{"id":4782,"date":"2025-03-31T16:53:59","date_gmt":"2025-03-31T07:53:59","guid":{"rendered":"https:\/\/www.ktech.biz\/jp\/?p=4782"},"modified":"2026-07-19T12:47:06","modified_gmt":"2026-07-19T03:47:06","slug":"11-quad","status":"publish","type":"post","link":"https:\/\/www.ktech.biz\/jp\/tutorial\/11-quad\/","title":{"rendered":"11. \u6570\u5024\u7a4d\u5206"},"content":{"rendered":"\n

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

\\(f(x)\\) \u306f\u533a\u9593 \\([a, b]\\) \u4e0a\u3067\u5b9a\u7fa9\u3055\u308c\u305f\u95a2\u6570\u3068\u3057, \u305d\u306e\u5b9a\u7a4d\u5206
\n\\[
\nI = \\int_a^b f(x) dx
\n\\]\n\u306e\u8fd1\u4f3c\u8a08\u7b97\u6cd5\u3092\u6570\u5024\u7a4d\u5206\u6cd5 (numerical intergation) \u307e\u305f\u306f \u6c42\u7a4d\u6cd5 (quadrature) \u3068\u3044\u3044\u307e\u3059.<\/p>\n

\u6570\u5024\u7a4d\u5206\u30d7\u30ed\u30b0\u30e9\u30e0\u306f\u5927\u304d\u304f 2 \u7a2e\u985e\u304c\u3042\u308a\u307e\u3059. 1 \u3064\u306f\u95a2\u6570\u5165\u529b\u30bf\u30a4\u30d7\u3067, \u88ab\u7a4d\u5206\u95a2\u6570\u306e\u5024\u304c\u8981\u6c42\u306b\u5fdc\u3058\u3066\u533a\u9593\u5185\u306e\u4efb\u610f\u306e\u70b9\u3067\u8a08\u7b97\u3067\u304d\u308b\u30d7\u30ed\u30b0\u30e9\u30e0\u3068\u3057\u3066\u4e0e\u3048\u3089\u308c\u308b\u5834\u5408\u3067\u3059. \u3082\u3046 1 \u3064\u306f\u30c7\u30fc\u30bf\u5165\u529b\u30bf\u30a4\u30d7\u3067, \u88ab\u7a4d\u5206\u95a2\u6570\u306e\u5024\u304c\u3068\u3073\u3068\u3073\u306e\u70b9\u306b\u304a\u3051\u308b\u30c7\u30fc\u30bf\u3068\u3057\u3066\u4e0e\u3048\u3089\u308c\u308b\u5834\u5408\u3067\u3059.<\/p>\n

11.2 \u57fa\u672c\u7684\u306a\u7a4d\u5206\u516c\u5f0f<\/h3>\n

\u6b21\u306e\u7a4d\u5206 I \u3092\u6570\u5024\u7a4d\u5206\u306b\u3088\u308a\u6c42\u3081\u308b\u4f8b\u3092\u4f7f\u3063\u3066\u8aac\u660e\u3057\u307e\u3059.
\n\\[
\n\\begin{align}
\n& f(x) = e^x cos(x) \\\\
\n& I = \\int_{0.2}^1 f(x) dx = 1.15816896857196 \\\\
\n\\end{align}
\n\\]\n

\u6700\u3082\u7c21\u5358\u306a\u65b9\u6cd5\u306f, \u7a4d\u5206\u7bc4\u56f2\u306e\u4e0b\u7aef\u3068\u4e0a\u7aef\u3092\u7d50\u3076\u76f4\u7dda\u3067\u56f2\u307e\u308c\u308b\u53f0\u5f62\u306e\u9818\u57df\u3067\u8fd1\u4f3c\u3059\u308b\u65b9\u6cd5\u3067\u3059. \u8a08\u7b97\u5f0f\u306f\u6b21\u306e\u3088\u3046\u306b\u306a\u308a\u307e\u3059.
\n\\[
\nI = \\frac{b – a}{2}(f(a) + f(b))
\n\\]\n\u3053\u308c\u3092\u53f0\u5f62\u5247\u3068\u3088\u3073\u307e\u3059.<\/p>\n

\u6b21\u306b, \u3082\u3063\u3068\u7cbe\u5ea6\u3092\u3088\u304f\u3059\u308b\u305f\u3081\u306b \\(a\\) \u3068 \\(b\\) \u306b\u52a0\u3048\u4e2d\u70b9 \\(x_1 = (a + b)\/2\\) \u306e 3 \u70b9\u3092\u4f7f\u3063\u3066, \u7a4d\u5206\u7bc4\u56f2\u306e\u4e0b\u7aef\u3068\u4e0a\u7aef\u3092\u7d50\u3073\u4e2d\u70b9\u3092\u901a\u308b\u653e\u7269\u7dda\u3067\u56f2\u307e\u308c\u308b\u9818\u57df\u3067\u8fd1\u4f3c\u3059\u308b\u65b9\u6cd5\u304c\u8003\u3048\u3089\u308c\u307e\u3059. \u8a08\u7b97\u5f0f\u306f\u6b21\u306e\u3088\u3046\u306b\u306a\u308a\u307e\u3059.
\n\\[
\nI = \\frac{b – a}{6}(f(a) + 4f(x_1) + f(b))
\n\\]\n\u3053\u308c\u3092\u30b7\u30f3\u30d7\u30bd\u30f3\u5247\u3068\u3088\u3073\u307e\u3059.<\/p>\n

