{
 "cells": [
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "# Numpy for arrays! "
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "import numpy as np"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "a = np.arange(0, 5)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "a"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Calculate\n",
    "$$\n",
    "m_i = \\frac{1}{2} (a_i + a_{i+1})\n",
    "$$"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## loops "
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "m = np.zeros(len(a)-1)\n",
    "# write with loop "
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## arrays "
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "subslide"
    }
   },
   "source": [
    "### Can we avoid the loop?"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "fragment"
    }
   },
   "source": [
    "* `a[i]` covers the elements `[0, 1, 2, 3, 4]`\n",
    "* `a[i+1]` covers the elements `[1, 2, 3, 4, 5]`"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Think about shifting the arrays:\n",
    "```\n",
    "[0 1 2 3 4]\n",
    "  [0 1 2 3 4]\n",
    "-------------\n",
    "   1 3 5 7\n",
    "               * 0.5\n",
    "=====================\n",
    "  0.5 1.5 2.5 3.5  \n",
    "```    "
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "fragment"
    }
   },
   "source": [
    "Can we get these numbers by slicing?"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "```\n",
    "  [1 2 3 4]  = a[1:]\n",
    "  [0 1 2 3]  = a[:-1]\n",
    "-----------\n",
    "   1 3 5 7   = a[1:] + a[:-1]\n",
    "```   "
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Think about these equivalences between loop and array approach:"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "fragment"
    }
   },
   "source": [
    "* `a[i]` is like `a[:-1]`\n",
    "* `a[i+1]` is like `a[1:]`"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### Use the sliced arrays "
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "# write with sliced arrays (NO loops!)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "subslide"
    }
   },
   "source": [
    "Using sliced arrays is\n",
    "* more readable (more elegant)\n",
    "* faster \n",
    "\n",
    "**Prefered way to work with arrays**"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### Speed-up"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "How fast is it?"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "%%timeit\n",
    "for i in range(len(a)-1):\n",
    "    m[i] = 0.5*(a[i] + a[i+1])"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "%%timeit\n",
    "m = 0.5*(a[:-1] + a[1:])"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "m"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "# Exercise\n",
    "Use *numpy arrays* to calculate for a vector $\\mathbf{b}$ of length $N$\n",
    "\n",
    "$$\n",
    "y_i = \\frac{1}{2}(b_{i-1} + b_{i+1})\n",
    "$$\n",
    "\n",
    "and set $y_0 = y_{N-1} = 100$."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "b = np.arange(11)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "b"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "# add your solution here"
   ]
  }
 ],
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