306e\u5b89\u5b9a\u6027, \u30b9\u30c6\u30a3\u30d5\u306a\u65b9\u7a0b\u5f0f, \u5fae\u5206\u4ee3\u6570\u65b9\u7a0b\u5f0f
\n\u5e38\u5fae\u5206\u65b9\u7a0b\u5f0f (4)<\/a>: \u5b9f\u7528\u30eb\u30fc\u30c1\u30f3\u306e\u30d9\u30f3\u30c1\u30de\u30fc\u30af (\u5b9f\u7528\u30eb\u30fc\u30c1\u30f3\u306e\u9078\u629e, \u30b9\u30c6\u30a3\u30d5\u3067\u306f\u306a\u3044\u554f\u984c, \u30b9\u30c6\u30a3\u30d5\u306a\u554f\u984c, \u5fae\u5206\u4ee3\u6570\u65b9\u7a0b\u5f0f)<\/p>\n

\u6982\u8981<\/span>1\u6bb5\u6cd5<\/span>\u591a\u6bb5\u6cd5<\/span>\u4f7f\u308f\u308c\u306a\u304f\u306a\u3063\u305f\u89e3\u6cd5<\/span><\/div>
\n

\u5e38\u5fae\u5206\u65b9\u7a0b\u5f0f\u306e\u521d\u671f\u5024\u554f\u984c\u306b\u3064\u3044\u3066<\/h2>\n

\u521d\u671f\u5024\u554f\u984c<\/h4>\n

1\u968e\u306e\u5e38\u5fae\u5206\u65b9\u7a0b\u5f0f\u306f\u6b21\u306e\u5f62\u306b\u66f8\u304f\u3053\u3068\u304c\u3067\u304d\u308b.<\/p>\n

  \r\n  y' = f(t, y(t))\r\n\r\n<\/pre>\n
 
\n\u3053\u3053\u3067, y(t) \u306f\u672a\u77e5\u306e\u95a2\u6570\u3067\u3042\u308b. \u5e38\u5fae\u5206\u65b9\u7a0b\u5f0f\u306e\u521d\u671f\u5024\u554f\u984c(initial value problem)\u306f, \u521d\u671f\u70b9 t0<\/sub> \u306b\u304a\u3051\u308b\u95a2\u6570\u5024 y(t0<\/sub>) \u304c\u4e0e\u3048\u3089\u308c\u305f\u3068\u304d, \u6700\u7d42\u70b9 tf<\/sub> \u306b\u304a\u3051\u308b\u5024 y(tf<\/sub>)\u3092\u6c42\u3081\u308b\u554f\u984c\u3067\u3042\u308b. \u521d\u671f\u70b9\u306b\u304a\u3051\u308b\u95a2\u6570\u5024\u3092\u4e0e\u3048\u308b\u6b21\u5f0f\u306f\u521d\u671f\u6761\u4ef6(initial condition)\u3068\u3088\u3070\u308c\u308b.<\/p>\n
  \r\n  y(t0<\/sub>) = y0<\/sub>\r\n\r\n<\/pre>\n
 <\/p>\n
1\u968e\u306e\u9023\u7acb\u65b9\u7a0b\u5f0f<\/h4>\n
1\u968e\u306e\u5e38\u5fae\u5206\u65b9\u7a0b\u5f0f\u306e\u521d\u671f\u5024\u554f\u984c\u306f, \u30b9\u30ab\u30e9\u30fc y \u3092\u30d9\u30af\u30c8\u30eb y<\/strong> \u306b\u5909\u3048\u3066\u305d\u306e\u307e\u307e\u9023\u7acb\u65b9\u7a0b\u5f0f\u306b\u62e1\u5f35\u3059\u308b\u3053\u3068\u304c\u3067\u304d\u308b.<\/p>\n
  \r\n  y'<\/strong> = f(t, y<\/strong>)\r\n  y<\/strong>(t0<\/sub>) = y0<\/sub><\/strong>\r\n\r\n<\/pre>\n
 
\n\u4f8b\u3048\u3070, 2\u5909\u6570(\u30d9\u30af\u30c8\u30eb\u306e\u8981\u7d20\u6570\u304c2)\u306e\u5834\u5408\u306b\u3064\u3044\u3066\u66f8\u304d\u4e0b\u3059\u3068\u6b21\u306e\u3088\u3046\u306b\u306a\u308b.<\/p>\n
  \r\n  y1<\/sub>' = f1<\/sub>(t, y1<\/sub>, y2<\/sub>)\r\n  y2<\/sub>' = f2<\/sub>(t, y1<\/sub>, y2<\/sub>)\r\n\r\n<\/pre>\n
 
\n\u3053\u306e\u5834\u5408, \u521d\u671f\u6761\u4ef6\u30822\u3064\u5fc5\u8981\u306b\u306a\u308b.<\/p>\n
  \r\n  y1<\/sub>(t0<\/sub>) = y1,0<\/sub>\r\n  y2<\/sub>(t0<\/sub>) = y2,0<\/sub>\r\n\r\n<\/pre>\n
 <\/p>\n
\u9ad8\u968e\u306e\u5e38\u5fae\u5206\u65b9\u7a0b\u5f0f<\/h4>\n
2\u968e\u306e\u5e38\u5fae\u5206\u65b9\u7a0b\u5f0f\u306f1\u968e\u306e\u9023\u7acb\u5e38\u5fae\u5206\u65b9\u7a0b\u5f0f\u306e\u5f62\u306b\u3057\u3066\u6271\u3046\u3053\u3068\u304c\u3067\u304d\u308b. \u4f8b\u3048\u3070, \u6b21\u306e\u65b9\u7a0b\u5f0f\u3092\u8003\u3048\u308b.<\/p>\n
  \r\n  y'' = f(t, y, y')\r\n\r\n<\/pre>\n
 
\n\u3053\u308c\u306f, y1<\/sub> = y, y2<\/sub> = y\u2019 \u3068\u304a\u3051\u3070, \u6b21\u306e\u3088\u3046\u306b1\u968e\u306e\u9023\u7acb\u5e38\u5fae\u5206\u65b9\u7a0b\u5f0f\u306b\u306a\u308b.<\/p>\n
  \r\n  y1' = y2\r\n  y2<\/sub>' = f(t, y1<\/sub>, y2<\/sub>)\r\n\r\n<\/pre>\n
 
\n\u3053\u306e\u5834\u5408, 2\u3064\u306e\u521d\u671f\u6761\u4ef6\u304c\u5fc5\u8981\u306b\u306a\u308b.<\/p>\n
  \r\n  y1<\/sub>(t0<\/sub>) = y(t0<\/sub>) = y0<\/sub>\r\n  y2<\/sub>(t0<\/sub>) = y'(t0<\/sub>) = y'0<\/sub>\r\n\r\n<\/pre>\n
 
\n\u540c\u69d8\u306b\u3057\u3066\u3055\u3089\u306b\u9ad8\u968e\u306e\u5e38\u5fae\u5206\u65b9\u7a0b\u5f0f\u3092\u89e3\u304f\u3053\u3068\u3082\u3067\u304d\u308b.<\/p>\n
\u5e38\u5fae\u5206\u65b9\u7a0b\u5f0f\u306e\u521d\u671f\u5024\u554f\u984c\u306e\u6570\u5024\u89e3\u6cd5<\/h2>\n
\u6570\u5024\u89e3\u6cd5\u306e\u5206\u985e<\/h3>\n
\u5909\u6570 t \u3092\u96e2\u6563\u5316\u3057\u3066, t0<\/sub>, t1<\/sub>, … , tn<\/sub>, … \u3068\u8868\u3059. \u3053\u308c\u3089\u306e\u70b9\u306e\u9593\u9694\u3092\u30b9\u30c6\u30c3\u30d7\u5e45\u3042\u308b\u3044\u306f\u30b9\u30c6\u30c3\u30d7\u5e45(step size)\u3068\u3044\u3044, hn<\/sub> \u3067\u8868\u3059.<\/p>\n
  \r\n  hn<\/sub> = tn+1<\/sub> - tn<\/sub>\r\n\r\n<\/pre>\n
 
\n\u30b9\u30c6\u30c3\u30d7\u5e45\u306f\u5fc5\u305a\u3057\u3082\u4e00\u5b9a\u3067\u306a\u304f\u3066\u3088\u3044\u304c, n \u306b\u3088\u3089\u305a\u4e00\u5b9a\u306e\u5834\u5408\u306b\u306f\u5358\u306b h \u3068\u8868\u3059\u3053\u3068\u304c\u3042\u308b. \u672a\u77e5\u95a2\u6570 y(t) \u306e tn<\/sub> \u306b\u304a\u3051\u308b\u5024 y(tn<\/sub>) \u306e\u8fd1\u4f3c\u5024\u3092 yn<\/sub> \u3068\u8868\u3059\u3053\u3068\u306b\u3059\u308b.<\/p>\n
\u5dee\u5206\u6cd5(\u5fae\u5206\u3092\u5dee\u5206\u3067\u8fd1\u4f3c\u3059\u308b\u65b9\u6cd5)\u306b\u3088\u308a\u6570\u5024\u89e3\u6cd5\u3092\u5c0e\u3044\u3066\u307f\u308b. \u4ee3\u8868\u7684\u306a\u5dee\u5206\u306e\u6c42\u3081\u65b9\u306b\u306f, \u6b21\u306e\u3088\u3046\u306a\u3082\u306e\u304c\u3042\u308b.<\/p>\n
    \n
  • \u524d\u9032\u5dee\u5206: (yn+1<\/sub> – yn<\/sub>) \/ h<\/li>\n
  • \u5f8c\u9000\u5dee\u5206: (yn<\/sub> – yn-1<\/sub>) \/ h<\/li>\n
  • \u4e2d\u5fc3\u5dee\u5206: (yn+1<\/sub> – yn-1<\/sub>) \/ 2h<\/li>\n<\/ul>\n
    \u307e\u305a, \u524d\u9032\u5dee\u5206\u3092\u4f7f\u3063\u3066\u5fae\u5206\u3092\u8fd1\u4f3c\u3059\u308b.<\/p>\n
      \r\n  y'n<\/sub> = f(tn<\/sub>, yn<\/sub>) \u2252 (yn+1<\/sub> - yn<\/sub>) \/ h\r\n\r\n<\/pre>\n
     
    \n\u3053\u308c\u3088\u308a\u6b21\u306e\u8fd1\u4f3c\u5f0f\u304c\u5f97\u3089\u308c\u308b.<\/p>\n
      \r\n  yn+1<\/sub> = yn<\/sub> + hf(tn<\/sub>, yn<\/sub>)\r\n\r\n<\/pre>\n
     
    \n\u3053\u306e\u5f0f\u3092\u4f7f\u3044, \u521d\u671f\u5024 y0<\/sub> \u304b\u3089\u59cb\u3081\u3066, y1<\/sub>, y2<\/sub>, … \u3068\u6b21\u3005\u306b\u5404\u70b9\u306b\u304a\u3051\u308b\u8fd1\u4f3c\u89e3\u3092\u6c42\u3081\u308b\u3053\u3068\u304c\u3067\u304d\u308b\u3053\u3068\u304c\u308f\u304b\u308b. \u3053\u308c\u3092\u30aa\u30a4\u30e9\u30fc\u6cd5<\/strong> (Euler method) \u3068\u3044\u3046.<\/p>\n
    \u5f8c\u9000\u5dee\u5206\u3092\u4f7f\u3063\u305f\u5834\u5408\u306b\u306f, \u540c\u69d8\u306b\u3057\u3066\u6b21\u5f0f\u306b\u3088\u308a\u8fd1\u4f3c\u89e3\u3092\u6c42\u3081\u308b\u3053\u3068\u304c\u3067\u304d\u308b.<\/p>\n
      \r\n  yn+1<\/sub> = yn<\/sub> + hf(tn+1<\/sub>, yn+1<\/sub>)\r\n\r\n<\/pre>\n
     
    \n\u3053\u308c\u3092\u5f8c\u9000\u30aa\u30a4\u30e9\u30fc\u6cd5<\/strong> (backward Euler method) \u3068\u3044\u3046.<\/p>\n
    \u5f8c\u9000\u30aa\u30a4\u30e9\u30fc\u6cd5\u3067\u306f\u53f3\u8fba\u306e f() \u306e\u4e2d\u306b yn+1<\/sub> \u304c\u542b\u307e\u308c\u3066\u304a\u308a, yn+1<\/sub> \u3092\u6c42\u3081\u308b\u305f\u3081\u306b\u65b9\u7a0b\u5f0f\u3092\u89e3\u304b\u306a\u3051\u308c\u3070\u306a\u3089\u306a\u3044(\u4e00\u822c\u7684\u306b\u306f\u975e\u7dda\u5f62\u65b9\u7a0b\u5f0f\u306b\u306a\u308b). \u3053\u306e\u3088\u3046\u306a\u89e3\u6cd5\u3092\u9670\u89e3\u6cd5<\/strong> (\u9670\u7684\u516c\u5f0f, implicit formula) \u3068\u3044\u3046. \u3053\u308c\u306b\u5bfe\u3057\u3066, \u30aa\u30a4\u30e9\u30fc\u6cd5\u306e\u5834\u5408\u306b\u306f\u65b9\u7a0b\u5f0f\u3092\u89e3\u304f\u3053\u3068\u306a\u304f\u76f4\u63a5 yn+1<\/sub> \u3092\u8a08\u7b97\u3059\u308b\u3053\u3068\u304c\u3067\u304d\u305f. \u305d\u306e\u3088\u3046\u306a\u89e3\u6cd5\u3092\u967d\u89e3\u6cd5<\/strong> (\u967d\u7684\u516c\u5f0f, explicit formula) \u3068\u3044\u3046.<\/p>\n
    \u6b21\u306b, \u4e2d\u5fc3\u5dee\u5206\u3092\u4f7f\u3063\u305f\u5834\u5408\u306b\u306f, \u6b21\u5f0f\u306b\u3088\u308a\u8fd1\u4f3c\u89e3\u3092\u6c42\u3081\u308b\u3053\u3068\u304c\u3067\u304d\u308b.<\/p>\n
      \r\n  yn+1<\/sub> = yn-1<\/sub> + 2hf(tn<\/sub>, yn<\/sub>)\r\n\r\n<\/pre>\n
     
