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ceptreee/Juliaの速さを体感する.ipynb

Created January 8, 2018 19:17
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Juliaの速さを体感する.ipynb
{
"cells": [
{
"cell_type": "markdown",
"metadata": {},
"source": [
"Qiitaの記事([Juliaの速さを体感する](https://qiita.com/ceptree/items/b5cfca180e85e61c42a8))をJupyter notebookに書き換えたものです。\n"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"# はじめに\n",
"年始からtwitter界隈では[@genkuroki](https://twitter.com/genkuroki)さんを中心にJuliaが急激な盛り上がりを見せています。 \n",
"Juliaを知らない方は、[@bicycle1885](https://twitter.com/bicycle1885)さんが書かれた下記の記事を参照下さい。\n",
"\n",
"[JuliaのQA](https://qiita.com/bicycle1885/items/ab8fff19a4ff0cdea287)\n",
"\n",
"注目された要因は色々あるかと思いますが、何と言っても１番は処理速度ではないでしょうか。 \n",
"Juliaは、JIT(Just-in-time)コンパイルを行うため高速に実行できるそうです。 \n",
"しかし、正直な話、私はコンパイラの知識がないので、\"なぜ速いか\"については詳しくは分かりません。 \n",
"そこで簡単な数値計算を通して、Juliaの速さを体感したいと思います。\n",
"\n",
"# 常微分方程式の数値計算\n",
"今回はこちらの常微分方程式の数値解を求める速さを比較したい思います。\n",
"$$\n",
"\\begin{eqnarray*}\n",
"y'&= -y \\\\\n",
"y_0&= 1\n",
"\\end{eqnarray*}\n",
"$$\n",
"\n",
"刻み幅$h$として、前進オイラー法で離散化します。\n",
"$$y_{n+1} = y_{n} - y_{n} * h$$\n",
"$y_n$でくくります。\n",
"$$y_{n+1}=(1-h)y_{n}$$\n",
"上記の式をJuliaで書くと、\n"
]
},
{
"cell_type": "code",
"execution_count": null,
"metadata": {},
"outputs": [],
"source": [
"y[n+1] = (1 - h) y[n]"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"となります。Juliaではカッコがあれば掛け算記号($*$)を省略できるので、数式とほとんど同じように書くことができます。\n",
"\n",
"まずは少ない点数できちんと計算できているか確認したいと思います。 \n",
"数値計算のプログラムは下記になります。 \n",
"点数$N$は1000点、刻み幅$h$は0.004としました。 "
]
},
{
"cell_type": "code",
"execution_count": 49,
"metadata": {},
"outputs": [],
"source": [
"# 点数\n",
"N = 1000\n",
"\n",
"# 刻み \n",
"h = 0.004\n",
"\n",
"# 初期化\n",
"y = zeros(N)\n",
"\n",
"# 初期値\n",
"y[1] = 1\n",
"\n",
"# 数値計算\n",
"for n = 1:N-1\n",
" y[n+1] = (1-h)y[n]\n",
"end"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"確認のために、数値解と解析解を比較したいと思います。上記の常微分方程式の解析解は下記になります。\n",
"\n",
"\n",
"$$f(x) = \\exp(-x)$$\n",
"\n",
"これをJuliaで書くとこうになります。\n",
"\n",
"\n"
]
},
{
"cell_type": "code",
"execution_count": 26,
"metadata": {},
"outputs": [
{
"data": {
"text/plain": [
"f (generic function with 1 method)"
]
},
"execution_count": 26,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"f(x) = exp.(-x)"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"expの後に(.)がついているのは、配列に関数を適用するためです。\n",
"またJuliaは、$f(x) = x$といった形式で関数を定義できるので、数式と同じように書くことができます。\n",
"それでは厳密解の値を求めてみます。"
]
},
{
"cell_type": "code",
"execution_count": 50,
"metadata": {},
"outputs": [
{
"data": {
"text/plain": [
"1000-element Array{Float64,1}:\n",
" 1.0 \n",
" 0.996008 \n",
" 0.992032 \n",
" 0.988072 \n",
" 0.984127 \n",
" 0.980199 \n",
" 0.976286 \n",
" 0.972388 \n",
" 0.968507 \n",
" 0.96464 \n",
" 0.960789 \n",
" 0.956954 \n",
