{"id":1045,"date":"2015-11-02T12:39:55","date_gmt":"2015-11-02T03:39:55","guid":{"rendered":"https:\/\/www.ktech.biz\/jp\/?p=1045"},"modified":"2020-11-01T21:57:50","modified_gmt":"2020-11-01T12:57:50","slug":"9-quad-2","status":"publish","type":"post","link":"https:\/\/www.ktech.biz\/jp\/num\/9-quad-2\/","title":{"rendered":"9. \u6570\u5024\u7a4d\u5206 (2) \u5909\u6570\u5909\u63db\u578b\u7a4d\u5206\u516c\u5f0f, \u81ea\u52d5\u7a4d\u5206"},"content":{"rendered":"

\u76ee\u6b21<\/h3>\n

\u6570\u5024\u7a4d\u5206 (1)<\/a>: \u6982\u8981, \u88dc\u9593\u578b\u7a4d\u5206\u516c\u5f0f, \u30ed\u30f3\u30d0\u30fc\u30b0\u7a4d\u5206, \u6570\u5024\u5b9f\u9a13, \u7a4d\u5206\u516c\u5f0f\u306e\u4fc2\u6570\u8868
\n\u6570\u5024\u7a4d\u5206 (2): \u53f0\u5f62\u5247\u306e\u8aa4\u5dee, \u5909\u6570\u5909\u63db\u578b\u7a4d\u5206\u516c\u5f0f, \u8aa4\u5dee\u306e\u63a8\u5b9a, \u81ea\u52d5\u7a4d\u5206, \u7a4d\u5206\u516c\u5f0f\u306e\u4fc2\u6570\u8868
\n\u6570\u5024\u7a4d\u5206 (3)<\/a>: \u5b9f\u7528\u30eb\u30fc\u30c1\u30f3\u306e\u30d9\u30f3\u30c1\u30de\u30fc\u30af (\u5b9f\u7528\u30eb\u30fc\u30c1\u30f3\u306e\u9078\u629e, \u6027\u80fd\u6bd4\u8f03)<\/p>\n

\u53f0\u5f62\u5247\u306e\u8aa4\u5dee<\/span>\u5909\u6570\u5909\u63db\u578b\u7a4d\u5206\u516c\u5f0f<\/span>\u8aa4\u5dee\u306e\u63a8\u5b9a<\/span>\u81ea\u52d5\u7a4d\u5206<\/span>\u7a4d\u5206\u516c\u5f0f\u306e\u4fc2\u6570\u8868<\/span><\/div>
\n

\u53f0\u5f62\u5247\u306e\u8aa4\u5dee<\/h2>\n

\u53f0\u5f62\u5247\u3067\u306f\u5206\u70b9\u6570\u3092\u500d(\u523b\u307f\u5e45 h \u30921\/2)\u306b\u3059\u308b\u3068\u8aa4\u5dee\u306f\u304a\u304a\u3088\u305d1\/4\u306b\u306a\u3063\u305f. \u3053\u308c\u306f, \u88ab\u7a4d\u5206\u95a2\u6570\u304c\u89e3\u6790\u7684\u306a\u5834\u5408\u53f0\u5f62\u5247\u306e\u8aa4\u5dee\u304c\u6b21\u306e\u3088\u3046\u306b\u8868\u3055\u308c\u308b\u3053\u3068\u306b\u3088\u308b(\u30aa\u30a4\u30e9\u30fc\u30fb\u30de\u30af\u30ed\u30fc\u30ea\u30f3\u5c55\u958b).<\/p>\n

  \r\n  Im<\/sub> - I = c2<\/sub>h2<\/sup>[f'(b) - f'(a)] + c4<\/sub>h4<\/sup>[f'''(b) - f'''(a)] + ... + c2p<\/sub>h2p<\/sup>[f(2p-1)<\/sup>(b) - f(2p-1)<\/sup>(a)] + O(h2p+2<\/sup>)\r\n\r\n<\/pre>\n
 
\n\u3053\u3053\u3067 ci<\/sub> \u306f h \u3084 f \u306b\u7121\u95a2\u4fc2\u306a\u5b9a\u6570\u3067\u3042\u308b.<\/p>\n
h4<\/sup> \u4ee5\u4e0a\u306e\u9805\u304c\u5341\u5206\u5c0f\u3055\u3044\u3068\u4eee\u5b9a\u3059\u308b\u3068\u8aa4\u5dee\u306f\u307b\u307c h2<\/sup> \u306b\u6bd4\u4f8b\u3059\u308b\u3068\u898b\u306a\u3059\u3053\u3068\u304c\u3067\u304d, h \u304c1\/2\u306b\u306a\u308b\u3068\u8aa4\u5dee\u306f1\/4\u306b\u306a\u308b. \u3057\u304b\u3057, \u5fae\u5206\u4e0d\u53ef\u80fd\u306a\u7279\u7570\u70b9\u304c\u3042\u3063\u305f\u308a\u3059\u308b\u3068\u3053\u308c\u304c\u6210\u308a\u7acb\u305f\u306a\u304f\u306a\u308b\u3053\u3068\u306f\u300c\u6570\u5024\u7a4d\u5206(1)\u306e\u6570\u5024\u5b9f\u9a13\u300d\u3067\u898b\u305f.<\/p>\n
\u6b21\u306b, \u7279\u5225\u306a\u5834\u5408\u306e\u4f8b\u3068\u3057\u3066\u6b21\u306e\u95a2\u6570\u306e[0, \u03c0\/2]\u3067\u306e\u7a4d\u5206\u3092\u6c42\u3081\u3066\u307f\u308b.<\/p>\n
  \r\n  I = \u222b1\/sqrt(1 - sin2<\/sup>x\/2) dx [0, \u03c0\/2] = 1.85407467730137\r\n\r\n<\/pre>\n
 
\n\"ex5f\"<\/p>\n
\u56f3\u304b\u3089\u3082\u308f\u304b\u308b\u3088\u3046\u306b\u3053\u308c\u306f\u5468\u671f\u95a2\u6570\u3067\u3042\u308b. \u7a2e\u3005\u306e\u7a4d\u5206\u516c\u5f0f\u3067\u5206\u70b9\u6570N(\u6a2a\u8ef8)\u3092\u5909\u3048\u3066\u8a08\u7b97\u3057, \u305d\u306e\u6642\u306e\u76f8\u5bfe\u8aa4\u5dee(\u7e26\u8ef8)\u3092\u5bfe\u6570\u30b0\u30e9\u30d5\u306b\u3059\u308b\u3068\u6b21\u306e\u3088\u3046\u306b\u306a\u3063\u305f.<\/p>\n
\"\"<\/p>\n
\u610f\u5916\u306b\u3082\u53f0\u5f62\u5247\u304c\u6700\u3082\u901f\u304f\u53ce\u675f\u3057\u3066\u3044\u308b. \u53f0\u5f62\u5247\u306e\u7d50\u679c\u3092\u8868\u306b\u3057\u3066\u307f\u308b\u3068, \u53ce\u675f\u7387 = 1\/4 \u3068\u6bd4\u8f03\u3059\u308b\u3068\u6841\u9055\u3044\u306b\u5c0f\u3055\u304f(\u53ce\u675f\u304c\u901f\u304f)\u306a\u3063\u3066\u3044\u308b\u3053\u3068\u304c\u308f\u304b\u308b. \u3053\u3046\u306a\u308b\u3068\u56f3\u306e\u3088\u3046\u306b\u30ed\u30f3\u30d0\u30fc\u30b0\u7a4d\u5206\u306f\u304b\u3048\u3063\u3066\u9045\u304f\u306a\u308b.
\n
\n\n\n\n\n\n\n\n
\u5206\u70b9\u6570<\/th>\n\u7a4d\u5206\u5024(\u8a08\u7b97\u5024)<\/th>\n\u7a4d\u5206\u5024(\u771f\u5024)<\/th>\n\u8aa4\u5dee<\/th>\n\u53ce\u675f\u7387<\/th>\n<\/tr>\n
2<\/td>\n1.85495913108563<\/td>\n1.85407467730137<\/td>\n8.8445378e-4<\/td>\n<\/td>\n<\/tr>\n
4<\/td>\n1.85407522776731<\/td>\n1.85407467730137<\/td>\n5.5046594e-7<\/td>\n1\/1607<\/td>\n<\/tr>\n
8<\/td>\n1.85407467730167<\/td>\n1.85407467730137<\/td>\n2.9465319e-13<\/td>\n1\/1.868e+6<\/td>\n<\/tr>\n
16<\/td>\n1.85407467730137<\/td>\n1.85407467730137<\/td>\n0<\/td>\n<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<\/div>\n\u3053\u308c\u306f, \u5468\u671f\u6027\u306e\u3042\u308b\u95a2\u6570\u306f\u305d\u306e\u4e21\u7aef\u3067\u5fae\u5206\u4fc2\u6570\u304c\u4e00\u81f4\u3057, \u4e0a\u5f0f(\u30aa\u30a4\u30e9\u30fc\u30fb\u30de\u30af\u30ed\u30fc\u30ea\u30f3\u5c55\u958b)\u3067\u8868\u3055\u308c\u308b\u8aa4\u5dee\u304c 0 \u306b\u8fd1\u304f\u306a\u308b\u305f\u3081\u3067\u3042\u308b. \u3053\u306e\u4f8b\u306f\u7a4d\u5206\u7bc4\u56f2\u304c\u534a\u5468\u671f\u3067\u3042\u308b\u304c\u5bfe\u79f0\u6027\u304c\u3042\u308b\u305f\u3081\u3053\u306e\u6761\u4ef6\u306b\u3042\u3066\u306f\u307e\u308b. \u3053\u306e\u3088\u3046\u306a\u5834\u5408\u306b\u306f\u53f0\u5f62\u5247\u304c\u6709\u52b9\u3067\u3042\u308b\u3053\u3068\u304c\u77e5\u3089\u308c\u3066\u3044\u308b.<\/p>\n
\u3082\u3046\u3072\u3068\u3064, \u7121\u9650\u7a4d\u5206\u306e\u4f8b\u3092\u898b\u3066\u307f\u308b.<\/p>\n
  \r\n  \u222bexp(-x^2)\/2 dx [0, \u221e] = 1.25331413731550\r\n\r\n<\/pre>\n
 
