{"id":4755,"date":"2025-03-31T13:51:36","date_gmt":"2025-03-31T04:51:36","guid":{"rendered":"https:\/\/www.ktech.biz\/jp\/?p=4755"},"modified":"2025-08-19T16:12:42","modified_gmt":"2025-08-19T07:12:42","slug":"7-nleq1","status":"publish","type":"post","link":"https:\/\/www.ktech.biz\/jp\/tutorial\/7-nleq1\/","title":{"rendered":"7. \u975e\u7dda\u5f62\u65b9\u7a0b\u5f0f (1 \u5909\u6570)"},"content":{"rendered":"\n

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

\u4e00\u822c\u306e 1 \u5909\u6570\u95a2\u6570 f(x) \u306e\u5b9f\u6570\u306e\u30bc\u30ed\u70b9\u3092\u6c42\u3081\u308b\u554f\u984c\u3092\u8003\u3048\u307e\u3059.
\n\\[
\nf(x) = 0
\n\\]\n\\(f(x)\\) \u304c\u591a\u9805\u5f0f\u3067\u8868\u3055\u308c\u308b\u5834\u5408\u306f\u4ee3\u6570\u65b9\u7a0b\u5f0f\u3068\u547c\u3070\u308c, \u305d\u308c\u306b\u7279\u5316\u3057\u305f\u89e3\u6cd5\u304c\u7528\u3044\u3089\u308c\u307e\u3059(6 \u7ae0\u3092\u53c2\u7167).<\/p>\n

\u305d\u3046\u3067\u306a\u3044\u5834\u5408\u306f\u975e\u7dda\u5f62\u65b9\u7a0b\u5f0f(\u3042\u308b\u3044\u306f, \u8d85\u8d8a\u65b9\u7a0b\u5f0f)\u3068\u547c\u3070\u308c\u307e\u3059. \u3053\u3053\u3067\u306f\u5b9f\u6570\u89e3\u304c\u5b58\u5728\u3059\u308b\u3082\u306e\u3068\u3057\u307e\u3059. \u89e3\u6790\u7684\u306b\u9006\u95a2\u6570\u304c\u6c42\u3081\u3089\u308c\u306a\u3044\u9650\u308a\u306f, \u9069\u5f53\u306a\u521d\u671f\u5024\u3092\u4e0e\u3048\u3066\u53cd\u5fa9\u6cd5\u306b\u3088\u3063\u3066\u8fd1\u4f3c\u89e3\u3092\u6c42\u3081\u308b\u3053\u3068\u306b\u306a\u308a\u307e\u3059.<\/p>\n

7.2 \u975e\u7dda\u5f62\u65b9\u7a0b\u5f0f (1 \u5909\u6570) \u306e\u89e3\u6cd5<\/h3>\n

\u4ee3\u8868\u7684\u306a\u89e3\u6cd5\u306f\u30cb\u30e5\u30fc\u30c8\u30f3\u6cd5\u3067\u3059. \u53ce\u675f\u304c\u901f\u304f, \u6570\u5024\u8a08\u7b97\u3067\u306f\u3053\u308c\u3092\u30d9\u30fc\u30b9\u306b\u3057\u305f\u89e3\u6cd5\u304c\u591a\u304f\u4f7f\u308f\u308c\u3066\u3044\u307e\u3059. \u4ed6\u306b, \u5b89\u5b9a\u3057\u305f\u89e3\u6cd5\u3068\u3057\u3066\u30bc\u30ed\u70b9\u3092\u542b\u3080\u533a\u9593\u3092\u7e2e\u5c0f\u3057\u3066\u3044\u304f\u4e8c\u5206\u6cd5\u304c\u4f7f\u308f\u308c\u307e\u3059.<\/p>\n

