{"id":5398,"date":"2026-05-26T10:23:09","date_gmt":"2026-05-26T01:23:09","guid":{"rendered":"https:\/\/www.ktech.biz\/jp\/?p=5398"},"modified":"2026-05-26T10:23:09","modified_gmt":"2026-05-26T01:23:09","slug":"6-1_inter","status":"publish","type":"post","link":"https:\/\/www.ktech.biz\/jp\/num\/6-1_inter\/","title":{"rendered":"6.1 \u591a\u9805\u5f0f\u88dc\u9593"},"content":{"rendered":"\n\u88dc\u9593\u5f0f\u3068\u3057\u3066\u591a\u9805\u5f0f\u3092\u4f7f\u7528\u3059\u308b\u3082\u306e\u3092\u591a\u9805\u5f0f\u88dc\u9593\u3068\u3044\u3046. \u3053\u3053\u3067\u306f, (\u8aa4\u5dee\u304c\u306a\u3051\u308c\u3070) \u540c\u3058\u7d50\u679c\u3092\u4e0e\u3048\u308b\u304c\u8a08\u7b97\u624b\u9806\u304c\u7570\u306a\u308b\u30e9\u30b0\u30e9\u30f3\u30b8\u30e5\u88dc\u9593\u516c\u5f0f\u3068\u30cb\u30e5\u30fc\u30c8\u30f3\u88dc\u9593\u516c\u5f0f\u3082\u5e83\u3044\u610f\u5473\u3067\u306f\u591a\u9805\u5f0f\u88dc\u9593\u3068\u3068\u3089\u3048\u3066\u304a\u304f\u3053\u3068\u306b\u3059\u308b.<\/p>\n

\u30e9\u30b0\u30e9\u30f3\u30b8\u30e5\u88dc\u9593\u516c\u5f0f\u306f\u3088\u304f\u4f7f\u308f\u308c\u308b\u4ee3\u8868\u7684\u306a\u88dc\u9593\u516c\u5f0f\u3067\u3042\u308b. \u30cb\u30e5\u30fc\u30c8\u30f3\u88dc\u9593\u516c\u5f0f\u306f\u3059\u3067\u306b\u6c42\u3081\u305f\u88dc\u9593\u5f0f\u306b\u65b0\u305f\u306b\u6a19\u672c\u70b9\u3092\u8ffd\u52a0\u3059\u308b\u3068\u304d\u306b\u9069\u3057\u3066\u3044\u308b.<\/p>\n

6.1.1 \u591a\u9805\u5f0f\u88dc\u9593<\/h3>\n

\\(n\\) \u6b21\u591a\u9805\u5f0f \\(p_n(x)\\) \u3092\u6b21\u306e\u3088\u3046\u306b\u5b9a\u7fa9\u3059\u308b.
\n\\[
\np_n(x) = a_0x^n + a_1x^{n-1} + \\dots + a_{n-1}x + a_n
\n\\]\n\\(n + 1\\) \u500b\u306e\u76f8\u7570\u306a\u308b\u70b9 \\(x_0, x_1, \\dots, x_n\\) \u306b\u304a\u3051\u308b\u95a2\u6570 \\(f(x)\\) \u306e\u5024 \\(f(x_0), f(x_1), \\dots, f(x_n)\\) \u304c \u304c\u4e0e\u3048\u3089\u308c\u3066\u3044\u308b\u3068\u304d, \u6b21\u306e\u9023\u7acb\u4e00\u6b21\u65b9\u7a0b\u5f0f\u3092\u89e3\u304f\u3053\u3068\u306b\u3088\u308a \\(f(x)\\) \u3092\u88dc\u9593\u3059\u308b \\(n\\) \u6b21\u591a\u9805\u5f0f \\(p_n(x)\\) \u306e\u4fc2\u6570 \\(a_0, a_1, \\dots, a_n\\) \u3092\u6c42\u3081\u308b\u3053\u3068\u304c\u3067\u304d\u308b.
\n\\[
\n\\boldsymbol{Va} = \\boldsymbol{f}
\n\\]\n\u305f\u3060\u3057,
\n\\[
\n\\boldsymbol{V} =
\n\\begin{pmatrix}
\nx_0^n & x_0^{n-1} & \\dots & 1 \\\\
\nx_1^n & x_1^{n-1} & \\dots & 1 \\\\
\n\\vdots & \\vdots & \\dots & \\vdots \\\\
\nx_n^n & x_n^{n-1} & \\dots & 1 \\\\
\n\\end{pmatrix}
\n,
\n\\boldsymbol{a} =
\n\\begin{pmatrix}
\na_0 \\\\
\na_1 \\\\
\n\\vdots \\\\
\na_n \\\\
\n\\end{pmatrix}
\n,
\n\\boldsymbol{f} =
\n\\begin{pmatrix}
\nf_0 \\\\
\nf_1 \\\\
\n\\vdots \\\\
\nf_n \\\\
\n\\end{pmatrix}
\n\\]\n\u3067\u3042\u308b.<\/p>\n