\u30b7\u30f3\u30d7\u30bd\u30f3\u5247\u3088\u308a\u3082\u3055\u3089\u306b\u7cbe\u5ea6\u306e\u826f\u3044\u65b9\u6cd5\u3068\u3057\u3066\u30ac\u30a6\u30b9\u516c\u5f0f (\u30ac\u30a6\u30b9\u30fb\u30eb\u30b8\u30e3\u30f3\u30c9\u30eb\u516c\u5f0f\u3068\u3082\u3044\u3046) \u304c\u3042\u308a\u307e\u3059.<\/p>\n

\u3053\u308c\u306f a \u3068 b \u306e\u9593\u306e\u30eb\u30b8\u30e3\u30f3\u30c9\u30eb\u591a\u9805\u5f0f\u306e N \u500b\u306e\u30bc\u30ed\u70b9\u3092\u6a19\u672c\u70b9\u3068\u3057\u3066\u88dc\u9593\u3057\u3066\u8fd1\u4f3c\u3059\u308b\u65b9\u6cd5\u3067, \u8a08\u7b97\u5f0f\u306f\u6b21\u306e\u3088\u3046\u306b\u306a\u308a\u307e\u3059.
\n\\[
\nI_N = \\frac{b – a}{2} \\sum_{i=1}^N w_i f(\\frac{a + b}{2} + \\frac{b – a}{2} x_i)
\n\\]\n\u3053\u306e\u516c\u5f0f\u306e \\(x_i\\) \u3068 \\(w_i\\) \u306f\u6559\u79d1\u66f8\u3084\u516c\u5f0f\u96c6\u306e\u6570\u8868\u306b\u8f09\u3063\u3066\u3044\u307e\u3059. n = 2 \u3068 3 \u306e\u5834\u5408\u306b\u306f\u6b21\u306e\u3088\u3046\u306b\u306a\u308a\u307e\u3059. \u305d\u308c\u305e\u308c, 2 \u70b9\u30ac\u30a6\u30b9\u516c\u5f0f, 3 \u70b9\u30ac\u30a6\u30b9\u516c\u5f0f\u3068\u3088\u3073\u307e\u3059.
\n\\[
\n\\begin{array}{ccccccc}
\nn & x_1 & x_2 & x_3 & w_1 & w_2 & w_3 \\\\ \\hline
\n2 & -\\sqrt{1\/3} & \\sqrt{1\/3} & – & 1 & 1 & – \\\\
\n3 & -\\sqrt{3\/5} & 0 & \\sqrt{3\/5} & 5\/9 & 8\/9 & 5\/9 \\\\
\n\\hline
\n\\end{array}
\n\\]\n\\(x_i\\) \u306b\u306f\u7a4d\u5206\u533a\u9593\u306e\u4e21\u7aef \\(a\\) \u3068 \\(b\\) \u304c\u542b\u307e\u308c\u307e\u305b\u3093.<\/p>\n

\u4ee5\u4e0a, 3 \u3064\u306e\u65b9\u6cd5\u3092\u56f3\u3067\u793a\u3059\u3068\u6b21\u306e\u3088\u3046\u306b\u306a\u308a\u307e\u3059. \u305f\u3060\u3057, \u30ac\u30a6\u30b9\u516c\u5f0f\u306f N = 2 \u306e\u5834\u5408\u3092\u793a\u3057\u3066\u3044\u307e\u3059.<\/p>\n

\"\"<\/p>\n

\u6b21\u306b, \u7a4d\u5206\u533a\u9593\u3092\u8907\u6570\u306e\u5c0f\u533a\u9593\u306b\u5206\u5272\u3057, \u305d\u306e\u5c0f\u533a\u9593\u3054\u3068\u306b\u516c\u5f0f\u3092\u9069\u7528\u3059\u308c\u3070\u5358\u4f53\u306e\u516c\u5f0f\u3088\u308a\u9ad8\u7cbe\u5ea6\u304c\u671f\u5f85\u3067\u304d\u305d\u3046\u3067\u3059. \u3053\u308c\u3092, \u8907\u5408\u516c\u5f0f\u3068\u3044\u3044\u307e\u3059. \u4f8b\u3048\u3070, \u4e0b\u56f3\u306e\u3088\u3046\u306b\u7a4d\u5206\u533a\u9593\u3092 4 \u5206\u5272\u3057\u305d\u308c\u305e\u308c\u306b\u53f0\u5f62\u5247\u3092\u9069\u7528\u3059\u308b\u3068\u3088\u3055\u305d\u3046\u3067\u3059.<\/p>\n

