{
 "cells": [
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "slide"
    }
   },
   "source": [
    "# NumPy Exercises\n",
    "\n",
    "Tamás Gál (tamas.gal@fau.de)\n",
    "\n",
    "The latest version of this notebook is available at [https://github.com/escape2020/school2021](https://github.com/escape2020/school2021)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 1,
   "metadata": {
    "slideshow": {
     "slide_type": "slide"
    }
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "Python version:  3.9.4 | packaged by conda-forge | (default, May 10 2021, 22:10:52) \n",
      "[Clang 11.1.0 ]\n",
      "NumPy version:   1.20.3\n",
      "Numba version:   0.53.1\n",
      "NumExpr version:   2.7.3\n"
     ]
    }
   ],
   "source": [
    "import numpy as np\n",
    "import numba as nb\n",
    "import numexpr as ne\n",
    "import sys\n",
    "\n",
    "print(f\"Python version:  {sys.version}\\n\"\n",
    "      f\"NumPy version:   {np.__version__}\\n\"\n",
    "      f\"Numba version:   {nb.__version__}\\n\"\n",
    "      f\"NumExpr version:   {ne.__version__}\")\n",
    "\n",
    "rng = np.random.default_rng(42)  # initialise our random number generator"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 2,
   "metadata": {
    "slideshow": {
     "slide_type": "fragment"
    }
   },
   "outputs": [],
   "source": [
    "%matplotlib inline\n",
    "import matplotlib.pyplot as plt\n",
    "plt.rcParams['figure.figsize'] = (16, 5)\n",
    "plt.rcParams['figure.dpi'] = 300"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "slide"
    }
   },
   "source": [
    "## Exercise 1: Extract the integer part of a random sample\n",
    "\n",
    "    rng.uniform(0, 10, 10)\n",
    "    \n",
    "e.g. `[23.5, 42.0, 500.3, 123.9] -> [23, 42, 500, 123]`"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "subslide"
    }
   },
   "source": [
    "### Solution:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 3,
   "metadata": {
    "slideshow": {
     "slide_type": "fragment"
    }
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "array([ 23.5,  42. , 500.3, 123.9])"
      ]
     },
     "execution_count": 3,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "a = np.array([23.5, 42.0, 500.3, 123.9])\n",
    "a"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 4,
   "metadata": {
    "slideshow": {
     "slide_type": "fragment"
    }
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "array([ 23.,  42., 500., 123.])"
      ]
     },
     "execution_count": 4,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "a - a%1"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 5,
   "metadata": {
    "slideshow": {
     "slide_type": "fragment"
    }
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "array([ 23.,  42., 500., 123.])"
      ]
     },
     "execution_count": 5,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "np.floor(a)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 6,
   "metadata": {
    "slideshow": {
     "slide_type": "fragment"
    }
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "array([ 23.,  41., 500., 123.])"
      ]
     },
     "execution_count": 6,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "np.ceil(a) - 1"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 7,
   "metadata": {
    "slideshow": {
     "slide_type": "fragment"
    }
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "array([ 23.,  42., 500., 123.])"
      ]
     },
     "execution_count": 7,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "np.trunc(a)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 8,
   "metadata": {
    "slideshow": {
     "slide_type": "fragment"
    }
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "array([ 23,  42, 500, 123])"
      ]
     },
     "execution_count": 8,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "a.astype(int)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "subslide"
    }
   },
   "source": [
    "### Further discussions"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 9,
   "metadata": {
    "slideshow": {
     "slide_type": "fragment"
    }
   },
   "outputs": [],
   "source": [
    "a = rng.uniform(0, 10, 10000)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 10,
   "metadata": {
    "slideshow": {
     "slide_type": "fragment"
    }
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "104 µs ± 3.75 µs per loop (mean ± std. dev. of 7 runs, 10000 loops each)\n"
     ]
    }
   ],
   "source": [
    "%timeit a - a%1"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 11,
   "metadata": {
    "slideshow": {
     "slide_type": "fragment"
    }
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "3.45 µs ± 22.8 ns per loop (mean ± std. dev. of 7 runs, 100000 loops each)\n"
     ]
    }
   ],
   "source": [
    "%timeit np.floor(a)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 12,
   "metadata": {
    "slideshow": {
     "slide_type": "fragment"
    }
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "5.78 µs ± 35.1 ns per loop (mean ± std. dev. of 7 runs, 100000 loops each)\n"
     ]
    }
   ],
   "source": [
    "%timeit np.ceil(a) - 1"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 13,
   "metadata": {
    "slideshow": {
     "slide_type": "fragment"
    }
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "3.48 µs ± 20.8 ns per loop (mean ± std. dev. of 7 runs, 100000 loops each)\n"
     ]
    }
   ],
   "source": [
    "%timeit np.trunc(a)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 14,
   "metadata": {
    "slideshow": {
     "slide_type": "fragment"
    }
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "1.26 µs ± 7.76 ns per loop (mean ± std. dev. of 7 runs, 1000000 loops each)\n"
     ]
    }
   ],
   "source": [
    "%timeit a.astype(int)  # the winner -> casting"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "slide"
    }
   },
   "source": [
    "## Exercise 2: Create a 5x5 matrix with 5's on its diagonal\n",
    "\n",
    "```\n",
    "5 0 0 0 0\n",
    "0 5 0 0 0\n",
    "0 0 5 0 0\n",
    "0 0 0 5 0\n",
    "0 0 0 0 5\n",
    "```"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "subslide"
    }
   },
   "source": [
    "### Solution: `np.eye()`"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 15,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "array([[1., 0., 0., 0., 0.],\n",
       "       [0., 1., 0., 0., 0.],\n",
       "       [0., 0., 1., 0., 0.],\n",
       "       [0., 0., 0., 1., 0.],\n",
       "       [0., 0., 0., 0., 1.]])"
