{"id":1003,"date":"2015-10-11T23:13:20","date_gmt":"2015-10-11T14:13:20","guid":{"rendered":"https:\/\/www.ktech.biz\/jp\/?p=1003"},"modified":"2021-01-14T11:37:43","modified_gmt":"2021-01-14T02:37:43","slug":"10-ode-2","status":"publish","type":"post","link":"https:\/\/www.ktech.biz\/jp\/num\/10-ode-2\/","title":{"rendered":"10. \u5e38\u5fae\u5206\u65b9\u7a0b\u5f0f (2) \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"},"content":{"rendered":"

\u76ee\u6b21<\/h3>\n

\u5e38\u5fae\u5206\u65b9\u7a0b\u5f0f (1)<\/a>: \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): \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

\u8aa4\u5dee\u306e\u63a8\u5b9a<\/span>\u88dc\u5916\u6cd5<\/span>\u30b9\u30c6\u30c3\u30d7\u5e45\u306e\u81ea\u52d5\u8abf\u7bc0<\/span>\u5bc6\u51fa\u529b<\/span>2\u968e\u5e38\u5fae\u5206\u65b9\u7a0b\u5f0f<\/span>\u9045\u5ef6\u5fae\u5206\u65b9\u7a0b\u5f0f<\/span><\/div>
\n

\u8aa4\u5dee\u306e\u63a8\u5b9a<\/h2>\n

\u6c42\u3081\u305f\u6570\u5024\u89e3\u306e\u8aa4\u5dee\u3092\u63a8\u5b9a\u3059\u308b\u65b9\u6cd5\u306b\u3064\u3044\u3066\u8aac\u660e\u3059\u308b. \u8a08\u7b97\u7d50\u679c\u304c\u5f97\u3089\u308c\u308c\u3070\u305d\u308c\u3067\u7d42\u308f\u308a\u3068\u3044\u3046\u306e\u3067\u306f\u306a\u304f, \u305d\u306e\u7d50\u679c\u306e\u8aa4\u5dee\u304c\u63a8\u5b9a\u3067\u304d\u3066\u3044\u308c\u3070, \u7279\u7570\u70b9\u306a\u3069\u306e\u305f\u3081\u306b\u7570\u5e38\u306a\u89e3\u304c\u5f97\u3089\u308c\u3066\u3044\u306a\u3044\u304b\u306a\u3069\u306e\u30c1\u30a7\u30c3\u30af\u304c\u3067\u304d\u8a08\u7b97\u7d50\u679c\u306e\u5229\u7528\u4fa1\u5024\u306f\u4e0a\u304c\u308b. \u307e\u305f, \u73fe\u5728\u4e3b\u6d41\u3068\u306a\u3063\u3066\u3044\u308b\u30b9\u30c6\u30c3\u30d7\u5e45\u306e\u81ea\u52d5\u8abf\u7bc0\u6a5f\u80fd\u4ed8\u304d\u30eb\u30fc\u30c1\u30f3\u306e\u5b9f\u88c5\u306b\u306f\u8aa4\u5dee\u63a8\u5b9a\u304c\u5fc5\u9808\u3067\u3042\u308b.<\/p>\n

1\u6bb5\u6cd5\u306e\u8aa4\u5dee<\/h3>\n

\u521d\u671f\u5024\u554f\u984c\u306e\u6570\u5024\u89e3\u306e\u8aa4\u5dee\u306e\u539f\u56e0\u306b\u306f\u6b21\u306e2\u3064\u304c\u3042\u308b.<\/p>\n

    \n
  • \u96e2\u6563\u5316\u8aa4\u5dee (\u6253\u3061\u5207\u308a\u8aa4\u5dee\u3068\u3082\u3044\u3046)
    \n\u6f14\u7b97\u304c\u3059\u3079\u3066\u7121\u9650\u7cbe\u5ea6\u3067\u884c\u308f\u308c\u305f\u3068\u3057\u305f\u3068\u304d\u306b, \u89e3\u6cd5\u306b\u8d77\u56e0\u3057\u3066\u751f\u3058\u308b\u8aa4\u5dee<\/p>\n
      \n
    • \u5c40\u6240\u7684\u96e2\u6563\u5316\u8aa4\u5dee
      \n\u305d\u308c\u307e\u3067\u306b\u5f97\u3089\u308c\u3066\u3044\u308b\u5024\u304c\u6b63\u78ba\u3067\u3042\u308b\u3068\u3057\u3066, \u6b21\u306e1\u30b9\u30c6\u30c3\u30d7\u3067\u751f\u3058\u308b\u8aa4\u5dee<\/li>\n
    • \u5927\u57df\u7684\u96e2\u6563\u5316\u8aa4\u5dee
      \n\u521d\u671f\u5024\u304b\u3089\u8a08\u7b97\u3055\u308c\u308b\u89e3\u3068\u771f\u306e\u89e3\u3068\u306e\u5dee<\/li>\n<\/ul>\n<\/li>\n
    • \u4e38\u3081\u8aa4\u5dee
      \n\u6570\u5024\u3092\u6d6e\u52d5\u5c0f\u6570\u3067\u8868\u3057\u6709\u9650\u6841\u6570\u3067\u6f14\u7b97\u3059\u308b\u3053\u3068\u306b\u3088\u308a\u751f\u305a\u308b\u8aa4\u5dee<\/li>\n<\/ul>\n

      \u3044\u307e, \u6b21\u306e\u521d\u671f\u5024\u554f\u984c\u3092\u8003\u3048\u308b.<\/p>\n

        \r\n  dy\/dt = f(t, y),  t = t0<\/sub> \u306b\u304a\u3044\u3066 y = y0<\/sub>\r\n\r\n<\/pre>\n
       
      \ntn<\/sub>\u306b\u304a\u3051\u308b\u8a08\u7b97\u5024\u3092yn<\/sub>\u3068\u3057, un<\/sub>(tn<\/sub>) = yn<\/sub> \u3068\u3059\u308b. \u3059\u306a\u308f\u3061, \u8a08\u7b97\u5024yn<\/sub>\u304b\u3089\u6c7a\u5b9a\u3055\u308c\u308b\u89e3\u66f2\u7dda\u3092un<\/sub>(t)\u3068\u3059\u308b. yn+1<\/sub>\u306f yn<\/sub>, tn<\/sub>, f \u3088\u308a\u8a08\u7b97\u3055\u308c\u308b\u63a8\u5b9a\u5024\u3067\u3042\u308b. \u4f8b\u3048\u3070, \u30aa\u30a4\u30e9\u30fc\u6cd5\u3092\u4f7f\u3046\u306a\u3089\u3070, \u30b9\u30c6\u30c3\u30d7\u5e45\u3092 h \u3068\u3057\u3066, yn+1<\/sub> = yn<\/sub> + hf(tn<\/sub>, yn<\/sub>) \u3068\u306a\u308b.<\/p>\n
      \u305d\u3046\u3059\u308b\u3068, \u5c40\u6240\u7684\u96e2\u6563\u5316\u8aa4\u5dee dn<\/sub> \u306f\u6b21\u306e\u3088\u3046\u306b\u8868\u3059\u3053\u3068\u304c\u3067\u304d\u308b.<\/p>\n
        \r\n  dn<\/sub> = yn+1<\/sub> - un<\/sub>(tn+1<\/sub>)\r\n\r\n<\/pre>\n
       
      \n\u5c40\u6240\u7684\u96e2\u6563\u5316\u8aa4\u5dee\u306f p\u6b21\u306e\u89e3\u6cd5\u306b\u304a\u3044\u3066 O(hp+1<\/sup>) \u3067\u3042\u308b. \u3059\u306a\u308f\u3061, \u6b21\u5f0f\u3092\u6e80\u305f\u3059\u6570 C \u304c\u5b58\u5728\u3059\u308b.<\/p>\n
        \r\n  |dn<\/sub>| \u2266 Chp+1<\/sup>\r\n\r\n<\/pre>\n
       
      \n\u5927\u57df\u7684\u96e2\u6563\u5316\u8aa4\u5dee en<\/sub> \u306f\u6b21\u306e\u3088\u3046\u306b\u8868\u3055\u308c\u308b.<\/p>\n
        \r\n  en<\/sub> = yn<\/sub> - y(tn<\/sub>)\r\n\r\n<\/pre>\n
       
      \n\u4ee5\u4e0a\u306e\u95a2\u4fc2\u3092\u4e0b\u306b\u56f3\u793a\u3059\u308b. \u3053\u308c\u306f, \u89e3\u66f2\u7dda\u304c\u5c11\u3057\u305a\u308c\u305f\u3068\u304d\u306b\u6642\u9593\u304c\u9032\u3080\u306b\u3064\u308c\u3066\u96e2\u308c\u3066\u3044\u304f\u95a2\u6570\u306e\u4f8b\u3067\u3042\u308b. \u3059\u306a\u308f\u3061, en<\/sub> > \u03a3di<\/sub> (i = 0 \uff5e n-1) \u3068\u306a\u308a, \u30b9\u30c6\u30c3\u30d7\u304c\u9032\u3080\u306b\u3064\u308c\u3066\u8aa4\u5dee\u304c\u62e1\u5927\u3057\u3066\u3044\u304f.<\/p>\n
      \"ode_fig-3\"<\/p>\n
      \u4e0b\u306b\u4ed6\u306e\u4f8b\u3092\u793a\u3059. \u3053\u308c\u306f, \u89e3\u66f2\u7dda\u304c\u5c11\u3057\u305a\u308c\u305f\u3068\u304d\u306b\u6642\u9593\u304c\u9032\u3080\u306b\u3064\u308c\u3066\u63a5\u8fd1\u3057\u3066\u3044\u304f\u95a2\u6570\u306e\u4f8b\u3067\u3042\u308b. \u3059\u306a\u308f\u3061, en<\/sub> < \u03a3di<\/sub> (i = 0 \uff5e n-1) \u3068\u306a\u308a, \u30b9\u30c6\u30c3\u30d7\u304c\u9032\u3080\u306b\u3064\u308c\u3066\u8aa4\u5dee\u306f\u7e2e\u5c0f\u3057\u3066\u3044\u304f.<\/p>\n
      \"ode_fig-4\"<\/p>\n
      2\u3064\u306e\u4f8b\u3092\u3042\u3052\u305f\u304c, \u95a2\u6570\u304c\u3069\u3061\u3089\u306e\u50be\u5411\u3092\u793a\u3059\u304b\u306f \u2202f\/\u2202y \u306e\u7b26\u53f7\u306b\u3088\u308a\u6c7a\u307e\u308a, \u6b63\u3067\u3042\u308c\u3070\u524d\u8005, \u8ca0\u3067\u3042\u308c\u3070\u5f8c\u8005\u306b\u306a\u308b. \u3082\u3057, f \u304c y \u306b\u4f9d\u5b58\u3057\u306a\u3044\u5834\u5408, \u3059\u306a\u308f\u3061, \u2202f\/\u2202y = 0 \u306a\u3089\u3070 en<\/sub> = \u03a3di<\/sub> \u3068\u306a\u308b. \u307e\u305f, \u2202f\/\u2202y \u306e\u7b26\u53f7\u304c\u533a\u9593\u5185\u3067\u5909\u5316\u3059\u308b\u3088\u3046\u306a\u8907\u96d1\u306a\u95a2\u6570\u3082\u3042\u308b. \u2202f\/\u2202y = 0 \u306b\u8fd1\u3051\u308c\u3070, \u5927\u57df\u7684\u96e2\u6563\u5316\u8aa4\u5dee\u306f\u304a\u304a\u3080\u306d O(hp<\/sup>) \u3068\u3044\u3048\u308b.<\/p>\n
      \u3053\u3053\u307e\u3067\u306f\u4e38\u3081\u8aa4\u5dee\u3092\u7121\u8996\u3057\u3066\u8003\u3048\u3066\u304d\u305f\u304c, \u6709\u9650\u6841\u3067\u8a08\u7b97\u3059\u308b\u9650\u308a\u5fc5\u305a\u4e38\u3081\u8aa4\u5dee \u03b5n<\/sub> \u304c\u767a\u751f\u3059\u308b. \u4f8b\u3048\u3070, \u30aa\u30a4\u30e9\u30fc\u6cd5\u306a\u3089\u3070, yn+1<\/sub> = yn<\/sub> + hf(yn<\/sub>, tn<\/sub>) + \u03b5n<\/sub> \u3068\u306a\u308b.<\/p>\n
      \u3053\u3053\u3067, \u4e38\u3081\u8aa4\u5dee \u03b5n<\/sub> \u306e\u30ce\u30eb\u30e0\u306f\u3042\u308b\u6570 \u03b5 \u4ee5\u4e0b\u3067\u3042\u308b\u3068\u3059\u308b.<\/p>\n
        \r\n  ||\u03b5n<\/sub>|| \u2266 \u03b5\r\n\r\n<\/pre>\n
       
      \n\u3044\u307e, \u6700\u7d42\u70b9\u3092 tN<\/sub> \u3068\u3059\u308b\u3068, \u30b9\u30c6\u30c3\u30d7\u6570 N = (tN<\/sub> – t0<\/sub>) \/ h \u3067\u3042\u308b. \u6700\u7d42\u7684\u306a\u5927\u57df\u7684\u96e2\u6563\u5316\u8aa4\u5dee\u3092 eN<\/sub> \u2252 \u03a3di<\/sub> (i = 0 \uff5e N-1) \u3068\u8fd1\u4f3c\u3057, \u4e38\u3081\u8aa4\u5dee\u3092 N\u03b5 \u3068\u8fd1\u4f3c\u3059\u308b\u3068, \u6700\u7d42\u7684\u306a\u5168\u8aa4\u5dee\u306f\u6b21\u306e\u3088\u3046\u306b\u8fd1\u4f3c\u3059\u308b\u3053\u3068\u304c\u3067\u304d\u308b.<\/p>\n
        \r\n  \u5168\u8aa4\u5dee \u2252 NChp+1<\/sup> + N\u03b5 = (tN<\/sub> - t0<\/sub>)(Chp<\/sup> + \u03b5\/h)\r\n\r\n<\/pre>\n
       <\/p>\n
      \u6570\u5024\u5b9f\u9a13 (10)<\/h3>\n
      <\/span>\u30aa\u30a4\u30e9\u30fc\u6cd5\u306e\u8aa4\u5dee<\/div>
      \n\u6b21\u306e\u521d\u671f\u5024\u554f\u984c\u3092\u30aa\u30a4\u30e9\u30fc\u6cd5\u306b\u3088\u308a\u89e3\u304f.<\/p>\n
        \r\n  y' = y, y(0) = 1\r\n\r\n<\/pre>\n
       
      \n\u30aa\u30a4\u30e9\u30fc\u6cd5\u306e\u5834\u5408 p = 1 \u3067\u3042\u308b\u304b\u3089, \u5168\u8aa4\u5dee\u306f\u6b21\u306e\u3088\u3046\u306b\u8fd1\u4f3c\u3067\u304d\u308b.<\/p>\n
        \r\n  \u5168\u8aa4\u5dee \u2252 (tN<\/sub> - t0<\/sub>)(Ch + \u03b5\/h)\r\n\r\n<\/pre>\n
       
