{"id":5557,"date":"2026-07-05T11:23:53","date_gmt":"2026-07-05T02:23:53","guid":{"rendered":"https:\/\/www.ktech.biz\/jp\/?p=5557"},"modified":"2026-07-05T12:07:05","modified_gmt":"2026-07-05T03:07:05","slug":"10-2_ode","status":"publish","type":"post","link":"https:\/\/www.ktech.biz\/jp\/num\/10-2_ode\/","title":{"rendered":"10.2 \u30b9\u30c6\u30c3\u30d7\u5e45\u306e\u81ea\u52d5\u8abf\u7bc0, \u5bc6\u51fa\u529b, 2\u968e\u5e38\u5fae\u5206\u65b9\u7a0b\u5f0f"},"content":{"rendered":"\n<h3>10.2.1 \u8aa4\u5dee\u306e\u63a8\u5b9a<\/h3>\n<p>\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<h4>10.2.1.1 1 \u6bb5\u6cd5\u306e\u8aa4\u5dee<\/h4>\n<p>\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<ul>\n<li>\u96e2\u6563\u5316\u8aa4\u5dee (\u6253\u3061\u5207\u308a\u8aa4\u5dee\u3068\u3082\u3044\u3046)<br \/>\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<ul>\n<li>\u5c40\u6240\u7684\u96e2\u6563\u5316\u8aa4\u5dee<br \/>\n  \u305d\u308c\u307e\u3067\u306b\u5f97\u3089\u308c\u3066\u3044\u308b\u5024\u304c\u6b63\u78ba\u3067\u3042\u308b\u3068\u3057\u3066, \u6b21\u306e 1 \u30b9\u30c6\u30c3\u30d7\u3067\u751f\u3058\u308b\u8aa4\u5dee<\/li>\n<li>\u5927\u57df\u7684\u96e2\u6563\u5316\u8aa4\u5dee<br \/>\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<li>\u4e38\u3081\u8aa4\u5dee<br \/>\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\u3058\u308b\u8aa4\u5dee<\/li>\n<\/ul>\n<p>\u3044\u307e, \u6b21\u306e\u521d\u671f\u5024\u554f\u984c\u3092\u8003\u3048\u308b.<br \/>\n\\[<br \/>\n  dy\/dt = f(t, y), \\space t = t_0 \u306b\u304a\u3044\u3066 y = y_0<br \/>\n\\]\n\\(t_n\\) \u306b\u304a\u3051\u308b\u8a08\u7b97\u5024\u3092 \\(y_n\\) \u3068\u3057, \\(u_n(t_n) = y_n\\) \u3068\u3059\u308b. \u3059\u306a\u308f\u3061, \u8a08\u7b97\u5024 \\(y_n\\) \u304b\u3089\u6c7a\u5b9a\u3055\u308c\u308b\u89e3\u66f2\u7dda\u3092 \\(u_n(t\\) )\u3068\u3059\u308b. \\(y_{n+1}\\) \u306f \\(y_n, t_n, 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, \\(y_{n+1} = y_n + hf(t_n, y_n)\\) \u3068\u306a\u308b.<\/p>\n<p>\u305d\u3046\u3059\u308b\u3068, \u5c40\u6240\u7684\u96e2\u6563\u5316\u8aa4\u5dee \\(d_n\\) \u306f\u6b21\u306e\u3088\u3046\u306b\u8868\u3059\u3053\u3068\u304c\u3067\u304d\u308b.<br \/>\n\\[<br \/>\n  d_n = y_{n+1} &#8211; u_n(t_{n+1})<br \/>\n\\]\n\u5c40\u6240\u7684\u96e2\u6563\u5316\u8aa4\u5dee\u306f \\(p\\) \u6b21\u306e\u89e3\u6cd5\u306b\u304a\u3044\u3066 \\(O(h^{p+1})\\) \u3067\u3042\u308b. \u3059\u306a\u308f\u3061, \u6b21\u5f0f\u3092\u6e80\u305f\u3059\u6570 \\(C\\) \u304c\u5b58\u5728\u3059\u308b.<br \/>\n\\[<br \/>\n  |d_n| \u2266 Ch^{p+1}<br \/>\n\\]\n\u5927\u57df\u7684\u96e2\u6563\u5316\u8aa4\u5dee \\(e_n\\) \u306f\u6b21\u306e\u3088\u3046\u306b\u8868\u3055\u308c\u308b.<br \/>\n\\[<br \/>\n  e_n = y_n &#8211; y(t_n)<br \/>\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, \\(e_n > \\sum_{i=0}^{n-1} d_i\\) \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<p><img loading=\"lazy\" decoding=\"async\" class=\"aligncenter wp-image-701 size-full\" src=\"https:\/\/www.ktech.biz\/jp\/wp-content\/uploads\/sites\/2\/2015\/01\/ode_fig-3.png\" alt=\"ode_fig-3\" width=\"500\" height=\"358\" srcset=\"https:\/\/www.ktech.biz\/jp\/wp-content\/uploads\/sites\/2\/2015\/01\/ode_fig-3.png 500w, https:\/\/www.ktech.biz\/jp\/wp-content\/uploads\/sites\/2\/2015\/01\/ode_fig-3-300x215.png 300w\" sizes=\"auto, (max-width: 500px) 100vw, 500px\" \/><\/p>\n<p>\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, \\(e_n < \\sum_{i=0}^{n-1} d_i\\) \u3068\u306a\u308a, \u30b9\u30c6\u30c3\u30d7\u304c\u9032\u3080\u306b\u3064\u308c\u3066\u8aa4\u5dee\u306f\u7e2e\u5c0f\u3057\u3066\u3044\u304f.\n\n<img loading=\"lazy\" decoding=\"async\" class=\"aligncenter wp-image-702 size-full\" src=\"https:\/\/www.ktech.biz\/jp\/wp-content\/uploads\/sites\/2\/2015\/01\/ode_fig-4.png\" alt=\"ode_fig-4\" width=\"500\" height=\"366\" srcset=\"https:\/\/www.ktech.biz\/jp\/wp-content\/uploads\/sites\/2\/2015\/01\/ode_fig-4.png 500w, https:\/\/www.ktech.biz\/jp\/wp-content\/uploads\/sites\/2\/2015\/01\/ode_fig-4-300x220.png 300w\" sizes=\"auto, (max-width: 500px) 100vw, 500px\" \/><\/p>\n<p>2 \u3064\u306e\u4f8b\u3092\u3042\u3052\u305f\u304c, \u95a2\u6570\u304c\u3069\u3061\u3089\u306e\u50be\u5411\u3092\u793a\u3059\u304b\u306f \\(\\partial f\/\\partial y\\) \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, \\(\\partial f\/\\partial y = 0\\) \u306a\u3089\u3070 \\(e_n = \\sum_{i=0}^{n-1} d_i\\) \u3068\u306a\u308b. \u307e\u305f, \\(\\partial f\/\\partial y\\) \u306e\u7b26\u53f7\u304c\u533a\u9593\u5185\u3067\u5909\u5316\u3059\u308b\u3088\u3046\u306a\u8907\u96d1\u306a\u95a2\u6570\u3082\u3042\u308b. \\(\\partial f\/\\partial y = 0\\) \u306b\u8fd1\u3051\u308c\u3070, \u5927\u57df\u7684\u96e2\u6563\u5316\u8aa4\u5dee\u306f\u304a\u304a\u3080\u306d \\(O(h^p)\\) \u3068\u3044\u3048\u308b.<\/p>\n<p>\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 \\(\\epsilon_n\\) \u304c\u767a\u751f\u3059\u308b. \u4f8b\u3048\u3070, \u30aa\u30a4\u30e9\u30fc\u6cd5\u306a\u3089\u3070, \\(y_{n+1} = y_n + hf(y_n, t_n) + \\epsilon_n\\) \u3068\u306a\u308b.<\/p>\n<p>\u3053\u3053\u3067, \u4e38\u3081\u8aa4\u5dee \\(\\epsilon_n\\) \u306e\u30ce\u30eb\u30e0\u306f\u3042\u308b\u6570 \\(\\epsilon\\) \u4ee5\u4e0b\u3067\u3042\u308b\u3068\u3059\u308b.<br \/>\n\\[<br \/>\n  \\|\\epsilon_n\\| \\le \\epsilon<br \/>\n\\]\n\u3044\u307e, \u6700\u7d42\u70b9\u3092 \\(t_N\\) \u3068\u3059\u308b\u3068, \u30b9\u30c6\u30c3\u30d7\u6570 \\(N = (t_N &#8211; t_0)\/h\\) \u3067\u3042\u308b. \u6700\u7d42\u7684\u306a\u5927\u57df\u7684\u96e2\u6563\u5316\u8aa4\u5dee\u3092 \\(e_N \\simeq \\sum_{i=0}^{N-1} d_i\\) \u3068\u8fd1\u4f3c\u3057, \u4e38\u3081\u8aa4\u5dee\u3092 \\(N\\epsilon\\) \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.<br \/>\n\\[<br \/>\n  \u5168\u8aa4\u5dee \\simeq NCh^{p+1} + N\\epsilon = (t_N &#8211; t_0)(Ch^p + \\epsilon\/h)<br \/>\n\\]\n<h4>\u6570\u5024\u5b9f\u9a13 (10)<\/h4>\n<p>\u6b21\u306e\u521d\u671f\u5024\u554f\u984c\u3092\u30aa\u30a4\u30e9\u30fc\u6cd5\u306b\u3088\u308a\u89e3\u304f.<br \/>\n\\[<br \/>\n  y&#8217; = y, \\space y(0) = 1<br \/>\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.<br \/>\n\\[<br \/>\n  \u5168\u8aa4\u5dee \\simeq (t_N &#8211; t_0)(Ch + \\epsilon\/h)<br \/>\n\\]\n\\(h\\) \u304c\u5927\u304d\u3044\u3068\u304d\u306f\u7b2c 1 \u9805 (\u96e2\u6563\u5316\u8aa4\u5dee) \u304c\u512a\u52e2\u3067, \\(h\\) \u304c\u5c0f\u3055\u3044\u3068\u304d\u306f\u7b2c 2 \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 \\(((\\epsilon\/C)^{-2})\\) \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<p>\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 32 \u30d3\u30c3\u30c8\u5358\u7cbe\u5ea6\u6f14\u7b97 (10 \u9032\u306e\u7cbe\u5ea6\u306f\u304a\u304a\u3088\u305d \\(10^{-7}\\)) \u3067\u884c\u3063\u305f.