1-whyjulia.ipynb

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   "source": [
    "<img style=\"float:left\" src=\"figs/julialogo-smaller.png\"/>\n",
    "\n",
    "<br>\n",
    "   * innovative open-source scientific programming language\n",
    "   * **easy as Matlab, fast as Fortran, flexible as Python, deep as LISP**\n",
    "   * leverages major C/Fortran science packages, e.g. LAPACK, MPI, FFTW... \n",
    "   * 5th language to run at 1 petaflops, after assembly, Fortran, C, C++ \n",
    "   * solves the \"two-language problem\"\n",
    "   * the future of scientific computing!\n",
    "   \n",
    "-- talk by John Gibson, Mathematics & Statistics, University of New Hampshire\n",
    "\n",
    "<br> "
   ]
  },
  {
   "cell_type": "markdown",
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   "source": [
    "# Background\n",
    "\n",
    "\n",
    "  * origins at MIT, UCSB applied math & computer science, 2010 \n",
    "    \n",
    "  * founders Viral Shah, Jeff Bezanson, Stefan Karpinsky (now at Julia Computing LLC), Alan Edelman (MIT)\n",
    "        \n",
    "  * ~10-person core language team, ~600 contributors, ~1000 registered packages \n",
    "    \n",
    "  * support from Intel, Microsoft, Wall St., Moore Foundation, NumFocus\n",
    "   \n",
    "  * julia-0.1 released 2012, julia-0.6 2017, julia-1.0 target August 2018\n",
    "  <br> <br> <br> <br> <br>"
   ]
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  {
   "cell_type": "code",
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   "source": [
    "using ODE, Plots, GenericSVD     # load packages"
   ]
  },
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   "source": [
    "# Julia: easy as Matlab\n",
    "\n",
    "Julia is a drop-in replacement for Matlab for interactive calculations with built-in numerical libraries and plotting. A few examples...\n",
    "\n",
    "### Linear algebra "
   ]
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     "data": {
      "text/plain": [
       "4×4 Array{Float64,2}:\n",
       "  2.04331    1.91726   -0.612682  -0.707393 \n",
       " -0.98854    0.253066   0.583844  -0.669205 \n",
       "  0.257819  -0.785724   0.224265   1.49101  \n",
       "  0.881065  -1.05563   -0.399439  -0.0234009"
      ]
     },
     "execution_count": 2,
     "metadata": {},
     "output_type": "execute_result"
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   ],
   "source": [
    "A = randn(4,4)        # 4 x 4 matrix of normally distributed random numbers"
   ]
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  {
   "cell_type": "code",
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   "outputs": [
    {
     "data": {
      "text/plain": [
       "4-element Array{Float64,1}:\n",
       "  2.04331 \n",
       " -0.98854 \n",
       "  0.257819\n",
       "  0.881065"
      ]
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     "execution_count": 3,
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   "source": [
    "A[:,1]                # familiar colon syntax --extract first column of A"
   ]
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   "source": [
    "b = randn(4)          \n",
    "x = A\\b               # solve Ax=b for x using backslash operator\n",
    "norm(A*x-b)"
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   "cell_type": "code",
   "execution_count": 5,
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   "outputs": [],
   "source": [
    "U, σ, V = svd(A);     # singular value decomposition --note unicode variable name σ"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 6,
   "metadata": {},
   "outputs": [
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       "2.1199870740062957e-15"
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       "2.1199870740062957e-15"
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   ],
   "source": [
    "Σ = diagm(σ)\n",
    "norm(A - U*Σ*V')      # compute error of SVD factorization A = U Σ V'"
   ]
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   "execution_count": 7,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/html": [
       "lu(A::<b>AbstractArray{T,2} where T</b>) at <a href=\"https://github.com/JuliaLang/julia/tree/9d11f62bcb124327831206089967f93020e84200/base/linalg/lu.jl#L190\" target=\"_blank\">linalg/lu.jl:190</a>"
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      "text/plain": [
       "lu(A::AbstractArray{T,2} where T) in Base.LinAlg at linalg/lu.jl:190"
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   "source": [
    "@which lu(A)"
   ]
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   "cell_type": "markdown",
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   "source": [
    "### Numerical integration of differential equation"
   ]
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       "  232.359,190.193 228.39,186.916 227.82,186.391 224.635,183.156 221.205,178.94 218.347,174.549 218.21,174.309 215.763,169.317 213.971,164.068 213.112,160.014 \n",
       "  212.931,158.675 212.725,153.275 213.392,148.026 213.727,146.65 214.958,143.089 217.396,138.616 219.436,136.004 220.649,134.742 224.629,131.575 228.911,129.305 \n",
       "  229.216,129.178 234.278,127.588 239.675,126.786 240.06,126.759 245.264,126.734 250.917,127.357 251.533,127.463 256.519,128.569 261.974,130.274 262.752,130.557 \n",
       "  267.204,132.372 272.155,134.775 273.334,135.408 276.784,137.397 281.072,140.169 282.894,141.455 285.005,143.031 288.584,145.934 290.852,147.944 291.818,148.844 \n",
       "  294.719,151.73 296.912,154.122 297.305,154.574 299.598,157.359 301.617,160.078 302.933,162.018 303.386,162.724 304.927,165.294 306.261,167.788 306.502,168.27 \n",
       "  307.408,170.204 308.388,172.545 308.954,174.059 309.216,174.812 309.91,177.009 310.397,178.788 310.484,179.136 310.951,181.197 311.222,182.616 311.322,183.196 \n",
       "  311.606,185.133 311.682,185.754 311.813,187.013 311.943,188.741 311.949,188.837 312.021,190.609 312.034,192.329 312.033,192.385 311.991,194.002 311.896,195.629 \n",
       "  311.864,196.034 311.751,197.211 311.556,198.752 311.414,199.669 311.313,200.252 311.021,201.713 310.679,203.138 310.635,203.302 310.285,204.527 309.837,205.883 \n",
       "  309.409,207.014 309.331,207.205 308.764,208.497 308.131,209.757 307.478,210.903 307.427,210.988 306.645,212.19 305.777,213.363 304.817,214.508 304.233,215.137 \n",
       "  303.754,215.623 302.579,216.71 301.28,217.764 299.845,218.786 298.875,219.404 298.259,219.771 296.51,220.717 294.577,221.616 292.445,222.462 291.566,222.772 \n",
       "  290.094,223.247 287.504,223.958 284.652,224.578 281.516,225.089 281.364,225.11 278.073,225.472 274.301,225.692 270.182,225.711 266.38,225.539 265.694,225.49 \n",
       "  260.82,224.98 255.562,224.113 249.93,222.812 245.451,221.501 243.934,221.001 237.605,218.589 231.026,215.478 224.473,211.684 224.288,211.566 217.503,206.756 \n",
       "  210.865,200.984 205.279,195.04 204.581,194.207 198.892,186.416 194.086,177.709 193.463,176.342 190.452,168.226 188.26,158.238 187.833,154.042 187.748,148.076 \n",
       "  189.064,138.152 190.182,134.145 192.261,128.908 197.27,120.764 199.098,118.594 203.91,114.078 211.9,109.077 212.49,108.799 220.9,105.871 227.663,104.664 \n",
       "  230.545,104.409 240.483,104.538 242.993,104.793 250.404,106.015 257.646,107.827 260.061,108.558 269.266,111.883 270.668,112.461 277.894,115.725 284.334,119.021 \n",
       "  285.877,119.863 293.182,124.115 299.806,128.346 299.818,128.354 305.803,132.477 311.183,136.434 315.73,139.967 316.004,140.185 320.318,143.708 324.18,147.002 \n",
       "  325.798,148.422 327.644,150.069 330.762,152.915 333.471,155.445 333.583,155.551 336.154,157.988 338.516,160.236 340.708,162.304 340.847,162.434 342.768,164.203 \n",
       "  344.725,165.937 345.848,166.888 346.611,167.512 348.452,168.934 350.272,170.203 350.596,170.414 352.091,171.322 353.929,172.29 355.59,173.021 355.802,173.105 \n",
       "  357.725,173.765 359.71,174.265 361.273,174.538 361.767,174.602 363.905,174.77 366.128,174.762 368.101,174.611 368.44,174.572 370.841,174.192 373.328,173.616 \n",
       "  375.893,172.837 376.517,172.62 378.53,171.849 381.221,170.648 383.946,169.236 386.681,167.612 386.72,167.587 389.402,165.778 392.069,163.748 394.642,161.542 \n",
       "  397.078,159.18 397.309,158.94 399.332,156.686 401.351,154.101 403.086,151.473 404.138,149.568 404.487,148.849 405.505,146.284 406.108,143.836 406.268,141.889 \n",
       "  406.262,141.566 405.943,139.531 405.153,137.779 404.218,136.641 403.897,136.353 402.191,135.289 400.077,134.6 398.52,134.358 397.598,134.293 394.807,134.363 \n",
       "  391.768,134.787 390.557,135.037 388.543,135.539 385.199,136.582 381.928,137.82 381.796,137.874 378.393,139.374 375.041,141.04 373.369,141.939 371.784,142.832 \n",
       "  368.661,144.715 365.698,146.659 365.265,146.957 362.923,148.635 360.348,150.622 357.985,152.605 357.976,152.613 355.842,154.566 353.919,156.499 352.215,158.394 \n",
       "  351.96,158.696 350.728,160.245 349.451,162.048 348.38,163.801 347.79,164.911 347.506,165.5 346.823,167.143 346.324,168.729 346.002,170.256 345.869,171.417 \n",
       "  345.851,171.721 345.865,173.121 346.041,174.455 346.374,175.717 346.863,176.905 347.108,177.372 347.503,178.015 348.296,179.04 349.241,179.976 350.338,180.815 \n",
       "  350.478,180.908 351.591,181.552 352.999,182.177 354.568,182.681 356.189,183.036 356.298,183.055 358.194,183.288 360.258,183.368 362.49,183.281 364.894,183.013 \n",
       "  365.211,182.966 367.469,182.551 370.213,181.879 373.116,180.981 376.168,179.843 378.529,178.832 379.361,178.449 382.679,176.783 386.089,174.84 389.553,172.617 \n",
       "  393.035,170.112 393.578,169.695 396.494,167.322 399.864,164.271 403.074,160.995 406.052,157.529 406.607,156.825 408.722,153.916 410.999,150.231 412.805,146.557 \n",
       "  413.536,144.666 414.061,142.979 414.703,139.592 414.69,136.493 414.548,135.613 413.985,133.777 412.588,131.52 410.741,129.92 410.524,129.784 407.829,128.614 \n",
       "  404.579,128.019 403.045,127.939 400.853,127.993 396.751,128.503 393.287,129.257 392.373,129.498 387.823,130.914 383.198,132.678 383.001,132.76 378.587,134.718 \n",
       "  374.064,136.965 372.843,137.611 369.697,139.352 365.532,141.826 363.185,143.309 361.608,144.342 357.952,146.862 354.576,149.361 354.364,149.524 351.492,151.818 \n",
       "  348.696,154.222 346.804,155.968 346.185,156.565 343.95,158.841 341.978,161.051 340.945,162.308 340.257,163.194 338.772,165.272 337.508,167.287 336.687,168.774 \n",
       "  336.45,169.241 335.584,171.136 334.898,172.974 334.38,174.756 334.154,175.753 334.018,176.484 333.802,178.156 333.726,179.774 333.746,180.762 333.781,181.336 \n",
       "  333.964,182.843 334.269,184.291 334.494,185.077 334.696,185.679 335.241,187.005 335.907,188.266 336.317,188.92 336.693,189.458 337.603,190.577 338.641,191.618 \n",
       "  339.438,192.291 339.812,192.576 341.121,193.445 342.577,194.216 344.188,194.882 344.494,194.989 345.963,195.433 347.912,195.858 350.045,196.142 352.374,196.272 \n",
       "  353.143,196.278 354.911,196.232 357.666,196.004 360.649,195.564 363.867,194.889 366.613,194.165 367.328,193.955 371.041,192.734 374.997,191.197 379.182,189.313 \n",
       "  383.584,187.052 384.883,186.332 388.188,184.385 392.948,181.288 397.798,177.752 402.674,173.77 403.047,173.445 407.497,169.338 412.148,164.501 416.495,159.323 \n",
       "  418.714,156.346 420.404,153.871 423.722,148.26 426.298,142.648 427.346,139.615 427.985,137.196 428.66,132.091 428.251,127.549 428.244,127.52 426.682,123.663 \n",
       "  423.998,120.656 423.224,120.095 420.247,118.607 415.548,117.55 413.598,117.41 410.054,117.475 403.946,118.31 401.568,118.827 397.417,119.942 390.655,122.231 \n",
       "  388.911,122.902 383.835,125.024 377.104,128.172 376.424,128.509 370.587,131.538 364.579,134.89 364.373,135.009 358.531,138.491 353.889,141.41 353.099,141.92 \n",
       "  348.098,145.249 344.952,147.437 343.531,148.452 339.391,151.514 335.658,154.435 334.626,155.274 332.31,157.217 329.319,159.869 326.653,162.403 326.21,162.842 \n",
       "  324.282,164.829 322.176,167.159 320.304,169.404 319.951,169.849 318.638,171.573 317.15,173.674 316.19,175.125 315.815,175.715 314.609,177.701 313.511,179.636 \n",
       "  313.289,180.042 312.5,181.524 311.559,183.368 311.277,183.935 310.67,185.171 309.817,186.933 309.702,187.172 308.987,188.657 308.197,190.28 308.166,190.343 \n",
       "  307.341,191.991 306.68,193.263 306.5,193.602 305.632,195.176 304.726,196.713 304.541,197.014 303.772,198.213 302.757,199.675 301.948,200.749 301.672,201.098 \n",
       "  300.505,202.48 299.244,203.821 298.582,204.469 297.879,205.117 296.396,206.366 294.782,207.563 294.018,208.078 293.024,208.703 291.107,209.782 289.016,210.789 \n",
       "  287.628,211.373 286.736,211.717 284.251,212.553 281.544,213.281 278.599,213.884 278.471,213.906 275.4,214.343 271.934,214.63 268.189,214.71 265.02,214.606 \n",
       "  264.157,214.552 259.827,214.115 255.212,213.347 250.328,212.19 245.96,210.854 245.193,210.59 239.838,208.48 234.336,205.797 228.771,202.479 226.741,201.098 \n",
       "  223.232,198.464 217.854,193.716 212.805,188.237 209.959,184.538 208.253,182.03 204.394,175.157 201.422,167.761 200.339,163.957 199.529,159.998 198.87,152.099 \n",
       "  199.323,145.731 199.552,144.338 201.633,137.004 204.773,130.841 205.074,130.38 209.792,124.725 215.187,120.483 215.612,120.218 222.327,116.981 228.21,115.313 \n",
       "  229.692,115.032 237.461,114.324 241.907,114.425 245.399,114.734 253.299,116.103 255.389,116.609 260.992,118.248 268.088,120.877 268.35,120.985 275.28,124.143 \n",
       "  279.215,126.181 281.727,127.573 287.662,131.15 290.847,133.241 293.081,134.78 297.992,138.388 302.423,141.928 303.498,142.83 306.402,145.358 309.966,148.661 \n",
       "  313.153,151.826 313.376,152.055 316.004,154.845 318.557,157.719 320.518,160.047 320.85,160.45 322.92,163.043 324.8,165.503 324.965,165.723 326.523,167.834 \n",
       "  328.117,170.043 328.629,170.757 329.612,172.132 331.031,174.107 331.69,175.013 332.399,175.971 333.736,177.726 334.561,178.763 335.064,179.374 336.401,180.915 \n",
       "  337.439,182.021 337.764,182.349 339.17,183.676 340.496,184.784 340.635,184.892 342.174,185.996 343.801,186.982 344.916,187.559 345.531,187.845 347.377,188.578 \n",
       "  349.353,189.173 350.509,189.439 351.472,189.619 353.745,189.907 356.184,190.022 357.983,189.992 358.8,189.95 361.603,189.674 364.599,189.175 367.79,188.433 \n",
       "  368.249,188.31 371.184,187.428 374.773,186.137 378.544,184.538 382.485,182.61 382.495,182.605 386.579,180.329 390.783,177.684 395.043,174.672 399.306,171.289 \n",
       "  399.573,171.064 403.51,167.536 407.557,163.453 411.343,159.094 414.1,155.465 414.763,154.515 417.7,149.8 420.037,145.069 421.511,140.983 421.661,140.45 \n",
       "  422.469,136.079 422.401,132.1 422.08,130.514 421.405,128.653 419.48,125.853 417.26,124.114 416.666,123.785 413.027,122.508 408.678,122.022 408.249,122.016 \n",
       "  403.744,122.304 398.372,123.282 397.116,123.59 392.71,124.868 386.903,126.953 385.458,127.533 381.081,129.424 375.356,132.171 373.967,132.882 369.822,135.096 \n",
       "  364.546,138.116 363.048,139.014 359.583,141.161 354.963,144.183 353.09,145.464 350.705,147.145 346.813,150.025 344.666,151.696 343.283,152.808 340.104,155.491 \n",
       "  337.846,157.523 337.259,158.072 334.726,160.556 332.484,162.948 331.203,164.42 330.509,165.256 328.778,167.486 327.267,169.644 325.962,171.725 325.955,171.737 \n",
       "  324.821,173.77 323.845,175.749 323.217,177.175 323.01,177.676 322.299,179.555 321.699,181.388 321.667,181.493 321.194,183.178 320.774,184.925 320.709,185.229 \n",
       "  320.43,186.633 320.15,188.3 320.116,188.528 319.929,189.928 319.769,191.412 319.759,191.518 319.634,193.07 319.592,193.759 319.55,194.584 319.503,196.062 \n",
       "  319.499,196.268 319.489,197.502 319.506,198.906 319.507,198.979 319.552,200.274 319.625,201.605 319.636,201.771 319.725,202.901 319.851,204.161 319.901,204.589 \n",
       "  320.004,205.385 320.183,206.574 320.333,207.427 320.39,207.728 320.627,208.847 320.894,209.93 320.998,210.312 321.194,210.978 321.53,211.99 321.905,212.967 \n",
       "  322.035,213.274 322.322,213.907 322.785,214.809 323.3,215.674 323.75,216.335 323.871,216.499 324.504,217.284 325.206,218.027 325.985,218.725 326.502,219.128 \n",
       "  326.849,219.375 327.806,219.976 328.869,220.522 330.047,221.009 330.28,221.093 331.354,221.432 332.804,221.784 334.414,222.055 335.479,222.177 336.199,222.237 \n",
       "  338.179,222.32 340.375,222.286 342.794,222.121 342.809,222.12 345.506,221.805 348.491,221.313 351.791,220.617 353.294,220.252 355.433,219.687 359.447,218.482 \n",
       "  363.857,216.955 368.686,215.055 368.773,215.019 373.96,212.727 379.682,209.902 385.844,206.507 389.892,204.079 392.439,202.47 399.428,197.715 406.719,192.182 \n",
       "  411.574,188.141 414.213,185.814 421.76,178.584 429.128,170.536 431.913,167.175 436.074,161.73 442.28,152.335 447.396,142.626 448.091,141.059 451.066,132.914 \n",
       "  452.969,123.638 453.171,120.183 452.83,115.264 450.506,108.244 449.409,106.507 445.978,102.975 439.401,99.7018 438.928,99.5672 431.049,98.5236 424.157,98.9144 \n",
       "  421.32,99.3294 410.654,101.871 407.882,102.74 399.498,105.782 391.666,109.091 388.251,110.652 377.244,116.08 376.426,116.505 366.725,121.719 363.701,123.403 \n",
       "  356.856,127.297 347.954,132.495 347.732,132.626 339.394,137.583 331.835,142.119 330.909,142.676 325.029,146.212 318.925,149.883 313.464,153.165 313.031,153.425 \n",
       "  308.586,156.096 304.227,158.72 302.334,159.863 300.323,161.08 296.817,163.212 294.265,164.772 293.653,165.147 290.779,166.916 288.151,168.535 285.894,169.921 \n",
       "  285.726,170.024 283.464,171.396 281.333,172.658 280.182,173.321 279.303,173.816 277.347,174.875 275.444,175.833 274.984,176.052 273.572,176.691 271.715,177.447 \n",
       "  269.862,178.095 269.748,178.131 267.998,178.631 266.118,179.051 264.217,179.347 264.095,179.361 262.29,179.512 260.34,179.54 258.371,179.425 257.76,179.359 \n",
       "  256.386,179.158 254.395,178.735 252.413,178.15 250.734,177.52 250.452,177.401 248.529,176.483 246.666,175.4 244.886,174.156 243.747,173.229 243.214,172.755 \n",
       "  241.674,171.208 240.294,169.532 239.101,167.747 238.66,166.966 238.12,165.872 237.375,163.937 236.884,161.974 236.673,160.21 236.664,160.015 236.728,158.096 \n",
       "  237.079,156.254 237.712,154.524 237.912,154.118 238.626,152.94 239.802,151.532 241.218,150.324 242.159,149.712 242.851,149.337 244.672,148.587 246.645,148.078 \n",
       "  248.365,147.841 248.737,147.812 250.911,147.788 253.133,147.993 255.08,148.347 255.371,148.414 257.59,149.036 259.765,149.838 261.615,150.675 261.87,150.801 \n",
       "  263.881,151.904 265.782,153.126 267.559,154.447 267.578,154.462 269.197,155.847 270.69,157.31 272.03,158.82 272.544,159.462 273.212,160.362 274.235,161.921 \n",
       "  275.097,163.488 275.799,165.052 275.943,165.425 276.341,166.603 276.725,168.132 276.952,169.632 277.026,171.097 276.991,172.057 276.949,172.52 276.722,173.894 \n",
       "  276.348,175.213 275.829,176.472 275.167,177.665 274.995,177.93 274.364,178.786 273.422,179.828 272.342,180.784 271.128,181.647 270.076,182.257 269.78,182.409 \n",
       "  268.302,183.063 266.697,183.598 264.972,184.005 263.131,184.273 262.467,184.332 261.181,184.393 259.129,184.352 256.992,184.14 254.784,183.747 252.736,183.224 \n",
