{"id":4484,"date":"2024-12-25T14:30:06","date_gmt":"2024-12-25T05:30:06","guid":{"rendered":"https:\/\/home.hirosaki-u.ac.jp\/relativity\/?page_id=4484"},"modified":"2024-12-25T14:32:02","modified_gmt":"2024-12-25T05:32:02","slug":"scipy-%e3%81%a7%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\/python-%e3%81%a7%e6%95%b0%e5%80%a4%e8%a7%a3%e6%9e%90\/scipy-%e3%81%a7%e6%95%b0%e5%80%a4%e8%a7%a3%e6%9e%90\/","title":{"rendered":"SciPy \u3067\u6570\u5024\u89e3\u6790"},"content":{"rendered":"
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Python \u3067\u6570\u5024\u89e3\u6790\u3059\u308b\u305f\u3081\u306e\u30e9\u30a4\u30d6\u30e9\u30ea\u3067\u3042\u308b SciPy<\/code> \u306e\u4f7f\u7528\u4f8b\u3092\u793a\u3057\u307e\u3059\u3002\u30b0\u30e9\u30d5\u306f Matplotlib<\/code> \u306e pyplot.plot()<\/code> \u3092\u4f7f\u3063\u3066\u63cf\u304d\u307e\u3059\u3002\u4e09\u89d2\u95a2\u6570\u306a\u3069\u306e\u6570\u5b66\u95a2\u6570\u306b\u3064\u3044\u3066\u306f\uff0cSymPy \u3092\u4f7f\u308f\u305a\uff0c\u5168\u3066 NumPy \u306e\u95a2\u6570\u3092\u4f7f\u3044\u307e\u3059\u3002<\/p>\n<\/div>\n<\/div>\n<\/div>\n

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\u30e9\u30a4\u30d6\u30e9\u30ea\u306e import \u3068\u8a2d\u5b9a<\/h3>\n<\/div>\n<\/div>\n<\/div>\n
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# NumPy \u3092\u4f7f\u3044\u307e\u3059<\/span>\r\nimport<\/span> numpy<\/span> as<\/span> np<\/span>\r\n# Matplotlib \u3067\u30b0\u30e9\u30d5\u3092\u63cf\u304d\u307e\u3059<\/span>\r\nimport<\/span> matplotlib.pyplot<\/span> as<\/span> plt<\/span>\r\n\r\n# \u30b0\u30e9\u30d5\u306e\u76ee\u76db\u8a2d\u5b9a\u7528<\/span>\r\nfrom<\/span> matplotlib.ticker<\/span> import<\/span> MultipleLocator<\/span>\r\n\r\n# \u4ee5\u4e0b\u306f\u30b0\u30e9\u30d5\u3092 SVG \u3067 Notebook \u306b\u30a4\u30f3\u30e9\u30a4\u30f3\u8868\u793a\u3055\u305b\u308b\u8a2d\u5b9a<\/span>\r\n%<\/span>config<\/span> InlineBackend.figure_formats = ['svg']\r\n\r\n# mathtext font \u306e\u8a2d\u5b9a<\/span>\r\nplt<\/span>.<\/span>rcParams<\/span>[<\/span>'mathtext.fontset'<\/span>]<\/span> =<\/span> 'cm'<\/span>\r\n<\/pre>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
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SciPy \u306b\u3088\u308b\u6570\u5024\u5fae\u5206<\/h3>\n
SciPy \u306b\u306f\u6570\u5024\u5fae\u5206\u3092\u3057\u3066\u304f\u308c\u308b\u95a2\u6570 scipy.misc.derivative<\/code> \u304c\u3042\u308a\u307e\u3057\u305f\u304c\uff0c\u975e\u63a8\u5968\u3068\u306a\u308a\uff0cSciPy 1.12.0 \u304b\u3089\u306f\u4f7f\u3048\u306a\u304f\u306a\u3063\u305f\u3088\u3046\u3067\u3059\u3002<\/p>\n
\u89e3\u6790\u7684\u306a\u5fae\u5206\u306e\u5b9a\u7fa9\u306f\uff08\u524d\u65b9\u5dee\u5206\u306e\u6975\u9650\u3068\u3057\u3066\uff09<\/p>\n
$$\\frac{df}{dx} \\equiv \\lim_{h \\rightarrow 0} \\frac{f(x+h) -f(x)}{h}$$<\/p>\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\u3067\u306f $ h\\rightarrow 0$ \u306e\u6975\u9650\u306f\u3068\u308c\u307e\u305b\u3093\u304b\u3089\u5341\u5206\u5c0f\u3055\u3044\u5024\u3068\u3057\u3066 h \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
\u95a2\u6570 $f(x)$ \u304c\u521d\u7b49\u95a2\u6570\uff08\u304a\u3088\u3073\u521d\u7b49\u95a2\u6570\u306e\u5408\u6210\u95a2\u6570\uff09\u3067\u3042\u308b\u5834\u5408\u306f\u89e3\u6790\u7684\u306b\u5fae\u5206\u3067\u304d\u307e\u3059\u304c\uff0c\u305d\u3046\u3067\u306a\u3044\u5834\u5408\uff0c\u4f8b\u3048\u3070 $f(x)$ \u304c\u6570\u5024\u7684\u306b\u3057\u304b\u4e0e\u3048\u3089\u308c\u3066\u3044\u306a\u3044\u3068\u304b\uff0c\u4f55\u304b\u89e3\u6790\u7684\u306b\u5fae\u5206\u3067\u304d\u306a\u3044\u5834\u5408\u306b\u306f\uff0c\u6570\u5024\u5fae\u5206\u306b\u3088\u3063\u3066\u8fd1\u4f3c\u7684\u306a\u5024\u3092\u6c42\u3081\u308b\u5fc5\u8981\u304c\u3042\u308a\u307e\u3059\u3002<\/p>\n
\u307e\u305f\uff0c\u89e3\u6790\u7684\u306b\u5fae\u5206\u3067\u304d\u305d\u3046\u3060\u304c\uff0c\u4eba\u529b\u3067\u306f\u3051\u3063\u3053\u3046\u5927\u5909\u305d\u3046\u306a\u5834\u5408\u306b\u3082\uff0c\u6570\u5024\u5fae\u5206\u306b\u3088\u3063\u3066\u8fd1\u4f3c\u7684\u306a\u5024\u3092\u4f7f\u3046\u307b\u3046\u304c\u4fbf\u5229\u306a\u5834\u5408\u304c\u3042\u308b\u304b\u3082\u3057\u308c\u307e\u305b\u3093\u3002<\/p>\n
\u4ee5\u4e0b\u306e\u3088\u3046\u306b\uff0c\u95a2\u6570 f(x)<\/code> \u3092\u6570\u5024\u5fae\u5206\u3057\u3066\u304f\u308c\u308b\u95a2\u6570 ndiff()<\/code> \u3092\u5b9a\u7fa9\u3057\u307e\u3059\u3002<\/p>\n<\/div>\n<\/div>\n<\/div>\n