<\/p>\n<p><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/www.ktech.biz\/jp\/wp-content\/uploads\/sites\/2\/2026\/06\/10-2_1_exp10.png\" alt=\"\" width=\"640\" height=\"480\" class=\"aligncenter size-full wp-image-5634\" srcset=\"https:\/\/www.ktech.biz\/jp\/wp-content\/uploads\/sites\/2\/2026\/06\/10-2_1_exp10.png 640w, https:\/\/www.ktech.biz\/jp\/wp-content\/uploads\/sites\/2\/2026\/06\/10-2_1_exp10-300x225.png 300w\" sizes=\"auto, (max-width: 640px) 100vw, 640px\" \/><\/p>\n<p>\u56f3\u4e2d\u306e\u9752\u8272\u306e\u30d7\u30ed\u30c3\u30c8\u306e\u3088\u3046\u306b, \u3053\u306e\u5834\u5408 \\(10^{-5}\\) \u4ed8\u8fd1\u3092\u5883\u306b\u4e38\u3081\u8aa4\u5dee\u304c\u652f\u914d\u7684\u306b\u306a\u308b\u3053\u3068\u304c\u308f\u304b\u308b.<\/p>\n<p>\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\u30aa\u30ec\u30f3\u30b8\u8272\u3067\u8868\u793a\u3057\u305f. \u4e38\u3081\u8aa4\u5dee\u304c\u306a\u3051\u308c\u3070 \\(h\\) \u306b\u6bd4\u4f8b\u3057\u3066\u8aa4\u5dee\u304c\u6e1b\u3063\u3066\u3044\u304f\u3053\u3068\u304c\u308f\u304b\u308b.<\/p>\n<h4>10.2.1.2 \u30ea\u30c1\u30e3\u30fc\u30c9\u30bd\u30f3\u306e\u88dc\u5916<\/h4>\n<p>\u4e0e\u3048\u3089\u308c\u305f\u70b9 \\(t_n\\) \u306b\u304a\u3051\u308b p \u6b21\u306e\u6570\u5024\u89e3\u6cd5\u306e\u5927\u57df\u7684\u96e2\u6563\u5316\u8aa4\u5dee\u306f \\(O(h^p)\\) \u3068\u306a\u308b\u306e\u3067, \u6b21\u306e\u3088\u3046\u306b\u8868\u3059\u3053\u3068\u304c\u3067\u304d\u308b.<br \/>\n\\[<br \/>\n  y_n &#8211; y(t_n) = ch^p + O(h^q), \\space q > p<br \/>\n\\]\n\u30b9\u30c6\u30c3\u30d7\u5e45 \\(h\\) \u3067\u8a08\u7b97\u3057\u305f \\(y_n\\) \u3092 \\(y_n^{(h)}\\) \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.<br \/>\n\\[<br \/>\n\\begin{align}<br \/>\n&#038; y_n^{(h)} = y(t_n) + ch^p + O(h^q) \\\\<br \/>\n&#038; y_n^{(2h)} = y(t_n) + c(2h)^p + O(h^q) \\\\<br \/>\n\\end{align}<br \/>\n\\]\n\u3053\u308c\u3089\u306e\u5dee\u3092\u3068\u308b\u3068,<br \/>\n\\[<br \/>\n  y_n^{(2h)} &#8211; y_n^{(h)} = (2^p &#8211; 1)ch^p + O(h^q)<br \/>\n\\]\n\u3068\u306a\u308a, \\(y_n^{(h)}\\) \u306e\u8aa4\u5dee\u306f \\(O(h^q)\\) \u3092\u7121\u8996\u3059\u308b\u3068\u6b21\u306e\u3088\u3046\u306b\u8fd1\u4f3c\u3067\u304d\u308b.<br \/>\n\\[<br \/>\n  y_n^{(h)} \u306e\u8aa4\u5dee \\simeq ch^p = \\frac{y_n^{(2h)} &#8211; y_n^{(h)}}{2^p &#8211; 1}<br \/>\n\\]\n\u3053\u308c\u306f, \u30ed\u30f3\u30d0\u30fc\u30b0\u7a4d\u5206\u3067\u4f7f\u308f\u308c\u308b<strong>\u30ea\u30c1\u30e3\u30fc\u30c9\u30bd\u30f3\u306e\u88dc\u5916<\/strong> (Richardson extrapolation) \u3068\u540c\u3058\u3067\u3042\u308b.<\/p>\n<h4>\u6570\u5024\u5b9f\u9a13 (11)<\/h4>\n<p>\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.<br \/>\n\\[<br \/>\n  y&#8217; = y, \\space y(0) = 1<br \/>\n\\]\n\u30b9\u30c6\u30c3\u30d7\u5e45 \\(h = 1\\) \u304b\u3089\u59cb\u3081\u3066, \\(h = 1\/2, 1\/4, 1\/8 \\dots, 1\/1024\\) \u3068\u3044\u3046\u3088\u3046\u306b\u30b9\u30c6\u30c3\u30d7\u5e45\u3092\u534a\u5206\u306b\u3057\u3066\u8a08\u7b97\u3092\u7e70\u308a\u8fd4\u3057\u305f. \u3053\u306e\u3068\u304d, \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, \u5b9f\u969b\u306e\u8aa4\u5dee\u3068\u5171\u306b\u30d7\u30ed\u30c3\u30c8\u3059\u308b.<\/p>\n<p><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/www.ktech.biz\/jp\/wp-content\/uploads\/sites\/2\/2026\/06\/10-2_2_exp11.png\" alt=\"\" width=\"640\" height=\"480\" class=\"aligncenter size-full wp-image-5636\" srcset=\"https:\/\/www.ktech.biz\/jp\/wp-content\/uploads\/sites\/2\/2026\/06\/10-2_2_exp11.png 640w, https:\/\/www.ktech.biz\/jp\/wp-content\/uploads\/sites\/2\/2026\/06\/10-2_2_exp11-300x225.png 300w\" sizes=\"auto, (max-width: 640px) 100vw, 640px\" \/><\/p>\n<p>\u3053\u306e\u4f8b\u3067\u306f, \u30ea\u30c1\u30e3\u30fc\u30c9\u30bd\u30f3\u306e\u88dc\u5916\u306b\u3088\u308b\u8aa4\u5dee\u306e\u63a8\u5b9a\u5024\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.<\/p>\n<h3>10.2.2 \u88dc\u5916\u6cd5<\/h3>\n<h4>10.2.2.1 \u88dc\u5916\u6cd5\u306e\u30a2\u30eb\u30b4\u30ea\u30ba\u30e0<\/h4>\n<p>\\(H = t &#8211; t_0\\) \u3068\u3059\u308b. \u6b63\u306e\u6574\u6570\u5217 \\(n_1 < n_2 < n_3 < \\dots\\) \u306b\u3064\u3044\u3066 \\(h_i = H\/n_i\\) \u3068\u3059\u308b\u3068, \u6574\u6570\u5217\u306b\u5bfe\u5fdc\u3059\u308b\u30b9\u30c6\u30c3\u30d7\u5e45 \\(h_1 > h_2 > h_3 > \\dots\\) \u304c\u5f97\u3089\u308c\u308b.<\/p>\n<p>\\(p\\) \u6b21\u306e\u6570\u5024\u89e3\u6cd5\u306b\u304a\u3044\u3066\u30b9\u30c6\u30c3\u30d7\u5e45 \\(h_j\\) \u3092\u4f7f\u3063\u3066 \\(n_j\\) \u30b9\u30c6\u30c3\u30d7\u8a08\u7b97\u3057\u3066\u6c42\u3081\u305f \\(t\\) \u306b\u304a\u3051\u308b\u89e3\u3092 \\(T_{j1}\\) \u3068\u3059\u308b. \\(T_{11} \\sim T_{k1} \\space (k \\le j)\\) \u306e\u5024\u3092\u4f7f\u3063\u3066\u9023\u7acb\u65b9\u7a0b\u5f0f\u3092\u89e3\u3051\u3070 \\(h\\) \u306e \\(k\\) \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<p>\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.<br \/>\n\\[<br \/>\n  T_{jk} = T_{j(k-1)} + (T_{j(k-1)} &#8211; T_{(j-1)(k-1)})\/(n_j\/n_{j-k+1} &#8211; 1)<br \/>\n\\]\n\u3053\u306e\u6f38\u5316\u5f0f\u3092\u4f7f\u3063\u3066\u6c42\u3081\u305f \\(T_{jk}\\) \u306f\u89e3\u306e\u826f\u3044\u8fd1\u4f3c\u3068\u306a\u3063\u3066\u3044\u308b. \u3053\u306e\u65b9\u6cd5\u306f \\(p + k &#8211; 1\\) \u6b21\u306e\u6570\u5024\u89e3\u6cd5\u3068\u306a\u3063\u3066\u304a\u308a, <strong>\u88dc\u5916\u6cd5<\/strong> (Extrapolation method) \u3068\u3088\u3070\u308c\u308b.<\/p>\n<p>\u88dc\u5916\u6cd5\u306e\u8a08\u7b97\u7d50\u679c\u3092\u8868\u3059\u6b21\u306e\u8868\u3092\u88dc\u5916\u8868\u3068\u3044\u3046.<br \/>\n\\[<br \/>\n\\begin{array}{c|cccc}<br \/>\nT_{11} \\\\<br \/>\nT_{21} &#038; T_{22} \\\\<br \/>\nT_{31} &#038; T_{32} &#038; T_{33} \\\\<br \/>\nT_{41} &#038; T_{42} &#038; T_{43} &#038; T_{44} \\\\<br \/>\n\\dots &#038; \\dots &#038; \\dots &#038; \\dots &#038; \\dots \\\\<br \/>\n\u6570\u5024\u89e3 &#038; \u88dc\u5916\u5024 \\dots \\\\<br \/>\n\\end{array}<br \/>\n\\]\n\\(T_{21}\\) \u306b\u3064\u3044\u3066\u306f \\(T_{22} (k = 2)\\) \u307e\u3067, \\(T_{31}\\) \u306f \\(T_{33} ((k = 3)\\) \u307e\u3067, \\(\\dots\\), \\(T_{j1}\\) \u306f \\(T_{jj} (k = j)\\) \u307e\u3067\u88dc\u5916\u3067\u304d\u308b.<\/p>\n<p>\u4e0a\u306b\u304a\u3044\u3066, \\(k = 2, n_1 = 1, n_2 = 2\\) \u306e\u5834\u5408\u306f\u30ea\u30c1\u30e3\u30fc\u30c9\u30bd\u30f3\u306e\u88dc\u5916\u306b\u4e00\u81f4\u3059\u308b.<\/p>\n<p>\u6574\u6570\u5217 \\(n_i\\) \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<p>\u30ed\u30f3\u30d0\u30fc\u30af\u5217: \\(1, 2, 4, 8, 16, 32, 64, 128, 256, 512, \\dots\\)<br \/>\n\u30d6\u30ea\u30a2\u30b7\u30e5\u5217: \\(1, 2, 3, 4, 6, 8, 12, 16, 24, 32, \\dots\\)<br \/>\n\u8abf\u548c\u6570\u5217: \\(1, 2, 3, 4, 5, 6, 7, 8, 9, 10, \\dots\\)<\/p>\n<h4>\u6570\u5024\u5b9f\u9a13 (12)<\/h4>\n<p>\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\u3059\u308b.<br \/>\n\\[<br \/>\n  y&#8217; = y, \\space y(0) = 1<br \/>\n\\]\n\u6570\u5024\u89e3\u6cd5\u3068\u3057\u3066\u306f\u30aa\u30a4\u30e9\u30fc\u6cd5 (p = 1), \u6574\u6570\u5217\u3068\u3057\u3066\u306f\u30ed\u30f3\u30d0\u30fc\u30af\u5217 1, 2, &#8230;, 128 \u3092\u4f7f\u3063\u305f.