{
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  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "# Notebook for exercise sheet 1"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "<span style=\"color:red; font-weight: bold\">Important: run the code cell below once when you just opened this notebook! It will make sure that plotting is enabled.</span>"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {
    "collapsed": false
   },
   "outputs": [],
   "source": [
    "# some setup\n",
    "%matplotlib inline\n",
    "import numpy as np # makes numpy routines and data types available as np.[name ouf routine or data type]\n",
    "import matplotlib.pyplot as plt # makes plotting command available as plot.[name of command]"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## H 3 (Newton interpolation)\n",
    "\n",
    "a) Write a python subroutine which computes the coefficients $c_1,\\dots,c_n$ in Sheet 1, H2. Do this by implementing a suitable divided differences scheme. Make sure that your implemented subroutine behaves according to the documentation given below."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {
    "collapsed": false
   },
   "outputs": [],
   "source": [
    "def divdiff(x,f):\n",
    "    \"\"\"\n",
    "    Computes the coefficient of the interpolation polynomial in Newton.\n",
    "    The coefficients are return as numpy.array.\n",
    "    \n",
    "    Keyword arguments:\n",
    "    x -- list or numpy.array of sampling points\n",
    "    f -- list or numpy.array of function values\n",
    "    \"\"\"\n",
    "    # your code goes here\n",
    "    \n",
    "divdiff([-5,-2,-1,0,1],[17,8,21,42, 35])"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "b) Implement a subroutine, which plots a function and the corresponding interpolation polynomial. For evaluating the interpolation polynomial you can use the subroutine <tt>evaluate_polynomial</tt> provided below. For the plot evaluate the the function and the interpolation polynomial at the points\n",
    "$$\n",
    " z_k = x_0 + (k-1) \\frac{x_n-x_0}{1000}, \\quad k = 1, \\dots, 1001,\n",
    "$$\n",
    "where $x_0$ and $x_n$ are the first and last interpolation point."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {
    "collapsed": false
   },
   "outputs": [],
   "source": [
    "def evaluate_polynomial(x,coef,z):\n",
    "    \"\"\"\n",
    "    Evaluates the Newton interpolation polynomial at the points given by z.\n",
    "    The values of the polynomial at the points z are returned as numpy.array.\n",
    "    \n",
    "    Keyword arguments:\n",
    "    x    -- list or numpy.array of sampling points\n",
    "    coef -- list or numpy.array of coefficients\n",
    "    z    -- list or numpy.array of points\n",
    "    \"\"\"\n",
    "    n = np.size(coef)\n",
    "    m = np.size(z)\n",
    "    p = coef[n-1] * np.ones(m)\n",
    "    for i in np.arange(n-1,-1,-1):\n",
    "        p = p*(z - x[i]*np.ones(m)) + coef[i]*np.ones(m)\n",
    "    return p\n",
    "\n",
    "def plotinterpol(x_sampl, f):\n",
    "    # your code goes here\n",
    "    return"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Use your implementation to generate plots for the function\n",
    "$$\n",
    " f(x) = \\frac{1}{1+25x^2}\n",
    "$$\n",
    "with interpolation points\n",
    "\n",
    "* 0.0, 0.5, 1.0\n",
    "* 0.0, 0.3, 0.6, 0.9, 1.0\n",
    "* 0.0 - 1.0 with step size 0.2\n",
    "* 0.0 - 1.0 with step size 0.1"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {
    "collapsed": false
   },
   "outputs": [],
   "source": [
    "f = lambda x: 1.0/(1+25*np.power(x,2))\n",
    "\n",
    "# your code goes here"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## H4 (Plotting of kernels)\n",
    "\n",
    "Generate plots of the Gauss kernel and the inverse multiquadrics for various parameter values. Also generate a plot of Wendland's kernel"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {
    "collapsed": true
   },
   "outputs": [],
   "source": [
    "# your code for generating plots of the Gauss kernel goes here"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {
    "collapsed": true
   },
   "outputs": [],
   "source": [
    "# your code for generating plots of the inverse multiquadrics goes here"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {
    "collapsed": true
   },
   "outputs": [],
   "source": [
    "# your code for generating plots of Wendland's kernel goes here"
   ]
  }
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