\u3053\u3053\u3067, \\(\\boldsymbol{V}\\) \u306f\u9023\u7acb\u4e00\u6b21\u65b9\u7a0b\u5f0f\u306e\u4fc2\u6570\u884c\u5217, \\(\\boldsymbol{f}\\) \u306f\u9023\u7acb\u4e00\u6b21\u65b9\u7a0b\u5f0f\u306e\u53f3\u8fba\u30d9\u30af\u30c8\u30eb (\u5404\u70b9\u306b\u304a\u3051\u308b\u95a2\u6570\u5024 \\(f_i = f(x_i)\\) \u3092\u8868\u3059), \\(\\boldsymbol{a}\\) \u306f\u591a\u9805\u5f0f\u306e\u4fc2\u6570\u3067, \u9023\u7acb\u4e00\u6b21\u65b9\u7a0b\u5f0f\u306e\u89e3\u30d9\u30af\u30c8\u30eb\u306b\u306a\u308b.<\/p>\n

\u3053\u306e\u9023\u7acb\u4e00\u6b21\u65b9\u7a0b\u5f0f\u306e\u4fc2\u6570\u884c\u5217 \\(\\boldsymbol{V}\\) \u306f\u30d5\u30a1\u30f3\u30c7\u30eb\u30e2\u30f3\u30c9\u884c\u5217\u3068\u547c\u3070\u308c\u308b\u5f62\u3092\u3057\u3066\u3044\u308b\u304c, \u30d5\u30a1\u30f3\u30c7\u30eb\u30e2\u30f3\u30c9\u884c\u5217\u306e\u884c\u5217\u5f0f\u306f\u6b21\u5f0f\u3067\u6c42\u3081\u3089\u308c\u308b\u3053\u3068\u304c\u77e5\u3089\u308c\u3066\u3044\u308b.
\n\\[
\ndet(\\boldsymbol{V}) = \\prod_{0 \\le i < j \\le n}(x_j - x_i)\n\\]\n\u3057\u305f\u304c\u3063\u3066, \\(x_0, x_1, \\dots, x_n\\) \u304c\u76f8\u7570\u306a\u308b\u70b9\u3067\u3042\u308c\u3070, \\(det(\\boldsymbol{V}) \\ne 0\\) \u3067\u3042\u308a, \u4e0a\u306e\u9023\u7acb\u4e00\u6b21\u65b9\u7a0b\u5f0f\u306f\u89e3\u3092\u6301\u3064. \u305d\u3057\u3066, \u89e3\u306f \\(n\\) \u6b21\u591a\u9805\u5f0f \\(p_n(x)\\) \u306e\u4fc2\u6570\u3067\u3042\u308b. \u3059\u306a\u308f\u3061, \u4e0a\u306e\u9023\u7acb\u4e00\u6b21\u65b9\u7a0b\u5f0f\u3092\u89e3\u304f\u3053\u3068\u306b\u3088\u308a\u88dc\u9593\u591a\u9805\u5f0f\u304c\u6c42\u3081\u3089\u308c\u308b.\n\n\n\n

6.1.2 \u6570\u5024\u5b9f\u9a13 (1)<\/h3>\n

\u533a\u9593 \\([-1, 1]\\) \u306b\u304a\u3044\u3066\u6b21\u306e\u6307\u6570\u95a2\u6570\u3092\u591a\u9805\u5f0f\u88dc\u9593\u3059\u308b.
\n\\[
\nf(x) = 2e^{x-1} – 1
\n\\]\n\\(n = 2 \\sim 5\\) \u3068\u3057\u3066 (\u3059\u306a\u308f\u3061, \\(2 \\sim 5\\) \u6b21\u591a\u9805\u5f0f\u3092\u7528\u3044\u3066) \u88dc\u9593\u3092\u884c\u3046. \u305f\u3060\u3057, \u533a\u9593 \\([-1, 1]\\) \u3092 \\(n\\) \u7b49\u5206\u3057\u3066, \u4e21\u7aef\u3092\u542b\u3081\u7b49\u9593\u9694\u306b\u4e26\u3093\u3060 \\(n + 1\\) \u70b9\u306b\u304a\u3051\u308b\u95a2\u6570\u5024\u3092\u30c7\u30fc\u30bf\u3068\u3057\u3066\u4f7f\u7528\u3059\u308b\u3053\u3068\u306b\u3059\u308b. \u591a\u9805\u5f0f\u88dc\u9593\u3068\u3057\u3066\u306f\u5404\u6a19\u672c\u70b9\u304c\u76f8\u7570\u306a\u3063\u3066\u3044\u308c\u3070\u3088\u304f\u7b49\u9593\u9694\u3067\u3042\u308b\u5fc5\u8981\u306f\u306a\u3044\u304c, \u6570\u8868\u306a\u3069\u3092\u60f3\u5b9a\u3059\u308c\u3070\u3053\u306e\u3088\u3046\u306b\u6a19\u672c\u70b9\u3092\u9078\u3076\u306e\u306f\u81ea\u7136\u3067\u3042\u308b.<\/p>\n