" 0.953134 \n",
" ⋮ \n",
" 0.0192162\n",
" 0.0191395\n",
" 0.0190631\n",
" 0.018987 \n",
" 0.0189112\n",
" 0.0188357\n",
" 0.0187605\n",
" 0.0186856\n",
" 0.018611 \n",
" 0.0185368\n",
" 0.0184628\n",
" 0.018389 "
]
},
"execution_count": 50,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"# x, start:step:stop\n",
"x = 0:h:(N-1)h \n",
"\n",
"# 厳密解\n",
"f(x) = exp.(-x)\n",
"\n",
"# y_exc\n",
"y_exc = f(x)"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"比較のためにプロットしてみます。Plotsというライブラリを使います。"
]
},
{
"cell_type": "code",
"execution_count": 51,
"metadata": {},
"outputs": [
{
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},
"execution_count": 51,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"# plot \n",
"using Plots\n",
"plot(x, y_exc)\n",
"plot!(x, y)"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"plot!のようにplotの後に!をつけると、プロットを追加することができます。ただこれでは、線が被っていてよくわからないので、線の太さ(linewidth)とスタイル(linestyle)を変更します。ついでに軸ラベル(xlabel, ylabel)もつけましょう。\n"
]
},
{
"cell_type": "code",
"execution_count": 52,
"metadata": {},
"outputs": [
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" 479.601,352.734 480.134,352.789 480.667,352.844 481.2,352.899 481.733,352.953 482.265,353.008 482.798,353.062 483.331,353.116 483.864,353.17 484.397,353.223 \n",
" 484.93,353.276 485.463,353.329 485.996,353.382 486.528,353.435 487.061,353.487 487.594,353.539 488.127,353.591 488.66,353.643 489.193,353.695 489.726,353.746 \n",
" 490.259,353.797 490.792,353.848 491.324,353.899 491.857,353.949 492.39,354 492.923,354.05 493.456,354.1 493.989,354.15 494.522,354.199 495.055,354.249 \n",
" 495.587,354.298 496.12,354.347 496.653,354.395 497.186,354.444 497.719,354.492 498.252,354.541 498.785,354.588 499.318,354.636 499.85,354.684 500.383,354.731 \n",
" 500.916,354.779 501.449,354.826 501.982,354.872 502.515,354.919 503.048,354.966 503.581,355.012 504.113,355.058 504.646,355.104 505.179,355.15 505.712,355.195 \n",
" 506.245,355.24 506.778,355.286 507.311,355.331 507.844,355.375 508.376,355.42 508.909,355.465 509.442,355.509 509.975,355.553 510.508,355.597 511.041,355.641 \n",
" 511.574,355.684 512.107,355.728 512.639,355.771 513.172,355.814 513.705,355.857 514.238,355.9 514.771,355.942 515.304,355.984 515.837,356.027 516.37,356.069 \n",
" 516.903,356.111 517.435,356.152 517.968,356.194 518.501,356.235 519.034,356.276 519.567,356.317 520.1,356.358 520.633,356.399 521.166,356.44 521.698,356.48 \n",
" 522.231,356.52 522.764,356.56 523.297,356.6 523.83,356.64 524.363,356.68 524.896,356.719 525.429,356.758 525.961,356.797 526.494,356.836 527.027,356.875 \n",
" 527.56,356.914 528.093,356.952 528.626,356.991 529.159,357.029 529.692,357.067 530.224,357.105 530.757,357.142 531.29,357.18 531.823,357.217 532.356,357.255 \n",