       "  252.52,183.16 250.216,182.37 247.901,181.369 245.605,180.155 243.357,178.725 242.34,177.987 241.19,177.076 239.14,175.215 237.249,173.16 235.558,170.93 \n",
       "  234.784,169.728 234.108,168.545 232.937,166.039 232.079,163.455 231.594,161.046 231.566,160.837 231.425,158.234 231.667,155.7 232.296,153.289 232.332,153.187 \n",
       "  233.313,151.053 234.698,149.04 236.425,147.288 236.619,147.126 238.464,145.837 240.771,144.706 243.299,143.906 243.488,143.861 245.998,143.446 248.818,143.314 \n",
       "  251.298,143.453 251.707,143.498 254.618,143.977 257.507,144.723 259.133,145.258 260.334,145.707 263.066,146.9 265.675,148.269 266.572,148.795 268.137,149.785 \n",
       "  270.435,151.419 272.557,153.146 273.244,153.758 274.491,154.942 276.234,156.787 277.783,158.663 278.692,159.9 279.138,160.554 280.3,162.447 281.273,164.332 \n",
       "  282.06,166.198 282.298,166.861 282.666,168.037 283.095,169.842 283.352,171.608 283.441,173.33 283.398,174.614 283.366,175.001 283.131,176.619 282.739,178.178 \n",
       "  282.191,179.674 281.652,180.805 281.49,181.102 280.637,182.458 279.632,183.737 278.475,184.931 277.426,185.833 277.166,186.035 275.705,187.042 274.091,187.943 \n",
       "  272.325,188.727 270.738,189.283 270.407,189.384 268.337,189.904 266.12,190.271 263.76,190.472 261.633,190.5 261.264,190.491 258.638,190.311 255.898,189.915 \n",
       "  253.064,189.285 250.513,188.525 250.154,188.404 247.192,187.252 244.216,185.818 241.267,184.094 238.67,182.287 238.387,182.071 235.625,179.744 233.039,177.128 \n",
       "  230.692,174.246 228.988,171.702 228.643,171.123 226.956,167.794 225.685,164.322 224.883,160.772 224.864,160.648 224.596,157.216 224.846,153.736 225.535,150.758 \n",
       "  225.646,150.418 226.997,147.345 228.869,144.591 230.351,143 231.226,142.221 234.015,140.288 237.168,138.82 238.224,138.458 240.614,137.836 244.273,137.331 \n",
       "  247.345,137.26 248.068,137.286 251.925,137.672 255.778,138.445 256.637,138.668 259.563,139.562 263.231,140.971 265.63,142.069 266.741,142.625 270.058,144.474 \n",
       "  273.161,146.478 273.982,147.056 276.032,148.596 278.661,150.795 281.047,153.047 281.314,153.318 283.186,155.327 285.085,157.616 286.749,159.9 287.112,160.441 \n",
       "  288.187,162.163 289.407,164.398 290.419,166.597 290.991,168.06 291.233,168.753 291.859,170.862 292.305,172.921 292.579,174.929 292.639,175.666 292.689,176.881 \n",
       "  292.64,178.778 292.438,180.617 292.086,182.397 292.055,182.522 291.588,184.117 290.944,185.774 290.154,187.368 289.71,188.131 289.219,188.894 288.135,190.351 \n",
       "  286.901,191.733 285.633,192.932 285.512,193.037 283.965,194.257 282.253,195.385 280.371,196.412 279.692,196.737 278.314,197.33 276.077,198.126 273.653,198.785 \n",
       "  271.677,199.186 271.04,199.292 268.23,199.629 265.229,199.774 262.041,199.7 258.673,199.382 256.456,199.044 255.13,198.794 251.427,197.902 247.602,196.677 \n",
       "  243.691,195.086 239.962,193.227 239.732,193.101 235.775,190.688 231.897,187.839 228.183,184.55 225.296,181.499 224.716,180.82 221.593,176.666 218.92,172.149 \n",
       "  216.8,167.337 216.52,166.551 215.332,162.313 214.596,157.2 214.597,152.784 214.652,152.133 215.544,147.255 217.266,142.709 218.052,141.24 219.796,138.637 \n",
       "  223.069,135.153 225.879,133.028 226.994,132.342 231.462,130.266 236.238,128.955 236.344,128.934 241.513,128.338 246.84,128.427 247.355,128.469 252.206,129.136 \n",
       "  257.51,130.383 258.425,130.65 262.663,132.079 267.599,134.138 269.032,134.814 272.265,136.474 276.626,139.015 278.824,140.434 280.662,141.696 284.363,144.462 \n",
       "  287.371,146.959 287.728,147.272 290.765,150.089 293.485,152.888 294.107,153.571 295.906,155.651 298.044,158.362 298.613,159.136 299.921,161.014 301.555,163.6 \n",
       "  302.281,164.856 302.967,166.116 304.176,168.562 304.411,169.078 305.2,170.937 305.98,173.019 306.057,173.241 306.762,175.476 307.112,176.763 307.328,177.645 \n",
       "  307.769,179.749 307.871,180.321 308.096,181.79 308.312,183.702 308.318,183.771 308.443,185.695 308.476,186.919 308.478,187.563 308.429,189.378 308.3,191.142 \n",
       "  308.281,191.329 308.093,192.857 307.811,194.525 307.593,195.561 307.455,196.148 307.023,197.728 306.516,199.265 306.345,199.726 305.93,200.761 305.262,202.217 \n",
       "  304.507,203.634 304.341,203.921 303.662,205.013 302.718,206.352 301.669,207.653 301.137,208.251 300.507,208.914 299.222,210.133 297.802,211.309 296.236,212.437 \n",
       "  295.593,212.858 294.512,213.515 292.616,214.537 290.53,215.494 288.24,216.376 286.301,217.005 285.727,217.173 282.975,217.873 279.965,218.454 276.678,218.892 \n",
       "  273.597,219.137 273.099,219.163 269.206,219.238 264.994,219.073 260.457,218.618 255.593,217.821 255.171,217.738 250.401,216.629 244.912,214.966 239.176,212.757 \n",
       "  234.575,210.613 233.242,209.926 227.184,206.391 221.136,202.092 215.593,197.309 215.241,196.972 209.661,190.989 204.615,184.183 200.327,176.62 200.238,176.435 \n",
       "  197.03,168.405 194.939,159.766 194.381,155.027 194.248,150.961 195.09,142.308 196.509,136.856 197.513,134.16 201.491,126.855 204.799,122.752 206.891,120.686 \n",
       "  213.517,115.872 217.3,113.971 221.106,112.526 229.369,110.659 231.485,110.417 238.016,110.186 245.859,110.812 246.781,110.944 255.432,112.729 259.686,113.945 \n",
       "  263.789,115.316 271.72,118.489 272.251,118.725 279.137,122.049 284.05,124.714 285.992,125.834 292.266,129.709 297.885,133.523 297.968,133.581 303.115,137.372 \n",
       "  307.745,141.041 311.454,144.172 311.896,144.557 315.613,147.901 318.941,151.07 320.565,152.679 321.926,154.062 324.612,156.881 327.04,159.533 327.606,160.163 \n",
       "  329.251,162.024 331.281,164.362 332.408,165.671 333.166,166.553 334.936,168.604 336.459,170.337 336.62,170.518 338.247,172.3 339.839,173.953 340.18,174.292 \n",
       "  341.421,175.477 343.012,176.875 343.864,177.562 344.633,178.144 346.3,179.284 347.752,180.14 348.031,180.29 349.84,181.159 351.742,181.885 352.045,181.984 \n",
       "  353.75,182.461 355.875,182.879 358.127,183.129 358.541,183.155 360.517,183.2 363.05,183.081 365.731,182.758 366.878,182.562 368.564,182.217 371.547,181.442 \n",
       "  374.674,180.421 377.935,179.137 377.952,179.13 381.321,177.575 384.805,175.728 388.353,173.589 391.93,171.156 392.207,170.956 395.498,168.425 398.994,165.416 \n",
       "  402.348,162.161 405.489,158.693 406.068,157.999 408.342,155.051 410.82,151.308 412.84,147.545 413.76,145.409 414.319,143.848 415.186,140.314 415.392,137.045 \n",
       "  415.292,135.924 414.894,134.139 413.683,131.683 411.87,129.814 411.776,129.744 409.205,128.379 406.04,127.603 404.393,127.445 402.361,127.419 398.266,127.798 \n",
       "  394.639,128.501 393.859,128.689 389.249,130.032 384.535,131.752 384.248,131.867 379.814,133.772 375.166,136.02 373.938,136.654 370.662,138.425 366.355,140.931 \n",
       "  364.098,142.325 362.287,143.486 358.487,146.051 355.061,148.532 354.972,148.599 351.754,151.104 348.83,153.557 347.24,154.977 346.198,155.946 343.849,158.267 \n",
       "  341.771,160.519 341.091,161.307 339.949,162.702 338.37,164.819 337.017,166.871 336.561,167.631 335.875,168.86 334.93,170.791 334.166,172.663 333.913,173.384 \n",
       "  333.572,174.48 333.136,176.243 332.847,177.951 332.769,178.632 332.696,179.606 332.677,181.207 332.782,182.754 332.834,183.184 333.007,184.245 333.35,185.679 \n",
       "  333.807,187.054 333.892,187.27 334.379,188.368 335.066,189.617 335.871,190.798 335.904,190.841 336.796,191.907 337.846,192.94 338.92,193.813 339.026,193.89 \n",
       "  340.343,194.753 341.806,195.519 343.423,196.181 344.306,196.474 345.203,196.729 347.158,197.154 349.298,197.439 351.636,197.57 353.656,197.553 354.184,197.531 \n",
       "  356.955,197.307 359.957,196.871 363.201,196.199 366.694,195.267 367.344,195.071 370.446,194.047 374.454,192.506 378.704,190.614 383.182,188.339 385.77,186.894 \n",
       "  387.872,185.65 392.738,182.516 397.713,178.928 402.725,174.878 404.121,173.664 407.7,170.359 412.518,165.407 417.04,160.089 419.971,156.164 421.127,154.475 \n",
       "  424.62,148.676 427.358,142.856 428.694,139.028 429.186,137.186 429.969,131.859 429.622,127.075 429.54,126.707 428.082,123.026 425.37,119.86 424.362,119.119 \n",
       "  421.54,117.691 416.716,116.562 414.534,116.4 411.059,116.461 404.758,117.314 402.287,117.852 398.017,119.001 391.034,121.373 389.417,121.997 383.993,124.268 \n",
       "  377.049,127.526 376.731,127.684 370.331,131.003 364.716,134.128 363.933,134.579 357.923,138.157 353.932,140.65 352.341,141.671 347.206,145.072 344.35,147.041 \n",
       "  342.521,148.334 338.277,151.444 334.451,154.403 333.548,155.126 331.02,157.214 327.953,159.887 325.217,162.437 324.883,162.76 322.781,164.874 320.613,167.211 \n",
       "  318.679,169.46 318.487,169.694 316.951,171.631 315.4,173.732 314.604,174.874 313.999,175.771 312.724,177.754 312.109,178.756 311.552,179.685 310.463,181.569 \n",
       "  309.968,182.448 309.435,183.407 308.453,185.202 308.065,185.914 307.498,186.956 306.557,188.67 306.246,189.228 305.614,190.343 304.656,191.977 304.404,192.394 \n",
       "  303.67,193.572 302.644,195.126 302.447,195.412 301.565,196.64 300.422,198.111 299.446,199.263 299.201,199.539 297.892,200.921 296.482,202.254 295.63,202.989 \n",
       "  294.957,203.536 293.306,204.761 291.513,205.924 290.476,206.528 289.566,207.019 287.449,208.037 285.148,208.968 283.289,209.603 282.649,209.799 279.934,210.517 \n",
       "  276.992,211.1 273.809,211.527 273.008,211.604 270.37,211.776 266.667,211.814 262.7,211.603 258.466,211.103 257.966,211.025 253.966,210.271 249.221,209.055 \n",
       "  244.271,207.397 239.153,205.241 238.814,205.081 233.914,202.527 228.65,199.206 223.471,195.241 220.887,192.948 218.49,190.594 213.857,185.261 209.742,179.299 \n",
       "  206.978,174.197 206.319,172.768 203.766,165.777 202.234,158.516 201.933,155.475 201.866,151.188 202.744,144.041 204.026,139.534 204.896,137.335 208.296,131.321 \n",
       "  211.744,127.248 212.837,126.21 218.375,122.169 223.284,119.803 224.709,119.279 231.625,117.568 236.322,117.061 238.902,116.977 246.328,117.405 249.538,117.87 \n",
       "  253.72,118.709 260.926,120.729 262.311,121.196 267.826,123.301 274.116,126.166 274.337,126.275 280.405,129.516 283.968,131.634 286.002,132.914 291.118,136.378 \n",
       "  295.382,139.546 295.765,139.844 299.958,143.258 303.726,146.59 306.321,149.05 307.1,149.818 310.114,152.925 312.805,155.908 314.249,157.601 315.208,158.764 \n",
       "  317.357,161.492 319.285,164.098 320.363,165.623 321.024,166.583 322.604,168.954 323.51,170.359 324.052,171.213 325.392,173.366 326.186,174.658 326.65,175.415 \n",
       "  327.845,177.365 328.58,178.55 328.999,179.219 330.13,180.978 330.852,182.058 331.255,182.644 332.39,184.217 333.142,185.192 333.552,185.699 334.753,187.088 \n",
       "  335.574,187.952 336.01,188.383 337.334,189.582 338.741,190.681 339.233,191.026 340.243,191.676 341.853,192.562 343.586,193.331 343.874,193.442 345.455,193.977 \n",
       "  347.473,194.488 349.653,194.854 350.117,194.91 352.011,195.062 354.558,195.097 357.307,194.94 358.781,194.782 360.27,194.573 363.457,193.974 366.874,193.118 \n",
       "  370.524,191.979 371.106,191.777 374.412,190.529 378.528,188.74 382.849,186.584 387.355,184.036 388.269,183.481 392.02,181.066 396.786,177.662 401.575,173.828 \n",
       "  406.056,169.808 406.309,169.567 410.897,164.892 415.205,159.873 419.097,154.593 420.599,152.278 422.433,149.141 425.065,143.658 426.856,138.312 427.193,136.841 \n",
       "  427.677,133.274 427.446,128.726 426.666,126.009 426.115,124.84 423.679,121.762 420.543,119.738 420.201,119.587 415.773,118.379 410.546,118.12 410.292,118.131 \n",
       "  404.686,118.764 398.377,120.203 398.158,120.264 391.804,122.31 385.642,124.726 385.137,124.941 378.529,127.949 373.405,130.518 372.1,131.202 365.949,134.583 \n",
       "  361.891,136.95 360.144,138.001 354.732,141.384 351.638,143.406 349.734,144.684 345.162,147.871 342.706,149.657 341.007,150.927 337.258,153.848 333.89,156.636 \n",
       "  332.958,157.441 330.879,159.298 328.197,161.842 325.813,164.28 325.346,164.781 323.698,166.622 321.822,168.879 320.158,171.06 319.612,171.818 318.678,173.173 \n",
       "  317.359,175.225 316.176,177.221 315.797,177.895 315.109,179.166 314.137,181.065 313.461,182.458 313.243,182.919 312.409,184.732 311.846,185.996 311.622,186.505 \n",
       "  310.866,188.24 310.463,189.17 310.13,189.937 309.401,191.598 308.941,192.623 308.667,193.223 307.919,194.812 307.232,196.198 307.146,196.366 306.337,197.885 \n",
       "  305.484,199.369 305.196,199.844 304.576,200.818 303.604,202.231 302.622,203.526 302.557,203.607 301.425,204.946 300.196,206.245 299.2,207.196 298.859,207.503 \n",
       "  297.402,208.717 295.812,209.883 294.475,210.754 294.074,210.997 292.175,212.053 290.099,213.043 287.829,213.957 287.776,213.976 285.348,214.786 282.64,215.513 \n",
       "  279.686,216.12 278.089,216.375 276.47,216.587 272.975,216.891 269.188,216.994 265.1,216.86 263.726,216.761 260.698,216.451 255.984,215.715 250.978,214.59 \n",
       "  245.7,213.013 243.86,212.367 240.174,210.919 234.462,208.234 228.659,204.891 224.396,201.982 222.857,200.824 217.186,195.975 211.823,190.341 207.411,184.602 \n",
       "  206.954,183.93 202.782,176.78 199.522,169.038 197.862,163.217 197.382,160.875 196.545,152.524 197.108,144.39 197.125,144.285 199.2,136.471 202.734,129.395 \n",
       "  202.944,129.072 207.64,123.344 213.731,118.519 213.791,118.481 220.785,115.057 227.215,113.228 228.535,112.979 236.712,112.232 241.276,112.339 245.061,112.68 \n",
       "  253.361,114.146 255.073,114.562 261.428,116.427 268.001,118.857 269.13,119.323 276.369,122.646 279.085,124.039 283.089,126.234 289.265,129.958 291.603,131.476 \n",
       "  294.893,133.715 299.989,137.43 304.581,141.055 304.994,141.394 308.702,144.55 312.393,147.899 315.288,150.686 315.697,151.091 318.655,154.121 321.309,156.993 \n",
       "  322.706,158.563 323.701,159.708 325.867,162.273 327.382,164.121 327.845,164.694 329.668,166.977 331.274,169.01 331.366,169.127 332.97,171.149 334.504,173.047 \n",
       "  334.7,173.286 335.994,174.825 337.462,176.483 337.925,176.984 338.929,178.025 340.414,179.449 341.173,180.12 341.935,180.754 343.51,181.939 344.635,182.684 \n",
       "  345.155,183 346.883,183.934 348.711,184.733 349.675,185.084 350.65,185.392 352.715,185.901 354.915,186.251 356.039,186.36 357.264,186.43 359.769,186.427 \n",
       "  362.439,186.225 364.497,185.944 365.279,185.81 368.294,185.164 371.481,184.27 374.835,183.11 375.964,182.67 378.351,181.663 382.012,179.914 385.788,177.851 \n",
       "  389.647,175.462 391.135,174.462 393.556,172.738 397.46,169.68 401.286,166.315 404.958,162.667 406.473,161.016 408.399,158.765 411.511,154.672 414.195,150.473 \n",
       "  416.148,146.706 416.35,146.257 417.88,142.127 418.709,138.203 418.838,135.723 418.77,134.606 418.023,131.451 416.467,128.835 416.08,128.406 414.117,126.844 \n",
       "  411.034,125.521 408.833,125.051 407.3,124.885 403.017,124.924 398.799,125.494 398.308,125.587 393.297,126.813 388.108,128.513 387.85,128.607 382.858,130.597 \n",
       "  377.649,132.973 376.879,133.348 372.57,135.554 367.689,138.266 366.321,139.066 363.062,141.042 358.725,143.834 356.523,145.328 354.702,146.604 351.006,149.323 \n",
       "  347.887,151.773 347.637,151.976 344.594,154.551 341.863,157.044 340.942,157.932 339.431,159.455 337.281,161.785 335.91,163.399 335.394,164.038 333.751,166.219 \n",
       "  332.333,168.332 331.895,169.043 331.122,170.381 330.099,172.37 329.655,173.339 329.248,174.304 328.554,176.183 328.15,177.482 328.003,178.012 327.582,179.792 \n",
       "  327.282,181.523 327.274,181.578 327.092,183.207 327.005,184.843 326.998,185.576 327.015,186.433 327.115,187.976 327.297,189.43 327.303,189.472 327.576,190.918 \n",
       "  327.933,192.315 328.181,193.111 328.373,193.662 328.897,194.956 329.507,196.195 329.742,196.616 330.206,197.378 330.999,198.501 331.891,199.562 332.243,199.933 \n",
       "  332.887,200.556 333.996,201.48 335.225,202.328 336.295,202.944 336.585,203.094 338.085,203.772 339.738,204.352 341.558,204.824 343.557,205.177 343.567,205.178 \n",
       "  345.751,205.401 348.156,205.477 350.788,205.386 353.665,205.107 355.125,204.902 356.804,204.619 360.221,203.896 363.928,202.902 367.937,201.603 371.146,200.407 \n",
       "  372.258,199.963 376.898,197.94 381.84,195.492 387.06,192.575 391.63,189.739 392.535,189.146 398.222,185.165 404.031,180.615 409.861,175.49 411.001,174.413 \n",
       "  415.602,169.786 421.085,163.565 426.122,156.926 427.766,154.487 430.515,149.977 434.047,142.911 436.51,135.96 436.732,135.112 437.704,129.374 437.499,123.429 \n",
       "  437.046,121.437 435.804,118.392 432.624,114.478 430.846,113.198 428.037,111.845 422.202,110.556 419.748,110.416 415.34,110.587 407.709,111.822 406.216,112.174 \n",
       "  399.581,114.085 392.123,116.794 391.219,117.157 382.856,120.808 378.334,122.979 374.683,124.82 366.85,129.006 365.458,129.781 359.463,133.213 354.557,136.132 \n",
       "  352.585,137.332 346.252,141.286 341.842,144.133 340.468,145.035 335.226,148.557 330.497,151.856 328.424,153.344 326.249,154.937 322.443,157.818 319.037,160.52 \n",
       "  318.296,161.126 315.987,163.063 313.252,165.467 310.942,167.606 310.791,167.75 308.563,169.928 306.533,172.013 306.25,172.312 304.667,174.015 302.932,175.944 \n",
       "  302.516,176.414 301.299,177.806 299.741,179.604 299.206,180.222 298.233,181.342 296.751,183.022 296.082,183.765 295.276,184.643 293.786,186.204 292.944,187.046 \n",
       "  292.263,187.704 290.691,189.14 289.62,190.049 289.052,190.508 287.332,191.804 285.517,193.02 284.631,193.56 283.592,194.151 281.545,195.187 279.364,196.118 \n",
       "  278.375,196.486 277.038,196.933 274.558,197.618 271.916,198.156 270.09,198.421 269.106,198.53 266.123,198.721 262.97,198.703 259.651,198.453 258.884,198.362 \n",
       "  256.17,197.942 252.545,197.143 248.805,196.024 244.979,194.556 244.001,194.126 241.098,192.707 237.219,190.454 233.413,187.785 229.75,184.691 228.573,183.562 \n",
       "  226.307,181.163 223.181,177.227 220.469,172.934 218.575,169.071 218.271,168.339 216.682,163.517 215.778,158.586 215.59,155.332 215.628,153.665 216.275,148.89 \n",
       "  217.722,144.398 218.082,143.6 219.96,140.328 222.93,136.79 225.101,134.911 226.557,133.88 230.739,131.66 235.001,130.241 235.359,130.153 240.295,129.361 \n",
       "  245.423,129.242 245.882,129.263 250.626,129.745 255.803,130.793 256.813,131.057 260.861,132.304 265.733,134.197 267.351,134.919 270.36,136.388 274.704,138.804 \n",
       "  277.149,140.325 278.74,141.381 282.453,144.062 285.813,146.779 285.841,146.803 288.906,149.566 291.659,152.326 292.826,153.589 294.112,155.058 296.283,157.749 \n",
       "  297.977,160.078 298.189,160.387 299.849,162.964 301.282,165.478 302.218,167.318 302.506,167.924 303.54,170.303 304.398,172.613 304.495,172.898 305.098,174.856 \n",
       "  305.652,177.034 305.86,178.009 306.072,179.148 306.37,181.2 306.537,182.94 306.554,183.193 306.633,185.128 306.613,187.008 306.582,187.684 306.499,188.834 \n",
       "  306.294,190.609 306.019,192.245 306.001,192.334 305.622,194.011 305.157,195.642 304.813,196.66 304.603,197.227 303.961,198.768 303.225,200.266 302.805,201.025 \n",
       "  302.391,201.72 301.455,203.131 300.41,204.499 299.6,205.439 299.247,205.822 297.96,207.099 296.536,208.327 294.966,209.502 294.225,210.002 293.237,210.622 \n",
