{"id":4706,"date":"2025-03-31T16:56:33","date_gmt":"2025-03-31T07:56:33","guid":{"rendered":"https:\/\/www.ktech.biz\/jp\/?p=4706"},"modified":"2025-05-26T12:13:01","modified_gmt":"2025-05-26T03:13:01","slug":"12-ode","status":"publish","type":"post","link":"https:\/\/www.ktech.biz\/jp\/tutorial\/12-ode\/","title":{"rendered":"12. \u5e38\u5fae\u5206\u65b9\u7a0b\u5f0f"},"content":{"rendered":"\n

12.1 \u6982\u8981<\/h3>\n

\u5e38\u5fae\u5206\u65b9\u7a0b\u5f0f (ODE) \u306f\u79d1\u5b66\u6280\u8853\u8a08\u7b97\u306b\u304a\u3044\u3066\u306f\u3057\u3070\u3057\u3070\u73fe\u308c\u307e\u3059. \u89e3\u6790\u7684\u306b\u89e3\u3051\u308b\u3053\u3068\u304c\u5c11\u306a\u304f, \u6570\u5024\u89e3\u6cd5\u306b\u983c\u308b\u3053\u3068\u304c\u591a\u304f\u306a\u308a\u307e\u3059.<\/p>\n

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

1 \u968e\u306e\u5e38\u5fae\u5206\u65b9\u7a0b\u5f0f\u306f\u6b21\u306e\u3088\u3046\u306b\u8868\u3055\u308c\u307e\u3059.
\n\\[
\ny’ = \\frac{dy}{dt} = f(t, y)
\n\\]\n\u3053\u3053\u3067, \\(y\\) \u306f \\(t\\) \u306e\u672a\u77e5\u306e\u95a2\u6570 \\(y(t)\\) \u3067\u3059. \u5e38\u5fae\u5206\u65b9\u7a0b\u5f0f\u306e\u521d\u671f\u5024\u554f\u984c\u306f, \u521d\u671f\u70b9 \\(t = t_0\\) \u306b\u304a\u3051\u308b\u95a2\u6570\u5024 \\(y = y_0 = y(t_0)\\) (\u521d\u671f\u6761\u4ef6\u3068\u3088\u3070\u308c\u307e\u3059) \u304c\u4e0e\u3048\u3089\u308c\u305f\u3068\u304d, \u6700\u7d42\u70b9 \\(t_f\\) \u306b\u304a\u3051\u308b\u95a2\u6570\u5024 \\(y(t_f)\\) \u3092\u6c42\u3081\u308b\u3053\u3068\u3092\u521d\u671f\u5024\u554f\u984c\u3068\u3044\u3044\u307e\u3059.<\/p>\n

12.1.2 1 \u968e\u306e\u9023\u7acb\u65b9\u7a0b\u5f0f<\/h4>\n

1 \u968e\u306e\u5e38\u5fae\u5206\u65b9\u7a0b\u5f0f\u306e\u521d\u671f\u5024\u554f\u984c\u306f, \u30b9\u30ab\u30e9\u30fc \\(y\\) \u3092\u30d9\u30af\u30c8\u30eb \\(\\boldsymbol{y}\\) \u306b\u4ee3\u3048\u3066\u3053\u306e\u307e\u307e\u9023\u7acb\u65b9\u7a0b\u5f0f\u306b\u62e1\u5f35\u3059\u308b\u3053\u3068\u304c\u3067\u304d\u307e\u3059.
\n\\[
\n\\boldsymbol{y’} = \\boldsymbol{f}(\\boldsymbol{y}, t) \\\\
\n\\boldsymbol{y_0} = \\boldsymbol{y}(t_0) \\\\
\n\\]\n\u4f8b\u3048\u3070, 2 \u5909\u6570\u306e\u5834\u5408\u306b\u3064\u3044\u3066\u30b9\u30ab\u30e9\u30fc\u5f0f\u306b\u66f8\u304d\u4e0b\u3059\u3068\u6b21\u306e\u3088\u3046\u306b\u306a\u308a\u307e\u3059.
\n\\[
\ny_1′ = f_1(y_1, y_2, t) \\\\
\ny_2′ = f_2(y_1, y_2, t) \\\\
\n\\]\n\u30b9\u30ab\u30e9\u30fc\u306e\u521d\u671f\u6761\u4ef6\u306f 2 \u3064\u5fc5\u8981\u3067, \u6b21\u306e\u3088\u3046\u306b\u306a\u308a\u307e\u3059.
\n\\[
\n{y_1}_0 = y_1(t_0) \\\\
\n{y_2}_0 = y_2(t_0) \\\\
\n\\]\n

12.1.3 \u9ad8\u968e\u306e\u5e38\u5fae\u5206\u65b9\u7a0b\u5f0f<\/h4>\n

\u9ad8\u968e\u306e\u5e38\u5fae\u5206\u65b9\u7a0b\u5f0f, \u3059\u306a\u308f\u3061, 2 \u968e\u4ee5\u4e0a\u306e\u5fae\u5206\u3092\u542b\u3093\u3060\u5e38\u5fae\u5206\u65b9\u7a0b\u5f0f\u306f, 1 \u968e\u306e\u9023\u7acb\u5e38\u5fae\u5206\u65b9\u7a0b\u5f0f\u306e\u5f62\u306b\u3057\u3066\u6271\u3046\u3053\u3068\u3082\u3067\u304d\u307e\u3059. \u4f8b\u3048\u3070, \u6b21\u306e\u5fae\u5206\u65b9\u7a0b\u5f0f\u3092\u8003\u3048\u307e\u3059.
\n\\[
\ny” = f(y, y’, t)
\n\\]\n\u3053\u308c\u306f, \\(y_1 = y, y_2 = y’\\) \u3068\u304a\u3051\u3070, \u6b21\u306e\u3088\u3046\u306b 1 \u968e\u306e\u9023\u7acb\u5e38\u5fae\u5206\u65b9\u7a0b\u5f0f\u306b\u3059\u308b\u3053\u3068\u304c\u3067\u304d\u307e\u3059.
\n\\[
\ny_1′ = y_2 \\\\
\ny_2′ = f(y_1, y_2, t) \\\\
\n\\]\n\u3053\u306e\u5834\u5408\u306b\u306f\u6b21\u306e\u3088\u3046\u306b 2 \u3064\u306e\u521d\u671f\u6761\u4ef6\u304c\u5fc5\u8981\u306b\u306a\u308a\u307e\u3059.
\n\\[
\ny_1(t_0) = y(t_0) = y_0 \\\\
\ny_2(t_0) = y'(t_0) = y’_0 \\\\
\n\\]\n\u540c\u69d8\u306b\u3057\u3066 3 \u968e\u4ee5\u4e0a\u306e\u9023\u7acb\u5e38\u5fae\u5206\u65b9\u7a0b\u5f0f\u3092\u89e3\u304f\u3053\u3068\u3082\u3067\u304d\u307e\u3059.<\/p>\n

12.2 \u5e38\u5fae\u5206\u65b9\u7a0b\u5f0f\u306e\u521d\u671f\u5024\u554f\u984c\u306e\u89e3\u6cd5<\/h3>\n

