18_Clenshaw_Curtis_Quadrature.ipynb

{
 "cells": [
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "# Lecture 18: Clenshaw-Curtis Quadrature\n",
    "\n",
    "Recall that the Newton-Cotes quadrature rules provide exponential convergence if\n",
    "the integrand is periodic and we integrate over a periodic domain:\n",
    "\n",
    "$$\n",
    "I = \\int_a^b f(x)\\, dx.\n",
    "$$"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 1,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/html": [
       "<script>\n",
       "// Immediately-invoked-function-expression to avoid global variables.\n",
       "(function() {\n",
       "    var warning_div = document.getElementById(\"webio-warning-18068085219610669455\");\n",
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       "        }\n",
       "        warning_div.style.display = \"none\";\n",
       "    };\n",
       "    if (typeof Jupyter !== \"undefined\") {\n",
       "        console.log(\"WebIO detected Jupyter notebook environment.\");\n",
       "        // Jupyter notebook.\n",
       "        var extensions = (\n",
       "            Jupyter\n",
       "            && Jupyter.notebook.config.data\n",
       "            && Jupyter.notebook.config.data.load_extensions\n",
       "        );\n",
       "        if (extensions && extensions[\"webio/main\"]) {\n",
       "            // Extension already loaded.\n",
       "            console.log(\"Jupyter WebIO nbextension detected; not loading ad-hoc.\");\n",
       "            hide();\n",
       "            return;\n",
       "        }\n",
       "        console.warn(\"Loading WebIO Jupyter extension on an ad-hoc basis. Consider enabling the WebIO nbextension for a stabler experience (this should happen automatically when building WebIO).\");\n",
       "        require([\"/assetserver/ea8282a8d723f49a68b1d36bce2394d3b9f3b531-dist/main.js\"], function (webIOModule) {\n",
       "            webIOModule.load_ipython_extension();\n",
       "        });\n",
       "        warning_div.innerHTML = \"<strong>Loading WebIO Jupyter extension on an ad-hoc basis. Consider enabling the WebIO nbextension for a stabler experience (this should happen automatically when building WebIO).</strong>\";\n",
       "    } else if (window.location.pathname.includes(\"/lab\")) {\n",
       "        // Guessing JupyterLa\n",
       "        console.log(\"Jupyter Lab detected; make sure the @webio/jupyter-lab-provider labextension is installed.\");\n",
       "        hide();\n",
       "        return;\n",
       "    }\n",
       "})();\n",
       "\n",
       "</script>\n",
       "<div\n",
       "    id=\"webio-warning-18068085219610669455\"\n",
       "    class=\"output_text output_stderr\"\n",
       "    style=\"padding: 1em; font-weight: bold;\"\n",
       ">\n",
       "    Unable to load WebIO. Please make sure WebIO works for your Jupyter client.\n",
       "    <!-- TODO: link to installation docs. -->\n",
       "</div>\n"
      ],
      "text/plain": [
       "HTML{String}(\"<script>\\n// Immediately-invoked-function-expression to avoid global variables.\\n(function() {\\n    var warning_div = document.getElementById(\\\"webio-warning-18068085219610669455\\\");\\n    var hide = function () {\\n        var script = document.getElementById(\\\"webio-setup-15570809841994579199\\\");\\n        var parent = script && script.parentElement;\\n        var grandparent = parent && parent.parentElement;\\n        if (grandparent) {\\n            grandparent.style.display = \\\"none\\\";\\n        }\\n        warning_div.style.display = \\\"none\\\";\\n    };\\n    if (typeof Jupyter !== \\\"undefined\\\") {\\n        console.log(\\\"WebIO detected Jupyter notebook environment.\\\");\\n        // Jupyter notebook.\\n        var extensions = (\\n            Jupyter\\n            && Jupyter.notebook.config.data\\n            && Jupyter.notebook.config.data.load_extensions\\n        );\\n        if (extensions && extensions[\\\"webio/main\\\"]) {\\n            // Extension already loaded.\\n            console.log(\\\"Jupyter WebIO nbextension detected; not loading ad-hoc.\\\");\\n            hide();\\n            return;\\n        }\\n        console.warn(\\\"Loading WebIO Jupyter extension on an ad-hoc basis. Consider enabling the WebIO nbextension for a stabler experience (this should happen automatically when building WebIO).\\\");\\n        require([\\\"/assetserver/ea8282a8d723f49a68b1d36bce2394d3b9f3b531-dist/main.js\\\"], function (webIOModule) {\\n            webIOModule.load_ipython_extension();\\n        });\\n        warning_div.innerHTML = \\\"<strong>Loading WebIO Jupyter extension on an ad-hoc basis. Consider enabling the WebIO nbextension for a stabler experience (this should happen automatically when building WebIO).</strong>\\\";\\n    } else if (window.location.pathname.includes(\\\"/lab\\\")) {\\n        // Guessing JupyterLa\\n        console.log(\\\"Jupyter Lab detected; make sure the @webio/jupyter-lab-provider labextension is installed.\\\");\\n        hide();\\n        return;\\n    }\\n})();\\n\\n</script>\\n<div\\n    id=\\\"webio-warning-18068085219610669455\\\"\\n    class=\\\"output_text output_stderr\\\"\\n    style=\\\"padding: 1em; font-weight: bold;\\\"\\n>\\n    Unable to load WebIO. Please make sure WebIO works for your Jupyter client.\\n    <!-- TODO: link to installation docs. -->\\n</div>\\n\")"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "using PyPlot\n",