      \nh \u304c\u5927\u304d\u3044\u3068\u304d\u306f\u7b2c1\u9805(\u96e2\u6563\u5316\u8aa4\u5dee)\u304c\u512a\u52e2\u3067, h \u304c\u5c0f\u3055\u3044\u3068\u304d\u306f\u7b2c2\u9805(\u4e38\u3081\u8aa4\u5dee)\u304c\u512a\u52e2\u306b\u306a\u308b\u3068\u8003\u3048\u3089\u308c\u308b. \u3059\u306a\u308f\u3061, \u30b9\u30c6\u30c3\u30d7\u5e45\u3092\u5c0f\u3055\u304f\u3057\u3066\u3044\u304f\u3068\u8aa4\u5dee\u306f\u5c0f\u3055\u304f\u306a\u3063\u3066\u3044\u304f\u304c, \u3042\u308b\u5024((\u03b5\/C)-2<\/sup>)\u304b\u3089\u4e0b\u3067\u306f\u4e38\u3081\u8aa4\u5dee\u306e\u305f\u3081\u306b\u304b\u3048\u3063\u3066\u8aa4\u5dee\u304c\u5927\u304d\u304f\u306a\u308b\u3068\u4e88\u60f3\u3055\u308c\u308b.<\/p>\n
      \u30b9\u30c6\u30c3\u30d7\u5e45\u3092\u5909\u5316\u3055\u305b\u3066, t = 1 \u306b\u304a\u3051\u308b\u76f8\u5bfe\u8aa4\u5dee\u3092\u30d7\u30ed\u30c3\u30c8\u3057\u305f\u7d50\u679c\u3092\u4e0b\u306b\u793a\u3059. \u6a2a\u8ef8\u306f\u30b9\u30c6\u30c3\u30d7\u5e45(\u53f3\u306b\u3044\u304f\u307b\u3069\u5c0f\u3055\u3044), \u7e26\u8ef8\u306f\u76f8\u5bfe\u8aa4\u5dee\u3067\u3042\u308b. \u306a\u304a, \u4e38\u3081\u8aa4\u5dee\u306e\u5f71\u97ff\u3092\u898b\u3084\u3059\u304f\u3059\u308b\u305f\u3081\u306b\u8a08\u7b97\u306f\u5358\u7cbe\u5ea6\u3067\u884c\u3063\u305f.<\/p>\n
      \"\"<\/p>\n
      \u56f3\u4e2d\u306e\u9752\u8272\u306e\u30d7\u30ed\u30c3\u30c8\u306e\u3088\u3046\u306b, \u3053\u306e\u5834\u5408 10-5<\/sup> \u4ed8\u8fd1\u3092\u5883\u306b\u4e38\u3081\u8aa4\u5dee\u304c\u652f\u914d\u7684\u306b\u306a\u308b\u3053\u3068\u304c\u308f\u304b\u308b.<\/p>\n
      \u30eb\u30f3\u30b2\u30fb\u30af\u30c3\u30bf\u30fb\u30b8\u30eb\u6cd5\u306e\u9805\u76ee\u3067\u4e38\u3081\u8aa4\u5dee\u3092\u6e1b\u3089\u3059\u65b9\u6cd5\u3092\u7d39\u4ecb\u3057\u305f\u304c, \u305d\u308c\u3092\u30aa\u30a4\u30e9\u30fc\u6cd5\u306b\u9069\u7528\u3057\u3066\u8a08\u7b97\u3057\u305f\u7d50\u679c\u3092\u8d64\u8272\u3067\u8868\u793a\u3057\u305f. \u4e38\u3081\u8aa4\u5dee\u304c\u306a\u3051\u308c\u3070h\u306b\u6bd4\u4f8b\u3057\u3066\u8aa4\u5dee\u304c\u6e1b\u3063\u3066\u3044\u304f\u3053\u3068\u304c\u308f\u304b\u308b.
      \n<\/div><\/div>\n
      \u30ea\u30c1\u30e3\u30fc\u30c9\u30bd\u30f3\u306e\u88dc\u5916<\/h3>\n
      p\u6b21\u306e\u6570\u5024\u89e3\u6cd5\u306e\u4e0e\u3048\u3089\u308c\u305f\u70b9 tn<\/sub> \u3067\u306e\u5927\u57df\u7684\u96e2\u6563\u5316\u8aa4\u5dee\u306f O(hp<\/sup>) \u3068\u306a\u308b\u306e\u3067, \u6b21\u306e\u3088\u3046\u306b\u8868\u3059\u3053\u3068\u304c\u3067\u304d\u308b.<\/p>\n
        \r\n  yn<\/sub> - y(tn<\/sub>) = chp<\/sup> + O(hq<\/sup>), q > p\r\n\r\n<\/pre>\n
       
      \n\u30b9\u30c6\u30c3\u30d7\u5e45 h \u3067\u8a08\u7b97\u3057\u305f yn<\/sub> \u3092 yn<\/sub>(h)<\/sup> \u3068\u66f8\u304f\u3053\u3068\u306b\u3057\u3066, \u30b9\u30c6\u30c3\u30d7\u5e45 2h \u3067\u3082\u8a08\u7b97\u3059\u308b\u3068\u6b21\u306e\u3088\u3046\u306b\u306a\u308b.<\/p>\n
        \r\n  yn<\/sub>(h)<\/sup> = y(tn<\/sub>) + chp<\/sup> + O(hq<\/sup>)\r\n  yn<\/sub>(2h)<\/sup> = y(tn<\/sub>) + c(2h)p<\/sup> + O(hq<\/sup>)\r\n\r\n<\/pre>\n
       
      \n\u3053\u308c\u3089\u306e\u5dee\u3092\u3068\u308b\u3068,<\/p>\n
        \r\n  yn<\/sub>(2h)<\/sup> - yn<\/sub>(h)<\/sup> = (2p<\/sup> - 1)chp<\/sup> + O(hq<\/sup>)\r\n\r\n<\/pre>\n
       
      \n\u3068\u306a\u308a, yn<\/sub>(h)<\/sup> \u306e\u8aa4\u5dee\u306f O(hq<\/sup>) \u3092\u7121\u8996\u3059\u308b\u3068\u6b21\u306e\u3088\u3046\u306b\u8fd1\u4f3c\u3067\u304d\u308b.<\/p>\n
        \r\n  yn<\/sub>(h)<\/sup>\u306e\u8aa4\u5dee \u2252 chp<\/sup> = (yn<\/sub>(2h)<\/sup> - yn<\/sub>(h)<\/sup>)\/(2p<\/sup> - 1)\r\n\r\n<\/pre>\n
       
      \n\u3053\u308c\u306f, \u30ed\u30f3\u30d0\u30fc\u30b0\u7a4d\u5206\u3067\u4f7f\u308f\u308c\u308b\u30ea\u30c1\u30e3\u30fc\u30c9\u30bd\u30f3\u306e\u88dc\u5916(Richardson extrapolation)\u3068\u540c\u3058\u3067\u3042\u308b.<\/p>\n
      \u6570\u5024\u5b9f\u9a13 (11)<\/h3>\n
      <\/span>\u30ea\u30c1\u30e3\u30fc\u30c9\u30bd\u30f3\u306e\u88dc\u5916<\/div>
      \n\u7c21\u5358\u306a\u5fae\u5206\u65b9\u7a0b\u5f0f\u306e\u521d\u671f\u5024\u554f\u984c\u306e\u4f8b\u3092\u30aa\u30a4\u30e9\u30fc\u6cd5, \u30db\u30a4\u30f3\u6cd5, \u30eb\u30f3\u30b2\u30fb\u30af\u30c3\u30bf\u6cd5\u3067\u8a08\u7b97\u3059\u308b. \u6b21\u306b, \u30ea\u30c1\u30e3\u30fc\u30c9\u30bd\u30f3\u306e\u88dc\u5916\u306b\u3088\u308a\u305d\u308c\u305e\u308c\u306e\u8a08\u7b97\u7d50\u679c\u306e\u8aa4\u5dee\u3092\u63a8\u5b9a\u3057\u3066\u30d7\u30ed\u30c3\u30c8\u3059\u308b.<\/p>\n
        \r\n  y' = y, y(0) = 1\r\n\r\n<\/pre>\n
       <\/p>\n
      \"\"<\/p>\n
      \u3053\u306e\u4f8b\u3067\u306f\u5168\u4f53\u3068\u3057\u3066\u5c11\u306a\u3081\u306b\u898b\u7a4d\u3082\u3089\u308c\u3066\u3044\u308b\u304c\u6bd4\u8f03\u7684\u8fd1\u3044\u5024\u304c\u5f97\u3089\u308c\u3066\u3044\u308b.
      \n<\/div><\/div>\n<\/div>\n
      \n
      \u88dc\u5916\u6cd5<\/h2>\n
      H = t – t0<\/sub> \u3068\u3059\u308b. \u6b63\u306e\u6574\u6570\u5217 n1<\/sub> < n2<\/sub> < n3<\/sub> < … \u306b\u3064\u3044\u3066 hi<\/sub> = H\/ni<\/sub> \u3068\u3059\u308b\u3068, \u6574\u6570\u5217\u306b\u5bfe\u5fdc\u3059\u308b\u30b9\u30c6\u30c3\u30d7\u5e45 h1<\/sub> > h2<\/sub> > h3<\/sub> > … \u304c\u5f97\u3089\u308c\u308b.<\/p>\n
      p\u6b21\u306e\u6570\u5024\u89e3\u6cd5\u306b\u304a\u3044\u3066\u30b9\u30c6\u30c3\u30d7\u5e45hj<\/sub>\u3092\u4f7f\u3063\u3066nj<\/sub>\u30b9\u30c6\u30c3\u30d7\u8a08\u7b97\u3057\u3066\u6c42\u3081\u305ft\u306b\u304a\u3051\u308b\u89e3\u3092Tj1<\/sub>\u3068\u3059\u308b. T11<\/sub> \uff5e Tk1<\/sub> (k <= j) \u306e\u5024\u3092\u4f7f\u3063\u3066\u9023\u7acb\u65b9\u7a0b\u5f0f\u3092\u89e3\u3051\u3070h\u306ek\u6b21\u88dc\u9593\u591a\u9805\u5f0f\u3092\u4f5c\u308b\u3053\u3068\u304c\u3067\u304d\u308b. \u305d\u3053\u3067 h = 0 \u3068\u3057\u3066\u88dc\u5916\u5024\u3092\u6c42\u3081\u308c\u3070\u3088\u3044\u8fd1\u4f3c\u3092\u5f97\u308b\u3053\u3068\u304c\u3067\u304d\u308b.<\/p>\n
      \u5b9f\u969b\u306e\u8a08\u7b97\u306b\u304a\u3044\u3066\u306f, \u88dc\u9593\u591a\u9805\u5f0f\u305d\u306e\u3082\u306e\u306f\u4e0d\u8981\u3067 h = 0 \u306b\u304a\u3051\u308b\u5024\u306e\u307f\u3042\u308c\u3070\u3088\u3044\u306e\u3067, \u6b21\u306e\u30cd\u30f4\u30a7\u30eb\u306e\u30a2\u30eb\u30b4\u30ea\u30ba\u30e0\u3068\u3088\u3070\u308c\u308b\u65b9\u6cd5\u3092\u4f7f\u3046\u3053\u3068\u304c\u3067\u304d\u308b.<\/p>\n
        \r\n  Tjk<\/sub> = Tj(k-1)<\/sub> + (Tj(k-1)<\/sub> - T(j-1)(k-1)<\/sub>)\/(nj<\/sub>\/nj-k+1<\/sub> - 1)\r\n\r\n<\/pre>\n
       
      \n\u3053\u306e\u6f38\u5316\u5f0f\u3092\u4f7f\u3063\u3066\u6c42\u3081\u305f Tjk<\/sub> \u306f\u89e3\u306e\u826f\u3044\u8fd1\u4f3c\u3068\u306a\u3063\u3066\u3044\u308b. \u3053\u306e\u65b9\u6cd5\u306fp+k-1\u6b21\u306e\u6570\u5024\u89e3\u6cd5\u3068\u306a\u3063\u3066\u304a\u308a, \u88dc\u5916\u6cd5<\/strong>(Extrapolation method)\u3068\u3088\u3070\u308c\u308b.<\/p>\n
      \u88dc\u5916\u6cd5\u306e\u8a08\u7b97\u7d50\u679c\u3092\u8868\u3059\u6b21\u306e\u8868\u3092\u88dc\u5916\u8868\u3068\u3044\u3046.<\/p>\n
        \r\n  T11<\/sub>   |\r\n  T21<\/sub>   |  T22<\/sub>\r\n  T31<\/sub>   |  T32<\/sub>  T33<\/sub>\r\n  T41<\/sub>   |  T42<\/sub>  T43<\/sub>  T44<\/sub>\r\n  ...   |  ...  ...  ...  ...\r\n \u6570\u5024\u89e3 |  k-1\u500b\u306e\u88dc\u5916\u5024 \u2192\r\n\r\n<\/pre>\n
       
      \nT21<\/sub>\u306b\u3064\u3044\u3066\u306fT22<\/sub>\u307e\u3067(k = 2), T31<\/sub>\u306fT33<\/sub>\u307e\u3067(k = 3), …, Tj1<\/sub>\u306fTjj<\/sub>\u307e\u3067(k = j)\u88dc\u5916\u3067\u304d\u308b. \u3059\u306a\u308f\u3061, \u6700\u5927\u306ek\u3092\u4f7f\u7528\u3059\u308b\u306a\u3089\u3070j\u304c\u5897\u3048\u308b\u307b\u3069\u6b21\u6570\u3082\u5927\u304d\u304f\u306a\u3063\u3066\u3044\u304f.<\/p>\n
      \u4e0a\u306b\u304a\u3044\u3066, k = 2, n1<\/sub> = 1, n2<\/sub> = 2 \u306e\u5834\u5408\u306f\u30ea\u30c1\u30e3\u30fc\u30c9\u30bd\u30f3\u306e\u88dc\u5916\u306b\u4e00\u81f4\u3059\u308b.<\/p>\n
      \u6574\u6570\u5217ni<\/sub>\u306e\u3068\u308a\u65b9\u3067\u3042\u308b\u304c, \u6b21\u306e\u3088\u3046\u306b\u3044\u304f\u3064\u304b\u306e\u65b9\u6cd5\u304c\u63d0\u6848\u3055\u308c\u3066\u3044\u308b.<\/p>\n
      \u30ed\u30f3\u30d0\u30fc\u30af\u5217: 1, 2, 4, 8, 16, 32, 64, 128, 256, 512, …
      \n\u30d6\u30ea\u30a2\u30b7\u30e5\u5217: 1, 2, 3, 4, 6, 8, 12, 16, 24, 32, …
      \n\u8abf\u548c\u6570\u5217: 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, …<\/p>\n
      \u6570\u5024\u5b9f\u9a13 (12)<\/h3>\n
      <\/span>\u88dc\u5916\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\u88dc\u5916\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\u6570\u5024\u89e3\u6cd5\u3068\u3057\u3066\u306f\u30aa\u30a4\u30e9\u30fc\u6cd5(p = 1\u6b21)\u3092\u4f7f\u3063\u305f. \u6574\u6570\u5217\u3068\u3057\u3066\u306f\u30ed\u30f3\u30d0\u30fc\u30af\u5217 1, 2, …, 128 \u3092\u4f7f\u3063\u305f.<\/p>\n
      \"\"<\/p>\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\u306e\u5bfe\u6570\u30d7\u30ed\u30c3\u30c8\u3068\u3057\u305f. \u88dc\u5916\u306b\u5fc5\u8981\u306a\u8a08\u7b97\u91cf\u306f\u7121\u8996\u3057\u305f. \u4e00\u756a\u5de6\u306e\u30e9\u30a4\u30f3\u304c\u30aa\u30a4\u30e9\u30fc\u6cd5\u306b\u3088\u308b\u6570\u5024\u89e3(Tj1<\/sub>)\u3067\u3042\u308b. \u4e00\u756a\u53f3\u306e\u8d64\u3044\u30e9\u30a4\u30f3\u306f\u6700\u9ad8\u6b21\u6570\u3067\u6c42\u3081\u305f\u88dc\u5916\u5024(Tjj<\/sub>)\u3067\u3042\u308b.
      \n<\/div><\/div>\n
      \u6570\u5024\u5b9f\u9a13 (13)<\/h3>\n
      <\/span>\u88dc\u5916\u6cd5\u306e\u8a08\u7b97\u4f8b (2)<\/div>
      \n\u540c\u3058\u7c21\u5358\u306a\u5fae\u5206\u65b9\u7a0b\u5f0f\u306e\u521d\u671f\u5024\u554f\u984c\u306e\u4f8b\u3067\u88dc\u5916\u6cd5\u306e\u30d1\u30e9\u30e1\u30fc\u30bf(k\u306e\u5024, \u6574\u6570\u5217)\u3092\u5909\u3048\u305f\u5834\u5408\u306e\u6319\u52d5\u3092\u78ba\u8a8d\u3059\u308b.<\/p>\n
        \r\n  y' = y, y(0) = 1\r\n\r\n<\/pre>\n
       
