{
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  {
   "cell_type": "markdown",
   "id": "1972fa2c",
   "metadata": {},
   "source": [
    "Quiz 6 MCS 471, on Friday 30 September 2022, due 10:50am."
   ]
  },
  {
   "cell_type": "markdown",
   "id": "3717cd15",
   "metadata": {},
   "source": [
    "Consider $p = x^2 - 5 x + 2$.\n",
    "\n",
    "1. Compute the table of divided differences to interpolate\n",
    "   at $(-2,p(-2))$, $(-1,p(-1))$, $(+1,p(+1))$, and $p(+2,p(+2))$.\n",
    "\n",
    "2. Explain why the highest order divided difference $f[-2,-1,+1,+2]$ is zero."
   ]
  },
  {
   "cell_type": "markdown",
   "id": "e9d38a6f",
   "metadata": {},
   "source": [
    "# answer to question 1"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 1,
   "id": "5ed3b16e",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "p (generic function with 1 method)"
      ]
     },
     "execution_count": 1,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "p(x) = x^2 - 5*x + 2"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "2eddebed",
   "metadata": {},
   "source": [
    "We grab the code from lecture 15."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 2,
   "id": "c9d731d9",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "divdif"
      ]
     },
     "execution_count": 2,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "\"\"\"\n",
    "    divdif(x::Array{Float64,1},f::Array{Float64,1})\n",
    "\n",
    "Returns the vector of divided differences for the\n",
    "Newton form of the interpolating polynomial.\n",
    "\n",
    "On entry are x and f;\n",
    "x contains the abscisses, given as a column vector;\n",
    "f contains the ordinates, given as a column vector.\n",
    "\n",
    "On return are the divided differences.\n",
    "\"\"\"\n",
    "function divdif(x::Array{Float64,1},f::Array{Float64,1})\n",
    "    n = length(x)\n",
    "    d = deepcopy(f)\n",
    "    for i=2:n\n",
    "        for j=1:i-1\n",
    "            d[i] = (d[j] - d[i])/(x[j] - x[i])\n",
    "        end\n",
    "    end\n",
    "    return d\n",
    "end"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 3,
   "id": "61549cbf",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "4-element Vector{Float64}:\n",
       " -2.0\n",
       " -1.0\n",
       "  1.0\n",
       "  2.0"
      ]
     },
     "execution_count": 3,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "x = [-2.0, -1.0, 1.0, 2.0]"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 4,
   "id": "b26b38b4",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "4-element Vector{Float64}:\n",
       " 16.0\n",
       "  8.0\n",
       " -2.0\n",
       " -4.0"
      ]
     },
     "execution_count": 4,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "y = [p(z) for z in x]"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 5,
   "id": "bdea33c9",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "4-element Vector{Float64}:\n",
       " 16.0\n",
       " -8.0\n",
       "  1.0\n",
       " -0.0"
      ]
     },
     "execution_count": 5,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "d = divdif(x, y)"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "ac0dfa50",
   "metadata": {},
   "source": [
    "# answer to question 2"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "edfcc43e",
   "metadata": {},
   "source": [
    "The last divided difference is zero because $p(x)$ is a quadratic polynomial and we use four points to interpolate $p(x)$.  \n",
    "\n",
    "1. Because the interpolation problem has a unique solution, the interpolating polynomial must equal $p(x)$.\n",
    "\n",
    "2. Because $p(x)$ equals the interpolating polynomial, the interpolating polynomial is also a quadratic and its leading term $f[-2, -1, 1, 2](x+2)(x+1)(x-1)$ must be zero, otherwise the interpolating polynomial would be a cubic."
   ]
  }
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