Learning Matlab

A Tutorial for Getting Started in MatLab
by Thomas M. Green

Natural & Applied Sciences Division (NAS)
Mathematics Department

Matlab is a computer program for doing numerical computation. It began as a "MATrix LABoratory" program in the late 1970s, intended to provide interactive access to the FORTRAN libraries Linpack and Eispack. These are carefully tested, high-quality programming packages for solving linear equations and eigenvalue problems. The goal of MatLab is to enable scientists to use matrix-based techniques to solve problems, using state-of-the-art code, without having to write programs in traditional languages like C and Fortran. More capabilities have been added as time has passed -- many more commands, a programming environment, and very fine graphics capabilities.

Matlab is available from The MathWorks, Inc. Natick, MA. Information can be found at "http://www.mathworks.com".

The following sessions will introduce you to MatLab and how you can use it to do calculations, solve problems, create plots and write your own functions. Use each session as a guided tour as you work through the examples presented therein while using MatLab. These sessions are designed to introduce the features of MatLab to a Pre-Calculus student so that MatLab can be used in the study of the Calculus and higher mathematics.

Note:

(Also, we offer a short-term, 6 week course, Math 175 - Intro. to MatLab, beginning in the Fall '98 semester and in subsequent semesters covering this material essentially at the pre-calculus level.)

Look at Outline

   (If you are viewing a printed copy of this document, then be advised that
   these sessions are available on-line over the Internet.  When one visits
   the Contra Costa College Home Page (www.contracosta.cc.ca.us), it's found
   via a link from the the Math Dept. Home Page. )

Sessions:

• Getting Started: This page.
• Session 1: MatLab Demos
• Session 2: Arithmetic Calculator
• Session 3: Built-In Functions
• Session 4: Polynomials, User Functions, M-Files
• Session 5: Graphics, 2-D Plots
• Session 6: Matrices, Systems of Equations
• Go To Math Dept. Home Page

		     Double click one of the following.

First Session

 ________________________________

MatLab Demos

Note:

In these sessions, rather than continually switching back and forth between this window and MatLab's window, it might be easier to print these sessions and use the printout to follow along while working in MatLab.

MatLab is installed in the NAS Computer Lab, or check with the Natural and Applied Science Division.

To start the program choose Math and then MatLab from the Start menu of Windows 95. The system should respond with the MatLab logo and then with the MatLab workspace as follows:

      This version is for educational classroom use only.

      To get started, type one of these commands: helpwin, helpdesk, or demo

      >> 

      >> quit

Before we quit, however, our first session will be to explore some demos to see examples of MatLab at work. Type:

	>> demo

Close the demo and select Klein Bottle Example from the third box for your next demo. Check out the Info button before closing this demo.

Next try the Vibration Movie.

Now, back at the MatLab Demos window select Gallery in the first box. Again read the second box and then select Knot in the third box and click the Run the "Demo button. Close that and try another selection or two before proceeding.

Back at the MatLab Demos window select Games in the first box. Again read the second box and then select one of the game demos in the third box and try it out.

Next, select Numerics in the first box of the MatLab Demos window, read the second box, and select 2-D Visual Solutions in the third box. Now click the Run the "Demo" button.
After viewing the demo, answer the following question?
Which is larger, e^pi or pi^e ?

		Write your answer on the printed copy.

Close the demo, close the MatLab Demos window, and enter the "quit" command at the prompt to terminate this MatLab session.

Finally, during the next week visit the MathWorks website several times for more information.
Matlab is available from The MathWorks, Inc. Natick, MA. Information can be found at "http://www.mathworks.com".

					  End of Session 1.

		     Double click one of the following.

Session 2

 ________________________________

Arithmetic Calculator

Some of the MatLab operations use fairly standard notation.

If you have the Student Edition of MatLab or your own copy, you can make a record of your sessions. The "diary" command saves your work in a file named "session2" as shown below: Type:

	>> diary session2

	>> diary off

	>> 2+3
	>> 3*4, 2-4

	>> 2+3*4^2

	>> 2+3*4^2/5

Note:

Repeated pressing of the arrow key scrolls back through prior commands. Similarly, pressing the arrow key scrolls forward through the commands. At any time pressing the arrow keys moves the cursor within the current command at the prompt. To correct an error in a previous command, use the arrow keys to place the cursor at the proper location, press "Delete" to remove a character, or enter a character to insert it, or both if necessary. Press "Enter" to tell MatLab to reevaluate the expression.

