03_polynomial_interpolation.ipynb

{
 "cells": [
  {
   "cell_type": "code",
   "execution_count": 1,
   "metadata": {},
   "outputs": [],
   "source": [
    "import numpy as np\n",
    "import scipy.linalg as la\n",
    "import matplotlib.pyplot as plt"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "# Polynomial Interpolation"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Given $d+1$ data points $(t_0,y_0), \\dots , (t_d,y_d)$ there exists a unique polynomial of degree $d$\n",
    "\n",
    "$$\n",
    "p(t) = c_0 + c_1 t + \\cdots + c_d t^d\n",
    "$$\n",
    "\n",
    "such that $p(t_k) = y_k$ for each $k=0,\\dots,d$. The coefficients $c_k$ are unknown and each data point gives us an equation\n",
    "\n",
    "\\begin{align*}\n",
    "c_0 + c_1t_1 + \\cdots + c_d t_1^d &= y_1 \\\\\n",
    "c_0 + c_1t_2 + \\cdots + c_d t_2^d &= y_2 \\\\\n",
    "& \\vdots \\\\\n",
    "c_0 + c_1t_d + \\cdots + c_d t_d^d &= y_d\n",
    "\\end{align*}\n",
    "\n",
    "This yields a system of equations $A \\boldsymbol{c} = \\boldsymbol{y}$\n",
    "$$\n",
    "\\begin{bmatrix}\n",
    "1 & t_1 & \\cdots & t_1^d \\\\\n",
    "1 & t_2 & \\cdots & t_2^d \\\\\n",
    "\\vdots & \\vdots & \\ddots & \\vdots \\\\\n",
    " 1 & t_d & \\cdots & t_d^d\n",
    "\\end{bmatrix}\n",
    "\\begin{bmatrix} c_0 \\\\ c_1 \\\\ \\vdots \\\\ c_d \\end{bmatrix}\n",
    "=\n",
    "\\begin{bmatrix} y_0 \\\\ y_1 \\\\ \\vdots \\\\ y_d \\end{bmatrix}\n",
    "$$\n",
    "The matrix $A$ is called a [Vandermonde matrix](https://en.wikipedia.org/wiki/Vandermonde_matrix)."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Example 1\n",
    "\n",
    "Start with data points for which we know the solution. Consider $(-1,1),(0,0),(1,1)$. The parabola $p(t) = t^2$ is the unique polynomial which interpolates these points. Let's create the system using [`numpy.vander`](https://numpy.org/doc/stable/reference/generated/numpy.vander.html) and solve with [`scipy.linalg.solve`](https://docs.scipy.org/doc/scipy/reference/generated/scipy.linalg.solve.html)."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 2,
   "metadata": {},
   "outputs": [],
   "source": [
    "t = np.array([-1,0,1])\n",
    "y = np.array([1,0,1])\n",
    "A = np.vander(t,increasing=True)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 3,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "[[ 1 -1  1]\n",
      " [ 1  0  0]\n",
      " [ 1  1  1]]\n"
     ]
    }
   ],
   "source": [
    "print(A)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 4,
   "metadata": {},
   "outputs": [],
   "source": [
    "c = la.solve(A,y)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 5,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "[0. 0. 1.]\n"
     ]
    }
   ],
   "source": [
    "print(c)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "The solution corresponds to the coefficients $c_0=0$, $c_1=0$ and $c_2=1$. That is $p(t) = t^2$."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Example 2\n",
    "\n",
    "Let's try a cubic polyniomial which interpolates $(0,0),(1,1),(2,0),(3,0)$."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 6,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "[ 0.   3.  -2.5  0.5]\n"
     ]
    }
   ],
   "source": [
    "t = np.array([0,1,2,3])\n",
    "y = np.array([0,1,0,0])\n",
    "A = np.vander(t,increasing=True)\n",
    "c = la.solve(A,y)\n",
    "print(c)"
   ]
  },
  {
   "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": [
    "T = np.linspace(-1,4,100)\n",
    "Y = c[0] + c[1]*T + c[2]*T**2 + c[3]*T**3\n",
    "plt.plot(T,Y,'b-',t,y,'r.',markersize=10)\n",
    "plt.grid(True)\n",
    "plt.show()"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Example 3\n",
    "\n",
    "What happens to the condition number as $N$ increases when $t_0,t_1,\\dots,t_N$ are $N+1$ equally spaced points from 0 to 1? That is, $t_k = k/N$, $k=0,\\dots,N$."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 8,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": 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\n",
      "text/plain": [
       "<Figure size 576x360 with 1 Axes>"
      ]
     },
     "metadata": {
      "needs_background": "light"
     },
     "output_type": "display_data"
    }
   ],
   "source": [
    "C = []\n",
    "Nmax = 11\n",
    "for N in range(2,Nmax):\n",
    "    t = np.linspace(0,1,N+1)\n",
    "    A = np.vander(t)\n",
    "    c = np.linalg.cond(A)\n",
    "    C.append(c)\n",
    "plt.figure(figsize=(8,5))\n",
    "plt.plot(range(2,Nmax),C,'b.-',markersize=10)\n",
    "plt.title('Condition Number of Vandermonde matrix of size N+1')\n",
    "plt.grid(True)\n",
    "plt.xlabel('N')\n",
    "plt.show()"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 9,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "115575244.52774188"
      ]
     },
     "execution_count": 9,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "C[-1]"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "The condition number for $N=10$ is around $10^8$. Yikes!"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Example 4\n",
    "\n",
    "Let's interpolate the points $\\sin(\\pi t_k)$ for $t_k = k/N$ for $N=15$."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 10,
   "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": [
    "N = 15\n",
    "t = np.linspace(0,1,N+1)\n",
    "y = np.sin(np.pi*t)\n",
    "A = np.vander(t,increasing=True)\n",
    "c = la.solve(A,y)\n",
    "T = np.linspace(0,1,100)\n",
    "Y = sum([c[k]*T**k for k in range(0,len(c))])\n",
    "plt.plot(T,Y,'b-',t,y,'r.',markersize=10)\n",
    "plt.grid(True)\n",
    "plt.show()"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "The interpolating polynomial is very nice! But what happens if we introduce some random noise in the data."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 11,
   "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": [
    "N = 15\n",
    "t = np.linspace(0,1,N+1)\n",
    "noise = 0.005*np.random.randn(t.size)\n",
    "y = np.sin(np.pi*t) + noise\n",
    "A = np.vander(t,increasing=True)\n",
    "c = la.solve(A,y)\n",
    "T = np.linspace(0,1,200)\n",
    "Y = sum([c[k]*T**k for k in range(0,len(c))])\n",
    "plt.plot(T,Y,'b-',t,y,'r.',markersize=10)\n",
    "plt.grid(True)\n",
    "plt.show()"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Yikes! The interpolating polynomial is very sensitive to small changes in the $y$ values! That's because the condition number is large!"
   ]
  }
 ],
 "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": "3.7.6"
  }
 },
 "nbformat": 4,
 "nbformat_minor": 4
}