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inducer · Dec 5, 2024 · 3d541d1 · 3d541d1
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diff --git a/cleared-demos/interpolation/Interpolation Error.ipynb b/cleared-demos/interpolation/Interpolation Error.ipynb
@@ -55,7 +55,7 @@
     },
     {
       "cell_type": "code",
-      "execution_count": 2,
+      "execution_count": 1,
       "metadata": {
         "collapsed": false,
         "jupyter": {
@@ -70,9 +70,12 @@
         "elif 0:\n",
         "    def f(x):\n",
         "        return np.sin(20*x)\n",
+        "elif 0:\n",
+        "    def f(x):\n",
+        "        return (x>=0.5).astype(np.int32).astype(np.float64) * (x-0.5)**2\n",
         "else:\n",
         "    def f(x):\n",
-        "        return (x>=0.5).astype(np.int).astype(np.float)\n",
+        "        return (x>=0.5).astype(np.int32).astype(np.float64)\n",
         "    "
       ]
     },
@@ -277,7 +280,7 @@
   ],
   "metadata": {
     "kernelspec": {
-      "display_name": "Python 3",
+      "display_name": "Python 3 (ipykernel)",
       "language": "python",
       "name": "python3"
     },
@@ -291,7 +294,7 @@
       "name": "python",
       "nbconvert_exporter": "python",
       "pygments_lexer": "ipython3",
-      "version": "3.8.3"
+      "version": "3.13.0"
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   "nbformat": 4,

diff --git a/cleared-demos/interpolation/Lebesgue Constant.ipynb b/cleared-demos/interpolation/Lebesgue Constant.ipynb
@@ -0,0 +1,252 @@
+{
+  "cells": [
+    {
+      "cell_type": "markdown",
+      "id": "55af40c4-ecf0-4a02-b7b4-9afb96786313",
+      "metadata": {},
+      "source": [
+        "# Lebesgue Constant\n",
+        "\n",
+        "Copyright (C) 2024 Andreas Kloeckner\n",
+        "\n",
+        "<details>\n",
+        "<summary>MIT License</summary>\n",
+        "Permission is hereby granted, free of charge, to any person obtaining a copy\n",
+        "of this software and associated documentation files (the \"Software\"), to deal\n",
+        "in the Software without restriction, including without limitation the rights\n",
+        "to use, copy, modify, merge, publish, distribute, sublicense, and/or sell\n",
+        "copies of the Software, and to permit persons to whom the Software is\n",
+        "furnished to do so, subject to the following conditions:\n",
+        "\n",
+        "The above copyright notice and this permission notice shall be included in\n",
+        "all copies or substantial portions of the Software.\n",
+        "\n",
+        "THE SOFTWARE IS PROVIDED \"AS IS\", WITHOUT WARRANTY OF ANY KIND, EXPRESS OR\n",
+        "IMPLIED, INCLUDING BUT NOT LIMITED TO THE WARRANTIES OF MERCHANTABILITY,\n",
+        "FITNESS FOR A PARTICULAR PURPOSE AND NONINFRINGEMENT. IN NO EVENT SHALL THE\n",
+        "AUTHORS OR COPYRIGHT HOLDERS BE LIABLE FOR ANY CLAIM, DAMAGES OR OTHER\n",
+        "LIABILITY, WHETHER IN AN ACTION OF CONTRACT, TORT OR OTHERWISE, ARISING FROM,\n",
+        "OUT OF OR IN CONNECTION WITH THE SOFTWARE OR THE USE OR OTHER DEALINGS IN\n",
+        "THE SOFTWARE.\n",
+        "</details>"
+      ]
+    },
+    {
+      "cell_type": "code",
+      "execution_count": 1,
+      "id": "a956ac5a-ccc1-4aec-8ebf-24b2e1541f5a",
+      "metadata": {},
+      "outputs": [],
+      "source": [
+        "import numpy as np\n",
+        "import numpy.linalg as la\n",
+        "import matplotlib.pyplot as plt"
+      ]
+    },
+    {
+      "cell_type": "code",
+      "execution_count": 2,
+      "id": "d734ec26-3d68-4a6f-ae16-5afbfaa005e8",
+      "metadata": {},
+      "outputs": [],
+      "source": [
+        "n = 10\n",
+        "nodes = np.linspace(-1, 1, n)"
+      ]
+    },
+    {
+      "cell_type": "markdown",
+      "id": "bfbe471d-cf99-4b09-9828-ee1cb0704835",
+      "metadata": {},
+      "source": [
+        "Create a Vandermonde matrix (as `vdm`) for monomial interpolation with `nodes`."