      ]
     },
     "execution_count": 15,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "np.eye(5)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 16,
   "metadata": {
    "slideshow": {
     "slide_type": "fragment"
    }
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "array([[5., 0., 0., 0., 0.],\n",
       "       [0., 5., 0., 0., 0.],\n",
       "       [0., 0., 5., 0., 0.],\n",
       "       [0., 0., 0., 5., 0.],\n",
       "       [0., 0., 0., 0., 5.]])"
      ]
     },
     "execution_count": 16,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "np.eye(5) * 5"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 17,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "array([[1., 0., 0., 0., 0.],\n",
       "       [0., 1., 0., 0., 0.],\n",
       "       [0., 0., 1., 0., 0.],\n",
       "       [0., 0., 0., 1., 0.],\n",
       "       [0., 0., 0., 0., 1.]])"
      ]
     },
     "execution_count": 17,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "np.eye(5)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "subslide"
    }
   },
   "source": [
    "### Alternative solutions and further discussions"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 18,
   "metadata": {
    "slideshow": {
     "slide_type": "fragment"
    }
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "634 µs ± 111 µs per loop (mean ± std. dev. of 7 runs, 1000 loops each)\n"
     ]
    }
   ],
   "source": [
    "%%timeit\n",
    "a = np.eye(1000) * 5"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 19,
   "metadata": {
    "slideshow": {
     "slide_type": "fragment"
    }
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "589 µs ± 12 µs per loop (mean ± std. dev. of 7 runs, 1000 loops each)\n"
     ]
    }
   ],
   "source": [
    "%%timeit\n",
    "a = np.eye(1000)\n",
    "np.multiply(a, 5, out=a)  # avoid creating a copy "
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 20,
   "metadata": {
    "slideshow": {
     "slide_type": "fragment"
    }
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "312 µs ± 5.57 µs per loop (mean ± std. dev. of 7 runs, 1000 loops each)\n"
     ]
    }
   ],
   "source": [
    "%%timeit\n",
    "a = np.zeros((1000, 1000))\n",
    "a[np.diag_indices_from(a)] = 5"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 21,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "(array([0, 1, 2, 3, 4, 5, 6, 7, 8, 9]), array([0, 1, 2, 3, 4, 5, 6, 7, 8, 9]))"
      ]
     },
     "execution_count": 21,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "a = np.zeros((10, 10))\n",
    "np.diag_indices_from(a)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 22,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "308 µs ± 9.53 µs per loop (mean ± std. dev. of 7 runs, 1000 loops each)\n"
     ]
    }
   ],
   "source": [
    "%timeit np.diag(np.full(1000, 5))"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 23,
   "metadata": {
    "slideshow": {
     "slide_type": "fragment"
    }
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "303 µs ± 5.82 µs per loop (mean ± std. dev. of 7 runs, 1000 loops each)\n"
     ]
    }
   ],
   "source": [
    "%timeit np.diag(np.ones(1000) * 5)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "subslide"
    }
   },
   "source": [
    "### Numba?"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 24,
   "metadata": {
    "slideshow": {
     "slide_type": "fragment"
    }
   },
   "outputs": [],
   "source": [
    "@nb.njit\n",
    "def diag_nb(n, value):\n",
    "    return np.diag(np.ones(n) * value)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 25,
   "metadata": {
    "slideshow": {
     "slide_type": "fragment"
    }
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "298 µs ± 5.3 µs per loop (mean ± std. dev. of 7 runs, 1000 loops each)\n"
     ]
    }
   ],
   "source": [
    "%timeit diag_nb(1000, 5)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 26,
   "metadata": {
    "slideshow": {
     "slide_type": "fragment"
    }
   },
   "outputs": [],
   "source": [
    "@nb.njit\n",
    "def diag2_nb(n, value):\n",
    "    mat = np.zeros((n, n))\n",
    "    for i in range(n):\n",
    "        mat[i,i] = value\n",
    "    return mat"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 27,
   "metadata": {
    "slideshow": {
     "slide_type": "fragment"
    }
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "296 µs ± 4.59 µs per loop (mean ± std. dev. of 7 runs, 1000 loops each)\n"
     ]
    }
   ],
   "source": [
    "%timeit diag2_nb(1000, 5)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "slide"
    }
   },
   "source": [
    "## Exercise 3: Calculate `c`, with:\n",
    "\n",
    "    a = rng.random(1234567)\n",
    "    b = rng.random(1234567)\n",
    "  \n",
    "so that\n",
    "\n",
    "$$\n",
    "c_i = \\tan(a_i) \\cdot b_i - a_i^{b_i}\n",
    "$$\n",
    "\n",
    "for $i \\in [0, 1234566]$"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 28,
   "metadata": {
    "slideshow": {
     "slide_type": "fragment"
    }
   },
   "outputs": [],
   "source": [
    "a = rng.random(1234567)\n",
    "b = rng.random(1234567)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "subslide"
    }
   },
   "source": [
    "### Solution:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 29,
