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quadde の性能評価と quadgk との比較
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Comparison_with_QuadGK.ipynb
{
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{
"cell_type": "code",
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"source": [
"using DoubleExponentialFormulas\n",
"using QuadGK\n",
"using Printf\n",
"using Plots\n",
"using LaTeXStrings\n",
"using SpecialFunctions"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"# 性能評価用の型とマクロの定義\n",
"\n",
"基本的に数値積分にかかる時間は、被積分関数の評価にかかる時間が大半を占めます。なので、実行時間を計っても被積分関数を評価するばかりで、数値積分関数自体の評価のためにはよい指標とは言えません。かわりに被積分関数 $f(x)$ を評価した回数を指標として用いることが一般的なようです。目標とする精度の積分値をより**少ない**回数で出せるほうがよい数値積分法といえます。\n",
"\n",
"---\n",
"\n",
"注: ただし、目標とする精度ぴったりを目指して計算することはほとんどの場合困難です。通常は、適当に反復をしながら目標精度を上回った時点で打ち切る、という方法をとるので反復のパラメータを微調整することで性能が劇的に改善することがあります。\n",
"\n",
"---\n",
"\n",
"注: ひとまず、以下では `quadde`, `quadgk` ともにデフォルトの精度での計算を行っていきます。これらは一致しており、およそ7～8桁の数字が厳密解と一致する程度を目標としています。\n",
"\n",
"---\n",
"\n",
"以下の `@printeval` マクロは数値積分を実行し、被積分関数の評価回数、積分値、推定誤差を画面に表示します。 `EvalCounter` は `@printeval` マクロの内部で使われる型です。"
]
},
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"data": {
"text/plain": [
"@printeval (macro with 1 method)"
]
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"source": [
"mutable struct EvalCounter <: Function\n",
" n::Int\n",
" f::Function\n",
" EvalCounter(f::Function) = new(0, f)\n",
"end\n",
"\n",
"function (ec::EvalCounter)(x)\n",
" ec.n += 1\n",
" ec.f(x)\n",
"end\n",
"\n",
"macro printeval(expr)\n",
" s = string(expr)\n",
" q = expr.args[1]\n",
" f = expr.args[2]\n",
" rest = expr.args[3:end]\n",
" quadexpr = Expr(:call, esc(q), :ec)\n",
" for part in rest\n",
" if part isa Expr || part isa Symbol\n",
" push!(quadexpr.args, esc(part))\n",
" else\n",
" push!(quadexpr.args, part)\n",
" end\n",
" end\n",
"\n",
" blk = Expr(:block)\n",
" push!(blk.args, :(ec = EvalCounter($(esc(f)))))\n",
" push!(blk.args, Expr(:(=), :ret, quadexpr))\n",
" push!(blk.args, :(println($(s) * \": Evaluation=\", @sprintf(\"%-5d\", ec.n), \n",
" \" I=\", @sprintf(\"%-1.15e\", ret[1]), \n",
" \" E=\", @sprintf(\"%-1.4e\", ret[2]))))\n",
" push!(blk.args, :ret)\n",
" return blk\n",
"end"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"# 多項式\n",
"\n",
"$$\n",
"\\begin{eqnarray*}\n",
" \\int_{0}^{1} x dx = \\frac{1}{2} \\\\\n",
" \\int_{-1}^{1} \\left(x^3 + x^2 + x + 1\\right) dx = \\frac{8}{3}\n",
"\\end{eqnarray*}\n",
"$$\n",
"\n",
"これらは明らかに不定積分が可能なので数値積分を使う必要はありませんが、ひとまず例として挙げます。　`quadgk` で使われているガウスークロンロッド積分法は低次の多項式に関しては、原理的にほぼ厳密解を返します（うろおぼえ）。 `quadgk` は驚異的に少ない評価回数で解を返しています。 `quadde` も二桁以内の回数に収めているので十分だと思います。\n",
"\n",
"---\n",
"\n",
"注: これら低次の多項式に限らず、以下で紹介する積分は複雑に見えても厳密解を得られるものが多いです。これは性能評価において、まず数値積分が正しく動いているかどうかを確かめる必要があるためです。しかし、現実に数値積分が必要な場合においては厳密解を得ることは不可能で、被積分関数 $f(x)$ はより複雑であると考えられます。\n",
"\n",
"---"
]
},
{
"cell_type": "code",
"execution_count": 3,
"metadata": {},
"outputs": [
{
"name": "stdout",
"output_type": "stream",
"text": [
"quadde(f, 0.0, 1.0): Evaluation=25 I=5.000000000000183e-01 E=1.1287e-10\n",
"quadgk(f, 0.0, 1.0): Evaluation=15 I=5.000000000000000e-01 E=0.0000e+00\n"
]
},
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"source": [
"let f(x) = x\n",
" @printeval quadde(f, 0.0, 1.0)\n",
" @printeval quadgk(f, 0.0, 1.0)\n",
" plot(f, label=L\"x\", xlim=[0.0, 1.0],\n",
" fillrange=0, fillalpha=0.1, fillcolor=:blue)\n",
"end"
]
},
{
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"execution_count": 4,
"metadata": {},
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"text": [
"quadde(f, -1.0, 1.0): Evaluation=51 I=2.666666666666667e+00 E=5.9246e-16\n",
"quadgk(f, -1.0, 1.0): Evaluation=15 I=2.666666666666667e+00 E=0.0000e+00\n"
]
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"metadata": {},
"output_type": "execute_result"
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],
"source": [
"let f(x) = x^3 + x^2 + x + 1\n",
" @printeval quadde(f, -1.0, 1.0)\n",
" @printeval quadgk(f, -1.0, 1.0)\n",
" plot(f, label=L\"x^3 + x^2 + x + 1\", xlim=[-1.0, 1.0],\n",
" fillrange=0, fillalpha=0.1, fillcolor=:blue)\n",
"end"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"# [ガウス積分](https://ja.wikipedia.org/wiki/%E3%82%AC%E3%82%A6%E3%82%B9%E7%A9%8D%E5%88%86)\n",
"\n",
"$$\n",
"\\begin{eqnarray*}\n",
" \\int_{-\\infty}^{\\infty} exp(-x^2) dx = \\sqrt{\\pi} \\\\\n",
" \\int_{0}^{\\infty} x^3 exp(-x^2) dx = \\frac{1}{2}\n",
"\\end{eqnarray*}\n",
"$$\n",
"\n",
"物理などの分野ではよく見られる積分ですね。これらは不定積分を求めることはできませんが、厳密な定積分値を得られることがよく知られています。 `quadde`, `quadgk` ともに目立った性能差はなさそうです。"
]
},
{
"cell_type": "code",
"execution_count": 5,
"metadata": {},
"outputs": [
{
"name": "stdout",
"output_type": "stream",
"text": [
"quadde(f, -Inf, Inf): Evaluation=227 I=1.772453850904036e+00 E=2.5628e-12\n",
"quadgk(f, -Inf, Inf): Evaluation=225 I=1.772453850905514e+00 E=6.4296e-09\n"
]
},
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"execution_count": 5,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"let f(x) = exp(-x^2)\n",
" @printeval quadde(f, -Inf, Inf)\n",
" @printeval quadgk(f, -Inf, Inf)\n",
" plot(f, label=L\"\\exp{\\left(-x^2\\right)}\", xlim=[-5.0, 5.0],\n",
" fillrange=0, fillalpha=0.1, fillcolor=:blue)\n",