" 532.889,357.292 533.422,357.329 533.955,357.366 534.487,357.402 535.02,357.439 535.553,357.475 536.086,357.511 536.619,357.548 537.152,357.583 537.685,357.619 \n",
" 538.218,357.655 538.75,357.69 539.283,357.726 539.816,357.761 540.349,357.796 540.882,357.831 541.415,357.866 541.948,357.901 542.481,357.935 543.014,357.97 \n",
" 543.546,358.004 544.079,358.038 544.612,358.072 545.145,358.106 545.678,358.14 546.211,358.173 546.744,358.207 547.277,358.24 547.809,358.273 548.342,358.306 \n",
" 548.875,358.339 549.408,358.372 549.941,358.405 550.474,358.437 551.007,358.469 551.54,358.502 552.072,358.534 552.605,358.566 553.138,358.598 553.671,358.63 \n",
" 554.204,358.661 554.737,358.693 555.27,358.724 555.803,358.755 556.335,358.786 556.868,358.817 557.401,358.848 557.934,358.879 558.467,358.91 559,358.94 \n",
" 559.533,358.971 560.066,359.001 560.598,359.031 561.131,359.061 561.664,359.091 562.197,359.121 562.73,359.15 563.263,359.18 563.796,359.209 564.329,359.239 \n",
" 564.862,359.268 565.394,359.297 565.927,359.326 566.46,359.355 566.993,359.383 567.526,359.412 568.059,359.441 568.592,359.469 569.125,359.497 569.657,359.525 \n",
" 570.19,359.553 570.723,359.581 571.256,359.609 571.789,359.637 572.322,359.664 572.855,359.692 573.388,359.719 573.92,359.747 574.453,359.774 574.986,359.801 \n",
" 575.519,359.828 576.052,359.855 576.585,359.881 577.118,359.908 577.651,359.934 578.183,359.961 578.716,359.987 579.249,360.013 579.782,360.039 580.315,360.065 \n",
" \n",
" \"/>\n",
"<polygon clip-path=\"url(#clip5800)\" points=\"\n",
"448.982,78.051 562.315,78.051 562.315,32.691 448.982,32.691 \n",
" \" fill=\"#ffffff\" fill-opacity=\"1\"/>\n",
"<polyline clip-path=\"url(#clip5800)\" style=\"stroke:#000000; stroke-width:1; stroke-opacity:1; fill:none\" points=\"\n",
" 448.982,78.051 562.315,78.051 562.315,32.691 448.982,32.691 448.982,78.051 \n",
" \"/>\n",
"<polyline clip-path=\"url(#clip5800)\" style=\"stroke:#000000; stroke-width:5; stroke-opacity:1; fill:none\" points=\"\n",
" 454.982,47.811 490.982,47.811 \n",
" \"/>\n",
"<g clip-path=\"url(#clip5800)\">\n",
"<text style=\"fill:#000000; fill-opacity:1; font-family:Arial,Helvetica Neue,Helvetica,sans-serif; font-size:12; text-anchor:start;\" transform=\"rotate(0, 496.982, 52.311)\" x=\"496.982\" y=\"52.311\">Exact</text>\n",
"</g>\n",
"<polyline clip-path=\"url(#clip5800)\" style=\"stroke:#ff0000; stroke-width:2; stroke-opacity:1; fill:none\" stroke-dasharray=\"8, 5\" points=\"\n",
" 454.982,62.931 490.982,62.931 \n",
" \"/>\n",
"<g clip-path=\"url(#clip5800)\">\n",
"<text style=\"fill:#000000; fill-opacity:1; font-family:Arial,Helvetica Neue,Helvetica,sans-serif; font-size:12; text-anchor:start;\" transform=\"rotate(0, 496.982, 67.431)\" x=\"496.982\" y=\"67.431\">Numerical</text>\n",
