LeastSqure.ipynb

{
 "cells": [
  {
   "cell_type": "code",
   "execution_count": 1,
   "metadata": {},
   "outputs": [],
   "source": [
    "import numpy as np\n",
    "import matplotlib.pyplot as plt\n",
    "%matplotlib inline"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 2,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "Enter the digree of polynomial: \n"
     ]
    },
    {
     "ename": "ValueError",
     "evalue": "invalid literal for int() with base 10: ''",
     "output_type": "error",
     "traceback": [
      "\u001b[1;31m---------------------------------------------------------------------------\u001b[0m",
      "\u001b[1;31mValueError\u001b[0m                                Traceback (most recent call last)",
      "\u001b[1;32m<ipython-input-2-503c6b551271>\u001b[0m in \u001b[0;36m<module>\u001b[1;34m\u001b[0m\n\u001b[1;32m----> 1\u001b[1;33m \u001b[0mN\u001b[0m \u001b[1;33m=\u001b[0m \u001b[0mint\u001b[0m\u001b[1;33m(\u001b[0m\u001b[0minput\u001b[0m\u001b[1;33m(\u001b[0m\u001b[1;34m'Enter the digree of polynomial: '\u001b[0m\u001b[1;33m)\u001b[0m\u001b[1;33m)\u001b[0m\u001b[1;33m\u001b[0m\u001b[1;33m\u001b[0m\u001b[0m\n\u001b[0m",
      "\u001b[1;31mValueError\u001b[0m: invalid literal for int() with base 10: ''"
     ]
    }
   ],
   "source": [
    "N = int(input('Enter the digree of polynomial: '))"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "m = eval(input('Enter number of points: '))"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": []
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "xs = np.zeros(m)\n",
    "ys = np.zeros(m)\n",
    "print('x')\n",
    "for i in range(m):\n",
    "    xs[i] = eval(input())\n",
    "print('Enter Y')\n",
    "for i in range(m):\n",
    "    ys[i] = eval(input())"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 8,
   "metadata": {},
   "outputs": [],
   "source": [
    "def least_squre(N, xs, ys):\n",
    "    M = np.zeros((N+1,N+1))\n",
    "    a = []\n",
    "    Y = []\n",
    "    for i in range(N+1):\n",
    "        c= []\n",
    "        for j in range(len(xs)):\n",
    "            c.append(ys[j]*(xs[j]**i))\n",
    "        Y.append(sum(c))\n",
    "    for i in range(2*N+1):\n",
    "        b = []\n",
    "        for j in range(len(xs)):\n",
    "            b.append(xs[j]**i)\n",
    "        a.append(sum(b))\n",
    "\n",
    "    for i in range(N+1):\n",
    "        for j in range(N+1):\n",
    "            M[i][j] = a[i+j]\n",
    "    m_inv = np.linalg.inv(M)\n",
    "\n",
    "    m_inv = np.array(m_inv)\n",
    "    Y = np.array(Y)\n",
    "    a = np.array(np.matmul(m_inv, Y))\n",
    "    #return a[0] + a[1]*x + a[2]*x**2 + a[3]*x**3\n",
    "    w = []\n",
    "    for i in range(N+1):\n",
    "       w.append(a[i])\n",
    "    return w\n",
    "\n",
    "def f(N,x):\n",
    "    ai = least_squre(N, xs, ys)\n",