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def<\/span> ndiff<\/span>(<\/span>f<\/span>,<\/span> x<\/span>,<\/span> h<\/span>=<\/span>1e-5<\/span>):<\/span>\r\n    # h \u3092\u7701\u7565\u3059\u308b\u3068 h=1e-5 \u3068\u3057\u3066\u8a08\u7b97\u3059\u308b<\/span>\r\n    return<\/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>*<\/span>h<\/span>)<\/span>\r\n<\/pre>\n<\/div>\n<\/div>\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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# h \u3092\u7701\u7565\u3057\u305f\u5834\u5408<\/span>\r\nprint<\/span>(<\/span>'       ndiff(f, 3) = <\/span>%.15f<\/span>'<\/span> %<\/span> ndiff<\/span>(<\/span>np<\/span>.<\/span>sin<\/span>,<\/span> 3<\/span>))<\/span>\r\n\r\nfor<\/span> h<\/span> in<\/span> [<\/span>1e-4<\/span>,<\/span> 1e-5<\/span>,<\/span> 1e-6<\/span>,<\/span> 1e-7<\/span>,<\/span> 1e-8<\/span>]:<\/span>\r\n    print<\/span>(<\/span>'h = <\/span>%3.1e<\/span>, ndiff = <\/span>%.15f<\/span>'<\/span> %<\/span> (<\/span>h<\/span>,<\/span> ndiff<\/span>(<\/span>np<\/span>.<\/span>sin<\/span>,<\/span> 3<\/span>,<\/span> h<\/span>)))<\/span>\r\n\r\nprint<\/span>(<\/span>' '<\/span>*<\/span>10<\/span> +<\/span> 'cos(<\/span>%3.1f<\/span>) = <\/span>%.15f<\/span>'<\/span> %<\/span> (<\/span>3<\/span>,<\/span> np<\/span>.<\/span>cos<\/span>(<\/span>3<\/span>)))<\/span>\r\n<\/pre>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
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       ndiff(f, 3) = -0.989992496590319\r\nh = 1.0e-04, ndiff = -0.989992494952602\r\nh = 1.0e-05, ndiff = -0.989992496590319\r\nh = 1.0e-06, ndiff = -0.989992496730485\r\nh = 1.0e-07, ndiff = -0.989992494926373\r\nh = 1.0e-08, ndiff = -0.989992490763036\r\n          cos(3.0) = -0.989992496600445\r\n<\/pre>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
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\u4ee5\u4e0a\u306e\u7d50\u679c\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\u306f\u306a\u3044\u3067\u3059\u3002<\/p>\n<\/div>\n<\/div>\n<\/div>\n
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\u25cb\u7df4\u7fd2\uff1a\u95a2\u6570\u306e\u6570\u5024\u5fae\u5206<\/h4>\n
$\\displaystyle f(x) = \\frac{x^3}{e^x -1}$ \u306e\u6975\u5927\u5024\u3092\u6c42\u3081\u308b\u305f\u3081\u306e\u6e96\u5099\u3068\u3057\u3066\uff0c\u307e\u305a\uff0c$f(x)$ \u3068\u305d\u306e\u6570\u5024\u5fae\u5206 ndiff(f, x)<\/code> \u3092 $0 \\leq x \\leq 5$ \u306e\u7bc4\u56f2\u3067\u30b0\u30e9\u30d5\u306b\u3059\u308b\u3002<\/p>\n<\/div>\n<\/div>\n<\/div>\n
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def<\/span> f<\/span>(<\/span>x<\/span>):<\/span>\r\n    return<\/span> x<\/span>**<\/span>3<\/span>\/<\/span>(<\/span>np<\/span>.<\/span>exp<\/span>(<\/span>x<\/span>)<\/span> -<\/span> 1<\/span>)<\/span>\r\n<\/pre>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
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# ax \u3092\u4f7f\u3046\u969b\u306e\u6700\u521d\u306e\u304a\u307e\u3058\u306a\u3044<\/span>\r\nfig<\/span>,<\/span> ax<\/span> =<\/span> plt<\/span>.<\/span>subplots<\/span>()<\/span>\r\n\r\n# x \u306e\u7bc4\u56f2<\/span>\r\nxrange<\/span> =<\/span> np<\/span>.<\/span>linspace<\/span>(<\/span>0.001<\/span>,<\/span> 5<\/span>,<\/span> 200<\/span>)<\/span>\r\n\r\n# y = f(x) \u306e\u30b0\u30e9\u30d5<\/span>\r\nax<\/span>.<\/span>plot<\/span>(<\/span>xrange<\/span>,<\/span> f<\/span>(<\/span>xrange<\/span>),<\/span> label<\/span> =<\/span> '$f(x)$'<\/span>)<\/span>\r\n# y = f'(x) \u306e\u30b0\u30e9\u30d5<\/span>\r\nax<\/span>.<\/span>plot<\/span>(<\/span>xrange<\/span>,<\/span> ndiff<\/span>(<\/span>f<\/span>,<\/span> xrange<\/span>),<\/span> label<\/span> =<\/span> '$f^{\\prime}(x)$'<\/span>)<\/span>\r\n\r\n# x \u8ef8\u306e\u30e9\u30d9\u30eb<\/span>\r\nax<\/span>.<\/span>set_xlabel<\/span>(<\/span>'$x$'<\/span>)<\/span>\r\n# \u30b0\u30ea\u30c3\u30c9<\/span>\r\nax<\/span>.<\/span>grid<\/span>()<\/span>\r\n# \u51e1\u4f8b\u306e\u8868\u793a<\/span>\r\nax<\/span>.<\/span>legend<\/span>();<\/span>\r\n<\/pre>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
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\u4e0a\u306e\u30b0\u30e9\u30d5\u304b\u3089\uff0c\u6570\u5024\u5fae\u5206\u3057\u305f\u5fae\u5206\u4fc2\u6570 $f^{\\prime}(x)$ \u306e\u5024\u304c\u30bc\u30ed\u306b\u306a\u308b $x$ \u306f\uff0c$2 < x < 3$ \u306e\u3042\u305f\u308a\u3067\u3042\u308b\u3053\u3068\uff0c\u305d\u306e\u3042\u305f\u308a\u3067\u78ba\u304b\u306b $f(x)$ \u304c\u6975\u5927\uff08\u6700\u5927\uff09\u3068\u306a\u3063\u3066\u3044\u308b\u3053\u3068\u304c\u898b\u3066\u3068\u308c\u307e\u3059\u3002<\/p>\n<\/div>\n<\/div>\n<\/div>\n
