Introduction

This lab will occupy three lab sessions. The first session will introduce the mechanics of using Matlab on the Linux PCs here in the lab. There is some reading to be completed before the second session and you can do that here in the lab or at any other computer with web access. The following two sessions will present exercises in Matlab use.

The discussion that follows assumes that you are basically familiar with using a Unix-like operating systems such as Linux, browsing the Web, and, to a lesser extent, using telnet (or ssh) to log in to campus computers. The next few paragraphs will give a brief introduction to Matlab and explain how to use it and those aspects of the environment that will be important to doing the labs.

Grading

The labs roughly follow the material presented in lecture, but are independent of the homework and other exercises presented in lecture. Lab grades count as 30% of your course grade.

Attendance is not required, but help is most readily available during the lab sessions.

You are encouraged to work together with other students, but you are expected to provide your own diary and summary files (explained further below).

Each lab will be given a grade of A+, A, B, or 0. These grades correspond with percentage grades of 99, 94, 88, and 0. At the end of the semester, your grades will be averaged and then integrated with your grade in lecture. A student who achieves an A+ on each of the labs will be given an overall grade of 100 rather than 99. The grading criterion is:

**A+**- The work is correct, or there are only very minor errors.
**A**- All portions of the lab work was attempted and most of it is correct. Some serious errors are present.
**B**- Not all portions of the lab work was attempted.
**Zero**- Lab was not submitted.

Each lab is due before the beginning of the next lab. Labs submitted after the day the following lab begins will have 1% deducted from the grade. An additional percent will be deducted for each further week they are late. The final due date for labs 2 through 6 is the last day of classes for the semester, and the final due date for labs 7 through 10 will be announced near the end of the semester. Labs that are not submitted before the final due date will be given a grade of zero.

Starting Up Matlab

In this section you will see how to start up recent versions of Matlab that use the Java windowing interface. These instructions are the same for the Unix and MS-Windows versions of Matlab. I will also give the command-line equivalents of many of the commands. These command-line equivalents are valid for all versions of Matlab, and many of them are valid for the Matlab clone named Octave. Generally speaking, anything you can do using a menu can also be done with command-line equivalents. You would use the equivalents when writing scripts and the menus when working interactively.

- Start up Matlab by (a) double-clicking the icon on the status bar
at the bottom (this is the fifth icon from the left in the default
configuration, and looks like this:
),
(b) choosing Matlab from the K (or Gnome or Start) menu
at the lower left corner, (c) choosing ``Run Command'' from the
K (or Gnome or Start) menu, or (d) typing the command
`matlab`at a command prompt. The first window that opens is a blank white window named ``Shell-Matlab,'' and you should minimize it using the dash at its top right. The second window is the ``splash'' that identifies Matlab, and it will go away by itself. Then the main Matlab window will open up on your screen. This window will look something like the following:Sometimes, the window will open up containing a large number of warning messages about files missing. These messages are ``normal'' and can be safely ignored.

The default window is divided into a larger pane on the right, and two panes, each with identification tabs at the bottoms, on the left. The right windowpane is a command windowpane and you will be typing Matlab commands in it. On the left side of this command windowpane somewhere (near the bottom in the figure) there will be a prompt of the form

`»`. Your typed commands go next to this prompt.There are two choices for display on each of the two windowpanes on the left. I suggest that you choose ``Workspace'' on the top and ``Command History'' on the bottom in general, although we will have occasion to use ``Current Directory'' on the bottom.

- It is a good idea to organize your work into directories
(or folders-they are the same thing). I suggest that you do
your work in your AFS (Andrew File System) directory. This is the
same directory that
`unixs`uses and it is permanent, unlike the default directory on the computers in GSCC. (Files left in the default home directory will disappear when you log out!) Use the Matlab ``Current Directory'' windowpane to navigate to your AFS directory. (Go first to the ``Desktop'' and then to ``AFS'' and then to ``private''.) Inside that directory, I suggest that you use a directory named`math2071`for all the work in this lab, and subdirectories`lab01`,`lab02`, ...,`lab10`for each of the labs. You will need to create these directories before you can use them, and the ``Current Directory'' windowpane has the appropriate buttons. After a directory has been created, you can also use the small box named ``Current Directory'' in the middle of the line of icons near the top of the Matlab window.If you wish to use the command line to create and switch directories, you can switch to your permanent ``AFS/private'' directory with the ``change directory'' commands

cd Desktop cd AFS cd private

You create a directory named`math2071`with the commandsmkdir math2071

and you can make it the current directory with the commandcd math2071

You can always find the name of your current directory with the commandpwd

and you can get a listing of the files in the current directory with either the commanddir

or the commandls

Before going on, create the directory

`math2071`, with subdirectories`lab01`,`lab02`, ...,`lab10`. - The ``Command History'' windowpane is convenient for recalling
what commands you have used recently. In addition, Matlab provides
the capability to keep a record of
*both*the commands*and*their output. The command to do this isdiary diary.txt

The name`diary.txt`is actually the name of the file that is created and you can use any name you like. It is a text file, and you should use`.txt`to name it. Before going on, type this command in the ``Command'' windowpane. Terminate the command by hitting the ``Enter'' key.You should also type a comment line that will help you identify your work later. It should start with the comment character, a percent sign