    \n\u3053\u308c\u306f\u4e2d\u70b9\u5247\u3068\u3088\u3070\u308c\u308b.<\/p>\n
    \u4e2d\u70b9\u5247\u3067\u306f, yn+1<\/sub> \u306e\u8a08\u7b97\u306b\u306f yn<\/sub> \u306e\u4ed6\u306b\u904e\u53bb\u306e\u5024 yn-1<\/sub> \u3082\u5fc5\u8981\u3067, 2\u3064\u306e\u5024\u3092\u4f7f\u3046\u306e\u30672\u6bb5\u6cd5\u3068\u3088\u3070\u308c, \u591a\u6bb5\u6cd5<\/strong> (\u591a\u6bb5\u968e\u6cd5, multistep method) \u3068\u3088\u3070\u308c\u308b\u89e3\u6cd5\u306e\u3072\u3068\u3064\u306b\u306a\u308b. \u3053\u308c\u306b\u5bfe\u3057\u3066, \u30aa\u30a4\u30e9\u30fc\u6cd5\u3068\u5f8c\u9000\u30aa\u30a4\u30e9\u30fc\u6cd5\u306f yn<\/sub> \u3060\u3051\u3057\u304b\u4f7f\u308f\u306a\u3044\u306e\u3067, \uff11\u6bb5\u6cd5<\/strong> (\uff11\u6bb5\u968e\u6cd5, one-step method) \u3068\u3088\u3070\u308c\u308b\u89e3\u6cd5\u306b\u5206\u985e\u3055\u308c\u308b.<\/p>\n
    \u307e\u3068\u3081\u308b\u3068, \u5e38\u5fae\u5206\u65b9\u7a0b\u5f0f\u306e\u521d\u671f\u5024\u554f\u984c\u306e\u89e3\u6cd5\u3092\u5927\u304d\u304f\u5206\u985e\u3059\u308b\u3068\uff11\u6bb5\u6cd5\u3068\u591a\u6bb5\u6cd5\u304c\u3042\u308a, \u305d\u308c\u305e\u308c\u306b\u967d\u89e3\u6cd5\u3068\u9670\u89e3\u6cd5\u304c\u3042\u308b. \u3053\u3053\u3067\u306f, \uff11\u6bb5\u6cd5\u306e\u967d\u89e3\u6cd5, \u304a\u3088\u3073, \u591a\u6bb5\u6cd5\u306b\u3064\u3044\u3066\u8aac\u660e\u3059\u308b. \uff11\u6bb5\u6cd5\u306e\u9670\u89e3\u6cd5\u306b\u3064\u3044\u3066\u306f\u30b9\u30c6\u30a3\u30d5\u306a\u65b9\u7a0b\u5f0f\u306e\u9805\u3067\u8aac\u660e\u3059\u308b.<\/p>\n
    \u6570\u5024\u89e3\u6cd5\u306e\u6b21\u6570<\/h3>\n
    \u95a2\u6570 y(t) (\u771f\u306e\u89e3) \u3092 tn<\/sub> \u306b\u304a\u3044\u3066\u30c6\u30a4\u30e9\u30fc\u5c55\u958b\u3059\u308b\u3068\u6b21\u306e\u3088\u3046\u306b\u306a\u308b.<\/p>\n
      \r\n  y(tn+1<\/sub>) = y(tn<\/sub>) + hy'(tn<\/sub>) + (1\/2)h2<\/sup>y''(tn<\/sub>) + (1\/6)h3<\/sup>y'''(tn<\/sub>) + ... + (1\/p)hp<\/sup>y(p)<\/sup>(tn<\/sub>) + ...\r\n          = y(tn<\/sub>) + hf(tn<\/sub>, y(tn<\/sub>)) + (1\/2)h2<\/sup>f'(tn<\/sub>, y(tn<\/sub>))+ (1\/6)h3<\/sup>f''(tn<\/sub>, y(tn<\/sub>)) + ... + (1\/p)hp<\/sup>f(p-1)<\/sup>(tn<\/sub>, y(tn<\/sub>)) + ...\r\n\r\n<\/pre>\n
     
    \n\u6570\u5024\u89e3\u6cd5\u306b\u3088\u308b\u8fd1\u4f3c\u89e3 yn+1<\/sub> \u306e\u53f3\u8fba\u304c\u4e0a\u306e\u30c6\u30a4\u30e9\u30fc\u5c55\u958b y(tn+1<\/sub>) \u306ep\u6b21\u306e\u9805\u307e\u3067\u4e00\u81f4\u3059\u308b\u3068\u304d, \u3059\u306a\u308f\u3061\u6b21\u5f0f\u304c\u6210\u308a\u7acb\u3064\u3068\u304d\u306b\u305d\u306e\u6570\u5024\u89e3\u6cd5\u306fp\u6b21\u306e\u65b9\u6cd5\u3067\u3042\u308b\u3068\u3044\u3046.<\/p>\n
      \r\n  yn+1<\/sub> - y(tn+1<\/sub>) = O(hp+1<\/sup>)\r\n\r\n<\/pre>\n
     
    \n\u30aa\u30a4\u30e9\u30fc\u6cd5\u306f\u30c6\u30a4\u30e9\u30fc\u5c55\u958b\u306e1\u6b21(h)\u306e\u9805\u307e\u3067\u3067\u6253\u3061\u5207\u3063\u305f\u3082\u306e\u306b\u7b49\u3057\u3044\u304b\u3089\u660e\u3089\u304b\u306b1\u6b21\u306e\u65b9\u6cd5\u3067\u3042\u308b. \u307e\u305f, \u5f8c\u9000\u30aa\u30a4\u30e9\u30fc\u6cd5\u306f1\u6b21, \u4e2d\u70b9\u5247\u306f2\u6b21\u306e\u65b9\u6cd5\u3067\u3042\u308b\u3053\u3068\u3092\u78ba\u304b\u3081\u308b\u3053\u3068\u304c\u3067\u304d\u308b.<\/p>\n
    \u6570\u5024\u5b9f\u9a13 (1)<\/h3>\n
    <\/span>\u6570\u5024\u89e3\u6cd5\u306e\u6b21\u6570<\/div>
    \n\u6700\u3082\u7c21\u5358\u306a\u5fae\u5206\u65b9\u7a0b\u5f0f\u306e\u521d\u671f\u5024\u554f\u984c\u306e\u4f8b\u3067\u3042\u308b.<\/p>\n
      \r\n  y' = y, y(0) = 1\r\n\r\n<\/pre>\n
     
    \n\u89e3\u6790\u89e3\u306f y = et<\/sup> \u3067\u3042\u308b. \u4e0a\u306e3\u3064\u306e\u516c\u5f0f\u3092\u7528\u3044\u3066\u30b9\u30c6\u30c3\u30d7\u5e45\u3092 0.1, 0.05, 0.01, … \u3068\u3044\u3046\u3088\u3046\u306b\u5c0f\u3055\u304f\u3057\u3066\u3044\u304d\u306a\u304c\u3089\u9806\u306b\u8a08\u7b97\u3059\u308b. t = 1 \u306b\u304a\u3044\u3066\u5f97\u3089\u308c\u305f\u8fd1\u4f3c\u89e3\u306e\u771f\u5024 y = e \u304b\u3089\u306e\u8aa4\u5dee\u3068\u5fc5\u8981\u3068\u3057\u305f\u30b9\u30c6\u30c3\u30d7\u6570\u306e\u5024\u3092\u5bfe\u6570\u30d7\u30ed\u30c3\u30c8\u3059\u308b\u3068\u4e0b\u56f3\u306e\u3088\u3046\u306b\u306a\u3063\u305f. \u6a2a\u8ef8\u306f\u76f8\u5bfe\u8aa4\u5dee(\u53f3\u306b\u884c\u304f\u307b\u3069\u7cbe\u5ea6\u304c\u3088\u3044), \u7e26\u8ef8\u306f\u30b9\u30c6\u30c3\u30d7\u6570(= \u95a2\u6570\u8a55\u4fa1(f()\u306e\u8a08\u7b97)\u56de\u6570)\u3067\u3042\u308b.
    \n 
    \n\"\"
    \n\u30aa\u30a4\u30e9\u30fc\u6cd5\u3068\u5f8c\u9000\u30aa\u30a4\u30e9\u30fc\u6cd5\u3067\u306f, 1,000\u30b9\u30c6\u30c3\u30d7(h = 1.0e-3)\u3067\u304a\u3088\u305d3\u6841\u534a\u306e\u7cbe\u5ea6, 1,000,000\u30b9\u30c6\u30c3\u30d7(h = 1.0e-6)\u3067\u304a\u3088\u305d6\u6841\u534a\u306e\u7cbe\u5ea6\u3068\u306a\u308a, 1\u6b21\u306e\u65b9\u6cd5\u3067\u3042\u308b\u3053\u3068\u304c\u78ba\u304b\u3081\u3089\u308c\u305f. \u4e00\u65b9, \u4e2d\u70b9\u5247\u3067\u306f, 1,000\u30b9\u30c6\u30c3\u30d7(h = 1.0e-3)\u3067\u304a\u3088\u305d6\u6841\u534a\u306e\u7cbe\u5ea6, 1,000,000\u30b9\u30c6\u30c3\u30d7(h = 1.0e-6)\u3067\u304a\u3088\u305d12\u6841\u534a\u306e\u7cbe\u5ea6\u304c\u3042\u308a, 2\u6b21\u306e\u65b9\u6cd5\u3068\u306a\u3063\u3066\u3044\u308b\u3053\u3068\u304c\u308f\u304b\u308b. \u540c\u3058\u8a08\u7b97\u91cf\u3067\u3042\u308c\u3070\u4e2d\u70b9\u5247\u306f\u30aa\u30a4\u30e9\u30fc\u6cd5\u306e\u500d\u306e\u6841\u6570\u3092\u6b63\u3057\u304f\u6c42\u3081\u308b\u3053\u3068\u304c\u3067\u304d\u308b.
    \n<\/div><\/div>\n<\/div>\n
    \n
    \uff11\u6bb5\u6cd5<\/h2>\n
    \u30c6\u30a4\u30e9\u30fc\u6cd5<\/h3>\n
    \u95a2\u6570 y(t) \u306e tn<\/sub> \u306b\u304a\u3051\u308b\u30c6\u30a4\u30e9\u30fc\u5c55\u958b\u3092\u518d\u5ea6\u6b21\u306b\u793a\u3059.<\/p>\n
      \r\n  y(tn+1<\/sub>) = y(tn<\/sub>) + hf(tn<\/sub>, y(tn<\/sub>)) + (1\/2)h2<\/sup>f'(tn<\/sub>, y(tn<\/sub>))+ (1\/6)h3<\/sup>f''(tn<\/sub>, y(tn<\/sub>)) + ... + (1\/p)hp<\/sup>f(p-1)<\/sup>(tn<\/sub>, y(tn<\/sub>)) + ...\r\n\r\n<\/pre>\n
     
    \n\u30aa\u30a4\u30e9\u30fc\u6cd5\u306f\u30c6\u30a4\u30e9\u30fc\u5c55\u958b\u30921\u6b21\u306e\u9805\u307e\u3067\u3067\u6253\u3061\u5207\u3063\u3066\u8fd1\u4f3c\u3057\u305f\u3082\u306e\u306b\u76f8\u5f53\u3057\u305f. \u305d\u3053\u3067, \u3088\u308a\u9ad8\u6b21\u306e\u9805\u307e\u3067\u4f7f\u3063\u305f\u8fd1\u4f3c\u5f0f\u3092\u4f7f\u3048\u3070\u30aa\u30a4\u30e9\u30fc\u6cd5\u3088\u308a\u7cbe\u5ea6\u306e\u3088\u3044, \u3059\u306a\u308f\u3061, \u3088\u308a\u6b21\u6570\u306e\u9ad8\u3044\u89e3\u6cd5\u3092\u5f97\u308b\u3053\u3068\u304c\u3067\u304d\u308b\u3068\u8003\u3048\u3089\u308c\u308b. \u3053\u308c\u3092\u30c6\u30a4\u30e9\u30fc\u6cd5\u3068\u3044\u3046.<\/p>\n
    \u4e0a\u5f0f\u3067f'(t, y(t))\u3068f”(t, y(t))\u306f\u5177\u4f53\u7684\u306b\u306f\u6b21\u306e\u3088\u3046\u306b\u8a08\u7b97\u3059\u308b.<\/p>\n
      \r\n  f'(t, y(t)) = ft<\/sub>(t, y(t)) + fy<\/sub>(t, y(t))\u30fbf(t, y(t))\r\n  f''(t, y(t)) = ftt<\/sub>(t, y(t)) + 2fty<\/sub>(t, y(t))\u30fbf(t, y(t)) + fyy<\/sub>(t, y(t))\u30fbf2<\/sup>(t, y(t)) + ft<\/sub>(t, y(t))\u30fbfy<\/sub>(t, y(t))\r\n    + fy<\/sub>2<\/sup>(t, y(t))\u30fbf(t, y(t))\r\n\r\n<\/pre>\n
     
    \n2\u6b21\u306e\u9805\u3092\u8a08\u7b97\u3059\u308b\u306e\u3067\u3055\u3048\u7d50\u69cb\u8907\u96d1\u306a\u3053\u3068\u304c\u308f\u304b\u308b. \u305d\u306e\u305f\u3081, f()\u306e\u9ad8\u968e\u306e\u5fae\u5206\u304c\u7c21\u5358\u306b\u8a08\u7b97\u3067\u304d\u308b\u95a2\u6570\u3067\u306a\u3044\u9650\u308a\u3053\u306e\u65b9\u6cd5\u3092\u5b9f\u969b\u306b\u9069\u7528\u3059\u308b\u306e\u306f\u96e3\u3057\u3044.<\/p>\n
    \u6570\u5024\u5b9f\u9a13 (2)<\/h3>\n
    <\/span>\u30c6\u30a4\u30e9\u30fc\u6cd5\u306b\u3088\u308b\u8a08\u7b97\u4f8b<\/div>
    \n\u6570\u5024\u89e3\u6cd5\u306e\u6b21\u6570\u306e\u5b9f\u9a13\u3068\u540c\u3058\u6700\u3082\u7c21\u5358\u306a\u5fae\u5206\u65b9\u7a0b\u5f0f\u306e\u521d\u671f\u5024\u554f\u984c\u306e\u4f8b\u3092\u4f7f\u3063\u30661\u6b21(\u30aa\u30a4\u30e9\u30fc\u6cd5), 2\u6b21, 3\u6b21\u306e\u30c6\u30a4\u30e9\u30fc\u6cd5\u306b\u3088\u308b\u8a08\u7b97\u3092\u5b9f\u969b\u306b\u884c\u3063\u3066\u6bd4\u8f03\u3057\u3066\u307f\u308b.<\/p>\n
      \r\n  y' = y, y(0) = 1\r\n\r\n<\/pre>\n
     