   "metadata": {
    "slideshow": {
     "slide_type": "fragment"
    }
   },
   "outputs": [],
   "source": [
    "def f(a, b):\n",
    "    return np.tan(a) * b - a**b"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 30,
   "metadata": {
    "slideshow": {
     "slide_type": "fragment"
    }
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "17 ms ± 218 µs per loop (mean ± std. dev. of 7 runs, 100 loops each)\n"
     ]
    }
   ],
   "source": [
    "%timeit f(a, b)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 31,
   "metadata": {
    "slideshow": {
     "slide_type": "fragment"
    }
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "4.1 ms ± 400 µs per loop (mean ± std. dev. of 7 runs, 100 loops each)\n"
     ]
    }
   ],
   "source": [
    "%timeit ne.evaluate(\"tan(a)*b - a**b\")"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "subslide"
    }
   },
   "source": [
    "### What about a Python loop?"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 32,
   "metadata": {
    "slideshow": {
     "slide_type": "subslide"
    }
   },
   "outputs": [],
   "source": [
    "def tanabab(a, b):\n",
    "    c = np.empty_like(a)\n",
    "    for i in range(len(a)):\n",
    "        c[i] = np.tan(a[i]) * b[i] - np.power(a[i], b[i])\n",
    "    return c"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 33,
   "metadata": {
    "slideshow": {
     "slide_type": "fragment"
    }
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "2.27 s ± 31.2 ms per loop (mean ± std. dev. of 7 runs, 1 loop each)\n"
     ]
    }
   ],
   "source": [
    "%timeit tanabab(a, b)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "subslide"
    }
   },
   "source": [
    "### Let's JIT it with `numba`!"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 34,
   "metadata": {
    "slideshow": {
     "slide_type": "fragment"
    }
   },
   "outputs": [],
   "source": [
    "@nb.jit\n",
    "def tanabab_nb(a, b):\n",
    "    c = np.empty_like(a)\n",
    "    for i in range(len(a)):\n",
    "        c[i] = np.tan(a[i]) * b[i] - np.power(a[i], b[i])\n",
    "    return c"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 35,
   "metadata": {
    "slideshow": {
     "slide_type": "subslide"
    }
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "CPU times: user 90.5 ms, sys: 2.38 ms, total: 92.9 ms\n",
      "Wall time: 92.2 ms\n"
     ]
    },
    {
     "data": {
      "text/plain": [
       "array([-0.01106144, -0.66248701, -0.32489978, ..., -0.21608449,\n",
       "       -0.13017545, -0.16956306])"
      ]
     },
     "execution_count": 35,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "%time tanabab_nb(a, b)  # first execution includes the compilation!"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 36,
   "metadata": {
    "slideshow": {
     "slide_type": "fragment"
    }
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "15.9 ms ± 328 µs per loop (mean ± std. dev. of 7 runs, 100 loops each)\n"
     ]
    }
   ],
   "source": [
    "%timeit tanabab_nb(a, b)  # the second is pure LLVM optimised code"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 37,
   "metadata": {
    "slideshow": {
     "slide_type": "fragment"
    }
   },
   "outputs": [],
   "source": [
    "@nb.jit\n",
    "def tanabab_nb_mutating_a(a, b):\n",
    "    for i in range(len(a)):\n",
    "        a[i] = np.tan(a[i]) * b[i] - np.power(a[i], b[i])"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 38,
   "metadata": {
    "slideshow": {
     "slide_type": "subslide"
    }
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "CPU times: user 56.2 ms, sys: 2.02 ms, total: 58.2 ms\n",
      "Wall time: 60.3 ms\n"
     ]
    }
   ],
   "source": [
    "%time tanabab_nb_mutating_a(a, b);  # first execution includes the compilation!"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 39,
   "metadata": {},
   "outputs": [],
   "source": [
    "a = rng.random(1234567)\n",
    "b = rng.random(1234567)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 40,
   "metadata": {
    "slideshow": {
     "slide_type": "fragment"
    }
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "7.67 ms ± 142 µs per loop (mean ± std. dev. of 7 runs, 100 loops each)\n"
     ]
    }
   ],
   "source": [
    "%timeit tanabab_nb_mutating_a(a, b);  # the second is pure LLVM optimised code"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "fragment"
    }
   },
   "source": [
    "Summary (your mileage may vary):\n",
    "- **~2200ms** (Python)\n",
    "- **~75ms** (reusing `a`, numba, inc. JIT comp.)\n",
    "- **~60ms** (numba, inc. JIT comp.)\n",
    "- **~17ms** (numpy)\n",
    "- **~16ms** (numba, JIT)\n",
    "- **~8ms** (reusing `a`, numba, JIT)\n",
    "- **~4ms** (numexpr)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "slide"
    }
   },
   "source": [
    "## Exercise 4: Given two arrays `a` anb `b`, check if they are (almost) equal\n",
    "\n",
    "    a = np.random.random(1234567)\n",
    "    b = a.copy()\n",
    "\n",
    "    b[-1] = 23  # artificially make them differ at the very end ;)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 41,
   "metadata": {
    "slideshow": {
     "slide_type": "skip"
    }
   },
   "outputs": [],
   "source": [
    "a = rng.random(1234567)\n",