       "  291.337,211.68 289.25,212.667 286.961,213.572 284.456,214.385 284.209,214.456 281.716,215.095 278.727,215.68 275.472,216.115 271.938,216.373 270.132,216.426 \n",
       "  268.108,216.428 263.971,216.242 259.532,215.764 254.795,214.942 250.644,213.96 249.766,213.722 244.462,212.043 238.941,209.832 233.267,207.022 230.728,205.571 \n",
       "  227.51,203.541 221.784,199.33 216.243,194.359 212.838,190.737 211.042,188.604 206.372,182.082 202.443,174.883 202.26,174.485 199.467,167.118 197.635,158.996 \n",
       "  197.223,155.2 197.116,150.758 198.021,142.695 199.31,137.937 200.388,135.124 204.187,128.346 207.357,124.499 209.295,122.619 215.532,118.14 219.483,116.193 \n",
       "  222.661,115.007 230.423,113.234 233.236,112.922 238.553,112.745 246.808,113.401 247.182,113.455 254.976,115.022 260.625,116.648 262.889,117.408 270.42,120.365 \n",
       "  272.934,121.493 277.484,123.712 283.808,127.17 284.031,127.3 290.037,131.001 295.508,134.727 296.845,135.691 300.455,138.402 304.91,141.981 308.908,145.437 \n",
       "  309.434,145.909 312.487,148.744 315.689,151.898 318.144,154.454 318.557,154.896 321.131,157.737 323.45,160.427 324.878,162.145 325.552,162.971 327.472,165.375 \n",
       "  329.047,167.391 329.243,167.644 330.895,169.783 332.456,171.798 332.511,171.868 333.952,173.692 335.406,175.468 335.674,175.786 336.841,177.128 338.277,178.673 \n",
       "  338.764,179.168 339.733,180.104 341.226,181.418 341.973,182.016 342.775,182.615 344.393,183.691 345.468,184.311 346.097,184.641 347.9,185.46 349.816,186.141 \n",
       "  350.868,186.439 351.859,186.675 354.04,187.053 356.369,187.263 357.793,187.302 358.858,187.292 361.516,187.126 364.347,186.749 367.123,186.197 367.357,186.143 \n",
       "  370.549,185.289 373.916,184.17 377.449,182.765 379.866,181.673 381.138,181.056 384.965,179.023 388.89,176.66 392.871,173.96 396.039,171.575 396.863,170.918 \n",
       "  400.808,167.538 404.622,163.854 408.214,159.911 410.532,157.043 411.497,155.752 414.365,151.448 416.716,147.104 418.336,143.162 418.449,142.824 419.469,138.726 \n",
       "  419.714,134.936 419.483,132.939 419.132,131.578 417.706,128.762 415.46,126.571 415.371,126.51 412.436,125.073 408.722,124.283 407.064,124.165 404.42,124.197 \n",
       "  399.654,124.772 396.503,125.435 394.554,125.939 389.25,127.613 385.338,129.092 383.864,129.697 378.507,132.095 374.287,134.187 373.271,134.716 368.233,137.476 \n",
       "  363.747,140.128 363.448,140.311 358.96,143.162 354.791,145.992 354.072,146.499 350.958,148.77 347.462,151.477 345.719,152.9 344.298,154.103 341.457,156.642 \n",
       "  339.263,158.755 338.923,159.096 336.679,161.465 334.704,163.754 334.671,163.794 332.978,165.967 331.482,168.111 331.2,168.546 330.196,170.189 329.198,172.015 \n",
       "  329.101,172.207 328.179,174.168 327.702,175.32 327.415,176.075 326.792,177.931 326.633,178.474 326.3,179.738 325.926,181.49 325.924,181.499 325.656,183.213 \n",
       "  325.529,184.372 325.486,184.882 325.408,186.505 325.401,187.125 325.414,188.084 325.501,189.619 325.633,190.87 325.665,191.107 325.903,192.55 326.214,193.946 \n",
       "  326.345,194.441 326.598,195.294 327.056,196.594 327.589,197.843 327.61,197.889 328.2,199.04 328.893,200.183 329.631,201.216 329.673,201.269 330.544,202.296 \n",
       "  331.515,203.26 332.594,204.157 332.87,204.361 333.787,204.982 335.106,205.73 336.563,206.393 338.168,206.964 338.517,207.069 339.936,207.435 341.881,207.795 \n",
       "  344.019,208.028 346.367,208.119 347.73,208.105 348.943,208.055 351.764,207.815 354.85,207.371 358.218,206.697 360.335,206.182 361.886,205.764 365.873,204.539 \n",
       "  370.184,202.977 374.823,201.036 378.693,199.221 379.795,198.673 385.095,195.836 390.683,192.482 396.511,188.567 399.358,186.493 402.53,184.05 408.643,178.91 \n",
       "  414.714,173.162 418.448,169.241 420.598,166.825 426.114,159.964 431.035,152.728 434.026,147.482 435.134,145.276 438.174,137.828 439.945,130.669 440.15,129.105 \n",
       "  440.254,124.093 438.998,118.394 438.184,116.666 436.134,113.841 431.73,110.624 429.989,109.886 425.928,108.852 418.955,108.519 417.503,108.613 411.084,109.53 \n",
       "  403.272,111.496 402.611,111.697 393.83,114.789 388.828,116.849 384.999,118.549 376.339,122.734 374.924,123.458 368.021,127.125 362.279,130.336 360.163,131.551 \n",
       "  352.843,135.882 350.418,137.356 346.099,140.03 339.941,143.948 336.49,146.2 334.358,147.613 329.325,151.021 324.803,154.188 322.687,155.711 320.75,157.131 \n",
       "  317.123,159.872 313.873,162.436 313.43,162.795 310.956,164.845 308.327,167.12 306.313,168.939 305.945,169.278 303.77,171.334 301.767,173.299 301.389,173.676 \n",
       "  299.9,175.184 298.138,176.993 297.246,177.916 296.454,178.733 294.821,180.406 293.448,181.783 293.214,182.013 291.611,183.555 289.993,185.028 289.694,185.29 \n",
       "  288.339,186.432 286.634,187.762 285.736,188.412 284.86,189.013 283.003,190.178 281.369,191.086 281.049,191.251 278.986,192.223 276.804,193.083 274.793,193.732 \n",
       "  274.493,193.818 272.044,194.418 269.452,194.864 266.715,195.14 266.373,195.161 263.829,195.227 260.801,195.105 257.639,194.749 255.234,194.325 254.353,194.136 \n",
       "  250.958,193.242 247.487,192.042 243.975,190.514 241.049,188.979 240.46,188.638 236.986,186.392 233.622,183.775 230.438,180.795 227.716,177.722 227.505,177.459 \n",
       "  224.903,173.786 222.714,169.834 221.016,165.669 220.672,164.583 219.888,161.361 219.386,157.011 219.551,152.728 219.572,152.563 220.415,148.625 221.966,144.813 \n",
       "  223.308,142.578 224.182,141.399 227.013,138.474 230.384,136.099 230.957,135.78 234.209,134.326 238.389,133.165 240.703,132.797 242.821,132.608 247.401,132.628 \n",
       "  250.993,133.006 252.036,133.172 256.638,134.184 261.135,135.595 261.162,135.604 265.464,137.337 269.58,139.345 270.836,140.029 273.447,141.558 277.043,143.921 \n",
       "  279.679,145.86 280.355,146.389 283.375,148.919 286.107,151.484 287.267,152.666 288.555,154.055 290.73,156.612 292.644,159.143 293.159,159.879 294.311,161.634 \n",
       "  295.746,164.078 296.962,166.47 297.445,167.534 297.976,168.806 298.801,171.083 299.449,173.301 299.714,174.402 299.932,175.459 300.263,177.558 300.447,179.598 \n",
       "  300.491,180.716 300.495,181.579 300.413,183.504 300.205,185.372 300.056,186.28 299.875,187.184 299.425,188.942 298.856,190.644 298.572,191.365 298.167,192.293 \n",
       "  297.358,193.887 296.424,195.425 296.047,195.979 295.362,196.906 294.166,198.329 292.83,199.69 292.393,200.093 291.346,200.986 289.707,202.212 287.901,203.362 \n",
       "  285.92,204.427 285.576,204.593 283.751,205.401 281.385,206.27 278.809,207.019 276.014,207.628 273.022,208.078 272.989,208.082 269.723,208.359 266.213,208.428 \n",
       "  262.46,208.253 258.57,207.817 258.463,207.802 254.224,207.034 249.77,205.903 245.136,204.36 240.559,202.45 240.356,202.356 235.477,199.84 230.586,196.772 \n",
       "  225.783,193.123 223.299,190.936 221.166,188.862 216.869,183.993 213.04,178.556 212.29,177.315 209.831,172.604 207.393,166.251 205.975,160.315 205.867,159.646 \n",
       "  205.376,152.96 205.99,146.412 206.349,144.708 207.753,140.223 210.637,134.609 212.412,132.161 214.573,129.764 219.436,125.839 222.712,123.974 225.066,122.924 \n",
       "  231.278,121.057 234.981,120.454 237.878,120.209 244.679,120.307 247.664,120.618 251.51,121.239 258.226,122.873 260.075,123.439 264.709,125.069 270.874,127.694 \n",
       "  271.754,128.111 276.657,130.624 282.025,133.753 282.132,133.82 286.958,136.994 291.456,140.279 291.514,140.323 295.529,143.553 299.198,146.782 301.677,149.136 \n",
       "  302.487,149.938 305.424,153.003 308.04,155.967 309.564,157.814 310.366,158.825 312.433,161.574 314.27,164.215 315.484,166.078 315.907,166.751 317.371,169.185 \n",
       "  318.686,171.52 318.825,171.774 319.877,173.759 320.963,175.907 321.466,176.932 321.966,177.968 322.904,179.943 323.323,180.836 323.792,181.837 324.647,183.651 \n",
       "  324.96,184.306 325.484,185.388 326.314,187.049 326.545,187.497 327.151,188.636 328.007,190.15 328.166,190.419 328.893,191.59 329.82,192.957 330.441,193.795 \n",
       "  330.799,194.25 331.841,195.466 332.957,196.604 333.248,196.875 334.159,197.662 335.458,198.634 336.866,199.517 336.939,199.559 338.395,200.304 340.06,200.989 \n",
       "  341.874,201.562 341.993,201.594 343.851,202.014 346.006,202.333 348.357,202.504 349.119,202.524 350.917,202.511 353.705,202.336 356.735,201.955 359.395,201.475 \n",
       "  360.022,201.344 363.582,200.476 367.423,199.319 371.549,197.837 374.586,196.597 375.965,195.994 380.672,193.751 385.639,191.07 390.834,187.915 394.254,185.637 \n",
       "  396.222,184.25 401.743,180.048 407.289,175.314 412.753,170.056 412.859,169.947 418.007,164.296 422.877,158.127 427.178,151.67 428.15,150.011 430.719,145.057 \n",
       "  433.308,138.498 434.697,132.717 434.769,132.222 434.952,126.471 433.782,121.483 433.488,120.816 431.223,117.478 427.336,114.604 426.273,114.118 422.235,112.957 \n",
       "  416.107,112.532 414.777,112.59 409.173,113.265 401.678,115.01 401.427,115.082 393.866,117.59 387.765,120.02 385.959,120.799 378.153,124.437 374.506,126.272 \n",
       "  370.6,128.323 363.416,132.305 362.216,132.995 356.679,136.265 351.987,139.145 350.434,140.12 344.702,143.815 339.764,147.131 339.481,147.325 334.758,150.634 \n",
       "  330.504,153.748 327.555,155.998 326.687,156.677 323.271,159.436 320.217,162.042 318.719,163.378 317.485,164.513 315.036,166.867 312.835,169.116 312.05,169.952 \n",
       "  310.845,171.275 309.033,173.353 308.267,174.266 307.368,175.361 305.822,177.303 305.134,178.189 304.368,179.187 302.982,181.015 302.33,181.877 301.64,182.789 \n",
       "  300.322,184.512 299.668,185.352 299.008,186.184 297.68,187.805 296.987,188.617 296.321,189.373 294.915,190.886 294.146,191.665 293.446,192.342 291.899,193.738 \n",
       "  290.26,195.068 289.794,195.419 288.515,196.327 286.65,197.51 284.652,198.607 284.312,198.778 282.509,199.611 280.207,200.509 277.736,201.287 276.988,201.488 \n",
       "  275.084,201.932 272.243,202.425 269.206,202.743 266.923,202.851 265.969,202.865 262.528,202.764 258.892,202.408 255.072,201.763 252.912,201.277 251.078,200.795 \n",
       "  246.933,199.465 242.686,197.737 238.387,195.577 235.749,194.025 234.085,192.948 229.856,189.823 225.801,186.202 222.021,182.092 220.762,180.512 218.625,177.501 \n",
       "  215.734,172.488 213.466,167.147 212.599,164.373 211.937,161.572 211.243,155.904 211.446,150.305 211.501,149.874 212.595,144.943 214.678,139.987 215.915,137.926 \n",
       "  217.659,135.598 221.457,131.904 224.682,129.699 225.957,128.998 231.026,126.936 235.772,125.831 236.514,125.719 242.269,125.319 247.483,125.592 248.15,125.667 \n",
       "  254.022,126.676 259.071,128.021 259.779,128.244 265.328,130.265 270.114,132.4 270.604,132.639 275.556,135.27 280.157,138.081 280.225,138.124 284.39,141 \n",
       "  288.253,143.976 288.839,144.456 291.751,146.963 294.898,149.931 295.873,150.912 297.713,152.855 300.216,155.719 302.431,158.513 302.772,158.968 304.383,161.228 \n",
       "  306.093,163.863 307.587,166.415 308.197,167.542 308.885,168.885 310.011,171.274 310.981,173.584 311.243,174.257 311.815,175.818 312.529,177.979 312.958,179.426 \n",
       "  313.137,180.07 313.653,182.093 314.038,183.813 314.088,184.051 314.453,185.948 314.738,187.665 314.757,187.785 315.008,189.566 315.19,191.096 315.212,191.294 \n",
       "  315.376,192.969 315.473,194.162 315.505,194.595 315.602,196.174 315.637,196.868 315.673,197.707 315.717,199.115 315.719,199.197 315.744,200.645 315.749,201.418 \n",
       "  315.749,202.053 315.736,203.422 315.726,203.926 315.705,204.755 315.659,206.051 315.638,206.523 315.597,207.314 315.519,208.544 315.472,209.167 315.425,209.742 \n",
       "  315.316,210.909 315.211,211.87 315.19,212.048 315.046,213.158 314.884,214.241 314.813,214.674 314.702,215.299 314.498,216.331 314.27,217.34 314.195,217.646 \n",
       "  314.017,218.327 313.734,219.291 313.421,220.235 313.182,220.879 313.072,221.159 312.685,222.065 312.254,222.952 311.776,223.823 311.584,224.144 311.245,224.678 \n",
       "  310.655,225.518 309.999,226.343 309.378,227.041 309.27,227.155 308.459,227.953 307.557,228.738 306.552,229.51 306.341,229.662 305.435,230.269 304.192,231.013 \n",
       "  302.807,231.741 302.099,232.079 301.266,232.451 299.55,233.14 297.64,233.803 296.121,234.263 295.513,234.432 293.148,235.022 290.517,235.558 287.607,236.025 \n",
       "  287.593,236.027 284.348,236.412 280.751,236.685 276.77,236.813 275.277,236.818 272.373,236.764 267.532,236.487 262.226,235.912 257.286,235.131 256.433,234.972 \n",
       "  250.139,233.585 243.366,231.635 236.147,229.002 235.502,228.737 228.527,225.558 220.627,221.158 213.715,216.496 212.599,215.663 204.635,208.937 197.029,200.916 \n",
       "  193.514,196.481 190.1,191.561 184.228,180.951 181.604,174.669 179.821,169.286 177.264,156.905 176.753,149.271 176.881,144.294 178.882,132.032 180.539,126.9 \n",
       "  183.317,120.756 190.054,111.054 191.324,109.687 198.8,103.404 206.617,99.0878 209.121,98.0663 220.521,95.1066 223.503,94.7205 232.492,94.3684 240.312,94.931 \n",
       "  244.577,95.5413 256.042,98.1269 256.402,98.2288 267.688,102.008 269.461,102.699 278.254,106.484 286.104,110.336 288,111.328 296.884,116.27 304.031,120.577 \n",
       "  304.919,121.131 312.139,125.774 318.602,130.131 319.905,131.03 324.372,134.156 329.519,137.839 332.1,139.712 334.112,141.182 338.218,144.201 341.115,146.337 \n",
       "  341.898,146.913 345.21,149.342 348.207,151.507 348.231,151.524 350.937,153.43 353.442,155.128 354.009,155.501 355.759,156.618 357.922,157.914 358.89,158.458 \n",
       "  359.96,159.028 361.898,159.97 363.756,160.747 364.173,160.903 365.555,161.368 367.306,161.838 369.022,162.161 369.202,162.187 370.711,162.343 372.38,162.388 \n",
       "  374.029,162.298 374.33,162.268 375.661,162.079 377.273,161.735 378.86,161.271 379.657,160.99 380.416,160.692 381.933,160.006 383.398,159.222 384.8,158.348 \n",
       "  384.95,158.247 386.126,157.395 387.36,156.378 388.488,155.31 389.393,154.325 389.493,154.208 390.359,153.088 391.074,151.969 391.624,150.871 391.886,150.184 \n",
       "  391.998,149.813 392.184,148.816 392.181,147.895 391.986,147.069 391.883,146.828 391.597,146.355 391.02,145.766 390.265,145.308 389.344,144.992 389.308,144.983 \n",
       "  388.268,144.822 387.057,144.798 385.731,144.917 384.665,145.099 384.312,145.174 382.82,145.563 381.278,146.074 379.708,146.695 379.304,146.871 378.133,147.415 \n",
       "  376.572,148.221 375.043,149.1 374.085,149.698 373.565,150.039 372.153,151.026 370.819,152.049 369.575,153.097 369.428,153.227 368.431,154.16 367.395,155.227 \n",
       "  366.472,156.292 365.77,157.203 365.668,157.346 364.986,158.38 364.43,159.389 363.999,160.366 363.694,161.306 363.685,161.344 363.518,162.203 363.468,163.051 \n",
       "  363.544,163.845 363.743,164.581 364.04,165.21 364.066,165.254 364.51,165.857 365.073,166.387 365.752,166.838 366.544,167.205 367.446,167.483 367.923,167.584 \n",
       "  368.455,167.667 369.567,167.751 370.775,167.733 372.073,167.606 373.454,167.368 374.102,167.222 374.912,167.013 376.438,166.538 378.017,165.944 379.638,165.23 \n",
       "  381.285,164.397 381.861,164.08 382.946,163.444 384.6,162.377 386.228,161.204 387.806,159.937 389.312,158.584 389.325,158.572 390.724,157.159 392.016,155.681 \n",
       "  393.164,154.172 394.144,152.653 394.215,152.531 394.931,151.148 395.508,149.684 395.859,148.286 395.962,147.352 395.969,146.982 395.827,145.798 395.436,144.754 \n",
       "  394.797,143.87 394.655,143.729 393.914,143.166 392.804,142.651 391.486,142.33 390.518,142.227 389.981,142.208 388.315,142.282 386.517,142.543 384.618,142.98 \n",
       "  384.604,142.984 382.648,143.582 380.638,144.33 378.616,145.208 378.294,145.359 376.612,146.197 374.648,147.28 372.747,148.438 372.222,148.778 370.93,149.654 \n",
       "  369.213,150.914 367.608,152.203 366.78,152.918 366.129,153.508 364.784,154.817 363.578,156.122 362.517,157.413 362.412,157.55 361.605,158.681 360.841,159.92 \n",
       "  360.228,161.125 359.764,162.288 359.719,162.421 359.449,163.404 359.282,164.469 359.26,165.477 359.383,166.424 359.602,167.185 359.647,167.303 360.052,168.111 \n",
       "  360.594,168.842 361.273,169.489 362.084,170.048 363.027,170.513 363.573,170.716 364.099,170.877 365.298,171.134 366.619,171.277 368.057,171.3 369.609,171.197 \n",
       "  369.916,171.162 371.27,170.96 373.032,170.583 374.884,170.062 376.814,169.393 378.666,168.635 378.812,168.57 380.864,167.591 382.948,166.456 385.042,165.172 \n",
       "  387.122,163.742 388.523,162.684 389.164,162.173 391.138,160.475 393.011,158.669 394.747,156.778 396.226,154.943 396.314,154.826 397.675,152.838 398.799,150.852 \n",
       "  399.654,148.904 399.992,147.887 400.212,147.031 400.449,145.274 400.357,143.667 400.039,142.517 399.924,142.248 399.149,141.047 398.047,140.087 396.66,139.394 \n",
       "  396.635,139.384 394.935,138.954 392.983,138.793 390.817,138.896 390.558,138.924 388.476,139.254 386.006,139.847 383.449,140.652 383.306,140.703 380.849,141.645 \n",
       "  378.245,142.798 375.955,143.937 375.673,144.085 373.167,145.479 370.754,146.956 368.962,148.145 368.458,148.494 366.3,150.071 364.292,151.672 362.706,153.046 \n",
       "  362.447,153.282 360.773,154.886 359.275,156.476 357.954,158.042 357.644,158.439 356.813,159.577 355.851,161.075 355.064,162.53 354.451,163.938 354.406,164.059 \n",
       "  354.009,165.294 353.734,166.594 353.623,167.834 353.672,169.01 353.791,169.749 353.879,170.117 354.24,171.151 354.754,172.107 355.419,172.979 356.142,173.685 \n",
       "  356.233,173.762 357.195,174.449 358.304,175.034 359.559,175.509 360.96,175.866 361.154,175.904 362.505,176.097 364.192,176.194 366.019,176.146 367.983,175.944 \n",
       "  368.736,175.83 370.079,175.579 372.302,175.041 374.64,174.321 377.081,173.411 378.595,172.769 379.613,172.304 382.219,170.992 384.871,169.477 387.541,167.76 \n",
       "  389.77,166.169 390.2,165.844 392.814,163.735 395.336,161.455 397.718,159.031 399.914,156.489 400.066,156.3 401.873,153.861 403.542,151.198 404.873,148.553 \n",
       "  405.468,147.056 405.818,145.978 406.338,143.533 406.408,141.276 406.216,139.997 406.006,139.264 405.128,137.548 403.787,136.165 402.913,135.588 402.004,135.148 \n",
       "  399.814,134.514 397.268,134.261 396.28,134.263 394.42,134.386 391.333,134.864 388.072,135.664 387.958,135.697 384.701,136.751 381.281,138.082 379.28,138.957 \n",
       "  377.87,139.614 374.52,141.306 371.271,143.119 370.806,143.393 368.162,145.017 365.219,146.971 362.897,148.633 362.466,148.954 359.918,150.945 357.584,152.928 \n",
       "  355.927,154.449 355.47,154.889 353.578,156.818 351.904,158.709 350.447,160.556 350.408,160.608 349.2,162.353 348.158,164.1 347.313,165.792 346.874,166.842 \n",
       "  346.657,167.429 346.184,169.008 345.887,170.528 345.762,171.984 345.801,173.376 345.85,173.827 346,174.701 346.356,175.955 346.866,177.133 347.53,178.232 \n",
       "  347.856,178.671 348.345,179.247 349.312,180.171 350.432,180.998 351.707,181.72 351.855,181.792 353.139,182.33 354.731,182.819 356.486,183.175 358.406,183.388 \n",
       "  358.436,183.39 360.495,183.447 362.755,183.337 365.185,183.045 367.784,182.555 368.727,182.336 370.553,181.853 373.486,180.922 376.567,179.748 379.782,178.318 \n",
       "  383.039,176.657 383.113,176.617 386.542,174.632 390.026,172.362 393.519,169.812 396.972,166.988 397.89,166.181 400.333,163.899 403.53,160.578 406.484,157.077 \n",
