{"id":4648,"date":"2023-12-18T10:38:42","date_gmt":"2023-12-18T01:38:42","guid":{"rendered":"https:\/\/home.hirosaki-u.ac.jp\/relativity\/?page_id=4648"},"modified":"2025-05-06T09:51:38","modified_gmt":"2025-05-06T00:51:38","slug":"maxima-%e3%81%a6%e3%82%99%e6%95%b0%e5%80%a4%e8%a7%a3%e6%9e%90","status":"publish","type":"page","link":"https:\/\/home.hirosaki-u.ac.jp\/relativity\/%e3%82%b3%e3%83%b3%e3%83%94%e3%83%a5%e3%83%bc%e3%82%bf%e6%bc%94%e7%bf%92\/%e5%8f%82%e8%80%83%ef%bc%9amaxima-%e7%b7%a8\/maxima-%e3%81%a6%e3%82%99%e6%95%b0%e5%80%a4%e8%a7%a3%e6%9e%90\/","title":{"rendered":"Maxima \u3066\u3099\u6570\u5024\u89e3\u6790"},"content":{"rendered":"
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\u3042\u3061\u3089\u306e\u300cMaxima \u3067\u6570\u5024\u89e3\u6790<\/a>\u300d\u3067\u306f\uff0cMaxima \u306e\u6570\u5024\u89e3\u6790\u7684\u306a\u95a2\u6570\u3092\u5229\u7528\u3059\u308b\u307b\u304b\u306b\uff0c\u89e3\u6790\u7684\u306a\u65b9\u6cd5\u3084\u81ea\u524d\u3067\u6570\u5024\u89e3\u6790\u7684\u30b3\u30fc\u30c9\u3092\u66f8\u304f\u65b9\u6cd5\u306a\u3069\u3082\u4f75\u305b\u3066\u7d39\u4ecb\u3057\u3066\u3044\u308b\u306e\u3067\uff0c\u3053\u3053\u3067\u306f\u6570\u5024\u89e3\u6790\u306a\u95a2\u6570\u306e\u5229\u7528\u306b\u3064\u3044\u3066\u629c\u7c8b\u3057\u3066\u7d39\u4ecb\u3059\u308b\u3002<\/p>\n<\/div>\n<\/div>\n<\/div>\n

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\u6570\u5024\u5fae\u5206<\/h3>\n

\u89e3\u6790\u7684\u306a\u5fae\u5206\u306e\u5b9a\u7fa9\u306f\uff08\u524d\u65b9\u5dee\u5206\u306e\u6975\u9650\u3068\u3057\u3066\uff09
\n$$\\frac{df}{dx} \\equiv \\lim_{h \\rightarrow 0} \\frac{f(x+h) -f(x)}{h}$$
\n\u3067\u3057\u305f\u3002\u6ed1\u3089\u304b\u306a\u95a2\u6570\u3067\u3042\u308c\u3070\uff0c\uff08\u5f8c\u65b9\u5dee\u5206\u306e\u6975\u9650\u3068\u3057\u3066\uff09
\n$$\\frac{df}{dx} = \\lim_{h \\rightarrow 0} \\frac{f(x) -f(x-h)}{h}$$
\n\u3068\u3057\u3066\u3082\u826f\u3044\u3067\u3059\u3002<\/p>\n

\u73fe\u5b9f\u4e16\u754c\u306e Maxima \u3067\u306f\uff0c$ h\\rightarrow 0$ \u306e\u6975\u9650\u306f\u3068\u308c\u307e\u305b\u3093\u304b\u3089\u5341\u5206\u5c0f\u3055\u3044\u5024\u3068\u3057\u3066 h<\/code> \u3092\u5b9a\u7fa9\u3057\uff0c\u8fd1\u4f3c\u7684\u306a\u6570\u5024\u5fae\u5206\u3092\uff08\u524d\u65b9\u5dee\u5206\u3068\u5f8c\u65b9\u5dee\u5206\u306e\u5e73\u5747\u3067\u3042\u308b\u4e2d\u5fc3\u5dee\u5206\u3068\u3057\u3066\uff09\u4ee5\u4e0b\u306e\u3088\u3046\u306b\u5b9a\u7fa9\u3057\u307e\u3057\u3087\u3046\u3002<\/p>\n

\\begin{eqnarray}
\n\\frac{df}{dx} &\\simeq& \\frac{1}{2} \\left\\{\\frac{f(x+h) -f(x)}{h} + \\frac{f(x) -f(x-h)}{h} \\right\\} \\\\
\n&=& \\frac{f(x+h) -f(x-h)}{2h}
\n\\end{eqnarray}<\/p>\n<\/div>\n<\/div>\n<\/div>\n

