phys512/derivatives_solutions.ipynb
2023-09-10 21:27:58 -04:00

133 lines
71 KiB
Text

{
"cells": [
{
"cell_type": "markdown",
"id": "77869faf",
"metadata": {},
"source": [
"## Derivatives"
]
},
{
"cell_type": "markdown",
"id": "9d468e8a",
"metadata": {},
"source": [
"#### Second order finite differences\n",
"\n",
"$$f(x+\\Delta x) \\approx f(x) + \\Delta x {df\\over dx} + {(\\Delta x)^2\\over 2}{d^2f\\over dx^2}$$\n",
"\n",
"$$f(x-\\Delta x) \\approx f(x) - \\Delta x {df\\over dx} + {(\\Delta x)^2\\over 2}{d^2f\\over dx^2}$$\n",
"\n",
"Adding and subtracting these equations gives the two expressions in the exercise."
]
},
{
"cell_type": "markdown",
"id": "2da5f3eb",
"metadata": {},
"source": [
"#### Optimal step size"
]
},
{
"cell_type": "code",
"execution_count": 46,
"id": "05710fd2",
"metadata": {},
"outputs": [
{
"data": {
"image/png": 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",
"text/plain": [
"<Figure size 640x480 with 1 Axes>"
]
},
"metadata": {},
"output_type": "display_data"
}
],
"source": [
"import numpy as np\n",
"import matplotlib.pyplot as plt\n",
"\n",
"def f(x):\n",
" return np.sin(x)\n",
"\n",
"dx_vals = 10**np.linspace(-13, -1, 100)\n",
"\n",
"x = 1.0 # Evaluate the derivatives \n",
"\n",
"err = np.zeros_like(dx_vals) # to store the fractional error from the 1st order difference\n",
"err2 = np.zeros_like(dx_vals) # same for the second order\n",
"\n",
"for i, dx in enumerate(dx_vals):\n",
" dfdx1 = (f(x + dx) - f(x)) / dx\n",
" err[i] = np.abs((dfdx1 - np.cos(x))/np.cos(x))\n",
"\n",
" dfdx2 = (f(x + dx) - f(x - dx)) / (2*dx)\n",
" err2[i] = np.abs((dfdx2 - np.cos(x))/np.cos(x))\n",
"\n",
"# Plot the results\n",
"plt.plot(dx_vals, err, label=\"First order\")\n",
"plt.plot(dx_vals, err2, label=\"Second order\")\n",
"\n",
"plt.ylim((1e-2*min(err2), 1.0))\n",
"\n",
"# and the analytic expressions\n",
"plt.plot(dx_vals, 1e-16 * f(x) / dx_vals, \":\", label=r'$\\epsilon f(x)/\\Delta x$')\n",
"plt.plot(dx_vals, dx_vals, \":\", label=r'$\\Delta x$')\n",
"plt.plot(dx_vals, dx_vals**2, \":\", label=r'$(\\Delta x)^2$')\n",
"\n",
"\n",
"plt.yscale('log')\n",
"plt.xscale('log')\n",
"plt.xlabel(r'$\\Delta x$')\n",
"plt.ylabel('Fractional error in derivative')\n",
"plt.legend()\n",
"plt.show()"
]
},
{
"cell_type": "markdown",
"id": "127d61b8",
"metadata": {},
"source": [
"The optimal step size for the 1st order expression is $\\sim \\epsilon^{1/2}\\sim 10^{-8}$ and the associated error is also $\\sim \\epsilon^{1/2}$. \n",
"\n",
"For the second order expression, the finite difference error is $\\propto (\\Delta x)^2$ rather than $\\Delta x$. Following the same argument as for the first order derivative leads to an optimal step size of order $\\epsilon^{1/3}\\sim 10^{-16/3}\\sim 10^{-5.3}$, and an error $\\epsilon^{2/3}\\sim 10^{-10.6}$.\n",
"\n",
"Notice that a higher order derivative allows us to get a better error with a larger step size."
]
},
{
"cell_type": "code",
"execution_count": null,
"id": "f7243ce5",
"metadata": {},
"outputs": [],
"source": []
}
],
"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": "3.11.4"
}
},
"nbformat": 4,
"nbformat_minor": 5
}