       "  409.115,153.447 409.471,152.903 411.342,149.747 413.084,146.065 414.271,142.495 414.482,141.585 414.832,139.13 414.726,136.067 413.972,133.482 413.931,133.395 \n",
       "  412.438,131.199 410.28,129.531 408.859,128.867 407.5,128.432 404.169,127.915 400.376,127.962 400.314,127.967 396.218,128.544 391.797,129.604 390.273,130.055 \n",
       "  387.218,131.076 382.576,132.888 379.961,134.034 377.959,134.967 373.443,137.243 369.9,139.186 369.087,139.652 364.943,142.14 361.044,144.665 360.44,145.073 \n",
       "  357.416,147.189 354.072,149.688 351.928,151.394 351.018,152.144 348.256,154.544 345.777,156.882 344.827,157.834 343.573,159.152 341.632,161.356 339.939,163.493 \n",
       "  339.471,164.131 338.48,165.565 337.241,167.574 336.206,169.523 335.566,170.922 335.362,171.413 334.695,173.246 334.194,175.024 333.848,176.746 333.803,177.042 \n",
       "  333.647,178.414 333.584,180.028 333.652,181.586 333.673,181.819 333.845,183.088 334.161,184.532 334.596,185.916 334.636,186.023 335.15,187.238 335.824,188.494 \n",
       "  336.618,189.682 336.683,189.769 337.535,190.797 338.58,191.834 339.758,192.787 340.106,193.034 341.074,193.651 342.537,194.418 344.156,195.079 345.664,195.551 \n",
       "  345.939,195.624 347.896,196.043 350.038,196.323 352.377,196.446 354.924,196.398 355.445,196.367 357.69,196.161 360.686,195.714 363.919,195.03 367.394,194.084 \n",
       "  369.159,193.527 371.119,192.85 375.093,191.298 379.301,189.398 383.722,187.119 387.263,185.088 388.34,184.434 393.12,181.312 397.994,177.748 402.887,173.738 \n",
       "  405.219,171.657 407.722,169.282 412.387,164.41 416.744,159.196 420.409,154.092 420.654,153.718 423.967,148.079 426.529,142.44 427.994,137.831 428.196,136.971 \n",
       "  428.839,131.856 428.385,127.283 428.106,126.289 426.779,123.434 424.043,120.445 422.42,119.388 420.24,118.42 415.487,117.397 412.349,117.257 409.943,117.357 \n",
       "  403.788,118.231 400.145,119.076 397.217,119.9 390.42,122.225 387.451,123.395 383.571,125.049 376.819,128.223 374.991,129.141 370.287,131.61 364.063,135.097 \n",
       "  363.232,135.581 358.215,138.591 352.781,142.028 352.721,142.067 347.78,145.361 343.441,148.405 343.216,148.567 339.08,151.629 335.351,154.549 333.161,156.351 \n",
       "  332.007,157.33 329.02,159.98 326.358,162.511 325.057,163.819 323.99,164.934 321.886,167.262 320.015,169.504 319.004,170.798 318.348,171.671 316.859,173.77 \n",
       "  315.521,175.808 315.502,175.838 314.31,177.791 313.207,179.724 312.753,180.553 312.189,181.61 311.238,183.452 310.831,184.262 310.338,185.252 309.473,187.012 \n",
       "  309.184,187.603 308.629,188.734 307.791,190.416 307.595,190.806 306.948,192.062 306.087,193.67 305.975,193.873 305.196,195.24 304.265,196.773 303.703,197.645 \n",
       "  303.282,198.269 302.237,199.725 301.118,201.142 300.911,201.39 299.914,202.517 298.614,203.849 297.248,205.097 297.204,205.136 295.673,206.373 294.007,207.556 \n",
       "  292.245,208.649 292.193,208.679 290.216,209.738 288.06,210.722 285.71,211.62 285.203,211.793 283.15,212.422 280.364,213.11 277.337,213.664 275.06,213.961 \n",
       "  274.053,214.063 270.497,214.281 266.662,214.281 262.543,214.025 260.083,213.751 258.133,213.472 253.437,212.574 248.482,211.269 243.301,209.497 240.478,208.349 \n",
       "  237.927,207.196 232.424,204.301 226.898,200.757 221.846,196.848 221.455,196.515 216.222,191.53 211.375,185.822 207.096,179.431 206.446,178.294 203.574,172.414 \n",
       "  201.001,164.929 199.686,158.316 199.551,157.17 199.373,149.367 200.54,141.803 200.722,141.112 203.091,134.766 206.961,128.523 207.785,127.509 212.04,123.31 \n",
       "  218.14,119.279 219.178,118.76 225.043,116.52 232.478,115.026 232.506,115.022 240.285,114.716 246.114,115.177 248.16,115.464 255.937,117.101 259.365,118.08 \n",
       "  263.463,119.444 270.624,122.316 271.668,122.786 277.34,125.554 281.988,128.099 283.567,129.019 289.283,132.599 294.086,135.915 294.491,136.208 299.202,139.777 \n",
       "  303.447,143.267 306.275,145.753 307.255,146.647 310.663,149.895 313.71,153.005 315.137,154.536 316.435,155.972 318.877,158.797 321.071,161.483 321.808,162.416 \n",
       "  323.055,164.034 324.859,166.455 325.552,167.407 326.517,168.752 328.055,170.929 328.724,171.883 329.5,172.99 330.877,174.94 331.553,175.882 332.209,176.781 \n",
       "  333.516,178.515 334.235,179.429 334.818,180.145 336.133,181.669 336.936,182.535 337.478,183.089 338.869,184.403 339.804,185.198 340.322,185.609 341.852,186.703 \n",
       "  343.473,187.68 344.133,188.029 345.198,188.536 347.042,189.263 349.019,189.852 349.6,189.992 351.14,190.295 353.418,190.579 355.866,190.69 356.915,190.683 \n",
       "  358.493,190.614 361.311,190.335 364.324,189.832 366.982,189.229 367.539,189.085 370.96,188.073 374.581,186.772 378.39,185.161 381.022,183.906 382.377,183.215 \n",
       "  386.524,180.911 390.783,178.239 395.105,175.192 398.255,172.741 399.441,171.762 403.722,167.953 407.846,163.809 411.711,159.377 413.377,157.25 415.21,154.709 \n",
       "  418.222,149.901 420.624,145.074 421.751,142.142 422.3,140.352 423.146,135.881 423.096,131.809 422.977,131.078 422.094,128.28 420.144,125.412 418.644,124.126 \n",
       "  417.279,123.298 413.571,121.994 409.91,121.527 409.135,121.501 404.102,121.797 398.799,122.764 398.622,122.806 392.849,124.436 387.04,126.53 386.931,126.574 \n",
       "  381.002,129.101 375.396,131.796 375.175,131.908 369.548,134.889 364.291,137.9 364.188,137.961 359.149,141.054 354.462,144.118 354.117,144.351 350.145,147.116 \n",
       "  346.202,150.027 345.447,150.605 342.627,152.836 339.408,155.54 338.221,156.587 336.527,158.139 333.962,160.638 331.69,163.043 331.196,163.595 329.686,165.361 \n",
       "  327.927,167.601 326.388,169.768 325.699,170.819 325.048,171.869 323.883,173.911 322.876,175.897 322.525,176.648 322.006,177.832 321.258,179.719 320.73,181.214 \n",
       "  320.616,181.561 320.066,183.36 319.597,185.117 319.584,185.171 319.198,186.835 318.858,188.513 318.818,188.727 318.57,190.154 318.326,191.757 318.296,191.977 \n",
       "  318.121,193.324 317.947,194.855 317.934,194.975 317.801,196.35 317.682,197.76 317.678,197.81 317.575,199.235 317.504,200.358 317.489,200.626 317.416,201.984 \n",
       "  317.379,202.791 317.356,203.308 317.306,204.6 317.289,205.076 317.263,205.86 317.228,207.089 317.225,207.227 317.199,208.287 317.178,209.256 317.175,209.454 \n",
       "  317.154,210.592 317.145,211.176 317.137,211.701 317.123,212.782 317.12,212.994 317.111,213.835 317.102,214.72 317.1,214.861 317.092,215.86 317.088,216.362 \n",
       "  317.084,216.834 317.078,217.782 317.077,217.924 317.072,218.706 317.068,219.412 317.067,219.606 317.062,220.482 317.06,220.823 317.058,221.336 317.054,222.147 \n",
       "  317.053,222.167 317.049,222.976 317.047,223.359 317.045,223.765 317.041,224.533 317.041,224.571 317.037,225.28 317.033,225.884 317.032,226.008 317.027,226.718 \n",
       "  317.023,227.268 317.022,227.408 317.016,228.081 317.01,228.718 317.01,228.736 317.003,229.374 316.995,229.995 316.992,230.226 316.986,230.601 316.977,231.19 \n",
       "  316.966,231.765 316.966,231.777 316.955,232.324 316.942,232.869 316.929,233.337 316.927,233.4 316.911,233.917 316.893,234.422 316.875,234.853 316.873,234.913 \n",
       "  316.85,235.392 316.825,235.859 316.798,236.283 316.796,236.314 316.764,236.758 316.729,237.191 316.689,237.61 316.689,237.614 316.644,238.026 316.594,238.429 \n",
       "  316.538,238.823 316.536,238.836 316.475,239.207 316.404,239.583 316.325,239.951 316.32,239.971 316.236,240.311 316.136,240.664 316.024,241.011 316.018,241.028 \n",
       "  315.898,241.351 315.756,241.685 315.596,242.013 315.592,242.022 315.417,242.337 315.216,242.657 314.994,242.968 314.99,242.973 314.735,243.287 314.449,243.597 \n",
       "  314.154,243.882 314.127,243.907 313.764,244.215 313.356,244.523 312.972,244.783 312.897,244.831 312.38,245.141 311.798,245.452 311.311,245.689 311.142,245.767 \n",
       "  310.404,246.084 309.573,246.405 308.973,246.618 308.637,246.731 307.583,247.06 306.395,247.393 305.68,247.576 305.057,247.728 303.551,248.065 301.854,248.4 \n",
       "  301.036,248.547 299.944,248.73 297.795,249.05 295.377,249.349 294.464,249.447 292.659,249.619 289.606,249.844 286.179,250.003 285.14,250.033 282.338,250.073 \n",
       "  278.041,250.016 273.245,249.786 271.971,249.696 267.906,249.33 261.985,248.573 255.453,247.416 254.274,247.171 248.281,245.756 240.479,243.448 233.192,240.782 \n",
       "  232.076,240.328 223.128,236.215 213.784,230.902 210.662,228.863 204.22,224.178 194.729,215.862 188.502,209.22 185.689,205.817 177.555,193.989 170.878,180.541 \n",
       "  170.063,178.464 166.226,165.726 164.117,150.123 164.043,147.196 164.984,134.456 168.419,121.167 169.014,119.62 176.172,106.524 180.534,101.187 186.119,95.943 \n",
       "  197.594,88.7172 198.308,88.3866 212.045,84.0352 216.341,83.3373 226.609,82.7138 234.86,83.141 241.345,83.9886 251.877,86.2181 255.724,87.2736 268.105,91.4535 \n",
       "  269.369,91.9427 282.047,97.4182 287.321,99.975 293.645,103.227 304.136,109.015 307.823,111.147 313.55,114.536 321.953,119.644 326.263,122.305 329.427,124.263 \n",
       "  336.058,128.365 340.508,131.092 341.935,131.959 347.138,135.071 350.732,137.163 351.743,137.741 355.818,140.011 358.712,141.557 359.421,141.925 362.605,143.527 \n",
       "  364.927,144.63 365.415,144.854 367.891,145.944 369.74,146.701 370.067,146.829 371.973,147.541 373.633,148.105 374.597,148.406 375.07,148.546 376.303,148.886 \n",
       "  377.348,149.144 377.868,149.262 378.22,149.338 378.933,149.485 379.497,149.597 379.813,149.662 379.926,149.687 380.228,149.768 380.415,149.846 380.495,149.932 \n",
       "  380.497,149.94 380.478,150.031 380.375,150.15 380.194,150.292 380.007,150.42 379.945,150.46 379.637,150.658 379.279,150.886 378.881,151.143 378.54,151.37 \n",
       "  378.452,151.43 378.001,151.745 377.536,152.087 377.065,152.453 376.595,152.84 376.538,152.889 376.136,153.246 375.692,153.666 375.271,154.098 374.878,154.536 \n",
       "  374.652,154.809 374.518,154.979 374.197,155.422 373.917,155.86 373.682,156.292 373.496,156.711 373.447,156.841 373.36,157.116 373.277,157.503 373.248,157.868 \n",
       "  373.273,158.208 373.352,158.521 373.401,158.641 373.486,158.802 373.675,159.05 373.917,159.262 374.209,159.436 374.55,159.57 374.937,159.661 375.008,159.673 \n",
       "  375.369,159.709 375.842,159.711 376.351,159.668 376.892,159.579 377.46,159.444 378.05,159.263 378.09,159.249 378.658,159.035 379.277,158.764 379.901,158.451 \n",
       "  380.522,158.098 381.133,157.71 381.398,157.528 381.73,157.287 382.302,156.835 382.845,156.359 383.349,155.865 383.81,155.357 383.961,155.175 384.218,154.842 \n",
       "  384.569,154.325 384.858,153.815 385.079,153.318 385.225,152.851 385.228,152.841 385.3,152.39 385.295,151.972 385.213,151.592 385.054,151.256 384.935,151.094 \n",
       "  384.817,150.97 384.504,150.737 384.122,150.561 383.674,150.442 383.167,150.384 383.017,150.378 382.605,150.387 381.996,150.45 381.349,150.572 380.672,150.75 \n",
       "  379.98,150.979 379.973,150.982 379.26,151.265 378.544,151.595 377.831,151.966 377.13,152.375 376.884,152.529 376.449,152.815 375.796,153.283 375.176,153.772 \n",
       "  374.597,154.278 374.256,154.603 374.064,154.795 373.583,155.317 373.156,155.841 372.788,156.36 372.483,156.871 372.464,156.906 372.242,157.368 372.069,157.847 \n",
       "  371.963,158.305 371.925,158.736 371.949,159.093 371.957,159.137 372.058,159.505 372.227,159.835 372.463,160.126 372.763,160.373 373.127,160.574 373.235,160.62 \n",
       "  373.552,160.726 374.035,160.827 374.572,160.875 375.158,160.869 375.789,160.807 376.382,160.705 376.46,160.688 377.166,160.512 377.9,160.278 378.655,159.989 \n",
       "  379.424,159.646 380.199,159.25 380.364,159.159 380.972,158.803 381.735,158.31 382.477,157.775 383.191,157.203 383.867,156.601 383.961,156.512 384.495,155.974 \n",
       "  385.066,155.33 385.573,154.677 386.006,154.024 386.259,153.577 386.359,153.379 386.623,152.751 386.795,152.15 386.872,151.584 386.849,151.061 386.847,151.046 \n",
       "  386.724,150.59 386.499,150.179 386.177,149.831 385.762,149.552 385.57,149.461 385.256,149.348 384.667,149.222 384.004,149.172 383.276,149.199 382.576,149.288 \n",
       "  382.491,149.303 381.661,149.482 380.797,149.731 379.909,150.046 379.01,150.423 378.874,150.485 378.109,150.856 377.219,151.339 376.348,151.864 375.506,152.427 \n",
       "  375.361,152.53 374.703,153.019 373.947,153.635 373.244,154.267 372.6,154.91 372.452,155.068 372.023,155.557 371.516,156.202 371.082,156.84 370.726,157.464 \n",
       "  370.559,157.81 370.45,158.071 370.256,158.654 370.144,159.21 370.116,159.733 370.17,160.219 370.208,160.376 370.309,160.664 370.53,161.064 370.831,161.416 \n",
       "  371.21,161.716 371.665,161.96 372.049,162.103 372.193,162.145 372.792,162.268 373.457,162.327 374.182,162.32 374.962,162.245 375.792,162.101 375.969,162.062 \n",
       "  376.666,161.885 377.577,161.598 378.515,161.242 379.471,160.817 380.437,160.326 380.804,160.123 381.404,159.77 382.359,159.154 383.292,158.483 384.191,157.765 \n",
       "  385.043,157.006 385.194,156.862 385.836,156.213 386.56,155.397 387.202,154.568 387.751,153.738 388.017,153.272 388.197,152.917 388.531,152.118 388.746,151.353 \n",
       "  388.838,150.633 388.826,150.171 388.8,149.971 388.632,149.377 388.334,148.86 387.912,148.428 387.492,148.151 387.368,148.088 386.708,147.846 385.944,147.703 \n",
       "  385.087,147.659 384.164,147.714 384.148,147.716 383.14,147.872 382.079,148.121 380.977,148.458 379.874,148.869 379.852,148.878 378.716,149.373 377.584,149.934 \n",
       "  376.469,150.555 375.648,151.056 375.385,151.225 374.343,151.937 373.352,152.683 372.423,153.453 371.942,153.883 371.564,154.239 370.783,155.035 370.084,155.832 \n",
       "  369.473,156.625 369.166,157.071 368.954,157.406 368.53,158.169 368.204,158.909 367.977,159.62 367.849,160.296 367.849,160.297 367.822,160.935 367.895,161.529 \n",
       "  368.067,162.074 368.336,162.567 368.702,163.002 368.735,163.034 369.162,163.375 369.714,163.682 370.354,163.92 371.078,164.086 371.882,164.175 372.368,164.19 \n",
       "  372.762,164.184 373.713,164.11 374.727,163.952 375.797,163.707 376.913,163.376 377.761,163.076 378.07,162.957 379.257,162.449 380.461,161.855 381.67,161.181 \n",
       "  382.871,160.428 383.675,159.875 384.051,159.601 385.194,158.707 386.286,157.756 387.309,156.76 388.249,155.728 388.42,155.524 389.088,154.673 389.812,153.61 \n",
       "  390.409,152.556 390.864,151.526 390.959,151.26 391.165,150.536 391.305,149.604 391.28,148.744 391.087,147.971 391.053,147.888 390.722,147.3 390.192,146.741 \n",
       "  389.504,146.302 388.688,145.995 388.668,145.989 387.694,145.81 386.598,145.762 385.399,145.843 384.318,146.01 384.113,146.05 382.759,146.379 381.358,146.82 \n",
       "  379.929,147.363 379.256,147.65 378.491,147.999 377.063,148.717 375.661,149.505 374.355,150.316 374.302,150.351 373,151.245 371.767,152.175 370.614,153.132 \n",
       "  370.03,153.654 369.551,154.105 368.586,155.086 367.724,156.066 366.97,157.037 366.715,157.4 366.33,157.993 365.806,158.926 365.398,159.831 365.109,160.702 \n",
       "  364.984,161.255 364.939,161.532 364.889,162.317 364.956,163.052 365.141,163.731 365.441,164.35 365.704,164.726 365.855,164.903 366.381,165.386 367.016,165.795 \n",
       "  367.757,166.124 368.6,166.369 369.542,166.525 369.687,166.54 370.579,166.587 371.705,166.552 372.913,166.417 374.195,166.178 375.544,165.832 375.778,165.763 \n",
       "  376.953,165.375 378.408,164.809 379.897,164.133 381.405,163.351 382.918,162.463 383.046,162.383 384.42,161.473 385.893,160.39 387.315,159.225 388.666,157.988 \n",
       "  389.511,157.136 389.925,156.692 391.07,155.352 392.08,153.989 392.936,152.622 393.419,151.702 393.616,151.274 394.103,149.966 394.386,148.722 394.455,147.563 \n",
       "  394.427,147.189 394.3,146.514 393.92,145.592 393.321,144.813 392.581,144.236 392.508,144.192 391.492,143.741 390.292,143.463 388.928,143.359 388.172,143.373 \n",
       "  387.42,143.428 385.794,143.666 384.078,144.062 382.424,144.561 382.298,144.604 380.48,145.279 378.651,146.073 376.833,146.969 376.527,147.13 375.052,147.951 \n",
       "  373.326,149.005 371.672,150.115 370.996,150.6 370.108,151.267 368.647,152.448 367.297,153.647 366.226,154.691 366.069,154.853 364.969,156.055 364.003,157.245 \n",
       "  363.172,158.417 362.707,159.165 362.48,159.562 361.929,160.673 361.519,161.746 361.247,162.775 361.118,163.715 361.115,163.754 361.121,164.678 361.264,165.542 \n",
       "  361.541,166.34 361.95,167.068 362.491,167.721 362.638,167.864 363.16,168.292 363.955,168.777 364.873,169.17 365.911,169.466 367.066,169.657 367.582,169.705 \n",
       "  368.334,169.739 369.711,169.705 371.189,169.55 372.761,169.27 374.419,168.859 374.738,168.767 376.156,168.312 377.958,167.626 379.811,166.802 381.696,165.839 \n",
       "  383.598,164.739 383.629,164.721 385.499,163.503 387.372,162.14 389.192,160.664 390.93,159.087 391.896,158.125 392.559,157.423 394.05,155.692 395.371,153.923 \n",
       "  396.494,152.143 396.97,151.261 397.389,150.381 398.032,148.668 398.404,147.039 398.491,145.587 398.489,145.525 398.273,144.16 397.757,142.97 396.947,141.977 \n",
       "  396.682,141.749 395.85,141.205 394.486,140.663 392.881,140.357 391.886,140.289 391.062,140.289 389.064,140.453 386.923,140.835 385.338,141.23 384.676,141.42 \n",
       "  382.362,142.191 380.016,143.124 378.412,143.843 377.673,144.197 375.366,145.386 373.12,146.671 371.714,147.542 370.962,148.03 368.912,149.442 366.987,150.893 \n",
       "  365.628,152 365.2,152.364 363.563,153.844 362.082,155.32 360.762,156.784 360.583,156.996 359.608,158.225 358.62,159.637 357.797,161.013 357.162,162.297 \n",
       "  357.139,162.349 356.645,163.637 356.313,164.875 356.138,166.057 356.12,167.178 356.156,167.61 356.255,168.234 356.542,169.22 356.977,170.131 357.559,170.962 \n",
       "  358.161,171.595 358.286,171.707 359.156,172.36 360.168,172.915 361.319,173.364 362.609,173.701 362.996,173.774 364.035,173.918 365.594,174.008 367.283,173.961 \n",
       "  369.096,173.771 370.478,173.54 371.03,173.429 373.077,172.925 375.226,172.255 377.463,171.413 379.776,170.392 380.153,170.211 382.148,169.188 384.557,167.8 \n",
       "  386.973,166.236 389.368,164.501 390.783,163.379 391.711,162.601 393.965,160.551 396.089,158.378 398.038,156.111 399.348,154.383 399.771,153.778 401.243,151.418 \n",
       "  402.414,149.077 403.247,146.803 403.367,146.368 403.705,144.643 403.769,142.648 403.425,140.863 403.242,140.378 402.666,139.332 401.5,138.088 399.951,137.154 \n",
       "  399.399,136.932 398.043,136.55 395.817,136.279 393.322,136.332 392.667,136.394 390.607,136.698 387.726,137.352 384.735,138.264 384.709,138.273 381.689,139.402 \n",
       "  378.634,140.73 376.606,141.71 375.619,142.214 372.682,143.82 369.856,145.518 368.822,146.178 367.17,147.279 364.646,149.079 362.299,150.9 361.72,151.374 \n",
       "  360.143,152.723 358.183,154.534 356.424,156.325 355.723,157.094 354.867,158.085 353.511,159.808 352.353,161.489 351.41,163.082 351.388,163.123 350.612,164.706 \n",
       "  350.02,166.236 349.606,167.709 349.365,169.122 349.361,169.164 349.293,170.473 349.385,171.757 349.637,172.97 350.007,174.018 350.047,174.11 350.613,175.171 \n",
       "  351.332,176.148 352.204,177.035 353.016,177.678 353.229,177.826 354.407,178.514 355.739,179.092 357.226,179.549 358.342,179.79 358.868,179.878 360.667,180.068 \n",
       "  362.624,180.109 364.736,179.988 366.024,179.84 367.004,179.693 369.425,179.213 371.99,178.534 374.693,177.645 375.987,177.159 377.523,176.532 380.465,175.186 \n",