\u4e0a\u306e\u9023\u7acb\u4e00\u6b21\u65b9\u7a0b\u5f0f\u3092\u89e3\u3044\u3066\u591a\u9805\u5f0f\u306e\u4fc2\u6570\u3092\u6c42\u3081, \u533a\u9593\u5185\u306b\u304a\u3051\u308b\u591a\u9805\u5f0f\u306e\u5024\u3092\u30d7\u30ed\u30c3\u30c8\u3059\u308b. \u6a2a\u8ef8\u306f \\(x\\), \u7e26\u8ef8\u306f\u95a2\u6570\u5024\u3067\u3042\u308b.<\/p>\n

\"\"<\/p>\n

\u7e26\u8ef8\u3092\u8aa4\u5dee (\u88dc\u9593\u5024 – \u53b3\u5bc6\u5024) \u3068\u3057\u3066\u30d7\u30ed\u30c3\u30c8\u3059\u308b\u3068\u6b21\u306e\u3088\u3046\u306b\u306a\u308b.<\/p>\n

\"\"<\/p>\n

\u591a\u9805\u5f0f\u306e\u6b21\u6570\u304c\u4e0a\u304c\u308b\u3068\u7cbe\u5ea6\u3082\u4e0a\u304c\u3063\u3066\u3044\u308b\u306e\u304c\u308f\u304b\u308b.<\/p>\n

6.1.3 \u6570\u5024\u5b9f\u9a13 (2)<\/h3>\n

\u591a\u9805\u5f0f\u88dc\u9593\u3067\u306f\u3069\u3093\u306a\u95a2\u6570\u306b\u3064\u3044\u3066\u3082\u7cbe\u5ea6\u3092\u4e0a\u3052\u305f\u3044\u3068\u304d\u306b\u306f\u6b21\u6570\u3092\u4e0a\u3052\u3066\u3044\u3051\u3070\u3088\u3044\u3068\u3044\u3046\u308f\u3051\u3067\u306f\u306a\u3044\u3053\u3068\u3092\u793a\u3059.<\/p>\n

\u533a\u9593 \\([-1, 1]\\) \u306b\u304a\u3044\u3066\u6b21\u306e\u95a2\u6570 (\u30eb\u30f3\u30b2\u306e\u95a2\u6570\u3068\u3044\u3046) \u3092\u591a\u9805\u5f0f\u88dc\u9593\u3059\u308b.
\n\\[
\nf(x) = 1\/(1 + 25x^2)
\n\\]\n\\(n = 4, 8, 12, 20\\) \u3068\u3057\u3066 (\u3059\u306a\u308f\u3061, \\(4, 8, 12, 20\\) \u6b21\u591a\u9805\u5f0f\u3092\u7528\u3044\u3066) \u88dc\u9593\u3092\u884c\u3046. \u305f\u3060\u3057, \u533a\u9593 \\([-1, 1]\\) \u3092 \\(n\\) \u7b49\u5206\u3057\u3066, \u4e21\u7aef\u3092\u542b\u3081\u7b49\u9593\u9694\u306b\u4e26\u3093\u3060 \\(n + 1\\) \u70b9\u306b\u304a\u3051\u308b\u95a2\u6570\u5024\u3092\u30c7\u30fc\u30bf\u3068\u3057\u3066\u4f7f\u7528\u3059\u308b.<\/p>\n

\u4e0a\u306e\u9023\u7acb\u4e00\u6b21\u65b9\u7a0b\u5f0f\u3092\u89e3\u3044\u3066\u591a\u9805\u5f0f\u306e\u4fc2\u6570\u3092\u6c42\u3081, \u533a\u9593\u5185\u306b\u304a\u3051\u308b\u591a\u9805\u5f0f\u306e\u5024\u3092\u30d7\u30ed\u30c3\u30c8\u3059\u308b. \u6a2a\u8ef8\u306f \\(x\\), \u7e26\u8ef8\u306f\u95a2\u6570\u5024\u3067\u3042\u308b.<\/p>\n

