Skip to content

Instantly share code, notes, and snippets.

@philzook58

philzook58/SOS Diff Eq.ipynb

Created December 12, 2019 03:53
Show Gist options
  • Star 0 You must be signed in to star a gist
  • Fork 0 You must be signed in to fork a gist
  • Save philzook58/2a42558f6f5bc417c0b68f930f0ef27c to your computer and use it in GitHub Desktop.
Save philzook58/2a42558f6f5bc417c0b68f930f0ef27c to your computer and use it in GitHub Desktop.
Download ZIP
Sum of squares for differential equations
Display the source blob
Display the rendered blob
Raw
SOS Diff Eq.ipynb
{
"cells": [
{
"cell_type": "code",
"execution_count": 1,
"metadata": {},
"outputs": [],
"source": [
"using JuMP\n",
"using SumOfSquares\n",
"using DynamicPolynomials\n",
"using SCS"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"## Minimax optimization of the residual of a differential equation. \n",
"\n",
"\n",
"We can't solve differential equations $Ly = 0$ exactly usually. We need to work in some finite subspace of the full function space $ y(t) = \\sum_i a_i f_i(t)$. A common criteria is to find a solution that is closest to obeying the differential equation in a least squares sense, say $ \\min (Ly)^2 $. This is nice because it leads to linear system of equations. However, a minimax objective $\\min \\max |Ly| $ is also feasible using the sum of squares method. See here for more http://www.philipzucker.com/deriving-the-chebyshev-polynomials-using-sum-of-squares-optimization-with-sympy-and-cvxpy/\n",
"\n",
"\n",
"We can write down the optimization problem using a finite polynomial parametrization of our solution. We relax the condition of being some of squares everywhere to instead just a region of interest by adding a term that makes the inequality stricter in the domain and looser outside the domain. The domain is described by a polynomial expression $t (1 - t) $ which is positive when $ 0 \\leq t \\leq 1$ and negative otherwise. \n",
"Here is an example for \n",
"\n",
"\n",
"$$ \\frac{d^2 y}{dt^2}=-y$$\n",
"$$y(0)=1$$ \n",
"$$y'(0) = 0$$\n",
"with exact solution $ \\cos(t) $\n"
]
},
{
"cell_type": "code",
"execution_count": 73,
"metadata": {},
"outputs": [
{
"name": "stdout",
"output_type": "stream",
"text": [
"----------------------------------------------------------------------------\n",
"\tSCS v2.0.2 - Splitting Conic Solver\n",
"\t(c) Brendan O'Donoghue, Stanford University, 2012-2017\n",
"----------------------------------------------------------------------------\n",
"Lin-sys: sparse-indirect, nnz in A = 194, CG tol ~ 1/iter^(2.00)\n",
"eps = 1.00e-05, alpha = 1.50, max_iters = 5000, normalize = 1, scale = 1.00\n",
"acceleration_lookback = 20, rho_x = 1.00e-03\n",
"Variables n = 78, constraints m = 96\n",
"Cones:\tprimal zero / dual free vars: 24\n",
"\tsd vars: 72, sd blks: 4\n",
"Setup time: 5.36e-05s\n",
"----------------------------------------------------------------------------\n",