       "  383.493,173.599 386.581,171.767 387.67,171.069 389.703,169.685 392.817,167.357 395.87,164.802 398.81,162.041 400.701,160.074 401.583,159.094 404.124,155.999 \n",
       "  406.365,152.814 408.239,149.598 408.731,148.612 409.678,146.414 410.627,143.34 411.041,140.455 411.054,139.634 410.879,137.837 410.127,135.558 408.791,133.675 \n",
       "  408.703,133.587 406.884,132.243 404.453,131.285 402.383,130.893 401.553,130.813 398.253,130.822 394.635,131.281 393.602,131.48 390.782,132.153 386.778,133.387 \n",
       "  384.056,134.386 382.704,134.926 378.635,136.713 374.634,138.69 374.531,138.744 370.76,140.805 367.053,143.014 365.426,144.042 363.55,145.274 360.276,147.556 \n",
       "  357.247,149.835 357.076,149.968 354.472,152.089 351.956,154.308 349.891,156.284 349.695,156.481 347.686,158.602 345.921,160.667 344.39,162.675 344.383,162.684 \n",
       "  343.082,164.624 341.987,166.515 341.094,168.348 340.734,169.206 340.392,170.123 339.87,171.839 339.52,173.497 339.347,174.913 339.333,175.096 339.302,176.635 \n",
       "  339.422,178.112 339.688,179.525 339.791,179.919 340.096,180.872 340.645,182.149 341.333,183.353 341.79,184.006 342.162,184.479 343.132,185.522 344.247,186.476 \n",
       "  345.318,187.217 345.511,187.335 346.927,188.091 348.501,188.735 350.241,189.257 350.495,189.319 352.152,189.646 354.242,189.89 356.517,189.974 357.536,189.958 \n",
       "  358.986,189.884 361.655,189.602 364.527,189.108 367.601,188.383 370.879,187.404 372.381,186.889 374.362,186.15 378.041,184.598 381.891,182.728 385.886,180.523 \n",
       "  389.724,178.144 390,177.963 394.196,175.033 398.408,171.74 402.558,168.099 405.82,164.91 406.573,164.127 410.358,159.86 413.8,155.373 415.753,152.451 \n",
       "  416.788,150.742 419.203,146.071 420.939,141.488 421.479,139.453 421.893,137.124 421.991,133.12 421.22,129.685 421.191,129.609 419.471,126.718 416.87,124.528 \n",
       "  415.762,123.94 413.443,123.105 409.292,122.462 406.387,122.447 404.54,122.578 399.325,123.398 395.194,124.423 393.794,124.834 388.089,126.785 383.616,128.589 \n",
       "  382.339,129.142 376.663,131.796 372.267,134.051 371.152,134.649 365.884,137.613 361.536,140.231 360.909,140.622 356.268,143.619 351.978,146.568 351.815,146.683 \n",
       "  348.05,149.44 344.48,152.224 343.695,152.862 341.26,154.91 338.374,157.498 337.181,158.629 335.803,159.989 333.525,162.389 331.518,164.705 330.899,165.467 \n",
       "  329.759,166.942 328.226,169.107 326.984,171.058 326.895,171.206 325.748,173.245 324.765,175.228 324.312,176.239 323.927,177.158 323.219,179.04 322.732,180.525 \n",
       "  322.626,180.875 322.136,182.666 321.787,184.17 321.736,184.414 321.417,186.121 321.24,187.262 321.169,187.787 320.985,189.413 320.959,189.698 320.859,190.999 \n",
       "  320.785,192.534 320.785,192.546 320.759,194.055 320.776,195.525 320.777,195.529 320.836,196.956 320.935,198.349 320.955,198.579 321.072,199.703 321.246,201.02 \n",
       "  321.342,201.629 321.459,202.297 321.709,203.536 321.983,204.674 321.999,204.737 322.33,205.898 322.705,207.019 322.978,207.738 323.126,208.1 323.598,209.14 \n",
       "  324.123,210.139 324.544,210.84 324.708,211.094 325.356,212.005 326.076,212.869 326.874,213.684 327.18,213.964 327.757,214.449 328.734,215.159 329.817,215.81 \n",
       "  331.016,216.398 331.368,216.548 332.342,216.918 333.809,217.362 335.433,217.721 337.068,217.969 337.231,217.988 339.218,218.15 341.415,218.193 343.845,218.099 \n",
       "  344.981,218.012 346.527,217.852 349.487,217.429 352.749,216.799 356.338,215.932 356.343,215.931 360.28,214.795 364.597,213.342 369.305,211.521 373.241,209.822 \n",
       "  374.423,209.282 379.964,206.565 385.909,203.303 392.234,199.432 394.761,197.768 398.911,194.883 405.861,189.602 412.96,183.557 415.789,180.942 420.08,176.72 \n",
       "  427.019,169.124 433.522,160.873 434.864,158.988 439.322,152.091 444.101,143.028 445.499,139.78 447.538,133.988 449.342,125.35 449.557,121.352 449.276,117.532 \n",
       "  447.204,110.936 445.715,108.597 443.11,105.914 437.118,102.694 435.431,102.205 429.469,101.374 421.078,101.797 420.506,101.876 410.62,103.998 405.324,105.615 \n",
       "  400.215,107.427 389.657,111.806 389.626,111.82 379.26,116.778 374.833,119.052 369.26,122.025 362.266,125.905 359.826,127.29 351.06,132.387 347.316,134.611 \n",
       "  343.012,137.194 335.689,141.646 331.001,144.527 329.072,145.718 323.125,149.413 317.795,152.758 314.843,154.626 313.026,155.783 308.762,158.525 304.942,161.021 \n",
       "  304.794,161.119 301.511,163.305 298.415,165.406 297.053,166.344 295.605,167.349 293.035,169.155 290.662,170.838 289.551,171.629 288.449,172.412 286.359,173.885 \n",
       "  284.364,175.262 284.219,175.361 282.433,176.547 280.544,177.74 279.208,178.536 278.675,178.84 276.805,179.844 274.92,180.748 273.998,181.148 273.005,181.547 \n",
       "  271.05,182.233 269.045,182.799 268.165,183.004 266.985,183.236 264.865,183.536 262.687,183.686 261.325,183.701 260.452,183.678 258.163,183.5 255.834,183.142 \n",
       "  253.475,182.595 253.231,182.527 251.099,181.846 248.73,180.892 246.394,179.726 244.216,178.414 244.116,178.348 241.926,176.754 239.861,174.955 237.959,172.966 \n",
       "  236.519,171.167 236.256,170.802 234.792,168.484 233.602,166.049 232.721,163.535 232.527,162.795 232.18,160.984 232.001,158.444 232.195,155.968 232.409,154.879 \n",
       "  232.773,153.608 233.729,151.414 235.045,149.432 235.893,148.47 236.701,147.704 238.665,146.264 240.893,145.131 242.249,144.636 243.343,144.324 245.966,143.845 \n",
       "  248.713,143.687 249.862,143.713 251.532,143.839 254.379,144.281 257.21,144.987 257.627,145.113 259.986,145.93 262.672,147.08 265.07,148.311 265.242,148.408 \n",
       "  267.671,149.882 269.942,151.477 271.822,152.98 272.041,153.167 273.957,154.927 275.687,156.74 277.227,158.585 277.449,158.874 278.574,160.448 279.732,162.314 \n",
       "  280.703,164.175 281.353,165.667 281.49,166.018 282.096,167.836 282.527,169.621 282.787,171.369 282.88,173.072 282.88,173.149 282.809,174.726 282.579,176.327 \n",
       "  282.191,177.87 281.648,179.35 281.591,179.481 280.953,180.762 280.106,182.102 279.107,183.364 277.957,184.542 277.789,184.696 276.657,185.63 275.205,186.619 \n",
       "  273.602,187.501 271.849,188.267 271.514,188.393 269.945,188.906 267.893,189.407 265.697,189.755 263.361,189.937 262.812,189.953 260.891,189.937 258.297,189.74 \n",
       "  255.596,189.328 252.805,188.683 252.031,188.465 249.941,187.789 247.032,186.63 244.118,185.196 241.235,183.476 240.332,182.868 238.423,181.46 235.735,179.152 \n",
       "  233.228,176.568 230.96,173.728 230.261,172.718 228.99,170.651 227.378,167.385 226.176,163.988 225.653,161.809 225.436,160.52 225.2,157.054 225.486,153.67 \n",
       "  225.855,151.935 226.31,150.45 227.666,147.474 229.526,144.811 230.216,144.043 231.858,142.527 234.605,140.668 237.702,139.261 237.765,139.238 241.08,138.325 \n",
       "  244.663,137.853 246.714,137.786 248.376,137.828 252.148,138.221 255.894,138.986 255.913,138.991 259.612,140.094 263.197,141.482 264.81,142.21 266.627,143.108 \n",
       "  269.869,144.927 272.902,146.897 273.114,147.045 275.707,148.979 278.277,151.143 280.433,153.184 280.608,153.36 282.697,155.606 284.55,157.862 286.172,160.113 \n",
       "  286.257,160.24 287.571,162.346 288.754,164.551 289.732,166.722 290.154,167.812 290.513,168.851 291.107,170.934 291.522,172.968 291.765,174.951 291.842,176.88 \n",
       "  291.842,177.015 291.762,178.752 291.527,180.567 291.141,182.322 290.676,183.823 290.606,184.016 289.925,185.646 289.097,187.212 288.12,188.708 287.748,189.208 \n",
       "  286.994,190.132 285.715,191.48 284.279,192.745 282.747,193.879 282.683,193.922 280.921,195.004 278.988,195.981 276.88,196.84 274.694,197.542 274.592,197.571 \n",
       "  272.117,198.161 269.455,198.59 266.607,198.837 263.574,198.882 261.788,198.81 260.356,198.702 256.962,198.27 253.414,197.556 249.736,196.53 245.954,195.163 \n",
       "  245.617,195.026 242.096,193.423 238.222,191.285 234.396,188.734 230.687,185.756 230.283,185.396 227.169,182.342 223.939,178.515 221.108,174.333 221.095,174.311 \n",
       "  218.735,169.774 216.955,164.994 215.938,160.674 215.84,160.068 215.469,155.112 215.88,150.262 216.355,148.044 217.1,145.658 219.118,141.436 221.684,137.943 \n",
       "  221.888,137.717 225.346,134.607 229.39,132.17 230.693,131.581 233.909,130.45 238.781,129.448 241.359,129.205 243.881,129.137 249.089,129.468 252.354,129.968 \n",
       "  254.298,130.367 259.413,131.757 263.111,133.059 264.358,133.552 269.072,135.67 273.267,137.893 273.514,138.034 277.65,140.575 281.467,143.238 282.464,143.987 \n",
       "  284.957,145.97 288.122,148.735 290.22,150.744 290.97,151.502 293.514,154.248 295.77,156.956 296.107,157.386 297.756,159.613 299.489,162.211 300.373,163.665 \n",
       "  300.99,164.746 302.278,167.214 303.288,169.423 303.369,169.614 304.281,171.945 305.029,174.209 305.145,174.6 305.628,176.407 306.091,178.54 306.278,179.603 \n",
       "  306.429,180.611 306.651,182.621 306.761,184.447 306.766,184.573 306.78,186.469 306.7,188.311 306.635,189.112 306.529,190.102 306.271,191.842 305.926,193.533 \n",
       "  305.909,193.605 305.496,195.178 304.98,196.777 304.524,197.975 304.376,198.332 303.681,199.843 302.892,201.311 302.269,202.331 302.003,202.737 301.008,204.12 \n",
       "  299.9,205.459 298.67,206.752 298.655,206.767 297.311,207.999 295.81,209.196 294.156,210.339 292.407,211.382 292.335,211.422 290.336,212.441 288.143,213.385 \n",
       "  285.739,214.242 283.108,214.998 281.391,215.403 280.234,215.64 277.1,216.15 273.692,216.497 269.996,216.649 266.283,216.589 266,216.577 261.691,216.242 \n",
       "  257.074,215.591 252.165,214.564 246.978,213.104 246.055,212.804 241.534,211.147 235.895,208.619 230.145,205.454 226.46,203.072 224.369,201.583 218.691,196.948 \n",
       "  213.282,191.539 209.189,186.548 208.317,185.35 203.992,178.419 200.523,170.866 200.152,169.862 198.122,162.84 196.964,154.579 196.887,150.939 197.196,146.353 \n",
       "  198.89,138.471 200.457,134.377 202.049,131.25 206.593,124.974 209.668,121.973 212.364,119.872 219.147,116.092 222.392,114.861 226.688,113.686 234.72,112.616 \n",
       "  236.309,112.555 242.985,112.772 250.224,113.779 251.256,113.983 259.341,116.056 263.5,117.436 267.093,118.79 274.41,121.995 275.431,122.489 281.224,125.505 \n",
       "  286.997,128.87 287.504,129.183 293.238,132.918 298.441,136.636 300.254,138.008 303.133,140.273 307.349,143.793 311.127,147.175 312,147.989 314.508,150.403 \n",
       "  317.534,153.473 320.087,156.209 320.247,156.385 322.688,159.142 324.894,161.75 326.629,163.874 326.902,164.213 328.747,166.538 330.46,168.731 331.177,169.652 \n",
       "  332.069,170.796 333.602,172.737 334.633,174.012 335.084,174.558 336.537,176.262 337.982,177.85 338.116,177.99 339.44,179.321 340.929,180.677 341.879,181.461 \n",
       "  342.465,181.914 344.066,183.031 345.745,184.023 346.273,184.299 347.519,184.885 349.401,185.611 351.403,186.192 351.747,186.274 353.538,186.62 355.819,186.884 \n",
       "  358.253,186.97 358.892,186.962 360.852,186.866 363.622,186.556 366.566,186.023 368.511,185.565 369.689,185.25 372.989,184.216 376.456,182.905 380.076,181.298 \n",
       "  381.614,180.543 383.837,179.376 387.711,177.124 391.652,174.54 395.61,171.623 397.82,169.841 399.538,168.372 403.362,164.803 406.994,160.964 410.34,156.905 \n",
       "  411.643,155.141 413.307,152.681 415.788,148.384 417.688,144.121 418.461,141.795 418.91,140.004 419.378,136.157 419.046,132.699 418.894,132.115 417.878,129.748 \n",
       "  415.891,127.392 414.212,126.232 413.123,125.7 409.641,124.709 405.55,124.412 405.548,124.412 400.959,124.788 395.999,125.768 394.903,126.049 390.798,127.277 \n",
       "  385.479,129.223 383.784,129.916 380.156,131.508 374.925,134.042 372.837,135.127 369.866,136.741 365.042,139.533 362.443,141.126 360.498,142.361 356.266,145.178 \n",
       "  352.959,147.518 352.359,147.956 348.788,150.669 345.548,153.306 344.853,153.898 342.632,155.859 340.026,158.328 338.658,159.712 337.712,160.713 335.673,163.017 \n",
       "  334.544,164.4 333.889,165.245 332.34,167.402 331.606,168.522 331.007,169.493 329.87,171.522 329.422,172.41 328.913,173.494 328.118,175.41 327.885,176.043 \n",
       "  327.472,177.275 326.96,179.09 326.861,179.493 326.57,180.857 326.293,182.577 326.267,182.773 326.118,184.251 326.04,185.879 326.039,185.888 326.05,187.462 \n",
       "  326.146,188.998 326.218,189.702 326.322,190.489 326.577,191.932 326.909,193.328 326.913,193.342 327.319,194.675 327.806,195.971 328.19,196.837 328.373,197.215 \n",
       "  329.023,198.405 329.76,199.539 330.245,200.191 330.588,200.614 331.514,201.627 332.545,202.573 333.533,203.339 333.689,203.449 334.954,204.249 336.352,204.967 \n",
       "  337.895,205.595 339.218,206.02 339.594,206.124 341.462,206.546 343.516,206.848 345.772,207.013 348.246,207.027 348.86,207.007 350.956,206.874 353.919,206.529 \n",
       "  357.154,205.965 360.676,205.153 362.054,204.785 364.503,204.065 368.649,202.661 373.115,200.9 377.899,198.737 381.043,197.166 383.004,196.128 388.411,193.025 \n",
       "  394.069,189.389 399.921,185.182 401.457,184 405.905,180.371 411.9,174.954 417.758,168.963 420.105,166.326 423.323,162.435 428.394,155.474 432.746,148.25 \n",
       "  434.539,144.685 436.153,140.943 438.396,133.805 439.281,127.458 439.291,127.119 438.686,121.177 436.533,116.241 436.374,116.01 432.836,112.541 427.717,110.202 \n",
       "  427.521,110.148 421.353,109.281 414.782,109.606 413.999,109.707 405.931,111.347 400.61,112.909 397.436,113.989 388.782,117.395 386.356,118.455 380.201,121.322 \n",
       "  372.712,125.109 371.879,125.549 363.956,129.887 360.414,131.913 356.525,134.193 349.641,138.364 348.482,139.083 343.328,142.335 337.583,146.074 334.713,147.991 \n",
       "  332.388,149.569 327.713,152.824 323.518,155.856 321.827,157.114 319.762,158.683 316.399,161.328 313.386,163.814 313.158,164.007 310.677,166.161 308.231,168.387 \n",
       "  306.306,170.219 306.009,170.507 303.974,172.535 302.091,174.48 301.031,175.6 300.328,176.351 298.655,178.152 297.176,179.749 297.047,179.888 295.477,181.56 \n",
       "  293.924,183.169 293.359,183.739 292.365,184.716 290.782,186.197 289.267,187.518 289.157,187.61 287.472,188.952 285.712,190.216 284.514,190.998 283.863,191.397 \n",
       "  281.91,192.488 279.842,193.478 278.629,193.986 277.648,194.359 275.316,195.118 272.84,195.741 271.01,196.089 270.213,196.212 267.429,196.515 264.489,196.628 \n",
       "  261.396,196.528 260.886,196.491 258.151,196.195 254.771,195.6 251.276,194.717 247.69,193.521 247.467,193.437 244.037,191.982 240.368,190.079 236.741,187.799 \n",
       "  233.213,185.13 232.296,184.349 229.85,182.06 226.736,178.607 223.957,174.802 223.418,173.952 221.601,170.683 219.755,166.318 218.497,161.797 218.357,161.074 \n",
       "  217.9,157.212 218.009,152.683 218.643,149.099 218.851,148.334 220.432,144.287 222.718,140.653 223.679,139.495 225.661,137.536 229.184,135.007 232.306,133.458 \n",
       "  233.189,133.112 237.573,131.875 242.224,131.28 242.559,131.261 247.03,131.297 251.892,131.87 253.135,132.099 256.714,132.935 261.421,134.419 263.481,135.202 \n",
       "  265.948,136.247 270.246,138.348 273.242,140.031 274.28,140.657 278.026,143.116 281.473,145.676 282.069,146.151 284.614,148.295 287.453,150.94 289.507,153.052 \n",
       "  290,153.587 292.263,156.214 294.257,158.808 295.199,160.146 296,161.356 297.505,163.851 298.79,166.29 299.492,167.791 299.872,168.668 300.765,170.984 \n",
       "  301.482,173.237 301.991,175.215 302.039,175.428 302.445,177.558 302.712,179.627 302.823,181.09 302.848,181.637 302.86,183.589 302.755,185.485 302.661,186.401 \n",
       "  302.537,187.327 302.209,189.114 301.772,190.85 301.62,191.363 301.229,192.533 300.576,194.166 299.813,195.748 299.63,196.09 298.937,197.279 297.942,198.759 \n",
       "  296.824,200.185 296.371,200.706 295.575,201.558 294.187,202.873 292.652,204.127 291.066,205.245 290.959,205.314 289.099,206.432 287.058,207.469 284.824,208.415 \n",
       "  282.384,209.259 281.461,209.533 279.725,209.99 276.832,210.59 273.696,211.033 270.308,211.29 266.66,211.334 264.016,211.227 262.742,211.136 258.55,210.658 \n",
       "  254.103,209.847 249.423,208.653 244.532,207.024 244.056,206.845 239.46,204.904 234.28,202.238 229.081,198.977 225.696,196.488 223.951,195.074 219.011,190.496 \n",
       "  214.418,185.262 210.329,179.398 210.324,179.39 206.915,172.956 204.346,166.075 203,160.323 202.782,158.916 202.36,151.678 203.155,144.609 203.349,143.726 \n",
       "  205.215,137.961 208.497,131.976 209.632,130.441 212.918,126.875 218.327,122.81 220.311,121.707 224.537,119.881 231.337,118.106 233.045,117.846 238.507,117.443 \n",
       "  245.843,117.791 246.209,117.833 253.158,119.015 259.077,120.576 260.302,120.958 267.154,123.462 271.143,125.194 273.63,126.375 279.673,129.566 281.696,130.737 \n",
       "  285.252,132.922 290.359,136.356 292.392,137.824 294.999,139.796 299.19,143.195 302.957,146.518 303.649,147.159 306.331,149.74 309.345,152.847 312.035,155.834 \n",
       "  312.146,155.962 314.435,158.695 316.579,161.432 318.497,164.044 318.501,164.049 320.231,166.547 321.798,168.932 322.149,169.482 323.23,171.208 324.552,173.378 \n",
       "  325.15,174.377 325.787,175.447 326.956,177.419 327.418,178.193 328.08,179.295 329.177,181.079 329.543,181.659 330.263,182.772 331.355,184.375 331.65,184.787 \n",
       "  332.469,185.889 333.618,187.313 333.849,187.582 334.816,188.646 336.077,189.887 337.156,190.824 337.415,191.032 338.841,192.078 340.371,193.02 341.287,193.505 \n",
       "  342.016,193.853 343.791,194.568 345.709,195.157 346.785,195.413 347.783,195.609 350.029,195.914 352.458,196.055 354.36,196.046 355.085,196.017 357.924,195.783 \n",
       "  360.984,195.33 364.274,194.635 365.084,194.433 367.806,193.674 371.58,192.418 375.59,190.837 379.829,188.903 380.488,188.579 384.289,186.584 388.935,183.853 \n",
       "  393.717,180.694 398.584,177.09 398.925,176.821 403.474,173.026 408.282,168.528 412.892,163.64 415.487,160.568 417.182,158.405 421.009,152.915 424.218,147.298 \n",
       "  424.571,146.587 426.654,141.696 428.182,136.287 428.634,132.922 428.682,131.257 428.07,126.796 426.411,123.195 426.323,123.068 423.448,120.222 419.529,118.334 \n",
       "  418.855,118.136 414.68,117.449 409.067,117.533 407.724,117.678 402.872,118.513 396.287,120.268 395.232,120.604 389.5,122.655 382.681,125.521 382.59,125.562 \n",
       "  375.977,128.716 370.367,131.64 369.501,132.11 363.345,135.59 359.02,138.175 357.567,139.07 352.205,142.485 349.251,144.449 347.276,145.796 342.781,148.976 \n",
       "  339.42,151.473 338.709,152.016 335.042,154.913 331.755,157.673 329.812,159.395 328.819,160.305 326.205,162.82 323.88,165.229 322.654,166.581 321.815,167.544 \n",
       "  319.978,169.776 318.343,171.933 317.196,173.56 316.882,174.024 315.569,176.055 314.389,178.02 314.382,178.032 313.299,179.959 312.3,181.839 312.097,182.233 \n",
       "  311.367,183.676 310.49,185.457 310.483,185.472 309.634,187.228 308.993,188.555 308.804,188.945 307.981,190.624 307.526,191.532 307.151,192.266 306.304,193.87 \n",
       "  306.019,194.391 305.428,195.438 304.511,196.969 303.74,198.169 303.544,198.462 302.514,199.917 301.412,201.332 300.932,201.905 300.225,202.707 298.943,204.039 \n",
       "  297.553,205.326 297.23,205.604 296.043,206.565 294.4,207.752 292.609,208.88 292.155,209.142 290.656,209.945 288.527,210.938 286.205,211.847 284.992,212.258 \n",