    "    s = 0\n",
    "    for i in range(N+1):\n",
    "        s += ai[i]*x**i\n",
    "    return s\n",
    "    \n",
    "def deviation(least_sqare,N, xs, ys):\n",
    "    sum = 0\n",
    "    for i in range(m):\n",
    "        sum += (f(N,xs[i]) - ys[i])**2\n",
    "    return sum\n",
    "\n"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 9,
   "metadata": {},
   "outputs": [],
   "source": [
    "def plot_bestfit(degree):\n",
    "    x = [i/100 for i in range(100)]\n",
    "    y = [f(degree,i) for i in x]\n",
    "    plt.plot(x,y, color='r', linewidth=2)\n",
    "    plt.plot(xs, ys, 'o');"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 10,
   "metadata": {},
   "outputs": [
    {
     "ename": "NameError",
     "evalue": "name 'xs' is not defined",
     "output_type": "error",
     "traceback": [
      "\u001b[1;31m---------------------------------------------------------------------------\u001b[0m",
      "\u001b[1;31mNameError\u001b[0m                                 Traceback (most recent call last)",
      "\u001b[1;32m<ipython-input-10-b4826c1b4a74>\u001b[0m in \u001b[0;36m<module>\u001b[1;34m\u001b[0m\n\u001b[1;32m----> 1\u001b[1;33m \u001b[0mplt\u001b[0m\u001b[1;33m.\u001b[0m\u001b[0mplot\u001b[0m\u001b[1;33m(\u001b[0m\u001b[0mxs\u001b[0m\u001b[1;33m,\u001b[0m\u001b[0mys\u001b[0m\u001b[1;33m,\u001b[0m \u001b[1;34m'o'\u001b[0m\u001b[1;33m)\u001b[0m\u001b[1;33m\u001b[0m\u001b[1;33m\u001b[0m\u001b[0m\n\u001b[0m\u001b[0;32m      2\u001b[0m \u001b[0mplot_bestfit\u001b[0m\u001b[1;33m(\u001b[0m\u001b[1;36m1\u001b[0m\u001b[1;33m)\u001b[0m\u001b[1;33m\u001b[0m\u001b[1;33m\u001b[0m\u001b[0m\n",
      "\u001b[1;31mNameError\u001b[0m: name 'xs' is not defined"
     ]
    }
   ],
   "source": [
    "plt.plot(xs,ys, 'o')\n",
    "plot_bestfit(1)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 11,
   "metadata": {},
   "outputs": [
    {
     "ename": "NameError",
     "evalue": "name 'xs' is not defined",
     "output_type": "error",
     "traceback": [
      "\u001b[1;31m---------------------------------------------------------------------------\u001b[0m",
      "\u001b[1;31mNameError\u001b[0m                                 Traceback (most recent call last)",
      "\u001b[1;32m<ipython-input-11-7e8e30477d92>\u001b[0m in \u001b[0;36m<module>\u001b[1;34m\u001b[0m\n\u001b[1;32m----> 1\u001b[1;33m \u001b[0mplot_bestfit\u001b[0m\u001b[1;33m(\u001b[0m\u001b[1;36m10\u001b[0m\u001b[1;33m)\u001b[0m\u001b[1;33m\u001b[0m\u001b[1;33m\u001b[0m\u001b[0m\n\u001b[0m",
