To get used with the new Help system, navigate as follows:

Contents-> MATLAB -> Getting Started ->Introduction->Product Overview->The MATLAB System (Read about the main parts of the MATLAB System)

Use the search window to find information about the MATLAB: det ( from determinant) function. Read the Description of d= det(X).

Use the search window to find information about format. Read about ``format'' on the the right part of the Help window.

Demos: Go back to Help->Product Help. From the the right part of the Help window choose ->MATLAB ->Mathematics - Matrix Manipulation or, if you are NOT in Ewing 207 room, choose Getting Started with MATLAB Video

>>diary lab1

From this point on, everything that appears in the Command Window will also be recorded in the text file lab1. Type the following list of commands (each one of them at a new prompt >>).

pi, exp(1), help exp, help elfun, diary off

FIND the text file lab1, you just have created. Learn how to open it, and erase or add more text in this file.

Erase from your diary file the output of the command help elfun

Start recording again by entering >>diary lab1

Add to your diary file, by typing in the Command Window (not by opening the file with another editor) a comment like "This class is great!, or something like that..." . It can be done by using the % sign, i.e.,

>>%This class is great, ...!

Now, type again >>diary off , and check the content of the file lab1. Make sure that you know how to e-mail this file to yourself, or you know how to copy the file on a portable device.

>> inc=pi/5
>> x=0:inc:2*pi
>> y=sin(x)
>> plot(x,y)

Use the Command History (arrow keys) to type:

>> inc=pi/100
>> x=0:inc:2*pi;
>> y=sin(x);
>> plot(x,y)

Notice the difference in the two plots.

You can save the plot as a .fig file by clicking on ``File'' and then on ''Save As..". Make sure that you save the file in your working directory.

You can print the plot by clicking on ``File'' and then on ``Print'' and then click on ``Print.'' DO NOT DO THIS NOW. For now, click ``Cancel.'' Print it at your convenience some other time.

Open a web browser and go to Examples of MATLAB files.

Open the file Plot1.m in the MATLAB editor. In the MATLAB window, click ``File'' and then ``Open..'' It should give you the list of the files in your mywork directory. Choose Plot1.m and click ``Open''.

Modify the value of n to n=410 , save, and re-execute (in the Command Window) the script file Plot1.m .

Click ``File'' and then ``New -M-File''

Type in the new window (MATLAB's editor):

% This is my first script file. It plots the sin function on [ 0, 2*pi], ``inc'' stands for increment here.

inc=pi/100;
x=0:inc:2*pi;
y=sin(x);
plot(x,y)

Click (use the editor's menu) ``File'' and then ``Save As''. Modify the default name untitled.m to, say my_plot.m . Make sure that you save the file in your working directory.

In the MATLAB Command Window type:

>> help my_plot

>> my_plot

Before you exit, click on ``Workspace'' ( find it on the MATLAB window) and notice the content of your current "Workspace". Then, type in the Command Window:

>> x=8

>>y=[1 2]

>>z=[ 1 2; 3 4]

Keep one eye on the "Workspace" window, and type

>> who

>> whos

>> clear x

>> z, y, x

>> clear

Make sure that you know exactly what each of the above commands does.

M-files (following ``Matlab Guide'', D. Higham & N. Higham).

An M-file is a text file that has a .m extension and contains MATLAB commands. There are two types of M-files:

Script M-files have no input or input arguments and operate only on the variables in the current workspace. A script file enables you to store a sequence of commands that are to be used repeatedly or will be used at a future time. One example of script file is my_plot.m from Lab #1. Other interesting example can be found at http://www.math.udel.edu/~bacuta/M353/MatlabCodes.html

under `` Warming up''.

Function M-files contain a function definition line and can accept input arguments and return output arguments. Their internal variables are local to the function. Function M-files enable you to extend the MATLAB language by writing your own functions.

Here is an example of Function file. Type the next lines in the MATLAB editor:

------------------------------------------------------------------

% Input : r - a positive number (or a vector with positive entries)

%Output1: A -area of the disc of radius r

%Output2: C -the length of the circumference of the circle of radius r

function [A, C] = circle(r);

A= pi *r.^2;

C=2*pi*r;

----------------------------------------------------------------------

For this function, ``r'' is the input argument, and ``A,C'' are output arguments.

Save the file as circle.m in mywork. In the Command Window, type:

>> help circle

>> circle(1)

>> [a, c] = circle(1)

What is the difference in output for the above two commands?

>> r=linspace(0,4,5)

>> [a, c] = circle(r)

The dot from r.^2 operation allows r to be a vector. If r is a vector, then r.^2 is a vector of the same dimension whose components are the 2nd power of the corresponding components of r.

Here is another example of MATLAB implementation of a simple real function of two variables (f(x,y)=3x + 2y^2 ). The name of the function is fun2.m.

----------------------------------------------------------------------

% function file for the two variables function z=fun2(x,y)=3x + 2y^2

%Input: two vectors x and y (of the same length)

%Output: a vector z of the same length with x and y.

function z = fun2(x, y);

u=3*x; w=2*y.^2;

z=u+ w;

----------------------------------------------------------------------

Save the file as fun2.m in mywork. In the Command Window, type:
>> zz=fun2(1,2)
>> fun2(1,2)
>> x=ones(1,4)
>> y=2*ones(1,4)
>> zz=fun2(x, y)
>> u

You should get ``??? Undefined function or variable 'u'.'' Why?