" Iter | pri res | dua res | rel gap | pri obj | dua obj | kap/tau | time (s)\n",
"----------------------------------------------------------------------------\n",
" 0| 4.78e+01 2.68e+01 9.76e-01 -7.62e+01 -4.32e-01 0.00e+00 7.52e-05 \n",
" 100| 3.50e-03 1.47e-03 2.78e-05 4.86e-03 4.89e-03 1.90e-16 6.21e-03 \n",
" 200| 9.02e-03 5.15e-03 4.34e-04 6.13e-03 6.57e-03 2.30e-17 1.28e-02 \n",
" 300| 3.15e-02 2.19e-02 3.84e-05 5.68e-03 5.64e-03 6.49e-16 1.92e-02 \n",
" 400| 9.99e-01 9.98e-01 2.19e-03 1.04e-03 -1.15e-03 9.14e-03 2.58e-02 \n",
" 500| 1.02e-02 6.49e-03 7.17e-04 1.03e-02 1.10e-02 8.30e-04 3.22e-02 \n",
" 600| 2.99e-03 1.53e-03 2.39e-04 1.22e-02 1.24e-02 2.99e-16 3.88e-02 \n",
" 700| 3.72e-03 1.42e-03 1.45e-04 1.26e-02 1.28e-02 1.75e-16 4.51e-02 \n",
" 800| 1.95e-03 7.17e-04 1.25e-04 1.30e-02 1.29e-02 7.31e-18 5.12e-02 \n",
" 900| 3.50e-03 1.47e-03 1.05e-04 1.27e-02 1.28e-02 1.04e-17 5.75e-02 \n",
" 1000| 1.88e-03 7.09e-04 3.37e-05 1.32e-02 1.31e-02 8.55e-17 6.39e-02 \n",
" 1100| 1.39e-03 1.30e-03 2.08e-04 1.35e-02 1.33e-02 2.78e-17 7.56e-02 \n",
" 1180| 4.01e-06 1.55e-06 5.25e-07 1.33e-02 1.33e-02 2.82e-16 8.12e-02 \n",
"----------------------------------------------------------------------------\n",
"Status: Solved\n",
"Timing: Solve time: 8.12e-02s\n",
"\tLin-sys: avg # CG iterations: 8.52, avg solve time: 9.38e-06s\n",
"\tCones: avg projection time: 3.61e-05s\n",
"\tAcceleration: avg step time: 2.04e-05s\n",
"----------------------------------------------------------------------------\n",
"Error metrics:\n",
"dist(s, K) = 2.6215e-09, dist(y, K*) = 1.7954e-09, s'y/|s||y| = -2.0203e-10\n",
"primal res: |Ax + s - b|_2 / (1 + |b|_2) = 4.0101e-06\n",
"dual res: |A'y + c|_2 / (1 + |c|_2) = 1.5531e-06\n",
"rel gap: |c'x + b'y| / (1 + |c'x| + |b'y|) = 5.2534e-07\n",
"----------------------------------------------------------------------------\n",
"c'x = 0.0133, -b'y = 0.0133\n",
"============================================================================\n"
]
}
],
"source": [
"@polyvar t\n",
"T = monomials(t, 0:4)\n",
"model = SOSModel(with_optimizer(SCS.Optimizer))\n",
"@variable(model, y, Poly(T))\n",
"@variable(model, α)\n",
"dy = differentiate(y, t)\n",
"ddy = differentiate(dy, t)\n",
"domain = t*(π/2-t)\n",
"@variable(model, λ_1 , SOSPoly(T))\n",
"@variable(model, λ_2 , SOSPoly(T))\n",
"@constraint(model, y(t => 0) == 1)\n",
"@constraint(model, dy(t => 0) == 0)\n",
"@constraint(model, ddy + y - λ_1*domain >= -α)\n",
"@constraint(model, α >= ddy + y + λ_2*domain)\n",
"\n",
"@objective(model, Min, α)\n",
"\n",
"optimize!(model)"
]
},
{
"cell_type": "code",
"execution_count": 74,
"metadata": {},
"outputs": [
{
"data": {
"text/latex": [
"$$ 0.027642031145745472t^{4} + 0.021799794207213376t^{3} - 0.5066442977156951t^{2} + 3.506190174561713e-8t + 1.0000000041335204 $$"
],
"text/plain": [
"0.027642031145745472t⁴ + 0.021799794207213376t³ - 0.5066442977156951t² + 3.506190174561713e-8t + 1.0000000041335204"