"end"
]
},
{
"cell_type": "code",
"execution_count": 6,
"metadata": {},
"outputs": [
{
"name": "stdout",
"output_type": "stream",
"text": [
"quadde(f, 0.0, Inf): Evaluation=145 I=4.999999999999333e-01 E=2.3797e-12\n",
"quadgk(f, 0.0, Inf): Evaluation=165 I=5.000000000000004e-01 E=2.7808e-09\n"
]
},
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"source": [
"let f(x) = x^3*exp(-x^2)\n",
" @printeval quadde(f, 0.0, Inf)\n",
" @printeval quadgk(f, 0.0, Inf)\n",
" plot(f, label=L\"x^3 \\exp{\\left(-x^2\\right)}\", xlim=[0.0, 5.0],\n",
" fillrange=0, fillalpha=0.1, fillcolor=:blue)\n",
"end"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"# 平方根を含む関数\n",
"\n",
"$$\n",
"\\begin{eqnarray*}\n",
" \\int_{0}^{1} \\sqrt{x} dx = \\frac{2}{3} \\\\\n",
" \\int_{0}^{1} \\frac{1}{\\sqrt{x}} dx = 2\n",
"\\end{eqnarray*}\n",
"$$\n",
"\n",
"理由はわかりませんが、被積分関数に平方根の計算 `sqrt` が含まれると数値積分の性能が悪化しやすくなるような気がします。（有効桁数の問題？）しかし、以下二つの簡単な例ではなぜか `quadde` が非常にいい結果を出しています。"
]
},
{
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"execution_count": 7,
"metadata": {},
"outputs": [
{
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"text": [
"quadde(f, 0.0, 1.0): Evaluation=25 I=6.666666666668387e-01 E=2.6111e-09\n",
"quadgk(f, 0.0, 1.0): Evaluation=315 I=6.666666670773991e-01 E=7.1095e-09\n"
]
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"source": [
"let f(x) = sqrt(x)\n",
" @printeval quadde(f, 0.0, 1.0)\n",
" @printeval quadgk(f, 0.0, 1.0)\n",
" plot(f, label=L\"\\sqrt{x}\", xlim=[0.0, 1.0],\n",
" fillrange=0, fillalpha=0.1, fillcolor=:blue)\n",
"end"
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"quadde(f, 0.0, 1.0): Evaluation=25 I=1.999999982931068e+00 E=9.0356e-12\n",
"quadgk(f, 0.0, 1.0): Evaluation=1305 I=1.999999984598392e+00 E=2.3763e-08\n"
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"execution_count": 8,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"let f(x) = 1/sqrt(x)\n",
" @printeval quadde(f, 0.0, 1.0)\n",
" @printeval quadgk(f, 0.0, 1.0)\n",
" plot(f, label=L\"\\frac{1}{\\sqrt{x}}\", xlim=[0.0, 1.0],\n",
" fillrange=0, fillalpha=0.1, fillcolor=:blue)\n",
"end"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"# 三角関数\n",
"\n",
"$$\n",
"\\begin{eqnarray*}\n",
" \\int_{0}^{\\pi} \\sin{x} dx = 2 \\\\\n",
" \\int_{0}^{\\pi/2} \\cos{x} dx = 1\n",
"\\end{eqnarray*}\n",
"$$\n",
"\n",
"一般的に言うと、振動する関数は数値積分が少し困難です。特に周期が極端に短い場合は非常に困難になり、積分区間を分割するなどの対策が必要になります。しかし、以下の簡単な例では、うまくいっているようですね。 `quadgk` の評価回数はここでも驚異的に少ないです。"
]
},
{
"cell_type": "code",
"execution_count": 9,
"metadata": {},
"outputs": [
{
"name": "stdout",
"output_type": "stream",
"text": [
"quadde(f, 0.0, π): Evaluation=51 I=2.000000000000000e+00 E=1.1712e-14\n",
"quadgk(f, 0.0, π): Evaluation=15 I=2.000000000000000e+00 E=1.7906e-12\n"
]
},
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"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"let f(x) = sin(x)\n",
" @printeval quadde(f, 0.0, π)\n",
" @printeval quadgk(f, 0.0, π)\n",
" plot(f, label=L\"\\sin{x}\", \n",
" xlim=[0.0, π], xticks=(0.0:π/2:π, [\"0\", \"\\\\pi/2\", \"\\\\pi\"]),\n",
" fillrange=0, fillalpha=0.1, fillcolor=:blue)\n",
"end"
]
},
{
"cell_type": "code",
"execution_count": 10,
"metadata": {},
"outputs": [
{
"name": "stdout",
"output_type": "stream",
"text": [
"quadde(f, 0.0, π / 2): Evaluation=51 I=1.000000000000000e+00 E=2.2393e-16\n",
"quadgk(f, 0.0, π / 2): Evaluation=15 I=1.000000000000000e+00 E=2.2204e-16\n"
]
},
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},
"execution_count": 10,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"let f(x) = cos(x)\n",
" @printeval quadde(f, 0.0, π/2)\n",
" @printeval quadgk(f, 0.0, π/2)\n",
" plot(f, label=L\"\\cos{x}\", \n",
" xlim=[0.0, π/2], xticks=([0.0, π/4, π/2], [\"0\", \"\\\\pi/4\", \"\\\\pi/2\"]),\n",
" fillrange=0, fillalpha=0.1, fillcolor=:blue)\n",
"end"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"# [アーベル変換](https://en.wikipedia.org/wiki/Abel_transform)\n",
"\n",
"$$\n",
"F(y) = 2 \\int_{|y|}^{\\infty} \\frac{f(r) r}{\\sqrt{r^2 - y^2}} dr \n",
"$$\n",
"\n",
"もう少し複雑な被積分関数の例として、アーベル変換を挙げてみます。変換を受ける関数 $f(x)$ によっては定積分の厳密解を計算することが不可能になり、数値計算が必要になります。"
]
},
{
"cell_type": "code",
"execution_count": 11,
"metadata": {},
"outputs": [
{
"name": "stdout",
"output_type": "stream",
"text": [
"quadde(g, y, Inf): Evaluation=89 I=1.772453810111211e+00 E=5.3962e-11\n",
"quadgk(g, y, Inf): Evaluation=1395 I=1.772453829887660e+00 E=4.3196e-09\n"
]
},
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},
"execution_count": 11,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"let f(r) = exp(-(r^2 - 1))\n",
" y = 1\n",
" g(r) = 2*f(r)*r/sqrt(r^2 - y^2)\n",
" @printeval quadde(g, y, Inf)\n",
" @printeval quadgk(g, y, Inf)\n",
" plot(g, label=L\"\\frac{2 f(r) r}{\\sqrt{r^2 - y^2}}\",\n",
" xlim=[y, 2.0], fillrange=0, fillalpha=0.1, fillcolor=:blue)\n",
"end"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"もとの式は $r = y$ となるすべての点で発散しますが、 $r = s^2 + |y|$ (y は実数)とおいて変数変換すると $r = y = 0$ の一点のみで発散する式に改良できます。\n",
"\n",
"$$\n",
"F(y) = 2 \\int_{|y|}^{\\infty} \\frac{f(r) r}{\\sqrt{r^2 - y^2}} dr = 2 \\int_{0}^{\\infty} \\frac{f(s^2 + |y|) (s^2 + |y|)}{\\sqrt{\\left( s^2 + |y| \\right)^2 - y^2 }} \\frac{dr}{ds} ds = 4 \\int_{0}^{\\infty} \\frac{f(s^2 + |y|) (s^2 + |y|)}{\\sqrt{s^2 + 2|y|}} ds = 4 \\int_{0}^{\\infty} \\frac{f(r) r}{\\sqrt{r + |y|}} ds\n",
