{ "cells": [ { "cell_type": "markdown", "id": "4c5c3cb8", "metadata": {}, "source": [ "## Numerisches Differenzieren" ] }, { "cell_type": "markdown", "id": "ac0594b5", "metadata": {}, "source": [ "*Notebook erstellt am 11.09.2022 von C. Rockstuhl, überarbeitet von Y. Augenstein*" ] }, { "cell_type": "markdown", "id": "f9e7de7c", "metadata": {}, "source": [ "### Ableitungen erster Ordnung\n", "\n", "In einer der ersten Vorlesung haben wir gelernt, dass der Differentialoperator definiert ist als der Grenzwert eines Differenzenoperators. Er kann entsprechend geschrieben werden als\n", "\n", "$$\n", "\\frac{\\mathrm{d}f(x)}{\\mathrm{d}x}=\\lim_{\\Delta x \\rightarrow 0} \\frac{f(x+\\Delta x)-f(x)}{\\Delta x}=\\lim_{\\Delta x \\rightarrow 0} \\frac{f(x)-f(x-\\Delta x)}{\\Delta x}.\n", "$$\n", "\n", "Wir wollen diese Definition benutzen, um im Folgenden zu lernen, wie man numerisch differenziert ohne eine analytische Lösung bemühen zu müssen. Wir schauen uns das für ein sehr einfaches Beispiel einer sinusförmigen Funktion an.\n", "\n", "$$\n", "f(x)=\\sin (x)\n", "$$\n", "\n", "und können das als eine Koordinate einer Kreisbahn verstehen, die wir in der Vorlesung gerade besprochen haben. Analytisch bekannt ist die Lösung der ersten und zweiten Ableitung\n", "\n", "$$\n", "f'(x)=\\cos (x)\n", "$$\n", "und\n", "$$\n", "f''(x)=-\\sin (x),\n", "$$\n", "\n", "gegen die wir numerische Näherungen dieser Ausdrücke immer vergleichen können.\n", "\n", "Diese numerische Evaluierung von Differenzenquotienten mag bei der Berechnung von Geschwindigkeiten und Beschleunigungen vielleicht nicht immer notwendig sein. Aber wir werden auf dieses Wissen zu einem späteren Zeitpunkt zurückgreifen, wenn wir numerische Lösungen von Bewegungsgleichungen studieren. Dies sind Differentialgleichungen, die wir näherungsweise in der Zeit sich entwickeln lassen. Und für genau diese näherungsweise Berechnung benötigen wir die Näherung der Differentialgleichung durch eine Differenzengleichung, die wir dann entsprechend passend durchführen werden.\n", "\n", "Als Zweites sei noch gesagt, dass das Konvergenzverhalten von verschiedenen Arten von Differenzenbildung im Allgemeinen wichtig ist. Genauso wie wir das schon diskutierten bei der Integralrechnung, ist es also nicht einfach nur unser Ziel, möglichst exakt zu rechnen, sondern möglichst exakt mit der kleinsten Anzahl an Rechenschritten zu rechnen. Dies reduziert den numerischen Aufwand insbesonders für komplexe Probleme.\n", "\n", "Wir beginnen mit einer einfachen Evaluierung und Visualisierung der relevanten Funktion." ] }, { "cell_type": "code", "execution_count": 1, "id": "1912689d", "metadata": {}, "outputs": [], "source": [ "import numpy as np\n", "import matplotlib.pyplot as plt\n", "\n", "N = 51\n", "a = 0\n", "b = 4 * np.pi\n", "\n", "x = np.linspace(a, b, N)\n", "y = np.sin(x)" ] }, { "cell_type": "code", "execution_count": 2, "id": "f314d4e0", "metadata": {}, "outputs": [ { "data": { "image/png": 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ex64eASaMi+Vdiybz4qFaLnR6HIvDBN6B6hYOn27lg1c4d36JCB9ZMZ3Dp1s5WnvOsTjM2HFNQgGWA6WqWq6qncAjwC2XWP924OGgRBYEjxVXkpeVxLIZaY7GcfPCHNo6Pbx6pM7ROExgPVZcSXxMFO9eNPZ9my7lnZdPIkrg2X3WRD0cuSmhTAH6dteu8i17GxFJBG4A/tBnsQIvishuEblrsIOIyF0iUiwixfX17mjRVFp3juITZ/hg4TTH53pfkZtORlIcz+yvcTQOEzjtXR7+tLeaGy+fRErCyCdoC4Ts5HGsyM3gmf01VuwVhtyUUAb6SzrYGfcuYEu/4q7VqroUb5HZ50RkzUAbquoDqlqoqoVZWWMzKJ6/Ht9dSXSU8N6l/s/lHWgx0VHcuGASL5fU0dbZ7XQ4JgBeOHia1vZuR4u7+lq/MIfy+vMcPm1z8YQbNyWUKqDvGT8VGOy++Db6FXep6infcx3wR7xFaK7X5enhD7uruWZONtnJ45wOB4D1CyZzocvDK4fdcQdnRufRXZVMS09gZW5g59QZqRsuFnvZXXC4cVNC2QXMFpFcEYnDmzSe7r+SiKQAa4Gn+ixLEpHk3tfA9cCBoEQ9Sq8eqafhXAcfcsnVI8Dy3HQyx8fz7H4r5w51lU1tbC1r5IPLphEVoBk/RytzfDyr8jN51oq9wo5rEoqqdgP3Ai8AJcBjqnpQRO4Wkbv7rHor8KKqnu+zbCKwWUTeAHYCz6rq88GKfTQe3eUdCPLqOe4ofgOIjhJuWjCJlw/Xcb7Dir1C2ePFlYjA+5YFb6iV4Vi/MIeKhvMcsmHtw4prEgqAqj6nqgWqmq+q3/Qtu19V7++zzq9V9bZ+25Wr6iLf47Lebd2urrWdV44EdyDI4Vq/IIf2rh42HrbWXqHK06M8sbuKNbOzmBykgSCH652XTSI6SnjGir3Cirv+ikWYv+w/jadHef9Sd109AhTOTCc7Od6ad4aw3SfOcKq53XV3J+CdgnpVfgbP7rNir3BiCcVBG0pqyctKYvbEZKdDeRtvsVcOrxyp55wVe4WkDSW1xEYL17ioOLWvmxfmcLKpjQPVVuwVLiyhOKS1vYvt5Y1cNy+4o7764+aFOXR297CxpNbpUMwIbCipZWVeBsnjnO17Mpjr508iJkp4xhp/hA1LKA557WgDXR51dUJZOj2NSRPGWTl3CCqvP0d5/fmgD1Pvj7SkOFbPyrRirzBiCcUhG0tqSU2MZen0VKdDGVSUr9hr05F6Wm1ipJCyscTbmOIdc7MdjuTS1i/MoerMBfZV2QjX4cASigO6PT28fKSOd8zJJsZlrbv6W78wh05PDxus2CukvFRSy