{"id":1639,"date":"2026-10-30T22:00:00","date_gmt":"2026-10-30T13:00:00","guid":{"rendered":"https:\/\/quantumsea.net\/?p=1639"},"modified":"2026-10-27T22:28:32","modified_gmt":"2026-10-27T13:28:32","slug":"qs-richardson-extrapolation-integration-20261025","status":"publish","type":"post","link":"https:\/\/quantumsea.net\/?p=1639","title":{"rendered":"Richardson\u5916\u633f\u3001\u7c97\u3044\u8a08\u7b97\u3068\u7d30\u304b\u3044\u8a08\u7b97\u3092\u7d44\u307f\u5408\u308f\u305b\u3066\u8aa4\u5dee\u3092\u6e1b\u3089\u3059"},"content":{"rendered":"<p>\u7a4d\u5206\u306e\u8a08\u7b97\u3067\u533a\u9593\u3092\u7d30\u304b\u304f\u5206\u3051\u308b\u3068\u3001\u7b54\u3048\u304c\u5c11\u3057\u305a\u3064\u5909\u308f\u3063\u3066\u3044\u304f\u3002\u305d\u306e\u3068\u304d\u3001\u524d\u306b\u4f7f\u3063\u305f\u7c97\u3044\u5206\u5272\u306e\u7b54\u3048\u306b\u3082\u4f7f\u3044\u9053\u304c\u3042\u308a\u307e\u3059\u3002\u7d30\u304b\u3044\u5206\u5272\u306e\u7b54\u3048\u3068\u306e\u5dee\u306b\u3001\u307e\u3060\u6b8b\u3063\u3066\u3044\u308b\u8aa4\u5dee\u306e\u624b\u639b\u304b\u308a\u304c\u542b\u307e\u308c\u3066\u3044\u308b\u305f\u3081\u3067\u3059\u3002<\/p>\n<p><strong>Richardson\u5916\u633f\uff08\u30ea\u30c1\u30e3\u30fc\u30c9\u30bd\u30f3\u5916\u633f\uff09<\/strong>\u306f\u3001\u523b\u307f\u5e45\u3068\u8aa4\u5dee\u306e\u95a2\u4fc2\u3092\u5229\u7528\u3057\u3001\u7570\u306a\u308b\u7d30\u304b\u3055\u306e\u8a08\u7b97\u7d50\u679c\u3092\u7d44\u307f\u5408\u308f\u305b\u3066\u7cbe\u5ea6\u3092\u4e0a\u3052\u308b\u65b9\u6cd5\u3067\u3059\u3002\u3053\u3053\u3067\u306f\u53f0\u5f62\u5247\u306b\u3088\u308b\u7a4d\u5206\u3092\u4f8b\u306b\u3001\u306a\u305c\u300c\u7d30\u304b\u3044\u7b54\u3048\u30924\u500d\u3057\u3066\u3001\u7c97\u3044\u7b54\u3048\u3092\u5f15\u304f\u300d\u306e\u304b\u3092\u8ffd\u3044\u307e\u3059\u3002\u52b9\u304d\u76ee\u3092\u6c7a\u3081\u308b\u306e\u306f\u3001\u8a08\u7b97\u56de\u6570\u3060\u3051\u3067\u306a\u304f\u3001\u8aa4\u5dee\u306e\u6e1b\u308a\u65b9\u304c\u60f3\u5b9a\u306b\u5408\u3063\u3066\u3044\u308b\u304b\u3069\u3046\u304b\u3067\u3059\u3002<\/p>\n<section class=\"influvetor-quick-summary\" aria-label=\"\u7c21\u5358\u306b\u307e\u3068\u3081\u308b\u3068\uff1f\">\n<p><strong>\u7c21\u5358\u306b\u307e\u3068\u3081\u308b\u3068\uff1f<\/strong><\/p>\n<ul>\n<li>\u6ed1\u3089\u304b\u306a\u95a2\u6570\u306e\u53f0\u5f62\u5247\u3067\u306f\u3001\u523b\u307f\u5e45\u3092\u534a\u5206\u306b\u3059\u308b\u3068\u4e3b\u8981\u306a\u8aa4\u5dee\u304c\u7d044\u5206\u306e1\u306b\u306a\u308a\u307e\u3059\u3002\u305d\u306e\u95a2\u4fc2\u3092\u4f7f\u3046\u3