"</g>\n",
"</svg>\n"
]
},
"execution_count": 52,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"# plot \n",
"using Plots\n",
"plot(x, y_exc, color=:black, linewidth=5, label=\"Exact\",xlabel=\"x\",ylabel=\"y\") \n",
"plot!(x, y, color=:red, linewidth=2, linestyle=:dash, label=\"Numerical\")"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"厳密解と数値解で一致していることが確認できました。\n",
"\n",
"\n",
"画像の保存はsavefig()です。"
]
},
{
"cell_type": "code",
"execution_count": 36,
"metadata": {},
"outputs": [],
"source": [
"savefig(\"ODE.png\")"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"値も見てみましょう。"
]
},
{
"cell_type": "code",
"execution_count": 37,
"metadata": {},
"outputs": [
{
"name": "stdout",
"output_type": "stream",
"text": [
" n Numerical Exact Error\n",
" 990 0.01898828 0.01913952 -1.51236228e-04\n",
" 991 0.01891233 0.01906311 -1.50784195e-04\n",
" 992 0.01883668 0.01898701 -1.50333360e-04\n",
" 993 0.01876133 0.01891122 -1.49883720e-04\n",
" 994 0.01868629 0.01883572 -1.49435274e-04\n",
" 995 0.01861154 0.01876053 -1.48988018e-04\n",
" 996 0.01853710 0.01868564 -1.48541950e-04\n",
" 997 0.01846295 0.01861105 -1.48097068e-04\n",
" 998 0.01838910 0.01853675 -1.47653370e-04\n",
" 999 0.01831554 0.01846275 -1.47210853e-04\n",
" 1000 0.01824228 0.01838905 -1.46769515e-04\n"
]
}
],
"source": [
"@printf(\"%12s %12s %12s %12s\\n\",\"n\" ,\"Numerical\" ,\"Exact\" , \"Error\")\n",
"for n = N-10:N\n",
" @printf(\"%12d %12.8f %12.8f %12.8e\\n\", n, y[n], y_exc[n], y[n]-y_exc[n])\n",
"end"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"誤差はありますが、同じような値になっていることが確認できました。\n",
"\n",
"ここまでのコードの全文です。"
]
},
{
"cell_type": "code",
"execution_count": null,
"metadata": {},
"outputs": [],
"source": [
"# 点数\n",
"N = 1000\n",
"\n",
"# 刻み \n",
"h = 0.004\n",
"\n",
"# 初期化\n",
"y = zeros(N)\n",
"\n",
"# 初期値\n",
"y[1] = 1\n",
"\n",
"# 数値計算\n",
"for n = 1:N-1\n",
" y[n+1] = (1-h)y[n]\n",
"end\n",
"\n",
"# x, start:step:stop\n",
"x = 0:h:(N-1)h \n",
"\n",
"# 厳密解\n",
"f(x) = exp.(-x)\n",
"\n",
"# y_exc\n",
"y_exc = f(x)\n",
"\n",
"# plot\n",
"using Plots\n",
"plot(x, y_exc, color=:black, linewidth=5, label=\"Exact\",xlabel=\"x\",ylabel=\"y\") \n",
"plot!(x, y, color=:red, linewidth=2, linestyle=:dash, label=\"Numerical\")\n",
"\n",
"# save\n",
"savefig(\"ODE.png\")\n",
"\n",
"# print\n",
"@printf(\"%12s %12s %12s %12s\\n\",\"n\",\"Numerical\",\"Exact\",\"Error\")\n",
"for n = N-10:N\n",
" @printf(\"%12d %12.8f %12.8f %12.8e\\n\", n, y[n],y_exc[n],y[n]-y_exc[n])\n",
"end"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"# 経過時間の計測"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"さて、それでは本題に移ります。経過時間の計測は@timeで出来ます。関数だけでなく、for文の頭につけることで、for文の経過時間も計測できます。\n"
]
},
{
"cell_type": "code",
"execution_count": 53,