<\/p>\n<p><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/www.ktech.biz\/jp\/wp-content\/uploads\/sites\/2\/2026\/06\/10-2_3_exp12.png\" alt=\"\" width=\"640\" height=\"480\" class=\"aligncenter size-full wp-image-5637\" srcset=\"https:\/\/www.ktech.biz\/jp\/wp-content\/uploads\/sites\/2\/2026\/06\/10-2_3_exp12.png 640w, https:\/\/www.ktech.biz\/jp\/wp-content\/uploads\/sites\/2\/2026\/06\/10-2_3_exp12-300x225.png 300w\" sizes=\"auto, (max-width: 640px) 100vw, 640px\" \/><\/p>\n<p>\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 \\(T_{j1} (k = 1)\\)\u3067\u3042\u308b. \u4e00\u756a\u53f3\u306e\u8d64\u3044\u30e9\u30a4\u30f3\u306f\u6700\u9ad8\u6b21\u6570\u3067\u6c42\u3081\u305f\u88dc\u5916\u5024 \\(T_{jj} (k = j)\\)\u3067\u3042\u308b.<\/p>\n<h4>\u6570\u5024\u5b9f\u9a13 (13)<\/h4>\n<p>\u540c\u3058\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\u304a\u3088\u3073\u6574\u6570\u5217\u306e\u7a2e\u985e) \u3092\u5909\u5316\u3055\u305b\u305f\u3068\u304d\u306e\u6319\u52d5\u3092\u78ba\u8a8d\u3059\u308b.<br \/>\n\\[<br \/>\n  y&#8217; = y, \\space y(0) = 1<br \/>\n\\]\n\u6570\u5024\u89e3\u6cd5\u3068\u3057\u3066\u306f\u30aa\u30a4\u30e9\u30fc\u6cd5 (p = 1) \u3092\u4f7f\u3063\u305f. \u30ed\u30f3\u30d0\u30fc\u30af\u5217\u3067 k = 1 (\u30aa\u30a4\u30e9\u30fc\u6cd5), 4, j \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><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/www.ktech.biz\/jp\/wp-content\/uploads\/sites\/2\/2026\/06\/10-2_4_exp13.png\" alt=\"\" width=\"640\" height=\"480\" class=\"aligncenter size-full wp-image-5640\" srcset=\"https:\/\/www.ktech.biz\/jp\/wp-content\/uploads\/sites\/2\/2026\/06\/10-2_4_exp13.png 640w, https:\/\/www.ktech.biz\/jp\/wp-content\/uploads\/sites\/2\/2026\/06\/10-2_4_exp13-300x225.png 300w\" sizes=\"auto, (max-width: 640px) 100vw, 640px\" \/><\/p>\n<p>\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\u3092 4 \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.<\/p>\n<h4>10.2.2.2 GBS \u30a2\u30eb\u30b4\u30ea\u30ba\u30e0<\/h4>\n<p>\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.<br \/>\n\\[<br \/>\n  T_{jk} = T_{j(k-1)} + \\frac{T_{j(k-1)} &#8211; T_{(j-1)(k-1)}}{(n_j\/n_{j-k+1})^2 &#8211; 1}<br \/>\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.<br \/>\n\\[<br \/>\n\\begin{align}<br \/>\n&#038; y_1 = y_0 + h_jf(t_0, y_0) \\space [\u30aa\u30a4\u30e9\u30fc\u6cd5] \\\\<br \/>\n&#038; y_{i+1} = y_{i-1} + 2h_jf(t_i, y_i) \\space (i = 1, 2, \\dots, n_j) \\space [\u4e2d\u70b9\u5247] \\\\<br \/>\n&#038; T_{j1} = (1\/4)(y_{n_{j-1}} + 2y_{n_j} + y_{n_{j+1}}) \\\\<br \/>\n\\end{align}<br \/>\n\\]\n\u3053\u306e\u3088\u3046\u306b\u3059\u308b\u3068 \\(T_{j1}\\) \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 10.1.5.2 \u3067\u8aac\u660e\u3057\u305f\u304c, \u3053\u306e\u5834\u5408\u306f\u554f\u984c\u306f\u751f\u3058\u306a\u3044.<\/p>\n<p>\u6574\u6570\u5217 \\(n_i\\) \u306f\u5076\u6570\u306b\u306a\u308b\u3088\u3046\u306b\u3059\u308b.<\/p>\n<p>\u30ed\u30f3\u30d0\u30fc\u30af\u5217: \\(2, 4, 8, 16, 32, 64, 128, 256, 512, 1024, \\dots\\)<br \/>\n\u30d6\u30ea\u30a2\u30b7\u30e5\u5217: \\(2, 4, 6, 8, 12, 16, 24, 32, 48, 64, \\dots\\)<br \/>\n\u8abf\u548c\u6570\u5217: \\(2, 4, 6, 8, 10, 12, 14, 16, 18, 20, \\dots\\)<\/p>\n<h4>\u6570\u5024\u5b9f\u9a13 (14)<\/h4>\n<p>\u6700\u3082\u7c21\u5358\u306a\u5fae\u5206\u65b9\u7a0b\u5f0f\u306e\u521d\u671f\u5024\u554f\u984c\u306e\u4f8b\u3067 GBS \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.<br \/>\n\\[<br \/>\n  y&#8217; = y, \\space y(0) = 1<br \/>\n\\]\n<p><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/www.ktech.biz\/jp\/wp-content\/uploads\/sites\/2\/2026\/06\/10-2_5_exp14.png\" alt=\"\" width=\"640\" height=\"480\" class=\"aligncenter size-full wp-image-5642\" srcset=\"https:\/\/www.ktech.biz\/jp\/wp-content\/uploads\/sites\/2\/2026\/06\/10-2_5_exp14.png 640w, https:\/\/www.ktech.biz\/jp\/wp-content\/uploads\/sites\/2\/2026\/06\/10-2_5_exp14-300x225.png 300w\" sizes=\"auto, (max-width: 640px) 100vw, 640px\" \/><\/p>\n<p>\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\u304c 0 \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.<\/p>\n<h3>10.2.3 \u30b9\u30c6\u30c3\u30d7\u5e45\u306e\u81ea\u52d5\u8abf\u7bc0<\/h3>\n<h4>10.2.3.1 \u57cb\u3081\u8fbc\u307f\u578b\u30eb\u30f3\u30b2\u30fb\u30af\u30c3\u30bf\u6cd5<\/h4>\n<p>\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\u3066 2 \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<p>\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.<br \/>\n\\[<br \/>\n\\begin{align}<br \/>\n&#038; k_i = f(y_n + h \\sum_{j=1}^s a_{ij}k_j, t_n + c_i h) \\space (i = 1, 2, \\dots, s) \\\\<br \/>\n&#038; y_{n+1} = y_n + h \\sum_{i=1}^s b_ik_i \\\\<br \/>\n\\end{align}<br \/>\n\\]\n\u540c\u3058 \\(a_{ij}\\) \u3068 \\(c_i\\) \u3092\u4f7f\u3044 (\u3059\u306a\u308f\u3061, \u540c\u3058 \\(k_i\\) \u3092\u4f7f\u3044), \\(b_i\\) \u3092 \\(b^*_i\\) \u306b\u5909\u3048\u305f\u6642\u306e\u8fd1\u4f3c\u5024 \\(y^*_{n+1}\\) \u304c \\(p^*\\) \u6b21 (\u4f8b\u3048\u3070, \\(p^* = p &#8211; 1, p^* = 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, \\(k_i\\) \u3092\u8a08\u7b97\u3057\u76f4\u3059\u3053\u3068\u306a\u304f\u6b21\u5f0f\u306b\u3088\u308a \\(p^*\\) \u6b21\u306e\u3068\u304d\u306e\u8fd1\u4f3c\u5024\u3092\u6c42\u3081\u308b\u3053\u3068\u304c\u3067\u304d\u308b.<br \/>\n\\[<br \/>\ny^*_{n+1} = y_n + h \\sum_{i=1}^s b^*_i k_i<br \/>\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.<br \/>\n\\[<br \/>\ny_{n+1} &#8211; y^*_{n+1} = h \\sum_{i=1}^s (b_i &#8211; b^*_i)k_i<br \/>\n\\]\n\u3053\u306e\u3088\u3046\u306a\u516c\u5f0f\u306f\u5b9f\u969b\u306b\u4f5c\u308b\u3053\u3068\u304c\u3067\u304d\u3066, <strong>\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<h4>10.2.3.1.1 \u30eb\u30f3\u30b2\u30fb\u30af\u30c3\u30bf\u30fb\u30de\u30fc\u30bd\u30f3\u6cd5<\/h4>\n<p>\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.<br \/>\n\\[<br \/>\n\\begin{array}{c|ccccc}<br \/>\n0 \\\\<br \/>\n1\/3 &#038; 1\/3 \\\\<br \/>\n1\/3 &#038; 1\/6 &#038; 1\/6 \\\\<br \/>\n1\/2 &#038; 1\/8 &#038; 0 &#038; 3\/8 \\\\<br \/>\n1 &#038; 1\/2 &#038; 0 &#038; -3\/2 &#038; 2 \\\\<br \/>\n\\hline<br \/>\ny &#038; 1\/6 &#038; 0 &#038; 0 &#038; 2\/3 &#038; 1\/6 \\\\<br \/>\ny^* &#038; 1\/2 &#038; 0 &#038; -3\/2 &#038; 2 &#038; 0 \\\\<br \/>\n\\end{array}<br \/>\n\\]\n\u901a\u5e38\u306e\u30eb\u30f3\u30b2\u30fb\u30af\u30c3\u30bf\u6cd5\u306e\u30d6\u30c3\u30c1\u30e3\u30fc\u914d\u5217\u306b\u5bfe\u3057\u3066 \\(y^*\\) \u306e\u305f\u3081\u306e\u4fc2\u6570\u304c 1 \u884c\u8ffd\u52a0\u3055\u308c\u3066\u3044\u308b.<\/p>\n<p>\u8a08\u7b97\u5f0f\u306f\u6b21\u306e\u3068\u304a\u308a\u3067\u3042\u308b.<br \/>\n\\[<br \/>\n\\begin{align}<br \/>\n&#038; k_1 = f(t_n, y_n) \\\\<br \/>\n&#038; k_2 = f(t_n + h\/3, y_n + hk_1\/3) \\\\<br \/>\n&#038; k_3 = f(t_n + h\/3, y_n + h(k_1\/6 + k_2\/6)) \\\\<br \/>\n&#038; k_4 = f(t_n + h\/2, y_n + h(k_1\/8 + 3k_3\/8)) \\\\<br \/>\n&#038; k_5 = f(t_n + h, y_n + h(k_1\/2 &#8211; 3k_3\/2 + 2k_4)) \\\\<br \/>\n&#038; y_{n+1} = y_n + h(k_1 + 4k_4 + k_5)\/6 \\space (4\u6b21) \\\\<br \/>\n&#038; y^*_{n+1} = y_n + h(k_1 &#8211; 3k_3 + 4k_4)\/2 \\space (\u8aa4\u5dee\u8a55\u4fa1\u7528 4\u6b21) \\\\<br \/>\n\\end{align}<br \/>\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 (y_{n+1} &#8211; y^*_{n+1})\\) \u3068\u306a\u308b.