\"\"<\/p>\n

\u56f3\u306e\u3088\u3046\u306b\u4e21\u7aef\u3067\u767a\u6563\u3057\u3066 (\u30eb\u30f3\u30b2\u306e\u73fe\u8c61\u3068\u3044\u3046) \u88dc\u9593\u5f0f\u3068\u3057\u3066\u306f\u4f7f\u3048\u306a\u3044. \u7279\u306b \\(n\\) \u304c\u5927\u304d\u304f\u306a\u308b\u307b\u3069\u4e21\u7aef\u3067\u306e\u767a\u6563\u304c\u5927\u304d\u304f\u306a\u3063\u305f. \u6700\u5927\u8aa4\u5dee\u3060\u3051\u3092\u898b\u308b\u3068\u3059\u3079\u3066\u306e \\(n\\) \u306e\u4e2d\u3067 \\(n = 4\\) \u306e\u3068\u304d\u304c\u6700\u5c0f\u306b\u306a\u308b. \u305f\u3060\u3057, \u56f3\u306e\u3088\u3046\u306b\u4e2d\u592e\u4ed8\u8fd1\u3067\u306f\u6b21\u6570\u304c\u4e0a\u304c\u308b\u307b\u3069\u3088\u3044\u8fd1\u4f3c\u3092\u3057\u3066\u3044\u308b\u3053\u3068\u304c\u308f\u304b\u308b\u306e\u3067, \u3053\u306e\u9818\u57df\u3067\u3060\u3051\u88dc\u9593\u5f0f\u3068\u3057\u3066\u4f7f\u3046\u3068\u3044\u3046\u65b9\u6cd5\u306f\u3042\u308b\u304b\u3082\u3057\u308c\u306a\u3044.<\/p>\n

\u3053\u306e\u554f\u984c\u306f\u6a19\u672c\u70b9\u3092\u7b49\u9593\u9694\u3067\u306f\u306a\u304f\u30c1\u30a7\u30d3\u30b7\u30a7\u30d5\u591a\u9805\u5f0f\u306e\u30bc\u30ed\u70b9\u306b\u3068\u308b\u3068\u89e3\u6c7a\u3059\u308b\u3053\u3068\u304c\u77e5\u3089\u308c\u3066\u3044\u308b. \u300c6.3 \u76f4\u4ea4\u591a\u9805\u5f0f\u88dc\u9593\u300d\u3092\u53c2\u7167\u305b\u3088.<\/p>\n

\u306a\u304a, \u4e00\u822c\u306b\u533a\u9593\u5185\u3067\u95a2\u6570\u5024\u306e\u5909\u5316\u304c\u5341\u5206\u3086\u308b\u3084\u304b\u3067\u3042\u308c\u3070\u3053\u306e\u3088\u3046\u306a\u632f\u52d5\u73fe\u8c61\u3092\u8d77\u3053\u3059\u3053\u3068\u306f\u306a\u304f, \u901a\u5e38\u306f\u7b49\u9593\u9694\u3067\u3042\u3063\u3066\u3082\u6a19\u672c\u70b9\u3092\u5897\u3084\u3059\u3068\u88dc\u9593\u306e\u7cbe\u5ea6\u306f\u5411\u4e0a\u3059\u308b.<\/p>\n

6.1.4 \u30e9\u30b0\u30e9\u30f3\u30b8\u30e5\u88dc\u9593<\/h3>\n

\u591a\u9805\u5f0f\u88dc\u9593\u3092\u884c\u3046\u306b\u306f, \u524d\u9805\u306e\u3088\u3046\u306b\u9023\u7acb\u4e00\u6b21\u65b9\u7a0b\u5f0f\u3092\u89e3\u3044\u3066\u591a\u9805\u5f0f\u306e\u4fc2\u6570\u3092\u76f4\u63a5\u6c42\u3081\u308b\u306e\u304c\u76f4\u89b3\u7684\u306a\u65b9\u6cd5\u3067\u306f\u3042\u308b\u304c, \u5b9f\u52d9\u7684\u306b\u306f\u65b9\u7a0b\u5f0f\u306e\u6761\u4ef6\u6570\u304c\u5927\u304d\u304f\u306a\u308a\u3084\u3059\u3044\u306a\u3069\u306e\u7406\u7531\u304b\u3089, \u7b49\u4fa1\u306a\u591a\u9805\u5f0f\u3067\u3042\u308b\u30e9\u30b0\u30e9\u30f3\u30b8\u30e5\u88dc\u9593\u516c\u5f0f\u304c\u7528\u3044\u3089\u308c\u308b.<\/p>\n

\u30e9\u30b0\u30e9\u30f3\u30b8\u30e5\u88dc\u9593\u516c\u5f0f\u306f\u6b21\u5f0f\u3067\u8868\u3055\u308c\u308b.
\n\\[
\nf(x) = \\sum_{k=0}^{n}L_k(x) f(x_k)
\n\\]\n\u305f\u3060\u3057
\n\\[
\nL_k(x) = \\prod_{i=0}^{n (i \\neq k)}(x – x_i)\/(x_k – x_i)
\n\\]\n\\(L_k(x_i)\\) \u306f \\(i = k\\) \u306e\u3068\u304d \\(1\\), \\(i \\ne k\\) \u306e\u3068\u304d \\(0\\) \u3067\u3042\u308b. \u3057\u305f\u304c\u3063\u3066, \u30e9\u30b0\u30e9\u30f3\u30b8\u30e5\u88dc\u9593\u516c\u5f0f\u306f\u5404\u6a19\u672c\u70b9\u306b\u304a\u3044\u3066\u4e0e\u3048\u3089\u308c\u305f\u30c7\u30fc\u30bf\u306b\u4e00\u81f4\u3059\u308b. \u307e\u305f, \u6a19\u672c\u70b9\u306e\u4e26\u3093\u3067\u3044\u308b\u9806\u5e8f\u306b\u95a2\u4fc2\u306a\u304f\u6210\u308a\u7acb\u3061, \u6a19\u672c\u70b9\u306f\u7b49\u9593\u9694\u3067\u306a\u304f\u3066\u3082\u3088\u3044.<\/p>\n