`%`, and include the lab number and date. This line will serve as an identifier when you look at the file. - Note that you can double-click a command in the ``Command History''
windowpane and it will be executed again. You can also drag a command
into the ``Command'' windowpane and then change it using the arrow
keys.
- You can also recover old commands by using the up-arrow key and
then change them using the right and left arrow keys.
- As will be described below, you will be sending your files
to me by email. Since there are sometimes a large number of these
files, it is inconvenient to attach them one by one, so I suggest
you create a ``zip'' file. Before you create the ``zip'' file,
exit your diary with the command
`diary`. If you do not, your diary will be incomplete in the ``zip'' file. You can create the ``zip'' file from the Matlab command line with the command!zip labfiles.zip *.m *.jpg *.txt

where ```labfiles.zip`'' is the name of the file and can be chosen as you wish. (The exclamation point is necessary to tell Matlab that the command is a ``system'' command and not a Matlab command.) You then only need to attach this one file to your email. You should do this only once, when you have completed all your work. - After you have completed a lab, exit from Matlab with
the command
`quit`or use the FileClose Matlab menu choice.

Using a USB drive

The storage in your AFS area is quite limited, amounting only to eight megabytes or so. This is plenty for the work we are doing, but if you try to keep many old files or files from other courses there, you will run out of space. The computers in GSCC support using USB drives (``thumb'' drives or ``jump'' drives) to save your files. You can insert a USB drive into one of the USB slots and wait a few seconds. A new icon will show up on the desktop and you can double-click on it to see the contents of the drive. You can drag and drop files from other directories into your USB drive directory, thus making portable copies of your files.

It is important to realize that you must not simply remove the USB drive from its slot! Instead, when you are ready, go to the icon on the Desktop representing your USB drive and right-click on it. Choose ``unmount'' or ``eject'' on the menu and give it a little time to get its act together. Then you may remove the drive. The reason for this rigmarole is that data does not get written immediately to the drive, but is written at the computer's leisure. Unmounting the drive forces the last data to be written. You may have noticed that MS-Windows offers the option to ``safely remove'' a USB drive, which accomplishes the same thing.

Using a browser to download files

Some of the labs require that you download files from the web in order to use them. The following exercise illustrates how to download files. The file you will download is a very simple script file.

**Exercise 1**: If you are not using the online version of this lab, please start it up by starting the browser and finding the online version of this page, beginning from my home page,`http://www.math.pitt.edu/~sussmanm`, clicking on Math2071, and then on Lab #1.Right-mouse click on the file demoscript.m to bring up a menu. Choose ``Save link as'' and a file save box will pop up. Navigate to the directory you made:

`math2071/lab01`

, and save the file with the name`demoscript.m`. You must use the`.m`extension to tell Matlab that the file contains Matlab commands. Return to the Matlab command window. The file should be visible to Matlab, a fact that you can confirm with the`dir`command or by its presence in the ``Current Directory'' windowpane.Edit the file by typing the command

edit demoscript.m

or by double-clicking on the file in the ``Current Directory'' windowpane, or using the ``FileOpen'' menu pick. An edit window will show up. Read through the file: the comments make it self-explanitory.You can tell Matlab to execute the file by typing its name, without the ``

`.m`'', on the command line, or by choosing ``Run'' from the Debug menu on the edit window. (The final statements in the file and comments refer to the following exercise.)

The following two exercises illustrate the use of the debugging capability of Matlab. Most of the time you will be able to see what is wrong from the Matlab error messages, but sometimes the error is not obvious. In Exercise 2 below, you will see what you might do when you just cannot see why something is wrong.

**Exercise 2**:- Turn on debugging with the commands
dbstop if error dbstop if naninf

or through the Breakpoints menu on the Edit window. - Now, look at
`demoscript.m`in the edit window. At the bottom there is a statement`%bad=1/(x-1);`. The percent is a comment character, so this statement is not executed. Make this statement active by deleting the percent character and save the changed file. - Execute the script file by typing its name without the
`.m`at the command line or by choosing ``Run'' from the Debug menu on the Edit window. - The division by zero caused an exception and Matlab popped up a window with the offending source line highlighted. You should also note the changed prompt in the command window.
- Place the mouse pointer on top of the
`x`in the expression`bad=1/(x-1)`and leave it there motionless for a second or so (this is called ``hovering''). The value of`x`should be displayed. You can then see why the error occurred. - Look at the prompt in the command window. It has changed
to
`K»`. You can do any legal Matlab command at this changed prompt. In particular, you can type the name`x`to get Matlab to print out its value. This is an alternative way to see the value of a variable in a debug situation. - You can exit from debugging mode either using the menu in the
edit window or with the command
dbquit

This action will return Matlab to its usual`»`prompt. - You can turn off debugging feature with the command
dbclear all

or from the debugging menu in the edit window.

- Turn on debugging with the commands

The following exercise illustrates how you can use the debugger to
trace execution. It uses the same `demoscript.m` file as before.