\n\u95a2\u6570\u5f62\u306f\u6b21\u306e\u3068\u304a\u308a\u3067\u3042\u308b. \u95a2\u6570\u5024\u306f x = 8 \u3067 10-14<\/sup> \u306e\u30aa\u30fc\u30c0\u30fc, x = 10 \u3067 10-22<\/sup> \u306e\u30aa\u30fc\u30c0\u30fc\u3068\u6025\u901f\u306b0\u306b\u8fd1\u3065\u304f\u95a2\u6570\u306b\u306a\u3063\u3066\u3044\u308b.<\/p>\n
\"ex6f\"<\/p>\n
\u3053\u3053\u3067\u306f\u7a4d\u5206\u7bc4\u56f2\u3092 [0, \u221e] \u306e\u4ee3\u308f\u308a\u306b [0, 10] \u3068\u3057\u3066\u8a08\u7b97\u3057\u3066\u307f\u308b\u3068\u6b21\u306e\u3088\u3046\u306b\u306a\u3063\u305f.<\/p>\n
\"\"<\/p>\n
\u3053\u3053\u3067\u3082\u53f0\u5f62\u5247\u304c\u4e00\u756a\u901f\u304f\u53ce\u675f\u3057, N = 15 \u3067\u6700\u9ad8\u7cbe\u5ea6\u306b\u9054\u3057\u305f.<\/p>\n
\u6e1b\u8870\u304c\u901f\u3044\u95a2\u6570\u3067\u306f\u7aef\u3067\u3059\u3050\u306b\u5fae\u5206\u4fc2\u6570\u304c0\u306b\u8fd1\u3065\u304f\u305f\u3081, \u4e0a\u5f0f\u3067\u8868\u3055\u308c\u308b\u8aa4\u5dee\u3082\u901f\u304f0\u306b\u8fd1\u3065\u304f\u3068\u8003\u3048\u3089\u308c\u308b. \u3053\u306e\u3088\u3046\u306a\u7121\u9650\u7a4d\u5206\u306b\u5bfe\u3057\u3066\u3082\u53f0\u5f62\u5247\u304c\u4ed6\u306e\u516c\u5f0f\u3088\u308a\u3082\u7cbe\u5ea6\u304c\u826f\u3044\u3053\u3068\u304c\u77e5\u3089\u308c\u3066\u3044\u308b.
\n<\/div>\n
\n
\u5909\u6570\u5909\u63db\u578b\u7a4d\u5206\u516c\u5f0f<\/h2>\n
\u53f0\u5f62\u5247\u306f\u5468\u671f\u6027\u306e\u3042\u308b\u95a2\u6570\u306e\u7a4d\u5206\u3084\u6e1b\u8870\u306e\u901f\u3044\u7121\u9650\u7a4d\u5206\u306b\u5bfe\u3057\u3066\u6709\u52b9\u3067\u3042\u308b\u3053\u3068\u304c\u308f\u304b\u3063\u305f\u304c, \u5b9f\u969b\u306b\u53f0\u5f62\u5247\u304c\u305d\u306e\u307e\u307e\u4f7f\u308f\u308c\u308b\u3053\u3068\u306f\u306a\u3044. \u3057\u304b\u3057, \u53f0\u5f62\u5247\u306e\u3053\u306e\u7279\u9577\u306f\u4ee5\u4e0b\u306b\u793a\u3059\u5909\u6570\u5909\u63db\u306b\u3088\u308b\u7a4d\u5206\u516c\u5f0f\u306b\u5fdc\u7528\u3055\u308c\u3066\u3044\u308b.<\/p>\n
\u4e00\u822c\u306e\u7a4d\u5206\u3092\u300c\u53f0\u5f62\u5247\u306e\u8aa4\u5dee\u300d\u306e2\u3064\u306e\u30b1\u30fc\u30b9\u306e\u3088\u3046\u306b\u53f0\u5f62\u5247\u304c\u901f\u304f\u306a\u308b\u3088\u3046\u306a\u7a4d\u5206\u306b\u306a\u308b\u3088\u3046\u306b\u5909\u6570\u5909\u63db\u3057, \u305d\u308c\u306b\u53f0\u5f62\u5247\u3092\u9069\u7528\u3059\u308c\u3070\u6027\u80fd\u306e\u826f\u3044\u7a4d\u5206\u516c\u5f0f\u3068\u306a\u308b\u3053\u3068\u304c\u671f\u5f85\u3067\u304d\u308b. \u3053\u306e\u30a2\u30a4\u30c7\u30a2\u306b\u57fa\u3065\u304f\u516c\u5f0f\u304c\u5909\u6570\u5909\u63db\u578b\u7a4d\u5206\u516c\u5f0f\u3067\u3042\u308b.<\/p>\n
\u7a4d\u5206\u5909\u6570\u3092\u5909\u63db\u3059\u308b\u516c\u5f0f\u306f\u6b21\u306e\u3068\u304a\u308a\u3067\u3042\u308b.<\/p>\n
  \r\n  \u222bf(x)dx [a, b] = \u222bg(t)dt [c, d]\r\n  \u305f\u3060\u3057 x = \u03c6(t), g(t) = f(\u03c6(t))\u03c6'(t)\r\n\r\n<\/pre>\n
 
\n\u3053\u3053\u3067 \u03c6(t) \u306f\u533a\u9593 [c, d] \u3067\u5358\u8abf\u3067, \u304b\u3064 a = \u03c6(c), b = \u03c6(d) \u3067\u3042\u308b. \u3053\u308c\u306b\u3088\u308a\u7a4d\u5206\u5024\u304c\u4e0d\u5909\u306e\u307e\u307e\u88ab\u7a4d\u5206\u95a2\u6570\u3092\u5909\u63db\u3059\u308b\u3053\u3068\u304c\u3067\u304d\u308b. \u9069\u5207\u306a\u03c6(t)\u3092\u898b\u3064\u3051\u308b\u3053\u3068\u304c\u3067\u304d\u308c\u3070\u52b9\u679c\u7684\u306a\u516c\u5f0f\u3092\u4f5c\u308b\u3053\u3068\u304c\u3067\u304d\u308b.<\/p>\n
IMT\u516c\u5f0f<\/h3>\n
\u03c6(t) \u3068\u3057\u3066\u6b21\u5f0f\u3092\u4f7f\u3046\u306e\u304cIMT\u516c\u5f0f\u3067\u3042\u308b.<\/p>\n
  \r\n  \u03c6(t) = (1\/Q)\u222b\u03c8(r)dr [0, t], Q = \u222b\u03c8(r)dr [0, 1]\r\n  \u305f\u3060\u3057 \u03c8(r) = exp(-1\/r - 1\/(1-r))\r\n\r\n<\/pre>\n
 
\n\u3053\u306e\u5909\u63db\u3092\u533a\u9593 [0, 1] \u306e\u7a4d\u5206\u306b\u9069\u7528\u3059\u308b\u3068, \u5909\u63db\u5f8c\u306e\u95a2\u6570 g(t) \u306f g(j)<\/sup>(0) = g(j)<\/sup>(1) = 0 (j = 0, 1, …) \u3068\u306a\u308b. \u3059\u306a\u308f\u3061, \u7a4d\u5206\u533a\u9593\u306e\u4e21\u7aef\u3067\u5fae\u5206\u4fc2\u6570\u304c0\u306b\u306a\u3063\u3066\u3044\u3066, \u53f0\u5f62\u5247\u306e\u8aa4\u5dee\u304c0\u306b\u8fd1\u304f\u306a\u308b\u6761\u4ef6\u3092\u6e80\u305f\u3057\u3066\u3044\u308b.<\/p>\n
\u3053\u306e\u5909\u63db\u3092\u65bd\u3057\u53f0\u5f62\u5247\u3092\u9069\u7528\u3059\u308b\u3068, \u5206\u70b9\u6570\u3092N\u3068\u3057\u3066\u6b21\u306e\u8a08\u7b97\u5f0f\u304c\u5f97\u3089\u308c\u308b.<\/p>\n
  \r\n  SN<\/sub> = (1\/N)\u03a3wi<\/sub>(N)<\/sup>f(xi<\/sub>(N)<\/sup>) (i = 1\uff5eN-1)\r\n    \u91cd\u307f wi<\/sub>(N)<\/sup> = \u03c6'(i\/N) = (1\/Q)\u03c8(i\/N)\r\n    \u5206\u70b9 xi<\/sub>(N)<\/sup> = \u03c6(i\/N) = (1\/Q)\u222b\u03c8(r)dr [0, i\/N]\r\n\r\n<\/pre>\n
 
\nIMT\u516c\u5f0f\u306e\u8a73\u7d30\u306f\u6587\u732e[1], [2]\u306b\u8a73\u3057\u3044.<\/p>\n
\u4e8c\u91cd\u6307\u6570\u578b\u7a4d\u5206\u516c\u5f0f(DE\u516c\u5f0f)<\/h3>\n
\u5bfe\u8c61\u3068\u3059\u308b\u7a4d\u5206\u3092\u6e1b\u8870\u306e\u901f\u3044\u7121\u9650\u7a4d\u5206\u306b\u5909\u63db\u3057\u3066\u53f0\u5f62\u5247\u306e\u8aa4\u5dee\u304c0\u306b\u8fd1\u304f\u306a\u308b\u6761\u4ef6\u3092\u6e80\u305f\u3059\u3053\u3068\u306b\u3088\u308a\u52b9\u7387\u3088\u304f\u7a4d\u5206\u3092\u8a08\u7b97\u3059\u308b\u306e\u304c\u4e8c\u91cd\u6307\u6570\u578b\u7a4d\u5206\u516c\u5f0f(DE\u516c\u5f0f)\u3067\u3042\u308b. \u5909\u63db\u95a2\u6570 \u03c6(t) \u306f\u7a4d\u5206\u533a\u9593\u306a\u3069\u306b\u3088\u3063\u3066\u3044\u304f\u3064\u304b\u306e\u7a2e\u985e\u304c\u63d0\u6848\u3055\u308c\u3066\u3044\u308b.<\/p>\n
DE\u516c\u5f0f\u306e\u8a73\u7d30\u306f\u6587\u732e[2], [3], [4]\u306b\u8a73\u3057\u3044. \u5909\u6570\u5909\u63db\u578b\u7a4d\u5206\u516c\u5f0f\u3068\u3057\u3066\u306fIMT\u516c\u5f0f\u304c\u306f\u3058\u3081\u306b\u63d0\u6848\u3055\u308c, \u305d\u306e\u5f8cDE\u516c\u5f0f\u304c\u63d0\u6848\u3055\u308c\u305f. \u73fe\u5728\u3067\u306f\u4e3b\u306bDE\u516c\u5f0f\u304c\u4f7f\u308f\u308c\u3066\u3044\u308b.<\/p>\n
(i) \u6709\u9650\u533a\u9593 [-1, 1] \u306e\u7a4d\u5206<\/h4>\n
\u03c6(t) \u3068\u3057\u3066\u6b21\u3092\u4f7f\u3046.<\/p>\n
  \r\n  \u03c6(t) = tanh((\u03c0\/2)sinh(t))\r\n  \u03c6'(t) = (\u03c0\/2)cosh(t) \/ cosh2<\/sup>((\u03c0\/2)sinh(t))\r\n\r\n<\/pre>\n
 