]
},
"execution_count": 74,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"value(y)"
]
},
{
"cell_type": "code",
"execution_count": 78,
"metadata": {},
"outputs": [
{
"data": {
"image/svg+xml": [
"<?xml version=\"1.0\" encoding=\"utf-8\"?>\n",
"<svg xmlns=\"http://www.w3.org/2000/svg\" xmlns:xlink=\"http://www.w3.org/1999/xlink\" width=\"600\" height=\"400\" viewBox=\"0 0 2400 1600\">\n",
"<defs>\n",
" <clipPath id=\"clip9800\">\n",
" <rect x=\"0\" y=\"0\" width=\"2400\" height=\"1600\"/>\n",
" </clipPath>\n",
"</defs>\n",
"<path clip-path=\"url(#clip9800)\" d=\"\n",
"M0 1600 L2400 1600 L2400 0 L0 0 Z\n",
" \" fill=\"#ffffff\" fill-rule=\"evenodd\" fill-opacity=\"1\"/>\n",
"<defs>\n",
" <clipPath id=\"clip9801\">\n",
" <rect x=\"480\" y=\"0\" width=\"1681\" height=\"1600\"/>\n",
" </clipPath>\n",
"</defs>\n",
"<path clip-path=\"url(#clip9800)\" d=\"\n",
"M180.66 1487.47 L2352.76 1487.47 L2352.76 47.2441 L180.66 47.2441 Z\n",
" \" fill=\"#ffffff\" fill-rule=\"evenodd\" fill-opacity=\"1\"/>\n",
"<defs>\n",
" <clipPath id=\"clip9802\">\n",
" <rect x=\"180\" y=\"47\" width=\"2173\" height=\"1441\"/>\n",
" </clipPath>\n",
"</defs>\n",
"<polyline clip-path=\"url(#clip9802)\" style=\"stroke:#000000; stroke-width:2; stroke-opacity:0.1; fill:none\" points=\"\n",
" 242.135,1487.47 242.135,47.2441 \n",
" \"/>\n",
"<polyline clip-path=\"url(#clip9802)\" style=\"stroke:#000000; stroke-width:2; stroke-opacity:0.1; fill:none\" points=\"\n",
" 894.729,1487.47 894.729,47.2441 \n",
" \"/>\n",
"<polyline clip-path=\"url(#clip9802)\" style=\"stroke:#000000; stroke-width:2; stroke-opacity:0.1; fill:none\" points=\"\n",
" 1547.32,1487.47 1547.32,47.2441 \n",
" \"/>\n",
"<polyline clip-path=\"url(#clip9802)\" style=\"stroke:#000000; stroke-width:2; stroke-opacity:0.1; fill:none\" points=\"\n",
" 2199.92,1487.47 2199.92,47.2441 \n",
" \"/>\n",
"<polyline clip-path=\"url(#clip9802)\" style=\"stroke:#000000; stroke-width:2; stroke-opacity:0.1; fill:none\" points=\"\n",
" 180.66,1447.8 2352.76,1447.8 \n",
" \"/>\n",
"<polyline clip-path=\"url(#clip9802)\" style=\"stroke:#000000; stroke-width:2; stroke-opacity:0.1; fill:none\" points=\"\n",
" 180.66,1107.85 2352.76,1107.85 \n",
" \"/>\n",
"<polyline clip-path=\"url(#clip9802)\" style=\"stroke:#000000; stroke-width:2; stroke-opacity:0.1; fill:none\" points=\"\n",
" 180.66,767.901 2352.76,767.901 \n",
" \"/>\n",
"<polyline clip-path=\"url(#clip9802)\" style=\"stroke:#000000; stroke-width:2; stroke-opacity:0.1; fill:none\" points=\"\n",
" 180.66,427.953 2352.76,427.953 \n",
" \"/>\n",
"<polyline clip-path=\"url(#clip9802)\" style=\"stroke:#000000; stroke-width:2; stroke-opacity:0.1; fill:none\" points=\"\n",
" 180.66,88.0053 2352.76,88.0053 \n",
" \"/>\n",
"<polyline clip-path=\"url(#clip9800)\" style=\"stroke:#000000; stroke-width:4; stroke-opacity:1; fill:none\" points=\"\n",