      \n\u6570\u5024\u89e3\u6cd5\u3068\u3057\u3066\u306f\u30aa\u30a4\u30e9\u30fc\u6cd5(p = 1\u6b21)\u3092\u4f7f\u3063\u305f. \u30ed\u30f3\u30d0\u30fc\u30af\u5217\u3067 k = j \u3068 k = 4 \u306e\u5834\u5408, \u30d6\u30ea\u30a2\u30b7\u30e5\u5217\u3067 k = j, \u8abf\u548c\u6570\u5217\u3067 k = j \u306e\u5834\u5408\u3092\u6bd4\u8f03\u3059\u308b.<\/p>\n
      \"\"<\/p>\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\u306e\u5bfe\u6570\u30d7\u30ed\u30c3\u30c8\u3068\u3057\u305f. \u88dc\u5916\u306b\u5fc5\u8981\u306a\u8a08\u7b97\u91cf\u306f\u7121\u8996\u3057\u305f. \u30ed\u30f3\u30d0\u30fc\u30af\u5217\u3067 k = 4 \u3068\u3057\u305f(\u88dc\u5916\u8868\u306e\u6a2a\u5e45\u30924\u306b\u5236\u9650\u3057\u305f)\u5834\u5408\u306b\u306f\u9014\u4e2d\u304b\u3089\u3053\u306e(\u5bfe\u6570)\u30d7\u30ed\u30c3\u30c8\u3067\u76f4\u7dda\u3068\u306a\u308a\u6b21\u6570\u304c\u5897\u3048\u306a\u304f\u306a\u3063\u305f\u306e\u304c\u308f\u304b\u308b. \u30d6\u30ea\u30a2\u30b7\u30e5\u5217\u3068\u8abf\u548c\u6570\u5217\u306f\u3053\u306e\u4f8b\u3067\u306f\u6700\u521d\u306f\u3088\u304b\u3063\u305f\u304c\u9014\u4e2d\u3067\u5931\u901f\u3057\u3066\u8aa4\u5dee\u304c\u6e1b\u3089\u306a\u304f\u306a\u3063\u305f.
      \n<\/div><\/div>\n
      GBS\u30a2\u30eb\u30b4\u30ea\u30ba\u30e0<\/h3>\n
      \u88dc\u9593\u591a\u9805\u5f0f\u304c\u5076\u6570\u306e\u3079\u304d\u4e57\u306e\u307f\u3067\u8868\u3055\u308c\u308b\u5834\u5408, \u30cd\u30f4\u30a7\u30eb\u306e\u30a2\u30eb\u30b4\u30ea\u30ba\u30e0\u306f\u6b21\u306e\u3088\u3046\u306b\u306a\u308a\u53ce\u675f\u304c\u901f\u304f\u306a\u308b.<\/p>\n
        \r\n  Tjk<\/sub> = Tj(k-1)<\/sub> + (Tj(k-1)<\/sub> - T(j-1)(k-1)<\/sub>)\/((nj<\/sub>\/nj-k+1<\/sub>)^2 - 1)\r\n\r\n<\/pre>\n
       
      \nGBS(Gragg-Bulirsch-Stoer)\u30a2\u30eb\u30b4\u30ea\u30ba\u30e0\u306f\u3053\u308c\u3092\u5229\u7528\u3057\u305f\u3082\u306e\u3067, \u6b21\u306e\u624b\u9806\u3067\u8a08\u7b97\u3092\u884c\u3046.<\/p>\n
        \r\n  y1<\/sub> = y0<\/sub> + hj<\/sub>f(t0<\/sub>, y0<\/sub>)                           [\u30aa\u30a4\u30e9\u30fc\u6cd5]\r\n  yi+1<\/sub> = yi-1<\/sub> + 2hj<\/sub>f(ti<\/sub>, yi<\/sub>) (i = 1, 2, ..., nj<\/sub>)  [\u4e2d\u70b9\u5247]\r\n\r\n  Tj1<\/sub> = (1\/4)(ynj<\/sub>-1<\/sub> + 2ynj<\/sub><\/sub> + ynj<\/sub>+1<\/sub>)\r\n\r\n<\/pre>\n
       
      \n\u3053\u306e\u3088\u3046\u306b\u3059\u308b\u3068 Tj1<\/sub> \u306f\u5076\u6570\u306e\u3079\u304d\u4e57\u306e\u307f\u3067\u8868\u3055\u308c, \u4e0a\u5f0f\u3092\u9069\u7528\u3057\u3066\u52b9\u7387\u3088\u304f\u88dc\u5916\u5024\u3092\u6c42\u3081\u308b\u3053\u3068\u304c\u3067\u304d\u308b. \u306a\u304a, \u4e2d\u70b9\u5247\u306b\u306f\u4e0d\u5b89\u5b9a\u6027\u306e\u554f\u984c\u304c\u3042\u308b\u3053\u3068\u3092\u300c\u4f7f\u308f\u308c\u306a\u304f\u306a\u3063\u305f\u89e3\u6cd5\u300d\u306e\u9805\u3067\u8aac\u660e\u3057\u305f\u304c, \u3053\u306e\u5834\u5408\u306f\u554f\u984c\u306f\u751f\u3058\u306a\u3044.<\/p>\n
      \u6574\u6570\u5217ni<\/sub>\u306f\u5076\u6570\u306b\u306a\u308b\u3088\u3046\u306b\u3059\u308b.<\/p>\n
      \u30ed\u30f3\u30d0\u30fc\u30af\u5217: 2, 4, 8, 16, 32, 64, 128, 256, 512, 1024, …
      \n\u30d6\u30ea\u30a2\u30b7\u30e5\u5217: 2, 4, 6, 8, 12, 16, 24, 32, 48, 64, …
      \n\u8abf\u548c\u6570\u5217: 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, …<\/p>\n
      \u6570\u5024\u5b9f\u9a13 (14)<\/h3>\n
      <\/span>GBS\u30a2\u30eb\u30b4\u30ea\u30ba\u30e0\u306b\u3088\u308b\u88dc\u5916\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\u3067GBS\u30a2\u30eb\u30b4\u30ea\u30ba\u30e0\u306b\u3088\u308b\u88dc\u5916\u6cd5\u3068\u901a\u5e38\u306e\u88dc\u5916\u6cd5\u306e\u7cbe\u5ea6\u3092\u6bd4\u8f03\u3059\u308b.<\/p>\n
        \r\n  y' = y, y(0) = 1\r\n\r\n<\/pre>\n
       
      \n\"\"<\/p>\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\u306e\u5bfe\u6570\u30d7\u30ed\u30c3\u30c8\u3067\u3042\u308b. GBS\u30a2\u30eb\u30b4\u30ea\u30ba\u30e0\u306e\u65b9\u304c\u660e\u3089\u304b\u306b\u901f\u304f\u8aa4\u5dee\u304c\u6e1b\u5c11\u3057\u3066\u3044\u308b. \u306a\u304a, \u56f3\u3067\u7dda\u304c\u5207\u308c\u3066\u3044\u308b\u3068\u3053\u308d\u306f\u8aa4\u5dee\u304c0\u306b\u306a\u3063\u305f\u305f\u3081\u30d7\u30ed\u30c3\u30c8\u3055\u308c\u3066\u3044\u306a\u3044\u533a\u9593\u3067\u3042\u308a, \u7cbe\u5ea6\u3044\u3063\u3071\u3044\u3067\u89e3\u304c\u6c42\u3081\u3089\u308c\u3066\u3044\u308b\u3053\u3068\u3092\u793a\u3057\u3066\u3044\u308b.
      \n<\/div><\/div>\n<\/div>\n
      \n
      \u30b9\u30c6\u30c3\u30d7\u5e45\u306e\u81ea\u52d5\u8abf\u7bc0<\/h2>\n
      \u57cb\u3081\u8fbc\u307f\u578b\u30eb\u30f3\u30b2\u30fb\u30af\u30c3\u30bf\u6cd5<\/h3>\n
      \u8aa4\u5dee\u306e\u63a8\u5b9a\u3092\u884c\u3046\u306e\u306b\u30ea\u30c1\u30e3\u30fc\u30c9\u30bd\u30f3\u306e\u88dc\u5916\u3092\u4f7f\u3046\u305f\u3081\u306b\u306f\u30b9\u30c6\u30c3\u30d7\u5e45\u3092\u5909\u3048\u30662\u56de\u8a08\u7b97\u3092\u884c\u308f\u306a\u3051\u308c\u3070\u306a\u3089\u305a, \u8a08\u7b97\u91cf\u304c\u5927\u304d\u304f\u306a\u308b\u306e\u304c\u6b20\u70b9\u3067\u3042\u308b. \u3057\u304b\u3057, \u30eb\u30f3\u30b2\u30fb\u30af\u30c3\u30bf\u6cd5\u81ea\u8eab\u306b\u8aa4\u5dee\u63a8\u5b9a\u6a5f\u80fd\u3092\u4ed8\u52a0\u3057\u305f\u516c\u5f0f\u3092\u4f5c\u308b\u3053\u3068\u304c\u3067\u304d\u308c\u3070, 1\u56de\u306e\u8a08\u7b97\u3067\u8aa4\u5dee\u63a8\u5b9a\u5024\u3082\u5f97\u308b\u3053\u3068\u304c\u3067\u304d\u6709\u7528\u3067\u3042\u308b. \u7279\u306b, \u3042\u3089\u304b\u3058\u3081\u8981\u6c42\u7cbe\u5ea6\u3092\u8a2d\u5b9a\u3057\u3066\u304a\u304d, \u305d\u308c\u306b\u5fc5\u8981\u306a\u30b9\u30c6\u30c3\u30d7\u5e45\u306b\u81ea\u52d5\u8abf\u7bc0\u3059\u308b\u30d7\u30ed\u30b0\u30e9\u30e0\u3092\u4f5c\u308b\u306e\u306b\u5fc5\u9808\u3067\u3042\u308b.<\/p>\n
      \u30eb\u30f3\u30b2\u30fb\u30af\u30c3\u30bf\u6cd5\u306f\u6b21\u306e\u3088\u3046\u306b\u8868\u3055\u308c\u305f. \u3053\u3053\u3067, \u3053\u306e\u516c\u5f0f\u306f p\u6b21\u3067\u3042\u308b\u3068\u3059\u308b.<\/p>\n
        \r\n  ki<\/sub> = f(yn<\/sub> + h\u03a3aij<\/sub>kj<\/sub>, tn<\/sub> + ci<\/sub>h) (\u03a3 \u306f j = 1, 2, ... , s) (i = 1, 2, ... , s)\r\n  yn+1<\/sub> = yn<\/sub> + h\u03a3bi<\/sub>ki<\/sub> (\u03a3 \u306f i = 1, 2, ... , s)\r\n\r\n<\/pre>\n
       
      \n\u540c\u3058 aij<\/sub> \u3068 ci<\/sub> \u3092\u4f7f\u3044 (\u3059\u306a\u308f\u3061, \u540c\u3058 ki<\/sub> \u3092\u4f7f\u3044), bi<\/sub> \u3092 b*<\/sup>i<\/sub> \u306b\u5909\u3048\u305f\u6642\u306e\u8fd1\u4f3c\u5024 y*<\/sup>n+1<\/sub> \u304cp*<\/sup>\u6b21 (\u4f8b\u3048\u3070, p*<\/sup> = p – 1, p*<\/sup> = p + 1 \u306a\u3069) \u306b\u306a\u3063\u3066\u3044\u308b\u3088\u3046\u306b\u5b9a\u6570\u3092\u6c7a\u3081\u308b\u3053\u3068\u304c\u3067\u304d\u308c\u3070, ki<\/sub> \u3092\u8a08\u7b97\u3057\u76f4\u3059\u3053\u3068\u306a\u304f\u6b21\u5f0f\u306b\u3088\u308a p*<\/sup>\u6b21\u306e\u5834\u5408\u306e\u8fd1\u4f3c\u5024\u3092\u6c42\u3081\u308b\u3053\u3068\u304c\u3067\u304d\u308b.<\/p>\n
        \r\n  y*<\/sup>n+1<\/sub> = yn<\/sub> + h\u03a3b*<\/sup>i<\/sub>ki<\/sub> (\u03a3 \u306f i = 1, 2, ... , s)\r\n\r\n<\/pre>\n
       
      \n\u3053\u306e\u3068\u304d, \u6b21\u6570\u304c\u4f4e\u3044\u65b9\u306e\u5c40\u6240\u96e2\u6563\u5316\u8aa4\u5dee\u3092\u6b21\u306e\u3088\u3046\u306b\u6c42\u3081\u308b\u3053\u3068\u304c\u3067\u304d\u308b.<\/p>\n
        \r\n  yn+1<\/sub> - y*<\/sup>n+1<\/sub> = h\u03a3(bi<\/sub> - b*<\/sup>i<\/sub>)ki<\/sub> (\u03a3 \u306f i = 1, 2, ... , s)\r\n\r\n<\/pre>\n
       
      \n\u3053\u306e\u3088\u3046\u306a\u516c\u5f0f\u306f\u5b9f\u969b\u306b\u4f5c\u308b\u3053\u3068\u304c\u3067\u304d\u3066, \u57cb\u3081\u8fbc\u307f\u578b\u30eb\u30f3\u30b2\u30fb\u30af\u30c3\u30bf\u6cd5<\/strong>(embedded Runge-Kutta method) \u3068\u3088\u3070\u308c\u308b. \u4e3b\u306a\u516c\u5f0f\u3092\u4ee5\u4e0b\u306b\u793a\u3059.<\/p>\n
      \u30eb\u30f3\u30b2\u30fb\u30af\u30c3\u30bf\u30fb\u30de\u30fc\u30bd\u30f3\u6cd5<\/h4>\n
      \u57cb\u3081\u8fbc\u307f\u578b\u30eb\u30f3\u30b2\u30fb\u30af\u30c3\u30bf\u6cd5\u3067\u6700\u521d\u306b\u63d0\u6848\u3055\u308c\u305f\u306e\u306f\u30de\u30fc\u30bd\u30f3\u306e\u516c\u5f0f (\u30eb\u30f3\u30b2\u30fb\u30af\u30c3\u30bf\u30fb\u30de\u30fc\u30bd\u30f3\u6cd5(Runge-Kutta-Merson method)) \u3068\u3055\u308c\u308b. \u4fc2\u6570\u306f\u6b21\u306e\u3068\u304a\u308a\u3067\u3042\u308b.<\/p>\n
        \r\n  0   |\r\n  1\/3 | 1\/3\r\n  1\/3 | 1\/6 1\/6\r\n  1\/2 | 1\/8  0  3\/8\r\n  1   | 1\/2  0 -3\/2   2\r\n  ---------------------------\r\n  y   | 1\/6  0   0   2\/3 1\/6 \r\n  y*  | 1\/2  0 -3\/2   2   0 \r\n    \r\n<\/pre>\n
       