7ycCUxNS3Q6lEt65/xJxEYLz9pQP2HB3X/NwtTuE2c429bFdS4ujui1ZFoqk1PGWa/mEHLmfCfFx5u4bp67704AUhJjudKKvcKGJRQHbDxcR2y0cNXsTKdDGdLFYq+j9TRfsGKvUPDq0Tp6FFfXz/W1fuFkqs9e4PXKs06HYkbJEooDNhxyd+ub/tYvzKHLo7x0yIq9QsGGQ3VkJ8ezYEqK06EMy7r5E4mLjrK74DBgCSXIyurPUd7g7tY3/S2elsqU1ATr5BgCOrt72HS0nmvnZbtm7K6hpCTEsqYgk+dsbK+QZwklyHr7dLi99U1fIsL1l01ka1kj7V02k6Ob7aho5FxHd8gUd/W6fv4kaprbbUj7EGcJJcg2lNSFROub/tYUZNHR3cPOiqahVzaO2VhSx7jYKFbPcn/9XF9XFXjjfe2oTZkQyiyhBFEotb7pb2VuBnExUWyyH7xrqXrrua6clcW42Ginw/FLTkoCcyYm2/kV4iyhBNErR0Kr9U1fCXHRrMhNtx+8ix0+3Ur12QshecECsHZOFruON9mUCSHMEkoQbSwJrdY3/a0tyKK07hzVZy84HYoZwMX6uRBNKGtmZ9HlUbaXNzodihkhVyUUEblBRI6ISKmI3DfA51eLSLOI7PU9/mW42zotFFvf9Le2wDtqrZVzu9NLJXUsnpbqmqmk/VU4M42E2Gi7Cw5hrkkoIhIN/BS4EZgP3C4i8wdY9a+qutj3+Dc/t3VMqLa+6WtW9nhyUsZZQnGhutZ23qg8G7LFXQDjYqMpys+w8yuEuSahAMuBUt/si53AI8AtQdg2KDYcqg3J1jd9iQhrC7LYfKyBLk+P0+GYPl72DQYZCsP5XMqa2Zkcb2zjeMP5oVc2ruOmhDIFqOzzvsq3rL8iEXlDRP4iIpf5ua0jVJUNJXUh2fqmv7UFWbR2dLPXhslwlQ0ldUxJ9baUCmVr53jvsF47ZncpochNCWWgioX+3Wb3ADNUdRHwY+BPfmzrXVHkLhEpFpHi+vrgnLS9rW/WzQ/d4oheq2ZlEh0lVizhIu1dHjaX1rNu/kREQrN+rtfMjESmpyfa+RWi3JRQqoBpfd5PBd4y1oeqtqjqOd/r54BYEckczrZ99vGAqhaqamFWVnCmRn3liLc44poQ6h0/mJSEWJZMS7WKUxfZVt5Ie1dPSI2+MBgRYU1BJlvLGunotlEZQo2bEsouYLaI5IpIHHAb8HTfFURkkvguwURkOd74G4ezrZO2lDYwd1JyyLa+6W9NQRb7q5tpPNfhdCgG2HKsgbiYKJbnpjsdSkCsLcimrdPD7uNnnA7F+Mk1CUVVu4F7gReAEuAxVT0oIneLyN2+1d4PHBCRN4AfAbep14DbBv9bvF17l4ddx8+EdGV8f2sLslCFzaUNTodigC1ljRTOSAv5+rleRfkZxEYLm6weJeS4JqGAtxhLVQtUNV9Vv+lbdr+q3u97/RNVvUxVF6nqSlXdeqlt3WD3iTN0dvdwZRgllAVTUkhPimPTEfvBO63hXAclNS1hdcEyPj6GZTPS7PwKQa5KKOFoS2kDMVESNsUR4J1068pZmbx2rIGeHhtu3Enbyry9ysMpoYC32Ovw6VZqW9qdDsX4wRLKGNtS2sCS6akkxcc4HUpArS3IouFcB4dqWpwOJaJtKW0geVxMyA7nMxgblSE0WUIZQ81tXeyvbmZVfnhdPcKbw41bay9nbSlroCgvg+gQHc5nMPNykslKjrfzK8RYQhlD28ob6VG4MgTmjvdXdvI45udMsCtIB51sbKOy6UJYnl8iwprZWWwubcBjxaohwxLKGNpa1kBiXDSLpqY6HcqYWDsni90nztDa3uV0KBFpS5m3lV043gGD9/w629bFvqqzTodihskSyhjaXNrAitx04mLC8595zewsunv0YsWwCa7NpQ1MmjCO/Kwkp0MZE1fNykQEXjtqzdNDRXj+pXOBmuYLlNefD7vWN30tm5FGUpwNN+6Enh5la2kDq2ZlhPxwK4NJS4pj4dRUNh2tczoUM0yWUMbIltLwbM7ZV1xMFEX5mTaQnwNKTrdwpq0rrPo3DWTt7Ez2Vp6l+YIVq4YCSyhjZEtpAxlJcSE/+utQVs/KoLLpAlVn2pwOJaJs8Y1SEM4XLOAdjLRHYWdFk9OhmGHwK6GISJKIzBORpSIybegtIpOqsqW0gVWzMkN2dsbhKsrPALB6lCDbUtrIrOzxTJwQHuPDDWbJ9FTiY6Ls/AoRQyYUERkvIveIyFbgDHAAKAaOi8hpEXnQN1Cj8SmtO0ddawerfX9sw1lBdjLpSXFss3nAg6azu4edFU0RcX7Fx0SzbEaanV8h4pIJRUQ+BxwHPgO8iHcWxMXAHGAV8M9AHLBRRJ4VkbyxDDZUREpxBHiHYVmZl872skZUrb9AMLx+8gwXujwRcX4BFOVlUFLTwpnznU6HYoYw1B3Ku4AbfYMxfk1V/6Kq+1X1mKruUNVfqupHgRzgVbxzuke8zaWNTE9PZFp6otOhBEVRXganmtupbLrgdCgRYUtpA1ECK/LC/w4F3ixW3WH1KK53yYSiqjeo6q6hdqKq51T1e6r608CFFpq6PT3sKG+MmKtH6FOPUm79BYJhS1kjC6emkpIQ63QoQbFwaioJsdFst2Iv1xt2pbyIDNp7yiro37SvupnWjm5Wz4qMq0eA/KzxZI6Pt4rTIGht72Jv5dmIOr/iYqIonJlm51cI8KeV1x4RWdx/oYi8F9gbqIBC3dbS8B4OYyAi3nqUbeVWjzLWdlY04enRiLoDBliZl8GR2labJdTl/EkoW4FtIvJ5ABGJF5GfAQ8D3wtEMCJyg4gcEZFSEblvgM8/IiL7fI+tIrKoz2fHRWS/iOwVkeJAxDMSm0sbmJ8zgfSkOKdCcERRfga1LR1UNJx3OpSwtrm0gfiYKJZOT3M6lKDqLVbdXm71KG427ISiqh8HPgl8Q0SewTuP+/XAVar67dEGIiLRwE/xVuzPB24Xkfn9VqsA1qrqQuAbwAP9Pr9GVRerauFo4xmJC50e9pw4G5ajvw6lKM9+8MGwtbSR5bnpYTPd73AtmJJCUpzVo7idXx0bVfX3eP/o3wTMBT6qqjsDFMtyoFRVy1W1E3gEbzPlvsffqqpnfG+3A1MDdOyA2HW8iU5PT8QVRwDkZiYxcUK89RcYQ3Wt7RypbY3I8ys2OorCmel2frmcP5XyE0TkUeBvgXuAPwAbROTTAYplClDZ532Vb9lgPgn8pc97BV4Ukd0ictdgG4nIXSJSLCLF9fWBHYNqa1kjsdHCFTMjqzgCvPUoRXkZbLP+KGOmt1J6VQR0aBxIUX6Gr9OwTQvsVv7cobyBt0PjFap6v6reDnwe+E8ReSwAsQw0RsmAf5lE5Bq8CeUf+yxerapL8RaZfU5E1gy0rao+oKqFqlqYlZU12pjfYnt5I4umppIYF17T/Q7XyrwMGs51UFZ/zulQwtL28iaS42O4bHJ4Tfc7XFas6n7+JJS/ACtU9XDvAlX9FVAIzApALFVA3+bHU4FT/VcSkYXAL4FbVPXi/a+qnvI91wF/xFuEFjTnO7rZX93Mirz0YB7WVWxcr7G1o6KRK3LTw2663+G6bPIEkuNj7PxyMX8q5e9R1be12VPVI8DKAMSyC5gtIrkiEgfcBjzddwURmQ48Cdyhqkf7LE8SkeTe13gbCxwIQEzDtvvEGTw9yorcyCyOAJiensjklHF2BTkG6lraKa8/z4rcyL1giYmO4orcdHZYPYprDTWWV8JwduKrRB/2+oPsoxu4F3gBKAEeU9WDInK3iNztW+1fgAzgv/s1D54IbBaRN4CdwLOq+vxIYxmJ7eWNxEQJy2ZEXv1JLxFhZX4G260/SsD1DjuyMkKGWxlMUV4G5Q3nqW2xehQ3GuoOpUxEPi8iqZdaSURWisiTwN+PJhhVfU5VC1Q1X1W/6Vt2v6re73v9KVVN8zUNvtg82NcybJHvcVnvtsG0o6KJBVNTSIqPzPqTXkV5GTSe7+RordWjBNKOikbGx8dw2eQJTofiKCtWdbehEsq7gfcDp0TkeRH5NxH5jIjcKSJfFJFfishR4HlgN/D9sQ7YjS50ethXdTaii7t69V5Bbyuzcb0CaUd5E8tmpBETHdlz4s3LmcCEcVaP4laXvJxW1WJgjYgUAh8BbgZygQSgHtgD/AD4napG7CXpnpNn6PJoRFfI95qWnsjUtAS2lTfysdW5TocTFhrOdXCs7hy3Lr1UK/rIEB0lrMjLsP4oLjWs8hlfYnFsOBO3217eSHSUUBjB9Sd9FeVl8FJJLT09GvYzVgbDTqs/eYuVeRm8dKiW6rMXmJI64mpbMwYi+/45QHaUN3H55Akkj4uM4cSHUpSfwdm2Lg6fbnU6lLCwo7yRxLhoFkyJzP4n/V3sj2LFXq7jVw2yiOQD78DbquotyUhV/y2AcYWM9i4PeyvP8rHVM50OxTXenB+lkfkRXokcCNt99SexEV5/0mvupGTSEmPZVt7I+5a5avSliDfshCIiHwUeAtqBWt7ai12BiEwor588S6enJ6L7B/SXk5LAzIxEtpU18skrrR5lNJrOd3KktpV3LcpxOhTXiIoSVuRmWMW8C/lzyfN1vK24UlV1lqrO7vMoGKP4XG97eSNRAoUzLaH0VZSfwY6KRjw91h9lNKz+ZGBF+RlUn71AZVOb06GYPvxJKJOAX6iqZ6yCCUU7KrzFOpEyHetwrcjNoLW9m8OnW5wOJaTtqGhkXGwUC6emOh2Kq/S2qLR55t3Fn4SyEVgyVoGEoo5uD6+ftP4nA7n4g7dhWEZle3kTS6enERdj9Sd9FWQnk5oYa8OwuIw/Z+n/At8Rkb8TkatFZFXfx1gF6GZvVDbT0d1jxREDyElJYHp6Ijsq7Ac/Us1tXRw+3WLn1wCiooTlM9PtDsVl/Gnl9ajv+YcDfKZAZE0hh7f+RASWW/3JgJbnprPR+qOM2M7jTahiDT4GsTw3nRcP1VLTfIGcFOuP4gb+3KHkXuKRF/jQ3G9HRSNzJ00gJdHqTwayIjedM21dHKuL2EEURmV7eSNxMVEsmpbqdCiu1HvnttPuUlzDn+HrT1zqMZZBulFndw+7T5yxq8dLePMHb8VeI7GjopEl01Ijbv744ZqXM4HkcTE2XYKLXLLIy1c3skNVPUPVk6jq1oBG5nL7q8/S3mX1J5cyNS3BOz9KRRN3FM10OpyQ0tLexaFTLfztO2Y7HYprRUcJV8xMt3o6FxmqDmUz3ubCdb7XyuBT9UbUZVTvVdFyu0MZlIiwPDedzaXe+VFErB5luIqPN9Gj2ICjQ1iRm87Lh+uoa20nO3mc0+FEvKGKvHLxjirc+zoPq0MBvOXbcyYmk54U53QorrbCN898ecN5p0MJKdvLm4iLjmLpdBtw9FJ6L+h2VZxxOBIDQyQUX/2I9nk9pnUoInKDiBwRkVIRuW+Az0VEfuT7fJ+ILB3utoHU5fHVn9jV45B665isP4p/dpQ3smhaitWfDOHyKSkkxkVbsZdLDLtSXkQWichlfd7fJCKPi8jXRGTU0xSKSDTwU+BGYD5wu4jM77fajcBs3+Mu4Gd+bBswB6qbaev0WP3JMORmJpGVHG8V835obe/iwCnrfzIcsdFRLJuRZhcsLuFPs+GfAwsARGQq8AQwHvg08O8BiGU5UOqbzrcTeAS4pd86twD/o17bgVQRyRnmtgHT25nK6k+G1luPsqOiyeaZH6biE2fw9KiNwDBMK/MyOFLbStP5TqdDCQm1Le184te7eKPybMD37U9CmQO87nv9XmCXqt4I/A3woQDEMgWo7PO+yrdsOOsMZ1sAROQuESkWkeL6+vqBVhlSV3cPhTPSyBwfP6LtI83K3HRqmtupbLrgdCghYUd5EzFRwtIZqU6HEhJ6i1WtP8rwbC9v5OXDdYxFGxl/Ekoc3qHrAa4G/uJ7fRRvS7DRGqz12HDWGc623oWqD6hqoaoWZmVl+Rmi199eO5snPhuRo82MyIreCZGs2GtYdlQ0smhaKolxoy5JjggLpqYQHxNlCWWYdlQ0kRwfw/ycwM9V5E9COQK8X0SmA+uADb7lOUAgmlhUAdP6vJ8KnBrmOsPZ1jhkdvZ40pPirJx7GM53dLO/qtk6zPohPiaapdPTrGJ+mHaUN1I4M42YMZiwzd/5UL4FVACbffPMA1zPm0Vho7ELmC0iuSISB9wGPN1vnaeBv/G19loJNKtqzTC3NQ4REa6YmcbO4/aDH8ruE2fo7tGLd3VmeFbkpXOopoXmC11Oh+Jqda3tlNWfH7Pzy5+hV54CpgPLgPV9PtoI/MNoA1HVbuBe4AWgBHhMVQ+KyN0icrdvteeAcqAU+AVwz6W2HW1MJnBW5GZQ2XSBU2etHuVSdlQ0Eh0lLJth/U/8sSI3A1Vvh1AzuLGesM2vQlpVrcU7/W/fZdsCFYyqPoc3afRddn+f1wp8brjbGvd4c0KkRm5dYvOAD2ZHeRMLpqQwPt7qT/yxZHoqcdHeepRr5010OhzX2lHeRFJcNJdPDnz9CfhX5IWI3C4iSf1fGzOUuZMmMGFcjNWjXMKFTg9vVJ21DrMjMC42mkXTUthuFfOXtL28kWUz08ek/gT8TCh4+6JMHOC1MZcUHfVmfxQzsD0nz9DlUVZa/5MRWZGbwYHqZs51dDsdiis1nuvgWN25MW3w4W9CkUFeGzOk5bnpVDScp66lfeiVI9CO8kaiBApnWv3JSKzIS8fTo+w+YeN6DWSs60/A/4RizIj19vy2u5SBba9o4vIpKSSPswnbRmLZjDRiosSG+RnEjoomEmKjWTg1ZcyOYQnFBM1lkycwPj7G+gsMoL3Lw96TZ63/ySgkxsWwYGqK1dMNYnt5I8tmpBE7RvUnYAnFBFGMDeQ3qNdPnqXT02Pjd43SitwM3qg6y4VOj9OhuMqZ850cPt3KyjFu8GEJxQTVirx0jtWdo+Fch9OhuMqOikZE4Aq7QxmVFXnpdHmU109aPUpfO339c8a6w6wlFBNUvRWCdpfyVjvKm5ifM4GUBKs/GY0rZqYTHSVsK7di1b52lDcRHxM1pvUnYAnFBNmCKSkkxUWzrbzB6VBco6Pbw56TZ6y4KwDGx8ewYEoK28osofS1vbyRpdPTiI8Z2wnb/E0ovwVaBnhtzLDERkdRODOd7XaHctEblc10dPeMefl2pFiZ561Haeu0/igAzW1dlJwOzoRtfiUUVf2sqjb0f22MP4ryMyitO0ddq/VHAW//ExGbsC1QivIz6PJYf5Reu443oUpQRmCwIi8TdEW986PYXQrg7R8wZ2IyqYlxTocSFgp9/VGs2MtrR0UjcTFRLJ6WOubHGvYIdCJSAFwD5AEJQD1QDGxS1baxCc+Eo8smTyA5PoZtZY28e9Fkp8NxVGd3D8UnmrjtiulOhxI2kuJjWDQt1SrmfbaXN7F4WirjYse2/gSGcYciIh8WkV3AYbxzolyHdwj7DwNPAadF5Ge+ibeMGVJMdBTLc9PZbj949lefpb3L6k8CrSgvg31VNq5XS3sXB081B6X+BIZIKCKyE/gK8HsgT1UnqeoyVV2tqvOACcD78N6xvC4i7x3ziE1YWJmXQUXDeU43R3Y9Sm+x33Jr4RVQK/My8PQouyJ8fpTdx8/Qo7AySPVzQ92h/KeqLlDVH6rq8f4fqmq7qr6kqh8DFgF1YxCjCUNF+b31KJF9l7K9vJGCid4pkk3geIcYEbZHeD3K9vJGYqOFJdODM+DoJROKqj4y3B2papWqbh5JECKSLiIvicgx3/Pbvr2ITBORV0SkREQOisjf9fnsayJSLSJ7fY+bRhKHCZ55Od75USK54rTL08PuE2eCVhwRSRLiolkyLc0uWCq89ScJcWNffwJ+tPISkRsu8dm9o4zjPmCjqs7GO6XwfQOs0w18yVfUthL4nIjM7/P5D1V1se9hMze6XHSUsCIvI6IrTg9UN9PW6bEOjWNkZX4G+6ubaWmPzHnmz3V0c6C6Oajnlz/Nhv8sIt8VkYupTkTSROQp4JujjOMW4De+178B3tN/BVWtUdU9vteteOeOnzLK4xoHFeVlcLKpjeoInWf+zfoTq5AfC0V5GfQo7IrQ6RKKjzfh6dGgzgDqT0K5Hm/Lrq0iMlNE1gD7gBy8rb5GY6Kq1oA3cQDZl1pZRGYCS4AdfRbfKyL7ROTBgYrM+mx7l4gUi0hxfX39KMM2o9Fb1BOp5dxbyxoomDierOR4p0MJS0umpxIXExWxxapbyxqJi46icIYLE4qqvgIsBhrwJpKXgEeA1apaOtT2IrJBRA4M8LjFn4BFZDzwB+ALqto79MvPgHxffDXADy7xPR5Q1UJVLczKyvLn0CbA5k5KJi0xNiKLvdq7POw63sTqWZlOhxK2xsVGs3R65PZH2VLawNIZwas/Af97yo8DkgGPb9vzeOs2hqSq16nq5QM8ngJqRSQHwPc8YGsxEYnFm0x+p6pP9tl3rap6VLUH+AWw3M/vZRwQFSWsyM2IyCvIPSfP0N7Vw+p8SyhjqSgvk0M1LZxt63Q6lKBqOt/JwVMtQT+//KmUvwV4A2gH5gHvBj4LvCoio63LeBq40/f6TrwdJvsfX4BfASWq+p/9Psvp8/ZW4MAo4zFBUpSfQfXZC1Q2RdZgC1tLG30NE6z+ZCwV5Weg+uZ86pGi9yJt9WyXJhTgceD7wDtV9bSq/gVv35MOvIlmNL4NrBORY8A633tEZLKI9LbYWg3cAbxjgObB3xWR/SKyD+/wMF8cZTwmSHr7o0RascTm0gYWTbX548faomkpjIuNisjzKzk+hoVTxnb+k/6GPZYXsFZVt/VdoKqngetF5MujCUJVG4FrB1h+CrjJ93ozIINsf8dojm+cMzt7PBlJcWwva+SDhdOcDicoWtq72Fd1lnuvmeV0KGEvPiaaZTPSIq5YdWtZAyvyMogZw/njB+JPpfy2S3z23cCEYyKNiLDS1x9FVZ0OJyi2lzXSo7DKKuSDoigvg8OnW2k6Hxn1KJVNbZxobGP1rOD3bxpqLK9hj80lIpNEpGj0IZlIszI/g5rmdk40RkY9ytayRhJio1kyPdXpUCJCb7Hqjggp9tpa5p2m6koHLliGukP5ZxHZIyKfEZFJ/T8UkRgRWS0iPwcOYh0NzQj0zo8SKeXcm0sbuCI3fcynYzVeC6emkhgXHUHnVyPZyfHMyh4f9GMPlVCWAT8F7gVOichJEdnmG1PrDbxTAL+Ity7mClV9YmzDNeEoPyuJrOT4iBh3qbalndK6c1zpQHFEpHpz2unwP796epStpQ2snpWJt2FscA1VKX8H8FtV/ZWILALWArm8OcHWHuAVVT07plGasHaxHqXMW4/ixA8hWLaUeosjVln/k6BamZfOd58/QsO5DjLHh+/IBEdqW2k83+lYh9mhEspDwF94M3nkqKoNUW8CblV+Bn9+4xRl9eeYlZ3sdDhjZktpI2mJsczPmeB0KBHFm8CPsDXMZwntvWBxokIehi7yqgdW+F4LEBnNcEzQ9VYgbjra4HAkY0dV2VLawKr8TKKiwvcuzI0WTEkhJSGWvx4N7/H7tpQ2kJeVRE5KgiPHHyqhPAg8LSJdeJNJtYh0DvQY+1BNOJuWnkheVhKvhfEPvrzhPKdb2m38LgdERwlXzs5k09H6sG2e3uXpYUdFk6PD+VyyyEtV/0lE/gQUAP8D/B+gOQhxmQi0tiCL3+84SXuXh3Gx4dcCyuniiEi3tiCLZ/fVcPh0K/PCsMhxb+VZ2jo9jl6wDNlTXlV3AjtF5FrgId9cJMYE3JqCLB7acpwdFU2sLQi/kaC3lDYwNS2B6emJTocSkdbM9p5Trx2tD8uEsqW0gSh5sxm+E/zpKf9xSyZmLK3MzSAuJiosi708Pcq2skZW5zvTnNPApJRxzJ2UzKYwPL/Am1AWTEkhJdG58eGCO9CLMZeQEBfNitz0sPzBH6hupqW9O+ijv5q3WluQxa7jTZzvGNasGyHjfEc3r5886/hwPpZQjKusLciitO5c2E0LvPli/xOrP3HSmoIsujwadp0cd1Y00d2jjgy30pclFOMqvXUn4VbstbWsgbmTksO6U10oKJyZRkJsdNjdBW8pbSAuJoplMwad/TwoLKEYV5mVPZ7JKePYdCR8fvDe6X7PWHNhF4iPiaYoPyPsLlg2lzZwxcw0x1tHuiKhiEi6iLwkIsd8zwOmWRE57ptIa6+IFPu7vXE/EWFNQRZbShvo8vQ4HU5A7D5xhs7uHseLI4zX2oIsjje2cbzhvNOhBETDuQ4On251xXA+rkgowH3ARlWdDWz0vR/MNaq6WFULR7i9cbm1BVm0dnSzt/Ks06EExF+PNRATJSzPtel+3eBiseqx8LhLebN/kyWUXrcAv/G9/g3wniBvb1xk1axMoqMkbIq9Xj5cyxUz00mK92eCVDNWZmYmMT09MWyKvV4+XEdGUhwLgjzd70DcklAmqmoNgO85e5D1FHhRRHaLyF0j2N6EgJSEWJZMSw2LK8iTjW0crT3HtfPslHSTtQVZbC1rpKPb43Qoo9Ll6eGVw3VcMzebaBeMDxe0hCIiG0TkwACPW/zYzWpVXQrcCHxORNaMII67RKRYRIrr60P/D1a4WluQxb6qZhrOdTgdyqhsKKkFYN38iQ5HYvpaW5BFW6eH3cfPOB3KqBQfP0NLezfXzXPH+RW0hKKq16nq5QM8ngJqRSQHwPc84BD5qnrK91wH/BFY7vtoWNv7tn1AVQtVtTArK/yG9wgXa+d4/282Hwvt0Yc3Hq5ldvZ4ZmQkOR2K6aMoP4PYaGFTiN8FbyipJS46iqtc0mHWLUVeTwN3+l7fCTzVfwURSRKR5N7XwPXAgeFub0LL5ZNTSE+KC+ly7uYLXewob+Jal1w9mjclxcdQOCM9pOvpVJUNJbWsmpXhmvo5tySUbwPrROQYsM73HhGZLCLP+daZCGz2TT28E3hWVZ+/1PYmdEVFCVfNzuS1Y/X09ITmcOObjtbT3aOsm2/1J260piCLw6dbqW1pdzqUESmrP8eJxjZXXbC4IqGoaqOqXquqs33PTb7lp1T1Jt/rclVd5HtcpqrfHGp7E9rWFmTRcK6TQzUtTocyIhtLaklPimPxNOsW5UahPirDhhJvyf51Lmrw4YqEYsxArvINNx6Kw2T0tr55h0ta35i3m5eTTFZyfEieXwAbDtVy+ZQJjs3OOBBLKMa1spLjuWzyhJD8wbut9Y15OxFhzewsNpc24AmxYtXGcx3sPnmGa+e66/yyhGJcbW1BFntOnKG1vcvpUPzittY3ZmBr52Rxtq2LfVVnnQ7FL68cqUfVfc3RLaEYV7t6TjbdPcqrIdQax42tb8zArvKNytDbXyhUbDhUy6QJ47hssrtmnrSEYlxt2Yw0spPjeXZfjdOhDJsbW9+YgaUlxbEqP4Nn99WgGhrFXu1dHl47Vs+187JdN/unJRTjatFRwk0LcnjlSB3nQmSWvZcOua/1jRnc+gU5HG9s4+Cp0GhNuL28kbZOjyvr5yyhGNdbvzCHju4eNoZIscTGEve1vjGDe+dlk4iOEp7dHxp3wRtKakmI9c7r4jaWUIzrLZuexsQJoVHs5dbWN2ZwaUlxrJ6VGRLFXqrKxpI61hRkOj6Z1kAsoRjXi/IVe716tN71rb3c2vrGXNrNC3I42dTGgWp3F3sdPNVCTXO7a+vnLKGYkHDzwhw6u3vYWDLouJ+u4NbWN+bSrr9sIjFRwjP7TzkdyiVtKKlFBN4x1531c5ZQTEhYMi2NnJRxPOPiYi83t74xl5aaGMeVs91f7LWxpI4l01LJHB/vdCgDsoRiQkJvsddrR+tpcWmx18XWN1bcFZLWL8ih6swF3qhqdjqUAdU0X2B/dbOrzy9LKCZk3Lwwh05PDxsOubO110uHakmMi6Yoz32tb8zQrr9sErHRwrP73Fns9eZgkJZQjBm1xdNSmZKa4MrWXu1dHp7ZV8O18ya6svWNGVpKQixrZme5ttjryT1VzMoez+zs8U6HMihLKCZkiAjrF+bw2rF6mtvcVez14qFami908aHCaU6HYkZh/cIcTjW383rlWadDeYtjta28fvIsHyqc5ur6OUsoJqSsX5BDl0d58dBpp0N5i8d2VTIlNYFVLuxsZobvuvkTiYuOct1d8GPFlcRECbcuneJ0KJfkioQiIuki8pKIHPM9v21GIhGZIyJ7+zxaROQLvs++JiLVfT67KehfwgTFwqkpTE1LcFWv5sqmNraUNfCBwqlE2dwnIW3CuFjWFGTx3P4a18wU2tndw5N7qrl2XrZrW3f1ckVCAe4DNqrqbGCj7/1bqOoRVV2sqouBZUAb8Mc+q/yw93NVfa7/9iY89BZ7bT7WwNm2TqfDAeCJ3VUAvH/ZVIcjMYFw88Icaprb2XPyjNOhAPDy4Toaz3fyoSvcX5zqloRyC/Ab3+vfAO8ZYv1rgTJVPTGWQRl3unnBZLp7lBcPOt/aq6dHeWJ3FVfOymRqWqLT4ZgAuHZeNnExUa7p8/RYcSXZyfGs8c1g6mZuSSgTVbUGwPc8VDfQ24CH+y27V0T2iciDAxWZ9RKRu0SkWESK6+tDZ44N86bLp0xgenoiz7ig2GtLWQPVZy/wQauMDxvJ42K52iXFXrUt7bx6pI73L5tKTLRb/lwPLmgRisgGETkwwOMWP/cTB7wbeLzP4p8B+cBioAb4wWDbq+oDqlqoqoVZWe7P+ObtRISbF+awpbSBM+edLfZ6rLiKlIRYG7srzNy8aDJ1rR0Un3C22OuJ3VX0KHwgRC5YgpZQVPU6Vb18gMdTQK2I5AD4ni81YNONwB5VvVjeoaq1qupR1R7gF8Dysfwuxnk3L5yMp0f5w54qx2I429bJCwdPc+uSKdb3JMxcOzebhNhoHt1V6VgMqsrjxZUsz00nNzPJsTj84ZZ7qKeBO32v7wSeusS6t9OvuKs3GfncChwIaHTGdeZPnsDy3HQe3FxBl6fHkRie2nuKzu4ePlBolfHhJik+hg8WTuXpN6o53dzuSAw7K5o43tgWUsWpbkko3wbWicgxYJ3vPSIyWUQuttgSkUTf50/22/67IrJfRPYB1wBfDE7YxkmfWZPHqeZ2x/oMPLqrksunTOCyySmOHN+MrU9dlYenR3loS