7.2.1 \u30cb\u30e5\u30fc\u30c8\u30f3\u6cd5<\/h4>\n

\\(f(x)\\) \u3092\u7b2c k \u8fd1\u4f3c \\(x_k\\) \u306e\u307e\u308f\u308a\u3067\u30c6\u30a4\u30e9\u30fc\u5c55\u958b\u3059\u308b\u3068\u6b21\u306e\u3088\u3046\u306b\u306a\u308a\u307e\u3059.
\n\\[
\nf(x) = f(x_k) + f'(x_k)(x – x_k) + (1\/2)f”(x_k)(x – x_k)^2 + \\dots
\n\\]\n\u3053\u3053\u3067, \u53f3\u8fba\u7b2c 2 \u9805\u307e\u3067\u3067\u8fd1\u4f3c\u3057 (2 \u6b21\u4ee5\u4e0a\u306e\u9805\u3092 0 \u3068\u307f\u306a\u3057), \\(f(x) = 0\\) \u3068\u3059\u308b\u3068\u6b21\u5f0f\u304c\u5f97\u3089\u308c\u307e\u3059.
\n\\[
\nx = x_k – f(x_k)\/f'(x_k)
\n\\]\n\u3053\u306e \\(x\\) \u3092\u7b2c k + 1 \u8fd1\u4f3c \\(x_{k+1}\\) \u3068\u3059\u308b\u3068\u6b21\u306e\u30cb\u30e5\u30fc\u30c8\u30f3\u6cd5\u306e\u53cd\u5fa9\u5f0f\u304c\u5f97\u3089\u308c\u307e\u3059.
\n\\[
\n\\begin{align}
\n& d_k = -f(x_k)\/f'(x_k) \\\\
\n& x_{k+1} = x_k + d_k \\\\
\n\\end{align}
\n\\]\n\u53cd\u5fa9\u306f, \u9069\u5f53\u306a\u521d\u671f\u8fd1\u4f3c \\(x_0\\) \u304b\u3089\u958b\u59cb\u3057, \\(|d_k|\\) \u304c\u5341\u5206\u306b\u5c0f\u3055\u304f\u306a\u3063\u305f\u3089\u53ce\u675f\u3068\u307f\u306a\u3057\u7d42\u4e86\u3057\u307e\u3059.<\/p>\n

\u5f0f\u306e\u8aa4\u5dee\u306f \\(d_k\\) \u306e\u4e8c\u4e57\u306e\u30aa\u30fc\u30c0\u30fc\u306b\u306a\u308a, \u3053\u306e\u89e3\u6cd5\u306f 2 \u6b21\u53ce\u675f\u3059\u308b\u3068\u3044\u3044\u307e\u3059.<\/p>\n

\u3053\u306e\u89e3\u6cd5\u306e\u5e7e\u4f55\u7684\u89e3\u91c8\u306f\u6b21\u306e\u3068\u304a\u308a\u3067\u3059.<\/p>\n

\"\"<\/p>\n

\\(x_0\\) \u304b\u3089\u59cb\u3081\u3066\u305d\u306e\u70b9\u306b\u304a\u3051\u308b\u63a5\u7dda\u304c \\(x\\) \u8ef8\u3068\u4ea4\u308f\u308b\u70b9\u3092 \\(x_1\\) \u3068\u3057\u307e\u3059. \u5fae\u5206\u5024 \\(f'(x_0)\\) \u306f\u63a5\u7dda\u306e\u50be\u304d\u3092\u8868\u3059\u306e\u3067, \u305d\u308c\u3092\u4f7f\u3063\u3066\u4ea4\u70b9 \\(x_1\\) \u3092\u6c42\u3081\u308b\u3053\u3068\u304c\u3067\u304d\u307e\u3059.<\/p>\n

\u540c\u69d8\u306b \\(x_2, x_3, \\dots\\) \u3068\u3044\u3046\u3088\u3046\u306b\u7e70\u308a\u8fd4\u3057\u6b21\u306e\u70b9\u3092\u6c42\u3081\u3066\u3044\u304f\u3068, \u6025\u901f\u306b\u89e3\u306b\u8fd1\u3065\u3044\u3066\u3044\u304f\u3053\u3068\u304c\u308f\u304b\u308a\u307e\u3059.<\/p>\n

\u3053\u306e\u89e3\u6cd5\u306f\u53ce\u675f\u304c\u901f\u3044\u305f\u3081\u3088\u304f\u4f7f\u308f\u308c\u307e\u3059. \u305f\u3060\u3057, \u5fae\u5206\u5024 \\(f'(x)\\) \u306e\u8a08\u7b97\u304c\u96e3\u3057\u3044\u5834\u5408\u306b\u306f\u9069\u7528\u3067\u304d\u306a\u3044\u3053\u3068\u304c\u3042\u308a\u307e\u3059.<\/p>\n