\n\u304d\u3056\u307f\u3092 h, \u5206\u70b9\u6570\u3092 N (= N+<\/sub> + N–<\/sub> + 1) \u3068\u3059\u308b\u3068, \u8a08\u7b97\u5f0f\u306f\u6b21\u306e\u3088\u3046\u306b\u306a\u308b.<\/p>\n
  \r\n  Sh<\/sub>(N)<\/sup> = h\u03a3wi<\/sub>(N)<\/sup>f(xi<\/sub>(N)<\/sup>) (i = -N-<\/sub>\uff5eN+<\/sub>)\r\n  \u91cd\u307f wi<\/sub>(N)<\/sup> = h\u03c6'(ih)\r\n  \u5206\u70b9 xi<\/sub>(N)<\/sup> = \u03c6(ih)\r\n\r\n<\/pre>\n
 
\n\u7121\u9650\u7a4d\u5206\u306e\u8a08\u7b97\u306f\u4e0b\u5074\u3092 -N–<\/sub>, \u4e0a\u5074\u3092N+<\/sub> \u3067\u6253\u3061\u5207\u308b. \u3053\u308c\u3089\u306e\u5024\u306f\u8a31\u5bb9\u8aa4\u5dee\u9650\u754c\u3092\u3082\u3068\u306b\u9069\u5207\u306a\u5024\u3092\u8a2d\u5b9a\u3059\u308b.<\/p>\n
\u7aef\u70b9\u306b\u7279\u7570\u70b9\u304c\u3042\u308b\u95a2\u6570\u306e\u4f8b\u3067\u53d6\u308a\u4e0a\u3052\u305f I = \u222b \u221a(1-x2<\/sup>)dx [0, 1] = \u03c0\/4 \u306e\u95a2\u6570\u306e\u30b0\u30e9\u30d5\u3092\u518d\u63b2\u3059\u308b. \u3053\u306e\u95a2\u6570\u306f\u4e0b\u306e\u6570\u5024\u5b9f\u9a13(2)\u3067\u3082\u4f7f\u7528\u3057\u3066\u3044\u308b.<\/p>\n
\"\"<\/p>\n
\u3053\u308c\u3092\u4e0a\u306e\u5909\u63db\u95a2\u6570 \u03c6(t) \u3092\u4f7f\u3063\u3066\u7121\u9650\u7a4d\u5206\u306b\u5909\u63db\u3059\u308b\u3068\u6b21\u306e\u3088\u3046\u306b\u306a\u308b.<\/p>\n
\"\"<\/p>\n
\u4e0a\u5074\u3082\u4e0b\u5074\u3082\u6e1b\u8870\u304c\u901f\u3044\u3053\u3068\u304c\u308f\u304b\u308b. \u304d\u3056\u307f\u5e45\u30920.2\u3068\u3057\u3066\u3084\u308b\u3068, \u4e0a\u5074\u30fb\u4e0b\u5074\u305d\u308c\u305e\u308c16\u70b9\u3067 10-16<\/sup> \u306e\u30aa\u30fc\u30c0\u30fc\u307e\u3067\u6e1b\u8870\u3057\u305f. \u3053\u308c\u306f\u53f0\u5f62\u5247\u3067\u7cbe\u5ea6\u3088\u304f\u7a4d\u5206\u3059\u308b\u3053\u3068\u304c\u3067\u304d\u308b.<\/p>\n
(ii) \u534a\u7121\u9650\u533a\u9593 [0, +\u221e] \u306e\u7a4d\u5206<\/h4>\n
\u534a\u7121\u9650\u533a\u9593\u306e\u7a4d\u5206\u306b\u304a\u3044\u3066\u88ab\u7a4d\u5206\u95a2\u6570\u304c\u3086\u308b\u3084\u304b\u306b\u6e1b\u8870\u3059\u308b\u3088\u3046\u306a\u5834\u5408\u306b\u3064\u3044\u3066\u3082\u6709\u9650\u533a\u9593\u306e\u5834\u5408\u3068\u540c\u69d8\u306e\u8003\u3048\u65b9\u3067DE\u516c\u5f0f\u3092\u5c0e\u304f\u3053\u3068\u304c\u3067\u304d\u308b. \u6587\u732e[4]\u306b\u793a\u3055\u308c\u3066\u3044\u308b\u30d7\u30ed\u30b0\u30e9\u30e0DEHINT\u3067\u306f\u88ab\u7a4d\u5206\u95a2\u6570\u306b\u3088\u308a3\u7a2e\u985e\u306e\u516c\u5f0f\u3092\u4f7f\u3044\u5206\u3051\u308b\u3088\u3046\u306b\u306a\u3063\u3066\u3044\u308b.<\/p>\n
(a) \u901a\u5e38\u306e\u95a2\u6570\u306e\u5834\u5408<\/p>\n
  \r\n  \u03c6(t) = exp(2sinh(t))\r\n  \u03c6'(t) = 2cosh(t)exp(2sinh(t))\r\n\r\n<\/pre>\n
 
\n(b) \u88ab\u7a4d\u5206\u95a2\u6570\u304c\u6307\u6570\u95a2\u6570\u7684\u306b\u6e1b\u8870\u3059\u308b\u5834\u5408 (f(x) = f1<\/sub>(x)exp(-x) \u306e\u5f62\u3092\u3057\u3066\u3044\u308b\u5834\u5408)<\/p>\n
  \r\n  \u03c6(t) = exp(t - exp(-t))\r\n  \u03c6'(t) = (1 + exp(-t))exp(t - exp(-t))\r\n\r\n<\/pre>\n
 
\n(c) \u88ab\u7a4d\u5206\u95a2\u6570\u306e\u6e1b\u8870\u304c\u3055\u3089\u306b\u6025\u6fc0\u306a\u5834\u5408 (f(x) = f2<\/sub>(x)exp(-x^2) \u306e\u5f62\u3092\u3057\u3066\u3044\u308b\u5834\u5408)<\/p>\n
  \r\n  \u03c6(t) = exp(t\/2 - exp(-t))\r\n  \u03c6'(t) = (1\/2 + exp(-t))exp(t\/2 - exp(-t))\r\n\r\n<\/pre>\n
 <\/p>\n
(iii) \u5168\u7121\u9650\u533a\u9593 [-\u221e, +\u221e] \u306e\u7a4d\u5206<\/h4>\n
\u88ab\u7a4d\u5206\u95a2\u6570\u304c\u3086\u308b\u3084\u304b\u306b\u6e1b\u8870\u3059\u308b\u3088\u3046\u306a\u5168\u7121\u9650\u533a\u9593\u306e\u7a4d\u5206\u306b\u3064\u3044\u3066\u3082\u540c\u69d8\u306bDE\u516c\u5f0f\u3092\u5c0e\u304f\u3053\u3068\u304c\u3067\u304d\u308b.<\/p>\n
  \r\n  \u03c6(t) = sinh((\u03c0\/2)sinh(t))\r\n  \u03c6'(t) = (\u03c0\/2)cosh(t)cosh((\u03c0\/2)sinh(t))\r\n\r\n<\/pre>\n
 <\/p>\n
(iv) \u30d5\u30fc\u30ea\u30a8\u578b\u7a4d\u5206<\/h4>\n
\u6b21\u306e\u534a\u7121\u9650\u533a\u9593\u306e\u7a4d\u5206(\u30d5\u30fc\u30ea\u30a8\u578b\u7a4d\u5206)\u3092\u6c42\u3081\u308b.<\/p>\n
  \r\n  Ic = \u222bf(x)cos(\u03c9x)dx [a, +\u221e] \u307e\u305f\u306f Is = \u222bf(x)sin(\u03c9x)dx [a, +\u221e]\r\n\r\n<\/pre>\n
 
\n\u6b21\u306e\u5909\u63db\u95a2\u6570\u3092\u4f7f\u7528\u3057\u3066\u7121\u9650\u533a\u9593 [-\u221e, +\u221e] \u306e\u7a4d\u5206\u306b\u5909\u63db\u3059\u308b.<\/p>\n
  \r\n  Ic\u306e\u5834\u5408: x = M\u03c6(t - \u03c0\/2M)\/\u03c9\r\n  Is\u306e\u5834\u5408: x = M\u03c6(t)\/\u03c9\r\n\r\n<\/pre>\n
 
\n\u03c6(t) \u306f t \u2192 -\u221e \u306e\u3068\u304d\u306b\u4e8c\u91cd\u6307\u6570\u95a2\u6570\u7684\u306b \u03c6'(t) \u2192 0, t \u2192 +\u221e \u306e\u3068\u304d\u306b\u4e8c\u91cd\u6307\u6570\u95a2\u6570\u7684\u306b \u03c6(t) \u2192 t \u3068\u306a\u308b\u3088\u3046\u306a\u95a2\u6570\u3067\u3042\u308b. \u304d\u3056\u307f\u5e45 h \u3068\u5b9a\u6570 M \u306f, Mh = \u03c0 \u304c\u6210\u308a\u7acb\u3064\u3088\u3046\u306b\u9078\u3070\u308c\u308b. \u3053\u306e\u5909\u63db\u306b\u3088\u308a, x\u304c\u5927\u304d\u304f\u306a\u308b\u3068\u304d\u306b\u516c\u5f0f\u306e\u5206\u70b9\u304c cos(\u03c9x) \u3042\u308b\u3044\u306f sin(\u03c9x) \u306e\u30bc\u30ed\u70b9\u306b\u4e8c\u91cd\u6307\u6570\u95a2\u6570\u7684\u306b\u8fd1\u3065\u304f\u3088\u3046\u306b\u306a\u308a, \u5927\u304d\u306ax\u306b\u3064\u3044\u3066\u306f\u95a2\u6570\u5024\u3092\u8a08\u7b97\u3057\u306a\u304f\u3066\u3082\u3088\u304f\u306a\u308b.<\/p>\n
\u03c6(t)\u306f\u3044\u304f\u3064\u304b\u63d0\u6848\u3055\u308c\u3066\u3044\u308b\u304c, \u4f8b\u3048\u3070\u6b21\u5f0f\u304c\u3042\u308b.<\/p>\n
  \r\n\u03c6(t) = t\/(1 - exp(-ksinh(t))), k = 6 (\u307e\u305f\u306f 2\u03c0) (\u6587\u732e[2], [3], [7])\r\n  \u307e\u305f\u306f\r\n\u03c6(t) = t\/(1 - exp(-2t -\u03b1(1 - exp(-t)) - \u03b2(exp(t) - 1))), \u03b2 = 1\/4, \u03b1 = \u03b2\/sqrt(1 + Mlog(1 + M)\/4\u03c0) (\u6587\u732e[8])\r\n\r\n<\/pre>\n
 