    "b = a.copy()\n",
    "c = a.copy()\n",
    "\n",
    "b[-1] = 23  # make them differ at the very end ;)\n",
    "c[0] = 23   # make them differ at the beginning"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "subslide"
    }
   },
   "source": [
    "### Solution:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 42,
   "metadata": {
    "slideshow": {
     "slide_type": "fragment"
    }
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "4.18 ms ± 168 µs per loop (mean ± std. dev. of 7 runs, 100 loops each)\n"
     ]
    }
   ],
   "source": [
    "%timeit np.allclose(a, b)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 43,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "4.16 ms ± 158 µs per loop (mean ± std. dev. of 7 runs, 100 loops each)\n"
     ]
    }
   ],
   "source": [
    "%timeit np.allclose(a, c)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "subslide"
    }
   },
   "source": [
    "### Using numba?"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 44,
   "metadata": {
    "slideshow": {
     "slide_type": "fragment"
    }
   },
   "outputs": [],
   "source": [
    "@nb.jit\n",
    "def allclose(a, b, tol=0.0001):\n",
    "    for i in range(len(a)):\n",
    "        if np.abs(a[i] - b[i]) > tol:\n",
    "            return False \n",
    "    return True"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 45,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "811 µs ± 13.2 µs per loop (mean ± std. dev. of 7 runs, 1000 loops each)\n"
     ]
    }
   ],
   "source": [
    "%timeit allclose(a, b)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 46,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "265 ns ± 10.3 ns per loop (mean ± std. dev. of 7 runs, 1000000 loops each)\n"
     ]
    }
   ],
   "source": [
    "%timeit allclose(a, c)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 47,
   "metadata": {
    "slideshow": {
     "slide_type": "fragment"
    }
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "635 µs ± 64.5 µs per loop (mean ± std. dev. of 7 runs, 1000 loops each)\n"
     ]
    }
   ],
   "source": [
    "%timeit np.count_nonzero(a == b) == a.size"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "slide"
    }
   },
   "source": [
    "## Exercise 5: Create the following matrix\n",
    "\n",
    "    1 2 3 4 5 6\n",
    "    1 2 3 4 5 6\n",
    "    1 2 3 4 5 6\n",
    "    1 2 3 4 5 6\n",
    "    1 2 3 4 5 6\n"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "subslide"
    }
   },
   "source": [
    "### Solution:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 48,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "array([[1., 1., 1., 1., 1., 1.],\n",
       "       [1., 1., 1., 1., 1., 1.],\n",
       "       [1., 1., 1., 1., 1., 1.],\n",
       "       [1., 1., 1., 1., 1., 1.],\n",
       "       [1., 1., 1., 1., 1., 1.]])"
      ]
     },
     "execution_count": 48,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "np.ones((5, 6))"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 49,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "array([1, 2, 3, 4, 5, 6])"
      ]
     },
     "execution_count": 49,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "np.arange(1, 7)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 50,
   "metadata": {
    "slideshow": {
     "slide_type": "fragment"
    }
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "array([[1., 2., 3., 4., 5., 6.],\n",
       "       [1., 2., 3., 4., 5., 6.],\n",
       "       [1., 2., 3., 4., 5., 6.],\n",
       "       [1., 2., 3., 4., 5., 6.],\n",
       "       [1., 2., 3., 4., 5., 6.]])"
      ]
     },
     "execution_count": 50,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "np.ones((5, 6)) * np.arange(1, 7)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 51,
   "metadata": {
    "slideshow": {
     "slide_type": "fragment"
    }
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "array([[1., 2., 3., 4., 5., 6.],\n",
       "       [1., 2., 3., 4., 5., 6.],\n",
       "       [1., 2., 3., 4., 5., 6.],\n",
       "       [1., 2., 3., 4., 5., 6.],\n",
       "       [1., 2., 3., 4., 5., 6.]])"
      ]
     },
     "execution_count": 51,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "np.ones(5)[:, np.newaxis] * np.arange(1, 7)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "subslide"
    }
   },
   "source": [
    "### Alternative solutions and further discussions"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 52,
   "metadata": {
    "slideshow": {
     "slide_type": "fragment"
    }
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "5.66 µs ± 133 ns per loop (mean ± std. dev. of 7 runs, 100000 loops each)\n"
     ]
    }
   ],
   "source": [
    "%timeit np.ones((500, 6)) * np.arange(1, 7)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 53,
   "metadata": {
    "slideshow": {
     "slide_type": "fragment"
    }
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "5.62 µs ± 87.4 ns per loop (mean ± std. dev. of 7 runs, 100000 loops each)\n"
     ]
    }
   ],
   "source": [
    "%%timeit\n",
    "a = np.ones((500, 6))\n",
    "np.multiply(a, np.arange(1, 7), out=a)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 54,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "array([1., 1., 1., 1., 1.])"