\u307e\u305f, \u89e3\u306b\u5341\u5206\u8fd1\u304f\u95a2\u6570\u304c\u6ed1\u3089\u304b\u3067\u3042\u308b\u3068\u307f\u306a\u305b\u308b\u9818\u57df\u306e\u521d\u671f\u5024\u3092\u4e0e\u3048\u308b\u5fc5\u8981\u304c\u3042\u308a\u307e\u3059. \u521d\u671f\u5024\u304c\u60aa\u3044\u305f\u3081\u306b\u5931\u6557\u3059\u308b\u4f8b\u306f\u6b21\u306e\u3088\u3046\u306a\u3082\u306e\u3067\u3059.<\/p>\n

\"\"<\/p>\n

\u3053\u306e\u4f8b\u3067\u306f 2 \u56de\u76ee\u306e\u53cd\u5fa9\u306b\u304a\u3044\u3066\u767a\u6563\u3057\u3066 (\u60f3\u5b9a\u5916\u306e \\(x\\) \u306b\u8df3\u3093\u3067) \u3044\u307e\u3059. \u7d50\u679c\u7684\u306b\u306f, \u30aa\u30fc\u30d0\u30fc\u30d5\u30ed\u30fc\u3092\u5f15\u304d\u8d77\u3053\u3057\u305f\u308a\u7121\u9650\u30eb\u30fc\u30d7\u306b\u9665\u3063\u305f\u308a\u3057\u307e\u3059.<\/p>\n

7.2.2 \u4e8c\u5206\u6cd5<\/h4>\n

\u521d\u671f\u5024\u3068\u3057\u3066 2 \u3064\u306e\u70b9 \\(a\\) \u3068 \\(b\\) \u3092\u4e0e\u3048\u307e\u3059. \u305d\u308c\u305e\u308c\u306b\u304a\u3051\u308b\u95a2\u6570\u5024\u306f \\(f(a)\\) \u3068 \\(f(b)\\) \u3067\u3059. \u533a\u9593 \\([a, b]\\) \u306b\u306f\u89e3\u304c 1 \u3064\u3060\u3051\u5165\u3063\u3066\u3044\u306a\u3051\u308c\u3070\u306a\u308a\u307e\u305b\u3093. \u5f93\u3063\u3066, \\(f(a)\\) \u306e\u7b26\u53f7 \u3068 \\(f(b)\\) \u306e\u7b26\u53f7\u306f\u53cd\u5bfe\u3067\u306a\u3051\u308c\u3070\u306a\u308a\u307e\u305b\u3093.<\/p>\n

\u4e2d\u70b9 \\(c = (a + b)\/2\\) \u3067\u95a2\u6570\u5024 \\(f(c)\\) \u3092\u8a08\u7b97\u3057\u307e\u3059.<\/p>\n

\"\"<\/p>\n

\\(f(c)\\) \u306e\u7b26\u53f7\u3068 \\(f(a)\\) \u306e\u7b26\u53f7\u304c\u540c\u3058\u306a\u3089\u3070, \\(c\\) \u3092\u65b0\u3057\u3044 \\(a\\) \u3068\u3057\u307e\u3059. \u305d\u3046\u3067\u306a\u3051\u308c\u3070, \\(c\\) \u3092\u65b0\u3057\u3044 \\(b\\) \u3068\u3057\u307e\u3059. \u305d\u3057\u3066\u65b0\u3057\u3044 \\(c = (a + b)\/2\\) \u3092\u6c42\u3081, \u7e70\u308a\u8fd4\u3057\u307e\u3059. <\/p>\n

\"\"<\/p>\n

\u3053\u306e\u3088\u3046\u306b\u3057\u3066\u53cd\u5fa9\u306e\u305f\u3073\u306b\u533a\u9593\u306e\u5e45\u3092\u534a\u5206\u306b\u3057\u3066\u3044\u304f\u89e3\u6cd5\u3092\u4e8c\u5206\u6cd5\u3068\u3044\u3044\u307e\u3059. \u533a\u9593\u306e\u5e45\u304c\u5341\u5206\u306b\u5c0f\u3055\u304f\u306a\u3063\u305f\u3089\u53cd\u5fa9\u3092\u7d42\u4e86\u3057\u307e\u3059.<\/p>\n