"$$\n",
"\n",
"被積分関数の変形によって `quadgk` の評価回数が **1/10** に改善されました。（`quadde` はなぜか変換後の関数のほうが苦手のようです）"
]
},
{
"cell_type": "code",
"execution_count": 12,
"metadata": {},
"outputs": [
{
"name": "stdout",
"output_type": "stream",
"text": [
"quadde(g, 0.0, Inf): Evaluation=161 I=1.772453850904258e+00 E=9.1114e-13\n",
"quadgk(g, 0.0, Inf): Evaluation=135 I=1.772453850905516e+00 E=4.0140e-09\n"
]
},
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"source": [
"let f(r) = exp(-(r^2 - 1))\n",
" y = 1\n",
" function g(s)\n",
" absy = abs(y)\n",
" r = s^2 + absy\n",
" 4*f(r)*r/sqrt(r + absy)\n",
" end\n",
" @printeval quadde(g, 0.0, Inf)\n",
" @printeval quadgk(g, 0.0, Inf)\n",
" plot(g, label=L\"\\frac{4 f(s^2 + |y|) (s^2 + |y|)}{\\sqrt{s^2 + 2|y|}}\",\n",
" xlim=[0.0, 2.0], fillrange=0, fillalpha=0.1, fillcolor=:blue)\n",
"end"
]
},
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"metadata": {},
"source": [
"# 積分区間の一部で急激に変化する関数(1)\n",
"\n",
"ほとんどの数値積分アルゴリズムは連続な関数 $f(x)$ の積分を離散値の重み付き和に近似させて計算します。\n",
"\n",
"$$\n",
"\\int_{a}^{b} f(x) dx \\approx \\Sigma^{k} f(x_k) w_k\n",
"$$\n",
"\n",
"ここで $x_k$ は離散値であり、非適応型の積分法では $x_k$ を一定の間隔でとります。結果的に、最も変化の激しい（＝スパイク）部分にあわせて全体を細かい間隔で評価するために、このような被積分関数に対しては $k$ の最大値が大きくなる場合があります。対して、適応型の積分法では $f(x)$ の変化の激しさによって $x_k$ の間隔を調整するためこの問題は幾分緩和されます。しかし、スパイクに含まれる $x_k$ を「発見」できるか、という困難は常に存在しています。このスパイクを発見できるか、という問題は適応型・非適応型問わずに存在する問題で、ユーザーが結果を見つつ対応しなければなりません。\n",
"\n",
"$$\n",
"\\int_{0}^{1} \\cosh^{-2}{(10x-2)} + \\cosh^{-4}{(100x-40)} + \\cosh^{-6}{(1000x-600)} dx = 0.2108027355005492773756\\dots\n",
"$$\n",
"\n",
"上の積分は `x = 0.4` および `x = 0.6` 付近にスパイクを含みます。適応型の `quadgk` は評価回数を抑えていますが 0.01 程度の誤差があります、どうやら一つのスパイクを見落としているようです。対して、非適応型の `quadde` は9桁を合わせていますが、評価回数は5桁に及んでしまっています。"
]
},
{
"cell_type": "code",
"execution_count": 13,
"metadata": {},
"outputs": [
{
"name": "stdout",
"output_type": "stream",
"text": [
"quadde(f, 0.0, 1.0): Evaluation=12893 I=2.108027351307217e-01 E=3.2400e-10\n",
"quadgk(f, 0.0, 1.0): Evaluation=315 I=2.097360711575962e-01 E=2.3612e-09\n"
]
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" L795.404 641.198 L797.453 650.926 L799.503 660.636 L801.552 670.323 L803.601 679.984 L805.65 689.617 L807.699 699.218 L809.748 708.784 L811.798 718.313 L813.847 727.8 \n",
" L815.896 737.245 L817.945 746.644 L819.994 755.995 L822.043 765.295 L824.092 774.542 L826.142 783.733 L828.191 792.867 L830.24 801.942 L832.289 810.956 L834.338 819.906 \n",
" L836.387 828.792 L838.436 837.61 L840.486 846.361 L842.535 855.042 L844.584 863.651 L846.633 872.188 L848.682 880.652 L850.731 889.04 L852.78 897.353 L854.83 905.588 \n",
" L856.879 913.746 L858.928 921.824 L860.977 929.823 L863.026 937.741 L865.075 945.579 L867.124 953.334 L869.174 961.008 L871.223 968.598 L873.272 976.106 L875.321 983.529 \n",
" L877.37 990.869 L879.419 998.125 L881.469 1005.3 L883.518 1012.38 L885.567 1019.39 L887.616 1026.31 L889.665 1033.14 L891.714 1039.89 L893.763 1046.55 L895.813 1053.13 \n",
" L897.862 1059.63 L899.911 1066.04 L901.96 1072.37 L904.009 1078.61 L906.058 1084.77 L908.107 1090.85 L910.157 1096.85 L912.206 1102.76 L914.255 1108.6 L916.304 1114.35 \n",
" L918.353 1120.02 L920.402 1125.61 L922.451 1131.12 L924.501 1136.55 L926.55 1141.9 L928.599 1147.18 L930.648 1152.38 L932.697 1157.5 L934.746 1162.54 L936.795 1167.51 \n",
" L938.845 1172.41 L940.894 1177.23 L942.943 1181.97 L944.992 1186.65 L947.041 1191.25 L949.09 1195.78 L951.14 1200.24 L953.189 1204.63 L955.238 1208.95 L957.287 1213.2 \n",
" L959.336 1217.39 L961.385 1221.51 L963.434 1225.56 L965.484 1229.54 L967.533 1233.47 L969.582 1237.33 L971.631 1241.12 L973.68 1244.85 L975.729 1248.52 L977.778 1252.13 \n",
" L979.828 1255.68 L981.877 1259.18 L983.926 1262.61 L985.975 1265.98 L988.024 1269.3 L990.073 1272.55 L992.122 1275.75 L994.172 1278.89 L996.221 1281.97 L998.27 1284.99 \n",
" L1000.32 1287.95 L1002.37 1290.83 L1004.42 1293.63 L1006.47 1296.34 L1008.52 1298.93 L1010.56 1301.38 L1012.61 1303.64 L1014.66 1305.65 L1016.71 1307.31 L1018.76 1308.48 \n",
" L1020.81 1308.96 L1022.86 1308.46 L1024.91 1306.57 L1026.96 1302.73 L1029.01 1296.14 L1031.06 1285.75 L1033.11 1270.16 L1035.15 1247.56 L1037.2 1215.77 L1039.25 1172.23 \n",
" L1041.3 1114.24 L1043.35 1039.25 L1045.4 945.519 L1047.45 832.843 L1049.5 703.511 L1051.55 563.092 L1053.6 420.784 L1055.65 288.924 L1057.7 181.434 L1059.74 111.348 \n",
" L1061.79 88.0053 L1063.84 114.806 L1065.89 188.35 L1067.94 299.301 L1069.99 434.625 L1072.04 580.401 L1074.09 724.294 L1076.14 857.107 L1078.19 973.271 L1080.24 1070.5 \n",
" L1082.28 1148.99 L1084.33 1210.51 L1086.38 1257.57 L1088.43 1292.91 L1090.48 1319.06 L1092.53 1338.23 L1094.58 1352.21 L1096.63 1362.4 L1098.68 1369.86 L1100.73 1375.39 \n",
" L1102.78 1379.56 L1104.83 1382.77 L1106.87 1385.31 L1108.92 1387.38 L1110.97 1389.13 L1113.02 1390.66 L1115.07 1392.02 L1117.12 1393.26 L1119.17 1394.43 L1121.22 1395.53 \n",
" L1123.27 1396.58 L1125.32 1397.59 L1127.37 1398.57 L1129.42 1399.52 L1131.46 1400.45 L1133.51 1401.36 L1135.56 1402.24 L1137.61 1403.11 L1139.66 1403.96 L1141.71 1404.8 \n",