**Exercise 3**:- Type the command
`clear`at the command line. This will return Matlab to its state just after starting up. No variables will show in the ``Workspace'' windowpane. - Click on the first executable line (
`x=1.3`) of the`demoscript.m`file. You will notice a column of dashes to the left of the code, and you can ``set'' a ``breakpoint'' by clicking on the dash. Alternatively, go to the ``Breakpoints'' menu and ``set'' a breakpoint there. One of the buttons at the top of the window will also set a breakpoint. When a breakpoint is set, a red dot will be placed next to the line number in the edit window. - Begin executing the file either by choosing ``Run'' from the
debug menu, by pressing the ``Run'' button, or by typing the name
`demoscript`at the command line. The edit window will show that the script is poised to execute the first line of the file. (That line has not yet been executed.) - Predict, in your mind, what will happen when the line is executed. In this case, the value of x will change to 1.3 and the result ``x=1.3'' will be printed because there is no semicolon at the end of the line.
- Choose ``Step'' from the Debug menu , press the ``Step'' button
(hovering over a button brings up its description)
or issue the command ``dbstep'' at the command line. This will cause
the current line (line 8) to be executed. You will see the variable
`x`appear in the Workspace windowpane and`x=1.3000`appear in the command window. - Predict what will happen on the next step. What will the
value of the variable
`xsquared`become? - Predict what will happen on the next step. What will the
value of the variable
`p`become? (An estimate is good enough.) - Continue using ``Step'' and watch the script execute, one line at a time.
- If the next line were to call a function, the button next to the ``Step'' button is the ``Step in'' button. The ``Step'' button goes on, using the value of the function while the ``Step in'' button jumps to the code inside the function and steps there.
- Pressing the ``Continue'' button causes Matlab to continue executing the script until another breakpoint is reached or until the end of the script is reached.
- Press the ``End Debugging'' button to exit from debug mode. Alternatively, choose Debug->Exit Debug Mode from the menus or use the command dbquit.

- Type the command

Getting help

It is important to be able to get help when you need it. Matlab provides
two help facilities from inside Matlab itself and a third on the web. The
easiest way to get help is to use the ``Help'' menu at the top of the
Matlab window. Command-line help is also available from
the Matlab prompt by typing ```help` command''. For
example,

help diaryYou will get a short description of how to use the command. You will also get a list of related commands near the bottom of the help description, and you will often find other appropriate commands there. When you write your own Matlab macro files, you should always include some special comments in the beginning of the file. The comments up to the first executable statement or blank line will be printed out in response to the

help demoscriptwill give a quick help message from the first three lines of

A second way to get help from the command prompt is the following.

helpdeskThis command brings up a comprehensive help facility, the same one that the Help menu brings up. This help facility is very similar to the one on the web from the URL:

http://www.mathworks.com/access/helpdesk/help/techdoc/ We will be looking at the help facility on the web later in this lab.

Quitting

You exit Matlab by typing `quit` at the command line or by
using the FileExit menu choice.

Lab summaries

You should complete a report of the results you obtained for each
completed lab. This report need not be elaborate. The report consists
of at least two files: the original `diary.txt` file(s) plus a
summary file. This summary file can be easily created as you do the
lab by keeping a text file up in the editor and copying parts of the
web page, your commands and output to the file as you work. Another
way is to start from the `diary.txt` file and delete all your
false starts and errors to produce a compact record. Click
here
to see a sample summary file.

This summary file is very important. It is what I will read first and, if it is well-written and the work is done correctly, I will not need to read anything else. Never put incorrect Matlab statements into your summary because it will take me a lot of time to discover you really didn't mean them. I expect to see

- Brief descriptions of the work you did for each exercise.
- Copies of the main Matlab commands you used for the exercises, along with the numerical results.
- Names of plot files
`.jpg`corresponding with the different exercises. - Explanations of anything unusual or interesting, or points of confusion that you were unable to resolve outside lab.

Summarizing your work is important not only for my convenience in this class but also for your own research work. In your research, you may be running very many different Matlab sessions and you will not remember from month to month what each one did. The idea of the summary is that you can easily refresh your memory on exactly what you did to accomplish some particular task.

Here is what I want to see in the summary file:

- Those parts of the answers to each exercise that I ask for.
- Explanations of what you did, in full sentences.
- I would like to see a few lines expressing your opinion of the point of each exercise.
- Easily identified answers to exercises, including numerical values. I do not want to look in your diary file to try to figure out what you did.

- Matlab error messages (unless I ask for them).
- False starts, mistyped commands, etc.
- Incorrect results that are corrected later.
- Duplications of anything, unless I explicitly ask for them.
- Large numbers of printed values, for example, the contents of a vector of length 100. I will not read all these numbers and they end up being like spam.

If you want to know how much detail to include, think of the following
scenario. You have completed this course and, a year from now, a
friend who is taking the course is having trouble. Your friend comes
to you and asks how you did a particular exercise. You have saved
your work, so you go look at it. The first place you will look is in
your summary to see what you did. If the summary file contains only
``Exercise 1.a: complete,'' you will then have to go re-read the
original lab and look for your script files, *etc.* The objective
of the summary is that you can read what you did and then explain it
in general to your friend without referring to other materials. If
your friend needs more detail, you can look at the other files you
wrote for the lab.