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\u4f8b\u3068\u3057\u3066\uff0c$f(x) = \\sin(x)$ \u306e $x=3$ \u306b\u304a\u3051\u308b\u6570\u5024\u5fae\u5206\u3092\u6c42\u3081\u3066\u307f\u307e\u3059\u3002\uff08\u7b54\u3048\u306f $\\cos(3)$ \u3067\u3059\u3002\uff09<\/p>\n<\/div>\n<\/div>\n<\/div>\n

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f<\/span>(<\/span>x<\/span>)<\/span>:=<\/span> sin<\/span>(<\/span>x<\/span>)<\/span>$\r\n\r\n\/* f(x) \u306e\u6570\u5024\u5fae\u5206 *\/<\/span>\r\n\/* Maxima \u3067\u306f\u6570\u5024\u5fae\u5206\u3059\u308b\u95a2\u6570\u306f\u306a\u3044\u306e\u3067\u81ea\u4f5c\u3059\u308b *\/<\/span>\r\ndiff_f<\/span>(<\/span>x<\/span>, h<\/span>)<\/span>:=<\/span> (<\/span>f<\/span>(<\/span>x<\/span>+<\/span>h<\/span>)<\/span> -<\/span> f<\/span>(<\/span>x<\/span>-<\/span>h<\/span>))<\/span>\/<\/span>(<\/span>2.<\/span>0*<\/span>h<\/span>)<\/span>$\r\n\r\nprintf<\/span>(<\/span>true<\/span>, \"\u6570\u5024\u5fae\u5206  ~f~%\"<\/span>, diff_f<\/span>(<\/span>3<\/span>, 1e-6<\/span>))<\/span>$\r\nprintf<\/span>(<\/span>true<\/span>, \"\u89e3\u6790\u89e3\u3000  ~f~%\"<\/span>, cos<\/span>(<\/span>3.0<\/span>))<\/span>$\r\n<\/pre>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
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\u6570\u5024\u5fae\u5206  -0.9899924967304852\r\n\u89e3\u6790\u89e3\u3000  -0.9899924966004454\r\n<\/pre>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
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\u4ee5\u4e0b\u306e\u4f8b\u3067\u306f\uff0c$f(x) = \\sin x$ \u306b\u5bfe\u3057\u3066 $f'(x) = \\cos x$ \u3068\uff0c\u4e0a\u8a18\u306e\u6570\u5024\u5fae\u5206\u3068\u306e\u8aa4\u5dee\u3092\u63cf\u3044\u3066\u3044\u307e\u3059\u3002<\/p>\n<\/div>\n<\/div>\n<\/div>\n
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In\u00a0[2]:<\/div>\n
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plot2d<\/span>(<\/span>cos<\/span>(<\/span>x<\/span>)<\/span> -<\/span> diff_f<\/span>(<\/span>x<\/span>, 1.<\/span>0E<\/span>-<\/span>6<\/span>)<\/span>, [<\/span>x<\/span>, -<\/span>2*<\/span>%pi<\/span>, 2*<\/span>%pi<\/span>]<\/span>, [<\/span>ylabel<\/span>,\"\"<\/span>])<\/span>$\r\n<\/pre>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
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\u4ee5\u4e0b\u306e\u4f8b\u304b\u3089\u308f\u304b\u308b\u3088\u3046\u306b\uff0ch<\/code> \u3092\u3084\u307f\u304f\u3082\u306b\u5c0f\u3055\u304f\u3059\u308c\u3070\u3059\u308b\u307b\u3069\uff0c\u6570\u5024\u5fae\u5206\u306e\u7cbe\u5ea6\u304c\u3042\u304c\u308b\uff0c\u3068\u3044\u3046\u308f\u3051\u3067\u3082\u306a\u3055\u305d\u3046\u3067\u3059\u3002<\/p>\n<\/div>\n<\/div>\n<\/div>\n
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In\u00a0[3]:<\/div>\n
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plot2d<\/span>(<\/span>cos<\/span>(<\/span>x<\/span>)<\/span> -<\/span> diff_f<\/span>(<\/span>x<\/span>, 1.<\/span>0E<\/span>-<\/span>7<\/span>)<\/span>, [<\/span>x<\/span>, -<\/span>2*<\/span>%pi<\/span>, 2*<\/span>%pi<\/span>]<\/span>, [<\/span>ylabel<\/span>,\"\"<\/span>])<\/span>$\r\n<\/pre>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
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\u6570\u5024\u7a4d\u5206<\/h3>\n
romberg()<\/code><\/h4>\n<\/div>\n<\/div>\n<\/div>\n
