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Math 1322, Fall 2002
Introduction to Matlab: Basic Commands
You should look at Sections 1.1-1.3 of the Matlab reference book ( byUsing Matlab in Calculus
Gary Jensen), preferably with Matlab running in front of you. Do not worry about remembering
each and every command. Try to get an overview of the basics and what topics are referred to in
the book so you can return to them when needed. The material there will probably be easier to
read if you go through these notes first.
Availability Matlab is available on PC's in the Artsci Computing Center in Eads and in the
Center for Engineering Computing. It should also be available on the computers at Cornerstone
and in the Residential Computing Facilities.
Starting Matlab on a PC
 If there is a Matlab icon on the desktop, double click on it
 If not, use the menus: Start, Programs, Matlab
 You will then be in the Matlab workspace where you can enter commands
Quitting Matlab
 You can use the menus: File, Exit Matlab
 OR, just type: quit
Help See p.1 of Using Matlab in Calculus
Numbers, Arrays, Variables
Matlab “thinks” in terms of numbers arraysand . For example
 a = [1 3 5 7] An array containing 4 numbers (called its or elements members
 or ). An array with a single row (or column)components
 is sometimes also called a . vector mostlyWe use arrays
 with just one row in Calculus I-II
 The same array in Matlab can also be entered with commas:
 a = [1,3,5,7]
 But when Matlab an array on screen, it always displays omits
 commas and brackets
 
 b = An array with 2 rows and 3 columns.1 3 52 1 4” •
Names
If we like, we can give arrays more imaginative than “a” or “b” . For examplenames
 evens = [8, 6, 4, 2, 0] or
 time_intervals = [1, 3, 5, 7, 8, 6, 4, 2, 0] (Blank spaces are not allowed in names: use
 an underscore character to connect the parts)
 “a” , “evens” and “time_intervals” are examples of in Matlab. A variablevariable names
 can refer either to a number or an array:
 b = 7 a = [1, 3, 5, 7]
 ( Actually, we can also think of 7 as being an array: the 1-element “array” [7] .)
Names in Matlab are “Evens” is different from “evens”case sensitive  for example, 
Names can be up to 19 characters, must start with a letter, and can't contain
 punctuation symbols.
A good name for a variable can remind you about what information it contains. For
example, if we measure the room temperature (F° each hour for 6 hours, we might enterÑ
the data in an array called “tempdata” :
 tempdata = [68,72,84,87,90,94]
In Matlab, there are certain built-in for examplespecial variable names 
 pi you can guess!
 i, j he complex number (which we won't use)both stand for t È  "
 ans the variable that “holds” the number or array which resulted
 from the most recent Matlab calculation
You can preempt these names and use them for your own variables if you want. For example,
you're allowed to define a variable pi = 17.634 but this isn't a wise practice!
Format
Matlab normally displays 4 digits (although it carries about 16 digits in its internal calculations).
The commands 
 
 format long will display more digits
 format short returns the display to the shorter display
 help format displays some of the other format options
Creating Arrays
You can in Matlab in several ways:create an array
 1) by directly typing it in 
 a = [1, 3, 5, 7] (or a = [1 3 5 7] without commas, if you like)
 
 a = [1,3,5 ; 7,8,9] to create The semicolon " ; " starts a new1 3 57 8 9” •
 row as you enter the numbers
 2) by specifying the “stepsize” from one element to the next
 a = [1:2:7] T : start with 1, increase in 2,his means steps of size
 and end with 7, giving a = [1, 3, 5, 7] again. If you like,
 you can leave off the brackets and simply enter
 a = 1:2:7 
 a = 1:4 If the stepsize is omitted, it is assumed to be 1, so
 creates the array a = [1, 2, 3, 4]
 For example, we could create
 b = [8, 6, 4, 2, 0] by typing it in directly, or by typing
 b = 8: 2:0 Negative stepsizes are allowed. This command means:
 start at 8, in steps of size 2, stop at 0decrease
 