      ]
     },
     "execution_count": 54,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "np.ones(5)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 55,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "(5,)"
      ]
     },
     "execution_count": 55,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "np.ones(5).shape"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 56,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "array([[1., 2., 3., 4., 5., 6.],\n",
       "       [1., 2., 3., 4., 5., 6.],\n",
       "       [1., 2., 3., 4., 5., 6.],\n",
       "       [1., 2., 3., 4., 5., 6.],\n",
       "       [1., 2., 3., 4., 5., 6.]])"
      ]
     },
     "execution_count": 56,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "np.ones(5)[:, np.newaxis] * np.arange(1, 7)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 57,
   "metadata": {
    "slideshow": {
     "slide_type": "fragment"
    }
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "6.78 µs ± 88.8 ns per loop (mean ± std. dev. of 7 runs, 100000 loops each)\n"
     ]
    }
   ],
   "source": [
    "%timeit np.ones(500)[:, np.newaxis] * np.arange(1, 7)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 58,
   "metadata": {
    "slideshow": {
     "slide_type": "fragment"
    }
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "2.13 µs ± 77.8 ns per loop (mean ± std. dev. of 7 runs, 100000 loops each)\n"
     ]
    }
   ],
   "source": [
    "%%timeit\n",
    "a = np.empty((500, 6))\n",
    "a[:] = np.arange(1, 7)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 59,
   "metadata": {
    "slideshow": {
     "slide_type": "subslide"
    }
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "array([1., 1., 1., 1., 1.])"
      ]
     },
     "execution_count": 59,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "np.ones(5)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 60,
   "metadata": {
    "slideshow": {
     "slide_type": "fragment"
    }
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "array([[1.],\n",
       "       [1.],\n",
       "       [1.],\n",
       "       [1.],\n",
       "       [1.]])"
      ]
     },
     "execution_count": 60,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "np.ones(5)[:, np.newaxis]  # adds a new dimension"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 61,
   "metadata": {
    "slideshow": {
     "slide_type": "fragment"
    }
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "(5, 1)"
      ]
     },
     "execution_count": 61,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "np.ones(5)[:, np.newaxis].shape"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 62,
   "metadata": {
    "slideshow": {
     "slide_type": "fragment"
    }
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "(6,)"
      ]
     },
     "execution_count": 62,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "np.arange(1, 7).shape"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 63,
   "metadata": {
    "slideshow": {
     "slide_type": "fragment"
    }
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "(5, 6)"
      ]
     },
     "execution_count": 63,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "# broadcasting will turn (5, 1) and (6,) into (5, 6)\n",
    "(np.ones(5)[:, np.newaxis] * np.arange(1, 7)).shape"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "subslide"
    }
   },
   "source": [
    "### Numba?"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 64,
   "metadata": {
    "slideshow": {
     "slide_type": "fragment"
    }
   },
   "outputs": [],
   "source": [
    "@nb.njit\n",
    "def grad_nb(n, m):\n",
    "    mat = np.empty((n, m))\n",
    "    for i in range(m):\n",
    "        for j in range(n):\n",
    "            mat[j,i] = i + 1\n",
    "    return mat"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 65,
   "metadata": {
    "slideshow": {
     "slide_type": "fragment"
    }
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "1.33 µs ± 15.5 ns per loop (mean ± std. dev. of 7 runs, 1000000 loops each)\n"
     ]
    }
   ],
   "source": [
    "%timeit grad_nb(500, 6)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 66,
   "metadata": {
    "slideshow": {
     "slide_type": "fragment"
    }
   },
   "outputs": [],
   "source": [
    "int_type = np.int32\n",
    "\n",
    "@nb.njit\n",
    "def grad_int_nb(n, m):\n",
    "    mat = np.empty((n, m), dtype=int_type)\n",
    "    for i in range(1, m):\n",
    "        for j in range(n):\n",
    "            mat[j,i] = i + 1\n",
    "    return mat"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 67,
   "metadata": {
    "slideshow": {
     "slide_type": "fragment"
    }
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "1.14 µs ± 6.06 ns per loop (mean ± std. dev. of 7 runs, 1000000 loops each)\n"
     ]
    }
   ],
   "source": [
    "%timeit grad_int_nb(500, 6)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "slide"
    }
   },
   "source": [
    "## Exercise 6: Roll two 6-sided dice 100 times and count each individual value"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "subslide"
    }
   },
   "source": [
    "### Solution:"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "fragment"
    }
   },
   "source": [
    "This is an ugly, hardcoded solution:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 68,
   "metadata": {
    "slideshow": {
     "slide_type": "fragment"
    }
   },
   "outputs": [],
   "source": [
    "def roll_dice(n):\n",
    "    dice_1 = rng.integers(1, 6, n)\n",
    "    dice_2 = rng.integers(1, 6, n)\n",
    "    sums = dice_1 + dice_2\n",
    "    return np.unique(sums, return_counts=True)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 69,
   "metadata": {
    "slideshow": {
     "slide_type": "fragment"
    }