" 180.66,1487.47 2352.76,1487.47 \n",
" \"/>\n",
"<polyline clip-path=\"url(#clip9800)\" style=\"stroke:#000000; stroke-width:4; stroke-opacity:1; fill:none\" points=\"\n",
" 180.66,1487.47 180.66,47.2441 \n",
" \"/>\n",
"<polyline clip-path=\"url(#clip9800)\" style=\"stroke:#000000; stroke-width:4; stroke-opacity:1; fill:none\" points=\"\n",
" 242.135,1487.47 242.135,1465.87 \n",
" \"/>\n",
"<polyline clip-path=\"url(#clip9800)\" style=\"stroke:#000000; stroke-width:4; stroke-opacity:1; fill:none\" points=\"\n",
" 894.729,1487.47 894.729,1465.87 \n",
" \"/>\n",
"<polyline clip-path=\"url(#clip9800)\" style=\"stroke:#000000; stroke-width:4; stroke-opacity:1; fill:none\" points=\"\n",
" 1547.32,1487.47 1547.32,1465.87 \n",
" \"/>\n",
"<polyline clip-path=\"url(#clip9800)\" style=\"stroke:#000000; stroke-width:4; stroke-opacity:1; fill:none\" points=\"\n",
" 2199.92,1487.47 2199.92,1465.87 \n",
" \"/>\n",
"<polyline clip-path=\"url(#clip9800)\" style=\"stroke:#000000; stroke-width:4; stroke-opacity:1; fill:none\" points=\"\n",
" 180.66,1447.8 213.242,1447.8 \n",
" \"/>\n",
"<polyline clip-path=\"url(#clip9800)\" style=\"stroke:#000000; stroke-width:4; stroke-opacity:1; fill:none\" points=\"\n",
" 180.66,1107.85 213.242,1107.85 \n",
" \"/>\n",
"<polyline clip-path=\"url(#clip9800)\" style=\"stroke:#000000; stroke-width:4; stroke-opacity:1; fill:none\" points=\"\n",
" 180.66,767.901 213.242,767.901 \n",
" \"/>\n",
"<polyline clip-path=\"url(#clip9800)\" style=\"stroke:#000000; stroke-width:4; stroke-opacity:1; fill:none\" points=\"\n",
" 180.66,427.953 213.242,427.953 \n",
" \"/>\n",
"<polyline clip-path=\"url(#clip9800)\" style=\"stroke:#000000; stroke-width:4; stroke-opacity:1; fill:none\" points=\"\n",
" 180.66,88.0053 213.242,88.0053 \n",
" \"/>\n",
"<g clip-path=\"url(#clip9800)\">\n",
"<text style=\"fill:#000000; fill-opacity:1; font-family:Arial,Helvetica Neue,Helvetica,sans-serif; font-size:48px; text-anchor:middle;\" transform=\"rotate(0, 242.135, 1541.47)\" x=\"242.135\" y=\"1541.47\">0.0</text>\n",
"</g>\n",
"<g clip-path=\"url(#clip9800)\">\n",
"<text style=\"fill:#000000; fill-opacity:1; font-family:Arial,Helvetica Neue,Helvetica,sans-serif; font-size:48px; text-anchor:middle;\" transform=\"rotate(0, 894.729, 1541.47)\" x=\"894.729\" y=\"1541.47\">0.5</text>\n",
"</g>\n",
"<g clip-path=\"url(#clip9800)\">\n",
"<text style=\"fill:#000000; fill-opacity:1; font-family:Arial,Helvetica Neue,Helvetica,sans-serif; font-size:48px; text-anchor:middle;\" transform=\"rotate(0, 1547.32, 1541.47)\" x=\"1547.32\" y=\"1541.47\">1.0</text>\n",
"</g>\n",
"<g clip-path=\"url(#clip9800)\">\n",
"<text style=\"fill:#000000; fill-opacity:1; font-family:Arial,Helvetica Neue,Helvetica,sans-serif; font-size:48px; text-anchor:middle;\" transform=\"rotate(0, 2199.92, 1541.47)\" x=\"2199.92\" y=\"1541.47\">1.5</text>\n",
"</g>\n",
"<g clip-path=\"url(#clip9800)\">\n",