068\u3001\u672a\u77e5\u306e\u8aa4\u5dee\u4fc2\u6570\u3092\u6d88\u305b\u307e\u3059\u3002<\/li>\n<li>\u7c97\u3044\u7d50\u679c\u3092T<sub>n<\/sub>\u3001\u5206\u5272\u6570\u3092\u500d\u306b\u3057\u305f\u7d50\u679c\u3092T<sub>2n<\/sub>\u3068\u3059\u308b\u3068\u3001\u5916\u633f\u5024\u306f\uff084T<sub>2n<\/sub>\u2212T<sub>n<\/sub>\uff09\/3\u3067\u3059\u3002\u3053\u308c\u306f\u540c\u3058\u70b9\u3092\u4f7f\u3046Simpson\u5247\u306b\u4e00\u81f4\u3057\u307e\u3059\u3002<\/li>\n<li>\u7aef\u70b9\u3067\u50be\u304d\u304c\u767a\u6563\u3059\u308b\u95a2\u6570\u3084\u3001\u4e38\u3081\u30fb\u6e2c\u5b9a\u30ce\u30a4\u30ba\u304c\u652f\u914d\u3059\u308b\u8a08\u7b97\u3078\u3001\u540c\u3058\u6539\u5584\u7387\u3092\u305d\u306e\u307e\u307e\u671f\u5f85\u3067\u304d\u307e\u305b\u3093\u3002\u5206\u5272\u3092\u91cd\u306d\u305f\u3068\u304d\u306e\u5909\u5316\u3082\u8abf\u3079\u307e\u3059\u3002<\/li>\n<\/ul>\n<\/section>\n<h2>\u53f0\u5f62\u3092\u7d30\u304b\u304f\u3057\u305f\u3068\u304d\u3001\u4f55\u304c\u6b8b\u308b\u306e\u304b<\/h2>\n<p>\u533a\u9593[a,b]\u3092n\u7b49\u5206\u3057\u3001\u96a3\u308a\u5408\u3046\u70b9\u3067\u306e\u95a2\u6570\u5024\u3092\u76f4\u7dda\u3067\u7d50\u3076\u3002\u5404\u90e8\u5206\u306e\u4e0b\u306b\u3067\u304d\u308b\u53f0\u5f62\u306e\u9762\u7a4d\u3092\u8db3\u3057\u305f\u3082\u306e\u304c<strong>\u8907\u5408\u53f0\u5f62\u5247<\/strong>\u3067\u3059\u3002\u523b\u307f\u5e45\u3092h=(b\u2212a)\/n\u3068\u3059\u308b\u3068\u3001\u4e21\u7aef\u306e\u95a2\u6570\u5024\u3060\u3051\u534a\u5206\u306e\u91cd\u307f\u3092\u4ed8\u3051\u3001\u5185\u90e8\u306e\u5024\u3068\u4e00\u7dd2\u306b\u8db3\u3057\u3066h\u3092\u639b\u3051\u307e\u3059\u3002[1]<\/p>\n<p>\u3053\u3053\u3067\u91cd\u8981\u306a\u306e\u306f\u3001\u76f4\u7dda\u3068\u672c\u6765\u306e\u66f2\u7dda\u306e\u9593\u306b\u9699\u9593\u304c\u6b8b\u308b\u3053\u3068\u3067\u3059\u3002\u95a2\u6570\u304c\u533a\u9593\u306e\u4e21\u7aef\u3092\u542b\u3081\u3066\u5341\u5206\u6ed1\u3089\u304b\u306a\u5834\u5408\u3001\u53f0\u5f62\u5247\u306e\u7d50\u679cT(h)\u306f\u3001\u771f\u306e\u7a4d\u5206\u5024I\u306b\u5bfe\u3057\u3066\u6b21\u306e\u5f62\u3067\u8868\u305b\u307e\u3059\u3002\u4ee5\u4e0b\u3067\u306fh\u306e4\u4e57\u306e\u9805\u307e\u3067\u6271\u3048\u308b\u6ed1\u3089\u304b\u3055\u3092\u4eee\u5b9a\u3057\u307e\u3059\u3002[1][2]<\/p>\n<p><strong>T(h) = I + c\u2082h\u00b2 + O(h\u2074)<\/strong><\/p>\n<p>c\u2082\u306f\u95a2\u6570\u3084\u7a4d\u5206\u533a\u9593\u3067\u6c7a\u307e\u308b\u4fc2\u6570\u3067\u3001\u540c\u3058\u554f\u984c\u306e\u523b\u307f\u5e45\u3092\u5909\u3048\u3066\u3082\u540c\u3058\u3067\u3059\u3002O(h\u2074)\u306f\u3001h\u304c\u5341\u5206\u5c0f\u3055\u3044\u3068\u304d\u306b\u6b8b\u308a\u306e\u5927\u304d\u3055\u3092h\u306e4\u4e57\u306b\u6bd4\u4f8b\u3059\u308b\u91cf\u3067\u6291\u3048\u3089\u308c\u308b\u3001\u3068\u3044\u3046\u610f\u5473\u3067\u3059\u3002\u5148\u982d\u306ec\u2082\u304c\u30bc\u30ed\u3067\u306a\u3051\u308c\u3070\u3001h\u3092\u534a\u5206\u306b\u3057\u305f\u4e3b\u8981\u306a\u8aa4\u5dee\u306fc\u2082h\u00b2\/4\u306b\u306a\u308a\u307e\u3059\u3002<\/p>\n<h2>4\u500d\u3057\u3066\u5f15\u304f\u3068\u3001\u672a\u77e5\u306e\u4fc2\u6570\u304c\u6d88\u3048\u308b<\/h2>\n<p>\u523b\u307f\u5e45\u304ch\u306e\u7d50\u679cT(h)\u3068\u3001h\/2\u306e\u7d50\u679cT(h\/2)\u3092\u4e26\u3079\u307e\u3059\u3002\u5f8c\u8005\u30924\u500d\u3059\u308b\u3068\u3001\u4e3b\u8981\u306a\u8aa4\u5dee\u306f4\u00d7c\u2082h\u00b2\/4=c\u2082h\u00b2\u3067\u3059\u3002\u305d\u3053\u304b\u3089T(h)\u3092\u5f15\u3051\u3070\u3001c\u2082\u306e\u5024\u3092\u77e5\u3089\u306a\u304f\u3066\u3082\u8aa4\u5dee\u306e\u5148\u982d\u9805\u304c\u76f8\u6bba\u3055\u308c\u307e\u3059\u3002\u4e00\u65b9\u3001\u771f\u306e\u5024\u306e\u90e8\u5206\u306f4I\u2212I=3I\u306a\u306e\u3067\u3001\u6700\u5f8c\u306b3\u3067\u5272\u308a\u307e\u3059\u3002<\/p>\n<p><strong>R = [4T(h\/2) \u2212 T(h)] \/ 3<\/strong><\/p>\n<p>\u3053\u3046\u3057\u3066\u6b8b\u308b\u8aa4\u5dee\u306fO(h\u2074)\u306b\u306a\u308a\u307e\u3059\u3002\u4fc2\u6570\u304c\u6d88\u3048\u306a\u3044\u901a\u5e38\u306e\u5834\u5408\u3001\u523b\u307f\u5e45\u3092\u3055\u3089\u306b\u534a\u5206\u306b\u3057\u305f\u3068\u304d\u306e\u8aa4\u5dee\u306f\u3001\u304a\u3088\u305d16\u5206\u306e1\u3067\u3059\u3002\u4e8c\u3064\u306e\u5024\u3092\u305f\u3060\u5e73\u5747\u3059\u308b\u306e\u3067\u306f\u306a\u304f\u3001\u8aa4\u5dee\u306e\u5f62\u306b\u5408\u308f\u305b\u305f\u91cd\u307f\u3092\u4ed8\u3051\u3066\u3044\u307e\u3059\u3002[2][3]<\/p>\n<p>\u53f0\u5f62\u5247\u3067\u306f\u3001\u3053\u306e1\u6bb5\u306e\u5916\u633f\u304c<strong>Simpson\u5247<\/strong>\u3068\u4e00\u81f4\u3057\u307e\u3059\u3002[1][2] \u7d30\u304b\u3044\u5074\u306e\u683c\u5b50\u306b\u306f\u3001\u7c97\u3044\u5074\u306e\u70b9\u3082\u542b\u307e\u308c\u3066\u3044\u307e\u3059\u3002\u7c97\u3044\u8a08\u7b97\u306e\u5024\u3092\u4fdd\u5b58\u3057\u3066\u304a\u3051\u3070\u3001\u65b0\u3057\u304f\u95a2\u6570\u3092\u8a08\u7b97\u3059\u308b\u306e\u306f\u5404\u533a\u9593\u306e\u4e2d\u70b9\u3060\u3051\u3067\u3059\u3002\u4f8b\u3048\u30708\u5206\u5272\u304b\u308916\u5206\u5272\u3078\u9032\u3081\u308b\u306a\u3089\u30019\u70b9\u306e\u65e2\u5b58\u5024\u306b8\u70b