\u30e9\u30b0\u30e9\u30f3\u30b8\u30e5\u88dc\u9593\u516c\u5f0f\u306f \\(n\\) \u6b21\u591a\u9805\u5f0f\u3067\u3042\u308a, \u30d0\u30e9\u30d0\u30e9\u306b\u3057\u3066\u6574\u7406\u3059\u308b\u3068\u305d\u306e\u4fc2\u6570\u306f\u9023\u7acb\u4e00\u6b21\u65b9\u7a0b\u5f0f\u3092\u89e3\u3044\u3066\u6c42\u3081\u305f\u591a\u9805\u5f0f\u306e\u4fc2\u6570\u3068\u4e00\u81f4\u3059\u308b\u306f\u305a\u3067\u3042\u308b. \u3057\u304b\u3057, \u88dc\u9593\u306b\u304a\u3044\u3066\u306f\u95a2\u6570\u5024\u304c\u6c42\u307e\u308c\u3070\u3088\u304f\u3066\u591a\u9805\u5f0f\u306e\u4fc2\u6570\u306f\u8a08\u7b97\u306b\u76f4\u63a5\u306f\u5fc5\u8981\u306a\u3044\u306e\u3067, \u3053\u306e\u5f62\u306e\u307e\u307e\u8a08\u7b97\u306b\u4f7f\u308f\u308c\u308b.<\/p>\n

\u30e9\u30b0\u30e9\u30f3\u30b8\u30e5\u88dc\u9593\u516c\u5f0f\u306e\u8a08\u7b97\u306f\u6b21\u306e\u3088\u3046\u306b\u3057\u3066\u884c\u3046\u3068\u3088\u3044.<\/p>\n

\u307e\u305a, \u4e0e\u3048\u3089\u308c\u305f\u30c7\u30fc\u30bf\u306b\u3064\u3044\u3066
\n\\[
\nw_k = f(x_k)\/\\prod_{i=0}^{n (i \\ne k)}(x_k – x_i)
\n\\]\n\u3092 \\(k = 0 \\sim n\\) \u306b\u3064\u3044\u3066\u8a08\u7b97\u3057\u3066\u304a\u304f.<\/p>\n

\u6b21\u306b, \u6c42\u3081\u305f\u3044\u70b9 \\(x\\) \u305d\u308c\u305e\u308c\u306b\u3064\u3044\u3066\u6b21\u306e\u3088\u3046\u306b\u3057\u3066\u88dc\u9593\u5024\u3092\u8a08\u7b97\u3059\u308b.
\n\\[
\n\\begin{align}
\n& w(x) = \\prod_{k=0}^{n}(x – x_k) \\\\
\n& p_n(x) = w(x)\\sum_{k=0}^{n}w_k\/(x – x_k) \\\\
\n\\end{align}
\n\\]\n\u305f\u3060\u3057, \\(x = x_k (k = 0, \\dots, n)\\) \u3067\u3042\u308c\u3070\u8a08\u7b97\u305b\u305a\u306b \\(p_n(x) = f(x_k)\\) \u3068\u3059\u308c\u3070\u3088\u3044.<\/p>\n

6.1.2 \u304a\u3088\u3073 6.1.3 \u306e\u6570\u5024\u5b9f\u9a13\u306b\u3064\u3044\u3066\u30e9\u30b0\u30e9\u30f3\u30b8\u30e5\u88dc\u9593\u516c\u5f0f\u306b\u3088\u308a\u8a08\u7b97\u3059\u308b\u3068\u591a\u9805\u5f0f\u88dc\u9593\u306e\u3068\u304d\u3068\u8aa4\u5dee\u306e\u7bc4\u56f2\u3067\u540c\u3058\u7d50\u679c\u304c\u5f97\u3089\u308c\u308b.<\/p>\n

6.1.5 \u5dee\u5206\u5546<\/h3>\n

\\(x = x_0, x_1\\) \u306b\u5bfe\u3059\u308b\u5dee\u5206\u5546\u3092\u6b21\u306e\u3088\u3046\u306b\u5b9a\u7fa9\u3059\u308b.
\n\\[
\nf[x_0, x_1] = (f(x_1) – f(x_0))\/(x_1 – x_0)
\n\\]\n\u3053\u308c\u306f \\(x_0\\) \u3068 \\(x_1\\) \u304c\u8fd1\u3051\u308c\u3070\u5fae\u5206\u5024 \\(f'(x_0)\\) \u3092\u8868\u3059. \u3042\u308b\u3044\u306f, \u533a\u9593 \\([x_0, x_1]\\) \u306b\u304a\u3051\u308b\u5fae\u5206\u5024\u306e\u5e73\u5747\u5024\u3068\u307f\u306a\u3059\u3053\u3068\u304c\u3067\u304d\u308b.<\/p>\n

