{
 "cells": [
  {
   "cell_type": "markdown",
   "id": "bf20212a-80f7-4066-a70e-58dea3f35a38",
   "metadata": {},
   "source": [
    "Submit your answers to gradescope before, or at 3:40pm."
   ]
  },
  {
   "cell_type": "markdown",
   "id": "e8cfd805-36b2-41f9-a7c6-64d881bdec40",
   "metadata": {},
   "source": [
    "# Question 1"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "03aceab3-b08f-47cb-82c0-b39d11a5bee0",
   "metadata": {},
   "source": [
    "Let $N = \\exp(\\sqrt{5})$.\n",
    "\n",
    "1. Compute a rational approximation for $N$, accurate to 32 decimal places.\n",
    "\n",
    "2. Verify the accuracy of your rational approximation by computing the relative error."
   ]
  },
  {
   "cell_type": "markdown",
   "id": "a483fc53-d094-4713-a7ce-fdfada77588b",
   "metadata": {},
   "source": [
    "# Question 2"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "fcc284b9-b3b2-4adc-8166-636a1c33f4f5",
   "metadata": {},
   "source": [
    "Consider the finite field with 7 elements.\n",
    "\n",
    "Show that $p = x^3 + 3 x^2 + 6$ is irreducible over this field.\n",
    "\n",
    "Declare $\\alpha$ as a formal root of $p$.\n",
    "What is $\\alpha^7$ in this finite field extension?"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "e0a6bc5d-b7ea-493f-bed7-9920fb512333",
   "metadata": {},
   "source": [
    "# Question 3"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "26124e9f-4afa-4105-9a0b-0c24765934eb",
   "metadata": {},
   "source": [
    "Compute a numeric factorization of $p = x^4 + 3 x^2 + x - 1$.\n",
    "\n",
    "Expand the factorization and compare the coefficients of the expanded form\n",
    "with the coefficients of $p$.  \n",
    "\n",
    "What is the largest error on the coefficients?"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "3f81a6c5-9c1c-4035-b199-22a29bc9875a",
   "metadata": {},
   "source": [
    "# Question 4"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "bf572878-6f86-47fd-ae4f-53cec12d625f",
   "metadata": {},
   "source": [
    "Draw the binary expression tree defined by the fast callable object for\n",
    "\n",
    "$$\n",
    "   (\\cos(b) - \\sin(c^2) + 2)/(b c - 2 a).\n",
    "$$"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "60c1573b-36ca-4211-a289-0abd8390a45b",
   "metadata": {},
   "source": [
    "# Question 5"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "139dd403-48c5-45f9-926a-c62830122fe6",
   "metadata": {},
   "source": [
    "What does the preparser in SageMath do?\n",
    "\n",
    "Give an application of `preparse(x)`, with a good example \n",
    "for its argument `x`."
   ]
  },
  {
   "cell_type": "markdown",
   "id": "36574236-42ec-4633-a58a-b031d9eb2f8a",
   "metadata": {},
   "source": [
    "# Question 6"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "194efe90-95c8-4221-a113-48adf8d36076",
   "metadata": {},
   "source": [
    "The\n",
    "``f = lambda n: float(sum([(k/n)*exp(k/n) for k in range(1,n)])/n)``\n",
    "\n",
    "computes the right hand side of\n",
    "$$\n",
    "    \\qquad \\int_0^1 x e^x dx \\approx \\frac{1}{n} \n",
    "    \\sum_{k=1}^{n-1} \\left( \\frac{k}{n} \\right) \\exp\\left( \\frac{k}{n} \\right).\n",
    "$$\n",
    "\n",
    "1. Time the execution of `f` for $n = 10000$.\n",
    "      \n",
    "   Explain why `f` is inefficient.\n",
    "\n",
    "2. Apply vectorization to improve the efficiency.  Verify the correctness.\n",
    "\n",
    "   Time the execution of the vectorized function for $n = 10000$, compare with timings of `f`."
   ]
  },
  {
   "cell_type": "markdown",
   "id": "6e022353-3cab-4db2-9db7-82ec1c9e34d3",
   "metadata": {},
   "source": [
    "# Question 7"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "9ca487b5-1e50-4fd2-96fc-bc533162709c",
   "metadata": {},
   "source": [
    "Let $q = (x^6 - 3 x^4 + 2 x + 1)/(x^6 + 2 x^5 - 3 x^4 + 3 x + 7)$.\n",
    "\n",
    "Compute a partial fraction decomposition of $q$ \n",
    "over the complex numbers.\n",
    "\n",
    "How many operations does it takes to evaluate $q$ compared\n",
    "the number of operations to evaluate its partial fraction decomposition?"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "4812edc4-e8c4-49d1-94ee-18bbd7c0c103",
   "metadata": {},
   "source": [
    "# Question 8"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "eea59726-9a68-4319-b3bf-c0301676f7ff",
   "metadata": {},
   "source": [
    "Transform\n",
    "      $\\displaystyle (x-y)^4 + \\frac{(x+y)^3}{(x-y)^2}$\n",
    "      into $\\displaystyle \\frac{(x-y)^6 + (x+y)^3}{(x-y)^2}$,\n",
    "      without retyping."
   ]
  },
  {
   "cell_type": "markdown",
   "id": "08631d7c-dc8f-4ff3-874b-50a4827bf323",
   "metadata": {},
   "source": [
    "# Question 9"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "ce87a84b-5368-4300-863b-26998c55947d",
   "metadata": {},
   "source": [
    "Are the expressions $\\displaystyle p = \\frac{x^2 + 7 x - 8}{x - 1}$ and $q = x + 8$ the same?\n",
    "\n",
    "Justify your answer by appropriate *symbolic* computations.\n",
    "\n",
    "Demonstrate the application of a *numerical* probability-one equality test."
   ]
  }
 ],
 "metadata": {
  "kernelspec": {
   "display_name": "SageMath 10.3",
   "language": "sage",
   "name": "sagemath"
  },
  "language_info": {
   "codemirror_mode": {
    "name": "ipython",
    "version": 3
   },
   "file_extension": ".py",
   "mimetype": "text/x-python",
   "name": "python",
   "nbconvert_exporter": "python",
   "pygments_lexer": "ipython3",
   "version": "3.10.14"
  }
 },
 "nbformat": 4,
 "nbformat_minor": 5
}