    \n\"\"
    \n\u6a2a\u8ef8\u306f\u76f8\u5bfe\u8aa4\u5dee(\u53f3\u306b\u884c\u304f\u307b\u3069\u7cbe\u5ea6\u304c\u3088\u3044), \u7e26\u8ef8\u306f\u30b9\u30c6\u30c3\u30d7\u6570\u306e\u5bfe\u6570\u30d7\u30ed\u30c3\u30c8\u3067\u3042\u308b. \u305d\u308c\u305e\u308c1\u6b21\uff5e3\u6b21\u306e\u50be\u304d\u3092\u793a\u3057\u3066\u3044\u308b. \u305f\u3060\u3057, 1\u30b9\u30c6\u30c3\u30d7\u306e\u8a08\u7b97\u306b\u306f\u4e0a\u306ef'(t, y(t))\u3084f”(t, y(t))\u306e\u8a08\u7b97\u304c\u542b\u307e\u308c\u308b\u306e\u3067, 1\u30b9\u30c6\u30c3\u30d7\u3042\u305f\u308a\u306e\u8a08\u7b97\u91cf\u306f, 2\u6b21\u306e\u30c6\u30a4\u30e9\u30fc\u6cd5\u3067\u306ff()\u306e\u8a08\u7b97\u304c3\u56de\u76f8\u5f53, 3\u6b21\u306e\u30c6\u30a4\u30e9\u30fc\u6cd5\u3067\u306ff()\u306e\u8a08\u7b97\u304c6\u56de\u76f8\u5f53\u3068\u3044\u3046\u3053\u3068\u306b\u306a\u308b. \u305d\u306e\u305f\u3081\u8a08\u7b97\u91cf\u3092\u6307\u6a19\u3068\u3059\u308b\u3068\u4e00\u822c\u7684\u306b\u306f\u30aa\u30a4\u30e9\u30fc\u6cd5\u3068\u306e\u5dee\u306f\u3084\u3084\u7e2e\u307e\u308b. \u305f\u3060\u3057, \u4e0a\u306e\u4f8b\u306e\u65b9\u7a0b\u5f0f\u3067\u306f\u5fae\u5206\u4fc2\u6570\u304c0\u304b1\u3068\u7c21\u5358\u306a\u305f\u30811\u30b9\u30c6\u30c3\u30d7\u3042\u305f\u308a\u306e\u8a08\u7b97\u91cf\u306e\u5dee\u306f\u5c11\u306a\u3044.
    \n<\/div><\/div>\n
    2\u6b21\u306e\u30eb\u30f3\u30b2\u30fb\u30af\u30c3\u30bf\u6cd5<\/h3>\n
    \u30eb\u30f3\u30b2\u30fb\u30af\u30c3\u30bf\u6cd5<\/strong> (Runge-Kutta method) \u306f, f() \u306e\u5fae\u5206\u3092\u8a08\u7b97\u3059\u308b\u3053\u3068\u306a\u304f\u9ad8\u6b21\u306e\u30c6\u30a4\u30e9\u30fc\u5c55\u958b\u306b\u76f8\u5f53\u3059\u308b\u7cbe\u5ea6\u3092\u5f97\u308b1\u6bb5\u6cd5\u3067\u3042\u308b. \u3044\u304f\u3064\u304b\u306et\u3068y\u306b\u3064\u3044\u3066f(t, y)\u3092\u8a08\u7b97\u3057\u305d\u308c\u3089\u306e\u91cd\u307f\u4ed8\u304d\u5e73\u5747\u3092\u4f5c\u3063\u3066\u30c6\u30a4\u30e9\u30fc\u5c55\u958b\u306ep\u6b21\u306e\u9805\u307e\u3067\u4e00\u81f4\u3055\u305b\u308b\u3053\u3068\u306b\u3088\u308ap\u6b21\u306e\u516c\u5f0f\u3092\u4f5c\u308b.<\/p>\n
    2\u6b21\u306e\u30eb\u30f3\u30b2\u30fb\u30af\u30c3\u30bf\u6cd5\u306e\u4e00\u822c\u5f62\u306f\u6b21\u5f0f\u306e\u3068\u304a\u308a\u3067\u3042\u308b. a, b1<\/sub>, b2<\/sub>, c \u306f\u30d1\u30e9\u30e1\u30fc\u30bf\u3067\u3042\u308b.<\/p>\n
      \r\n  k1<\/sub> = f(tn<\/sub>, yn<\/sub>)\r\n  k2<\/sub> = f(tn<\/sub> + ch, yn<\/sub> + hak1<\/sub>)\r\n  yn+1<\/sub> = yn<\/sub> + h(b1<\/sub>k1<\/sub> + b2<\/sub>k2<\/sub>)\r\n\r\n<\/pre>\n
     
    \nk2<\/sub>\u3092 (tn<\/sub>, yn<\/sub>)\u306e\u307e\u308f\u308a\u3067\u30c6\u30a4\u30e9\u30fc\u5c55\u958b\u3059\u308b\u3068\u6b21\u306e\u3088\u3046\u306b\u306a\u308b.<\/p>\n
      \r\n  k2<\/sub> = fn<\/sub> + chft,n<\/sub> + ak1<\/sub>fy,n<\/sub> + ...\r\n\r\n<\/pre>\n
     
    \n\u3053\u308c\u3092\u4e0a\u306e\u5f0f\u306b\u4ee3\u5165\u3057\u3066, \u6b21\u5f0f\u304c\u5f97\u3089\u308c\u308b.<\/p>\n
      \r\n  yn+1<\/sub> = yn<\/sub> + (b1<\/sub> + b2<\/sub>)hfn<\/sub> + b2<\/sub>h2<\/sup>(cft,n<\/sub> + afy,n<\/sub>fn<\/sub> + ... )\r\n\r\n<\/pre>\n
     
    \n\u3053\u306e\u5f0f\u3068\u30c6\u30a4\u30e9\u30fc\u5c55\u958b\u306e\u5f0f\u3092\u6bd4\u8f03\u3057\u3066, h\u3068h2<\/sup>\u306e\u4fc2\u6570\u304c\u7b49\u3057\u3044\u3068\u3059\u308b\u3068\u6b21\u5f0f\u304c\u5f97\u3089\u308c\u308b.<\/p>\n
      \r\n  b1<\/sub> + b2<\/sub> = 1\r\n  b2<\/sub>c = 1\/2\r\n  b2<\/sub>a = 1\/2\r\n\r\n<\/pre>\n
     
    \n\u672a\u77e5\u6570\u304c4\u500b\u3067\u5f0f\u304c3\u672c\u306a\u306e\u3067\u81ea\u7531\u5ea6\u304c1\u3042\u308b. \u3088\u304f\u4f7f\u308f\u308c\u308b\u3082\u306e\u3068\u3057\u3066\u306f, c = 1 \u3068\u3057\u305f\u3082\u306e\u3092\u30db\u30a4\u30f3\u6cd5<\/strong>(Heun method)\u3068\u3088\u3076. \u307e\u305f, c = 1\/2 \u3068\u3057\u305f\u3082\u306e\u3092\u4fee\u6b63\u30aa\u30a4\u30e9\u30fc\u6cd5<\/strong>\u3068\u3088\u3076\u3053\u3068\u304c\u3042\u308b (2\u6b21\u306e\u30eb\u30f3\u30b2\u30fb\u30af\u30c3\u30bf\u6cd5\u3068\u306f\u3042\u307e\u308a\u3088\u3070\u308c\u306a\u3044\u304c, \u3053\u308c\u3089\u306e\u7dcf\u79f0\u3068\u3068\u3089\u3048\u3066\u3088\u3044).<\/p>\n
    \u30db\u30a4\u30f3\u6cd5\u306e\u8a08\u7b97\u5f0f\u306f\u6b21\u306e\u3068\u304a\u308a.<\/p>\n
      \r\n  k1<\/sub> = f(tn<\/sub>, yn<\/sub>)\r\n  k2<\/sub> = f(tn<\/sub> + h, yn<\/sub> + hk1<\/sub>)\r\n  yn+1<\/sub> = yn<\/sub> + h(k1<\/sub> + k2<\/sub>)\/2\r\n\r\n<\/pre>\n
     
    \n\u4fee\u6b63\u30aa\u30a4\u30e9\u30fc\u6cd5\u306e\u8a08\u7b97\u5f0f\u306f\u6b21\u306e\u3068\u304a\u308a.<\/p>\n
      \r\n  k1<\/sub> = f(tn<\/sub>, yn<\/sub>)\r\n  k2<\/sub> = f(tn<\/sub> + h\/2, yn<\/sub> + hk1<\/sub>\/2)\r\n  yn+1<\/sub> = yn<\/sub> + hk2<\/sub>\r\n\r\n<\/pre>\n
     <\/p>\n
    \u6570\u5024\u5b9f\u9a13 (3)<\/h3>\n
    <\/span>\u30db\u30a4\u30f3\u6cd5\u3068\u4fee\u6b63\u30aa\u30a4\u30e9\u30fc\u6cd5\u306e\u8a08\u7b97\u4f8b<\/div>
    \n\u518d\u3073\u6700\u3082\u7c21\u5358\u306a\u5fae\u5206\u65b9\u7a0b\u5f0f\u306e\u521d\u671f\u5024\u554f\u984c\u306e\u4f8b\u3067\u30db\u30a4\u30f3\u6cd5\u3068\u4fee\u6b63\u30aa\u30a4\u30e9\u30fc\u6cd5\u306e\u7cbe\u5ea6\u3092\u78ba\u8a8d\u3057\u3066\u307f\u308b.<\/p>\n
      \r\n  y' = y, y(0) = 1\r\n\r\n<\/pre>\n
     
    \n\"\"
    \n\u6a2a\u8ef8\u306f\u76f8\u5bfe\u8aa4\u5dee(\u53f3\u306b\u884c\u304f\u307b\u3069\u7cbe\u5ea6\u304c\u3088\u3044), \u7e26\u8ef8\u306f\u30b9\u30c6\u30c3\u30d7\u6570\u306e\u5bfe\u6570\u30d7\u30ed\u30c3\u30c8\u3067\u3042\u308b. \u30db\u30a4\u30f3\u6cd5\u3068\u4fee\u6b63\u30aa\u30a4\u30e9\u30fc\u6cd5\u306e\u8a08\u7b97\u7d50\u679c\u306f\u307b\u307c\u540c\u3058\u3067\u3042\u308b. 2\u6b21\u306e\u50be\u304d\u3092\u78ba\u8a8d\u3067\u304d, \u6700\u521d\u306e\u65b9\u3067\u793a\u3057\u305f\u4e2d\u70b9\u5247\u3068\u4f3c\u305f\u30b0\u30e9\u30d5\u306b\u306a\u3063\u305f. \u305f\u3060\u3057, \u8a08\u7b97\u5f0f\u304b\u3089\u308f\u304b\u308b\u3088\u3046\u306b\u30db\u30a4\u30f3\u6cd5\u3068\u4fee\u6b63\u30aa\u30a4\u30e9\u30fc\u6cd5\u3067\u306f1\u30b9\u30c6\u30c3\u30d7\u3042\u305f\u308a2\u56de\u306e\u95a2\u6570\u8a55\u4fa1(f()\u306e\u8a08\u7b97)\u3092\u884c\u3063\u3066\u3044\u308b\u306e\u306b\u5bfe\u3057\u3066, \u4e2d\u70b9\u5247\u3067\u306f1\u30b9\u30c6\u30c3\u30d7\u3042\u305f\u308a1\u56de\u306e\u95a2\u6570\u8a55\u4fa1\u3057\u304b\u884c\u308f\u306a\u304f\u3066\u3088\u3044.
    \n<\/div><\/div>\n
    \u30eb\u30f3\u30b2\u30fb\u30af\u30c3\u30bf\u6cd5\u306e\u4e00\u822c\u5f62<\/h3>\n
    \u4e0a\u3068\u540c\u69d8\u306b3\u6b21, 4\u6b21\u30fb\u30fb\u30fb\u3068\u9ad8\u6b21\u306e\u30eb\u30f3\u30b2\u30fb\u30af\u30c3\u30bf\u6cd5\u306e\u516c\u5f0f\u3092\u4f5c\u308b\u3053\u3068\u304c\u3067\u304d, \u591a\u304f\u306e\u516c\u5f0f\u304c\u63d0\u6848\u3055\u308c\u3066\u3044\u308b.<\/p>\n
    \u30eb\u30f3\u30b2\u30fb\u30af\u30c3\u30bf\u6cd5\u306f\u4e00\u822c\u7684\u306b\u6b21\u306e\u3088\u3046\u306b\u8868\u3055\u308c\u308b.<\/p>\n
      \r\n  yn+1<\/sub> = yn<\/sub> + h\u03a3bi<\/sub>ki<\/sub> (\u03a3 \u306f i = 1, 2, ... , s)\r\n  ki<\/sub> = f(tn<\/sub> + ci<\/sub>h, yn<\/sub> + h\u03a3aij<\/sub>kj<\/sub>) (\u03a3 \u306f j = 1, 2, ... , s) (i = 1, 2, ... , s)\r\n\r\n<\/pre>\n
     
    \ns \u306f\u6bb5\u6570<\/strong>(stage)\u3068\u3088\u3070\u308c, 1\u30b9\u30c6\u30c3\u30d7\u9032\u3080\u306e\u306b\u5fc5\u8981\u306a f(t, y) \u306e\u8a08\u7b97\u56de\u6570\u3092\u8868\u3059(1\u6bb5\u6cd5\u3084\u591a\u6bb5\u6cd5\u306e”\u6bb5”\u3068\u306f\u7570\u306a\u308b). aij<\/sub>, bi<\/sub> \u304a\u3088\u3073 ci<\/sub> \u306f\u30d1\u30e9\u30e1\u30fc\u30bf\u3067, bi<\/sub> \u304a\u3088\u3073 ci<\/sub> \u306f\u6b21\u5f0f\u3092\u6e80\u305f\u3059.<\/p>\n
      \r\n  ci<\/sub> = \u03a3aij<\/sub> (\u03a3 \u306f j = 1, 2, ... , s) (i = 1, 2, ... , s)\r\n  \u03a3bi<\/sub> = 1 (\u03a3 \u306f i = 1, 2, ... , s)\r\n\r\n<\/pre>\n
     
    \n\u30eb\u30f3\u30b2\u30fb\u30af\u30c3\u30bf\u6cd5\u306e\u30d1\u30e9\u30e1\u30fc\u30bf\u306b\u306f\u6b21\u306e\u3088\u3046\u306a\u8868\u8a18\u6cd5\u304c\u3042\u308b (\u30d6\u30c3\u30c1\u30e3\u30fc\u914d\u5217<\/strong>\u3068\u3044\u3046).<\/p>\n
      \r\n  c1<\/sub> | a11<\/sub>  a12<\/sub> ... a1s<\/sub>\r\n  c2<\/sub> | a21<\/sub>  a22<\/sub> ... a2s<\/sub>\r\n  c3<\/sub> | a31<\/sub>  a32<\/sub> ... a3s<\/sub>\r\n  .  | .  .\r\n  cs<\/sub> | as1<\/sub>  as2<\/sub> ... ass<\/sub>\r\n  ----------------------------\r\n     | b1<\/sub>   b2<\/sub>  ... bs<\/sub>  \r\n    \r\n<\/pre>\n
     