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SciPy \u306b\u3088\u308b\u6570\u5024\u7a4d\u5206<\/h3>\n
scipy.integrate.quad()<\/code><\/h4>\n
SciPy \u306b\u306f\u6570\u5024\u7a4d\u5206\u3057\u3066\u304f\u308c\u308b\u95a2\u6570 scipy.integrate.quad()<\/code> \u304c\u7528\u610f\u3055\u308c\u3066\u3044\u307e\u3059\u3002\u305d\u308c\u3092\u4f7f\u3063\u3066\u307f\u307e\u3057\u3087\u3046\u3002\u7d50\u679c\u306b\u306f\u7a4d\u5206\u5024\u3068\u63a8\u5b9a\u8aa4\u5dee\u304c\u51fa\u529b\u3055\u308c\u307e\u3059\u3002<\/p>\n<\/div>\n<\/div>\n<\/div>\n
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$\\displaystyle \\int_0^{\\pi} \\sin(\\sin x)\\, dx$ \u306e\u8a08\u7b97\u3092\u3057\u307e\u3059\u3002<\/p>\n
\u6570\u5024\u8a08\u7b97\u306e\u969b\u306f\uff0c\u4e09\u89d2\u95a2\u6570\u7b49\u306f NumPy \u306e\u95a2\u6570\u3092\u4f7f\u3044\uff0c\u5186\u5468\u7387 $\\pi$ \u3082 NumPy \u306e np.pi<\/code> \u3092\u4f7f\u3044\u307e\u3059\u3002<\/p>\n<\/div>\n<\/div>\n<\/div>\n
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from<\/span> scipy.integrate<\/span> import<\/span> quad<\/span>\r\n\r\n# quad \u3092\u4f7f\u3046\u3068\u304d\u306f\u5909\u6570\u540d x \u6c7a\u3081\u6253\u3061<\/span>\r\ndef<\/span> sinsin<\/span>(<\/span>x<\/span>):<\/span>\r\n    return<\/span> np<\/span>.<\/span>sin<\/span>(<\/span>np<\/span>.<\/span>sin<\/span>(<\/span>x<\/span>))<\/span>\r\n\r\n# quad \u3067\u6570\u5024\u7a4d\u5206<\/span>\r\n# quad(f, a, b): \u88ab\u7a4d\u5206\u95a2\u6570 f(x) \u3092 x = a \u304b\u3089 b \u307e\u3067\u6570\u5024\u7a4d\u5206<\/span>\r\nquad<\/span>(<\/span>sinsin<\/span>,<\/span> 0<\/span>,<\/span> np<\/span>.<\/span>pi<\/span>)<\/span>\r\n<\/pre>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
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(1.7864874819500525, 2.4687274069628184e-11)<\/pre>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
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scipy.integrate.quad()<\/code> \u306f\u4e0a\u8a18\u306e\u3088\u3046\u306b\uff0c\u6570\u5024\u7a4d\u5206\u306e\u5024\u3068\u7d76\u5bfe\u8aa4\u5dee\u306e\u63a8\u5b9a\u5024\u3092\u30bf\u30d7\u30eb (tuple) \u3067\u8fd4\u3057\u307e\u3059\u3002<\/p>\n
\u6570\u5024\u7a4d\u5206\u5024\uff08\u3068\u8aa4\u5dee\uff09\u3092\u500b\u5225\u306b\u53d6\u308a\u51fa\u3059\u306b\u306f\uff0c\u4f8b\u3048\u3070\u4ee5\u4e0b\u306e\u3088\u3046\u306b\u3002<\/p>\n<\/div>\n<\/div>\n<\/div>\n
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y<\/span>,<\/span> abserr<\/span> =<\/span> quad<\/span>(<\/span>sinsin<\/span>,<\/span> 0<\/span>,<\/span> np<\/span>.<\/span>pi<\/span>)<\/span>\r\n\r\nprint<\/span>(<\/span>\"\u7a4d\u5206\u5024\u306f\"<\/span>,<\/span> y<\/span>)<\/span>\r\nprint<\/span>(<\/span>\"\u8aa4\u5dee\u306f\"<\/span>,<\/span> abserr<\/span>)<\/span>\r\n<\/pre>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
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\u7a4d\u5206\u5024\u306f 1.7864874819500525\r\n\u8aa4\u5dee\u306f 2.4687274069628184e-11\r\n<\/pre>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
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\u53c2\u8003\uff1a\u30b7\u30f3\u30d7\u30bd\u30f3\u6cd5\u306b\u3088\u308b\u6570\u5024\u7a4d\u5206<\/h4>\n
\u30b7\u30f3\u30d7\u30bd\u30f3\u6cd5\u3067 $\\displaystyle \\int_a^b f(x)\\, dx$ \u3092\u6c42\u3081\u308b\u3002<\/p>\n
\u7a4d\u5206\u533a\u9593 $[a, b]$ \u3092 $N (=2n)$ \u7b49\u5206\uff08\u5076\u6570\u7b49\u5206\uff09\u3057\uff0c<\/p>\n
$$h = \\frac{b-a}{N}, \\quad f_i = f(a + i\\,h)$$<\/p>\n
\u3068\u3059\u308b\u3002\u30b7\u30f3\u30d7\u30bd\u30f3\u6cd5\u306e\u516c\u5f0f\u306f<\/p>\n
\\begin{eqnarray}
\n\\int_a^b f(x)\\, dx &\\simeq&
\n\\frac{h}{3}( f_0 + 4 f_1 + f_2) + \\frac{h}{3}( f_2 + 4 f_3 + f_4) + \\\\
\n&& \\frac{h}{3}( f_4 + 4 f_5 + f_6) + \\frac{h}{3}( f_6 + 4 f_7 + f_8)+ \\cdots\\\\
\n&&+ \\frac{h}{3}( f_{N-2} + 4 f_{N-1} + f_N)\\\\
\n&=&\\frac{h}{3}
\n\\left(f_0 + 2 \\sum_{i=1}^{n-1} f_{2i} + 4 \\sum_{i=1}^{n} f_{2i-1}+ f_{2n}\\right)
\n\\end{eqnarray}<\/p>\n
