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Example 1: A root finding environment.

In this example, we will implement a simple interactive graphic application in MATLAB. This will illustrate many of the graphics concepts we have introduced in this part while at the same time demonstrate the simplest of MATLAB's GUI capabilities, function ginput.

The problem we want to solve is the following:

Usually, in applications of mathematics to engineering one wants to find all roots of a continuous function on a given interval. Recall that a root of a function f on the interval [a,b] is a number x in this interval such that f(x) == 0. MATLAB of course has built-in procedures for finding a root of a function on a given interval. Alternatively, one can implement a simple procedure such as the Bisection method that always finds a root of a continuous function f on an interval [L,R] provided that f(L) and f(R) have opposite signs. However, the root returned by such procedures can be any of the many possible roots in the interval and the user has little control over which of them is returned. Also, it is a priori impossible to have the procedure return all roots contained in the interval.

One way around this problem is to let the user specify a series narrower intervals containing only one root each and invoque the root finding procedure on each of them individually. Such intervals are termed bracketing intervals. The problem with this approach is that it places the burden back on the user, as she has to come up with such bracketing intervals and to call the root finding procedure on each of them.

In what follows, we will develop an environment to streamline the procedure of finding all bracketing intervals and the roots on each of them, thus greatly alleviating the burden on the user side.

The basic idea is to let the user locate bracketing intervals by visual exploration in a zoom-in/zoom-out process driven by mouseclicks on a carefully designed "bisection window". This bisection window, shown in the figure below, displays at any point in time a plot of the function in some interval [xMin, xMax] along with the roots found so far (marked with black dots in the figure). It also displays a candidate bracketing interval [L,R] for a root.

Example of Bisection window display.

This window is interactive in that the user can click on any of a number of different zones and this causes the application to perform one of the possible actions:

  1. Try to find a root on the interval [L,R] using the Bisection method. This is triggered by clicking on the blue zone. Recall that for the bisection method to be applicable f(L) and f(R) should have opposite signs.
  2. Zoom-out to a wider interval, triggered by clicking to the left of L or the right of R .
  3. Zoom-in to a narrower interval, triggered by clicking on a point with x-coordinate between L and R.
  4. Quit the application, triggered by clicking on a point with x coordinate outside of [xMin, xMax] .

To try out the application yourself download the following files to a directory on your computer and run the command FindRoots(@MyF, 0,10) FindRoots.m BWindow.m Bisection.m MyF.m DrawRect.m

A description of each of these files follows:

Function FindRoots

This is driver, i.e. main, function. It takes as a input a reference to a function whose roots are to be found along with an the limits for an initial interval [xMin, xMax]. The root finding procedure will be constrained to this interval.

A reference to a function is constructed by prepending @ to the name of any built-in or user-defined function. We have provided an example function MyF but this could be replaced by any continuous function.

FindRoots runs a loop, executing actions according to the user clicks as described above. At each iteration it calls function BWindow to refresh the bisection window and to get a new click from the user. When exiting, it returns all roots found.

Function BWindow

This function refreshes the bisection window of our application and to get a new mouse-click from the user.

The contents of the window look like the figure above. The grid lines and coloring identify the important regions in the figure window. Here are the key quantities:

xMin the first component of x
xMax the last component of x
L one-quarter the way from xMin to xMax
R three-quarters the way from xMin to xMax
yMax the largest value in abs(y)
yMin -yMax

Refer to the rectangle defined by xMin, xMax, yMin and yMax as the display rectangle. The figure window is slightly larger than the display rectangle. In particular, we let delx = (xMax-xMin)/20 and dely = (yMax-yMin)/20 and set the axes by axis([xMin-delx xMax+delx yMin-dely yMax+dely])

Other features of the Bisection window are listed below. As an exercise, find the command that implement each of these.

  • There are four labelled ticks along the x-axis: xMin, L, R, and xMax.
  • There are two labelled ticks along the y-axis: yMin and yMax.
  • The grid is on to "chop up" the window according to the ticks.
  • The title shows the current values of L and R through six decimals, along with a message indicating what the last action was.
  • The x-axis should be drawn as a thick red line between xMin and L , a thick green line between L and R , and a thick red line between R and xMax.
  • Center a (R-L) -by- dely blue rectangle on the bottom edge of the figure window.
  • It plots y versus x and highlight the already computed roots that are input via FoundRoots.
  • The window is drawn big.

Function Bisection

This function implements the bisection root finding method itself, that finds f on [L,R] given that f(L) and f(R) have opposite signs. Do not worryif you do not understand its inner workings of these function as the point of this example is to illustrate the use of advanced graphics.