\u3053\u306e\u89e3\u6cd5\u306f, \u53ce\u675f\u306f\u901f\u304f\u3042\u308a\u307e\u305b\u3093\u304c, \u89e3\u3092\u306f\u3055\u3080\u533a\u9593\u304c\u308f\u304b\u3063\u3066\u3055\u3048\u3044\u308c\u3070\u78ba\u5b9f\u306b\u53ce\u675f\u3059\u308b\u7279\u9577\u304c\u3042\u308a\u307e\u3059. \u307e\u305f, \u5fae\u5206\u5024\u306e\u8a08\u7b97\u304c\u4e0d\u8981\u306a\u306e\u3082\u30e1\u30ea\u30c3\u30c8\u3068\u306a\u308a\u307e\u3059.<\/p>\n

7.3 XLPack \u3092\u4f7f\u3063\u305f 1 \u5909\u6570\u975e\u7dda\u5f62\u65b9\u7a0b\u5f0f\u306e\u89e3\u304d\u65b9<\/h3>\n

VBA \u30b5\u30d6\u30eb\u30fc\u30c1\u30f3 Dfzero<\/strong> \u306f\u89e3\u3092\u306f\u3055\u3080\u533a\u9593\u3092\u4e0e\u3048\u308b\u3068\u4e8c\u5206\u6cd5\u3088\u308a\u3082\u9ad8\u901f\u306b\u89e3\u3092\u898b\u3064\u3051\u308b\u30d7\u30ed\u30b0\u30e9\u30e0\u3067\u3059. \u53cd\u5fa9\u306b\u304a\u3044\u3066 \\(a, b, c\\) \u306e 3 \u70b9\u3092\u4f7f\u3063\u3066\u88dc\u9593\u3092\u884c\u3044\u30bc\u30ed\u70b9\u3092\u63a8\u5b9a\u3057\u307e\u3059. \u305d\u308c\u304c\u59a5\u5f53\u3067\u3042\u308c\u3070\u63a1\u7528\u3057, \u305d\u3046\u3067\u306a\u3051\u308c\u3070\u4e8c\u5206\u6cd5\u306e\u53cd\u5fa9\u3092\u884c\u3044\u307e\u3059. \u5168\u4f53\u3068\u3057\u3066\u4e8c\u5206\u6cd5\u3088\u308a\u3082\u901f\u304f\u53ce\u675f\u3059\u308b\u3053\u3068\u306b\u306a\u308a\u307e\u3059. Dfzero \u306f XLPack \u30bd\u30eb\u30d0\u30fc\u304b\u3089\u4f7f\u3046\u3053\u3068\u3082\u3067\u304d\u307e\u3059.<\/p>\n

\u4f8b\u984c<\/h5>\n

\u6b21\u306e\u65b9\u7a0b\u5f0f\u3092\u89e3\u304f.
\n\\[
\nf(x) = x^3 – 2x – 5 = 0
\n\\]\n\\(f(x)\\) \u306e\u5f62\u3092\u78ba\u8a8d\u3059\u308b\u305f\u3081\u306b\u30b0\u30e9\u30d5\u3092\u63cf\u3044\u3066\u307f\u307e\u3059.<\/p>\n

\"\"<\/p>\n

\u30bc\u30ed\u70b9\u306f \\(x = 2\\) \u4ed8\u8fd1\u306b\u3042\u308b\u3053\u3068\u304c\u308f\u304b\u308b\u306e\u3067, \u30bc\u30ed\u70b9\u3092\u306f\u3055\u3080\u533a\u9593\u3068\u3057\u3066 \\([1, 3]\\) \u3092\u63a1\u7528\u3059\u308b\u3053\u3068\u306b\u3057\u307e\u3059.<\/p>\n