"<text style=\"fill:#000000; fill-opacity:1; font-family:Arial,Helvetica Neue,Helvetica,sans-serif; font-size:48px; text-anchor:end;\" transform=\"rotate(0, 156.66, 1465.3)\" x=\"156.66\" y=\"1465.3\">0.00</text>\n",
"</g>\n",
"<g clip-path=\"url(#clip9800)\">\n",
"<text style=\"fill:#000000; fill-opacity:1; font-family:Arial,Helvetica Neue,Helvetica,sans-serif; font-size:48px; text-anchor:end;\" transform=\"rotate(0, 156.66, 1125.35)\" x=\"156.66\" y=\"1125.35\">0.25</text>\n",
"</g>\n",
"<g clip-path=\"url(#clip9800)\">\n",
"<text style=\"fill:#000000; fill-opacity:1; font-family:Arial,Helvetica Neue,Helvetica,sans-serif; font-size:48px; text-anchor:end;\" transform=\"rotate(0, 156.66, 785.401)\" x=\"156.66\" y=\"785.401\">0.50</text>\n",
"</g>\n",
"<g clip-path=\"url(#clip9800)\">\n",
"<text style=\"fill:#000000; fill-opacity:1; font-family:Arial,Helvetica Neue,Helvetica,sans-serif; font-size:48px; text-anchor:end;\" transform=\"rotate(0, 156.66, 445.453)\" x=\"156.66\" y=\"445.453\">0.75</text>\n",
"</g>\n",
"<g clip-path=\"url(#clip9800)\">\n",
"<text style=\"fill:#000000; fill-opacity:1; font-family:Arial,Helvetica Neue,Helvetica,sans-serif; font-size:48px; text-anchor:end;\" transform=\"rotate(0, 156.66, 105.505)\" x=\"156.66\" y=\"105.505\">1.00</text>\n",
"</g>\n",
"<polyline clip-path=\"url(#clip9802)\" style=\"stroke:#009af9; stroke-width:4; stroke-opacity:1; fill:none\" points=\"\n",
" 242.135,88.0053 255.187,88.0733 268.238,88.2773 281.29,88.6172 294.342,89.093 307.394,89.7047 320.446,90.4522 333.498,91.3355 346.55,92.3544 359.602,93.5088 \n",
" 372.654,94.7986 385.706,96.2238 398.757,97.7841 411.809,99.4794 424.861,101.31 437.913,103.274 450.965,105.374 464.017,107.607 477.069,109.975 490.121,112.476 \n",
" 503.173,115.111 516.224,117.879 529.276,120.78 542.328,123.814 555.38,126.98 568.432,130.278 581.484,133.708 594.536,137.269 607.588,140.962 620.64,144.785 \n",
" 633.691,148.738 646.743,152.822 659.795,157.035 672.847,161.376 685.899,165.847 698.951,170.446 712.003,175.172 725.055,180.026 738.107,185.007 751.158,190.113 \n",
" 764.21,195.346 777.262,200.704 790.314,206.186 803.366,211.793 816.418,217.523 829.47,223.376 842.522,229.352 855.574,235.45 868.625,241.669 881.677,248.008 \n",
" 894.729,254.467 907.781,261.046 920.833,267.744 933.885,274.559 946.937,281.492 959.989,288.541 973.041,295.706 986.092,302.987 999.144,310.382 1012.2,317.891 \n",
" 1025.25,325.512 1038.3,333.246 1051.35,341.092 1064.4,349.048 1077.46,357.114 1090.51,365.289 1103.56,373.572 1116.61,381.963 1129.66,390.46 1142.72,399.063 \n",
" 1155.77,407.771 1168.82,416.583 1181.87,425.498 1194.92,434.515 1207.97,443.633 1221.03,452.852 1234.08,462.171 1247.13,471.588 1260.18,481.103 1273.23,490.714 \n",
" 1286.29,500.421 1299.34,510.223 1312.39,520.118 1325.44,530.106 1338.49,540.186 1351.55,550.357 1364.6,560.618 1377.65,570.967 1390.7,581.404 1403.75,591.928 \n",