<\/p>\n<p>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\u306e 4 \u6b21\u306e\u30eb\u30f3\u30b2\u30fb\u30af\u30c3\u30bf\u6cd5\u306b\u6bd4\u3079 1 \u30b9\u30c6\u30c3\u30d7\u3042\u305f\u308a 1 \u56de\u4f59\u8a08\u306b\u95a2\u6570\u8a55\u4fa1\u3092\u884c\u3046\u3068\u3044\u3046\u72a0\u7272\u3092\u6255\u3063\u3066\u3044\u308b.<\/p>\n<h4>10.2.3.1.2 5(4) \u6b21 \u30eb\u30f3\u30b2\u30fb\u30af\u30c3\u30bf\u30fb\u30d5\u30a7\u30fc\u30eb\u30d9\u30eb\u30b0\u6cd5<\/h4>\n<p>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. \u4fc2\u6570\u306f\u6b21\u306e\u3068\u304a\u308a\u3067\u3042\u308b.<br \/>\n\\[<br \/>\n\\begin{array}{c|cccccc}<br \/>\n0 \\\\<br \/>\n1\/4 &#038; 1\/4 \\\\<br \/>\n3\/8 &#038; 3\/32 &#038; 9\/32 \\\\<br \/>\n12\/13 &#038; 1932\/2197 &#038;-7200\/2197 &#038; 7296\/2197 \\\\<br \/>\n1 &#038; 439\/216 &#038; -8 &#038; 3680\/513 &#038; -845\/4104 \\\\<br \/>\n1\/2 &#038; -8\/27 &#038; 2 &#038; -3544\/2565 &#038; 1859\/4104 &#038; -11\/40 \\\\<br \/>\n\\hline<br \/>\ny &#038; 16\/135 &#038; 0 &#038; 6656\/12825 &#038; 28561\/56430 &#038; -9\/50 &#038; 2\/55 \\\\<br \/>\ny^* &#038; 25\/216 &#038; 0 &#038; 1408\/2565 &#038; 2197\/4104 &#038; -1\/5 &#038; 0 \\\\<br \/>\n\\end{array}<br \/>\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<h4>10.2.3.1.3 5(4) \u6b21 \u30c9\u30eb\u30de\u30f3\u30fb\u30d7\u30ea\u30f3\u30b9\u6cd5<\/h4>\n<p>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\u3067\u3042\u308b.<br \/>\n\\[<br \/>\n\\begin{array}{c|ccccccc}<br \/>\n0 \\\\<br \/>\n1\/5 &#038; 1\/5 \\\\<br \/>\n3\/10 &#038; 3\/40 &#038; 9\/40 \\\\<br \/>\n4\/5 &#038; 44\/45 &#038; -56\/15 &#038; 32\/9 \\\\<br \/>\n8\/9 &#038; 19372\/6561 &#038; -25360\/2187 &#038; 64448\/6561 &#038; -212\/729 \\\\<br \/>\n1 &#038; 9017\/3168 &#038; -355\/33 &#038; 46732\/5247 &#038; 49\/176 &#038; -5103\/18656 \\\\<br \/>\n1 &#038; 35\/384 &#038; 0 &#038; 500\/1113 &#038; 125\/192 &#038; -2187\/6784 &#038; 11\/84 \\\\<br \/>\n\\hline<br \/>\ny &#038; 35\/384 &#038; 0 &#038; 500\/1113 &#038; 125\/192 &#038; -2187\/6784 &#038; 11\/84 &#038; 0 \\\\<br \/>\ny^* &#038; 5179\/57600 &#038; 0 &#038; 7571\/16695 &#038; 393\/640 &#038; -92097\/339200 &#038; 187\/2100 &#038; 1\/40 \\\\<br \/>\n\\end{array}<br \/>\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\u9577\u306f, 7 \u756a\u76ee (\\(k_7\\)) \u306e\u4fc2\u6570 \\(a_{7j}\\) \u304c\u8fd1\u4f3c\u5024 \\(y\\) \u3092\u8a08\u7b97\u3059\u308b\u305f\u3081\u306e\u4fc2\u6570 \\(b_i\\) \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, \\(k_7\\) \u304c\u6b21\u306e \\(k_1 = f(y_{n+1}, t_{n+1})\\) \u306b\u7b49\u3057\u304f\u306a\u308b\u3088\u3046\u5de7\u5999\u306b\u4f5c\u3089\u308c\u3066\u3044\u308b.<\/p>\n<h4>10.2.3.1.4 6(5) \u6b21 \u30eb\u30f3\u30b2\u30fb\u30af\u30c3\u30bf\u30fb\u30f4\u30a1\u30fc\u30ca\u30fc\u6cd5<\/h4>\n<p>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\u3067\u3042\u308b.<br \/>\n\\[<br \/>\n\\begin{array}{c|cccccccc}<br \/>\n0 \\\\<br \/>\n1\/6 &#038; 1\/6 \\\\<br \/>\n4\/15 &#038; 4\/75 &#038; 16\/75 \\\\<br \/>\n2\/3 &#038; 5\/6 &#038; -8\/3 &#038; 5\/2 \\\\<br \/>\n5\/6 &#038; -165\/64 &#038; 55\/6 &#038; -425\/64 &#038; 85\/96 \\\\<br \/>\n1 &#038; 12\/5 &#038; -8 &#038; 4015\/612 &#038; -11\/36 &#038; 88\/255 \\\\<br \/>\n1\/15 &#038; -8263\/15000 &#038; 124\/75 &#038; -643\/680 &#038; -81\/250 &#038; 2484\/10625 &#038; 0 \\\\<br \/>\n1 &#038; 3501\/1720 &#038; -300\/43 &#038; 297275\/52632 &#038; -319\/2322 &#038; 24068\/84065 &#038; 0 &#038; 3850\/26703 \\\\<br \/>\n\\hline<br \/>\ny &#038; 3\/40 &#038; 0 &#038; 875\/2244 &#038; 23\/72 &#038; 264\/1955 &#038; 0 &#038; 125\/11592 &#038; 43\/616 \\\\<br \/>\ny^* &#038; 13\/160 &#038; 0 &#038; 2375\/5984 &#038; 5\/16 &#038; 12\/85 &#038; 3\/44 &#038; 0 &#038; 0 \\\\<br \/>\n\\end{array}<br \/>\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. \u53e4\u304f\u304b\u3089\u5e83\u304f\u4f7f\u308f\u308c\u3066\u3044\u308b\u30b5\u30d6\u30eb\u30fc\u30c1\u30f3 DVERK \u306b\u63a1\u7528\u3055\u308c\u3066\u3044\u308b\u516c\u5f0f\u3067\u3042\u308b.<\/p>\n<h4>10.2.3.2 \u591a\u6bb5\u6cd5\u306e\u8aa4\u5dee<\/h4>\n<p>\u591a\u6bb5\u6cd5\u306b\u304a\u3051\u308b\u5c40\u6240\u8aa4\u5dee\u306f\u6b21\u306e\u3088\u3046\u306b\u6c42\u3081\u3089\u308c\u308b.<\/p>\n<table width=\"500\">\n<tbody>\n<tr>\n<td>\u89e3\u6cd5<\/td>\n<td>\u6b21\u6570<\/td>\n<td>\u5c40\u6240\u8aa4\u5dee<\/td>\n<\/tr>\n<tr>\n<td>\u30a2\u30c0\u30e0\u30b9-\u30d0\u30b7\u30e5\u30d5\u30a9\u30fc\u30b9\u6cd5<\/td>\n<td>\\(k\\)<\/td>\n<td>\\(\\gamma_k h^{k+1} y^{k+1}\\)<\/td>\n<\/tr>\n<tr>\n<td>\u30a2\u30c0\u30e0\u30b9-\u30e0\u30eb\u30c8\u30f3\u6cd5<\/td>\n<td>\\(k + 1\\)<\/td>\n<td>\\(\\gamma^*_{k+1} h^{k+2} y^{k+2}\\)<\/td>\n<\/tr>\n<tr>\n<td>\u5f8c\u9000\u5fae\u5206\u516c\u5f0f (BDF \u6cd5)<\/td>\n<td>\\(k\\)<\/td>\n<td>\\(-1\/(k+1) h^{k+1} y^{k+1}\\)<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<p>\u4f8b\u3048\u3070, 4 \u6b21\u306e\u30a2\u30c0\u30e0\u30b9\u6cd5\u3067\u306f, \u4e88\u6e2c\u5b50 (\u53f3\u80a9\u306b P) \u3068\u4fee\u6b63\u5b50 (\u53f3\u80a9\u306b C) \u306e\u305d\u308c\u305e\u308c\u306e\u5c40\u6240\u8aa4\u5dee\u306f\u5177\u4f53\u7684\u306b\u306f\u6b21\u306e\u3088\u3046\u306b\u306a\u308b.<br \/>\n\\[<br \/>\n\\begin{align}<br \/>\n&#038; y(t_{n+1}) &#8211; y^P_{n+1} = \\gamma_4h^5y^5 + O(h^6) = 251\/720 h^5y^5 + O(h^6) \\\\<br \/>\n&#038; y(t_{n+1}) &#8211; y^C_{n+1} = \\gamma^*_4h^5y^5 + O(h^6) = -19\/720 h^5y^5 + O(h^6) \\\\<br \/>\n\\end{align}<br \/>\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\u5229\u7528\u3057\u3066\u8aa4\u5dee\u306e\u63a8\u5b9a\u3092\u884c\u3046\u3053\u3068\u304c\u3067\u304d\u308b.<\/p>\n<p>\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.<br \/>\n\\[<br \/>\ny^C_{n+1} &#8211; y^P_{n+1} \\simeq 3\/8 h^5y^5<br \/>\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.<br \/>\n\\[<br \/>\ny(t_{n+1}) &#8211; y^C_{n+1} \\simeq -19\/27(y^C_{n+1} &#8211; y^P_{n+1})<br \/>\n\\]\n<h4>10.2.3.3 \u30b9\u30c6\u30c3\u30d7\u5e45\u306e\u81ea\u52d5\u8abf\u7bc0\u6a5f\u80fd\u306e\u5b9f\u88c5<\/h4>\n<p>\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<p>\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. \u305d\u306e\u51e6\u7406\u624b\u9806\u306f\u304a\u304a\u307e\u304b\u306b\u306f\u6b21\u306e\u3068\u304a\u308a\u3067\u3042\u308b.<\/p>\n<ul>\n<li>\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<li>\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)<\/li>\n<li>\u63a8\u5b9a\u8aa4\u5dee\u304c\u5927\u304d\u3059\u304e\u308b\u306a\u3089\u3070, \u30b9\u30c6\u30c3\u30d7\u5e45\u3092\u5c0f\u3055\u304f\u3057\u3066\u3084\u308a\u76f4\u3059 (\u30b9\u30c6\u30c3\u30d7\u5931\u6557)<\/li>\n<li>\u63a8\u5b9a\u8aa4\u5dee\u304c\u5c0f\u3055\u3059\u304e\u308b\u306a\u3089\u3070, \u30b9\u30c6\u30c3\u30d7\u6570\u304c\u4e0d\u8981\u306b\u5897\u3048\u3066\u3057\u307e\u308f\u306a\u3044\u3088\u3046\u6b21\u30b9\u30c6\u30c3\u30d7\u304b\u3089\u30b9\u30c6\u30c3\u30d7\u5e45\u3092\u5927\u304d\u304f\u3059\u308b<\/li>\n<\/ul>\n<p>\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\u3088\u3046\u306b\u3059\u308b.