{"id":4093,"date":"2022-10-24T18:23:48","date_gmt":"2022-10-24T09:23:48","guid":{"rendered":"https:\/\/home.hirosaki-u.ac.jp\/relativity\/?p=4093"},"modified":"2023-03-14T16:58:07","modified_gmt":"2023-03-14T07:58:07","slug":"gnuplot-%e3%81%a7%e3%82%b1%e3%83%97%e3%83%a9%e3%83%bc%e6%96%b9%e7%a8%8b%e5%bc%8f%e3%81%ae%e8%bf%91%e4%bc%bc%e8%a7%a3%e3%81%a8%e6%95%b0%e5%80%a4%e8%a7%a3","status":"publish","type":"post","link":"https:\/\/home.hirosaki-u.ac.jp\/relativity\/4093\/","title":{"rendered":"gnuplot \u3067\u30b1\u30d7\u30e9\u30fc\u65b9\u7a0b\u5f0f\u306e\u8fd1\u4f3c\u89e3\u3068\u6570\u5024\u89e3"},"content":{"rendered":"

\u300cMaxima \u3067\u30b1\u30d7\u30e9\u30fc\u65b9\u7a0b\u5f0f\u306e\u8fd1\u4f3c\u89e3\u3068\u6570\u5024\u89e3<\/a>\u300d\u306e gnuplot \u7248\u3002<\/p>\n

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\u30d9\u30c3\u30bb\u30eb\u95a2\u6570\u306b\u3088\u308b\u30b1\u30d7\u30e9\u30fc\u65b9\u7a0b\u5f0f\u306e\u30d5\u30fc\u30ea\u30a8\u7d1a\u6570\u89e3<\/h3>\n

\u30b1\u30d7\u30e9\u30fc\u65b9\u7a0b\u5f0f<\/p>\n

$$u – e \\sin u = \\omega t$$<\/p>\n

\u306e\u30d5\u30fc\u30ea\u30a8\u7d1a\u6570\u89e3\u306f\uff0c\u30d9\u30c3\u30bb\u30eb\u95a2\u6570 $J_n(x)$ \u3092\u4f7f\u3063\u3066\u4ee5\u4e0b\u306e\u3088\u3046\u306b\u66f8\u3051\u308b\u3002<\/p>\n

$$u = \\omega t+ \\sum_{n=1}^{\\infty} \\frac{2}{n} J_n(n e) \\sin(n\\, \\omega t)$$<\/p>\n

\u73fe\u5b9f\u4e16\u754c\u3067\u306f\u7121\u9650\u5927\u307e\u3067\u8db3\u3057\u7b97\u306f\u3067\u304d\u306a\u3044\u306e\u3067\uff0c\u4ee5\u4e0b\u306e\u3088\u3046\u306b $N$ \u307e\u3067\u306e\u548c\u3092\u3068\u308b\u3053\u3068\u306b\u3059\u308b\u3002<\/p>\n

$$u(N, e, \\omega t) = \\omega t+ \\sum_{n=1}^{N} \\frac{2}{n} J_n(n e) \\sin(n\\, \\omega t)$$<\/p>\n

\u3053\u308c\u3092 gnuplot \u306e\u95a2\u6570\u3068\u3057\u3066\u5b9a\u7fa9\u3059\u308b\u3002<\/p>\n<\/div>\n<\/div>\n<\/div>\n

