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   "source": [
    "## Homework 2"
   ]
  },
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   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "This homework covers the material in Lessons 3 & 4.  It is due **Thurday, June 15**.  Please submit your **answers as a PDF**."
   ]
  },
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   "metadata": {},
   "source": [
    "1\\. Modify the LU method (without pivoting) to work **in-place**.  That is, it should not allocate any new memory for L or U, but instead overwrite A."
   ]
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   "source": [
    "def LU(A):\n",
    "    U = np.copy(A)\n",
    "    m, n = A.shape\n",
    "    L = np.eye(n)\n",
    "    for k in range(n-1):\n",
    "        for j in range(k+1,n):\n",
    "            L[j,k] = U[j,k]/U[k,k]\n",
    "            U[j,k:n] -= L[j,k] * U[k,k:n]\n",
    "    return L, U"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "2\\. Modify our LU method from class to add pivoting, as described in the lesson.  *Hint: the swap method below will be useful* "
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {
    "collapsed": true
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   "outputs": [],
   "source": [
    "def swap(a,b):\n",
    "    temp = np.copy(a)\n",
    "    a[:] = b\n",
    "    b[:] = temp"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "3\\. For each of the following sets of dimensions, either\n",
    "  - give the dimensions of the output of an operation on A and B, **or** \n",
    "  - answer *incompatible* if the dimensions are incompatible according to the [numpy rules of broadcasting](https://docs.scipy.org/doc/numpy-1.10.0/user/basics.broadcasting.html)."
   ]
  },
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   "cell_type": "markdown",
   "metadata": {
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   "source": [
    "    a.  A      (2d array):  3 x 3\n",
    "        B      (1d array):      1\n",
    "\n",
    "    b.  A      (2d array):      2 x 1\n",
    "        B      (3d array):  6 x 4 x 2\n",
    "\n",
    "    c.  A      (2d array):  5 x 4\n",
    "        B      (1d array):      4\n",
    "\n",
    "    d.  A      (3d array):  32 x 64 x 8\n",
    "        B      (3d array):   32 x 1 x 8\n",
    "\n",
    "    e.  A      (3d array):       64 x 1\n",
    "        B      (3d array):  32 x 1 x 16\n",
    "\n",
    "    f.  A      (3d array):  32 x 64 x 2\n",
    "        B      (3d array):   32 x 1 x 8"
   ]
  },
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   "cell_type": "markdown",
   "metadata": {
    "collapsed": true
   },
   "source": [
    "4\\. Write how this matrix would be stored in compressed row format:\n",
    "\n",
    "\\begin{pmatrix}\n",
    "  1 & & & & -2 & -3 \\\\\n",
    "  & 3 &  & & & -9 \\\\\n",
    "  &  &  & -7 & 4 & \\\\ \n",
    "  -1 & 2 & & 7 & & \\\\\n",
    "  -3 & & & 26 & &\n",
    " \\end{pmatrix}"
   ]
  }
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