SciPy \u3067\u306f scipy.integrate.quad()<\/code> \u3092\u4f7f\u3063\u3066\u6570\u5024\u7a4d\u5206\u3059\u308b\u3053\u3068\u304c\u3067\u304d\u308b\u306e\u3067\u308f\u3056\u308f\u3056\u81ea\u4f5c\u306e\u95a2\u6570\u3092\u3064\u304f\u308b\u5fc5\u8981\u306f\u306a\u3044\u306e\u3067\u3059\u304c\uff0c\u6559\u80b2\u7684\u898b\u5730\u304b\u3089\u3042\u3048\u3066\uff0c\u30b7\u30f3\u30d7\u30bd\u30f3\u6cd5\u306e\u516c\u5f0f\u3092\u4f7f\u3063\u3066\u6570\u5024\u7a4d\u5206\u306e\u5024\u3092\u8fd4\u3059\u95a2\u6570 simpson()<\/code> \u3092\u4ee5\u4e0b\u306e\u3088\u3046\u306b\u5b9a\u7fa9\u3057\u3066\u307f\u307e\u3059\u3002<\/p>\n<\/div>\n<\/div>\n<\/div>\n
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In\u00a0[8]:<\/div>\n
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def<\/span> simpson<\/span>(<\/span>f<\/span>,<\/span> a<\/span>,<\/span> b<\/span>,<\/span> Ndiv<\/span>):<\/span>\r\n    h<\/span> =<\/span> (<\/span>b<\/span> -<\/span> a<\/span>)<\/span>\/<\/span>Ndiv<\/span>\r\n    simp<\/span> =<\/span> 0<\/span>\r\n    for<\/span> i<\/span> in<\/span> range<\/span>(<\/span>0<\/span>,<\/span> Ndiv<\/span>,<\/span> 2<\/span>):<\/span>\r\n        simp<\/span> +=<\/span> f<\/span>(<\/span>a<\/span> +<\/span> i<\/span>*<\/span>h<\/span>)<\/span> +<\/span> 4<\/span>*<\/span>f<\/span>(<\/span>a<\/span> +<\/span> (<\/span>i<\/span>+<\/span>1<\/span>)<\/span>*<\/span>h<\/span>)<\/span> +<\/span> f<\/span>(<\/span>a<\/span> +<\/span> (<\/span>i<\/span>+<\/span>2<\/span>)<\/span>*<\/span>h<\/span>)<\/span>\r\n    return<\/span> simp<\/span>*<\/span>h<\/span>\/<\/span>3<\/span>\r\n<\/pre>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
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N<\/code> \u306e\u5024\u3092\u5909\u3048\u3066 $\\displaystyle \\int_0^{\\pi} \\sin(\\sin x)\\, dx$ \u3092 simpson()<\/code> \u3067\u6570\u5024\u7a4d\u5206\u3057\u3066\uff0c\u7cbe\u5ea6\u3092\u78ba\u8a8d\u3057\u3066\u307f\u307e\u3059\u3002<\/p>\n<\/div>\n<\/div>\n<\/div>\n
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In\u00a0[9]:<\/div>\n
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def<\/span> f<\/span>(<\/span>x<\/span>):<\/span>\r\n    return<\/span> np<\/span>.<\/span>sin<\/span>(<\/span>np<\/span>.<\/span>sin<\/span>(<\/span>x<\/span>))<\/span>\r\n\r\nfor<\/span> N<\/span> in<\/span> [<\/span>10<\/span>*<\/span>2<\/span>**<\/span>i<\/span> for<\/span> i<\/span> in<\/span> range<\/span>(<\/span>9<\/span>)]:<\/span>\r\n    print<\/span>(<\/span>'N = <\/span>%4d<\/span>, <\/span>%.15f<\/span>'<\/span> %<\/span>(<\/span>N<\/span>,<\/span> simpson<\/span>(<\/span>f<\/span>,<\/span> 0<\/span>,<\/span> np<\/span>.<\/span>pi<\/span>,<\/span> N<\/span>)))<\/span>\r\n<\/pre>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
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N =   10, 1.786721213744079\r\nN =   20, 1.786501256275661\r\nN =   40, 1.786488331266680\r\nN =   80, 1.786487534856180\r\nN =  160, 1.786487485253951\r\nN =  320, 1.786487482156504\r\nN =  640, 1.786487481962955\r\nN = 1280, 1.786487481950862\r\nN = 2560, 1.786487481950104\r\n<\/pre>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
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\u4ee5\u4e0a\u306e\u7d50\u679c\u304b\u3089\uff0c\u3069\u3093\u306a\u3053\u3068\u304c\u308f\u304b\u308b\u304b\u306a\uff1f<\/p>\n<\/div>\n<\/div>\n<\/div>\n
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SciPy \u306b\u3088\u308b\u65b9\u7a0b\u5f0f\u306e\u6570\u5024\u89e3<\/h3>\n<\/div>\n<\/div>\n<\/div>\n
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\u4f8b\u3068\u3057\u3066\u30d9\u30c3\u30bb\u30eb\u95a2\u6570 $J_0(x)$ \u306e\u30bc\u30ed\u70b9\u3092\u6c42\u3081\u3066\u307f\u307e\u3059\u3002SciPy<\/code> \u306b\u304a\u3051\u308b\u30d9\u30c3\u30bb\u30eb\u95a2\u6570 $J_0(x)$ \u306f scipy.special.jv(0,x)<\/code> \u3067\u3059\u3002<\/p>\n
\u307e\u305a\u306f $J_0(x) = 0$ \u3068\u306a\u308b $x$ \u306e\u3042\u305f\u308a\u3092\u3064\u3051\u308b\u305f\u3081\u306b\u30b0\u30e9\u30d5\u3092\u63cf\u3044\u3066\u307f\u307e\u3059\u3002<\/p>\n<\/div>\n<\/div>\n<\/div>\n
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In\u00a0[10]:<\/div>\n