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# \u6700\u8fd1\u306e gnuplot \u306b\u306f\u30d9\u30c3\u30bb\u30eb\u95a2\u6570 Jn(x) \u3082\u3042\u308b<\/span>\r\nhelp<\/span> besjn<\/span>\r\n<\/pre>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
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Out[1]:<\/div>\n
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 \u95a2\u6570 `besjn(n,x)` \u306f\u5f15\u6570\u306e Jn \u30d9\u30c3\u30bb\u30eb\u95a2\u6570 (n \u6b21\u306e\u7b2c 1 \u7a2e\u5186\u67f1\u95a2\u6570 Jn\u3001\r\n n \u6b21\u30d9\u30c3\u30bb\u30eb\u95a2\u6570) \u306e\u5024\u3092\u8fd4\u3057\u307e\u3059\u3002\u5f15\u6570 x \u306f\u30e9\u30b8\u30a2\u30f3\u3067\u4e0e\u3048\u307e\u3059\u3002\r\n<\/pre>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
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\u4ee5\u4e0b\u306e\u3088\u3046\u306b\u3057\u3066\uff0c\u4e09\u9805\u6f14\u7b97\u5b50\u3092\u4f7f\u3063\u3066\u518d\u5e30\u7684\u306b\u5b9a\u7fa9\u3057\u307e\u3059\u3002
\n\\begin{eqnarray}
\nu(0, e, \\omega t) &=& \\omega t \\\\
\nu(N, e, \\omega t) &=& \\omega t+ \\sum_{n=1}^{N} \\frac{2}{n} J_n(n e) \\sin(n\\, \\omega t) \\\\
\n&=& u(N-1, e, \\omega t) + \\frac{2}{N} J_N(N e) \\sin(N\\, \\omega t)
\n\\end{eqnarray}<\/p>\n<\/div>\n<\/div>\n<\/div>\n
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In\u00a0[2]:<\/div>\n
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# \u4e09\u9805\u6f14\u7b97\u5b50\u3092\u4f7f\u3063\u305f\u518d\u5e30\u5b9a\u7fa9\u306b\u3088\u308b sum2 \u306e\u5b9a\u7fa9\u4f8b\r\n# i**2 \u3092 i=1 \u304b\u3089 N \u307e\u3067\u8db3\u3059\u4f8b\r\n# sum2(N) = (N==1 ? 1**2 : sum2(N-1) + N**2)\r\n<\/span>\r\nUbes<\/span>(N, e, omegat) =<\/span> (<\/span>N<\/span> ==<\/span> 0<\/span> ?<\/span> \\\r\n   omegat<\/span> \\\r\n   :<\/span> Ubes<\/span>(<\/span>N<\/span>-1<\/span>,<\/span> e<\/span>,<\/span> omegat<\/span>)<\/span> +<\/span> 2.<\/span>\/<\/span>N<\/span> *<\/span> besjn<\/span>(<\/span>N<\/span>,<\/span> N<\/span>*<\/span>e<\/span>)<\/span>*<\/span>sin<\/span>(<\/span>N<\/span>*<\/span>omegat<\/span>))<\/span>\r\n<\/pre>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
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In\u00a0[3]:<\/div>\n
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# \u305f\u3068\u3048\u3070 N=10, e=0.5, omegat = 1.0 \u306e\u5024 <\/span>\r\nprint<\/span> Ubes<\/span>(<\/span>10<\/span>,<\/span> 0.5<\/span>,<\/span> 1<\/span>)<\/span>\r\n<\/pre>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
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Out[3]:<\/div>\n
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1.49885975062147\r\n<\/pre>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
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\u9010\u6b21\u8fd1\u4f3c\u6cd5\u306b\u3088\u308b\u30b1\u30d7\u30e9\u30fc\u65b9\u7a0b\u5f0f\u306e\u8fd1\u4f3c\u89e3<\/h3>\n
\u96e2\u5fc3\u7387 $e$ \u306f $0 \\leq e < 1$ \u3067\u3042\u308b\u3053\u3068\u304b\u3089\uff0c\u30b1\u30d7\u30e9\u30fc\u65b9\u7a0b\u5f0f
\n$$u – e \\sin u = \\omega t$$
\n\u3092\u4ee5\u4e0b\u306e\u3088\u3046\u306b\u9010\u6b21\u8fd1\u4f3c\u7684\u306b\u89e3\u3044\u3066\u307f\u307e\u3059\u3002<\/p>\n
\\begin{eqnarray}
\nu &=& \\omega t + e \\sin u \\\\
\nu_0 &=& \\omega t\\\\
\nu_1 &=& \\omega t + e \\sin u_0 = \\omega t + e \\sin \\omega t\\\\
\nu_2 &=& \\omega t + e \\sin u_1 =\\omega t + e \\sin\\left(\\omega t + e \\sin \\omega t\\right) \\\\
\nu_3 &=& \\omega t + e \\sin u_2 =\\omega t + e \\sin\\left\\{\\omega t + e \\sin\\left(\\omega t + e \\sin \\omega t\\right)\\right\\} \\\\
\n&\\vdots&\\\\
\nu_{n} &=& \\omega t + e \\sin u_{n-1} = \\dots\\\\
\n\\end{eqnarray}<\/p>\n