 Notice that
 b = [8: 2 1] produces the array b = [8,6,4,2,0] since adding À  same 
 2 to 0 would go past the stopping value 1  Þ
 last number created before the stopping
 value 1
 c = 0:50:100000 creates the array c = [0,50,100,150, ... ,100000]
 (The “ ... ” is not part of Matlab: in these notes it just
 means “and so on.”) It would be very time-consuming to
 enter the whole array by actually typing all the entries.
This array “c” is very long (how many elements does it have?). If all these elements are displayed
on the screen, it's a lot of clutter you don't need to see. How to avoid this? When there's a
semicolon “ ; ” at the end of a command, Matlab creates the variable in memory but doesn't
produce a display. A wiser command, therefore, would probably be:
 c = 0:50:100000;
 
 
 3) By specifying the number of elements in the array
In 2) we specified the stepsize (and the number of elements in the array is whatever it turns out
to be). An alternative is to specify the number of (equally-spaced) elements in the array (and
then the stepsize is whatever it turns out to be). 
The command
 a = linspace(1,5,7) creates an array of equally spaced ("linearly7 points
 spaced") from 1 to 5. Notice that these seven numbers
 divide the interval [ 1,5 ] into equal subintervals.six
 Rounded to 4 decimal places,
 a = [1.0000 1.6667 2.3333 3.0000 3.6667 4.3333 5.0000]
 (As usual, Matlab omits the brackets in its display.)
 3) By “piecing together” two arrays you have already defined
If you already have a = [1, 3, 5] and b = [2, 4, 6], you can use the commands:
 c = [a b] to create c = [1, 3, 5, 2, 4, 6] or
 c = [b a] to create [2, 4, 6, 1, 3, 5] or
 c = [b ; a] to create (when a,b have the same length)2 4 61 3 5” •
During a long Matlab session you might have defined many variables. The following commands
are useful to “know what you've got” and to “clean up”:
The command:
 
 who lists all the current user-defined variables
 clear a causes Matlab to “forget a” or “undefine” a
 clear a b causes Matlab to undefine a and b (note: no comma!)
 clear causes Matlab to “undefine” user-definedall 
 variables: CAUTION!
Exercise You want to divide the interval [1, 7] into 10 equal subintervals. In differentthree
ways, create an array called “endpoints” which lists, in order of increasing size, the endpoints of
all the subintervals.
 
Arithmetic using numbers and arrays
We can multiply an array by a number:
 
 If a = [2, 4, 6] then 2*a = [4, 8, 12]
If a and b have the e can add and subtract arrays:same number of elements, w
 If a = [2, 4, 6] and b = [1, 3, 7] then a + b = [3, 1 ,13]
 and a b = [1, 7 , 1] 
 We are also able to multiply and divide in arrays by using thethe elements
 commands “ .* ” and “ ./ ” (note the dots (periods) in front of * and / )
 If a = [2, 4, 6] and b = [1, 3, 7] then a.*b = [2, 12, 42] 
 and a./b = [2.000, 1.333 , 0.8571]
 (rounded to 4 places)
 We can also exponentiate the elements (note the “dot” again in the command)
 If a = [2, 4, 6] then a .^2 = [4 ,16, 36] 
 and 2 .^a = [4,16,64]
 
 If b = [1, 2, 3] then a .^b = [2,16,216] 
 and b .^a = [1, 16, 729]
Note: Between , we do not need the “dot”: 2*3 2/3 and 2^3 work fine (although 2.*3numbers ß
2./3 and 2.^3 will also work since 2 and 3 can be thought of as the arrays [2] and [3] ).  For the
arrays a,b above, the commands and a^b (without the “dot”) will produce errora*ba/bß
messages. The operations * and / (without the “dot”) are used between arrays in Matlabcertain
to do something quite different and irrelevant for now. )
Exercise Let a = [1,2,3]. In which of the following can the “ ” be safely omitted? What do youÞ
get in each case?
 2 a a .^2 2 .^a 2 .*a a ./2Þ 
 2 /a max(a) .*aÞ
 
Built-in Functions
 
Matlab has the usual list of built-in functions, just like a calculator. These include, for example:
 
 trig functions sin, cos, tan, etc. (when using trig functions, Matlab
 works in radians.)always
 inverse trig functions asin, acos, atan, etc.
 
