{ "cells": [ { "cell_type": "markdown", "id": "75ecc8b2", "metadata": {}, "source": [ "Quiz 10 MCS 471, on Friday 4 November 2022, due 10:50am." ] }, { "cell_type": "markdown", "id": "ca788b1d", "metadata": {}, "source": [ "Consider the application of Euler's method to\n", "\n", "$$\n", " y' = -33 y, \\quad y(0) = 1.\n", "$$" ] }, { "cell_type": "markdown", "id": "942b85d1", "metadata": {}, "source": [ "# Question 1" ] }, { "cell_type": "markdown", "id": "9c91c799", "metadata": {}, "source": [ "What is a good choice for the step size if we want to compute $y(1)$ with three decimal places of accuracy?\n", "\n", "Justify your answer." ] }, { "cell_type": "markdown", "id": "86cbe642", "metadata": {}, "source": [ "## Answer to Question 1" ] }, { "cell_type": "markdown", "id": "302287d3", "metadata": {}, "source": [ "Euler's method has a global error of $O(h)$. If we want three decimal places of accuracy, or a global error of $10^{-3}$, then we must take $h = 10^{-3}$." ] }, { "cell_type": "markdown", "id": "96528a71", "metadata": {}, "source": [ "Let us verify this with an experiment, adjusting the code of Lecture 28." ] }, { "cell_type": "code", "execution_count": 1, "id": "450aa83a", "metadata": {}, "outputs": [ { "data": { "text/plain": [ "eulerexp" ] }, "execution_count": 1, "metadata": {}, "output_type": "execute_result" } ], "source": [ "using Printf\n", "\n", "\"\"\"\n", " eulerexp(n::Int64,verbose::Bool=true)\n", "\n", "Euler's method with n evaluations\n", "on the interval [0,1] on the test\n", "equation y' = -33*y, y(0) = 1.\n", "\"\"\"\n", "function eulerexp(n::Int64,verbose::Bool=true)\n", " h = 1.0/n\n", " if verbose\n", " println(\" i x approx exact error\")\n", " end\n", " y = 0.0\n", " for i=0:n\n", " x = i*h\n", " y = (1.0-33*h)^i\n", " exact = exp(-33*x)\n", " if verbose\n", " stri = @sprintf(\"%3d\", i)\n", " strx = @sprintf(\"%.2f\", x)\n", " stry = @sprintf(\"%.6e\", y)\n", " strexp = @sprintf(\"%.6e\", exact)\n", " strerr = @sprintf(\"%.2e\", abs(y-exact))\n", " println(\"$stri $strx $stry $strexp $strerr\")\n", " end\n", " end\n", " return y\n", "end" ] }, { "cell_type": "markdown", "id": "404e9036", "metadata": {}, "source": [ "If the step size has to be $10^{-3} = 1/1000$, then we need to take 1000 steps." ] }, { "cell_type": "code", "execution_count": 2, "id": "f6f47614", "metadata": {}, "outputs": [ { "name": "stdout", "output_type": "stream", "text": [ " i x approx exact error\n", " 0 0.00 1.000000e+00 1.000000e+00 0.00e+00\n", " 1 0.00 9.670000e-01 9.675386e-01 5.39e-04\n", " 2 0.00 9.350890e-01 9.361309e-01 1.04e-03\n", " 3 0.00 9.042311e-01 