    \naij<\/sub>\u306e\u5024\u306b\u3088\u3063\u30663\u3064\u306e\u5834\u5408\u306b\u5206\u3051\u3089\u308c\u308b.<\/p>\n
    (1) aij<\/sub> = 0 (i \u2264 j) \u306e\u5834\u5408 (\u53f3\u4e0a\u4e09\u89d2\u90e8\u5206\u3068\u5bfe\u89d2\u90e8\u5206\u304c0)
    \n\u967d\u89e3\u6cd5. (\u5358\u306b\u300c\u30eb\u30f3\u30b2\u30fb\u30af\u30c3\u30bf\u6cd5\u300d\u3042\u308b\u3044\u306f\u300c\u967d\u7684(explicit)\u30eb\u30f3\u30b2\u30fb\u30af\u30c3\u30bf\u6cd5<\/strong>\u300d\u3068\u3044\u3046)
    \nc1<\/sub> = 0 \u3067\u3042\u308b.
    \n\u30d6\u30c3\u30c1\u30e3\u30fc\u914d\u5217\u3067\u306faij<\/sub>\u306e\u53f3\u4e0a\u4e09\u89d2\u90e8\u5206\u3068\u5bfe\u89d2\u90e8\u5206\u3092\u66f8\u304b\u306a\u3044(\u7a7a\u767d\u306b\u3059\u308b).<\/p>\n
    (2) aij<\/sub> = 0 (i < j) \u306e\u5834\u5408 (\u53f3\u4e0a\u4e09\u89d2\u90e8\u5206\u304c0)
    \n\u534a\u9670\u89e3\u6cd5. (\u534a\u9670\u7684(semi-implicit)\u30eb\u30f3\u30b2\u30fb\u30af\u30c3\u30bf\u6cd5\u3068\u3044\u3046)<\/p>\n
    (3) aij<\/sub> \u2260 0 (i < j) \u306e\u5834\u5408 (\u53f3\u4e0a\u4e09\u89d2\u90e8\u5206\u306b0\u3067\u306a\u3044\u3082\u306e\u304c\u3042\u308b)
    \n\u9670\u89e3\u6cd5. (\u9670\u7684(implicit)\u30eb\u30f3\u30b2\u30fb\u30af\u30c3\u30bf\u6cd5<\/strong>\u3068\u3044\u3046)<\/p>\n
    \u3053\u3053\u3067\u306f(1)\u306b\u3064\u3044\u3066\u306e\u307f\u8aac\u660e\u3059\u308b. (2), (3)\u306b\u3064\u3044\u3066\u306f\u5225\u9014\u30b9\u30c6\u30a3\u30d5\u306a\u65b9\u7a0b\u5f0f\u306e\u9805\u3067\u89e6\u308c\u308b.<\/p>\n
    \u4e0a\u3067\u793a\u3057\u305f\u30db\u30a4\u30f3\u6cd5\u3068\u4fee\u6b63\u30aa\u30a4\u30e9\u30fc\u6cd5\u3092\u30d6\u30c3\u30c1\u30e3\u30fc\u914d\u5217\u3067\u8868\u3059\u3068\u6b21\u306e\u3088\u3046\u306b\u306a\u308b.<\/p>\n
    \u30db\u30a4\u30f3\u6cd5<\/h4>\n
      \r\n  0   |\r\n  1   | 1\r\n  ----------------\r\n      | 1\/2  1\/2  \r\n    \r\n<\/pre>\n
     <\/p>\n
    \u4fee\u6b63\u30aa\u30a4\u30e9\u30fc\u6cd5<\/h4>\n
      \r\n  0   |\r\n  1\/2 | 1\/2\r\n  -------------\r\n      | 0   1  \r\n    \r\n<\/pre>\n
     
    \n(\u967d\u7684)\u30eb\u30f3\u30b2\u30fb\u30af\u30c3\u30bf\u6cd5\u3067\u306f\u3067\u304d\u308b\u3060\u3051\u9ad8\u6b21\u306e\u516c\u5f0f\u306b\u306a\u308b\u3088\u3046\u306b\u30d1\u30e9\u30e1\u30fc\u30bf\u3092\u6c7a\u3081\u308b\u304c, s\u6bb5\u306a\u3089\u3070s\u6b21\u306b\u306a\u308b\u308f\u3051\u3067\u306f\u306a\u304f, s\u6bb5\u306e\u516c\u5f0f\u304c\u9054\u6210\u3057\u3046\u308b\u6700\u5927\u306e\u6b21\u6570p\u306f\u6b21\u306e\u3088\u3046\u306b\u306a\u308b\u3053\u3068\u304c\u308f\u304b\u3063\u3066\u3044\u308b.<\/p>\n\n\n\n\n
    \u6bb5\u6570 s<\/td>\n1<\/td>\n2<\/td>\n3<\/td>\n4<\/td>\n5<\/td>\n6<\/td>\n7<\/td>\n8<\/td>\n9<\/td>\n10<\/td>\n11<\/td>\n<\/tr>\n
    \u6700\u5927\u6b21\u6570 p<\/td>\n1<\/td>\n2<\/td>\n3<\/td>\n4<\/td>\n4<\/td>\n5<\/td>\n6<\/td>\n6<\/td>\n7<\/td>\n7<\/td>\n8<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n
    4\u6b21\u306e\u30eb\u30f3\u30b2\u30fb\u30af\u30c3\u30bf\u6cd5<\/h3>\n
    4\u6bb54\u6b21\u306e\u516c\u5f0f\u3092\u4f5c\u308d\u3046\u3068\u3059\u308b\u3068, \u30d1\u30e9\u30e1\u30fc\u30bf\u304c12\u500b\u3067\u81ea\u7531\u5ea6\u304c4\u306a\u306e\u3067\u69d8\u3005\u306a\u516c\u5f0f\u3092\u4f5c\u308b\u3053\u3068\u304c\u53ef\u80fd\u3067\u3042\u308b. \u305d\u306e\u4e2d\u3067\u6700\u826f\u306e\u3082\u306e\u3092\u9078\u3076\u305f\u3081\u306e\u591a\u304f\u306e\u7814\u7a76\u304c\u884c\u308f\u308c\u305f. \u4e2d\u3067\u3082\u30af\u30c3\u30bf\u304c\u9078\u3093\u3060\u4ee5\u4e0b\u306e2\u7d44\u306e\u4fc2\u6570\u304c\u6709\u540d\u3067\u3042\u308b.<\/p>\n
    \u53e4\u5178\u7684\u30eb\u30f3\u30b2\u30fb\u30af\u30c3\u30bf\u6cd5<\/h4>\n
    \u5358\u306b\u300c\u30eb\u30f3\u30b2\u30fb\u30af\u30c3\u30bf\u6cd5\u300d\u3068\u3044\u3046\u3068\u3053\u306e\u516c\u5f0f\u3092\u6307\u3059.<\/p>\n
      \r\n  0   |\r\n  1\/2 | 1\/2\r\n  1\/2 | 0   1\/2\r\n  1   | 0   0    1\r\n  -----------------------\r\n      | 1\/6 1\/3 1\/3 1\/6 \r\n    \r\n<\/pre>\n
     
    \n\u8a08\u7b97\u5f0f\u306f\u6b21\u306e\u3068\u304a\u308a.<\/p>\n
      \r\n  k1<\/sub> = f(tn<\/sub>, yn<\/sub>)\r\n  k2<\/sub> = f(tn<\/sub> + h\/2, yn<\/sub> + hk1<\/sub>\/2)\r\n  k3<\/sub> = f(tn<\/sub> + h\/2, yn<\/sub> + hk2<\/sub>\/2)\r\n  k4<\/sub> = f(tn<\/sub> + h, yn<\/sub> + hk3<\/sub>)\r\n  yn+1<\/sub> = yn<\/sub> + h(k1<\/sub> + 2k2<\/sub> + 2k3<\/sub> + k4<\/sub>)\/6\r\n\r\n<\/pre>\n
     <\/p>\n
    \u30af\u30c3\u30bf\u306e3\/8\u516c\u5f0f<\/h4>\n
      \r\n  0   |\r\n  1\/3 |  1\/3\r\n  2\/3 | -1\/3  1\r\n  1   |  1   -1   1\r\n  -----------------------\r\n      |  1\/8 3\/8 3\/8 1\/8 \r\n    \r\n<\/pre>\n
     
    \n\u8a08\u7b97\u5f0f\u306f\u6b21\u306e\u3068\u304a\u308a.<\/p>\n
      \r\n  k1<\/sub> = f(tn<\/sub>, yn<\/sub>)\r\n  k2<\/sub> = f(tn<\/sub> + h\/2, yn<\/sub> + hk1<\/sub>\/3)\r\n  k3<\/sub> = f(tn<\/sub> + h\/2, yn<\/sub> + h(-k1<\/sub>\/3 + k2<\/sub>))\r\n  k4<\/sub> = f(, tn<\/sub> + h, yn<\/sub> + h(k1<\/sub> - k2<\/sub> + k3<\/sub>))\r\n  yn+1<\/sub> = yn<\/sub> + h(k1<\/sub> + 3k2<\/sub> + 3k3<\/sub> + k4<\/sub>)\/8\r\n\r\n<\/pre>\n
     
    \n\u30eb\u30f3\u30b2\u30fb\u30af\u30c3\u30bf\u6cd5\u4e00\u65cf\u306b\u304a\u3044\u3066\u6700\u3082\u3088\u304f\u4f7f\u308f\u308c\u3066\u3044\u308b\u306e\u306f4\u6b21\u306e\u53e4\u5178\u7684\u30eb\u30f3\u30b2\u30fb\u30af\u30c3\u30bf\u6cd5\u3067\u3042\u308b. \u3044\u304f\u3064\u304b\u3042\u308b4\u6bb54\u6b21\u306e\u516c\u5f0f\u9593\u3067\u306e\u7cbe\u5ea6\u306e\u5dee\u306f\u307b\u3068\u3093\u3069\u306a\u3044\u304c, \u3053\u306e\u516c\u5f0f\u306f\u4fc2\u6570aij<\/sub>\u306b0\u304c\u591a\u304f, \u3059\u3079\u3066\u306e\u4fc2\u6570\u304c\u6b63\u3067\u3042\u308b\u306a\u3069\u306e\u7406\u7531\u306b\u3088\u308a\u4ed6\u306e\u3082\u306e\u3088\u308a\u4f7f\u3044\u3084\u3059\u3044. \u307e\u305f, 5\u6b21\u306e\u516c\u5f0f\u306f6\u6bb5\u4ee5\u4e0a\u3067\u306a\u3044\u3068\u4f5c\u308c\u306a\u3044\u305f\u3081, \u3053\u308c\u4ee5\u4e0a\u9ad8\u6b21\u306e\u3082\u306e\u3092\u6c42\u3081\u308b\u3068\u8907\u96d1\u306b\u306a\u308b\u305f\u3081\u3053\u308c\u304c\u666e\u53ca\u3057\u305f\u306e\u304b\u3082\u3057\u308c\u306a\u3044.<\/p>\n
    \u6570\u5024\u5b9f\u9a13 (4)<\/h3>\n
    <\/span>4\u6b21\u306e\u30eb\u30f3\u30b2\u30fb\u30af\u30c3\u30bf\u6cd5\u306e\u8a08\u7b97\u4f8b<\/div>
    \n\u6700\u3082\u7c21\u5358\u306a\u5fae\u5206\u65b9\u7a0b\u5f0f\u306e\u521d\u671f\u5024\u554f\u984c\u306e\u4f8b\u30674\u6b21\u306e\u53e4\u5178\u7684\u30eb\u30f3\u30b2\u30fb\u30af\u30c3\u30bf\u6cd5\u304a\u3088\u30733\/8\u516c\u5f0f\u306e\u7cbe\u5ea6\u3092\u78ba\u8a8d\u3057\u3066\u307f\u308b.<\/p>\n
      \r\n  y' = y, y(0) = 1\r\n\r\n<\/pre>\n
     
    \n\"\"
    \n\u6a2a\u8ef8\u306f\u76f8\u5bfe\u8aa4\u5dee(\u53f3\u306b\u884c\u304f\u307b\u3069\u7cbe\u5ea6\u304c\u3088\u3044), \u7e26\u8ef8\u306f\u30b9\u30c6\u30c3\u30d7\u6570\u306e\u5bfe\u6570\u30d7\u30ed\u30c3\u30c8\u3067\u3042\u308b. \u53e4\u5178\u7684\u30eb\u30f3\u30b2\u30fb\u30af\u30c3\u30bf\u6cd5\u304a\u3088\u30733\/8\u516c\u5f0f\u306e\u8a08\u7b97\u7d50\u679c\u306f\u307b\u307c\u540c\u3058\u3067\u3042\u308b. \u4e21\u65b9\u3068\u30824\u6b21\u306e\u50be\u304d\u3092\u3057\u3066\u3044\u308b\u3053\u3068\u304c\u78ba\u8a8d\u3067\u304d\u308b. \u305f\u3060\u3057, 1\u30b9\u30c6\u30c3\u30d7\u3042\u305f\u308a4\u56de\u306e\u95a2\u6570\u8a55\u4fa1(f()\u306e\u8a08\u7b97)\u304c\u5fc5\u8981\u3067\u3042\u308b.
    \n<\/div><\/div>\n<\/div>\n
    \n
    \u591a\u6bb5\u6cd5<\/h2>\n
    1\u6bb5\u6cd5\u306f yn+1<\/sub> \u306e\u8a08\u7b97\u306b\u73fe\u5728\u306e\u5024 yn<\/sub> \u306e\u307f\u3092\u4f7f\u3046\u89e3\u6cd5\u3067\u3042\u3063\u305f\u304c, \u904e\u53bb\u306e\u5024(\u305d\u308c\u307e\u3067\u306b\u8a08\u7b97\u6e08\u306e\u5024) yn-1<\/sub>, yn-2<\/sub>, … \u3082\u4f7f\u3063\u305f\u65b9\u304c\u3088\u3044\u7d50\u679c\u304c\u5f97\u3089\u308c\u305d\u3046\u306b\u601d\u308f\u308c\u308b. \u3053\u306e\u30a2\u30a4\u30c7\u30a2\u3092\u5b9f\u73fe\u3057\u305f\u306e\u304c\u591a\u6bb5\u6cd5\u3067\u3042\u308b.<\/p>\n
    1\u968e\u306e\u5e38\u5fae\u5206\u65b9\u7a0b\u5f0f\u306e\u7a4d\u5206\u5f62\u3092\u8003\u3048\u308b.<\/p>\n
      \r\n  y(tn+1<\/sub>) = y(tn<\/sub>) + \u222b f(t, y(t)) dt [tn<\/sub>, tn+1<\/sub>]\r\n\r\n<\/pre>\n
     