\"\"<\/p>\n

\u3053\u308c\u3092\u8907\u5408\u53f0\u5f62\u5247\u3068\u3044\u3044\u307e\u3059. \u305f\u3060\u3057, \u53f0\u5f62\u5247\u306e\u5834\u5408\u306b\u306f\u5b9f\u7528\u4e0a\u8907\u5408\u516c\u5f0f\u3092\u4f7f\u3046\u65b9\u304c\u591a\u3044\u305f\u3081\u3053\u308c\u3092\u5358\u306b\u53f0\u5f62\u5247\u3068\u3044\u3046\u306e\u304c\u666e\u901a\u3067\u3059. \u30b7\u30f3\u30d7\u30bd\u30f3\u5247\u3082\u540c\u69d8\u306b\u8907\u5408\u30b7\u30f3\u30d7\u30bd\u30f3\u5247\u306e\u3053\u3068\u3092\u6307\u3059\u3053\u3068\u304c\u3042\u308a\u307e\u3059. m \u5206\u5272\u3057\u305f\u3068\u304d\u306e\u8907\u5408\u53f0\u5f62\u5247\u306e\u8a08\u7b97\u5f0f\u306f\u6b21\u306e\u3088\u3046\u306b\u306a\u308a\u307e\u3059.
\n\\[
\nI_m = h(\\frac{f(x_0)}{2} + f(x_1) + f(x_2) + \\dots + \\frac{f(x_m)}{2}), \\space h = \\frac{b – a}{m}
\n\\]\nh \u306f\u5c0f\u533a\u9593\u306e\u5e45\u3092\u8868\u3057\u307e\u3059. \u3053\u3053\u3067\u306f h \u3092\u4e00\u5b9a\u3068\u3057\u3066\u3044\u307e\u3059\u304c\u5c0f\u533a\u9593\u3054\u3068\u306b\u53ef\u5909\u306b\u3059\u308b\u65b9\u6cd5\u3082\u3042\u308a\u307e\u3059.<\/p>\n

\u3053\u308c\u3089\u306e\u516c\u5f0f\u3092\u4f7f\u3063\u3066\u5b9f\u969b\u306b\u6700\u521d\u306b\u3042\u3052\u305f\u4f8b\u984c\u3092\u8a08\u7b97\u3057\u3066\u307f\u308b\u3068\u6b21\u306e\u3088\u3046\u306b\u306a\u308a\u307e\u3059.<\/p>\n

\n\n\n\n\n\n\n
\u516c\u5f0f<\/th>\n\u8a08\u7b97\u5024<\/th>\n\u8aa4\u5dee<\/th>\n\u8a08\u7b97\u91cf\u306e\u76ee\u5b89<\/th>\n<\/tr>\n
\u53f0\u5f62\u5247<\/td>\n1.0663<\/td>\n-7.9%<\/td>\n2<\/td>\n<\/tr>\n
\u30b7\u30f3\u30d7\u30bd\u30f3\u5247<\/td>\n1.1575<\/td>\n-0.06%<\/td>\n3<\/td>\n<\/tr>\n
\u30ac\u30a6\u30b9\u516c\u5f0f (N = 2)<\/td>\n1.15862<\/td>\n0.04%<\/td>\n2<\/td>\n<\/tr>\n
\u8907\u5408\u53f0\u5f62\u5247 (m = 4)<\/td>\n1.1523<\/td>\n-0.5%<\/td>\n5<\/td>\n<\/tr>\n<\/table>\n<\/div>\n

\u8a08\u7b97\u91cf\u304c 3 \u306e\u30b7\u30f3\u30d7\u30bd\u30f3\u516c\u5f0f\u3084\u8a08\u7b97\u91cf\u304c 5 \u306e\u8907\u5408\u53f0\u5f62\u5247\u3088\u308a\u3082, \u8a08\u7b97\u91cf\u304c 2 \u306e\u30ac\u30a6\u30b9\u516c\u5f0f\u306e\u307b\u3046\u304c\u7cbe\u5ea6\u304c\u3088\u3044\u3053\u3068\u304c\u308f\u304b\u308a\u307e\u3059.<\/p>\n

11.3 \u9069\u5fdc\u6c42\u7a4d\u30d7\u30ed\u30b0\u30e9\u30e0<\/h3>\n

\u5f97\u3089\u308c\u305f\u7a4d\u5206\u5024\u306e\u8aa4\u5dee\u304c\u63a8\u5b9a\u3067\u304d\u305f\u3068\u3059\u308b\u3068, \u5fc5\u8981\u306a\u7cbe\u5ea6\u3092\u6e80\u305f\u3059\u3088\u3046\u306b\u90e8\u5206\u533a\u9593\u306e\u6570\u3092\u5897\u6e1b\u3055\u305b\u308b\u306a\u3069\u3057\u3066\u81ea\u52d5\u7684\u306b\u8a08\u7b97\u91cf\u3092\u6700\u9069\u5316 (\u8981\u6c42\u7cbe\u5ea6\u3092\u6e80\u305f\u3059\u304c\u8a08\u7b97\u91cf\u3092\u3067\u304d\u308b\u3060\u3051\u5c11\u306a\u304f\u3059\u308b\u3053\u3068) \u3067\u304d\u307e\u3059. \u3053\u306e\u3088\u3046\u306a\u30d7\u30ed\u30b0\u30e9\u30e0\u3092\u9069\u5fdc\u6c42\u7a4d\u30d7\u30ed\u30b0\u30e9\u30e0\u3068\u3044\u3044\u307e\u3059.<\/p>\n

\u8aa4\u5dee\u3092\u63a8\u5b9a\u3059\u308b\u305f\u3081\u306b\u306f, \u4f8b\u3048\u3070\u90e8\u5206\u533a\u9593\u306e\u6570\u3092\u5909\u3048\u3066\u8a08\u7b97\u3057\u7d50\u679c\u3092\u6bd4\u8f03\u3059\u308b\u65b9\u6cd5\u306a\u3069\u304c\u8003\u3048\u3089\u308c\u307e\u3059. \u53f0\u5f62\u516c\u5f0f\u3084\u30b7\u30f3\u30d7\u30bd\u30f3\u516c\u5f0f\u3067\u7b49\u9593\u9694\u306e\u90e8\u5206\u533a\u9593\u306b\u5206\u5272\u3059\u308b\u3053\u3068\u3057\u305f\u5834\u5408, \u5206\u5272\u6570\u304c\u500d (\u90e8\u5206\u533a\u9593\u306e\u5e45\u304c\u534a\u5206) \u306b\u306a\u3063\u305f\u3068\u304d\u306b\u534a\u5206\u306e\u70b9\u3067\u306f \\(f(x)\\) \u306e\u8a08\u7b97\u6e08\u306e\u7d50\u679c\u3092\u518d\u5229\u7528\u3067\u304d\u308b\u305f\u3081\u8a08\u7b97\u91cf\u3092\u6e1b\u3089\u3059\u3053\u3068\u304c\u3067\u304d\u307e\u3059.<\/p>\n