" L1143.76 1405.61 L1145.81 1406.42 L1147.86 1407.2 L1149.91 1407.97 L1151.96 1408.73 L1154.01 1409.47 L1156.05 1410.2 L1158.1 1410.91 L1160.15 1411.61 L1162.2 1412.29 \n",
" L1164.25 1412.97 L1166.3 1413.63 L1168.35 1414.27 L1170.4 1414.91 L1172.45 1415.53 L1174.5 1416.14 L1176.55 1416.74 L1178.59 1417.32 L1180.64 1417.9 L1182.69 1418.46 \n",
" L1184.74 1419.02 L1186.79 1419.56 L1188.84 1420.09 L1190.89 1420.61 L1192.94 1421.12 L1194.99 1421.63 L1197.04 1422.12 L1199.09 1422.6 L1201.14 1423.07 L1203.18 1423.54 \n",
" L1205.23 1423.99 L1207.28 1424.44 L1209.33 1424.88 L1211.38 1425.3 L1213.43 1425.72 L1215.48 1426.14 L1217.53 1426.54 L1219.58 1426.94 L1221.63 1427.33 L1223.68 1427.71 \n",
" L1225.73 1428.08 L1227.77 1428.45 L1229.82 1428.81 L1231.87 1429.16 L1233.92 1429.5 L1235.97 1429.84 L1238.02 1430.17 L1240.07 1430.5 L1242.12 1430.82 L1244.17 1431.13 \n",
" L1246.22 1431.44 L1248.27 1431.74 L1250.31 1432.03 L1252.36 1432.32 L1254.41 1432.61 L1256.46 1432.88 L1258.51 1433.16 L1260.56 1433.42 L1262.61 1433.68 L1264.66 1433.94 \n",
" L1266.71 1434.19 L1268.76 1434.44 L1270.81 1434.68 L1272.86 1434.92 L1274.9 1435.15 L1276.95 1435.38 L1279 1435.6 L1281.05 1435.82 L1283.1 1436.04 L1285.15 1436.25 \n",
" L1287.2 1436.45 L1289.25 1436.66 L1291.3 1436.85 L1293.35 1437.05 L1295.4 1437.24 L1297.45 1437.43 L1299.49 1437.61 L1301.54 1437.79 L1303.59 1437.97 L1305.64 1438.14 \n",
" L1307.69 1438.31 L1309.74 1438.47 L1311.79 1438.64 L1313.84 1438.8 L1315.89 1438.95 L1317.94 1439.1 L1319.99 1439.25 L1322.04 1439.4 L1324.08 1439.55 L1326.13 1439.69 \n",
" L1328.18 1439.83 L1330.23 1439.96 L1332.28 1440.1 L1334.33 1440.23 L1336.38 1440.35 L1338.43 1440.48 L1340.48 1440.6 L1342.53 1440.72 L1344.58 1440.84 L1346.62 1440.96 \n",
" L1348.67 1441.07 L1350.72 1441.18 L1352.77 1441.29 L1354.82 1441.4 L1356.87 1441.51 L1358.92 1441.61 L1360.97 1441.71 L1363.02 1441.81 L1365.07 1441.9 L1367.12 1442 \n",
" L1369.17 1442.09 L1371.21 1442.18 L1373.26 1442.27 L1375.31 1442.36 L1377.36 1442.45 L1379.41 1442.53 L1381.46 1442.61 L1383.51 1442.7 L1385.56 1442.78 L1387.61 1442.85 \n",
" L1389.66 1442.93 L1391.71 1443 L1393.76 1443.08 L1395.8 1443.15 L1397.85 1443.22 L1399.9 1443.29 L1401.95 1443.36 L1404 1443.42 L1406.05 1443.49 L1408.1 1443.55 \n",
" L1410.15 1443.61 L1412.2 1443.68 L1414.25 1443.74 L1416.3 1443.79 L1418.34 1443.85 L1420.39 1443.91 L1422.44 1443.96 L1424.49 1444.02 L1426.54 1444.07 L1428.59 1444.12 \n",
" L1430.64 1444.18 L1432.69 1444.23 L1434.74 1444.28 L1436.79 1444.32 L1438.84 1444.37 L1440.89 1444.42 L1442.93 1444.46 L1444.98 1444.51 L1447.03 1444.55 L1449.08 1444.59 \n",
" L1451.13 1444.64 L1453.18 1444.68 L1455.23 1444.72 L1457.28 1444.76 L1459.33 1444.79 L1461.38 1444.83 L1463.43 1444.87 L1465.48 1444.91 L1467.52 1444.49 L1469.57 1350.97 \n",
" L1471.62 175.963 L1473.67 1351.04 L1475.72 1444.63 L1477.77 1445.11 L1479.82 1445.14 L1481.87 1445.17 L1483.92 1445.2 L1485.97 1445.23 L1488.02 1445.26 L1490.07 1445.29 \n",
" L1492.11 1445.32 L1494.16 1445.35 L1496.21 1445.37 L1498.26 1445.4 L1500.31 1445.43 L1502.36 1445.45 L1504.41 1445.48 L1506.46 1445.5 L1508.51 1445.53 L1510.56 1445.55 \n",
" L1512.61 1445.57 L1514.65 1445.6 L1516.7 1445.62 L1518.75 1445.64 L1520.8 1445.66 L1522.85 1445.68 L1524.9 1445.7 L1526.95 1445.72 L1529 1445.74 L1531.05 1445.76 \n",
" L1533.1 1445.78 L1535.15 1445.8 L1537.2 1445.82 L1539.24 1445.83 L1541.29 1445.85 L1543.34 1445.87 L1545.39 1445.88 L1547.44 1445.9 L1549.49 1445.92 L1551.54 1445.93 \n",
" L1553.59 1445.95 L1555.64 1445.96 L1557.69 1445.98 L1559.74 1445.99 L1561.79 1446.01 L1563.83 1446.02 L1565.88 1446.04 L1567.93 1446.05 L1569.98 1446.06 L1572.03 1446.07 \n",
" L1574.08 1446.09 L1576.13 1446.1 L1578.18 1446.11 L1580.23 1446.12 L1582.28 1446.14 L1584.33 1446.15 L1586.38 1446.16 L1588.42 1446.17 L1590.47 1446.18 L1592.52 1446.19 \n",
" L1594.57 1446.2 L1596.62 1446.21 L1598.67 1446.22 L1600.72 1446.23 L1602.77 1446.24 L1604.82 1446.25 L1606.87 1446.26 L1608.92 1446.27 L1610.96 1446.28 L1613.01 1446.29 \n",
" L1615.06 1446.29 L1617.11 1446.3 L1619.16 1446.31 L1621.21 1446.32 L1623.26 1446.33 L1625.31 1446.33 L1627.36 1446.34 L1629.41 1446.35 L1631.46 1446.36 L1633.51 1446.36 \n",
" L1635.55 1446.37 L1637.6 1446.38 L1639.65 1446.38 L1641.7 1446.39 L1643.75 1446.4 L1645.8 1446.4 L1647.85 1446.41 L1649.9 1446.41 L1651.95 1446.42 L1654 1446.43 \n",
" L1656.05 1446.43 L1658.1 1446.44 L1660.14 1446.44 L1662.19 1446.45 L1664.24 1446.45 L1666.29 1446.46 L1668.34 1446.46 L1670.39 1446.47 L1672.44 1446.47 L1674.49 1446.48 \n",
" L1676.54 1446.48 L1678.59 1446.49 L1680.64 1446.49 L1682.68 1446.5 L1684.73 1446.5 L1686.78 1446.5 L1688.83 1446.51 L1690.88 1446.51 L1692.93 1446.52 L1694.98 1446.52 \n",
" L1697.03 1446.52 L1699.08 1446.53 L1701.13 1446.53 L1703.18 1446.54 L1705.23 1446.54 L1707.27 1446.54 L1709.32 1446.55 L1711.37 1446.55 L1713.42 1446.55 L1715.47 1446.56 \n",
" L1717.52 1446.56 L1719.57 1446.56 L1721.62 1446.57 L1723.67 1446.57 L1725.72 1446.57 L1727.77 1446.57 L1729.82 1446.58 L1731.86 1446.58 L1733.91 1446.58 L1735.96 1446.58 \n",
" L1738.01 1446.59 L1740.06 1446.59 L1742.11 1446.59 L1744.16 1446.59 L1746.21 1446.6 L1748.26 1446.6 L1750.31 1446.6 L1752.36 1446.6 L1754.41 1446.61 L1756.45 1446.61 \n",
