{"id":5988,"date":"2026-07-17T11:04:24","date_gmt":"2026-07-17T02:04:24","guid":{"rendered":"https:\/\/www.ktech.biz\/jp\/?p=5988"},"modified":"2026-07-17T10:03:04","modified_gmt":"2026-07-17T01:03:04","slug":"9_quad","status":"publish","type":"post","link":"https:\/\/www.ktech.biz\/jp\/num\/9_quad\/","title":{"rendered":"9. \u6570\u5024\u7a4d\u5206"},"content":{"rendered":"\n

\u306f\u3058\u3081\u306b<\/h3>\n

\u6709\u9650\u533a\u9593\u3042\u308b\u3044\u306f\u7121\u9650\u533a\u9593\u306b\u304a\u3051\u308b\u5b9a\u7a4d\u5206\u306e\u8fd1\u4f3c\u8a08\u7b97\u6cd5\u3092\u6570\u5024\u7a4d\u5206\u6cd5 (numerical intergation) \u3068\u3044\u3046. \u307e\u305f\u306f \u6c42\u7a4d\u6cd5 (quadrature) \u3068\u3044\u3046\u3053\u3068\u3082\u3042\u308b.<\/p>\n

\u5b9f\u969b\u306b\u6570\u5024\u7a4d\u5206\u306e\u8a08\u7b97\u3092\u884c\u3046\u969b\u306b\u306f, \u7a4d\u5206\u3092\u8a08\u7b97\u3059\u308b\u95a2\u6570 (\u88ab\u7a4d\u5206\u95a2\u6570) \u306e\u5229\u7528\u53ef\u80fd\u306a\u5024\u304c, \u3068\u3073\u3068\u3073\u306e\u70b9 (\u7b49\u9593\u9694\u306e\u5834\u5408\u3068\u4e0d\u7b49\u9593\u9694\u306e\u5834\u5408\u304c\u3042\u308b) \u306b\u304a\u3051\u308b\u30c7\u30fc\u30bf\u3068\u3057\u3066\u4e0e\u3048\u3089\u308b\u5834\u5408\u3068, \u4efb\u610f\u306e\u70b9\u306b\u304a\u3051\u308b\u95a2\u6570\u5024\u304c\u8a08\u7b97\u3067\u304d\u308b\u30b5\u30d6\u30eb\u30fc\u30c1\u30f3\u3068\u3057\u3066\u4e0e\u3048\u3089\u308c\u308b\u5834\u5408\u304c\u3042\u308b. \u305d\u308c\u305e\u308c\u306e\u5834\u5408\u3067\u4f7f\u7528\u3067\u304d\u308b\u516c\u5f0f\u3084\u30d7\u30ed\u30b0\u30e9\u30e0\u304c\u7570\u306a\u308b\u304c, \u3053\u3053\u3067\u306f\u57fa\u672c\u7684\u306b\u306f\u5f8c\u8005\u3092\u60f3\u5b9a\u3059\u308b. \u524d\u8005\u306b\u3064\u3044\u3066\u306f\u300c6. \u88dc\u9593\u6cd5\u300d\u3067\u8aac\u660e\u4e88\u5b9a\u306e\u30b9\u30d7\u30e9\u30a4\u30f3\u88dc\u9593\u304c\u53c2\u8003\u306b\u306a\u308b.<\/p>\n

\u6b21\u306e\u7a4d\u5206\u3092\u6570\u5024\u8a08\u7b97\u3067\u6c42\u3081\u308b.
\n\\[
\nI = \\int_{0.2}^1 e^x cos(x) dx = 1.1582
\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\u3042\u308b. \u8a08\u7b97\u5f0f\u306f\u6b21\u306e\u3068\u304a\u308a\u3067\u3042\u308b.
\n\\[
\nI = \\frac{b – a}{2}(f(a) + f(b))
\n\\]\n\u3068\u306a\u308a, \u53f0\u5f62\u5247\u3068\u3088\u3070\u308c\u308b.<\/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 x1 = (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\u308b. \u5f0f\u306f\u6b21\u306e\u3068\u304a\u308a\u3067\u3042\u308b.
\n\\[
\nI = \\frac{b – a}{6}(f(a) + 4f(x1) + f(b))
\n\\]\n\u3053\u308c\u306f\u30b7\u30f3\u30d7\u30bd\u30f3\u5247\u3068\u3088\u3070\u308c\u308b.<\/p>\n