Do not write a summary of the work in this lab so far. Instead, please read the following information about Matlab commands from either the PC here in the lab or from another computer on the web.

More on Matlab

The Mathworks, maker of Matlab, includes a short tutorial on using
Matlab called
Getting Started
This tutorial is also available from the Matlab command prompt with
the command `helpdesk` and also from the Help menu, and, if you
have your own copy of the Matlab manuals, it comprises the ``Getting
Started'' book.

The beginning of the ``Getting Started'' tutorial is the best presentation of the general capabilities of Matlab that I have come across. In order to have an overview of Matlab, browse through the first sections of the tutorial. There are only the equivalent of about 35 pages of material here, mostly very easy to understand. The two most important chapters are Desktop Tools that covers use of the Matlab windows, and Manipulating Matrices that covers use of Matlab as a tool for mathematics.

Begin this tutorial now, during your first lab session. Read as much of it as you can now, and complete it at home or from any convenient computer connected to the web. The list of topics that follows (roughly in order of appearance) are topics that will be necessary for this semester's labs. The sections following this one will address some of these topics, but you will be well-served if you have seen the introduction.

- What is a matrix in Matlab? What is a vector? How are they different from plain numbers?
- How to you refer to the elements of a matrix or vector in Matlab?
- How to you use a colon to represent a range of numbers or subscripts?
- What symbols are associated with the elementary arithmetic operations? (Addition, multiplication, exponentiation, etc.)
- What meanings do the special constants
`pi`,`i`,`j`,`inf`(infinity), and`NaN`(not-a-number) have? - How do you use the ellipsis
`...`to put a single statement on more than one line? - What are script m-files? What are function m-files? What is the difference?
- What does
`format long`mean and what other options might be useful? - How are the arrow keys used to edit commands in the command window?
- What is the form of the
`plot`command? - What are the
`hold on`and`hold off`commands used for? - How do you indicate matrix sum, product, inverse, and transpose?
- What do the element-by-element matrix operations such as product mean and how are they denoted?
- What is an
`if ... else ... end`statement used for? Can you give an example? - How do you use the
`for`statement to write a loop? - How is the
`break`statement used in a loop? - What is the
`feval`statement used for?

The following sections discuss more of the use of Matlab. I will expect a summary of your work for the rest of this lab.

Matlab files

The best way to use Matlab is to use its scripting facility. With sequences of Matlab commands contained in files, it is easy to see what calculations were done to produce a certain result, and it is easy to show that the correct values were used in producing a graph. It is terribly embarrassing to produce a very nice plot that you show to your advisor only to discover later that you cannot reproduce it or anything like it for similar conditions or parameters. When the commands are in clear text files, with easily read, well-commented code, you have a very good idea of how a particular result was obtained. And you will be able to reproduce it and similar calculations as often as you please.

The Matlab comment character is a percent sign (`%`). That is,
lines starting with `%` are not read as Matlab commands
and can contain any text. Similarly, any text on a line
after a `%` can contain textual comments and not Matlab commands.

A Matlab script file is a text file with the extension `.m`.
Matlab script files should *always* start off with comments that
identify the author, the date, and a brief description of the intent
of the calculation that the file performs. Matlab script files
are invoked by typing their names at the Matlab command line or
by using their names inside another Matlab file.

Matlab function files are also text files with the extension `.m`,
but the first non-comment line must start with the word
`function` and be of the form

`function`output variable = function name`(`parameters`)`

- The first line following the signature should repeat the signature (I often leave out the word ``function'') to provide a reminder of the usage of the function.
- Brief description of the mathematical task the function performs.
- Description of all the input parameters.
- Description of all the output parameters.
- Your name and the date.

The key difference between function and script files is that functions are intended to be used repetitively. They can accept parameters and variables used inside a function are invisible outside the function. When I am working on a task, I often start out using script files. As I discover just what tasks are repetitive or when I start to need the same calculation repeated for different parameters, or when I have many intermediate variables that might have the same names as variables in other parts of the calculation, I switch to function files. In these labs, I will specify function file or script file when it is important, and you are free to use what you like when I do not specify.

Because function files are intended to be used multiple times, it is a bad idea to have them print or plot things. Imagine what happens if you have a function that prints just one line of information that you think might be useful, and you put it into a loop that is executed a thousand times. Do you plan to read those lines?

Matlab commands are sometimes terminated with a semicolon (`;`)
and sometimes not. The difference is that the result of a
calculation is printed to the screen when there is no semicolon
but no printing is done when there is a semicolon. It is a good
idea to put semicolons at the ends of all calculational lines
in a function file.

Matlab also supports data files. The Matlab `save` command
will cause every variable that Matlab currently knows about to be saved
in a file called ```matlab.mat`''. You can also name the file
with the command `save filename` that will put everything into
a file named ```filename.mat`''. This command has many other options,
and you can find more about it using the help facility. The inverse
of the `save` command is `load`.

**Note**: Matlab function names are case-sensitive.
This means that the function cos is different from Cos,
coS, and COS. File names in Unix and Linux are also case-sensitive,
but file names in MS-Windows may not be. In order that no
confusion arises, all file names and all function names
in this course will be lower-case.