      "\u001b[1;32m<ipython-input-9-4b645280ee3e>\u001b[0m in \u001b[0;36mplot_bestfit\u001b[1;34m(degree)\u001b[0m\n\u001b[0;32m      1\u001b[0m \u001b[1;32mdef\u001b[0m \u001b[0mplot_bestfit\u001b[0m\u001b[1;33m(\u001b[0m\u001b[0mdegree\u001b[0m\u001b[1;33m)\u001b[0m\u001b[1;33m:\u001b[0m\u001b[1;33m\u001b[0m\u001b[1;33m\u001b[0m\u001b[0m\n\u001b[0;32m      2\u001b[0m     \u001b[0mx\u001b[0m \u001b[1;33m=\u001b[0m \u001b[1;33m[\u001b[0m\u001b[0mi\u001b[0m\u001b[1;33m/\u001b[0m\u001b[1;36m100\u001b[0m \u001b[1;32mfor\u001b[0m \u001b[0mi\u001b[0m \u001b[1;32min\u001b[0m \u001b[0mrange\u001b[0m\u001b[1;33m(\u001b[0m\u001b[1;36m100\u001b[0m\u001b[1;33m)\u001b[0m\u001b[1;33m]\u001b[0m\u001b[1;33m\u001b[0m\u001b[1;33m\u001b[0m\u001b[0m\n\u001b[1;32m----> 3\u001b[1;33m     \u001b[0my\u001b[0m \u001b[1;33m=\u001b[0m \u001b[1;33m[\u001b[0m\u001b[0mf\u001b[0m\u001b[1;33m(\u001b[0m\u001b[0mdegree\u001b[0m\u001b[1;33m,\u001b[0m\u001b[0mi\u001b[0m\u001b[1;33m)\u001b[0m \u001b[1;32mfor\u001b[0m \u001b[0mi\u001b[0m \u001b[1;32min\u001b[0m \u001b[0mx\u001b[0m\u001b[1;33m]\u001b[0m\u001b[1;33m\u001b[0m\u001b[1;33m\u001b[0m\u001b[0m\n\u001b[0m\u001b[0;32m      4\u001b[0m     \u001b[0mplt\u001b[0m\u001b[1;33m.\u001b[0m\u001b[0mplot\u001b[0m\u001b[1;33m(\u001b[0m\u001b[0mx\u001b[0m\u001b[1;33m,\u001b[0m\u001b[0my\u001b[0m\u001b[1;33m,\u001b[0m \u001b[0mcolor\u001b[0m\u001b[1;33m=\u001b[0m\u001b[1;34m'r'\u001b[0m\u001b[1;33m,\u001b[0m \u001b[0mlinewidth\u001b[0m\u001b[1;33m=\u001b[0m\u001b[1;36m2\u001b[0m\u001b[1;33m)\u001b[0m\u001b[1;33m\u001b[0m\u001b[1;33m\u001b[0m\u001b[0m\n\u001b[0;32m      5\u001b[0m     \u001b[0mplt\u001b[0m\u001b[1;33m.\u001b[0m\u001b[0mplot\u001b[0m\u001b[1;33m(\u001b[0m\u001b[0mxs\u001b[0m\u001b[1;33m,\u001b[0m \u001b[0mys\u001b[0m\u001b[1;33m,\u001b[0m \u001b[1;34m'o'\u001b[0m\u001b[1;33m)\u001b[0m\u001b[1;33m;\u001b[0m\u001b[1;33m\u001b[0m\u001b[1;33m\u001b[0m\u001b[0m\n",
      "\u001b[1;32m<ipython-input-9-4b645280ee3e>\u001b[0m in \u001b[0;36m<listcomp>\u001b[1;34m(.0)\u001b[0m\n\u001b[0;32m      1\u001b[0m \u001b[1;32mdef\u001b[0m \u001b[0mplot_bestfit\u001b[0m\u001b[1;33m(\u001b[0m\u001b[0mdegree\u001b[0m\u001b[1;33m)\u001b[0m\u001b[1;33m:\u001b[0m\u001b[1;33m\u001b[0m\u001b[1;33m\u001b[0m\u001b[0m\n\u001b[0;32m      2\u001b[0m     \u001b[0mx\u001b[0m \u001b[1;33m=\u001b[0m \u001b[1;33m[\u001b[0m\u001b[0mi\u001b[0m\u001b[1;33m/\u001b[0m\u001b[1;36m100\u001b[0m \u001b[1;32mfor\u001b[0m \u001b[0mi\u001b[0m \u001b[1;32min\u001b[0m \u001b[0mrange\u001b[0m\u001b[1;33m(\u001b[0m\u001b[1;36m100\u001b[0m\u001b[1;33m)\u001b[0m\u001b[1;33m]\u001b[0m\u001b[1;33m\u001b[0m\u001b[1;33m\u001b[0m\u001b[0m\n\u001b[1;32m----> 3\u001b[1;33m     \u001b[0my\u001b[0m \u001b[1;33m=\u001b[0m \u001b[1;33m[\u001b[0m\u001b[0mf\u001b[0m\u001b[1;33m(\u001b[0m\u001b[0mdegree\u001b[0m\u001b[1;33m,\u001b[0m\u001b[0mi\u001b[0m\u001b[1;33m)\u001b[0m \u001b[1;32mfor\u001b[0m \u001b[0mi\u001b[0m \u001b[1;32min\u001b[0m \u001b[0mx\u001b[0m\u001b[1;33m]\u001b[0m\u001b[1;33m\u001b[0m\u001b[1;33m\u001b[0m\u001b[0m\n\u001b[0m\u001b[0;32m      4\u001b[0m     \u001b[0mplt\u001b[0m\u001b[1;33m.