\n\u4e0b\u306e\u6570\u5024\u5b9f\u9a13(3)\u3067\u4f7f\u7528\u3059\u308b I = \u222blog(x)sin(x)dx [0, +\u221e] = -\u03b3 \u306e\u95a2\u6570\u306e\u30b0\u30e9\u30d5\u3092\u793a\u3059.<\/p>\n
\"\"<\/p>\n
\u6587\u732e[7]\u306e\u516c\u5f0f(K = 2\u03c0)\u3092\u4f7f\u3044, Tol = 10-5<\/sup> \u3068\u3057\u305f\u3068\u304d\u306e\u5206\u70b9\u3092\u8d64\u4e38\u3067\u793a\u3057\u305f. x\u304c\u5927\u304d\u304f\u306a\u308b\u3068\u5206\u70b9\u304c\u95a2\u6570\u306e\u30bc\u30ed\u70b9\u306b\u4e00\u81f4\u3057\u3066\u3044\u304f\u306e\u304c\u308f\u304b\u308b.<\/p>\n
\u3053\u306e\u7bc0\u306e\u53c2\u8003\u6587\u732e:
\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] Ooura and Mori, “A robust double exponential formula for Fourier-type integrals”, J. Comput. Appl. Math., 112 (1999), 229-241.<\/p>\n
\u6570\u5024\u5b9f\u9a13 (1)<\/h3>\n
<\/span>\u6027\u8cea\u306e\u3088\u3044\u95a2\u6570<\/div>
\n\u300c\u6570\u5024\u7a4d\u5206(1)\u300d\u3067\u5b9f\u9a13\u3057\u305f\u6027\u8cea\u306e\u3088\u3044\u95a2\u6570\u3092\u7528\u3044\u3066\u5909\u6570\u5909\u63db\u578b\u516c\u5f0f\u306e\u6027\u80fd\u3092\u898b\u3066\u307f\u308b.
\n\u6b21\u306e\u7a4d\u5206\u3092\u8a08\u7b97\u3059\u308b\u3002<\/p>\n
  \r\n  I = \u222b ex<\/sup> cos x dx [0, 1] = 1.37802461354736\r\n\r\n<\/pre>\n
 
\nDE\u516c\u5f0f\u3067\u306f\u30d7\u30ed\u30b0\u30e9\u30e0\u306e\u4f5c\u308a\u306b\u3088\u3063\u3066\u95a2\u6570\u8a55\u4fa1\u56de\u6570\u304c\u5909\u308f\u3063\u3066\u304f\u308b\u304c, \u3053\u3053\u3067\u306f\u304d\u3056\u307f\u307f\u5e45\u3092\u5909\u5316\u3055\u305b\u3066\u305d\u306e\u3068\u304d\u306e\u5b9f\u969b\u306e\u7cbe\u5ea6\u306b\u307f\u3042\u3063\u305f\u3068\u3053\u308d\u3067\u7121\u9650\u7a4d\u5206\u3092\u6253\u3061\u5207\u308b\u3088\u3046\u306b\u3057\u305f(\u81ea\u52d5\u7a4d\u5206\u3067\u306f\u306a\u3044).<\/p>\n
\"\"<\/p>\n
\u6bd4\u8f03\u306e\u305f\u3081\u53f0\u5f62\u5247, \u30b7\u30f3\u30d7\u30bd\u30f3\u5247, \u30ac\u30a6\u30b9\u516c\u5f0f\u3082\u30d7\u30ed\u30c3\u30c8\u3057\u305f.<\/p>\n
IMT\u516c\u5f0f\u306f\u307b\u307c\u30b7\u30f3\u30d7\u30bd\u30f3\u5247\u306a\u307f\u306e\u6027\u80fd\u306b\u898b\u3048\u308b. DE\u516c\u5f0f\u306fIMT\u516c\u5f0f\u3088\u308a\u306f\u901f\u3044\u304c\u30ac\u30a6\u30b9\u516c\u5f0f\u307b\u3069\u3067\u306f\u306a\u3044.
\n<\/div><\/div>\n
\u6570\u5024\u5b9f\u9a13 (2)<\/h3>\n
<\/span>\u7aef\u70b9\u306b\u7279\u7570\u70b9\u3092\u6301\u3064\u95a2\u6570<\/div>
\n\u300c\u6570\u5024\u7a4d\u5206(1)\u300d\u3067\u5b9f\u9a13\u3057\u305f\u7aef\u70b9\u306b\u7279\u7570\u70b9\u3092\u6301\u3064\u95a2\u6570\u306e\u5834\u5408\u306b\u3064\u3044\u3066\u898b\u3066\u307f\u308b.<\/p>\n
  \r\n  I = \u222b \u221a(1-x2<\/sup>)dx [0, 1] = \u03c0\/4\r\n\r\n<\/pre>\n
 
\nx = 1 \u306b\u7279\u7570\u70b9\u304c\u3042\u308b.<\/p>\n
\"\"<\/p>\n
\u524d\u306b\u3082\u793a\u3057\u305f\u3088\u3046\u306b\u88dc\u9593\u578b\u516c\u5f0f\u3067\u306f\u3069\u308c\u3082\u307e\u3068\u3082\u306a\u7d50\u679c\u304c\u5f97\u3089\u308c\u306a\u3044\u304c, IMT\u516c\u5f0f\u3068DE\u516c\u5f0f\u306f\u7279\u7570\u70b9\u306e\u5f71\u97ff\u3092\u53d7\u3051\u305a\u306b\u6027\u8cea\u306e\u3088\u3044\u95a2\u6570\u306e\u3068\u304d\u3068\u307b\u307c\u540c\u3058\u3088\u3046\u306b\u554f\u984c\u306a\u304f\u8a08\u7b97\u3067\u304d\u305f. \u3053\u306e\u3088\u3046\u306b, \u7aef\u70b9\u306b\u7279\u7570\u70b9\u304c\u3042\u308b(\u7aef\u70b9\u3067\u5fae\u5206\u4fc2\u6570\u304c\u5b9a\u7fa9\u3067\u304d\u306a\u3044)\u5834\u5408\u306b\u3082\u9069\u7528\u3067\u304d\u308b\u3053\u3068\u306fIMT\u516c\u5f0f\u3068DE\u516c\u5f0f\u306e\u7279\u9577\u306e\u3072\u3068\u3064\u3067\u3042\u308b.
\n<\/div><\/div>\n
\u6570\u5024\u5b9f\u9a13 (3)<\/h3>\n
<\/span>\u30d5\u30fc\u30ea\u30a8\u578b\u7a4d\u5206<\/div>
\n\u6b21\u306e\u30d5\u30fc\u30ea\u30a8\u578b\u7a4d\u5206\u3092\u8a08\u7b97\u3059\u308b.<\/p>\n
  \r\n  I = \u222blog(x)sin(x)dx [0, +\u221e] = -\u03b3\r\n\r\n<\/pre>\n
 
\n\"\"<\/p>\n
\u3053\u3053\u3067\u306f\u6587\u732e[7]\u306e\u516c\u5f0f\u3092\u4f7f\u7528\u3057\u305f(K = 2\u03c0). \u8981\u6c42\u7cbe\u5ea6 Tol \u3092\u9069\u5f53\u306b\u5909\u5316\u3055\u305b, M = -\u03c0log(Tol)\/2 \u3068\u56fa\u5b9a\u3057\u3066\u8a08\u7b97\u3057\u305f. \u3061\u306a\u307f\u306b, XLPack\u306b\u53ce\u9332\u3055\u308c\u3066\u3044\u308b Deoint \u3067\u306f\u63a8\u5b9a\u8aa4\u5dee\u3092\u307f\u3066 M \u306e\u5024\u3092\u8abf\u7bc0\u3059\u308b\u3088\u3046\u306b\u306a\u3063\u3066\u3044\u308b.
\n<\/div><\/div>\n<\/div>\n
\n
\u8aa4\u5dee\u306e\u63a8\u5b9a<\/h2>\n
\u300c\u6570\u5024\u7a4d\u5206(1) \u88dc\u5b8c\u578b\u7a4d\u5206\u516c\u5f0f\u300d\u3067\u898b\u305f\u3088\u3046\u306b\u95a2\u6570\u8a55\u4fa1\u56de\u6570\u304c\u540c\u7a0b\u5ea6\u3067\u3042\u308c\u3070\u7a2e\u3005\u306e\u516c\u5f0f\u306e\u4e2d\u3067\u30ac\u30a6\u30b9\u516c\u5f0f\u304c\u6700\u3082\u7cbe\u5ea6\u304c\u826f\u304b\u3063\u305f. \u5358\u306b\u8a08\u7b97\u3059\u308b\u306e\u3067\u3042\u308c\u3070\u3053\u308c\u3067\u826f\u3044\u304c, \u5b9f\u52d9\u7684\u306b\u306f\u7a4d\u5206\u5024\u3068\u540c\u6642\u306b\u63a8\u5b9a\u8aa4\u5dee\u3082\u77e5\u308a\u305f\u3044. \u307e\u305f, \u5f8c\u3067\u793a\u3059\u81ea\u52d5\u7a4d\u5206\u306e\u30d9\u30fc\u30b9\u3068\u3057\u3066\u4f7f\u3046\u5834\u5408\u306b\u306f\u8aa4\u5dee\u63a8\u5b9a\u304c\u3067\u304d\u306a\u3051\u308c\u3070\u306a\u3089\u306a\u3044. \u30ac\u30a6\u30b9\u516c\u5f0f\u306e\u5f31\u70b9\u306f\u8aa4\u5dee\u306e\u63a8\u5b9a\u304c\u3057\u306b\u304f\u3044\u3053\u3068\u3067\u3042\u308b.<\/p>\n
\u53f0\u5f62\u5247\u306e\u8aa4\u5dee\u63a8\u5b9a<\/h3>\n
\u53f0\u5f62\u5247\u306e\u8a08\u7b97\u7d50\u679c\u306e\u8aa4\u5dee\u3092\u63a8\u5b9a\u3059\u308b\u305f\u3081\u306b\u306f, \u4f8b\u3048\u3070, N\u5206\u70b9\u3067\u8a08\u7b97\u3057\u305f\u3082\u306e\u30682N\u5206\u70b9\u3067\u8a08\u7b97\u3057\u305f\u3082\u306e\u3092\u6bd4\u8f03\u3059\u308b\u306a\u3069\u3059\u308c\u3070\u3088\u3044. \u6b21\u306e\u4f8b(\u6ed1\u3089\u304b\u306a\u95a2\u6570)\u3067\u53f0\u5f62\u5247\u306e\u8a08\u7b97\u4f8b\u3092\u898b\u3066\u307f\u308b.<\/p>\n
  \r\n  I = \u222b ex<\/sup> cos x dx [0, 1] = 1.37802461354736\r\n\r\n<\/pre>\n
 