Parentheses may be used to group terms, or to make them more readable.

	>> (2 + 3*4^2)/5

Variable names consist of a letter, followed by any number of letters, digits, or underscores. MATLAB uses only the first 31 characters of a variable name. Names are case sensitive; MatLab distinguishes between uppercase and lowercase letters. A and a are not the same variable.

The "=" sign is used to assign values to variables and the "==" sign is used to compare variables.

	>> x = 4
	>> y = x^3
	>> y/x
	>> X = 1;
	>> X == x
	>> Area = x*y

If no other name is given, an answer is saved in a variable named ans.

	>> 4 + 6
	>> ans, z=ans^2, ans 

To get a list of your variables, use

	>> who

	>> x, y, ans, X, Area

Note:

More than one command may be entered on a single line, if they are seperated by commas. Names are case sensitive; MatLab distinguishes between uppercase and lowercase letters. A semicolon placed at the end of a MatLab statement suppresses the printing of the results. If no other name is given, an answer is saved in a variable named ans.

	>> save myvars

	>> load myvars

There are other variations of the save command and you can either look these up in the text, the reference manuals, use the help save command, or use the helpdesk command. There's also a "lookfor" command, so that you don't have to guess the topic name precisely (more on this in Session 3). >> help >> help save >> help sqrt >> helpdesk >> dir

All arithmetic is done to double precision [about 16 decimal digits], even though results are normally displayed in a shorter form. MATLAB uses conventional decimal notation for numbers, with an optional decimal point and leading plus or minus sign. Scientific notation uses the letter e to specify a power-of-ten scale factor. Imaginary numbers use either i or j as a suffix. Some examples of legal numbers are

3              -99            0.0001
9.6397238      1.60210e-20    6.02252e23
1i             -3.14159j      3e5i

	>> -4e3
	>> a=sqrt(2)
	>> format long, b=sqrt(2)
	>> a-b
	>> 10/30, format rat, 10/30
	>> format short
	>> ans

Suppose we want to solve the quadratic equation ax² + bx + c = 0. (In Session 4 we will learn about a special built-in function called roots for solving polynomial equations in general. For now we will use the quadratic formula.) The roots are given by

       x1 = (-b + sqrt(b2 - 4ac))/(2a)
     and
       x2 = (-b - sqrt(b2 - 4ac))/(2a)

	>> a=1, b=8, c=12       %set the coeficients to 1, 8, and 12
        >> x1 = (-b + sqrt(b^2 - 4*a*c))/(2*a)
        >> x2 = (-b - sqrt(b^2 - 4*a*c))/(2*a)
	>> a*x1^2+b*x1+c   %substitute x1 to check answer
	>> a*x2^2+b*x2+c   %substitute x2 to check answer
				 %What should ans equal now? 

		Write your answers on the printed copy.

Matlab statements, like the above, can be prepared with any editor, and stored in a file for later use. The file is referred to as a script, or an M-File (since they must have names of the form filename.m). We can also use the MatLab special editor. From the File menu choose New and select M-File.
This brings up the text editor window where we can enter our statements. Save the file with the extension .m, for example, quadrat.m. Then in the MatLab workspace just enter the prefix file name to have MatLab load and execute the statements.

	>> quadrat

	>> what

	>> edit quadrat.m

Finally, to stop a Matlab session and return to the operating system, use

	>> quit

					  End of Session 2.

		     Double click one of the following.

Session 3

 ________________________________

Built-In Functions

MATLAB provides a large number of standard elementary mathematical functions, including abs, sqrt, exp, and sin.

Taking the square root or logarithm of a negative number is not an error; the appropriate complex result is produced automatically.

	>> a=sqrt(-1)
	>> b=sqrt(a)
	>> c=b^2
	>> b/a
	>> b\a
	>> a/b

	>> a=1, b=1, c=1
        >> x1 = (-b + sqrt(b^2 - 4*a*c))/(2*a)
        >> x2 = (-b - sqrt(b^2 - 4*a*c))/(2*a)
	>> a*x1^2+b*x1+c   %substitute x1 to check answer
	>> a*x2^2+b*x2+c   %substitute x2 to check answer
				 %What should ans equal now? 

		Write your answers on the printed copy.