"metadata": {},
"outputs": [
{
"name": "stdout",
"output_type": "stream",
"text": [
" 0.000437 seconds (5.46 k allocations: 101.000 KiB)\n"
]
}
],
"source": [
"@time for n = 1:N-1\n",
" y[n+1] = (1-h)y[n]\n",
"end"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"どうせならということで、点数は1億点、刻み幅は$40 \\times 10^{-9}$に変更しました。"
]
},
{
"cell_type": "code",
"execution_count": 41,
"metadata": {},
"outputs": [
{
"name": "stdout",
"output_type": "stream",
"text": [
" 34.899070 seconds (700.00 M allocations: 11.921 GiB, 5.13% gc time)\n"
]
}
],
"source": [
"# 点数\n",
"N = 100000000\n",
"\n",
"# 刻み \n",
"h = 0.00000004\n",
"\n",
"# 初期化\n",
"y = zeros(N)\n",
"\n",
"# 初期値\n",
"y[1] = 1\n",
"\n",
"# 数値計算\n",
"@time for n = 1:N-1\n",
" y[n+1] = (1-h)y[n]\n",
"end\n"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"結果は約35秒となりました。参考に、同様の計算をPythonで行ってみました。"
]
},
{
"cell_type": "code",
"execution_count": 2,
"metadata": {},
"outputs": [
{
"name": "stdout",
"output_type": "stream",
"text": [
"Elapsed Time(Python): 49.52 [s]\n"
]
}
],
"source": [
"import numpy as np \n",
"import time\n",
"\n",
"# 点数\n",
"N = 100000000\n",
"\n",
"# 刻み \n",
"h = 0.00000004\n",
"\n",
"# 初期化\n",
"y = np.zeros(N)\n",
"\n",
"# 初期値\n",
"y[0] = 1\n",
"\n",
"# 数値計算\n",
"start = time.time()\n",
"\n",
"for n in range(N-1):\n",
" y[n+1] = (1-h)*y[n]\n",
"\n",
"elapsed_time = time.time() - start\n",
"print(\"Elapsed Time(Python): %.2f [s]\" % elapsed_time)"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"結果は約50秒となりました。Pythonに比べて、Juliaの方が15秒速い程度ですね。\n",
"\n",
"...というようなことをtwitterで呟いたら@bicycle1885さんに、「Juliaは、関数内に書かないと速度が出ない」と[指摘](https://twitter.com/bicycle1885/status/948209611802255362)して頂きました。 \n",
"それでは、先程のコードを関数内に書いてみましょう。"
]
},
{
"cell_type": "code",
"execution_count": 2,
"metadata": {},
"outputs": [
{
"data": {
"text/plain": [
"main (generic function with 1 method)"
]
},
"execution_count": 2,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"function main()\n",
" # 点数\n",
" N = 100000000\n",
" \n",
" # 刻み \n",
" h = 0.00000004\n",
" \n",
" # 初期化\n",
" y = zeros(N)\n",
" \n",
" # 初期値\n",
" y[1] = 1\n",
" \n",
" # 数値計算\n",
" @time for n = 1:N-1\n",
" y[n+1] = (1-h)y[n]\n",
" end\n",
"end"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"それでは実行してみます。"
]
},
{
"cell_type": "code",
"execution_count": 3,
"metadata": {},
"outputs": [
{
"name": "stdout",
"output_type": "stream",
"text": [
" 0.373891 seconds\n"
]
}
],
"source": [
"main()"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"ん？　間違えたかな、もう一度。"
]
},
{
"cell_type": "code",
"execution_count": 4,
"metadata": {},
"outputs": [
{
"name": "stdout",
"output_type": "stream",
"text": [
" 0.330274 seconds\n"
]
}
],
"source": [
"main()"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"Oh...。なんと35秒から0.33秒と、約100倍も速くなりました。 \n",