12.2.1 \u30aa\u30a4\u30e9\u30fc\u6cd5<\/h4>\n

\u5909\u6570 \\(t\\) \u3092\u96e2\u6563\u5316\u3057\u3066, \\(t_0, t_1, \\dots , t_n, \\dots\\) \u3068\u8868\u3057\u307e\u3059. \u3053\u308c\u3089\u306e\u70b9\u306e\u9593\u9694\u3092\u30b9\u30c6\u30c3\u30d7\u5e45\u3068\u3044\u3044, \\(h_n\\) \u3067\u8868\u3057\u307e\u3059.
\n\\[
\nh_n = t_{n+1} – t_n
\n\\]\n\u30b9\u30c6\u30c3\u30d7\u5e45\u306f\u5fc5\u305a\u3057\u3082\u4e00\u5b9a\u3067\u306a\u304f\u3066\u3082\u3088\u3044\u306e\u3067\u3059\u304c, \u3053\u3053\u3067\u306f \\(n\\) \u306b\u3088\u3089\u305a\u4e00\u5b9a\u306b\u3059\u308b\u3053\u3068\u306b\u3057\u3066\u5358\u306b \\(h\\) \u3068\u8868\u3057\u307e\u3059. \u672a\u77e5\u95a2\u6570 \\(y(t)\\) \u306e \\(t_n\\) \u306b\u304a\u3051\u308b\u5024 \\(y(t_n)\\) \u306e\u8fd1\u4f3c\u5024\u3092 \\(y_n\\) \u3068\u8868\u3057\u307e\u3059.<\/p>\n

\u3053\u3053\u3067, \u5fae\u5206\u3092\u5dee\u5206\u8fd1\u4f3c\u3059\u308b\u3068\u6b21\u306e\u3088\u3046\u306b\u8868\u3055\u308c\u307e\u3059.
\n\\[
\ny’_n = f(t_n, y_n) \\simeq (y_{n+1} – y_n) \/ h
\n\\]\n\u3053\u308c\u3088\u308a\u6b21\u306e\u8fd1\u4f3c\u5f0f\u304c\u5f97\u3089\u308c\u307e\u3059.
\n\\[
\ny_{n+1} = y_n + hf(t_n, y_n)
\n\\]\n\u3053\u306e\u5f0f\u3092\u4f7f\u3063\u3066\u521d\u671f\u5024 \\(y_0\\) \u304b\u3089\u59cb\u3081\u3066, \\(y_1, y_2, \\dots\\) \u3068\u6b21\u3005\u306b\u5404\u70b9\u306b\u304a\u3051\u308b\u8fd1\u4f3c\u89e3\u3092\u6c42\u3081\u308b\u65b9\u6cd5\u3092\u30aa\u30a4\u30e9\u30fc\u6cd5\u3068\u3044\u3044\u307e\u3059.<\/p>\n

\u30aa\u30a4\u30e9\u30fc\u6cd5\u306e\u5f0f\u306f \\(y\\) \u3092\u30c6\u30a4\u30e9\u30fc\u5c55\u958b\u3057\u305f\u3068\u304d\u306b 2 \u6b21\u4ee5\u4e0a\u306e\u9805\u3092\u7701\u7565\u3057\u305f\u3082\u306e\u306b\u76f8\u5f53\u3059\u308b\u306e\u3067 1 \u6b21\u306e\u89e3\u6cd5\u3068\u547c\u3070\u308c, \u8aa4\u5dee\u306f \\(h^2\\) \u306e\u30aa\u30fc\u30c0\u30fc (\\(O(h^2)\\)) \u306b\u306a\u308a\u307e\u3059.<\/p>\n

12.2.2 \u30eb\u30f3\u30b2\u30fb\u30af\u30c3\u30bf\u6cd5<\/h4>\n

\u30eb\u30f3\u30b2\u30fb\u30af\u30c3\u30bf\u6cd5\u306f \\(f()\\) \u306e\u5fae\u5206\u3092\u8a08\u7b97\u3059\u308b\u3053\u3068\u306a\u304f\u9ad8\u6b21\u306e\u30c6\u30a4\u30e9\u30fc\u5c55\u958b\u306b\u76f8\u5f53\u3059\u308b\u7cbe\u5ea6\u3092\u5f97\u308b\u65b9\u6cd5\u3067\u3059. \u3044\u304f\u3064\u304b\u306e \\(t\\) \u3068 \\(y\\) \u306b\u3064\u3044\u3066 \\(f(t, y)\\) \u3092\u8a08\u7b97\u3057, \u305d\u308c\u3089\u306e\u91cd\u307f\u4ed8\u304d\u5e73\u5747\u3092\u4f5c\u3063\u3066\u30c6\u30a4\u30e9\u30fc\u5c55\u958b\u306e \\(p\\) \u6b21\u306e\u9805\u307e\u3067\u4e00\u81f4\u3055\u305b\u308b\u3053\u3068\u306b\u3088\u308a \\(p\\) \u6b21\u306e\u516c\u5f0f\u3092\u4f5c\u308a\u307e\u3059.<\/p>\n

\u30eb\u30f3\u30b2\u30fb\u30af\u30c3\u30bf\u6cd5\u306e\u4e00\u822c\u5f62\u306f\u6b21\u306e\u3088\u3046\u306b\u306a\u308a\u307e\u3059.
\n\\[
\n\\begin{align}
\n& y_{n+1} = y_n + h\\sum_{i=1}^s b_ik_i \\\\
\n& k_i = f(t_n + c_ih, y_n + h\\sum_{j=1}^s a_{ij}k_j) \\\\
\n\\end{align}
\n\\]\n\\(a_{ij}, b_i, c_i\\) \u306f\u30d1\u30e9\u30e1\u30fc\u30bf\u3067\u3059. \u3053\u3053\u3067\u306f \\(a_{ij} = 0 (j >= i), c_i = \\sum_{j=1}^s a_{ij}\\) \u3068\u3057\u307e\u3059. \u306a\u304a, \\(s\\) \u306f\u6bb5\u6570\u3068\u547c\u3070\u308c\u307e\u3059.<\/p>\n

12.2.2.1 \u30db\u30a4\u30f3\u6cd5<\/h5>\n

\\(s = 2\\) \u306e\u5834\u5408, \u30c6\u30a4\u30e9\u30fc\u5c55\u958b\u5f0f\u306e 2 \u6b21\u306e\u9805\u307e\u3067\u4e00\u81f4\u3059\u308b\u3088\u3046\u306b\u3067\u304d\u3066\u6b21\u5f0f\u304c\u6210\u308a\u7acb\u3061\u307e\u3059.
\n\\[
\n\\begin{align}
\n& b_1 + b_2 = 1 \\\\
\n& b_2c_2 = 1\/2 \\\\
\n& b_2a_{21} = 1\/2 \\\\
\n\\end{align}
\n\\]\n\u672a\u77e5\u6570\u304c 4 \u500b\u3067\u5f0f\u304c 3 \u672c\u306a\u306e\u3067\u81ea\u7531\u5ea6\u304c 1 \u3042\u308a\u307e\u3059. \u3088\u304f\u4f7f\u308f\u308c\u308b\u3082\u306e\u3068\u3057\u3066\u306f, \\(c_2 = 1\\) \u3068\u3057\u305f\u6b21\u5f0f\u3092\u30db\u30a4\u30f3\u6cd5\u3068\u3044\u3044\u307e\u3059. \u901a\u5e38\u306f 2 \u6b21\u306e\u30eb\u30f3\u30b2\u30fb\u30af\u30c3\u30bf\u6cd5\u3068\u306f\u547c\u3073\u307e\u305b\u3093.
\n\\[
\n\\begin{align}
\n& y_{n+1} = y_n + h(k_1 + k_2)\/2 \\\\
\n& k_1 = f(t_n, y_n) \\\\
\n& k_2 = f(t_n + h, y_n + hk_1) \\\\
\n\\end{align}
\n\\]\n\u3053\u308c\u306f 2 \u6b21\u306e\u516c\u5f0f\u3067, \u8aa4\u5dee\u306f \\(O(h^3)\\) \u306b\u306a\u308a\u307e\u3059.<\/p>\n