MATLAB also provides many more advanced mathematical functions, including Bessel and gamma functions. Most of these functions accept complex arguments. For a list of the elementary mathematical functions, type >> help elfun

Some of the functions, like sqrt and sin, are built-in. They are part of the MATLAB core so they are very efficient, but the computational details are not readily accessible. Other functions, like gamma and sinh, are implemented in M-files (We'll examine M-files further in Session 4). You can see the code and even modify it if you want.

Try the following to determine the effect of the functions round, ceil and floor.

	>> a=sqrt(2)
	>> b=sqrt(3)
	>> round(a), round(b)
	>> ceil(a), ceil(b)
	>> floor(a), floor(b)

		Write your answers on the printed copy.

Several special functions provide values of useful constants.

pi	3.14159265
i	imaginary unit, -1
j	same as i
eps	floating-point relative precision, 2-52
realmin	smallest floating-point number, 2-1022
realmax	largest floating-point number, (2-)21023
inf	infinity
nan	Not-a-number

Type the following and take note of each result:

	>> a=1+3i
	>> b=4-3i
	>> c=a + b
	>> d=a*b
	>> b\a
	>> a/b
	>> i
	>> i == sqrt(-1)

	>> lookfor complex

        >> b=4-3i
	>> b, real(b), imag(b)
	>> conj(b)
	>> abs(b)

        >> format long, pi      _____________________________________

        >> realmax              _____________________________________

        >> realmin              _____________________________________

        >> format rat, pi       _____________________________________

        >> 2*pi                 _____________________________________

        >> format short         _____________________________________

        >> pi                   _____________________________________

        >> 0/3                  _____________________________________

        >> 3/0                  _____________________________________

        >> 1 + realmax          _____________________________________

        >> ans + ans            _____________________________________

        >> 0/0                  _____________________________________ 

Infinity is generated by dividing a nonzero value by zero, or by evaluating well defined mathematical expressions that overflow, i.e., exceed realmax. Not-a-number is generated by trying to evaluate expressions like 0/0 or inf-inf that do not have well defined mathematical values.
Note: The function names are not reserved. It is possible to overwrite any of them with a new variable, such as

eps = 1.e-6

clear eps

Try the following exponential and logarithmic function expressions:

	>> log(1), log(10), log(100), log(1000)
	>> log10(1), log10(10), log10(100), log10(1000)
	>> log(0), log10(0)
        >> exp(-1), exp(0), exp(1) ____________________________________

        >> exp(-1), 1/exp(1)       ____________________________________
	>> 10^log10(100)
	>> exp(log(10)), log(exp(10))
	>> x = 2.345, (exp(x)+exp(-x))/2, cosh(x)
	>> acosh(x), cosh(acosh(x))
        >> realmax, log10(realmax) ____________________________________

		Write your answer on the printed copy.

	>> e = exp(1)
	>> e^pi
	>> pi^e

           Write your answer on the printed copy.

The trigonometric functions work in radian angle measure.

	    180 degrees = pi radians, 

	>> sin(pi), cos(pi), tan(pi) 
	>> sin(pi/2), cos(pi/2), tan(pi/2)
	>> sin(pi/4), cos(pi/4), tan(pi/4)
	>> pi/4, asin(sin(pi/4))

        >> x = sqrt(2)/2      ____________________________________

        >> y = asin(x)        ____________________________________

        >> y_deg = y*180/pi   ____________________________________

	>> sin(pi/2), sqrt(4)/2
	>> sin(pi/3), sqrt(3)/2
	>> sin(pi/4), sqrt(2)/2
	>> sin(pi/6), sqrt(1)/2

		Write your answers on the printed copy.

MatLab also provides a set of function functions. By that, we mean functions that work with arguments which are themselves functions. As an example, we will examine the function fzero:

	>> x = fzero( 'sin', 3)

        >> fzero( 'log', 3)   ____________________________________

        >> fzero( 'exp', -1)  ____________________________________

        >> fzero( 'sqrt', 1)  ____________________________________

        >> fzero( 'tan', 3)   ____________________________________

Another function function is fmin which returns one of the x-values in the domain specified where the minimum value of the input function can be found. The next example finds the x-value where the sine function has a minimum value over the interval from 0 to 2*pi.

	>> x = fmin( 'sin', 0, 2*pi)

        >> y_min = sin(x)

					  End of Session 3.

		     Double click one of the following.