\n
\n\n\n\n\n\n\n\n\n
\u5206\u70b9\u6570<\/th>\n\u7a4d\u5206\u5024(\u8a08\u7b97\u5024)<\/th>\n\u7a4d\u5206\u5024(\u771f\u5024)<\/th>\n\u8aa4\u5dee<\/th>\n\u8aa4\u5dee\u63a8\u5b9a\u5024<\/th>\n<\/tr>\n
2<\/td>\n1.34061800327106<\/td>\n1.37802461354736<\/td>\n0.037406610<\/td>\n0.035423678<\/td>\n<\/tr>\n
4<\/td>\n1.36858238253106<\/td>\n1.37802461354736<\/td>\n0.009442231<\/td>\n0.009321460<\/td>\n<\/tr>\n
8<\/td>\n1.37565843490021<\/td>\n1.37802461354736<\/td>\n0.002366179<\/td>\n0.002358684<\/td>\n<\/tr>\n
16<\/td>\n1.37743271822098<\/td>\n1.37802461354736<\/td>\n0.000591895<\/td>\n0.000591428<\/td>\n<\/tr>\n
32<\/td>\n1.37787661780930<\/td>\n1.37802461354736<\/td>\n0.000147996<\/td>\n0.000147967<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<\/div>\n\u8aa4\u5dee\u63a8\u5b9a\u5024\u306f\u30ed\u30f3\u30d0\u30fc\u30b0\u7a4d\u5206\u306e\u6642\u3068\u540c\u3058\u3088\u3046\u306b\u6b21\u306e\u5f0f\u304b\u3089\u8a08\u7b97\u3057\u305f. \u5b9f\u969b\u306e\u8aa4\u5dee\u3068\u63a8\u5b9a\u5024\u306f\u3088\u304f\u4e00\u81f4\u3057\u3066\u3044\u308b.<\/p>\n
  \r\n  \u63a8\u5b9a\u8aa4\u5dee = (4I - Iprev<\/sub>)\/3\r\n\r\n<\/pre>\n
 
\n\u3053\u3053\u3067\u306f\u5206\u70b9\u6570\u3092\u500d\u3005\u306b\u3059\u308b\u3088\u3046\u306a\u9806\u5e8f(\u5206\u5272\u5e45\u306f\u7b49\u9593\u9694\u3067\u534a\u5206)\u3067\u8a08\u7b97\u3092\u884c\u3063\u305f. I\u306f\u7a4d\u5206\u306e\u8a08\u7b97\u5024, Iprev<\/sub>\u306f\u5206\u70b9\u6570\u304c\u534a\u5206\u306e\u6642\u306e(\u3064\u307e\u308a1\u3064\u524d\u306e)\u8a08\u7b97\u5024\u3067\u3042\u308b. \u3053\u306e\u5834\u5408, \u5206\u70b9\u6570\u3092\u500d\u306b\u3057\u305f\u6642\u306b\u3082\u3046\u4e00\u56de\u6700\u521d\u304b\u3089\u5168\u90e8\u306e\u5206\u70b9\u3067\u8a08\u7b97\u3059\u308b\u5fc5\u8981\u306f\u306a\u304f, \u534a\u5206\u306f\u3059\u3067\u306b\u8a08\u7b97\u6e08\u306e\u5024\u304c\u305d\u306e\u307e\u307e\u4f7f\u3048\u308b.<\/p>\n
\u8aa4\u5dee\u63a8\u5b9a\u6a5f\u80fd\u4ed8\u304d\u306e\u30ac\u30a6\u30b9\u516c\u5f0f<\/h3>\n
\u53f0\u5f62\u5247\u3068\u306f\u7570\u306a\u308a, \u30ac\u30a6\u30b9\u516c\u5f0f\u3067\u306f\u5206\u70b9\u304c\u7b49\u9593\u9694\u3067\u306a\u3044\u305f\u3081, \u5206\u70b9\u6570\u3092\u500d\u306b\u3059\u308b\u3068\u3082\u3046\u4e00\u56de\u6700\u521d\u304b\u3089\u5168\u90e8\u306e\u5206\u70b9\u3067\u8a08\u7b97\u3059\u308b\u5fc5\u8981\u304c\u3042\u308b\u306e\u304c\u5f31\u70b9\u3067\u3042\u308b.<\/p>\n
\u3053\u308c\u3092\u89e3\u6c7a\u3059\u308b\u305f\u3081\u306b, N\u70b9\u30ac\u30a6\u30b9\u516c\u5f0f\u306e\u8a08\u7b97\u5024\u3092\u751f\u304b\u3057\u305f\u307e\u307e 2N+1\u70b9\u3092\u4f7f\u3063\u305f\u7a4d\u5206\u516c\u5f0f\u304c\u30ac\u30a6\u30b9\u30fb\u30af\u30ed\u30f3\u30ed\u30c3\u30c9\u516c\u5f0f<\/strong>\u3067\u3042\u308b.<\/p>\n
N\u70b9\u30ac\u30a6\u30b9\u516c\u5f0f\u306b\u3088\u308b\u7a4d\u5206\u3092 GN<\/sub> \u3067\u8868\u3059(\u7a4d\u5206\u533a\u9593\u3092 [-1, 1] \u3068\u3059\u308b).<\/p>\n
  \r\n  GN<\/sub> = \u03a3wi<\/sub>f(xi<\/sub>) (i = 1\uff5eN)\r\n\r\n<\/pre>\n
 
\n\u30ac\u30a6\u30b9\u30fb\u30af\u30ed\u30f3\u30ed\u30c3\u30c9\u516c\u5f0f\u3067\u306f\u540c\u3058N\u500b\u306e\u5206\u70b9 xi<\/sub> \u3068\u65b0\u305f\u306b\u8a08\u7b97\u304c\u5fc5\u8981\u306a N+1\u500b\u306e\u5206\u70b9 yi<\/sub> \u3092\u4f7f\u3063\u3066\u6c42\u3081\u308b\u7a4d\u5206 K2N+1<\/sub> \u3092\u6b21\u5f0f\u3067\u8868\u3059.<\/p>\n
  \r\n  K2N+1<\/sub> = \u03a3ai<\/sub>f(xi<\/sub>) + \u03a3bj<\/sub>f(yj<\/sub>)  (i = 1\uff5eN, j = 1\uff5eN+1)\r\n\r\n<\/pre>\n
 
\nN\u500b\u306e\u5206\u70b9 xi<\/sub> \u306b\u304a\u3051\u308b\u4fc2\u6570 ai<\/sub> \u306f\u30ac\u30a6\u30b9\u516c\u5f0f\u306e\u4fc2\u6570 wi<\/sub> \u3068\u306f\u7570\u306a\u308b\u5024\u306b\u306a\u308b.<\/p>\n
N\u70b9\u30ac\u30a6\u30b9\u516c\u5f0f GN<\/sub> \u306f 2N-1\u6b21\u591a\u9805\u5f0f\u307e\u3067\u6b63\u3057\u3044\u89e3\u3092\u4e0e\u3048\u308b(\u3059\u306a\u308f\u3061 2N-1\u6b21\u3067\u3042\u308b)\u304c, \u30ac\u30a6\u30b9\u30fb\u30af\u30ed\u30f3\u30ed\u30c3\u30c9\u516c\u5f0f K2N+1<\/sub> \u306f 3N+1\u6b21(N\u304c\u5076\u6570\u306e\u3068\u304d\u306f 3N+2\u6b21)\u3068\u306a\u308b\u3088\u3046\u306b\u5206\u70b9\u3068\u91cd\u307f\u3092\u9078\u3076.<\/p>\n
xi<\/sub> \u306f N\u6b21\u306e\u30eb\u30b8\u30e3\u30f3\u30c9\u30eb\u591a\u9805\u5f0f PN<\/sub>(x) \u306e\u30bc\u30ed\u70b9\u3067\u3042\u308b\u304c, \u6b8b\u308a\u306e\u5206\u70b9 yi<\/sub> \u306f n+1\u6b21\u306e\u591a\u9805\u5f0f EN+1<\/sub>(x) \u306e\u30bc\u30ed\u70b9\u306b\u306a\u308b\u3068\u3059\u308b\u3068, EN+1<\/sub>(x) \u306f\u6b21\u306e\u76f4\u4ea4\u95a2\u4fc2\u3092\u6e80\u305f\u3059.<\/p>\n
  \r\n  \u222b PN<\/sub>(x)EN+1<\/sub>(x)xk<\/sup>dx = 0 (k = 0, ..., N)\r\n\r\n<\/pre>\n
 
\n\u4e0a\u306e\u6761\u4ef6\u3092\u6e80\u305f\u3059\u591a\u9805\u5f0f EN+1<\/sub>(x) \u306e\u30bc\u30ed\u70b9\u306f\u9023\u7acb\u65b9\u7a0b\u5f0f\u3092\u89e3\u3044\u3066\u6c42\u3081\u308b\u3053\u3068\u304c\u3067\u304d, \u533a\u9593(-1, 1)\u306b\u5b58\u5728\u3057, PN<\/sub>(x) \u306e\u30bc\u30ed\u70b9\u3068\u4ea4\u4e92\u306b\u73fe\u308c\u308b\u3053\u3068\u304c\u308f\u304b\u3063\u3066\u3044\u308b. \u30af\u30ed\u30f3\u30ed\u30c3\u30c9\u306f\u3053\u306e\u9023\u7acb\u65b9\u7a0b\u5f0f\u3092\u89e3\u3044\u3066\u6c42\u3081\u305f\u304c, \u3053\u306e\u65b9\u7a0b\u5f0f\u306f\u975e\u5e38\u306b\u60aa\u6761\u4ef6\u3067\u89e3\u304d\u306b\u304f\u3044\u305f\u3081, \u5f8c\u306b\u3044\u304f\u3064\u304b\u306e\u8a08\u7b97\u6cd5\u304c\u63d0\u6848\u3055\u308c\u3066\u3044\u308b.<\/p>\n
N = 2\uff5e10 \u306e\u3068\u304d\u306e\u5206\u70b9\u304a\u3088\u3073\u91cd\u307f\u306e\u5024\u3092\u300c\u7a4d\u5206\u516c\u5f0f\u306e\u4fc2\u6570\u8868\u300d\u306b\u63b2\u8f09\u3057\u305f. \u3053\u3053\u3067\u306f, \u4e0b\u8a18\u6587\u732e[9]\u306b\u5f93\u3044, \u30e4\u30b3\u30d3\u30fb\u30af\u30ed\u30f3\u30ed\u30c3\u30c9\u884c\u5217\u3068\u3088\u3070\u308c\u308b\u5bfe\u79f0\u4e09\u91cd\u5bfe\u89d2\u884c\u5217\u3092\u4f5c\u308a\u305d\u306e\u56fa\u6709\u5024\u3092\u6c42\u3081\u308b\u65b9\u6cd5\u306b\u3088\u308a\u8a08\u7b97\u3057\u305f.<\/p>\n
\u30ac\u30a6\u30b9\u30fb\u30af\u30ed\u30f3\u30ed\u30c3\u30c9\u516c\u5f0f\u3067\u306f, 2N+1\u56de\u306e\u95a2\u6570\u8a55\u4fa1\u3067 2N+1\u6b21\u306e GN<\/sub> \u3068 3N+1\u6b21\u306e K2N+1<\/sub> \u306e\u4e21\u65b9\u3092\u6c42\u3081\u308b\u3053\u3068\u304c\u3067\u304d, \u3053\u306e2\u3064\u3092\u6bd4\u3079\u308b\u3053\u3068\u306b\u3088\u308a\u8aa4\u5dee\u3092\u63a8\u5b9a\u3059\u308b\u3053\u3068\u304c\u3067\u304d\u308b. \u63a8\u5b9a\u8aa4\u5dee\u306f\u7d4c\u9a13\u7684\u306b\u6b21\u5f0f\u3067\u8868\u3055\u308c\u308b.<\/p>\n
  \r\n  \u63a8\u5b9a\u8aa4\u5dee = (200*|GN<\/sub> - K2N+1<\/sub>|)1.5<\/sup>\r\n\r\n<\/pre>\n
 