9.057427e-01 1.51e-03\n", " 4 0.00 8.743914e-01 8.763410e-01 1.95e-03\n", " 5 0.01 8.455365e-01 8.478937e-01 2.36e-03\n", " 6 0.01 8.176338e-01 8.203699e-01 2.74e-03\n", " 7 0.01 7.906519e-01 7.937395e-01 3.09e-03\n", " 8 0.01 7.645604e-01 7.679735e-01 3.41e-03\n", " 9 0.01 7.393299e-01 7.430440e-01 3.71e-03\n", " 10 0.01 7.149320e-01 7.189237e-01 3.99e-03\n", " 11 0.01 6.913393e-01 6.955864e-01 4.25e-03\n", " 12 0.01 6.685251e-01 6.730067e-01 4.48e-03\n", " 13 0.01 6.464637e-01 6.511599e-01 4.70e-03\n", " 14 0.01 6.251304e-01 6.300223e-01 4.89e-03\n", " 15 0.01 6.045011e-01 6.095709e-01 5.07e-03\n", " 16 0.02 5.845526e-01 5.897834e-01 5.23e-03\n", " 17 0.02 5.652623e-01 5.706381e-01 5.38e-03\n", " 18 0.02 5.466087e-01 5.521144e-01 5.51e-03\n", " 19 0.02 5.285706e-01 5.341920e-01 5.62e-03\n", " 20 0.02 5.111278e-01 5.168513e-01 5.72e-03\n", " 21 0.02 4.942606e-01 5.000736e-01 5.81e-03\n", " 22 0.02 4.779500e-01 4.838405e-01 5.89e-03\n", " 23 0.02 4.621776e-01 4.681343e-01 5.96e-03\n", " 24 0.02 4.469258e-01 4.529380e-01 6.01e-03\n", " 25 0.03 4.321772e-01 4.382350e-01 6.06e-03\n", " 26 0.03 4.179154e-01 4.240093e-01 6.09e-03\n", " 27 0.03 4.041241e-01 4.102453e-01 6.12e-03\n", " 28 0.03 3.907880e-01 3.969281e-01 6.14e-03\n", " 29 0.03 3.778920e-01 3.840433e-01 6.15e-03\n", " 30 0.03 3.654216e-01 3.715767e-01 6.16e-03\n", " 31 0.03 3.533627e-01 3.595148e-01 6.15e-03\n", " 32 0.03 3.417017e-01 3.478444e-01 6.14e-03\n", " 33 0.03 3.304256e-01 3.365529e-01 6.13e-03\n", " 34 0.03 3.195215e-01 3.256279e-01 6.11e-03\n", " 35 0.04 3.089773e-01 3.150575e-01 6.08e-03\n", " 36 0.04 2.987811e-01 3.048303e-01 6.05e-03\n", " 37 0.04 2.889213e-01 2.949351e-01 6.01e-03\n", " 38 0.04 2.793869e-01 2.853611e-01 5.97e-03\n", " 39 0.04 2.701671e-01 2.760978e-01 5.93e-03\n", " 40 0.04 2.612516e-01 2.671353e-01 5.88e-03\n", " 41 0.04 2.526303e-01 2.584637e-01 5.83e-03\n", " 42 0.04 2.442935e-01 2.500736e-01 5.78e-03\n", " 43 0.04 2.362318e-01 2.419559e-01 5.72e-03\n", " 44 0.04 2.284362e-01 2.341016e-01 5.67e-03\n", " 45 0.04 2.208978e-01 2.265023e-01 5.60e-03\n", " 46 0.05 2.136081e-01 2.191497e-01 5.54e-03\n", " 47 0.05 2.065591e-01 2.120358e-01 5.48e-03\n", " 48 0.05 1.997426e-01 2.051528e-01 5.41e-03\n", " 49 0.05 1.931511e-01 1.984933e-01 5.34e-03\n", " 50 0.05 1.867771e-01 1.920499e-01 5.27e-03\n", " 51 0.05 1.806135e-01 1.858157e-01 5.20e-03\n", " 52 0.05 1.746532e-01 1.797838e-01 5.13e-03\n", " 53 0.05 1.688897e-01 1.739478e-01 5.06e-03\n", " 54 0.05 1.633163e-01 1.683012e-01 4.98e-03\n", " 55 0.06 1.579269e-01 1.628379e-01 4.91e-03\n", " 56 0.06 1.527153e-01 1.575520e-01 4.84e-03\n", " 57 0.06 