Moved the exercises for lecture 4 into the lecture notes.

Author: Christian Jacobs

notebook/Lecture-4-Introduction-to-programming-for-geoscientists.ipynb

@@ -1,18 +1,18 @@
 {
  "metadata": {
-  "name": ""
+  "name": "Lecture-4-Introduction-to-programming-for-geoscientists"
  },
  "nbformat": 3,
  "nbformat_minor": 0,
  "worksheets": [
   {
    "cells": [
     {
-     "cell_type": "heading",
-     "level": 1,
+     "cell_type": "markdown",
      "metadata": {},
      "source": [
-      "Introduction to programming for Geoscientists (through Python)"
+      "#Introduction to programming for Geoscientists (through Python)\n",
+      "###[Gerard Gorman](http://www.imperial.ac.uk/people/g.gorman), [Christian Jacobs](http://www.imperial.ac.uk/people/c.jacobs10)"
      ]
     },
     {
@@ -21,15 +21,18 @@
      "source": [
       "#Lecture 4: Array computing and curve plotting\n",
       "\n",
-      "[Gerard J. Gorman](http://www.imperial.ac.uk/people/g.gorman) <g.gorman@imperial.ac.uk>\n",
+      "Learning objectives: \n",
       "\n",
-      "Learning objectives: You will learn how to compute using arrays, *i.e.* vectorise code, and generate 2D graphs."
+      "* Learn how to compute using arrays, *i.e.* vectorise code.\n",
+      "* Learn how to generate 2D graphs."
      ]
     },
     {
      "cell_type": "markdown",
      "metadata": {},
      "source": [
+      "##Vectors and arrays\n",
+      "\n",
       "You have known **vectors** since high school mathematics, *e.g.*, point $(x,y)$ in the plane, point $(x,y,z)$ in space. In general, we can describe a vector $v$ as an $n$-tuple of numbers: $v=(v_0,\\ldots,v_{n-1})$. One way to store vectors in Python is by using *lists*: $v_i$ is stored as *v[i]*."
      ]
     },
@@ -348,7 +351,79 @@
      "cell_type": "markdown",
      "metadata": {},
      "source": [
-      "###Generalized array indexing\n",
+      "## <span style=\"color:blue\">Exercise 1: Fill lists and arrays with function values</span>\n",
+      "A function with many applications in science is defined as:</br></br>\n",
+      "$h(x) = \\frac{1}{\\sqrt{2\\pi}}\\exp(-0.5x^2)$</br></br>\n",
+      "\n",
+      "* Fill two lists *xlist* and *hlist* with *x* and *h(x)* values for uniformly spaced *x* coordinates in [\u22124, 4]. You may adapt the first example in the lecture 4 notes.\n",
+      "\n",
+      "* Fill two arrays *x* and *y* with *x* and *h(x)* values, respectively, where *h(x)* is defined above. Let the *x* values be uniformly spaced in [\u22124, 4]. Use list comprehensions to create the *x* and *y* arrays.\n",
+      "\n",
+      "* Vectorize the code by creating the *x* values using the *linspace* function and by evaluating *h(x)* for an array argument."
+     ]
+    },
+    {
+     "cell_type": "code",
+     "collapsed": false,
+     "input": [],
+     "language": "python",
+     "metadata": {},
+     "outputs": []
+    },
+    {
+     "cell_type": "markdown",
+     "metadata": {},
+     "source": [
+      "## <span style=\"color:blue\">Exercise 2: Apply a function to a vector</span>\n",
+      "Given a vector $v = (2, 3, \u22121)$ and a function $f(x) = x^3 + xe^x + 1$, apply $f$ to each element in $v$. Then calculate $f(v)$ as $v^3 + ve^v + 1$ using vector computing rules. Show that the two results are equal."
+     ]
+    },
+    {
+     "cell_type": "code",
+     "collapsed": false,
+     "input": [],
+     "language": "python",
+     "metadata": {},
+     "outputs": []
+    },
+    {
+     "cell_type": "markdown",
+     "metadata": {},
+     "source": [
+      "## <span style=\"color:blue\">Exercise 3: Simulate by hand a vectorized expression</span>\n",
+      "Suppose *x* and *t* are two arrays of the same length, entering a vectorized expression:"
+     ]
+    },
+    {
+     "cell_type": "code",
+     "collapsed": false,
+     "input": [
+      "y = cos(sin(x)) + exp(1/t)"
+     ],
+     "language": "python",
+     "metadata": {},
+     "outputs": []
+    },
+    {
+     "cell_type": "markdown",
+     "metadata": {},
+     "source": [
+      "If *x* holds two elements, 0 and 2, and *t* holds the elements 1 and 1.5, calculate by hand (using a calculator) the *y* array. Thereafter, write a program that mimics the series of computations you did by hand (use explicit loops, but at the end you can use NumPy functionality to check the results)."
+     ]
+    },
+    {
+     "cell_type": "code",
+     "collapsed": false,
+     "input": [],
+     "language": "python",
+     "metadata": {},
+     "outputs": []
+    },
+    {
+     "cell_type": "markdown",
+     "metadata": {},
+     "source": [
+      "##Generalised array indexing\n",
