{
 "cells": [
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "# Introduction to programming for Geoscientists (through Python)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "# Lecture 3 solutions\n",
    "## Gerard J. Gorman (g.gorman@imperial.ac.uk) http://www.imperial.ac.uk/people/g.gorman"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## <span style=\"color:blue\">Exercise 3.1: Make a table of function values</span>\n",
    "\n",
    "Write a program that prints a table with $t$ values in the first column and the corresponding $y(t) = v_0 t − 0.5gt^2$ values in the second column.\n",
    "Use $n$ uniformly spaced $t$ values throughout the interval [0, $2v_0/g$]. Set $v0 = 1$, $g = 9.81$, and $n = 11$."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 2,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "0.000000, 0.000000\n",
      "0.020387, 0.018349\n",
      "0.040775, 0.032620\n",
      "0.061162, 0.042813\n",
      "0.081549, 0.048930\n",
      "0.101937, 0.050968\n",
      "0.122324, 0.048930\n",
      "0.142712, 0.042813\n",
      "0.163099, 0.032620\n",
      "0.183486, 0.018349\n",
      "0.203874, 0.000000\n"
     ]
    }
   ],
   "source": [
    "# Make a table of function values\n",
    "\n",
    "v0 = 1.0\n",
    "g = 9.81\n",
    "n = 11\n",
    "\n",
    "# The step size between each uniformly spaced value. \n",
    "# The end-points of the interval are included.\n",
    "step = (2*v0/g)/(n-1) \n",
    "\n",
    "for i in range(0, n):\n",
    "    t = step*i\n",
    "    y = v0*t - 0.5*g*t**2\n",
    "    print(\"%f, %f\" % (t, y))"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Modify the program so that the $t$ and $y$ values are stored in two lists *t* and *y*. Thereafter, transverse the lists with a *for* loop and write out a nicely formatted table of *t* and *y* values using a *zip* construction."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 7,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "0.000000, 0.000000\n",
      "0.020387, 0.018349\n",
      "0.040775, 0.032620\n",
      "0.061162, 0.042813\n",
      "0.081549, 0.048930\n",
      "0.101937, 0.050968\n",
      "0.122324, 0.048930\n",
      "0.142712, 0.042813\n",
      "0.163099, 0.032620\n",
      "0.183486, 0.018349\n",
      "0.203874, 0.000000\n"
     ]
    }
   ],
   "source": [
    "# Use list comprehension to build the lists\n",
    "t = [step*i for i in range(n)]\n",
    "y = [v0*ti - 0.5*g*ti**2 for ti in t]\n",
    "\n",
    "for (T,Y) in zip(t,y):\n",
    "    print(\"%f, %f\" % (T, Y))"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## <span style=\"color:blue\">Exercise 3.2: Implement a Gaussian function</span>\n",
    "\n",
    "Make a Python function *gauss*( *x*, *m*=0, *s*=1) for computing the Gaussian function \n",
    "$$f(x)=\\frac{1}{\\sqrt{2\\pi}s}\\exp\\left(-\\frac{1}{2} \\left(\\frac{x-m}{s}\\right)^2\\right)$$\n",
    "Call the function and print out the result for x equal to −5, −4.9, −4.8, ..., 4.8, 4.9, 5, using default values for *m* and *s*."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 7,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "For x = -5.000000, gaussian(x) = 0.000001\n",
      "For x = -4.900000, gaussian(x) = 0.000002\n",
      "For x = -4.800000, gaussian(x) = 0.000004\n",
      "For x = -4.700000, gaussian(x) = 0.000006\n",
      "For x = -4.600000, gaussian(x) = 0.000010\n",