\u3053\u308c\u306b\u5bfe\u3057\u3066, \u30ac\u30a6\u30b9\u516c\u5f0f\u306e\u5834\u5408\u306b\u306f \\(n\\) \u3092\u5909\u3048\u308b\u3068\u5168\u90e8\u306e\u70b9\u3067 \\(f(x)\\) \u306e\u8a08\u7b97\u3092\u3057\u76f4\u3059\u5fc5\u8981\u304c\u3042\u308a, \u8a08\u7b97\u52b9\u7387\u304c\u826f\u304f\u3042\u308a\u307e\u305b\u3093. \u305d\u3053\u3067, 1 \u3064\u306e\u516c\u5f0f\u3067 \\(n\\) \u304c\u7570\u306a\u308b 2 \u3064\u306e\u5024\u3092\u5f97\u308b\u3053\u3068\u304c\u3067\u304d\u308b\u30ac\u30a6\u30b9\u30fb\u30af\u30ed\u30f3\u30ed\u30c3\u30c9\u516c\u5f0f\u304c\u63d0\u6848\u3055\u308c\u307e\u3057\u305f. \u3053\u308c\u306f\u6b21\u306e\u3088\u3046\u306a\u3082\u306e\u3067\u3059.
\n\\[
\nI = \\frac{b – a}{2}(\\sum_{i=1}^n a_i f(\\frac{a + b}{2} + \\frac{b – a}{2}x_i) + \\sum_{j=1}^{n+1} b_j f(\\frac{a + b}{2} + \\frac{b – a}{2}y_j))
\n\\]\n\u3053\u3053\u3067, \\(a_i\\) \u3068 \\(b_j\\) \u306f\u30ac\u30a6\u30b9\u30fb\u30af\u30ed\u30f3\u30ed\u30c3\u30c9\u516c\u5f0f\u306e\u4fc2\u6570\u3067\u3059. \u59cb\u70b9\u30fb\u7d42\u70b9\u3092\u8868\u3059 \\(a, b\\) \u3068\u306f\u5225\u306e\u3082\u306e\u3067\u3059.<\/p>\n

\\(x_i\\) \u306f\u30ac\u30a6\u30b9\u516c\u5f0f\u3068\u540c\u3058\u3082\u306e\u3067\u3059. \u3057\u305f\u304c\u3063\u3066, \u30ac\u30a6\u30b9\u516c\u5f0f\u306e \\(w_i\\) \u3068\u7d44\u307f\u5408\u308f\u305b\u308b\u3068 \\(n\\) \u70b9\u30ac\u30a6\u30b9\u516c\u5f0f\u3092\u4f7f\u3063\u305f\u7a4d\u5206\u5024 (\\(G_n\\) \u3068\u8868\u3059) \u3092\u6c42\u3081\u308b\u3053\u3068\u304c\u3067\u304d\u307e\u3059. \u306a\u304a, \\(a_i\\) \u3068 \\(w_i\\) \u306e\u5024\u306f\u7570\u306a\u308a\u307e\u3059.<\/p>\n

\u5168\u4f53\u3067\u306f \\(2n + 1\\) \u70b9\u30ac\u30a6\u30b9\u30fb\u30af\u30ed\u30f3\u30ed\u30c3\u30c9\u516c\u5f0f\u3092\u4f7f\u3063\u305f\u7a4d\u5206\u5024 (\\(K_{2n+1}\\) \u3068\u8868\u3059) \u3092\u6c42\u3081\u308b\u3053\u3068\u304c\u3067\u304d\u307e\u3059.<\/p>\n

\\(K_{2n+1}\\) \u306f \\(f(x)\\) \u306e\u8a08\u7b97\u56de\u6570\u304c\u540c\u3058 \\(G_{2n+1}\\) \u3088\u308a\u306f\u7cbe\u5ea6\u304c\u60aa\u304f\u306a\u308a\u307e\u3059. \u3064\u307e\u308a, \u30ac\u30a6\u30b9\u30fb\u30af\u30ed\u30f3\u30ed\u30c3\u30c9\u516c\u5f0f\u306f, \u30ac\u30a6\u30b9\u516c\u5f0f\u3088\u308a\u306f\u3084\u3084\u7cbe\u5ea6\u304c\u843d\u3061\u308b\u4ee3\u308f\u308a\u306b\u8aa4\u5dee\u63a8\u5b9a\u6a5f\u80fd\u3092\u8ffd\u52a0\u3057\u305f\u3082\u306e\u3068\u3044\u3048\u307e\u3059.<\/p>\n