       "  283.674,212.661 280.919,213.366 277.922,213.94 274.67,214.363 274.653,214.364 271.146,214.61 267.34,214.646 263.249,214.43 259.356,213.994 258.867,213.924 \n",
       "  254.191,213.079 249.248,211.836 244.07,210.133 239.645,208.338 238.693,207.909 233.168,205.094 227.6,201.634 222.098,197.477 221.031,196.572 216.783,192.576 \n",
       "  211.825,186.934 207.411,180.595 205.752,177.714 203.73,173.612 200.979,166.122 199.334,158.325 199.254,157.676 198.957,150.442 199.932,142.756 200.514,140.47 \n",
       "  202.302,135.561 206.021,129.132 207.782,126.923 210.979,123.713 217,119.474 219.323,118.286 223.863,116.513 231.325,114.834 232.706,114.664 239.137,114.372 \n",
       "  246.394,114.904 247.074,114.997 254.933,116.542 259.673,117.874 262.557,118.823 269.823,121.659 271.972,122.616 276.648,124.881 282.168,127.876 282.983,128.348 \n",
       "  288.803,131.939 294.109,135.568 294.475,135.831 298.913,139.161 303.242,142.677 306.733,145.726 307.128,146.083 310.606,149.356 313.717,152.489 315.589,154.48 \n",
       "  316.499,155.478 318.992,158.321 321.232,161.024 322.277,162.332 323.257,163.589 325.099,166.022 326.106,167.388 326.79,168.328 328.359,170.512 329.226,171.728 \n",
       "  329.833,172.579 331.237,174.533 332.023,175.608 332.593,176.375 333.924,178.11 334.681,179.053 335.248,179.738 336.585,181.259 337.355,182.074 337.952,182.674 \n",
       "  339.365,183.981 340.185,184.668 340.839,185.178 342.39,186.261 344.033,187.226 344.558,187.498 345.78,188.067 347.647,188.778 349.645,189.348 350.055,189.443 \n",
       "  351.789,189.769 354.089,190.029 356.558,190.114 357.389,190.101 359.206,190.009 362.042,189.697 365.071,189.159 367.455,188.6 368.3,188.375 371.73,187.324 \n",
       "  375.354,185.983 379.159,184.329 381.442,183.22 383.134,182.341 387.257,179.998 391.48,177.29 395.752,174.213 398.537,172.008 400.023,170.76 404.219,166.941 \n",
       "  408.24,162.802 411.983,158.394 413.386,156.558 415.344,153.771 418.2,149.032 420.436,144.299 421.342,141.8 421.938,139.697 422.612,135.367 422.397,131.453 \n",
       "  422.313,131.04 421.245,128.095 419.168,125.4 417.818,124.344 416.204,123.455 412.432,122.304 409.018,121.958 407.965,121.946 402.935,122.351 397.949,123.326 \n",
       "  397.491,123.439 391.781,125.121 386.283,127.149 385.947,127.285 380.12,129.816 374.751,132.43 374.407,132.607 368.898,135.56 363.768,138.53 363.66,138.595 \n",
       "  358.741,141.646 354.17,144.666 353.719,144.974 349.965,147.62 346.126,150.489 345.16,151.24 342.649,153.259 339.52,155.928 338.317,157.008 336.723,158.495 \n",
       "  334.235,160.966 332.034,163.346 331.541,163.91 330.097,165.641 328.4,167.86 326.92,170.009 326.184,171.172 325.634,172.093 324.523,174.118 323.567,176.089 \n",
       "  323.261,176.778 322.749,178.009 322.052,179.881 321.623,181.184 321.462,181.709 320.966,183.493 320.607,184.989 320.553,185.235 320.212,186.937 319.971,188.362 \n",
       "  319.934,188.6 319.713,190.224 319.586,191.349 319.54,191.809 319.411,193.357 319.375,193.904 319.321,194.868 319.27,196.202 319.266,196.342 319.242,197.779 \n",
       "  319.243,198.789 319.248,199.18 319.28,200.545 319.319,201.495 319.338,201.875 319.42,203.169 319.51,204.249 319.526,204.428 319.657,205.652 319.811,206.841 \n",
       "  319.837,207.022 319.991,207.996 320.196,209.116 320.347,209.838 320.429,210.203 320.692,211.254 320.986,212.271 321.131,212.723 321.315,213.254 321.68,214.202 \n",
       "  322.088,215.114 322.39,215.713 322.54,215.99 323.042,216.829 323.599,217.63 324.217,218.391 324.503,218.706 324.902,219.112 325.662,219.79 326.506,220.421 \n",
       "  327.442,221.004 327.529,221.053 328.48,221.535 329.633,222.008 330.913,222.419 331.658,222.611 332.333,222.76 333.91,223.025 335.661,223.203 337.414,223.279 \n",
       "  337.605,223.282 339.762,223.252 342.155,223.093 344.809,222.785 345.573,222.674 347.748,222.309 351,221.636 354.594,220.73 357.374,219.925 358.558,219.557 \n",
       "  362.922,218.073 367.706,216.223 372.932,213.947 374.809,213.068 378.618,211.185 384.765,207.862 391.346,203.902 396.495,200.506 398.333,199.229 405.669,193.768 \n",
       "  413.225,187.473 417.995,183.103 420.865,180.306 428.391,172.274 435.525,163.473 437.777,160.372 441.972,154.027 447.378,144.189 449.304,139.887 451.374,134.294 \n",
       "  453.618,124.762 453.998,120.005 453.824,116.07 451.814,108.691 450.339,106.146 447.552,103.045 441.166,99.4118 439.884,99.0051 432.923,97.9168 425.052,98.2299 \n",
       "  423.216,98.4738 412.492,100.848 408.664,102.016 401.213,104.672 392.323,108.381 389.795,109.526 378.587,114.995 376.964,115.833 367.851,120.711 364.153,122.765 \n",
       "  357.766,126.388 348.431,131.821 348.26,131.922 339.895,136.879 332.153,141.505 331.06,142.159 325.181,145.674 318.929,149.405 313.335,152.732 312.718,153.099 \n",
       "  308.34,155.694 303.878,158.336 301.84,159.543 299.885,160.702 296.3,162.831 293.682,164.39 293.069,164.756 290.138,166.506 287.461,168.103 285.179,169.456 \n",
       "  284.996,169.564 282.703,170.904 280.548,172.132 279.373,172.783 278.501,173.254 276.536,174.274 274.63,175.193 274.145,175.414 272.762,176.011 270.917,176.725 \n",
       "  269.082,177.332 268.932,177.377 267.245,177.829 265.399,178.209 263.539,178.467 263.367,178.485 261.663,178.598 259.771,178.594 257.868,178.451 257.21,178.368 \n",
       "  255.959,178.162 254.052,177.722 252.162,177.129 250.495,176.465 250.301,176.379 248.485,175.471 246.736,174.408 245.074,173.197 243.976,172.265 243.522,171.843 \n",
       "  242.105,170.356 240.846,168.754 239.77,167.057 239.36,166.285 238.899,165.284 238.255,163.463 237.852,161.625 237.706,159.896 237.704,159.799 237.824,158.019 \n",
       "  238.21,156.318 238.859,154.728 239.148,154.202 239.765,153.28 240.913,152 242.28,150.908 243.46,150.212 243.843,150.024 245.577,149.362 247.449,148.924 \n",
       "  249.426,148.711 249.537,148.706 251.476,148.722 253.568,148.946 255.671,149.37 256,149.454 257.754,149.98 259.794,150.758 261.766,151.686 262.224,151.927 \n",
       "  263.649,152.744 265.428,153.913 267.088,155.175 267.817,155.789 268.618,156.513 270.007,157.909 271.253,159.35 272.335,160.802 272.349,160.822 273.291,162.31 \n",
       "  274.08,163.805 274.714,165.297 275.177,166.707 275.196,166.776 275.524,168.233 275.702,169.661 275.731,171.054 275.613,172.405 275.463,173.238 275.351,173.708 \n",
       "  274.945,174.955 274.399,176.142 273.715,177.261 272.893,178.307 271.938,179.272 271.545,179.613 270.85,180.151 269.632,180.936 268.287,181.619 266.821,182.19 \n",
       "  265.263,182.636 265.236,182.642 263.536,182.966 261.731,183.151 259.83,183.187 257.843,183.065 256.441,182.884 255.781,182.773 253.658,182.302 251.496,181.644 \n",
       "  249.317,180.793 247.145,179.743 246.194,179.214 245.004,178.488 242.925,177.028 240.945,175.374 239.098,173.536 237.805,172.02 237.421,171.524 235.951,169.357 \n",
       "  234.723,167.067 233.771,164.69 233.488,163.766 233.128,162.259 232.818,159.82 232.853,157.421 233.055,155.974 233.248,155.11 234.005,152.936 235.108,150.942 \n",
       "  236.168,149.575 236.543,149.171 238.286,147.659 240.299,146.428 242.192,145.617 242.543,145.497 244.975,144.879 247.548,144.568 249.574,144.534 250.216,144.559 \n",
       "  252.933,144.836 255.658,145.378 257.147,145.781 258.351,146.162 260.975,147.159 263.504,148.343 264.418,148.827 265.91,149.685 268.174,151.158 270.282,152.738 \n",
       "  271.014,153.341 272.219,154.401 273.978,156.124 275.556,157.892 276.502,159.081 276.95,159.687 278.157,161.493 279.182,163.301 280.025,165.098 280.285,165.743 \n",
       "  280.69,166.876 281.181,168.626 281.5,170.343 281.653,172.02 281.665,173.096 281.643,173.651 281.473,175.23 281.146,176.755 280.664,178.218 280.177,179.325 \n",
       "  280.03,179.616 279.245,180.943 278.309,182.194 277.224,183.362 276.096,184.358 275.991,184.442 274.608,185.425 273.078,186.304 271.402,187.068 269.579,187.707 \n",
       "  269.484,187.736 267.612,188.213 265.506,188.57 263.266,188.766 260.899,188.787 260.457,188.77 258.413,188.617 255.823,188.24 253.148,187.642 250.408,186.806 \n",
       "  249.542,186.494 247.626,185.718 244.833,184.366 242.072,182.744 239.382,180.849 238.222,179.919 236.807,178.676 234.398,176.236 232.213,173.555 230.305,170.658 \n",
       "  229.814,169.788 228.731,167.577 227.541,164.365 226.78,161.083 226.577,159.485 226.487,157.796 226.689,154.577 227.391,151.501 227.801,150.355 228.596,148.642 \n",
       "  230.284,146.069 232.417,143.836 232.93,143.406 234.953,141.998 237.833,140.579 240.778,139.648 240.991,139.597 244.36,139.056 247.868,138.939 249.606,139.028 \n",
       "  251.449,139.222 255.039,139.873 258.493,140.821 258.584,140.85 262.031,142.11 265.343,143.611 267.017,144.48 268.486,145.308 271.436,147.163 274.175,149.139 \n",
       "  274.827,149.65 276.691,151.204 278.978,153.329 281.036,155.494 281.515,156.039 282.864,157.675 284.467,159.858 285.851,162.03 286.528,163.229 287.023,164.181 \n",
       "  287.992,166.3 288.765,168.382 289.35,170.423 289.484,171 289.755,172.417 289.988,174.36 290.053,176.251 289.969,177.957 289.957,178.087 289.704,179.865 \n",
       "  289.298,181.584 288.739,183.239 288.354,184.154 288.031,184.831 287.173,186.354 286.163,187.806 285.001,189.181 284.796,189.399 283.685,190.476 282.21,191.685 \n",
       "  280.574,192.799 279.224,193.575 278.772,193.811 276.801,194.711 274.657,195.487 272.336,196.125 271.478,196.314 269.837,196.611 267.158,196.927 264.303,197.053 \n",
       "  261.445,196.978 261.276,196.967 258.081,196.646 254.735,196.064 251.263,195.193 247.69,194.007 245.915,193.305 244.042,192.479 240.361,190.581 236.711,188.302 \n",
       "  233.154,185.63 231.116,183.857 229.755,182.554 226.596,179.082 223.767,175.252 221.359,171.108 220.904,170.18 219.46,166.701 218.153,162.124 217.503,157.482 \n",
       "  217.472,156.909 217.574,152.885 218.385,148.462 219.428,145.459 219.942,144.337 222.223,140.63 225.169,137.437 225.878,136.83 228.708,134.843 232.743,132.892 \n",
       "  235.318,132.05 237.166,131.605 241.863,130.976 245.836,130.923 246.724,130.965 251.641,131.525 256.45,132.564 256.524,132.583 261.289,134.069 265.875,135.905 \n",
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      ]
     },
     "execution_count": 8,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "# define Lorenz dynamics dx/dt = lorenz(t,x)\n",
    "function lorenz(t, x)\n",
    "    σ = 10\n",
    "    β = 8/3\n",
    "    ρ = 28\n",
    "    [σ*(x[2]-x[1]); x[1]*(ρ-x[3]); x[1]*x[2] - β*x[3]]\n",
    "end\n",
    "\n",
    "# integrate dx/dt = lorenz(t,x) numerically from t=0 to t=50 and starting point x₀\n",
    "tspan = 0:0.01:50.0\n",
    "x₀ = [2.0; 0.0; 0.0]\n",
    "t,x = ode45(lorenz, x₀, tspan)\n",
    "y = x\n",
    "x = hcat(x...)';\n",
    "\n",
    "# plot Lorenz attractor\n",
    "plot(x[:,1], x[:,2], x[:,3], l=:path3d, xlab=\"x\", ylab=\"y\", zlab=\"z\", label=\"\")"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### Don't try this in Matlab..."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 9,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "5 ⊕ 1 = 17\n",
      "\n",
      "2 // 5 ⊕ (3 - 7im) = 36//5 - 14//1*im\n",
      "\n",
      "BigFloat(π) = 3.141592653589793238462643383279502884197169399375105820974944592307816406286208998628034825342117067982148086513282306647093844609550582231725359408128481117450284102701938521105559644622948954930381964428810975665933446128475648233786783165271201909145648566923460348610454326648213393607260249141273724586997\n"
     ]
    }
   ],
   "source": [
    "⊕ = (x,y) -> 3x + 2y       # define ⊕ operator (unicode program text)\n",
    "\n",
    "@show 5 ⊕ 1;   println()   # apply ⊕ to two Int64s\n",
    "\n",
    "@show 2//5 ⊕ (3 - 7im);    # Rational{Int64} ⊕ Complex{Int64} -> Complex{Rational{Int64}\n",
    "\n",
    "println(); setprecision(1024);\n",
    "\n",
    "@show BigFloat(π);         # π as a 1024-bit floating-point number"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "<br> <br> <br> <br> <br>\n",
    "\n",
    "\n",
    "# Julia: fast as Fortran\n",
    "\n",
    "Unlike Matlab, Julia runs as fast as Fortran. A demo on a nonlinear PDE:\n",
    "\n",
    "### Kuramoto-Sivashinsky equation\n",
    "\n",
    "The Kuramoto-Sivashinsky (KS) equation is a nonlinear time-evolving partial differential equation (PDE) on a 1d spatial domain.\n",
    "\n",
    "\\begin{equation*}\n",
    "u_t = - u_{xx} - u_{xxxx} - u u_x\n",
    "\\end{equation*}\n",
    "\n",
    "where $x$ is space, $t$ is time, and subscripts indicate differentiation. We assume a spatial domain $x \\in [0, L_x]$ with periodic boundary conditions and initial condition $u(x,0) = u_0(x)$.\n",
    "\n",
    "<img style=\"float:center\" src=\"figs/ksdynamics.png\"/>"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "\n",
    "### Numerical integration algorithm: KS-CNAB2\n",
    "\n",
    "Represent the linear terms as $Lu = (-D^2 -D^4)\\, u$ and the nonlinear term as $N(u) = 1/2\\, D \\,u^2$ where $D = \\partial/\\partial x$.\n",
    "\n",
    "\\begin{equation*}\n",
    "u_t = Lu + N(u)\n",
    "\\end{equation*}\n",
    "\n",
    "Discretize time by letting $u^n(x) = u(x, n\\Delta t)$ for some small $\\Delta t$, and use Crank-Nicolson, Adams-Bashforth (CNAB2) timestepping. \n",
    "\n",
    "\\begin{equation*}\n",
    "\\frac{u^{n+1} - u^n}{\\Delta t} = L\\left(\\frac{u^{n+1} + u^n}{2} \\right) + \\frac{3}{2} N(u^n) - \\frac{1}{2} N(u^{n-1})\n",
    "\\end{equation*}\n",
    "\n",
    "\n",
    "Put the unknowns (future) $u^{n+1}$'s on the LHS and knowns (past) $u^{n}$ and past $u^{n+1}$ on the RHS.\n",
    "\n",
    "\\begin{equation*}\n",
    "\\left(I  - \\frac{\\Delta t}{2} L \\right) u^{n+1} = \\left(I  + \\frac{\\Delta t}{2}L \\right) u^{n} + \\frac{3 \\Delta t}{2} N(u^n) - \\frac{\\Delta t}{2} N(u^{n-1})\n",
    "\\end{equation*}\n",
    "\n",
    "Discretize space with a finite Fourier transform, so that $u$ becomes a vector of Fourier coefficients, $(I  \\pm \\Delta t/2 \\, L)$ becomes the diagonal matrices $A$ and $B$, and $N^n$ is the Fourier transform of $N(u^n)$.\n",
    "\n",
    "\\begin{equation*}\n",
    "A \\, u^{n+1} = B \\, u^n + \\frac{3 \\Delta t}{2} N^n -  \\frac{\\Delta t}{2}N^{n-1}\n",
    "\\end{equation*}\n",
    "\n",
    "This is an $Ax=b$ problem whose iteration approximates the time-evolution of the Kuramoto-Sivashinksy PDE. \n",
    "\n",
    "### Julia code for KS-CNAB2 algorithm"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 10,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "ksintegrate (generic function with 1 method)"
      ]
     },
     "execution_count": 10,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "function ksintegrate(u₀, Lₓ, Δt, Nsteps, nsave);\n",
    "    uⁿ = (1+0im)*u₀                      # force u to be complex\n",
    "    Nₓ = length(uⁿ)                       # number of gridpoints\n",
    "    kₓ = vcat(0:Nₓ/2-1, 0:0, -Nₓ/2+1:-1)  # integer wavenumbers: exp(2πikₓ x/L)\n",
    "    α  = 2*pi*kₓ/Lₓ                       # real wavenumbers:    exp(iαx)\n",
    "    D  = 1im*α                           # spectral D = d/dx operator \n",
    "    L  = α.^2 - α.^4                     # spectral L = -D^2 - D^4 operator\n",
    "    G  = -0.5*D                          # spectral -1/2 D operator\n",
    "    \n",
    "    Nsave = div(Nsteps, nsave)+1         # number of saved u's, including t=0\n",
    "    t = (0:Nsave-1)*(Δt*nsave)           # times of saved u's\n",
    "    U = zeros(Nsave, Nₓ)                 # matrix of u(tᵢ, xⱼ) values\n",
    "    U[1,:] = real(u₀)                    # assign initial condition to U\n",
    "    s = 2                                # counter for saved data\n",
    "    \n",
    "    # construct diagonal matrices for LHS and RHS linear operators\n",
    "    I   = ones(Nₓ)\n",
    "    A⁻¹ = (I - Δt/2*L).^(-1)             \n",
    "    B   = I + Δt/2*L\n",
    "    \n",
    "    # compute in-place FFTW plans\n",
    "    FFT!  = plan_fft!(uⁿ,  flags=FFTW.ESTIMATE)\n",
    "    IFFT! = plan_ifft!(uⁿ, flags=FFTW.ESTIMATE)\n",
    "\n",
    "    # initialize nonlinear terms \n",
    "    Nuⁿ  = G.*fft(uⁿ.^2)      # set Nuⁿ = -1/2 d/dx (uⁿ)² = -u uₓ \n",
    "    Nuⁿ⁻¹ = copy(Nuⁿ)         # set Nuⁿ⁻¹ = Nuⁿ for first time step\n",
    "    FFT!*uⁿ\n",
    "    \n",
    "    # timestepping loop\n",
    "    for n = 0:Nsteps-1\n",
    "        \n",
    "        Nuⁿ⁻¹ .= Nuⁿ        # shift nonlinear term in time\n",
    "        Nuⁿ  .= uⁿ         # store uⁿ in Nuⁿ in prep for collocation computation\n",
    "        \n",
    "        IFFT!*Nuⁿ          # transform to gridpt values of u\n",
    "        Nuⁿ .= Nuⁿ.*Nuⁿ    # collocation calculation of u²\n",
    "        FFT!*Nuⁿ           # transform u² back to spectral coeffs\n",
    "\n",
    "        Nuⁿ .= G.*Nuⁿ      # compute  Nuⁿ = -1/2 d/dx u² = -u uₓ\n",
    "\n",
    "        # Loop fusion! Julia unrolls this line into a single for-loop over vector elems,\n",
    "        # avoiding the generation and allocation of temporary vectors you'd get in Matlab\n",
    "        uⁿ .= A⁻¹ .* (B .* uⁿ .+ 3Δt/2.*Nuⁿ .- Δt/2.*Nuⁿ⁻¹)\n",
    "        \n",
    "        if mod(n, nsave) == 0\n",
    "            U[s,:] = real(ifft(uⁿ))  # save into return matrix U \n",
    "            s += 1            \n",
    "        end\n",
    "    end\n",
    "   \n",
    "    t,U\n",
    "end"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### Execute Julia KS-CNAB2 code"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 12,
   "metadata": {},
   "outputs": [
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     },
     "execution_count": 12,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "# set discretization parameters\n",
    "Lx = 64π\n",
    "Nx = 1024\n",
    "Δt = 1/16\n",
    "nsave = 8\n",
    "Nsteps = 3200\n",
    "\n",
    "# set initial condition\n",
    "x = Lx*(0:Nx-1)/Nx\n",
    "u₀ = cos.(x) + 0.1*sin.(x/8) + 0.002*cos.(x/32);\n",
    "\n",
    "# compile the integrator function by running it one time step\n",
    "t,U = ksintegrate(u₀, Lx, Δt, 8, nsave)\n",
    "\n",
    "# integrate the PDE\n",
    "t,U = ksintegrate(u₀, Lx, Δt, Nsteps, nsave) \n",
    "\n",
    "# plot results\n",
    "heatmap(x,t, U, xlim=(x[1], x[end]), ylim=(t[1], t[end]), \n",
    "    xlabel=\"x\", ylabel=\"t\", title=\"u(x,t)\", fillcolor=:bluesreds\n",
    ")"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### KS-CNAB2 benchmark: CPU time of identical algorithm in six languages. \n",
    "\n",
    "Takeaway: Julia is nearly as fast as C and Fortran, within a few percent. Codes: [ks.py](https://github.com/johnfgibson/whyjulia/tree/master/ks-codes/ksbenchmark.py#48)\n",
    " [ks.m](https://github.com/johnfgibson/whyjulia/tree/master/ks-codes/ksbenchmark.m#46) \n",
    " [ks.jl](https://github.com/johnfgibson/whyjulia/tree/master/ks-codes/ksbenchmark.jl#L129) [ks.c](https://github.com/johnfgibson/whyjulia/tree/master/ks-codes/ksbenchmark.c#L82) [ks.cpp](https://github.com/johnfgibson/whyjulia/tree/master/ks-codes/ksbenchmark.cpp#L87) [ks.f90](https://github.com/johnfgibson/whyjulia/tree/master/ks-codes/ksbenchmark.f90#L84) \n",
    "\n",
    "<img style=\"float:left\" src=\"figs/ks_cpu_scaling.png\"/>"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### KS-CNAB2 benchmark: CPU time versus lines of code \n",
    "\n",
    "Takeaway: The Julia PDE code is almost as compact as Matlab/Python, almost as fast as C/Fortran.\n",