Variables and values

Values in Matlab are always ``floating point'' numbers with about sixteen decimal digits of accuracy. When Matlab prints values, however, it will truncate a number to about four digits to the right of the decimal point, or less if appropriate. Values that are integers are usually printed without a decimal point. Remember, however, that when Matlab prints a number, it may not be telling you all it knows about that number.

When Matlab prints values, it often uses a notation similar to scientific
notation, but written without the exponent. For example, Avogadro's
number is
in usual scientific notation, but
Matlab would write this as `6.022e+23`. The `e` denotes
. Similarly, Matlab would write `1/2048=4.8828e-04`.
You can change the number of digits displayed with the `format`
command. (See `help format` for details.)

Matlab uses variable names to represent data. A variable name
represents a matrix containing complex double-precision data. Of
course, if you simply tell Matlab `x=1`, Matlab will understand
that you mean a matrix and it is smart enough to
print `x` out without its decimal and imaginary parts, but make
no mistake: they are there. And `x` can just as easily
turn into a matrix.

A variable can represent some important value in a program, or it can
represent some sort of dummy or temporary value. Important quantities
should be given names longer than a few letters, and the names should
indicate the meaning of the quantity. For example, if you were using
Matlab to generate a matrix containing a table of squares of numbers,
you might name the table `tableOfSquares`. (The rule I am using
here is that the first part of the variable name should be a noun and
it should be lower case. Modifying words follow with upper case
letters separating the words. This rule comes from the officially
recommended naming of Java variables.)

Once you have used a variable name, it is bad practice to re-use it to mean something else. It is sometimes necessary to do so, however, and the statement

clear firstVariable secondVariableshould be used to clear the two variables

Matlab has a few reserved variable names. You should not
use these variables in your m-files. If you do use
variables such as `i`, they will lose their special
meaning until you clear them. Reserved names include

`ans`: The result of the previous calculation.`computer`: The type of computer you are on.`eps`: The smallest positive number that satisfies the expression . This value indicates the size of ``roundoff.'' (Note: The name`eps`is a special name, and it means a particular value. It is not the same as a variable named`epsilon`, which could take any value.)`i, j`: The imaginary unit (). In this course you should avoid using`i`or`j`as subscripts or loop indices.`inf`: Infinity (). This will be the result of dividing 1 by 0.`NaN`: ``Not a number.'' This will be the result of dividing 0 by 0,`inf`by`inf`,*etc.*`pi`:`realmax, realmin`: The largest and smallest real numbers that can be represented on this computer.`version`: The version of Matlab you are running. (The`ver`command gives more detailed information.)

**Exercise 4**: Start up Matlab and use it to answer the following questions.- What are the values of the reserved variables
`pi`,`eps`,`realmax`, and`realmin`? - Choose a value and set the variable
`x`to that value. - What is the square of
`x`? Its cube? - Choose an angle (pick one whose and you
know) and set the variable
`theta`to its value. - What is ? ? Is the angle interpreted as degrees or radians?
- Use the
`save`command to save all your variables. - Use the
`clear`command. Check that there are no variables left in the ``current workspace'' (windowpane is empty). - Restore all the variables with
`load`and check that the variables have been restored to the ``Current workspace'' windowpane.

- What are the values of the reserved variables

Variables are matrices

We said that Matlab treats all its variables as though they were matrices. Important subclasses of matrices include row vectors (matrices with a single row and possibly several columns) and column vectors (matrices with a single column and possibly several rows). One important thing to remember is that you don't have to declare the size of your variable; Matlab decides how big the variable is when you try to put a value in it. The easiest way to define a row vector is to list its values inside of square brackets, and separated by spaces or commas:

rowVector = [ 0, 1, 3, 6, 10 ]The easiest way to define a column vector is to list its values inside of square brackets, separated by semicolons or line breaks.

columnVector1 = [ 0; 1; 3; 6; 10 ] columnVector2 = [ 0 1 9 36 100 ](It is not necessary to line the entries up as I have done, but it makes it look nicer.)

Matlab has a special notation for generating a set of equally spaced values, which can be useful for plotting and other tasks. The format is:

start : increment : finishor

start : finishin which case the increment is understood to be 1. Both of these expressions result in row vectors. So we could define all the integers between 10 and 20 by:

integers = 10 : 20and just the even values from 10 to 20 by:

evens = 10 : 2 : 20

Sometimes, you'd prefer to specify the *number* of items in
the list, rather than their spacing. In that case, you can use the
`linspace` function, which has the form

linspace( firstValue, lastValue, numberOfValues )in which case we could generate fifty numbers with the command:

x = linspace ( 10, 20, 50 )As a general rule, use the colon notation when the increment is an integer or when you know what the increment is and use

Another nice thing about Matlab vector variables is that they are
flexible. If you decide you want to add another entry to a
vector, it's very easy to do so. To add the value 22 to the end of
our `evens` vector:

evens = [ evens, 22 ]and you could just as easily have inserted a value before the other entries, as well.