    \n\u3053\u3053\u3067, \u88ab\u7a4d\u5206\u95a2\u6570f()\u3092\u88dc\u9593\u591a\u9805\u5f0f\u3067\u8fd1\u4f3c\u3057\u3066\u53f3\u8fba\u306e\u7a4d\u5206\u3092\u6c42\u3081\u308b\u65b9\u6cd5\u3092\u4e00\u822c\u306b\u30a2\u30c0\u30e0\u30b9\u6cd5<\/strong>(Adams method)\u3068\u3044\u3046.<\/p>\n
    \u30a2\u30c0\u30e0\u30b9\u30fb\u30d0\u30b7\u30e5\u30d5\u30a9\u30fc\u30b9\u6cd5<\/h3>\n
    y(tn+1<\/sub>) \u3068 y(tn<\/sub>)\u306f\u8fd1\u4f3c\u5024 yn+1<\/sub> \u3068 yn<\/sub> \u3092\u4f7f\u3044, \u53f3\u8fba\u306e\u7a4d\u5206\u306f (ti<\/sub>, fi<\/sub>) (i = n – k +1, …, n) \u3092\u901a\u308b\u88dc\u9593\u591a\u9805\u5f0f p(t) \u3067 f(t, y(t)) \u3092\u7f6e\u304d\u63db\u3048\u308b\u3068, \u6b21\u306e\u3088\u3046\u306b k\u6bb5\u306e\u967d\u89e3\u6cd5\u306e\u516c\u5f0f\u304c\u5f97\u3089\u308c, \u30a2\u30c0\u30e0\u30b9\u30fb\u30d0\u30b7\u30e5\u30d5\u30a9\u30fc\u30b9\u6cd5<\/strong> (Adams-Bashforth method) \u3068\u3088\u3070\u308c\u308b. \u305f\u3060\u3057, fi<\/sub> = f(ti<\/sub>, yi<\/sub>) \u3092\u8868\u3057, yi<\/sub> (i = n – k + 1, …, n) \u304c\u8a08\u7b97\u6e08\u306a\u306e\u3067 fi<\/sub> \u306f\u3059\u3050\u306b\u6c42\u3081\u308b\u3053\u3068\u304c\u3067\u304d\u308b. \u3053\u306e\u516c\u5f0f\u306f, k\u6b21\u306e\u516c\u5f0f\u306b\u306a\u308b.<\/p>\n
      \r\n  yn+1<\/sub> = yn<\/sub> + \u222b p(t) dt [tn<\/sub>, tn+1<\/sub>]\r\n       = yn<\/sub> + h\u03a3\u03b3j<\/sub>\u25bdj<\/sup>fn<\/sub>  (\u03a3\u306f j = 0, ... , k-1)\r\n\r\n<\/pre>\n
     
    \n\u305f\u3060\u3057,<\/p>\n
      \r\n  \u25bd0<\/sup>fn<\/sub> = fn<\/sub>,  \u25bdj+1<\/sup>fn<\/sub> = \u25bdj<\/sup>fn<\/sub> - \u25bdj<\/sup>fn-1<\/sub>  (\u5f8c\u9000\u5dee\u5206)\r\n    \u03b3j<\/sub> = (-1)j<\/sup> \u222b C(-s, j) ds [0, 1]\r\n\r\n<\/pre>\n
     
    \n\u3067\u3042\u308b. \u305f\u3060\u3057, C(n, k)\u306f\u4e8c\u9805\u4fc2\u6570\u3067\u3042\u308b. \u306a\u304a, \u03b3j<\/sub> \u306f\u76f4\u63a5\u8a08\u7b97\u3059\u308b\u3088\u308a\u3082\u6b21\u306e\u6f38\u5316\u5f0f\u3092\u7528\u3044\u3066\u6c42\u3081\u308b\u3068\u3088\u3044.<\/p>\n
      \r\n  \u03b30<\/sub> = 1,  \u03b3j<\/sub> + (1\/2)\u03b3j-1<\/sub> + (1\/3)\u03b3j-2<\/sub> + ... + (1\/(j+1))\u03b30<\/sub> = 1\r\n\r\n<\/pre>\n
     
    \nj = 0 \uff5e 7 \u306e\u3068\u304d\u306e\u5024\u306f\u6b21\u306e\u3068\u304a\u308a\u3067\u3042\u308b.<\/p>\n\n\n\n\n
    j<\/td>\n0<\/td>\n1<\/td>\n2<\/td>\n3<\/td>\n4<\/td>\n5<\/td>\n6<\/td>\n7<\/td>\n<\/tr>\n
    \u03b3j<\/sub><\/td>\n1<\/td>\n1\/2<\/td>\n5\/12<\/td>\n3\/8<\/td>\n251\/720<\/td>\n95\/288<\/td>\n19087\/60480<\/td>\n5257\/17280<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n
    \u3053\u306e\u65b9\u6cd5\u3092\u56f3\u306b\u793a\u3059\u3068\u6b21\u306e\u3088\u3046\u306b\u306a\u308b. \u533a\u9593 [tn<\/sub>, tn+1<\/sub>] \u306e\u7a4d\u5206\u3092\u8a08\u7b97\u3059\u308b\u305f\u3081\u306b, \u533a\u9593 [tn-k+1<\/sub>, tn<\/sub>] \u306e\u88dc\u9593\u30c7\u30fc\u30bf\u3092\u4f7f\u7528\u3057\u305f\u88dc\u9593\u95a2\u6570 p(t) \u3092\u4f7f\u3046. \u56f3\u3067, \u9ed2\u4e38\u306f\u88dc\u9593\u70b9, \u8272\u3064\u304d\u306e\u90e8\u5206\u306f\u7a4d\u5206\u7bc4\u56f2\u3092\u8868\u3059.
    \n 
    \n\"ode_fig-1\"<\/p>\n
    k = 1 \uff5e 5 \u306b\u3064\u3044\u3066, fn<\/sub>, fn-1<\/sub>, fn-2<\/sub>, … \u306b\u3064\u3044\u3066\u6574\u7406\u3059\u308b\u3068\u6b21\u5f0f\u306e\u3088\u3046\u306b\u306a\u308b.<\/p>\n
      \r\n  k = 1 : yn+1<\/sub> = yn<\/sub> + hfn<\/sub>\r\n  k = 2 : yn+1<\/sub> = yn<\/sub> + h\/2(3fn<\/sub> - fn-1<\/sub>)\r\n  k = 3 : yn+1<\/sub> = yn<\/sub> + h\/12(23fn<\/sub> - 16fn-1<\/sub> + 5fn-2<\/sub>)\r\n  k = 4 : yn+1<\/sub> = yn<\/sub> + h\/24(55fn<\/sub> - 59fn-1<\/sub> + 37fn-2<\/sub> - 9fn-3<\/sub>)\r\n  k = 5 : yn+1<\/sub> = yn<\/sub> + h\/720(1901fn<\/sub> - 2774fn-1<\/sub> + 2616fn-2<\/sub> - 1274fn-3<\/sub> + 251fn-4<\/sub>)\r\n\r\n<\/pre>\n
     
    \nk = 1 \u306e\u5834\u5408\u306f1\u6bb5\u6cd5\u3068\u307f\u308b\u3053\u3068\u304c\u3067\u304d\u30aa\u30a4\u30e9\u30fc\u6cd5\u306b\u4e00\u81f4\u3059\u308b.
    \n\u306a\u304a, \u30a2\u30c0\u30e0\u30b9\u30fb\u30d0\u30b7\u30e5\u30d5\u30a9\u30fc\u30b9\u6cd5\u306fk\u6b21\u3067\u3042\u308b.<\/p>\n
    \u30a2\u30c0\u30e0\u30b9\u30fb\u30e2\u30eb\u30c8\u30f3\u6cd5<\/h3>\n
    \u30a2\u30c0\u30e0\u30b9\u30fb\u30d0\u30b7\u30e5\u30d5\u30a9\u30fc\u30b9\u6cd5\u3067\u306f\u88dc\u9593\u95a2\u6570\u3092\u4f7f\u3063\u3066\u5916\u633f\u306b\u3088\u308a\u7a4d\u5206\u3092\u6c42\u3081\u305f\u304c, \u7cbe\u5ea6\u3092\u4e0a\u3052\u308b\u305f\u3081\u306b tn+1<\/sub>\u3092\u52a0\u3048\u3066\u533a\u9593 [tn-k+1<\/sub>, tn+1<\/sub>] \u306e\u88dc\u9593\u30c7\u30fc\u30bf\u3092\u4f7f\u3046, \u3059\u306a\u308f\u3061, (ti<\/sub>, fi<\/sub>) (i = n – k + 1, …, n, n + 1) \u3092\u901a\u308b\u88dc\u9593\u591a\u9805\u5f0f p*(t) \u3092\u4f7f\u3044, \u5916\u633f\u3059\u308b\u3053\u3068\u3092\u907f\u3051\u305f\u306e\u304c\u30a2\u30c0\u30e0\u30b9\u30fb\u30e2\u30eb\u30c8\u30f3\u6cd5<\/strong> (Adams-Moulton method) \u3067\u3042\u308b. \u3053\u3061\u3089\u306e\u65b9\u304c\u7cbe\u5ea6\u304c\u3088\u3055\u305d\u3046\u3067\u3042\u308b\u304c, \u305d\u306e\u4ee3\u308f\u308a\u53f3\u8fba\u306b yn+1<\/sub> \u304c\u73fe\u308c\u9670\u89e3\u6cd5\u306b\u306a\u308b. \u5f0f\u306f\u6b21\u306e\u3068\u304a\u308a\u3067\u3042\u308b. \u3053\u306e\u516c\u5f0f\u306f, k+1\u6b21\u306e\u516c\u5f0f\u306b\u306a\u308b.<\/p>\n
      \r\n  yn+1<\/sub> = yn<\/sub> + \u222b p*(x) dx [tn<\/sub>, tn+1<\/sub>]\r\n          = yn<\/sub> + h\u03a3\u03b3*<\/sup>j<\/sub>\u25bdj<\/sup>fn+1<\/sub>  (\u03a3\u306f j = 0, ... , k)\r\n\r\n<\/pre>\n
     
    \n\u305f\u3060\u3057,<\/p>\n
      \r\n  \u25bd0<\/sup>fn<\/sub> = fn<\/sub>,  \u25bdj+1<\/sup>fn<\/sub> = \u25bdj<\/sup>fn<\/sub> - \u25bdj<\/sup>fn-1<\/sub>  (\u5f8c\u9000\u5dee\u5206)\r\n  \u03b3*<\/sup>j<\/sub> = (-1)j<\/sup> \u222b C(-s+1, j) ds [0, 1]\r\n\r\n<\/pre>\n
     
    \n\u3067\u3042\u308b. \u305f\u3060\u3057, C(n, k)\u306f\u4e8c\u9805\u4fc2\u6570\u3067\u3042\u308b. \u306a\u304a, \u03b3*<\/sup>j<\/sub> \u306f\u6b21\u306e\u6f38\u5316\u5f0f\u3092\u7528\u3044\u3066\u8a08\u7b97\u3059\u308b\u3068\u3088\u3044.<\/p>\n
      \r\n  \u03b3*<\/sup>0<\/sub> = 1,  \u03b3*<\/sup>j<\/sub> + (1\/2)\u03b3*<\/sup>j-1<\/sub> + (1\/3)\u03b3*<\/sup>j-2<\/sub> + ... + (1\/(j+1))\u03b3*<\/sup>0<\/sub> = 0\r\n\r\n<\/pre>\n
     
    \nj = 0 \uff5e 7 \u306e\u3068\u304d\u306e\u5024\u306f\u6b21\u306e\u3068\u304a\u308a\u3067\u3042\u308b.<\/p>\n\n\n\n\n
    j<\/td>\n0<\/td>\n1<\/td>\n2<\/td>\n3<\/td>\n4<\/td>\n5<\/td>\n6<\/td>\n7<\/td>\n<\/tr>\n
    \u03b3*<\/sup>j<\/sub><\/td>\n1<\/td>\n-1\/2<\/td>\n-1\/12<\/td>\n-1\/24<\/td>\n-19\/720<\/td>\n-3\/160<\/td>\n-863\/60480<\/td>\n-275\/24192<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n
    \u3053\u306e\u65b9\u6cd5\u3092\u56f3\u306b\u793a\u3059\u3068\u6b21\u306e\u3088\u3046\u306b\u306a\u308b. \u4e0b\u56f3\u3067, \u9ed2\u4e38\u306f\u88dc\u9593\u70b9, \u8272\u3064\u304d\u306e\u90e8\u5206\u306f\u7a4d\u5206\u7bc4\u56f2\u3092\u8868\u3059. \u7a4d\u5206\u7bc4\u56f2\u306f\u88dc\u9593\u533a\u9593\u306b\u542b\u307e\u308c\u3066\u304a\u308a, \u5916\u633f\u3055\u308c\u3066\u3044\u306a\u3044.<\/p>\n
    \"ode_fig-2\"<\/p>\n
     
    \nk = 0 \uff5e 5 \u306b\u3064\u3044\u3066, fn+1<\/sub>, fn<\/sub>, fn-1<\/sub>, … \u306b\u3064\u3044\u3066\u6574\u7406\u3059\u308b\u3068\u6b21\u5f0f\u306e\u3088\u3046\u306b\u306a\u308b.<\/p>\n
      \r\n  k = 0 : yn+1<\/sub> = yn<\/sub> + hfn+1<\/sub>\r\n  k = 1 : yn+1<\/sub> = yn<\/sub> + h\/2(fn+1<\/sub> + fn<\/sub>)\r\n  k = 2 : yn+1<\/sub> = yn<\/sub> + h\/12(5fn+1<\/sub> + 8fn<\/sub> - fn-1<\/sub>)\r\n  k = 3 : yn+1<\/sub> = yn<\/sub> + h\/24(9fn+1<\/sub> + 19fn<\/sub> - 5fn-1<\/sub> + fn-2<\/sub>)\r\n  k = 4 : yn+1<\/sub> = yn<\/sub> + h\/720(251fn+1<\/sub> + 646fn<\/sub> - 264fn-1<\/sub> + 106fn-2<\/sub> - 19fn-3<\/sub>)\r\n  k = 5 : yn+1<\/sub> = yn<\/sub> + h\/1440 (475fn+1<\/sub> + 1427fn<\/sub> - 798fn-1<\/sub> + 482fn-2<\/sub> - 173fn-3<\/sub> + 27fn-4<\/sub>)\r\n\r\n<\/pre>\n
     
