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What is math lab

MATLAB, which stands for MATrix LABoratory, is a state-of-the-art mathematical software package, which is used extensively in both academia and industry. It is an interactive program for numerical computation and data visualization, which along with its programming capabilities provides a very useful tool for almost all areas of science and engineering. Unlike other mathematical packages, such as MAPLE or MATHEMATICA, MATLAB cannot perform symbolic manipulations without the use of additional Toolboxes. It remains however, one of the leading software packages for numerical computation.


The basic features

Let us start with something simple, like defining a row vector with components the numbers 1, 2, 3, 4, 5 and assigning it a variable name, say x.
» x = [1 2 3 4 5]
x = 1 2 3 4 5
Note that we used the equal sign for assigning the variable name x to the vector, brackets to enclose its entries and spaces to separate them. (Just like you would using the linear algebra notation). We could have used commas ( , ) instead of spaces to separate the entries, or even a combination of the two. The use of either spaces or commas is essential!
To create a column vector (MATLAB distinguishes between row and column vectors, as it should) we can either use semicolons ( ; ) to separate the entries, or first define a row vector and take its transpose to obtain a column vector. Let us demonstrate this by defining a column vector y with entries 6, 7, 8, 9, 10 using both techniques.
» y = [6;7;8;9;10]
y = 6 7 8 9 10
» y = [6,7,8,9,10]
y = 6 7 8 9 10
» y’
ans = 6 7 8 9 10
Let us make a few comments. First, note that to take the transpose of a vector (or a matrix for that matter) we use the single quote ( ’ ). Also note that MATLAB repeats (after it processes) what we typed in. Sometimes, however, we might not wish to “see” the output of a specific command. We can suppress the output by using a semicolon ( ; ) at the end of the command line. Finally, keep in mind that MATLAB automatically assigns the variable name ans to anything that has not been assigned a name. In the example above, this means that a new variable has been created with the column vector entries as its value. The variable ans, however, gets recycled and every time we type in a command without assigning a variable, ans gets that value.
It is good practice to keep track of what variables are defined and occupy our workspace. Due to the fact that this can be cumbersome, MATLAB can do it for us. The command whos gives all sorts of information on what variables are active.
» whos
Name Size Elements Bytes Density Complex
ans 5 by 1 5 40 Full No x 1 by 5 5 40 Full No y 1 by 5 5 40 Full No
Grand total is 15 elements using 120 bytes
A similar command, called who, only provides the names of the variables that are active.
» who
Your variables are:
ans x y
If we no longer need a particular variable we can “erase” it from memory using the command clear variable_name. Let us clear the variable ans and check that we indeed did so.
» clear ans » who
Your variables are:
x y
The command clear used by itself, “erases” all the variables from the memory. Be careful, as this is not reversible and you do not have a second chance to change your mind.
You may exit the program using the quit command. When doing so, all variables are lost. However, invoking the command save filename before exiting, causes all variables to be written to a binary file called filename.mat. When we start MATLAB again, we may retrieve the information in this file with the command load filename. We can also create an ascii (text) file containing the entire MATLAB session if we use the command diary filename at the beginning and at the end of our session. This will create a text file called filename (with no extension) that can be edited with any text editor, printed out etc. This file will include everything we typed into MATLAB during the session (including error messages but excluding plots). We could also use the command save filename at the end of our session to create the binary file described above as well as the text file that includes our work.
One last command to mention before we start learning some more interesting things about MATLAB, is the help command. This provides help for any existing MATLAB command. Let us try this command on the command who.
» help who
WHO List current variables. WHO lists the variables in the current workspace. WHOS lists more information about each variable. WHO GLOBAL and WHOS GLOBAL list the variables in the global workspace.
Try using the command help on itself!
On a PC, help is also available from the Window Menus. Sometimes it is easier to look up a command from the list provided there, instead of using the command line help.