<\/p>\n<p>\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<p>\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\u5e45 BDF \u6cd5\u306a\u3069\u3092\u4f7f\u3046\u5fc5\u8981\u304c\u3042\u308b.<\/p>\n<h4>\u6570\u5024\u5b9f\u9a13 (15)<\/h4>\n<p>\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\u52d5\u4f5c\u3092\u78ba\u8a8d\u3059\u308b.<\/p>\n<p>RKM \u306e\u8a08\u7b97\u30a2\u30eb\u30b4\u30ea\u30ba\u30e0\u306f\u6b21\u306e\u3088\u3046\u306b\u3054\u304f\u7c21\u5358\u306a\u3082\u306e\u3068\u3057\u305f.<\/p>\n<ul>\n<li>\u30de\u30fc\u30bd\u30f3\u306e\u516c\u5f0f\u3067 1 \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\u3092 1\/2 \u306b\u3057\u3066\u3084\u308a\u76f4\u3059<\/li>\n<li>\u63a8\u5b9a\u8aa4\u5dee\u304c\u8a31\u5bb9\u9650\u754c\u306e 1\/32 \u3088\u308a\u5c0f\u3055\u3044\u3068\u304d\u306b\u306f\u6b21\u30b9\u30c6\u30c3\u30d7\u304b\u3089\u30b9\u30c6\u30c3\u30d7\u5e45\u3092 2 \u500d\u306b\u3059\u308b<\/li>\n<li>\u305d\u308c\u4ee5\u5916\u306a\u3089\u3070 1 \u30b9\u30c6\u30c3\u30d7\u9032\u3081\u308b<\/li>\n<\/ul>\n<p>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\u3073 5(4) \u6b21\u306e\u30eb\u30f3\u30b2\u30fb\u30af\u30c3\u30bf\u30fb\u30d5\u30a7\u30fc\u30eb\u30d9\u30eb\u30b0\u6cd5 (RKF45: XLPack \u306e Derkfa \u3092\u4f7f\u7528) \u306b\u3088\u308b\u8a08\u7b97\u7d50\u679c\u3082\u793a\u3059.<\/p>\n<p>\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 \u3068 RKF45 \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\u3092 50*Tol \u3068\u3057\u305f.<\/p>\n<h4>(\u4f8b 1)<\/h4>\n<p>\u5909\u5316\u304c\u5c11\u306a\u3044\u95a2\u6570\u306e\u4f8b\u3067, \u89e3\u6790\u89e3\u306f \\(y = e^t\\) \u3067\u3042\u308b. \\(t = 1\\) \u306b\u304a\u3051\u308b\u5024 (\\(e\\)) \u3092\u6c42\u3081\u8aa4\u5dee\u3092\u5bfe\u6570\u30d7\u30ed\u30c3\u30c8\u3057\u305f.<br \/>\n\\[<br \/>\n  y&#8217; = y, \\space y(0) = 1<br \/>\n\\]\n<p><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/www.ktech.biz\/jp\/wp-content\/uploads\/sites\/2\/2026\/06\/10-2_6_exp15.png\" alt=\"\" width=\"640\" height=\"480\" class=\"aligncenter size-full wp-image-5643\" srcset=\"https:\/\/www.ktech.biz\/jp\/wp-content\/uploads\/sites\/2\/2026\/06\/10-2_6_exp15.png 640w, https:\/\/www.ktech.biz\/jp\/wp-content\/uploads\/sites\/2\/2026\/06\/10-2_6_exp15-300x225.png 300w\" sizes=\"auto, (max-width: 640px) 100vw, 640px\" \/><\/p>\n<h4>(\u4f8b 2)<\/h4>\n<p>\u6587\u732e[5]\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\u8aa4\u5dee\u3092\u5bfe\u6570\u30d7\u30ed\u30c3\u30c8\u3057\u305f.<br \/>\n\\[<br \/>\n\\begin{align}<br \/>\n&#038; y_1&#8242; = 1 + y_1^2 y_2 &#8211; 4y_1 \\\\<br \/>\n&#038; y_2&#8242; = 3y_1 &#8211; y_1^2 y_2 \\\\<br \/>\n&#038; y_1(0) = 1.5, y_2(0) = 3 \\\\<br \/>\n\\end{align}<br \/>\n\\]\n<p><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/www.ktech.biz\/jp\/wp-content\/uploads\/sites\/2\/2026\/06\/10-2_7_exp15.png\" alt=\"\" width=\"640\" height=\"480\" class=\"aligncenter size-full wp-image-5644\" srcset=\"https:\/\/www.ktech.biz\/jp\/wp-content\/uploads\/sites\/2\/2026\/06\/10-2_7_exp15.png 640w, https:\/\/www.ktech.biz\/jp\/wp-content\/uploads\/sites\/2\/2026\/06\/10-2_7_exp15-300x225.png 300w\" sizes=\"auto, (max-width: 640px) 100vw, 640px\" \/><\/p>\n<p>RKM \u3068 RK4 \u306f\u3069\u3061\u3089\u3082 4 \u6b21\u3067\u3042\u308b\u304c, \u4f8b 1 \u3067\u306f RK4 \u306e\u65b9\u304c\u901f\u304f, \u4f8b 2 \u3067\u306f RKM \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.<\/p>\n<p>\u6b21\u306b, \u4e0a\u306e 2 \u3064\u306e\u4f8b\u3092\u8a08\u7b97\u3057\u305f\u3068\u304d\u306e RKM \u306e\u52d5\u4f5c\u3092\u8a73\u7d30\u306b\u898b\u308b.<\/p>\n<h4>(\u4f8b 1)<\/h4>\n<p>\\(Tol = 10^{-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.<\/p>\n<p><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/www.ktech.biz\/jp\/wp-content\/uploads\/sites\/2\/2026\/06\/10-2_8_exp15-2.png\" alt=\"\" width=\"640\" height=\"480\" class=\"aligncenter size-full wp-image-5645\" srcset=\"https:\/\/www.ktech.biz\/jp\/wp-content\/uploads\/sites\/2\/2026\/06\/10-2_8_exp15-2.png 640w, https:\/\/www.ktech.biz\/jp\/wp-content\/uploads\/sites\/2\/2026\/06\/10-2_8_exp15-2-300x225.png 300w\" sizes=\"auto, (max-width: 640px) 100vw, 640px\" \/><\/p>\n<p>\u95a2\u6570\u304c\u6ed1\u3089\u304b\u306a\u306e\u3067 \\(h = 0.082\\) \u306b\u306a\u3063\u305f\u5f8c\u306f\u30b9\u30c6\u30c3\u30d7\u5e45\u306f\u5909\u66f4\u3055\u308c\u306a\u304b\u3063\u305f. \u30b9\u30c6\u30c3\u30d7\u5e45\u306e\u521d\u671f\u5024 (50*Tol) \u304c\u5c0f\u3055\u3059\u304e\u305f\u3088\u3046\u306a\u306e\u3067\u5de5\u592b\u304c\u5fc5\u8981\u3067\u3042\u308d\u3046. \u306a\u304a, \u6700\u5f8c\u306e 1 \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\u3069 1 \u306b\u5408\u308f\u305b\u308b\u305f\u3081\u306b\u8abf\u6574\u3055\u308c\u305f\u3082\u306e\u3067\u3042\u308b.<\/p>\n<h4>(\u4f8b 2)<\/h4>\n<p>\\(Tol = 10^{-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.<\/p>\n<p><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/www.ktech.biz\/jp\/wp-content\/uploads\/sites\/2\/2026\/06\/10-2_9_exp15-2.png\" alt=\"\" width=\"640\" height=\"480\" class=\"aligncenter size-full wp-image-5646\" srcset=\"https:\/\/www.ktech.biz\/jp\/wp-content\/uploads\/sites\/2\/2026\/06\/10-2_9_exp15-2.png 640w, https:\/\/www.ktech.biz\/jp\/wp-content\/uploads\/sites\/2\/2026\/06\/10-2_9_exp15-2-300x225.png 300w\" sizes=\"auto, (max-width: 640px) 100vw, 640px\" \/><\/p>\n<p>\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\u304c 97 \u56de\u3067\u8a08\u7b97\u3092\u7d42\u3048\u3066\u3044\u308b\u304c, \u5931\u6557\u30b9\u30c6\u30c3\u30d7\u304c 25 \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\u3082 RK4 \u3088\u308a\u306f\u52b9\u7387\u304c\u3088\u304b\u3063\u305f\u3053\u3068\u306b\u306a\u308b.<\/p>\n<p>\u3053\u306e\u4f8b\u3067 RK4 \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\u5168\u4f53\u3068\u3057\u3066\u306f\u7121\u99c4\u306a\u8a08\u7b97\u304c\u591a\u304b\u3063\u305f\u305f\u3081\u3068\u8003\u3048\u3089\u308c\u308b.<\/p>\n<h3>10.2.4 \u5bc6\u51fa\u529b<\/h3>\n<p>\u30b9\u30c6\u30c3\u30d7\u5e45\u3092\u81ea\u52d5\u8abf\u7bc0\u3059\u308b\u30d7\u30ed\u30b0\u30e9\u30e0\u306b\u304a\u3044\u3066\u51fa\u529b\u70b9\u306e\u9593\u9694\u304c\u72ed\u3044\u3068\u304d\u306b\u52b9\u7387\u304c\u60aa\u304f\u306a\u308b\u3053\u3068\u304c\u3042\u308b. \u305d\u306e\u3088\u3046\u306a\u5834\u5408\u306b\u7121\u99c4\u306a\u8a08\u7b97\u3092\u907f\u3051\u308b\u65b9\u6cd5\u306b\u3064\u3044\u3066\u8ff0\u3079\u308b.<\/p>\n<h4>\u6570\u5024\u5b9f\u9a13 (16)<\/h4>\n<p>\u6b21\u306e\u521d\u671f\u5024\u554f\u984c\u306e\u89e3\u3092\u6c42\u3081\u308b.<br \/>\n\\[<br \/>\n  y&#8217; = y, \\space y(0) = 1<br \/>\n\\]\n\u89e3\u6790\u89e3\u306f \\(y = e^t\\) \u3067\u3042\u308b.