4Ujx3+suIrx8THctGCSI8cfCVckFFVtVNVrVXW277nJt/yUqt7UZ702Vc1Q1eZ+29+hqgtUdaGqvru3gt+Et2vmZDMrezw/f6086ENlHKhu5lBNS0hdPRr/TEtP5MYFOfx+x8mgz8PT2t7Fc/treNeiHBLjYoJ67NFwRUIxZiSiooS7rsqjpKaFzaUNQT32Y8WVxMVEccsid/dcNqPzmTV5tHZ08/DOk0E97jP7arjQ5Qm5CxZLKCak3bJkMtnJ8fx8U3nQjtne5eFPr1dzw2WTSEmMDdpxTfAtnJpKUV4GD24+Tmd38OrqHiuuZHb2eBZPSw3aMQPBEooJafEx0Xx8dS6bSxs4UB2ceSxeOHialvbukOi5bEbvrrV5nG5p589vBGdY+4sDQV7h7oEgB2IJxYS8D6+YTlJcNL/469jfpagq/7PtBFPTEmzekwhxdUEWcyYm80CQ6up+s+04MVHCe5aEXnGqJRQT8lISYrl9+XSe2VdD1Zm2MT3Wc/tPs/vEGe5em28DQUYIEeHTa/I4UtvKq0fHdnSNo7WtPLyzktuWT3P9QJADsYRiwsInrsxFgAc3Hx+zY7R3efjWcyXMnZTM7cunj9lxjPu8e9FkJk0YxwNjWFenqnzjmUMkxUXz9+vmjNlxxpIlFBMWJqcm8K5Fk3lk18kxm3zrF6+VU332Av/yrvlE291JRImLieITV85kW3kj+8dozvmNJXX89VgDX7iugPSkuDE5xlizhGLCxqevyqOt08NvdwR+EOrTze3896tl3HDZJFblZwZ8/8b9bl8+neT4GH7+WlnA993R7eHfnz1EflYSdxTNCPj+g8USigkb8ydP4KrZmTy05TjtXZ6A7vs7zx/Go8o/3TQvoPs1oSN5XCwfXjGd5/bXUNkU2Lq632w9zvHGNv7vzfOJDYFRhQcTupEbM4C71+bTcK6DX20O3HAZe06e4Y+vV/OpK3OZnmFznkSyj6/OJTpK+M7zhwPW4qu+tYMfbyzlmjlZXD1nqJk73M0Sigkrq/IzWL8whx+8eIS/Hht9i5yeHuXf/nyI7OR47rlmVgAiNKFsUso4vnBdAc/sq+GhLccDss8fvHiEC10e/vnm+QHZn5MsoZiwIiJ8930LmZ2dzN8+/Pqoiyb+tLeavZVn+fINcxkfHzpjKpmx89m1+Vw/fyLffK6E7eWNo9rXgepmHi2u5M5VM8nPGh+gCJ1jCcWEnaT4GH5+xzI8Pcrdv9094vqU8x3dfOf5wyyamsJ7Q7CTmRkbUVHCDz64iJkZiXzud3s4dfbCiPaj6r37TUuM4/PXzg5wlM6whGLC0szMJP7rtsUcqmnhn57cP6Ly7h+/XEptSwf/8q7LrBOjeYvkcbH8/I5COrp7+OwIL1qefuMUO4838aXrC0hJCI8x4SyhmLD1jrkT+eJ1BTz5ejW/2Xp82Nu1d3n4ypP7uH9TGe9fNpVlM9LGLkgTsmZlj+cHH1zEG1XN/OtTB4d90dLTo9y/qYy/f+wNFk5N4bYrwqeTrBUKm7B27zWz2FfVzL8/W8L8ySksz02/5PonGs/z2d/u4VBNC5+7Jp8vXlcQpEhNKHrnZZO495pZ/OSVUhZOS+EjKy7dh6S5rYsvPb6XDSV1rF+Yw3fetzCsOsm64g5FRD4gIgdFpEdECi+x3g0ickRESkXkvj7L00XkJRE55nu2S0oDeMu7//NDi5iensg9v9vNzoqmQacMfuHgaW7+8Waqz17gwY8V8g/vnEtMCPcJMMHxxXUFrC3I4mtPH+TZfTVc6By4+Gtf1VnW//ivbDpaz9feNZ+f3L4k7Bp6SLBnuhswCJF5QA/wc+D/qGrxAOtEA0fxTgFcBewCblfVQyLyXaBJVb/tSzRpqvqPQx23sLBQi4vfdigThkrrWrn1v7fS2t5NUlw0V+SmU5SXQVF+BgUTk/nBi0f4xV8rWDQ1hZ98eCnT0q2/iRm+5rYu3vPfW6hoOE9stLBkWhor8zMoystgyfRUHt9dxTf+fIjM8XH89CNLWTI9tK95RWS3qr7t4t8VCaWXiLzK4AmlCPiaqr7T9/4rAKr6HyJyBLhaVWtEJAd4VVWHHF3NEkpkOdvWydayRraVNbK1rIGy+vMAxEYLXR7lb4pm8NX184iPiXY4UhOKLnR62FHhPb+2lTdyoLqZHn3z/FpbkMX/+9Bi0kJ0nK6+BksooXS/NQWo7PO+Cljhez2xdx55X1IZtLupiNwF3AUwfXr4VIaZoaUmxnHTghxuWpADQG1LO9vLG9l94gxFeRnc6FtuzEgkxEVz9Zzsi73dmy90sbOiie3ljUxLS+BvimaGfWvBoCUUEdkATBrgo6+q6lPD2cUAy/y+vVLVB4AHwHuH4u/2JnxMnDCOWxZP4ZbF1sfEBF5KQizr5k9k3fyJTocSNEFLKKp63Sh3UQX0nXN1KtA7J2etiOT0KfKqG+WxjDHG+CmUmrDsAmaLSK6IxAG3AU/7PnsauNP3+k5gOHc8xhhjAsgVCUVEbhWRKqAIeFZEXvAtnywizwGoajdwL/ACUAI8pqoHfbv4NrBORI7hbQX27WB/B2OMiXSuauUVbNbKyxhj/DdYKy9X3KEYY4wJfZZQjDHGBIQlFGOMMQFhCcUYY0xARHSlvIjUAydGuHkm0BDAcJwQ6t/B4ndeqH+HUI8fnPkOM1Q1q//CiE4ooyEixQO1cgglof4dLH7nhfp3CPX4wV3fwYq8jDHGBIQlFGOMMQFhCWXkHnA6gAAI9e9g8Tsv1L9DqMcPLvoOVodijDEmIOwOxRhjTEBYQjHGGBMQllBGQERuEJEjIlLqm8M+ZIjINBF5RURKROSgiPyd0zGNhIhEi8jrIvKM07GMhIikisgTInLY939R5HRM/hCRL/rOnwMi8rCIjHM6pqGIyIMiUiciB/osSxeRl0TkmO/Z1ZO9D/Idvuc7j/aJyB9FJNWp+Cyh+ElEooGfAjcC84HbRWS+s1H5pRv4kqrOA1YCnwux+Hv9Hd5pDELVfwHPq+pcYBEh9F1EZArweaBQVS8HovHOT+R2vwZu6LfsPmCjqs4GNvreu9mveft3eAm4XFUXAkeBrwQ7qF6WUPy3HChV1XJV7QQeAW5xOKZhU9UaVd3je92K9w9ZSM2BKyJTgfXAL52OZSREZAKwBvgVgKp2qupZR4PyXwyQICIxQCJvzp7qWqr6GtDUb/EtwG98r38DvCeYMflroO+gqi/65osC2I53NltHWELx3xSgss/7KkLsD3IvEZkJLAF2OByKv/4f8GWgx+E4RioPqAce8hXb/VJEkpwOarhUtRr4PnASqAGaVfVFZ6MasYmqWgPeiy0g2+F4RusTwF+cOrglFP/JAMtCru21iIwH/gB8QVVbnI5nuETkZqBOVXc7HcsoxABLgZ+p6hLgPO4varnIV89wC5ALTAaSROSjzkZlROSreIu0f+dUDJZQ/FcFTOvzfiohcLvfl4jE4k0mv1PVJ52Ox0+rgXeLyHG8xY3vEJHfOhuS36qAKlXtvTN8Am+CCRXXARWqWq+qXcCTwCqHYxqpWhHJAfA91zkcz4iIyJ3AzcBH1MHOhZZQ/LcLmC0iuSISh7cy8mmHYxo2ERG8ZfclqvqfTsfjL1X9iqpOVdWZeP/tX1bVkLo6VtXTQKWIzPEtuhY45GBI/joJrBSRRN/5dC0h1Kign6eBO32v7wSecjCWERGRG4B/BN6tqm1OxmIJxU++yq97gRfw/ogeU9WDzkbll9XAHXiv7Pf6Hjc5HVQE+lvgdyKyD1gMfMvZcIbPd2f1BLAH2I/374hrhv8YjIg8DGwD5ohIlYh8Evg2sE5EjgHrfO9da5Dv8BMgGXjJ93u+37H4bOgVY4wxgWB3KMYYYwLCEooxxpiAsIRijDEmICyhGGOMCQhLKMYYYwLCEooxxpiAsIRijDEmICyhGGOMCQhLKMa4hIhkiEiliPxXn2XZIlIjIt9xMjZjhsN6yhvjIiKyBu9ET+8FngGeB1KAq3wDMRrjWjFOB2CMeZOqviYi/w48hHfCpxXAEksmJhTYHYoxLiMiUcBmoAi4TVUfdTgkY4bF6lCMcZ8coADw+J6NCQl2h2KMi/juTjbivdj7IfA4cI2qbnY0MGOGwepQjHGXrwILgMWqWuWb2+J3IrJYVc84HJsxl2R3KMa4hIisAjYB71PVp33L4oGdQKmqvs/J+IwZiiUUY4wxAWGV8sYYYwLCEooxxpiAsIRijDEmICyhGGOMCQhLKMYYYwLCEooxxpiAsIRijDEmICyhGGOMCYj/D23k8DLs85JfAAAAAElFTkSuQmCC\n", 