" L1758.5 1446.61 L1760.55 1446.61 L1762.6 1446.61 L1764.65 1446.62 L1766.7 1446.62 L1768.75 1446.62 L1770.8 1446.62 L1772.85 1446.62 L1774.9 1446.63 L1776.95 1446.63 \n",
" L1778.99 1446.63 L1781.04 1446.63 L1783.09 1446.63 L1785.14 1446.63 L1787.19 1446.64 L1789.24 1446.64 L1791.29 1446.64 L1793.34 1446.64 L1795.39 1446.64 L1797.44 1446.64 \n",
" L1799.49 1446.64 L1801.54 1446.65 L1803.58 1446.65 L1805.63 1446.65 L1807.68 1446.65 L1809.73 1446.65 L1811.78 1446.65 L1813.83 1446.65 L1815.88 1446.65 L1817.93 1446.66 \n",
" L1819.98 1446.66 L1822.03 1446.66 L1824.08 1446.66 L1826.13 1446.66 L1828.17 1446.66 L1830.22 1446.66 L1832.27 1446.66 L1834.32 1446.66 L1836.37 1446.67 L1838.42 1446.67 \n",
" L1840.47 1446.67 L1842.52 1446.67 L1844.57 1446.67 L1846.62 1446.67 L1848.67 1446.67 L1850.71 1446.67 L1852.76 1446.67 L1854.81 1446.67 L1856.86 1446.67 L1858.91 1446.67 \n",
" L1860.96 1446.68 L1863.01 1446.68 L1865.06 1446.68 L1867.11 1446.68 L1869.16 1446.68 L1871.21 1446.68 L1873.26 1446.68 L1875.3 1446.68 L1877.35 1446.68 L1879.4 1446.68 \n",
" L1881.45 1446.68 L1883.5 1446.68 L1885.55 1446.68 L1887.6 1446.68 L1889.65 1446.68 L1891.7 1446.69 L1893.75 1446.69 L1895.8 1446.69 L1897.85 1446.69 L1899.89 1446.69 \n",
" L1901.94 1446.69 L1903.99 1446.69 L1906.04 1446.69 L1908.09 1446.69 L1910.14 1446.69 L1912.19 1446.69 L1914.24 1446.69 L1916.29 1446.69 L1918.34 1446.69 L1920.39 1446.69 \n",
" L1922.44 1446.69 L1924.48 1446.69 L1926.53 1446.69 L1928.58 1446.69 L1930.63 1446.69 L1932.68 1446.69 L1934.73 1446.69 L1936.78 1446.7 L1938.83 1446.7 L1940.88 1446.7 \n",
" L1942.93 1446.7 L1944.98 1446.7 L1947.02 1446.7 L1949.07 1446.7 L1951.12 1446.7 L1953.17 1446.7 L1955.22 1446.7 L1957.27 1446.7 L1959.32 1446.7 L1961.37 1446.7 \n",
" L1963.42 1446.7 L1965.47 1446.7 L1967.52 1446.7 L1969.57 1446.7 L1971.61 1446.7 L1973.66 1446.7 L1975.71 1446.7 L1977.76 1446.7 L1979.81 1446.7 L1981.86 1446.7 \n",
" L1983.91 1446.7 L1985.96 1446.7 L1988.01 1446.7 L1990.06 1446.7 L1992.11 1446.7 L1994.16 1446.7 L1996.2 1446.7 L1998.25 1446.7 L2000.3 1446.7 L2002.35 1446.7 \n",
" L2004.4 1446.7 L2006.45 1446.7 L2008.5 1446.7 L2010.55 1446.7 L2012.6 1446.7 L2014.65 1446.7 L2016.7 1446.71 L2018.74 1446.71 L2020.79 1446.71 L2022.84 1446.71 \n",
" L2024.89 1446.71 L2026.94 1446.71 L2028.99 1446.71 L2031.04 1446.71 L2033.09 1446.71 L2035.14 1446.71 L2037.19 1446.71 L2039.24 1446.71 L2041.29 1446.71 L2043.33 1446.71 \n",
" L2045.38 1446.71 L2047.43 1446.71 L2049.48 1446.71 L2051.53 1446.71 L2053.58 1446.71 L2055.63 1446.71 L2057.68 1446.71 L2059.73 1446.71 L2061.78 1446.71 L2063.83 1446.71 \n",
" L2065.88 1446.71 L2067.92 1446.71 L2069.97 1446.71 L2072.02 1446.71 L2074.07 1446.71 L2076.12 1446.71 L2078.17 1446.71 L2080.22 1446.71 L2082.27 1446.71 L2084.32 1446.71 \n",
" L2086.37 1446.71 L2088.42 1446.71 L2090.47 1446.71 L2092.51 1446.71 L2094.56 1446.71 L2096.61 1446.71 L2098.66 1446.71 L2100.71 1446.71 L2102.76 1446.71 L2104.81 1446.71 \n",
" L2106.86 1446.71 L2108.91 1446.71 L2110.96 1446.71 L2113.01 1446.71 L2115.05 1446.71 L2117.1 1446.71 L2119.15 1446.71 L2121.2 1446.71 L2123.25 1446.71 L2125.3 1446.71 \n",
" L2127.35 1446.71 L2129.4 1446.71 L2131.45 1446.71 L2133.5 1446.71 L2135.55 1446.71 L2137.6 1446.71 L2139.64 1446.71 L2141.69 1446.71 L2143.74 1446.71 L2145.79 1446.71 \n",
" L2147.84 1446.71 L2149.89 1446.71 L2151.94 1446.71 L2153.99 1446.71 L2156.04 1446.71 L2158.09 1446.71 L2160.14 1446.71 L2162.19 1446.71 L2164.23 1446.71 L2166.28 1446.71 \n",
" L2168.33 1446.71 L2170.38 1446.71 L2172.43 1446.71 L2174.48 1446.71 L2176.53 1446.71 L2178.58 1446.71 L2180.63 1446.71 L2182.68 1446.71 L2184.73 1446.71 L2186.78 1446.71 \n",
" L2188.82 1446.71 L2190.87 1446.71 L2192.92 1446.71 L2194.97 1446.71 L2197.02 1446.71 L2199.07 1446.71 L2201.12 1446.71 L2203.17 1446.71 L2205.22 1446.71 L2207.27 1446.71 \n",
" L2209.32 1446.71 L2211.36 1446.71 L2213.41 1446.71 L2215.46 1446.71 L2217.51 1446.71 L2219.56 1446.71 L2221.61 1446.71 L2223.66 1446.71 L2225.71 1446.71 L2227.76 1446.71 \n",
" L2229.81 1446.71 L2231.86 1446.71 L2233.91 1446.71 L2235.95 1446.71 L2238 1446.71 L2240.05 1446.71 L2242.1 1446.71 L2244.15 1446.71 L2246.2 1446.71 L2248.25 1446.71 \n",
" L2250.3 1446.71 L2252.35 1446.71 L2254.4 1446.71 L2256.45 1446.71 L2258.5 1446.71 L2260.54 1446.71 L2262.59 1446.71 L2264.64 1446.71 L2266.69 1446.71 L2268.74 1446.71 \n",
" L2270.79 1446.71 L2272.84 1446.71 L2274.89 1446.71 L2276.94 1446.71 L2278.99 1446.71 L2281.04 1446.71 L2283.08 1446.71 L2285.13 1446.71 L2287.18 1446.71 L2289.23 1446.71 \n",
" L2291.28 1446.71 L2291.28 1446.71 L2289.23 1446.71 L2287.18 1446.71 L2285.13 1446.71 L2283.08 1446.71 L2281.04 1446.71 L2278.99 1446.71 L2276.94 1446.71 L2274.89 1446.71 \n",
" L2272.84 1446.71 L2270.79 1446.71 L2268.74 1446.71 L2266.69 1446.71 L2264.64 1446.71 L2262.59 1446.71 L2260.54 1446.71 L2258.5 1446.71 L2256.45 1446.71 L2254.4 1446.71 \n",
" L2252.35 1446.71 L2250.3 1446.71 L2248.25 1446.71 L2246.2 1446.71 L2244.15 1446.71 L2242.1 1446.71 L2240.05 1446.71 L2238 1446.71 L2235.95 1446.71 L2233.91 1446.71 \n",
" L2231.86 1446.71 L2229.81 1446.71 L2227.76 1446.71 L2225.71 1446.71 L2223.66 1446.71 L2221.61 1446.71 L2219.56 1446.71 L2217.51 1446.71 L2215.46 1446.71 L2213.41 1446.71 \n",