12.2.2.2 4 \u6b21\u306e\u30eb\u30f3\u30b2\u30fb\u30af\u30c3\u30bf\u6cd5<\/h5>\n

\\(s = 4\\) \u306e\u5834\u5408, 4 \u6b21\u306e\u516c\u5f0f\u3092\u4f5c\u308b\u3053\u3068\u304c\u3067\u304d\u307e\u3059. \u3053\u306e\u5834\u5408, \u30d1\u30e9\u30e1\u30fc\u30bf\u306f 12 \u500b\u3067\u5f0f\u304c 8 \u672c\u306a\u306e\u3067\u81ea\u7531\u5ea6\u304c 4 \u306b\u306a\u308a, \u591a\u69d8\u306a\u516c\u5f0f\u3092\u4f5c\u308b\u3053\u3068\u304c\u3067\u304d\u307e\u3059. \u6700\u3082\u3088\u304f\u4f7f\u308f\u308c\u3066\u3044\u308b\u306e\u306f\u6b21\u5f0f\u3067\u3059.
\n\\[
\n\\begin{align}
\n& y_{n+1} = y_n + h(k_1 + 2k_2 + 2k_3 + k_4)\/6 \\\\
\n& k_1 = f(t_n, y_n) \\\\
\n& k_2 = f(t_n + h\/2, y_n + hk_1\/2) \\\\
\n& k_3 = f(t_n + h\/2, y_n + hk_2\/2) \\\\
\n& k_4 = f(t_n + h, y_n + hk_3) \\\\
\n\\end{align}
\n\\]\n\u3053\u308c\u306f, \u5358\u306b\u30eb\u30f3\u30b2\u30fb\u30af\u30c3\u30bf\u6cd5\u3068\u547c\u3070\u308c\u308b\u3053\u3068\u304c\u591a\u3044 4 \u6b21\u306e\u516c\u5f0f\u3067, \u8aa4\u5dee\u306f \\(O(h^5)\\) \u306b\u306a\u308a\u307e\u3059.<\/p>\n

12.2.3 \u57cb\u3081\u8fbc\u307f\u578b\u30eb\u30f3\u30b2\u30fb\u30af\u30c3\u30bf\u6cd5<\/h4>\n

\u3053\u3053\u307e\u3067\u793a\u3057\u305f\u89e3\u6cd5\u306e\u8aa4\u5dee\u306f \\(O(h^p)\\) \u306a\u306e\u3067 \\(h\\) \u3092\u5c0f\u3055\u304f\u3059\u308b\u307b\u3069\u7cbe\u5ea6\u304c\u4e0a\u304c\u308a\u307e\u3059. \u3057\u304b\u3057, \\(h\\) \u3092\u5c0f\u3055\u304f\u3059\u308b\u307b\u3069\u76ee\u7684\u306e\u89e3\u3092\u6c42\u3081\u308b\u307e\u3067\u306e \\(f()\\) \u306e\u8a08\u7b97\u56de\u6570\u304c\u5897\u3048\u308b\u3068\u3044\u3046\u30c8\u30ec\u30fc\u30c9\u30aa\u30d5\u306e\u95a2\u4fc2\u306b\u3042\u308a\u307e\u3059.<\/p>\n

\u3082\u3057, \u5f97\u3089\u308c\u305f\u7d50\u679c\u306e\u7cbe\u5ea6\u3092\u77e5\u308b\u3053\u3068\u304c\u3067\u304d\u308c\u3070, \u8981\u6c42\u7cbe\u5ea6\u306b\u5bfe\u3057\u3066\u9069\u5207\u306a(\u5fc5\u8981\u5341\u5206\u306a)\u30b9\u30c6\u30c3\u30d7\u5e45 \\(h\\) \u3092\u9078\u3076\u3053\u3068\u304c\u3067\u304d\u7121\u99c4\u306a\u8a08\u7b97\u6642\u9593\u3092\u8cbb\u3084\u3055\u306a\u304f\u3066\u3088\u304f\u306a\u308a\u307e\u3059.<\/p>\n

\u305d\u3053\u3067, \u89e3\u3092\u6c42\u3081\u308b\u3060\u3051\u3067\u306f\u306a\u304f\u8aa4\u5dee\u306e\u63a8\u5b9a\u3082\u52b9\u7387\u3088\u304f\u884c\u3048\u308b\u516c\u5f0f\u304c\u958b\u767a\u3055\u308c\u3066\u3044\u307e\u3059. \u57cb\u3081\u8fbc\u307f\u578b\u30eb\u30f3\u30b2\u30fb\u30af\u30c3\u30bf\u6cd5\u306f 1 \u3064\u306e\u516c\u5f0f\u3067 2 \u3064\u306e\u7570\u306a\u3063\u305f\u6b21\u6570\u306e\u8fd1\u4f3c\u89e3\u3092\u6c42\u3081\u308b\u3053\u3068\u304c\u3067\u304d\u308b\u3082\u306e\u3067\u3059. \u6b21\u6570\u306e\u9ad8\u3044\u65b9\u306e\u89e3\u306f\u8a08\u7b97\u7d50\u679c\u3068\u3057\u3066, \u3082\u3046\u4e00\u3064\u306f\u8aa4\u5dee\u63a8\u5b9a\u306e\u305f\u3081\u306b\u4f7f\u308f\u308c\u307e\u3059.<\/p>\n

\u3053\u308c\u3092\u4f7f\u3046\u3068\u63a8\u5b9a\u8aa4\u5dee\u3092\u3082\u3068\u306b\u30b9\u30c6\u30c3\u30d7\u5e45 \\(h\\) \u3092\u81ea\u52d5\u7684\u306b\u8abf\u7bc0\u3057\u306a\u304c\u3089\u89e3\u3092\u6c42\u3081\u308b\u300c\u9069\u5fdc\u30d7\u30ed\u30b0\u30e9\u30e0\u300d\u3092\u4f5c\u308b\u3053\u3068\u304c\u3067\u304d\u307e\u3059.<\/p>\n