       "We can select a slice of an array using *a[start:stop:inc]*, where the slice *start:stop:inc* implies a set of indices starting from *start*, up to *stop* in increments of *inc*. In fact, any integer list or array can be used to indicate a set of indices:"
      ]
     },
@@ -458,6 +533,22 @@
      ],
      "prompt_number": 16
     },
+    {
+     "cell_type": "markdown",
+     "metadata": {},
+     "source": [
+      "## <span style=\"color:blue\">Exercise 4: Demonstrate array slicing</span>\n",
+      "Create an array *w* with values 0, 0.1, 0.2, ..., 3. Write out *w[:]*, *w[:-2]*, *w[::5]*, *w[2:-2:6]*. Convince yourself in each case that you understand which elements of the array are printed."
+     ]
+    },
+    {
+     "cell_type": "code",
+     "collapsed": false,
+     "input": [],
+     "language": "python",
+     "metadata": {},
+     "outputs": []
+    },
     {
      "cell_type": "markdown",
      "metadata": {},
@@ -570,6 +661,43 @@
      "cell_type": "markdown",
      "metadata": {},
      "source": [
+      "## <span style=\"color:blue\">Exercise 5: Plot a formula</span>\n",
+      "* Make a plot of the function $y(t) = v_0t \u2212 0.5gt^2$ for $v_0 = 10$, $g = 9.81$, and $t \\in [0, 2v_0/g]$. The label on the *x* axis should be 'time (s)' and the label on the *y* axis should be 'height (m)'.\n",
+      "* Extend the program such that the minimum and maximum *x* and *y* values are computed, and use the extreme values to specify the extent of the *x* and *y* axes. Add some space above the heighest curve."
+     ]
+    },
+    {
+     "cell_type": "code",
+     "collapsed": false,
+     "input": [],
+     "language": "python",
+     "metadata": {},
+     "outputs": []
+    },
+    {
+     "cell_type": "markdown",
+     "metadata": {},
+     "source": [
+      "## <span style=\"color:blue\">Exercise 6: Plot another formula</span>\n",
+      "The function</br></br>\n",
+      "$f(x, t) = \\exp(-(x - 3t)^2)\\sin(3\\pi(x - t))$\n",
+      "</br></br>\n",
+      "describes, for a fixed value of *t*, a wave localized in space. Make a program that visualizes this function as a function of *x* on the interval [\u22124, 4] when *t* = 0."
+     ]
+    },
+    {
+     "cell_type": "code",
+     "collapsed": false,
+     "input": [],
+     "language": "python",
+     "metadata": {},
+     "outputs": []
+    },
+    {
+     "cell_type": "markdown",
+     "metadata": {},
+     "source": [
+      "##Multiple curves in one plot\n",
       "We can also plot several curves in one plot:"
      ]
     },
@@ -644,6 +772,22 @@
       "For further examples check out the [PyLab website](http://scipy.org/PyLab)."
      ]
     },
+    {
+     "cell_type": "markdown",
+     "metadata": {},
+     "source": [
+      "## <span style=\"color:blue\">Exercise 7: Plot a formula for several parameters</span>\n",
+      "Make a program that reads a set of $v_0$ values using raw_input and plots the corresponding curves $y(t) = v_0t \u2212 0.5gt^2$ in the same figure (set $g = 9.81$). Let $t \\in [0, 2v_0/g$] for each curve, which implies that you need a different vector of $t$ coordinates for each curve."
+     ]
+    },
+    {
+     "cell_type": "code",
+     "collapsed": false,
+     "input": [],
+     "language": "python",
+     "metadata": {},
+     "outputs": []
+    },
     {
      "cell_type": "markdown",
      "metadata": {},
@@ -974,6 +1118,44 @@
       }
      ],
      "prompt_number": 30
+    },
+    {
+     "cell_type": "markdown",
+     "metadata": {},
+     "source": [
+      "## <span style=\"color:blue\">Exercise 8: Implement matrix-vector multiplication</span>\n",
+      "A matrix $\\mathbf{A}$ and a vector $\\mathbf{b}$, represented in Python as a 2D array and a 1D array respectively, are given by:\n",
+      "\n",
+      "$$\n",
+      "\\mathbf{A} = \\left\\lbrack\\begin{array}{ccc}\n",
+      "0 & 12 & -1\\cr\n",
+      "-1 & -1 & -1\\cr\n",
+      "11 & 5 & 5\n",
+      "\\end{array}\\right\\rbrack\n",
+      "$$\n",
+      "\n",
+      "$$\n",
+      "\\mathbf{b} = \\left\\lbrack\\begin{array}{c}\n",
+      "-2\\cr\n",
+      "1\\cr\n",
+      "7\n",
+      "\\end{array}\\right\\rbrack\n",
+      "$$\n",
+      "\n",
+      "Multiplying a matrix by a vector results in another vector $\\mathbf{c}$, whose components are defined by the general rule\n",
+      "\n",
+      "$$\\mathbf{c}_i = \\sum_j\\mathbf{A}_{i,j}\\mathbf{b}_j$$\n",
+      "\n",
+      "Define $\\mathbf{A}$ and $\\mathbf{b}$ as NumPy arrays, and multiply them together using the above rule."
+     ]
+    },
+    {
+     "cell_type": "code",
+     "collapsed": false,
+     "input": [],
+     "language": "python",
+     "metadata": {},
+     "outputs": []
     }
    ],
    "metadata": {}