+      ]
+    },
+    {
+      "cell_type": "code",
+      "execution_count": 3,
+      "id": "7ba02567-c5df-452e-8be0-205b403b22b0",
+      "metadata": {},
+      "outputs": [],
+      "source": []
+    },
+    {
+      "cell_type": "markdown",
+      "id": "523271cd-e6b3-46ec-8f42-24d58029de87",
+      "metadata": {},
+      "source": [
+        "Create a \"fine\" set of nodes (as `nodes_fine`) and a corresponding Vandermonde matrix with the monomials (as `vdm_fine`):"
+      ]
+    },
+    {
+      "cell_type": "code",
+      "execution_count": 4,
+      "id": "db776c8d-28e3-42d9-b756-58326cb0cfa0",
+      "metadata": {},
+      "outputs": [],
+      "source": [
+        "nodes_fine = np.linspace(-1, 1, 10_000)\n",
+        "\n",
+        "vdm_fine = np.array([\n",
+        "    nodes_fine**i for i in range(n)\n",
+        "]).T"
+      ]
+    },
+    {
+      "cell_type": "markdown",
+      "id": "d1002ba4-9297-486d-b947-08d66c26cfb0",
+      "metadata": {},
+      "source": [
+        "Generate 300 random functions with nodal data $\\|\\boldsymbol y\\|_\\infty \\le 1$ and plot their interpolants on the fine grid:"
+      ]
+    },
+    {
+      "cell_type": "code",
+      "execution_count": 10,
+      "id": "7dcf5af9-90b3-4d30-b351-cd67ebdbbcd1",
+      "metadata": {},
+      "outputs": [],
+      "source": [
+        "for _ in range(300):\n",
+        "    y = 2*np.random.rand(n) - 1\n",
+        "    coeffs = la.solve(vdm, y)\n",
+        "    yfine = vdm_fine @ coeffs\n",
+        "    plt.plot(nodes_fine, yfine)"
+      ]
+    },
+    {
+      "cell_type": "markdown",
+      "id": "4727b2e4-219d-4941-b71e-7534411bda56",
+      "metadata": {},
+      "source": [
+        "Systematically estimate the Lebesgue constant:"
+      ]
+    },
+    {
+      "cell_type": "code",
+      "execution_count": 11,
+      "id": "3dbbe752-7892-48f0-8e7d-0e86540aa758",
+      "metadata": {},
+      "outputs": [],
+      "source": []
+    },
+    {
+      "cell_type": "markdown",
+      "id": "70caa6e1-063a-4cbc-8903-4b761a88ca98",
+      "metadata": {},
+      "source": [
+        "Plug in all-ones data, disregarding signs:"
+      ]
+    },
+    {
+      "cell_type": "code",
+      "execution_count": 12,
+      "id": "a83fbe81-bf4d-40e1-bbed-cfc45f8576c6",
+      "metadata": {},
+      "outputs": [],
+      "source": [
+        "interp = vdm_fine @ la.inv(vdm)\n",
+        "worst = np.sum(np.abs(interp), axis=1)\n",
+        "plt.plot(nodes_fine, worst)"
+      ]
+    },
+    {
+      "cell_type": "markdown",
+      "id": "68a17a7d-026a-426a-a869-54c61da720f9",
+      "metadata": {},
+      "source": [
+        "Now experiment:\n",
+        "\n",
+        "- Increase `n`.\n",
+        "- Can you come up with better `nodes`?"
+      ]
+    },
+    {
+      "cell_type": "code",
+      "execution_count": 46,
+      "id": "2636686b-f5ff-4d80-9111-b1b5165aa5f0",
+      "metadata": {},
+      "outputs": [],
+      "source": []
+    },
+    {
+      "cell_type": "markdown",
+      "id": "5d53ac63-0ded-489b-8788-b9a39c1efaee",
+      "metadata": {},
+      "source": [
+        "### Chebyshev nodes\n",
+        "\n",
+        "Extremal/second kind:"
+      ]
+    },
+    {
+      "cell_type": "code",
+      "execution_count": 7,
+      "id": "474e4729-5076-445e-b728-edf855d46d20",
+      "metadata": {},
+      "outputs": [],
+      "source": [
+        "k = n - 1\n",
+        "i = np.arange(0, k+1)\n",
+        "\n",
+        "def f(x):\n",
+        "    return np.cos(k*np.arccos(x))\n",
+        "\n",
+        "nodes = np.cos(i/k*np.pi)"
+      ]
+    },
+    {
+      "cell_type": "markdown",
+      "id": "3875152e-1583-471e-a618-2ebe032c92e7",
+      "metadata": {},
+      "source": [
+        "Roots/first kind:"
+      ]
+    },
+    {
+      "cell_type": "code",
+      "execution_count": 14,
+      "id": "9b52b06b-37cf-4ee5-a8c9-abccf455cbdc",
+      "metadata": {},
+      "outputs": [],
+      "source": [
+        "i = np.arange(1, n+1)\n",
+        "x = np.linspace(-1, 1, 3000)\n",
+        "\n",
+        "def f(x):\n",
+        "    return np.cos(n*np.arccos(x))\n",
+        "\n",
+        "nodes = np.cos((2*i-1)/(2*n)*np.pi)\n"
+      ]
+    },
+    {
+      "cell_type": "code",
+      "execution_count": null,
+      "id": "1bc26ebc-5c42-4e98-890c-83728a410390",
+      "metadata": {},
+      "outputs": [],
+      "source": []
+    }
+  ],
+  "metadata": {
+    "kernelspec": {
+      "display_name": "Python 3 (ipykernel)",
+      "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": "3.13.0"
+    }
+  },
+  "nbformat": 4,
+  "nbformat_minor": 5
+}