\u4f8b\u3068\u3057\u3066, \u57cb\u3081\u8fbc\u307f\u578b\u30eb\u30f3\u30b2\u30fb\u30af\u30c3\u30bf\u6cd5\u306e\u516c\u5f0f\u3067\u3042\u308b 5(4)\u6b21 \u30eb\u30f3\u30b2\u30fb\u30af\u30c3\u30bf\u30fb\u30d5\u30a7\u30fc\u30eb\u30d9\u30eb\u30b0\u6cd5\u306e\u516c\u5f0f\u3092\u4e0b\u306b\u793a\u3057\u307e\u3059. \u3053\u308c\u3092\u4f7f\u3063\u305f\u9069\u5fdc\u30d7\u30ed\u30b0\u30e9\u30e0 RKF45 \u306f\u53e4\u304f\u304b\u3089\u5e83\u304f\u4f7f\u308f\u308c\u3066\u3044\u307e\u3059.
\n\\[
\n\\begin{align}
\n& y_{n+1} = y_n + h((16\/135)k_1 + (6656\/12825)k_3 + (28561\/56430)k_4 – (9\/50)k_5 + (2\/55)k_6) \\space (\u8a08\u7b97\u7528: 5\u6b21\u306e\u516c\u5f0f) \\\\
\n& y^*_{n+1} = y_n + h((25\/216)k_1 + (1408\/2565)k_3 + (2197\/4104)k_4 – (1\/5)k_5) \\space (\u8aa4\u5dee\u8a55\u4fa1\u7528: 4\u6b21\u306e\u516c\u5f0f) \\\\
\n& k_1 = f(t_n, y_n) \\\\
\n& k_2 = f(t_n + (1\/4)h, y_n + (1\/4)hk_1) \\\\
\n& k_3 = f(t_n + (3\/8)h, y_n + (1\/32)h(3k_1 + 9k_2)) \\\\
\n& k_4 = f(t_n + (12\/13)h, y_n + (1\/2197)h(1932k_1 – 7200k_2 + 7296k_3)) \\\\
\n& k_5 = f(t_n + h, y_n + h((439\/216)k_1 – 8k_2 + (3680\/513)k_3 – (845\/4104)k_4)) \\\\
\n& k_6 = f(t_n + (1\/2)h, y_n + h((-8\/27)k_1 + 2k_2 – (3544\/2565)k_3 + (1859\/4104)k_4 – (11\/40)k_5)) \\\\
\n\\end{align}
\n\\]\n\u8a08\u7b97\u7528\u306b\u306f 5 \u6b21\u306e\u516c\u5f0f\u3092\u4f7f\u3044\u307e\u3059. \\(h\\) \u306e\u81ea\u52d5\u8abf\u7bc0\u306f\u8aa4\u5dee\u8a55\u4fa1\u7528\u306e 4 \u6b21\u306e\u516c\u5f0f\u3068\u306e\u5dee \\(\\delta_{n+1} = ||y_{n+1} – y^*_{n+1}||\\) \u306e\u5024\u3092\u4f7f\u3063\u3066\u884c\u3044\u307e\u3059.<\/p>\n

12.2.4 \u5404\u89e3\u6cd5\u306e\u6bd4\u8f03<\/h4>\n

\u4e0a\u306b\u8aac\u660e\u3057\u305f\u5404\u89e3\u6cd5\u306b\u3088\u308a\u8a08\u7b97\u3092\u884c\u3063\u3066\u307f\u307e\u3059.<\/p>\n

\u4f8b\u984c 1<\/h5>\n

\u6b21\u306e\u521d\u671f\u5024\u554f\u984c\u3092\u89e3\u304f.
\n\\[
\n\\frac{dy}{dt} = 1\/(2 – t)^2 \\space (y(0) = 0.5)
\n\\]\n\u3053\u306e\u4f8b\u984c\u306f\u4ee3\u6570\u7684\u306b\u89e3\u3092\u6c42\u3081\u308b\u3053\u3068\u304c\u3067\u304d, \u6b21\u306e\u3088\u3046\u306b\u306a\u308a\u307e\u3059.
\n\\[
\ny = 1\/(2 – t)
\n\\]\n\u4e0b\u56f3\u3067\u306f\u6a2a\u8ef8\u306b\u8aa4\u5dee (\u53f3\u306b\u884c\u304f\u307b\u3069\u7cbe\u5ea6\u304c\u3088\u3044), \u7e26\u8ef8\u306b\u8a08\u7b97\u91cf (= \u95a2\u6570\u547c\u3073\u51fa\u3057\u56de\u6570 (\\(f(t, y)\\) \u3092\u4f55\u56de\u547c\u3073\u51fa\u3057\u305f\u304b)) \u3092\u5bfe\u6570\u76ee\u76db\u308a\u3067\u793a\u3057\u307e\u3059. \u30b0\u30e9\u30d5\u304c\u53f3\u4e0b\u306b\u3044\u304f\u307b\u3069\u6027\u80fd\u304c\u3088\u3044\u3053\u3068\u306b\u306a\u308a\u307e\u3059.<\/p>\n

\"\"<\/p>\n

\u30aa\u30a4\u30e9\u30fc\u6cd5\u306f 1 \u6b21, \u30db\u30a4\u30f3\u6cd5\u306f 2 \u6b21, \u30eb\u30f3\u30b2\u30fb\u30af\u30c3\u30bf\u6cd5\u306f 4 \u6b21\u306e\u516c\u5f0f\u3067\u3059. \u305d\u308c\u305e\u308c\u306f (\u5bfe\u6570\u30d7\u30ed\u30c3\u30c8\u3067) \u76f4\u7dda\u95a2\u4fc2\u3092\u793a\u3057\u307e\u3059\u304c, \u50be\u304d\u304c\u6b21\u6570\u3092\u53cd\u6620\u3057\u3066\u3044\u307e\u3059.<\/p>\n

Derkfa<\/strong> \u306f XLPack \u3067\u4f7f\u308f\u308c\u3066\u3044\u308b 5(4)\u6b21 \u30eb\u30f3\u30b2\u30fb\u30af\u30c3\u30bf\u30fb\u30d5\u30a7\u30fc\u30eb\u30d9\u30eb\u30b0\u6cd5\u306e\u9069\u5fdc\u30d7\u30ed\u30b0\u30e9\u30e0\u3067\u3059. \\(h\\) \u306e\u81ea\u52d5\u8abf\u7bc0\u306e\u305f\u3081\u306e\u30aa\u30fc\u30d0\u30fc\u30d8\u30c3\u30c9\u304c\u3042\u308b\u305f\u3081\u7d14\u7c8b\u306a 5 \u6b21\u306e\u516c\u5f0f\u3088\u308a\u306f\u3084\u3084\u52b9\u7387\u304c\u60aa\u304f\u306a\u308a\u307e\u3059.<\/p>\n

12.2.5 2 \u968e\u306e\u5e38\u5fae\u5206\u65b9\u7a0b\u5f0f<\/h4>\n

2 \u968e\u306e\u5e38\u5fae\u5206\u65b9\u7a0b\u5f0f\u3067\u6b21\u306e\u3088\u3046\u306a\u7279\u5225\u306a\u5f62\u304c\u73fe\u308c\u308b\u3053\u3068\u304c\u3088\u304f\u3042\u308a\u307e\u3059.
\n\\[
\ny” = f(y, t) \\space (y’ \u306b\u4f9d\u5b58\u3057\u306a\u3044)
\n\\]\n2 \u968e\u306e\u5e38\u5fae\u5206\u65b9\u7a0b\u5f0f\u306f 1 \u968e\u306e\u9023\u7acb\u5e38\u5fae\u5206\u65b9\u7a0b\u5f0f\u306e\u5f62\u306b\u3057\u3066\u89e3\u304f\u3053\u3068\u304c\u3067\u304d\u308b\u3068\u8aac\u660e\u3057\u307e\u3057\u305f\u304c, \u3053\u306e\u3088\u3046\u306a\u7279\u5225\u306a\u5f62\u306e\u5834\u5408\u306b\u306f 2 \u968e\u306e\u307e\u307e\u76f4\u63a5\u89e3\u304f\u516c\u5f0f, \u4f8b\u3048\u3070\u30cb\u30e5\u30b9\u30c8\u30ec\u30e0\u6cd5\u304c\u958b\u767a\u3055\u308c\u3066\u3044\u307e\u3059.<\/p>\n