 exponential and log functions exp(x) is ex
 “log” in Matlab means “base e.” Matlab's “log” isalways
 called “ ln ” in most elementary calculus texts)
 To find a logarithm in base 10,
 use the command log10(x)
 
 exp(1) = ? Exercises (by hand): log(exp( )) = ?#
 exp(log10(1)) = ? exp(log( )) = ?$
 
 absolute value abs(x) is xl l
 square root sqrt( x ) is the nonnegative square root of x.
 help elfun displays a more complete list of built-in functions
 (“elfun” = “elementary functions” )
Using the built-in functions
The built-in functions can be applied to numbers (like sin(2)) just as on a calculator, but toalso
arrays, giving a new array as the answer.
 if a = [1,4,9] then sin(a) = [sin(1),sin(4),sin(9)]
 = [.8415, .7568, .4121] (rounded)
 and
 sqrt(a) = [1 2 3]
 if a = linspace(0, 2*pi, 5) then sin(a) = [0,1,0, 1,0] ( ) explain why
 
 if a = linspace(0, 5, 7) then (sin(a)) .^2 + (cos(a)) .^2 = [1,1,1,1,1,1,1] ( )explain
 if a = [ 3,4, 1, 5] then abs(a) = [3,4,1,5]  
 
The are some other useful functions that also apply to arrays:
 if a = [ 7,3,5]
 sum adds the elements: sum(a) = 1
 length counts the elements: length(a) = 3
 max gives the biggest element: max(a)=5
 min gives the smallest element: min(a)= 7
 abs kills the “ ” signs: abs(a) = [7, 3, 5]
 mean averages the elements: mean(a) =
 0.3333 (rounded)
 sort sorts into increasing order: sort( a) = [ 5, 3, 7]  
 
Exercises
1. Suppose a = [1,3, 7,5]. By hand, find
 length(a)
 max(abs(a))
 max(a)*min(a)
 length(max(a))
 Check your answers by using Matlab.
2. Suppose you give the following commands:
 a = linspace(1,4,4) à
 b = 4: 2: 3   à
 max(b .^a) min (b.^a); ans .^2 Note: you can put more than one command on
 a line they are separated by a semicolon
What then would be the values of 
 ans.^2 ?
 sum(a.*b) ?
3. Let a = 1:5. How do you create b = the array of reciprocals of elements of a ?
 