1.476757e-01 1.524376e-01 4.76e-03\n", " 58 0.06 1.428024e-01 1.474892e-01 4.69e-03\n", " 59 0.06 1.380899e-01 1.427015e-01 4.61e-03\n", " 60 0.06 1.335330e-01 1.380692e-01 4.54e-03\n", " 61 0.06 1.291264e-01 1.335873e-01 4.46e-03\n", " 62 0.06 1.248652e-01 1.292509e-01 4.39e-03\n", " 63 0.06 1.207446e-01 1.250552e-01 4.31e-03\n", " 64 0.06 1.167601e-01 1.209957e-01 4.24e-03\n", " 65 0.07 1.129070e-01 1.170680e-01 4.16e-03\n", " 66 0.07 1.091811e-01 1.132678e-01 4.09e-03\n", " 67 0.07 1.055781e-01 1.095910e-01 4.01e-03\n", " 68 0.07 1.020940e-01 1.060335e-01 3.94e-03\n", " 69 0.07 9.872490e-02 1.025915e-01 3.87e-03\n", " 70 0.07 9.546698e-02 9.926125e-02 3.79e-03\n", " 71 0.07 9.231657e-02 9.603909e-02 3.72e-03\n", " 72 0.07 8.927012e-02 9.292152e-02 3.65e-03\n", " 73 0.07 8.632421e-02 8.990515e-02 3.58e-03\n", " 74 0.07 8.347551e-02 8.698670e-02 3.51e-03\n", " 75 0.07 8.072082e-02 8.416299e-02 3.44e-03\n", " 76 0.08 7.805703e-02 8.143094e-02 3.37e-03\n", " 77 0.08 7.548115e-02 7.878757e-02 3.31e-03\n", " 78 0.08 7.299027e-02 7.623001e-02 3.24e-03\n", " 79 0.08 7.058159e-02 7.375548e-02 3.17e-03\n", " 80 0.08 6.825240e-02 7.136127e-02 3.11e-03\n", " 81 0.08 6.600007e-02 6.904478e-02 3.04e-03\n", " 82 0.08 6.382207e-02 6.680349e-02 2.98e-03\n", " 83 0.08 6.171594e-02 6.463495e-02 2.92e-03\n", " 84 0.08 5.967931e-02 6.253681e-02 2.86e-03\n", " 85 0.09 5.770990e-02 6.050677e-02 2.80e-03\n", " 86 0.09 5.580547e-02 5.854263e-02 2.74e-03\n", " 87 0.09 5.396389e-02 5.664226e-02 2.68e-03\n", " 88 0.09 5.218308e-02 5.480357e-02 2.62e-03\n", " 89 0.09 5.046104e-02 5.302456e-02 2.56e-03\n", " 90 0.09 4.879583e-02 5.130331e-02 2.51e-03\n", " 91 0.09 4.718556e-02 4.963793e-02 2.45e-03\n", " 92 0.09 4.562844e-02 4.802661e-02 2.40e-03\n", " 93 0.09 4.412270e-02 4.646760e-02 2.34e-03\n", " 94 0.09 4.266665e-02 4.495919e-02 2.29e-03\n", " 95 0.10 4.125865e-02 4.349975e-02 2.24e-03\n", " 96 0.10 3.989712e-02 4.208769e-02 2.19e-03\n", " 97 0.10 3.858051e-02 4.072146e-02 2.14e-03\n", " 98 0.10 3.730736e-02 3.939958e-02 2.09e-03\n", " 99 0.10 3.607621e-02 3.812062e-02 2.04e-03\n", "100 0.10 3.488570e-02 3.688317e-02 2.00e-03\n", "101 0.10 3.373447e-02 3.568589e-02 1.95e-03\n", "102 0.10 3.262123e-02 3.452747e-02 1.91e-03\n", "103 0.10 3.154473e-02 3.340666e-02 1.86e-03\n", "104 0.10 3.050376e-02 3.232223e-02 1.82e-03\n", "105 0.10 2.949713e-02 3.127301e-02 1.78e-03\n", "106 0.11 2.852373e-02 3.025784e-02 1.73e-03\n", "107 0.11 2.758244e-02 2.927563e-02 1.69e-03\n", "108 0.11 2.667222e-02 2.832530e-02 1.65e-03\n", "109 0.11 2.579204e-02 2.740582e-02 1.61e-03\n", "110 0.11 2.494090e-02 2.651618e-02 