12.3 XLPack \u3092\u4f7f\u3063\u305f\u5e38\u5fae\u5206\u65b9\u7a0b\u5f0f\u306e\u89e3\u304d\u65b9<\/h3>\n

Derkfa<\/strong> \u306f 5(4)\u6b21\u30eb\u30f3\u30b2\u30fb\u30af\u30c3\u30bf\u30fb\u30d5\u30a7\u30fc\u30eb\u30d9\u30eb\u30b0\u6cd5\u306b\u3088\u308b\u9069\u5fdc\u30d7\u30ed\u30b0\u30e9\u30e0\u3067, RKF45 \u3068\u307b\u307c\u540c\u3058\u3082\u306e\u3067\u3059\u304c\u5bc6\u51fa\u529b\u6a5f\u80fd (\u5f8c\u8ff0) \u3092\u8ffd\u52a0\u3057\u3066\u3044\u307e\u3059.<\/p>\n

Dopn43<\/strong> \u306f 4(3)\u6b21\u30eb\u30f3\u30b2\u30fb\u30af\u30c3\u30bf\u30fb\u30cb\u30e5\u30b9\u30c8\u30ec\u30e0\u6cd5\u306b\u3088\u308b\u9069\u5fdc\u30d7\u30ed\u30b0\u30e9\u30e0\u3067\u3059. \\(y’\\) \u306b\u4f9d\u5b58\u3057\u306a\u3044\u7279\u5225\u306a\u5f62\u306e 2 \u968e\u9023\u7acb\u5e38\u5fae\u5206\u65b9\u7a0b\u5f0f\u306b\u4f7f\u7528\u3057\u307e\u3059.<\/p>\n

Derkfa<\/strong> \u304a\u3088\u3073 Dopn43<\/strong> \u306f XLPack \u30bd\u30eb\u30d0\u30fc\u304b\u3089\u4f7f\u3046\u3053\u3068\u3082\u3067\u304d\u307e\u3059.<\/p>\n

12.3.1 \u5358\u4e00\u306e\u5e38\u5fae\u5206\u65b9\u7a0b\u5f0f<\/h4>\n

\u5358\u4e00\u306e (\u9023\u7acb\u3067\u306f\u306a\u3044) \u5e38\u5fae\u5206\u65b9\u7a0b\u5f0f\u306e\u4f8b\u3092\u4f8b\u984c 1 \u3092\u4f7f\u3063\u3066\u8aac\u660e\u3057\u307e\u3059.<\/p>\n

12.3.1.1 VBA \u30d7\u30ed\u30b0\u30e9\u30e0\u3092\u4f7f\u7528\u3057\u305f\u89e3\u304d\u65b9 (1)<\/h5>\n

Derkfa<\/strong> \u3092\u4f7f\u3063\u305f\u30d7\u30ed\u30b0\u30e9\u30e0\u4f8b\u3092\u793a\u3057\u307e\u3059.<\/p>\n