Session 4

 ________________________________

Polynomials, User Functions, M-Files

Polynomials & Vectors

Interestingly, in MatLab a polynomial is represented by a vector! The elements of the vector are the coefficients of the polynomial in descending order.
For example, the polynomial x² + 8x + 12 is entered as

	>> p = [1 8 12]

	>> q = [1 -12 0 25 116]

Note:

Vectors in MatLab are arrays (ordered lists) of elements, numbers in this case. The elements are separated from one another by a space (or spaces) and not commas. The elements are enclosed in brackets, [ ]. (The same is true of matrices in MatLab, but matrices will be discussed at another time.) Terms with zero coefficents must be included.

One of the most useful functions for polynomials, and a very powerful function, is roots for finding the roots of a polynomial.

	>> roots_of_p = roots(p)
	>> roots_of_q = roots(q)

Given the roots of a polynomial, it is also possible to construct the associated polynomial by using the poly function.

	>> original_p = poly(roots_of_p)
	>> original_q = poly(roots_of_q)

	>> original_p = real(original_p)  %discard imag. part
	>> original_q = real(original_q)

A polynomial can be evaluated at a value of the independent variable using polyval. The polynomial p, given above for instance would be evaluated at, say -6, one of its roots, as follows:

	>> value_of_p = polyval( p, -6)

	>> value_of_q = polyval( q, -1-i)
	>> value_of_q = polyval( q, [2 3])

The function polyval is similar to a function function (Session 3), however, the arguments for polyval are two vectors. Most of MatLab's utility is based on using vectors or matrices for input and output, making it a very powerful mathematical tool. The built-in elementary functions, for instance, all accept a vector input. Try the following:

	>> x = [ 0 pi/6 pi/4 pi/3 pi/2 ] ;
	>> y = sin(x)

The colon operator,:, in MatLab is used to create vectors. The expression 1:10 is the same as [1 2 3 4 5 6 7 8 9 10]. Suppose we wanted to evaluate the polynomial q over the integers from -5 to 5. In MatLab enter the following:

	>> vector = -5:5
	>> value_of_q = polyval( q, vector)

	>> vector = -5:0.1:5
	>> value_of_q = polyval( q, vector)

	>> vector = linspace(-5, 5, 100) ;  %100 points between -5 and 5
	>> value_of_q = polyval( q, vector)

In MatLab the multiplication of two polynomials is accomplished by the convolution of two arrays. The conv function is used and takes two arguments, the polynomials to be multiplied. For example, to multiply the polynomial x² + 8x + 12 and the polynomial x⁴ - 12x³ + 25x + 116 enter the following:

	>> p = [1 8 12]; q = [1 -12 0 25 116] ;
	>> product = conv(p, q)

		Write your answers on the printed copy.

Division is accomplished with the function deconv which also takes two arguments, but this time the order of the arguments matters. The first argument is the dividend and the second is the divisor. Also, not all divisions come out exact, there can be a nonzero remainder. The following examples will illustrate this (using p and q from above):

	>> [quot, remain] = deconv(q, p)   %dividing q by p
	>> [quot, remain] = deconv(p, q)   %dividing p by q

There are no functions in MatLab to add or subtract polynomials. However, polynomials are represented by vectors and these can be added, subtracted and multiplied by a scalar if desired. The difference is that vectors must be of the same size in order to be added or subtracted. Thus, if v1 = [1 2 3] and v2 = [3 -4 5], then

	>> v1=[1 2 3]; v2=[3 -4 5]; 
	>> Sum=v1 + v2

		Write your answers on the printed copy.

Note:

Although there are no functions in basic MatLab to add or subtract polynomials, there are tools in the Symbolic Math Toolbox (another MathWorks product) to enter symbolic expressions, symbolic equations, etc., and to carry out all of the fundamental algebraic operations (including addition and subtraction, and solving equations symbolically), as well as those processes and operations of the calculus, linear algebra, differential equations, and more. The handling of symbolic expressions, which are character strings or arrays of character strings, is the topic of another set of tutorial lessons.

User Functions and M-Files

We will use the factorial function as an example of a user-defined function.

	function f = fact(n)
	%FACT Factorial
	%FACT(n) returns the factorial of n, denoted n!
	%The built-in function, prod, is used because
	%FACT(n) = prod(1:n).  See Also: prod

	f = prod(1:n) ;  %uses the colon : operator

Note:

Line 1: A function definition or header line. This line defines the function name and the number and order of the input and output argumnents. Line 2: A H1 line. H1 stands for "help 1" line. This line is used by the online MatLab help system when help or lookfor command are invoked. Line 2 starts with a % symbol indicating that this line is a comment, not an executable statement. Lines 3-5: Additional comments providing more of the "help" text, all beginning with the % symbol. Last Line(s): A function body. May consist of many lines. This part of the function contains the statements to perform the computations and assign values to any output arguments.