\u540c\u69d8\u306b, \\(x = x_0, x_1, x_2\\) \u306b\u5bfe\u3059\u308b 2 \u968e\u306e\u5dee\u5206\u5546\u306f\u6b21\u306e\u3088\u3046\u306b\u5b9a\u7fa9\u3067\u304d\u308b.
\n\\[
\nf[x_0, x_1, x_2] = (f[x_1, x_2] – f[x_0, x_1])\/(x_2 – x_0)
\n\\]\n\u3059\u306a\u308f\u3061, 2 \u968e\u306e\u5dee\u5206\u5546\u306f 1 \u968e\u306e\u5dee\u5206\u5546\u306e\u5dee\u5206\u5546\u3067\u3042\u308b.<\/p>\n

\u4fbf\u5b9c\u4e0a \\(f[x_i] = f(x_i)\\) \u3068\u8868\u3059\u3068, \u9ad8\u968e\u306e\u5dee\u5206\u5546\u307e\u3067\u542b\u3081\u305f\u4e00\u822c\u5f62\u306f n + 1 \u500b\u306e\u6a19\u672c\u70b9 \\(x_0, x_1, \\dots, x_n\\) \u306b\u3064\u3044\u3066\u6b21\u306e\u3088\u3046\u306b\u8868\u3059\u3053\u3068\u304c\u3067\u304d\u308b.
\n\\[
\nf[x_i, \\dots, x_{i+k}] = (f[x_{i+1}, \\dots, x_{i+k}] – f[x_i, \\dots, x_{i+k-1}])\/(x_{i+k} – x_i) (k = 1 \\sim n, i = 0 \\sim n – k)
\n\\]\n

\u3053\u3053\u3067, \u95a2\u6570 \\(f(x)\\) \u3092\u5dee\u5206\u5546\u3092\u7528\u3044\u3066\u8868\u3059\u3053\u3068\u3092\u8003\u3048\u308b.<\/p>\n

\\(x, x_0\\) \u306b\u5bfe\u3059\u308b 1 \u968e\u306e\u5dee\u5206\u5546
\n\\[
\nf[x, x_0] = (f(x_0) – f(x))\/(x_0 – x)
\n\\]\n\u3092\u5909\u5f62\u3059\u308b\u3068\u6b21\u306e\u3088\u3046\u306b\u306a\u308b.
\n\\[
\nf(x) = f(x_0) + (x – x_0)f[x, x_0]\n\\]\n\u3055\u3089\u306b, \\(x, x_0, x_1\\) \u306b\u5bfe\u3059\u308b 2 \u968e\u306e\u5dee\u5206\u5546
\n\\[
\nf[x, x_0, x_1] = (f[x_0, x_1] – f[x, x_0])\/(x_1 – x)
\n\\]\n\u3092 \\(f[x, x_0]\\) \u306b\u3064\u3044\u3066\u89e3\u304d, \u4ee3\u5165\u3059\u308b\u3068\u6b21\u306e\u3088\u3046\u306b\u306a\u308b.
\n\\[
\nf(x) = f(x_0) + (x – x_0)f[x_0, x_1] + (x – x_0)(x – x_1)f[x, x_0, x_1]\n\\]\n\u540c\u69d8\u306e\u4ee3\u5165\u3092\u7e70\u308a\u8fd4\u3059\u3068, n + 1 \u500b\u306e\u6a19\u672c\u70b9 \\(x_0, x_1, \\dots, x_n\\) \u306b\u3064\u3044\u3066\u6b21\u5f0f\u304c\u5f97\u3089\u308c\u308b (\\(f[x_i] = f(x_i)\\) \u3068\u8868\u3057\u305f).
\n\\[
\n\\begin{align}
\nf(x) & = f[x_0] \\\\
\n& + (x – x_0)f[x_0, x_1] \\\\
\n& + (x – x_0)(x – x_1)f[x_0, x_1, x_2] \\\\
\n& + \\dots \\\\
\n& + (x – x_0)(x – x_1) \\dots (x – x_{n-1})f[x_0, x_1, \\dots, x_n] \\\\
\n& + (x – x_0)(x – x_1) \\dots (x – x_{n-1})(x – x_n)f[x, x_0, x_1, \\dots, x_n] \\\\
\n\\end{align}
\n\\]\n\u3053\u3053\u3067, \u6700\u5f8c\u306e\u9805\u306f \\(f[x, x_0, x_1, \\dots, x_n]\\) \u306b\u5909\u6570 \\(x\\) \u3092\u542b\u3080\u305f\u3081, \u901a\u5e38\u306f\u8a08\u7b97\u3059\u308b\u3053\u3068\u304c\u96e3\u3057\u3044.<\/p>\n