\\(n = 2\\) \u306e\u5834\u5408 (5 \u70b9\u30ac\u30a6\u30b9\u30fb\u30af\u30ed\u30f3\u30ed\u30c3\u30c9\u516c\u5f0f) \u306e\u4fc2\u6570\u306f\u6b21\u306e\u3088\u3046\u306b\u306a\u308a\u307e\u3059.
\n\\[
\n\\begin{array}{cccc}
\n\\hline
\nx_i & -0.5773502691896258 & 0.5773502691896258 \\\\
\na_i & 0.4909090909090909 & 0.4909090909090909 \\\\
\ny_j & -0.9258200997725515 & 0 & 0.9258200997725515 \\\\
\nb_j & 0.1979797979797980 & 0.6222222222222222 & 0.1979797979797980 \\\\
\n\\hline
\n\\end{array}
\n\\]\n\u6c42\u3081\u3089\u308c\u305f \\(G_n\\) \u3068 \\(K_{2n+1}\\) \u304b\u3089\u8aa4\u5dee\u3092\u63a8\u5b9a\u3059\u308b\u3053\u3068\u304c\u3067\u304d\u307e\u3059.<\/p>\n

\u73fe\u5728\u3067\u306f\u6570\u5024\u7a4d\u5206\u306b\u306f\u7cbe\u5ea6\u3092\u3042\u308b\u7a0b\u5ea6\u4fdd\u8a3c\u3067\u304d\u308b\u3088\u3046\u306b\u9069\u5fdc\u6c42\u7a4d\u30d7\u30ed\u30b0\u30e9\u30e0\u3092\u4f7f\u3046\u306e\u304c\u4e00\u822c\u7684\u306b\u306a\u3063\u3066\u3044\u307e\u3059. \u7279\u306b, \u5404\u90e8\u5206\u533a\u9593\u306b\u30ac\u30a6\u30b9\u30fb\u30af\u30ed\u30f3\u30ed\u30c3\u30c9\u516c\u5f0f\u3092\u9069\u7528\u3057\u3066\u8aa4\u5dee\u3092\u63a8\u5b9a\u3057, \u5404\u90e8\u5206\u533a\u9593\u306e\u5e45\u3092\u6700\u9069\u5316\u3057\u3066\u7cbe\u5ea6\u3092\u4fdd\u3063\u305f\u307e\u307e\u8a08\u7b97\u91cf\u3092\u6e1b\u3089\u3059\u3088\u3046\u306b\u3057\u305f\u30d7\u30ed\u30b0\u30e9\u30e0\u304c\u3088\u304f\u4f7f\u308f\u308c\u3066\u3044\u307e\u3059.<\/p>\n

11.4 \u30c7\u30fc\u30bf\u5165\u529b\u30bf\u30a4\u30d7\u306e\u6570\u5024\u7a4d\u5206\u30d7\u30ed\u30b0\u30e9\u30e0<\/h3>\n

\u88ab\u7a4d\u5206\u95a2\u6570\u306e\u5024\u304c\u3068\u3073\u3068\u3073\u306e\u70b9\u306b\u304a\u3051\u308b\u30c7\u30fc\u30bf\u3068\u3057\u3066\u3057\u304b\u4e0e\u3048\u3089\u308c\u3066\u3044\u306a\u3044\u5834\u5408\u306b\u306f, \u4e0a\u8a18\u306e\u7a4d\u5206\u516c\u5f0f\u304c\u4f7f\u3048\u306a\u3044\u5834\u5408\u304c\u307b\u3068\u3093\u3069\u3067\u3059. \u305f\u3060\u3057, \u7b49\u9593\u9694\u306b\u5341\u5206\u306a\u30c7\u30fc\u30bf\u304c\u4e0e\u3048\u3089\u308c\u308c\u3070\u8907\u5408\u53f0\u5f62\u5247\u3084\u30b7\u30f3\u30d7\u30bd\u30f3\u5247\u3092\u4f7f\u3048\u305d\u3046\u3067\u3059.<\/p>\n

\u4e00\u822c\u7684\u306b\u306f, \u5165\u529b\u3055\u308c\u305f\u30c7\u30fc\u30bf\u70b9\u306e\u88dc\u9593\u3092\u884c\u3044, \u305d\u306e\u88dc\u9593\u5f0f\u306e\u7a4d\u5206\u3092\u8a08\u7b97\u3059\u308b\u3053\u3068\u306b\u3088\u308a\u7a4d\u5206\u5024\u3092\u63a8\u5b9a\u3057\u307e\u3059.<\/p>\n

\u591a\u9805\u5f0f\u88dc\u9593\u3059\u308b\u3082\u306e\u3084\u30b9\u30d7\u30e9\u30a4\u30f3\u88dc\u9593\u3059\u308b\u3082\u306e\u306a\u3069\u7a2e\u3005\u306e\u30d7\u30ed\u30b0\u30e9\u30e0\u304c\u3042\u308a\u307e\u3059. \u305f\u3060\u3057, \u5165\u529b\u30c7\u30fc\u30bf\u306b\u3082\u3088\u308a\u307e\u3059\u304c, \u4e00\u822c\u7684\u306b\u306f\u9ad8\u3044\u7cbe\u5ea6\u304c\u671f\u5f85\u3067\u304d\u308b\u3082\u306e\u3067\u306f\u3042\u308a\u307e\u305b\u3093.<\/p>\n