The uncommented part of the function can be simplified to:

----------------------------------------------------------------------

function z = fun2(x, y);

z=3*x + 2*y.^2;

----------------------------------------------------------------------

In many situations one function needs to be passed as argument to another function. One way to do that is to pass the function name as an argument to another function in a string. One simple example is:
>>ezplot('cos')

Here, the name of the trig function cos is passed as an argument of the function ezplot .

A better way to pass a function as an argument to another function is through a function handle. A function handle is a MATLAB data type that contains all the information necessary to evaluate the function to be passed as an argument. It is created by putting the @ character before the function name. Try:
>>ezplot(@sin)
>> h=@tan
>>ezplot(h)

The MATLAB function fzero.m (>>X = fzero(FUN,x0) tries to find a zero of the function FUN near x0. We can apply this function to find the roots of a simple polynomial function like f(x) =x^2-4. The function f(x) =x^2-4 is not one of the elementary functions already implemented in MATLAB. Thus, you have to create a .m function named, say fun4u.m . The body of the function could be as follows:

----------------------------------------------------------------------

%nice comments here.

function y=fun4u(x)

y=x.^2 -4;

----------------------------------------------------------------------

(type the above three commands in the MATLAB editor and save them as fun4u.m

Next, in the Command Window type
>>fzero(@fun4u,1)
>>fzero(@fun4u,-1)
>> fzero('fun4u',4)
>> fzero('fun4u',-5)

To avoid writing a .m file implementing the function f(x) =x^2-4, another alternative is to call the function as an ``inline" expression:
>> fun4me='x.^2-4';

OR >>fun4me= inline( 'x.^2-4', 'x')
>> root=fzero(fun4me,8)
> ezplot(fun4me)

Note that, in this case of functions defined ``inline" there is NO NEED to use @ or ' ' when you pass the function as argument to another function.

MATLAB supports the usual programming structures ``for'', ``if'' , ``while'' and ``switch.'' The syntax is a bit different from C syntax, but the use is much the same. Information about any of them may be obtained from the help command. The following ``for'' statement sums 0.1 n times (assuming that n is given).

sum=0;
for k=1:n
    sum=sum+0.1;
end;

Incorporate the above lines in a function named, say mysum:

----------------------------------------------------------------------

function S=mysum(n)

S=0;

for k=1:n

    S=S+0.1;

end

----------------------------------------------------------------------

What should be the name of the function? Now type:
>>format long
>>S=mysum(10);
>>S=mysum(1000);

Why is the answer NOT S=100?

Other examples of functions which contain the ``for'' structure are Table1.m and fixpt.m Examples of MATLAB files The function ``fixpt.m'' for implementing fixed point iterations contains the ``if'' statement.

Here is a simple example of how to use the ``while'' structure. Type:
>> n=101;
>> while n>0   %hit enter
disp(sprintf('at this step n=%d and sqrt(n)=%2.5f ', n, sqrt(n))) %hit enter
n=n-4;   %hit enter
end   %hit enter

Comments? Make sure that you understand what each of the above command lines do.

To illustrate the ``if-else'' structure, type:
>> help if

Copy the statements given by MATLAB help for ``if''- after the ``Example'' line - and create the following function named stiff.m.

----------------------------------------------------------------------

function A=stiff(n)

%%Building a square matrix which has 2 on the main diagonal,

%-1 on the diagonals above and bellow the main diagonal, and

% 0 off the first three diagonals.

for I=1:n

    for J=1:n

          if I==J

                A(I,J) = 2;

          elseif abs(I-J) == 1

                A(I,J) = -1;

          else

                A(I,J) = 0;

          end

    end

end

---------------------------------------------------------------------- Now type
>>A=stiff(8);

Do you get a tridiagonal matrix?

Make sure that you understand what each of the above command lines do.

Feel free to ask (many) questions.        

----------------------------------------------------------------------

Finding fixed points by using a fix-point theorem

Let us try to use the fpi.m function, to find the fixed point of the function g=exp(x/4) in the interval (0, 2). First you need to write or download the function fpi.m.

Second, the function g(x) = exp(x/4) is not one of the elementary functions already implemented in MATLAB. Thus, you have to create a .m function named, say myexp.m . The body of the function could be as follows:

----------------------------------------------------------------------

%nice comments here.

function y=myexp(x)

y=exp(x/4);

----------------------------------------------------------------------

Third, you will need to pass some values for the rest of the arguments of the function:

fpi.m

Type >>help fpi . For the initial guess, you should plot y=g(x) and y=x, on (0,2) in the same system of coordinates. One fast way to do so is:
>>x=linspace(0,2);
>>y=myexp(x);
>> plot(x,y,x,x)

From the plot one can easily guess that a good initial guess is x0= 1.

To find the fixed point of myexp.m using fpi.m, type
>> xc=fpi(@myexp, 1,10) IF you do not to create a .m file for y=exp(x/4), the you can go to Command Window and type:
>> sameexp = inline('exp(x/4)')
>> >> fpi(sameexp,1,10)

Note the difference in the way the first argument is called for the function myexp.m vs. the inline function sameexp.m Finally, to find the fixed point using fixptpt.m type the following:
>>tol=1e-8
>> maxIter=100
[k, xc, err, x]=fixpt(@myexp,1, tol, maxIter)

How many iterations were performed?

Finding eigenvalues (lambda's ) and the condition number for a symmetric matrix A. Example:
>>A=[ 4 2 1; 2 6 3; 1 3 9] %symmetric 3 by 3 matrix
>>Lam=eig(A) % finds the vector of eigenvalues of A
>>cond(A,2) % finds the condition number of A using the 2 (euclidian) norms
>> Lam(3)/Lam(1) % Check that cond(A,2) = lambda_{max}(A) / lambda_{min}(A)

----------------------------------------------------------------------