      \n\u901a\u5e38\u306e\u30eb\u30f3\u30b2\u30fb\u30af\u30c3\u30bf\u6cd5\u306e\u30d6\u30c3\u30c1\u30e3\u30fc\u914d\u5217\u306b\u5bfe\u3057\u3066y*\u306e\u305f\u3081\u306e\u4fc2\u6570\u304c1\u884c\u8ffd\u52a0\u3055\u308c\u3066\u3044\u308b.<\/p>\n
      \u8a08\u7b97\u5f0f\u306f\u6b21\u306e\u3068\u304a\u308a\u3067\u3042\u308b.<\/p>\n
        \r\n  k1<\/sub> = f(tn<\/sub>, yn<\/sub>)\r\n  k2<\/sub> = f(tn<\/sub> + h\/3, yn<\/sub> + hk1<\/sub>\/3)\r\n  k3<\/sub> = f(tn<\/sub> + h\/3, yn<\/sub> + h(k1<\/sub>\/6 + k2<\/sub>\/6))\r\n  k4<\/sub> = f(tn<\/sub> + h\/2, yn<\/sub> + h(k1<\/sub>\/8 + 3k3<\/sub>\/8))\r\n  k5<\/sub> = f(tn<\/sub> + h, yn<\/sub> + h(k1<\/sub>\/2 - 3k3<\/sub>\/2 + 2k4<\/sub>))\r\n  yn+1<\/sub> = yn<\/sub> + h(k1<\/sub> + 4k4<\/sub> + k5<\/sub>)\/6  (4\u6b21)\r\n  y*<\/sup>n+1<\/sub> = yn<\/sub> + h(k1<\/sub> - 3k3<\/sub> + 4k4<\/sub>)\/2  (\u8aa4\u5dee\u8a55\u4fa1\u7528 4\u6b21)\r\n\r\n<\/pre>\n
       
      \n\u3053\u306e\u516c\u5f0f\u306b\u3088\u308b\u63a8\u5b9a\u8aa4\u5dee\u306f, f(t, y) \u304c t \u3068 y \u306b\u95a2\u3057\u3066\u7dda\u5f62\u306a\u3089\u3070 1\/5 (yn+1<\/sub> – y*<\/sup>n+1<\/sub>) \u3068\u306a\u308b.<\/p>\n
      1\u30b9\u30c6\u30c3\u30d7\u3042\u305f\u308a 5\u56de\u306e\u95a2\u6570\u8a55\u4fa1\u304c\u5fc5\u8981\u3067\u3042\u308b. \u3059\u306a\u308f\u3061, \u8aa4\u5dee\u8a55\u4fa1\u3092\u884c\u3046\u305f\u3081\u306b, \u901a\u5e38\u306e4\u6b21\u306e\u30eb\u30f3\u30b2\u30fb\u30af\u30c3\u30bf\u6cd5\u306b\u6bd4\u30791\u30b9\u30c6\u30c3\u30d7\u3042\u305f\u308a1\u56de\u4f59\u8a08\u306b\u95a2\u6570\u8a55\u4fa1\u3092\u884c\u3046\u3068\u3044\u3046\u72a0\u7272\u3092\u6255\u3063\u3066\u3044\u308b.<\/p>\n
      5(4)\u6b21 \u30eb\u30f3\u30b2\u30fb\u30af\u30c3\u30bf\u30fb\u30d5\u30a7\u30fc\u30eb\u30d9\u30eb\u30b0\u6cd5<\/h4>\n
      5(4)\u6b21 \u30eb\u30f3\u30b2\u30fb\u30af\u30c3\u30bf\u30fb\u30d5\u30a7\u30fc\u30eb\u30d9\u30eb\u30b0\u6cd5(Runge-Kutta-Fehlberg method)\u306f\u57cb\u3081\u8fbc\u307f\u578b\u30eb\u30f3\u30b2\u30fb\u30af\u30c3\u30bf\u6cd5\u306e\u4e2d\u3067\u3082\u6709\u540d\u306a\u516c\u5f0f\u3067\u3042\u308b. RKF45\u3068\u3044\u3046\u30d7\u30ed\u30b0\u30e9\u30e0\u304c\u5e83\u304f\u4f7f\u308f\u308c\u3066\u3044\u308b.<\/p>\n
        \r\n  0     |\r\n  1\/4   | 1\/4\r\n  3\/8   | 3\/32       9\/32\r\n  12\/13 | 1932\/2197 -7200\/2197 7296\/2197\r\n  1     | 439\/216    -8        3680\/513   -845\/4104\r\n  1\/2   | -8\/27       2       -3544\/2565  1859\/4104 -11\/40\r\n  ----------------------------------------------------------------\r\n  y     | 16\/135      0       6656\/12825 28561\/56430 -9\/50  2\/55 \r\n  y*    | 25\/216      0       1408\/2565   2197\/4104  -1\/5    0 \r\n    \r\n<\/pre>\n
       
      \n\u3053\u306e\u516c\u5f0f\u3067\u306f, \u8fd1\u4f3c\u5024 y \u306e\u8a08\u7b97\u3092 5\u6b21\u3067\u884c\u3044, \u8aa4\u5dee\u8a55\u4fa1\u7528\u306e\u5024 y* \u306e\u8a08\u7b97\u3092 4\u6b21\u3067\u884c\u3046. \u3053\u308c\u3092, 5(4)\u6b21\u306e\u516c\u5f0f\u3068\u66f8\u304f. 1\u30b9\u30c6\u30c3\u30d7\u3042\u305f\u308a 6\u56de\u306e\u95a2\u6570\u8a55\u4fa1\u304c\u5fc5\u8981\u3067\u3042\u308b.<\/p>\n
      5(4)\u6b21 \u30c9\u30eb\u30de\u30f3\u30fb\u30d7\u30ea\u30f3\u30b9\u6cd5<\/h4>\n
      5(4)\u6b21 \u30c9\u30eb\u30de\u30f3\u30fb\u30d7\u30ea\u30f3\u30b9\u6cd5(Dormand-Prince method)\u306e\u4fc2\u6570\u306f\u6b21\u306e\u3068\u304a\u308a.<\/p>\n
        \r\n  0    |\r\n  1\/5  |     1\/5\r\n  3\/10 |     3\/40        9\/40\r\n  4\/5  |    44\/45      -56\/15      32\/9\r\n  8\/9  | 19372\/6561 -25360\/2187 64448\/6561  -212\/729\r\n  1    |  9017\/3168   -355\/33   46732\/5247    49\/176  -5103\/18656\r\n  1    |    35\/384        0       500\/1113   125\/192  -2187\/6784    11\/84\r\n  --------------------------------------------------------------------------------\r\n  y    |    35\/384        0       500\/1113   125\/192  -2187\/6784    11\/84     0\r\n  y*   |  5179\/57600      0      7571\/16695  393\/640 -92097\/339200 187\/2100  1\/40\r\n    \r\n<\/pre>\n
       
      \n5(4)\u6b21\u306e\u516c\u5f0f\u3067, 1\u30b9\u30c6\u30c3\u30d7\u3042\u305f\u308a 7\u56de\u306e\u95a2\u6570\u8a55\u4fa1\u304c\u5fc5\u8981\u3067\u3042\u308b. \u3053\u306e\u516c\u5f0f\u306e\u7279\u5fb4\u306f, 7\u756a\u76ee(k7<\/sub>)\u306e\u4fc2\u6570 a7j<\/sub> \u304c\u8fd1\u4f3c\u5024 y \u3092\u8a08\u7b97\u3059\u308b\u305f\u3081\u306e\u4fc2\u6570 bi<\/sub> \u306b\u7b49\u3057\u3044\u3053\u3068\u3067\u3042\u308b. \u3053\u308c\u306f FSAL (First Same As Last) \u3068\u3088\u3070\u308c\u308b\u30a2\u30a4\u30c7\u30a2\u3067, k7<\/sub> \u304c\u6b21\u306e k1<\/sub> = f(yn+1<\/sub>, tn+1<\/sub>) \u306b\u7b49\u3057\u304f\u306a\u308b\u3088\u3046\u5de7\u5999\u306b\u4f5c\u3089\u308c\u3066\u3044\u308b.<\/p>\n
      6(5)\u6b21 \u30eb\u30f3\u30b2\u30fb\u30af\u30c3\u30bf\u30fb\u30f4\u30a1\u30fc\u30ca\u30fc\u6cd5<\/h4>\n
      6(5)\u6b21 \u30eb\u30f3\u30b2\u30fb\u30af\u30c3\u30bf\u30fb\u30f4\u30a1\u30fc\u30ca\u30fc\u6cd5(Runge-Kutta-Verner method)\u306e\u4fc2\u6570\u306f\u6b21\u306e\u3068\u304a\u308a.<\/p>\n
        \r\n  0    |\r\n  1\/6  |     1\/6\r\n  4\/15 |     4\/75     16\/75\r\n  2\/3  |     5\/6      -8\/3       5\/2\r\n  5\/6  |  -165\/64     55\/6    -425\/64      85\/96\r\n  1    |    12\/5       -8     4015\/612    -11\/36     88\/255\r\n  1\/15 | -8263\/15000 124\/75   -643\/680    -81\/250  2484\/10625 0\r\n  1    |  3501\/1720 -300\/43 297275\/52632 -319\/2322 24068\/84065  0  3850\/26703\r\n  -----------------------------------------------------------------------------------\r\n  y    |     3\/40       0     875\/2244     23\/72     264\/1955   0  125\/11592  43\/616\r\n  y*   |    13\/160      0    2375\/5984      5\/16      12\/85    3\/44   0      0\r\n    \r\n<\/pre>\n
       
      \n6(5)\u6b21\u306e\u516c\u5f0f\u3067, 1\u30b9\u30c6\u30c3\u30d7\u3042\u305f\u308a 8\u56de\u306e\u95a2\u6570\u8a55\u4fa1\u304c\u5fc5\u8981\u3067\u3042\u308b. \u9577\u3044\u9593\u5e83\u304f\u4f7f\u308f\u308c\u3066\u3044\u308b\u30b5\u30d6\u30eb\u30fc\u30c1\u30f3DVERK\u306b\u63a1\u7528\u3055\u308c\u3066\u3044\u308b\u516c\u5f0f\u3067\u3042\u308b.<\/p>\n
      \u591a\u6bb5\u6cd5\u306e\u8aa4\u5dee<\/h3>\n
      \u591a\u6bb5\u6cd5\u306b\u304a\u3051\u308b\u5c40\u6240\u8aa4\u5dee\u306f\u6b21\u306e\u3088\u3046\u306b\u6c42\u3081\u3089\u308c\u308b.<\/p>\n\n\n\n\n\n\n
      \u89e3\u6cd5<\/td>\n\u6b21\u6570<\/td>\n\u5c40\u6240\u8aa4\u5dee<\/td>\n<\/tr>\n
      \u30a2\u30c0\u30e0\u30b9-\u30d0\u30b7\u30e5\u30d5\u30a9\u30fc\u30b9\u6cd5<\/td>\nk<\/td>\n\u03b3k<\/sub> hk+1<\/sup> y(k+1)<\/sup><\/td>\n<\/tr>\n
      \u30a2\u30c0\u30e0\u30b9-\u30e0\u30eb\u30c8\u30f3\u6cd5<\/td>\nk+1<\/td>\n\u03b3*<\/sup>k+1<\/sub> hk+2<\/sup> y(k+2)<\/sup><\/td>\n<\/tr>\n
      \u5f8c\u9000\u5fae\u5206\u516c\u5f0f (BDF\u6cd5)<\/td>\nk<\/td>\n-1\/(k+1) hk+1<\/sup> y(k+1)<\/sup><\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n
      \u4f8b\u3048\u3070, 4\u6b21\u306e\u30a2\u30c0\u30e0\u30b9\u6cd5\u3067\u306f, \u4e88\u6e2c\u5b50(\u53f3\u80a9\u306bP)\u3068\u4fee\u6b63\u5b50(\u53f3\u80a9\u306bC)\u306e\u305d\u308c\u305e\u308c\u306e\u5c40\u6240\u8aa4\u5dee\u306f\u5177\u4f53\u7684\u306b\u306f\u6b21\u306e\u3088\u3046\u306b\u306a\u308b.<\/p>\n
        \r\n  y(tn+1<\/sub>) - yP<\/sup>n+1<\/sub> = \u03b34<\/sub>h5<\/sup>y(5)<\/sup> + O(h6<\/sup>) = 251\/720 h5<\/sup>y(5)<\/sup> + O(h6<\/sup>)\r\n  y(tn+1<\/sub>) - yC<\/sup>n+1<\/sub> = \u03b3*<\/sup>4<\/sub>h5<\/sup>y(5)<\/sup> + O(h6<\/sup>) = -19\/720 h5<\/sup>y(5)<\/sup> + O(h6<\/sup>)\r\n\r\n<\/pre>\n
       
      \n\u3053\u306e\u3088\u3046\u306b\u591a\u6bb5\u6cd5\u3067\u306f\u4e88\u6e2c\u5b50\u3068\u4fee\u6b63\u5b50\u3067\u5c40\u6240\u8aa4\u5dee\u304c\u7570\u306a\u308b\u306e\u3067, \u305d\u308c\u3092\u4f7f\u3044\u8aa4\u5dee\u306e\u63a8\u5b9a\u3092\u3059\u308b\u3053\u3068\u304c\u3067\u304d\u308b.<\/p>\n
      \u4e21\u8fba\u306e\u5f15\u304d\u7b97\u3092\u3057\u3066\u9ad8\u6b21\u306e\u9805\u3092\u7121\u8996\u3059\u308b\u3068\u6b21\u306e\u8fd1\u4f3c\u5f0f\u304c\u5f97\u3089\u308c\u308b.<\/p>\n
        \r\n  yC<\/sup>n+1<\/sub> - yP<\/sup>n+1<\/sub> \u2252 3\/8 h5<\/sup>y(5)<\/sup>\r\n\r\n<\/pre>\n
       