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "(array([ 2,  3,  4,  5,  6,  7,  8,  9, 10]),\n",
       " array([ 6,  9,  9, 18, 18, 20,  7,  9,  4]))"
      ]
     },
     "execution_count": 69,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "roll_dice(100)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "slide"
    }
   },
   "source": [
    "## Exercise 7: Roll five 12-sided dice 123456 times and count each individual value"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "fragment"
    }
   },
   "source": [
    "If you did it right, you now only need to change 2 parameters of your previous code ;)\n",
    "\n",
    "If not, write an appropriate function.\n",
    "\n",
    "Create a histogram of the values!"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "subslide"
    }
   },
   "source": [
    "### Solution:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 70,
   "metadata": {
    "slideshow": {
     "slide_type": "fragment"
    }
   },
   "outputs": [],
   "source": [
    "def roll_dice(n_rolls, n_sides, n_die):\n",
    "    rolls = np.sum(rng.integers(1, n_sides+1, n_rolls*n_die)\n",
    "                   .reshape(n_die, n_rolls), axis=0)\n",
    "    return np.unique(rolls, return_counts=True)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 71,
   "metadata": {
    "slideshow": {
     "slide_type": "fragment"
    }
   },
   "outputs": [],
   "source": [
    "rolls = roll_dice(123456, 12, 5)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 72,
   "metadata": {
    "slideshow": {
     "slide_type": "fragment"
    }
   },
   "outputs": [
    {
     "data": {
      "image/png": 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\n",
      "text/plain": [
       "<Figure size 4800x1500 with 1 Axes>"
      ]
     },
     "metadata": {
      "needs_background": "light"
     },
     "output_type": "display_data"
    }
   ],
   "source": [
    "plt.hist(range(len(rolls[1])), bins=rolls[0], weights=rolls[1]);"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "slide"
    }
   },
   "source": [
    "## Exercise 8: Find the value closest to a given number in an array\n",
    "\n",
    "    a = rng.random(10)\n",
    "    target = 0.23"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "subslide"
    }
   },
   "source": [
    "### Solution:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 73,
   "metadata": {
    "slideshow": {
     "slide_type": "fragment"
    }
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "array([0.88422871, 0.48558074, 0.9052416 , 0.0752361 , 0.06916456,\n",
       "       0.05940674, 0.59183429, 0.76920539, 0.80313491, 0.91361757])"
      ]
     },
     "execution_count": 73,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "a = rng.random(10)\n",
    "target = 0.23\n",
    "a"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 74,
   "metadata": {
    "slideshow": {
     "slide_type": "fragment"
    }
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "0.07523610127287217"
      ]
     },
     "execution_count": 74,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "a[np.argmin(np.abs(a - target))]"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "subslide"
    }
   },
   "source": [
    "### Numba?"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 75,
   "metadata": {},
   "outputs": [],
   "source": [
    "a = rng.random(1000)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 76,
   "metadata": {
    "slideshow": {
     "slide_type": "fragment"
    }
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "2.23 µs ± 9.72 ns per loop (mean ± std. dev. of 7 runs, 100000 loops each)\n"
     ]
    }
   ],
   "source": [
    "%timeit a[np.argmin(np.abs(a - target))]"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 77,
   "metadata": {
    "slideshow": {
     "slide_type": "fragment"
    }
   },
   "outputs": [],
   "source": [
    "@nb.jit\n",
    "def find_closest(arr, target):\n",
    "    idx = 0\n",
    "    delta = np.nan\n",
    "    for i in range(len(arr)):\n",
    "        _delta = abs(arr[i] - target)\n",
    "        if _delta < delta:\n",
    "            delta = _delta\n",
    "            idx = i\n",
    "    return arr[idx]"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 78,
   "metadata": {
    "slideshow": {
     "slide_type": "fragment"
    }
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "1.43 µs ± 11.5 ns per loop (mean ± std. dev. of 7 runs, 1000000 loops each)\n"
     ]
    }
   ],
   "source": [
    "%timeit find_closest(a, 0.23)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "slide"
    }
   },
   "source": [
    "## Exercise 9: Create a random array with 10 elements and replace its largest value with 0\n",
    "\n",
    "## Bonus: Replace the second largest value with 0"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "subslide"
    }
   },
   "source": [
    "### Solution:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 79,
   "metadata": {
    "slideshow": {
     "slide_type": "fragment"
    }
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "array([25, 56, 23, 62,  9, 87, 54, 75, 75, 11])"
      ]
     },
     "execution_count": 79,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "a = rng.integers(0, 100, 10)\n",
    "a"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 80,
   "metadata": {
    "slideshow": {
     "slide_type": "fragment"
    }
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "array([25, 56, 23, 62,  9,  0, 54, 75, 75, 11])"
      ]
     },
     "execution_count": 80,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "np.argmax(a)  # gives the index of the maximum\n",
    "a[np.argmax(a)] = 0\n",
    "a"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "subslide"
    }
   },
   "source": [
    "### Solution for the bonus:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 81,