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from<\/span> scipy.special<\/span> import<\/span> jv<\/span>\r\n\r\ndef<\/span> J0<\/span>(<\/span>x<\/span>):<\/span>\r\n    return<\/span> jv<\/span>(<\/span>0<\/span>,<\/span>x<\/span>)<\/span>\r\n\r\n# ax \u3092\u4f7f\u3046\u969b\u306e\u6700\u521d\u306e\u304a\u307e\u3058\u306a\u3044<\/span>\r\nfig<\/span>,<\/span> ax<\/span> =<\/span> plt<\/span>.<\/span>subplots<\/span>()<\/span>\r\n\r\nxrange<\/span> =<\/span> np<\/span>.<\/span>linspace<\/span>(<\/span>0<\/span>,<\/span> 10<\/span>,<\/span> 200<\/span>)<\/span>\r\nax<\/span>.<\/span>plot<\/span>(<\/span>xrange<\/span>,<\/span> J0<\/span>(<\/span>xrange<\/span>))<\/span>\r\n\r\n# x \u8ef8<\/span>\r\nax<\/span>.<\/span>plot<\/span>(<\/span>xrange<\/span>,<\/span> 0<\/span>*<\/span>xrange<\/span>)<\/span>\r\n\r\nax<\/span>.<\/span>set_xlabel<\/span>(<\/span>'x'<\/span>)<\/span>\r\nax<\/span>.<\/span>set_ylabel<\/span>(<\/span>'J0(x)'<\/span>)<\/span>\r\nax<\/span>.<\/span>grid<\/span>()<\/span>\r\nax<\/span>.<\/span>set_xlim<\/span>(<\/span>0<\/span>,<\/span> 10<\/span>)<\/span>\r\nax<\/span>.<\/span>set_title<\/span>(<\/span>'\u30d9\u30c3\u30bb\u30eb\u95a2\u6570 J0(x)'<\/span>);<\/span>\r\n<\/pre>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
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\"\"<\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
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scipy.optimize.root_scalar<\/code><\/h4>\n
\u65b9\u7a0b\u5f0f\u306e\u6570\u5024\u89e3\u3092\u6c42\u3081\u3066\u304f\u308c\u308b\u95a2\u6570\u306f scipy.optimize.root_scalar<\/code> \u3067\u3059\u3002$2 < x < 3$ \u306e\u3042\u305f\u308a\u3067 $x$ \u8ef8\u3068\u4ea4\u308f\u3063\u3066\u3044\u307e\u3059\u304b\u3089\uff0c\u3053\u306e\u3042\u305f\u308a\u306e\u89e3\u3092\u63a2\u3057\u307e\u3059\u3002<\/p>\n<\/div>\n<\/div>\n<\/div>\n
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In\u00a0[11]:<\/div>\n
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from<\/span> scipy.optimize<\/span> import<\/span> root_scalar<\/span>\r\n\r\n# x \u304c 2 ~ 3 \u306e\u9593\u3067 j0(x) = 0 \u306e\u6839\u3092\u63a2\u3059<\/span>\r\nroot_scalar<\/span>(<\/span>J0<\/span>,<\/span> bracket<\/span> =<\/span> [<\/span>2<\/span>,<\/span> 3<\/span>])<\/span>\r\n<\/pre>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
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Out[11]:<\/div>\n
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      converged: True\r\n           flag: 'converged'\r\n function_calls: 8\r\n     iterations: 7\r\n           root: 2.404825557695807<\/pre>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
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scipy.optimize.root_scalar()<\/code> \u306f\u7d50\u679c\u3092\u4e0a\u8a18\u306e\u3088\u3046\u306b\u8868\u793a\u3057\u307e\u3059\u3002\u89e3 root<\/code> \u306e\u307f\u3092\u8868\u793a\u3057\u305f\u3044\u5834\u5408\u306f\u4ee5\u4e0b\u306e\u3088\u3046\u306b\u3057\u307e\u3059\u3002<\/p>\n<\/div>\n<\/div>\n<\/div>\n
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In\u00a0[12]:<\/div>\n
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# \u6839\u306e\u307f\u3092\u51fa\u529b\u3059\u308b\u306b\u306f...<\/span>\r\nans<\/span> =<\/span> root_scalar<\/span>(<\/span>J0<\/span>,<\/span> bracket<\/span> =<\/span> [<\/span>2<\/span>,<\/span> 3<\/span>])<\/span>\r\nprint<\/span>(<\/span>ans<\/span>.<\/span>root<\/span>)<\/span>\r\n<\/pre>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
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2.404825557695807\r\n<\/pre>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
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In\u00a0[13]:<\/div>\n
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# \u307b\u3093\u3068\u306b\u6839\u306b\u306a\u3063\u3066\u308b\u304b\u78ba\u8a8d\u3002\u30bc\u30ed\u306b\u8fd1\u3051\u308c\u3070\u3088\u3044\u3002<\/span>\r\nJ0<\/span>(<\/span>ans<\/span>.<\/span>root<\/span>)<\/span>\r\n<\/pre>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
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Out[13]:<\/div>\n
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-1.7918655681235457e-14<\/pre>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
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In\u00a0[14]:<\/div>\n
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# \u3069\u3046\u3057\u3066\u3082\u3082\u3063\u3068\u7cbe\u5ea6\u304c\u6b32\u3057\u3044\u3068\u3044\u3046\u306e\u3067\u3042\u308c\u3070...<\/span>\r\n# xtol \u306b\uff0c\u5c0f\u3055\u3044\u5024\u3092\u3044\u308c\u3066\u307f\u308b<\/span>\r\nans<\/span> =<\/span> root_scalar<\/span>(<\/span>J0<\/span>,<\/span> bracket<\/span> =<\/span> [<\/span>2<\/span>,<\/span> 3<\/span>],<\/span> xtol<\/span>=<\/span>1e-15<\/span>)<\/span>\r\nans<\/span>.<\/span>root<\/span>\r\n<\/pre>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
