Lagrange Polynomial problems & Runge phenomenon.ipynb

{
 "cells": [
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "# Interpolation and Lagrange Polynomial"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 1,
   "metadata": {},
   "outputs": [],
   "source": [
    "import numpy as np\n",
    "from math import factorial, pi\n",
    "from matplotlib import pyplot as plt\n",
    "%matplotlib inline"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 2,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "3.141592653589793\n",
      "3.141592653589793\n"
     ]
    }
   ],
   "source": [
    "x = np.linspace(-pi,pi,10)\n",
    "x #10 elements, the last element is x[9]. Also, there is backwards index, where the last element is x[-1].\n",
    "print(x[9])\n",
    "print(x[-1])"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 3,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "[[-3.14159265 -2.44346095 -1.74532925 -1.04719755 -0.34906585  0.34906585\n",
      "   1.04719755  1.74532925  2.44346095  3.14159265]\n",
      " [ 2.02612013 -0.0120131  -0.8592331  -0.85580078 -0.34197708  0.34197708\n",
      "   0.85580078  0.8592331   0.0120131  -2.02612013]\n",
      " [ 0.          0.          0.          0.          0.          0.\n",
      "   0.          0.          0.          0.        ]]\n",
      "[[-3.14159265 -2.44346095 -1.74532925 -1.04719755 -0.34906585  0.34906585\n",
      "   1.04719755  1.74532925  2.44346095  3.14159265]\n",
      " [ 2.02612013 -0.0120131  -0.8592331  -0.85580078 -0.34197708  0.34197708\n",
      "   0.85580078  0.8592331   0.0120131  -2.02612013]\n",
      " [-0.52404391 -0.73786124 -0.99419326 -0.86629528 -0.34202027  0.34202027\n",
      "   0.86629528  0.99419326  0.73786124  0.52404391]]\n"
     ]
    }
   ],
   "source": [
    "x = np.linspace(-pi,pi,10)\n",
    "pSin = np.zeros([3,10])\n",
    "pSin[0,:] = x\n",
    "for n in range(1,3):\n",
    "    pSin[n,:] = pSin[n-1,:] + (-1)**n/(factorial(2*n+1))*x**(2*n+1)\n",
    "    print(pSin)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 4,
   "metadata": {},
   "outputs": [],
   "source": [
    "def L(xvals,kk,x):\n",
    "    lval = np.ones(x.size)\n",
    "    n = xvals.size\n",
    "    for ii in range(0,n):\n",
    "        if ii != kk:\n",
    "            lval *= (x-xvals[ii])/(xvals[kk]-xvals[ii])\n",
    "    return lval"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 17,
   "metadata": {},
   "outputs": [],
   "source": [
    "def p_Lagrange(xvals,fvals,x):\n",
    "    n = fvals.size\n",
    "    pLagrange = np.zeros(x.size)\n",
    "    for kk in range(0,n):\n",
    "        pLagrange += yvals[kk]*L(xvals,kk,x)\n",
    "    return pLagrange"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 6,
   "metadata": {},
   "outputs": [],
   "source": [
    "def LpSum(xvals,kk,x):\n",
    "    lval = np.ones(x.size)\n",
    "    n = xvals.size\n",
    "    for ii in range(0,n):\n",
    "        if ii != kk:\n",
    "            lval += 1/(x-xvals[ii])\n",
    "    return lval\n",
    "\n",
    "def p_LagrangeP(xvals,fvals,x):\n",
    "    n = fvals.size\n",
    "    pLagrange = np.zeros(x.size)\n",
    "    for kk in range(0,n):\n",
    "        pLagrange += yvals[kk]*L(xvals,kk,x)*LpSum(xvals,kk,x)\n",
    "    return pLagrange"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 7,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": 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\n",
      "text/plain": [
       "<Figure size 432x288 with 1 Axes>"
      ]
     },
     "metadata": {
      "needs_background": "light"
     },
     "output_type": "display_data"
    }
   ],
   "source": [
    "xvals = np.array([-9.,-4.,-1.,7.])\n",
    "yvals = np.array([5.,2.,-2.,9.])\n",
    "plt.plot(xvals,yvals,'o',markersize=10)\n",
    "plt.grid()"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 20,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "array([ 5.        ,  5.10414532,  5.18969573,  5.2571861 ,  5.30715131,\n",
       "        5.34012621,  5.35664568,  5.35724458,  5.34245779,  5.31282016,\n",
       "        5.26886658,  5.2111319 ,  5.14015099,  5.05645872,  4.96058997,\n",
       "        4.85307959,  4.73446246,  4.60527344,  4.4660474 ,  4.31731921,\n",
