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Upload bisection.ipynb
Browse files- bisection.ipynb +151 -0
bisection.ipynb
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{
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"cells": [
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{
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"cell_type": "code",
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"execution_count": 10,
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"metadata": {},
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"outputs": [],
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"source": [
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"import mercury as mr\n",
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"import numpy as np\n",
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"import matplotlib.pyplot as plt\n",
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"%matplotlib inline"
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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": null,
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"metadata": {},
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"outputs": [],
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"source": [
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"name = mr.Text(value=\"Piotr\", label=\"What is your name?\")\n",
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"print(f\"Hello {name.value}\")\n",
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"# set application properites with App object\n",
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"app = mr.App(show_code = True)"
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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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"Consider one of the most basic problems:\n",
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"\n",
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"Finding the root of a function, i.e: $f(x) = 0$ .\n",
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"\n",
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"## Technique:\n",
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"\n",
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"We start with a boundary $[a,b]$ and then hope that there exists a point $p$ in that boundary where $f(p) = 0$. We half the end points of the boundary depending on if $f(a_i) \\text{ or } f(b_i)$ is negative or positive, until we reach the point $p$\n",
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"\n",
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"## Example\n",
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"Suppose we want to look for the zero for $f(x)= \\sin(x) + 0.5$ between $[-1,2]$. "
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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": 42,
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"metadata": {},
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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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"26\n"
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]
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},
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{
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"data": {
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"text/plain": [
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"-0.523598775267601"
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]
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},
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"execution_count": 42,
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"metadata": {},
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"output_type": "execute_result"
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}
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],
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"source": [
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"def f(x): \n",
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" return np.sin(x)+0.5\n",
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"\n",
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"def validate_interval(f,x0,x1):\n",
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" return f(x0)*f(x1) < 0\n",
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"\n",
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"\n",
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"def bisection(f, interval, n, tol):\n",
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" x0, x1 = interval[0], interval[1] #extract interval \n",
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" if not validate_interval(f, x0, x1): #check interval can be solved for roots\n",
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" return \"Not valid interval\"\n",
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"\n",
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" counter = 1\n",
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" while True:\n",
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" p = x0 + ((x1-x0)/2)\n",
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" y = f(p)\n",
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" if -tol < y < tol:\n",
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" print(counter)\n",
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" return p\n",
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" if validate_interval(f,x0,p):\n",
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" x1 = p\n",
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" else:\n",
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" x0 = p\n",
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" counter += 1\n",
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"\n",
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"\n",
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"\n",
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"bisection(f,[-1,2], 50, 0.000000001) "
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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": 12,
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"metadata": {},
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"outputs": [],
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"source": [
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"def graph(x):\n",
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" #need function\n",
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" #need tangents\n",
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" #root\n",
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" fig, ax = plt.subplots(figsize=(10, 10))\n",
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" y=f(x)\n",
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" plt.plot(x,y)\n",
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" \n",
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" plt.ylabel('some numbers')\n",
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" plt.axis('tight')\n",
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" plt.grid(True)\n",
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" plt.show()\n",
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" ax.spines['top'].set_visible(False)\n",
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" ax.spines['right'].set_visible(False)\n",
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"\n",
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"graph(np.arange(-np.pi,np.pi,np.pi/32))"
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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": null,
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"metadata": {},
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"outputs": [],
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"source": []
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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",
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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": "3.10.13"
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},
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"orig_nbformat": 4
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},
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"nbformat": 4,
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"nbformat_minor": 2
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}
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