\n
Const Ndata = 19, RowR = 5, ColR = 1\r\nDim Neval As Long\r\n\r\nSub F(N As Long, T As Double, Y() As Double, Yp() As Double)\r\n    Yp(0) = 1 \/ (2 - T) ^ 2\r\n    Neval = Neval + 1\r\nEnd Sub\r\n\r\nSub Start()\r\n    Const N = 1\r\n    Dim Y(N - 1) As Double, T As Double, Tout As Double, Tend As Double\r\n    Dim RTol(0) As Double, ATol(0) As Double\r\n    Dim Mode As Long, Info As Long, I As Long\r\n    RTol(0) = 0.00000001 '** 1.0e-8\r\n    ATol(0) = RTol(0)\r\n    Mode = 2\r\n    T = Cells(RowR, ColR)\r\n    Y(0) = Cells(RowR, ColR + 1)\r\n    Tend = Cells(RowR + Ndata, ColR)\r\n    Info = 0\r\n    For I = 1 To Ndata\r\n        Tout = Cells(RowR + I, ColR)\r\n        Neval = 0\r\n        Call Derkfa(N, AddressOf F, T, Y(), Tout, Tend, RTol(), ATol(), Mode, Info)\r\n        If Info <> 0 And Info <> Mode Then\r\n            Cells(RowR + I, ColR + 2) = \"Error: \" + Str(Info)\r\n            Exit For\r\n        End If\r\n        Cells(RowR + I, ColR + 1) = Y(0)\r\n        Cells(RowR + I, ColR + 2) = Neval\r\n    Next\r\nEnd Sub<\/code><\/pre>\n<\/div>\n
Derkfa<\/strong> \u306f\u76ee\u6a19\u7cbe\u5ea6\u3092\u4e0e\u3048\u308b\u3068\u30b9\u30c6\u30c3\u30d7\u5e45\u3092\u81ea\u52d5\u7684\u306b\u8abf\u7bc0\u3057\u3066\u89e3\u3092\u6c42\u3081\u307e\u3059. \u5fae\u5206\u65b9\u7a0b\u5f0f\u3092\u30b5\u30d6\u30eb\u30fc\u30c1\u30f3 F \u3068\u3057\u3066\u7528\u610f\u3057, \u521d\u671f\u5024\u3068\u8981\u6c42\u7cbe\u5ea6\u3092\u6307\u5b9a\u3057\u3066\u547c\u3073\u51fa\u3059\u3068\u89e3 (\\(y\\) \u306e\u5024) \u3092\u6c42\u308b\u3053\u3068\u304c\u3067\u304d\u307e\u3059.<\/p>\n
\u3053\u306e\u5834\u5408, \u9023\u7acb\u65b9\u7a0b\u5f0f\u3067\u306f\u306a\u3044\u306e\u3067 Y() \u3068 Yp() \u306f\u914d\u5217\u3067\u306f\u306a\u304f\u3066\u5358\u7d14\u306a\u5909\u6570\u3067\u3088\u3055\u305d\u3046\u3067\u3059\u304c, VBA \u306e\u5f15\u6570\u306e\u6574\u5408\u6027\u306e\u305f\u3081\u306b\u914d\u5217\u3067\u5ba3\u8a00\u3057\u3066 Y(0) \u3068 Yp(0) \u306e\u307f\u3092\u4f7f\u7528\u3057\u307e\u3059.<\/p>\n
Mode = 2 \u306f\u901a\u5e38\u306e\u52d5\u4f5c\u30e2\u30fc\u30c9\u3067, T \u306e\u521d\u671f\u5024\u304b\u3089\u59cb\u3081\u3066\u6700\u7d42\u5024 Tend \u307e\u3067\u8a08\u7b97\u3092\u884c\u3044\u307e\u3059. \u30b9\u30c6\u30c3\u30d7\u5e45\u306f\u8981\u6c42\u7cbe\u5ea6\u306b\u5408\u308f\u305b\u3066\u6700\u9069\u306b\u306a\u308b\u3088\u3046\u306b\u81ea\u52d5\u7684\u306b\u8abf\u7bc0\u3055\u308c\u307e\u3059. \u9014\u4e2d Tout \u304c\u6700\u9069\u30b9\u30c6\u30c3\u30d7\u5e45\u3088\u308a\u3082\u8fd1\u304f\u306b\u304d\u305f\u3089, \u6b21\u306e\u30b9\u30c6\u30c3\u30d7\u304c\u3061\u3087\u3046\u3069 Tout \u306b\u304f\u308b\u3088\u3046\u306b\u30b9\u30c6\u30c3\u30d7\u5e45\u3092\u5909\u66f4\u3057\u3066\u304b\u3089\u8a08\u7b97\u3057\u9014\u4e2d\u7d4c\u904e\u3092\u8fd4\u3059\u305f\u3081\u306b\u623b\u308a\u307e\u3059. \u305d\u306e\u5f8c Tout \u3092\u518d\u8a2d\u5b9a\u3057\u3066\u518d\u547c\u3073\u51fa\u3057\u3059\u308b\u3053\u3068\u306b\u3088\u308a Tend \u307e\u3067\u306e\u8a08\u7b97\u3092\u7d99\u7d9a\u3057\u307e\u3059.<\/p>\n
Neval \u306f\u95a2\u6570\u8a55\u4fa1\u56de\u6570\u3092\u30ab\u30a6\u30f3\u30c8\u3059\u308b\u305f\u3081\u3060\u3051\u306e\u5909\u6570\u3067\u3059.<\/p>\n
\u3053\u306e\u30d7\u30ed\u30b0\u30e9\u30e0\u3092\u5b9f\u884c\u3059\u308b\u3068, \u6b21\u306e\u7d50\u679c\u304c\u5f97\u3089\u308c\u307e\u3057\u305f. <\/p>\n
\"\"<\/p>\n
\u8981\u6c42\u7cbe\u5ea6\u306e 8 \u6841\u3067\u6b63\u3057\u3044\u7d50\u679c\u304c\u5f97\u3089\u308c\u3066\u3044\u307e\u3059. \u306a\u304a, \u30b9\u30bf\u30fc\u30c8\u76f4\u5f8c\u3068 \\(t\\) \u304c\u5927\u304d\u304f\u306a\u308a\u5909\u5316\u304c\u6025\u306b\u306a\u3063\u3066\u304d\u305f\u3068\u304d\u306b, \u7cbe\u5ea6\u3092\u4fdd\u3064\u305f\u3081\u306b\u95a2\u6570\u8a55\u4fa1\u56de\u6570\u304c\u5897\u3048\u3066\u3044\u308b(\u30b9\u30c6\u30c3\u30d7\u5e45\u3092\u5c0f\u3055\u304f\u3057\u305f)\u306e\u304c\u308f\u304b\u308a\u307e\u3059.<\/p>\n
12.3.1.2 VBA \u30d7\u30ed\u30b0\u30e9\u30e0\u3092\u4f7f\u7528\u3057\u305f\u89e3\u304d\u65b9 (2)<\/h5>\n
\u30ea\u30d0\u30fc\u30b9\u30b3\u30df\u30e5\u30cb\u30b1\u30fc\u30b7\u30e7\u30f3\u7248 (RCI) \u306e Derkfa_r<\/strong> \u3092\u4f7f\u3063\u305f\u30d7\u30ed\u30b0\u30e9\u30e0\u4f8b\u3092\u793a\u3057\u307e\u3059.<\/p>\n
\n
Sub Start_r()\r\n    Const N = 1\r\n    Dim Y(N - 1) As Double, T As Double, Tout As Double, Tend As Double\r\n    Dim RTol(0) As Double, ATol(0) As Double\r\n    Dim Mode As Long, Info As Long, I As Long\r\n    Dim TT As Double, YY(0) As Double, YYp(0) As Double, IRev As Long\r\n    RTol(0) = 0.00000001 '** 1.0e-8\r\n    ATol(0) = RTol(0)\r\n    Mode = 2\r\n    T = Cells(RowR, ColR)\r\n    Y(0) = Cells(RowR, ColR + 1)\r\n    Tend = Cells(RowR + Ndata, ColR)\r\n    Info = 0\r\n    For I = 1 To Ndata\r\n        Tout = Cells(RowR + I, ColR)\r\n        Neval = 0\r\n        IRev = 0\r\n        Do\r\n            Call Derkfa_r(N, T, Y(), Tout, Tend, RTol(), ATol(), Mode, Info, TT, YY(), YYp(), IRev)\r\n            If IRev = 1 Then\r\n                YYp(0) = 1 \/ (2 - TT) ^ 2\r\n                Neval = Neval + 1\r\n            End If\r\n        Loop While IRev <> 0\r\n        If Info <> 0 And Info <> Mode Then\r\n            Cells(RowR + I, ColR + 2) = \"Error: \" + Str(Info)\r\n            Exit For\r\n        End If\r\n        Cells(RowR + I, ColR + 1) = Y(0)\r\n        Cells(RowR + I, ColR + 2) = Neval\r\n    Next\r\nEnd Sub<\/code><\/pre>\n<\/div>\n