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$\\displaystyle \\int_0^{\\pi} \\sin (\\sin x) \\, dx$ \u306e\u3088\u3046\u306b\u89e3\u6790\u7684\u306b\u7a4d\u5206\u3067\u304d\u306a\u3044\u5834\u5408\u306f\uff0cromberg()<\/code> \u3068\u3044\u3046\u7d44\u8fbc\u95a2\u6570\u3067\u6570\u5024\u7a4d\u5206\u3067\u304d\u307e\u3059\u3002<\/p>\n<\/div>\n<\/div>\n<\/div>\n
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In\u00a0[4]:<\/div>\n
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romberg<\/span>(<\/span>sin<\/span>(<\/span>sin<\/span>(<\/span>x<\/span>))<\/span>, x<\/span>, 0<\/span>, %pi<\/span>)<\/span>;\r\n<\/pre>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
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\\[\\tag{${\\it \\%o}_{9}$}1.786487487137068\\]<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
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\u6570\u5024\u7a4d\u5206 romberg \u306e\u7cbe\u5ea6\u306f\uff0c\u5927\u57df\u5909\u6570 rombergtol<\/code> \u3066\u3099\u6c7a\u307e\u308a\u307e\u3059\u3002<\/p>\n
rombergtol<\/code> \u306e\u5024\u3092\u5c0f\u3055\u304f\u3059\u308b\u3053\u3068\u3066\u3099\u7cbe\u5ea6\u3092\u3042\u3051\u3099\u308b\u3053\u3068\u304b\u3099\u3066\u3099\u304d\u307e\u3059\u3002<\/p>\n
\u30c7\u30d5\u30a9\u30eb\u30c8\u306e rombergtol<\/code> \u306e\u5024\u306f…<\/p>\n<\/div>\n<\/div>\n<\/div>\n
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rombergtol<\/span>;\r\n<\/pre>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
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\\[\\tag{${\\it \\%o}_{10}$}1.0 \\times 10^{-4}\\]<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
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rombergtol<\/code> \u306e\u5024\u3092\u5909\u3048\u3066\u8a08\u7b97\u3057\uff0c\u7cbe\u5ea6\u3092\u78ba\u8a8d\u3057\u307e\u3059\u3002\u4ee5\u4e0b\u306e\u7d50\u679c\u304b\u3089\uff0c<\/p>\n
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rombergtol<\/span>:<\/span> 1e<\/span>-<\/span>11$\r\n<\/pre>\n<\/div>\n
\u3067\u5c0f\u6570\u70b9\u4ee5\u4e0b15\u6841\u7a0b\u5ea6\u306e\u7cbe\u5ea6\u304c\u3042\u308b\u3053\u3068\u304c\u308f\u304b\u308a\u307e\u3059\u3002<\/p>\n<\/div>\n<\/div>\n<\/div>\n
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In\u00a0[6]:<\/div>\n
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for<\/span> h<\/span> in<\/span> [<\/span>1e-8<\/span>, 1e-9<\/span>, 1e-10<\/span>, 1e-11<\/span>, 1e-12<\/span>, 1e<\/span>-<\/span>13]<\/span> do<\/span>(<\/span>\r\n  rombergtol<\/span>:<\/span> h<\/span>,\r\n  printf<\/span>(<\/span>true<\/span>, \"~17,15f~%\"<\/span>, romberg<\/span>(<\/span>sin<\/span>(<\/span>sin<\/span>(<\/span>x<\/span>))<\/span>, x<\/span>, 0<\/span>, %pi<\/span>))<\/span>\r\n)<\/span>$\r\n<\/pre>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
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1.786487481932776\r\n1.786487481950060\r\n1.786487481950060\r\n1.786487481950052\r\n1.786487481950052\r\n1.786487481950052\r\n<\/pre>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
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quad_qags()<\/code><\/h4>\n
Maxima \u306b\u306f\u6570\u5024\u7a4d\u5206\u3092\u884c\u3046\u95a2\u6570\u3068\u3057\u3066 quad_qags()<\/code> \u3082\u3042\u308a\u307e\u3059\u3002romberg()<\/code> \u306b\u5bfe\u3059\u308b quad_qags()<\/code> \u306e\u5229\u70b9\u306b\u3064\u3044\u3066\u306f\uff0c\u4ee5\u4e0b\u306e\u30ea\u30f3\u30af\u306b\u7c21\u5358\u306b\u307e\u3068\u3081\u3066\u3044\u307e\u3059\u3002<\/p>\n