      \n\u3053\u308c\u3092\u4e0a\u306e\u4fee\u6b63\u5b50\u306e\u5f0f\u306b\u4ee3\u5165\u3057\u3066\u9ad8\u6b21\u306e\u9805\u3092\u7121\u8996\u3059\u308b\u3068\u6b21\u306e\u3088\u3046\u306a\u8aa4\u5dee\u8a55\u4fa1\u5f0f\u304c\u5f97\u3089\u308c\u308b.<\/p>\n
        \r\n  y(tn+1<\/sub>) - yC<\/sup>n+1<\/sub> \u2252 -19\/27(yC<\/sup>n+1<\/sub> - yP<\/sup>n+1<\/sub>)\r\n\r\n<\/pre>\n
       <\/p>\n
      \u30b9\u30c6\u30c3\u30d7\u5e45\u306e\u81ea\u52d5\u8abf\u7bc0<\/h2>\n
      \u4e00\u5b9a\u306e\u30b9\u30c6\u30c3\u30d7\u5e45\u3067\u8a08\u7b97\u3059\u308b\u306e\u3067\u306f\u306a\u304f, \u76ee\u6a19\u8aa4\u5dee\u3092\u6e80\u305f\u3059\u3088\u3046\u306b\u6700\u9069\u306a\u30b9\u30c6\u30c3\u30d7\u5e45\u306b\u81ea\u52d5\u8abf\u7bc0\u3057\u306a\u304c\u3089\u8a08\u7b97\u3059\u308b\u30d7\u30ed\u30b0\u30e9\u30e0\u304c\u73fe\u5728\u306e\u4e3b\u6d41\u3067\u3042\u308b. \u3053\u308c\u306f, \u76ee\u6a19\u8aa4\u5dee\u306b\u5bfe\u3057\u3066\u8a08\u7b97\u91cf\u3092\u6700\u5c0f\u5316\u3059\u308b\u3053\u3068\u3068, \u8a08\u7b97\u306e\u5b89\u5b9a\u6027\u3092\u640d\u306a\u308f\u306a\u3044\u3088\u3046\u306b\u30b9\u30c6\u30c3\u30d7\u5e45\u3092\u7dad\u6301\u3059\u308b\u3053\u3068\u306b\u52b9\u679c\u304c\u3042\u308b.<\/p>\n
      \u4e0a\u306b\u8aac\u660e\u3057\u305f\u8aa4\u5dee\u63a8\u5b9a\u6a5f\u80fd\u3092\u6301\u3064\u516c\u5f0f\u3092\u4f7f\u3048\u3070\u305d\u306e\u3088\u3046\u306a\u30d7\u30ed\u30b0\u30e9\u30e0\u3092\u5b9f\u88c5\u3059\u308b\u3053\u3068\u304c\u3067\u304d\u308b.<\/p>\n
      \u30b9\u30c6\u30c3\u30d7\u5e45\u306e\u81ea\u52d5\u8abf\u7bc0\u6a5f\u80fd\u306e\u5b9f\u88c5<\/h3>\n
      \u304a\u304a\u3056\u3063\u3071\u306b\u3044\u3048\u3070, \u30b9\u30c6\u30c3\u30d7\u5e45\u306e\u521d\u671f\u5024\u3068\u5c40\u6240\u96e2\u6563\u5316\u8aa4\u5dee\u306e\u8a31\u5bb9\u9650\u754c\u3092\u4e0e\u3048\u8a08\u7b97\u3092\u958b\u59cb\u3057, \u30b9\u30c6\u30c3\u30d7\u3054\u3068\u306b\u63a8\u5b9a\u8aa4\u5dee\u3092\u30c1\u30a7\u30c3\u30af\u3057, \u8a31\u5bb9\u5185\u3067\u3042\u308c\u3070\u6b21\u30b9\u30c6\u30c3\u30d7\u306b\u9032\u307f\u8a08\u7b97\u3092\u7d9a\u3051\u308b(\u30b9\u30c6\u30c3\u30d7\u6210\u529f). \u63a8\u5b9a\u8aa4\u5dee\u304c\u5927\u304d\u3059\u304e\u308b\u5834\u5408, \u30b9\u30c6\u30c3\u30d7\u5e45\u3092\u5c0f\u3055\u304f\u3057\u3066\u3084\u308a\u76f4\u3059(\u30b9\u30c6\u30c3\u30d7\u5931\u6557). \u307e\u305f, \u63a8\u5b9a\u8aa4\u5dee\u304c\u5c0f\u3055\u3059\u304e\u308b\u3068\u30b9\u30c6\u30c3\u30d7\u6570\u304c\u4e0d\u8981\u306b\u5897\u3048\u3066\u3057\u307e\u3046\u306e\u3067, \u6b21\u30b9\u30c6\u30c3\u30d7\u304b\u3089\u30b9\u30c6\u30c3\u30d7\u5e45\u3092\u5927\u304d\u304f\u3059\u308b\u306a\u3069\u306e\u8abf\u6574\u3092\u884c\u3046. \u3053\u306e\u3088\u3046\u306a\u51e6\u7406\u3092\u7e70\u308a\u8fd4\u3057, \u76ee\u6a19\u7cbe\u5ea6\u3092\u6e80\u305f\u3057\u3064\u3064\u6700\u5c0f\u306e\u30b9\u30c6\u30c3\u30d7\u6570\u3067\u8a08\u7b97\u3092\u5b8c\u4e86\u3059\u308b\u3053\u3068\u3092\u76ee\u6307\u3059. \u305f\u3060\u3057, \u5b9f\u969b\u306b\u5b9f\u7528\u306b\u4f9b\u3059\u308b\u30d7\u30ed\u30b0\u30e9\u30e0\u3092\u5b9f\u88c5\u3059\u308b\u306e\u306f\u7c21\u5358\u3067\u306f\u306a\u304f\u591a\u304f\u306e\u7d30\u304b\u3044\u5de5\u592b\u304c\u5fc5\u8981\u3067\u3042\u308b.<\/p>\n
      \u591a\u6bb5\u6cd5\u306e\u5834\u5408\u306b\u306f, \u30b9\u30c6\u30c3\u30d7\u5e45\u3092\u5909\u66f4\u3059\u308b\u3068\u4ee5\u524d\u306e\u5024\u306e\u9593\u9694\u304c\u65b0\u3057\u3044\u30b9\u30c6\u30c3\u30d7\u5e45\u3068\u5408\u308f\u306a\u304f\u306a\u308a, \u516c\u5f0f\u304c\u305d\u306e\u307e\u307e\u3067\u306f\u4f7f\u3048\u306a\u304f\u306a\u308b. \u305d\u3053\u3067, \u4e0d\u7b49\u9593\u9694\u306e\u30b9\u30c6\u30c3\u30d7\u70b9\u306b\u5bfe\u3059\u308b\u516c\u5f0f\u306b\u5909\u66f4\u3057\u305f\u53ef\u5909\u30b9\u30c6\u30c3\u30d7\u5e45\u30a2\u30c0\u30e0\u30b9\u6cd5\u3084\u53ef\u5909\u30b9\u30c6\u30c3\u30d7\u5e45BDF\u6cd5\u306a\u3069\u3092\u4f7f\u3046\u5fc5\u8981\u304c\u3042\u308b.<\/p>\n
      \u6570\u5024\u5b9f\u9a13 (15)<\/h3>\n
      <\/span>\u30b9\u30c6\u30c3\u30d7\u5e45\u81ea\u52d5\u8abf\u7bc0\u6a5f\u80fd\u4ed8\u304d\u30d7\u30ed\u30b0\u30e9\u30e0\u306e\u8a08\u7b97\u4f8b<\/div>
      \n\u30eb\u30f3\u30b2\u30fb\u30af\u30c3\u30bf\u30fb\u30de\u30fc\u30bd\u30f3\u6cd5\u3092\u4f7f\u3044\u7c21\u5358\u306a\u30b9\u30c6\u30c3\u30d7\u5e45\u81ea\u52d5\u8abf\u7bc0\u6a5f\u80fd\u4ed8\u304d\u30d7\u30ed\u30b0\u30e9\u30e0(RKM)\u3092\u4f5c\u6210\u3057\u3066\u5b9f\u9a13\u3057\u3066\u307f\u308b.<\/p>\n
      \u8a08\u7b97\u30a2\u30eb\u30b4\u30ea\u30ba\u30e0\u306f\u6b21\u306e\u3088\u3046\u306b\u3057\u305f.<\/p>\n
        \n
      • \u30b9\u30c6\u30c3\u30d7\u5e45\u306e\u521d\u671f\u5024\u3068\u5c40\u6240\u96e2\u6563\u5316\u8aa4\u5dee\u306e\u8a31\u5bb9\u9650\u754c\u3092\u4e0e\u3048\u8a08\u7b97\u3092\u958b\u59cb\u3059\u308b<\/li>\n
      • \u30de\u30fc\u30bd\u30f3\u306e\u516c\u5f0f\u30671\u30b9\u30c6\u30c3\u30d7\u8a08\u7b97\u5f8c, \u63a8\u5b9a\u8aa4\u5dee\u304c\u8a31\u5bb9\u9650\u754c\u3088\u308a\u5927\u304d\u3051\u308c\u3070\u30b9\u30c6\u30c3\u30d7\u5e45\u30921\/2\u306b\u3057\u3066\u3084\u308a\u76f4\u3059(\u5931\u6557\u30b9\u30c6\u30c3\u30d7)<\/li>\n
      • \u63a8\u5b9a\u8aa4\u5dee\u304c\u8a31\u5bb9\u9650\u754c\u3088\u308a\u5c0f\u3055\u3051\u308c\u30701\u30b9\u30c6\u30c3\u30d7\u9032\u3081\u308b(\u6210\u529f\u30b9\u30c6\u30c3\u30d7). \u305f\u3060\u3057, \u63a8\u5b9a\u8aa4\u5dee\u304c\u8a31\u5bb9\u9650\u754c\u306e1\/32\u3088\u308a\u5c0f\u3055\u3044\u3068\u304d\u306b\u306f\u6b21\u30b9\u30c6\u30c3\u30d7\u304b\u3089\u30b9\u30c6\u30c3\u30d7\u5e45\u30922\u500d\u306b\u3059\u308b<\/li>\n<\/ul>\n
        2\u7a2e\u985e\u306e\u5fae\u5206\u65b9\u7a0b\u5f0f\u306e\u521d\u671f\u5024\u554f\u984c\u4f8b\u3092\u4f7f\u3063\u3066\u30b9\u30c6\u30c3\u30d7\u5e45\u306e\u81ea\u52d5\u8abf\u7bc0\u6a5f\u80fd\u306e\u52d5\u4f5c\u3092\u78ba\u8a8d\u3059\u308b. \u6bd4\u8f03\u306e\u305f\u3081\u306b, 4\u6b21\u306e\u30eb\u30f3\u30b2\u30fb\u30af\u30c3\u30bf\u6cd5(RK4)\u304a\u3088\u30735(4)\u6b21\u306e\u30eb\u30f3\u30b2\u30fb\u30af\u30c3\u30bf\u30fb\u30d5\u30a7\u30fc\u30eb\u30d9\u30eb\u30b0\u6cd5(RKF45)(XLPack\u306eDerkf\u3092\u4f7f\u7528)\u306b\u3088\u308b\u8a08\u7b97\u7d50\u679c\u3082\u793a\u3059.<\/p>\n
        \u6a2a\u8ef8\u306f\u76f8\u5bfe\u8aa4\u5dee(\u53f3\u306b\u884c\u304f\u307b\u3069\u7cbe\u5ea6\u304c\u3088\u3044. n = 2 \u306e\u5834\u5408\u306f\u305d\u308c\u305e\u308c\u306e\u5024\u306e\u5e73\u5747), \u7e26\u8ef8\u306f\u95a2\u6570\u8a55\u4fa1\u56de\u6570\u306e\u5bfe\u6570\u30d7\u30ed\u30c3\u30c8\u3067\u3042\u308b. RKM\u3068RKF45\u3067\u306f\u8981\u6c42\u7cbe\u5ea6(Tol)\u3092\u5c11\u3057\u3065\u3064\u5909\u3048\u3066\u8a08\u7b97\u3092\u884c\u3044, \u5f97\u3089\u308c\u305f\u7d50\u679c\u306e\u5b9f\u969b\u306e\u8aa4\u5dee\u3068\u95a2\u6570\u8a55\u4fa1\u56de\u6570\u3092\u30d7\u30ed\u30c3\u30c8\u3057\u305f. RK4\u3067\u306f\u30b9\u30c6\u30c3\u30d7\u5e45\u3092\u5909\u3048\u3066\u3044\u304d, \u305d\u306e\u3068\u304d\u306e\u8aa4\u5dee\u3068\u95a2\u6570\u8a55\u4fa1\u56de\u6570\u3092\u30d7\u30ed\u30c3\u30c8\u3057\u305f. \u306a\u304a, RKM\u3067\u306f\u30b9\u30c6\u30c3\u30d7\u5e45\u306e\u521d\u671f\u5024\u309250*Tol\u3068\u3057\u305f.<\/p>\n
        (\u4f8b1)<\/h4>\n
          \r\n\r\n  y' = y, y(0) = 1\r\n\r\n<\/pre>\n
         
        \n\u4f8b1\u306f\u5909\u5316\u304c\u5c11\u306a\u3044\u65b9\u7a0b\u5f0f\u306e\u4f8b\u3067, \u89e3\u6790\u89e3\u306f y = et<\/sup> \u3067\u3042\u308b. t = 1 \u306b\u304a\u3051\u308b\u5024 (= e) \u3092\u6c42\u3081\u308b.<\/p>\n
        \"\"<\/p>\n
        (\u4f8b2)<\/h4>\n
          \r\n  y1<\/sub>' = 1 + y1<\/sub>2<\/sup>y2<\/sub> - 4y1<\/sub>\r\n  y2<\/sub>' = 3y1<\/sub> - y1<\/sub>2<\/sup>y2<\/sub>\r\n  y1<\/sub>(0) = 1.5, y2<\/sub>(0) = 3\r\n\r\n<\/pre>\n
         