The function definition line, function f = fact(n), has five parts:

1. the keyword, function.
2. the output argument, f.
3. the equal sign, =.
4. the function name, fact.
5. the input argument, n.

        function [x,y] = polarconv(r, theta)

Now that we have seen a MatLab function, we need to put it in action. User defined functions, such as the example above, are placed in special MatLab files called M-Files. These files can be created using a text editor or the MatLab special editor, just as was the case with MatLab scripts discussed in Session 2.
From the File menu choose New and select M-File. This brings up the text editor window where we can enter our functions. When the file is saved it should be saved with a filename the same as the function name, followed by .m . In the example above, the fact function we would type in each of the lines indicated, including the comment lines, and then save the file with the name fact.m .

We can then use the function in the MatLab workspace as follows:

	>> a = fact(10) 
	>> x = 14
	>> y = fact(x)
	>> z = fact(x)/(fact(10)*fact(x - 10))

Create the function y = f(x) where f(x) = x² - 1/x and evaluate the function at x equal to the values -1, -0.5, 0, 0.5, and 1. (The function should be undefined at x = 0.)

Now create the same function, f(x), so that it will accept a vector input. In order to do this we need to learn more about vector operations.

	>> s = 3 
	>> U = 1:4
	>> V = [5 10 15 20]
	>> V + s
	>> s*V 
	>> U + V         % U - V also works
	>> U.*V
	>> U./V
	>> U.\V
	>> V.^s
	>> s.^V
	>> V.^U
	>> V.'

New functions can be created in terms of existing functions in MatLab. For additional practice, create a function called, cat that takes one input argument, x, and returns a value equal to

	(exp(x) + exp(-x))/2

The expression

	fact(n)/(fact(r)*fact(n-r))

	comb(4,4), comb(4,3), comb(4,2), comb(4,1), comb(4,0)

Note:

MatLab also provides the common programming constructs, such as, logical operators, if_then_else decision making, for_ loops, and while_ loops. These can be used in creating your own functions or directly in the MatLab workspace. Here is an example, creating a "program loop." >> total=0 ; >> for i=1:100 >> total = total + i ; >> end >> total However, these are topics for another tutorial session.

					  End of Session 4.

		     Double click one of the following.

Session 5

 ________________________________

Graphics, 2-D Plots

The graphics capabilites of MatLab are extensive, including both 2 and 3 dimensional plots. Begin by trying the following:

	>> x = linspace(-pi, pi, 40) ;
	>> y = sin(x) ; 
	>> plot(x, y)

The graph was generated by MatLab as a plot of y vs. x ( both x and y are vectors (arrays)) by placing 40 points in the figure window and connecting them by straight lines. In this example x is a vector of 40 entries evenly positioned between -pi and pi. The plot function can take two or more arguments, the first two of which are vectors. (plot can also take just one vector argument, see the online help system for more information.)
The following generates a graph of x vs. y and adds a grid system:

	>> plot(y, x)
	>> grid
	>> figure(1)

After our first plot, we need to recall the figure window with figure(1).
MatLab directs all graphics output to a figure window. When we add a feature, we can redisplay the figure window in several ways, one of which is to type the statement

	>> figure(1)

We may have several figure windows available in any given session, and the figure function returns the selected figure window to view. The other way to view a figure window is to 'click' on its icon in the status bar at the bottom of the workspace window, or the 'Window' pull-down menu from the menu bar near the top of the workspace window. figure by itself opens a new figure window: >> figure The command gcf gets the current figure window handle. We will use this frequently in our graphic scripts. There are many more properties for the figure function that we won't go into here. Also the set function can specify properties, such as, background colors, axes colors, window resizing, and more. See the online help system for more information.

When we generate a new plot, the old plot is cleared unless one of the following conditions hold:

• a new figure window is selected, for instance, >> figure(2), or
• the statement >> hold on is issued.

	>> plot(x, y)
	>> hold on
	>> plot(y, x)
	>> hold off
        >> figure(gcf)

	>> plot(x, y, y, x)
	>> figure(1)

Sometimes we just want to start over with a figure window. Use the command clf to clear the current figure window.