\u001b[0m\u001b[0mplot\u001b[0m\u001b[1;33m(\u001b[0m\u001b[0mx\u001b[0m\u001b[1;33m,\u001b[0m\u001b[0my\u001b[0m\u001b[1;33m,\u001b[0m \u001b[0mcolor\u001b[0m\u001b[1;33m=\u001b[0m\u001b[1;34m'r'\u001b[0m\u001b[1;33m,\u001b[0m \u001b[0mlinewidth\u001b[0m\u001b[1;33m=\u001b[0m\u001b[1;36m2\u001b[0m\u001b[1;33m)\u001b[0m\u001b[1;33m\u001b[0m\u001b[1;33m\u001b[0m\u001b[0m\n\u001b[0;32m      5\u001b[0m     \u001b[0mplt\u001b[0m\u001b[1;33m.\u001b[0m\u001b[0mplot\u001b[0m\u001b[1;33m(\u001b[0m\u001b[0mxs\u001b[0m\u001b[1;33m,\u001b[0m \u001b[0mys\u001b[0m\u001b[1;33m,\u001b[0m \u001b[1;34m'o'\u001b[0m\u001b[1;33m)\u001b[0m\u001b[1;33m;\u001b[0m\u001b[1;33m\u001b[0m\u001b[1;33m\u001b[0m\u001b[0m\n",
      "\u001b[1;32m<ipython-input-8-8eb3e7672ccb>\u001b[0m in \u001b[0;36mf\u001b[1;34m(N, x)\u001b[0m\n\u001b[0;32m     29\u001b[0m \u001b[1;33m\u001b[0m\u001b[0m\n\u001b[0;32m     30\u001b[0m \u001b[1;32mdef\u001b[0m \u001b[0mf\u001b[0m\u001b[1;33m(\u001b[0m\u001b[0mN\u001b[0m\u001b[1;33m,\u001b[0m\u001b[0mx\u001b[0m\u001b[1;33m)\u001b[0m\u001b[1;33m:\u001b[0m\u001b[1;33m\u001b[0m\u001b[1;33m\u001b[0m\u001b[0m\n\u001b[1;32m---> 31\u001b[1;33m     \u001b[0mai\u001b[0m \u001b[1;33m=\u001b[0m \u001b[0mleast_squre\u001b[0m\u001b[1;33m(\u001b[0m\u001b[0mN\u001b[0m\u001b[1;33m,\u001b[0m \u001b[0mxs\u001b[0m\u001b[1;33m,\u001b[0m \u001b[0mys\u001b[0m\u001b[1;33m)\u001b[0m\u001b[1;33m\u001b[0m\u001b[1;33m\u001b[0m\u001b[0m\n\u001b[0m\u001b[0;32m     32\u001b[0m     \u001b[0ms\u001b[0m \u001b[1;33m=\u001b[0m \u001b[1;36m0\u001b[0m\u001b[1;33m\u001b[0m\u001b[1;33m\u001b[0m\u001b[0m\n\u001b[0;32m     33\u001b[0m     \u001b[1;32mfor\u001b[0m \u001b[0mi\u001b[0m \u001b[1;32min\u001b[0m \u001b[0mrange\u001b[0m\u001b[1;33m(\u001b[0m\u001b[0mN\u001b[0m\u001b[1;33m+\u001b[0m\u001b[1;36m1\u001b[0m\u001b[1;33m)\u001b[0m\u001b[1;33m:\u001b[0m\u001b[1;33m\u001b[0m\u001b[1;33m\u001b[0m\u001b[0m\n",
      "\u001b[1;31mNameError\u001b[0m: name 'xs' is not defined"
     ]
    }
   ],
   "source": [
    "plot_bestfit(10)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 12,
   "metadata": {},
   "outputs": [
    {
     "ename": "NameError",
     "evalue": "name 'xs' is not defined",
     "output_type": "error",
     "traceback": [
      "\u001b[1;31m---------------------------------------------------------------------------\u001b[0m",
      "\u001b[1;31mNameError\u001b[0m                                 Traceback (most recent call last)",
      "\u001b[1;32m<ipython-input-12-34f00e9e2e01>\u001b[0m in \u001b[0;36m<module>\u001b[1;34m\u001b[0m\n\u001b[0;32m      1\u001b[0m \u001b[1;32mfor\u001b[0m \u001b[0mi\u001b[0m \u001b[1;32min\u001b[0m \u001b[0mrange\u001b[0m\u001b[1;33m(\u001b[0m\u001b[1;36m21\u001b[0m\u001b[1;33m)\u001b[0m\u001b[1;33m:\u001b[0m\u001b[1;33m\u001b[0m\u001b[1;33m\u001b[0m\u001b[0m\n\u001b[1;32m----> 2\u001b[1;33m     \u001b[0mprint\u001b[0m\u001b[1;33m(\u001b[0m\u001b[0mi\u001b[0m \u001b[1;33m,\u001b[0m \u001b[1;34m'\\t'\u001b[0m\u001b[1;33m,\u001b[0m\u001b[0mdeviation\u001b[0m\u001b[1;33m(\u001b[0m\u001b[0mleast_squre\u001b[0m\u001b[1;33m,\u001b[0m\u001b[0mi\u001b[0m\u001b[1;33m,\u001b[0m \u001b[0mxs\u001b[0m\u001b[1;33m,\u001b[0m \u001b[0mys\u001b[0m\u001b[1;33m)\u001b[0m\u001b[1;33m)\u001b[0m\u001b[1;33m\u001b[0m\u001b[1;33m\u001b[0m\u001b[0m\n\u001b[0m",