        \n\u4f8b2\u306f\u6587\u732e[3]\u306e\u30d6\u30e9\u30bb\u30ec\u30fc\u30bf\u3068\u3088\u3070\u308c\u308b\u5909\u5316\u306e\u6fc0\u3057\u3044\u65b9\u7a0b\u5f0f\u306e\u4f8b\u3067 n = 2 \u3067\u3042\u308b. t = 16 \u306b\u304a\u3051\u308b\u5024\u3092\u6c42\u3081\u308b.<\/p>\n
        \"\"<\/p>\n
        RKM\u3068RK4\u306f\u3069\u3061\u3089\u30824\u6b21\u3067\u3042\u308b\u304c, \u4f8b1\u3067\u306fRK4\u306e\u65b9\u304c\u901f\u304f, \u4f8b2\u3067\u306fRKM\u306e\u65b9\u304c\u901f\u3044. \u3053\u308c\u306f, \u5909\u5316\u304c\u5c11\u306a\u3044\u95a2\u6570\u306b\u304a\u3044\u3066\u306f\u30b9\u30c6\u30c3\u30d7\u5e45\u306e\u8abf\u7bc0\u3092\u884c\u3046\u5fc5\u8981\u304c\u306a\u3044\u304c, RKM\u3067\u306f\u8abf\u7bc0\u306e\u305f\u3081\u306b\u7121\u99c4\u306a\u30b9\u30c6\u30c3\u30d7\u3092\u8cbb\u3084\u3057\u3066\u3044\u308b\u305f\u3081\u3068\u601d\u308f\u308c\u308b. \u6570\u5024\u5b9f\u9a13(16)\u3067\u8a73\u7d30\u3092\u78ba\u8a8d\u3059\u308b.
        \n<\/div><\/div>\n
        \u6570\u5024\u5b9f\u9a13 (16)<\/h3>\n
        <\/span>\u30b9\u30c6\u30c3\u30d7\u5e45\u81ea\u52d5\u8abf\u7bc0\u306e\u52d5\u4f5c\u306e\u8a73\u7d30<\/div>
        \n\u4e0a\u306e2\u3064\u306e\u4f8b\u3092\u8a08\u7b97\u3057\u305f\u3068\u304d\u306eRKM\u306e\u52d5\u4f5c\u3092\u8a73\u7d30\u306b\u898b\u3066\u307f\u308b.<\/p>\n
        (\u4f8b1)<\/h4>\n
        Tol = 1.0e-7 \u3068\u3057\u305f\u3068\u304d\u306e\u52d5\u4f5c. \u4e0a\u6bb5\u306f\u8a08\u7b97\u5024, \u4e0b\u6bb5\u306f\u30b9\u30c6\u30c3\u30d7\u5e45\u3067\u3042\u308b. \u30b9\u30c6\u30c3\u30d7\u5e45\u306e\u521d\u671f\u5024\u306f 50*Tol \u3068\u3057\u305f. \u4e38\u306f\u5404\u30b9\u30c6\u30c3\u30d7\u3092\u793a\u3059.<\/p>\n
        \"\"<\/p>\n
        \u95a2\u6570\u304c\u6ed1\u3089\u304b\u306a\u306e\u3067 h = 0.082 \u306b\u306a\u3063\u305f\u5f8c\u306f\u5909\u66f4\u3055\u308c\u306a\u304b\u3063\u305f. \u30b9\u30c6\u30c3\u30d7\u5e45\u306e\u521d\u671f\u5024\u304c\u5c0f\u3055\u3059\u304e\u305f\u3088\u3046\u306a\u306e\u3067\u5de5\u592b\u304c\u5fc5\u8981\u3067\u3042\u308d\u3046. \u306a\u304a, \u6700\u5f8c\u306e1\u30b9\u30c6\u30c3\u30d7\u3067\u30b9\u30c6\u30c3\u30d7\u5e45\u304c\u5909\u5316\u3057\u3066\u3044\u308b\u306e\u306f\u8aa4\u5dee\u306e\u554f\u984c\u3067\u306f\u306a\u304f, \u7d42\u70b9\u3092\u3061\u3087\u3046\u30691\u306b\u5408\u308f\u305b\u308b\u305f\u3081\u306b\u8abf\u6574\u3055\u308c\u305f\u3082\u306e\u3067\u3042\u308b.<\/p>\n
        (\u4f8b2)<\/h4>\n
        Tol = 1.0e-4 \u3068\u3057\u305f\u3068\u304d\u306e\u52d5\u4f5c. \u4e0a\u6bb5\u306f\u8a08\u7b97\u5024, \u4e0b\u6bb5\u306f\u30b9\u30c6\u30c3\u30d7\u5e45\u3067\u3042\u308b. \u30b9\u30c6\u30c3\u30d7\u5e45\u306e\u521d\u671f\u5024\u306f 50*Tol \u3068\u3057\u305f. \u4e38\u306f\u5404\u30b9\u30c6\u30c3\u30d7\u3092\u793a\u3059.<\/p>\n
        \"\"<\/p>\n
        \u6025\u306a\u5909\u5316\u3092\u3059\u308b\u3068\u3053\u308d\u3067\u306f\u30b9\u30c6\u30c3\u30d7\u5e45\u3092\u5c0f\u3055\u304f, \u3086\u308b\u3084\u304b\u306a\u5909\u5316\u3092\u3059\u308b\u3068\u3053\u308d\u3067\u306f\u30b9\u30c6\u30c3\u30d7\u5e45\u3092\u5927\u304d\u304f\u8abf\u7bc0\u3057\u3066\u3044\u308b\u306e\u304c\u308f\u304b\u308b. \u6210\u529f\u30b9\u30c6\u30c3\u30d7\u304c97\u56de\u3067\u8a08\u7b97\u3092\u7d42\u3048\u3066\u3044\u308b\u304c, \u5931\u6557\u30b9\u30c6\u30c3\u30d7\u304c25\u56de\u3042\u3063\u305f. \u3059\u306a\u308f\u3061, 2\u5272\u307b\u3069\u7121\u99c4\u30b9\u30c6\u30c3\u30d7\u3092\u8e0f\u3093\u3067\u3044\u308b\u304c\u305d\u308c\u3067\u3082RK4\u3088\u308a\u306f\u52b9\u7387\u304c\u3088\u304b\u3063\u305f\u3053\u3068\u306b\u306a\u308b.<\/p>\n
        \u3053\u306e\u4f8b\u3067RK4\u306e\u65b9\u304c\u9045\u304b\u3063\u305f\u306e\u306f, \u5168\u4f53\u306e\u7cbe\u5ea6\u3092\u78ba\u4fdd\u3059\u308b\u305f\u3081\u306b\u306f\u6025\u306a\u5909\u5316\u3092\u3059\u308b\u3068\u3053\u308d\u3067\u5fc5\u8981\u306a\u30b9\u30c6\u30c3\u30d7\u5e45\u306b\u5408\u308f\u305b\u308b\u5fc5\u8981\u304c\u3042\u308a, \u3086\u308b\u3084\u304b\u306a\u5909\u5316\u3092\u3059\u308b\u3068\u3053\u308d\u3067\u5fc5\u8981\u4ee5\u4e0a\u306b\u30b9\u30c6\u30c3\u30d7\u5e45\u304c\u5c0f\u3055\u304f\u306a\u308a\u7121\u99c4\u306a\u8a08\u7b97\u3092\u884c\u3063\u305f\u305f\u3081\u3068\u8003\u3048\u3089\u308c\u308b.
        \n<\/div><\/div>\n<\/div>\n
        \n
        \u5bc6\u51fa\u529b<\/h2>\n
        \u6b21\u306e\u521d\u671f\u5024\u554f\u984c\u306e\u89e3\u3092\u6c42\u3081\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.<\/p>\n
        \u30b9\u30c6\u30c3\u30d7\u5e45\u3092\u81ea\u52d5\u8abf\u7bc0\u3059\u308b\u30d7\u30ed\u30b0\u30e9\u30e0 Dopri5(5(4)\u6b21 \u30c9\u30eb\u30de\u30f3\u30fb\u30d7\u30ea\u30f3\u30b9\u6cd5) \u3068 Derkf(5(4)\u6b21 \u30eb\u30f3\u30b2\u30fb\u30af\u30c3\u30bf\u30fb\u30d5\u30a7\u30fc\u30eb\u30d9\u30eb\u30b0\u6cd5) \u3092\u4f7f\u3063\u3066, \u8981\u6c42\u7cbe\u5ea6(RTol, ATol) = 1.0e-7 \u3068\u3057\u3066, t = 0 \uff5e 1 \u3067 0.05 \u3054\u3068\u306b 0.0 \uff5e 0.05, 0.05 \uff5e 0.1, …, 0.95 \uff5e 1.0 \u3068\u3044\u3046\u3088\u3046\u306b\u7d42\u70b9\u3092\u5909\u3048\u306a\u304c\u3089\u7e70\u308a\u8fd4\u3057\u89e3\u3092\u6c42\u3081\u305f. \u3053\u306e\u3068\u304d\u306e\u8aa4\u5dee\u3092\u30d7\u30ed\u30c3\u30c8\u3059\u308b\u3068\u6b21\u306e\u3088\u3046\u306b\u306a\u3063\u305f. \u9752\u306e\u5b9f\u7dda\u3068\u5de6\u5074\u306e\u76ee\u76db\u306f\u95a2\u6570\u5f62\u3092\u8868\u3059. \u70b9\u7dda\u3068\u53f3\u5074\u306e\u76ee\u76db\u306f\u5404\u89e3\u6cd5\u306b\u3088\u308b\u8aa4\u5dee\u3092\u8868\u3059.<\/p>\n
        \"\"<\/p>\n
        Dopri5\u3082Derkf\u3082 0.05 \u5e45\u306e\u5404\u533a\u9593\u3067\u306f1\u30b9\u30c6\u30c3\u30d7\u3067\u89e3\u3092\u6c42\u3081, \u8aa4\u5dee\u306f 1.0e-10 \u4ed8\u8fd1\u3068\u8981\u6c42\u7cbe\u5ea6\u3088\u308a\u304b\u306a\u308a\u3088\u3044\u7cbe\u5ea6\u3067\u89e3\u3092\u6c42\u3081\u305f. \u3053\u308c\u306f, \u81ea\u52d5\u8abf\u7bc0\u3055\u308c\u308b\u3079\u304d\u8981\u6c42\u7cbe\u5ea6\u306b\u898b\u5408\u3063\u305f\u30b9\u30c6\u30c3\u30d7\u5e45\u3088\u308a\u3082 0.05 \u3068\u3044\u3046\u5e45\u304c\u5c0f\u3055\u304f, \u7d42\u70b9\u306b\u5408\u308f\u305b\u308b\u305f\u3081\u306b\u30b9\u30c6\u30c3\u30d7\u5e45\u304c\u5e38\u306b 0.05 \u306b\u8abf\u7bc0\u3055\u308c\u305f\u305f\u3081\u3067\u3042\u308b. \u3059\u306a\u308f\u3061, \u8981\u6c42\u7cbe\u5ea6\u306b\u6bd4\u3079\u3066\u516c\u5f0f\u306e\u7cbe\u5ea6\u304c\u826f\u3059\u304e\u308b\u306e\u3067\u3042\u308b. \u6bd4\u8f03\u306e\u305f\u3081\u306b4\u6b21\u306e\u30eb\u30f3\u30b2\u30fb\u30af\u30c3\u30bf\u6cd5(RK4)\u3067\u30b9\u30c6\u30c3\u30d7\u5e45\u3092 0.05 \u3068\u3057\u3066\u8a08\u7b97\u3057\u3066\u307f\u308b\u3068, \u8aa4\u5dee\u306f\u3061\u3087\u3046\u3069 1.0e-7 \u4ed8\u8fd1\u306b\u306a\u308a, RK4\u306e\u7cbe\u5ea6\u7a0b\u5ea6\u3067\u3088\u3044\u3053\u3068\u304c\u308f\u304b\u308b. \u95a2\u6570\u8a55\u4fa1\u56de\u6570\u306f, Dopri5\u304c128\u56de, Derkf\u304c124\u56de\u3067, RK4\u306f80\u56de\u3068\u306a\u3063\u305f. Dopri5\u3068Derkf\u306f\u3053\u306e\u5834\u5408\u306b\u306f\u9ad8\u7cbe\u5ea6\u306a\u3086\u3048\u306b\u7121\u99c4\u306a\u8a08\u7b97\u3092\u3057\u3066\u3044\u308b\u3053\u3068\u306b\u306a\u308b.<\/p>\n
        \u305d\u3053\u3067, \u30b9\u30c6\u30c3\u30d7\u5e45\u306f\u5fc5\u8981\u7cbe\u5ea6\u306e\u9762\u304b\u3089\u6700\u9069\u306a\u3082\u306e\u3092\u9078\u3073, \u30b9\u30c6\u30c3\u30d7\u306b\u4e00\u81f4\u3057\u306a\u3044\u51fa\u529b\u70b9\u306b\u304a\u3044\u3066\u306f\u8a08\u7b97\u3092\u884c\u308f\u305a\u306b\u88dc\u9593\u5024\u3092\u4f7f\u3046\u3053\u3068\u306b\u3088\u308a\u7121\u99c4\u306a\u8a08\u7b97\u3092\u907f\u3051\u308b\u65b9\u6cd5\u304c\u8003\u3048\u3089\u308c\u308b. \u3053\u308c\u3092\u5bc6\u51fa\u529b(dense output)<\/strong>\u3068\u3088\u3076.<\/p>\n
        \u591a\u6bb5\u6cd5\u306e\u5834\u5408<\/h3>\n
        \u30a2\u30c0\u30e0\u30b9\u30fb\u30e0\u30eb\u30c8\u30f3\u6cd5\u3092\u4f8b\u306b\u3068\u308b. \u30a2\u30c0\u30e0\u30b9\u30fb\u30e0\u30eb\u30c8\u30f3\u6cd5\u306e\u8aac\u660e\u3067\u4f7f\u7528\u3057\u305f\u56f3\u3092\u518d\u63b2\u3059\u308b.<\/p>\n
        \"\"<\/p>\n
        tn<\/sub>\u306e\u6b21\u306e\u30b9\u30c6\u30c3\u30d7 tn+1<\/sub> = tn<\/sub> + h \u306b\u304a\u3051\u308b\u5024\u3092\u6c42\u3081\u308b\u969b\u306b\u306f\u56f3\u306e\u3088\u3046\u306b\u88dc\u9593\u591a\u9805\u5f0fp*(t)\u3092\u6c42\u3081, \u3053\u308c\u3092\u533a\u9593[tn<\/sub>, tn+1<\/sub>]\u3067\u7a4d\u5206\u3059\u308b\u3053\u3068\u306b\u3088\u308a\u5024\u3092\u6c42\u3081\u305f.<\/p>\n
          \r\n  yn+1<\/sub> = yn<\/sub> + \u222b p*<\/sup>(x) dx [tn<\/sub>, tn+1<\/sub>]\r\n\r\n<\/pre>\n
         