After we generate a plot, we add features, such as demonstrated in the following: (this is a good place to create a script m-file!)

        >> clf
	>> x = linspace(0, 4*pi, 50);
	>> y = 1.5*sin(x-1) ;
	>> plot(x, y)
	>> grid
	>> axis([0,15,-2,2])
	>> title('Harmonic Motion')
	>> xlabel('Time in Seconds')
	>> ylabel('Displacement in Feet')
	>> text(3*pi/4+1, sin(3*pi/4), '\leftarrowdescending')
        >> figure(gcf)

	>> axis auto
	>> figure(1)

The plot function also has an optional string argument, called linetype, where we can specify color, linestyle, and marker for the given plot. For example:

	>> plot(x, y, 'red : pentagram')
        >> figure(gcf)

The following code plots the sine function twice, once with the default linetype built into MatLab, the second with the data points marked with green asterisks :

        >> clf
	>> x = linspace(0, 2*pi, 30) ;
	>> y = sin(x) ;
        >> plot(x, y, x, y, 'g *')
        >> figure(gcf)

	>> plot( x1, y1, 'linetype1', x2, y2, 'linetype2', ... )

The print command sends the contents of the current figure window to the default printer. We can either orient the output in landscape or portrait mode.

        >> clf
	>> x = 0:0.05:2*pi ;
	>> y = 3*sin(x./2) ;
	>> plot(x, y)
        >> figure(gcf)
	>> orient landscape
	>> print

We can interactively zoom in or out at various points of a 2-D plot in order to view a specific region in more detail. The command zoom on turns on the zoom mode and zoom off turns it off. While in zoom mode we are prompted for the zoom center with the mouse. We can zoom in by pressing the left mouse button and zoom out by pressing the right mouse button. The command zoom out zooms out fully, and just using the command zoom toggles the zoom mode.

We can also create polar coordinate plots where the independent variable, t in the following example, is measured in radians:

        >> clf
	>> t = 0:0.05:2*pi ;
	>> r = sin(2*t) ;
	>> polar(t, r, 'red')
        >> figure(gcf)

        >> clf
	>> t = 0:0.05:2*pi ;
	>> x = 5*sin(t) ;
	>> y = 5*cos(t) ;
	>> plot(x, y), title('Cartesian')
        >> figure(gcf)
        >> pause
	>> figure
	>> [theta rho] = cart2pol(x, y) ;
	>> polar(theta, rho), title('Polar')
        >> figure(gcf)

The preceding example also illustrates the plotting of parametric equations, with the parameter being t in this case for plotting xy coordinates for a circle. Using parametric equations, we can plot many possibilities. Here is an example of an ellipse:

        >> clf
	>> t = 0:0.05:2*pi ;
	>> x = 3*sin(t) ;
	>> y = 5*cos(t) ;
	>> plot(x, y), title('Ellipse')
        >> figure(gcf)

Similar to parametric equations, we can also plot various polygons. Here is an example of a pentagon (5 vertices):

        >> clf
        >> t = linspace(0, 2*pi, 6) ;
	>> x = sin(t) ;
	>> y = cos(t) ;
        >> hold on
        >> fill(x, y, 'red')
	>> plot(x, y, 'yellow'), title('Pentagon')
        >> axis off
        >> axis square
        >> figure(gcf)
        >> hold off

Another function function in MatLab that plots functions is fplot. This is especially useful for plotting user defined function (M-Files). For example, the function from Session 4, f(x) = x² - 1/x, called f could be plotted as follows:

        >> clf
	>> fplot('f', [-1 1])
        >> figure(gcf)

	>> fplot('f(x)', [-1 1 -5 5])
        >> figure(gcf)

        >> clf
	>> y = '2*exp(-x).*cos(x)' ;
	>> fplot( y, [-2*pi 2*pi], 'red')
	>> title(y), xlabel('x')
        >> figure(gcf)

Another use of fplot is the following:
[x y] = fplot(...)
where the abscissae and ordinates for the function are returned in the column vectors x and y. No plot is drawn on the screen. The function can then be plotted with plot(x,y).