       "        4.15962374,  3.99349586,  3.81947042,  3.63808231,  3.44986639,\n",
       "        3.25535753,  3.05509058,  2.84960044,  2.63942195,  2.42508999,\n",
       "        2.20713943,  1.98610513,  1.76252197,  1.5369248 ,  1.30984851,\n",
       "        1.08182795,  0.85339799,  0.62509351,  0.39744936,  0.17100043,\n",
       "       -0.05371843, -0.27617235, -0.49582645, -0.71214587, -0.92459575,\n",
       "       -1.13264121, -1.33574739, -1.53337942, -1.72500242, -1.91008154,\n",
       "       -2.08808191, -2.25846865, -2.4207069 , -2.57426179, -2.71859846,\n",
       "       -2.85318203, -2.97747763, -3.09095041, -3.19306549, -3.283288  ,\n",
       "       -3.36108307, -3.42591585, -3.47725145, -3.51455502, -3.53729168,\n",
       "       -3.54492656, -3.5369248 , -3.51275154, -3.47187189, -3.413751  ,\n",
       "       -3.337854  , -3.24364601, -3.13059217, -2.99815762, -2.84580748,\n",
       "       -2.67300689, -2.47922097, -2.26391487, -2.02655371, -1.76660262,\n",
       "       -1.48352674, -1.1767912 , -0.84586113, -0.49020167, -0.10927794,\n",
       "        0.29744493,  0.73050179,  1.19042752,  1.67775699,  2.19302507,\n",
       "        2.73676661,  3.3095165 ,  3.9118096 ,  4.54418077,  5.20716488,\n",
       "        5.90129681,  6.62711142,  7.38514357,  8.17592814,  9.        ])"
      ]
     },
     "execution_count": 20,
     "metadata": {},
     "output_type": "execute_result"
    },
    {
     "data": {
      "image/png": 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\n",
      "text/plain": [
       "<Figure size 432x288 with 1 Axes>"
      ]
     },
     "metadata": {
      "needs_background": "light"
     },
     "output_type": "display_data"
    }
   ],
   "source": [
    "x = np.linspace(min(xvals),max(xvals),int(1e2))\n",
    "y = p_Lagrange(xvals,yvals,x)\n",
    "plt.plot(xvals,yvals,'o',markersize=10)\n",
    "plt.plot(x,y,color='k')\n",
    "plt.xlabel('$x$')\n",
    "plt.ylabel(\"$P_{3}(x)$\")\n",
    "plt.grid()"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 15,
   "metadata": {},
   "outputs": [
    {
     "name": "stderr",
     "output_type": "stream",
     "text": [
      "c:\\users\\dionh\\appdata\\local\\programs\\python\\python36\\lib\\site-packages\\ipykernel_launcher.py:6: RuntimeWarning: divide by zero encountered in true_divide\n",
      "  \n",
      "c:\\users\\dionh\\appdata\\local\\programs\\python\\python36\\lib\\site-packages\\ipykernel_launcher.py:13: RuntimeWarning: invalid value encountered in multiply\n",
      "  del sys.path[0]\n"
     ]
    },
    {
     "data": {
      "image/png": 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\n",
      "text/plain": [
       "<Figure size 432x288 with 1 Axes>"
      ]
     },
     "metadata": {
      "needs_background": "light"
     },
     "output_type": "display_data"
    }
   ],
   "source": [
    "x = np.linspace(min(xvals),max(xvals),int(1e2))\n",
    "y = p_Lagrange(xvals,yvals,x);\n",
    "yp = p_LagrangeP(xvals,yvals,x)\n",
    "plt.plot(xvals,yvals,'o',markersize=10)\n",
    "plt.plot(x[1:len(yp)-2],yp[1:len(yp)-2],color='k')\n",
    "plt.xlabel('$x$')\n",
    "plt.ylabel(\"$P_{3}(x)$\")\n",
    "plt.grid()"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "$a_0 = 2, b_0 = 3/4, c_0 = 0, d_0 = 1/4, a_1 = 3, b_1 = 3/2, c_1 = 3/4, d_1 = -1/4$"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 10,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "[<matplotlib.lines.Line2D at 0x215e55664a8>]"
      ]
     },
     "execution_count": 10,
     "metadata": {},
     "output_type": "execute_result"
    },
    {
     "data": {
      "image/png": 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\n",
      "text/plain": [
       "<Figure size 432x288 with 1 Axes>"
      ]
     },
     "metadata": {
      "needs_background": "light"
     },
     "output_type": "display_data"
    }
   ],
   "source": [
    "x = np.linspace(1,3,100)\n",
    "S = (2 + (3/4)*(x-1) + (1/4)*(x-1)**3)*(x<2) + (3 + (3/2)*(x-2) + (3/4)*(x-2)**2 + (-1/4)*(x-2)**3)*(x>=2)\n",
    "xV = np.array([1, 2, 3])\n",
    "yV = np.array([2, 3, 5])\n",
    "plt.plot(x,S,\".\")\n",
    "plt.plot(xV,yV, \"o\")"
   ]
  },
  {
   "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": "3.6.7"
  }
 },
 "nbformat": 4,
 "nbformat_minor": 2
}