Making Simple Plots
If you want to plot the graph of a function y = f(x) by hand, one way is
 make a list of the x-values
 compute the corresponding y-values
 plot the pairs (x,y) on a set of coordinate axes
 connect the points
Every graphing device does it just like that, connecting the plotted points with straight line
segments. If the x-values are very close together, then you can't see that the curve really
consists of straight line segments it looks like a circle, a parabola, or whatever. Matlab does
the same thing.
To plot the graph of over (say) the interval [ ], we can do the followingC œ 0ÐBÑ œ  #ß #B"B#
commands one after the other:
 x = linspace( 2,2,5) pick an array of 5 equally spaced x-values
 between 2 to 2
 y = x ./ (1 + x .^2) compute the y values: note the “ . ”
 plot (x,y) plot the points and connect them
Matlab automatically picks an appropriate window size and scale on the axes and supplies tick
marks. The graph appears in a new "figure" window. This graph looks pretty bad (angular)
because we used so few points. It looks fine if we use more points, as
 x = linspace( 2,2,50) pick an array of 50 x-values; using a lot more
 than 50 wouldn't improve the picture noticeably
 y = x ./ (1 + x .^2) compute the y values: after changing x, you
 need to enter this command again to compute
 the y valuesnew
 plot (x,y) plot the curve
The plot command can contain several functions, all to be graphed on the same screen. If we
want the graphs of f(x) = and g(x) = on the same screen, we also definex x1+x 1+x# #
#
 z = x.^2 ./ (1+x .^2) Notice we use a different letter for the output
 array z because we don't want to overwrite
 the array y that we also need
 plot (x,y,x,z)
The command
 plot ( x, y, '--',x,z) creates a dashed plot for the curve y = f(x).first
 Replacing ' --' with ' : ' makes a dotted graph.
 This is handy to distinguish the two graphs if you
 don't have a color printer.
The command
 help plot shows you some of the extra plotting “bells and
 whistles”
There are many additional commands for enhancing the picture. Here are a few.
 x=linspace( 2,2,50);
 z=x.^2./(1+x.^2);
 plot (x,y,x,z,':')
 xlabel ('x=time') ; ylabel('y') adds the text in quotes to axes; DO THIS ON
 ALL YOUR GRAPHS! Note there are two
 commands here one for each axis.
 title ('Ron Freiwald,Two Graphs') adds text in quotes as a title at the
 top of the figure.
 ALWAYS PUT YOUR NAME AND A TITLE
 ON YOUR GRAPHS
 grid on adds a horizontal and vertical grid to the
 picture, and
 grid off removes them
 legend('x/(1+x^2)','x^2/(1+x^2)') labels the graphs (keep the commands in the
 same order as they were graphed)
You can also by using the buttons onadd text, arrows, and straight line segments to your graphs
the toolbar in the figure window. Experiment !
If you want to put several graphs on the same screen but with separate commands, you have to
tell Matlab to "hold" the current screen, using the command
 hold on holds the current graphing window for more graphs
 hold off “releases” the current graphing window so that it can
 be overwritten.
 clf erases the last picture (“clear last figure”)
Printing graphs Use the printer icon on the toolbar, or the menu commands: file, print
Example
 x=linspace(0,2*pi,50);
 y=sin(x);
 plot(x,y,'+') plots only the data points (without connections),
 using a “+” for each point
 plot(x,y,'+',x,y) plots the data points as +'s; graphs the curve bythen
 connecting the points. You end up with the curve with
 the data points highlighted as well
Example 
 t=linspace(0,2*pi,50);
 x=cos(t);
 y=sin(t);
 plot(x,y); Here x,y are given in terms of a 3rd variable or
 "parameter" t. The points (x,y) are evaluated
 for each t value and then the points (x,y) are
 plotted. The axes are x and y: t doesn't appear
 in the picture. If you were only interested in the
 picture, you could omit the definitions of x and y and,
 after line 1, simply command plot(cos(t),sin(t))
The graph is to be a circle. (Why? For any t, if you compute x^2+y^2 by hand, whatsupposed 
do you get? )
However, the picture an ellipse! That's because of Matlab's scaling: the units are oflooks like
unequal lengths on the axes. Sometimes that's very convenient, but it does distort shapes.
The unequal scaling also distorts slopes. If you give the commands
 hold on
 plot (t,t)
 
you add a line of slope 1 (why?) to the picture but it doesn't look like it's at 45° inclination to the
horizontal.
If you want shapes and slopes not distorted, then
 axis equal creates equal scales on the axes, and
 axis normal returns to the Matlab's original scaling.
To see other axis commands that are available, use 
 help axis
Also see , especially the reference on p. 17 to"axis tick marks."Using Matlab in Calculus
Exercise
 1) Plot the graph of the function y = f(x) = 3x+5 for x over the interval [ 2,4].
 2) How many points did you use in your array of x-values? Was this the most efficient
choice? Why or why not?
Making a table of values of a function
We may want to make a table of values for a function, or to put into a table the data collected
from some experiment. For example, suppose we want to make a table of the values of
y = f(x) = x for x = 1,2,...5.#
We can start by making a chart with the x-values in one row and the y-values in the second row:
 x = 1:10 ;
 y = x.^2 ;
 chart = [x ; y] Note the semicolon “;” used to start a second row in chart
The result is
chart = (1 2 3 4 51 4 9 16 25” • As usual, Matlab displays on screen without the brackets.)
If we'd prefer a table with vertical columns, use the Matlab command
 table = chart ' The operation prime “ ' ” creates the transposed array; the
 first row of chart column becomes the first of table, etc., so
 table = 
1 1
2 4
3 9
4 16
5 25
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Writing M-files (script files)
It can be awkward to correct errors made in the Matlab workspace, especially if a calculation is
long and previous commands have scrolled off the screen as you executed new ones. You have
probably noticed already that you can't just use the mouse to move to an old command and edit
it. The best you can do is to use the “up-arrow” key “ ” to move back through the “history list” ofÅ
earlier commands until you “resurrect” the old command you'd like to edit.
Therefore, except for very simple jobs, we usually write our instructions to Matlab in a separate
file and then have Matlab run the whole sequence of commands (the “program”) at one time. If
an error turns up, we go back to the file, make and save the corrections, and execute the file
again.
These files are called “script files” because they are written in ordinary text: you could, in fact,
create them on any word processor, you save the file as a file. When you createprovided text
and save these files through Matlab, you provide a name (like, say, myfile1 ) and Matlab
automatically adds extension “.m” to the name to identify the file as a Matlab file. Its full, official
name would be and so these files are also referred to as “m-files.”myfile1.m
To create an m-file in Matlab use the “blank white sheet" icon at the upper left of the toolbar to
open a new (empty) Matlab m-file. (Or, use the menu commands: file, new, m-file )
A window will open for Matlab Editor/Debugger. Simply type your commands there. For
example, enter the lines
 t = linspace(0,2*pi,50); %Choose 50 equally spaced points between 0,2*pi
 x = cos (t); %Compute cos of each point
 y = sin (t); %Compute sin of each point
 plot (x,y); %Plot the points and connect the dots
(Note: Matlab ignores anything on a line following a “ % ” symbol. This lets you add comments on
some of the lines in your m-file as illustrated above.)
 