6.1.6 \u30cb\u30e5\u30fc\u30c8\u30f3\u88dc\u9593<\/h3>\n

\u95a2\u6570 \\(f(x)\\) \u3092\u5dee\u5206\u5546\u3092\u7528\u3044\u3066\u8868\u3057\u305f\u4e0a\u5f0f\u306b\u304a\u3044\u3066\u6700\u5f8c\u306e\u9805\u3092 \\(0\\) \u3068\u304a\u3044\u305f\u6b21\u5f0f\u3092\u30cb\u30e5\u30fc\u30c8\u30f3\u88dc\u9593\u516c\u5f0f\u3068\u3044\u3046.
\n\\[
\n\\begin{align}
\nf(x) & = f[x_0] \\\\
\n& + (x – x_0)f[x_0, x_1] \\\\
\n& + (x – x_0)(x – x_1)f[x_0, x_1, x_2] \\\\
\n& + \\dots \\\\
\n& + (x – x_0)(x – x_1) \\dots (x – x_{n-1})f[x_0, x_1, \\dots, x_n] \\\\
\n\\end{align}
\n\\]\n\u3053\u306e\u5f0f\u306f \\(f(x)\\) \u306e\u8fd1\u4f3c\u5f0f\u306b\u306a\u3063\u3066\u304a\u308a, \\(0\\) \u3068\u304a\u3044\u305f\u6700\u5f8c\u306e\u9805\u306f\u5270\u4f59\u9805\u3068\u547c\u3070\u308c\u8fd1\u4f3c\u8aa4\u5dee\u3092\u8868\u3059.<\/p>\n

\u30cb\u30e5\u30fc\u30c8\u30f3\u88dc\u9593\u516c\u5f0f\u306f, \u30e9\u30b0\u30e9\u30f3\u30b8\u30e5\u88dc\u9593\u516c\u5f0f\u3092 \\(1, (x – x_0), (x – x_0)(x – x_1), \\dots\\) \u306b\u3064\u3044\u3066\u5c55\u958b\u3057\u305f\u5f0f\u306b\u76f8\u5f53\u3059\u308b. \u3059\u3067\u306b\u6c42\u3081\u305f\u88dc\u9593\u5f0f\u306b\u65b0\u305f\u306b\u6a19\u672c\u70b9\u3092\u8ffd\u52a0\u3059\u308b\u3068\u304d, \u30e9\u30b0\u30e9\u30f3\u30b8\u30e5\u88dc\u9593\u516c\u5f0f\u3067\u306f\u5168\u90e8\u8a08\u7b97\u3057\u76f4\u3059\u5fc5\u8981\u304c\u3042\u308b\u304c, \u30cb\u30e5\u30fc\u30c8\u30f3\u88dc\u9593\u516c\u5f0f\u3067\u306f\u65b0\u305f\u306a\u9805\u3092\u8ffd\u52a0\u3059\u308b\u3060\u3051\u3067\u3088\u3044.<\/p>\n

\u30cb\u30e5\u30fc\u30c8\u30f3\u88dc\u9593\u516c\u5f0f\u306e\u8a08\u7b97\u306f\u6b21\u306e\u3088\u3046\u306b\u3057\u3066\u884c\u3046\u3068\u3088\u3044.<\/p>\n

\u307e\u305a, \u4e0e\u3048\u3089\u308c\u305f\u30c7\u30fc\u30bf\u306b\u3064\u3044\u3066\u5dee\u5206\u5546\u3092\u4e0b\u56f3\u306e\u3088\u3046\u306b\u5de6\u304b\u3089\u53f3\u306e\u9806\u306b\u8a08\u7b97\u3057\u3066\u304a\u304f. \u5b9f\u969b\u306b\u88dc\u9593\u5024\u306e\u8a08\u7b97\u306b\u5fc5\u8981\u306a\u306e\u306f\u5bfe\u89d2\u90e8\u5206\u3060\u3051\u3067\u3042\u308b\u304c, \u6a19\u672c\u70b9\u3092\u8ffd\u52a0\u3059\u308b\u3068\u304d\u306b\u4e00\u756a\u4e0b\u306e\u884c\u304c\u5fc5\u8981\u306b\u306a\u308b.
\n\\[
\n\\begin{align}
\n& f[x_0] \\\\
\n& & \\ddots \\\\
\n& f[x_1] & \\dots & \\quad f[x_0, x_1] \\\\
\n& & \\ddots & & \\ddots \\\\
\n& f[x_2] & \\dots & \\quad f[x_1, x_2] & \\dots & \\quad f[x_0, x_1, x_2] \\\\
\n& & \\ddots & & \\ddots \\\\
\n& \\vdots \\\\
\n& & \\ddots & & \\ddots & & & & \\ddots \\\\
\n& f[x_n] & \\dots & \\quad f[x_{n-1}, x_n] & \\dots & \\quad f[x_{n-2}, x_{n-1}, x_n] & \\dots & \\quad & \\dots & \\quad f[x_0, \\dots, x_n] \\\\
\n\\end{align}
\n\\]\n\u6b21\u306b, \u6c42\u3081\u305f\u3044\u70b9 \\(x\\) \u305d\u308c\u305e\u308c\u306b\u3064\u3044\u3066, \u591a\u9805\u5f0f\u3092\u8a08\u7b97\u3059\u308b\u8981\u9818\u3067\u30cb\u30e5\u30fc\u30c8\u30f3\u88dc\u9593\u516c\u5f0f\u306b\u3088\u308a\u88dc\u9593\u5024\u3092\u8a08\u7b97\u3059\u308b.<\/p>\n