$n$ \u304c\u5927\u304d\u304f\u306a\u308b\u3068\uff0c\u5165\u308c\u5b50\u306b\u306a\u3063\u3066\u3044\u308b\u9805\u304c\u3069\u3093\u3069\u3093\u5897\u6b96\u3057\u3066\u3044\u304d\u307e\u3059\u304c\uff0c$u_3$ \u306e\u3042\u305f\u308a\u307e\u3067\u306f\uff0c\u8fd1\u4f3c\u7684\u306b $u$ \u306f $t$ \u306e\u967d\u95a2\u6570\u3068\u3057\u3066\u3042\u3089\u308f\u3055\u308c\u3066\u3044\u308b\u306a\u3041… \u3068\u3044\u3046\u898b\u305f\u76ee\u304c\u3057\u307e\u3059\u3002<\/p>\n
\u4e0a\u306e\u5f0f\u306b\u305d\u3063\u3066\uff0c\u4ee5\u4e0b\u306e\u3088\u3046\u306b\u95a2\u6570 $u(N, e, \\omega t)$ \u3092\u4e09\u9805\u6f14\u7b97\u5b50\u3092\u4f7f\u3063\u3066\u518d\u5e30\u7684\u306b\u5b9a\u7fa9\u3057\u307e\u3059\u3002<\/p>\n
\\begin{eqnarray}
\nN = 0 \\ \\ &\\Rightarrow \\ \\ u_0 =& \\omega t\\\\
\nN > 0 \\ \\ &\\Rightarrow \\ \\ u_{N} =& \\omega t + e \\sin u_{N-1} \\\\
\n\\end{eqnarray}<\/p>\n<\/div>\n<\/div>\n<\/div>\n
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In\u00a0[4]:<\/div>\n
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# gnuplot \u306e\u4e09\u9805\u6f14\u7b97\u5b50\u3092\u4f7f\u3063\u305f\u95a2\u6570\u306e\u5b9a\u7fa9\u4f8b \r\n# N==0 \u306a\u3089 f(N)=A, \u305d\u3046\u3067\u306a\u3044\u306a\u3089 f(N)=B \r\n# f(N) = (N==0 ? A : B) <\/span>\r\n\r\nUite<\/span>(N, e, omegat) =<\/span> (<\/span>N<\/span> ==<\/span> 0<\/span> ?<\/span> \\\r\n  omegat<\/span> \\\r\n  :<\/span> omegat<\/span> +<\/span> e<\/span>*<\/span>sin<\/span>(<\/span>Uite<\/span>(<\/span>N<\/span>-1<\/span>,<\/span> e<\/span>,<\/span> omegat<\/span>)))<\/span>\r\n<\/pre>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
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In\u00a0[5]:<\/div>\n
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# \u305f\u3068\u3048\u3070 N=10, e=0.5, omegat = 1.0 \u306e\u5024 <\/span>\r\nprint<\/span> Uite<\/span>(<\/span>10<\/span>,<\/span> 0.5<\/span>,<\/span> 1<\/span>)<\/span>\r\n<\/pre>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
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Out[5]:<\/div>\n
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1.49870113351784\r\n<\/pre>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
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\u65b9\u7a0b\u5f0f\u306e\u6570\u5024\u89e3\u6cd5\u306b\u3088\u308b\u30b1\u30d7\u30e9\u30fc\u65b9\u7a0b\u5f0f\u306e\u6570\u5024\u89e3<\/h3>\n
$\\omega t$ \u3092\u4e0e\u3048\u305f\u3068\u304d\u306b\uff0c\u30b1\u30d7\u30e9\u30fc\u65b9\u7a0b\u5f0f\u3092\u6570\u5024\u7684\u306b\u89e3\u3044\u3066 $u(\\omega t)$ \u3092\u6c42\u3081\u308b\u3002\u30b1\u30d7\u30e9\u30fc\u65b9\u7a0b\u5f0f\u306f\u8d85\u8d8a\u65b9\u7a0b\u5f0f\u3067\u3042\u308b\u306e\u3067\uff0c\u30cb\u30e5\u30fc\u30c8\u30f3\u6cd5\u3092\u4f7f\u3063\u3066\u6570\u5024\u306b\u89e3\u304f\u3002<\/p>\n<\/div>\n<\/div>\n<\/div>\n
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In\u00a0[6]:<\/div>\n
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f<\/span>(<\/span>e<\/span>,<\/span> omt<\/span>,<\/span> u<\/span>)<\/span> =<\/span> u<\/span> -<\/span> e<\/span>*<\/span>sin<\/span>(<\/span>u<\/span>)<\/span> -<\/span> omt<\/span>\r\n\r\n# \u518d\u5e30\u5b9a\u7fa9\u306b\u3088\u308b\u89e3<\/span>\r\n# x \u306b\u306f\u63a2\u7d22\u306e\u521d\u671f\u5024\u3092\u5165\u308c\u308b<\/span>\r\n# h \u306f\u89e3\u306e\u53ce\u675f\u6761\u4ef6\u3067\u3042\u308b\u5fae\u5c0f\u91cf\u3068\u6570\u5024\u5fae\u5206\u3092\u3059\u308b\u3068\u304d\u306b\u4f7f\u3046\u5fae\u5c0f\u91cf\u3092\u517c\u306d\u308b\u3002<\/span>\r\nnewton_f<\/span>(e, omt, x, h) =<\/span> \\\r\n  (<\/span>abs<\/span>(<\/span>f<\/span>(<\/span>e<\/span>,<\/span> omt<\/span>,<\/span> x<\/span>))<\/span> ><\/span> h<\/span> ?