#!/bin/bash# cannon.sh: Approximating PI by firing cannonballs.# This is a very simple instance of a "Monte Carlo" simulation:#+ a mathematical model of a real-life event,#+ using pseudorandom numbers to emulate random chance.#  Consider a perfectly square plot of land, 10000 units on a side.#  This land has a perfectly circular lake in its center,#+ with a diameter of 10000 units.#  The plot is actually mostly water, except for land in the four corners.#  (Think of it as a square with an inscribed circle.)##  We will fire iron cannonballs from an old-style cannon#+ at the square.#  All the shots impact somewhere on the square,#+ either in the lake or on the dry corners.#  Since the lake takes up most of the area,#+ most of the shots will SPLASH! into the water.#  Just a few shots will THUD! into solid ground#+ in the four corners of the square.##  If we take enough random, unaimed shots at the square,#+ Then the ratio of SPLASHES to total shots will approximate#+ the value of PI/4.##  The reason for this is that the cannon is actually shooting#+ only at the upper right-hand quadrant of the square,#+ i.e., Quadrant I of the Cartesian coordinate plane.#  (The previous explanation was a simplification.)##  Theoretically, the more shots taken, the better the fit.#  However, a shell script, as opposed to a compiled language#+ with floating-point math built in, requires a few compromises.#  This tends to lower the accuracy of the simulation, of course.DIMENSION=10000  # Length of each side of the plot.                 # Also sets ceiling for random integers generated.MAXSHOTS=1000    # Fire this many shots.                 # 10000 or more would be better, but would take too long.PMULTIPLIER=4.0  # Scaling factor to approximate PI.get_random (){SEED=$(head -1 /dev/urandom | od -N 1 | awk '{ print $2 }')RANDOM=$SEED                                  #  From "seeding-random.sh"                                              #+ example script.let "rnum = $RANDOM % $DIMENSION"             #  Range less than 10000.echo $rnum}distance=        # Declare global variable.hypotenuse ()    # Calculate hypotenuse of a right triangle.{                # From "alt-bc.sh" example.distance=$(bc -l << EOFscale = 0sqrt ( $1 * $1 + $2 * $2 )EOF)#  Setting "scale" to zero rounds down result to integer value,#+ a necessary compromise in this script.#  This diminshes the accuracy of the simulation, unfortunately.}# main() {# Initialize variables.shots=0splashes=0thuds=0Pi=0while [ "$shots" -lt  "$MAXSHOTS" ]           # Main loop.do  xCoord=$(get_random)                        # Get random X and Y coords.  yCoord=$(get_random)  hypotenuse $xCoord $yCoord                  #  Hypotenuse of right-triangle =                                              #+ distance.  ((shots++))  printf "#%4d   " $shots  printf "Xc = %4d  " $xCoord  printf "Yc = %4d  " $yCoord  printf "Distance = %5d  " $distance         #  Distance from                                               #+ center of lake --                                              #  the "origin" --                                              #+ coordinate (0,0).  if [ "$distance" -le "$DIMENSION" ]  then    echo -n "SPLASH!  "    ((splashes++))  else    echo -n "THUD!    "    ((thuds++))  fi  Pi=$(echo "scale=9; $PMULTIPLIER*$splashes/$shots" | bc)  # Multiply ratio by 4.0.  echo -n "PI ~ $Pi"  echodoneechoecho "After $shots shots, PI looks like approximately $Pi."# Tends to run a bit high . . . # Probably due to round-off error and imperfect randomness of $RANDOM.echo# }exit 0#  One might well wonder whether a shell script is appropriate for#+ an application as complex and computation-intensive as a simulation.##  There are at least two justifications.#  1) As a proof of concept: to show it can be done.#  2) To prototype and test the algorithms before rewriting#+    it in a compiled high-level language.