1.58e-03\n", "111 0.11 2.411785e-02 2.565543e-02 1.54e-03\n", "112 0.11 2.332196e-02 2.482262e-02 1.50e-03\n", "113 0.11 2.255234e-02 2.401684e-02 1.46e-03\n", "114 0.11 2.180811e-02 2.323722e-02 1.43e-03\n", "115 0.12 2.108844e-02 2.248291e-02 1.39e-03\n", "116 0.12 2.039252e-02 2.175308e-02 1.36e-03\n", "117 0.12 1.971957e-02 2.104694e-02 1.33e-03\n", "118 0.12 1.906883e-02 2.036373e-02 1.29e-03\n", "119 0.12 1.843955e-02 1.970269e-02 1.26e-03\n", "120 0.12 1.783105e-02 1.906311e-02 1.23e-03\n", "121 0.12 1.724262e-02 1.844430e-02 1.20e-03\n", "122 0.12 1.667362e-02 1.784557e-02 1.17e-03\n", "123 0.12 1.612339e-02 1.726628e-02 1.14e-03\n", "124 0.12 1.559132e-02 1.670579e-02 1.11e-03\n", "125 0.12 1.507680e-02 1.616349e-02 1.09e-03\n", "126 0.13 1.457927e-02 1.563880e-02 1.06e-03\n", "127 0.13 1.409815e-02 1.513115e-02 1.03e-03\n", "128 0.13 1.363291e-02 1.463997e-02 1.01e-03\n", "129 0.13 1.318303e-02 1.416473e-02 9.82e-04\n", "130 0.13 1.274799e-02 1.370493e-02 9.57e-04\n", "131 0.13 1.232730e-02 1.326004e-02 9.33e-04\n", "132 0.13 1.192050e-02 1.282960e-02 9.09e-04\n", "133 0.13 1.152713e-02 1.241314e-02 8.86e-04\n", "134 0.13 1.114673e-02 1.201019e-02 8.63e-04\n", "135 0.14 1.077889e-02 1.162032e-02 8.41e-04\n", "136 0.14 1.042319e-02 1.124311e-02 8.20e-04\n", "137 0.14 1.007922e-02 1.087814e-02 7.99e-04\n", "138 0.14 9.746606e-03 1.052502e-02 7.78e-04\n", "139 0.14 9.424968e-03 1.018336e-02 7.58e-04\n", "140 0.14 9.113944e-03 9.852796e-03 7.39e-04\n", "141 0.14 8.813184e-03 9.532960e-03 7.20e-04\n", "142 0.14 8.522349e-03 9.223506e-03 7.01e-04\n", "143 0.14 8.241112e-03 8.924098e-03 6.83e-04\n", "144 0.14 7.969155e-03 8.634409e-03 6.65e-04\n", "145 0.14 7.706173e-03 8.354124e-03 6.48e-04\n", "146 0.15 7.451869e-03 8.082937e-03 6.31e-04\n", "147 0.15 7.205957e-03 7.820553e-03 6.15e-04\n", "148 0.15 6.968161e-03 7.566687e-03 5.99e-04\n", "149 0.15 6.738211e-03 7.321061e-03 5.83e-04\n", "150 0.15 6.515851e-03 7.083409e-03 5.68e-04\n", "151 0.15 6.300827e-03 6.853471e-03 5.53e-04\n", "152 0.15 6.092900e-03 6.630998e-03 5.38e-04\n", "153 0.15 5.891834e-03 6.415746e-03 5.24e-04\n", "154 0.15 5.697404e-03 6.207482e-03 5.10e-04\n", "155 0.15 5.509390e-03 6.005978e-03 4.97e-04\n", "156 0.16 5.327580e-03 5.811015e-03 4.83e-04\n", "157 0.16 5.151770e-03 5.622381e-03 4.71e-04\n", "158 0.16 4.981761e-03 5.439871e-03 4.58e-04\n", "159 0.16 4.817363e-03 5.263285e-03 4.46e-04\n", "160 0.16 4.658390e-03 5.092431e-03 4.34e-04\n", "161 0.16 4.504663e-03 4.927123e-03 4.22e-04\n", "162 0.16 4.356009e-03 4.767182e-03 4.11e-04\n", "163 0.16 4.212261e-03 4.612432e-03 4.00e-04\n", "164 