"ほんとにきちんと計算できているんでしょうか。厳密解と比較してみましょう。\n"
]
},
{
"cell_type": "code",
"execution_count": 47,
"metadata": {},
"outputs": [
{
"name": "stdout",
"output_type": "stream",
"text": [
" 0.322447 seconds\n",
" n Numerical Exact Error\n",
" 99999990 0.01831565 0.01831565 -1.42668458e-09\n",
" 99999991 0.01831564 0.01831565 -1.42668453e-09\n",
" 99999992 0.01831564 0.01831565 -1.42668450e-09\n",
" 99999993 0.01831564 0.01831564 -1.42668445e-09\n",
" 99999994 0.01831564 0.01831564 -1.42668441e-09\n",
" 99999995 0.01831564 0.01831564 -1.42668436e-09\n",
" 99999996 0.01831564 0.01831564 -1.42668433e-09\n",
" 99999997 0.01831564 0.01831564 -1.42668428e-09\n",
" 99999998 0.01831564 0.01831564 -1.42668423e-09\n",
" 99999999 0.01831564 0.01831564 -1.42668419e-09\n",
" 100000000 0.01831564 0.01831564 -1.42668415e-09\n"
]
}
],
"source": [
"function main(N,h,y0)\n",
" # 初期化\n",
" y = zeros(N)\n",
" \n",
" # 初期値\n",
" y[1] = y0\n",
" \n",
" # 数値計算\n",
" @time for n = 1:N-1\n",
" y[n+1] = (1-h)y[n]\n",
" end\n",
" return y\n",
"end\n",
"# 点数\n",
"N = 100000000\n",
"\n",
"# 刻み \n",
"h = 0.00000004\n",
"\n",
"# 初期値\n",
"y0 = 1\n",
"\n",
"# main\n",
"y = main(N,h,y0)\n",
"\n",
"# x, start:step:stop\n",
"x = 0:h:(N-1)h \n",
"\n",
"# 厳密解\n",
"f(x) = exp.(-x)\n",
"\n",
"# y_exc\n",
"y_exc= f(x)\n",
"\n",
"# print\n",
"@printf(\"%12s %12s %12s %12s\\n\",\"n\",\"Numerical\",\"Exact\",\"Error\")\n",
"for n = N-10:N\n",
" @printf(\"%12d %12.8f %12.8f %12.8e\\n\", n, y[n],y_exc[n],y[n]-y_exc[n])\n",
"end"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"きちんとできていますね！すごい！\n",
"上記の結果を表にしてみました。\n",
"\n",
"| |Python(参考) |Julia(TopLeve)|Julia (Function) |\n",
"|-------------|:-------:|:--------:|:--------:|\n",
"|経過時間 [s] |41.74 | 31.96 |0.313|\n"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"# おわりに\n",
"いかがでしたでしょうか。簡単な数値計算の結果ではありましたが、Juliaの速さを体感して頂けたでしょうか。\n",
"\n",
"「Juliaだけ最適化してせこい！PythonだってnumbaやCythonを使えばもっと速くなる！」という声もあるかと思います。 \n",
"numbaについては試しているので、時間があればそちらの記事も書きたいと思います。\n",
"\n",
"他にも、コンパイラ言語との比較も気になる所だと思います。これに関してはFortranとの比較を行いましたので、同じ時に書きたいと思います。\n",
"\n",
"最後までお読み頂きありがとうございました。"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"# 環境\n",
"OS : macOS High Sierra Version 10.13.2 \n",
"PC : MacBook Pro (Retina, 15-inch, Early 2013) \n",
"CPU : Intel Core i7 2.7 GHz \n",
"Memory : 16 GB 1600 MHz DDR3\n",
"\n",
"Julia : 0.6.2\n",
"\n",
"Python : 3.6.3 \n",
"numpy : 1.13.3"
]
}
],
"metadata": {
"kernelspec": {
"display_name": "Julia 0.6.2",
"language": "julia",
"name": "julia-0.6"
},
"language_info": {
"file_extension": ".jl",
"mimetype": "application/julia",
"name": "julia",
"version": "0.6.2"
}
},
"nbformat": 4,
"nbformat_minor": 2
}
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