9\u3092\u8ffd\u52a0\u3057\u3001\u5408\u8a0817\u70b9\u306e\u5024\u3067\u5916\u633f\u307e\u3067\u884c\u3048\u307e\u3059\u3002<\/p>\n<h2>\u540c\u305817\u70b9\u3067\u3082\u3001\u7b54\u3048\u306e\u4f7f\u3044\u65b9\u3067\u7cbe\u5ea6\u304c\u5909\u308f\u308b<\/h2>\n<p>\u72ec\u81ea\u8a08\u7b97\u3068\u3057\u3066\u30010\u304b\u30891\u307e\u3067\u306ee<sup>x<\/sup>\u306e\u7a4d\u5206\u3092\u6c42\u3081\u307e\u3057\u305f\u3002\u771f\u306e\u5024\u306fe\u22121\u22521.718281828459\u3067\u3059\u3002\u53f0\u5f62\u5247\u3067\u306f8\u5206\u5272\u3067\u7d041.720518592164\u300116\u5206\u5272\u3067\u7d041.718841128580\u306b\u306a\u308a\u307e\u3057\u305f\u300216\u5206\u5272\u3060\u3051\u3092\u4f7f\u3063\u305f\u7d76\u5bfe\u8aa4\u5dee\u306f\u7d045.59\u00d710<sup>\u22124<\/sup>\u3067\u3059\u3002<\/p>\n<p>\u3053\u306e\u4e8c\u3064\u304b\u3089\u5916\u633f\u3059\u308b\u3068\u3001\u7d50\u679c\u306f\u7d041.718281974052\u3001\u7d76\u5bfe\u8aa4\u5dee\u306f\u7d041.46\u00d710<sup>\u22127<\/sup>\u306b\u306a\u308a\u307e\u3059\u3002\u95a2\u6570\u5024\u306e\u53d6\u5f97\u70b9\u309217\u70b9\u3088\u308a\u5897\u3084\u3057\u305f\u308f\u3051\u3067\u306f\u3042\u308a\u307e\u305b\u3093\u3002\u7c97\u3044\u683c\u5b50\u306e\u60c5\u5831\u3082\u4f7f\u3046\u3053\u3068\u3067\u300116\u5206\u5272\u306e\u53f0\u5f62\u5247\u306b\u6b8b\u3063\u3066\u3044\u305f\u504f\u308a\u3092\u5927\u5e45\u306b\u5c0f\u3055\u304f\u3067\u304d\u307e\u3057\u305f\u3002<\/p>\n<p>\u3055\u3089\u306b16\u5206\u5272\u306832\u5206\u5272\u3092\u7d44\u307f\u5408\u308f\u305b\u305f\u5916\u633f\u3067\u306f\u3001\u8aa4\u5dee\u306f\u7d049.10\u00d710<sup>\u22129<\/sup>\u306b\u306a\u308a\u3001\u5148\u307b\u3069\u306e\u7d0416\u5206\u306e1\u3067\u3059\u3002\u56f3\u306e\u4e0a\u6bb5\u306f\u3001\u3053\u306e\u6e1b\u308a\u65b9\u3092\u793a\u3057\u3066\u3044\u307e\u3059\u3002\u6570\u5024\u306f\u5341\u903260\u6841\u3067\u8a08\u7b97\u3057\u300180\u6841\u3067\u306e\u518d\u8a08\u7b97\u3068\u3001Simpson\u5247\u306e\u91cd\u307f\u3092\u76f4\u63a5\u8db3\u3059\u5225\u306e\u5f0f\u3067\u7167\u5408\u3057\u307e\u3057\u305f\u3002\u5b9f\u6e2c\u30c7\u30fc\u30bf\u306e\u7cbe\u5ea6\u3092\u8a55\u4fa1\u3057\u305f\u3082\u306e\u3067\u306f\u3042\u308a\u307e\u305b\u3093\u3002<\/p>\n<figure><img decoding=\"async\" src=\"https:\/\/quantumsea.net\/wp-content\/uploads\/2026\/10\/qs-richardson-extrapolation-integration-20261025-figure-b7ad88e00d02.png\" style=\"max-width:100%;height:auto\" 