    "PyPlot.svg(true);\n",
    "\n",
    "using Interact"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Error analysis for the trapezoid rule"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 2,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "trapezoid_rule (generic function with 1 method)"
      ]
     },
     "execution_count": 2,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "function trapezoid_rule(f, a, b, N)\n",
    "    Δ = (b - a)/N\n",
    "    \n",
    "    I = 0.5*Δ*f(a) + 0.5*Δ*f(b)\n",
    "    for n=1:N-1\n",
    "        I += Δ*f(a + n*Δ)\n",
    "    end\n",
    "    \n",
    "    return I\n",
    "end"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 3,
   "metadata": {},
   "outputs": [],
   "source": [
    "f(x) = x^2 - x                    # Not periodic\n",
    "g(x) = exp((cos(12x)*sin(2x))^4)  # Periodic\n",
    "\n",
    "a = -π\n",
    "b = π\n",
    "\n",
    "f_exact = 2π^3/3\n",
    "g_exact = 7.467289053463321;"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 4,
   "metadata": {},
   "outputs": [],
   "source": [
    "# analyze errors\n",
    "Ns = 10.0.^(0:0.5:7)\n",
    "\n",
    "f_error = [(trapezoid_rule(f, a, b, Int64(trunc(N))) - f_exact)/f_exact\n",
    "        for N in Ns]\n",
    "\n",
    "g_error = [(trapezoid_rule(g, a, b, Int64(trunc(N))) - g_exact)/g_exact\n",
    "        for N in Ns];"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 5,
   "metadata": {},
   "outputs": [
    {
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    "figure()\n",
    "\n",
    "loglog(Ns, abs.(f_error), \"o-\", label=L\"non-periodic $f(x)$\")\n",
    "loglog(Ns, abs.(g_error), \"o-\", label=L\"periodic $g(x)$\")\n",
    "\n",
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   "source": [
    "## Clenshaw-Curtis quadrature\n",
    "\n",
    "Let's say we want to integrate\n",
    "$$\n",
    "I = \\int_{-1}^1 f(x)\\, dx.\n",
    "$$\n",
    "\n",
    "Turn into an integral over a periodic function by introducing $x=\\cos\\theta$,\n",
    "\n",
    "$$\n",
    "I = \\int_{0}^\\pi f(\\cos\\theta)\\,\\sin\\theta\\, d\\theta.\n",
    "$$"
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  {
   "cell_type": "code",
   "execution_count": 6,
   "metadata": {},
   "outputs": [
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       "   <g id=\"patch_3\">\n",
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       "   </g>\n",
       "   <g id=\"patch_4\">\n",
       "    <path d=\"M 400.90125 276.812 \n",
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       "    <path d=\"M 43.78125 276.812 \n",
       "L 400.90125 276.812 \n",
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       "   <g id=\"patch_6\">\n",
       "    <path d=\"M 43.78125 10.7 \n",
       "L 400.90125 10.7 \n",
       "\" style=\"fill:none;stroke:#000000;stroke-linecap:square;stroke-linejoin:miter;stroke-width:0.8;\"/>\n",
       "   </g>\n",
       "  </g>\n",
       " </g>\n",
       " <defs>\n",
       "  <clipPath id=\"pb986b9496b\">\n",
       "   <rect height=\"266.112\" width=\"357.12\" x=\"43.78125\" y=\"10.7\"/>\n",
       "  </clipPath>\n",
       " </defs>\n",
       "</svg>\n"
      ],
      "text/plain": [
       "Figure(PyObject <Figure size 640x480 with 1 Axes>)"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    },
    {
     "data": {
      "text/plain": [
       "PyObject Text(26.000000000000007, 0.5, '$f(x)$')"
      ]
     },
     "execution_count": 6,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "xs = LinRange(-1, 1, 200)\n",
    "θs = LinRange(0, 2π, 200)\n",
    "\n",
    "fx = f.(xs)\n",
    "fcosθ = f.(cos.(θs))\n",
    "figure()\n",
    "\n",
    "plot(xs, fx)\n",
    "xlabel(L\"x\")\n",
    "ylabel(L\"f(x)\")"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 7,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": 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       "L 384.668523 237.833513 \n",
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       "   </g>\n",
       "   <g id=\"patch_3\">\n",
       "    <path d=\"M 43.78125 276.812 \n",
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       "    <path d=\"M 43.78125 10.7 \n",
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       "\" style=\"fill:none;stroke:#000000;stroke-linecap:square;stroke-linejoin:miter;stroke-width:0.8;\"/>\n",