      "\u001b[1;31mNameError\u001b[0m: name 'xs' is not defined"
     ]
    }
   ],
   "source": [
    "for i in range(21):\n",
    "    print(i , '\\t',deviation(least_squre,i, xs, ys))"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "# Weight Modification "
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 5,
   "metadata": {},
   "outputs": [
    {
     "ename": "NameError",
     "evalue": "name 'm' is not defined",
     "output_type": "error",
     "traceback": [
      "\u001b[1;31m---------------------------------------------------------------------------\u001b[0m",
      "\u001b[1;31mNameError\u001b[0m                                 Traceback (most recent call last)",
      "\u001b[1;32m<ipython-input-5-038a169138f1>\u001b[0m in \u001b[0;36m<module>\u001b[1;34m\u001b[0m\n\u001b[0;32m      1\u001b[0m \u001b[1;31m#Taking weight from the user\u001b[0m\u001b[1;33m\u001b[0m\u001b[1;33m\u001b[0m\u001b[1;33m\u001b[0m\u001b[0m\n\u001b[1;32m----> 2\u001b[1;33m \u001b[0mweights\u001b[0m \u001b[1;33m=\u001b[0m \u001b[0mnp\u001b[0m\u001b[1;33m.\u001b[0m\u001b[0mzeros\u001b[0m\u001b[1;33m(\u001b[0m\u001b[1;33m[\u001b[0m\u001b[0mm\u001b[0m\u001b[1;33m]\u001b[0m\u001b[1;33m)\u001b[0m\u001b[1;33m\u001b[0m\u001b[1;33m\u001b[0m\u001b[0m\n\u001b[0m\u001b[0;32m      3\u001b[0m \u001b[0mprint\u001b[0m\u001b[1;33m(\u001b[0m\u001b[1;34m'Enter Weights: '\u001b[0m\u001b[1;33m)\u001b[0m\u001b[1;33m\u001b[0m\u001b[1;33m\u001b[0m\u001b[0m\n\u001b[0;32m      4\u001b[0m \u001b[1;32mfor\u001b[0m \u001b[0mi\u001b[0m \u001b[1;32min\u001b[0m \u001b[0mrange\u001b[0m\u001b[1;33m(\u001b[0m\u001b[0mm\u001b[0m\u001b[1;33m)\u001b[0m\u001b[1;33m:\u001b[0m\u001b[1;33m\u001b[0m\u001b[1;33m\u001b[0m\u001b[0m\n\u001b[0;32m      5\u001b[0m     \u001b[0minpt\u001b[0m \u001b[1;33m=\u001b[0m \u001b[0meval\u001b[0m\u001b[1;33m(\u001b[0m\u001b[0minput\u001b[0m\u001b[1;33m(\u001b[0m\u001b[1;33m)\u001b[0m\u001b[1;33m)\u001b[0m\u001b[1;33m\u001b[0m\u001b[1;33m\u001b[0m\u001b[0m\n",
      "\u001b[1;31mNameError\u001b[0m: name 'm' is not defined"
     ]
    }
   ],
   "source": [
    "#Taking weight from the user\n",
    "weights = np.zeros([m])\n",
    "print('Enter Weights: ')\n",
    "for i in range(m):\n",
    "    inpt = eval(input())\n",
    "    weights[i] = inpt\n",
    "def least_squre_weight(N, x):\n",
    "    M = np.zeros((N+1,N+1))\n",
    "    a = []\n",
    "    Y = []\n",
    "    for i in range(N+1):\n",
    "        c= []\n",
    "        for j in range(m):\n",
    "            c.append(ys[j]*(xs[j]**i)*weights[i])\n",