\u76ee\u7684\u95a2\u6570\u3092\u5916\u90e8\u30b5\u30d6\u30eb\u30fc\u30c1\u30f3\u3068\u3057\u3066\u4e0e\u3048\u308b\u306e\u3067\u306f\u306a\u304f, IRev = 1 \u306e\u3068\u304d\u306b TT \u304a\u3088\u3073 YY \u306e\u5024\u3092\u4f7f\u3063\u3066\u95a2\u6570\u5024\u3092\u8a08\u7b97\u3057 YYp() \u306b\u5165\u308c\u3066\u518d\u5ea6 Derkfa_r<\/strong> \u3092\u547c\u3073\u51fa\u3057\u307e\u3059. RCI \u306e\u8a73\u7d30\u306b\u3064\u3044\u3066\u306f \u3053\u3061\u3089<\/a> \u3092\u53c2\u7167\u3057\u3066\u304f\u3060\u3055\u3044. <\/p>\n
\u3053\u306e\u30d7\u30ed\u30b0\u30e9\u30e0\u3092\u5b9f\u884c\u3059\u308b\u3068\u4e0a\u3068\u540c\u3058\u7d50\u679c\u304c\u5f97\u3089\u308c\u307e\u3059.<\/p>\n
12.3.1.3 VBA \u30d7\u30ed\u30b0\u30e9\u30e0\u3092\u4f7f\u7528\u3057\u305f\u89e3\u304d\u65b9 (3)<\/h5>\n
Derkfa<\/strong> \u306b\u3088\u308b\u5bc6\u51fa\u529b\u6a5f\u80fd\u3092\u4f7f\u3063\u305f\u4f8b\u3092\u793a\u3057\u307e\u3059. \u5bc6\u51fa\u529b\u6a5f\u80fd\u3068\u306f, \u9014\u4e2d\u7d4c\u904e\u51fa\u529b\u306e\u305f\u3081\u306b\u623b\u308b\u3068\u304d\u306b\u30b9\u30c6\u30c3\u30d7\u5e45\u306e\u5909\u66f4\u3092\u884c\u308f\u305a, \u6700\u9069\u30b9\u30c6\u30c3\u30d7\u5e45\u3092\u30ad\u30fc\u30d7\u3057\u3066\u30c8\u30fc\u30bf\u30eb\u306e\u95a2\u6570\u8a55\u4fa1\u56de\u6570\u3092\u6e1b\u3089\u3059\u6a5f\u80fd\u3067\u3059. \u9014\u4e2d\u7d4c\u904e\u306e\u89e3\u306f\u88dc\u9593\u306b\u3088\u308a\u6c42\u3081\u3089\u308c\u307e\u3059.<\/p>\n
\u6ce8 – \u5bc6\u51fa\u529b\u306b\u3064\u3044\u3066\u8a73\u3057\u304f\u306f \u3053\u3061\u3089<\/a> \u3092\u53c2\u7167\u3057\u3066\u304f\u3060\u3055\u3044(\u76ee\u6b21\u306e\u4e0b\u306e\u4e2d\u592e\u4ed8\u8fd1\u306e\u300c\u5bc6\u51fa\u529b\u300d\u30bf\u30d6\u3092\u30af\u30ea\u30c3\u30af\u3057\u3066\u304f\u3060\u3055\u3044).<\/p>\n
\u5bc6\u51fa\u529b\u6a5f\u80fd\u3092\u4f7f\u7528\u3059\u308b\u305f\u3081\u306b\u306f\u30d7\u30ed\u30b0\u30e9\u30e0\u306b\u304a\u3044\u3066 Mode = 2 \u306e\u4ee3\u308f\u308a\u306b Mode = 3 \u3068\u3059\u308b\u3060\u3051\u3067\u3059. 12.3.1.1 \u306e\u30d7\u30ed\u30b0\u30e9\u30e0\u3067 Mode = 3 \u3068\u5909\u66f4\u3057\u305f\u3082\u306e\u3092\u5b9f\u884c\u3059\u308b\u3068\u6b21\u306e\u7d50\u679c\u304c\u5f97\u3089\u308c\u307e\u3057\u305f.<\/p>\n
\"\"<\/p>\n
\u95a2\u6570\u8a55\u4fa1\u56de\u6570 Neval = 0 \u306e\u533a\u9593\u304c\u3042\u308a\u307e\u3059\u304c, \u3053\u3046\u3044\u3046\u3068\u3053\u308d\u3067\u306f\u88dc\u9593\u306b\u3088\u308a\u89e3\u3092\u6c42\u3081\u3066\u3044\u308b\u3053\u3068\u304c\u306f\u3063\u304d\u308a\u3068\u308f\u304b\u308a\u307e\u3059.<\/p>\n
\u5bc6\u51fa\u529b\u6a5f\u80fd\u306f, \u4f8b\u3048\u3070\u6ed1\u3089\u304b\u306a\u30b0\u30e9\u30d5\u3092\u63cf\u304f\u305f\u3081\u306a\u3069, \u8981\u6c42\u7cbe\u5ea6\u304c\u9ad8\u304f\u306f\u306a\u3044\u304c\u9014\u4e2d\u7d4c\u904e\u306e\u51fa\u529b\u9593\u9694\u3092\u72ed\u304f\u3057\u305f\u3044\u5834\u5408\u306a\u3069\u306b\u52b9\u679c\u3092\u767a\u63ee\u3057\u307e\u3059.<\/p>\n
12.3.1.4 \u30bd\u30eb\u30d0\u30fc\u3092\u4f7f\u7528\u3057\u305f\u89e3\u304d\u65b9<\/h5>\n
XLPack \u30bd\u30eb\u30d0\u30fc\u30a2\u30c9\u30a4\u30f3\u306e\u300c1 \u968e\u5e38\u5fae\u5206\u65b9\u7a0b\u5f0f\u300d\u3092\u4f7f\u3063\u3066\u89e3\u304f\u3053\u3068\u3082\u3067\u304d\u307e\u3059. B6\u30bb\u30eb\u306b\u6570\u5f0f (=1\/(2-B4)^2) \u304c\u5165\u529b\u3055\u308c\u3066\u3044\u307e\u3059.<\/p>\n
\"\"<\/p>\n
\u30bd\u30eb\u30d0\u30fc\u306b\u3064\u3044\u3066\u306f \u3053\u3061\u3089<\/a> \u3082\u53c2\u7167\u304f\u3060\u3055\u3044.<\/p>\n
12.3.2 \u9023\u7acb\u5e38\u5fae\u5206\u65b9\u7a0b\u5f0f<\/h3>\n
\u9023\u7acb\u5e38\u5fae\u5206\u65b9\u7a0b\u5f0f\u306e\u4f8b\u3092\u8aac\u660e\u3057\u307e\u3059.<\/p>\n
\u4f8b\u984c 2<\/h4>\n
\u6b21\u306e\u521d\u671f\u5024\u554f\u984c\u3092\u89e3\u3051.
\n\\[
\n\\frac{d^2y1}{dt^2} = -y1\/r \\\\
\n\\frac{d^2y2}{dt^2} = -y2\/r \\\\
\n\\]\n\u305f\u3060\u3057,
\n\\[
\nr = (y1^2 + y2^2)^{3\/2} \\\\
\n\\]\n\\(t = 0\\) \u306b\u304a\u3044\u3066,
\n\\[
\ny_1 = 1, y_2 = 0, dy_1\/dt = 0, dy_2\/dt = \\sqrt{3}
\n\\]\n
12.3.2.1 VBA \u30d7\u30ed\u30b0\u30e9\u30e0\u3092\u4f7f\u7528\u3057\u305f\u89e3\u304d\u65b9 (1)<\/h5>\n
\u3053\u306e\u4f8b\u984c\u306f 2 \u5143\u9023\u7acb\u65b9\u7a0b\u5f0f\u3067\u3059\u304c, Derkfa<\/strong> \u3092\u4f7f\u3063\u3066\u89e3\u304f\u305f\u3081\u306b\u306f 12.1.3 \u3067\u8aac\u660e\u3057\u305f\u3088\u3046\u306b, \\(y_3 = y_1′, y_4 = y_2’\\) \u306e\u3088\u3046\u306b\u3057\u3066 4 \u5143\u9023\u7acb\u65b9\u7a0b\u5f0f\u3068\u3057\u3066\u6271\u3044\u307e\u3059.<\/p>\n
\u30d7\u30ed\u30b0\u30e9\u30e0\u4f8b\u3092\u793a\u3057\u307e\u3059.<\/p>\n
\n
Const Ndata = 19, RowR = 5, ColR = 1\r\nDim Neval As Long\r\n\r\nSub F2(N As Long, T As Double, Y() As Double, Yp() As Double)\r\n    Dim R As Double\r\n    R = (Y(0) ^ 2 + Y(1) ^ 2) ^ (3 \/ 2)\r\n    Yp(0) = Y(2)\r\n    Yp(1) = Y(3)\r\n    Yp(2) = -Y(0) \/ R\r\n    Yp(3) = -Y(1) \/ R\r\n    Neval = Neval + 1\r\nEnd Sub\r\n\r\nSub Start_2()\r\n    Const N = 4\r\n    Dim Y(N - 1) As Double, T As Double, Tout As Double, Tend As Double\r\n    Dim RTol(0) As Double, ATol(0) As Double\r\n    Dim Mode As Long, Info As Long, I As Long\r\n    RTol(0) = 0.00000001 '**1e-8\r\n    ATol(0) = RTol(0)\r\n    Mode = 2\r\n    T = Cells(RowR, ColR)\r\n    Y(0) = Cells(RowR, ColR + 1)\r\n    Y(1) = Cells(RowR, ColR + 2)\r\n    Y(2) = Cells(RowR, ColR + 3)\r\n    Y(3) = Cells(RowR, ColR + 4)\r\n    Tend = Cells(RowR + Ndata, ColR)\r\n    Info = 0\r\n    For I = 1 To Ndata\r\n        Tout = Cells(RowR + I, ColR)\r\n        Neval = 0\r\n        Call Derkfa(N, AddressOf F2, T, Y(), Tout, Tend, RTol(), ATol(), Mode, Info)\r\n        If Info <> 0 And Info <> Mode Then\r\n            Cells(RowR + I, ColR + 2) = \"Error: \" + Str(Info)\r\n            Exit For\r\n        End If\r\n        Cells(RowR + I, ColR + 1) = Y(0)\r\n        Cells(RowR + I, ColR + 2) = Y(1)\r\n        Cells(RowR + I, ColR + 3) = Y(2)\r\n        Cells(RowR + I, ColR + 4) = Y(3)\r\n        Cells(RowR + I, ColR + 5) = Neval\r\n    Next\r\nEnd Sub<\/code><\/pre>\n<\/div>\n