    \nk = 0 \u306e\u5834\u5408\u306f1\u6bb5\u6cd5\u3068\u307f\u308b\u3053\u3068\u304c\u3067\u304d\u5f8c\u9000\u30aa\u30a4\u30e9\u30fc\u6cd5\u306b\u4e00\u81f4\u3059\u308b. k = 1 \u306e\u5834\u5408\u306f\u53f0\u5f62\u5247\u3068\u3088\u3070\u308c\u308b\u516c\u5f0f\u306b\u4e00\u81f4\u3059\u308b.
    \n\u306a\u304a, \u30a2\u30c0\u30e0\u30b9\u30fb\u30e2\u30eb\u30c8\u30f3\u6cd5\u306fk+1\u6b21\u3067\u3042\u308b.<\/p>\n
    \u6570\u5024\u5b9f\u9a13 (5)<\/h3>\n
    <\/span>\u30a2\u30c0\u30e0\u30b9\u30fb\u30d0\u30b7\u30e5\u30d5\u30a9\u30fc\u30b9\u6cd5\u3068\u30a2\u30c0\u30e0\u30b9\u30fb\u30e2\u30eb\u30c8\u30f3\u6cd5\u306e\u8a08\u7b97\u4f8b<\/div>
    \n\u6700\u3082\u7c21\u5358\u306a\u5fae\u5206\u65b9\u7a0b\u5f0f\u306e\u521d\u671f\u5024\u554f\u984c\u306e\u4f8b\u3067 k = 4 \u306e\u30a2\u30c0\u30e0\u30b9\u30fb\u30d0\u30b7\u30e5\u30d5\u30a9\u30fc\u30b9\u6cd5(\u967d\u89e3\u6cd5)\u3068\u30a2\u30c0\u30e0\u30b9\u30fb\u30e2\u30eb\u30c8\u30f3\u6cd5(\u9670\u89e3\u6cd5)\u5358\u4f53(\u6ce8)\u3067\u8a08\u7b97\u3092\u884c\u3044\u7cbe\u5ea6\u3092\u78ba\u8a8d\u3059\u308b. (\u6ce8: \u5b9f\u52d9\u3067\u306f\u5f8c\u3067\u8aac\u660e\u3059\u308b\u3088\u3046\u306b\u7d44\u307f\u5408\u308f\u305b\u3066\u4f7f\u7528\u3057, \u5358\u4f53\u3067\u306f\u7528\u3044\u306a\u3044)<\/p>\n
      \r\n  y' = y, y(0) = 1\r\n\r\n<\/pre>\n
     
    \n\"\"
    \n\u6a2a\u8ef8\u306f\u76f8\u5bfe\u8aa4\u5dee(\u53f3\u306b\u884c\u304f\u307b\u3069\u7cbe\u5ea6\u304c\u3088\u3044), \u7e26\u8ef8\u306f\u30b9\u30c6\u30c3\u30d7\u6570\u306e\u5bfe\u6570\u30d7\u30ed\u30c3\u30c8\u3067\u3042\u308b. 4\u6b21\u306e\u30eb\u30f3\u30b2\u30fb\u30af\u30c3\u30bf\u6cd5\u306e\u8a08\u7b97\u7d50\u679c\u3082\u793a\u3057\u305f. \u591a\u6bb5\u6cd5\u3067\u306f\u9670\u89e3\u6cd5\u306e\u65b9\u304c\u7cbe\u5ea6\u304c\u3088\u3044\u3053\u3068\u304c\u78ba\u8a8d\u3067\u304d\u308b.\u3000\u591a\u6bb5\u6cd5\u306e\u307b\u3046\u304c\u540c\u3058\u6b21\u6570\u3067\u306f\u5c11\u3057\u7cbe\u5ea6\u304c\u60aa\u3044\u3088\u3046\u306b\u611f\u3058\u3089\u308c\u308b\u304c, \u30b9\u30c6\u30c3\u30d7\u6570\u3067\u8868\u793a\u3057\u3066\u3044\u308b\u7e26\u8ef8\u3092\u95a2\u6570\u8a55\u4fa1\u56de\u6570(\u8a08\u7b97\u91cf)\u3067\u8868\u793a\u3057\u76f4\u3059\u3068\u307b\u3068\u3093\u3069\u5dee\u304c\u306a\u304f\u306a\u308b.<\/p>\n
    \u591a\u6bb5\u6cd5\u3067\u306f, \u306f\u3058\u3081\u306b\u51fa\u767a\u5024 y1<\/sub>, …, yk-1<\/sub> \u306e\u5024\u3092\u6c42\u3081\u3066\u304a\u304f\u5fc5\u8981\u304c\u3042\u308b\u304c, \u305d\u306e\u305f\u3081\u306b1\u6bb5\u89e3\u6cd5\u306a\u3069\u3092\u88dc\u52a9\u7684\u306b\u4f7f\u308f\u306a\u3051\u308c\u3070\u306a\u3089\u306a\u3044. \u3053\u306e\u7cbe\u5ea6\u304c\u60aa\u3044\u3068\u4ee5\u964d\u306e\u8a08\u7b97\u5024\u306b\u5f71\u97ff\u3059\u308b\u305f\u3081, \u4f7f\u7528\u3059\u308b\u591a\u6bb5\u6cd5\u306e\u516c\u5f0f\u306e\u7cbe\u5ea6\u306b\u5341\u5206\u898b\u5408\u3046\u3060\u3051\u306e\u7cbe\u5ea6\u3092\u6301\u3063\u305f\u65b9\u6cd5\u3092\u4f7f\u7528\u3059\u308b\u5fc5\u8981\u304c\u3042\u308b. \u3053\u3053\u3067\u306f\u6700\u521d\u306e4\u30b9\u30c6\u30c3\u30d7\u76ee\u307e\u3067\u306f\u30eb\u30f3\u30b2\u30fb\u30af\u30c3\u30bf\u6cd5(4\u6b21)\u306b\u3088\u308a\u8a08\u7b97\u3092\u884c\u3063\u305f.
    \n<\/div><\/div>\n
    \u4e88\u6e2c\u5b50\u30fb\u4fee\u6b63\u5b50\u6cd5<\/h3>\n
    \u591a\u6bb5\u6cd5\u306b\u304a\u3044\u3066\u306f, \u967d\u89e3\u6cd5\u306f\u8a08\u7b97\u304c\u5bb9\u6613\u3067\u3042\u308b\u304c\u6e80\u8db3\u306e\u3044\u304f\u7cbe\u5ea6\u304c\u5f97\u3089\u308c\u306a\u3044. \u4e00\u65b9, \u9670\u89e3\u6cd5\u306f\u7cbe\u5ea6\u306f\u3088\u3044\u304c\u4e00\u822c\u7684\u306b\u306f\u975e\u7dda\u5f62\u65b9\u7a0b\u5f0f\u3092\u6bce\u56de\u89e3\u304b\u306a\u3051\u308c\u3070\u306a\u3089\u306a\u3044\u305f\u3081\u8a08\u7b97\u304c\u96e3\u3057\u3044.<\/p>\n
    \u305d\u3053\u3067\u5e83\u304f\u4f7f\u308f\u308c\u3066\u3044\u308b\u65b9\u6cd5\u304c\u4e88\u6e2c\u5b50\u30fb\u4fee\u6b63\u5b50\u6cd5(predictor-corrector method)\u3067\u3042\u308b. \u307e\u305a\u4e88\u6e2c\u5b50(predictor)\u3068\u3088\u3070\u308c\u308b\u967d\u89e3\u6cd5\u306b\u3088\u308ayn+1<\/sub>\u306e\u7b2c1\u8fd1\u4f3c\u3092\u6c42\u3081\u308b. \u6b21\u306b\u305d\u306e\u5024\u3092\u4f7f\u3063\u3066\u4fee\u6b63\u5b50(corrector)\u3068\u3088\u3070\u308c\u308b\u9670\u89e3\u6cd5\u30921\u56de\uff5e\u6570\u56de\u9069\u7528\u3057\u3053\u308c\u3092\u6539\u826f\u3059\u308b\u65b9\u6cd5\u3067\u3042\u308b.<\/p>\n
    \u4e88\u6e2c\u5b50\u3068\u4fee\u6b63\u5b50\u306e\u7d44\u307f\u5408\u308f\u305b\u306b\u306f\u30d0\u30ea\u30a8\u30fc\u30b7\u30e7\u30f3\u304c\u3042\u308b\u304c, \u4e88\u6e2c\u5b50\u306b\u30a2\u30c0\u30e0\u30b9\u30fb\u30d0\u30b7\u30e5\u30d5\u30a9\u30fc\u30b9\u6cd5, \u4fee\u6b63\u5b50\u306b\u30a2\u30c0\u30e0\u30b9\u30fb\u30e2\u30eb\u30c8\u30f3\u6cd5\u3092\u4f7f\u3063\u305f\u4e88\u6e2c\u5b50\u30fb\u4fee\u6b63\u5b50\u6cd5\u3092\u30a2\u30c0\u30e0\u30b9\u30fb\u30d0\u30b7\u30e5\u30d5\u30a9\u30fc\u30b9\u30fb\u30e2\u30eb\u30c8\u30f3\u6cd5(\u3042\u308b\u3044\u306f\u5358\u306b\u30a2\u30c0\u30e0\u30b9\u6cd5)\u3068\u3088\u3073, \u5e83\u304f\u4f7f\u308f\u308c\u3066\u3044\u308b.<\/p>\n
    \u4e88\u6e2c\u5b50\u30fb\u4fee\u6b63\u5b50\u6cd5\u306e\u8a08\u7b97\u624b\u9806\u306f, (1) \u4e88\u6e2c\u5b50\u3092\u9069\u7528\u3057\u3066 Yn+1<\/sub>\u3092\u6c42\u3081\u308b(P), (2) \u305d\u308c\u3092\u4f7f\u3063\u3066\u95a2\u6570\u5024 fn+1<\/sub>\u3092\u8a08\u7b97\u3059\u308b(E), (3) \u4fee\u6b63\u5b50\u3092\u9069\u7528\u3057\u3066 Yn+1<\/sub>\u3092\u66f4\u65b0\u3059\u308b(C), (4) \u66f4\u65b0\u3057\u305f Yn+1<\/sub>\u3092\u4f7f\u3063\u3066 fn+1<\/sub>\u3092\u66f4\u65b0\u3059\u308b(E) \u3068\u306a\u308b. \u3053\u308c\u3092 PECE \u3068\u8868\u3059\u3053\u3068\u306b\u3059\u308b\u3068, \u6700\u5f8c\u306e E \u3092\u7701\u7565\u3057\u3066 PEC \u3068\u3059\u308b\u65b9\u6cd5, \u4fee\u6b63\u5b50\u306e\u9069\u7528\u3092 2\u56de\u7e70\u308a\u8fd4\u3057\u3066 PECECE \u3068\u3059\u308b\u65b9\u6cd5\u306a\u3069, \u30d0\u30ea\u30a8\u30fc\u30b7\u30e7\u30f3\u304c\u3042\u308b.<\/p>\n
    \u6570\u5024\u5b9f\u9a13 (6)<\/h3>\n
    <\/span>\u4e88\u6e2c\u5b50\u30fb\u4fee\u6b63\u5b50\u6cd5(\u30a2\u30c0\u30e0\u30b9\u30fb\u30d0\u30b7\u30e5\u30d5\u30a9\u30fc\u30b9\u30fb\u30e2\u30eb\u30c8\u30f3\u6cd5)\u306e\u8a08\u7b97\u4f8b<\/div>
    \n\u6700\u3082\u7c21\u5358\u306a\u5fae\u5206\u65b9\u7a0b\u5f0f\u306e\u521d\u671f\u5024\u554f\u984c\u306e\u4f8b\u3067 k = 4 \u306e\u4e88\u6e2c\u5b50\u30fb\u4fee\u6b63\u5b50\u6cd5(\u30a2\u30c0\u30e0\u30b9\u30fb\u30d0\u30b7\u30e5\u30d5\u30a9\u30fc\u30b9\u30fb\u30e2\u30eb\u30c8\u30f3\u6cd5)\u306e\u7cbe\u5ea6\u3092\u78ba\u8a8d\u3057\u3066\u307f\u308b.<\/p>\n
      \r\n  y' = y, y(0) = 1\r\n\r\n<\/pre>\n
     
    \n\u3053\u3053\u3067\u306fPEC\u3068PECE\u306e2\u7a2e\u985e\u306b\u3064\u3044\u3066\u8a66\u3057\u3066\u307f\u305f. \u51fa\u767a\u5024\u3092\u6c42\u3081\u308b\u305f\u3081\u306b\u306f4\u6b21\u306e\u30eb\u30f3\u30b2\u30fb\u30af\u30c3\u30bf\u6cd5\u3092\u4f7f\u7528\u3057\u305f.
    \n 
    \n\"\"
    \n\u6a2a\u8ef8\u306f\u76f8\u5bfe\u8aa4\u5dee(\u53f3\u306b\u884c\u304f\u307b\u3069\u7cbe\u5ea6\u304c\u3088\u3044), \u7e26\u8ef8\u306f\u4eca\u56de\u306f\u95a2\u6570\u8a55\u4fa1\u56de\u6570(\u30b9\u30c6\u30c3\u30d7\u6570\u3067\u306f\u306a\u3044)\u306e\u5bfe\u6570\u30d7\u30ed\u30c3\u30c8\u3068\u3057\u305f. \u30b9\u30c6\u30c3\u30d7\u3042\u305f\u308a\u306e\u95a2\u6570\u8a55\u4fa1\u56de\u6570\u304cPEC\u3068PECE\u3067\u7570\u306a\u308b\u305f\u3081, \u7dcf\u5408\u7684\u306b\u8a55\u4fa1\u3059\u308b\u3088\u3046\u306b\u7e26\u8ef8\u306e\u30d7\u30ed\u30c3\u30c8\u3092\u5909\u3048\u305f.
    \n\u305d\u306ePEC\u3068PECE\u3067\u3042\u308b\u304c, \u3053\u306e\u4f8b\u306f\u7c21\u5358\u306a\u95a2\u6570\u3067\u3042\u308b\u305f\u3081\u304b\u307b\u3068\u3093\u3069\u8a08\u7b97\u7d50\u679c\u306b\u5dee\u304c\u51fa\u306a\u304b\u3063\u305f. \u305d\u306e\u305f\u3081, 2\u56de\u76ee\u306eE\u306e\u8a08\u7b97\u306e\u3076\u3093\u3060\u3051PECE\u306e\u95a2\u6570\u8a55\u4fa1\u56de\u6570\u304c\u5897\u3048\u305f\u3060\u3051\u3068\u3044\u3046\u7d50\u679c\u306b\u306a\u3063\u305f. \u306a\u304a, \u5225\u9014\u96e3\u3057\u3044\u95a2\u6570\u3067\u8a66\u3057\u3066\u307f\u308b\u3068PECE\u306e\u307b\u3046\u304c\u3088\u304f\u306a\u308b\u3053\u3068\u304c\u3042\u3063\u305f.
    \n<\/div><\/div>\n<\/div>\n
    \n
    \u4f7f\u308f\u308c\u306a\u304f\u306a\u3063\u305f\u89e3\u6cd5<\/h2>\n
    \u3088\u304f\u77e5\u3089\u308c\u3066\u3044\u308b\u304c\u73fe\u5728\u3067\u306f\u4f7f\u308f\u308c\u306a\u304f\u306a\u3063\u305f\u89e3\u6cd5\u306b\u3064\u3044\u3066\u3053\u3053\u3067\u89e6\u308c\u3066\u304a\u304f.<\/p>\n
    \u30eb\u30f3\u30b2\u30fb\u30af\u30c3\u30bf\u30fb\u30b8\u30eb\u6cd5<\/h3>\n
    \u30eb\u30f3\u30b2\u30fb\u30af\u30c3\u30bf\u30fb\u30b8\u30eb\u6cd5(Runge-Kutta-Gill method)\u306f\u4fc2\u6570\u304c\u6b21\u306e\u3088\u3046\u306b\u8868\u3055\u308c\u308b4\u6b21\u306e\u30eb\u30f3\u30b2\u30fb\u30af\u30c3\u30bf\u6cd5\u3067\u3042\u308b.<\/p>\n
      \r\n  0   |\r\n  1\/2 |    1\/2\r\n  1\/2 | (\u221a2-1)\/2 (2-\u221a2)\/2\r\n  1   |     0     (-\u221a2)\/2  1+(\u221a2)\/2\r\n  ---------------------------------------\r\n      |    1\/6   (2-\u221a2)\/6  (2+\u221a2)\/6  1\/6 \r\n    \r\n<\/pre>\n
     