" L2211.36 1446.71 L2209.32 1446.71 L2207.27 1446.71 L2205.22 1446.71 L2203.17 1446.71 L2201.12 1446.71 L2199.07 1446.71 L2197.02 1446.71 L2194.97 1446.71 L2192.92 1446.71 \n",
" L2190.87 1446.71 L2188.82 1446.71 L2186.78 1446.71 L2184.73 1446.71 L2182.68 1446.71 L2180.63 1446.71 L2178.58 1446.71 L2176.53 1446.71 L2174.48 1446.71 L2172.43 1446.71 \n",
" L2170.38 1446.71 L2168.33 1446.71 L2166.28 1446.71 L2164.23 1446.71 L2162.19 1446.71 L2160.14 1446.71 L2158.09 1446.71 L2156.04 1446.71 L2153.99 1446.71 L2151.94 1446.71 \n",
" L2149.89 1446.71 L2147.84 1446.71 L2145.79 1446.71 L2143.74 1446.71 L2141.69 1446.71 L2139.64 1446.71 L2137.6 1446.71 L2135.55 1446.71 L2133.5 1446.71 L2131.45 1446.71 \n",
" L2129.4 1446.71 L2127.35 1446.71 L2125.3 1446.71 L2123.25 1446.71 L2121.2 1446.71 L2119.15 1446.71 L2117.1 1446.71 L2115.05 1446.71 L2113.01 1446.71 L2110.96 1446.71 \n",
" L2108.91 1446.71 L2106.86 1446.71 L2104.81 1446.71 L2102.76 1446.71 L2100.71 1446.71 L2098.66 1446.71 L2096.61 1446.71 L2094.56 1446.71 L2092.51 1446.71 L2090.47 1446.71 \n",
" L2088.42 1446.71 L2086.37 1446.71 L2084.32 1446.71 L2082.27 1446.71 L2080.22 1446.71 L2078.17 1446.71 L2076.12 1446.71 L2074.07 1446.71 L2072.02 1446.71 L2069.97 1446.71 \n",
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" 733.93,360.866 735.979,369.178 738.028,377.608 740.077,386.151 742.127,394.801 744.176,403.554 746.225,412.404 748.274,421.346 750.323,430.375 752.372,439.485 \n",
" 754.421,448.673 756.471,457.932 758.52,467.258 760.569,476.646 762.618,486.091 764.667,495.587 766.716,505.131 768.765,514.718 770.815,524.342 772.864,534 \n",
" 774.913,543.687 776.962,553.397 779.011,563.128 781.06,572.875 783.109,582.633 785.159,592.398 787.208,602.167 789.257,611.935 791.306,621.699 793.355,631.454 \n",
" 795.404,641.198 797.453,650.926 799.503,660.636 801.552,670.323 803.601,679.984 805.65,689.617 807.699,699.218 809.748,708.784 811.798,718.313 813.847,727.8 \n",
" 815.896,737.245 817.945,746.644 819.994,755.995 822.043,765.295 824.092,774.542 826.142,783.733 828.191,792.867 830.24,801.942 832.289,810.956 834.338,819.906 \n",
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" 856.879,913.746 858.928,921.824 860.977,929.823 863.026,937.741 865.075,945.579 867.124,953.334 869.174,961.008 871.223,968.598 873.272,976.106 875.321,983.529 \n",
" 877.37,990.869 879.419,998.125 881.469,1005.3 883.518,1012.38 885.567,1019.39 887.616,1026.31 889.665,1033.14 891.714,1039.89 893.763,1046.55 895.813,1053.13 \n",
" 897.862,1059.63 899.911,1066.04 901.96,1072.37 904.009,1078.61 906.058,1084.77 908.107,1090.85 910.157,1096.85 912.206,1102.76 914.255,1108.6 916.304,1114.35 \n",
" 918.353,1120.02 920.402,1125.61 922.451,1131.12 924.501,1136.55 926.55,1141.9 928.599,1147.18 930.648,1152.38 932.697,1157.5 934.746,1162.54 936.795,1167.51 \n",
" 938.845,1172.41 940.894,1177.23 942.943,1181.97 944.992,1186.65 947.041,1191.25 949.09,1195.78 951.14,1200.24 953.189,1204.63 955.238,1208.95 957.287,1213.2 \n",
" 959.336,1217.39 961.385,1221.51 963.434,1225.56 965.484,1229.54 967.533,1233.47 969.582,1237.33 971.631,1241.12 973.68,1244.85 975.729,1248.52 977.778,1252.13 \n",
" 979.828,1255.68 981.877,1259.18 983.926,1262.61 985.975,1265.98 988.024,1269.3 990.073,1272.55 992.122,1275.75 994.172,1278.89 996.221,1281.97 998.27,1284.99 \n",
" 1000.32,1287.95 1002.37,1290.83 1004.42,1293.63 1006.47,1296.34 1008.52,1298.93 1010.56,1301.38 1012.61,1303.64 1014.66,1305.65 1016.71,1307.31 1018.76,1308.48 \n",
" 1020.81,1308.96 1022.86,1308.46 1024.91,1306.57 1026.96,1302.73 1029.01,1296.14 1031.06,1285.75 1033.11,1270.16 1035.15,1247.56 1037.2,1215.77 1039.25,1172.23 \n",
" 1041.3,1114.24 1043.35,1039.25 1045.4,945.519 1047.45,832.843 1049.5,703.511 1051.55,563.092 1053.6,420.784 1055.65,288.924 1057.7,181.434 1059.74,111.348 \n",
" 1061.79,88.0053 1063.84,114.806 1065.89,188.35 1067.94,299.301 1069.99,434.625 1072.04,580.401 1074.09,724.294 1076.14,857.107 1078.19,973.271 1080.24,1070.5 \n",
" 1082.28,1148.99 1084.33,1210.51 1086.38,1257.57 1088.43,1292.91 1090.48,1319.06 1092.53,1338.23 1094.58,1352.21 1096.63,1362.4 1098.68,1369.86 1100.73,1375.39 \n",
" 1102.78,1379.56 1104.83,1382.77 1106.87,1385.31 1108.92,1387.38 1110.97,1389.13 1113.02,1390.66 1115.07,1392.02 1117.12,1393.26 1119.17,1394.43 1121.22,1395.53 \n",
" 1123.27,1396.58 1125.32,1397.59 1127.37,1398.57 1129.42,1399.52 1131.46,1400.45 1133.51,1401.36 1135.56,1402.24 1137.61,1403.11 1139.66,1403.96 1141.71,1404.8 \n",
" 1143.76,1405.61 1145.81,1406.42 1147.86,1407.2 1149.91,1407.97 1151.96,1408.73 1154.01,1409.47 1156.05,1410.2 1158.1,1410.91 1160.15,1411.61 1162.2,1412.29 \n",
" 1164.25,1412.97 1166.3,1413.63 1168.35,1414.27 1170.4,1414.91 1172.45,1415.53 1174.5,1416.14 1176.55,1416.74 1178.59,1417.32 1180.64,1417.9 1182.69,1418.46 \n",
" 1184.74,1419.02 1186.79,1419.56 1188.84,1420.09 1190.89,1420.61 1192.94,1421.12 1194.99,1421.63 1197.04,1422.12 1199.09,1422.6 1201.14,1423.07 1203.18,1423.54 \n",
" 1205.23,1423.99 1207.28,1424.44 1209.33,1424.88 1211.38,1425.3 1213.43,1425.72 1215.48,1426.14 1217.53,1426.54 1219.58,1426.94 1221.63,1427.33 1223.68,1427.71 \n",