We will conclude this session with a discussion of the subplot function. Above we saw plots in multiple figure windows, and we saw plots of several functions on one coordinate axes system in one figure window. Now, subplot breaks the figure window into an m-by-n matrix or array of small rectangular panes, each capable of containing its own coordinate axes and plots. In the following example we will plot the Cartesian and polar functions (demonstrated above) in the same figure window, but on different coordinate systems:

        >> clf
	>> t = 0:0.01:2*pi ;
	>> x = sin(2*t) ;
	>> y = cos(3*t) ;
	>> subplot(1,2,1)
	>> plot(x, y), title('Cartesian')
	>> [theta rho] = cart2pol(x, y) ;
	>> subplot(1,2,2)
	>> polar(theta, rho), title('Polar')
        >> figure(gcf)

					  End of Session 5.

		     Double click one of the following.

Session 6

 ________________________________

Matrices, Systems of Equations

The underlying data structure for most of MatLab's operations and functions is the matrix. A matrix is a rectangular array of numbers: for example,

	[ 1 2 3 ]
	[ 4 5 6 ]

To enter a matrix in MatLab we use the brackets, [ ], and semi-colons, ;, to separate the rows. A matrix is entered in "row-major order", i.e., all of the first row, then all of the second row, etc. The elements are separated from each other by spaces or by commas. Here is a 2 row by 3 column matrix assigned to a:

	>> a = [1,2,3; 4 5 6]

        >> a(1,2)     %element in row 1 and column 2

	>> a(2,3) = 10

Without explaining everything about matrices at this time (you may take a course in Linear Algebra at some point in your studies where the study of matrices is rigorously examined) we will use matrices in MatLab to solve systems of linear equations. To illustrate consider the following example:

     John works part time for an establishment that is open 24 hours
     per day.  Each week he may be asked to work days, swing shift,
     or grave yard and he is paid different rates for each shift.
     Last week he worked 10 hours on days, 5 hours
     swing shift, and 5 hours grave yard and earned $152.50.  The
     week before he worked 12 hours on days, 6 hours swing shift,
     and 2 hours grave yard and earned $147.  The week before that
     John worked 8 hours on days, 12 hours swing shift, no
     grave yard hours, and earned $146.  How much does John earn
     per hour on days? on swing shift? on grave yard?

To solve this problem, write each statement as an equation. If we let x₁ be his hourly rate on days, x₂ his hourly rate on swing shift, and x₃ his rate on grave yard, then

     10x1 +  5x2 + 5x3 = 152.50
     12x1 +  6x2 + 2x3 = 147.00
      8x1 + 12x2 + 0x3 = 146.00

       [10  5  5]     [x1]     [152.50]
       [12  6  2]  *  [x2]  =  [147.00]
       [ 8 12  0]     [x3]     [146.00]

        >> x = A \ b

Here is how we solve the above system of equations:

        >> A = [10 5 5; 12 6 2; 8 12 0]
        >> b = [152.50; 147.00; 146]   % a column vector
        >> x = A \ b

		Write your answers on the printed copy.

Not all systems of equations have a solution and sometimes they have more than one (unique) solution. Systems like the above have a unique solution wherever the determinant of the coefficient matrix, A, is nonzero. In MatLab we can test this with the det function as follows:

        >> det(A)

        >> x = inv(A)*b    % inverse of A times b

        >> x = A \ b       % matrix left division 

        >> Identity = A * inv(A)
        >> I = eye(3)

For more practice, you should solve the following problems:

     1x1 +  2x2 + 3x3 = 12
     4x1 +  5x2 + 6x3 = 36
     7x1 +  8x2 + 0x3 = 33

     Joanna works part time for an establishment that is open 24 hours
     per day.  Each week she may be asked to work days, swing shift,
     or grave yard and she is paid different rates for each shift.
     Last week she worked 14 hours on days, 4 hours
     swing shift, and 2 hours grave yard and earned $150.50.  The
     week before she worked 8 hours on days, 6 hours swing shift,
     and 6 hours grave yard and earned $157.  The week before that
     Joanna worked 10 hours on days, no hours swing shift, 10
     grave yard hours, and earned $157.50.  How much does Joanna earn
     per hour on days? on swing shift? on grave yard?

     A parabola given by the equation y = ax2 + bx + c
     passes through the points (0, -3), (1, -2), and (-1, -6).
     Find the exact equation of the parabola. (Hint: substitute the
     coordinates of the points into the generic equation to get
     a system of three equations in three unknowns, one for each
     point; the unknowns will be a, b, and c which
     you can then substitute in the generic equation to get your
     answer.)

					  End of Session 6.

		     Double click one of the following.