    \nGill\u306e\u63d0\u6848\u306f\u4fc2\u6570\u306e\u9055\u3044\u3060\u3051\u3092\u898b\u3066\u3082\u3088\u304f\u308f\u304b\u3089\u306a\u3044\u304c, \u305d\u306e\u7279\u9577\u306f\u6b21\u306e2\u70b9\u3067\u3042\u308b.<\/p>\n
    (1) \u666e\u901a\u306e4\u6b21\u306e\u30eb\u30f3\u30b2\u30fb\u30af\u30c3\u30bf\u6cd5\u3067\u306f\u8a08\u7b97\u306e\u305f\u3081\u306b4\u500b\u306e\u5909\u6570(\u9023\u7acb\u65b9\u7a0b\u5f0f\u306e\u5834\u5408\u306b\u306f\u914d\u5217)\u304c\u5fc5\u8981\u306a\u306e\u306b\u5bfe\u3057, \u3053\u306e\u65b9\u6cd5\u3067\u306f\u4fc2\u6570\u3092\u5de5\u592b\u3057\u30663\u500b\u306e\u5909\u6570(\u914d\u5217)\u3060\u3051\u3067\u8a08\u7b97\u3067\u304d\u308b\u3088\u3046\u306b\u306a\u3063\u3066\u3044\u3066\u305d\u306e\u624b\u9806\u3082\u793a\u3055\u308c\u3066\u3044\u308b. [\u30e1\u30e2\u30ea\u306e\u7bc0\u7d04]\n
    (2) \u4e38\u3081\u8aa4\u5dee\u3092\u6e1b\u3089\u3059\u305f\u3081\u306e\u624b\u9806\u304c\u542b\u307e\u308c\u3066\u3044\u308b(\u3053\u308c\u306f\u4ed6\u306e\u30eb\u30f3\u30b2\u30fb\u30af\u30c3\u30bf\u6cd5\u306b\u3082\u9069\u7528\u53ef\u80fd). [\u7cbe\u5ea6\u306e\u5411\u4e0a]\n
    \u30e1\u30e2\u30ea\u304c\u5c11\u306a\u304f\u304a\u305d\u3089\u304f\u5358\u7cbe\u5ea6\u8a08\u7b97\u304c\u4e3b\u6d41\u3067\u3042\u3063\u305f\u521d\u671f\u306e\u8a08\u7b97\u6a5f\u306b\u304a\u3051\u308b\u30d7\u30ed\u30b0\u30e9\u30e0\u306b\u9069\u3057\u305f\u65b9\u6cd5\u3060\u3063\u305f\u3068\u601d\u308f\u308c\u308b. \u304b\u3064\u3066\u5e83\u304f\u4f7f\u308f\u308c\u3066\u3044\u305f\u304c, \u30e1\u30e2\u30ea\u304c\u8c4a\u5bcc\u3067\u500d\u7cbe\u5ea6\u8a08\u7b97\u304c\u3042\u305f\u308a\u307e\u3048\u306e\u73fe\u4ee3\u3067\u306f\u5fc5\u8981\u6027\u304c\u8584\u308c\u4f7f\u308f\u308c\u306a\u304f\u306a\u3063\u305f.<\/p>\n
    \u3053\u3053\u3067, (2)\u306b\u3064\u3044\u3066\u5c11\u3057\u8aac\u660e\u3059\u308b.
    \n\u5e38\u5fae\u5206\u65b9\u7a0b\u5f0f\u306e\u57fa\u672c\u7684\u306a\u6570\u5024\u89e3\u6cd5\u3067\u306f\u5404\u30b9\u30c6\u30c3\u30d7\u306b\u304a\u3044\u3066\u95a2\u6570\u5024\u306e\u66f4\u65b0\u5dee\u5206(\u3053\u3053\u3067\u306f k\u3068\u8868\u3059)\u3092\u8a08\u7b97\u3057, \u6b21\u306e\u3088\u3046\u306b\u3057\u3066\u95a2\u6570\u5024\u3092\u66f4\u65b0\u3059\u308b.<\/p>\n
      \r\n  yn+1<\/sub> = yn<\/sub> + k\r\n\r\n<\/pre>\n
     
    \n\u3053\u306e\u8a08\u7b97\u3092\u8a08\u7b97\u6a5f\u306e\u6d6e\u52d5\u5c0f\u6570\u30c7\u30fc\u30bf(\u30ef\u30fc\u30c9\u9577\u306f\u6709\u9650)\u3067\u884c\u3046\u3068\u4e0b\u56f3\u306e\u3088\u3046\u306b\u306a\u308b.<\/p>\n
    \"ode_fig-5\"<\/p>\n
    \u3053\u3053\u3067, k \u3092\u30ef\u30fc\u30c9\u9577\u3044\u3063\u3071\u3044\u6b63\u3057\u304f\u6c42\u3081\u305f\u3068\u3057\u3066\u3082, \u3053\u306e\u66f4\u65b0\u306b\u4f7f\u7528\u3055\u308c\u308b\u306e\u306f\u8272\u306e\u3064\u3044\u305f\u4e00\u90e8\u3060\u3051\u3067\u6b8b\u308a\u306e\u90e8\u5206\u306f\u6368\u3066\u3089\u308c\u3066\u3057\u307e\u3046(\u4e38\u3081\u8aa4\u5dee\u3068\u3044\u3046). \u305d\u3053\u3067, \u3053\u306e\u6368\u3066\u3089\u308c\u308b\u90e8\u5206\u3092\u5225\u306e\u5834\u6240\u306b\u7d2f\u7a4d\u3057\u3066\u304a\u304d, \u8272\u306e\u3064\u3044\u305f\u90e8\u5206\u306b\u982d\u304c\u51fa\u308b\u307b\u3069\u306b\u8caf\u307e\u308b\u305f\u3073\u306bk\u306b\u52a0\u3048\u3066\u95a2\u6570\u5024\u306e\u66f4\u65b0\u306b\u4f7f\u7528\u3059\u308b\u3053\u3068\u306b\u3088\u308a\u4e38\u3081\u8aa4\u5dee\u3092\u6e1b\u3089\u3059\u65b9\u6cd5\u304c\u8003\u6848\u3055\u308c\u305f. \u3053\u306e\u65b9\u6cd5\u304c\u30eb\u30f3\u30b2\u30fb\u30af\u30c3\u30bf\u30fb\u30b8\u30eb\u6cd5\u306b\u4f7f\u308f\u308c\u3066\u3044\u308b.<\/p>\n
    \u6570\u5024\u5b9f\u9a13 (7)<\/h3>\n
    <\/span>\u30eb\u30f3\u30b2\u30fb\u30af\u30c3\u30bf\u30fb\u30b8\u30eb\u6cd5\u306e\u8a08\u7b97\u4f8b<\/div>
    \n\u6700\u3082\u7c21\u5358\u306a\u5fae\u5206\u65b9\u7a0b\u5f0f\u306e\u521d\u671f\u5024\u554f\u984c\u306e\u4f8b\u3067\u30eb\u30f3\u30b2\u30fb\u30af\u30c3\u30bf\u30fb\u30b8\u30eb\u6cd5\u306e\u7cbe\u5ea6\u3092\u78ba\u8a8d\u3057\u3066\u307f\u308b.<\/p>\n
      \r\n  y' = y, y(0) = 1\r\n\r\n<\/pre>\n
     
    \n\"\"
    \n\u6a2a\u8ef8\u306f\u76f8\u5bfe\u8aa4\u5dee(\u53f3\u306b\u884c\u304f\u307b\u3069\u7cbe\u5ea6\u304c\u3088\u3044), \u7e26\u8ef8\u306f\u95a2\u6570\u8a55\u4fa1\u56de\u6570\u3092\u5bfe\u6570\u30d7\u30ed\u30c3\u30c8\u3068\u3057\u305f. \u6bd4\u8f03\u306e\u305f\u3081\u666e\u901a\u306e\u30eb\u30f3\u30b2\u30fb\u30af\u30c3\u30bf\u6cd5\u3082\u30d7\u30ed\u30c3\u30c8\u3057\u3066\u3044\u308b.<\/p>\n
    \u76f8\u5bfe\u8aa4\u5dee\u304c1e-14\u4ed8\u8fd1\u307e\u3067\u306f\u5dee\u304c\u306a\u3044\u304c, \u666e\u901a\u306e\u30eb\u30f3\u30b2\u30fb\u30af\u30c3\u30bf\u6cd5\u3067\u306f1e-15\u3042\u305f\u308a\u3067\u982d\u6253\u3061\u306b\u306a\u308a, \u305d\u306e\u5148\u306f\u523b\u307f\u5e45\u3092\u5c0f\u3055\u304f\u3059\u308b\u3068\u3080\u3057\u308d\u8aa4\u5dee\u304c\u5927\u304d\u304f\u306a\u3063\u3066\u3057\u307e\u3046. \u3053\u308c\u306f, 64\u30d3\u30c3\u30c8\u500d\u7cbe\u5ea6\u3067\u306f1\u56de\u306e\u6f14\u7b97\u306e\u7cbe\u5ea6\u306e\u9650\u754c\u306f2e-16\u304f\u3089\u3044\u306a\u306e\u3067, \u3053\u306e\u5834\u5408\u306f\u7dcf\u5408\u7684\u306b\u306f1e-15\u304f\u3089\u3044\u304c\u6700\u9ad8\u7cbe\u5ea6\u3060\u3068\u601d\u308f\u308c, \u305d\u308c\u4ee5\u4e0a\u8a08\u7b97\u3092\u7e70\u308a\u8fd4\u3059\u3068\u4e38\u3081\u8aa4\u5dee\u304c\u7a4d\u307f\u91cd\u306a\u3063\u3066\u3044\u304d\u304b\u3048\u3063\u3066\u7cbe\u5ea6\u304c\u843d\u3061\u3066\u3057\u307e\u3046\u3068\u8003\u3048\u3089\u308c\u308b.<\/p>\n
    \u4e00\u65b9\u3067, \u30eb\u30f3\u30b2\u30fb\u30af\u30c3\u30bf\u30fb\u30b8\u30eb\u6cd5\u3067\u306f\u6700\u5f8c\u306e\u307b\u3046\u306e\u8aa4\u5dee\u306f1.6e-16\u307e\u305f\u306f0\u3068\u306a\u3063\u305f(\u56f3\u3067\u7dda\u304c\u5207\u308c\u3066\u3044\u308b\u3068\u3053\u308d\u306f\u5b8c\u5168\u306b0\u306e\u305f\u3081\u30d7\u30ed\u30c3\u30c8\u3055\u308c\u3066\u3044\u306a\u3044). \u3064\u307e\u308a, \u6700\u7d421\u30d3\u30c3\u30c8\u304c\u9055\u3046\u304b\u3069\u3046\u304b\u306e\u3081\u3044\u3063\u3071\u3044\u306e\u7cbe\u5ea6\u3067\u8a08\u7b97\u3055\u308c\u305f\u3053\u3068\u3092\u793a\u3057\u3066\u3044\u308b.
    \n<\/div><\/div>\n
    \u30df\u30eb\u30f3\u6cd5<\/h3>\n
    \u30a2\u30c0\u30e0\u30b9\u6cd5\u3068\u4f3c\u3066\u3044\u308b\u304c, \u7a4d\u5206\u7bc4\u56f2\u304c\u7570\u306a\u308b\u6b21\u306e\u7a4d\u5206\u5f62\u3092\u8003\u3048\u308b.<\/p>\n
      \r\n  y(tn+1<\/sub>) = y(tn-1<\/sub>) + \u222b f(t, y(t)) dt [tn-1<\/sub>, tn+1<\/sub>]\r\n\r\n<\/pre>\n
     
    \n\u30a2\u30c0\u30e0\u30b9\u6cd5\u3068\u540c\u69d8\u306b (ti<\/sub>, fi<\/sub>) (i = n – k + 1, …, n) \u3092\u901a\u308b\u88dc\u9593\u591a\u9805\u5f0fp(t)\u3067\u7a4d\u5206\u3092\u8fd1\u4f3c\u3059\u308b\u3068, \u6b21\u306e\u967d\u89e3\u6cd5\u306e\u516c\u5f0f\u304c\u5f97\u3089\u308c\u308b.<\/p>\n
      \r\n  yn+1<\/sub> = yn-1<\/sub> + h\u03a3\u03baj<\/sub>\u25bdj<\/sup>fn<\/sub>  (\u03a3\u306f j = 0, ... , k-1)\r\n\r\n<\/pre>\n
     
    \n\u305f\u3060\u3057,<\/p>\n
      \r\n  \u03baj<\/sub> = (-1)j<\/sup> \u222b C(-s, j) ds [-1, 1]\r\n\r\n<\/pre>\n
     