      "For x = -4.500000, gaussian(x) = 0.000016\n",
      "For x = -4.400000, gaussian(x) = 0.000025\n",
      "For x = -4.300000, gaussian(x) = 0.000039\n",
      "For x = -4.200000, gaussian(x) = 0.000059\n",
      "For x = -4.100000, gaussian(x) = 0.000089\n",
      "For x = -4.000000, gaussian(x) = 0.000134\n",
      "For x = -3.900000, gaussian(x) = 0.000199\n",
      "For x = -3.800000, gaussian(x) = 0.000292\n",
      "For x = -3.700000, gaussian(x) = 0.000425\n",
      "For x = -3.600000, gaussian(x) = 0.000612\n",
      "For x = -3.500000, gaussian(x) = 0.000873\n",
      "For x = -3.400000, gaussian(x) = 0.001232\n",
      "For x = -3.300000, gaussian(x) = 0.001723\n",
      "For x = -3.200000, gaussian(x) = 0.002384\n",
      "For x = -3.100000, gaussian(x) = 0.003267\n",
      "For x = -3.000000, gaussian(x) = 0.004432\n",
      "For x = -2.900000, gaussian(x) = 0.005953\n",
      "For x = -2.800000, gaussian(x) = 0.007915\n",
      "For x = -2.700000, gaussian(x) = 0.010421\n",
      "For x = -2.600000, gaussian(x) = 0.013583\n",
      "For x = -2.500000, gaussian(x) = 0.017528\n",
      "For x = -2.400000, gaussian(x) = 0.022395\n",
      "For x = -2.300000, gaussian(x) = 0.028327\n",
      "For x = -2.200000, gaussian(x) = 0.035475\n",
      "For x = -2.100000, gaussian(x) = 0.043984\n",
      "For x = -2.000000, gaussian(x) = 0.053991\n",
      "For x = -1.900000, gaussian(x) = 0.065616\n",
      "For x = -1.800000, gaussian(x) = 0.078950\n",
      "For x = -1.700000, gaussian(x) = 0.094049\n",
      "For x = -1.600000, gaussian(x) = 0.110921\n",
      "For x = -1.500000, gaussian(x) = 0.129518\n",
      "For x = -1.400000, gaussian(x) = 0.149727\n",
      "For x = -1.300000, gaussian(x) = 0.171369\n",
      "For x = -1.200000, gaussian(x) = 0.194186\n",
      "For x = -1.100000, gaussian(x) = 0.217852\n",
      "For x = -1.000000, gaussian(x) = 0.241971\n",
      "For x = -0.900000, gaussian(x) = 0.266085\n",
      "For x = -0.800000, gaussian(x) = 0.289692\n",
      "For x = -0.700000, gaussian(x) = 0.312254\n",
      "For x = -0.600000, gaussian(x) = 0.333225\n",
      "For x = -0.500000, gaussian(x) = 0.352065\n",
      "For x = -0.400000, gaussian(x) = 0.368270\n",
      "For x = -0.300000, gaussian(x) = 0.381388\n",
      "For x = -0.200000, gaussian(x) = 0.391043\n",
      "For x = -0.100000, gaussian(x) = 0.396953\n",
      "For x = -0.000000, gaussian(x) = 0.398942\n",
      "For x = 0.100000, gaussian(x) = 0.396953\n",
      "For x = 0.200000, gaussian(x) = 0.391043\n",
      "For x = 0.300000, gaussian(x) = 0.381388\n",
      "For x = 0.400000, gaussian(x) = 0.368270\n",
      "For x = 0.500000, gaussian(x) = 0.352065\n",
      "For x = 0.600000, gaussian(x) = 0.333225\n",
      "For x = 0.700000, gaussian(x) = 0.312254\n",
      "For x = 0.800000, gaussian(x) = 0.289692\n",
      "For x = 0.900000, gaussian(x) = 0.266085\n",
      "For x = 1.000000, gaussian(x) = 0.241971\n",
      "For x = 1.100000, gaussian(x) = 0.217852\n",
      "For x = 1.200000, gaussian(x) = 0.194186\n",
      "For x = 1.300000, gaussian(x) = 0.171369\n",
      "For x = 1.400000, gaussian(x) = 0.149727\n",
      "For x = 1.500000, gaussian(x) = 0.129518\n",
      "For x = 1.600000, gaussian(x) = 0.110921\n",
      "For x = 1.700000, gaussian(x) = 0.094049\n",
      "For x = 1.800000, gaussian(x) = 0.078950\n",
      "For x = 1.900000, gaussian(x) = 0.065616\n",