\u6a19\u672c\u70b9\u3092\u8ffd\u52a0\u3059\u308b\u3068\u304d\u306f\u5dee\u5206\u5546\u306e\u4e00\u756a\u4e0b\u306e\u884c\u3092\u8ffd\u52a0\u8a08\u7b97\u3057, \u30cb\u30e5\u30fc\u30c8\u30f3\u88dc\u9593\u516c\u5f0f\u306e\u6700\u5f8c\u306e\u9805\u3092\u88dc\u9593\u5024\u306b\u52a0\u3048\u308b.<\/p>\n

6.1.2 \u304a\u3088\u3073 6.1.3 \u306e\u6570\u5024\u5b9f\u9a13\u306b\u3064\u3044\u3066\u30cb\u30e5\u30fc\u30c8\u30f3\u88dc\u9593\u516c\u5f0f\u306b\u3088\u308a\u8a08\u7b97\u3059\u308b\u3068\u591a\u9805\u5f0f\u88dc\u9593\u304a\u3088\u3073\u30e9\u30b0\u30e9\u30f3\u30b8\u30e5\u88dc\u9593\u516c\u5f0f\u3068\u8aa4\u5dee\u306e\u7bc4\u56f2\u3067\u540c\u3058\u7d50\u679c\u304c\u5f97\u3089\u308c\u308b.<\/p>\n

6.1.7 \u53cd\u5fa9\u88dc\u9593\u6cd5<\/h3>\n

\u88dc\u9593\u306b\u3088\u308a \\(x\\) \u306b\u304a\u3051\u308b\u95a2\u6570\u5024\u3092\u8a08\u7b97\u3059\u308b\u3068\u304d\u306b, \u306f\u3058\u3081\u306f\u5c11\u306a\u3044\u6a19\u672c\u70b9\u3092\u4f7f\u3063\u3066\u8a08\u7b97\u3057, \u6b21\u7b2c\u306b\u6a19\u672c\u70b9\u306e\u6570\u3092\u5897\u3084\u3057\u3066\u8a08\u7b97\u3092\u7e70\u308a\u8fd4\u3057\u76ee\u6a19\u7cbe\u5ea6\u306b\u9054\u3057\u305f\u3089 (\u6a19\u672c\u70b9\u3092\u5897\u3084\u3059\u524d\u5f8c\u306e\u5dee\u304c\u8a31\u5bb9\u7bc4\u56f2\u306b\u5165\u3063\u305f\u3089) \u53ce\u675f\u3068\u307f\u306a\u3057\u3066\u7d42\u4e86\u3059\u308b\u65b9\u6cd5\u3092\u53cd\u5fa9\u88dc\u9593\u6cd5\u3068\u3044\u3046.<\/p>\n

\u8a08\u7b97\u306f, \u30cb\u30e5\u30fc\u30c8\u30f3\u88dc\u9593\u516c\u5f0f\u3092\u4f7f\u7528\u3057\u3066\u6a19\u672c\u70b9\u3092\u8ffd\u52a0\u3057\u3066\u3044\u304f\u3053\u3068\u306b\u3088\u308a\u884c\u3046. \u30cb\u30e5\u30fc\u30c8\u30f3\u88dc\u9593\u516c\u5f0f\u306f\u6a19\u672c\u70b9\u306e\u4e26\u3073\u9806\u306b\u95a2\u4fc2\u306a\u304f\u9069\u7528\u3067\u304d\u308b\u304c, \u53cd\u5fa9\u88dc\u9593\u6cd5\u306b\u9069\u7528\u3059\u308b\u5834\u5408\u306b\u306f, \u6a19\u672c\u70b9\u3092 \\(x\\) \u306b\u8fd1\u3044\u9806\u306b\u8ffd\u52a0\u3059\u308b\u3088\u3046\u306b\u3057, \u3057\u304b\u3082\u6a19\u672c\u70b9\u306e\u6570\u304c \\(x\\) \u306e\u4e21\u5074\u3067\u307b\u307c\u7b49\u3057\u304f\u306a\u308b\u3088\u3046\u306b\u3059\u308b\u306e\u304c\u53ce\u675f\u306e\u901f\u3055\u306e\u70b9\u3067\u6709\u5229\u3067\u3042\u308b.<\/p>\n","protected":false},"excerpt":{"rendered":"

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