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" ] }, "metadata": { "needs_background": "light" }, "output_type": "display_data" } ], "source": [ "plt.title(\"Originalfunktion\", size=\"x-large\")\n", "plt.ylabel(\"f(x)=sin(x)\", size=\"x-large\")\n", "plt.xlabel(\"x\", size=\"x-large\")\n", "plt.plot(x, y)\n", "plt.show()" ] }, { "cell_type": "markdown", "id": "eb6aa4b3", "metadata": {}, "source": [ "Jetz haben wir die Funktion bereits diskretisiert und an diesen diskreten Stützstellen evaluiert, um sie passend darzustellen. Zur näheren Illustration der Diskretisierung der Funktionswerte an Raumpunkten, soll uns das folgende Schema helfen:\n", "\n", "\"Diskretisierung\n", "\n", "\n", "\n", "Wir benötigen also jetzt im Folgenden den Funktionswert an der Stelle, an dem die Ableitung ausgerechnet wird sowie den Funktionswert an einer um die Diskretisierung in die positive $x$-Richtung und in die negative $x$-Richtung verschobenen Stelle. Die Ableitung berechnet sich dann aus der Steigung der Funktion an dem Raumpunkt, einmal unter Berücksichtigung des Vorgängers und einmal unter Berücksichtigung des Nachfolgers. Man unterscheidet dann entsprechend zwischen einem Vorwärts- und einem Rückwärtsdifferenzenquotuenten. \n", "\n", "\"Berechnung\n", "\n", "\n", "Der Differentialoperator wird dann näherungsweise als Differenzenquotient berechnet. Man spricht hier von einer ersten Ordnung:\n", "\n", "$$\n", "\\frac{\\mathrm{d} f}{\\mathrm{d} x}\\approx\\frac{\\Delta f_1}{\\Delta x}=\\frac{f_3-f_2}{\\Delta x}\n", "$$\n", "\n", "$$\n", "\\frac{\\mathrm{d} f}{\\mathrm{d} x}\\approx\\frac{\\Delta f_2}{\\Delta x}=\\frac{f_2-f_1}{\\Delta x}.\n", "$$\n", "\n", "\n", "In beiden Fällen macht man natürlich einen Fehler. In einem Fall überschätzt man die Ableitung und im anderen unterschätzt man die Ableitung. Eine bessere Näherung erreichen wir, in dem wir den Mittelwert beider Ableitungen berechnen\n", "$$\n", "\\frac{\\mathrm{d} f}{\\mathrm{d} x} \\approx \\frac{\\frac{\\Delta f_1}{\\Delta x}+\\frac{\\Delta f_2}{\\Delta x}}{2}\n", "\\approx \\frac{f_3-f_2+f_2-f_1}{2 \\Delta x}\n", "\\approx \\frac{f_3-f_1}{2 \\Delta x}\n", "$$\n", "Da man hier für die Ableitung an einem Raumpunkt nur die beiden nächsten Nabarn berücksichtigt, spricht man hier von einem zentralen Differenzenquotienten.\n", "\n", "Alle drei Differenzenquotienten werden wir im folgenden berechnen und die auftretenden Fehler vergleichen.\n" ] }, { "cell_type": "code", "execution_count": 3, "id": "bc8c112b", "metadata": {}, "outputs": [], "source": [ "# Beachten Sie bitte im Folgenden, wie wir auf die Elemente der Liste zugreifen.\n", "# Mit [1:] greifen wir vom ersten Element bis zum letzten Element auf die Liste zu. Das nullte Element wird \n", "#nicht berücksichtigt. In dem Vorwärtsdifferenzenoperator ist das also der entsprechende Nachfolger.\n", "# Mit [:-1] greifen wir vom nullten Element bis zum vorletzten Element auf die Liste zu. Das letzte Element wird \n", "# nicht berücksichtigt. In dem Vorwärtsdifferenzenoperator ist das also der entsprechende Raumpunk, \n", "# an dem wir den Differentialoperator näheren.\n", "# Entsprechend müssen wir dann auch die Raumkoordinate extrahieren, an denen der \n", "# Differentialoperator angenähert wird.\n", "\n", "forward_difference = (y[1:] - y[:-1]) / (x[1] - x[0])\n", "x_forward = x[:-1]\n", "\n", "# Bei dem Rückwärtsdifferenzialoperator ist die Raumkoordinate entsprechend um einen Raumpunkt verschoben. \n", "# Dies ist ein kleines Detail aber an der Stelle wichtig.\n", "backward_difference = (y[1:] - y[:-1]) / (x[1] - x[0])\n", "x_backward = x[1:]\n", "\n", "# Und bei dem zentralen Differenzenquotienten mitteln wir einfach beide Werte. Technisc, rechnen wir die Ableitung\n", "# hier also nicht mehr bei dem Raumpunkt aus, an dem wir die Funktion evaluiert haben, sondern an einem Raumpunkt\n", "# der genau dazwischen liegt.