\u9023\u7acb\u3067\u306f\u306a\u3044\u5358\u4e00\u306e\u5e38\u5fae\u5206\u65b9\u7a0b\u5f0f(\u4f8b\u984c 1)\u306e\u5834\u5408\u3068\u9055\u3046\u306e\u306f\u914d\u5217 Y() \u3068 Yp() \u306b\u8907\u6570(\u3053\u306e\u5834\u5408\u306f 4 \u3064)\u306e\u5f0f\u306e\u5024\u3092\u8a2d\u5b9a\u3059\u308b\u3053\u3068\u3060\u3051\u3067\u3059.<\/p>\n
\u3053\u306e\u30d7\u30ed\u30b0\u30e9\u30e0\u3092\u5b9f\u884c\u3059\u308b\u3068, \u6b21\u306e\u7d50\u679c\u304c\u5f97\u3089\u308c\u307e\u3057\u305f.<\/p>\n
\"\"<\/p>\n
12.3.2.2 VBA \u30d7\u30ed\u30b0\u30e9\u30e0\u3092\u4f7f\u7528\u3057\u305f\u89e3\u304d\u65b9 (2)<\/h5>\n
\u4f8b\u984c 2 \u306b\u306f 1 \u968e\u306e\u9805\u304c\u73fe\u308c\u306a\u3044\u306e\u3067 Dopn43<\/strong> \u3092\u4f7f\u3046\u3053\u3068\u304c\u3067\u304d, 2 \u5143\u9023\u7acb\u65b9\u7a0b\u5f0f\u306e\u307e\u307e\u6271\u3046\u3053\u3068\u304c\u3067\u304d\u307e\u3059. \u30d7\u30ed\u30b0\u30e9\u30e0\u4f8b\u3092\u793a\u3057\u307e\u3059.<\/p>\n
\n
Const Ndata = 19, RowR = 5, ColR = 1\r\nDim Neval As Long\r\n\r\nSub F3(N As Long, T As Double, Y() As Double, Yp2() As Double)\r\n    Dim R As Double\r\n    R = (Y(0) ^ 2 + Y(1) ^ 2) ^ (3 \/ 2)\r\n    Yp2(0) = -Y(0) \/ R\r\n    Yp2(1) = -Y(1) \/ R\r\n    Neval = Neval + 1\r\nEnd Sub\r\n\r\nSub Start_3()\r\n    Const N = 2\r\n    Dim T As Double, Y(N - 1) As Double, Yp(N - 1) As Double, Tout As Double, Tend As Double\r\n    Dim RTol(0) As Double, ATol(0) As Double\r\n    Dim Mode As Long, Info As Long, I As Long\r\n    RTol(0) = 0.00000001 '1.0e-8\r\n    ATol(0) = RTol(0)\r\n    Mode = 2\r\n    T = Cells(RowR, ColR)\r\n    Y(0) = Cells(RowR, ColR + 1)\r\n    Y(1) = Cells(RowR, ColR + 2)\r\n    Yp(0) = Cells(RowR, ColR + 3)\r\n    Yp(1) = Cells(RowR, ColR + 4)\r\n    Tend = Cells(RowR + Ndata, ColR)\r\n    Tend = 12\r\n    Info = 0\r\n    For I = 1 To Ndata\r\n        Tout = Cells(RowR + I, ColR)\r\n        Neval = 0\r\n        Call Dopn43(N, AddressOf F3, T, Y(), Yp(), Tout, Tend, RTol(), ATol(), Mode, Info)\r\n        If Info <> 0 And Info <> Mode Then\r\n            Cells(RowR + I, ColR + 2) = \"Error: \" + Str(Info)\r\n            Exit For\r\n        End If\r\n        Cells(RowR + I, ColR + 1) = Y(0)\r\n        Cells(RowR + I, ColR + 2) = Y(1)\r\n        Cells(RowR + I, ColR + 3) = Yp(0)\r\n        Cells(RowR + I, ColR + 4) = Yp(1)\r\n        Cells(RowR + I, ColR + 5) = Neval\r\n    Next\r\nEnd Sub<\/code><\/pre>\n<\/div>\n
Dopn43<\/strong> \u306f\u968e\u6570\u304c\u7570\u306a\u308b\u3053\u3068\u4ee5\u5916, Derkfa \u3068\u540c\u3058\u4f7f\u3044\u65b9\u306b\u306a\u308a\u307e\u3059.<\/p>\n
\u3053\u306e\u30d7\u30ed\u30b0\u30e9\u30e0\u3092\u5b9f\u884c\u3059\u308b\u3068, \u6b21\u306e\u7d50\u679c\u304c\u5f97\u3089\u308c\u307e\u3057\u305f. <\/p>\n
\"\"<\/p>\n
12.3.2.3 \u30bd\u30eb\u30d0\u30fc\u3092\u4f7f\u7528\u3057\u305f\u89e3\u304d\u65b9<\/h5>\n
XLPack \u30bd\u30eb\u30d0\u30fc\u30a2\u30c9\u30a4\u30f3\u306e\u300c2 \u968e\u5e38\u5fae\u5206\u65b9\u7a0b\u5f0f\u300d\u3092\u4f7f\u3063\u3066\u89e3\u304f\u3053\u3068\u3082\u3067\u304d\u307e\u3059.<\/p>\n
\"\"<\/p>\n
\u30bd\u30eb\u30d0\u30fc\u306b\u3064\u3044\u3066\u306f \u3053\u3061\u3089<\/a> \u3082\u53c2\u7167\u304f\u3060\u3055\u3044.<\/p>\n
\u306a\u304a, \u30c0\u30a6\u30f3\u30ed\u30fc\u30c9\u3067\u304d\u308b\u4f8b\u984c\u30ef\u30fc\u30af\u30b7\u30fc\u30c8\u306b\u306f\u30bd\u30eb\u30d0\u30fc\u30a2\u30c9\u30a4\u30f3\u306e\u300c1 \u968e\u5e38\u5fae\u5206\u65b9\u7a0b\u5f0f\u300d\u3092\u4f7f\u3063\u305f\u4f8b\u3082\u5165\u3063\u3066\u3044\u307e\u3059.<\/p>\n","protected":false},"excerpt":{"rendered":"
\u5e38\u5fae\u5206\u65b9\u7a0b\u5f0f\u306e\u521d\u671f\u5024\u554f\u984c\u306e\u89e3\u6cd5\u3068XLPack\u3092\u4f7f\u3063\u305f\u89e3\u304d\u65b9<\/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":[10],"tags":[],"class_list":["post-4706","post","type-post","status-publish","format-standard","hentry","category-tutorial"],"aioseo_notices":[],"_links":{"self":[{"href":"https:\/\/www.ktech.biz\/jp\/wp-json\/wp\/v2\/posts\/4706","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=4706"}],"version-history":[{"count":5,"href":"https:\/\/www.ktech.biz\/jp\/wp-json\/wp\/v2\/posts\/4706\/revisions"}],"predecessor-version":[{"id":5074,"href":"https:\/\/www.ktech.biz\/jp\/wp-json\/wp\/v2\/posts\/4706\/revisions\/5074"}],"wp:attachment":[{"href":"https:\/\/www.ktech.biz\/jp\/wp-json\/wp\/v2\/media?parent=4706"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/www.ktech.biz\/jp\/wp-json\/wp\/v2\/categories?post=4706"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/www.ktech.biz\/jp\/wp-json\/wp\/v2\/tags?post=4706"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}