\u8a73\u7d30\u306f\u5f8c\u8ff0\u3059\u308b\u304c, a \u3068 b \u306e\u9593\u306e N \u70b9\u3092\u901a\u308b\u30eb\u30b8\u30e3\u30f3\u30c9\u30eb\u591a\u9805\u5f0f\u3067\u88dc\u9593\u3059\u308b\u9ad8\u7cbe\u5ea6\u306e\u65b9\u6cd5\u304c\u3042\u308a, \u8a08\u7b97\u5f0f\u306f\u6b21\u306e\u3068\u304a\u308a\u3067\u3042\u308b.
\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\u308c\u306f (N \u70b9) \u30ac\u30a6\u30b9\u516c\u5f0f\u3068\u3088\u3070\u308c\u308b.<\/p>\n

\u4ee5\u4e0a\u3092\u56f3\u3067\u793a\u3059\u3068\u6b21\u306e\u3088\u3046\u306b\u306a\u308b. \u305f\u3060\u3057, \u30ac\u30a6\u30b9\u516c\u5f0f\u306f N = 2 \u306e\u5834\u5408\u3092\u793a\u3057\u305f.<\/p>\n

\"\"<\/p>\n

\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\u308b\u3082\u306e\u3092\u8907\u5408\u516c\u5f0f<\/strong>\u3068\u3044\u3046. \u4f8b\u3048\u3070, \u7a4d\u5206\u533a\u9593\u3092\u4e0b\u56f3\u306e\u3088\u3046\u306b 4 \u5206\u5272\u3057\u305d\u308c\u305e\u308c\u306b\u53f0\u5f62\u5247\u3092\u9069\u7528\u3059\u308c\u3070\u5358\u4f53\u306e\u53f0\u5f62\u5247\u3088\u308a\u9ad8\u7cbe\u5ea6\u304c\u671f\u5f85\u3067\u304d\u305d\u3046\u3067\u3042\u308b.<\/p>\n

\"\"<\/p>\n

\u3053\u308c\u3092\u8907\u5408\u53f0\u5f62\u5247\u3068\u3044\u3046\u304c, \u53f0\u5f62\u5247\u306e\u5834\u5408\u306b\u306f\u5b9f\u7528\u4e0a\u8907\u5408\u516c\u5f0f\u3092\u4f7f\u3046\u306e\u304c\u666e\u901a\u3067\u3042\u308b\u305f\u3081\u3053\u308c\u3092\u5358\u306b\u53f0\u5f62\u5247\u3068\u3044\u3046\u3053\u3068\u304c\u591a\u3044 (\u30b7\u30f3\u30d7\u30bd\u30f3\u5247\u3082\u540c\u69d8\u306b\u8907\u5408\u30b7\u30f3\u30d7\u30bd\u30f3\u5247\u3092\u6307\u3059\u3053\u3068\u304c\u3042\u308b). m \u5206\u5272\u3057\u305f\u3068\u304d\u306e\u8907\u5408\u53f0\u5f62\u5247\u306e\u8a08\u7b97\u5f0f\u306f\u6b21\u306e\u3088\u3046\u306b\u306a\u308b.
\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\u3067\u3042\u308b. \u3053\u3053\u3067\u306f h \u3092\u4e00\u5b9a\u3068\u3057\u3066\u3044\u308b\u304c\u5c0f\u533a\u9593\u3054\u3068\u306b\u53ef\u5909\u306b\u3059\u308b\u65b9\u6cd5\u3082\u8003\u3048\u3089\u308c\u308b.<\/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\u3063\u305f.<\/p>\n

\n\n\n\n\n\n\n
\u516c\u5f0f<\/th>\n\u8a08\u7b97\u5024 S<\/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.<\/p>\n