11.5 XLPack \u3092\u4f7f\u3063\u305f\u6570\u5024\u7a4d\u5206\u306e\u6c42\u3081\u65b9<\/h3>\n

VBA \u30b5\u30d6\u30eb\u30fc\u30c1\u30f3 Qk15<\/strong> \u306f\u5206\u70b9\u56fa\u5b9a (15\u70b9) \u306e\u30ac\u30a6\u30b9\u30fb\u30af\u30ed\u30f3\u30ed\u30c3\u30c9\u516c\u5f0f\u306b\u3088\u308b\u6570\u5024\u7a4d\u5206\u30d7\u30ed\u30b0\u30e9\u30e0\u3067\u3059. 15 \u56de\u306e\u95a2\u6570\u547c\u3073\u51fa\u3057\u3067\u7a4d\u5206\u5024\u3068\u305d\u306e\u63a8\u5b9a\u8aa4\u5dee\u3092\u8a08\u7b97\u3057\u307e\u3059. VBA \u30b5\u30d6\u30eb\u30fc\u30c1\u30f3 Qag<\/strong> \u306f\u30ac\u30a6\u30b9\u30fb\u30af\u30ed\u30f3\u30ed\u30c3\u30c9\u516c\u5f0f\u306b\u3088\u308b\u9069\u5fdc\u6c42\u7a4d\u30d7\u30ed\u30b0\u30e9\u30e0\u3067\u3059. \u88ab\u7a4d\u5206\u95a2\u6570\u304c\u6ed1\u3089\u304b\u306a\u5834\u5408\u306b\u306f Qk15 \u3067\u5341\u5206\u306a\u7cbe\u5ea6\u304c\u5f97\u3089\u308c\u307e\u3059. \u88ab\u7a4d\u5206\u95a2\u6570\u304c\u6ed1\u3089\u304b\u3067\u306a\u3044\u5834\u5408\u3084, \u95a2\u6570\u547c\u3073\u51fa\u3057\u304c\u5897\u3048\u3066\u3067\u3082\u6240\u5b9a\u306e\u76ee\u7684\u7cbe\u5ea6\u3092\u5f97\u305f\u3044\u5834\u5408\u306b\u306f, Qag \u3092\u4f7f\u3046\u3068\u3088\u3044\u3067\u3057\u3087\u3046.<\/p>\n

\u7a4d\u5206\u7bc4\u56f2\u306e\u7247\u65b9\u3042\u308b\u3044\u306f\u4e21\u65b9\u304c\u6709\u9650\u3067\u306a\u3044\u5834\u5408\u306b\u306f VBA \u30b5\u30d6\u30eb\u30fc\u30c1\u30f3 Qagi<\/strong> \u3092\u4f7f\u3046\u3053\u3068\u304c\u3067\u304d\u307e\u3059. \u3053\u308c\u3082\u30ac\u30a6\u30b9\u30fb\u30af\u30ed\u30f3\u30ed\u30c3\u30c9\u516c\u5f0f\u306b\u3088\u308b\u9069\u5fdc\u6c42\u7a4d\u5206\u30d7\u30ed\u30b0\u30e9\u30e0\u3067\u3059. \u534a\u7121\u9650\u7a4d\u5206\u3092\u6709\u9650\u533a\u9593 [0, 1] \u306e\u7a4d\u5206\u306b\u5909\u63db\u3057\u3066\u8a08\u7b97\u3057\u307e\u3059.<\/p>\n

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

3 \u6b21\u30b9\u30d7\u30e9\u30a4\u30f3\u88dc\u9593\u4fc2\u6570\u3092\u6c42\u3081\u308b VBA \u30b5\u30d6\u30eb\u30fc\u30c1\u30f3 Pchse<\/strong> \u3042\u308b\u3044\u306f\u30ef\u30fc\u30af\u30b7\u30fc\u30c8\u95a2\u6570 WPchse<\/strong> (\u300c5. \u88dc\u9593\u6cd5\u300d\u53c2\u7167) \u306b VBA \u30b5\u30d6\u30eb\u30fc\u30c1\u30f3 Pchia<\/strong> \u3042\u308b\u3044\u306f\u30ef\u30fc\u30af\u30b7\u30fc\u30c8\u95a2\u6570 WPchia<\/strong> \u3092\u7d44\u307f\u5408\u308f\u305b\u308b\u3053\u3068\u306b\u3088\u308a\u30c7\u30fc\u30bf\u5165\u529b\u30bf\u30a4\u30d7\u306e\u6570\u5024\u7a4d\u5206\u30d7\u30ed\u30b0\u30e9\u30e0\u3068\u3057\u3066\u4f7f\u7528\u3059\u308b\u3053\u3068\u304c\u3067\u304d\u307e\u3059.<\/p>\n

\u4ee5\u4e0b, \u6b21\u306e\u7a4d\u5206\u3092\u6570\u5024\u7a4d\u5206\u306b\u3088\u308a\u6c42\u3081\u308b\u4f8b\u3092\u4f7f\u3063\u3066\u8aac\u660e\u3057\u307e\u3059.
\n\\[
\n\\begin{align}
\nI = \\int_0^4 \\frac{1}{x + 1} dx = 1.6094379124341
\n \\\\
\n\\end{align}
\n\\]\n

11.3.1 VBA \u30d7\u30ed\u30b0\u30e9\u30e0\u3092\u4f7f\u7528\u3057\u305f\u89e3\u304d\u65b9 (\u95a2\u6570\u5165\u529b) (1)<\/h4>\n

Qag<\/strong> \u304a\u3088\u3073 Qk15<\/strong> \u3092\u4f7f\u3063\u305f\u30d7\u30ed\u30b0\u30e9\u30e0\u4f8b\u3092\u793a\u3057\u307e\u3059.<\/p>\n