Even though the number of elements in a vector can change, Matlab
always knows how many there are. You can request this value at any
time by using the `length` function. For instance,

length ( evens )should yield the value 7 (the 6 original values of 10, 12, ... 20, plus the value 22 tacked on later). You can also view the length of vectors by looking in the Workspace windowpane. In the case of matrices with more than one nontrivial dimension, the

To specify an individual entry, you need to use index notation, which
uses **round** parentheses enclosing the index of an entry.
**The first element of an array has index 1** (as in Fortran,
but not C and Java). Thus, if I want to alter the third element of
`evens`, I could say

x(3) = 7

**Exercise 5**:- Use the
`linspace`function to create a row vector called`meshPoints`containing exactly 100 elements with values evenly spaced between -1 and 1. - What expression will yield the value of the
element of
`meshPoints`? What is this value? - Double-click on the variable
`meshPoints`in the ``Current workspace'' windowpane to view it as a vector and confirm its length is 100. - Produce a plot of a sinusoid on the interval using
the command
plot(meshPoints, sin(2*pi*meshPoints))

Please save (export) this plot as a jpeg (.jpg) file and include it with your summary. - Create a file named
`exer5.m`(you can use the Matlab command`edit`, type the commands into the window and use ''Save as'' to give it a name). The first lines of the file should be the following:% Lab 1, exercise 5 % A sample script file. % Your name and the date

Follow the header comments with the commands containing exactly the commands you used in the earlier parts of this exercise. Test your script by using`clear`to clear your results and then execute the script from the command line by typing`exer5`, by using the ``Run'' button (looks like a sheet of paper with an arrow pointing down) at the top of the edit window, or by using DebugRun from the edit window menus.

- Use the

Vector Operations

Matlab provides a large assembly of tools for matrix and vector manipulation. We will investigate a few of these by trying them out.

**Exercise 6**: Define the following vectors and matrices:rowVec1 = [ -1 -4 -9] colVec1 = [ 2 4 -8 ] mat1 = [ 1 3 5 7 9 2 4 6 8 ]

- You can multiply vectors by constants. Compute
colVec2 = (pi/4) * colVec1

- The cosine function can be applied to a
vector to yield a vector of cosines.
Compute
colVec2 = cos( colVec2 )

Note that the values of`colVec2`have been overwritten. Are these the values you expect? - You can add vectors and multiply by scalars. Compute
colVec3 = colVec1 + 2 * colVec2

- Matlab will not allow you to do illegal
operations! Try to compute
illegal = colVec1 + rowVec1;

Look carefully at the error message. You must recognize from the message what went wrong when you see it in the future. Unfortunately, Matlab error messages are not always clear. - You can do row-column matrix multiplication. Compute
colvec4 = mat1 * colVec1

- A single quote following a matrix or vector indicates
a (Hermitian) transpose.
mat1Transpose = mat1' rowVec1 = colVec3'

Warning: The single quote really means the complex-conjugate transpose (or Hermitian adjoint). If you want a true transpose applied to a complex matrix you must use ```.'`''. - Transposes allow the usual operations. You might
find
a useful expression to
compute the dot (inner) product
(although there is a
`dot`function in Matlab).mat2 = mat1 * mat1' % mat2 is symmetric rowVec2 = rowVec1 * mat1 dotProduct = colVec3' * colVec1 euclideanNorm = sqrt( colVec2' * colVec2 )

- Matrix operations such as determinant and trace
are available, too.
determinant = det( mat1 ) traceOfMat1 = trace( mat1 )

- You can pick certain elements out of a vector, too.
Find the smallest element in a vector
`rowVec1`.min(rowVec1)

- The
`min`and`max`functions work along one dimension at a time. They produce vectors when applied to matrices.max(mat1)

- You can compose vector and matrix functions. For
example, use the following expression to compute the max norm
of a vector.
max(abs(colVec1))

- How would you find the single smallest element of a matrix?
- As you know, a magic square is a matrix all of whose row sums,
column sums and the sums of the two diagonals are the same. (One
diagonal of a matrix goes from the top left to the bottom right, the
other diagonal goes from top right to bottom left.) Show by
direct computation that if
A=magic(100); % please do not print all 10,000 entries.

then the largest and smallest row sums, the largest and smallest column sums, and the sums of the two diagonals are all the same, and hence that all row, column and diagonal sums are equal to each other.

Hints:- Use the Matlab
`min`and`max`functions. - Recall that
`sum`applied to a matrix yields a row vector whose values are the sums of the columns. - The
Matlib function
`diag`extracts the diagonal of a matrix, and the composed function`sum(diag(fliplr(A)))`computes the sum of the other diagonal.

- Use the Matlab
- Suppose we want a table of integers from 0 to 10, their squares
and cubes. We could start with
integers = 0 : 10

but now we'll get an error when we try to multiply the entries of`integers`by themselves.squareIntegers = integers * integers

Realize that Matlab deals with vectors, and the default multiplication operation with vectors is row by column vector multiplication. In order to do

*element-by-element*multiplication, we need to place a*period*in front of the operator:squareIntegers = integers .* integers

Now we can define

`cubeIntegers`and`fourthIntegers`in a similar way.cubeIntegers = squareIntegers .* integers fourthIntegers = squareIntegers .* squareIntegers

Finally, we would like to print them out as a table.