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307e\u305f\u3001\u5404\u95a2\u6570\u5024\u306b\u6e2c\u5b9a\u30ce\u30a4\u30ba\u304c\u3042\u308b\u5834\u5408\u3084\u3001\u975e\u5e38\u306b\u5c0f\u3055\u3044\u5dee\u3092\u6709\u9650\u6841\u306e\u6570\u3067\u8a08\u7b97\u3059\u308b\u5834\u5408\u306f\u3001\u5dee\u306b\u542b\u307e\u308c\u308b\u3082\u306e\u304c\u53f0\u5f62\u8fd1\u4f3c\u306e\u8aa4\u5dee\u3060\u3051\u3067\u306f\u306a\u304f\u306a\u308a\u307e\u3059\u3002\u7d30\u304b\u304f\u3057\u3066\u3082\u5024\u304c\u5b89\u5b9a\u3057\u306a\u3044\u6bb5\u968e\u3067\u306f\u3001\u5916\u633f\u3092\u91cd\u306d\u308b\u524d\u306b\u3001\u5165\u529b\u5024\u3068\u6570\u5024\u8a08\u7b97\u306e\u7cbe\u5ea6\u3092\u898b\u76f4\u3059\u3053\u3068\u304c\u5fc5\u8981\u3067\u3059\u3002Richardson\u5916\u633f\u306e\u4f7f\u3044\u3069\u3053\u308d\u306f\u3001\u524d\u306e\u8a08\u7b97\u3092\u6368\u3066\u305a\u3001\u8aa4\u5dee\u304c\u898f\u5247\u7684\u306b\u6e1b\u3063\u3066\u3044\u308b\u7bc4\u56f2\u3067\u6b21\u306e\u8a08\u7b97\u3078\u751f\u304b\u3059\u3053\u3068\u306b\u3042\u308a\u307e\u3059\u3002<\/p>\n<h2>\u51fa\u5178<\/h2>\n<ol>\n<li>NIST Digital Library of Mathematical Functions\u3001<a href=\"https:\/\/dlmf.nist.gov\/3.5\">\u00a73.5 Quadrature<\/a>\u3001(i)\u301c(iii)\u3002\u53f0\u5f62\u5247\u3001Simpson\u5247\u3001\u6ed1\u3089\u304b\u3055\u3092\u4eee\u5b9a\u3057\u305f\u8aa4\u5dee\u5c55\u958b\u30022026\u5e7410\u670825\u65e5\u4fdd\u5b58\u3002<\/li>\n<li>Tobin A. Driscoll\u30fbRichard J. Braun\u3001<a href=\"https:\/\/fncbook.com\/integration\/\">Fundamentals of Numerical Computation, 5.6 Numerical integration<\/a>\u30015.6.1\u301c5.6.3\u3002\u5916\u633f\u3068\u683c\u5b50\u70b9\u306e\u518d\u5229\u7528\u3002\u53d6\u5f97\u6642\u70b9\u306eWeb\u7248\u3002<\/li>\n<li>UBC CLP-2 Integral Calculus\u3001<a href=\"https:\/\/personal.math.ubc.ca\/~CLP\/CLP2\/clp_2_ic\/ap_Richardson.html\">Richardson Extrapolation<\/a>\u3002\u523b\u307f\u5e45\u3092\u534a\u5206\u306b\u3057\u305f\u7d50\u679c\u306e\u7d44\u5408\u305b\u3002\u6570\u5024\u4f8b\u3068\u56f3\u306f\u672c\u6587\u306b\u8a18\u3057\u305f\u72ec\u81ea\u8a08\u7b97\u3002<\/li>\n<\/ol>\n","protected":false},"excerpt":{"rendered":"<p>\u7a4d\u5206\u306e\u8a08\u7b97\u3067\u533a\u9593\u3092\u7d30\u304b\u304f\u5206\u3051\u308b\u3068\u3001\u7b54\u3048\u304c\u5c11\u3057\u305a\u3064\u5909\u308f\u3063\u3066\u3044\u304f\u3002\u305d\u306e\u3068\u304d\u3001\u524d\u306b\u4f7f\u3063\u305f\u7c97\u3044\u5206\u5272\u306e\u7b54\u3048\u306b\u3082\u4f7f\u3044\u9053\u304c\u3042\u308a\u307e\u3059\u3002\u7d30\u304b\u3044\u5206\u5272\u306e\u7b54\u3048\u3068\u306e\u5dee\u306b\u3001\u307e\u3060\u6b8b\u3063\u3066\u3044\u308b\u8aa4\u5dee\u306e\u624b\u639b\u304b\u308a\u304c\u542b\u307e\u308c\u3066\u3044\u308b\u305f\u3081\u3067\u3059\u3002 