<\/p>\n<p>\u30b9\u30c6\u30c3\u30d7\u5e45\u3092\u81ea\u52d5\u8abf\u7bc0\u3059\u308b\u30d7\u30ed\u30b0\u30e9\u30e0 Dopri5a (5(4)\u6b21 \u30c9\u30eb\u30de\u30f3\u30fb\u30d7\u30ea\u30f3\u30b9\u6cd5) \u3068 Derkfa (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 = \\(10^{-7}\\) \u3068\u3057\u3066, t = 0 \uff5e 1 \u3067 0.05 \u3054\u3068\u306b 0.0 \uff5e 0.05, 0.05 ~ 0.1, &#8230;, 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. \u8d64\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><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/www.ktech.biz\/jp\/wp-content\/uploads\/sites\/2\/2026\/06\/10-2_10_exp16.png\" alt=\"\" width=\"640\" height=\"480\" class=\"aligncenter size-full wp-image-5647\" srcset=\"https:\/\/www.ktech.biz\/jp\/wp-content\/uploads\/sites\/2\/2026\/06\/10-2_10_exp16.png 640w, https:\/\/www.ktech.biz\/jp\/wp-content\/uploads\/sites\/2\/2026\/06\/10-2_10_exp16-300x225.png 300w\" sizes=\"auto, (max-width: 640px) 100vw, 640px\" \/><\/p>\n<p>Dopri5a \u3082 Derkfa \u3082 0.05 \u5e45\u306e\u5404\u533a\u9593\u3067\u306f 1 \u30b9\u30c6\u30c3\u30d7\u3067\u89e3\u3092\u6c42\u3081, \u8aa4\u5dee\u306f \\(10^{-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\u306b 4 \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 \\(10^{-7}\\) \u4ed8\u8fd1\u306b\u306a\u308a, RK4 \u306e\u7cbe\u5ea6\u3067\u5341\u5206\u3067\u3042\u308b\u3053\u3068\u304c\u308f\u304b\u308b. \u95a2\u6570\u8a55\u4fa1\u56de\u6570\u306f, Dopri5a \u304c 128 \u56de, Derkfa \u304c 124 \u56de\u3067, RK4 \u306f 80 \u56de\u3068\u306a\u3063\u305f. Dopri5a \u3068 Derkfa \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<p>\u305d\u3053\u3067, \u30b9\u30c6\u30c3\u30d7\u5e45\u306f\u5fc5\u8981\u7cbe\u5ea6\u306e\u89b3\u70b9\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<strong>\u5bc6\u51fa\u529b<\/strong> (dense output) \u3068\u3088\u3076.<\/p>\n<h4>10.2.4.1 \u591a\u6bb5\u6cd5\u306e\u5834\u5408<\/h4>\n<p>\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><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/www.ktech.biz\/jp\/wp-content\/uploads\/sites\/2\/2015\/01\/ode_fig-2-300x150.png\" alt=\"\" width=\"300\" height=\"150\" class=\"aligncenter size-medium wp-image-711\" srcset=\"https:\/\/www.ktech.biz\/jp\/wp-content\/uploads\/sites\/2\/2015\/01\/ode_fig-2-300x150.png 300w, https:\/\/www.ktech.biz\/jp\/wp-content\/uploads\/sites\/2\/2015\/01\/ode_fig-2.png 501w\" sizes=\"auto, (max-width: 300px) 100vw, 300px\" \/><\/p>\n<p>\\(t_n\\) \u306e\u6b21\u306e\u30b9\u30c6\u30c3\u30d7 \\(t_{n+1} = t_n + h\\) \u306b\u304a\u3051\u308b\u5024\u3092\u6c42\u3081\u308b\u969b\u306b\u306f\u56f3\u306e\u3088\u3046\u306b\u88dc\u9593\u591a\u9805\u5f0f \\(p^*(t)\\) \u3092\u6c42\u3081, \u3053\u308c\u3092\u533a\u9593 \\([t_n, t_{n+1}]\\) \u3067\u7a4d\u5206\u3059\u308b\u3053\u3068\u306b\u3088\u308a\u5024\u3092\u6c42\u3081\u305f.<br \/>\n\\[<br \/>\ny_{n+1} = y_n + \\int_{t_n}^{t_{n+1}} p^*(x) dx<br \/>\n\\]\n\u540c\u69d8\u306b, \\(t_n + \\theta h (0 \\le \\theta \\le 1)\\) \u306b\u304a\u3051\u308b\u5024 \\(y(t_n + \\theta h)\\) \u306f\u540c\u3058\u88dc\u9593\u5f0f\u3092\u533a\u9593 \\([t_n, t_n + \\theta h]\\) \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<h4>\u6570\u5024\u5b9f\u9a13 (17)<\/h4>\n<p>\u4e0a\u306e\u4f8b\u984c\u306b\u304a\u3044\u3066 4 \u6b21\u30a2\u30c0\u30e0\u30b9\u30fb\u30e0\u30eb\u30c8\u30f3\u6cd5\u3067 h = 0.1 \u3068\u3057\u3066 t = 0.4, 0.5, &#8230;, 1.0 \u306b\u304a\u3051\u308b y \u306e\u5024\u3092\u6c42\u3081\u308b. t = 0.1, 0.2, 0.3 \u306b\u304a\u3051\u308b\u5024 (\u51fa\u767a\u5024) \u306f 4 \u6b21\u306e\u30eb\u30f3\u30b2\u30fb\u30af\u30c3\u30bf\u6cd5\u3067\u6c42\u3081\u305f. t = 0.35, 0.45, &#8230;, 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><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/www.ktech.biz\/jp\/wp-content\/uploads\/sites\/2\/2026\/06\/10-2_11_exp17.png\" alt=\"\" width=\"640\" height=\"480\" class=\"aligncenter size-full wp-image-5648\" srcset=\"https:\/\/www.ktech.biz\/jp\/wp-content\/uploads\/sites\/2\/2026\/06\/10-2_11_exp17.png 640w, https:\/\/www.ktech.biz\/jp\/wp-content\/uploads\/sites\/2\/2026\/06\/10-2_11_exp17-300x225.png 300w\" sizes=\"auto, (max-width: 640px) 100vw, 640px\" \/><\/p>\n<p>\u9752\u3044\u70b9\u304c\u30a2\u30c0\u30e0\u30b9\u30fb\u30e0\u30eb\u30c8\u30f3\u6cd5\u306b\u3088\u308b\u8a08\u7b97\u5024\u3067 \\(y = e^t\\) \u306e\u30b0\u30e9\u30d5\u306b\u306a\u308b. \u30aa\u30ec\u30f3\u30b8\u8272\u306e\u70b9\u306f\u88dc\u9593\u306b\u3088\u308a\u6c42\u3081\u305f\u70b9\u3092\u793a\u3059. \u7dd1\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\u306b, \u8aa4\u5dee\u304c\u5897\u3048\u308b\u3053\u3068\u306a\u304f\u88dc\u9593\u5024\u304c\u6c42\u3081\u3089\u308c\u3066\u3044\u308b.<\/p>\n<h4>10.2.4.2 \u9023\u7d9a\u30eb\u30f3\u30b2\u30fb\u30af\u30c3\u30bf\u6cd5<\/h4>\n<p>\u30eb\u30f3\u30b2\u30fb\u30af\u30c3\u30bf\u6cd5\u306f 1 \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, \\(t_n + \\theta h (0 \\le \\theta \\le 1)\\) \u306b\u304a\u3051\u308b\u5024 \\(y(t_n + \\theta h)\\) \u3092\u6c42\u3081\u308b\u6b21\u306e\u5f62\u306e\u5f0f\u3067\u3042\u308b.<br \/>\n\\[<br \/>\n\\begin{align}<br \/>\n&#038; y(t_n + \\theta h) = y_n + h \\sum_{i=1}^{s^*} b_i(\\theta)k_i \\\\<br \/>\nk&#038; _i = f(t_n + c_ih, y_n + h \\sum_{j=1}^{i-1} a_{ij}k_j) \\space (i = 1, 2, \\dots, s^*) \\\\<br \/>\n\\end{align}<br \/>\n\\]\n\u30eb\u30f3\u30b2\u30fb\u30af\u30c3\u30bf\u6cd5\u306e\u516c\u5f0f\u3068\u540c\u3058\u5f62\u3092\u3057\u3066\u3044\u308b\u304c, \u4fc2\u6570 \\(b_i\\) \u306f\u5b9a\u6570\u3067\u306f\u306a\u304f \\(\\theta\\) \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<strong>\u9023\u7d9a\u30eb\u30f3\u30b2\u30fb\u30af\u30c3\u30bf\u6cd5<\/strong> (Continuous Runge-Kutta (CRK) method) \u3068\u3044\u3046.<\/p>\n<p>4 \u6b21\u306e\u30eb\u30f3\u30b2\u30fb\u30af\u30c3\u30bf\u6cd5\u306b\u3064\u3044\u3066\u306f, \\(s^* = s = 4\\) \u306b\u5bfe\u3057\u3066\u5bc6\u51fa\u529b\u516c\u5f0f\u306e\u6b21\u6570 \\(p^* = 3\\) \u3068\u306a\u308b\u6b21\u306e\u89e3\u304c\u5f97\u3089\u308c\u3066\u3044\u308b.<br \/>\n\\[<br \/>\n\\begin{align}<br \/>\n&#038; b_1(\\theta) = \\theta &#8211; (3\/2)\\theta^2 + (2\/3)\\theta^3 \\\\<br \/>\n&#038; b_2(\\theta) = b_3(\\theta) = \\theta^2 &#8211; (2\/3)\\theta^3 \\\\<br \/>\n&#038; b_4(\\theta) = -(1\/2)\\theta^2 + (2\/3)\\theta^3 \\\\<br \/>\n\\end{align}<br \/>\n\\]\n\\(k_i\\) \u306f 4 \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^* = 4\\) \u306b\u306f\u3067\u304d\u306a\u3044.<\/p>\n<p>\u901a\u5e38\u306f \\(p^* = p &#8211; 1\\) \u3067\u3042\u308c\u3070\u5341\u5206\u3067\u3042\u308b\u304c, \u5c0e\u95a2\u6570 \\(y'(t_n + \\theta h)\\) \u304c\u6b63\u78ba\u306a\u5fc5\u8981\u304c\u3042\u308b\u5834\u5408\u306b\u306f \\(p^* = p\\) \u306b\u306a\u308b\u3088\u3046\u306b\u3057\u306a\u3051\u308c\u3070\u306a\u3089\u306a\u3044.<\/p>\n<h4>\u6570\u5024\u5b9f\u9a13 (18)<\/h4>\n<p>\u4e0a\u306e\u4f8b\u984c\u306b\u304a\u3044\u3066 4 \u6b21\u30eb\u30f3\u30b2\u30fb\u30af\u30c3\u30bf\u6cd5\u3067 h = 0.1 \u3068\u3057\u3066 t = 0.1, 0.2, &#8230;, 1.0 \u306b\u304a\u3051\u308b y \u306e\u5024\u3092\u6c42\u3081\u308b. t = 0.05, 0.15, &#8230;, 0.95 \u306b\u304a\u3051\u308b\u5024\u306f\u4e0a\u3067\u793a\u3057\u305f 3 \u6b21\u306e\u5bc6\u51fa\u529b\u516c\u5f0f\u3092\u4f7f\u3063\u3066\u6c42\u3081\u308b.