" 1416.8,602.537 1429.86,613.23 1442.91,624.008 1455.96,634.867 1469.01,645.808 1482.06,656.829 1495.12,667.929 1508.17,679.107 1521.22,690.362 1534.27,701.693 \n",
" 1547.32,713.098 1560.38,724.577 1573.43,736.128 1586.48,747.75 1599.53,759.443 1612.58,771.204 1625.64,783.032 1638.69,794.928 1651.74,806.888 1664.79,818.913 \n",
" 1677.84,831 1690.89,843.15 1703.95,855.359 1717,867.628 1730.05,879.955 1743.1,892.339 1756.15,904.778 1769.21,917.272 1782.26,929.818 1795.31,942.417 \n",
" 1808.36,955.066 1821.41,967.764 1834.47,980.51 1847.52,993.303 1860.57,1006.14 1873.62,1019.02 1886.67,1031.95 1899.72,1044.92 1912.78,1057.92 1925.83,1070.97 \n",
" 1938.88,1084.05 1951.93,1097.17 1964.98,1110.33 1978.04,1123.52 1991.09,1136.74 2004.14,1149.99 2017.19,1163.28 2030.24,1176.59 2043.3,1189.92 2056.35,1203.29 \n",
" 2069.4,1216.68 2082.45,1230.09 2095.5,1243.52 2108.56,1256.97 2121.61,1270.45 2134.66,1283.94 2147.71,1297.44 2160.76,1310.97 2173.81,1324.5 2186.87,1338.05 \n",
" 2199.92,1351.61 2212.97,1365.18 2226.02,1378.75 2239.07,1392.34 2252.13,1405.93 2265.18,1419.52 2278.23,1433.12 2291.28,1446.71 \n",
" \"/>\n",
"<polyline clip-path=\"url(#clip9802)\" style=\"stroke:#e26f46; stroke-width:4; stroke-opacity:1; fill:none\" points=\"\n",
" 242.135,88.0053 255.187,88.0742 268.238,88.2807 281.29,88.6245 294.342,89.1056 307.394,89.7237 320.446,90.4786 333.498,91.37 346.55,92.3978 359.602,93.5616 \n",
" 372.654,94.8612 385.706,96.2964 398.757,97.8669 411.809,99.5724 424.861,101.413 437.913,103.387 450.965,105.496 464.017,107.738 477.069,110.114 490.121,112.623 \n",
" 503.173,115.265 516.224,118.04 529.276,120.946 542.328,123.984 555.38,127.153 568.432,130.453 581.484,133.884 594.536,137.445 607.588,141.136 620.64,144.956 \n",
" 633.691,148.904 646.743,152.981 659.795,157.186 672.847,161.519 685.899,165.978 698.951,170.564 712.003,175.276 725.055,180.114 738.107,185.077 751.158,190.164 \n",
" 764.21,195.375 777.262,200.709 790.314,206.167 803.366,211.747 816.418,217.448 829.47,223.271 842.522,229.215 855.574,235.278 868.625,241.461 881.677,247.763 \n",
" 894.729,254.183 907.781,260.721 920.833,267.376 933.885,274.147 946.937,281.034 959.989,288.035 973.041,295.152 986.092,302.381 999.144,309.724 1012.2,317.179 \n",
" 1025.25,324.746 1038.3,332.424 1051.35,340.211 1064.4,348.108 1077.46,356.114 1090.51,364.228 1103.56,372.449 1116.61,380.776 1129.66,389.209 1142.72,397.747 \n",
" 1155.77,406.389 1168.82,415.134 1181.87,423.981 1194.92,432.93 1207.97,441.98 1221.03,451.13 1234.08,460.379 1247.13,469.726 1260.18,479.17 1273.23,488.711 \n",
" 1286.29,498.348 1299.34,508.079 1312.39,517.904 1325.44,527.821 1338.49,537.831 1351.55,547.932 1364.6,558.123 1377.65,568.403 1390.7,578.771 1403.75,589.226 \n",
" 1416.8,599.768 1429.86,610.395 1442.91,621.106 1455.96,631.9 1469.01,642.777 1482.06,653.734 1495.12,664.772 1508.17,675.889 1521.22,687.085 1534.27,698.357 \n",