0.16 4.073256e-03 4.462706e-03 3.89e-04\n", "165 0.17 3.938839e-03 4.317840e-03 3.79e-04\n", "166 0.17 3.808857e-03 4.177677e-03 3.69e-04\n", "167 0.17 3.683165e-03 4.042063e-03 3.59e-04\n", "168 0.17 3.561621e-03 3.910852e-03 3.49e-04\n", "169 0.17 3.444087e-03 3.783900e-03 3.40e-04\n", "170 0.17 3.330432e-03 3.661069e-03 3.31e-04\n", "171 0.17 3.220528e-03 3.542226e-03 3.22e-04\n", "172 0.17 3.114250e-03 3.427240e-03 3.13e-04\n", "173 0.17 3.011480e-03 3.315987e-03 3.05e-04\n", "174 0.17 2.912101e-03 3.208345e-03 2.96e-04\n", "175 0.18 2.816002e-03 3.104198e-03 2.88e-04\n", "176 0.18 2.723074e-03 3.003431e-03 2.80e-04\n", "177 0.18 2.633213e-03 2.905935e-03 2.73e-04\n", "178 0.18 2.546317e-03 2.811604e-03 2.65e-04\n", "179 0.18 2.462288e-03 2.720336e-03 2.58e-04\n", "180 0.18 2.381033e-03 2.632030e-03 2.51e-04\n", "181 0.18 2.302458e-03 2.546590e-03 2.44e-04\n", "182 0.18 2.226477e-03 2.463924e-03 2.37e-04\n", "183 0.18 2.153004e-03 2.383942e-03 2.31e-04\n", "184 0.18 2.081954e-03 2.306555e-03 2.25e-04\n", "185 0.18 2.013250e-03 2.231681e-03 2.18e-04\n", "186 0.19 1.946813e-03 2.159238e-03 2.12e-04\n", "187 0.19 1.882568e-03 2.089146e-03 2.07e-04\n", "188 0.19 1.820443e-03 2.021329e-03 2.01e-04\n", "189 0.19 1.760369e-03 1.955714e-03 1.95e-04\n", "190 0.19 1.702276e-03 1.892229e-03 1.90e-04\n", "191 0.19 1.646101e-03 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5.48e-14\n", "898 0.90 8.184153e-14 1.349327e-13 5.31e-14\n", "899 0.90 7.914076e-14 1.305526e-13 5.14e-14\n", "900 0.90 7.652911e-14 1.263147e-13 4.98e-14\n", "901 0.90 7.400365e-14 1.222143e-13 4.82e-14\n", "902 0.90 7.156153e-14 1.182471e-13 4.67e-14\n", "903 0.90 6.920000e-14 1.144086e-13 4.52e-14\n", "904 0.90 6.691640e-14 1.106947e-13 4.38e-14\n", "905 0.91 6.470816e-14 1.071014e-13 4.24e-14\n", "906 0.91 6.257279e-14 1.036248e-13 4.11e-14\n", "907 0.91 6.050789e-14 1.002610e-13 3.98e-14\n", "908 0.91 5.851113e-14 9.700635e-14 3.85e-14\n", "909 0.91 5.658026e-14 9.385738e-14 3.73e-14\n", "910 0.91 5.471311e-14 9.081063e-14 3.61e-14\n", "911 0.91 5.290758e-14 8.786279e-14 3.50e-14\n", "912 0.91 5.116163e-14 8.501064e-14 3.38e-14\n", "913 0.91 4.947330e-14 8.225107e-14 3.28e-14\n", "914 0.91 4.784068e-14 7.958108e-14 3.17e-14\n", "915 0.92 4.626193e-14 7.699776e-14 3.07e-14\n", "916 0.92 4.473529e-14 7.449831e-14 2.98e-14\n", "917 0.92 4.325903e-14 7.207998e-14 2.88e-14\n", "918 0.92 4.183148e-14 6.974016e-14 2.79e-14\n", "919 0.92 4.045104e-14 6.747630e-14 2.70e-14\n", "920 0.92 3.911615e-14 6.528592e-14 2.62e-14\n", "921 0.92 3.782532e-14 6.316665e-14 2.53e-14\n", "922 0.92 3.657709e-14 6.111616e-14 