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Out[14]:<\/div>\n
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2.4048255576957724<\/pre>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
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In\u00a0[15]:<\/div>\n
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# \u307b\u3093\u3068\u306b\u6839\u306b\u306a\u3063\u3066\u308b\u304b\u78ba\u8a8d\u3002\u30bc\u30ed\u306b\u8fd1\u3051\u308c\u3070\u3088\u3044\u3002<\/span>\r\nJ0<\/span>(<\/span>_<\/span>)<\/span>\r\n<\/pre>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
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Out[15]:<\/div>\n
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1.535884772676773e-16<\/pre>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
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\u25cb\u7df4\u7fd2\uff1a\u95a2\u6570\u306e\u6975\u5927\u5024<\/h4>\n
    \n
  1. $\\displaystyle f(x) = \\frac{x^3}{e^x -1}$ \u304c\u6975\u5927\u5024\u3092\u3068\u308b\u3068\u304d\u306e $x$ \u306e\u5024\uff0c\u304a\u3088\u3073\u305d\u306e\u3068\u304d\u306e\u6975\u5927\u5024\u3092\u6c42\u3081\u308b\u3002<\/li>\n
  2. $\\displaystyle g(x) = \\frac{x^5}{e^x -1}$ \u304c\u6975\u5927\u5024\u3092\u3068\u308b\u3068\u304d\u306e $x$ \u306e\u5024\uff0c\u304a\u3088\u3073\u305d\u306e\u3068\u304d\u306e\u6975\u5927\u5024\u3082\u540c\u69d8\u306b\u3057\u3066\u6c42\u3081\u3066\u307f\u3088\u3002<\/li>\n<\/ol>\n
    \u307e\u305a $f(x)$ \u306e\u6570\u5024\u5fae\u5206 ndiff(f, x)<\/code> \u3092\u30b0\u30e9\u30d5\u306b\u3057\u3066\u307f\u308b\u3002<\/p>\n<\/div>\n<\/div>\n<\/div>\n
    \n
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    In\u00a0[16]:<\/div>\n
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    def<\/span> f<\/span>(<\/span>x<\/span>):<\/span>\r\n    return<\/span> x<\/span>**<\/span>3<\/span>\/<\/span>(<\/span>np<\/span>.<\/span>exp<\/span>(<\/span>x<\/span>)<\/span> -<\/span> 1<\/span>)<\/span>\r\n\r\n# ax \u3092\u4f7f\u3046\u969b\u306e\u6700\u521d\u306e\u304a\u307e\u3058\u306a\u3044<\/span>\r\nfig<\/span>,<\/span> ax<\/span> =<\/span> plt<\/span>.<\/span>subplots<\/span>()<\/span>\r\n\r\nxrange<\/span> =<\/span> np<\/span>.<\/span>linspace<\/span>(<\/span>0<\/span>,<\/span> 5<\/span>,<\/span> 200<\/span>)<\/span>\r\nax<\/span>.<\/span>plot<\/span>(<\/span>xrange<\/span>,<\/span> ndiff<\/span>(<\/span>f<\/span>,<\/span> xrange<\/span>),<\/span> label<\/span> =<\/span> '$f^{\\prime}(x)$'<\/span>)<\/span>\r\n\r\n# x \u8ef8<\/span>\r\nax<\/span>.<\/span>plot<\/span>(<\/span>xrange<\/span>,<\/span> 0<\/span>*<\/span>xrange<\/span>)<\/span>\r\n\r\nax<\/span>.<\/span>set_xlabel<\/span>(<\/span>'$x$'<\/span>)<\/span>\r\nax<\/span>.<\/span>set_ylabel<\/span>(<\/span>' '<\/span>)<\/span>\r\nax<\/span>.<\/span>grid<\/span>()<\/span>\r\nax<\/span>.<\/span>legend<\/span>();<\/span>\r\n<\/pre>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
    \n
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    \"\"<\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
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    <\/div>\n
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    ndiff(f, x)<\/code> \u306f $2 < x <3$ \u306e\u3042\u305f\u308a\u3067\u30bc\u30ed\u306b\u306a\u3063\u3066\u3044\u308b\u306e\u3067\uff0c\u305d\u306e\u3042\u305f\u308a\u3092 root_scalar()<\/code> \u3067\u63a2\u3057\u3066\u307f\u308b\u3002<\/p>\n<\/div>\n<\/div>\n<\/div>\n
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    In\u00a0[17]:<\/div>\n
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    def<\/span> df<\/span>(<\/span>x<\/span>):<\/span>\r\n    return<\/span> ndiff<\/span>(<\/span>f<\/span>,<\/span> x<\/span>)<\/span>\r\n\r\nans<\/span> =<\/span> root_scalar<\/span>(<\/span>df<\/span>,<\/span> bracket<\/span> =<\/span> [<\/span>2<\/span>,<\/span> 3<\/span>])<\/span>\r\nans<\/span>.<\/span>root<\/span>\r\n<\/pre>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
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    Out[17]:<\/div>\n
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    2.8214393721317643<\/pre>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
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    In\u00a0[18]:<\/div>\n