   "metadata": {
    "slideshow": {
     "slide_type": "fragment"
    }
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "array([25, 28, 96, 30, 39, 38, 69, 77, 95, 53])"
      ]
     },
     "execution_count": 81,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "a = rng.integers(0, 100, 10)\n",
    "a"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 82,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "array([25, 28, 96, 30, 39, 38, 69, 77,  0, 53])"
      ]
     },
     "execution_count": 82,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "idx = a.argsort()[-2]  # index of the second largest value\n",
    "a[idx] = 0\n",
    "a"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 83,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "array([25, 28, 96, 30, 39, 38, 69,  0,  0, 53])"
      ]
     },
     "execution_count": 83,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "a[np.argpartition(a, -2)[-2]] = 0\n",
    "a"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "This `np.argpartition` thing is a bit tricky, let's examine this..."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 84,
   "metadata": {},
   "outputs": [],
   "source": [
    "a = np.array([5, 4, 7, 9])"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "`np.partition` will \"partition\" the array, so that the it guarantees that the element at the specified index will sit in the correct position **and** every element to the left is less or equal and every element to the right is greater or equal to it (in undefined order)."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 85,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "array([4, 5, 7, 9])"
      ]
     },
     "execution_count": 85,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "np.partition(a, 1)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Here, you can see that if the array \"was\" sorted (it would be 4, 5, 7, 9), the `5` should sit at the position 1 (counting from 0). This can be much quicker than sorting the full array."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "In the solution above, we pass `-2` which means that the second last element should sit in place, so that we catch the two \"largest\" values (they are either the same or the last one is the largest)."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "`np.argpartition` will return the indices instead, which we can use to pick the position of the n-th largest element and set it to zero."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 86,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "array([1, 0, 2, 3])"
      ]
     },
     "execution_count": 86,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "np.argpartition(a, 1)  # the index of five (0) is sitting on position 1"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "subslide"
    }
   },
   "source": [
    "### Numba?"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 87,
   "metadata": {
    "slideshow": {
     "slide_type": "fragment"
    }
   },
   "outputs": [],
   "source": [
    "a = np.random.random(10000000)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 88,
   "metadata": {
    "slideshow": {
     "slide_type": "fragment"
    }
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "3.29 ms ± 49.2 µs per loop (mean ± std. dev. of 7 runs, 100 loops each)\n"
     ]
    }
   ],
   "source": [
    "%timeit a[np.argmax(a)] = 0"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 89,
   "metadata": {
    "slideshow": {
     "slide_type": "fragment"
    }
   },
   "outputs": [],
   "source": [
    "@nb.njit\n",
    "def zero_largest(arr):\n",
    "    idx = 0\n",
    "    largest = arr[0]\n",
    "    for i in range(len(arr)):\n",
    "        if arr[i] > largest:\n",
    "            largest = arr[i]\n",
    "            idx = i\n",
    "    arr[idx] = 0.0"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 90,
   "metadata": {
    "slideshow": {
     "slide_type": "fragment"
    }
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "42.6 ms ± 434 µs per loop (mean ± std. dev. of 7 runs, 10 loops each)\n"
     ]
    }
   ],
   "source": [
    "%timeit zero_largest(a)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "subslide"
    }
   },
   "source": [
    "### Using numba to solve the bonus exercise"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 91,
   "metadata": {
    "slideshow": {
     "slide_type": "fragment"
    }
   },
   "outputs": [],
   "source": [
    "a = rng.random(10000000)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 92,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "1.28 s ± 42.2 ms per loop (mean ± std. dev. of 7 runs, 1 loop each)\n"
     ]
    }
   ],
   "source": [
    "%%timeit\n",
    "idx = np.argsort(a)[-2]\n",
    "a[idx] = 0"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 93,
   "metadata": {
    "slideshow": {
     "slide_type": "fragment"
    }
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "105 ms ± 9.28 ms per loop (mean ± std. dev. of 7 runs, 10 loops each)\n"
     ]
    }
   ],
   "source": [
    "%%timeit \n",
    "a[np.argpartition(a, -2)[-2]] = 0"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 94,
   "metadata": {
    "slideshow": {
     "slide_type": "fragment"
    }
   },
   "outputs": [],
   "source": [
    "@nb.njit\n",
    "def zero_second_largest(arr):\n",
    "    idx_largest = 0\n",
    "    idx_second_largest = 0\n",
    "    largest = arr[0]\n",
    "    second_largest = 0\n",
    "    for i in range(len(arr)):\n",
    "        if arr[i] > largest:\n",
    "            second_largest, largest = largest, arr[i]\n",
    "            idx_second_largest, idx_largest = idx_largest, i\n",
    "    arr[idx_second_largest] = 0"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 95,
   "metadata": {