\n\u3053\u306e\u5f0f\u306b\u3088\u308b\u63a8\u5b9a\u8aa4\u5dee\u306f\u5b89\u5168\u5074\u306b\u6c42\u3081\u3089\u308c\u308b\u3068\u3055\u308c\u3066\u3044\u308b.<\/p>\n
\u30ac\u30a6\u30b9\u30fb\u30af\u30ed\u30f3\u30ed\u30c3\u30c9\u516c\u5f0f K2N+1<\/sub> \u306f 3N+1\u6b21\u3067\u3042\u308b\u304c, \u540c\u3058\u95a2\u6570\u8a55\u4fa1\u56de\u6570\u3067\u8a08\u7b97\u3067\u304d\u308b G2N+1<\/sub> \u306f 4N+1\u6b21\u3068\u306a\u308b. \u30ac\u30a6\u30b9\u30fb\u30af\u30ed\u30f3\u30ed\u30c3\u30c9\u516c\u5f0f\u306f\u7cbe\u5ea6\u7684\u306b\u306f\u30ac\u30a6\u30b9\u516c\u5f0f\u306b\u52a3\u308b\u304c, \u8aa4\u5dee\u306e\u63a8\u5b9a\u304c\u3067\u304d\u308b\u5229\u70b9\u306e\u65b9\u304c\u5927\u304d\u3044\u5834\u5408\u306b\u6709\u52b9\u3067\u3042\u308b.<\/p>\n
\u3053\u306e\u7bc0\u306e\u53c2\u8003\u6587\u732e:
\n[9] D. P. Laurie, “Calculation of Gauss-Kronrod quadrature rules”, Mathematics of Computation, Vol.66 No.219 (1997).<\/p>\n
\u6570\u5024\u5b9f\u9a13 (4)<\/h3>\n
<\/span>\u30ac\u30a6\u30b9\u30fb\u30af\u30ed\u30f3\u30ed\u30c3\u30c9\u516c\u5f0f<\/div>
\n\u30ac\u30a6\u30b9\u30fb\u30af\u30ed\u30f3\u30ed\u30c3\u30c9\u516c\u5f0f\u3092\u7528\u3044\u3066\u6b21\u306e\u7a4d\u5206\u3092\u6c42\u3081\u3066\u307f\u308b\u3002<\/p>\n
  \r\n  I = \u222b ex<\/sup> cos x dx [0, 1] = 1.37802461354736\r\n\r\n<\/pre>\n
 
\n\u3053\u308c\u306f\u524d\u306b\u3082\u4f8b\u984c\u306b\u4f7f\u7528\u3057\u305f\u6027\u8cea\u306e\u3088\u3044\u95a2\u6570\u306e\u4f8b\u3067\u3042\u308a, \u8a08\u7b97\u7d50\u679c\u306f\u6b21\u306e\u3088\u3046\u306b\u306a\u3063\u305f.<\/p>\n
\"\"<\/p>\n
\u6a2a\u8ef8\u306b\u95a2\u6570\u8a55\u4fa1\u56de\u6570, \u7e26\u8ef8\u306b\u8aa4\u5dee\u3092\u30d7\u30ed\u30c3\u30c8\u3057\u305f. \u7dd1\u8272\u306e\u30c7\u30fc\u30bf\u306f\u524d\u9805\u306e\u5f0f\u3067\u6c42\u3081\u305f\u63a8\u5b9a\u8aa4\u5dee\u3067\u3042\u308b. \u5404\u30c7\u30fc\u30bf\u70b9\u306b\u8868\u793a\u3057\u305f\u6570\u5b57\u306f\u5206\u70b9\u6570(\u30ac\u30a6\u30b9\u516c\u5f0f\u3067\u306f N, \u30ac\u30a6\u30b9\u30fb\u30af\u30ed\u30f3\u30ed\u30c3\u30c9\u516c\u5f0f\u3067\u306f 2N + 1 \u306e\u5024)\u3067\u3042\u308b.<\/p>\n
\u63a8\u5b9a\u8aa4\u5dee\u306f\u304b\u306a\u308a\u5b89\u5168\u5074\u306b\u7b97\u51fa\u3055\u308c\u3066\u3044\u308b\u306e\u304c\u308f\u304b\u308b.
\n<\/div><\/div>\n
<\/span>\u30ac\u30a6\u30b9\u30fb\u30af\u30ed\u30f3\u30ed\u30c3\u30c9\u516c\u5f0f(\u7279\u7570\u6027\u306e\u3042\u308b\u5834\u5408)<\/div>
\n\u4e0a\u306e\u4f8b\u984c\u3067\u306f\u53ce\u675f\u304c\u901f\u304f\u3059\u3050\u306b\u4e38\u3081\u8aa4\u5dee\u306e\u9818\u57df\u306b\u5165\u3063\u3066\u3044\u3066\u3046\u307e\u304f\u6bd4\u8f03\u3067\u304d\u306a\u3044\u306e\u3067\u3082\u3046\u5c11\u3057\u96e3\u3057\u3044\u6b21\u306e\u7a4d\u5206\u3092\u8a08\u7b97\u3057\u3066\u307f\u308b.<\/p>\n
  \r\n  I = \u222b 1\/(1+x)dx [0, 4] = ln(1+4) = ln(5)\r\n\r\n<\/pre>\n
 