Richardson\u5916 [&hellip;]<\/p>\n","protected":false},"author":3,"featured_media":1637,"comment_status":"closed","ping_status":"closed","sticky":false,"template":"","format":"standard","meta":{"_jetpack_newsletter_access":"","_jetpack_dont_email_post_to_subs":true,"_jetpack_newsletter_tier_id":0,"_jetpack_memberships_contains_paywalled_content":false,"_jetpack_feature_clip_id":0,"_jetpack_memberships_contains_paid_content":false,"footnotes":"","jetpack_publicize_message":"","jetpack_publicize_feature_enabled":false,"jetpack_social_post_already_shared":true,"jetpack_social_options":{"image_generator_settings":{"template":"highway","default_image_id":0,"font":"","enabled":false},"version":2},"jetpack_post_was_ever_published":false},"categories":[31],"tags":[332],"class_list":["post-1639","post","type-post","status-publish","format-standard","has-post-thumbnail","hentry","category-ai-software","tag-richardson-extrapolation"],"jetpack_publicize_connections":[],"jetpack_sharing_enabled":true,"jetpack_featured_media_url":"https:\/\/quantumsea.net\/wp-content\/uploads\/2026\/10\/qs-richardson-extrapolation-integration-20261025-featured-be4e8ced0fdd.png","_links":{"self":[{"href":"https:\/\/quantumsea.net\/index.php?rest_route=\/wp\/v2\/posts\/1639","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/quantumsea.net\/index.php?rest_route=\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/quantumsea.net\/index.php?rest_route=\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/quantumsea.net\/index.php?rest_route=\/wp\/v2\/users\/3"}],"replies":[{"embeddable":true,"href":"https:\/\/quantumsea.net\/index.php?rest_route=%2Fwp%2Fv2%2Fcomments&post=1639"}],"version-history":[{"count":1,"href":"https:\/\/quantumsea.net\/index.php?rest_route=\/wp\/v2\/posts\/1639\/revisions"}],"predecessor-version":[{"id":1640,"href":"https:\/\/quantumsea.net\/index.php?rest_route=\/wp\/v2\/posts\/1639\/revisions\/1640"}],"wp:featuredmedia":[{"embeddable":true,"href":"https:\/\/quantumsea.net\/index.php?rest_route=\/wp\/v2\/media\/1637"}],"wp:attachment":[{"href":"https:\/\/quantumsea.net\/index.php?rest_route=%2Fwp%2Fv2%2Fmedia&parent=1639"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/quantumsea.net\/index.php?rest_route=%2Fwp%2Fv2%2Fcategories&post=1639"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/quantumsea.net\/index.php?rest_route=%2Fwp%2Fv2%2Ftags&post=1639"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}