2.45e-14\n", "923 0.92 3.537004e-14 5.913225e-14 2.38e-14\n", "924 0.92 3.420283e-14 5.721273e-14 2.30e-14\n", "925 0.93 3.307414e-14 5.535552e-14 2.23e-14\n", "926 0.93 3.198269e-14 5.355860e-14 2.16e-14\n", "927 0.93 3.092726e-14 5.182001e-14 2.09e-14\n", "928 0.93 2.990666e-14 5.013786e-14 2.02e-14\n", "929 0.93 2.891974e-14 4.851031e-14 1.96e-14\n", "930 0.93 2.796539e-14 4.693560e-14 1.90e-14\n", "931 0.93 2.704253e-14 4.541200e-14 1.84e-14\n", "932 0.93 2.615013e-14 4.393786e-14 1.78e-14\n", "933 0.93 2.528718e-14 4.251158e-14 1.72e-14\n", "934 0.93 2.445270e-14 4.113159e-14 1.67e-14\n", "935 0.94 2.364576e-14 3.979640e-14 1.62e-14\n", "936 0.94 2.286545e-14 3.850455e-14 1.56e-14\n", "937 0.94 2.211089e-14 3.725464e-14 1.51e-14\n", "938 0.94 2.138123e-14 3.604530e-14 1.47e-14\n", "939 0.94 2.067565e-14 3.487521e-14 1.42e-14\n", "940 0.94 1.999335e-14 3.374311e-14 1.37e-14\n", "941 0.94 1.933357e-14 3.264776e-14 1.33e-14\n", "942 0.94 1.869556e-14 3.158797e-14 1.29e-14\n", "943 0.94 1.807861e-14 3.056258e-14 1.25e-14\n", "944 0.94 1.748202e-14 2.957047e-14 1.21e-14\n", "945 0.95 1.690511e-14 2.861057e-14 1.17e-14\n", "946 0.95 1.634724e-14 2.768183e-14 1.13e-14\n", "947 0.95 1.580778e-14 2.678324e-14 1.10e-14\n", "948 0.95 1.528613e-14 2.591382e-14 1.06e-14\n", "949 0.95 1.478168e-14 2.507262e-14 1.03e-14\n", "950 0.95 1.429389e-14 2.425873e-14 9.96e-15\n", "951 0.95 1.382219e-14 2.347125e-14 9.65e-15\n", "952 0.95 1.336606e-14 2.270934e-14 9.34e-15\n", "953 0.95 1.292498e-14 2.197216e-14 9.05e-15\n", "954 0.95 1.249845e-14 2.125892e-14 8.76e-15\n", "955 0.96 1.208600e-14 2.056882e-14 8.48e-15\n", "956 0.96 1.168717e-14 1.990113e-14 8.21e-15\n", "957 0.96 1.130149e-14 1.925511e-14 7.95e-15\n", "958 0.96 1.092854e-14 1.863006e-14 7.70e-15\n", "959 0.96 1.056790e-14 1.802530e-14 7.46e-15\n", "960 0.96 1.021916e-14 1.744017e-14 7.22e-15\n", "961 0.96 9.881926e-15 1.687404e-14 6.99e-15\n", "962 0.96 9.555822e-15 1.632629e-14 6.77e-15\n", "963 0.96 9.240480e-15 1.579631e-14 6.56e-15\n", "964 0.96 8.935544e-15 1.528354e-14 6.35e-15\n", "965 0.96 8.640671e-15 1.478741e-14 6.15e-15\n", "966 0.97 8.355529e-15 1.430739e-14 5.95e-15\n", "967 0.97 8.079797e-15 1.384295e-14 5.76e-15\n", "968 0.97 7.813163e-15 1.339359e-14 5.58e-15\n", "969 0.97 7.555329e-15 1.295882e-14 5.40e-15\n", "970 0.97 7.306003e-15 1.253815e-14 5.23e-15\n", "971 0.97 7.064905e-15 1.213115e-14 5.07e-15\n", "972 0.97 6.831763e-15 1.173735e-14 4.91e-15\n", "973 0.97 6.606315e-15 1.135634e-14 4.75e-15\n", "974 0.97 6.388307e-15 1.098770e-14 4.60e-15\n", "975 0.97 6.177493e-15 1.063102e-14 4.45e-15\n", "976 0.98 