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    # \u305d\u306e\u3068\u304d\u306e f(x) \u306e\u6975\u5927\u5024\u306f...<\/span>\r\nf<\/span>(<\/span>_<\/span>)<\/span>\r\n<\/pre>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
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    Out[18]:<\/div>\n
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    1.4214354727477367<\/pre>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
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    <\/div>\n
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    SciPy \u306b\u3088\u308b1\u968e\u5e38\u5fae\u5206\u65b9\u7a0b\u5f0f\u306e\u6570\u5024\u89e3\u6cd5<\/h3>\n
    scipy.integrate.solve_ivp()<\/code><\/h4>\n
    SciPy \u306b\u306f\u5e38\u5fae\u5206\u65b9\u7a0b\u5f0f\u3092\u6570\u5024\u7684\u306b\u89e3\u3044\u3066\u304f\u308c\u308b\u95a2\u6570 scipy.integrate.solve_ivp()<\/code> \u304c\u7528\u610f\u3055\u308c\u3066\u3044\u307e\u3059\u3002\u305d\u308c\u3092\u4f7f\u3063\u3066\u307f\u307e\u3057\u3087\u3046\u3002<\/p>\n<\/div>\n<\/div>\n<\/div>\n
    \n
    <\/div>\n
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    \u4f8b\u3068\u3057\u3066\uff0c\u7c21\u5358\u306a\u30ed\u30b8\u30b9\u30c6\u30a3\u30c3\u30af\u65b9\u7a0b\u5f0f<\/p>\n
    $$ \\frac{dx}{dt} = f(x) = (1-x)\\,x$$<\/p>\n
    \u3092\uff0c\u521d\u671f\u6761\u4ef6 $t_0 = 0$ \u3067 $x_0 = x(t_0) = 0.1$\uff0c$t = t_0 = 0$ \u304b\u3089 $t_1 = 10$ \u307e\u3067\u89e3\u304d\u307e\u3059\u3002<\/p>\n
    solve_ivp()<\/code> \u3092\u4f7f\u3046\u5834\u5408\u306f\u5909\u6570\u540d\u304c\u6c7a\u3081\u6253\u3061\u3067 t, y<\/code> \u3067\u3059\u3002\u9023\u7acb\u5e38\u5fae\u5206\u65b9\u7a0b\u5f0f\u306e\u5834\u5408\u306f y<\/code> \u304c\u30ea\u30b9\u30c8\u306b\u306a\u308a\u307e\u3059\u3002<\/p>\n<\/div>\n<\/div>\n<\/div>\n
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    In\u00a0[19]:<\/div>\n
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    from<\/span> scipy.integrate<\/span> import<\/span> solve_ivp<\/span>\r\n\r\n# scipy.integrate.solve_ivp() \u306f<\/span>\r\n# dy\/dt = f(t, y) \u306e\u5f62\u3092\u89e3\u304f<\/span>\r\n# solve_ivp \u3092\u4f7f\u3046\u969b\u306f\u5909\u6570\u540d t, y \u306f\u6c7a\u3081\u6253\u3061<\/span>\r\n# solve.ivp() \u306b\u6e21\u3059\u95a2\u6570\u306e\u5f15\u6570\u306e\u9806\u756a\u306b\u6ce8\u610f\u3002t \u304c\u5148<\/span>\r\n# \u9023\u7acb\u3067\u306a\u3044\u3068\u304d\u306f\uff0c1\u6b21\u5143\u30ea\u30b9\u30c8\u3068\u3057\u3066\u8fd4\u3059<\/span>\r\n\r\ndef<\/span> f<\/span>(<\/span>t<\/span>,<\/span> y<\/span>):<\/span>\r\n    x<\/span> =<\/span> y<\/span>[<\/span>0<\/span>]<\/span>\r\n    return<\/span> [(<\/span>1<\/span>-<\/span>x<\/span>)<\/span>*<\/span>x<\/span>]<\/span>\r\n\r\n# t \u306e\u7bc4\u56f2<\/span>\r\ntini<\/span> =<\/span> 0<\/span>\r\ntend<\/span> =<\/span> 10<\/span>\r\n# \u8a08\u7b97\u3059\u308b\u70b9\u306f t \u306e\u7bc4\u56f2\u3092 N \u7b49\u5206\u3057\u3066 N+1 \u70b9<\/span>\r\nN<\/span> =<\/span> 20<\/span>\r\nt_list<\/span> =<\/span> np<\/span>.<\/span>linspace<\/span>(<\/span>tini<\/span>,<\/span> tend<\/span>,<\/span> N<\/span>+<\/span>1<\/span>)<\/span>\r\n# y \u306e\u521d\u671f\u5024<\/span>\r\ny_ini<\/span> =<\/span> [<\/span>0.1<\/span>]<\/span>\r\n\r\nanswer<\/span> =<\/span> solve_ivp<\/span>(<\/span>f<\/span>,<\/span> [<\/span>tini<\/span>,<\/span> tend<\/span>],<\/span> y_ini<\/span>,<\/span> t_eval<\/span> =<\/span> t_list<\/span>,<\/span> \r\n                   # \u30c7\u30d5\u30a9\u30eb\u30c8\u306e\u8a31\u5bb9\u8aa4\u5dee\u306f\u5927\u304d\u3081\u306a\u306e\u3067<\/span>\r\n                   # \u5c0f\u3055\u3044\u5024\u306b\u8a2d\u5b9a<\/span>\r\n                   rtol<\/span> =<\/span> 1.e-12<\/span>,<\/span> \r\n                   atol<\/span> =<\/span> 1.e-15<\/span>)<\/span>\r\nprint<\/span>(<\/span>answer<\/span>)<\/span>\r\n<\/pre>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
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      message: The solver successfully reached the end of the integration interval.\r\n  success: True\r\n   status: 0\r\n        t: [ 0.000e+00  5.000e-01 ...  9.500e+00  1.000e+01]\r\n        y: [[ 1.000e-01  1.548e-01 ...  9.993e-01  9.996e-01]]\r\n      sol: None\r\n t_events: None\r\n y_events: None\r\n     nfev: 1790\r\n     njev: 0\r\n      nlu: 0\r\n<\/pre>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
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    In\u00a0[20]:<\/div>\n