    "slideshow": {
     "slide_type": "fragment"
    }
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "40.9 ms ± 268 µs per loop (mean ± std. dev. of 7 runs, 10 loops each)\n"
     ]
    }
   ],
   "source": [
    "%timeit zero_second_largest(a)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "slide"
    }
   },
   "source": [
    "## Exercise 10: Create a checkerboard (8x8, 0s and 1s)\n",
    "\n",
    "    0 1 0 1 0 1 0 1\n",
    "    1 0 1 0 1 0 1 0\n",
    "    0 1 0 1 0 1 0 1\n",
    "    1 0 1 0 1 0 1 0\n",
    "    0 1 0 1 0 1 0 1\n",
    "    1 0 1 0 1 0 1 0\n",
    "    0 1 0 1 0 1 0 1\n",
    "    1 0 1 0 1 0 1 0"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "subslide"
    }
   },
   "source": [
    "### Solution:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 96,
   "metadata": {
    "slideshow": {
     "slide_type": "fragment"
    }
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "array([[0, 1, 0, 1, 0, 1, 0, 1],\n",
       "       [1, 0, 1, 0, 1, 0, 1, 0],\n",
       "       [0, 1, 0, 1, 0, 1, 0, 1],\n",
       "       [1, 0, 1, 0, 1, 0, 1, 0],\n",
       "       [0, 1, 0, 1, 0, 1, 0, 1],\n",
       "       [1, 0, 1, 0, 1, 0, 1, 0],\n",
       "       [0, 1, 0, 1, 0, 1, 0, 1],\n",
       "       [1, 0, 1, 0, 1, 0, 1, 0]], dtype=int32)"
      ]
     },
     "execution_count": 96,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "checkerboard = np.zeros((8, 8), dtype='i')\n",
    "checkerboard[::2, 1::2] = 1\n",
    "checkerboard[1::2, ::2] = 1\n",
    "checkerboard"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 97,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "<matplotlib.image.AxesImage at 0x12cf42a30>"
      ]
     },
     "execution_count": 97,
     "metadata": {},
     "output_type": "execute_result"
    },
    {
     "data": {
      "image/png": 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\n",
      "text/plain": [
       "<Figure size 4800x1500 with 1 Axes>"
      ]
     },
     "metadata": {
      "needs_background": "light"
     },
     "output_type": "display_data"
    }
   ],
   "source": [
    "plt.imshow(checkerboard)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "subslide"
    }
   },
   "source": [
    "###  Numba?"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 98,
   "metadata": {
    "slideshow": {
     "slide_type": "fragment"
    }
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "757 ns ± 6.99 ns per loop (mean ± std. dev. of 7 runs, 1000000 loops each)\n"
     ]
    }
   ],
   "source": [
    "%%timeit\n",
    "checkerboard = np.zeros((8, 8), dtype='i')\n",
    "checkerboard[::2, 1::2] = 1\n",
    "checkerboard[1::2, ::2] = 1\n",
    "checkerboard"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 99,
   "metadata": {},
   "outputs": [],
   "source": [
    "import numpy as np\n",
    "import numba as nb"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 100,
   "metadata": {
    "slideshow": {
     "slide_type": "fragment"
    }
   },
   "outputs": [],
   "source": [
    "@nb.njit\n",
    "def checkerboard(n):\n",
    "    board = np.zeros((n, n), dtype='i')\n",
    "    for i in range(n):\n",
    "        for j in range(n):\n",
    "            k = i + j*2\n",
    "            while k >= n:\n",
    "                k = k - n\n",
    "            board[k,i] = 1\n",
    "            \n",
    "    return board"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 101,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "array([[1, 0, 1, 0, 1, 0, 1, 0],\n",
       "       [0, 1, 0, 1, 0, 1, 0, 1],\n",
       "       [1, 0, 1, 0, 1, 0, 1, 0],\n",
       "       [0, 1, 0, 1, 0, 1, 0, 1],\n",
       "       [1, 0, 1, 0, 1, 0, 1, 0],\n",
       "       [0, 1, 0, 1, 0, 1, 0, 1],\n",
       "       [1, 0, 1, 0, 1, 0, 1, 0],\n",
       "       [0, 1, 0, 1, 0, 1, 0, 1]], dtype=int32)"
      ]
     },
     "execution_count": 101,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "checkerboard(8)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 102,
   "metadata": {
    "slideshow": {
     "slide_type": "fragment"
    }
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "336 ns ± 4.69 ns per loop (mean ± std. dev. of 7 runs, 1000000 loops each)\n"
     ]
    }
   ],
   "source": [
    "%timeit checkerboard(8)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "slide"
    }
   },
   "source": [
    "## Exercise 11: Calculate the diagonal of a dot product"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 103,
   "metadata": {
    "slideshow": {
     "slide_type": "fragment"
    }
   },
   "outputs": [],
   "source": [
    "A = rng.random((5, 5))\n",
    "B = rng.random((5, 5))"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "subslide"
    }
   },
   "source": [
    "### Solution:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 104,
   "metadata": {
    "slideshow": {
     "slide_type": "fragment"
    }
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "1.91 µs ± 24.9 ns per loop (mean ± std. dev. of 7 runs, 1000000 loops each)\n"
     ]
    }
   ],
   "source": [
    "%timeit np.diag(np.dot(A, B))"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 105,
   "metadata": {
    "slideshow": {
     "slide_type": "fragment"
    }
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "2.52 µs ± 44.2 ns per loop (mean ± std. dev. of 7 runs, 100000 loops each)\n"
     ]
    }
   ],
   "source": [
    "%timeit np.sum(A * B.T, axis=1)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 106,
   "metadata": {
    "slideshow": {
     "slide_type": "fragment"
    }
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "1.18 µs ± 10.4 ns per loop (mean ± std. dev. of 7 runs, 1000000 loops each)\n"
     ]
    }
   ],
   "source": [
    "%timeit np.einsum(\"ij,ji->i\", A, B)"
   ]
  }
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