 
To save your m-file,
 
 Use the “Save” icon on the toolbar (looks like a “diskette”), or use the menu
 commands: file, save 
 Either way, for a new file, you'll be asked to give the file a name for example  myfile1
 (Don't include “.m” as part of the name; Matlab will automatically add the extension “.m” )
 You should save the work in the suggested folder "Work," since that's where Matlab
 .normally looks to find m-files
 
Note: files left in the Work directory on computers in Eads 14 (and perhaps elsewhere on
campus) will be each day. This is to prevent a buildup of clutter on the hard drive, anderased
also as antiviral protection. Anything you want to keep should be insaved on your own diskette
the a: drive
You can copy the files you want from the Work directory to the a: drive before you leave. Or you
can also choose to save files directly to the a: drive during the original save process.
To run your m-file (program), from the toolbar select:
 tools, run
Alternately, you can return to the Matlab workspace window and simply enter the name
of your file to execute it: say,
 myfile1
Note: If you do this, Matlab , , . If itwill by default look in the Work folder to find this file
can't find it because, say, you saved the file instead on your a: drive then you'll get 
an error message. If possible, keep your m-files in the Work folder during a working
session.
You see your instructor to learn how to direct Matlab to look instead on the a: drive for
your m-files.
Or, to make Matlab go to the a: drive to find this file, you can use the command
 a:myfile1
The script will in the Matlab workspace area. You will get an error message (sometimesrun
rather cryptic!) if something's wrong somewhere in the script you wrote. In that case, find the
error, correct it in the script file, , and run it again.save it again then
If a program isn't working and you run out of time, it's useful to save a copy of the file. It's very
hard for anyone else to try to help if they can't see the exact file that was creating the problem.
Writing script files
 1) lets you keep the sequence of commands right in front of you
 2) makes it easier to correct mistakes
 3) makes it easier to modify the script to run a slightly different version another time
 4) allows you to save your work and open it again later.
From inside Matlab, you can open an m-file you've previously saved by using the “open” icon
(looks like an opening folder” on the toolbar, or by using the menu commands: file, open
You should then see the available list of files in the “Work” folder and can select the file you want.
If you forget the name of a file and are in the Matlab workspace, the command
 what lists all the m-files in the current directory ( the "Work" directoryœ
 unless you have changed it).
 
Interrupting Matlab
Sometimes an error of a misjudgment sends Matlab into a long calculation that you want to
interrupt: Use the key combination Ctrl-C to stop Matlab's calculation.
Miscellaneous
 If you're entering a line that's too long to fit on the screen, you can continue it to the next
line: use periods ( ... ) at end of your typing to continue to the next line.three