mirror of
https://github.com/vale981/phys512
synced 2025-03-04 09:11:37 -05:00
177 lines
76 KiB
Text
177 lines
76 KiB
Text
{
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"cells": [
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{
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"cell_type": "markdown",
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"id": "77869faf",
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"metadata": {},
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"source": [
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"## Derivatives exercises"
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]
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},
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{
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"cell_type": "markdown",
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"id": "9d468e8a",
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"metadata": {},
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"source": [
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"#### Second order finite differences\n",
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"\n",
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"To keep track of the error terms, write out the Taylor expansions up to the 4th order terms:\n",
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"\n",
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"$$f(x+\\Delta x) \\approx f(x) + \\Delta x {df\\over dx} + {(\\Delta x)^2\\over 2}{d^2f\\over dx^2} + {(\\Delta x)^3\\over 6}{d^3f\\over dx^3} + {(\\Delta x)^4\\over 24}{d^4f\\over dx^4}$$\n",
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"\n",
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"$$f(x-\\Delta x) \\approx f(x) - \\Delta x {df\\over dx} + {(\\Delta x)^2\\over 2}{d^2f\\over dx^2}-{(\\Delta x)^3\\over 6}{d^3f\\over dx^3} + {(\\Delta x)^4\\over 24}{d^4f\\over dx^4}$$\n",
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"\n",
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"Then add and subtract:\n",
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"\n",
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"$$f(x+\\Delta x) + f(x-\\Delta x) \\approx 2f(x) + (\\Delta x)^2{d^2f\\over dx^2} + {(\\Delta x)^4\\over 12}{d^4f\\over dx^4}$$\n",
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"\n",
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"$$f(x+\\Delta x) - f(x-\\Delta x) \\approx 2\\Delta x {df\\over dx} + {(\\Delta x)^3\\over 3}{d^3f\\over dx^3}$$\n",
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"\n",
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"Therefore\n",
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"\n",
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"$${d^2f\\over dx^2}\\approx {f(x+\\Delta x) + f(x-\\Delta x) -2f(x)\\over (\\Delta x)^2} - {(\\Delta x)^2\\over 12}{d^4f\\over dx^4}$$\n",
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"\n",
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"$${df\\over dx} \\approx {f(x+\\Delta x) - f(x-\\Delta x)\\over 2\\Delta x} - {(\\Delta x)^2\\over 6}{d^3f\\over dx^3}$$\n",
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"\n",
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"For comparison, the first order derivative with the error term kept in is\n",
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"\n",
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"$${df\\over dx} \\approx {f(x+\\Delta x) - f(x)\\over \\Delta x} - {\\Delta x\\over 2}{d^2f\\over dx^2}$$\n",
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"\n"
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]
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},
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{
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"cell_type": "markdown",
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"id": "2da5f3eb",
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"metadata": {},
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"source": [
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"#### Optimal step size"
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]
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},
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{
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"cell_type": "code",
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"execution_count": 2,
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"id": "05710fd2",
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"metadata": {},
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"outputs": [
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{
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"data": {
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"image/png": 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",
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"text/plain": [
|
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"<Figure size 640x480 with 1 Axes>"
|
|
]
|
|
},
|
|
"metadata": {},
|
|
"output_type": "display_data"
|
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}
|
|
],
|
|
"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 at this value of x\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",
|
|
"plt.title(r'$f(x) = \\sin x, \\ \\ \\ x=%lg$' % (x,))\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": "markdown",
|
|
"id": "b5969ebb",
|
|
"metadata": {},
|
|
"source": [
|
|
"#### Automatic derivatives\n",
|
|
"\n",
|
|
"Inspecting the code, we can see that the operators that are implemented inside the class return a new instance of Var that is initialized with (1) the value obtained from the operation, and (2) a set of children that give the partial derivatives with respect to the variables involved in the operation. \n",
|
|
"\n",
|
|
"The multiplication operator is instructive to look at. For the operation `x*y`, there are two variables `x` and `y`, so we need to return (1) the operation value `x*y`, (2) the partial derivative with respect to `x`, which is $\\partial(xy)/\\partial x = y$, and (3) the partial derivative with respect to `y`, which is $\\partial(xy)/\\partial y = x$. The corresponding function definition is\n",
|
|
"\n",
|
|
"```\n",
|
|
"def __mul__(self, other):\n",
|
|
" return Var(self.value * other.value, [(other.value, self), (self.value, other)])\n",
|
|
"```\n",
|
|
"\n",
|
|
"For $exp(x)$, there is only one argument to the function so we just have to return one partial derivative $(\\partial/\\partial x)\\exp(x) = \\exp(x)$. So we can add \n",
|
|
"```\n",
|
|
"def exp(self):\n",
|
|
" return Var(math.exp(self.value), [(math.exp(self.value), self)])\n",
|
|
"```\n",
|
|
"\n",
|
|
"which we use for example as \n",
|
|
"\n",
|
|
"`f= x * y + x.exp()`\n"
|
|
]
|
|
},
|
|
{
|
|
"cell_type": "code",
|
|
"execution_count": null,
|
|
"id": "6cbe50bd",
|
|
"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": "3.11.5"
|
|
}
|
|
},
|
|
"nbformat": 4,
|
|
"nbformat_minor": 5
|
|
}
|