<\/span> \\\r\n   newton_f<\/span>(<\/span>e<\/span>,<\/span>omt<\/span>,<\/span>x<\/span>-<\/span>f<\/span>(<\/span>e<\/span>,<\/span>omt<\/span>,<\/span>x<\/span>)<\/span>*<\/span>2.<\/span>*<\/span>h<\/span>\/<\/span>(<\/span>f<\/span>(<\/span>e<\/span>,<\/span>omt<\/span>,<\/span>x<\/span>+<\/span>h<\/span>)<\/span>-<\/span>f<\/span>(<\/span>e<\/span>,<\/span>omt<\/span>,<\/span>x<\/span>-<\/span>h<\/span>))<\/span>,<\/span>h<\/span>)<\/span> \\\r\n   :<\/span> x<\/span>)<\/span>\r\n\r\nUnum<\/span>(e, omegat) =<\/span> newton_f<\/span>(<\/span>e<\/span>,<\/span> omegat<\/span>,<\/span> omegat<\/span>,<\/span> 1E-15<\/span>)<\/span>\r\n<\/pre>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
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In\u00a0[7]:<\/div>\n
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# \u305f\u3068\u3048\u3070 e=0.5, omegat = 1.0 \u306e\u5024<\/span>\r\nprint<\/span> Unum<\/span>(<\/span>0.5<\/span>,<\/span> 1<\/span>)<\/span>\r\n<\/pre>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
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Out[7]:<\/div>\n
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1.49870113351785\r\n<\/pre>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
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\u6570\u5024\u89e3\u3068\u8fd1\u4f3c\u89e3\u3068\u306e\u6bd4\u8f03<\/h3>\n<\/div>\n<\/div>\n<\/div>\n
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In\u00a0[8]:<\/div>\n
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# \u6570\u5024\u89e3 Unum \u3068\u9010\u6b21\u8fd1\u4f3c\u89e3 Uite, \u30d5\u30fc\u30ea\u30a8\u7d1a\u6570\u89e3 Ubes \u306e\u5dee<\/span>\r\ne<\/span> =<\/span> 0.5<\/span>\r\nN<\/span> =<\/span> 10<\/span>\r\nprint<\/span> \"omega t    Unum               Uite-Unum          Ubes-Unum\"<\/span>\r\n\r\ndo<\/span> for<\/span> [<\/span>i<\/span>=<\/span>1<\/span>:<\/span> 9<\/span>]<\/span> {<\/span>\r\n  omti<\/span> =<\/span> pi<\/span>\/<\/span>10<\/span>*<\/span>i<\/span>\r\n  print<\/span> sprintf<\/span>(<\/span>\"%7.5f %18.15f %18.15f %18.15f\"<\/span>,<\/span> \\\r\n                 omti<\/span>,<\/span> \\\r\n                 Unum<\/span>(<\/span>e<\/span>,<\/span> omti<\/span>)<\/span>,<\/span> \\\r\n                 Uite<\/span>(<\/span>N<\/span>,<\/span> e<\/span>,<\/span> omti<\/span>)<\/span>-<\/span>Unum<\/span>(<\/span>e<\/span>,<\/span> omti<\/span>)<\/span>,<\/span> \\\r\n                 Ubes<\/span>(<\/span>N<\/span>,<\/span> e<\/span>,<\/span> omti<\/span>)<\/span>-<\/span>Unum<\/span>(<\/span>e<\/span>,<\/span> omti<\/span>))<\/span>\r\n}<\/span>\r\n<\/pre>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
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Out[8]:<\/div>\n
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omega t    Unum               Uite-Unum          Ubes-Unum\r\n0.31416  0.593999023813606 -0.000048370871787  0.000205334548065\r\n0.62832  1.065940683889791 -0.000000479808531 -0.000233511559215\r\n0.94248  1.438080909968085 -0.000000000003061  0.000199446542732\r\n1.25664  1.748741781633489  0.000000000005287 -0.000159160439054\r\n1.57080  2.020979938089771 -0.000000056633643  0.000123562038968\r\n1.88496  2.268208852924499 -0.000003478720316 -0.000093184551056\r\n2.19911  2.498822425235399 -0.000028291765788  0.000066804145119\r\n2.51327  2.718544855625697 -0.000076315203530 -0.000043154412395\r\n2.82743  2.931640124182721 -0.000080693322920  0.000021180282106\r\n<\/pre>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
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\u6570\u5024\u89e3\u3068\u306e\u5dee\u306e\u4e8c\u4e57\u5e73\u5747\u5e73\u65b9\u6839<\/h3>\n