    \n\u307e\u305f, (ti<\/sub>, fi<\/sub>) (i = n – k +1, …, n+1) \u3092\u901a\u308b\u88dc\u9593\u591a\u9805\u5f0f p*(x) \u3067\u7a4d\u5206\u3092\u8fd1\u4f3c\u3059\u308b\u3068, \u6b21\u306e\u9670\u89e3\u6cd5\u306e\u516c\u5f0f\u304c\u5f97\u3089\u308c\u308b.<\/p>\n
      \r\n  yn+1<\/sub> = yn-1<\/sub> + h\u03a3\u03ba*j<\/sub>\u25bdj<\/sup>fn+1<\/sub>  (\u03a3\u306f j = 0, ... , k)\r\n\r\n<\/pre>\n
     
    \n\u305f\u3060\u3057,<\/p>\n
      \r\n  \u03ba*j<\/sub> = (-1)j<\/sup> \u222b C(-s+1, j) ds [-1, 1]\r\n\r\n<\/pre>\n
     
    \n\u4e88\u6e2c\u5b50\u30fb\u4fee\u6b63\u5b50\u6cd5\u306e\u4e00\u7a2e\u306b\u30df\u30eb\u30f3\u6cd5(Milne method)\u3068\u3088\u3070\u308c\u308b\u65b9\u6cd5\u304c\u3042\u308a, \u4e0a\u306e\u9670\u89e3\u6cd5\u306e\u5f0f\u306b\u304a\u3044\u3066 k = 2 \u3068\u3057\u305f\u6b21\u306e\u5f0f\u304c\u305d\u306e\u4fee\u6b63\u5b50\u3068\u3057\u3066\u4f7f\u308f\u308c\u3066\u3044\u308b.<\/p>\n
      \r\n  yn+1<\/sub> = yn-1<\/sub> + h\/3 (fn+1<\/sub> + 4fn<\/sub> + fn-1<\/sub>)\r\n\r\n<\/pre>\n
     
    \n\u30df\u30eb\u30f3\u6cd5\u306e\u4e88\u6e2c\u5b50\u306b\u306f, \u7a4d\u5206\u7bc4\u56f2\u3092\u3055\u3089\u306b [tn-3<\/sub>, tn+1<\/sub>] \u306b\u5e83\u3052\u305f\u7a4d\u5206\u5f62\u3088\u308a\u5c0e\u304b\u308c\u308b\u6b21\u5f0f\u304c\u4f7f\u308f\u308c\u3066\u3044\u308b.<\/p>\n
      \r\n  yn+1<\/sub> = yn-3<\/sub> + h\/3 (8fn<\/sub> - 4fn-1<\/sub> + 8fn-2<\/sub>)\r\n\r\n<\/pre>\n
     
    \n\u30df\u30eb\u30f3\u6cd5\u306f4\u6b21\u306e\u516c\u5f0f\u3067, \u30b3\u30f3\u30d4\u30e5\u30fc\u30bf\u304c\u666e\u53ca\u3059\u308b\u4ee5\u524d\u306e\u624b\u8a08\u7b97\u306e\u6642\u4ee3\u306b\u3088\u304f\u4f7f\u308f\u308c\u3066\u3044\u305f. \u3057\u304b\u3057, \u30b3\u30f3\u30d4\u30e5\u30fc\u30bf\u3067\u4f7f\u308f\u308c\u308b\u3088\u3046\u306b\u306a\u308b\u3068\u5b89\u5b9a\u6027\u306e\u9762\u3067\u554f\u984c\u304c\u3042\u308b\u3053\u3068\u304c\u308f\u304b\u308a, \u73fe\u5728\u3067\u306f\u4f7f\u308f\u308c\u306a\u304f\u306a\u3063\u305f.<\/p>\n
    \u306a\u304a, \u5b89\u5b9a\u6027\u306e\u554f\u984c\u3092\u89e3\u6c7a\u3059\u308b\u305f\u3081\u306b, \u30df\u30eb\u30f3\u6cd5\u306e\u4e88\u6e2c\u5b50\u3068\u7d44\u307f\u5408\u308f\u305b\u3066\u4f7f\u3046\u5b89\u5b9a\u306a\u4fee\u6b63\u5b50\u3068\u3057\u3066\u6b21\u306e\u30cf\u30df\u30f3\u30b0(Hamming) \u306e\u4fee\u6b63\u5b50\u304c\u63d0\u6848\u3055\u308c\u3066\u3044\u308b.<\/p>\n
      \r\n  yn+1<\/sub> = 9\/8 yn<\/sub> - 1\/8 yn-2<\/sub> + h\/8 (3fn+1<\/sub> + 6fn<\/sub> - 3fn-1<\/sub>)\r\n\r\n<\/pre>\n
     <\/p>\n
    \u6570\u5024\u5b9f\u9a13 (8)<\/h3>\n
    <\/span>\u30df\u30eb\u30f3\u6cd5\u306e\u8a08\u7b97\u4f8b<\/div>
    \n\u6700\u3082\u7c21\u5358\u306a\u5fae\u5206\u65b9\u7a0b\u5f0f\u306e\u521d\u671f\u5024\u554f\u984c\u306e\u4f8b\u3067\u30df\u30eb\u30f3\u6cd5\u306e\u7cbe\u5ea6\u3092\u78ba\u8a8d\u3057\u3066\u307f\u308b.<\/p>\n
      \r\n  y' = y, y(0) = 1\r\n\r\n<\/pre>\n
     
    \n\"\"
    \n\u6a2a\u8ef8\u306f\u76f8\u5bfe\u8aa4\u5dee(\u53f3\u306b\u884c\u304f\u307b\u3069\u7cbe\u5ea6\u304c\u3088\u3044), \u7e26\u8ef8\u306f\u95a2\u6570\u8a55\u4fa1\u56de\u6570\u3092\u5bfe\u6570\u30d7\u30ed\u30c3\u30c8\u3068\u3057\u305f. \u6bd4\u8f03\u306e\u305f\u3081\u30eb\u30f3\u30b2\u30fb\u30af\u30c3\u30bf\u6cd5\u3068\u30a2\u30c0\u30e0\u30b9\u30fb\u30d0\u30b7\u30e5\u30d5\u30a9\u30fc\u30b9\u30fb\u30e2\u30eb\u30c8\u30f3\u6cd5\u3082\u30d7\u30ed\u30c3\u30c8\u3057\u3066\u3044\u308b. \u306a\u304a, \u51fa\u767a\u5024\u3092\u6c42\u3081\u308b\u305f\u3081\u306b\u306f4\u6b21\u306e\u30eb\u30f3\u30b2\u30fb\u30af\u30c3\u30bf\u6cd5\u3092\u4f7f\u7528\u3057\u305f.<\/p>\n
    \u30df\u30eb\u30f3\u6cd5\u306f\u516c\u5f0f\u304c\u6bd4\u8f03\u7684\u7c21\u5358\u3067\u306f\u3042\u308b\u304c\u56f3\u306e\u3088\u3046\u306b\u6027\u80fd\u304c\u3088\u3044\u89e3\u6cd5\u3067\u3042\u308b.
    \n<\/div><\/div>\n
    \u6570\u5024\u5b9f\u9a13 (9)<\/h3>\n
    <\/span>\u30df\u30eb\u30f3\u6cd5\u306e\u4e0d\u5b89\u5b9a\u6027<\/div>
    \n\u6b21\u306e\u521d\u671f\u5024\u554f\u984c\u3092\u30df\u30eb\u30f3\u6cd5\u306b\u3088\u308a\u89e3\u304f.<\/p>\n
      \r\n  y' = -y + 1, y(0) = 2\r\n\r\n<\/pre>\n
     
    \n\u3053\u306e\u65b9\u7a0b\u5f0f\u306e\u89e3\u306f y(t) = exp(-t) + 1 \u3067\u3042\u308b. \u30b9\u30c6\u30c3\u30d7\u5e45\u3092 0.001 \u304a\u3088\u3073 0.0001 \u3068\u3057\u3066\u8a08\u7b97\u3057\u305f\u7d50\u679c, \u6b21\u306e\u3088\u3046\u306b\u306a\u3063\u305f(\u30b0\u30e9\u30d5\u4f5c\u6210\u306e\u90fd\u5408\u4e0a\u30c7\u30fc\u30bf\u70b9\u3092\u9593\u5f15\u304d\u3057\u3066\u8868\u793a).<\/p>\n
    \"ode_test-9\"<\/p>\n
    \u30b9\u30c6\u30c3\u30d7\u5e45\u304c\u5341\u5206\u5c0f\u3055\u3044\u306b\u3082\u304b\u304b\u308f\u3089\u305a\u8a08\u7b97\u304c\u9032\u3093\u3067\u30b9\u30c6\u30c3\u30d7\u6570\u304c\u591a\u304f\u306a\u308b\u3068\u632f\u52d5\u3092\u59cb\u3081\u767a\u6563\u3057\u305f. \u3053\u308c\u306f, \u89e3\u6cd5\u306e\u5dee\u5206\u65b9\u7a0b\u5f0f\u306e\u4e00\u822c\u89e3\u304c\u771f\u306e\u89e3\u306e\u8fd1\u4f3c(\u6709\u52b9\u6210\u5206)\u306e\u4ed6\u306b\u5fc5\u8981\u306e\u306a\u3044\u6210\u5206\u3092\u542b\u307f, \u6761\u4ef6\u306b\u3088\u3063\u3066\u305d\u308c\u304c\u6709\u52b9\u6210\u5206\u3088\u308a\u3082\u5927\u304d\u304f\u306a\u3063\u305f\u3068\u304d\u306b\u8d77\u304d\u308b\u73fe\u8c61\u3067\u3042\u308b.<\/p>\n
    \u30df\u30eb\u30f3\u6cd5\u3067\u306f\u3053\u306e\u4e0d\u5b89\u5b9a\u6027\u306e\u5bfe\u7b56\u3068\u3057\u3066\u30cf\u30df\u30f3\u30b0\u306e\u4fee\u6b63\u5b50\u304c\u3042\u308b\u306e\u3067, \u305d\u308c\u3092\u4f7f\u3063\u3066\u540c\u3058\u8a08\u7b97\u3092\u884c\u3046\u3068\u6b21\u306e\u3088\u3046\u306b\u306a\u3063\u305f(\u30b0\u30e9\u30d5\u4f5c\u6210\u306e\u90fd\u5408\u4e0a\u30c7\u30fc\u30bf\u70b9\u3092\u9593\u5f15\u304d\u3057\u3066\u8868\u793a).<\/p>\n
    \"ode_test-9_2\"<\/p>\n
     <\/p>\n
    \u30cf\u30df\u30f3\u30b0\u306e\u4fee\u6b63\u5b50\u3092\u4f7f\u3046\u3053\u3068\u306b\u3088\u308a\u4e0d\u5b89\u5b9a\u6027\u304c\u53ce\u307e\u308b\u3053\u3068\u304c\u78ba\u8a8d\u3055\u308c\u305f.<\/p>\n
    \u6570\u5024\u5b9f\u9a13(1)\u3067\u4e2d\u70b9\u5247\u306e\u6027\u80fd\u304c\u3088\u3044\u3053\u3068\u306b\u6c17\u3065\u3044\u305f\u304b\u3082\u3057\u308c\u306a\u3044\u304c, \u4e2d\u70b9\u5247\u306b\u3082\u3053\u306e\u7a2e\u306e\u4e0d\u5b89\u5b9a\u6027\u304c\u3042\u308b\u3053\u3068\u304c\u308f\u304b\u3063\u3066\u3044\u308b. \u306a\u304a, 1\u6bb5\u6cd5\u3067\u306f\u3053\u306e\u7a2e\u306e\u4e0d\u5b89\u5b9a\u6027\u306f\u751f\u3058\u306a\u3044.
    \n<\/div><\/div>\n<\/div><\/div><\/div>\n
    \n
    \u53c2\u8003\u6587\u732e<\/h4>\n[1] \u540d\u53d6\u4eae\u300c\u6570\u5024\u89e3\u6790\u3068\u305d\u306e\u5fdc\u7528\u300d(1990) \u30b3\u30ed\u30ca\u793e
    \n[2] \u4e09\u4e95\u658c\u53cb\u300c\u5e38\u5fae\u5206\u65b9\u7a0b\u5f0f\u306e\u6570\u5024\u89e3\u6cd5\u300d(2003) \u5ca9\u6ce2\u66f8\u5e97
    \n[3] E. Hairer, \u4ed6\u300c\u5e38\u5fae\u5206\u65b9\u7a0b\u5f0f\u306e\u6570\u5024\u89e3\u6cd5 I\u300d(2007) \u30b7\u30e5\u30d7\u30ea\u30f3\u30ac\u30fc\u30fb\u30b8\u30e3\u30d1\u30f3<\/p>\n","protected":false},"excerpt":{"rendered":"
    \u5e38\u5fae\u5206\u65b9\u7a0b\u5f0f\u306e\u521d\u671f\u5024\u554f\u984c\u306e\u57fa\u672c\u7684\u306a\u89e3\u6cd5\u306b\u3064\u3044\u3066\u8aac\u660e\u3057\u307e\u3059\u3002<\/p>\n","protected":false},"author":3,"featured_media":0,"comment_status":"closed","ping_status":"closed","sticky":false,"template":"","format":"standard","meta":{"_monsterinsights_skip_tracking":false,"_monsterinsights_sitenote_active":false,"_monsterinsights_sitenote_note":"","_monsterinsights_sitenote_category":0,"footnotes":""},"categories":[13],"tags":[],"class_list":["post-1001","post","type-post","status-publish","format-standard","hentry","category-num"],"aioseo_notices":[],"_links":{"self":[{"href":"https:\/\/www.ktech.biz\/jp\/wp-json\/wp\/v2\/posts\/1001","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/www.ktech.biz\/jp\/wp-json\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/www.ktech.biz\/jp\/wp-json\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/www.ktech.biz\/jp\/wp-json\/wp\/v2\/users\/3"}],"replies":[{"embeddable":true,"href":"https:\/\/www.ktech.biz\/jp\/wp-json\/wp\/v2\/comments?post=1001"}],"version-history":[{"count":5,"href":"https:\/\/www.ktech.biz\/jp\/wp-json\/wp\/v2\/posts\/1001\/revisions"}],"predecessor-version":[{"id":4284,"href":"https:\/\/www.ktech.biz\/jp\/wp-json\/wp\/v2\/posts\/1001\/revisions\/4284"}],"wp:attachment":[{"href":"https:\/\/www.ktech.biz\/jp\/wp-json\/wp\/v2\/media?parent=1001"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/www.ktech.biz\/jp\/wp-json\/wp\/v2\/categories?post=1001"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/www.ktech.biz\/jp\/wp-json\/wp\/v2\/tags?post=1001"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}