5.973635e-15 1.028592e-14 4.31e-15\n", "977 0.98 5.776505e-15 9.952028e-15 4.18e-15\n", "978 0.98 5.585881e-15 9.628971e-15 4.04e-15\n", "979 0.98 5.401547e-15 9.316401e-15 3.91e-15\n", "980 0.98 5.223296e-15 9.013977e-15 3.79e-15\n", "981 0.98 5.050927e-15 8.721371e-15 3.67e-15\n", "982 0.98 4.884246e-15 8.438262e-15 3.55e-15\n", "983 0.98 4.723066e-15 8.164344e-15 3.44e-15\n", "984 0.98 4.567205e-15 7.899318e-15 3.33e-15\n", "985 0.98 4.416487e-15 7.642895e-15 3.23e-15\n", "986 0.99 4.270743e-15 7.394795e-15 3.12e-15\n", "987 0.99 4.129809e-15 7.154749e-15 3.02e-15\n", "988 0.99 3.993525e-15 6.922496e-15 2.93e-15\n", "989 0.99 3.861739e-15 6.697782e-15 2.84e-15\n", "990 0.99 3.734301e-15 6.480362e-15 2.75e-15\n", "991 0.99 3.611069e-15 6.270000e-15 2.66e-15\n", "992 0.99 3.491904e-15 6.066467e-15 2.57e-15\n", "993 0.99 3.376671e-15 5.869541e-15 2.49e-15\n", "994 0.99 3.265241e-15 5.679007e-15 2.41e-15\n", "995 0.99 3.157488e-15 5.494658e-15 2.34e-15\n", "996 1.00 3.053291e-15 5.316294e-15 2.26e-15\n", "997 1.00 2.952532e-15 5.143719e-15 2.19e-15\n", "998 1.00 2.855099e-15 4.976747e-15 2.12e-15\n", "999 1.00 2.760880e-15 4.815194e-15 2.05e-15\n", "1000 1.00 2.669771e-15 4.658886e-15 1.99e-15\n" ] }, { "data": { "text/plain": [ "2.6697714408441038e-15" ] }, "execution_count": 2, "metadata": {}, "output_type": "execute_result" } ], "source": [ "eulerexp(1000)" ] }, { "cell_type": "markdown", "id": "1c197a0c", "metadata": {}, "source": [ "While the error is small, observe that the exact value is just as small ... With 1000 steps, Euler's method manages only to keep up with the rapid decay of the solution." ] }, { "cell_type": "markdown", "id": "11e629ae", "metadata": {}, "source": [ "# Question 2" ] }, { "cell_type": "markdown", "id": "5e30b325", "metadata": {}, "source": [ "Considering the long term behavior of the solution,\n", "\n", "what is the condition on the step size for Euler's method to converge?" ] }, { "cell_type": "markdown", "id": "3d1061e8", "metadata": {}, "source": [ "## Answer to Question 2" ] }, { "cell_type": "markdown", "id": "9bf05413", "metadata": {}, "source": [ "As the long term behavior is decaying, the computed solution must decay as well. \n", "\n", "The condition on the step size is then $|1 - 33h| < 1$ or $-1 < 1 - 33h < 1$ or $-2 < -33h < 0$, or $2 > 33h > 0$.\n", "\n", "So $h < 2/33$." ] } ], "metadata": { "kernelspec": { "display_name": "Julia 1.8.0", "language": "julia", "name": "julia-1.8" }, "language_info": { "file_extension": ".jl", "mimetype": "application/julia", "name": "julia", "version": "1.8.0" } }, "nbformat": 4, "nbformat_minor": 5 }