`integers`,`squareIntegers`and`cubeIntegers`are row vectors, so make a matrix whose**columns**consist of these vectors and allow Matlab to print out the whole matrix at once.tableOfPowers= ... [integers', squareIntegers', cubeIntegers', fourthIntegers']

- Watch out when you use vectors. The multiplication, division and
exponentiation operators all have two possible forms, depending on
whether you want to operate on the arrays, or on the elements in the
arrays. In all these cases, you need to use the
**period**notation to force elementwise operations. Compute`squareIntegers`alternatively using the exponentiation operator as:squareIntegers = integers .^ 2

These problems*never*come up with addition or subtraction; nor do they occur with division or multiplication by a scalar. - Matlab will allow you to be sloppy with addition. You can
add a constant to each component of a vector in the following way.
squaresPlus1 = squareIntegers + 1

- Beware when you use division with vectors or matrices!
Matlab will ``divide'' one matrix by another using the Moore-Penrose
pseudoinverse, and the result is rarely what you expect!
A common mistake is to forget the dot in front of a division
symbol, giving an incorrect result without any error messages.
Look at the difference between the following two commands.
cubeIntegers ./ squaresPlus1 cubeIntegers / squaresPlus1

- The index notation can also be used to refer to a subset of
elements of the array. With the
*start:increment:finish*notation, we can refer to a range of indices. Two-dimensional vectors and matrices can be constructed by leaving out some elements of our three-dimensional ones. For example, submatrices an be constructed from`tableOfPowers`. (The`end`function in Matlab means the last value of that dimension.)tableOfCubes = tableOfPowers(:,[1,3]) tableOfOddCubes = tableOfPowers(2:2:end,[1,3]) tableOfEvenFourths = tableOfPowers(1:2:end,1:3:4)

- Use the Matlab function
`magic`it to construct a matrix:A = magic(10)

What commands would be needed to generate the four matrices in the upper left quarter, the upper right quarter, the lower left quarter, and the lower right quarter of`A`?

- You can multiply vectors by constants. Compute

Flow control

It is critical to be able to ask questions and to perform repetitive
calculations in m-files. These topics are examples of ``flow
control'' constructs in programming languages. Matlab provides
two basic looping (repetition) constructs: `for` and `while`,
and the `if` construct for asking questions. These statements
each surround several Matlab statements with `for`, `while`
or `if` at the top and `end` at the bottom.

**Note:** It is an excellent idea to indent the statements between
the `for`, `while`, or `if` lines and the `end` line.
This indentation strategy makes code immensely more readable.
Your m-files will be expected to follow this convention.

The syntax of a `for` loop is

for control-variable=start : increment : end |

Matlab statement ... |

... |

end |

The syntax of a `while` loop is

Matlab statement initializing a control variable |

while logical condition involving the control variable |

Matlab statement ... |

... |

Matlab statement changing the control variable |

end |

The syntax of a simple `if` statement is

if logical condition |

Matlab statement ... |

... |

end |

The syntax of a compound `if` statement is

if logical condition |

Matlab statement ... |

... |

elseif logical condition |

... |

else |

... |

end |

Note that `elseif` is one word! Using two words `else if`
changes the statement into two nested `if` statements with
possibly a *very* different meaning, and a different number of
`end` statements.

**Exercise 7**: The ``max'' or ``sup'' or ``infinity'' norm of a vector is given as the maximum of the absolute values of the components of the vector. Suppose is a vector in , then the infinity norm is given as

If`v`is a Matlab vector, then the Matlab function`length`gives its length, and the following code will compute the infinity norm. Note how indentation helps make the code understandable.% find the infinity norm of a vector v N=length(v); norm=abs(v(1)); for n=2:N if abs(v(n))>norm norm=abs(v(n)); % largest value up to now end end norm % no semicolon means value is printed

- Define a vector as
v=[ -5 2 0 6 8 -1 -3 -10 ];

- How many elements does
`v`have? Does that agree with the result of the`length`function? - Use cut-and-paste to put the above code into the Matlab command windowpane and execute it.
- What is the first value that
`norm`takes on? - How many times is the statement with the comment
``
`largest value up to now`'' executed? - What are all the values taken by the variable
`norm`? - What is the final value of
`norm`?

- Define a vector as

M-files

If you have to type everything at the command line, you will not get very far. You need some sort of scripting capability to save the trouble of typing, to make editing easier, and to provide a record of what you have done. You also need the capability of making functions or your scripts will become too long to maintain. In this section we will consider first script files and later function m-files.

**Exercise 8**:- Use cut-and-paste to copy the code given above for the infinity
norm into a file named
`exer8a.m`. Recall you can get an editor window from the FileNewM-file menu or from the`edit`command in the command windowpane. - Add your name and the date below the introductory comment. You should always place your name and the date in an m-file. Don't forget to save the file.
- Redefine the vector
v = [-35 -20 38 49 4 -42 -9 0 -44 -34];

- Execute the script m-file you just created by typing just its name
(
`exer8a`) without the`.m`extension in the command windowpane. What is the infinity norm of this vector? - The usual Euclidean or 2- norm is defined as

Copy the following Matlab code to compute the 2-norm into a file named`exer8b.m`. Be sure to add your name and the date to the comments.% find the two norm of a vector v N=length(v); norm=v(1)^2; for n=2:N norm = norm + v(n)^2; end norm=sqrt(norm) % no semicolon means value is printed

- Using the same vector
`v`, execute the script`exer8b`. What are the first four values the variable`norm`takes on? What is its final value? - Look carefully at the mathematical expression (2) and
the Matlab code in
`exer8b.m`. This is the way one translates mathematical summations into Matlab code.