" 1225.73,1428.08 1227.77,1428.45 1229.82,1428.81 1231.87,1429.16 1233.92,1429.5 1235.97,1429.84 1238.02,1430.17 1240.07,1430.5 1242.12,1430.82 1244.17,1431.13 \n",
" 1246.22,1431.44 1248.27,1431.74 1250.31,1432.03 1252.36,1432.32 1254.41,1432.61 1256.46,1432.88 1258.51,1433.16 1260.56,1433.42 1262.61,1433.68 1264.66,1433.94 \n",
" 1266.71,1434.19 1268.76,1434.44 1270.81,1434.68 1272.86,1434.92 1274.9,1435.15 1276.95,1435.38 1279,1435.6 1281.05,1435.82 1283.1,1436.04 1285.15,1436.25 \n",
" 1287.2,1436.45 1289.25,1436.66 1291.3,1436.85 1293.35,1437.05 1295.4,1437.24 1297.45,1437.43 1299.49,1437.61 1301.54,1437.79 1303.59,1437.97 1305.64,1438.14 \n",
" 1307.69,1438.31 1309.74,1438.47 1311.79,1438.64 1313.84,1438.8 1315.89,1438.95 1317.94,1439.1 1319.99,1439.25 1322.04,1439.4 1324.08,1439.55 1326.13,1439.69 \n",
" 1328.18,1439.83 1330.23,1439.96 1332.28,1440.1 1334.33,1440.23 1336.38,1440.35 1338.43,1440.48 1340.48,1440.6 1342.53,1440.72 1344.58,1440.84 1346.62,1440.96 \n",
" 1348.67,1441.07 1350.72,1441.18 1352.77,1441.29 1354.82,1441.4 1356.87,1441.51 1358.92,1441.61 1360.97,1441.71 1363.02,1441.81 1365.07,1441.9 1367.12,1442 \n",
" 1369.17,1442.09 1371.21,1442.18 1373.26,1442.27 1375.31,1442.36 1377.36,1442.45 1379.41,1442.53 1381.46,1442.61 1383.51,1442.7 1385.56,1442.78 1387.61,1442.85 \n",
" 1389.66,1442.93 1391.71,1443 1393.76,1443.08 1395.8,1443.15 1397.85,1443.22 1399.9,1443.29 1401.95,1443.36 1404,1443.42 1406.05,1443.49 1408.1,1443.55 \n",
" 1410.15,1443.61 1412.2,1443.68 1414.25,1443.74 1416.3,1443.79 1418.34,1443.85 1420.39,1443.91 1422.44,1443.96 1424.49,1444.02 1426.54,1444.07 1428.59,1444.12 \n",
" 1430.64,1444.18 1432.69,1444.23 1434.74,1444.28 1436.79,1444.32 1438.84,1444.37 1440.89,1444.42 1442.93,1444.46 1444.98,1444.51 1447.03,1444.55 1449.08,1444.59 \n",
" 1451.13,1444.64 1453.18,1444.68 1455.23,1444.72 1457.28,1444.76 1459.33,1444.79 1461.38,1444.83 1463.43,1444.87 1465.48,1444.91 1467.52,1444.49 1469.57,1350.97 \n",
" 1471.62,175.963 1473.67,1351.04 1475.72,1444.63 1477.77,1445.11 1479.82,1445.14 1481.87,1445.17 1483.92,1445.2 1485.97,1445.23 1488.02,1445.26 1490.07,1445.29 \n",
" 1492.11,1445.32 1494.16,1445.35 1496.21,1445.37 1498.26,1445.4 1500.31,1445.43 1502.36,1445.45 1504.41,1445.48 1506.46,1445.5 1508.51,1445.53 1510.56,1445.55 \n",
" 1512.61,1445.57 1514.65,1445.6 1516.7,1445.62 1518.75,1445.64 1520.8,1445.66 1522.85,1445.68 1524.9,1445.7 1526.95,1445.72 1529,1445.74 1531.05,1445.76 \n",
" 1533.1,1445.78 1535.15,1445.8 1537.2,1445.82 1539.24,1445.83 1541.29,1445.85 1543.34,1445.87 1545.39,1445.88 1547.44,1445.9 1549.49,1445.92 1551.54,1445.93 \n",
" 1553.59,1445.95 1555.64,1445.96 1557.69,1445.98 1559.74,1445.99 1561.79,1446.01 1563.83,1446.02 1565.88,1446.04 1567.93,1446.05 1569.98,1446.06 1572.03,1446.07 \n",
" 1574.08,1446.09 1576.13,1446.1 1578.18,1446.11 1580.23,1446.12 1582.28,1446.14 1584.33,1446.15 1586.38,1446.16 1588.42,1446.17 1590.47,1446.18 1592.52,1446.19 \n",
" 1594.57,1446.2 1596.62,1446.21 1598.67,1446.22 1600.72,1446.23 1602.77,1446.24 1604.82,1446.25 1606.87,1446.26 1608.92,1446.27 1610.96,1446.28 1613.01,1446.29 \n",
" 1615.06,1446.29 1617.11,1446.3 1619.16,1446.31 1621.21,1446.32 1623.26,1446.33 1625.31,1446.33 1627.36,1446.34 1629.41,1446.35 1631.46,1446.36 1633.51,1446.36 \n",
" 1635.55,1446.37 1637.6,1446.38 1639.65,1446.38 1641.7,1446.39 1643.75,1446.4 1645.8,1446.4 1647.85,1446.41 1649.9,1446.41 1651.95,1446.42 1654,1446.43 \n",
" 1656.05,1446.43 1658.1,1446.44 1660.14,1446.44 1662.19,1446.45 1664.24,1446.45 1666.29,1446.46 1668.34,1446.46 1670.39,1446.47 1672.44,1446.47 1674.49,1446.48 \n",
" 1676.54,1446.48 1678.59,1446.49 1680.64,1446.49 1682.68,1446.5 1684.73,1446.5 1686.78,1446.5 1688.83,1446.51 1690.88,1446.51 1692.93,1446.52 1694.98,1446.52 \n",
" 1697.03,1446.52 1699.08,1446.53 1701.13,1446.53 1703.18,1446.54 1705.23,1446.54 1707.27,1446.54 1709.32,1446.55 1711.37,1446.55 1713.42,1446.55 1715.47,1446.56 \n",
" 1717.52,1446.56 1719.57,1446.56 1721.62,1446.57 1723.67,1446.57 1725.72,1446.57 1727.77,1446.57 1729.82,1446.58 1731.86,1446.58 1733.91,1446.58 1735.96,1446.58 \n",
" 1738.01,1446.59 1740.06,1446.59 1742.11,1446.59 1744.16,1446.59 1746.21,1446.6 1748.26,1446.6 1750.31,1446.6 1752.36,1446.6 1754.41,1446.61 1756.45,1446.61 \n",
" 1758.5,1446.61 1760.55,1446.61 1762.6,1446.61 1764.65,1446.62 1766.7,1446.62 1768.75,1446.62 1770.8,1446.62 1772.85,1446.62 1774.9,1446.63 1776.95,1446.63 \n",
" 1778.99,1446.63 1781.04,1446.63 1783.09,1446.63 1785.14,1446.63 1787.19,1446.64 1789.24,1446.64 1791.29,1446.64 1793.34,1446.64 1795.39,1446.64 1797.44,1446.64 \n",
" 1799.49,1446.64 1801.54,1446.65 1803.58,1446.65 1805.63,1446.65 1807.68,1446.65 1809.73,1446.65 1811.78,1446.65 1813.83,1446.65 1815.88,1446.65 1817.93,1446.66 \n",
" 1819.98,1446.66 1822.03,1446.66 1824.08,1446.66 1826.13,1446.66 1828.17,1446.66 1830.22,1446.66 1832.27,1446.66 1834.32,1446.66 1836.37,1446.67 1838.42,1446.67 \n",
" 1840.47,1446.67 1842.52,1446.67 1844.57,1446.67 1846.62,1446.67 1848.67,1446.67 1850.71,1446.67 1852.76,1446.67 1854.81,1446.67 1856.86,1446.67 1858.91,1446.67 \n",
" 1860.96,1446.68 1863.01,1446.68 1865.06,1446.68 1867.11,1446.68 1869.16,1446.68 1871.21,1446.68 1873.26,1446.68 1875.3,1446.68 1877.35,1446.68 1879.4,1446.68 \n",