    "        Y.append(sum(c))\n",
    "    for i in range(2*N+1):\n",
    "        b = []\n",
    "        for j in range(m):\n",
    "            b.append((xs[j]**i)*weights[i])\n",
    "        a.append(sum(b))\n",
    "\n",
    "    for i in range(N+1):\n",
    "        for j in range(N+1):\n",
    "            M[i][j] = a[i+j]\n",
    "    m_inv = np.linalg.inv(M)\n",
    "\n",
    "    m_inv = np.array(m_inv)\n",
    "    Y = np.array(Y)\n",
    "    a = np.matmul(m_inv, Y)\n",
    "    #return a[0] + a[1]*x + a[2]*x**2 + a[3]*x**3\n",
    "    s = 0\n",
    "    for i in range(N+1):\n",
    "        s += a[i]*x**i\n",
    "    return s\n",
    "def plot_bestfit(degree):\n",
    "    x = [i/100 for i in range(100)]\n",
    "    y = [least_squre_weight(degree, i) for i in x]\n",
    "    plt.plot(x,y, color='r', linewidth=2)\n",
    "    plt.plot(xs, ys, 'o');"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 6,
   "metadata": {},
   "outputs": [
    {
     "ename": "NameError",
     "evalue": "name 'weights' is not defined",
     "output_type": "error",
     "traceback": [
      "\u001b[1;31m---------------------------------------------------------------------------\u001b[0m",
      "\u001b[1;31mNameError\u001b[0m                                 Traceback (most recent call last)",
      "\u001b[1;32m<ipython-input-6-6d902e3f479b>\u001b[0m in \u001b[0;36m<module>\u001b[1;34m\u001b[0m\n\u001b[1;32m----> 1\u001b[1;33m \u001b[0ma\u001b[0m \u001b[1;33m=\u001b[0m \u001b[0mweights\u001b[0m\u001b[1;33m\u001b[0m\u001b[1;33m\u001b[0m\u001b[0m\n\u001b[0m\u001b[0;32m      2\u001b[0m \u001b[0ma\u001b[0m\u001b[1;33m\u001b[0m\u001b[1;33m\u001b[0m\u001b[0m\n",
      "\u001b[1;31mNameError\u001b[0m: name 'weights' is not defined"
     ]
    }
   ],
   "source": [
    "a = weights\n",
    "a"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 7,
   "metadata": {},
   "outputs": [
    {
     "ename": "NameError",
     "evalue": "name 'plot_bestfit' is not defined",
     "output_type": "error",
     "traceback": [
      "\u001b[1;31m---------------------------------------------------------------------------\u001b[0m",
      "\u001b[1;31mNameError\u001b[0m                                 Traceback (most recent call last)",
      "\u001b[1;32m<ipython-input-7-5119bbf2f0a3>\u001b[0m in \u001b[0;36m<module>\u001b[1;34m\u001b[0m\n\u001b[1;32m----> 1\u001b[1;33m \u001b[0mplot_bestfit\u001b[0m\u001b[1;33m(\u001b[0m\u001b[1;36m3\u001b[0m\u001b[1;33m)\u001b[0m\u001b[1;33m\u001b[0m\u001b[1;33m\u001b[0m\u001b[0m\n\u001b[0m",
      "\u001b[1;31mNameError\u001b[0m: name 'plot_bestfit' is not defined"
     ]
    }
   ],
   "source": [
    "plot_bestfit(3)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": []
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": []
  }
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