- Use cut-and-paste to copy the code given above for the infinity
norm into a file named

Script files are very convenient, but they have drawbacks. For example,
if you had two different vectors, `v` and `w`, for which you
wanted norms, it would be inconvenient to use `exer8a` or
`exer8b`. It would be especially inconvenient if you wanted
to get, for example,
.
The solution to this inconvenience is to use function m-files.
Function m-files define your own functions that can be used just
like Matlab functions such as sin(x), etc.

**Exercise 9**:- Copy the file
`exer8a.m`to a file named`infinityNorm.m`. (You can use ``save as'' or cut-and-paste to do this.) Add the following lines to the beginning of the file:function norm=infinityNorm(v) % norm=infinityNorm(v) % v is a vector % norm is its infinity norm

- The first line of a function m-file is called the ``signature''
of the function.
The first comment line repeats the signature in order to explain
the ``usage'' of the funciton. Subsequent comments explain the parameters
(such as
`v`) and the output (such as`norm`) and, if possible, briefly explain the methods used. You should have one line with your name and the date. The function name and the file name must agree. - Place a semicolon on the last line of the file so that nothing will normally be printed by the function.
- Invoke the function in the command windowpane by typing
infinityNorm(v)

- Repeat the above steps to define a function named
`twoNorm.m`from the code in`exer8b.m`. Be sure to put comments in correctly. - Define two vectors
a = [ -43 -37 24 27 37 -33 -19 29 43 -49 ]; b = [ -5 -4 -29 -29 30 33 20 31 42 14];

and find the value of infinity norm of`a`and the two norm of`b`with the commandsaInfinity = infinityNorm(a) bTwo = twoNorm(b)

Note that you no longer need to use the letter`v`to denote the vector, and it is easy to manipulate the values of the norms. - What Matlab expression would yield the value ? What is this value?

- Copy the file

Ordinary differential equations and graphics

In this section you will see how to use plotting to enhance work that is
focussed on something else, like solving a differential equation.
The differential equation

with initial condition has an exact solution . It also has an approximate numerical solution defined by Euler's formula (see below for more detail) as

In some sense, We are going to look at how this expression evolves for .

**Exercise 10**: Copy the following text into a file named`exer10.m`and then answer the questions about the code.% compute the solution of the differential equation % y'+y=sin(x) % starting at y=0 at x=0 using Euler's method STEPSIZE=.5; NTERMS=30; clear x y exactSolution y(1)=0; x(1)=0; exactSolution(1)=0; for k=1:NTERMS x(k+1)=x(k)+STEPSIZE; y(k+1)=y(k)+STEPSIZE*(-y(k)+sin(x(k))); exactSolution(k+1)=.5*(exp(-x(k+1))+sin(x(k+1))-cos(x(k+1))); plot(x,y); % default line color is blue axis([0,16,-1.1,1.1]); hold on plot(x,exactSolution,'g'); % g for green line legend('Euler solution','Exact solution') hold off disp('Press a key to continue.') pause end

It is always good programming practice to define constants symbolically at the beginning of a program and then to use the symbols within the program. Sometimes these special constants are called ``magic numbers.'' By convention, symbolic constants are named using all upper case.- Add your name and the date to the comments at the beginning of the file.
- How is the Matlab variable
`x`related to the dummy variable in Equation (3)? (Please use no more than one sentence for the answer.) - How is the Matlab statement that begins
`y(k+1)=...`related to the expression in Equation (3)? (Please use no more than one sentence for the answer.) - In your own words, what does the Matlab function
`clear`do? You can use the Matlab command`help clear`to get this information. - Use the Matlab help facility to see what the
`plot`commands, the two`hold`commands, the`axis`command, the`pause`command, and the`legend`command do. - Execute the script by typing its name
`exer10`at the command line. The script displays a plot and waits for you to hit the ``enter'' key to track evolution of the solution. You do not have to send me these plots.

In general, a first-order ordinary differential equation can be written
in the form

This method can be compared with the following expression appearing in

y(k+1)=y(k)+STEPSIZE*(-y(k)+sin(x(k)));

A slightly more complicated method for (4) is
Euler's implicit method, which can be written in the following way.

**Exercise 11**:- Copy
`exer10.m`to a new script m-file named`exer11.m`. Modify it so that Euler's implict method is used instead of Euler's explicit method. Be sure to modify your comments as necessary. - What are the final values of
`x`and`y`? (You can print the final value of the vector`y`using the special Matlab subscript ```end`'' by writing`y(end)`. - Please include the final plot with your summary. You can save
(export) it as a ``jpeg'' file from the file menu on the plot window
or by using the command
print -djpeg exer11.jpg

where ```exer11`'' can be any file name, but the extension should be`.jpg`.

- Copy

Back to MATH2071 page.

Mike Sussman 2009-01-03