\n\u3053\u308c\u3082\u524d\u306b\u4f8b\u984c\u306b\u4f7f\u7528\u3057\u305f\u95a2\u6570\u3067, \u7a4d\u5206\u533a\u9593\u5916\u3067\u3042\u308b\u304c\u8fd1\u304f\u306b\u7279\u7570\u70b9\u3092\u6301\u3064\u4f8b\u3067\u3042\u308b.<\/p>\n
\"\"<\/p>\n
\u3053\u308c\u3092\u898b\u308b\u3068\u30ac\u30a6\u30b9\u30fb\u30af\u30ed\u30f3\u30ed\u30c3\u30c9\u516c\u5f0f\u3092\u4f7f\u3063\u3066\u3082\u305d\u308c\u307b\u3069\u5927\u304d\u306a\u30da\u30ca\u30eb\u30c6\u30a3\u3092\u6255\u308f\u305a\u306b\u8aa4\u5dee\u3092\u63a8\u5b9a\u3067\u304d\u308b\u3053\u3068\u304c\u308f\u304b\u308b.
\n<\/div><\/div>\n<\/div>\n
\n
\u81ea\u52d5\u7a4d\u5206<\/h2>\n
\u81ea\u52d5\u7a4d\u5206\u30eb\u30fc\u30c1\u30f3\u3068\u306f, \u7a4d\u5206\u533a\u9593\u3068\u8981\u6c42\u7cbe\u5ea6\u3092\u4e0e\u3048\u308b\u3068\u8aa4\u5dee\u63a8\u5b9a\u5024\u304c\u8981\u6c42\u7cbe\u5ea6\u3092\u6e80\u305f\u3059\u307e\u3067\u304d\u3056\u307f\u5e45\u3092\u9806\u6b21\u5c0f\u3055\u304f\u3059\u308b\u306a\u3069\u3057\u3066\u7a4d\u5206\u3092\u7e70\u308a\u8fd4\u3059\u30d7\u30ed\u30b0\u30e9\u30e0\u3067\u3042\u308b.<\/p>\n
\u9069\u5fdc\u81ea\u52d5\u7a4d\u5206\u30eb\u30fc\u30c1\u30f3\u306f\u3053\u308c\u3092\u4e00\u6b69\u9032\u3081\u3066, \u7a4d\u5206\u533a\u9593\u3092\u90e8\u5206\u533a\u9593\u306b\u5206\u5272\u3057, \u8aa4\u5dee\u63a8\u5b9a\u5024\u304c\u8981\u6c42\u3055\u308c\u305f\u7cbe\u5ea6\u3092\u6e80\u305f\u3059\u3088\u3046\u306b\u90e8\u5206\u533a\u9593\u306e\u5e45\u3092\u81ea\u52d5\u7684\u306b\u8abf\u7bc0\u3059\u308b. \u88ab\u7a4d\u5206\u95a2\u6570\u304c\u306a\u3081\u3089\u304b\u306b\u3086\u3063\u304f\u308a\u5909\u5316\u3059\u308b\u3088\u3046\u306a\u533a\u9593\u3067\u306f\u5927\u304d\u306a\u5e45\u3092\u7528\u3044, \u8aa4\u5dee\u63a8\u5b9a\u5024\u304c\u5927\u304d\u3044\u90e8\u5206\u3067\u306f\u7d30\u304b\u3044\u533a\u9593\u306b\u5206\u5272\u3057\u3066\u304d\u3056\u307f\u5e45\u3092\u5c0f\u3055\u304f\u3057\u3066\u8a08\u7b97\u3059\u308b\u3053\u3068\u306b\u3088\u308a, \u6700\u5c0f\u306e\u8a08\u7b97\u91cf\u3067\u8981\u6c42\u7cbe\u5ea6\u306e\u7d50\u679c\u3092\u5f97\u3088\u3046\u3068\u3059\u308b\u3082\u306e\u3067\u3042\u308b.<\/p>\n
\u9069\u5fdc\u81ea\u52d5\u7a4d\u5206\u3068\u3057\u3066\u306f, \u8aa4\u5dee\u63a8\u5b9a\u3092\u884c\u3046\u305f\u3081\u3068\u8a08\u7b97\u6e08\u306e\u95a2\u6570\u5024\u3092\u518d\u5229\u7528\u3059\u308b\u305f\u3081\u306b, \u5206\u70b9\u304c\u7b49\u9593\u9694\u306a\u30b7\u30f3\u30d7\u30bd\u30f3\u5247\u3084\u30cb\u30e5\u30fc\u30c8\u30f3\u30fb\u30b3\u30fc\u30c4\u516c\u5f0f\u304c\u4f7f\u308f\u308c\u3066\u304d\u305f. \u3057\u304b\u3057, \u73fe\u5728\u3067\u306f\u7cbe\u5ea6\u3092\u8ffd\u6c42\u3057\u3066, \u8aa4\u5dee\u63a8\u5b9a\u6a5f\u80fd\u3092\u6301\u3064\u30ac\u30a6\u30b9\u30fb\u30af\u30ed\u30f3\u30ed\u30c3\u30c9\u516c\u5f0f\u306e\u3088\u3046\u306a\u4e0d\u7b49\u9593\u9694\u5206\u70b9\u3067\u306f\u3042\u308b\u304c\u9ad8\u7cbe\u5ea6\u306e\u516c\u5f0f\u3092\u4f7f\u3063\u305f\u3082\u306e\u304c\u4f7f\u308f\u308c\u3066\u3044\u308b.<\/p>\n
\u9069\u5fdc\u81ea\u52d5\u7a4d\u5206\u30d7\u30ed\u30b0\u30e9\u30e0 QAG<\/h3>\n
\u9069\u5fdc\u81ea\u52d5\u7a4d\u5206\u30eb\u30fc\u30c1\u30f3\u306e\u4f8b\u3068\u3057\u3066 QUADPACK (\u6587\u732e[6]) \u306e QAG \u306b\u3064\u3044\u3066\u7c21\u5358\u306b\u8aac\u660e\u3059\u308b. QAG \u306f\u8981\u6c42\u7cbe\u5ea6\u3092\u6e80\u305f\u3059\u307e\u3067\u81ea\u52d5\u7684\u306b\u7a4d\u5206\u533a\u9593\u3092\u90e8\u5206\u533a\u9593\u306b\u5206\u5272\u3057\u306a\u304c\u3089\u30ac\u30a6\u30b9\u30fb\u30af\u30ed\u30f3\u30ed\u30c3\u30c9\u516c\u5f0f\u306e\u56fa\u5b9a\u5206\u70b9\u306e\u30eb\u30fc\u30c1\u30f3(QK15, …, QK61)\u3092\u547c\u3073\u51fa\u3059\u3053\u3068\u306b\u3088\u308a\u81ea\u52d5\u7a4d\u5206\u3092\u884c\u3046. XLPack\u306b\u306fQag\u3068\u3057\u3066\u53ce\u9332\u3055\u308c\u3066\u304a\u308a, \u901a\u5e38\u306e\u6570\u5024\u7a4d\u5206\u30d7\u30ed\u30b0\u30e9\u30e0\u3068\u3057\u3066\u306f\u6700\u3082\u983b\u7e41\u306b\u4f7f\u7528\u3055\u308c\u308b.<\/p>\n
\u3053\u306e\u30d7\u30ed\u30b0\u30e9\u30e0\u306e\u52d5\u4f5c\u306e\u6982\u8981\u306f\u6b21\u306e\u3068\u304a\u308a\u3067\u3042\u308b.<\/p>\n
\u4e0e\u3048\u3089\u308c\u305f\u53ce\u675f\u6761\u4ef6\u3092\u300c\u63a8\u5b9a\u8aa4\u5dee <= Tol\u300d\u3068\u3059\u308b.<\/p>\n
    \n
  1. \u307e\u305a, \u5168\u533a\u9593\u3092\u30ac\u30a6\u30b9\u30fb\u30af\u30ed\u30f3\u30ed\u30c3\u30c9\u516c\u5f0f\u306e\u56fa\u5b9a\u5206\u70b9\u30eb\u30fc\u30c1\u30f3\u306b\u3088\u308a\u7a4d\u5206\u3092\u884c\u3044\u7a4d\u5206\u5024 S \u3068\u63a8\u5b9a\u8aa4\u5dee E \u3092\u6c42\u3081\u308b.<\/li>\n
  2. E > Tol \u3067\u3042\u308b\u9650\u308a\u4ee5\u4e0b\u3092\u7e70\u308a\u8fd4\u3059.\n
      \n
    1. \u63a8\u5b9a\u8aa4\u5dee\u304c\u6700\u3082\u5927\u304d\u3044\u90e8\u5206\u533a\u9593\u3092\u534a\u5206\u306b\u5206\u5272\u3057\u3066\u305d\u308c\u305e\u308c\u3092\u56fa\u5b9a\u5206\u70b9\u30eb\u30fc\u30c1\u30f3\u306b\u3088\u308a\u7a4d\u5206\u3057, \u7a4d\u5206\u5024 Si<\/sub> \u3068\u63a8\u5b9a\u8aa4\u5dee Ei<\/sub> \u3092\u6c42\u3081\u308b(i<\/sub> \u306f\u90e8\u5206\u533a\u9593\u3054\u3068\u306e\u5024\u3092\u793a\u3059).<\/li>\n
    2. \u65b0\u305f\u306b\u69cb\u6210\u3055\u308c\u305f\u3082\u306e\u3092\u542b\u3081\u305f\u3059\u3079\u3066\u306e\u90e8\u5206\u533a\u9593\u306b\u304a\u3051\u308b\u7a4d\u5206\u5024\u3068\u63a8\u5b9a\u8aa4\u5dee\u306e\u305d\u308c\u305e\u308c\u306e\u7dcf\u548c S \u3068 E \u3092\u66f4\u65b0\u3059\u308b.<\/li>\n<\/ol>\n<\/li>\n
    3. E <= Tol \u306b\u306a\u3063\u305f\u3089\u305d\u306e\u6642\u306e\u7a4d\u5206\u5024\u306e\u7dcf\u548c S \u3092\u6700\u7d42\u7d50\u679c\u3068\u3057\u3066\u8fd4\u3059.<\/li>\n<\/ol>\n
      QAG \u3067\u306f\u30d9\u30fc\u30b9\u306e\u56fa\u5b9a\u5206\u70b9\u30eb\u30fc\u30c1\u30f3\u3068\u3057\u3066 QK15 \uff5e QK61 \u306e\u3046\u3061\u30e6\u30fc\u30b6\u30fc\u304c\u6307\u5b9a\u3057\u305f\u3082\u306e\u3092\u4f7f\u7528\u3057, \u3053\u308c\u3092\u81ea\u52d5\u7684\u306b\u5909\u66f4\u3059\u308b\u3053\u3068\u306f\u3057\u306a\u3044. \u307e\u305f, \u533a\u9593\u3092\u5206\u5272\u3059\u308b\u969b\u306b\u8a08\u7b97\u6e08\u306e\u95a2\u6570\u5024\u3092\u518d\u5229\u7528\u3059\u308b\u3053\u3068\u3082\u3057\u306a\u3044.<\/p>\n
      \u4f8b\u3068\u3057\u3066, \u533a\u9593[0,1]\u306b\u304a\u3044\u3066\u6b21\u306e\u95a2\u6570(\u6587\u732e[5]\u53c2\u7167)\u3092\u7a4d\u5206\u3057\u305f\u6642\u306e QAG \u306e\u6319\u52d5\u3092\u4e0b\u56f3\u306b\u793a\u3059.<\/p>\n
        \r\n  f(x) = 1\/((x - 0.3)2<\/sup> + 0.01) + 1\/((x - 0.9)2<\/sup> + 0.04) - 6\r\n\r\n<\/pre>\n
       <\/p>\n
      \"\"<\/p>\n
      \u3053\u3053\u3067\u306f, XLPack\u306b\u53ce\u9332\u3055\u308c\u3066\u3044\u308b Qag \u3092\u4f7f\u7528\u3057, \u76ee\u7684\u8aa4\u5dee\u3092 10-5<\/sup> \u3068\u3057\u3066\u56fa\u5b9a\u5206\u70b9\u30eb\u30fc\u30c1\u30f3\u306f QK15 \u3092\u6307\u5b9a\u3057\u3066\u8a08\u7b97\u3057\u305f. \u533a\u9593\u306f5\u3064\u306b\u5206\u5272\u3055\u308c\u305f(0~0.25, 0.25~0.375, 0.375~0.5, 0.5~0.75, 0.75~1). \u30c8\u30fc\u30bf\u30eb\u306e\u95a2\u6570\u8a55\u4fa1\u56de\u6570\u306f135\u56de\u3067\u3042\u3063\u305f. \u6700\u3082\u6025\u5cfb\u306a\u5909\u5316\u306e\u3068\u3053\u308d\u3067\u5206\u5272\u3055\u308c\u3066\u3044\u308b\u306e\u304c\u308f\u304b\u308b.<\/p>\n
      \u540c\u3058\u6761\u4ef6\u3067 QK61 \u3092\u6307\u5b9a\u3057\u3066\u307f\u308b\u3068\u6b21\u306e\u3088\u3046\u306b\u306a\u3063\u305f.<\/p>\n
      \"\"<\/p>\n
      2\u5206\u5272\u3067\u8a08\u7b97\u3057\u3066\u304a\u308a(0~0.5, 0.5~1), \u30c8\u30fc\u30bf\u30eb\u306e\u95a2\u6570\u8a55\u4fa1\u56de\u6570\u306f183\u56de\u306b\u306a\u3063\u305f. \u5909\u5316\u304c\u6fc0\u3057\u3044\u5834\u5408\u306b\u306f QK15 \u3092\u9078\u629e\u3057\u5206\u5272\u6570\u3092\u591a\u304f\u3057, \u305d\u3046\u3067\u306a\u3044\u5834\u5408\u306b\u306f QK61 \u306e\u3088\u3046\u306b\u9ad8\u7cbe\u5ea6\u306e\u3082\u306e\u3092\u9078\u3073\u5206\u5272\u6570\u3092\u5c11\u306a\u304f\u3059\u308b\u3068\u3088\u3055\u305d\u3046\u3067\u3042\u308b.
      \n<\/div>\n
      \n
      \u7a4d\u5206\u516c\u5f0f\u306e\u4fc2\u6570\u8868<\/h2>\n
      \u30ac\u30a6\u30b9\u30fb\u30af\u30ed\u30f3\u30ed\u30c3\u30c9\u516c\u5f0f<\/h4>\n
        \r\n  K2N+1<\/sub> = \u03a3 ai<\/sub> f(xi<\/sub>) + \u03a3 bj<\/sub> f(yj<\/sub>)  (\u7b2c1\u9805\u306e\u03a3\u306f i = 1\uff5eN, \u7b2c2\u9805\u306e\u03a3\u306f j = 1\uff5eN+1)\r\n\r\n<\/pre>\n
       