7.3.1 VBA \u30d7\u30ed\u30b0\u30e9\u30e0\u3092\u4f7f\u7528\u3057\u305f\u89e3\u304d\u65b9 (1)<\/h4>\n

Dfzero<\/strong> \u3092\u4f7f\u3063\u305f\u30d7\u30ed\u30b0\u30e9\u30e0\u4f8b\u3092\u793a\u3057\u307e\u3059.<\/p>\n

\n
Function F(X As Double) As Double\r\n    F = X ^ 3 - 2 * X - 5\r\nEnd Function\r\n\r\nSub Start()\r\n    Dim Ax As Double, Bx As Double, Rx As Double, Re As Double, Ae As Double, Info As Long\r\n    '--- Input data\r\n    Ax = 1\r\n    Bx = 3\r\n    Rx = Ax\r\n    Re = 0.000000000000001 '1e-15\r\n    Ae = Re\r\n    '--- Compute zero of equation\r\n    Call Dfzero(AddressOf F, Ax, Bx, Rx, Re, Ae, Info)\r\n    '--- Output zero\r\n    MsgBox Str(Ax) & \" (Info = \" & Str(Info) & \")\"\r\nEnd Sub<\/code><\/pre>\n<\/div>\n
\u76ee\u7684\u95a2\u6570\u3092\u5916\u90e8\u95a2\u6570\u3068\u3057\u3066\u7528\u610f\u3057, \u533a\u9593\u3068\u8981\u6c42\u7cbe\u5ea6\u3092\u6307\u5b9a\u3057\u3066\u547c\u3073\u51fa\u3057\u307e\u3059.<\/p>\n
Dfzero<\/strong> \u306b\u306f\u78ba\u5b9f\u306b\u30bc\u30ed\u70b9\u3092\u306f\u3055\u3080\u533a\u9593\u3092\u4e0e\u3048\u308b\u3088\u3046\u306b\u6ce8\u610f\u304c\u5fc5\u8981\u3067\u3059. \u306a\u304a, \u4e0e\u3048\u3089\u308c\u305f\u533a\u9593\u306b 2 \u3064\u4ee5\u4e0a\u306e\u30bc\u30ed\u70b9\u304c\u542b\u307e\u308c\u3066\u3044\u308b\u5834\u5408, \u3069\u306e\u70b9\u306b\u53ce\u675f\u3059\u308b\u304b\u306f\u308f\u304b\u308a\u307e\u305b\u3093.<\/p>\n
\u3053\u306e\u30d7\u30ed\u30b0\u30e9\u30e0\u3092\u5b9f\u884c\u3059\u308b\u3068, \\(x = 2.0945514815\\) \u304c\u5f97\u3089\u308c\u307e\u3059.<\/p>\n
\"\"<\/p>\n
7.3.2 VBA \u30d7\u30ed\u30b0\u30e9\u30e0\u3092\u4f7f\u7528\u3057\u305f\u89e3\u304d\u65b9 (2)<\/h4>\n
\u30ea\u30d0\u30fc\u30b9\u30b3\u30df\u30e5\u30cb\u30b1\u30fc\u30b7\u30e7\u30f3\u7248 (RCI) \u306e Dfzero_r<\/strong> \u3092\u4f7f\u3063\u305f\u30d7\u30ed\u30b0\u30e9\u30e0\u4f8b\u3092\u793a\u3057\u307e\u3059.<\/p>\n
\n
Sub Start()\r\n    Dim Ax As Double, Bx As Double, Rx As Double, Re As Double, Ae As Double, Info As Long\r\n    Dim XX As Double, YY As Double, IRev As Long\r\n    '--- Input data\r\n    Ax = 1\r\n    Bx = 3\r\n    Rx = Ax\r\n    Re = 0.000000000000001 '1e-15\r\n    Ae = Re\r\n    '--- Compute zero of equation\r\n    IRev = 0\r\n    Do\r\n        Call Dfzero_r(Ax, Bx, Rx, Re, Ae, Info, XX, YY, IRev)\r\n        If IRev <> 0 Then YY = XX ^ 3 - 2 * XX - 5\r\n    Loop While IRev <> 0\r\n    '--- Output zero\r\n    MsgBox Str(Ax) & \" (Info = \" & Str(Info) & \")\"\r\nEnd Sub<\/code><\/pre>\n<\/div>\n
