192 lines
32 KiB
Plaintext
192 lines
32 KiB
Plaintext
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{
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"cells": [
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{
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"cell_type": "markdown",
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"metadata": {},
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"source": [
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"\n",
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"# Minima and roots of a function\n",
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"\n",
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"Demos finding minima and roots of a function.\n"
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]
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},
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{
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"cell_type": "markdown",
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"metadata": {},
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"source": [
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"## Define the function\n",
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"\n"
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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": 1,
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"metadata": {
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"collapsed": false,
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"jupyter": {
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"outputs_hidden": false
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}
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},
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"outputs": [],
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"source": [
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"import numpy as np\n",
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"\n",
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"x = np.arange(-10, 10, 0.1)\n",
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"\n",
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"\n",
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"def f(x):\n",
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" return x**2 + 10 * np.sin(x)"
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]
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},
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{
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"cell_type": "markdown",
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"metadata": {},
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"source": [
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"## Find minima\n",
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"\n"
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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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"metadata": {
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"collapsed": false,
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"jupyter": {
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"outputs_hidden": false
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}
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},
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"outputs": [
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{
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"name": "stdout",
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"output_type": "stream",
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"text": [
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"Global minima found [-1.30641113]\n",
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"Local minimum found 3.8374671194983834\n"
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]
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}
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],
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"source": [
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"import scipy as sp\n",
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"\n",
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"# Global optimization\n",
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"grid = (-10, 10, 0.1)\n",
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"xmin_global = sp.optimize.brute(f, (grid,))\n",
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"print(f\"Global minima found {xmin_global}\")\n",
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"\n",
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"# Constrain optimization\n",
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"xmin_local = sp.optimize.fminbound(f, 0, 10)\n",
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"print(f\"Local minimum found {xmin_local}\")"
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]
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},
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{
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"cell_type": "markdown",
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"metadata": {},
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"source": [
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"## Root finding\n",
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"\n"
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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": 3,
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"metadata": {
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"collapsed": false,
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"jupyter": {
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"outputs_hidden": false
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}
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},
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"outputs": [
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{
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"name": "stdout",
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"output_type": "stream",
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"text": [
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"First root found [0.]\n",
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"Second root found [-2.47948183]\n"
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]
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}
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],
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"source": [
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"root = sp.optimize.root(f, 1) # our initial guess is 1\n",
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"print(f\"First root found {root.x}\")\n",
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"root2 = sp.optimize.root(f, -2.5)\n",
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"print(f\"Second root found {root2.x}\")"
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]
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},
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{
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"cell_type": "markdown",
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"metadata": {},
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"source": [
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"## Plot function, minima, and roots\n",
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"\n"
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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": 4,
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"metadata": {
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"collapsed": false,
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"jupyter": {
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"outputs_hidden": false
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}
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},
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"outputs": [
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{
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"data": {
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"image/png": "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"text/plain": [
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"<Figure size 600x400 with 1 Axes>"
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]
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},
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"metadata": {},
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"output_type": "display_data"
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}
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],
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"source": [
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"import matplotlib.pyplot as plt\n",
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"\n",
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"fig = plt.figure(figsize=(6, 4))\n",
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"ax = fig.add_subplot(111)\n",
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"\n",
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"# Plot the function\n",
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"ax.plot(x, f(x), \"b-\", label=\"f(x)\")\n",
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"\n",
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"# Plot the minima\n",
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"xmins = np.array([xmin_global[0], xmin_local])\n",
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"ax.plot(xmins, f(xmins), \"go\", label=\"Minima\")\n",
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"\n",
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"# Plot the roots\n",
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"roots = np.array([root.x, root2.x])\n",
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"ax.plot(roots, f(roots), \"kv\", label=\"Roots\")\n",
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"\n",
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"# Decorate the figure\n",
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"ax.legend(loc=\"best\")\n",
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"ax.set_xlabel(\"x\")\n",
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"ax.set_ylabel(\"f(x)\")\n",
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"ax.axhline(0, color=\"gray\")\n",
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"plt.show()"
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]
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}
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],
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"metadata": {
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"kernelspec": {
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"display_name": "Python 3 (ipykernel)",
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"language": "python",
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"name": "python3"
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},
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"language_info": {
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"codemirror_mode": {
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"name": "ipython",
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"version": 3
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},
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"file_extension": ".py",
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"mimetype": "text/x-python",
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"name": "python",
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"nbconvert_exporter": "python",
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"pygments_lexer": "ipython3",
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"version": "3.12.11"
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}
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},
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"nbformat": 4,
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"nbformat_minor": 4
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}
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