    "<img style=\"float:left\" src=\"figs/ks_cpu_vs_lines_1024.png\"/>\n"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### Benchmarks on common code pattterns\n",
    "\n",
    "Takeaway: Julia timings are clustered around C timing, Matlab/Python/R orders of magnitude slower.\n",
    "<img style=\"float:left\" src=\"figs/benchmarks.png\"/>"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "<br> <br> <br>\n",
    "\n",
    "# Julia: easy, dynamic, and fast. How? \n",
    "\n",
    "\n",
    " * **Just-in-time compilation** (JIT)\n",
    "    - user-level code is compiled to machine code on-the-fly  \n",
    "    \n",
    "    \n",
    " * **Meticulous type system**\n",
    "    - designed to maximize impact of JIT\n",
    "    - type inference: compiler determines types of variables\n",
    "    \n",
    "    \n",
    " * **Multiple dispatch**\n",
    "    - functions are compiled for each set of argument types\n",
    "    - function dispatch determined at compile time when possible, run time when not\n",
    " \n",
    " \n",
    " <br> <br>\n",
    "\n",
    "### Just-in-time compilation\n",
    "\n",
    "Functions are compiled to machine code when first run. Subsequent runs are as fast as compiled C, C++, Fortran.\n",
    "\n",
    "#### Logistic map example $f(x) = 4\\, x\\, (1-x)$.\n"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 13,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "  0.004713 seconds (497 allocations: 29.339 KiB)\n",
      "  0.000002 seconds (5 allocations: 176 bytes)\n"
     ]
    }
   ],
   "source": [
    "f(x) = 4x*(1-x)     # define logistic map\n",
    "@time f(0.3);       # first run is slow\n",
    "@time f(0.4);       # second run is a thousand times faster"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "#### Observe the compilation of $f(x)$ by stages\n",
    "\n",
    "user Julia -> generic Julia expression -> typed Julia expression -> intermediate compiler language -> machine code"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 14,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "CodeInfo(:(begin \n",
       "        nothing\n",
       "        return (4 * x) * (1 - x)\n",
       "    end))"
      ]
     },
     "execution_count": 14,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "@code_lowered f(0.3)  # show f(x) as generic Julia expression"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 15,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "CodeInfo(:(begin \n",
       "        return (Base.mul_float)((Base.mul_float)((Base.sitofp)(Float64, 4)::Float64, x)::Float64, (Base.sub_float)((Base.sitofp)(Float64, 1)::Float64, x)::Float64)::Float64\n",
       "    end))=>Float64"
      ]
     },
     "execution_count": 15,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "@code_typed f(0.3)    # show f(x) as Julia expr with inferred types, based on the arg types"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 16,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "\n",
      "define double @julia_f_64842(double) #0 !dbg !5 {\n",
      "top:\n",
      "  %1 = fmul double %0, 4.000000e+00\n",
      "  %2 = fsub double 1.000000e+00, %0\n",
      "  %3 = fmul double %1, %2\n",
      "  ret double %3\n",
      "}\n"
     ]
    }
   ],
   "source": [
    "@code_llvm f(0.3)    # show f(x) in intermediate compiler language (LLVM)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 17,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "\t.text\n",
      "Filename: In[13]\n",
      "\tpushq\t%rbp\n",
      "\tmovq\t%rsp, %rbp\n",
      "\tmovabsq\t$139635269663760, %rax  # imm = 0x7EFF5EA57C10\n",
      "\tmovabsq\t$139635269663768, %rcx  # imm = 0x7EFF5EA57C18\n",
      "Source line: 1\n",
      "\tmovsd\t(%rcx), %xmm1           # xmm1 = mem[0],zero\n",
      "\tsubsd\t%xmm0, %xmm1\n",
      "\tmulsd\t(%rax), %xmm0\n",
      "\tmulsd\t%xmm1, %xmm0\n",
      "\tpopq\t%rbp\n",
      "\tretq\n",
      "\tnop\n",
      "\tnop\n",
      "\tnop\n",
      "\tnop\n",
      "\tnop\n",
      "\tnop\n"
     ]
    }
   ],
   "source": [
    "@code_native f(0.3) # show f(x) in IA-64 assembly language"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### Millionth iterate of logistic map: comparison of Julia, C, Matlab\n",
    "\n",
    "Given $f(x) = 4\\,x \\, (1-x)$, \n",
    "\n",
    "let $~~f^N(x) = f(\\,f(\\,...(\\,f(\\,f(x)))))$ be the millionth iterate of $f$, with $N=10^6$."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "#### $f^N$ in Julia: run-time code generation & compilation!"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 18,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/html": [
       "0.10116885334547539"
      ],
      "text/plain": [
       "0.10116885334547539"
      ]
     },
     "execution_count": 18,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "# define function that, given function g(x), \n",
    "# returns iterated function gᴺ(x) = g(g(...(g(g(x)))))\n",
    "function iterator(g, N)\n",
    "    \n",
    "    # construct gᴺ, the Nth iterate of g\n",
    "    function gᴺ(x)\n",
    "       for i ∈ 1:N             \n",
    "          x = g(x)\n",
    "       end        \n",
    "       return x\n",
    "    end    \n",
    "    \n",
    "    return gᴺ\n",
    "end\n",
    "\n",
    "\n",
    "fᴺ = iterator(f, 10^6);  # generate millionth iterate funcion fᴺ(x) for f(x) = 4x(1-x)\n",
    "\n",
    "fᴺ(0.67)                 # run fᴺ(x) once to compile it"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "#### $f^N$ in C++, compiled"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 19,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "#include <stdlib.h>\n",
      "#include <iostream>\n",
      "#include <iomanip>\n",
      "#include <ctime>\n",
      "\n",
      "using namespace std;\n",
      "\n",
      "inline double f(double x) {\n",
      "  return 4*x*(1-x);\n",
      "}\n",
      "\n",
      "int main(int argc, char* argv[]) {\n",
      "  double x = argc > 1 ? atof(argv[1]) : 0.0;\n",
      "\n",
      "  double t0 = clock();\n",
      "  for (int n=0; n<1000000; ++n)\n",
      "    x = f(x);\n",
      "  double t1 = clock();\n",
      "\n",
      "  cout << \"t = \" << (t1-t0)/CLOCKS_PER_SEC << \" seconds\" << endl;\n",
      "  cout << setprecision(17);\n",
      "  cout << \"x = \" << x << endl;\n",
      "  \n",
      "  return 0;\n",
      "}\n",
      "  \n"
     ]
    }
   ],
   "source": [
    "; cat codes/fN.cpp"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    " Compile fN.cpp in Unix shell"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 20,
   "metadata": {},
   "outputs": [],
   "source": [
    "; g++ -O3 -o fN codes/fN.cpp -lm "
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### Compare execution time of $f^N$ in Julia, C++, Matlab\n",
    "\n",
    "#### Julia"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 21,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "t=  0.003011 seconds (5 allocations: 176 bytes)\n",
      "x = 0.10116885334547539\n"
     ]
    }
   ],
   "source": [
    "print(\"t=\");\n",
    "@time x = fᴺ(0.67);         # run the Julia fᴺ on x = 0.67\n",
    "@show x;"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "#### C++"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 24,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "t = 0.00383 seconds\n",
      "x = 0.10116885334547539\n"
     ]
    }
   ],
   "source": [
    "; fN 0.67 "
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Julia and C++ get the same result `x` in roughly same execution time `t`. Sometimes Julia is faster, sometimes C++."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "#### Matlab\n",
    "```\n",
    ">> tic(); x=fN(0.67); t=toc();\n",
    ">> t, x\n",
    "t = 0.048889000000000\n",
    "x = 0.101168853345475\n",
    "```\n",
    "Matlab gets the same result $x$, but ten to twenty times slower than Julia or C++."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Type inference\n",
    "\n",
    "Julia reasons about types and infers types of variables from context, enabling compilation to hardware instructions. \n",
    "\n",
    "Language and type system carefully designed maximize effectiveness of just-in-time compilation. Matlab, Python syntax precludes this.\n",
    "\n",
    "### Type inference on $f(x)$"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 25,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "f (generic function with 1 method)"
      ]
     },
     "execution_count": 25,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "# recall definition of f(x): no types are mentioned!\n",
    "f(x) = 4x*(1-x)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 26,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "\n",
      "define double @julia_f_65027(double) #0 !dbg !5 {\n",
      "top:\n",
      "  %1 = fmul double %0, 4.000000e+00\n",
      "  %2 = fsub double 1.000000e+00, %0\n",
      "  %3 = fmul double %1, %2\n",
      "  ret double %3\n",
      "}\n"
     ]
    }
   ],
   "source": [
    "@code_llvm f(0.3)   # intermediate compiler code for f applied to Float64 argument"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 27,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "\n",
      "define i64 @julia_f_65029(i64) #0 !dbg !5 {\n",
      "top:\n",
      "  %1 = shl i64 %0, 2\n",
      "  %2 = sub i64 1, %0\n",
      "  %3 = mul i64 %1, %2\n",
      "  ret i64 %3\n",
      "}\n"
     ]
    }
   ],
   "source": [
    "@code_llvm f(2)     # intermediate compiler code for f applied to Int64 argument"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### Type inference on $f^N(x)$\n",
    "\n",
    "No types are specified in the definition of the iterated function $f^N(x)$.\n",
    "\n",
    "The compiler figures out that the loop index $i$ is an integer, and that arithmetic temporaries are Float64."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 28,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "\n",
      "define double @\"julia_g\\E1\\B4\\BA_65031\"(%\"#g\\E1\\B4\\BA#16\"* nocapture readonly dereferenceable(8), double) #0 !dbg !5 {\n",
      "top:\n",
      "  %2 = getelementptr inbounds %\"#g\\E1\\B4\\BA#16\", %\"#g\\E1\\B4\\BA#16\"* %0, i64 0, i32 1\n",
      "  %3 = load i64, i64* %2, align 8\n",
      "  %4 = icmp slt i64 %3, 1\n",
      "  br i1 %4, label %L14, label %if.preheader\n",
      "\n",
      "if.preheader:                                     ; preds = %top\n",
      "  br label %if\n",
      "\n",
      "if:                                               ; preds = %if.preheader, %if\n",
      "  %x.03 = phi double [ %8, %if ], [ %1, %if.preheader ]\n",
      "  %\"#temp#.02\" = phi i64 [ %5, %if ], [ 1, %if.preheader ]\n",
      "  %5 = add i64 %\"#temp#.02\", 1\n",
      "  %6 = fmul double %x.03, 4.000000e+00\n",
      "  %7 = fsub double 1.000000e+00, %x.03\n",
      "  %8 = fmul double %6, %7\n",
      "  %9 = icmp eq i64 %\"#temp#.02\", %3\n",
      "  br i1 %9, label %L14.loopexit, label %if\n",
      "\n",
      "L14.loopexit:                                     ; preds = %if\n",
      "  br label %L14\n",
      "\n",
      "L14:                                              ; preds = %L14.loopexit, %top\n",
      "  %x.0.lcssa = phi double [ %1, %top ], [ %8, %L14.loopexit ]\n",
      "  ret double %x.0.lcssa\n",
      "}\n"
     ]
    }
   ],
   "source": [
    "# Recall definition of fᴺ\n",
    "function iterator(g, N)\n",
    "    \n",
    "    # construct gᴺ, the Nth iterate of g\n",
    "    function gᴺ(x)\n",
    "       for i ∈ 1:N             \n",
    "          x = g(x)\n",
    "       end        \n",
    "       return x\n",
    "    end    \n",
    "    \n",
    "    return gᴺ\n",
    "end\n",
    "\n",
    "fᴺ = iterator(f, 10^6);  # generate million iterate function fᴺ(x) for f(x) = 4x(1-x)\n",
    "\n",
    "@code_llvm fᴺ(0.67)      # show intermediate compiler code for fᴺ applied to Float64 argument"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "<br> <br> <br> <br>\n",
    "\n",
    "# Julia: flexible as Python\n",
    "\n",
    "\n",
    "### Comprehensive built-in numeric type system"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 29,
   "metadata": {
    "scrolled": false
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "typeof(x) = Int64, f(x) = 0\n",
      "\n",
      "typeof(x) = Float32, f(x) = 1.0\n",
      "\n",
      "typeof(x) = Float64, f(x) = 1.0\n",
      "\n",
      "typeof(x) = Rational{Int64}, f(x) = 8//9\n",
      "\n",
      "typeof(x) = Complex{Int64}, f(x) = 28 - 36im\n",
      "\n",
      "typeof(x) = Complex{Float64}, f(x) = 64.72 - 55.04im\n",
      "\n",
      "typeof(x) = Complex{Rational{Int64}}, f(x) = 776//225 + 16//15*im\n",
      "\n",
      "typeof(x) = BigFloat, f(x) = 8.888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888877e-01\n",
      "\n",
      "typeof(x) = Complex{BigFloat}, f(x) = 3.448888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888893 - 1.066666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666683im\n",
      "\n"
     ]
    }
   ],
   "source": [
    "# define a set of various numeric types\n",
    "X = (1, 0.5f0, 0.5, 2//3, 2+3im, 2.1 + 4.3im, 2//3 + 4//5im, BigFloat(2//3), BigFloat(2//3) + BigFloat(4//5)*im);\n",
    "\n",
    "# loop over elements of X, print type and compute value of logistic map f(x)\n",
    "for x ∈ X\n",
    "    println(\"typeof(x) = $(typeof(x)), f(x) = $(f(x))\\n\")\n",
    "end"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 30,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "x = f⁷(big(2 // 3)) = 674838593766753484487654913831820720085359667709963001167872//11790184577738583171520872861412518665678211592275841109096961\n"
     ]
    }
   ],
   "source": [
    "# exact arithmetic: compute 7th iterate of logisitic map on Rational{BigInt}\n",
    "f⁷ = iterator(f, 7)\n",
    "@show x = f⁷(big(2//3));"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 31,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "Rational{BigInt}"
      ]
     },
     "execution_count": 31,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "typeof(x)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### Singular values of Hilbert matrix\n",
    "\n",
    "The Hilbert matrix $A_{ij} = (i+j-1)^{-1}$ is notoriously ill-conditioned. "
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 32,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "8×8 Array{Rational{Int64},2}:\n",
       " 1//1  1//2  1//3   1//4   1//5   1//6   1//7   1//8 \n",
       " 1//2  1//3  1//4   1//5   1//6   1//7   1//8   1//9 \n",
       " 1//3  1//4  1//5   1//6   1//7   1//8   1//9   1//10\n",
       " 1//4  1//5  1//6   1//7   1//8   1//9   1//10  1//11\n",
       " 1//5  1//6  1//7   1//8   1//9   1//10  1//11  1//12\n",
       " 1//6  1//7  1//8   1//9   1//10  1//11  1//12  1//13\n",
       " 1//7  1//8  1//9   1//10  1//11  1//12  1//13  1//14\n",
       " 1//8  1//9  1//10  1//11  1//12  1//13  1//14  1//15"
      ]
     },
     "execution_count": 32,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "m = 8\n",
    "A = [1//(i+j-1) for i=1:m, j=1:m]    # 8 x 8 Hilbert matrix of Rationals"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 33,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/html": [
       "2.220446049250313e-16"
      ],
      "text/plain": [
       "2.220446049250313e-16"
      ]
     },
     "execution_count": 33,
     "metadata": {},
     "output_type": "execute_result"
    },
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "cond(A) = 7.865467778431645e17\n",
      "eps(Float64) = 2.220446049250313e-16\n"
     ]
    }
   ],
   "source": [
    "# 16 x 16 Hilbert is too ill-conditioned for 64-bit arithmetic \n",
    "m = 16\n",
    "A = [1/(i+j-1) for i=1:m, j=1:m]    # 16 x 16 Hilbert matrix of Float64s\n",
    "@show cond(A)\n",
    "@show eps(Float64)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 34,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "16-element Array{Float64,1}:\n",
       " 1.86004    \n",
       " 0.440131   \n",
       " 0.0611143  \n",
       " 0.00629782 \n",
       " 0.000514675\n",
       " 3.41276e-5 \n",
       " 1.85298e-6 \n",
       " 8.25314e-8 \n",
       " 3.00458e-9 \n",
       " 8.86171e-11\n",
       " 2.08593e-12\n",
       " 3.8278e-14 \n",
       " 5.39491e-16\n",
       " 1.22941e-17\n",
       " 1.06575e-17\n",
       " 2.36481e-18"
      ]
     },
     "execution_count": 34,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "σ = svdvals(A)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 35,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "A[1, 1] = 1.000000000000000000000000000000000000000000000000000000000000000000000000000000\n",
      "A[2, 1] = 5.000000000000000000000000000000000000000000000000000000000000000000000000000000e-01\n",
      "A[3, 1] = 3.333333333333333333333333333333333333333333333333333333333333333333333333333348e-01\n"
     ]
    }
   ],
   "source": [
    "# Make 32 x 32 Hilbert matrix of 256-bit BigFloats and show a few elements\n",
    "setprecision(256)\n",
    "m = 32\n",
    "A = [BigFloat(1//(i+j-1)) for i=1:m, j=1:m]; \n",
    "@show A[1,1]\n",
    "@show A[2,1]\n",
    "@show A[3,1]\n",
    ";"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 36,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "32-element Array{BigFloat,1}:\n",
       " 1.998434811479943656818393069323257960640895672475752391825234214538557962751571    \n",
       " 5.862023373833755450036553769152732128253887467482456889981306320710605366320407e-01\n",
       " 1.10847523452293641165712019076916986852940670606685497550561288825334330556402e-01 \n",
       " 1.673319658958179140677531975263979754746187466404676472020306095015480020192925e-02\n",
       " 2.15575491554152836754878398453770830758654707824868046467567877302736921411345e-03 \n",
       " 2.430276596697422938228473142877594100259722632966877505528956749837836051990004e-04\n",
       " 2.429495582927391325328945583959494719139547266197054564322197618596025730380946e-05\n",
       " 2.171522505963389703148618411661941671277063234196349157743888780845209557391788e-06\n",
       " 1.744940782252377496458626015544168389499120927266562258019751784656307464036591e-07\n",
       " 1.265263755175621542941323081140106746391741410303876700785990407921028286492275e-08\n",
       " 8.299361818990081080483039727963430527712415734657494874743550521365933062548887e-10\n",
       " 4.932225502812551689960154090376861688463115611337698460243274517118047150319048e-11\n",
       " 2.657755729647246294347676628661052219189258529886244896408806499357584161032459e-12\n",
       " ⋮                                                                                   \n",
       " 4.964677392194419412771534581275505498952853244849643235235676004222587223023033e-24\n",
       " 1.03288583195413057734257523516093016847970149985762475466914356244274156684127e-25 \n",
       " 1.890883111336726781245519957991374953500469426332054678531832304542085160296988e-27\n",
       " 3.018453206643979489193726584612186032852633067862793078989280812098241737561698e-29\n",
       " 4.153453716256919797662583088756555826364140174243653830139302954047500817619929e-31\n",
       " 4.853640532096687295414900053614393688050108363016130784162374386131134009630752e-33\n",
       " 4.722253487722886629537566561063426718193936108783761901779058653423762612571911e-35\n",
       " 3.721728303529685301163286716629296819149422130979579196306836707182776329082812e-37\n",