<\/p>\n<p><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/www.ktech.biz\/jp\/wp-content\/uploads\/sites\/2\/2026\/06\/10-2_12_exp18.png\" alt=\"\" width=\"640\" height=\"480\" class=\"aligncenter size-full wp-image-5649\" srcset=\"https:\/\/www.ktech.biz\/jp\/wp-content\/uploads\/sites\/2\/2026\/06\/10-2_12_exp18.png 640w, https:\/\/www.ktech.biz\/jp\/wp-content\/uploads\/sites\/2\/2026\/06\/10-2_12_exp18-300x225.png 300w\" sizes=\"auto, (max-width: 640px) 100vw, 640px\" \/><\/p>\n<p>\u9752\u3044\u70b9\u304c\u30eb\u30f3\u30b2\u30fb\u30af\u30c3\u30bf\u6cd5\u306b\u3088\u308b\u8a08\u7b97\u5024\u3067 \\(y = e^t\\) \u306e\u30b0\u30e9\u30d5\u3068\u306a\u308b. \u30aa\u30ec\u30f3\u30b8\u8272\u306e\u70b9\u306f\u88dc\u9593\u306b\u3088\u308a\u6c42\u3081\u305f\u70b9\u3092\u793a\u3059. \u7dd1\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\u306e\u5834\u5408\u306f\u5c11\u3057\u8aa4\u5dee\u304c\u5897\u3048\u308b.<\/p>\n<h4>10.2.4.3 \u30b9\u30c6\u30c3\u30d7\u5e45\u306e\u81ea\u52d5\u8abf\u7bc0\u3092\u884c\u3046\u5834\u5408<\/h4>\n<p>\u4e0a\u306e 2 \u3064\u306e\u6570\u5024\u5b9f\u9a13\u3067\u307f\u305f\u88dc\u9593\u3092\u4f7f\u3046\u5bc6\u51fa\u529b\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<p>\u591a\u6bb5\u6cd5\u306e\u5834\u5408, \u4f8b\u3048\u3070, XLPack \u306b\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, \u7279\u306b\u6307\u5b9a\u3057\u306a\u3044\u9650\u308a\u88dc\u9593\u5024\u3092\u4f7f\u3046\u3088\u3046\u306b\u4f5c\u3089\u308c\u3066\u3044\u308b.<\/p>\n<p>\u30eb\u30f3\u30b2\u30fb\u30af\u30c3\u30bf\u6cd5\u306e\u5834\u5408, \u4f8b\u3048\u3070, XLPack \u306e Derkfa \u306b\u306f 5 \u6b21\u306e\u88dc\u9593\u516c\u5f0f\u3092\u4f7f\u3063\u305f\u5bc6\u51fa\u529b\u6a5f\u80fd\u304c\u7d44\u307f\u8fbc\u307e\u308c\u3066\u304a\u308a, Dopri5a \u306b\u306f 4 \u6b21\u306e\u88dc\u9593\u516c\u5f0f\u3092\u4f7f\u3063\u305f\u5bc6\u51fa\u529b\u6a5f\u80fd\u304c\u7d44\u307f\u8fbc\u307e\u308c\u3066\u3044\u308b. \u3053\u308c\u3089\u306e\u30d7\u30ed\u30b0\u30e9\u30e0\u3067\u306f\u5bc6\u51fa\u529b\u306e\u305f\u3081\u306e\u8ffd\u52a0\u95a2\u6570\u8a55\u4fa1\u3092\u5fc5\u8981\u3068\u3057\u306a\u3044\u88dc\u9593\u516c\u5f0f\u304c\u63a1\u7528\u3055\u308c\u3066\u3044\u308b. XLPack \u306b\u53ce\u9332\u3055\u308c\u3066\u3044\u308b\u4ed6\u306e\u591a\u304f\u306e\u30d7\u30ed\u30b0\u30e9\u30e0\u306b\u3082\u5bc6\u51fa\u529b\u6a5f\u80fd\u304c\u7d44\u307f\u8fbc\u307e\u308c\u3066\u3044\u308b.<\/p>\n<h4>\u6570\u5024\u5b9f\u9a13 (19)<\/h4>\n<p>\u6570\u5024\u5b9f\u9a13 (16) \u306e\u4f8b\u306b\u3064\u3044\u3066, Derkfa, Dopri5a \u304a\u3088\u3073 Deabm \u3092\u4f7f\u3063\u3066\u8981\u6c42\u7cbe\u5ea6 RTol = ATol = \\(10^{-7}\\) \u3092\u6307\u5b9a\u3057\u3066\u8a08\u7b97\u3092\u884c\u3044, 0.05 \u3054\u3068\u306b\u7d50\u679c\u306e\u51fa\u529b\u3092\u884c\u3046. \u305d\u306e\u969b\u306b\u5bc6\u51fa\u529b\u3092\u30aa\u30f3\u306b\u3059\u308b\u3068\u3069\u3046\u5909\u308f\u308b\u304b\u78ba\u8a8d\u3059\u308b.<\/p>\n<p><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/www.ktech.biz\/jp\/wp-content\/uploads\/sites\/2\/2026\/06\/10-2_14_exp19_2.png\" alt=\"\" width=\"640\" height=\"480\" class=\"aligncenter size-full wp-image-5651\" srcset=\"https:\/\/www.ktech.biz\/jp\/wp-content\/uploads\/sites\/2\/2026\/06\/10-2_14_exp19_2.png 640w, https:\/\/www.ktech.biz\/jp\/wp-content\/uploads\/sites\/2\/2026\/06\/10-2_14_exp19_2-300x225.png 300w\" sizes=\"auto, (max-width: 640px) 100vw, 640px\" \/><\/p>\n<p><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/www.ktech.biz\/jp\/wp-content\/uploads\/sites\/2\/2026\/06\/10-2_13_exp19.png\" alt=\"\" width=\"640\" height=\"480\" class=\"aligncenter size-full wp-image-5650\" srcset=\"https:\/\/www.ktech.biz\/jp\/wp-content\/uploads\/sites\/2\/2026\/06\/10-2_13_exp19.png 640w, https:\/\/www.ktech.biz\/jp\/wp-content\/uploads\/sites\/2\/2026\/06\/10-2_13_exp19-300x225.png 300w\" sizes=\"auto, (max-width: 640px) 100vw, 640px\" \/><\/p>\n<p><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/www.ktech.biz\/jp\/wp-content\/uploads\/sites\/2\/2026\/06\/10-2_15_exp19_3.png\" alt=\"\" width=\"640\" height=\"480\" class=\"aligncenter size-full wp-image-5652\" srcset=\"https:\/\/www.ktech.biz\/jp\/wp-content\/uploads\/sites\/2\/2026\/06\/10-2_15_exp19_3.png 640w, https:\/\/www.ktech.biz\/jp\/wp-content\/uploads\/sites\/2\/2026\/06\/10-2_15_exp19_3-300x225.png 300w\" sizes=\"auto, (max-width: 640px) 100vw, 640px\" \/><\/p>\n<p>\u305d\u308c\u305e\u308c\u306e\u56f3\u306e\u4e0a\u6bb5\u306f\u95a2\u6570\u8a55\u4fa1\u56de\u6570\u306e\u30d7\u30ed\u30c3\u30c8\u3067\u3042\u308b. \u9752\u306f\u901a\u5e38\u52d5\u4f5c, \u30aa\u30ec\u30f3\u30b8\u8272\u306f\u5bc6\u51fa\u529b\u3092\u884c\u3063\u305f\u5834\u5408\u3067\u3042\u308b. \u3069\u306e\u30d7\u30ed\u30b0\u30e9\u30e0\u3067\u3082\u5bc6\u51fa\u529b\u3092\u884c\u3063\u305f\u5834\u5408\u306b\u306f\u95a2\u6570\u8a55\u4fa1\u56de\u6570\u304c\u304a\u304a\u3088\u305d\u534a\u5206\u306b\u6e1b\u3063\u3066\u3044\u308b.<\/p>\n<p>\u4e0b\u6bb5\u306f\u8a08\u7b97\u7d50\u679c\u306e\u76f8\u5bfe\u8aa4\u5dee\u3092\u8868\u3059. \u901a\u5e38\u52d5\u4f5c\u3067\u306f, \u8981\u6c42\u7cbe\u5ea6 \\(10^{-7}\\) \u3092\u6e80\u305f\u3059\u30b9\u30c6\u30c3\u30d7\u5e45\u304c 0.05 \u3088\u308a\u5927\u304d\u304f\u306a\u308b\u305f\u3081, \u7d50\u679c\u7684\u306b\u30b9\u30c6\u30c3\u30d7\u306f\u305d\u306e\u7d42\u308f\u308a\u304c\u51fa\u529b\u3092\u884c\u3046\u70b9\u306b\u8abf\u6574\u3055\u308c\u3066 0.05 \u306b\u30ab\u30c3\u30c8\u3055\u308c\u3066\u3057\u307e\u3044, \u5fc5\u8981\u4ee5\u4e0a\u306e\u7cbe\u5ea6 \\(10^{-9} \\sim 10^{-11}\\) \u304c\u51fa\u3066\u3044\u308b. \u5bc6\u51fa\u529b\u3092\u884c\u3063\u305f\u5834\u5408\u306b\u306f\u8981\u6c42\u7cbe\u5ea6\u306b\u5bfe\u3059\u308b\u6700\u9069\u30b9\u30c6\u30c3\u30d7\u5e45\u306b\u8abf\u6574\u3055\u308c\u3066\u3044\u308b\u305f\u3081, \u8981\u6c42\u3069\u304a\u308a \\(10^{-7}\\) \u4ed8\u8fd1\u306b\u306a\u3063\u3066\u3044\u308b.<\/p>\n<p>\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.<\/p>\n<p>\u306a\u304a, Deabm \u306e\u5834\u5408\u306b\u306f\u6307\u5b9a\u3057\u306a\u3044\u9650\u308a\u5bc6\u51fa\u529b\u3092\u884c\u3046\u3088\u3046\u306a\u5b9f\u88c5\u306b\u306a\u3063\u3066\u3044\u308b.<\/p>\n<h3>10.2.5 2 \u968e\u5e38\u5fae\u5206\u65b9\u7a0b\u5f0f<\/h3>\n<p>\u5b9f\u969b\u306b\u5fdc\u7528\u3067\u73fe\u308c\u308b\u65b9\u7a0b\u5f0f\u306f 2 \u968e\u5e38\u5fae\u5206\u65b9\u7a0b\u5f0f\u304c\u591a\u3044.<br \/>\n\\[<br \/>\n\\begin{align}<br \/>\n&#038; y^{\\prime\\prime} = f(t, y, y&#8217;) \\\\<br \/>\n&#038; y(t_0) = y_0, \\space y'(t_0) = y&#8217;_0 \\space (\u521d\u671f\u6761\u4ef6) \\\\<br \/>\n\\end{align}<br \/>\n\\]\n\u3053\u308c\u306f\u300c\u306f\u3058\u3081\u306b\u300d\u306e\u9805\u3067\u8aac\u660e\u3057\u305f\u3088\u3046\u306b 1 \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<p>\u3057\u304b\u3057, \u6b21\u306e\u3088\u3046\u306b \\(f()\\) \u304c \\(y&#8217;\\) \u306b\u4f9d\u5b58\u3057\u306a\u3044\u3088\u3046\u306a\u7279\u5225\u306a\u5f62\u3092\u3057\u305f\u5834\u5408\u306b\u306f 1 \u968e\u306e\u9023\u7acb\u5e38\u5fae\u5206\u65b9\u7a0b\u5f0f\u306b\u5909\u63db\u3059\u308b\u3053\u3068\u306a\u304f\u76f4\u63a5\u89e3\u304f\u3053\u3068\u304c\u3067\u304d\u308b\u65b9\u6cd5\u304c\u3042\u308b.