\n
Function F(X As Double) As Double\r\n    F = 1 \/ (1 + X)\r\nEnd Function\r\n\r\nSub Start()\r\n    Dim A As Double, B As Double, S As Double, Info As Long\r\n    Dim AbsErr As Double, Neval As Long\r\n    '--- Input data\r\n    A = Cells(4, 2): B = Cells(5, 2)\r\n    '--- Compute integration by Qag\r\n    Call Qag(AddressOf F, A, B, S, Info, AbsErr, Neval)\r\n    Cells(11, 2) = Info\r\n    If Info = 0 Then\r\n        Cells(8, 2) = S\r\n        Cells(9, 2) = AbsErr\r\n        Cells(10, 2) = Neval\r\n    End If\r\n    '--- Compute integration by Qk15\r\n    Call Qk15(AddressOf F, A, B, S, AbsErr)\r\n    Cells(8, 3) = S\r\n    Cells(9, 3) = AbsErr\r\nEnd Sub<\/code><\/pre>\n<\/div>\n
\u76ee\u7684\u95a2\u6570\u3092\u5916\u90e8\u95a2\u6570\u3068\u3057\u3066\u7528\u610f\u3057, \u533a\u9593\u3068\u5fc5\u8981\u306b\u5fdc\u3058\u3066\u4ed6\u306e\u30d1\u30e9\u30e1\u30fc\u30bf\u3092\u6307\u5b9a\u3057\u3066\u547c\u3073\u51fa\u3057\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\u3059.<\/p>\n
\"\"<\/p>\n
Qag<\/strong> \u306f, \u88ab\u7a4d\u5206\u95a2\u6570\u547c\u3073\u51fa\u3057 75 \u56de (\u5185\u90e8\u3067 Qk15 \u3092 5 \u56de\u547c\u3073\u51fa\u3057) \u3067\u63a8\u5b9a\u8aa4\u5dee \\(7.38 \\times 10^{-13}\\) \u306e\u7d50\u679c\u304c\u5f97\u3089\u308c\u307e\u3057\u305f. Qk15<\/strong> \u306f\u88ab\u7a4d\u5206\u95a2\u6570\u547c\u3073\u51fa\u3057 15 \u56de\u3067\u63a8\u5b9a\u8aa4\u5dee \\(2.07 \\times 10^{-5}\\) \u3068\u306a\u308a\u307e\u3057\u305f. \u63a8\u5b9a\u8aa4\u5dee\u306f\u5b89\u5168\u65b9\u5411\u306b\u7b97\u51fa\u3055\u308c\u3066\u304a\u308a, \u5b9f\u969b\u306b\u306f\u305d\u308c\u305e\u308c \\(2.22 \\times 10^{-16}, 1.20 \\times 10^{-11}\\) \u3067\u3057\u305f.<\/p>\n
11.3.2 VBA \u30d7\u30ed\u30b0\u30e9\u30e0\u3092\u4f7f\u7528\u3057\u305f\u89e3\u304d\u65b9 (\u95a2\u6570\u5165\u529b) (2)<\/h4>\n
\u30ea\u30d0\u30fc\u30b9\u30b3\u30df\u30e5\u30cb\u30b1\u30fc\u30b7\u30e7\u30f3\u7248 (RCI) \u306e Qag_r<\/strong> \u304a\u3088\u3073 Qk15_r<\/strong> \u3092\u4f7f\u3063\u305f\u30d7\u30ed\u30b0\u30e9\u30e0\u4f8b\u3092\u793a\u3057\u307e\u3059.<\/p>\n
\n
Sub Start()\r\n    Dim A As Double, B As Double, S As Double, Info As Long\r\n    Dim AbsErr As Double, Neval As Long\r\n    Dim XX As Double, YY As Double, IRev As Long\r\n    '--- Input data\r\n    A = Cells(4, 2): B = Cells(5, 2)\r\n    '--- Compute integration by Qag_r\r\n    IRev = 0\r\n    Do\r\n        Call Qag_r(A, B, S, Info, XX, YY, IRev, AbsErr, Neval)\r\n        If IRev = 1 Then YY = 1 \/ (1 + XX ^ 2)\r\n    Loop While IRev <> 0\r\n    Cells(11, 2) = Info\r\n    If Info = 0 Then\r\n        Cells(8, 2) = S\r\n        Cells(9, 2) = AbsErr\r\n        Cells(10, 2) = Neval\r\n    End If\r\n    '--- Compute integration by Qk15_r\r\n    IRev = 0\r\n    Do\r\n        Call Qk15_r(A, B, S, XX, YY, IRev, AbsErr)\r\n        If IRev = 1 Then YY = 1 \/ (1 + XX ^ 2)\r\n    Loop While IRev <> 0\r\n    Cells(8, 3) = S\r\n    Cells(9, 3) = AbsErr\r\nEnd Sub<\/code><\/pre>\n<\/div>\n
\u76ee\u7684\u95a2\u6570\u3092\u5916\u90e8\u95a2\u6570\u3068\u3057\u3066\u4e0e\u3048\u308b\u306e\u3067\u306f\u306a\u304f, IRev = 1 \u306e\u3068\u304d\u306b XX \u306e\u5024\u3092\u4f7f\u3063\u3066\u95a2\u6570\u5024\u3092\u8a08\u7b97\u3057 YY \u306b\u5165\u308c\u3066\u518d\u5ea6 Qag_r<\/strong> \u3042\u308b\u3044\u306f Qk15_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