\n", "central_difference = (forward_difference + backward_difference) / 2\n", "x_central = (x_forward + x_backward) / 2" ] }, { "cell_type": "code", "execution_count": 4, "id": "439d0001", "metadata": {}, "outputs": [ { "data": { "image/png": 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\n", 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" ] }, "metadata": { "needs_background": "light" }, "output_type": "display_data" }, { "name": "stdout", "output_type": "stream", "text": [ "Der Fehler im Forward beträgt 0.08960240840718171\n", "Der Fehler im Backward beträgt 0.08960240840718176\n", "Der Fehler im Central beträgt 0.00187844087678339\n" ] } ], "source": [ "plt.title(\"Ableitungen\", size=\"x-large\")\n", "plt.ylabel(\"f'(x)\", size=\"x-large\")\n", "plt.xlabel(\"y\", size=\"x-large\")\n", "plt.plot(x_forward, forward_difference, label=\"Forward\")\n", "plt.plot(x_backward, backward_difference, label=\"Backward\")\n", "plt.plot(x_central, central_difference, label=\"Central\")\n", "plt.legend()\n", "plt.tight_layout()\n", "plt.show()\n", "\n", "error_forward = np.std(forward_difference - np.cos(x_forward), ddof=1)\n", "error_backward = np.std(backward_difference - np.cos(x_backward), ddof=1)\n", "error_central = np.std(central_difference - np.cos(x_central), ddof=1)\n", "\n", "print(f\"Der Fehler im Forward beträgt {error_forward}\")\n", "print(f\"Der Fehler im Backward beträgt {error_backward}\")\n", "print(f\"Der Fehler im Central beträgt {error_central}\")" ] }, { "cell_type": "markdown", "id": "62f9ee88", "metadata": {}, "source": [ "Der Fehler, evaluiert hier als die Standardabweichung, ist gleich groß bei dem Vorwärts- und Rückwärtsdifferenzenquotienten (nicht weiter verwunderlich) aber wesentlich größer im Vergleich zu dem Fehler bei dem zentralen Differenzenquotienten.\n", "\n", "Eine exakte Analyse (das machen wir hier nicht) würde ergeben, dass der Fehler bei der erstgenannten Art der Differenzenbildung mindestents linear von der Schrittweite abhängt. Sie können also den Fehler um den Faktor zwei reduzieren, wenn Sie die Schrittweite halbieren.\n", "\n", "Eine exakte Analyse würde außerdem ergeben, dass der Fehler bei der zweitgenannten Art der Differenzenbildung mindestents quadratisch von der Schrittweite abhängt. Sie können also den Fehler um den Faktor vier reduzieren, wenn Sie die Schrittweite halbieren!\n", "\n", "Das wollen wir im Folgenden evaluieren (exemplarisch an dem konkreten Beispiel). Wir evaluieren hier nur den Fehler des Vorwärtsdifferenzenoperators. Zur Quantisierung des Fehlers benutzen wir die Standardabweichung der Differenz." ] }, { "cell_type": "code", "execution_count": 5, "id": "5ab9bbf0", "metadata": {}, "outputs": [], "source": [ "error_forward = []\n", "error_central = []\n", "resolution = []\n", "\n", "for N in range(10, 510, 10): # Wir lassen hier also de facto einfach eine Schleife durchlaufen, in der wir die\n", " # Anzahl der Stützstellen ändern, mit denen wir ein vorhandenes Interval \n", " # diskretisieren. Das inverse dieser Zahl wäre eine Maßzahl für die Diskretisierung.\n", " x = np.linspace(a, b, N)\n", " y = np.sin(x)\n", " forward_difference = (y[1:] - y[:-1]) / (x[1] - x[0])\n", " x_forward = x[:-1]\n", " backward_difference = (y[1:] - y[:-1]) / (x[1] - x[0])\n", " x_backward = x[1:]\n", " central_difference = (forward_difference + backward_difference) / 2\n", " x_central = (x_forward + x_backward) / 2\n", " error_forward.append(np.std(forward_difference - np.cos(x_forward), ddof=1)) # Mit dem Befehl append hängen Sie\n", " # ein neues Element an eine Liste an. \n", " error_central.append(np.std(central_difference - np.cos(x_central), ddof=1))\n", " resolution.append(N)\n", "resolution = np.asarray(resolution) # Der Befehl konvertiert die Eingabe in einer Liste, die äquivalent zu\n", " # einer von NumPy erzeugten Liste ist. " ] }, { "cell_type": "code", "execution_count": 6, "id": "2a2f91e6", "metadata": {}, "outputs": [ { "data": { "image/png": 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\n", "text/plain": [ "
" ] }, "metadata": { "needs_background": "light" }, "output_type": "display_data" } ], "source": [ "plt.plot(resolution, error_forward, \"b-\", label=\"Forward difference\")\n", "plt.plot(resolution, error_central, \"r-\", label=\"Central difference\")\n", "plt.plot(\n", " resolution,\n", " resolution[0] * error_forward[0] / resolution,\n", " \"b--\",\n", " label=r\"$O(\\Delta x)$\",\n", ")\n", "plt.plot(\n", " resolution,\n", " resolution[0] ** 2 * error_central[0] / resolution**2,\n", " \"r--\",\n", " label=r\"$O(\\Delta x^2)$\",\n", ")\n", "plt.xscale(\"log\")\n", "plt.yscale(\"log\")\n", "plt.title(\"Fehlerrate als Funktion der inversen Schrittweite\", size=\"x-large\")\n", "plt.ylabel(\"Fehlerrate\", size=\"x-large\")\n", "plt.xlabel(\"Inverse Schrittweite\", size=\"x-large\")\n", "plt.legend()\n", "plt.tight_layout()\n", "plt.show()" ] }, { "cell_type": "markdown", "id": "c6061f59", "metadata": {}, "source": [ "Dieses Verfahren lässt sich noch weiter verbessern, indem man weitere Operatoren hinzufügt, die den Differentialquotienten immer besser annähern. Man benötigt dann lediglich mehr Stützstellen zur Berechnung. Den quadratischen Fehlerterm kann man eliminieren durch die folgende Näherung:\n", "$$\n", "\\frac{\\mathrm{d} f}{\\mathrm{d} x}\\approx\\frac{1}{12 \\Delta x}\\left[-f(x+2\\Delta x)+8f(x+\\Delta x)-8f(x-\\Delta x)+f(x-2\\Delta x)\\right]\n", "$$\n", "Hier approximiert man den Differentialoperator mit einem Differenzenoperator, bei dem der Fehler mindestens quartisch mit der Schrittweite reduziert wird. Das heißt, bei einer Halbierung der Schrittweite reduziert sich der Fehler um den Faktor sechzehn. Sie können das beliebig verfeinern." ] }, { "cell_type": "markdown", "id": "bece51d9", "metadata": {}, "source": [ "### Ableitungen zweiter Ordnung\n", "\n", "Die Berechnung einer Ableitung zweiter Ordnung verlangt die diskrete Berechnung der Änderung der Änderung eines Funktionswertes. Mit dem oben bereits eingeführten Vorwärts- und Rückwärtsdifferenzenquotienten ist dies sehr einfach, da wir lediglich die Änderug dieser Quotienten noch einmal berechnen müssen.\n", "\n", "Im Speziellen haben wir:\n", "\n", "$$\n", "\\frac{\\mathrm{d}^2 f(x)}{\\mathrm{d}x^2}\\approx \\frac{\\frac{\\Delta f_1}{\\Delta x}-\\frac{\\Delta f_2}{\\Delta x}}{\\Delta x}\n", "\\approx \\frac{f_3-f_2-f_2+f_1}{\\Delta x^2}\\approx \\frac{f(x+\\Delta x)-2f(x)+f(x-\\Delta x)}{\\Delta x^2}\n", "$$\n", "\n", "Dieser Differenzenoperator konvergiert in zweiter Ordnung mit der Schrittweite. Halbierung der Schrittweite reduziert den Fehler mindestens um den Faktor vier. " ] }, { "cell_type": "code", "execution_count": 7, "id": "1a454bdb", "metadata": {}, "outputs": [], "source": [ "error_second_order_difference = []\n", "resolution = []\n", "a = 0\n", "b = 4 * np.pi\n", "\n", "for N in range(10, 510, 10):\n", " x = np.linspace(a, b, N)\n", " h = x[1] - x[0]\n", "\n", " xplush = x + h\n", " xminush = x - h\n", "\n", " second_order_difference = (\n", " 1 / (h**2) * (np.sin(xminush) - 2 * np.sin(x) + np.sin(xplush))\n", " )\n", " error_second_order_difference.append(\n", " np.std(second_order_difference + np.sin(x), ddof=1)\n", " )\n", " resolution.append(N)\n", "resolution = np.asarray(resolution)" ] }, { "cell_type": "code", "execution_count": 8, "id": "1e7062a5", "metadata": {}, "outputs": [ { "data": { "image/png": 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\n", "text/plain": [ "
" ] }, "metadata": { "needs_background": "light" }, "output_type": "display_data" } ], "source": [ "plt.title(\"Zweite Ableitung\", size=\"x-large\")\n", "plt.plot(x, np.sin(x), label=r\"$f(x)$\")\n", "plt.plot(x, second_order_difference, label=r\"$f''(x)$\")\n", "plt.ylabel(r\"$f''(x)=-\\sin(x)$\", size=\"x-large\")\n", "plt.xlabel(\"x\", size=\"x-large\")\n", "plt.legend()\n", "plt.tight_layout()\n", "plt.show()" ] }, { "cell_type": "code", "execution_count": 9, "id": "84848ee1", "metadata": {}, "outputs": [ { "data": { "image/png": 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\n", 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" ] }, "metadata": { "needs_background": "light" }, "output_type": "display_data" } ], "source": [ "plt.plot(resolution, error_second_order_difference, \"b-\", label=r\"Fehler in $f''(x)$\")\n", "plt.plot(\n", " resolution,\n", " resolution[0] ** 2 * error_second_order_difference[0] / resolution**2,\n", " \"b--\",\n", " label=r\"$O(\\Delta x^2)$\",\n", ")\n", "plt.xscale(\"log\")\n", "plt.yscale(\"log\")\n", "plt.title(\"Fehlerrate als Funktion der inversen Schrittweite\", size=\"x-large\")\n", "plt.ylabel(\"Fehlerrate\", size=\"x-large\")\n", "plt.xlabel(\"Inverse Schrittweite\", size=\"x-large\")\n", "plt.legend()\n", "plt.tight_layout()\n", "plt.show()" ] } ], "metadata": { "kernelspec": { "display_name": "Python 3 (ipykernel)", "language": "python", "name": "python3" }, "language_info": { "codemirror_mode": { "name": "ipython", "version": 3 }, "file_extension": ".py", "mimetype": "text/x-python", "name": "python", "nbconvert_exporter": "python", "pygments_lexer": "ipython3", "version": "3.9.12" }, "vscode": { "interpreter": { "hash": "77c156d20ad9e0a33269d50c8fafe7044ce4815250dfaf90496dfc42f56ab107" } } }, "nbformat": 4, "nbformat_minor": 5 }