" 1547.32,709.705 1560.38,721.128 1573.43,732.625 1586.48,744.195 1599.53,755.836 1612.58,767.548 1625.64,779.329 1638.69,791.178 1651.74,803.095 1664.79,815.077 \n",
" 1677.84,827.124 1690.89,839.235 1703.95,851.409 1717,863.643 1730.05,875.938 1743.1,888.292 1756.15,900.703 1769.21,913.171 1782.26,925.694 1795.31,938.271 \n",
" 1808.36,950.9 1821.41,963.582 1834.47,976.313 1847.52,989.094 1860.57,1001.92 1873.62,1014.8 1886.67,1027.72 1899.72,1040.68 1912.78,1053.68 1925.83,1066.73 \n",
" 1938.88,1079.82 1951.93,1092.94 1964.98,1106.11 1978.04,1119.3 1991.09,1132.54 2004.14,1145.8 2017.19,1159.1 2030.24,1172.42 2043.3,1185.78 2056.35,1199.16 \n",
" 2069.4,1212.57 2082.45,1226.01 2095.5,1239.46 2108.56,1252.94 2121.61,1266.44 2134.66,1279.96 2147.71,1293.49 2160.76,1307.04 2173.81,1320.6 2186.87,1334.18 \n",
" 2199.92,1347.77 2212.97,1361.36 2226.02,1374.97 2239.07,1388.58 2252.13,1402.2 2265.18,1415.82 2278.23,1429.44 2291.28,1443.06 \n",
" \"/>\n",
"<path clip-path=\"url(#clip9800)\" d=\"\n",
"M1989.93 312.204 L2280.76 312.204 L2280.76 130.764 L1989.93 130.764 Z\n",
" \" fill=\"#ffffff\" fill-rule=\"evenodd\" fill-opacity=\"1\"/>\n",
"<polyline clip-path=\"url(#clip9800)\" style=\"stroke:#000000; stroke-width:4; stroke-opacity:1; fill:none\" points=\"\n",
" 1989.93,312.204 2280.76,312.204 2280.76,130.764 1989.93,130.764 1989.93,312.204 \n",
" \"/>\n",
"<polyline clip-path=\"url(#clip9800)\" style=\"stroke:#009af9; stroke-width:4; stroke-opacity:1; fill:none\" points=\"\n",
" 2013.93,191.244 2157.93,191.244 \n",
" \"/>\n",
"<g clip-path=\"url(#clip9800)\">\n",
"<text style=\"fill:#000000; fill-opacity:1; font-family:Arial,Helvetica Neue,Helvetica,sans-serif; font-size:48px; text-anchor:start;\" transform=\"rotate(0, 2181.93, 208.744)\" x=\"2181.93\" y=\"208.744\">y1</text>\n",
"</g>\n",
"<polyline clip-path=\"url(#clip9800)\" style=\"stroke:#e26f46; stroke-width:4; stroke-opacity:1; fill:none\" points=\"\n",
" 2013.93,251.724 2157.93,251.724 \n",
" \"/>\n",
"<g clip-path=\"url(#clip9800)\">\n",
"<text style=\"fill:#000000; fill-opacity:1; font-family:Arial,Helvetica Neue,Helvetica,sans-serif; font-size:48px; text-anchor:start;\" transform=\"rotate(0, 2181.93, 269.224)\" x=\"2181.93\" y=\"269.224\">y2</text>\n",
"</g>\n",
"</svg>\n"
]
},
"execution_count": 78,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"using Plots\n",
"xs = 0:0.01:π/2; exact_y = cos.(xs); approx_y = map(x -> value(y)(t => x), xs)# These are the plotting data\n",
"plot(xs,[exact_y,approx_y])\n"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"Nice"
]
}
],
"metadata": {
"kernelspec": {
"display_name": "Julia 1.2.0",
"language": "julia",
"name": "julia-1.2"
},
"language_info": {
"file_extension": ".jl",
"mimetype": "application/julia",
"name": "julia",
"version": "1.2.0"
}
},
"nbformat": 4,
"nbformat_minor": 2
}
Sign up for free to join this conversation on GitHub. Already have an account? Sign in to comment