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 "cells": [
  {
   "cell_type": "markdown",
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   "source": [
    "# Notebook for Exercise sheet 2"
   ]
  },
  {
   "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 2 (Dirichlet and Fej&eacute;r kernel)\n",
    "\n",
    "Plot the Dirichlet and Fej&eacute;r kernel for various values of n. Also generate a plot where you plot both kernels in one diagram."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {
    "collapsed": true
   },
   "outputs": [],
   "source": [
    "# Code for Dirichlet plots"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {
    "collapsed": true
   },
   "outputs": [],
   "source": [
    "#code for Fejer plots"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## H3 (Interpolation with kernels)\n",
    "\n",
    "This exercise is about interpolation with the Gauss kernel $k(x,y)=\\exp(-\\gamma\\|x-y\\|_2^2)$.\n",
    "\n",
    "a) You are given a set of sampling points $X = \\{x_1,\\dots,x_n\\}$ with $x_i \\in \\mathbb R^2$ and function values at these points $f|_X = \\{f(x_1),\\dots,f(x_n)\\}$. Write some code which solves\n",
    "$$\n",
    " A_{X,X}\\alpha = f|_X,\n",
    "$$\n",
    "where $A_{X,X} = (k(x_i,x_j))_{i,j=1,\\dots,n}$ is the kernel matrix. You can use any of the linear algebra routines that come along with numpy. Also write a subroutine which evaluates the interpolant $s_{X,f|_X} = \\sum_{j=1}^n \\alpha_j k(x_j,\\cdot)$ at a given set of points $\\{y_1,\\dots,y_m\\}$."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {
    "collapsed": true
   },
   "outputs": [],
   "source": [
    "# your code goes here"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "b) For each of the following type of sampling points, write a subroutine which generates, for given $m \\in \\mathbb N$, $m^2$ sampling points:\n",
    "* uniform grid points in $[0,1]^2$\n",
    "* tensor-product Chebyshev points on $[0,1]^2$ given by\n",
    "$$\n",
    " (x_i,x_j) = \\left( \\cos\\left(\\frac{2i-1}{2m} \\pi\\right), \\cos\\left(\\frac{2j-1}{2m} \\pi\\right) \\right), \\quad i,j=1,\\dots,m\n",
    "$$\n",
    "* Halton points\n",
    "\n",
    "For the Halton sequence you can use the <a href=../../../../_https_/github.com/fmder/ghalton/_.html>ghalton package</a>. It is not in the <tt>conda</tt> package index, but in the <tt>pip</tt> package index."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {
    "collapsed": true
   },
   "outputs": [],
   "source": [
    "# Uniform grid points"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {
    "collapsed": true
   },
   "outputs": [],
   "source": [
    "# tensor-product Chebyshev points"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {
    "collapsed": true
   },
   "outputs": [],
   "source": [
    "# Halton points"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "c) Use your code to interpolate the function\n",
    "$$\n",
    "f(x,y) = \n",
    "3(1-x)^2 \\exp(-x^2 - (y+1)^2)\\\\\n",
    "   - 10(x/5 - x^3 - y^5) \\exp(-x^2-y^2)\\\\ \n",
    "   - \\exp(-(x+1)^2 - y^2)/3\n",
    "$$\n",
    "on $[0,1]$ at $m^2$ uniform grid points, tensor-product Chebyshev points and Halton points for $m=10,20,30,40$ and Gauss parameter $\\gamma = 1,2$. Generate plots which show the pointwise error\n",
    "$$\n",
    " |f(y) - s_{X,f|_X}(y)| \n",
    "$$\n",
    "for each of type of sampling points. To this end, evaluate $f$ and $s_{X,f|_X}$ on a uniform grid with mesh size $1/1000$."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {
    "collapsed": true
   },
   "outputs": [],
   "source": [
    "# your code goes here"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "d) For each the sampling points used in c) plot the power function\n",
    "$$\n",
    " P_X^2(x) = k(x,x) - k(x,\\cdot)(x)\n",
    "$$\n",
    "Again, use a uniform grid with mesh size $1/1000$ to generate the plots. Compare the plots with the plots obtained in c)."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {
    "collapsed": true
   },
   "outputs": [],
   "source": [
    "# your code goes here"
   ]
  }
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