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{
"cells": [
{
"cell_type": "code",
"execution_count": 10,
"metadata": {},
"outputs": [],
"source": [
"import mercury as mr\n",
"import numpy as np\n",
"import matplotlib.pyplot as plt\n",
"%matplotlib inline"
]
},
{
"cell_type": "code",
"execution_count": null,
"metadata": {},
"outputs": [],
"source": [
"name = mr.Text(value=\"Piotr\", label=\"What is your name?\")\n",
"print(f\"Hello {name.value}\")\n",
"# set application properites with App object\n",
"app = mr.App(show_code = True)"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"Consider one of the most basic problems:\n",
"\n",
"Finding the root of a function, i.e: $f(x) = 0$ .\n",
"\n",
"## Technique:\n",
"\n",
"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",
"\n",
"## Example\n",
"Suppose we want to look for the zero for $f(x)= \\sin(x) + 0.5$ between $[-1,2]$. "
]
},
{
"cell_type": "code",
"execution_count": 42,
"metadata": {},
"outputs": [
{
"name": "stdout",
"output_type": "stream",
"text": [
"26\n"
]
},
{
"data": {
"text/plain": [
"-0.523598775267601"
]
},
"execution_count": 42,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"def f(x): \n",
" return np.sin(x)+0.5\n",
"\n",
"def validate_interval(f,x0,x1):\n",
" return f(x0)*f(x1) < 0\n",
"\n",
"\n",
"def bisection(f, interval, n, tol):\n",
" x0, x1 = interval[0], interval[1] #extract interval \n",
" if not validate_interval(f, x0, x1): #check interval can be solved for roots\n",
" return \"Not valid interval\"\n",
"\n",
" counter = 1\n",
" while True:\n",
" p = x0 + ((x1-x0)/2)\n",
" y = f(p)\n",
" if -tol < y < tol:\n",
" print(counter)\n",
" return p\n",
" if validate_interval(f,x0,p):\n",
" x1 = p\n",
" else:\n",
" x0 = p\n",
" counter += 1\n",
"\n",
"\n",
"\n",
"bisection(f,[-1,2], 50, 0.000000001) "
]
},
{
"cell_type": "code",
"execution_count": 12,
"metadata": {},
"outputs": [],
"source": [
"def graph(x):\n",
" #need function\n",
" #need tangents\n",
" #root\n",
" fig, ax = plt.subplots(figsize=(10, 10))\n",
" y=f(x)\n",
" plt.plot(x,y)\n",
" \n",
" plt.ylabel('some numbers')\n",
" plt.axis('tight')\n",
" plt.grid(True)\n",
" plt.show()\n",
" ax.spines['top'].set_visible(False)\n",
" ax.spines['right'].set_visible(False)\n",
"\n",
"graph(np.arange(-np.pi,np.pi,np.pi/32))"
]
},
{
"cell_type": "code",
"execution_count": null,
"metadata": {},
"outputs": [],
"source": []
}
],
"metadata": {
"kernelspec": {
"display_name": "Python 3",
"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": "3.10.13"
},
"orig_nbformat": 4
},
"nbformat": 4,
"nbformat_minor": 2
}
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