       "   </g>\n",
       "  </g>\n",
       " </g>\n",
       " <defs>\n",
       "  <clipPath id=\"p9d8d30cbf8\">\n",
       "   <rect height=\"266.112\" width=\"357.12\" x=\"43.78125\" y=\"10.7\"/>\n",
       "  </clipPath>\n",
       " </defs>\n",
       "</svg>\n"
      ],
      "text/plain": [
       "Figure(PyObject <Figure size 640x480 with 1 Axes>)"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    },
    {
     "data": {
      "text/plain": [
       "3-element Array{PyCall.PyObject,1}:\n",
       " PyObject Text(0.0, 0, '0')               \n",
       " PyObject Text(3.141592653589793, 0, 'π') \n",
       " PyObject Text(6.283185307179586, 0, '2π')"
      ]
     },
     "execution_count": 7,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "figure()\n",
    "\n",
    "plot(θs, fcosθ)\n",
    "xlabel(L\"θ\")\n",
    "ylabel(L\"f(\\cos\\theta)\")\n",
    "xticks([0, π, 2π])\n",
    "gca().set_xticklabels([\"0\", \"π\", \"2π\"])"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Clenshaw-Curtis quadrature removes discontinuities at the ends of the integration interal\n",
    "and expresses the numerical integral $I$ through the Fourier series coefficients:\n",
    "\n",
    "$$\n",
    "I^{CC} = a_0 + 2\\sum_{\\nu=1, \\nu\\text{ even}}^N \\frac{a_\\nu}{1 - \\nu^2},\n",
    "$$\n",
    "\n",
    "where\n",
    "\n",
    "$$\n",
    "a_\\nu = \\frac{2}{\\pi}\\int_0^\\pi f(\\cos\\theta) \\cos(\\nu\\theta)\\, d\\theta\n",
    "$$\n",
    "\n",
    "is evaluated either by FFT or by a Newton-Cotes rule."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 8,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "clenshaw_curtis (generic function with 1 method)"
      ]
     },
     "execution_count": 8,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "function clenshaw_curtis(f, N)\n",
    "    \"\"\" Naive implementation of Clenshaw-Curtis\n",
    "    using the trapezoid rule for the Fourier coefficients.\n",
    "    \"\"\"     \n",
    "    # compute even Fourier coefficients\n",
    "    a = zeros(N+1)\n",
    "    \n",
    "    θs = collect(0:N).*π./N\n",
    "    ts = cos.(θs)\n",
    "    fts = f.(ts)\n",
    "    \n",
    "    # Trapezoid rule approximation for the even Fourier coefficients\n",
    "    for ν=1:2:N+1\n",
    "        a[ν] = fts[1]/N + fts[end]/N + (2/N)*sum(fts[2:N].*cos.((ν-1)*θs[2:N]))\n",
    "    end\n",
    "    \n",
    "    # compute integral\n",
    "    I = a[1]\n",
    "    for ν=3:N+1\n",
    "        I += 2a[ν]/(1 - (ν-1)^2)\n",
    "    end\n",
    "    \n",
    "    return I\n",
    "end"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 9,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "3.0170690539542693"
      ]
     },
     "execution_count": 9,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "f(x) = x^2 - x + exp(x)\n",
    "\n",
    "f_exact = 2.0/3.0 - exp(-1) + exp(1)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 13,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "3.038231843764906"
      ]
     },
     "execution_count": 13,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "# 2 correct digits\n",
    "trapezoid_rule(f, -1, 1, 10)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 14,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "3.0170690539431275"
      ]
     },
     "execution_count": 14,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "# Almost exact\n",
    "clenshaw_curtis(f, 10)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 15,
   "metadata": {},
   "outputs": [],
   "source": [
    "# Convergence analysis\n",
    "\n",
    "Ns = Int64.(2.0.^(1:8))\n",
    "\n",
    "trap_error = [(trapezoid_rule(f, -1, 1, N) - f_exact)/f_exact\n",
    "        for N in Ns]\n",
    "\n",
    "cc_error = [(clenshaw_curtis(f, N) - f_exact)/f_exact\n",
    "        for N in Ns];"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 16,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": 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    "figure()\n",
    "\n",
    "loglog(Ns, abs.(trap_error), \"o-\", label=\"trapezoid rule\")\n",
    "loglog(Ns, abs.(cc_error), \"o-\", label=\"Clenshaw-Curtis\")\n",
    "\n",
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   "source": [
    "fig, ax = subplots()\n",
    "\n",
    "\n",
    "@manipulate for N=slider(1:20, value=1, label=\"N\")\n",
    "    withfig(fig, clear=false) do\n",
    "        ax.cla()\n",
    "        \n",
    "        θs = collect(cos.((0:N)*π/N))\n",
    "        \n",
    "        ax.plot(θs, zeros(length(θs)), \"xr\")\n",
    "\n",
    "        ax.set_xlim(-1.02, 1.02)\n",
    "        ax.set_ylim(-1, 1)\n",
    "        ax.set_yticks([])\n",
    "\n",
    "        ax.set_xlabel(L\"$x$\")\n",
    "        ax.set_title(\"Clenshaw-Curtis interpolation points (Chebyshev points)\")\n",
    "    end\n",
    "end"
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