        \n\u540c\u69d8\u306b, tn<\/sub> + \u03b8h (0 \u2266 \u03b8 \u2266 1) \u306b\u304a\u3051\u308b\u5024 y(tn<\/sub> + \u03b8h) \u306f\u540c\u3058\u88dc\u9593\u5f0f\u3092\u533a\u9593[tn<\/sub>, tn<\/sub> + \u03b8h]\u3067\u7a4d\u5206\u3059\u308c\u3070\u6c42\u3081\u308b\u3053\u3068\u304c\u3067\u304d\u308b. \u65b0\u305f\u306b\u95a2\u6570\u8a55\u4fa1\u3092\u884c\u3046\u5fc5\u8981\u304c\u306a\u3044\u3057\u7cbe\u5ea6\u304c\u843d\u3061\u308b\u3053\u3068\u3082\u306a\u3044.<\/p>\n
        \u6570\u5024\u5b9f\u9a13 (17)<\/h3>\n
        <\/span>\u30a2\u30c0\u30e0\u30b9\u30fb\u30e0\u30eb\u30c8\u30f3\u6cd5\u306b\u3088\u308b\u5bc6\u51fa\u529b<\/div>
        \n\u4e0a\u306e\u4f8b\u984c\u306b\u304a\u3044\u30664\u6b21\u30a2\u30c0\u30e0\u30b9\u30fb\u30e0\u30eb\u30c8\u30f3\u6cd5\u3067 h = 0.1 \u3068\u3057\u3066 t = 0.4, 0.5, …, 1.0 \u306b\u304a\u3051\u308by\u306e\u5024\u3092\u6c42\u3081\u308b. t = 0.1, 0.2, 0.3 \u306b\u304a\u3051\u308b\u5024(\u51fa\u767a\u5024)\u306f4\u6b21\u306e\u30eb\u30f3\u30b2\u30fb\u30af\u30c3\u30bf\u6cd5\u3067\u6c42\u3081\u305f. t = 0.35, 0.45, …, 0.95 \u306b\u304a\u3051\u308b\u5024\u306f\u4e0a\u3067\u8aac\u660e\u3057\u305f\u88dc\u9593\u306b\u3088\u308a\u6c42\u3081\u308b.<\/p>\n
        \"\"<\/p>\n
        \u9752\u3044\u70b9\u304c\u30a2\u30c0\u30e0\u30b9\u30fb\u30e0\u30eb\u30c8\u30f3\u6cd5\u306b\u3088\u308b\u8a08\u7b97\u5024\u3067 y = exp(t) \u306e\u30b0\u30e9\u30d5\u3068\u306a\u308b. \u7dd1\u306e\u70b9\u306f\u88dc\u9593\u306b\u3088\u308a\u6c42\u3081\u305f\u70b9\u3092\u793a\u3059. \u8d64\u306f\u8a08\u7b97\u5024\u304a\u3088\u3073\u88dc\u9593\u5024\u306e\u76f8\u5bfe\u8aa4\u5dee\u3092\u53f3\u5074\u306e\u5bfe\u6570\u76ee\u76db\u308a\u306b\u3088\u308a\u8868\u3059. \u671f\u5f85\u901a\u308a\u8aa4\u5dee\u304c\u5897\u3048\u308b\u3053\u3068\u306a\u304f\u88dc\u9593\u5024\u304c\u6c42\u3081\u3089\u308c\u3066\u3044\u308b.
        \n<\/div><\/div>\n
        \u30eb\u30f3\u30b2\u30fb\u30af\u30c3\u30bf\u6cd5\u306e\u5834\u5408(\u9023\u7d9a\u30eb\u30f3\u30b2\u30fb\u30af\u30c3\u30bf\u6cd5)<\/h3>\n
        \u30eb\u30f3\u30b2\u30fb\u30af\u30c3\u30bf\u6cd5\u306f1\u6bb5\u6cd5\u3067\u3042\u308a\u904e\u53bb\u306e\u5024\u306f\u4f7f\u7528\u3057\u306a\u3044\u305f\u3081\u3059\u3050\u306b\u4f7f\u3048\u308b\u88dc\u9593\u5f0f\u306f\u306a\u3044. \u305d\u3053\u3067, \u8ca0\u62c5\u306e\u5c11\u306a\u3044, \u3059\u306a\u308f\u3061\u3053\u306e\u305f\u3081\u306e\u8ffd\u52a0\u306e\u95a2\u6570\u8a55\u4fa1\u304c\u306a\u3044\u307e\u305f\u306f\u3054\u304f\u5c11\u6570\u56de\u3067\u3042\u308b\u3088\u3046\u306a, \u5bc6\u51fa\u529b\u306e\u305f\u3081\u306e\u88dc\u9593\u516c\u5f0f\u304c\u7814\u7a76\u3055\u308c\u3066\u304d\u305f. \u4f8b\u3048\u3070, tn<\/sub> + \u03b8h (0 \u2266 \u03b8 \u2266 1) \u306b\u304a\u3051\u308b\u5024 y(tn<\/sub> + \u03b8h) \u3092\u6c42\u3081\u308b\u6b21\u306e\u5f62\u306e\u5f0f\u3067\u3042\u308b.<\/p>\n
          \r\n  y(tn<\/sub> + \u03b8h) = yn<\/sub> + h\u03a3bi<\/sub>(\u03b8)ki<\/sub> (\u03a3 \u306f i = 1, 2, ... , s*<\/sup>)\r\n  ki<\/sub> = f(tn<\/sub> + ci<\/sub>h, yn<\/sub> + h\u03a3aij<\/sub>kj<\/sub>) (\u03a3 \u306f j = 1, 2, ... , i - 1) (i = 1, 2, ... , s*<\/sup>)\r\n\r\n<\/pre>\n
         
        \n\u30eb\u30f3\u30b2\u30fb\u30af\u30c3\u30bf\u6cd5\u306e\u516c\u5f0f\u3068\u540c\u3058\u5f62\u3092\u3057\u3066\u3044\u308b\u304c, \u4fc2\u6570bi<\/sub>\u306f\u5b9a\u6570\u3067\u306f\u306a\u304f\u03b8\u306b\u4f9d\u5b58\u3059\u308b. \u3053\u306e\u3088\u3046\u306a\u30eb\u30f3\u30b2\u30fb\u30af\u30c3\u30bf\u6cd5\u306e\u81ea\u7136\u306a\u62e1\u5f35\u3092\u9023\u7d9a\u30eb\u30f3\u30b2\u30fb\u30af\u30c3\u30bf\u6cd5(Continuous Runge-Kutta (CRK) method)<\/strong>\u3068\u3082\u3044\u3046.<\/p>\n
        4\u6b21\u306e\u30eb\u30f3\u30b2\u30fb\u30af\u30c3\u30bf\u6cd5\u306b\u3064\u3044\u3066\u306f, s*<\/sup> = s = 4 \u306b\u5bfe\u3057\u3066\u5bc6\u51fa\u529b\u516c\u5f0f\u306e\u6b21\u6570 p*<\/sup> = 3 \u3068\u306a\u308b\u6b21\u306e\u89e3\u304c\u5f97\u3089\u308c\u3066\u3044\u308b.<\/p>\n
          \r\n  b1<\/sub>(\u03b8) = \u03b8 - (3\/2)\u03b82<\/sup> + (2\/3)\u03b83<\/sup>\r\n  b2<\/sub>(\u03b8) = b3<\/sub>(\u03b8) = \u03b82<\/sup> - (2\/3)\u03b83<\/sup>\r\n  b4<\/sub>(\u03b8) = -(1\/2)\u03b82<\/sup> + (2\/3)\u03b83<\/sup>\r\n\r\n<\/pre>\n
         
        \nki<\/sub>\u306f4\u6b21\u30eb\u30f3\u30b2\u30fb\u30af\u30c3\u30bf\u6cd5\u306e\u516c\u5f0f\u3068\u540c\u3058\u3067\u3042\u308b. \u3059\u306a\u308f\u3061, \u65b0\u305f\u306a\u95a2\u6570\u8a55\u4fa1\u3092\u5fc5\u8981\u3068\u3057\u306a\u3044. \u306a\u304a, 4\u6b21\u30eb\u30f3\u30b2\u30fb\u30af\u30c3\u30bf\u6cd5\u3067\u306f\u95a2\u6570\u8a55\u4fa1\u3092\u8ffd\u52a0\u3057\u306a\u3051\u308c\u3070 p*<\/sup> = 4 \u3068\u306f\u306a\u3089\u306a\u3044.<\/p>\n
        \u901a\u5e38\u306f p*<\/sup> = p – 1 \u3067\u3042\u308c\u3070\u5341\u5206\u3067\u3042\u308b. \u5c0e\u95a2\u6570 y'(tn<\/sub> + \u03b8h) \u304c\u6b63\u78ba\u306a\u5fc5\u8981\u304c\u3042\u308b\u5834\u5408\u306b\u306f p*<\/sup> = p \u3067\u306a\u3051\u308c\u3070\u306a\u3089\u306a\u3044.<\/p>\n
        \u6570\u5024\u5b9f\u9a13 (18)<\/h3>\n
        <\/span>\u9023\u7d9a\u30eb\u30f3\u30b2\u30fb\u30af\u30c3\u30bf\u6cd5\u306b\u3088\u308b\u5bc6\u51fa\u529b<\/div>
        \n\u4e0a\u306e\u4f8b\u984c\u306b\u304a\u3044\u30664\u6b21\u30eb\u30f3\u30b2\u30fb\u30af\u30c3\u30bf\u6cd5\u3067 h = 0.1 \u3068\u3057\u3066 t = 0.1, 0.2, …, 1.0 \u306b\u304a\u3051\u308by\u306e\u5024\u3092\u6c42\u3081\u308b. t = 0.05, 0.15, …, 0.95 \u306b\u304a\u3051\u308b\u5024\u306f\u4e0a\u3067\u793a\u3057\u305f3\u6b21\u306e\u5bc6\u51fa\u529b\u516c\u5f0f\u3092\u4f7f\u3063\u3066\u6c42\u3081\u308b.<\/p>\n
        \"\"<\/p>\n
        \u9752\u3044\u70b9\u304c\u30eb\u30f3\u30b2\u30fb\u30af\u30c3\u30bf\u6cd5\u306b\u3088\u308b\u8a08\u7b97\u5024\u3067 y = exp(t) \u306e\u30b0\u30e9\u30d5\u3068\u306a\u308b. \u7dd1\u306e\u70b9\u306f\u88dc\u9593\u306b\u3088\u308a\u6c42\u3081\u305f\u70b9\u3092\u793a\u3059. \u8d64\u306f\u8a08\u7b97\u5024\u304a\u3088\u3073\u88dc\u9593\u5024\u306e\u76f8\u5bfe\u8aa4\u5dee\u3092\u53f3\u5074\u306e\u5bfe\u6570\u76ee\u76db\u308a\u306b\u3088\u308a\u8868\u3059. \u88dc\u9593\u5024\u306b\u306f\u5c11\u3057\u8aa4\u5dee\u304c\u751f\u3058\u308b.
        \n<\/div><\/div>\n
        \u30b9\u30c6\u30c3\u30d7\u5e45\u306e\u81ea\u52d5\u8abf\u7bc0\u3092\u884c\u3046\u5834\u5408<\/h3>\n
        \u4e0a\u306e2\u3064\u306e\u4f8b\u3067\u307f\u305f\u88dc\u9593\u3092\u4f7f\u3046\u65b9\u6cd5\u306f, \u30b9\u30c6\u30c3\u30d7\u5e45\u306e\u81ea\u52d5\u8abf\u7bc0\u3092\u884c\u3046\u30d7\u30ed\u30b0\u30e9\u30e0\u306b\u9069\u7528\u3059\u308b\u3068\u52b9\u7387\u3092\u4e0a\u3052\u308b\u3053\u3068\u304c\u3067\u304d\u308b. \u3059\u306a\u308f\u3061, \u30b9\u30c6\u30c3\u30d7\u5e45\u306f\u5fc5\u8981\u7cbe\u5ea6\u306e\u9762\u304b\u3089\u6700\u9069\u306a\u3082\u306e\u3092\u9078\u3073, \u30b9\u30c6\u30c3\u30d7\u306b\u4e00\u81f4\u3057\u306a\u3044\u51fa\u529b\u70b9\u306b\u304a\u3044\u3066\u306f\u88dc\u9593\u5024\u3092\u4f7f\u3046\u3053\u3068\u306b\u3088\u308a\u7121\u99c4\u306a\u95a2\u6570\u8a08\u7b97\u3092\u907f\u3051\u308b\u3053\u3068\u304c\u3067\u304d\u308b.<\/p>\n
        \u591a\u6bb5\u6cd5\u306e\u4f8b\u3068\u3057\u3066, XLPack\u306b\u3082\u53ce\u9332\u3055\u308c\u3066\u3044\u308b Deabm (\u30a2\u30c0\u30e0\u30b9\u30fb\u30d0\u30b7\u30e5\u30d5\u30a9\u30fc\u30b9\u30fb\u30e0\u30eb\u30c8\u30f3\u6cd5) \u306f, \u88dc\u9593\u5024\u3092\u4f7f\u3046\u3053\u3068\u304c\u3067\u304d\u308b\u5834\u5408\u306b\u306f\u88dc\u9593\u5024\u3092\u4f7f\u3046\u3088\u3046\u306b\u4f5c\u3089\u308c\u3066\u3044\u308b.<\/p>\n
        \u30eb\u30f3\u30b2\u30fb\u30af\u30c3\u30bf\u6cd5\u306e\u4f8b\u3068\u3057\u3066, Dopri5\u306b\u306f4\u6b21\u306e\u88dc\u9593\u516c\u5f0f\u3092\u4f7f\u3063\u305f\u5bc6\u51fa\u529b\u6a5f\u80fd\u304c\u7d44\u307f\u8fbc\u307e\u308c\u3066\u304a\u308a, XLPack\u306eDerkf\u306b\u306f5\u6b21\u306e\u88dc\u9593\u516c\u5f0f\u3092\u4f7f\u3063\u305f\u5bc6\u51fa\u529b\u6a5f\u80fd\u304c\u7d44\u307f\u8fbc\u307e\u308c\u3066\u3044\u308b. Dopri5\u3068Derkf\u306e\u5834\u5408\u306b\u306f\u5bc6\u51fa\u529b\u306e\u305f\u3081\u306e\u8ffd\u52a0\u95a2\u6570\u8a55\u4fa1\u3092\u5fc5\u8981\u3068\u3057\u306a\u3044\u516c\u5f0f\u3092\u63a1\u7528\u3057\u3066\u3044\u308b. XLPack\u306b\u53ce\u9332\u3055\u308c\u3066\u3044\u308b\u4ed6\u306e\u591a\u304f\u306e\u30d7\u30ed\u30b0\u30e9\u30e0\u3067\u3082\u5bc6\u51fa\u529b\u6a5f\u80fd\u304c\u7d44\u307f\u8fbc\u307e\u308c\u3066\u3044\u308b.<\/p>\n
        Dopri5\u3068Derkf\u3092\u4f7f\u3063\u3066\u8981\u6c42\u7cbe\u5ea6(RTol, ATol) = 1.0e-7 \u3067\u8a08\u7b97\u3092\u884c\u3044, 0.02\u3054\u3068\u306b\u7d30\u304b\u304f\u7d50\u679c\u306e\u51fa\u529b\u3092\u884c\u3063\u305f\u4f8b\u3092\u4ee5\u4e0b\u306b\u793a\u3059.<\/p>\n
        \"\"<\/p>\n
        \"\"<\/p>\n
        \u7dd1\u306e\u4e38\u304c\u8a08\u7b97\u30b9\u30c6\u30c3\u30d7\u3067, \u7d2b\u306e\u4e09\u89d2\u304c\u305d\u306e\u6642\u306e\u8aa4\u5dee\u3067\u3042\u308b. \u8981\u6c42\u7cbe\u5ea6\u3092\u6e80\u305f\u3059\u306b\u306f\u30b9\u30c6\u30c3\u30d7\u5e45\u306f0.1\uff5e0.15\u7a0b\u5ea6\u3067\u5341\u5206\u3067\u3042\u308b\u3053\u3068\u304c\u308f\u304b\u308b. Dopri5\u306e\u5834\u5408\u306f\u3053\u306e\u4f8b\u3067\u306f\u7cbe\u5ea6 1.0e-8 \u7a0b\u5ea6\u3092\u9054\u6210\u3057\u3066\u304a\u308a, \u3084\u3084\u5b89\u5168\u5074\u306b\u51fa\u3066\u3044\u308b. Derkf\u3067\u306f\u3061\u3087\u3046\u3069 1.0e-7 \u7a0b\u5ea6\u306b\u8abf\u7bc0\u3055\u308c\u3066\u3044\u308b. \u5fc5\u8981\u3068\u3057\u305f\u30b9\u30c6\u30c3\u30d7\u6570\u306f, Dopri5\u304c9\u30b9\u30c6\u30c3\u30d7, Derkf\u304c7\u30b9\u30c6\u30c3\u30d7\u3067\u3042\u3063\u305f. \u95a2\u6570\u8a55\u4fa1\u56de\u6570\u306f, Dopri5\u304c56\u56de, Derkf\u304c46\u56de\u3067\u3042\u3063\u305f.<\/p>\n
        \u9752\u306e\u4e38\u306f\u88dc\u9593\u5024(\u5bc6\u51fa\u529b), \u8d64\u306e\u4e09\u89d2\u306f\u305d\u306e\u8aa4\u5dee\u3067\u3042\u308b. \u88dc\u9593\u5024\u3082\u5341\u5206\u306a\u7cbe\u5ea6\u3067\u6c42\u3081\u3089\u308c\u3066\u3044\u308b. \u5bc6\u51fa\u529b\u306b\u304a\u3044\u3066\u306f\u88dc\u9593\u5024\u3092\u3069\u308c\u3060\u3051\u7d30\u304b\u306a\u9593\u9694\u3067\u6c42\u3081\u3066\u3082\u95a2\u6570\u8a55\u4fa1\u56de\u6570\u306f\u5897\u3048\u306a\u3044(\u88dc\u9593\u8a08\u7b97\u7a0b\u5ea6\u306e\u8a08\u7b97\u91cf\u3057\u304b\u5897\u3048\u306a\u3044)\u306e\u3067, \u7cbe\u5bc6\u306a\u30b0\u30e9\u30d5\u3092\u63cf\u304f\u305f\u3081\u306b\u7d30\u304b\u304f\u5024\u3092\u51fa\u529b\u3057\u305f\u3044\u5834\u5408\u306a\u3069\u306b\u6709\u52b9\u3067\u3042\u308b. \u9006\u306b, \u5fc5\u8981\u306a\u30b9\u30c6\u30c3\u30d7\u5e45\u304c\u51fa\u529b\u70b9\u306e\u9593\u9694\u3088\u308a\u3082\u5c0f\u3055\u304f\u306a\u308b\u5834\u5408\u306b\u306f\u5bc6\u51fa\u529b\u306e\u52b9\u679c\u306f\u5c0f\u3055\u304f\u306a\u308b\u306e\u3067, \u5358\u7d14\u306b\u7e70\u308a\u8fd4\u3057\u8a08\u7b97\u3092\u884c\u3063\u3066\u3082\u3088\u3044.
        \n<\/div>\n
        \n
        2\u968e\u5e38\u5fae\u5206\u65b9\u7a0b\u5f0f<\/h2>\n
        \u5b9f\u969b\u306b\u5fdc\u7528\u3067\u73fe\u308c\u308b\u65b9\u7a0b\u5f0f\u306f2\u968e\u5e38\u5fae\u5206\u65b9\u7a0b\u5f0f\u304c\u591a\u3044.<\/p>\n
          \r\n  y'' = f(t, y, y')\r\n  y(t0<\/sub>) = y0<\/sub>, y'(t0<\/sub>) = y'0<\/sub> (\u521d\u671f\u6761\u4ef6)\r\n\r\n<\/pre>\n
         