       " 2.282607735977945944697879155526163440375905564402972710884859678322712924025411e-39\n",
       " 1.022045231388037386957792253667740236529498419265717624536070287487104900963478e-41\n",
       " 2.971550767330245214028010047204120157838584421268951012254754735920458055015361e-44\n",
       " 4.210099032402693745703330910061443236619344029786752208732102155291009952659503e-47"
      ]
     },
     "execution_count": 36,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "# Compute singular values of 32 x 32 Hilbert matrix in 256-bit arithmetic\n",
    "\n",
    "σ = svdvals(A)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## User-defined type: Galois field GF(p)\n",
    "\n",
    "Finite scalar field GF(p) is $\\mathbb{Z}/p$, the integers modulo p, where p is prime. Example by Andreas Noack, Julia Computing."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 37,
   "metadata": {},
   "outputs": [],
   "source": [
    "# Type definition: Galois fields GF(p), where p is prime modulus, T is integer type\n",
    "immutable GF{p,T} <: Number where {p,T<:Integer}\n",
    "    rep::T  # representative integer which holds the value of a GF(p) variable\n",
    "    function GF{p,T}(x::Integer) where {p,T<:Integer}\n",
    "        return new(mod(x, p))\n",
    "    end\n",
    "end\n",
    "GF{p}(x::T) where {p,T<:Integer} = GF{p,T}(x)\n",
    "\n",
    "\n",
    "# Define some basic methods for GF(p) that all Julia Numbers must have\n",
    "import Base: convert, inv, one, promote_rule, show, zero\n",
    "\n",
    "function call{p}(::Type{GF{p}}, x::Integer)\n",
    "    if !isprime(p)\n",
    "        throw(ArgumentError(\"p must be prime\"))\n",
    "    end\n",
    "    return GF{p,typeof(x)}(mod(x, p))\n",
    "end\n",
    "\n",
    "convert{p,T}(::Type{GF{p,T}}, x::Integer) = GF{p}(x)\n",
    "convert{p}(::Type{GF{p}}, x::Integer) = GF{p}(x)\n",
    "convert{p,T}(::Type{GF{p,T}}, x::GF{p}) = GF{p,T}(x.rep)\n",
    "promote_rule{p,T1,T2<:Integer}(::Type{GF{p,T1}}, ::Type{T2}) = GF{p,promote_type(T1,T2)}\n",
    "show(io::IO, x::GF) = show(io, x.rep);\n",
    "\n",
    "\n",
    "# Define arithmetic operations on GF(p)\n",
    "import Base: +, -, *, /\n",
    "\n",
    "for op in (:+, :-, :*)   # loop over ops, defining each in terms of integer ops mod p\n",
    "    @eval begin\n",
    "        ($op){p,T}(x::GF{p,T}, y::GF{p,T}) = GF{p,T}($(op)(x.rep, y.rep))\n",
    "    end\n",
    "end\n",
    "\n",
    "# Define inverse and ÷. Requires more care than +, -, * because dividing by zero throws error\n",
    "function inv{p,T}(x::GF{p,T})\n",
    "    if x == zero(x)\n",
    "        throw(DivideError())\n",
    "    end\n",
    "    r, u, v = gcdx(x.rep, p)\n",
    "    GF{p,T}(u)\n",
    "end\n",
    "(/){p}(x::GF{p}, y::GF{p}) = x*inv(y);"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 38,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "3-element Array{Union{DataType, UnionAll},1}:\n",
       " Complex\n",
       " GF     \n",
       " Real   "
      ]
     },
     "execution_count": 38,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "subtypes(Number)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 39,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "x = 2\n",
      "y = 5\n",
      "x + y = 0\n",
      "x - y = 4\n",
      "x * y = 3\n",
      "x / y = 6\n"
     ]
    }
   ],
   "source": [
    "# Create some GF(7) variables and do arithmetic\n",
    "\n",
    "x = GF{7}(9)   # x =  9 mod 7 = 2\n",
    "y = GF{7}(12)  # y = 12 mod 7 = 5\n",
    "@show x\n",
    "@show y\n",
    "@show x + y     # 2 + 5 mod 7 = 0\n",
    "@show x - y     # 2 - 5 mod 7 = 4\n",
    "@show x * y     # 2 * 5 mod 7 = 3\n",
    "@show x / y     # 2 ÷ 5 mod 7 = 6, because 2 = 6*5 mod 7\n",
    ";"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### Linear algebra over Galois field"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 40,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "4×4 Array{GF{7,Int64},2}:\n",
       " 3  5  3  3\n",
       " 6  6  2  0\n",
       " 2  2  6  4\n",
       " 2  6  1  2"
      ]
     },
     "execution_count": 40,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "srand(1234)\n",
    "A = [GF{7}(rand(0:6)) for i = 1:4, j = 1:4] # matrix of random GF(7) elems"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 41,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "4-element Array{GF{7,Int64},1}:\n",
       " 1\n",
       " 4\n",
       " 3\n",
       " 0"
      ]
     },
     "execution_count": 41,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "b = [GF{7}(rand(0:6)) for i = 1:4]          # random vector b for Ax=b problem"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 42,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "4-element Array{GF{7,Int64},1}:\n",
       " 5\n",
       " 5\n",
       " 0\n",
       " 1"
      ]
     },
     "execution_count": 42,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "x = A\\b   # solve Ax=b over GF(p)!"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 43,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "4-element Array{GF{7,Int64},1}:\n",
       " 0\n",
       " 0\n",
       " 0\n",
       " 0"
      ]
     },
     "execution_count": 43,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "A*x - b   # check that x satisfies Ax=b"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "**Whoa! Built-in backslash operator on matrix of GF(p) worked!** \n",
    "\n",
    "Julia generated and compiled a GF(p)-specific version of its generic LU decomp function. \n",
    "\n",
    "All it needs is definitions of $+, -, \\times, \\div$."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "<br> <br> <br> <br>\n",
    "\n",
    "\n",
    "\n",
    "\n",
    "# Julia: deep as Lisp\n",
    "\n",
    "Julia is *homoiconic*: it can represent its own expressions as data structures and operate on them. "
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### Expressions"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 44,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "\"for n ∈ 1:N begin println(n^2) end end\""
      ]
     },
     "execution_count": 44,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "program = \"for n ∈ 1:N begin println(n^2) end end\""
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 45,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       ":(for n = 1:N # none, line 1:\n",
       "        begin  # none, line 1:\n",
       "            println(n ^ 2)\n",
       "        end\n",
       "    end)"
      ]
     },
     "execution_count": 45,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "expression = parse(program)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 46,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "Expr"
      ]
     },
     "execution_count": 46,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "typeof(expression)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 47,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "Expr\n",
      "  head: Symbol for\n",
      "  args: Array{Any}((2,))\n",
      "    1: Expr\n",
      "      head: Symbol =\n",
      "      args: Array{Any}((2,))\n",
      "        1: Symbol n\n",
      "        2: Expr\n",
      "          head: Symbol :\n",
      "          args: Array{Any}((2,))\n",
      "            1: Int64 1\n",
      "            2: Symbol N\n",
      "          typ: Any\n",
      "      typ: Any\n",
      "    2: Expr\n",
      "      head: Symbol block\n",
      "      args: Array{Any}((2,))\n",
      "        1: Expr\n",
      "          head: Symbol line\n",
      "          args: Array{Any}((2,))\n",
      "            1: Int64 1\n",
      "            2: Symbol none\n",
      "          typ: Any\n",
      "        2: Expr\n",
      "          head: Symbol block\n",
      "          args: Array{Any}((2,))\n",
      "            1: Expr\n",
      "              head: Symbol line\n",
      "              args: Array{Any}((2,))\n",
      "                1: Int64 1\n",
      "                2: Symbol none\n",
      "              typ: Any\n",
      "            2: Expr\n",
      "              head: Symbol call\n",
      "              args: Array{Any}((2,))\n",
      "                1: Symbol println\n",
      "                2: Expr\n",
      "              typ: Any\n",
      "          typ: Any\n",
      "      typ: Any\n",
      "  typ: Any\n"
     ]
    }
   ],
   "source": [
    "dump(expression)   # show expression as an abstract syntax tree"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 48,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "\"for n ∈ 1:N begin println(n^2) end end\""
      ]
     },
     "execution_count": 48,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "program"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 49,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "1\n",
      "4\n",
      "9\n"
     ]
    }
   ],
   "source": [
    "N = 3; eval(parse(program))"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 50,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "1\n",
      "4\n",
      "9\n",
      "16\n",
      "25\n"
     ]
    }
   ],
   "source": [
    "N = 5; eval(parse(program))"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### Macros: code that transforms code"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 51,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "3 * A[:, 1] + 7 * A[:, 2] = GF{7,Int64}[2, 4, 6, 6]\n"
     ]
    }
   ],
   "source": [
    "# @show macro prints expression, an equals sign, then evaluates the expr and prints its value\n",
    "@show 3*A[:,1] + 7*A[:,2];"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 53,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/html": [
       "8//9"
      ],
      "text/plain": [
       "8//9"
      ]
     },
     "execution_count": 53,
     "metadata": {},
     "output_type": "execute_result"
    },
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "  0.000014 seconds (6 allocations: 224 bytes)\n"
     ]
    }
   ],
   "source": [
    "# @time macro inserts timing and memory profiling into expression, then evaluates, and prints\n",
    "@time f(2//3)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 54,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/html": [
       "exp(x::<b>Real</b>) at <a href=\"https://github.com/JuliaLang/julia/tree/9d11f62bcb124327831206089967f93020e84200/base/math.jl#L243\" target=\"_blank\">math.jl:243</a>"
      ],
      "text/plain": [
       "exp(x::Real) in Base.Math at math.jl:243"
      ]
     },
     "execution_count": 54,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "# @which macro determines which function is called, provides link to source code on GitHub\n",
    "@which exp(π)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 55,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/html": [
       "exp(z::<b>Complex</b>) at <a href=\"https://github.com/JuliaLang/julia/tree/9d11f62bcb124327831206089967f93020e84200/base/complex.jl#L502\" target=\"_blank\">complex.jl:502</a>"
      ],
      "text/plain": [
       "exp(z::Complex) in Base at complex.jl:502"
      ]
     },
     "execution_count": 55,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "@which exp(π*im)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 56,
   "metadata": {},
   "outputs": [],
   "source": [
    "# run-time code generation and evaluation using @eval macro (from prior example, Galois field GF(p))\n",
    "\n",
    "for op in (:+, :-, :*)   # loop over ops, defining each in terms of integer ops mod p\n",
    "    @eval begin\n",
    "        ($op){p,T}(x::GF{p,T}, y::GF{p,T}) = GF{p,T}($(op)(x.rep, y.rep))\n",
    "    end\n",
    "end"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Macros enable run-time code generation and transformation. Applications\n",
    "\n",
    "  * generation and execution of boilerplate code\n",
    "  * run-time generation and optimization of algorithms, e.g. FFTW, ATLAS\n",
    "  * symbolic mathematics, automatic differentiation\n",
    "  * *all written like high-level Python, running like compiled C !!!*"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Parallelism in Julia: MPI plus native constructs\n",
    "\n",
    "Some very trivial examples of Julia's built-in parallelism"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 57,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "1-element Array{Int64,1}:\n",
       " 1"
      ]
     },
     "execution_count": 57,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "workers()"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 58,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "2-element Array{Int64,1}:\n",
       " 2\n",
       " 3"
      ]
     },
     "execution_count": 58,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "addprocs(2)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "#### Remote procedure calls"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 59,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "function count_heads(n)\n",
      "    c::Int = 0\n",
      "    for i=1:n\n",
      "        c += rand(Bool)\n",
      "    end\n",
      "    c\n",
      "end\n"
     ]
    }
   ],
   "source": [
    "; cat codes/count_heads.jl"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 60,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/html": [
       "10000312"
      ],
      "text/plain": [
       "10000312"
      ]
     },
     "execution_count": 60,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "@everywhere include(\"codes/count_heads.jl\")\n",
    "\n",
    "a = @spawn count_heads(10000000)\n",
    "b = @spawn count_heads(10000000)\n",
    "fetch(a)+fetch(b)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### Parallel loops with reduction"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 61,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/html": [
       "99998494"
      ],
      "text/plain": [
       "99998494"
      ]
     },
     "execution_count": 61,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "nheads = @parallel (+) for i=1:200000000\n",
    "  Int(rand(Bool))\n",
    "end"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### Shared arrays"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 62,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "2-element Array{Future,1}:\n",
       " Future(2, 1, 17, #NULL)\n",
       " Future(3, 1, 18, #NULL)"
      ]
     },
     "execution_count": 62,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "a = SharedArray{Float64}(10)\n",
    "@parallel for i=1:10\n",
    "  a[i] = i\n",
    "end"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 63,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "10-element SharedArray{Float64,1}:\n",
       "  1.0\n",
       "  2.0\n",
       "  3.0\n",
       "  4.0\n",
       "  5.0\n",
       "  6.0\n",
       "  7.0\n",
       "  8.0\n",
       "  9.0\n",
       " 10.0"
      ]
     },
     "execution_count": 63,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "a"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### Distributed arrays"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 64,
   "metadata": {},
   "outputs": [],
   "source": [
    "@everywhere using DistributedArrays"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 65,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "5×5 DistributedArrays.DArray{Int64,2,Array{Int64,2}}:\n",
       " 2  3  4  5   6\n",
       " 3  4  5  6   7\n",
       " 4  5  6  7   8\n",
       " 5  6  7  8   9\n",
       " 6  7  8  9  10"
      ]
     },
     "execution_count": 65,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "A = @DArray [i+j for i = 1:5, j = 1:5]"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 66,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/html": [
       "sum(d::<b>DistributedArrays.DArray</b>) at <a href=\"https://github.com/JuliaParallel/DistributedArrays.jl/tree/6a376d1e568e98b2efa501d1592a5eede585f727//src/mapreduce.jl#L150\" target=\"_blank\">/home/gibson/.julia/v0.6/DistributedArrays/src/mapreduce.jl:150</a>"
      ],
      "text/plain": [
       "sum(d::DistributedArrays.DArray) in DistributedArrays at /home/gibson/.julia/v0.6/DistributedArrays/src/mapreduce.jl:150"
      ]
     },
     "execution_count": 66,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "@which sum(A)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 67,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "5×5 DistributedArrays.DArray{Float64,2,Array{Float64,2}}:\n",
       " -1.21951    0.206616  -0.833779   0.93749   -2.19312 \n",
       "  0.522166   0.297829  -0.996103   1.04825   -0.15237 \n",
       " -1.86705   -0.753513  -0.912036   0.997582  -0.484843\n",
       "  0.953774   1.53082   -1.39017    1.87275   -0.835136\n",
       " -1.61665    1.18274    2.4475    -0.149276  -1.46184 "
      ]
     },
     "execution_count": 67,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "B = drandn(5,5)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 68,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "5×5 DistributedArrays.DArray{Float64,2,Array{Float64,2}}:\n",
       " 0.780489  3.20662   3.16622  5.93749  3.80688\n",
       " 3.52217   4.29783   4.0039   7.04825  6.84763\n",
       " 2.13295   4.24649   5.08796  7.99758  7.51516\n",
       " 5.95377   7.53082   5.60983  9.87275  8.16486\n",
       " 4.38335   8.18274  10.4475   8.85072  8.53816"
      ]
     },
     "execution_count": 68,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "C = A+B"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 69,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/html": [
       "4.24648742731781"
      ],
      "text/plain": [
       "4.24648742731781"
      ]
     },
     "execution_count": 69,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "C[3,2]"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "# Conclusions\n",
    "\n",
    "Julia\n",
    "  * **easy as Matlab, fast as Fortran, flexible as Python, deep as Lisp**\n",
    "  * poised to take over scientific computing, from interactive exploration to HPC\n",
    "\n",
    "Not covered\n",
    "  * large-scale programming, development ecosystem, environments, debuggers, etc.\n",
    "  * function calls to C, Fortran, Python (direct, no glue code)\n",
    "  * interfaces to major C/Fortran libraries, e.g. LAPACK, MPI, FFTW, PETSc, etc.   \n",
    "  * rough edges: plotting, package loading times, pre-1.0 syntax flux\n",
    "\n",
    "Thanks\n",
    "  * Julia team, Julia community\n",
    "  * David Sanders, National University of Mexico City\n",
    "  * Andreas Noack, Julia Computing \n",
    "  * Mahtab Lak, UNH Math Ph.D. student\n",
    "\n",
    "  \n",
    "Julia website: http://www.julialang.org, this talk: https://github.com/johnfgibson/whyjulia\n",
    "<br> <br> <br>  <br> <br> <br> <br> <br> <br> <br> <br> <br> <br> <br> <br> <br>"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "#### Preload and run some functions"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "using ODE, Plots, GenericSVD     # load some packages \n",
    "gr();                            # set plotting backend\n",
    "t = linspace(0,4π)\n",
    "x,y,z = cos.(t), sin.(t), t\n",
    "plot(x, y, z, l=:path3d, xlab=\"x\", ylab=\"y\", zlab=\"z\", label=\"\")"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": []
  }
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