\u76ee\u7684\u95a2\u6570\u3092\u5916\u90e8\u95a2\u6570\u3068\u3057\u3066\u4e0e\u3048\u308b\u306e\u3067\u306f\u306a\u304f, IRev = 1\u306e\u3068\u304d\u306b XX \u306e\u5024\u3092\u4f7f\u3063\u3066\u95a2\u6570\u5024\u3092\u8a08\u7b97\u3057 YY \u306b\u5165\u308c\u3066\u518d\u5ea6 Dfzero_r<\/strong> \u3092\u547c\u3073\u51fa\u3057\u307e\u3059. RCI \u306e\u8a73\u7d30\u306b\u3064\u3044\u3066\u306f \u3053\u3061\u3089<\/a> \u3092\u53c2\u7167\u3057\u3066\u304f\u3060\u3055\u3044.<\/p>\n
\u3053\u306e\u30d7\u30ed\u30b0\u30e9\u30e0\u3092\u5b9f\u884c\u3059\u308b\u3068\u4e0a\u3068\u540c\u3058\u7d50\u679c\u304c\u5f97\u3089\u308c\u307e\u3059.<\/p>\n
7.3.3 \u30bd\u30eb\u30d0\u30fc\u3092\u4f7f\u7528\u3057\u305f\u89e3\u304d\u65b9<\/h4>\n
XLPack\u30bd\u30eb\u30d0\u30fc\u30a2\u30c9\u30a4\u30f3\u306e\u300c\uff11\u5909\u6570\u975e\u7dda\u5f62\u65b9\u7a0b\u5f0f\u300d\u3092\u4f7f\u3063\u3066\u89e3\u304f\u3053\u3068\u3082\u3067\u304d\u307e\u3059. B7\u30bb\u30eb\u306b\u6570\u5f0f (=B6^3-2*B6-5) \u304c\u5165\u529b\u3055\u308c\u3066\u3044\u307e\u3059.<\/p>\n
\"\"<\/p>\n
\u30bd\u30eb\u30d0\u30fc\u306b\u3064\u3044\u3066\u306f \u3053\u3061\u3089<\/a> \u3082\u53c2\u7167\u304f\u3060\u3055\u3044.<\/p>\n","protected":false},"excerpt":{"rendered":"
\u975e\u7dda\u5f62\u65b9\u7a0b\u5f0f\u306e\u89e3\u6cd5\u3068XLPack\u3092\u4f7f\u3063\u305f\u89e3\u304d\u65b9<\/p>\n","protected":false},"author":3,"featured_media":0,"comment_status":"closed","ping_status":"closed","sticky":false,"template":"","format":"standard","meta":{"_monsterinsights_skip_tracking":false,"_monsterinsights_sitenote_active":false,"_monsterinsights_sitenote_note":"","_monsterinsights_sitenote_category":0,"footnotes":""},"categories":[10],"tags":[],"class_list":["post-4755","post","type-post","status-publish","format-standard","hentry","category-tutorial"],"aioseo_notices":[],"_links":{"self":[{"href":"https:\/\/www.ktech.biz\/jp\/wp-json\/wp\/v2\/posts\/4755","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/www.ktech.biz\/jp\/wp-json\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/www.ktech.biz\/jp\/wp-json\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/www.ktech.biz\/jp\/wp-json\/wp\/v2\/users\/3"}],"replies":[{"embeddable":true,"href":"https:\/\/www.ktech.biz\/jp\/wp-json\/wp\/v2\/comments?post=4755"}],"version-history":[{"count":5,"href":"https:\/\/www.ktech.biz\/jp\/wp-json\/wp\/v2\/posts\/4755\/revisions"}],"predecessor-version":[{"id":5131,"href":"https:\/\/www.ktech.biz\/jp\/wp-json\/wp\/v2\/posts\/4755\/revisions\/5131"}],"wp:attachment":[{"href":"https:\/\/www.ktech.biz\/jp\/wp-json\/wp\/v2\/media?parent=4755"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/www.ktech.biz\/jp\/wp-json\/wp\/v2\/categories?post=4755"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/www.ktech.biz\/jp\/wp-json\/wp\/v2\/tags?post=4755"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}