\u4e00\u822c\u7684\u306b, \u6570\u5024\u7a4d\u5206\u306b\u304a\u3044\u3066\u306f\u7a4d\u5206\u533a\u9593 [a, b] \u3092 n \u5206\u5272\u3059\u308b\u9069\u5f53\u306a\u5206\u70b9\u5217 \\(a = x_0 < x_1 < \\dots < x_n = b\\) \u3092\u3068\u308a, \u5206\u70b9 (\u6a19\u672c\u70b9\u3068\u3082\u3044\u3046) \u306b\u304a\u3051\u308b\u95a2\u6570\u5024\\(f(x_i)\\) \u306e\u91cd\u307f\u4ed8\u304d\u306e\u548c\u3092\u7a4d\u5206\u306e\u8fd1\u4f3c\u5024\u3068\u3059\u308b\u6b21\u5f0f\u304c\u4f7f\u7528\u3055\u308c\u308b.\n\\[\nI = \\int_a^b f(x)dx \\simeq I_n = \\sum_{i=0}^N w_i f(x_i)\n\\]\n\u3053\u3053\u3067 \\(w_i\\) \u306f \\(x_i\\) \u306b\u304a\u3051\u308b\u91cd\u307f\u3067\u3042\u308b.\n\n\u306a\u304a, \u4e21\u7aef\u70b9\u3092\u542b\u307e\u305a \\(a < x_1 < x_2 < \\dots < x_N < b \\) \u3068\u306a\u308b N \u500b\u306e\u5206\u70b9\u3092\u4f7f\u7528\u3059\u308b\u516c\u5f0f\u3082\u3042\u308b.\n\n\u3053\u306e\u3088\u3046\u306b, \u5206\u70b9 \\(x_i\\) \u306b\u304a\u3044\u3066\u95a2\u6570\u5024 \\(f(x_i)\\) \u3092\u3068\u308b\u88dc\u5b8c\u591a\u9805\u5f0f\u3092\u6c42\u3081\u305d\u306e\u7a4d\u5206\u3068\u3044\u3046\u5f62\u3067\u4e0a\u306e\u516c\u5f0f\u3092\u4f5c\u308b\u3082\u306e\u3092\u88dc\u9593\u578b\u7a4d\u5206\u516c\u5f0f<\/strong>\u3068\u3088\u3076. \u4e0a\u306e\u53f0\u5f62\u5247\u3068\u30b7\u30f3\u30d7\u30bd\u30f3\u5247\u306f\u88dc\u9593\u578b\u7a4d\u5206\u516c\u5f0f\u306e\u4ee3\u8868\u7684\u306a\u4f8b\u3067\u3042\u308b.<\/p>\n

\u6570\u5024\u7a4d\u5206\u3092\u52b9\u7387\u3088\u304f\u884c\u3046\u305f\u3081\u306b\u7a2e\u3005\u306e\u6280\u6cd5\u3068\u3053\u308c\u3089\u306e\u516c\u5f0f\u3092\u7d44\u307f\u5408\u308f\u305b\u3066\u4f7f\u3046\u3053\u3068\u3082\u591a\u3044. \u4f8b\u3048\u3070, \u52a0\u901f\u6cd5\u3068\u53f0\u5f62\u5247\u3092\u7d44\u307f\u5408\u308f\u305b\u308b\u30ed\u30f3\u30d0\u30fc\u30b0\u7a4d\u5206<\/strong>, \u5909\u6570\u5909\u63db\u3068\u53f0\u5f62\u5247\u3092\u7d44\u307f\u5408\u308f\u305b\u308b\u5909\u6570\u5909\u63db\u578b\u7a4d\u5206\u516c\u5f0f<\/strong>, \u8907\u5408\u516c\u5f0f (\u533a\u9593\u5206\u5272) \u3068\u9ad8\u6b21\u516c\u5f0f\u3092\u7d44\u307f\u5408\u308f\u305b\u308b\u9069\u5fdc\u81ea\u52d5\u7a4d\u5206<\/strong>\u306a\u3069\u304c\u3042\u308b.<\/p>\n