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    # ax \u3092\u4f7f\u3046\u969b\u306e\u6700\u521d\u306e\u304a\u307e\u3058\u306a\u3044<\/span>\r\nfig<\/span>,<\/span> ax<\/span> =<\/span> plt<\/span>.<\/span>subplots<\/span>()<\/span>\r\n\r\nax<\/span>.<\/span>scatter<\/span>(<\/span>answer<\/span>.<\/span>t<\/span>,<\/span> answer<\/span>.<\/span>y<\/span>[<\/span>0<\/span>],<\/span> s<\/span> =<\/span> 10<\/span>)<\/span>\r\n\r\nax<\/span>.<\/span>grid<\/span>()<\/span>\r\nax<\/span>.<\/span>set_xlabel<\/span>(<\/span>\"$t$\"<\/span>)<\/span>\r\nax<\/span>.<\/span>set_ylabel<\/span>(<\/span>\"$x(t)$\"<\/span>)<\/span>\r\nax<\/span>.<\/span>set_title<\/span>(<\/span>\"\u30ed\u30b8\u30b9\u30c6\u30a3\u30c3\u30af\u65b9\u7a0b\u5f0f\u306e\u6570\u5024\u89e3\"<\/span>);<\/span>\r\n<\/pre>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
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    \"\"<\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
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    \u53c2\u8003\u307e\u3067\u306b\uff0c\u89e3\u6790\u89e3\u306f<\/p>\n
    $$x(t) = \\frac{\\exp(t)}{9 + \\exp(t)}$$<\/p>\n
    \u3067\u3059\u3002\uff08\u89e3\u6790\u89e3\u304c\u308f\u304b\u3063\u3066\u3044\u308b\u3068\u304d\u306b\u306f\uff0c\u308f\u3056\u308f\u3056\u6570\u5024\u8a08\u7b97\u3059\u308b\u5fc5\u8981\u306f\u306a\u3044\u306e\u3067\u3059\u304c\uff0c\u7df4\u7fd2\u3068\u3044\u3046\u3053\u3068\u3067\u3002\uff09<\/p>\n<\/div>\n<\/div>\n<\/div>\n
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    In\u00a0[21]:<\/div>\n
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    # \u89e3\u6790\u89e3<\/span>\r\ndef<\/span> xsol<\/span>(<\/span>t<\/span>):<\/span>\r\n    return<\/span> np<\/span>.<\/span>exp<\/span>(<\/span>t<\/span>)<\/span>\/<\/span>(<\/span>9<\/span>+<\/span>np<\/span>.<\/span>exp<\/span>(<\/span>t<\/span>))<\/span>\r\n\r\n# ax \u3092\u4f7f\u3046\u969b\u306e\u6700\u521d\u306e\u304a\u307e\u3058\u306a\u3044<\/span>\r\nfig<\/span>,<\/span> ax<\/span> =<\/span> plt<\/span>.<\/span>subplots<\/span>()<\/span>\r\n\r\n# \u89e3\u6790\u89e3<\/span>\r\nts<\/span> =<\/span> np<\/span>.<\/span>linspace<\/span>(<\/span>tini<\/span>,<\/span> tend<\/span>,<\/span> 200<\/span>)<\/span>\r\nax<\/span>.<\/span>plot<\/span>(<\/span>ts<\/span>,<\/span> xsol<\/span>(<\/span>ts<\/span>),<\/span> c<\/span> =<\/span> 'tab:blue'<\/span>,<\/span> label<\/span>=<\/span>'\u89e3\u6790\u89e3'<\/span>)<\/span>\r\n# \u6570\u5024\u89e3<\/span>\r\nax<\/span>.<\/span>scatter<\/span>(<\/span>answer<\/span>.<\/span>t<\/span>,<\/span> answer<\/span>.<\/span>y<\/span>[<\/span>0<\/span>],<\/span> zorder<\/span> =<\/span> 2<\/span>,<\/span>\r\n           c<\/span> =<\/span> 'tab:orange'<\/span>,<\/span> s<\/span> =<\/span> 10<\/span>,<\/span> label<\/span> =<\/span> '\u6570\u5024\u89e3'<\/span>)<\/span>\r\n\r\nax<\/span>.<\/span>grid<\/span>()<\/span>\r\nax<\/span>.<\/span>set_xlabel<\/span>(<\/span>\"$t$\"<\/span>)<\/span>\r\nax<\/span>.<\/span>set_ylabel<\/span>(<\/span>\"$x(t)$\"<\/span>)<\/span>\r\nax<\/span>.<\/span>set_title<\/span>(<\/span>\"\u30ed\u30b8\u30b9\u30c6\u30a3\u30c3\u30af\u65b9\u7a0b\u5f0f\u306e\u89e3\"<\/span>)<\/span>\r\nax<\/span>.<\/span>legend<\/span>();<\/span>\r\n<\/pre>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
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    SciPy \u306b\u3088\u308b2\u968e\u5e38\u5fae\u5206\u65b9\u7a0b\u5f0f\u306e\u6570\u5024\u89e3\u6cd5<\/h3>\n
    scipy.integrate.solve_ivp()<\/code><\/h4>\n
    SciPy \u306b\u306f\u5e38\u5fae\u5206\u65b9\u7a0b\u5f0f\u3092\u6570\u5024\u7684\u306b\u89e3\u3044\u3066\u304f\u308c\u308b\u95a2\u6570 scipy.integrate.solve_ivp()<\/code> \u304c\u7528\u610f\u3055\u308c\u3066\u3044\u307e\u3059\u3002\u3053\u306e\u51fd\u6570\u306f\uff0c1\u968e\u5e38\u5fae\u5206\u65b9\u7a0b\u5f0f\u3060\u3051\u3067\u306a\u304f\uff0c\u9023\u7acb1\u968e\u5e38\u5fae\u5206\u65b9\u7a0b\u5f0f\u3082\uff0c\u5f93\u3063\u3066\u9ad8\u968e\u306e\u5e38\u5fae\u5206\u65b9\u7a0b\u5f0f\u3082\u89e3\u3044\u3066\u304f\u308c\u308b\u306e\u3067\uff0c\u305d\u308c\u3092\u4f7f\u3063\u3066\u307f\u307e\u3057\u3087\u3046\u3002<\/p>\n<\/div>\n<\/div>\n<\/div>\n
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    \u4f8b\u3068\u3057\u3066\uff0c\u7c21\u5358\u306a\u6e1b\u8870\uff0b\u5f37\u5236\u632f\u52d5\u306e\u65b9\u7a0b\u5f0f<\/p>\n
    $$ \\frac{d^2 x}{dt^2} = -x – a \\frac{dx}{dt} + b \\cos(t) $$<\/p>\n
    \u3092\uff0c<\/p>\n