Vectors and matrices

We have already seen how to define a vector and assign a variable name to it. Often it is useful to define vectors (and matrices) that contain equally spaced entries. This can be done by specifying the first entry, an increment, and the last entry. MATLAB will automatically figure out how many entries you need and their values. For example, to create a vector whose entries are 0, 1, 2, 3, …, 7, 8, you can type
» u = [0:8]
u = 0 1 2 3 4 5 6 7 8
Here we specified the first entry 0 and the last entry 8, separated by a colon ( : ). MATLAB automatically filled-in the (omitted) entries using the (default) increment 1. You could also specify an increment as is done in the next example.
To obtain a vector whose entries are 0, 2, 4, 6, and 8, you can type in the following line:
» v = [0:2:8]
v = 0 2 4 6 8
Here we specified the first entry 0, the increment value 2, and the last entry 8. The two colons ( : ) “tell” MATLAB to fill in the (omitted) entries using the specified increment value.
MATLAB will allow you to look at specific parts of the vector. If you want, for example, to only look at the first 3 entries in the vector v, you can use the same notation you used to create the vector:
» v(1:3)
ans = 0 2 4
Note that we used parentheses, instead of brackets, to refer to the entries of the vector. Since we omitted the increment value, MATLAB automatically assumes that the increment is 1. The following command lists the first 4 entries of the vector v, using the increment value 2 :
» v(1:2:4)
ans =
0 4
Defining a matrix is similar to defining a vector. To define a matrix A, you can treat it like a column of row vectors. That is, you enter each row of the matrix as a row vector (remember to separate the entries either by commas or spaces) and you separate the rows by semicolons ( ; ).
» A = [1 2 3; 3 4 5; 6 7 8]
A = 1 2 3 3 4 5 6 7 8
We can avoid separating each row with a semicolon if we use a carriage return instead. In other words, we could have defined A as follows
» A = [ 1 2 3 3 4 5 6 7 8]
A = 1 2 3 3 4 5 6 7 8
which is perhaps closer to the way we would have defined A by hand using the linear algebra notation.
You can refer to a particular entry in a matrix by using parentheses. For example, the number 5 lies in the 2nd row, 3rd column of A, thus
» A(2,3)
ans = 5
The order of rows and columns follows the convention adopted in the linear algebra notation. This means that A(2,3) refers to the number 5 in the above example and A(3,2) refers to the number 7, which is in the 3rd row, 2nd column.
Note MATLAB’s response when we ask for the entry in the 4th row, 1st column.
» A(4,1) ??? Index exceeds matrix dimensions.
As expected, we get an error message. Since A is a 3-by-3 matrix, there is no 4th row and MATLAB realizes that. The error messages that we get from MATLAB can be quite informative
when trying to find out what went wrong. In this case MATLAB told us exactly what the problem was.
We can “extract” submatrices using a similar notation as above. For example to obtain the submatrix that consists of the first two rows and last two columns of A we type
» A(1:2,2:3)
ans = 2 3 4 5
We could even extract an entire row or column of a matrix, using the colon ( : ) as follows. Suppose we want to get the 2nd column of A. We basically want the elements [A(1,2) A(2,2) A(3,2)]. We type
» A(:,2)
ans = 2 4 7
where the colon was used to tell MATLAB that all the rows are to be used. The same can be done when we want to extract an entire row, say the 3rd one.
» A(3,:)
ans = 6 7 8
Define now another matrix B, and two vectors s and t that will be used in what follows.
» B = [ -1 3 10 -9 5 25 0 14 2]
B = -1 3 10 -9 5 25 0 14 2
» s = [-1 8 5]
s = -1 8 5
» t = [7;0;11]
t = 7 0 11
The real power of MATLAB is the ease in which you can manipulate your vectors and matrices. For example, to subtract 1 from every entry in the matrix A we type
» A-1
ans = 0 1 2 2 3 4 5 6 7
It is just as easy to add (or subtract) two compatible matrices (i.e. matrices of the same size).
» A+B
ans = 0 5 13 -6 9 30 6 21 10
The same is true for vectors.
» s-t ??? Error using ==> - Matrix dimensions must agree.
This error was expected, since s has size 1-by-3 and t has size 3-by-1. We will not get an error if we type
» s-t’
ans = -8 8 -6
since by taking the transpose of t we make the two vectors compatible.
We must be equally careful when using multiplication.
» Bs ??? Error using ==> * Inner matrix dimensions must agree.
» B
ans =
103 212 22
Another important operation that MATLAB can perform with ease is “matrix division”. If M is an invertible† square matrix and b is a compatible vector then
x = M\b is the solution of M x = b and x = b/M is the solution of x M = b.
Let us illustrate the first of the two operations above with M = B and b = t.
» x=B\t
x = 2.4307 0.6801 0.7390
x is the solution of B x = t as can be seen in the multiplication below.
» B*x
ans = 7.0000 0.0000 11.0000
Since x does not consist of integers, it is worth while mentioning here the command format long. MATLAB only displays four digits beyond the decimal point of a real number unless we use the command format long, which tells MATLAB to display more digits.
» format long
» x
x = 2.43071593533487 0.68013856812933 0.73903002309469
On a PC the command format long can also be used through the Window Menus.
There are many times when we want to perform an operation to every entry in a vector or matrix. MATLAB will allow us to do this with “element-wise” operations.
For example, suppose you want to multiply each entry in the vector s with itself. In other words, suppose you want to obtain the vector s2 = [s(1)s(1), s(2)s(2), s(3)s(3)].
The command s
s will not work due to incompatibility. What is needed here is to tell MATLAB to perform the multiplication element-wise. This is done with the symbols ".
". In fact, you can put a period in front of most operators to tell MATLAB that you want the operation to take place on each entry of the vector (or matrix).
» s
s ??? Error using ==> * Inner matrix dimensions must agree.
» s.*s
ans = 1 64 25
The symbol " .^ " can also be used since we are after all raising s to a power. (The period is needed here as well.)
» s.^2
ans = 1 64 25
The table below summarizes the operators that are available in MATLAB.

  • addition - subtraction * multiplication ^ power ’ transpose \ left division / right division
    Remember that the multiplication, power and division operators can be used in conjunction with a period to specify an element-wise operation.


Create a diary session called sec2_2 in which you should complete the following exercises. Define
[ ]A b a=

   

   


   

   
= − −
2 9 0 0 0 4 1 4 7 5 5 1 7 8 7 4
1 6 0 9 3 2 4 5, ,

  1. Calculate the following (when defined)
    (a) A ⋅ b (b) a + 4 © b ⋅ a (d) a ⋅ bT (e) A ⋅ aT
  2. Explain any differences between the answers that MATLAB gives when you type in A*A, A^2 and A.^2.
  3. What is the command that isolates the submatrix that consists of the 2nd to 3rd rows of the matrix A?
  4. Solve the linear system A x = b for x. Check your answer by multiplication.
    Edit your text file to delete any errors (or typos) and hand in a readable printout.
    2.3 Built-in functions
    There are numerous built-in functions (i.e. commands) in MATLAB. We will mention a few of them in this section by separating them into categories.
    Scalar Functions
    Certain MATLAB functions are essentially used on scalars, but operate element-wise when applied to a matrix (or vector). They are summarized in the table below.
    sin trigonometric sine cos trigonometric cosine tan trigonometric tangent asin trigonometric inverse sine (arcsine) acos trigonometric inverse cosine (arccosine) atan trigonometric inverse tangent (arctangent)


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