EXM_2015
Experiments with Matlab
Senior Seminar in Scientific Computing

http://people.sc.fsu.edu/~jburkardt/classes/exm_2015/exm_2015.html

EXM_2015 is the home page for the class ISC4932, "Experiments with Matlab", Senior Seminar in Scientific Computing, an undergraduate seminar class offered by the Department of Scientific Computing at Florida State University, Fall Session 2015.

This is a 1-credit seminar class, intended to introduce the students to ideas, algorithms, and programs common to Scientific Computing. The topic this semester will be numerical linear algebra.

The course is based on the book "Experiments with Matlab" by Cleve Moler. Copies of this book may be downloaded from the MathWorks at http://www.mathworks.com/moler/exm/chapters.html .

The class meets on ?days, from ?to?, in room ?.

Weekly class presentations will be chosen from the following topic list:

• Iteration: Iteration is a key element in much of technical computation. Examples involving the Golden Ratio introduce the Matlab assignment statement, for and while loops, and the plot function.
• Fibonacci Numbers: Fibonacci numbers introduce vectors, functions and recursion.
• Calendars and Clocks: Computations involving time, dates, biorhythms and Easter.
• Matrices: Matlab began as a matrix calculator. Here we consider how matrices can be regarded as rules for combining vectors, and then look at how a matrix can be used to describe simple operations on a picture, such as shifting it to the left, enlarging it, or rotating it.
• Linear Equations: The most important tool in scientific computing involves setting up and solving systems of linear equations. The solution of such a system is a vector, and the coefficients are a matrix. This allows us to look at such problems in a very general way, and to describe techniques for solving a system, or for determining that a system can't be solved.
• Fractal Fern: As an example of a linear transformation, we demonstrate how a beautiful picture of a fern can be created by a simple iteration.
• Google PageRank: In order to quickly deliver the most relevant pages in response to a search request, Google has to solve the world's largest matrix problem.
• Exponential Function: The exponential function is often defined using a limit process, which Matlab can allow us to observe. The exponential function is its own derivative, and we can use Matlab to try to check this fact out. The exponential function can be used to model population growth. When the function is used with a complex argument, we get some new and interesting behavior.
• T Puzzle: A simple puzzle is made of four flat pieces of wood that can be arranged to form the shape of the letter T. Since we know how linear transformations work, we can actually ask the computer to appropriately rotate, flip, or shift our initial layout so that it can solve the puzzle for us.
• Magic Squares: Tables of numbers can be created whose rows and columns add up to the same value. Because it can be quite difficult to create such a table, they were often thought to have magical significance. Matlab can quickly create magic squares of almost any size, and because they are treated as matrices, it's possible to make some interesting numerical observations on them.
• TicTacToe Magic: Three simple games are related in a surprising way. It's possible to write programs that allow a user to interactively make the choices at each step.
• Game of Life: Here is a game that allows you to place a pattern of dots on a grid, and then watch as the dots "grow" or "die" following a few simple rules. Once you get the hang of the rules, you can create shapes that don't change, or that rotate, or even a glider that gradually flies across the page.
• Mandelbrot Set: Fractals, topology, complex arithmetic and fascinating computer graphics come together in what starts out as a simple exercise involving the repeated squaring of numbers in the complex plane.
• Sudoku: Given a sudoku puzzle, a human will scan the grid for simple patterns; an efficient computer program, however, will methodically ask a long series of hypothetical questions that start out with "Suppose I put a 1 here?" and keep going until the puzzle is solved.
• Ordinary Differential Equations: Things change! For many cases, there are mathematical models that allow us to predict the future based on the current state of affairs, and our understanding of how these laws of change work. Here, we provide a simple introduction to such laws and how they can be worked out on a computer.
• Predator-Prey Model: The simple exponential model of growth is unrealistic; for one thing, eventually we will outrun our food supply. But another effect arises when there are two species, one of which preys on the other. In that case, the size of the two populations can vary over time like dueling roller coasters.
• Orbits: One of the first clues to the laws of physics came from watching the motion of planets in their orbits. Although the planets seem to stay in their orbits without any help, a mathematical computation of that same orbit can quickly pick up errors that send the planet crashing into the sun. In this topic, we look at how to compute orbits, how errors can creep in, and how we can ensure that the planet stays on its correct path.
• Shallow Water Equations: The motion of a fluid is a very complex process; however, a reasonable model exists for cases where the fluid is not very deep; in that case, the shallow water equations can be used to show how waves will travel across the surface.
• Morse Code: Morse code was once used for telegraphic communication; we can draw a simple picture that illustrates the relationship between the letters and the sequence of dots and dashes. The fact that the symbols will be of different lengths can make it tricky to efficiently store them (the problem becomes worse when trying to store individual words from a text). We look at how Matlab's cell array feature can handle this problem elegantly.
• Music: There is a mathematical system underlying the arrangement of the keys on a piano. Using Matlab, we can actually see the sound signal emitted by each key, and understand what harmony means.