" 1881.45,1446.68 1883.5,1446.68 1885.55,1446.68 1887.6,1446.68 1889.65,1446.68 1891.7,1446.69 1893.75,1446.69 1895.8,1446.69 1897.85,1446.69 1899.89,1446.69 \n",
" 1901.94,1446.69 1903.99,1446.69 1906.04,1446.69 1908.09,1446.69 1910.14,1446.69 1912.19,1446.69 1914.24,1446.69 1916.29,1446.69 1918.34,1446.69 1920.39,1446.69 \n",
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"metadata": {},
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],
"source": [
"let f(x) = 1/cosh((10x-2))^2+1/cosh((100x-40))^4+1/cosh((1000x-600))^6\n",
" @printeval quadde(f, 0.0, 1.0)\n",
" @printeval quadgk(f, 0.0, 1.0)\n",
" x = 0.0:0.001:1.0\n",
" plot(x, f.(x), label=\"\", xlim=[0.0, 1.0], xticks=0.0:0.2:1.0,\n",
" fillrange=0, fillalpha=0.1, fillcolor=:blue)\n",
"end"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"これらの問題に対する万能の解決策というものはないのですが、例えば積分区間を分割することで改善することがあります。以下の例では積分区間を分割し、`quadgk` では積分値の精度を、`quadde` では評価回数を改善しています。"
]
},
{
"cell_type": "code",
"execution_count": 14,
"metadata": {},
"outputs": [
{
"name": "stdout",
"output_type": "stream",
"text": [
"quadde(f, 0.0, 0.3, 0.5, 0.7, 1.0): Evaluation=6976 I=2.108027355006400e-01 E=1.6566e-12\n",
"quadgk(f, 0.0, 0.3, 0.5, 0.7, 1.0): Evaluation=690 I=2.108027355005494e-01 E=1.9724e-09\n"
]
}
],
"source": [
"let f(x) = 1/cosh((10x-2))^2+1/cosh((100x-40))^4+1/cosh((1000x-600))^6\n",
" @printeval quadde(f, 0.0, 0.3, 0.5, 0.7, 1.0)\n",
" @printeval quadgk(f, 0.0, 0.3, 0.5, 0.7, 1.0)\n",
"end\n",
";"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"# 積分区間の一部で急激に変化する関数(2)\n",
"\n",
"次は $x=0$ 付近で急激に変化する関数です。\n",
"\n",
"$$\n",
"\\int_{0}^{\\infty} \\left( K_0(x) \\right)^2 dx = \\frac{\\pi^2}{4} = 2.46740110027233\\dots\n",
"$$\n",
"\n",
"`quadgk` は評価回数は多いものの、7桁目まで一致、9桁目を四捨五入すると8桁目まで正答しています。対して、`quadde` は評価回数は少ないものの、 4桁まで一致と `x = 0` 付近の急な変化を見落としているようです。"
]
},
{
"cell_type": "code",
"execution_count": 15,
"metadata": {},
"outputs": [
{
"name": "stdout",
"output_type": "stream",
"text": [
"quadde(f, 0.0, Inf): Evaluation=47 I=2.467369673162837e+00 E=1.9604e-08\n",
"quadgk(f, 0.0, Inf): Evaluation=765 I=2.467401095296906e+00 E=2.7633e-08\n"
]
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},
"execution_count": 15,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"let f(x) = besselk(0, x)^2\n",
" @printeval quadde(f, 0.0, Inf)\n",
" @printeval quadgk(f, 0.0, Inf)\n",
" plot(f, label=L\"\\left( K_0(x) \\right)^2\", xlim=[0.0, 2.0],\n",
" fillrange=0, fillalpha=0.1, fillcolor=:blue)\n",
"end"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"これも、やはり積分区間を区切ることで改善します。もとの積分区間 (0 ~ ∞) よりも狭い範囲 (0 ~ 2) を具体的に指定することで急な変化を見つけることができたようです。"
]
},
{
"cell_type": "code",
"execution_count": 16,
"metadata": {},
"outputs": [
{
"name": "stdout",
"output_type": "stream",
"text": [
"quadde(f, 0.0, 2.0, Inf): Evaluation=79 I=2.467401098347747e+00 E=1.7313e-11\n",
"quadgk(f, 0.0, 2.0, Inf): Evaluation=750 I=2.467401093715945e+00 E=3.3688e-08\n"
]
}
],
"source": [
"let f(x) = besselk(0, x)^2\n",
" @printeval quadde(f, 0.0, 2.0, Inf)\n",
" @printeval quadgk(f, 0.0, 2.0, Inf)\n",
"end\n",
";"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"# 積分区間の一部で急激に変化する関数(3)\n",
"\n",
"最後に積分区間に不連続な点を含む関数です。これは数値積分法にとっては悪夢のような問題といえます。\n",
"\n",
"$$\n",
"\\int_{0}^{1} \\lfloor \\min \\left(\\frac{x}{0.3}, 1\\right) \\rfloor dx\n",
"$$\n",
"\n",
"とにかく、不連続に変化している $x = 0.3$ を発見できるかによって精度が大きく左右されます。この一点を見つけられなければまともな精度は得られません。\n",
"\n",
"`quadgk` は数百程度の評価回数で8桁を一致させています。`quadde` は苦戦しており数万回の評価を経てなお4,5桁程度の一致にとどまります。"
]
},
{
"cell_type": "code",
"execution_count": 17,
"metadata": {},
"outputs": [
{
"name": "stdout",
"output_type": "stream",
"text": [
"quadde(f, 0.0, 1.0): Evaluation=25787 I=6.999612521325285e-01 E=2.7833e-07\n",
"quadgk(f, 0.0, 1.0): Evaluation=675 I=7.000000081693309e-01 E=9.2649e-09\n"
]
},
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},
"execution_count": 17,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"let f(x) = floor(min(x/3*10, one(x)))\n",
" @printeval quadde(f, 0.0, 1.0)\n",
" @printeval quadgk(f, 0.0, 1.0)\n",
" plot(f, label=L\"\\lfloor \\min \\left(\\frac{x}{0.3}, 1\\right) \\rfloor\", \n",
" xlim=[0.0, 1.0], xticks=0.0:0.1:1.0,\n",
" fillrange=0, fillalpha=0.1, fillcolor=:blue)\n",
"end"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"$x = 0.3$ で不連続になることが分かっていれば、ここを端点とすることで問題を回避できます。"
]
},
{
"cell_type": "code",
"execution_count": 18,
"metadata": {},
"outputs": [
{
"name": "stdout",
"output_type": "stream",
"text": [
"quadde(f, 0.0, 0.3, 1.0): Evaluation=38 I=7.000000000000256e-01 E=3.1603e-10\n",
"quadgk(f, 0.0, 0.3, 1.0): Evaluation=30 I=7.000000000000000e-01 E=0.0000e+00\n"
]
}
],
"source": [
"let f(x) = floor(min(x/3*10, one(x)))\n",
" @printeval quadde(f, 0.0, 0.3, 1.0)\n",
" @printeval quadgk(f, 0.0, 0.3, 1.0)\n",
"end\n",
";"
]
}
],
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"lastCommId": null,
"lastKernelId": null
},
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"display_name": "Julia 1.4.0",
"language": "julia",
"name": "julia-1.4"
},
"language_info": {
"file_extension": ".jl",
"mimetype": "application/julia",
"name": "julia",
"version": "1.4.0"
}
},
"nbformat": 4,
"nbformat_minor": 2
}
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