      "For x = 2.000000, gaussian(x) = 0.053991\n",
      "For x = 2.100000, gaussian(x) = 0.043984\n",
      "For x = 2.200000, gaussian(x) = 0.035475\n",
      "For x = 2.300000, gaussian(x) = 0.028327\n",
      "For x = 2.400000, gaussian(x) = 0.022395\n",
      "For x = 2.500000, gaussian(x) = 0.017528\n",
      "For x = 2.600000, gaussian(x) = 0.013583\n",
      "For x = 2.700000, gaussian(x) = 0.010421\n",
      "For x = 2.800000, gaussian(x) = 0.007915\n",
      "For x = 2.900000, gaussian(x) = 0.005953\n",
      "For x = 3.000000, gaussian(x) = 0.004432\n",
      "For x = 3.100000, gaussian(x) = 0.003267\n",
      "For x = 3.200000, gaussian(x) = 0.002384\n",
      "For x = 3.300000, gaussian(x) = 0.001723\n",
      "For x = 3.400000, gaussian(x) = 0.001232\n",
      "For x = 3.500000, gaussian(x) = 0.000873\n",
      "For x = 3.600000, gaussian(x) = 0.000612\n",
      "For x = 3.700000, gaussian(x) = 0.000425\n",
      "For x = 3.800000, gaussian(x) = 0.000292\n",
      "For x = 3.900000, gaussian(x) = 0.000199\n",
      "For x = 4.000000, gaussian(x) = 0.000134\n",
      "For x = 4.100000, gaussian(x) = 0.000089\n",
      "For x = 4.200000, gaussian(x) = 0.000059\n",
      "For x = 4.300000, gaussian(x) = 0.000039\n",
      "For x = 4.400000, gaussian(x) = 0.000025\n",
      "For x = 4.500000, gaussian(x) = 0.000016\n",
      "For x = 4.600000, gaussian(x) = 0.000010\n",
      "For x = 4.700000, gaussian(x) = 0.000006\n",
      "For x = 4.800000, gaussian(x) = 0.000004\n",
      "For x = 4.900000, gaussian(x) = 0.000002\n",
      "For x = 5.000000, gaussian(x) = 0.000001\n"
     ]
    }
   ],
   "source": [
    "# Implement a Gaussian function\n",
    "\n",
    "from math import exp, pi, sqrt\n",
    "\n",
    "def gaussian(x, m = 0, s = 1):\n",
    "    # Note: 'm' is the mean, and 's' is the standard deviation.\n",
    "    coefficient = 1/(sqrt(2*pi)*s)\n",
    "    result = coefficient*exp(-0.5*(float(x - m)/s)**2) # Be careful here. x, m, and s are supplied as integers - let's cast the numerator to a float to ensure we don't encounter integer division problems.\n",
    "    return result\n",
    "\n",
    "x = -5.0\n",
    "while x <= 5.0:\n",
    "    print(\"For x = %f, gaussian(x) = %f\" % (x, gaussian(x)))\n",
    "    x = x + 0.1 # Increment x by a step size of 0.1."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## <span style=\"color:blue\">Exercise 3.3: Express a step function as a Python function</span>\n",
    "\n",
    "The following \"step\" function is known as the Heaviside function and\n",
    "is widely used in mathematics:\n",
    "$$H(x)=\\begin{cases}0, & \\text{if $x<0$}.\\\\\\\\\n",
    "1, & \\text{if $x\\ge 0$}.\\end{cases}$$\n",
    "Write a Python function H(x) that computes H(x)."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 8,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "1\n"
     ]
    }
   ],
   "source": [
    "# Express a step function as a Python function\n",
    "\n",
    "def step(x):\n",
    "    if(x < 0):\n",
    "        H = 0\n",
    "    else: # Otherwise x must be greater than or equal to zero\n",
    "        H = 1\n",
    "    return H # Return the result\n",
    "\n",
    "x = 0.5\n",
    "print(step(x)) # The value that is returned by the function will be printed out here. For the case of x = 0.5, this should print '1'."
   ]
  }
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