        \n\u3053\u308c\u306f\u300c\u6982\u8981\u300d\u306e\u9805\u3067\u8aac\u660e\u3057\u305f\u3088\u3046\u306b1\u968e\u306e\u9023\u7acb\u5e38\u5fae\u5206\u65b9\u7a0b\u5f0f\u306b\u5909\u63db\u3057\u3066\u89e3\u304f\u3053\u3068\u304c\u3067\u304d\u308b.<\/p>\n
        \u3053\u3053\u3067\u306f, \u6b21\u306e\u3088\u3046\u306bf()\u304cy’\u306b\u4f9d\u5b58\u3057\u306a\u3044\u7279\u6b8a\u306a\u5834\u5408\u306b\u52b9\u7387\u306e\u3088\u3044\u65b9\u6cd5\u3092\u7d39\u4ecb\u3059\u308b.<\/p>\n
          \r\n  y'' = f(t, y)\r\n  y(t0<\/sub>) = y0<\/sub>, y'(t0<\/sub>) = y'0<\/sub> (\u521d\u671f\u6761\u4ef6)\r\n\r\n<\/pre>\n
         <\/p>\n
        \u30cb\u30e5\u30b9\u30c8\u30ec\u30e0\u6cd5<\/h3>\n
        \u30eb\u30f3\u30b2\u30fb\u30af\u30c3\u30bf\u6cd5\u306b\u4f3c\u3066\u3044\u308b\u304c\u4fc2\u6570\u306e\u6c7a\u3081\u65b9\u304c\u7570\u306a\u308b\u30cb\u30e5\u30b9\u30c8\u30ec\u30e0\u6cd5(Nystrom method)\u304c\u77e5\u3089\u308c\u3066\u304a\u308a, \u6b21\u306e\u3088\u3046\u306b\u8a08\u7b97\u3092\u884c\u3046.<\/p>\n
          \r\n  k'i<\/sub> = f(tn<\/sub> + ci<\/sub>h, yn<\/sub> + ci<\/sub>hy'n<\/sub> + h2<\/sup>\u03a3a<\/u>ij<\/sub>k'j<\/sub>) (\u03a3 \u306f j = 1, 2, ... , s) (i = 1, 2, ... , s)\r\n  yn+1<\/sub> = yn<\/sub> + hy'n<\/sub> + h2<\/sup>\u03a3b<\/u>i<\/sub>k'i<\/sub> (\u03a3 \u306f i = 1, 2, ... , s)\r\n  y'n+1<\/sub> = y'n<\/sub> + h\u03a3bi<\/sub>k'i<\/sub> (\u03a3 \u306f i = 1, 2, ... , s)\r\n\r\n<\/pre>\n
         
        \n\u4fc2\u6570\u306f\u6b21\u306e\u3068\u304a\u308a\u3067\u3042\u308b.<\/p>\n
        4\u6b21\u30cb\u30e5\u30b9\u30c8\u30ec\u30e0\u6cd5<\/h4>\n
          \r\n  0   |\r\n  1\/2 | 1\/8         a<\/u>ij<\/sub>\r\n  2\/3 |  0   1\/2\r\n  ---------------------\r\n  b<\/u>i<\/sub>  | 1\/6  1\/3  0 \r\n  bi<\/sub>  | 1\/6  4\/6  1\/6\r\n    \r\n<\/pre>\n
         <\/p>\n
        5\u6b21\u30cb\u30e5\u30b9\u30c8\u30ec\u30e0\u6cd5<\/h4>\n
          \r\n  0   |\r\n  1\/5 |  1\/50         a<\/u>ij<\/sub>\r\n  2\/3 | -1\/27   7\/27\r\n  1   |  3\/10  -2\/35    9\/35\r\n  -------------------------------------\r\n  b<\/u>i<\/sub>  | 14\/336 100\/336 54\/336  0 \r\n  bi<\/sub>  | 14\/336 125\/336 162\/336 35\/336\r\n    \r\n<\/pre>\n
         
        \n\u30eb\u30f3\u30b2\u30fb\u30af\u30c3\u30bf\u6cd5\u306b\u6bd4\u3079\u3066\u6bb5\u6570\u304c\u5c11\u306a\u3044\u3053\u3068\u304c\u308f\u304b\u308b(4\u6b21: 4\u6bb5\u21923\u6bb5, 5\u6b21: 6\u6bb5\u21924\u6bb5).<\/p>\n
        \u88dc\u5916\u6cd5<\/h3>\n
        1\u968e\u306e\u9023\u7acb\u5e38\u5fae\u5206\u65b9\u7a0b\u5f0f\u306b\u5909\u63db\u3057\u3066GBS\u30a2\u30eb\u30b4\u30ea\u30ba\u30e0\u3092\u9069\u7528\u3059\u308b.<\/p>\n
          \r\n  y1<\/sub> = y0<\/sub> + hj<\/sub>y'0<\/sub>\r\n  y'1<\/sub> = y'0<\/sub> + hj<\/sub>f(t0<\/sub>, y0<\/sub>)\r\n\r\n  yi+1<\/sub> = yi-1<\/sub> + 2hj<\/sub>y'i<\/sub> (i = 1, 2, ..., nj<\/sub>)\r\n  y'i+1<\/sub> = y'i-1<\/sub> + 2hj<\/sub>f(ti<\/sub>, yi<\/sub>) (i = 1, 2, ..., nj<\/sub>)\r\n\r\n  Tj1<\/sub> = (1\/4)(ynj<\/sub>-1<\/sub> + 2ynj<\/sub><\/sub> + ynj<\/sub>+1<\/sub>)\r\n  T'j1<\/sub> = (1\/4)(y'nj<\/sub>-1<\/sub> + 2y'nj<\/sub><\/sub> + y'nj<\/sub>+1<\/sub>)\r\n\r\n<\/pre>\n
         
        \n\u3053\u306e\u5834\u5408\u5b9f\u306f\u6b21\u306e\u3088\u3046\u306b\u306a\u308b\u306e\u3067\u534a\u5206\u3060\u3051(\u5947\u6570\u6dfb\u5b57\u306e\u3068\u304d\u3060\u3051)\u95a2\u6570\u8a55\u4fa1\u3059\u308c\u3070\u3088\u304f\u306a\u308b.<\/p>\n
          \r\n  Tj1<\/sub> = ynj<\/sub><\/sub>\r\n  T'j1<\/sub> = (1\/2)(y'nj<\/sub>-1<\/sub> + y'nj<\/sub>+1<\/sub>)\r\n\r\n<\/pre>\n
         <\/p>\n
        \u6570\u5024\u5b9f\u9a13 (19)<\/h3>\n
        <\/span>2\u968e\u5e38\u5fae\u5206\u65b9\u7a0b\u5f0f(y'' = f(t, y))\u306e\u8a08\u7b97\u4f8b<\/div>
        \n\u6b21\u306e2\u968e\u5e38\u5fae\u5206\u65b9\u7a0b\u5f0f\u306e\u521d\u671f\u5024\u554f\u984c\u306e\u4f8b\u30924\u6b21, 5\u6b21\u30cb\u30e5\u30b9\u30c8\u30ec\u30e0\u6cd5\u304a\u3088\u3073\u88dc\u5916\u6cd5(GBS, \u30ed\u30f3\u30d0\u30fc\u30af\u5217)\u306b\u3088\u308a\u8a08\u7b97\u3059\u308b.<\/p>\n
          \r\n  y'' = y\r\n  y(0) = 1, y'(0) = 1\r\n\r\n<\/pre>\n
         <\/p>\n
        \"\"<\/p>\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.<\/p>\n
        4\u6b21\u30cb\u30e5\u30b9\u30c8\u30ec\u30e0\u6cd5\u306f\u6bb5\u6570\u306f\u5c11\u306a\u3044\u304c4\u6b21\u306e\u30eb\u30f3\u30b2\u30fb\u30af\u30c3\u30bf\u6cd5\u3068\u540c\u7a0b\u5ea6\u306e\u7cbe\u5ea6\u3067\u3042\u308b\u3053\u3068\u304c\u78ba\u8a8d\u3067\u304d\u308b. \u95a2\u6570\u8a55\u4fa1\u56de\u6570\u304c\u5c11\u306a\u3044\u5206\u3060\u3051\u30d7\u30ed\u30c3\u30c8\u306f\u4e0b\u306b\u304d\u3066\u3044\u308b.
        \n<\/div><\/div>\n<\/div>\n
        \n
        \u9045\u5ef6\u5fae\u5206\u65b9\u7a0b\u5f0f<\/h2>\n
        \u9045\u5ef6\u5fae\u5206\u65b9\u7a0b\u5f0f(Delay differential equations (DDE))\u3068\u306f\u6b21\u306e\u3088\u3046\u306a\u5fae\u5206\u65b9\u7a0b\u5f0f\u3092\u3044\u3046.<\/p>\n
          \r\n  y'(t) = f(t, y(t), y(t - \u03c4))\r\n\r\n<\/pre>\n
         
        \n\u03c4\u306f\u9045\u5ef6\u6642\u9593\u3092\u8868\u3059\u6b63\u5b9a\u6570\u3067\u3042\u308b. \u8907\u6570\u306e\u9045\u5ef6\u6642\u9593\u304c\u542b\u307e\u308c\u308b\u306a\u3069\u3088\u308a\u4e00\u822c\u7684\u306a\u3082\u306e\u3082\u8003\u3048\u3089\u308c\u308b.<\/p>\n
        \u3053\u308c\u3092\u30eb\u30f3\u30b2\u30fb\u30af\u30c3\u30bf\u6cd5\u3067\u89e3\u304f\u305f\u3081\u306b\u306f\u30b9\u30c6\u30c3\u30d7\u5e45\u3092 \u03c4 = kh (k\u306f\u6574\u6570)\u3068\u306a\u308b\u3088\u3046\u306b\u9078\u3093\u3067, \u8a08\u7b97\u3057\u305fy(t – \u03c4)\u306e\u5024\u3092\u3068\u3063\u3066\u304a\u304f\u3088\u3046\u306b\u3059\u308c\u3070\u3088\u3044. \u3057\u304b\u3057, \u30b9\u30c6\u30c3\u30d7\u5e45\u3092\u81ea\u52d5\u8abf\u7bc0\u3059\u308b\u30eb\u30fc\u30c1\u30f3\u306e\u5834\u5408\u306b\u306f\u3046\u307e\u304f\u3044\u304b\u306a\u3044. \u3053\u308c\u306f\u5bc6\u51fa\u529b\u3092\u884c\u3046\u65b9\u6cd5\u3068\u540c\u3058\u304f, \u591a\u6bb5\u6cd5\u304b\u5bc6\u51fa\u529b\u6a5f\u80fd\u3092\u6301\u3063\u305f\u30eb\u30f3\u30b2\u30fb\u30af\u30c3\u30bf\u6cd5\u306a\u3089\u3070\u8a08\u7b97\u3067\u304d\u308b. XLPack\u306b\u53ce\u9332\u3055\u308c\u3066\u3044\u308bRetard\u306f\u30c9\u30eb\u30de\u30f3\u30fb\u30d7\u30ea\u30f3\u30b9\u6cd5\u30eb\u30fc\u30c1\u30f3Dopri5\u3092\u9045\u5ef6\u5fae\u5206\u65b9\u7a0b\u5f0f\u7528\u306b\u4fee\u6b63\u3057\u305f\u3082\u306e\u3067\u3042\u308b. Dopri5\u3067\u306f1\u30b9\u30c6\u30c3\u30d7\u524d\u307e\u3067\u306e\u904e\u53bb\u306e\u5024\u3092\u6c42\u3081\u3089\u308c\u305f\u304c, Retard\u3067\u306f\u305a\u3063\u3068\u524d\u307e\u3067\u306e\u5024\u3092\u5f97\u308b\u3053\u3068\u304c\u3067\u304d\u308b.<\/p>\n
        \u9045\u5ef6\u5fae\u5206\u65b9\u7a0b\u5f0f\u306e\u8a08\u7b97\u4f8b\u306f\u300c\u5b9f\u7528\u30eb\u30fc\u30c1\u30f3\u306e\u30d9\u30f3\u30c1\u30de\u30fc\u30af\u300d\u3092\u53c2\u7167\u305b\u3088.
        \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\u6570\u5024\u89e3\u6cd5\u306b\u304a\u3051\u308b\u8aa4\u5dee\u306e\u63a8\u5b9a\u6cd5\u304a\u3088\u3073\u305d\u308c\u3092\u4f7f\u3063\u305f\u81ea\u52d5\u7a4d\u5206\u306b\u3064\u3044\u3066\u8aac\u660e\u3057\u307e\u3059\u3002\u307e\u305f\u3001\u89e3\u6cd5\u306e\u5b89\u5b9a\u6027\u3068\u30b9\u30c6\u30a3\u30d5\u306a\u65b9\u7a0b\u5f0f\u306e\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-1003","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\/1003","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=1003"}],"version-history":[{"count":5,"href":"https:\/\/www.ktech.biz\/jp\/wp-json\/wp\/v2\/posts\/1003\/revisions"}],"predecessor-version":[{"id":3199,"href":"https:\/\/www.ktech.biz\/jp\/wp-json\/wp\/v2\/posts\/1003\/revisions\/3199"}],"wp:attachment":[{"href":"https:\/\/www.ktech.biz\/jp\/wp-json\/wp\/v2\/media?parent=1003"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/www.ktech.biz\/jp\/wp-json\/wp\/v2\/categories?post=1003"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/www.ktech.biz\/jp\/wp-json\/wp\/v2\/tags?post=1003"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}