Adds exercise for interp 2

Author: fruzsinaagocs

poly_interp2.ipynb

@@ -368,10 +368,36 @@
     "         continuous_update = False) "
    ]
   },
+  {
+   "cell_type": "markdown",
+   "id": "b38ed7a0",
+   "metadata": {},
+   "source": [
+    "```{admonition} Exercise\n",
+    ":class: danger\n",
+    "\n",
+    "This question is about estimating the error in interpolation with analytic formulae and by brute force.\n",
+    "\n",
+    "For a function $f$ analytic in $[-1, 1]$ and analytically continuable to the \"stadium\" $E_{\\rho}$, where it satisfies $f \\leq M$ for some $M$, [Trefethen's ATAP]() derives the following error bound on the Chebyshev interpolant (with $n+1$ nodes):\n",
+    "$$ ||f - p_n|| \\leq \\frac{4M\\rho^{-n}}{\\rho - 1}. $$\n",
+    "\n",
+    "The stadium $E_{\\rho}$ is defined as follows: if we have a circle of radius $\\rho$ in the complex $z$-plane, then make the subsitution $x = (z + z^{-1})/2$, then we get the stadium $E_{\\rho}$ (known as a Bernstein ellipse) in the complex $x$-plane.\n",
+    "\n",
+    "- a) Show that we do indeed get an ellipse in the complex $x$-plane. What are its semi-major and semi-minor axes?\n",
+    "- b) Take the function $f(x) = \\frac{1}{1 + (x/x_0)^2}$. What $\\rho$ parameter does the Bernstein ellipse going through its poles correspond to? \n",
+    "- c) Plot the expected convergence rate (ommitting anything that doesn't depend on $n$) of barycentric interpolation with respect to $n$, and the actual convergence rate, for $x_0 = 2$. What type of convergence is this? Comment on the plot.\n",
+    "\n",
+    "_Hints_:\n",
+    "* Use an $L_{\\infty}$ norm to measure the error (i.e. take the maximum over dense, equally spaced \"target\" points distributed on the interval).\n",
+    "* Use a logarithmic $y$ and linear $x$ axis (and label them!).\n",
+    "* Label the curves and add a legend.\n",
+    "```"
+   ]
+  },
   {
    "cell_type": "code",
    "execution_count": null,
-   "id": "76e9cef1",
+   "id": "4fce701e",
    "metadata": {},
    "outputs": [],
    "source": []