<br \/>\n\\[<br \/>\n\\begin{align}<br \/>\n&#038; y^{\\prime\\prime} = f(t, y) \\\\<br \/>\n&#038; y(t_0) = y_0, \\space y'(t_0) = y&#8217;_0 \\space (\u521d\u671f\u6761\u4ef6) \\\\<br \/>\n\\end{align}<br \/>\n\\]\n<h4>10.2.5.1 \u30cb\u30e5\u30b9\u30c8\u30ec\u30e0\u6cd5<\/h4>\n<p>\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.<br \/>\n\\[<br \/>\n\\begin{align}<br \/>\n&#038; k&#8217;_i = f(t_n + c_ih, y_n + c_ihy&#8217;_n + h^2 \\sum_{j=1}^s \\bar{a}_{ij}k&#8217;_j) \\space (i = 1, 2, \\dots, s) \\\\<br \/>\n&#038; y_{n+1} = y_n + hy&#8217;_n + h^2 \\sum_{i=1}^s \\bar{b}_ik&#8217;_i \\\\<br \/>\n&#038; y&#8217;_{n+1} = y&#8217;_n + h \\sum_{i=1}^s b_i k&#8217;_i \\\\<br \/>\n\\end{align}<br \/>\n\\]\n\u4fc2\u6570\u306f\u4ee5\u4e0b\u306e\u3068\u304a\u308a\u3067\u3042\u308b.<\/p>\n<h4>4 \u6b21\u30cb\u30e5\u30b9\u30c8\u30ec\u30e0\u6cd5<\/h4>\n<p>\\[<br \/>\n\\begin{array}{c|ccc}<br \/>\n0 &#038; &#038; &#038; \\bar{a}_{ij} \\\\<br \/>\n1\/2 &#038; 1\/8 \\\\<br \/>\n2\/3 &#038; 0 &#038; 1\/2 \\\\<br \/>\n\\hline<br \/>\n\\bar{b}_i &#038; 1\/6 &#038; 1\/3 &#038; 0 \\\\<br \/>\nb_i &#038; 1\/6 &#038; 4\/6 &#038; 1\/6 \\\\<br \/>\n\\end{array}<br \/>\n\\]\n<h4>5 \u6b21\u30cb\u30e5\u30b9\u30c8\u30ec\u30e0\u6cd5<\/h4>\n<p>\\[<br \/>\n\\begin{array}{c|cccc}<br \/>\n0 &#038; &#038; &#038; &#038; \\bar{a}_{ij} \\\\<br \/>\n1\/5 &#038; 1\/50 \\\\<br \/>\n2\/3 &#038; -1\/27 &#038; 7\/27 \\\\<br \/>\n1 &#038; 3\/10 &#038; -2\/35 &#038; 9\/35 \\\\<br \/>\n\\hline<br \/>\n\\bar{b}_i &#038; 14\/336 &#038; 100\/336 &#038; 54\/336 &#038; 0 \\\\<br \/>\nb_i &#038; 14\/336 &#038; 125\/336 &#038; 162\/336 &#038; 35\/336 \\\\<br \/>\n\\end{array}<br \/>\n\\]\n\u30cb\u30e5\u30b9\u30c8\u30ec\u30e0\u6cd5\u306f\u30eb\u30f3\u30b2\u30fb\u30af\u30c3\u30bf\u6cd5\u306b\u6bd4\u3079\u3066\u6bb5\u6570\u304c\u5c11\u306a\u304f\u3066\u3059\u3080 (4\u6b21: 4\u6bb5 \u2192 3\u6bb5, 5\u6b21: 6\u6bb5 \u2192 4\u6bb5).<\/p>\n<h4>10.2.5.2 \u88dc\u5916\u6cd5<\/h4>\n<p>1 \u968e\u306e\u9023\u7acb\u5e38\u5fae\u5206\u65b9\u7a0b\u5f0f\u306b\u5909\u63db\u3057\u3066 GBS \u30a2\u30eb\u30b4\u30ea\u30ba\u30e0\u3092\u9069\u7528\u3059\u308b.<br \/>\n\\[<br \/>\n\\begin{align}<br \/>\n&#038; y_1 = y_0 + h_jy&#8217;_0 \\\\<br \/>\n&#038; y&#8217;_1 = y&#8217;_0 + h_jf(t_0, y_0) \\\\<br \/>\n\\\\<br \/>\n&#038; y_{i+1} = y_{i-1} + 2h_jy&#8217;_i \\space (i = 1, 2, \\dots, n_j) \\\\<br \/>\n&#038; y&#8217;_{i+1} = y&#8217;_{i-1} + 2h_jf(t_i, y_i) \\space (i = 1, 2, \\dots, n_j) \\\\<br \/>\n\\\\<br \/>\n&#038; T_{j1} = (1\/4)(y_{n_{j-1}} + 2y_{n_j} + y_{n_{j+1}}) \\\\<br \/>\n&#038; T&#8217;_{j1} = (1\/4)(y&#8217;_{n_{j-1}} + 2y&#8217;_{n_j} + y&#8217;_{n_{j+1}}) \\\\<br \/>\n\\end{align}<br \/>\n\\]\n\u3053\u306e\u5834\u5408, \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\u3092\u3059\u308c\u3070\u3088\u304f\u306a\u308b.<br \/>\n\\[<br \/>\n\\begin{align}<br \/>\n&#038; T_{j1} = y_{n_j} \\\\<br \/>\n&#038; T&#8217;_{j1} = (1\/2)(y&#8217;_{n_{j-1}} + y&#8217;_{n_{j+1}}) \\\\<br \/>\n\\end{align}<br \/>\n\\]\n<h4>\u6570\u5024\u5b9f\u9a13 (20)<\/h4>\n<p>\u6b21\u306e 2 \u968e\u5e38\u5fae\u5206\u65b9\u7a0b\u5f0f\u306e\u521d\u671f\u5024\u554f\u984c\u306e\u4f8b\u3092 4 \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.<br \/>\n\\[<br \/>\n\\begin{align}<br \/>\n&#038; y^{\\prime\\prime} = y \\\\<br \/>\n&#038; y(0) = 1, y'(0) = 1 \\\\<br \/>\n\\end{align}<br \/>\n\\]\n<p><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/www.ktech.biz\/jp\/wp-content\/uploads\/sites\/2\/2026\/06\/10-2_16_exp20.png\" alt=\"\" width=\"640\" height=\"480\" class=\"aligncenter size-full wp-image-5655\" srcset=\"https:\/\/www.ktech.biz\/jp\/wp-content\/uploads\/sites\/2\/2026\/06\/10-2_16_exp20.png 640w, https:\/\/www.ktech.biz\/jp\/wp-content\/uploads\/sites\/2\/2026\/06\/10-2_16_exp20-300x225.png 300w\" sizes=\"auto, (max-width: 640px) 100vw, 640px\" \/><\/p>\n<p>\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. \u6bd4\u8f03\u306e\u305f\u3081 4 \u6b21\u306e\u30eb\u30f3\u30b2\u30fb\u30af\u30c3\u30bf\u6cd5\u306e\u8a08\u7b97\u7d50\u679c\u3082\u793a\u3057\u305f.<\/p>\n<p>4 \u6b21\u30cb\u30e5\u30b9\u30c8\u30ec\u30e0\u6cd5\u306f\u6bb5\u6570\u306f\u5c11\u306a\u3044\u304c 4 \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.<\/p>\n<h3>10.2.6 \u9045\u5ef6\u5fae\u5206\u65b9\u7a0b\u5f0f<\/h3>\n<p><strong>\u9045\u5ef6\u5fae\u5206\u65b9\u7a0b\u5f0f<\/strong> (Delay differential equations (DDE)) \u3068\u306f\u6b21\u306e\u3088\u3046\u306a\u5fae\u5206\u65b9\u7a0b\u5f0f\u3092\u3044\u3046.<br \/>\n\\[<br \/>\ny'(t) = f(t, y(t), y(t &#8211; \\tau))<br \/>\n\\]\n\\(\\tau\\) \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<p>\u3053\u308c\u3092\u30eb\u30f3\u30b2\u30fb\u30af\u30c3\u30bf\u6cd5\u3067\u89e3\u304f\u305f\u3081\u306b\u306f\u30b9\u30c6\u30c3\u30d7\u5e45\u3092 \\(\\tau = kh\\) (k \u306f\u6574\u6570) \u3068\u306a\u308b\u3088\u3046\u306b\u9078\u3093\u3067, \u8a08\u7b97\u3057\u305f \\(y(t &#8211; \\tau)\\) \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\u30d7\u30ed\u30b0\u30e9\u30e0\u3067\u306f\u3046\u307e\u304f\u3044\u304b\u306a\u3044. \u305d\u306e\u5834\u5408, \u591a\u6bb5\u6cd5\u304b\u5bc6\u51fa\u529b\u6a5f\u80fd\u3092\u6301\u3063\u305f\u30eb\u30f3\u30b2\u30fb\u30af\u30c3\u30bf\u6cd5\u3067\u904e\u53bb\u306e\u8a08\u7b97\u5024\u3092\u3068\u3063\u3066\u304a\u304f\u3088\u3046\u306b\u3059\u308c\u3070\u8a08\u7b97\u3067\u304d\u308b.<\/p>\n<p>XLPack \u306b\u53ce\u9332\u3055\u308c\u3066\u3044\u308b Retarda \u306f\u30c9\u30eb\u30de\u30f3\u30fb\u30d7\u30ea\u30f3\u30b9\u6cd5\u30d7\u30ed\u30b0\u30e9\u30e0 Dopri5a \u3092\u9045\u5ef6\u5fae\u5206\u65b9\u7a0b\u5f0f\u7528\u306b\u4fee\u6b63\u3057\u305f\u3082\u306e\u3067\u3042\u308b. Dopri5a \u3067\u306f 1 \u30b9\u30c6\u30c3\u30d7\u524d\u307e\u3067\u306e\u904e\u53bb\u306e\u5024\u3092\u6c42\u3081\u3089\u308c\u305f\u304c, Retarda \u3067\u306f\u305a\u3063\u3068\u524d\u307e\u3067\u306e\u5024\u3092\u5f97\u308b\u3053\u3068\u304c\u3067\u304d\u308b.<\/p>\n<p>\u9045\u5ef6\u5fae\u5206\u65b9\u7a0b\u5f0f\u306e\u8a08\u7b97\u4f8b\u306f\u300c10.4 \u5b9f\u7528\u30d7\u30ed\u30b0\u30e9\u30e0\u306e\u30d9\u30f3\u30c1\u30de\u30fc\u30af\u300d\u3092\u53c2\u7167\u305b\u3088.<\/p>\n","protected":false},"excerpt":{"rendered":"<p>10.2.1 \u8aa4\u5dee\u306e\u63a8\u5b9a \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 [&hellip;]<\/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,"footnotes":""},"categories":[13],"tags":[],"class_list":["post-5557","post","type-post","status-publish","format-standard","hentry","category-num"],"aioseo_notices":[],"aioseo_head":"\n\t\t<!-- All in One SEO 5.0.1.1 - aioseo.com -->\n\t<meta name=\"description\" content=\"10.2.1 \u8aa4\u5dee\u306e\u63a8\u5b9a \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\" \/>\n\t<meta name=\"robots\" content=\"max-snippet:-1, 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