\u76ee\u6b21<\/h2>\n

9.1 \u6570\u5024\u7a4d\u5206\u306e\u57fa\u672c\u89e3\u6cd5 (\u57fa\u672c\u7684\u306a\u7a4d\u5206\u516c\u5f0f, \u30ed\u30f3\u30d0\u30fc\u30b0\u7a4d\u5206)<\/a>
\n9.2 \u5909\u6570\u5909\u63db\u578b\u7a4d\u5206\u516c\u5f0f, \u9069\u5fdc\u81ea\u52d5\u7a4d\u5206<\/a>
\n9.3 \u5b9f\u7528\u30d7\u30ed\u30b0\u30e9\u30e0\u306e\u30d9\u30f3\u30c1\u30de\u30fc\u30af<\/a><\/p>\n\n

\u672c\u7ae0\u306e\u53c2\u8003\u6587\u732e<\/h4>\n[1] Forsythe\u4ed6\u300c\u8a08\u7b97\u6a5f\u306e\u305f\u3081\u306e\u6570\u5024\u8a08\u7b97\u6cd5\u300d\u79d1\u5b66\u6280\u8853\u51fa\u7248\u793e (1978)
\n[2] \u68ee\u6b63\u6b66\u300cFORTRAN77\u6570\u5024\u8a08\u7b97\u30d7\u30ed\u30b0\u30e9\u30df\u30f3\u30b0(\u5897\u88dc\u7248)\u300d\u5ca9\u6ce2\u66f8\u5e97 (1987)
\n[3] \u68ee\u53e3\u7e41\u4e00\u300c\u6570\u5024\u8a08\u7b97\u5de5\u5b66\u300d\u5ca9\u6ce2\u66f8\u5e97 (1989)
\n[4] \u6749\u539f\u6b63\u9855\u3001\u5ba4\u7530\u4e00\u96c4\u300c\u6570\u5024\u8a08\u7b97\u6cd5\u306e\u6570\u7406\u300d\u5ca9\u6ce2\u66f8\u5e97 (1994)
\n[5] \u68ee\u6b63\u6b66\u300c\u6570\u5024\u89e3\u6790 (\u7b2c2\u7248)\u300d(2002) \u5171\u7acb\u51fa\u7248
\n[6] Piessens, de Doncker-Kapenga, Uberhuber, Kahaner, \u201cQUADPACK: A subroutine package for automatic integration\u201d, Springer-Verlag. (1983)
\n[7] Ooura and Mori, “The double exponential formula for oscillatory functions over the half infinite interval”, J. Comput. Appl. Math., 38 (1991), 353-360.
\n[8] D. P. Laurie, “Calculation of Gauss-Kronrod quadrature rules”, Mathematics of Computation, Vol.66 No.219 (1997).
\n[9] Ooura and Mori, “A robust double exponential formula for Fourier-type integrals”, J. Comput. Appl. Math., 112 (1999), 229-241.<\/p>\n","protected":false},"excerpt":{"rendered":"

\u306f\u3058\u3081\u306b \u6709\u9650\u533a\u9593\u3042\u308b\u3044\u306f\u7121\u9650\u533a\u9593\u306b\u304a\u3051\u308b\u5b9a\u7a4d\u5206\u306e\u8fd1\u4f3c\u8a08\u7b97\u6cd5\u3092\u6570\u5024\u7a4d\u5206\u6cd5 (numerical intergation) \u3068\u3044\u3046. \u307e\u305f\u306f \u6c42\u7a4d\u6cd5 (quadrature) \u3068\u3044\u3046\u3053\u3068\u3082\u3042\u308b. \u5b9f\u969b\u306b\u6570\u5024\u7a4d\u5206\u306e\u8a08\u7b97\u3092\u884c\u3046 […]<\/p>\n","protected":false},"author":3,"featured_media":0,"comment_status":"closed","ping_status":"closed","sticky":false,"template":"","format":"standard","meta":{"_monsterinsights_skip_tracking":false,"_monsterinsights_sitenote_active":false,"_monsterinsights_sitenote_note":"","_monsterinsights_sitenote_category":0,"footnotes":""},"categories":[13],"tags":[],"class_list":["post-5988","post","type-post","status-publish","format-standard","hentry","category-num"],"aioseo_notices":[],"aioseo_head":"\n\t\t\n\t\n\t\n\t\n\t\n\t\n\t\n\t\n\t\t