      \n\u30ac\u30a6\u30b9\u30fb\u30af\u30ed\u30f3\u30ed\u30c3\u30c9\u516c\u5f0f\u306b\u304a\u3051\u308b\u5206\u70b9 xi<\/sub>, yj<\/sub> \u304a\u3088\u3073\u91cd\u307f ai<\/sub>, bj<\/sub> \u3092\u4ee5\u4e0b\u306b\u793a\u3059(xi<\/sub> \u306f\u30ac\u30a6\u30b9\u516c\u5f0f\u306e xi<\/sub> \u3068\u540c\u3058). \u8a08\u7b97\u306f\u62e1\u5f35\u500d\u7cbe\u5ea6(80\u30d3\u30c3\u30c8\u6d6e\u52d5\u5c0f\u6570)\u306b\u3088\u308a\u884c\u3044, \u7d50\u679c\u306e16\u6841\u3092\u8868\u793a\u3057\u305f.<\/p>\n
      N \u306f\u5bfe\u5fdc\u3059\u308b\u30ac\u30a6\u30b9\u516c\u5f0f\u306e N \u306e\u5024\u3092\u793a\u3059. \u30ac\u30a6\u30b9\u30fb\u30af\u30ed\u30f3\u30ed\u30c3\u30c9\u516c\u5f0f\u3068\u3057\u3066\u306e\u5206\u70b9\u6570\u306f 2N + 1 \u3068\u306a\u308b (\u4f8b. N = 7 \u306f 15\u5206\u70b9).<\/p>\n
        \r\nN = 2\r\n  xi: -5.773502691896258E-0001  5.773502691896258E-0001\r\n  ai:  4.909090909090909E-0001  4.909090909090909E-0001\r\n  yi: -9.258200997725515E-0001  0.000000000000000E+0000  9.258200997725515E-0001\r\n  bi:  1.979797979797980E-0001  6.222222222222222E-0001  1.979797979797980E-0001\r\nN = 3\r\n  xi: -7.745966692414834E-0001  0.000000000000000E+0000  7.745966692414834E-0001\r\n  ai:  2.684880898683334E-0001  4.509165386584741E-0001  2.684880898683334E-0001\r\n  yi: -9.604912687080203E-0001 -4.342437493468026E-0001  4.342437493468026E-0001\r\n       9.604912687080203E-0001\r\n  bi:  1.046562260264673E-0001  4.013974147759622E-0001  4.013974147759622E-0001\r\n       1.046562260264673E-0001\r\nN = 4\r\n  xi: -8.611363115940526E-0001 -3.399810435848563E-0001  3.399810435848563E-0001\r\n       8.611363115940526E-0001\r\n  ai:  1.700536053357227E-0001  3.269491896014516E-0001  3.269491896014516E-0001\r\n       1.700536053357227E-0001\r\n  yi: -9.765602507375731E-0001 -6.402862174963100E-0001  0.000000000000000E+0000\r\n       6.402862174963100E-0001  9.765602507375731E-0001\r\n  bi:  6.297737366547301E-0002  2.667983404522844E-0001  3.464429818901364E-0001\r\n       2.667983404522844E-0001  6.297737366547301E-0002\r\nN = 5\r\n  xi: -9.061798459386640E-0001 -5.384693101056831E-0001  0.000000000000000E+0000\r\n       5.384693101056831E-0001  9.061798459386640E-0001\r\n  ai:  1.152333166224734E-0001  2.410403392286476E-0001  2.829874178574912E-0001\r\n       2.410403392286476E-0001  1.152333166224734E-0001\r\n  yi: -9.840853600948425E-0001 -7.541667265708492E-0001 -2.796304131617832E-0001\r\n       2.796304131617832E-0001  7.541667265708492E-0001  9.840853600948425E-0001\r\n  bi:  4.258203675108183E-0002  1.868007965564927E-0001  2.728498019125589E-0001\r\n       2.728498019125589E-0001  1.868007965564927E-0001  4.258203675108183E-0002\r\nN = 6\r\n  xi: -9.324695142031520E-0001 -6.612093864662645E-0001 -2.386191860831969E-0001\r\n       2.386191860831969E-0001  6.612093864662645E-0001  9.324695142031520E-0001\r\n  ai:  8.369444044690663E-0002  1.810719943231376E-0001  2.337708641169944E-0001\r\n       2.337708641169944E-0001  1.810719943231376E-0001  8.369444044690663E-0002\r\n  yi: -9.887032026126789E-0001 -8.213733408650279E-0001 -4.631182124753046E-0001\r\n       0.000000000000000E+0000  4.631182124753046E-0001  8.213733408650279E-0001\r\n       9.887032026126789E-0001\r\n  bi:  3.039615411981977E-0002  1.373206046344469E-0001  2.132096522719623E-0001\r\n       2.410725801734648E-0001  2.132096522719623E-0001  1.373206046344469E-0001\r\n       3.039615411981977E-0002\r\nN = 7\r\n  xi: -9.491079123427585E-0001 -7.415311855993944E-0001 -4.058451513773972E-0001\r\n       0.000000000000000E+0000  4.058451513773972E-0001  7.415311855993944E-0001\r\n       9.491079123427585E-0001\r\n  ai:  6.309209262997855E-0002  1.406532597155259E-0001  1.903505780647854E-0001\r\n       2.094821410847278E-0001  1.903505780647854E-0001  1.406532597155259E-0001\r\n       6.309209262997855E-0002\r\n  yi: -9.914553711208126E-0001 -8.648644233597691E-0001 -5.860872354676911E-0001\r\n      -2.077849550078985E-0001  2.077849550078985E-0001  5.860872354676911E-0001\r\n       8.648644233597691E-0001  9.914553711208126E-0001\r\n  bi:  2.293532201052922E-0002  1.047900103222502E-0001  1.690047266392679E-0001\r\n       2.044329400752989E-0001  2.044329400752989E-0001  1.690047266392679E-0001\r\n       1.047900103222502E-0001  2.293532201052923E-0002\r\nN = 8\r\n  xi: -9.602898564975362E-0001 -7.966664774136267E-0001 -5.255324099163290E-0001\r\n      -1.834346424956498E-0001  1.834346424956498E-0001  5.255324099163290E-0001\r\n       7.966664774136267E-0001  9.602898564975362E-0001\r\n  ai:  4.943939500213931E-0002  1.116463708268396E-0001  1.566526061681884E-0001\r\n       1.814000250680346E-0001  1.814000250680346E-0001  1.566526061681884E-0001\r\n       1.116463708268396E-0001  4.943939500213931E-0002\r\n  yi: -9.933798758817162E-0001 -8.941209068474564E-0001 -6.723540709451587E-0001\r\n      -3.607010979281320E-0001  0.000000000000000E+0000  3.607010979281320E-0001\r\n       6.723540709451587E-0001  8.941209068474564E-0001  9.933798758817162E-0001\r\n  bi:  1.782238332071036E-0002  8.248229893135833E-0002  1.362631092551722E-0001\r\n       1.720706085552113E-0001  1.844464057446916E-0001  1.720706085552113E-0001\r\n       1.362631092551722E-0001  8.248229893135833E-0002  1.782238332071036E-0002\r\nN = 9\r\n  xi: -9.681602395076261E-0001 -8.360311073266358E-0001 -6.133714327005904E-0001\r\n      -3.242534234038089E-0001  0.000000000000000E+0000  3.242534234038089E-0001\r\n       6.133714327005904E-0001  8.360311073266358E-0001  9.681602395076261E-0001\r\n  ai:  3.963189516026125E-0002  9.079068168872639E-0002  1.300014068553412E-0001\r\n       1.564135277884839E-0001  1.648960128283494E-0001  1.564135277884839E-0001\r\n       1.300014068553412E-0001  9.079068168872639E-0002  3.963189516026126E-0002\r\n  yi: -9.946781606773402E-0001 -9.149635072496779E-0001 -7.344867651839338E-0001\r\n      -4.754624791124599E-0001 -1.642235636149868E-0001  1.642235636149868E-0001\r\n       4.754624791124599E-0001  7.344867651839338E-0001  9.149635072496779E-0001\r\n       9.946781606773402E-0001\r\n  bi:  1.430477564383894E-0002  6.651815594027414E-0002  1.117891346844183E-0001\r\n       1.452395883843662E-0001  1.628628274401151E-0001  1.628628274401151E-0001\r\n       1.452395883843662E-0001  1.117891346844183E-0001  6.651815594027414E-0002\r\n       1.430477564383894E-0002\r\nN = 10\r\n  xi: -9.739065285171717E-0001 -8.650633666889845E-0001 -6.794095682990244E-0001\r\n      -4.333953941292472E-0001 -1.488743389816312E-0001  1.488743389816312E-0001\r\n       4.333953941292472E-0001  6.794095682990244E-0001  8.650633666889845E-0001\r\n       9.739065285171717E-0001\r\n  ai:  3.255816230796473E-0002  7.503967481091995E-0002  1.093871588022976E-0001\r\n       1.347092173114733E-0001  1.477391049013385E-0001  1.477391049013385E-0001\r\n       1.347092173114733E-0001  1.093871588022976E-0001  7.503967481091995E-0002\r\n       3.255816230796473E-0002\r\n  yi: -9.956571630258081E-0001 -9.301574913557082E-0001 -7.808177265864169E-0001\r\n      -5.627571346686047E-0001 -2.943928627014602E-0001  0.000000000000000E+0000\r\n       2.943928627014602E-0001  5.627571346686047E-0001  7.808177265864169E-0001\r\n       9.301574913557082E-0001  9.956571630258081E-0001\r\n  bi:  1.169463886737187E-0002  5.475589657435200E-0002  9.312545458369761E-0002\r\n       1.234919762620659E-0001  1.427759385770601E-0001  1.494455540029169E-0001\r\n       1.427759385770601E-0001  1.234919762620659E-0001  9.312545458369761E-0002\r\n       5.475589657435200E-0002  1.169463886737187E-0002\r\n\r\n<\/pre>\n<\/div><\/div><\/div>\n
      \n
      \u53c2\u8003\u6587\u732e<\/h4>\n[1] \u68ee\u53e3\u7e41\u4e00\u300c\u6570\u5024\u8a08\u7b97\u5de5\u5b66\u300d\u5ca9\u6ce2\u66f8\u5e97 (1989)
      \n[2] \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[3] \u68ee\u6b63\u6b66\u300c\u6570\u5024\u89e3\u6790 (\u7b2c2\u7248)\u300d(2002) \u5171\u7acb\u51fa\u7248
      \n[4] \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[5] 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[6] Piessens, de Doncker-Kapenga, \u00dcberhuber, Kahaner, \u201cQUADPACK: A subroutine package for automatic integration\u201d, Springer-Verlag. (1983)<\/p>\n","protected":false},"excerpt":{"rendered":"
      \u5b9f\u7528\u30eb\u30fc\u30c1\u30f3\u306b\u4f7f\u308f\u308c\u3066\u3044\u308b\u5909\u6570\u5909\u63db\u306b\u3088\u308b\u7a4d\u5206\u516c\u5f0f\u3001\u8aa4\u5dee\u306e\u63a8\u5b9a\u3001\u81ea\u52d5\u7a4d\u5206\u306b\u3064\u3044\u3066\u8aac\u660e\u3057\u307e\u3059\u3002<\/p>\n","protected":false},"author":3,"featured_media":0,"comment_status":"closed","ping_status":"closed","sticky":false,"template":"","format":"standard","meta":{"_monsterinsights_skip_tracking":false,"_monsterinsights_sitenote_active":false,"_monsterinsights_sitenote_note":"","_monsterinsights_sitenote_category":0,"footnotes":""},"categories":[13],"tags":[],"class_list":["post-1045","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\n\t\t