11.3.3 \u30ef\u30fc\u30af\u30b7\u30fc\u30c8\u95a2\u6570\u3092\u4f7f\u7528\u3057\u305f\u89e3\u304d\u65b9 (\u30c7\u30fc\u30bf\u5165\u529b)<\/h4>\n
\u30c7\u30fc\u30bf\u5165\u529b\u306e\u6570\u5024\u7a4d\u5206\u30eb\u30fc\u30c1\u30f3\u3068\u3057\u3066, \u30b9\u30d7\u30e9\u30a4\u30f3\u88dc\u9593\u306b\u3088\u308b\u7a4d\u5206\u306e\u6c42\u3081\u65b9\u3092\u8aac\u660e\u3057\u307e\u3059.<\/p>\n
\u4e0a\u3068\u540c\u3058\u4f8b\u306b\u304a\u3044\u3066, \u95a2\u6570\u5f62\u306f\u4e0d\u660e\u3067, \\(x = 0, 0.5, 1, \\dots, 4\\) \u306b\u304a\u3051\u308b\u95a2\u6570\u5024\u306e\u307f\u304c\u4e0e\u3048\u3089\u308c\u3066\u3044\u308b\u3082\u306e\u3068\u3057\u307e\u3059.
\n\\[
\n\\begin{array}{cccc}
\n\\hline
\nx & f(x) \\\\ \\hline
\n0 & 1 \\\\
\n0.5 & 0.666667 \\\\
\n1 & 0.5 \\\\
\n1.5 & 0.4 \\\\
\n2 & 0.333333 \\\\
\n2.5 & 0.285714 \\\\
\n3 & 0.25 \\\\
\n3.5 & 0.222222 \\\\
\n4 & 0.2 \\\\
\n\\hline
\n\\end{array}
\n\\]\n\u307e\u305a\u306f, WPchse<\/strong> \u95a2\u6570\u3092\u4f7f\u3063\u3066\u30b9\u30d7\u30e9\u30a4\u30f3\u95a2\u6570\u306e\u4fc2\u6570\u3092\u6c42\u3081\u307e\u3059(\u300c5. \u88dc\u9593\u300d\u3092\u53c2\u7167). <\/p>\n
\"\"<\/p>\n
\u3044\u3063\u305f\u3093\u4fc2\u6570\u304c\u6c42\u3081\u3089\u308c\u308c\u3070, WPchia<\/strong> \u95a2\u6570\u3092\u4f7f\u3063\u3066\u7a4d\u5206\u5024\u3092\u8a08\u7b97\u3059\u308b\u3053\u3068\u304c\u3067\u304d\u307e\u3059. WPchia<\/strong> \u306e\u5fc5\u8981\u306a\u30d1\u30e9\u30e1\u30fc\u30bf\u306f A, B, N, X, Y, D \u3067\u3059. A \u3068 B \u306f\u7a4d\u5206\u7bc4\u56f2(\u3053\u306e\u4f8b\u3067\u306f 0 \u3068 4). N \u306f\u30c7\u30fc\u30bf\u6570(\u3053\u306e\u4f8b\u3067\u306f 9), X \u3068 Y \u306f\u88dc\u9593\u30c7\u30fc\u30bf\u306e\u30bb\u30eb\u7bc4\u56f2\u3067\u3059. D \u306f\u30b9\u30d7\u30e9\u30a4\u30f3\u4fc2\u6570\u306e\u30bb\u30eb\u7bc4\u56f2\u3067\u3059.<\/p>\n
\"\"<\/p>\n
Enter \u3092\u62bc\u3059\u3068, 1.61082… \u3067, 9\u70b9\u306e\u30c7\u30fc\u30bf\u304b\u30893\u6841\u7a0b\u5ea6\u306e\u7cbe\u5ea6\u3067\u7a4d\u5206\u5024\u3092\u6c42\u3081\u308b\u3053\u3068\u304c\u3067\u304d\u307e\u3057\u305f.<\/p>\n
\"\"<\/p>\n
\u6ce8 – \u4f8b\u984c\u30ef\u30fc\u30af\u30b7\u30fc\u30c8\u306b\u306f VBA \u3092\u4f7f\u3063\u305f\u30b9\u30d7\u30e9\u30a4\u30f3\u88dc\u9593\u306b\u3088\u308b\u7a4d\u5206\u4f8b\u3082\u5165\u3063\u3066\u3044\u307e\u3059.<\/p>\n
11.3.4 \u30bd\u30eb\u30d0\u30fc\u3092\u4f7f\u7528\u3057\u305f\u89e3\u304d\u65b9<\/h4>\n
XLPack \u30bd\u30eb\u30d0\u30fc\u30a2\u30c9\u30a4\u30f3\u306e\u300c\u6570\u5024\u7a4d\u5206 (\u6709\u9650\u533a\u9593)\u300d\u3092\u4f7f\u3063\u3066\u89e3\u304f\u3053\u3068\u3082\u3067\u304d\u307e\u3059. B4\u30bb\u30eb\u306b\u88ab\u7a4d\u5206\u95a2\u6570\u5f0f (=1\/(1+B3^2)) \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","protected":false},"excerpt":{"rendered":"
\u6570\u5024\u7a4d\u5206\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,"footnotes":""},"categories":[10],"tags":[],"class_list":["post-4782","post","type-post","status-publish","format-standard","hentry","category-tutorial"],"aioseo_notices":[],"aioseo_head":"\n\t\t\n\t\n\t\n\t\n\t\n\t\n\t\n\t\n\t\t